Available online at www.sciencedirect.com Automatica 40 (2004) 65–72 www.elsevier.com/locate/automatica Brief paper Delay-dependent robust H ∞ control for uncertain systems with a state-delay  Y.S. Lee a , Y.S. Moon b , W.H. Kwon a ; ∗ , P.G. Park c a Control Information Systems Lab., School of Electrical Engineering and Computer Science, Seoul National University, San 56-1, Shilim-dong, Kwanak-gu, Seoul 151-742, South Korea b Suprema Inc., Yangjae-dong, Seocho-Ku, Seoul 137-130, South Korea c Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Pohang, Kyung-Book 790-784, South Korea Received 24 June 2002; received in revised form 11 February 2003; accepted 19 July 2003 Abstract A robust H∞ control for uncertain linear systems with a state-delay is described. Systems with norm-bounded parameter uncertainties are considered and linear memoryless state feedback controllers are obtained. Firstly, a delay-dependent bounded real lemma for systems with a state-delay is presented in terms of linear matrix inequalities (LMIs). By taking a new Lyapunov–Krasovsii functional, neither model transformation nor bounding for cross terms is required to obtain delay-dependent results. Secondly, based on the bounded real lemma obtained, delay-dependent condition for the existence of robust H∞ control is presented in terms of nonlinear matrix inequalities. In order to solve these nonlinear matrix inequalities, an iterative algorithm involving convex optimization is proposed. Numerical examples show that the proposed methods are much less conservative than existing results. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Time-delay systems; Delay-dependent; Bounded real lemma; H∞ control; Uncertain systems 1. Introduction There have been considerable research eorts on robust H ∞ control for uncertain time-delay systems, in particular, with parameter uncertainties. The existing results for H ∞ control of time-delay systems deal with either one of two types of stabilization: delay-independent stabilization (Choi & Chung, 1997; Kapila & Haddad, 1998; Zribi & Mahmoud, 1999; Kim & Park, 1999) and delay-dependent stabilization (deSouza&Li,1999; Fridman & Shaked, 2001; Lee, Moon, & Kwon, 2001; Fridman & Shaked, 2002). Recent research eort is focused more on delay-dependent stabilization. The main objective of the delay-dependent H ∞ control is to ob- tain a controller that allows a maximum delay size for a xed H ∞ performance bound or achieves a minimum H ∞ performance bound for a xed delay size. The conservatism  This paper was not presented at the IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hitay Ozbay under the direction of Editor Tamer Ba sar. This work was supported by Brain Korea 21. * Corresponding author. Tel.: +82-2-880-7307; fax: +82-2-871-7010. E-mail address: whkwon@cisl.snu.ac.kr (W.H. Kwon). in the delay-dependent H ∞ control is hence measured by the allowable delay size or performance bound obtained. In the past few years, there have been various approaches to reduce the conservatism of delay-dependent conditions by using new bounding for cross terms or choosing new Lyapunov–Krasovskii functional. The delay-dependent sta- bility criterion of Park, Moon, and Kwon (1998) and Park (1999) is based on a so-called Park’s inequality for bound- ing cross terms. This stability criterion was later extended to controller synthesis (Moon, 1998). However, major draw- back in using the bounding of Park et al. (1998) and Park (1999) is that some matrix variables should be limited to a certain structure to obtain controller synthesis conditions in terms of LMIs. This limitation introduces some con- servatism. In Moon, Park, Kwon, and Lee (2001) a new inequality, which is more general than the Park’s inequality, was introduced for bounding cross terms and controller synthesis conditions were presented in terms of nonlinear matrix inequalities in order to reduce the conservatism. An iterative algorithm was developed to solve the nonlinear ma- trix inequalities (Moon et al., 2001). Stabilization method of Moon et al. (2001) was later extended to H ∞ control 0005-1098/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2003.07.004