Hybrid Systems and Control, Spring 2008

• Title: Hybrid systems and control

• Organization: HYCON-EECI Graduate School on Control

• Period: April 14-18 2008

• Instructors: Christophe Prieur, LAAS-CNRS, Toulouse and Andrew Teel, University of California at Santa-Barbara

• Web page: Course web-page

• Contact: Christophe.Prieur at laas.fr

• Course description:
  Hybrid systems are systems modeled by discrete dynamics (e.g. a resetting subordinated to logic-even) combined with continuous dynamics (e.g. a physical parameter). Also, although many control systems are usually described by either a continuous-time model or a discrete-time model, when designing a stabilizing feedback, it may be necessary (or more efficient) to consider a hybrid feedback law (e.g with some flows and some jumps). In this context, the system in closed loop becomes hybrid.
  In the first part of this course, we recall some notions and some tools for the stabilization of control systems by means of (dis)continuous state-feedbacks. In the second part, we introduce a class of hybrid systems and define a notion of solution for such systems. This leads to several engineering applications with the hybrid systems framework. We study the stability theories by focusing on the KLL stability, robust KLL stability, and necessary and sufficient conditions for stability of hybrid systems. Several examples illustrate this part. In the third part of this course, we consider the design problem of hybrid feedbacks for nonlinear control systems. Some theoretic applications of these technics are given, including robust stabilization problems or optimal control theory. Also some applications are considered in robotics or in embedded systems. In the last part of this course, the problem of the output regulation by means of hybrid controllers is studied.
  The course is suitable for engineering and mathematics students who are familiar with basic linear control system theory.

• Program:
  • Stability of nonlinear control systems;
  • Definition of hybrid systems, notion of solutions, stability with respect to perturbations;
  • Design of hybrid state or output feedback laws;
  • Applications.

• Timetable: The course is divided into 14 sessions. Each slot is 1.5 hour long. The following timetable is tentative. 'AT' (resp. 'CP') means that the instructor is Andrew Teel (resp. Christophe Prieur)

  	Monday	Thuesday	Wednesday	Thursday	Friday
  9:00am - 10:30am		Slot 3: CP	Slot 7: AT	Slot 9: CP	Slot 13: AT
  11:00am - 12:30pm		Slot 4: AT	Slot 8: CP	Slot 10: CP	Slot 14: AT
  Lunch
  2:00pm - 3:30pm	Slot 1: CP	Slot 5: AT		Slot 11: CP
  4:00pm - 5:30pm	Slot 2: CP	Slot 6: AT		Slot 12: AT

• Tentative schedule and keywords
  
  Part 1: Classical system theory [CP]
  Stability of differential equations; Stabilization by means of continuous feedbacks; Control Lyapunov functions; Stabilization by means of discontinuous feedbacks
  
  Part 2: Hybrid systems theory [AT]
  Basic notions of hybrid dynamics; Robustness issue, generalized solutions for hybrid systems; Stability of hybrid systems; Lyapunov functions
  
  Part 3: Hybrid stabilizers [CP]
  Robust asymptotic stabilization of asymp. controllable systems; Nearly-optimal robust stabilization; Patchy control Lyapunov functions;
  
  Part 4: Further control applications [CP and AT]
  Reset systems; Output regulation by means of hybrid feedbacks

Classical systems theory

• A.F. Filippov, Differential equations with discontinuous right-hand sides, Transl. from the Russian, Kluwer Academic Publishers, 1988.
• O. Hájek, Discontinuous differential equations, part I, J. Diff. Equations, vol. 32, pp. 149-170, 1979.
• H. Hermes, Discontinuous vector fields and feedback control, Differential Equations and Dynamic Systems, J. K. Hale and J.P. LaSalle, Academic Press, New York and London, 1967.
• H. K. Khalil, Nonlinear Systems, 2nd edition, Prentice Hall, 1996.

Modeling

• M. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automatic Control, vol. 43, 4, pp. 475-482, 1998.
• D. Liberzon, Switching in systems and control, Systems Control: Foundations and Applications series, Birkhauser, Boston, 2003.
• H. Ye, A. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Trans. Automatic Control, vol. 43, 4, pp. 461-474, 1998.

Stability and stabilization

• K. Åström, K. Furuta, Swinging up a pendulum by energy control, Automatica, vol. 36, pp. 287-295, 2000.
• R.A. Decarlo, M.S. Branicky, S. Pettersson, B. Lennartson, Perspectives and results on the stability and stabilizability of hybrid systems, Proc. of the IEEE, vol. 88, 7, pp. 1069-1082, 2000.
• J. Hespanha, Uniform stability of switched linear systems: Extensions of LaSalle's Invariance Principle, IEEE Trans. on Automat. Contr., vol. 49, 4, pp. 470-482, 2004.
• J. Hespanha, D. Liberzon, A. S. Morse, Overcoming the limitations of adaptive control by means of logic-based switching, Syst. & Contr. Lett., vol. 49, 1, pp. 49-65, 2003.
• J.P. Hespanha, A.S. Morse, Stabilization of nonholonomic integrators via logic-based switching, Automatica, vol. 35, 3, pp. 385-393, 1999.
• J. Hespanha, A. S. Morse, Switching between stabilizing controllers, Automatica, 38, 11, pp.385-393, 2002.
• D. Liberzon, A. S. Morse, Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, vol. 19, 5, pp. 59-70, 1999.
• A.N. Michel, B. Hu, Towards a stability theory of general hybrid dynamical systems, Automatica, vol. 35, 3, pp. 371-384, 1999.
• J. Lygeros, K.H. Johansson, S.N. Simic, J. Zhang, S.S. Sastry, Dynamical properties of hybrid automata, IEEE Trans. Automat. Control, vol. 48, 1, pp. 2-17, 2003.
• H. Ye, A.N. Michel, L. Hou, Stability analysis of systems with impulse effects, IEEE Trans. Automat. Control, vol. 43, 12, pp. 1719-1723, 1998.

