- Final Value Theorem Definition: The Final Value Theorem predicts the ultimate behavior of a function as time approaches infinity, using its Laplace Transform.
- Laplace Transform Basics: Laplace Transform converts time-domain functions into the s-domain, simplifying the analysis of systems over time.
- Proof Techniques: The theorem’s proof involves conditions on the location of poles and the behavior of the transform as s approaches zero.
- Application Examples: Practical examples in the article show how to apply the theorem to determine the behavior of functions at infinity.
- Initial Value Theorem: This theorem complements the Final Value Theorem by providing insights into the behavior of functions at the initial time point.
In network and control analysis, the steady value of a signal may matter more than its complete transient. The final value theorem can obtain a finite limit of f(t) from its Laplace Transform F(s) without first finding the full inverse transform. The theorem applies only after its existence and pole conditions are checked.
The limit being calculated
Given F(s), the required final value is f(t) as t approaches infinity, not F(∞). The final value theorem evaluates sF(s) as s approaches zero. The complementary initial value theorem uses a different limit and has different conditions.
Definition of Final Value Theorem of Laplace Transform
For the standard rational control-system form, a sufficient condition is that every pole of sF(s), after cancellation, lies in the open left half-plane. Equivalently, F(s) may have at most one simple pole at the origin, while all other poles lie strictly in the left half-plane. Then the finite final value is given below.
Proof of Final Value Theorem of Laplace Transform
Start with the one-sided Laplace differentiation property:
Initial-time convention
The value 0– accounts for any impulse or jump contribution at t = 0 under the one-sided convention.
As s approaches zero through values in the transform’s region of convergence, e-st approaches 1. Under the theorem’s hypotheses, the transient modes decay and the limiting relation becomes
Check these conditions before evaluating the limit:
- Confirm that F(s) represents the signal of interest and that the required one-sided transforms exist.
- Factor the reduced sF(s) and confirm that every remaining pole lies in the open left half-plane.
A right-half-plane pole produces a growing mode. [Example 3]
Nonzero conjugate poles on the imaginary axis produce sustained oscillation. [Example 4]
A pole remaining at the origin in sF(s) usually represents an unbounded polynomial term such as a ramp. [Example 5]
- Only after the pole test passes, evaluate
Examples of Final Value Theorem of Laplace Transform
For each F(s), inspect the poles of the reduced sF(s) before calculating any final value.
Answer
Answer
Note
When the pole condition passes, the theorem can give the final value even if a full inverse Laplace transform would require more algebra.
Answer
Pole check
Examples 1 and 2 satisfy the condition. In Example 3, the reduced sF(s) has a right-half-plane pole because its denominator has a positive real root.
The growing mode means the finite Final Value Theorem cannot be used.
Answer
Pole check
In Example 4, sF(s) has poles at +2i and -2i on the imaginary axis.
The corresponding sustained oscillation has no final value, so the Final Value Theorem cannot be used.
Answer
Pole check
In Example 5, sF(s) retains a pole at the origin after cancellation.
The finite Final Value Theorem therefore does not apply.
Final check
Do not replace the pole test with a check that lim sF(s) is bounded near zero. For an oscillating signal, that s-domain limit can exist even though f(t) has no final value. Inspect every pole of the reduced sF(s) before using the Final Value Theorem.
