Approximate Bisimulations for Constrained Linear Systems

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Approximate Bisimulations for Constrained Linear
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
constrained linear 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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Constrained Linear Systems 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? Cx
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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 Constrained Linear
Systems
is a bisimulation function between S1 and S2 if
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Quadratic Bisimulation Functions for
Deterministic Linear Systems
is a bisimulation function if
Two stable deterministic linear systems are
approximately bisimilar (the precision can be
very bad)
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Quadratic Bisimulation Functions for Constrained
Linear Systems
is a bisimulation function between S1 and S2
Then, S1 and S2 have exactly the same asymptotic
behaviors (V(0)0).
?Too restrictive
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Truncated Quadratic Bisimulation Functions for
Constrained Linear Systems
Search a bisimulation function of the form
holds for the approximation of the transient phase
holds for the approximation of the asymptotic
phase
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Truncated Quadratic Bisimulation Functions for
Constrained Linear Systems
is a bisimulation function between S1 and S2 if
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Truncated Quadratic Bisimulation Functions for
Constrained Linear Systems

• Truncated quadratic bisimulation functions are
  universal for stable constrained linear systems
• Extension to non-stable constrained linear
  systems - decompose systems into
  stable/unstable subsystems
• - bisimulation function of the form

Two stable constrained linear systems are
approximately bisimilar (the precision can be
very bad)
Two constrained linear systems with exactly
bisimilar unstable subsystems are approximately
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
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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 constrained linear systems
• - Lyapunov like characterization of bisimulation
  functions
• - Computations based on LMIs Games
• - Implemented in the toolbox MATISSE
• - Useful to simplify safety verification