Hybrid Systems

Course guarantor:

Zdeněk Hurák

Lecturer:

Zdeněk Hurák

Tutor:

Zdeněk Hurák

Supervisor:

Department of Control Engineering

Synopsis:

Hybrid dynamical systems, sometimes also referred to as cyber-physical systems, contain both subsystems governed by physical laws and subsystems behaving according to logical rules and regulations, often encoded in the form of algorithms and implemented in software.

The behaviour of the former can be described by real quantities whose evolution in continuous or discrete time is commonly modelled by differential or difference equations. The behaviour of the latter is commonly described by quantities taking on a countable or finite number of values (or even just two in the case of binary quantities), whose evolution is modelled by logical models such as finite state automata or Petri nets. In the modelling and analysis of hybrid systems and the design of control systems for them, these two classes of models intersect.

However, the control system itself can also be hybrid. And the industrial reality is that practical control systems contain, in addition to the continuous subsystems represented by PID controllers or Kalman filters, a subsystem or component evaluating the satisfaction of logic conditions. Switched linear controllers (gain scheduling), supervisory control, sliding mode control or reset control are examples of such controllers with hybrid dynamics. Hybrid control methods are also becoming particularly important in a networked environment, where measurements or controls are sent over the network only when some condition is met, in order to minimize network traffic (event triggered control).

Hybrid dynamical systems thus represent a suitable theoretical and extremely practical framework for modelling, analysis and synthesis of a large number of practical control systems.

The aim of this advanced course is to help students acquire basic competences (knowledge but also practical design/computational skills) in this practically very relevant and theoretically still intensively developed area.

Requirements:

1) basics of the theory of dynamical systems: state-space models, solutions, stability; 2) basics of automatic feedback control: (state) feedback control.

Syllabus of lectures:

1. Discrete-event systems (as a prequel to hybrid systems) and their models: (State) automata

2. Discrete-event systems and their models: Petri nets

3. Discrete-event systems and their models: Max-plus algebra and Max-plus systems as computational frameworks for a subclass of discrete-event systems

4. Models of hybrid systems: Hybrid (state) automata

5. Models of hybrid systems: Hybrid (state) equations

6. Some special classes of hybrid systems: Reset systems, Switched systems, Piecewise affine (PWA) systems

7. Solutions of hybrid systems: concepts, types, intricacies

8. Stability of hybrid systems: Common Lyapunov Function approach using Linear Matrix Inequalities (LMI) and Sum of Squares (SOS) programming

9. Stability of hybrid systems: Piecewise Lyapunov Function approach via LMI and SOS

10. Complementarity dynamical systems: yet another framework for special classes of hybrid systems such as mechanical systems with impacts and electronic circuits with switching. Using complementarity optimization problems.

11. Mixed logical dynamical (MLD) systems: yet another framework for modeling hybrid systems that turns the logical part into integer arithmetics making it suitable for optimization

12. Model predictive control (MPC) for MLD systems: based on Mixed integer optimization

13. Formal verification of hybrid systems: techniques based Reachability analysis, Barrier certificate function, and Temporal logics.

Syllabus of tutorials:

The exercises will follow the lectures and will serve to develop some problem-solving skills.

Study Objective:

Study materials:

The field of hybrid systems is vast and spans several areas of science and engineering. It is therefore difficult, if not impossible, to find a single comprehensive reference to recommend to students for this subject. The more so that our choice of topics is inevitably determined by personal preferences. Therefore, the key learning resource will be the lecturer's own lecture notes. And we will always provide a list of recommended readings for further study when studying particular topics.

Among the texts that provide motivation for studying hybrid systems as well as some introduction into theoretical and computational frameworks, we recommend [1], which is also available on the authors webpage. Yet another overview, which is also available online, is [2]. And yet another is [3], which is available on the authors web page. The quartet of recommended online resources is concluded by [4].

