ECE 7850 Hybrid Systems: Theory and Applications (Spring 2017)

Course Info

Instructor: Wei Zhang, 404 Dreese Labs
Time: TuTh 11:10AM - 12:30PM
Location: Caldwell Labs 0119
Webpage: ECE 7850 webpage

Course Description

Hybrid System

Hybrid dynamical systems are characterized by coupled continuous and discrete dynamics. It switches between many operating modes where each mode is governed by its own characteristic dynamical law. Mode transitions are triggered by variables crossing specific thresholds (state events), by the elapse of certain time periods (time events), or by external inputs (input events). This course will introduce the students to the recent theoretical and application advances in modeling, control, and estimation of hybrid systems. The main topics include stability analysis and stabilization, optimal control, reachability analysis, safety verification, and estimation of hybrid systems. Emphasis will be placed on connecting the state-of-the-art theories and methods to real world applications in electrical, mechanical, and aerospace engineering.

Special Notes

  • Prerequisite: ECE 5750 - Linear System Theory or consent of instructor
  • While some prior exposure to control theory is certainly helpful, the course is sufficiently self-contained. Motivated students specializing in other technical areas with a solid math background should be able to follow course.
  • This is not a seminar type of course. Students are required to understand the mathematical derivations and proofs for most of the results covered in the class.

Grading Policy

The students will be evaluated based on several homework assignments (30%), a Midterm Exam (30%) and a final project (40%). The homework will be designed to solidify the students' understanding in the existing theories and tools in the literature, while the final project is primarily for the students to extend the theory to solve real world problems of their interest.

Tentative Topics

  1. Introduction to Hybrid Systems
    1. Examples and modeling frameworks of hybrid systems
    2. Solution and execution:
      1. Background: Basic nonsmooth analysis, differential inclusion
      2. Hybrid state trajectory, Filippov solution, zeno phenomena
  2. Stability Analysis and Stabilization
    1. Background: Lyapunov theory, Linear Matrix Inequality, S-procedure
    2. Stability analysis: stability under arbitrary switching, stability under constrained switching, Multiple-Lyapunov function
    3. Stabilization: LMI based synthesis using multiple-Lyapunov function; control-Lyapunov function approach
  3. Discrete Time Optimal Control
    1. Background: dynamic programming, Model Predictive Control (MPC), multi-parametric programming
    2. Switched LQR problem
    3. MPC of switched Piecewise Affine Systems
    4. Infinite-horizon optimal control and its connection to stability/stabilization
  4. Reachability Analysis and Computation
    1. Background: Differential games and Hamilton-Jacobi-Isaacs (HJI) equations
    2. HJI based reachability analysis
    3. Discrete-time reachability through polyhedral operations
    4. Applications in safety verification in robotics and intelligent transportation
  5. Continuous Time Optimal Control
    1. Background: Hilbert space, weak topology, chattering lemma, basic calculus of variation
    2. Theory of numerical optimization in infinite-dimensional space
      1. Optimality functions and master algorithms
      2. Epi-convergence and consistent approximation
    3. Applications to optimal control of switched nonlinear systems

