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Definitions of incremental stability for hybrid systems

Authors: Postoyan, R.; Biemond, J.J.B.; Heemels, W.P.M.H.; van de Wouw, N.;

Definitions of incremental stability for hybrid systems

Abstract

International audience; The analysis of incremental stability properties typically involves measuring the distance between any pair of solutions of a given dynamical system, corresponding to different initial conditions, at the same time instant. This approach is not directly applicable for hybrid systems in general. Indeed, hybrid systems generate solutions that are defined with respect to hybrid times, which consist of both the continuous time elapsed and the discrete time, that is the number of jumps the solution has experienced. Two solutions of a hybrid system do not a priori have the same time domain, and we may therefore not be able to compare them at the same hybrid time instant. To overcome this issue, we invoke graphical closeness concepts. We present definitions for incremental stability depending on whether incremental asymptotic stability properties hold with respect to the hybrid time, the continuous time, or the discrete time, respectively. Examples are provided throughout the paper to illustrate these definitions, and the relations between these three incremental stability notions are investigated. The definitions are shown to be consistent with those available in the literature for continuous-time and discrete-time systems.

Countries
France, Netherlands
Subjects by Vocabulary

Microsoft Academic Graph classification: Mathematical optimization Stability (learning theory) Discrete time and continuous time Exponential stability Hybrid system Convergence (routing) A priori and a posteriori Time domain Dynamical system (definition) Mathematics

Keywords

Hybrid systems, Incremental stability, [SPI.AUTO]Engineering Sciences [physics]/Automatic, [SPI.AUTO] Engineering Sciences [physics]/Automatic

25 references, page 1 of 3

[1] D. Angeli. A Lyapunov approach to incremental stability properties. IEEE Transactions on Automatic Control, 47(3):410-421, 2002.

[2] J.J.B. Biemond, N. van de Wouw, W.P.M.H. Heemels, and H. Nijmeijer. Tracking control for hybrid systems with state-triggered jumps. IEEE Transactions on Automatic Control, 58(4):876-890, 2013.

[3] C. Cai, A.R. Teel, and R. Goebel. Smooth Lyapunov functions for hybrid systems - Part II: (pre)asymptotically stable compact sets. IEEE Transactions on Automatic Control, 53(3):734-748, 2008.

[4] B.P. Demidovich. Lectures on Stability Theory. Nauka (in Russian), 1967.

[5] F. Forni and R. Sepulchre. A differential Lyapunov framework for contraction analysis. IEEE Trans. on Aut. Control, 59:614-628, 2014.

[6] F. Forni, A.R. Teel, and L. Zaccarian. Follow the bouncing ball: global results on tracking and state estimation with impacts. IEEE Transactions on Automatic Control, 58:1470-1485, 2013. [OpenAIRE]

[7] V. Fromion, S. Monaco, and D. Normand-Cyrot. A link between inputoutput stability and Lyapunov stability. Systems & Control Letters, 27:243-248, 1996. [OpenAIRE]

[8] R. Goebel, R.G. Sanfelice, and A.R. Teel. Hybrid dynamical systems. Princeton University Press, Princeton, U.S.A., 2012.

[9] R. Leine and N. van de Wouw. Stability and convergence of mechanical systems with unilateral constraints. Lecture Notes in Applied and Computational Mechanics, vol. 36. Springer-Verlag, Berlin, 2008.

[10] G.A. Leonov, I.M. Burkin, and A.I. Shepeljavyi. Frequency methods in oscillation theory. Kluwer Academic Publishers, Dordrecht, 1996.

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  • citations
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    7
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
7
Average
Average
Average
Funded by
NWO| Wireless Control Systems: A New Frontier in Automation
Project
  • Funder: Netherlands Organisation for Scientific Research (NWO) (NWO)
  • Project Code: 11382
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