Research ArticleAdaptive Neural Network Control for Nonlinear HydraulicServo-System with Time-Varying State Constraints

Shu-Min Lu1 and Dong-Juan Li2

1College of Science Liaoning University of Technology Jinzhou Liaoning 121001 China2School of Chemical and Environmental Engineering Liaoning University of Technology Jinzhou Liaoning 121001 China

Correspondence should be addressed to Dong-Juan Li lidongjuanlivecom

Received 7 July 2017 Accepted 29 August 2017 Published 18 October 2017

Academic Editor Junpei Zhong

Copyright copy 2017 Shu-Min Lu and Dong-Juan Li This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

An adaptive neural network control problem is addressed for a class of nonlinear hydraulic servo-systems with time-varying stateconstraints In view of the low precision problem of the traditional hydraulic servo-system which is caused by the tracking errorssurpassing appropriate bound the previous works have shown that the constraint for the system is a good way to solve the lowprecision problemMeanwhile comparedwith constant constraints the time-varying state constraints aremore general in the actualsystemsTherefore when the states of the system are forced to obey bounded time-varying constraint conditions the high precisiontracking performance of the system can be easily realized In order to achieve this goal the time-varying barrier Lyapunov function(TVBLF) is used to prevent the states from violating time-varying constraints By the backstepping design the adaptive controllerwill be obtained A radial basis function neural network (RBFNN) is used to estimate the uncertainties Based on analyzing thestability of the hydraulic servo-system we show that the error signals are bounded in the compacts sets the time-varying stateconstrains are never violated and all singles of the hydraulic servo-system are bounded The simulation and experimental resultsshow that the tracking accuracy of system is improved and the controller has fast tracking ability and strong robustness

1 Introduction

Throughout history the hydraulic servo-system has beenalways widely applied in many aspects such as aerospaceaviation navigation weapon mining and metallurgy due tothe advantages of high response high power large stiffnessstrong robustness capability small volume and so on How-ever serious nonlinear behavior such as control input satu-ration [1] state constraint valve opening nonlinear friction[2] and model uncertainty [3] (load changes the parametersvariation and the element parameter uncertainty [4] causedby the abrasive containing external disturbance [5 6] leak-age and other uncertain nonlinear elements) restricts thedevelopment of high-performance closed loop system con-troller [7 8]

Recently in order to solve the uncertain nonlinear prob-lem in the system the most authors adopted the fuzzylogical system or adaptive neural network (NN) control[9 10] to approximate the external disturbance [11] and

model uncertainty [12 13] In [14 15] the different stabilityproblems of switched nonlinear systems are solved by usingthe fuzzy adaptive controlThe paper [14] solved the discrete-time switch system tracking control problem and [15 16]presented decentralized controller for switched uncertainnonlinear large-scale systems with dead zones based onobserver [17] The article [18] solved the control problemof the nonlinear system with dynamic uncertainties andinput dead zone Similarly the authors in [19] used the NNto deal with the uncertain problem of nonlinear nonstrictfeedback discrete-time systems [20] The article [21] adoptedgeneral projection neural network (GPN) to iteratively solvea quadratic programming (QP) problem which updatesthe algorithm of NN to develop the new application Twoneural networks which include the critic NN and the actionNN are used to approximate the strategic utility functionand to minimize both the strategic utility function and thetracking error in [22] respectivelyThe articles [23ndash26] adoptthe adaptive NN in quantized nonlinear systems nonlinear

HindawiComplexityVolume 2017 Article ID 6893521 11 pageshttpsdoiorg10115520176893521

2 Complexity

time-delay systems and uncertain nonlinear systems Thesearticles extend the theoretical application of neural networkIn [27] an adaptive neural control scheme is presented to takethe unknown output hysteresis and computational efficiencyinto account The motionforce performances limits of therobotic systemwill be relaxedTheNN turns out to be a clevertheory to solve the uncertain information of the systems fromthe above articles

At present the control problemof hydraulic servo-systemhas been an active research field by designing the adaptivecontrol technique The authors in [28] have presented anadaptive controller based on the robust integral of the errorsignal to deal with modeling uncertainties of the systemand the feedback gain is reduced In [29] the robust adap-tive control ensures uniform boundedness of the thrusterassisted position mooring system which is in the transversedirection under the ocean current disturbance The robustadaptive controller has been designed for single-rod non-linear hydraulic servo-system where the element parameteruncertainty and nonlinear uncertainty behavior exist in [30]The article in [31] presents output feedback nonlinear robustcontrol based on an extended state observer (ESO) to solvemismatched modeling uncertainties problem which is veryimportant for high-accuracy tracking control of hydraulicservo-systems In [32] considering the influence of frictionon the system an adaptive controller is designed to solve theproblemof system friction and parameter uncertainty In factdue to the high bearing capacity and high stiffness proper-ties of the hydraulic servo-system so the state constraintsproblems are ignored in the environment and the interactionof measurement unit tests process which lead to too largedisplacement velocity or acceleration causing damage to themeasuring equipment during testing process [33] In tests andexperiments if the initial conditions of the system do notmatch it is possible that the displacement velocity or accel-eration of the system is oversized At the same time the con-straint for the system is a good way to achieve high accuracytracking performance However none of the above articlesconsidered constraints condition And considering that thestate variables varywith time the constraint condition shouldalso be considered the factor of time changing Meanwhilecompared with constant constraints the time-varying stateconstraints are more general in the actual systems

In fact some systems are often affected by constraints[34] such as temperature constraints in chemical reac-tions and the speed or acceleration constraints when somemechanical systems suffer physical failure [35] The authorsfirst present the theory of nonlinear system with outputconstrains and time-varying output constrains by usingBarrier Lyapunov in [36 37] However those articles do notconsider the interference of unknown functions In [38]considering a class of special nonlinear systems with thehysteretic output mechanism and the unmeasured statesadaptive neural output feedback control is presented toguarantee the prescribed convergence of the tracking error[39]The article in [40] has studied adaptive neural control ofuncertain stochastic nonlinear systems with dead zone andoutput constraint The continuous stirred tank reactor [41]flexible crane system [42] and robot system [43 44] all design

adaptiveNN controller based on output constrains conditionIt is essential that the theories of constraint are combinedwith the actual systems [45] Similarly the state constrainshave also been researched by many scientific researchersFor example the article in [46] presents adaptive fuzzy NNcontrol using impedance learning for a constrained robotThe authors considering time-delay condition investigate theroboticmanipulator system [47] nonlinearMIMOunknownsystems [48] and common nonlinear systems [49] with fullstate constrains [50]The authors further research the nonlin-ear pure-feedback systems Nussbaum gain adaptive controland ABLF adaptive controller to apply them on stochasticnonlinear systems and wheeled mobile robotic system byusing the theory of integral barrier Lyapunov function or stateconstrains in [51] [52] [53] [54] [55] and [56] respectively

Based on the above descriptions this article presentsan adaptive NN tracking control for hydraulic servo-systembased on time-varying state constraint and hydraulic systemadaptive control is first designed with the time-varying stateconstraints Finally when the states of the system are forced toobey bounded time-varying constraint conditions the highprecision tracking performance of the system can be easilyrealized Meanwhile it can be proved that the tracking errorsignal converges to zero asymptotically and all singles of thehydraulic servo-system are bounded by using the Lyapunovanalysis The experimental results show the availability of thedesign controller

2 System Descriptions andProblem Preliminaries

A class of hydraulic servo-system dynamics of the inertialload can be described as the following form

119898 = 119875119890119860 minus 120588 minus 119889 (119909 119905) (1)

where 119909 represents the displacement of the inertial load119898 isthemass ofmoving parts119875119890 = 1198751minus11987521198751 and1198752 are pressuresof the left and right inside the hydraulic cylinder respectively119860 is the corresponding areas of the above-mentioned cham-ber 120588 represents the viscous friction coefficient 119889(119909 119905) isdefined as the unconsidered disturbance

Considering the compressibility of the hydraulic oil andignoring leakage of hydraulic cylinder the pressure dynamicequation of both chambers is as follows

1198811198904119861V 119890 = minus119860 minus 119862119898119875119890 minus 119908 + 119876 (2)

where 119881119890 = 1198811199041 + 1198811199042 is the total effective cylinder volume1198811199041 = 1198811198681 + 119860119909 and 1198811199042 = 1198811198682 minus 119860119909 represent the totalvolume of the left and the right chamber inside hydrauliccylinder respectively 1198811198681 and 1198811198682 are the corresponding theinitial volume 119861V is the effective bulk modulus of hydrauliccylinder 119862119898 represents the total internal leakage coefficientof hydraulic cylinder119908 represents themodeling error of bothchambers 119876 = (1198761 + 1198762)2 is the load-flow of hydrauliccylinder 1198761 and 1198762 respectively represent the supplied oilflow rate to the left chamber and the returned oil flow rate tothe right chamber

Complexity 3

Consider that the flow rates 1198761 and 1198762 are the functionof the spool displacement 119909119904 The flow equation of hydrauliccylinder is written as follows

1198761 = 119896119891119909119904ℎ1 (119909119904 1198751) 1198762 = 119896119891119909119904ℎ2 (119909119904 1198752) (3)

with

ℎ1 (119909119904 1198751) = (119875119904 minus 1198751)12 119909119904 ge 0(1198751 minus 119875119903)12 119909119904 lt 0

ℎ2 (119909119904 1198752) = (1198752 minus 119875119903)12 119909119904 ge 0(119875119904 minus 1198752)12 119909119904 lt 0

(4)

where 119896119891 denotes the flow gain coefficient 119875119904 is the supplypressure of the oil 119875119903 represents the return oil pressure

We assume that the spool displacement 119909119904 is directproportion with control input voltage 119906 in the situation ofignoring the high frequency that is 119909119904 = 119896119904119906 119896119904 gt 0 is thesmall gain Then (3) can be rewritten as

1198761 = 119870119891119906ℎ1 (119906 1198751) 1198762 = 119870119891119906ℎ2 (119906 1198752) (5)

where 119870119891 = 119896119891119896119904 represents the total flow gain coefficientThen we can conclude that the load-flow of hydrauliccylinder is

119876 = 119870119891119906 (119875119904 minus sgn (119906) 119875119890)12 (6)

where sgn(119906) represents the step function as

sgn (119906) = 1 119906 ge 0minus1 119906 lt 0

119875119904 = 1198751 + 1198752119875119903 = 0119875119890 = 1198751 minus 1198752(7)

Define state variables 119909 = [119909 ]119879 = [1199091 1199092 1199093]119879 thedynamics of the hydraulic servo-system (1) can be translatedinto the following the state space expression based on thepressure dynamic equation (2) and the flow equations (5)-(6)

1 = 11990922 = 11990933 = minus11989111199092 minus 1198912 (119909) + 1198913119892119906(8)

where

1198911 = 41198602119861V119898119881119890 minus 1205882119898 1198913 = 4119860119861V119870119891119898119881119890 119892 = radic (119875119904 minus sgn (119906) 119875119890)2

1198912 (119909) = 4119860119861V119862119898119875119890119898119881119890 + 4119860119861V119908119898119881119890 + 1205881198601198751198901198982 minus 120588119898119889 (119909)+ 119889 (119909)119898

(9)

1198911(119909) 1198912(119909) and 1198913(119909) are unknown smooth parameterfunctions In this paper the state variable 119909119894 is constrainedin the following time-varying compact sets1003816100381610038161003816119909119894 (119905)1003816100381610038161003816 le 119896119887119894 (119905) 119894 = 1 2 3 (10)

where 119896119887119894(119905) represents the time-varying smooth functions

Assumption 1 (see [37]) There exist the functions 1198913(119909) and1198913(119909) which satisfy the inequality 0 lt 119891

3le 1198913 le 1198913

Assumption 2 (see [43]) There are functions1198840(119905) 1198841(119905) and1198842(119905) We assume that the functions satisfy the inequalities|119884119894minus1(119905)| le |119896119887119894(119905)| 119894 = 1 2 3 Then the smooth desiredtrajectory 119910119889(119905) and its time derivatives respectively satisfy|119910119889(119905)| le 1198840(119905) and |119910(119894)119889 (119905)| le 119884119894+1(119905) 119894 = 1 2 for all 119905 gt 0Lemma 3 (see [54]) For any time-varying function 119896119886(119905) thefollowing inequality holds for all error function 119911(119905) in theinterval |119911(119905)| le 119896119886(119905)

log1198962119886 (119905)1198962119886 (119905) minus 1199112 (119905) le 1199112 (119905)1198962119886 (119905) minus 1199112 (119905) (11)

The above-mentioned unknown smooth functions 11989111198912(119909) and 1198913 are approximated by the radial basis functionNN as

119891119894 (119883) = 119882119879119894 Φ119894 (119883) 119894 = 1 2 3 (12)

where119882119894 isin 119877119897119894 denotes the neural network weight vector andΦ119894(119883) = [1205931198941 120593119894119897119894] isin 119877119897119894 are the basis function vectorswith input vector 119883 isin 119877119899119894 and the NN node number 119897119894 gt 1Gaussian functions 120593119894119895(119883) are chosen in the following forms

