On observability and reachability in a class of discrete-time switched linear systems

Abstract

We study the class of switched linear systems with constant dynamics, i.e., with switching in only the measurement or control matrices (which correspond to plants with several sensory and actuation modes), and establish that the length of the shortest mode path achieving observability or reachability, if one exists, is at most the square of the dimension of the state space. This not only proves that observability/reachability is decidable for such systems, but also gives a direct way of finding the observable/reachable mode path.

Switching Stabilizability for Continuous-Time Uncertain Switched Linear Systems

“…Then, the polyhedral Lyapunov-like function candidate from the polyhedron can be defined as (10) Note that by definition, and that is continuous and piecewise linear for , and by definition. However, does not imply that .…”

“…Properties of this type of model have been studied for the past fifty years to consider engineering systems that contain relays and/or hysteresis. Due to its success in application and importance in theory, the last decade has seen increasing research activities in stability [8], [19], [22], controllability [34], [35], observability [1], [10], [14], stabilization [16], [25], [33], and switching optimal control [2], [39] of switched systems.…”

Observability of Switched Linear Systems

“…The authors presented observability characterizations under arbitrary discrete state sequences, and defined the concept of discernability of discrete state sequences to present mode observability, which is similar to the state distinguishability notion presented in [3] and [4]. In [9], the authors extended the results from [7] and [8] to the continuous-time case. In that paper, the characterization of control-discernability (discernability of nonautonomous systems) is focused on the existence of a continuous input allowing to distinguish between discrete states by their continuous outputs.…”

“…. (8) where , and the transfer matrix has a left inverse. These conditions imply that once the marking is computed, every future marking will be known.…”

“…In [7] and [8], an observability characterization for non autonomous discrete-time SLS is presented. The authors presented observability characterizations under arbitrary discrete state sequences, and defined the concept of discernability of discrete state sequences to present mode observability, which is similar to the state distinguishability notion presented in [3] and [4].…”