# Instability Condition Derivation for Hydraulic AGC System under Pressure Closed-Loop Control

Authors: Yong Zhu; Guangpeng Li; Shengnan Tang; Wanlu Jiang; Pengfei Qian; Zhi Zheng; Zhijian Zheng
Journal: Shock and Vibration
 (2021)

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## Abstract

In strip rolling, hydraulic automatic gauge control (HAGC) system is the key element to guarantee the precision of strip gauge. The stability of the kernel pressure closed loop (PCL) in the HAGC system plays an essential role in guaranteeing the rolling process with high performance. Nevertheless, there is some difficulty in exploring the instability mechanism of the HAGC system due to the fact that the PCL is a representative nonlinear closed-loop control system. In this work, for each component of the HAGC system, the mathematical model was established. And on the basis of the linking relation of various elements, we derived the incremental transfer model of the PCL system. Furthermore, in accordance with the deduced information transfer relation, the transfer block diagram of disturbing variable of the PCL system was obtained. Moreover, for the purpose of deriving the instability condition of the PCL system, the Popov frequency criterion was employed. The instability conditions of the HAGC system were obtained under PCL control. Furthermore, the derived instability conditions of the HAGC system were experimentally verified under various working conditions. The research results provide a fundamental foundation for studying the instability mechanism of the HAGC system.


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## Body

## 1. Introduction

In the rolling industry, hydraulic automatic gauge control (HAGC) system is a pivotal component to guarantee the precision of strip thickness. The reliability of its work is the kernel of ensuring the rolling process with stabilization. Once the system fails, the downtime will occur or the quality of products will be greatly influenced [1–3]. More seriously, the entire production line will be paralyzed, which will result in huge economic losses. However, mass production practices show that there are frequently some interference factors or parameter variations in the work of rolling mill. When the HAGC system is disturbed by the disturbing force in the stable rolling process, the intrinsic balance relationship of forces will be damaged and the chattering phenomenon will occur [4, 5]. The vibration of the HAGC system will lead to the bright and dark chatter mark on the surface of the rolled strip (as shown in Figure 1), which affects the accuracy of rolling. When the vibration is severe, it will also bring about problems such as strip breakage and steel-heaping and even cause production suspension and equipment damage in serious cases [6–8]. Therefore, for the rolling production process, it is urgent to thoroughly investigate the instability mechanism of the HAGC system and take effective control measures in real time.Figure 1
Photos of chatter mark on the strip surface.The HAGC system is an intricate dynamic system formed by mechanical, electrical, and hydraulic elements. It has the features such as nonlinear, time-varying, strong coupling, and large hysteresis. The existence of multivarious interference elements makes it hard to research the instability mechanism of the HAGC system [9, 10]. Currently, many researchers both domestically and overseas have been committed to the vibration characteristics and instability mechanism of rolling mill. The models for the third octave chatter of vibrational instability during rolling were presented and discussed by Meehan [11]. Niroomand et al. [12] introduced wave propagation theory to explain chatter mechanism in tandem cold rolling mills. Kapil et al. [13] used finite-element method to develop the kinematic equation of a four-high rolling mill and research the oscillation of working roll under various parameters. Fan et al. [14] developed the single degree of freedom (DOF) model of vertical vibration system of the F3 mill rollers, and some hysteretic features of rolled strip were further investigated. Zhang et al. [15] researched the electromechanical coupling resonance of the main drive system of mill, the cycloconverter and the synchronous motor were modeled and simulated by the matrix and laboratory (MATLAB)/Simulink, and then the natural characteristics of the mechanical drive system were calculated by ANSYS. Zeng et al. [16] established a coupling vibration structure model through analyzing the features of vertical or torsional vibrations. The effects of initial thickness and rolling reduction on thickness fluctuation were researched by Wang et al. [17]. Huang et al. [18] investigated the effects of asymmetric structure parameters on the stability of rolling mill via the numerical approach. Ji [19] studied the chatter in cold rolling mill in view of the interaction among vertical or torsional vibrations of upper work roll and longitudinal oscillation of rolled strip. A time-varying criterion was developed to estimate the stabilization of tandem cold rolling mill by Lu et al. [20]. By using the Routh-Hurwitz determinant and Hopf bifurcation theorem, the bifurcation features of mill vibration system were researched [21]. Chen et al. [22] presented a novel hybrid aeroelastic-pressure balance (HAPB) technique for measuring unsteady aerodynamics on a Pentium prism. Mannini et al. [23] researched the interference between vortex-induced vibration (VIV) and lateral degree of agreement in a wind tunnel. Chen and Tse [24] implemented an improved method for the physical nonlinearity of the weakly nonlinear spring suspension system, which is successfully used in the HAPB system. Wang et al. [25–27] investigated the effects of exciting forces on pump oscillation. The influences of key elements such as valve on system stability were studied by Qian et al. [28–30]. Zhang et al. explored the influence law of spring stiffness on pump motion via analyzing the coupling characteristics of pump and plunger motion based on dynamic grid and UDF method [31]. Wu et al. probed into the variation of pressure pulsation in pump under different flow conditions [32]. Wang et al. [33–35] and Bai et al. [36–39] studied the pump vibration under several statuses. Based on linear matrix inequality and Lyapunov-Krasovskii stabilization theory, Mobayen and Tchier [40] proposed a feedback control method with good transient performance. Moreover, Mobayen [41] proposed an adaptive global terminal sliding mode control method and proved its effectiveness and applicability via numerical simulation. Mobayen [42] also constructed a controller, which is helpful for the stability and robustness of the system. According to structural flexibility, the control strategy of the hydraulic shaking table was researched by Zhang et al. [43]. Lots of the above scholars have carried out many useful researches, which can provide theoretical direction for the further analysis on oscillation mechanism of rolling mill. However, there are still some problems. The effect of the HAGC system is usually neglected by most of research results when the oscillation mechanism of rolling mill is investigated. The instability mechanism of the HAGC system on the oscillation of rolling mill is rarely observed in-depth.Formerly, the Lyapunov method was often adopted by scholars to research the absolute stability of nonlinear closed-loop system. Hua and Yu [44] gave a rigorous verification by utilizing Lyapunov stability theory to obtain the exponential stability of HAGC. Liu et al. [45] studied regenerative chatter mechanism in rolling through stability analysis by using integral criterion and Lyapunov method. Ding and Zheng [46] developed the explicit feedback controller for a nonlinear multiple-input affine system by assuming that a control Lyapunov function exists. Liu et al. [47] utilized Lyapunov analysis to verify the stability in finite time of closed-loop sliding mode control system. Zhang et al. [48, 49] used the Lyapunov functions to restrict the state variables to enhance the performance of nonlinear system. However, this Lyapunov method is a sufficient condition for evaluating the system stability, so it possesses certain conservatism. It is difficult to establish the required Lyapunov function in application. Until 1960, a frequency discrimination method was proposed to distinguish the absolute stability of nonlinear closed-loop system by Popov V.M. This method can depend on the classical transfer function and break away the predicament of rebuilding decision function. It represents a high application value [50]. The HAGC system is a representative nonlinear closed-loop system with various influencing factors. The instability vibration of the HAGC system may be produced in certain working status [51–53]. Hence, it is very significant to investigate the instability mechanism of the HAGC system from origin. It is a novel research way to theoretically deduct the instability mechanism of HAGC via utilizing the Popov criterion, which needs an in-depth study.In this study, three important contributions are as follows:(1)
The mathematical model of the HAGC system is established. The incremental transfer model of the key pressure closed loop (PCL) in HAGC is deduced. Moreover, the transfer block diagram of the perturbation of the PCL system is instituted.(2)
The Popov criterion method is employed to derive the instability condition of the pivotal PCL system in HAGC. The instability mechanism of the HAGC system on the oscillation of rolling mill is observed in-depth.(3)
The derived instability conditions of the HAGC system are experimentally verified under various working conditions. The research results lay a fundamental foundation for the investigation on the instability mechanism of the HAGC system.In the work, the contents of the article are arranged as follows. Firstly, the mathematical model of the PCL system is instituted in Section2. Secondly, the incremental transfer model of the PCL system is derived in Section 3. Thirdly, the absolute stable frequency criterion for nonlinear closed-loop system is introduced in Section 4. Further, the instability condition for the PCL system is derived in Section 5. Finally, conclusions are obtained in Section 6.


