Research Article Robust Adaptive Fuzzy Control for a Class of...

Research ArticleRobust Adaptive Fuzzy Control for a Class ofUncertain MIMO Nonlinear Systems with Input Saturation

Shenglin Wen and Ye Yan

College of Aerospace Science and Engineering National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Shenglin Wen wenshenglin1108gmailcom

Received 11 November 2014 Revised 30 January 2015 Accepted 3 February 2015

Academic Editor Yan-Jun Liu

Copyright copy 2015 S Wen and Y Yan This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the robust adaptive fuzzy control design problem for a class of uncertain multiple-input and multiple-output(MIMO) nonlinear systems in the presence of actuator amplitude and rate saturation In the control scheme fuzzy logic systems areused to approximate unknown nonlinear systems To compensate the effect of input saturations an auxiliary system is constructedand the actuator saturations then can be augmented into the controllerThemodified tracking error is introduced and used in fuzzyparameter update laws Furthermore in order to deal with fuzzy approximation errors for unknown nonlinear systems and externaldisturbances a robust compensation control is designed It is proved that the closed-loop system obtains119867

infintracking performance

through Lyapunov analysis Steady and transient modified tracking errors are analyzed and the bound of modified tracking errorscan be adjusted by tuning certain design parameters The proposed control scheme is applicable to uncertain nonlinear systemsnot only with actuator amplitude saturation but also with actuator amplitude and rate saturation Detailed simulation results ofa rigid body satellite attitude control system in the presence of parametric uncertainties external disturbances and control inputconstraints have been presented to illustrate the effectiveness of the proposed control scheme

1 Introduction

In most practical control applications such as those inrobot manipulation and aerospace industry the performanceof the controller is directly related to the accuracy of themathematical model and external disturbances However itis difficult to establish an appropriate mathematical modelfor a large number of nonlinear systems when the systemsare complex and highly coupled nonlinear with structureduncertainties and external disturbances [1] To tackle withthis problem fuzzy logic systems and neural networks havebeen extensively used in complex and ill-defined nonlinearsystems due to their approximation ability of dealing with thenonlinear smooth functions [2] Many adaptive fuzzy controland adaptive neural network control schemes have beendeveloped for single-input and single-output (SISO) non-linear systems [3ndash7] MIMO nonlinear systems [8ndash17] andSISOMIMO nonlinear systems with immeasurable states[18ndash20] respectively Generally these adaptive fuzzy andneural network control approaches can achieve nice control

performance without control saturation If physical actuatorssaturation such as magnitude and rate constraints is consid-ered the adaptive intelligent control approaches mentionedabove can not be implemented [21]

As we know in many practical dynamic systems physicalactuators saturation on hardware indicates an inevitableconstraint of themagnitude and rate limitations of the controlsignal For example due to physical limitation momentumexchange devices or thrusters as actuator for the satelliteattitude control system fail to render infinite control torqueand thus the actuator can only provide limited control torqueswithin a limited rate [22] Control saturation is one ofthe most common nonsmooth nonlinearity that should beexplicitly considered in the control design The controllersthat ignore actuator limitations may give rise to undesirableinaccuracy severely degrade the performance of system oreven damage the stability of system [23]Hence the controllerdesign subjected to the control saturation while simulta-neously achieving higher performance is a very practicalproblem

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 561397 14 pageshttpdxdoiorg1011552015561397

2 Mathematical Problems in Engineering

The design of tracking controllers for uncertain MIMOnonlinear systems with actuator constraints is a challengingproblem During the past decades there have been extensiveresearches on the control of nonlinear systems with variousconstraints Analysis and design of control systems with con-trol saturation have been widely studied in [24ndash42] Farrell etal [24ndash26] have presented an adaptive backstepping approachfor unknown nonlinear systems with known magnituderate and bandwidth constraints on intermediate states oractuators without disturbance To tackle with the physical sat-uration an auxiliary systemwith the same order as that of theplant was constructed to compensate the effect of saturationThe control input saturation is investigated through onlineapproximation based control for uncertain nonlinear systemsin [26] An adaptive control and the constrained adaptivecontrol in combination with the backstepping technique areproposed in [29] A direct adaptive fuzzy control approachfor uncertain nonlinear systems with input saturation ispresented in [30] in which a Nussbaum function is usedto compensate for the nonlinear term arising from theinput saturation In [31] an adaptive fuzzy output feedbackcontrol algorithm for a class of output constrained uncertainnonlinear systems with input saturation is developed byemploying a barrier Lyapunov function and an auxiliarysystem Adaptive backstepping tracking control based onfuzzy neural networks is investigated for unknown chaoticsystems in [32] and with control input constraints in [33]Neural network based adaptive control schemeswith externaldisturbances and actuator saturations are presented in [35]in which auxiliary systems are added to attenuate the effectsof input saturation It is apparent that the presence of inputsaturation constraint substantially increases the complexityof control system design for uncertain MIMO nonlinearsystems In the constrained adaptive control the key problemis how to handle the constraint effect of the actuatorrsquos physicalconstraints To this end we introduce an auxiliary designsystem to handle the constraint effect in this paper Basedon the states of the auxiliary design system constrainedadaptive control is investigated for a class of uncertainMIMOnonlinear systems with input constraints using robust fuzzycontrol technique

In this paper a robust adaptive fuzzy tracking controlscheme is presented to handle the external disturbances andactuator physical constraints for uncertain MIMO nonlinearsystems In control design fuzzy logic systems are used toapproximate unknown nonlinear systems Note that inputsaturations are nonsmooth functions but the adaptive fuzzycontrol technique requires all functions differentiable [6]To compensate the effect of input saturations an auxiliarysystem is constructed and the actuator saturations then canbe augmented into the controller The modified trackingerror is introduced and used in fuzzy parameter update lawsBesides in order to deal with fuzzy approximation errorsfor unknown nonlinear systems and external disturbancesa robust compensation control is designed It is proved thatthe proposed control approach can guarantee that all thesignals of the resulting closed-loop system are bounded andthe closed-loop system obtains 119867


through Lyapunov analysis The transient modified tracking

errors performance is derived to be explicit functions ofdesign parameters and thus bounds of modified trackingerrors can be adjusted by tuning design parameters

The rest of this paper is organized as followsThe descrip-tion of the uncertain MIMO nonlinear system under consid-eration and necessary preliminaries are given in Section 2 InSection 3 the robust adaptive fuzzy control without controlsaturation is firstly designed When the actuators have phys-ical limitations this approach may not be able to be success-fully implemented In order to solve this problem a robustadaptive fuzzy control scheme with input saturation is inves-tigated The simulation results of satellite attitude controlare presented to demonstrate the effectiveness of proposedcontroller in Section 4 Section 5 contains the conclusion

2 Problem Formulation and Preliminary

21 Problem Formulation Consider a class of uncertainMIMO affine nonlinear system

x = f (x) +

119898

sum

119894=1

g119894(x) 119906119894+ d1015840

1199101= ℎ1(x)

119910119898

= ℎ119898

(x)

(1)

where x = [1199091 119909

119899] isin 119877

119899 is the state vector available formeasurement u = [119906

1 119906

119898] isin 119877

119898 is the control vectorwith |119906

119894| le 119906119894max where 119906

119894max denote the actuator amplitudey = [119910

1 119910

119898] isin 119877

119898 is the output vector ℎ1(x) ℎ

119898(x)

are smooth functions defined on the open set of R119899 f(x)g1(x) g

119898(x) are continuous unknown but smooth vector

fields d1015840 = [1198891015840

1 119889

1015840

119899]119879

isin R119899 is external disturbance vectorwhere 119889

1015840

1is unknown but bounded

Define that 119871 fℎ119894 is the Lie derivative of ℎ119894along vector

field f(x) and 119871119896

fℎ119894 is recursively as 119871119896

fℎ119894 = 119871 f(119871119896minus1

f ℎ119894) A

multivariable uncertain nonlinear system of the form (1) hasa vector relative degree [119903

1 1199032 119903

119898] at a point x

0if

119871g119895119871119896

fℎ119894 (x) = 0 forall1 le 119895 le 119898 119896 le 119903119894minus 1 (2)

for all x in a neighborhood of x0

Denote

G (x) =

[[[[[[

[

119871g11198711199031minus1

f ℎ1(x) sdot sdot sdot 119871g119898119871

1199031minus1

f ℎ1(x)

119871g11198711199032minus1


1199032minus1

f ℎ2(x)

sdot sdot sdot d sdot sdot sdot

119871g1119871119903119898minus1

f ℎ119898

(x) sdot sdot sdot 119871g119898119871119903119898minus1

f ℎ119898

(x)

]]]]]]

]

=

[[[[

[

11989211

(x) sdot sdot sdot 1198921119898

(x) d

1198921198981

(x) sdot sdot sdot 119892119898119898

(x)

]]]]

]

(3)

Mathematical Problems in Engineering 3

Note that G(x) is nonsingular at x = x0

Using feedback linearization the nonlinear system (1) canbe transformed into the following form [43]

y(r) = F (x) + G (x) u + d (4)

where

y(r) = [119910(1199031)

1 119910(1199032)

2 119910

(119903119898)

119898]119879

(5)

F (x) =

[[[[[[[

[

1198711199031

f ℎ1 (x)

1198711199032

f ℎ2 (x)

119871119903119898

f ℎ119898

(x)

]]]]]]]

]

=

[[[[[[

[

1198911(x)

1198912(x)

119891119898

(x)

]]]]]]

]

(6)

d =

[[[[[[[[[[[[[

[

1199031

sum

119894=1

119871119894minus1

x 11987111988910158401198711199031minus119894

f ℎ1(x)

1199032

sum

119894=1

119871119894minus1

x 11987111988910158401198711199032minus119894

f ℎ2(x)

119903119898

sum

119894=1

119871119894minus1

x 1198711198891015840119871119903119898minus119894

f ℎ119898

(x)

]]]]]]]]]]]]]

]

=

[[[[[[

[

1198891

1198892

119889119898

]]]]]]

]

(7)

x = [1199101 119910

(1199031)

1 1199102 119910

(1199032)

2 119910

119898 119910

(119903119898)

119898] (8)

In system (4) d still represents unknown boundedexternal disturbance vector The relative degree of the systemis assumed to be equal to the order of the system and isexpressed as sum

119898

119894=1119903119894= 119899 which implies that the system does

not have any zero dynamicsLet the desired output trajectory be given by

y119889= [1199101198891

1199101198892

119910119889119898

]119879

(9)

This paper aims to develop a robust adaptive fuzzy track-ing control scheme such that all closed-loop system signalsasymptotically converge to a compact set in the presence ofinput saturation system uncertainties and external distur-bances and ensure the system outputs y track the desiredtrajectory y

119889 For system (4) to be controllable the following

assumptions are made

Assumption 1 The desired trajectory 119910119889119894

and its 119899th orderderivatives are known and bounded

Assumption 2 For x in certain controllability regionU119888isin 119877119899

120590(G(x)) = 0 where 120590(G(x)) denotes the minimum singularvalue of the matrix G(x)

22 Fuzzy Logic Systems In this paper fuzzy logic systemsare used to approximate unknown nonlinear functions F(x)and G(x) Following the approximation presentation offuzzy logic systems is given Without loss of generality theunknown uncertainty is assumed as 119891(x) A fuzzy logic sys-tem consists of four parts the knowledge base the fuzzifier

the fuzzy inference engine working on fuzzy rules and thedefuzzifier The fuzzy inference engine uses fuzzy IF-THENrules to perform a mapping from an input linguistic vectorx = [119909

1 1199092 119909

119899]119879

isin 119877119899 to an output variable 119910 isin 119877 Then

the 119894th fuzzy rule can be represented as [3]

