Generalization of the Filippov method for systems with a large periodic input

Using the Filippov method, the stability of a nominal cyclic steady state of a nonlinear dynamic system (a buck dc-dc converter) is investigated. A common approach to this study is based upon a complete clock period, and assumes that the input is from a regulated dc power supply. In reality, this is usually not the case: converters are mostly fed from a rectified and filtered source. This dc voltage will then contain ripples (i.e. the peak-to-peak input voltage is not zero). Therefore, we consider the input as a sinusoidal voltage. Its frequency is chosen as a submultiple T of the converter's clock and our objective is to analyze, clarify and predict some of the nonlinear behaviors that these circuits may exhibit, when the input voltage frequency changes in time. This input frequency's parameter T determines the number of the switching instants over a whole clock cycle, obtained as Newton-Raphson solutions. Then, for the considered buck converter, we develop a mathematical model in a compact form of the Jacobian matrix with a variable dimension proportional to the input voltage harmonics. Finally, the Floquet multipliers of the monodromy matrix are used to predict the system stability. Numerical examples illustrate how these multipliers cross the unit cycle causing various bifurcations.