Approximation Metrics for Discrete, Continuous and Hybrid Systems'

1
Approximation Metrics for Discrete, Continuous
and Hybrid Systems.

• Antoine Girard
• (joint work with George J. Pappas)

agirard_at_seas.upenn.edu
Chess Seminar University of California at
BerkeleyNovember 29th, 2005
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Approximation Theories

• Approximation theories for systems
• Model reduction (control)
• - for continuous systems
• - approximation of input/output mappings -
  quantification of error (H? norm)
• - not suitable for many problems (e.g.
  reachability)
• System refinement and equivalence (computer
  science)
• - initially for discrete systems
• - extends to continuous and hybrid systems -
  based on language inclusion and equivalence
• - restrictive (binary notion) and not robust

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A New Approximation Theory

• A new notion of system approximation
• based on approximate language inclusion and
  equivalence - extends usual notions of exact
  system refinement and equivalence -
  distances between languages - useful for
  reachability question (L? norm)
• which applies to discrete/continuous/hybrid
  systems
• which can be used for practical problems -
  computational framework - compositional reasoning

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Outline of the Talk
1. Exact system refinement and equivalence -
Transition systems framework - Simulation and
bisimulation relations 2. Approximation metrics
for transition systems - Hierarchy of
approximation metrics - Simulation and
bisimulation functions 3. Approximation of
continuous and hybrid systems
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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 labels S
• The transition relation
• A set of observations ?
• The observation map ?q? p
• The sets Q, S, and ? may be infinite.

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Transition Systems

• A state trajectory of S (Q,Q0,S,?,?,?.?) is
• Automata-like semantics. We assume S is
  non-blocking, possibly non-deterministic.
• The associated external trajectory is noted
• The set of external trajectories is the language
  of S (noted L(S)).
• Reach(S) ? ? is the set of points reachable by
  external trajectories

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Continuous Dynamics as Transition Systems
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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System Approximation from Computer Science
Given a (complicated) system S1, we consider a
(simple) system S2
All the trajectories of S1 are trajectories of
S2. (i.e. L(S1) ? L(S2)).
Application to safety verification
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System Approximation from Computer Science
Given a (complicated) system S1, we consider a
(simple) system S2
All the trajectories of S2 are trajectories of
S1. (i.e. L(S2) ? L(S1)).
Application to synthesis
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Simulation Relations

• Language inclusion is difficult to verify (even
  for discrete systems)
• Simulation relations pointwise
  characterization of language inclusion
• Consider two transition systems
• R ? Q1 x Q2 is a simulation relation of S1 by
  S2 if it
• 1. respects observations if (q1,q2) ? R then
  ?q1?1 ?q2?2
• 2. respects transitions if (q1,q2) ? R then

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Simulation Relations

• If R ? Q1 x Q2 is a simulation relation of S1
  by S2 and
• then we say that S2 simulates S1 (noted S1 ?
  S2)
• Approximation result
• If S1 ? S2 then L(S1) ? L(S2)

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Bisimulation Relations

• Bisimulation relations symmetric simulation
  relations
• Consider two transition systems
• R ? Q1 x Q2 is a bisimulation relation between
  S1 and S2 if it
• 1. respects observations if (q1,q2) ? R then
  ?q1?1 ?q2?2
• 2. respects transitions if (q1,q2) ? R then

and
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Bisimulation Relations

• If R ? Q1 x Q2 is a bisimulation relation
  between S1 and S2 and
• then we say that S1 and S2 are bisimilar
  (noted S1 ? S2)
• Equivalence result
• If S1 ? S2 then L(S1) L(S2)

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Bi-simulation Relationsfor Continuous and Hybrid
Systems

• Continuous systems
• Unifying discrete and continuous notions
• Extensions to hybrid systems

- G.J. Pappas, Bisimilar linear systems,
Automatica, December 2003. - P. Tabuada and G.J.
Pappas, Bisimilar control affine systems, Systems
and Control Letters, May 2004. - G.J. Pappas and
S.Simic, Consistent abstractions of affine
control systems, IEEE TAC 2002. - P. Tabuada and
G.J. Pappas, Abstractions of Hamiltonian systems,
Automatica, 2003. - K. Grasse, Admissibility of
trajectories in F-related systems, MCSS 2003. -
A. van der Schaft, Bisimulations of dynamical
systems, Hybrid Systems Computation and
Control, 2004.
- E. Hagverdi, P.Tabuada, G.J. Pappas,
Bisimulations of discrete, continuous, and hybrid
systems, TCS, 2005. - A.A.Julius, A.J. van der
Schaft, A behavioral framework for
compositionality, MTNS 2004.
- P. Tabuada, G.J. Pappas, P. Lima, Composing
abstractions of hybrid systems, DEDS, 2004. - A.
van der Schaft, Bisimulations of dynamical
systems, Hybrid Systems Computation and
Control, 2004. - G. Pola, A. van der Schaft, M.
di Bennedeto, Equivalence of switching linear
systems by bisimulation, CDC 2004.
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Hierarchy of System Relationships
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 Exact to Approximate

