Approximation Metrics for Discrete and Continuous Systems

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Approximation Metrics forDiscrete and Continuous
Systems

• Antoine Girard and George J. Pappas

Antoine.Girard_at_imag.fr, pappasg_at_ee.upenn.edu
Workshop Topics in Computation and
ControlMarch 27th 2006, Santa Barbara, CA, USA
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Safety Verification
A general system S with observations

• Language of S set of observed trajectories of
  S.
• Reachable set of S subset of observations
  reached by trajectories of S.
• Safety verification problem or Reachability
  problem

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What is Abstraction?
Given a (complicated) system S1, we compute a
(simple) system S2
All the trajectories of S1 are trajectories of
S2. (i.e. L(S1) ? L(S2)). Then, Reach(S1) ?
Reach(S2).
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Hierarchy of Abstraction
Bisimulation relation S1 ? S2
Simulation relation S1 ? S2
Language equivalence L(S1) L(S2)
Language inclusion L(S1) ? L(S2)
Reachability equivalence Reach(S1) Reach(S2)
Reachability inclusion Reach(S1) ? Reach(S2)
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From Abstraction to Approximation

• The previous notions of abstraction are all
  exact
• When dealing with continuous and hybrid
  systems
• - Uncertain parameters,
• - Noisy inputs.
• Notions of abstraction become restrictive and
  not robust.
• Notions of approximation seem more appropriate.
• Notions of approximation need metrics.

Each trajectory of S1 is a trajectory of S2.
Each trajectory of S1 has a neighboring
trajectory of S2.
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Outline of the Talk

• Approximation metrics for transition systems
• - Hierarchy of approximation metrics
• - Computational framework
• 2. Applications to safety verification
• - Approximation of continuous systems
• - Safety verification using simulation

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Transition Systems

• A transition system
• consists of
• A set of states Q
• A subset of initial states Q0 ? Q
• A set of events S
• The transition relation
• A set of observations ?
• The observation map ?q? p
• We assume systems to be non-blocking, possibly
  nondeterministic.
• The sets Q, S, and ? may be infinite.
• Modeling framework for discrete, continuous and
  hybrid systems.

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Metric Transition Systems

• A quantitative theory of approximations
  requires metrics.
• A transition system
• is a called metric transition system if
• The set of states has a metric dQ Q x Q ? R
• The set of events has the discrete metric
• The set of observations has a metric d?
  Q x Q ? R
• some regularity assumptions.

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Reachability Metrics

• Relevant question for the safety verification
  problem
• Since Reach(S1), Reach(S2) ? ? which is a
  metric space
• where h?, h denote Hausdorff distances.

How well Reach(S1) is approximated by Reach(S2) ?
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Application to Safety Verification
Reach(S1) ? N(Reach(S2),d) where d dR?(S1,S2)
Reach(S2) ? N(?U,d) ? ? Reach(S1) ? ?U ?
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Language Metrics

• More complex properties language approximation
  is more appropriate.
• Lifting the metric d? to sequences (in the
  infinity sense)
• Reachability and language metrics are useful
  but difficult to compute.

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Approximate Simulation

• Consider two transition systems and let d ? 0
  be given
• R ? Q1 x Q2 is a d - approximate simulation
  relation if it
• 1. respects observations if (q1,q2) ? R then
  d?(?q1?1, ?q2?2) ? d
• 2. respects transitions if (q1,q2) ? R then
• For d 0, we recover the usual notion of exact
  simulation.

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Simulation Metric

• If ? q1 ? Q10, ? q2 ? Q20 such that (q1,q2) ? R
  then we say that
• Tightest precision with which S2 approximately
  simulates S1
• ? Simulation metric
• Under some regularity assumptions

S2 approximately simulates S1 with precision d
S1 ?d S2
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Bisimulation Metric

• Symmetric version of approximate simulation
  approximate bisimulation
• Tightest precision with which S1 and S2 are
  approximately bisimilar
• ? Bisimulation metric
• Under some regularity assumptions

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Hierarchy of Approximation Metrics
Bisimulation metric dB(S1,S2)
Simulation metric dS?(S1,S2)
Undirected language metric dL(S1,S2)
Directed language metric dL?(S1,S2)
Undirected reachability metric dR(S1,S2)
Directed reachability metric dR?(S1,S2)
A. Girard, G.J. Pappas, Approximation metrics for
discrete and continuous systems, TAC, accepted.
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Zero Sections
Bisimulation relation S1 ? S2
Simulation relation S1 ? S2
Language equivalence cl(L(S1)) cl(L(S2))
Language inclusion cl(L(S1)) ? cl(L(S2))
Reachability equivalence cl(Reach(S1))
cl(Reach(S2))
Reachability inclusion cl(Reach(S1)) ?
cl(Reach(S2))
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Computational Framework

