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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 698050 15 pageshttpdxdoiorg1011552013698050
Research ArticleAdaptive Speed Control Design for Brushed Permanent MagnetDC Motor Based on Worst-Case Analysis Approach
Sheng Zeng
Research and Development Department of Critical Care Engineering CareFusion Yorba Linda CA 92886 USA
Correspondence should be addressed to Sheng Zeng shengzengcarefusioncom
Received 30 April 2012 Revised 23 July 2012 Accepted 23 August 2012
Academic Editor Pedro Ribeiro
Copyright copy 2013 Sheng Zeng This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents the adaptive controller design for brushed permanent magnet DCmotor used in velocity-tracking applicationsbased on worst-case approach We first formulate the robust adaptive control problem as a nonlinear119867infin-control problem underimperfect state measurement and then solve it using game-theoretic approach The controller guarantees the boundedness ofclosed-loop signals with bounded exogenous disturbances and achieves desired disturbance attenuation level with respect to theunmeasured exogenous disturbance inputs and the measured disturbance inputs The strong robustness properties are illustratedby a simulation example
1 Introduction
Permanentmagnet brushedDC (PMBDC)motors are widelyused in real world applications and particularly in the high-volume commercial products which is due to the PMBDCmotorsrsquo better cost-to-performance ratio than most othermotors However the structure of these permanent mag-netic motors induces several uncertainties such as unmod-elled nonlinear dynamics and undesired commutationdenttorques The magnets losedegrade their magnetic propertiesover time and the motor constant varies with the changesof temperature and operating conditions Moreover the everincreasing control demands for high-efficiency and low-costrequired to run the PMBDC motor at its technical limit Allabove design challenges call for a robust adaptive controllerformotion control applications over awide range of operatingconditions
Adaptive control attracted a lot of research attention incontrol theory since 1970sThe classic adaptive control designbased on the certainty equivalence approach leads to struc-turally simple adaptive controllers [1 2] and its effectivenessfor linear systems with or without stochastic disturbanceinputs has been demonstrated when long-term asymptoticperformance is considered [3] However early designs basedon this approach were not robust to exogenous disturbance
inputs and unmodeled dynamics [4] Then the stability andthe performance of the closed-loop system becomes animportant issue whichmotivated the study of robust adaptivecontrol in the 1980s and 1990s
Robust adaptive control design is one of the most impor-tant research topics in control theory which addresses thedesign of controllers that are robust to model uncertain-ties and insensitive to the exogenous disturbances Variousadaptive controllers were modified to render the closed-loopsystems robust [5ndash10] Despite their successes they howeverfell short of addressing the disturbance attenuation propertyof the closed-loop system Worst-case analysis-based robustadaptive-control design was motivated by the success of thegame-theoretic approach to 119867
infin-optimal control problems[11] in late 1990s which addresses the disturbance attenuationproperty directly In this approach the robust adaptive con-trol problem is formulated as a nonlinear 119867infin-control prob-lem under imperfect state measurements By cost-to-comefunction analysis it is converted into an119867infin-control problemwith full information measurements This full informationmeasurements problem is then solved using nonlinear designtools for a suboptimal solution This design paradigm hasbeen applied to worst-case parameter identification problems[12] which has led to new classes of parametrized identifiersfor linear and nonlinear systems [13ndash16] It has also been
2 Mathematical Problems in Engineering
applied to adaptive control problems [17ndash21] and offered apromising tool to system subjected to uncertainties
In this paper we study the adaptive control design forpermanent magnet DC motor based on worst-case analysisapproach We first model the permanent magnet DC motorservo system which is linear in all of the uncertainties Wethen formulate the robust adaptive control problem as anonlinear 119867infin-control problem under imperfect state mea-surements and apply the cost-to-come function analysis toderive the worst-case identifier and state estimator The con-trol design of the plant subsystem follows [22] and the adap-tive controller can be obtained by the integrator backstep-ping methodology The robust adaptive controller achievesasymptotic tracking if the disturbances are bounded andof finite energy and guarantees the stability of the closed-loop system with respect to the bounded disturbance inputsand the initial conditions Furthermore the close1d-loop sys-tem admits a guaranteed disturbance attenuation level withrespect to the exogenous disturbance inputs and the mea-sured disturbances where ultimate lower bound for theachievable attenuation performance level is only related tothe noise intensity in the measurement channel of the plantsystem
The balance of the paper is organized as follows InSection 2 we present the formulation of the adaptive controlproblem and discuss the general solution methodology thenwe obtain parameter identifier and state estimator usingthe cost-to-come function analysis in Section 3 In Section 4we derive the adaptive control law based on backsteppingmethod and present the main results on the robustness ofthe system in Section 5 The effectiveness of the proposedapproach is illustrated with an example in Section 6 and thepaper ends with some concluding remarks in Section 7
2 Problem Formulation
We consider a velocity-control problem of a brushed perma-nent magnet DC motor which is described by the followingdynamics
=
[[[
[
minus119863
119897
119869
119870119905
119869
minus119870
119890
119871minus119877
119871
]]]
]
119909 + [
0
1
119871
] 119906 + [
[
1
119869
0
]
]
119879+ (1a)
119910 = [1 0] 119909 + (1b)
where isin R2 is the state vector and represents load shaftangular speed and motor current respectively 119906 isin R is thescalar control input 119910 isin R is the load shaft angular speedmeasurement output 119871 isin R is the motor inductor 119877 isin R
is the armature resistance 119870119890is the back-emf constant 119870
119905is
the motor torque constant 119869 is the motor system inertia119863119897isin
R is the motor-system damping factor = [119879119908
120596
119879119891]1015840
is the arbitrary disturbance vector which is composed ofarbitrary disturbance torque 119879
119908isin R arbitrary disturbance
at the output measurement channel 120596
isin R and frictiondisturbance torque 119879
119891isin R 119879
isin R is the measured or
estimated disturbance torque In this paper we assume the
variables 119870119905 119870
119890 119869 119863
119897 119877 and 119871 are completely unknown or
partially unknownIt is easy to check that the true system is observable con-
trollable a minimal phase system and the transfer functionfrom 119906 to 119910 has relative degree 119903 = 2 And then we canfind a state diffeomorphism 119909 = and a disturbance trans-formation 119908 = by following [22] such that system (1a)and (1b) can be transformed into the following form in the 119909coordinates
119909 = 119860119909 + (11991011986021+ 119906119860
22+ 119860
23) 120579 + 119861119906 + 119863119908 + (2a)
119910 = 119862119909 + 119864119908 (2b)
where 119909 isin R2 is the state vector 119909 = [11990911199092]1015840 = 119879
120579 isin R120590 120590 isin N is the 120590-dimensional vector of unknownparameters of the true system the matrices 119860 119860
21 119860
22 119860
23
119863 119862 and 119864 are of appropriate dimensions known andhave the structure as below
119860 = [11988611
1
11988621
11988622
] 11986022
= [01times120590
11986022 0
]
119861 = [0
1198871199010
] 119862 = [1 0]
(3)
where 11986022 0
is a 120590-dimensional row vector and 1198871199010
isin R Thenthe high-frequency gain of (2a) and (2b) is 119887
0= 119887
1199010+ 119860
22 0120579
Since we consider a trajectory tracking control designproblem we make the following assumption about the refe-rence signal 119910
119889
Assumption 1 The reference trajectory 119910119889 is two times con-
tinuously differentiable Define vector119884119889= [119910
(0)
119889119910(1)
119889119910(2)
119889]1015840
and1198841198890
= [119910119889(0) 119910
(1)
119889(0)]
1015840
The signal119884119889is available for con-
trol design
To guarantee the stability of the closed-loop system andthe boundedness of the estimate of 120579 we assume there is an apriori convex compact set where the parameter vector 120579 liesin
Assumption 2 There exists a known smooth nonnegativeradially unbounded strictly convex function 119875 R120590
rarr Rsuch that the true value 120579 isin Θ = 120579 isin R120590
119875(120579) le 1 For all120579 isin Θ 119887
0= 119887
1199010+ 119860
22 0120579 gt 0
The control objective is to design a robust adaptive con-troller for system (1) such that the output 119910(119905) tracks adesired reference signal 119910
119889(119905) while rejecting the uncertain-
ties (119909(0) 120579 [0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W = R2
times Θ times
C times C times R2
times C comprises the initial state the truevalues of unknown parameters unmeasured disturbancesand themeasured disturbances while all signals in the closed-loop system are uniformly bounded We make the controlobjective precise as follows
Mathematical Problems in Engineering 3
Definition 3 A controller 120583 is said to achieve disturbanceattenuation level 120574 if there exist nonnegative functions119897(119905 120579 119909 119910
[0119905] 119879
[0119905]) and 119897
0(
0 120579
0) such that
sup(119909(0)120579119908
120596[0infin)119879119891[0infin)
119879119908[0infin)
119879[0infin)
)
119869119905
le 0 (4)
where
119869119905
= int
119905
0
((119862119909 minus 119910119889)2
+ 119897 minus 12057421003816100381610038161003816119879119908
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119879119891
10038161003816100381610038161003816
2
minus12057421003816100381610038161003816119908120596
1003816100381610038161003816
2
minus 1205742 1003816100381610038161003816119879
1003816100381610038161003816
2
) 119889120591
minus 120574210038161003816100381610038161003816120579 minus 120579
0
10038161003816100381610038161003816
2
1198760
minus 12057421003816100381610038161003816119909 (0) minus
0
1003816100381610038161003816
2
Πminus1
0
minus 1198970
(5)
where 1205790isin Θ is the initial guess of the unknown parameter
vector 1198760gt 0 is the quadratic weighting on the error bet-
ween the true value of 120579 and the initial guess 1205790quantifying
the level of confidence in the estimate 1205790
0is the initial guess
of the unknown initial state 119909(0) Πminus1
0gt 0 is the weighting
on the initial state-estimation error quantifying the level ofconfidence in the estimate
0 |119911|
119876denotes 119911
119879
119876119911 for anyvector 119911 and any symmetric matrix 119876
The control law to system (2a) (2b) is generated by thefollowing control law
119906 (119905) = 120583 (119905 119910[0119905]
[0119905]
) (6)
where 120583 [0infin) times L2times L
2rarr R We denote the class of
these admissible controllers byMClearly when the inequality (4) is achieved we have
1003817100381710038171003817119862119909 minus 119910119889
1003817100381710038171003817
2
2(1003817100381710038171003817119879119908
1003817100381710038171003817
2
2+10038171003817100381710038171003817119879119891
10038171003817100381710038171003817
2
2
+1003817100381710038171003817119908120596
1003817100381710038171003817
2
2+1003817100381710038171003817119879
1003817100381710038171003817
2
2
+10038161003816100381610038161003816120579 minus 120579
0
10038161003816100381610038161003816
2
1198760
+1003816100381610038161003816119909 (0) minus
0
1003816100381610038161003816
2
Πminus1
0
+ 1198620)
minus1
le 1205742
(7)
where sdot2denotesL
2norm and119862
0is a constantWhen
2
and 2are finite 119862119909 minus 119910
1198892is also finite which implies
lim119905rarrinfin
|119862119909 minus 119910119889| = 0 under additional assumptions
The following notation will be used throughout thispaper denotes the estimate of the current state of the sys-tem 119909 denotes the state-estimation error 119909minus 120579 denotes theestimate of the parameter vector 120579 120579 denotes the estimationerror 120579 minus 120579 any function symbol with an ldquoover barrdquo willdenote a function defined in the terms of the transformedstate variables such as 119891(119911) being the identical function as119891(119909) for any matrix 119872 the vector 997888rarr119872 is formed by stackingup its column vectors 119890
119895119894denotes a 119895-dimensional column
vector all of its elements are 0 except its 119894th row is 1 such as11989022
= [0 1]1015840
Let 120585 denote the expanded state vector 120585 = [1205791015840
1199091015840
]1015840 we
have the following expanded dynamics for system (2a) (2b)in view of
120579 = 0
120585 = [0 0
11991011986021+ 119906119860
22+ 119860
23 119860
] 120585
+ [0119861] 119906 + [
0119863]119908 + [
0]
= 119860120585 + 119861119906 + 119863119908 +
(8a)
119910 = [0 119862] 120585 + 119864119908 = 119862120585 + 119864119908 (8b)
The worst-case optimization of the cost function (4) canbe carried out in two steps as depicted in the followingequations
sup(119909(0)120579119908
[0infin)[0infin)
)isinW
119869119905
= sup119910[0infin)
[0infin)
sup(119909(0)120579119908
[0infin))|119910[0infin)
[0infin)
119869119905
(9)
The right-hand supremum operator will be carried out firstIt is the identification design step which will be presented inSection 3 Succinctly stated in this step we will calculate themaximum cost that is consistent with the givenmeasurementwaveform
The left-hand supremum operator will be carried outsecond It is the controller design step whichwill be discussedin Section 4 In this step we use a backstepping method todesign the control input 119906 and prove that all state variablesof the closed-loop system are uniformly bounded in time forany uniformly bounded disturbance input waveforms
This completes the formulation of the robust adaptivecontrol problem Next we turn to the identification designstep in the next section
3 Estimation Design
In this section we present the identification design for theadaptive control problem formulated
In this step themeasurement waveform 119910[0infin)
and [0infin)
are assumed to be known Since the control input is a causalfunction of 119910 and then it is known We ignore termsconsidered to be constant in the estimation design step andset 119897 in (5) to be |120585 minus 120585|
2
119876
+ 2(120585 minus 120585)1015840
1198972+ The equivalent cost
function of (5) is then given by
119869120574119905119891 =int
119905119891
0
(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+ 2(120585 minus 120585)1015840
1198972+ minus 120574
2
|119908|2
) d120591minus1205742100381610038161003816100381610038161205850
10038161003816100381610038161003816
2
1198760
(10)
where119876 is amatrix-valuedweighting function 120585 is the worst-case estimates for the expanded state 120585 119897
2is a design function
and is considered to be constant in the estimation designstep The cost function is then of a linear quadratic structureand the robust adaptive control problem becomes an 119867
infin-control of affine quadratic problem which admits a finitedimensional solution
4 Mathematical Problems in Engineering
We introduce the value function119882 = |120585minus 120585|2
Σminus1 and then
we can obtain the dynamics of state estimator 120585 and worst-case covariance matrix Σ as below
Σ = (119860 minus 1205772
119871119862)Σ + Σ(119860 minus 1205772
119871119862)1015840
+1
1205742119863119863
1015840
minus1
12057421205772
119871 1198711015840
minus Σ (1205742
1205772
1198621015840
119862 minus 1198621015840
119862 minus 119876)Σ
Σ (0) =1
1205742[119876
00
0119899times120590
Πminus1
0
]
minus1
(11a)
120585 = (119860 + Σ (1198621015840
119862 + 119876)) 120585 + 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585)
+ 119861119906 + minus Σ (1198621015840
119910119889+ 119876120585)
120585 (0) = [1205790
0
]
(11b)
where 120577 = 1(1198641198641015840
)12 and 119871 is defined as 119871 = [0 119871
1015840
]1015840 where
119871 = 1198631198641015840
Then the cost function (5) can be equivalently written as
119869119905
= minus10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σminus1(119905)
+ int
119905
0
(10038161003816100381610038161003816119862 120585 minus 119910
119889
10038161003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
120577210038161003816100381610038161003816119910 minus 119862 120585
10038161003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
)119889120591
(12)
where 120585119888= 120585 minus 120585 which will be determined later to improve
the performance of the adaptive system 119908lowast is the worst-casedisturbance given by
119908lowast
(120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)
= 1205772
1198641015840
(119910 minus 119862120585) +1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
(13)
We partition Σ as
Σ = [Σ Σ
12
Σ21
Σ22
] (14)
and introduceΦ = Σ21Σ
minus1 and Π = 1205742
(Σ22minus Σ
21Σ
minus1
Σ12)
For the boundedness ofΣ wemake the following assump-tion on the weighting matrix 119876
Assumption 4 The weighting matrix 119876 of function 119897 in (5) isgiven by
119876 = Σminus1
[0 00 Δ
]Σminus1
+ [120598Φ
1015840
1198621015840
(1205742
1205772
minus 1)119862Φ 00 0] (15)
where Δ is 2 times 2 positive-definite matrix and 120598 is a scalarfunction defined by
120598 (120591) =
Tr (Σminus1
(120591))
119870119888
119870119888ge 120574
2 Tr (1198760) 120591 ge 0 (16)
Then we have the following differential equation of ΣΦand Π
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ
Σ (0) =1
1205742119876
minus1
0
(17a)
Φ = (119860 minus 1205772
119871119862 minus1
1205742Π119862
1015840
(1205742
1205772
minus 1)119862)Φ
+ 11991011986021+ 119906119860
22+ 119860
23 Φ (0) = 0
(17b)
Π = (119860 minus 1205772
119871119862)Π+Π(119860 minus 1205772
119871119862)1015840
minusΠ1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ Π (0) = Π0
(17c)
The matrix Σ will play the role of worst-case covariancematrix of the parameter estimation error Assumption 4 guar-antees thatΣ is uniformly bounded fromabove anduniformlybounded frombelow away from 0 as depicted in the followinglemma and its proof is given in [14]
Lemma 5 Consider the dynamic equation (17a) for thecovariance mat-rix Σ Let Assumption 4 hold and 120574 ge 120577
minus1Then Σ is uniformly upper and lower bounded as follows
1
119870119888
le Σ (120591) le Σ (0) =1
1205742119876
minus1
0
1205742 Tr (119876
0) le Tr (Σminus1
(120591)) le 119870119888 forall120591 isin [0 119905]
(18)
We define 119904Σ(119905) = Tr((Σ(119905))minus1) and its dynamic is given
by
119904Σ= 120574
2
1205772
(1 minus 120598) 119862ΦΦ1015840
1198621015840
119904Σ(0) = 120574
2 Tr (1198760) (19)
Then 120598(120591) = 119870minus1
119888119904minus1
Σ(120591) which does not require the inversion
of ΣFrom Assumption 4 and (17a) we note that 120574 ge 120577
minus1 Thismeans the quantity 120577
minus1 is the ultimate lower bound on theachievable performance level for the adaptive system usingthe design method proposed in this paper
Assumption 6 If the matrix 119860 minus 1205772
119871119862 is Hurwitz thenthe desired disturbance attenuation level 120574 ge 120577
minus1 If thematrix119860minus120577
2
119871119862 is not Hurwitz then the desired disturbanceattenuation level 120574 gt 120577
minus1
Mathematical Problems in Engineering 5
Assumption 7 The initial weightingmatrixΠ0in (17c) is cho-
sen as the unique positive definite solution to the followingalgebraic Riccati equation
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)1015840
