## 2.4. Initial- and Final-Value Theorems

Two theorems are now presented that can be used to find the values of the time-domain function at two extremes, t = 0 and t = ?, without having to do the inverse transform. In control, we use the final-value theorem quite often. The initial-value theorem is less useful. As we have seen from our first example in Section 2.1, the problems that we solve are defined to have exclusively zero initial conditions.

• Initial-Value Theorem:

 (2.23)
• Final-Value Theorem:

 (2.24)

The final-value theorem is valid provided that a final-value exists. The proofs of these theorems are straightforward. We will do the one for the final-value theorem. The proof of the initial-value theorem is in the Review Problems.

Consider the definition of the Laplace transform of a derivative. If we take the limit as s approaches zero, we find

If the infinite integral exists, [9] we can interchange the limit and the integration on the LHS to give

Now if we equate the RHSs of the previous two steps, we have

We arrive at the final-value theorem after we cancel the f(0) terms on both sides.

##### Example 2.1:

Consider the Laplace transform F( s) = {[6( s - 2)( s + 2)] /[ s( s + 1) ( s + 3)( s + 4)]}. What is f( t = ?)?

##### Example 2.2:

Consider the Laplace transform F

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2.5. Partial-Fraction Expansion Because we rely on a look-up table to do a reverse Laplace transform, we need the skill to reduce a complex function down to simpler parts that match our table. In...

This chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and...

This chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and...

This chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and...

This chapter begins with an introduction to the Laplace transformation, definitions, and properties of the Laplace transformation. The initial value and final value theorems are also discussed and...

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