Lecture Notes

  • Lecture 1: Course Info and Hybrid System Examples. [Blank Version] [Annotated Version]
  • Lecture 2: Modeling Frameworks for Hybrid Systems. [Blank Version] [Annotated Version]
    Finite state automaton, differential equation/inclusion, Caratheodory solutions, hybrid automaton, other hybrid system models
  • Lecture 3: Solution Notions of Hybrid Systems. [Blank Version] [Annotated Version]
    Execution of hybrid automaton, Zeno phenomenon, Filippov solution, existence/uniques/computation of sliding motion
  • Lecture 4: Basic Lyapunov Stability. [Blank Version] [Annotated Version]
    Basic stability concepts, Lyapunov theorems, Lyapunov equation for linear system, converse Lyapunov theorems
  • Lecture 5: Semidefinite Programming for Stability Analysis [Blank Version] [Annotated Version]
    Properties of Symmetric matrices, Shur complement lemma, Linear Matrix Inequalities, semidefinite programming, S-procedure
  • Lecture 6: Stability Analysis of Switched and Hybrid Systems [Blank Version] [Annotated Version]
    Stability under arbitrary switching, stability under slow switching, stability under state-dependent switching, Multiple Lyapunov Functions, Computation of piecewise quadratic Lyapunov functions for piecewise linear systems
  • Lecture 7: Switching Stabilization via Control-Lyapunov Function [Blank Version]
    Classical Control-Lyapunov Function Approach, Switching Stabilization Problem, Switching Stabilization via Control Lyapunov Function, Special Case: Quadratic Switching Stabilization, Special Case: Piecewise Quadratic Switching Stabilization
  • Lecture 8: Discrete Time Optimal Control and Dynamic Programming [Blank Version]
    Discrete-time optimal control problems, Dynamic Programming, Bellman Equation, Convergence of Value Iterations, Connection between Optimal Control and Stabilization Problems
  • Lecture 9: Model Predictive Control of Linear and Hybrid Systems: Basic Formulation and Algorithms [Blank Version] [Annotated Version]
    Formulation of General MPC Problems, Linear MPC Problems, Linear MPC Example: Cessna Citation Aircraft, Discrete-Time Hybrid System Models, MPC of Hybrid Systems
  • Lecture 10: Explicit Model Predictive Control [Blank Version]
    Online MPC vs. explicit MPC, Introduction Multiparametric Programming, Linear Explicit MPC, Hybrid Explicit MPC
  • Lecture 11: Model Predictive Control: Theoretical Aspects [Blank Version] [Annotated Version]
    Theoretical issues of model predictive controller, Persistent Feasibility of MPC, Stability of MPC, Stability Without Terminal Constraint/Cost, Connection to Unconstrained Problem
  • Lecture 12: Continuous-Time Optimal Control (Background) [Blank Version] [Annotated Version]
    Basics on finite dimentionsl optimization, directional derivative, calculus of variation, Euler-Lagrange equation
  • Lecture 13: Continuous-Time Switched Optimal Control [Blank Version] [Annotated Version]



Other hybrid system courses
  1. UCSB (Prof. Hespanha)

  2. UC Berkeley (Prof. Sastry and Prof. Tomlin)

  3. Cornell (Prof. Kress-Gazit)

  4. RPI (Prof. Julius)

  5. TU Delft (Prof. Schutter)

  6. University of Porto

  1. “ Predictive Control for linear and hybrid systems”, F. Borrelli, A. Bemporad and M. Morari, 2013.

  2. “Hybrid Dynamical Systems: modeling, stability, and robustness”, R. Goebel, R. G. Sanfelice and A. R. Teel, 2012.

  3. “Switching in systems and control”, D. Liberzon, 2003.

  4. Hybrid Systems: Foundations, advanced topics and applications”, J. Lygeros, S. Sastry and C. Tomlin, 2012.

Papers for Model and Solution
  1. Discontinuous Dynamical Systems: A tutorial on solutions, nonsmooth analysis, and stability”, J. Cortes, ArXiv e-prints, 2009. [C09]

  2. “Variable structure control of nonlinear multivariable systems: a tutorial”, R. DeCarlo, S. Zak and G. Matthews, Proceedings of the IEEE, 1988. [DZM88]

  3. “Hybrid dynamical systems”, R. Goebel, R. Sanfelice and A. Teel, Control Systems, IEEE, 2009. [GST09]

  4. “Variable structure systems with sliding modes”, V. Utkin, Automatic Control, IEEE Transactions on, 1977. [U77]

  5. Zeno hybrid systems, J. Zhang, K. H. Johansson, J. Lygeros and S. Sastry, International Journal of Robust and Nonlinear Control, 2001. [ZJLS01]

Papers for Stability Analysis
  1. “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems”, M. Branicky, Automatic Control, IEEE Transactions on, 1998. [B98]

