EE 428
PROBLEM SET 4
DUE: 12 OCT 2007
Reading assignment: Please read chapter 4 sections 4.2 through 4.4.
Problem 14:
(20 points)
This problem considers the effects of a single finite zero on the transient response of the secondorder system
Y
(
s
)
U
(
s
)
=
(
s/αζω
n
) + 1
(
s/ω
n
)
2
+ 2
ζ
(
s/ω
n
) + 1
.
The zero is located at
s
=
−
αζω
n
.
If
α
is large, than the zero is far removed from the poles and will have little effect on the response. In fact, as
α
→ ∞
,
we have
lim
α
→∞
Y
(
s
)
U
(
s
)
=
1
(
s/ω
n
)
2
+ 2
ζ
(
s/ω
n
) + 1
.
1. (8 points) Let
ζ
= 0
.
5 and
ω
n
= 1 rad/sec. Using MATLAB, plot the unitstep response for
α
= 1
,
2
,
4
,
and
100 in a single figure.
Label the individual responses using the
legend
command .
In order to obtain the
Greek symbol
α
in the legend, use the MATLAB symbol
\alpha
. Add your name and section using
gtext
, an
include a copy of the mfile specifying the commands used to generate the plots.
2. (7 points) For each system in part 1, sketch the polezero map, and using MATLAB, determine the percent peak
overshoot, the timetopeak, risetime, and settling time. (You may wish to use the function from Problem Set
2, Problem 9). Based on these time response characteristics, summarize the effect of moving the zero towards
the imaginary axis in two or three short sentences.
3. (5 points) In part 1 the zero is always located in the lefthalf plane. Now consider the affect of an
unstable zero
,
that is, a zero located in the righthalf plane. With
ζ
= 0
.
5 and
ω
n
= 1 rad/sec, simulate the stepresponse
for
α
=
−
1 using MATLAB. In comparison to the results obtained in part 1, what is the salient feature of the
stepresponse obtained with a system that has an unstable zero?
Problem 15:
(15 points)
Consider the following second order system with an added pole
H
(
s
) =
1
(
s/p
+ 1)(
s
2
+
s
+ 1)
.
1. (8 points) Let
p
=
α/
2 and plot the unitstep response using MATLAB for
α
= 0
.
1
,
1
,
and 10 in a single figure.
Label the responses using the
legend
command and add your name and section number using
gtext
. Include
a copy of an mfile showing the MATLAB commands used to generate the plots.
2. (7 points) For each system in part 1, sketch the polezero map, and using MATLAB, determine the percent peak
overshoot, the timetopeak, risetime, and settling time. (You may wish to use the function from Problem Set
2, Problem 9). Based on these time response characteristics, summarize the effect of moving the pole towards
the imaginary axis in two or three short sentences.
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Problem 16:
(15 points) Many physical systems are represented as the cascade of a fast and slow subsystem as
shown in Figure 1. This situation often arises in electromechanical systems, such as the DC motor. In this problem
we show that the behavior of the cascade system can be adequately described by a reduced order model. Suppose
that
H
f
(
s
)
=
4
sτ
f
+ 1
H
s
(
s
)
=
0
.
5
sτ
s
+ 1
,
where the
fast
system has a time constant
τ
f
= 0
.
1 s while the
slow
system has a time constant
τ
s
= 1
.
0 s.
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 Fall '07
 SCHIANO
 Static Equilibrium, equilibrium state, linearized, static equilibrium state

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