A New Approach to Multilinear Dynamical Systems and Control
This work addresses a foundational gap in systems theory for researchers and engineers dealing with complex multilinear models, representing a novel method rather than an incremental improvement.
The paper tackles the problem of analyzing and controlling multilinear dynamical systems by introducing a tensor eigenvalue decomposition based on tensor algebra, enabling the extension of linear systems theory techniques to multilinear counterparts.
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.