TY - JOUR
T1 - Stability and completion of Zeno equilibria in Lagrangian hybrid systems
AU - Or, Yizhar
AU - Ames, Aaron D.
N1 - Funding Information: Manuscript received December 30, 2008; revised September 03, 2009; accepted August 28, 2010. Date of publication September 27, 2010; date of current version June 08, 2011. This work was supported by a Fulbright Postdoctoral Fellowship and Bikura Scholarship of the Israeli Science Foundation. Recommended by Associate Editor H. Ishii. Y. Or is with the Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]). A. D. Ames is with the Mechanical Engineering Department, Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2010.2080790
PY - 2011/6
Y1 - 2011/6
N2 - This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts. This class of systems naturally display Zeno behavior - an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint.
AB - This paper studies Lagrangian hybrid systems, which are a special class of hybrid systems modeling mechanical systems with unilateral constraints that are undergoing impacts. This class of systems naturally display Zeno behavior - an infinite number of discrete transitions that occur in finite time, leading to the convergence of solutions to limit sets called Zeno equilibria. This paper derives simple conditions for stability of Zeno equilibria. Utilizing these results and the constructive techniques used to prove them, the paper introduces the notion of a completed hybrid system which is an extended hybrid system model allowing for the extension of solutions beyond Zeno points. A procedure for practical simulation of completed hybrid systems is outlined, and conditions guaranteeing upper bounds on the incurred numerical error are derived. Finally, we discuss an application of these results to the stability of unilaterally constrained motion of mechanical systems under perturbations that violate the constraint.
KW - Langrangian hybrid systems
KW - Zeno behavior
KW - stability
KW - unilateral contact constraints
UR - http://www.scopus.com/inward/record.url?scp=79958244598&partnerID=8YFLogxK
U2 - 10.1109/TAC.2010.2080790
DO - 10.1109/TAC.2010.2080790
M3 - مقالة
SN - 0018-9286
VL - 56
SP - 1322
EP - 1336
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 6
M1 - 5586645
ER -