DocumentCode :
1365390
Title :
Zeno Stability of the Set-Valued Bouncing Ball
Author :
Or, Yizhar ; Teel, Andrew R.
Author_Institution :
Fac. of Mech. Eng., Technion - Israel Inst. of Technol., Haifa, Israel
Volume :
56
Issue :
2
fYear :
2011
Firstpage :
447
Lastpage :
452
Abstract :
Hybrid dynamical systems consist of both continuous-time and discrete-time dynamics. A fundamental phenomenon that is unique to hybrid systems is Zeno behavior, where the solution involves an infinite number of discrete transitions occurring in finite time, as best illustrated in the classical example of a bouncing ball. In this note, we study the hybrid system of the set-valued bouncing ball, for which the continuous-time dynamics has a set-valued right-hand side. This system is typically used for deriving bounds on the solution of nonlinear single-valued hybrid systems in a small neighborhood of a Zeno equilibrium point in order to establish its local stability. We utilize methods of Lyapunov analysis and optimal control to derive a necessary and sufficient condition for Zeno stability of the set-valued bouncing ball system and to obtain a tight bound on the Zeno time as a function of initial conditions.
Keywords :
Lyapunov methods; continuous time systems; discrete time systems; nonlinear dynamical systems; optimal control; stability; Lyapunov analysis; Zeno equilibrium point; Zeno stability; continuous-time dynamics; discrete transitions; discrete-time dynamics; hybrid dynamical systems; nonlinear single-valued hybrid systems; optimal control; set-valued bouncing ball system; Hybrid systems; Lyapunov stability; Zeno solutions;
fLanguage :
English
Journal_Title :
Automatic Control, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9286
Type :
jour
DOI :
10.1109/TAC.2010.2090411
Filename :
5613918
Link To Document :
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