Stability from Nyquist plot. The complete Nyquist plot: Plot G ( j ω ) for ω = 0 + to + ∞ Get complex conjugate of plot, that’s G ( j ω ) for ω = 0 – to – ∞ If G(s) has pole on j ω -axis, treat separately Mark direction of ω increasing Locate point: –1.
The completeNyquist plot:
# (unstable poles of closed-loop)Z= # (unstable poles of open-loop)P+ # encirclementNor: Z = P + NTo have closed-loop stable:need Z = 0, i.e. N = –P
from ω = –∞ to 0–,
to 0+, to +∞, you go
around “–1” c.c.w.
# encirclement N = – 1.
# unstable pole P = 1
# encirclement N = _____
# open-loop unstable poles P = _____
Z = P + N = ________
= # closed-loop unstable poles.
closed-loop stability: _______
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? ______
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 ∞?
Where’s “–1” ?
# encirclement N = _______
Z = P + N = _______
Make sense? _______
Where is“–1” ?
# encir.N = _____
Z = P + N= _______
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.
N = 0
Z = P + N = 0
N = 2
Z = P + N= 2
Closedloop has two unstable poles
For stable & non-minimum phase systems,
# encirclement N = ?
If “–1” is to the left of A
i.e.A > –1
thenN = 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
For A > –1N = ______
Z = P + N
For A < –1N = ______
Z = ______
No contradiction. This is the correct way.
Example: G(s) stable, minimum phase
P = 0
G(jω) as given:
Connect ω = 0–
to ω = 0+,
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)
If C < –1 < B :N = ______Z = P + N = ______closed-loop stability: ______
If –1 < C :N = ______Z = P + N = ______closed-loop stability: ______