# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
###### Abstract  
The availability of synthetic indicators of the degree and type of nonlinearity in systems is used in various fields to assess system quality or to highlight possible malfunctions. Different distortion or damage indexes are synthetic measures designed (and standardized) to evaluate the frequency trend of specific aspects resulting from the nonlinear behavior of the system under consideration. The different measures of deviation from linear behavior quantitatively consider the system and its nonlinearity characteristics; they were defined according to practically feasible measurement methodologies and the various aspects of the system's nonlinearity that needed to be highlighted. In parallel, techniques for representing and modeling nonlinear systems  
In parallel, techniques for representing and modeling nonlinear systems have been defined, capable of describing the system being tested in a more general way, attempting to capture its input-output characteristics as the level of stress to which the system is subjected changes. Numerous modeling techniques have been proposed, aimed at representing the nonlinear behavior of physical devices. For each of these procedures, techniques for identifying the parameters that characterize the model have been proposed. Modeling techniques can be grouped into three main categories, which differ in the level of knowledge of the physical phenomenon that is represented by the model itself: white box-type approaches require in-depth knowledge of the physics that govern the nonlinear system. Among white box-type modeling approaches, we can mention those that make use of differential equations [1] or use wave digital filters for the representation of the nonlinear system [2; 3]. The white box-type approach often proves to be too difficult to be defined for the representation of physical systems of significant complexity. In these cases, simplifications are used, and depending on the degree of simplification, the modeling procedures are denoted as gray box-type models, which imply partial knowledge of the physical phenomenon [4; 5], or black box-type models, which are based only on knowledge of the input-output relationship and imply no knowledge of the physics of the system. Among the latter models, the most widely adopted are the Volterra series [6; 7] or models derived by simplifying the Volterra structure, such as the Hammerstein or the Wiener model. Numerous procedures for the identification of Volterra-like models have been proposed in the literature; in particular, the case of the Hammerstein model, a technique based on the use of exponential swept sine (chirp) signals and pulse compression, has been proposed [8; 9; 10; 11; 12], which has been shown to be particularly effective in a number of areas, such as acoustics and nondestructive diagnostics [13; 14; 15]. In the present work, we will show that for the vast class of nonlinear systems that can be adequately represented through a Hammerstein model, we can deduce global parameter estimators that represent the characteristics of the nonlinear system directly from the parameters of the identified model. The global parameters are useful in different application contexts: we will consider as examples (i) the degree of distortion introduced by the nonlinear system, widely used in the field of signal analysis, and (ii) an index that measures the degree of damage suffered by a metallic or concrete structure due to repeated stresses. After a review of the identification technique of the Hammerstein model based on swept sinusoidal signals, we show how the nonlinear model of the system can be used to obtain estimates of the distortion or damage measures, by adopting a technique based on the use of exponential swept sine signals as an input. A variation of this technique is also recalled, which by making use of an appropriate input signal, allows the estimation of intermodulation distortion.