Global CLF Stabilization of Nonlinear Systems. Part II: An Approximation Approach---Closed CVS

Abstract

The aim of this paper is to design regular feedback controls for the global asymptotic stabilization gas) of affine control systems with closed (convex) control value sets cvs) $U\subseteq\mathbb{R}^{m}$ with $0\in{int}U$, in the framework of Artstein and Sontag's control Lyapunov function clf) approach. We can consider this work as a continuation and complement of our paper [SIAM J. Control Optim., 51 (2013), pp. 2152--2175]. First, we resume some of the results achieved in that paper. Then, based on a method for approximation by compact convex sets, we propose an explicit formula for regular feedback controls for the gas of affine control systems with general compact cvs $U$ with $0\in{int}U$, provided an appropriate clf is known, but at the expense of small overflows in the control values. Moreover, if we assume smoothness on the system and the clf, a redesign yields practically smooth feedback controls. Consequently, we approximately solve the synthesis problem entailed by Artstein's theorem and also (in some degree) a Sontag's open problem with respect to general compact cvs $U$ with $0\in{int}U$. Finally, tools from convex analysis and the approximation method by compact convex sets allow us to address the problem of the global clf stabilization of affine control systems with respect to unbounded closed cvs $U$ with $0\in{int}U$, and also to the design of regular feedback controls. We illustrate the results with some examples.