119877119897

If 1199091is 119865119897

1and sdot sdot sdot and 119909

119899is 119865119897

119899 then 119910 is 119866

119897

(119897 = 1 2 119903)

(10)

where 119865119897

119894and 119866

119897 are fuzzy sets characterized with the fuzzymembership functions 120583

119865119897119894(119909119894) and 120583

119866119897(119910) respectively and 119903

is the number of fuzzy rulesThrough singleton function center average defuzzifier

and product inference the fuzzy logic system can beexpressed as follows

119910 (x) =

sum119903

119897=1119910119897

(prod119899

119894=1120583119865119897119894(119909119894))

sum119903

119897=1prod119899

119894=1120583119865119897119894(119909119894)

(11)

where 119910119897

= max119910isin119877

120583119866119897(119910)

By introducing the concept of fuzzy basis function vectorthe final output of the fuzzy logic system can be expressed as

119910 (x) = 120579119879

120585 (x) (12)

where 120579 = [1199101

119910119903

]119879 is the adjustable parameter vector

and 120585(x) = [1205851(x) 120585

119903(x)]119879 is the fuzzy basis function

vector The fuzzy basis function is defined as follows

120585119897(x) =

sum119903

119897=1(prod119899

119894=1120583119865119897119894(119909119894))

sum119903

119897=1prod119899

119894=1120583119865119897119894(119909119894)

(13)

Lemma 3 Let 119891(x) be a continuous function defined on acompact set U

119888 Then for any constants 120576 gt 0 there exists a

fuzzy logic system such as [2]

supxisinU119888

10038161003816100381610038161003816119891 (x) minus 120579

119879

120585 (x)10038161003816100381610038161003816 le 120576 (14)

Define the optimal parameter vector 120579lowast as

120579lowast

= arg min120579isinΩ119891

[supxisinU119888

10038161003816100381610038161003816119891 (x) minus 120579

119879

120585 (x)10038161003816100381610038161003816] (15)

Under the optimization parameter vector the unknownfunction 119891(x) can be written as

119891 (x) = 120579lowast119879

120585 (x) + 120576 (16)

where 120576 is the fuzzy minimum approximation errorIn this work F(x) and G(x) are unknown vector and

matrix respectively and the fuzzy logic systems are used toapproximate the unknown functions Let 119891

119894(x) (119894 = 1 119898)

and 119892119894119895(x) (119894 119895 = 1 119898) be approximated by fuzzy logic

systems as follows

119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 119898

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 119898

(17)


where 120579119891119894

isin R119872119891119894 and 120579119892119894119895

isin R119872119892119894119895 are parameter vectors120585119891119894(x) isin R119872119891119894 and 120585

119892119894119895(x) isin R119872119892119894119895 are fuzzy basis function vec-

tors and 119872119891119894and 119872

119892119894119895are the corresponding dimensions of

the basis vectorsDenote

F (x | 120579119865) =

[[[[

[

1198911(x | 120579

1198911)

119891119898

(x | 120579119891119898

)

]]]]

]

G (x | 120579119866) =

[[[[

[

11989211

(x | 12057911989211

) sdot sdot sdot 1198921119898

(x | 1205791198921119898

)

d

1198921198981

(x | 1205791198921198981

) sdot sdot sdot 119892119898119898

(x | 120579119892119898119898

)

]]]]

]

(18)

Then F(x | 120579119865) and G(x | 120579

119866) can be used as the approxi-

mation of F(x) and G(x) respectivelyDefine the optimal approximation weight vectors for

119891119894(119894 = 1 119898) and 119892

119894119895(119894 119895 = 1 119898) as follows

120579lowast

119891119894= arg min

120579119891119894isinΩF

supxisinU119888

10038161003816100381610038161003816119891119894(x | 120579

119891119894) minus 119891119894(x)10038161003816100381610038161003816

120579lowast

119892119894119895= arg min

120579119892119894119895isinΩG

supxisinU119888

100381610038161003816100381610038161003816119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)

100381610038161003816100381610038161003816

(19)

where ΩF ΩG and U119888denote the sets of suitable bounds on

120579119891119894 120579119892119894119895 and x respectively 120579lowast

119891119894(119894 = 1 119898) and 120579lowast

119892119894119895(119894 119895 =

1 119898) are constants vectorsThe unknown function 119891

119894and 119892

119894119895can be expressed as

119891119894(x) = 120579

lowast119879

119891119894120585119891119894

(x) + 120576119891119894

119894 = 1 119898

119892119894119895(x) = 120579

lowast119879

119892119894119895120585119892119894119895

(x) + 120576119892119894119895

119894 119895 = 1 119898

(20)

where 120576119891119894and 120576119892119894119895

are the smallest approximation errors of thefuzzy logic systems

3 Adaptive Fuzzy Robust Control Designs

In this section we first design the adaptive fuzzy approx-imation based control problem without control saturationand then consider the case where actuators have physicallimitations such as magnitude and rate constraints

31 Adaptive Fuzzy Robust Control The tracking errors aredefined as

119890119894= 119910119889119894

minus 119910119894

119894 = 1 2 119898 (21)

Define 119906119897119894

(119894 = 1 119898) as follows

119906119897119894= 119910(119903119894)

119889119894+

119903119894

sum

119895=1

119896119894119895119890(119895minus1)

119894119894 = 1 119898 (22)

where 1198961198941 119896

119894119903119894are parameters to be chosen such that the

roots of the equation 119904119903119894 + 119896119894119903119894119904119903119894minus1 + sdot sdot sdot + 119896

1198941= 0 in the open

left-half complex planeIf F(x) and G(x) are known external disturbances are

ignored then according to nonlinear dynamic inversioncontrol techniques the control law is given by

u = Gminus1 (x) [minusF (x) + u119897] (23)

where u119897= [1199061198971 119906

119897119898]119879

Because F(x) and G(x) are unknown 119889119894

= 0 thecontrol law (23) cannot be implemented in practice Usingthe approximation ofF(x) andG(x) and considering externaldisturbances the controller is modified as follows

u = Gminus1 (x | 120579119866) [minusF (x | 120579

119865) + u119897+ u119889] (24)

where u119889is the robust compensation term which is used to

attenuate the effect of external disturbances and approxima-tion errors

Substituting (24) into system (4) yields

y(119903) = F (x) minus F (x | 120579119865) + [G (x) minus G (x | 120579

119866)] u + u

119897

+ u119889+ d

(25)

Considering (22) the 119894th subsystem of (25) can be rewrit-ten as

e119894= A119894e119894+ B119894

[119891119894(x | 120579

119891119894) minus 119891119894(x)]

+

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus B119894119906119889119894

minus B119894119889119894

(26)

where 119906119889119894is the 119894th element of u

119889and

A119894=

[[[[[[[[[

[

0 1 0 sdot sdot sdot 0


d


minus1198961198941

minus1198961198942

minus1198961198943

sdot sdot sdot minus119896119894119903119894

]]]]]]]]]

]

B119894=

[[[[[[[[[

[

0

0

0

1

]]]]]]]]]

]

e119894=

[[[[[[[[[[

[

119890119894

119890119894

119890(119903119894minus2)

119894

119890(119903119894minus1)

119894

]]]]]]]]]]

]

(27)

Define the minimum approximation error as

1205961015840

119894= [119891119894(x | 120579

lowast

119891119894) minus 119891119894(x)] +

119898

sum

119895=1

[119892119894119895(x | 120579

lowast

119892119894119895) minus 119892119894119895(x)] 119906

119895

(28)


According to fuzzy system theory the following assump-tion is reasonable

Assumption 4 The minimum approximation error is squareintegrable that is int119879

0

1205961015840119879

1198941205961015840

119894119889119905 lt infin

Using the definition (28) and the optimal approximationfor 119891119894(x) and 119892

119894119895(x) (26) can be rewritten in the following

form

e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(29)

where 119891119894

= 120579119891119894

minus 120579lowast

119891119894 119892119894119895

= 120579119892119894119895

minus 120579lowast

119892119894119895 and 120596

119894= 1205961015840

119894minus 119889119894

Theorem 5 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance If the controller ischosen as (24) the parameters update laws and 119906

119889119894are adopted

as follows

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (30)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (31)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (32)

Then the following 119867infin

tracking performance can beobtained

int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(33)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895

(119894 119895 = 1 119898) are positiveadaptive scalar 120588

119894(119894 = 1 119898) are positive parameters

representing for prescribed disturbance attenuation levelse = [e119879

1 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin

R119898times119898 (119894 = 1 119898) are arbitrary symmetric positive definitematrices and P = diag[P

1 P

119898] and P

119894isin R119898times119898 (119894 =

1 119898) are symmetric positive definite solution of thefollowing equations

P119894A119894+ A119879119894P119894= minusQ119894 (34)

where Q119894

isin R119898times119898 (119894 = 1 119898) are arbitrary symmetricpositive definite matrices

Proof For the 119894th subsystem of (4) consider the followingLyapunov function candidate

119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(35)

Differentiating (35) and considering (29) yield

119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(36)

Substituting (30)ndash(32) into (36) we obtain

119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(37)

Integrating both sides of the above inequality from 0 to 119879

yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (38)

Since 119881119894(119879) is nonnegative according to the definition of

119881119894 the following inequality is obtained

int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(39)

Considering the Lyapunov function 119881 = sum119898

119894=1119881119894 it is

easy to obtain the 119867infin

tracking performance index (55) Thiscompletes the proof

The controller proposed by (24) is able to guarantee theLyapunov stability of the closed-loop system and attenuatethe effect of system uncertainties and external disturbancesHowever no saturation on actuators is taken into accountwhich is rather important for practical applications (includ-ing satellite systems) Assume that the control input u is con-straint by saturation functions it can be readily shown thatthe control law (24) the parameters update laws (30) (31)and the robust compensation term (32) with saturation limitscannot guarantee the stability of the closed-loop system

It is expected that during saturation the magnitude of thetracking error will increase since the control signal is notbeing achieved This tracking error is not the result of func-tion approximation error therefore we need to be careful so


that the approximator does not cause ldquounlearningrdquo duringthe period when the actuators are saturated Clearly theparameters update laws (30) and (31) depend on the trackingerror e

119894 thus if the tracking error increases due to saturation

the parameters update laws may cause a significant change intheweights in response to the increase in tracking error Nextwe develop a robust fuzzy adaptive control scheme to addressthe input saturation problem

32 Adaptive Fuzzy Control with Input Saturation When theactuators have physical constraints such as themagnitude andrate limitations the above approach may not be able to besuccessfully implemented Considering the magnitude andrate limitations on the actuator controller (24) is modified as

u119888= Gminus1 (x | 120579

119866) [minusF (x | 120579

119865) + u119897+ u119886119908

+ u119889]

u = 119878119872119877

(u119888)

(40)

where u119888is obtained by certainty equivalence principle u

119886119908

is the auxiliary control term which is used to compensate theeffect of input saturation 119878

119872119877(sdot) is a function including the

magnitude and rate constraints which can produce a limitedoutput

The state space representation of each component of u119888is

[

1199031198941

1199031198942

] = [

1199031198942

119878119877(120596119894(119878119872

(119906119888119894) minus 1199031198941))

]

[

119906119894

119894

] = [

1199031198941

1199031198942

]

(41)


119888 and 119906

119894is the 119894th element

of u 120596119894is the natural frequency and 119878

119872(sdot) and 119878

119877(sdot) are the

saturation functions corresponding to magnitude and raterespectively The function 119878

119872(sdot) is defined as

119878119872

(sdot) =

119872 if 119909 ge 119872

119909 if |119909| lt 119872

minus119872 if 119909 le minus119872

(42)

and 119878119877(sdot) has the same definition

To compensate the effects of input limitations an auxil-iary system is introduced as follows [35]

