Yamin YAN,Jie HUANG

Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong,Shatin,Hong Kong.Received 31 October 2015;revised 25 November 2015;accepted 25 November 2015

Output regulation problem for discrete-time linear time-delay systems by output feedback control

Yamin YAN,Jie HUANG?

Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong,Shatin,Hong Kong.Received 31 October 2015;revised 25 November 2015;accepted 25 November 2015

In this paper,we study the output regulation problem of discrete linear time-delay systems by output feedback control.We have established some results parallel to those for the output regulation problem of continuous linear time-delay systems.

Output regulation,discrete linear systems,time-delay

1 Introduction

Theoutputregulationproblemofcontinuous-timelinear systems was thoroughly studied in[1-4],to just name a few.The output regulation problem of discretetime linear systems was also studied in[4]and chapter 1 ofthebook[5].Itisknownthatthesolvabilityoftheoutput regulation problem for either continuous-time linear systems or discrete-time linear systems depends on the solvability of a set of matrix equations called the regulator equations[4,5].Since the regulator equations for continuous-time linear systems and discrete-time linear systems are exactly the same,the results on the output regulation problem for continuous-time linear systems can be directly applied to the output regulation problem for discrete-time linear systems.

Since the 2000s,there have been some studies on the output regulation problem of continuous-time linear time-delay systems[6-11].While references[6]and[7]studied the output regulation problem of linear time-delay systems using the operator approach,references[8-11]studied the output regulation problem of continuous-time linear time-delay systems using the finite-dimensional linear state space techniques.In particular,references[8]and[9]studied a quite general class of continuous-time linear time-delay systems and established the solvability conditions of the problem in terms of some matrix equations such as the regulator equations.

In this paper,we will consider the output regulation problem for discrete-time linear time-delay systems of the form(1)to be introduced in the next section.The topic of this paper can be viewed as the discrete counterpart of the topic in[8]in which the output regu-lation problem of continuous-time linear systems with both state and input time-delays was studied.In order to simplify the notation,we only consider the input time-delay in this paper.It will be seen that,unlike the case without time-delay,certain key matrix equations such as the so-called discrete regulator equations associated with the problem in this paper are somehow different from those equations associated with the continuous-time case in[8].Thus,it is of interest to give the discrete-time case an independent treatment.Another motivation for this paper arises from studying the cooperative output regulation problem for discrete-time linear time-delay multi-agent systems which is quite different from its continuous-time counterpart.This paper will pave the way for studying this problem.

The rest of this paper is organized as follows.Section2 gives the problem formulation and some preliminaries.Section3 establishes three lemmas for laying the foundation of the problem.Section4 presents our main result.An example is used to illustrate our design in Section5.The paper is closed with some concluding remarks in Section6.Finally,we note that the preliminary version of this paper appeared in[12]where the output regulation problem for a system somehow different from(1)was studied by both the state feedback control and the output feedback control.

Notationσ(A)denotes the spectrum of a square matrixA.ForXi∈Rni,i=1,...,m,col(X1,...,Xm)=[XT1···XTm]T.ForX=[X1···Xm]whereXi∈ Rn×1,vec(X)=col(X1,...,Xm).For some nonnegative integerr,I[-r,0]denotes the set of integers{-r,-r+1,...,0}and C(I[-r,0],Rn)denotes the set of functions mapping the integer setI[-r,0]into Rn.Z+={0,1,...}.

2 Problem formulation and preliminaries

We consider the output regulation problem for discrete-time linear time-delay systems of the following form:

wherex∈Rnis the state,u∈Rmthe input,e∈Rpthe error output,ym∈Rpmthe measurement output,randrl,l=0,...,h,are nonnegative integers satisfying 0=r0＜r1＜r2＜...＜rh=r＜∞,v∈Rqis the measurable exogenous signal such as the reference input to be tracked,andw∈Rsis the unmeasurable exogenous signal such as the external disturbance.We assume thatvandware generated,respectively,by the exosystem of the following form:

withS0∈ Rq×qa constant matrix,and

withQ∈ Rs×sa constant matrix.

We will consider the output feedback control law as follows:

wherez∈ R(n+s),K1∈ Rm×n,K2w∈ Rm×sandK2v∈Rm×qare some constant matrices to be specified later.

