Stability of switched systems with average dwell-time

TLDR

The study demonstrates that scale‑independent hysteresis can generate slow‑on‑the‑average switching, allowing stability analysis of hysteresis‑based adaptive control systems. This is achieved by employing scale‑independent hysteresis within a supervisory control framework to produce the desired switching behavior. It is proved that switching among stable linear systems remains exponentially stable when the average switching rate is sufficiently low, that this stability is uniform across all such switchings, that input‑to‑state norms are uniformly bounded for systems with inputs, and that analogous results hold for certain classes of nonlinear switched systems.

Abstract

It is shown that switching among stable linear systems results in a stable system provided that switching is "slow-on-the-average". In particular, it is proved that exponential stability is achieved when the number of switches in any finite interval grows linearly with the length of the interval, and the growth rate is sufficiently small. Moreover, the exponential stability is uniform over all switchings with the above property. For switched systems with inputs this guarantees that several input-to-state induced norms are bounded uniformly over all slow-on-the-average switchings. These results extend to classes of nonlinear switched systems that satisfy suitable uniformity assumptions. In this paper it is also shown that, in a supervisory control context, scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of hysteresis-based adaptive control systems.