Optimal Stabilization using Lyapunov Measures

TL;DR

This paper develops a numerical method for optimal feedback stabilization of discrete-time systems using Lyapunov measures and linear programming, enabling stabilization of complex orbits.

Contribution

It introduces a set-theoretic Lyapunov measure approach combined with linear programming for optimal stabilization, extending traditional methods.

Findings

Finite-dimensional linear program yields stabilizing feedback controls.

Conditions for existence of stabilizing controls are established.

Method successfully stabilizes a period-two orbit in a controlled standard map.

Abstract

Numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this paper. Set-theoretic notion of almost everywhere stability introduced by the Lyapunov measure, weaker than conventional Lyapunov function-based stabilization methods, is used for optimal stabilization. The linear Perron-Frobenius transfer operator is used to pose the optimal stabilization problem as an infinite dimensional linear program. Set-oriented numerical methods are used to obtain the finite dimensional approximation of the linear program. We provide conditions for the existence of stabilizing feedback controls and show the optimal stabilizing feedback control can be obtained as a solution of a finite dimensional linear program. The approach is demonstrated on stabilization of period two orbit in a controlled standard map.