Hardware Reference
In-Depth Information
To change the slope of -40 dB/decade of the uncompensated nominal plant
to -60 dB/decade slope for the compensated plant in the frequencies between
ω 1 and ω 2 , we need a lag compensator
1
ω 2 s +1
1
C g (s)=
ω 1 s +1 ,
(3.6)
where ω 2 is a few times higher than ω 1 . Similarly to change the slope from
-40 dB/decade to -20 dB/decade in the frequency range between ω 3 and ω 4 ,a
1
ω 3 s +1
1
C d (s)=
ω 4 s +1 ,
(3.7)
can be used, where ω 4 is a few times higher than ω 3 . The open loop 0-dB
crossover frequency f v should be somewhere in the middle of ω 3 and ω 4 .Be-
cause of 2 poles and 2 zeros in the combination of these two compensators,
it results in 0 dB/decade slope in the high frequency. The required high fre-
quency roll off -40 dB/decade at frequencies higher than ω 4 is achieved by the
actuator model itself.
The combined lag-lead compensator G c (s)iswrittenas
G c (s)=k c ( ω 2 s +1)
( ω 1 s +1)
( ω 3 s +1)
( ω 4 s +1) ,
(3.8)
¯
¯
( ω 1 s +1)
( ω 2 s +1)
( ω 4 s +1)
( ω 3 s +1)
s 2
k
k c =
,
(3.9)
s=j2πf v
whichmakestheopenlooptransferfunctionG c (s)G p (s) crossing the 0 dB line
at frequency f v with a slope of -20 dB/decade.
The design can be further simplified by assigning pre-defined relations be-
tween frequencies of different poles and zeros of the compensator, and the de-
sired cross-over frequency f v .Letω 3 = f v /3, ω 4 =3f v , ω 1 =20π,ω 2 = f v /5.2.
With these assumptions,
( f v s +1)
(
G c (s)=k c ( f v
5.2 s +1)
3f v s +1) ,
(3.10)
1
1
(
20π s +1)
¯
¯
1
1
(
3f v s +1)
( f v s +1)
(
20π s +1)
( f v
s 2
k
k c =
(3.11)
5.2 s +1)
s=j2πf v
Thisdesignresultsinanopenloopbandwidthoff v ,41 phase margin and
infinite gain margin for the
k
s 2 model. Figures 3.2-3.5 show the bode plots of
different transfer functions with this nominal controller designed for an example
system. These plots show the response for two cases - rigid body plant (solid
line) and plant with resonance (dashed line). The models of the rigid body             Search WWH ::

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