# transfer function

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^{}

$\dot{x}(t)$ | $=$ | $Ax(t)+Bu(t)$ | ||

$y(t)$ | $=$ | $Cx(t)+Du(t)$ |

with input $u:R\mapsto {R}^{n}$, output $y:R\mapsto {R}^{m}$ and state $x:R\mapsto {R}^{p}$, and $(A,B,C,D)$ are constant matrices of conformable sizes.

The frequency domain representation is

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

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

In the case of single-input-single-output systems ($m=n=1$), the transfer function is commonly expressed as a rational function of $s$:

$$H(s)=\frac{{\prod}_{i=0}^{Z}(s-{z}_{i})}{{\prod}_{i=0}^{P}(s-{p}_{i})}.$$ |

The values ${z}_{i}$ are called the zeros of $H(s)$, and the values ${p}_{i}$ 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*.

Title | transfer function |
---|---|

Canonical name | TransferFunction |

Date of creation | 2013-03-22 14:02:41 |

Last modified on | 2013-03-22 14:02:41 |

Owner | lha (3057) |

Last modified by | lha (3057) |

Numerical id | 13 |

Author | lha (3057) |

Entry type | Definition |

Classification | msc 93A10 |

Defines | frequency domain |

Defines | stable |

Defines | unstable |