Abstract
<p>In this paper, we introduce a new class of discrete multilinear time-invariant system under the t-product (tDMLTI system). The t-product is a tensor multiplication which allows a third-order tensor to be written as a product of third-order tensors. The definition of t-product can be extended to higher order tensors through recursion and we focus on third-order tensors. Basic definitions about matrices can be extended to tensors using the t-product, such as the transpose, the identity, and the inverse. Using these definitions, we provide the tensor algebraic conditions for the stability, the reachability, and the observability of the tDMLTI systems, which are analogous to the conditions for traditional linear time-invariant systems. We also discuss the balanced realization and give an error bound for the balanced truncation. Besides, a theorem about the pole assignment under the state feedback control is proved. Finally, we mention the definition of the higher order t-product and some numerical experiments are given to illustrate our theories.</p>