120593119894119895 (119883) = exp(minus10038171003817100381710038171003817119883 minus 1205831198941198951003817100381710038171003817100381721205782119894119895 ) 119894 = 1 2 3 119895 = 1 119897119894

(13)

where 120583119894119895 = [1205871198941198951 120587119894119895119899119894]119879 denotes the center of NNfunction and 120578119894119895 is the width of the Gaussian function

4 Complexity

Define the unknown optimal weight vector as the follow-ing form

119882lowast119894 = argmin119882119894isin119877

119897119894

[ sup119883isin119877119899119894

10038161003816100381610038161003816119882119879119894 Φ119894 (119883) minus 119891119894 (119883)10038161003816100381610038161003816] 119894 = 1 2 3 (14)

Then there exists the unknown approximation errors 120576119894(119883)which has supremum 120576119894 that is |120576119894(119883)| le 120576119894 Equation (12)will be rewritten in the approximation equations as follows

119891119894 (119883) = 119882lowast119879119894 Φ119894 (119883) + 120576119894 (119883) 119894 = 1 2 3 (15)

3 The Adaptive NN Controller Design

Define the track trajectory 119910119889 and the virtual controllers 1205721and1205722 we have the track error 1199111 = 1199091minus119910119889Then let 1199112 = 1199092minus1205721 and 1199113 = 1199093 minus 1205722 Choose the barrier Lyapunov functioncandidate as

119881 = 3sum119894=1

119881119894 + 119881119873119881119894 = 12 log 11989621198861198941198962119886119894 minus 1199112119894 119881119873 =

123sum119895=1

119879119895 Θminus1119895 119895(16)

where log(sdot) represents the natural logarithm 119896119886119894(119905) = 119896119887119894(119905)minus119863119894(119905) is boundaries of error vector 119911119894 from subsequent feasi-bility analysis Θ119895 119895 = 1 2 3 are the positive gain matrices119894 is the neural network weight error with 119894 = 119894 minus 119882lowast119894 and note that 119881 is the positive definite and continuouslydifferentiable in the set |119911119894(119905)| le 119896119886119894(119905) for 119894 = 1 2 3

Based on (16) the time derivative of 119881 is

= 3sum119894=1

119894 + 119873= 3sum119894=1

119911119894(1198962119886119894 minus 1199112119894 ) (119894 minus119886119894119896119886119894 119911119894) +

3sum119895=1

119879119895 Θminus1119895 119882119895(17)

Design the virtual controllers as

1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 119910119889 (18)

1205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111 (19)

where the time-varying gains are given by

120574119894 (119905) = radic(119886119894119896119886119894 )2 + 120575119894 119894 = 1 2 3 (20)

for all positive constants 120575119894 and 120574119894When any 119886119894 is zero 120575119894 willensure that the time derivatives of virtual controllers 120572119894 andcontroller 119906 are bounded

Substituting (8) and (18)ndash(20) into (17) we can obtain

= minus 2sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 ) +

11991121199113(11989621198862 minus 11991122)+ 1199113(11989621198863 minus 11991123) (3 minus 2 minus

11988631198961198863 1199113)+ 3sum119895=1

119879119895 Θminus1119895 119882119895(21)

where

1 = 12059712057211205971199091 1 +1sum119896=0

(1205971205721119910(119896)119889

119910(119896+1)119889 + 1205971205721119896(119896)1198861 119896(119896+1)1198861 ) 2 = 2sum119895=1

1205971205722120597119909119895 119895+ 2sum119896=0

(1205971205722119910(119896)119889

119910(119896+1)119889 + 1205971205722119896(119896)1198861 119896(119896+1)1198861 + 1205971205722119896(119896)1198862 119896(119896+1)1198862 ) (22)

According to approximation equations (15) (21) becomes

= minus 2sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 )

+ 3sum119895=1

119879119895 Θminus1119895 119882119895 + 1199113(11989621198863 minus 11991123) [[minus119882lowast1198791 Φ1 (119883) 1199092

minus119882lowast1198792 Φ2 (119883) +119882lowast1198793 Φ3 (119883) 119892119906 + 119863 (119909 119905) minus 2+ (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11988631198961198863 1199113]]

(23)

where119863(119909 119905) represents the systematic disturbance compen-sation as the following form

119863(119909 119905) = minus12057611199092 minus 1205762 + 1205763119892119906 (24)

Choose the controller as follows

119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092+ 1198792 Φ2 (119909) + 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123)) (25)

Complexity 5

Adopting the equations 119882lowast119894 = 119894 minus 119894 119894 = 1 2 3 andcontroller (25) the expression for (23) can be rewritten by theabove controller as

= minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 )

+ 1198791 (Θminus11 1198821 + Φ1 (119883) 11990921199113(11989621198863 minus 11991123))+ 1198792 (Θminus12 1198822 + Φ2 (119883) 1199113(11989621198863 minus 11991123))+ 1198793 (Θminus13 1198823 minus Φ3 (119883) 1199113119892119906(11989621198863 minus 11991123))+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2

(26)

Based on the above-mentioned definition of 120574119894(119905) we canobtain that

120574119894 (119905) + 119886119894119896119886119894 ge 0 119894 = 1 2 3 (27)

Introducing the projection algorithm of adaptive controlwe design the adaptive law from (26) as follows

1198821=

minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123) + 12059011]] Θ1 le 0 1 gt 1198721198911

1198751198911 (1 Φ1) Θ1 gt 0 1 = 1198721198911(28)

where 1198751198911(1 Φ1) represents projection operator as thefollowing form

1198751198911 (1 Φ1) = minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123)

minus Φ1 (119883) 1198791 111990921199113100381710038171003817100381710038171100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059011]]

(29)

1198721198911 is the minuteness positive constant to ensure the adap-tive law 1198821 ge 0 in any situation

Similarly the other adaptive laws can obtain the forms as

1198822 = minusΘ2 [[Φ2 (119883)

1199113(11989621198863 minus 11991123) + 12059022]] Θ2 le 0 2 gt 11987211989121198751198912 (2 Φ2) Θ2 gt 0 2 = 1198721198912

(30)

1198751198912 (2 Φ2) = minusΘ2 [[Φ2 (119883)1199113(11989621198863 minus 11991123) minus Φ2 (119883)

1198792 21199113100381710038171003817100381710038172100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059022]] (31)

1198823 = Θ3 [[Φ3 (119883)

1199113119892119906(11989621198863 minus 11991123) minus 12059033]] Θ3 le 0 3 gt 11987211989131198751198913 (3 Φ3) Θ3 gt 0 3 = 1198721198913

(32)

1198751198913 (3 Φ3) = minusΘ3 [[Φ3 (119883)1199113119892119906(11989621198863 minus 11991123) minus Φ3 (119883)

1198793 31199113119892119906100381710038171003817100381710038173100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059033]] (33)

In particular when component 3119894 of 3 is a small con-stant 1198981198913119894 that is 3 = 1198721198913 we adopt the following com-ponent of adaptive law

1198823119894 = minusΘ3119894 [[Φ3119894 (119883)

1199113119892119906(11989621198863 minus 11991123) + 12059033119894]] 0(34)

Remark 4 Based on the above description (34) we canmakethe following conclusion If 3119894 = 1198981198913119894 we can get 1198823119894 gt 0

that is 3119894 ge 1198981198913119894 gt 0 for all any 3119894 Then there does notexist insignificance function for controller 119906

Substituting (28) (30) and (32) into (26) we obtain

le minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 ) minus

3sum119894=1

120590119894119879119894 119894+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2 (35)

6 Complexity

Using Youngrsquos inequality in the last two terms of weseem to easily get

minus120590119894119879119894 119894 = minus120590119894119879119894 (119894 +119882lowast119894 )le minus120590119894 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817minus119882lowast119894 10038171003817100381710038172+ 1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172

= minus1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 119863 (119909 119905) 1199113(11989621198863 minus 11991123) le

121198632 (119909 119905) + 119911232 (11989621198863 minus 11991123)2

(36)

Based on definition (27) and using inequalities (36) and(11) in Lemma 3 (35) can be rewritten as

le minus 3sum119894=1

120574119894 log 1198962119886119894(1198962119886119894 minus 1199112119894 ) minus3sum119894=1

1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 121198632 (119909 119905)+ 3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (37)

Theorem 5 A class of hydraulic servo-system is described by(1)ndash(3) under Assumptions 1 and 2 If the initial displacement1199091(0) the initial velocity 1199092(0) and the initial acceleration1199093(0) satisfy |119909119894(0)| le 119896119887119894(0) 119894 = 1 2 3 then the proposedcontrol method has the following properties

(1) The error signals are bounded in the following sets

119911119894 (119905) isin Ω119911119894 119894 = 1 2 3 (38)

where Ω119911119894 fl 119911119894 isin R |119911119894(119905)| le 119896119886119894(119905)radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595noting that the tracking error signal converges to zero asymp-totically

(2) The time-varying state constrains are never violatedthat is |119909119894(119905)| le 119896119887119894(119905) forall119905 ge 0 119894 = 1 2 3

(3) All singles of the hydraulic servo-system are bounded

Proof Please see the Appendix

4 Simulation Example

Simulation has been performed to demonstrate the per-formance of the proposed approach The values of systemparameters which are used in the simulation are given inTable 1

The initial states of the hydraulic servo-system are givenas 1199091(0) = 0 1199091(0) = 0 and 1199091(0) = 0 The time-varyingconstraint functions are 1198961198871(119905) = 2 sin(5119905) + 40 1198961198872(119905) =2 sin(5119905) + 60 and 1198961198873(119905) = 4 sin(5119905) + 150 The desiredtrajectory tracking periodic reciprocation of the displace-ment is given 119910119889 = 40 arctan(sin(04120587119905))(1 minus 119890minus119905) with 119905 isin[0 119905119891] and 119905119891 = 50 s The modeling error disturbance of the

Table 1 The system parameters used in the simulation

Name Value119898(kg) 2200119875119904 (Pa) 20119875119903 (Pa) 0119860 (m3) 2 times 10minus3120588 (Nsm) 5001198811198681 (m3) 5 times 10minus41198811198682 (m3) 5 times 10minus3119861V (Pa) 200119862119898 (m5Ns) 2 times 10minus5119870119891 (m3Ns) 8 times 10minus2both chambers is 119908 = sin(119905) + 2 We choose the followingcontroller1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 1199101198891205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092 + 1198792 Φ2+ 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123))

(39)

where we choose design parameters as 1198840 = 20 1198841 = 101198842 = 50 1205741 = 10 1205742 = 3 1205743 = 6 1205751 = 08 1205752 = 07 1205753 = 06the initial weight vector estimation 1 = minus100 2 = 203 = 10 Θ1 = 3 Θ2 = 2 Θ3 = 2 1205901 = 2 1205902 = 03 1205903 = 05Φ1 = 02 Φ2 = 04 Φ3 = 006

Figures 1ndash6 are illustrated to show the simulation resultsFigures 1ndash3 show the better tracking trajectories performanceof the system variable state and minor error curve of theerrors on the time-varying constraints respectively It can beeasily obtained that the time-varying state constrains are notviolated In Figure 4 we can clearly get the conclusion that thetracking errors are all bounded and the tracking error signalcurve converges to zero Figures 5 and 6 show the trajectoriesof estimation and the controller respectively We make theconclusion that the adaptive laws and the controller of thesystem are all ensured boundedness

5 Conclusion

An adaptive control algorithm for hydraulic servo-systemconsidering time-varying state constraints has been designedin this paper The uncertainty and time-varying stateconstraints problem of the system have been consideredin the controller designing process The uncertainties andthe accurate estimation of disturbance have been solved byRBFNN By designing barrier Lyapunov function to solve thetime-varying state constraints problemof system the stabilityof the control strategy has been proved Besides the proposedmethod has proved that the error signals are bounded in the

Complexity 7

0 10 20 30 40 50minus50

0

50

Time (s)

kb1x1

yd

minuskb1

(a)

Time (s)0 10 20 30 40 50

minus40

minus20

0

20

40

ka1z1minuska1

(b)

Figure 1 (a) Trajectories of 1199091 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199111

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

kb2x2 minuskb2

yd

(a)

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

ka2z2minuska2

(b)

Figure 2 (a) Trajectories of 1199092 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199112small compacts sets noting that the tracking error signal con-verges to zero asymptotically the time-varying state con-strains are never violated and all singles of the hydraulicservo-system are bounded The experimental results haveshowed the effectiveness of the proposed method

Appendix

Proof ofTheorem 5 Based on the state constrains (10) and thedefinition of errors 119911119894 the initial error signals are boundedbetween the sets as |119911119894(0)| le 119896119886119894(0) From (37) the followingcan be obtained

le minus120595119881 + 119862 (A1)

where

120595 = min 21205741 21205742 2 119862 = 121198632 (119909 119905) +

3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (A2)

