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Observer Design for Switched Linear Systems with State Jumps

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Hybrid Dynamical Systems

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

Abstract

An observer design for switched linear systems with state resets is proposed based on the geometric conditions for large-time observability from our recent work. Without assuming the observability of individual subsystems, the basic idea is to combine the maximal information available from each mode to obtain a good estimate of the state after a certain time interval (over which the switched system is observable) has passed. We first study systems where state reset maps at switching instants are invertible, in which case it is possible to collect all the observable and unobservable information separately at one time instant. One can then annihilate the unobservable component of all the modes and obtain an estimate of the state by introducing an error correction map at that time instant. However, for the systems with non-invertible jump maps, this approach needs to be modified and a recursion-based error correction scheme is proposed. In both approaches, the criterion for choosing the output injection matrices is given, which leads to the asymptotic recovery of the system state.

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Notes

  1. 1.

    See Definition 7.1 for precise meaning.

  2. 2.

    Note that, \(A(\fancyscript{V}_1 \cap \fancyscript{V}_1) \subset A\fancyscript{V}_1 \cap A\fancyscript{V}_2\), and the equality does not hold in general. The necessary and sufficient condition for equality to hold is that \((\fancyscript{V}_1+\fancyscript{V}_2) \cap \ker A = \fancyscript{V}_1 \cap \ker A + \fancyscript{V}_2 \cap \ker A\), which is the case when \(A\) is invertible. For systems with non-invertible jump maps, the flow matrix \(\varPhi _i^j\) is not necessarily invertible and (7.4) does not hold in general.

  3. 3.

    With slight abuse of notation, the vectors \(z_j\) in (7.9) will be replaced by \(z_j(t_j^-)\), so that the notation \(z_j\) will be used to denote a function.

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Author information

Authors and Affiliations

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

    Aneel Tanwani

  2. ASRI School of Electrical Engineering, Seoul National University, Seoul, South Korea

    Hyungbo Shim

  3. Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Daniel Liberzon

Authors
  1. Aneel Tanwani
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  2. Hyungbo Shim
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  3. Daniel Liberzon
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Correspondence to Aneel Tanwani .

Editor information

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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Tanwani, A., Shim, H., Liberzon, D. (2015). Observer Design for Switched Linear Systems with State Jumps. 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_7

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  • DOI: https://doi.org/10.1007/978-3-319-10795-0_7

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  • Print ISBN: 978-3-319-10794-3

  • Online ISBN: 978-3-319-10795-0

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