{{Short description|Relation between frequency- and time-domain behavior at large time}} {{Context|date=January 2022}}
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.<ref name="RWang2010">{{cite web |url=http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html |title=Initial and Final Value Theorems |first=Ruye |last=Wang |date=2010-02-17 |accessdate=2011-10-21 |archive-date=2017-12-26 |archive-url=https://web.archive.org/web/20171226033147/http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html |url-status=dead }}</ref><ref name="OppenheimWillskyNawab1997">{{cite book |isbn=0-13-814757-4 |title=Signals & Systems |author1=Alan V. Oppenheim |author2=Alan S. Willsky |author3=S. Hamid Nawab |location=New Jersey, USA |publisher=Prentice Hall |year=1997}}</ref><ref name="Schiff1999">{{cite book |last1=Schiff |first1=Joel L. |title=The Laplace Transform: Theory and Applications |date=1999 |publisher=Springer |location=New York |isbn=978-1-4757-7262-3}}</ref><ref name="Graf2004">{{cite book |last1=Graf |first1=Urs |title=Applied Laplace Transforms and z-Transforms for Scientists and Engineers |date=2004 |publisher=Birkhäuser Verlag |location=Basel |isbn=3-7643-2427-9}}</ref> Mathematically, if <math>f(t)</math> in continuous time has (unilateral) Laplace transform <math>F(s)</math>, then a final value theorem establishes conditions under which <math display="block">\lim_{t\,\to\,\infty}f(t) = \lim_{s\,\to\, 0}{sF(s)}.</math> Likewise, if <math>f[k]</math> in discrete time has (unilateral) Z-transform <math>F(z)</math>, then a final value theorem establishes conditions under which <math display="block">\lim_{k\,\to\,\infty}f[k] = \lim_{z\,\to\, 1}{(z-1)F(z)}.</math>
An Abelian final value theorem makes assumptions about the time-domain behavior of <math>f(t) \text{ (or }f[k])</math> to calculate <math display="inline">\lim_{s\,\to\, 0}{sF(s)}.</math> Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of <math>F(s)</math> to calculate <math>\lim_{t\to\infty}f(t)</math> <math>\text{(or }\lim_{k\to\infty}f[k])</math> (see Abelian and Tauberian theorems for integral transforms).
== Final value theorems for the Laplace transform ==
=== Deducing {{math|lim<sub>''t'' → ∞</sub> ''f''(''t'')}} ===
In the following statements, the notation <math>\text{‘}s \to 0\text{’}</math> means that <math>s</math> approaches 0, whereas <math>\text{‘}s \downarrow 0\text{’}</math> means that <math>s</math> approaches 0 through the positive numbers.
==== Standard Final Value Theorem ====
Suppose that every pole of <math>F(s)</math> is either in the open left half plane or at the origin, and that <math>F(s)</math> has at most a single pole at the origin. Then <math>\lim_{t\to\infty}f(t) = \lim_{s\to 0}sF(s).</math><ref name="ChenLundbergDavisonBernstein2007">{{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}}</ref>
==== Final Value Theorem using Laplace transform of the derivative ====
Suppose that <math>f(t)</math> and <math>f'(t)</math> both have Laplace transforms that exist for all <math>s > 0.</math> If <math>\lim_{t\to\infty}f(t)</math> exists and <math>\lim_{s\,\to\, 0}{sF(s)}</math> exists then <math>\lim_{t\to\infty}f(t) = \lim_{s\,\to\, 0}{sF(s)}.</math>{{r|"Schiff1999"|page=Theorem 2.36}}{{r|"Graf2004"|page=20}}<ref>{{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}}</ref>
''Remark''
