Robust control for discrete-time singular large-scale systems with parameter uncertainty

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Abstract

This paper addresses the problem of robust stabilization for uncertain discrete-time singular large-scale systems with parameter uncertainties. The system under consideration is not necessarily regular. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the robust stabilization problem is to design state feedback controllers such that, for all admissible uncertainties, the closed-loop system is regular, causal and stable. In terms of strict LMIs, sufficient conditions for the solvability of the problem is presented, and the parameterization of desired state feedback controllers is also given. A numerical example is given to demonstrate the applicability of the proposed design approach.

Introduction

During the past years, the problems of robust stability and robust stabilization for state-space systems with parameter uncertainties have attracted a lot of attention and significant advances have been made on these topics. For example, necessary and sufficient conditions for quadratic stabilizability are given in [1], while Refs. [2], [3] present some sufficient conditions for the existence of stabilizing state feedback controllers in terms of linear algebraic equations. These results are also extended to uncertain systems with time delays in [4], where the robust H control problem is studied, and state feedback controllers and observer-based controllers are designed, respectively. The corresponding results on the discrete-time case can be found in [5].

On the other hand, the control theory based on singular systems has been extensively studied for many years since singular system models have much wider applications than state-space systems in physical systems. Many notions and results in state-space systems have been extended to singular systems [6]. Very recently, lots of progress about the topics of robust stabilization and H control for singular systems has been reported in the literature [7]. It should be pointed out that the robust stability problem for singular systems is much complicated than that for state-space systems because it requires to consider not only stability and robustness, but also regularity and impulse immunity (for continuous singular systems) and causality (for discrete-time singular systems) simultaneously. The H control problem and robust stabilization for singular systems are investigated in [7], [8], [9], [10], [11], [12], [15]. The conditions for existence of desired dynamic controllers are presented in terms of non-strict LMIs and generalized algebraic Riccati equations, respectively. Similar to the case for state-space systems, in practical applications, parameter uncertainty in discrete-time singular systems is unavoidable. However, for discrete-time singular large-scale systems with parameter uncertainty, few results on the problems of robust stability analysis and robust stabilization have been reported so far; the study of such problems, however, is of both practical and theoretical importance.

In this paper, we investigate the problem of robust stabilization for uncertain discrete-time singular large-scale systems. The parameter uncertainties are assumed to be time invariant and unknown but norm-bounded. The objective is the design of state feedback controllers such that, for all admissible uncertainties, the close-loop system is regular, causal as well as stable. Sufficient conditions for the existence of robust controllers are obtained in terms of strict LMIs. When these LMIs are feasible, the parameterization of desired state feedback gains is also given.

Notation: Throughout this paper, for symmetric matrices X and Y, the notation XY (respectively, X>Y) means that the matrix XY is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension. The superscripts “T” represent the transpose. D(0, 1) is the open unit disk with center at origin. Matrices, if not explicitly stated, are assumed to have compatible dimensions. ΦiTRni×(ni-ri) denotes a matrix with the properties of EiTΦiT=0 and rank ΦiT=ni-ri. ((M)ij) denotes an n×n dimensional matrix, which has the form of((M)ij)=[0000M0000]n×n,where MRni×nj, 0 is a zero matrix with appropriate dimensions and i=1Nni=n.

Section snippets

Definitions and problem formulation

Consider a linear discrete-time singular large-scale system with parameter uncertainties described byEixi(k+1)=[Aii+ΔAii]xi(k)+j=1jiN[Aij+ΔAij]xj(k)+[Bi+ΔBi]ui(k),where i=1,2,…,N; xi(k)Rni is the state, ui(k)Rmi is the control input. The matrix EiRni×ni may be singular, we shall assume that rank Ei=rini and i=1Nni=n,i=1Nri=rn. Aii, Aij, Bi are known real constant matrices with appropriate dimensions. ΔAii, ΔAij, ΔBi are time-invariant matrices representing norm-bounded parameter

Robust stabilization

In this section, we shall first give a sufficient condition for the robust stability. Then a solution to robust stabilization problem for uncertain discrete-time singular large-scale system (1) is proposed, and an LMI approach will be developed.

Theorem 4

The uncertain discrete-time singular large-scale system (1) with Bi=0 and ΔBi=0 is robustly stable if there exist positive definite matrices Pi, symmetric matrices Si and a scalar ε>0 such that(AiiT(Pi-ΦiTSiΦi)Aii-EiTPiEi+QiAiiT(Pi-ΦiTSiΦi)(Pi-ΦiTSiΦi)Aii

Numerical example

In this section, we provide an example to demonstrate the effectiveness of the proposed method.

Consider an uncertain discrete-time singular large-scale system (1) with parameters as follows:E1=[10425211547],A11=[0.50.20.30.250.10.250.60.201],D11=[0.0020.0010.0010.0010.0020.0010.0010.0010.002],A12=[0.01-0.010.01-0.01-0.010.01],D12=[0.0010.0010.0010.00100.001],B1=[0.10.050.100.050.1],H1=[0.0010.0010.0010.00100.001],E2=[0.1000],A22=[0.050.05-0.050.1],D2=[0.0020.0010.0010.002],B2=[0.10.05],A21=[0.01

Conclusions

In this paper, we have studied the problem of robust stabilization for uncertain discrete-time singular large-scale systems with parameter uncertainties. Attention has been focused on the design of state feedback controllers. An LMI design approach has been developed. An example has been presented to demonstrate the proposed method.

References (15)

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This work is supported by NSF of China under Grant 60474078, Foundation for UKT by Ministry of Education, and EYTP of MDE, PR China.

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