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Lie Brackets and Stabilizing Feedback Controls

  • Mikhail I. Krastanov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4310)

Abstract

The relation between a class of high-order control variations and the asymptotic stabilizability of a smooth control system is briefly discussed. Assuming that there exist high-order control variations ”pointing” to a closed set at every point of some its neighborhood, an approach for constructing stabilizing feedback controls is proposed. Two illustrative examples are also presented.

Keywords

Smooth Vector Switching Linear System Admissible Trajectory Piecewise Linear System Geometric Control Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agrachev, A., Gamkrelidze, R.: Local controllability and semigroups of diffeomorphisms. Acta Applicandae Mathematicae 32, 1–57 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brockett, R.: Asymptotic stability and feedback stabilization. In: Brockett, R., Millmann, R., Sussmann, H. (eds.) Differential Geometric Control Theory. Progr. Math., vol. 27, pp. 181–191. Birkhäuser, Basel (1983)Google Scholar
  3. 3.
    Clarke, F.H., et al.: Nonsmooth analysis and control theory. Graduate Text in Mathematics, vol. 178. Springer, New York (1998)zbMATHGoogle Scholar
  4. 4.
    Frankowska, H.: Local controllability of control systems with feedback. J. Optimiz. Theory Appl. 60, 277–296 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hermes, H.: On the synthesis of a stabilizing feedback control via Lie algebraic methods. SIAM J. Control Optimiz. 16, 715–727 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hermes, H.: Lie algebras of vector fields and local approximation of attainable sets. SIAM J. Control Optimiz. 18, 352–361 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Krastanov, M., Quincampoix, M.: Local small-time controllability and attainability of a set for nonlinear control systems. ESAIM: Control. Optim. Calc. Var. 6, 499–516 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Krastanov, M.I.: A sufficient condition for small-time local attainability of a set. Control and Cybernetics 31(3), 739–750 (2002)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Krastanov, M.I., Veliov, V.M.: On the controllability of switching linear systems. Automatica 41, 663–668 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Sussmann, H.J.: A general theorem on local controllability. SIAM J. Control Optimiz. 25, 158–194 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Veliov, V.M., Krastanov, M.I.: Controllability of piecewise linear systems. Systems & Control Letters 7, 335–341 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Veliov, V.: On the controllability of control constrained linear systems. Math. Balk., New Ser. 2, 147–155 (1988)MathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Mikhail I. Krastanov
    • 1
  1. 1.Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 SofiaBulgaria

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