Generalized sampling solutions for discontinuous stabilization of nonlinear systems

Abstract

Sample-and-hold solution and sample stability are useful tools for analyzing stability of nonlinear control systems with discontinuous state feedbacks; for nonlinear control systems on Euclidean spaces, open-loop asymptotic controllability and sample stability of closed-loop systems are equivalent. Moreover, several sample stabilizing state feedback controllers based on nonsmooth control Lyapunov functions (CLFs) are proposed.However, analyzing sample stability based on the nonsmooth CLFs are too complex tasks in general. This is a serious obstacle to introduce more advanced properties such as invariance principle and its application to nonlinear adaptive control or finite-time stability for the sample stability framework.In this paper, we introduce the concept of generalized sampling solution, which is defined as the uniform limit of a sequence of sample-and-hold solutions. Moreover, we define asymptotic stability for generalized sampling solutions.The main contribution of this paper is to prove the existence and the asymptotic stability under the assumption that a locally semiconcave practical CLF (LS-PFCLF) is given.We also briefly mention an invariance principle based on a generalized sampling solution and an LS-PCLF.