Let us multiply both sides of the above inequality by 119890120595119905(A1) can be rewritten as

119890120595119905 + 120595119881119890120595119905 le 119862119890120595119905 (A3)

And thenwe integrate both sides of inequality (A3) over [0 119905]to obtain

0 le 119881 (119905) le 119881 (0) 119890minus120595119905 + 119862120595 (A4)

Substituting (16) into (A4) we can get

log11989621198861198941198962119886119894 minus 1199112119894 le 119881 (119905) le 119881 (0) 119890minus120595119905 +

119862120595 (A5)

Thus we have the following inequality based on the transitiv-ity of the inequality from (A5)

11989621198861198941198962119886119894 minus 1199112119894 le 119890119881(0)119890minus120595119905+119862120595 (A6)

Then (A6) can be rewritten as

1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 le 119896119886119894 (119905) radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595 (A7)

Furthermore it can be easy proved that the error signal 119911119894converges to zero asymptotically based on appropriate para-meter design

Based on Assumption 2 we can get |119910119889(119905)| le 1198840(119905) and|119910(119894)119889(119905)| le 119884119894+1(119905) 119894 = 1 2 Then from |1199111| le 1198961198861 and

8 Complexity

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

kb3x3 minuskb3

yd

(a)

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

ka3z3minuska3

(b)

Figure 3 (a) Trajectories of 1199093 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199113

minus5

0

5

minus50

0

50minus150

minus100

minus50

0

50

100

z 3

z2z1

Figure 4 The phase portrait of tracking errors 1199111 1199111 and 1199113

0 10 20 30 40 50minus300

minus200

minus100

Time (s)

W1

(a)

0 10 20 30 40 50minus500

0

500

Time (s)

W2

(b)

0 10 20 30 40 500

10

20

Time (s)

W3

(c)

Figure 5 (a) The trajectory of 1 (b) the trajectory of 2 (c) the trajectory of 3

1199091 = 1199111 + 119910119889 we can get the inequality |1199091(119905)| le 1198961198861(119905) +1198840(119905) le 1198961198871(119905) In (18) the victual controller 1205721 can bebounded from the boundedness of 1199091 and 119910119889 It can be seenthat there exists the supremum 1205721 of 1205721 From |1199112| le 1198961198862and 1199092 = 1199112 + 1205721 it has |1199092| le 1198961198862 + 1205721 le 1198961198872 In the sameway we can get the boundedness supremum 1205722 based on the

definition of 1205722 in (19) Thus the state 1199093 can be proved bythe function 1198961198873 from the above proof of boundedness So thetime-varying state constrains are never violated

From (A4) and adaptive laws in (28) (30) and (32)we can get that the neural network weight vector errors 119894are bounded Because the weight vectors 119882lowast119894 are bounded

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 3

Consider that the flow rates 1198761 and 1198762 are the functionof the spool displacement 119909119904 The flow equation of hydrauliccylinder is written as follows

1198761 = 119896119891119909119904ℎ1 (119909119904 1198751) 1198762 = 119896119891119909119904ℎ2 (119909119904 1198752) (3)

with

ℎ1 (119909119904 1198751) = (119875119904 minus 1198751)12 119909119904 ge 0(1198751 minus 119875119903)12 119909119904 lt 0

ℎ2 (119909119904 1198752) = (1198752 minus 119875119903)12 119909119904 ge 0(119875119904 minus 1198752)12 119909119904 lt 0

(4)

where 119896119891 denotes the flow gain coefficient 119875119904 is the supplypressure of the oil 119875119903 represents the return oil pressure

We assume that the spool displacement 119909119904 is directproportion with control input voltage 119906 in the situation ofignoring the high frequency that is 119909119904 = 119896119904119906 119896119904 gt 0 is thesmall gain Then (3) can be rewritten as

1198761 = 119870119891119906ℎ1 (119906 1198751) 1198762 = 119870119891119906ℎ2 (119906 1198752) (5)

where 119870119891 = 119896119891119896119904 represents the total flow gain coefficientThen we can conclude that the load-flow of hydrauliccylinder is

119876 = 119870119891119906 (119875119904 minus sgn (119906) 119875119890)12 (6)

where sgn(119906) represents the step function as

sgn (119906) = 1 119906 ge 0minus1 119906 lt 0

119875119904 = 1198751 + 1198752119875119903 = 0119875119890 = 1198751 minus 1198752(7)

Define state variables 119909 = [119909 ]119879 = [1199091 1199092 1199093]119879 thedynamics of the hydraulic servo-system (1) can be translatedinto the following the state space expression based on thepressure dynamic equation (2) and the flow equations (5)-(6)

1 = 11990922 = 11990933 = minus11989111199092 minus 1198912 (119909) + 1198913119892119906(8)

where

1198911 = 41198602119861V119898119881119890 minus 1205882119898 1198913 = 4119860119861V119870119891119898119881119890 119892 = radic (119875119904 minus sgn (119906) 119875119890)2

1198912 (119909) = 4119860119861V119862119898119875119890119898119881119890 + 4119860119861V119908119898119881119890 + 1205881198601198751198901198982 minus 120588119898119889 (119909)+ 119889 (119909)119898

(9)

1198911(119909) 1198912(119909) and 1198913(119909) are unknown smooth parameterfunctions In this paper the state variable 119909119894 is constrainedin the following time-varying compact sets1003816100381610038161003816119909119894 (119905)1003816100381610038161003816 le 119896119887119894 (119905) 119894 = 1 2 3 (10)

where 119896119887119894(119905) represents the time-varying smooth functions

Assumption 1 (see [37]) There exist the functions 1198913(119909) and1198913(119909) which satisfy the inequality 0 lt 119891

3le 1198913 le 1198913

Assumption 2 (see [43]) There are functions1198840(119905) 1198841(119905) and1198842(119905) We assume that the functions satisfy the inequalities|119884119894minus1(119905)| le |119896119887119894(119905)| 119894 = 1 2 3 Then the smooth desiredtrajectory 119910119889(119905) and its time derivatives respectively satisfy|119910119889(119905)| le 1198840(119905) and |119910(119894)119889 (119905)| le 119884119894+1(119905) 119894 = 1 2 for all 119905 gt 0Lemma 3 (see [54]) For any time-varying function 119896119886(119905) thefollowing inequality holds for all error function 119911(119905) in theinterval |119911(119905)| le 119896119886(119905)

log1198962119886 (119905)1198962119886 (119905) minus 1199112 (119905) le 1199112 (119905)1198962119886 (119905) minus 1199112 (119905) (11)

The above-mentioned unknown smooth functions 11989111198912(119909) and 1198913 are approximated by the radial basis functionNN as

119891119894 (119883) = 119882119879119894 Φ119894 (119883) 119894 = 1 2 3 (12)

where119882119894 isin 119877119897119894 denotes the neural network weight vector andΦ119894(119883) = [1205931198941 120593119894119897119894] isin 119877119897119894 are the basis function vectorswith input vector 119883 isin 119877119899119894 and the NN node number 119897119894 gt 1Gaussian functions 120593119894119895(119883) are chosen in the following forms

120593119894119895 (119883) = exp(minus10038171003817100381710038171003817119883 minus 1205831198941198951003817100381710038171003817100381721205782119894119895 ) 119894 = 1 2 3 119895 = 1 119897119894

(13)

where 120583119894119895 = [1205871198941198951 120587119894119895119899119894]119879 denotes the center of NNfunction and 120578119894119895 is the width of the Gaussian function

4 Complexity

Define the unknown optimal weight vector as the follow-ing form

119882lowast119894 = argmin119882119894isin119877

119897119894

[ sup119883isin119877119899119894

10038161003816100381610038161003816119882119879119894 Φ119894 (119883) minus 119891119894 (119883)10038161003816100381610038161003816] 119894 = 1 2 3 (14)

Then there exists the unknown approximation errors 120576119894(119883)which has supremum 120576119894 that is |120576119894(119883)| le 120576119894 Equation (12)will be rewritten in the approximation equations as follows

119891119894 (119883) = 119882lowast119879119894 Φ119894 (119883) + 120576119894 (119883) 119894 = 1 2 3 (15)

3 The Adaptive NN Controller Design

Define the track trajectory 119910119889 and the virtual controllers 1205721and1205722 we have the track error 1199111 = 1199091minus119910119889Then let 1199112 = 1199092minus1205721 and 1199113 = 1199093 minus 1205722 Choose the barrier Lyapunov functioncandidate as

119881 = 3sum119894=1

119881119894 + 119881119873119881119894 = 12 log 11989621198861198941198962119886119894 minus 1199112119894 119881119873 =

123sum119895=1

119879119895 Θminus1119895 119895(16)

where log(sdot) represents the natural logarithm 119896119886119894(119905) = 119896119887119894(119905)minus119863119894(119905) is boundaries of error vector 119911119894 from subsequent feasi-bility analysis Θ119895 119895 = 1 2 3 are the positive gain matrices119894 is the neural network weight error with 119894 = 119894 minus 119882lowast119894 and note that 119881 is the positive definite and continuouslydifferentiable in the set |119911119894(119905)| le 119896119886119894(119905) for 119894 = 1 2 3

Based on (16) the time derivative of 119881 is

= 3sum119894=1

119894 + 119873= 3sum119894=1

119911119894(1198962119886119894 minus 1199112119894 ) (119894 minus119886119894119896119886119894 119911119894) +

3sum119895=1

119879119895 Θminus1119895 119882119895(17)

Design the virtual controllers as

1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 119910119889 (18)

1205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111 (19)

where the time-varying gains are given by

120574119894 (119905) = radic(119886119894119896119886119894 )2 + 120575119894 119894 = 1 2 3 (20)

for all positive constants 120575119894 and 120574119894When any 119886119894 is zero 120575119894 willensure that the time derivatives of virtual controllers 120572119894 andcontroller 119906 are bounded

Substituting (8) and (18)ndash(20) into (17) we can obtain

= minus 2sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 ) +

11991121199113(11989621198862 minus 11991122)+ 1199113(11989621198863 minus 11991123) (3 minus 2 minus

11988631198961198863 1199113)+ 3sum119895=1

119879119895 Θminus1119895 119882119895(21)

where

1 = 12059712057211205971199091 1 +1sum119896=0

(1205971205721119910(119896)119889

119910(119896+1)119889 + 1205971205721119896(119896)1198861 119896(119896+1)1198861 ) 2 = 2sum119895=1

1205971205722120597119909119895 119895+ 2sum119896=0

(1205971205722119910(119896)119889

119910(119896+1)119889 + 1205971205722119896(119896)1198861 119896(119896+1)1198861 + 1205971205722119896(119896)1198862 119896(119896+1)1198862 ) (22)

According to approximation equations (15) (21) becomes

= minus 2sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 )

+ 3sum119895=1

119879119895 Θminus1119895 119882119895 + 1199113(11989621198863 minus 11991123) [[minus119882lowast1198791 Φ1 (119883) 1199092

minus119882lowast1198792 Φ2 (119883) +119882lowast1198793 Φ3 (119883) 119892119906 + 119863 (119909 119905) minus 2+ (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11988631198961198863 1199113]]

(23)

where119863(119909 119905) represents the systematic disturbance compen-sation as the following form

119863(119909 119905) = minus12057611199092 minus 1205762 + 1205763119892119906 (24)

Choose the controller as follows

119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092+ 1198792 Φ2 (119909) + 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123)) (25)

Complexity 5

Adopting the equations 119882lowast119894 = 119894 minus 119894 119894 = 1 2 3 andcontroller (25) the expression for (23) can be rewritten by theabove controller as

= minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 )

+ 1198791 (Θminus11 1198821 + Φ1 (119883) 11990921199113(11989621198863 minus 11991123))+ 1198792 (Θminus12 1198822 + Φ2 (119883) 1199113(11989621198863 minus 11991123))+ 1198793 (Θminus13 1198823 minus Φ3 (119883) 1199113119892119906(11989621198863 minus 11991123))+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2

(26)

Based on the above-mentioned definition of 120574119894(119905) we canobtain that

120574119894 (119905) + 119886119894119896119886119894 ge 0 119894 = 1 2 3 (27)

Introducing the projection algorithm of adaptive controlwe design the adaptive law from (26) as follows

1198821=

minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123) + 12059011]] Θ1 le 0 1 gt 1198721198911

1198751198911 (1 Φ1) Θ1 gt 0 1 = 1198721198911(28)

where 1198751198911(1 Φ1) represents projection operator as thefollowing form

1198751198911 (1 Φ1) = minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123)

minus Φ1 (119883) 1198791 111990921199113100381710038171003817100381710038171100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059011]]

(29)

1198721198911 is the minuteness positive constant to ensure the adap-tive law 1198821 ge 0 in any situation

Similarly the other adaptive laws can obtain the forms as

1198822 = minusΘ2 [[Φ2 (119883)

