# Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification
## 1 Introduction
Note that the matrix factors \(\mathbf{U}\) and \(\mathbf{V}\) which are column-orthonormal, contain the left and right singular vectors, respectively, whereas the diagonal matrix \(\mathbf{\Sigma}\) contains the nonzero singular values, and \(R\) denotes the rank of the matrix. A historical review of the theory of matrices and tensors, with basic decompositions and applications, can be found in [20]. Similarly, from the system modeling point of view, linear models of dynamic systems in the form of input-output relationships or state space equations have given rise to nonlinear and multilinear models to take into account nonlinearities inherent in physical systems. This explains why nonlinear models are appropriate in many engineering applications. Consequently, standard parameter estimation and filtering methods for linear systems, such as the least-squares (LS) algorithm and the Kalman filter (KF), first proposed by [PERSON] 1805 [21] and [PERSON] in 1960 [22], respectively, were extended for parameter and state estimation of nonlinear systems. Thus, the alternating least-squares (ALS) algorithm [13] and the extended Kalman filter (EKF) [23] were developed, respectively, for estimating the parameters of a PARAFAC decomposition and applying the KF to nonlinear systems. In Table 2, we present two examples of standard linear models, namely the single-input single-output (SISO) finite impulse response (FIR) model and the memoryless multi-input multi-output (MIMO) model, often used for modeling a communication channel between \(n_{T}\) transmit antennas and \(n_{R}\) receive antennas, where \(h_{i,j}\) is the fading coefficient between the \(j\)th transmit antenna and the \(i\)th receiver antenna. The FIR model is one of the most used for modeling linear time-invariant (LTI) systems, i.e., systems which satisfy the constraints of linearity and time-invariance, which means that the system output \(y(t)\) can be obtained from the input via a convolution \(y(t)=(h\star u)(t)\), where \(h(. )\) is the system's impulse response (IR), and \(\star\) denotes the convolution operator. The notion of linear dynamical system has been generalized to multilinear dynamical systems in [24] to model tensor time series data, i.e., time series in which input and output data are tensors. In this paper, the multilinear operator is chosen in the form of a Kronecker product of matrices, and the parameters are estimated by means of an expectation-maximization algorithm, with application to various real datasets. Then, the notion of LTI system has been extended to multilinear LTI (MLTI) systems by [25] using the [PERSON] product of even-order paired tensors, with an extension of the classical stability, reachability, and observability criteria to the case of MLTI systems. In Table 2, four examples of nonlinear (NL) and multilinear (ML) models are introduced, namely the polynomial, truncated Volterra, tensor-input tensor-output (TITO), and multilinear tensor-input single-output (TISO) models, which will be studied in more detail in Sections 5 and 6, as mentioned in Table 2.  
System modeling and identification is a fundamental problem in engineering applications. Real-life systems being often nonlinear in nature, NL models are very useful for various application areas. Parameter estimation using measurements of input and output (I/O) signals is at the heart of identification methods. In this paper, two main families of NL models are considered: (i) discrete-time Volterra models, also called truncated Volterra series expansions; (ii) block-oriented (Wiener, Hammerstein, Wiener-Hammerstein) models. In the sequel, we assume that the systems to be modeled are time invariant, i.e., their properties and consequently the parameters of their model do not depend on time.