Stabilization with Closed-loop DOA Enlargement: An Interval Analysis Approach

TL;DR

This paper introduces an interval analysis-based method to enlarge the domain of attraction for nonlinear discrete-time systems by approximating invariant sets and optimizing Lyapunov functions.

Contribution

It proposes a novel approach to estimate and enlarge the closed-loop DOA using interval analysis without designing structured controllers.

Findings

Effective approximation of invariant sets using interval analysis.

Enlargement of the closed-loop DOA through Lyapunov function optimization.

Applicable to general nonlinear discrete-time systems.

Abstract

In this paper, the stabilization problem with closed-loop domain of attraction (DOA) enlargement for discrete-time general nonlinear plants is solved. First, a sufficient condition for asymptotic stabilization and estimation of the closed-loop DOA is given. It shows that, for a given Lyapunov function, the negative-definite and invariant set in the state-control space is a stabilizing controller set and its projection along the control space to the state space can be an estimate of the closed-loop DOA. Then, an algorithm is proposed to approximate the negative-definite and invariant set for the given Lyapunov function, in which an interval analysis algorithm is used to find an inner approximation of sets as precise as desired. Finally, a solvable optimization problem is formulated to enlarge the estimate of the closed-loop DOA by selecting an appropriate Lyapunov function from a positive-definite function set. The proposed method try to find a unstructured controller set (namely, the negative-definite and invariant set) in the state-control space rather than design parameters of a structured controller in traditional synthesis methods.