非线性复系统X^m-BX-C=0的Hermite正定解

Hermite Positive Definite Solution of Nonlinear Complex System X^m-BX-C=0

非线性矩阵方程在控制理论、动态规划、梯形网络和随机过滤等领域有广泛应用.在一定条件下讨论了m次非对称复系统X^m-BX-C=0(m≥2)的Hermite正定解问题.给出了该系统存在正定解的充要条件,并利用系数矩阵的极大极小特征值所构成代数方程的根以及Brouwer不动点定理,获得了其正定解的存在区间.为迭代求出该系统的正定解,给出了与原方程同解且具对称结构的非线性系统.然后针对系数矩阵B分别为正定、负定、不定3种情况构造出相应的迭代格式,并根据相关代数方程的特征性质,分别证明了所建立的3种迭代格式的收敛性.与此同时根据每种迭代的特点,给出了迭代初始矩阵的选取方法.运用一个5次复系统的数值算例,检验了所给方法的有效性及可行性.

Nonlinear matrix equations are widely used in fields of control theory,dynamic programming,ladder networks,and stochastic filtering.The Hermite positive definite solution of the m-th asymmetric complex system X m-BX-C=0(m≥2)under the certain conditions has been discussed.The necessary and sufficient conditions for the existence of positive definite solutions of this system have been given,and the roots of two algebraic equations been used on the based on minimax eigenvalues of the coefficient matrices and Brouwer fixed point theorem,the existence interval of positive definite solution is obtained.In order to use the iterative method to find the positive definite solution of the system,a nonlinear system has been given with the same solution and symmetric structure.Then,the three iterative schemes corresponding to B are positive definite,negative definite and indeterminate are constructed respectively,and by the properties of the relative algebraic equations,the convergence of these iterations are proved.At the same time,according to the features of each iteration,the selection method of iteration initial matrix is given.Finally,a simulation example is given to illustrate the validity and feasibility of the method.