526 RWTH Publication No: 841574        2021       
TITLE Reduced Order Model Predictive Control for Parametrized Parabolic Partial Differential Equations
AUTHORS Saskia Dietze, Martin A. Grepl
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.
KEYWORDS model predictive control, suboptimality, asymptotic stability, reduced basis method, a posteriori error estimation, model order reduction, partial differential equations, parameter dependent systems
DOI 10.1016/j.amc.2023.128044
PUBLICATION Applied Mathematics and Computation, Volume 453, 15 September 2023, 128044