Conference Agenda

The controller design for both nonlinear and large-scale dynamical systems is a challenge and, up to now, no generally applicable and feasible computational approach has been established.
In this talk, we will consider approximative LPV embeddings that allow us to parametrize and reduce the system's structural complexity (so that controller design becomes easier) while leaving the state space untouched (so that the model expressiveness is preserved).
For such models with a reduced and parametrized nonlinear structure but a high-dimensional state space, several computational approaches are available for controller design. For all of them, however, a low-dimensional parametrization is important. For that we present the concept of polytopic autoencoders that reliably outperform standard model order reduction like POD (Proper Orthogonal Decomposition) at very low dimensions and that provide additional beneficial structures in the LPV parametrization. The main idea is that reconstruction happens in a polytope rather than in a linear space and the realization is done in specially developed neural network architectures.
Owing to a particular structure in the neural network design, the concept of polytopic autoencoders comes with the beneficial analytical property of preserving the exact Jacobian in the points of interest.
The general concept will be illustrated with numerical example simulations.
References:
1. Heiland, Jan / Kim, Yongho: Polytopic autoencoders with smooth clustering for reduced-order modeling of flows (2025) https://doi.org/10.1016/j.jcp.2024.113526 — https://arxiv.org/abs/2403.18044
2. Heiland, Jan / Kim, Yongho / Werner, Steffen W. R.: Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback controller design (2025) https://doi.org/10.1007/s10444-025-10269-1 — https://arxiv.org/abs/2401.10620

Model Predictive Control (MPC) is a well established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is, in general, infeasible, the method determines a suboptimal feedback control by repeatedly solving finite time optimal control problems.

In this talk, we consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis method (RB) as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as the feedback control for the MPC of the original large-scale system. We briefly recall the RB-MPC approach for linear systems which guarantees asymptotic stability of the closed-loop system. We then extend these results to nonlinear sytems based on the Schlögl model and problems with distributed controls. Numerical results are presented to validate our approach.

We address the stabilization of linear time-varying parabolic PDEs using model predictive control (MPC) based on reduced-order models (ROMs). We first prove exponential stability and suboptimality of the full-order MPC scheme in Hilbert spaces. Since MPC requires the repeated solution of finite-horizon optimal control subproblems, and since the dynamics typically become simpler close to the equilibrium, model order reduction is a natural approach to accelerate the online computations.

We then introduce a Galerkin reduced-order approach together with a rigorous a posteriori error analysis for the associated finite-horizon optimal control problems. This leads to a ROM-based MPC algorithm that adaptively constructs reduced-order controls, ensures exponential stability of the full-order closed-loop state, and provides computable performance bounds with respect to the infinite-horizon full-order control problem. The resulting ROM-MPC scheme combines error estimation, adaptive optimization, and closed-loop control in a certified pipeline, and can efficiently update the ROM when the dynamics change unpredictably online.

Numerical experiments with a nonsmooth cost functional involving the squared $\ell^1$-norm confirm the effectiveness of the method, showing substantial computational savings together with low approximation errors, even for exponentially unstable systems.

We present a short introduction to Funnel MPC, a model predictive control algorithm allowing for tracking of a smooth reference signal within a prescribed error bound. The optimal control problem solved in each time step resembles that of a barrier function in optimization. We first motivate our approach by discussing finite dimensional systems with stable internal dynamics before showing initial and recursive feasibility for special boundary controlled heat equations.