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Necessary optimality conditions for switched systems

  • Computer Methods
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Journal of Computer and Systems Sciences International Aims and scope

Abstract

A dynamic system that models the operation of a switching device (switch) is considered. During the operation, the system changes its state a finite number of times. The change of state (switching) is described by a recurrent inclusion, which corresponds to the representation of the switch by a dynamic finite state machine with memory; instantaneous multiple switchings are admitted. The instants of time at which switchings are made and the number of switchings are not given in advance. They are found by optimizing a functional in which the number of switchings and the cost of each of them are taken into account. Necessary optimality conditions for such systems are proved. Different versions of the optimality conditions for different types of constraints are given. In particular, under additional convexity conditions, conditions that are similar to the maximum principle for discrete systems are obtained. The application of the optimality conditions is illustrated by examples.

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References

  1. R. W. Brockett, “Hybrid models for motion control systems,” in Perspectives in the Theory and Its Applications (Birkhauser, Boston, 1993), pp. 29–53.

    Google Scholar 

  2. M. S. Branicky, V. S. Borkar, and S. K. Mitter, “A unified framework for hybrid control: model and optimal control theory,” IEEE Trans. Aut. Control 43, 31–45 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Hedlund and A. Rantzer, “Optimal control of hybrid systems,” in Proceedings of the 38th IEEE Conference on Decision and Controk, Phoenix, AZ, 1999, p. 3972–3977.

    Google Scholar 

  4. C. G. Cassandras, D. L. Pepyne, and Y. Wardi, “Optimal control of a class of hybrid systems,” IEEE Trans. Autom. Control 46 (3), 398–415 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  5. V. I. Gurman, “Models and optimality conditions for hybrid controlled systems,” J. Comput. Syst. Sci. Int. 43, 560 (2004).

    MathSciNet  Google Scholar 

  6. D. Liberzzon, Switching in Systems and Control (Springer, Berlin, 2003).

    Book  Google Scholar 

  7. Z. Li, Y. Soh, and C. Wen, Switched and Impulsive Systems: Analysis, Design and Applications (Springer, Berlin, 2005).

    Book  MATH  Google Scholar 

  8. X. Xu and P. J. Antsaklis, “On time optimal control of integrator switched systems with state constrains,” J. Nonlin. Anal., Spec. Iss. Hybrid Syst. 62, 1453–1465 (2005).

    MathSciNet  Google Scholar 

  9. H. Axelsson, M. Boccadoro, M. Egerstedt, et al., “Optimal mode-switching for hybrid systems with varying initial states,” J. Nonlin. Anal.: Hybrid Syst. Appl. 2, 765–772 (2008).

    MathSciNet  MATH  Google Scholar 

  10. S. N. Vasil’ev and A. I. Malikov, “On some results on stability of switched and hybride systems,” in Actual Problems of Continuous Media Mechanics (Foliant, Kazan, 2001), Vol. 1 [in Russian].

    Google Scholar 

  11. A. S. Bortnikovskii and A. V. Panteleev, “Sufficient conditions for optimal control of continuous-discrete systems,” Avtom. Telemekh., No. 7, 57–66 (1987).

    Google Scholar 

  12. S. N. Vasil’ev, A. K. Zherlov, E. A. Fedosov, et al., Intelligent Control of Dynamic Systems (Fizmatlit, Moscow, 2000) [in Russian].

    Google Scholar 

  13. K. D. Zhuk and A. A. Timchenko, Automated Design of Logical-Dynamical Systems (Nauk. Dumka, Kiev, 1981) [in Russian].

    MATH  Google Scholar 

  14. K. D. Zhuk, A. A. Timchenko, and T. I. Dalenko, Investigation of Structures and Modeling of Logical-Dynamical Systems (Nauk. dumka, Kiev, 1975) [in Russian].

    Google Scholar 

  15. V. V. Semenov, “Dinamical programming in the synthesis of logical-dynamical systems,” Priborostroenie, No. 9, 71–77 (1984).

    Google Scholar 

  16. A. S. Bortakovskii, “Sufficient conditions of optimal control of determinated logical-dynamical systems,” Informat., Ser. A: Avtomatiz. Proektir., Nos. 2–3, 72–79 (1992).

    Google Scholar 

  17. Proceedings of the IFAC Workshop on Modelling and Analysis of Logic Controlled Dynamic Systems (Inst. Dinamiki Sustem Teor. Upravl., Sib. Otdel. RAN, Irkutsk, 2003).

