Controllable Sequences of Minimal Length for Discrete-Time Switched Linear Control Systems

TL;DR

This paper introduces a new way to characterize the reachable states of discrete-time switched linear systems and provides criteria for controllability, including minimal switching sequences, with bounds on the time needed for full state reachability.

Contribution

It offers a novel characterization of the reachable set and a Kalman-type controllability criterion for switched systems, including estimates on minimal controllability time.

Findings

Existence of switching sequences covering the entire state space

Bounds on minimal time for controllability based on system parameters

Tightness of estimates in relevant cases

Abstract

In this paper, we provide a novel characterization of the reachable set of discrete-time switched linear control systems and a Kalman-type criterion for controllability, assuming that the switching parameter can be used as a control parameter in addition to the actual control variable. For controllable switched linear control systems it turns out that there always exists a switching sequence such that the reachable set of the corresponding linear time-variant system covers the whole state space after a sufficiently large time. We provide estimates on the minimal time guaranteeing this property in terms of the state dimension, number of modes and rank of the control matrices, and show that such estimates are actually tight in some relevant cases.