正则周期矩阵对分离度的估计

正则周期矩阵对在分析和设计线性离散周期控制系统中有重要应用．正则周期矩阵对的分离度是测量周期矩阵对的周期收缩子空间敏感性的一个重要指标，因此，计算这个分离度显得非常重要．然而，这需要的太多的计算量．目前对单个矩阵和矩阵对的分离度的估计，常用的处理方法有二种：一种是利用矩阵的Schur分解，另一种则是利用矩阵的约当分解．本文应用正则周期矩阵对的周期Schur分解，给出这个分离度的上界和下界．这与计算精确的分离度相比较可大大减少运算量．另外，这些界可以看成是两正则矩阵对分离度的上、下界的推广．最后，数值例子验证了所给的下界和上界．

Estimation of the Separation Degree of Regular Periodic Matrix Pairs

The regular periodic matrix pair has some important applications in the analysis and design of linear discrete time periodic control systems. The separation degree between two regular periodic matrix pairs is an important quantity that measures the sensitivity of periodic deflating subspaces of regular periodic matrix pairs. So it is important to compute this quantity. However, this requires a lot of floating point arithmetic operations. Up to now, there are two different methods for estimating the separation degree of two matrices or two regular matrix pairs. One is based on the Schur decomposition of a matrix, the other is based on the Jordan decomposition of a matrix. In this paper, we apply the periodic Schur decompositions of regular periodic matrix pairs to derive lower and upper bounds of the separation degree. Comparing with the exact separation degree computation, estimating these bounds requires much less floating point arithmetic operations. In addition, these lower and upper bounds can be regarded as a generalization of those of the separation degree for two regular matrix pairs. Finally, lower and upper bounds are illustrated by a numerical example.