## 2. Mathematical Model of the PCL System

In order to be readable, the nomenclature that includes all the used variables in this paper is summarized as follows:NomenclatureTd: time constant of differentialTi: time constant of integralKp: proportionality factors: LaplacianKa: amplification coefficient, A/VU: input voltage, VI: output current, Axv: spool displacement, mmQL: load flow, l/minW: area gradient of valve port, mCd: flow factor of valve portps: supply pressure, MPapt: return pressure, MPapL: operating pressure of rodless chamber, MPaρ: density of hydraulic oil, kg/m3Ic: input current, AKsv: amplification factor of spool displacement on input current, m/Aωsv: specific angular frequency of servo valve, rad/sξsv: damping factor of servo valve, N·s/mIN: rated current of servo valve, Ax1: displacement of piston rod, mmAp: valid working area of piston, m2Cep: coefficient of external leakage, m3⋅s−1⋅Pa−1Cip: coefficient of internal leakage, m3⋅s−1⋅Pa−1pb: operating pressure of rod chamber, MPaβe: bulk modulus of oil, MPaV0: incipient volume of control chamber, m3m1: equivalent mass of moving elements of upper roll system (URS), kgm2: equivalent mass of the moving elements of lower roll system (LRS), kgx1: displacement of URS, mmx2: displacement of LRS, mmc1: linear damping coefficient of moving elements of URS, N·s/mc2: linear damping coefficient of moving elements of LRS, N·s/mk1: linear stiffness coefficient between upper frame beam and moving elements of URS, N/mk2: linear stiffness coefficient between lower frame beam and moving elements of LRS, N/mAb: effective operating area of piston of rod chamber, m2FL: load force acting on roll system, NKf: amplification coefficient, V/PaTf: time constant of pressure sensorQLA: the value of load flow at the working point A, l/minpLA: the value of working pressure at point A, MPaxvA: the value of spool displacement at point A, mmx1A: the value of piston displacement at point A, mmΔQL: perturbation quantity of load flow at point A, l/minΔpL: perturbation quantity of working pressure at point A, MPaΔxv: perturbation quantity of spool displacement at point A, mmΔx: perturbation quantity of piston displacement at point A, mmKq: flow gainKc: flow-pressure coefficient, m3⋅s−1⋅Pa−1Kce: total flow-pressure coefficient, m3⋅s−1⋅Pa−1In order to achieve the adjustment of rolling pressure and working roll gap, hydraulic cylinder and electrohydraulic servo valve are primarily utilized to control the HAGC system. In consideration of function, an integrated HAGC system is formed by several automatic control systems. The following three closed loops are regarded to be the most important, control loop of rolling pressure, control loop of cylinder position, and control loop of thickness gauge monitoring, as displayed in Figure2.Figure 2
Function schematic of HAGC system.Rolling PCL is considered as the significant foundation for gauge control, which is used to control the pressure timely when the rolling conditions change so that the working roll gap can be well controlled. In the PCL system, pressure sensor is often used to measure the working chamber pressure of the cylinder to acquire real-time data. And the obtained pressure value will be transferred to the signal input end and used to make a comparison with the given pressure signal. If a deviation occurs, it will be adjusted by the pressure adjuster, and the power amplifier will be used to convert it into current signal and send it to electrohydraulic servo valve. When the current signal is gained by the servo valve, the flow into the working chamber of cylinder will be controlled via the motion of valve spool, and then the working chamber pressure of cylinder will be modulated until the feedback value is equivalent to the set value.


### 2.1. Controller

Currently, PID regulator is used as the controller, whose dynamic transfer function can be represented as(1)Gcs=Kp1+1Tis+Tds,where Td and Ti are the time constants of differential and integral, Kp is the proportionality factor, and s is Laplacian.


### 2.2. Servo Amplifier

For the purpose of flow regulation by controlling the servo valve, servo amplifier is used to transform voltage into electric current. The transfer function of servo amplifier is expressed as(2)Ka=IU,where Ka is the amplification coefficient (A/V), U is the input voltage (V), and I is the output current (A).


