A robust self-learning PID control system design for nonlinear systems using a particle optimization

ORIGINAL ARTICLEA robust self-learning PID control system design for nonlinear systems using a particle swarm optimization algorithmChih-Min Lin •Ming-Chia Li •Ang-Bung Ting •Ming-Hung LinReceived:23February 2010/Accepted:18May 2011/Published online:22June 2011ÓSpringer-Verlag 2011Abstract This study presents a robust self-learning pro-portional-integral-derivative (RSPID)control system design for nonlinear systems.This RSPID control system comprises a self-learning PID (SPID)controller and a robust controller.The gradient descent method is utilized to derive the on-line tuning laws of SPID controller;and the H 1control technique is applied for the robust con-troller design so as to achieve robust tracking performance.Moreover,in order to achieve fast learning of PID con-troller,a particle swarm optimization (PSO)algorithm is adopted to search the optimal learning-rates of PID adap-tive gains.Finally,two nonlinear systems,a two-link manipulator and a chaotic system are examined to illustrate the effectiveness of the proposed control algorithm.Sim-ulation results show that the proposed control system can achieve favorable control performance for these nonlinear systems.Keywords PID control ÁParticle swarm optimization (PSO)ÁH ?control1IntroductionThe proportional-integral-derivative (PID)controller has been practically applied in industries for 60years due to its simple architecture and easy design properties.Until today,the PID controller is still used in different control appli-cations,even though lots of new control techniques have been proposed.However,the traditional PID controller needs some manual tuning before it is used to practical application in industries.When the PID controllers are applied to complicated systems such as nonlinear systems,different tuning algorithms of PID controllers have been proposed [1–5].In recent decade,several intelligent tuning algorithms have been applied to tune the PID controllers.The PID controller automatic tuning methods have been proposed by using a genetic algorithm [6],an immune algorithm [7]and a fuzzy-genetic algorithm [8].However,the learning speed of these algorithms is slow;thus,they are not suitable for real time control systems.Harinath and Mann [9]proposed a fuzzy PID controller for multivariable process systems.However,it takes a two-level tuning algorithm;thus,it also takes too much computation time.Therefore,in this paper,another simple optimization algorithm called particle swam optimization (PSO)algo-rithm will be used for the optimal parameter search of the PID controller.The PSO algorithm,a new evolutionary computation technique,is proposed by Kennedy and Eberhart [10].It was developed by the research of the social behavior of animals,e.g.,bird flocking.Unlike a typical GA,PSO algorithms have memorial capability without complicated evolutionary process such as crossover and mutation in GA,and each particle can memorize its best solution.In addition,if another particle discovers a better solution,it will be shared among other particles.The best solution isC.