Final value theorem explained

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

f(t)

in continuous time has (unilateral) Laplace transform

F(s)

, then a final value theorem establishes conditions under which

\lim_f(t) = \lim_.

Likewise, if

f[k]

in discrete time has (unilateral) Z-transform

F(z)

, then a final value theorem establishes conditions under which

\lim_f[k] = \lim_.

An Abelian final value theorem makes assumptions about the time-domain behavior of

f(t)(orf[k])

to calculate

\lim_.

Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of

F(s)

to calculate

\lim_t\toinftyf(t)

(or\lim_k\toinftyf[k])

(see Abelian and Tauberian theorems for integral transforms).

Final value theorems for the Laplace transform

Deducing

In the following statements, the notation

‘s\to0’

means that

approaches 0, whereas

‘s\downarrow0’

means that

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)\toL\inR

s\to0,

and

\lim_t\toinftyf(t)=L.

^[5]

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.

\lim_t\toinftyf(t)

exists and

\lim_s\to{sF(s)}

exists then

\lim_t\toinftyf(t)=\lim_s\to{sF(s)}.

^[6]

Remark

Both limits must exist for the theorem to hold. For example, if

f(t)=\sin(t)

then

\lim_t\toinftyf(t)

does not exist, but

\lim_ = \lim_ = 0.

Improved Tauberian converse Final Value Theorem

Suppose that

f:(0,infty)\toC

is bounded and differentiable, and that

tf'(t)

is also bounded on

(0,infty)

. If

sF(s)\toL\inC

s\to0

then

\lim_t\toinftyf(t)=L.

^[7]

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)\toL\inR

s\downarrow0,

and

\lim_t\toinftyf(t)=L.

sF(s)\to+infty\inR

s\downarrow0,

and

f(t)\to+infty

t\toinfty.

sF(s)\to-infty\inR

s\downarrow0,

and

f(t)\to-infty

t\toinfty.

In particular, if

s=0

is a multiple pole of

F(s)

then case 2 or 3 applies

(f(t)\to+inftyorf(t)\to-infty).

^[5]

Generalized Final Value Theorem

Suppose that

f(t)

is Laplace transformable. Let

λ>-1

. If

\lim_\frac

exists and

\lim_

exists then

\lim_t\toinfty

	f(t)
	t^λ

	1
	\Gamma(λ+1)

\lim_s\downarrow0{s^λ+1F(s)},

where

\Gamma(x)

denotes the Gamma function.^[5]

Applications

Final value theorems for obtaining

\lim_t\toinftyf(t)

have applications in establishing the long-term stability of a system.

Deducing

Abelian Final Value Theorem

Suppose that

f:(0,infty)\toC

is bounded and measurable and

\lim_t\toinftyf(t)=\alpha\inC.

Then

F(s)

exists for all

s>0

and

\lim_s\downarrow{sF(s)}=\alpha.

^[7]

Elementary proof^[7]

Suppose for convenience that

|f(t)|\le1

(0,infty),

and let

\alpha=\lim_t\toinftyf(t)

. Let

\epsilon>0,

and choose

so that

|f(t)-\alpha|<\epsilon

for all

t>A.

Since

s\int_0^\infty e^\,\mathrm dt=1,

for every

s>0

we have

	infty(f(t)-\alpha)e
sF(s)-\alpha=s\int
	0

^-stdt;

hence

|sF(s)-\alpha|\le

	A\|f(t)-\alpha\|e
s\int
	0

^-st

	infty \|f(t)-\alpha\|e
dt+s\int
	A

^-st

	Ae
dt \le2s\int
	0

^-st

	infty
dt+\epsilons\int
	A

e^-stdt\equivI+II.

Now for every

s>0

we have

II<\epsilon

	infty
s\int
	0

e^-stdt=\epsilon.

On the other hand, since

A<infty

is fixed it is clear that

\lim_s\toI=0

, and so

|sF(s)-\alpha|<\epsilon

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)\toC

is continuously differentiable and both

and

have a Laplace transform

is absolutely integrable - that is,

	infty
\int
	0

|f'(\tau)|d\tau

is finite

\lim_t\toinftyf(t)

exists and is finiteThen^[8]

\lim_ sF(s) = \lim_ f(t).

Remark

The proof uses the dominated convergence theorem.^[8]

Final Value Theorem for the mean of a function

Let

f:(0,infty)\toC

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

\lim_T\toinfty

	1
	T

	T
\int
	0

f(t)dt=\alpha\inC

Then

\lim_s\to{sF(s)}=\alpha.

^[9]

Final Value Theorem for asymptotic sums of periodic functions

Suppose that

f:[0,infty)\toR

is continuous and absolutely integrable in

[0,infty).

Suppose further that

is asymptotically equal to a finite sum of periodic functions

f_as,

that is

|f(t)-f_as(t)|<\phi(t),

where

\phi(t)

is absolutely integrable in

[0,infty)

and vanishes at infinity. Then

\lim_ssF(s)=\lim_t

	1
	t

	t
\int
	0

f(x)dx.

^[10]

Final Value Theorem for a function that diverges to infinity

Let

f(t):[0,infty)\toR

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^(k)(t)

has a Laplace transform for all non-negative integers

f(t)

diverges to infinity as

t\toinfty

Then

sF(s)

diverges to infinity as

s\downarrow0.

^[11]

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

Let

h:[0,infty)\toR

be measurable and such that the (possibly improper) integral

f(x):=

	x
\int
	0

h(t)dt

converges for

x\toinfty.

Then

\int_0^\infty h(t)\, \mathrm dt := \lim_ f(x) = \lim_\int_0^\infty e^h(t)\,\mathrm dt.

This is a version of Abel's theorem.

