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Journal of Nonlinear Science

, Volume 29, Issue 6, pp 2845–2875 | Cite as

Two-Parameter Boundary Equilibrium Bifurcations in 3D-Filippov Systems

  • Rony Cristiano
  • Daniel J. PaganoEmail author
Article
  • 110 Downloads

Abstract

This work concerns the analysis of boundary equilibrium bifurcations (BEBs) in 3D piecewise-smooth systems of Filippov type. In particular, we consider a family whose vector fields are linear on both sides of the switching boundary and which exhibit two parallel tangency lines each containing a cusp point. This configuration is observed in piecewise-linear control systems in which the control action is discontinuous such as the sliding mode control. We consider a general system of this class and then derive a canonical form to reduce the number of system parameters. The general objective in this work is, from the canonical form, to perform an analysis of the equilibria, stability, sliding dynamics and BEBs. The main result is the classification of the BEBs and its unfoldings in the sliding vector field. This and others results obtained on the existence and stability of equilibria are applied in a practical example involving the control of DC–DC buck power converter.

Keywords

Filippov systems Boundary equilibrium Pseudo-equilibrium Sliding vector field Bifurcations Buck converter Sliding mode control 

Mathematics Subject Classification

34A36 34C20 34C23 34C60 34H05 

Notes

Acknowledgements

Daniel J. Pagano acknowledges CNPq/BRAZIL for partially funding its work under Project 302229/2018-3. Rony Cristiano acknowledges the CAPES/Brazil for funding its work.

