1. Modeling, Simulation, and Control of a Real System Robert Throne Electrical and Computer Engineering Rose-Hulman Institute of Technology

2. Introduction Models of physical systems are widely used in undergraduate science and engineering education. Students erroneously believe even simple models are exact.

3. Introduction Obtained ECP Model 210a rectilinear mass, spring, damper systems for use in both system dynamics and controls systems labs. Models for these systems are easy to develop and students have seen these types of models in a variety of courses.

4. Introduction ( mass, springs, and encoder)

5. Introduction (motor, rack and pinion, damper, and spring connecting to first cart)

6. Introduction We developed four groups of labs for the ECE introductory controls class for a one degree of freedom system: Time domain system identification. Frequency domain system identification. Closed loop plant gain estimation. Controller design based on the model.

7. Parameters to Identify In the transfer function model we need to determine the gain the damping ratio the natural frequency

8. Time Domain System Identification Log decrement analysis Fitting the step response of a second order system to the measured step response

11. Frequency Domain System Identification Determine steady state frequency response by exciting the system at different frequencies. Compare to predicted frequency response. Optimize transfer function model to best fit measured frequency response.

12. Model/Actual Frequency Response (from log-decrement)

13. Model/Actual Frequency Response (from fitting step response)

14. Model From Frequency Response

15. Closed Loop Plant Gain Estimation We model the motor as a gain,, and assume it is part of the plant We use a proportional controller with gain The closed loop system is The closed loop plant gain is then

16. Closed Loop Plant Gain Estimation Input step of amplitude A Steady state output The closed loop plant gain is given by

17. Results with Controllers After identifying the system, I, PI, PD, and PID controllers were designed using Matlab’s sisotool to control the position of the mass (the first cart). Both predicted (model based) responses and actual (real system) responses are plotted on the same graph.

18. Integral (I) Controller

19. ``It doesn’t work!’’

20. Proportional+Derivative (PD) Controller

21. PID Controller (complex conjugate zeros)

22. PID Controller ( real zeros)

23. State Variable Feedback

24. Conclusions Students learn: Simple, commonly used models are not exact, but still very useful. Simple models are a reasonable starting point for design. Motors have limitations which must be incorporated into designs.

25. Conclusions We have extended these labs to include Model matching ITAE quadratic optimal polynomial equation (Diophantine) 2 and 3 DOF state variable models

26. Acknowledgement This material is based upon work supported by the National Science Foundation under Grant No. 0310445