To see this, notice that

f'(t)=h(t)

and apply the final value theorem to

after an integration by parts: For

s>0,

	infty
s\int
	0

e^-stf(t)dt=[-e^-st

	infty
f(t)]
	t=o

	infty
\int
	0

e^-stf'(t)dt=

	infty
\int
	0

e^-sth(t)dt.

By the final value theorem, the left-hand side converges to

\lim_x\toinftyf(x)

for

s\to0.

To establish the convergence of the improper integral

\lim_x\toinftyf(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_s\to{sF(s)}

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

and let

\rho(s)

be the Laplace–Stieltjes transform of

R(x).

Then the

-th moment of

can be calculated as

E[X^n] = (-1)^n\left.\frac\right|_.

The strategy is to write

\frac = \mathcal\bigl(G_1(s), G_2(s), \dots, G_k(s), \dots\bigr),

where

l{F}(...)

is continuous and for each

G_k(s)=sF_k(s)

for a function

F_k(s).

For each

put

f_k(t)

as the inverse Laplace transform of

F_k(s),

obtain

\lim_t\toinftyf_k(t),

and apply a final value theorem to deduce

\lim_s\to{G_k(s)}=\lim_s\to{sF_k(s)}=\lim_t\toinftyf_k(t).

Then

\left.	d^n\rho(s)
	dsⁿ

\right|_s=0=l{F}l(\lim_s\toG_1(s),\lim_s\toG_2(s),...,\lim_s\toG_k(s),...r),

and hence

E[X^n]

is obtained.

Examples

Example where FVT holds

For example, for a system described by transfer function

H(s)=

	6
	s+2

the impulse response converges to

\lim_th(t)=\lim_s

	6s
	s+2

=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)=

	1
	s

	6
	s+2

and so the step response converges to

\lim_tg(t)=\lim_s

	s
	s

	6
	s+2

	6
	2

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)=

	9
	s²+9

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:

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

\lim_k\toinftyf[k]

exists and

\lim_z\to{(z-1)F(z)}

exists then

\lim_k\toinftyf[k]=\lim_z\to{(z-1)F(z)}.

Final value of linear systems

Continuous-time LTI systems

Final value of the system

•

(t)=Ax(t)+Bu(t)

y(t)=Cx(t)

in response to a step input

u(t)

with amplitude

is:

\lim_t\toinftyy(t)=-CA^-1BR

Sampled-data systems

The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times

t_i,i=1,2,...

is the discrete-time system

}(t_) = \mathbf(h_) \mathbf(t_) + \mathbf(h_) \mathbf(t_)

y(t_i)=Cx(t_i)

where

h_i=t_i+1-t_i

and

\Phi(h_i

	Ah_i
)=e

\Gamma(h_i

	h_i
)=\int
	0

e^Asds

The final value of this system in response to a step input

u(t)

with amplitude

is the same as the final value of its original continuous-time system.^[12]

Notes

Web site: Initial and Final Value Theorems . Ruye . Wang . 2010-02-17 . 2011-10-21 . 2017-12-26 . https://web.archive.org/web/20171226033147/http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html . dead .
Book: 0-13-814757-4 . Signals & Systems . Alan V. Oppenheim . Alan S. Willsky . S. Hamid Nawab . New Jersey, USA . Prentice Hall . 1997.
Book: Schiff . Joel L. . The Laplace Transform: Theory and Applications . 1999 . Springer . New York . 978-1-4757-7262-3.
Book: Graf . Urs . Applied Laplace Transforms and z-Transforms for Scientists and Engineers . 2004 . Birkhäuser Verlag . Basel . 3-7643-2427-9.
Chen . Jie . Lundberg . Kent H. . Davison . Daniel E. . Bernstein . Dennis S. . The Final Value Theorem Revisited - Infinite Limits and Irrational Function . IEEE Control Systems Magazine . June 2007 . 27 . 3 . 97–99 . 10.1109/MCS.2007.365008.
Web site: Final Value Theorem of Laplace Transform . ProofWiki . 12 April 2020.
Web site: Ullrich . David C. . The tauberian final value Theorem . Math Stack Exchange . 2018-05-26.
Web site: Sopasakis . Pantelis . A proof for the Final Value theorem using Dominated convergence theorem . Math Stack Exchange . 2019-05-18.
Web site: Murthy . Kavi Rama . Alternative version of the Final Value theorem for Laplace Transform . Math Stack Exchange . 2019-05-07.
Gluskin . Emanuel . Let us teach this generalization of the final-value theorem . European Journal of Physics . 1 November 2003 . 24 . 6 . 591–597 . 10.1088/0143-0807/24/6/005.
Web site: Hew . Patrick . Final Value Theorem for function that diverges to infinity? . Math Stack Exchange . 2020-04-22 .
Mohajeri . Kamran . Madadi . Ali . Tavassoli . Babak . Tracking Control with Aperiodic Sampling over Networks with Delay and Dropout . International Journal of Systems Science . 2021 . 52 . 10 . 1987–2002 . 10.1080/00207721.2021.1874074.

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 : 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

Final value theorem explained

Final value theorems for the Laplace transform

Deducing

Standard Final Value Theorem

Final Value Theorem using Laplace transform of the derivative

Improved Tauberian converse Final Value Theorem

Extended Final Value Theorem

Generalized Final Value Theorem

Applications

Deducing

Abelian Final Value Theorem

Final Value Theorem using Laplace transform of the derivative

Final Value Theorem for the mean of a function

Final Value Theorem for asymptotic sums of periodic functions

Final Value Theorem for a function that diverges to infinity

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

Applications

Examples

Example where FVT holds

Example where FVT does not hold

Final value theorems for the Z transform

Deducing

Final Value Theorem

Final value of linear systems

Continuous-time LTI systems

Sampled-data systems

See also

Notes

External links