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Final value theorem

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. Mathematically, if $f(t)$ in continuous time has (unilateral) Laplace transform $F(s)$ then a final value theorem establishes conditions under which

$\lim _{t\to \infty }f(t)=\lim _{s\,\to \,0}{sF(s)}$

Likewise, if $f[k]$ in discrete time has (unilateral) Z-transform $F(z)$ then a final value theorem establishes conditions under which

$\lim _{k\to \infty }f[k]=\lim _{z\to 1}{(z-1)F(z)}$

An Abelian final value theorem makes assumptions about the time-domain behavior of $f(t)$ (or $f[k]$) to calculate $\lim _{s\,\to \,0}{sF(s)}$. Conversely a Tauberian final value theorem makes assumptions about the frequency-domain behaviour of $F(s)$ to calculate $\lim _{t\to \infty }f(t)$ (or $\lim _{k\to \infty }f[k]$) (see Abelian and Tauberian theorems for integral transforms).

Final value theorems for the Laplace transform

Deducing $\lim _{t\to \infty }f(t)$

In the following statements, the notation '$s\to 0$' means that $s$ approaches 0, whereas '$s\downarrow 0$' means that $s$ approaches 0 through the positive numbers.

Standard Final Value Theorem

Suppose that every pole of $F(s)$ is either in the open left half plane or at the origin, and that $F(s)$ has at most a single pole at the origin. Then $sF(s)\to L\in \mathbb {R}$ as $s\to 0$, and $\lim _{t\to \infty }f(t)=L$.

Final Value Theorem using Laplace Transform of the Derivative

Suppose that $f(t)$ and $f'(t)$ both have Laplace transforms that exist for all $s>0$. If $\lim _{t\to \infty }f(t)$ exists and $\lim _{s\,\to \,0}{sF(s)}$ exists then $\lim _{t\to \infty }f(t)=\lim _{s\,\to \,0}{sF(s)}$.

Remark

Both limits must exist for the theorem to hold. For example, if $f(t)=\sin(t)$ then $\lim _{t\to \infty }f(t)$ does not exist, but $\lim _{s\,\to \,0}{sF(s)}=\lim _{s\,\to \,0}{\frac {s}{s^{2}+1}}=0$.

Improved Tauberian Converse Final Value Theorem

Suppose that $f:(0,\infty )\to \mathbb {C}$ is bounded and differentiable, and that $tf'(t)$ is also bounded on $(0,\infty )$. If $sF(s)\to L\in \mathbb {C}$ as $s\to 0$ then $\lim _{t\to \infty }f(t)=L$.

Extended Final Value Theorem

Suppose that every pole of $F(s)$ is either in the open left half plane or at the origin. Then one of the following occurs:

1. $sF(s)\to L\in \mathbb {R}$ as $s\downarrow 0$, and $\lim _{t\to \infty }f(t)=L$.
2. $sF(s)\to +\infty \in \mathbb {R}$ as $s\downarrow 0$, and $f(t)\to +\infty$ as $t\to \infty$.
3. $sF(s)\to -\infty \in \mathbb {R}$ as $s\downarrow 0$, and $f(t)\to -\infty$ as $t\to \infty$.

In particular, if $s=0$ is a multiple pole of $F(s)$ then case 2 or 3 applies ($f(t)\to +\infty$ or $f(t)\to -\infty$).

Generalized Final Value Theorem

Suppose that $f(t)$ is Laplace transformable. Let $\lambda >-1$. If $\lim _{t\to \infty }{\frac {f(t)}{t^{\lambda }}}$ exists and $\lim _{s\downarrow 0}{s^{\lambda +1}F(s)}$ exists then

$\lim _{t\to \infty }{\frac {f(t)}{t^{\lambda }}}={\frac {1}{\Gamma (\lambda +1)}}\lim _{s\downarrow 0}{s^{\lambda +1}F(s)}$

where $\Gamma (x)$ denotes the Gamma function.

