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Structural stabilization of linear 2D discrete systems using equivalence transformations

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Abstract

We consider stability and stabilization issues for linear two-dimensional (2D) discrete systems. We give a general definition of structural stability for all linear 2D discrete systems which coincides with the existing definitions in the particular cases of the classical Roesser and Fornasini–Marchesini discrete models. We study the preservation of the structural stability by equivalence transformations in the sense of the algebraic analysis approach to linear systems theory. This allows us to use recent works both on the stabilization of linear 2D Roesser models and on the equivalence of linear multidimensional systems in order to develop a stabilization method for linear 2D discrete Fornasini–Marchesini models.

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Notes

  1. Note that the isomorphism between the D-modules associated to \(R_s\,x=0\) and \(R_s'\,x'=0\) is defined by the matrix \(P \in D^{(d_x+d_u) \times (d_{x'}+d_{u'})}\) defining the isomorphism between the D-modules associated to (11) and (12).

  2. Note however that this does not necessarily imply that the Fornasini model (4) cannot be stabilized.

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Acknowledgments

The authors are grateful to all the members of the ANR-13-BS03-0005 (MSDOS) project. In particular, we thank Alban Quadrat for helpful discussions on this subjects and Yacine Bouzidi for providing us with his Maple implementation of the algorithm developed in Bouzidi et al. (2015). Finally, we warmly thank the referees for their comments and suggestions which have enhanced the clarity of the paper.

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Authors and Affiliations

  1. LIAS-ENSIP, University of Poitiers, Bâtiment B25, 2 rue Pierre Brousse, TSA 41105, 86073, Poitiers Cedex 9, France

    Olivier Bachelier, Ronan David & Nima Yeganefar

  2. CNRS XLIM UMR 7252, University of Limoges, 123 avenue Albert Thomas, 87060, Limoges Cedex, France

    Thomas Cluzeau

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  1. Olivier Bachelier
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  2. Thomas Cluzeau
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  3. Ronan David
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  4. Nima Yeganefar
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Correspondence to Olivier Bachelier.

Additional information

This work was supported by the ANR-13-BS03-0005 (MS-DOS).

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Bachelier, O., Cluzeau, T., David, R. et al. Structural stabilization of linear 2D discrete systems using equivalence transformations. Multidim Syst Sign Process 28, 1629–1652 (2017). https://doi.org/10.1007/s11045-016-0439-7

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