Published in IET Control Theory and Applications Received on 26th July 2007 Revised on 21st December 2007 doi: 10.1049/iet-cta:20070277 ISSN 1751-8644 Robust root-clustering analysis in a union of subregions for descriptor systems O. Rejichi 1,2 O. Bachelier 2 M. Chaabane 1 D. Mehdi 2 1 UCPI, ENIS, University of Sfax, Route de Soukra, km 3.5, Sfax 3018, Tunisia 2 LAII, ESIP, University of Poitiers, 40 Avenue du Recteur Pineau, Poitiers Cedex 86022, France E-mail: ouiem.rejichi@etu.univ-poitiers.fr Abstract: A robust matrix root-clustering analysis for descriptor systems is considered. It states a necessary and sufficient condition for a descriptor system to be regular, impulse free and to have its finite poles in a specified region of the complex plane. This region is a union of convex and possibly disjoint and non-symmetric subregions. A sufficient condition to preserve those properties in the presence of polytopic and norm- bounded uncertainties affecting the state matrix is also established. This condition is based upon the implicit derivation of parameter-dependent Lyapunov functions. All the conditions are expressed in terms of linear matrix inequalities. 1 Introduction Singular systems [1, 2] also known as descriptor systems [3] or generalised systems [4] are an important class of dynamic system models from both theoretical and practical point of views since they can take algebraic constraints between physical variables into account. Actually, descriptor representation of dynamical systems is more general and often more natural than conventional state–space models. It is particularly useful to model and to handle systems such as mechanical structures [5], electrical networks [6], interconnected systems [7], and so on [1, 2]. In this paper we consider either continuous or discrete systems in descriptor form characterised by a pair of matrices (E, A) where fE; Ag [ fR nn g 2 . Matrix E may be singular so that it is assumed that rank(E) ¼ r  n. Because of the possible singularity of E, the matrix pencil associated with (E, A) has finite eigenvalues but might also have eigenvalues at infinity. All those eigenvalues are the poles of the corresponding descriptor system. The asymptotic stability of such a system, which concerns only the finite poles, is obviously of high importance. But in addition, there are two major concerns in descriptor system analysis that are the regularity and the impulse freeness. A system is said to be regular when it is well posed or in other words when there exists a unique solution to state– space equation. Moreover, a system is said to be impulse free when its response does not involve impulsive modes, which might occur because of infinite poles. These properties can be formulated by conditions on the pair (E, A) as in [1, 2]: the pair (E, A) is said to be † stable in the sense of Hurwitz (resp. in the sense of Schur) if det(sE 2 A) = 0, 8 s [ C þ (resp. 8s [ D þ ), where C þ stands for the closed right half complex plane (resp. where D þ stands for the closed exterior of the unit disc). † regular if det (sE 2 A) is not identically zero, † impulse free if deg fdet(sE 2 A)g ¼ rank(E) A pair (E, A) is said to be admissible if it is regular, impulse free and stable. Many notions and results developed for state – space systems have been successfully generalised to singular systems. The stability problem has been studied by many authors (see for example papers [2, 8–11]). More recently, finite root-clustering, also referred to as matrix pencil D-stability, also became a topic of interest. Indeed, even if the stability is a sine qua none requirement, there is no doubt that a level of performance is usually needed. For IET Control Theory Appl., 2008, Vol. 2, No. 7, pp. 615–624 615 doi: 10.1049/iet-cta:20070277 & The Institution of Engineering and Technology 2008 www.ietdl.org