The constant pressure control system for the hydraulic system of the leveler

Jun WANG, Yu-xi SHEN

(College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China)

Abstract: For medium plate leveler, the constant pressure control can improve the quality and precision of the plates, at the same time, the rack can be protected. A constant pressure control system is put forward for the screw-down system of the leveler, which converts the position control system to the pressure control system at a specific point. For the characteristics of nonlinearity, time-varying and disturbance of the hydraulic system of levelers, the control strategy of auto disturbance rejection with disturbance feedforward is adopted, which can reduce the influences of disturbance forces on the straightening force, improving the control precision of the straightening force and achieving constant pressure control. This control strategy is verified by the Matlab simulation and the hydraulic leveler prototype. The results indicate that the influences of disturbance force are brought down and the stability of the leveler hydraulic system is significantly improved.

Key words: Auto disturbance rejection control, Disturbance feedforward compensation, Hydraulic system, Constant pressure control, Leveler

1 Introduction

Along with the rapid development of industrialization, the steel industry is in deeply structural adjustment. The medium plates are widely used in machine manufacturing, building and military affairs, etc. The leveler is the last step of the plate production which determines whether the quality of the production is good or not[1-3]. So, straightening is an essential process of the plate shape control. The most important part of the cold leveler is the screw-down system, which determines the strip quality. While its precision is determined by the hydraulic system. In this paper, a hybrid control system of position and pressure is applied for constant pressure control, which switches the control system from the position control to the pressure control at a specific pressure value. However, because of the nonlinearity and time-varying of electro-hydraulic control system, coupling with the mechanical error and mechanical vibration of levelers, and uneven material and complex influences of disturbance, the constant pressure control is difficult to achieve.

There are a lot of research on how to improve the control precision of the hydraulic system to compensate the nonlinearity of system, Wei Wei Gu uses output feedback model predictive control with the integration of an extended state observer to estimate not only the unmeasured system states but also the disturbances[4]. Wang improved the nonlinear energy conservation control, which self-adaptive control the discharge pressure supplied by a variable displacement pump, and the displacement is controlled by a nonlinear cascaded control strategy[5]. F.C. Yin describes a multivariable active disturbance rejection control to improve the hot strip mill precision[6]. Daun Jeong proposed an output feedback fuzzy controller that does not require a mathematical model to reduce camber including the lateral bar movement called side-slipping by tilting the roll[7]. Baraka presents a new nonlinear control scheme incorporating a state observer, a fuzzy neural network, and a new Nussbaum function for strict-feedback nonlinear systems by considering several challenges[8]. Hou puts forward a new adaptive tracking controller which uses a backstepping strategy improving the capacity of resisting disturbance and dynamic property[9]. Yao also uses an adaptive backstepping controller for precise tracking control of the hydraulic systems[10].Zhen Zhang proposed a quantitative feedback theory controller aiming at the problem of parameter time-varying and external load force change in the electro-hydrostatic actuator[11]. H. Y. Yang proposes a thrust hydraulic system compounding the pressure and flow control, which can maintain the pressure constantly under the complex excavating conditions[12]. In this paper, the hybrid control system of position and pressure is proposed, and the auto disturbance rejection with disturbance feedforward is applied to achieve the pressure constant. The external disturbance is compensated by the disturbance force feedforward compensation controller, and the internal disturbance is suppressed by the auto disturbance rejection controller (ADRC). Duc-Thien Tran used the super twisting algorithm backstepping control (STABC) to compensate for the unknown perturbations of the electro-hydraulic system[13].

The hydraulic system of the leveler consists of four cylinders, the fig.1 shows the electro-hydraulic servo system of one of the cylinders. During the straightening, if the cleaning between the rolls is on-line adjustment, the needed straightening force may beyond the capacity of the leveler, and the overload protection is not in time, the rack will be broken. For this reason, taking advantage of the fast response of the position control system, this paper proposes a paralleled position and pressure control system which switches from the position control system to pressure control system at a specific pressure value. At beginning of the process of screw-down, the control system is position closed-loop, the position signal through the multipath analog switch and feedback to the computer, then compared with the instruction signal to get the error to control the servo valve moving, when the straightening roll contacts the plates, the pressure sensors collet the pressure signals of the cylinder, when the straightening force is beyond the threshold value, the control system is switched from the position control to the pressure control, along with the auto disturbance rejection with disturbance feedforward controller achieves the constant pressure control, protecting the rack will not be broken by the overload, and improving the quality of the plate shape.

