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Surveys in Differential-Algebraic Equations IV pp 161–219Cite as

Observability of Linear Differential-Algebraic Systems: A Survey

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Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

We investigate different concepts related to observability of linear constant coefficient differential-algebraic equations. Regularity, which, loosely speaking, guarantees existence and uniqueness of solutions for any inhomogeneity, is not required in this article. Concepts like impulse observability, observability at infinity, behavioral observability, strong and complete observability are described and defined in the time-domain. Special emphasis is placed on a normal form under output injection, state space and output space transformation. This normal form together with duality is exploited to derive Hautus-type criteria for observability. We also discuss geometric criteria, Kalman decompositions and detectability. Some new results on stabilization by output injection are proved.

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Notes

  1. 1.

    For singular DAEs it is however not true that all \(x(0^{-}) \in \mathbb{R}^{n}\) are feasible for an ITP. For example, the overdetermined DAE \(\dot{x} = 0\), 0 = x has no ITP solution with x(0−) ≠ 0, because then x(0+) = 0 and \(0 =\dot{ x}[0] = (x(0^{+}) - x(0^{-}))\delta _{0}\) are conflicting.

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Acknowledgements

We thank the referees of this article for their valuable comments which very much helped to improve the manuscript.

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  1. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Thomas Berger & Timo Reis

  2. Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653, Kaiserslautern, Germany

    Stephan Trenn

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    Achim Ilchmann

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    Timo Reis

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Berger, T., Reis, T., Trenn, S. (2017). Observability of Linear Differential-Algebraic Systems: A Survey. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations IV. Differential-Algebraic Equations Forum. Springer, Cham. https://doi.org/10.1007/978-3-319-46618-7_4

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