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. Mathematically, if f(t) in continuous time has (unilateral) Laplace transform F(s), then a final value theorem establishes conditions under which Likewise, if f[k] in discrete time has (unilateral) Z-transform F(z), then a final value theorem establishes conditions under which An Abelian final value theorem makes assumptions about the time-domain behavior of to calculate Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of F(s) to calculate (see Abelian and Tauberian theorems for integral transforms).

Final value theorems for the Laplace transform

Deducing

lim{{sub|t → ∞}} f(t) In the following statements, the notation means that s approaches 0, whereas means that s approaches 0 through the positive numbers.

Standard Final Value Theorem

Suppose that every pole of F(s) is either in the open left half plane or at the origin, and that F(s) has at most a single pole at the origin. Then as s \to 0, and

Final Value Theorem using Laplace transform of the derivative

Suppose that f(t) and f'(t) both have Laplace transforms that exist for all s > 0. If exists and exists then Remark Both limits must exist for the theorem to hold. For example, if then does not exist, but

Improved Tauberian converse Final Value Theorem

Suppose that is bounded and differentiable, and that t f'(t) is also bounded on (0,\infty). If as s \to 0 then

Extended Final Value Theorem

Suppose that every pole of F(s) is either in the open left half-plane or at the origin. Then one of the following occurs: In particular, if s = 0 is a multiple pole of F(s) then case 2 or 3 applies

Generalized Final Value Theorem

Suppose that f(t) is Laplace transformable. Let. If exists and exists then where \Gamma(x) denotes the Gamma function.

Applications

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

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lim{{sub|s → 0}} s F(s)

Abelian Final Value Theorem

Suppose that is bounded and measurable and Then F(s) exists for all s > 0 and Elementary proof Suppose for convenience that |f(t)|\le1 on (0,\infty), and let. Let \epsilon>0, and choose A so that for all t > A. Since for every s>0 we have hence Now for every s>0 we have On the other hand, since A<\infty is fixed it is clear that, and so if s>0 is small enough.

Final Value Theorem using Laplace transform of the derivative

Suppose that all of the following conditions are satisfied: Then Remark The proof uses the dominated convergence theorem.

Final Value Theorem for the mean of a function

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

Final Value Theorem for asymptotic sums of periodic functions

Suppose that is continuous and absolutely integrable in [0,\infty). Suppose further that f is asymptotically equal to a finite sum of periodic functions that is where \phi(t) is absolutely integrable in [0,\infty) and vanishes at infinity. Then

Final Value Theorem for a function that diverges to infinity

Let and F(s) be the Laplace transform of f(t). Suppose that f(t) satisfies all of the following conditions: Then sF(s) diverges to infinity as

Final Value Theorem for improperly integrable functions (Abel's theorem for integrals)

Let be measurable and such that the (possibly improper) integral converges for x\to\infty. Then This is a version of Abel's theorem. To see this, notice that and apply the final value theorem to f after an integration by parts: For s > 0, By the final value theorem, the left-hand side converges to for s\to 0. To establish the convergence of the improper integral in practice, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.

Applications

Final value theorems for obtaining have applications in probability and statistics to calculate the moments of a random variable. Let R(x) be cumulative distribution function of a continuous random variable X and let \rho(s) be the Laplace–Stieltjes transform of R(x). Then the n-th moment of X can be calculated as The strategy is to write where is continuous and for each k, for a function F_k(s). For each k, put f_k(t) as the inverse Laplace transform of F_k(s), obtain and apply a final value theorem to deduce Then and hence E[X^n] is obtained.

Examples

Example where FVT holds

For example, for a system described by transfer function 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 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 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: Rule 1 was not satisfied in this example, in that the roots of the denominator are 0+j3 and 0-j3.

Final value theorems for the Z transform

Deducing

lim{{sub|k → ∞}} f[k]

Final Value Theorem

If exists and exists then

Final value of linear systems

Continuous-time LTI systems

Final value of the system in response to a step input with amplitude R is:

Sampled-data systems

The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times is the discrete-time system where and The final value of this system in response to a step input with amplitude R is the same as the final value of its original continuous-time system.