-M.Lin (&)ÁM.-C.Li ÁA.-B.Ting ÁM.-H.LinDepartment of Electrical Engineering,Yuan Ze University,No.135,Yuan-Tung Road,Chung-Li,Tao-Yuan 320,Taiwan,ROCe-mail:cml@.tw M.-C.Lie-mail:s929101@.tw A.-B.Tinge-mail:s968501@.tw M.-H.Line-mail:s958505@.twInt.J.Mach.Learn.&Cyber.(2011)2:225–234DOI 10.1007/s13042-011-0021-4thus memorized and every particle will move toward this best solution.In past decade,PSO algorithms have been widely used to solve the modeling and control problems of application systems[11–14].Moreover,the PID control based on PSO algorithms have been proposed in[15,16]. However,it is difficult to solve the inequality for the optimal solution in[15];and it has not given the stability analysis in[16].In this paper,a robust self-learning PID(RSPID) control system is proposed for nonlinear systems.This RSPID control system comprises a self-learning PID (SPID)controller and a robust controller.The SPID controller is utilized to approximate an ideal controller, and the robust controller is designed to recover the residual approximation error between the ideal controller and SPID controller.The gradient descent method and H1control technique are utilized to derive the on-line tuning laws of SPID controller and robust controller,so that the robust stability of the system can be obtained. Furthermore,the PSO algorithm is adopted to auto-search the optimal learning-rates of PID controller to increase the learning speed.Finally,two nonlinear systems are presented to support the validity of the proposed control method.This study is organized as follows.Problem formulation is described in Sect.2.The PSO algorithm is briefly reviewed in Sect.3.The design procedure of the proposed RSPID control system is constructed in Sect.4.In Sect.5, simulations are performed to verify the effectiveness of the proposed control method.Finally,conclusions are drawn in Sect.6.2Problem formulationConsider a class of n th-order multi-input multi-output (MIMO)nonlinear systems expressed in the following form:xðnÞðtÞ¼fðxðtÞÞþGðxÞðtÞuðtÞþdðtÞy¼xðtÞð1ÞwhereuðtÞ¼½u1ðtÞ;u2ðtÞ;...;u mðtÞ T2<m the controlinput vectorof the system y¼xðtÞ¼½x1ðtÞ;x2ðtÞ;...;x mðtÞ T2<m the systemoutput vector xðtÞ¼½x TðtÞ;_x TðtÞ;...;xðnÀ1ÞTðtÞ T2<mn the statevector of thesystemfðxðtÞÞ2<m unknownboundednonlinearfunctionGðxðtÞÞ2<mÂm unknownboundednonlinearmatrixdðtÞ¼½d1ðtÞ;d2ðtÞ;...;d mðtÞ T2<m unknownboundedexternaldisturbance.When neglecting the modeling uncertainty and external disturbance,the nominal system of Eq.1can be obtained as xðnÞðtÞ¼f nðxðtÞÞþG n uðtÞð2Þwhere f nðxðtÞÞ2<m is the nominal function of fðxðtÞÞ, and the constant matrix G n¼diagðg n1;g n2;...;g nmÞ2 <mÂm is the nominal functions of GðxðtÞÞ.Without losing generality,assume the constant g ni!0for i¼1...m. Assume that the nonlinear system of Eq.2is controllable and GÀ1n exists for all xðtÞ.If the external disturbance and modeling uncertainty exist,the MIMO nonlinear systems Eq.1can be reformulated asxðnÞðtÞ¼f nðxðtÞÞþG n uðtÞþlðxðtÞ;tÞð3Þwhere lðxðtÞ;tÞis referred to as the lumped