Control of linear systems using piecewise continuous systems

Control of linear systems using piecewise continuous systemsV.Koncar and C.VasseurAbstract:A control method is presented based on the use of piecewise continuous systems(PCS)as a control vector generator over sampling periods.The PCS has two inputs and one output whichis equivalent to the control vector.Each input is corresponding to one specific time space,thefirst tothe discrete time space defined by S¼{t k;k¼0;1;2;…}and the second to the continuous timespace t2=2S with=¼{t2½0;1½}:The control is generated in the time space=:Thearchitecture involving the PCS contains two feedback loops,based on the concept of double timeand input spaces.These inputs are computed using two transformation maps C and F defined infunctions of the control strategy adopted to achieve the desired control properties.The control lawdepends on the PCS properties and on the method used to generate PCS controller inputs.Twoexamples are given:in thefirst the PCS controller is working in open-loop configuration during asample period,therefore ripple is possible during sampling period.This limited control strategyassures imposed trajectory tracking only at sampling instants.In the second example the PCS isgenerating the control that satisfies the antiripple control strategy.The plant state is observed incontinuous time and the imposed state trajectory must be compared with the plant statecontinuously.The PCS controller initial state is generated according to‘optimal control’law thatminimises the quadratic criterion concerning the error-control compromise.This method appears tohave several advantages over dynamic controllers:it provides control system designers with morefreedom,and it requires an important computation only at the beginning of control process.Examples are given.1IntroductionFeedback control is generally used for the following purposes:to stabilise or improve the stability of a system; to cause a system to have a specified input–output behaviour;to decrease the sensitivity of a system to noise; model/plant mismatch and disturbances;to ensure satisfac-tory performance and stability of a system despite parameter variations and unmodelled dynamics.For single-input, single-output,time-invariant systems,classical control theory provides methods for designing control systems which meet these four objectives.These methods are well established and understood[1–4].Classical control methods are limited to single-input, single-output time-invariant systems.As the physical system treated by the control engineer became more complex,modern control methods were developed in partial answer to the limitations of classical methods[5–7].Central to modern control is the use of state-space formalism. Several state-space control system design techniques have been developed to implement the objectives for multi-input, multioutput systems.The two most widely used methods are quadratic optimisation and eigenstructure assignment[5].In both of these methods the control has the form of a state feedback. Many interesting papers have been published in the area of state control;Kabamba[8]investigated the use of generalised sampled-data hold function(GSHF)in the control of linear time-invariant systems.The idea of GSHF is to periodically sample the output of the system,and generate the control by means of a hold function applied to the resulting sequence.The hold function is chosen based on the dynamics of the system to be controlled.This method has the efficiency of state feedback without the requirement