基于观测器的自适应控制机械手(英)

Observer-based adaptive control of robot manipulators:Fuzzy systems approachNoureddine Gole´a a,*,Amar Gole´a b,Kamel Barra a,Tarek Bouktir aa Electrical Engineering Institute,Oum El-Bouaghi University,04000Oum El-Bouaghi,Algeriab Electrical Engineering Department,Biskra University,07000Biskra,AlgeriaReceived30January2006;received in revised form3April2007;accepted24May2007Available online15June2007AbstractThis paper presents a fuzzy adaptive control suitable for motion control of multi-link robot manipulators with structured and unstructured uncertainties.When joint velocities are available,full state fuzzy adaptive feedback control is designed to ensure the stability of the closed loop dynamic.If the joint velocities are not measurable,an observer is introduced and an adaptive output feedback control is designed based on the estimated velocities.In the proposed control scheme,we need not derive the linear formulation of robot dynamic equation and tune the parameters. To reduce the number of fuzzy rules of the fuzzy controller,we consider the properties of robot dynamics and the decomposition of the uncertainties terms.The proposed controller is robust against uncertainties and external disturbance.Further,it is shown that required stability conditions,in both cases,can be formulated as LMI problems and solved using dedicated software.The validity of the control scheme is demonstrated by computer simulations on a two-link robot manipulator.#2007Elsevier B.V.All rights reserved.Keywords:Robot manipulators;Fuzzy systems;Adaptive control;Observer;Stability1.IntroductionRobots are one of the most important pieces of machinery for industrial automation nowadays.Design of robust adaptive controllers suitable for real-time control of robot manipulators is one of the most challenging tasks for many control engineers, especially when manipulators are required to maneuver very quickly under external disturbances.Motion control of robot manipulators has been studied using various approaches(see e.g.,[1–8]and references therein).The traditional proportional and derivative(PD)controller is very simple and does not require any knowledge of the robot dynamics.However,it requires very large actuation to achieve precise control,which is not practical but highly demanded in many cases.Robot manipulators are multivariable nonlinear coupling systems and are frequently subjected to structured and/or unstructured uncertainties even in a well-structured setting for industrial use. Structured uncertainties are mainly caused by imprecision in the manipulator link properties,unknown loads,and so on. Unstructured uncertainties are caused by unmodeled dynamics, e.g.,nonlinear friction,disturbances,and high-frequency models of the dynamics.As a result,it is difficult to obtain an accurate mathematical model so that computed torque controllers[1,6]or other model-based controllers[3,8]can be accurately applied.Although adaptive controllers[1,5,7]can achievefine control and compensate for partially unknown manipulator dynamics(i.e.,structured uncertainties),they often suffer from incapacity to deal with unstructured uncertainties. Hence,there is a need for model-free adaptive control strategies.The application of fuzzy systems to robots dynamic control