Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations
Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations
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DOI：
10.1016/j.cam.2011.02.011
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发表时间：
2011-04
期刊：
影响因子：
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通讯作者：
Haijun Peng;Zhigang Wu;W. Zhong
中科院分区：
文献类型：
--
作者：
Haijun Peng;Zhigang Wu;W. Zhong
将傅里叶级数展开与递归矩阵公式相结合，提出了计算周期Riccati和Lyapunov矩阵微分方程的周期、非负、定稳定解的可靠新算法。首先，将周期系数用傅里叶级数展开，求解时变周期Riccati微分方程，并用正弦和余弦级数精确求出相关哈密顿系统的状态转移矩阵。通过引入Riccati变换方法，推导出求解由四块状态转移矩阵组成的周期Riccati微分方程的递归矩阵公式。其次，基于傅里叶级数展开和递推矩阵公式，提出了求解时变周期系数Lyapunov微分方程的两种数值子方法。前一种算法是维数展开法，后一种算法是齐次周期Riccati微分方程的解。最后，通过四个算例验证了所提算法的有效性和可靠性。
Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.