Determining if Final Value Exists for Final Value Theorem

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Discussion Overview

The discussion revolves around the final value theorem in control systems, specifically addressing how to determine if a final value exists for a given function before applying the theorem. Participants explore the implications of stability and the relationship between the transfer function and the time response of a system.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant questions how to check if the final value exists for a function without first finding its inverse Laplace transform, suggesting that this would negate the need for the final value theorem.
  • Another participant mentions that analyzing the poles of the transfer function can indicate system stability, which affects whether the output converges to a final value or diverges.
  • A different viewpoint suggests that the final value theorem can be used to verify if the time function converges to a real value by solving the limit as s approaches zero, indicating that if this limit exists, the transfer function is valid.
  • Some participants propose that solving the limit as s approaches zero is often easier due to cancellations in the transfer function, which may simplify the analysis.

Areas of Agreement / Disagreement

Participants express differing views on the utility of the final value theorem and the methods for determining the existence of a final value, indicating that there is no consensus on the best approach to take.

Contextual Notes

Limitations include the dependence on the stability of the system and the nature of the transfer function, which may affect the convergence of the output response.

salman213
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I had a question about the final value theorem:

lim sF(s)
S->0

=

lim F(t)
t->infinity


I was told I can only use this if I know that a final value exists for f, or in other words that f(infinity) exists.

How can I check if it exists? If I have F(s), the only way I can think to check is to find the inverse Laplace and then see. If I do that, then there is no use of me applying the final value theorem as I will already know what it is.

Can some explain how one can determine whether or not the final value exists before applying the theorem?

Thanks!
 
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Well, if you have the transfer function of a system, by analyzing the poles you can determine whether the system is a stable or an unstable one. Giving an appropriate input to a stable system results in an output response that converges to a particular value as t tends to infinity while for an unstable system the output goes to infinity (i.e. it is not bounded)
 
I think you USE the final value theorem to verify IF the time function converges to a real value. Basically you solve lim sF(s) for s->0, and if this value exists then the transfer function is good and works because lim sF(s) for s->0 = lim f(t) for t->infinity. If the transfer function is good you would get the same value when solving either limit(if you know the time function). The difference would be that solving lim sF(s) for s->0 is easier because many terms of the transfer function will cancel out because of the s->0.

Hope you got it!
 

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