Certified Model Predictive Control for Switched Evolution Equations using Model Order Reduction
For researchers in PDE-constrained optimization and control, this work provides certified ROM-MPC algorithms with rigorous error bounds, though it is an incremental extension of existing ROM-MPC techniques to switched systems.
The paper presents an MPC framework for linear switched evolution equations from parabolic PDEs, using ROM to reduce computational cost. It derives a-posteriori estimates guaranteeing that the ROM-MPC trajectory stays within a computable neighborhood of the true MPC trajectory.
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.