关键词: 正则周期矩阵对, 周期Schur分解, 广义周期Sylvester方程, 分离度, Frobenius范数

Abstract: 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.

Key words: regular periodic matrix pair, periodic Schur decomposition, generalized periodic Sylvester equation, separation, Frobenius norm