Final value theorem
   HOME

TheInfoList



OR:

In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the final value theorem (FVT) is one of several similar theorems used to relate
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
expressions to the
time domain Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
behavior as time approaches infinity. Mathematically, if f(t) in continuous time has (unilateral)
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
F(s), then a final value theorem establishes conditions under which :\lim_f(t) = \lim_ Likewise, if f /math> in discrete time has (unilateral)
Z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...
F(z), then a final value theorem establishes conditions under which :\lim_f = \lim_ An Abelian final value theorem makes assumptions about the time-domain behavior of f(t) (or f /math>) to calculate \lim_. Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of F(s) to calculate \lim_f(t) (or \lim_f /math>) (see Abelian and Tauberian theorems for integral transforms).


Final value theorems for the Laplace transform


Deducing

In the following statements, the notation 's \to 0' means that s approaches 0, whereas 's \downarrow 0' 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 sF(s) \to L \in \mathbb as s \to 0, and \lim_f(t) = L.


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 \lim_f(t) exists and \lim_ exists then \lim_f(t) = \lim_.
''Remark'' Both limits must exist for the theorem to hold. For example, if f(t) = \sin(t) then \lim_f(t) does not exist, but \lim_ = \lim_ = 0.


Improved Tauberian converse Final Value Theorem

Suppose that f : (0,\infty) \to \mathbb is bounded and differentiable, and that t f'(t) is also bounded on (0,\infty). If sF(s) \to L \in \mathbb as s \to 0 then \lim_f(t) = L.


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: # sF(s) \to L \in \mathbb as s \downarrow 0, and \lim_f(t) = L. # sF(s) \to +\infty \in \mathbb as s \downarrow 0, and f(t) \to +\infty as t \to \infty. # sF(s) \to -\infty \in \mathbb as s \downarrow 0, and f(t) \to -\infty as t \to \infty. In particular, if s = 0 is a multiple pole of F(s) then case 2 or 3 applies (f(t) \to +\infty or f(t) \to -\infty).


Generalized Final Value Theorem

Suppose that f(t) is Laplace transformable. Let \lambda > -1. If \lim_\frac exists and \lim_ exists then :\lim_\frac = \frac \lim_ where \Gamma(x) denotes the
Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.


Applications

Final value theorems for obtaining \lim_f(t) have applications in establishing the long-term stability of a system.


Deducing


Abelian Final Value Theorem

Suppose that f : (0,\infty) \to \mathbb is bounded and measurable and \lim_f(t) = \alpha \in \mathbb. Then F(s) exists for all s > 0 and \lim_ = \alpha. ''Elementary proof'' Suppose for convenience that , f(t), \le1 on (0,\infty), and let \alpha=\lim_f(t). Let \epsilon>0, and choose A so that , f(t)-\alpha, <\epsilon for all t>A. Since s\int_0^\infty e^\,dt=1, for every s>0 we have :sF(s)-\alpha=s\int_0^\infty(f(t)-\alpha)e^\,dt; hence :, sF(s)-\alpha, \le s\int_0^A, f(t)-\alpha, e^\,dt+s\int_A^\infty , f(t)-\alpha, e^\,dt \le2s\int_0^Ae^\,dt+\epsilon s\int_A^\infty e^\,dt=I+II. Now for every s>0 we have :II<\epsilon s\int_0^\infty e^\,dt=\epsilon. On the other hand, since A<\infty is fixed it is clear that \lim_I=0, and so , sF(s)-\alpha, < \epsilon if s>0 is small enough.


Final Value Theorem using Laplace transform of the derivative

Suppose that all of the following conditions are satisfied: # f : (0,\infty) \to \mathbb is continuously differentiable and both f and f' have a Laplace transform # f' is absolutely integrable - that is, \int_^ , f'(\tau) , \, d\tau is finite # \lim_ f(t) exists and is finite Then :\lim_ sF(s) = \lim_ f(t). ''Remark'' The proof uses the
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary t ...
.


Final Value Theorem for the mean of a function

Let f : (0,\infty) \to \mathbb be a continuous and bounded function such that such that the following limit exists :\lim_ \frac \int_^ f(t) \, dt = \alpha \in \mathbb Then \lim_ = \alpha.


Final Value Theorem for asymptotic sums of periodic functions

Suppose that f : [0,\infty) \to \mathbb is continuous and absolutely integrable in [0,\infty). Suppose further that f is asymptotically equal to a finite sum of periodic functions f_, that is :, f(t) - f_(t) , < \phi(t) where \phi(t) is absolutely integrable in [0,\infty) and vanishes at infinity. Then :\lim_sF(s) = \lim_ \frac \int_^ f(x) \, dx.


