# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 1 Introduction
\begin{table}
\begin{tabular}{c} \hline \hline
**Linear models** \\ \hline SISO FIR model \\ \(y(t)=\sum_{i}^{n_{k}}h_{i}u(t-i)\) \\ \hline Memoryless MIMO model \\ \(y_{i}(t)=\sum_{j=1}^{n}h_{i,j}u_{j}(t),\;\;i\in[1,n_{R}]\) \\ \(\mathbf{y}(t)=\mathbf{H}\mathbf{u}(t),\;\;\mathbf{y}(t)\in\mathbb{R}^{n_{R}}, \;\mathbf{u}(t)\in\mathbb{R}^{n_{T}},\mathbf{H}\in\mathbb{R}^{n_{R}\times n_{T}}\) \\ \hline
**Nonlinear models** \\ \hline Polynomial model (Section 5.1) \\ \(y(t)=\sum_{p=1}^{p}f_{p}[u(t),\cdots,u(t-n_{u}),y(t-1),\cdots,y(t-n_{y})]\) \\ \(f_{p}(.)=p\)th-degree polynomial in the system input \((u)\) and output \((y)\) signals \\ \hline Truncated Volterra model (Section 5.2) \\ \(y(t)=h_{0}+\sum_{p=1}^{p}\sum_{m_{1}=1}^{M_{p}}\cdots\sum_{m_{p}=1}^{M_{p}}h_{ m_{1},\cdots,m_{p}}^{(p)}\prod_{q=1}^{p}u(t-m_{q}+1)\) \\ \(h_{m_{1},\cdots,m_{p}}^{(p)}=p\)th-order Volterra kernel with memory \(M_{p}\) \\ \hline
**Multilinear models** \\ \hline TITO model (Section 6.2) \\ \(y_{i_{1},\cdots,i_{p}}(t)=\sum_{j_{1}=1}^{l_{1}}\cdots\sum_{j_{N}=1}^{l_{N}}h_{ i_{1},\cdots,i_{p}j_{N},i_{p}j_{N}}^{\text{t}}u_{j_{1},\cdots,j_{N}}(t)\) \\ \(\mathcal{U}(t)\in\mathbb{R}^{l_{N}\times\cdots\times l_{N}},\mathcal{Y}(t)\in \mathbb{R}^{l_{N}\times\cdots\times l_{N}}\) \\ \hline Multilinear TISO model (Sections 6.3 and 6.4) \\ \(y(t)=\sum_{j_{1}=1}^{l_{1}}\cdots\sum_{j_{N}=1}^{l_{N}}\Big{(}\prod\limits_{n =1}^{N}h_{i_{1}}^{(n)}\Big{)}u_{i_{1},\cdots,j_{N}}(t)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Some examples of linear, nonlinear and multilinear models. (LTI) dynamic blocks and static NL blocks. The linear blocks may be parametric (transfer functions, FIR models, state-space representations) or nonparametric (impulse responses), whereas the NL blocks may be with memory or memoryless. The different blocks are concatenated in series leading to the so-called Hammerstein (NL-LTI) and Wiener (LTI-NL) models, extended to the Wiener-Hammerstein (LTI-NL-LTI) and Hammerstein-Wiener (NL-LTI-NL) models, abbreviated W-H and H-W, respectively. To extend the modeling potential of block-oriented models, several W-H and H-W models can also be interconnected in parallel. Although such models are simpler but less general than Volterra models, they allow us to represent numerous nonlinear systems. One of the first applications of block-oriented NL models was for modeling biological systems [44]. A lot of papers have been devoted to the identification of block-oriented models and their applications. For more details, the reader is referred to the book [45] and the survey papers [46; 47]. In Section 5.5, we show that the Wiener, Hammerstein and W-H models are equivalent to structured Volterra models. This equivalence is at the origin of the structure identification method for block-oriented systems, which will be presented in Section 5.5.4. Tensor-based methods using this equivalence have been developed to estimate the parameters of block-oriented nonlinear systems [48; 49; 50; 51]. These methods are generally composed of two steps. In the first one, the Volterra kernel associated with a particular block-oriented system is used to estimate the LTI component(s). Note that there exist closed-form solutions for estimating only the Volterra kernel of interest. Such a solution is proposed in [52; 53] for a third-order and fifth-order kernel, respectively. Then, in a second step, the nonlinear block is estimated using the LS method. An example of a tensor-based method for identifying a nonlinear communication channel represented by means of a W-H model was proposed in [54] using the associated third-order Volterra kernel. On the other hand, multilinear models are useful for modeling coupled dynamical systems in engineering, biology, and physics. Tensor-based approaches have been proposed for solving and identifying multilinear systems [24; 55; 56].