### 2.3. Hydraulic Power Unit

It can be primarily achieved by utilizing electrohydraulic servo valve to regulate the action of hydraulic cylinder, which is hydraulic power unit of the HAGC system. Its working principle is represented in Figure3. With a view to the system, for purpose of improving the response performance, servo valve is utilized to regulate the rodless chamber of cylinder, and constant pressure oil is provided into the rod chamber of the cylinder.Figure 3
Principle of hydraulic cylinder controlled by servo valve.


#### 2.3.1. Flow Equation of Servo Valve

By controlling the movement of valve spool with weak current signal, high power hydraulic energy can be controlled, which is the action of servo valve. Servo valve possesses the following special advantages, including high power amplification, small volume, quick response, and high dynamic property.On the basis of the operating mechanism of servo valve, when spool displacementxv is taken as input and load flow QL is used as output, the flow equation of servo valve can be gained:(3)QL=fxv,pL=CdWxv2ps−pLρ,xv≥0,CdWxv2pL−ptρ,xv<0,where W is the area gradient of valve port (m), Cd is the flow factor of valve port, ps and pt are the supply and return pressure (MPa), pL is the operating pressure of rodless chamber (MPa), and ρ is the density of hydraulic oil (kg/m3).With respect to the connection between input current and spool displacement, it can be given by(4)Gvs=xvIc=Ksvs2/ωsv+2ξsv/ωsvs+1,where Ic is the input current (A), Ksv is the amplification factor of spool displacement on input current (m/A), ωsv is the specific angular frequency of servo valve (rad/s), and ξsv is the damping factor of servo valve (N⋅s/m).In regard to the characteristics of the servo valve, nonlinear saturation cannot be ignored, and the input current is restricted by(5)Ic=I,I<IN,IN,I≥IN,where IN is the rated current of servo valve (A).


#### 2.3.2. Flow Equation of Hydraulic Cylinder

The flow from servo valve into hydraulic cylinder satisfies the following requirements: firstly, it can drive the piston; then it can compensate for the leakage of cylinder; in addition, it can compensate for oil compression and chamber deformation.On account of the rodless chamber of cylinder, the flow continuity equation can be represented by(6)QL=Apx˙1+CippL−pb+CeppL+V0+Apx1βep˙L,where x1 is the displacement of piston rod (mm), Ap is the valid working area of piston (m2), Cep and Cip are the coefficients of external leakage and internal leakage (m3⋅s−1⋅Pa−1), pb is the operating pressure of rod chamber (MPa), βe is the bulk modulus of oil (MPa) [54], and V0 is the incipient volume of control chamber (m3).In consideration of the tiny change of piston displacement when system works stably, that is,Apx1≪V0, then it can be concluded that the cumulative volume of control chamber is approximately equivalent to incipient volume. Additionally, in view of the actual system, it can be ignored for this reason that the external leakage is small. Hence, with regard to the flow continuous equation of the control chamber of cylinder, it can be further expressed as(7)QL=Apx˙1+CippL−pb+V0βep˙L.


### 2.4. Load

The load of HAGC system is composed of multiple sets of rolls which possess symmetric structures. Four-high mill is commonly used in the load roll system, whose basic structure is shown in Figure4.Figure 4
Schematic of load roll of four-high mill.In current times, for the purpose of the convenient study, the load rolls is primarily classified as the following two models: one is single DOF model according to the lumped model, and the other is multi-DOF mass distribution model according to the distribution parameter model. Furthermore, it has been demonstrated by enormous researches that the stiffness is asymmetrical in view of the URS and LRS of rolling mill. In accordance with the two-DOF mass distribution load model, the corresponding analysis for the HAGC system is more aligned with that in the actual operation conditions [55, 56].To get closer to the authentic operating conditions, the URS and LRS are utilized as two different mass systems, respectively. Then the two-DOF mechanical model of load roll system is instituted, as represented in Figure5.Figure 5
Two-DOF mechanics model.On the basis of Newton’s second law, the force balance equation of load of the HAGC system can be given by(8)pLAp−pbAb=m1x˙1+c1x˙1+k1x1+FL,FL=m2x¨2+c2x˙2+k2x2,where m1 and m2 are the equivalent mass of moving elements of URS and LRS (kg); x1 and x2 are the displacement of URS and LRS (mm); c1 and c2 are the linear damping coefficient of moving elements of URS and LRS (N·s/m); k1 is the linear stiffness coefficient between upper frame beam and moving elements of URS (N/m); k2 is the linear stiffness coefficient between lower frame beam and moving elements of LRS (N/m); Ab is the effective operating area of piston of rod chamber (m2); FL is the load force acting on roll system (N).


### 2.5. Sensor

The pressure sensor is considered to be the primary feedback element of the PCL system. In view of the transfer function of pressure sensor, it is expressed as(9)Gps=KfTfs+1,where Kf is the amplification coefficient (V/Pa) and Tf is the time constant of pressure sensor.


## 2.1. Controller

Currently, PID regulator is used as the controller, whose dynamic transfer function can be represented as(1)Gcs=Kp1+1Tis+Tds,where Td and Ti are the time constants of differential and integral, Kp is the proportionality factor, and s is Laplacian.


## 2.2. Servo Amplifier

For the purpose of flow regulation by controlling the servo valve, servo amplifier is used to transform voltage into electric current. The transfer function of servo amplifier is expressed as(2)Ka=IU,where Ka is the amplification coefficient (A/V), U is the input voltage (V), and I is the output current (A).


## 2.3. Hydraulic Power Unit

It can be primarily achieved by utilizing electrohydraulic servo valve to regulate the action of hydraulic cylinder, which is hydraulic power unit of the HAGC system. Its working principle is represented in Figure3. With a view to the system, for purpose of improving the response performance, servo valve is utilized to regulate the rodless chamber of cylinder, and constant pressure oil is provided into the rod chamber of the cylinder.Figure 3
Principle of hydraulic cylinder controlled by servo valve.


