15.3.1. Definition of Final Value Theorem (FVT)

If a function f(t) has a Laplace transform F(s), and the limit lim t→∞f(t) exists, then the Final Value Theorem states:

lim f(t) = lim sF(s)
t→∞ s→0

The Final Value Theorem (FVT) is a tool used in control systems and engineering to find the long-term behavior of a function without fully reversing the Laplace transform. It states that if 'f(t)' has a corresponding Laplace transform 'F(s)', and the limit of 'f(t)' as 't' approaches infinity exists, then it can be determined by taking the limit of 'sF(s)' as 's' approaches zero.

Here's the procedure broken down:

1. Compute the Laplace transform of the function 'f(t)', resulting in 'F(s)'.
2. Multiply 'F(s)' by 's' to form 'sF(s)'.
3. Find the limit of 'sF(s)' as 's' approaches zero to obtain the steady-state value of 'f(t)' as 't' approaches infinity.

Think of a car slowing down as it approaches a stoplight. The car's speed decreases over time and eventually settles at zero. The Final Value Theorem is like a formula that helps you predict the exact moment the car reaches rest as you observe its speed over time.

The Final Value Theorem can only be applied if:

1. All poles of sF(s) lie in the left half of the complex plane, except possibly at the origin.
2. The function f(t) converges to a finite value as t→∞.

If f(t) has oscillatory or diverging behavior, the FVT does not apply.

The application of the Final Value Theorem has specific conditions that must be met:

1. All poles of the expression 'sF(s)' must lie in the left half of the complex plane, which guarantees stability in the system being studied. The only exception is if there is a pole exactly at the origin.
2. The behavior of 'f(t)' as 't' approaches infinity must be stable, meaning it converges to a finite value instead of oscillating indefinitely or diverging, as such behaviors invalidate the use of FVT.

Imagine trying to predict the final destination of a bouncing ball. If the ball continues to bounce indefinitely or starts spinning wildly without coming to rest, you cannot accurately determine where it will settle. This is similar to the conditions in FVT—predicting a steady outcome requires that the system behaves consistently rather than chaotically.

To apply the Final Value Theorem:

1. Find F(s) – the Laplace transform of f(t).
2. Multiply by s to get sF(s).
3. Take the limit as s→0.

Applying the Final Value Theorem involves a systematic three-step process:

1. First, compute the Laplace transform of the given function 'f(t)', resulting in 'F(s)'. This converts the time-domain function into a frequency-domain representation.
2. Next, multiply the Laplace transform 'F(s)' by 's' to create 'sF(s)'. This operation is crucial for preparing to find the limit.
3. Finally, take the limit of 'sF(s)' as 's' approaches zero. This limit gives the steady-state value of 'f(t)' as time goes towards infinity, allowing us to understand the long-term behavior of the system.

Think of baking a cake using a recipe. The steps to find out how it turns out are analogous to the FVT process:

1. Finding the right ingredients ('F(s)') is like computing the Laplace transform.
2. Mixing the ingredients ('sF(s)') represents the step of preparing for the final bake.
3. Finally, watching the cake bake and evaluating its final appearance ('s→0') gives you the final product, just as the limit provides the final steady-state value.

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