Computing Solutions to the Polynomial-Polynomial Regulator Problem

TL;DR

This paper introduces a method for computing polynomial feedback laws for nonlinear control systems, improving control costs and stabilization regions over traditional linear methods, demonstrated on aircraft and PDE models.

Contribution

It develops explicit formulas and scalable software for polynomial value function approximation in nonlinear optimal control, extending beyond linear-quadratic regulators.

Findings

PPR control outperforms LQR in aircraft stall stabilization.

PPR reduces control costs by approximately 75% in PDE example.

Method is scalable to high-dimensional systems.

Abstract

We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, resulting in lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas -- and a scalable, general purpose software implementation -- for computing the polynomial approximation to the value function that solves the optimal control problem. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of dimension $n=129$ of a partial differential equation, for which the PPR control reduces the control cost by approximately 75% compared to LQR for the initial condition of interest.