## 3 Methods  
### Fuzzy Control  
According to the theory in [37] the fuzzy control system can be assimilated to a state feedback control system. Physical state variables \(e_{\theta}\), \(d\omega/dt\), and \(\omega\) become input variables of the fuzzy controller, where \(e_{\theta}=\theta^{*}-\theta\). The torque \(M\) is the output variable of the fuzzy controller. The universes of discourse of the fuzzy variables are:  
\[U_{M}\in[-M_{m},M_{m}],\ U_{\omega}\in[-\omega_{m},\omega_{m}],\ U_{\theta} \in[-\theta_{m},\theta_{m}] \tag{17}\]  
where \(M_{m}=1000\) Nm, \(\omega_{m}=0.3\) rad/s, and \(\theta_{m}=\pi\).  
The fuzzy controller was designed based on previous experiments [19; 20; 21].  
[PERSON]'s structure was chosen for the fuzzy block. The membership functions chosen for the fuzzy variables are presented in Figure 6, where \(x\) symbolizes the above physical variables, and \(x_{m}\) represents their maximum values from Equation (17). Three fuzzy values were chosen: negative N, zero Z, and positive P.  
The rule base has 27 rules, resulting from the reasoning of working with the process. The rules are presented in Table 1[38; 39; 40; 41].  
Some examples of fuzzy reasoning on which the rule base was built are given below.  
* If \(e_{\theta}\) is zero and \(d\omega/dt\) is zero and \(\omega\) is zero then \(M\) is zero.
* If \(e_{\theta}\) is negative and \(d\omega/dt\) is zero and \(\omega\) is zero then \(M\) is negative.
* If \(e_{\theta}\) is positive and \(d\omega/dt\) is positive and \(\omega\) is positive then \(M\) is zero.  
An example of firing rules is presented in Figure 7.  
The inference is min-max and defuzzification is with the center of gravity. The use of only three fuzzy values and the min-max inference ensures the strongest nonlinear character at the fuzzy controller [38; 39; 40; 41].  
The surfaces of the input-output characteristics of the fuzzy block are presented in Figures 8-10.  
The structure of the fuzzy control system is presented in Figure 11. It is a state feedback fuzzy control system.  
The structure of the fuzzy controller is presented in Figure 12.  
The fuzzy controller has three inputs. Each input is limited at its maximum value.  
The fuzzy block was implemented using the fuzzy logic designer from MATLAB/Simulink. It has Mamdani's structure characteristics presented above [38; 39; 40; 41].  
Figure 6: The membership functions for the fuzzy values of the fuzzy controller.  
Figure 7: The membership functions and the variable values of 0 for rule (\(Z,Z,Z,Z\)).  
\begin{table}
\begin{tabular}{c c c c} \hline \(e_{0}\) & \(d\omega/dt\) & \(\omega\) & \(M\) \\ \hline Z & Z & Z & Z \\ Z & Z & N & P \\ Z & Z & P & N \\ Z & N & Z & P \\ Z & N & N & P \\ Z & N & P & Z \\ Z & P & Z & N \\ Z & P & N & Z \\ Z & P & P & N \\ N & Z & N & N \\ N & Z & N & N \\ N & Z & N & N \\ N & N & Z & N \\ N & N & Z & N \\ N & N & N & Z \\ N & N & P & N \\ P & Z & N & P \\ P & Z & N & P \\ P & Z & N & P \\ P & N & Z & P \\ P & N & Z & P \\ P & N & P & Z \\ \hline \end{tabular}
\end{table}
Table 1: Rule base.  
Figure 8: Surface \(M=f(e_{\theta},d\omega)\).  
Figure 10: Surface \(M=f(d\omega,\omega)\).  
The dynamic behavior of the fuzzy control system was adjusted with the coefficients \(c_{e}\), \(c_{dw}\), \(c_{w}\) and \(c_{M}\).  
The characteristics obtained by simulation for the variables, namely position error \(e_{\theta}\), torque \(M\), rotation speed \(\omega\), and position \(\theta\), from the fuzzy control system, are presented, respectively, in Section 4.