Four Quadrant Hybrid Control Oriented Dynamical System Model of Digital Displacement^® Units

Proper feedback control of dynamical systems requires models that enables stability analysis, from where control laws may be established. Development of control oriented models for digital displacement (DD) fluid power units is complicated by the non-smooth behavior, which is considered the core reason for the greatly simplified state of the art control strategies for these machines. The DD unit comprises numerous pressure chambers in a modular construction, such that the power throughput is determined by the sequence of activated pressure chambers. The dynamics of each pressure chamber is governed by non-linear differential equations, while the binary input (active or inactive) is updated discretely as function of the shaft angle. Simple dynamical approximations based on continuous or discrete system theory is often inaccurate and is not applicable for such system when it is to operate in all four quadrants. Therefore, a method of applying hybrid dynamical system theory, comprising both continuous and discrete elements is proposed in this paper. The paper presents a physical oriented hybrid model accurately describing the machine dynamics. Since development of stabilizing control laws for hybrid dynamical systems is a complicated task, a simpler hybrid model only including the fundamental machine characteristics is beneficial. Therefore, a discussion and several proposals are made on how a simpler DD hybrid model may be established and used for feedback control development.