离散对偶代数Riccati方程异类约束解的双迭代算法

数学物理学报 ›› 2014, Vol. 34 ›› Issue (6): 1440-1449.

离散对偶代数Riccati方程异类约束解的双迭代算法

宋卫红|张凯院|聂玉峰

西北工业大学应用数学系西安 710072

收稿日期:2013-04-25 修回日期:2014-03-20 出版日期:2014-12-25 发布日期:2014-12-25
基金资助:
国家自然科学基金(11071196)资助

Double Iterative Algorithm for Different Constrained Solution of Discrete Coupled Algebraic Riccati Equation

SONG Wei-Hong, ZHANG Kai-Yuan, NIE Yu-Feng

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072

Received:2013-04-25 Revised:2014-03-20 Online:2014-12-25 Published:2014-12-25
Supported by:
国家自然科学基金(11071196)资助

摘要/Abstract

摘要：

利用逆矩阵的Neumann级数形式,将在离散时间跳跃线性二次控制问题中遇到的含未知矩阵之逆的离散对偶代数Riccati方程(DCARE)转化为高次多项式矩阵方程组, 然后采用牛顿算法求高次多项式矩阵方程组的异类约束解,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组的异类约束解或者异类约束最小二乘解,建立求DCARE的异类约束解的双迭代算法.双迭代算法仅要求DCARE有异类约束解,不要求它的异类约束解唯一, 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: 离散对偶代数Riccati方程, 异类约束解, 牛顿算法, 修正共轭梯度法, 双迭代算法

Abstract:

By using Neumann series of inverse matrix,discrete coupled algebraic Riccati equation with unknown matrix inverse in discrete-time jump linear quadratic control problems can be transformed into the high degree polynomial matrix equations.Then Newton's method is applied to find different constrained solution of polynomial matrix equations, and the modified conjugate gradient method is used to solve different constrained solution or different constrained least square solution of linear matrix equations derived from each iterative step of Newton's method.In this way, a double iterative method is established to solve for different constrained solution of discrete coupled algebraic Riccati equation.Different constrained solution of discrete coupled algebraic Riccati equation is only required by double iterative algorithm.But it may not be unique.Besides there are not additional limits to its coefficient matrix of discrete coupled algebraic Riccati equation.The effectiveness of the double iterative method is demonstrated by numerical examples.

Key words: Discrete coupled algebraic Riccati equation, Different constrained solution, Newton´s method, Modified conjugate gradient method, Double iterative method

中图分类号:

49M15

宋卫红, 张凯院, 聂玉峰. 离散对偶代数Riccati方程异类约束解的双迭代算法[J]. 数学物理学报, 2014, 34(6): 1440-1449.

SONG Wei-Hong, ZHANG Kai-Yuan, NIE Yu-Feng. Double Iterative Algorithm for Different Constrained Solution of Discrete Coupled Algebraic Riccati Equation[J]. Acta mathematica scientia,Series A, 2014, 34(6): 1440-1449.

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