Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time‐delay systems

Abstract

In this paper, an improved linear matrix inequality (LMI)-based robust delay-dependent stability test is introduced to ensure a larger upper bound for time-varying delays affecting the state vector of an uncertain continuous-time system with norm-bounded-type uncertainties. A quasi-full-size Lyapunov-Krasovskii functional is chosen and free-weighting matrix approach is employed. Less restrictive sufficient conditions are derived for robust stability of time-varying delay systems with norm-bounded-type uncertainties. Moreover, the investigation of the stabilization problem with memoryless state-feedback control is presented such that the stabilizability criteria are obtained in terms of matrix inequalities, which can be solved via utilizing a cone complementarity minimization algorithm. Finally, the problem of output feedback stabilization for square systems is also taken into consideration. The output feedback stabilizability criteria are derived in the form of linear matrix inequalities, which are convex and can be easily solved using interior point algorithms. A plenty of numerical examples are presented indicating that the proposed stability and stabilization methods effectively improve the existing results. modelling errors or some ignored factors. The subject of uncertain time-delay systems, thus, has received a considerable amount of interest from researchers. See, for example, References [4][5][6][7][8][9][10][11] and the references therein. A number of linear time-invariant control (LTI) system analysis and a framework for a numerical search of quadratic Lyapunov functions are considered in Reference [12]. The stability analysis or stabilization synthesis for time-delay systems can be achieved delay-independently or delay-dependently. Several authors, see, e.g. References [6,8,9], introduce delay-independent results. Since there is no upper limit to time-delay, often delayindependent results can be regarded as conservative in practice where unbounded time delays are not so realistic. Therefore, stability results that depend on the size of the time-delay receive quite a lot of interest. See, for example, some of the delay-dependent results given in References [13][14][15][16][17][18][19][20]. The problems of delay-dependent robust stability analysis and robust control with memoryless state-feedback for a class of uncertain linear systems with time-varying delayed state and norm-bounded uncertainty are considered in Reference [13] based on Razumikhin stability theorem. Robust stabilization and robust H 1 control for uncertain linear constant state-delayed systems is investigated in Reference [14]. Delay-dependent stability conditions are derived in Reference [15] for a class of neutral linear systems in terms of neutral-type model transformation. A non-convex delay-dependent robust stabilization condition is obtained in Reference [16] for uncertain state-delayed systems with a generalized form of inequality to bind the cross-product terms. The delay-dependent robust stability for time-delay systems is inves...

State feedback stabilization of uncertain linear time-delay systems: A nonlinear matrix inequality approach

“…that used in [8]. Thus, the resulting criterion is of lesser-dimension and with few numbers of free variables than that of those proposed in [1]. This dimensional and variable reduction in the criterion seems to be advantageous since these can be solved more efficiently.…”

“…where A and A d are constant matrices, A(t) and A d (t) are time-varying matrices which model the parametric uncertainties in the system that are Lebesgue measurable and norm-bounded [1,12,15]. Further, these may possibly be decomposed by exploiting their structural description as…”

“…These can be solved in two ways: (a) using suitable iterative algorithms as have been used in [1,13] or (b) by approximating some of the matrices in terms of others but with some scalar scaling factor such that the resulting criterion becomes linear with only few tuning parameters [12,14,16]. Clearly, the former approach may yield less conservative results in terms of obtaining the maximum tolerable delay bound for which the system remains stable.…”

“…This motivated several researchers to develop approximate but computationally efficient methods [7] to analyze time-delay systems. Several stability and stabilization criteria [8][9][10] have been proposed in the recent years gradually improving the effectiveness in the approximated results, while robust delay-dependent stability and (or robust stabilization) criterion for time-delay system has been investigated in [1,[11][12][13][14][15] and references therein.…”

“…inherently possess time-delay, which is often a source of poor performance and even instability [1][2][3][4][5][6]. Moreover, presence of time-delay in a system model induces infinite roots in its characteristic equation forcing the exact analysis to be computationally difficult.…”