Among the high-quality printed books, for which we are not aware of legally available online versions, the slim book [5] can be regarded as the classic. The handbook [6] contains a wealth of contributions from several authors (in fact two of the online resources linked above are chapters from this book). The latest textbook on the topic of hybrid systems is [7]. The book was probably the prime candidate for the book for this course, however we wanted a slightly different emphasis on each topic. Another relatively recent book is [8]. Although it is very well written and is certainly recommendable, it follows a particular framework that is not the most common one in the literature on hybrid systems the framework of hybrid equations. But we are certainly going to introduce their approach in our course. The more so that it is supported by a freely available Matlab toolbox. The book [9] can be regarded as a predecessor and/or complement of the just mentioned [8]. Although the book is not available online, a short version appears as an article [10] in the popular IEEE Control Systems magazine. Last but not least, MPC methodology is specialized to hybrid systems in [11]. Unlike the other books in this list, this one is freely available on the authors webpage.

[1] W. P. M. H. Heemels, D. Lehmann, J. Lunze, and B. De Schutter, Introduction to hybrid systems, in Handbook of Hybrid Systems Control: Theory, Tools, Applications, J. Lunze and F. Lamnabhi-Lagarrigue, Eds., Cambridge University Press, 2009, pp. 330. doi: 10.1017/CBO9780511807930.002.

[2] K. H. Johansson, Hybrid control systems, in UNESCO Encyclopedia of Life Support Systems (EOLSS), UNESCO, 2004. Accessed: Jun. 02, 2022. [Online]. Available: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-90411

[3] B. De Schutter, W. P. M. H. Heemels, J. Lunze, and C. Prieur, Survey of modeling, analysis, and control of hybrid systems, in Handbook of Hybrid Systems Control: Theory, Tools, Applications, F. Lamnabhi-Lagarrigue and J. Lunze, Eds., Cambridge: Cambridge University Press, 2009, pp. 3156. doi: 10.1017/CBO9780511807930.003.

[4] J. Lygeros, Lecture Notes on Hybrid Systems. 2004. Available: https://people.eecs.berkeley.edu/~sastry/ee291e/lygeros.pdf

[5] A. J. van der Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems. in Lecture Notes in Control and Information Sciences, no. 251. London: Springer-Verlag, 2000. Accessed: Dec. 11, 2018. [Online]. Available: https://doi.org/10.1007/BFb0109998

[6] J. Lunze and F. Lamnabhi-Lagarrigue, Eds., Handbook of Hybrid Systems Control: Theory, Tools, Applications, 1 edition. Cambridge, UK ; New York: Cambridge University Press, 2009.

[7] H. Lin and P. J. Antsaklis, Hybrid Dynamical Systems: Fundamentals and Methods. in Advanced Textbooks in Control and Signal Processing. Cham: Springer, 2022. Accessed: Jul. 09, 2022. [Online]. Available: https://doi.org/10.1007/978-3-030-78731-8

[8] R. G. Sanfelice, Hybrid Feedback Control. Princeton University Press, 2021. Accessed: Sep. 23, 2020. [Online]. Available: https://press.princeton.edu/books/hardcover/9780691180229/hybrid-feedback-control

[9] R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness. Princeton University Press, 2012. Available: https://press.princeton.edu/books/hardcover/9780691153896/hybrid-dynamical-systems

[10] R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid dynamical systems, IEEE Control Systems Magazine, vol. 29, no. 2, pp. 2893, Apr. 2009, doi: 10.1109/MCS.2008.931718.

[11] F. Borrelli, A. Bemporad, and M. Morari, Predictive Control for Linear and Hybrid Systems. Cambridge, New York: Cambridge University Press, 2017. Available: http://cse.lab.imtlucca.it/~bemporad/publications/papers/BBMbook.pdf

Note:

Further information:

https://hurak.github.io/hys/

Time-table for winter semester 2025/2026:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon	roomKN:E-26 Hurák Z. 12:45–14:15 (lecture parallel1 parallel nr.101) Karlovo nám.
Tue
Wed	roomKN:A-313 Hurák Z. 11:00–12:30 (lecture parallel1) Karlovo nám.
Thu
Fri

Time-table for summer semester 2025/2026:

Time-table is not available yet

The course is a part of the following study plans: