Pathwise observability through arithmetic progressions and non-pathological sampling

We consider the problem of establishing pathwise observability for a class of switched linear systems with constant, autonomous dynamics, but with switched measurement equations. Using Van der Waerden's theorem, a standard result in Ramsey theory, we give a sufficient condition on the components of the system for it to be pathwise observable. This first result then enables us to extend the KaIman-Bertram criterion, which concerns the conservation of observability after the introduction of sampling, to switched linear systems. We then dualize these results to pathwise controllability.