In the recent control literature there has been a great interest in controller design techniques which are robust with respect to plant perturbations.

Crucial for all robustness studies is of course the availability of a “good metric” deﬁned on the set of all plants. A metric is considered good if it can be easily computed and if it gives a measure of numerical robustness.

On the set of proper transfer functions several metrics have been proposed. Most prominently we want to mention the gap metric introduced by Zames and El-Sakkary [15], the graph metric introduced by Vidyasager [13] and the pointwise gap metric introduced recently by Qiu and Davison [7]. All those metrics have in common that they induce the same topology on the set of proper transfer functions with a ﬁxed McMillan degree. For recent contributions to the sub ject of gap metric and graph metric we refer to [3, 7, 11] and in particular to the survey article of Glüsing-Lüerßen [4], where a comparative study between those metrics is provided. A natural generalization of the set of proper transfer functions is the set of autoregressive systems. The main contribution of this short paper is a new metric on the space of autoregressive systems. Because those systems are not so widely known and for the convenience of the reader we summarize in the following section the relevant notions. For simplicity we will develop the theory over the real numbers R.

Rosenthal, J; Wang, X (1993). *What is the distance between two autoregressive systems?* In: Bowers, K L; Lund, J. Computation and control, III (Bozeman, MT, 1992). Boston, MA: Birkhäuser, 333-340.

## Abstract

In the recent control literature there has been a great interest in controller design techniques which are robust with respect to plant perturbations.

Crucial for all robustness studies is of course the availability of a “good metric” deﬁned on the set of all plants. A metric is considered good if it can be easily computed and if it gives a measure of numerical robustness.

On the set of proper transfer functions several metrics have been proposed. Most prominently we want to mention the gap metric introduced by Zames and El-Sakkary [15], the graph metric introduced by Vidyasager [13] and the pointwise gap metric introduced recently by Qiu and Davison [7]. All those metrics have in common that they induce the same topology on the set of proper transfer functions with a ﬁxed McMillan degree. For recent contributions to the sub ject of gap metric and graph metric we refer to [3, 7, 11] and in particular to the survey article of Glüsing-Lüerßen [4], where a comparative study between those metrics is provided. A natural generalization of the set of proper transfer functions is the set of autoregressive systems. The main contribution of this short paper is a new metric on the space of autoregressive systems. Because those systems are not so widely known and for the convenience of the reader we summarize in the following section the relevant notions. For simplicity we will develop the theory over the real numbers R.

## Citations

## Altmetrics

## Additional indexing

Other titles: | Proceedings of the third Bozeman conference, Bozeman, MT, USA, August 5-11, 1992 |
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Item Type: | Book Section, refereed, original work |

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | polynomial matrix; autoregressive system; gap metric; pole-zero cancellations |

Language: | English |

Date: | 1993 |

Deposited On: | 22 Mar 2010 14:49 |

Last Modified: | 05 Apr 2016 13:28 |

Publisher: | Birkhäuser |

Series Name: | Progress in Systems and Control Theory |

Number: | 15 |

ISBN: | 0-8176-3656-0 |

Free access at: | Related URL. An embargo period may apply. |

Official URL: | http://www.springer.com/new+%26+forthcoming+titles+%28default%29/book/978-0-8176-3656-2 |

Related URLs: | http://www.nd.edu/~rosen/Paper/metric-Final.pdf (Author) http://www.ams.org/mathscinet-getitem?mr=1247462 http://opac.nebis.ch:80/F/?local_base=NEBIS&con_lng=GER&func=find-b&find_code=SYS&request=000905132 http://www.zentralblatt-math.org/zmath/en/search/?q=an:0821.93028&format=complete |

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