1. Stability from Nyquist plot
G(s)
Get complete Nyquist plot
Obtain the # of encirclement of “-1”
# (unstable poles of closed-loop) Z = # (unstable poles of open-loop) P + # encirclement N
To have closed-loop stable: need Z = 0, i.e. N = –P

2. Here we are counting only poles with positive real part as “unstable poles”
jw-axis poles are excluded
Completing the NP when there are jw-axis poles in the open-loop TF G(s):
If jwo is a non-repeated pole, NP sweeps 180 degrees in clock-wise direction as w goes from wo- to wo+.
If jwo is a double pole, NP sweeps 360 degrees in clock-wise direction as w goes from wo- to wo+.

3. In most cases, stability of this closed-loop
Margins on Bode plots
In most cases, stability of this closed-loop
can be determined from the Bode plot of G:
Phase margin > 0
Gain margin > 0
G(s)

6. Margins on Nyquist plot
Suppose:
Draw Nyquist plot G(jω) & unit circle
They intersect at point A
Nyquist plot cross neg. real axis at –k

7. System type, steady state tracking
C(s)
Gp(s)

8. 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

9. Type 1: magnitude plot becomes -20 dB/dec as w  0
Asymptotic straight line
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

10. Type 2: magnitude plot becomes -40 dB/dec as w  0
Asymptotic straight line Ka
Sqrt(Ka)
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
= w2 at which asymptotic line crosses 0 dB horizontal line

12. Prototype 2nd order system frequency response
For small zeta,
resonance freq
is about wn
BW ranges from
0.5wn to 1.5wn
For good z range,
BW is 0.8 to 1.1 wn
So take BW = wn
z=0.1
0.2
0.3
No resonance
for z <= 0.7
Mr=1dB for z=0.6
Mr=3dB for z=0.5
Mr=7dB for z=0.4

13. 0.2
z=0.1
0.3
0.4
wgc
In the range of good zeta,
wgc is about 0.65 times to 0.8 times wn
w/wn

14. In the range of good zeta,
PM is about 100*z
z=0.1
0.2
0.3
0.4
w/wn

15. Important relationships
Prototype wn, open-loop wgc, closed-loop BW are all very close to each other
When there is visible resonance peak, it is located near or just below wn,
This happens when z <= 0.6
When z >= 0.7, no resonance
z determines phase margin and Mp: z
PM deg ≈100z
Mp %

16. Important relationships
wgc determines wn and bandwidth
As wgc ↑, ts, td, tr, tp, etc ↓
Low frequency gain determines steady state tracking:
L.F. magnitude plot slope/(-20dB/dec) = type
L.F. asymptotic line evaluated at w = 1: the value gives Kp, Kv, or Ka, depending on type
High frequency gain determines noise immunity

17. Desired Bode plot shape