Automatica 44 (2008) 792 – 798 www.elsevier.com/locate/automatica Brief paper Robust filtering for uncertain linear discrete-time descriptor systems  Carlos E. de Souza a , ∗ , Karina A. Barbosa a , Minyue Fu b a Department of Systems and Control, Laboratório Nacional de Computação Científica (LNCC/MCT), Av. GetúlioVargas 333, Petrópolis, RJ 25651-075, Brazil b School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia Received 8 August 2006; received in revised form 12 February 2007; accepted 13 July 2007 Abstract This paper is concerned with the problem of robust filtering for uncertain linear discrete-time descriptor systems. The matrices of the system state-space model are uncertain, belonging to a given polytope. A linear matrix inequality based method is proposed for designing a linear stationary filter that guarantees the asymptotic stability of the estimation error and gives an optimized upper bound on the asymptotic error variance, irrespective of the parameter uncertainty. The proposed robust filter design is based on a parameter-dependent Lyapunov function, which is shown to outperform parameter-independent ones.  2007 Elsevier Ltd. All rights reserved. Keywords: Robust Kalman filters; Descriptor systems; Uncertain linear systems; Parameter-dependent Lyapunov functions; Discrete-time systems 1. Introduction Descriptor systems (also known as singular, implicit or differential-/difference-algebraic systems) are an important class of dynamic systems from both a theoretical and practical points of view due to their capacity to describe algebraic con- straints between physical variables (see, e.g., Xu & Lam, 2006 and the reference therein). A great deal of interest has been devoted in the last decade or so to Kalman filtering methods for linear discrete-time de- scriptor systems; see, for instance, Bianco, Ishihara, and Terra (2005), Dai (1989), Deng and Liu (1999), Ishihara, Terra, and Campos (2005), Nikoukhah, Willsky, and Levy (1992), Nikoukhah, Campbell, and Delebecque (1999) and Zhang, Xie, and Soh (1999). The aforementioned methods rely on the knowledge of a perfect system model and they may fail to provide a guaranteed error variance when only an approximate model is available. In the context of robust Kalman filtering,  A preliminary version of this paper was presented at the 14th IFAC Symposium on System Identification, held in Newcastle, Australia. This paper was recommended for publication in revised form by Associate Editor Ben M. Chen under the direction of Editor Ian Petersen. ∗ Corresponding author. Tel.: +55 24 22336012; fax: +55 24 22336141. E-mail addresses: csouza@lncc.br (C.E. de Souza), karinab@lncc.br (K.A. Barbosa), minyue.fu@newcastle.edu.au (M. Fu). 0005-1098/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2007.07.006 only very recently this problem was addressed in Ishihara, Terra, and Campos (2004) for descriptor systems with norm- bounded parameter uncertainties, where a Riccati equation ap- proach was proposed. Although norm-bounded parameter un- certainties are important to consider, most uncertain system models are much better described by polytopic structures; see, e.g., Boyd, El Ghaoui, Feron, and Balakrishnan (1994). Indeed, polytopic structures arise naturally when there are multiple real- valued uncertain parameters. Using norm-bounded structures typically over-estimates the uncertainties in the system. To the best of the authors’ knowledge, the problem of robust Kalman filtering for linear discrete-time descriptor systems with poly- topic uncertainty has not yet been investigated. This paper addresses the problem of robust filter design for linear discrete-time descriptor systems with polytopic-type un- certainties, namely the matrices in the system state-space model are uncertain and assumed to belong to a given polytopic set. We develop a linear matrix inequality (LMI) method for de- signing a linear stationary filter that provides the asymptotic stability of the estimation error and an optimized upper bound on the asymptotic error variance, irrespective of the parame- ter uncertainty. The proposed robust filter design is based on a parameter-dependent Lyapunov function to achieve improved performance. An example is presented in the paper to illustrate this feature.