# Multilinear Time Invariant System Theory

## TL;DR

This paper introduces a novel multilinear time invariant (MLTI) system framework using tensor algebra, extending classical LTI concepts to tensor-based dynamic systems for applications in biological and engineering contexts.

## Contribution

It proposes a new MLTI system model based on Einstein product and paired tensors, extending classical system theory concepts to tensor representations.

## Key findings

- Extended stability, reachability, and observability notions to MLTI systems.
- Utilized recent tensor algebra advances for system analysis.
- Provided a framework for tensor-based dynamic system modeling.

## Abstract

In biological and engineering systems, structure, function and dynamics are highly coupled. Such interactions can be naturally and compactly captured via tensor based state space dynamic representations. However, such representations are not amenable to the standard system and controls framework which requires the state to be in the form of a vector. In order to address this limitation, recently a new class of multiway dynamical systems has been introduced in which the states, inputs and outputs are tensors. We propose a new form of multilinear time invariant (MLTI) systems based on the Einstein product and even-order paired tensors. We extend classical linear time invariant (LTI) system notions including stability, reachability and observability for the new MLTI system representation by leveraging recent advances in tensor algebra.