Abstract: To address the impact of nonlinearity and parameter uncertainty on electro-hydraulic position servo systems, a two-degree-of-freedom internal model control method is proposed based on internal model control theory. The designed controller has only two adjustable parameters. By adjusting these two parameters, the system can simultaneously possess good setpoint tracking characteristics, disturbance suppression characteristics, and robustness, improving upon the shortcomings of conventional internal model control. Furthermore, the control algorithm is simple, and parameter adjustment is convenient. Theoretical analysis and simulation results demonstrate its effectiveness. 1 Introduction Electro-hydraulic servo control systems are among the most common control devices in mechatronic equipment. They organically combine electronics and hydraulics, forming a closed-loop control system with electrical signals as input and hydraulic signals as output. They possess both fast, easily adjustable, and high-precision response capabilities, and can control large inertia to achieve high-power motion output, thus finding wide application. Among these, electro-hydraulic position servo systems are the most widely used, such as positioning and machining trajectory control of CNC machine tools, tracking control of radar antennas, and displacement control of simulated vibration test benches. All of these applications require high-precision tracking of a specified trajectory and good disturbance suppression characteristics. However, electro-hydraulic position servo systems are typical nonlinear systems and suffer from parameter uncertainties. Meanwhile, the dynamic characteristics of the control system are very complex, making it difficult to obtain an accurate mathematical model. Traditional control algorithms are insufficient to meet the requirements of high performance. In recent years, genetic algorithms and particle swarm optimization algorithms have been used to optimize the parameters of PID controllers, which have overcome the nonlinearity and parameter uncertainty of the system to a certain extent. However, the system response has a certain overshoot and a relatively slow response speed. This paper addresses the impact of nonlinearity and parameter uncertainty on electro-hydraulic position servo systems by combining internal model control theory with two-degree-of-freedom control methods. A two-degree-of-freedom internal model control method is proposed. Based on the characteristic of the internal model principle that the system model requirements are relatively low, the controller can be designed by simplifying the model of the electro-hydraulic servo system. The designed controller has only two adjustable parameters. By adjusting these two parameters, it can simultaneously have good setpoint following characteristics, disturbance suppression characteristics, and robustness, improving the shortcomings of conventional internal model control. It can meet the performance requirements of high-precision electro-hydraulic servo systems, and the control algorithm is simple and the parameters are easy to adjust. Theoretical analysis and simulation results demonstrate its effectiveness. 2 Electro-hydraulic Servo Position System Model The research object is the electro-hydraulic position servo system, as shown in Figure 1. This system is a typical nonlinear system. Due to the inherent nonlinear characteristics of the electro-hydraulic servo system, conventional control methods are insufficient to simultaneously achieve good static and dynamic performance and fast response speed. It is easily affected by external disturbances and changes in system parameters, resulting in poor robustness. [align=center] Figure 1 Schematic diagram of the electro-hydraulic position servo control system[/align] In this system, both the amplifier and displacement sensor can be considered proportional elements. The characteristics of the servo valve depend on the system bandwidth and the servo valve bandwidth. The servo hydraulic cylinder, assumed to be a purely inertial load and ignoring external force interference, has the following dynamic characteristics: Where: is the undamped natural frequency of the inherent oscillating element of the system, is the damping ratio of the inherent oscillating element of the system, is the area of ​​the hydraulic cylinder, and is the flow rate. When the system bandwidth is 10–20 Hz, the servo valve can be considered as an amplification element. Therefore, the electro-hydraulic servo system can be approximately considered as a third-order system composed of an integral element plus an oscillating element. Its open-loop transfer function is as follows: Where: is the open-loop amplification coefficient, Where: is the amplifier gain, is the servo valve gain, and is the sensor gain. From the equivalent transfer function of the system, it can be seen that the electro-hydraulic servo system belongs to a high-order system. Since the coefficients of the high-order terms are relatively small, and considering the simplicity of the controller design, the system can be simplified to 3. Internal Model Controller Design Principle The internal model control method was proposed by Garcia and Morari. Due to its simple design principle, intuitive parameter