# Final value theorem

Final value theorem This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (Janeiro 2022) (Saiba como e quando remover esta mensagem de modelo) 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, E se {estilo de exibição f(t)} in continuous time has (unilateral) Laplace transform {estilo de exibição F(s)} , then a final value theorem establishes conditions under which {displaystyle lim _{tto infty }f(t)=lim_{s,para ,0}{sF(s)}} Da mesma maneira, E se {estilo de exibição f[k]} in discrete time has (unilateral) Z-transform {estilo de exibição F(z)} , then a final value theorem establishes conditions under which {displaystyle lim _{kto infty }f[k]=lim_{zto 1}{(z-1)F(z)}} An Abelian final value theorem makes assumptions about the time-domain behavior of {estilo de exibição f(t)} (ou {estilo de exibição f[k]} ) to calculate {displaystyle lim _{s,para ,0}{sF(s)}} . Por outro lado, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of {estilo de exibição F(s)} to calculate {displaystyle lim _{tto infty }f(t)} (ou {displaystyle lim _{kto infty }f[k]} ) (see Abelian and Tauberian theorems for integral transforms).

Conteúdo 1 Final value theorems for the Laplace transform 1.1 Deducing limt → ∞ f(t) 1.1.1 Standard Final Value Theorem 1.1.2 Final Value Theorem using Laplace transform of the derivative 1.1.3 Improved Tauberian converse Final Value Theorem 1.1.4 Extended Final Value Theorem 1.1.5 Generalized Final Value Theorem 1.1.6 Formulários 1.2 Deducing lims → 0 s F(s) 1.2.1 Abelian Final Value Theorem 1.2.2 Final Value Theorem using Laplace transform of the derivative 1.2.3 Final Value Theorem for the mean of a function 1.2.4 Final Value Theorem for asymptotic sums of periodic functions 1.2.5 Final Value Theorem for a function that diverges to infinity 1.2.6 Final Value Theorem for improperly integrable functions (Abel's theorem for integrals) 1.2.7 Formulários 1.3 Exemplos 1.3.1 Example where FVT holds 1.3.2 Example where FVT does not hold 2 Final value theorems for the Z transform 2.1 Deducing limk → ∞ f[k] 2.1.1 Final Value Theorem 3 Final value of linear systems 3.1 Continuous-time LTI systems 3.2 Sampled-data systems 4 Veja também 5 Notas 6 External links Final value theorems for the Laplace transform Deducing limt → ∞ f(t) In the following statements, the notation ' {displaystyle sto 0} ' means that {estilo de exibição s} approaches 0, whereas ' {displaystyle sdownarrow 0} ' means that {estilo de exibição s} approaches 0 through the positive numbers.

Standard Final Value Theorem Suppose that every pole of {estilo de exibição F(s)} is either in the open left half plane or at the origin, and that {estilo de exibição F(s)} has at most a single pole at the origin. Então {displaystyle sF(s)to Lin mathbb {R} } Como {displaystyle sto 0} , e {displaystyle lim _{tto infty }f(t)=L} .[5] Final Value Theorem using Laplace transform of the derivative Suppose that {estilo de exibição f(t)} e {estilo de exibição f'(t)} both have Laplace transforms that exist for all {displaystyle s>0} . Se {displaystyle lim _{tto infty }f(t)} exists and {displaystyle lim _{s,para ,0}{sF(s)}} exists then {displaystyle lim _{tto infty }f(t)=lim_{s,para ,0}{sF(s)}} .[3]: Theorem 2.36 [4]: 20 [6] Remark Both limits must exist for the theorem to hold. Por exemplo, E se {estilo de exibição f(t)=sin(t)} então {displaystyle lim _{tto infty }f(t)} não existe, mas {displaystyle lim _{s,para ,0}{sF(s)}=lim_{s,para ,0}{fratura {s}{s^{2}+1}}=0} .[3]: Example 2.37 [4]: 20  Improved Tauberian converse Final Value Theorem Suppose that {estilo de exibição f:(0,infty )para mathbb {C} } is bounded and differentiable, and that {displaystyle tf'(t)} is also bounded on {estilo de exibição (0,infty )} . Se {displaystyle sF(s)to Lin mathbb {C} } Como {displaystyle sto 0} então {displaystyle lim _{tto infty }f(t)=L} .[7] Extended Final Value Theorem Suppose that every pole of {estilo de exibição F(s)} is either in the open left half-plane or at the origin. Then one of the following occurs: {displaystyle sF(s)to Lin mathbb {R} } Como {displaystyle sdownarrow 0} , e {displaystyle lim _{tto infty }f(t)=L} . {displaystyle sF(s)to +infty in mathbb {R} } Como {displaystyle sdownarrow 0} , e {estilo de exibição f(t)para +infty } Como {estilo de exibição para infty } . {displaystyle sF(s)to -infty in mathbb {R} } Como {displaystyle sdownarrow 0} , e {estilo de exibição f(t)to -infty } Como {estilo de exibição para infty } .

