Observability and Detectability of Linear Switching Systems: A Structural Approach

TL;DR

This paper investigates the conditions under which the internal state of linear switching systems can be reconstructed or approximated from outputs, providing computational criteria and linking detectability to hybrid system stability.

Contribution

It introduces a necessary and sufficient condition for observability, characterizes control inputs for observability, and relates detectability to hybrid system stability.

Findings

Derived a computationally verifiable condition for observability.

Characterized control inputs that ensure observability.

Linked detectability to the asymptotic stability of related hybrid systems.

Abstract

We define observability and detectability for linear switching systems as the possibility of reconstructing and respectively of asymptotically reconstructing the hybrid state of the system from the knowledge of the output for a suitable choice of the control input. We derive a necessary and sufficient condition for observability that can be verified computationally. A characterization of control inputs ensuring observability of switching systems is given. Moreover, we prove that checking detectability of a linear switching system is equivalent to checking asymptotic stability of a suitable switching system with guards extracted from it, thus providing interesting links to Kalman decomposition and the theory of stability of hybrid systems.