## 2 Control System Composition Principle and Description  
A swashplate axial piston variable pump is the type of variable pump being explored. This type of pump is used in hydraulic systems due to its adaptability and efficiency. It works on the swashplate mechanism, in which the angle of the swashplate affects the displacement of the pistons and, as a result, the flow rate of the pump. Furthermore, the swashplate axial piston variable pump is a four-quadrant pump, referring to its ability to operate in four separate operating situations or quadrants. The combination of positive/negative flow and positive/negative pressure defines these quadrants. Positive flow and positive pressure, negative flow and positive pressure, or negative flow and negative pressure can all be generated by the pump. The variable pump's capacity to work in all four quadrants provides more flexibility and adaptability in a variety of hydraulic applications. It can deal with a variety of load requirements, reverse flow events, and even energy recovery scenarios. As a result, the swashplate axial piston variable pump is an excellent solution for a variety of industrial and mobile hydraulic systems [25]. An axial piston pump is seen schematically in Figure 1. The control system, as illustrated in Figure 2, employs a typical hydraulic valve to regulate the closed-loop control of the inflow volume, with an angle sensor determining whether normal operating conditions have been met. The diagram depicts the operation of the variable pump control system [23]. Sensors detect and recognize the position of the piston in the variable cylinder. The controller algorithm changes the direction of the force delivered to the proportional directional valve by adjusting the output drive current of the proportional solenoid based on the detected position information. When the proportional reversing valve is pushed to the left, hydraulic fluid enters the piston and pushes it to be displaced, causing the L4 VG variable pump to change. Through the pull fork, the piston returns a force to the proportional directional valve, causing the spool to reset. After the proportional directional valve's spool returns to zero, the spool can be in a balanced state when the size of the current signal input force received by the proportional directional valve equals the force generated by the feedback spring, forming a closed-loop system of feedback control of the piston position. Because the swashplate and variable cylinder of the L4 VG variable pump are mechanically connected, any change in variable piston position will result in a change in swashplate inclination. As a result, the variable mechanism displacement of the variable pump changes after an operational procedure [26]. This mechanism change can be performed by combining it with feedback control of piston position to produce accurate control and regulation of the hydraulic system to fulfill the needs under various working situations. The variable pump may alter and adapt to the working conditions of the hydraulic system during the working process using this closed-loop control system based on the real application requirements. Mathematical Modeling of the Variable Mechanism of L4 VG Variable Pump  
A proportional directional valve is the core component of the servo control system; its proportional solenoid can accept the electrical signal and convert it into a mechanical signal, while generating electromagnetic force \(F_{m}\), because the thrust of the proportional solenoid is proportional to the current in the solenoid. Therefore, let the current through the proportional solenoid be \(I\), and then the equation of electromagnetic force and current is:  
\[F_{m}=k_{I}\cdot I \tag{1}\]  
Figure 1: Variable mechanism. Figure 2: Control system block diagram. In electromagnets, this is a frequent depiction. When F denotes magnetic force, k denotes a quantity known as the coefficient of magnetism or the coefficient of magnetic induction, and I denotes current.