Em particular, E se {displaystyle s=0} is a multiple pole of {estilo de exibição F(s)} then case 2 ou 3 applies ( {estilo de exibição f(t)para +infty } ou {estilo de exibição f(t)to -infty } ).[5] Generalized Final Value Theorem Suppose that {estilo de exibição f(t)} is Laplace transformable. Deixar {displaystyle lambda >-1} . Se {displaystyle lim _{tto infty }{fratura {f(t)}{t^{lambda }}}} exists and {displaystyle lim _{sdownarrow 0}{s^{lambda +1}F(s)}} exists then {displaystyle lim _{tto infty }{fratura {f(t)}{t^{lambda }}}={fratura {1}{Gama (lambda +1)}}lim_{sdownarrow 0}{s^{lambda +1}F(s)}} Onde {displaystyle Gama (x)} denotes the Gamma function.[5] Applications Final value theorems for obtaining {displaystyle lim _{tto infty }f(t)} have applications in establishing the long-term stability of a system.

Deducing lims → 0 s F(s) Abelian Final Value Theorem Suppose that {estilo de exibição f:(0,infty )para mathbb {C} } is bounded and measurable and {displaystyle lim _{tto infty }f(t)=alpha in mathbb {C} } . Então {estilo de exibição F(s)} exists for all {displaystyle s>0} e {displaystyle lim _{s,para ,0^{+}}{sF(s)}= alfa } .[7] Elementary proof[7] Suppose for convenience that {estilo de exibição |f(t)|leq 1} sobre {estilo de exibição (0,infty )} , e deixar {displaystyle alpha =lim _{tto infty }f(t)} . Deixar {displaystyle epsilon >0} , and choose {estilo de exibição A} de modo a {estilo de exibição |f(t)-alfa |UMA} . Desde {displaystyle sint _{0}^{infty }e^{-st},dt=1} , para cada {displaystyle s>0} temos {displaystyle sF(s)-alpha =sint _{0}^{infty }(f(t)-alfa )e^{-st},dt;} por isso {estilo de exibição |sF(s)-alfa |leq sint _{0}^{UMA}|f(t)-alfa |e^{-st},dt+sint _{UMA}^{infty }|f(t)-alfa |e^{-st},dtleq 2sint _{0}^{UMA}e^{-st},dt+epsilon sint _{UMA}^{infty }e^{-st},dt=I+II.} Now for every {displaystyle s>0} temos {displaystyle II0} is small enough.

