System type, steady state tracking

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System type, steady state tracking
C(s)
Gp(s)
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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
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Asymptotic straight line, -20dB/dec
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 8, Ka 0 Kv height of asymptotic line
at w 1 w at which asymptotic line
crosses 0 dB horizontal line
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Asymptotic straight line, -40dB/dec
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 8, Kv 8 Ka height of asymptotic line
at w 1 w2 at which asymptotic line
crosses 0 dB horizontal line
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• 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)
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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

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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

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• 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.

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• 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

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• 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? ________

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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? ______
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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?
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• 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 _______

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Correct
Incorrect
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• 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?

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Choice a) Wheres 1 ? encirclement N
_______ Z P N _______ Make sense? _______
Incorrect
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Choice b) Where is1 ? encir.N _____ Z
P N _______ closed-loopstability _______
Correct
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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.
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Incorrect
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• Example G(s) stable, P 0
• Get conjugatefor ? lt 0
• Connect ? 0to ? 0.Needs to goone full
  circlewith radius 8.Two choices.

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Choice a) N 0 Z P N
0 closed-loopstable
Incorrect!
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Choice b) N 2 Z P N 2 Closedloop has
two unstable poles
Correct!
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Which way is correct? For stable non-minimum
phase systems,
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• 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.

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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
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If connect c.w. For A gt 1N ______ Z P
N ______ For A lt 1N ______ Z ______ No
contradiction. This is the correct way.
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Example G(s) stable, minimum phase P
0 G(j?) as given get conjugate. Connect ?
0 to ? 0,
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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)
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If C lt 1 lt B N ______ Z P N
______ closed-loop stability ______ If 1 lt C
N ______ Z P N ______ closed-loop
stability ______