Convex computation of the maximum controlled invariant set for polynomial control systems

TL;DR

This paper presents a convex optimization approach using semidefinite programming and sum-of-squares techniques to compute the maximum controlled invariant set for polynomial control systems, providing converging outer approximations.

Contribution

It introduces a hierarchy of LMI relaxations and SOS problems that efficiently approximate the MCI set for polynomial systems, with convergence guarantees.

Findings

Converging outer approximations of the MCI set are obtained via a single SDP.

The method applies to both discrete and continuous-time polynomial systems.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.