References

  1. Benadero, L., Cristiano, R., Pagano, D.J., Ponce, E.: Nonlinear analysis of interconnected power converters: a case study. IEEE J. Emerg. Sel. Topics Circuits Syst. 5(3), 326–335 (2015)CrossRefGoogle Scholar
  2. Benadero, L., Ponce, E., Aroudi, A.E., Torres, F.: Limit cycle bifurcations in resonant LC power inverters under zero current switching strategy. Nonlinear Dyn. 91(2), 1145–1161 (2018)CrossRefGoogle Scholar
  3. Carvalho, T., Tonon, D.J.: Normal forms for codimension one planar piecewise smooth vector fields. Int. J. Bifurc. Chaos 24(07), 1450090 (2014)MathSciNetCrossRefGoogle Scholar
  4. Colombo, A., Lamiani, P., Benadero, L., Di Bernardo, M.: Two-parameter bifurcation analysis of the buck converter. SIAM J. Appl. Dyn. Syst. 8(4), 1507–1522 (2009)MathSciNetCrossRefGoogle Scholar
  5. Cristiano, R., Pagano, D.J., Benadero, L., Ponce, E.: Bifurcation analysis of a DC–DC bidirectional power converter operating with constant power load. Int. J. Bifur. Chaos 26(4), 1630010 (2016). (18 pages)CrossRefGoogle Scholar
  6. Cristiano, R., Carvalho, T., Tonon, D.J., Pagano, D.J.: Hopf and homoclinic bifurcations on the sliding vector field of switching systems in \({\mathbb{R}}^3\): a case study in power electronics. Phys. D 347, 12–20 (2017)MathSciNetCrossRefGoogle Scholar
  7. Cunha, F., Pagano, J.: Bifurcation analysis of the Lotka–Volterra model subject to variable structure control. IFAC Proc. Vol. 35(1), 101–106 (2002). 15th IFAC World CongressCrossRefGoogle Scholar
  8. Dercole, F., Rossa, F.D., Colombo, A., Kuznetsov, Y.A.: Two degenerate boundary equilibrium bifurcations in planar Filippov systems. SIAM J. Appl. Dyn. Syst. 10(4), 1525–1553 (2011)MathSciNetCrossRefGoogle Scholar
  9. Di Bernardo, M., Hogan, S.J.: Discontinuity-induced bifurcations of piecewise smooth dynamical systems. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 368(1930), 4915–4935 (2010)MathSciNetCrossRefGoogle Scholar
  10. Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer, Berlin (2008a)zbMATHGoogle Scholar
  11. Di Bernardo, M., Nordmark, A., Olivar, G.: Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems. Phys. D 237(1), 119–136 (2008b)MathSciNetCrossRefGoogle Scholar
  12. Di Bernardo, M., Pagano, D.J., Ponce, E.: Nonhyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach. Int. J. Bifurc. Chaos 18(5), 1377–1392 (2008c)MathSciNetCrossRefGoogle Scholar
  13. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer Academic Publishers, Dordrecht (1988)CrossRefGoogle Scholar
  14. Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear Dyn. 78(1), 251–263 (2014)MathSciNetCrossRefGoogle Scholar
  15. Glendinning, P.A.: Classification of boundary equilibrium bifurcations in planar Filippov systems. Chaos Interdiscip. J. Nonlinear Sci. 26(1), 013108 (2016)MathSciNetCrossRefGoogle Scholar
  16. Glendinning, P.A.: Shilnikov chaos, Filippov sliding and boundary equilibrium bifurcations. Eur. J. Appl. Math. 29(5), 757–777 (2018)MathSciNetCrossRefGoogle Scholar
  17. Guardia, M., Seara, T., Teixeira, M.: Generic bifurcations of low codimension of planar Filippov systems. J. Differ. Equ. 250(4), 1967–2023 (2011)MathSciNetCrossRefGoogle Scholar
  18. Hogan, S.J., Homer, M.E., Jeffrey, M.R., Szalai, R.: Piecewise smooth dynamical systems theory: the case of the missing boundary equilibrium bifurcations. J. Nonlinear Sci. 26(5), 1161–1173 (2016)MathSciNetCrossRefGoogle Scholar
  19. Jacquemard, A., Teixeira, M.A., Tonon, D.J.: Stability conditions in piecewise smooth dynamical systems at a two-fold singularity. J. Dyn. Control Syst. 19(1), 47–67 (2013)MathSciNetCrossRefGoogle Scholar
  20. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, New York (2004)CrossRefGoogle Scholar
  21. Kuznetsov, Y.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. J. Bifurc. Chaos 13(8), 2157–2188 (2003)MathSciNetCrossRefGoogle Scholar
  22. Lee, H.C., Abed, E.H.: Washout filter in the bifurcation control of high alpha flight dynamics. In: Proceedings of the American Control Conference, Boston, vol. 1, pp. 206–211 (1991)Google Scholar
  23. Leine, R.I., Campen, D.H.V., de Vrande, B.L.V.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23(2), 105–164 (2000)MathSciNetCrossRefGoogle Scholar
  24. Pagano, D.J., Ponce, E.: On the robustness of the DC–DC Boost converter under Washout SMC. In: Power Electronics Conference, Brazil 2009, pp. 110–115 (2009)Google Scholar
  25. Pagano, D.J., Ponce, E.: Sliding mode controllers design through bifurcation analysis. In: Preprints of the 8th IFAC on Nonlinear Control Systems, Bologna, Italy, pp. 1284–1289 (2010)Google Scholar
  26. Pagano, D.J., Ponce, E., Torres, F.: On double boundary equilibrium bifurcations in piecewise smooth planar systems. Qual. Theory Dyn. Syst. 10(2), 277–301 (2011)MathSciNetCrossRefGoogle Scholar
  27. Simpson, D.J.W.: A general framework for boundary equilibrium bifurcations of Filippov systems. Chaos Interdiscip. J. Nonlinear Sci. 28(10), 103114 (2018)MathSciNetCrossRefGoogle Scholar
  28. Tahim, A.P.N., Pagano, D.J., Heldwein, M.L., Ponce, E.: Control of interconnected power electronic converters in dc distribution systems. In: XI Brazilian Power Electronics Conference, pp. 269–274 (Sept 2011)Google Scholar
  29. Teixeira, M.A.: Stability conditions for discontinuos vector fields. J. Differ. Equ. 88, 15–29 (1990)CrossRefGoogle Scholar
  30. Utkin, V.I.: Sliding modes and their application in variable structure systems, volume (Translated from the Russian). MiR (1978)Google Scholar
  31. Utkin, V.I., Guldner, J., Shi, J.: Sliding Mode Control in Electro-Mechanical Systems. Automation and Control Engineering. CRC Press, Boca Raton (2009)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Automation and SystemsFederal University of Santa CatarinaFlorianópolisBrazil

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