• How do we compute of the simulation and
  bisimulation metrics ?
• Dual approach to the relations based on
  functions
• A (bi)-simulation function is a function V Q1 x
  Q2 ? R ? ? ,
• RV(d) (q1,q2) V (q1,q2) ? d
• is a d-approximate (bi)-simulation relation
• Then, the (bi)-simulation metrics can be bounded
  by

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Bisimulation functions

• Characterization of bisimulation functions
• Minimal bisimulation function smallest function
  satisfying equation
• For the minimal bisimulation function
• Minimal bisimulation function hard to compute for
  infinite state systems.

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Outline of the Talk

• Approximation metrics for transition systems
• - Hierarchy of approximation metrics
• - Computational framework
• 2. Applications to safety verification
• - Approximation of continuous systems
• - Safety verification using simulation

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Continuous Dynamics
S generates the transition system T (Q, Q0, S,
?, ?, ?.? ) where The set of states Q Rn
The subset of initial states Q0 I The set
of labels S R The transition relation is
given by The set of observations ? Rp The
observation map ?x? g(x)
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Bisimulation functions
is a bisimulation function if and only if
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Example
Bisimulation function
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Example
Indeed, And Then, Since
,
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Constrained Linear Systems
For bisimulation functions of the form
we get
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Truncated Quadratic Functions

• We search bisimulation functions of the form
• Decomposition transient/asymptotic error
• Characterization

For some ? 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 constrained linear
  systems
• Two stable constrained linear systems are
  approximately bisimilar.(but the precision can
  be very bad!)
• Characterization allows to derive
  computationally effective algorithms.
• Generalizable to non-stable systems
• two systems are approximately bisimilar ifftheir
  unstable subsystems are exactly bisimilar.

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MATISSE
Metrics for Approximate TransItion Systems
Simulation and Equivalence

• MATLAB toolbox
• Functionalities
• - Computes a bisimulation function between a
  system and its projection.
• - Evaluates the bisimulation distance between
  a system and its projection.
• - Finds a good projection of a system (given
  the desired dimension).
• - Performs reachability computations using
  zonotopes.
• Available at
• 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
ten-dimensional system
10-dimensionaloriginal system
5-dimensionalapproximation
7-dimensionalapproximation
A. Girard, G.J. Pappas, Approximate bisimulation
relations for constrained linear systems,
Submitted 2005.
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Extensions

• Computational method for nonlinear autonomous
  systems (SOS)
• Characterization of approximate simulation for
  hybrid systems
• Theoretical framework, computational methods
  for stochastic linear dynamical/hybrid
  systems (with stochastic jumps)

A. Girard, G.J. Pappas, Approximate bisimulations
for nonlinear dynamical systems, CDC 2005.
A. Girard, A.A Julius, G.J. Pappas, Approximate
simulation relations for hybrid systems, ADHS
2006.
A.A. Julius, A. Girard, G.J. Pappas, Approximate
bisimulation for a class of stochastic hybrid
systems, ACC 2006. Talk on Wednesday A.A.
Julius, Approximate abstraction of stochastic
hybrid automata, HSCC 2006.
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Outline of the Talk

• Approximation metrics for transition systems
• - Hierarchy of approximation metrics
• - Computational framework
• 2. Applications to safety verification
• - Approximation of continuous systems
• - Safety verification using simulation

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Back to Transition Systems

• Let us consider a metric transition system
• A pseudo-metric dB on the set of states Q
• dB(q, q) 0
• dB(q1, q2) dB(q2,q1)
• dB(q1, q3) ? dB(q1, q2) dB(q2, q3)
• is a bisimulation metric if there exists ? gt 1
• Bisimulation metric ? pseudo-metric
  bisimulation function.

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Simulation-based Reachability

• The bisimulation metric allows to sample
  subsets of Q
• Simulation-based reachability
• - sample the set of initial states
• - sample of the successor operators

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Simulation-based Reachability

• Simulation-based reachability let d ? d/? e

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Simulation-based Reachability

• Because d is a bisimulation metric we can show
  that
• Then, it follows that

Talk on Friday A. Girard, G.J. Pappas,
Verification using Simulation, HSCC 2006.
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Conclusion

• Unified (discrete/continuous) framework for
  system approximation.
• Approximation as a relaxation of abstraction-
  metrics instead of relations.- more significant
  complexity reduction.
• Approach based on bisimulation functions-
  Lyapunov like characterization- computational
  methods (LMIs, SOS, Games)
• Robustness of the safety of the original system
  is critical for the amount of approximations that
  can be done.