Applications

Final value theorems for obtaining $\lim _{t\to \infty }f(t)$ have applications in establishing the long-term stability of a system.

Deducing $\lim _{s\,\to \,0}{sF(s)}$

Abelian Final Value Theorem

Suppose that $f:(0,\infty )\to \mathbb {C}$ is bounded and measurable and $\lim _{t\to \infty }f(t)=\alpha \in \mathbb {C}$. Then $F(s)$ exists for all $s>0$ and $\lim _{s\,\to \,0^{+}}{sF(s)}=\alpha$.

Elementary proof

Suppose for convenience that $|f(t)|\leq 1$ on $(0,\infty )$, and let $\alpha =\lim _{t\to \infty }f(t)$. Let $\epsilon >0$, and choose $A$ so that $|f(t)-\alpha |<\epsilon$ for all $t>A$. Since $s\int _{0}^{\infty }e^{-st}\,dt=1$, for every $s>0$ we have

$sF(s)-\alpha =s\int _{0}^{\infty }(f(t)-\alpha )e^{-st}\,dt;$

hence

$|sF(s)-\alpha |\leq s\int _{0}^{A}|f(t)-\alpha |e^{-st}\,dt+s\int _{A}^{\infty }|f(t)-\alpha |e^{-st}\,dt\leq 2s\int _{0}^{A}e^{-st}\,dt+\epsilon s\int _{A}^{\infty }e^{-st}\,dt=I+II.$

Now for every $s>0$ we have

$II<\epsilon s\int _{0}^{\infty }e^{-st}\,dt=\epsilon$.

On the other hand, since $A<\infty$ is fixed it is clear that $\lim _{s\to 0}I=0$, and so $|sF(s)-\alpha |<\epsilon$ if $s>0$ is small enough.

Final Value Theorem using Laplace Transform of the Derivative

Suppose that all of the following conditions are satisfied:

1. $f:(0,\infty )\to \mathbb {C}$ is continuously differentiable and both $f$ and $f'$ have a Laplace Transform
2. $f'$ is absolutely integrable, that is $\int _{0}^{\infty }|f'(\tau )|\,d\tau$ is finite
3. $\lim _{t\to \infty }f(t)$ exists and is finite

Then

$\lim _{s\to 0^{+}}sF(s)=\lim _{t\to \infty }f(t)$.

Remark

The proof uses the Dominated Convergence Theorem.

Final Value Theorem for Mean of a Function

Let $f:(0,\infty )\to \mathbb {C}$ be a continuous and bounded function such that such that the following limit exists

$\lim _{T\to \infty }{\frac {1}{T}}\int _{0}^{T}f(t)\,dt=\alpha \in \mathbb {C}$

Then $\lim _{s\,\to \,0,\,s>0}{sF(s)}=\alpha$.

Final Value Theorem for Asymptotic Sums of Periodic Functions

Suppose that $f:[0,\infty )\to \mathbb {R}$ is continuous and absolutely integrable in $[0,\infty )$. Suppose further that $f$ is asymptotically equal to a finite sum of periodic functions $f_{\mathrm {as} }$, that is

$|f(t)-f_{\mathrm {as} }(t)|<\phi (t)$

where $\phi (t)$ is absolutely integrable in $[0,\infty )$ and vanishes at infinity. Then

$\lim _{s\to 0}sF(s)=\lim _{t\to \infty }{\frac {1}{t}}\int _{0}^{t}f(x)\,dx$.

Final Value Theorem for a Function that diverges to infinity

Let $f(t):[0,\infty )\to \mathbb {R}$ and $F(s)$ be the Laplace transform of $f(t)$. Suppose that $f(t)$ satisfies all of the following conditions:

1. $f(t)$ is infinitely differentiable at zero
2. $f^{(k)}(t)$ has a Laplace transform for all non-negative integers $k$
3. $f(t)$ diverges to infinity as $t\to \infty$

Then $sF(s)$ diverges to infinity as $s\to 0^{+}$.

