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3.9 – 1
Solution to problem 3.9
a) As explained in section 3.4.1 in Seborg
et al
. 2004, the Final Value Theorem can only be
applied if the
[ ]
lim
( )
s
p
sX s
→
exists for all
p
with
Re( )
0
p
≥
. This limit does not exist when
i
p
cancels the denominator of a fraction, or in other words, when
i
p
is a root of the polynomial
in the denominator (
i
p
is a pole
1
). Therefore, we need to check whether the poles of
( )
sX s
have positive or zero real parts, in which case the Final Value Theorem would not be
applicable.
( )
( )( )
( )
( )( )( )
2
6
2
6
2
( )
4
5
4
9
20
4
s
s
s X s
s
s
s
s
s
s
s
s
+
+
⋅
= ⋅
= ⋅
+
+
+
+
+
+
The poles of
( )
sX s
are
1
2
4
p
p
=
= 
and
3
5
p
= 
. Note that the
[ ]
lim
( )
s
p
sX s
→
does not exist
only when
p
is equal to 4 or 5. For any other value of
p
this limit exists (and in particular for
any value of
p
with
Re( )
0
p
≥
), therefore the necessary condition for the validity of this
theorem is satisfied for this example.
Using the initial value theorem we obtain:
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 Spring '09
 RafiqulGani

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