# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 5 Tensor-Based Approaches for Nonlinear Systems
### Truncated Volterra Models  
When the polynomial functions \(f_{p}(.)\) in (9) are independent from the output signal, i.e., without output feedback, the polynomial model is called a nonrecursive polynomial model or a discrete-time Volterra model. A \(P\)th-order Volterra model for a causal, stable, finite-memory, time-invariant SISO system is described by the following I/O relationship:  
\[\hat{\mathbf{y}}(t)=h_{0}+\sum_{p=1}^{P}\sum_{m_{1}=1}^{M_{p}}\cdots\sum_{m_{ p}=1}^{M_{p}}h_{m_{1},\cdots,m_{p}}^{(p)}\prod_{q=1}^{p}u(t-m_{q}+1)=h_{0}+ \sum_{p=1}^{p}\hat{y}^{(p)}(t), \tag{12}\]  
where \(h_{0}\) is the offset, \(M_{p}\) is the memory of the \(p\)th-order homogeneous term \(\hat{y}^{(p)}(t)\), and \(h_{m_{1},\cdots,m_{p}}^{(p)}\) is a coefficient of the \(p\)th-order Volterra kernel, assumed to be real-valued.  
Note that a truncated Volterra model can be seen as a truncated Taylor series expansion for approximating a given smooth nonlinear function (around 0 by convention).  
Equation (12) can also be written as a polynomial regression model linear in its parameters and composed of monomials in previous samples of the input signal  
\[\hat{\mathbf{y}}(t)=h_{0}+\boldsymbol{\varphi}^{T}(\mathbf{u}(t))\boldsymbol{ \theta}, \tag{13}\]  
where \(\mathbf{u}(t)=[u(t),\cdots,u(t-M)]^{T}\), with \(M=max_{p}(M_{p})\), and \(\boldsymbol{\theta}\) is the parameter vector containing all the kernel coefficients, and the vector \(\boldsymbol{\varphi}\) contains all possible monomials in \(u\) up to degree \(P\). In the sequel, we assume that all memories \(M_{p}\) are equal to \(M\). The coefficient \(h_{m_{1},\cdots,m_{p}}^{(p)}\) being characterized by \(p\) indices can be viewed as an element of a tensor \(\mathcal{H}^{(p)}\in\mathbb{R}^{[p,M]}\), of order \(p\), characterized by \(M^{p}\) entries, which is a number growing very fast with the kernel order \(p\). The \(p\)th-order homogeneous term \(\hat{y}^{(p)}(t)\) can then be written using the Tucker product as  
\[\hat{y}^{(p)}(t)=\mathcal{H}^{(p)}\underset{q=1}{\overset{p}{\times}}\mathbf{ u}^{T}(t), \tag{14}\]  
which is a homogeneous polynomial of degree \(p\) in the components of the input vector.  
Several adaptive and nonadaptive methods have been proposed to identify truncated Volterra models from I/O measurements, both in the time and frequency domains. Frequency methods are based on the use of input signal cumulants, which requires estimating high-order statistics of the input signal, up to order \(2P\) for a \(P\)th-order Volterra model. Such an approach is mainly interesting with a Gaussian input signal, since the input cumulants of order higher than two are then zero, which implies a significant simplification of frequency methods. In the time domain, we can distinguish the optimal minimum mean-square error (MMSE) estimator, based on the use of input signal statistics, the nonrecursive least-squares (LS) algorithm, which can be viewed as an approximation of the MMSE solution, and adaptive methods. Note that estimating the parameters of an homogeneous \(P\)th-order Volterra kernel, with memory \(M\), using the MMSE and nonrecursive LS solutions requires inverting an autocorrelation matrix of size \(M^{P}\times M^{P}\), which is a time consuming and numerically difficult task.  
Adaptive methods are often associated with adaptive Volterra filters used for representing NL time-varying signals and systems as encountered in echo cancellation, for instance. Parameter estimation of adaptive Volterra filters is carried out using the well-known least-mean-square (LMS) or recursive LS (RLS) algorithms. See the book [28] and the references therein for an overview of the methods briefly introduced above.  
In the next section, we present an approach for identifying reduced complexity Volterra models which is based on a PARAFAC decomposition of symmetrized kernels, leading to the so-called Volterra-PARAFAC models.