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 = ?)?