1205851198941

= 1205851198942

minus 12058211989411205851198941

sdot sdot sdot 120585119894119903119894minus1

= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(43)

where 120582119894119895

(119894 = 1 119898 119895 = 1 119903119894) are positive design

parameters

Define the modified tracking error as

119890119894= 119910119889119894

minus 119910119894minus 1205851198941

119894 = 1 2 119898 (44)

From (4) (40) and (43) we obtain

119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119886119908119894

minus 119906119889119894

minus 119889119894minus 120585(119903119894)

1198941minus

119903119894

sum

119895=1

119896119894119895120585(119895minus1)

1198941

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895)

(45)

where 119906119886119908119894

is the 119894th element of u119886119908

From (43) the term 120585(119903119894)


120585(119903119894)

1198941= minus

119903119894

sum

119895=1

120582119894119895120585(119903119894minus119895)

119894119895+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) (46)

Let

119906119886119908119894

=

119903119894

sum

119895=1

(120582119894119895120585(119903119894minus119895)

119894119895minus 119896119894119895120585(119895minus1)

119894119895) (47)

With (46) and (47) (45) becomes

119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119889119894

minus 119889119894

(48)

Define e119894

= [119890119894 119890119894 119890

(119903119894minus1)

119894]119879 and using the optimal

approximation for 119891119894(x) and 119892

119894119895(x) (48) can be rewritten as

e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(49)

Theorem 6 Consider the uncertain MIMO nonlinear systempresented by (4) with external disturbance and input satura-tion if the controller is chosen as (40) the auxiliary control


term u119886119908 the parameters update laws and 119906

119889119894are adopted as

follows

1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(50)

u119886119908

= [

[

1199031

sum

119895=1

(1205821119895120585(1199031minus119895)

1119895minus 1198961119895120585(119895minus1)

1119895)

sdot sdot sdot

119903119898

sum

119895=1

(120582119898119895

120585(119903119898minus119895)

119898119895minus 119896119898119895

120585(119895minus1)

119898119895)]

]

119879

(51)

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (52)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (53)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (54)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(55)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895


119894(119894 = 1 119898) are positive parameters rep-

resenting for prescribed disturbance attenuation levels e =

[e1198791 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin R119898times119898 (119894 =

1 119898) are arbitrary symmetric positive definite matricesand P = diag[P

1 P

119898] and P

119894isin R119898times119898 (119894 = 1 119898)

are symmetric positive definite solution of the followingequations

P119894A119894+ A119879119894P119894= minusQ119894 (56)


119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(57)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(58)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(59)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (60)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(61)


119894=1119881119894 it is



From the above analysis it is concluded that in the caseof no control saturations the signals 120585

119894119895(119894 = 1 119898 119895 =

1 119903119894) remain zeros and the control law becomes the same

as the standard robust adaptive control law described in theprevious section In the presence of control saturations 120585

119894119895

is nonzero thus giving rise to a modified tracking error119890119894= 119910119889119894

minus 119910119894minus 1205851198941 which is used in fuzzy parameter update

laws The auxiliary control term u119886119908

in (40) will be usedto compensate the effects of control saturations Note thatthe fuzzy parameter update laws (52) and (53) are similar tothe corresponding update laws (30) and (31) derived in thestandard fuzzy approximation based control problem withtracking error 119890

119894being replaced by the modified tracking

error 119890119894 The use of the modified tracking error in the fuzzy

update laws is crucial in preventing actuator constraints inonline approximation schemes


Corollary 7 For the 119894th subsystem of (4) it is assumed thatint119879

0

1198892

119894119889119905 lt infin If the control law (40) the auxiliary control term

(51) and the parameter update laws (52)ndash(54) are adoptedthen the following statements hold

(i) The closed loop system is stable and the signals e119894 120579119891119894

120579119892119894119895 and 119906

119894are bounded

(ii) Thesteadymodified tracking error satisfies lim119905rarrinfin

119890119894=

0 and the bound of the transient modified trackingerror will be given as follows

10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)

Proof From (59) it can be obtained that

119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894

= min120582V 1120574119891119894 1120574119892119894119895 120582V = mininf 120582min(Q119894)sup 120582max(Q119894) 119872

119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894

and 120582min(Q119894) and 120582max(Q119894) are the minimum and maximumeigenvalue ofQ

119894 respectively

From inequality (63) all the signals e119894 120579119891119894 120579119892119894119895 and 119906

119894are

bounded by using Barbalatrsquos lemma [8]On the other hand from (59) it can be obtained that

119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)

From the above inequality one obtains

10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)

According to the definition of 119881119894 (62) is obtained

Assumption 4 implies int119879

0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2

119889119905 lt infin Therefore 120596119894isin 1198712 From (66) we can

conclude that 119890119894isin 1198712 Using Barbalatrsquos lemma it follows that

lim119905rarrinfin

119890119894(119905) = 0 This completes the proof

Remark 8 According to Theorem 6 the 119894th subsystem of(4) achieves a 119867

infin tracking performance with a prescribeddisturbance attenuation level 120588

119894 that is the 119871

2gain from 120596

119894

to the modified tracking error 119890119894is equal or less than 120588

119894

Remark 9 As indicated from Corollary 7 the steady modi-fied tracking error 119890

119894will converge to zero The bound of the

transient modified tracking error 119890119894is an explicit function

of the design parameters and the external disturbance andfuzzy approximation error 120596

119894 The bound can be decreased

by choosing the initial estimates 120579119891119894(0) 120579

119892119894119895(0) closing to

the true values 120579lowast119891119894 120579lowast119892119894119895 The effects of parameter initial

estimate errors on the transient tracking performance canbe reduced by increasing the adaptive gain values 120574

119891119894 120574119892119894119895

and 120582min(Q119894) Furthermore the effect of external disturbanceand fuzzy approximation error 120596

119894on the transient tracking

performance can be reduced by decreasing 120588119894and increasing

120582min(Q119894) Small 120588119894implies high disturbance attenuation level

4 Simulation Examples

The attitude tracking control problem of a rigid body satellitesystem is simulated in this section to illustrate the effective-ness of the robust adaptive fuzzy controllers proposed inthis paper The mathematical model of the satellite attitudesystem can be reformulated to the general form of uncertainnonlinear MIMO system as follows [42 44]

y = F (x) + G (x) u + d (67)

where x = [q120596]119879 is the state vector where q =

[1199020 1199021 1199022 1199023]119879 is the attitude quaternion in the body-fixed

reference frame relative to the inertial frame satisfying 1199022

0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902

0is chosen as 119902

0= radic1 minus 119902

2

1minus 1199022

2minus 1199022

3

120596 = [120596119909 120596119910 120596119911]119879 is the angular velocity of the body-fixed

reference relative to the inertial frame y = [1199021 1199022 1199023]119879 is the

output vector u = [1199061 1199062 1199063]119879 is the control torque vector

d1015840 isin R3 denotes the bounded external disturbance torquesand d ≜ G(x)d1015840 F(x) and G(x) are expressed as follows

1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868

119911are the principal central moments of

inertia of the satelliteWithin this simulation the nonlinear functions F(x) and

G(x) are assumed completely unknown that is the fuzzyadaptive controllers do not require the knowledge of thesystemrsquos model In fact the dynamic model of the satelliteattitude system is only required for simulation purpose

Since the components of F(x) and G(x) are assumedunknown three fuzzy logic systems in the form of (12) areused to approximate the elements of F(x) and nine are usedto approximate the elements ofG(x) The fuzzy logic systemsused to describe F(x) have 119902

1 1199022 1199023 1205961 1205962 and 120596

3as inputs

and the ones used to describe G(x) have 1199021 1199022 and 119902

3as

inputs For each state variable x = [1199021 1199022 1199023 1205961 1205962 1205963]119879 we

define seven Gaussian membership functions as

1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)

The fuzzy basis function vectors are chosen as follows

120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)

Then we obtain fuzzy logic systems

119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)

Using (71) approximate the unknown functions119891119894and119892119894119895

respectivelyFor the given coefficients 119896

1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)

SelectingQ119894= diag[05 05] and solving (34) we obtain

P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)

The parameters of the adaptive fuzzy controllers arechosen as 120574

119891119894= 1 times 10

minus3 120574119892119894119895

= 1 times 10minus3 and 120588

119894= 05

119894 119895 = 1 2 3The initial attitude quaternion is q(0) = [09014 025

025 025]119879 and the initial value of the angular velocity

is 120596(0) = [0005 0005 0005]119879 rads The parameters

of system are given as 119868119909

= 1875 kgsdotm2 119868119910

= 4685 kgsdotm2and 119868

119911= 4685 kgsdotm2 The external disturbance torque

vector is given by d1015840 = [03 sin(005119905) 03 cos(005119905)minus03 sin(005119905)]119879Nsdotm The initial values of the parameters120579119891119894and 120579

119892119894119895are set to random values uniformly distributed

between [0 1] The desired output trajectory is chosen asy119889

= [05 sin(0005119905) 05 sin(0005119905) 05 sin(0005119905)]119879 Thecontrol objective is to force the system output y to track thedesired output trajectory y

119889

41 Without Input Saturation In this section the trackingcontrol problem is simulated to demonstrate the effective-ness of robust adaptive fuzzy controller (24) proposed inSection 31

Using the control law (24) and (30)ndash(32) simulationresults are presented in Figures 1 and 2 Figure 1 shows thecurves of the system outputs and its reference trajectorieswhich indicates that the robust adaptive fuzzy controllerachieves a good performance in tracking control problemand the effects of fuzzy approximation errors and externaldisturbances on tracking errors are effectively attenuatedFigure 2 shows that the control inputs can be carried outfeasibly without any constraints The control signals areobtained by certainty equivalence principle Therefore theydo not satisfy the control input limitations naturally


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3

Figure 1 Trajectories of outputs without saturation

42 Actuator Amplitude Saturation In order to demonstratethat the proposed adaptive control scheme (40) can workeffectively under actuator amplitude saturation numericalsimulations have been performed and presented in thissection

Consider satellite attitude model (67) with the same dis-turbances and system initial conditions mentioned aboveThe control input vector u = [119906

1 1199062 1199063]119879 has amplitude lim-

its |119906119894| le 1Nsdotm 119894 = 1 2 3The auxiliary system is constructed

as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =

1 2 3 119895 = 1 2)Using the control law (40) the auxiliary control term

(51) the parameters update laws (52)-(53) and the robustcompensation term (54) we can get the simulations inFigures 3 and 4

The system outputs and its reference trajectories areshown in Figure 3 which indicate that the outputs tracktheir reference trajectories well in spite of the externaldisturbances system uncertainties and actuator amplitudesaturation Figure 4 shows the trajectories of the control

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3

Figure 2 Trajectories of control inputs without saturation

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3

Figure 3 Trajectories of outputs with amplitude saturation


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3

Figure 4 Trajectories of control inputs with amplitude saturation

inputs with actuator amplitude saturation From these sim-ulations it is obvious that proposed control scheme (40)can achieve a good tracking performance when the actuatoramplitude limits are considered

43 Actuator Amplitude and Rate Saturation For furtheranalysis actuator amplitude and rate saturations are con-sidered The control input vector u = [119906

1 1199062 1199063]119879 has the

amplitude and rate limitation |119906119894| le 1 |

119894| le 2 119894 = 1 2 3

The other initial parameters are the same as mentioned inSection 42

According to (41) the dynamics of control inputs areexpressed as follows

119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)

Using control law (40) the actuator amplitude and rateconstraints (41) the auxiliary control term (51) the parame-ters update laws (52)-(53) and the robust compensation term(54) simulation results are presented in Figures 5ndash8 Figure 5shows the curves of outputs and their reference trajectorieswhich indicate that a good tracking control performance isstill achieved under actuator amplitude and rate constraintconditionsThe control signals and their derivatives are givenin Figures 6 and 7 It is observed that the control signals satisfythe amplitude and rate limitations That is the proposedrobust control scheme for the satellite attitude control systemcan prevent the control signals from reaching amplitude and

