# Parameter Optimization for an Accurate Swept-Sine Identification Procedure of Nonlinear Systems
## 2 Materials and Methods
### Hammerstein Model and Pulse-Compression Identification Procedure  
Assume that the representation of the nonlinear system is validly carried out through a Hammerstein model of order \(N_{H}\), the schematic description of which is shown in Figure 1. The physical system is designed to operate in the frequency band between \(f_{MIN}\) and \(f_{MAX}\). A specific technique must be followed to identify the Hammerstein structure filter kernels that allow the model to have behavior equivalent to that of the nonlinear system being modeled. Among the identification techniques, those based on correlation involve using a specific input signal to the nonlinear system and analyzing the corresponding outputs of the system to define the individual filter kernels that identify the model. The identification technique based on pulse compression (PuC) belongs to this category [16; 17; 18; 19]; it relies on the ability to define a couple of signals, \(x(t)\) and \(\psi(t)\), of which one represents the excitation signal and the other the impulse response of the matched filter, respectively. They  
Figure 1: The Hammerstein model. are defined such that the convolution between the two is the best possible approximation of the mathematical pulse:  
\[o(t)=x(t)\otimes\psi(t)\cong\delta(t) \tag{1}\]  
Since the signals have to be necessarily band limited, their convolution can only approximate the mathematical pulse. If the two signals are chosen in such a way that, in the frequency band from \(f_{MIN}\) to the frequency \(f_{MAX}\), the transform of their convolution has a nearly constant magnitude, the time course of this convolution will be of the type [23]:  
\[\text{f}(t)\propto\text{ f}_{\text{MAX}}\text{ sin}[2\pi\text{ f}_{\text{MAX}}t]/[2\pi\text{f}_{\text{MAX}}t]-f_{MIN}\text{ sin}[2\pi\text{ f}_{MIN}\text{ }t]/[2\pi\text{ }f_{MIN}\text{ }t] \tag{2}\]  
For the input-output relation of the system, if defined in the time domain, the following expression will hold:  
\[y_{H}(t)=x(t)\otimes h_{1}(t)+x(t)^{2}\otimes h_{2}(t)+\ldots+x(t)^{N_{H}} \otimes h_{N_{H}}(t) \tag{3}\]  
in which the symbol \(\otimes\) denotes convolution. The \(N_{H}\) impulsive functions \(h_{i}(t)\) completely characterize the model, and therefore, model identification coincides with the identification of these functions. To apply the PuC procedure for identification, the swept-sine signal of unit amplitude can be adopted as input, described by \(\text{x}(t)=\text{Cos}(\phi(t))\). In this case, the previous equation can be expressed in compact form as:  
\[y_{H}(t)=\left[\left[\text{Cos}(\phi(t))\right]^{k}\right]^{T}\otimes\left[h(t)\right] \tag{4}\]  
where the entities in square brackets represent vectors; \(\left[h(t)\right]\) is the vector of the different kernels \(h_{k}(t)\), \(k=1,\ \ldots,N_{H}\); \(\left[\left[\text{Cos}(\phi(t))\right]^{k}\right]^{T}\) is the transpose of the vector of powers of the input signal. The Chebyshev polynomials of the first kind allow the vector of powers of the cosine functions to be expressed through the harmonics of the same functions [26], therefore, the above expression can be rewritten in an alternative form as:  
\[y_{H}(t)=\left[\left[A_{c}\right]^{-1}\text{Cos}(k\text{ }\phi(t))\right]^{T}\otimes\left[h(t)\right]=\left[\text{Cos}(k\text{ }\phi(t))\right]^{T}\otimes\left[g(t)\right] \tag{5}\]  
in which \(\left[A_{c}\right]\) is the matrix of coefficients of the Chebyshev polynomials of the first kind and the vector and \(\left[g(t)\right]\) contains \(N_{H}\) impulsive patterns directly related to the kernels \(\left[h(t)\right]\), through the relation \(\left[g(t)\right]=\left[\left[A_{c}\right]^{-1}\right]^{T}\left[h(t)\right]\)[19; 21]. A comparison of Relations (4) and (5) shows that the output of the Hammerstein model can be expressed alternatively through the sum of the convolutions between the powers of the input signal and the \(\left[h(t)\right]\) kernels or through the sum of the convolutions between the harmonics of the cosine functions and the \(\left[g(t)\right]\) kernels.