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Final value theorem

The final value theorem gives the steady-state value directly from the Laplace expression, without working out the whole time response. It is useful for finding steady-state error in control systems, but only holds when the system is stable.

\lim_{t\to\infty} y(t) = \lim_{s\to 0} s\,Y(s)

final value

Y(s) = G(s)\frac{1}{s}

response to a unit step

Symbols

$Y(s)$	Laplace transform of the output
$G(s)$	transfer function

Example

$G(s) = \frac{4}{2s + 1}$ with a unit step:

$\lim_{s\to 0} s\cdot\frac{4}{2s+1}\cdot\frac1s = 4$ .

Check stability first. For an unstable system the theorem gives a meaningless number.

Practise step response and first-order systems for free →

← Rise and settling time

Part of Control Engineering: Step response and first-order systems.