Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback controller design

Abstract Polytopic autoencoders provide low-dimensional parametrizations of states in a polytope. For nonlinear partial differential equations (PDEs), this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control applications and show how it improves on standard linear approaches in view of LPV approximations of nonlinear systems. We discuss how the particular architecture enables exact representations of target states and higher-order series expansions of the nonlinear feedback law at little extra computational effort in the online phase. In the offline phase, a system of linear though high-dimensional and nonstandard Lyapunov equations has to be solved. Here, we expand on how to adapt state-of-the-art methods for the efficient numerical treatment. In a numerical study, we illustrate the procedure and how this approach can reliably outperform the standard linear-quadratic regulator design.