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Brief Comments for Doubts in Filippov Method

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Abstract

Different methods of solution continuation for the system with discontinuous control and sliding modes are discussed in the paper. First, the systems with one discontinuity surface are discussed when the right-hand side of differential equation can take two values only for the selected state x and time t. According to Filippov method the right-hand side of sliding mode equation can be found in convex hull of these two values. Other five methods of solution continuation are developed under additional assumption about values of control on discontinuity surface. They led to sliding mode equations different from Filippov’s equations both for affine systems and systems nonlinear with respect to control. However, if we apply Filippov method based on the procedure of passing over to a convex hull considering additional assumptions of each method then the sliding mode equation is the same as for this method.

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References

  • Aizerman, M. A., & Pyatnitskii, Y. S. (1974). Foundations of the theory of discontinuous systems. II. Automation and Remote Control, 35(8), 39–61 (in Russian).

    MathSciNet  MATH  Google Scholar 

  • André, J., & Seibert, P. (1956). Über stückweise lineare differentialgleichungen, die bei regelungsproblemen auftreten I, II. Archiv der Mathematik. 7, 148–156, 157–164. https://doi.org/10.1007/BF01899832

    Article  MATH  Google Scholar 

  • Barbashin, Y. (1967). Introduction to stability theory. Nauka.

    Google Scholar 

  • Cortes, J. (2008). Discontinuous dynamical systems. IEEE Control Systems Magazine, 28(3), 36–73.

    Article  MathSciNet  MATH  Google Scholar 

  • Filippov, A. (1960). Differential Equations with discontinuous right-hand side, Set of math. papers, Publisher MGU, 51(93), #1, 99–128 (in Russian).

  • Filippov, A. (1961). Application of the theory of differential equations with discontinuous right-hand sides to non-linear problems in automatic control. In Proceedings of the First International Congress of the International Federation of Automatic Control, Moscow, 1960 (vol. II, pp. 923–925). Butterworths.

    Google Scholar 

  • Filippov, A. F. (1988). Differential Equations with Discontinuous Right-hand Sides. Kluwer Academic Publishers.

    Book  MATH  Google Scholar 

  • Gelig, A., Leonov, G., & Yakubovich, V. (1978). Stability of nonlinear systems with non-unique equilibrium point. Nauka. (in Russian).

    MATH  Google Scholar 

  • Jeffrey, M. R., Seidman, T. I., Teixeira, M. A., & Utkin, V. I. (2022). Into higher dimensions for nonsmooth dynamical systems. Phisica D: Nonlinear Phenomena, 434, 133222.

  • Leonov, G. A., Kuznetsov, N. V., Kiseleva, N. A., & Mokaev, R. N. (2017). Global problems for differential inclusions. Kalman and Vyshnegradskii problems and Chua circuits. Differential Equations, 53, 1671–1702.

    Article  MathSciNet  MATH  Google Scholar 

  • Neimark, Y. I. (1957). On sliding process in control relay systems. Automation and Remote Control, 18(1), 27–33 (in Russian).

  • Neimark, Y. (1961). Discussion of the paper by A. Filippov. In Proceedings of the First International Congress of the International Federation of Automatic Control, Moscow, 1960 (vol. II, pp. 926–927). Butterworths.

  • Polyakov, A., & Fridman, L. (2014). Stability notion and Lyapunov functions for sliding mode control systems. Journal of Franklin Institute, 351(4), 1831–1865.

    Article  MathSciNet  MATH  Google Scholar 

  • Rockafeller, R. T., & Wets, R. J. B. (1998). Variational analysis. Berlin Heidelberg: Springer-Verlag, p. 117.

  • Tsypkin, Y. (1956). Theory of relay automatic regulation. State Publisher of Engineering Literature (in Russian).

  • Utkin, V. I. (1971). Concerning equations of sliding motions in discontinuous systems I. Automation and Remote Control, 32(12), 1897–1907 (in Russian).

  • Utkin, V. I. (1972). On sliding mode equations in discontinuous systems II. Automation and Remote Control, 32(2), 211–219 (in Russian).

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  1. Ohio State University, Columbus, OH, USA

    Vadim Utkin

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  1. Vadim Utkin
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Utkin, V. Brief Comments for Doubts in Filippov Method. J Control Autom Electr Syst 33, 1628–1632 (2022). https://doi.org/10.1007/s40313-022-00952-9

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