Tensor-Based Approaches for Nonlinear and Multilinear Systems Modeling and Identification

Nonlinear (NL) and multilinear (ML) systems play a fundamental role in engineering and science. Over the last two decades, active research has been carried out on exploiting the intrinsically multilinear structure of input-output signals and/or models in order to develop more efficient identification algorithms. This has been achieved using the notion of tensors, which are the central objects in multilinear algebra, giving rise to tensor-based approaches. The aim of this paper is to review such approaches for modeling and identifying NL and ML systems using input-output data, with a reminder of the tensor operations and decompositions needed to render the presentation as self-contained as possible. In the case of NL systems, two families of models are considered: the Volterra models and block-oriented ones. Volterra models, frequently used in numerous fields of application, have the drawback to be characterized by a huge number of coefficients contained in the so-called Volterra kernels, making their identification difficult. In order to reduce this parametric complexity, we show how Volterra systems can be represented by expanding high-order kernels using the parallel factor (PARAFAC) decomposition or generalized orthogonal basis (GOB) functions, which leads to the so-called Volterra-PARAFAC, and Volterra-GOB models, respectively. The extended Kalman filter (EKF) is presented to estimate the parameters of a Volterra-PARAFAC model. Another approach to reduce the parametric complexity consists in using block-oriented models such as those of Wiener, Hammerstein and Wiener-Hammerstein. With the purpose of estimating the parameters of such models, we show how the Volterra kernels associated with these models can be written under the form of structured tensor decompositions. In the last part of the paper, the notion of tensor systems is introduced using the Einstein product of tensors. Discrete-time memoryless tensor-input tensor-output (TITO) systems are defined by means of a relation between an Nth-order tensor of input signals and a Pth-order tensor of output signals via a (P+N)th-order transfer tensor. Such systems generalize the standard memoryless multi-input multi-output (MIMO) system to the case where input and output data define tensors of order higher than two. The case of a TISO system is then considered assuming the system transfer is a rank-one Nth-order tensor viewed as a global multilinear impulse response (IR) whose parameters are estimated using the weighted least-squares (WLS) method. A closed-form solution is proposed for estimating each individual IR associated with each mode-n subsystem.