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