1-Servo valve; 2, 5-Pressure measurement node; 3, 6-Pressure senor; 4-Proportional pressure relief valve; 7-Hydraulic cylinder; 8-Displacement sensor

2 The mathematical model of full-hydraulic leveler control system

A more comprehensive and correct mathematical model of position and pressure hybrid control system of full-hydraulic leveler is built based on the control mechanisms and characteristics of the full-hydraulic leveler. The hydraulic system of the leveler adopts the asymmetric valve to control the asymmetric cylinder.

The flow equation of servo valve is as following[14]:

(1)

Where:q1is the flow rate of the blind end of the cylinder,q2is the flow rate of the rod end of the cylinder,Kq1is the flow gain of the blind end of the cylinder,Kq2is the flow gain of the rod end of the cylinder,Kc1is the flow-pressure coefficient of the blind end of the cylinder,Kc2is the flow-pressure coefficient of the rod end of the cylinder,Xvis the displacement of the spool,Psis the supply pressure,P1is the pressure of the blind end of the cylinder,P2is the pressure of the rod end of the cylinder.

The flow continuity equations of the cylinder are as follows:

(2)

Where:A1is the area of the piston,A2is the area of the piston rod,Xpis the piston displacement,Cipis the internal leakage coefficient,Cepis the external leakage coefficient,βeis the oil modulus of elasticity,V1is the effective volume of the blind end of the cylinder,V2is the effective volume of the rod end of the cylinder.

We define that：

λis the area ratio of the cylinder piston (λ<1),Vtis the effective volume of the cylinder.

The loss of flow caused by cylinder leakage is much less than the flow of volume effects of the piston movement and the flow of the cylinder. Therefore, when the hydraulic cylinder is in motion process, the effect of the cylinder leaks can be ignored, the flow continuity equation of the cylinder is approximate as follows:

(3)

The dynamics equation of the cylinder is

A1PL=mts2XP+BPsXp+KXp+FL

(4)

Where:mtis the total mass on piston,Bpis the viscosity damping coefficient,Kis the spring stiffness of load,FLis the external load,PL=P1-λP2.

The equations (1)-(4) are simultaneous, the transfer function of output displacement of the cylinder is as follows:

(5)

The transfer function of the output force of the cylinder is as follows:

(6)

The transfer function of the servo valve:

(7)

Where:Ksvis the flow gain of the servo valve, I is the current input,wswis the break frequency of the servo valve.

The transfer function of the servo amplifier is

(8)

Where:Ue=Ur-Uf,Uris the signal of the instruction voltage,Ufis the signal of the feedforward voltage.

The transfer function of the displacement sensor:

(9)

Where:Kfis the gain of the displacement sensor.

The transfer function of the pressure sensor is

(10)

Where:KFis the gain of the pressure sensor.

The equations (1)-(10) are simultaneous, based on which the block diagram of the electro-hydraulic position and pressure control system is constructed as Fig.2, whereU1is the instruction displacement signal,U2is the instruction pressure signal.

Fig.2 Block diagram of the leveler position/pressure control system

3 Auto-disturbance rejection control with disturbance feedforward compensation

In order to improve the precision of the straightening force, and reduce the influence of the external and internal disturbances, this paper puts forward the auto-disturbance rejection control[15] (ADRC) with disturbance feedforward compensation, which consists of the disturbance feedforward compensation controller and the auto disturbance rejection controller. The disturbance feedforward compensation controller can counteract the influences of the uncertain external disturbance by adding a control signal. The internal disturbance is estimated and compensated by ADRC, which reduces the effects of the external disturbance at the same time. The block diagram of the controller of ADRC with disturbance feedforward compensation is shown in Fig.3.

Fig.3 Block diagram of the controller of ADRC with disturbance feedforward compensation

Uris given voltage signal;Vi(i=1～5) are state variables;Zi(i=1～5) is the estimation of the system state;Z6is the total disturbance estimate value of system;eiis deviation valve;uis control valve;u0is compensation factor.

3.1 Disturbance feedforward compensation

The disturbance feedforward compensation controller has the advantages that can correct the error immediately, keep the system stable and reduce the steady-state error effectively, which is used to compensate the external disturbance force, improves the control performance of the system. When the external load is constant value without any external disturbance, the transfer function of the straightening force (according to the Fig.2) is as the equation (11):

(11)

When the system existing external disturbance, the straightening force is as the equation (12):

(12)

Where: ΔF1is the disturbance of the external load force.

