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

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月4日

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  • Michael Kartmann,
  • Mattia Manucci,
  • Benjamin Unger,
  • Stefan Volkwein

摘要

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. This 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.