Perspectives and results on the stability and stabilizability of hybrid systems

TLDR

Hybrid systems combine continuous and discrete dynamics, yet guaranteeing stability—particularly for switched stable systems and designing controllers for switched unstable systems—remains challenging. The paper surveys Lyapunov stability results for finite‑dimensional hybrid systems and examines specialized results for switched linear systems. It reviews the literature and presents key results as linear matrix inequalities. The authors demonstrate stability analyses for aircraft angle‑of‑attack regulation and vehicle PI control, and provide additional illustrative examples.

Abstract

This paper introduces the concept of a hybrid system and some of the challenges associated with the stability of such systems, including the issues of guaranteeing stability of switched stable systems and finding conditions for the existence of switched controllers for stabilizing switched unstable systems. In this endeavour, this paper surveys the major results in the (Lyapunov) stability of finite-dimensional hybrid systems and then discusses the stronger, more specialized results of switched linear (stable and unstable) systems. A section detailing how some of the results can be formulated as linear matrix inequalities is given. Stability analyses on the regulation of the angle of attack of an aircraft and on the PI control of a vehicle with an automatic transmission are given. Other examples are included to illustrate various results in this paper.