Lyapunov stability theory of nonsmooth systems

Abstract

Abstract-This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajec· tories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right-hand side such as in nonsmooth dynamic systems or variable structure control.

Finite-time stability with respect to a closed invariant set for a class of discontinuous systems

“…v). Lemma 1.1 [3,18] (Chain rule) Let x(t) ∈ S(x 0 ) be a Filippov solution of system (1). and V : D → R be a Lipschitz continuous and regular function.…”

“…Definition 1.6 [3,18] (Clarke s generalized gradient) For a locally Lipschitz function V : D → R, define the generalized gradient of V at x by…”

“…For example, He [2] presented the Lasalle principle and stability for discontinuous systems. In the framework of the Filippov solutions, Shevitz and Paden [3] developed the Lyapunov stability theory for nonsmooth systems and Lasalle's invariance principle for discontinuous autonomous systems in 1994. In 1999, Bacciottic and Ceragioli [4] defined a class of new set-value derivative and generalized the results in the more general context of differential inclusions.…”

A smooth switching adaptive controller for linearizable systems with improved transient performance

“…& Remarks 1. The differential equation in (20), owing to the proposed control (15), is actually discontinuous with respect to the hyperplane e T PB ¼ 0: To take this fact into account, the derivative ' V in (34) should be replaced with the so-called generalized gradient for a rigorous analysis, even though the conclusion remains unaltered [11,12]. However, for the ease of access, the derivation here is still based on the smooth Lyapunov theory.…”

The equilibrium stability for a smooth and discontinuous oscillator with dry friction

“…[11], the sufficient conditions for attractivity and unstability of the equilibrium sets were given by employing the Lyapunov stability theory and Lasalle's invariance principle. The stability properties of the equilibrium set for the nonsmooth dynamic systems can be determined by Lyapunov stability theory [12,13]. By using the Coulomb friction law, the set of equilibrium states of a mass-spring system with unilateral contact and Coulomb friction were investigated, and the stability properties of all these equilibrium states are obtained with numerical experiments in Ref.…”