# Research on the Control Method of Force Servo System of Liquid Hydrostatic Guide Oil Film Thickness Test Bench for Machine Tools

Authors: Xu Fang; Youmin Wang; Bo Zhu
Journal: Shock and Vibration
 (2022)

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## Abstract

The test bench was designed to solve the problem of measuring the oil film thickness of the liquid hydrostatic guideway of machine tools. And based on the fast overshoot of the classical PID control, introducing the self-antidisturbance control algorithm, this study established a mathematical model of the test bench electro-hydraulic servo control system. At the same time, this paper also designed a specific structure of the self-turbulence controller, which adopts optimal quadratic control, pole configuration, PID control, and self-turbulence control to study the control methods of the test bench electro-hydraulic servo system, respectively, and uses the SIMULINK module to establish the system simulation model of the above-mentioned four control methods of the test bench. Simulation results of the test bench control system show that the response speed of the self-turbulence control is 0.4 seconds, 0.2 seconds, and 4.4 seconds faster than PID control, pole configuration, and optimal quadratic control, respectively, and the overshoot is significantly reduced. Finally, the test bench was built to complete the experiments, and the experimental results showed that the followability of the loading force and thickness detection accuracy of the test bench were better after adding the self-turbulence control algorithm and the stability of the oil film thickness was significantly improved by the test bench electro-hydraulic servo control system.


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## Body

## 1. Introduction

In practice, the hydrostatic guide will be subject to various variable loads so that the oil film thickness is unstable, and the oil film thickness instability will have a negative effect on the oil film stiffness and bearing capacity [1]. Maintaining the oil film thickness in ideal conditions is an important factor in improving the accuracy of the guide rail [2]. The traditional machine tool liquid hydrostatic guide oil film thickness detection test bench mainly uses fixed-point detection to detect the oil film thickness. It cannot simulate the real-time changes of cutting force in machine tool processing, so the detection obtained is neither real-time nor has automatic detection capability. In addition, the hydraulic control system of the test bench is mainly quadratic optimal control, pole configuration, PID control, etc., which can no longer meet the requirements of the existing test bench to detect the oil film thickness. Therefore, it is necessary to propose a new test bench hydraulic control system for detecting oil film thickness test bench.Wu [3] and others have studied the state stochastic linear quadratic optimal control and proposed that the optimal control strategy is the segmental affine function of the system state. Dario [4] proposed a new method of inverse resonance assignment and regional pole configuration for linear time-invariant vibration systems. Although many scholars have used quadratic optimal control and pole configuration to solve relevant practical engineering problems, they have not yet introduced this control method into the design of hydraulic guideways. Yang [5] and Ji-min [6] mainly used classical PID control of the hydraulic system to meet the usage requirements. However, this control has a relatively poor anti-interference capability. Since the electro-hydraulic servo system model has indeterminacy for some more important equipment, the traditional control algorithm no longer meets the requirements of the system use [7]. Guo [8] used anti-interference control to achieve accurate control of the servo system. This control has been studied academically in several aspects, but only at the theoretical level, and has not yet been applied in practice. Goforth et al. [9] and Wang et al. [10]. used a self-anti-disturbance control algorithm to optimize the system performance, which comes with a specific structure and error feedback mechanism for the control technique. After that, Jingqing Han found that the integrator series type structure of the control system was available for both linear and nonlinear systems in the corresponding feedback case and pointed out that the feedback can be taken to identify the system, and they regarded the system function as state feedback and proposed the establishment method of the expansion state observer [11]. In summary, on the basis of the testbed control model establishment, many intelligent algorithms have emerged, and the key to their performance is whether they can suppress disturbances well.In this paper, in order to study the optimal control method of the test bench for liquid hydrostatic guideway oil film thickness inspection of machine tools, the general scheme design of the test bench was designed and the three-dimensional model of the test bench was established by NX. Based on the hydraulic control valve flow equation, the cylinder flow continuity equation, and the force balance equation between the cylinder and the load, this study obtained the mathematical model of the test bench control system and simulated the system. This paper used quadratic optimal control, pole configuration, PID control, and self-anti-disturbance control to optimize the electro-hydraulic servo system of the test bench, respectively. The SIMULINK module was also used to simulate the test bench control system, and the response curves of the four control strategies were obtained. The results showed that the response speed of the self-turbulence control is 0.4 sec, 0.2 sec, and 4.4 sec faster than that of the PID control, pole configuration, and quadratic optimal control, respectively, and the overshoot is significantly reduced. Finally, this paper built a test bench to complete the experiment, and the experimental results show that the anti-interference ability of the test bench electro-hydraulic force servo control system has been improved after adding the self-antidisturbance control algorithm, at which time the test bench loading force following and thickness detection accuracy is the highest, which also leads to a significant improvement in the oil film thickness stability.


## 2. Overall Design of the Test Bench

This paper designs a test bench for detecting oil film thickness to study the optimal control method for the test bench for detecting oil film thickness of liquid hydrostatic guideways of machine tools. The overall scheme of this test bench mainly includes the design of the mechanical structure and the design of the hydraulic control system.


### 2.1. Mechanical Structure Design

The mechanical structure of the oil film thickness inspection test bench is mainly divided into (1) a horizontal moving module and (2) a vertical moving module, which is used to simulate the variation of the cutting force of the machine tool [3], and the three-dimensional model of the test bench mechanical structure is shown in Figure 1. Lab table rails have certain errors in their directions of width and length when they are designed. In addition, when the hydraulic guide consists of opposite pads, the oil film stiffness plays a crucial role in the error averaging effect [12].Figure 1
3D model of the mechanical structure of the test bench.The horizontal movement module consists of an electric motor, a first coupling, a first ball screw, a dynamic guide, a static guide fixed to the base, and left and right bearing supports assembled on both ends of the static guide. When the first motor is energized, the dynamic guide can make horizontal reciprocating movements along with the static guide.The vertical movement module consists of a handle, a handle bracket, a second ball screw, a first slider, a second slider, a linkage, a column, and a hydraulic cylinder. The second slider is assembled on the column and can move reciprocally along the column groove, and it is connected to the first slider by a connecting rod.


### 2.2. Design of the Hydraulic Control System of the Test Bench

The hydraulic control system design of the test stand contains (a) an electro-hydraulic servo control system (b) and an oil film thickness detection system. The schematic diagram of the hydraulic control system is shown in Figure2.Figure 2
Principle diagram of hydraulic control system of test bench. 1: tank; 2: accumulator; 3: relief valve; 4: electro-hydraulic servo valve; 5: hydraulic cylinder; 6: check valve; 7: quantitative pump; 8: filter; 9: motor; 10: pressure sensing; 11: throttle valve.
(a)(b)The size of the cutting force of the machine is simulated by the electro-hydraulic force servo control system of the test bench. When the electro-hydraulic servo control circuit works, this study takes the spool opening and direction to adjust the piston rod load and the cutting force of the guide [13]. The oil film thickness detection system is connected to the computer through the pressure sensor, and the measurement data are processed by the computer. According to the schematic diagram of the hydraulic control system of the test bench, after calculating the selection of each component in the hydraulic system, the hydraulic control system components of the test bench model are as listed in Table 1. Simulation of hydraulic control systems using AMESim (Advanced Modeling Environment for performing Simulations of engineering systems) is used to continuously optimize the dynamic performance of the system by adjusting the parameters of each component, thus improving the efficiency of the equipment [14, 15]. The system is set at a rated pressure of 6 MPa, a nominal pressure of 10 KN, and a piston rod stroke of 100 mm left and right.Table 1
Hydraulic control system component models of the test bench.
Hydraulic component nameHydraulic component parametersHydraulic component modelsHydraulic cylinderD = 63 mm d = 35 mm—Hydraulic pumps—CBGF1018 gear pumpsElectric motorsPower rating 5.5kwY2-132S-4Electro-hydraulic servo valveRated flow rate 204WSE2EM6-2XOne-way valveRated flow rate 30RVP10-30BServo amplifier—NB2000Relief valveRated flow rate 30 L/minDBDS6G10/200Pressure sensorsMeasurement range 0∼1000 kg, voltage 0∼10 VSeamless steel pipe: inner diameter 16 mm, wall thickness 2 mm


## 2.1. Mechanical Structure Design

The mechanical structure of the oil film thickness inspection test bench is mainly divided into (1) a horizontal moving module and (2) a vertical moving module, which is used to simulate the variation of the cutting force of the machine tool [3], and the three-dimensional model of the test bench mechanical structure is shown in Figure 1. Lab table rails have certain errors in their directions of width and length when they are designed. In addition, when the hydraulic guide consists of opposite pads, the oil film stiffness plays a crucial role in the error averaging effect [12].Figure 1
3D model of the mechanical structure of the test bench.The horizontal movement module consists of an electric motor, a first coupling, a first ball screw, a dynamic guide, a static guide fixed to the base, and left and right bearing supports assembled on both ends of the static guide. When the first motor is energized, the dynamic guide can make horizontal reciprocating movements along with the static guide.The vertical movement module consists of a handle, a handle bracket, a second ball screw, a first slider, a second slider, a linkage, a column, and a hydraulic cylinder. The second slider is assembled on the column and can move reciprocally along the column groove, and it is connected to the first slider by a connecting rod.


## 2.2. Design of the Hydraulic Control System of the Test Bench

The hydraulic control system design of the test stand contains (a) an electro-hydraulic servo control system (b) and an oil film thickness detection system. The schematic diagram of the hydraulic control system is shown in Figure2.Figure 2
Principle diagram of hydraulic control system of test bench. 1: tank; 2: accumulator; 3: relief valve; 4: electro-hydraulic servo valve; 5: hydraulic cylinder; 6: check valve; 7: quantitative pump; 8: filter; 9: motor; 10: pressure sensing; 11: throttle valve.
(a)(b)The size of the cutting force of the machine is simulated by the electro-hydraulic force servo control system of the test bench. When the electro-hydraulic servo control circuit works, this study takes the spool opening and direction to adjust the piston rod load and the cutting force of the guide [13]. The oil film thickness detection system is connected to the computer through the pressure sensor, and the measurement data are processed by the computer. According to the schematic diagram of the hydraulic control system of the test bench, after calculating the selection of each component in the hydraulic system, the hydraulic control system components of the test bench model are as listed in Table 1. Simulation of hydraulic control systems using AMESim (Advanced Modeling Environment for performing Simulations of engineering systems) is used to continuously optimize the dynamic performance of the system by adjusting the parameters of each component, thus improving the efficiency of the equipment [14, 15]. The system is set at a rated pressure of 6 MPa, a nominal pressure of 10 KN, and a piston rod stroke of 100 mm left and right.Table 1
Hydraulic control system component models of the test bench.
Hydraulic component nameHydraulic component parametersHydraulic component modelsHydraulic cylinderD = 63 mm d = 35 mm—Hydraulic pumps—CBGF1018 gear pumpsElectric motorsPower rating 5.5kwY2-132S-4Electro-hydraulic servo valveRated flow rate 204WSE2EM6-2XOne-way valveRated flow rate 30RVP10-30BServo amplifier—NB2000Relief valveRated flow rate 30 L/minDBDS6G10/200Pressure sensorsMeasurement range 0∼1000 kg, voltage 0∼10 VSeamless steel pipe: inner diameter 16 mm, wall thickness 2 mm


## 3. Modeling and Simulation of the Electro-Hydraulic Servo Control System of the Test Bench

In this paper, to get the mathematical model of the control system of the test bench, the hydraulic control valve flow equation, the cylinder flow continuity equation, and the force balance equation between the cylinder and the load are established, respectively, combined with the control system deviation voltage signal equation and the feedback link pressure sensor equation to get the transfer function of the system, and finally the simulation analysis of the system.


### 3.1. Modeling of the Electro-Hydraulic Servo Control System of the Test Bench

The electro-hydraulic force servo control is used to simulate the cutting force during the actual operation of the machine tool and add variable load to the dynamic guide. The schematic diagram of the electro-hydraulic servo control system of the test bench is shown in Figure3. The block diagram of the machine’s hydrostatic guide control system is shown in Figure 4.Figure 3
Electro-hydraulic servo control system schematic.Figure 4
Block diagram of the machine’s hydrostatic guide control system.This paper establishes the hydraulic control valve flow equation, the cylinder flow continuity equation, and the force balance equation between the cylinder and the load, respectively.(1)qL=Kqxv−KcpL,(2)qL=Apdxpdt+CtppL+Vt4βedpLdt,(3)AppL=mtd2xpdt2+Bpdxpdt+Kxp.In the formula,qL is the load flow (m3/s); Kp is the flow gain coefficient (m3/s); xv is the valve spool displacement (m); Kc is the low-pressure gain coefficient (m5/N∙s); PL is the load pressure (Pa); Ap is the effective working area of hydraulic cylinder (m2); xp is the hydraulic rod displacement (m); Ctp is the total leakage coefficient of hydraulic cylinder (m5/N∙s); Vt is the total compressed volume (m3); βe is the effective bulk modulus of elasticity (Pa); mt is the load quality (kg); Bp is the load damping factor (N∙s/m); and K is the load spring stiffness (N/m).The Laplace transform of equations (1)–(3) yields(4)QL=KqXv−KcPL,QL=ApsXp+CtpPL+Vt4βesPL,ApPL=mts2Xp+BpsXp+KXp.This paper establishes the control system deviation voltage signal equation and feedback link pressure sensor equation, respectively.(5)Ue=Ur−Uf,(6)Uf=KfFFg.In the formula,Ur is the input voltage signal (V); Uf is the feedback voltage signal (V); kfF is the pressure sensor gain (V/N); and Fg is the output force of the cylinder (N).The output current of the amplifier, considering only the quiescent performance, is(7)ΔI=KaUe.In the formula,Ka is the servo amplifier gain (A/V).According to the above-given equation, the electro-hydraulic servo valve transfer function is(8)XvΔI=KsvGsvs.In the formula,Xv is valve spool displacement (m); Ksv is hydraulic valve gain (m3/s∙A); and Gsv (s) is Ksv = 1.The block diagram of the system is shown in Figure5 based on equations (4)–(8). Here, Kce = Kc + Ctp.Figure 5
Block diagram of the control system.Servo valves consider dynamic performance. The system model is complex with fifth-order, fourth-order, and third-order functions, because the electro-hydraulic servo valve fast response generally only considers the static performance directly as a proportional link. According to the stage of valve spool displacement to cylinder output force in Figure5 to simplify the control system block diagram, the system mathematical model is simplified as(9)Gs=KaKsvKq/KceApKfFs2/wm2+1s/wr+1s2/w02+2ξ0/w0s+1.The parameter assignment and simplification of equation (9) give the transfer function(10)Gs=4.07×10−3s2+3.255.87×10−5s3+5.75×10−3s2+3.50s+1.


### 3.2. Simulation of the Electro-Hydraulic Servo Control System of the Test Bench

Simulation of the system transfer function using SIMULINK yields the system step response curve shown in Figure6.Figure 6
System step response curve.As can be seen from Figure6, the control performance of the system is very unsatisfactory. When the input unit step signal is applied, the system can be stabilized in about 20.5 s, which is a long time, and its input and output signals do not match, which needs to be optimized.


## 3.1. Modeling of the Electro-Hydraulic Servo Control System of the Test Bench

The electro-hydraulic force servo control is used to simulate the cutting force during the actual operation of the machine tool and add variable load to the dynamic guide. The schematic diagram of the electro-hydraulic servo control system of the test bench is shown in Figure3. The block diagram of the machine’s hydrostatic guide control system is shown in Figure 4.Figure 3
Electro-hydraulic servo control system schematic.Figure 4
Block diagram of the machine’s hydrostatic guide control system.This paper establishes the hydraulic control valve flow equation, the cylinder flow continuity equation, and the force balance equation between the cylinder and the load, respectively.(1)qL=Kqxv−KcpL,(2)qL=Apdxpdt+CtppL+Vt4βedpLdt,(3)AppL=mtd2xpdt2+Bpdxpdt+Kxp.In the formula,qL is the load flow (m3/s); Kp is the flow gain coefficient (m3/s); xv is the valve spool displacement (m); Kc is the low-pressure gain coefficient (m5/N∙s); PL is the load pressure (Pa); Ap is the effective working area of hydraulic cylinder (m2); xp is the hydraulic rod displacement (m); Ctp is the total leakage coefficient of hydraulic cylinder (m5/N∙s); Vt is the total compressed volume (m3); βe is the effective bulk modulus of elasticity (Pa); mt is the load quality (kg); Bp is the load damping factor (N∙s/m); and K is the load spring stiffness (N/m).The Laplace transform of equations (1)–(3) yields(4)QL=KqXv−KcPL,QL=ApsXp+CtpPL+Vt4βesPL,ApPL=mts2Xp+BpsXp+KXp.This paper establishes the control system deviation voltage signal equation and feedback link pressure sensor equation, respectively.(5)Ue=Ur−Uf,(6)Uf=KfFFg.In the formula,Ur is the input voltage signal (V); Uf is the feedback voltage signal (V); kfF is the pressure sensor gain (V/N); and Fg is the output force of the cylinder (N).The output current of the amplifier, considering only the quiescent performance, is(7)ΔI=KaUe.In the formula,Ka is the servo amplifier gain (A/V).According to the above-given equation, the electro-hydraulic servo valve transfer function is(8)XvΔI=KsvGsvs.In the formula,Xv is valve spool displacement (m); Ksv is hydraulic valve gain (m3/s∙A); and Gsv (s) is Ksv = 1.The block diagram of the system is shown in Figure5 based on equations (4)–(8). Here, Kce = Kc + Ctp.Figure 5
Block diagram of the control system.Servo valves consider dynamic performance. The system model is complex with fifth-order, fourth-order, and third-order functions, because the electro-hydraulic servo valve fast response generally only considers the static performance directly as a proportional link. According to the stage of valve spool displacement to cylinder output force in Figure5 to simplify the control system block diagram, the system mathematical model is simplified as(9)Gs=KaKsvKq/KceApKfFs2/wm2+1s/wr+1s2/w02+2ξ0/w0s+1.The parameter assignment and simplification of equation (9) give the transfer function(10)Gs=4.07×10−3s2+3.255.87×10−5s3+5.75×10−3s2+3.50s+1.


