# Simulating Zeno Hybrid Systems Beyond Their Zeno Points

### Haiyang Zheng

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2006-114

September 8, 2006

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-114.pdf

In this report a technique is proposed to extend the simulation of a Zeno hybrid system beyond its Zeno time point. A Zeno hybrid system model is a hybrid system with an execution that takes an infinite number of discrete transitions during a finite time interval. Some classical Zeno models that incompletely describe the dynamics of the system being modeled are revisited to show that the presence of Zeno behavior indicates that the hybrid system model is incomplete. This motivates the systematic development of a method for completing hybrid system models through the introduction of new post-Zeno states, where the completed hybrid system transitions to these post-Zeno states at the Zeno time point. In practice, simulating a Zeno hybrid system is challenging in that simulation effectively halts near the Zeno time point. Moreover, due to unavoidable numerical errors, it is not practical to exactly simulate a Zeno hybrid system. Therefore, a method for constructing approximations of Zeno models by leveraging the completed hybrid system model is proposed. Using these approximation, a Zeno hybrid system model can be simulated beyond its Zeno point and so the complete dynamics of the system being modeled is revealed.

**Advisor:** Edward A. Lee

BibTeX citation:

@mastersthesis{Zheng:EECS-2006-114, Author = {Zheng, Haiyang}, Title = {Simulating Zeno Hybrid Systems Beyond Their Zeno Points}, School = {EECS Department, University of California, Berkeley}, Year = {2006}, Month = {Sep}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-114.html}, Number = {UCB/EECS-2006-114}, Abstract = { In this report a technique is proposed to extend the simulation of a Zeno hybrid system beyond its Zeno time point. A Zeno hybrid system model is a hybrid system with an execution that takes an infinite number of discrete transitions during a finite time interval. Some classical Zeno models that incompletely describe the dynamics of the system being modeled are revisited to show that the presence of Zeno behavior indicates that the hybrid system model is incomplete. This motivates the systematic development of a method for completing hybrid system models through the introduction of new post-Zeno states, where the completed hybrid system transitions to these post-Zeno states at the Zeno time point. In practice, simulating a Zeno hybrid system is challenging in that simulation effectively halts near the Zeno time point. Moreover, due to unavoidable numerical errors, it is not practical to exactly simulate a Zeno hybrid system. Therefore, a method for constructing approximations of Zeno models by leveraging the completed hybrid system model is proposed. Using these approximation, a Zeno hybrid system model can be simulated beyond its Zeno point and so the complete dynamics of the system being modeled is revealed.} }

EndNote citation:

%0 Thesis %A Zheng, Haiyang %T Simulating Zeno Hybrid Systems Beyond Their Zeno Points %I EECS Department, University of California, Berkeley %D 2006 %8 September 8 %@ UCB/EECS-2006-114 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2006/EECS-2006-114.html %F Zheng:EECS-2006-114