Remark 1The control law(4)is based on a combined controller and observer design.This control law can be put in the following more standard form:

A special case of the measurement output feedback control is the dynamic error output feedback control whenym=e.In many cases,the error outputeis not the only measurement variable available for feedback control.Using the measurement output feedback control allows us to solve the output regulation problem for systems that cannot be solved by the error output feedback control.

The composition of system(1),and the control law(5)is called the closed-loop system and can be put in the following form:

wherexc=col(x,z),?v=col(v,w),and various matrices are given by

Now we describe our problem as follows.

De fi nition 1Linear output regulation problem:Design a control law of the form(4)such that the closedloop system(7)satisfies the following two properties.

Property 1The closed-loop system(7)is exponentially stable whenv=0 andw=0,i.e.,the systemis exponentially stable.

Property 2For any initial conditions andw0∈Rs,the trajectory of system(7)satisfies

Clearly,the above problem is a generalization of the outputregulationproblemforlinearsystemswithoutdelay as studied in[1,2,5].In order to solve the problem,we list some assumptions as follows.

Assumption 1S0andQhave no eigenvalues with modulus smaller than 1.

Assumption 2There exists a matrixK1∈ Rm×nsuch that the systemx(t+1)=is exponentially stable.

Assumption 3The pairis detectable.

Assumption 4There exists a pair of matrices(X,U)that satisfies the following matrix equations:

Remark 2Since the modes associated with the eigenvalues ofS0andQwith modulus smaller than 1 will not contribute to the steady-state of the closed-loop system satisfying Property1,Assumption 1 will not lose the generality of our result.It is made only for establishing the necessary condition of our main result.

Remark 3It is known that the system

is exponentially stable if and only if all the roots of the following characteristic polynomial of(11)

have modulus smaller than 1.We call the equation Δ(λ)=0 the characteristic equation of(11).As a result,Assumption 2 is satisfied if and only if there exists a matrixK1∈ Rm×nsuch that all the roots of the following polynomial:

have modulus smaller than 1.

Remark 4Whenh=0,Equations(10)reduces to the regulator equations associated with linear systems without time delay as can be found,say,in[5,Chapter 1].However,in the presence of the time-delay,Equations(10)are somehow different from the regulator equations associated with continuous-time linear time-delay systems[8].In what follows,we call(10)the discrete regulator equations.It will play the same role as what the regulator equations do for continuous-time linear time-delay systems.

3 Three lemmas

In this section,we will first establish three lemmas for laying the foundation for studying the output regulation problem of discrete-time linear time-delay systems.

Lemma 1Suppose all the eigenvalues of the matrixS∈Rˉq×ˉqhave no modulus smaller than 1 and the systemis exponentially stable.Then,for anythe following matrix equation:

has one unique solutionX∈Rn×ˉq.

ProofUsing the properties of the Kronecker product,equation(14)can be transformed into the following standard form:

Thus,equation(14)has a unique solution for any matrixˉBif and only ifis nonsingular.Denote the eigenvalues ofSby λi,i=1,..and let Δ(λ)=Toobtaintheconditionunderwhichis nonsingular,similar to the proof of Theorem 1.9 of[5],we assume,without loss of generality,thatSis in the following Jordan form:

whereJihas dimensionnisuch thatn1+n2+...+nk=ˉqand is given by

where λi∈ σ(S).A simple calculation shows,for any integerm＞0,is an upper triangular matrix with its diagonal elements being.Thus,the matrixQˉ is a block lower triangular matrix ofkblocks with itsith,1≤i≤k,diagonal block having the form

Lemma 2Under Assumption 1,consider the controller(4).Assume the closed-loop system(7)has Property1.Then,the controller solves the linear output regulation problem if and only if there exists a unique matrixXc∈ Rnc×(q+s)withnc=2n+sthatsatisfies thefollowing matrix equations:

ProofSince the closed-loop system(7)has Property1,by Lemma 1,the first equation of(19)has a unique solutionXc.Letˉxc(t)=xc(t)-Xc?v(t).Then,

That is,the controller solves the linear output regulation problem.

for all?v(t)=St?v(0)with any?v(0)∈R(q+s).Under Assumption 1,?v(t)will not decay to zero.Therefore,

Remark 5From the proof of Lemma 1,the solvability of(14)does not require that none of the eigenvalues of the matrixShave modulus smaller than 1.It suffices to require that the eigenvalues of the matrixSdo not coincide with the roots of Δ(λ).Thus,Assumption 1 is not necessary for the validity of Lemma 2.

We close this section by providing the solvability condition for the discrete regulator equations(10).

