On the Left-invertibility of Switched Linear Systems

The main object of study in this paper is the class of switched linear systems, i.e. a collection of linear input/state/output systems and a set of switching signals that determine the active subsystem from the collection at any given time instant. We address the problem of left-invertibility for these systems. Left-invertible switched linear systems are those for which one can recover the switching signal (up to an equivalence class) and the input uniquely from a given output of the system. After introducing an equivalence relation for switching signals, we give precise definition of left-invertibility based on this equivalence relation. By taking a geometric approach, the paper presents a set of necessary and sufficient conditions for left-invertibility. These conditions are linear in nature and can be checked by using the techniques from geometric approach to linear systems.