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