Data-driven Analysis of T-Product-based Dynamical Systems

TL;DR

This letter introduces a novel framework for data-driven analysis of T-product-based dynamical systems (TPDSs), where the system evolution is governed by the T-product between a third-order dynamic tensor and a third-order state tensor.

Abstract

A wide variety of data can be represented using third-order tensors, spanning applications in chemometrics, psychometrics, and image processing. However, traditional data-driven frameworks are not naturally equipped to process tensors without first unfolding or flattening the data, which can result in a loss of crucial higher-order structural information. In this article, we introduce a novel framework for the data-driven analysis of T-product-based dynamical systems (TPDSs), where the system evolution is governed by the T-product between a third-order dynamic tensor and a third-order state tensor. In particular, we examine the data informativity of TPDSs concerning system identification, stability, controllability, and stabilizability and illustrate significant computational improvements over traditional approaches by leveraging the unique properties of the T-product. The effectiveness of our framework is demonstrated through numerical examples.

Figures (3)

• Figure 1: Computational time comparison in determining the data informativity for system identification between the unfolding-based and T-product-based approaches. a. Log-log plot of computational time with respect to the dimension of the third mode $r$. b. Ratio of time to the dimension of the third mode $r$ (i.e., time/$r^3$ for the unfolding-based approach and time/$r$ for the T-product-based approach).
• Figure 2: Computational time comparison in determining the data informativity for stability between the unfolding-based and T-product-based approaches. a. Log-log plot of computational time with respect to the dimension of the third mode $r$. b. Ratio of time to the dimension of the third mode $r$ (i.e., time/$r^3$ for the unfolding-based approach and time/$r$ for the T-product-based approach).
• Figure 3: Computational time comparison in determining the data informativity for controllability between the unfolding-based and T-product-based approaches. a. Log-log plot of computational time with respect to the dimension of the third mode $r$. b. Ratio of time to the dimension of the third mode $r$ (i.e., time/$r^3$ for the unfolding-based approach and time/$r$ for the T-product-based approach).

Theorems & Definitions (25)

• Definition 1: T-product
• Definition 2: T-eigenvalue decomposition
• Definition 3: T-singular value decomposition
• Definition 4: System identification
• Proposition 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof