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1st order compensators
1) Phase lead element
901
1arcsinmax
1Tmax
controller-PD:01,0usually
1,11
atleadphaseimum
TsTs
sC
G
Bode plot of lead compensators
Other representation: T
m
m
s
s
sGz
h
zh
zR
1,1,
1
1
Maximum phase lead at mh z
Effect: d can be increased because phase shift is reduced
G0 unchanged for zT
1
usually the maximal phase lead is used at d
Caution: gain has to be adjusteddecay of the gain plot is
reduced
sensor noise is amplified esp. if is small use in a region where
the decay is 40 dB/decadeproblematic for non-minimum phase
plants
z then has to be made large small effect at d
When to use a lead compensator:
stabilization
improvement of speed of response and/or damping
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2) Gain reducing compensator (phase lag)
0
5
10
15
20
10-2
10-1
100
101
-60
-50
-40
-30
-20
-10
0
Phase/Grad
= 0.10
Betrag/dB
T
= 0.50
= 0.17 = 0.25
Bode plot of lag compensators
=Tatphasetheofminimum
1,1(s)G R sTsT
Other form:
T
msG
nss
ms
s
R mn
ns1
,1
,1
1
PI-Controller: sTsGR 11 (45 phase shift at 1
T)
Effect: - Increase the gain at low frequencies
- better tracking/ better disturbance rejection at low
frequencies
- additional phase lag up to 10/T
thus reduced phase margin unless d T 10 /
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Higher order compensators
- multiple lag elements not convenient, phase too negative
for
small , creeping of the response
- multiple lead elements necessary for large phase lead
(>90)
problems with sensor noise
(approximate multiple differentiation)
- lead-lag element (real PID) standard compensator
sTsT
sTsTKsG RC
2
2
1
1
11
11
PID: DVV
D
I
RC TTsT
sT
sTKsG
,
1
11
0
-90
j (log)
(log)
Zeros of the real PID-controller:
DVInn
DVnn
TTTTT
TTTT
21
21
Often it is reasonable to compensate the smallest pole of the
plant by a
controller zero. Using a lead-lag-compensator, both the
bandwidth and
the damping and the behaviour at low frequencies can be
improved
independently.
