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Article

  • Title

    Speeded search of the periodic processes of dynamical systems

  • Authors

    Shpak Olexandr L.

  • Subject

    ENERGETICS. HEAT ENGINEERING. ELECTRICAL ENGINEERING

  • Year 2016
    Issue 1(48)
    UDC 621.31:517.938
    DOI 10.15276/opu.1.48.2016.10
    Pages 54-57
  • Abstract

    Modelling of periodic processes of electromagnetic devices is a complex problem. The task of finding of periodic solutions of nonlinear differential equations is more complex than the Cauchy task of integrating of these equations because it imposes another condition on the solution – the condition of periodicity, that is it becomes a point-to-point T-periodic boundary value problem of nonlinear differential equations. Aim: The aim of the work is to simplify the computing method of the sensitivity matrix elements of the object parameters to their initial conditions by analytical solution of the equations of first variation. Materials and Methods: The author proposed to calculate the matrix elements of the sensitivity model to the initial conditions of the variational equations not through their joint numerical integration with the original equations in the period but to search for the common solution of these equations by means of using a transition state matrix. In this case the order of the equations of the object remains the same, restrictions on the method use for analysis of complex dynamical systems are removed and its efficiency improves significantly. Results: Due to the independence of differential equations in variations from external influence, they become a homogeneous system of linear differential equations. This approach greatly simplifies the analysis and removes the restrictions on the application of the method of sensitivity to initial conditions during the study of dynamical systems of any complexity.

  • Keywords

    sensitivity model to initial conditions, variational equations, periodic process

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  • References

    Література
    1.    Skelboe, S. Computation of the periodic steady-state response of nonlinear networks by extrapolation methods / S. Skelboe // IEEE Transactions on Circuits and Systems. — 1980. — Vol. 27, Issue 3. — PP. 161—175.
    2.    Nakhla, M.S. Determining the periodic response of nonlinear systems by a gradient method / M.S. Nakhla, F.H. Branin Jr. // International Journal of Circuit Theory and Applications. — 1977. — Vol. 5, Issue 3. — PP. 255—273.
    3.    Aprille, T.J. Steady-state analysis of nonlinear circuits with periodic inputs / T.J. Aprille, T.N. Trick // Proceedings of the IEEE. — 1972. — Vol. 60, Issue 1. — PP. 108—114.
    4.    Kalman, R.E. Topics in mathematical system theory / R.E. Kalman, P.L. Falb, M.A. Arbib. — New Delhi: McGraw-Hill, 1974. — 358 p.

    References
    1.    Skelboe, S. (1980). Computation of the periodic steady-state response of nonlinear networks by extrapolation methods. IEEE Transactions on Circuits and Systems, 27(3), 161—175. DOI:10.1109/TCS.1980.1084794
    2.    Nakhla, M.S., & Branin Jr., F.H. (1977). Determining the periodic response of nonlinear systems by a gradient method. International Journal of Circuit Theory and Applications, 5(3), 255—273. DOI:10.1002/cta.4490050307
    3.    Aprille, T.J., & Trick, T.N. (1972). Steady-state analysis of nonlinear circuits with periodic inputs. Proceedings of the IEEE, 60(1), 108—114. DOI:10.1109/PROC.1972.8563
    4.    Kalman, R.E., Falb, P.L., & Arbib, M.A. (1974). Topics in Mathematical System Theory. New Delhi: McGraw-Hill.

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