**Abstract: Aiming at the experimental hydraulic servo-driven continuous casting mold vibration simulation device, a mathematical model of the electro-hydraulic servo system was established by comprehensively considering the servo valve flow nonlinearity and the resonance of the load structure, and its correctness was proved through experiments. In order to suppress its parameter perturbation, a variable structure controller is designed for the position closed-loop control system.**

**Keywords: continuous casting mold; electro-hydraulic servo system; variable structure controller**

**1 Introduction**

In recent years, a large number of high and new technologies have been used in metallurgical industrial production to adapt to increasingly fierce product competition, and high-efficiency continuous casting technology has also been rapidly developed and applied. The mold is the main equipment in the continuous casting process of steel billet, and its non-sinusoidal vibration control is one of the key technologies in the continuous casting production process. Non-sinusoidal vibration can prompt the mold slag to be filled in time between the mold and the billet shell, forming a lubricating layer with a certain thickness, preventing the mold from adhering to the molten steel, reducing leakage and cracking of the billet shell, and having a significant impact on the surface quality of the steel billet. improvement. Compared with traditional mechanical devices, the continuous casting machine mold vibration device driven by the electro-hydraulic servo system has the advantages of accurately realizing various desired vibration signals, modifying vibration parameters online and being flexible in layout. Therefore, the research has good reliability and high control accuracy. And the electro-hydraulic servo control system with fast response speed has important practical significance.

This paper uses the mechanism modeling method, comprehensively considers the nonlinearity of servo valve flow and load structure resonance, establishes a mathematical model of the hydraulic servo system, and designs a variable structure controller, which has certain reference significance for the design of similar products.

**2. Composition of continuous casting copper mould tube experimental bench**

The continuous casting mold experimental vibration table is designed to simulate the vibration load of the billet continuous casting mold. The vibration device adopts a special four-link vibration mechanism, with leaf springs replacing the connecting rods, eliminating the need for bearings. The weight of the entire experimental bench is greatly reduced, and its schematic diagram is shown in Figure 1.

**Figure 1 Schematic diagram of continuous casting copper mould tube experimental device**

The electro-hydraulic servo control system mainly consists of a hydraulic pump station, PID controller, electro-hydraulic servo valve, servo cylinder, load, etc. The oil source adopts a constant pressure axial piston variable pump plus a safety valve. The pump output pressure is stable and the flow pulsation is small. The servo valve uses a dual-nozzle baffle force feedback two-stage electro-hydraulic servo valve, and the power mechanism adopts a valve-controlled double-rod piston cylinder; the displacement sensor is installed at the piston rod, which is a semi-closed loop system.

**3 Mathematical model of electro-hydraulic servo system**

1) PID controller

2) Electro-hydraulic servo valve

The electro-hydraulic servo valve can be regarded as a second-order oscillation link, and its transfer function of valve core displacement to current is:

The flow equation of the servo valve is a nonlinear equation:

3) The hydraulic cylinder flow continuity equation is:

4) As can be seen from Figure 1, due to the existence of leaf springs, the load of the dynamic mechanism of the continuous casting mold test bench is a multi-level resonant load with concentrated masses connected by a flexible structure. This article simplifies it to a two-degree-of-freedom concentrated load. as shown in picture 2.

**Figure 2 Load simplified diagram**

Therefore, the force balance equations of the hydraulic cylinder and the load are:

5) Displacement sensor

The displacement sensor is installed at the piston rod and can be regarded as an inertia link:

After the models of each link above are established respectively, the simulation model of the entire electro-hydraulic servo system is established according to Figure 3.

**Figure 3 System transfer function block diagram**

The step response curve of the simulation model is shown in Figure 4a, from which it can be seen that the system adjustment time is 0.43s, the overshoot is 11%, and the system is an under-damped system with strong oscillation. Since the mass converted from the load to the piston is small, the stiffness of the hydraulic spring is very large, and the bandwidth of the entire system is much smaller than the hydraulic natural frequency, which mainly depends on the structural flexibility of the system.

