The Connection Between Sliding Mode Analysis and Singular Perturbation Theory for Modelling Fast Hydraulically Fed-Back Switching Valves

In most cases modeling of fast switching valves in hydraulics results in fast and slow subsystems. System equations which incorporate fast and slow dynamics are called stiff systems and one can apply singular perturbation theory to reduce the system order and get handy approximate expressions in a lower-order description. This is not only useful to reduce complexity to numerically solve the system efficiently, but also to understand the system’s key parameters and how they affect the behavior which is of great interest during design phase. In a previous paper the authors published an approach which uses switched systems and sliding modes to get a reduced system description of hydraulically fed-back switching. There, one models a hydraulic valve as either completely open or closed. A partially opened valve is then modelled as a sliding mode which can be interpreted as a pulse-width modulation of a fast switching digital valve. Even though, the resulting sliding mode dynamics approximation does not preserve topological properties of the full system dynamics an advantage of this approach is that the system incorporates the nonlinearities which arise due to end-stops of valves in a very natural way. Therefore, it is capable of describing system dynamics which results from such non-smooth properties. In this paper the authors work out the naturally suggested - even though not obvious - connection between both approaches for reducing systems with hydraulically fed-back switching valves.