# Computing effectively stabilizing controllers for a class of $n$D systems

1 NON-A - Non-Asymptotic estimation for online systems
Inria Lille - Nord Europe, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189
3 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimen-sional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc U n = {z = (z 1 ,. .. , z n) ∈ C n | |z 1 | 1,. .. , |z n | 1}. It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal I = p 1 ,. .. , p r that can be explicitly described in terms of the transfer matrix of the plant. More precisely the system is stabilizable if and only if V (I) = {z ∈ C n | p 1 (z) = · · · = p r (z) = 0} ∩ U n = ∅. In the present article, we consider the specific class of linear nD systems (which includes the class of 2D systems) for which the ideal I is zero-dimensional, i.e., the p i 's have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if V (I) ∩ U n = ∅, as well as for computing, if it exists, a stable polynomial p ∈ I which allows the effective computation of a stabilizing controller. We illustrate our algorithms through an example and finally provide running times of prototype implementations for 2D and 3D systems.
Keywords :
Type de document :
Article dans une revue
IFAC-PapersOnLine, Elsevier, 2017, 50 (1), pp.1847 - 1852. 〈10.1016/j.ifacol.2017.08.200〉
Domaine :
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https://hal.archives-ouvertes.fr/hal-01667161
Contributeur : Guillaume Moroz <>
Soumis le : mardi 19 décembre 2017 - 10:35:33
Dernière modification le : vendredi 19 janvier 2018 - 10:13:29

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### Citation

Yacine Bouzidi, Thomas Cluzeau, Guillaume Moroz, Alban Quadrat. Computing effectively stabilizing controllers for a class of $n$D systems. IFAC-PapersOnLine, Elsevier, 2017, 50 (1), pp.1847 - 1852. 〈10.1016/j.ifacol.2017.08.200〉. 〈hal-01667161〉

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