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The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We analyze the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the $p$ -radius of the system and the entropy of the language of allowed switching sequences. We end this paper with a method to reduce the computation of the JSR of low-rank matrices to the computation of the constrained JSR of matrices of small dimension. [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 addresses the stability problem for linear continuous-time Markovian switching singular systems. Considering the inherent state jump behavior at the switching instants, a necessary and sufficient condition of exponential stability in the mean square sense for the Markovian switching singular system is established in terms of linear matrix inequalities by means of a stochastic Lyapunov approach. Based on the obtained stability result, sufficient conditions of exponential stability in the mean square sense for the Markovian switching singular system with uncertain and partly unknown transition probability are presented. Numerical examples are presented to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper introduces a model for freeway traffic dynamics under stochastic capacity-reducing incidents, and provides insights for freeway incident management by analyzing long-time (stability) properties of the proposed model. Incidents on a multicell freeway are modeled by reduction in capacity at the affected freeway sections, which occur and clear according to a Markov chain. We develop conditions under which the traffic queue induced by stochastic incidents is bounded. A necessary condition is that the demand must not exceed the time-average capacity adjusted for spillback. A sufficient condition, in the form of a set of bilinear inequalities, is also established by constructing a Lyapunov function and applying the classical Foster–Lyapunov drift condition. Both conditions can be easily verified for realistic instances of the stochastic incident model. Our analysis relies on the construction of a globally attracting invariant set, and exploits the properties of the traffic flow dynamics. We use our results to analyze the impact of stochastic capacity fluctuation (frequency, intensity, and spatial correlation) on the throughput of a freeway segment. [ABSTRACT FROM AUTHOR]

This paper generalizes the LaSalle–Yoshizawa Theorem to switched nonsmooth systems. The Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A common candidate Lyapunov function that has a negative semidefinite generalized time derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle–Yoshizawa-like results for the switched system. Of independent interest, are the results on approximate continuity and Filippov regularization of set-valued maps, reduction of differential inclusions using Lipschitz continuous regular functions, and comparative remarks on different generalizations of the time derivative along the trajectories of a nonsmooth system. [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]

The goal of this paper is to study the exponential stabilization problem for autonomous discrete-time switched linear systems (SLSs), where only the discrete mode can be controlled. Our approach is based on periodic control Lyapunov functions whose value decreases periodically instead of at each time step as in the classical control Lyapunov functions. Using periodic control Lyapunov functions, we develop stabilizability analysis and controller synthesis conditions that are less conservative than existing results in that they apply to a larger class of SLSs. Utilizing recent results on the switched optimal control problems, a constructive way to find periodic control Lyapunov functions is presented. [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]