Hydraulic Switching Control with a Nonlinear Converter

In this paper, the hydraulic analogy of the so called `Buck Converter` of power electronics is studied. It consists of a discrete inductivity, a capacity, and a valve. In our hydraulic `Buck Converter`, the capacity is realised by a standard gas spring hydraulic accumulator with a nonlinear behaviour and the inductivity by a hydraulic pipe which may exhibit complex wave propagation phenomena. The task to be solved here is to work out a reduced model for this converter configuration, which describes the essence of its behaviour, and, additionally to find the ranges in parameter space of the complex system for which this reduced model is valid. For small accumulator capacities, the reduced model exhibits a one dimensional invariant manifold in each switching state which characterises a slow dynamics of the system. The remaining part of the phase space is "foliated" by trajectories representing the fast dynamics of the reduced system. Approximate simple formulas describe both type of trajectories and the flow on them. With these results, the reduced system can be evaluated w.r.t. the technical requirements. In a second step the question is investigated, under which circumstances the disturbed system, i.e. the system taking into account the complex dynamics of wave propagation in the pipe, the finite capacity of the large accumulator, the finite switching time of the valves, and the dynamical properties of an attached hydraulic cylinder, is close to the reduced system. This is carried out by a numerical analysis.