一类离散时间代数Riccati矩阵方程对称解的双迭代算法

J4 ›› 2015, Vol. 37 ›› Issue (02): 329-334.

一类离散时间代数Riccati矩阵方程对称解的双迭代算法

张凯院，宁倩芝，牛婷婷

(西北工业大学应用数学系，陕西西安 710072)

收稿日期:2013-06-13 修回日期:2013-10-27 出版日期:2015-02-25 发布日期:2015-02-25
基金资助:
国家自然科学基金资助项目（11071196）

A double iterative algorithm for symmetric solution todiscrete time algebraic Riccati matrix equation

ZHANG Kaiyuan，NING Qianzhi，NIU Tingting

(Department of Applied Mathematics,Northwestern Polytechnical University,Xi’an 710072,China)

Received:2013-06-13 Revised:2013-10-27 Online:2015-02-25 Published:2015-02-25

摘要/Abstract

摘要：

利用逆矩阵的Neumann级数形式，将在线性二次优化问题中遇到的含未知矩阵之逆的离散时间代数Riccati矩阵方程（DTARME）转化为高次多项式矩阵方程，然后采用牛顿算法求高次多项式矩阵方程的对称解，并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解，建立求DTARME的对称解的双迭代算法。双迭代算法仅要求DTARME有对称解，不要求它的对称解唯一，也不对它的系数矩阵做附加限定。数值算例表明双迭代算法是有效的。

关键词: Riccati矩阵方程, 对称解, 牛顿算法, 修正共轭梯度法, 双迭代算法

Abstract:

Using Neumann series of inverse matrix,the discrete time algebraic Riccati matrix equation (DTARME) with unknown matrix inversion,which occurs in connection with the linear quadratic optimization problem,can be transformed into a highorder polynomial matrix equation.Then Newton’s method can be applied to find the symmetric solution to the highorder polynomial matrix equation.And from here,the modified conjugate gradient method can be used to find the symmetric solution or the symmetric leastsquare solution to the linear matrix equation derived from each iterative step of Newton's method.In this way,a double iterative algorithm is established to find the symmetric solution to DTARME.Having symmetric solution is the only requisite for DTARME based on the double iterative algorithm,and the solution may not necessarily be unique. Besides,there are no additional limits to the coefficient matrix of DTARME.Numerical experiments show that the double iterative algorithm is effective.

Key words: Riccati matrix equation;symmetric solution;Newton’s method;modified conjugate gradient method;double iterative algorithm

张凯院，宁倩芝，牛婷婷. 一类离散时间代数Riccati矩阵方程对称解的双迭代算法[J]. J4, 2015, 37(02): 329-334.

ZHANG Kaiyuan，NING Qianzhi，NIU Tingting. A double iterative algorithm for symmetric solution todiscrete time algebraic Riccati matrix equation [J]. J4, 2015, 37(02): 329-334.