tuning, strong robustness, and good control performance, it has been applied in process control systems in recent years. The internal model control structure is shown in Figure 2. [align=center] Figure 2 Internal Model Control System[/align] In Figure 2, P(s) is the controlled object, M(s) is the nominal mathematical model of the controlled object, i.e., the internal model, Q(s) is the internal model controller, and R(s), Y(s) and D(s) are the input, output and disturbance signals of the control system, respectively. From Figure 2, we can obtain that under ideal conditions: Q(s) = M-1(s) (6) Then Y(s) = R(s) (7) That is, the output of the system is always equal to the input and is not affected by any disturbance. However, in practical applications, there are some non-ideal situations. The model is decomposed into an all-pass part M+(s) and a minimum-phase part M-(s), that is: M(s) = M+(s)M-(s) (8) In the formula, M+(s) contains the pure time delay element in M(s) and the zero point of the right half of the S-plane. Usually, M+(s) has the following form: (9) Where H represents complex conjugate. However, the ideal control shown in (6) is generally not easy to achieve. For this reason, a low-pass filter F(s) is added to the internal model controller to make (6) feasible and to achieve robust control of the system. Thus, the internal model controller has the following form: Q(s) = M-1(s)F(s) (10) Generally, F(s) takes the simplest form as follows: (11) Where the order n depends on the order of M-(s) to make the controller feasible, and λ is the filter time constant. If the object model is accurate, the system output is: (12) 4 Two-degree-of-freedom internal model tuning method In order to overcome the shortcomings of conventional internal model control, this paper proposes a two-degree-of-freedom internal model control structure. The system structure diagram is shown in Figure 3. [align=center] Figure 3 Two-degree-of-freedom internal model control system structure diagram[/align] In the figure, r is the setpoint, y is the system output, d is the disturbance signal, P is the process object, GMP(s) is the reference model of the setpoint following characteristic, and Q1(s) is the internal model controller. is the feedforward controller of the system. In this two-degree-of-freedom control structure, the role of is to adjust the setpoint following characteristic of the system, and the role of is to suppress disturbances and ensure that the system has good robust stability. As shown in Figure 3, the expression of the closed-loop transfer function is as follows: (13) (14) [align=center] Figure 4 Equivalent structure diagram of two-degree-of-freedom internal model control system[/align] As shown in Figure 4: (15) Here, the process object generally includes two parts: P(s) = PM+(s)PM-(s) (16) GMP = GMP+(s)GMP-(s) (17) In the formula, PM+(s) is the minimum phase part, PM-(s) is the non-minimum phase part and the time delay part. If PM-(s) = GMP-(s) is designed and GFP(s) = PM+-1(s)GMP+(s) is considered, then we have: (18) From equations (10) and (11), the internal model filter transfer function can be selected as: (19) In the formula, λ1 is the filtering time constant. Therefore, Q1(s) can be selected as: (20) Substituting equation (20) into equation (15), we get: (21) Considering the feasibility of the feedforward controller, the setpoint tracking characteristic transfer function can be selected as: (22) Equations (13)(17)(21)(22) show that adjusting λ2 can make the system have good setpoint following characteristics, and adjusting λ1 can make the system have good interference suppression characteristics and robust stability. 5 Simulation Study The mathematical model selected in this paper is simulated using MATLAB software. The system transfer function is: (23) From equation (22), it can be seen that the system transfer function can be simplified to: (24) Where K=1.15, T=0.0025. According to the structure in Figure 4, the step response curves with amplitude of 1 when λ1=0.1, λ2=0.1, λ1=0.05, λ2=0.1, λ1=0.1, λ2=0.1, λ1=0.1, λ2=0.3 are shown in Figures 5 and 6. [align=center] Figure 5 Simulation curves when λ1 is different and λ2 is the same Figure 6 Simulation curves when λ1 is the same and λ2 is different[/align] As can be seen from Figures 5 and 6, changing λ2 can adjust the setpoint following performance of the system, and changing λ1 can adjust the interference suppression characteristics of the system. Moreover, changing λ1 does not affect the setpoint following performance of the system, and changing λ1 and λ2 does not affect the interference suppression characteristics of the system. By selecting appropriate parameters, the system can simultaneously possess good setpoint following performance and disturbance suppression performance, thus overcoming the shortcomings of conventional internal model control. 6. Conclusion The two-degree-of-freedom internal model control method designed in this paper has two adjustable parameters, enabling the system to simultaneously possess good setpoint following characteristics, disturbance suppression characteristics, and robust stability, overcoming the shortcomings of conventional internal model control. The proposed control algorithm is simple, the parameters are easy to adjust, and it is easy to implement in engineering.