Approximate Bisimulations for Nonlinear Dynamical Systems

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Approximate Bisimulations for Nonlinear
Dynamical Systems

• Antoine Girard George J. Pappas

Department of Electrical and Systems
Engineering University of Pennsylvania
CDC ECC 2005 Seville, SpainDecember 12-15,
2005
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Abstractions of Systems

• Notion of approximation of systems (Computer
  Science)
• Based on language inclusion and equivalence
• Useful to reduce complexity of
• - safety verification
• - controller synthesis
• Initially, for purely discrete systems
• Extended to continuous and hybrid systems

- G.J. Pappas, Bisimilar linear systems,
Automatica, 2003. - A. van der Schaft,
Equivalence of dynamical systems by bisimulation,
IEEE TAC, 2004. - E. Hagverdi, P.Tabuada, G.J.
Pappas, Bisimulations of discrete, continuous,
and hybrid systems, TCS, 2005.
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From Abstraction to Approximation

• Continuous and hybrid systems - natural
  metrics on the state space
• Language inclusion and equivalence become
• - restrictive (binary) - not robust
• More general approach based on distance between
  languages
• More significant complexity reduction for
• - safety verification
• - controller synthesis

A. Girard, G.J. Pappas, Approximation metrics for
discrete and continuous systems, IEEE TAC,
submitted 2005.
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Outline of the Talk
1. Usual abstraction framework for systems -
Transition systems - Bisimulation
relations 2. Approximation of systems -
Approximate bisimulation relations -
Bisimulation functions 3. Approximation of
nonlinear dynamical 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
• A transition relation
• A set of observations ?
• An 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
• Similar to a possibly non-deterministic
  automaton.
• The associated external (observed) trajectory is
  noted
• The set of external trajectories is the language
  of S (noted L(S)).

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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 is time S R The transition
relation is given by The set of observations
? Rp The observation map ?x? g(x)
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Bisimulation Relations

• Language equivalence is difficult to verify
  (even for discrete systems)
• Bisimulation relations pointwise
  characterization of language equivalence
• 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

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Bisimilar Systems

• 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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From Exact to Approximate

• The previous notion is exact
• For continuous systems natural metric d? on
  the set of observations ?Rp.
• Notion of approximate language equivalence

Each trajectory of S1 is a trajectory of S2 (and
conversely).
Each trajectory of S1 has a neighboring
trajectory of S2 (and conversely).
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Approximate Bisimulation Relations

• Consider two transition systems and d ? 0
• R ? Q1 x Q2 is a d approximate
  bisimulation 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
  bisimulation relation.

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Approximately Bisimilar Systems

• If R is a d approximate bisimulation
  relation and
• then S1 and S2 are approximately bisimilar
  with precision d (S1 ?d S2)
• If S1 and S2 are approximately bisimilar with
  precision d then

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Application to Safety Verification
If S1 ?d S2 then Reach(S1) ? N(Reach(S2),d)
Reach(S2) ? N(?F,d) ? ? Reach(S1) ? ?F ?
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Computational Framework

• How do we compute
• - approximate bisimulation relations
• - an evaluation of the bisimulation metric
  between two systems
• An effective approach based on functions
• A function V Q1 x Q2 ? R ? ? is a
  bisimulation function if
• RV(d) (q1,q2) V (q1,q2) ? d
• is a d-approximate bisimulation relation
• A bisimulation function defines a parameterized
  family of approximate bisimulation relations.

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Bisimulation Functions

• Intuitively, a bisimulation function
• - bounds the distance between the observations
  - does not increase under the evolution of the
  systems
• Characterization of bisimulation functions
• Bound on the bisimulation metric between S1 and
  S2

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Bisimulation Functions for Continuous Systems
is a bisimulation function between S1 and S2 if
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Bisimulation Functions for Deterministic
Continuous Systems
is a bisimulation function if
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Sum of Squares Relaxation

• A (multivariate) polynomial p(x) is a sum of
  squares if
• A sum of squares is a positive polynomial but
• p(x) is a sum of squares more tractable than
  p(x) ? 0
• is a bisimulation
  function if

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Sum of Squares Programming

• A sum of squares program is an optimization of
  the form
• Can be solved using semidefinite programming via
  SOSTOOLS
• is a
  bisimulation function if
• Difficulty choices of a(x1,x2), ?.

S. Prajna, A. Papachristodoulou, P. Seiler and
P.A. Parrilo, SOSTOOLS, sum of squares
optimization toolbox for MATLAB, 2004.
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Example

• Search a bisimulation function of the form
• Then,
• We choose ? 0 , 0 , 1 , 4 .

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Example

• Bisimulation function obtained by SOSTOOLS
• Then,
• S1 and S2 are approximately bisimilar with
  precision 0.590

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Example

• Application to safety verification

Reachability analysis of S2 precision d 0.590
? S1 is safe.
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Conclusion

• A new framework for system approximation.
• Approximate versions of usual notions of
  abstraction
• - approximate language inclusion.
• - more robust, more significant complexity
  reduction.
• Computational framework based on bisimulation
  functions
• Approximation of nonlinear systems
• - Lyapunov like characterization of bisimulation
  functions
• - Computations based on sum of squares
  programming
• - Useful to simplify safety verification

Talk on Wednesday Approximation of linear
systems with constrained inputs