Controller design and value function approximation for nonlinear dynamical systems

Milan Korda (1, 2) , Didier Henrion (3, 4) , Colin N. Jones (1, 2)

1 Laboratoire d'Automatique
2 EPFL - Ecole Polytechnique Fédérale de Lausanne
3 CTU - Czech Technical University in Prague
4 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande

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.

Mots clés: Optimal control; value function approximation; nonlinear control; occupation measures; semidefinite programming; sum-of-squares

Domaines: Optimisation et contrôle [math.OC]

Milan Korda, Didier Henrion, Colin N. Jones. Controller design and value function approximation for nonlinear dynamical systems. Automatica, 2016, 67 (5), pp.54-66. ⟨hal-01136669⟩