# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 5 Tensor-Based Approaches for Nonlinear Systems
### Volterra-GOB Models  
Under stability and causality conditions, a Volterra kernel \(\mathcal{H}^{(p)}\) can be expanded on a basis of orthogonal functions [27]. Various functions have been introduced in the literature ([PERSON], [PERSON], generalized orthogonal basis functions (GOBF), etc.). Selection of such a basis has been widely studied (see [38] for instance). Defining by \(b^{(j,p)}_{k_{j}}(. )\), \(k_{j}=1,2,\cdots\), a set of orthogonal basis functions for expanding the \(p\)th-order Volterra kernel along its \(j\)th mode, \(j\in\langle p\rangle\), the GOB expansion of this Volterra kernel in such a basis is given by  
\[h^{(p)}_{m_{1},m_{2},\cdots,m_{p}}=\sum_{k_{1}=1}^{\infty}\sum_{k_{2}=1}^{ \infty}\cdots\sum_{k_{p}=1}^{\infty}g^{(p)}_{k_{1},k_{2},\cdots,k_{p}}\prod_{j =1}^{p}b^{(j,p)}_{k_{j}}(m_{j})\,\ m_{j}\in\langle M_{p}\rangle\,\ j\in\langle p\rangle, \tag{42}\]  
where \(g^{(p)}_{k_{1},k_{2},\cdots,k_{p}}\) are the coefficients of the expansion, also called Fourier coefficients, and the GOB functions \(b^{(j,p)}_{k_{j}}(. )\) in the time domain can be derived from the inverse \(z\)-transform  
Figure 4: Normalized mean square error in steady state for a sum of sines. Figure 5: Normalized mean square error in steady state for a random input. of some transfer function [86]. This expansion is often truncated to a given order \(K_{p}\) for practical reasons, leading to the following truncated development  
\[h_{m_{1},m_{2},\cdots,m_{p}}^{(p)}=\sum_{k_{1}=1}^{K_{p}}\sum_{k_{2}=1}^{K_{p}} \cdots\sum_{k_{p}=1}^{K_{p}}g_{k_{1},k_{2},\cdots,k_{p}}^{(p)}\prod_{j=1}^{p}b_ {k_{j}}^{(j,p)}(m_{j}), \tag{43}\]  
where \(b_{k_{j}}^{(j,p)}(m_{j})\) is the \(m_{j}\)th entry of the \(k_{j}\)th column \(\mathbf{B}_{k_{j}}^{(j,p)}\in\mathbb{R}^{M_{p}}\) of the matrix factor \(\mathbf{B}^{(j,p)}\in\mathbb{R}^{M_{p}\times K_{p}}\), associated with mode \(j\in\langle p\rangle\). The development (43) of the \(p\)th-order Volterra kernel, viewed as a \(p\)th-order tensor \(\mathcal{H}^{(p)}=\left[h_{m_{1},m_{2},\cdots,m_{p}}^{(p)}\right]\in\mathbb{R} ^{M_{p}\times\cdots\times M_{p}}\), can be interpreted as the following Tucker model:  
\[\mathcal{H}^{(p)}=\mathcal{G}^{(p)}\underset{j=1}{\overset{p}{\times}}\mathbf{ B}^{(j,p)}, \tag{44}\]  
where the core tensor \(\mathcal{G}^{(p)}\in\mathbb{R}^{K_{p}\times\cdots\times K_{p}}\) contains the Fourier coefficients. Consider the FIR linear filter \(B_{k_{j}}^{(j,p)}(q^{-1})=\sum_{m_{j}=1}^{M_{p}}b_{k_{j}}^{(j,p)}(m_{j})q^{-m_ {j}}\), where \(q^{-1}\) is the unit delay operator. This filter, with memory \(M_{p}\), is associated with mode \(j\) of the tensor \(\mathcal{H}^{(p)}\). Now, let us define the filtered input \(s_{k_{j}}^{(j,p)}(t)\), for \(k_{j}\in\langle K_{p}\rangle\) and \(j\in\langle p\rangle\), as  
\[s_{k_{j}}^{(j,p)}(t)=B_{k_{j}}^{(j,p)}(q^{-1})u(t)=\sum_{m_{j}=1}^{M_{p}}b_{k_ {j}}^{(j,p)}(m_{j})u(t-m_{j}). \tag{45}\]  
Using the truncated expansion (43) of the \(p\)th Volterra kernel and the filtered inputs (45), the input-output relationship for the \(p\)th-order homogeneous Volterra-GOB term can then be written as  
\[\hat{y}^{(p)}(t) =\sum_{m_{1}=1}^{M_{p}}\sum_{m_{2}=1}^{M_{p}}\cdots\sum_{m_{p}=1} ^{M_{p}}h_{m_{1},m_{2},\cdots,m_{p}}^{(p)}\prod_{j=1}^{p}u(t-m_{j}) \tag{46}\] \[=\sum_{k_{1}=1}^{K_{p}}\sum_{k_{2}=1}^{K_{p}}\cdots\sum_{k_{p}=1 }^{K_{p}}g_{k_{1},k_{2},\cdots,k_{p}}^{(p)}\prod_{j=1}^{p}s_{k_{j}}^{(j,p)}(t). \tag{47}\]  
Taking the Tucker model (44) of the \(p\)th order kernel tensor \(\mathcal{H}^{(p)}\) into account, and defining the vector \(\mathbf{s}^{(j,p)}(t)=\left[s_{1}^{(j,p)}(t),\cdots,s_{K_{p}}^{(j,p)}(t)\right] ^{T}\in\mathbb{R}^{K_{p}}\), the input-output equation for the Volterra-GOB model then becomes  
\[\hat{y}(t)=h_{0}+\sum_{p=1}^{p}\mathcal{G}^{(p)}\underset{j=1}{\overset{p}{ \times}}[\mathbf{s}^{(j,p)}(t)]^{T}. \tag{48}\]  
Figure 6 illustrates a third-order Volterra-GOB model.