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In this paper, the stability and $L_2$ -gain properties of linear impulsive delay systems with delayed impulses are studied. Commonly employed techniques, in which the delayed impulses are treated using Newton–Leibniz formula, may not be applicable to $L_2$ -gain analysis, since they make the disturbance input appear in the impulse part. In order to circumvent the difficulty, we first augment the considered system to a time-delay system with switching nondelayed impulses. Due to the absence of delayed impulses, this new approach has advantages in constructing Lyapunov functions and handling the effects of impulse delays on the system performance. Switching-based time-dependent Lyapunov functions are introduced to deal with the resultant switching impulses of the augmented system. Sufficient conditions for exponential stability and $L_2$ -gain properties are derived in terms of linear matrix inequalities. Numerical examples are provided to illustrate the efficiency of the new approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 output-maps. 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 weak zero-state 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 the 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 semiquasi-Z-source inverter [ABSTRACT FROM AUTHOR]

It is proved in this paper that the existence of suitable multiple Lyapunov functions is a necessary and sufficient condition for a discrete-time nonlinear switching system, with given switches digraph, to be globally asymptotically stable. The same result is provided for the input-to-state stability. The less is the number of edges in the switches digraph, the less is the number of inequalities that are involved in the provided necessary and sufficient Lyapunov conditions. [ABSTRACT FROM AUTHOR]

This paper considers the input-to-state stability (ISS) of time-varying switched systems with time delays, where the upper bound estimation for the operator of Lyapunov function (UBEOL) is assumed to be time varying and mode dependent. The ISS and integral ISS are investigated for time-varying switched systems with time delays by using the Lyapunov–Razumikhin and comparison theorem methods. Since the coefficient in the UBEOL is time varying and takes a positive/negative value, the subsystems consist of both ISS and non-ISS subsystems, simultaneously. It is shown that our presented results have wider applications than some existing works. Two examples, including one of the consensus for time-varying multiagent systems with cooperative and competitive protocols, are presented to demonstrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

This paper is concerned with dynamic output feedback control for a class of switched systems with mode-dependent average dwell-time switching. By constructing a quasi-time-dependent Lyapunov function, the issues of global uniform asymptotic stability and $\ell _{2}$ -gain analysis for the switched system are addressed first. Then, a set of reduced-order output feedback controllers is designed, which is both mode-dependent and quasi-time-dependent. Compared with time-independent criteria, the new results greatly reduce the conservatism. The effectiveness and merits of the proposed method are illustrated with a numerical example. [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]

Multiple Lyapunov functions (MLFs)-based small-gain theorems are presented for switched interconnected nonlinear systems with unstable subsystems, which extend the small-gain technique from its original non-switched nonlinear version to a switched nonlinear version. Each low dimensional subsystem does not necessarily have the input-to-state stability (ISS) property in the whole state space, and it only has individual ISS property in some subregions of the state space. The novelty of this paper is that integral-type MLFs and small-gain techniques are utilized to establish some MLFs-based small-gain theorems for switched interconnected nonlinear systems, which derive various stability results under some novel switching laws designed and construct integral-type MLFs. The small-gain theorems proposed cover several recent results as special cases, which also permit removal of a common restriction in which all low dimensional subsystems in switched interconnected systems are ISS or only some are ISS and others are not. Finally, two illustrative examples are presented to demonstrate the effectiveness of the results provided. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of designing time-varying quantizers for the stabilization of switched linear systems with quantized output and switching delays. The detection delays of switches are assumed to be time-varying but bounded, and the dwell time of the switching signal is assumed to be larger than the maximum delay. Given a switching controller, we analyze reachable sets of the closed-loop state by using a common Lyapunov function and then construct a quantizer that guarantees asymptotic stability. A sufficient condition for the existence of such a quantizer is characterized by the maximum switching delay and the dwell time. [ABSTRACT FROM AUTHOR]

In this paper we consider the class of discrete-time switched systems switching between p autonomous positive subsystems. First, sufficient conditions for testing stability, based on the existence of special classes of common Lyapunov functions, are investigated, and these conditions are mutually related, thus proving that if a linear copositive common Lyapunov function can be found, then a quadratic positive definite common function can be found, too, and this latter, in turn, ensures the existence of a quadratic copositive common function. Secondly, stabilizability is introduced and characterized. It is shown that if these systems are stabilizable, they can be stabilized by means of a periodic switching sequence, which asymptotically drives to zero every positive initial state. Conditions for the existence of state-dependent stabilizing switching laws, based on the values of a copositive (linear/quadratic) Lyapunov function, are investigated and mutually related, too. [ABSTRACT FROM PUBLISHER]

This paper considers asymptotic stability and transient performance of discrete-time piecewise affine systems. We propose a procedure to construct a nested sequence of finite-state symbolic models, each of which abstracts the original piecewise affine system and leads to linear matrix inequalities for guaranteed stability and performance levels. This sequence is in the order of decreasing conservatism, and hence gives us the option to pay more computational cost and analyze a finer symbolic model within the sequence in return for less conservative results. Moreover, in the special case where this sequence is finite, an exact analysis of stability and performance is achieved via semidefinite programming. [ABSTRACT FROM PUBLISHER]