1199113(11989621198863 minus 11991123) + 12059022]] Θ2 le 0 2 gt 11987211989121198751198912 (2 Φ2) Θ2 gt 0 2 = 1198721198912

(30)

1198751198912 (2 Φ2) = minusΘ2 [[Φ2 (119883)1199113(11989621198863 minus 11991123) minus Φ2 (119883)

1198792 21199113100381710038171003817100381710038172100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059022]] (31)

1198823 = Θ3 [[Φ3 (119883)

1199113119892119906(11989621198863 minus 11991123) minus 12059033]] Θ3 le 0 3 gt 11987211989131198751198913 (3 Φ3) Θ3 gt 0 3 = 1198721198913

(32)

1198751198913 (3 Φ3) = minusΘ3 [[Φ3 (119883)1199113119892119906(11989621198863 minus 11991123) minus Φ3 (119883)

1198793 31199113119892119906100381710038171003817100381710038173100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059033]] (33)

In particular when component 3119894 of 3 is a small con-stant 1198981198913119894 that is 3 = 1198721198913 we adopt the following com-ponent of adaptive law

1198823119894 = minusΘ3119894 [[Φ3119894 (119883)

1199113119892119906(11989621198863 minus 11991123) + 12059033119894]] 0(34)

Remark 4 Based on the above description (34) we canmakethe following conclusion If 3119894 = 1198981198913119894 we can get 1198823119894 gt 0

that is 3119894 ge 1198981198913119894 gt 0 for all any 3119894 Then there does notexist insignificance function for controller 119906

Substituting (28) (30) and (32) into (26) we obtain

le minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 ) minus

3sum119894=1

120590119894119879119894 119894+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2 (35)

6 Complexity

Using Youngrsquos inequality in the last two terms of weseem to easily get

minus120590119894119879119894 119894 = minus120590119894119879119894 (119894 +119882lowast119894 )le minus120590119894 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817minus119882lowast119894 10038171003817100381710038172+ 1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172

= minus1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 119863 (119909 119905) 1199113(11989621198863 minus 11991123) le

121198632 (119909 119905) + 119911232 (11989621198863 minus 11991123)2

(36)

Based on definition (27) and using inequalities (36) and(11) in Lemma 3 (35) can be rewritten as

le minus 3sum119894=1

120574119894 log 1198962119886119894(1198962119886119894 minus 1199112119894 ) minus3sum119894=1

1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 121198632 (119909 119905)+ 3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (37)

Theorem 5 A class of hydraulic servo-system is described by(1)ndash(3) under Assumptions 1 and 2 If the initial displacement1199091(0) the initial velocity 1199092(0) and the initial acceleration1199093(0) satisfy |119909119894(0)| le 119896119887119894(0) 119894 = 1 2 3 then the proposedcontrol method has the following properties

(1) The error signals are bounded in the following sets

119911119894 (119905) isin Ω119911119894 119894 = 1 2 3 (38)

where Ω119911119894 fl 119911119894 isin R |119911119894(119905)| le 119896119886119894(119905)radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595noting that the tracking error signal converges to zero asymp-totically

(2) The time-varying state constrains are never violatedthat is |119909119894(119905)| le 119896119887119894(119905) forall119905 ge 0 119894 = 1 2 3

(3) All singles of the hydraulic servo-system are bounded

Proof Please see the Appendix

4 Simulation Example

Simulation has been performed to demonstrate the per-formance of the proposed approach The values of systemparameters which are used in the simulation are given inTable 1

The initial states of the hydraulic servo-system are givenas 1199091(0) = 0 1199091(0) = 0 and 1199091(0) = 0 The time-varyingconstraint functions are 1198961198871(119905) = 2 sin(5119905) + 40 1198961198872(119905) =2 sin(5119905) + 60 and 1198961198873(119905) = 4 sin(5119905) + 150 The desiredtrajectory tracking periodic reciprocation of the displace-ment is given 119910119889 = 40 arctan(sin(04120587119905))(1 minus 119890minus119905) with 119905 isin[0 119905119891] and 119905119891 = 50 s The modeling error disturbance of the

Table 1 The system parameters used in the simulation

Name Value119898(kg) 2200119875119904 (Pa) 20119875119903 (Pa) 0119860 (m3) 2 times 10minus3120588 (Nsm) 5001198811198681 (m3) 5 times 10minus41198811198682 (m3) 5 times 10minus3119861V (Pa) 200119862119898 (m5Ns) 2 times 10minus5119870119891 (m3Ns) 8 times 10minus2both chambers is 119908 = sin(119905) + 2 We choose the followingcontroller1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 1199101198891205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092 + 1198792 Φ2+ 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123))

(39)

where we choose design parameters as 1198840 = 20 1198841 = 101198842 = 50 1205741 = 10 1205742 = 3 1205743 = 6 1205751 = 08 1205752 = 07 1205753 = 06the initial weight vector estimation 1 = minus100 2 = 203 = 10 Θ1 = 3 Θ2 = 2 Θ3 = 2 1205901 = 2 1205902 = 03 1205903 = 05Φ1 = 02 Φ2 = 04 Φ3 = 006

Figures 1ndash6 are illustrated to show the simulation resultsFigures 1ndash3 show the better tracking trajectories performanceof the system variable state and minor error curve of theerrors on the time-varying constraints respectively It can beeasily obtained that the time-varying state constrains are notviolated In Figure 4 we can clearly get the conclusion that thetracking errors are all bounded and the tracking error signalcurve converges to zero Figures 5 and 6 show the trajectoriesof estimation and the controller respectively We make theconclusion that the adaptive laws and the controller of thesystem are all ensured boundedness

5 Conclusion

An adaptive control algorithm for hydraulic servo-systemconsidering time-varying state constraints has been designedin this paper The uncertainty and time-varying stateconstraints problem of the system have been consideredin the controller designing process The uncertainties andthe accurate estimation of disturbance have been solved byRBFNN By designing barrier Lyapunov function to solve thetime-varying state constraints problemof system the stabilityof the control strategy has been proved Besides the proposedmethod has proved that the error signals are bounded in the

Complexity 7

0 10 20 30 40 50minus50

0

50

Time (s)

kb1x1

yd

minuskb1

(a)

Time (s)0 10 20 30 40 50

minus40

minus20

0

20

40

ka1z1minuska1

(b)

Figure 1 (a) Trajectories of 1199091 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199111

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

kb2x2 minuskb2

yd

(a)

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

ka2z2minuska2

(b)

Figure 2 (a) Trajectories of 1199092 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199112small compacts sets noting that the tracking error signal con-verges to zero asymptotically the time-varying state con-strains are never violated and all singles of the hydraulicservo-system are bounded The experimental results haveshowed the effectiveness of the proposed method

Appendix

Proof ofTheorem 5 Based on the state constrains (10) and thedefinition of errors 119911119894 the initial error signals are boundedbetween the sets as |119911119894(0)| le 119896119886119894(0) From (37) the followingcan be obtained

le minus120595119881 + 119862 (A1)

where

120595 = min 21205741 21205742 2 119862 = 121198632 (119909 119905) +

3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (A2)

Let us multiply both sides of the above inequality by 119890120595119905(A1) can be rewritten as

119890120595119905 + 120595119881119890120595119905 le 119862119890120595119905 (A3)

And thenwe integrate both sides of inequality (A3) over [0 119905]to obtain

0 le 119881 (119905) le 119881 (0) 119890minus120595119905 + 119862120595 (A4)

Substituting (16) into (A4) we can get

log11989621198861198941198962119886119894 minus 1199112119894 le 119881 (119905) le 119881 (0) 119890minus120595119905 +

119862120595 (A5)

Thus we have the following inequality based on the transitiv-ity of the inequality from (A5)

11989621198861198941198962119886119894 minus 1199112119894 le 119890119881(0)119890minus120595119905+119862120595 (A6)

Then (A6) can be rewritten as

1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 le 119896119886119894 (119905) radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595 (A7)

Furthermore it can be easy proved that the error signal 119911119894converges to zero asymptotically based on appropriate para-meter design

Based on Assumption 2 we can get |119910119889(119905)| le 1198840(119905) and|119910(119894)119889(119905)| le 119884119894+1(119905) 119894 = 1 2 Then from |1199111| le 1198961198861 and

8 Complexity

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

kb3x3 minuskb3

yd

(a)

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

ka3z3minuska3

(b)

Figure 3 (a) Trajectories of 1199093 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199113

minus5

0

5

minus50

0

50minus150

minus100

minus50

0

50

100

z 3

z2z1

Figure 4 The phase portrait of tracking errors 1199111 1199111 and 1199113

0 10 20 30 40 50minus300

minus200

minus100

Time (s)

W1

(a)

0 10 20 30 40 50minus500

0

500

Time (s)

W2

(b)

0 10 20 30 40 500

10

20

Time (s)

W3

(c)

Figure 5 (a) The trajectory of 1 (b) the trajectory of 2 (c) the trajectory of 3

1199091 = 1199111 + 119910119889 we can get the inequality |1199091(119905)| le 1198961198861(119905) +1198840(119905) le 1198961198871(119905) In (18) the victual controller 1205721 can bebounded from the boundedness of 1199091 and 119910119889 It can be seenthat there exists the supremum 1205721 of 1205721 From |1199112| le 1198961198862and 1199092 = 1199112 + 1205721 it has |1199092| le 1198961198862 + 1205721 le 1198961198872 In the sameway we can get the boundedness supremum 1205722 based on the

definition of 1205722 in (19) Thus the state 1199093 can be proved bythe function 1198961198873 from the above proof of boundedness So thetime-varying state constrains are never violated

From (A4) and adaptive laws in (28) (30) and (32)we can get that the neural network weight vector errors 119894are bounded Because the weight vectors 119882lowast119894 are bounded

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

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Stochastic AnalysisInternational Journal of

4 Complexity

Define the unknown optimal weight vector as the follow-ing form

119882lowast119894 = argmin119882119894isin119877

119897119894

[ sup119883isin119877119899119894

10038161003816100381610038161003816119882119879119894 Φ119894 (119883) minus 119891119894 (119883)10038161003816100381610038161003816] 119894 = 1 2 3 (14)

Then there exists the unknown approximation errors 120576119894(119883)which has supremum 120576119894 that is |120576119894(119883)| le 120576119894 Equation (12)will be rewritten in the approximation equations as follows

119891119894 (119883) = 119882lowast119879119894 Φ119894 (119883) + 120576119894 (119883) 119894 = 1 2 3 (15)

3 The Adaptive NN Controller Design

Define the track trajectory 119910119889 and the virtual controllers 1205721and1205722 we have the track error 1199111 = 1199091minus119910119889Then let 1199112 = 1199092minus1205721 and 1199113 = 1199093 minus 1205722 Choose the barrier Lyapunov functioncandidate as

119881 = 3sum119894=1

119881119894 + 119881119873119881119894 = 12 log 11989621198861198941198962119886119894 minus 1199112119894 119881119873 =

123sum119895=1

119879119895 Θminus1119895 119895(16)

where log(sdot) represents the natural logarithm 119896119886119894(119905) = 119896119887119894(119905)minus119863119894(119905) is boundaries of error vector 119911119894 from subsequent feasi-bility analysis Θ119895 119895 = 1 2 3 are the positive gain matrices119894 is the neural network weight error with 119894 = 119894 minus 119882lowast119894 and note that 119881 is the positive definite and continuouslydifferentiable in the set |119911119894(119905)| le 119896119886119894(119905) for 119894 = 1 2 3

Based on (16) the time derivative of 119881 is

= 3sum119894=1

119894 + 119873= 3sum119894=1

119911119894(1198962119886119894 minus 1199112119894 ) (119894 minus119886119894119896119886119894 119911119894) +

3sum119895=1

119879119895 Θminus1119895 119882119895(17)

Design the virtual controllers as

1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 119910119889 (18)

1205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111 (19)

where the time-varying gains are given by

120574119894 (119905) = radic(119886119894119896119886119894 )2 + 120575119894 119894 = 1 2 3 (20)

for all positive constants 120575119894 and 120574119894When any 119886119894 is zero 120575119894 willensure that the time derivatives of virtual controllers 120572119894 andcontroller 119906 are bounded

Substituting (8) and (18)ndash(20) into (17) we can obtain

= minus 2sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 ) +

11991121199113(11989621198862 minus 11991122)+ 1199113(11989621198863 minus 11991123) (3 minus 2 minus

11988631198961198863 1199113)+ 3sum119895=1

119879119895 Θminus1119895 119882119895(21)

where

1 = 12059712057211205971199091 1 +1sum119896=0

(1205971205721119910(119896)119889

119910(119896+1)119889 + 1205971205721119896(119896)1198861 119896(119896+1)1198861 ) 2 = 2sum119895=1

1205971205722120597119909119895 119895+ 2sum119896=0

(1205971205722119910(119896)119889

119910(119896+1)119889 + 1205971205722119896(119896)1198861 119896(119896+1)1198861 + 1205971205722119896(119896)1198862 119896(119896+1)1198862 ) (22)

