Stability Analysis of Discrete LPV Systems Subject to Rate-Bounded Parameters ⋆ Chia-Po Wei * and Li Lee * * Electrical Engineering Department, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, (e-mail: leeli@mail.ee.nsysu.edu.tw) Abstract: This paper considers the stability analysis of the feedback connection of a discrete LTI system and time-varying parameters whose variation intervals and bounds of variation rates are assumed known. To tackle the problem, the robust D-admissibility of uncertain descriptor systems is first analyzed. Based on this result, we derive a necessary and sufficient LMI condition for the existence of a parameter dependent Lyapunov function to ensure the robust stability of the considered LPV system. In view of the infinitely many LMIs involved due to the uncertainty description, three sufficient conditions in finite number of LMIs are derived by means of the vertex separator, the D-G scaling, and the SOS relaxation techniques. Finally, a simple example is used to illustrate the effectiveness of the proposed method. 1. INTRODUCTION Linear parameter-varying (LPV) systems (Iwasaki and Shibata [2001]; Scherer [2001]) have received considerable attention because of their wide applicability in various fields such as gain scheduling (Rugh and Shamma [2000]) or model predictive control (Kothare et al. [1996]). Hence, there is a strong need for research on stability analysis of LPV systems. In this paper, we consider the stability analysis problem of a special class of LPV systems described by the feedback connection of a linear time-invariant system and time- varying parameters. Both information of the variation intervals and the bounds for the variation rate of these parameters are assumed available. Our approach is similar to that taken by Iwasaki and Shibata [2001]. The difference is that we focus on discrete-time systems, while they focus on continuous-time systems. From the considered LPV system, an augmented system is constructed so that the information of variation rate of each parameter can be exploited. The augmented system can be viewed as an uncertain descriptor system. This motivates us to consider the robust D-admissibility analy- sis of uncertain descriptor systems, which can be reduced to robust admissibility problems for continuous/discrete- time descriptor systems by choosing suitable D-regions. (The robust admissibility problems for continuous-time de- scriptor systems is studied in Iwasaki and Shibata [2001], and the D-admissibility problem is discussed in Wei and Lee [2007].) Based on an equivalent characterization of robust D-admissibility of the uncertain descriptor system, a sufficient condition in LMIs is first derived to ensure this property. To link with the stability issue of the discrete ⋆ This work was supported by National Science Council of Taiwan, R.O.C., under grant no. NSC 96-2221-E-110-087-MY2. LPV system, this sufficient condition is shown further to imply the exponential stability of the uncertain descriptor system. This result leads directly to a set of new suf- ficient condition which implies the exponential stability of the system augmented from the considered discrete LPV system. By relaxing the positive definite requirement of P solved from the new sufficient condition, we show that the new condition is equivalent to the existence of a parameter dependent Lyapunov function for the discrete LPV system, which depends on the parameters in a lin- ear fractional manner. Since the new sufficient condition involves infinitely many LMIs, three sufficient conditions in finite number of LMIs are derived by means of the vertex separator and the D-G scaling, proposed in Iwasaki and Shibata [2001], and the SOS relaxation, proposed in Scherer [2006], respectively. Finally, a simple example is used to compare our results with those from related studies (Amato [2006]; Daafouz and Bernussou [2001]; Oliveira et al. [1999]). The following notations are used in the sequel. N denotes the set of positive integers, and S n denotes the set of symmetric matrices of dimensions n × n. For a subset D in the complex plane, D c denotes the complement of D. For a matrix D, its transpose is denoted as D T and, when it is full-column rank, D † is used to denote any left inverse of it. For matrices M and N having the same number of columns, [M ; N ] is used to mean [M T N T ] T . Finally, the symbol ⊗ denotes the Kronecker product between two matrices. 2. ROBUST D-ADMISSIBILITY FOR RECTANGULAR DESCRIPTOR SYSTEMS In this section, the robust D-admissibility of rectangular descriptor systems is analyzed. The result will be applied in the next section to the stability analysis problem of Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 6383 10.3182/20080706-5-KR-1001.3380