ROBUST ABSOLUTE STABILITY ANALYSIS OF MULTIPLE TIME-DELAY LUR’E SYSTEMS WITH PARAMETRIC UNCERTAINTIES A. Kazemy and M. Farrokhi ABSTRACT The problem of robust absolute stability for time-delay Lur’e systems with parametric uncertainties is investigated in this paper. The nonlinear part of the Lur’e system is assumed to be both time-invariant and time-varying. The structure of uncertainty is a general case that includes norm-bounded uncertainty. Based on the Lyapunov–Krasovskii stability theory, some delay- dependent sufficient conditions for the robust absolute stability of the Lur’e system will be derived and expressed in the form of linear matrix inequalities (LMIs). These conditions reduce the conservativeness in computing the upper bound of the maximum allowed delay in many cases. Numerical examples are given to show that the proposed stability criteria are less conservative than those reported in the established literatures. Key Words: Time-delayed systems, Lur’e systems, delay-dependent stability. I. INTRODUCTION Since the notion of absolute stability was introduced by Lur’e [1], the problem of the absolute stability of Lur’e control systems has been widely studied for several decades [2–5]. The Lur’e control system has attracted attention since many nonlinear control systems can be represented as the feedback connection of a linear dynamic system and a non- linear element, where the nonlinear element satisfies certain sector constraints [6]. Examples of such nonlinear systems are continuous stirred-tank reactor (CSTR) and Chua’s circuit [7,8]. As time-delays are encountered in many types of control systems and the stability properties are strongly affected by them [9], the problem of absolute stability of Lur’e systems with a time-delay has been attracting a great deal of attention [10–13]. In general, stability conditions for time-delay systems are classified into two categories according to their dependence on the size of delays: delay- independent and delay-dependent stability criteria [9]. Delay- dependent conditions usually give less conservative results since they use delay information in their conditions. In addi- tion, most of the engineering practices are subjected to finite time-delays. Hence, the delay-dependent conditions are more practical. Recently, practical considerations such as model uncer- tainties are considered for stability analysis of Lur’e systems [14–17]. A challenging problem is to find the maximum allowable time-delay that can guarantee the absolute stability of uncertain Lur’e systems with time-delay [18–21]. For stability analysis of time-delay systems, researchers usually use the Lyapunov-Krasovskii theorem [19,20]. The direct application of the Lyapunov-Krasovskii theorem is seriously complicated by difficulties arising in its construc- tion. For example, one has to look for a functional such as vx t t t s t s ds t T T h () = () () + () () + ( ) - ∫ x Px x Q x 2 0 where matrix Q(s) is a time varying weighting matrix. Analysis of its posi- tivity is a challenging task. Therefore, the theorem is usually used with particular types of functional [22]. Matrix Q(s) is usually replaced by a constant matrix Q [19]. Xu et al. intro- duced a more general Lyapunov functional and showed that their results were less conservative [19]. Wu et al. used the well-known Lyapunov discretization method [23] which led to better results [20]. In this method, the delay interval [0,h] is discretized into N-equal segments and matrix Q(s) is substi- tuted with N-constant matrices Qi(i = 1, . . ., N). The main advantage of this method is better approximation of Q(s) and giving more weighting matrices to the Lyapunov functional. However, this segmentation method doesn’t select the segment intervals optimally and may lead to many redundant weighting matrices in its conditions. A better method is to select the segment intervals where the matrix Q(s) has the maximum variations [24, 25]. In this paper, it will be shown that in many cases, two segmentations with an unequal length can lead to better results than the conventional segmentations with more than three segments. To this end, by employing the idea of discretizing the delay interval into two segments with selectable length, the problem of delay-dependent robust absolute stability of uncertain multiple time-delayed Lur’e systems with sector-bounded nonlinearity is investigated. The nonlinear part of the Lur’e system is assumed to be both Manuscript received April 27, 2011; revised August 23, 2011; accepted November 6, 2011. The authors are with Iran University of Science and Technology, Department of Electrical Engineering, Tehran. Mohammad Farrokhi is the corresponding author (e-mail: farrokhi@iust.ac.ir). Asian Journal of Control, Vol. 15, No. 1, pp. 203–213, January 2013 Published online 14 March 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.503 © 2012 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society