**Figure 4 System unit step response curve**

**4 Experimental verification of mathematical model of electro-hydraulic servo system**

In order to verify the correctness of the established model, a 1250 frequency characteristic tester was used to conduct actual measurements on the position control objects of the hydraulic servo-driven continuous casting mold. The 1250 frequency response analyzer has the advantages of simple operation and strong functionality. It can measure the amplitude-frequency characteristics and phase-frequency characteristics of the system by giving a known sinusoidal signal to the system under test. During actual measurement, the sinusoidal signal amplitude of the input signal generator is 500 mV, the system oil supply pressure is 5 MPa, and the frequency sweep range is 0.5~ 40 Hz, integration time 100 ms, and then connected to the printer through the standard interface for printout. The open-loop logarithmic amplitude-frequency characteristic curve is obtained as shown in Figure 5. Through the theoretical analysis of this logarithmic amplitude-frequency characteristic curve, it is concluded that the actual transfer function of the position loop controlled object is composed of an integral link, an oscillation link and a second-order differential link. Moreover, the difference between the turning frequency of the differential link and the oscillation link is very small, and it has the characteristics of a typical multi-degree-of-freedom load. Its open-loop transfer function is as follows:

**Figure 5 Open-loop Bode plot of the measured system**

The closed-loop step response curve of the above formula is shown in Figure 4b. It can be seen that the adjustment time is 0.4 s, the overshoot is 12%, and the oscillation is strong. Comparing with Figure 4a, we can see that the waveforms of the two are very similar, indicating that the theoretical modeling is correct.

**5 Design of variable structure controller**

It can be seen from theoretical models and experiments that the damping of the system is relatively small and will change greatly with changes in working conditions. Since the damping ratio has a great impact on system performance, its perturbation needs to be suppressed. Variable structure has many essential advantages such as fast response, insensitivity to changes in parameters and external interference, no need for online system identification, and simple physical implementation. Therefore, the mathematical model of the hydraulic servo system is designed as a variable structure controller, and equation (8) is written as an error equation:

Define the error vector

; e =x- x 0, x 0 is the expected output value of the system for the step input.

The switching function is taken as s(X) =cX, c=[ c 1 c2 c 3], because the highest vibration frequency of this system is less than 400 times, so the expected bandwidth of the system is taken as 10 Hz. Therefore, the row vector c is:

Exponential approach control:

The variable structure control is:

The -εsgn s term in the control law is the restricted control part, which generates the sliding mode and plays the role of “variable structure”. The -kcE option generates new object dynamics to enable the system to meet dynamic performance requirements. – The cAE term plays the role of eliminating the original dynamics.

The selection of parameters ε and k in the reaching law has a great influence on the performance of the closed-loop system and the final generation of chattering on the sliding mode surface. Because k plays a major role at the beginning of normal operation of the system, and the role of k weakens when approaching the switching surface, and ε plays a major role.

According to the actual situation of the system, take ε=46 and k=154. The controller equation is:

The designed variable structure controller is applied to the theoretical model of the system, and its step response curve is shown in Figure 4c. Its response time reaches 0.15s, and there is no overshoot, and the performance meets the requirements.

Increasing the elastic and viscous loads of the theoretical model by 20% is equivalent to system parameter perturbation, and its step response curve is shown in Figure 4d. It can be seen from the figure that after applying the variable structure controller, the system has strong robustness and adaptability to parameter perturbations, fully meeting the system design indicators.

**6 Conclusion**

(1) The load of the hydraulic servo system can be simplified to a double two-degree-of-freedom resonant load. By comparison with the experiment, the correctness of the established mathematical model of the hydraulic servo system is verified.

(2) When the parameters of the controlled object model are perturbed, the sliding mode control has good output performance, strong robustness and adaptability, and can ensure good dynamic characteristics of the system.