Dipartimento di Ingegneria Elettrica ed Elettronica
UniversitÓ di Cagliari, Italy

Course: Hybrid Systems
Credits: Ph.D. course, spring semester, 24 hours
Teacher: Alessandro Giua - email: giua@unica.it
Ufficio: DIEE pad B, 3║ piano.   Tel: 070-675-5751
Web site: https://www.alessandro-giua.it/UNICA/SI/index.html

Prospective students should register sending an email to the instructor by March of each year. The course will start in mid April.


The course is devoted to the study of hybrid systems, i.e., systems where a discrete event dynamics (modeled, say, by a finite state automaton) is strictly coupled with a time-driven dynamics (modeled, say, by a differential equation). Systems of this type can be found in many application domains: automotive systems (automatic transmission, cruise control systems, semi-active suspensions), mechanical systems (gears, friction models, gain switching systems), chemical systems (batch reactors), electrical systems (static converters, systems with switches or non linearities), communications (buffer control, wireless networks), automation (control using PLC, supervisory control, manufacturing systems), transportation (modeling and control of urban traffic and railways networks), embedded systems.

Hybrid systems have a great modeling power, being capable of describing physical systems at different level of abstraction, and their study presents many challenging theoretical problems: this is why the interest in hybrid systems has constantly been growing in the last years, both in academia and in industry.

The reference model adopted in this course are hybrid automata. The students will learn to model complex systems, taken from different application domains, and simulate their behavior using software tools based on the programming language Modelica. The verification of hybrid automata is addressed by means of a more abstract model called state transitions systems, by means of theoretical approaches such as (finite) bisimulations and quotient automata. Finally this general formalism will be applied to study the reachability of interesting classes of rectangular automata including timed automata.

The course, targeted to PhD students, is offered in English.


  • Hybrid systems and hybrid automata (12h lecture+ 2h homework + 2h lab)
    State variable models of time-driven systems. Examples of hybrid systems. Autonomous hybrid automata and generalizations. Hybrid automata with inputs. Evolution of a hybrid automaton. Pathological cases of continuous and hybrid evolutions.
  • Reachability analysis of hybrid systems (12h lecture + 2h homework)
    State transition systems (STSs). STS associated with a hybrid automaton. Reachability of a STS. Equivalences between states of a STS. Bisimulation between states of an STS and quotient system. Classes of rectangular automata. Timed automata: regions, equivalence between states and region graph. Initialized rectangular automata and reduction to timed automata. Elements of model checking.


Evaluation Three graded homeworks and a final project.