Ying Luo CSOIS / ECE / USU 17 / 2 / 2008

1. A Fractional Order Proportional and Derivative (FOPD) Controller Tuning Algorithm Simulation and Experiment Ying Luo CSOIS / ECE / USU 17 / 2 / 2008

2. Outline • Tuning Algorithm • Simulation • Repeating the simulation results in the paper • Simulation with dynamometer model • Experiment on dynamometer

3. Tuning Algorithm • FOPD controller transfer function • Typical second-order plant transfer function • System open loop transfer function

4. Tuning Algorithm How to fix the parameters of the FOPD controller , and ? Rule (i) : Phase margin specification

5. Tuning Algorithm Rule (ii) : Robustness to variation in the gain of the plant

6. Tuning Algorithm Rule (iii) : Gain crossover frequency specification

7. Tuning Algorithm According to the three equations above in rule (i) (ii) (iii), we can get the three parameters of FOPD controller:

8. Tuning Algorithm • Design Procedures • Given , the gain crossover frequency; • (2) Given , the desired phase margin; • (3) Plot the curve 1, Kd w. r. t μ, according to rule (i) • (4) Plot the curve 2, Kd w. r. t μ, according to rule (ii); • (5) Obtain the μ and Kd from the intersection point on the • above two curves; • (6) Calculate the Kp from rule (iii).

9. Outline • Tuning Algorithm • Simulation • Repeating the simulation results in the paper • Simulation with dynamometer model • Experiment on dynamometer

10. Simulation • Repeat the simulation results in the paper (1) IOP Controller with the model in the paper. • IOP Controller transfer function: • Typical second-order plant : • Given optimum proportional parameter :

11. Simulation (2) Repeating the algorithm and getting FOPD controller parameters , and with the model in the paper. • FOPD controller transfer function: • Typical second-order plant : • Given condition:

12. Simulation • Numerical Solution = 0.053 = 0.86 = 84.89

13. Simulation • Bode diagram with FOPD controller

14. Simulation (3) Simulation Results IOP Controller: K = [8,9,10,11,12] FOPD Controller: Frequency range = [0.0001, 10000] Kp= [67.90 ,76.40, 84.88, 93.35, 101.80]

15. Outline • Tuning Algorithm • Simulation • Repeating the simulation results in the paper • Simulation with dynamometer model • Experiment on dynamometer

16. Experiment (1) IOP Controller with the model in the paper. • IOP Controller transfer function: • Real Dynamometer plant • Given optimum proportional parameter :

17. Experiment (2) Repeating the algorithm and getting , and with dynamometer model • FOPD transfer function: • Real Dynamometer model plant • Given condition:

18. Experiment • Numerical Solution = 1.109 = 0.80 = 13.775

19. Experiment (3) IOP Experiment Results IOP Controller: K = [0.39604, 0.445545, 0.49505, 0.544555, 0.59406]

20. Experiment (3) FOPD Experiment Results (A) Frequency range [0.02, 200] FOPD Controller: Kp= [11.0200 ,12.3975, 13.7750, 15.1525, 16.5300]

21. Experiment (3) FOPD Experiment Results (B) Frequency range [10/10, 10*10] = [1, 100] FOPD Controller: Kp= [11.0200 ,12.3975, 13.7750, 15.1525, 16.5300]