Robust control for discrete-time singular large-scale systems with parameter uncertainty☆
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 X⩾Y (respectively, X>Y) means that the matrix X−Y 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. denotes a matrix with the properties of and rank . ((M)ij) denotes an n×n dimensional matrix, which has the form ofwhere , 0 is a zero matrix with appropriate dimensions and .
Section snippets
Definitions and problem formulation
Consider a linear discrete-time singular large-scale system with parameter uncertainties described bywhere i=1,2,…,N; is the state, is the control input. The matrix may be singular, we shall assume that rank Ei=ri⩽ni and . 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
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:
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)
- et al.
Robust observer-based H∞ controller design for linear uncertain time-delay system
Automatica
(1997) - et al.
H∞ Control of discrete-time-time linear systems with norm-bounded uncertainties and time delay in state
Automatica
(1998) - et al.
Stabilization of discrete-time-time singular systems: a matrix inequality approach
Automatica
(1999) - et al.
Robust stabilization for uncertain discrete-time singular systems
Automatica
(2001) Necessary and sufficient conditions for quadratic stabilizability of an uncertain system
J. Optim. Theory Appl.
(1985)- et al.
Decentralized robust control for linear uncertain interconnected systems
Inform. Control
(1998) - et al.
Stabilization of an uncertain linear dynamic system by state and output feedback: a quadratic stabilizability approach
Int. J. Control
(1996)
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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.