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3

Figure 5 Trajectories of outputs with amplitude and rate saturation

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3

Figure 6 Trajectories of control inputs with amplitude and ratesaturation


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3

Figure 7 Derivatives of control inputs

0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883

rate saturation limits Figure 8 shows the signals 119906119888119894

(119894 =

1 2 3) Obviously they do not satisfy the control inputlimitations

From the aforementioned simulations it is demonstratedthat the robust adaptive fuzzy tracking controller proposedin this paper not only can generate control inputs thatsatisfy actuator amplitude and rate saturations but also caneffectively attenuate the effects of approximation error andextern disturbance on tracking errors Thus the proposedrobust adaptive control scheme is valid for satellite attitudecontrol system with actuator amplitude and rate saturation

5 Conclusions

In this work a robust fuzzy tracking control approach hasbeen presented for a class of uncertain nonlinear MIMOsystems in the presence of input saturation and externaldisturbances Fuzzy logic systems are used to approximatethe unknown system nonlinear terms An auxiliary system isconstructed to compensate the effects of actuator saturationsand then the actuator saturations can be augmented into thecontroller The modified tracking error is introduced andused in fuzzy parameter update laws A robust compensationcontrol is designed to attenuate the effects of external distur-bances and fuzzy approximation errors The stability prop-erties and tracking performance of the closed-loop systemare obtained through Lyapunov analysis Steady and transientmodified tracking errors are analyzed and the bound ofmodified tracking errors can be adjusted by tuning certaindesign parametersTheproposed control scheme is applicableto uncertain nonlinear systems not only with actuator ampli-tude saturation but also with actuator amplitude and ratesaturation The simulation results of satellite attitude controlsystem are presented to demonstrate the effectiveness of pro-posed controller In this paper the control system relies on theoutput derivatives up to 119903

119894minus 1 order as in (22) which is also

practically restrictive for high order systems Future researchwill be concentrated on an observer-based robust adaptivefuzzy control of uncertain nonlinear systems with actuatorsaturations based on the results of [18ndash20] and this paper

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is supported by the Fund of Innovation byGraduate School of National University of Defense Technol-ogy under Grant B140106

References

[1] L-X Wang ldquoAutomatic design of fuzzy controllersrdquo Automat-ica vol 35 no 8 pp 1471ndash1475 1999

[2] S C Tong J T Tang and T Wang ldquoFuzzy adaptive control ofmultivariable nonlinear systemsrdquo Fuzzy Sets and Systems vol111 no 2 pp 153ndash167 2000


[3] B-S Chen C-H Lee and Y-C Chang ldquo119867infin

tracking designof uncertain nonlinear SISO systems adaptive fuzzy approachrdquoIEEE Transactions on Fuzzy Systems vol 4 no 1 pp 32ndash431996

[4] T Zhang S S Ge and C C Hang ldquoAdaptive neural networkcontrol for strict-feedback nonlinear systems using backstep-ping designrdquo Automatica vol 36 no 12 pp 1835ndash1846 2000

[5] S-C Tong X-L He and H-G Zhang ldquoA combined backstep-ping and small-gain approach to robust adaptive fuzzy outputfeedback controlrdquo IEEE Transactions on Fuzzy Systems vol 17no 5 pp 1059ndash1069 2009

[6] Y Pan Y Zhou T Sun and M J Er ldquoComposite adaptivefuzzy 119867

infintracking control of uncertain nonlinear systemsrdquo

Neurocomputing vol 99 pp 15ndash24 2013[7] Y-T Kim and Z Z Bien ldquoRobust adaptive fuzzy control in

the presence of external disturbance and approximation errorrdquoFuzzy Sets and Systems vol 148 no 3 pp 377ndash393 2004

[8] S C Tong and H-X Li ldquoFuzzy adaptive sliding-mode controlfor MIMO nonlinear systemsrdquo IEEE Transactions on FuzzySystems vol 11 no 3 pp 354ndash360 2003

[9] Y-C Chang ldquoRobust tracking control for nonlinear MIMOsystems via fuzzy approachesrdquo Automatica vol 36 no 10 pp1535ndash1545 2000

[10] Y-J Liu C L P Chen G-X Wen and S Tong ldquoAdaptiveneural output feedback tracking control for a class of uncertaindiscrete-time nonlinear systemsrdquo IEEE Transactions on NeuralNetworks vol 22 no 7 pp 1162ndash1167 2011

[11] S C Tong B Chen and Y F Wang ldquoFuzzy adaptive outputfeedback control for MIMO nonlinear systemsrdquo Fuzzy Sets andSystems vol 156 no 2 pp 285ndash299 2005

[12] Y-J Liu S-C Tong and W Wang ldquoAdaptive fuzzy outputtracking control for a class of uncertain nonlinear systemsrdquoFuzzy Sets and Systems vol 160 no 19 pp 2727ndash2754 2009

[13] Y-J Liu W Wang S-C Tong and Y-S Liu ldquoRobust adaptivetracking control for nonlinear systems based on bounds of fuzzyapproximation parametersrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems and Humans vol 40 no1 pp 170ndash184 2010

[14] Y N Yang J Wu and W Zheng ldquoAdaptive fuzzy sliding modecontrol for robotic airship with model uncertainty and externaldisturbancerdquo Journal of Systems Engineering and Electronics vol23 no 2 pp 250ndash255 2012

[15] C L P Chen Y-J Liu and G-X Wen ldquoFuzzy neural network-based adaptive control for a class of uncertain nonlinearstochastic systemsrdquo IEEE Transactions on Cybernetics vol 44no 5 pp 583ndash593 2014

[16] C L P Chen G-X Wen Y-J Liu and F-Y Wang ldquoAdaptiveconsensus control for a class of nonlinearmultiagent time-delaysystems using neural networksrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 25 no 6 pp 1217ndash12262014

[17] Y J Liu and S C Tong ldquoAdaptive neural network trackingcontrol of uncertain nonlinear discrete-time systems withnonaffine dead-zone inputrdquo IEEE Transactions on Cyberneticsvol 45 no 3 pp 497ndash505 2015

[18] Y-J Liu S Tong and C L P Chen ldquoAdaptive fuzzy controlvia observer design for uncertain nonlinear systems withunmodeled dynamicsrdquo IEEETransactions on Fuzzy Systems vol21 no 2 pp 275ndash288 2013

[19] S C Tong Y Li YM Li andY J Liu ldquoObserver-based adaptivefuzzy backstepping control for a class of stochastic nonlinear

strict-feedback systemsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart B Cybernetics vol 41 no 6 pp 1693ndash1704 2011

[20] S C Tong B Y Huo and Y M Li ldquoObserver-based adaptivedecentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failuresrdquo IEEE Transactions onFuzzy Systems vol 22 no 1 pp 1ndash15 2014

[21] J FarrellM Polycarpou andM SharmaAdaptive Backsteppingwith Magnitude Rate and Bandwidth Constraints AircraftLongitude Control CaliforniaUniversity Riverside Departmentof Electrical Engineering 2006

[22] X P Shi G P Yuan and L Li ldquoAdaptive fuzzy attitudetracking control of spacecraft with input magnitude and rateconstraintsrdquo in Proceedings of the 31st Chinese Control Confer-ence (CCC rsquo12) pp 842ndash846 July 2012

[23] A Leonessa W M Haddad T Hayakawa and Y MorelldquoAdaptive control for nonlinear uncertain systems with actuatoramplitude and rate saturation constraintsrdquo International Journalof Adaptive Control and Signal Processing vol 23 no 1 pp 73ndash96 2009

[24] J A Farrell M Sharma and M Polycarpou ldquoLongitudinalflight-path control using online function approximationrdquo Jour-nal of Guidance Control and Dynamics vol 26 no 6 pp 885ndash897 2003

[25] J Farrell M Sharma and M Polycarpou ldquoBackstepping-basedflight control with adaptive function approximationrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1089ndash11022005

[26] M Polycarpou J Farrell and M Sharma ldquoOn-line approx-imation control of uncertain nonlinear systems issues withcontrol input saturationrdquo in Proceedings of the American ControlConference pp 543ndash548 June 2003

[27] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011

[28] Y-L Zhou and M Chen ldquoSliding mode control for NSVswith input constraint using neural network and disturbanceobserverrdquo Mathematical Problems in Engineering vol 2013Article ID 904830 12 pages 2013

[29] Y M Li T S Li and S C Tong ldquoAdaptive fuzzy modularbackstepping output feedback control of uncertain nonlinearsystems in the presence of input saturationrdquo InternationalJournal of Machine Learning and Cybernetics vol 4 no 5 pp527ndash536 2013

[30] Y M Li S C Tong and T S Li ldquoDirect adaptive fuzzy back-stepping control of uncertain nonlinear systems in the presenceof input saturationrdquoNeural Computing andApplications vol 23no 5 pp 1207ndash1216 2013

[31] Y M Li S C Tong and T S Li ldquoAdaptive fuzzy output-feedback control for output constrained nonlinear systems inthe presence of input saturationrdquo Fuzzy Sets and Systems vol248 pp 138ndash155 2014

[32] D Lin X YWang F Z Nian and Y L Zhang ldquoDynamic fuzzyneural networks modeling and adaptive backstepping trackingcontrol of uncertain chaotic systemsrdquo Neurocomputing vol 73no 16ndash18 pp 2873ndash2881 2010

[33] D Lin X Wang and Y Yao ldquoFuzzy neural adaptive trackingcontrol of unknown chaotic systems with input saturationrdquoNonlinear Dynamics vol 67 no 4 pp 2889ndash2897 2012

[34] W Z Gao andR R Selmic ldquoNeural network control of a class ofnonlinear systems with actuator saturationrdquo IEEE Transactionson Neural Networks vol 17 no 1 pp 147ndash156 2006


[35] R Y Yuan J Q Yi W S Yu and G L Fan ldquoAdaptive controllerdesign for uncertain nonlinear systems with input magnitudeand rate limitationsrdquo in Proceedings of the American ControlConference (ACC rsquo11) pp 3536ndash3541 July 2011

[36] R Y Yuan X M Tan G L Fan and J Q Yi ldquoRobustadaptive neural network control for a class of uncertain non-linear systems with actuator amplitude and rate saturationsrdquoNeurocomputing vol 125 pp 72ndash80 2014

[37] B Xiao Q-L Hu and G Ma ldquoAdaptive sliding mode backstep-ping control for attitude tracking of flexible spacecraft underinput saturation and singularityrdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engineeringvol 224 no 2 pp 199ndash214 2010

[38] C Y Wen J Zhou Z T Liu and H Y Su ldquoRobust adaptivecontrol of uncertain nonlinear systems in the presence of inputsaturation and external disturbancerdquo IEEE Transactions onAutomatic Control vol 56 no 7 pp 1672ndash1678 2011

[39] S Sui S C Tong and Y M Li ldquoAdaptive fuzzy backsteppingoutput feedback tracking control of MIMO stochastic pure-feedback nonlinear systems with input saturationrdquo Fuzzy Setsand Systems vol 254 pp 26ndash46 2014

[40] X Wang T S Li L Y Fang and B Lin ldquoAdaptive NN controlfor a class of strict-feedback discrete-time nonlinear systemswith input saturationrdquo in Advances in Neural NetworksmdashISNN2013 vol 7952 of Lecture Notes in Computer Science pp 70ndash78Springer 2013

[41] Z Zhu Y Q Xia and M Y Fu ldquoAdaptive sliding modecontrol for attitude stabilization with actuator saturationrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4898ndash4907 2011