minus Π1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(20)
Then we note that the unique positive-definite solutionof (17c) is time-invariant and equal to the initial value Π
0
Remark 8 To simplify the estimator structure we can choose120598 = 1 so that Σ will be a constant positive-definite matrixand 119904
Σwill be a finite positive constant To further simplify
the identifier the initial weighting matrix Π0is chosen as
the unique positive-definite solutions to its algebraic Riccatiequation (17c) which also implies Σ gt 0 in view of Σ gt 0
To guarantee the boundedness of estimated parameterswithout persistently exciting signals we introduce soft pro-jection design on the parameter estimate We define
120588 = inf 119875 (120579) | 120579 isin R120590
1198871199010
+ 11986022 0
120579 = 0 (21)
By Assumption 2 and Lemma 2 in [23] we have 1 lt 120588 le infinFor any fixed 120588
119900isin (1 120588) we define the open set
Θ119900= 120579 isin R
120590
| 119875 (120579) lt 120588119900 (22)
Our control design will guarantee that the estimate 120579 lies inΘ
119900 which immediately implies |119887
1199010+ 119860
22 0
120579| gt 1198880
gt 0for some 119888
0gt 0 Moreover the convexity of 119875 implies the
following inequality
120597119875
120597120579( 120579) (120579 minus 120579) lt 0 forall 120579 isin R
120590
Θ (23)
We set 1198972= [minus(119875
119903( 120579))
1015840
0]1015840
where
119875119903( 120579) =
1198901(1minus119875(
120579))
((120597119875120597120579) ( 120579))1015840
(120588119900minus 119875 ( 120579))
3forall120579 isin Θ
119900 Θ
0 forall120579 isin Θ
(24)
Then we obtain
120585 = minus Σ[(119875119903( 120579))
1015840
0]1015840
+ 119860 120585 + 119861119906 minus Σ119876120585119888
+ 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585) +
120585 (0) = [ 1205791015840
01015840
0]1015840
(25)
where 120585119888= 120585 minus 120585
We summarize the estimation dynamics equations below
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)
minus Π1198621015840
(1205772
minus1
1205742)119862Π + 119863119863
1015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(26a)
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ Σ (0) =1
1205742119876
minus1
0
(26b)
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) 119862ΦΦ1015840
1198621015840
119904120590(0) = 120574
2 Tr (1198760)
(26c)
120598 =1
119870119888119904Σ
(26d)
119860119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus1
1205742) (26e)
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (26f)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus[Σ ΣΦ1015840
] 119876120585119888+120574
2
1205772
ΣΦ1015840
1198621015840
(119910 minus 119862) 120579 (0)= 1205790
(26g)
= minus ΦΣ119875119903( 120579) + 119860 minus (
1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862)
minus [ΦΣ1
1205742Π + ΦΣΦ
1015840
]119876120585119888
+ (11991011986021+ 119906119860
22+ 119860
23) 120579
+ 1205772
(Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871)
times (119910 minus 119862) + + 119861119906 (0) = 0
(26h)For the controller structure simplification the dynamics
for Φ can be implemented as below First we observe thematrix 119860
119891has the same structure as the matrix 119860 Then we
introduce the matrix119872
119891= [119860
11989111990121199012] (27)
where1199012is a 2-dimensional vector such that the pair (119860
119891 119901
2)
is controllable which implies that119872119891is invertible Then the
following prefiltering system for 119910 119906 and generates the Φonline
120578 = 119860119891120578 + 119901
2119910 120578 (0) = 0 (28a)
120582 = 119860119891120582 + 119901
2119906 120582 (0) = 0 (28b)
120578
= 119860119891120578
+ 1199012 120578
(0) = 0 (28c)
Φ = [119860119891120578 120578]119872
minus1
119891119860
21+ [119860
119891120582 120582]119872
minus1
119891119860
22
+ [119860119891120578
120578
]119872minus1
119891119860
23
(28d)
6 Mathematical Problems in Engineering
Associated with the above identifier introduce the valuefunction
119882(119905 120585 (119905) 120585 (119905) Σ (119905))
=10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σ
minus1
(119905)
=10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
+ 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
(29)
whose time derivative is given by
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 1205742
|119908|2
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
+ 2(120579 minus 120579)1015840
119875119903( 120579)
(30)
We note that the last term in
119882 is nonpositive zero on the setΘ and approaches minusinfin as 120579 approaches the boundary of theset Θ which guarantees the boundness of 120579
Then the cost function can be equivalently written as
119869119905
= 119869119905
+119882(0) minus119882 (119905) + int
119905
0
119882119889120591
= minus10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
minus 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
+ int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
times (120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591
(31)
This completes the identification design step
4 Control Design
In this section we describe the controller design for theuncertain system under consideration Note that we ignoredsome terms in the cost function (5) in the identification stepsince they are constant when 119910 and are given In the controldesign step we will include such terms Then based on thecost function (5) in the Section 2 the controller design is to
guarantee that the following supremum is less than or equalto zero for all measurement waveforms
sup(119909(0)120579119908[0infin)[0infin))isinW
119869119905
le sup119910[0infin)
[0infin)
int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591 minus 1198970
(32)
where function (120591 119910[0120591]
[0120591]
) is part of the weighting func-tion 119897(120591 120579 119909 119910
[0120591]
[0120591]) to be designed which is a constant
in the identifier design step and is therefore neglectedBy (32) we observe that the cost function is expressed
in term of the states of the estimator we derived whosedynamics are driven by the measurement 119910 input 119906 mea-sured disturbance and the worst-case estimate for theexpanded state vector 120585 which are signals we either measureor can constructThis is then a nonlinear119867infin-optimal controlproblem under full information measurements Instead ofconsidering 119910 and as the maximizing variable we canequivalently deal with the transformed variable
119907 = [120577 (119910 minus 119862)
] (33)
Then we have
120578 = 119860119891120578 + 119901
2119862 + 119901
2(1198901015840
21119907
120577) (34)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus [Σ ΣΦ1015840
] 119876120585119888+ 120574
2
ΣΦ1015840
1198621015840
1205771198901015840
21119907
(35)
= 119860 minus (1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862) + 119860
21
120579119862
minus ΦΣ119875119903( 120579) minus [ΦΣ
1
1205742Π + ΦΣΦ
1015840
]119876120585119888+ 119861119906
+ 11986022
120579119906 + ((120577minus2
11986021
120579 + Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871) 1205771198901015840
21
+11986023
1205791198901015840
22+ [0
119899times1] ) 119907
(36)
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
+ 2(120579 minus 120579)1015840
119875119903( 120579)
+ 1205742
||2
+ 1205742
|119908|2
minus 1205742
|119907|2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
(37)
Mathematical Problems in Engineering 7
The variables to be designed at this stage include 119906 and120585119888 The design for 120585
119888will be carried out last Note that the
structure of 119860 in the dynamics is in strict-feedback formwe will use the backstepping methodology [24] to designthe control input 119906 which will guarantee the global uniformboundedness of the closed-loop system states and the asymp-totic convergence of tracking error
Consider the dynamics of Φ
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (38)
For ease of the ensuing study we will separate Φ as the sumof several matrices as follows
Φ = Φ119906
+ Φ119910
+ Φ
(39a)
Φ119910
= [119860119891120578 120578]119872
minus1
119891119860
21= [
1205781015840
1198791
1205781015840
1198792
] (39b)
Φ119906
= 119860119891Φ
119906
+ 11990611986022 Φ
119906
(0) = 0 (39c)
Φ
= 119860119891Φ
+ 11986023 Φ
(0) = 0 (39d)
where 119879119894 119894 = 1 2 are 2 times 1-dimensional constant matrices
depending on119860119891119872
119891 and119860
21 ExpressΦ119906 andΦ in terms
of their row vectorsΦ119906
= [Φ1199061015840
1Φ
1199061015840
2]
1015840
andΦ
= [Φ1015840
1Φ
1015840
2]1015840
Then 119862Φ119910
= 1205781015840
1198791 119862Φ119906
= Φ119906
1 and 119862Φ
= Φ
1
We summarized the dynamics for backstepping design inthe following where we have emphasized the dependence ofvarious functions on the independent variables
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
119906
1+ Φ
1)
times (1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
(40a)
120598 =1
119870119888119904Σ
(40b)
Σ = minus (1 minus 120598) Σ(1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
times (1205742
1205772
minus 1) (1205781015840
1198791+ Φ
119906
1+ Φ
1) Σ
(40c)
120579 = 120575 (119910119889minus
1 120578 Φ
1 Φ
119906
1 120579
997888rarrΣ)
+ 120593(120578997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ 120581 (120578Φ
1 Φ
119906
1997888rarrΣ) 119907
(40d)
120578 = 119860119891120578 + 119901
21+ 119901
2(1198901015840
21119907
120577) (40e)
1=
2+ 119891
1(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(40f)
2= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
+ 1198912(119910
119889minus
1
1
2 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ)
+ 9848582(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ2( 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ) 119907
(40g)
Φ119906
1= 120595
1(Φ
119906
1) + Φ
119906
2 (40h)
Φ
1= 120603
1(Φ
1) + Φ
2+ 119890
1015840
21119860
231198901015840
22119907 (40i)
where the nonlinear functions 120575 1198911 and 119891
2are smooth as
long as 120579 isin Θ119900 the nonlinear functions 120593 120581 984858
1 984858
2 ℎ
1 ℎ
2
1205951 and 120603
1are smooth Here we use Φ
119906
1 Φ119906
2 Φ
1 and Φ
2
as independent variables instead of 120582 1205781 for the clarity of
ensuing analysisWe observe that the above dynamics is linear in 120585
119888 which
will be optimatized after backstepping design Σ 119904Σ Φ and
120579 will always be bounded by the design in Section 3 thenthey will not be stabilized in the control design Φ119906 is notnecessary bounded since the control input 119906 appeared intheir dynamics it can not stabilzed in conjunction with
using backstepping Hence we assume it is bounded andprove later that it is indeed so under the derived control law
The following backstepping design will achieve the 120574 levelof disturbance attenuation with respect to the disturbance 119907
Step 1 In this step we try to stabilize 120578 by virtual control law1= 119910
119889 Introduce variable 120578
119889 as the desired trajectory of 120578
which satisfies the dynamics
120578119889= 119860
119891120578119889+ 119901
2119910119889 120578
119889(0) = 0
2 times 1 (41)
Define the error variable 120578 = 120578 minus 120578119889 Then 120578 satisfies the
dynamics
120578 = 119860119891120578 + 119901
2(1198901015840
21119907
120577) + 119901
2(
1minus 119910
119889) (42)
By [14] the following holds
Lemma 9 Given any Hurwitz matrix 119860119891 there exists a
positive-definite matrix 119884 such that the following generalizedalgebraic Riccati equation admits a positive-definite solution119885
1198601015840
119891119885 + 119885119860
119891+
1
12057421205772119885119901
21199011015840
2119885 + 119884 = 0 (43)
Note that 119860119891in (42) is a Hurwitz matrix then we define
the following value function in terms of the positive-definitematrix 119885
1198810(120578) =
10038161003816100381610038161205781003816100381610038161003816
2
119885 (44)
Then its time derivative is given by
1198810= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781015840
119885119901119899(
1minus 119910
119889) (45)
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
applied to adaptive control problems [17ndash21] and offered apromising tool to system subjected to uncertainties
In this paper we study the adaptive control design forpermanent magnet DC motor based on worst-case analysisapproach We first model the permanent magnet DC motorservo system which is linear in all of the uncertainties Wethen formulate the robust adaptive control problem as anonlinear 119867infin-control problem under imperfect state mea-surements and apply the cost-to-come function analysis toderive the worst-case identifier and state estimator The con-trol design of the plant subsystem follows [22] and the adap-tive controller can be obtained by the integrator backstep-ping methodology The robust adaptive controller achievesasymptotic tracking if the disturbances are bounded andof finite energy and guarantees the stability of the closed-loop system with respect to the bounded disturbance inputsand the initial conditions Furthermore the close1d-loop sys-tem admits a guaranteed disturbance attenuation level withrespect to the exogenous disturbance inputs and the mea-sured disturbances where ultimate lower bound for theachievable attenuation performance level is only related tothe noise intensity in the measurement channel of the plantsystem
The balance of the paper is organized as follows InSection 2 we present the formulation of the adaptive controlproblem and discuss the general solution methodology thenwe obtain parameter identifier and state estimator usingthe cost-to-come function analysis in Section 3 In Section 4we derive the adaptive control law based on backsteppingmethod and present the main results on the robustness ofthe system in Section 5 The effectiveness of the proposedapproach is illustrated with an example in Section 6 and thepaper ends with some concluding remarks in Section 7
2 Problem Formulation
We consider a velocity-control problem of a brushed perma-nent magnet DC motor which is described by the followingdynamics
=
[[[
[
minus119863
119897
119869
119870119905
119869
minus119870
119890
119871minus119877
119871
]]]
]
119909 + [
0
1
119871
] 119906 + [
[
1
119869
0
]
]
119879+ (1a)
119910 = [1 0] 119909 + (1b)
where isin R2 is the state vector and represents load shaftangular speed and motor current respectively 119906 isin R is thescalar control input 119910 isin R is the load shaft angular speedmeasurement output 119871 isin R is the motor inductor 119877 isin R
is the armature resistance 119870119890is the back-emf constant 119870
119905is
the motor torque constant 119869 is the motor system inertia119863119897isin
R is the motor-system damping factor = [119879119908
120596
119879119891]1015840
is the arbitrary disturbance vector which is composed ofarbitrary disturbance torque 119879
119908isin R arbitrary disturbance
at the output measurement channel 120596
isin R and frictiondisturbance torque 119879
119891isin R 119879
isin R is the measured or
estimated disturbance torque In this paper we assume the
variables 119870119905 119870
119890 119869 119863
119897 119877 and 119871 are completely unknown or
partially unknownIt is easy to check that the true system is observable con-
trollable a minimal phase system and the transfer functionfrom 119906 to 119910 has relative degree 119903 = 2 And then we canfind a state diffeomorphism 119909 = and a disturbance trans-formation 119908 = by following [22] such that system (1a)and (1b) can be transformed into the following form in the 119909coordinates
119909 = 119860119909 + (11991011986021+ 119906119860
22+ 119860
23) 120579 + 119861119906 + 119863119908 + (2a)
119910 = 119862119909 + 119864119908 (2b)
where 119909 isin R2 is the state vector 119909 = [11990911199092]1015840 = 119879
120579 isin R120590 120590 isin N is the 120590-dimensional vector of unknownparameters of the true system the matrices 119860 119860
21 119860
22 119860
23
119863 119862 and 119864 are of appropriate dimensions known andhave the structure as below
119860 = [11988611
1
11988621
11988622
] 11986022
= [01times120590
11986022 0
]
119861 = [0
1198871199010
] 119862 = [1 0]
(3)
where 11986022 0
is a 120590-dimensional row vector and 1198871199010
isin R Thenthe high-frequency gain of (2a) and (2b) is 119887
0= 119887
1199010+ 119860
22 0120579
Since we consider a trajectory tracking control designproblem we make the following assumption about the refe-rence signal 119910
119889
Assumption 1 The reference trajectory 119910119889 is two times con-
tinuously differentiable Define vector119884119889= [119910
(0)
119889119910(1)
119889119910(2)
119889]1015840
and1198841198890
= [119910119889(0) 119910
(1)
119889(0)]
1015840
The signal119884119889is available for con-
trol design
To guarantee the stability of the closed-loop system andthe boundedness of the estimate of 120579 we assume there is an apriori convex compact set where the parameter vector 120579 liesin
Assumption 2 There exists a known smooth nonnegativeradially unbounded strictly convex function 119875 R120590
rarr Rsuch that the true value 120579 isin Θ = 120579 isin R120590
119875(120579) le 1 For all120579 isin Θ 119887
0= 119887
1199010+ 119860
22 0120579 gt 0
The control objective is to design a robust adaptive con-troller for system (1) such that the output 119910(119905) tracks adesired reference signal 119910
119889(119905) while rejecting the uncertain-
ties (119909(0) 120579 [0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W = R2
times Θ times
C times C times R2
times C comprises the initial state the truevalues of unknown parameters unmeasured disturbancesand themeasured disturbances while all signals in the closed-loop system are uniformly bounded We make the controlobjective precise as follows
Mathematical Problems in Engineering 3
Definition 3 A controller 120583 is said to achieve disturbanceattenuation level 120574 if there exist nonnegative functions119897(119905 120579 119909 119910
[0119905] 119879
[0119905]) and 119897
0(
0 120579
0) such that
sup(119909(0)120579119908
120596[0infin)119879119891[0infin)
119879119908[0infin)
119879[0infin)
)
119869119905
le 0 (4)
where
119869119905
= int
119905
0
((119862119909 minus 119910119889)2
+ 119897 minus 12057421003816100381610038161003816119879119908
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119879119891
10038161003816100381610038161003816
2
minus12057421003816100381610038161003816119908120596
1003816100381610038161003816
2
minus 1205742 1003816100381610038161003816119879
1003816100381610038161003816
2
) 119889120591
minus 120574210038161003816100381610038161003816120579 minus 120579
0
10038161003816100381610038161003816
2
1198760
minus 12057421003816100381610038161003816119909 (0) minus
0
1003816100381610038161003816
2
Πminus1
0
minus 1198970
(5)
where 1205790isin Θ is the initial guess of the unknown parameter
vector 1198760gt 0 is the quadratic weighting on the error bet-
ween the true value of 120579 and the initial guess 1205790quantifying
the level of confidence in the estimate 1205790
0is the initial guess
of the unknown initial state 119909(0) Πminus1
0gt 0 is the weighting
on the initial state-estimation error quantifying the level ofconfidence in the estimate
0 |119911|
119876denotes 119911
119879
119876119911 for anyvector 119911 and any symmetric matrix 119876
The control law to system (2a) (2b) is generated by thefollowing control law
119906 (119905) = 120583 (119905 119910[0119905]
[0119905]
) (6)
where 120583 [0infin) times L2times L
2rarr R We denote the class of
these admissible controllers byMClearly when the inequality (4) is achieved we have
1003817100381710038171003817119862119909 minus 119910119889
1003817100381710038171003817
2
2(1003817100381710038171003817119879119908
1003817100381710038171003817
2
2+10038171003817100381710038171003817119879119891
10038171003817100381710038171003817
2
2
+1003817100381710038171003817119908120596
1003817100381710038171003817
2
2+1003817100381710038171003817119879
1003817100381710038171003817
2
2
+10038161003816100381610038161003816120579 minus 120579
0
10038161003816100381610038161003816
2
1198760
+1003816100381610038161003816119909 (0) minus
