Closed-loop Analysis-III

1. Closed-loop Analysis-III ----Frequency-domain approach

2. All zeros of the zero polynomial lie on LHP Plane. Stability requirement Characteristic equation:

3. No zero lies in here

4. F(s) Conformal Mapping in Complex planes A mapping that preserve shapes

5. S1-Zo Zo S-plane F- plane

6. S1-Zo Zo S2 S1 S-Zo S-plane F(s) = S-Zo F- plane

7. S1-Zo Zo S2 S1 S-Zo S-plane S3 F(s) = S-Zo F- plane

8. Zo F(s) = S-Zo Zo

9. Contour encircles Zo S-plane Zo S F(s) = S-Zo F- plane Zo S-Zo

10. Contour not encircles Zo S-plane Zo S F(s) = S-Zo F- plane Zo S-Zo

11. Conclusions on mapping of f(s)=s-z • When contour on s-plane encircles z, then map encircles origin of F-plane. The direction of encirclement is the same as that of the contour on s-plane. • When contour on s-plane does not encircle z, then map of function F(s) = s-z does not encircle origin of F-plane.

12. Similar conclusion can be drawn for mapping the contour on s-plane onto F-plane with function: The encirclement around the origin of F-plane depends on whether the contour on s-plane encircles zo or not. The direction of map on F-plane is reversed from that of the contour on s-plane.

13. Nyquist theorem • Consider a contour on s-plane be mapped through a function F(s) of the following into the F-plane. If inside the contour there are Z number of zeros and P number of poles of F(s), the net encirclement of the map in F-plane around the origin will be:

14. Application to stability Nyquist contour F(s) F-plane The contour encloses the whole RHP plane, including the imaginary axis

15. Thus, if there is Z number of zeros of F(s)=1+Gc(s)Gp(s) on the RHP Plane, the map of F(s) has net encirclements around the origin Z-P times. Where, P is the number of poles of F(s) on RHP plane. In other words, the number of zero of F(s) on the RHP plane is:Z=N+P

16. Thus, to test for the stability of a closed loopSystem, we need to: • Draw the map of F(s) for the Nyquist contour. • Examine the number of poles of F(s) that lie on RHP plane. • Count the net encirclement of the map around the origin, i.e. N. • The number of RHP zero of the system is: Z=N+P. Notice that the number P equals the number of RHP poles of GcGp(s).

17. How to draw Nyquist map of F(s) Nyquist contour =C1+C2+C3 C1 C2 C3

18. F(s)

19. Relation to open-loop Nyquist map ---I F- plane G- plane 0,0 -1,0 1,0 0,0 G(jw) G(jw) 1+G(jw) 1+G(jw)

21. Relation to open-loop Nyquist map ---II • The map of Nyquist contour on F- plane has identical shape as that on G-plane. The only difference is the shift of the origin of F- plane to point (-1, 0) on the G-plane. • The encirclement of Nyquist map around the origin of F-plane becomes that around (-1,0) on the G-plane. • The stability test can thus use map on G-plane instead of using map on F-plane. The check point becomes (-1,0) instead of point (0,0).

22. Bode criterion for stabililty • The Bode’criterion for stability is a direct result from the Nyquist stability criterion. • It simply focuses on the map along C1 path. • The criterion applies to open-loop stable loop process. • When applies to control loop with integral mode, the Nyquist contour should be modified.

23. Bode criterion for stabililty • For a control system to be stable, the amplitude ratio must be less than unity when the phase is -180o. • If |Gloop(jw)|<1 at -180o phase angle, the system is stable. • If |Gloop(jw)|>1 at -180o phase angle, the system is unstable.

24. Relation between open-loop Nyquist plot and frequency response --- 1 G- plane -1,0 0,0 Plot the value of the |G| and phase against w on a log-log scale or Semi-log scale coordinates to become a Bode diagram G(jw) Map along C1

25. Relation between open-loop Nyquist plot and frequency response--- 2 • In stability analysis, G(jw) is just a part of Nyquist maps of G(s). • On the other hand, for a given process G(s), G(jw) is known as the frequency response of the process. This terminology is mainly owing to that G(jw) stands for the particular solution of G(s), which is subjected to a sinusoidal input.

28. y G(s) Frequency response

29. Representations of frequency response • Polar plot --- the same as the map of part C1 of the Nyquist contour on the s-plane. • Bode’ diagram --- consists of two parts: one for b vs. w on a log-log coordinates and f vs. w on a semi-log coordinates. • Nichol’s plot --- a plot of b vs. f.

30. Polar plot G(jw)

32. Bode’s diagram for first order process Corner frequency=1/t slope=-20db/decade |G|

33. Bode’s diagram for pure dead time

34. Bode’s diagram for 2nd order process slope=-40db/decade

35. Bode’s diagram for 1st order lead process

36. Bode’s diagram for integrator process

37. Bode’ diagram of PID controller --- PI

38. Bode’ diagram of PID controller --- PD

39. Bode’ diagram of PID controller --- PID(1)

40. Bode’ diagram of PID controller --- PID(2)

41. G1 G2 G3 …

43. … G3 Gc G1

44. Stability using Bode’ diagram a a < 1?

45. Application of Bode’s criterion

49. Measure of relative stability 1/a a KC,u MR

50. 1/a 1/a a MR