Towards Optimal Spatio-Temporal Decomposition of Control-Related Sum-of-Squares Programs

TL;DR

This paper introduces an optimization-based approach to automatically determine the optimal spatio-temporal decomposition for sum-of-squares programs, improving the calculation of the Region of Attraction in nonlinear systems.

Contribution

It eliminates the need for ad-hoc split selection by optimizing the splits via conic differentiation, enhancing accuracy and robustness in ROA computation.

Findings

The method effectively computes ROA with improved accuracy.

The approach guarantees no duality gap in the optimization.

Numerical examples demonstrate practical effectiveness.

Abstract

This paper presents a method for calculating the Region of Attraction (ROA) of nonlinear dynamical systems, both with and without control. The ROA is determined by solving a hierarchy of semidefinite programs (SDPs) defined on a splitting of the time and state space. Previous works demonstrated that this splitting could significantly enhance approximation accuracy, although the improvement was highly dependent on the ad-hoc selection of split locations. In this work, we eliminate the need for this ad-hoc selection by introducing an optimization-based method that performs the splits through conic differentiation of the underlying semidefinite programming problem. We provide the differentiability conditions for the split ROA problem, prove the absence of a duality gap, and demonstrate the effectiveness of our method through numerical examples.