Revisiting the Teixeira Singularity Bifurcation Analysis: Application to the Control of Power Converters

Abstract

For a discontinuous piecewise smooth (DPWS) dynamical system in [Formula: see text], whose state space is divided into two open regions by a plane acting as the switching manifold, there generically appear two lines of quadratic tangency, one for each involved vector field. When these two tangency lines have a transversal intersection, such a point is called a two-fold singularity. If furthermore, both tangencies are of invisible type, then the two-fold point is known as a Teixeira singularity (TS). The Teixeira singularity can undergo an interesting bifurcation, namely when a pseudo-equilibrium point crosses the two-fold singularity, passing from the attractive sliding region to the repulsive sliding region (or vice versa) and, simultaneously, a crossing limit cycle (CLC) arises. After deriving carefully a local canonical form, we revisit the previous works regarding this bifurcation thus correcting some detected misconceptions. Furthermore, we provide by means of a more direct approach the critical coefficients characterizing the bifurcation, also giving computational procedures for them. The achieved results are applied to some illustrative examples, within the realm of discontinuous piecewise linear (DPWL) systems. This family acts like a normal form for the bifurcation, since DPWL systems are able to reproduce all the unfolded dynamics. The study of TS-points in electronic DC-DC Boost power converters under a sliding mode control (SMC) strategy is addressed. Apart from being a relevant application, it allows to show the real usefulness of the analysis done.