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. (Gennaio 2022) (Scopri come e quando rimuovere questo messaggio modello) 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, Se {stile di visualizzazione f(t)} in continuous time has (unilateral) Laplace transform {stile di visualizzazione F(S)} , then a final value theorem establishes conditions under which {displaystyle lim _{tto infty }f(t)=lim _{S,a ,0}{sF(S)}} Allo stesso modo, Se {stile di visualizzazione f[K]} in discrete time has (unilateral) Z-transform {stile di visualizzazione 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 {stile di visualizzazione f(t)} (o {stile di visualizzazione f[K]} ) to calculate {displaystyle lim _{S,a ,0}{sF(S)}} . al contrario, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of {stile di visualizzazione F(S)} to calculate {displaystyle lim _{tto infty }f(t)} (o {displaystyle lim _{kto infty }f[K]} ) (see Abelian and Tauberian theorems for integral transforms).
Contenuti 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 Applicazioni 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 Applicazioni 1.3 Esempi 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 Guarda anche 5 Appunti 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 {stile di visualizzazione s} approaches 0, whereas ' {displaystyle sdownarrow 0} ' means that {stile di visualizzazione s} approaches 0 through the positive numbers.
Standard Final Value Theorem Suppose that every pole of {stile di visualizzazione F(S)} is either in the open left half plane or at the origin, e quello {stile di visualizzazione F(S)} has at most a single pole at the origin. Quindi {displaystyle sF(S)to Lin mathbb {R} } come {displaystyle sto 0} , e {displaystyle lim _{tto infty }f(t)=L} .[5] Final Value Theorem using Laplace transform of the derivative Suppose that {stile di visualizzazione f(t)} e {stile di visualizzazione 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,a ,0}{sF(S)}} exists then {displaystyle lim _{tto infty }f(t)=lim _{S,a ,0}{sF(S)}} .[3]: Theorem 2.36 [4]: 20 [6] Remark Both limits must exist for the theorem to hold. Per esempio, Se {stile di visualizzazione f(t)=sin(t)} poi {displaystyle lim _{tto infty }f(t)} non esiste, ma {displaystyle lim _{S,a ,0}{sF(S)}=lim _{S,a ,0}{frac {S}{s^{2}+1}}=0} .[3]: Example 2.37 [4]: 20 Improved Tauberian converse Final Value Theorem Suppose that {stile di visualizzazione f:(0,infty )a matematicabb {C} } is bounded and differentiable, e quello {displaystyle tf'(t)} is also bounded on {stile di visualizzazione (0,infty )} . Se {displaystyle sF(S)to Lin mathbb {C} } come {displaystyle sto 0} poi {displaystyle lim _{tto infty }f(t)=L} .[7] Extended Final Value Theorem Suppose that every pole of {stile di visualizzazione 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} } come {displaystyle sdownarrow 0} , e {displaystyle lim _{tto infty }f(t)=L} . {displaystyle sF(S)to +infty in mathbb {R} } come {displaystyle sdownarrow 0} , e {stile di visualizzazione f(t)a +infty } come {displaystyle tto infty } . {displaystyle sF(S)to -infty in mathbb {R} } come {displaystyle sdownarrow 0} , e {stile di visualizzazione f(t)to -infty } come {displaystyle tto infty } .