According to approximation equations (15) (21) becomes

= minus 2sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 )

+ 3sum119895=1

119879119895 Θminus1119895 119882119895 + 1199113(11989621198863 minus 11991123) [[minus119882lowast1198791 Φ1 (119883) 1199092

minus119882lowast1198792 Φ2 (119883) +119882lowast1198793 Φ3 (119883) 119892119906 + 119863 (119909 119905) minus 2+ (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11988631198961198863 1199113]]

(23)

where119863(119909 119905) represents the systematic disturbance compen-sation as the following form

119863(119909 119905) = minus12057611199092 minus 1205762 + 1205763119892119906 (24)

Choose the controller as follows

119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092+ 1198792 Φ2 (119909) + 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123)) (25)

Complexity 5

Adopting the equations 119882lowast119894 = 119894 minus 119894 119894 = 1 2 3 andcontroller (25) the expression for (23) can be rewritten by theabove controller as

= minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 )

+ 1198791 (Θminus11 1198821 + Φ1 (119883) 11990921199113(11989621198863 minus 11991123))+ 1198792 (Θminus12 1198822 + Φ2 (119883) 1199113(11989621198863 minus 11991123))+ 1198793 (Θminus13 1198823 minus Φ3 (119883) 1199113119892119906(11989621198863 minus 11991123))+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2

(26)

Based on the above-mentioned definition of 120574119894(119905) we canobtain that

120574119894 (119905) + 119886119894119896119886119894 ge 0 119894 = 1 2 3 (27)

Introducing the projection algorithm of adaptive controlwe design the adaptive law from (26) as follows

1198821=

minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123) + 12059011]] Θ1 le 0 1 gt 1198721198911

1198751198911 (1 Φ1) Θ1 gt 0 1 = 1198721198911(28)

where 1198751198911(1 Φ1) represents projection operator as thefollowing form

1198751198911 (1 Φ1) = minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123)

minus Φ1 (119883) 1198791 111990921199113100381710038171003817100381710038171100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059011]]

(29)

1198721198911 is the minuteness positive constant to ensure the adap-tive law 1198821 ge 0 in any situation

Similarly the other adaptive laws can obtain the forms as

1198822 = minusΘ2 [[Φ2 (119883)

1199113(11989621198863 minus 11991123) + 12059022]] Θ2 le 0 2 gt 11987211989121198751198912 (2 Φ2) Θ2 gt 0 2 = 1198721198912

(30)

1198751198912 (2 Φ2) = minusΘ2 [[Φ2 (119883)1199113(11989621198863 minus 11991123) minus Φ2 (119883)

1198792 21199113100381710038171003817100381710038172100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059022]] (31)

1198823 = Θ3 [[Φ3 (119883)

1199113119892119906(11989621198863 minus 11991123) minus 12059033]] Θ3 le 0 3 gt 11987211989131198751198913 (3 Φ3) Θ3 gt 0 3 = 1198721198913

(32)

1198751198913 (3 Φ3) = minusΘ3 [[Φ3 (119883)1199113119892119906(11989621198863 minus 11991123) minus Φ3 (119883)

1198793 31199113119892119906100381710038171003817100381710038173100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059033]] (33)

In particular when component 3119894 of 3 is a small con-stant 1198981198913119894 that is 3 = 1198721198913 we adopt the following com-ponent of adaptive law

1198823119894 = minusΘ3119894 [[Φ3119894 (119883)

1199113119892119906(11989621198863 minus 11991123) + 12059033119894]] 0(34)

Remark 4 Based on the above description (34) we canmakethe following conclusion If 3119894 = 1198981198913119894 we can get 1198823119894 gt 0

that is 3119894 ge 1198981198913119894 gt 0 for all any 3119894 Then there does notexist insignificance function for controller 119906

Substituting (28) (30) and (32) into (26) we obtain

le minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 ) minus

3sum119894=1

120590119894119879119894 119894+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2 (35)

6 Complexity

Using Youngrsquos inequality in the last two terms of weseem to easily get

minus120590119894119879119894 119894 = minus120590119894119879119894 (119894 +119882lowast119894 )le minus120590119894 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817minus119882lowast119894 10038171003817100381710038172+ 1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172

= minus1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 119863 (119909 119905) 1199113(11989621198863 minus 11991123) le

121198632 (119909 119905) + 119911232 (11989621198863 minus 11991123)2

(36)

Based on definition (27) and using inequalities (36) and(11) in Lemma 3 (35) can be rewritten as

le minus 3sum119894=1

120574119894 log 1198962119886119894(1198962119886119894 minus 1199112119894 ) minus3sum119894=1

1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 121198632 (119909 119905)+ 3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (37)

Theorem 5 A class of hydraulic servo-system is described by(1)ndash(3) under Assumptions 1 and 2 If the initial displacement1199091(0) the initial velocity 1199092(0) and the initial acceleration1199093(0) satisfy |119909119894(0)| le 119896119887119894(0) 119894 = 1 2 3 then the proposedcontrol method has the following properties

(1) The error signals are bounded in the following sets

119911119894 (119905) isin Ω119911119894 119894 = 1 2 3 (38)

where Ω119911119894 fl 119911119894 isin R |119911119894(119905)| le 119896119886119894(119905)radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595noting that the tracking error signal converges to zero asymp-totically

(2) The time-varying state constrains are never violatedthat is |119909119894(119905)| le 119896119887119894(119905) forall119905 ge 0 119894 = 1 2 3

(3) All singles of the hydraulic servo-system are bounded

Proof Please see the Appendix

4 Simulation Example

Simulation has been performed to demonstrate the per-formance of the proposed approach The values of systemparameters which are used in the simulation are given inTable 1

The initial states of the hydraulic servo-system are givenas 1199091(0) = 0 1199091(0) = 0 and 1199091(0) = 0 The time-varyingconstraint functions are 1198961198871(119905) = 2 sin(5119905) + 40 1198961198872(119905) =2 sin(5119905) + 60 and 1198961198873(119905) = 4 sin(5119905) + 150 The desiredtrajectory tracking periodic reciprocation of the displace-ment is given 119910119889 = 40 arctan(sin(04120587119905))(1 minus 119890minus119905) with 119905 isin[0 119905119891] and 119905119891 = 50 s The modeling error disturbance of the

Table 1 The system parameters used in the simulation

Name Value119898(kg) 2200119875119904 (Pa) 20119875119903 (Pa) 0119860 (m3) 2 times 10minus3120588 (Nsm) 5001198811198681 (m3) 5 times 10minus41198811198682 (m3) 5 times 10minus3119861V (Pa) 200119862119898 (m5Ns) 2 times 10minus5119870119891 (m3Ns) 8 times 10minus2both chambers is 119908 = sin(119905) + 2 We choose the followingcontroller1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 1199101198891205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092 + 1198792 Φ2+ 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123))

(39)

where we choose design parameters as 1198840 = 20 1198841 = 101198842 = 50 1205741 = 10 1205742 = 3 1205743 = 6 1205751 = 08 1205752 = 07 1205753 = 06the initial weight vector estimation 1 = minus100 2 = 203 = 10 Θ1 = 3 Θ2 = 2 Θ3 = 2 1205901 = 2 1205902 = 03 1205903 = 05Φ1 = 02 Φ2 = 04 Φ3 = 006

Figures 1ndash6 are illustrated to show the simulation resultsFigures 1ndash3 show the better tracking trajectories performanceof the system variable state and minor error curve of theerrors on the time-varying constraints respectively It can beeasily obtained that the time-varying state constrains are notviolated In Figure 4 we can clearly get the conclusion that thetracking errors are all bounded and the tracking error signalcurve converges to zero Figures 5 and 6 show the trajectoriesof estimation and the controller respectively We make theconclusion that the adaptive laws and the controller of thesystem are all ensured boundedness

5 Conclusion

An adaptive control algorithm for hydraulic servo-systemconsidering time-varying state constraints has been designedin this paper The uncertainty and time-varying stateconstraints problem of the system have been consideredin the controller designing process The uncertainties andthe accurate estimation of disturbance have been solved byRBFNN By designing barrier Lyapunov function to solve thetime-varying state constraints problemof system the stabilityof the control strategy has been proved Besides the proposedmethod has proved that the error signals are bounded in the

Complexity 7

0 10 20 30 40 50minus50

0

50

Time (s)

kb1x1

yd

minuskb1

(a)

Time (s)0 10 20 30 40 50

minus40

minus20

0

20

40

ka1z1minuska1

(b)

Figure 1 (a) Trajectories of 1199091 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199111

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

kb2x2 minuskb2

yd

(a)

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

ka2z2minuska2

(b)

Figure 2 (a) Trajectories of 1199092 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199112small compacts sets noting that the tracking error signal con-verges to zero asymptotically the time-varying state con-strains are never violated and all singles of the hydraulicservo-system are bounded The experimental results haveshowed the effectiveness of the proposed method

Appendix

Proof ofTheorem 5 Based on the state constrains (10) and thedefinition of errors 119911119894 the initial error signals are boundedbetween the sets as |119911119894(0)| le 119896119886119894(0) From (37) the followingcan be obtained

le minus120595119881 + 119862 (A1)

where

120595 = min 21205741 21205742 2 119862 = 121198632 (119909 119905) +

3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (A2)

Let us multiply both sides of the above inequality by 119890120595119905(A1) can be rewritten as

119890120595119905 + 120595119881119890120595119905 le 119862119890120595119905 (A3)

And thenwe integrate both sides of inequality (A3) over [0 119905]to obtain

0 le 119881 (119905) le 119881 (0) 119890minus120595119905 + 119862120595 (A4)

Substituting (16) into (A4) we can get

log11989621198861198941198962119886119894 minus 1199112119894 le 119881 (119905) le 119881 (0) 119890minus120595119905 +

119862120595 (A5)

Thus we have the following inequality based on the transitiv-ity of the inequality from (A5)

11989621198861198941198962119886119894 minus 1199112119894 le 119890119881(0)119890minus120595119905+119862120595 (A6)

Then (A6) can be rewritten as

1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 le 119896119886119894 (119905) radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595 (A7)

Furthermore it can be easy proved that the error signal 119911119894converges to zero asymptotically based on appropriate para-meter design

Based on Assumption 2 we can get |119910119889(119905)| le 1198840(119905) and|119910(119894)119889(119905)| le 119884119894+1(119905) 119894 = 1 2 Then from |1199111| le 1198961198861 and

8 Complexity

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

kb3x3 minuskb3

yd

(a)

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

ka3z3minuska3

(b)

Figure 3 (a) Trajectories of 1199093 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199113

minus5

0

5

minus50

0

50minus150

minus100

minus50

0

50

100

z 3

z2z1

Figure 4 The phase portrait of tracking errors 1199111 1199111 and 1199113

0 10 20 30 40 50minus300

minus200

minus100

Time (s)

W1

(a)

0 10 20 30 40 50minus500

0

500

Time (s)

W2

(b)

0 10 20 30 40 500

10

20

Time (s)

W3

(c)

Figure 5 (a) The trajectory of 1 (b) the trajectory of 2 (c) the trajectory of 3

1199091 = 1199111 + 119910119889 we can get the inequality |1199091(119905)| le 1198961198861(119905) +1198840(119905) le 1198961198871(119905) In (18) the victual controller 1205721 can bebounded from the boundedness of 1199091 and 119910119889 It can be seenthat there exists the supremum 1205721 of 1205721 From |1199112| le 1198961198862and 1199092 = 1199112 + 1205721 it has |1199092| le 1198961198862 + 1205721 le 1198961198872 In the sameway we can get the boundedness supremum 1205722 based on the

definition of 1205722 in (19) Thus the state 1199093 can be proved bythe function 1198961198873 from the above proof of boundedness So thetime-varying state constrains are never violated

From (A4) and adaptive laws in (28) (30) and (32)we can get that the neural network weight vector errors 119894are bounded Because the weight vectors 119882lowast119894 are bounded

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 5

Adopting the equations 119882lowast119894 = 119894 minus 119894 119894 = 1 2 3 andcontroller (25) the expression for (23) can be rewritten by theabove controller as

= minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 )

+ 1198791 (Θminus11 1198821 + Φ1 (119883) 11990921199113(11989621198863 minus 11991123))+ 1198792 (Θminus12 1198822 + Φ2 (119883) 1199113(11989621198863 minus 11991123))+ 1198793 (Θminus13 1198823 minus Φ3 (119883) 1199113119892119906(11989621198863 minus 11991123))+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2

(26)

Based on the above-mentioned definition of 120574119894(119905) we canobtain that

120574119894 (119905) + 119886119894119896119886119894 ge 0 119894 = 1 2 3 (27)

Introducing the projection algorithm of adaptive controlwe design the adaptive law from (26) as follows

1198821=

minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123) + 12059011]] Θ1 le 0 1 gt 1198721198911