of state estimation.Several other works have been realised in thefield of deadbeat control concerning our developments.Urikura and Nagata[9]developed the ripple-free deadbeat control method for sampled-data system.The control objective is to settle the error to zero for all time after somefinite settling time i.e.to eliminate the ripples between the sampling instants in deadbeat control of sampled-data systems. Another interesting work in thefield has been realised by Yamamoto[10]whose work handles a new framework for hybrid sampled-data control systems.Instead of considering the state only at sampling instants,Yamamoto introduced a function piece during the sampling period as the state and gives an infinite-dimensional model with such a state space. This gives the advantage of a sampled-data system with built-in intersample behaviour that can be regarded as a linear,time-invariant,discrete time system.Tracking problems can be studied in this setting in a simple and unified way,and ripples are completely characterised as a mismatch between the intersample reference signal and transmission zero directions.q IEE,2003IEE Proceedings online no.20030964doi:10.1049/ip-cta:20030964V.Koncar is with GEMTEX/ENSAIT9rue de1’Ermitage,BP30329, 59056ROUBAIX Cedex01FranceC.Vasseur is with Laboratoire d’Automatique I3D,Universite´des Sciences et Technologies de Lille,Cite´Scientifique,59655Villeneuve d’Ascq, FrancePaper received18th September2002The work developed in this paper is based on the works introducing the compound control realised by Vasseur[11] and Laurent[12].The state-space approach is well adapted to represent system dynamics.This approach is used in both continuous and discrete-time spaces that lead to continuous or discrete dynamical systems.In the two cases,a state-space approach uses three vector spaces,n-dimensional state space denoted S n;r-dimensional input space denoted U r and m-dimensional output space denoted Y m:Let=denote the time space(continuous or discrete)and let,respectively, xðtÞ2S n;uðtÞ2U r and yðtÞ2Y m denote the state,the input and the output as functions of time t2=;then the system dynamics can be expressed by the transform=Â=ÂS nÂU r!S n;hence xðt lÞ¼gðt k;t l;xðt kÞ;u½tk ;t l Þð1aÞwhere t l>t k and u½tk ;t ldescribes the input over a timeinterval½t k;t l ;=ÂS n!Y m;hence yðt lÞ¼hðt l;xðt lÞÞð1bÞExpression(1a),called the state transition equation, expresses the causality and constitutes the solution of a differential or difference equation according to continuous or discrete time space.Expression(1b),called the output equation,is a memoryless equation.The main idea expressed in this paper is to define a state-space model using two input spaces and two time spaces.Thefirst time space=is identical to the previously defined time space.The second time space denoted S;is defined as a subspace of=;S being always discrete:S&= and S¼f t k;k¼0;1;2;...g:S is called the switching set. Thefirst input space U r is the same as previously defined, the second is denoted V s and its dimension is s.Under these conditions,if vðtÞ2V s is the second input,it is possible to define a transformation as follows:SÂV s!S n;hence xðtþkÞ¼zðt k;vðt kÞÞ;8t k2Sð2Þwhere xðtþk Þ¼lim t!tk;t>t k½xðtÞ :By combining(1)and(2)the new system evolution can be described by(3)xðt lÞ¼gðt k;t l;xðtþkÞ;u½tk ;t lÞ;8t l2 t k;t kþ1½and8t k2Sð3aÞxðtþkÞ¼zðt k;vðt kÞÞ;8t k2Sð3bÞyðtÞ¼hðt;xðtÞÞ;8t2=ÀSð3cÞThese equations define a piecewise system evolution,each time piece being described by t k;t kþ1½that is defined by twosuccessive instants of