An efficient iteration for the extremal solutions of discrete-time algebraic Riccati equations

TL;DR

This paper introduces an accelerated fixed-point iteration method for efficiently computing extremal solutions of discrete-time algebraic Riccati equations, which are crucial in control applications.

Contribution

It develops a new AFPI method based on the semigroup property, with proven R-superlinear convergence for extremal solutions of AREs.

Findings

The AFPI converges at least R-superlinearly under mild conditions.

Numerical examples demonstrate the method's efficiency and feasibility.

The approach effectively computes extremal solutions in control problems.

Abstract

Algebraic Riccati equations (AREs) have been extensively applicable in linear optimal control problems and many efficient numerical methods were developed. The most attention of numerical solutions is the (almost) stabilizing solution in the past works. Nevertheless, it is an interesting and challenging issue in finding the extremal solutions of AREs which play a vital role in the applications. In this paper, based on the semigroup property, an accelerated fixed-point iteration (AFPI) is developed for solving the extremal solutions of the discrete-time algebraic Riccati equation. In addition, we prove that the convergence of the AFPI is at least R-suplinear with order $r>1$ under some mild assumptions. Numerical examples are shown to illustrate the feasibility and efficiency of the proposed algorithm.