The equations (11) and (12) are simultaneous, the change value of straightening force is obtained as follows:

(13)

The external disturbances will affect the normal work of the levelers, so it is essential to add a controller to eliminate the effect of the external disturbance. The straightening force with the added feedforward compensation controller is as the equation (14)

(14)

According to the principle of external invariance, equation (15) is equal to zero. The external disturbance force is compensated by the signal which is converted by the external disturbance force to suppress the effect of external disturbance force on the dynamic performance of the system.

1-F(s)G1G7(Kq1G3-Kq2G4)=0

(15)

The transfer function of the controller of disturbance feedforward compensation is

(16)

3.2 Auto disturbance rejection control (ADRC)

ADRC is only decided by the bearing capacity and change of the controlled variables of the controlled system, which doesn’t depend on the exact mathematical models of the controlled object. The object uncertainty, non-modeling dynamic and total external disturbances of the system are estimated to compensate the system by ADRC effectively. Because of the strong anti-disturbance and robustness of ADRC, it is widely used in the nonlinear systems. ADRC is a nonlinear controller which consists of three parts: tracking differential (TD), extended state observer (ESO), and nonlinear law of state error feedback (NLSEF).

The higher the order of the system is, the more difficult it is to analysis and control the nonlinear system. When the order is more than three, many complex problems will appear in the structure design of higher order. Since there are few parameters in the lower order system, it is easy to tune. Superior control performance can be obtained by ADRC when the order is less than three. The hydraulic system of the leveler is a fourth-order system, which needs a fifth-order extended state observer. This paper uses two third-order ESOs to install a fifth-order ESO in series to achieve dynamic compensation for the disturbances. According to the range of the system model variables, the ESOs in serial can adjust the controller parameters and obtain a good control performance. It is much more convenient to apply in engineering.

The design of tracking differentiator (TD) is based on the theory of classical differentiator. The differential signal is collected firstly, then the differential signal is used to arrange the transient process reasonably, and the control signal is provided. The parameter is bigger, the trace ability of TD is more power, the phase-lag becomes smaller, and the adjustment speed of the control system becomes faster. The contradiction between the response speed and the overshoot is settled by TD compensating model and external disturbance. The expression of the tracking differentiator (TD) is shown as follows:

(17)

Where:βiis undetermined parameters of TD.

As the key of the ADRC, extended state observer (ESO) uses the output to observe the all-order derivatives. The ESO utilizes the output state of each order of the system and internal and external disturbances under the unknown disturbances to estimate and compensate the disturbances, changing controlled object into integral in serial, which is a standard linear system. Because of the fast adjustment, no overshoot, and strong robustness, the ESO can effectively deal with the uncertainties of the control system. The system of the leveler is a four-order system, which needs a five-order ESO. The hydraulic system of levelers is a fourth-order system, which needs a fifth-order ESO. Two third-order state observers are arranged that: the input of the ESO2 is the third output of ESO1. The outputs of the two ESO are the extension state of the controlled object. The structures and parameters of the two ESO are the same, but they are independence with each other. The state equation of ESO is:

(18)

Where:

(19)

Where:k1,k2,k3are the undetermined parameters of ESO.

Nonlinear state error feedback (NLSEF) combines the tracking signal of TD and system state estimation of ESO to be a nonlinear function, which finds the best combination of the nonlinear functions. According to its nonlinear structure, NLSEF provides the control strategy and control signal of the controlled object. The equations of NLSEF is:

(20)

Where:eiis an undetermined parameter of NLSEF.

According to the above analysis of the ADRC with disturbance feedforward compensation, the block diagram of it is shown in the Fig. 3.

4 Simulations

According to the mathematical model of levelers hydraulic system and its control strategy, the hybrid control system for the hydraulic leveler is simulated by the MTALAB software.

The control system of the hydraulic system of the leveler uses the position and pressure control in parallel with the controller PID, the Fig.4 shows the position and straightening force curves of one of the screw cylinders.

Fig.4 Displacement and force curves of the position and pressure hybrid control

When the hydraulic cylinder stretching out 4 mm, straightening rollers begin to contact the plate. As long as the value of straightening force reaches 4.5×106N, the control system will switch to the pressure closed-loop control system. The hybrid control system of position and pressure reflects the merits both position control and pressure control, whose transient response speed is 3 times faster than that of the position control system, and the pressure step response time is only 0.472 1 s. When the straightening rollers contact the plate, the control system still remains the high screwing down speed, which improves the working efficiency of the leveler. The switch value is decided according to the actual working conditions. While, when the load changes suddenly, the straightening force will appear concussion, which will lead to the force beyond the range of the straightening ability, breaking the rack of the leveler. Therefore, the ADRC with disturbance feedforward compensation will be used to reduce the influence of the disturbance.

