基于求线性矩阵方程约束解的修正共轭梯度法的基本思想,研究在最优控制系统中遇到的离散时间代数Riccati矩阵方程(DTARME)对称解的数值计算问题.首先对DTARME中的逆矩阵采用矩阵级数方法进行等价转化,然后运用牛顿算法将DTARME的对称解问题转化为线性矩阵方程的对称解或者对称最小二乘解问题,最后采用修正共轭梯度法进行计算.由此,可建立求DTARME的对称解的双迭代算法,并给出相应的收敛性结论.数值算例表明,双迭代算法是有效的.

Based on the modified conjugate gradient method in computing constrained solution of general linear matrix equation,  an iterative method is proposed to compute the symmetric  solution of discrete time algebra Riccati matrix equation  (DTARME) in optimal control system. Firstly, DTARME is processed
 by matrix series. Then Newton's method is applied to find symmetric solution of DTARME, and the symmetric solution  or symmetric least-square solution of the linear matrix equation  is solved. Finally, the modified conjugate gradient method  is applied to solve the derived linear matrix equation.  Therefore, a double iterative method is established to find  symmetric solution of DTARME, and the corresponding convergence  conclusion is given. Finally, numerical experiments show that the double  iterative algorithm is effective.