Multilinear Time Invariant System Theory

Abstract

In this paper, we provide a system theoretic treatment of a new class of multilinear time invariant (MLTI) systems in which the states, inputs and outputs are tensors, and the system evolution is governed by multilinear operators. The MLTI system representation is based on the Einstein product and even-order paired tensors. There is a particular tensor unfolding which gives rise to an isomorphism from this tensor space to the general linear group, i.e. group of invertible matrices. By leveraging this unfolding operation, one can extend classical linear time invariant (LTI) system notions including stability, reachability and observability to MLTI systems. While the unfolding based formulation is a powerful theoretical construct, the computational advantages of MLTI systems can only be fully realized while working with the tensor form, where hidden patterns/structures (e.g. redundancy/correlations) can be exploited for efficient representations and computations. Along these lines, we establish new results which enable one to express tensor unfolding based stability, reachability and observability criteria in terms of more standard notions of tensor ranks/decompositions. In addition, we develop the generalized CANDECOMP/PARAFAC decomposition and tensor train decomposition based model reduction framework, which can significantly reduce the number of MLTI system parameters. Further, we provide a review of relevant tensor numerical methods to facilitate computations associated with MLTI systems without requiring unfolding. We demonstrate our framework with numerical examples.