15.3: Derivation of the Final-Value Theorem

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Consider a continuous physical function \(f(t)\), with continuous derivative \(d f / d t\), and with Laplace transform \(L[f(t)]=F(s)\). The final-value theorem expresses the final, steady-state value of \(f(t)\) in terms of \(F(s)\) as:

\[\lim _{t \rightarrow \infty} f(t)=\lim _{s \rightarrow 0}[s F(s)]\label{eqn:15.15} \]

This theorem is useful for finding the final value because it is almost always easier to derive the Laplace transform and evaluate the limit on the right-hand side, than to derive the equation for \(f(t)\) and evaluate the limit on the left-hand side. Final-value theorem Equation \(\ref{eqn:15.15}\) is valid provided that \(\lim _{t \rightarrow \infty} f(t)\) exists (i.e., is a finite, constant value). But we must apply Equation \(\ref{eqn:15.15}\) with care, because the theorem itself fails to distinguish between functions for which the limit exists and functions which have no limit. Indeed, the theorem can predict falsely that an unstable system has a limit when, in fact, there is none, i.e., that \(\lim _{t \rightarrow \infty} f(t) \rightarrow \pm \infty\).

Derivation of the final-value theorem is based upon definition Equation 2.2.5 of a Laplace transform and the Laplace transform Equation 2.2.9 of a derivative:

\[L\left[\frac{d}{d t} f(t)\right]=\int_{t=0}^{t=\infty} e^{-s t}\left[\frac{d}{d t} f(t)\right] d t=s F(s)-f(0) \nonumber \]

Taking the limit of all terms as \(s \rightarrow 0\) gives

\[\lim _{s \rightarrow 0} L\left[\frac{d}{d t} f(t)\right]=\int_{t=0}^{t=\infty} 1 \times\left[\frac{d}{d t} f(t)\right] d t=\lim _{s \rightarrow 0}[s F(s)]-f(0) \nonumber \]

Now the integral is evaluated easily:

\[\int_{t=0}^{t=\infty}\left[\frac{d}{d t} f(t)\right] d t \equiv f(\infty)-f(0)=\lim _{s \rightarrow 0}[s F(s)]-f(0) \nonumber \]

\[\Rightarrow \quad f(\infty) \equiv \lim _{t \rightarrow \infty} f(t)=\lim _{s \rightarrow 0}[s F(s)] \nonumber \]

This completes the derivation of final-value theorem Equation \(\ref{eqn:15.15}\), which was applied earlier to re-derive result Equation 15.2.12 and in Equation 15.2.18.