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. (January 2022) (Learn how and when to remove this template 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, if {displaystyle f(t)} in continuous time has (unilateral) Laplace transform {displaystyle F(s)} , then a final value theorem establishes conditions under which {displaystyle lim _{tto infty }f(t)=lim _{s,to ,0}{sF(s)}} Likewise, if {displaystyle f[k]} in discrete time has (unilateral) Z-transform {displaystyle 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 {displaystyle f(t)} (or {displaystyle f[k]} ) to calculate {displaystyle lim _{s,to ,0}{sF(s)}} . Conversely, a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of {displaystyle F(s)} to calculate {displaystyle lim _{tto infty }f(t)} (or {displaystyle lim _{kto infty }f[k]} ) (see Abelian and Tauberian theorems for integral transforms).

Contents 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 Examples 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 See also 5 Notes 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 {displaystyle s} approaches 0, whereas ' {displaystyle sdownarrow 0} ' means that {displaystyle s} approaches 0 through the positive numbers.

Standard Final Value Theorem Suppose that every pole of {displaystyle F(s)} is either in the open left half plane or at the origin, and that {displaystyle F(s)} has at most a single pole at the origin. Then {displaystyle sF(s)to Lin mathbb {R} } as {displaystyle sto 0} , and {displaystyle lim _{tto infty }f(t)=L} .[5] Final Value Theorem using Laplace transform of the derivative Suppose that {displaystyle f(t)} and {displaystyle f'(t)} both have Laplace transforms that exist for all {displaystyle s>0} . If {displaystyle lim _{tto infty }f(t)} exists and {displaystyle lim _{s,to ,0}{sF(s)}} exists then {displaystyle lim _{tto infty }f(t)=lim _{s,to ,0}{sF(s)}} .[3]: Theorem 2.36 [4]: 20 [6] Remark Both limits must exist for the theorem to hold. For example, if {displaystyle f(t)=sin(t)} then {displaystyle lim _{tto infty }f(t)} does not exist, but {displaystyle lim _{s,to ,0}{sF(s)}=lim _{s,to ,0}{frac {s}{s^{2}+1}}=0} .[3]: Example 2.37 [4]: 20  Improved Tauberian converse Final Value Theorem Suppose that {displaystyle f:(0,infty )to mathbb {C} } is bounded and differentiable, and that {displaystyle tf'(t)} is also bounded on {displaystyle (0,infty )} . If {displaystyle sF(s)to Lin mathbb {C} } as {displaystyle sto 0} then {displaystyle lim _{tto infty }f(t)=L} .[7] Extended Final Value Theorem Suppose that every pole of {displaystyle 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} } as {displaystyle sdownarrow 0} , and {displaystyle lim _{tto infty }f(t)=L} . {displaystyle sF(s)to +infty in mathbb {R} } as {displaystyle sdownarrow 0} , and {displaystyle f(t)to +infty } as {displaystyle tto infty } . {displaystyle sF(s)to -infty in mathbb {R} } as {displaystyle sdownarrow 0} , and {displaystyle f(t)to -infty } as {displaystyle tto infty } .

In particular, if {displaystyle s=0} is a multiple pole of {displaystyle F(s)} then case 2 or 3 applies ( {displaystyle f(t)to +infty } or {displaystyle f(t)to -infty } ).[5] Generalized Final Value Theorem Suppose that {displaystyle f(t)} is Laplace transformable. Let {displaystyle lambda >-1} . If {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)}} where {displaystyle 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 {displaystyle f:(0,infty )to mathbb {C} } is bounded and measurable and {displaystyle lim _{tto infty }f(t)=alpha in mathbb {C} } . Then {displaystyle F(s)} exists for all {displaystyle s>0} and {displaystyle lim _{s,to ,0^{+}}{sF(s)}=alpha } .[7] Elementary proof[7] Suppose for convenience that {displaystyle |f(t)|leq 1} on {displaystyle (0,infty )} , and let {displaystyle alpha =lim _{tto infty }f(t)} . Let {displaystyle epsilon >0} , and choose {displaystyle A} so that {displaystyle |f(t)-alpha |A} . Since {displaystyle sint _{0}^{infty }e^{-st},dt=1} , for every {displaystyle s>0} we have {displaystyle sF(s)-alpha =sint _{0}^{infty }(f(t)-alpha )e^{-st},dt;} hence {displaystyle |sF(s)-alpha |leq sint _{0}^{A}|f(t)-alpha |e^{-st},dt+sint _{A}^{infty }|f(t)-alpha |e^{-st},dtleq 2sint _{0}^{A}e^{-st},dt+epsilon sint _{A}^{infty }e^{-st},dt=I+II.} Now for every {displaystyle s>0} we have {displaystyle II0} is small enough.

