Determining if Final Value Exists for Final Value Theorem

[salman213]

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!

 

[salman213]

anyone?

 

[Lunat1c]

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)

 

[MightyJayJay]

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!

 

[DeeNos]

I understand your confusion about determining if the final value exists for the final value theorem. The final value theorem is a mathematical tool used to calculate the limit of a function as the independent variable approaches infinity. In order for the final value theorem to be applicable, the function must have a finite limit as the independent variable approaches infinity, meaning that the final value exists.

One way to determine if the final value exists is to analyze the behavior of the function as the independent variable approaches infinity. For example, if the function approaches a constant value or oscillates around a certain value, then the final value exists. However, if the function diverges or approaches infinity, then the final value does not exist.

Another way to check for the existence of the final value is to look at the properties of the function itself. For example, if the function is continuous and bounded, then the final value exists. On the other hand, if the function has discontinuities or is unbounded, then the final value does not exist.

In some cases, it may be necessary to use other mathematical tools or techniques to determine the existence of the final value. These could include techniques such as the Cauchy criterion or the Dirichlet test. It may also be helpful to consult with a mathematician or use software programs to assist with the analysis.

In conclusion, the determination of whether or not the final value exists for a function is an important step in applying the final value theorem. By analyzing the behavior and properties of the function, one can determine if the final value exists and if the final value theorem can be applied. I hope this explanation has helped to clarify the concept for you.