Simultaneous and Robust Stabilization of Nonlinear Systems by Means of Continuous and Time-Varying Feedback

Abstract

In this dissertation, we address the stabilization of uncertain systems described by finite, countably infinite or uncountable families of systems. We adopt an approach that enables us to consider control systems with merely continuous dynamics as well as continuous time-invariant and time-varying feedback laws.We show that for any countable family of asymptotically stabilizable systems, there exists a continuous nonlinear time-invariant controller that simultaneously stabilizes (not asymptotically) the family. Although these controllers do not achieve simultaneous asymptotic stabilization in the general case, we manage to modify our construction in order to design continuous time-invariant feedback laws that simultaneously asymptotically stabilize certain pairs of systems in the plane.By introducing continuous time-varying feedback laws, we then prove that an finite family of linear time-invariant (LTI) systems is simultaneously asymptotically stabilizable by means of continuous nonlinear time-varying feedback if each system of the family is asymptotically stabilizable by a LTI controller. We also provide sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of LTI systems, by means of continuous time-varying feedback.We then obtain sufficient conditions for the existence of a continuous time- varying feedback law that simultaneously asymptotically stabilizes a finite family on nonlinear systems. We illustrate these results by establishing the simultaneous asymptotic stabilizability of the elements of a class of pairs of homogeneous nonlinear systems We finally consider a class of parameterized families of systems in the plane [where the parameter lies in an uncountable set] that are not robustly asymptotically stabilizable by means of C1 feedback. We solve their robust asymptotic stabilization by means of continuous feedback, through a new approach where a robust asymptotic stabilizer is considered as a feedback law that simultaneously robustly asymptotically stabilizes two sub-families of the family under consideration.