Robust control analysis and design for discrete-time singular systems

-In this paper, we propose a simple approach to analyse stability robustness of discrete-time singular systems under structured perturbations. The developed robustness criteria are then applied to solve robust regional poleassignment problems of singular systems. A robust control design algorithm, via state feedback, is also given. The robust stability problem of singular systems is more complicated than that of regular systems. Not only stability robustness but system regularity and impulse elimination should be considered simultaneously. Since the results of robust control and analysis for singular systems is not available in the literature as much as other fields, the paper may be viewed as a complementary result in this field. Although only discrete-time case is discussed, several results can be directly applied to continuous-time systems as well. 1. NOTATION THE FOLLOWINO notation will be used throughout the paper p(M): spectral radius of M E C "×" IMIm: modulus matrix of M ~ C nx" IMl: determinant of matrix M E C "×" JZ]: magnitude of complex number z D(-h, f): a disk centered at point h +jO with radius f *Received 4 August 1992; revised 10 August 1993; received in final form 10 February 1994. The original version of this paper was presented at the 12th IFAC World Congress which was held in Sydney, Australia, during 18-23 July 1993. The Published Proceedings of this IFAC Meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, U.K. This paper was recommended for publication in revised form by Associate Editor Vladimir Ku~era under the direction of Editor Huibert Kwakernaak. Corresponding author Dr L. Lee. Fax 886 7 5615137; E-mail [email protected]. t This work was supported by the National Science Council of Taiwan, R.O.C. under Grant No. NSC-82-0404-E151-025. Department of Electrical Engineering, National Kaohsiung Institute of Technology, Kaohsiung 807, Taiwan, R.O.C. § Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, R.O.C. II Department of Electrical Engineering, National Taiwan University, Taipei 104, Taiwan, R.O.C. 1741 mij: the (i, j)th element of matrix M M>-N: mij, nijER and mij>-n~j for i , j = 1, 2 , . . . , n AAA: end of statement in quoted lemmas Q.E.D.: end of proof. 2. INTRODUCTION In the past ten years, there has been a growing interest in the system-theoretic problems of singular systems due to the extensive applications of singular systems to large-scale systems, circuits, economics, polynomial matrices and other areas (Luenberger, 1977; Verghese et al., 1981; Lewis, 1986; Dai, 1989; Fang and Chang, 1991, 1992). Sometimes the system is called generalized state-space systems, or implicit systems, or descriptor systems or semistate systems (Lewis, 1986). Several important and fundamental results in standard state-space systems have been successfully extended to singular systems (Cobb, 1984; Bender and Laub, 1987; Shayman, 1987; Zhou et al., 1987; Kucera and Zagalak, 1988; Fang and Chang, 1990). Many excellent design methods for singular system control have also been well developed recently (Kucera and Zagalak, 1991; Paraskevopoulos and Koumboulis, 1991). However, little effort has been devoted to studying the robust control problems of singular systems. In this paper, we will investigate such problems. Consider a discrete-time perturbed singular where E, A e R " x perturbations. singular. We system Ex(k + 1) = Ax(k) + AAx(k), Ex(O) = Exo, (1) and AA stands for the Here the matrix E may be will assume rank E r -< n. In 1742 CHtm-HsIUNG FANG et al. system (1), AA denotes an n × n time-invariant structured perturbation matrix. The perturbations in the various elements of the system matrix A are independent of one another (Sobel et al., 1989; Tesi and Vicino, 1990; Juang, 1991). In a practical situation, the perturbation matrix is not known exactly but the magnitude of the deviation which can be expected in the entries of A may be known. The perturbation considered in this paper is described as