Certified Model Predictive Control for Switched Evolution Equations using Model Order Reduction

TL;DR

This paper develops a certified model predictive control framework for switched evolution equations derived from PDEs, utilizing model order reduction to ensure computational efficiency and provide explicit bounds on control accuracy.

Contribution

It introduces a novel ROM-based MPC approach with recursive a-posteriori error estimates for switched PDE systems, enabling certified control within a low-dimensional surrogate model.

Findings

ROM-MPC trajectories stay close to true MPC trajectories within a quantifiable neighborhood.

Explicit bounds on the deviation between ROM-MPC and true MPC are derived.

Two algorithms with guaranteed performance are proposed for switched PDE control.

Abstract

We present a model predictive control (MPC) framework for linear switched evolution equations arising from a parabolic partial differential equation (PDE). First-order optimality conditions for the resulting finite-horizon optimal control problems are derived. The analysis allows for the incorporation of convex control constraints and sparse regularization. Then, to mitigate the computational burden of the MPC procedure, we employ Galerkin reduced-order modeling (ROM) techniques to obtain a low-dimensional surrogate for the state-adjoint systems. We derive recursive a-posteriori estimates for the ROM feedback law and the ROM-MPC closed-loop state and show that the ROM-MPC trajectory evolves within a neighborhood of the true MPC trajectory, whose size can be explicitly computed and is controlled by the quality of the ROM. Such estimates are then used to formulate two ROM-MPC algorithms with closed-loop certification.