I. Introduction
Research on hydraulic valves has been ongoing for a long time. The electrohydraulic servo valve appeared in the 1940s. Novel actuation systems for such valves were developed during the 1950s. In the early 2000s, academic research on hydraulic valves increased due to the availability of high-speed microprocessors, which motivated the design of novel hydraulic valves. There are monographs with physical models of a wide range of hydraulic valves [1] [2]. Two-stage pro-portional cartridge valves with a poppet-type construction are used in applications with fast response, large flow, and high-pressure requirements [3] [4]. Xu et al. modeled a directional valve with three stages from physical principles, where two proportional poppet-type cartridge valves were in the second stage [5]. Han et al. proposed an optimal design for a two-stage proportional cartridge valve with a poppet construction type [6]. This type of valve is controlled by proportional-integral-derivative (PID) controller with a feedback loop, using signals continuous in amplitude. In contrast, other types of valves are controlled by a model predictive control (MPC) based controller using pulse width modulation signals in [7].