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Inspired by the idea of multiple Lyapunov functions ($\mathtt {MLFs}$), we use piecewise continuous scalar functions to investigate the stabilizability of time-varying switched systems. Starting with time-varying switched linear systems, we first combine the idea of $\mathtt {MLFs}$ with the existence of asymptotically (exponentially, uniformly exponentially) stable functions to provide necessary and sufficient conditions for their asymptotic (exponential, uniform exponential) stabilizability. Compared to traditional differential Lyapunov inequalities, we release the requirement on negative definiteness of the derivatives of $\mathtt {MLFs}$. Successively, the above results are extended to time-varying switched nonlinear systems. Then, two illustrative examples are given to show the applicability of our theoretical results. In the end, we consider the computation issue of our current results for a special class of nonautonomous switched systems, i.e., rational nonautonomous switched systems. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of common zeroing-output systems (CZOS) to analyze the stability of switched systems. The concept of CZOS allows one to verify weak zero-state detectability. It characterizes a common behavior of any individual subsystem when the output signal for each subsystem is “approaching” zero. Heuristically speaking, it removes the effect of switching behavior, and thus enables one to analyze stability properties in systems with complex switching signals. With the help of CZOS, the Krasovskii–LaSalle theorem can be extended to switched nonlinear time-varying systems with both arbitrary switching and more general restricted switching cases. For switched nonlinear time-invariant systems, the needed detectability condition is further simplified, leading to several new stability results. Particularly, when a switched linear time-invariant system is considered, it is possible to generate a recursive method, which combines a Krasovskii–LaSalle result and a nested Matrosov result, to find a CZOS if it exists. The power of the proposed CZOS is demonstrated by consensus problems in literature to obtain a stronger convergence result with weaker conditions. [ABSTRACT FROM AUTHOR]

This article investigates the stability of positive switched linear systems. We start from motivating examples and focus on the case when each switched subsystem is marginally stable (in the sense that all the eigenvalues of the subsystem matrix are in the closed left-half plane with those on the imaginary axis simple) instead of asymptotically stable. A weak excitation condition is first proposed such that the considered positive switched linear system is exponentially stable. An extension to the case without dwell time assumption is also presented. Then, we study the influence of time-varying delay on the stability of the considered positive switched linear system. We show that the proposed weak excitation condition for the delay-free case is also sufficient for the asymptotic stability of the positive switched linear system under unbounded time-varying delay. In addition, it is shown that the convergence rate is exponential if there exists an upper bound for the delay, irrespective of the magnitude of this bound. The motivating examples are revisited to illustrate the theoretical results. [ABSTRACT FROM PUBLISHER]

In this note, the stabilization problem for a class of discrete-time switched linear systems with additive disturbances is investigated. The considered switching signals are of mode-dependent persistent dwell-time (MPDT) property and the disturbances are assumed to be amplitude-bounded. By constructing a quasi-time-varying (QTV) Lyapunov function, a QTV stabilizing controller is designed for the nominal system such that the resulting closed-loop system is globally uniformly asymptotically stable. In the presence of bounded additive disturbances, a MPDT robust positive invariant set is determined for the error system between the nominal system and disturbed system. A concept of generalized robust positive invariant (GRPI) set under admissible MPDT switching is further proposed for the error system. It is demonstrated that the disturbed system is also asymptotically stable in the sense of converging to the MPDT GRPI set that can be regarded as the cross section of a uniform tube of the disturbed system. A numerical example is provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]