0
1003816100381610038161003816
2
Πminus1
0
+ 1198620)
minus1
le 1205742
(7)
where sdot2denotesL
2norm and119862
0is a constantWhen
2
and 2are finite 119862119909 minus 119910
1198892is also finite which implies
lim119905rarrinfin
|119862119909 minus 119910119889| = 0 under additional assumptions
The following notation will be used throughout thispaper denotes the estimate of the current state of the sys-tem 119909 denotes the state-estimation error 119909minus 120579 denotes theestimate of the parameter vector 120579 120579 denotes the estimationerror 120579 minus 120579 any function symbol with an ldquoover barrdquo willdenote a function defined in the terms of the transformedstate variables such as 119891(119911) being the identical function as119891(119909) for any matrix 119872 the vector 997888rarr119872 is formed by stackingup its column vectors 119890
119895119894denotes a 119895-dimensional column
vector all of its elements are 0 except its 119894th row is 1 such as11989022
= [0 1]1015840
Let 120585 denote the expanded state vector 120585 = [1205791015840
1199091015840
]1015840 we
have the following expanded dynamics for system (2a) (2b)in view of
120579 = 0
120585 = [0 0
11991011986021+ 119906119860
22+ 119860
23 119860
] 120585
+ [0119861] 119906 + [
0119863]119908 + [
0]
= 119860120585 + 119861119906 + 119863119908 +
(8a)
119910 = [0 119862] 120585 + 119864119908 = 119862120585 + 119864119908 (8b)
The worst-case optimization of the cost function (4) canbe carried out in two steps as depicted in the followingequations
sup(119909(0)120579119908
[0infin)[0infin)
)isinW
119869119905
= sup119910[0infin)
[0infin)
sup(119909(0)120579119908
[0infin))|119910[0infin)
[0infin)
119869119905
(9)
The right-hand supremum operator will be carried out firstIt is the identification design step which will be presented inSection 3 Succinctly stated in this step we will calculate themaximum cost that is consistent with the givenmeasurementwaveform
The left-hand supremum operator will be carried outsecond It is the controller design step whichwill be discussedin Section 4 In this step we use a backstepping method todesign the control input 119906 and prove that all state variablesof the closed-loop system are uniformly bounded in time forany uniformly bounded disturbance input waveforms
This completes the formulation of the robust adaptivecontrol problem Next we turn to the identification designstep in the next section
3 Estimation Design
In this section we present the identification design for theadaptive control problem formulated
In this step themeasurement waveform 119910[0infin)
and [0infin)
are assumed to be known Since the control input is a causalfunction of 119910 and then it is known We ignore termsconsidered to be constant in the estimation design step andset 119897 in (5) to be |120585 minus 120585|
2
119876
+ 2(120585 minus 120585)1015840
1198972+ The equivalent cost
function of (5) is then given by
119869120574119905119891 =int
119905119891
0
(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+ 2(120585 minus 120585)1015840
1198972+ minus 120574
2
|119908|2
) d120591minus1205742100381610038161003816100381610038161205850
10038161003816100381610038161003816
2
1198760
(10)
where119876 is amatrix-valuedweighting function 120585 is the worst-case estimates for the expanded state 120585 119897
2is a design function
and is considered to be constant in the estimation designstep The cost function is then of a linear quadratic structureand the robust adaptive control problem becomes an 119867
infin-control of affine quadratic problem which admits a finitedimensional solution
4 Mathematical Problems in Engineering
We introduce the value function119882 = |120585minus 120585|2
Σminus1 and then
we can obtain the dynamics of state estimator 120585 and worst-case covariance matrix Σ as below
Σ = (119860 minus 1205772
119871119862)Σ + Σ(119860 minus 1205772
119871119862)1015840
+1
1205742119863119863
1015840
minus1
12057421205772
119871 1198711015840
minus Σ (1205742
1205772
1198621015840
119862 minus 1198621015840
119862 minus 119876)Σ
Σ (0) =1
1205742[119876
00
0119899times120590
Πminus1
0
]
minus1
(11a)
120585 = (119860 + Σ (1198621015840
119862 + 119876)) 120585 + 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585)
+ 119861119906 + minus Σ (1198621015840
119910119889+ 119876120585)
120585 (0) = [1205790
0
]
(11b)
where 120577 = 1(1198641198641015840
)12 and 119871 is defined as 119871 = [0 119871
1015840
]1015840 where
119871 = 1198631198641015840
Then the cost function (5) can be equivalently written as
119869119905
= minus10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σminus1(119905)
+ int
119905
0
(10038161003816100381610038161003816119862 120585 minus 119910
119889
10038161003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
120577210038161003816100381610038161003816119910 minus 119862 120585
10038161003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
)119889120591
(12)
where 120585119888= 120585 minus 120585 which will be determined later to improve
the performance of the adaptive system 119908lowast is the worst-casedisturbance given by
119908lowast
(120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)
= 1205772
1198641015840
(119910 minus 119862120585) +1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
(13)
We partition Σ as
Σ = [Σ Σ
12
Σ21
Σ22
] (14)
and introduceΦ = Σ21Σ
minus1 and Π = 1205742
(Σ22minus Σ
21Σ
minus1
Σ12)
For the boundedness ofΣ wemake the following assump-tion on the weighting matrix 119876
Assumption 4 The weighting matrix 119876 of function 119897 in (5) isgiven by
119876 = Σminus1
[0 00 Δ
]Σminus1
+ [120598Φ
1015840
1198621015840
(1205742
1205772
minus 1)119862Φ 00 0] (15)
where Δ is 2 times 2 positive-definite matrix and 120598 is a scalarfunction defined by
120598 (120591) =
Tr (Σminus1
(120591))
119870119888
119870119888ge 120574
2 Tr (1198760) 120591 ge 0 (16)
Then we have the following differential equation of ΣΦand Π
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ
Σ (0) =1
1205742119876
minus1
0
(17a)
Φ = (119860 minus 1205772
119871119862 minus1
1205742Π119862
1015840
(1205742
1205772
minus 1)119862)Φ
+ 11991011986021+ 119906119860
22+ 119860
23 Φ (0) = 0
(17b)
Π = (119860 minus 1205772
119871119862)Π+Π(119860 minus 1205772
119871119862)1015840
minusΠ1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ Π (0) = Π0
(17c)
The matrix Σ will play the role of worst-case covariancematrix of the parameter estimation error Assumption 4 guar-antees thatΣ is uniformly bounded fromabove anduniformlybounded frombelow away from 0 as depicted in the followinglemma and its proof is given in [14]
Lemma 5 Consider the dynamic equation (17a) for thecovariance mat-rix Σ Let Assumption 4 hold and 120574 ge 120577
minus1Then Σ is uniformly upper and lower bounded as follows
1
119870119888
le Σ (120591) le Σ (0) =1
1205742119876
minus1
0
1205742 Tr (119876
0) le Tr (Σminus1
(120591)) le 119870119888 forall120591 isin [0 119905]
(18)
We define 119904Σ(119905) = Tr((Σ(119905))minus1) and its dynamic is given
by
119904Σ= 120574
2
1205772
(1 minus 120598) 119862ΦΦ1015840
1198621015840
119904Σ(0) = 120574
2 Tr (1198760) (19)
Then 120598(120591) = 119870minus1
119888119904minus1
Σ(120591) which does not require the inversion
of ΣFrom Assumption 4 and (17a) we note that 120574 ge 120577
minus1 Thismeans the quantity 120577
minus1 is the ultimate lower bound on theachievable performance level for the adaptive system usingthe design method proposed in this paper
Assumption 6 If the matrix 119860 minus 1205772
119871119862 is Hurwitz thenthe desired disturbance attenuation level 120574 ge 120577
minus1 If thematrix119860minus120577
2
119871119862 is not Hurwitz then the desired disturbanceattenuation level 120574 gt 120577
minus1
Mathematical Problems in Engineering 5
Assumption 7 The initial weightingmatrixΠ0in (17c) is cho-
sen as the unique positive definite solution to the followingalgebraic Riccati equation
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)1015840
minus Π1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(20)
Then we note that the unique positive-definite solutionof (17c) is time-invariant and equal to the initial value Π
0
Remark 8 To simplify the estimator structure we can choose120598 = 1 so that Σ will be a constant positive-definite matrixand 119904
Σwill be a finite positive constant To further simplify
the identifier the initial weighting matrix Π0is chosen as
the unique positive-definite solutions to its algebraic Riccatiequation (17c) which also implies Σ gt 0 in view of Σ gt 0
To guarantee the boundedness of estimated parameterswithout persistently exciting signals we introduce soft pro-jection design on the parameter estimate We define
120588 = inf 119875 (120579) | 120579 isin R120590
1198871199010
+ 11986022 0
120579 = 0 (21)
By Assumption 2 and Lemma 2 in [23] we have 1 lt 120588 le infinFor any fixed 120588
119900isin (1 120588) we define the open set
Θ119900= 120579 isin R
120590
| 119875 (120579) lt 120588119900 (22)
Our control design will guarantee that the estimate 120579 lies inΘ
119900 which immediately implies |119887
1199010+ 119860
22 0
120579| gt 1198880
gt 0for some 119888
0gt 0 Moreover the convexity of 119875 implies the
following inequality
120597119875
120597120579( 120579) (120579 minus 120579) lt 0 forall 120579 isin R
120590
Θ (23)
We set 1198972= [minus(119875
119903( 120579))
1015840
0]1015840
where
119875119903( 120579) =
1198901(1minus119875(
120579))
((120597119875120597120579) ( 120579))1015840
(120588119900minus 119875 ( 120579))
3forall120579 isin Θ
119900 Θ
0 forall120579 isin Θ
(24)
Then we obtain
120585 = minus Σ[(119875119903( 120579))
1015840
0]1015840
+ 119860 120585 + 119861119906 minus Σ119876120585119888
+ 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585) +
120585 (0) = [ 1205791015840
01015840
0]1015840
(25)
where 120585119888= 120585 minus 120585
We summarize the estimation dynamics equations below
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)
minus Π1198621015840
(1205772
minus1
1205742)119862Π + 119863119863
1015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(26a)
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ Σ (0) =1
1205742119876
minus1
0
(26b)
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) 119862ΦΦ1015840
1198621015840
119904120590(0) = 120574
2 Tr (1198760)
(26c)
120598 =1
119870119888119904Σ
(26d)
119860119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus1
1205742) (26e)
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (26f)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus[Σ ΣΦ1015840
] 119876120585119888+120574
2
1205772
ΣΦ1015840
1198621015840
(119910 minus 119862) 120579 (0)= 1205790
(26g)
= minus ΦΣ119875119903( 120579) + 119860 minus (
1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862)
minus [ΦΣ1
1205742Π + ΦΣΦ
1015840
]119876120585119888
+ (11991011986021+ 119906119860
22+ 119860
23) 120579
+ 1205772
(Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871)
times (119910 minus 119862) + + 119861119906 (0) = 0
(26h)For the controller structure simplification the dynamics
for Φ can be implemented as below First we observe thematrix 119860
119891has the same structure as the matrix 119860 Then we
introduce the matrix119872
119891= [119860
11989111990121199012] (27)
where1199012is a 2-dimensional vector such that the pair (119860
119891 119901
2)
is controllable which implies that119872119891is invertible Then the
following prefiltering system for 119910 119906 and generates the Φonline
120578 = 119860119891120578 + 119901
2119910 120578 (0) = 0 (28a)
120582 = 119860119891120582 + 119901
2119906 120582 (0) = 0 (28b)
120578
= 119860119891120578
+ 1199012 120578
(0) = 0 (28c)
Φ = [119860119891120578 120578]119872
minus1
119891119860
21+ [119860
119891120582 120582]119872
minus1
119891119860
22
+ [119860119891120578
120578
]119872minus1
119891119860
23
(28d)
6 Mathematical Problems in Engineering
Associated with the above identifier introduce the valuefunction
119882(119905 120585 (119905) 120585 (119905) Σ (119905))
=10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σ
minus1
(119905)
=10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
+ 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
(29)
whose time derivative is given by
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 1205742
|119908|2
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
+ 2(120579 minus 120579)1015840
119875119903( 120579)
(30)
We note that the last term in
119882 is nonpositive zero on the setΘ and approaches minusinfin as 120579 approaches the boundary of theset Θ which guarantees the boundness of 120579
Then the cost function can be equivalently written as
119869119905
= 119869119905
+119882(0) minus119882 (119905) + int
119905
0
119882119889120591
= minus10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
minus 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
+ int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
times (120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591
(31)
This completes the identification design step
4 Control Design
In this section we describe the controller design for theuncertain system under consideration Note that we ignoredsome terms in the cost function (5) in the identification stepsince they are constant when 119910 and are given In the controldesign step we will include such terms Then based on thecost function (5) in the Section 2 the controller design is to
guarantee that the following supremum is less than or equalto zero for all measurement waveforms
sup(119909(0)120579119908[0infin)[0infin))isinW
119869119905
le sup119910[0infin)
[0infin)
int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591 minus 1198970
(32)
where function (120591 119910[0120591]
[0120591]
) is part of the weighting func-tion 119897(120591 120579 119909 119910
[0120591]
[0120591]) to be designed which is a constant
in the identifier design step and is therefore neglectedBy (32) we observe that the cost function is expressed
in term of the states of the estimator we derived whosedynamics are driven by the measurement 119910 input 119906 mea-sured disturbance and the worst-case estimate for theexpanded state vector 120585 which are signals we either measureor can constructThis is then a nonlinear119867infin-optimal controlproblem under full information measurements Instead ofconsidering 119910 and as the maximizing variable we canequivalently deal with the transformed variable
119907 = [120577 (119910 minus 119862)
] (33)
Then we have
120578 = 119860119891120578 + 119901
2119862 + 119901
2(1198901015840
21119907
120577) (34)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus [Σ ΣΦ1015840
] 119876120585119888+ 120574
2
ΣΦ1015840
1198621015840
1205771198901015840
21119907
(35)
= 119860 minus (1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862) + 119860
21
120579119862
minus ΦΣ119875119903( 120579) minus [ΦΣ
1
1205742Π + ΦΣΦ
1015840
]119876120585119888+ 119861119906
+ 11986022
120579119906 + ((120577minus2
11986021
120579 + Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871) 1205771198901015840
21
+11986023
1205791198901015840
22+ [0
119899times1] ) 119907
(36)
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
+ 2(120579 minus 120579)1015840
119875119903( 120579)
+ 1205742
||2
+ 1205742
|119908|2
minus 1205742
|119907|2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
(37)
Mathematical Problems in Engineering 7
The variables to be designed at this stage include 119906 and120585119888 The design for 120585
119888will be carried out last Note that the
structure of 119860 in the dynamics is in strict-feedback formwe will use the backstepping methodology [24] to designthe control input 119906 which will guarantee the global uniformboundedness of the closed-loop system states and the asymp-totic convergence of tracking error
Consider the dynamics of Φ
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (38)
For ease of the ensuing study we will separate Φ as the sumof several matrices as follows
Φ = Φ119906
+ Φ119910
+ Φ
(39a)
Φ119910
= [119860119891120578 120578]119872
minus1
119891119860
21= [
1205781015840
1198791
1205781015840
1198792
] (39b)
Φ119906
= 119860119891Φ
119906
+ 11990611986022 Φ
119906
(0) = 0 (39c)
Φ
= 119860119891Φ
+ 11986023 Φ
(0) = 0 (39d)
where 119879119894 119894 = 1 2 are 2 times 1-dimensional constant matrices
depending on119860119891119872
119891 and119860
21 ExpressΦ119906 andΦ in terms
of their row vectorsΦ119906
= [Φ1199061015840
1Φ
1199061015840
2]
1015840
andΦ
= [Φ1015840
1Φ
1015840
2]1015840
Then 119862Φ119910
= 1205781015840
1198791 119862Φ119906
= Φ119906
1 and 119862Φ
= Φ
1
We summarized the dynamics for backstepping design inthe following where we have emphasized the dependence ofvarious functions on the independent variables
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
119906
1+ Φ
1)
times (1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
(40a)
120598 =1
119870119888119904Σ
(40b)
Σ = minus (1 minus 120598) Σ(1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
times (1205742
1205772
minus 1) (1205781015840
1198791+ Φ
119906
1+ Φ
1) Σ
(40c)
120579 = 120575 (119910119889minus
1 120578 Φ
1 Φ
119906
1 120579
997888rarrΣ)
+ 120593(120578997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ 120581 (120578Φ
1 Φ
119906
1997888rarrΣ) 119907
(40d)
120578 = 119860119891120578 + 119901
21+ 119901
2(1198901015840
21119907
120577) (40e)
1=
2+ 119891
1(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(40f)
2= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
+ 1198912(119910
119889minus
1
1
2 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ)
+ 9848582(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ2( 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ) 119907
(40g)
Φ119906
1= 120595
1(Φ
119906
1) + Φ
119906
2 (40h)
Φ
1= 120603
1(Φ
1) + Φ
2+ 119890
1015840
21119860
231198901015840
22119907 (40i)
where the nonlinear functions 120575 1198911 and 119891
2are smooth as
long as 120579 isin Θ119900 the nonlinear functions 120593 120581 984858
1 984858
2 ℎ
1 ℎ
2
1205951 and 120603
1are smooth Here we use Φ
119906
1 Φ119906
2 Φ
1 and Φ
2
as independent variables instead of 120582 1205781 for the clarity of
ensuing analysisWe observe that the above dynamics is linear in 120585
119888 which
will be optimatized after backstepping design Σ 119904Σ Φ and
120579 will always be bounded by the design in Section 3 thenthey will not be stabilized in the control design Φ119906 is notnecessary bounded since the control input 119906 appeared intheir dynamics it can not stabilzed in conjunction with
using backstepping Hence we assume it is bounded andprove later that it is indeed so under the derived control law
The following backstepping design will achieve the 120574 levelof disturbance attenuation with respect to the disturbance 119907
Step 1 In this step we try to stabilize 120578 by virtual control law1= 119910
119889 Introduce variable 120578
119889 as the desired trajectory of 120578
which satisfies the dynamics
120578119889= 119860
119891120578119889+ 119901
2119910119889 120578
119889(0) = 0
2 times 1 (41)
Define the error variable 120578 = 120578 minus 120578119889 Then 120578 satisfies the
dynamics
120578 = 119860119891120578 + 119901
2(1198901015840
21119907
120577) + 119901
2(
1minus 119910
119889) (42)
By [14] the following holds
Lemma 9 Given any Hurwitz matrix 119860119891 there exists a
positive-definite matrix 119884 such that the following generalizedalgebraic Riccati equation admits a positive-definite solution119885
1198601015840
119891119885 + 119885119860
119891+
1
12057421205772119885119901
21199011015840
2119885 + 119884 = 0 (43)
Note that 119860119891in (42) is a Hurwitz matrix then we define
the following value function in terms of the positive-definitematrix 119885
1198810(120578) =
10038161003816100381610038161205781003816100381610038161003816
2
119885 (44)
Then its time derivative is given by
1198810= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781015840
119885119901119899(
1minus 119910
119889) (45)
