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. 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
Likewise, in discrete time
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.
Suppose for convenience that on , and let . Let , and choose so that for all . Since , for every we have
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
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,
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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 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:
- 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 .
- http://wikis.controltheorypro.com/index.php?title=Final_Value_Theorem[permanent dead link]
- 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
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