Théorème de la valeur finale
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. (Janvier 2022) (Découvrez comment et quand supprimer ce modèle de message) 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, si {style d'affichage f(t)} in continuous time has (unilateral) Laplace transform {style d'affichage F(s)} , then a final value theorem establishes conditions under which {style d'affichage lim _{tto infty }F(t)=lim _{s,à ,0}{sF(s)}} De même, si {style d'affichage f[k]} in discrete time has (unilateral) Z-transform {style d'affichage F(z)} , then a final value theorem establishes conditions under which {style d'affichage 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 {style d'affichage f(t)} (ou {style d'affichage f[k]} ) to calculate {style d'affichage lim _{s,à ,0}{sF(s)}} . inversement, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of {style d'affichage F(s)} to calculate {style d'affichage lim _{tto infty }F(t)} (ou {style d'affichage lim _{kto infty }F[k]} ) (see Abelian and Tauberian theorems for integral transforms).
Contenu 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 Applications 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 Applications 1.3 Exemples 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 Voir également 5 Remarques 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 {style d'affichage s} approaches 0, whereas ' {displaystyle sdownarrow 0} ' means that {style d'affichage s} approaches 0 through the positive numbers.
Standard Final Value Theorem Suppose that every pole of {style d'affichage F(s)} is either in the open left half plane or at the origin, et cela {style d'affichage F(s)} has at most a single pole at the origin. Alors {displaystyle sF(s)to Lin mathbb {R} } comme {displaystyle sto 0} , et {style d'affichage lim _{tto infty }F(t)=L} .[5] Final Value Theorem using Laplace transform of the derivative Suppose that {style d'affichage f(t)} et {style d'affichage f'(t)} both have Laplace transforms that exist for all {displaystyle s>0} . Si {style d'affichage lim _{tto infty }F(t)} exists and {style d'affichage lim _{s,à ,0}{sF(s)}} exists then {style d'affichage lim _{tto infty }F(t)=lim _{s,à ,0}{sF(s)}} .[3]: Theorem 2.36 [4]: 20 [6] Remark Both limits must exist for the theorem to hold. Par exemple, si {style d'affichage f(t)=sin(t)} alors {style d'affichage lim _{tto infty }F(t)} n'existe pas, mais {style d'affichage lim _{s,à ,0}{sF(s)}=lim _{s,à ,0}{frac {s}{s ^{2}+1}}=0} .[3]: Example 2.37 [4]: 20 Improved Tauberian converse Final Value Theorem Suppose that {style d'affichage f:(0,infime )à mathbb {C} } is bounded and differentiable, et cela {displaystyle tf'(t)} is also bounded on {style d'affichage (0,infime )} . Si {displaystyle sF(s)to Lin mathbb {C} } comme {displaystyle sto 0} alors {style d'affichage lim _{tto infty }F(t)=L} .[7] Extended Final Value Theorem Suppose that every pole of {style d'affichage 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} } comme {displaystyle sdownarrow 0} , et {style d'affichage lim _{tto infty }F(t)=L} . {displaystyle sF(s)to +infty in mathbb {R} } comme {displaystyle sdownarrow 0} , et {style d'affichage f(t)to +infty } comme {displaystyle tto infty } . {displaystyle sF(s)to -infty in mathbb {R} } comme {displaystyle sdownarrow 0} , et {style d'affichage f(t)to -infty } comme {displaystyle tto infty } .
