Necessary and Sufficient Lyapunov-Like Conditions for Robust Nonlinear Stabilization

TL;DR

This paper develops a comprehensive framework of Lyapunov-like conditions that are both necessary and sufficient for stabilizing feedback in general nonlinear systems, including time-varying and delay systems.

Contribution

It extends the Control Lyapunov Function method to cover broader classes of nonlinear systems with disturbances, partial stability, and delay structures.

Findings

Provides necessary and sufficient conditions for stabilization.

Introduces a new tool for designing robust controllers for delay systems.

Addresses partial stability with respect to output variables.

Abstract

In this work, we propose a methodology for the expression of necessary and sufficient Lyapunov-like conditions for the existence of stabilizing feedback laws. The methodology is an extension of the well-known Control Lyapunov Function (CLF) method and can be applied to very general nonlinear time-varying systems with disturbance and control inputs, including both finite- and infinite-dimensional systems. The generality of the proposed methodology is also reflected upon by the fact that partial stability with respect to output variables is addressed. In addition, it is shown that the generalized CLF method can lead to a novel tool for the explicit design of robust nonlinear controllers for a class of time-delay nonlinear systems with a triangular structure.