is not new[9–11].Though the proposed methods have been practically successful,it has proved extremely difficult to develop a general analysis and design theory for conventional fuzzy control systems.During the last few years,a number of papers have been presented to deal with the problem of robot adaptive control[12–16].The basic idea of these methods is to design the feedback controller based on the computed torque principle,and to use an adaptive fuzzy system to approximate the robot nonlinearities involved in the control input design. However,most of the above designs present two drawbacks./locate/asocApplied Soft Computing8(2008)778–787*Corresponding author.E-mail addresses:nour_golea@yahoo.fr(N.Gole´a),goleaamar1@yahoo.fr(A.Gole´a).1568-4946/$–see front matter#2007Elsevier B.V.All rights reserved.doi:10.1016/j.asoc.2007.05.011First,the robot dynamic is presented as single nonlinearityapproximated by a single fuzzy system with the robot real and desired positions and velocities as inputs,which results in large number of rules with lot of parameters to be tuned.Second,the feedback control design is setup under the constraint that the states of the robot are available,which is not true in many practical cases where the joint velocities are not measurable.In this paper,we develop an adaptive fuzzy control for rigid robot manipulators,with reduced complexity.This work can be classed in the context of observer-based fuzzy adaptive control of MIMO nonlinear systems where few approaches were proposed (see e.g.,[17–19]).Ref.[17]proposes a hybrid direct–indirect fuzzy adaptive approach,while [17,18]investigate the indirect one.In [17],a linear observer is used to estimate the tracking errors,while in [18,19]a nonlinear observer,based on the system estimated fuzzy model,is used to estimate the system state vector.The three approaches share the introduction of robustifying terms to ensure the convergence.Further,[17,19]share the assumption that strict positive real (SPR)condition is fulfilled for the estimation error dynamic,which is not true due to the triplet (A ,B ,C )structure.This assumption is meet in [18]by appropriately filtering the estimation error dynamic.The present work differs from [17–19]in several aspects.The first one is that this work is dedicated to robot manipulators dynamic control and exploits the robots dynamic structure in the control design,while [17,18]are of general purpose.The second aspect is that,compared to [17–19]where all input gain matrix elements are estimated,only the diagonal elements are considered here which reduces the used fuzzy approximators number and complexity.The third aspect is that no robustifying term is used,and the tracking and stabilization is ensured by the fuzzy adaptive controller.The last aspect is that,as in [17],a linear observer is used to estimate the tracking errors,which in turn are used in the parameters update laws.This technique avoids the SPR condition problem encountered in [17,19]and the filtering approach used in [18].The paper is organized as follows.In Section 2,the rigid robot control problem is formulated,and a brief description of fuzzy systems is presented.In Section 3,based on full state information assumption,a fuzzy adaptive state feedback is designed and it’s