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MAE143B Homework Set 2  Solutions
Question 1
FPEN Question 3.33: Sketch the step response of a system with transfer function
G
(
s
) =
s/
2 + 1
(
s/
40 + 1)[(
s/
4)
2
+
s/
4 + 1]
.
Justify your answer on the basis of locations of poles and zeros. (Do not ﬁnd the inverse Laplace
transform. [Although feel free to compute approximate residue values in your answer.]) To assist
you, I have plotted the response using matlab. Describe as much as you can about the curve and
the evidence for this from the poles and zero positions.
Figure 1: Step response for Question 1.
The system has three poles; one at
s
=

40
and the others as a complex conjugate pair at
s
=

2
.
0000
±
3
.
4641
j.
But I choose to leave these poles described by
s
2
+ 2
ζω
n
s
+
ω
2
n
,
with undamped
natural frequency
ω
n
= 4
and damping ratio
ζ
= 1
/
2
.
The system has a zero at
s
=

2
. The point of
this question is that the pole at
s
= 40
can effectively be neglected in the step response, since with a
step input the residue at
s
= 40
will be roughly
20
/
(100
×
40)
, which is pretty small. It is clear from the
way the transfer function is factored that the steady state response to a step will be one. Thus, the
system is dominated by the complex conjugate pair of poles.
We can appeal to the approximate expressions for rise time (
τ
r
≈
1
.
8
ω
n
= 0
.
45
s), overshoot
M
p
=
e

πζ/
√
1

ζ
2
= 0
.
16
or 16 percent, peak time
t
p
=
π/ω
d
=
π/
[
ω
n
p
1

ζ
2
] = 0
.
9069
s, settling time
t
s
= 4
.
6
/
(
ζω
n
) = 2
.
3
s.
This is, of course, all approximate and fails to take account of the role of the zero in the calculation.
Hence, the overshoot is poorly estimated. The peak time is a little better and the setting time is very
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 Fall '09
 CALLAFON

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