uncertainty, including system’s uncertainty and external disturbance.The control problem is tofind a suitable controller for the MIMO nonlinear systems Eq.1so that the system output vector xðtÞcan track desired reference trajectory vector x dðtÞ¼½x d1ðtÞ;x d2ðtÞ;...;x dmðtÞ T2<m closely.A lot of control techniques have been presented to achieve reference trajectory tracking.However,in this paper,a simple adaptive PID control scheme will be pro-posed to deal with the uncertain MIMO nonlinear system. Moreover,the H1control technique will be used to guarantee the robust tracking performance.Define the tracking error aseðtÞD x dðtÞÀxðtÞ2<mð4Þthen the system tracking error vector is defined ase D½e TðtÞ;_e TðtÞ;...;eðnÀ1ÞTðtÞ T2<mnð5ÞIf the system dynamics f nðxðtÞÞand G n,and the lumped uncertainty lðx;tÞare exactly known,an ideal controller can be designed asuÃðtÞ¼GÀ1n½xðnÞdðtÞÀf nðxðtÞÞÀlðxðtÞ;tÞþH T eðtÞ ð6Þwhere H¼½H n;...;H2;H1 T2<mnÂm is the feedback gain matrix which contains real numbers.H i¼diagðh i1;h i2;...;h imÞ2<mÂm is a nonzero positive constant diagonal matrix.Substituting the ideal controller Eq.6 into Eq.3,gives the error dynamic equationeðnÞþH T e¼0ð7ÞIn Eq.7,if H is chosen to correspond to the coefficients of a Hurwitz polynomial,it implies limt!1jj eðtÞjj¼0. However,in practical application,the system uncertainties and external disturbance of nonlinear systems are generally unknown,so that the idea controller uÃin Eq.6is always unobtainable.Thus,a SPID controller is designed to mimic the idea controller.And then,based on the H1control technique,the robust controller is developed to attenuate the effect of the approximation error between SPID controller and the ideal controller so that the robust tracking performance can be achieved.3Robust self-learning PID(RSPID)control system designThe block diagram of the nonlinear control system is shown in Fig.1.The RSPID control system is assumed to take the following form:u RSPIDðtÞ¼u SPIDðtÞþu RðtÞð8Þwhere u SPIDðtÞis a self-learning PID controller utilized to approximate the ideal controller uÃ,and u RðtÞis the robust controller designed to suppress the influence of residual approximation error between the ideal controller and SPID controller.3.1SPID controller designThe SPID controller can be described asu SPIDðtÞ¼^K P eðtÞþ^K IZ t0eðsÞd sþ^K Dd eðtÞdtð9Þwhere^K P;^K I and^K D are the adaptive parameters ofproportional gain,integral gain and derivative gain matri-ces,respectively;and^K P¼diagð^k P1;^k P2;...;^k PmÞ2<mÂm;^KI¼diagð^k I1;^k I2;...;^k ImÞ2<mÂm;^K D¼diagð^k D1;^k D2;...^kDmÞ2<mÂm:An integrated error function is defined assðe;tÞ eðnÀ1ÞþH1eðnÀ2ÞþÁÁÁþH nZ teðsÞd sð10Þwhere sðe;tÞ¼½s1ðtÞ;s2ðtÞ;...;s mðtÞ T.From Eq.9,thecontrol law Eq.8can be rewritten asu RSPIDðtÞ¼u SPIDð^K P;^K I;^K D;tÞþu RðtÞð11ÞTaking the time derivative of both sides of Eq.10andusing Eq.3,it can be obtained that_sðe;tÞ¼eðnÞþH T e¼Àf nðxðtÞÞÀG n uðtÞþxðnÞdÀlðxðtÞ;tÞþH T eð12ÞSubstituting Eq.9into Eq.12and multiplying both sidesby s Tðe;tÞ,yieldss Tðe;tÞ_sðe;tÞ¼Às Tðe;tÞf nðxðtÞÞÀs Tðe;tÞG n½u SPIDð^K P;^K I;^K D;tÞþu RðtÞþs