SIMULATIONS
The controller parameters are
Euler integration is used for the simulation and the time step is taken as 10ms. Initial angle is selected as 60 degrees measured from the equilibrium point in the counter clockwise direction. The system reaches equilibrium point in 3 seconds as can be seen in Figure 1.
Figure 1. Step Response -Angular Position of the Suspended Pendulum-.
In Figure 2, phase portrait is given. The two phases of the dynamics can easily be observed in Figure 2. The states first reach the sliding surface and then are constrained to the sliding surface. Instead of sliding, the states fluctuate around the sliding surface. When the states reach to a close neighborhood of the sliding surface they don’t stay on the sliding surface, they continuously cross the sliding surface. This phenomenon is called chattering. This is due to the fact that control input is not in infinite frequency.
Figure 2 Phase Portrait of the Suspended Pendulum.
Figure 3. Control Input.
The control signals starts switching when the sliding surface is reached. Because the time step is selected as 10ms, the frequency of the control input is 100Hz. Sliding mode control is insensitive to matched uncertainty, non-linearity or disturbances when the frequency of the control signal is infinite as mentioned before. This is practically not possible, especially in mechanical systems. Therefore the insensitivity property of the sliding mode control is not guaranteed. Furthermore, the states will be only constrained to a small neighborhood of the sliding surface as discussed before.
The discontinuous term of the sliding mode control can be made continuous by approximating the sign function with a continuous one. The pseudo sliding mode control is hence
,
where δ is a small positive constant. It is selected as 0.01 and 0.1 in this simulation.
Figure 4. Step Responses -Angular Position of the Suspended Pendulum-.
As can be seen from Figure 4, changing the discontinuous term in the sliding mode control with an approximation does not change the response much for the suspended pendulum. However as δ increases, the states move away from the sliding surface (Figure 5).
Figure 5. Phase Portraits of the Suspended Pendulum.
Figure 6. Control Inputs.
Increasing δ helps to get rid of the chattering in the control signal as can be seen in Figure 6. When δ=0.1 the control signal does not have any chattering, it is also equivalent to the filtered version of the original control signal.
In this section, imperfections of the real system, such as delay and unidirectional control signal are included in the simulations. It can be seen that the system can still be controlled. Next delay is added to the system. It is applied by simply holding the control input for 100ms and 250ms.
The following figures show the simulation results. The system can still be successfully controlled with pseudo sliding mode control when the time delay is 100ms and the control signal is constrained to have only positive values.
Figure 7. Step Response -Angular Position of the Suspended Pendulum-.
Figure 8. Phase Portrait of the Suspended Pendulum.
Figure 9. Control Input.
Following are the simulation results when 100ms and 250 ms delays are added to the system that can only apply positive control signals.
Figure 10. Step Responses -Angular Position of the Suspended Pendulum-.
In Figure 10, it is observed that when the delay is increased, the system starts to oscillate around a point higher than the equilibrium point. The high delay is responsible for the oscillations and the only positive control signal is responsible for the offset of the oscillations.
Figure 11. Phase Portraits of the Suspended Pendulum.
In Figure 11, it is observed that the system moves away from the sliding surface due to the increased delay.
Figure 12. Control Inputs.
In this part of the simulations, the pendulum is commanded to 60°. The step response is shown below in Figure 13. The sliding mode control is designed with the error signal instead of position signal. Error signal is the difference between the actual angle and the commanded angle. This is similar to either shifting the phase portrait by an amount of commanded angle to the left or shifting the sliding surface to the right by the same amount. This phenomenon can be observed in Figure 14. Two models are used in this part of the simulations. One is the almost real model (pseudo sliding mode control, time delay and unidirectional control signal are included in the model.) and the other is the theoretical model.
Figure 13. Step Responses -Angular Position of the Suspended Pendulum-.
From Figure 13, the almost real model has steady state error. This was expected because the system is critically damped and control signal is unidirectional. Furthermore the system does not have any integrators. Steady state error is negligible in the theoretical model, however it becomes higher as more imperfections are included in the system.
Figure 14. Phase Portrait of the Suspended Pendulum.
Figure 15. Control Inputs.
For questions about these simulations, please feel free to contact vefa@drexel.edu.
