On the robust stability of interval time-delay systems - An application of the upper solution bounds of the Lyapunov equation

Abstract In this paper, the robust stability test problem of the interval time-delay systems is discussed. We first study the upper solution bounds of the continuous algebraic Lyapunov equation (the CALE). By using linear algebraic techniques, a simple approach is proposed to derive new upper bounds of the CALE. Compared to existing works on this topic, these newly obtained bounds are less restrictive and easier to calculate. Then, we apply them to solve the mentioned problem. By using Lyapunov equation approach associated with these upper bounds, new concise criteria for the robust stability and decay rate test problem of the mentioned systems are presented. Comparing to some well-known results of the time-delay systems, the obtained criteria are sharper. An interesting consequence of these results is that it is not necessary to solve any Lyapunov equation although the Lyapunov equation approach is used.

[1]  Bor-Sen Chen,et al.  Robust stability of uncertain time-delay systems , 1987 .

[2]  Hans Schneider,et al.  Some theorems on the inertia of general matrices , 1962 .

[3]  T. Mori,et al.  A brief summary of the bounds on the solution of the algebraic matrix equations in control theory , 1984 .

[4]  Takehiro Mori,et al.  On solution bounds for three types of Lyapunov matrix equations: continuous, discrete and unified equations , 2002, IEEE Trans. Autom. Control..

[5]  T. Mori,et al.  Simple stability criteria for single and composite linear systems with time delays , 1981 .

[6]  Richard Davies,et al.  New lower matrix bounds for the solution of the continuous algebraic lyapunov equation , 2008 .

[7]  M. T. Qureshi,et al.  Lyapunov Matrix Equation in System Stability and Control , 2008 .

[8]  C. D. Souza,et al.  Robust exponential stability of uncertain systems with time-varying delays , 1998, IEEE Trans. Autom. Control..

[9]  Yannis A. Phillis,et al.  Trace bounds on the covariances of continuous-time systems with multiplicative noise , 1993, IEEE Trans. Autom. Control..

[10]  Tzuu-Hseng S. Li,et al.  A new approach for the robust stability of perturbed systems with a class of noncommensurate time delays , 1993 .

[11]  J. Yan.  Robust stability analysis of uncertain time delay systems with delay-dependence , 2001 .

[12]  E. Shapiro.  On the Lyapunov matrix equation , 1974 .

[13]  Yong He,et al.  Delay-dependent stability criteria for linear systems with multiple time delays , 2006 .

[14]  T. Mori,et al.  On an estimate of the decay rate for stable linear delay systems , 1982 .

[15]  Han Ho Choi,et al.  Lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations , 2002, Autom..

[16]  Rama K. Yedavalli,et al.  Robust root clustering for linear uncertain systems using generalized lyapunov theory , 1993, Autom..

[17]  Tsung-Lieh Hsien,et al.  New sufficient conditions for the stability of continuous and discrete time-delay interval systems , 1997 .

[18]  Chien-Hua Lee,et al.  On the estimation of solution bounds of the generalized Lyapunov equations and the robust root clustering for the linear perturbed systems , 2001 .

[19]  C. H. Lee.  Solution Bounds of the Continuous and Discrete Lyapunov Matrix Equations , 2004 .

[20]  Nejat Olgaç,et al.  Stability Robustness Analysis of Multiple Time- Delayed Systems Using “Building Block” Concept , 2007, IEEE Transactions on Automatic Control.

[21]  On bounds of Lyapunov's matrix equation , 1979 .

[22]  A. R. Amir-Moéz.  Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations , 1956 .

[23]  T. Mori,et al.  Explicit solution and eigenvalue bounds in the Lyapunov matrix equation , 1986 .

[24]  Ninoslav Truhar,et al.  Bounds on the trace of a solution to the Lyapunov equation with a general stable matrix , 2007, Syst. Control. Lett..

[25]  F. Kung,et al.  Upper and lower matrix bounds of the solutions for the continuous and discrete Lyapunov equations , 1997 .

[26]  Huijun Gao,et al.  Novel Robust Stability Criteria for Stochastic Hopfield Neural Networks With Time Delays , 2009, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).