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Transfer Function and Poles

A transfer function $G(s)$ describes the ratio between a system's output and input in the Laplace domain. The poles are the values of $s$ that make the denominator zero, and they determine whether the system is stable: if the real part of every pole is negative, they lie in the left half-plane and the system is stable. The zeros shape the response, but do not determine stability.

G(s)=\dfrac{Y(s)}{U(s)}

transfer function, output over input

D(s)=0

characteristic equation; its roots are the poles

Symbols

$G(s)$	transfer function
$Y(s),\ U(s)$	Laplace-transformed output/input
$D(s)$	the denominator (characteristic polynomial)

Example

$G(s)=\dfrac{5}{s+2}$ :

Pole at $s=-2$ , which lies in the left half-plane $\Rightarrow$ stable system.

It is the poles, not the zeros, that determine whether a system is stable.

Practise laplace and transfer functions for free →

First-Order System →

Part of Control Engineering: Laplace and transfer functions.