The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

TL;DR

This paper introduces a Toeplitz-structured closed form solution for algebraic Riccati equations, enabling efficient large-scale computation using FFT without complex assumptions or high-order matrix calculations.

Contribution

The paper presents a novel Toeplitz-based closed form for Riccati solutions and a fast FFT-based algorithm applicable to large-scale problems, improving efficiency and simplicity.

Findings

Efficient algorithm for large-scale Riccati equations using Toeplitz structure and FFT.

Theoretical equivalence of the proposed method with existing algorithms.

Numerical results demonstrating the method's effectiveness.

Abstract

In this paper we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the found form and fast Fourier transform, we propose a new algorithm for solving both discrete-time and continuous-time large-scale algebraic Riccati equations with low-rank structure. It works without unnecessary assumptions, complicated shift selection strategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting.