Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Hybrid Systems: Computation and Control

HSCC 2008: Hybrid Systems: Computation and Control pp 30–42Cite as

On the Maximum Principle for Impulsive Hybrid Systems

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4981))

Abstract

In this contribution, we consider a class of hybrid systems with continuous dynamics and jumps in the continuous state (impulsive hybrid systems). By using a newly elaborated version of the Pontryagin-type Maximum Principle (MP) for optimal control processes governed by hybrid dynamics with autonomous location transitions, we extend the necessary optimality conditions to a class of Impulsive Hybrid Optimal Control Problems (IHOCPs). For these problems, we obtain a concise characterization of the Impulsive Hybrid MP (IHMP), namely, the corresponding boundary-value problem and some additional relations. As in the classical case, the proposed IHMP provides a basis for diverse computational algorithms for the treatment of IHOCPs.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arutyunov, A.V., Aseev, S.M.: Investigation of the degeneracy phenomenon in the maximum principle for optimal control with state constraints. SIAM Journal on Control and Optimization 35, 930–952 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Attia, S.A., Alamir, M., Canudas de Wit, C.: Suboptimal control of switched nonlinear systems unde location and switching constraints. In: Proceedings of the 16th IFAC World Congress, Prague (2005)

    Google Scholar 

  3. Attia, S.A., Azhmyakov, V., Raisch, J.: State jump optimization for a class of hybrid autonomous systems. In: Proceedings of the 2007 IEEE Multi-conference on Systems and Control, Singapore, pp. 1408–1413 (2007)

    Google Scholar 

  4. Attia, S.A., Azhmyakov, V., Raisch, J.: On gradient methods for hybrid systems optimization (submitted, 2007)

    Google Scholar 

  5. Azhmyakov, V., Raisch, J.: A gradient-based approach to a class of hybrid optimal control problems. In: Proceedings of the 2nd IFAC Conference on Analysis and Design of Hybrid Systems, Alghero, pp. 89–94 (2006)

    Google Scholar 

  6. Azhmyakov, V., Attia, S.A., Gromov, D., Raisch, J.: Necessary optimality conditions for a class of hybrid optimal control problems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 637–640. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  7. Azhmyakov, V.: Optimal control of hybrid and switched systems. In: Proceedings of the IX Chetaev Conference ”Analytical Mechanics, Stability and Control of Motion”, Irkutsk, pp. 308–317 (2007)

    Google Scholar 

  8. Baiocchi, C., Capello, A.: Variational and Quasivariational Inequalities: Application to Free Boundary Problems. Wiley, New York (1984)

    Google Scholar 

  9. Berkovitz, L.D.: Optimal Control Theory. Springer, New York (1974)

    MATH  Google Scholar 

  10. Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unifed framework for hybrid control: model and optimal control theory. IEEE Transactions on Automatic Control 43, 31–45 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bryson, A.E., Ho, Y.C.: Applied Optimal Control. Blaisdell Publishing Company, Waltham (1969)

    Google Scholar 

  12. Caines, P., Shaikh, M.S.: Optimality zone algorithms for hybrid systems computation and control: From exponential to linear complexity. In: Proceedings of the 13th Mediterranean Conference on Control and Automation, Limassol, pp. 1292–1297 (2005)

    Google Scholar 

  13. Caines, P., Shaikh, M.S.: Convergence analysis of Hybrid Maximum Principle (HMP) optimal control algorithms. In: Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, pp. 2083–2088 (2006)

    Google Scholar 

  14. Cassandras, C., Pepyne, D.L., Wardi, Y.: Optimal control of a class of hybrid systems. IEEE Transactions on Automatic Control 46, 398–415 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Clarke, F., Vinter, R.: Optimal multiprocesses. SIAM Journal on Control and Optimization 27, 1072–1090 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  16. Clarke, F.H., Vinter, R.B.: Applications of optimal multiprocesses. SIAM Journal on Control and Optimization 27, 1048–1071 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dmitruk, A.V., Kaganovivh, A.M.: The Hybrid Maximum Principle is a Consequence of Pontryagin Maximum Principle, http://www.optimization-online.org/