## 3.2. Simulation of the Electro-Hydraulic Servo Control System of the Test Bench

Simulation of the system transfer function using SIMULINK yields the system step response curve shown in Figure6.Figure 6
System step response curve.As can be seen from Figure6, the control performance of the system is very unsatisfactory. When the input unit step signal is applied, the system can be stabilized in about 20.5 s, which is a long time, and its input and output signals do not match, which needs to be optimized.


## 4. Test Bench Electro-Hydraulic Servo Control System Optimization Method Comparison Analysis

The article is based on the comprehensive optimization of classical PID control based on the self-turbulence control algorithm. This study designs the specific structure of the self-turbulence controller and uses quadratic optimal control, pole configuration, PID control, and self-turbulence control to optimize the electro-hydraulic servo system of the test bench, respectively. Finally, the SIMULINK module is used to simulate the test bench control system and obtain the response curves under the four control strategies.


### 4.1. Linear Quadratic Optimal Control

The principle of optimal control is to find a control variableu(t) that has a small value and can satisfy the minimum system error x(t), so that the system output quantity follows the input quantity quickly and with low energy consumption [16]. According to Pontryagin’s principle, to be able to obtain the index minimal value, the essence of quadratic optimal control is to approximate the feedback K(t) added to the original system. In this paper, the feedback of the optimal regulator is used to approximate the optimization, then the simplified optimal control rate is(11)ut=−Ktxt=R−1tBTtPtxt.The algebraic equation of the Riccati matrix is(12)PA+ATP−PBR−1BTP+Q=0.In the formula,A is the system matrix; B is the control matrix; Q-R is the weighted matrix; and P is the equation solution.The optimal weight matrix selected in this paper isQ = diag{275000,1,1} and R = 0.0001.The optimized system matrix is(13)A−BK=010001−55138−59727−141.Using MATLAB to simulate the transfer function under quadratic optimal control, the step response curve of the system after linear quadratic optimal control is as shown in Figure7.Figure 7
Step response curve of the system after quadratic optimal control.As can be seen from Figure7, the system reaches stability at 5.6 s. After the optimization of the oil film thickness control system by quadratic optimal control, the system stabilization time is shortened by 14.9 s. The control performance is greatly improved, but the system reaches its stabilization time is longer.


### 4.2. Pole Optimization Configuration

After adding the state feedback gain matrixK = [k1 k2 k3], the characteristic polynomial is(14)αλ=detλI−A+bK=λ000λ000λ−010001−17036−59625−98+000000k1k2k3=λ3+k3+98λ2+k2+59625λ+k1+17036.The key to controlling system performance is determined by the dominant pole, while the role of the far pole is negligible [17]. Therefore, the testbed control system can be equivalently replaced by a second-order control system containing the dominant pole pair.From the dynamic indicatorsσp% and ts to determine the location of the expected dominant pole,(15)σp%=e−πξ/1−ξ2×100%.In the formula,σp% is the maximum overshoot and ξ is the damping (0 < ξ < 1).Let the allowable error of the control system be 5%; then,(16)ts=3ξωn.In the formula,ts is the adjustment time (s) and ωn is the undamped frequency (rad/s).The expected dominant pole of the system is(17)λ1,2=−ξωn±jωn1−ξ2.In the formula,λ1,2 are the expectations that dominate the poles.Because of the maximum overshoot of the systemσp% ≤5%, adjustment time ts ≤ 0.5 s; therefore, the expected dominant pole of the control system is obtained from the following equation with λ1 and λ2.(18)σp%=e−πξ/1−ξ2×100%≤5%,ts=3ξωn≤0.5.The solution is(19)ξ≥0.69,ξωn≥6.Takingξ = 0.707 and ξωn = 6, then(20)λ1,2=−ξωn±jωn1−ξ2=−6±j6.Make the third poleλ3 = 10Re[λ1] = −60. Therefore, there isα∗s=s+60s+6−j6s+6+j6=s3+72s2+792s+4320. By αs=α∗s, we obtain(21)K=k1k2k3=−26−58833−12716.The step response curve of the system after the pole configuration is obtained using SIMULINK module simulation as shown in Figure8.Figure 8
Step response curve of the system after pole configuration.According to Figure8, it can be seen that the system reached stability at 1.4 seconds. After the oil film thickness control system was optimized by the pole configuration, the stabilization time was shortened by 19.1 seconds, and the control performance was significantly improved, but the control system showed an overshoot phenomenon.


### 4.3. PID Control

The classical PID control has been widely used because of its simple structure and easy parameter positive determination [18, 19]. PID control starts by determining the proportionality factor Kp. Let the integral and differential constants be 0. Gradually increase the proportional gain from zero until the response curve oscillates and then slowly decrease the data until the oscillation stops. Take 30%–70% of the proportional gain as the system proportionality factor. Here, 60% is taken. After the proportionality factor is decided, the larger value is chosen as the initial value of the integration constant Ki of the system. By repeatedly adjusting the system to stop the oscillation, 150%–180% of the integration constant is taken as the final value of the integration constant of the system, which is 165% at this time. In the case of the system output force, the differential constant Kd is usually selected as 0. Finally, the parameters of the PID controller are selected as Kp = 11.17, Ki = 63.11, and Kd = 0.The PID control step response curve of the system using SIMULINK simulation is shown in Figure9.Figure 9
System PID control step response curve.As can be seen from Figure9, the system performance is improved somewhat by using PID control. The system can reach stability at 1.6 s when a unit step signal is an input, and its input and output signals match, but the system is accompanied by a certain amount of overshoot.


### 4.4. Self-Antidisturbance Control

In order to further improve the control effect of the test bench control system, a self-antidisturbance control algorithm is introduced based on the fast overshoot of the PID control [20], and the specific structure of the self-antidisturbance controller is analyzed and designed. The block diagram of the self-antidisturbance control designed in this paper is shown in Figure 10.Figure 10
Control block diagram of the third-order self-tampering controller.