stability is analyzed.In Section 4,a linear observer is used to estimate the joint velocities,and an observer-based output-feedback control law is developed.In Section 5,simulation tests for a two-link robot under uncertainties and disturbances are presented to confirm the effectiveness and applicability of the proposed method.Finally,conclusions are included in Section 6.2.PreliminariesStandard notation is used in this paper.Let R be the real number set,R n be the n -dimensional vector space,R n Ân be the n Ân real matrix space.The norm of vector x 2R n and that of matrix A 2R n Ân are defined,respectively,as j x j ¼ffiffiffiffiffiffiffix T x p and j A j 2¼tr ½A TA .If y is a scalar,then j y j denotes the absolute value.2.1.Robot control problemThe dynamic equations of the robot manipulator are a set of highly nonlinear coupled differential ing the Lagrange–Euler formulation,the dynamic equation of an n -joint robot arm can be expressed asM ðq Þ¨q þc ðq ;˙q Þþg ðq Þþt c ðq ;˙qÞþt d ðq ;˙q Þ¼u (1)where M ðq Þ2R n Ân is the bounded positive definite inertia matrix;c ðq ;˙q Þ2R n the vector representing centrifugal and Coriolis effects;g ðq Þ2R n the vector representing gravita-tional torques;and t c ðq ;˙q Þ2R n ,t d ðq ;˙q Þ2R n are the vectors representing the dynamic effects as nonlinear frictions,small joint and link elasticities,backlash and bounded torque disturbances.Here the uncertainties effect is decom-posed as continuous part t c ðq ;˙q Þand discontinuous part t d ðq ;˙q Þ.u 2R n is the vector of joint torques supplied by the actuators;q 2R n the vector of joint positions;˙q 2R n the vector of joint velocities;and ¨q 2R n is the vector of joint accelerations.Taking as state vector x T ¼½x T 1ÁÁÁx T n with x Ti ¼½q i ˙q i ;the robot model (1)can be rewritten as ˙x ¼Ax þB ½F ðq ;˙q ÞþG ðq Þu þd ðq ;˙q Þ (2)whereF ðq ;˙q Þ¼f 1ðq ;˙q Þ...f n ðq ;˙q Þ264375:¼ÀM À1ðq Þ½c ðq ;˙q Þþg ðq Þþt c ðq ;˙q Þ G ðq Þ¼g 11ðq ÞÁÁÁg 1n ðq Þ...}...g n 1ðq ÞÁÁÁg nn ðq Þ264375:¼M À1ðq Þd ðq ;˙q Þ¼d 1ðq ;˙q Þ...d n ðq ;˙q Þ264375:¼ÀM À1ðq Þt d ðq ;˙q Þand A ¼diag ½A 1;...;A n ,B ¼diag ½b 1;...;b n with A i ¼0100 ;b i ¼01;i ¼1;...;n The tracking control problem can be stated as:for given bounded reference trajectories q r ,˙q r and ¨q r 2R n ,(with j ¨q r j q 0form some known constant q 0)design the control input torques u such as the robot’s states follow the reference trajectories,with all involved signals in closed loop remain bounded.2.2.Fuzzy systemsA fuzzy system (FS)consists of four parts:the knowledge base,the fuzzifier,the fuzzy inference engine working on fuzzy rules,and the defuzzifier.The knowledge base for the FSN.Gole ´a et al./Applied Soft Computing 8(2008)778–787779comprises a collection of fuzzy If–then rules of the following form:R j:If z1is Z j1and z2is Z j2and;...;and z p is Z j m then y is Y j;j¼1;...;m(3) where z¼ðz1;...;z pÞ2R p,y2R are the fuzzy system inputand output,respectively.The fuzzy sets Z jl and Y j,associatedwith the membership functions mZ jl ðz jÞand m Y jðyÞ,respec-tively,operate fuzzy partitions of the fuzzy system input and output spaces Z and Y,respectively.m is the rules number.The output of the FS with weighted-center defuzzifier,product inference,and singleton fuzzifier is of the following form:y¼X mj¼1f jðzÞw