Ph.D. Course

Hybrid systems

Ph.D. in Information Technology

Dipartimento di Elettronica, Informazione e Bioingegneria (DEIB)

Politecnico di Milano


Dates: January 25 - 29, 2021



Maria Prandini - Politecnico di Milano



Karl Henrik Johansson - KTH, Stockholm

John Lygeros - ETH Zurich

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 fueled 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 with connections to networked control and scheduling.

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 equations, in a unifying framework.


Lectures will take place online via MS Teams.

Link to the MS Teams virtual class will be provided in due course to registrants.


To register to the course, please, visit this link here.


Structure of the course

Monday, January 25
11.30 - 13.00 Introduction. Definition of hybrid automaton
14.30 - 16.00 Execution of a hybrid automaton

Tuesday, January 26 rescheduled to Monday February 15
09.30 - 11.00 Lyapunov Stability
11.30 - 13.00 Stability for switched linear systems
15.00 - 16.30 Observer design

Wednesday, January 27
11.30 - 13.00 Reachability analysis: definition and deductive methods
14.30 - 16.00 Model checking

Thursday, January 28
09.30 - 11.00 Reachability with inputs: a viability theory perspective
11.30 - 13.00 Control design for hybrid systems
14.30 - 17.00 Hybrid control strategies and scheduling

Friday, January 29
09.30 - 11.00 Hybrid control in networked systems
11.30 - 13.00 Hybrid control in networked systems



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.