Perspectives and results on the stability and stabilizability of hybrid systems

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 endeavor, 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.

Stabilization by means of state space depending switching rules

“…There are apparent analogies between Theorem 1 of this paper and the main result of [15] (see also Theorem 11 of [13] and Theorem 4.1 of [8]): however, both the assumptions and the conclusions differ. The result of [15] requires the existence of a Hurwitz convex combination of A 1 , A 2 , while in Theorem 1 a neutrally stable convex combination suffices.…”

“…In an open loop philosophy, under the additional requirement that the map t → u(t) is piecewise constant, systems of this form are usually called (linear) switched systems: a great effort has been recently done in order to characterize stability of switched systems under arbitrary switched signals (see the survey papers [13], [8] and the references therein). As a matter of fact, in the spirit of geometric control theory, system (1) is equivalent to the assignment of the family of linear vector fields {A 1 x, A 2 x}; the investigation of the asymptotic behavior of families of vector fields under arbitrary switching (polysystems) was actually initiated in the early paper [4]).…”

Controle de sistemas lineares com comutação

“…O objetivo consiste na determinação de uma regra de comutação σ(·) de tal forma a assegurar a estabilidade assintó-tica global. Em (Liberzon e Morse, 1999), (Liberzon, 2003), (DeCarlo et al, 2000), (Wirth, 2005), (Shorten et al, 2007) o leitor pode encontrar conjuntos de resultados bastante completos sobre sistemas lineares com comutação a tempo contí-nuo, em especial, para o caso de comutação envolvendo apenas dois subsistemas lineares. O artigo (Hespanha, 2005) utiliza extensões do Princípio da Invariância de LaSalle e fornece uma discussão interessante sobre resultados referentes à estabilidade uniforme de sistemas com comutação.…”

Min-switching local stabilization for discrete-time switching systems with nonlinear modes

“…Switched systems are a subclass of hybrid systems and consist in the combination of a finite set of modal dynamic systems and a switching law, which indicates at each time the active mode among them [8,9]. The issue of the stability and stabilization of switched systems, which has been clearly formalized in [10], has generated many contributions to the control theory literature (see [11,12,13,14,15,16] and references therein).…”