Dipartimento di Ingegneria Elettrica ed Elettronica
UniversitÓ di Cagliari, Italy

Course: Hybrid Systems
Credits: Ph.D. / Master's course, spring semester, 20 hours
Teacher: Alessandro Giua - email: giua@diee.unica.it
Ufficio: DIEE pad B, 3║ piano.   Tel: 070-675-5751
Web site: http://www.diee.unica.it/giua/SI (Italiano)
http://www.diee.unica.it/giua/SI/index_en.html (English)


  • The course is in English.

  • Schedule 2016:
    Tuesday April 12h 11-13Room Beta
    Friday April 15h 11-13Room Beta
    Monday April 18h 11-13Room M
    Tuesday April 26h 11-13Room G
    Friday April 29h 11-13Room G
    Tuesday May 3h 11-13Room G
    Friday May 6h 11-13Room G
    Tuesday May 10h 11-13Room G
    Friday May 13h 11-13Room G
    Tuesday May 17h 11-13Room G


    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 models studied in this course are: hybrid automata, timed automata, and switched systems. The students will learn to: (a) model complex systems, taken from different application domains, using hybrid automata and simulated their behavior using software tools; (b) study the reachability set of timed automata and related models; (c) analyze the stability of switched systems and design stabilizing switching laws for this class of systems.

    The course, targeted to PhD and Master students, is offered in English. It can be offered in Italian if all registered students request it.


    • Systems classification: time-driven systems, discrete event systems, hybrid systems. (2 h)
    • Examples of hybrid systems. (2 h)
    • Hybrid automata: deterministic and nondeterministic; controlled and autonomous (2 h)
    • Evolution of hybrid automata: existence and uniqueness of a solution; zenonicity, spatial regularization and minimal dwell-time; simulation using SIMULINK and MODELICA. (4 h)
    • State transition systems: reachability, equivalence classes and bisimulation. Timed automata: region graph and determination of the reachability set. Classes of hybrid automata: multirate automata; rectangular automata; initialized automata and decidability of properties by reduction to timed automata. (6 h)
    • Switched linear systems and stability: common Lyapunov function, multiple Lyapunov functions, quadratic stability, stability under slow switching. (4 h)


    Evaluation Three graded homeworks and a final project.