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

Michael Kartmann, Mattia Manucci, Benjamin Unger, Stefan Volkwein

TL;DR

This paper develops a certified model predictive control framework for linear switched parabolic PDEs by integrating Galerkin model order reduction to create low-dimensional surrogates for forward and adjoint dynamics. It derives the finite-horizon optimality conditions, accommodates convex and non-differentiable regularizations, and introduces two ROM-MPC algorithms with recursive a-posteriori error certification to bound deviations from the full-order MPC trajectory. The approach is validated on a two-room heat-conduction example using POD-based ROMs, demonstrating that ROM-MPC trajectories remain in a computable neighborhood of the true MPC trajectory, with neighborhood size governed by ROM quality and implementable in real time with limited FOM updates. The work advances certified MOR-MPC for infinite-dimensional switched systems, enabling reliable, fast MPC for complex PDE dynamics and offering practical guidance for ROM updates and error estimation in receding-horizon 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.

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

TL;DR

This paper develops a certified model predictive control framework for linear switched parabolic PDEs by integrating Galerkin model order reduction to create low-dimensional surrogates for forward and adjoint dynamics. It derives the finite-horizon optimality conditions, accommodates convex and non-differentiable regularizations, and introduces two ROM-MPC algorithms with recursive a-posteriori error certification to bound deviations from the full-order MPC trajectory. The approach is validated on a two-room heat-conduction example using POD-based ROMs, demonstrating that ROM-MPC trajectories remain in a computable neighborhood of the true MPC trajectory, with neighborhood size governed by ROM quality and implementable in real time with limited FOM updates. The work advances certified MOR-MPC for infinite-dimensional switched systems, enabling reliable, fast MPC for complex PDE dynamics and offering practical guidance for ROM updates and error estimation in receding-horizon 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.

Paper Structure

This paper contains 26 sections, 16 theorems, 115 equations, 7 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main results
  3. Literature-review and state-of-the-art
  4. Optimal control and MPC for switched systems
  5. MPC and MOR
  6. MOR for switched systems
  7. Structure of the paper
  8. Notation
  9. Problem formulation and well-posedness
  10. MPC problem
  11. Well-posedness of the switched system
  12. Guiding example
  13. Well-posedness of the optimal control problem
  14. Guiding example (continued)
  15. Certified approximations for model predictive control
  16. ...and 11 more sections

Key Result

Proposition 2.4

Let the switched system eqn:switchedPDE with external switching signal $\sigma\in\mathbb{S}$ satisfy ass:switchedSystemass:transitionOperators. Then, for every $f\in L^2(\mathbb{T}_n,V')$ there exists a unique solution $\theta\in{\mathscr X}_{n,\sigma}$. If in addition $\mathcal{K}_{i,j} = \mathcal{

Figures (7)

  • Figure 1: The domain $\Omega$ and the two rooms $\Omega_1$, $\Omega_3$. The switching signal $\sigma$ acts on the door $\Omega_2$ between the walls $\Omega_4$ and the control $u$ acts on $\Gamma_c$.
  • Figure 2: Decay of the error estimator $\Delta_\theta, \Delta_p$ and effectivities against the basis size $r$ for the open-loop problem
  • Figure 3: Decay of the error estimators $\Delta_A, \Delta_B, \tilde{\Delta}_B$ (left) and corresponding effectiveness rate (right) with respect to the basis size $r$ for the open-loop problem
  • Figure 4: FOM-ROM-MPC error estimators (left) and corresponding effectiveness rate (right) over the MPC steps $n$
  • Figure 5: ROM-ROM-MPC error estimators (left) and corresponding effectiveness rate (right) over the MPC steps $n$
  • ...and 2 more figures

Theorems & Definitions (39)

  • Remark 2.2
  • Proposition 2.4: Well-posedness of the switched system
  • proof
  • Corollary 2.5
  • Definition 2.6: Solution operator
  • Remark 2.7
  • Theorem 2.9: Unique solution of the OCP
  • proof
  • Theorem 2.10: First-order optimality condition
  • Corollary 2.11: Well-posedness of the adjoint equation
  • ...and 29 more