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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 | |||
:<math>\lim_{t\to\infty}f(t)</math> | |||
has a finite limit, then | |||
:<math>\lim_{t\to\infty}f(t) = \lim_{s\to 0}{sF(s)}</math> | |||
where <math>F(s)</math> is the (unilateral) [[Laplace transform]] of <math>f(t)</math>.<ref name="RWang2010">{{cite web |url=http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html |title=Initial and Final Value Theorems |first=Ruye |last=Wang |date=2010-02-17 |accessdate=2011-10-21}}</ref><ref name="OppenheimWillskyNawab1997">{{cite book |isbn=0-13-814757-4 |title=Signals & Systems |author=Alan V. Oppenheim, Alan S. Willsky, S. Hamid Nawab |location=New Jersey, USA |publisher=Prentice Hall |year=1997}}</ref> | |||
== Example where FVT holds == | |||
{{Unreferenced section|date=October 2011}} | |||
For example, for a system described by [[transfer function]] | |||
:<math>H(s) = \frac{ 6 }{s + 2},</math> | |||
and so the [[impulse response]] converges to | |||
:<math>\lim_{t \to \infty} h(t) = \lim_{s \to 0} \frac{6s}{s+2} = 0.</math> | |||
That is, the system returns to zero after being disturbed by a short impulse. However, the Laplace transform of the [[step response|unit step response]] is | |||
:<math>G(s) = \frac{1}{s} \frac{6}{s+2}</math> | |||
and so the step response converges to | |||
:<math>\lim_{t \to \infty} g(t) = \lim_{s \to 0} \frac{s}{s} \frac{6}{s+2} = \frac{6}{2} = 3</math> | |||
and so a zero-state system will follow an exponential rise to a final value of 3. | |||
== Example where FVT does not hold == | |||
{{Unreferenced section|date=October 2011}} | |||
However, for a system described by the transfer function | |||
:<math>H(s) = \frac{9}{s^2 + 9},</math> | |||
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 roots of the denominator of <math>H(s)</math> must have negative real parts. | |||
# <math>H(s)</math> 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 <math>+j3</math> and <math>-j3</math>. | |||
==See also== | |||
* [[Initial value theorem]] | |||
* [[Z-transform]] | |||
* [[Laplace Transform]] | |||
==Notes== | |||
<references /> | |||
==External links== | |||
*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 | |||
*http://www.engr.iupui.edu/~skoskie/ECE595s7/handouts/fvt_proof.pdf Final value proof for Z-transforms | |||
[[Category:Theorems in analysis]] | |||
[[Category:Fourier analysis]] | |||
{{mathanalysis-stub}} | |||
Revision as of 00:26, 14 December 2012
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
has a finite limit, then
where is the (unilateral) Laplace transform of .[1][2]
Example where FVT holds
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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
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However, 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 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 .
See also
Notes
- ↑ Template:Cite web
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External links
- 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
- http://www.engr.iupui.edu/~skoskie/ECE595s7/handouts/fvt_proof.pdf Final value proof for Z-transforms