1198751198911 (1 Φ1) Θ1 gt 0 1 = 1198721198911(28)

where 1198751198911(1 Φ1) represents projection operator as thefollowing form

1198751198911 (1 Φ1) = minusΘ1 [[Φ1 (119883)11990921199113(11989621198863 minus 11991123)

minus Φ1 (119883) 1198791 111990921199113100381710038171003817100381710038171100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059011]]

(29)

1198721198911 is the minuteness positive constant to ensure the adap-tive law 1198821 ge 0 in any situation

Similarly the other adaptive laws can obtain the forms as

1198822 = minusΘ2 [[Φ2 (119883)

1199113(11989621198863 minus 11991123) + 12059022]] Θ2 le 0 2 gt 11987211989121198751198912 (2 Φ2) Θ2 gt 0 2 = 1198721198912

(30)

1198751198912 (2 Φ2) = minusΘ2 [[Φ2 (119883)1199113(11989621198863 minus 11991123) minus Φ2 (119883)

1198792 21199113100381710038171003817100381710038172100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059022]] (31)

1198823 = Θ3 [[Φ3 (119883)

1199113119892119906(11989621198863 minus 11991123) minus 12059033]] Θ3 le 0 3 gt 11987211989131198751198913 (3 Φ3) Θ3 gt 0 3 = 1198721198913

(32)

1198751198913 (3 Φ3) = minusΘ3 [[Φ3 (119883)1199113119892119906(11989621198863 minus 11991123) minus Φ3 (119883)

1198793 31199113119892119906100381710038171003817100381710038173100381710038171003817100381710038172 (11989621198863 minus 11991123) + 12059033]] (33)

In particular when component 3119894 of 3 is a small con-stant 1198981198913119894 that is 3 = 1198721198913 we adopt the following com-ponent of adaptive law

1198823119894 = minusΘ3119894 [[Φ3119894 (119883)

1199113119892119906(11989621198863 minus 11991123) + 12059033119894]] 0(34)

Remark 4 Based on the above description (34) we canmakethe following conclusion If 3119894 = 1198981198913119894 we can get 1198823119894 gt 0

that is 3119894 ge 1198981198913119894 gt 0 for all any 3119894 Then there does notexist insignificance function for controller 119906

Substituting (28) (30) and (32) into (26) we obtain

le minus 3sum119894=1

(120574119894 + 120574119894 (119905) + 119886119894119896119886119894 )1199112119894(1198962119886119894 minus 1199112119894 ) minus

3sum119894=1

120590119894119879119894 119894+ 119863 (119909 119905) 1199113(11989621198863 minus 11991123) minus

119911232 (11989621198863 minus 11991123)2 (35)

6 Complexity

Using Youngrsquos inequality in the last two terms of weseem to easily get

minus120590119894119879119894 119894 = minus120590119894119879119894 (119894 +119882lowast119894 )le minus120590119894 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817minus119882lowast119894 10038171003817100381710038172+ 1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172

= minus1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 119863 (119909 119905) 1199113(11989621198863 minus 11991123) le

121198632 (119909 119905) + 119911232 (11989621198863 minus 11991123)2

(36)

Based on definition (27) and using inequalities (36) and(11) in Lemma 3 (35) can be rewritten as

le minus 3sum119894=1

120574119894 log 1198962119886119894(1198962119886119894 minus 1199112119894 ) minus3sum119894=1

1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 121198632 (119909 119905)+ 3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (37)

Theorem 5 A class of hydraulic servo-system is described by(1)ndash(3) under Assumptions 1 and 2 If the initial displacement1199091(0) the initial velocity 1199092(0) and the initial acceleration1199093(0) satisfy |119909119894(0)| le 119896119887119894(0) 119894 = 1 2 3 then the proposedcontrol method has the following properties

(1) The error signals are bounded in the following sets

119911119894 (119905) isin Ω119911119894 119894 = 1 2 3 (38)

where Ω119911119894 fl 119911119894 isin R |119911119894(119905)| le 119896119886119894(119905)radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595noting that the tracking error signal converges to zero asymp-totically

(2) The time-varying state constrains are never violatedthat is |119909119894(119905)| le 119896119887119894(119905) forall119905 ge 0 119894 = 1 2 3

(3) All singles of the hydraulic servo-system are bounded

Proof Please see the Appendix

4 Simulation Example

Simulation has been performed to demonstrate the per-formance of the proposed approach The values of systemparameters which are used in the simulation are given inTable 1

The initial states of the hydraulic servo-system are givenas 1199091(0) = 0 1199091(0) = 0 and 1199091(0) = 0 The time-varyingconstraint functions are 1198961198871(119905) = 2 sin(5119905) + 40 1198961198872(119905) =2 sin(5119905) + 60 and 1198961198873(119905) = 4 sin(5119905) + 150 The desiredtrajectory tracking periodic reciprocation of the displace-ment is given 119910119889 = 40 arctan(sin(04120587119905))(1 minus 119890minus119905) with 119905 isin[0 119905119891] and 119905119891 = 50 s The modeling error disturbance of the

Table 1 The system parameters used in the simulation

Name Value119898(kg) 2200119875119904 (Pa) 20119875119903 (Pa) 0119860 (m3) 2 times 10minus3120588 (Nsm) 5001198811198681 (m3) 5 times 10minus41198811198682 (m3) 5 times 10minus3119861V (Pa) 200119862119898 (m5Ns) 2 times 10minus5119870119891 (m3Ns) 8 times 10minus2both chambers is 119908 = sin(119905) + 2 We choose the followingcontroller1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 1199101198891205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092 + 1198792 Φ2+ 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123))

(39)

where we choose design parameters as 1198840 = 20 1198841 = 101198842 = 50 1205741 = 10 1205742 = 3 1205743 = 6 1205751 = 08 1205752 = 07 1205753 = 06the initial weight vector estimation 1 = minus100 2 = 203 = 10 Θ1 = 3 Θ2 = 2 Θ3 = 2 1205901 = 2 1205902 = 03 1205903 = 05Φ1 = 02 Φ2 = 04 Φ3 = 006

Figures 1ndash6 are illustrated to show the simulation resultsFigures 1ndash3 show the better tracking trajectories performanceof the system variable state and minor error curve of theerrors on the time-varying constraints respectively It can beeasily obtained that the time-varying state constrains are notviolated In Figure 4 we can clearly get the conclusion that thetracking errors are all bounded and the tracking error signalcurve converges to zero Figures 5 and 6 show the trajectoriesof estimation and the controller respectively We make theconclusion that the adaptive laws and the controller of thesystem are all ensured boundedness

5 Conclusion

An adaptive control algorithm for hydraulic servo-systemconsidering time-varying state constraints has been designedin this paper The uncertainty and time-varying stateconstraints problem of the system have been consideredin the controller designing process The uncertainties andthe accurate estimation of disturbance have been solved byRBFNN By designing barrier Lyapunov function to solve thetime-varying state constraints problemof system the stabilityof the control strategy has been proved Besides the proposedmethod has proved that the error signals are bounded in the

Complexity 7

0 10 20 30 40 50minus50

0

50

Time (s)

kb1x1

yd

minuskb1

(a)

Time (s)0 10 20 30 40 50

minus40

minus20

0

20

40

ka1z1minuska1

(b)

Figure 1 (a) Trajectories of 1199091 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199111

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

kb2x2 minuskb2

yd

(a)

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

ka2z2minuska2

(b)

Figure 2 (a) Trajectories of 1199092 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199112small compacts sets noting that the tracking error signal con-verges to zero asymptotically the time-varying state con-strains are never violated and all singles of the hydraulicservo-system are bounded The experimental results haveshowed the effectiveness of the proposed method

Appendix

Proof ofTheorem 5 Based on the state constrains (10) and thedefinition of errors 119911119894 the initial error signals are boundedbetween the sets as |119911119894(0)| le 119896119886119894(0) From (37) the followingcan be obtained

le minus120595119881 + 119862 (A1)

where

120595 = min 21205741 21205742 2 119862 = 121198632 (119909 119905) +

3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (A2)

Let us multiply both sides of the above inequality by 119890120595119905(A1) can be rewritten as

119890120595119905 + 120595119881119890120595119905 le 119862119890120595119905 (A3)

And thenwe integrate both sides of inequality (A3) over [0 119905]to obtain

0 le 119881 (119905) le 119881 (0) 119890minus120595119905 + 119862120595 (A4)

Substituting (16) into (A4) we can get

log11989621198861198941198962119886119894 minus 1199112119894 le 119881 (119905) le 119881 (0) 119890minus120595119905 +

119862120595 (A5)

Thus we have the following inequality based on the transitiv-ity of the inequality from (A5)

11989621198861198941198962119886119894 minus 1199112119894 le 119890119881(0)119890minus120595119905+119862120595 (A6)

Then (A6) can be rewritten as

1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 le 119896119886119894 (119905) radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595 (A7)

Furthermore it can be easy proved that the error signal 119911119894converges to zero asymptotically based on appropriate para-meter design

Based on Assumption 2 we can get |119910119889(119905)| le 1198840(119905) and|119910(119894)119889(119905)| le 119884119894+1(119905) 119894 = 1 2 Then from |1199111| le 1198961198861 and

8 Complexity

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

kb3x3 minuskb3

yd

(a)

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

ka3z3minuska3

(b)

Figure 3 (a) Trajectories of 1199093 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199113

minus5

0

5

minus50

0

50minus150

minus100

minus50

0

50

100

z 3

z2z1

Figure 4 The phase portrait of tracking errors 1199111 1199111 and 1199113

0 10 20 30 40 50minus300

minus200

minus100

Time (s)

W1

(a)

0 10 20 30 40 50minus500

0

500

Time (s)

W2

(b)

0 10 20 30 40 500

10

20

Time (s)

W3

(c)

Figure 5 (a) The trajectory of 1 (b) the trajectory of 2 (c) the trajectory of 3

1199091 = 1199111 + 119910119889 we can get the inequality |1199091(119905)| le 1198961198861(119905) +1198840(119905) le 1198961198871(119905) In (18) the victual controller 1205721 can bebounded from the boundedness of 1199091 and 119910119889 It can be seenthat there exists the supremum 1205721 of 1205721 From |1199112| le 1198961198862and 1199092 = 1199112 + 1205721 it has |1199092| le 1198961198862 + 1205721 le 1198961198872 In the sameway we can get the boundedness supremum 1205722 based on the

definition of 1205722 in (19) Thus the state 1199093 can be proved bythe function 1198961198873 from the above proof of boundedness So thetime-varying state constrains are never violated

From (A4) and adaptive laws in (28) (30) and (32)we can get that the neural network weight vector errors 119894are bounded Because the weight vectors 119882lowast119894 are bounded

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Complexity

Using Youngrsquos inequality in the last two terms of weseem to easily get

minus120590119894119879119894 119894 = minus120590119894119879119894 (119894 +119882lowast119894 )le minus120590119894 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817minus119882lowast119894 10038171003817100381710038172+ 1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172

= minus1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 119863 (119909 119905) 1199113(11989621198863 minus 11991123) le

121198632 (119909 119905) + 119911232 (11989621198863 minus 11991123)2

(36)

Based on definition (27) and using inequalities (36) and(11) in Lemma 3 (35) can be rewritten as

le minus 3sum119894=1

120574119894 log 1198962119886119894(1198962119886119894 minus 1199112119894 ) minus3sum119894=1

1205901198942 10038171003817100381710038171003817119894100381710038171003817100381710038172 + 121198632 (119909 119905)+ 3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (37)

Theorem 5 A class of hydraulic servo-system is described by(1)ndash(3) under Assumptions 1 and 2 If the initial displacement1199091(0) the initial velocity 1199092(0) and the initial acceleration1199093(0) satisfy |119909119894(0)| le 119896119887119894(0) 119894 = 1 2 3 then the proposedcontrol method has the following properties

(1) The error signals are bounded in the following sets

119911119894 (119905) isin Ω119911119894 119894 = 1 2 3 (38)

where Ω119911119894 fl 119911119894 isin R |119911119894(119905)| le 119896119886119894(119905)radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595noting that the tracking error signal converges to zero asymp-totically

(2) The time-varying state constrains are never violatedthat is |119909119894(119905)| le 119896119887119894(119905) forall119905 ge 0 119894 = 1 2 3

(3) All singles of the hydraulic servo-system are bounded

Proof Please see the Appendix

4 Simulation Example

Simulation has been performed to demonstrate the per-formance of the proposed approach The values of systemparameters which are used in the simulation are given inTable 1

The initial states of the hydraulic servo-system are givenas 1199091(0) = 0 1199091(0) = 0 and 1199091(0) = 0 The time-varyingconstraint functions are 1198961198871(119905) = 2 sin(5119905) + 40 1198961198872(119905) =2 sin(5119905) + 60 and 1198961198873(119905) = 4 sin(5119905) + 150 The desiredtrajectory tracking periodic reciprocation of the displace-ment is given 119910119889 = 40 arctan(sin(04120587119905))(1 minus 119890minus119905) with 119905 isin[0 119905119891] and 119905119891 = 50 s The modeling error disturbance of the

