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)


  • Classes for the 2018 course will start on April 18, h. 17-19, in Aula Mocci Pad.A.
  • Prospective students should register sending an email to the instructor by April 16.
  • The course is in English.

  • Schedule 2018:
    Thursday 26 Apr h 15-17, Room M
    Friday 27 Aprh 16-18, Room R
    Wednesday 2 Mayh 16-18, Room B
    Thursday 3 Mayh 15-17, Room M
    Wednesday 9 Mayh 16-18, Room B
    Thursday 10 Mayh 15-17, Room M
    Friday 11 Mayh 16-18, Room R
    Wednesday 14 Mayh 11-13, Room B
    Monday 16 Mayh 16-18, Room B


    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 or a final project.