Characterization of stability transitions and practical stability of planar singularly perturbed linear switched systems

Abstract

This paper is concerned with the stability of planar linear singularly perturbed switched systems in continuous time. Based on a necessary and sufficient stability condition, we characterize all possible stability transitions for this class of switched systems and we propose a practical stability result. We answer the questions related to what happen as ∈, the singular perturbation parameter, grows and how many times the system can change its stability behavior (asymptotic stability, stability, instability) and which transitions are possible. Moreover, we analyze practical stability from the viewpoint of Tikhonov approach and in particular based on existing results obtained in the context of differential inclusions. We show that these approaches can be applied to singularly perturbed switched systems allowing to prove practical stability in some cases. This kind of stability focuses on the behavior of the system on compact time-intervals as ∈ tends to 0 (in particular, it does not ensure the asymptotic stability towards the origin). It is therefore different from the stability criteria where ∈ is fixed (arbitrarily small) and the asymptotic behavior for large times is considered. For planar systems, it turns out that when practical stability can be deduced from Tikhonov-type results, then global uniform asymptotic stability (for ∈ > 0 small) holds true. It is an open question whether this is still true for higher dimensional singularly perturbed switched systems.