Robust stability of discrete-time systems under parametric perturbations

Abstract

of time varying linear systems," IEEE Trans. Automat. Contr., vol. 33, pp. 150-155, 1988. B. E. Ydstie, "Stability of discrete model reference control revisited," Syst. Contr. Lett., vol. 13, pp. 429438, 1989. -, "Stability of the direct self-tuning regulator," in Foundations of Adaptive Control, P. V. Kokotovic, Ed., 1991, pp. 201-238 [61-[81).The main objective of this paper is to link the stability robustness problem of discrete-time systems to that of continuous-time systems. We show, using two different approaches, that stability robustness of a discrete-time system can be reformulated as that of an auxiliary continuous-time system. One of these approaches makes use of Lyapunov theory and yields a sufficient condition. The second approach, which is based on the properties of Kronecker products, provides a necessary and sufficient condition at the expense of an increase in the dimensionality. This is a pleasing development, since it allows for a direct application of the known results on stability robustness bounds for continuous-time systems to discrete-time systems. The results are applied to stability analysis of interconnected systems, where the interconnections are treated as perturbations on a collection of stable subsystems. This demonstrates how a knowledge of the structure of perturbations can be exploited to obtain simple robustness bounds. PROBLEM STATEMENT Robust Stability of Discrete-Time Systems Under Parametric PerturbationsConsider a discrete-time system under additive multiparameter perturbations, which is described as Mehmet Karan, M. Erol Sezer, and Ogan Ocali Absfract-Stability robustness analysis of a system under parametric perturbations is concerned with characterizing a region in the parameter space in which the system remains stable. In this paper, two methods are presented to estimate the stability robustness region of a linear, time-invariant, discrete-time system under multiparameter additive perturbations. An inherent difficulty, which originates from the nonlinear appearance of the perturbation parameters in the inequalities defining the robustness region, is resolved by transforming the problem to stability of a higher order continuous-time system. This allows for application of the available results on stability robustness of continuous-time systems to discrete-time systems. The results are also applied to stability analysis of discrete-time interconnected systems, where the interconnections are treated as perturbations on decoupled stable subsystems.

Controller Synthesis on Non-uniform and Uncertain Discrete–Time Domains

“…Such conditions give admissible values for the upper bound ∆ on the sampling period variations (4) for which the performances are guaranteed. Furthermore, (20) and (21) provide the radius of a ball BÃ with centerĀ C that contains the time-varying dynamic matrix of system (18). Then, a sufficient condition can be obtained from [C1] by considering any polytope PÃ that contains the ball BÃ.…”

“…The design problem on non-uniform discrete-time domains can be successfully approached by exploiting interesting results on robust stability (see [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]) for perturbed systems of type…”

“…Further robust stability conditions have been obtained using LMI techniques (see [20,22,23,25]) and H 1 and H ∞ formulations (see [19,21,24]). Such approaches will be evaluated in future work.…”