Digital Control Systems
Final Exam (Part 2)
Instruction
The e
xam in
this part contains
3 problems,
each of which has 15 points.
Do only 2 problems.
All the problems are solvable by hand computation. However,
some problems can also be solved
by MATLAB.
If you solve a
problem using MATLAB
,
you are required to list all MATLAB commands used in
the problem, and to attach
all
numerical and
/or
grap
hical
results obtain
ed
from MATLAB
in
your solution sheets.
Problem 1
We are given the continuous time plant
10( 1)
( ).
5
s
G s
s
First,
suppose that the plant
( )
G s
is controlled by a
gain
K
(
both negative and positive
)
in
the closed

loop system below
:
a)
Let
max
K
and
min
K
be
t
he maximum
and minimum
possible value
s
, respectively,
of the
gain
K
for
which the above system is stable. Compute
min
K
and
max
K
. (4 points)
Now consider t
he digital control system below:
b)
Let
max
K
and
min
K
be
t
he maximum
and minimum
possible value
s
, respectively,
of the
gain
K
for
which the digital control
sy
stem
is stable.
Express
min
K
and
max
K
in terms of the sampling period
T
.
(5 points)
c)
C
alculate
min
K
and
max
K
in p
roblem 1b)
when
0.1.
T s
(2 points)
Now consid
er the closed

loop system below. Note that there is one sample delay in the feedback path
.
The sampling period is assumed to be 0.1
sec.
d)
Let
max
K
and
min
K
be
t
he maximum
and minimum
possible value
s
, respectively,
of the
gain
K
for
w
hich the above
sy
stem
is stable.
Calculate
min
K
and
max
K
, and then
discuss the differences betwee
n
this answer and
those of p
roblems
1a) and 1c). (4 points)
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