# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 1 Introduction
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Matrices** & \(\vdots\) & **Third-Order Tensors** \\ \hline \(\mathbf{X}\in\mathbb{R}^{I\times J}\) & \(\vdots\) & \(\mathcal{X}\in\mathbb{R}^{I\times J\times K}\) \\ \hline Reduced SVD & \(\vdots\) & PARAFAC/CPD \\ \(x_{i,j}=\sum_{r=1}^{R}\sigma_{r}u_{i,r}v_{j,r}\iff\mathbf{X}=\mathbf{U} \mathbf{\Sigma}\mathbf{V}^{\mathrm{T}}\) & \(\vdots\) & \(x_{i,j,k}=\sum_{r=1}^{R}a_{i,r}b_{j,r}c_{k,r}\) \\ \(\mathbf{U}\in\mathbb{R}^{I\times R},\mathbf{V}\in\mathbb{R}^{I\times R}, \mathbf{\Sigma}\in\mathbb{R}^{R\times R}\) & \(\vdots\) & \(\mathbf{A}\in\mathbb{R}^{I\times R},\mathbf{B}\in\mathbb{R}^{I\times R}, \mathbf{C}\in\mathbb{R}^{K\times R}\) \\ \hline \multirow{4}{*}{TD} & \(\vdots\) & TD \\  & \(\vdots\) & \(x_{i,j,k}=\sum_{p=1}^{p}\sum_{q=1}^{Q}\sum_{s=1}^{S}g_{p,qs}a_{i,p}b_{j,q}c_{ks}\) \\  & \(\vdots\) & \(\mathbf{A}\in\mathbb{R}^{I\times P},\mathbf{B}\in\mathbb{R}^{I\times Q}, \mathbf{C}\in\mathbb{R}^{K\times S},\mathcal{G}\in\mathbb{R}^{P\times Q\times S}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Reduced SVD, PARAFAC/CPD, and TD. Volterra models are frequently used due to the fact that they allow approximating any fading memory nonlinear systems with an arbitrary precision, as shown in [26]. They represent a direct nonlinear extension of the very popular FIR linear model, with guaranteed stability in the bounded-input bounded-output (BIBO) sense, and they have the advantage to be linear in their parameters, the kernel coefficients [27]. The nonlinearity of a \(P\)th-order truncated Volterra model is due to products of up to \(P\) samples of delayed inputs. Moreover, they are interpretable in terms of multidimensional convolutions which makes the derivation of their z-transform and Fourier transform representations easy[28]. Among the numerous application areas of Volterra models, we can mention chemical and biochemical processes [29], radio-over-fiber (RoF) wireless communication systems (due to optical/electrical (O/E) conversion) [30; 31], high-power amplifiers (HPA) in satellite communications [32; 33], physiological systems [34], vibrating structures and more generally mechatronic systems like robots [35], and acoustic echo cancellation [36]. The main drawback of Volterra models is their parametric complexity implying the need to estimate a huge number of parameters which exponentially grows with the order and memory of the kernels. So, several complexity reduction approaches for Volterra models have been developed using symmetrization or triangularization of Volterra kernels, or their expansion on orthogonal bases like Laguerre and Kautz ones, or generalized orthogonal bases (GOB). Considering Volterra kernels as tensors, they can also be decomposed using a PARAFAC decomposition or a tensor train (TT). These approaches lead to the so-called Volterra-Laguerre, Volterra-GOB-Tucker, Volterra-PARAFAC and Volterra-TT models [37; 38; 39; 40; 41; 42]. In Sections 5.3 and 5.4, we review the Volterra-PARAFAC and Volterra-GOB-Tucker models. Note that a model-pruning approach can also be employed to adjust the complexity reduction in considering only nearly diagonal coefficients of the kernels and removing the other ones which correspond to more delayed input values whose influence decreases when the delay increases [43]. Another approach for ensuring a reduced parametric complexity consists in considering block-oriented NL models, composed of two types of blocks: linear time-invariant