Controllability and Stabilizability in Multi-Pair Systems

Abstract

In this paper discrete-time systems of the form \[x_{k + 1} = C_p x_k + D_p U_k ,\] in which the pair $(C_p ,D_p )$ is selected from a finite set $\{ (C_i ,D_i )\} _{i = 1}^N $, are studied. Such systems, called “mufti-pair systems,” arise naturally in the study of multi-rate sampled-data systems. It is shown that the set of points reachable from zero (the controllable set) is a subspace under certain hypotheses, but not always. When this is the case, an extended version of the controllability canonical form is obtained, and it is applied to the study of state deadbeat response and more general forms of stabilizability.