System type, steady state tracking. C(s). G p (s). Type 0: magnitude plot becomes flat as w 0 phase plot becomes 0 deg as w 0 Kv = 0, Ka = 0 Kp = flat magnitude height near w 0. Asymptotic straight line, -20dB/ dec. Kv in dB. Kv in Value.
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phase plot becomes 0 deg as w 0
Kv = 0, Ka = 0
Kp = flat magnitude height near w 0
Kv in dB
Kv in Value
Type 1: magnitude plot becomes -20 dB/dec as w 0
phase plot becomes -90 deg as w 0
Kp = ∞, Ka = 0
Kv = height of asymptotic line at w = 1
= w at which asymptotic line crosses 0 dB horizontal line
Ka in dB
Sqrt(Ka) in Value
Type 2: magnitude plot becomes -40 dB/dec as w 0
phase plot becomes -180 deg as w 0
Kp = ∞, Kv = ∞
Ka = height of asymptotic line at w = 1
= w2at which asymptotic line crosses 0 dB horizontal line
In most cases, stability of this closed-loop
can be determined from the Bode plot of G:
To have closed-loop stable: need Z = 0, i.e. N = –P
Here we are counting only poles with positive real part as “unstable poles”
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,
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
For A > –1N = ______
Z = P + N
For A < –1N = ______
Z = ______
No contradiction. This is the correct way.
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: ______