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Checkable Conditions for Contraction After Small Transients in Time and Amplitude

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Feedback Stabilization of Controlled Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 473))

Abstract

Contraction theory is a powerful tool for proving asymptotic properties of nonlinear dynamical systems including convergence to an attractor and entrainment to a periodic excitation. We consider generalizations of contraction with respect to a norm that allow contraction to take place after small transients in time and/or amplitude. These generalized contractive systems (GCSs) are useful for several reasons. First, we show that there exist simple and checkable conditions guaranteeing that a system is a GCS, and demonstrate their usefulness using several models from systems biology. Second, allowing small transients does not destroy the important asymptotic properties of contractive systems like convergence to a unique equilibrium point, if it exists, and entrainment to a periodic excitation. Third, in some cases as we change the parameters in a contractive system it becomes a GCS just before it looses contractivity with respect to a norm. In this respect, generalized contractivity is the analogue of marginal stability in Lyapunov stability theory.

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Notes

  1. 1.

    Note that the proof that IC implies ST used a result for time-invariant systems, but an analogous argument holds for the time-varying case as well.

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Acknowledgements

We thank Zvi Artstein and George Weiss for helpful comments. EDS’s work is supported in part by grants NIH 1R01GM100473, AFOSR FA9550-14-1-0060, and ONR N00014-13-1-0074. The research of MM and TT is partly supported by a research grant from the Israeli Ministry of Science, Technology and Space. The research of EDS, MM and TT is also supported by a research grant from the US-Israel Binational Science Foundation. We thank an anonymous reviewer for carefully reading this chapter and providing us several useful comments.

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Authors and Affiliations

  1. School of Electrical Engineering-Systems and the Sagol School of Neuroscience, Tel Aviv University, 69978, Tel Aviv, Israel

    Michael Margaliot

  2. Department of Biomedical Engineering and the Sagol School of Neuroscience, Tel-Aviv University, 69978, Tel-aviv, Israel

    Tamir Tuller

  3. Department of Mathematics and the Center for Quantitative Biology, Rutgers University, Piscataway, NJ, 08854, USA

    Eduardo D. Sontag

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  1. Michael Margaliot
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  3. Eduardo D. Sontag
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Correspondence to Michael Margaliot .

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  1. Centre Automatique et Systèmes CAS, MINES ParisTech, PSL Research University, Centre Automatique et Systèmes CAS, Paris Cedex 06, France

    Nicolas Petit

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Margaliot, M., Tuller, T., Sontag, E.D. (2017). Checkable Conditions for Contraction After Small Transients in Time and Amplitude. In: Petit, N. (eds) Feedback Stabilization of Controlled Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 473. Springer, Cham. https://doi.org/10.1007/978-3-319-51298-3_11

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  • DOI: https://doi.org/10.1007/978-3-319-51298-3_11

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