  2. “Perspectives and results on the stability and stabilizability of hybrid systems”, R. DeCarlo, M. Branicky, S. Pettersson and B. Lennartson, Proceedings of the IEEE, 2000. [DBPL00]

  3. Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications”, J. Hu, J. Shen and W. Zhang, Automatic Control, IEEE Transactions on, 2011. [HSZ11]

  4. “Computation of piecewise quadratic Lyapunov functions for hybrid systems”, M. Johansson and A. Rantzer, Automatic Control, IEEE Transactions on, 1998. [JR98]

  5. “Basic problems in stability and design of switched systems”, D. Liberzon and A. Morse, Control Systems, IEEE, 1999. [LM99]

  6. “Stability and stabilizability of switched linear systems: a survey of recent results”, H. Lin and P. J. Antsaklis, Automatic control, IEEE Transactions on, 2009. [LA09]

  7. “Help on SOS”, A. Packard, U. Topcu, P. Seiler and G. Balas, Control Systems, IEEE, 2010. [PTSB10]

  8. “A tutorial on sum of squares techniques for systems analysis”, A. Papachristodoulou and S. Prajna, American Control Conference, 2005. Proceedings of the 2005, 2005. [PP05]

  9. Stabilization of hybrid systems using a min-projection strategy, S. Pettersson and B. Lennartson, American Control Conference, 2001. Proceedings of the 2001, 2001. [PL01]

  10. SOSTOOLS: sum of squares optimization toolbox for MATLAB--user's guide, S. Prajna, A. Papachristodoulou and P. A. Parrilo, Control and Dynamical Systems, California Institute of Technology, Pasadena, CA, 2013. [PPP13]

  11. “Stability results for switched controller systems”, E. Skafidas, R. J. Evans, A. V. Savkin and I. R. Petersen, Automatica, 1999. [SESP99]

  12. “A tutorial on linear and bilinear matrix inequalities”, J. G. VanAntwerp and R. D. Braatz, Journal of Process Control , 2000. [JR00]

  13. Exponential stabilization of discrete-time switched linear systems, W. Zhang, A. Abate, J. Hu and M. Vitus, Automatica, 2009. [ZAHV09]

  14. A path-following method for solving BMI problems in control, A. Hassibi, J. How and S. Boyd, American Control Conference, 1999. Proceedings of the 1999, 1999. [HHB99]

  15. “Stabilization of switched systems via composite quadratic functions”, T. Hu, L. Ma and Z. Lin, Automatic Control, IEEE Transactions on, 2008. [HML08]

Papers for Optimal Control
  1. “Optimal control of switching systems ”, S. C. Bengea and R. A. DeCarlo, Automatica , 2005. [SR05]

  2. “Dynamic programming for constrained optimal control of discrete-time linear hybrid systems”, F. Borrelli, M. Baotic, A. Bemporad and M. Morari, Automatica, 2005. [BBBM05]

  3. “Transition-time optimization for switched-mode dynamical systems”, M. Egerstedt, Y. Wardi and H. Axelsson, Automatic Control, IEEE Transactions on, 2006. [EWA06]

  4. “Piecewise linear quadratic optimal control”, A. Rantzer and M. Johansson, Automatic Control, IEEE Transactions on, 2000. [RJ00]

  5. “Consistent Approximations for the Optimal Control of Constrained Switched Systems---Part 2: An Implementable Algorithm”, R. Vasudevan, H. Gonzalez, R. Bajcsy and S. Sastry, SIAM Journal on Control and Optimization, 2013. [VGBS13]

  6. “Consistent Approximations for the Optimal Control of Constrained Switched Systems---Part 1: A Conceptual Algorithm”, R. Vasudevan, H. Gonzalez, R. Bajcsy and S. Sastry, SIAM Journal on Control and Optimization, 2013. [VGBS13_2]

