Stability for a Multi-Rate Sampled-Data System

Abstract

A sampled-data system with sampling interval lengths selected from a finite set is considered. Stabilizability of the system via feedbacks associated with sampling interval lengths is studied, and conditions for stabilizability involving “pre-contractiveness”, “contractiveness” and “positive definiteness” of a finite set of matrices are given. Included in these results is a generalization of a theorem by P. Stein stating that for a real square matrix H, $\lim _{n \to \infty } H^n = 0$ if and only if there is a symmetric matrix Q such that $Q - H^T QH$ is positive definite. Finally, some results concerning a choice of feedbacks which will produce stability are presented.