Generic and generalized boundary operating points in piecewise-linear (discontinuous) control systems

Switching control strategies introducing a discontinuity in the closed-loop equations are widely used in different fields of science and engineering. When an operating point that is viable for one of the available control actions attains a switching boundary, as a process or control parameter is varied, two generic scenarios are possible: either the operating point turns into a pseudo-equilibrium on the boundary (persistence), so the stationary solution persists through the bifurcation and a sliding mode control is initiated; or it collides and disappears together with a previously coexisting pseudo-equilibrium (nonsmooth-fold), triggering a transient toward another attractor of the control system. In this paper we analyze the degenerate situation separating these two scenarios-the codimension-2 bifurcation called Generalized Boundary Equilibrium (GBE) in the literature on Filippov systems-in the special case of piecewise-linear control systems. We apply our results in power electronics, where the desired operating point often works in sliding mode. In particular, we analyze a three-dimensional model of DC-DC boost converter, where the GBE bifurcation-though occurring for non operative values of the control parameters-plays a critical role in identifying the operative regions of the electrical and control parameters.