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International Journal of Control

Publication date: 2018-01-01
Volume: 91 Pages: 2692 - 2704
Publisher: Taylor & Francis

Author:

Dreesen, Philippe
Batselier, Kim ; De Moor, Bart

Keywords:

SISTA, DYSCO, Science & Technology, Technology, Automation & Control Systems, Multidimensional systems, realisation theory, difference equations, eigenvalue problem, polynomial system solving, STATE-SPACE REALIZATION, LINEAR-SYSTEM, TIME-SERIES, GROBNER BASES, EQUATIONS, MODEL, STADIUS-15-145, C16/15/059#53326574, 0102 Applied Mathematics, 0906 Electrical and Electronic Engineering, 0913 Mechanical Engineering, Industrial Engineering & Automation, 4901 Applied mathematics

Abstract:

© 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group. Multidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional systems is linked to multivariate polynomials, and is therefore more difficult than the well-known analysis of one-dimensional systems, which is linked to univariate polynomials. In the current paper, we relate the realisation theory for overdetermined autonomous multidimensional systems to the problem of solving a system of polynomial equations. We show that basic notions of linear algebra suffice to analyse and solve the problem. The difference equations are associated with a Macaulay matrix formulation, and it is shown that the null space of the Macaulay matrix is a multidimensional observability matrix. Application of the classical shift trick from realisation theory allows for the computation of the corresponding system matrices in a multidimensional state-space setting. This reduces the task of solving a system of polynomial equations to computing an eigenvalue decomposition. We study the occurrence of multiple solutions, as well as the existence and analysis of solutions at infinity, which allow for an interpretation in terms of multidimensional descriptor systems.