# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 3 Results
It confirms the considerations made in correspondence to Figure 7, which show that, in this case, the different functions, \(G_{i}(\omega)\), are characterized in frequency bands that are different from each other, and therefore, only up to \(f_{MAX}\) are all the components that must add up present, while in higher frequency bands the lower order functions \(G_{i}(\omega)\) are not defined and the combination occurs in the absence of some contributions that would be essential. A second aspect that is evidenced by examining the panels from \"k\" to \"o,\" is that only some of the components enter into the combination, as evidenced by the structure of matrix \([\mathrm{A}_{C}^{\mathrm{T}}]\) in Relation (12). The \(G_{i}(\omega)\) functions that combine to give rise to the first order function, \(H_{1}(\omega)\), are all the \(G_{i}(\omega)\) components of odd index starting from \(G_{1}(\omega)\), and thus, the frequencies at which the components will be involved are odd integer multiples of \(f_{MAX}\) (10 kHz, 30 kHz,... ); for \(H_{k}(\omega)\) of higher orders, given the upper triangular structure of the \([\mathrm{A}_{C}^{\mathrm{T}}]\) combination matrix, the \(G_{i}(\omega)\) functions that will combine will be those of indices k to OrdMax, and therefore, discontinuities in the combination will occur at higher frequencies. For example, in \(H_{2}(\omega)\), the first discontinuity occurs at \(2f_{MAX}\) (20 kHz) and the next discontinuity occurs at \(4f_{MAX}\) (40 kHz); in the case of \(H_{3}(\omega)\), the first discontinuity is found at \(3f_{MAX}\) (30 kHz) and the next discontinuutiy is found at \(5f_{MAX}\) (50 kHz). Figure 9: Panels from (**a**–**e**) show the amplitude responses of the filters on the five branches of the Hammerstein model considered as reference behaviors in the experiment. Panels from (**f**–**j**) show the corresponding reference impulse responses. In the case of identification Procedure 3 (panels \"p\" to \"t\"), the trends faithfully reflect throughout the frequency band of interest the ideal trends. It is evident that the result comes from the combination of several components and, in fact, at frequency multiples of \(f_{MAX}\), small irregularities are found; however, the overall trend is consistent with the ideal reference, and this denotes that there is a correct combination of all the necessary components. The only aspect that merits some further investigation concerns the peak found at the minimum frequency \(f_{MIN}\). The corresponding result obtained in the time domain is very interesting. For ease of reading, in each of Figures 11a, 12a and 13a are reported the reference trend of each of the three functions \(h_{1}(t)\), \(h_{2}(t)\), and \(h_{3}(t)\) considered, respectively, shown as compared with those obtained by the identification Procedures 1, 2, and 3, plotted in Figures 11b-d, 12b-d and 13b-d, respectively. Figure 10: Panels from (**a–e**) show the reference amplitude responses of the filters on the five branches of the Hammerstein model in the experiment. Panels from (**f–j**) show the amplitude responses estimated with the Procedure 1 (swept-sine range from \(f_{MIN}\) to \(f_{MAX}\); matched filter range from \(f_{MIN}\) to \(f_{MAX}\)). Panels from (**k–o**) show the result applying Procedure 2 (swept-sine range from \(f_{MIN}\) to \(f_{MAX}\); matched filter range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\)). Panels from (**p–t**) show the amplitude responses estimated with the Procedure 3 (swept-sine range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\); matched filter range from \(f_{MIN}\) to \(OrdMax*f_{MAX}\)).