# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 5 Tensor-Based Approaches for Nonlinear Systems
### Volterra-PARAFAC Models  
As each permutation of the indices \(m_{1},\cdots,m_{p}\) corresponds to the same product \(\prod_{q=1}^{p}u(t-m_{q}+1)\) of delayed inputs, we can sum all the coefficients associated with these permutations to get a symmetric kernel given by  
\[h_{m_{1},\cdots,m_{p}}^{(p,sym)}=\frac{1}{p!}\sum_{\pi(.)}h_{m_{\pi(1)}, \cdots,m_{\pi(p)}}^{(p)},\]  
where \((\pi(1),\cdots,\pi(p))\) denotes a permutation of \((1,\cdots,p)\). So, in the sequel, without loss of generality, the Volterra kernels of order \(p\geq 2\) will be considered in symmetric form. Assuming all the kernels have the same memory \(M\), the number of independent coefficients contained in the symmetric \(p\)th-order kernel is equal to \(C_{p}^{M+p-1}=\frac{(M+p-1)!}{p!\,(M-1)! }\), showing that this number, and consequently the parametric complexity of the Volterra model, grows quickly with \(M\) even for moderate \(p\). In order to reduce the complexity of Volterra models, a PARAFAC decomposition of symmetrized kernels was exploited in [40; 41]. The symmetrized \(p\)th-order Volterra kernel can then be decomposed using a symmetric PARAFAC decomposition, with symmetric rank \(r_{p}\) and matrix factor \(\mathbf{A}^{(p)}\in\mathbb{R}^{M\times r_{p}}\), for \(p\in\langle P\rangle\), as [77]  
\[h_{m_{1},\cdots,m_{p}}^{(p,sym)}=\sum_{r=1}^{r_{p}}\prod_{q=1}^{p}a_{m_{q}}^{ (p)}\,\ m_{q}=1,\cdots M. \tag{15}\]  
**Remark 1**. : _Note that a \(p\)th-order Volterra kernel is said to be separable if it can be written as the product of \(p\) first-order kernels [28], i.e.,_  
\[h_{m_{1},\cdots,m_{p}}^{(p)}=\prod_{q=1}^{p}a_{m_{q}}^{(p)}\,\ m_{q}=1,\cdots M \tag{16}\]  
_which corresponds to a rank-one PARAFAC decomposition (15)._The kernel decomposition (15) allows rewritting the \(p\)th-order homogeneous term as follows:  
\[\hat{y}^{(p)}(t) =\sum_{m_{1}=1}^{M}\cdots\sum_{m_{p}=1}^{M}h_{m_{1},\cdots,m_{p}}^{ (p,sym)}\prod_{q=1}^{p}u(t-m_{q}+1) \tag{17}\] \[=\sum_{m_{1}=1}^{M}\cdots\sum_{m_{p}=1}^{M}\big{(}\sum_{r=1}^{r_{ p}}\prod_{q=1}^{p}a_{m_{q},r}^{(p)}\big{)}\prod_{q=1}^{p}u(t-m_{q}+1), \tag{18}\]  
or equivalently  
\[\hat{y}^{(p)}(t)=\sum_{r=1}^{r_{p}}\prod_{q=1}^{p}\Big{(}\sum_{m_{q}=1}^{M}a_{ m_{q},r}^{(p)}u(t-m_{q}+1)\Big{)}=\sum_{r=1}^{r_{p}}\big{(}\mathbf{u}^{T}(t) \mathbf{A}_{r}^{(p)}\big{)}^{p}. \tag{19}\]  
We then obtain a homogeneous polynomial of degree \(p\) expressed as a sum of powers of linear forms, which is directly connected to the Waring problem. Note that a Waring's decomposition consists in expressing a homogeneous polynomial of degree \(p\) in \(n\) variables (i.e., a quantics), associated with a symmetric tensor, as a sum of \(p\)th powers of linear forms [84]. This \(p\)th-order homogeneous term can therefore be carried out in parallelizing \(r_{p}\) Wiener models. As introduced later (see Section 5.5.2), each Wiener model is composed of a FIR linear filter whose coefficients are the components of a column \(\mathbf{A}_{r}^{(p)}\in\mathbb{R}^{M}\) of the matrix factor \(\mathbf{A}^{(p)}\), in cascade with a static nonlinearity equal to the power \((.)^{p}\). Consequently, the Volterra model output (12) is obtained as the sum of the offset term \(h_{0}\), and the outputs of \(\sum_{p=1}^{p}r_{p}\) Wiener models in parallel, as illustrated in Figure 1 for a cubic Volterra-PARAFAC model, where \(\mathbf{A}_{1}^{(1)}=\left[h_{1}^{(1)},\cdots,h_{M}^{(1)}\right]\) and \(r_{1}=1\). It is worth noting that such a Volterra-PARAFAC model provides a very attractive modular and parallel architecture for approximating nonlinear systems with a low computational complexity. Figure 1: Realization of a third-order Volterra–PARAFAC model as Wiener models in parallel.