#### 4.4.1. Analysis of Self-Tampering Controllers

The self-resistant controller consists of three modules, the tracking differentiator (TD), the expanded state observer (ESO), and the control rate (CL). What can be learned from the system state space expression is that the controlled object is in the third order, so the tracking differentiator is third order and the dilated state observer is fourth-order.To improve the dynamic characteristics of this energy control system, the required poles are configured using state feedback. If the zero-point configuration of the control system can be achieved, the dynamic performance of the control system can be better improved. However, the zero-point configuration requires the use of differential signals, and the general differential cannot filter the noise well and cannot be used for zero-point configuration, while the tracking differential can filter the noise well and its differential signal can be used for zero-point configuration [21].The tracking differentiator follows the given signal and obtains its differential equation. Arranging the transition process through the tracking differentiator prevents overshoot due to sudden changes in the given signal and makes the controlled object approach the target smoothly, thus improving the stability of the control system. By eliminating the poles of the system with the configured zeros, the system can be viewed as a control system approximating 1. Let the string-level object be(22)y=1p1s1p2su.Assume thatP2(s) is known and let the virtual control volume be(23)Ut=1p2su.Then, the control system becomes(24)y=1p1sUt.If the virtual control quantityU(t) is determined, the actual control quantity is(25)u=p2sUt.In the formula,P1(s) and P2(s) are polynomials.The virtual control volumeU(t) is determined by applying a self-turbulent controller design approach to the control subsystem y = U(t)/P1(s), which thus allows for a certain range of uncertainties to exist.Let the mathematical model of the controlled object contain zeros and poles. Its general form is(26)y=qspsu.Let the virtual control volumeU(t) = q(s)u; then, the system becomes y = U(t)/P(s); hence, the actual control volume becomes u = U(t)/q(s).If the controlled object conforms to the minimum phase system, thenq(s) belongs to a stable polynomial and is known. At this point, a self-antidisturbance controller can be used to determine the virtual control quantity U(t), and then the problem can be solved by considering U(t) as the input and solving the system u = U(t)/q(s) to obtain the actual control quantity u(t).Based on the above-mentioned analysis, the zero-pole diagram of the controlled object is derived using MATLAB as shown in Figure11.Figure 11
Zero-pole diagram of the controlled object.From Figure11, it can be seen that the system model has no zeros and poles in the positive part of the complex plane, so it belongs to the minimum phase system. In this case, it is possible to determine the virtual control quantity U(t) by a self-antidisturbance controller, and then the U(t) is considered as an input to solve the system u = U(t)/q(s) to obtain the actual control quantity.From the control system transfer function is known to satisfy the cubic polynomialp3(s) = s3 + a3s2 + a2s + a1, and at this time, the parameters a1, a2, and a3 are known; the algorithm of the third-order tracking differentiator can be obtained as(27)fs=−rrrv1−vt+3v2+3v3,v1=v1+hv2,v2=v2+hv3,v3=v3+hfs,Ut=v1t.In the formula,r is the fast factor and h is the sampling step size.The expansive state observer follows the control system output, observes in real time the state conditions of each order of the object of study and the total system disturbance, and compensates for the disturbance accordingly [22]. If the system is subject to unknown disturbances, the control object is uncertain [23, 24]. The uncertain objects are(28)xn=fx,x•,⋅⋅⋅,xn−1,t+wt+bu,y=xt.In the formula,f(x, x′,...,x(n−1), t) is the system unknown function; ω(t) is the unknown perturbation; u, y are system inputs and outputs; and b is the control gain.The dilation state observer in this paper is fourth-order, and under linear conditions, the control system fourth-order dilation state observer algorithm is(29)e=z1−y,z1=z1+hz2−β01e,z2=z2+hz3−β02e,z3=z3+hz4−β03e+bu,z4=z4+h−β04e.In the formula,β01, β02, β03, and β04 are expansion state observer parameters and e is error term. In the control system, the ESO contains four output variables, z1 following the system output y, z2 following y′, z3 following y″, and z4 following the system integrated disturbance, using feed-forward methods to compensate for the disturbance [23].The transfer relation from input to the output of the expansion state observer is(30)z1=w1sy=β01s3+β02s2+β03s+β04s4+β01s3+β02s2+β03s+β04y,z2=w2sy=β02s3+β03s2+β04ss4+β01s3+β02s2+β03s+β04y,z3=w3sy=β03s3+β04s2s4+β01s3+β02s2+β03s+β04y,z4=w4sy=β04s3s4+β01s3+β02s2+β03s+β04y.In order to achieve a certain estimation accuracy, it is necessary to choose larger gain coefficients, gain coefficientsβ01, β02, β03, and β04 which need larger values so as to satisfy the high gain observer mode.The error feedback control rate compares the state variables obtained from the transition process, each order of differentiation, and the dilated state observer for a given signal of the system, and then the deviation signal is algorithmically derived from the control quantity. The algorithm is(31)e1=v1−z1,e2=v2−z2,e3=v3−z3,u0=β11e1+β12e2+β13e3.In the formula,β11 is the control rate parameter and ei = vi−zi is the state error.So the algorithm for the control volume is(32)u=u0−z4b.


#### 4.4.2. Creation of the Self-Tampering Controller Subsystem

When the controller used for system optimization is complex, a subsystem is used to split the complex self-resistant controller module into a tracking differentiator submodule, an expansion state observer submodule, and an error feedback control rate submodule, thus making the complex self-resistant controller seen more clearly and simply.The tracking differentiator subsystem is created, where the tracking differentiator subsystem module is specifically set with the following settings: function name TD3_L; function parameters arer, h; function module is TD3_L; and the TD3_L. m program is added inside Edit.In the TD3_L subsystem packaging process, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“TD3_L”); Prompt in the Parameters property; the modular composition of the tracking differentiator subsystem is shown in Figure12, and the TD3_L subsystem module after the package is completed is shown in Figure 13.Figure 12
Modular composition of the tracking differentiator subsystem.Figure 13
The TD3_L subsystem module after packaging.We created the dilated state observer subsystem, where the tracking differentiator subsystem module is specifically set with the following settings: function name ESO4_L; function parameters Beta01, Beta02, Beta03, and Beta04; function module ESO4_L. We added the ESO4_L. m program inside Edit.In the ESO4_L subsystem wrapping process, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“ESO4_L”). Parameters property in the Parameters property, prompt and variable are set to Beta01, Beta02, Beta03, and Beta04, and their types are edited; the mask type in the documentation property is set to ESO4_L.The modular composition of the expansion state observer subsystem is shown in Figure14, and the completed ESO4_L subsystem module of the package is shown in Figure 15.Figure 14
Modular composition of the ESO subsystem.Figure 15
Packaged ESO4_L subsystem module.The error feedback control rate subsystem is created, where the specific settings of the tracking differentiator subsystem module are function name CL3_L; function parameters are Beta11, Beta12, and Beta13; function module is CL3_L; and the CL3_L. m program is added inside the edit.During the packaging of the CL3_L subsystem, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“CL3_L”); Prompt in the Parameters property and the modular composition of the error feedback control rate subsystem is shown in Figure16, and the completed CL3_L subsystem module is shown in Figure 17.Figure 16
Modular composition of the CL subsystem.Figure 17
Packaged CL3_L subsystem module.


#### 4.4.3. Simulation of Self-Anti-Disturbance Control of the Test Bench Electro-Hydraulic Servo Control System

Combined with the SIMULINK module, we establish the control system self-anti-disturbance control simulation model, as shown in Figure18.Figure 18
Simulation model of control system self-immunity control.The parameterr in the tracking differentiator has the greatest impact on its performance. r too large or too small will change the time of arranging the transition process and the amount of overshoot of the system, and the tracking effect of the system will become poor. The specifics of the change in the performance of the control system with the parameter r are shown in Figures 19 and 20, respectively.Figure 19
Response curve at 2 times r.Figure 20
Response curve at 0.1 times r.The expansion state observer parametersβ01, β02, β03, and β04 have an impact on its dynamic characteristics. When β01 increases, the system does not have much effect; when β01 decreases, the system will have divergent oscillations. Usually, the value of β01 is of the same order of magnitude as the reciprocal of the sampling time h. The specifics of how the performance of the control system changes with parameter β01 are shown in Figures 21 and 22, respectively.Figure 21
Response curve at 10 timesβ01.Figure 22
Response curve at 0.1 timesβ01.Whenβ02 becomes large, the system emits a noise signal, resulting in reduced system control performance; when β02 becomes small, the system will increase the number of oscillations, and the amplitude will increase. The details of the change in the performance of the control system with parameter β02 are shown in Figures 23 and 24, respectively.Figure 23
Response curve at 10 timesβ02.Figure 24
Response curve at 0.1 timesβ02.Whenβ03 becomes large, it increases the number of oscillations and the amplitude becomes larger. As β03 becomes smaller, the system tracking time becomes longer and the oscillation amplitude gradually becomes smaller. The details of the change in the performance of the control system with parameter β03 are shown in Figures 25 and 26, respectively.Figure 25
Response curve at 10 timesβ03.Figure 26
Response curve at 0.1 timesβ03.Whenβ04 becomes large, the system oscillation amplitude becomes larger and the tracking speed slows down; when β04 decreases, the tracking speed of the system slows down substantially and can even cause phase lag. The details of the change in the performance of the control system with parameter β04 are shown in Figures 27 and 28, respectively.Figure 27
Response curve at 10 timesβ04.Figure 28
Response curve at 0.1 timesβ04.The parametersβ11,β12, and β13 of the error feedback control rate have an impact on its performance. When β11 becomes large, the system overshoot increases, making its control quality lower; when β11 becomes small, the system response is lower than the given signal and the tracking effect is poor. The details of the change in the performance of the control system with parameter β11 are shown in Figures 29 and 30, respectively.Figure 29
Response curve at 10 timesβ11.Figure 30
Response curve at 0.1 timesβ11.Whenβ12 becomes large, the response is lower than the given signal and the tracking effect is poor; When β12 becomes small, the system oscillates and the amplitude decreases slowly. The details of the change in the performance of the control system with parameter β12 are shown in Figures 31 and 32, respectively.Figure 31
Response curve at 2 timesβ12.Figure 32
Response curve at 0.1 timesβ12.Whenβ13 becomes large, the system responds well; when β13 is reduced, the system is slightly overshoot. The specifics of how the performance of the control system changes with parameter β13 are shown in Figures 33 and 34, respectively.Figure 33
Response curve at 10 timesβ13.Figure 34
Response curve at 0.05 timesβ13.Setting the parameters of the three modules of the controller separately: TD:h = 0.1; ESO: β01 = 10, β02 = 680, β03 = 6900, β04 = 22500; CL: β11 = 20, β12 = 70, β13 = 1, r = 20.By SIMULINK simulation, the control system self-antidisturbance control step response curve is shown in Figure35.Figure 35
Self-antidisturbance control step response curve of control system.As can be seen from Figure35, the servo system self-turbulence control performance is significantly improved. The control system is stable at 1.2 s when a unit step signal is an input and its input and output signals match. As a result, the performance of the servo system is greatly improved after the self-turbulence optimization.


