Research Article Adaptive Sliding Mode Robust Control...
Transcript of Research Article Adaptive Sliding Mode Robust Control...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 343851 9 pageshttpdxdoiorg1011552013343851
Research ArticleAdaptive Sliding Mode Robust Control for VirtualCompound-Axis Servo System
Yan Ren12 Zhenghua Liu2 Le Chang2 and Nuan Wen2
1 Information Engineering School Inner Mongolia University of Science and Technology Baotou 014010 China2 School of Automation Science and Electrical Engineering Beihang University Beijing 100191 China
Correspondence should be addressed to Yan Ren renyanry163com
Received 26 July 2013 Revised 12 October 2013 Accepted 20 October 2013
Academic Editor Xudong Zhao
Copyright copy 2013 Yan Ren et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A structure mode of virtual compound-axis servo system is proposed to improve the tracking accuracy of the ordinary optoelectrictracking platform It is based on the structure and principles of compound-axis servo system A hybrid position control schemecombining the PD controller and feed-forward controller is used in subsystem to track the tracking error of the main systemThispaper analyzes the influences of the equivalent disturbance in main system and proposes an adaptive sliding mode robust controlmethod based on the improved disturbance observer The sliding mode technique helps this disturbance observer to deal with theuncompensated disturbance in high frequency by making use of the rapid switching control value which is based on the subtleerror of disturbance estimation Besides the high-frequency chattering is alleviated effectively in this proposalThe effectiveness ofthe proposal is confirmed by experiments on optoelectric tracking platform
1 Introduction
Optoelectric tracking (OET) platform is applied to ensurethe stability of line of sight (LOS) and achieve the automatictracking of the maneuvering target under the maneuvermotion of carrier and the external disturbance When thetarget distance is farther the small deviation will lead to alarge change of the target position Accordingly the hightracking accuracy is very important in OET system Thecompound-axis system is a form of dual-dimension asso-ciation control system in the multivariable control systemsIt is demonstrated effectively to improve the accuracy andbandwidth of the OET system by inserting a fast steeringmirror (FSM) with the high resonant frequency in the largeinertia tracking frame of themain optical path [1] It has beenwidely applied in high-precision OET systems satellite laserranging laser communications and space remote sensingdetection and so forth [2ndash5] But there are some ordinaryOET systems without the FSM Therefore it is valuableto research how to improve the tracking accuracy of theordinary OET system
High-precision motion control is the key of OET devicewhich directly influences the accuracy of LOS tracking
As a typical kind of servo motor system the robustnessagainst system uncertainties and external disturbances isurgently required without considering mechanical factorsThe disturbance observer (DOB) approach has been widelyused as an effective robust method to compensate the distur-bance and parameter variations from both environment andsystem [6 7] But DOB only deals with the disturbances inthe low-frequency domain High-frequency components ofthe disturbances such as sudden changes in external forcesand Coulomb friction can degrade the control effect of aDOB based tracking control Therefore some researchershave tried to design the fuzzy disturbance observer [8 9]nonlinear disturbance observer [10 11] and extended stateobserver [12 13]
As one of the most significant approaches sliding modecontrol (SMC) has been proven to be an effective robust con-trol strategy for the systems with large uncertainties nonlin-earities and bounded external disturbances Consequentlysome researchers have actively developed and researchedSMC which is used in uncertain systems [14 15] time-delay systems [16 17] fuzzy systems [18 19] and Markovianjump systems [20 21] Besides a sliding mode disturbanceobserver (SDOB) was employed in [22] It can deal with
2 Mathematical Problems in Engineering
high-frequency disturbance through selecting appropriateswitching control value which is greater than the upperbound of the disturbance Meanwhile the switching controlvalue also causes chattering phenomenon For alleviatingchattering [23] proposed an SDOB with an adaptive law thatrequires only a small switching gain However this methodneeds a model of unknown disturbance which is difficult toobtain in engineering practice As a result some intelligentmethods have been devoted to estimate the upper bound [2425] However most of these intelligent units are not realizedeasily in engineering practice and they are not adequatelysensitive to the chattering of control input
In this paper the characteristics and structure com-pound-axis system are analyzed at first For improving thetracking accuracy of the ordinary OET servo system withoutthe FSM the design scheme and the realization means ofvirtual compound-axis are presented In this method theadaptive slidingmode controller (ASMC) is designed inmainsystem to reduce the tracking error and deal with high-frequency disturbance by making use of its switching controlvalue and virtual subsystem employs a hybrid control schemecombining the PD controller (PDC) and feed-forward con-troller (FFC) simultaneously The improved DOB compen-sates disturbance and helps to acquire a small switchinggain of ASMC to alleviate the chattering Experiments areimplemented to confirm the validity of this method
The rest of the paper is organized as follows Section 2introduces the OET servo system including compound-axissystem and virtual compound-axis system Section 3 analyzesthe compound control scheme based on virtual compound-axis servo system Then the structure and design of DOBbased on ASMC are proposed Experimental results areshown in Section 4 Finally the conclusion and future workare given in Section 5
2 OET Servo System
21 Compound-Axis OET Servo System The compound-axisservo system improves the tracking accuracy of optoelectricaltracking system greatly which is an effective method to lockthe beam and target on one point A typical compound-axis control system uses double detectors structure includ-ing the main-axis system and subaxis system The mainsystem mainly realizes the coarse tracking and its error isthe subsystem input The subsystem is applied to adjustthe tracking residuals of the main system to realize high-precision tracking
22 Virtual Compound-Axis OET Servo System Accordingto compound-axis control principle the conception of thevirtual compound-axis servo system is put forward in OETsystem without the FSM To simplify the problem this paperanalyses a uniaxial platform as an example at firstWe assumea coaxial virtual platform that has the same position withthe physical platform and a virtual platform owns the virtualtracking detector Virtual platform connects to the physicalplatform which connects to foundation bed The spatial
Virtual detector line of sightTracking detector line of sight
Tracking platform
Foundation bed
Virtual platform
120579ref
120579n e
e
m
Δe
120579
Target
Figure 1 Virtual compound-axis servo system
120579ref
120579n
n
minus
minus Virtualtrackingdetector
Tracking detector 120579p
120579
eminus120591s
K2
K1
P2
P1
Figure 2 Structure diagramof virtual compound-axis servo system
position is overlapped between the tracking detector andvirtual tracking detector
The relationship of all the parts of the virtual compound-axis servo system is shown in Figure 1 where 119890 denotes theLOS deviation of the tracking detector and the target Δ119890
denotes the LOS deviation of the virtual tracking detectorand the target and 119890
119898denotes the LOS deviation of the
tracking detector and the virtual tracking detector Thetracking platform tracks the target roughly while the virtualplatform tracks 119890 finely Therefore the cooperated trackingmode which is similar to the compound-axis servo system isrealized through the combination of main axis and subaxisThe tracking platform is in motion relative to foundation bedand the virtual platform is in motion relative to the trackingplatform in motion
The structure of the virtual compound-axis system isshown in Figure 2 where 119870
1(119904) 119870
2(119904) 119875
1(119904) and 119875
2(119904)
denote the tracking loop controller virtual loop controllerthe equivalent characteristic of the tracking platform and theequivalent characteristic of the virtual platform respectively119890minus120591119904 is the tracking detector delay 119899 is the measuring noise
120579ref is the maneuvering target position in inertia space 120579 isthe LOS position of the tracking platform 120579
119901is the LOS of
the tracking detector and 120579119899is the LOS of the virtual tracking
detectorThe delay factor only affects the phase-frequency char-
acteristic of the system but does not affect the gain of thesystem In order to simplify the analysis the delay factorand the transfer function of the virtual tracking detector are
Mathematical Problems in Engineering 3
ignoredThen the closed-loop transfer function of the systemis expressed as
119866 (119904)
=
1198701(119904) 1198751(119904) + 119870
2(119904) 1198752(119904) + 119870
1(119904) 1198751(119904) 1198702(119904) 1198752(119904)
[1 + 1198701(119904) 1198751(119904)] [1 + 119870
2(119904) 1198752(119904)]
(1)
The transfer functions of main system and subsystem aredescribed as follows respectively
1198661(119904) =
1198701(119904) 1198751(119904)
1 + 1198701(119904) 1198751(119904)
1198662(119904) =
1198702(119904) 1198752(119904)
1 + 1198702(119904) 1198752(119904)
(2)
From (1)-(2) the poles of the entire system are the polesof the main system and subsystem Only if the main systemand subsystem are stable the entire system is stable
Suppose 1198901(119904) as the error of the main system and 119890
2(119904)
as the error of the subsystem The error of subsystem can beobtained as
1198902(119904) =
1198901(119904)
1 + 1198702(119904) 1198752(119904)
=
120579ref (119904)
[1 + 1198701(119904) 1198751(119904)] [1 + 119870
2(119904) 1198752(119904)]
(3)
From (3) the tracking accuracy of the virtual compound-axis system can be improved by controlling the main systemand subsystem And the error of the system is the errorof subsystem Therefore the virtual compound-axis systemowns the high tracking accuracy
23 Virtual Compound-Axis Servo System ImplementationTwo problems should be solved as virtual compound-axisservo system
(1) problem of virtual objects property namely howto determine the transfer function of the virtualplatform
(2) problem of implementing the virtual compound-axisservo system namely how to achieve the coarse-finetracking mode on a single tracking platform
In order to implement virtual compound-axis servosystem its control structure diagram is shown in Figure 3where the delay of the tracking detector and virtual detectoris ignored In Figure 3 main system is tracking platformsubsystem is virtual tracking platform and 119890 is the inputsignal of the subsystem In order to simplify the design thetransfer function of virtual platform uses the nominal modelof the practical system As the tracking accuracy of virtualplatform is better than that of the tracking platform thetracking platform tracks virtual platform to decrease 119890
119898for
getting high tracking accuracy In Figure 3 1198701and 119870
2are
the position controllers of the main system and subsystem
P2 nominalplant
P1 plant
Subsystem
DOB120579ref minus
minus
e
Main system
K1 adaptivesliding mode
control
K2 PD+feed-forward
control
un(t)
uc(t)
d
u
em
120579n
120579
minus
Figure 3 Control realization diagram of virtual compound-axisservo system
In order to get the satisfactory control effect the ASMC isused in themain system and PDC and FFC are adopted in thesystem The output of PDC and FFC which can be regardedas a control component of the main system provides controlvariable to the actual plant
3 Compound Control Scheme of VirtualCompound-Axis Servo System
31 Controller Design of Subsystem Because a LOS stabi-lization servo system is driven by a DC torque motor thedynamics of the plant can be described as
119869
120579 + 119861
120579 = 119906 + 119891 (sdot) (4)
