## 4 Construction of a Control System Model and Test Results
### Simulation Tests of a Control System with an Inertial Block (\(F_{c1}\)) of the Actuation System  
The control system with an inertial block (\(F_{c1}\)) of the actuation system was connected according to the diagram shown in Figure 7 in order to perform tests. As before, two control systems were tested: one with a PID controller and the other with a TR controller. In each test, the range of the actuation system was limited in accordance with Table 1. Three tests were performed for the TR controller. In each test, the \(k_{c}\) parameter was set to 500. The tests showed the influence of the inertial block (\(F_{c1}\)) on the quality of control. As before, during the tests, the value of \(\varphi_{min}=0.05\) was set to be constant, while the initial value of \(\varphi_{0}\) was changed; in subsequent tests, value of \(\varphi_{0}=1\) (57.3\({}^{\circ}\) ), \(\varphi_{0}=0.5\) (28.65\({}^{\circ}\)) and \(\varphi_{0}=0.2\) (11.46\({}^{\circ}\)) were assumed. For the PID controller, two tests are presented, showing the influence of the inertial block (\(F_{c1}\)) on the quality of control. The PID controller is a linear controller; therefore, the controller settings selected in previous tests were ineffective. After attaching the inertial block (\(F_{c1}\)), the system became unstable. It was necessary to reselect the controller settings and carry out a time-consuming procedure. From the obtained results, one set of PID settings was selected (Table 4). For the PID, tests were performed, showing the influence of the inertial block (\(F_{c1}\)) for two initial values. Tests were conducted with settings of \(\varphi_{0}=0.5\) (28.65\({}^{\circ}\)) and \(\varphi_{0}=0.2\) (11.46\({}^{\circ}\)). For \(\varphi_{0}=1\) (57.3\({}^{\circ}\)), no settings were found that would enable the stab  
Figure 7: Diagram of the control system with an inertial block. The results of the simulation tests of the TR controller are shown in the graphs in Figure 8, while the results of the simulation tests of the PID controller are shown in the graphs in Figure 9.  
As can be seen from the graphs (Figures 8 and 9), the TR and PID controllers successfully stabilised the inverted pendulum at the equilibrium point with settings of \(\varphi_{0}=0.5\) (\(28.65^{\circ}\)) and \(\varphi_{0}=0.2\) (\(11.46^{\circ}\)). However, for \(\varphi_{0}=1\) (\(57.3^{\circ}\)), a stable result was obtained only for the TR controller. The influence of the inertial term is visible in the graphs. Immediately after the start, the control error increases for a while, and only when the force that rectifies the pendulum increases to a value that allows it to counteract the force of gravity do we observe a decrease in the control error. In this respect, the reactions of the two controllers are quite similar. As in the previous test, the advantage of the TR controller is the lack of overshoot and, to a small extent, the speed of reaching the set value. In the PID result graph in Figure 9, a smoother result can be observed when equilibrium is reached. As shown in Section 4.1, the accuracy of the stabilisation of the TR controller depends on the value of the \(\varphi_{min}\) parameter, which, for this test, was set to a value of \(\varphi_{min}=0.05\) (\(2.86^{\circ}\)). As in the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**P** & **I** & **D** & **N** \\ \hline
3934.905 & 2948.614 & 1248.676 & 1473.653 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: PID controller settings for an inverted pendulum with an inertial block. Figure 8: The results of the simulation test of the TR controller with an inertial block (\(F_{\text{c1}}\)). Figure 9: The results of the simulation test of the PID controller with an inertial block (\(F_{\text{c1}}\)). previous test, the accuracy of achieving the set value for the TR controller was within the range of 40% of the \(\varphi_{min}\) value.