En particulier, si {displaystyle s=0} is a multiple pole of {style d'affichage F(s)} then case 2 ou 3 applies ( {style d'affichage f(t)to +infty } ou {style d'affichage f(t)to -infty } ).[5] Generalized Final Value Theorem Suppose that {style d'affichage f(t)} is Laplace transformable. Laisser {displaystyle lambda >-1} . Si {style d'affichage lim _{tto infty }{frac {F(t)}{t ^{lambda }}}} exists and {style d'affichage lim _{sdownarrow 0}{s ^{lambda +1}F(s)}} exists then {style d'affichage lim _{tto infty }{frac {F(t)}{t ^{lambda }}}={frac {1}{Gamma (lambda +1)}}lim _{sdownarrow 0}{s ^{lambda +1}F(s)}} où {style d'affichage Gamma (X)} denotes the Gamma function.[5] Applications Final value theorems for obtaining {style d'affichage 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 {style d'affichage f:(0,infime )à mathbb {C} } is bounded and measurable and {style d'affichage lim _{tto infty }F(t)=alpha in mathbb {C} } . Alors {style d'affichage F(s)} exists for all {displaystyle s>0} et {style d'affichage lim _{s,à ,0^{+}}{sF(s)}=alpha } .[7] Elementary proof[7] Suppose for convenience that {style d'affichage |F(t)|leq 1} sur {style d'affichage (0,infime )} , et laissez {displaystyle alpha =lim _{tto infty }F(t)} . Laisser {displaystyle epsilon >0} , and choose {style d'affichage A} pour que {style d'affichage |F(t)-alpha |
Final Value Theorem using Laplace transform of the derivative Suppose that all of the following conditions are satisfied: {style d'affichage f:(0,infime )à mathbb {C} } is continuously differentiable and both {style d'affichage f} et {style d'affichage f'} have a Laplace transform {style d'affichage f'} is absolutely integrable - C'est, {style d'affichage entier _{0}^{infime }|F'(oui )|,dtau } est fini {style d'affichage lim _{tto infty }F(t)} exists and is finite Then {style d'affichage 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 {style d'affichage f:(0,infime )à mathbb {C} } be a continuous and bounded function such that such that the following limit exists {style d'affichage lim _{Tto infty }{frac {1}{J}}entier _{0}^{J}F(t),dt=alpha in mathbb {C} } Alors {style d'affichage lim _{s,à ,0,,s>0}{sF(s)}=alpha } .[9] Final Value Theorem for asymptotic sums of periodic functions Suppose that {style d'affichage f:[0,infime )à mathbb {R} } is continuous and absolutely integrable in {style d'affichage [0,infime )} . Suppose further that {style d'affichage f} is asymptotically equal to a finite sum of periodic functions {style d'affichage f_{mathrm {comme} }} , C'est {style d'affichage |F(t)-F_{mathrm {comme} }(t)|
To establish the convergence of the improper integral {style d'affichage lim _{xto infty }F(X)} en pratique, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.
Applications Final value theorems for obtaining {style d'affichage lim _{s,à ,0}{sF(s)}} have applications in probability and statistics to calculate the moments of a random variable. Laisser {style d'affichage R(X)} be cumulative distribution function of a continuous random variable {style d'affichage X} et laissez {style d'affichage rho (s)} be the Laplace–Stieltjes transform of {style d'affichage R(X)} . Then the {displaystyle n} -th moment of {style d'affichage X} can be calculated as {style d'affichage E[X^{n}]=(-1)^{n}la gauche.{frac {d^{n}Rho (s)}{ds^{n}}}droit|_{s=0}} The strategy is to write {style d'affichage {frac {d^{n}Rho (s)}{ds^{n}}}={mathématique {F}}{soudain (}G_{1}(s),G_{2}(s),des points ,G_{k}(s),des points {plus grand )}} où {style d'affichage {mathématique {F}}(des points )} is continuous and for each {style d'affichage k} , {style d'affichage G_{k}(s)=sF_{k}(s)} pour une fonction {style d'affichage F_{k}(s)} . For each {style d'affichage k} , put {style d'affichage f_{k}(t)} as the inverse Laplace transform of {style d'affichage F_{k}(s)} , obtain {style d'affichage lim _{tto infty }F_{k}(t)} , and apply a final value theorem to deduce {style d'affichage lim _{s,à ,0}{G_{k}(s)}=lim _{s,à ,0}{sF_{k}(s)}=lim _{tto infty }F_{k}(t)} . Alors {displaystyle left.{frac {d^{n}Rho (s)}{ds^{n}}}droit|_{s=0}={mathématique {F}}{Gros (}lim _{s,à ,0}G_{1}(s),lim _{s,à ,0}G_{2}(s),des points ,lim _{s,à ,0}G_{k}(s),des points {Plus grand )}} et donc {style d'affichage E[X^{n}]} is obtained.
