# Hurwitz matrix

A square matrix^{} $A$ is called a Hurwitz matrix if all eigenvalues^{} of $A$ have strictly negative real part, $$; $A$ is also called a stability matrix, because the feedback system

$$\dot{x}=Ax$$ |

is stable.

If $G(s)$ is a (matrix-valued) transfer function, then $G$ is called Hurwitz if the poles of all elements of $G$ have negative real part. Note that it is not necessary that $G(s)$, for a specific argument $s$, be a Hurwitz matrix — it need not even be square. The connection is that if $A$ is a Hurwitz matrix, then the dynamical system^{}

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

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

has a Hurwitz transfer function.

Reference: Hassan K. Khalil, Nonlinear Systems, Prentice Hall, 2002

Title | Hurwitz matrix |
---|---|

Canonical name | HurwitzMatrix |

Date of creation | 2013-03-22 14:02:45 |

Last modified on | 2013-03-22 14:02:45 |

Owner | lha (3057) |

Last modified by | lha (3057) |

Numerical id | 4 |

Author | lha (3057) |

Entry type | Definition |

Classification | msc 93D99 |

Defines | Hurwitz transfer function |

Defines | stability matrix |