On Approximating Polynomial-Quadratic Regulator Problems

TL;DR

This paper introduces a novel approach for approximating solutions to polynomial-quadratic regulator problems involving polynomial nonlinearities, using low-degree polynomial feedback controls and efficient algorithms, demonstrated on various nonlinear systems.

Contribution

It extends LQR and QQR frameworks to polynomial nonlinear systems, proposing an efficient algorithm and software for low-degree polynomial feedback control approximation.

Findings

Effective feedback control approximation for polynomial nonlinear systems.

Successful numerical demonstrations on Lorenz, van der Pol, and Burgers equations.

Software implementation available on Github.

Abstract

Feedback control problems involving autonomous polynomial systems are prevalent, yet there are limited algorithms and software for approximating their solution. This paper represents a step forward by considering the special case of the regulator problem where the state equation has polynomial nonlinearity, control costs are quadratic, and the feedback control is approximated by low-degree polynomials. As this represents the natural extension of the linear-quadratic regulator (LQR) and quadratic-quadratic regulator (QQR) problems, we denote this class as polynomial-quadratic regulator (PQR) problems. The present approach is amenable to feedback approximations with low degree polynomials and to problems of modest model dimension. This setting can be achieved in many problems using modern model reduction methods. The Al'Brekht algorithm, when applied to polynomial nonlinearities represented as Kronecker products leads to an elegant formulation. The terms of the feedback control lead to large linear systems that can be effectively solved with an N-way generalization of the Bartels-Stewart algorithm. We demonstrate our algorithm with numerical examples that include the Lorenz equations, a ring of van der Pol oscillators, and a discretized version of the Burgers equation. The software described here is available on Github.