PlanetMath: transfer function

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transfer function

(Definition)

The transfer function of a linear dynamical system is the ratio of the Laplace transform of its output to the Laplace transform of its input. In systems theory, the Laplace transform is called the “frequency domain” representation of the system.

Consider a canonical dynamical system

$\displaystyle \dot x(t)$	$\displaystyle =$	$\displaystyle A x(t) + B u(t)$
$\displaystyle y (t)$	$\displaystyle =$	$\displaystyle C x(t) + D u(t)$

with input $u: R \mapsto R^n$ , output $y: R \mapsto R^m$ and state $x:R \mapsto R^p$ , and

are constant matrices of conformable sizes.

The frequency domain representation is

$\displaystyle y(s) = (D + C(sI - A)^{-1}B)u(s),$

and thus the transfer function matrix is $D + C(sI - A)^{-1}B$ .

In the case of single-input-single-output systems (), the transfer function is commonly expressed as a rational function of :

$\displaystyle H(s) = \frac{\prod_{i=0}^Z (s - z_i)}{\prod_{i=0}^P (s - p_i)}.$

The values

are called the zeros of

, and the values

are called the poles. If any of the poles has positive real part, then the transfer function is termed unstable; if all of the poles have strictly negative real part, it is stable.

"transfer function" is owned by lha.

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Also defines:

frequency domain, stable, unstable

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Cross-references: negative, strictly, real part, positive, poles, rational function, sizes, matrices, canonical, representation, theory, Laplace transform, ratio, dynamical system
There are 15 references to this entry.

This is version 10 of transfer function, born on 2003-10-16, modified 2006-10-08.
Object id is 5394, canonical name is TransferFunction.
Accessed 12337 times total.

Classification:

AMS MSC:

93A10 (Systems theory; control :: General :: General systems)

Pending Errata and Addenda

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