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On the reduction of 2-D polynomial systems into first order equivalent models

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Abstract

In this paper we propose a novel approach for the reduction of a 2-D rectangular polynomial matrix of arbitrary degree, to first-order matrix pencils of the form \(sE_{1}+zE_{2}+A\), utilizing the framework of zero coprime equivalence (ZC-E). The proposed approach is in turn employed to derive a series of ZC-E matrix pencils, which can be obtained “by inspection” of the coefficients of the original bivariate polynomial matrix. Improving similar constructions of first order pencils available in the literature, our approach results in matrix pencils whose size increases linearly with the degrees of the indeterminates of the original polynomial matrix. From a system-theoretic point of view, the proposed method, provides the algebraic tools to transform a high order bivariate linear system, into a zero coprime system equivalent first order representation. Notably, one of the proposed transformation techniques gives rise to generalized 2\(-D\) Roesser models.

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Acknowledgements

The authors wish to acknowledge financial support provided by the Special Account for Research Funds of the Technological Education Institute of Central Macedonia, Greece, under Grant SAT/IE/130917-129/16. The authors would like to thank the two anonymous reviewers for their constructive comments and suggestions, which helped improving the quality of the material presented in the paper. We are particularly grateful to reviewer #2 for suggesting the alternative proof of Theorem 1 given in Remark 2, establishing this way the relation between the proposed ZC-E and the module isomorphism based approach.

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Correspondence to Efstathios Antoniou.

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Antoniou, E., Vologiannidis, S. On the reduction of 2-D polynomial systems into first order equivalent models. Multidim Syst Sign Process 31, 249–268 (2020). https://doi.org/10.1007/s11045-019-00661-8

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  • DOI: https://doi.org/10.1007/s11045-019-00661-8

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