# Ball and beam system

Lars Cremean, 29 September 2003

## System Description

A ball is placed on a beam, see figure below, where it is allowed to roll with 1 degree of freedom along the length of the beam. A motor is connected to the beam at the center, and produces a torque that causes the beam to rotate. When the angle is changed from the horizontal position, gravity causes the ball to roll along the beam. A controller will be designed for this system so that the ball's position can be manipulated.

For this problem, we will assume that the ball rolls without slipping and friction between the beam and ball is negligible. The constants and variables for this example are defined as follows:

M | mass of the ball | 0.11 kg |

R | radius of the ball | 0.015 m |

g | gravitational acceleration | 9.8 m/s^2 |

J | ball's moment of inertia | 9.99e-6 kgm^2 |

r | ball position coordinate | |

alpha | beam angle coordinate | |

A state space representation of the system can be found to be

This representation is included in the Simulink model provided, along with full-state (i.e. all of the state variables are available to the controller) feedback control with reference input.

### Instructions

Right-click {{{2}}} and save it to a local drive. Open it with MATLAB (Simulink) and follow the instructions given above.

### Notes

- This problem was adapted from Control Tutorials for MATLAB
- The time trace of the full state for each run is saved to the MATLAB workspace as a structure called "ballbeam". You can access the elements of a structure using a period (.), e.g. ballbeam.signals.
- The settling time is the time it takes for the system error to settle to within the desired value plus or minus 5%. Any method of finding these values to produce your plots is acceptable, but you may like to investigate the MATLAB "find" command.
- Include at least ten values in each of your plots.
- Use the MATLAB "plot" command to produce your plots. Type "help plot" in the MATLAB command window for help on using this function.
- Include a title, axis labels (with units), and reasonable axis limits on all plots. See MATLAB commands "title", "axis", "ylabel" and "xlabel".