1. Examples of Control Systems Application

2. Modeling the Ball and Beam Experiment

3. Problem Setup M mass of the ball 0.11 kg R radius of the ball 0.015 m d lever arm offset 0.03 m g gravitational acceleration 9.8 m/s^2 L length of the beam 1.0 m J ball's moment of inertia 9.99e-6 kgm^2 r ball position coordinate

4. The design criteria for this problem Settling time less than 3 seconds Overshoot less than 5%

5. System Equations Linearization of this equation about the beam angle, alpha = 0, gives us the following linear approximation of the system

6. System Equations The equation which relates the beam angle to the angle of the gear can be approximated as linear by the equation below Substituting this into the previous equation, we get

7. Transfer Function

8. State-Space

9. for our state-space example we will be using a slightly different model. The same equation for the ball still applies but instead of controlling the position through the gear angle, theta, we will control alpha-doubledot. This is essentially controlling the torque of the beam. Below is the representation of this system

10. State-Space

11. Matlab Representation and Open- Loop Response Transfer Function

12. Matlab Representation and Open- Loop Response State-Space

13. Closed-loop Representation

14. PID controller Kp = Proportional gain KI = Integral gain Kd = Derivative gain

15. The characteristics of P, I, and D controllers CL RESPONSE RISE TIME OVERSHOOTSETTLING TIME S-S ERROR KpDecreaseIncreaseSmall Change Decrease KiDecreaseIncrease Eliminate KdSmall Change Decrease Small Change

16. Proportional Control kp = 1

17. Proportional-Derivative Control kp = 10; kd = 10

18. Proportional-Derivative Control kp = 10, kd = 20

19. Proportional-Derivative Control kp = 15 and kd = 40

20. Ball & Beam Problem Using Root Locus Method Open-loop Root Locus

21. Ball & Beam Problem Using Root Locus Method From these equation damping ratio and natural frequency were found to be 0.7 and 1.9 respectively.

22. Ball & Beam Problem Using Root Locus Method The design criteria can also be plotted onto the root locus using the sgrid command. This command generates a grid of constant damping ratio and natural frequency

23. Animation

24. Questions