Two-Parameter bifurcation Curves in Power Electronic converters

In many control problems, the control strategy is phrased in terms of average quantities. In this paper we prove the general result that, given a linear system ẋ = Ax + u where A is hyperbolic, u is piecewise linear and L-periodic, such that R L 0 u(t)dt = 0, then there exists a unique Lperiodic solution x = xp(t) such that R L 0 xp(t)dt = 0. We then consider a DC/DC buck (step-down) converter controlled by the ZAD (zero-average dynamics) strategy. The ZAD strategy sets the duty cycle, d (the length of time the input voltage is applied across an inductance), by ensuring that, on average, a function of the state variables is always zero. The two control parameters are vref , a reference voltage that the circuit is required to follow, and ks, a time constant which controls the approach to the zero average. We show how to calculate d exactly for a periodic system response, without knowledge of the state space solutions. In particular we show that for a T -periodic response d is independent of ks. We calculate curves in (vref , ks) space at which a T -periodic response of the system undergoes period doubling and corner collision bifurcations, the latter occurring when the duty cycle saturates and is unable to switch. We also show the presence of a codimension two bifurcation in this system when a corner collision bifurcation and a saddle node bifurcation collide, to produce stable unsaturated 2T -periodic responses which can be obtained either in the presence or absence of the stable T -periodic response. Abbreviated Title: Bifurcations in power electronic converters