Pole of a transfer function
High school
Consider a fraction of polynomials, after canceling common factors. A pole of the fraction is a value where the denominator is zero, so that the function blows up near that point.
Undergraduate
Consider a transfer function $G(s)$, a function of the complex variable $s$. A pole of $G$ is a value of $s$ at which $G(s)$ is infinite: for a rational function, a root of the denominator after common factors are canceled.
Examples
Worked examples
- $G(s) = \frac{1}{s + 2}$ : $s = -2$ High school
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$G(s) = \frac{s - 1}{(s + 2)(s + 3)}$
: $s = -2$ and $s = -3$
High school
The poles of $G$ are $-2$ and $-3$, where its denominator is zero.
-
$G(s) = \frac{s + 1}{(s + 1)(s + 2)}$
: $s = -2$
High school
The pole of $G(s)$ is $-2$, because \(G(s) = \frac{1}{s + 2}\) after canceling $s + 1$.
Properties
- For a minimal (controllable and observable) realization $\dot{x} = A x + B u$, $y = C x + D u$, the poles of $G(s) = C (s I - A)^{-1} B + D$ are exactly the eigenvalues of $A$, with multiplicity.
- A linear time-invariant system is BIBO stable if and only if every pole of its transfer function has negative real part.