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Analytic Interpolation With a Degree Constraint for Matrix-Valued Functions

机译：具有度约束的矩阵值函数的解析插值

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摘要

We consider a Nehari problem for matrix-valued, positive-real functions, and characterize the class of (generically) minimal-degree solutions. Analytic interpolation problems (such as the one studied herein) for positive-real functions arise in time-series modeling and system identification. The degree of positive-real interpolants relates to the dimension of models and to the degree of matricial power-spectra of vector-valued time-series. The main result of the paper generalizes earlier results in scalar analytic interpolation with a degree constraint, where the class of (generically) minimal-degree solutions is characterized by an arbitrary choice of “spectral-zeros”. Naturally, in the current matricial setting, there is freedom in assigning the Jordan structure of the spectral-zeros of the power spectrum, i.e., the spectral-zeros as well as their respective invariant subspaces. The characterization utilizes Rosenbrock''s theorem on assignability of dynamics via linear state feedback.
机译：我们考虑矩阵值正实函数的Nehari问题，并刻画（一般）最小度解的类别。正实函数的解析插值问题（例如本文研究的问题）出现在时间序列建模和系统识别中。正实插值的程度与模型的维数以及向量值时间序列的矩阵幂谱的程度有关。本文的主要结果概括了带有度约束的标量解析插值的早期结果，其中（一般）最小度解的类别具有“光谱零”的任意选择的特征。自然地，在当前的矩阵设置中，自由地分配功率谱的频谱零的约旦结构，即频谱零以及它们各自的不变子空间。该表征利用Rosenbrock定理，通过线性状态反馈确定动力学的可分配性。

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