  7. “Optimal control of switched systems based on parameterization of the switching instants”, X. Xu and P. Antsaklis, Automatic Control, IEEE Transactions on, 2004. [XA04]

  8. Infinite-Horizon Switched LQR Problems in Discrete Time: A Suboptimal Algorithm With Performance Analysis”, W. Zhang, J. Hu and A. Abate, Automatic Control, IEEE Transactions on, 2012. [ZHA12]

Papers for Reachability Analysis
  1. “Computational techniques for hybrid system verification”, A. Chutinan and B. Krogh, Automatic Control, IEEE Transactions on, 2003. [CK03]

  2. Hybrid Systems in Robotics: Toward Reachability-Based Controller Design”, J. Ding, J. H. Gillulay, H. Huang, M. P. Vitus, W. Zhang and C. Tomlin, Robotics Automation Magazine, IEEE, 2011. [DGHVZT11]

  3. Reachability Calculations for Vehicle Safety During Manned/Unmanned Vehicle Interaction, J. Ding, J. Sprinkle, C. J. Tomlin, S. S. Sastry and J. L. Paunicka, Journal of Guidance, Control, and Dynamics, 2012. [DSTSP12]

  4. “A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games”, I. Mitchell, A. Bayen and C. Tomlin, Automatic Control, IEEE Transactions on, 2005. [MBT05]

  5. “Safety Verification of Hybrid Systems Using Barrier Certificates”, S. Prajna and A. Jadbabaie, Hybrid Systems: Computation and Control, 2004. [PJ04]

  6. “A game theoretic approach to controller design for hybrid systems”, C. Tomlin, J. Lygeros and S. Sastry, Proceedings of the IEEE, 2000. [TLS00]

Papers for Estimation
  1. “Design of Luenberger observers for a class of hybrid linear systems”, A. Alessandri and P. Coletta, Hybrid systems: computation and control, 2001. [AC01]

  2. “On a stochastic sensor selection algorithm with applications in sensor scheduling and sensor coverage”, V. Gupta, T. H. Chung, B. Hassibi and R. M. Murray, Automatica, 2006. [GCHM06]

  3. Scheduling Kalman Filters in Continuous Time”, J. Le Ny, E. Feron and M. A. Dahleh, ArXiv e-prints, 2008. [LFD08]

  4. “Stochastic Linear Hybrid Systems: Modeling, Estimation, and Application in Air Traffic Control”, C. Seah and I. Hwang, Control Systems Technology, IEEE Transactions on, 2009. [SH09]

  5. Observability of linear hybrid systems, R. Vidal, A. Chiuso, S. Soatto and S. Sastry, Hybrid systems: computation and control, 2003. [VCSS03]

  6. On efficient sensor scheduling for linear dynamical systems, M. P. Vitus, W. Zhang, A. Abate, J. Hu and C. J. Tomlin, Automatica, 2012. [VZAHT12]

Papers for Applications
  1. Hybrid Systems in Robotics: Toward Reachability-Based Controller Design”, J. Ding, J. H. Gillulay, H. Huang, M. P. Vitus, W. Zhang and C. Tomlin, Robotics Automation Magazine, IEEE, 2011. [DGHVZT11]

  2. “Low complexity model predictive control in power electronics and power systems”, T. Geyer, 2005. [G05]

  3. A model for stochastic hybrid systems with application to communication networks, J. P. Hespanha, Nonlinear Analysis: Theory, Methods & Applications, 2005. [H05]

  4. Stochastic models for chemically reacting systems using polynomial stochastic hybrid systems, J. P. Hespanha and A. Singh, International Journal of robust and nonlinear control, 2005. [HS05]

  5. A unifying Lyapunov-based framework for the event-triggered control of nonlinear systems, R. Postoyan, A. Anta, D. Nesic and P. Tabuada, Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, 2011. [PANT11]

  6. Dynamic Buffer Management Using Optimal Control of Hybrid Systems, W. Zhang and J. Hu, Automatica, 2008. [ZH08]