where 119869 119861 120579 and 119906 denote the inertia mass the damping theangular position response and the control input respectively119891(sdot) denotes the disturbance such as nonlinear friction theforce from the environment the carrier disturbance andunknown time-varying and nonlinear dynamics which isdifficult to model Consider that 119869
119898le 119869 le 119869
119872 119861119898
le 119861 le
119861119872
is satisfied where 119869119898 119869119872 119861119898 and 119861
119872are positive real
numberBy assembling the parameter mismatch external distur-
bance and unknown dynamics into an equivalent distur-bance 119889 (4) can be written as
119869119899
120579 + 119861119899
120579 = 119906 + 119889 (5)
where 119869119899and 119861
119899denote the nominal mass and the nominal
damping respectivelyThe equivalent disturbance is given by119889 = (119869
119899minus 119869)
120579 + (119861
119899minus 119861)
120579 + 119891(sdot)
Define a tracking error of the system as 119890 = 120579ref minus 120579where 120579ref is a given reference signal Using the structure ofvirtual compound-axis for system (5) the control value 119906 canbe described as
119906 = 119906119899+ 119906119888minus
119889 (6)
where 119906119899and 119906
119888are the control value of subsystem and main
system respectively 119889 is the estimated disturbance through
DOBWhen the equivalent disturbance and parameters meetrequirements the task is to design 119906
119888and 119906
119899to make 119890 rarr 0
4 Mathematical Problems in Engineering
wu
d P1 plant
P(s)
dDOBQ(s)
120585
un(t)
uc(t)
minus
minus +
+ ++
++
1205791
s
Pminus1n (s)
Figure 4 Basic diagram of DOB
The subsystem nominal model is described as follows
119869119899
120579 + 119861119899
120579 = 119906119899 (7)
The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as
119906119875119863
= 119896119875119890 + 119896119863
119890
119906FFC = 119869119899
120579ref + 119861
119899
120579ref
(8)
where 119896119875and 119896
119863denote proportional coefficient and differ-
ential coefficient respectivelyAn approximate method for difference to estimate the
reference values of velocity and acceleration is introducedThis method is expressed by [26]
120579ref =
119892119904
119904 + 119892
120579ref
120579ref = (
119892119904
119904 + 119892
)
2
120579ref =119892119904
119904 + 119892
120579ref
(9)
where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0
Thence the output of controller is written as (8)
119906119899= 119906119875119863
+ 119906FFC = 119896119875119890 + 119896119863
119890 + 119869119899
120579ref + 119861
119899
120579ref (10)
32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively
In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875
119899(119904)
and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and
119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple
119908 (119904) = 119866119880119882
(119904) 119906 (119904) + 119866119863119882
(119904) 119889 (119904) + 119866120585119882
(119904) 120585 (119904) (11)
where
119866119880119882
(119904) =
119875 (119904) 119875119899(119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866119863119882
(119904) =
119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866120585119882
(119904) =
119875 (119904) 119876 (119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
(12)
Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At
low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910
therefore 119866119880119882
(119904) asymp 119875119899(119904) 119866
119863119882(119904) asymp 0 and 119866
120585119882(119904) asymp 1
This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state
In high-frequency domain if frequency 119891 gt 1198910 then
119876(119904) asymp 0 Consequently 119866119880119882
(119904) = 119866119863119882
(119904) = 119875(119904) and119866120585119882
(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise
The simplified nominal velocity model is procured asfollows
119875119899(119904) =
1
119869119899119904 + 119861119899
(13)
119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1
119899(119904) in order to make119876(119904)119875
minus1
119899(119904) regular According to the
nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above
119876 (119904) =
119892
119904 + 119892
(14)
where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as
119889 = 119876 (119904) 119889 =
119892
119904 + 119892
119889 (15)
Taking into account the physical realization of 119869119899119904
119889 isrepresented as follows
119889 = 119876 (119904) [(119869
119899119904 + 119861119899) 119908 minus 119906]
=
119892
119904 + 119892
[(119869119899119904 + 119861119899) 119908 minus 119906]
= 119869119899119892119908 minus
119892
119904 + 119892
[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]
(16)
Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation
Mathematical Problems in Engineering 5
wu
dun(t)
uc(t)
d
120579
Improved DOBJng
1
Js + B
Jng minus JnBn
1
s
P1 plant
minus
minus
minus
+
+
+
+
s
g
Figure 5 Structural diagram of improved DOB
The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as
119891 (sdot) =119889 + 120575 (17)
where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet
|120575| lt 120593 (18)
where 120593 gt 0 120593 = 120593 minus 120593 and = minus
120593 are introduced
to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33
33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890
119898= 120579 minus 120579
119899
where 120579119899is an output value of subsystem Define a sliding
mode 119911 as follows
119911 = 119890119898
+ 120582119890119898 120582 =
119861119899
119869119899
(19)
Then 119906119888is designed as
119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (20)
where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows
sgn (119911) =
1 119911 gt 0
0 119911 = 0
minus1 119911 lt 0
(21)
The adaptive law is adopted
= Proj
119869(minus1199031(
1
119869119899
119906119899minus 120582
120579) 119911)
= Proj
119861(minus1199032
120579119911)
= Proj
120593[1199033119911 sgn (119911)]
ℎ = 120593
(22)
where 1199031 1199032 and 119903
3are positive real constants The function
Proj∙(V) is defined as
Proj∙(V) =
V ∙119898
lt ∙ lt ∙119872
0 ∙ = ∙119872
and V gt 0
0 ∙ = ∙119898and V lt 0
(23)
where ∙119898is the minimum value of ∙ and ∙
119872is the maximum
value of ∙
Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890
119898asymptotically decays to
zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile
Proof Define a positive-definite Lyapunov candidate
119881 (119911) =
1
2
1198691199112
+
1
21199031
1198692
+
1
21199032
1198612
+
1
21199033
1205932
(24)
119869 = 119869 [(
120579 minus
120579119899) + 120582 (
120579 minus
120579119899)]
= (119869
120579 + 119861
120579) minus
119869
119869119899
(119869119899
120579119899+ 119861119899
120579119899) minus 119861
120579 + 120582119869
120579
= 119906119888+ 119891 (sdot) minus
119889 + 119906119899minus
119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
= 119906119888+ 120575 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
(25)
Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus
= minus
Substitute (20) into (25)
119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899
+ 119861
120579 minus 120582119869
120579 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (
1
119869119899
119906119899minus 120582
120579)
+ (119861 minus 119861)
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (
1
119869119899
119906119899minus 120582
120579) minus 119861
120579 + 120575
(119911) = minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869119911 (
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
(26)
6 Mathematical Problems in Engineering
According to (22) the time derivative of Lyapunovfunction becomes
(119911) = minus 1198701199112
minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575
le minus 1198701199112
minus 120593 |119911| + |119911| |120575|
= minus 1198701199112
minus |119911| (120593 minus |120575|)
(27)
Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is
metAccording to (24) there is
1199112
(119905) le 119911(0)2 exp(minus
119870
119869
119905) (28)
which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890
119898+120582119890119898 119890119898asymptotically
decays to zero If 119890119898(0) and 119890
119898(0) are bounded then 119890
119898(119905)
and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)
being uniformly bounded If 120579119899(0) and
120579119899(0) are bounded
then 120579119899(119905) and
120579119899(119905) are bounded According to 119890
119898= 120579 minus 120579
119899
120579 and 120579 are bounded Consequently the state variables of the
system are boundedTheorem 1 is proved
In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further
sat(119909
Δ
) =
1 119909 gt Δ
1
Δ
119909 |119909| le Δ
minus1 119909 lt minusΔ
(29)
where Δ is positive real constant namely the width of aboundary layer
Introducing sat(sdot) control law (20) becomes
119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911
4120576
) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (30)
where 120576 is a small positive constant Then there is thefollowingTheorem 2
Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then
(1) 119911 exponentially decays to zero and lim119905rarrinfin
|119911(119905)| le
radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim
119905rarrinfin|119890119898(119905)| le
(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim
119905rarrinfin| 119890119898(119905)| le
2radic120576119870 is met
Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as
(119911) = minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869119911(
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
(31)
Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911
2 is metTherefore 119911 exponentially converges until
lim119905rarrinfin
|119911 (119905)| lt
4120576
ℎ
(32)
If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas
(119911) le minus 1198701199112
minus 120593
ℎ|119911|2
4120576
+ |119911| |120575|
le minus 1198701199112
minus
1205932
|119911|2
4120576
+ 120593 |119911|
le minus 1198701199112
minus
1
120576
(
1
2
120593 |119911| minus 120576)
2
+ 120576
le minus 1198701199112
+ 120576 le minus
2119870
119869
119881 + 120576
(33)
Consequently
119881 (119905) le expminus(2119870119869)119905119881 (0) +
119869120576
2119870
(1 minus expminus(2119870119869)119905) (34)
From (34) yield to
lim119905rarrinfin
|119911 (119905)| le radic
120576
119870
(35)
It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890
119898(119905) and
119890119898(119905) are satisfied as follows
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le
1
120582
radic
120576
119870
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le 2radic
120576
119870
(36)
Mathematical Problems in Engineering 7Ph
ase (
deg)
Log
mag
nitu
de
Actual plantNominal plant
100 101 102 103 104
102
100
10minus2
10minus4
minus400minus300minus200minus100
0
Frequency (rads)
100 101 102 103 104
Frequency (rads)
Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model
which means that 119890119898(119905) and 119890
119898(119905) asymptotically converge to
the bounded regionTheorem 2 is proved
4 Experiment Results
This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform
Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869
119899=
0000125 kg and 119861119899
= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6
The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896
119875= 045 119896
119863= 012 119870 = 004 119903
1= 001 119903
2=
0001 1199033= 0005 and 120576 = 00002 Considering the modeling
mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads
Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898
better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890
119898again whose gain is
greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme
In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application
5 Conclusions
To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
high-frequency disturbance through selecting appropriateswitching control value which is greater than the upperbound of the disturbance Meanwhile the switching controlvalue also causes chattering phenomenon For alleviatingchattering [23] proposed an SDOB with an adaptive law thatrequires only a small switching gain However this methodneeds a model of unknown disturbance which is difficult toobtain in engineering practice As a result some intelligentmethods have been devoted to estimate the upper bound [2425] However most of these intelligent units are not realizedeasily in engineering practice and they are not adequatelysensitive to the chattering of control input
