Robust Absolute Stability Criteria for Uncertain Lurie System With Interval Time-Varying Delay

Abstract

This paper addresses the problem of absolute stability of Lurie system with interval time-varying delay. The delay range is divided into two equal segments and an appropriate Lyapunov–Krasovskii functional (LKF) is defined. A tighter bounding technique for the derivative of LKF is developed. This bounding technique in combination with the Wirtinger inequality is used to develop the absolute stability criterion in terms of linear matrix inequalities (LMIs). The stability analysis is also extended to the Lurie system with norm-bounded parametric uncertainties. The effectiveness of the proposed approach has been illustrated through a numerical example and Chua's oscillator.

Further improvement on delay-range-dependent robust absolute stability for Lur׳e uncertain systems with interval time-varying delays

“…However, in many practical systems, for instance networked control systems (NCSs) [17], the lower delay bound may be different from zero. Recently, the stability of time delay systems with interval time-varying delay has been widely investigated [4,6,7,16,17,20,22,28,30,31,33,34,37,41,42]. Furthermore, stability of systems with interval time-varying delay, which means that the lower bound of the delay parameter is not restricted to zero but varies in a time interval, has been an important research topic [6,7,17,34,37,38,42].…”

“…Therefore, many methods have been proposed in these results to reduce the conservatism of the stability criteria, such as Jensen integral inequality [10], free-weighting matrix method [6,33,39,41], Wirtinger-based inequality method [28,30], Finsler lemma method [36], delay-partitioning approach [29,34,35], delaydividing approach [32], generalized convex combination [20] and delayed decomposition approach [42]. In [10], using Jensen integral inequality, Han [10] obtain the less conservative absolute stability criteria where both model transformation and a bounding technique for the cross terms are avoided, since model transformation sometimes will induce additional dynamics [8].…”

“…As time-delays are frequently encountered in practical systems and are often a source of instability and poor performance, the problem of absolute stability of Lur'e systems with time-delay has been attracting much attention [2,[3][4][5][6][9][10][11][12]19,20,25,[28][29][30][33][34][35][36]41,42]. In stability analysis of system with time delays, delay-dependent methods [3][4][5][6][10][11][12]20,25,[28][29][30][33][34][35][36]41,42] have been paid more attentions than delay-independent one [2,9,19] because of the fact that the sufficient conditions with delaydependent method provide maximum delay bounds for guaranteeing the asymptotic stability of the concerned system and are generally less conservative than delay-independent ones. During the last decade, increasing attention has been paid to the development of the technologies for delay-dependent stability analysis.…”

“…The Lur'e system is one of a significant class of nonlinear systems and has a nonlinear element satisfying certain sector bounded constraints. Since the Lur'e system and absolute stability were firstly introduced by [26,27], the study of the absolute stability for Lur'e system has attracted many researchers [2,[3][4][5][6][9][10][11][12]14,15,19,20,25,[28][29][30][32][33][34][35][36][39][40][41][42]. Also, in practice, the systems almost present some uncertainties because it is very difficult to obtain an exact mathematical model due to environmental noise, uncertain or slowly varying parameters.…”

“…Li et al [20] investigated the problem of delay-dependent absolute and robust stability for time-delay Lur'e system and the relaxed conditions were presented some previously ignored terms when estimating the triple integral Lyapunov-Krasovskii functional terms' derivative. In [28], an improved bounding technique in combination with Wirtinger-based inequality inequality method is used to derive the delay-dependent stability criterion in terms of LMIs. By choosing a modified LKF and employing the novel integral-inequality lemma proposed in [5] and Wirtinger-based inequality method [30], some improved delaydependent robust stability criteria are derived in terms of LMIs without using the general free-weighting matrix method.…”