### 4.5. Simulation Comparison of Electro-Hydraulic Servo Control System of Test Bench

To compare the performance advantages and disadvantages of classical PID control, quadratic optimal control, pole configuration, and self-antidisturbance control, SIMULINK is used to build the control system simulation model of the above four optimal control methods, as shown in Figure36, and the step response curve of the unit step signal under the four control strategies is derived as shown in Figure 37.Figure 36
Simulation model of control system based on four control strategies.Figure 37
Response curve of control system based on four control strategies.From Figure37, it can be seen that the system has the most stable performance after self-anti-disturbance control. The response speed of PID control is 0.4 seconds slower than that of self-tampering control, and the response is accompanied by a large amount of overshoot; the response time of the pole configuration is 0.2 seconds longer than that of the self-tampering control, and although the overshoot is significantly reduced compared to the PID control, there is still overshoot; neither the secondary type of optimal control nor the self-rejecting control showed overshoot, but the response time of the self-rejecting control was shorter than its response time by 4.4 seconds.


## 4.1. Linear Quadratic Optimal Control

The principle of optimal control is to find a control variableu(t) that has a small value and can satisfy the minimum system error x(t), so that the system output quantity follows the input quantity quickly and with low energy consumption [16]. According to Pontryagin’s principle, to be able to obtain the index minimal value, the essence of quadratic optimal control is to approximate the feedback K(t) added to the original system. In this paper, the feedback of the optimal regulator is used to approximate the optimization, then the simplified optimal control rate is(11)ut=−Ktxt=R−1tBTtPtxt.The algebraic equation of the Riccati matrix is(12)PA+ATP−PBR−1BTP+Q=0.In the formula,A is the system matrix; B is the control matrix; Q-R is the weighted matrix; and P is the equation solution.The optimal weight matrix selected in this paper isQ = diag{275000,1,1} and R = 0.0001.The optimized system matrix is(13)A−BK=010001−55138−59727−141.Using MATLAB to simulate the transfer function under quadratic optimal control, the step response curve of the system after linear quadratic optimal control is as shown in Figure7.Figure 7
Step response curve of the system after quadratic optimal control.As can be seen from Figure7, the system reaches stability at 5.6 s. After the optimization of the oil film thickness control system by quadratic optimal control, the system stabilization time is shortened by 14.9 s. The control performance is greatly improved, but the system reaches its stabilization time is longer.


## 4.2. Pole Optimization Configuration

After adding the state feedback gain matrixK = [k1 k2 k3], the characteristic polynomial is(14)αλ=detλI−A+bK=λ000λ000λ−010001−17036−59625−98+000000k1k2k3=λ3+k3+98λ2+k2+59625λ+k1+17036.The key to controlling system performance is determined by the dominant pole, while the role of the far pole is negligible [17]. Therefore, the testbed control system can be equivalently replaced by a second-order control system containing the dominant pole pair.From the dynamic indicatorsσp% and ts to determine the location of the expected dominant pole,(15)σp%=e−πξ/1−ξ2×100%.In the formula,σp% is the maximum overshoot and ξ is the damping (0 < ξ < 1).Let the allowable error of the control system be 5%; then,(16)ts=3ξωn.In the formula,ts is the adjustment time (s) and ωn is the undamped frequency (rad/s).The expected dominant pole of the system is(17)λ1,2=−ξωn±jωn1−ξ2.In the formula,λ1,2 are the expectations that dominate the poles.Because of the maximum overshoot of the systemσp% ≤5%, adjustment time ts ≤ 0.5 s; therefore, the expected dominant pole of the control system is obtained from the following equation with λ1 and λ2.(18)σp%=e−πξ/1−ξ2×100%≤5%,ts=3ξωn≤0.5.The solution is(19)ξ≥0.69,ξωn≥6.Takingξ = 0.707 and ξωn = 6, then(20)λ1,2=−ξωn±jωn1−ξ2=−6±j6.Make the third poleλ3 = 10Re[λ1] = −60. Therefore, there isα∗s=s+60s+6−j6s+6+j6=s3+72s2+792s+4320. By αs=α∗s, we obtain(21)K=k1k2k3=−26−58833−12716.The step response curve of the system after the pole configuration is obtained using SIMULINK module simulation as shown in Figure8.Figure 8
Step response curve of the system after pole configuration.According to Figure8, it can be seen that the system reached stability at 1.4 seconds. After the oil film thickness control system was optimized by the pole configuration, the stabilization time was shortened by 19.1 seconds, and the control performance was significantly improved, but the control system showed an overshoot phenomenon.


## 4.3. PID Control

The classical PID control has been widely used because of its simple structure and easy parameter positive determination [18, 19]. PID control starts by determining the proportionality factor Kp. Let the integral and differential constants be 0. Gradually increase the proportional gain from zero until the response curve oscillates and then slowly decrease the data until the oscillation stops. Take 30%–70% of the proportional gain as the system proportionality factor. Here, 60% is taken. After the proportionality factor is decided, the larger value is chosen as the initial value of the integration constant Ki of the system. By repeatedly adjusting the system to stop the oscillation, 150%–180% of the integration constant is taken as the final value of the integration constant of the system, which is 165% at this time. In the case of the system output force, the differential constant Kd is usually selected as 0. Finally, the parameters of the PID controller are selected as Kp = 11.17, Ki = 63.11, and Kd = 0.The PID control step response curve of the system using SIMULINK simulation is shown in Figure9.Figure 9
System PID control step response curve.As can be seen from Figure9, the system performance is improved somewhat by using PID control. The system can reach stability at 1.6 s when a unit step signal is an input, and its input and output signals match, but the system is accompanied by a certain amount of overshoot.


## 4.4. Self-Antidisturbance Control

In order to further improve the control effect of the test bench control system, a self-antidisturbance control algorithm is introduced based on the fast overshoot of the PID control [20], and the specific structure of the self-antidisturbance controller is analyzed and designed. The block diagram of the self-antidisturbance control designed in this paper is shown in Figure 10.Figure 10
Control block diagram of the third-order self-tampering controller.


### 4.4.1. Analysis of Self-Tampering Controllers

The self-resistant controller consists of three modules, the tracking differentiator (TD), the expanded state observer (ESO), and the control rate (CL). What can be learned from the system state space expression is that the controlled object is in the third order, so the tracking differentiator is third order and the dilated state observer is fourth-order.To improve the dynamic characteristics of this energy control system, the required poles are configured using state feedback. If the zero-point configuration of the control system can be achieved, the dynamic performance of the control system can be better improved. However, the zero-point configuration requires the use of differential signals, and the general differential cannot filter the noise well and cannot be used for zero-point configuration, while the tracking differential can filter the noise well and its differential signal can be used for zero-point configuration [21].The tracking differentiator follows the given signal and obtains its differential equation. Arranging the transition process through the tracking differentiator prevents overshoot due to sudden changes in the given signal and makes the controlled object approach the target smoothly, thus improving the stability of the control system. By eliminating the poles of the system with the configured zeros, the system can be viewed as a control system approximating 1. Let the string-level object be(22)y=1p1s1p2su.Assume thatP2(s) is known and let the virtual control volume be(23)Ut=1p2su.Then, the control system becomes(24)y=1p1sUt.If the virtual control quantityU(t) is determined, the actual control quantity is(25)u=p2sUt.In the formula,P1(s) and P2(s) are polynomials.The virtual control volumeU(t) is determined by applying a self-turbulent controller design approach to the control subsystem y = U(t)/P1(s), which thus allows for a certain range of uncertainties to exist.Let the mathematical model of the controlled object contain zeros and poles. Its general form is(26)y=qspsu.Let the virtual control volumeU(t) = q(s)u; then, the system becomes y = U(t)/P(s); hence, the actual control volume becomes u = U(t)/q(s).If the controlled object conforms to the minimum phase system, thenq(s) belongs to a stable polynomial and is known. At this point, a self-antidisturbance controller can be used to determine the virtual control quantity U(t), and then the problem can be solved by considering U(t) as the input and solving the system u = U(t)/q(s) to obtain the actual control quantity u(t).Based on the above-mentioned analysis, the zero-pole diagram of the controlled object is derived using MATLAB as shown in Figure11.Figure 11
Zero-pole diagram of the controlled object.From Figure11, it can be seen that the system model has no zeros and poles in the positive part of the complex plane, so it belongs to the minimum phase system. In this case, it is possible to determine the virtual control quantity U(t) by a self-antidisturbance controller, and then the U(t) is considered as an input to solve the system u = U(t)/q(s) to obtain the actual control quantity.From the control system transfer function is known to satisfy the cubic polynomialp3(s) = s3 + a3s2 + a2s + a1, and at this time, the parameters a1, a2, and a3 are known; the algorithm of the third-order tracking differentiator can be obtained as(27)fs=−rrrv1−vt+3v2+3v3,v1=v1+hv2,v2=v2+hv3,v3=v3+hfs,Ut=v1t.In the formula,r is the fast factor and h is the sampling step size.The expansive state observer follows the control system output, observes in real time the state conditions of each order of the object of study and the total system disturbance, and compensates for the disturbance accordingly [22]. If the system is subject to unknown disturbances, the control object is uncertain [23, 24]. The uncertain objects are(28)xn=fx,x•,⋅⋅⋅,xn−1,t+wt+bu,y=xt.In the formula,f(x, x′,...,x(n−1), t) is the system unknown function; ω(t) is the unknown perturbation; u, y are system inputs and outputs; and b is the control gain.The dilation state observer in this paper is fourth-order, and under linear conditions, the control system fourth-order dilation state observer algorithm is(29)e=z1−y,z1=z1+hz2−β01e,z2=z2+hz3−β02e,z3=z3+hz4−β03e+bu,z4=z4+h−β04e.In the formula,β01, β02, β03, and β04 are expansion state observer parameters and e is error term. In the control system, the ESO contains four output variables, z1 following the system output y, z2 following y′, z3 following y″, and z4 following the system integrated disturbance, using feed-forward methods to compensate for the disturbance [23].The transfer relation from input to the output of the expansion state observer is(30)z1=w1sy=β01s3+β02s2+β03s+β04s4+β01s3+β02s2+β03s+β04y,z2=w2sy=β02s3+β03s2+β04ss4+β01s3+β02s2+β03s+β04y,z3=w3sy=β03s3+β04s2s4+β01s3+β02s2+β03s+β04y,z4=w4sy=β04s3s4+β01s3+β02s2+β03s+β04y.In order to achieve a certain estimation accuracy, it is necessary to choose larger gain coefficients, gain coefficientsβ01, β02, β03, and β04 which need larger values so as to satisfy the high gain observer mode.The error feedback control rate compares the state variables obtained from the transition process, each order of differentiation, and the dilated state observer for a given signal of the system, and then the deviation signal is algorithmically derived from the control quantity. The algorithm is(31)e1=v1−z1,e2=v2−z2,e3=v3−z3,u0=β11e1+β12e2+β13e3.In the formula,β11 is the control rate parameter and ei = vi−zi is the state error.So the algorithm for the control volume is(32)u=u0−z4b.