j(4)where f jðzÞ¼Q pl¼1mZ jlðz lÞand w j is the point in Y at whichm Y jðyÞachieves its maximum value.In more compact form,(4) can be arranged asy¼u T fðzÞ(5) where u T¼½w1...w m and f TðzÞ¼½f1ðzÞ...f mðzÞ .In this work,the membership functions are chosen as gaussian function;that ism Z jl ðz lÞ¼expÀðz lÀc jlÞ22sjl(6)where c jl and s jl are the membership function center and variance,respectively.The above membership function could be replaced by any other function,but as was shown in[20],this function has the best approximation property.To maintain consistent performance of the fuzzy systems in situations where there is a large uncertainty or unknown variation in plant parameters and structures,the fuzzy systems should be adaptive.To reduce the processing requirements,the membership functions parameters arefixed(i.e.,regular membership functions distribution is used),and the vector u is chosen as the free adjustable parameter.3.Fuzzy adaptive state feedbackIn this section,it is assumed that robot joint velocities are available to measurement.Then,adaptive fuzzy state feedback can be designed using full state vector information.Theoretical results[20,21]have shown that fuzzy systems (5)are universal approximators,i.e.,they can approximate any smooth function on a compact space.Due to this approximation capability,we can assume that the nonlinear terms in(2)can be approximated asf iðq;˙qÞ¼uÃT fif iðq;˙qÞþe iðq;˙qÞ;g iiðqÞ¼uÃT gi c iðqÞþe iðqÞ;i¼1;...;n(7)where uÃT fi f iðq;˙qÞand uÃT gic iðqÞare fuzzy systems of the from(5),and e iðq;˙qÞ,e iðqÞare the inherent approximation errors due to thefinite number of rules in the fuzzy systems.The optimal weights uÃfiand uÃgidefined above are quantities required only for analytical purpose.Typically uÃfiand uÃgiare chosen to minimize e iðq;˙qÞand e iðqÞover the compact regions V f and V g respectively,that isuÃf i¼arg min f supq;˙q2V fj f iðq;˙qÞÀu T fi f iðq;˙qÞjguÃg i¼arg min f supq2V gj g iiðqÞÀu T gi c iðqÞjgAssumption1.The fuzzy systems approximation errors are bounded by j e iðq;˙qÞj e0i and j e iðqÞj e0i,i¼1;...;n,for some constants e0i and e0i.Assumption1results from the universal approximation property of fuzzy systems,that can approximate any well-defined function over a compact space withfinite approxima-tion error.Using(7)in(2),the robot dynamic can be written as˙x¼AxþB½QÃf Fðq;˙qÞþQÃg CðqÞuþHðqÞuþvðq;˙qÞ (8) whereFðq;˙qÞ¼block-diag½f1ðq;˙qÞ;...;f nðq;˙qÞ ;-Cðq;˙qÞ¼block-diag½c1ðq;˙qÞ;...;c nðq;˙qÞ ,QÃf¼block-diag½uÃT f1;...;uÃTf n,QÃg¼block-diag½uÃT g1;...;uÃT gn,vðq;˙qÞ¼eþdðq;˙qÞ,with e T¼½e1ÁÁÁe n ,andHðqÞ¼e1g12ðqÞÁÁÁg1nðqÞg21ðqÞ}......}gðnÀ1ÞnðqÞg n1ðqÞÁÁÁg nðnÀ1ÞðqÞe n266664377775 Based on(7)and(8),the control inputs are defined asu¼½Q g CðqÞ À1½ÀQ f Fðq;˙qÞþ¨q rþK e (9)where e T¼½ðqrÀqÞTð˙q rÀ˙qÞT is the tracking error vec-tor,Q g;Q f are the estimated fuzzy systems parameters,and K¼diag½K1;...;K n with K i2R2is PD regulator gain vector, chosen such as the matrix A c¼AÀBK is Hurwitz.Then,introducing the control input(9)in(8)yields˙e¼A c eÀB½¯Q f Fðq;˙qÞþ¯Q g CðqÞuþHðqÞuþvðq;˙qÞ(10) where¯Q f¼QÃfÀQ f and¯Q g¼QÃgÀQ g are the parameters estimation errors.From(10),it can be seen that the tracking error vector is driven by the coupling terms and thefinite approximation accuracy effects