[42] Q L Hu B Xiao and M I Friswell ldquoRobust fault-tolerantcontrol for spacecraft attitude stabilisation subject to inputsaturationrdquo IET Control Theory amp Applications vol 5 no 2 pp271ndash282 2011

[43] G Feng and R Lozano Adaptive Control Systems NewnesOxford UK 1999

[44] Y N Yang J Wu and W Zheng ldquoAttitude control for a stationkeeping airship using feedback linearization and fuzzy slidingmode controlrdquo International Journal of Innovative ComputingInformation and Control vol 8 no 12 pp 8299ndash8310 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The design of tracking controllers for uncertain MIMOnonlinear systems with actuator constraints is a challengingproblem During the past decades there have been extensiveresearches on the control of nonlinear systems with variousconstraints Analysis and design of control systems with con-trol saturation have been widely studied in [24ndash42] Farrell etal [24ndash26] have presented an adaptive backstepping approachfor unknown nonlinear systems with known magnituderate and bandwidth constraints on intermediate states oractuators without disturbance To tackle with the physical sat-uration an auxiliary systemwith the same order as that of theplant was constructed to compensate the effect of saturationThe control input saturation is investigated through onlineapproximation based control for uncertain nonlinear systemsin [26] An adaptive control and the constrained adaptivecontrol in combination with the backstepping technique areproposed in [29] A direct adaptive fuzzy control approachfor uncertain nonlinear systems with input saturation ispresented in [30] in which a Nussbaum function is usedto compensate for the nonlinear term arising from theinput saturation In [31] an adaptive fuzzy output feedbackcontrol algorithm for a class of output constrained uncertainnonlinear systems with input saturation is developed byemploying a barrier Lyapunov function and an auxiliarysystem Adaptive backstepping tracking control based onfuzzy neural networks is investigated for unknown chaoticsystems in [32] and with control input constraints in [33]Neural network based adaptive control schemeswith externaldisturbances and actuator saturations are presented in [35]in which auxiliary systems are added to attenuate the effectsof input saturation It is apparent that the presence of inputsaturation constraint substantially increases the complexityof control system design for uncertain MIMO nonlinearsystems In the constrained adaptive control the key problemis how to handle the constraint effect of the actuatorrsquos physicalconstraints To this end we introduce an auxiliary designsystem to handle the constraint effect in this paper Basedon the states of the auxiliary design system constrainedadaptive control is investigated for a class of uncertainMIMOnonlinear systems with input constraints using robust fuzzycontrol technique

In this paper a robust adaptive fuzzy tracking controlscheme is presented to handle the external disturbances andactuator physical constraints for uncertain MIMO nonlinearsystems In control design fuzzy logic systems are used toapproximate unknown nonlinear systems Note that inputsaturations are nonsmooth functions but the adaptive fuzzycontrol technique requires all functions differentiable [6]To compensate the effect of input saturations an auxiliarysystem is constructed and the actuator saturations then canbe augmented into the controller The modified trackingerror is introduced and used in fuzzy parameter update lawsBesides in order to deal with fuzzy approximation errorsfor unknown nonlinear systems and external disturbancesa robust compensation control is designed It is proved thatthe proposed control approach can guarantee that all thesignals of the resulting closed-loop system are bounded andthe closed-loop system obtains 119867


through Lyapunov analysis The transient modified tracking

errors performance is derived to be explicit functions ofdesign parameters and thus bounds of modified trackingerrors can be adjusted by tuning design parameters

The rest of this paper is organized as followsThe descrip-tion of the uncertain MIMO nonlinear system under consid-eration and necessary preliminaries are given in Section 2 InSection 3 the robust adaptive fuzzy control without controlsaturation is firstly designed When the actuators have phys-ical limitations this approach may not be able to be success-fully implemented In order to solve this problem a robustadaptive fuzzy control scheme with input saturation is inves-tigated The simulation results of satellite attitude controlare presented to demonstrate the effectiveness of proposedcontroller in Section 4 Section 5 contains the conclusion

2 Problem Formulation and Preliminary

21 Problem Formulation Consider a class of uncertainMIMO affine nonlinear system

x = f (x) +

119898

sum

119894=1

g119894(x) 119906119894+ d1015840

1199101= ℎ1(x)

119910119898

= ℎ119898

(x)

(1)

where x = [1199091 119909

119899] isin 119877

119899 is the state vector available formeasurement u = [119906

1 119906

119898] isin 119877

119898 is the control vectorwith |119906

119894| le 119906119894max where 119906

119894max denote the actuator amplitudey = [119910

1 119910

119898] isin 119877

119898 is the output vector ℎ1(x) ℎ

119898(x)

are smooth functions defined on the open set of R119899 f(x)g1(x) g

119898(x) are continuous unknown but smooth vector

fields d1015840 = [1198891015840

1 119889

1015840

119899]119879

isin R119899 is external disturbance vectorwhere 119889

1015840

1is unknown but bounded

Define that 119871 fℎ119894 is the Lie derivative of ℎ119894along vector

field f(x) and 119871119896

fℎ119894 is recursively as 119871119896

fℎ119894 = 119871 f(119871119896minus1

f ℎ119894) A

multivariable uncertain nonlinear system of the form (1) hasa vector relative degree [119903

1 1199032 119903

119898] at a point x

0if

119871g119895119871119896

fℎ119894 (x) = 0 forall1 le 119895 le 119898 119896 le 119903119894minus 1 (2)

for all x in a neighborhood of x0

Denote

G (x) =

[[[[[[

[

119871g11198711199031minus1


1199031minus1

f ℎ1(x)

119871g11198711199032minus1


1199032minus1

f ℎ2(x)

sdot sdot sdot d sdot sdot sdot

119871g1119871119903119898minus1

f ℎ119898

(x) sdot sdot sdot 119871g119898119871119903119898minus1

f ℎ119898

(x)

]]]]]]

]

=

[[[[

[

11989211

(x) sdot sdot sdot 1198921119898

(x) d

1198921198981

(x) sdot sdot sdot 119892119898119898

(x)

]]]]

]

(3)




y(r) = F (x) + G (x) u + d (4)

where

y(r) = [119910(1199031)

1 119910(1199032)

2 119910

(119903119898)

119898]119879

(5)

F (x) =

[[[[[[[

[

1198711199031

f ℎ1 (x)

1198711199032

f ℎ2 (x)

119871119903119898

f ℎ119898

(x)

]]]]]]]

]

=

[[[[[[

[

1198911(x)

1198912(x)

119891119898

(x)

]]]]]]

]

(6)

d =

[[[[[[[[[[[[[

[

1199031

sum

119894=1

119871119894minus1

x 11987111988910158401198711199031minus119894

f ℎ1(x)

1199032

sum

119894=1

119871119894minus1

x 11987111988910158401198711199032minus119894

f ℎ2(x)

119903119898

sum

119894=1

119871119894minus1

x 1198711198891015840119871119903119898minus119894

f ℎ119898

(x)

]]]]]]]]]]]]]

]

=

[[[[[[

[

1198891

1198892

119889119898

]]]]]]

]

(7)

x = [1199101 119910

(1199031)

1 1199102 119910

(1199032)

2 119910

119898 119910

(119903119898)

119898] (8)


119898



y119889= [1199101198891

1199101198892

119910119889119898

]119879

(9)










1 1199092 119909

119899]119879



119877119897

If 1199091is 119865119897


119899is 119865119897

119899 then 119910 is 119866

119897

(119897 = 1 2 119903)

(10)

where 119865119897

119894and 119866


119865119897119894(119909119894) and 120583




119910 (x) =

sum119903

119897=1119910119897

(prod119899

119894=1120583119865119897119894(119909119894))

sum119903

119897=1prod119899

119894=1120583119865119897119894(119909119894)

(11)

where 119910119897

= max119910isin119877

120583119866119897(119910)


119910 (x) = 120579119879

120585 (x) (12)

where 120579 = [1199101

119910119903


and 120585(x) = [1205851(x) 120585



120585119897(x) =

sum119903

119897=1(prod119899

119894=1120583119865119897119894(119909119894))

sum119903

119897=1prod119899

119894=1120583119865119897119894(119909119894)

(13)




supxisinU119888

10038161003816100381610038161003816119891 (x) minus 120579

119879

120585 (x)10038161003816100381610038161003816 le 120576 (14)


120579lowast


[supxisinU119888

10038161003816100381610038161003816119891 (x) minus 120579

119879

120585 (x)10038161003816100381610038161003816] (15)


119891 (x) = 120579lowast119879

120585 (x) + 120576 (16)



119894(x) (119894 = 1 119898)


systems as follows

119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 119898

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 119898

(17)


where 120579119891119894

isin R119872119891119894 and 120579119892119894119895



tors and 119872119891119894and 119872



F (x | 120579119865) =

[[[[

[

1198911(x | 120579

1198911)

119891119898

(x | 120579119891119898

)

]]]]

]

G (x | 120579119866) =

[[[[

[

11989211

(x | 12057911989211


(x | 1205791198921119898

)

d

1198921198981

(x | 1205791198921198981

) sdot sdot sdot 119892119898119898

(x | 120579119892119898119898

)

]]]]

]

(18)

Then F(x | 120579119865) and G(x | 120579



119891119894(119894 = 1 119898) and 119892

119894119895(119894 119895 = 1 119898) as follows

120579lowast

119891119894= arg min

120579119891119894isinΩF

supxisinU119888

10038161003816100381610038161003816119891119894(x | 120579

119891119894) minus 119891119894(x)10038161003816100381610038161003816

120579lowast

119892119894119895= arg min

120579119892119894119895isinΩG

supxisinU119888

100381610038161003816100381610038161003816119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)

100381610038161003816100381610038161003816

(19)



119891119894(119894 = 1 119898) and 120579lowast

119892119894119895(119894 119895 =


119894and 119892


119891119894(x) = 120579

lowast119879

119891119894120585119891119894

(x) + 120576119891119894

119894 = 1 119898

119892119894119895(x) = 120579

lowast119879

119892119894119895120585119892119894119895

(x) + 120576119892119894119895

119894 119895 = 1 119898

(20)

where 120576119891119894and 120576119892119894119895





119890119894= 119910119889119894

minus 119910119894

119894 = 1 2 119898 (21)

Define 119906119897119894

(119894 = 1 119898) as follows

119906119897119894= 119910(119903119894)

119889119894+

119903119894

sum

119895=1

119896119894119895119890(119895minus1)

119894119894 = 1 119898 (22)

where 1198961198941 119896







where u119897= [1199061198971 119906

119897119898]119879




119865) + u119897+ u119889] (24)





119866)] u + u

119897

+ u119889+ d

(25)


e119894= A119894e119894+ B119894

[119891119894(x | 120579

119891119894) minus 119891119894(x)]

+

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus B119894119906119889119894

minus B119894119889119894

(26)


119889and

A119894=

[[[[[[[[[

[



d


minus1198961198941

minus1198961198942

minus1198961198943


]]]]]]]]]

]

B119894=

[[[[[[[[[

[

0

0

0

1

]]]]]]]]]

]

e119894=

[[[[[[[[[[

[

119890119894

119890119894

119890(119903119894minus2)

119894

119890(119903119894minus1)

119894

]]]]]]]]]]

]

(27)


1205961015840

119894= [119891119894(x | 120579

lowast

119891119894) minus 119891119894(x)] +

119898

sum

119895=1

[119892119894119895(x | 120579

lowast

119892119894119895) minus 119892119894119895(x)] 119906

119895

(28)




0

1205961015840119879

1198941205961015840

119894119889119905 lt infin



form

e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(29)

where 119891119894

= 120579119891119894

minus 120579lowast

119891119894 119892119894119895

= 120579119892119894119895

minus 120579lowast

119892119894119895 and 120596

119894= 1205961015840

119894minus 119889119894



as follows

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (30)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (31)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (32)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(33)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