Tðe;tÞðxðnÞdÀlðxðtÞ;tÞþH T eÞð13ÞBy defining12s Tðe;tÞsðe;tÞas a cost function,then itsderivative is s Tðe;tÞ_sðe;tÞ.According to the gradientdescent method,the gains of^K P;^K I and^K D are updatedby the following tuning laws_^kPi¼Àgo s Tðe;tÞ_sðe;tÞSPID io u SPIDiðtÞ^Pig P s iðtÞg ni e iðtÞð14Þ_^kIi¼Àg Io s Tðe;tÞ_sðe;tÞSPID io u SPIDiðtÞ^Ii¼g I s iðtÞg niZ te iðtÞd sð15Þ_^kDi¼Àg Do s Tðe;tÞ_sðe;tÞo u SPIDiðtÞo u SPIDiðtÞo^k Di¼g D s iðtÞg nide iðtÞð16Þwhere u SPIDiis the i th element of u SPID;g P;g I and g D arethe learning-rates,which will be auto-searched by PSOalgorithm.3.2Robust controller designIn case of the existence of an approximation error,the idealcontroller can be reformulated as the summation of SPIDcontroller and the approximation error:uÃðtÞ¼u SPIDð^K P;^K I;^K D;tÞþeðtÞð17Þwhere eðtÞ¼½e1ðtÞ;e2ðtÞ;...;e mðtÞ T2<m denotes the approximation error.Substituting Eq.8into Eq.3,yieldsxðnÞðtÞ¼f nðxðtÞÞþG n½u SPIDðtÞþu RðtÞ þlðxðtÞ;tÞð18ÞFrom Eqs.6,10,18and after some straightforward manipulation,it can be obtained thateðnÞþH T e¼G n½uÃðtÞÀu SPIDðtÞÀu RðtÞ ¼_sðe;tÞð19ÞNow,the robust controller can be developed to attenuate the effect of the approximation error between the ideal controller and SPID controller so that the H1tracking performance can be achieved.In case of the existence of eðtÞ;consider a specified H1tracking performance[17]X m i¼1Z Ts2iðtÞdtX mi¼1½s2ið0Þ=g ni þX mi¼1r2iZ Te2iðtÞdtð20Þwhere r i is a prescribed attenuation constant.The robust controller is designed asu RðtÞ¼ð2R2ÞÀ1ðR2þIÞsðe;tÞð21Þwhere R¼diagðr1;r2;...;r mÞ2<mÂm:Then the follow-ing theorem can be stated and proven.Theorem1:Consider the nth-order MIMO nonlinear systems represented by Eq.1.The RSPID control law is designed as Eq.11,where u SPIDðtÞis given in Eq.9with the on-line parameter tuning algorithms given as Eqs.14–16, and the robust controller is designed as Eq.21.Then the desired H1tracking performance in Eq.20can be achieved for the specified attenuation levels r i;i¼1;2;...;m: Proof:The Lyapunov function is given byVðsðe;tÞÞ¼12s Tðe;tÞsðe;tÞð22ÞTaking the derivative of the Lyapunov function and using Eqs.17,19and21,yields_Vðsðe;tÞÞ¼s Tðe;tÞ_sðe;tÞ¼s Tðe;tÞG n½eðtÞÀð2R2ÞÀ1ðR2þIÞsðe;tÞ¼X mi¼1g ni s iðtÞe iðtÞÀs2iðtÞr2iþ12r2i!¼X mi¼1g ni s iðtÞe iðtÞÀs2iðtÞ2Às2iðtÞ2r2i!¼X mi¼1g niÀs2iðtÞ2À12s iðtÞr iÀr i e iðtÞ2þr2ie2iðtÞ2 "#X m i¼1g niÀs2iðtÞ2þr2ie2iðtÞ2!ð23ÞAssuming e iðtÞ2L2½0;T ;8T2½0;1Þ;integrating theabove equation from t=0to t=T,yieldsVðTÞÀVð0ÞX mi¼1g niÀ12Z Ts2iðtÞtðÞdtþr2i2Z Te2iðtÞdt2435ð24ÞSince VðTÞ!0,the above inequality implies the followinginequality12X mi¼1g niZ Ts2iðtÞdt Vð0Þþ12X mi¼1g ni r2iZ Te2iðtÞdtð25ÞUsing Eq.22,the above inequality is equivalent to thefollowingX mi¼1Z Ts2iðtÞdtX mi¼1½s2ið0Þ=g ni þX mi¼1r2iZ Te2iðtÞdtð26ÞThus the proof is completed.4Particle swarm optimization(PSO)algorithmThe learning-rates of the tuning laws in SPID controller areusually selected by trial-and-error process.In order toachieve the best learning speed,the PSO algorithm isadopted to search the optimal learning-rates g P;g I and g Din the SPID controller.In1995,Kennedy