Port-Hamiltonian structures in infinite-dimensional optimal control: Primal-Dual gradient method and control-by-interconnection

TL;DR

This paper explores port-Hamiltonian structures in infinite-dimensional optimal control, demonstrating stability and convergence of primal-dual gradient methods and proposing a port-Hamiltonian based control strategy.

Contribution

It introduces a novel class of nonlinear monotone port-Hamiltonian systems and applies port-Hamiltonian modeling to analyze and design stable optimal control and receding horizon control schemes.

Findings

Primal-dual gradient method viewed as an infinite-dimensional nonlinear pH system.

Exponential stability of the primal-dual dynamics towards optimality.

Closed-loop system exhibits local exponential convergence.

Abstract

In this note, we consider port-Hamiltonian structures in numerical optimal control of ordinary differential equations. By introducing a novel class of nonlinear monotone port-Hamiltonian (pH) systems, we show that the primal-dual gradient method may be viewed as an infinite-dimensional nonlinear pH system. The monotonicity and the particular block structure arising in the optimality system is used to prove exponential stability of the dynamics towards its equilibrium, which is a critical point of the first-order optimality conditions. Leveraging the port-based modeling, we propose an optimization-based controller in a suboptimal receding horizon control fashion. To this end, the primal-dual gradient based optimizer-dynamics is coupled to a pH plant dynamics in a power-preserving manner. We show that the resulting model is again monotone pH system and prove that the closed-loop exhibits local exponential convergence towards the equilibrium.