### Rafael Wisniewski

* Department of Control Engineering, Aalborg University, Denmark.
*
**Time: Wednesday 26.4.2006, 15:00**

**Place: Room B2-109, Fr. Bajersvej 7**

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Towards Modelling of Hybrid Systems

By a model of a hybrid system we understand a mathematical tool for
predicting, with certain accuracy, the future behaviour of the system. The
models depend to a great extend on a type of enquiry we wish to undertake.
There are models with very little structure who take merely the underlying
topological spaces into ccount. They can be applied for majority of hybrid
systems. However, they answer only very general questions such as: Is the
given system Zeno. Adding the structure of continuous dynamics can help
answering more specific questions of reachability and safety. Finally, for
control synthesis it might be advantageous to consider the continuous
dynamics restricted to piecewise affine dynamical systems.

The program is to understand the space of trajectories/traces of a hybrid
system. I will introduce during the presentation a categorical version of a
discrete system and a labelled discrete system. To work with the space of
trajectories I use the notion of a directed topological space of Marco
Grandis. A directed topological space (X,dX) is a topological space X
equipped with a directed structure dX, that is a set of continuous paths.
The hybrid system is then a number of directed topological spaces, which are
glued together in a way encoded in a discrete system.