In this paper the characteristics and structure com-pound-axis system are analyzed at first For improving thetracking accuracy of the ordinary OET servo system withoutthe FSM the design scheme and the realization means ofvirtual compound-axis are presented In this method theadaptive slidingmode controller (ASMC) is designed inmainsystem to reduce the tracking error and deal with high-frequency disturbance by making use of its switching controlvalue and virtual subsystem employs a hybrid control schemecombining the PD controller (PDC) and feed-forward con-troller (FFC) simultaneously The improved DOB compen-sates disturbance and helps to acquire a small switchinggain of ASMC to alleviate the chattering Experiments areimplemented to confirm the validity of this method
The rest of the paper is organized as follows Section 2introduces the OET servo system including compound-axissystem and virtual compound-axis system Section 3 analyzesthe compound control scheme based on virtual compound-axis servo system Then the structure and design of DOBbased on ASMC are proposed Experimental results areshown in Section 4 Finally the conclusion and future workare given in Section 5
2 OET Servo System
21 Compound-Axis OET Servo System The compound-axisservo system improves the tracking accuracy of optoelectricaltracking system greatly which is an effective method to lockthe beam and target on one point A typical compound-axis control system uses double detectors structure includ-ing the main-axis system and subaxis system The mainsystem mainly realizes the coarse tracking and its error isthe subsystem input The subsystem is applied to adjustthe tracking residuals of the main system to realize high-precision tracking
22 Virtual Compound-Axis OET Servo System Accordingto compound-axis control principle the conception of thevirtual compound-axis servo system is put forward in OETsystem without the FSM To simplify the problem this paperanalyses a uniaxial platform as an example at firstWe assumea coaxial virtual platform that has the same position withthe physical platform and a virtual platform owns the virtualtracking detector Virtual platform connects to the physicalplatform which connects to foundation bed The spatial
Virtual detector line of sightTracking detector line of sight
Tracking platform
Foundation bed
Virtual platform
120579ref
120579n e
e
m
Δe
120579
Target
Figure 1 Virtual compound-axis servo system
120579ref
120579n
n
minus
minus Virtualtrackingdetector
Tracking detector 120579p
120579
eminus120591s
K2
K1
P2
P1
Figure 2 Structure diagramof virtual compound-axis servo system
position is overlapped between the tracking detector andvirtual tracking detector
The relationship of all the parts of the virtual compound-axis servo system is shown in Figure 1 where 119890 denotes theLOS deviation of the tracking detector and the target Δ119890
denotes the LOS deviation of the virtual tracking detectorand the target and 119890
119898denotes the LOS deviation of the
tracking detector and the virtual tracking detector Thetracking platform tracks the target roughly while the virtualplatform tracks 119890 finely Therefore the cooperated trackingmode which is similar to the compound-axis servo system isrealized through the combination of main axis and subaxisThe tracking platform is in motion relative to foundation bedand the virtual platform is in motion relative to the trackingplatform in motion
The structure of the virtual compound-axis system isshown in Figure 2 where 119870
1(119904) 119870
2(119904) 119875
1(119904) and 119875
2(119904)
denote the tracking loop controller virtual loop controllerthe equivalent characteristic of the tracking platform and theequivalent characteristic of the virtual platform respectively119890minus120591119904 is the tracking detector delay 119899 is the measuring noise
120579ref is the maneuvering target position in inertia space 120579 isthe LOS position of the tracking platform 120579
119901is the LOS of
the tracking detector and 120579119899is the LOS of the virtual tracking
detectorThe delay factor only affects the phase-frequency char-
acteristic of the system but does not affect the gain of thesystem In order to simplify the analysis the delay factorand the transfer function of the virtual tracking detector are
Mathematical Problems in Engineering 3
ignoredThen the closed-loop transfer function of the systemis expressed as
119866 (119904)
=
1198701(119904) 1198751(119904) + 119870
2(119904) 1198752(119904) + 119870
1(119904) 1198751(119904) 1198702(119904) 1198752(119904)
[1 + 1198701(119904) 1198751(119904)] [1 + 119870
2(119904) 1198752(119904)]
(1)
The transfer functions of main system and subsystem aredescribed as follows respectively
1198661(119904) =
1198701(119904) 1198751(119904)
1 + 1198701(119904) 1198751(119904)
1198662(119904) =
1198702(119904) 1198752(119904)
1 + 1198702(119904) 1198752(119904)
(2)
From (1)-(2) the poles of the entire system are the polesof the main system and subsystem Only if the main systemand subsystem are stable the entire system is stable
Suppose 1198901(119904) as the error of the main system and 119890
2(119904)
as the error of the subsystem The error of subsystem can beobtained as
1198902(119904) =
1198901(119904)
1 + 1198702(119904) 1198752(119904)
=
120579ref (119904)
[1 + 1198701(119904) 1198751(119904)] [1 + 119870
2(119904) 1198752(119904)]
(3)
From (3) the tracking accuracy of the virtual compound-axis system can be improved by controlling the main systemand subsystem And the error of the system is the errorof subsystem Therefore the virtual compound-axis systemowns the high tracking accuracy
23 Virtual Compound-Axis Servo System ImplementationTwo problems should be solved as virtual compound-axisservo system
(1) problem of virtual objects property namely howto determine the transfer function of the virtualplatform
(2) problem of implementing the virtual compound-axisservo system namely how to achieve the coarse-finetracking mode on a single tracking platform
In order to implement virtual compound-axis servosystem its control structure diagram is shown in Figure 3where the delay of the tracking detector and virtual detectoris ignored In Figure 3 main system is tracking platformsubsystem is virtual tracking platform and 119890 is the inputsignal of the subsystem In order to simplify the design thetransfer function of virtual platform uses the nominal modelof the practical system As the tracking accuracy of virtualplatform is better than that of the tracking platform thetracking platform tracks virtual platform to decrease 119890
119898for
getting high tracking accuracy In Figure 3 1198701and 119870
2are
the position controllers of the main system and subsystem
P2 nominalplant
P1 plant
Subsystem
DOB120579ref minus
minus
e
Main system
K1 adaptivesliding mode
control
K2 PD+feed-forward
control
un(t)
uc(t)
d
u
em
120579n
120579
minus
Figure 3 Control realization diagram of virtual compound-axisservo system
In order to get the satisfactory control effect the ASMC isused in themain system and PDC and FFC are adopted in thesystem The output of PDC and FFC which can be regardedas a control component of the main system provides controlvariable to the actual plant
3 Compound Control Scheme of VirtualCompound-Axis Servo System
31 Controller Design of Subsystem Because a LOS stabi-lization servo system is driven by a DC torque motor thedynamics of the plant can be described as
119869
120579 + 119861
120579 = 119906 + 119891 (sdot) (4)
where 119869 119861 120579 and 119906 denote the inertia mass the damping theangular position response and the control input respectively119891(sdot) denotes the disturbance such as nonlinear friction theforce from the environment the carrier disturbance andunknown time-varying and nonlinear dynamics which isdifficult to model Consider that 119869
119898le 119869 le 119869
119872 119861119898
le 119861 le
119861119872
is satisfied where 119869119898 119869119872 119861119898 and 119861
119872are positive real
numberBy assembling the parameter mismatch external distur-
bance and unknown dynamics into an equivalent distur-bance 119889 (4) can be written as
119869119899
120579 + 119861119899
120579 = 119906 + 119889 (5)
where 119869119899and 119861
119899denote the nominal mass and the nominal
damping respectivelyThe equivalent disturbance is given by119889 = (119869
119899minus 119869)
120579 + (119861
119899minus 119861)
120579 + 119891(sdot)
Define a tracking error of the system as 119890 = 120579ref minus 120579where 120579ref is a given reference signal Using the structure ofvirtual compound-axis for system (5) the control value 119906 canbe described as
119906 = 119906119899+ 119906119888minus
119889 (6)
where 119906119899and 119906
119888are the control value of subsystem and main
system respectively 119889 is the estimated disturbance through
DOBWhen the equivalent disturbance and parameters meetrequirements the task is to design 119906
119888and 119906
119899to make 119890 rarr 0
4 Mathematical Problems in Engineering
wu
d P1 plant
P(s)
dDOBQ(s)
120585
un(t)
uc(t)
minus
minus +
+ ++
++
1205791
s
Pminus1n (s)
Figure 4 Basic diagram of DOB
The subsystem nominal model is described as follows
119869119899
120579 + 119861119899
120579 = 119906119899 (7)
The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as
119906119875119863
= 119896119875119890 + 119896119863
119890
119906FFC = 119869119899
120579ref + 119861
119899
120579ref
(8)
where 119896119875and 119896
119863denote proportional coefficient and differ-
ential coefficient respectivelyAn approximate method for difference to estimate the
reference values of velocity and acceleration is introducedThis method is expressed by [26]
120579ref =
119892119904
119904 + 119892
120579ref
120579ref = (
119892119904
119904 + 119892
)
2
120579ref =119892119904
119904 + 119892
120579ref
(9)
where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0
Thence the output of controller is written as (8)
119906119899= 119906119875119863
+ 119906FFC = 119896119875119890 + 119896119863
119890 + 119869119899
120579ref + 119861
119899
120579ref (10)
32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively
In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875
119899(119904)
and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and
119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple
119908 (119904) = 119866119880119882
(119904) 119906 (119904) + 119866119863119882
(119904) 119889 (119904) + 119866120585119882
(119904) 120585 (119904) (11)
where
119866119880119882
(119904) =
119875 (119904) 119875119899(119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866119863119882
(119904) =
119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866120585119882
(119904) =
119875 (119904) 119876 (119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
(12)
Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At
low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910
therefore 119866119880119882
(119904) asymp 119875119899(119904) 119866
119863119882(119904) asymp 0 and 119866
120585119882(119904) asymp 1
This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state
In high-frequency domain if frequency 119891 gt 1198910 then
119876(119904) asymp 0 Consequently 119866119880119882
(119904) = 119866119863119882
(119904) = 119875(119904) and119866120585119882
(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise
The simplified nominal velocity model is procured asfollows
119875119899(119904) =
1
119869119899119904 + 119861119899
(13)
119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1
119899(119904) in order to make119876(119904)119875
minus1
119899(119904) regular According to the
nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above
119876 (119904) =
119892
119904 + 119892
(14)
where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as
119889 = 119876 (119904) 119889 =
119892
119904 + 119892
119889 (15)
Taking into account the physical realization of 119869119899119904
119889 isrepresented as follows
119889 = 119876 (119904) [(119869