Lemma 3For any matricesEandF,the regulator equations(10)are solvable if and only if,for all λ ∈ σ(S),

ProofThe idea of the proof is similar to the proof of[5,Theorem 1.9].Like the proof of Lemma 1,using the properties of the Kronecker product,equation(10)can be transformed to the following standard form:

Thus,equation(24)hasauniquesolution forany?bifand only if?Qis nonsingular.Similar to the proof of Lemma1,we assumeSis in the Jordan form(16)and thusJiis in the form(17).A simple calculation shows that the matrix?Qis a block lower triangular matrix ofkblocks with itsith,1≤i≤k,diagonal block having the form

4 Main result

In this section,we will present our main result.

Theorem 1Under Assumptions 2 and 3,the linear output regulation problem is solvable by the output feedback controller of the form(4)if Assumption 4 is satisfied,andundertheadditionalAssumption1,thelinear output regulation problem is solvable by the output feedback controller of the form(4)only if Assumption 4 is satisfied.

Proof(If part) For system(1),lump the statexand the disturbanceswtogether to obtain the following system:

Let(X,U)satisfythediscreteregulatorequations(10),K2=U-K1X,and partitionK2asK2=[K2vK2w]whereK2v∈Rm×qandK2w∈Rm×s.Letˉx=x-X?v,ˉu=u-U?v,ze=z-col(x,w).Then it can be verified that

Substituting(28)into(29)gives

Denote the characteristic polynomial of the systemand system(30)by Δx(λ)and Δz(λ),respectively.Then it is ready to see thatthecharacteristicpolynomialoftheclosed-loopsystem composed of(32)and(30)is given by Δx(λ)Δz(λ).Under Assumption 2,all the roots of Δx(λ)have modulus smaller than 1,and,by our design ofL,all the roots of Δz(λ)have modulus smaller than 1.Thus,all the roots of Δx(λ)Δz(λ)have modulus smaller than 1.Thus,by Remark 3,the closed-loop system composed of(32)and(30)is exponentially stable.Thus,we have,Therefore,from(28),we haveFinally,from(31),we haveThus,the proof is completed.

(Only if part) Under the control law(4),the closedloop system can be put in the form(7).Since the linear output regulation problem is solved by the control law(4),by Lemma 2,equations(19)has a unique solutionLetXc=col(X,Z)whereandZDecompose equations(19)into the following form:

LettingU=KzZ+[K2v0]in the first and the third equations of(33)completes the proof. □

5 An example

Consider the discrete time-delay systems of the form(1)with

The exogenous signalsvandware generated by(2)and,respectively,by(3)with

LettingK1=[-0.075-0.465]gives the roots of the polynomial det(λI2-A-B0K1-B1K1λ-1)as follows:

which are all inside the unit circle.Thus,Assumption 2 is satisfied.Assumption 3 is also satisfied.In fact,letting

With random initial conditions,the control input is as shown in Fig.1 which is bounded,and the error outputeof the system is as shown in Fig.2 which approaches the origin asymptotically.

Fig.1 The control input u(t)of the system.

Fig.2 The tracking error e(t)of the closed-loop system.

6 Conclusions

In this paper,we have studied the output regulation problem of discrete-time linear time-delay systems by measurement output feedback control law where the delay is assumed to be known and constant.We will further consider the random delays as studied in,say,[13]and[14].

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DOI10.1007/s11768-016-5110-1

?Corresponding author.

E-mail:jhuang@mae.cuhk.edu.hk.Tel.:+852-39438473;fax:+852-26036002.

This work was supported in part by the Research Grants Council of the Hong Kong Special Administration Region(No.412813)and in part by the National Natural Science Foundation of China(No.61174049).

her B.Eng.degree in 2013 from Sichuan University,Chengdu,China.She is currently a Ph.D.candidate in the Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong,Hong Kong,China.Her research interests include discrete-time systems,outputregulation,and time-delaysystems.E-mail:ymyan@mae.cuhk.edu.hk.

Jie HUANGis Choh-Ming Li professor and chairman of the Department of Mechanical and Automation Engineering,The Chinese University of Hong Kong,Hong Kong,China.Hisresearchinterestsincludenonlinear control theory and applications,multiagent systems,and flight guidance and control.Dr.Huang is a Fellow of IEEE,a Fellow of IFAC,and a Fellow of CAA.E-mail:jhuang@mae.cuhk.edu.hk.