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Definition 3 A controller 120583 is said to achieve disturbanceattenuation level 120574 if there exist nonnegative functions119897(119905 120579 119909 119910
[0119905] 119879
[0119905]) and 119897
0(
0 120579
0) such that
sup(119909(0)120579119908
120596[0infin)119879119891[0infin)
119879119908[0infin)
119879[0infin)
)
119869119905
le 0 (4)
where
119869119905
= int
119905
0
((119862119909 minus 119910119889)2
+ 119897 minus 12057421003816100381610038161003816119879119908
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119879119891
10038161003816100381610038161003816
2
minus12057421003816100381610038161003816119908120596
1003816100381610038161003816
2
minus 1205742 1003816100381610038161003816119879
1003816100381610038161003816
2
) 119889120591
minus 120574210038161003816100381610038161003816120579 minus 120579
0
10038161003816100381610038161003816
2
1198760
minus 12057421003816100381610038161003816119909 (0) minus
0
1003816100381610038161003816
2
Πminus1
0
minus 1198970
(5)
where 1205790isin Θ is the initial guess of the unknown parameter
vector 1198760gt 0 is the quadratic weighting on the error bet-
ween the true value of 120579 and the initial guess 1205790quantifying
the level of confidence in the estimate 1205790
0is the initial guess
of the unknown initial state 119909(0) Πminus1
0gt 0 is the weighting
on the initial state-estimation error quantifying the level ofconfidence in the estimate
0 |119911|
119876denotes 119911
119879
119876119911 for anyvector 119911 and any symmetric matrix 119876
The control law to system (2a) (2b) is generated by thefollowing control law
119906 (119905) = 120583 (119905 119910[0119905]
[0119905]
) (6)
where 120583 [0infin) times L2times L
2rarr R We denote the class of
these admissible controllers byMClearly when the inequality (4) is achieved we have
1003817100381710038171003817119862119909 minus 119910119889
1003817100381710038171003817
2
2(1003817100381710038171003817119879119908
1003817100381710038171003817
2
2+10038171003817100381710038171003817119879119891
10038171003817100381710038171003817
2
2
+1003817100381710038171003817119908120596
1003817100381710038171003817
2
2+1003817100381710038171003817119879
1003817100381710038171003817
2
2
+10038161003816100381610038161003816120579 minus 120579
0
10038161003816100381610038161003816
2
1198760
+1003816100381610038161003816119909 (0) minus
0
1003816100381610038161003816
2
Πminus1
0
+ 1198620)
minus1
le 1205742
(7)
where sdot2denotesL
2norm and119862
0is a constantWhen
2
and 2are finite 119862119909 minus 119910
1198892is also finite which implies
lim119905rarrinfin
|119862119909 minus 119910119889| = 0 under additional assumptions
The following notation will be used throughout thispaper denotes the estimate of the current state of the sys-tem 119909 denotes the state-estimation error 119909minus 120579 denotes theestimate of the parameter vector 120579 120579 denotes the estimationerror 120579 minus 120579 any function symbol with an ldquoover barrdquo willdenote a function defined in the terms of the transformedstate variables such as 119891(119911) being the identical function as119891(119909) for any matrix 119872 the vector 997888rarr119872 is formed by stackingup its column vectors 119890
119895119894denotes a 119895-dimensional column
vector all of its elements are 0 except its 119894th row is 1 such as11989022
= [0 1]1015840
Let 120585 denote the expanded state vector 120585 = [1205791015840
1199091015840
]1015840 we
have the following expanded dynamics for system (2a) (2b)in view of
120579 = 0
120585 = [0 0
11991011986021+ 119906119860
22+ 119860
23 119860
] 120585
+ [0119861] 119906 + [
0119863]119908 + [
0]
= 119860120585 + 119861119906 + 119863119908 +
(8a)
119910 = [0 119862] 120585 + 119864119908 = 119862120585 + 119864119908 (8b)
The worst-case optimization of the cost function (4) canbe carried out in two steps as depicted in the followingequations
sup(119909(0)120579119908
[0infin)[0infin)
)isinW
119869119905
= sup119910[0infin)
[0infin)
sup(119909(0)120579119908
[0infin))|119910[0infin)
[0infin)
119869119905
(9)
The right-hand supremum operator will be carried out firstIt is the identification design step which will be presented inSection 3 Succinctly stated in this step we will calculate themaximum cost that is consistent with the givenmeasurementwaveform
The left-hand supremum operator will be carried outsecond It is the controller design step whichwill be discussedin Section 4 In this step we use a backstepping method todesign the control input 119906 and prove that all state variablesof the closed-loop system are uniformly bounded in time forany uniformly bounded disturbance input waveforms
This completes the formulation of the robust adaptivecontrol problem Next we turn to the identification designstep in the next section
3 Estimation Design
In this section we present the identification design for theadaptive control problem formulated
In this step themeasurement waveform 119910[0infin)
and [0infin)
are assumed to be known Since the control input is a causalfunction of 119910 and then it is known We ignore termsconsidered to be constant in the estimation design step andset 119897 in (5) to be |120585 minus 120585|
2
119876
+ 2(120585 minus 120585)1015840
1198972+ The equivalent cost
function of (5) is then given by
119869120574119905119891 =int
119905119891
0
(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+ 2(120585 minus 120585)1015840
1198972+ minus 120574
2
|119908|2
) d120591minus1205742100381610038161003816100381610038161205850
10038161003816100381610038161003816
2
1198760
(10)
where119876 is amatrix-valuedweighting function 120585 is the worst-case estimates for the expanded state 120585 119897
2is a design function
and is considered to be constant in the estimation designstep The cost function is then of a linear quadratic structureand the robust adaptive control problem becomes an 119867
infin-control of affine quadratic problem which admits a finitedimensional solution
4 Mathematical Problems in Engineering
We introduce the value function119882 = |120585minus 120585|2
Σminus1 and then
we can obtain the dynamics of state estimator 120585 and worst-case covariance matrix Σ as below
Σ = (119860 minus 1205772
119871119862)Σ + Σ(119860 minus 1205772
119871119862)1015840
+1
1205742119863119863
1015840
minus1
12057421205772
119871 1198711015840
minus Σ (1205742
1205772
1198621015840
119862 minus 1198621015840
119862 minus 119876)Σ
Σ (0) =1
1205742[119876
00
0119899times120590
Πminus1
0
]
minus1
(11a)
120585 = (119860 + Σ (1198621015840
119862 + 119876)) 120585 + 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585)
+ 119861119906 + minus Σ (1198621015840
119910119889+ 119876120585)
120585 (0) = [1205790
0
]
(11b)
where 120577 = 1(1198641198641015840
)12 and 119871 is defined as 119871 = [0 119871
1015840
]1015840 where
119871 = 1198631198641015840
Then the cost function (5) can be equivalently written as
119869119905
= minus10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σminus1(119905)
+ int
119905
0
(10038161003816100381610038161003816119862 120585 minus 119910
119889
10038161003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
120577210038161003816100381610038161003816119910 minus 119862 120585
10038161003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
)119889120591
(12)
where 120585119888= 120585 minus 120585 which will be determined later to improve
the performance of the adaptive system 119908lowast is the worst-casedisturbance given by
119908lowast
(120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)
= 1205772
1198641015840
(119910 minus 119862120585) +1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
(13)
We partition Σ as
Σ = [Σ Σ
12
Σ21
Σ22
] (14)
and introduceΦ = Σ21Σ
minus1 and Π = 1205742
(Σ22minus Σ
21Σ
minus1
Σ12)
For the boundedness ofΣ wemake the following assump-tion on the weighting matrix 119876
Assumption 4 The weighting matrix 119876 of function 119897 in (5) isgiven by
119876 = Σminus1
[0 00 Δ
]Σminus1
+ [120598Φ
1015840
1198621015840
(1205742
1205772
minus 1)119862Φ 00 0] (15)
where Δ is 2 times 2 positive-definite matrix and 120598 is a scalarfunction defined by
120598 (120591) =
Tr (Σminus1
(120591))
119870119888
119870119888ge 120574
2 Tr (1198760) 120591 ge 0 (16)
Then we have the following differential equation of ΣΦand Π
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ
Σ (0) =1
1205742119876
minus1
0
(17a)
Φ = (119860 minus 1205772
119871119862 minus1
1205742Π119862
1015840
(1205742
1205772
minus 1)119862)Φ
+ 11991011986021+ 119906119860
22+ 119860
23 Φ (0) = 0
(17b)
Π = (119860 minus 1205772
119871119862)Π+Π(119860 minus 1205772
119871119862)1015840
minusΠ1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ Π (0) = Π0
(17c)
The matrix Σ will play the role of worst-case covariancematrix of the parameter estimation error Assumption 4 guar-antees thatΣ is uniformly bounded fromabove anduniformlybounded frombelow away from 0 as depicted in the followinglemma and its proof is given in [14]
Lemma 5 Consider the dynamic equation (17a) for thecovariance mat-rix Σ Let Assumption 4 hold and 120574 ge 120577
minus1Then Σ is uniformly upper and lower bounded as follows
1
119870119888
le Σ (120591) le Σ (0) =1
1205742119876
minus1
0
1205742 Tr (119876
0) le Tr (Σminus1
(120591)) le 119870119888 forall120591 isin [0 119905]
(18)
We define 119904Σ(119905) = Tr((Σ(119905))minus1) and its dynamic is given
by
119904Σ= 120574
2
1205772
(1 minus 120598) 119862ΦΦ1015840
1198621015840
119904Σ(0) = 120574
2 Tr (1198760) (19)
Then 120598(120591) = 119870minus1
119888119904minus1
Σ(120591) which does not require the inversion
of ΣFrom Assumption 4 and (17a) we note that 120574 ge 120577
minus1 Thismeans the quantity 120577
minus1 is the ultimate lower bound on theachievable performance level for the adaptive system usingthe design method proposed in this paper
Assumption 6 If the matrix 119860 minus 1205772
119871119862 is Hurwitz thenthe desired disturbance attenuation level 120574 ge 120577
minus1 If thematrix119860minus120577
2
119871119862 is not Hurwitz then the desired disturbanceattenuation level 120574 gt 120577
minus1
Mathematical Problems in Engineering 5
Assumption 7 The initial weightingmatrixΠ0in (17c) is cho-
sen as the unique positive definite solution to the followingalgebraic Riccati equation
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)1015840
minus Π1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(20)
Then we note that the unique positive-definite solutionof (17c) is time-invariant and equal to the initial value Π
0
Remark 8 To simplify the estimator structure we can choose120598 = 1 so that Σ will be a constant positive-definite matrixand 119904
Σwill be a finite positive constant To further simplify
the identifier the initial weighting matrix Π0is chosen as
the unique positive-definite solutions to its algebraic Riccatiequation (17c) which also implies Σ gt 0 in view of Σ gt 0
To guarantee the boundedness of estimated parameterswithout persistently exciting signals we introduce soft pro-jection design on the parameter estimate We define
120588 = inf 119875 (120579) | 120579 isin R120590
1198871199010
+ 11986022 0
120579 = 0 (21)
By Assumption 2 and Lemma 2 in [23] we have 1 lt 120588 le infinFor any fixed 120588
119900isin (1 120588) we define the open set
Θ119900= 120579 isin R
120590
| 119875 (120579) lt 120588119900 (22)
Our control design will guarantee that the estimate 120579 lies inΘ
119900 which immediately implies |119887
1199010+ 119860
22 0
120579| gt 1198880
gt 0for some 119888
0gt 0 Moreover the convexity of 119875 implies the
following inequality
120597119875
120597120579( 120579) (120579 minus 120579) lt 0 forall 120579 isin R
120590
Θ (23)
We set 1198972= [minus(119875
119903( 120579))
1015840
0]1015840
where
119875119903( 120579) =
1198901(1minus119875(
120579))
((120597119875120597120579) ( 120579))1015840
(120588119900minus 119875 ( 120579))
3forall120579 isin Θ
119900 Θ
0 forall120579 isin Θ
(24)
Then we obtain
120585 = minus Σ[(119875119903( 120579))
1015840
0]1015840
+ 119860 120585 + 119861119906 minus Σ119876120585119888
+ 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585) +
120585 (0) = [ 1205791015840
01015840
0]1015840
(25)
where 120585119888= 120585 minus 120585
We summarize the estimation dynamics equations below
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)
minus Π1198621015840
(1205772
minus1
1205742)119862Π + 119863119863
1015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(26a)
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ Σ (0) =1
1205742119876
minus1
0
(26b)
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) 119862ΦΦ1015840
1198621015840
119904120590(0) = 120574
2 Tr (1198760)
(26c)
120598 =1
119870119888119904Σ
(26d)
119860119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus1
1205742) (26e)
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (26f)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus[Σ ΣΦ1015840
] 119876120585119888+120574
2
1205772
ΣΦ1015840
1198621015840
(119910 minus 119862) 120579 (0)= 1205790
(26g)
= minus ΦΣ119875119903( 120579) + 119860 minus (
1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862)
minus [ΦΣ1
1205742Π + ΦΣΦ
1015840
]119876120585119888
+ (11991011986021+ 119906119860
22+ 119860
23) 120579
+ 1205772
(Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871)
times (119910 minus 119862) + + 119861119906 (0) = 0
(26h)For the controller structure simplification the dynamics
for Φ can be implemented as below First we observe thematrix 119860
119891has the same structure as the matrix 119860 Then we
introduce the matrix119872
119891= [119860
11989111990121199012] (27)
where1199012is a 2-dimensional vector such that the pair (119860
119891 119901
2)
is controllable which implies that119872119891is invertible Then the
following prefiltering system for 119910 119906 and generates the Φonline
120578 = 119860119891120578 + 119901
2119910 120578 (0) = 0 (28a)
120582 = 119860119891120582 + 119901
2119906 120582 (0) = 0 (28b)
120578
= 119860119891120578
+ 1199012 120578
(0) = 0 (28c)
Φ = [119860119891120578 120578]119872
minus1
119891119860
21+ [119860
119891120582 120582]119872
minus1
119891119860
22
+ [119860119891120578
120578
]119872minus1
119891119860
23
(28d)
6 Mathematical Problems in Engineering
Associated with the above identifier introduce the valuefunction
119882(119905 120585 (119905) 120585 (119905) Σ (119905))
=10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σ
minus1
(119905)
=10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
+ 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
(29)
whose time derivative is given by
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 1205742
|119908|2
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
+ 2(120579 minus 120579)1015840
119875119903( 120579)
(30)
We note that the last term in
119882 is nonpositive zero on the setΘ and approaches minusinfin as 120579 approaches the boundary of theset Θ which guarantees the boundness of 120579
Then the cost function can be equivalently written as
119869119905
= 119869119905
+119882(0) minus119882 (119905) + int
119905
0
119882119889120591
= minus10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
minus 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
+ int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
times (120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591
(31)
This completes the identification design step
4 Control Design
In this section we describe the controller design for theuncertain system under consideration Note that we ignoredsome terms in the cost function (5) in the identification stepsince they are constant when 119910 and are given In the controldesign step we will include such terms Then based on thecost function (5) in the Section 2 the controller design is to
guarantee that the following supremum is less than or equalto zero for all measurement waveforms
sup(119909(0)120579119908[0infin)[0infin))isinW
119869119905
le sup119910[0infin)
[0infin)
int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591 minus 1198970
(32)
where function (120591 119910[0120591]
[0120591]
) is part of the weighting func-tion 119897(120591 120579 119909 119910
[0120591]
[0120591]) to be designed which is a constant
in the identifier design step and is therefore neglectedBy (32) we observe that the cost function is expressed
in term of the states of the estimator we derived whosedynamics are driven by the measurement 119910 input 119906 mea-sured disturbance and the worst-case estimate for theexpanded state vector 120585 which are signals we either measureor can constructThis is then a nonlinear119867infin-optimal controlproblem under full information measurements Instead ofconsidering 119910 and as the maximizing variable we canequivalently deal with the transformed variable
119907 = [120577 (119910 minus 119862)
] (33)
Then we have
120578 = 119860119891120578 + 119901
2119862 + 119901
2(1198901015840
21119907
120577) (34)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus [Σ ΣΦ1015840
] 119876120585119888+ 120574
2
ΣΦ1015840
1198621015840
1205771198901015840
21119907
(35)
= 119860 minus (1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862) + 119860
21
120579119862
minus ΦΣ119875119903( 120579) minus [ΦΣ
1
1205742Π + ΦΣΦ
1015840
]119876120585119888+ 119861119906
+ 11986022
120579119906 + ((120577minus2
11986021
120579 + Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871) 1205771198901015840
21
+11986023
1205791198901015840
22+ [0
119899times1] ) 119907
(36)
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
+ 2(120579 minus 120579)1015840
119875119903( 120579)
+ 1205742
||2
+ 1205742
|119908|2
minus 1205742
|119907|2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
(37)
Mathematical Problems in Engineering 7
The variables to be designed at this stage include 119906 and120585119888 The design for 120585
119888will be carried out last Note that the
structure of 119860 in the dynamics is in strict-feedback formwe will use the backstepping methodology [24] to designthe control input 119906 which will guarantee the global uniformboundedness of the closed-loop system states and the asymp-totic convergence of tracking error
Consider the dynamics of Φ
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (38)
For ease of the ensuing study we will separate Φ as the sumof several matrices as follows
Φ = Φ119906
+ Φ119910
+ Φ
(39a)
Φ119910
= [119860119891120578 120578]119872
minus1
119891119860
21= [
1205781015840
1198791
1205781015840
1198792
] (39b)
Φ119906
= 119860119891Φ
119906
+ 11990611986022 Φ
119906
(0) = 0 (39c)
Φ
= 119860119891Φ
+ 11986023 Φ
(0) = 0 (39d)
where 119879119894 119894 = 1 2 are 2 times 1-dimensional constant matrices
depending on119860119891119872
119891 and119860
21 ExpressΦ119906 andΦ in terms
of their row vectorsΦ119906
= [Φ1199061015840
1Φ
1199061015840
2]
1015840
andΦ
= [Φ1015840
1Φ
1015840
2]1015840
Then 119862Φ119910
= 1205781015840
1198791 119862Φ119906
= Φ119906
1 and 119862Φ
= Φ
1
We summarized the dynamics for backstepping design inthe following where we have emphasized the dependence ofvarious functions on the independent variables
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
119906
1+ Φ
1)
times (1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
(40a)
120598 =1
119870119888119904Σ
(40b)
Σ = minus (1 minus 120598) Σ(1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
times (1205742
1205772
minus 1) (1205781015840
1198791+ Φ
119906
1+ Φ
1) Σ
(40c)
120579 = 120575 (119910119889minus
1 120578 Φ
1 Φ
119906
1 120579
997888rarrΣ)
+ 120593(120578997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ 120581 (120578Φ
1 Φ
119906
1997888rarrΣ) 119907
(40d)
120578 = 119860119891120578 + 119901
21+ 119901
2(1198901015840
21119907
120577) (40e)
1=
2+ 119891
1(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(40f)
2= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
+ 1198912(119910
119889minus
1
1
2 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ)
+ 9848582(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ2( 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ) 119907
(40g)
Φ119906
1= 120595
1(Φ
119906
1) + Φ
119906
2 (40h)
Φ
1= 120603