S.2Statement of the problem and basic concepts 2.1Definition of piecewise continuous system(PCS)Define a piecewise continuous,finite-dimensional,strictly causal linear time-invariant system S pðS;A;B c;B d;CÞ(S p in further notation)byS¼f t k;k¼0;1;2;...j t kþ1>t k g;t k2=:switching set A2R nÂn;B c2R nÂr;B d2R nÂs;C2R mÂn:real,time-invariant matrices.Such a system is described byx0ðtÞ¼AxðtÞþB c uðtÞ;8t=2Sð4aÞxðtþkÞ¼B d vðt kÞ;8t k2Sð4bÞyðtÞ¼CxðtÞ;8t=2Sð4cÞwhere xðtÞ2S n is the state vector,uðtÞ2U r is the bounded input vector,vðt kÞ2V s is the bounded discrete controlvector,yðtÞ2Y m is the output vector,and S n is n-dimensional state space,U r is r-dimensional input space,V s is s-dimensional‘condition’space,and Y m is m-dimensional output space.In this case it is obvious that= describes the continuous time space.Equation(4a) describes the continuous evolution of system over the time interval t k;t kþ1½;8t k2S:Equation(4b)gives the right limit value of xðtÞ;8t k2S:Equation(4c)is the output equation.The following Section shows that the previous definition is sufficient to completely describe system dynamics.2.2Dynamics of PCSConsidering a time interval t k;t kþ1½;(4a)leads to xðtÞ¼expðAðtÀt kÞÞxðtþkÞþZ tt kexpðAðtÀtÞÞB c uðtÞd t;8t2 t k;t kþ1½or by using(4b)xðtÞ¼expðAðtÀt kÞÞB d vðt kÞþZ tt kexpðAðtÀtÞÞB c uðtÞd t;8t2 t k;t kþ1½ð5ÞThe left limit value of xðtÞat t¼t k;noted xðtÀkÞ;is obtained from(5)over the interval t kÀ1;t k½and by approaching t to t k from leftxðtÀkÞ¼expðAðt kÀt kÀ1ÞÞB d vðt kÀ1ÞþZ t kt kÀ1expðAðtÀtÞÞB c uðtÞd tIn the general case:xðtÀkÞ¼xðtþkÞ:From previous definitions and expressions the PCS response can be schematised as in Fig.1.2.3Realisation of PCSThe PCS defined by S pðS;A;B c;B d;CÞrealisation is given in Fig.2a.This system is characterised by two inputs:uðtÞ; called the continuous input,and vðtÞ;called the input to be discretised,giving vðt kÞusing discretising defined by S. PCS evolution is possible only if uðtÞis bounded8t2= and if uðtÞis continuous and bounded8t¼t k2S:In these conditions,at each moment t k2S;the integrator output isx(t)B d v(t1)B d v(t2)B d v(t)tt1t2timet k t k+1t l t l+1B d v(t k)B d v(t k+1)x(t–1)x(t–2)x(t–k)x(t–k+1)Fig.1State evolution of PCSswitched to x ðt þk Þ¼B d v ðt k Þ;then the system evolutes from x ðt þkÞ;controlled by u ðt Þ;as in (5).At t k þ1the integrator output is switched to a new value x ðt þk þ1Þ¼B dv ðt k þ1Þ;and so on.In Fig.2b PCS inputs are symbolised with a point for v ðt Þ;to be discretised using S ,and the input is u ðt Þis marked by its usual symbol.2.4Particular casesIn this Section we give particular forms to S and to matricesA ;B c ;B d andC to model some relevant well-known systems.2.4.1Zero-order hold (ZOH):In this case define:A ¼0;B d ¼C ¼I n ;C ¼I n ;ðs ¼m ¼n Þ;I n is an n -dimensional unit matrix.Moreover,let u ðt Þ 0:Accordingto (4c )and (5),y ðt Þhas the expressiony ðt Þ¼v ðt k Þ;8t 2 t k ;t k þ1½In this case S p realises the ZOH of v ðt Þ:If S ¼f kT ;k ¼0;1;2;...g with T sampling period,S p is a constant sampling period ZOH.2.4.2Generalised sampled-data hold func-tion (GSHF):In this case we impose S ¼f kT ;k ¼0;1;2;...g ;where T is the sampling period.Moreover,u ðt Þ 0;8t 2=;that is equivalent to B c 0:Finally,to simplify note that v ðkT Þ¼v k :Under these conditions and according to (4c )and (5),y ðt Þis defined asy ðt Þ¼C exp ðA ðt ÀkT ÞÞB d v k ;8t 2 kT ;ðk þ1ÞT ½and 8k Note thatF ðt Þ¼X 1k ¼0G ðt ;k Þ;8twith G ðt ;k Þ¼C exp ðA ðt ÀkT ÞÞB d 8t 2 kT ;ðk þ1ÞT ½and G ðt ;k Þ 0otherwise.Therefore F ðt Þ¼F ðt þT Þand F ðt Þis integrable.From