Final value theorem

In the following statements, the notation $\backslash text\{‘\}s\; \backslash to\; 0\backslash text\{’\}$ means that $s$ approaches 0, whereas $\backslash text\{‘\}s\; \backslash downarrow\; 0\backslash text\{’\}$ means that $s$ approaches 0 through the positive numbers.

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 $sF(s)\; \backslash to\; L\; \backslash in\; \backslash mathbb\{R\}$ as $s\; \backslash to\; 0,$ and $\backslash lim\_\{t\backslash to\backslash infty\}f(t)\; =\; L.${{cite journal |last1=Chen |first1=Jie |last2=Lundberg |first2=Kent H. |last3=Davison |first3=Daniel E. |last4=Bernstein |first4=Dennis S. |title=The Final Value Theorem Revisited - Infinite Limits and Irrational Function |journal=IEEE Control Systems Magazine |date=June 2007 |volume=27 |issue=3 |pages=97–99 |doi=10.1109/MCS.2007.365008}}

Suppose that $f(t)$ and $f\text{'}(t)$ both have Laplace transforms that exist for all $s\; >\; 0.$ If $\backslash lim\_\{t\backslash to\backslash infty\}f(t)$ exists and $\backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\{sF(s)\}$ exists then $\backslash lim\_\{t\backslash to\backslash infty\}f(t)\; =\; \backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\{sF(s)\}.${{r|"Schiff1999"|page=Theorem 2.36}}{{r|"Graf2004"|page=20}}{{cite web |title=Final Value Theorem of Laplace Transform |url=https://proofwiki.org/wiki/Final_Value_Theorem_of_Laplace_Transform |website=ProofWiki |accessdate=12 April 2020}}

Remark

Both limits must exist for the theorem to hold. For example, if $f(t)\; =\; \backslash sin(t)$ then $\backslash lim\_\{t\backslash to\backslash infty\}f(t)$ does not exist, but{{r|"Schiff1999"|page=Example 2.37}}{{r|"Graf2004"|page=20}}

$$\backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\{sF(s)\}\; =\; \backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\{\backslash frac\{s\}\{s^2+1\}\}\; =\; 0.$$

Suppose that $f\; :\; (0,\backslash infty)\; \backslash to\; \backslash mathbb\{C\}$ is bounded and differentiable, and that

$t\; f\text{'}(t)$ is also bounded on $(0,\backslash infty)$. If $sF(s)\; \backslash to\; L\; \backslash in\; \backslash mathbb\{C\}$ as $s\; \backslash to\; 0$ then $\backslash lim\_\{t\backslash to\backslash infty\}f(t)\; =\; L.${{cite web |last1=Ullrich |first1=David C. |title=The tauberian final value Theorem |url=https://math.stackexchange.com/q/2795640 |website=Math Stack Exchange |date=2018-05-26}}

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:

1. $sF(s)\; \backslash to\; L\; \backslash in\; \backslash mathbb\{R\}$ as $s\; \backslash downarrow\; 0,$ and $\backslash lim\_\{t\backslash to\backslash infty\}f(t)\; =\; L.$
2. $sF(s)\; \backslash to\; +\backslash infty\; \backslash in\; \backslash mathbb\{R\}$ as $s\; \backslash downarrow\; 0,$ and $f(t)\; \backslash to\; +\backslash infty$ as $t\; \backslash to\; \backslash infty.$
3. $sF(s)\; \backslash to\; -\backslash infty\; \backslash in\; \backslash mathbb\{R\}$ as $s\; \backslash downarrow\; 0,$ and $f(t)\; \backslash to\; -\backslash infty$ as $t\; \backslash to\; \backslash infty.$

In particular, if $s\; =\; 0$ is a multiple pole of $F(s)$ then case 2 or 3 applies $(f(t)\; \backslash to\; +\backslash infty\backslash text\{\; or\; \}f(t)\; \backslash to\; -\backslash infty).$

Suppose that $f(t)$ is Laplace transformable. Let $\backslash lambda\; >\; -1$. If $\backslash lim\_\{t\backslash to\backslash infty\}\backslash frac\{f(t)\}\{t^\backslash lambda\}$ exists and $\backslash lim\_\{s\backslash downarrow0\}\{s^\{\backslash lambda+1\}F(s)\}$ exists then

:$\backslash lim\_\{t\backslash to\backslash infty\}\backslash frac\{f(t)\}\{t^\backslash lambda\}\; =\; \backslash frac\{1\}\{\backslash Gamma(\backslash lambda+1)\}\; \backslash lim\_\{s\backslash downarrow0\}\{s^\{\backslash lambda+1\}F(s)\},$

where $\backslash Gamma(x)$ denotes the Gamma function.