and Eberhart[10]initially proposedthe particle swarm concept and PSO algorithm;this algo-rithm is one of optimization methods.It has been proven tobe efficient in solving optimization problem.In the PSOalgorithm,each particle represents a candidate solution tothe optimization problem.The particle keeps track of itscoordinates in the problem space which are associated withthe personal best solution.Another is the global best valuethat is tracked by the global version of the particle swarmoptimizer.At each time step,the PSO algorithm consists ofchanging the velocity that accelerates each particle towardits personal best and global best locations.Acceleration isweighted by a random term with separate random numbersbeing generated for acceleration toward personal best andglobal best locations,respectively[11].From then on,several PSO algorithms have been proposed with slightlydifferent versions[12–16].Theflowchart of the utilized PSO algorithm is drawn inFig.2.4.1Fitness functionIn order to maintain the control characteristic of SPIDcontroller,afitness function is chosen asfit ¼10:1þjj e ðt Þjjð27ÞIt means that if the error states e ðt Þis forced to zero then the expected value of fitness will be fit ¼10.4.2Velocity and position update lawIn PSO algorithm,a population of particles is randomly generated initially;each particle adjusts self-position with velocity according to its own experience and the experi-ences of other particles.The particle velocity and position update law is adopted as [16]v l q ðn þ1Þ¼v l q ðn ÞÂiw þn 1Ârand 1ðÁÞ½L l best q Àp l q ðn Þþn 2Ârand 2ðÁÞ½G l best qÀp l q ðn Þþn 3Ârand 3ðÁÞ½S l best q Àp l q ðn Þð28Þp l q ðn þ1Þ¼p l q ðn Þþv l q ðn þ1Þwhere iw is called the inertia weight which balances the global and local search,v l q ðn Þand p l q ðn Þdenote current velocity and current position,respectively;rand 1ðÁÞ,rand 2ðÁÞand rand 3ðÁÞdenote random variables between 0and 1;n 1,n 2,and n 3denote acceleration factor_1,acceleration factor_2and acceleration factor_3,respectively;L l best q ,G l best q ,and S l best q are the personal best index,the global best index,and the sub-population best index of the q th particle,respectively.Additionally,q ¼1;2;...;n p ,in which n p is the population size and‘¼1;2;...;n d ,in which n d is the dimension of each particle and iw is given by iw ¼iw max Àiw max Àiw minN maxÂN nð29Þwhere iw max ,iw min ,N max and N n are iteration maximum value,iteration minimum value,total iteration number and current iteration number of inertia weight.This PSO algorithm is used to on-line tune the learning-rates g P ,g I and g D in Eqs.14–16to achieve fast learning speed of PID gains.5Simulation resultsTwo uncertain nonlinear systems,a two-link manipulator control system and a unified chaotic system are examined to illustrate the effectiveness of the proposed design method.The parameters of PSO are set as:the population size z =20,the dimension of the particle h =2,the acceleration factors n 1=0.75,n 2=3.25,n 3=0.1,the total iteration number N max =10,the iteration maximum value iw max =0.9,the iteration minimum value iw