Feedback Control of Pulse-Density Modulated Digital Displacement Transmission using a Continuous Approximation

Feedback control of digital displacement machines is complicated due to the nonsmooth digital behavior. Full-stroke operated digital displacement machines are characterized by delivering a discrete volumetric output based on the ratio of activated cylinder chambers. The binary input decision (active or inactive) is made discretely with an update rate proportional to the speed of the machine. For a digital fluid power transmission with two digital displacement machines with varying and different speeds and which dynamics greatly influence each other through the pressurized fluid line, the control task is further complicated. To overcome this problem, this article presents a continuous approximation of a pulse-density-modulated digital displacement machine, which allows for dynamic analysis and control design. This article shows that linear feedback control theory is adequate to show stability if the number of cylinders, displacement throughput, and rotational speed of the machine are sufficiently high. Additionally, the excitation frequencies must be sufficiently low to not excite the discrete behavior. An optimal state feedback controller is synthesized and tested in a nonlinear simulation model, which represents the physical digital hydraulic transmission. Simulation results shows great tracking performance similar to a transmission with ideal fluid power machines, but with noticeable fluctuations due to the digital machine characteristics.