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EE 429
FINAL EXAM
6 May 1998
Name:
ID#:
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO
Problem
Weight
Score
1
25
2
25
3
25
4
25
Total
100
This test consists of four problems. Answer each problem on the exam itself; if you use additional
paper, repeat the identifying information above, and staple it to the rest of your exam when you
hand it in. The quality of your analysis and evaluation is as important as your answers. Your
reasoning must be precise and clear; your complete
English sentences should convey what you are
doing.
For your convenience, a zTransform table and ruler are provided on pages 16 and 17, respec
tively.
1
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View Full DocumentProblem 1:
(25 points)
Happy Valley Technologies, Inc. has been contracted to replace an existing analog PI control system with
a digital controller as shown in Figure 1, where
α
and
β
are the proportional and integral control gains
respectively.
+

Σ
R(z)
α+β
1
z 1
Y(z)
E(z)
z  0.5
Plant
Controller
U(z)
1
Figure 1: Sampleddata control system.
1. (5 points) Find the closedloop transfer function
Y
(
z
)
/R
(
z
) in terms of the control gains
α
and
β
.
2. (10 points) Suppose that the sample period
T
= 1 s and
β
=3
/
2. Determine the steadystate error
e
ss
due to a ramp input
r
(
k
)=
kTu
o
(
k
). Does the steadystate error depend on
α
?
3. (10 points) With
β
set to 3/2, use the Jury stability test to determine the range of
α
for which the
closedloop system is BIBO stable.
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