Linear Hybrid Dynamical Control of Digital Displacement Units

This paper concerns development of control strategies for energy efficient fluid power digital displacement machines R(DDM). The DDM technology yields an efficient reduction in displacement levels by deactivating independently cylinder pressure chambers by electromagnetically controlled on-off valves. Since the continuous dynamics of each pressure chamber is activated/deactivated discretely at fixed shaft positions, the DDM dynamics belongs to the class of hybrid dynamical systems. However, control development for hybrid system is in general very complex due to the use of Lyapunov stability theory for both continuous and discrete systems. This paper shows stability based on hybrid dynamical theory for a linear continuous plant actuated by a DDM, which dynamics has been approximated by a linear discrete model. It is shown that the control design problem is identically to that of a fully discrete description of the system based on a zero-order-hold
approximation of the continuous design. Furthermore, the problem of having a nonlinear plant, multiple DDMs or an angle dependent asynchronous control update rate for a variable speed DDM is addressed, where measures of stability analysis is discussed.