• The previous notions are all exact
• When dealing with continuous and hybrid
  systems
• - Uncertain parameters,
• - Noisy inputs.
• System relationships become restrictive and not
  robust.
• An approximate version seems more appropriate.
• Approximate versions 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
1. Exact system refinement and equivalence -
Transition systems framework - Simulation and
bisimulation relations 2. Approximation metrics
for transition systems - Hierarchy of
approximation metrics - Simulation and
bisimulation functions 3. Approximation of
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 reachability problems
• 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(?F,d) ? ? Reach(S1) ? ?F ?
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Language Metrics

• More complex properties language approximation
  is needed.
• 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 relation.

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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, submitted 2005.
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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

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Exact Metric Computation

• Minimal (bi)-simulation function smallest
  function satisfying equation
• Then, the (bi)-simulation metrics can be computed
  by solving

- L. de Alfaro, M. Faella, M. Stoelinga, Linear
and branching metrics for quantitative transition
systems, ICALP 2004. - A. Girard, G.J. Pappas,
Approximation metrics for discrete and continuous
systems, submitted 2005.
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Approximate Metric Computation

• Minimal (bi)-simulation function may be hard to
  compute for continuous and hybrid systems (HJB
  equation).
• Characterization of (bi)-simulation functions
• Then, the (bi)-simulation metrics can be bounded
  by solving

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Outline of the Talk
1. Exact system refinement and equivalence -
Transition systems framework - Simulation and
bisimulation relations 2. Approximation metrics
for transition systems - Hierarchy of
approximation metrics - Simulation and
bisimulation functions 3. Approximation of
continuous and hybrid systems
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Bisimulation functions for continuous systems
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

A. Girard, G.J. Pappas, Approximate bisimulation
relations for constrained linear systems,
Submitted 2005. A. Girard, Reachability of
uncertain linear systems using zonotopes, HSCC
2005. A. Girard, C. Le Guernic, O. Maler,
Efficient computation of reachable sets of linear
time-invariant systems with inputs, HSCC 2006.
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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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MATISSE
Metrics for Approximate TransItion Systems
Simulation and Equivalence
Example of application safety verification of a
hundred-dimensional system
100-dimensionaloriginal system
6-dimensionalapproximation
10-dimensionalapproximation
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Extensions

• Done
• Computational method for nonlinear autonomous
  systems (SOS)
• Theoretical framework, computational methods
  for stochastic linear dynamical/hybrid systems
  (with stochastic jumps)
• Ongoing work
• Computational methods for nonlinear systems
  with inputs.

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, submitted to ACC 2006. A.A. Julius,
Approximate abstraction of stochastic hybrid
automata, HSCC 2006.
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Approximation of Hybrid Systems
Hybrid automaton H1 of the type
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Approximation of Hybrid Systems

• Approximation H2 of the hybrid automaton H1
• metric on the set of observation
• this topology implies that
• - H1 and H2 have the same discrete structure
  -gt same underlying automaton -gt
  approximation on the continuous dynamics
• - the transitions occurs synchronously in H1 and
  H2

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

• In each location l?L, the continuous dynamics
  of H1 is approximated.
• We compute a simulation function
• Let us define

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Approximate Simulation Relationsfor Hybrid
Systems
Let d1, , dL be positive scalars such that
Then, the relation is an approximate simulation
of H1 by H2 with precision dmax(d1, ,
dL). Moreover, H1 ?d H2.
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Conclusion

• Unified (discrete/continuous/hybrid) framework
  for system approximation.
• Approximation as a relaxation of abstraction-
  metrics instead of relations.- more significant
  complexity reduction.
• Approach based on bi-simulation 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.

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Ongoing and Future Work

• Explore properties of approximations under
  composition operators
• Computational methods for bisimulation functions
• - for nonlinear systems
• - for hybrid systems
• Simulation-based verification of robust
  properties
• - A. Girard, G.J. Pappas, Verification using
  Simulation, HSCC 2006
• Use an approximate bisimulation-based approach
  for hierarchical control
• Include these functionalities in MATISSE.