reflected by HðqÞand the uncertainty term vðq;˙qÞ.To design the fuzzy systems parameters update laws and to ensure boundedness of the involved signals in the closed loop robot control,the following assumptions are used:Assumption2.The diagonal elements of GðqÞare bounded such as g m diag½g11ðqÞ;...;g nnðqÞ g M,the matrix HðqÞisN.Gole´a et al./Applied Soft Computing8(2008)778–787 780bounded by j HðqÞj h0,and the disturbance term vðq;˙qÞis bounded by j vðq;˙qÞj v0.Assumption3.The fuzzy systems parameters are bounded by the constraint sets V f and V g such that:V f¼f QÃf jj QÃf j f M g and V g¼f QÃg j g m j QÃg j g M g;respectively,where f M,g m,and g M are some known constants.Thefirst part of Assumption2follows from the fact that MðqÞis bounded positive definite matrix,the second part follows from the boundedness of MðqÞand e iðqÞ.Finally,the third part follows from boundedness of MðqÞ,t d and e iðq;˙qÞ. The bounds used in Assumption3result from the Assumptions 1and2and are used to ensure the boundedness of the fuzzy systems outputs.In order to constraint the parameters Q f and Q g within the sets V f and V g,respectively,we use the following parameter projection algorithm:˙Qf ¼Àg1W fif j Q f j<f M orðj Q f j¼f M and tr½W f Q T f !0ÞÀg1W fþg1tr½W f Q T f1þj Q f jf M2Q fif j Q f j¼f M and tr½W f Q T f <08>>>>>><>>>>>>:(11)and˙Q g ¼Àg2W gif j Q g j<g M orðj Q g j¼g M and tr½W g Q T g !0ÞÀg2W gþg2tr½W g Q T g1þj Q g jg M2Q gif j Q g j¼g M and tr½W g Q T g 08>>>>>>>><>>>>>>>>:(12)where g1;g2>0are design parameters,W f¼B T Pe F Tðq;˙qÞ, W g¼B T Peu T C TðqÞ,and P¼P T>0is the solution,for a given Q¼Q T>0,of the Lyapunov equation:A TcPþPA c¼ÀQ(13) Moreover,in order to guarantee j Q g j!g m such that an inverse of Q g CðqÞalways exists,we use the following law to adjust the parameter Q g.1.Whenever any element½Q gi j¼g m use½˙Q gi j ¼Àg2½W gi jif½W gi j<00if½W gi j!0(14)2.Otherwise,use(12).where½Ai j stands for the ij th element of the matrix A.The stability properties of the proposed fuzzy adaptive state feedback are summarized by the following theorem. Theorem1.The robot adaptive control composed by the robot dynamic(2),the control input(9),the update laws (11)–(12)and(14)verifying Assumptions1–3,guarantees the following:1.j Q f j f M and g m j Q g j g M2.j e j2L13.j u j2L1Proof.1.To prove that g m j Q g j g M,let V g¼12g1tr½Q T g Q g ,then ˙Vg¼tr½˙Q T g Q g .If thefirst line of(12)is true,we have either j Q g j<g M or˙V g¼Àtr½W T g Q g 0when j Q g j¼g M,that is, we get always j Q g j g M.If the second line of(12)is true, we have j Q g j¼g M and˙Vg¼trÀW T g Q gþtr½W g Q T g1þj Qgjg M2Q TgQ g¼Àtr½W T g Q g þtr½W g Q T g1þj Qgjg M2tr½Q T g Q g (15)since j Q g j¼g M;we get˙Vg¼tr½W g Q T g g2M0(16) that is,j Q g j g M.Therefore,we have j Q g j g M,8t!0.From(14)we see that if j Q g j i j¼g m,then½˙Q gi j!0;that is,we have that j Q g j!g ing the same analysis,we can prove that j Q f j f M,8t!0.2.Consider the Lyapunov function:V¼1e T P eþ11tr½¯Q T f¯Q f þ12tr½¯Q T g¯Q g (17)The differentiation of(17)along(10)yields˙V¼À12e T Q eÀe T PB½¯Qf Fðq;˙qÞþ¯Qg CðqÞuþHðqÞuþvðq;˙qÞ þ1g1tr½˙¯Q T f¯Q f þ1g2tr½˙¯Q T g¯Q g (18)which,using matrices traces properties,can be arranged as˙V¼À12e T Q eÀe T PB½HðqÞuþvðq;˙qÞþ1g1tr½ð˙¯Q T fÀg1Fðq;˙qÞe T PBÞ¯Q fþ1g2tr½ð˙¯Q T