基于求线性矩阵方程约束解的修正共轭梯度法, 针对源于低增益反馈设计中的一类参量连续代数~Riccati~方程, 建立求其非零对称解的两种互为补充的迭代算法, 称之为 变换-MCG~算法和牛顿-MCG~算法. 在一定条件下, 当~Riccati~方程 存在可逆对称解或唯一对称正定解时, 由变换-MCG~算法所得对称 解具备可逆性或正定性. 牛顿-MCG~算法仅要求~Riccati~方程存 在非零对称解, 对系数矩阵等没有附加限定, 但所得对称解不能 保证可逆性或正定性. 数值算例表明, 两种迭代算法是有效的.

Concerning the continuous algebraic Riccati equation with a parameter arising from low gain feedback design, this note proposes two algorithms for the nonzero symmetric solution of the Riccati equation. These two algorithms, based on the modified conjugate gradient method for the constrained solution of linear matrix equation, are called Transforming-MCG algorithm and Newton-MCG algorithm. Two algorithms are complementary with each other. Under suitable assumptions that the Riccati equation has the reversible symmetric or unique symmetric positive definite solution, the nonzero symmetric solution by the former algorithm can promise the above character, while the solution by the latter algorithm cannot. However, Newton-MCG algorithm has no other limits to the coefficient matrix of the Riccati equation except for the existence of nonzero symmetric solution. Numerical experiments confirm that these two algorithms are effective.