# Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality
## 3 Results
### Synthetic Experiments  
The first set of experiments reported are related to synthetic experiments, which are used in the present work as they allow to quantitatively verify the effectiveness and reliability of the proposed procedures, and in particular the procedure that allows to use different methods of estimation of the Hammerstein model of the nonlinear system to obtain the different types of synthetic parameter that have been considered (distortion measures, damage indexes used in nondestructive testing). The first verification that we report the results of consists of a comparison between the values of these parameters, calculated analytically with the values obtained through estimates made by means of the PuC technique. In particular, we consider a nonlinear synthetic system of polynomial type, characterized by instantaneous nonlinearity of the following type:  
\[\mathrm{y}(t)=x(t)+Coef\;x(t)^{2}+Coef\;x(t)^{3} \tag{20}\]  
If the input to the nonlinear system is of the harmonic type, it is possible to calculate analytically the value of the harmonic distortions: if the input is a simple sine wave: \(x(t)=R\;Cos[\omega_{1}t]\) it is possible to calculate analytically the value of the total harmonic distortion and the distortions of second or third harmonics. In our experiments, we considered a signal of unit amplitude (R = 1) and developed the powers of the sinusoidal signal, represented these powers through the corresponding harmonics, and grouped the terms related to the different harmonics. In this way we were able to calculate the components that go into the definition of \(THD\), \(HD_{2}\) and \(HD_{3}\). We observe that, since the nonlinear system is instantaneous, the frequency trend of the value of the estimated parameters is completely flat, since the parameters that characterize the nonlinear system and, therefore, the parameters \(THD\), \(HD_{2}\) and \(HD_{3}\) that we are estimating, do not depend on frequency. For our calculations we used unit amplitude of the input sinusoidal signal (R = 1) and adopted for the parameter characterizing the degree of nonlinearity of the system, the value Coef = 0.125. We also calculated the value of the distortions produced by the same synthetic nonlinear system, which would occur if the signal amplitude were R = 0.5. These values constitute the references for verifying the extrapolation technique. The values obtained analytically for the distortion were as follows:  
\[ \begin{array}{llll}\text{R}=1&THD=6.38\%;&HD_{2}=5.71\%;&HD_{3}=2.85\%\\ \text{R}=0.5&THD=3.15\%;&HD_{2}=3.05\%;&HD_{3}=0.763\%\end{array}  \tag{21}\]  
These values, obtained analytically, will be useful references for carrying out our verifications of the procedures for estimating the same quantities through the techniques proposed in this paper. Using the reference parameters (R = 1; Coef = 0.125) we simulated the harmonic distortion estimation procedure that makes use of the PuC technique. We generated an exponential chirp signal characterized by fMin = 10 [Hz]; fMax = 1000 [Hz]; T = 0.47 [s] which was input to the nonlinear system. The output generated by the nonlinear system was used to feed the filter matched to the chirp signal input. From the output of the matched filter, we took the different impulse functions of the vector \([g(t)]\). Combining the transforms of these functions according to (16), we obtained the trends of Figure 2a as estimates of \(THD\), \(HD_{2}\) and \(HD_{3}\). It is evident from Figure 2 that there is an almost constant frequency trend of the estimated value, apart from the very low frequencies, and that these values estimated through the PuC technique are very good estimates of the analytical reference values.