gÀg2CðqÞu e T PBÞ¯Q g (19)Then,using(11)and(12)and the fact that˙¯Q f¼À˙Q f (˙¯Q g¼À˙Q g),(19)can be written as˙V¼À12e T Q eÀe T PB½HðqÞuþvðq;˙qÞÀ1g1tr½ð˙Q T fþg1W T fÞ¯Q f À1g2tr½ð˙Q T gþg2W T gÞ¯Q g(20)We now prove that the last two terms in(20)are always0.If thefirst lines of(11)and(12)are true the result isN.Gole´a et al./Applied Soft Computing8(2008)778–787781trivial.If the second lines are true,then we get ˙V ¼À12e T Q e Àe T PB ½H ðq Þu þv ðq ;˙q ÞÀtr ½W f Q T f 1þj Q f j f M2tr ½Q T f ¯Q f Àtr ½W g Q T g 1þj Q g j g M2tr ½Q T g ¯Q g (21)On the other hand,we havetr ½Q T f ¯Qf ¼12tr ½Q ÃT f Q Ãf À12tr ½Q T f Q f À12tr ½¯Q T f ¯Q f (22)Since tr ½Q ÃT f Q Ãf f 2M,tr ½Q T f Q f ¼f 2M and tr ½¯Q T f ¯Q f !0we get tr ½Q T f ¯Qf 0,which,with tr ½W f Q T f <0,means that third term in (21)is 0.The same analysis can be used to show that last term in (21)is also 0.Then,(21)can be arranged as˙V À12e T Q e Àe T PB ½H ðq Þu þv ðq ;˙q Þ(23)Further,introducing the control law (9)in (23)yields˙VÀ1e T Q e Àe T PB z 1e Àe T PB z 2(24)with the bounded terms:z 1¼H ðq Þ½Q g C ðq Þ À1Kz 2¼H ðq Þ½Q g C ðq Þ À1½ÀQ f F ðq ;˙q Þþ¨q r þv ðq ;˙q ÞMore,(24)can be upper bounded by ˙VÀ12e T Q e Àe T PB z 1e þ12e T PBB T P e þ12j z 2j 2(25)which can be arranged as˙VÀ12e T ðQ þ2PB z 1ÀPBB T P Þe þ12j z 2j 2(26)Then,if Q and P are chosen such that the followinginequalityQ þ2PB z 1ÀPBB T P !Q 1(27)holds for some positive definite matrix Q 1,then we get˙VÀ12e T Q 1e þ12j z 2j 2(28)which can be upper bounded by ˙VÀ12l min ðQ 1Þj e j 2þþ12j z 2j 2(29)where l min ðQ 1Þis the smallest eigenvalue of Q 1.Then ˙V0whenever the tracking error is outside the region given by j e j j z 2jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil min ðQ 1Þp (30)which implies that j e j 2L 1.3.Consider that (9)can be upper bounded by u j½Q g C ðq Þ À1j½j Q f F ðq ;˙q Þj þj ¨q r j þj K jj e j (31)and the results 1,2and (31)yieldsj u j 1g mf M þq 0þj K j j z 2j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil min ðQ 1Þp (32)this implies that j u j 2L 1.&4.Observer-based fuzzy adaptive controlIn Section 3,the joint velocities were supposed to beavailable for feedback.This assumption limits the applica-tion of the proposed approach,because,in many practical situations only the joint positions are measured.In this section,fuzzy adaptive output feedback is investigated by introducing an observer to reconstruct the robot state vector.Let’s define the following linear state observer:˙ˆx ¼A ˆx þL¯q (33)ˆq¼C ˆx (34)where ˆx T ¼½ˆq T ˙ˆq Tare the estimated positions and veloci-ties,¯q ¼q Àˆqare the positions estimation errors.L T¼diag ½L 1;...;L n ,with L i 2R 2,is the observer gain vector,and C ¼diag ½c 1;...;c n with c i ¼½10 .The state and output estimation errors,using (8),(33)and (34),are given by˙¯x ¼A¯x þB ½Q Ãf F ðq ;˙q ÞþQ Ãg C ðq Þu þH ðq Þu þv ðq ;˙qÞ ÀL¯q (35)¯q ¼C¯x (36)where ¯x ¼x Àˆxis the state estimation error.Based on robot estimated state vector,the control input (9)is redefined asu ¼½Q g C ðq Þ À1½ÀQ f F ðq ;˙ˆq Þþ¨q r þK ˆe (37)where ˆeis the estimated tracking error.Then,introducing (37)in (35)yields˙¯x ¼A¯x þB ½Q Ãf F ðq ;˙q ÞÀQ f F ðq ;˙ˆq Þþ¯Q g C ðq Þu þ¨q rþK ˆe þH ðq Þu þd ðq ;˙q Þ ÀL¯q (38)Using (36)and the fact that ˆe ¼e þ¯x ,(38)can be arrangedas˙¯x ¼A 0¯x þB ½Q