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Dynamics and Control of Trajectory Tubes pp 395–429Cite as

Birkhäuser

Verification: Hybrid Systems

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Part of the book series: Systems & Control: Foundations & Applications ((SCFA,volume 85))

Abstract

This chapter deals with a specific class of hybrid systems which combine controlled continuous dynamics through switching from one available motion to another due to discrete-time logical commands. Solutions to the reachability problem and their verification are indicated, followed by computational schemes, The application of impulse controls to the switching process is described. Examples of various difficulty are worked out. The chapter is to demonstrate applicability of methods of this book to hybrid systems.

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Notes

  1. 1.

    The part with memory may be important for making the decision—“to switch” or “not to switch,” for example, if the number of switchings is restricted.

  2. 2.

    From here on it is important to emphasize the dependence of ellipsoids, reach sets, and value functions on l. Therefore we further include l in the arguments of respective items.

  3. 3.

    In more complicated problems, for example, with complex constraints on the number of switchings or other outputs of resets, this memorized part may be logically controlled and its knowledge may be important. Such components are also important in the design of feedback controls and computation of backward reach sets for hybrid systems. Situations mentioned in this footnote mostly lie beyond the scope of this book.

  4. 4.

    This example was worked out by P.A. Tochilin.

  5. 5.

    As before \(\mathcal{I}(x\;\vert \;D) = 0\), if x ∈ D and + ∞ otherwise.

References

  1. Bensoussan, A., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi-variationnelles. Dunod, Paris (1982)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Brockett, R.W.: Hybrid models for motion control systems. In: Trentelman, H., Willems, J. (eds.) Essays in Control: Perspectives in the Theory and Its Applications, pp. 29–53. Birkhäuser, Boston (1993)

    Chapter  Google Scholar 

  4. Goebel, R., Sanfelice, R.G., Teel, A.R.: Hybrid Dynamical Systems. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  5. Johansson, M.: Piecewise Linear Control Systems. Lecture Notes in Control and Information Sciences, vol. 284. Springer, Berlin (2003)

    Google Scholar 

  6. Kurzhanski, A.B., Tochilin, P.A.: Weakly invariant sets of hybrid systems. Differ. Equ. 44(11), 1585–1594 (2008)

    Article  MathSciNet  Google Scholar 

  7. Kurzhanski, A.B., Tochilin, P.A.: Impulse controls in models of hybrid systems. Differ. Equ. 45(5), 731–742 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. In: Proceedings of the Pittsburg Conference “Hybrid Systems-2000”. Lecture Notes in Control Science, vol. 1790, pp. 202–214. Springer, Berlin (2000)

    Google Scholar 

  9. Lygeros, J., Tomlin, C., Sastry, S.: Controllers for reachability specifications for hybrid systems. Automatica 35(3), 349–370 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Puri, A., Varaiya, P.: Decidability of hybrid systems with rectangular differential inclusions. In: Proceedings of the 6th Workshop on Computer-Aided Verification. Lecture Notes in Control Science, vol. 818, pp. 95–104. Springer, New York (1994)

    Google Scholar 

  11. Puri, A., Borkar, V.S., Varaiya, P.: \(\upvarepsilon\)-Approximation of differential inclusions. In: Hybrid Systems. Lecture Notes in Control Science, vol. 1066, pp. 362–376. Springer, New York (1996)

    Google Scholar 

  12. Riesz, F., Sz-.Nagy, B.: Leçons d’analyse fonctionnelle. Akadémiai Kiadó, Budapest (1972)

    Google Scholar 

  13. Van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Lecture Notes in Control and Information Sciences, vol. 25. Springer, Berlin (2000)

    Google Scholar 

  14. Varaiya, P.: Reach set computation using optimal control. In: Proceedings of the KIT Workshop on Verification of Hybrid Systems, Grenoble, October 1998

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State (Lomonosov) University, Moscow, Russia

    Alexander B. Kurzhanski

  2. Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA

    Alexander B. Kurzhanski & Pravin Varaiya

Authors
  1. Alexander B. Kurzhanski
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  2. Pravin Varaiya
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Kurzhanski, A.B., Varaiya, P. (2014). Verification: Hybrid Systems. In: Dynamics and Control of Trajectory Tubes. Systems & Control: Foundations & Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10277-1_11

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