Sum of squares approximations to energy functions

TL;DR

This paper develops sum of squares polynomial approximations for energy functions in nonlinear systems, providing a computationally feasible alternative to solving complex HJB equations, and validates their effectiveness through case studies.

Contribution

It introduces a sum of squares polynomial approach with least-squares collocation to approximate energy functions, ensuring non-negativity and applicability to large-scale nonlinear systems.

Findings

Accurate polynomial approximations validated against known solutions

Effective control of van der Pol oscillator ring demonstrated

Successful application to discretized Burgers equation

Abstract

Energy functions offer natural extensions of controllability and observability Gramians to nonlinear systems, enabling various applications such as computing reachable sets, optimizing actuator and sensor placement, performing balanced truncation, and designing feedback controllers. However, these extensions to nonlinear systems depend on solving Hamilton-Jacobi-Bellman (HJB) partial differential equations, which are infeasible for large-scale systems. Polynomial approximations are a viable alternative for modest-sized systems, but conventional polynomial approximations may yield negative values of the energy away from the origin. To address this issue, we explore polynomial approximations expressed as a sum of squares to ensure non-negative approximations. In this study, we focus on a reduced sum of squares polynomial where the coefficients are found through least-squares collocation -- minimizing the HJB residual at sample points within a desired neighborhood of the origin. We validate the accuracy of these approximations through a case study with a closed-form solution and assess their effectiveness for controlling a ring of van der Pol oscillators with a Laplacian-like coupling term and discretized Burgers equation with source terms.