Both limits must exist for the theorem to hold. For example, if <math>f(t) = \sin(t)</math> then <math>\lim_{t\to\infty}f(t)</math> does not exist, but{{r|"Schiff1999"|page=Example 2.37}}{{r|"Graf2004"|page=20}} <math display="block">\lim_{s\,\to\, 0}{sF(s)} = \lim_{s\,\to\, 0}{\frac{s}{s^2+1}} = 0.</math>
==== Improved Tauberian converse Final Value Theorem ====
Suppose that <math>f : (0,\infty) \to \mathbb{C} </math> is bounded and differentiable, and that <math>t f'(t)</math> is also bounded on <math>(0,\infty)</math>. If <math>sF(s) \to L \in \mathbb{C}</math> as <math>s \to 0</math> then <math>\lim_{t\to\infty}f(t) = L.</math><ref name="UllrichTauberian">{{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}}</ref>
==== Extended Final Value Theorem ====
Suppose that <math>F</math> is a proper rational function and that every pole of <math>F</math> is either in the open left half-plane or at the origin. Then one of the following occurs: # <math>sF(s) \to L \in \mathbb{R}</math> as <math>s \downarrow 0,</math> and <math>\lim_{t\to\infty}f(t) = L.</math> # <math>sF(s) \to +\infty \in \mathbb{R}</math> as <math>s \downarrow 0,</math> and <math>f(t) \to +\infty</math> as <math>t \to \infty.</math> # <math>sF(s) \to -\infty \in \mathbb{R}</math> as <math>s \downarrow 0,</math> and <math>f(t) \to -\infty</math> as <math>t \to \infty.</math> In particular, if <math>s = 0</math> is a multiple pole of <math>F(s)</math> then case 2 or 3 applies <math>(f(t) \to +\infty\text{ or }f(t) \to -\infty).</math><ref name="ChenLundbergDavisonBernstein2007"/>
==== Generalized Final Value Theorem ====
Suppose that <math>f(t)</math> is Laplace transformable. Let <math>\lambda > -1</math>. If <math display="inline">\lim_{t\to\infty}\frac{f(t)}{t^\lambda}</math> exists and <math display="inline">\lim_{s\downarrow0}{s^{\lambda+1}F(s)}</math> exists then :<math>\lim_{t\to\infty}\frac{f(t)}{t^\lambda} = \frac{1}{\Gamma(\lambda+1)} \lim_{s\downarrow0}{s^{\lambda+1}F(s)},</math> where <math>\Gamma(x)</math> denotes the Gamma function.<ref name="ChenLundbergDavisonBernstein2007"/>
==== Applications ====
Final value theorems for obtaining <math>\lim_{t\to\infty}f(t)</math> have applications in establishing the long-term stability of a system.
=== Deducing {{math|lim<sub>''s'' → 0</sub> ''s'' ''F''(''s'')}} ===
==== Abelian Final Value Theorem ====
Suppose that <math>f : (0,\infty) \to \mathbb{C} </math> is bounded and measurable and <math>\lim_{t\to\infty}f(t) = \alpha \in \mathbb{C}.</math> Then <math>F(s)</math> exists for all <math>s > 0</math> and <math>\lim_{s\,\downarrow\, 0}{sF(s)} = \alpha.</math><ref name="UllrichTauberian"/>
''Elementary proof''<ref name="UllrichTauberian"/>
Suppose for convenience that <math>|f(t)|\le1</math> on <math>(0,\infty),</math> and let <math>\alpha=\lim_{t\to\infty}f(t)</math>. Let <math>\epsilon>0,</math> and choose <math>A</math> so that <math>|f(t)-\alpha|<\epsilon</math> for all <math>t > A.</math> Since <math display="block">s\int_0^\infty e^{-st}\,\mathrm dt=1,</math> for every <math>s>0</math> we have
:<math>sF(s)-\alpha=s\int_0^\infty(f(t)-\alpha)e^{-st}\,\mathrm dt;</math> hence :<math>|sF(s)-\alpha|\le s\int_0^A|f(t)-\alpha|e^{-st}\,\mathrm dt+s\int_A^\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.</math>
Now for every <math>s>0</math> we have :<math>II<\epsilon s\int_0^\infty e^{-st}\,\mathrm dt=\epsilon.</math> On the other hand, since <math>A<\infty</math> is fixed it is clear that <math>\lim_{s\to 0}I=0</math>, and so <math>|sF(s)-\alpha| < \epsilon</math> if <math>s>0</math> is small enough.