Final Value Theorem using Laplace transform of the derivative Suppose that all of the following conditions are satisfied: {displaystyle f:(0,infty )to mathbb {C} } is continuously differentiable and both {displaystyle f} and {displaystyle f'} have a Laplace transform {displaystyle f'} is absolutely integrable - that is, {displaystyle int _{0}^{infty }|f'(tau )|,dtau } is finite {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 {displaystyle f:(0,infty )to mathbb {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} } Then {displaystyle lim _{s,to ,0,,s>0}{sF(s)}=alpha } .[9] Final Value Theorem for asymptotic sums of periodic functions Suppose that {displaystyle f:[0,infty )to mathbb {R} } is continuous and absolutely integrable in {displaystyle [0,infty )} . Suppose further that {displaystyle f} is asymptotically equal to a finite sum of periodic functions {displaystyle f_{mathrm {as} }} , that is {displaystyle |f(t)-f_{mathrm {as} }(t)|0} , {displaystyle sint _{0}^{infty }e^{-st}f(t),dt={Big [}-e^{-st}f(t){Big ]}_{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)} for {displaystyle sto 0} .

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

Applications Final value theorems for obtaining {displaystyle lim _{s,to ,0}{sF(s)}} have applications in probability and statistics to calculate the moments of a random variable. Let {displaystyle R(x)} be cumulative distribution function of a continuous random variable {displaystyle X} and let {displaystyle rho (s)} be the Laplace–Stieltjes transform of {displaystyle R(x)} . Then the {displaystyle n} -th moment of {displaystyle X} can be calculated as {displaystyle E[X^{n}]=(-1)^{n}left.{frac {d^{n}rho (s)}{ds^{n}}}right|_{s=0}} The strategy is to write {displaystyle {frac {d^{n}rho (s)}{ds^{n}}}={mathcal {F}}{bigl (}G_{1}(s),G_{2}(s),dots ,G_{k}(s),dots {bigr )}} where {displaystyle {mathcal {F}}(dots )} is continuous and for each {displaystyle k} , {displaystyle G_{k}(s)=sF_{k}(s)} for a function {displaystyle F_{k}(s)} . For each {displaystyle k} , put {displaystyle f_{k}(t)} as the inverse Laplace transform of {displaystyle F_{k}(s)} , obtain {displaystyle lim _{tto infty }f_{k}(t)} , and apply a final value theorem to deduce {displaystyle lim _{s,to ,0}{G_{k}(s)}=lim _{s,to ,0}{sF_{k}(s)}=lim _{tto infty }f_{k}(t)} . Then {displaystyle 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 )}} and hence {displaystyle E[X^{n}]} is obtained.

Examples This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2011) (Learn how and when to remove this template message) Example where FVT holds For example, for a system described by transfer function {displaystyle 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.} That is, the system returns to zero after being disturbed by a short impulse. However, the Laplace transform of the unit step response is {displaystyle 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 {displaystyle 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. 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 {displaystyle H(s)} must have negative real parts. {displaystyle 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} and {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,to ,1}{(z-1)F(z)}} exists then {displaystyle lim _{kto infty }f[k]=lim _{z,to ,1}{(z-1)F(z)}} .[4]: 101  Final value of linear systems Continuous-time LTI systems Final value of the system {displaystyle {dot {mathbf {x} }}(t)=mathbf {A} mathbf {x} (t)+mathbf {B} mathbf {u} (t)} {displaystyle mathbf {y} (t)=mathbf {C} mathbf {x} (t)} in response to a step input {displaystyle mathbf {u} (t)} with amplitude {displaystyle R} is: {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 {displaystyle t_{i},i=1,2,...} is the discrete-time system {displaystyle {mathbf {x} }(t_{i+1})=mathbf {Phi } (h_{i})mathbf {x} (t_{i})+mathbf {Gamma } (h_{i})mathbf {u} (t_{i})} {displaystyle mathbf {y} (t_{i})=mathbf {C} mathbf {x} (t_{i})} where {displaystyle h_{i}=t_{i+1}-t_{i}} and {displaystyle mathbf {Phi } (h_{i})=e^{mathbf {A} h_{i}}} , {displaystyle mathbf {Gamma } (h_{i})=int _{0}^{h_{i}}e^{mathbf {A} s},ds} The final value of this system in response to a step input {displaystyle mathbf {u} (t)} with amplitude {displaystyle 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". Retrieved 2011-10-21. ^ Alan V. Oppenheim; Alan S. Willsky; S. Hamid Nawab (1997). Signals & Systems. New Jersey, USA: Prentice Hall. ISBN 0-13-814757-4. ^ Jump up to: a b c Schiff, Joel L. (1999). The Laplace Transform: Theory and Applications. New York: Springer. ISBN 978-1-4757-7262-3. ^ Jump up to: a b c d Graf, Urs (2004). Applied Laplace Transforms and z-Transforms for Scientists and Engineers. Basel: Birkhäuser Verlag. ISBN 3-7643-2427-9. ^ Jump up to: a b c Chen, Jie; Lundberg, Kent H.; Davison, Daniel E.; Bernstein, Dennis S. (June 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. Retrieved 12 April 2020. ^ Jump up to: a b c Ullrich, David C. (2018-05-26). "The tauberian final value Theorem". Math Stack Exchange. ^ Jump up to: 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 November 2003). "Let us teach this generalization of the final-value theorem". European Journal of Physics. 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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