Optimal switching control design for polynomial systems: an LMI approach

TL;DR

This paper introduces an LMI-based method for designing optimal switching sequences in polynomial systems, providing guarantees of global optimality and a practical way to approximate solutions.

Contribution

It presents a novel LMI approach that relaxes the problem to a hierarchy of convex LMIs, ensuring asymptotic convergence to the global optimum.

Findings

Hierarchy of LMIs converges to the global optimum.

Method guarantees asymptotic optimality with vanishing conservatism.

Provides a procedure to construct nearly optimal switching sequences.

Abstract

We propose a new LMI approach to the design of optimal switching sequences for polynomial dynamical systems with state constraints. We formulate the switching design problem as an optimal control problem which is then relaxed to a linear programming (LP) problem in the space of occupation measures. This infinite-dimensional LP can be solved numerically and approximately with a hierarchy of convex finite-dimensional LMIs. In contrast with most of the existing work on LMI methods, we have a guarantee of global optimality, in the sense that we obtain an asympotically converging (i.e. with vanishing conservatism) hierarchy of lower bounds on the achievable performance. We also explain how to construct an almost optimal switching sequence.