Applications

Final value theorems for obtaining $\lim _{s\,\to \,0}{sF(s)}$ have applications in probability & statistics to calculate the moments of a random variable. Let $R(x)$ be cumulative distribution function of a continuous random variable $X$ and let $\rho (s)$ be the Laplace-Stieltjes transform of $R(x)$. Then the $n$th moment of $X$ can be calculated as

$E[X^{n}]=(-1)^{n}\left.{\frac {d^{n}\rho (s)}{ds^{n}}}\right|_{s=0}$

The strategy is to write

${\frac {d^{n}\rho (s)}{ds^{n}}}={\mathcal {F}}{\bigl (}G_{1}(s),G_{2}(s),\dots ,G_{k}(s),\dots {\bigr )}$

where ${\mathcal {F}}(\dots )$ is continuous and for each $k$, $G_{k}(s)=sF_{k}(s)$ for a function $F_{k}(s)$. For each $k$, put $f_{k}(t)$ as the inverse Laplace transform of $F_{k}(s)$, obtain $\lim _{t\to \infty }f_{k}(t)$, and apply a final value theorem to deduce $\lim _{s\,\to \,0}{G_{k}(s)}=\lim _{s\,\to \,0}{sF_{k}(s)}=\lim _{t\to \infty }f_{k}(t)$. Then

$\left.{\frac {d^{n}\rho (s)}{ds^{n}}}\right|_{s=0}={\mathcal {F}}{\Bigl (}\lim _{s\,\to \,0}G_{1}(s),\lim _{s\,\to \,0}G_{2}(s),\dots ,\lim _{s\,\to \,0}G_{k}(s),\dots {\Bigr )}$

and hence $E[X^{n}]$ is obtained.

Examples

Example where FVT holds

For example, for a system described by transfer function

$H(s)={\frac {6}{s+2}},$

and so the impulse response converges to

$\lim _{t\to \infty }h(t)=\lim _{s\,\searrow \,0}{\frac {6s}{s+2}}=0.$

That is, the system returns to zero after being disturbed by a short impulse. However, the Laplace transform of the unit step response is

$G(s)={\frac {1}{s}}{\frac {6}{s+2}}$

and so the step response converges to

$\lim _{t\to \infty }g(t)=\lim _{s\,\searrow \,0}{\frac {s}{s}}{\frac {6}{s+2}}={\frac {6}{2}}=3$

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

$H(s)={\frac {9}{s^{2}+9}},$

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:

1. All non-zero roots of the denominator of $H(s)$ must have negative real parts.
2. $H(s)$ 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 $0+j3$ and $0-j3$.

Final value theorems for the Z transform

Deducing $\lim _{t\to \infty }f[k]$

Final Value Theorem

If $\lim _{k\to \infty }f[k]$ exists and $\lim _{z\,\to \,1}{(z-1)F(z)}$ exists then $\lim _{k\to \infty }f[k]=\lim _{z\,\to \,1}{(z-1)F(z)}$.

Other things

Carbon based life

In the movie adaptation of Arthur C. Clarke's "2010" (1984) a character argues, "Whether we are based on carbon or on silicon makes no fundamental difference; we should each be treated with appropriate respect". This quote may be the basis of Steve Job's quip in 1998 when he introduced Carbon within MacOS X, "Carbon. All life forms will be based on it".

Calculating the variance

Let S_n be the sum of n random variables. Many central limit theorems provide conditions such that S_n/√Var(S_n) converges in distribution to N(0,1) (the normal distribution with mean 0, variance 1) as n→ ∞. In some cases, it is possible to find a constant σ² and function f(n) such that S_n/(σ√n⋅f(n)) converges in distribution to N(0,1) as n→ ∞.

