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

TL;DR

The paper develops a polytopic autoencoder with smooth clustering to obtain a low-dimensional, differentiable LPV representation of nonlinear, high-dimensional systems and couples this with higher-order series expansions of state-dependent Riccati equations (SDRE) for nonlinear feedback design. It provides offline strategies for solving large Lyapunov and Riccati equations using low-rank and ADI-based methods, enabling efficient online evaluation of expanded SDRE laws. Numerical experiments on flow past a cylinder show that the nonlinear LPV parametrization with a few clusters achieves superior reconstruction at low dimensions and that second-order SDRE expansions widen the domain of stable operation compared to standard LQR. Overall, the approach offers a scalable route to accurate, low-complexity nonlinear feedback design in high-dimensional systems, with potential for LMIs and further optimization.

Abstract

Polytopic autoencoders provide low-di\-men\-sion\-al parametrizations of states in a polytope. For nonlinear 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 representation of target states and higher order series expansions of the nonlinear feedback law at little extra computational effort in the online phase and how the linear though high-dimensional and nonstandard Lyapunov equations are efficiently computed during the offline phase. In a numerical study, we illustrate the procedure and how this approach can reliably outperform the standard linear-quadratic regulator design.

Figures (6)

• Figure 1: Spatial domain and snapshots of the flow behind a cylinder in the unstable steady state (left) and in the periodic vortex shedding regime (right); cf. morHeiW23.
• Figure 2: Trajectory with pointwise reconstruction errors $\|x(t^{(i)}) - \tilde{x}(t^{(i)})\|_{M}$. The improvement of using $3$ clusters is clearly visible as is the initial alignment with the nonlinear approach PAE-3.5 (and the subsequent deviation from it) of the first-order expansion.
• Figure 3: Grid search for the selection of $r$ and $q$ with lowest averaged reconstruction error. A trade-off between the number of clusters and the reduced dimension becomes clearly visible.
• Figure 4: Differences to the exact SDRE feedback along a sample trajectory for feedback approximations through the LPV approximation of the nonlinearity by PAE-q.5, for $q = 1, 3$, and the corresponding zeroth, first and second-order series expansions of the SDRE feedback. The LPV approximation provides the most accurate approximation to the true SDRE feedback in its training regime up to $0.5$. The first and second-order approximations also strongly improve over the classical LQR feedback.
• Figure 5: Performance index \ref{['eq:fb-performance-index']} map of the linear and nonlinear feedbacks for PAE reduced-order models with $r=5$, $q = 1, 3$, the SDRE feedback law expansion \ref{['eq:sdre_fblaw_xpanded']} for $p = 0, 1, 2$ and varying penalization parameter $\gamma$ and startup times $t_s$. The darker the color, the smaller (i.e., the better) the performance index. At the blank spaces, the system blew up. The trajectories of the data marked with * are plotted in \ref{['fig:fb-trjcheck']}.

Theorems & Definitions (4)

• Remark 1
• Lemma 1
• proof
• Remark 2