Table 1 The system parameters used in the simulation

Name Value119898(kg) 2200119875119904 (Pa) 20119875119903 (Pa) 0119860 (m3) 2 times 10minus3120588 (Nsm) 5001198811198681 (m3) 5 times 10minus41198811198682 (m3) 5 times 10minus3119861V (Pa) 200119862119898 (m5Ns) 2 times 10minus5119870119891 (m3Ns) 8 times 10minus2both chambers is 119908 = sin(119905) + 2 We choose the followingcontroller1205721 = minus (1205741 + 1205741 (119905)) 1199111 + 1199101198891205722 = minus (1205742 + 1205742 (119905)) 1199112 + 1 minus 11989621198862 minus 1199112211989621198861 minus 11991121 1199111119906 = 11198793 Φ3119892 (minus (1205743 + 1205743 (119905)) 1199113 + 1198791 Φ11199092 + 1198792 Φ2+ 2 minus (11989621198863 minus 11991123)(11989621198862 minus 11991122)1199112 minus

11991132 (11989621198863 minus 11991123))

(39)

where we choose design parameters as 1198840 = 20 1198841 = 101198842 = 50 1205741 = 10 1205742 = 3 1205743 = 6 1205751 = 08 1205752 = 07 1205753 = 06the initial weight vector estimation 1 = minus100 2 = 203 = 10 Θ1 = 3 Θ2 = 2 Θ3 = 2 1205901 = 2 1205902 = 03 1205903 = 05Φ1 = 02 Φ2 = 04 Φ3 = 006

Figures 1ndash6 are illustrated to show the simulation resultsFigures 1ndash3 show the better tracking trajectories performanceof the system variable state and minor error curve of theerrors on the time-varying constraints respectively It can beeasily obtained that the time-varying state constrains are notviolated In Figure 4 we can clearly get the conclusion that thetracking errors are all bounded and the tracking error signalcurve converges to zero Figures 5 and 6 show the trajectoriesof estimation and the controller respectively We make theconclusion that the adaptive laws and the controller of thesystem are all ensured boundedness

5 Conclusion

An adaptive control algorithm for hydraulic servo-systemconsidering time-varying state constraints has been designedin this paper The uncertainty and time-varying stateconstraints problem of the system have been consideredin the controller designing process The uncertainties andthe accurate estimation of disturbance have been solved byRBFNN By designing barrier Lyapunov function to solve thetime-varying state constraints problemof system the stabilityof the control strategy has been proved Besides the proposedmethod has proved that the error signals are bounded in the

Complexity 7

0 10 20 30 40 50minus50

0

50

Time (s)

kb1x1

yd

minuskb1

(a)

Time (s)0 10 20 30 40 50

minus40

minus20

0

20

40

ka1z1minuska1

(b)

Figure 1 (a) Trajectories of 1199091 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199111

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

kb2x2 minuskb2

yd

(a)

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

ka2z2minuska2

(b)

Figure 2 (a) Trajectories of 1199092 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199112small compacts sets noting that the tracking error signal con-verges to zero asymptotically the time-varying state con-strains are never violated and all singles of the hydraulicservo-system are bounded The experimental results haveshowed the effectiveness of the proposed method

Appendix

Proof ofTheorem 5 Based on the state constrains (10) and thedefinition of errors 119911119894 the initial error signals are boundedbetween the sets as |119911119894(0)| le 119896119886119894(0) From (37) the followingcan be obtained

le minus120595119881 + 119862 (A1)

where

120595 = min 21205741 21205742 2 119862 = 121198632 (119909 119905) +

3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (A2)

Let us multiply both sides of the above inequality by 119890120595119905(A1) can be rewritten as

119890120595119905 + 120595119881119890120595119905 le 119862119890120595119905 (A3)

And thenwe integrate both sides of inequality (A3) over [0 119905]to obtain

0 le 119881 (119905) le 119881 (0) 119890minus120595119905 + 119862120595 (A4)

Substituting (16) into (A4) we can get

log11989621198861198941198962119886119894 minus 1199112119894 le 119881 (119905) le 119881 (0) 119890minus120595119905 +

119862120595 (A5)

Thus we have the following inequality based on the transitiv-ity of the inequality from (A5)

11989621198861198941198962119886119894 minus 1199112119894 le 119890119881(0)119890minus120595119905+119862120595 (A6)

Then (A6) can be rewritten as

1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 le 119896119886119894 (119905) radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595 (A7)

Furthermore it can be easy proved that the error signal 119911119894converges to zero asymptotically based on appropriate para-meter design

Based on Assumption 2 we can get |119910119889(119905)| le 1198840(119905) and|119910(119894)119889(119905)| le 119884119894+1(119905) 119894 = 1 2 Then from |1199111| le 1198961198861 and

8 Complexity

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

kb3x3 minuskb3

yd

(a)

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

ka3z3minuska3

(b)

Figure 3 (a) Trajectories of 1199093 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199113

minus5

0

5

minus50

0

50minus150

minus100

minus50

0

50

100

z 3

z2z1

Figure 4 The phase portrait of tracking errors 1199111 1199111 and 1199113

0 10 20 30 40 50minus300

minus200

minus100

Time (s)

W1

(a)

0 10 20 30 40 50minus500

0

500

Time (s)

W2

(b)

0 10 20 30 40 500

10

20

Time (s)

W3

(c)

Figure 5 (a) The trajectory of 1 (b) the trajectory of 2 (c) the trajectory of 3

1199091 = 1199111 + 119910119889 we can get the inequality |1199091(119905)| le 1198961198861(119905) +1198840(119905) le 1198961198871(119905) In (18) the victual controller 1205721 can bebounded from the boundedness of 1199091 and 119910119889 It can be seenthat there exists the supremum 1205721 of 1205721 From |1199112| le 1198961198862and 1199092 = 1199112 + 1205721 it has |1199092| le 1198961198862 + 1205721 le 1198961198872 In the sameway we can get the boundedness supremum 1205722 based on the

definition of 1205722 in (19) Thus the state 1199093 can be proved bythe function 1198961198873 from the above proof of boundedness So thetime-varying state constrains are never violated

From (A4) and adaptive laws in (28) (30) and (32)we can get that the neural network weight vector errors 119894are bounded Because the weight vectors 119882lowast119894 are bounded

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 7

0 10 20 30 40 50minus50

0

50

Time (s)

kb1x1

yd

minuskb1

(a)

Time (s)0 10 20 30 40 50

minus40

minus20

0

20

40

ka1z1minuska1

(b)

Figure 1 (a) Trajectories of 1199091 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199111

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

kb2x2 minuskb2

yd

(a)

0 10 20 30 40 50minus100

minus50

0

50

100

Time (s)

ka2z2minuska2

(b)

Figure 2 (a) Trajectories of 1199092 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199112small compacts sets noting that the tracking error signal con-verges to zero asymptotically the time-varying state con-strains are never violated and all singles of the hydraulicservo-system are bounded The experimental results haveshowed the effectiveness of the proposed method

Appendix

Proof ofTheorem 5 Based on the state constrains (10) and thedefinition of errors 119911119894 the initial error signals are boundedbetween the sets as |119911119894(0)| le 119896119886119894(0) From (37) the followingcan be obtained

le minus120595119881 + 119862 (A1)

where

120595 = min 21205741 21205742 2 119862 = 121198632 (119909 119905) +

3sum119894=1

1205901198942 1003817100381710038171003817119882lowast119894 10038171003817100381710038172 (A2)

Let us multiply both sides of the above inequality by 119890120595119905(A1) can be rewritten as

119890120595119905 + 120595119881119890120595119905 le 119862119890120595119905 (A3)

And thenwe integrate both sides of inequality (A3) over [0 119905]to obtain

0 le 119881 (119905) le 119881 (0) 119890minus120595119905 + 119862120595 (A4)

Substituting (16) into (A4) we can get

log11989621198861198941198962119886119894 minus 1199112119894 le 119881 (119905) le 119881 (0) 119890minus120595119905 +

119862120595 (A5)

Thus we have the following inequality based on the transitiv-ity of the inequality from (A5)

11989621198861198941198962119886119894 minus 1199112119894 le 119890119881(0)119890minus120595119905+119862120595 (A6)

Then (A6) can be rewritten as

1003816100381610038161003816119911119894 (119905)1003816100381610038161003816 le 119896119886119894 (119905) radic1 minus 119890minus119881(0)119890minus120595119905minus119862120595 (A7)

Furthermore it can be easy proved that the error signal 119911119894converges to zero asymptotically based on appropriate para-meter design

Based on Assumption 2 we can get |119910119889(119905)| le 1198840(119905) and|119910(119894)119889(119905)| le 119884119894+1(119905) 119894 = 1 2 Then from |1199111| le 1198961198861 and

8 Complexity

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

kb3x3 minuskb3

yd

(a)

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

ka3z3minuska3

(b)

Figure 3 (a) Trajectories of 1199093 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199113

minus5

0

5

minus50

0

50minus150

minus100

minus50

0

50

100

z 3

z2z1

Figure 4 The phase portrait of tracking errors 1199111 1199111 and 1199113

0 10 20 30 40 50minus300

minus200

minus100

Time (s)

W1

(a)

0 10 20 30 40 50minus500

0

500

Time (s)

W2

(b)

0 10 20 30 40 500

10

20

Time (s)

W3

(c)

Figure 5 (a) The trajectory of 1 (b) the trajectory of 2 (c) the trajectory of 3

1199091 = 1199111 + 119910119889 we can get the inequality |1199091(119905)| le 1198961198861(119905) +1198840(119905) le 1198961198871(119905) In (18) the victual controller 1205721 can bebounded from the boundedness of 1199091 and 119910119889 It can be seenthat there exists the supremum 1205721 of 1205721 From |1199112| le 1198961198862and 1199092 = 1199112 + 1205721 it has |1199092| le 1198961198862 + 1205721 le 1198961198872 In the sameway we can get the boundedness supremum 1205722 based on the

definition of 1205722 in (19) Thus the state 1199093 can be proved bythe function 1198961198873 from the above proof of boundedness So thetime-varying state constrains are never violated

From (A4) and adaptive laws in (28) (30) and (32)we can get that the neural network weight vector errors 119894are bounded Because the weight vectors 119882lowast119894 are bounded

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Complexity

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

kb3x3 minuskb3

yd

(a)

0 10 20 30 40 50minus200

minus100

0

100

200

Time (s)

ka3z3minuska3

(b)

Figure 3 (a) Trajectories of 1199093 (dashed) and 119910119889 (solid) (b) the trajectory of tracking error 1199113

minus5

0

5

minus50

0

50minus150

minus100

minus50

0

50

100

z 3

z2z1

Figure 4 The phase portrait of tracking errors 1199111 1199111 and 1199113

0 10 20 30 40 50minus300

minus200

minus100

Time (s)

W1

(a)

0 10 20 30 40 50minus500

0

500

Time (s)

W2

(b)

0 10 20 30 40 500

10

20

Time (s)

W3

(c)

Figure 5 (a) The trajectory of 1 (b) the trajectory of 2 (c) the trajectory of 3

1199091 = 1199111 + 119910119889 we can get the inequality |1199091(119905)| le 1198961198861(119905) +1198840(119905) le 1198961198871(119905) In (18) the victual controller 1205721 can bebounded from the boundedness of 1199091 and 119910119889 It can be seenthat there exists the supremum 1205721 of 1205721 From |1199112| le 1198961198862and 1199092 = 1199112 + 1205721 it has |1199092| le 1198961198862 + 1205721 le 1198961198872 In the sameway we can get the boundedness supremum 1205722 based on the

definition of 1205722 in (19) Thus the state 1199093 can be proved bythe function 1198961198873 from the above proof of boundedness So thetime-varying state constrains are never violated

From (A4) and adaptive laws in (28) (30) and (32)we can get that the neural network weight vector errors 119894are bounded Because the weight vectors 119882lowast119894 are bounded

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 9

0 10 20 30 40 50minus200

0

200

400

600

800

1000

1200

u

Time (s)

Figure 6 The trajectory of 119906the weight vector estimations 119894 = 119894 + 119882lowast119894 are boundedBased on the above identified process the actual controller 119906consists of the bounded functions119910119889 119910119889 119910(2)119889 119910(3)

119889 1205721 1205722 and119911119894 119894 = 1 2 3Then it is easy to obtain that 119906 is boundedThus

all the signals of the hydraulic servo-system are boundedThe proof is completed

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The work is supported by the National Natural ScienceFoundation of China (61603164 61473139 and 61622303) andthe project for Distinguished Professor of Liaoning Province

References

[1] W He Y Dong and C Sun ldquoAdaptive neural impedance con-trol of a roboticmanipulator with input saturationrdquo IEEETrans-actions on Systems Man and Cybernetics Systems vol 46 no3 pp 334ndash344 2016

