Drexel University Mechanical Eng

REAL TIME CONTROL

Real time sliding mode control of the suspended pendulum is performed by using Labview 7.1. Labview file that was used to realize real time sliding mode control of the suspended pendulum can be downloaded here. In the previous section, virtual sliding mode control of the suspended pendulum was performed by using Matlab 7.0. Forcing function that was manipulated by sliding mode control law is equal to:

F is the force that is generated at the bottom end of the suspended pendulum by DC motor and propeller system. DC motor is going to be driven with the control voltage that is generated by the labview code. The transfer function between the input voltage of the DC motor and the force that is generated by the propeller is not known for sure. Therefore it must be modeled. This is done simply by obtaining the gain between the input voltage applied to the DC motor and the steady state angular position of the suspended pendulum. Input voltages between 0.5 V and 2.5 V with 0.25 V increments are sent to the suspended pendulum system and the steady state angular position of the suspended pendulum is stored. Then the following graph is obtained.

Figure 1. Least squares line fit to the "Teta" versus "Input Voltage" data.

On the upper right corner of Figure 1, the line equation that gives the relationship between the input voltage and the angular position of the suspended pendulum is shown. This line is forced to pass from origin, however due to the internal friction of the DC motor this was not necessary.

When steady state is reached the static equilibrium equations can be used to obtain the force generated at the end of the suspended pendulum. Following equation is found by equating the total moment acting on the suspended pendulum about the pin point.

Least squares line fit to the "Control Input" versus "Input Voltage" data.

In the up right corner of the graph, the line equation that gives the relationship between the control input and the input voltage is shown. Again, it is not necessary to force the line to pass from origin.

Now the system is ready to be tested to see the efficiency of the pseudo sliding mode control when applied to suspended pendulum.

For reference the equation of motion of the suspended pendulum is:

and sliding mode control that is going to be applied to the system is:

where, k₁ = m*fi, k₂ = m + fi - a₂, ρ = a₁ + η (eta). m and fi are the poles of the second order system. Suspended pendulum is matched to that second order system with the control law given above with an additional disturbance or parameter uncertainty rejection term.

First test to be done is the step input. The system parameters are a₁ = 2.74, a₂ = 0.01. The controller parameters are m = 2, fi = 2, η = 0.1, δ (delta) = 0.1. Time delay is chosen as 100ms. It was observed that when the time delay is 100ms the system was somewhat more stable than the systems with different time delays.

To compare the real time results with the simulation results, the angular position of the suspended pendulum is written to a file. However, due to the delay that added to the system, the tests are performed without writing the data. The results when the angular position data is written is shown below with the previous simulation results.