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# Identification and matrix rational $H^2$ approximation : a gradient algorithm based on Schur analysis

Abstract : This report deals with rational approximation of any specified order $n$ of transfer functions. Transfer functions are assumed to be matrices whose entries belong to the Hardy space for the complement of the closed unit disk endowed with the $L^2$-norm. A new approach is developed leading to an original algorithm, the first one to our knowledge which concerns matrix transfer functions. This approach generalizes the ideas developed in the scalar case, but involves substantial new difficulties. The inner-unstable factorization of transfer functions allows to express the criterion in terms of inner matrices of Mac-Millan degree $n$. These matrices form a differential manifold. Based on a tangential Schur algorithm, an atlas of this manifold is given for which the coordinates vary in $n$ copies of the unit ball. Then a gradient algorithm can be used to solve this problem. The different cases which can arise while processing the algorithm are studied~:~ how to switch to another chart of the atlas, what has to be done when a boundary point is reached. In the neighbourhood of a boundary point, the criterion can be smoothly extended. Moreover, such a point can be considered as an initial point for the research of a lower degree approximant. It is explained how to cope with the decrease and the increase of the degree. The convergence of the algorithm to a local minimum of appropriate degree is proved and demonstrated on a simple example.
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https://hal.inria.fr/inria-00074158
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Submitted on : Wednesday, May 24, 2006 - 2:38:37 PM
Last modification on : Friday, February 4, 2022 - 3:17:25 AM
Long-term archiving on: : Tuesday, April 12, 2011 - 3:33:31 PM

### Identifiers

• HAL Id : inria-00074158, version 1

### Citation

Pascale Fulcheri, Martine Olivi. Identification and matrix rational $H^2$ approximation : a gradient algorithm based on Schur analysis. RR-2520, INRIA. 1995. ⟨inria-00074158⟩

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