Motivations

1
Alessandro Abate Aaron Ames Shankar Sasty
Stochastic Approximations of Deterministic Hybrid
systems
Simulations
http//chess.eecs.berkeley.edu
Idea

• Motivations
• In a deterministic setting
• Difficult to assess global properties (stability,
  reachability)
• Model glitches Zenoness
• De-abstaction is the solution? Not
  always! No a-priori check
• Ill-posedness Grazing events
• Problematic to simulate them
• Event detection can be hard

Event Detection
Two-tanks system Eliminating Zeno
Consider the ODE dx/dt f(x). Examine the IVP I
(f,t0,tF,x0) on t0,tF with xt0x0. Usually
solved by Numerical Integration approximated
solution xn(t) on t0,tF, such that xn(t0)x0.
Step size is h. Assume a numerical scheme
produces a solution that is accurate of order
M(t,h) M(t,0)M(0,h)0. Global bound on the
error 9 a constant CI such that x(t) -
xn(t) CI M(t-t0,h).
HS are hard! 2 Examples
decreasing step-size h
The original trajectory is Zeno the
approximated isnt.
After proper guard linearization, assume
that At every point in time, the probability
that the actual solution switches from the
current domain to the one identified by a guard
is given by the proportion of the volume sphere
centered around the numerical solution that lies
beyond the guard, by the volume of this
sphere. Given an initial condition for the
execution, we have the following
knowledge Pij(t) P q(t)jq(t0)i, 8 t
t0. This way, build a transition probability
matrix P(t). Define time-dependent jump
intensities G(t) dP(t)/dt, 8 t t0. The
trajectories of the original Deterministic Hybrid
System HS can be handled as continuous-time
Markov processes, solutions of the approximated
Stochastic Hybrid System SHSh.

• Setting
• Define a (deterministic) Hybrid System as a
  6-tuple HS (Q,E,D,G,R,F)
• Q 1,...,m ½ Z set of discrete states-finite,
  subset of the integers.
• E ½ Q Q set of edges which define relations
  between the domains. For e (i,j) 2 E
  denote its source by s(e) i and its target by
  t(e) j the edges in E can be indexed, E
  e1,,eE.
• D Dii 2 Q set of domains where Di µ Rn.
• G Gee 2 E set of guards, where Ge µ Ds(e).
• R Ree 2 E set of reset maps, continuous from
  Ge µ Ds(e) to
• Re(Ge) µ Dt(e) and Lipschitz.
• F fii 2 Q set of vector fields or ordinary
  differential equations (ODEs),
• such that fi is Lipschitz on Rn. The solution to
  the ODE
• fi with i.c. x0 2 Di is xi(t) where xi(t_0)
  x_0.
• The guards are spacial, given by the zero level
  sets of smooth functions gee 2 E such that Ge
  x ge(x) 0 we also assume that ge(x) 0
  for all x 2 Ds(e)\ Ge.
• In this work we shall introduce stochasticity on
  the Reset Maps.

Propagation of the error cones.
Conjecture. Given a hybrid system HS, there
exists a non-trivial stochastic hybrid system SHS
whose probabilistic behavior encompasses the
deterministic behavior of the hybrid system HS
B(HS) µ B(SHS), where B(HS) is the behavior
of the hybrid system HS. Moreover, SHS yields
itself more easily to analysis.

• References
• A. Abate and A. D. Ames and S. Sastry
  Stochastic Approximations of Hybrid Systems".
  Proceedings of the ACC 2005.
• A. Abate and A. D. Ames and S. Sastry
  Characterizing the Behavior of Deterministic
  Hybrid Systems
• through Stochastic Approximations". In
  preparation.
• A. Abate Analysis of Stochastic Hybrid Systems.
  MS Thesis, EECS Department, UC Berkeley, May
  2004.
• A.Abate, L. Shi, S. Simic, S. Sastry A
  Stability Criterion for Stochastic Hybrid
  Systems". Proceedings
• of the MTNS 2004. Leuven, BG, July 2004.

Theorem Given a hybrid system HS, the non
trivial stochastic hybrid system SHSh, dependent
on a parameter h (the integration step), verifies
the following limh -gt 0SHSh HS.
Theorem Given a hybrid system HS, the non
trivial stochastic hybrid system SHSh admits no
Zeno behavior, 8 h gt 0.
May 11th, 2005
Contact aabate_at_eecs.berkeley.edu