Final value theorems for obtaining $\backslash lim\_\{t\backslash to\backslash infty\}f(t)$ have applications in establishing the long-term stability of a system.

Suppose that $f\; :\; (0,\backslash infty)\; \backslash to\; \backslash mathbb\{C\}$ is bounded and measurable and $\backslash lim\_\{t\backslash to\backslash infty\}f(t)\; =\; \backslash alpha\; \backslash in\; \backslash mathbb\{C\}.$ Then $F(s)$ exists for all $s\; >\; 0$ and $\backslash lim\_\{s\backslash ,\backslash downarrow\backslash ,\; 0\}\{sF(s)\}\; =\; \backslash alpha.$

Elementary proof

Suppose for convenience that $|f(t)|\backslash le1$ on $(0,\backslash infty),$ and let $\backslash alpha=\backslash lim\_\{t\backslash to\backslash infty\}f(t)$. Let $\backslash epsilon>0,$ and choose $A$ so that for all $t\; >\; A.$ Since

$$s\backslash int\_0^\backslash infty\; e^\{-st\}\backslash ,\backslash mathrm\; dt=1,$$

for every $s>0$ we have

:$sF(s)-\backslash alpha=s\backslash int\_0^\backslash infty(f(t)-\backslash alpha)e^\{-st\}\backslash ,\backslash mathrm\; dt;$

hence

:$|sF(s)-\backslash alpha|\backslash le\; s\backslash int\_0^A|f(t)-\backslash alpha|e^\{-st\}\backslash ,\backslash mathrm\; dt+s\backslash int\_A^\backslash infty$

|f(t)-\alpha|e^{-st}\,\mathrm dt

\le2s\int_0^Ae^{-st}\,\mathrm dt+\epsilon s\int_A^\infty e^{-st}\,\mathrm dt \equiv I+II.

Now for every $s>0$ we have

:

On the other hand, since is fixed it is clear that $\backslash lim\_\{s\backslash to\; 0\}I=0$, and so if $s>0$ is small enough.

Suppose that all of the following conditions are satisfied:

1. $f:(0,\backslash infty)\; \backslash to\; \backslash mathbb\{C\}$ is continuously differentiable and both $f$ and $f\text{'}$ have a Laplace transform
2. $f\text{'}$ is absolutely integrable - that is, $\backslash int\_\{0\}^\{\backslash infty\}\; |\; f\text{'}(\backslash tau)\; |\; \backslash ,\; \backslash mathrm\; d\backslash tau$ is finite
3. $\backslash lim\_\{t\backslash to\backslash infty\}\; f(t)$ exists and is finite

Then{{cite web |last1=Sopasakis |first1=Pantelis |title=A proof for the Final Value theorem using Dominated convergence theorem |url=https://math.stackexchange.com/q/3218593 |website=Math Stack Exchange |date=2019-05-18}}

$$\backslash lim\_\{s\; \backslash to\; 0^\{+\}\}\; sF(s)\; =\; \backslash lim\_\{t\backslash to\backslash infty\}\; f(t).$$

Remark

The proof uses the dominated convergence theorem.

Let $f\; :\; (0,\backslash infty)\; \backslash to\; \backslash mathbb\{C\}$ be a continuous and bounded function such that such that the following limit exists

:$\backslash lim\_\{T\backslash to\backslash infty\}\; \backslash frac\{1\}\{T\}\; \backslash int\_\{0\}^\{T\}\; f(t)\; \backslash ,\; dt\; =\; \backslash alpha\; \backslash in\; \backslash mathbb\{C\}$

Then $\backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0,\; \backslash ,\; s>0\}\{sF(s)\}\; =\; \backslash alpha.${{cite web |last1=Murthy |first1=Kavi Rama |title=Alternative version of the Final Value theorem for Laplace Transform |url=https://math.stackexchange.com/questions/3216837/alternative-version-of-the-final-value-theorem-for-laplace-transform |website=Math Stack Exchange |date=2019-05-07}}

Suppose that $f\; :\; [0,\backslash infty)\; \backslash to\; \backslash mathbb\{R\}$ is continuous and absolutely integrable in $[0,\backslash infty).$ Suppose further that $f$ is asymptotically equal to a finite sum of periodic functions $f\_\{\backslash mathrm\{as\}\},$ that is