119899119904 + 119861119899) 119908 minus 119906]
=
119892
119904 + 119892
[(119869119899119904 + 119861119899) 119908 minus 119906]
= 119869119899119892119908 minus
119892
119904 + 119892
[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]
(16)
Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation
Mathematical Problems in Engineering 5
wu
dun(t)
uc(t)
d
120579
Improved DOBJng
1
Js + B
Jng minus JnBn
1
s
P1 plant
minus
minus
minus
+
+
+
+
s
g
Figure 5 Structural diagram of improved DOB
The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as
119891 (sdot) =119889 + 120575 (17)
where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet
|120575| lt 120593 (18)
where 120593 gt 0 120593 = 120593 minus 120593 and = minus
120593 are introduced
to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33
33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890
119898= 120579 minus 120579
119899
where 120579119899is an output value of subsystem Define a sliding
mode 119911 as follows
119911 = 119890119898
+ 120582119890119898 120582 =
119861119899
119869119899
(19)
Then 119906119888is designed as
119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (20)
where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows
sgn (119911) =
1 119911 gt 0
0 119911 = 0
minus1 119911 lt 0
(21)
The adaptive law is adopted
= Proj
119869(minus1199031(
1
119869119899
119906119899minus 120582
120579) 119911)
= Proj
119861(minus1199032
120579119911)
= Proj
120593[1199033119911 sgn (119911)]
ℎ = 120593
(22)
where 1199031 1199032 and 119903
3are positive real constants The function
Proj∙(V) is defined as
Proj∙(V) =
V ∙119898
lt ∙ lt ∙119872
0 ∙ = ∙119872
and V gt 0
0 ∙ = ∙119898and V lt 0
(23)
where ∙119898is the minimum value of ∙ and ∙
119872is the maximum
value of ∙
Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890
119898asymptotically decays to
zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile
Proof Define a positive-definite Lyapunov candidate
119881 (119911) =
1
2
1198691199112
+
1
21199031
1198692
+
1
21199032
1198612
+
1
21199033
1205932
(24)
119869 = 119869 [(
120579 minus
120579119899) + 120582 (
120579 minus
120579119899)]
= (119869
120579 + 119861
120579) minus
119869
119869119899
(119869119899
120579119899+ 119861119899
120579119899) minus 119861
120579 + 120582119869
120579
= 119906119888+ 119891 (sdot) minus
119889 + 119906119899minus
119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
= 119906119888+ 120575 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
(25)
Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus
= minus
Substitute (20) into (25)
119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899
+ 119861
120579 minus 120582119869
120579 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (
1
119869119899
119906119899minus 120582
120579)
+ (119861 minus 119861)
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (
1
119869119899
119906119899minus 120582
120579) minus 119861
120579 + 120575
(119911) = minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869119911 (
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
(26)
6 Mathematical Problems in Engineering
According to (22) the time derivative of Lyapunovfunction becomes
(119911) = minus 1198701199112
minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575
le minus 1198701199112
minus 120593 |119911| + |119911| |120575|
= minus 1198701199112
minus |119911| (120593 minus |120575|)
(27)
Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is
metAccording to (24) there is
1199112
(119905) le 119911(0)2 exp(minus
119870
119869
119905) (28)
which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890
119898+120582119890119898 119890119898asymptotically
decays to zero If 119890119898(0) and 119890
119898(0) are bounded then 119890
119898(119905)
and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)
being uniformly bounded If 120579119899(0) and
120579119899(0) are bounded
then 120579119899(119905) and
120579119899(119905) are bounded According to 119890
119898= 120579 minus 120579
119899
120579 and 120579 are bounded Consequently the state variables of the
system are boundedTheorem 1 is proved
In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further
sat(119909
Δ
) =
1 119909 gt Δ
1
Δ
119909 |119909| le Δ
minus1 119909 lt minusΔ
(29)
where Δ is positive real constant namely the width of aboundary layer
Introducing sat(sdot) control law (20) becomes
119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911
4120576
) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (30)
where 120576 is a small positive constant Then there is thefollowingTheorem 2
Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then
(1) 119911 exponentially decays to zero and lim119905rarrinfin
|119911(119905)| le
radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim
119905rarrinfin|119890119898(119905)| le
(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim
119905rarrinfin| 119890119898(119905)| le
2radic120576119870 is met
Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as
(119911) = minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869119911(
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
(31)
Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911
2 is metTherefore 119911 exponentially converges until
lim119905rarrinfin
|119911 (119905)| lt
4120576
ℎ
(32)
If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas
(119911) le minus 1198701199112
minus 120593
ℎ|119911|2
4120576
+ |119911| |120575|
le minus 1198701199112
minus
1205932
|119911|2
4120576
+ 120593 |119911|
le minus 1198701199112
minus
1
120576
(
1
2
120593 |119911| minus 120576)
2
+ 120576
le minus 1198701199112
+ 120576 le minus
2119870
119869
119881 + 120576
(33)
Consequently
119881 (119905) le expminus(2119870119869)119905119881 (0) +
119869120576
2119870
(1 minus expminus(2119870119869)119905) (34)
From (34) yield to
lim119905rarrinfin
|119911 (119905)| le radic
120576
119870
(35)
It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890
119898(119905) and
119890119898(119905) are satisfied as follows
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le
1
120582
radic
120576
119870
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le 2radic
120576
119870
(36)
Mathematical Problems in Engineering 7Ph
ase (
deg)
Log
mag
nitu
de
Actual plantNominal plant
100 101 102 103 104
102
100
10minus2
10minus4
minus400minus300minus200minus100
0
Frequency (rads)
100 101 102 103 104
Frequency (rads)
Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model
which means that 119890119898(119905) and 119890
119898(119905) asymptotically converge to
the bounded regionTheorem 2 is proved
4 Experiment Results
This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform
Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869
119899=
0000125 kg and 119861119899
= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6
The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896
119875= 045 119896
119863= 012 119870 = 004 119903
1= 001 119903
2=
0001 1199033= 0005 and 120576 = 00002 Considering the modeling
mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads
Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898
better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890
119898again whose gain is
greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme
In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application
5 Conclusions
To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
ignoredThen the closed-loop transfer function of the systemis expressed as
119866 (119904)
=
1198701(119904) 1198751(119904) + 119870
2(119904) 1198752(119904) + 119870
1(119904) 1198751(119904) 1198702(119904) 1198752(119904)
[1 + 1198701(119904) 1198751(119904)] [1 + 119870
2(119904) 1198752(119904)]
(1)
The transfer functions of main system and subsystem aredescribed as follows respectively
1198661(119904) =
1198701(119904) 1198751(119904)
1 + 1198701(119904) 1198751(119904)
1198662(119904) =
1198702(119904) 1198752(119904)
1 + 1198702(119904) 1198752(119904)
(2)
From (1)-(2) the poles of the entire system are the polesof the main system and subsystem Only if the main systemand subsystem are stable the entire system is stable
Suppose 1198901(119904) as the error of the main system and 119890
2(119904)
as the error of the subsystem The error of subsystem can beobtained as
1198902(119904) =
1198901(119904)
1 + 1198702(119904) 1198752(119904)
=
120579ref (119904)
[1 + 1198701(119904) 1198751(119904)] [1 + 119870
2(119904) 1198752(119904)]
(3)
From (3) the tracking accuracy of the virtual compound-axis system can be improved by controlling the main systemand subsystem And the error of the system is the errorof subsystem Therefore the virtual compound-axis systemowns the high tracking accuracy
23 Virtual Compound-Axis Servo System ImplementationTwo problems should be solved as virtual compound-axisservo system
(1) problem of virtual objects property namely howto determine the transfer function of the virtualplatform
(2) problem of implementing the virtual compound-axisservo system namely how to achieve the coarse-finetracking mode on a single tracking platform
In order to implement virtual compound-axis servosystem its control structure diagram is shown in Figure 3where the delay of the tracking detector and virtual detectoris ignored In Figure 3 main system is tracking platformsubsystem is virtual tracking platform and 119890 is the inputsignal of the subsystem In order to simplify the design thetransfer function of virtual platform uses the nominal modelof the practical system As the tracking accuracy of virtualplatform is better than that of the tracking platform thetracking platform tracks virtual platform to decrease 119890
119898for
getting high tracking accuracy In Figure 3 1198701and 119870
2are
the position controllers of the main system and subsystem
P2 nominalplant
P1 plant
Subsystem
DOB120579ref minus
minus
e
Main system
K1 adaptivesliding mode
control
K2 PD+feed-forward
control
un(t)
uc(t)
d
u
em
120579n
120579
minus
Figure 3 Control realization diagram of virtual compound-axisservo system
In order to get the satisfactory control effect the ASMC isused in themain system and PDC and FFC are adopted in thesystem The output of PDC and FFC which can be regardedas a control component of the main system provides controlvariable to the actual plant
3 Compound Control Scheme of VirtualCompound-Axis Servo System
31 Controller Design of Subsystem Because a LOS stabi-lization servo system is driven by a DC torque motor thedynamics of the plant can be described as
119869
120579 + 119861
120579 = 119906 + 119891 (sdot) (4)
where 119869 119861 120579 and 119906 denote the inertia mass the damping theangular position response and the control input respectively119891(sdot) denotes the disturbance such as nonlinear friction theforce from the environment the carrier disturbance andunknown time-varying and nonlinear dynamics which isdifficult to model Consider that 119869
119898le 119869 le 119869
119872 119861119898
le 119861 le
119861119872
is satisfied where 119869119898 119869119872 119861119898 and 119861
119872are positive real
numberBy assembling the parameter mismatch external distur-
bance and unknown dynamics into an equivalent distur-bance 119889 (4) can be written as
119869119899
120579 + 119861119899
120579 = 119906 + 119889 (5)
where 119869119899and 119861
119899denote the nominal mass and the nominal
damping respectivelyThe equivalent disturbance is given by119889 = (119869
119899minus 119869)
120579 + (119861
119899minus 119861)
120579 + 119891(sdot)
Define a tracking error of the system as 119890 = 120579ref minus 120579where 120579ref is a given reference signal Using the structure ofvirtual compound-axis for system (5) the control value 119906 canbe described as
119906 = 119906119899+ 119906119888minus
119889 (6)
where 119906119899and 119906
119888are the control value of subsystem and main
system respectively 119889 is the estimated disturbance through
DOBWhen the equivalent disturbance and parameters meetrequirements the task is to design 119906
119888and 119906
119899to make 119890 rarr 0
4 Mathematical Problems in Engineering
wu
d P1 plant
P(s)
dDOBQ(s)
120585
un(t)
uc(t)
minus
minus +
+ ++
++
1205791
s
Pminus1n (s)
Figure 4 Basic diagram of DOB
The subsystem nominal model is described as follows
119869119899
120579 + 119861119899
120579 = 119906119899 (7)
