Ball and Beam: Controlling the Position of a Ball along a Beam with an LQR Controller

Introduction

This model studies a ball rolling on top of a beam. The ball translational acceleration will be dependent on how the beam is angled. This example studies two different control schemes, the PID regulator and the LQ regulator, which can be used to control the position of the ball along the beam, using the beam angle as input.

In order to get the full experience of this example, you need the following:

• A desktop Wolfram Language product; a free trial download is available at www.wolfram.com/mathematica/trial/
• The BallAndBeam.nb notebook that came in the same bundle as this model

Dynamics

This model is different from the main model in that it uses a linear-quadratic regulator to control the ball position along the beam, instead of a PID controller, as in the example model. The gains for the controller have been calculated using the accompanying Wolfram Language notebook.

Simulation

To simulate the model, perform this step:

• Click the Simulate button in the top toolbar .

Plot the results

Explore how the actual ball position differs from the reference ball position. Do this by plotting the variables x and y. The first variable describes the reference position and the latter describes the measured position. This plot will be displayed immediately upon simulation.

You should now see the following plot:

Visualize

Multibody systems have automatic visualizers to show what a real-world system would look like.

In this example, custom CAD models have been loaded to better represent the system. To see a 3D representation of the system, follow the steps below:

• From the Simulation Center, click the Animation button in the toolbar.
• Use your mouse or trackpad to drag the animation to a good angle and zoom in with your scroll wheel or by using the trackpad. Then click the Play button to play the animation.

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