1 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin


1 P

119898] and P

119894isin R119898times119898 (119894 =


P119894A119894+ A119879119894P119894= minusQ119894 (34)

where Q119894



119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(35)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(36)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(37)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (38)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(39)


119894=1119881119894 it is










u119888= Gminus1 (x | 120579

119866) [minusF (x | 120579

119865) + u119897+ u119886119908

+ u119889]

u = 119878119872119877

(u119888)

(40)


119886119908





[

1199031198941

1199031198942

] = [

1199031198942

119878119877(120596119894(119878119872

(119906119888119894) minus 1199031198941))

]

[

119906119894

119894

] = [

1199031198941

1199031198942

]

(41)


119888 and 119906



119872(sdot) and 119878




119878119872

(sdot) =

119872 if 119909 ge 119872

119909 if |119909| lt 119872

minus119872 if 119909 le minus119872

(42)



1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(43)

where 120582119894119895


parameters


119890119894= 119910119889119894

minus 119910119894minus 1205851198941

119894 = 1 2 119898 (44)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119886119908119894

minus 119906119889119894

minus 119889119894minus 120585(119903119894)

1198941minus

119903119894

sum

119895=1

119896119894119895120585(119895minus1)

1198941

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895)

(45)

where 119906119886119908119894


From (43) the term 120585(119903119894)


120585(119903119894)

1198941= minus

119903119894

sum

119895=1

120582119894119895120585(119903119894minus119895)

119894119895+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) (46)

Let

119906119886119908119894

=

119903119894

sum

119895=1

(120582119894119895120585(119903119894minus119895)

119894119895minus 119896119894119895120585(119895minus1)

119894119895) (47)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119889119894

minus 119889119894

(48)

Define e119894

= [119890119894 119890119894 119890

(119903119894minus1)




e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(49)





follows

1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(50)

u119886119908

= [

[

1199031

sum

119895=1

(1205821119895120585(1199031minus119895)

1119895minus 1198961119895120585(119895minus1)

1119895)

sdot sdot sdot

119903119898

sum

119895=1

(120582119898119895

120585(119903119898minus119895)

119898119895minus 119896119898119895

120585(119895minus1)

119898119895)]

]

119879

(51)

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (52)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (53)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (54)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(55)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




[e1198791 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin R119898times119898 (119894 =


1 P

119898] and P

119894isin R119898times119898 (119894 = 1 119898)


P119894A119894+ A119879119894P119894= minusQ119894 (56)


119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(57)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(58)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(59)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (60)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(61)


119894=1119881119894 it is




119894119895(119894 = 1 119898 119895 =



119894119895










0

1198892




120579119892119894119895 and 119906

119894are bounded


119890119894=


10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)


119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894


119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894


119894 respectively


119894are


119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)


10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)



0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2



lim119905rarrinfin




119894 that is the 119871

2gain from 120596

119894


119894







119892119894119895(0) closing to



119891119894 120574119892119894119895







y = F (x) + G (x) u + d (67)




0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902



2

1minus 1199022

2minus 1199022

3





1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












y(r) = F (x) + G (x) u + d (4)

where

y(r) = [119910(1199031)

1 119910(1199032)

2 119910

(119903119898)

119898]119879

(5)

F (x) =

[[[[[[[

[

1198711199031

f ℎ1 (x)

1198711199032

f ℎ2 (x)

119871119903119898

f ℎ119898

(x)

]]]]]]]

]

=

[[[[[[

[

1198911(x)

1198912(x)

119891119898

(x)

]]]]]]

]

(6)

d =

[[[[[[[[[[[[[

[

1199031

sum

119894=1

119871119894minus1

x 11987111988910158401198711199031minus119894

f ℎ1(x)

1199032

sum

119894=1

119871119894minus1

x 11987111988910158401198711199032minus119894

f ℎ2(x)

119903119898

sum

119894=1

119871119894minus1

x 1198711198891015840119871119903119898minus119894

f ℎ119898

(x)

]]]]]]]]]]]]]

]

=

[[[[[[

[

1198891

1198892

119889119898

]]]]]]

]

(7)

x = [1199101 119910

(1199031)

1 1199102 119910

(1199032)

2 119910

119898 119910

(119903119898)

119898] (8)


119898



y119889= [1199101198891

1199101198892

119910119889119898

]119879

(9)










1 1199092 119909

119899]119879



119877119897

If 1199091is 119865119897


119899is 119865119897

119899 then 119910 is 119866

119897

(119897 = 1 2 119903)

(10)

where 119865119897

119894and 119866


119865119897119894(119909119894) and 120583




119910 (x) =

sum119903

119897=1119910119897

(prod119899

119894=1120583119865119897119894(119909119894))

sum119903

119897=1prod119899

119894=1120583119865119897119894(119909119894)

(11)

where 119910119897

= max119910isin119877

120583119866119897(119910)


119910 (x) = 120579119879

120585 (x) (12)

where 120579 = [1199101

119910119903


and 120585(x) = [1205851(x) 120585



120585119897(x) =

sum119903

119897=1(prod119899

119894=1120583119865119897119894(119909119894))

sum119903

119897=1prod119899

119894=1120583119865119897119894(119909119894)

(13)




supxisinU119888

10038161003816100381610038161003816119891 (x) minus 120579

119879

120585 (x)10038161003816100381610038161003816 le 120576 (14)


120579lowast


[supxisinU119888

10038161003816100381610038161003816119891 (x) minus 120579

119879

120585 (x)10038161003816100381610038161003816] (15)


119891 (x) = 120579lowast119879

120585 (x) + 120576 (16)



119894(x) (119894 = 1 119898)


systems as follows

119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 119898

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 119898

(17)


where 120579119891119894

isin R119872119891119894 and 120579119892119894119895



tors and 119872119891119894and 119872



F (x | 120579119865) =

[[[[

[

1198911(x | 120579

1198911)

119891119898

(x | 120579119891119898

)

]]]]

]

G (x | 120579119866) =

[[[[

[

11989211

(x | 12057911989211


(x | 1205791198921119898

)

d

1198921198981

(x | 1205791198921198981

) sdot sdot sdot 119892119898119898

(x | 120579119892119898119898

)

]]]]

]

(18)

Then F(x | 120579119865) and G(x | 120579



119891119894(119894 = 1 119898) and 119892

119894119895(119894 119895 = 1 119898) as follows

120579lowast

119891119894= arg min

120579119891119894isinΩF

supxisinU119888

10038161003816100381610038161003816119891119894(x | 120579

119891119894) minus 119891119894(x)10038161003816100381610038161003816

120579lowast

119892119894119895= arg min

120579119892119894119895isinΩG

supxisinU119888

100381610038161003816100381610038161003816119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)

100381610038161003816100381610038161003816

(19)



119891119894(119894 = 1 119898) and 120579lowast

119892119894119895(119894 119895 =


119894and 119892


119891119894(x) = 120579

lowast119879

119891119894120585119891119894

(x) + 120576119891119894

119894 = 1 119898

119892119894119895(x) = 120579

lowast119879

119892119894119895120585119892119894119895

(x) + 120576119892119894119895

119894 119895 = 1 119898

(20)

where 120576119891119894and 120576119892119894119895





119890119894= 119910119889119894

minus 119910119894

119894 = 1 2 119898 (21)

Define 119906119897119894

(119894 = 1 119898) as follows

119906119897119894= 119910(119903119894)

119889119894+

119903119894

sum

119895=1

119896119894119895119890(119895minus1)

119894119894 = 1 119898 (22)

where 1198961198941 119896







where u119897= [1199061198971 119906

119897119898]119879




119865) + u119897+ u119889] (24)





119866)] u + u

119897

+ u119889+ d

(25)


e119894= A119894e119894+ B119894

[119891119894(x | 120579

119891119894) minus 119891119894(x)]

+

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus B119894119906119889119894

minus B119894119889119894

(26)


119889and

A119894=

[[[[[[[[[

[



d


minus1198961198941

minus1198961198942

minus1198961198943


]]]]]]]]]

]

B119894=

[[[[[[[[[

[

0

0

0

1

]]]]]]]]]

]

e119894=

[[[[[[[[[[

[

119890119894

119890119894

119890(119903119894minus2)

119894

119890(119903119894minus1)

119894

]]]]]]]]]]

]

(27)


1205961015840

119894= [119891119894(x | 120579

lowast

119891119894) minus 119891119894(x)] +

119898

sum

119895=1

[119892119894119895(x | 120579

lowast

119892119894119895) minus 119892119894119895(x)] 119906

119895

(28)




0

1205961015840119879

1198941205961015840

119894119889119905 lt infin



form

e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(29)

where 119891119894

= 120579119891119894

minus 120579lowast

119891119894 119892119894119895

= 120579119892119894119895

minus 120579lowast

119892119894119895 and 120596

119894= 1205961015840

119894minus 119889119894



as follows

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (30)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (31)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (32)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(33)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




1 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin


1 P

119898] and P

119894isin R119898times119898 (119894 =


P119894A119894+ A119879119894P119894= minusQ119894 (34)

where Q119894



119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(35)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(36)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(37)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (38)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(39)


119894=1119881119894 it is










u119888= Gminus1 (x | 120579

119866) [minusF (x | 120579

119865) + u119897+ u119886119908

+ u119889]

u = 119878119872119877

(u119888)

(40)


119886119908





[

1199031198941

1199031198942

] = [

1199031198942

119878119877(120596119894(119878119872

(119906119888119894) minus 1199031198941))

]

[

119906119894

119894

] = [

1199031198941

1199031198942

]

(41)


119888 and 119906



119872(sdot) and 119878




119878119872

(sdot) =

119872 if 119909 ge 119872

119909 if |119909| lt 119872

minus119872 if 119909 le minus119872

(42)



1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(43)

where 120582119894119895


parameters


119890119894= 119910119889119894

minus 119910119894minus 1205851198941

119894 = 1 2 119898 (44)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119886119908119894

minus 119906119889119894

minus 119889119894minus 120585(119903119894)

1198941minus

119903119894

sum

119895=1

119896119894119895120585(119895minus1)

1198941

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895)

(45)

where 119906119886119908119894


From (43) the term 120585(119903119894)


120585(119903119894)

1198941= minus

119903119894

sum

119895=1

120582119894119895120585(119903119894minus119895)

119894119895+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) (46)

Let

119906119886119908119894

=

119903119894

sum

119895=1

(120582119894119895120585(119903119894minus119895)

119894119895minus 119896119894119895120585(119895minus1)

119894119895) (47)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119889119894

minus 119889119894

(48)

Define e119894

= [119890119894 119890119894 119890

(119903119894minus1)




e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(49)





follows

1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(50)

u119886119908

= [

[

1199031

sum

119895=1

(1205821119895120585(1199031minus119895)

1119895minus 1198961119895120585(119895minus1)

1119895)

sdot sdot sdot

119903119898

sum

119895=1

(120582119898119895

120585(119903119898minus119895)

119898119895minus 119896119898119895

120585(119895minus1)

119898119895)]

]

119879

(51)

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (52)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (53)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (54)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(55)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




[e1198791 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin R119898times119898 (119894 =


1 P

119898] and P

119894isin R119898times119898 (119894 = 1 119898)


P119894A119894+ A119879119894P119894= minusQ119894 (56)


119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(57)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(58)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(59)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (60)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(61)


119894=1119881119894 it is




119894119895(119894 = 1 119898 119895 =



119894119895










0

1198892




120579119892119894119895 and 119906

119894are bounded


119890119894=


10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)


119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894


119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894


119894 respectively


119894are


119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)


10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)



0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2



lim119905rarrinfin




119894 that is the 119871

2gain from 120596

119894


119894







119892119894119895(0) closing to



119891119894 120574119892119894119895







y = F (x) + G (x) u + d (67)




