## 4 Results  
### Takagi-Sugeno Model of the DC Motor Based on Linearized Models  
The design technique was implemented and experimentally evaluated on a DC motor with gear and load. The experiments were carried out with the help of a Quanser rotary servomotor, model SRV02. To measure motor velocity and position, this type includes an optical encoder and a tachometer. The Q8-USB data acquisition board was utilized in the Windows environment for real-time data collection and control using Matlab/Simulink and QUARC@ real-time control software. The local linearized (affine) models in the consequent rules, Equation (2), are produced by employing identification methods based on measured input-output data using MATLAB 'System Identification Toolbox'. Utilized identification methods were based on the plant's step response. The nominal points are chosen so that the TS model represents the DC motor model on the considered range [0-5] [V]. The first and third nominal points are taken near the ends of the range, and the second in the middle of the range; that is, \(U_{N1}=0.5\), \(U_{N2}=2.5\) and \(U_{N3}=4.5\) [V]. The corresponding values of the nominal angular velocities are \(\Omega_{N1}=0.62\), \(\Omega_{N2}=4.02\) and \(\Omega_{N3}=7.56\) [rad/s]. Table 1 shows the nominal angular velocities \(\Omega_{Ni}\), nominal voltages \(U_{Ni}\), and associated identified discrete transfer functions for the sampling time \(T=0.002\) [s]. Based on the identified models, different equations that describe the DC motor in the vicinity of the selected nominal points can be obtained. Below is the procedure for the nominal point (\(U_{N1}\), \(\Omega_{N1}\)), as an example. If \(u(k)\) and \(\omega(k)\) denote the deviations of the control variable and the angular velocity, respectively, the following is obtained:  
\[G_{1}(z)=\frac{\Omega(z)}{U(z)}=\frac{0.1023}{z-0.9398}, \tag{10}\]  
\[\omega(k+1)-0.9398\omega(k)=0.1023u(k). \tag{11}\]  
The output variable is selected as the state variable, \(x(k)=y(k)=\omega(k)\). The plant's discrete state equation and discrete output equation are derived by substituting the state variable into the preceding discrete equation,  
\[x(k+1)=0.9398x(k)+0.1023u(k)=a_{1}x(k)+b_{1}u(k), \tag{12}\]  
\[y(k)=x(k). \tag{13}\]  
If \(X(k)\) denotes angular velocity value in total coordinates, then in the vicinity of the nominal point \((U_{N1},\Omega_{N1})\), it is valid that \(X(k)=x(k)+X_{N1}=x(k)+\Omega_{N1}\) and \(U(k)=u(k)+U_{N1}\). From Equations (12) and (13) the first affine local model for nominal point \((U_{N1},\Omega_{N1})\) (i.e., affine local model for the first rule in Equation (5)) is created:  
\[X(k+1)-X_{N1}=a_{1}(X(k)-X_{N1})+b_{1}(U(k)-U_{N1}), \tag{14}\]  
\[X(k+1)=a_{1}X(k)+b_{1}U(k)+\alpha_{1},\quad\alpha_{1}=X_{N1}-a_{1}X_{N1}-b_{1 }U_{N1}, \tag{15}\]  
\[Y(k)=X(k). \tag{16}\]  
Similar procedure was followed for the other two discrete equations. Table 1 shows the constants for the state space plant models. Based on the linear subsystems, the following affine TS fuzzy model is built:  
Model rule \(i\):  
\[\text{IF }X(k)\text{ is }M_{i}\text{ THEN}\bigg{\{} \begin{matrix}X(k+1)=a_{i}X(k)+b_{i}U(k)+ \alpha_{i},\ i=1,2,3,\\ Y(k)=X(k).\end{matrix}  \tag{17}\]  
The implied membership functions, which correspond to the nominal points \(\Omega_{Ni}\), are depicted in Figure 1. Due to simplicity, triangular membership functions were chosen. Their parameters are clearly defined with centers in nominal points. The triangular membership function \(M_{2}\) is determined by three parameters \(\{\Omega_{N1},\Omega_{N2},\Omega_{N3}\}\), as follows: \begin{table}
\begin{tabular}{c c c c c c c c} \hline i & \(\Omega_{Ni}\) & \(U_{Ni}\) [V] & \(G_{i}(z)\) & \(a_{i}\) & \(b_{i}\) & \(\alpha_{i}\) & \(\beta_{i}\) \\ \hline
1 & 0.62 & 0.5 & \(\frac{0.1023}{z-0.9398}\) & 0.9398 & 0.1023 & \(-0.0138\) & 0 \\
2 & 4.02 & 2.5 & \(\frac{0.0953}{z-0.9444}\) & 0.9444 & 0.0953 & \(-0.0149\) & 0 \\
3 & 7.56 & 4.5 & \(\frac{0.1042}{z-0.9443}\) & 0.9443 & 0.1042 & \(-0.0477\) & 0 \\ \hline \end{tabular}
\end{table}