A New Approach to Multilinear Dynamical Systems and Control

TL;DR

This paper introduces a novel tensor eigenvalue decomposition framework for multilinear dynamical systems, extending classical linear system techniques to tensors using tensor decompositions and circulant algebra.

Contribution

It develops a tensor eigenvalue decomposition method based on circulant algebra, enabling multilinear system analysis and control analogous to linear systems.

Findings

Tensor eigenvalue decomposition analogous to matrix eigen-decomposition.

Extension of linear system techniques to multilinear systems.

Framework based on tensor-exponential for system analysis.

Abstract

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.