1(Φ
1) + Φ
2+ 119890
1015840
21119860
231198901015840
22119907 (40i)
where the nonlinear functions 120575 1198911 and 119891
2are smooth as
long as 120579 isin Θ119900 the nonlinear functions 120593 120581 984858
1 984858
2 ℎ
1 ℎ
2
1205951 and 120603
1are smooth Here we use Φ
119906
1 Φ119906
2 Φ
1 and Φ
2
as independent variables instead of 120582 1205781 for the clarity of
ensuing analysisWe observe that the above dynamics is linear in 120585
119888 which
will be optimatized after backstepping design Σ 119904Σ Φ and
120579 will always be bounded by the design in Section 3 thenthey will not be stabilized in the control design Φ119906 is notnecessary bounded since the control input 119906 appeared intheir dynamics it can not stabilzed in conjunction with
using backstepping Hence we assume it is bounded andprove later that it is indeed so under the derived control law
The following backstepping design will achieve the 120574 levelof disturbance attenuation with respect to the disturbance 119907
Step 1 In this step we try to stabilize 120578 by virtual control law1= 119910
119889 Introduce variable 120578
119889 as the desired trajectory of 120578
which satisfies the dynamics
120578119889= 119860
119891120578119889+ 119901
2119910119889 120578
119889(0) = 0
2 times 1 (41)
Define the error variable 120578 = 120578 minus 120578119889 Then 120578 satisfies the
dynamics
120578 = 119860119891120578 + 119901
2(1198901015840
21119907
120577) + 119901
2(
1minus 119910
119889) (42)
By [14] the following holds
Lemma 9 Given any Hurwitz matrix 119860119891 there exists a
positive-definite matrix 119884 such that the following generalizedalgebraic Riccati equation admits a positive-definite solution119885
1198601015840
119891119885 + 119885119860
119891+
1
12057421205772119885119901
21199011015840
2119885 + 119884 = 0 (43)
Note that 119860119891in (42) is a Hurwitz matrix then we define
the following value function in terms of the positive-definitematrix 119885
1198810(120578) =
10038161003816100381610038161205781003816100381610038161003816
2
119885 (44)
Then its time derivative is given by
1198810= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781015840
119885119901119899(
1minus 119910
119889) (45)
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
We introduce the value function119882 = |120585minus 120585|2
Σminus1 and then
we can obtain the dynamics of state estimator 120585 and worst-case covariance matrix Σ as below
Σ = (119860 minus 1205772
119871119862)Σ + Σ(119860 minus 1205772
119871119862)1015840
+1
1205742119863119863
1015840
minus1
12057421205772
119871 1198711015840
minus Σ (1205742
1205772
1198621015840
119862 minus 1198621015840
119862 minus 119876)Σ
Σ (0) =1
1205742[119876
00
0119899times120590
Πminus1
0
]
minus1
(11a)
120585 = (119860 + Σ (1198621015840
119862 + 119876)) 120585 + 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585)
+ 119861119906 + minus Σ (1198621015840
119910119889+ 119876120585)
120585 (0) = [1205790
0
]
(11b)
where 120577 = 1(1198641198641015840
)12 and 119871 is defined as 119871 = [0 119871
1015840
]1015840 where
119871 = 1198631198641015840
Then the cost function (5) can be equivalently written as
119869119905
= minus10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σminus1(119905)
+ int
119905
0
(10038161003816100381610038161003816119862 120585 minus 119910
119889
10038161003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
120577210038161003816100381610038161003816119910 minus 119862 120585
10038161003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
)119889120591
(12)
where 120585119888= 120585 minus 120585 which will be determined later to improve
the performance of the adaptive system 119908lowast is the worst-casedisturbance given by
119908lowast
(120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)
= 1205772
1198641015840
(119910 minus 119862120585) +1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
(13)
We partition Σ as
Σ = [Σ Σ
12
Σ21
Σ22
] (14)
and introduceΦ = Σ21Σ
minus1 and Π = 1205742
(Σ22minus Σ
21Σ
minus1
Σ12)
For the boundedness ofΣ wemake the following assump-tion on the weighting matrix 119876
Assumption 4 The weighting matrix 119876 of function 119897 in (5) isgiven by
119876 = Σminus1
[0 00 Δ
]Σminus1
+ [120598Φ
1015840
1198621015840
(1205742
1205772
minus 1)119862Φ 00 0] (15)
where Δ is 2 times 2 positive-definite matrix and 120598 is a scalarfunction defined by
120598 (120591) =
Tr (Σminus1
(120591))
119870119888
119870119888ge 120574
2 Tr (1198760) 120591 ge 0 (16)
Then we have the following differential equation of ΣΦand Π
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ
Σ (0) =1
1205742119876
minus1
0
(17a)
Φ = (119860 minus 1205772
119871119862 minus1
1205742Π119862
1015840
(1205742
1205772
minus 1)119862)Φ
+ 11991011986021+ 119906119860
22+ 119860
23 Φ (0) = 0
(17b)
Π = (119860 minus 1205772
119871119862)Π+Π(119860 minus 1205772
119871119862)1015840
minusΠ1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ Π (0) = Π0
(17c)
The matrix Σ will play the role of worst-case covariancematrix of the parameter estimation error Assumption 4 guar-antees thatΣ is uniformly bounded fromabove anduniformlybounded frombelow away from 0 as depicted in the followinglemma and its proof is given in [14]
Lemma 5 Consider the dynamic equation (17a) for thecovariance mat-rix Σ Let Assumption 4 hold and 120574 ge 120577
minus1Then Σ is uniformly upper and lower bounded as follows
1
119870119888
le Σ (120591) le Σ (0) =1
1205742119876
minus1
0
1205742 Tr (119876
0) le Tr (Σminus1
(120591)) le 119870119888 forall120591 isin [0 119905]
(18)
We define 119904Σ(119905) = Tr((Σ(119905))minus1) and its dynamic is given
by
119904Σ= 120574
2
1205772
(1 minus 120598) 119862ΦΦ1015840
1198621015840
119904Σ(0) = 120574
2 Tr (1198760) (19)
Then 120598(120591) = 119870minus1
119888119904minus1
Σ(120591) which does not require the inversion
of ΣFrom Assumption 4 and (17a) we note that 120574 ge 120577
minus1 Thismeans the quantity 120577
minus1 is the ultimate lower bound on theachievable performance level for the adaptive system usingthe design method proposed in this paper
Assumption 6 If the matrix 119860 minus 1205772
119871119862 is Hurwitz thenthe desired disturbance attenuation level 120574 ge 120577
minus1 If thematrix119860minus120577
2
119871119862 is not Hurwitz then the desired disturbanceattenuation level 120574 gt 120577
minus1
Mathematical Problems in Engineering 5
Assumption 7 The initial weightingmatrixΠ0in (17c) is cho-
sen as the unique positive definite solution to the followingalgebraic Riccati equation
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)1015840
minus Π1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(20)
Then we note that the unique positive-definite solutionof (17c) is time-invariant and equal to the initial value Π
0
Remark 8 To simplify the estimator structure we can choose120598 = 1 so that Σ will be a constant positive-definite matrixand 119904
Σwill be a finite positive constant To further simplify
the identifier the initial weighting matrix Π0is chosen as
the unique positive-definite solutions to its algebraic Riccatiequation (17c) which also implies Σ gt 0 in view of Σ gt 0
To guarantee the boundedness of estimated parameterswithout persistently exciting signals we introduce soft pro-jection design on the parameter estimate We define
120588 = inf 119875 (120579) | 120579 isin R120590
1198871199010
+ 11986022 0
120579 = 0 (21)
By Assumption 2 and Lemma 2 in [23] we have 1 lt 120588 le infinFor any fixed 120588
119900isin (1 120588) we define the open set
Θ119900= 120579 isin R
120590
| 119875 (120579) lt 120588119900 (22)
Our control design will guarantee that the estimate 120579 lies inΘ
119900 which immediately implies |119887
1199010+ 119860
22 0
120579| gt 1198880
gt 0for some 119888
0gt 0 Moreover the convexity of 119875 implies the
following inequality
120597119875
120597120579( 120579) (120579 minus 120579) lt 0 forall 120579 isin R
120590
Θ (23)
We set 1198972= [minus(119875
119903( 120579))
1015840
0]1015840
where
119875119903( 120579) =
1198901(1minus119875(
120579))
((120597119875120597120579) ( 120579))1015840
(120588119900minus 119875 ( 120579))
3forall120579 isin Θ
119900 Θ
0 forall120579 isin Θ
(24)
Then we obtain
120585 = minus Σ[(119875119903( 120579))
1015840
0]1015840
+ 119860 120585 + 119861119906 minus Σ119876120585119888
+ 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585) +
120585 (0) = [ 1205791015840
01015840
0]1015840
(25)
where 120585119888= 120585 minus 120585
We summarize the estimation dynamics equations below
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)
minus Π1198621015840
(1205772
minus1
1205742)119862Π + 119863119863
1015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(26a)
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ Σ (0) =1
1205742119876
minus1
0
(26b)
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) 119862ΦΦ1015840
1198621015840
119904120590(0) = 120574
2 Tr (1198760)
(26c)
120598 =1
119870119888119904Σ
(26d)
119860119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus1
1205742) (26e)
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (26f)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus[Σ ΣΦ1015840
] 119876120585119888+120574
2
1205772
ΣΦ1015840
1198621015840
(119910 minus 119862) 120579 (0)= 1205790
(26g)
= minus ΦΣ119875119903( 120579) + 119860 minus (
1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862)
minus [ΦΣ1
1205742Π + ΦΣΦ
1015840
]119876120585119888
+ (11991011986021+ 119906119860
22+ 119860
23) 120579
+ 1205772
(Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871)
times (119910 minus 119862) + + 119861119906 (0) = 0
(26h)For the controller structure simplification the dynamics
for Φ can be implemented as below First we observe thematrix 119860
119891has the same structure as the matrix 119860 Then we
introduce the matrix119872
119891= [119860
11989111990121199012] (27)
where1199012is a 2-dimensional vector such that the pair (119860
119891 119901
2)
is controllable which implies that119872119891is invertible Then the
following prefiltering system for 119910 119906 and generates the Φonline
120578 = 119860119891120578 + 119901
2119910 120578 (0) = 0 (28a)
120582 = 119860119891120582 + 119901
2119906 120582 (0) = 0 (28b)
120578
= 119860119891120578
+ 1199012 120578
(0) = 0 (28c)
Φ = [119860119891120578 120578]119872
minus1
119891119860
21+ [119860
119891120582 120582]119872
minus1
119891119860
22
+ [119860119891120578
120578
]119872minus1
119891119860
23
(28d)
6 Mathematical Problems in Engineering
Associated with the above identifier introduce the valuefunction
119882(119905 120585 (119905) 120585 (119905) Σ (119905))
=10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σ
minus1
(119905)
=10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
+ 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
(29)
whose time derivative is given by
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 1205742
|119908|2
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
+ 2(120579 minus 120579)1015840
119875119903( 120579)
(30)
We note that the last term in
119882 is nonpositive zero on the setΘ and approaches minusinfin as 120579 approaches the boundary of theset Θ which guarantees the boundness of 120579
Then the cost function can be equivalently written as
119869119905
= 119869119905
+119882(0) minus119882 (119905) + int
119905
0
119882119889120591
= minus10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
minus 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
+ int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
times (120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591
(31)
This completes the identification design step
4 Control Design
In this section we describe the controller design for theuncertain system under consideration Note that we ignoredsome terms in the cost function (5) in the identification stepsince they are constant when 119910 and are given In the controldesign step we will include such terms Then based on thecost function (5) in the Section 2 the controller design is to
guarantee that the following supremum is less than or equalto zero for all measurement waveforms
sup(119909(0)120579119908[0infin)[0infin))isinW
119869119905
le sup119910[0infin)
[0infin)
int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591 minus 1198970
(32)
where function (120591 119910[0120591]
[0120591]
) is part of the weighting func-tion 119897(120591 120579 119909 119910
[0120591]
[0120591]) to be designed which is a constant
in the identifier design step and is therefore neglectedBy (32) we observe that the cost function is expressed
in term of the states of the estimator we derived whosedynamics are driven by the measurement 119910 input 119906 mea-sured disturbance and the worst-case estimate for theexpanded state vector 120585 which are signals we either measureor can constructThis is then a nonlinear119867infin-optimal controlproblem under full information measurements Instead ofconsidering 119910 and as the maximizing variable we canequivalently deal with the transformed variable
119907 = [120577 (119910 minus 119862)
] (33)
Then we have
120578 = 119860119891120578 + 119901
2119862 + 119901
2(1198901015840
21119907
120577) (34)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus [Σ ΣΦ1015840
] 119876120585119888+ 120574
2
ΣΦ1015840
1198621015840
1205771198901015840
21119907
(35)
= 119860 minus (1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862) + 119860
21
120579119862
minus ΦΣ119875119903( 120579) minus [ΦΣ
1
1205742Π + ΦΣΦ
1015840
]119876120585119888+ 119861119906
+ 11986022
120579119906 + ((120577minus2
11986021
120579 + Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871) 1205771198901015840
21
+11986023
1205791198901015840
22+ [0
119899times1] ) 119907
(36)
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
+ 2(120579 minus 120579)1015840
119875119903( 120579)
+ 1205742
||2
+ 1205742
|119908|2
minus 1205742
|119907|2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
(37)
Mathematical Problems in Engineering 7
The variables to be designed at this stage include 119906 and120585119888 The design for 120585
119888will be carried out last Note that the
structure of 119860 in the dynamics is in strict-feedback formwe will use the backstepping methodology [24] to designthe control input 119906 which will guarantee the global uniformboundedness of the closed-loop system states and the asymp-totic convergence of tracking error
Consider the dynamics of Φ
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (38)
For ease of the ensuing study we will separate Φ as the sumof several matrices as follows
Φ = Φ119906
+ Φ119910
+ Φ
(39a)
Φ119910
= [119860119891120578 120578]119872
minus1
119891119860
21= [
1205781015840
1198791
1205781015840
1198792
] (39b)
Φ119906
= 119860119891Φ
119906
+ 11990611986022 Φ
119906
(0) = 0 (39c)
Φ
= 119860119891Φ
+ 11986023 Φ
(0) = 0 (39d)
where 119879119894 119894 = 1 2 are 2 times 1-dimensional constant matrices
depending on119860119891119872
119891 and119860
21 ExpressΦ119906 andΦ in terms
of their row vectorsΦ119906
= [Φ1199061015840
1Φ
1199061015840
2]
1015840
andΦ
= [Φ1015840
1Φ
1015840
2]1015840
Then 119862Φ119910
= 1205781015840
1198791 119862Φ119906
= Φ119906
1 and 119862Φ
= Φ
1
We summarized the dynamics for backstepping design inthe following where we have emphasized the dependence ofvarious functions on the independent variables
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
119906
1+ Φ
1)
times (1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
(40a)
120598 =1
119870119888119904Σ
(40b)
Σ = minus (1 minus 120598) Σ(1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
times (1205742
1205772
minus 1) (1205781015840
1198791+ Φ
119906
1+ Φ
1) Σ
(40c)
120579 = 120575 (119910119889minus
1 120578 Φ
1 Φ
119906
1 120579
997888rarrΣ)
+ 120593(120578997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ 120581 (120578Φ
1 Φ
119906
1997888rarrΣ) 119907
(40d)
120578 = 119860119891120578 + 119901
21+ 119901
2(1198901015840
21119907
120577) (40e)
1=
2+ 119891
1(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(40f)
2= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
+ 1198912(119910
119889minus
1
1
2 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ)
+ 9848582(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ2( 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ) 119907
(40g)
Φ119906
1= 120595
1(Φ
119906
1) + Φ
119906
2 (40h)
Φ
1= 120603
1(Φ
1) + Φ
2+ 119890
1015840
21119860
231198901015840
22119907 (40i)
where the nonlinear functions 120575 1198911 and 119891
2are smooth as
long as 120579 isin Θ119900 the nonlinear functions 120593 120581 984858
1 984858
2 ℎ
1 ℎ
2
1205951 and 120603
1are smooth Here we use Φ
119906
1 Φ119906
2 Φ
1 and Φ
2
as independent variables instead of 120582 1205781 for the clarity of
ensuing analysisWe observe that the above dynamics is linear in 120585
119888 which
will be optimatized after backstepping design Σ 119904Σ Φ and
120579 will always be bounded by the design in Section 3 thenthey will not be stabilized in the control design Φ119906 is notnecessary bounded since the control input 119906 appeared intheir dynamics it can not stabilzed in conjunction with
using backstepping Hence we assume it is bounded andprove later that it is indeed so under the derived control law
The following backstepping design will achieve the 120574 levelof disturbance attenuation with respect to the disturbance 119907
Step 1 In this step we try to stabilize 120578 by virtual control law1= 119910
119889 Introduce variable 120578
119889 as the desired trajectory of 120578
which satisfies the dynamics
120578119889= 119860
119891120578119889+ 119901
2119910119889 120578
119889(0) = 0
2 times 1 (41)
Define the error variable 120578 = 120578 minus 120578119889 Then 120578 satisfies the
dynamics
120578 = 119860119891120578 + 119901
2(1198901015840
21119907
120577) + 119901
2(
1minus 119910
119889) (42)
By [14] the following holds
Lemma 9 Given any Hurwitz matrix 119860119891 there exists a
positive-definite matrix 119884 such that the following generalizedalgebraic Riccati equation admits a positive-definite solution119885
1198601015840
119891119885 + 119885119860
119891+
1
12057421205772119885119901
21199011015840
2119885 + 119884 = 0 (43)
Note that 119860119891in (42) is a Hurwitz matrix then we define
the following value function in terms of the positive-definitematrix 119885
1198810(120578) =
10038161003816100381610038161205781003816100381610038161003816
2
119885 (44)
Then its time derivative is given by
1198810= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781015840
119885119901119899(
1minus 119910
119889) (45)
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Assumption 7 The initial weightingmatrixΠ0in (17c) is cho-
sen as the unique positive definite solution to the followingalgebraic Riccati equation
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)1015840
minus Π1198621015840
(1205772
minus1
1205742)119862Π
+ 1198631198631015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(20)
Then we note that the unique positive-definite solutionof (17c) is time-invariant and equal to the initial value Π
0
Remark 8 To simplify the estimator structure we can choose120598 = 1 so that Σ will be a constant positive-definite matrixand 119904
Σwill be a finite positive constant To further simplify
the identifier the initial weighting matrix Π0is chosen as
the unique positive-definite solutions to its algebraic Riccatiequation (17c) which also implies Σ gt 0 in view of Σ gt 0
To guarantee the boundedness of estimated parameterswithout persistently exciting signals we introduce soft pro-jection design on the parameter estimate We define
120588 = inf 119875 (120579) | 120579 isin R120590
1198871199010
+ 11986022 0
120579 = 0 (21)
By Assumption 2 and Lemma 2 in [23] we have 1 lt 120588 le infinFor any fixed 120588
119900isin (1 120588) we define the open set
Θ119900= 120579 isin R
120590
| 119875 (120579) lt 120588119900 (22)
Our control design will guarantee that the estimate 120579 lies inΘ
119900 which immediately implies |119887
1199010+ 119860
22 0
120579| gt 1198880