The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as
119906119875119863
= 119896119875119890 + 119896119863
119890
119906FFC = 119869119899
120579ref + 119861
119899
120579ref
(8)
where 119896119875and 119896
119863denote proportional coefficient and differ-
ential coefficient respectivelyAn approximate method for difference to estimate the
reference values of velocity and acceleration is introducedThis method is expressed by [26]
120579ref =
119892119904
119904 + 119892
120579ref
120579ref = (
119892119904
119904 + 119892
)
2
120579ref =119892119904
119904 + 119892
120579ref
(9)
where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0
Thence the output of controller is written as (8)
119906119899= 119906119875119863
+ 119906FFC = 119896119875119890 + 119896119863
119890 + 119869119899
120579ref + 119861
119899
120579ref (10)
32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively
In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875
119899(119904)
and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and
119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple
119908 (119904) = 119866119880119882
(119904) 119906 (119904) + 119866119863119882
(119904) 119889 (119904) + 119866120585119882
(119904) 120585 (119904) (11)
where
119866119880119882
(119904) =
119875 (119904) 119875119899(119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866119863119882
(119904) =
119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866120585119882
(119904) =
119875 (119904) 119876 (119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
(12)
Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At
low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910
therefore 119866119880119882
(119904) asymp 119875119899(119904) 119866
119863119882(119904) asymp 0 and 119866
120585119882(119904) asymp 1
This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state
In high-frequency domain if frequency 119891 gt 1198910 then
119876(119904) asymp 0 Consequently 119866119880119882
(119904) = 119866119863119882
(119904) = 119875(119904) and119866120585119882
(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise
The simplified nominal velocity model is procured asfollows
119875119899(119904) =
1
119869119899119904 + 119861119899
(13)
119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1
119899(119904) in order to make119876(119904)119875
minus1
119899(119904) regular According to the
nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above
119876 (119904) =
119892
119904 + 119892
(14)
where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as
119889 = 119876 (119904) 119889 =
119892
119904 + 119892
119889 (15)
Taking into account the physical realization of 119869119899119904
119889 isrepresented as follows
119889 = 119876 (119904) [(119869
119899119904 + 119861119899) 119908 minus 119906]
=
119892
119904 + 119892
[(119869119899119904 + 119861119899) 119908 minus 119906]
= 119869119899119892119908 minus
119892
119904 + 119892
[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]
(16)
Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation
Mathematical Problems in Engineering 5
wu
dun(t)
uc(t)
d
120579
Improved DOBJng
1
Js + B
Jng minus JnBn
1
s
P1 plant
minus
minus
minus
+
+
+
+
s
g
Figure 5 Structural diagram of improved DOB
The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as
119891 (sdot) =119889 + 120575 (17)
where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet
|120575| lt 120593 (18)
where 120593 gt 0 120593 = 120593 minus 120593 and = minus
120593 are introduced
to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33
33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890
119898= 120579 minus 120579
119899
where 120579119899is an output value of subsystem Define a sliding
mode 119911 as follows
119911 = 119890119898
+ 120582119890119898 120582 =
119861119899
119869119899
(19)
Then 119906119888is designed as
119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (20)
where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows
sgn (119911) =
1 119911 gt 0
0 119911 = 0
minus1 119911 lt 0
(21)
The adaptive law is adopted
= Proj
119869(minus1199031(
1
119869119899
119906119899minus 120582
120579) 119911)
= Proj
119861(minus1199032
120579119911)
= Proj
120593[1199033119911 sgn (119911)]
ℎ = 120593
(22)
where 1199031 1199032 and 119903
3are positive real constants The function
Proj∙(V) is defined as
Proj∙(V) =
V ∙119898
lt ∙ lt ∙119872
0 ∙ = ∙119872
and V gt 0
0 ∙ = ∙119898and V lt 0
(23)
where ∙119898is the minimum value of ∙ and ∙
119872is the maximum
value of ∙
Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890
119898asymptotically decays to
zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile
Proof Define a positive-definite Lyapunov candidate
119881 (119911) =
1
2
1198691199112
+
1
21199031
1198692
+
1
21199032
1198612
+
1
21199033
1205932
(24)
119869 = 119869 [(
120579 minus
120579119899) + 120582 (
120579 minus
120579119899)]
= (119869
120579 + 119861
120579) minus
119869
119869119899
(119869119899
120579119899+ 119861119899
120579119899) minus 119861
120579 + 120582119869
120579
= 119906119888+ 119891 (sdot) minus
119889 + 119906119899minus
119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
= 119906119888+ 120575 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
(25)
Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus
= minus
Substitute (20) into (25)
119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899
+ 119861
120579 minus 120582119869
120579 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (
1
119869119899
119906119899minus 120582
120579)
+ (119861 minus 119861)
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (
1
119869119899
119906119899minus 120582
120579) minus 119861
120579 + 120575
(119911) = minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869119911 (
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
(26)
6 Mathematical Problems in Engineering
According to (22) the time derivative of Lyapunovfunction becomes
(119911) = minus 1198701199112
minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575
le minus 1198701199112
minus 120593 |119911| + |119911| |120575|
= minus 1198701199112
minus |119911| (120593 minus |120575|)
(27)
Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is
metAccording to (24) there is
1199112
(119905) le 119911(0)2 exp(minus
119870
119869
119905) (28)
which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890
119898+120582119890119898 119890119898asymptotically
decays to zero If 119890119898(0) and 119890
119898(0) are bounded then 119890
119898(119905)
and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)
being uniformly bounded If 120579119899(0) and
120579119899(0) are bounded
then 120579119899(119905) and
120579119899(119905) are bounded According to 119890
119898= 120579 minus 120579
119899
120579 and 120579 are bounded Consequently the state variables of the
system are boundedTheorem 1 is proved
In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further
sat(119909
Δ
) =
1 119909 gt Δ
1
Δ
119909 |119909| le Δ
minus1 119909 lt minusΔ
(29)
where Δ is positive real constant namely the width of aboundary layer
Introducing sat(sdot) control law (20) becomes
119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911
4120576
) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (30)
where 120576 is a small positive constant Then there is thefollowingTheorem 2
Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then
(1) 119911 exponentially decays to zero and lim119905rarrinfin
|119911(119905)| le
radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim
119905rarrinfin|119890119898(119905)| le
(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim
119905rarrinfin| 119890119898(119905)| le
2radic120576119870 is met
Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as
(119911) = minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869119911(
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
(31)
Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911
2 is metTherefore 119911 exponentially converges until
lim119905rarrinfin
|119911 (119905)| lt
4120576
ℎ
(32)
If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas
(119911) le minus 1198701199112
minus 120593
ℎ|119911|2
4120576
+ |119911| |120575|
le minus 1198701199112
minus
1205932
|119911|2
4120576
+ 120593 |119911|
le minus 1198701199112
minus
1
120576
(
1
2
120593 |119911| minus 120576)
2
+ 120576
le minus 1198701199112
+ 120576 le minus
2119870
119869
119881 + 120576
(33)
Consequently
119881 (119905) le expminus(2119870119869)119905119881 (0) +
119869120576
2119870
(1 minus expminus(2119870119869)119905) (34)
From (34) yield to
lim119905rarrinfin
|119911 (119905)| le radic
120576
119870
(35)
It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890
119898(119905) and
119890119898(119905) are satisfied as follows
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le
1
120582
radic
120576
119870
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le 2radic
120576
119870
(36)
Mathematical Problems in Engineering 7Ph
ase (
deg)
Log
mag
nitu
de
Actual plantNominal plant
100 101 102 103 104
102
100
10minus2
10minus4
minus400minus300minus200minus100
0
Frequency (rads)
100 101 102 103 104
Frequency (rads)
Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model
which means that 119890119898(119905) and 119890
119898(119905) asymptotically converge to
the bounded regionTheorem 2 is proved
4 Experiment Results
This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform
Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869
119899=
0000125 kg and 119861119899
= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6
The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896
119875= 045 119896
119863= 012 119870 = 004 119903
1= 001 119903
2=
0001 1199033= 0005 and 120576 = 00002 Considering the modeling
mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads
Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898
better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890
119898again whose gain is
greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme
In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application
5 Conclusions
To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
wu
d P1 plant
P(s)
dDOBQ(s)
120585
un(t)
uc(t)
minus
minus +
+ ++
++
1205791
s
Pminus1n (s)
Figure 4 Basic diagram of DOB
The subsystem nominal model is described as follows
119869119899
120579 + 119861119899
120579 = 119906119899 (7)
The controller of subsystem uses a hybrid control schemecombining PDC and FFC which can be described as
119906119875119863
= 119896119875119890 + 119896119863
119890
119906FFC = 119869119899
120579ref + 119861
119899
120579ref
(8)
where 119896119875and 119896
119863denote proportional coefficient and differ-
ential coefficient respectivelyAn approximate method for difference to estimate the
reference values of velocity and acceleration is introducedThis method is expressed by [26]
120579ref =
119892119904
119904 + 119892
120579ref
120579ref = (
119892119904
119904 + 119892
)
2
120579ref =119892119904
119904 + 119892
120579ref
(9)
where 119904 is used to concatenate one low-pass filter whose cut-off frequency is 119892 gt 0
Thence the output of controller is written as (8)
119906119899= 119906119875119863
+ 119906FFC = 119896119875119890 + 119896119863
119890 + 119869119899
120579ref + 119861
119899
120579ref (10)
32 Disturbance Observer-BasedMain System Control In theOET system nonlinear dynamic and uncertain elements aredifficult to compensate by accurate model Robust closed-loop control method based on DOB which is widely appliedin the high-precision servo system has simple design processIt can inhibit the variety of external disturbances and param-eters effectively
In order to realize disturbances suppression the equalcompensation is introduced into control input by means ofthe estimation of the DOB improved in this paper The basicidea of DOB is shown in Figure 4 In Figure 4 119875(119904) 119875
119899(119904)
and 119876(119904) represent the velocity model of the actual plantthe nominal velocity model a filter respectively 119904 meansLaplace operator 119908 120585 and
119889 are the actual velocity outputthe measurement noise and the estimated disturbancesrespectively Let 119906 119889 and 120585 be system input The velocityresponse can be acquired on the basis of superpositionprinciple
