Hybrid Systems

PhD course, October 2015

Tentatively 4 ECTS Credits


Maria Prandini
Dipartimento di Elettronica, Informazione e Bioingegneria
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. Control of hybrid systems will also be addressed focusing on the class of switched linear 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.

Structure and schedule:

Monday October 5, 2015

                      10.30 – 12.00   Introduction and motivation
                      13.30 – 15.00   Modeling 

Tuesday October 6, 2015

                      09.30 – 12.00   Stability analysis
                      13.30 – 15.00   Reachability analysis

Wednesday October 7, 2015

                      10.30 – 12.00   Reachability analysis
                      13.30 – 15.00   Observability
                      15.30 – 16.15   Observability

Thursday October 8, 2015

                      09.30 – 12.00   Control design
                      13:30 – 15:00   Control design

The lectures will take place in M:2112B, that is, in the Department of Automatic Control's seminar room on the second floor on the M-building.

Reference material:

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.

Remark: Additional references will be provided within the slides of the course.


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