  18. Egerstedt, M., Wardi, Y., Axelsson, H.: Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control 51, 110–115 (2006)

    Article  MathSciNet  Google Scholar 

  19. Ferreira, M.M.A., Fontes, F.A.C.C., Vinter, R.B.: Nondegenerate necessary conditions for nonconvex optimal control problems with state constraints. Journal of Mathematical Analysis and Applications 233, 116–129 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers, Dordrecht (1988)

    MATH  Google Scholar 

  21. Garavello, M., Piccoli, B.: Hybrid necessary priniple. SIAM Journal on Control and Optimization 43, 1867–1887 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ioffe, A.D., Tichomirov, V.M.: Theory of Extremal Problems. North Holland, Amsterdam (1979)

    MATH  Google Scholar 

  23. Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (2007)

    MATH  Google Scholar 

  24. Kaplan, A., Tichatschke, R.: Stable Methods for Ill-Posed Variational Problems. Akademie Verlag, Berlin (1994)

    MATH  Google Scholar 

  25. Lygeros, J.: Lecture Notes on Hyrid Systems. University of Cambridge, Cambridge (2003)

    Google Scholar 

  26. Lygeros, J., Quincampoix, M., Rzezuchowski, T.: Impulse differential inclusions driven by discrete measures. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 385–398. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  27. Piccoli, B.: Hybrid systems and optimal control. In: Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, pp. 13–18 (1998)

    Google Scholar 

  28. Piccoli, B.: Necessary conditions for hybrid optimization. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, pp. 410–415 (1999)

    Google Scholar 

  29. Polak, E.: Optimization. Springer, New York (1997)

    MATH  Google Scholar 

  30. Rantzer, A.: On relaxed dynamic programming in switching systems. In: IEE Proceedings on Control Theory and Applications, vol. 153, pp. 567–574 (2006)

    Google Scholar 

  31. Shaikh, M.S., Caines, P.E.: On the Hybrid Optimal Control Problem: Theory and Algorithms. IEEE Trans. On Aut. Control 52, 1587–1603 (2007)

    Article  MathSciNet  Google Scholar 

  32. Sussmann, H.J.: A maximum principle for hybrid optimization. In: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, pp. 425–430 (1999)

    Google Scholar 

  33. Verriest, E., Delmotte, F., Egerstedt, M.: Optimal impulsive control of point delay systems with refractory period. In: Proceedings of the 5th IFAC workshop on Time Delay Systems, Leuven, Belgium (2004)

    Google Scholar 

  34. Verriest, E., Delmotte, F., Egerstedt, M.: Control of epidemics by vaccination. In: Proceedings of the American Control Conference, Portland, pp. 985–990 (2005)

    Google Scholar 

  35. Xu, X., Antsaklis, P.J.: Optimal Control of Hybrid autonomous Systems with State Jumps. In: Proceedings of the American Control Conference, Denver, pp. 5191–5196 (2003)

    Google Scholar 

  36. Xu, X., Antsaklis, P.J.: Results and Perspectives on Computational Methods for Optimal Control of Switched Systems. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 540–555. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Control Automatico, CINVESTAV, A.P. 14-740, Av. Instituto Politecnico Nacional No. 2508, C.P. 07360, Mexico D.F., Mexico

    Vadim Azhmyakov

  2. Fachgebiet Regelungssysteme, Technische Universität Berlin, Einsteinufer 17, D-10587, Berlin, Germany

    Sid Ahmed Attia & Jörg Raisch

  3. Systems and Control Theory Group, MPI for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106, Magdeburg, Germany

    Jörg Raisch

Authors
  1. Vadim Azhmyakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sid Ahmed Attia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jörg Raisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Magnus Egerstedt Bud Mishra

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Azhmyakov, V., Attia, S.A., Raisch, J. (2008). On the Maximum Principle for Impulsive Hybrid Systems. In: Egerstedt, M., Mishra, B. (eds) Hybrid Systems: Computation and Control. HSCC 2008. Lecture Notes in Computer Science, vol 4981. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78929-1_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-78929-1_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-78928-4

  • Online ISBN: 978-3-540-78929-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation