# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 4 Discussion and Conclusions  
Having tools to highlight the presence of nonlinearity in a system, and analyze its nonlinearity characteristics, allows us to evaluate the quality of the system itself or the presence of anomalies or defects in the system structure, in many applications. These aspects are often highlighted by global indicators: examples in this sense are the measures of the distortion introduced by a system (electronics, telecommunications [17; 18]), or some damage indexes used to evaluate the presence or onset of anomalies in structures subjected to strong stresses (buildings, bridges, and viaducts [19; 20; 21]). These global indicators can be evaluated by analyzing the system response to known type of stresses, according to the requirements of the various regulations. The development of increasingly efficient and sophisticated techniques for the modeling representation of the nonlinear system allows us to define a different approach for the evaluation of the same indicators. For example, it has been demonstrated that, within the limits of validity of Hammerstein models, an effective representation of the nonlinear system can be obtained, and from the parameters of the system representation, reliable estimates of the global indicators can be obtained [14; 15]. Figure 7: Trend of the signal energy ratio for different input signal amplitudes; in both panels the dotted lines represent linear interpolations on the first group of points (from 1 to 5) and on the remaining 15 points: (**a**) estimates obtained from 20 independent measures. (**b**) blue dots represent four estimates obtained from 20 independent measures; red dots are estimates obtained from each of the four measured values by extrapolating for lower signal amplitudes. The system represented obviously must have the characteristics that allow its representation through a Volterra model, or a model derived from it, such as the Hammerstein model. The approximate nonlinear system must be characterized by \"fading memory\"; that is, the system response must depend less and less on past inputs as we move away in time. It follows that this constraint excludes systems whose output depends substantially on the initial conditions and on input values far in the past, such as those characterized by chaotic behaviors. Furthermore, as far as input signals are concerned, representability through Volterra series requires that the inputs are sufficiently \"smooth\" and do not show variations that are too abrupt, which excludes, for example, piecewise constant input signals [22]. A very wide variety of systems of engineering interest can therefore be validly represented in terms of their nonlinear characteristics by models such as the Hammerstein model, adopted in the present work, whose identification can be effectively obtained through the PuC technique, as is widely verified in the technical literature [8; 9; 10; 11; 12]. The identification technique we use refers directly to the structure of the Hammerstein model shown in Figure 1: variants to this structure have been proposed in an attempt to use a modeling technique derived from [PERSON]'s also to nonlinear systems of chaotic type [23; 24]. The technique proposed in this paper, valid in most applications, is not applicable to the special case of modeling nonlinear systems of chaotic type. In the present work we have proposed two original contributions related to the estimation of global parameters starting from the Hammerstein model: the first is related to the possibility, through a variant of the PuC technique for the identification of the Hammerstein model, of obtaining estimates of global parameters, as the distortion or damage indexes, relating to signal levels different from the one actually used for the experiments. This possibility allows us to reduce the number of laboratory experiments necessary to obtain a picture of the effects on the nonlinearity of the signal input levels.