Final Value Theorem using Laplace transform of the derivative Suppose that all of the following conditions are satisfied: {estilo de exibição f:(0,infty )para mathbb {C} } is continuously differentiable and both {estilo de exibição f} e {estilo de exibição f'} have a Laplace transform {estilo de exibição f'} is absolutely integrable - isso é, {estilo de exibição int _{0}^{infty }|f'(sim )|,dtau } é finito {displaystyle lim _{tto infty }f(t)} exists and is finite Then {displaystyle lim _{sto 0^{+}}sF(s)=lim_{tto infty }f(t)} .[8] Remark The proof uses the Dominated Convergence Theorem.[8] Final Value Theorem for the mean of a function Let {estilo de exibição f:(0,infty )para mathbb {C} } be a continuous and bounded function such that such that the following limit exists {displaystyle lim _{Tto e cinquenta }{fratura {1}{T}}int_{0}^{T}f(t),dt=alpha in mathbb {C} } Então {displaystyle lim _{s,para ,0,,s>0}{sF(s)}= alfa } .[9] Final Value Theorem for asymptotic sums of periodic functions Suppose that {estilo de exibição f:[0,infty )para mathbb {R} } is continuous and absolutely integrable in {estilo de exibição [0,infty )} . Suppose further that {estilo de exibição f} is asymptotically equal to a finite sum of periodic functions {estilo de exibição f_{matemática {Como} }} , isso é {estilo de exibição |f(t)-f_{matemática {Como} }(t)|0} , {displaystyle sint _{0}^{infty }e^{-st}f(t),dt={Grande [}-e^{-st}f(t){Grande ]}_{t=o}^{infty }+int_{0}^{infty }e^{-st}f'(t),dt=int _{0}^{infty }e^{-st}h(t),dt.} By the final value theorem, the left-hand side converges to {displaystyle lim _{xto infty }f(x)} por {displaystyle sto 0} .

To establish the convergence of the improper integral {displaystyle lim _{xto infty }f(x)} na prática, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.

Applications Final value theorems for obtaining {displaystyle lim _{s,para ,0}{sF(s)}} have applications in probability and statistics to calculate the moments of a random variable. Deixar {estilo de exibição R(x)} be cumulative distribution function of a continuous random variable {estilo de exibição X} e deixar {estilo de exibição rho (s)} be the Laplace–Stieltjes transform of {estilo de exibição R(x)} . Then the {estilo de exibição m} -th moment of {estilo de exibição X} can be calculated as {estilo de exibição E[X^{n}]=(-1)^{n}deixei.{fratura {d^{n}rho (s)}{ds^{n}}}certo|_{s=0}} The strategy is to write {estilo de exibição {fratura {d^{n}rho (s)}{ds^{n}}}={matemática {F}}{De repente (}G_{1}(s),G_{2}(s),pontos ,G_{k}(s),pontos {maior )}} Onde {estilo de exibição {matemática {F}}(pontos )} is continuous and for each {estilo de exibição k} , {estilo de exibição G_{k}(s)=sF_{k}(s)} para uma função {estilo de exibição F_{k}(s)} . For each {estilo de exibição k} , put {estilo de exibição f_{k}(t)} as the inverse Laplace transform of {estilo de exibição F_{k}(s)} , obtain {displaystyle lim _{tto infty }f_{k}(t)} , and apply a final value theorem to deduce {displaystyle lim _{s,para ,0}{G_{k}(s)}=lim_{s,para ,0}{sF_{k}(s)}=lim_{tto infty }f_{k}(t)} . Então {displaystyle left.{fratura {d^{n}rho (s)}{ds^{n}}}certo|_{s=0}={matemática {F}}{Grande (}lim_{s,para ,0}G_{1}(s),lim_{s,para ,0}G_{2}(s),pontos ,lim_{s,para ,0}G_{k}(s),pontos {Maior )}} e, portanto {estilo de exibição E[X^{n}]} is obtained.

Examples This section does not cite any sources. Please help improve this section by adding citations to reliable sources. O material sem fonte pode ser contestado e removido. (Outubro 2011) (Saiba como e quando remover esta mensagem de modelo) Example where FVT holds For example, for a system described by transfer function {estilo de exibição H(s)={fratura {6}{s+2}},} and so the impulse response converges to {displaystyle lim _{tto infty }h(t)=lim_{sto 0}{fratura {6s}{s+2}}=0.} Aquilo é, the system returns to zero after being disturbed by a short impulse. No entanto, the Laplace transform of the unit step response is {estilo de exibição G(s)={fratura {1}{s}}{fratura {6}{s+2}}} and so the step response converges to {displaystyle lim _{tto infty }g(t)=lim_{sto 0}{fratura {s}{s}}{fratura {6}{s+2}}={fratura {6}{2}}=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 {estilo de exibição H(s)={fratura {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. No entanto, neither time-domain limit exists, and so the final value theorem predictions are not valid. Na verdade, both the impulse response and step response oscillate, e (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 {estilo de exibição H(s)} must have negative real parts. {estilo de exibição 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 {displaystyle 0+j3} e {displaystyle 0-j3} .

