Approximate Abstraction for Verification of Continuous and Hybrid Systems

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Approximate Abstraction for Verification of
Continuous andHybrid Systems

• Antoine Girard

Antoine.Girard_at_imag.fr
Guest lectureESE601 Hybrid Systems03/22/2006
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Hybrid Systems

• General modeling framework for complex systems
  
• - continuous dynamics (ode, pde, sde)
• - discrete dynamics (automata, Markov processes)
• Several applications including embedded systems
  
• - design computer automata, continuous
  environment
• - implementation integrated circuits,
  analogical et numerical components
• These systems are generally
• - structured (hierarchical modeling/architecture)
  
• - large scale systems (numerous continuous
  variables)
• - safety critical (plane, subway, nuclear power
  plant)

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Algorithmic Verification

• Algorithmic proof of the safety of a system
• No trajectory of the system can reacha set of
  unsafe states.
• Initially on the software part 1980 -
  - verification of discrete systems, Model
  Checking
• - for some properties, one cannot ignore the
  continuous dynamics
• Verification of continuous and hybrid systems
  1995 - - exhaustive simulation of
  systems using set valued computations
  techniques. - central notion reachable set
  subset of the state space, reachable by the
  trajectories of the system from a subset of
  initial states.

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Reachability Analysis

• Computation of the reachable set
• - exactly for some very simple classes of
  systems Piecewise constant differential
  inclusions, some linear systems
• - approximately for other classes
  (over-approximation algorithms)
• Over-approximation algorithms
• Set-based simulation numerical errors
• - Polytopes Asarin, Dang, Maler Krogh et.al.
  Girard
• - Ellipsoids Kurzhanski, Varayia

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Complexity Barrier
Computational cost of the reachable set is a
major issue !
Complex system
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Abstraction

• Notion of system approximation
• S2 is an abstraction of S1 iffevery trajectory
  of S1 is also a trajectory of S2.
• Hybridization Approximation of complex
  continuous dynamics by simpler hybrid
  dynamics. Asarin, Dang, Girard Lefebvre,
  Gueguen Frehse
• Dimension reduction Pappas et.al. van der
  Schaft
• If S2 is safe then S1 is safe

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Analysis of complex systems
Abstraction methods for complexity reduction of
systems.
Complex system
Dimension reduction
Hybridization
Abstraction
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Outline
1. Abstraction and Approximation - Simulation
relation - Approximate simulation
relation 2. Approximate simulation relations for
continuous systems. 3. Approximate simulation
relations for hybrid systems.
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Simulation Relations

• Local characterization of trajectories
  inclusion.
• Simulation relation R ? X1 x X2
• If for all initial state x1 of S1 there exists
  an initial state x2 of S2 such that (x1,x2) ?
  R then S2 is an abstraction of S1.

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From Abstraction to Approximation

• Trajectories inclusion is well suited to
  discrete systems.
• For continuous and hybrid systems, it is
  restrictive
• Natural topology on the state space
• ?
• Distance between the trajectories seems more
  appropriate
• Thus, S2 is an approximate abstraction or
  approximation of S1 if
• For every trajectory of S1, there exists a
  trajectory of S2 such that the distance between
  the trajectories remains bonded by ?
• ? is the precision of the approximation (? 0,
  abstraction).

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A Useful Notion for Verification

• If S2 is an approximation of S1 of precision ?
• Therefore,
• The safety of S1 can be proved using an
  approximation S2.

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Approximate Simulation Relation

• Local characterization of the notion of
  approximation.
• Approximate simulation relation of precision ?,
  R ? X1 x X2
• If for every initial state x1 of S1 there
  exists an initial state x2 of S2 such that
  (x1,x2) ? R, then S2 is an approximation of S1 of
  precision ?.

