Multidimensional realisation theory and polynomial system solving

ABSTRACT

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.

KEYWORDS:

• Multidimensional systems
• realisation theory
• difference equations
• eigenvalue problem
• polynomial system solving

Acknowledgments

Philippe Dreesen is a postdoctoral researcher at Vrije Universiteit Brussel, Department of ELEC and a free researcher at KU Leuven, Department of ESAT-STADIUS. Kim Batselier is a postdoctoral researcher at The University of Hong Kong, Department of EEE. Bart De Moor is a full professor at KU Leuven, Department of ESAT-STADIUS. This work was supported in part by: Belgian Federal Science Policy Office: IUAP P7/19 (DYSCO, Dynamical systems, control and optimisation, 2012–2017); Flemish Government: IWT/FWO PhD grants; KU Leuven Internal Funds C16/15/059, C32/16/013; imec strategic funding 2017; Fund for Scientific Research (FWO-Vlaanderen); the Flemish Government (Methusalem); the Belgian Government through the Inter university Poles of Attraction (IAP VII) Program; ERC advanced grant SNLSID, under contract 320378; FWO grants G028015N and G090117N. Part of this work was done when Philippe Dreesen held a PhD grant of Flanders’ Agency for Innovation by Science and Technology (IWT Vlaanderen).

The scientific responsibility is assumed by the authors.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Federaal Wetenschapsbeleid [grant number IUAP P7/19 (DYSCO, Dynamical systems, control and optimization, 2012-2017)]; Fonds Wetenschappelijk Onderzoek (FWO) [grant number G028015N, G090117N]; H2020 European Research Council [grant number SNLSID (contract 320378)]; Research council KU Leuven [grant number C16/15/059, C32/16/013, imec strategic funding 2017]; Flemish Government [Methusalem].