==== Final Value Theorem using Laplace transform of the derivative ====
Suppose that all of the following conditions are satisfied: # <math>f:(0,\infty) \to \mathbb{C} </math> is continuously differentiable and both <math>f</math> and <math>f'</math> have a Laplace transform # <math>f'</math> is absolutely integrable - that is, <math> \int_{0}^{\infty} | f'(\tau) | \, \mathrm d\tau </math> is finite # <math>\lim_{t\to\infty} f(t)</math> exists and is finite Then<ref name="SopasakisUsingDominatedConvergenceTheorem">{{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}}</ref> <math display="block">\lim_{s \to 0^{+}} sF(s) = \lim_{t\to\infty} f(t).</math>
''Remark''
The proof uses the dominated convergence theorem.<ref name="SopasakisUsingDominatedConvergenceTheorem"/>
==== Final Value Theorem for the mean of a function ====
Let <math>f : (0,\infty) \to \mathbb{C} </math> be a continuous and bounded function such that such that the following limit exists :<math>\lim_{T\to\infty} \frac{1}{T} \int_{0}^{T} f(t) \, dt = \alpha \in \mathbb{C}</math> Then <math>\lim_{s\,\to\, 0, \, s>0}{sF(s)} = \alpha.</math><ref name="KaviRamaMurthy">{{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}}</ref>
==== Final Value Theorem for asymptotic sums of periodic functions ====
Suppose that <math>f : [0,\infty) \to \mathbb{R} </math> is continuous and absolutely integrable in <math>[0,\infty).</math> Suppose further that <math>f</math> is asymptotically equal to a finite sum of periodic functions <math>f_{\mathrm{as}},</math> that is :<math>| f(t) - f_{\mathrm{as}}(t) | < \phi(t),</math> where <math>\phi(t)</math> is absolutely integrable in <math>[0,\infty)</math> and vanishes at infinity. Then :<math>\lim_{s \to 0}sF(s) = \lim_{t \to \infty} \frac{1}{t} \int_{0}^{t} f(x) \, \mathrm dx.</math><ref>{{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}}</ref>
==== Final Value Theorem for a function that diverges to infinity ====
Let <math>f(t) : [0,\infty) \to \mathbb{R}</math> satisfy all of the following conditions: # <math>f(t)</math> is infinitely differentiable at zero # <math>f^{(k)}(t)</math> has a Laplace transform for all non-negative integers <math>k</math> # <math>f(t)</math> diverges to infinity as <math>t \to \infty</math> Let <math>F(s)</math> be the Laplace transform of <math>f(t)</math>. Then <math>sF(s)</math> diverges to infinity as <math>s \downarrow 0.</math><ref name="HewDivergesToInfinity">{{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}}</ref>
==== Final Value Theorem for improperly integrable functions (Abel's theorem for integrals) ====
Let <math>h : [0,\infty) \to \mathbb{R}</math> be measurable and such that the (possibly improper) integral <math>f(x) := \int_0^x h(t)\,\mathrm dt</math> converges for <math>x\to\infty.</math> Then <math display="block">\int_0^\infty h(t)\, \mathrm dt := \lim_{x\to\infty} f(x) = \lim_{s\downarrow 0}\int_0^\infty e^{-st}h(t)\,\mathrm dt.</math> This is a version of Abel's theorem.
To see this, notice that <math>f'(t) = h(t)</math> and apply the final value theorem to <math>f</math> after an integration by parts: For <math>s > 0,</math>
:<math> 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. </math>
By the final value theorem, the left-hand side converges to <math>\lim_{x\to\infty} f(x)</math> for <math>s\to 0.</math>
To establish the convergence of the improper integral <math>\lim_{x\to\infty}f(x)</math> in practice, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.
