周期系统是控制领域研究的一个重要课题，而周期离散时间代数Riccati方程在周期离散时间线性系统最优控制问题的稳定性分析中具有决定性的作用。本课题首次从双重结构---周期结构与symplectic结构或J-正交结构的角度出发研究周期离散时间代数Riccati方程的数值方法。 主要研究内容包括：(1) 用有理矩阵函数系统地建立周期离散时间代数Riccati方程的几何理论；(2) 通过研究带有symplectic结构或J-正交结构的周期矩阵的Schur分解方法获得周期离散时间代数Riccati方程的保结构的算法；(3) 研究周期线性矩阵方程扰动分析与数值算法。研究结果将为周期系统最优控制问题提供理论依据和稳定的数值方法。

The periodic system is one of the most important topics in the control field and periodic discrete-time algebraic Riccati equations play a key role in stability analysis of the periodic discrete-time linear optimal control prolem. This project firstly studies numerical methods for periodic discrete-time algebraic Riccati equations in terms of the dual structure, that is, periodic structure and symplectic structure or J-orthognal structure. The main research contents include: (1) The geometry theory of the periodic discrete-time algebraic Riccati equations is established systematically with rational matrix functions; (2) Structure-preserving algorithms for periodic discrete-time algebraic Riccati equations are obtained by establishing periodic Schur decompositions with symplectic or J-orthognal structures; (3) perturbation analysis and numerical methods for periodic linear matrix equations are studied; The research results will provide a theoretical basis and stable numerical methods for the periodic discrete-time linear optimal control problem.