previous definitionsF ðt Þ¼G ðt ;k Þ;8t 2 kT ;ðk þ1ÞT ½and y ðt Þ¼F ðt Þv k Hence F ðt Þis T a periodic and integrable matrix defined in the space R m Âs :This implies that S p represents a generalised sampled-data hold function (GSHF)as defined by Kabamba [8].2.4.3Continuous system with imposed initial state:Define S ¼f t 0g and B d ¼I n ðs ¼n Þ:S p isa continuous system with the initial state defined byx ðt þÞ¼v ðt 0Þ;Fig.3.From (4b )and (5)we obtain x ðt Þ¼exp ðA ðt Àt 0ÞÞv ðt 0ÞþZ tt 0exp ðA ðt Àt ÞÞB c u ðt Þd t ;8t >t 02.4.4Plant model:A linear time-invariant systemwith unknown initial condition (Fig.4),can be modelled by S ¼f t 0g ;and B d ¼I n ðs ¼n Þ:Moreover,v ðt Þ x ðt Þ:In this case the initial state is unknown and the state vector evolution can be described from (4b )and (5)by the relationx ðt Þ¼exp ðA ðt Àt 0ÞÞx ðt 0ÞþZ tt 0exp ðA ðt Àt ÞÞB c u ðt Þd t ;8t >t 0It is obvious that x ðt Þis continuous 8t and x ðt þk Þ¼x ðt Àk Þ;8t k :2.5ConclusionIn the general case the system defined by S p has two inputspaces,the continuous one U r and the discrete one V s ;and two time spaces ðS and =Þ:Therefore S p has double character,both continuous and discrete.This characteristic enables the use of continuous and discrete sampled system properties.This is the object of the following Section where we propose to use this kind of system to define a new control architecture called piecewise continuous control,defined in Section3.a bFig.2Making of PCSa Detailed representationb SymbolicrepresentationFig.3Continuous system with imposed initial stateFig.4Continuous plant model3Piecewise continuous controllersLinear time-invariant continuous plant control is examined.In the first time the control architecture involving PCS is presented.Two particular cases are examined,the first one is a discrete tracking algorithm.In the second example the antiripple discrete tracking method is studied.3.1Control architectureThe control proposed is applied to continuous linear time-invariant plant.The architecture presented in Fig.5shows.Piecewise continuous controller defined by S p ðS ;a ;b c ;b d ;g Þwith S ¼f t k ;k ¼0;1;2;...j t k þ1>t k g and where l ðt Þis the state vector,c ðt Þis the state set points vector, ðt Þand ’ðt Þare bounded continuous input vectors,and u ðt Þis the output vector..Plant,as defined in Section 2.4.4by S p ðS A ;A ;B ;I n ;C Þwith S A ¼f t 0g ;B d ¼I n ðs ¼n Þ;v ðt Þ¼x ðt Þ;and by real-time invariant matrices A 2R n Ân ;B c 2R n Âr ;C 2R m Ân ;where x ðt Þ2R n is the state vector,a ðt Þ2R r is a bounded continuous input vector,and y ðt Þ2R m is the output vector.The PCS controller output is the plant input a ðt Þ¼u ðt Þ:.Therefore a double feedback configuration is realised:First feedback loop called the continuous loop,defined by the transformation’ðt Þ¼F ðx ðt Þ;c ðt ÞÞ:Second feedback loop called the discrete loop,defined by the transformationðt Þ¼C ðx ðt Þ;c ðt ÞÞ:Maps C and F are defined as the function of the control strategy adopted to achieve the desired control properties.This is developed in the following Sections.3.2Discrete trackingThe control strategy described by (6)is examinedx ððk þ1ÞT e Þ¼c ðkT e Þ;8k ¼0;1;2;...