### 4.4.2. Creation of the Self-Tampering Controller Subsystem

When the controller used for system optimization is complex, a subsystem is used to split the complex self-resistant controller module into a tracking differentiator submodule, an expansion state observer submodule, and an error feedback control rate submodule, thus making the complex self-resistant controller seen more clearly and simply.The tracking differentiator subsystem is created, where the tracking differentiator subsystem module is specifically set with the following settings: function name TD3_L; function parameters arer, h; function module is TD3_L; and the TD3_L. m program is added inside Edit.In the TD3_L subsystem packaging process, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“TD3_L”); Prompt in the Parameters property; the modular composition of the tracking differentiator subsystem is shown in Figure12, and the TD3_L subsystem module after the package is completed is shown in Figure 13.Figure 12
Modular composition of the tracking differentiator subsystem.Figure 13
The TD3_L subsystem module after packaging.We created the dilated state observer subsystem, where the tracking differentiator subsystem module is specifically set with the following settings: function name ESO4_L; function parameters Beta01, Beta02, Beta03, and Beta04; function module ESO4_L. We added the ESO4_L. m program inside Edit.In the ESO4_L subsystem wrapping process, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“ESO4_L”). Parameters property in the Parameters property, prompt and variable are set to Beta01, Beta02, Beta03, and Beta04, and their types are edited; the mask type in the documentation property is set to ESO4_L.The modular composition of the expansion state observer subsystem is shown in Figure14, and the completed ESO4_L subsystem module of the package is shown in Figure 15.Figure 14
Modular composition of the ESO subsystem.Figure 15
Packaged ESO4_L subsystem module.The error feedback control rate subsystem is created, where the specific settings of the tracking differentiator subsystem module are function name CL3_L; function parameters are Beta11, Beta12, and Beta13; function module is CL3_L; and the CL3_L. m program is added inside the edit.During the packaging of the CL3_L subsystem, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“CL3_L”); Prompt in the Parameters property and the modular composition of the error feedback control rate subsystem is shown in Figure16, and the completed CL3_L subsystem module is shown in Figure 17.Figure 16
Modular composition of the CL subsystem.Figure 17
Packaged CL3_L subsystem module.


### 4.4.3. Simulation of Self-Anti-Disturbance Control of the Test Bench Electro-Hydraulic Servo Control System

Combined with the SIMULINK module, we establish the control system self-anti-disturbance control simulation model, as shown in Figure18.Figure 18
Simulation model of control system self-immunity control.The parameterr in the tracking differentiator has the greatest impact on its performance. r too large or too small will change the time of arranging the transition process and the amount of overshoot of the system, and the tracking effect of the system will become poor. The specifics of the change in the performance of the control system with the parameter r are shown in Figures 19 and 20, respectively.Figure 19
Response curve at 2 times r.Figure 20
Response curve at 0.1 times r.The expansion state observer parametersβ01, β02, β03, and β04 have an impact on its dynamic characteristics. When β01 increases, the system does not have much effect; when β01 decreases, the system will have divergent oscillations. Usually, the value of β01 is of the same order of magnitude as the reciprocal of the sampling time h. The specifics of how the performance of the control system changes with parameter β01 are shown in Figures 21 and 22, respectively.Figure 21
Response curve at 10 timesβ01.Figure 22
Response curve at 0.1 timesβ01.Whenβ02 becomes large, the system emits a noise signal, resulting in reduced system control performance; when β02 becomes small, the system will increase the number of oscillations, and the amplitude will increase. The details of the change in the performance of the control system with parameter β02 are shown in Figures 23 and 24, respectively.Figure 23
Response curve at 10 timesβ02.Figure 24
Response curve at 0.1 timesβ02.Whenβ03 becomes large, it increases the number of oscillations and the amplitude becomes larger. As β03 becomes smaller, the system tracking time becomes longer and the oscillation amplitude gradually becomes smaller. The details of the change in the performance of the control system with parameter β03 are shown in Figures 25 and 26, respectively.Figure 25
Response curve at 10 timesβ03.Figure 26
Response curve at 0.1 timesβ03.Whenβ04 becomes large, the system oscillation amplitude becomes larger and the tracking speed slows down; when β04 decreases, the tracking speed of the system slows down substantially and can even cause phase lag. The details of the change in the performance of the control system with parameter β04 are shown in Figures 27 and 28, respectively.Figure 27
Response curve at 10 timesβ04.Figure 28
Response curve at 0.1 timesβ04.The parametersβ11,β12, and β13 of the error feedback control rate have an impact on its performance. When β11 becomes large, the system overshoot increases, making its control quality lower; when β11 becomes small, the system response is lower than the given signal and the tracking effect is poor. The details of the change in the performance of the control system with parameter β11 are shown in Figures 29 and 30, respectively.Figure 29
Response curve at 10 timesβ11.Figure 30
Response curve at 0.1 timesβ11.Whenβ12 becomes large, the response is lower than the given signal and the tracking effect is poor; When β12 becomes small, the system oscillates and the amplitude decreases slowly. The details of the change in the performance of the control system with parameter β12 are shown in Figures 31 and 32, respectively.Figure 31
Response curve at 2 timesβ12.Figure 32
Response curve at 0.1 timesβ12.Whenβ13 becomes large, the system responds well; when β13 is reduced, the system is slightly overshoot. The specifics of how the performance of the control system changes with parameter β13 are shown in Figures 33 and 34, respectively.Figure 33
Response curve at 10 timesβ13.Figure 34
Response curve at 0.05 timesβ13.Setting the parameters of the three modules of the controller separately: TD:h = 0.1; ESO: β01 = 10, β02 = 680, β03 = 6900, β04 = 22500; CL: β11 = 20, β12 = 70, β13 = 1, r = 20.By SIMULINK simulation, the control system self-antidisturbance control step response curve is shown in Figure35.Figure 35
Self-antidisturbance control step response curve of control system.As can be seen from Figure35, the servo system self-turbulence control performance is significantly improved. The control system is stable at 1.2 s when a unit step signal is an input and its input and output signals match. As a result, the performance of the servo system is greatly improved after the self-turbulence optimization.