119908 (119904) = 119866119880119882
(119904) 119906 (119904) + 119866119863119882
(119904) 119889 (119904) + 119866120585119882
(119904) 120585 (119904) (11)
where
119866119880119882
(119904) =
119875 (119904) 119875119899(119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866119863119882
(119904) =
119875 (119904) 119875119899(119904) [1 minus 119876 (119904)]
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
119866120585119882
(119904) =
119875 (119904) 119876 (119904)
119875119899(119904) + [119875 (119904) minus 119875
119899(119904)] 119876 (119904)
(12)
Assume that the bandwidth of ideal filter 119876(119904) is 1198910 At
low frequencies there is 119876(119904) asymp 1 when frequency 119891 le 1198910
therefore 119866119880119882
(119904) asymp 119875119899(119904) 119866
119863119882(119904) asymp 0 and 119866
120585119882(119904) asymp 1
This means that DOB makes the characteristic of the actualplant approximately the same as that of the nominal model inlow-frequency domain Therefore the system has powerfulinhibiting effect against external disturbances DOB is verysensitive to low-frequency noise In practical applications itis necessary to consider that appropriate measures are takento reduce the low-frequency noise in the measurement of themotion state
In high-frequency domain if frequency 119891 gt 1198910 then
119876(119904) asymp 0 Consequently 119866119880119882
(119904) = 119866119863119882
(119904) = 119875(119904) and119866120585119882
(119904) = 0 This means that DOB has no inhibiting effectson external disturbances while having great inhibiting abilityagainst the high-frequency measurement noise
The simplified nominal velocity model is procured asfollows
119875119899(119904) =
1
119869119899119904 + 119861119899
(13)
119876(119904) design is significant in DOB design The designmethod of low-pass filter119876(119904)was proposed [27]The relativedegree of 119876(119904) must be equal to or greater than that of119875minus1
119899(119904) in order to make119876(119904)119875
minus1
119899(119904) regular According to the
nominal model shown in (13) the following is chosen for thisresearch which satisfies the property stated above
119876 (119904) =
119892
119904 + 119892
(14)
where 119892 is the cut-off frequency of the low-pass filter (LPF)in DOB Then the estimated disturbance is described as
119889 = 119876 (119904) 119889 =
119892
119904 + 119892
119889 (15)
Taking into account the physical realization of 119869119899119904
119889 isrepresented as follows
119889 = 119876 (119904) [(119869
119899119904 + 119861119899) 119908 minus 119906]
=
119892
119904 + 119892
[(119869119899119904 + 119861119899) 119908 minus 119906]
= 119869119899119892119908 minus
119892
119904 + 119892
[(119869119899119892 minus 119869119899119861119899) 119908 + 119906]
(16)
Therefore the improved DOB can be obtained as shownin Figure 5 The improved DOB has only one differentialelement simple structure and small calculation
Mathematical Problems in Engineering 5
wu
dun(t)
uc(t)
d
120579
Improved DOBJng
1
Js + B
Jng minus JnBn
1
s
P1 plant
minus
minus
minus
+
+
+
+
s
g
Figure 5 Structural diagram of improved DOB
The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as
119891 (sdot) =119889 + 120575 (17)
where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet
|120575| lt 120593 (18)
where 120593 gt 0 120593 = 120593 minus 120593 and = minus
120593 are introduced
to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33
33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890
119898= 120579 minus 120579
119899
where 120579119899is an output value of subsystem Define a sliding
mode 119911 as follows
119911 = 119890119898
+ 120582119890119898 120582 =
119861119899
119869119899
(19)
Then 119906119888is designed as
119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (20)
where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows
sgn (119911) =
1 119911 gt 0
0 119911 = 0
minus1 119911 lt 0
(21)
The adaptive law is adopted
= Proj
119869(minus1199031(
1
119869119899
119906119899minus 120582
120579) 119911)
= Proj
119861(minus1199032
120579119911)
= Proj
120593[1199033119911 sgn (119911)]
ℎ = 120593
(22)
where 1199031 1199032 and 119903
3are positive real constants The function
Proj∙(V) is defined as
Proj∙(V) =
V ∙119898
lt ∙ lt ∙119872
0 ∙ = ∙119872
and V gt 0
0 ∙ = ∙119898and V lt 0
(23)
where ∙119898is the minimum value of ∙ and ∙
119872is the maximum
value of ∙
Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890
119898asymptotically decays to
zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile
Proof Define a positive-definite Lyapunov candidate
119881 (119911) =
1
2
1198691199112
+
1
21199031
1198692
+
1
21199032
1198612
+
1
21199033
1205932
(24)
119869 = 119869 [(
120579 minus
120579119899) + 120582 (
120579 minus
120579119899)]
= (119869
120579 + 119861
120579) minus
119869
119869119899
(119869119899
120579119899+ 119861119899
120579119899) minus 119861
120579 + 120582119869
120579
= 119906119888+ 119891 (sdot) minus
119889 + 119906119899minus
119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
= 119906119888+ 120575 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
(25)
Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus
= minus
Substitute (20) into (25)
119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899
+ 119861
120579 minus 120582119869
120579 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (
1
119869119899
119906119899minus 120582
120579)
+ (119861 minus 119861)
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (
1
119869119899
119906119899minus 120582
120579) minus 119861
120579 + 120575
(119911) = minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869119911 (
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
(26)
6 Mathematical Problems in Engineering
According to (22) the time derivative of Lyapunovfunction becomes
(119911) = minus 1198701199112
minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575
le minus 1198701199112
minus 120593 |119911| + |119911| |120575|
= minus 1198701199112
minus |119911| (120593 minus |120575|)
(27)
Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is
metAccording to (24) there is
1199112
(119905) le 119911(0)2 exp(minus
119870
119869
119905) (28)
which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890
119898+120582119890119898 119890119898asymptotically
decays to zero If 119890119898(0) and 119890
119898(0) are bounded then 119890
119898(119905)
and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)
being uniformly bounded If 120579119899(0) and
120579119899(0) are bounded
then 120579119899(119905) and
120579119899(119905) are bounded According to 119890
119898= 120579 minus 120579
119899
120579 and 120579 are bounded Consequently the state variables of the
system are boundedTheorem 1 is proved
In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further
sat(119909
Δ
) =
1 119909 gt Δ
1
Δ
119909 |119909| le Δ
minus1 119909 lt minusΔ
(29)
where Δ is positive real constant namely the width of aboundary layer
Introducing sat(sdot) control law (20) becomes
119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911
4120576
) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (30)
where 120576 is a small positive constant Then there is thefollowingTheorem 2
Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then
(1) 119911 exponentially decays to zero and lim119905rarrinfin
|119911(119905)| le
radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim
119905rarrinfin|119890119898(119905)| le
(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim
119905rarrinfin| 119890119898(119905)| le
2radic120576119870 is met
Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as
(119911) = minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869119911(
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
(31)
Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911
2 is metTherefore 119911 exponentially converges until
lim119905rarrinfin
|119911 (119905)| lt
4120576
ℎ
(32)
If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas
(119911) le minus 1198701199112
minus 120593
ℎ|119911|2
4120576
+ |119911| |120575|
le minus 1198701199112
minus
1205932
|119911|2
4120576
+ 120593 |119911|
le minus 1198701199112
minus
1
120576
(
1
2
120593 |119911| minus 120576)
2
+ 120576
le minus 1198701199112
+ 120576 le minus
2119870
119869
119881 + 120576
(33)
Consequently
119881 (119905) le expminus(2119870119869)119905119881 (0) +
119869120576
2119870
(1 minus expminus(2119870119869)119905) (34)
From (34) yield to
lim119905rarrinfin
|119911 (119905)| le radic
120576
119870
(35)
It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890
119898(119905) and
119890119898(119905) are satisfied as follows
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le
1
120582
radic
120576
119870
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le 2radic
120576
119870
(36)
Mathematical Problems in Engineering 7Ph
ase (
deg)
Log
mag
nitu
de
Actual plantNominal plant
100 101 102 103 104
102
100
10minus2
10minus4
minus400minus300minus200minus100
0
Frequency (rads)
100 101 102 103 104
Frequency (rads)
Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model
which means that 119890119898(119905) and 119890
119898(119905) asymptotically converge to
the bounded regionTheorem 2 is proved
4 Experiment Results
This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform
Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869
119899=
0000125 kg and 119861119899
= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6
The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896
119875= 045 119896
119863= 012 119870 = 004 119903
1= 001 119903
2=
0001 1199033= 0005 and 120576 = 00002 Considering the modeling
mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads
Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898
better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890
119898again whose gain is
greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme
In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application
5 Conclusions
To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
wu
dun(t)
uc(t)
d
120579
Improved DOBJng
1
Js + B
Jng minus JnBn
1
s
P1 plant
minus
minus
minus
+
+
+
+
s
g
Figure 5 Structural diagram of improved DOB
The nonlinear function 119891(sdot) can be modeled by theimproved DOB system as
119891 (sdot) =119889 + 120575 (17)
where 120575 is the approximation error of the DOB Considering120575 is a bounded variable 120593 is defined to meet
|120575| lt 120593 (18)
where 120593 gt 0 120593 = 120593 minus 120593 and = minus
120593 are introduced
to assist in estimating the switching gain of the sliding modecontroller which is designed in Section 33
33 Adaptive Sliding Mode Controller Based Main SystemDesign Define the tracking error of model as 119890
119898= 120579 minus 120579
119899
where 120579119899is an output value of subsystem Define a sliding
mode 119911 as follows
119911 = 119890119898
+ 120582119890119898 120582 =
119861119899
119869119899
(19)
Then 119906119888is designed as
119906119888= minus119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (20)
where 119870 is a positive constant gain and ℎ is a positiveswitching gain 119869 and 119861 are the estimation values of 119869 and119861 respectively sgn(119911) is a sign function that is defined asfollows
sgn (119911) =
1 119911 gt 0
0 119911 = 0
minus1 119911 lt 0
(21)
The adaptive law is adopted
= Proj
119869(minus1199031(
1
119869119899
119906119899minus 120582
120579) 119911)
= Proj
119861(minus1199032
120579119911)
= Proj
120593[1199033119911 sgn (119911)]
ℎ = 120593
(22)
where 1199031 1199032 and 119903
3are positive real constants The function
Proj∙(V) is defined as
Proj∙(V) =
V ∙119898
lt ∙ lt ∙119872
0 ∙ = ∙119872
and V gt 0
0 ∙ = ∙119898and V lt 0
(23)
where ∙119898is the minimum value of ∙ and ∙
119872is the maximum
value of ∙
Theorem 1 For system (4) if the condition |120575| lt 120593 is satisfied119911 exponentially decays to zero and 119890
119898asymptotically decays to
zero using the control law as (10) (15) (20) and (22) The statevariables of the system are bounded meanwhile
Proof Define a positive-definite Lyapunov candidate
119881 (119911) =
1
2
1198691199112
+
1
21199031
1198692
+
1
21199032
1198612
+
1
21199033
1205932
(24)
119869 = 119869 [(
120579 minus
120579119899) + 120582 (
