1. Global Stability Results for Switched Systems Based on Weak Lyapunov Functions
- Author
-
Rafael Garcia, Hernan Haimovich, and José Luis Mancilla-Aguilar
- Subjects
Lyapunov function ,0209 industrial biotechnology ,INGENIERÍAS Y TECNOLOGÍAS ,02 engineering and technology ,Lyapunov exponent ,symbols.namesake ,LYAPUNOV METHODS ,020901 industrial engineering & automation ,Exponential stability ,Control theory ,Stability theory ,0202 electrical engineering, electronic engineering, information engineering ,Lyapunov equation ,SWITCHED SYSTEMS ,Electrical and Electronic Engineering ,Lyapunov redesign ,Ingeniería Eléctrica, Ingeniería Electrónica e Ingeniería de la Información ,Control-Lyapunov function ,Mathematics ,Ingeniería de Sistemas y Comunicaciones ,INPUT-TO-STATE STABILITY ,NONLINEAR DYNAMICAL SYSTEMS ,Computer Science Applications ,Nonlinear system ,TIME-VARYING SYSTEMS ,Control and Systems Engineering ,symbols ,020201 artificial intelligence & image processing ,ASYMPTOTIC STABILITY - Abstract
In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like part and a perturbation-like one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-time-varying. We provide two types of ISS results: Standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm. Fil: Mancilla Aguilar, Jose Luis. Instituto Tecnológico de Buenos Aires; Argentina Fil: Haimovich, Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentina Fil: Garcia Galiñanes, Rafael Antonio. Instituto Tecnológico de Buenos Aires; Argentina
- Published
- 2017