## 1 Introduction  
The behavior of physical systems is very often modeled using linear techniques. In reality, only in very particular cases physical systems can behave as linear, so if we want to obtain high quality results when mathematical models of physical systems are involved, it becomes of remarkable importance to have models that can represent the behavior of the system also in a non-linear regime. The modeling of nonlinear dynamical systems is one of the most challenging research areas in the field of system representation. Much of the research developed in recent years has increasingly focused on predominantly data-driven approaches, including neural nets or fuzzy logic [1], methodologies that assume the availability of large amounts of data. The amount of available data is not always adequate to represent the complexity of the system and the computing power is sufficient to handle them. For this reason, they continue to be widely diffused in the white-box approaches and the gray-box approaches, which place side by side to a number of data-driven only, black box methods: Such techniques integrate in different measures the information derived from the knowledge of the physics of the system and information extracted from the data produced by the nonlinear system. The white-box approaches require an accurate knowledge of the system. These techniques rely, for example, on wave digital filters [2, 3] or differential equations [4] to obtain an accurate model of the real system. If one has detailed knowledge of the physical system, these models can be highly accurate. If it is not possible to adequately represent the overall physical system by equations, the gray and black-box approaches can be applied, as they require only partial knowledge of the system or only the knowledge of input and output signals. A partial knowledge of the physics of the system allows to fractionate it into connected subsystems, each representable in a simpler way by adopting a gray-box approach [5, 6]. Among the black-box approaches, the methods derived from the Volterra series are in a prominent position, both in terms of accuracy and historically, having been defined long before the neural network-based approaches. Modeling techniques based on this infinite series are obtained by truncating them to a finite number of terms [7]. It is however important to note that, even considering a truncation to a low order of the Volterra series, the number of coefficients required to define the relevant model quickly becomes very large. This is an important limitation, because it makes it possible in practice to define adequate models by following this path only in the case where the system shows a degree of nonlinearity of small entity. This has prompted the development of simplified expansion models of the Volterra series [8]. Among these, the Hammerstein and Wiener models are among the most popular. Having available accurate models and effective techniques for their identification has allowed the successful application of these nonlinear modeling techniques of physical systems in many engineering fields, notably including acoustics and nondestructive testing [9; 10; 11; 12]. Among the different types of black-box nonlinear models of physical structures, the parallel Hammerstein model is particularly interesting as it can be shown that the problem of identifying its parameters reduces to a linear problem, so classical least squares methods can be used for their estimate [13]. An efficient technique was proposed in recent years for the identification of the parallel Hammerstein model [14; 15; 16]: Its relative simplicity of implementation, and the excellent results obtained both in laboratory experiments and in practical applications, make it one among the most competitive techniques for the identification of the Hammerstein model: A swept sine signal is input to the nonlinear system to be modelled and the corresponding output is acquired.