### 2.3.1. Flow Equation of Servo Valve

By controlling the movement of valve spool with weak current signal, high power hydraulic energy can be controlled, which is the action of servo valve. Servo valve possesses the following special advantages, including high power amplification, small volume, quick response, and high dynamic property.On the basis of the operating mechanism of servo valve, when spool displacementxv is taken as input and load flow QL is used as output, the flow equation of servo valve can be gained:(3)QL=fxv,pL=CdWxv2ps−pLρ,xv≥0,CdWxv2pL−ptρ,xv<0,where W is the area gradient of valve port (m), Cd is the flow factor of valve port, ps and pt are the supply and return pressure (MPa), pL is the operating pressure of rodless chamber (MPa), and ρ is the density of hydraulic oil (kg/m3).With respect to the connection between input current and spool displacement, it can be given by(4)Gvs=xvIc=Ksvs2/ωsv+2ξsv/ωsvs+1,where Ic is the input current (A), Ksv is the amplification factor of spool displacement on input current (m/A), ωsv is the specific angular frequency of servo valve (rad/s), and ξsv is the damping factor of servo valve (N⋅s/m).In regard to the characteristics of the servo valve, nonlinear saturation cannot be ignored, and the input current is restricted by(5)Ic=I,I<IN,IN,I≥IN,where IN is the rated current of servo valve (A).


### 2.3.2. Flow Equation of Hydraulic Cylinder

The flow from servo valve into hydraulic cylinder satisfies the following requirements: firstly, it can drive the piston; then it can compensate for the leakage of cylinder; in addition, it can compensate for oil compression and chamber deformation.On account of the rodless chamber of cylinder, the flow continuity equation can be represented by(6)QL=Apx˙1+CippL−pb+CeppL+V0+Apx1βep˙L,where x1 is the displacement of piston rod (mm), Ap is the valid working area of piston (m2), Cep and Cip are the coefficients of external leakage and internal leakage (m3⋅s−1⋅Pa−1), pb is the operating pressure of rod chamber (MPa), βe is the bulk modulus of oil (MPa) [54], and V0 is the incipient volume of control chamber (m3).In consideration of the tiny change of piston displacement when system works stably, that is,Apx1≪V0, then it can be concluded that the cumulative volume of control chamber is approximately equivalent to incipient volume. Additionally, in view of the actual system, it can be ignored for this reason that the external leakage is small. Hence, with regard to the flow continuous equation of the control chamber of cylinder, it can be further expressed as(7)QL=Apx˙1+CippL−pb+V0βep˙L.


## 2.3.1. Flow Equation of Servo Valve

By controlling the movement of valve spool with weak current signal, high power hydraulic energy can be controlled, which is the action of servo valve. Servo valve possesses the following special advantages, including high power amplification, small volume, quick response, and high dynamic property.On the basis of the operating mechanism of servo valve, when spool displacementxv is taken as input and load flow QL is used as output, the flow equation of servo valve can be gained:(3)QL=fxv,pL=CdWxv2ps−pLρ,xv≥0,CdWxv2pL−ptρ,xv<0,where W is the area gradient of valve port (m), Cd is the flow factor of valve port, ps and pt are the supply and return pressure (MPa), pL is the operating pressure of rodless chamber (MPa), and ρ is the density of hydraulic oil (kg/m3).With respect to the connection between input current and spool displacement, it can be given by(4)Gvs=xvIc=Ksvs2/ωsv+2ξsv/ωsvs+1,where Ic is the input current (A), Ksv is the amplification factor of spool displacement on input current (m/A), ωsv is the specific angular frequency of servo valve (rad/s), and ξsv is the damping factor of servo valve (N⋅s/m).In regard to the characteristics of the servo valve, nonlinear saturation cannot be ignored, and the input current is restricted by(5)Ic=I,I<IN,IN,I≥IN,where IN is the rated current of servo valve (A).


## 2.3.2. Flow Equation of Hydraulic Cylinder

The flow from servo valve into hydraulic cylinder satisfies the following requirements: firstly, it can drive the piston; then it can compensate for the leakage of cylinder; in addition, it can compensate for oil compression and chamber deformation.On account of the rodless chamber of cylinder, the flow continuity equation can be represented by(6)QL=Apx˙1+CippL−pb+CeppL+V0+Apx1βep˙L,where x1 is the displacement of piston rod (mm), Ap is the valid working area of piston (m2), Cep and Cip are the coefficients of external leakage and internal leakage (m3⋅s−1⋅Pa−1), pb is the operating pressure of rod chamber (MPa), βe is the bulk modulus of oil (MPa) [54], and V0 is the incipient volume of control chamber (m3).In consideration of the tiny change of piston displacement when system works stably, that is,Apx1≪V0, then it can be concluded that the cumulative volume of control chamber is approximately equivalent to incipient volume. Additionally, in view of the actual system, it can be ignored for this reason that the external leakage is small. Hence, with regard to the flow continuous equation of the control chamber of cylinder, it can be further expressed as(7)QL=Apx˙1+CippL−pb+V0βep˙L.


## 2.4. Load

The load of HAGC system is composed of multiple sets of rolls which possess symmetric structures. Four-high mill is commonly used in the load roll system, whose basic structure is shown in Figure4.Figure 4
Schematic of load roll of four-high mill.In current times, for the purpose of the convenient study, the load rolls is primarily classified as the following two models: one is single DOF model according to the lumped model, and the other is multi-DOF mass distribution model according to the distribution parameter model. Furthermore, it has been demonstrated by enormous researches that the stiffness is asymmetrical in view of the URS and LRS of rolling mill. In accordance with the two-DOF mass distribution load model, the corresponding analysis for the HAGC system is more aligned with that in the actual operation conditions [55, 56].To get closer to the authentic operating conditions, the URS and LRS are utilized as two different mass systems, respectively. Then the two-DOF mechanical model of load roll system is instituted, as represented in Figure5.Figure 5
Two-DOF mechanics model.On the basis of Newton’s second law, the force balance equation of load of the HAGC system can be given by(8)pLAp−pbAb=m1x˙1+c1x˙1+k1x1+FL,FL=m2x¨2+c2x˙2+k2x2,where m1 and m2 are the equivalent mass of moving elements of URS and LRS (kg); x1 and x2 are the displacement of URS and LRS (mm); c1 and c2 are the linear damping coefficient of moving elements of URS and LRS (N·s/m); k1 is the linear stiffness coefficient between upper frame beam and moving elements of URS (N/m); k2 is the linear stiffness coefficient between lower frame beam and moving elements of LRS (N/m); Ab is the effective operating area of piston of rod chamber (m2); FL is the load force acting on roll system (N).


