二次自伴矩阵多项式阵特征值界的数值计算

Computing Bound of Quadratic Self-Adjoint Matrix Polynomial′s Eigenvalues

在参数不确定性线性系统的鲁棒控制研究中,常用到的一个指标就是使不确定性系统在输出反馈或状态反馈控制下的闭环系统在H∞-范数界γ的条件下的二次稳定.是否二次稳定,一般要验证能否找到一个正常数,ε使相应的R iccati方程有正定解.而R iccati方程一般情况下求解相当困难.本文通过具体的分析,提出了一种在给定正定矩阵的条件下,找使此正定阵是R iccati方程的解相对应的正常数ε的可能范围的方法,即求解二次自伴矩阵多项式阵特征值界的方法.文中详细给出了所用理论及算法.给出了求正常数ε范围的一个实例.

In the research of robust H∞ control problem with parameter uncertainty, one of goals with the closed-loop system under state or output feedback control is： quadratically stable and guarantee an H∞-norm bound constraint on disturbance attenuation for all admissible uncertainties. Finding a suitable positive constant ε and making its Riccati equation have a positive definite matrix solution is a rule that can judge the plant is quadratic stable or not. But it is difficult to solve the Riccati equation in generally. This note provide a method of computing the possible range of positive constant ε to the given positive definite matrix. Theory of the method and algorithm is given in detail. A numerical example is provided to explain the method.