Kybernetika 50 no. 1, 5-18, 2014

Optimal control processes associated with a class of discontinuous control systems: applications to sliding mode dynamics

Arturo Enrique Gil García, Vadim Azhmyakov and Michael V. BasinDOI: 10.14736/kyb-2014-1-0005


This paper presents a theoretical approach to optimal control problems (OCPs) governed by a class of control systems with discontinuous right-hand sides. A possible application of the framework developed in this paper is constituted by the conventional sliding mode dynamic processes. The general theory of constrained OCPs is used as an analytic background for designing numerically tractable schemes and computational methods for their solutions. The proposed analytic method guarantees consistency of the resulting approximations related to the original infinite-dimensional optimization problem and leads to specific implementable algorithms.


sliding mode, nonlinear systems, absolute continuous approximations


93E12, 62A10


  1. C. D. Aliprantis and K. C. Border: Infinite Dimensional Analysis. Springer, Berlin 1999.   CrossRef
  2. K. Atkinson and W. Han: Theoretical Numerical Analysis. Springer, New York 2005.   CrossRef
  3. D. Gómez-Gutiérrez, S. Čelikovský, A. Ramírez-Trevino, J. Ruiz-León and S. Di Gennaro: Robust regulation via sliding modes of a rotary inverted pendulum. In: Preprints 3rd IFAC Symposium on Robust Control Design, ÚTIA AV ČR, Praha 2000.   CrossRef
  4. S. A. Attia, V. Azhmyakov and J. Raisch: On an optimization problem for a class of impulsive hybrid systems. In: Discrete Event Dynamical Systems, 2009.   CrossRef
  5. V. Azhmyakov and J. Raisch: Convex control systems and convex optimal control problems with constraints. IEEE Trans. Automat. Control 53 (2008), 993-998.   CrossRef
  6. V. Azhmyakov, V. G. Boltyanski and A. Poznyak: Optimal control of impulsive hybrid systems. Nonlinear Anal.: Hybrid Systems 2 (2008), 1089-1097.   CrossRef
  7. V. Azhmyakov, R. Galvan-Guerra and M. Egerstedt: Hybrid LQ-optimization using dynamic programming. In: Proc. 2009 American Control Conference, St. Louis 2009, pp. 3617-3623.   CrossRef
  8. V. Azhmyakov, M. Egerstedt, L. Fridman and A. Poznyak: Continuity properties of nonlinear affine control systems: applications to hybrid and sliding mode dynamics. In: Proc. 2009 IFAC Conference on Analysis and Design of Hybrid Systems, Zaragoza 2009, pp. 204-209.   CrossRef
  9. V. Azhmyakov, V. G. Boltyanski and A. Poznyak.: The dynamic programming approach to multi-model robust optimization. Nonlinear Anal.: Theory, Methods Applications 72 (2010), 1110-1119.   CrossRef
  10. V. Azhmyakov: Optimal control of sliding mode processes: A general approach. In: Proc. 11th International Workshop on Variable Structure Systems, Mexico City 2010, pp. 504-509.   CrossRef
  11. V. Azhmyakov M. Basin and A. E. Gil García: A general approach to optimal control processes associated with a class of discontinuous control systems: Applications to the sliding mode dynamics. In: Proc. 2012 IEEE International Conference on Control Applications, Dubrovnik 2012, pp. 1154-1159.   CrossRef
  12. G. Bartolini, L. Fridman, A. Pisano and E. Usai (eds.): Modern Sliding Mode Control Theory. Lecture Notes in Control and Inform. Sci. 375, Springer, Berlin 2008.   CrossRef
  13. L. D. Berkovitz: Optimal Control Theory. Springer, New York 1974.   CrossRef
  14. I. Boiko: Discontinuous Control Systems Frequency-Domain Analysis and Design. Birkhauser, New York 2009.   CrossRef
  15. I. Boiko, L. Fridman, A. Pisano and E. Usai: On the transfer properties of the ``generalized sub-optimal" second-order sliding mode control algorithm. IEEE Trans. Automat. Control 54 (2009), 399-403.   CrossRef
  16. S. Čelikovský: Numerical algorithm for nonsmooth stabilization of triangular form systems. Kybernetika 32 (1996), 261-274.   CrossRef