  18. V. A. Baturin, E. V. Goncharova, and N. S. Maltugueva, “Iterative methods for solution of problems of optimal control of logic–dynamic systems,” J. Comput. Syst. Sci. Int. 49, 731 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. V. I. Gurman, Expansion Principle in Control Problems (Nauka, Moscow, 1985) [in Russian].

    MATH  Google Scholar 

  20. I. V. Rasina, “Discrete-continuous models and optimization of controlled processes,” Program. Sist.: Teor. Prilozh., No. 5 (9), 49–72 (2012).

    Google Scholar 

  21. V. I. Gurman, “Models and optimality conditions for hybrid controlled systems,” J. Comput. Syst. Sci. Int. 43, 560 (2004).

    MathSciNet  Google Scholar 

  22. B. M. Miller and E. Ya. Rubinovich, Optimization of Dynamic Systems with Impulsive Controls (Nauka, Moscow, 2004) [in Russian].

    Google Scholar 

  23. A. I. Propoi, Theory Elements of Optimal Discrete Processes (Nauka, Moscow, 1973).

    MATH  Google Scholar 

  24. V. G. Boltyanskii, Optimal Control of Discrete Systems (Nauka, Moscow, 1973) [in Russian].

    MATH  Google Scholar 

  25. A. S. Bortakovskii, “Synthesis of optimal switched systems,” J. Comput. Syst. Sci. Int. 54, 715 (2015).

    Article  MathSciNet  Google Scholar 

  26. A. S. Bortakovskii, “Analytical design of optimal controllers in the class of logic-dynamic (hybrid) systems,” Autom. Remote Control 72, 2425 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  27. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, et al., Mathematical Theory of Optimal Processes (Fizmatgiz, Moscow, 1961) [in Russian].

    Google Scholar 

  28. S. T. Zavalishchin and A. N. Sesekin, Impulse Processes: Models and Applications (Nauka, Moscow, 1991) [in Russian].

    MATH  Google Scholar 

  29. V. A. Dykhta and O. N. Samsonyuk, Optimal Impulse Control with Applications (Fizmatlit, Moscow, 2000) [in Russian].

    MATH  Google Scholar 

  30. A. B. Kurzhanskii and P. A. Tochilin, “Impulse controls in models of hybrid systems,” Differ. Equations 45, 731 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  31. B. T. Polyak, Introduction to Optimization, Translations Series in Mathematics and Engineering (Nauka, Moscow, 1983; Optimization Software Inc., New York, 1987).

    Google Scholar 

  32. V. V. Vasil’ev, Methods of Extreme Problems Solution (Nauka, Moscow, 1981) [in Russian].

    Google Scholar 

  33. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974; North-Holland, Amsterdam, 1979).

    Google Scholar 

  34. I. P. Natanson, Theory of Functions of a Real Variable (Gostekhteorizdat, Moscow, 1957; Literary Licensing, 2013).

    Google Scholar 

  35. Yu. G. Borisovich, B. D. Gel’man, A. D. Myshkis, and V. V. Obukhovskii, Introduction to the Theory of Multivalued Maps and Differential Inclusions (Voronezh. Gos. Univ., Voronezh, 1986) [in Russian].

    MATH  Google Scholar 

  36. S. F. Krotov and V. I. Gurman, Methods and Problems of Optimal Control (Nauka, Moscow, 1973) [in Russian].

    Google Scholar 

  37. A. T. Fuller, “Optimization of relay regulation systems by different quality criteria,” in Proceedings of the 1st International Congress of International Federation of Automatic Control IFAC (Akad. Nauk SSSR, Moscow, 1961), pp. 584–605.

    Google Scholar 

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Authors and Affiliations

  1. Moscow Institute of Aviation (National Research University), Moscow, 125993, Russia

    A. S. Bortakovskii

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  1. A. S. Bortakovskii
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Correspondence to A. S. Bortakovskii.

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Original Russian Text © A.S. Bortakovskii, 2016, published in Izvestiya Akademii Nauk, Teoriya i Sistemy Upravleniya, 2016, No. 5, pp. 34–46.

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Bortakovskii, A.S. Necessary optimality conditions for switched systems. J. Comput. Syst. Sci. Int. 55, 712–724 (2016). https://doi.org/10.1134/S1064230716050051

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  • DOI: https://doi.org/10.1134/S1064230716050051

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