In particolare, Se {displaystyle s=0} is a multiple pole of {stile di visualizzazione F(S)} then case 2 o 3 applies ( {stile di visualizzazione f(t)a +infty } o {stile di visualizzazione f(t)to -infty } ).[5] Generalized Final Value Theorem Suppose that {stile di visualizzazione f(t)} is Laplace transformable. Permettere {displaystyle lambda >-1} . Se {displaystyle lim _{tto infty }{frac {f(t)}{t^{lambda }}}} exists and {displaystyle lim _{sdownarrow 0}{s^{lambda +1}F(S)}} exists then {displaystyle lim _{tto infty }{frac {f(t)}{t^{lambda }}}={frac {1}{Gamma (lambda +1)}}lim _{sdownarrow 0}{s^{lambda +1}F(S)}} dove {stile di visualizzazione Gamma (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 {stile di visualizzazione f:(0,infty )a matematicabb {C} } is bounded and measurable and {displaystyle lim _{tto infty }f(t)=alpha in mathbb {C} } . Quindi {stile di visualizzazione F(S)} exists for all {displaystyle s>0} e {displaystyle lim _{S,a ,0^{+}}{sF(S)}=alpha } .[7] Prova elementare[7] Suppose for convenience that {stile di visualizzazione |f(t)|leq 1} Su {stile di visualizzazione (0,infty )} , e lascia {displaystyle alpha =lim _{tto infty }f(t)} . Permettere {displaystyle epsilon >0} , e scegli {stile di visualizzazione A} affinché {stile di visualizzazione |f(t)-alfa |
Final Value Theorem using Laplace transform of the derivative Suppose that all of the following conditions are satisfied: {stile di visualizzazione f:(0,infty )a matematicabb {C} } is continuously differentiable and both {stile di visualizzazione f} e {stile di visualizzazione f'} have a Laplace transform {stile di visualizzazione f'} is absolutely integrable - questo è, {displaystyle int _{0}^{infty }|f'(sì )|,dtau } è 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 {stile di visualizzazione f:(0,infty )a matematicabb {C} } be a continuous and bounded function such that such that the following limit exists {displaystyle lim _{Tto infty }{frac {1}{T}}int _{0}^{T}f(t),dt=alpha in mathbb {C} } Quindi {displaystyle lim _{S,a ,0,,s>0}{sF(S)}=alpha } .[9] Final Value Theorem for asymptotic sums of periodic functions Suppose that {stile di visualizzazione f:[0,infty )a matematicabb {R} } is continuous and absolutely integrable in {stile di visualizzazione [0,infty )} . Suppose further that {stile di visualizzazione f} is asymptotically equal to a finite sum of periodic functions {stile di visualizzazione f_{matematica {come} }} , questo è {stile di visualizzazione |f(t)-f_{matematica {come} }(t)|
To establish the convergence of the improper integral {displaystyle lim _{xto infty }f(X)} in pratica, Dirichlet's test for improper integrals is often helpful. An example is the Dirichlet integral.
Applications Final value theorems for obtaining {displaystyle lim _{S,a ,0}{sF(S)}} have applications in probability and statistics to calculate the moments of a random variable. Permettere {stile di visualizzazione R(X)} be cumulative distribution function of a continuous random variable {stile di visualizzazione X} e lascia {stile di visualizzazione rho (S)} be the Laplace–Stieltjes transform of {stile di visualizzazione R(X)} . Then the {stile di visualizzazione n} -th moment of {stile di visualizzazione X} can be calculated as {stile di visualizzazione E[X^{n}]=(-1)^{n}sinistra.{frac {d^{n}rho (S)}{ds^{n}}}Giusto|_{s=0}} The strategy is to write {stile di visualizzazione {frac {d^{n}rho (S)}{ds^{n}}}={matematico {F}}{all'improvviso (}G_{1}(S),G_{2}(S),punti ,G_{K}(S),punti {più grande )}} dove {stile di visualizzazione {matematico {F}}(punti )} is continuous and for each {stile di visualizzazione k} , {stile di visualizzazione G_{K}(S)=sF_{K}(S)} per una funzione {stile di visualizzazione F_{K}(S)} . For each {stile di visualizzazione k} , put {stile di visualizzazione f_{K}(t)} as the inverse Laplace transform of {stile di visualizzazione F_{K}(S)} , obtain {displaystyle lim _{tto infty }f_{K}(t)} , and apply a final value theorem to deduce {displaystyle lim _{S,a ,0}{G_{K}(S)}=lim _{S,a ,0}{sF_{K}(S)}=lim _{tto infty }f_{K}(t)} . Quindi {displaystyle left.{frac {d^{n}rho (S)}{ds^{n}}}Giusto|_{s=0}={matematico {F}}{Grande (}lim _{S,a ,0}G_{1}(S),lim _{S,a ,0}G_{2}(S),punti ,lim _{S,a ,0}G_{K}(S),punti {Più grande )}} e quindi {stile di visualizzazione E[X^{n}]} is obtained.