## 2.5. Sensor

The pressure sensor is considered to be the primary feedback element of the PCL system. In view of the transfer function of pressure sensor, it is expressed as(9)Gps=KfTfs+1,where Kf is the amplification coefficient (V/Pa) and Tf is the time constant of pressure sensor.


## 3. Incremental Transfer Model of the PCL System

### 3.1. Hydraulic Transmission Element

On the basis of the above mathematical model and information transfer relationship, when the system is in equilibrium at the working point A, the balance equations of hydraulic transmission part of the HAGC system can be deduced by(10)QLA=fxvA,pLA,(11)QLA=Apx˙1A+CippLA−pb+V0βep˙LA,(12)pLAAp−pbAb=m1x¨1A+c1x˙1A+k1x1A+FLA,where QLA is the value of QL at the working point A; pLA is the value of pL at point A; xvA is the value of xv at point A; and x1A is the value of x1 at point A.When small perturbation occurs in system close to the operating point A, all the variables of system will change around the equilibrium point; there are(13)QL=QLA+ΔQL,xv=xvA+Δxv,pL=pLA+ΔpL,x1=x1A+Δx,where ΔQL and ΔpL are the perturbation quantity of QL and pL at point A; Δxv and Δx are the perturbation quantity of xv and x1 at point A.By the use of the Taylor series, the load flow of servo valve is unfolded close to the operating point A. Furthermore, the high-order minor terms are eliminated; there is(14)QL=QLA+∂QL∂xv|AΔxv+∂QL∂pL|AΔpL.Then, when a small disturbance motion appears in the system near the operating point A, the approximate equation of perturbation flow can be deduced.(15)ΔQL=QL−QLA=∂QL∂xv|AΔxv+∂QL∂pL|AΔpL=KqΔxv−KcΔpL,where Kq is the flow gain, Kq=∂QL/∂xv, and Kc is the flow-pressure coefficient, Kc=−∂QL/∂pL.When small disturbance motion happens in the system close to the operating point A, the flow continuity equation of cylinder can be represented as(16)QLA+ΔQL=Apx˙1A+Δx˙+CippLA+ΔpL−pb+V0βep˙LA+Δp˙L.Combined with equation (16) and equation (11), there is(17)ΔQL=ApΔx˙+CipΔpL+V0βeΔp˙L.When small disturbance motion occurs in the system close to the operating point A, the load force equilibrium equation can be given by(18)pLA+ΔpLAp−pbAb=m1x¨1A+Δx¨+c1x˙1A+Δx˙+k1x1A+Δx+FLA.In combination with equation (18) and equation (12), there is(19)ΔpLAp=m1Δx¨+c1Δx˙+k1Δx.In combination with equations (15), (17), and (19), when small disturbance motion happens in the system close to the operating point A, the incremental equations of the hydraulic transmission element can be derived.(20)ΔQL=KqΔxv−KcΔpL,ΔQL=ApΔx˙+CipΔpL+V0βeΔp˙L,ΔpL=m1Δx¨+c1Δx˙+k1ΔxAp.Laplace transformation is performed on equation (20); the following equations are obtained:(21)ΔQL=KqΔxv−KcΔpL,ΔQL=ApsΔx+Cip+V0βesΔpL,ΔpL=m1s2+c1s+k1ΔxAp.Through the further organization of equation (21), as for the connection between ΔpL and Δxv, it can be deduced that(22)ΔpL=1sAp2/m1s2+c1s+k1+V0/βe+KceKqΔxv,where Kce is the total flow-pressure coefficient (m3⋅s−1⋅Pa−1), Kce=Cip+Kc.Assume that(23)G2s=1sAp2/m1s2+c1s+k1+V0/βe+Kce.Additionally, in accordance with the aforementioned theoretical formula (3), there is(24)Kq=∂QL∂xv=CdW2ps−pLρ,xv≥0,CdW2pL−ptρ,xv<0.From equations (22)–(24), the information transfer connection between ΔpL and Δxv can be acquired, which is linked by the transfer function G2s and nonlinear mathematical formula Kq.


### 3.2. Feedback and Control Element

When PCL is adopted in the HAGC system, on the basis of the mathematical model of pressure feedback and control, the connection betweenΔxv and ΔpL can be derived.(25)Δxv=GcsKaGvsGpsΔpL=Kp1+1/Tis+TdsKaKfKsvTfs+1s2/ωsv+2ξsv/ωsvs+1ΔpL.Suppose that(26)G4s=Kp1+1/Tis+TdsKaKfKsvTfs+1s2/ωsv+2ξsv/ωsvs+1.From formulas (25) and (26), the information connection between Δxv and ΔpL is linked by the transfer function G4s. Additionally, in accordance with the limitation condition of input current described in equation (5), it can be seen that G4s presents a nonlinear saturation feature and is considered to be a nonlinear transfer function.


