## 5 Conclusions  
The proposed algorithms based on the fuzzy c-means technique, along with the PDC controller and the PSO metaheuristic optimization method, demonstrated effectiveness in modelling, identifying, and controlling a complex system. This capability was verified through numerical tests conducted on the obtained power generator system model.  
Taking advantage of experimental data from the open-loop plant, the optimization strategy utilizing the PSO algorithm enabled the determination of the optimal fuzzy parameter \(m\) for the c-means algorithm. It was demonstrated that, by optimizing the value of \(m\), the learning data pertaining to the premises of fuzzy rules are classified with greater similarity. This leads to an enhanced interpretation and reduced approximation error of the fuzzy model. Additionally, it facilitated the selection of poles within the unit circle for each closed-loop rule's consequent, aiding in the fine-tuning of fuzzy controllers via state feedback.  
By utilizing input and output data, the process of identifying a fuzzy model for the power generation system bypasses the necessity to derive and identify a mathematical model through conventional techniques. This is particularly advantageous given the  
Figure 6: Comparison between the PDC strategy and a classic PD controller.  
significant challenge posed by the complexity of the entire system, which encompasses both an internal combustion engine and a mechanically coupled electric generator.  
The comparison of the proposed control strategies with a classical control law demonstrated that classical controllers are not able to reach the desired voltages at different step inputs due to the controller's gains only being tuned for a certain region of operation. However, when the desired voltage is required to be defined at a different region of operation, the proposed PDC controller modifies the corresponding controller's gains thorough the obtained firing value, \(\mu_{ik}\), from the fuzzy logic strategy.  
In our upcoming research work, we aim to deploy the developed algorithms onto a physical power generation system. This step is crucial to validate their efficacy in addressing the inherent complexities characteristic of such systems. These complexities include dead zones stemming from the actuator utilized in the internal combustion engine's accelerator, voltage fluctuations, and non-linearities within the electric load and its associated electronic speed controller.  
The obtained quality-of-fit indexes between the regulated voltage and the identified voltage are \(QoF=3.4763\times 10^{-4}\) and \(QoF=6.1264\times 10^{-4}\), respectively. For this index, values much smaller than 1 are considered a good performance.  
Conceptualization, C.R. and E.S.E.; methodology, C.R., J.C.R.-F. and E.S.E.; simulations, C.R., J.C.R.-F. and E.S.E.; dynamics modelling and control, C.R., J.C.R.-F. and E.S.E.; investigation, C.R., J.C.R.-F. and E.S.E.; writing--original draft preparation, C.R., E.S.E. and J.C.R.-F.; writing--review and editing, C.R., J.C.R.-F., R.L. and E.S.E.; supervision, E.S.E., J.C.R.-F. and R.L.; project administration, E.S.E. All authors have read and agreed to the published version of the manuscript.  
This work was partially supported by the Mexican National Council for Humanities, Sciences, and Technologies through Project: Laboratorio Nacional en Vehiculos Autonomos y Exoesqueletos.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
[PERSON] thanks [PERSON] in recognition of the invaluable support provided throughout his PhD studies at the Center for Research and Advanced Studies of the National Polytechnic Institute Zacatenco, Mexico.  
The authors declare no conflicts of interest.