:

where $\backslash phi(t)$ is absolutely integrable in $[0,\backslash infty)$ and vanishes at infinity. Then

:$\backslash lim\_\{s\; \backslash to\; 0\}sF(s)\; =\; \backslash lim\_\{t\; \backslash to\; \backslash infty\}\; \backslash frac\{1\}\{t\}\; \backslash int\_\{0\}^\{t\}\; f(x)\; \backslash ,\; \backslash mathrm\; dx.${{cite journal |last1=Gluskin |first1=Emanuel |title=Let us teach this generalization of the final-value theorem |journal=European Journal of Physics |date=1 November 2003 |volume=24 |issue=6 |pages=591–597 |doi=10.1088/0143-0807/24/6/005}}

Let $f(t)\; :\; [0,\backslash infty)\; \backslash to\; \backslash mathbb\{R\}$ satisfy all of the following conditions:

1. $f(t)$ is infinitely differentiable at zero
2. $f^\{(k)\}(t)$ has a Laplace transform for all non-negative integers $k$
3. $f(t)$ diverges to infinity as $t\; \backslash to\; \backslash infty$

Let $F(s)$ be the Laplace transform of $f(t)$.

Then $sF(s)$ diverges to infinity as $s\; \backslash downarrow\; 0.${{cite web |last1=Hew |first1=Patrick |title=Final Value Theorem for function that diverges to infinity? |url=https://math.stackexchange.com/a/5019946/190548 |website=Math Stack Exchange |date=2025-01-06}}

Let $h\; :\; [0,\backslash infty)\; \backslash to\; \backslash mathbb\{R\}$ be measurable and such that the (possibly improper) integral $f(x)\; :=\; \backslash int\_0^x\; h(t)\backslash ,\backslash mathrm\; dt$ converges for $x\backslash to\backslash infty.$ Then

$$\backslash int\_0^\backslash infty\; h(t)\backslash ,\; \backslash mathrm\; dt\; :=\; \backslash lim\_\{x\backslash to\backslash infty\}\; f(x)\; =\; \backslash lim\_\{s\backslash downarrow\; 0\}\backslash int\_0^\backslash infty\; e^\{-st\}h(t)\backslash ,\backslash mathrm\; dt.$$

This is a version of Abel's theorem.

To see this, notice that $f\text{'}(t)\; =\; h(t)$ and apply the final value theorem to $f$ after an integration by parts: For $s\; >\; 0,$

:$$

s\int_0^\infty e^{-st}f(t)\, \mathrm dt = \Big[- e^{-st}f(t)\Big]_{t=o}^\infty + \int_0^\infty e^{-st} f'(t) \, \mathrm dt = \int_0^\infty e^{-st} h(t) \, \mathrm dt.

By the final value theorem, the left-hand side converges to $\backslash lim\_\{x\backslash to\backslash infty\}\; f(x)$ for $s\backslash to\; 0.$

To establish the convergence of the improper integral $\backslash lim\_\{x\backslash to\backslash infty\}f(x)$ in practice, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.

Final value theorems for obtaining $\backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\{sF(s)\}$ 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 $X$ and let $\backslash rho(s)$ be the Laplace–Stieltjes transform of $R(x).$ Then the $n$-th moment of $X$ can be calculated as

$$E[X^n]\; =\; (-1)^n\backslash left.\backslash frac\{d^n\backslash rho(s)\}\{ds^n\}\backslash right|\_\{s=0\}.$$

The strategy is to write

$$\backslash frac\{d^n\backslash rho(s)\}\{ds^n\}\; =\; \backslash mathcal\{F\}\backslash bigl(G\_1(s),\; G\_2(s),\; \backslash dots,\; G\_k(s),\; \backslash dots\backslash bigr),$$

where $\backslash mathcal\{F\}(\backslash dots)$ 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 of $F\_k(s),$ obtain

$\backslash lim\_\{t\backslash to\backslash infty\}f\_k(t),$ and apply a final value theorem to deduce

$\backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\{G\_k(s)\}\; =\backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\{sF\_k(s)\}\; =\; \backslash lim\_\{t\backslash to\backslash infty\}f\_k(t).$ Then

:$\backslash left.\backslash frac\{d^n\backslash rho(s)\}\{ds^n\}\backslash right|\_\{s=0\}\; =\; \backslash mathcal\{F\}\backslash Bigl(\backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\; G\_1(s),\; \backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\; G\_2(s),\; \backslash dots,\; \backslash lim\_\{s\backslash ,\backslash to\backslash ,\; 0\}\; G\_k(s),\; \backslash dots\backslash Bigr),$

and hence $E[X^n]$ is obtained.

