The convergence analysis of an accelerated iteration for solving algebraic Riccati equations

TL;DR

This paper analyzes the convergence properties of an accelerated fixed-point iteration method for solving discrete-time algebraic Riccati equations, providing new theoretical insights and practical verification.

Contribution

It offers new stability analysis and reduces convergence conditions for AFPI in solving DARE, with proofs and numerical validation.

Findings

AFPI converges R-superlinearly when spectral radius > 1

Theoretical stability properties of DARE solutions are established

Numerical examples confirm the effectiveness of the proposed analysis

Abstract

The discrete-time algebraic Riccati equation (DARE) have extensive applications in optimal control problems. We provide new theoretical supports to the stability properties of solutions to the DARE and reduce the convergence conditions under which the accelerated fixed-point iteration (AFPI) can be applied to compute the numerical solutions of DARE. In particular, we verify that the convergence of AFPI is R-superlinear when the spectral radius of the closed-loop matrix is greater than 1, which is shown by mild assumption and only using primary matrix theories. Numerical examples are shown to illustrate the consistency and effectiveness of our theoretical results.