FEL3230 PhD Course on Hybrid Systems, HT21

Credits: The course is worth 7.5 credits. Grading is based on P/F system.


Course responsible: Dimos Dimarogonas, dimos@kth.se.


Lecturers: Dimos Dimarogonas, Maryam Sharifi, Pian Yu, Jana Tumova




Hybrid systems are dynamical systems that exhibit both continuous and discrete behavior and as such, they allow to model complex dynamic phenomena in real-world systems, such as cyber-physical systems, with application examples varying from automotive industry, to consumer electronics, power systems, smart buildings, and transportation systems. For all of these examples, properties such as stability or correctness with respect to design specifications are crucial. However, the rich expressiveness of hybrid systems requires special techniques to analyze and derive these properties.  This course will focus on selected related topics in hybrid systems, with a special focus on their stability, stabilization, abstraction and formal verification.




Learning outcomes


After the course, the student should be able to:

       know the essential theoretical tools to model hybrid control systems and cope with related verification problems

       know the established problems and results in the area

       apply the theoretical tools to problems in the area

       contribute to the research frontier in the area


Course main content


A preliminary structure and exact dates are given below. Reading material for each lecture will be updated as the course evolves.


Lecture 1: Course outline. Introduction to hybrid systems. Motivating examples. Modelling and hybrid Automata. Zeno behavior (November 9, 1-3 pm)

Reading material for lecture 1:

1.     J. Lygeros, K. H. Johansson, S. Simic, J. Zhang, and S. Sastry, Dynamical properties of hybrid automata, IEEE Transactions on Automatic Control, 48:1, 2-17, 2003.


Lecture 2: Stability of hybrid systems. Multiple Lyapunov Functions. (November 10, 3-5 pm)

Reading material for lecture 2:

  1. D. Liberzon, Switching in Systems and Control, Birkhauser, 2003, Part II.
  2. M. S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems (Links to an external site.), IEEE Transactions on Automatic Control, Vol. 43, No. 4, pp. 475-482, April 1998.


Lecture 3: Stabilization of hybrid systems. Quantized and event-based control. (November 12, 10-12 am)

Reading material for lecture 3:

  1. D. Liberzon, Switching in Systems and Control, Birkhauser, 2003, Part III.
  2. P. Tabuada, Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks, IEEE Transactions on Automatic Control, Vol. 52, No. 9, pp. 1680-1685, Sept. 2007.


Lecture 4: Transitions systems, simulation and bisimulation relations, reachability and safety, temporal logic specifications (November 16, 1-3 pm)

Reading material for lecture 4:

  1. C. Belta, B. Yordanov and E.A. Gol, Formal Methods for Discrete-Time Dynamical Systems, Springer, 2017, Chapters 1-3.


Lecture 5: Timed automata, bisimilar linear systems. Given by Prof. Jana Tumova. (November 19, 10-12 pm)


Reading material for lecture 5:

1.      G. Pappas, Bisimilar linear systems, Automatica, 39(12):2035-2047, 2003.

2.      R. Alur, Timed Automata, Conference on Computer Aided Verification (CAV), pp. 379-395, 1999.


Lecture 6: Abstraction based LTL control synthesis. Given by Dr. Pian Yu. (November 22, 10-12 am)

Reading material for lecture 6:

1.      A. Girard, G. Pola  and P. Tabuada (2009). Approximately bisimilar symbolic models for incrementally stable switched systems. IEEE Transactions on Automatic Control, 55(1), 116-126. https://ieeexplore.ieee.org/abstract/document/5342460

2.     P. Yu, D. V. Dimarogonas (2021). Robust approximate symbolic models for a class of continuous-time uncertain nonlinear systems via a control interface, https://arxiv.org/abs/2103.09024

3.      P. Yu, D. V. Dimarogonas (2021). Distributed motion coordination for multi-robot systems under LTL specifications. IEEE Transactions on Robotics, DOI: 10.1109/TRO.2021.3088764. https://ieeexplore.ieee.org/document/9473029



Lecture 7: Control barrier functions, finite-time stability, finite-time control barrier functions. Given by Dr. Maryam Sharifi. (November 24, 3-5 pm)

Reading material for lecture 6:

1.      Aaron D. Ames, Xiangru Xu, Jessy W. Grizzle, and Paulo Tabuada,  Control Barrier Function Based Quadratic Programs for Safety Critical Systems, IEEE Transactions on Automatic Control 62, no.8, pp.3861-3876, 2017.

2.      Sanjay P. Bhat, and Dennis S. Bernstein, Finite-Time Stability of Continuous Autonomous Systems, SIAM Journal on Control and Optimization 38, no. 3, pp. 751-766, 2000.

3.      Kunal Garg, Ehsan Arabi, and Dimitra Panagou, Fixed-time Control under Spatiotemporal and Input Constraints: A Quadratic Program Based Approach ,  arXiv preprint arXiv:1906.10091, 2019.


Lecture 8: Signal Temporal Logic. Control of systems under STL tasks using control barrier functions and finite-time control barrier functions. Given by Dr. Maryam Sharifi. (November 26, 10-12 pm)

Reading material for lecture 7:

1.    Ezio Bartocci, Jyotirmoy Deshmukh, Alexandre Donze, Georgios Fainekos, Oded Maler, Dejan Nickovic, and Sriram SankaranarayananSpecification-Based Monitoring of Cyber-Physical Systems: A Survey on Theory, Tools and Applications, In Lectures on Runtime Verification, pp. 135-175. Springer, Cham, 2018.

2.      Lars Lindemann, and Dimos V. Dimarogonas, Control Barrier Functions for Signal Temporal Logic Tasks, IEEE control systems letters 3, no. 1, pp. 96-101, 2018.

3.      Maryam Sharifi, and Dimos V. Dimarogonas, Fixed-Time Convergent Control Barrier Functions for Coupled Multi-Agent Systems Under STL Tasks, ECC 2021, arXiv preprint arXiv:2103.00986.



Lecture 9: Special topics on hybrid control. (December 2, 1-3 pm)

Reading material for lecture 9:



Project presentations: (December 7, 1-5 pm)


Course disposition


Lectures, course literature.




Basic courses on Automatic Control, Linear Algebra. At least one advance course in automatic control will be of help, but not compulsory.


Requirements for final grade


Passing Grade based on homework and final project/take-home exam.