多线性动态系统与控制的一种新方法
A New Approach to Multilinear Dynamical Systems and Control
摘要：
本文提出了一种新的方法来分析和控制多线性动力系统。该方法是基于张量分解和一个新定义的循环代数的最新发展。特别地，它表明，在右张量乘法算子下，三阶张量可以被写为三阶张量的乘积，这类似于传统的矩阵特征值分解，其中“特征向量”成为特征矩阵，“特征值”成为特征元组。这一新的发展允许一个适当的张量特征值分解被定义，并通过\textit{tensor-exponential}自然延伸到线性系统理论。通过这个框架，我们扩展了许多传统的技术在线性系统理论中使用的多线性对应。
英文摘要：
The current paper presents a new approach to multilinear dynamical systems analysis and control. The approach is based upon recent developments in tensor decompositions and a newly defined algebra of circulants. In particular, it is shown that under the right tensor multiplication operator, a third order tensor can be written as a product of third order tensors that is analogous to a traditional matrix eigenvalue decomposition where the "eigenvectors" become eigenmatrices and the "eigenvalues" become eigen-tuples. This new development allows for a proper tensor eigenvalue decomposition to be defined and has natural extension to linear systems theory through a \textit{tensor-exponential}. Through this framework we extend many of traditional techniques used in linear system theory to their multilinear counterpart.
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