Invertibility of switched linear systems

Abstract

We address a new problem-the invertibility problem for continuous-time switched linear systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the subsystems should be invertible and there should be no singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs.

Invertibility and nonsingularity of Boolean control networks

“…Hirschorn, 1979, Nijmeijer, 1982, Singh, 1982, and switched systems (cf. Tanwani & Liberzon, 2010, Vu & Liberzon, 2008. However, to the best of our knowledge, there has been no result on the invertibility problem of BCNs so far.…”

A Mixed-Boolean Hybrid Mathematical Model with Discontinuous States

“…Switching systems "comprise a family of dynamical subsystems together with a switching signal determining the active system at a current time" (Vu and Liberzon, 2008). They are a subset of hybrid systems, where there is some discontinuous behaviour modelled by an on/off switch or other binary signal.…”

Stability of A Switched Linear System

“…It was later proved by that for switched linear systems, a basic switching sequence exists such that the controllable state set of this basic switching sequence is equal to the controllable state set of the system. In contrast, a different emphasis of research was taken by Vu and Liberzon (2006), who were interested in a new issue in switched systems. They focused on the invertibility problem of a pair of subsystems for continuous-time linear switched systems.…”