Low-complexity linear parameter-varying approximations of incompressible Navier-Stokes equations for truncated state-dependent Riccati feedback

TL;DR

This paper introduces a method to approximate nonlinear Navier-Stokes equations with linear parameter-varying models, enabling efficient feedback control design for high-dimensional systems.

Contribution

It presents a novel approach to approximate nonlinear systems using LPV models with series expansions, reducing computational complexity for large-scale systems.

Findings

Successful stabilization of Navier-Stokes equations demonstrated

Efficient numerical solution of matrix equations confirmed

Approach applicable to high-dimensional nonlinear systems

Abstract

Nonlinear feedback design via state-dependent Riccati equations is well established but unfeasible for large-scale systems because of computational costs. If the system can be embedded in the class of linear parameter-varying (LPV) systems with the parameter dependency being affine-linear, then the nonlinear feedback law has a series expansion with constant and precomputable coefficients. In this work, we propose a general method to approximating nonlinear systems such that the series expansion is possible and efficient even for high-dimensional systems. We lay out the stabilization of incompressible Navier-Stokes equations as application, discuss the numerical solution of the involved matrix-valued equations, and confirm the performance of the approach in a numerical example.