The sinusoidal disturbance force and random disturbance force are used to simulate the actual working condition. Sinusoidal disturbance force stands for wave defect of the plates, and random disturbance force simulates the plate unevenness. The sine wave is 10 Hz in frequency and 0.4×103mm in amplitude treated as the signal of external disturbance force. The curve of straightening force response is shown in the Fig.5 under the sinusoidal disturbance force by using PID and ADRC respectively.

As shown in the Fig. 5, the maximum overshoot of the straightening force is 2.53% when using the PID controller, and the straightening force oscillates sharply and maintains sustainable. While, when the ADRC with disturbance feedforward compensation is used, the change of straightening force oscillates in a small range and the maximum overshoot is only 1.31%, and the pressure tracking error is less than using the PID control obviously. The system can achieve better control effect when using ADRC with disturbance feedforward compensation, because of its higher reliability and stronger anti disturbance performance.

Fig.5 Straightening force curves under the sinusoidal disturbance

The maximum peak value of the random signal is 0.4KN, which is taken as the external disturbance force. The curves of pressure response using ADRC with disturbance feedforward compensation and PID are shown in the Fig.6.

Fig.6 Straightening force curves under the random disturbance

As shown in the Fig.6, the maximum overshoot of straightening force is about 10.56% when using the PID controller. The overshoot when using the controller of ADRC with disturbance feedforward compensation is 1/20 as less as that when using the PID controller. So, the proposed system has rapid response and better robustness, the external and internal disturbance can be suppressed well.

In a world, position and pressure control system has merits such as the rapid response speed, no overshoot, and the uncertain disturbance is estimated and compensated by the control strategy of ADRC with disturbance feedforward compensation.

5 Experiment

To further verify the actual control effect of the proposed control strategy, the controller of ADRC with disturbance feedforward compensation is applied to the 11 rollers full-hydraulic experimental leveler, which is shown in the Fig.7. The parameters of servo system of hydraulic system of full-hydraulic experimental leveler are shown in the Table. 1. The experimental steel plate is AH36 with 1 000 mm in length, 750 mm in width, and 30 mm in thickness. The yielding strength of plates is less than 355 N/mm2and tensile strength is 490～620 N/mm2.

Fig.7 11 Rollers full-hydraulic leveler of laboratory

Table 1 The main performance of 11 rollers full-hydraulic leveler in the laboratory

The thickness deviation of the experimental steel plates are 0.3 mm and 0.7 mm respectively, the crown is 100 mm. The thickness deviation of plates is taken as the interference signal in the experiment. The Fig.8 and Fig.9 show the curves of pressure response and displacement response respectively.

Fig.8 The position and pressure curves with 0.3 mm plate deviation

Fig.9 The position and pressure curves with 0.7 mm plate deviation

When the plate thickness deviation is 0.3 mm, the maximum overshoot of pressure response is 3.65% using the PID controller. The curve shows vibration and the control precision is low. While, there is almost no overshoot when using the controller of ADRC with disturbance feedforward compensation.

When plate thickness deviation is 0.7 mm, the maximum overshoot of pressure response is 6.30% using the PID controller. The maximum overshoot is 2.21% using the controller of ADRC with disturbance feedforward compensation. The results indicate that the straightening force error can be kept within 0.35 kN using the ADRC with disturbance feedforward compensation. Although the precision is worse than straightening the plate with 0.03 mm thickness deviation, which is higher than using the PID control strategy. The control precision of straightening force is improved and the straightening force is maintained a constant value using ADRC with disturbance feedforward compensation. The results of the experiment verify the precision of the control strategy, and the reliability and stabilization of the new control system. This control strategy can also be applied in other electro-hydraulic force control system with a heavy load.

6 Conclusions

For the hydraulic system of the leveler, this paper puts forward a new controller of ADRC with disturbance force feedforward compensation, which can make the straightening force constant. By conducting the simulations and experiments, the following conclusions are drawn as follows:

(1) The controller of ADRC with disturbance feedforward compensation can be independent on the specific mathematical model of controlled object. The general describing of the controlled object with the exerted the extreme version disturbance can be taken as the controlled object model. The experiment results and computer simulation can be applied to the actual system directly.

(2) The external disturbance can be suppressed by disturbance feedforward compensation effectively. Based on this, ADRC reduces internal interference. The external and internal disturbance are estimated and compensated by the controller of ADRC with disturbance feedforward compensation. the straightening force is maintained a constant value under different disturbance load and the quality of production are improved obviously. The controller can also be applied to other nonlinear control system, which has important theoretical significance and practical value for other control systems.