gt 0for some 119888
0gt 0 Moreover the convexity of 119875 implies the
following inequality
120597119875
120597120579( 120579) (120579 minus 120579) lt 0 forall 120579 isin R
120590
Θ (23)
We set 1198972= [minus(119875
119903( 120579))
1015840
0]1015840
where
119875119903( 120579) =
1198901(1minus119875(
120579))
((120597119875120597120579) ( 120579))1015840
(120588119900minus 119875 ( 120579))
3forall120579 isin Θ
119900 Θ
0 forall120579 isin Θ
(24)
Then we obtain
120585 = minus Σ[(119875119903( 120579))
1015840
0]1015840
+ 119860 120585 + 119861119906 minus Σ119876120585119888
+ 1205772
(1205742
Σ1198621015840
+ 119871) (119910 minus 119862 120585) +
120585 (0) = [ 1205791015840
01015840
0]1015840
(25)
where 120585119888= 120585 minus 120585
We summarize the estimation dynamics equations below
(119860 minus 1205772
119871119862)Π + Π(119860 minus 1205772
119871119862)
minus Π1198621015840
(1205772
minus1
1205742)119862Π + 119863119863
1015840
minus 1205772
1198711198711015840
+ 1205742
Δ = 0
(26a)
Σ = minus (1 minus 120598) ΣΦ1015840
1198621015840
(1205742
1205772
minus 1)119862ΦΣ Σ (0) =1
1205742119876
minus1
0
(26b)
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) 119862ΦΦ1015840
1198621015840
119904120590(0) = 120574
2 Tr (1198760)
(26c)
120598 =1
119870119888119904Σ
(26d)
119860119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus1
1205742) (26e)
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (26f)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus[Σ ΣΦ1015840
] 119876120585119888+120574
2
1205772
ΣΦ1015840
1198621015840
(119910 minus 119862) 120579 (0)= 1205790
(26g)
= minus ΦΣ119875119903( 120579) + 119860 minus (
1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862)
minus [ΦΣ1
1205742Π + ΦΣΦ
1015840
]119876120585119888
+ (11991011986021+ 119906119860
22+ 119860
23) 120579
+ 1205772
(Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871)
times (119910 minus 119862) + + 119861119906 (0) = 0
(26h)For the controller structure simplification the dynamics
for Φ can be implemented as below First we observe thematrix 119860
119891has the same structure as the matrix 119860 Then we
introduce the matrix119872
119891= [119860
11989111990121199012] (27)
where1199012is a 2-dimensional vector such that the pair (119860
119891 119901
2)
is controllable which implies that119872119891is invertible Then the
following prefiltering system for 119910 119906 and generates the Φonline
120578 = 119860119891120578 + 119901
2119910 120578 (0) = 0 (28a)
120582 = 119860119891120582 + 119901
2119906 120582 (0) = 0 (28b)
120578
= 119860119891120578
+ 1199012 120578
(0) = 0 (28c)
Φ = [119860119891120578 120578]119872
minus1
119891119860
21+ [119860
119891120582 120582]119872
minus1
119891119860
22
+ [119860119891120578
120578
]119872minus1
119891119860
23
(28d)
6 Mathematical Problems in Engineering
Associated with the above identifier introduce the valuefunction
119882(119905 120585 (119905) 120585 (119905) Σ (119905))
=10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σ
minus1
(119905)
=10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
+ 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
(29)
whose time derivative is given by
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 1205742
|119908|2
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
+ 2(120579 minus 120579)1015840
119875119903( 120579)
(30)
We note that the last term in
119882 is nonpositive zero on the setΘ and approaches minusinfin as 120579 approaches the boundary of theset Θ which guarantees the boundness of 120579
Then the cost function can be equivalently written as
119869119905
= 119869119905
+119882(0) minus119882 (119905) + int
119905
0
119882119889120591
= minus10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
minus 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
+ int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
times (120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591
(31)
This completes the identification design step
4 Control Design
In this section we describe the controller design for theuncertain system under consideration Note that we ignoredsome terms in the cost function (5) in the identification stepsince they are constant when 119910 and are given In the controldesign step we will include such terms Then based on thecost function (5) in the Section 2 the controller design is to
guarantee that the following supremum is less than or equalto zero for all measurement waveforms
sup(119909(0)120579119908[0infin)[0infin))isinW
119869119905
le sup119910[0infin)
[0infin)
int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591 minus 1198970
(32)
where function (120591 119910[0120591]
[0120591]
) is part of the weighting func-tion 119897(120591 120579 119909 119910
[0120591]
[0120591]) to be designed which is a constant
in the identifier design step and is therefore neglectedBy (32) we observe that the cost function is expressed
in term of the states of the estimator we derived whosedynamics are driven by the measurement 119910 input 119906 mea-sured disturbance and the worst-case estimate for theexpanded state vector 120585 which are signals we either measureor can constructThis is then a nonlinear119867infin-optimal controlproblem under full information measurements Instead ofconsidering 119910 and as the maximizing variable we canequivalently deal with the transformed variable
119907 = [120577 (119910 minus 119862)
] (33)
Then we have
120578 = 119860119891120578 + 119901
2119862 + 119901
2(1198901015840
21119907
120577) (34)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus [Σ ΣΦ1015840
] 119876120585119888+ 120574
2
ΣΦ1015840
1198621015840
1205771198901015840
21119907
(35)
= 119860 minus (1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862) + 119860
21
120579119862
minus ΦΣ119875119903( 120579) minus [ΦΣ
1
1205742Π + ΦΣΦ
1015840
]119876120585119888+ 119861119906
+ 11986022
120579119906 + ((120577minus2
11986021
120579 + Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871) 1205771198901015840
21
+11986023
1205791198901015840
22+ [0
119899times1] ) 119907
(36)
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
+ 2(120579 minus 120579)1015840
119875119903( 120579)
+ 1205742
||2
+ 1205742
|119908|2
minus 1205742
|119907|2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
(37)
Mathematical Problems in Engineering 7
The variables to be designed at this stage include 119906 and120585119888 The design for 120585
119888will be carried out last Note that the
structure of 119860 in the dynamics is in strict-feedback formwe will use the backstepping methodology [24] to designthe control input 119906 which will guarantee the global uniformboundedness of the closed-loop system states and the asymp-totic convergence of tracking error
Consider the dynamics of Φ
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (38)
For ease of the ensuing study we will separate Φ as the sumof several matrices as follows
Φ = Φ119906
+ Φ119910
+ Φ
(39a)
Φ119910
= [119860119891120578 120578]119872
minus1
119891119860
21= [
1205781015840
1198791
1205781015840
1198792
] (39b)
Φ119906
= 119860119891Φ
119906
+ 11990611986022 Φ
119906
(0) = 0 (39c)
Φ
= 119860119891Φ
+ 11986023 Φ
(0) = 0 (39d)
where 119879119894 119894 = 1 2 are 2 times 1-dimensional constant matrices
depending on119860119891119872
119891 and119860
21 ExpressΦ119906 andΦ in terms
of their row vectorsΦ119906
= [Φ1199061015840
1Φ
1199061015840
2]
1015840
andΦ
= [Φ1015840
1Φ
1015840
2]1015840
Then 119862Φ119910
= 1205781015840
1198791 119862Φ119906
= Φ119906
1 and 119862Φ
= Φ
1
We summarized the dynamics for backstepping design inthe following where we have emphasized the dependence ofvarious functions on the independent variables
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
119906
1+ Φ
1)
times (1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
(40a)
120598 =1
119870119888119904Σ
(40b)
Σ = minus (1 minus 120598) Σ(1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
times (1205742
1205772
minus 1) (1205781015840
1198791+ Φ
119906
1+ Φ
1) Σ
(40c)
120579 = 120575 (119910119889minus
1 120578 Φ
1 Φ
119906
1 120579
997888rarrΣ)
+ 120593(120578997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ 120581 (120578Φ
1 Φ
119906
1997888rarrΣ) 119907
(40d)
120578 = 119860119891120578 + 119901
21+ 119901
2(1198901015840
21119907
120577) (40e)
1=
2+ 119891
1(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(40f)
2= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
+ 1198912(119910
119889minus
1
1
2 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ)
+ 9848582(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ2( 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ) 119907
(40g)
Φ119906
1= 120595
1(Φ
119906
1) + Φ
119906
2 (40h)
Φ
1= 120603
1(Φ
1) + Φ
2+ 119890
1015840
21119860
231198901015840
22119907 (40i)
where the nonlinear functions 120575 1198911 and 119891
2are smooth as
long as 120579 isin Θ119900 the nonlinear functions 120593 120581 984858
1 984858
2 ℎ
1 ℎ
2
1205951 and 120603
1are smooth Here we use Φ
119906
1 Φ119906
2 Φ
1 and Φ
2
as independent variables instead of 120582 1205781 for the clarity of
ensuing analysisWe observe that the above dynamics is linear in 120585
119888 which
will be optimatized after backstepping design Σ 119904Σ Φ and
120579 will always be bounded by the design in Section 3 thenthey will not be stabilized in the control design Φ119906 is notnecessary bounded since the control input 119906 appeared intheir dynamics it can not stabilzed in conjunction with
using backstepping Hence we assume it is bounded andprove later that it is indeed so under the derived control law
The following backstepping design will achieve the 120574 levelof disturbance attenuation with respect to the disturbance 119907
Step 1 In this step we try to stabilize 120578 by virtual control law1= 119910
119889 Introduce variable 120578
119889 as the desired trajectory of 120578
which satisfies the dynamics
120578119889= 119860
119891120578119889+ 119901
2119910119889 120578
119889(0) = 0
2 times 1 (41)
Define the error variable 120578 = 120578 minus 120578119889 Then 120578 satisfies the
dynamics
120578 = 119860119891120578 + 119901
2(1198901015840
21119907
120577) + 119901
2(
1minus 119910
119889) (42)
By [14] the following holds
Lemma 9 Given any Hurwitz matrix 119860119891 there exists a
positive-definite matrix 119884 such that the following generalizedalgebraic Riccati equation admits a positive-definite solution119885
1198601015840
119891119885 + 119885119860
119891+
1
12057421205772119885119901
21199011015840
2119885 + 119884 = 0 (43)
Note that 119860119891in (42) is a Hurwitz matrix then we define
the following value function in terms of the positive-definitematrix 119885
1198810(120578) =
10038161003816100381610038161205781003816100381610038161003816
2
119885 (44)
Then its time derivative is given by
1198810= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781015840
119885119901119899(
1minus 119910
119889) (45)
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
Associated with the above identifier introduce the valuefunction
119882(119905 120585 (119905) 120585 (119905) Σ (119905))
=10038161003816100381610038161003816120585 (119905) minus 120585 (119905)
10038161003816100381610038161003816
2
Σ
minus1
(119905)
=10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
+ 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
(29)
whose time derivative is given by
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 1205742
|119908|2
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
+ 2(120579 minus 120579)1015840
119875119903( 120579)
(30)
We note that the last term in
119882 is nonpositive zero on the setΘ and approaches minusinfin as 120579 approaches the boundary of theset Θ which guarantees the boundness of 120579
Then the cost function can be equivalently written as
119869119905
= 119869119905
+119882(0) minus119882 (119905) + int
119905
0
119882119889120591
= minus10038161003816100381610038161003816120579 minus 120579 (119905)
10038161003816100381610038161003816
2
Σminus1
(119905)
minus 120574210038161003816100381610038161003816119909 (119905) minus (119905) minus Φ (119905) (120579 minus 120579 (119905))
10038161003816100381610038161003816
2
Πminus1
+ int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
times (120591 119906[0120591]
119910[0120591]
[0120591]
120585[0120591]
120585[0120591]
120585[0120591]
)10038161003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591
(31)
This completes the identification design step
4 Control Design
In this section we describe the controller design for theuncertain system under consideration Note that we ignoredsome terms in the cost function (5) in the identification stepsince they are constant when 119910 and are given In the controldesign step we will include such terms Then based on thecost function (5) in the Section 2 the controller design is to
guarantee that the following supremum is less than or equalto zero for all measurement waveforms
sup(119909(0)120579119908[0infin)[0infin))isinW
119869119905
le sup119910[0infin)
[0infin)
int
119905
0
(1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
minus 1205742
12057721003816100381610038161003816119910 minus 119862
1003816100381610038161003816
2
+ minus 1205742
||2
) 119889120591 minus 1198970
(32)
where function (120591 119910[0120591]
[0120591]
) is part of the weighting func-tion 119897(120591 120579 119909 119910
[0120591]
[0120591]) to be designed which is a constant
in the identifier design step and is therefore neglectedBy (32) we observe that the cost function is expressed
in term of the states of the estimator we derived whosedynamics are driven by the measurement 119910 input 119906 mea-sured disturbance and the worst-case estimate for theexpanded state vector 120585 which are signals we either measureor can constructThis is then a nonlinear119867infin-optimal controlproblem under full information measurements Instead ofconsidering 119910 and as the maximizing variable we canequivalently deal with the transformed variable
119907 = [120577 (119910 minus 119862)
] (33)
Then we have
120578 = 119860119891120578 + 119901
2119862 + 119901
2(1198901015840
21119907
120577) (34)
120579 = minus Σ119875119903( 120579) minus ΣΦ
1015840
1198621015840
(119910119889minus 119862)
minus [Σ ΣΦ1015840
] 119876120585119888+ 120574
2
ΣΦ1015840
1198621015840
1205771198901015840
21119907
(35)
= 119860 minus (1
1205742Π + ΦΣΦ
1015840
)1198621015840
(119910119889minus 119862) + 119860
21
120579119862
minus ΦΣ119875119903( 120579) minus [ΦΣ
1
1205742Π + ΦΣΦ
1015840
]119876120585119888+ 119861119906
+ 11986022
120579119906 + ((120577minus2
11986021
120579 + Π1198621015840
+ 1205742
ΦΣΦ1015840
1198621015840
+ 119871) 1205771198901015840
21
+11986023
1205791198901015840
22+ [0
119899times1] ) 119907
(36)
119882 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+1003816100381610038161003816119862 minus 119910
119889
1003816100381610038161003816
2
+1003816100381610038161003816120585119888
1003816100381610038161003816
2
119876
+ 2(120579 minus 120579)1015840
119875119903( 120579)
+ 1205742
||2
+ 1205742
|119908|2
minus 1205742
|119907|2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
(37)
Mathematical Problems in Engineering 7
The variables to be designed at this stage include 119906 and120585119888 The design for 120585
119888will be carried out last Note that the
structure of 119860 in the dynamics is in strict-feedback formwe will use the backstepping methodology [24] to designthe control input 119906 which will guarantee the global uniformboundedness of the closed-loop system states and the asymp-totic convergence of tracking error
Consider the dynamics of Φ
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (38)
For ease of the ensuing study we will separate Φ as the sumof several matrices as follows
Φ = Φ119906
+ Φ119910
+ Φ
(39a)
Φ119910
= [119860119891120578 120578]119872
minus1
119891119860
21= [
1205781015840
1198791
1205781015840
1198792
] (39b)
Φ119906
= 119860119891Φ
119906
+ 11990611986022 Φ
119906
(0) = 0 (39c)
Φ
= 119860119891Φ
+ 11986023 Φ
(0) = 0 (39d)
where 119879119894 119894 = 1 2 are 2 times 1-dimensional constant matrices
depending on119860119891119872
119891 and119860
21 ExpressΦ119906 andΦ in terms
of their row vectorsΦ119906
= [Φ1199061015840
1Φ
1199061015840
2]
1015840
andΦ
= [Φ1015840
1Φ
1015840
2]1015840
Then 119862Φ119910
= 1205781015840
1198791 119862Φ119906
= Φ119906
1 and 119862Φ
= Φ
1
We summarized the dynamics for backstepping design inthe following where we have emphasized the dependence ofvarious functions on the independent variables
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
119906
1+ Φ
1)
times (1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
(40a)
120598 =1
119870119888119904Σ
(40b)
Σ = minus (1 minus 120598) Σ(1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
times (1205742
1205772
minus 1) (1205781015840
1198791+ Φ
119906
1+ Φ
1) Σ
(40c)
120579 = 120575 (119910119889minus
1 120578 Φ
1 Φ
119906
1 120579
997888rarrΣ)
+ 120593(120578997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ 120581 (120578Φ
1 Φ
119906
1997888rarrΣ) 119907
(40d)
120578 = 119860119891120578 + 119901
21+ 119901
2(1198901015840
21119907
120577) (40e)
1=
2+ 119891
1(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(40f)
2= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
+ 1198912(119910
119889minus
1
1
2 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ)
+ 9848582(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ2( 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ) 119907
(40g)
Φ119906
1= 120595
1(Φ
119906
1) + Φ
119906
2 (40h)
Φ
1= 120603
1(Φ
1) + Φ
2+ 119890
1015840
21119860
231198901015840
22119907 (40i)
where the nonlinear functions 120575 1198911 and 119891
2are smooth as
long as 120579 isin Θ119900 the nonlinear functions 120593 120581 984858
1 984858
2 ℎ
1 ℎ
2
1205951 and 120603
1are smooth Here we use Φ
119906
1 Φ119906
2 Φ
1 and Φ
2
as independent variables instead of 120582 1205781 for the clarity of
ensuing analysisWe observe that the above dynamics is linear in 120585
119888 which
will be optimatized after backstepping design Σ 119904Σ Φ and
120579 will always be bounded by the design in Section 3 thenthey will not be stabilized in the control design Φ119906 is notnecessary bounded since the control input 119906 appeared intheir dynamics it can not stabilzed in conjunction with
using backstepping Hence we assume it is bounded andprove later that it is indeed so under the derived control law
The following backstepping design will achieve the 120574 levelof disturbance attenuation with respect to the disturbance 119907
Step 1 In this step we try to stabilize 120578 by virtual control law1= 119910
119889 Introduce variable 120578
119889 as the desired trajectory of 120578
which satisfies the dynamics
120578119889= 119860
119891120578119889+ 119901
2119910119889 120578
119889(0) = 0
2 times 1 (41)
Define the error variable 120578 = 120578 minus 120578119889 Then 120578 satisfies the
dynamics
120578 = 119860119891120578 + 119901
2(1198901015840
21119907
120577) + 119901
2(
1minus 119910
119889) (42)
By [14] the following holds
Lemma 9 Given any Hurwitz matrix 119860119891 there exists a
positive-definite matrix 119884 such that the following generalizedalgebraic Riccati equation admits a positive-definite solution119885
1198601015840
119891119885 + 119885119860
119891+
1
12057421205772119885119901
21199011015840
2119885 + 119884 = 0 (43)
Note that 119860119891in (42) is a Hurwitz matrix then we define
the following value function in terms of the positive-definitematrix 119885
1198810(120578) =
10038161003816100381610038161205781003816100381610038161003816
2
119885 (44)
Then its time derivative is given by
1198810= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781015840
119885119901119899(
1minus 119910
119889) (45)
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
The variables to be designed at this stage include 119906 and120585119888 The design for 120585
119888will be carried out last Note that the
