Stability from Nyquist plot

1. Stability from Nyquist plot The completeNyquist plot: • Plot G(jω) for ω = 0+ to +∞ • Get complex conjugate of plot, that’s G(jω) for ω = 0– to –∞ • If G(s) has pole on jω-axis, treat separately • Mark direction of ω increasing • Locate point: –1

3. Encirclement of the -1 point • As you follow along the G(jω) curve for one complete cycle, you may “encircle” the –1 point • Going around in clock wise direction once is +1 encirclement • Counter clock wise direction once is –1 encirclement

7. Nyquist Criterion Theorem # (unstable poles of closed-loop) Z= # (unstable poles of open-loop) P + # encirclement Nor: Z = P + NTo have closed-loop stable: need Z = 0, i.e. N = –P

8. That is: G(jω) needs to encircle the “–1” point counter clock wise P times. • If open loop is stable to begin with, G(jω) cannot encircle the “–1” point for closed-loop stability • In previous example: • No encirclement, N = 0. • Open-loop stable, P = 0 • Z = P + N = 0, no unstable poles in closed-loop, stable

9. Example:

10. As you move around from ω = –∞ to 0–, to 0+, to +∞, you go around “–1” c.c.w. once. # encirclement N = – 1. # unstable pole P = 1

11. i.e. # unstable poles of closed-loop = 0 • closed-loop system is stable. • Check: • c.l. pole at s = –3, stable.

12. Example: • Get G(jω) forω = 0+ to +∞ • Use conjugate to get G(jω) forω = –∞ to 0– • How to go from ω = 0– to ω = 0+? At ω ≈ 0 :

15. # encirclement N = _____ # open-loop unstable poles P = _____ Z = P + N = ________ = # closed-loop unstable poles. closed-loop stability: _______

16. Example: • Given: • G(s) is stable • With K = 1, performed open-loop sinusoidal tests, and G(jω) is on next page • Q: 1. Find stability margins • 2. Find Nyquist criterion to determine closed-loop stability

18. Solution: • Where does G(jω) cross the unit circle? ________ Phase margin ≈ ________Where does G(jω) cross the negative real axis? ________ Gain margin ≈ ________Is closed-loop system stable withK = 1? ________

19. Note that the total loop T.F. is KG(s). If K is not = 1, Nyquist plot of KG(s) is a scaling of G(jω). e.g. If K = 2, scale G(jω) by a factor of 2 in all directions. Q: How much can K increase before GM becomes lost? ________ How much can K decrease? ______

20. Some people say the gain margin is 0 to 5 in this example Q: As K is increased from 1 to 5, GM is lost, what happens to PM? What’s the max PM as K is reduced to 0 and GM becomes ∞?

21. To use Nyquist criterion, need complete Nyquist plot. • Get complex conjugate • Connect ω = 0– to ω = 0+ through an infinite circle • Count # encirclement N • Apply: Z = P + N • o.l. stable, P = _______ • Z = _______ • c.l. stability: _______

22. Correct Incorrect

23. Example: • G(s) stable, P = 0 • G(jω) for ω > 0 as given. • Get G(jω) forω < 0 by conjugate • Connect ω = 0– to ω = 0+.But how?

24. Choice a) : Where’s “–1” ? # encirclement N = _______ Z = P + N = _______ Make sense? _______ Incorrect

25. Choice b) : Where is“–1” ? # encir.N = _____ Z = P + N= _______ closed-loopstability _______ Correct

26. Note: If G(jω) is along –Re axis to ∞ as ω→0+, it means G(s) has in it. when s makes a half circle near ω = 0, G(s) makes a full circle near ∞. choice a) is impossible,but choice b) is possible.

27. Incorrect

28. Example: G(s) stable, P = 0 • Get conjugatefor ω < 0 • Connect ω = 0–to ω = 0+.Needs to goone full circlewith radius ∞.Two choices.

29. Choice a) : N = 0 Z = P + N = 0 closed-loopstable Incorrect!

30. Choice b) : N = 2 Z = P + N= 2 Closedloop has two unstable poles Correct!

31. Which way is correct? For stable & non-minimum phase systems,

32. Example: G(s) has one unstable pole • P = 1, no unstable zeros • Get conjugate • Connectω = 0–to ω = 0+.How?One unstablepole/zeroIf connect in c.c.w.

33. # encirclement N = ? If “–1” is to the left of A i.e. A > –1 then N = 0 Z = P + N = 1 + 0 = 1 but if a gain is increased, “–1” could be inside, N = –2 Z = P + N = –1 c.c.w. is impossible

34. If connect c.w.: For A > –1N = ______ Z = P + N = ______ For A < –1N = ______ Z = ______ No contradiction. This is the correct way.

35. Example: G(s) stable, minimum phase P = 0 G(jω) as given: get conjugate. Connect ω = 0– to ω = 0+,

36. If A < –1 < 0 :N = ______Z = P + N = ______ stability of c.l. : ______ If B < –1 < A : A=-0.2, B=-4, C=-20N = ______Z = P + N = ______ closed-loop stability: ______ Gain margin: gain can be varied between (-1)/(-0.2) and (-1)/(-4), or can be less than (-1)/(-20)

37. If C < –1 < B :N = ______Z = P + N = ______ closed-loop stability: ______ If –1 < C :N = ______Z = P + N = ______ closed-loop stability: ______