Ph.D. Course

Hybrid systems

Ph.D. in Information Technology

Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB)

Politecnico di Milano


Dates: February 20 – 24, 2017

Location: Sala Seminari DEIB, Milano



Maria Prandini – Politecnico di Milano



Goran Frehse – University Grenoble Alpes

Olaf Stursberg –  Universitδt Kassel

Maria Prandini – Politecnico di Milano


Course description:

The aim of this course is to introduce the student to the area of hybrid systems, that is dynamical systems characterized by the interaction of different types of dynamics, both continuous and discrete.

The interest in a systematic study of hybrid systems has been fuelled by recent technological innovations, which led to the pervasive diffusion of increasingly complex digital system for the control and supervision of physical systems (“embedded” systems represent a significant example in this respect).

The study of hybrid systems is generally more challenging than that of purely discrete or purely continuous systems, because of the interaction between dynamics of different nature. In this course, we shall introduce models for hybrid systems, and describe general methods for investigating properties such as reachability, observability and stability. Verification and control of hybrid systems will also be addressed, with a special focus on the class of piecewise affine systems. Analysis and design methods will be demonstrated on examples from different application contexts.

Students attending the course should be able to appreciate the diversity of phenomena that arise in hybrid systems, and understand how concepts that are classical in the theory of discrete systems, modelled by automata, can coexist with concepts that are classical in the theory of continuous systems, modelled by differential or difference equations, in a unifying framework


To register to the course, please, send an e-mail to


Lectures will take place at Sala Seminari of the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, via Ponzio 34/5, 20133 Milano.

See for directions on how to reach the Dipartimento di Elettronica, Informazione e Bioingegneria.


Structure of the course:

Monday February 20 [file pdf]

14.30 – 15.15   Introduction and motivation 

15.15 – 16.00   Modeling

16.30 – 18.00   Modeling

Tuesday February 21      

11.00 – 12.30   Stability [file pdf]

14.30 – 16.00   Stability 

16.30 – 17. 15  Observability [file pdf]

Wednesday February 22

11.00 – 12.30   Observability [file pdf]

14.30 – 16.00   Reachability  [file pdf]

16.30 – 18.00   Reachability [file pdf]

Thursday February 23

10.00 – 12.30     Reachability [file pdf]    

14.30 – 16. 00     Control design [file pdf]

16.30 – 18.00      Control design

Friday February 24

10.00 – 12.30     Control design

12.30 – 12.45     Conclusions



§  The above time schedule does not include breaks but refers to effective lecture-time.

§  The slides of the course will be posted in due course on this website.


Relevant references:

John Lygeros, Lecture Notes on Hybrid Systems.

H. Lin and P.J. Antsaklis, Hybrid Dynamical Systems: An introduction to Control and Verification.

Daniel Liberzon, Switching in Systems and Control.