Dynamical Low-Rank Approximation Strategies for Nonlinear Feedback Control Problems

TL;DR

This paper introduces low-rank approximation methods to efficiently solve high-dimensional nonlinear feedback control problems, significantly reducing computational costs while maintaining accuracy.

Contribution

It proposes the Dynamical Low-Rank Approximation framework with two novel algorithms, cascade-Newton-Kleinman and R-DLRA, for real-time, parametric feedback control solutions.

Findings

R-DLRA outperforms standard DLRA in speed and accuracy.

The methods effectively handle high-dimensional, parametric control problems.

Validation on nonlinear test cases demonstrates practical efficiency.

Abstract

This paper addresses the stabilization of dynamical systems in the infinite horizon optimal control setting using nonlinear feedback control based on State-Dependent Riccati Equations (SDREs). While effective, the practical implementation of such feedback strategies is often constrained by the high dimensionality of state spaces and the computational challenges associated with solving SDREs, particularly in parametric scenarios. To mitigate these limitations, we introduce the Dynamical Low-Rank Approximation (DLRA) methodology, which provides an efficient and accurate framework for addressing high-dimensional feedback control problems. DLRA dynamically constructs a compact, low-dimensional representation that evolves with the problem, enabling the simultaneous resolution of multiple parametric instances in real-time. We propose two novel algorithms to enhance numerical performances: the cascade-Newton-Kleinman method and Riccati-based DLRA (R-DLRA). The cascade-Newton-Kleinman method accelerates convergence by leveraging Riccati solutions from the nearby parameter or time instance, supported by a theoretical sensitivity analysis. R-DLRA integrates Riccati information into the DLRA basis construction to improve the quality of the solution. These approaches are validated through nonlinear one-dimensional and two-dimensional test cases showing transport-like behavior, demonstrating that R-DLRA outperforms standard DLRA and Proper Orthogonal Decomposition-based model order reduction in both speed and accuracy, offering a superior alternative to Full Order Model solutions.