# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 5 Tensor-Based Approaches for Nonlinear Systems
### Polynomial Models  
Polynomial models are a direct extension of linear models. For a single-input single-output (SISO) system, the output of a recursive polynomial model, at discrete-time instant \(t\), is given by  
\[\hat{y}(t)=\sum_{p=1}^{P}f_{p}[u(t),\cdots,u(t-n_{u}),y(t-1),\cdots,y(t-n_{y})], \tag{9}\]  
where \(f_{p}(. )\) is a \(p\)th-degree polynomial in the system input \((u)\) and output \((y)\) signals, \(P\) is the nonlinearity order, and \(M=max(n_{u},n_{y})\) is the memory of the model. In the sequel, all the signals will be assumed to be real-valued. The input/output (I/O) relationship (9) is also called a nonlinear autoregressive with exogenous input (NARX) model [83], or a one-step prediction model, i.e., a model whose output \(\hat{y}(t)\) at time \(t\) depends on past values \(y(t-n)\) (for \(n\in\langle n_{y}\rangle\)) of the system output, and current and past values \(u(t-n)\) (for \(n=0,1,\cdots,n_{u}\)) of the system input. This model is an extension of the standard autoregressive with exogenous input (ARX) model, frequently used to study discrete time series, due to the presence of nonlinear terms in the input-output signals, which explains its success in many industrial applications. Equation (9) can also be written as a regression model which is linear in its parameters, namely the polynomial coefficients, and nonlinear in the I/O signals:  
\[\hat{y}(t)=\boldsymbol{\sigma}^{T}(\mathbf{u}(t),\mathbf{y}(t-1))\boldsymbol{ \theta}, \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Third-Order Tensor** & **Nth-Order Tensor** \\ \hline \(\mathcal{X}\in\mathbb{K}^{\mathbb{I}\times\mathbb{I}\times\mathbb{K}}\) & \(\mathcal{X}\in\mathbb{K}^{\mathbb{I}\times\mathbb{I}}\) \\ \hline \(\mathcal{G}\in\mathbb{K}^{P\times Q\times S},\mathbf{A}\in\mathbb{K}^{\mathbb{I }\times P},\) & \(\mathcal{G}\in\mathbb{K}^{\mathbb{E}\times},\,\mathbf{A}^{(n)}\in\mathbb{K}^{ \mathbb{I}\times\mathbb{R}_{u}},n\in\langle N\rangle\) \\ \(\mathbf{B}\in\mathbb{K}^{\mathbb{I}\times Q},\mathbf{C}\in\mathbb{K}^{\mathbb{K }\times S}\) & \\ \hline \(x_{ijk}=\sum\limits_{p=1}^{P}\sum\limits_{q=1}^{Q}\sum\limits_{s=1}^{S}\sum \limits_{s=1}^{R_{pq}}a_{p}b_{p}a_{p}b_{q}c_{\mathbb{I}s}\) & [PERSON] writing \(x_{ik}=\sum\limits_{\tau_{1}=1}^{R_{1}}\sum\limits_{\tau_{2}=1}^{R_{N}}s_{ \tau_{1}\cdots\tau_{N}}\prod_{n=1}^{N}a_{n}^{(n)}\) \\ \hline \(\mathcal{X}=\mathcal{G}\times_{1}\mathbf{A}\times_{2}\mathbf{B}\times_{3} \mathbf{C}\) & with mode-\(n\) products \(\mathcal{X}=\mathcal{G}\times_{1}\mathbf{A}^{(1)}\times_{2}\mathbf{A}^{(2)} \times_{3}\cdots\times_{N}\mathbf{A}^{(N)}\) \\ \hline \(\mathcal{X}=\sum\limits_{p=1}^{P}\sum\limits_{q=1}^{Q}\sum\limits_{s=1}^{S}s_{ \rho_{pq}}\mathbf{A}_{p}\circ\mathbf{B}_{q}\circ\mathbf{C}_{s}\) & with outer products \(\mathcal{X}=\sum\limits_{\tau_{1}=1}^{R_{1}}\cdots\sum\limits_{\tau_{N}=1}^{R_ {N}}s_{\tau_{1}\cdots\tau_{N}}\sum\limits_{n=1}^{N}\mathbf{A}_{\tau_{s}}^{(n)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Tucker decomposition of a tensor of order three and order \(N\). where \(\mathbf{u}(t)=[u(t)\cdots u(t-n_{u})]^{T}\), \(\mathbf{y}(t-1)=[y(t-1)\cdots y(t-n_{y})]^{T}\), and \(\boldsymbol{\varphi}\) is the nonlinear regressor vector whose components are monomials in (i.e., products of) previous system outputs and previous and current system inputs contained in the vectors \(\mathbf{y}(t-1)\) and \(\mathbf{u}(t)\), and \(\boldsymbol{\theta}\) is the parameter vector containing the polynomial coefficients. If the previous system outputs \(y(t-1),\cdots,y(t-n_{y})\) are replaced by previous model outputs \(\hat{y}(t-1),\cdots,\hat{y}(t-n_{y})\), the polynomial model (9) is then called a simulation or nonlinear output error (NOE) model, defined as  
\[\hat{y}(t)=\sum_{p=1}^{p}f_{p}[u(t),\cdots,u(t-n_{u}),\hat{y}(t-1),\cdots,\hat {y}(t-n_{y})]=\boldsymbol{\varphi}^{T}(\mathbf{u}(t),\hat{\mathbf{y}}(t-1)) \boldsymbol{\theta}, \tag{11}\]  
with \(\hat{\mathbf{y}}(t-1)=[\hat{y}(t-1)\cdots\hat{y}(t-n_{y})]^{T}\).