ð6Þwhere T e is the sampling period.Thus,the discrete tracking of the c ðt Þtrajectory is realised by x ðt Þ;with one sampling period delay.3.2.1Development:To satisfy (6),the followingsettings are defined:.Controller definition:dim ðl ðt ÞÞ¼dim ðx ðt ÞÞ¼n anddim ðu ðt ÞÞ¼dim ða ðt ÞÞ¼r ;hence dim ða Þ¼dim ðA Þ¼n Ân ;dim ðB Þ¼n Âr and dim ðg Þ¼r Ân ;S ¼f kT e ;k ¼0;1;2;...g :.Continuous feedback:’ðt Þ 0;or b c ¼0which is equivalent..To simplify notation we usex ðkT e Þ¼x k ; ðkT e Þ¼ k and c ðkT e Þ¼c kl ðkT þe Þ¼l þk and l ðkT Àe Þ¼l ÀkThen the evolution of the close-loop system over the time interval kT e ;ðk þ1ÞT e ½;is described byx 0ðt Þ¼Ax ðt ÞþBu ðt Þ;8t 2 kT e ;ðk þ1ÞT e ½ð7a Þl 0ðt Þ¼al ðt Þ;8t 2 kT e ;ðk þ1ÞT e ½ð7b Þu ðt Þ¼g ·l ðt Þ;8t 2 kT e ;ðk þ1ÞT e ½ð7c Þl þk ¼b dk ;8k ¼0;1;2;...ð7d ÞThenx k þ1¼exp ðAT e Þx k þM ðA ;B ;g ;a ;T e Þb d kð8ÞwithM ðA ;B ;g ;a ;T e Þ¼exp ðAT e ÞZT e0exp ðA ðÀt ÞÞB g exp ðat Þd tð9a ÞorM ðA ;B ;g ;a ;T e Þ¼exp ðAT e Þ~M ðA ;B ;g ;a ;T e Þð9b Þwith~MðA ;B ;g ;a ;T e Þ¼ZT e0exp ðA ðÀt ÞÞB g exp ðat Þd tNote that M ðA ;B ;g ;a ;T e Þ¼M and ~MðA ;B ;g ;a ;T e Þ¼~M ;when variables are well defined.M is defined in the space R n Ân :If M is invertible,the expression x k þ1¼c k can be achieved if we impose,according to (8),b d k ¼M À1½c k Àexp ðAT e Þx kð10ÞEquation (10)leads tob d ¼M À1ð11a Þandðt Þ¼c ðt ÞÀexp ðAT e Þx ðt Þð11b Þðt Þhas to be continuous,as x ðt Þis continuous c ðt Þhas also to be continuous.Matrices M À1and exp ðAT e Þare time invariant and finally the control architecture showed in Fig.6can be defined.The controller is completely defined by S p ðf kT e g ;a ;0;M À1;g Þ;with M defined by (9).The continuous feedback is not included in the control architecture.The discrete feedback is defined by (11b ).3.2.2Conditions of M À1ðA ;B ;g ;a ;T e Þmatrix existence:The conditions of M À1existence are thesame as for the ~MÀ1matrix.(i)Computation of ~M As ~M ¼R T e0exp ðA ðÀt ÞÞB g exp ðat Þd t ;letFig.5Piecewise continuous controllerexp ðA ðÀt ÞÞ¼X n À1j ¼0p j ðÀt ÞA jð12a Þexp ðat Þ¼X n À1j ¼0q j ðt Þa jð12b Þwhere real coefficients p j ðÀt Þand q j ðt Þðj ¼0;...;n À1Þ;denoted simply by p j and q j ;make two vectors P ðÀt Þand Q ðt Þ;denoted P and Q and defined byP T ¼½p 0;p 2;...;p n À1 ð13a ÞQ T ¼½q 0;q 1;...;q n À1ð13b ÞAccording to (12a )and (12b )~M ¼Z T e 0X n À1j ¼0p j A j "#B g Xn À1j ¼0q j a j "#d tor~M¼ZT e0X n À1j ¼0A jB ðp j I r Þ"#X n À1j ¼0ðq j I r Þga j "#d tthen from (13a )and (13b )~M ¼Z T eK ðP I r ÞðQ T I r ÞO d tð14Þwhere is the Kronecker multiplication operator andK ¼½B j AB j ...j A n À1B ;K 2R n Ânrð15a Þis the controlability matrix of the system (A,B)andO ¼½g T j a T g T j ...jða T Þn À1g T T ;O 2R nr Ânð15b Þis the observability matrix of the system ðg ;a Þ:Equation(14)can be rewritten~M ¼Z T eK ðPQ T I r ÞO d t ¼K Z T eðPQ T I r Þd t !ODefineP ¼ZT eP ðÀt ÞQ T ðt Þd tð16a Þand^P¼P I r ð16b Þthen~M¼K ^P O ð17Þ(ii)Conditions of existence for M À1matrixTheorem 3.1:Define a square matrix ðn Ân ÞM ðA ;B ;g ;a ;T e Þ;with A ðn Ân Þ;B ðn Âr Þ;g ðr Ân Þand a ðn Ân ÞandM ðA ;B ;g ;a ;T e Þ¼exp ðAT e ÞR T eexp ðA ðÀt ÞÞB g exp ðat Þd t :If we noteK ¼½B j AB j ...j A n À1B ;controllability matrix of the pair ðA ;B ÞO ¼½g T j a T g T j ...jða T Þn À1g T T ;observability matrix of the pair ðg ;a ÞP ¼R T e0P ðÀt ÞQ T ðt Þd t ;where P ðt Þand Q ðt Þare,respectively,co-ordinate vectors of exp ðAt Þand exp ða t Þover A i and a i ði ¼0;...;n À1Þand^P ¼P I r then M ðA ;B ;g ;a ;T e Þis invertible if and only if (a )Matrices K ,P and O verifyrank ðK Þ¼n (pair (A,B)is controllable),rank ðO Þ¼n (pair ðg ;a Þis observable),rank ðP Þ¼r with r r !n :(b )Ker f M ðA ;B ;g ;a ;T e Þg ¼f 0gKer ð^PÞ\Im ðO Þ¼f 0g ;Ker ðK Þ\Im ð^PÞ\Im ðO Þ¼f 0g ;Ker ðK Þ\Im ð^PO Þ¼f 0g Proof:Rank of matrices K,P and O ~M¼K ^O P :Therefore the rank of ~M verifies rank ð~MÞ min f rank ðK Þ;rank ð^P Þ;rank ðO Þg The rank of ~Mwill be equal to n ,if rank ðK Þ!n rank ð^PÞ!n