[2] L Chen and Q Wang ldquoAdaptive robust control for a classof uncertain MIMO non-affine nonlinear systemsrdquo IEEECAAJournal of Automatica Sinica vol 3 no 1 pp 105ndash112 2016

[3] X Fu Y Kang and P Li ldquoSampled-data observer design for aclass of stochastic nonlinear systems based on the approximatediscrete-time modelsrdquo IEEECAA Journal of Automatica Sinicavol 4 no 3 pp 507ndash511 2017

[4] W He S S Ge B V How Y S Choo and K-S Hong ldquoRobustadaptive boundary control of a flexible marine riser with vesseldynamicsrdquo Automatica A Journal of IFAC the InternationalFederation of Automatic Control vol 47 no 4 pp 722ndash732 2011

[5] H C Lu and W C Lin ldquoRobust controller with disturbancerejection for hydraulic servo systemsrdquo IEEE Transactions onIndustrial Electronics vol 40 no 1 pp 157ndash162 1993

[6] M Chen ldquoRobust tracking control for self-balancing mobilerobots using disturbance observerrdquo IEEECAA Journal of Auto-matica Sinica vol 4 no 3 pp 458ndash465 2017

[7] C Yang Y Jiang Z Li W He and C-Y Su ldquoNeural controlof bimanual robots with guaranteed global stability and motionprecisionrdquo IEEE Transactions on Industrial Informatics 2017

[8] W He S Nie T Meng and Y-J Liu ldquoModeling and vibrationcontrol for a moving beam with application in a drilling riserrdquoIEEE Transactions on Control Systems Technology vol 25 no 3pp 1036ndash1043 2017

[9] M Yue L J Wang and T Ma ldquoNeural network based terminalsliding mode control for WMRs affected by an augmentedground friction with slippage effectrdquo IEEECAA Journal ofAutomatica Sinica vol 4 no 3 pp 498ndash506 2017

[10] HWang P X Liu S Li and DWang ldquoAdaptive neural output-feedback control for a class of nonlower triangular nonlinearsystems with unmodeled dynamicsrdquo IEEE Transactions onNeural Networks and Learning Systems pp 1ndash11

[11] N Aguila-Camacho and M A Duarte-Mermoud ldquoImprovingthe control energy inmodel reference adaptive controllers usingfractional adaptive lawsrdquo IEEECAA Journal of AutomaticaSinica vol 3 no 3 pp 332ndash337 2016

[12] C Yang X Wang L Cheng and H Ma ldquoNeural-learning-based telerobot control with guaranteed performancerdquo IEEETransactions on Cybernetics 2017

[13] B Xu Z Shi C Yang and F Sun ldquoComposite neural dynamicsurface control of a class of uncertain nonlinear systems instrict-feedback formrdquo IEEE Transactions on Cybernetics vol 44no 12 pp 2626ndash2634 2014

[14] H Wang Z Wang Y-J Liu and S Tong ldquoFuzzy trackingadaptive control of discrete-time switched nonlinear systemsrdquoFuzzy Sets and Systems An International Journal in InformationScience and Engineering vol 316 pp 35ndash48 2017

[15] S Tong L Zhang and Y Li ldquoObserved-based adaptive fuzzydecentralized tracking control for switched uncertain nonlinearlarge-scale systems with dead zonesrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 1 pp 37ndash472015

[16] M Chen and G Tao ldquoAdaptive fault-tolerant control of uncer-tain nonlinear large-scale systems with unknown dead zonerdquoIEEE Transactions on Cybernetics vol 46 no 8 pp 1851ndash18622016

[17] S Sui Y Li and S Tong ldquoObserver-based adaptive fuzzycontrol for switched stochastic nonlinear systems with partialtracking errors constrainedrdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 46 no 12 pp 1605ndash16172016

[18] HWang H R Karimi P X Liu and H Yang ldquoAdaptive neuralcontrol of nonlinear systems with unknown control directionsand input dead-zonerdquo IEEE Transactions on Systems Man andCybernetics Systems pp 1ndash11

[19] Y-J Liu S Li S Tong and C L Chen ldquoNeural approximation-based adaptive control for a class of nonlinear nonstrictfeedback discrete-time systemsrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 28 no 7 pp 1531ndash1541 2017

[20] Z Wang L Liu H Zhang and G Xiao ldquoFault-tolerant con-troller design for a class of nonlinear MIMO discrete-time sys-tems via online reinforcement learning algorithmrdquo IEEE Trans-actions on Systems Man and Cybernetics Systems vol 46 no5 pp 611ndash622 2016

[21] Z LiHXiaoC Yang andY Zhao ldquoModel predictive control ofnonholonomic chained systems using general projection neuralnetworks optimizationrdquo IEEE Transactions on Systems Manand Cybernetics Systems vol 45 no 10 pp 1313ndash1321 2015

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Complexity

[22] B Xu C Yang and Z Shi ldquoReinforcement learning outputfeedback NN control using deterministic learning techniquerdquoIEEE Transactions on Neural Networks and Learning Systemsvol 25 no 3 pp 635ndash641 2014

[23] F Wang B Chen C Lin J Zhang and X Meng ldquoAdaptiveneural network finite-time output feedback control of quantizednonlinear systemsrdquo IEEE Transactions on Cybernetics pp 1ndash10

[24] HWang P X Liu and S Liu ldquoAdaptive neural synchronizationcontrol for bilateral teleoperation systems with time delayand backlash-like hysteresisrdquo IEEE Transactions on Cybernetics2017

[25] M Chen and S Ge ldquoAdaptive neural output feedback controlof uncertain nonlinear systems with unknown hysteresis usingdisturbance observerrdquo IEEE Transactions on Industrial Electron-ics 2015

[26] F Wang B Chen C Lin and X Li ldquoDistributed adaptiveneural control for stochastic nonlinear multiagent systemsrdquoIEEE Transactions on Cybernetics vol 47 no 7 pp 1795ndash18032017

[27] Z Liu C Chen Y Zhang and C L P Chen ldquoAdaptive neuralcontrol for dual-arm coordination of humanoid robot withunknown nonlinearities in output mechanismrdquo IEEE Transac-tions on Cybernetics vol 45 no 3 pp 521ndash532 2015

[28] J y Yao Z x Jiao D w Ma and L Yan ldquoHigh-accuracy track-ing control of hydraulic rotary actuators with modelling uncer-taintiesrdquo IEEEASME Transactions on Mechatronics vol 19 no2 pp 633ndash641 2014

[29] W He S Zhang and S S Ge ldquoRobust adaptive control of athruster assisted positionmooring systemrdquoAutomatica A Jour-nal of IFAC the International Federation of Automatic Controlvol 50 no 7 pp 1843ndash1851 2014

[30] B Yao F p Bu J Reedy and C T C Chiu ldquoAdaptive robustmotion control of single-rod hydraulic actuators theory andexperimentsrdquo IEEEASME Transactions on Mechatronics vol 5no 1 pp 79ndash91 2000

[31] J Y Yao Z X Jiao and D W Ma ldquoExtended-state-observer-based output feedback nonlinear robust control of hydraulicsystems with backsteppingrdquo IEEE Transactions on IndustrialElectronics vol 61 no 11 pp 6285ndash6293 2014

[32] J Yao W Deng and Z Jiao ldquoAdaptive control of hydraulicactuators with LuGre model-based friction compensationrdquoIEEE Transactions on Industrial Electronics vol 62 no 10 pp6469ndash6477 2015

[33] S Tong J Fang and Y Zhang ldquoOutput tracking control of ahydrogen-air PEM fuel cellrdquo IEEECAA Journal of AutomaticaSinica vol 4 no 2 pp 273ndash279 2017

[34] Z Fu W Xie S Rakheja and J Na ldquoObserver-based adaptiveoptimal control for unknown singularly perturbed nonlinearsystems with input constraintsrdquo IEEECAA Journal of Automat-ica Sinica vol 4 no 1 pp 48ndash57 2017

[35] Z J Li S T Xiao S Z S Ge and H Su ldquoConstrained multi-legged robot systemmodeling and fuzzy control with uncertainkinematics and dynamics incorporating foot force optimiza-tionrdquo IEEE Transactions on Systems Man amp Cybernetics Sys-tems vol 46 no 1 pp 1ndash15 2016

[36] K P Tee S S Ge and E H Tay ldquoBarrier lyapunov functions forthe control of output-constrained nonlinear systemsrdquoAutomat-ica A Journal of IFAC the International Federation of AutomaticControl vol 45 no 4 pp 918ndash927 2009

[37] K P Tee B Ren and S S Ge ldquoControl of nonlinear systemswith time-varying output constraintsrdquoAutomatica A Journal of

IFAC the International Federation of Automatic Control vol 47no 11 pp 2511ndash2516 2011

[38] Z Liu G Lai Y Zhang andC L Chen ldquoAdaptive neural outputfeedback control of output-constrained nonlinear systems withunknown output nonlinearityrdquo IEEE Transactions on NeuralNetworks and Learning Systems vol 26 no 8 pp 1789ndash18022015

[39] Z Li J Deng R Lu Y Xu J Bai andC Su ldquoTrajectory-trackingcontrol of mobile robot systems incorporating neural-dynamicoptimized model predictive approachrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 46 no 6 pp 740ndash749 2016

[40] H Li L Bai L Wang Q Zhou and H Wang ldquoAdaptive neuralcontrol of uncertain nonstrict-feedback stochastic nonlinearsystems with output constraint and unknown dead zonerdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2048ndash2059 2017

[41] D-J Li and D-P Li ldquoAdaptive controller design-based neuralnetworks for output constraint continuous stirred tank reactorrdquoNeurocomputing vol 153 pp 159ndash163 2015

[42] W He S Zhang and S S Ge ldquoAdaptive control of a flexiblecrane system with the boundary output constraintrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4126ndash4133 2014

[43] Y J Liu S M Lu and S C Tong ldquoNeural network controllerdesign for an uncertain robot with time-varying output con-straintrdquo IEEE Transactions on Systems Man and CyberneticsSystems vol 47 no 8 pp 2060ndash2068 2017

[44] Z Li Q GeW Ye and P Yuan ldquoDynamic balance optimizationand control of quadruped robot systems with flexible jointsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 46 no 10 pp 1338ndash1351 2016

[45] M Chen ldquoConstrained control allocation for overactuated air-craft using a neurodynamic modelrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1630ndash1641 2016

[46] W He and Y Dong ldquoAdaptive fuzzy neural network control fora constrained robot using impedance learningrdquo IEEE Transac-tions on Neural Networks and Learning Systems no 99 pp 1ndash132017

[47] D P Li and D J Li ldquoAdaptive neural tracking control foran uncertain state constrained robotic manipulator with time-varying delaysrdquo IEEE Transactions on Systems Man and Cyber-netics Systems 2017

[48] D P Li D J Li Y J Liu S Tong and C L P Chen ldquoApproxi-mation-based adaptive neural tracking control of nonlinearmimo unknown time-varying delay systems with full state con-straintsrdquo IEEE Transactions on Cybernetics vol 47 no 10 pp3100ndash3109 2017

[49] D Li and D Li ldquoAdaptive neural tracking control for nonlineartime-delay systems with full state constraintsrdquo IEEE Transac-tions on Systems Man and Cybernetics Systems vol 47 no 7pp 1590ndash1601 2017

[50] Z-L Tang S S Ge K P Tee and W He ldquoRobust adaptiveneural tracking control for a class of perturbed uncertain non-linear systems with state constraintsrdquo IEEE Transactions on Sys-tems Man and Cybernetics Systems vol 46 no 12 pp 1618ndash1629 2016

[51] Y-J Liu and S Tong ldquoBarrier Lyapunov functions for Nuss-baum gain adaptive control of full state constrained nonlinearsystemsrdquo Automatica A Journal of IFAC the International Fed-eration of Automatic Control vol 76 pp 143ndash152 2017

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 11

[52] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-basedadaptive control for a class of nonlinear pure-feedback systemswith full state constraintsrdquo Automatica A Journal of IFAC theInternational Federation of Automatic Control vol 64 pp 70ndash75 2016

[53] Y J Liu S Tong C L P Chen and D J Li ldquoAdaptiveNN control using integral barrier lyapunov functionals foruncertain nonlinear block-triangular constraint systemsrdquo IEEETransactions on Cybernetics

[54] Y J Liu S M Lu D J Li and S C Tong ldquoAdaptive controllerdesign-based ABLF for a class of nonlinear time-varying stateconstraint systemsrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 47 no 7 pp 1546ndash1553 2017

[55] Y-J Liu and S Tong ldquoBarrier Lyapunov functions-based adapt-ive control for a class of nonlinear pure-feedback systems withfull state constraintsrdquo Automatica vol 64 pp 70ndash75 2016

[56] L Ding S Li Y J Liu H B Gao C Chen and Z Q DengldquoAdaptive neural network-based tracking control for full-stateconstrained wheeled mobile robotic systemrdquo IEEE Transactionson Systems Man and Cybernetics Systems vol 47 no 8 pp2410ndash2419 2017

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

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 201

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

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of