# Complex-Order Models: A System Identification Point of View
## 6 Conclusions  
This work proposed integro-differential fractional operators of complex order for application in modeling and control problems. The previous literature has shown some examples in which complex-order operators can be used for robust control, as well as many cases in which real, not simply fractional but not complex, non-integer order operators are fruitfully used. Moreover, it is remarkable that models based on non-integer order operators--often misnamed as fractional models--allow a compact representation, reducing the number of model parameters with respect to integer-order models. This last important property is further extended in this paper by taking advantage of the new complex-order operators, which have the important potential to describe the amplitude of the frequency response, independent of phase, and vice versa. In fact, compared to modeling frameworks based solely on integer- and/or real-order models, the proposed framework facilitates the development of models with simpler structures (a smaller number of factors) and even a reduced number of parameters.  
The benefits of the introduced complex-order operators are shown on a testbed that is the injection system of an automotive engine fueled by compressed gas. The system is highly nonlinear, time-varying, and subject to uncertainties, such that control of the common-rail pressure, the main output variable, is really a hard task. The model is identified through the optimization of a cost index, and is compared with a classical ARX model of high-integer order. Results indicate that the model based on complex-order operators fits experimental data and can be effectively used for control.  
Data curation, P.L.; Software, M.N.K.; Supervision, Z.D.J.; Writing--original draft, M.R.R.; Writing--review & editing, G.M. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(K\) & \(\omega_{1,1}\) & \(\alpha_{1,1}\) & \(\omega_{1,2}\) & \(\alpha_{1,2}\) & \(\omega_{1,3}\) & \(\alpha_{1,3}\) & \(\gamma_{1,1}\) \\ \hline
16 pars. & 2.5186 & 0.0043 & 2.2748 & 195.7408 & 4.6835 & 1.3786 & 1.8735 & 0.0193 \\ \hline  & \(\gamma_{1,2}\) & \(\gamma_{1,3}\) & \(\omega_{2,1}\) & \(\alpha_{2,1}\) & \(\omega_{2,2}\) & \(\alpha_{2,2}\) & \(\omega_{2,3}\) & \(\alpha_{2,3}\) \\ \hline  & \(-\)39,713.78 & \(-\)0.9435 0.0052 & 1.1469 & 0.6288 & 2.3355 & 1.9707 & 0.64830 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters of the fitted 16-parameter real-order model.  
Figure 11: Comparison of the frequency responses of the fitted high-order ARX model, complex-order model, and the real-order models \(G_{2,1}\), \(G_{2,2}\).  
**Acknowledgments:** This research has been supported by the Ministry of Education, Science and Technological Development of Serbia through project no. 451-03-68/2022-14/200156 \"Innovative scientific and artistic research from the FTS (activity) domain\". Moreover, [PERSON], [PERSON], and [PERSON] wish to acknowledge the Visiting Professors program by the Polytechnic University of Bari, which supported their scientific collaboration.  
**Conflicts of Interest:** The authors declare no conflict of interest.