Analysis of hybrid systems’ dynamics using the common Lyapunov functions and multiple homomorphisms (original) (raw)

References

  1. DeCarlo, R., Branicky, M., Pettersson, S., and Lennartson, B., Perspectives and Results on the Stability and Stabilizability of Hybrid Systems, Proc. IEEE., 2000, vol. 88(7), pp. 1069–1082.
    Article Google Scholar
  2. Haddad, W.M., Chellaboina, V., and Nersesov, S.G., Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton: Princeton Univ. Press, 2006.
    MATH Google Scholar
  3. Emel’yanov, S.V., A Method of Getting Complex Control Laws Using Only the Signal of Error and Its First Derivative, Avtom. Telemekh., 1957, vol. 18, no. 10, pp. 873–885.
    MathSciNet Google Scholar
  4. Liberzon, D. and Morse, A.S., Basic Problems in Stability and Design of Switched Systems, IEEE Control Syst. Mag., 1999, vol. 19(5), pp. 59–70.
    Article Google Scholar
  5. Shorten, R., Wirth, F., Mason, O., Wulf, K., et al., Stability Criteria for Switched and Hybrid Systems, SIAM Rev., 2007, vol. 49, no. 4, pp. 545–592.
    Article MathSciNet MATH Google Scholar
  6. Lin, H. and Antsaklis, P.J., Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results, IEEE Trans. Automat. Control, 2009, vol. 54, no. 2, pp. 308–322.
    Article MathSciNet Google Scholar
  7. Pakshin, P.V. and Pozdyaev, V.V., Existence Criterion for the Common Quadratic Lyapunov Function of the Set of Linear Second-order Systems, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2005, no. 4, pp. 22–27.
  8. Unsolved Problems in Mathematical Systems and Control Theory, Blondel, V.D. and Megretski, A., Eds., Princeton: Princeton Univ. Press, 2004.
    MATH Google Scholar
  9. Hankey, A. and Stanley, H.E., Systematic Application of Generalized Homogeneous Functions to Static Scaling, Dynamic Scaling, and Universality, Physical Rev. B, 1972, vol. 6, no. 9, pp. 3515–3542.
    Article Google Scholar
  10. Zubov, V.I., Matematicheskie metody issledovaniya sistem avtomaticheskogo regulirovaniya (Mathematical Methods for Studying Automatic Control Systems), Leningrad: Mashinostroenie, 1974.
    Google Scholar
  11. Kozlov, V.V. and Furta, S.D., Asimptotiki reshenii sil’no nelineinykh sistem differentsial’nykh uravnenii (Asymptotics of Solution of Strongly Nonlinear System of Differential Equations), Izhevsk: Regulyarnaya i Khaoticheskaya Dinamika, 2009.
    Google Scholar
  12. Matrosov, V.M., Vassilyev, S.N., Kozlov, R.I., et al., Algoritmy vyvoda teorem metoda vektornykh funktsii Lyapunova (Algorithms to Derive Theorems of the Method of Vector Lyapunov Functions), Novosibirsk: Nauka, 1981.
    Google Scholar
  13. Vassilyev, S.N., Method of Reduction and Qualitative System Analysis, Izv. Ross. Akad. Nauk, Teor. Sist. Upravlen., 2006, no. 1, pp. 21–29; no. 2. pp. 5–17.
  14. Matrosov, V.M., Comparison Method in System Dynamics, in Equat. Different. Fonction. Lineair., Paris: Herman, 1973, pp. 407–445.
    Google Scholar
  15. Matrosov, V.M., Anapol’skii, L.Yu., and Vassilyev, S.N., Metod sravneniya v matematicheskoi teorii sistem (Method of Comparison in the Mathematical System Theory), Novosibirsk: Nauka, 1980.
    Google Scholar
  16. Šhiljak, D., Decentralized Control for Complex Systems, Cambridge: Academic, 1991. Translated under the title Detsentralizovannoe upravlenie slozhnymi sistemami, Matrosov, V.M. and Savastyuk, S.V., Eds., Moscow: Mir, 1994.
    Google Scholar
  17. Mil’man, V.D. and Myshkis, A.D., On Motion Stability in the Presence of Impacts, Sib. Mat. Zh., 1960, vol. 1, no. 2, pp. 233–237.
    Google Scholar
  18. Myshkis, A.D. and Samoilenko, A.M., Systems with Impacts at Given Time Instants, Mat. Sb., 1967, vol. 74(116), no. 2, pp. 202–208.
    Google Scholar
  19. Hermes, H., Discontinuous Vector Fields and Feedback Control, in Differential Equations and Dynamic Systems, Hale, J.K. and LaSalle, J.P., Eds., New York: Academic, 1967.
    Google Scholar
  20. Lakshmikantham, V., Bainov, D.D., and Simeonov, P.S., Theory of Impulsive Differential Equations, Singapore: World Scientific, 1989.
    MATH Google Scholar
  21. Sivasundaram, S. and Vassilyev, S.N., Stability and Attractivity of Differential Equations with Impulses at Fixed Times, J. Appl. Math. Stochast. Anal., 2000, vol. 13, no. 1, pp. 77–84.
    Article MathSciNet MATH Google Scholar
  22. Vassilyev, S.N., Homomorphisms of Impulsive Differential Equations with Impulses at Unfixed Times and Comparison Method, Int. J. Hybrid Syst., 2002, vol. 2, no. 3, pp. 289–296.
    Google Scholar
  23. Bobylev, N.A., Il’in, A.V., Korovin, S.K., and Fomichev, V.V., On Stability of Families of Dynamic Systems, Differ. Uravn., 2002, no. 4, pp. 447–452.
  24. Matrosov, V.M., Metod vektornykh funktsii Lyapunova: analiz dinamicheskikh svoistv nelineinykh sistem (Method of Lyapunov Vector Functions: Analysis of Dynamic Characteristics of Nonlinear Systems), Moscow: Fizmatlit, 2001.
    Google Scholar
  25. Chernikov, S.N., Lineinye neravenstva (Linear Inequaltities), Moscow: Nauka, 1968.
    Google Scholar
  26. Michel, A., Wang, K., and Hu, B., Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings, New York: Marcel Dekker, 2001.
    MATH Google Scholar
  27. Sirazetdinov, T.K. and Aminov, A.B., On the Problem of Constructing Lyapunov Functions in the Studies of Stability on the Whole of the Solution of Systems with Polynomial Right Side, in Metod funktsii Lyapunova i ego prilozheniya (Method of Lyapunov Functions and Its Applications), Matrosov, V.M. and Vassilyev, S.N., Eds., Novosibirsk: Nauka, 1981, pp. 72–87.
    Google Scholar

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