## 4.4.1. Analysis of Self-Tampering Controllers

The self-resistant controller consists of three modules, the tracking differentiator (TD), the expanded state observer (ESO), and the control rate (CL). What can be learned from the system state space expression is that the controlled object is in the third order, so the tracking differentiator is third order and the dilated state observer is fourth-order.To improve the dynamic characteristics of this energy control system, the required poles are configured using state feedback. If the zero-point configuration of the control system can be achieved, the dynamic performance of the control system can be better improved. However, the zero-point configuration requires the use of differential signals, and the general differential cannot filter the noise well and cannot be used for zero-point configuration, while the tracking differential can filter the noise well and its differential signal can be used for zero-point configuration [21].The tracking differentiator follows the given signal and obtains its differential equation. Arranging the transition process through the tracking differentiator prevents overshoot due to sudden changes in the given signal and makes the controlled object approach the target smoothly, thus improving the stability of the control system. By eliminating the poles of the system with the configured zeros, the system can be viewed as a control system approximating 1. Let the string-level object be(22)y=1p1s1p2su.Assume thatP2(s) is known and let the virtual control volume be(23)Ut=1p2su.Then, the control system becomes(24)y=1p1sUt.If the virtual control quantityU(t) is determined, the actual control quantity is(25)u=p2sUt.In the formula,P1(s) and P2(s) are polynomials.The virtual control volumeU(t) is determined by applying a self-turbulent controller design approach to the control subsystem y = U(t)/P1(s), which thus allows for a certain range of uncertainties to exist.Let the mathematical model of the controlled object contain zeros and poles. Its general form is(26)y=qspsu.Let the virtual control volumeU(t) = q(s)u; then, the system becomes y = U(t)/P(s); hence, the actual control volume becomes u = U(t)/q(s).If the controlled object conforms to the minimum phase system, thenq(s) belongs to a stable polynomial and is known. At this point, a self-antidisturbance controller can be used to determine the virtual control quantity U(t), and then the problem can be solved by considering U(t) as the input and solving the system u = U(t)/q(s) to obtain the actual control quantity u(t).Based on the above-mentioned analysis, the zero-pole diagram of the controlled object is derived using MATLAB as shown in Figure11.Figure 11
Zero-pole diagram of the controlled object.From Figure11, it can be seen that the system model has no zeros and poles in the positive part of the complex plane, so it belongs to the minimum phase system. In this case, it is possible to determine the virtual control quantity U(t) by a self-antidisturbance controller, and then the U(t) is considered as an input to solve the system u = U(t)/q(s) to obtain the actual control quantity.From the control system transfer function is known to satisfy the cubic polynomialp3(s) = s3 + a3s2 + a2s + a1, and at this time, the parameters a1, a2, and a3 are known; the algorithm of the third-order tracking differentiator can be obtained as(27)fs=−rrrv1−vt+3v2+3v3,v1=v1+hv2,v2=v2+hv3,v3=v3+hfs,Ut=v1t.In the formula,r is the fast factor and h is the sampling step size.The expansive state observer follows the control system output, observes in real time the state conditions of each order of the object of study and the total system disturbance, and compensates for the disturbance accordingly [22]. If the system is subject to unknown disturbances, the control object is uncertain [23, 24]. The uncertain objects are(28)xn=fx,x•,⋅⋅⋅,xn−1,t+wt+bu,y=xt.In the formula,f(x, x′,...,x(n−1), t) is the system unknown function; ω(t) is the unknown perturbation; u, y are system inputs and outputs; and b is the control gain.The dilation state observer in this paper is fourth-order, and under linear conditions, the control system fourth-order dilation state observer algorithm is(29)e=z1−y,z1=z1+hz2−β01e,z2=z2+hz3−β02e,z3=z3+hz4−β03e+bu,z4=z4+h−β04e.In the formula,β01, β02, β03, and β04 are expansion state observer parameters and e is error term. In the control system, the ESO contains four output variables, z1 following the system output y, z2 following y′, z3 following y″, and z4 following the system integrated disturbance, using feed-forward methods to compensate for the disturbance [23].The transfer relation from input to the output of the expansion state observer is(30)z1=w1sy=β01s3+β02s2+β03s+β04s4+β01s3+β02s2+β03s+β04y,z2=w2sy=β02s3+β03s2+β04ss4+β01s3+β02s2+β03s+β04y,z3=w3sy=β03s3+β04s2s4+β01s3+β02s2+β03s+β04y,z4=w4sy=β04s3s4+β01s3+β02s2+β03s+β04y.In order to achieve a certain estimation accuracy, it is necessary to choose larger gain coefficients, gain coefficientsβ01, β02, β03, and β04 which need larger values so as to satisfy the high gain observer mode.The error feedback control rate compares the state variables obtained from the transition process, each order of differentiation, and the dilated state observer for a given signal of the system, and then the deviation signal is algorithmically derived from the control quantity. The algorithm is(31)e1=v1−z1,e2=v2−z2,e3=v3−z3,u0=β11e1+β12e2+β13e3.In the formula,β11 is the control rate parameter and ei = vi−zi is the state error.So the algorithm for the control volume is(32)u=u0−z4b.


## 4.4.2. Creation of the Self-Tampering Controller Subsystem

When the controller used for system optimization is complex, a subsystem is used to split the complex self-resistant controller module into a tracking differentiator submodule, an expansion state observer submodule, and an error feedback control rate submodule, thus making the complex self-resistant controller seen more clearly and simply.The tracking differentiator subsystem is created, where the tracking differentiator subsystem module is specifically set with the following settings: function name TD3_L; function parameters arer, h; function module is TD3_L; and the TD3_L. m program is added inside Edit.In the TD3_L subsystem packaging process, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“TD3_L”); Prompt in the Parameters property; the modular composition of the tracking differentiator subsystem is shown in Figure12, and the TD3_L subsystem module after the package is completed is shown in Figure 13.Figure 12
Modular composition of the tracking differentiator subsystem.Figure 13
The TD3_L subsystem module after packaging.We created the dilated state observer subsystem, where the tracking differentiator subsystem module is specifically set with the following settings: function name ESO4_L; function parameters Beta01, Beta02, Beta03, and Beta04; function module ESO4_L. We added the ESO4_L. m program inside Edit.In the ESO4_L subsystem wrapping process, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“ESO4_L”). Parameters property in the Parameters property, prompt and variable are set to Beta01, Beta02, Beta03, and Beta04, and their types are edited; the mask type in the documentation property is set to ESO4_L.The modular composition of the expansion state observer subsystem is shown in Figure14, and the completed ESO4_L subsystem module of the package is shown in Figure 15.Figure 14
Modular composition of the ESO subsystem.Figure 15
Packaged ESO4_L subsystem module.The error feedback control rate subsystem is created, where the specific settings of the tracking differentiator subsystem module are function name CL3_L; function parameters are Beta11, Beta12, and Beta13; function module is CL3_L; and the CL3_L. m program is added inside the edit.During the packaging of the CL3_L subsystem, Transparency in the Icon & Ports property is set to Transparent, and the Drawing commands section is disp (“CL3_L”); Prompt in the Parameters property and the modular composition of the error feedback control rate subsystem is shown in Figure16, and the completed CL3_L subsystem module is shown in Figure 17.Figure 16
Modular composition of the CL subsystem.Figure 17
Packaged CL3_L subsystem module.


## 4.4.3. Simulation of Self-Anti-Disturbance Control of the Test Bench Electro-Hydraulic Servo Control System

Combined with the SIMULINK module, we establish the control system self-anti-disturbance control simulation model, as shown in Figure18.Figure 18
Simulation model of control system self-immunity control.The parameterr in the tracking differentiator has the greatest impact on its performance. r too large or too small will change the time of arranging the transition process and the amount of overshoot of the system, and the tracking effect of the system will become poor. The specifics of the change in the performance of the control system with the parameter r are shown in Figures 19 and 20, respectively.Figure 19
Response curve at 2 times r.Figure 20
Response curve at 0.1 times r.The expansion state observer parametersβ01, β02, β03, and β04 have an impact on its dynamic characteristics. When β01 increases, the system does not have much effect; when β01 decreases, the system will have divergent oscillations. Usually, the value of β01 is of the same order of magnitude as the reciprocal of the sampling time h. The specifics of how the performance of the control system changes with parameter β01 are shown in Figures 21 and 22, respectively.Figure 21
Response curve at 10 timesβ01.Figure 22
Response curve at 0.1 timesβ01.Whenβ02 becomes large, the system emits a noise signal, resulting in reduced system control performance; when β02 becomes small, the system will increase the number of oscillations, and the amplitude will increase. The details of the change in the performance of the control system with parameter β02 are shown in Figures 23 and 24, respectively.Figure 23
Response curve at 10 timesβ02.Figure 24
Response curve at 0.1 timesβ02.Whenβ03 becomes large, it increases the number of oscillations and the amplitude becomes larger. As β03 becomes smaller, the system tracking time becomes longer and the oscillation amplitude gradually becomes smaller. The details of the change in the performance of the control system with parameter β03 are shown in Figures 25 and 26, respectively.Figure 25
Response curve at 10 timesβ03.Figure 26
Response curve at 0.1 timesβ03.Whenβ04 becomes large, the system oscillation amplitude becomes larger and the tracking speed slows down; when β04 decreases, the tracking speed of the system slows down substantially and can even cause phase lag. The details of the change in the performance of the control system with parameter β04 are shown in Figures 27 and 28, respectively.Figure 27
Response curve at 10 timesβ04.Figure 28
Response curve at 0.1 timesβ04.The parametersβ11,β12, and β13 of the error feedback control rate have an impact on its performance. When β11 becomes large, the system overshoot increases, making its control quality lower; when β11 becomes small, the system response is lower than the given signal and the tracking effect is poor. The details of the change in the performance of the control system with parameter β11 are shown in Figures 29 and 30, respectively.Figure 29
Response curve at 10 timesβ11.Figure 30
Response curve at 0.1 timesβ11.Whenβ12 becomes large, the response is lower than the given signal and the tracking effect is poor; When β12 becomes small, the system oscillates and the amplitude decreases slowly. The details of the change in the performance of the control system with parameter β12 are shown in Figures 31 and 32, respectively.Figure 31
Response curve at 2 timesβ12.Figure 32
Response curve at 0.1 timesβ12.Whenβ13 becomes large, the system responds well; when β13 is reduced, the system is slightly overshoot. The specifics of how the performance of the control system changes with parameter β13 are shown in Figures 33 and 34, respectively.Figure 33
Response curve at 10 timesβ13.Figure 34
Response curve at 0.05 timesβ13.Setting the parameters of the three modules of the controller separately: TD:h = 0.1; ESO: β01 = 10, β02 = 680, β03 = 6900, β04 = 22500; CL: β11 = 20, β12 = 70, β13 = 1, r = 20.By SIMULINK simulation, the control system self-antidisturbance control step response curve is shown in Figure35.Figure 35
Self-antidisturbance control step response curve of control system.As can be seen from Figure35, the servo system self-turbulence control performance is significantly improved. The control system is stable at 1.2 s when a unit step signal is an input and its input and output signals match. As a result, the performance of the servo system is greatly improved after the self-turbulence optimization.


