Elsevier

European Journal of Control

Volume 63, January 2022, Pages 40-47
European Journal of Control

Spectrum assignment and stabilization by static output feedback of linear difference equations with state variable delays

https://doi.org/10.1016/j.ejcon.2021.08.001Get rights and content

Abstract

We study the problem of arbitrary eigenvalue spectrum assignment for a linear discrete-time control system with delays defined by a linear difference equation of the nth order with multiple inputs and multiple outputs via static output feedback with delays. Necessary and sufficient conditions are obtained for arbitrary eigenvalue spectrum assignment by linear static output feedback with delays. These conditions are expressed in terms of system coefficients. A corollary on stabilization is derived. An illustrative example is presented.

Introduction

Time delays often arise in many practical systems such as process control systems, mechanical systems, manufacturing, communication networks, automotive engineering control and chemical processes (see, e.g., [15], [38], [39]). The introduction of time delays in the system models naturally leads to stability problems. Stability analysis of time-delay systems has attracted a lot of attention during the last decades [13]. The main approach for stability analysis relies on the use of Lyapunov–Krasovskii functionals and linear matrix inequality (LMI) approach for constructing a common Lyapunov function. In the paper [37] asymptotic stability problem is studied for switched linear discrete systems with interval time-varying delays by using improved Lyapunov–Krasovskii functionals combined with LMI technique. In [42] a novel periodic Lyapunov–Krasovskii functional is proposed for stability analysis of discrete-time linear systems with time-varying delays. The approach of Lyapunov matrix equation presented in [11], [12], [18], [19] for continuous-time systems is used for solving the problem of exponential stability for delayed linear discrete-time systems in [22]. Some results on stability analysis for discrete time-delay systems based on new finite-sum inequalities are obtained in [40]. In [14], the output-feedback stabilization problem is solved for discrete-time systems with time-varying delay in the state within the linear matrix inequality (LMI) framework. In [8] sufficient conditions are obtained for global stabilization via dynamic output feedback of semi-linear system with delay in state. The paper [49] studies the delayed output feedback of linear discrete-time systems with input and output delays. The problem of dynamic output feedback stabilization of discrete-time linear systems with bounded input delay is studied in [47].

Another method for studying stabilization problems is in the problems of controlling over the eigenvalue spectrum of the system, that is, it is within the framework of the first Lyapunov method. The classical problem of spectrum assignment (for continuous-time systems without delays) is as follows. Consider a linear time-invariant control systemx˙=Ax+Bu,xKn is a state, uKm is a control (here K=C or K=R). The controller has the form of linear static state feedbacku=Qx.The closed-loop system (1), (2) has the formx˙=(A+BQ)x.One needs to construct, for any γiK, i=1,n¯, a gain matrix Q such that the characteristic polynomial of the matrix A+BQ of the system (3) coincides with the polynomialλn+γ1λn1++γn.This problem was solved in [34] (for K=C) and in [48] (for K=R): it was proved that complete controllability of (1) is necessary and sufficient for arbitrary spectrum assignability of system (3). The corresponding results are true for discrete-time systems.

One of the first works on spectrum assignment for systems with delays was [32]. One of the possible formulation of this problem is as follows (see, e.g., [2], [20]). Consider a linear time-invariant control system with (commensurate lumped) delaysx˙(t)=i=0s(Aix(tih)+Biu(tih)),t>0,with initial conditions x(ζ)=μ(ζ)Kn, ζ[sh,0]; here xKn is a state, uKm is a control, h>0 is a constant delay. Let the controller have the form of linear static state feedback with delaysu(t)=j=0βQjx(tjh),x(t)=0, t<sh. The closed-loop system (5), (6) has the formx˙(t)=i=0s(Aix(tih)+j=0βBiQjx(tihjh)).The characteristic function φ(λ) of system (7) is a quasipolynomialλn+k=0n=0ξδkλkeλh.Here integer ξ0 and numbers δk depend on coefficients of system (7). The set Λ={λC:φ(λ)=0} forms the spectrum of system (7). If ΛC:={λC:Reλ<0}, then system (7) is exponentially stable. The spectrum of system (7) consist of an infinite set of numbers, in general. It is uniquely determined by the coefficients of system (7). If for any integer ξ0 and any numbers δkK there exist β0 and Qj, j=1,β¯ such that the characteristic function of the closed-loop system (7) coincides with (8), then system (5) is called arbitrary spectrum assignable by feedback (6). If the characteristic function of the closed-loop system (7) can be reduced to the polynomial (4) with any pregiven coefficients, then system (5) is said to be arbitrary finite spectrum assignable by feedback (6). It is clear that arbitrary spectrum assignability implies exponential stabilizability of the system with any pregiven decay of rate.

