Stability from Nyquist plot

1
Stability from Nyquist plot

• The completeNyquist plot
• Plot G(j?) for ? 0 to 8
• Get complex conjugate of plot, thats G(j?) for
  ? 0 to 8
• If G(s) has pole on j?-axis, treat separately
• Mark direction of ? increasing
• Locate point 1

2
(No Transcript)
3
Encirclement of the -1 point

• As you follow along the G(j?) 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

4
(No Transcript)
5
(No Transcript)
6
(No Transcript)
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(j?) needs to encircle the 1 point
  counter clock wise P times.
• If open loop is stable to begin with, G(j?)
  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 ? 8 to 0, to 0, to
8, 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(j?) for? 0 to 8
• Use conjugate to get G(j?) for? 8 to 0
• How to go from ? 0 to ? 0? At ? 0

13
(No Transcript)
14
(No Transcript)
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(j?) is on next page
• Q 1. Find stability margins
• 2. Find Nyquist criterion to determine
  closed-loop stability

17
(No Transcript)
18

• Solution
• Where does G(j?) cross the unit circle?
  ________ Phase margin ________Where does
  G(j?) 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(j?). e.g. If K 2, scale G(j?) 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? Whats the max
PM as K is reduced to 0 and GM becomes 8?
21

• To use Nyquist criterion, need complete Nyquist
  plot.
• Get complex conjugate
• Connect ? 0 to ? 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(j?) for ? gt 0 as given.
• Get G(j?) for? lt 0 by conjugate
• Connect ? 0 to ? 0.But how?

24
Choice a) Wheres 1 ? encirclement N
_______ Z P N _______ Make sense? _______
Incorrect
25
Choice b) Where is1 ? encir.N _____ Z
P N _______ closed-loopstability _______
Correct
26
Note If G(j?) is along Re axis to 8 as ??0,
it means G(s) has in it. when s makes a half
circle near ? 0, G(s) makes a full circle near
8. choice a) is impossible,but choice b) is
possible.
27
Incorrect
28

• Example G(s) stable, P 0
• Get conjugatefor ? lt 0
• Connect ? 0to ? 0.Needs to goone full
  circlewith radius 8.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
• Connect? 0to ? 0.How?One
  unstablepole/zeroIf connect in c.c.w.

33
encirclement N ? If 1 is to the left of
A i.e. A gt 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 gt 1N ______ Z P
N ______ For A lt 1N ______ Z ______ No
contradiction. This is the correct way.
35
Example G(s) stable, minimum phase P
0 G(j?) as given get conjugate. Connect ?
0 to ? 0,
36
If A lt 1 lt 0 N ______ Z P N
______ stability of c.l. ______ If B lt 1 lt A
A-0.2, B-4, C-20 N ______ 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 lt 1 lt B N ______ Z P N
______ closed-loop stability ______ If 1 lt C
N ______ Z P N ______ closed-loop
stability ______