# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 6 Tensor-Based Approaches for Multilinear Systems
### Estimation of the System Transfer Tensor from I/O Data  
Let us define the lexicographical vectorization \(\mathbf{u}(t)\triangleq\mathrm{vec}(\mathcal{X}(t))\in\mathbb{R}^{\Pi I_{N}}\) such as  
\[u_{\hat{l}^{n}\cdots\hat{l}_{N}}(t)=x_{j_{1},\cdots,j_{N}}(t), \tag{93}\]and the (multilinear) global impulse response (GIR) \(\mathbf{h}\) as the vectorized form of the system transfer tensor \(\mathcal{A}\)  
\[\mathbf{h}=\operatorname{vec}(\mathop{\circ}\limits_{n=1}^{N}\mathbf{h}^{(n)})= \mathop{\otimes}\limits_{n=1}^{N}\mathbf{h}^{(n)}=\mathbf{h}^{(1)}\otimes \mathbf{h}^{(2)}\otimes\cdots\otimes\mathbf{h}^{(N)}\in\mathbb{R}^{\Pi I_{N}}. \tag{94}\]  
The output of the multilinear model can then be rewritten as  
\[\hat{y}(t)=\mathbf{u}^{T}(t)\,\mathbf{h}. \tag{95}\]  
Considering noisy output measurements on the time interval \([1,T]\), the noisy output vector \(\mathbf{y}(T)\in\mathbb{R}^{T}\) is given by  
\[\mathbf{y}(T)\triangleq\left[ \begin{array}{c}y(1)\\ \vdots\\ y(T)\end{array} \right]=\left[ \begin{array}{c}\mathbf{u}^{T}(1)\\ \vdots\\ \mathbf{u}^{T}(T)\end{array} \right]\mathbf{h}+\left[ \begin{array}{c}e(1)\\ \vdots\\ e(T)\end{array} \right]\triangleq\mathbf{U}(T)\,\mathbf{h}+\mathbf{e}(T), \tag{96}\]  
where \(e(t)\) is a zero-mean AWGN, at the time instant \(t\), and \(\mathbf{U}(T)\in\mathbb{R}^{T\times\Pi I_{N}}\). We now determine the weighted least-squares (WLS) estimate of the vectorized form \(\mathbf{h}\) of the GIR tensor, which minimizes the following cost fun \(\min_{\mathbf{h}}\lVert\mathbf{e}(T)\rVert_{\mathbf{W}}^{2}\), with  
\[\lVert\mathbf{e}(T)\rVert_{\mathbf{W}}^{2}\triangleq\mathbf{e}^{T}(T)\mathbf{ W}\mathbf{e}(T)=\sum_{t=1}^{T}w_{t}\,e^{2}(t)=\lVert\mathbf{y}(T)-\mathbf{U}(T) \mathbf{h}\rVert_{\mathbf{W}}^{2}, \tag{97}\]  
where \(\mathbf{W}\triangleq\text{diag}(w_{1},\cdots,w_{T})\) is a diagonal weighting matrix, with \(w_{t}>0\) for all \(t\in[1,T]\). The WLS criterion (97) can be developed as  
\[\lVert\mathbf{e}(T)\rVert_{\mathbf{W}}^{2}=\lVert\mathbf{y}(T)\rVert_{ \mathbf{W}}^{2}-2\mathbf{h}^{T}\mathbf{U}^{T}(T)\mathbf{W}\mathbf{y}(T)+ \mathbf{h}^{T}\mathbf{U}^{T}(T)\mathbf{W}\mathbf{U}(T)\mathbf{h}. \tag{98}\]  
It is a quadratic cost function with respect to the unknown parameters vector \(\mathbf{h}\). The Hessian (\(2\mathbf{U}^{T}(T)\mathbf{W}\mathbf{U}(T)\)) being a nonnegative definite matrix, this criterion has a unique global minimum obtained in canceling its gradient with respect to \(\mathbf{h}\), which gives  
\[\Big{(}\mathbf{U}^{T}(T)\mathbf{W}\mathbf{U}(T)\Big{)}\mathbf{\hat{h}}(T)= \mathbf{U}^{T}(T)\mathbf{W}\mathbf{y}(T). \tag{99}\]  
Assuming the matrix \(\mathbf{U}^{T}(T)\mathbf{W}\mathbf{U}(T)\) is nonsingular, the WLS estimate of \(\mathbf{h}\) is given by  
\[\mathbf{\hat{h}}(T)=\Big{(}\mathbf{U}^{T}(T)\mathbf{W}\mathbf{U}(T)\Big{)}^{ -1}\mathbf{U}^{T}(T)\mathbf{W}\mathbf{y}(T). \tag{100}\]  
As the diagonal weighting matrix \(\mathbf{W}\) is positive definite, a condition for ensuring the uniqueness of the WLS estimate is that \(\mathbf{U}(T)\) be full column rank, which implies the necessary condition \(T\geq\Pi I_{N}\). When the weighting matrix is chosen as the identity matrix, we obtain the standard LS estimate of the GIR given by  
\[\mathbf{\hat{h}}(T)=\mathbf{U}^{\dagger}(T)\mathbf{y}(T). \tag{101}\]  
In [56], an iterative Wiener filter and LMS-based algorithms are proposed to identify multilinear systems as described in (91). Tensorizing the GIR vector estimate \(\mathbf{\hat{h}}\) as a \(N\)th-order rank-one tensor \(\hat{\mathcal{H}}\in\mathbb{R}^{I_{N}}\), an estimate \(\mathbf{\hat{h}}^{(n)}\) of each individual IR \(\mathbf{h}^{(n)}\) can be obtained by using the high order singular value decomposition (HOSVD) of \(\hat{\mathcal{H}}\), i.e., calculating the left singular vector associated with the largest singular value of the matrix unfolding \(\mathbf{\hat{H}}_{I_{n}\times I_{1}\cdots I_{n-1}I_{n}+1\cdots I_{N}}\).