Final value theorem

Final value theorem
In mathematical analysis, the final value theorem (FVT) 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 (unilateral) Laplace transform , then a final value theorem establishes conditions under which
Likewise, if in discrete time has (unilateral) Z-transform , then a final value theorem establishes conditions under which
An Abelian final value theorem makes assumptions about the time-domain behavior of (or ) to calculate .
Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of to calculate (or )
(see Abelian and Tauberian theorems for integral transforms).
Final value theorems for the Laplace transform
Deducing
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:
as , and .
as , and as .
as , and as .
In particular, if is a multiple pole of then case 2 or 3 applies (or ).
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
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:
is continuously differentiable and both and have a Laplace transform
is absolutely integrable - that is, is finite
exists and is finite
Then
.
Remark
The proof uses the dominated convergence theorem.
Final Value Theorem for the mean of a function
Let be a continuous and bounded function such that such that the following limit exists
Then .
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:
is infinitely differentiable at zero
has a Laplace transform for all non-negative integers
diverges to infinity as
Then diverges to infinity as .
Final Value Theorem for improperly integrable functions (Abel's theorem for integrals)
Let be measurable and such that the (possibly improper) integral converges for . Then
This is a version of Abel's theorem.
To see this, notice that and apply the final value theorem to after an integration by parts: For ,
By the final value theorem, the left-hand side converges to for .
To establish the convergence of the improper integral in practice, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.
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
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
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 (in this special case) 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:
All non-zero roots of the denominator of must have negative real parts.
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
Final Value Theorem
If exists and exists then .
Final value of linear systems
Continuous-time LTI systems
Final value of the system
in response to a step input with amplitude is:
Sampled-data systems
The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times is the discrete-time system
where and
,
The final value of this system in response to a step input with amplitude is the same as the final value of its original continuous-time system.
See also
Initial value theorem
Z-transform
Laplace Transform
Abelian and Tauberian theorems
Notes
External links
https://web.archive.org/web/20101225034508/http://wikis.controltheorypro.com/index.php?title=Final_Value_Theorem
http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html : final value for Laplace
https://web.archive.org/web/20110719222313/http://www.engr.iupui.edu/~skoskie/ECE595s7/handouts/fvt_proof.pdf: final value proof for Z-transforms
Theorems in Fourier analysis
it:Teorema del valore iniziale