and rank ðO Þ!n :The highest possible rank of K and O is n ,in fact K and O are rectangular matrices with the smallest size equal to n .Therefore it is necessary that rank ðK Þ¼n and rank ðO Þ¼n :Moreover,according to the structure of ^Pgiven by (16b ),if rank ðP Þ¼r ;then rank ð^PÞ¼r r :From this statement the new condition is obtained rank ðP Þ¼r and r r !n :(iii)Kernel of ~MIf the previous conditions are verified,rank ð~MÞ¼n if and only if Ker ðK ^PO Þ¼f 0g :It is well known that rank ðXY Þ¼dim ðIm ðY ÞÞÀdim ðIm ðY Þ\Ker ðX ÞÞ;whereXY is the product of two matrices [13].Therefore for K ^PO ;we obtainrank ððK ^PÞO Þ¼dim ðIm ðO ÞÞÀdim ðIm ðO Þ\Ker ðK ^P ÞÞrank ðK ð^PO ÞÞ¼dim ðIm ð^P O ÞÞÀdim ðIm ð^P O Þ\Ker ðK ÞÞKer ðK ^PO Þ¼Ker ðO Þ[½Ker ð^P Þ\Im ðO Þ [½Ker ðK Þ\Im ð^PÞ\Im ðO Þ whereKer ðK ^PO Þ¼Ker ðO Þ[½Ker ð^P Þ\Im ðO Þ [½Ker ðK Þ\Im ð^PO Þ ;according to the notation K ^PO ¼ðK ^P ÞO or K ^P O ¼K ð^PO Þ:As O 2R nr Ân rank ðO Þ¼n :Then Ker ðO Þ¼f 0g :ConditionsareFig.6Discrete trackingKer ð^PÞ\Im ðO Þ¼f 0g Ker ðK Þ\Im ð^PÞ\Im ðO Þ¼f 0g Ker ðK Þ\Im ð^PO Þ¼f 0g As rank ðM Þ¼rank ð~MÞthis ends the proof.3.2.3Numerical example:The second-ordercontinuous unstable plant is controlled by the PCS controller;all simulations have been realised with MATLAB software.Plant state space modelA ¼012À1 !;B ¼02 !;state vector x ¼x 1x 2 !PCS controller state space modela ¼À28À710 !;b ¼00!;g ¼1010½ ;state vector l ¼l 1l 2!In Fig.7a the plant state evolution from the initial state 00ÂÃand set points (imposed trajectory)defined asc ðt Þ¼À0:62Ãt 2þ5:7Ãt À5À0:17Ãt 3þ2:5Ãt 2À10Ãt!is given.The sampling period T e ¼1s :In Fig.7b the control generated by the PCS controller is given.In Fig.8the same example is simulated with the sampling period T e ¼0:1s and the plant initial state À510ÂÃ:The simulation is realised within 10s as in the previous example.3.3Antiripple discrete trackingThe architecture proposed in the previous Section uses,a generalised sampled-data hold device (cf .Section 2.4.2),with b c 0:In this case the continuous feedback is not used and the control system is functioning in open-loopconfiguration between sampling instants.This kind of control even if it assures tracking at sampling instants can generate important ripple.The solution proposed in this Section overcomes this problem and exploits all the possibilities of piecewise continuous control strategy.Therefore both continuous and discrete feedback are used in parallel and the proper choice of piecewise continuous system matrices enables optimal control tracking between sampling instants.3.3.1Development:The continuous plant is thesame as in the previous Section (cf.Section 3.1).Set points (imposed trajectory)are noted c ðt Þ:The control strategy is defined byx k þ1¼c k ;8k ¼0;1;2;...ð18a Þerror between c ðt ÀT e Þand x ðt Þto be as small aspossible over kT e ;ðk þ1ÞT e ½ð18b ÞNote w ðt Þ¼c ðt ÀT e Þ:Thus trajectory tracking is assured at sampling instants and error is minimised using an optimal control method.Condition (18a )is satisfied by the discrete and continuous feedback used to satisfy condition (18b ).Under these conditions the PCS satisfying all constraints and defined over kT e ;ðk þ1ÞT e ½†for the control systemx 0ðt Þ¼Ax ðt ÞþBa ðt Þð19Þ†to satisfy (18b )we use the criterionJ ¼12Zðk þ1ÞT ekT e½ðw ðt ÞÀx ðt ÞÞT E ðw ðt ÞÀx ðt ÞÞþa T ðt ÞGa ðt Þ d tð20Þwhere E 2R n Ân and G 2R r Âr are two symmetric positive024baT e 6810a (t )w 1(t )w 2(t )x 2(t )x 1(t )200150100500–50–20–10Fig.7a Tracking example,state and set points T e ¼1sb Tracking example,control T e ¼1s。