0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902



2

1minus 1199022

2minus 1199022

3





1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 120579119891119894

isin R119872119891119894 and 120579119892119894119895



tors and 119872119891119894and 119872



F (x | 120579119865) =

[[[[

[

1198911(x | 120579

1198911)

119891119898

(x | 120579119891119898

)

]]]]

]

G (x | 120579119866) =

[[[[

[

11989211

(x | 12057911989211


(x | 1205791198921119898

)

d

1198921198981

(x | 1205791198921198981

) sdot sdot sdot 119892119898119898

(x | 120579119892119898119898

)

]]]]

]

(18)

Then F(x | 120579119865) and G(x | 120579



119891119894(119894 = 1 119898) and 119892

119894119895(119894 119895 = 1 119898) as follows

120579lowast

119891119894= arg min

120579119891119894isinΩF

supxisinU119888

10038161003816100381610038161003816119891119894(x | 120579

119891119894) minus 119891119894(x)10038161003816100381610038161003816

120579lowast

119892119894119895= arg min

120579119892119894119895isinΩG

supxisinU119888

100381610038161003816100381610038161003816119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)

100381610038161003816100381610038161003816

(19)



119891119894(119894 = 1 119898) and 120579lowast

119892119894119895(119894 119895 =


119894and 119892


119891119894(x) = 120579

lowast119879

119891119894120585119891119894

(x) + 120576119891119894

119894 = 1 119898

119892119894119895(x) = 120579

lowast119879

119892119894119895120585119892119894119895

(x) + 120576119892119894119895

119894 119895 = 1 119898

(20)

where 120576119891119894and 120576119892119894119895





119890119894= 119910119889119894

minus 119910119894

119894 = 1 2 119898 (21)

Define 119906119897119894

(119894 = 1 119898) as follows

119906119897119894= 119910(119903119894)

119889119894+

119903119894

sum

119895=1

119896119894119895119890(119895minus1)

119894119894 = 1 119898 (22)

where 1198961198941 119896







where u119897= [1199061198971 119906

119897119898]119879




119865) + u119897+ u119889] (24)





119866)] u + u

119897

+ u119889+ d

(25)


e119894= A119894e119894+ B119894

[119891119894(x | 120579

119891119894) minus 119891119894(x)]

+

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus B119894119906119889119894

minus B119894119889119894

(26)


119889and

A119894=

[[[[[[[[[

[



d


minus1198961198941

minus1198961198942

minus1198961198943


]]]]]]]]]

]

B119894=

[[[[[[[[[

[

0

0

0

1

]]]]]]]]]

]

e119894=

[[[[[[[[[[

[

119890119894

119890119894

119890(119903119894minus2)

119894

119890(119903119894minus1)

119894

]]]]]]]]]]

]

(27)


1205961015840

119894= [119891119894(x | 120579

lowast

119891119894) minus 119891119894(x)] +

119898

sum

119895=1

[119892119894119895(x | 120579

lowast

119892119894119895) minus 119892119894119895(x)] 119906

119895

(28)




0

1205961015840119879

1198941205961015840

119894119889119905 lt infin



form

e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(29)

where 119891119894

= 120579119891119894

minus 120579lowast

119891119894 119892119894119895

= 120579119892119894119895

minus 120579lowast

119892119894119895 and 120596

119894= 1205961015840

119894minus 119889119894



as follows

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (30)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (31)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (32)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(33)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




1 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin


1 P

119898] and P

119894isin R119898times119898 (119894 =


P119894A119894+ A119879119894P119894= minusQ119894 (34)

where Q119894



119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(35)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(36)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(37)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (38)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(39)


119894=1119881119894 it is










u119888= Gminus1 (x | 120579

119866) [minusF (x | 120579

119865) + u119897+ u119886119908

+ u119889]

u = 119878119872119877

(u119888)

(40)


119886119908





[

1199031198941

1199031198942

] = [

1199031198942

119878119877(120596119894(119878119872

(119906119888119894) minus 1199031198941))

]

[

119906119894

119894

] = [

1199031198941

1199031198942

]

(41)


119888 and 119906



119872(sdot) and 119878




119878119872

(sdot) =

119872 if 119909 ge 119872

119909 if |119909| lt 119872

minus119872 if 119909 le minus119872

(42)



1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(43)

where 120582119894119895


parameters


119890119894= 119910119889119894

minus 119910119894minus 1205851198941

119894 = 1 2 119898 (44)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119886119908119894

minus 119906119889119894

minus 119889119894minus 120585(119903119894)

1198941minus

119903119894

sum

119895=1

119896119894119895120585(119895minus1)

1198941

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895)

(45)

where 119906119886119908119894


From (43) the term 120585(119903119894)


120585(119903119894)

1198941= minus

119903119894

sum

119895=1

120582119894119895120585(119903119894minus119895)

119894119895+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) (46)

Let

119906119886119908119894

=

119903119894

sum

119895=1

(120582119894119895120585(119903119894minus119895)

119894119895minus 119896119894119895120585(119895minus1)

119894119895) (47)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119889119894

minus 119889119894

(48)

Define e119894

= [119890119894 119890119894 119890

(119903119894minus1)




e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(49)





follows

1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(50)

u119886119908

= [

[

1199031

sum

119895=1

(1205821119895120585(1199031minus119895)

1119895minus 1198961119895120585(119895minus1)

1119895)

sdot sdot sdot

119903119898

sum

119895=1

(120582119898119895

120585(119903119898minus119895)

119898119895minus 119896119898119895

120585(119895minus1)

119898119895)]

]

119879

(51)

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (52)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (53)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (54)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(55)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




[e1198791 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin R119898times119898 (119894 =


1 P

119898] and P

119894isin R119898times119898 (119894 = 1 119898)


P119894A119894+ A119879119894P119894= minusQ119894 (56)


119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(57)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(58)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(59)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (60)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(61)


119894=1119881119894 it is




119894119895(119894 = 1 119898 119895 =



119894119895










0

1198892




120579119892119894119895 and 119906

119894are bounded


119890119894=


10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)


119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894


119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894


119894 respectively


119894are


119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)


10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)



0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2



lim119905rarrinfin




119894 that is the 119871

2gain from 120596

119894


119894







119892119894119895(0) closing to



119891119894 120574119892119894119895







y = F (x) + G (x) u + d (67)




0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902



2

1minus 1199022

2minus 1199022

3





1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0

1205961015840119879

1198941205961015840

119894119889119905 lt infin



form

e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(29)

where 119891119894

= 120579119891119894

minus 120579lowast

119891119894 119892119894119895

= 120579119892119894119895

minus 120579lowast

119892119894119895 and 120596

119894= 1205961015840

119894minus 119889119894



as follows

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (30)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (31)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (32)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(33)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




1 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin


1 P

119898] and P

119894isin R119898times119898 (119894 =


P119894A119894+ A119879119894P119894= minusQ119894 (34)

where Q119894



119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(35)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(36)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(37)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (38)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(39)


119894=1119881119894 it is










u119888= Gminus1 (x | 120579

119866) [minusF (x | 120579

119865) + u119897+ u119886119908

+ u119889]

u = 119878119872119877

(u119888)

(40)


119886119908





[

1199031198941

1199031198942

] = [

1199031198942

119878119877(120596119894(119878119872

(119906119888119894) minus 1199031198941))

]

[

119906119894

119894

] = [

1199031198941

1199031198942

]

(41)


119888 and 119906



119872(sdot) and 119878




119878119872

(sdot) =

119872 if 119909 ge 119872

119909 if |119909| lt 119872

minus119872 if 119909 le minus119872

(42)



1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(43)

where 120582119894119895


parameters


119890119894= 119910119889119894

minus 119910119894minus 1205851198941

119894 = 1 2 119898 (44)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119886119908119894

minus 119906119889119894

minus 119889119894minus 120585(119903119894)

1198941minus

119903119894

sum

119895=1

119896119894119895120585(119895minus1)

1198941

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895)

(45)

where 119906119886119908119894


From (43) the term 120585(119903119894)


120585(119903119894)

1198941= minus

119903119894

sum

119895=1

120582119894119895120585(119903119894minus119895)

119894119895+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) (46)

Let

119906119886119908119894

=

119903119894

sum

119895=1

(120582119894119895120585(119903119894minus119895)

119894119895minus 119896119894119895120585(119895minus1)

119894119895) (47)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119889119894

minus 119889119894

(48)

Define e119894

= [119890119894 119890119894 119890

(119903119894minus1)




e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(49)





follows

1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(50)

u119886119908

= [

[

1199031

sum

119895=1

(1205821119895120585(1199031minus119895)

1119895minus 1198961119895120585(119895minus1)

1119895)

sdot sdot sdot

119903119898

sum

119895=1

(120582119898119895

120585(119903119898minus119895)

119898119895minus 119896119898119895

120585(119895minus1)

119898119895)]

]

119879

(51)

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (52)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (53)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (54)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(55)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




[e1198791 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin R119898times119898 (119894 =


1 P

119898] and P

119894isin R119898times119898 (119894 = 1 119898)


P119894A119894+ A119879119894P119894= minusQ119894 (56)


119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(57)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(58)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(59)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (60)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(61)


119894=1119881119894 it is




119894119895(119894 = 1 119898 119895 =



119894119895










0

1198892




120579119892119894119895 and 119906

119894are bounded


119890119894=


10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)


119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894


119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894


119894 respectively


119894are


119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)


10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)



0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2



lim119905rarrinfin




119894 that is the 119871

2gain from 120596

119894


119894







119892119894119895(0) closing to



119891119894 120574119892119894119895







y = F (x) + G (x) u + d (67)




0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902



2

1minus 1199022

2minus 1199022

3





1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














u119888= Gminus1 (x | 120579

119866) [minusF (x | 120579

119865) + u119897+ u119886119908

+ u119889]

u = 119878119872119877

(u119888)

(40)


119886119908





[

1199031198941

1199031198942

] = [

1199031198942

119878119877(120596119894(119878119872

(119906119888119894) minus 1199031198941))

]

[

119906119894

119894

] = [

1199031198941

1199031198942

]

(41)


119888 and 119906



119872(sdot) and 119878




119878119872

(sdot) =

119872 if 119909 ge 119872

119909 if |119909| lt 119872

minus119872 if 119909 le minus119872

(42)



1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(43)

where 120582119894119895


parameters


119890119894= 119910119889119894

minus 119910119894minus 1205851198941

119894 = 1 2 119898 (44)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119886119908119894

minus 119906119889119894

minus 119889119894minus 120585(119903119894)

1198941minus

119903119894

sum

119895=1

119896119894119895120585(119895minus1)

1198941

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895)

(45)

where 119906119886119908119894


From (43) the term 120585(119903119894)


120585(119903119894)

1198941= minus

119903119894

sum

119895=1

120582119894119895120585(119903119894minus119895)

119894119895+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) (46)

Let

119906119886119908119894

=

119903119894

sum

119895=1

(120582119894119895120585(119903119894minus119895)

119894119895minus 119896119894119895120585(119895minus1)

119894119895) (47)


119890(119903119894)

119894+

119903119894

sum

119895=1

119896119894119895119890(119903119894minus1)

119894

= 119891119894(x | 120579

119891119894) minus 119891119894(x) +

119898

sum

119895=1

[119892119894119895(x | 120579

119892119894119895) minus 119892119894119895(x)] 119906

119895

minus 119906119889119894

minus 119889119894

(48)

Define e119894

= [119890119894 119890119894 119890

(119903119894minus1)




e119894= A119894e119894+ B119894

[

[

119879

119891119894120585119891119894

(x) +

119898

sum

119895=1

119879

119892119894119895120585119892119894119895

(x) 119906119895

]

