Automatica 43 (2007) 491 – 498 www.elsevier.com/locate/automatica Brief paper Comparisons among robust stability criteria for linear systems with affine parameter uncertainties  Guang-HongYang a , ∗, 1 , Kai-Yew Lum b a College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China b Temasek Laboratories, National University of Singapore, 5 Sports Drive 2, Singapore 117508, Republic of Singapore. Received 30 July 2004; received in revised form 3 September 2006; accepted 20 September 2006 Abstract This paper is concerned with the problem of comparisons among robust stability criteria for a class of uncertain linear systems, where the system state matrices considered are affinely dependent on the uncertain parameters. At first, a robust stability criterion for the class of systems to be affinely quadratically stable (AQS) is derived based on the vertex separator approach, where affine parameter-dependent Lyapunov functions are exploited to prove stability. Then comparison results between the robust stability criterion and the existing tests for AQS are given in terms of degree of conservatism. A numerical example is given to illustrate the results.  2007 Elsevier Ltd. All rights reserved. Keywords: Linear systems; Robust stability; Parameter-dependent Lyapunov function; Linear matrix inequalities; Parameter uncertainty 1. Introduction Robust stability analysis for control systems with parame- ter uncertainties is one of the fundamental issues in systems theory, many important advances have been achieved, see de Oliveira, Oliveira, Leite, Montagner, and Peres (2002), Dettori and Scherer (2000), Gahinet, Nemirovski, Laub, and Chilali (1995), Peaucelle and Arzelier (2001) and Zhou, Doyle, and Glover (1996) and the references therein. For linear sys- tems with affine parameter uncertainties, a comprehensive  This work is supported under Grant DSTA POD1820, Singapore, Pro- gram for New Century Excellent Talents in University (NCET-04-0283), the Funds for Creative Research Groups of China (No. 60521003), Program for Changjiang Scholars and Innovative Research Team in University (No. IRT0421) and the State Key Program of National Natural Science of China (Grant No. 60534010), and the Funds of National Natural Science of China (Grant No. 60674021). This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kenko Uchida under the direction of Editor I. Petersen. 1 He is also with the Key Laboratory of Integrated Automation of Process Industry (Northeastern University), Ministry of Education, Shenyang 110004, PR China. ∗ Corresponding author. E-mail addresses: yangguanghong@ise.neu.edu.cn (G.-H. Yang), tsllumky@nus.edu.sg (K.-Y. Lum). 0005-1098/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.09.011 development via the notion of quadratic stability (i.e., use of a single quadratic Lyapunov function of the form V (x) = x T Px to prove stability Barmish, 1982) is presented by the linear matrix inequality (LMI) approach in Boyd, El Ghaoui, Feron, and Balakrishan (1994). However, the use of a single Lya- punov function for robust stability often leads to overly con- servative results. To overcome this, affine parameter-dependent Lyapunov functions which depend on the real uncertain param- eters are used to establish robust stability criteria in Gahinet, Apkarian, and Chilali (1996), which is less conservative than quadratic stability. In Feron, Apkarian, and Gahinet (1996),a sufficient condition for the existence of an affine parameter- dependent Lyapunov function is presented in terms of LMIs, and the resulting stability criterion is less conservative than Popov’s stability criterion. Recently, more general parameter-dependent Lyapunov functions, the projection lemma (Gahinet & Apkarian, 1994) and the multiplier technique have been exploited to develop less conservative robust stability criteria (Dettori & Scherer, 1998, 2000; Geromel, de Oliveira, & Hsu, 1998; Iwasaki, 1997, 1998; Peaucelle & Arzelier, 2001; Scherer, 1997). The multiplier technique consists of parameter-dependent multipliers (Dettori & Scherer, 2000; Iwasaki, 1998) and parameter-independent multipliers (Dettori & Scherer, 1998; Iwasaki & Shibata, 2001;