## 5 Results  
The results presented in this section are based on several numerical experiments performed with different forms of disturbance of the damping coefficient and additional loading torque, both increasing almost stepwise; low and, respectively, high frequency; and various controllers to obtain data of a comparative nature. We investigate three types of controllers working in an open and closed model reference control system:  
* classical PID--see Section 4.2,
* FOPID (PID*)--see Section 4.3,
* adaptive PID with PI action at the second stage of error compensation--see Section 4.4. Results of all numerical experiments are visualized (Figures 5-12) and assessed with the use of Integral Time-Weighted Absolute Error (ITAE) performance index summarized in Table 2. Performance of each control system, as well as robustness of the best ones, with respect to variations of selected electric parameters of the mechatronic drive, is discussed in Section 6. The selected motor model is described by the electromechanical parameters given in Table 1.  
Numerical simulation of the investigated dynamical system for the motor parameters listed in Table 1 is conducted in time \(t_{f}=1.75\) (s) at a step of integration \(h=2\cdot 10^{-5}\) (s). It brings an interesting, but some anticipated solutions shown in Figure 5a,b. The dashed line represents a pyramid stepwise shaped variation of damping coefficient (Figure 5a) and variation of an additional loading torque (Figure 5b) in the motion of the driven load inertia; the green line is the time trajectory of velocity to be followed by the inertia, and the red line shows its actual position. We see that both actual solutions are far from the required ones. It is important to note that also in both cases the disturbances are very harmful to the process, producing large temporary errors of the regulation. Together with the uncertainties of the parameter \(b_{la}(t)\) and the loading torque \(T_{la}(t)\), the control system has to deal with the stepwise second-order inertia increase of command velocity; see the green line. At these times, the motor's actual velocity represented by the black line experiences some oscillations that incorporate some portion of error into the temporary states initiated by the smoothed steps of command rotational velocity. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ITAE** & **Controller** & **Control Loop** & **Disturbed Parameter** & **Disturbance Frequency** & **Figure** \\ \hline
0.0368 & PID & closed & \(T_{l}\) & low & Figure 6b \\
0.0551 & Adaptive PID & closed & \(T_{l}\) & low & Figure 8b \\
0.0839 & Adaptive PID & closed & \(b_{l}\) & low & Figure 8a \\
0.0962 & FOPID & closed & \(T_{l}\) & low & Figure 7b \\
0.1104 & PID & closed & \(T_{l}\) & high & Figure 10b \\
0.1313 & Adaptive PID & closed & \(T_{l}\) & high & Figure 12b \\
0.2210 & PID & closed & \(b_{l}\) & high & Figure 10a \\
0.2263 & Adaptive PID & closed & \(b_{l}\) & high & Figure 12a \\
0.2383 & FOPID & closed & \(T_{l}\) & high & Figure 11b \\
0.4780 & FOPID & closed & \(b_{l}\) & low & Figure 7a \\
0.7453 & – & open & \(T_{l}\) & high & Figure 9b \\
1.2115 & FOPID & closed & \(b_{l}\) & high & Figure 11a \\
1.4949 & – & open & \(T_{l}\) & low & Figure 5b \\
1.5168 & PID & closed & \(b_{l}\) & low & Figure 6a \\
1.9720 & – & open & \(b_{l}\) & high & Figure 9a \\
2.4874 & – & open & \(b_{l}\) & low & Figure 5a \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The Integral Time-Weighted Absolute Error (ITAE) performance index for various numerical experiments. Figure 5: The open-loop reference model control system response at the presence of the additional time-dependent disturbances of: (**a**) damping of load \(b_{la}(t)\)—see on the right-hand vertical axis; (**b**) load \(T_{la}(t)\) with adaptation of PID parameters and PI compensation of error dynamics. Green line represents the desired rotational velocity trajectory, while the red line the actual velocity of the load (low-frequency case).