]

minus B119894119906119889119894

+ B119894120596119894

(49)





follows

1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(50)

u119886119908

= [

[

1199031

sum

119895=1

(1205821119895120585(1199031minus119895)

1119895minus 1198961119895120585(119895minus1)

1119895)

sdot sdot sdot

119903119898

sum

119895=1

(120582119898119895

120585(119903119898minus119895)

119898119895minus 119896119898119895

120585(119895minus1)

119898119895)]

]

119879

(51)

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (52)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (53)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (54)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(55)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




[e1198791 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin R119898times119898 (119894 =


1 P

119898] and P

119894isin R119898times119898 (119894 = 1 119898)


P119894A119894+ A119879119894P119894= minusQ119894 (56)


119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(57)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(58)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(59)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (60)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(61)


119894=1119881119894 it is




119894119895(119894 = 1 119898 119895 =



119894119895










0

1198892




120579119892119894119895 and 119906

119894are bounded


119890119894=


10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)


119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894


119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894


119894 respectively


119894are


119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)


10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)



0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2



lim119905rarrinfin




119894 that is the 119871

2gain from 120596

119894


119894







119892119894119895(0) closing to



119891119894 120574119892119894119895







y = F (x) + G (x) u + d (67)




0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902



2

1minus 1199022

2minus 1199022

3





1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












follows

1205851198941

= 1205851198942

minus 12058211989411205851198941


= 120585119894119903119894

minus 120582119894119903119894minus1

120585119894119903119894minus1

120585119894119903119894

= minus120582119894119903119894120585119894119903119894

+

119898

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 119898

(50)

u119886119908

= [

[

1199031

sum

119895=1

(1205821119895120585(1199031minus119895)

1119895minus 1198961119895120585(119895minus1)

1119895)

sdot sdot sdot

119903119898

sum

119895=1

(120582119898119895

120585(119903119898minus119895)

119898119895minus 119896119898119895

120585(119895minus1)

119898119895)]

]

119879

(51)

119891119894

= minus120574119891119894120585119891119894

(x)B119879119894P119894e119894

119894 = 1 119898 (52)

119892119894119895

= minus120574119892119894119895120585119892119894119895

(x)B119879119894P119894e119894119906119895

119894 119895 = 1 119898 (53)

119906119889119894

=1

21205882

119894

B119879119894P119894e119894

119894 = 1 119898 (54)



int

119879

0

e119879Qe 119889119905 le e119879 (0)Pe (0) +

119898

sum

119894=1

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894119895=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) +

119898

sum

119894=1

1205882

119894int

119879

0

1205962

119894119889119905

(55)

where 120574119891119894

(119894 = 1 119898) and 120574119892119894119895




[e1198791 e119879

119898]119879 Q = diag[Q

1 Q

119898] and Q

119894isin R119898times119898 (119894 =


1 P

119898] and P

119894isin R119898times119898 (119894 = 1 119898)


P119894A119894+ A119879119894P119894= minusQ119894 (56)


119881119894=

1

2e119879119894P119894e119894+

1

2120574119891119894

119879

119891119894119891119894

+1

2

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895119892119894119895

(57)


119894=

1

2e119879119894P119894e119894+

1

2e119879119894P119894e119894+

1

120574119891119894

120579

119879

119891119894119891119894

+

119898

sum

119894=1

1

120574119892119894119895

120579

119879

119892119894119895119892119894119895

=1

2e119879119894(P119894A119894+ A119879119894P119894) e119894+

1

2(minus119906119889119894

+ 120596119894) (B119879119894P119894e119894+ e119879119894P119894B119894)

+1

120574119891119894

[120574119891119894e119879119894P119894B119894120585119879

119891119894(x) +

120579

119879

119891119894] 119891119894

+

119898

sum

119894=1

1

120574119892119894119895

[120574119892119894119895e119879119894P119894B119894120585119879

119892119894119895(x) 119906119895+

120579

119879

119892119894119895] 119892119894119895

(58)


119894= minus

1

2e119879119894Q119894e119894+

1

2120596119894(B119879119894P119894e119894+ e119879119894P119894B119894) minus

1

21205882

119894

e119879119894P119894B119894B119879119894P119894e119894

= minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894minus

1

2(

1

120588119894

e119879119894P119894B119894minus 120588119894120596119894)

2

le minus1

2e119879119894Q119894e119894+

1

21205882

1198941205962

119894

(59)


yields

1

2int

119879

0

e119879119894Q119894e119894119889119905 le 119881

119894(0) minus 119881

119894(119879) +

1205882

119894

2int

119879

0

1205962

119894119889119905 (60)



int

119879

0

e119879119894Q119894e119894119889119905 le e119879

119894(0)P119894e119894(0) +

1

120574119891119894

119879

119891119894(0) 119891119894

(0)

+

119898

sum

119894=1

1

120574119892119894119895

119879

119892119894119895(0) 119892119894119895

(0) + 1205882

119894int

119879

0

1205962

119894119889119905

(61)


119894=1119881119894 it is




119894119895(119894 = 1 119898 119895 =



119894119895










0

1198892




120579119892119894119895 and 119906

119894are bounded


119890119894=


10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)


119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894


119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894


119894 respectively


119894are


119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)


10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)



0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2



lim119905rarrinfin




119894 that is the 119871

2gain from 120596

119894


119894







119892119894119895(0) closing to



119891119894 120574119892119894119895







y = F (x) + G (x) u + d (67)




0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902



2

1minus 1199022

2minus 1199022

3





1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

1198892




120579119892119894119895 and 119906

119894are bounded


119890119894=


10038171003817100381710038171198901198941003817100381710038171003817

2

le

2 [(1120574119891119894) 119879

119891119894(0) 119891119894

(0) + sum119898

119894=1(1120574119892119894119895

) 119879

119892119894119895(0) 119892119894119895

(0)]

120582min (Q119894)

+

2e119879119894(0)P119894e119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(62)


119894le minus119888119894119881119894+ 120583119894 (63)

where 119888119894


119894= max(120579

119891119894) 119872119894119895

= max(120579119892119894119895

) 120583119894

=

1198722

1198942120574119891119894

+ sum119898

119895=1(12120574119892119894119895

)1198722

119894119895+ (12)120588

2

1198941205962

119894 120596119894

= sup 120596119894


119894 respectively


119894are


119894le minus

1

2120582min (Q

119894)1003817100381710038171003817e119894

1003817100381710038171003817

2

+1

21205882

1198941205962

119894 (64)


10038171003817100381710038171198901198941003817100381710038171003817

2

le1003817100381710038171003817e119894

1003817100381710038171003817

2

leminus2119894+ 1205882

1198941205962

119894

120582min (119876) (65)

Hence

10038171003817100381710038171198901198941003817100381710038171003817

2

= int

119879

0

1198902

119894119889119905 le

2 [119881119894(0) minus 119881

119894(119879)] + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

le

2119881119894(0) + 120588

2

119894int119879

0

1205962

119894119889119905

120582min (Q119894)

(66)



0

12059610158402

119894119889119905 lt infin then int

119879

0

1205962

119894119889119905 =

int119879

0

(1205961015840

119894minus 119889119894)2



lim119905rarrinfin




119894 that is the 119871

2gain from 120596

119894


119894







119892119894119895(0) closing to



119891119894 120574119892119894119895







y = F (x) + G (x) u + d (67)




0+

1199022

1+ 1199022

2+ 1199022

3= 1 here 119902



2

1minus 1199022

2minus 1199022

3





1198911(x) = minus

1

41199021(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199022120596119909120596119910

+119868119909minus 119868119911

2119868119910

1199023120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199020120596119910120596119911

1198912(x) = minus

1

41199022(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119910minus 119868119909

2119868119911

1199021120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199020120596119909120596119911+

119868119910minus 119868119911

2119868119909

1199023120596119910120596119911


1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198913(x) = minus

1

41199023(1205962

119909+ 1205962

119910+ 1205962

119911) +

119868119909minus 119868119910

2119868119911

1199020120596119909120596119910

+119868119911minus 119868119909

2119868119910

1199021120596119909120596119911+

119868119911minus 119868119910

2119868119909

1199022120596119910120596119911

G (x) =

[[[[[[[

[

1199020

2119868119909

minus1199023

2119868119910

1199022

2119868119911

1199023

2119868119909

1199020

2119868119910

minus1199021

2119868119911

minus1199022

2119868119909

1199021

2119868119910

1199020

2119868119911

]]]]]]]

]

(68)

where 119868119909 119868119910 and 119868





1 1199022 1199023 1205961 1205962 and 120596

3as inputs


3as



1205831198651119894(119909119894) =

1

1 + exp (minus5 (119909119894+ 06))

1205831198652119894(119909119894) = exp (minus05 (119909

119894+ 04)

2

)

1205831198653119894(119909119894) = exp (minus05 (119909

119894+ 02)

2

)

1205831198654119894(119909119894) = exp (minus05119909

119894

2

)

1205831198655119894(119909119894) = exp (minus05 (119909

119894minus 02)

2

)

1205831198656119894(119909119894) = exp (minus05 (119909

119894minus 04)

2

)

1205831198657119894(119909119894) =

1

1 + exp (minus5 (119909119894minus 06))

119894 = 1 2 3 4 5 6

(69)


120585119891119894

= [

[

prod6

119894=11205831198651119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

prod6

119894=11205831198657119894(119909119894)

sum7

119895=1prod6

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 = 1 2 3

120585119892119894119895

= [

[

prod3

119894=11205831198651119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

prod3

119894=11205831198657119894(119909119894)

sum7

119895=1prod3

119894=1120583119865119895

119894

(119909119894)

]

]

119879

119894 119895 = 1 2 3

(70)


119891119894(x | 120579

119891119894) = 120579119879

119891119894120585119891119894

(x) 119894 = 1 2 3

119892119894119895(x | 120579

119892119894119895) = 120579119879

119892119894119895120585119892119894119895

(x) 119894 119895 = 1 2 3

(71)



1198941= 025 (119894 = 1 2 3) and 119896

1198942=

05 (119894 = 1 2 3) we have

A1= A2= A3= [

0 1

minus025 minus05] B

1= B2= B3= [

0

1]

(72)


P119894= [

1625 05

05 3] 119894 = 1 2 3 (73)


119891119894= 1 times 10

minus3 120574119892119894119895


119894= 05





= 1875 kgsdotm2 119868119910







119889




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3






as follows1205851198941

= 1205851198942

minus 12058211989411205851198941

1205851198942

= minus 12058211989421205851198942

+

3

sum

119895=1

119892119894119895(x | 120579

119892119894119895) (119906119895minus 119906119888119895) 119894 = 1 2 3

(74)

where 120582119894119895

= 1 (119894 = 1 2 3 119895 = 1 2) and 120585119894119895(0) = 0 (119894 =




0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus20

0

20

u1

minus20

0

20

u2

minus20

0

20

u3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3




1 1199062 1199063]119879 has the


119894| le 2 119894 = 1 2 3



119894= 1198781(1205961198941198781(119906119888119894) minus 119906119894) 119894 = 1 2 3 (75)

where 120596119894= 205 (119894 = 1 2 3)


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

ReferenceResponse

minus1

0

1

q1

minus1

0

1

q2

minus1

0

1

q3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus1

0

1

u1

minus1

0

1

u2

minus1

0

1

u3



0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus2

0

2

du1

minus2

0

2

du2

minus2

0

2

du3


0 500 1000 1500

0 500 1000 1500

0 500 1000 1500Time (s)

minus50

0

50

uc1

minus50

0

50

uc2

minus50

0

50

uc3

Figure 8 1199061198881 1199061198882 and 119906

1198883


(119894 =



5 Conclusions






Acknowledgment


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Robust Adaptive Fuzzy Control for a Class of...

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