Low-Rank Tensor Decompositions for Nonlinear System Identification: A Tutorial with Examples

Tensor decompositions can be a powerful tool when faced with the curse of dimensionality and have been applied in myriad applications. Their application to problems in the control community remains largely unexplored. This article aims at filling this gap by introducing tensor decompositions, where the key idea is always to exploit structure in the problem to lift the curse of dimensionality. This structure leads to the notion of low rank, which can be intuitively understood as parameters in the problem being correlated. The potential of low-rank tensor decompositions is illustrated by means of three applications, specifically in nonlinear system identification. The parametric identifiation of both Volterra systems and state-space models with polynomial inputs is discussed.