Final value theorem

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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. A final value theorem allows the time domain behavior to be directly calculated by taking a limit of a frequency domain expression, as opposed to converting to a time domain expression and taking its limit.

Mathematically, if is bounded on and

has a finite limit, then

where is the (unilateral) Laplace transform of .[1][2]

Likewise, in discrete time

where is the (unilateral) Z-transform of .[2]

Proofs[edit]

The first proof below has the virtue of being totally elementary and self-contained. If one is willing to use not-quite-so-elementary convergence theorems one can give a one-line proof using the Dominated Convergence Theorem.

Elementary proof[edit]

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.

Proof using the dominated convergence theorem[edit]

Let . A change of variable in the integral defining shows that

Since is bounded, there exists such that is dominated by the integrable function

Thus, by the Dominated Convergence Theorem,

Examples[edit]

Example where FVT holds[edit]

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[edit]

For a system described by the transfer function

the final value theorem appears to predict the final value of the impulse response to be 1 and the final value of the step response to be 0. 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. 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 .

See also[edit]

Notes[edit]

  1. ^ Wang, Ruye (2010-02-17). "Initial and Final Value Theorems". Retrieved 2011-10-21. 
  2. ^ a b Alan V. Oppenheim; Alan S. Willsky; S. Hamid Nawab (1997). Signals & Systems. New Jersey, USA: Prentice Hall. ISBN 0-13-814757-4. 

External links[edit]