Final value theorem


In mathematical analysis, the final value theorem is one of several similar theorems used to relate frequency domain expressions to the time domain behavior as time approaches infinity.
Mathematically, if in continuous time has Laplace transform then a final value theorem establishes conditions under which
Likewise, if in discrete time has Z-transform then a final value theorem establishes conditions under which
An Abelian final value theorem makes assumptions about the time-domain behavior of to calculate.
Conversely a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of to calculate
.

Final value theorems for the Laplace transform

Deducing \lim_{t\to\infty}f(t)

In the following statements, the notation means that approaches 0, whereas means that approaches 0 through the positive numbers.

Standard Final Value Theorem

Suppose that every pole of is either in the open left half plane or at the origin, and that has at most a single pole at the origin. Then as, and.

Final Value Theorem using Laplace Transform of the Derivative

Suppose that and both have Laplace transforms that exist for all. If exists and exists then.
Remark
Both limits must exist for the theorem to hold. For example, if then does not exist, but

Improved Tauberian Converse Final Value Theorem

Suppose that is bounded and differentiable, and that
is also bounded on. If as then.

Extended Final Value Theorem

Suppose that every pole of is either in the open left half plane or at the origin. Then one of the following occurs:
  1. as, and.
  2. as, and as.
  3. as, and as.
In particular, if is a multiple pole of then case 2 or 3 applies.

Generalized Final Value Theorem

Suppose that is Laplace transformable. Let. If exists and exists then
where denotes the Gamma function.

Applications

Final value theorems for obtaining have applications in establishing the long-term stability of a system.

Deducing \lim_{s\,\to\, 0}{sF(s)}

Abelian Final Value Theorem

Suppose that is bounded and measurable and
. Then exists for all and.
Elementary proof
Suppose for convenience that on, and let. Let,
and choose so that for all
. Since, for every
we have
hence
Now for every we have
On the other hand, since is fixed it is clear that, and
so if is small enough.

Final Value Theorem using Laplace Transform of the Derivative

Suppose that all of the following conditions are satisfied:
  1. is continuously differentiable and both and have a Laplace Transform
  2. is absolutely integrable, that is is finite
  3. exists and is finite
Then
Remark
The proof uses the Dominated Convergence Theorem.

Final Value Theorem for Asymptotic Sums of Periodic Functions

Suppose that is continuous and absolutely integrable in. Suppose further that is asymptotically equal to a finite sum of periodic functions, that is
where is absolutely integrable in and vanishes at infinity. Then

Final Value Theorem for a function that diverges to infinity

Let and be the Laplace transform of. Suppose that satisfies all of the following conditions:
  1. is infinitely differentiable at zero
  2. has a Laplace transform for all non-negative integers
  3. diverges to infinity as
Then diverges to infinity as.

Applications

Final value theorems for obtaining have applications in probability and statistics to calculate the moments of a random variable. Let be cumulative distribution function of a continuous random variable and let be the Laplace-Stieltjes transform of. Then the -th moment of can be calculated as
The strategy is to write
where is continuous and
for each, for a function. For each, put as the inverse Laplace transform of, obtain
, and apply a final value theorem to deduce
. Then
and hence is obtained.

Examples

Example where FVT holds

For example, for a system described by transfer function
and so the impulse response converges to
That is, the system returns to zero after being disturbed by a short impulse. However, the Laplace transform of the unit step response is
and so the step response converges to
and so a zero-state system will follow an exponential rise to a final value of 3.

Example where FVT does not hold

For a system described by the transfer function
the final value theorem appears to predict the final value of the impulse response to be 0 and the final value of the step response to be 1. However, neither time-domain limit exists, and so the final value theorem predictions are not valid. In fact, both the impulse response and step response oscillate, and the final value theorem describes the average values around which the responses oscillate.
There are two checks performed in Control theory which confirm valid results for the Final Value Theorem:
  1. All non-zero roots of the denominator of must have negative real parts.
  2. must not have more than one pole at the origin.
Rule 1 was not satisfied in this example, in that the roots of the denominator are and.

Final value theorems for the Z transform

Deducing \lim_{k\to\infty}fk

Final Value Theorem

If exists and exists then.