Hardware Reference
In-Depth Information
Example 2: Consider the system shown in Figure 3.37. Suppose the
plant is a double integrator i.e. G
p
(s)=
s
2
and the controller is an ideal PD
controller G
c
(s)=K
p
(1 + T
d
s). The open-loop transfer function is
G
c
(s)G
p
(s)=
1
s
2
K
p
(1 + T
d
s).
(3.64)
And the closed-loop transfer function from r to y:
K
p
(1 + T
d
s)
s
2
1+K
p
(1 + T
d
s)
s
2
G
c
(s)G
p
(s)
G
c
(s)G
p
(s)+1
=
K
p
T
d
s + K
p
s
2
+ K
p
T
d
s + K
p
.
=
(3.65)
p
The n
atu
ral frequency of the closed loop is ω =
K
p
and damping ratio is
√
K
p
T
d
2
.Figure3.38showsthestepresponseofthisclosedloop.
We can use input shaping controller I
s
to improve the performance during
step response. Let
ζ =
I
s
=
s
2
+ K
p
T
d
s + K
p
K
p
T
d
s + K
p
1
T
s
s +1
,
(3.66)
where T
s
is a desirable time constant, the command input response from r
1
T
s
s+1
. T
s
can be selected according the
the limitations on control current. No change has been made in the parameters
of the feedback controller K
p
and T
d
to achieve the desirable step response of
a first order system.
Step responses of
(r = I
s
r)toy through r now becomes
K
p
(1+T
d
s)
s
2
+K
p
(1+T
d
s)
for K
p
=1withdifferent values of T
d
are
showninFigure3.38.WhenT
d
increases, overshoot decreases but can not be
eliminated even if the damping ratio of the closed-loop system is above 1. How-
ever, when the input shaper I
s
=
s
2
+0.5s+1
0.05s
2
+0.6s+1
, designed using equation 3.66
with T
s
=0.1andT
d
=0.5, is used, step response shows rapid change with-
out overshoot regardless of the feedback loop's overshoot and damping. This
example also illustrates the fact that the closed-loop step response is affected
not only by the closed-loop poles but also by the zeros.
It is easy to verify that the closed-loop transfer function from a to y is:
G
p
(s)
G
p
(s)G
c
(s)+1
=
1
s
2
+ K
p
T
d
s + K
p
.
(3.67)
√
K
p
T
d
2
One can easily select T
d
such that ζ =
> 1 hence no overshoot in step
response from a to y.
Comparison between equations 3.65 and 3.67 reveals that even if the open-
loop transfer functions are the same, the closed-loop behaviours may be differ-
ent because different transmission zeros. It is well known that state feedback
control can arbitrarily place the closed-loop poles for controllable plants but
