Order reduction approaches for the algebraic Riccati equation and the LQR problem

TL;DR

This paper introduces Petrov-Galerkin order reduction methods for efficiently solving the algebraic Riccati equation and the LQR problem, especially for large matrices, by combining system reduction and approximate solutions.

Contribution

It proposes a novel Petrov-Galerkin framework that generalizes existing Galerkin methods for simultaneous system and Riccati equation reduction.

Findings

Enhanced efficiency for large matrix problems

Superior performance over classical methods

Effective approximation of LQR solutions

Abstract

We explore order reduction techniques for solving the algebraic Riccati equation (ARE), and investigating the numerical solution of the linear-quadratic regulator problem (LQR). A classical approach is to build a surrogate low dimensional model of the dynamical system, for instance by means of balanced truncation, and then solve the corresponding ARE. Alternatively, iterative methods can be used to directly solve the ARE and use its approximate solution to estimate quantities associated with the LQR. We propose a class of Petrov-Galerkin strategies that simultaneously reduce the dynamical system while approximately solving the ARE by projection. This methodology significantly generalizes a recently developed Galerkin method by using a pair of projection spaces, as it is often done in model order reduction of dynamical systems. Numerical experiments illustrate the advantages of the new class of methods over classical approaches when dealing with large matrices.