Stability robustness bounds for linear systems with delayed perturbations H. Trinh M. Aldeen zyxwvutsrqpon Indexing corns: Linear systems, Stability zyxwvutsrqponm Abstract: The paper presents sufficient conditions for delay-independent asymptotic stability for linear systems with multiple time-varying delayed perturbations. It demonstrates that zyxwvuts judicious selection of Lyapunov functionals leads to improved bounds on allowed perturbations. It is shown that the bounds obtained for delayed per- turbations are better than those derived in the lit- erature for nondelayed perturbations. Numerical examples are given to illustrate the results. List of symbols R" zyxwvutsrqpo = real vector space of dimension zyxwvuts n R""" = real matrix space of dimension n x rn x, = segment of the function x(~) on [t - h, t], i.e. x,: [ - h, 01 + R" defined by x,(O) A x(t + e), -h<O<O AmAA), zyxwvutsrqponm Amin(A) = maximum, minimum eigenvalue of matrix A, respectively AT = trannose of matrix A 1 1 ~ 1 1 =matrix norm of A, 1 1 ~ 1 1 zyxwvutsrqponmlkjihgfedcbaZ PaM#) = A. [Amy TA)] "* = symmetric part of matrix A, A. 0 OS(A + AT) p(A) z = complex number; z = exp (jw) = cos (0) +jsin(o)VwE[O 2x1 then 1z1=1, where i=J(-1) = matrix measure of A, AA) 4 051,,(AT + A) A > 0 = matrix A is symmetric positive definite 1, Vi = for all i = identity matrix of order n 1 Introduction The stability robustness of linear time-invariant systems subject to time-varying perturbations has attracted the attention of many researchers for the past few years. As a result, explicit bounds for both structured and unstruc- tured time-varying perturbations have been derived (see, for example [l-81). More recently, considerable interest has been shown in the problem of robust stability of linear systems subject to delayed time-varying pertur- bations [9-113. zyxwvutsr 0 IEE, 1995 Paper 1963D (CS), fiat received 7th November 1994 and in final revised form 6th March I995 The authors are with the Control and Power Systems Group, Depart- ment of Electrical and Electronic Engineering, The University of Mel- bourne, Parkville, Victoria, 3052, Australia IEE Proc.-Control Theory Appl., Vol. 142, No. 4, July 1995 In [lo, 111 it is shown that less conservative bounds than those of [9] can be obtained and the bounds are the same as those derived in [1-41 for nondelayed unstruc- tured perturbations. When the delayed perturbations are time-varying and structured, the derived bounds derived in [lo] are close to those derived in [7l for the nonde- layed structured perturbations. The results in [9, 111 are based on the Razumikhin-type theorem and apply to unstructured delayed perturbations. The results in [lo] are based on the Lyapunov functionals theory, and both structured and unstructured delayed perturbations are considered. In this paper we use the Lyapunov functionals theory and show that: (i) the results in [lo, 111 can be further improved, i.e. better bounds than those reported in the literature can be obtained for both unstructured and structured nondelayed perturbations 11-73 ; and zyx (U) the results of (i) can be extended to include multiple time- varying delays. First, let us consider the following system: Yt) = Ax@) + E(tMt - h) (1) where x(t) E R" is the state vector, h 2 0 is a constant and represents the delay duration, A E R" " is a real constant stable matrix and E(t) E R" '" is the time-varying pertur- bation in the delayed state. We consider two cases. Case I: First we consider systems with time-varying and unstructured perturbations where E(t) is assumed to be bounded, i.e. IIE(t)ll G zyxwvu rl (2) and q is a positive constant number. Here we show that better bounds than those reported for nondelayed unstructured perturbations [l-61 can be obtained. Further, we show, through a numerical example, that for the case where E(t) is time-invariant, i.e. E(t) = E, less conservative bounds than those reported in the literature [12-141 can be obtained. Case 2: Secondly, we consider systems with time-varying and structured perturbations, where E(t) is assumed to take the form m i=l E(Q = C kdt)Ei (3) where Ei E R""" are real constant matrices, m is the number of uncertain parameters and kdt) are time- varying uncertain parameters. Here we show, through a numerical example, that our derived bounds are better 345