Examples This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Le matériel non sourcé peut être contesté et supprimé. (Octobre 2011) (Découvrez comment et quand supprimer ce modèle de message) Example where FVT holds For example, for a system described by transfer function {style d'affichage H(s)={frac {6}{s+2}},} and so the impulse response converges to {style d'affichage lim _{tto infty }h(t)=lim _{sto 0}{frac {6s}{s+2}}=0.} C'est-à-dire, the system returns to zero after being disturbed by a short impulse. Cependant, the Laplace transform of the unit step response is {style d'affichage G(s)={frac {1}{s}}{frac {6}{s+2}}} and so the step response converges to {style d'affichage lim _{tto infty }g(t)=lim _{sto 0}{frac {s}{s}}{frac {6}{s+2}}={frac {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 {style d'affichage H(s)={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. Cependant, neither time-domain limit exists, and so the final value theorem predictions are not valid. En réalité, both the impulse response and step response oscillate, et (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 {style d'affichage H(s)} must have negative real parts. {style d'affichage 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} et {displaystyle 0-j3} .
Final value theorems for the Z transform Deducing limk → ∞ f[k] Final Value Theorem If {style d'affichage lim _{kto infty }F[k]} exists and {style d'affichage lim _{z,à ,1}{(z-1)F(z)}} exists then {style d'affichage lim _{kto infty }F[k]=lim _{z,à ,1}{(z-1)F(z)}} .[4]: 101 Final value of linear systems Continuous-time LTI systems Final value of the system {style d'affichage {point {mathbf {X} }}(t)= mathbf {UN} mathbf {X} (t)+mathbf {B} mathbf {tu} (t)} {style d'affichage mathbf {y} (t)= mathbf {C} mathbf {X} (t)} in response to a step input {style d'affichage mathbf {tu} (t)} with amplitude {style d'affichage R} est: {style d'affichage lim _{tto infty }mathbf {y} (t)= mathbf {Californie} ^{-1}mathbf {B} R} Sampled-data systems The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times {style d'affichage t_{je},i=1,2,...} is the discrete-time system {style d'affichage {mathbf {X} }(t_{je+1})= mathbf {Phi } (h_{je})mathbf {X} (t_{je})+mathbf {Gamma } (h_{je})mathbf {tu} (t_{je})} {style d'affichage mathbf {y} (t_{je})= mathbf {C} mathbf {X} (t_{je})} où {style d'affichage h_{je}=t_{je+1}-t_{je}} et {style d'affichage mathbf {Phi } (h_{je})=e^{mathbf {UN} h_{je}}} , {style d'affichage mathbf {Gamma } (h_{je})=int _{0}^{h_{je}}e ^{mathbf {UN} s},dès} The final value of this system in response to a step input {style d'affichage mathbf {tu} (t)} with amplitude {style d'affichage 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". Récupéré 2011-10-21. ^ Alan V. Oppenheim; Alan S. Willsky; S. Hamid Nawab (1997). Signals & Systems. New Jersey, Etats-Unis: Prentice Hall. ISBN 0-13-814757-4. ^ Sauter à: a b c Schiff, Joel L. (1999). The Laplace Transform: Théorie et applications. New York: Springer. ISBN 978-1-4757-7262-3. ^ Sauter à: a b c d Graf, Urs (2004). Applied Laplace Transforms and z-Transforms for Scientists and Engineers. Bâle: Birkhäuser Verlag. ISBN 3-7643-2427-9. ^ Sauter à: a b c Chen, Jie; Lundberg, Kent H.; Davison, Daniel E.; Bernstein, Dennis S. (Juin 2007). "The Final Value Theorem Revisited - Infinite Limits and Irrational Function". IEEE Control Systems Magazine. 27 (3): 97–99. est ce que je:10.1109/MCS.2007.365008. ^ "Final Value Theorem of Laplace Transform". ProofWiki. Récupéré 12 Avril 2020. ^ Sauter à: a b c Ullrich, David C.. (2018-05-26). "The tauberian final value Theorem". Math Stack Exchange. ^ Sauter à: 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, Emmanuel (1 Novembre 2003). "Let us teach this generalization of the final-value theorem". Journal européen de physique. 24 (6): 591–597. est ce que je: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. est ce que je: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: Théorèmes de l'analyse de Fourier
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