## 4.5. Simulation Comparison of Electro-Hydraulic Servo Control System of Test Bench

To compare the performance advantages and disadvantages of classical PID control, quadratic optimal control, pole configuration, and self-antidisturbance control, SIMULINK is used to build the control system simulation model of the above four optimal control methods, as shown in Figure36, and the step response curve of the unit step signal under the four control strategies is derived as shown in Figure 37.Figure 36
Simulation model of control system based on four control strategies.Figure 37
Response curve of control system based on four control strategies.From Figure37, it can be seen that the system has the most stable performance after self-anti-disturbance control. The response speed of PID control is 0.4 seconds slower than that of self-tampering control, and the response is accompanied by a large amount of overshoot; the response time of the pole configuration is 0.2 seconds longer than that of the self-tampering control, and although the overshoot is significantly reduced compared to the PID control, there is still overshoot; neither the secondary type of optimal control nor the self-rejecting control showed overshoot, but the response time of the self-rejecting control was shorter than its response time by 4.4 seconds.


## 5. Test Bench Construction and Experiments

Completed the manufacturing and construction of the test stand, as well as the preparation of the oil film stiffness and thickness adjustment before the experiment, and conducted relevant testing experiments on the test stand.


### 5.1. Test Bench Construction

The basic configuration of the test equipment has a Mitsubishi PLC, touch screen, static floating, and power loading hydraulic control system. The whole hydraulic control system is controlled by a programmable programmer and oil film thickness testing test bench as shown in Figure38.Figure 38
General assembly of the test bench.


### 5.2. Testing Test

First, the oil film stiffness and thickness were adjusted, and then the loading cylinder was controlled by PLC to complete the loading experiment. The detected experimental values are shown in Table2.Table 2
Testing experimental values.
aaaaabcdeInput force (kg)Output force (kg)Micrometer oil film thickness (μm)Display oil film thickness (μm)Self-antidisturbance control of oil film thickness (μm)Display oil film thickness (μm)Display oil film thickness (μm)Display oil film thickness (μm)Play oil film thickness (μm)400402525346535452535004994546424547464760060040403941394041700702363536343636358007983131343133323190090127283227282829100099926263126252427Based on the inspection test datasheet, the following analysis was performed:(1)
The force following case is shown in Figure39 for the input force as the reference, and the force following effect graph is shown in Figure 39 for the comparison of input force and output force at point a. As can be seen from Figure 39, the deviation of the output force from the input force at point a is between (−2∼2) kg, and the force follows well.(2)
Oil film thickness detection accuracy, that is, through the display to observe the oil film thickness at point a, while using a micrometer to measure the oil film thickness at this time. With the micrometer measurement value as the reference, the oil film thickness detection accuracy follows the effect graph as shown in Figure40. As can be seen from Figure 40, by comparing the detected oil film thickness in both cases, it can be seen that the accuracy of oil film thickness detection is higher.(3)
Analyze the relationship between the loading force and the oil film thickness. For the same action point, it can be seen from the point of Figure40, with the loading force gradually increasing, the oil film thickness gradually becomes smaller, but the trend of becoming smaller is slowing down. For the analysis of different action points, the oil film thickness of five points in Table 2 shows that the oil film thickness does not differ much when the same loading force is loaded at different points.(4)
The change curve of oil film thickness before and after the self-turbulence control is shown in Figure41. As can be seen from Figure 41, after adding the self-turbulence controller, the change in the oil film thickness of the rail is much slower when it is under load, making the oil film thickness more stable.Figure 39
Force following effect.Figure 40
Detection accuracy following effect graph.Figure 41
Oil film thickness variation curve.


## 5.1. Test Bench Construction

The basic configuration of the test equipment has a Mitsubishi PLC, touch screen, static floating, and power loading hydraulic control system. The whole hydraulic control system is controlled by a programmable programmer and oil film thickness testing test bench as shown in Figure38.Figure 38
General assembly of the test bench.


## 5.2. Testing Test

First, the oil film stiffness and thickness were adjusted, and then the loading cylinder was controlled by PLC to complete the loading experiment. The detected experimental values are shown in Table2.Table 2
Testing experimental values.
aaaaabcdeInput force (kg)Output force (kg)Micrometer oil film thickness (μm)Display oil film thickness (μm)Self-antidisturbance control of oil film thickness (μm)Display oil film thickness (μm)Display oil film thickness (μm)Display oil film thickness (μm)Play oil film thickness (μm)400402525346535452535004994546424547464760060040403941394041700702363536343636358007983131343133323190090127283227282829100099926263126252427Based on the inspection test datasheet, the following analysis was performed:(1)
The force following case is shown in Figure39 for the input force as the reference, and the force following effect graph is shown in Figure 39 for the comparison of input force and output force at point a. As can be seen from Figure 39, the deviation of the output force from the input force at point a is between (−2∼2) kg, and the force follows well.(2)
Oil film thickness detection accuracy, that is, through the display to observe the oil film thickness at point a, while using a micrometer to measure the oil film thickness at this time. With the micrometer measurement value as the reference, the oil film thickness detection accuracy follows the effect graph as shown in Figure40. As can be seen from Figure 40, by comparing the detected oil film thickness in both cases, it can be seen that the accuracy of oil film thickness detection is higher.(3)
Analyze the relationship between the loading force and the oil film thickness. For the same action point, it can be seen from the point of Figure40, with the loading force gradually increasing, the oil film thickness gradually becomes smaller, but the trend of becoming smaller is slowing down. For the analysis of different action points, the oil film thickness of five points in Table 2 shows that the oil film thickness does not differ much when the same loading force is loaded at different points.(4)
The change curve of oil film thickness before and after the self-turbulence control is shown in Figure41. As can be seen from Figure 41, after adding the self-turbulence controller, the change in the oil film thickness of the rail is much slower when it is under load, making the oil film thickness more stable.Figure 39
Force following effect.Figure 40
Detection accuracy following effect graph.Figure 41
Oil film thickness variation curve.


## 6. Conclusion

The main research elements of this paper are as follows:(1)
This paper designed the overall plan of the test bench, which includes the test bench mechanical structure, electro-hydraulic force servo control system, oil film thickness detection system, and electro-hydraulic force servo control system original parts selection. This study established a three-dimensional model of the test bench by NX and obtained the transfer function of the system through the hydraulic control valve flow equation, the cylinder flow continuity equation, the force balance equation between the cylinder and the load, and combined it with the control system deviation voltage signal equation and the feedback link pressure sensor equation, and finally simulated and analyzed the system.(2)
Using the fast overshoot of PID control, this study introduced a self-anti-disturbance control algorithm and designed the specific structure of the self-anti-disturbance controller. Also, the creation and packaging of the subsystem were completed using zero-pole phase elimination, algorithms for each module, corresponding programming, and customization of the module library. Then the electro-hydraulic servo system of the test bench was optimized by using quadratic optimal control, pole configuration, PID control, and self-turbulence control, respectively. The SIMULINK module was also used to simulate the test bench control system, and then the response curves of the four control strategies were obtained. The results showed that the response speed of the self-tampering control was 0.4 sec, 0.2 sec, and 4.4 sec faster than that of the PID control, pole configuration, and quadratic optimal control, respectively, and the overshoot was significantly reduced.(3)
According to the overall scheme design of the test bench and the design requirements of the relevant technical scheme parameters, this study completed the assembly of the test equipment. The experimental results showed that after adding the self-anti-disturbance control algorithm, the anti-disturbance ability of the electro-hydraulic servo control system of the test bench was improved, the followability of the loading force and the thickness detection accuracy of the test bench were the best, and the stability of the rail oil film thickness was significantly improved.



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*Source: 6221067-2022-10-06.xml*