Final value theorems for the Z transform Deducing limk → ∞ f[k] Final Value Theorem If {displaystyle lim _{kto infty }f[k]} exists and {displaystyle lim _{z,para ,1}{(z-1)F(z)}} exists then {displaystyle lim _{kto infty }f[k]=lim_{z,para ,1}{(z-1)F(z)}} .[4]: 101  Final value of linear systems Continuous-time LTI systems Final value of the system {estilo de exibição {ponto {mathbf {x} }}(t)= mathbf {UMA} mathbf {x} (t)+mathbf {B} mathbf {você} (t)} {estilo de exibição mathbf {y} (t)= mathbf {C} mathbf {x} (t)} in response to a step input {estilo de exibição mathbf {você} (t)} with amplitude {estilo de exibição R} é: {displaystyle lim _{tto infty }mathbf {y} (t)= mathbf {CA} ^{-1}mathbf {B} R} Sampled-data systems The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times {estilo de exibição t_{eu},i=1,2,...} is the discrete-time system {estilo de exibição {mathbf {x} }(t_{i+1})= mathbf {Phi } (h_{eu})mathbf {x} (t_{eu})+mathbf {Gama } (h_{eu})mathbf {você} (t_{eu})} {estilo de exibição mathbf {y} (t_{eu})= mathbf {C} mathbf {x} (t_{eu})} Onde {estilo de exibição h_{eu}=t_{i+1}-t_{eu}} e {estilo de exibição mathbf {Phi } (h_{eu})=e^{mathbf {UMA} h_{eu}}} , {estilo de exibição mathbf {Gama } (h_{eu})=int_{0}^{h_{eu}}e^{mathbf {UMA} s},ds} The final value of this system in response to a step input {estilo de exibição mathbf {você} (t)} with amplitude {estilo de exibição R} is the same as the final value of its original continuous-time system. [12] See also Initial value theorem Z-transform Laplace Transform Abelian and Tauberian theorems Notes ^ Wang, Ruye (2010-02-17). "Initial and Final Value Theorems". Recuperado 2011-10-21. ^ Alan V. Oppenheim; Alan S. Willsky; S. Hamid Nawab (1997). Signals & Systems. Nova Jersey, EUA: Prentice Hall. ISBN 0-13-814757-4. ^ Saltar para: a b c Schiff, Joel L. (1999). The Laplace Transform: Teoria e aplicações. Nova york: Springer. ISBN 978-1-4757-7262-3. ^ Saltar para: a b c d Graf, Urs (2004). Applied Laplace Transforms and z-Transforms for Scientists and Engineers. Basileia: Birkhäuser Verlag. ISBN 3-7643-2427-9. ^ Saltar para: a b c Chen, Jie; Lundberg, Kent H.; Davison, Daniel E.; Bernstein, Dennis S. (Junho 2007). "The Final Value Theorem Revisited - Infinite Limits and Irrational Function". IEEE Control Systems Magazine. 27 (3): 97-99. doi:10.1109/MCS.2007.365008. ^ "Final Value Theorem of Laplace Transform". ProofWiki. Recuperado 12 abril 2020. ^ Saltar para: a b c Ullrich, David C. (2018-05-26). "The tauberian final value Theorem". Math Stack Exchange. ^ Saltar para: a b Sopasakis, Pantelis (2019-05-18). "A proof for the Final Value theorem using Dominated convergence theorem". Math Stack Exchange. ^ Murthy, Kavi Rama (2019-05-07). "Alternative version of the Final Value theorem for Laplace Transform". Math Stack Exchange. ^ Gluskin, Emanuel (1 novembro 2003). "Let us teach this generalization of the final-value theorem". Revista Europeia de Física. 24 (6): 591–597. doi:10.1088/0143-0807/24/6/005. ^ Hew, Patrick (2020-04-22). "Final Value Theorem for function that diverges to infinity?". Math Stack Exchange. ^ Mohajeri, Kamran; Madadi, Ali; Tavassoli, Babak (2021). "Tracking Control with Aperiodic Sampling over Networks with Delay and Dropout". International Journal of Systems Science. 52 (10): 1987–2002. doi: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 Categories: Theorems in Fourier analysis

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