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.[1][2][3][4] 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[edit]

Deducing [edit]

In the following statements, the notation '' means that approaches 0, whereas '' means that approaches 0 through the positive numbers.

Standard Final Value Theorem[edit]

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 .[5]

Final Value Theorem using Laplace Transform of the Derivative[edit]

Suppose that and both have Laplace transforms that exist for all . If exists and exists then .[3]:Theorem 2.36[4]:20[6]

Remark

Both limits must exist for the theorem to hold. For example, if then does not exist, but .[3]:Example 2.37[4]:20

Improved Tauberian Converse Final Value Theorem[edit]

Suppose that is bounded and differentiable, and that is also bounded on . If as then .[7]

Extended Final Value Theorem[edit]

Suppose that every pole of is either in the open left half plane or at the origin. Then one of the following occurs:

  1. as , and .
  2. as , and as .
  3. as , and as .

In particular, if is a multiple pole of then case 2 or 3 applies ( or ).[5]

Generalized Final Value Theorem[edit]

Suppose that is Laplace transformable. Let . If exists and exists then

where denotes the Gamma function.[5]

Applications[edit]

Final value theorems for obtaining have applications in establishing the long-term stability of a system.

Deducing [edit]

Abelian Final Value Theorem[edit]

Suppose that is bounded and measurable and . Then exists for all and .[7]

Elementary proof[7]

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[edit]

Suppose that all of the following conditions are satisfied:

  1. is continuously differentiable and both and have a Laplace Transform
  2. is absolutely integrable, that is is finite
  3. exists and is finite

Then

.[8]

Remark

The proof uses the Dominated Convergence Theorem.[8]

Final Value Theorem for the Mean of a Function[edit]

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

Then .[9]

Final Value Theorem for Asymptotic Sums of Periodic Functions[edit]

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

.[10]

Final Value Theorem for a Function that Diverges to Infinity[edit]

Let and be the Laplace transform of . Suppose that satisfies all of the following conditions:

  1. is infinitely differentiable at zero
  2. has a Laplace transform for all non-negative integers
  3. diverges to infinity as

Then diverges to infinity as .[11]

Applications[edit]

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[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 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. 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 .

Final value theorems for the Z transform[edit]

Deducing [edit]

Final Value Theorem[edit]

If exists and exists then .[4]:101

See also[edit]

Notes[edit]

  1. ^ Wang, Ruye (2010-02-17). "Initial and Final Value Theorems". Retrieved 2011-10-21.
  2. ^ Alan V. Oppenheim; Alan S. Willsky; S. Hamid Nawab (1997). Signals & Systems. New Jersey, USA: Prentice Hall. ISBN 0-13-814757-4.
  3. ^ a b c Schiff, Joel L. (1999). The Laplace Transform: Theory and Applications. New York: Springer. ISBN 978-1-4757-7262-3.
  4. ^ a b c d Graf, Urs (2004). Applied Laplace Transforms and z-Transforms for Scientists and Engineers. Basel: Birkhäuser Verlag. ISBN 3-7643-2427-9.
  5. ^ a b c Chen, Jie; Lundberg, Kent H.; Davison, Daniel E.; Bernstein, Dennis S. (June 2007). "The Final Value Theorem Revisited - Infinite Limits and Irrational Function". IEEE Control Systems Magazine. 27 (3): 97–99. doi:10.1109/MCS.2007.365008.
  6. ^ "Final Value Theorem of Laplace Transform". ProofWiki. Retrieved 12 April 2020.
  7. ^ a b c Ullrich, David C. (2018-05-26). "The tauberian final value Theorem". Math Stack Exchange.
  8. ^ a b Sopasakis, Pantelis (2019-05-18). "A proof for the Final Value theorem using Dominated convergence theorem". Math Stack Exchange.
  9. ^ Murthy, Kavi Rama (2019-05-07). "Alternative version of the Final Value theorem for Laplace Transform". Math Stack Exchange.
  10. ^ Gluskin, Emanuel (1 November 2003). "Let us teach this generalization of the final-value theorem". European Journal of Physics. 24 (6): 591–597. doi:10.1088/0143-0807/24/6/005.
  11. ^ Hew, Patrick (2020-04-22). "Final Value Theorem for function that diverges to infinity?". Math Stack Exchange.

External links[edit]