  17. M. S. Branicky, V. S. Borkar and S. K. Mitter: A unifed framework for hybrid control: model and optimal control theory. IEEE Trans. Automat. Control 43 (1998), 31-45.   CrossRef
  18. P. Caines, M. Egerstedt, R. Malhame and A. Schoellig: A hybrid Bellman equation for bimodal systems. Lecture Notes in Computer Sci. 4416, Springer, Berlin 2007, pp. 656-659.   CrossRef
  19. K. Deimling: Multivalued Differential Equations. de Gruyter, Berlin 1992.   CrossRef
  20. R. E. Edwards: Functional Analysis. Dover, New York 1995.   CrossRef
  21. M. Egerstedt, Y. Wardi and H. Axelsson: Transition-time optimization for switched-mode dynamical systems. IEEE Trans. Automat. Control 51 (2006), 110-115.   CrossRef
  22. H. O. Fattorini: Infinite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge 1999.   CrossRef
  23. A. Ferrara and M. Rubbagoti: A sub-optimal second order sliding mode controller for systems with saturating actuators. IEEE Trans. Automat. Control 54 (2009), 1082-1087.   CrossRef
  24. A. F. Filippov: Differential Equations with Discontinuous Right-Hand Sides. Kluwer, Dordrecht 1988.   CrossRef
  25. A. G. Gallardo-Hern{á}ndez, L. Fridman, S. Islas-Andrade and Y. Shtessel: Quasi-continuous high order sliding modes controllers applied to glucose-insulin regulatory system models. In: Proc. 47th IEEE Conference on Decision and Control, Cancun 2008, pp. 2208-2213.   CrossRef
  26. R. Gamkrelidze: Principles of Optimal Control Theory. Plenum Press, London 1978.   CrossRef
  27. D. Gómez-Gutiérrez, S. Čelikovský, A. Ramírez-Trevino, J. Ruiz-León and S. Di Gennaro: Sliding mode observer for switched linear systems. In: Proc. 2011 IEEE Conference on Automation Science and Engineering, IEEE Conference on Automation Science and Engineering, Trieste 2011.   CrossRef
  28. W. M. Haddad and V. Chellaboina: Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton University Press, New Jersey 2008.   CrossRef
  29. J. Hale: Ordinary Differential Equations. J. Wiley, New York 1969.   CrossRef
  30. C. J. Himmelberg: Measurable relations. Fund. Math. 87 (1975), 53-72.   CrossRef
  31. A. Isidori: Nonlinear Control Systems. Springer, New York 1989.   CrossRef
  32. J. Jahn: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin 2007.   CrossRef
  33. H. K. Khalil: Nonlinear Systems. Prentice Hall, New Jersey 2001.   CrossRef
  34. A. Levant: Universal SISO sliding-mode controllers with finite-time convergence. IEEE Trans. Automat. Control 46 (2001), 1447-1451.   CrossRef
  35. D. Liberzon: Switching in Systems and Control. Birkhäuser, Boston 2003.   CrossRef
  36. Y. Orlov: Discontinuous Systems: Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions. Springer, New York 2008.   CrossRef
  37. B. E. Paden and S. S. Sastry: Calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulator. IEEE Trans. Circuits Systems 34 (1987), 73-82.   CrossRef
  38. A. Poznyak: Advanced Mathematical Tools for Automatic Control Engineers. Elsevier, Amsterdam 2008.   CrossRef
  39. V. G. Boltyanski and A. Poznyak: The Robust Maximum Principle. Birkhäuser, Boston 2011.   CrossRef
  40. R. Pytlak: Numerical Methods for Optimal Control Problems with State Constraints. Springer-Verlag, Berlin 1999.   CrossRef
  41. R. T. Rockafellar: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14 (1976), 877-898.   CrossRef
  42. L. E. Ramos-Velasco, J. J. Ruiz-León and S. Čelikovský: Rotary inverted pendulum: Trajectory tracking via nonlinear control techniques. Kybernetika 38 (2002), 217-232.   CrossRef
  43. T. Roubíček: Approximation theory for generalized young measures. Numer. Funct. Anal. Optim. 16 (1995), 1233-1253.   CrossRef
  44. M. S. Shaikh and P. E. Caines: On the hybrid optimal control problem: theory and algorithms. IEEE Trans. Automat. Control 52 (2007), 1587-1603.   CrossRef
  45. V. Utkin: Sliding Modes in Control and Optimization. Springer, Berlin 1992.   CrossRef