1. سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي - دکتر سید مجید اسما عیل زاده

2. Design a lag controller for the following system such that the phase margin be 45° and the ramp error constant be 100. + - Step 1: Consider with as a phase-lag controller. Note: If the plant has another gain k, let Step 2: Try to fix k according to the performance request, otherwise let k=1 Example 2:

3. Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller. Step 4: Find the system PM and if it is not sufficient choose the new gain crossover frequency such that the PM is ok (reduce it a little). PM= -22°

4. Step 5: Find the gain of the system at the new gain crossover frequency and let: Step 6: Put the right corner of the controller sufficiently far from

5. Step 7: Check the designed controller.

6. 6 Design a lag controller for the following system such that the phase margin be 45° and the open loop bandwidth be 10 rad/sec + - It is not possible explain why? Example 3:

7. 7 0 0 φmφm 20 log a PM=25° How can the controller help us? PM=40° Design fundamental of Lead controller:

8. 8 1- lead controller increases the BW. It makes the system faster. 2- It exaggerates the noise effect. Increases the noise affect on the output. Design fundamental of Lead the controller:

9. 9 Phase (deg) Magnitude (db) Design fundamental of Lead the controller:

10. Lead controller design

11. Design procedure of a phase-Lead controller in the frequency domain Step 1: Consider with as a phase-Lead controller. Step 2: Try to fix k according to the performance request, otherwise let k=1 Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller. Note: If the plant has another gain k, let 0 0 φmφm 20 log a

12. Step 4: Find the system PM and if it is not sufficient choose the required phase by: ? Step 5: Put the center of the controller in the new gain crossover frequency: Step 6: Check the controller.

13. Design a lead controller for the following system such that the phase margin be 45° and the ramp error constant be 100. Find the M p of overall system. + - Step 1: Consider with as a phase-lead controller. Note: If the plant has another gain k, let Step 2: Try to fix k according to the performance request, otherwise let k=1 Example 4:

14. Step 3: Sketch the Bode plot of the system (with the fixed k ) without controller. Step 4: Find the system PM and if it is not sufficient choose the required phase by: PM=25°

15. Step 5: Put the center of the controller in the new gain crossover frequency:

16. Step 6: Check the controller.

17. Finding M p Clearly M p = 2 db + -

18. Find the step response of example 4 and compare it with example 1. + - Lead controller of example 4 Lag controller of example 1 Closed loop transfer function without controller Closed loop transfer function with a phase-lead controller Closed loop transfer function with a phase-lag controller Example 5:

19. + - Lead controller of example 4 Lag controller of example 1 Without controller Phase-lag controller Phase-lead controller Check the speed of different controllers and compare it with the controller properties

20. Design a lead controller for the following system such that the phase margin be 45° and the ramp error constant be 100. + - Step 1: Consider with as a phase-lead controller. Note: If the plant has another gain k, let Step 2: Try to fix k according to performance request, otherwise let k=1 Example 6:

21. Step 3: Sketch the Bode plot of system (with the fixed k ) without controller. Step 4: Find the system PM and if it is not sufficient choose the required phase by: PM= -20°

22. Step 5: Put the center of controller in the new gain crossover frequency: PM= -20°

23. Step 6: Check the controller. Note: Design is not possible! Why?

24. Design a lead controller for the following system such that the phase margin be 45° and the open loop bandwidth be 10 rad/sec + - Step 1: Consider with as a phase-lag controller. Note: If the plant has another gain k, let Step 2: Try to fix k according to performance request, otherwise let k=1 Why? Example 7:

25. Step 4: Find the system PM and if it is not sufficient choose the required phase by: Step 3: Sketch the Bode plot of system (with the fixed k ) without controller.

26. Step 5: Put the center of controller in the new gain crossover frequency:

27. Step 6: Check the controller. Controller is not ok Try again

28. Step 2: Try to fix k according to performance request, otherwise let k=1 Open loop bandwidth is near to gain crossover frequency so: Design a lead controller for the following system such that the phase margin be 45° and the open loop bandwidth be 10 rad/sec + - Example 7(Continue):

29. Step 3: Sketch the Bode plot of system (with the fixed k ) without controller. Step 4: Find the system PM and if it is not sufficient choose the required phase by:

30. Step 5: Put the center of controller in the new gain crossover frequency:

31. Step 6: Check the controller. Controller is not ok Try again

32. Step 2: Try to fix k according to performance request, otherwise let k=1 Open loop bandwidth is near to gain crossover frequency so: Design a lead controller for the following system such that the phase margin be 45° and the open loop bandwidth be 10 rad/sec + - Example 7(Continue):

33. Step 3: Sketch the Bode plot of system (with the fixed k ) without controller. Step 4: Find the system PM and if it is not sufficient choose the required phase by:

34. Step 5: Put the center of controller in the new gain crossover frequency:

35. Step 6: Check the controller. Controller is ok