## 4 PDC Fuzzy Control  
To achieve the regulation of the T-S system, the control technique based on the distributed parallel compensator was used. In this control strategy, the consequent of each fuzzy rule of the model (6) is replaced by a control law with the form of linear state feedback ([29; 35]). In the PDC approach with pole assignment, the eigenvalues of \((A_{i}-B_{i}K_{i})\) are proposed. The consequent for each PDC fuzzy rule is given by:  
\[u_{i}(k)=-K_{i}x(k) \tag{13}\]  
The global output of the PDC, in other words, the fuzzy aggregation of all control rules, has the following form:  
\[u(k)=\frac{\sum_{i=1}^{c}\mu_{ik}u_{i}(k)}{\sum_{i=1}^{c}\mu_{ik}} \tag{14}\]  
where \(K_{i}\) is the vector of gains of the \(i\)-th rule, as shown in Table 6, and \(k_{r}=1.38\) is a precompensation gain, as shown in Figure 1. For obtaining the PDC gains, the method by pole assignment was used, as obtained by the PSO algorithm.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Optimal** **Poles** & **Bounds** & **\# of Particles** & **Inertia Coefficient**\(\omega\) & **\$1.52** & **\# of** **Iterations** \\ \hline
0.3, 0.8 & \([0.01,0.9]\) & 300 & 0.3820 & 0.9549 & 100 \\ \hline \end{tabular}
\end{table}
Table 4: Parameters used in the PSO algorithm and obtained values for the poles.  
\begin{table}
\begin{tabular}{c c c} \hline
**ISE** & **IAE** & **ITAE** \\ \hline \(\sum_{k=0}^{N}e^{2}(k)T_{s}\) & \(\sum_{k=0}^{N}|e(k)|T_{s}\) & \(\sum_{k=0}^{N}k|e(k)|T_{s}\) \\ With PSO & 2.6677 \(\times\)\(10^{3}\) & 51.6501 & 8.02 \(\times\)\(10^{5}\) \\ \hline Without PSO & 6.9970 \(\times\)\(10^{6}\) & 2.6452 \(\times\)\(10^{3}\) & 2.1061 \(\times\)\(10^{9}\) \\ \hline \end{tabular} Where \(N=301\) is the total number of samples, \(T_{s}=1\) s is the sampling time, and \(e(k)\) is the system output tracking error.  
\end{table}
Table 5: Error indexes using the algorithm PSO and without the algorithm PSO.  
Figure 5 shows the simulation results of the voltage regulation at different reference values, in which it can be seen that the PDC controller regulates the generated voltage, defined by reference steps of 30, 40, and 25 volts. These reference values are typical voltage values in applications that employs LiPo batteries. In such applications, 25, 40, and 30 volts are nominal voltages of LiPo batteries with different numbers of cells.  
Furthermore, Figure 5 shows the control signal applied to the PGS, which is bounded to values in a range of \([0,1]\). This variable regulates the amount of gasoline in the explosion chamber by manipulating the throttle unit (carburettor) with a servomotor. It is seen that the PDC fuzzy controller regulates the voltage in the simulation test to the reference voltage value.  
Algorithm 2 describes the required steps for the optimization of the controller's gains as well as the implementation of the PDC controller. The fuzzy model tracking of the closed-loop power generation system achieves a QoF = 3.4763 \(\times\)\(10^{-4}\), given by Equation (9). The QoF is much less than 1, even though the PGS is subjected to noise in its generated voltage, natural dead zones of the control actuator in the throttle body (carburettor servo motor), and non-linear dynamically varying electrical disturbances (load).  
\begin{table}
\begin{tabular}{c c} \hline
**Gains** & **Values** \\ \hline \(K_{1}\) & \(0.0448,-0.0559\) \\ \(K_{2}\) & \(0.2130,-0.2118\) \\ \(K_{3}\) & \(0.0529,-0.0632\) \\ \(K_{4}\) & \(0.0720,-0.0809\) \\ \hline \end{tabular}
\end{table}
Table 6: Defined gain values for each submodel.  
Figure 5: Simulation results of the voltage regulation using the obtained model of the power generator system.