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Advances of Implicit Description Techniques in Modelling and Control of Switched Systems

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Structural Methods in the Study of Complex Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 482))

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Abstract

Our contribution is devoted to a constructive overview of the implicit system approach in modern control of switched dynamic models. We study a class of non-stationary autonomous switched systems and formally establish the existence of solution. We next incorporate the implicit systems approach into our consideration. At the beginning of the contribution, we also develop a specific system example that is used for illustrations of various system aspects that we consider. Our research involves among others a deep examination of the reachability property in the framework of the implicit system framework that we propose. Based on this methodology, we finally propose a resulting robust control design for the switched systems under consideration and the proposed control strategy is implemented in the context of the illustrative example.

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Notes

  1. 1.

    In the case where there is no output equation \(y = C x\), we simply write \(\varSigma ^{imp}(E, A, B)\).

  2. 2.

    Geerts [18] considered the linear combinations of impulsive and smooth distributions, with \(\mu \) coordinates, denoted by \({\mathscr {C}}_{\mathrm {imp}}^{\mu }\), as the signal sets. The set \({\mathscr {C}}_{\mathrm {imp}}^{\mu }\) is a subalgebra and it can be decomposed as \({\mathscr {C}}_{\mathrm {p-imp}}^{\mu }\oplus \mathscr {C}_{\mathrm {sm}}^{\mu }\), where \({\mathscr {C}}_{\mathrm {p-imp}}^{\mu }\) and \({\mathscr {C}}_{\mathrm {sm}}^{\mu }\) denote the subalgebras of pure impulses and smooth distributions, respectively [41]. The unit element of this subalgebra is the Dirac delta distribution, \(\delta \). Any linear combination of \(\delta \) and its distributional derivatives \(\delta ^{(\ell )}\), \(\ell > 1\), is called impulsive.

  3. 3.

    \(E{x_{0}}\) stands for \(E{x_{0}}\delta \), \({x_{0}}\in {\mathscr {X}}_{d}\) being the initial condition, and pE x stands for \(\delta ^{(1)}*E{x}\) (\(*\) denotes convolution); if pE x is smooth and \(E\dot{x}\) stands for the distribution that can be identified with the ordinary derivative, \(E{\mathrm {d}{x}}/{\mathrm {d}{t}}\), then \(pE{x} = E\dot{x} + E{x_{0^{+}}}\).

  4. 4.

    We restrict our discussion to subspaces of finite-dimensional vector spaces. In [16] and in [2], these definitions are stated in the more general framework of closed sets of normed vector spaces.

  5. 5.

    Let us note that (7.31) implies that (cf. Fig. 7.2): \(E{{\mathscr {V}}}_{{{\mathscr {X}}}_{d}^{*}} + \mathrm {Im}\,{B} = \underline{{\mathscr {X}}}_{eq}\) and \({{\mathscr {V}}}_{{{\mathscr {X}}}_{d}^{*}} = {{\mathscr {X}}}_{d}\), hence (7.64) takes the form (7.71).

  6. 6.

    Recall that two representations are externally equivalent when the sets of all possible trajectories for their external signals (here u and y) are identical (see [37, 44, 45]).

  7. 7.

    Since Theorem 7.5 is satisfied, one can also assign the output dynamics.

  8. 8.

    This region is obtained from \(\det \left[ \begin{array}{ccc} {\mathrm {s}} &{} -1 &{} -1 \\ 0 &{} 0 &{} ({\mathrm {s}} + 1/{\tau })\\ \hline -{\alpha } &{} ({\beta }+2) &{} 1 \end{array}\right] = -(({\beta }+2){\mathrm {s}} - {\alpha })({\mathrm {s}} + 1/{\tau })\).

  9. 9.

    This region is obtained following the methodology of [42], namely, we solve two Lyapunov equations for the two cases: (i) \(\beta \ne -2\) and (ii) \(\alpha \ne 0\) (with \(\beta =-2\)), with a common positive definite matrix P.

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Acknowledgements

This research was conducted in the framework of the regional programme “Atlanstic 2020, Research, Education and Innovation in Pays de la Loire”, supported by the French Region Pays de la Loire and the European Regional Development Fund.

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Authors and Affiliations

  1. CINVESTAV-IPN, Control Automático, UMI 3175 CINVESTAV -CNRS, A.P. 14-740, 07000, México City, México

    Moisés Bonilla Estrada

  2. CNRS, LS2N (Laboratoire des Sciences du Numérique de Nantes) UMR 6004, B.P. 92101, 44321 Nantes, Cedex 03, France

    Michel Malabre

  3. Department of Mathematical Science, Universidad EAFIT, Medellin, Colombia

    Vadim Azhmyakov

Authors
  1. Moisés Bonilla Estrada
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  2. Michel Malabre
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  3. Vadim Azhmyakov
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Corresponding author

Correspondence to Michel Malabre .

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Editors and Affiliations

  1. Department of Electrical, Electronic and Information Engineering “G. Marconi”, Alma Mater Studiorum Università di Bologna, Bologna, Bologna, Italy

    Elena Zattoni

  2. Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, ANCONA, Ancona, Italy

    Anna Maria Perdon

  3. Dipartimento di Ingegneria dell’Informazione, Università Politecnica delle Marche, ANCONA, Ancona, Italy

    Giuseppe Conte

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Bonilla Estrada, M., Malabre, M., Azhmyakov, V. (2020). Advances of Implicit Description Techniques in Modelling and Control of Switched Systems. In: Zattoni, E., Perdon, A., Conte, G. (eds) Structural Methods in the Study of Complex Systems. Lecture Notes in Control and Information Sciences, vol 482. Springer, Cham. https://doi.org/10.1007/978-3-030-18572-5_7

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