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Robust Hybrid Control Systems: An Overview of Some Recent Results

  • Andrew R. Teel
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 353)

Abstract

This paper gives an overview of a framework for analyzing hybrid dynamical systems. The emphasis is on modeling assumptions that guarantee robustness. These conditions lead to a general invariance principle and to results on the existence of smooth Lyapunov functions (converse theorems) for hybrid systems. In turn, the stability analysis tools motivate novel hybrid control algorithms for nonlinear systems.

Keywords

Hybrid Systems Robustness Stability Theory Hybrid Control Systems 

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References

  1. 1.
    Ancona F, Bressan A (1999) Patchy vector fields and asymptotic stabilization, ESAIM: Control, Optimisation and Calculus of Variations, 4:445–471zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Byrnes C I, Martin C F (1995) An integral-invariance principle for nonlinear systems, IEEE Transactions on Automatic Control, 40:983–994zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cai C, Teel A R, Goebel R (2007) Results on existence of smooth Lyapunov functions for asymptotically stable hybrid systems with nonopen basin of attraction, submitted to the 2007 American Control ConferenceGoogle Scholar
  4. 4.
    Cai C, Teel A R, Goebel R (2006) Smooth Lyapunov functions for hybrid systems Part I: Existence is equivalent to robustness, Part II: (Pre-)asymptotically stable compact sets, submittedGoogle Scholar
  5. 5.
    Cai C, Teel A R, Goebel R (2005) Converse Lyapunov theorems and robust asymptotic stability for hybrid systems, Proceedings of 24th American Control Conference, 12–17Google Scholar
  6. 6.
    Chellaboina V, Bhat S P, Haddad WH (2003) An invariance principle for nonlinear hybrid and impulsive dynamical systems, Nonlinear Analysis, 53:527–550zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Collins P (2004) A trajectory-space approach to hybrid systems, Proceedings of 16th International Symposium on Mathematical Theory of Networks and SystemsGoogle Scholar
  8. 8.
    Coron J M, Rosier L (1994) A relation between continuous time-varying and discontinuous feedback stabilization, Journal of Mathematical Systems Estimation and Control, 4:67–84zbMATHMathSciNetGoogle Scholar
  9. 9.
    Filippov A F (1988) Differential Equations with Discontinuous Righthand Sides. KluwerGoogle Scholar
  10. 10.
    Goebel R, Hespanha J, Teel A R, Cai C, Sanfelice R G (2004) Hybrid systems: generalized solutions and robust stability, Proceedings of 6th IFAC Symposium on Nonlinear Control SystemsGoogle Scholar
  11. 11.
    Goebel R, Prieur C, Teel A R (2006) smooth patchy control Lyapunov functions, Proceedings of 45th IEEE Conference on Decision and ControlGoogle Scholar
  12. 12.
    Goebel R, Teel A R (2006) Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica, 42:573–587zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hàjek O (1979) Discontinuous differential equations I, Journal Differential Equations, 32:149–170zbMATHCrossRefGoogle Scholar
  14. 14.
    Hermes H (1967) Discontinuous vector fields and feedback control, in Differential Equations and Dynamical Systems, Academic Press, 155–165Google Scholar
  15. 15.
    Hespanha J P, Liberzon D, Teel A R (2005) On input-to-state stability of impulsive systems, Proceedings of 44th IEEE Conference on Decision and Control, 3992–3997Google Scholar
  16. 16.
    Hespanha J P, Morse A S (1999) Stability of switched systems with average dwell-time, Proceedings of 38th IEEE Conference on Decision and Control, 2655–2660Google Scholar
  17. 17.
    Hespanha J P, Morse A S (1999) Stabilization of nonholonomic integrators via logic-based switching, Automatica, 35:385–393zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    LaSalle J P (1967) An invariance principle in the theory of stability, in Differential equations and dynamical systems, Academic Press, New YorkGoogle Scholar
  19. 19.
    LaSalle J P (1976) The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAMGoogle Scholar
  20. 20.
    Lygeros J, Johansson K H, Simić S N, Zhang J, Sastry S S (2003) Dynamical properties of hybrid automata, IEEE Transactions on Automatic Control, 48:2–17CrossRefGoogle Scholar
  21. 21.
    Malisoff M, Rifford L, Sontag E D (2004) Global asymptotic controllability implies input-to-state stabilization, SIAM Journal on Control and Optimization, 42:2221–2238zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Prieur C (2001) Uniting local and global controllers with robustness to vanishing noise, Mathematics Control Signals Systems, 14:143–172zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Rockafellar R T, Wets R J B (1998) Variational Analysis. Springer VerlagGoogle Scholar
  24. 24.
    Ryan E P (1998) An integral invariance principle for differential inclusions with applications in adaptive control, SIAM Journal on Control and Optimization, 36:960–980zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sanfelice R G, Goebel R, Teel A R (2005) Results on convergence in hybrid systems via detectability and an invariance principle, Proceedings of 2005 American Control Conference, 551–556Google Scholar
  26. 26.
    Sanfelice R G, Goebel R, Teel A R (2006) A feedback control motivation for generalized solutions to hybrid systems, In J. P. Hespanha and A. Tiwari, editors, Hybrid Systems: Computation and Control: 9th International Workshop, volume LNCS 3927, 522–536Google Scholar
  27. 27.
    Sanfelice R G, Messina M J, Tuna S E, Teel A R (2006) Robust hybrid controllers for continuous-time systems with applications to obstacle avoidance and regulation to disconnected set of points, Proceedings of 2006 American Control Conference, 3352–3357Google Scholar
  28. 28.
    Sanfelice R G, Teel A R (2007) A “throw-and-catch” hybrid control strategy for robust stabilization of nonlinear systems, submitted to the 2007 American Control Conference.Google Scholar
  29. 29.
    Sontag E (1989) Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, 34:435–443zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Spong M W, Block D J (1995) The pendubot: A mechatronic system for control research and education, Proceedings of 34th Conference on Decision and Control, 555–556Google Scholar
  31. 31.
    Tuna S E, Sanfelice R G, Messina M J, Teel A R (2005) Hybrid MPC: Open-minded but not easily swayed, in L. Biegler R. Findeisen, F. Allgower, editor, Preprints of the International Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control, Freudenstadt-Lauterbad, Germany, 169–180Google Scholar
  32. 32.
    Ye H, Michel A N, Hou L (1998) Stability theory for hybrid dynamical systems. IEEE Transactions on Automatic Control, 43:461–474zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Andrew R. Teel
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of CaliforniaSanta Barbara

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