Is there Life after Zeno?

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Is there Life after Zeno?

• Aaron D. Ames Robert D. Gregg
• Haiyang Zheng
• Shankar Sastry

• Completed Hybrid Systems
• A simple hybrid Lagrangian is defined to be the
  tuple
• , where
• Q is the configuration space
• L is a (hyperregular)
  Lagrangian
• h provides unilateral constraints
  on the configuration space h-1(0) is assumed a
  manifold
• Associated to a hybrid Lagrangian is a hybrid
  system
• , where
• Zeno executions converge towards an infinite
  number of discrete transitions in a finite amount
  of time. A hybrid system can be completed so that
  it goes beyond Zeno

Abstract Understanding Zeno phenomena
plays an important role in understanding hybrid
systems an intriguing question is what happens
after a Zeno point? We propose a method for
extending Zeno trajectories past a Zeno point for
a class of hybrid systems Lagrangian hybrid
systems. We argue that after the Zeno point is
reached, the hybrid system should switch to a
holonomically constrained dynamical system, where
the holonomic constraints are based on the
unilateral constraints on the original hybrid
systems configuration space.
Inverted Pendulum on Ground
Simulations implemented on PtolemyII 5.0
Association
Bouncing Ball on Sinusoidal Surface
Inverted Pendulum on Cart
Bouncing ball without hybrid completion
Completion
Completing the hybrid system results in a
post-Zeno execution.
At the Zeno point the pendulums collisions with
the top of the cart become negligible, but the
frictionless cart continues moving.
Near the Zeno point, the simulator fails as the
number of computation steps explodes.
With hybrid completion
The post-Zeno dynamical system is considerably
different than the pre-Zeno hybrid system.