Solution_to_problem_3.9

# Solution_to_problem_3.9 - Solution to problem 3.9 a As...

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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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Solution_to_problem_3.9 - Solution to problem 3.9 a As...

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