EECS 291e and ME 291S

Hybrid Systems: Computation and Control

Shankar Sastry

with Jonathan Sprinkle and Mikael Eklund

285 Cory Hall and occassionally 540A Cory Hall
Tu-Th 2 -3:30 pm.

The multi-disciplinary research field of hybrid systems has emerged over the last decade and lies at the boundary of computer science, control engineering and applied mathematics. In general, a hybrid system can be defined as a system built from atomic discrete components and continuous components by parallel and/or serial composition, arbitrarily nested. The behaviors and interactions of components are governed by models of computation.

Hybrid phenomena captured by such mathematical models are manifested in a great diversity of complex engineering applications such as real-time systems, embedded software, robotics, mechatronics, aeronautics, and process control. The high-profile and safety-critical nature of such applications has fostered a large and growing body of work on formal methods for hybrid systems: mathematical logic, computational models and methods and automated reasoning tools supporting the formal specification and verification of performance requirements for hybrid systems, and the design and synthesis of control programs for hybrid systems that are provably correct with respect to formal specifications.

This course investigates modeling, analysis and verification of various classes of hybrid systems.  Special attention  is paid to computational and simulation tools for hybrid systems. Applications ranging from networked sensors, power electronics, avionics, autonomous vehicles will be covered.  The course consists of lectures, a handful of homework assignments, and a final project.

2005 Course Outline

Office Hours

Office: 514 Cory Hall

Office Hours: Monday Wednesday 3:00pm-4:00pm




Homework 30%
Project 70%


Students should understand basic concepts in differential equations, dynamical systems and logic. They should know how to program in some language: for example, Matlab, Mathematica, Java or C. Familiarity with control theory and/or automata theory is useful. Thus a course like EECS 221A or CS 170 is desirable, but not essential.

Class email alias

hscc_ee291e at

Problem Sets

Problem Set on Hybrid Systems by John Lygeros

Issued March 1st 2005, Due March 29th, 2005

Solution Set to Problem Set on Hybrid Systems by John Lygeros

Project Topics

Potential project topics are maintained by Jonathan Sprinkle at Projects List

Key Deadlines
Feburary 28th Preliminary Proposals Due
March 18th Preliminary Proposal Outline Due
May 9th, 10th Final Project Presentations

Recommended Reading

J. Lygeros Lecture Notes on Hybrid Systems, Notes for an ENSIETA short course, Feb 2004.

J. Lygeros, C. Tomlin and S. Sastry Art of Hybrid Systems, Compendium of Lecture Notes for the Hybrid Systems Class, 2002.

P. J. Mosterman An Overview of Hybrid Simulation Phenomena and their support by simulation packages, in HYbrid Systems: COmputation and COntrol, W. VaanDrager and J. van Schuppen (editors), Springer Verlag Lectures Notes in Computer Science, Vol. LNCS-1569, 1999, pp. 178-192.

Luca Carloni, Maria D. Di Benedetto, Alessandro Pinto and Alberto Sangiovanni-Vincentelli Modeling Techniques, Programming Languages Design Toolsets and Interchange Formats for Hybrid Systems EU-IST Columbus Project, DHS3 Report, 2004

G. Lafferiere, G. J. Pappas and S. Sastry O-Minimal Hybrid Systems Mathematics of Control, Signals and Systems, Vol. 13. No. 1, pp. 1-21, March 2000

G. J. Pappas and co-authors Papers on Extensions of Model Checking Methods to Many Interesting Classes of Systems

I. M. Mitchell, A, M. Bayen, C. Tomlin A Time Dependent Hamilton-Jacobi Equation Formulation of Reachable Sets for Continuous Dynamical Systems To appear in the IEEE Transaction in Automatic Control, 2005 (month tbd)

H. de Jong, J.-L. Gouzé, C. Hernandez, M. Page, T. Sari, J. Geiselmann (2003), Hybrid modeling and simulation of genetic regulatory networks: A qualitative approach, Hybrid Systems: Computation and Control, HSCC 2003, Lecture Notes in Computer Science 2623, 267-282.

C. Belta, P. Finin, L.C.G.J.M. Habets, A. Halasz, M. Imielinski, V.Kumar, and H. Rubin, Understanding the bacterial stringent response using reachability analysis of hybrid systems Hybrid Systems: Computation and Control, HSCC 2004, Lecture Notes in Computer Science 2993, 111-126.

