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Page 1 of 2
12/8/2005
(Spring 1999)
Find the total response of the system
5
( )
x
x
u t
+
=
d
,
( )
0
2
x
= 
,
for the input shown to the right.
Note that
( )
1
5
G s
s
=
+
and that the homogeneous solution is
( )
5
t
x t
Ae

=
.
Next write out an analytical form of the input function,
( ) ( ) ( )
10
5
1
5
3
s
r
r
u t
u
u t
u t
=


+

.
Alternatively note that the following inputs occur,
( )
{ }
10; 0
1,10 5
1 ;1
3,0; 3
u
t
t
t
t
=
≤ ≤


≤ ≤
≤
.
As the Laplace Transform of a unit step is 1/
s
,
for the first time region
( ) ( ) ( )
1
1
10
10
5
5
a
b
x s
G s u s
s
s
s
s
=
=
=
+
+
+
i
i
.
Solving for
a
and
b
by
multiplying both sides of the equation by the denominator of the left hand side gives
( )
1
5
a s
bs
=
+
+
.
Equating coefficients of like powers of
s
gives 1 5
a
=
and 0
a b
=
+
.
Thus,
1/5
a
=
and
1/5
b
a
=  = 
.
Therefore
( )
10 1
1
5
5
x s
s
s
=

+
and
( ) ( )
5
2 1
t
p
x
t
e

=

.
The total
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 Spring '08
 Perreira

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