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Wikipedia / 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. in continuous time has (unilateral) Laplace transform in discrete time has (unilateral) Z-transform

(see Abelian and Tauberian theorems for integral transforms).

Final value theorems for the Laplace transform

Deducing limt → ∞ f(t)

approaches 0 through the positive numbers.

Standard Final Value Theorem

Final Value Theorem using Laplace transform of the derivative

Remark

does not exist, but

Improved Tauberian converse Final Value Theorem

Extended Final Value Theorem

is either in the open left half-plane or at the origin. Then one of the following occurs:

Generalized Final Value Theorem

exists then

denotes the Gamma function.

Applications

have applications in establishing the long-term stability of a system.

Deducing lims → 0 s F(s)

Abelian Final Value Theorem

Elementary proof

we have

hence

we have

is small enough.

Final Value Theorem using Laplace transform of the derivative

Suppose that all of the following conditions are satisfied:

  1. have a Laplace transform

  2. is finite

  3. exists and is finite

Then

Remark

The proof uses the dominated convergence theorem.

Final Value Theorem for the mean of a function

be a continuous and bounded function such that such that the following limit exists

Final Value Theorem for asymptotic sums of periodic functions

that is

and vanishes at infinity. Then

Final Value Theorem for a function that diverges to infinity

satisfy all of the following conditions:

  1. is infinitely differentiable at zero

Final Value Theorem for improperly integrable functions (Abel's theoremAbel's theorem for integrals)

This is a version of Abel's theorem.

after an integration by parts

in practice, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.

Applications

have applications in probability and statistics to calculate the moments of a random variable be cumulative distribution function be the Laplace–Stieltjes transform as the inverse Laplace transform Then

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:

  1. must have negative real parts.

  2. must not have more than one pole at the origin.

Final value theorems for the Z transform

Deducing limk → ∞ f[k]

Final Value Theorem

Final value of linear systems

Continuous-time LTI systems

Final value of the system

is:

Sampled-data systems

is the discrete-time system

and

is the same as the final value of its original continuous-time system.

See also

Notes

  • 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

it:Teorema del valore iniziale