Controller design and value function approximation for nonlinear dynamical systems

TL;DR

This paper introduces a sum-of-squares based method for approximating solutions to infinite-time optimal control problems in nonlinear polynomial systems, providing converging controllers and value functions with explicit suboptimality bounds.

Contribution

It develops a novel SOS-based approach that produces converging rational controllers and value functions for nonlinear systems, with explicit suboptimality estimates.

Findings

Sequence of SOS approximations converges to the optimal value function.

Method yields asymptotically optimal rational controllers.

Numerical examples validate the effectiveness of the approach.

Abstract

This work considers the infinite-time discounted optimal control problem for continuous time input-affine polynomial dynamical systems subject to polynomial state and box input constraints. We propose a sequence of sum-of-squares (SOS) approximations of this problem obtained by first lifting the original problem into the space of measures with continuous densities and then restricting these densities to polynomials. These approximations are tightenings, rather than relaxations, of the original problem and provide a sequence of rational controllers with value functions associated to these controllers converging (under some technical assumptions) to the value function of the original problem. In addition, we describe a method to obtain polynomial approximations from above and from below to the value function of the extracted rational controllers, and a method to obtain approximations from below to the optimal value function of the original problem, thereby obtaining a sequence of asymptotically optimal rational controllers with explicit estimates of suboptimality. Numerical examples demonstrate the approach.