Further references of A. Teel (and coworkers)

The downloads are protected.
• R.W. O’Flaherty, R.G. Sanfelice, A.R. Teel, Hybrid control strategy for robust global swing-up of the pendubot, Am. Control Conf. (ACC08), Seattle, WA, 2008. pdf-file
• R.G. Sanfelice, A.R. Teel, A nested Matrosov theorem for hybrid systems, Am. Control Conf. (ACC08), Seattle, WA, 2008. pdf-file
• A.R. Teel, R.G. Sanfelice, On robust, global stabilization of the attitude of an underactuated rigid body using hybrid feedback, Am. Control Conf. (ACC08), Seattle, WA, 2008. pdf-file
• R. Goebel, C. Prieur, A.R. Teel, Smooth patchy control Lyapunov functions, submitted, 2008. pdf-file
• A.R. Teel, R.G. Sanfelice, R. Goebel, Hybrid Control Systems. Encyclopedia of Complexity and Systems Science, Springer, 2008. pdf-file
• C. Cai, A.R. Teel, R. Goebel, Smooth Lyapunov functions for hybrid systems - part II: (pre)asymptotically stable compact sets, IEEE Transactions on Automatic Control, vol. 53, 3, pp. 734-748, 2008. pdf-file
• R. Goebel, A.R. Teel, Direct design of robustly asymptotically stabilizing hybrid feedback ESAIM: COCV, DOI: 10.1051/cocv:2008023, 2008. pdf-file
• R.G. Sanfelice, R. Goebel, A.R. Teel, Generalized solutions to hybrid dynamical systems ESAIM: COCV, DOI: 10.1051/cocv:2008008, 2008.) pdf-file
• R.G. Sanfelice, R. Goebel, A.R. Teel, Invariance principles for hybrid systems with connections to detectability and asymptotic stability, IEEE Transactions on Automatic Control, vol. 52, 12, pp. 2282-2297, 2007. pdf-file
• C. Prieur, R. Goebel, A.R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems, IEEE Transactions on Automatic Control, vol.52, 11, pp. 2103-2117, 2007. pdf-file
• C. Cai, A.R. Teel, R. Goebel, Smooth Lyapunov functions for hybrid systems - part I: existence is equivalent to robustness, IEEE Transactions on Automatic Control, vol. 52, 7, pp. 1264-1277, 2007. pdf-file
• C. Cai, R. Goebel, R.G. Sanfelice, A.R. Teel, Complex hybrid systems: stability analysis for omega limit sets, 26th Chinese Control Conference, Zhangjiajie, Hunan, China, 2007. pdf-file
• A.R. Teel, Robust hybrid control systems: an overview of some recent results, Bonivento, C.; Isidori, A.; Marconi, L.; Rossi, C. (Eds.), Advances in Control Theory and Applications, vol. 353, Springer. pp. 279-302, 2007. pdf-file
• D. Carnevale, A.R. Teel, D. Nesic, A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems, IEEE Transactions on Automatic Control, vol. 52, 5, pp. 892-897, 2007. pdf-file
• R. Goebel, A.R. Teel, Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica, vol. 42, 4, pp. 573-587, 2006. pdf-file
• R. Goebel, J. Hespanha, A. Teel, C. Cai, R. Sanfelice. Hybrid Systems: Generalized Solutions and Robust Stability. In Proc. of the 6th IFAC Symp. on Nonlinear Contr. Systems, Sep. 2004. pdf-file

Further references of C. Prieur (and coworkers)

The downloads are protected.
• R. Goebel, C. Prieur, A.R. Teel, Smooth patchy control Lyapunov functions, submitted, 2008. pdf-file
• C. Prieur, R. Goebel, A.R. Teel, Hybrid feedback control and robust stabilization of nonlinear systems, IEEE Transactions on Automatic Control, vol.52, 11, pp. 2103-2117, 2007. pdf-file
• C. Prieur, E. Trélat, Quasi-optimal Robust Stabilization of Control Systems, SIAM J. Control Opt., vol. 45, 5, pp. 1875-1897, 2006. pdf-file
• C. Prieur, E. Trélat, Hybrid robust stabilization in the Martinet case, Control and Cybernetics, vol. 35, 4, pp. 923-945, 2006. pdf-file
• C. Prieur, Robust stabilization of nonlinear control systems by means of hybrid feedbacks, Rend. Sem. Mat. Univ. Pol. Torino, vol. 64, 1, pp. 25-38, 2006. pdf-file
• C. Prieur, E. Trélat, Robust optimal stabilization of the Brockett integrator via a hybrid feedback, Math. Control Signals Systems, vol. 17, 3, pp. 201-216, 2005. pdf-file
• C. Prieur, Asymptotic controllability and robust asymptotic stabilizability, SIAM J. Control Opt., vol.43, 5, pp. 1888-1912, 2005. pdf-file
• C. Prieur, A. Astolfi, Robust stabilization of chained systems via hybrid control, IEEE Trans. Auto. Control, vol. 48, 10, pp. 1768-1772, 2003. pdf-file
• C. Prieur, Uniting local and global controllers with robustness to vanishing noise, Math. Control Signals Systems, vol. 14, pp. 143-172, 2001. pdf-file