Invariance-Like Results for Nonautonomous Switched Systems

Abstract

This paper generalizes the LaSalle-Yoshizawa Theorem to switched nonsmooth systems. 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 candidate common Lyapunov function that has a negative semidefinite derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle-Yoshizawa results for the switched system. Results for regular and non-regular candidate Lyapunov functions are developed via appropriate generalization of the notion of a time derivative. The developed generalization is motivated by adaptive control of switched systems where the derivative of the candidate Lyapunov function is typically negative semidefinite.Index Terms switched systems, differential inclusions, adaptive systems, nonlinear systems Rushikesh Kamalapurkar is with the School 2 benefit from adaptive methods where the controller adapts to the uncertain dynamics without strictly relying on high gain or high frequency feedback often associated with robust control methods that can lead to overstimulation.Switched dynamics are inherent in a variety of modern adaptation strategies. For example, in sparse neural networks [5], the use of different approximation architectures for different regions of the state-space introduce switching via the feedforward part of the controller. In adaptive gain scheduling methods [6], switching is introduced due to changing feedback gains. Switching is also utilized as a tool to improve transient response of adaptive controllers by selecting between multiple estimated models of stable plants (see, e.g., [7]-[16]). In addition, switched systems theory can be utilized to extend the scope of existing adaptive solutions to more complex circumstances that involve switched dynamics.Lyapunov-based stability analysis of switched nonautonomous adaptive systems is challenging because adaptive update laws typically result in non-strict Lyapunov functions for the individual subsystems. For each subsystem, convergence of the error signal to the origin is typically established using Barbȃlat's lemma (e.g., [17, Lemma 8.2]).In traditional methods that utilize multiple Lyapunov functions (e.g., [18, Theorem 3.2]) the class of admissible switching signals is restricted based on the rate of decay of the cLf (cf. [18, Eq. 3.10]). Since Barbȃlat's lemma provides no information about the rate of decay of the cLf, it alone is insufficient to establish stability of a switched system using multiple Lyapunov functions. Approaches based on common cLfs have been developed for systems with negative definite Lyapunov derivatives; however, common cLf-based approaches do not trivially extend to systems with non-strict Lyapunov functions (cf. [19]-[21] and [18, Example 2.1]). Because of complications resulting from a negative semidefinite Lyapunov derivative, few results are available in literature that examine adaptive control of uncertain nonlinear switched systems. An...

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