## 6 Experimental Results
### Pneumatic Arm  
Figure 15 shows the experimental platform used in the case of the pneumatic arm. In this case, a study of the control of the position of the arm is accomplished. The chosen reference signal is a step with an amplitude of 1 V and a frequency of 0.05 Hz. On the other hand, a damping ratio \(\zeta=1\) and a natural frequency \(\omega\)n = 1 rad/s are chosen as control specifications. The time in which the conventional controller acts is 25 s, and the simulation period is 60 s.  
Figure 14: Responses of the DC motor speed for different \(\zeta\) with \(\omega\)n = 3 rad/s.  
Figure 13: Values of the identified parameters of the DC motor with \(\omega\)n = 10 y, \(\zeta\) = 1.5 y, and k variable.  
Figure 16 shows the values identified from the pneumatic arm. It is notable how the identified parameters stabilize at t = 12 s.  
Figure 17 shows the values for the gains of the controller. In the DC motor case, a sudden variation appears. At second 24, due to the change from the conventional controller to the adaptive one, the gain parameters \(\mathrm{K_{p}}\) and \(\mathrm{K_{i}}\) stabilize their values to obtain the imposed specifications.  
Figure 16: Values of identified parameters of the DC motor with \(\mathrm{\omega n}=1\) rad/s and \(\zeta=0.5\).  
Figure 17: Evolution of the values of the controller gains pneumatic circuit with \(\mathrm{\omega n}=1\) rad/s and \(\zeta=0.5\).  
Figure 15: Pneumatic arm used in the experiments.  
Finally, Figure 18 shows the system response to the step input. At second 24, when the conventional controller switches to the adaptive one, the output signal presents a peak because the adaptive controller is still tuning the parameters.  
Although the identifier needs time to estimate the parameters, the controller keeps the system under control, guaranteeing that the system reaches the control specifications.  
Figure 19 illustrates the case of an overdamped system, where the adaptive controller requires more time in order to correctly perform the tuning of the parameters. This effect needs to be considered depending on the safety measures.  
The results from the experiments allow us to establish the following:  
* The implemented identification system reaches the model convergence of the parameters even with sudden changes in the dynamics of the plant.
* It is possible to control the dynamics of the plant through the configuration of the damping and natural frequency. This allows for a much faster system with different responses, namely, overdamped, critically damped, and underdamped.
* Prior knowledge of the plant for the regulators (both conventional and adaptive) is necessary.
* For changes in the operating point of the plant, the laws of adaptation can recalculate the regulator gains. Therefore, the dynamic behavior of the system is always the same.
* Fast implementation is thanks to MBD hardware support and the easy configuration of the model in achieving the desired dynamic.
* The embedded algorithm can run the program independently from the computer in a low-cost controller.  
Figure 19: Input and output of the pneumatic circuit with \(\omega\)n = 1 rad/s and \(\zeta\) = 1.2.  
Figure 18: Input and output of the pneumatic circuit with \(\omega\)n = 1 rad/s and \(\zeta\) = 0.5.  
* The real-time graphical interface on the computer allows for interaction with the embedded controller, allowing for both the monitoring and modification of system variables.
* The critical safety evaluation scheme for MBD for the developed controllers is correctly tested. The controller behaves as expected. It should be pointed out that each scenario (e.g., industrial or aeronautics systems) requires different approaches to safety measurements. Further research in this area is recommended due to discrepancies between the critical safety concepts.