Ãf F ðq ;˙q ÞÀQ f F ðq ;˙ˆq Þþ¯Q g C ðq Þu þ¨q rþH ðq Þu þd ðq ;˙q Þ þBK e (39)where A 0¼ðA þBK ÀLC Þ.N.Gole´a et al./Applied Soft Computing 8(2008)778–787782From the definition of the membership functions (6),F ðq ;˙q Þcan be decomposed as F ðq ;˙q Þ¼F ðq ;˙ˆqÞþS ðq ;˙¯q Þ(40)where S ðq ;˙q Þrepresents the high order terms of Taylor devel-opment of F ðq ;˙q Þin the neighborhood of ˙ˆq.Note,that since j F ðq ;˙q Þj ;j F ðq ;˙ˆqÞj 1then S ðq ;˙¯q Þis also bounded by j S ðq ;˙¯q Þj 1.Substituting (40)in (39)yields ˙¯x ¼A 0¯x þB ½¯Q f F ðq ;˙ˆq Þþ¯Q g C ðq Þu þ¨q r þH ðq Þu þ&ðq ;˙q ÞþBK e (41)where &ðq ;˙q Þ¼Q Ãf S ðq ;˙qÞþv ðq ;˙q Þ.Then,using the control input (37),the tracking error dynamic is given by˙e ¼A c e ÀB ½¯Q f F ðq ;˙ˆq Þþ¯Q g C ðq Þu þH ðq Þu þ&ðq ;˙q ÞÀBK ¯x (42)In order to constraint the parameters Q f and Q g within thesets V f and V g ,respectively,the same update laws (11),(12)and (14)are used with W f ¼B T P ˆeF T ðq ;˙ˆq Þand W g ¼B T P ˆe u T C Tðq Þ.Theorem 2.The observer-based fuzzy adaptive robot con-trol,composed by the robot dynamic (2),the observer (33)and (34),the control input (37)and the update laws (11),(12)and (14)(with the indicated changes),guarantees the follow-ing :1.j Q f j f M and g m j Q g j g M2.j e j 2L 1;j ¯x j 2L 13.j u j 2L 1Proof.1.The proof follows the same line as in Theorem 12.Consider the Lyapunov function:V ¼12e T P e þ12¯x T P 0¯x þ12g 1tr ½¯Q T f ¯Q f þ12g 2tr ½¯Q T g ¯Q g (43)where P 0¼P T 0>0is the solution,for a given Q 0¼Q T 0>0,of the Lyapunov equationA T 0P 0þP 0A 0¼ÀQ 0(44)The differentiation of (43)along (41)and (42)yields ˙V ¼À12¯x T Q 0¯x þ¯x T P 0B ½¯Qf F ðq ;˙ˆq Þþ¯Qg C ðq Þu þ¨q r þH ðq Þu þ&ðq ;˙qÞ À12e T Q e Àe T PB ½¯Qf F ðq ;˙ˆq Þþ¯Qg C ðq Þu þH ðq Þu þ&ðq ;˙q Þ þ¯xT P 0BK e Àe T PBK ¯x þ1g 1tr ½˙¯Q T f ¯Q f þ1g 2tr ½˙¯Q T g ¯Q g (45)Then,using the fact that e ¼ˆe À¯x in (45)becomes ˙V ¼À12¯x T Q 0¯x À12e T Q e ÀˆeT PB ½¯Q f F ðq ;˙ˆq Þþ¯Q g C ðq Þu Àe T PB ½H ðq Þu þ&ðq ;˙q Þ þ¯x T P 0B ½H ðq Þu þ¨q r þ&ðq ;˙q Þ þ¯x T P 0B ½¯Qf F ðq ;˙ˆq Þþ¯Qg C ðq Þu þ¯x T PB ½¯Q f F ðq ;˙ˆq Þþ¯Q g C ðq Þu þ¯x T P 0BK e Àe T PBK ¯x þ1g 1tr ½˙¯QT f ¯Q f þ1g 2tr ½˙¯Q T g ¯Q g (46)Further,using the update laws (11)and (12)(with the indicated changes)in (46)yields˙V À12¯x T Q 0¯x À12e T Q e Àe T PB ½H ðq Þu þ&ðq ;˙q Þþ¯x T P 0B ½H ðq Þu þ¨q r þ&ðq ;˙q Þ þ¯x T P 0B ½¯Qf F ðq ;˙ˆq Þþ¯Qg C ðq Þu þ¯x T PB ½¯Q f F ðq ;˙ˆq Þþ¯Q g C ðq Þu þ¯x T P 0BK e Àe T PBK ¯x (47)which,using the control input (37),can be arranged as ˙V À12¯x T Q 0¯x À12e T Q e þ¯x T ½2P 0þP B L 1K ¯x Àe T PB L 1K e Àe T PB ½L 1ÀI n K¯x þ¯x T ½2P 0B ðL 1þI n ÞþPB L 1 K e þ¯x T ½P 0B ðL 3þL 4ÞþPB L 4 Àe T PB L 2(48)where L 1¼H ðq Þ½Q g C ðq Þ À1;L 2¼L 1½ÀQ f F ðq ;˙ˆq Þþ¨q r þ&ðq ;˙q Þ;L 3¼L 2þ¨qr ;L 4¼¯Q f F ðq ;˙ˆq Þþ¯Q g C ðq Þ½Q g C ðq Þ À1½ÀQ f F ðq ;˙ˆq Þþ¨q r .Further,adopting the following notation:z ¼¯x e ;Q ¼Q 0À½2P 0þP B L 1K À½2P 0B ðL 1þI n ÞþPB L 1 K PB ½L 1ÀI n K Q þPB L 1K ;G ¼½P 0B ðL 3þL 4ÞþPB L 4 ÀPB L 2Eq.(48)can be rewritten as ˙VÀz T Q z þz T G (49)Hence,if there exist definite matrices P and P 0such thatQ is positive definite matrix,then (49)can be upper bounded by˙VÀl min ðQÞj z j 2þj z jj G j (50)where l min ðQÞis the smallest eigenvalue of Q .N.Gole ´a et al./Applied Soft Computing 8(2008)778–787783。