stability from nyquist plot
DESCRIPTION
Stability from Nyquist plot. Plot G ( j ω ) for ω > 0 Flip about real axis to get plot of G ( j ω ) for ω < 0 If G(s) has pole(s) on j ω -axis, make a small detour around these pole(s), so that G ( j ω ) is connected Count the #encirclement of –1 by G ( j ω ) - PowerPoint PPT PresentationTRANSCRIPT
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Stability from Nyquist plot
• Plot G(jω) for ω > 0• Flip about real axis to get plot of G(jω) for
ω < 0• If G(s) has pole(s) on jω-axis, make a
small detour around these pole(s), so that G(jω) is connected
• Count the #encirclement of –1 by G(jω)• Nyquist criterion: Z=P+N
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Example:
G(s) stable, P = 0
G(jω) for ω > 0 as given.
1. Get G(jω) forω < 0 by conjugating
2. Connect ω = 0– to ω = 0+.But how?
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Choice a) :
Where’s “–1” ?
# encirclement N = _______
Z = P + N = _______
Make sense? _______
Incorrect
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Choice b) :
Where is“–1” ?
# encir.N = _____
Z = P + N= _______
closed-loopstability _______
Correct
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Incorrect
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Example: G(s) stable, P = 0
1. Get conjugatefor ω < 0
2. Connect ω = 0–
to ω = 0+.
Needs to goone full circlewith radius ∞.
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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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Example: G(s) has one unstable pole
P = 1, no unstable zeros
1. Get conjugate
2. Connectω = 0–
to ω = 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 > –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 > –1N = ______
Z = P + N
= ______
For A < –1N = ______
Z = ______
No contradiction. This is the correct way.
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Example: G(s) stable, minimum phase
P = 0
G(jω) as given:
get conjugate.
Connect ω = 0–
to ω = 0+,00 Kdirection c.w.
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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)
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If C < –1 < B :N = ______Z = P + N = ______closed-loop stability: ______
If –1 < C :N = ______Z = P + N = ______closed-loop stability: ______
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G(s)
Open vs Closed Loop Frequency Response And Frequency Domain Specifications
C(s)
Goal: 1) Define typical “good” frequency response shape for closed-loop 2) Relate closed-loop freq response shape to step response shape 3) Relate closed-loop freq shape to open-loop freq response shape 4) Design C(s) to make C(s)G(s) into “good” shape.
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Mr and BW are widely used
Closed-loop phase resp. rarely used
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10-0.6
10-0.4
10-0.2
100
100.2
-15
-10
-5
0
5
10
15 =0.1
=0.2
=0.3
=0.4
=0.1
0.2
0.3
No resonancefor <= 0.7
Mr=0.3dB for =0.6
Mr=1.2dB for =0.5
Mr=2.6dB for =0.4
For small zeta,resonance freqis about n
BW ranges from0.5n to 1.5n
For good range,BW is 0.8~1.25 n
So take BW ≈ n
Prototype 2nd order system closed-loop frequency response
n
-3dB
BW
1.580.63
0.79 1.26
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Closed-loop BW to n ratio
BW≈n
BW/n
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100
-20
-15
-10
-5
0
5
10
15
=0.1
0.2
0.3
No resonancefor <= 0.7
Mr<0.5 dB for =0.6
Mr=1.2 dB for =0.5
Mr=2.6 dB for =0.4
When >=0.6no visible resonance peak
Prototype 2nd order system closed-loop frequency response
n
When <=0.5 visible resonance peak near n
n
Since we design for >=0.5, Mr and r are of less value
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0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750
0.5
1
1.5
2
2.5
3
Mr in value
Mr in dB
Prototype 2nd order system closed-loop frequency response
Mr vs
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0.4 0.45 0.5 0.55 0.6 0.650
0.5
1
1.5
2
2.5
3
Mr in dB
20.7
50.4
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0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.750
5
10
15
20
25
30
Percentage Overshoot in closed-loop step response
> 0.5 is good
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1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.40
5
10
15
20
25
30
Mr
Percentage Overshoot in closed-loop step response
Mr < 15% is good,
>40% not tolerable
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0 0.5 1 1.5 2 2.5 30
5
10
15
20
25
30
Percentage Overshoot in closed-loop step response
Mr in dB
Mr < 1 dB is good, >3 dB not tolerable
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10-2
10-1
100
101
102
-200
-150
-100
-50
0
50
100
150
n
=0.1 0.2 0.3
0.4
gc
In the range of good zeta,gc is about 0.7 times n
Open loop frequency response
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Open-loop gc to n ratio
gc≈0.7n
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10-2
10-1
100
101
102
-180
-170
-160
-150
-140
-130
-120
-110
-100
-90
n
=0.10.2
0.3
0.4
In the range of good zeta,PM is about 100*
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0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.930
35
40
45
50
55
60
65
70
Phase Margin
PM = 100
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30 35 40 45 50 55 60 65 700
5
10
15
20
25
30
35
40
Phase Margin in degrees: PM in deg
Percentage Overshoot in closed-loop step response
PM+Mp =70 line
Pe
rce
nta
ge
ove
rsh
oot:
Mp
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Important relationships• Closed-loop BW are very close to n
• Open-loop gain cross over gc ≈ (0.65~0.8)*
n,
• When <= 0.6, r and n are close
• When >= 0.7, no resonance• determines phase margin and Mp:
0.4 0.5 0.6 0.7
PM 44 53 61 67 deg ≈100 Mp 25 16 10 5 %
PM+Mp ≈70
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Mid frequency requirements• gc is critically important
– It is approximately equal to closed-loop BW
– It is approximately equal ton • Hence it determines tr, td directly
• PM at gc controls – Mp 70 – PM
• PM and gc together controls and d
– Determines ts, tp
• Need gc at the right frequency, and need sufficient PM at gc
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Low frequency requirements• Low freq gain slope and/or phase
determines system type
• Height of at low frequency determine error constants Kp, Kv, Ka
• Which in turn determine ess
• Need low frequency gain plot to have sufficient slope and sufficient height
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High frequency requirements• Noise is always present in any system
• Noise is rich in high frequency contents
• To have better noise immunity, high frequency gain of system must be low
• Need loop gain plot to have sufficient slope and sufficiently small value at high frequency
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Desired Bode plot shape
Ess requirement
Noise requirement
0
-90
-180
0dB
gcd
High low-freq-gain for steady state trackingLow high-freq-gain for noise attenuationSufficient PM near gc for stability
PMd
Mid frequency
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C(s) Gp(s)
21
21)(psps
zszsKsC
Controller design with Bode
From specs: => desired Bode shape of Gol(s)Make Bode plot of Gp(s) Add C(s) to change Bode shape as desiredGet closed loop systemRun step response, or sinusoidal responseModify controller as needed