## 3.1. Hydraulic Transmission Element

On the basis of the above mathematical model and information transfer relationship, when the system is in equilibrium at the working point A, the balance equations of hydraulic transmission part of the HAGC system can be deduced by(10)QLA=fxvA,pLA,(11)QLA=Apx˙1A+CippLA−pb+V0βep˙LA,(12)pLAAp−pbAb=m1x¨1A+c1x˙1A+k1x1A+FLA,where QLA is the value of QL at the working point A; pLA is the value of pL at point A; xvA is the value of xv at point A; and x1A is the value of x1 at point A.When small perturbation occurs in system close to the operating point A, all the variables of system will change around the equilibrium point; there are(13)QL=QLA+ΔQL,xv=xvA+Δxv,pL=pLA+ΔpL,x1=x1A+Δx,where ΔQL and ΔpL are the perturbation quantity of QL and pL at point A; Δxv and Δx are the perturbation quantity of xv and x1 at point A.By the use of the Taylor series, the load flow of servo valve is unfolded close to the operating point A. Furthermore, the high-order minor terms are eliminated; there is(14)QL=QLA+∂QL∂xv|AΔxv+∂QL∂pL|AΔpL.Then, when a small disturbance motion appears in the system near the operating point A, the approximate equation of perturbation flow can be deduced.(15)ΔQL=QL−QLA=∂QL∂xv|AΔxv+∂QL∂pL|AΔpL=KqΔxv−KcΔpL,where Kq is the flow gain, Kq=∂QL/∂xv, and Kc is the flow-pressure coefficient, Kc=−∂QL/∂pL.When small disturbance motion happens in the system close to the operating point A, the flow continuity equation of cylinder can be represented as(16)QLA+ΔQL=Apx˙1A+Δx˙+CippLA+ΔpL−pb+V0βep˙LA+Δp˙L.Combined with equation (16) and equation (11), there is(17)ΔQL=ApΔx˙+CipΔpL+V0βeΔp˙L.When small disturbance motion occurs in the system close to the operating point A, the load force equilibrium equation can be given by(18)pLA+ΔpLAp−pbAb=m1x¨1A+Δx¨+c1x˙1A+Δx˙+k1x1A+Δx+FLA.In combination with equation (18) and equation (12), there is(19)ΔpLAp=m1Δx¨+c1Δx˙+k1Δx.In combination with equations (15), (17), and (19), when small disturbance motion happens in the system close to the operating point A, the incremental equations of the hydraulic transmission element can be derived.(20)ΔQL=KqΔxv−KcΔpL,ΔQL=ApΔx˙+CipΔpL+V0βeΔp˙L,ΔpL=m1Δx¨+c1Δx˙+k1ΔxAp.Laplace transformation is performed on equation (20); the following equations are obtained:(21)ΔQL=KqΔxv−KcΔpL,ΔQL=ApsΔx+Cip+V0βesΔpL,ΔpL=m1s2+c1s+k1ΔxAp.Through the further organization of equation (21), as for the connection between ΔpL and Δxv, it can be deduced that(22)ΔpL=1sAp2/m1s2+c1s+k1+V0/βe+KceKqΔxv,where Kce is the total flow-pressure coefficient (m3⋅s−1⋅Pa−1), Kce=Cip+Kc.Assume that(23)G2s=1sAp2/m1s2+c1s+k1+V0/βe+Kce.Additionally, in accordance with the aforementioned theoretical formula (3), there is(24)Kq=∂QL∂xv=CdW2ps−pLρ,xv≥0,CdW2pL−ptρ,xv<0.From equations (22)–(24), the information transfer connection between ΔpL and Δxv can be acquired, which is linked by the transfer function G2s and nonlinear mathematical formula Kq.


## 3.2. Feedback and Control Element

When PCL is adopted in the HAGC system, on the basis of the mathematical model of pressure feedback and control, the connection betweenΔxv and ΔpL can be derived.(25)Δxv=GcsKaGvsGpsΔpL=Kp1+1/Tis+TdsKaKfKsvTfs+1s2/ωsv+2ξsv/ωsvs+1ΔpL.Suppose that(26)G4s=Kp1+1/Tis+TdsKaKfKsvTfs+1s2/ωsv+2ξsv/ωsvs+1.From formulas (25) and (26), the information connection between Δxv and ΔpL is linked by the transfer function G4s. Additionally, in accordance with the limitation condition of input current described in equation (5), it can be seen that G4s presents a nonlinear saturation feature and is considered to be a nonlinear transfer function.


## 4. Frequency Criterion for Absolute Stability of the Nonlinear Closed-Loop System

According to modern control theory and nonlinear vibration theory, if the nonlinear closed-loop system is subjected to arbitrary large initial disturbance, all the state point of the system can return to the equilibrium point; then the balance point of the system is global asymptotic stability or absolute stability [57, 58].The transfer block diagram of the nonlinear closed-loop feedback system is revealed in Figure6. The transfer function Gs of open loop is often a rational fraction. According to its pole distribution, the nonlinear closed-loop control system can be classified into two categories:(1)
Formal system: all poles ofGs have negative real parts, which is equivalent to the asymptotic stability of the open-loop system.(2)
Critical system: at least one pole of the transfer functionGs is on the imaginary axis, and the other poles have negative real parts, which is the critical stability of the open-loop system.Figure 6
Transfer block diagram of nonlinear closed-loop feedback system.Popov’s theorem: if the nonlinear feature functionfet of the nonlinear closed-loop feedback system can satisfy one of the following two conditions, then the balance point of the system is global asymptotic stability or absolute stability.Condition 1.
For formal system, it must be satisfied as(27)f0=0,0≤fetet≤P.Condition 2.
For critical system, it must be satisfied as(28)f0=0,0<ε≤fetet≤P.
And, for allω≥0 and any small parameters ε>0, any finite constants η can make the following inequalities true:(29)Re1+iηωGiω+1P>0.
Based on the above theorem, Popov deduced the geometric criterion to estimate the absolute stability of nonlinear closed-loop system. For practical application, the derivation process of this method is described as follows [59].
First, the open-loop frequency characteristicGiω is decomposed into forms that contain real and imaginary parts:(30)Giω=Reω+iImω.
Equation (30) is substituted into equation (29); then the real domain inequality which guarantees the absolute stability of the system is derived:(31)Reω−ηωImω+1P>0.
Secondly, the formula of the modified frequency featureG∗iω is defined as(32)G∗iω=Xω+iYω,(33)Xω=Reω,Yω=ωImω.
Then, the characteristic curve ofG∗iω can be drawn on a complex plane according to equation (32).
By substituting equation (33) into equation (31), the absolute stability condition of nonlinear closed-loop system can be expressed as(34)Xω−ηYω+1P>0.
If the inequality sign in equation (34) is changed into an equal sign, then the linear equation on the plane G∗iω is(35)Xω−ηYω+1P=0.
TheG∗iω plane is established with Xω as the horizontal axis and Yω as the vertical axis. When the point Q of Xω=−P−1 is taken on the real axis, then the line l which passes through the point Q−P−1,0 with slope 1/η and is satisfied with equation (35) is called the Popov line. The G∗iω plane is divided into two areas by Popov line: stable region and unstable region, as shown in Figure 7. Because the slope 1/η of line l can take any finite value, so the characteristic curve of G∗iω is firstly analyzed. Furthermore, the appropriate value of P is determined according to the nonlinear characteristic curve fet by equations (27) or (28). Usually, the value of P needs to be taken as small as possible. Finally, the Popov line l is drawn through the point Q−P−1,0 so that the characteristic curve G∗iω is located in right stable region of line l. Then the nonlinear system is absolute stability.
The above description of the Popov frequency criterion is further explained by using a set of legends below. The corrected frequency characteristic curvesG1∗iω, G2∗iω, and G3∗iω of three different nonlinear closed-loop systems are plotted in Figure 8. Firstly, P1, P2, and P3 are determined according to the nonlinear characteristic curves f1et, f2et, and f3et. Then, the positions of Q1−P1−1,0, Q2−P2−1,0, and Q3−P3−1,0 are determined. Furthermore, according to Popov’s theorem, the stability of these three kinds of nonlinear systems can be evaluated.
As displayed in Figure8(a), the intersection of G1∗iω and real axis Xω is on the right side of Q1−P1−1,0. And the characteristic curve G1∗iω is also on the right side of Q1. Through the point Q1, it is easy to plot a line l so that the characteristic curve G1∗iω is located in the right stable region. Hence, the system is absolutely stable.
As shown in Figure8(b), the intersection of G2∗iω and real axis Xω is at the right side of Q2−P2−1,0. However, over the point Q2, we cannot draw a line l that makes the characteristic curve G2∗iω be completely located in the right stable region. So the system is unstable.
As demonstrated in Figure8(c), the intersection of G3∗iω and real axis Xω is at the left side of Q3−P3−1,0. Therefore, through the point Q3, the line l cannot be obtained so that the characteristic curve G3∗iω is completely in the right stable region. So the system is also unstable.
In summary, according to the Popov frequency criterion, the stabilization of nonlinear closed-loop system can be easily discriminated.Figure 7
Relationship between Popov linel and frequency characteristic curve.Figure 8
Modified frequency characteristic curves of various systems. (a) Absolute stability system. (b) Unstable system. (c) Unstable system.
(a)(b)(c)