For time-delay systems, there are various formulations of problems of eigenvalue assignment (or stabilization): delays can be lumped and/or distributed, commensurate and/or noncommensurate; delays can be in state, and/or in control, and/or in feedback; the controller can take various forms; the system can be of retarded or neutral type etc. Every type of the system like (5) with some controller like (6) generates a separate formulation of the problem (see, e.g., [7], [24], [25], [26]). The problems of spectrum assignment by static state feedback for various classes of regulators have been studied in [23], [27], [28], [30], [31], [41], [45], [46]. In [16], necessary and sufficient conditions have been obtained for strong stabilizability of neutral functional differential equations by proportional and derivative state feedback.

For discrete-time systems with delays like (5), (6), the spectrum assignment problem by static state feedback can be reduced to the corresponding problem for augmented systems without delays (see the transformation, e.g., in [13, Section 6.1, 6.2]). So, this problem is solved by using methods of the well-known theory of eigenvalue assignment with static state feedback (see, e.g., [1, Section 4.2]). Note that in the recent works [3], [4], [5] this theory was extended to linear time-varying discrete-time systems.

The problem of eigenvalue assignment (in particular, of stabilization) by static output feedback is one of the most important and difficult open questions in control theory [36]. In the continuous-time formulation, this problem is as follows. For the systemz˙=Fz+Gu,ξ=Hz,zKn, uKm, ξKk, one needs to construct a linear static output feedback control u=Qξ such that the characteristic polynomial of the matrix F+GQH of the closed-loop systemz˙=(F+GQH)zcoincides with the polynomial (4) with an arbitrary pregiven γiK, i=1,n¯. This problem has been studied by various authors for many years. The most essential results have been obtained in [9] for K=C and in [35], [43] for K=R. Some new results on eigenvalue assignment by static output feedback were obtained in recent works [6], [21], [33], [44], [52], [53], [56]. Although there is a huge amount of papers on static output feedback, however, as noted in [36], “so far, there has been no exact solution to this prominent problem which can guarantee the design of static output feedback or determine that such a feedback does not exist”. All this also applies to discrete-time systems.

The corresponding problems for time-delay systems are much more difficult. Consider a linear time-invariant control system with (e.g., noncommensurate lumped) delaysz˙(t)=i=0s(Fiz(thi)+Giu(thi)),t>0,ξ(t)=j=0qHjz(tdj),z(ζ)=z0(ζ)C([dqhs,0],Kn), u(ζ)=u0(ζ)C([dqhs,0],Km), 0=h0<h1<<hs and 0=d0<d1<<dq are constant delays; zKn is a state vector, uKm is the input vector, ξKk is the output vector. The problems of spectrum assignment and stabilization of system (9) by means of dynamic output feedback (for some partial cases of system coefficients) in various classes of regulators were studied, e.g, in [10], [29], [57] and, for the similar discrete-time systems, in [47], [49], [55]. The problems of spectrum assignment or stabilization of system (9) by means of static output feedback in any class of regulators looks insuperable in a general case. Therefore, any partial progress in this direction is essential. In the paper [54], the problem of arbitrary spectrum assignment by means of static output feedback with delays have been solved for a time-invariant control system defined by a linear differential equation with multiple non-commensurate lumped delays in state with multiple inputs and multiple outputs. In the present paper we study the problem of arbitrary spectrum assignment for discrete-time control system defined by a difference equation of nth order with delays with multiple inputs and multiple outputs by static output feedback with delays.

