## 2 Theoretical Aspects
### Model Identification by Swept Sine Excitation  
If we assume that a valid representation of the nonlinear system is the Hammerstein model, a schematic description of which is shown in Figure 1, the input-output relation of the system will be:  
\[y_{H}(t)=x(t)\otimes h_{1}(t)+x(t)^{2}\otimes h_{2}(t)+\ldots+x(t)^{N_{H}} \otimes h_{N_{H}}(t) \tag{1}\]  
where we indicated with the symbol \(\otimes\) the convolution operation. The output \(y_{H}(t)\) is the sum of the powers up to order \(N_{H}\) of the input signal \(x(t)\), convolved with appropriate impulsive functions \(h_{i}(t)\); such impulsive functions \(h_{i}(t)\) are the kernels that identify the Hammerstein model. Assume the input is a swept sine, i.e., an harmonic signal \(x(t)=R\,\text{Cos}(\phi(t))\) of amplitude \(R\), with instantaneous angular frequency \(\omega(t)=d\phi(t)/dt\), which varies over  
Figure 1: The Hammerstein model. The input \(x(t)\) and each of its powers up to order \(N_{H}\) pass through a different linear filter: \((x(t))^{k}\) is convolved with \(h_{k}(t)\). The order \(N_{H}\) and the kernels \(h_{k}(t)\) completely characterize the model. time. If we represent vectors and matrices by square brackets, we can write for the output of the Hammerstein model of order \(N_{H}\):  
\[y_{H}(t)=\left[\left[R\ Cos(\phi(t))\right]^{k}\right]^{T}\otimes\left[h(t)\right] \tag{2}\]  
where \([h(t)]\) is the vector of the different kernels \(h_{k}(t),\ k=1,\ldots,N_{H}\), and \(\left[\left[R\ Cos(\phi(t))\right]^{k}\right]^{T}\) is the transpose of the vector of powers of the input signal. By means of the Chebyshev polynomials of the first kind, we can express the harmonics \(\ Cos(k\ \phi(t))\) of the swept sine function \(\ Cos(\phi(t))\), as functions of the powers \(\left[R\ Cos(\phi(t))\right]^{k}\) of the input signal [16]:  
\[\left[\ Cos(k\ \phi(t))\right]=\left[A_{c}\right]\left[\ Cos(\phi(t))\right]^{k }=\left[A_{c}\right]\left[R_{c}\right]^{-1}\left[R\ Cos(\phi(t))\right]^{k} \tag{3}\]  
The entries of matrix \([A_{c}]\) are the coefficients of Chebyshev polynomials of the first type, and \([R_{c}]\) is the diagonal matrix in which the index term \(\{i,i\}\) is the \(i-th\) power of the amplitude \(R\). From expression (3), we can derive the expression of \(\left[R\ Cos(\phi(t))\right]^{k}\) as a function of the harmonics \(\ Cos(k\ \phi(t))\) which, substituted in (2), gives:  
\[y_{H}(t) =\left[\left[R_{c}\right]\left[A_{c}\right]^{-1}\ Cos(k\ \phi(t))\right]^{T}\otimes\left[h(t)\right]=\left[\ Cos(k\ \phi(t))\right]^{T}\left[\left[R_{c}\right]\left[A_{c}\right]^{-1}\right]^{T} \otimes\left[h(t)\right]=\] (4) \[=\left[\ Cos(k\ \phi(t))\right]^{T}\otimes\left(\left[\left[A_{c} \right]^{-1}\right]^{T}\left[R_{c}\right]\right)\left[h(t)\right]=\left[\ Cos(k\ \phi(t))\right]^{T}\otimes\left[g(t)\right]\] where we considered that \([R_{c}]\) is a diagonal matrix, and we defined the vector \([g(t)]\) containing the functions \(g_{i}(t),\ i=1,\ldots N_{H}\), by means of \(\left[g(t)\right]=\left[\left[A_{c}\right]^{-1}\right]^{T}\left[R_{c}\right] \ [h(t)]\). A comparison between Equations (2) and (4) tells us that the output \(y_{H}(t)\) of the nonlinear system, obtained when the swept sine signal \(x(t)=R\ Cos(\phi(t))\) is input, can be expressed either as the sum of the convolutions between the powers of the harmonic signal itself and the functions in \([h(t)]\), or, alternatively, as the sum of the convolutions between the harmonics \(\ Cos(k\ \phi(t))\) of the signal \(\ Cos(\phi(t))\) and the impulsive functions \([g(t)]\). The functions \([g(t)]\) are related to the functions \([h(t)]\) through a linear transformation; the transformation matrix \(\left[\left[A_{c}\right]^{-1}\right]^{T}\left[R_{c}\right]\) depends on the amplitude \(R\) of the input signal and on the matrix \([A_{c}]\), whose entries are the coefficients of the Chebyshev polynomials of the first kind.