min =0.4,the initial states of velocity v l q ðn Þand position p l q ðn Þof each particle are randomly generated.Example 1.Two-link manipulator systemIn this example,the proposed control system is applied to control a two-link robot manipulator.Figure 3depicts this two-link manipulator,the angles of the second and the third links were considered to be h 1and h 2,respectivelyFig.2The flowchart of PSO algorithm[18].In addition,the numerical values of parameters of the robot model were specified as that in [19].The dynamic equation is given as followsM ðh Þ€h þA m ðh ;_hÞ_h þB ðh ÞþZ _h þs d ¼s ð30Þwhere h 2<n is the joint position vector;M ðh Þ2<n Ân is asymmetric positive definite inertia matrix;A m ðh ;_hÞis a vector of Coriolos and centripetal torques;B ðh Þ2<n representing the gravitational torques;Z ¼K x þV f 2<n Ân is a diagonal matrix consisting of the back emf coefficient matrix K x and the viscous friction coefficient matrix V f ;s d 2<n is the unmodeled disturbances vector;s 2<n is the vector of control input torques.By defining the state vector x ðt Þ¼½x 1ðt Þ;x 2ðt Þ T ¼½h 1;h 2 T ,the dynamic Eq.30can be expressed as €x ðt Þ¼f ðx ðt ÞÞþG ðx ðt ÞÞu ðt Þþd ðt Þð31Þwhere the unknown nonlinear function G ðx ðt ÞÞ¼M À1ðh Þ;and f ðx ðt ÞÞ¼M À1ðh ÞÀA m ðh ;_h Þ_h ÀB ð_h ÞÀZ _h Às d ÂÃ;where A m ðh ;_h Þ¼sin ðh 2Þða 2þp 2a 9ÞÀ_h 2Àð_h 1þ_h 2Þ_h 10 !;B ðh Þ¼a 200a 8!;Z ¼a 5cos ðh 1Þþa 6cos ðh 1þh 2Þþp 4½cos ðh 1Þþp 5cos ðh 1þh 2Þa 9a 6cos ðh 1þh 2Þþp 5cos ðh 1þh 2Þa 9 ;M ¼M 11M 12M 21M 22!where M 11¼a 1þ2a 2cos ðh 2Þþðp 1þ2p 2cos ðh 2Þa 9Þ;M 12¼ða 3þa 2cos ðh 2ÞÞþðp 3þp 2cos ðh 2Þa 9Þ;M 21¼ða 3þa 2cos ðh 2ÞÞþðp 3þp 2cos ðh 2Þa 9Þ;M 22¼a 1þp 3a 9[19].In this example,the initial conditions of the states are given as a 1¼6:33;a 2¼0:14;a 3¼0:11;a 4¼27:6;a 5¼31:9;a 6¼3:3;a 7¼0:94;a 8¼4:54;a 9¼1:25;p 1¼0:37;p 2¼0:18;p 3¼0:18;p 4¼4:23;p 5¼4:15;h 1¼50p =180;h 2¼10p =180;_h1¼0;_h 2¼0:For demonstrating the tracking performance of the proposed control system,the desired trajectories for x d 1ðt Þand x d 2ðt Þare set as x d ðt Þ¼x d 1ðt Þx d 2ðt Þ!¼0:5þ0:2ðsin ðt Þþsin ð2t ÞÞ1:3À0:1ðsin ðt Þþsin ð2t ÞÞ!ð32Þwhere x d ðt Þ¼x d 1ðt Þ;x d 2ðt Þ½ T ¼h d 1;h d 2½ T is the reference trajectory vector.For the proposed RSPID control system,the feedback gain matrix is designed as H ¼diag ð40;4Þ;the initial conditions of the learning-rates are chosen as g P =400,g I =300and g D =400,respectively;all the PID gains are set as zero initially;the initial states of two-link manipulator are specified as x 1ð0Þ¼20p =180;x 2ð0Þ¼10p =180;_x1ð0Þ¼0and _x 2ð0Þ¼0;and the initial states of desired trajectories x d 1ð0Þ¼0;x d 2ð0Þ¼0;_xd 1ð0Þ¼0and _xd 2ð0Þ¼0:In order to study the robustness of the proposed control system,assume that the two-link manip-ulator control system has external disturbance s d ¼½40;25 at t ¼5s.Besides,the attenuation level is chosen as R ¼diag ð0:5;0:5ÞIn order to compare the control performance,the adaptive fuzzy control (AFC)presented in [18]is also applied to this manipulator ing this control system,the tracking trajectories are shown in Fig.4a,b,respectively.The simulation