Final Value Theorem for a function that diverges to infinity

Let f(t) : [0,\infty) \to \mathbb and F(s) be the Laplace transform of f(t). Suppose that f(t) satisfies all of the following conditions: # f(t) is infinitely differentiable at zero # f^(t) has a Laplace transform for all non-negative integers k # f(t) diverges to infinity as t \to \infty Then sF(s) diverges to infinity as s \to 0^.


Final Value Theorem for improperly integrable functions (

Abel's theorem In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Theorem Let the Taylor series G (x) = \sum_^\infty a_k x^k be a pow ...
for integrals)

Let h : [0,\infty) \to \mathbb be measurable and such that the (possibly improper) integral f(x) := \int_0^x h(t)\, dt converges for x\to\infty. Then :\int_0^\infty h(t)\, dt := \lim_ f(x) = \lim_\int_0^\infty e^h(t)\, dt. This is a version of
Abel's theorem In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel. Theorem Let the Taylor series G (x) = \sum_^\infty a_k x^k be a pow ...
. To see this, notice that f'(t) = h(t) and apply the final value theorem to f after an integration by parts: For s > 0, : s\int_0^\infty e^f(t)\, dt = \Big[- e^f(t)\Big]_^\infty + \int_0^\infty e^ f'(t) \, dt = \int_0^\infty e^ h(t) \, dt. By the final value theorem, the left-hand side converges to \lim_ f(x) for s\to 0. To establish the convergence of the improper integral \lim_f(x) in practice, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.


Applications

Final value theorems for obtaining \lim_ 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 The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is ...
of R(x). Then the n-th moment of X can be calculated as :E ^n= (-1)^n\left.\frac\_ The strategy is to write :\frac = \mathcal\bigl(G_1(s), G_2(s), \dots, G_k(s), \dots\bigr) where \mathcal(\dots) is continuous and for each k, G_k(s) = sF_k(s) for a function F_k(s). For each k, put f_k(t) as the
inverse Laplace transform In mathematics, the inverse Laplace transform of a function ''F''(''s'') is the piecewise-continuous and exponentially-restricted real function ''f''(''t'') which has the property: :\mathcal\(s) = \mathcal\(s) = F(s), where \mathcal denotes the ...
of F_k(s), obtain \lim_f_k(t), and apply a final value theorem to deduce \lim_ =\lim_ = \lim_f_k(t). Then :\left.\frac\_ = \mathcal\Bigl(\lim_ G_1(s), \lim_ G_2(s), \dots, \lim_ G_k(s), \dots\Bigr) and hence E ^n/math> is obtained.


Examples


Example where FVT holds

For example, for a system described by
transfer function In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output for each possible input. They are widely used ...
:H(s) = \frac, and so the impulse response converges to :\lim_ h(t) = \lim_ \frac = 0. That is, the system returns to zero after being disturbed by a short impulse. However, the Laplace transform of the unit step response is :G(s) = \frac \frac and so the step response converges to :\lim_ g(t) = \lim_ \frac \frac = \frac = 3 and 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 :H(s) = \frac, 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 Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
which confirm valid results for the Final Value Theorem: # All non-zero roots of the denominator of H(s) must have negative real parts. # H(s) 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 0+j3 and 0-j3.


Final value theorems for the Z transform


Deducing


Final Value Theorem

If \lim_f /math> exists and \lim_ exists then \lim_f = \lim_.


Final value of linear systems


Continuous-time LTI systems

Final value of the system :\dot(t) = \mathbf \mathbf(t) + \mathbf \mathbf(t) :\mathbf(t) = \mathbf \mathbf(t) in response to a step input \mathbf(t) with amplitude R is: :\lim_\mathbf(t) = \mathbf^\mathbfR


Sampled-data systems

The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times t_, i=1,2,... is the discrete-time system :(t_) = \mathbf(h_) \mathbf(t_) + \mathbf(h_) \mathbf(t_) :\mathbf(t_) = \mathbf \mathbf(t_) where h_ = t_-t_ and :\mathbf(h_)=e^, \mathbf(h_)=\int_0^ e^ \,ds The final value of this system in response to a step input \mathbf(t) with amplitude R is the same as the final value of its original continuous-time system.


See also

* Initial value theorem *
Z-transform In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-tim ...
*
Laplace Transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the '' time domain'') to a function of a complex variable s (in the ...
*
Abelian and Tauberian theorems In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing th ...


Notes


External links

*https://web.archive.org/web/20101225034508/http://wikis.controltheorypro.com/index.php?title=Final_Value_Theorem *http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html {{Webarchive, url=https://web.archive.org/web/20171226033147/http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html , date=2017-12-26 : final value for Laplace *https://web.archive.org/web/20110719222313/http://www.engr.iupui.edu/~skoskie/ECE595s7/handouts/fvt_proof.pdf: final value proof for Z-transforms Theorems in Fourier analysis it:Teorema del valore iniziale