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Optimality Zone Algorithms for Hybrid Systems: Efficient Algorithms for Optimal Location and Control Computation

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Hybrid Systems: Computation and Control (HSCC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3927))

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Abstract

A general Hybrid Minimum Principle (HMP) for hybrid optimal control problems (HOCPs) is presented in [1, 2, 3, 4] and in [4, 5], a class of efficient, provably convergent Hybrid Minimum Principle (HMP) algorithms were obtained based upon the HMP. The notion of optimality zones (OZs) ([3, 4]) provides a theoretical framework for the computation of optimal location (i.e. discrete state) schedules for HOCPs (i.e. discrete state sequences with the associated switching times and states). This paper presents the algorithm HMPOZ which fully integrates the prior computation of the OZs into the HMP algorithms class. Summing (a) the computational investment in the construction of the OZs for a given HOCP, and (b) the complexity of (i) the computation of the optimal schedule, (ii) the optimal switching time and optimal switching state sequence, and (iii) the optimal continuous control input, yields a complexity estimate for the algorithm HMPOZ which is linear (i.e. O(L)) in the number of switching times L.

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References

  1. Shaikh, M.S., Caines, P.E.: On trajectory optimization for hybrid systems: Theory and algorithms for fixed schedules. In: Proc. 41st IEEE Int. Conf. Decision and Control, Las Vegas, pp. 1997–1998 (2002)

    Google Scholar 

  2. Shaikh, M.S., Caines, P.E.: On the optimal control of hybrid systems: Optimization of trajectories, switching times, and location schedules. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 466–481. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  3. Shaikh, M.S., Caines, P.E.: On the optimal control of hybrid systems: Analysis and algorithms for trajectory and schedule optimization. In: Proc. 42nd IEEE Int. Conf. Decision and Control, Maui, Hawaii, pp. 2144–2149 (2003)

    Google Scholar 

  4. Shaikh, M.S.: Optimal Control of Hybrid Systems: Theory and Algorithms. PhD thesis, Department of Electrical and Computer Engineering, McGill University, Montréal, Canada (2004), Available: http://www.cim.mcgill.ca/~msshaikh/publications/thesis.pdf

  5. Shaikh, M.S., Caines, P.E.: On the hybrid optimal control problem: Theory and algorithms. revised for IEEE Trans. Automat. Contr (2005)

    Google Scholar 

  6. Shaikh, M.S., Caines, P.E.: On the optimal control of hybrid systems: Optimization of switching times and combinatoric location schedules. In: Proc. American Control Conference, Denver, pp. 2773–2778 (2003)

    Google Scholar 

  7. Xu, X., Antsaklis, P.J.: Optimal control of switched systems based on parameterization of the switching instants. IEEE Trans. Automat. Contr. 49(1), 2–16 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Branicky, M.S., Borkar, V.S., Mitter, S.K.: A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Automat. Contr. 43(1), 31–45 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Branicky, M.S., Mitter, S.K.: Algorithms for optimal hybrid control. In: Proc. 34th IEEE Int. Conf. Decision and Control, New Orleans, pp. 2661–2666 (1995)

    Google Scholar 

  10. Sussmann, H.: A maximum principle for hybrid optimal control problems. In: Proc. 38th IEEE Int. Conf. Decision and Control, Phoenix, pp. 425–430 (1999)

    Google Scholar 

  11. Riedinger, P., Kratz, F., Iung, C., Zanne, C.: Linear quadratic optimization of hybrid systems. In: Proc. 38th IEEE Int. Conf. Decision and Control, Phoenix, pp. 3059–3064 (1999)

    Google Scholar 

  12. Tomiyama, K.: Two-stage optimal control problems and optimality conditions. J. Economic Dynamics and Control 9(3), 317–337 (1985)

    Article  MathSciNet  Google Scholar 

  13. Clarke, F.H., Vinter, R.B.: Optimal multiprocesses. SIAM J. Control and Optimization 27(5), 1072–1079 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Clarke, F.H., Vinter, R.B.: Applications of optimal multiprocesses. SIAM J. Control and Optimization 27(5), 1048–1071 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Egerstedt, M., Wardi, Y., Delmotte, F.: Optimal control of switching times in switched dynamical systems. In: Proc. 42nd IEEE Int. Conf. Decision and Control, Maui, HI, pp. 2138–2143 (2003)

    Google Scholar 

  16. Wardi, Y., Egerstedt, M., Boccadoro, M., Verriest, E.: Optimal control of switching surfaces. In: Proc. 43rd IEEE Int. Conf. Decision and Control, Atlantis, Paradise Island, Bahamas, pp. 1854–1859 (2004)

    Google Scholar 

  17. Axelsson, H., Egerstedt, M., Wardi, Y., Vachtsevanos, G.: Algorithm for switching-time optimization in hybrid dynamical systems. In: Proc. 2005 International Symposium on Intelligent Control/13th Mediterranean Conference on Control and Automation, Cypres, pp. 256–261 (2005)

    Google Scholar 

  18. Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Trans. Automat. Contr. 40(9), 1528–1538 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sethian, J.A., Vladimirsky, A.: Ordered upwind methods for hybrid control. In: Tomlin, C.J., Greenstreet, M.R. (eds.) HSCC 2002. LNCS, vol. 2289, pp. 393–406. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  20. Giua, A., Seatzu, C., Mee, C.V.D.: Optimal control of autonomous linear systems switched with a pre-assigned finite sequence. In: Proc. 2001 IEEE International Symposium on Intelligent Control, pp. 144–149 (2001)

    Google Scholar 

  21. Giua, A., Seatzu, C., Mee, C.V.D.: Optimal control of switched autonomous linear systems. In: Proc. 40th IEEE Int. Conf. Decision and Control, Orlando, pp. 2472–2477 (2001)

    Google Scholar 

  22. Bemporad, A., Giua, A., Seatzu, C.: A master-slave algorithm for the optimal control of continuous-time switched affine systems. In: Proc. 41st IEEE Int. Conf. Decision and Control, Las Vegas, pp. 1976–1981 (2002)

    Google Scholar 

  23. Shaikh, M.S., Caines, P.E.: Optimality zone algorithms for hybrid systems computation and control. Technical report, ECE Department, McGill University (2005)

    Google Scholar 

  24. Caines, P.E., Shaikh, M.S.: Optimality zone algorithms for hybrid control systems. IEEE Trans. Automat. Contr (submitted to, 2005)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical & Computer Engineering, Centre for Intelligent Machines, McGill University, Montréal, Québec, H3A 2A7, Canada

    Peter E. Caines & M. Shahid Shaikh

Authors
  1. Peter E. Caines
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  2. M. Shahid Shaikh
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Editor information

Editors and Affiliations

  1. Center for Control, Dynamical Systems and Computation, University of California, 93106, Santa Barbara, CA, USA

    João P. Hespanha

  2. SRI International, 333 Ravenswood Ave, Menlo Park, CA, USA

    Ashish Tiwari

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© 2006 Springer-Verlag Berlin Heidelberg

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Caines, P.E., Shaikh, M.S. (2006). Optimality Zone Algorithms for Hybrid Systems: Efficient Algorithms for Optimal Location and Control Computation. In: Hespanha, J.P., Tiwari, A. (eds) Hybrid Systems: Computation and Control. HSCC 2006. Lecture Notes in Computer Science, vol 3927. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11730637_12

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  • DOI: https://doi.org/10.1007/11730637_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-33170-4

  • Online ISBN: 978-3-540-33171-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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