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

Abstract

This paper is concerned with the study of both, local and global, uniform asymptotic stability for switched nonlinear time-varying (NLTV) systems through the detectability of outputmaps. With this aim the notion of reduced limiting control systems for switched NLTV systems whose switchings verify time/state dependent constraints, and the concept of weakly zerostate detectability for those reduced limiting systems are introduced. Necessary and sufficient conditions for the (global)uniform asymptotic stability of families of trajectories of the switched system are obtained in terms of this detectability property. These sufficient conditions in conjunction with the existence of multiple weak Lyapunov functions, yield a criterion for the (global) uniform asymptotic stability of families of trajectories of the switched system. This criterion can be seen as an extension of the classical Krasovskii-LaSalle theorem. An interesting feature of the results is that no dwell-time assumptions are made. Moreover, they can be used for establishing the global uniform asymptotic stability of switched NLTV system under arbitrary switchings. The effectiveness of the proposed results is illustrated by means of various interesting examples, including the stability analysis of a semi-quasi-Z-source inverter.

Invariance Principles and Observability in Switched Systems With an Application in Consensus

“…Let s = (2i n + 1)n + t + Ini(x) for some T n ≤ i n ≤ 2nM n + T n − 1 satisfying that Therefore, 1+α 1 (t)−e t s α3(τ )dτ (1+α 1 (s)) ≤ 0 for all t ≥ s, which results in (32). Now integrating both sides of (31) and by using (32), one gets for all t ≥ s. By taking the limit, (33) holds. This completes the proof.…”

“…Similarly, ( 12) can be replaced by ζ∈Λ λζ (t) hζ (t, x(t)) = 0 for almost all t ∈ R, see [5], [33], [38] for some discussion on this.…”

“…It is worthwhile to highlight that some early attempts 1 A weak Lyapunov function is a positive definite function whose derivative along the trajectories of dynamics is negative semi-definite. arXiv:2006.02021v1 [eess.SY] 3 Jun 2020 2 [25], [26], [33] on a switched NLTV system followed this direction with the assumption that the switched NTLV system has a weak Lyapunov function, and gave the generalization of the so-called Krasovskii-LaSalle theorem, which requires stronger assumptions than that in the classical LaSalle invariance principle. For switched NLTV systems, such results can only guarantee the convergence of an equilibrium point (or a compact set).…”

Finite‐time asynchronous control of linear time‐varying switched systems

“…24 Necessary and sufficient conditions for the uniform asymptotic stability of switched nonlinear time-varying systems were obtained by the extension of the classical Krasovskii-LaSalle theorem. 25 The input-to-state stability of time-varying switched systems was investigated in Reference 26 and further developed for the case of time delay. 27 In view of the present situation, we find that in contract to the time invariant case, when the conventional Lyapunov function method is used to solve the stability and stabilization problem of LTV switched systems, it may either lead to matrix Riccati differential equations or indefinite LMIs which makes the solving process become complex, even the solution cannot be obtained.…”