## 4 Results
### Simulation and Experimental Results  
From the three previously identified linear models of the plant, a nonlinear TS fuzzy model is formed. Additionally, based on optimal parameters of individual first-order PS control systems that are ideal for the appropriate linear models of the plant, PDC control is formed.  
Both the simulation and the experimental results for the servo motor closed loop control system operating with the optimal \(\alpha\), \(\beta\) parameter values are provided.  
Figure 5: The constant damping coefficient curve with optimal values applied to it for nonzero and zero initial conditions.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**i** & \(\mathbf{u_{i}(0)}\) & \(\mathbf{u_{i}(1)}\) & \(\mathbf{y_{i}(0)}\) & \(\mathbf{r_{i}(0)}\) & \(\mathbf{I_{i~{}min~{}zero}}\) & \(\mathbf{K_{i}}\) & \(\mathbf{K_{Si}}\) \\ \hline
1 & 0 & 0 & 0 & 0 & 1491.02 & 8.7564 & 9.8506 \\
2 & 0 & 0 & 0 & 0 & 2500.6 & 8.0313 & 9.0201 \\
3 & 0 & 0 & 0 & 0 & 4351.55 & 7.3831 & 8.3161 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Zero initial conditions and optimal values.  
Figure 6 depicts the system simulation and experimental step responses for both scenarios (nonzero and zero initial conditions, corresponding to the novel and the traditional concept). All responses start from the same nonzero initial conditions.  
To begin with, a 2 [s] long pre-experiment is run to achieve nonzero initial conditions, and after they are obtained, the system is switched to the closed-loop mode of operation.  
The \"black\" system's simulated behavior is derived by the new theory optimal parameters, which are appropriate only for the particular initial conditions used. As a result, the \"magenta\" system's simulated response is derived by the traditional theory optimal parameters, which are \"universally optimal\" and \"valid\" for any initial conditions. As a consequence, we started the \"magenta\" system response from the identical nonzero initial conditions as the \"black\" one in order to compare them. Again, the \"black\" system response is far better than the \"magenta\" system response, as indicated by numerical values of the sum of squared errors as a metric of performance,  
\[SSE_{nonzero}=404.528,\quad SSE_{zero}=1401.0306. \tag{62}\]  
As can be seen in Figure 6, the experimental results coincide perfectly with the simulation ones. Again, the \"red\" experimental response of the system is determined by the new theory optimal parameters, which are just applicable for the specific initial conditions applied. So, the \"blue\" experimental response of the system is derived by the traditional theory optimal parameters, where zero initial conditions are assumed, but it started from the same nonzero initial conditions as the \"red\" one to allow for comparison. The response resulting from taking into account nonzero initial conditions has a shorter rise time, as well as a shorter settling time.  
A comparison of control signals is given in Figure 7. It is noticeable that the control signal is more \"aggressive\" in the case where nonzero initial conditions are present, which results in, as recently mentioned, improvements depicted in the step response.  
Figure 6: Simulation and experimental results of designed PDC controllers that take into account zero and nonzero initial conditions.