structure of 119860 in the dynamics is in strict-feedback formwe will use the backstepping methodology [24] to designthe control input 119906 which will guarantee the global uniformboundedness of the closed-loop system states and the asymp-totic convergence of tracking error
Consider the dynamics of Φ
Φ = 119860119891Φ + 119910119860
21+ 119906119860
22+ 119860
23 Φ (0) = 0 (38)
For ease of the ensuing study we will separate Φ as the sumof several matrices as follows
Φ = Φ119906
+ Φ119910
+ Φ
(39a)
Φ119910
= [119860119891120578 120578]119872
minus1
119891119860
21= [
1205781015840
1198791
1205781015840
1198792
] (39b)
Φ119906
= 119860119891Φ
119906
+ 11990611986022 Φ
119906
(0) = 0 (39c)
Φ
= 119860119891Φ
+ 11986023 Φ
(0) = 0 (39d)
where 119879119894 119894 = 1 2 are 2 times 1-dimensional constant matrices
depending on119860119891119872
119891 and119860
21 ExpressΦ119906 andΦ in terms
of their row vectorsΦ119906
= [Φ1199061015840
1Φ
1199061015840
2]
1015840
andΦ
= [Φ1015840
1Φ
1015840
2]1015840
Then 119862Φ119910
= 1205781015840
1198791 119862Φ119906
= Φ119906
1 and 119862Φ
= Φ
1
We summarized the dynamics for backstepping design inthe following where we have emphasized the dependence ofvarious functions on the independent variables
119904Σ= (120574
2
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
119906
1+ Φ
1)
times (1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
(40a)
120598 =1
119870119888119904Σ
(40b)
Σ = minus (1 minus 120598) Σ(1205781015840
1198791+ Φ
119906
1+ Φ
1)1015840
times (1205742
1205772
minus 1) (1205781015840
1198791+ Φ
119906
1+ Φ
1) Σ
(40c)
120579 = 120575 (119910119889minus
1 120578 Φ
1 Φ
119906
1 120579
997888rarrΣ)
+ 120593(120578997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ 120581 (120578Φ
1 Φ
119906
1997888rarrΣ) 119907
(40d)
120578 = 119860119891120578 + 119901
21+ 119901
2(1198901015840
21119907
120577) (40e)
1=
2+ 119891
1(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(40f)
2= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
+ 1198912(119910
119889minus
1
1
2 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ)
+ 9848582(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888
+ ℎ2( 120579 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
2997888rarrΣ) 119907
(40g)
Φ119906
1= 120595
1(Φ
119906
1) + Φ
119906
2 (40h)
Φ
1= 120603
1(Φ
1) + Φ
2+ 119890
1015840
21119860
231198901015840
22119907 (40i)
where the nonlinear functions 120575 1198911 and 119891
2are smooth as
long as 120579 isin Θ119900 the nonlinear functions 120593 120581 984858
1 984858
2 ℎ
1 ℎ
2
1205951 and 120603
1are smooth Here we use Φ
119906
1 Φ119906
2 Φ
1 and Φ
2
as independent variables instead of 120582 1205781 for the clarity of
ensuing analysisWe observe that the above dynamics is linear in 120585
119888 which
will be optimatized after backstepping design Σ 119904Σ Φ and
120579 will always be bounded by the design in Section 3 thenthey will not be stabilized in the control design Φ119906 is notnecessary bounded since the control input 119906 appeared intheir dynamics it can not stabilzed in conjunction with
using backstepping Hence we assume it is bounded andprove later that it is indeed so under the derived control law
The following backstepping design will achieve the 120574 levelof disturbance attenuation with respect to the disturbance 119907
Step 1 In this step we try to stabilize 120578 by virtual control law1= 119910
119889 Introduce variable 120578
119889 as the desired trajectory of 120578
which satisfies the dynamics
120578119889= 119860
119891120578119889+ 119901
2119910119889 120578
119889(0) = 0
2 times 1 (41)
Define the error variable 120578 = 120578 minus 120578119889 Then 120578 satisfies the
dynamics
120578 = 119860119891120578 + 119901
2(1198901015840
21119907
120577) + 119901
2(
1minus 119910
119889) (42)
By [14] the following holds
Lemma 9 Given any Hurwitz matrix 119860119891 there exists a
positive-definite matrix 119884 such that the following generalizedalgebraic Riccati equation admits a positive-definite solution119885
1198601015840
119891119885 + 119885119860
119891+
1
12057421205772119885119901
21199011015840
2119885 + 119884 = 0 (43)
Note that 119860119891in (42) is a Hurwitz matrix then we define
the following value function in terms of the positive-definitematrix 119885
1198810(120578) =
10038161003816100381610038161205781003816100381610038161003816
2
119885 (44)
Then its time derivative is given by
1198810= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781015840
119885119901119899(
1minus 119910
119889) (45)
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
where
1205840(120578) =
1
1205742120577119890211199011015840
2119885120578 (46)
If 1is control input then we may choose the control law
1= 119910
119889 (47)
and the design achieves attenuation level 120574 from the distur-bance 119907 to the output 11988412
(120578 minus 120578119889) This completes the virtual
control design for the 120578 dynamics
Step 2 Define the transformed variable
1199111=
1minus 119910
119889 (48)
which is the deviation of 1from its desired trajectory 119910
119889
Then the time derivative of 1199111is given by
1199111= 119891
1(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) +
2minus 119910
(1)
119889
+ 9848581(120578
997888rarrΦ
997888rarrΦ
119906
997888rarrΣ)119876120585
119888+ ℎ
1( 120579 120578 Φ
1 Φ
119906
1997888rarrΣ) 119907
(49)
where the function 1198911is defined as
1198911(119911
1 119910
119889 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
= 1198911(119910
119889minus
1
1 120579 120578 Φ
1 Φ
119906
1997888rarrΣ)
(50)
Introduce the value function for this step
1198811= 119881
0+1
21199112
1(51)
whose derivative is given by
1198811= minus
10038161003816100381610038161205781003816100381610038161003816
2
119884+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
0
1003816100381610038161003816
2
+ 21205781198851199011198991199111
+ 1199111(
2minus 119910
(1)
119889+ 119891
1+ 984858
1119876120585
119888+ ℎ
1119907)
= minus1199112
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 119911
11199112+ 120589
1015840
1119876120585
119888
+ 1205742
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
1
1003816100381610038161003816
2
(52)
where
1199112=
2minus 119910
(1)
119889minus 120572
1 (53a)
1205841(119911
1 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ) = 120584
0+
1
21205742ℎ1015840
11199111 (53b)
1205721(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) = minus 119911
1minus 120573
11199111minus 2119901
1015840
119899119885120578
minus 1198911minusℎ
11205840minus
1
41205742ℎ1ℎ1015840
11199111
(53c)
1205731(119911
1 119910
119889 120579 120578 120578 Φ
1 Φ
119906
1997888rarrΣ 119904) ge 119888
1205731
gt 0 (53d)
1205891(119911
1 119910
119889 120578 120578 Φ
1 Φ
119906
997888rarrΣ) = 984858
1015840
11199111 (53e)
where 1198881205731
is any positive constant and the nonlinear function1205731is to be chosen by the designer Note that the function 120572
1is
smooth as long as 120579 isin Θ119900 If
2were the actual controls then
we would choose the following control law
2= 119910
(1)
119889+ 120572
1 (54)
and set 120585119888= 0 to guarantee the dissipation inequality with
supply rate
minus10038161003816100381610038161
minus 119910119889
1003816100381610038161003816
2
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus 120573
11199112
1+ 120574
2
1199072
(55)
This completes the second step of backstepping design
Step 3 At this step the actual control appears in the derivativeof 119911
2 which is given by
1199112= 119886
222+ (119887
1199010+ 119860
220
120579) 119906
minus 119910(2)
119889+ 120594
21+ 2120574
2
12059422119907 + 120594
23119876120585
119888
(56)
where 12059421 120594
22 and 120594
23are given as follows
12059421
= 1198912minus120597120572
1
1205971
(1198911+
2) minus
1205971205721
120597119910119889
119910(1)
119889
minus120597120572
1
120597 120579
120575 minus120597120572
1
120597120578(119860
119891120578 + 119901
21199111)
minus120597120572
1
120597120578(119860
119891120578 + 119901
21) minus
1205971205721
120597Φ
1
(Φ
2+ 120603
1)1015840
minus120597120572
1
120597Φ119906
1
(Φ119906
2+ 120595
1)1015840
minus120597120572
1
120597997888rarrΣ
(120598 minus 1)
times
997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr
(Σ(1205781015840
1198791+Φ
1+ Φ
119906
1)1015840
(1205742
1205772
minus1) (1205781015840
1198791+Φ
1+Φ
119906
1) Σ)
minus120597120572
1
120597119904Σ
(1205742
1205772
minus 1) (1 minus 120598) (1205781015840
1198791+ Φ
1+ Φ
119906
1)
times (1205781015840
1198791+ Φ
1+ Φ
119906
1)1015840
12059422
=1
21205742(ℎ
2minus120597120572
1
1205971
ℎ1minus120597120572
1
120597 120579
120581 minus120597120572
1
120597120578
11990121198901015840
21
120577
minus120597120572
1
120597120578
11990121198901015840
21
120577minus
1205971205721
120597Φ
1
1198601015840
23119890221198901015840
22)
12059423
= 9848582minus120597120572
1
1205971
9848581minus120597120572
1
120597 120579
120593
(57)
Introduce the following value function for this step
1198812= 119881
1+1
21199112
2 (58)
Its derivative can be written as
1198812= minus119911
2
1minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119907|2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
+ 1205891015840
2119876120585
119888
(59)
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
with the control law defined by
119906 = 120583 (1199111 119911
2
1
2 119910
119889 119910
(1)
119889
120579 120578 120578 Φ
1 Φ
2 Φ
119906
1 Φ
119906
119903
997888rarrΣ 119904
Σ)
= minus1
1198871199010
+ 119860220
120579
(119886222minus 119910
(2)
119889minus 120572
2)
(60)
where
1205722= minus 120594
21minus 2120574
2
120594222
minus 21205742
120594221
1198901015840
211205841
minus 1205742
1205942
2211199112minus 120573
21199112minus 119911
1
(61)
1205842= 120584
1+ 119890
21120594221
1199112 (62)
where 12059422
= [120594221
120594222
] Clearly the functions 120583 12059421 120594
22
12059423 120584
2 and 120589
2are smooth as long as 120579 isin Θ
119900
This completes the backstepping design procedure
For the closed-loop adaptive nonlinear system we havethe following value function
119880 = 1198812+119882 =
10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Σminus1+ 120574
210038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
+10038161003816100381610038161205781003816100381610038161003816
2
119885+1
2
2
sum
119895=1
(119895minus 119910
(119895minus1)
119889minus 120572
119895minus1)2
(63)
where we have introduced 1205720= 0 for notational consistency
The time derivative of this function is given by
119880 = minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
10038161003816100381610038161205851198881003816100381610038161003816
2
119876
+ 1205891015840
119903119876120585
119888minus10038161003816100381610038161205781003816100381610038161003816
2
119884
minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
= minus1003816100381610038161003816119862119909 minus 119910
119889
1003816100381610038161003816
2
minus 120574410038161003816100381610038161003816119909 minus 119909 minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
minus 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
+ 2(120579 minus 120579)1015840
119875119903( 120579) +
1003816100381610038161003816100381610038161003816120585119888+1
21205892
1003816100381610038161003816100381610038161003816
2
119876
minus1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119895=1
1205731198951199112
119895+ 120574
2
|119908|2
+ 1205742
||2
minus 120574210038161003816100381610038161003816119908 minus 119908
lowast
(119905 119906[0119905]
119910[0119905]
[0119905]
120585[0119905]
120585[0119905]
120585[0119905]
)10038161003816100381610038161003816
2
minus 12057421003816100381610038161003816119907 minus 120584
2
1003816100381610038161003816
2
(64)
Then the optimal choice for the variables 120585119888and 120585 are
120585lowast
119888= minus
1
21205892lArrrArr 120585
lowast
= 120585 minus1
21205892 (65)
which yields that the closed-loop system is dissipative withstorage function 119880 and supply rate
minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 1205742
|119908|2
+ 1205742
||2
(66)
Furthermore the worst case disturbance with respect to thevalue function 119880 is given by
119908opt = 1205771198641015840
1198901015840
211205842+
1
1205742(119868 minus 120577
2
1198641015840
119864)1198631015840
Σminus1
(120585 minus 120585)
+ 1205772
1198641015840
119862 ( minus 119909)
(67)
opt = 119890221205842 (68)
5 Main Result
For the adaptive control law with 120585119888chosen according to (65)
the closed-loop system dynamics are
119883 = 119865 (119883 119910(2)
119889) + 119866 (119883)119908 + 119866
(119883) (69)
119883 is the state vector of the close-loop system and given by
119883 = [1205791015840
1199091015840
119904Σ
1205791015840
1015840
1205781015840
1205781015840
1198891205781015840
997888rarrΦ
119906
1015840 larr997888Σ
1015840
119910119889
119910(1)
119889
]
1015840
(70)
which belongs to the setD = 119883 | Σ gt 0 119904Σgt 0 120579 isin Θ
119900119865
and119866 are smoothmapping ofDtimesR andD respectively andwith the initial condition 119883(0) = 119883
0isin D
0= 119883
0isin D | 120579 isin
Θ 1205790isin Θ Σ(0) = 120574
minus2
119876minus1
0gt 0Tr((Σ(0))minus1) le 119870
119888 119904
Σ(0) =
1205742 Tr(119876
0)
Since (64) holds the value function119880 satisfies Hamilton-Jacobi-Isaacs equation for all119883 isin D for all 119910(2)
119889isin R
120597119880
120597119883(119883) 119865 (119883 119910
(2)
119889) +
1
41205742
120597119880
120597119883(119883) [119866 (119883) 119866
119908(119883)]
sdot [119866 (119883) 119866119908(119883)]
1015840
(120597119880
120597119883(119883))
1015840
+ 119876 (119883 119910(2)
119889) = 0
(71)
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
where 119876 D timesR rarr R is smooth and given by
119876(119883 119910(2)
119889) =
100381610038161003816100381611990911minus 119910
119889
1003816100381610038161003816
2
+(10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
+119875119903( 120579)
10038161003816100381610038161205781003816100381610038161003816
2
119884minus2(120579 minus 120579)
1015840
times119875119903( 120579)+
2
sum
119895=1
1205731198951199112
119895+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876)
(72)
Although the value function 119880 satisfies an Hamilton-Jacobi-Isaacs equation we cannot deduce the stability androbustness properties of the closed-loop system directly from(64) since119880 is not a positive-definite function of the closed-loop state vector 119883 We will use the following theorem toprecisely state the strong stability properties of the closed-loop adaptive system
Theorem 10 Consider the robust adaptive control problemformulated in Section 2 with Assumptions 1ndash7 holding Therobust adaptive controller 120583 defined by (60) with the optimalchoice for the worst-case estimate 120585 defined by (65) achievesthe following strong robustness properties for the closed-loopsystem
(1) The controller 120583 achieves disturbance attenuationlevel 120574 for any uncertainty quadruple (119909(0) 120579 119908
[0infin)
[0infin)
1198841198890 119910
(2)
119889) isin W
(2) Given a 119888119908
gt 0 there exists a constant 119888119888gt 0 and a
compact set Θ119888sub Θ
119900 such that for any uncertainty
(119909(0) 120579 [0infin)
[0infin)
119884119889) with
|119909 (0)| le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119908 forall119905 isin [0infin)
(73)
all closed-loop state variables 119909 120579 Σ 119904Σ 120578 120578 120578
119889
and 120582 are bounded as follows for all 119905 isin [0infin)
|119909 (119905)| le 119888119888 | (119905)| le 119888
119888 120579 (119905) isin Θ
119888
1003816100381610038161003816120578 (119905)1003816100381610038161003816 le 119888
119888
1003816100381610038161003816120578119889 (119905)1003816100381610038161003816 le 119888
119888 |120582 (119905)| le 119888
119888
1003816100381610038161003816100381612057810038161003816100381610038161003816le 119888
119888
1
119870119888
119868 le Σ (119905) le1
1205742119876
minus1
0
1
119870119888
le 119904Σ(119905) le
1
1205742 Tr (1198760)
(74)
(3) For any uncertainty quadruple (119909(0) 120579 [0infin)
[0infin)
119884119889[0infin)
) with [0infin)
isin L2capL
infin
[0infin)isin L
2capL
infin
and 119884119889[0infin)
isin Linfin the output of the system 119909
1
asymptoti-cally tracks the reference trajectory 119910119889 that
is
lim119905rarrinfin
(1199091(119905) minus 119910
119889(119905)) = 0 (75)
Proof For the frits statement if we define
1198970(
0 120579
0) = 119881
2(0) =
1
2
2
sum
119895=1
1199112
119895(0)
119897 (120591 120579 119909 119910[0119905]
[0119905]
) =10038161003816100381610038161003816120585 minus 120585
10038161003816100381610038161003816
2
119876
minus 2(120579 minus 120579)1015840
119875119903( 120579)
+1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
= 120574410038161003816100381610038161003816(119909 minus 119909) minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 120598 (1205742
1205772
minus 1)10038161003816100381610038161003816120579 minus 120579
10038161003816100381610038161003816
2
Φ10158401198621015840119862Φ
minus 2(120579 minus 120579)1015840
119875119903( 120579) +
1
4
100381610038161003816100381612058921003816100381610038161003816
2
119876
+10038161003816100381610038161205781003816100381610038161003816
2
119884+
2
sum
119895=1
1205731198951199112
119895
(76)
then we have
119869119905
= 119869119905
+ int
119905
0
119880119889120591 minus 119880 (119905) + 119880 (0)
le minus119880 (119905) le 0
(77)
It follows thatsup
(119909(0)120579119908[0infin)
[0infin)
)isinW
119869119905
le 0 (78)
This establishes the first statementNext we will prove the second statement Define [0 119905
119891)
to be the maximal interval on which the closed-loop systemadmits a solution We will show that 119905
119891is alwaysinfin
Fix 119888119908
ge 0 and 119888119889
ge 0 consider any uncertainty(119909
0 120579
[0infin)
[0infin) 119884
119889(119905)) that satisfies
10038161003816100381610038161199090
1003816100381610038161003816 le 119888119908 | (119905)| le 119888
119908 | (119905)| le 119888
119908
1003816100381610038161003816119884119889(119905)
1003816100381610038161003816 le 119888119889
forall119905 isin [0infin)
(79)
We define [0 119879119891) to be the maximal length interval on which
for the closed system there exists a solution that lies in itsdefinition Furthemore from the estiamtion design step Σand 119904
Σare uniformly upper bounded and uniformly bounded
away from 0 as desiredIntroduce the vector of variables
119883119890= [ 120579
1015840
(119909 minus Φ120579)1015840
1205781015840
11991111199112]
1015840
(80)
and two nonnegtive and continuous functions defined onR6+120590
119880119872(119883
119890) = 119870
119888
1003816100381610038161003816100381612057910038161003816100381610038161003816
2
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
119880119898(119883
119890) = 120574
21003816100381610038161003816100381612057910038161003816100381610038161003816
2
1198760
+ 120574210038161003816100381610038161003816119909 minus Φ120579
10038161003816100381610038161003816
2
Πminus1+10038161003816100381610038161205781003816100381610038161003816
2
119885+
2
sum
119895=1
1
21199112
119895
(81)
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
then we have
119880119898(119883
119890)le119880 (119905 119883
119890)le119880
119872(119883
119890) forall (119905 119883
119890)isin [0 119879
119891)timesR
6+120590
(82)
Since119880119898(119883
119890) is continuous nonnegative definite and radially
unbounded then for all 120572 isin R the set 1198781120572
= 119883119890isin R6+120590
|
119880119898(119883
119890) le 120572 is compact or empty Since |(119905)| le 119888
119908 and
|(119905)| le 119888119908 for all 119905 isin [0infin) we have the following inequality
for the derivative of 119880
119880 le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1+ 2 (120579 minus 120579)
1015840
119875119903( 120579)
minus10038161003816100381610038161205781003816100381610038161003816
2
119884minus
2
sum
119894=1
1205731198941199112
119894+ 120574
21003817100381710038171003817100381710038171003817100381710038171003817
2
2
1198882
119908+ 120574
2
1198882
119908
(83)
Since minus(1205744
2)|119909minusminusΦ(120579minus 120579)|2
Πminus1
ΔΠminus1 minus|120578|
2
119884+2 (120579 minus 120579)
