矩阵方程X+A*（R+B*XB）-tA=Q的Hermite正定解

矩阵方程X+A^（R+B^XB）^-tA=Q的Hermite正定解

摘要

摘要:非线性矩阵方程 X+ A ^*( R+ B ^* XB) ^-t A= Q( t≥1)来源于离散时间代数Riccati方程.本文给出该方程存在Hermite正定解的充分条件及上下界估计,构造了求解该矩阵方程的不动点迭代和免逆迭代算法,运用单调有界定理证明了算法的收敛性,最后通过数值算例说明所提算法对求解该矩阵方程的有效性及可行性.

Abstract:The nonlinear matrix equation X+ A ^*( R+ B ^* XB) ^-t A= Q( t≥ 1) is derived from the discrete-time algebraic Riccati equation. The sufficient conditions for the existence of Hermite positive definite solutions and the upper and lower bounds are given. The fixed point iteration and inverse-free iteration algorithms for solving the equation are constructed and the convergence of the algorithm is proved by using the monotone boundedness theorem. Finally, two numerical examples are given to illustrate the effectiveness and feasibility of the proposed algorithm for solving the matrix equation.