Stability from Nyquist plot

1
Stability from Nyquist plot
G(s)

• Get completeNyquist 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

• Margins on Bode plots
• In most cases, stability of this closed-loop
• can be determined from the Bode plot of G
• Phase margin gt 0
• Gain margin gt 0

G(s)
4
(No Transcript)
5
(No Transcript)
6
Margins on Nyquist plot

• Suppose
• Draw Nyquist plot G(j?) 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
Asymptotic straight line
Type 1 magnitude plot becomes -20 dB/dec as w ?
0 phase plot becomes -90 deg as w ?
0 Kp 8, Ka 0 Kv height of asymptotic line
at w 1 w at which asymptotic line
crosses 0 dB horizontal line
10
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 8, Kv 8 Ka height of asymptotic line
at w 1 w2 at which asymptotic line
crosses 0 dB horizontal line
11
(No Transcript)
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
z0.1
0.2
0.3
No resonance for z lt 0.7 Mr1dB for
z0.6 Mr3dB for z0.5 Mr7dB for z0.4
13
0.2
z0.1
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 100z
z0.1
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 lt 0.6
• When z gt 0.7, no resonance
• z determines phase margin and Mp
• z 0.4 0.5 0.6 0.7
• PM 44 53 61 67 deg 100z
• Mp 25 16 10 5

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
18
(No Transcript)
19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
(No Transcript)