• Professor Claire Tomlin, Durand 028A
• tomlin at stanford.edu
• Office hours: T Th 1 - 2

• Steven Waslander
• stevenw@stanford.edu
• Office hours: Wed 1-3 pm, Durand 032

• Sherann Ellsworth, Durand 028
• sheranne@stanford.edu
• Phone: 650.723.3389

The revolution in digital technology has fueled a need for design techniques that can guarantee safety and performance specifications of embedded systems, or systems that couple discrete logic with the analog physical environment.

Hybrid systems are dynamical systems with interacting continuous-time dynamics (modeled by differential equations) and discrete-event dynamics (modeled by automata). They are important in applications in CAD, real-time software, robotics and automation, mechatronics, aeronautics, air and ground transportation systems, process control, and have recently been at the center of intense research activity in the control theory, computer-aided verification, and artificial intelligence communities. In the past several years, methodologies have been developed to model hybrid systems, to analyze their behavior, and to synthesize controllers that guarantee closed-loop safety and performance specifications. These advances have been complemented by computational tools for the automatic verification and simulation of hybrid systems.
This course will present the recent advances in modeling, analysis, control, and verification of hybrid systems. Topics will include:

• continuous-time and discrete-event models
• safety specifications and model checking
• optimal control theory and differential games
• Lyapunov stability analysis and verification tools
• numerical simulation
• a range of engineering examples

• 3/31 Course Syllabus (PDF)
• 3/31 Lecture 1 (PDF)
• 4/4 Lecture 2 (PDF)
• 4/5 Lecture 3 (PDF)
• 4/7 Lecture 4 (PDF)
• 4/11 Lecture 5 (PDF)
• 4/11 Homework 1 (PDF)
• 4/21 Lecture 6 (PDF)
• 5/2 MILP Seminar Presentation (PPT)
• 5/3 Homework 2 (PDF)
• 5/4 Homework 1 Solutions (PDF)
• 5/10 Lecture 7 (PDF)
• 5/12 Lecture 8 (PDF)
• 5/19 Lecture 9 (PDF)
• 5/20 Homework 3 (PDF)
• 5/23 Lecture 10 (PDF)
• 5/26 PPT supplement to Lecture 10 (PPT)

• Online Controls Tutorial in MATLAB
, developed by Professor Dawn Tilbury of the Department of Mechanical Engineering and Applied Mechanics at the University of Michigan, and Professor Bill Messner of the Department of Mechanical Engineering at Carnegie Mellon University.
• MATLAB Documentation Help Desk
• ODE Software for MATLAB, (by Professor Polking at Rice University)
• Reachability Analysis Tools for MATLAB , (by Professor Ian Mitchell at University of British Columbia)
• MS&E 212 Class Website, (Combinatorial Optimization with Professor Ashish Goel)
• MILP Example filesfor AMPL-CPLEX

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E205, E209A, EE263
MATLAB will be used

Homework 40%
Final project 60%

The course is based on a set of lecture notes and articles which will be made available throughout the term. A reading list of relevant articles follows:

Hybrid Modeling - Reachability Analysis

1. M. Oishi, I. Mitchell, A. Bayen, C. Tomlin, and A. Degani, Hybrid Verification of an Interface for an Automatic Landing, In the Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, December 2002.
2. Ian M. Mitchell, Alexandre M. Bayen, and Claire J. Tomlin, A Time-Dependent Hamilton-Jacobi Formulation of Reachable Sets for Continuous Dynamic Games, To appear in the IEEE Transactions on Automatic Control, June 2005.
3. 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, pp. 986-1001, July 2003.
4. A Game Theoretic Approach to Controller Design for Hybrid Systems
   Claire Tomlin, John Lygeros, and Shankar Sastry.
   Proceedings of the IEEE, Volume 88, Number 7, July 2000.
5. Validating a Hamilton-Jacobi Approximation to Hybrid System Reachable Sets
   Ian Mitchell, Alexandre Bayen, and Claire Tomlin
   ©Springer-Verlag 2001, may not be further reproduced without their permission, and is published in Volume 2034 of the LNCS series.
   

1. 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.
2. 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.
3. 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.
4. 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.
5. R. Ghosh and C. J. Tomlin, Symbolic reachable set computation of piecewise affine hybrid automata and its application to biological modeling: Delta-Notch protein signaling, IEE Transactions on Systems Biology, Volume 1, Number 1, pp. 170-183, June 2004.

1. Raymond A. DeCarlo, Michael S. Branicky, Stefan Pettersson, Bengt Lennartson, Perspectives and Results on the Stability and Stabilizability of Hybrid Systems , Proceedings of IEEE, Special Issue on Hybrid Systems, July 2000.

1. Ian Mitchell and C. J. Tomlin, Overapproximating Reachable Sets by Hamilton-Jacobi Projections, Journal of Scientific Computation, Volume 19, Number 1, pp. 323-346, December 2003.
2. T. Mehta and M. Egerstedt. Learning Multi-Modal Control Programs. To appear in Hybrid Systems: Computation and Control, Springer-Verlag, Zurich, Switzerland, March 2005
3. Y. Wardi, M. Egerstedt, M. Boccadoro, and E. Verriest. Control of Switching Surfaces. IEEE Conference on Decision and Control, Atlantis, Bahamas, Dec. 2004.
4. Rantzer, Anders, Approximate dynamic programming in switching systems, IEE Proceedings special issue on Hybrid Systems, Invited paper, to appear in 2005.

1. Cook, Cunningham, Pulleybank, and Schrijver, Combinatorial Optimization, Wiley 1997.
2. Robert Fourer, David M. Gay, Brian W. Kernighan, AMPL: A modeling language for mathematical programming, Brooks Cole, 2003
3. CPLEX Manuals available at /usr/pubsw/doc/Sweet/ilog.