- A. Girard, G.J. Pappas, Approximation metrics
for discrete and continuous systems, IEEE TAC,
accepted 2006.
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Outline
1. Abstraction and Approximation - Simulation
relation - Approximate simulation
relation 2. Approximate simulation relations for
continuous systems. 3. Approximate simulation
relations for hybrid systems.
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Simulation Functions
A. Girard, G.J. Pappas, Approximate bisimulations
for constrained linear systems, CDC 2005. A.
Girard, G.J. Pappas, Approximate bisimulations
for nonlinear dynamical systems, CDC 2005.
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Simulation Functions

• Simulation functions define approximate
  simulation relations
• Particularly,
• Let
• then S2 is an approximation of S1 of precision
  ?.

- A. Girard, G.J. Pappas, Approximation metrics
for discrete and continuous systems, IEEE TAC,
accepted 2006.
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Example
Simulation function
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Example
Indeed, and Then, Since Reach(S2)
(-1,8.5,
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Linear Systems
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Truncated Quadratic Functions

• We look for simulation functions of the form
• Decomposition of the approximation error
  transient /asymptotic
• Characterization

For a ? gt 0.
A. Girard, G.J. Pappas, Approximate bisimulations
for constrained linear systems, CDC 2005.
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Truncated Quadratic Functions

• Universal for stable linear systems
• Two stable linear systemsare approximations of
  each other.(though the precision may be very
  bad)
• Characterisation allows algorithmic
  computation of simulation functions.
• Generalizable to non-stable systems
• Two linear systems with identical unstable
  subsystemsare approximations of each other.

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MATISSE
Metrics for Approximate TransItion Systems
Simulation and Equivalence

• MATLAB toolbox
• Functionalities
• - Computation of a simulation function between
  a system and its projection. - Evaluates the
  precision of the approximation of a system by its
  projection.
• - Finds a good projection of a system (for a
  given dimension).
• - Reachability computations based on
  zonotopes.
• Available from
• http//www.seas.upenn.edu/agirard/Software/MATISS
  E/index.html

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MATISSE
Metrics for Approximate TransItion Systems
Simulation and Equivalence
Example of application safety verification of a
10 dimensional system
10 dimensionaloriginal system
5 dimensionalapproximation
7 dimensional approximation
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Outline
1. Abstraction and Approximation - Simulation
relation - Approximate simulation
relation 2. Approximate simulation relations for
continuous systems. 3. Approximate simulation
relations for hybrid systems.
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Hybrid Systems
Hybrid automaton H1 of the type
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Approximation of Hybrid Systems

• Approximation H2 of the hybrid automaton H1
• Metrics on the set of observations
• H1 et H2 have the same discrete structure -
  same underlying automaton - approximation of the
  continuous dynamics

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Approximation of Hybrid Systems
H2 approximation of H1 of the form
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Approximation of the Continuous Dynamics

• For each mode l?L, the continuous dynamics of
  H1 is approximated.
• We compute a simulation function
• We define a notion of neighborhood

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Approximate Simulation Relationsfor Hybrid
Systems

• Simulation relation of the form
• of precision dmax(d1, , dL).
• Sufficient conditions
• If then H2 is an approximation of
  H1 of precision dmax(d1, ,
  dL).

A. Girard, A.A. Julius, G.J. Pappas, Approximate
simulation relations for hybrid systems, ADHS
2006, submitted.
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Example
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Example
The first dynamics (dimension 4) is approximated
by a 2 dimensional dynamics.
Original system
Approximation
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Extensions

• Methods for the computation simulation
  functions for continuous nonlinear systems (SOS
  programs)
• Theoretical framework and aglorithms for
  approximation of stochastic hybrid systems

A. Girard, G.J. Pappas, Approximate bisimulations
for nonlinear dynamical systems, CDC 2005.
A.A. Julius, A. Girard, G.J. Pappas, Approximate
bisimulation for a class of stochastic hybrid
systems, ACC 2006. A.A. Julius, Approximate
abstraction of stochastic hybrid automata, HSCC
2006.
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Conclusion

• Unified (discrete/continuous/hybrid) framework
  for system approximation.
• Approximation as a relaxation of the notion of
  abstraction- distance between trajectories
  rather than an inclusion relation.- allows
  additional simplifications.
• Approach based on simulation functions-
  Lyapunov-like characterization - Algorithms
  (LMIs, SOS, Optimization)
• Framework suitable for safety verification of
  complex systems.