Notations. N is the set of natural numbers; N0:={0}N; K=C or K=R; a¯ is the complex conjugate number to aK; Mn,m(K) is the space of n×m-matrices over K; Mn(K):=Mn,n(K); T is the transposition of the matrix; * is the Hermitian conjugation, i.e., A*=A¯T; IMn(K) is the identity matrix; [e1,,en]:=I; A0:=I for any AMn(K); J:={ϑij}Mn(K) where ϑij=1 for j=i+1 and ϑij=0 for ji+1; SpQ is the trace of a matrix QMn(K).

Section snippets

Problem statement

Consider a linear discrete-time control system defined by a linear difference equation of nth orderx(t+n)+a1x(t+n1)+a2x(t+n2)++anx(t)=u(t),tN0, with initial conditions x(ζ)=μ(ζ)K, ζ=0,,n1; here xK is the state variable, uK is the control variable, aiK, i=1,n¯. For any u(t), tN0, the initial value problem for Eq. (10) has the unique solution x(t), tN0 (see [17, Theorem 3.2]). System (10) can be rewritten in the vector form (see [17, Section 4.1]): definezi(t):=x(t+i1),i=1,n¯,z(t):=

Proof of Theorem 3

From (25), we haveyβ(t)=ν=1pc¯νβx(t+ν1),β=1,k¯.Hence, by (23) and (26),uα(t)=ξ=0θβ=1kqαβξ(ν=1pc¯νβx(t+ν1ξn)),α=1,m¯.The right-hand side of (24) has the form l=pnα=1mblαuα(t+nl). Thus, the closed-loop system (24), (25), (23) takes the formx(t+n)+i=1n+haix(t+ni)l=pnα=1mblαξ=0θβ=1kqαβξ×(ν=1pc¯νβx(t+nl+ν1ξn))=0.Let ψ(λ) be the characteristic function of the closed-loop system (27). Thenψ(λ)=λn+i=1n+haiλnil=pnα=1mblα(ξ=0θβ=1kqαβξ×(ν=1pc¯νβλnl+ν1ξn)).

Consider the

Discussion

Remark 2

Theorem 1 follows from Theorem 3 for h=0, σ=0, τ=0, θ=0. So, Theorem 3 is an extension of Theorem 1 to systems with delays.

Remark 3

System (20), (21) is a partial case of system (24), (25), (23) if m=1, k=n, p=n,B=enMn,1(K),C=[en,,e1]Mn(K).If the coefficients of system (24), (25) have the form (42) then it is easy to check that the matrices (19) are linearly independent. Hence, by Theorem 3, system (24), (25) is arbitrary spectrum assignable by static output feedback (23). So, Theorem 3 is an

Example

Let K=R. Consider the following system (24), (25):x(t+4)2x(t+1)x(t)+2x(t3)=u1(t+2)+u1(t+1)u2(t+1)+u2(t),y1(t)=x(t)+x(t+1),y2(t)=x(t)x(t+1),xR, u=col(u1,u2)R2, y=col(y1,y2)R2. We have: n=4, m=2, k=2, p=2, h=3; a1=a2=0, a3=2, a4=1, a5=a6=0, a7=2, B=[00101101], and C=[11110000]. Note that the free system is not stable: the roots λj of the characteristic function ψ(λ)=λ42λ1+2λ3 do not belong to Ω. Construct matrices (19). We haveC*B=[1010],C*JB=[2101],C*J2B=[1012],C*J3B=[0101].

Conclusion

In this paper we have studied the problem of arbitrary eigenvalue spectrum assignment for a control system defined by a linear difference equation of the nth order with delays with a multiple inputs and multiple outputs via static output feedback with delays. We have obtained necessary and sufficient conditions for arbitrary eigenvalue spectrum assignment by linear static output feedback with delays. These conditions are expressed in terms of system coefficients. Note that the proof of the main

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The research of the first and second authors were supported by the National Science Centre in Poland under Grant DEC-2017/25/B/ST7/02888. The research of the third author was funded by the Polish National Agency for Academic Exchange NAWA (the Ulam program) granted according to the decision No. PPN/ULM/2019/1/00287/DEC/1. The research of the fourth author was partially funded by the Russian Foundation for Basic Research (project number 20–01–00293) and by the Ministry of Science and Higher

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