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
lead section of
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