120579 minus
120579119899)]
= (119869
120579 + 119861
120579) minus
119869
119869119899
(119869119899
120579119899+ 119861119899
120579119899) minus 119861
120579 + 120582119869
120579
= 119906119888+ 119891 (sdot) minus
119889 + 119906119899minus
119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
= 119906119888+ 120575 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579
(25)
Define 119869 = 119869 minus 119869 119861 = 119861 minus 119861 then = minus
= minus
Substitute (20) into (25)
119869 = minus 119870119911 minus ℎ sdot sgn (119911) minus
119869119899minus 119869
119869119899
119906119899
+ 119861
120579 minus 120582119869
120579 +
119869119899minus 119869
119869119899
119906119899minus 119861
120579 + 120582119869
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) + (119869 minus 119869) (
1
119869119899
119906119899minus 120582
120579)
+ (119861 minus 119861)
120579 + 120575
= minus 119870119911 minus ℎ sdot sgn (119911) minus 119869 (
1
119869119899
119906119899minus 120582
120579) minus 119861
120579 + 120575
(119911) = minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869119911 (
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sgn (119911) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
(26)
6 Mathematical Problems in Engineering
According to (22) the time derivative of Lyapunovfunction becomes
(119911) = minus 1198701199112
minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575
le minus 1198701199112
minus 120593 |119911| + |119911| |120575|
= minus 1198701199112
minus |119911| (120593 minus |120575|)
(27)
Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is
metAccording to (24) there is
1199112
(119905) le 119911(0)2 exp(minus
119870
119869
119905) (28)
which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890
119898+120582119890119898 119890119898asymptotically
decays to zero If 119890119898(0) and 119890
119898(0) are bounded then 119890
119898(119905)
and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)
being uniformly bounded If 120579119899(0) and
120579119899(0) are bounded
then 120579119899(119905) and
120579119899(119905) are bounded According to 119890
119898= 120579 minus 120579
119899
120579 and 120579 are bounded Consequently the state variables of the
system are boundedTheorem 1 is proved
In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further
sat(119909
Δ
) =
1 119909 gt Δ
1
Δ
119909 |119909| le Δ
minus1 119909 lt minusΔ
(29)
where Δ is positive real constant namely the width of aboundary layer
Introducing sat(sdot) control law (20) becomes
119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911
4120576
) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (30)
where 120576 is a small positive constant Then there is thefollowingTheorem 2
Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then
(1) 119911 exponentially decays to zero and lim119905rarrinfin
|119911(119905)| le
radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim
119905rarrinfin|119890119898(119905)| le
(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim
119905rarrinfin| 119890119898(119905)| le
2radic120576119870 is met
Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as
(119911) = minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869119911(
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
(31)
Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911
2 is metTherefore 119911 exponentially converges until
lim119905rarrinfin
|119911 (119905)| lt
4120576
ℎ
(32)
If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas
(119911) le minus 1198701199112
minus 120593
ℎ|119911|2
4120576
+ |119911| |120575|
le minus 1198701199112
minus
1205932
|119911|2
4120576
+ 120593 |119911|
le minus 1198701199112
minus
1
120576
(
1
2
120593 |119911| minus 120576)
2
+ 120576
le minus 1198701199112
+ 120576 le minus
2119870
119869
119881 + 120576
(33)
Consequently
119881 (119905) le expminus(2119870119869)119905119881 (0) +
119869120576
2119870
(1 minus expminus(2119870119869)119905) (34)
From (34) yield to
lim119905rarrinfin
|119911 (119905)| le radic
120576
119870
(35)
It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890
119898(119905) and
119890119898(119905) are satisfied as follows
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le
1
120582
radic
120576
119870
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le 2radic
120576
119870
(36)
Mathematical Problems in Engineering 7Ph
ase (
deg)
Log
mag
nitu
de
Actual plantNominal plant
100 101 102 103 104
102
100
10minus2
10minus4
minus400minus300minus200minus100
0
Frequency (rads)
100 101 102 103 104
Frequency (rads)
Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model
which means that 119890119898(119905) and 119890
119898(119905) asymptotically converge to
the bounded regionTheorem 2 is proved
4 Experiment Results
This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform
Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869
119899=
0000125 kg and 119861119899
= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6
The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896
119875= 045 119896
119863= 012 119870 = 004 119903
1= 001 119903
2=
0001 1199033= 0005 and 120576 = 00002 Considering the modeling
mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads
Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898
better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890
119898again whose gain is
greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme
In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application
5 Conclusions
To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
According to (22) the time derivative of Lyapunovfunction becomes
(119911) = minus 1198701199112
minus 120593 sdot 119911 sgn (119911) minus |119911| 120593 + 119911120575
le minus 1198701199112
minus 120593 |119911| + |119911| |120575|
= minus 1198701199112
minus |119911| (120593 minus |120575|)
(27)
Since |120575| lt 120593 then |119911|(120593 minus |120575|) gt 0 Therefore (119911) le minus1198961199112 is
metAccording to (24) there is
1199112
(119905) le 119911(0)2 exp(minus
119870
119869
119905) (28)
which implies that 119911 exponentially decays to zero Thenaccording to the definition of 119911 = 119890
119898+120582119890119898 119890119898asymptotically
decays to zero If 119890119898(0) and 119890
119898(0) are bounded then 119890
119898(119905)
and 119890119898(119905) are bounded to arbitrary 119905 gt 0 because of 119911(119905)
being uniformly bounded If 120579119899(0) and
120579119899(0) are bounded
then 120579119899(119905) and
120579119899(119905) are bounded According to 119890
119898= 120579 minus 120579
119899
120579 and 120579 are bounded Consequently the state variables of the
system are boundedTheorem 1 is proved
In this method the switching gain ℎ is only requiredto be greater than |120575| which is a small variable Thisdesign alleviates the chattering phenomenon of sliding modecontroller In engineering practice since sign function sgn(sdot)in control law (20) can cause the frequent switching of thecontrol variable and result in the output chattering it is easyto damage the power amplifier In order to avoid frequentswitching of the control output the sgn(sdot) is replaced by thesaturation function (29) to weaken the chattering further
sat(119909
Δ
) =
1 119909 gt Δ
1
Δ
119909 |119909| le Δ
minus1 119909 lt minusΔ
(29)
where Δ is positive real constant namely the width of aboundary layer
Introducing sat(sdot) control law (20) becomes
119906119888= minus119870119911 minus ℎ sdot sat(ℎ119911
4120576
) minus
119869119899minus 119869
119869119899
119906119899+ 119861
120579 minus 120582119869
120579 (30)
where 120576 is a small positive constant Then there is thefollowingTheorem 2
Theorem 2 For system (4) if the condition |120575| lt 120593 is satisfiedand the control law as (10) (15) (22) and (30) is adopted then
(1) 119911 exponentially decays to zero and lim119905rarrinfin
|119911(119905)| le
radic120576119870 is met(2) 119890119898asymptotically decays to zero and lim
119905rarrinfin|119890119898(119905)| le
(1120582)radic120576119870 is met(3) 119890119898(119905) asymptotically converges and lim
119905rarrinfin| 119890119898(119905)| le
2radic120576119870 is met
Proof Select the positive-definite function119881(119911) as (24) thenits time derivative is calculated as
(119911) = minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869119911(
1
119869119899
119906119899minus 120582
120579)
minus 119861119911
120579 minus
1
1199031
119869
minus
1
1199032
119861
minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus ℎ sdot 119911 sat(ℎ119911
4120576
) minus 119869 [119911(
1
119869119899
119906119899minus 120582
120579) +
1
1199031
]
minus 119861(119911
120579 +
1
1199032
) minus
1
1199033
120593
+ 119911120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
= minus 1198701199112
minus 120593119911 sat(ℎ119911
4120576
) + |119911| 120575
(31)
Since |119911| ge 4120576ℎ sat(ℎ1199114120576) = sgn(119911) According to theproof of Theorem 1 when 120593 ge |120575| (119911) le minus119896119911
2 is metTherefore 119911 exponentially converges until
lim119905rarrinfin
|119911 (119905)| lt
4120576
ℎ
(32)
If |119911| lt 4120576ℎ then sat(ℎ1199114120576) = ℎ1199114120576 (119911) is describedas
(119911) le minus 1198701199112
minus 120593
ℎ|119911|2
4120576
+ |119911| |120575|
le minus 1198701199112
minus
1205932
|119911|2
4120576
+ 120593 |119911|
le minus 1198701199112
minus
1
120576
(
1
2
120593 |119911| minus 120576)
2
+ 120576
le minus 1198701199112
+ 120576 le minus
2119870
119869
119881 + 120576
(33)
Consequently
119881 (119905) le expminus(2119870119869)119905119881 (0) +
119869120576
2119870
(1 minus expminus(2119870119869)119905) (34)
From (34) yield to
lim119905rarrinfin
|119911 (119905)| le radic
120576
119870
(35)
It is similar to Theorem 1 that the state variables arebounded in this system From (35) 119911 asymptotically con-verges to the bounded region According to (19) 119890
119898(119905) and
119890119898(119905) are satisfied as follows
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le
1
120582
radic
120576
119870
lim119905rarrinfin
1003816100381610038161003816119890119898
(119905)1003816100381610038161003816le 2radic
120576
119870
(36)
Mathematical Problems in Engineering 7Ph
ase (
deg)
Log
mag
nitu
de
Actual plantNominal plant
100 101 102 103 104
102
100
10minus2
10minus4
minus400minus300minus200minus100
0
Frequency (rads)
100 101 102 103 104
Frequency (rads)
Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model
which means that 119890119898(119905) and 119890
119898(119905) asymptotically converge to
the bounded regionTheorem 2 is proved
4 Experiment Results
This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform
Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869
119899=
0000125 kg and 119861119899
= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6
The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896
119875= 045 119896
119863= 012 119870 = 004 119903
1= 001 119903
2=
0001 1199033= 0005 and 120576 = 00002 Considering the modeling
mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads
Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898
better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890
119898again whose gain is
greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme
In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application
5 Conclusions
To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7Ph
ase (
deg)
Log
mag
nitu
de
Actual plantNominal plant
100 101 102 103 104
102
100
10minus2
10minus4
minus400minus300minus200minus100
0
Frequency (rads)
100 101 102 103 104
Frequency (rads)
Figure 6 Fitting curves for frequency characteristics of actual plantand nominal model
which means that 119890119898(119905) and 119890
119898(119905) asymptotically converge to
the bounded regionTheorem 2 is proved
4 Experiment Results
This section investigates the feasibility and effectiveness ofthe proposed compound controller based on the structureof the virtual compound-axis servo system by experimentsThe tracking experiments are implemented through usingsome type of OET platform The position sensor of thedevice uses the optical-electrical encoder with resolution of00007 degrees The control algorithm program is writtenwith C language based on Windows-RTX real-time systemin an industrial computer which adopts Advantech IPC610 and connects with the servo drivers by a 16-bit DAconvertor of PCI bus The control cycle is 0001 s It is worthmentioning that the adaptive online adjustment will consumelots of computational resource With the development of theadvanced technologies on computer the problem of lots ofcomputation has been solved The proposed adaptive slidingcontroller is proved to be feasible in a practical system by thisexperimental platform
