On the Stability of Multilinear Dynamical Systems

2021/05/03 by Chen, Can
#FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · doi:10.48550/arxiv.2105.01041

Abstract

This paper investigates the stability properties of discrete-time multilinear dynamical systems via tensor spectral theory. In particular, if the dynamic tensor of a multilinear dynamical system is orthogonally decomposable (odeco), we can construct its explicit solution by exploiting tensor Z-eigenvalues and Z-eigenvectors. Based on the form of the explicit solution, we illustrate that the Z-eigenvalues of the dynamic tensor play a significant role in the stability analysis, offering necessary and sufficient conditions. In addition, by utilizing the upper bounds of Z-spectral radii, we are able to determine the asymptotic stability of the multilinear dynamical system efficiently. Furthermore, we extend the stability results to the multilinear dynamical systems with non-odeco dynamic tensors by exploiting tensor singular values. We demonstrate our results via numerical examples.

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