Model Reduction in Hydraulics by Singular Perturbation Techniques

The requirement to keep mathematical models as simple as possible is contrasted by the power of modern computers by which complex models can be simulated. Particularly in conceptual design the total view must be preserved. The interpretation of simulation results will benefit from a thorough understanding of the system under study, i.e. from the role the diverse physical effects and the main design parameters play for system performance. This requires simple models with a small number of system parameters. Such models either are `designed` properly or are derived from more complex models by proper reduction techniques. For extremely small or big values of some parameters certain effects might practically vanish or reduce to small layers, so called boundary or transition layers. A mathematical technique which focuses on the analysis of such behaviour is perturbation theory, in particular singular perturbation theory. The small fluid compressibility, for instance, makes many hydraulic state equations singularly perturbed systems. Compressibility effects create boundary or transition layers in the time signals especially of pressure. Such layers are triggered either by initial conditions or by a change of the active network topology of the hydraulic circuit due to a switching of a valve. Away from such layers, system behaviour is only weakly affected by compressibility. This view of singular perturbation methods is helpful to figure out the origin and parameter dependence of some effects. The use of such techniques is highlighted by two examples of hydraulic drive systems.