Examples This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Il materiale non fornito può essere contestato e rimosso. (ottobre 2011) (Scopri come e quando rimuovere questo messaggio modello) Example where FVT holds For example, for a system described by transfer function {stile di visualizzazione H(S)={frac {6}{s+2}},} and so the impulse response converges to {displaystyle lim _{tto infty }h(t)=lim _{sto 0}{frac {6S}{s+2}}=0.} Questo è, the system returns to zero after being disturbed by a short impulse. Tuttavia, the Laplace transform of the unit step response is {stile di visualizzazione G(S)={frac {1}{S}}{frac {6}{s+2}}} and so the step response converges to {displaystyle 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 {stile di visualizzazione 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. Tuttavia, neither time-domain limit exists, and so the final value theorem predictions are not valid. Infatti, 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 {stile di visualizzazione H(S)} must have negative real parts. {stile di visualizzazione 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,a ,1}{(z-1)F(z)}} exists then {displaystyle lim _{kto infty }f[K]=lim _{z,a ,1}{(z-1)F(z)}} .[4]: 101 Final value of linear systems Continuous-time LTI systems Final value of the system {stile di visualizzazione {punto {mathbf {X} }}(t)=mathbf {UN} mathbf {X} (t)+mathbf {B} mathbf {tu} (t)} {displaystyle mathbf {y} (t)=mathbf {C} mathbf {X} (t)} in response to a step input {displaystyle mathbf {tu} (t)} with amplitude {stile di visualizzazione R} è: {displaystyle lim _{tto infty }mathbf {y} (t)=mathbf {circa} ^{-1}mathbf {B} R} Sampled-data systems The sampled-data system of the above continuous-time LTI system at the aperiodic sampling times {stile di visualizzazione t_{io},i=1,2,...} is the discrete-time system {stile di visualizzazione {mathbf {X} }(t_{io+1})=mathbf {Phi } (h_{io})mathbf {X} (t_{io})+mathbf {Gamma } (h_{io})mathbf {tu} (t_{io})} {displaystyle mathbf {y} (t_{io})=mathbf {C} mathbf {X} (t_{io})} dove {stile di visualizzazione h_{io}=t_{io+1}-t_{io}} e {displaystyle mathbf {Phi } (h_{io})=e^{mathbf {UN} h_{io}}} , {displaystyle mathbf {Gamma } (h_{io})=int _{0}^{h_{io}}e^{mathbf {UN} S},ds} The final value of this system in response to a step input {displaystyle mathbf {tu} (t)} with amplitude {stile di visualizzazione 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". Recuperato 2011-10-21. ^ Alan V. Oppenheim; Alan S. Willsky; S. Hamid Nawab (1997). Signals & Systems. New Jersey, Stati Uniti d'America: Sala dell'Apprendista. ISBN 0-13-814757-4. ^ Salta su: a b c Schiff, Joel L. (1999). The Laplace Transform: Teoria e applicazioni. New York: Springer. ISBN 978-1-4757-7262-3. ^ Salta su: a b c d Graf, Urs (2004). Applied Laplace Transforms and z-Transforms for Scientists and Engineers. Basilea: Birkhauser Verlag. ISBN 3-7643-2427-9. ^ Salta su: a b c Chen, Jie; Lundberg, Kent H.; Davison, Daniel E.; Bernstein, Dennis S. (Giugno 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. Recuperato 12 aprile 2020. ^ Salta su: a b c Ullrich, David C. (2018-05-26). "The tauberian final value Theorem". Math Stack Exchange. ^ Salta su: 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, Emanuele (1 novembre 2003). "Let us teach this generalization of the final-value theorem". Giornale europeo di fisica. 24 (6): 591–597. doi:10.1088/0143-0807/24/6/005. ^ Hew, Patrizio (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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