## 1 Introduction  
Currently, energy and environmental pollution are the most pressing problems in the world, while fossil fuels are depleting, the demand for use is increasing. Moreover, the high urbanization rate leads to environmental problems and global warming due to CO\({}_{2}\) emissions, which are emitted from activities of machinery equipment in the fields such as construction, mining, and other industries [1; 2; 3; 4]. Among them, earth-moving machinery is most commonly used, especially in the construction industry [5; 6; 7]. The earth-moving machines such as hydraulic excavators consume a huge amount of fossil fuels and emit a lot of CO\({}_{2}\) emissions during operation [8; 9; 10]. Therefore, reducing energy consumption in hydraulic excavators saves fossil energy sources and protects the environment. Conventional excavators use a lot of energy to work, but only about 50% of the energy is delivered to the cylinder, the remaining 50% of the energy is lost in the pump, pipes, and valve [11; 12; 13]. As a widely used device in the excavators to control flow, a conventional four-way proportional directional control valve is one of the main problems causing losses. To overcome this problem, the independent metering valve (IMV) has been proposed and proved its effectiveness compared with conventional proportional directional valves [14; 15; 16]. By replacing the conventional valves with four 2/2-way proportional valves, the IMV system allows the flow rate at the inlet and outlet of the cylinder to be independently managed according to five metering modes specially designed for the hydraulic excavators [17; 18; 19]. In particular, when applied to the boom excavator, the IMV system can save up to 44% of energy by using gravitational potential energy [20; 21; 22]. However, the IMV system has two problems that make it meet an obstacle for applying in practice. Firstly, the cost of the proportional valve causes a high cost for the system [23]. Secondly, controlling four valves is particularly challenging because of various types of nonlinearities in hydraulic systems [24]. Therefore, reducing the number of proportional valves and developing control methods are current research trends [25; 26]. The solution of proportional valves is always more expensive than that of conventional proportional directional valves, while the conventional IMV (CIMV) system uses up to four proportional valves [27]. Thus, reducing the number of proportional valves in the CIMV system leads to cost savings in the system. [PERSON] et al. [28] proposed a configuration of IMV system that used three electro-hydraulic poppet valves ((EHPV) a kind of 2/2-way proportional valve) and applied the equivalent valve coefficient equation to estimate the valve conductance of those valves based on the metering modes. Although, the cost of this system was cheaper and saved more energy compared with the CIMV system due to removing one proportional valve and a check valve, the approach with the equivalent valve coefficient made the control system complicated. Moreover, during the operation, friction and leakage occurred, leading to low tracking performance of the boom cylinder. [PERSON] et al. [29] used two 4/3-way proportional direction valves for the IMV system and proposed the velocity feedforward and position feedback (VFPB) for the control method. With the same approach, [PERSON] et al. [30] used two 3/3-way proportional direction valves to replace four proportional valves in the IMV system. Moreover, the system was controlled by velocity and position of the boom cylinder based on the difference between the feedback signal and the target signal control. Their results showed that the control method helped the system operate smoothly and achieve high position precision. However, the use of two proportional direction valves made the system become more expensive due to their high cost. In a different approach, [PERSON] et al.