==== Applications ====
Final value theorems for obtaining <math>\lim_{s\,\to\, 0}{sF(s)}</math> have applications in probability and statistics to calculate the moments of a random variable. Let <math>R(x)</math> be cumulative distribution function of a continuous random variable <math>X</math> and let <math>\rho(s)</math> be the Laplace–Stieltjes transform of <math>R(x).</math> Then the <math>n</math>-th moment of <math>X</math> can be calculated as <math display="block">E[X^n] = (-1)^n\left.\frac{d^n\rho(s)}{ds^n}\right|_{s=0}.</math> The strategy is to write <math display="block">\frac{d^n\rho(s)}{ds^n} = \mathcal{F}\bigl(G_1(s), G_2(s), \dots, G_k(s), \dots\bigr),</math> where <math>\mathcal{F}(\dots)</math> is continuous and for each <math>k,</math> <math>G_k(s) = sF_k(s)</math> for a function <math>F_k(s).</math> For each <math>k,</math> put <math>f_k(t)</math> as the inverse Laplace transform of <math>F_k(s),</math> obtain <math>\lim_{t\to\infty}f_k(t),</math> and apply a final value theorem to deduce <math>\lim_{s\,\to\, 0}{G_k(s)} =\lim_{s\,\to\, 0}{sF_k(s)} = \lim_{t\to\infty}f_k(t).</math> Then :<math>\left.\frac{d^n\rho(s)}{ds^n}\right|_{s=0} = \mathcal{F}\Bigl(\lim_{s\,\to\, 0} G_1(s), \lim_{s\,\to\, 0} G_2(s), \dots, \lim_{s\,\to\, 0} G_k(s), \dots\Bigr),</math> and hence <math>E[X^n]</math> is obtained.
=== Examples === {{Unreferenced section|date=October 2011}}
==== Example where FVT holds ====
For example, for a system described by transfer function :<math>H(s) = \frac{ 6 }{s + 2},</math> the impulse response converges to :<math>\lim_{t \to \infty} h(t) = \lim_{s \to 0} \frac{6s}{s+2} = 0.</math> That is, the system returns to zero after being disturbed by a short impulse. However, the Laplace transform of the unit step response is :<math>G(s) = \frac{1}{s} \frac{6}{s+2}</math> and so the step response converges to :<math>\lim_{t \to \infty} g(t) = \lim_{s \to 0} \frac{s}{s} \frac{6}{s+2} = \frac{6}{2} = 3</math> 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
:<math>H(s) = \frac{9}{s^2 + 9},</math> 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 <math>H(s)</math> must have negative real parts. # <math>H(s)</math> 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 <math>0+j3</math> and <math>0-j3.</math>
== Final value theorems for the Z transform ==
=== Deducing {{math|lim<sub>''k'' → ∞</sub> ''f''[''k'']}} ===
==== Final Value Theorem ====
If <math>\lim_{k\to\infty}f[k]</math> exists and <math>\lim_{z\,\to\, 1}{(z-1)F(z)}</math> exists then <math>\lim_{k\to\infty}f[k] = \lim_{z\,\to\, 1}{(z-1)F(z)}.</math>{{r|"Graf2004"|page=101}}
== Final value of linear systems ==
=== Continuous-time LTI systems === Final value of the system :<math>\dot{\mathbf{x}}(t) = \mathbf{A} \mathbf{x}(t) + \mathbf{B} \mathbf{u}(t)</math> :<math>\mathbf{y}(t) = \mathbf{C} \mathbf{x}(t)</math>
in response to a step input <math>\mathbf{u}(t)</math> with amplitude <math>R</math> is:
:<math>\lim_{t\to\infty}\mathbf{y}(t) = -\mathbf{CA}^{-1}\mathbf{B}R</math>
=== Sampled-data systems ===
The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times <math>t_{i}, i=1,2,...</math> is the discrete-time system :<math>{\mathbf{x}}(t_{i+1}) = \mathbf{\Phi}(h_{i}) \mathbf{x}(t_{i}) + \mathbf{\Gamma}(h_{i}) \mathbf{u}(t_{i})</math> :<math>\mathbf{y}(t_{i}) = \mathbf{C} \mathbf{x}(t_{i})</math> where <math>h_{i} = t_{i+1}-t_{i}</math> and :<math>\mathbf{\Phi}(h_{i})=e^{\mathbf{A}h_{i}}</math>, <math>\mathbf{\Gamma}(h_{i})=\int_0^{h_{i}} e^{\mathbf{A}s} \,\mathrm ds</math> The final value of this system in response to a step input <math>\mathbf{u}(t)</math> with amplitude <math>R</math> is the same as the final value of its original continuous-time system.<ref name="MohajeriMadadiTavassoli2021">{{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}}</ref>
==See also== * Initial value theorem * Z-transform * Laplace Transform *Abelian and Tauberian theorems
==Notes== <references />
==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
Category:Theorems in Fourier analysis