{{Unreferenced section|date=October 2011}}

For example, for a system described by transfer function

:$H(s)\; =\; \backslash frac\{\; 6\; \}\{s\; +\; 2\},$

the impulse response converges to

:$\backslash lim\_\{t\; \backslash to\; \backslash infty\}\; h(t)\; =\; \backslash lim\_\{s\; \backslash to\; 0\}\; \backslash frac\{6s\}\{s+2\}\; =\; 0.$

That is, the system returns to zero after being disturbed by a short impulse. However, the Laplace transform of the unit step response is

:$G(s)\; =\; \backslash frac\{1\}\{s\}\; \backslash frac\{6\}\{s+2\}$

and so the step response converges to

:$\backslash lim\_\{t\; \backslash to\; \backslash infty\}\; g(t)\; =\; \backslash lim\_\{s\; \backslash to\; 0\}\; \backslash frac\{s\}\{s\}\; \backslash frac\{6\}\{s+2\}\; =\; \backslash frac\{6\}\{2\}\; =\; 3$

So a zero-state system will follow an exponential rise to a final value of 3.

For a system described by the transfer function

:$H(s)\; =\; \backslash frac\{9\}\{s^2\; +\; 9\},$

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:

1. All non-zero roots of the denominator of $H(s)$ must have negative real parts.
2. $H(s)$ 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 $0-j3.$

If $\backslash lim\_\{k\backslash to\backslash infty\}f[k]$ exists and $\backslash lim\_\{z\backslash ,\backslash to\backslash ,\; 1\}\{(z-1)F(z)\}$ exists then $\backslash lim\_\{k\backslash to\backslash infty\}f[k]\; =\; \backslash lim\_\{z\backslash ,\backslash to\backslash ,\; 1\}\{(z-1)F(z)\}.${{r|"Graf2004"|page=101}}

Final value of the system

:$\backslash dot\{\backslash mathbf\{x\}\}(t)\; =\; \backslash mathbf\{A\}\; \backslash mathbf\{x\}(t)\; +\; \backslash mathbf\{B\}\; \backslash mathbf\{u\}(t)$

:$\backslash mathbf\{y\}(t)\; =\; \backslash mathbf\{C\}\; \backslash mathbf\{x\}(t)$

in response to a step input $\backslash mathbf\{u\}(t)$ with amplitude $R$ is:

:$\backslash lim\_\{t\backslash to\backslash infty\}\backslash mathbf\{y\}(t)\; =\; -\backslash mathbf\{CA\}^\{-1\}\backslash mathbf\{B\}R$

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

:$\{\backslash mathbf\{x\}\}(t\_\{i+1\})\; =\; \backslash mathbf\{\backslash Phi\}(h\_\{i\})\; \backslash mathbf\{x\}(t\_\{i\})\; +\; \backslash mathbf\{\backslash Gamma\}(h\_\{i\})\; \backslash mathbf\{u\}(t\_\{i\})$

:$\backslash mathbf\{y\}(t\_\{i\})\; =\; \backslash mathbf\{C\}\; \backslash mathbf\{x\}(t\_\{i\})$

where $h\_\{i\}\; =\; t\_\{i+1\}-t\_\{i\}$ and

:$\backslash mathbf\{\backslash Phi\}(h\_\{i\})=e^\{\backslash mathbf\{A\}h\_\{i\}\}$, $\backslash mathbf\{\backslash Gamma\}(h\_\{i\})=\backslash int\_0^\{h\_\{i\}\}\; e^\{\backslash mathbf\{A\}s\}\; \backslash ,\backslash mathrm\; ds$

The final value of this system in response to a step input $\backslash mathbf\{u\}(t)$ with amplitude $R$ is the same as the final value of its original continuous-time system.{{cite journal |last1=Mohajeri |first1=Kamran |last2=Madadi |first2=Ali |last3=Tavassoli |first3=Babak |title= Tracking Control with Aperiodic Sampling over Networks with Delay and Dropout |journal= International Journal of Systems Science |date=2021 |volume=52 |issue=10 |pages= 1987–2002 |doi=10.1080/00207721.2021.1874074}}

• Initial value theorem
• Z-transform
• Laplace Transform
• Abelian and Tauberian theorems

• 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

Category:Theorems in Fourier analysis

it:Teorema del valore iniziale