## 5. Instability Condition of the PCL System

In accordance with the above deduced transitive relation, the transfer block diagram is established about the perturbation of the PCL system, as displayed in Figure9. The mathematical model is used to research the absolute stability of system via the frequency method.Figure 9
Transfer block diagram of the perturbation of PCL system.In the work, in order to estimate the absolute stability of the PCL of the HAGC system, the Popov frequency criterion is employed. For this, in transfer functionG2s, assume that s=iω; then the frequency characteristic is expressed as(36)G2iω=Re2ω+iIm2ω.Substitute the expression (23) of G2s into equation (36); then the features of real frequency and imaginary frequency can be represented as(37)Re2ω=Kce+Ap2ω2c1k1−m1ω22+c12ω2×Kce+Ap2ω2c1k1−m1ω22+c12ω22+V0ωβe+Ap2ωk1−m1ω2k1−m1ω22+c12ω22−1,(38)Im2ω=−V0ωβe−Ap2ωk1−m1ω2k1−m1ω22+c12ω2×Kce+Ap2ω2c1k1−m1ω22+c12ω22+V0ωβe+Ap2ωk1−m1ω2k1−m1ω22+c12ω22−1.The following definition is obtained about the expression of corrected frequency characteristicG2∗iω:(39)G2∗iω=X2ω+iY2ω,(40)X2ω=Re2ω,Y2ω=ωIm2ω.Then, on the basis of equations (37), (38), and (40), the modified features of real frequency and imaginary frequency can be expressed as(41)X2ω=Kce+Ap2ω2c1k1−m1ω22+c12ω2×Kce+Ap2ω2c1k1−m1ω22+c12ω22+V0ωβe+Ap2ωk1−m1ω2k1−m1ω22+c12ω22−1,(42)Y2ω=−ω2V0βe+Ap2k1−m1ω2k1−m1ω22+c12ω2×Kce+Ap2ω2c1k1−m1ω22+c12ω22+V0ωβe+Ap2ωk1−m1ω2k1−m1ω22+c12ω22−1.The intersection betweenG2∗iω and the real axis is considered to be the critical point of the Popov criterion. The coordinate is defined as −P2−1,0. By using equations (41) and (42), the abscissa value of the critical point can be acquired:(43)X2ω∗=m1βem1βeAp2−V0c12+V0c12−m1βeAp22−4m1k1V02c12Kcem1βe+V0c1m1βeAp2−V0c12+V0c12−m1βeAp22−4m1k1V02c12+2c1m1k1V02.Then through the definition of the Popov line, it can be found that(44)P2=−1X2ω∗=−Kcem1βe+V0c1m1βeAp2−V0c12+V0c12−m1βeAp22−4m1k1V02c12+2c1m1k1V02m1βem1βeAp2−V0c12+V0c12−m1βeAp22−4m1k1V02c12=2c1k1V02βeV0c12−m1βeAp2−V0c12−m1βeAp22−4m1k1V02c12−c1V0m1βe−Kce.In accordance with Popov’s theorem, if equation (27) or (28) is satisfied by the nonlinear characteristic function f2Δe=G4sKqΔe of the PCL system, the balance point of the system is absolute stability; that is,(45)f0=0,0<f2ΔeΔe≤P2.From equation (45), it can be found that if the feature curve of nonlinear transfer function G4sKq is situated in the sector area, the PCL system is global asymptotic stability. The sector area consists of the horizontal axis and the line l2 which traverse the area with a slope P2, as displayed in Figure 10(a). On the contrary, if the characteristic curve of G4sKq is beyond the sector area (Figure 10(b)), it can be determined that the PCL system is unstable. At this time, as the system parameters change, intricate dynamic behavior potentially happens.Figure 10
Relationship betweenl2 and the nonlinear feature curve of PCL system. (a) Absolute stability system. (b) Unstable system.
(a)(b)Based on the above analysis, as for the absolute stability conditions of the PCL system, they can be deduced:(46)G4sKq≤2c1k1V02βeV0c12−m1βeAp2−V0c12−m1βeAp22−4m1k1V02c12−c1V0m1βe−Kce.Then, in terms of the instability conditions of the PCL system, it can be further obtained:(47)2c1k1V02βeV0c12−m1βeAp2−V0c12−m1βeAp22−4m1k1V02c12−c1V0m1βe−Kce<G4sKq.Substitute the formula ofG4s and Kq into equation (47); when the spool displacement is positivexv≥0, then the instability condition of the PCL system can be expressed as(48)2c1k1V02βeV0c12−m1βeAp2−V0c12−m1βeAp22−4m1k1V02c12−c1V0m1βe−Kce<Kp1+1/Tis+TdsKaKfKsvTfs+1s2/ωsv+2ξsv/ωsvs+1CdW2ps−pLρ.As the spool displacement is negativexv<0, with regard to the instability condition of the PCL system, it can be represented as(49)2c1k1V02βeV0c12−m1βeAp2−V0c12−m1βeAp22−4m1k1V02c12−c1V0m1βe−Kce<Kp1+1/Tis+TdsKaKfKsvTfs+1s2/ωsv+2ξsv/ωsvs+1CdW2pL−ptρ.


