# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 3 Results
Consequently, the exponential swept-sine signal duration from \(f_{MIN}\) to \(f_{MAX}\) is calculated to be T = 0.5815 [s] while the duration up to the frequency OrdMax\(*f_{MAX}=50\) kHz is T = 0.8484 [s]. On the five branches of the nonlinear system simulated through a Hammerstein model, bandpass filters of different types and order were placed: on branches 1, 3, and 5, Butterworth-aligned bandpass filters of orders 6, 4, and 2, respectively; on branches 2 and 4, Chebyshev-aligned bandpass filters of orders 5 and 3, respectively. The cutoff frequencies of all five bandpass filters are 300 Hz and 20 kHz for the lower and upper cutoff frequencies; the high end cutoff frequency is twice \(f_{MAX}\), therefore, that there are significant components of the amplitude responses up to the frequency OrdMax\(*f_{MAX}\). All parameters used to define the simulated system in accordance with Hammerstein model are given in Table 1. The simulations were carried out using the software MathematicaTM. Figure 9 shows, on the first line, the five amplitude responses of the filters that were inserted into the branches of the Hammerstein structure that represent the nonlinear system in the simulation. The second line of Figure 9 shows the corresponding five impulse responses. The curves on the two rows are our references in the frequency and time domains, respectively. Figure 10 compares the results obtained by applying the PuC procedure of model identification, viewed in terms of estimating the frequency responses of the filters on the five branches of the model itself, according to the three different procedures described above for the choice of frequency bands adopted for the exponential swept-sine signal at the input and for the corresponding matched filter. The three procedures can be summarized as follows: All have as their starting point the frequency range from \(f_{MIN}\) to \(f_{MAX}\), within which the nonlinear system must operate. Specifically, for Procedure 1, the exponential swept-sine input test signal is defined in the range from \(f_{MIN}\) to \(f_{MAX}\) and the matched filter operates in the same frequency range; for Procedure 2, the \begin{table}
\begin{tabular}{c c c c c c} \hline
**Model Order** & **Filter Name** & **Filter Alignment** & **Filter Order** & **Low End Cutoff (Hz)** & **High End Cutoff (Hz)** \\ \hline
1 & \(H_{1}(\omega)\) & Butterworth & 6 & 300 & 20.000 \\
2 & \(H_{2}(\omega)\) & Chebyshev & 5 & 300 & 20.000 \\
3 & \(H_{3}(\omega)\) & Butterworth & 4 & 300 & 20.000 \\
4 & \(H_{4}(\omega)\) & Chebyshev & 3 & 300 & 20.000 \\
5 & \(H_{5}(\omega)\) & Butterworth & 2 & 300 & 20.000 \\ \hline \end{tabular}
\end{table} 
Table 1: Filter parameters in the five branches of the nonlinear system simulator. swept-sine input test signal goes from \(f_{MIN}\) to \(f_{MAX}\), while the matched filter covers the frequency range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\); for Procedure 3, both the exponential swept-sine test signal at the input and the impulse response of the matched filter cover the frequency range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\). In the case of Procedure 1 (panels \"f\" to \"j\"), it is evident that the amplitude response estimation is strictly limited to the maximum frequency \(f_{MAX}\) (10 kHz in the experiment) common to the exponential swept-sine signal and the matched filter, and this is in line with what was assumed following the observation of Figure 6.  
In the case of Procedure 2 (panels \"k\" to \"o\"), the trends exhibit discontinuities at frequency multiple integers of \(f_{MAX}\). For example, panel \"k\" shows a discontinuity at \(f_{MAX}\) (10 kHz) and an additional discontinuity at \(3f_{MAX}\) (30 kHz). The trends between one discontinuity and the next are not in line with the desired ideal trend of the amplitude response (first row). This result confirms two aspects highlighted earlier.