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On the Teixeira singularity bifurcation in a DC–DC power electronic converter

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Abstract

The electronic circuit of a DC–DC boost power converter under a specific sliding mode control strategy is analyzed. This circuit is modeled as a discontinuous piecewise linear three-dimensional dynamical system, whose state space is divided into two open regions by a plane acting as the switching manifold. This system displays sliding motion confined to the switching manifold and limited by two straight lines where lie the points of tangency, one for each involved linear vector field. Such tangency lines can intersect transversally on a invisible two-fold point known as Teixeira singularity. Our goal is to investigate an interesting bifurcation that occurs on the Teixeira singularity, involving both a pseudo-equilibrium point and a crossing periodic orbit. This bifurcation, named TS-bifurcation, occurs when a pseudo-equilibrium point of the discontinuous piecewise linear system, capable of moving between the attractive and repulsive sliding regions as a result of the change in a parameter, collides with the Teixeira singularity. Simultaneously, a periodic orbit arises from the Teixeira singularity crossing the switching manifold in two points. Apart from the TS-bifurcation characterization in the DC–DC converter, we have numerically detected other non-local phenomena like a saddle-node bifurcation of crossing limit cycles. Experimental results to illustrate the effects of the TS-bifurcation on the dynamics of a power electronic converter are also shown.

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Notes

  1. High-order Lie derivatives can be obtained by writing \(L^m_{{\mathbf {F}}^{\pm }}h = \langle \nabla L_{{\mathbf {F}}^{\pm }}^{m-1} h, {\mathbf {F}}^{\pm }\rangle \).

  2. The converter operates with a non-null inductance current at any time, i.e., \(i_L>0\).

  3. The trivial equilibrium point of (21)–(22) is always in a crossing region or collides with the double tangency if \(k=\omega y_r\) (then, \(\widehat{{\mathbf {x}}}={\mathbf {0}}\)). We do not worry about this trivial equilibrium.

  4. The flight time is the time required for a trajectory, started in \(\varSigma \), to return to \(\varSigma \).

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Acknowledgements

Rony Cristiano acknowledges CNPQ/Brazil for funding its work under process 200891/2015-4. Daniel J. Pagano acknowledges CNPq/Brazil for funding his work under Grant PDE-201256/2015-0. Enrique Ponce is partially supported by the Ministerio de Economía y Competitividad, in the frame of project MTM2015-65608-P, and by the Consejería de Economía y Conocimiento de la Junta de Andalucía under Grant P12-FQM-1658.

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Authors and Affiliations

  1. Department of Automation and Systems, Federal University of Santa Catarina, Florianópolis, SC, 88040-900, Brazil

    Rony Cristiano, Daniel J. Pagano & Mathieu Granzotto

  2. Department of Applied Mathematics, ETSI, University of Seville, 41092, Seville, Spain

    Enrique Ponce

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  1. Rony Cristiano
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Correspondence to Daniel J. Pagano.

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Cristiano, R., Ponce, E., Pagano, D.J. et al. On the Teixeira singularity bifurcation in a DC–DC power electronic converter. Nonlinear Dyn 96, 1243–1266 (2019). https://doi.org/10.1007/s11071-019-04851-8

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