Based on what has been mentioned above the pitch axisis chosen herein to verify the method because each axisof the OET platform can be designed independently Theparameters of the nominal model are identified as 119869
119899=
0000125 kg and 119861119899
= 0003125N sdot s sdot mminus1 by a white noisefrequency sweep method The fitting curves for frequencycharacteristics of actual plant and nominal model are shownin Figure 6
The experiments are separated into two parts First theexperiment is completed under the proposed compound con-trol scheme in which sign function uses saturation functionSecond in order to verify the effect of the proposedmethod acomparative experiment is achieved by the traditional controlschemewith PDC + FFC + DOBOther parameters are givenas follows 119896
119875= 045 119896
119863= 012 119870 = 004 119903
1= 001 119903
2=
0001 1199033= 0005 and 120576 = 00002 Considering the modeling
mismatch and the robustness of the system the value ofparameter 119892 in improved DOB is chosen as 1000 rads
Figure 7 compares the tracking error and the controlvalue of the two control schemes when the reference inputsignal is a sinusoidal signal where the amplitude is 2 degreesand frequency is 05HzThe results indicate that the proposedcontrol scheme could achieve a better position tracking per-formance than traditional control laws a maximum trackingerror decreases from 0052 degrees to 0023 degrees withthe help of the ASMC ASMC achieves the adjustment to119890119898
better From Figure 7(a) we can see that the trackingerror reaches the maximum value at zero velocity and thetracking error of the proposed control scheme is smallerCompared with traditional control scheme the compoundcontrol scheme based on the virtual compound-axis systemhas better tracking performance and robustness The distur-bance is sufficiently compensated by using improved DOBwhich simplifies design process and decreases calculationtime Meanwhile the accuracy of the system is guaranteedby introducing ASMC to adjust 119890
119898again whose gain is
greatly related to the upper bound on the error of thedisturbance estimation and the reduction of switching gainhelps to alleviate the chattering Figure 7(b) illustrates that thecontroller output curve of the compound control scheme issmoother than that of traditional control scheme
In order to test the dynamic performance of the trackingobject further the tracking command signal is employed assin(2120587 sdot 3119905) Figure 8(a) shows that the maximum trackingerror decreases from 014 degrees to 006 degrees underthe reference signal Compared to the traditional controlscheme the systemwith compound control schemehas betterdynamic performanceThe testing standard of OET platformstipulates that the maximum tracking error cannot exceed10 of the amplitude of the reference signal in working bandObviously the tracking errors of Figures 7(a) and 8(a) meetthis standard under the proposed method Because it has aswitching control the adaptive sliding mode technique canreject the impact of nonlinear disturbance in high-frequencydomain which is not compensated by the DOB From thispoint of view the proposed control scheme enhances thesystem robustness Furthermore according to Figures 7(b)and 8(b) the control value of the proposed method is farless than the limit of the DA converter The control valuedoes not exhibit obvious chattering phenomenon becauseASMC with the improved DOB can alleviate the chatteringphenomenon Consequently the proposed scheme not onlyensures strong robustness against system uncertainties andsmall tracking error but also suppresses the high-frequencychattering at control input effectively The results indicatethat the compound controller can be reliably performed inpractical application
5 Conclusions
To obtain the high performance and good robustness forordinary OET system without the FSM the design methodof virtual compound-axis servo system is proposed in thispaperThe proposed compound control scheme based on the
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8Time (s)
minus006
minus004
minus002
0
002
004
006
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8Time (s)
Compound control schemeScheme with PDC + FFC + DOB
minus04minus03minus02minus01
001
020304
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 7 Comparison curves under two schemes with sinusoidal input (119860 = 2∘ 119891 = 05Hz)
0 1 2 3 4 5 6 7 8 9 10minus02
minus01
0
01
02
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Trac
king
erro
r (∘)
(a) Comparison of tracking error curves
0 1 2 3 4 5 6 7 8 9 10minus03minus02minus01
001020304
Time (s)
Compound control schemeScheme with PDC + FFC + DOB
Con
trol v
alue
(V)
(b) Comparison of control value
Figure 8 Comparison curves under two schemes with sinusoidal input (119860 = 1∘ 119891 = 3Hz)
structure of the virtual compound-axis system can maintainthe robustness of OET system against various disturbancesin the whole control period and then the tracking errorof OET system can be limited within an expected levelin the existence of sustained uncertainties Although thevirtual compound-axis does not perform as well as the realcompound-axis it can also improve the tracking precisionof the OET system to some extent Therefore the proposedmethod can be used as a new control technology applied inOET system In addition because OET system is a typicalservo motion system the theoretic results are able to beextended to other relational fields
However the velocity and acceleration command signalin FFC cannot be measured directly and easily in practiceTherefore some more excellent differential methods shouldbe considered in further work Simultaneously in order totrack the maneuvering target with higher speed furtherstudies aimed at better performance at higher frequenciesare needed Besides with the development of advancedcomputer and information technologies massive amountof measurement data can be utilized to extract the usefulinformation about the current state of the OET process As
a result the excellent method of the process monitoring andfault diagnosis such as data-driven techniques [28] can beused as references in future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was supported by the National Basic Research Pro-gram (Grant no 2012CB821200) and the Scientific Researchfor Colleges and Universities of Inner Mongolia (Grant noNJZY13143) in China
References
[1] W Thanmas ldquoDigital laser ranging and tracking using acompound axis servomechanismrdquo Applied Optics vol 5 no 4pp 497ndash505 1966
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
[2] J M Hilkert ldquoInertially stabilized platform technology con-cepts and principlesrdquo IEEE Control Systems Magazine vol 28no 1 pp 26ndash46 2008
[3] Y Song ldquoVariable structure control technology of the finetracking assembly in airborne laser communication systemrdquoInfrared and Laser Engineering vol 39 no 5 pp 934ndash938 2010(Chinese)
[4] M Guelman A Kogan A Kazarian A Livne M Orensteinand H Michalik ldquoAcquisition and pointing control for inter-satellite laser communicationsrdquo IEEE Transactions onAerospaceand Electronic Systems vol 40 no 4 pp 1239ndash1248 2004
[5] QWang ldquoSingle detector compound axis control based on real-time predicted trajectory correcting methodrdquo Opto-ElectronicEngineering vol 34 no 4 pp 17ndash21 2007
[6] T R Grochmal and A F Lynch ldquoPrecision tracking of arotating shaft with magnetic bearings by nonlinear decoupleddisturbance observersrdquo IEEE Transactions on Control SystemsTechnology vol 15 no 6 pp 1112ndash1121 2007
[7] K Kyung-Soo R Keun-Ho and K Soohyun ldquoDisturbanceobserver for estimating higher order disturbances in time seriesexpansionrdquo IEEETransactions onAutomatic Control vol 55 no8 pp 1905ndash1911 2010
[8] E Kim ldquoA fuzzy disturbance observer and its application tocontrolrdquo IEEE Transactions on Fuzzy Systems vol 10 no 1 pp77ndash84 2002
[9] Y Wu Y Liu and D Tian ldquoA compound fuzzy disturbanceobserver based on sliding modes and its applicaton on flightsimulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 913538 8 pages 2013
[10] F-J Lin P-H Chou and Y-S Kung ldquoRobust fuzzy neuralnetwork controller with nonlinear disturbance observer fortwo-axis motion control systemrdquo IET Control Theory andApplications vol 2 no 2 pp 151ndash167 2008
[11] X Chen C-Y Su and T Fukuda ldquoA nonlinear disturbanceobserver for multivariable systems and its application to mag-netic bearing systemsrdquo IEEE Transactions on Control SystemsTechnology vol 12 no 4 pp 569ndash577 2004
[12] S Li J Yang W-H Chen and X Chen ldquoGeneralized extendedstate observer based control for systems with mismatcheduncertaintiesrdquo IEEE Transactions on Industrial Electronics vol59 no 12 pp 4792ndash4802 2012
[13] B-Z Guo and Z-L Zhao ldquoOn convergence of non-linearextended state observer for multi-input multi-output systemswith uncertaintyrdquo IET ControlTheory amp Applications vol 6 no15 pp 2375ndash2386 2012
[14] Q Khan A I Bhatti M Iqbal and Q Ahmed ldquoDynamicintegral sliding mode control for SISO uncertain nonlinearsystemsrdquo International Journal of Innovative Computing Infor-mation and Control vol 8 no 7 pp 4621ndash4633 2012
[15] T H Yan B He X D Chen and X S Xu ldquoThe discrete-timesliding mode control with computation time delay for repeat-able run-out compensation of hard disk drivesrdquo MathematicalProblems in Engineering vol 2013 Article ID 505846 13 pages2013
[16] L Wu and W X Zheng ldquoPassivity-based sliding mode controlof uncertain singular time-delay systemsrdquo Automatica vol 45no 9 pp 2120ndash2127 2009
[17] Y Niu J Lam X Wang and D W C Ho ldquoObserver-basedsliding mode control for nonlinear state-delayed systemsrdquoInternational Journal of Systems Science vol 35 no 2 pp 139ndash150 2004
[18] N Yagiz Y Hacioglu and Y Taskin ldquoFuzzy sliding-modecontrol of active suspensionsrdquo IEEE Transactions on IndustrialElectronics vol 55 no 11 pp 3883ndash3890 2008
[19] J H Zhang P Shi andYQ Xia ldquoRobust adaptive sliding-modecontrol for fuzzy systems with mismatched uncertaintiesrdquo IEEETransactions on Fuzzy Systems vol 18 no 4 pp 700ndash711 2010
[20] Y Niu D W C Ho and XWang ldquoSliding mode control for Itostochastic systems with Markovian switchingrdquo Automatica vol43 no 10 pp 1784ndash1790 2007
[21] L Wu W X Zheng and H Gao ldquoDissipativity-based slidingmode control of switched stochastic systemsrdquo IEEE Transac-tions on Automatic Control vol 58 no 3 pp 785ndash791 2012
[22] R Yan L Zhenghua andZ Rui ldquoApplication of low speed opto-electronic tracking systems based on sliding mode distutbanceobserverrdquo Journal of Beihang University of Aeronautics andAstronautics vol 39 no 6 pp 835ndash840 2013 (Chinese)
[23] Y Lu ldquoSliding-mode disturbance observer with switching-gainadaptation and its application to optical disk drivesrdquo IEEETransactions on Industrial Electronics vol 56 no 9 pp 3743ndash3750 2009
[24] M Zhihong X H Yu K Eshraghian and M Palaniswami ldquoArobust adaptive sliding mode tracking control using an RBFneural network for robotic manipulatorsrdquo in Proceedings of theIEEE International Conference on Neural Networks pp 2403ndash2408 Perth Australia December 1995
[25] X Liu YWu andB Liu ldquoThe research of adaptive slidingmodecontroller for motor servo system using fuzzy upper bound ondisturbancesrdquo International Journal of Control Automation andSystems vol 10 no 5 pp 1064ndash1069 2012
[26] Y Wu X Liu and D Tian ldquoResearch of compound controllerfor flight simulator with disturbance observerrdquo Chinese Journalof Aeronautics vol 24 no 5 pp 613ndash621 2011
[27] H S LEE Robust digital tracking controllers for high-speedhigh-accuracy positioning systems [PhD thesis] University ofCalifor-nia Berkeley Calif USA 1994
[28] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark TennesseeEastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of