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

Yacine Bouzidi 1 Thomas Cluzeau 2 Guillaume Moroz 3 Alban Quadrat 1
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
LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry, Inria Nancy - Grand Est
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.
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Submitted on : Tuesday, December 19, 2017 - 10:35:33 AM
Last modification on : Wednesday, April 3, 2019 - 1:23:03 AM


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Yacine Bouzidi, Thomas Cluzeau, Guillaume Moroz, Alban Quadrat. Computing effectively stabilizing controllers for a class of $n$D systems. The 20th World Congress of the International Federation of Automatic Control, Jul 2017, Toulouse, France. pp.1847 - 1852, ⟨10.1016/j.ifacol.2017.08.200⟩. ⟨hal-01667161⟩



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