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The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

Zhen-Chen Guo, Xin Liang

TL;DR

The paper tackles the challenge of computing the d-stabilizing solution to large-scale discrete-time and continuous-time algebraic Riccati equations when the state matrix may be unstable. It derives a Toeplitz-structured closed form and develops an FFT-based Toeplitz-structured approximation (FTA) that dramatically reduces computational cost by exploiting low displacement rank and block-Toeplitz structure, avoiding heavy shift strategies. By utilizing Cayley transformations and an incorporation (defect-correction) technique, the authors unify several existing Riccati solvers (e.g., SDA, RADI, Cayley-transformed methods) under a common framework and show equivalence under the same initial guess and shifts. The approach yields scalable time and space complexities, demonstrates robustness to spectral properties of $A$, and delivers competitive accuracy on large benchmarks, making it practical for large-scale control design with low-rank inputs. Overall, the work provides both a theoretical closed form and a practical, FFT-enabled solver that broadens the applicability of Riccati equation techniques to massive systems.

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.

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

TL;DR

The paper tackles the challenge of computing the d-stabilizing solution to large-scale discrete-time and continuous-time algebraic Riccati equations when the state matrix may be unstable. It derives a Toeplitz-structured closed form and develops an FFT-based Toeplitz-structured approximation (FTA) that dramatically reduces computational cost by exploiting low displacement rank and block-Toeplitz structure, avoiding heavy shift strategies. By utilizing Cayley transformations and an incorporation (defect-correction) technique, the authors unify several existing Riccati solvers (e.g., SDA, RADI, Cayley-transformed methods) under a common framework and show equivalence under the same initial guess and shifts. The approach yields scalable time and space complexities, demonstrates robustness to spectral properties of , and delivers competitive accuracy on large benchmarks, making it practical for large-scale control design with low-rank inputs. Overall, the work provides both a theoretical closed form and a practical, FFT-enabled solver that broadens the applicability of Riccati equation techniques to massive systems.

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.
Paper Structure (19 sections, 17 theorems, 124 equations, 1 figure, 2 algorithms)

This paper contains 19 sections, 17 theorems, 124 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. DARE
  4. In the operator view
  5. In the matrix view
  6. Efficient method
  7. Parameter and output
  8. Time complexity
  9. Space complexity
  10. Arbitrary initial term
  11. CARE
  12. FTA and its equivalence to many other methods
  13. Parameter and output
  14. Time complexity
  15. Space complexity
  16. ...and 4 more sections

Key Result

Lemma 2.1

Given $Y\in \mathbb{R}^{p_1\times p_2}, D_{t-1}\in \mathbb{R}^{p_1(t-1)\times p_2}$, let Then

Figures (1)

  • Figure 5.1: accuracy vs. time

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Remark 3.1
  • proof
  • ...and 24 more