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Observability of Switched Linear Systems

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Book cover Hybrid Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 457))

Abstract

Observability of switched linear systems has been well studied during the past decade and depending on the notion of observability, several criteria have appeared in the literature. The main difference in these approaches is how the switching signal is viewed: Is it a fixed and known function of time, is it an unknown external signal, is it the result of a discrete dynamical system (an automaton) or is it controlled and is therefore an input? We will focus on the recently introduced geometric characterization of observability, which assumes knowledge of the switching signal. These geometric conditions depend on computing the exponential of the matrix and require the exact knowledge of switching times. To relieve the computational burden, some relaxed conditions that do not rely on the switching times are given; this also allows for a direct comparison of the different observability notions. Furthermore, the generalization of the geometric approach to linear switched differential algebraic systems is possible and presented as well.

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Notes

  1. 1.

    Note that, for simplicity, we are misusing the notation by writing \(u(t)=e^{2t}+\delta _{-1}+\delta _0\) because \(u\) is a piecewise-smooth distribution and therefore only the evaluations \(u(t-)\), \(u(t+)\), \(u[t]\) are well defined. The correct way of writing would be to write \(\hat{u}(t)=e^{2t}\) and \(u=\hat{u}_\mathbb {D}+\delta _{-1}+\delta _0\).

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Author information

Authors and Affiliations

  1. Department of Automatic Control and Computer Science, Ecole des Mines de Douai, 941, rue Charles Bourseul, 59508, Douai, France

    Mihaly Petreczky

  2. Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany

    Aneel Tanwani & Stephan Trenn

Authors
  1. Mihaly Petreczky
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  2. Aneel Tanwani
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  3. Stephan Trenn
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Correspondence to Stephan Trenn .

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Editors and Affiliations

  1. LAMIH CNRS UMR 8201, University of Valenciennes and Hainaut-Cambresis, Valenciennes, France

    Mohamed Djemai

  2. LAMIH UMR-CNRS 8201, University of Valenciennes and Hainaut-Cambresis, Valenciennes, France

    Michael Defoort

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Petreczky, M., Tanwani, A., Trenn, S. (2015). Observability of Switched Linear Systems. In: Djemai, M., Defoort, M. (eds) Hybrid Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 457. Springer, Cham. https://doi.org/10.1007/978-3-319-10795-0_8

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  • DOI: https://doi.org/10.1007/978-3-319-10795-0_8

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