Abstract
Zeno behaviors are one of the (perhaps unintended) features of many hybrid models of physical systems. They have no counterpart in traditional dynamical systems or automata theory and yet they have remained relatively unexplored over the years. In this paper we address the stability properties of a class of Zeno equilibria, and we introduce a necessary paradigm shift in the study of hybrid stability. Motivated by the peculiarities of Zeno equilibria, we consider a form of asymptotic stability that is global in the continuous state, but local in the discrete state. We provide sufficient conditions for stability of these equilibria, resulting in sufficient conditions for the existence of Zeno behavior.
This research is supported by the National Science Foundation (award numbers CCR-0225610 and CSR-EHS-0509313).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Ames, A.D., Zheng, H., Gregg, R.D., Sastry, S.: Is there life after Zeno? Taking executions past the breaking (Zeno) point (Submitted to the 2006 American Control Conference)
Brogliato, B.: Nonsmooth Mechanics. Springer, Heidelberg (1999)
Ames, A.D., Abate, A., Sastry, S.: Sufficient conditions for the existence of Zeno behavior. In: 44th IEEE Conference on Decision and Control and European Control Conference ECC (2005)
Branicky, M.S.: Stability of hybrid systems: State of the art. In: Proceedings of the 36th IEEE Conference on Decision and Control, San Diego (1997)
Heymann, M., Lin, F., Meyer, G., Resmerita, S.: Analysis of Zeno behaviors in hybrid systems. In: Proceedings of the 41st IEEE Conference on Decision and Control, Las Vagas (2002)
Johansson, K.H., Lygeros, J., Sastry, S., Egerstedt, M.: Simulation of Zeno hybrid automata. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix (1999)
Zhang, J., Johansson, K.H., Lygeros, J., Sastry, S.: Zeno hybrid systems. Int. J. Robust and Nonlinear Control 11(2), 435–451 (2001)
Zheng, H., Lee, E.A., Ames, A.D.: Beyond Zeno: Get on with it (To appear in Hybrid Systems: Computation and Control, 2006)
Ames, A.D., Sastry, S.: A homology theory for hybrid systems: Hybrid homology. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 86–102. Springer, Heidelberg (2005)
Ames, A.D., Sangiovanni-Vincentelli, A., Sastry., S.: Homogenous semantic preserving deployments of heterogenous networks of embedded systems. In: Workshop on Networked Embedded Sensing and Control, Notre Dame (2005)
Lane, S.M.: Categories for the Working Mathematician. Graduate Texts in Mathematics, vol. 5. Springer, Heidelberg (1998)
Ames, A.D., Tabuada, P., Sastry, S.: H-categories and graphs (Technical Note)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ames, A.D., Tabuada, P., Sastry, S. (2006). On the Stability of Zeno Equilibria. In: Hespanha, J.P., Tiwari, A. (eds) Hybrid Systems: Computation and Control. HSCC 2006. Lecture Notes in Computer Science, vol 3927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11730637_6
Download citation
DOI: https://doi.org/10.1007/11730637_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33170-4
Online ISBN: 978-3-540-33171-1
eBook Packages: Computer ScienceComputer Science (R0)