Bisimulation Relation

A lecture overE. Hagherdi, P. Tabuada, G. J. Pappas

Bisimulation relation for dynamical, control, and hybrid systems

Rafael WisniewskiAalborg University

Ph.D. course November 2005

Please ask as much as possible. I would be happy for all relevant to the topic questions.

Labeled Transition Systems

Product and Pullback

Product of C1 and C2

Pullback

Product of Transition Systems

Strong Bisimulation

Whenever commutes then commutes

Open Maps:

BranL Open Maps

BranL is a full subcategory of TL of all synchrinization trees with a single finite branch.

P-bisimilarity:

Generalization of P-open mapsWe generalize P-open maps to the category Dyn of dynamical systems and Hyb the category of hybrid dynamical systems.

The path category P as the full subcategory of Dynwith objects P : I → TI, where P(t) = (t, 1) and I is an open interval of Rcontaining the origin.

Morphism:

P-open Maps

P-bisimilarity for dynamical systems

Pullback in the category of P-open surjective submersions:

Bisimilarity of Dynamical Systems

Example

Consider the vector field X on M = RR2 defined

Also consider the vector field Y on N = R R defined by

is a Dyn-morphismThen

Hybrid Dynamical Systems

Category HybRecall a time transition system from Henzinger

The state space is

Transition relation like in Henzinger

Path Category in HybThe path category P is the full subcategory of Hyb:

t0 t1 t2 tk-1 tk

dx/dt = 1

Example of a pathConsider a path

This path is represented by the path object P which has states l0, l1, l2

P-open Maps for Hyb

Characterization of bisimulation in Hyb

is said to be a bisimulation relation iff for all

implies

▒▒

Bisimulation Characterization

Future Work• Extension of the bisimulation notion from the article from timed transition systems to time abstract transition systems. This can be done by identify a whole flow line with a point in the space of flow lines.

• The strong simulation is too strong equivalence relation on dynamical systems is too strong. Try to use weaker equivalence relation some form of topological equivalency.

On Friday 18th Nov. try to understand the definitions and go through proofs in the section dealing with the dynamical systems.If you understand P-open maps and bisimulation in the category of dynamical systems the generalization to hybrid systems seems natural.