Reduced Order Model Predictive Control for Parametrized Parabolic Partial Differential Equations

TL;DR

This paper introduces a reduced basis-based Model Predictive Control method for large-scale parametrized parabolic PDE systems, providing error bounds and stability guarantees with an adaptive prediction horizon strategy.

Contribution

It develops a reduced order MPC framework for parametrized parabolic PDEs, including error analysis, stability guarantees, and an adaptive horizon selection method.

Findings

Efficient offline-online error bounds for control and cost functional

Guaranteed asymptotic stability of the closed-loop system

Numerical results demonstrating the approach's effectiveness

Abstract

Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems -- arising, e.g., through discretization of partial differential equations -- requires the solution of high-dimensional optimal control problems and thus poses immense computational effort. We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis (RB) method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach in several practical scenarios. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.