Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

TL;DR

This paper presents a method to construct a CPA contraction metric for exponentially stable periodic orbits using semidefinite optimization, enabling verification of stability and basin of attraction.

Contribution

It formulates the construction of a CPA contraction metric as a semidefinite feasibility problem, providing a systematic approach to stability analysis.

Findings

Feasible solutions yield CPA contraction metrics for stable periodic orbits.

The optimization problem is always feasible if the orbit is exponentially stable and triangulation is sufficiently fine.

An objective function can bound the Floquet exponent of the orbit.

Abstract

A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem. In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices. The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.