Measures and LMIs for optimal control of piecewise-affine systems

TL;DR

This paper develops a hierarchy of LMI relaxations for solving optimal control problems of piecewise-affine systems, providing polynomial approximations of the value function and stabilizing suboptimal policies.

Contribution

It introduces a novel approach using LMI relaxations and occupation measures to approximate solutions to PWA optimal control problems with constraints.

Findings

The method yields stabilizing suboptimal feedback laws.

The approach effectively approximates the value function solving the HJB equation.

Results demonstrate the method's ability to handle constrained PWA systems.

Abstract

This paper considers the class of deterministic continuous-time optimal control problems (OCPs) with piecewise-affine (PWA) vector field, polynomial Lagrangian and semialgebraic input and state constraints. The OCP is first relaxed as an infinite-dimensional linear program (LP) over a space of occupation measures. This LP, a particular instance of the generalized moment problem, is then approached by an asymptotically converging hierarchy of linear matrix inequality (LMI) relaxations. The relaxed dual of the original LP returns a polynomial approximation of the value function that solves the Hamilton-Jacobi-Bellman (HJB) equation of the OCP. Based on this polynomial approximation, a suboptimal policy is developed to construct a state feedback in a sample-and-hold manner. The results show that the suboptimal policy succeeds in providing a stabilizing suboptimal state feedback law that drives the system relatively close to the optimal trajectories and respects the given constraints.