Lemma. Suppose $X_{1},X_{2},\dots$ is a sequence of real-valued and strictly stationary random variables with $\mathbb {E} (X_{i})=0$ for all $i$, $g:[0,1]\rightarrow \mathbb {R}$, and $S_{n}=\sum _{i=1}^{n}g({\tfrac {i}{n}})X_{i}$. Construct

$\sigma ^{2}=\mathbb {E} (X_{1}^{2})+2\sum _{i=1}^{\infty }\mathbb {E} (X_{1}X_{1+i})$

1. If $\sum _{i=1}^{\infty }\mathbb {E} (X_{1}X_{1+i})$ is absolutely convergent, $\left|\int _{0}^{1}g(x)g'(x)\,dx\right|<\infty$, and $0<\int _{0}^{1}(g(x))^{2}dx<\infty$ then $\mathrm {Var} (S_{n})/(n\gamma _{n})\rightarrow \sigma ^{2}$ as $n\rightarrow \infty$ where $\gamma _{n}={\frac {1}{n}}\sum _{i=1}^{n}(g({\tfrac {i}{n}}))^{2}$.
2. If in addition $\sigma >0$ and $S_{n}/{\sqrt {\mathrm {Var} (S_{n})}}$ converges in distribution to ${\mathcal {N}}(0,1)$ as $n\rightarrow \infty$ then $S_{n}/(\sigma {\sqrt {n\gamma _{n}}})$ also converges in distribution to ${\mathcal {N}}(0,1)$ as $n\rightarrow \infty$.

Alternately, replace the 3n+1 with n' / H(n') where n' = 3n+1 and H(n') is the highest power of 2 that divides n' (with no remainder). The resulting function f maps from odd numbers to odd numbers. Now suppose that for some odd number n, applying this operation k times yields the number 1 (that is, $f^{k}(n)=1$). Then in binary, the number n can be written as the concatenation of strings w_k w_k-1 … w₁ where each w_h is a finite and contiguous extract from the representation of 1 / 3^h . The representation of n therefore holds the repetends of 1 / 3^h , where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs.

Artificial systems and moral responsibility

The emergence of robotics and automation prompted the question, 'Can an artificial system can be morally responsible?' The question has a closely-related variant, 'When (if ever) does moral responsibility transfer from its human creator(s) to the system?'

Arguments against the possibility of artificial systems being morally responsible

Batya Friedman and Peter Kahn Jr posited that intentionality is a necessary condition for moral responsibility, and that computer systems as conceivable in 1992 in material and structure could not have intentionality.

Arthur Kuflik asserted that humans must bear the ultimate moral responsibility for a computer's decisions, as it is humans who design the computers and write their programs. He further proposed that humans can never relinquish oversight of computers.

Frances Grodzinsky et al considered artificial systems that could be modelled as finite state machines. They posited that if the machine had a fixed state transition table, then it could not be morally responsible. If the machine could modify its table, then the machine's designer still retained some moral responsibility.

Patrick Hew argued that for an artificial system to be morally responsible, its rules for behaviour and the mechanisms for supplying those rules must not be supplied entirely by external humans. He further argued that such systems are a substantial departure from technologies and theory as extant in 2014. An artificial system based on those technologies will carry zero responsibility for its behaviour. Moral responsibility is apportioned to the humans that created and programmed the system.

(Further arguments may be found in .)

Arguments that artificial systems can be morally responsible

Colin Allen et al proposed that an artificial system may be morally responsible if its behaviours are functionally indistinguishable from a moral person, coining the idea of a 'Moral Turing Test'. They subsequently disavowed the Moral Turing Test in recognition of controversies surrounding the Turing test.

Andreas Matthias described a 'responsibility gap' where to hold humans responsible for a machine would be an injustice, but to hold the machine responsible would challenge 'traditional' ways of ascription. He proposed three cases where the machine's behaviour ought to be attributed to the machine and not its designers or operators. First, he argued that modern machines are inherently unpredictable (to some degree), but perform tasks that need to be performed yet cannot be handled by simpler means. Second, that there are increasing 'layers of obscurity' between manufacturers and system, as hand coded programs are replaced with more sophisticated means. Third, in systems that have rules of operation that can be changed during the operation of the machine.

(Further arguments may be found in .)