# Complex-Order Models: A System Identification Point of View
## 5 Identification Results
with \(\rho>0\) being a scale factor used to tune the relative importance of amplitude and phase errors, and where \(G(s)\) is the transfer function of the fitted ARX model, and \(\omega_{\ell}\), for \(\ell=1,\ldots,N\), with \(N=100\), are adjacent, uniformly logarithmic distributed angular frequencies between 0.1 and 1000 rad/s. Note that \(\text{arg}^{*}\) denotes the _unwrapped_ argument. Identification of the First Transfer Function  
The shape of the amplitude response shown in Figure 7 (left) strongly indicates that a proper model could be obtained using a pair of aperiodic elements, one at the numerator and one at the denominator of the transfer function. Based on the shape of the frequency response, we expect that the numerator corner frequency is several orders of magnitude higher than the denominator one. Considering the slopes of the amplitude response, we also expect that the real part of the order of the denominator will be close to 1 (since the the slope of the second segment is approximately \(-20\) dB/decade), and that the real part of the numerator will be close to 1.5 (since the slope in the third segment is approximately 10 dB/decade). Finally, considering the shape of the phase response, we expect the numerator element to be a non-minimal-phase one. All these assumptions will be confirmed by the optimization process, as seen in Figure 6.4  
We focus on the transfer functions of the form:  
Figure 6.5 Figure 6.6  
The given transfer function was selected based on the observation that the amplitude characteristic changes slope two times: firstly, just above 0.1 rad/s, the amplitude slope decreases, and then at about 50 rad/s, the slope increases (see Figure 9 (upper part)). This would indicate that we need two factors, the first one being a generalization of a pole, and the second one being the generalization of a zero. We also noticed that at the second critical frequency, instead of presenting the expected increase, the phase actually decreases further, indicating non-minimum phase behavior. The choice of the specific complex-order elements, of the first or second kind, was performed by means of the optimization procedure itself. Here, we explicitly present the kind of each element in order to illustrate that both of them are practically significant (see Figure 9 (lower part)). Figure 9 compares the frequency responses of this model with that of the previously identified ARX model. The plots clearly show that the more compact complex-order model can replace the high-order ARX model. Table 1 provides the parameters of the identified complex-order model, which was obtained with \(\rho=5\). The optimization has shown that the numerator element should be selected to be of the second type. This can be seen from the fact that a negative imaginary part of the order is obtained. The resulting transfer function is proper, since the numerator has a lower order than the denominator. Identification of the Second Transfer Function  
We focus on the transfer functions of the form:  
Figure 6.7 Figure 6.8  
Figure 10 compares the frequency responses of this model with that of the previously identified ARX model. Table 2 gives the parameters of the complex-order fractional model. Again, the results indicate agreement between the two models and the accuracy of the complex-order model. In this particular case, the actual structure of the model was obtained empirically: by trial and error. First, a non-minimum phase real-order resonant element was placed at the numerator in order to capture the resonant peak, visible at approximately 200 rad/s. Other elements were added iteratively until a proper fit was obtained. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(K\) & \(\omega_{1}\) & \(\alpha_{1}\) & \(\beta_{1}\) & \(\omega_{2}\) & \(\alpha_{2}\) & \(\beta_{2}\) \\ \hline
7 pars. & 0.9740 & 0.2016 & 1.0223 & 0.0970 & 44.2719 & 1.5781 & 0.4348 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the fitted 7-parameter complex-order model.