[Paper Review] On the Stability of Multilinear Dynamical Systems

Can Chen|arXiv (Cornell University)|May 3, 2021

Tensor decomposition and applications40 references4 citations

TL;DR

This paper establishes necessary and sufficient conditions for the asymptotic stability of discrete-time multilinear dynamical systems using tensor Z-eigenvalues when the dynamic tensor is orthogonally decomposable (odeco). By deriving an explicit solution via Z-eigenvalues and Z-eigenvectors, the authors show that stability depends on the spectral radius of the dynamic tensor, and provide efficient upper bounds for Z-spectral radii to determine stability, extending results to non-odeco tensors using singular values.

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.

Motivation & Objective

• To investigate the stability properties of discrete-time multilinear dynamical systems using tensor spectral theory.
• To develop explicit solution formulas for systems with orthogonally decomposable (odeco) dynamic tensors using Z-eigenvalues and Z-eigenvectors.
• To establish necessary and sufficient stability conditions based on the Z-spectral radius of the dynamic tensor.
• To extend stability analysis to non-odeco dynamic tensors using tensor singular values and Frobenius norms.
• To explore future directions in stabilizability, reachability, and Lyapunov theory for multilinear control systems.

Proposed method

• The paper derives an explicit solution for discrete-time multilinear dynamical systems with odeco dynamic tensors by leveraging tensor Z-eigenvalues and Z-eigenvectors.
• It formulates stability conditions based on the magnitude of Z-eigenvalues, showing that asymptotic stability occurs if and only if the Z-spectral radius is less than one.
• The authors provide an upper bound for the Z-spectral radius of even-order supersymmetric tensors to enable efficient stability verification.
• For non-odeco tensors, stability is analyzed using the tensor Frobenius norm and p-mode singular values as proxies for spectral behavior.
• Theoretical results are validated through numerical examples demonstrating the effectiveness of the proposed bounds and conditions.
• The paper conjectures a reachability criterion for multilinear control systems based on iterative span expansion of control subspaces.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for asymptotic stability in discrete-time multilinear dynamical systems with odeco dynamic tensors?
• RQ2 How can tensor Z-eigenvalues and Z-eigenvectors be used to construct explicit solutions for such systems?
• RQ3 Can upper bounds on the Z-spectral radius be used to efficiently determine stability without full eigenvalue computation?
• RQ4 How can stability analysis be extended to multilinear systems with non-odeco dynamic tensors?
• RQ5 What is the relationship between reachability and the structure of the dynamic tensor and control inputs in multilinear control systems?

Key findings

• For odeco dynamic tensors, the asymptotic stability of the multilinear system is determined by the Z-spectral radius: the system is asymptotically stable if and only if the Z-spectral radius is strictly less than one.
• An explicit solution formula is derived using Z-eigenvalues and Z-eigenvectors, enabling direct analysis of system trajectories and stability.
• An upper bound for the Z-spectral radius of even-order supersymmetric tensors is provided, allowing efficient stability checks without computing all Z-eigenvalues.
• For non-odeco tensors, stability is assessed using the tensor Frobenius norm and p-mode singular values, offering a practical alternative to full spectral analysis.
• The results are applicable to homogeneous polynomial dynamical systems by constructing equivalent multilinear representations.
• A conjecture is proposed for reachability of multilinear control systems: the system is reachable if the iterative span of control subspaces generates the full space R^n, particularly when the system degree k is even.