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.

翻译：在本文中,我们为离散代数比方形,特别是状态矩阵不稳定的情况,得出了特普利茨结构化的封闭式半确定性半确定性稳定解决方案形式。基于发现的形式和快速的Fourier变异,我们提出了一种新的算法,用低级结构解决离散时间和连续的大型代数立卡蒂方程式。它没有不必要的假设、复杂的转移选择策略或对问题规模的立方体顺序矩阵计算。提供了数字示例来说明其特征。此外,我们表明,在理论上它等同于文献中存在的几种算法,即它们都在同一参数设置下产生相同的序列。