results of the proposed RSPID control system for this two-link manipulator control system are shown in Figs.4,5,6,7.Figure 4c,d represents the tracking responses of x 1ðt Þand x 2ðt Þby the proposed RSPID control scheme,respectively.Moreover,the control inputs and tracking errors of x 1ðt Þand x 2ðt Þare plotted in Fig.5a–d,respectively.In addition,the fitness function and learning-rates are shown in Fig.6a–d,respectively.The PID gains K P ;K I and K D are shown in Fig.7a–c,paring Fig.4c,d with Fig.4a,b,it can be seen the tracking error have been much reduced by using the RSPID controller.Moreover,from Fig.6,it is also seen that the learning-rates of PID controller converge after the 5th second.The simulation results also show that the pro-posed RSPID control system can effectively achieve parameter tuning and favorable control for the two-link manipulator control system.Example 2.Unified chaotic systemConsider a general master–slave unified chaotic sys-tems;the master system and slave system are given as [20]._xd 1ðt Þ¼ð25h c þ10Þðx d 2ðt ÞÀx d 1ðt ÞÞMaster _xd 2ðt Þ¼ð28À35h c Þx d 1ðt ÞÀx d 1ðt Þx d 3ðt Þþð29h c À1Þx d 2ðt Þ_x d 3ðt Þ¼x d 1ðt Þx d 2ðt ÞÀ8þh c3 x d 3ðt Þð33Þ_x1ðt Þ¼ð25h c þ10Þðx 2ðt ÞÀx 1ðt ÞÞþd 1ðt Þþu 1ðt ÞSlave _x2ðt Þ¼ð28À35h c Þx 1ðt ÞÀx 1ðt Þx 3ðt Þþð29h c À1Þx 2ðt Þþd 2ðt Þþu 2ðt Þ_x 3ðt Þ¼x 1ðt Þx 2ðt ÞÀ8þh c3x 3ðt Þþd 3ðt Þþu 3ðt Þð34Þwhere x di (t )and x i (t )are the system states variables of master system and slave system,respectively;d i ,i =1,2,3denote the disturbances and u i ,i =1,2,3are the control inputs.Assume a 1¼ð25h c þ10Þ;a 2¼ð28À35h c Þ;a 3¼ð29h c À1Þand a 4¼8þh c 3ÀÁ;then the master system Eq.33and the slave system Eq.34can be rewritten as_xd 1ðt Þ¼a 1ðx d 2ðt ÞÀx d 1ðt ÞÞ_xd 2ðt Þ¼a 2x d 1ðt ÞÀx d 1ðt Þx d 3ðt Þþa 3x d 2ðt Þ_xd 3ðt Þ¼x d 1ðt Þx d 2ðt ÞÀa 4x d 3ðt Þð35Þ_x1ðt Þ¼a 1ðx 2ðt ÞÀx 1ðt ÞÞþd 1ðt Þþu 1ðt Þ_x2ðt Þ¼a 2x 1ðt ÞÀx 1ðt Þx 3ðt Þþa 3x 2ðt Þþd 2ðt Þþu 2ðt Þ_x3ðt Þ¼x 1ðt Þx 2ðt ÞÀa 4x 3ðt Þþd 3ðt Þþu 3ðt Þð36Þthe master–slave system can be expressed as _x d ðt Þ¼f ðx d ðt ÞÞð37Þand_xðt Þ¼f ðx ðt ÞÞþG ðx ðt ÞÞu ðt Þþd ðt Þð38Þwhere x d ðt Þ¼½x d 1ðt Þ;x d 2ðt Þ;x d 3ðt Þ T ;x ðt Þ¼½x 1ðt Þ;x 2ðt Þ;x 3ðt Þ T ;f ðx d ðt ÞÞ¼½a 1ðx d 2ðt ÞÀx d 1ðt ÞÞ;a 2x d 1ðt ÞÀx d 1ðt Þx d 3ðt Þþa 3x d 2ðt Þ;x d 1ðt Þx d 2ðt ÞÀa 4x d 3ðt Þ T ,f ðx ðt ÞÞ¼½a 1ðx 2ðt ÞÀx 1ðt ÞÞ;a 2x 1ðt ÞÀx 1ðt Þx 3ðt Þþa 3x 2ðt Þ;x 1ðt Þx 2ðt ÞÀa 4x 3ðt Þ T ;G ðx ðt ÞÞ¼diag ½1;1;1 ,d ðt Þ¼½d 1ðt Þ;d 2ðt Þ;d 3ðt Þ T and u ðt Þ¼½u 1ðt Þ;u 2ðt Þ;u 3ðt Þ T .The control objective is to find a suitable control law u ðt Þ,so that the state trajectories of slave chaotic system x ðt Þcan follow the master chaotic system x d ðt Þunder different initial conditions and subject todisturbances.Fig.5The control inputs and tracking errors of the two-link manipulator system a control input 1,b control input 2,c tracking error 1,d tracking error2Fig.4The state responses of AFC and RSPID.a state response 1of AFC,b state response 2of AFC,c state response 1of RSPID,d state response 2of RSPID (dashed line desired trajectory,continuous line state trajectory)。