Lincoln, P. and Tiwari, A., Symbolic systems biology: Hybrid modeling and analysis of biological networks Hybrid Systems: Computation and Control, HSCC 2004, Lecture Notes in Computer Science 2993, 660-672

Joshi, K. Neogi, N., and W. Sanders, Dynamic Partitioning of Large Discrete Event Biological Systems for Hybrid Simulation and Analysis Hybrid Systems: Computation and Control, HSCC 2004, Lecture Notes in Computer Science 2993

R. Alur, T.A. Henzinger, H. Wong-Toi. Symbolic analysis of hybrid systems. Proceedings of the 37th IEEE Conference on Decision and Control, Invited survey, 1997.

Thomas A. Henzinger, The Symbolic Approach to Hybrid Systems,  (CAV '02), UC Berkeley,

E. Asarin, O. Bournez, T. Dang, O. Maler, and A. Pnueli, Effective Synthesis of Switching Controllers for Linear Systems,
In Proceedings of the IEEE, 88, Special Issue Hybrid System: Theory & Applications, 1011-1025, 2000.

Thao Dang, Alexandre Donze, and Oded Maler, Verification of Analog and Mixed-Signal Circuits using Hybrid Systems Techniques, Submitted to DAC'04 - Design Automation Conference, June 2004.

B. H. Krogh and O. Stursberg, On efficient representation and computation of reachable sets for hybrid systems, in Hybrid Systems: Computation and Control (HSCC'03), Lecture Notes in Computer Science (LNCS), Springer..

Paulo Tabuada and George J. Pappas, Linear temporal logic control of linear systems,  IEEE Transactions on Automatic Control, Submitted February 2004.

Gerardo Lafferriere, George J. Pappas, and Sergio Yovine, Symbolic reachability computations for families of linear vector fields,  Journal of Symbolic Computation, 32(3):231-253, September 2001.

C. Tomlin, I. Mitchell, A. Bayen, and M. Oishi, Computational Techniques for the Verification and Control of Hybrid Systems,
Proceedings of the IEEE, Volume 91, Number 7, July 2003.

Claire Tomlin, John Lygeros, and Shankar Sastry, A Game Theoretic Approach to Controller Design for Hybrid Systems, Proceedings of the IEEE, Volume 88, Number 7, July 2000.

Matthew Senesky, Gabriel Eirea, and T. John Koo, Hybrid Modelling and Control of Power Electronics, Hybrid Systems: Computation and Control April,  Lecture Notes in Computer Science, Vol. 2623, pp. 450-465, Springer-Verlag, 2003.

T. J. Koo, S. Sastry, Bisimulation Based Hierarchical System Architecture for Single-Agent Multi-Modal Systems, Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, Vol. 2289, pp. 281-293, Springer-Verlag, 2002.


Class Schedule from John Koo's offering in Spring 2004 at Vanderbilt University: I will use this material interspersed in the lecture notes

Date Possible Topic(s) Downloads
January 18 Introduction   Introduction
  Examples: Hybrid Automata  
  Modeling: Finite State Machine Finite state Machines
  Modeling: Timed Automata
Tool: HyTech
  Modeling: Ordinary Differential Equations
Tool: Matlab
  Modeling: Hybrid Automata
Tool: HyVisual
  Analysis: Reachability - Discrete Discete Reachability
  Analysis: Reachability - Continuous Continuous Reachability
  Analysis: Reachability - Hybrid Hybrid Reachability
  Analysis: Reachability - Hybrid
Tool: Requiem

Tool: Ptolemy II - Multi-Modal Systems

  Tool: Ptolemy II - Multi-Modal Systems
  Presentation: Paper  
  Presentation: Paper  
March 22 No Class  
March 24 No Class  
  Computation: Hybrid Automata
Tool: d/dt
  Discussion: Projects  
  Verification: Temporal Logic  
  Verification: Model Checking  
  Discussion: Projects  
  Verification: Time Automaton
Application: Sensor Network
  Verification: Timed Automata Model Checking for Timed Automata
  Verification: Time Automaton  
  Verification: Time Automaton  
  Verification: Time Automaton  
  Summary: Hybrid Automaa  
  Computation: Linear Systems
Tool: Requiem,d/dt ,Checkmate
  Computation: HSIF  
  Games and Maximal Invariant Sets Church and Hamilton Jacobi based Invariant Set Computation
  Overview of Hybrid Systems with applications Hybrid Systems: Modeling, Analysis and Control
  Eckman Award Plenary talk by Professor Claire Tomlin , American Control Conference, July 2004 Hybrid Control: from Air Traffic to Fly Wings
  Guest lecture by Professor Alex Bayen , Computing Reach Sets Using A Modified Hamilton Jacobi Equation
  Guest lecture by Professor Alex Bayen , A Viability Theory Approach to Hybrid Systems See also the lecture notes by Lygeros





























Useful Links:

Thanks to John Koo of Vanderbilt University for helping develop this home page.