1. LINEAR CONTROL SYSTEMS Ali Karimpour Associate Professor Ferdowsi University of Mashhad

2. Dr. Ali Karimpour Apr 2013 Lecture 18 2 Nyquist stability criteria. Topics to be covered include: v Frequency domain analysis of control systems. v Frequency domain specification. u Peak of resonance and resonance frequency. u Bandwidth. v Nyquist stability criteria.

3. Dr. Ali Karimpour Apr 2013 Lecture 18 3 One degree-of-freedom configuration T(s) S(s) S(s) is : Sensitivity Function T(s) is : Complementary Sensitivity Function Why? Since S+T=1

4. Dr. Ali Karimpour Apr 2013 Lecture 18 4 One degree-of-freedom configuration Where T(s) and S(s) are How to derive command tracking How to derive disturbance rejection How to derive noise attenuation

5. Dr. Ali Karimpour Apr 2013 Lecture 18 5 One degree-of-freedom configuration Derive command tracking and disturbance rejection

6. Dr. Ali Karimpour Apr 2013 Lecture 18 6 One degree-of-freedom configuration Derive command tracking and disturbance rejection How to derive noise attenuation

7. Dr. Ali Karimpour Apr 2013 Lecture 18 7 One degree-of-freedom configuration Ideal |T| for command tracking, disturbance rejection and noise attenuation Ideal |L| for command tracking, disturbance rejection and noise attenuation Actual |T| for command tracking, disturbance rejection and noise attenuation Actual |L| for command tracking, disturbance rejection and noise attenuation

8. Dr. Ali Karimpour Apr 2013 Lecture 18 8 A prototype second order system. یک سیستم نمونه درجه 2 + - c e r c r Frequency (rad/s)

9. Dr. Ali Karimpour Apr 2013 Lecture 18 9 Frequency domain specification Peak of resonance ( ) Resonance frequency ( )

10. Dr. Ali Karimpour Apr 2013 Lecture 18 10 Frequency domain specification Closed-loop bandwidth ( ) -3 Open - loop bandwidth ( ) Gain crossover frequency ( )

11. Dr. Ali Karimpour Apr 2013 Lecture 18 11 Introducing a prototype second order system. + - c e r c r Resonance frequency ( ) Peak of resonance ( )

12. Dr. Ali Karimpour Apr 2013 Lecture 18 12 Introducing a prototype second order system. + - c e r c r Closed-loop bandwidth ( ) -3

13. Dr. Ali Karimpour Apr 2013 Lecture 18 13 Introducing a prototype second order system. + - c e r c r

14. Dr. Ali Karimpour Apr 2013 Lecture 18 14 Mapping نگاشت b' b d c a a' c' d' 2 4

15. Dr. Ali Karimpour Apr 2013 Lecture 18 15 Mapping نگاشت s=0.5 1 -2 0.2

16. Dr. Ali Karimpour Apr 2013 Lecture 18 16 Mapping نگاشت r=0.5 1 -2

17. Dr. Ali Karimpour Apr 2013 Lecture 18 17 Mapping نگاشت 1 -2 r=0.5 -5 7

18. Dr. Ali Karimpour Apr 2013 Lecture 18 18 1 -2 Mapping نگاشت r=2.5

19. Dr. Ali Karimpour Apr 2013 Lecture 18 19 1+kf(s)=0 : تعداد قطبهای RHP، 1+kf(s) Nyquist path Nyquist fundamental : تعداد صفرهای RHP ، 1+kf(s)

20. Dr. Ali Karimpour Apr 2013 Lecture 18 20 Nyquist path Nyquist fundamental

21. Dr. Ali Karimpour Apr 2013 Lecture 18 21 Nyquist path Nyquist plot Nyquist fundamental

22. Dr. Ali Karimpour Apr 2013 Lecture 18 22 Example 1: Check the stability of the following system by Nyquist method. مثال 1 : پایداری سیستم را توسط روش نایکوئیست بررسی کنید. + -

23. Dr. Ali Karimpour Apr 2013 Lecture 18 23 Example 1: Check the stability of the following system by Nyquist method. مثال 1 : پایداری سیستم را توسط روش نایکوئیست بررسی کنید. + -

24. Dr. Ali Karimpour Apr 2013 Lecture 18 24 Nyquist path Nyquist plot Nyquist fundamental

25. Dr. Ali Karimpour Apr 2013 Lecture 18 25 More study مطالعه بیشتر + -

26. Dr. Ali Karimpour Apr 2013 Lecture 18 26 Exercises تمرینها 1- Assume the following control system Find the value of k which make the system stable. Answer: k>1 + - + - R(s)C(s) 2- Consider the following control system Find the value of k which make the system stable. + G(s) - K

27. Dr. Ali Karimpour Apr 2013 Lecture 18 27 Exercises تمرینها 3- Polar plot of an open loop transfer function with two RHP poles is shown in the following: W=0 -1+j0 Discuss about the stability of system. Answer: It is unstable and it has 4 unstable poles.

28. Dr. Ali Karimpour Apr 2013 Lecture 18 28 Exercises تمرینها 4- Discuss the stability of following system by Nyquist criteria. + + - 5- Discuss the stability of following system by Nyquist criteria by use of following Nyquist path(Final 1391). + + -

29. Dr. Ali Karimpour Apr 2013 Lecture 18 29 + - Very important Example 2: Check the stability of following system by Nyquist method. مثال 2 : پایداری سیستم را توسط روش نایکوئیست بررسی کنید.

30. Dr. Ali Karimpour Apr 2013 Lecture 18 30 Example 2: Check the stability of following system by Nyquist method. پایداری سیستم را توسط روش نایکوئیست بررسی کنید. + - Very important How to calculate?

31. Dr. Ali Karimpour Apr 2013 Lecture 18 31 Example 2: Check the stability of following system by Nyquist method. پایداری سیستم را توسط روش نایکوئیست بررسی کنید. + -

32. Dr. Ali Karimpour Apr 2013 Lecture 18 32 Example 2: Check the stability of following system by Nyquist method. پایداری سیستم را توسط روش نایکوئیست بررسی کنید. + -

33. Dr. Ali Karimpour Apr 2013 Lecture 18 33 + - More study مطالعه بیشتر