1015840
119875119903( 120579)minus
sum2
119895=11205731198951199112
119895will tend tominusinfinwhen119883
119890approaches the boundary
ofΘ119900timesR6 then there exists a compact setΩ
1(119888
119908) sub Θ
119900timesR6
such that
119880 lt 0 for for all 119883119890
isin Θ119900times R6
Ω1 Then
119880(119905 119883119890(119905)) le 119888
1 and 119883
119890(119905) is in the compact set 119878
11198881
sube R6+120590for all 119905 isin [0 119879
119891) It follows that the signal 119883
119890is uniformly
bounded namely 120579 119909 minus Φ120579 120578 1199111 and 119911
2are uniformly
boundedBased on the dynamics of 120578
119889 we have 120578
119889is uniformly
bounded Since 120578 = 120578 minus 120578119889is uniformly bounded then 120578 is
also uniformly bounded Furthermore there is a particularlinear combination of the components of 120578 denoted by 120578
119871
120578 = 119860119891120578 + 119901
2119910
120578119871= 119879
119871120578
(84)
which is strictly minimum phase and has relative degree 1with respect to 119910Then the signal 120578
119871has relative degree 3with
respect to the input 119906 and is uniformly boundedNote Φ = Φ
119910
+ Φ119906
+ Φ Since Φ
119910 and Φ are
uniformly bounded to proveΦ is bounded we need to proveΦ
119906 is uniformly bounded Define the following equations toseparate Φ119906 into two parts
Φ119906
= Φ119906119904
+ 120582119887119860
22 0
120582119887= [
1205821198871
1205821198872
]
120582119887= 119860
119891120582119887+ 119890
22119906 120582
119887(0) = 0
2times1
Φ119906119904
= [Φ
1199061199041
Φ1199061199042
]
Φ119906119904
= 119860119891Φ
119906119904
Φ119906119904
(0) = Φ119906 0
(85)
ClearlyΦ119906119904
is uniformly bounded because119860119891is HurwitzThe
first-row element of 119909 minus Φ120579 is
1199091minus Φ
1199061199041120579 minus 120582
1198871119860
22 0120579 minus Φ
1120579 minus 120578
10158401198791
120579
(86)
We can conclude that 1199091minus120582
1198871119860
22 0120579 is uniformly bounded in
view of the boundedness of 119909 minus Φ120579 120579 Φ119906119904
Φ and 120578 Since1199111=
1minus 119910
119889 and 119911
1 119910
119889are both uniformly bounded
1is
also uniformly boundedNotice that 119860
119891= 119860 minus 120577
2
119871119862 minus Π1198621015840
119862(1205772
minus 120574minus2
) and 1198870=
1198871199010
+ 11986022 0
120579 we generated the signal 1199091minus 119887
01205821198871by
119909 minus 1198870
120582119887= 119860
119891(119909 minus 119887
0120582119887) + 119860
21120579119910 + 119863 + 119860
23120579
+ (1205772
119871 + Π1198621015840
(1205772
minus1
1205742)) (119910 minus 119864) +
1199091minus 119887
01205821198871
= 119862 (119909 minus 1198870120582119887)
(87)
Since 1199091minus 119887
01205821198871has relative degree at least 1 with respect to
119910 take 120578119871and 119910 as output and input of the reference system
we conclude 1199091minus 119887
01205821198871
is uniformly bounded by boundinglemma It follows that
1minus120582
1198871(119887
1199010+119860
212 0
120579) is also uniformlybounded Since
1is uniformly bounded and 120579 is uniformly
bounded away from 0 we have 1205821198871
is uniformly boundedThat further implies that Φ
1 that is 119862Φ is uniformly
bounded Furthermore since 1199091minus 119887
01205821198871 and are
bounded we have that the signals of 1199091and 119910 are uniformly
bounded It further implies the uniform boundedness of119909 minus 119887
0120582119887since 119860
119891is a Hurwitz matrix By a similar line of
reasoning above we have 1199092 120582
1198872are uniformly bounded
Thenwe can conclude thatΦ119906119904andΦ are uniformly bounded
Next we need to prove the existence of a compact setΘ119888sub
Θ119900such that 120579(119905) isin Θ
119888 for all 119905 isin [0 119879
119891) First introduce the
function
Υ = 119880 + (120588119900minus 119875 ( 120579))
minus1
119875 ( 120579) (88)
We notice that when 120579 approaches the boundary of Θ119900 119875( 120579)
approaches 120588119900 Then Υ approaches infin as 119883
119890approaches the
boundary of Θ119900times R6 We introduce two nonnegative and
continuous functions defined on Θ119900timesR4
Υ119872
= 119880119872(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
Υ119898= 119880
119898(119883
119890) + (120588
119900minus 119875 ( 120579))
minus1
119875 ( 120579)
(89)
Then by the previous analysis we have
Υ119898(119883
119890) le Υ (119905 119883
119890) le Υ
119872(119883
119890)
forall (119905 119883119890) isin [0 119879
119891) times Θ
119900timesR
6
(90)
Note that the set 1198782120572
= 119883119890isin Θ
119900times R6
| Υ119898(119883
119890) le 120572
is a compact set or empty Then we consider the derivative
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
of Υ as follows
Υ =
119880 + (120588119900minus 119875 ( 120579))
minus2
120588119900
120597119875
120597120579( 120579)
120579
le minus1205744
2
10038161003816100381610038161003816119909 minus minus Φ (120579 minus 120579)
10038161003816100381610038161003816
2
Πminus1
ΔΠminus1
+ 2 (120579 minus 120579)1015840
119875119903( 120579) minus
10038161003816100381610038161205781003816100381610038161003816
2
119884minus
119903
sum
119895=1
119888120573119895
1199112
119895
minus
100381610038161003816100381610038161003816100381610038161003816
(120597119875
120597120579( 120579))
1015840100381610038161003816100381610038161003816100381610038161003816
2
(120588119900minus 119875 ( 120579))
minus4
times (119870minus1
119888120588119900119901119903( 120579) (120588
119900minus 119875 ( 120579))
2
minus 119888) + 119888
(91)
where 119888 isin R is a positive constant Since
Υ will tend to minusinfin
when 119883119890approaches the boundary of Θ
119900times R4 there exists a
compact setΩ2(119888
119908) sub Θ
119900timesR4 such that for all119883
119890isin Θ
119900timesR4
Ω2
Υ(119883119890) lt 0Then there exists a compact setΘ
119888sub Θ
119900 such
that 120579(119905) isin Θ119888 for all 119905 isin [0 119879
119891) Moreover Υ(119905 119883
119890(119905)) le 119888
2
and 119883119890(119905) is in the compact set 119878
21198882
sube Θ119900times R6 for all 119905 isin
[0 119879119891) It follows that 119875
119903( 120579) is also uniformly bounded
Also 120578 120582 are some stably filtered signals of 119906 and 119910 theyare uniformly bounded Since 120578
is uniformly bounded Φis uniformly bounded Then we can conclude is uniformlybounded from the boundedness of 119909 minus Φ120579 This furtherimplies that the control input 119906 is uniformly bounded
Then we can get the conclusion that the complete systemstates and 119906 are uniformly bounded on [0 119905
119891) Σ 119904
Σare
uniformly bounded and bounded away from 0 and 120579 isuniformly bounded away from the boundary of the set Θ
119900
Therefore it follows that 119905119891= infin and the complete system
states are uniformly bounded on [0infin)Last we will establish the third statement By the follow-
ing inequality
int
infin
0
119880119889120591 le int
infin
0
(minus10038161003816100381610038161199091
minus 119910119889
1003816100381610038161003816
2
+ 120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 (92)
it follows that
int
infin
0
10038161003816100381610038161199091minus 119910
119889
1003816100381610038161003816
2
119889120591
le int
infin
0
(120574210038161003816100381610038161003816
10038161003816100381610038161003816
2
+ 1205742
||2
) 119889120591 + 119880 (0) lt +infin
(93)
By the second statement we notice that
sup0le119905ltinfin
1003816100381610038161003816
1199091minus
119910119889
1003816100381610038161003816 lt infin (94)
Then we have
lim119905rarrinfin
10038161003816100381610038161199091(119905) minus 119910
119889(119905)
1003816100381610038161003816 = 0 (95)
This complete the proof of the theorem
6 Example
In this section we present one example to illustrate the mainresults of this paper The designs were carried out usingMATLAB symbolic computation tools and the closed-loopsystems were simulated using SIMULINK
The example was based on a four-pole-permanent-magnet brushed DC motor We assume that the nominalvalues of 119870
119905 119870
119890 119869 119877 and 119871 are given as below and the
variations can be lumped into the arbitrary disturbance 119870
119905= 001 N-cmAmp
119870119890= 1 Voltrads
119869 = 001 N-cmrads2119877 = 1 Ohm119871 = 01 L
The value of 119863 is unknown and with true value 001N-cmradsThen the true system is of the following state-spacerepresentation
[
120596
119894] = [
120579 1
minus10 minus10] [
120596
119894] + [
0
10] 119906 + [
1
0]119879
+ [1 0 1
0 0 0][
[
119879119908
119908120596
119879119891
]
]
[120596 (0)
119894 (0)] = [
0
0]
119910 = [1 0] [120596
119894] + [0 1 0] [
[
119879119908
119908120596
119879119891
]
]
(96)
where 120596 is the motor speed in rads 119894 is the motor current inamp 119906 is control input in volt 119910 is the motor speed measu-rement in rads 119879
is the estimated disturbance torque in
N-cm 119879119908is the arbitrary disturbance torque in N-cm 119879
119891is
the friction torque in N-cm 119908120596is the measurement channel
noise in rads 120579 is the 1-dimensional unknown parameterwith the true value 120579lowast = minus1 belonging to the interval [minus2 0]
The control objective is to have the systemoutput trackingvelocity reference trajectory 119910
119889 which is generated by the
following linear system
119910119889=
119889
1199043 + 21199042 + 2119904+3 (97)
where 119889 is the command input signalIntroduce the following state and disturbance transfor-
mation
119909 = [1 0
10 1] [
120596
119894] 119908 = [
1 minus120579 1
0 1 0][
[
119879119908
119908120596
119879119891
]
]
(98)
We obtain the design model
119909 = [minus10 1
minus10 0] 119909 + [
1
10] 119910120579
+ [0
10] 119906 + [
1
10] + [
1 0
10 0]119908
119910 = [1 0] 119909 + [0 1]119908
(99)
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
0 5 10 15 20 25 30minus1
minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
Time (s)
(a)
0 5 10 15 20 25 30minus15
minus10
minus5
0
5
10
15Control input
u
Time (s)
(b)
0
0
5 10 15 20 25 30minus2
minus18minus16minus14minus12minus1
minus08minus06minus04minus02
Parameter estimation
Time (s)
θ
(c)
0 5 10 15 20 25minus04minus035minus03minus025minus02minus015minus01minus005
000501
Time (s)
State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 5 10 15 20 25 30minus4
minus35minus3
minus25minus2
minus15minus1
minus050
051
Time (s)
State-estimation errormdashx2St
ate
esti
mat
ion
erro
rmdashx
2
(e)
0 5 10 15 20 25 300
005
01
015
02
025Cost function
Cos
t fun
ctio
n
Time (s)
(f)
Figure 1 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= 0 119879
119908= 0 119908
120596= 0 and 119879
= 0 (a) Tracking error (b)
control input (c) parameter estimate (d) state-estimation error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus
1205742
|119908|2
minus 1205742
||2d)120591
The ultimate performance lower bound for this system is 1with respect to 119908 For the adaptive control design we set thedesired disturbance attenuation level 120574 = radic2 The parameter120579 is assumed to belong to the set [minus2 0] with the projectionfunction 119875(120579) chosen as
119875 (120579) = (120579 + 1)2
(100)
For other design and simulation parameters we select
0= [
01
05] 120579
0= minus05
1198760= 1 119870
119888= 100 Δ = [
1 0
0 1]
1205731= 120573
2= 05 119884 = [
1592262 minus170150
minus170150 18786]
(101)
Then we obtain
119860119891= [
minus102993 10000
minus122882 0] 119885 = [
88506 minus09393
minus09393 01229]
Π = [05987 45764
45764 431208]
(102)
We present two sets of simulation results in this exampleIn the first set of simulation we set
119879119891= 0 N-cm
119879119908= 0 N-cm
119908120596= 0 rads
119879= 0 N-cm
This simulation is to demonstrate the regulatory behaviour ofthe adaptive controllerThe results are shown in Figures 1(a)ndash1(f) We observe from Figure 1 that the parameter estimateof minus119863119869 asymptotically converges to its true value minus1 theoutput-tracking error and state-estimation error asymptoti-cally converge to zeros and 119905 within 20 second The controlinput is bounded by 12 and the transient of the system is wellbehaved
The second set of simulation results is to demonstratethe robustness of the adaptive controller to unmodeledexogenous disturbance inputs We set
119879119891= minus001 times sgn(120596) N-cm
119879119908= 004 sin (119905) N-cm
119908120596= White noise signal with power 001 sample 119889 at
1 HZ rads119879= 005 sin (4119905) N-cm
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
0 20 40 60 80 100
Time (s)
minus1minus08minus06minus04minus02
002040608
1Tracking error
Tra
ckin
g er
ror
(a)
0 20 40 60 80 100
Control input
minus15
minus10
minus5
0
5
10
15
u
Time (s)
(b)
0 20 40 60 80 100minus11
minus1
minus09
minus08
minus07
minus06
minus05
minus04
Time (s)
θ
Parameter estimation
(c)
0 20 40 60 80 100Time (s)
minus1minus08minus06minus04minus02
002040608
1State-estimation errormdashx1
Stat
e es
tim
atio
n er
rormdash
x1
(d)
0 20 40 60 80 100minus3
minus25
minus2
minus15
minus1
minus05
0
05
1
Time (s)
State-estimation errormdashx2
Stat
e es
tim
atio
n er
rormdash
x2
(e)
0 20 40 60 80 100minus025minus02minus015minus01
minus0050
00501
01502
025
Time (s)
Cost function
Cos
t fun
ctio
n
(f)
Figure 2 System response for example under command input 119889(119905) = 5 sin(119905) 119879119891= minus001 times sgn(120596) and 119879
119908= 004 sin (119905) 119908
120596= white noise
signal with power 001 sample 119889 at 1HZ 119879= 005 sin(4119905) (a) Tracking error (b) control input (c) parameter estimate (d) state-estimation
error 1minus 119909
1 (e) state-estimation error
2minus 119909
2 (f) cost function int
119905
0
((1199091minus 119910
119889)2
minus 1205742
|119908|2
minus 1205742
||2d)120591
The simulation results are presented in Figures 2(a)ndash2(f)We observe that the the parameter estimate of minus119863119869
no longer converges to the true value minus1 but itrsquos sta-bilized around the true value The output-tracking errorand state-estimation error no longer converge to zerosbut output-tracking error satisfies the targeted attenuationlevel based on Figure 2(f) and the state-estimation errorsasymptotically oscillate around zeros The control input isagain bounded by 12 and the transient of the system is wellbehaved as well
7 Conclusions
In this paper we studied the permanent magnet brushed DCadaptive control design for velocity tracking applications Weformulate the robust adaptive control problem as a nonlinear119867
infin-control problem under imperfect state measurementsand then use cost-to-come function analysis and the integratorbackstepping methodology to obtain the controller Thecontroller then achieves the desired disturbance attenuationlevel with the ultimate lower bound of the attenuation levelbeing the noise intensity in the measurement channel It alsoguarantees the total stability of the closed-loop system andachieves asymptotic tracking of the reference trajectory whenthe disturbance is of finite energy and uniformly bounded
References
[1] G C Goodwin and K S Sin Adaptive Filtering Prediction andControl Prentice-Hall Englewood Cliffs NJ USA 1984
[2] G C Goodwin and D Q Mayne ldquoA parameter estimation per-spective of continuous time model reference adaptive controlrdquoAutomatica vol 23 no 1 pp 57ndash70 1987
[3] P R Kumar ldquoA survey of some results in stochastic adaptivecontrolrdquo SIAM Journal on Control and Optimization vol 23 no3 pp 329ndash380 1985
[4] C E Rohrs L Valavani M Athans and G Stein ldquoRobustnessof continuous-time adaptive control algorithms in the presenceof unmodeled dynamicsrdquo IEEE Transactions on AutomaticControl vol 30 no 9 pp 881ndash889 1985
[5] ADatta andPA Ioannou ldquoPerformance analysis and improve-ment in model reference adaptive controlrdquo IEEE Transactionson Automatic Control vol 39 no 12 pp 2370ndash2387 1994
[6] P A Ioannou and J SunRobust Adaptive Control PrenticeHallUpper Saddle River NJ USA 1996
[7] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996
[8] E Mosca and T Agnoloni ldquoInference of candidate loop per-formance and data filtering for switching supervisory controlrdquoAutomatica vol 37 no 4 pp 527ndash534 2001
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
[9] A Bilbao-Guillerna M De la Sen A Ibeas and S Alonso-Quesada ldquoRobustly stable multiestimation scheme for adaptivecontrol and identificationwithmodel reduction issuesrdquoDiscreteDynamics in Nature and Society no 1 pp 31ndash67 2005
[10] N Luo M de la Sen and J Rodellar ldquoRobust stabilization ofa class of uncertain time delay systems in sliding moderdquo Inter-national Journal of Robust and Nonlinear Control vol 7 no 1pp 59ndash74 1997
[11] T Basar and P Bernhard Hinfin-Optimal Control and RelatedMinimax Design Problems Systems amp Control Foundations ampApplications Birkhauser Boston Inc Boston MA Secondedition 1995 A dynamic game approach
[12] Z Pan and T Basar ldquoParameter identification for uncertainlinear systems with partial state measurements under an 119867
infin
criterionrdquo IEEE Transactions on Automatic Control vol 41 no9 pp 1295ndash1311 1996
[13] I E Tezcan and T Basar ldquoDisturbance attenuating adaptivecontrollers for parametric strict feedback nonlinear systemswith output measurementsrdquo Journal of Dynamic Systems Mea-surement and Control Transactions of the ASME vol 121 no 1pp 48ndash57 1999
[14] Z Pan and T Basar ldquoAdaptive controller design and distur-bance attenuation for SISO linear systems with noisy outputmeasurementsrdquo CSL Report University of Illinois at Urbana-Champaign Urbana Ill USA 2000
[15] G Arslan and T Basar ldquoDisturbance attenuating controllerdesign for strict-feedback systems with structurally unknowndynamicsrdquo Automatica vol 37 no 8 pp 1175ndash1188 2001
[16] S Zeng and E Fernandez ldquoAdaptive controller design anddisturbance attenuation for sequentially interconnected SISOlinear systems under noisy output measurementsrdquo IEEE Trans-actions on Automatic Control vol 55 no 9 pp 2123ndash2129 2010
[17] Q Zhao Z Pan and E Fernandez ldquoConvergence analysis forreduced-order adaptive controller design of uncertain SISOlinear systems with noisy output measurementsrdquo InternationalJournal of Control vol 82 no 11 pp 1971ndash1990 2009
[18] Q Zhao Z Pan and E Fernandez ldquoReduced-order robustadaptive control design of uncertain SISO linear systemsrdquo Inter-national Journal of Adaptive Control and Signal Processing vol22 no 7 pp 663ndash704 2008
[19] S Zeng ldquoAdaptive controller design and disturbance attenu-ation for a general class of sequentially interconnected SISOlinear systems with noisy output measurementsrdquo in Proceedingsof the 49th IEEE Conference on Decision and Control (CDC rsquo10)pp 2608ndash2613 Atlanta Ga USA December 2010
[20] S Zeng ldquoAdaptive controller design and disturbance attenua-tion for a general class of sequentially interconnected siso linearsystems with noisy output measurements and partly measureddisturbancesrdquo in Proceedings of the IEEE International Sympo-sium on Computer-Aided Control System Design (CACSD) Partof 2011 IEEEMulti-Conference on Systems andControl pp 1050ndash1055 Denver Colo USA 2011
[21] S Zeng ldquoWorst-case analysis based adaptive control design forsiso linear systems with plant and actuation uncertaintiesrdquo inProceedings of the 50th IEEEConference onDecision and Controland European Control Conference (CDC-ECC rsquo11) pp 6349ndash6354 Orlando Fla USA 2011
[22] S Zeng and Z Pan ldquoAdaptive controls design and disturbanceattenuation for SISO linear systems with noisy output measure-ments and partly measured disturbancesrdquo International Journalof Control vol 82 no 2 pp 310ndash334 2009
[23] S Zeng Z Pan and E Fernandez ldquoAdaptive controller designand disturbance attenuation for SISO linear systems with zerorelative degree under noisy output measurementsrdquo Interna-tional Journal of Adaptive Control and Signal Processing vol 24no 4 pp 287ndash310 2010
[24] M Krstic I Kanellakopoulos and P V Kokotovic NonlinearandAdaptive Control Design JohnWileyamp Sons NewYork NYUSA 1995
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of