Global Stability Results for Switched Systems Based on Weak Lyapunov Functions

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 inputto-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.

Necessary and sufficient conditions for stabilisability of discrete‐time time‐varying switched systems

“…Inspired by the idea of using scalar functions in [24] and [41] and the time-varying Lyapunov functions in [14] and [27], we are dedicated to analysing the stabilisability of time-varying discrete-time switched systems. Firstly, resorting to asymptotically (exponentially, uniformly exponentially) stable scalar functions, we explore necessary and sufficient conditions for the asymptotic (exponential, uniform exponential) stabilisability of time-varying SLSs, respectively.…”

Average dwell time based stability analysis for nonautonomous continuous‐time switched systems

“…However, the stability analysis of nonautonomous switched systems has been relatively less extensively studied . Liu et al proposed stability conditions for switched systems with both stable and unstable subsystems via multiple Lyapunov functions (MLF s for short) and the average dwell time based method, which required the MLF s to be nonincreasing.…”

“…[29][30][31][32][33][34][35][36][37][38][39][40] Liu et al 31 proposed stability conditions for switched systems with both stable and unstable subsystems via multiple Lyapunov functions (MLFs for short) and the average dwell time based method, which required the MLFs to be nonincreasing. Mancilla-Aguilar et al 32 studied nonautonomous switched nonlinear systems (SNSs for short) with perturbations via decomposing the system dynamics into a nominal-like part and a perturbation-like one. Especially, Chen and Yang 34 considered the problems of input-to-state stability and global uniform asymptotic stability of nonautonomous SNSs by employing the methods of MLFs, minimum dwell time, and infinite switchings, which allowed the derivatives of MLFs to be indefinite.…”

Uniform Asymptotic Stability of Switched Nonlinear Time-Varying Systems and Detectability of Reduced Limiting Control Systems

“…Opposite to the time-invariant case, results based on weak Lyapunov functions valid for switched time-varying systems are scarce. A GUAS result was obtained in [20] by using a perturbation approach. A refinement of Matrosov's theorem can be found in [21].…”