矩阵值有理逼近的向量拟合
Vector Fitting for Matrix-valued Rational Approximation
AI总结：
本文研究了向量拟合方法在矩阵值有理逼近中的应用，提出控制麦卡伦度数的新机制，并通过数值实验验证了改进的拟合精度。
摘要：
向量拟合（VF）是一种构造有理近似的流行方法，能提供最小二乘拟合频率响应测量。在先前的工作中，我们分析了VF在标量有理函数中的应用，并建立了与最优H_2逼近的联系。本文在此基础上扩展了之前的框架，以构造有效的矩阵值函数有理近似，这是一个在标量情况下不出现的重大挑战。多输入/多输出（MIMO）动力系统相关的传递函数属于我们考虑的函数类别。其他人也考虑了将VF扩展到矩阵值函数的相关数值实现。然而，到目前为止，关于数值问题的详细分析尚不存在。我们在此提供这样的分析，包括关键的实现细节。对于矩阵值函数的VF，一个重要的问题仍未得到充分解决，即控制所得到的有理近似器的麦卡伦度数；在输入/输出维度较大的情况下，麦卡伦度数可能变得非常大。我们引入了两种新的机制来控制最终近似器的麦卡伦度数，一种基于交替最小二乘最小化，另一种基于辅助系统理论的缩减方法。受我们之前在标量VF问题上的工作以及最近计算最优H_2逼近的创新启发，我们建立了与最优H_2逼近的联系，并通过数值求积显著提高了VF的保真度，几乎不增加成本或复杂度。我们提供了多个数值示例来支持理论讨论和所提出的算法。
英文摘要：
Vector Fitting (VF) is a popular method of constructing rational approximants that provides a least squares fit to frequency response measurements. In an earlier work, we provided an analysis of VF for scalar-valued rational functions and established a connection with optimal $H_2$ approximation. We build on this work and extend the previous framework to include the construction of effective rational approximations to matrix-valued functions, a problem which presents significant challenges that do not appear in the scalar case. Transfer functions associated with multi-input/multi-output (MIMO) dynamical systems typify the class of functions that we consider here. Others have also considered extensions of VF to matrix-valued functions and related numerical implementations are readily available. However to our knowledge, a detailed analysis of numerical issues that arise does not yet exist. We offer such an analysis including critical implementation details here. One important issue that arises for VF on matrix-valued functions that has remained largely unaddressed is the control of the McMillan degree of the resulting rational approximant; the McMillan degree can grow very high in the case of large input/output dimensions. We introduce two new mechanisms for controlling the McMillan degree of the final approximant, one based on alternating least-squares minimization and one based on ancillary system-theoretic reduction methods. Motivated in part by our earlier work on the scalar VF problem as well as by recent innovations for computing optimal $H_2$ approximation, we establish a connection with optimal $H_2$ approximation, and are able to improve significantly the fidelity of VF through numerical quadrature, with virtually no increase in cost or complexity. We provide several numerical examples to support the theoretical discussion and proposed algorithms.
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