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On the stability and the stabilization of linear discrete repetitive processes

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Abstract

This article deals with stability and stabilization issues for linear two-dimensional (2D) discrete models. More precisely, we focus on repetitive processes and Roesser models. Within the algebraic analysis approach to linear systems theory, we first show how a given linear repetitive process can be transformed into an equivalent linear Roesser model. We then prove that the structural stability is preserved by this equivalence transformation. This enables us to design new approaches for stabilizing linear repetitive processes by means of existing methods for computing a state feedback control law stabilizing a linear 2D discrete Roesser model. We also show that we can interpret the stability along the pass of a linear repetitive process as its structural stability. This implies that one of our new approaches can be applied to stabilize along the pass a linear repetitive process which is only stable from pass to pass.

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Notes

  1. In a more general case, T can be a non-square matrix: see, e.g., Bachelier et al. (2017a). Nevertheless, it would bring nothing in the present article so T is here assumed to be square.

  2. The dimensions of the blocks are naturally consistent with the splittings of \(\eta \) and \(\eta '\). In particular, this block structure implies \(y'=y\), i.e., the two systems have the same output.

  3. All the numerical values given in the example can be considered as exact, i.e., neither truncated nor rounded. It means that we consider \({\mathbb K}={\mathbb {Q}}\).

  4. The LMI solver works with floating-point numbers but the entries of the matrices exposed here can be considered as (exact) rational numbers since we check the results by performing an a posteriori stability analysis with the algorithms proposed in Bouzidi et al. (2015), Bouzidi and Rouillier (2016) and dedicated to rational values.

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Acknowledgements

The authors are grateful to all the members of the ANR-13-BS03-0005 (MSDOS).

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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 & 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. Nima Yeganefar
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Correspondence to Olivier Bachelier.

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

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Bachelier, O., Cluzeau, T. & Yeganefar, N. On the stability and the stabilization of linear discrete repetitive processes. Multidim Syst Sign Process 30, 963–987 (2019). https://doi.org/10.1007/s11045-018-0583-3

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