EEL4657  Dr. Haniph Latchman
Chapter 4: Example Problems
1.
The transfer function for a system with unity negative feedback is given
by:
G
(
s
) =
1
(
s
+ 1)(
s
+ 3)
a) Find the closedloop transfer function
T
(
s
) =
Y
(
s
)
R
(
s
)
.
b) Find the time response
y
(
t
) for a step input of
r
(
t
) =
A.
c) Use the final value theorem to find the steadystate value of y(t).
d) Find the steadystate error.
Solution
a) We can use the formula for the closedloop transfer function given in sec
tion 4.4.1 in the book.
T
(
s
) =
G
(
s
)
1 +
G
(
s
)
=
1
(
s
+1)(
s
+3)
1 +
1
(
s
+1)(
s
+3)
=
1
(
s
+ 1)(
s
+ 3) + 1
=
1
s
2
+ 4
s
+ 3 + 1
=
1
s
2
+ 4
s
+ 4
T
(
s
) =
1
(
s
+ 2)
2
1
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b) The output
Y
(
s
) is given by
Y
(
s
) =
R
(
s
)
T
(
s
)
and we use the fact that the Laplace transform of the step input
r
(
t
) =
A
is
R
(
s
) =
A
s
.
Y
(
s
) =
R
(
s
)
T
(
s
)
=
A
·
1
s
·
1
(
s
+ 2)
2
Y
(
s
) =
A
1
s
(
s
+ 2)
2
Use partial fractions method to transform
Y
(
s
), we can remove the term
A
for now, realizing that it is a constant and we can include it later in the final
equation
Y
(
s
) =
1
s
(
s
+ 2)
2
=
a
s
+
b
s
+ 2
+
c
(
s
+ 2)
2
by multiplying both sides by
s
(
s
+ 2)
2
and combining the like terms, we can
solve for
a
,
b
, and
c
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 Spring '08
 Staff
 Laplace, closedloop transfer function

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