# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 3 Results  
The consequences of the aspects that were highlighted in the previous section need to be verified experimentally. To this end, a specific synthetic experiment, described later in this section, was defined to verify the correctness of the observations made in the previous section regarding the bandwidth of the signals involved in the identification step carried out by means of the PuC technique, and to analyze what consequences the choices made on the frequency bands of these signals have on the quality of the Hammerstein model identification result. The choice was made that the simulated nonlinear physical system defined for the experiment was constructed following the exact Hammerstein model. The reason for this choice is that, as a result, the reference transfer functions, \(H_{k}(\omega)\), of each branch, or, equivalently, the corresponding functions, \(h_{k}(t)\), are known to us. The ideal parameters to which the identification procedure should strive are known. The PuC identification technique is applied to this simulated physical system in order to obtain, through the identification procedure, an estimate of the different branch functions of the model. Knowledge of the ideal reference trend of the \(H_{k}(\omega)\) or \(h_{k}(t)\) functions allows us to compare these reference trends with the corresponding trends obtained through the identification technique. Thus, it is possible to verify the impact of different choices of the parameters of the identification procedure on the quality of the result obtained. The nonlinear physical system was defined by simulating it through a Hammerstein structure. Then, since the ideal functions \(h_{k}(t)\) were known, it was possible to calculate the  
Figure 8: Frequency overlap in the case in which both the matched filter and the input swept signal range between \(f_{MIN}\) and \(OrdMax*f_{MAX}\). The solid lines represent the swept-sine signal (blue line) its second harmonic (orange line) its third harmonic (green line) and its fourth harmonic (red line). The dotted line represents the frequency band covered by the adapted filter. The four panels represent the cases when the matched filter correlates with (**a**) the fundamental frequency; (**b**) the second harmonic; (**c**) the third harmonic; (**d**) the fourth harmonic. output corresponding to the exponential swept-sine signal at the input. Through the PuC technique, the \(\widehat{g_{i}}(t)\) functions were estimated, their equivalent \(\widehat{G_{i}}(\omega)\) functions evaluated in the frequency domain and, from these, the \(\widehat{H_{k}}(\omega)\) functions (and thus the \(\widehat{h_{k}}(t)\)). The PuC estimate of the ideal \(H_{k}(\omega)\) is obtained through Relation (12), which is the inverse of Relation (11):  
\[ \begin{bmatrix}\mathrm{H_{1}}(\omega)\\ \mathrm{H_{2}}(\omega)\\ \mathrm{H_{3}}(\omega)\\ \mathrm{H_{4}}(\omega)\end{bmatrix} =\left[\mathrm{A}_{\mathrm{C}}^{\top} \right]\cdot \begin{bmatrix}\mathrm{G_{1}}(\omega)\\ \mathrm{G_{2}}(\omega)\\ \mathrm{G_{3}}(\omega)\\ \mathrm{G_{4}}(\omega)\end{bmatrix} = \begin{bmatrix}1&0&-3&0\\ 0&2&0&-8\\ 0&0&4&0\\ 0&0&0&8\end{bmatrix} \cdot \begin{bmatrix}\mathrm{G_{1}}(\omega)\\ \mathrm{G_{2}}(\omega)\\ \mathrm{G_{3}}(\omega)\\ \mathrm{G_{4}}(\omega)\end{bmatrix}  \tag{12}\]  
A comparison of the amplitude responses of the estimated \(\widehat{H_{k}}(\omega)\) functions obtained from \(\widehat{G_{i}}(\omega)\) by applying (12) on the results obtained in simulation, comparing the moduli, and the ideal \(H_{k}(\omega)\) or a comparison of the corresponding time functions \(h_{k}(t)\) and \(\widehat{h_{k}}(t)\) give us an indication of the effects of the choice of parameters adopted in the PuC identification technique. The system considered in our simulation experiment is a fifth-order Hammerstein system at the input of which an exponential swept-sine signal was input, sampled at 300 kHz, extending in the frequency range from 300 Hz to 10 kHz, with a growth rate L = 0.16583.