## 6. Experiment Research

### 6.1. Test System

The 650 4/6-roll cold rolling mill from the “National Engineering Research Center for Equipment and Technology of Cold Strip Rolling” is employed as a test object. Some physical photos of the test system are displayed in Figure11. The test mill can provide good experimental conditions for this study, including complete basic automation, process automation, HAGC, and other control systems.Figure 11
Photos of test system. (a) Test platform. (b) Computer control and data acquisition system.
(a)(b)The acquisition system of vibration data in the test is mainly composed of PCB vibration acceleration sensors and NI data acquisition equipment. Due to the working principle of load rollers and the installation layout, the vibration acceleration sensors are installed on the bearing seat of each work roller. The vibration responses of the load of the HAGC system are measured by the sensors and collected by the data acquisition system.


### 6.2. Setting of Steady State

The oil supply pressure is set to 18 MPa, and the back pressure is set to 1 MPa. The system is switched to PCL control, and the unilateral rolling force is set to 350 kN. Then, a square wave signal (amplitude 50 kN; frequency 0.1 Hz) and a sine signal (amplitude 20 kN; frequency 0.1 Hz) are input, respectively. Meanwhile, the parameters of the PID controller are adjusted to achieve the best state of the dynamic performance of the HAGC system as far as possible. The final set of parameters areKp = 0.1, Ti = 4000 ms, and Td = 0. The following effect of rolling force when the square wave signal is input is shown in Figure 12, and the following effect of rolling force when the sine signal is input is displayed in Figure 13. It can be seen that the rolling force can follow the typical signal input well. This indicates that the system possesses good dynamic performance under this controller parameters combination and can be used as the initial steady state of the tests.Figure 12
Following effect of rolling force when a square wave signal is input. (a) Operation side. (b) Drive side.
(a)(b)Figure 13
Following effect of rolling force when a sine signal is input. (a) Operation side. (b) Drive side.
(a)(b)


### 6.3. Collection of Test Data

With regard to the derived instability condition expressions (48) and (49), the parameters that often change in the practical working process are the proportional factor Kp of controller and the operating pressure pL of rodless chamber. Hence, in the following signal acquisition scheme, the change of Kp is mainly reflected by the variation of the proportional coefficient of controller. The change of pL is mirrored by changing the rolling force. Then, through feature extraction and analysis for system state signals under various working conditions, the influences of the aforementioned different factors on system vibration are explored.The dynamic tests are carried out according to the various working conditions set up as below. Meanwhile, the vibration responses of the load of the HAGC system are collected by the data acquisition system. The sampling frequency is 1 kHz.The proportional coefficient of the controller is changed by adjusting the PID parameters. The setting values of the proportional coefficient of the controller are shown in Table1.Table 1
Proportional coefficient of controller.
NameProportional coefficient of controllerValue0.10.150.20.250.30.35The setting values of rolling force are represented in Table2.Table 2
Rolling force.
NameRolling force (kN)Value200280360440520600680760840920


### 6.4. Analysis of Test Results

When the proportional coefficient of controller is changed, the variation curve of vibration severity of the load of the HAGC system is displayed in Figure14. It can be seen that, with the increase of the proportional coefficient Kp of controller, the vibration severity of the load of the HAGC system gradually increases. It means that as Kp increases, the vibration intensity of the HAGC system will increase, and the system will be likely to be unstable. Therefore, if the magnitude of the proportionality coefficient Kp is appropriately controlled, it is beneficial to the stability of the system. The result reflects the effect of the gain coefficient, which is consistent with the theoretically derived law of the influence of the gain coefficient on the system instability.Figure 14
Variation curve of vibration severity under different proportion coefficient.When rolling force changes, the variation curve of vibration severity of load of HAGC system is represented in Figure15. It can be found that with the increase of rolling force, the vibration severity of the load of the HAGC system increases gradually. It indicates that the vibration strength of the HAGC system will augment with the increase of rolling force. This result reflects the effect of the operating pressure pL, which shows agreement with the influence of pL on the system stability derived from the instability condition.Figure 15
Variation curve of vibration severity under different roll force.


## 6.1. Test System

The 650 4/6-roll cold rolling mill from the “National Engineering Research Center for Equipment and Technology of Cold Strip Rolling” is employed as a test object. Some physical photos of the test system are displayed in Figure11. The test mill can provide good experimental conditions for this study, including complete basic automation, process automation, HAGC, and other control systems.Figure 11
Photos of test system. (a) Test platform. (b) Computer control and data acquisition system.
(a)(b)The acquisition system of vibration data in the test is mainly composed of PCB vibration acceleration sensors and NI data acquisition equipment. Due to the working principle of load rollers and the installation layout, the vibration acceleration sensors are installed on the bearing seat of each work roller. The vibration responses of the load of the HAGC system are measured by the sensors and collected by the data acquisition system.


## 6.2. Setting of Steady State

The oil supply pressure is set to 18 MPa, and the back pressure is set to 1 MPa. The system is switched to PCL control, and the unilateral rolling force is set to 350 kN. Then, a square wave signal (amplitude 50 kN; frequency 0.1 Hz) and a sine signal (amplitude 20 kN; frequency 0.1 Hz) are input, respectively. Meanwhile, the parameters of the PID controller are adjusted to achieve the best state of the dynamic performance of the HAGC system as far as possible. The final set of parameters areKp = 0.1, Ti = 4000 ms, and Td = 0. The following effect of rolling force when the square wave signal is input is shown in Figure 12, and the following effect of rolling force when the sine signal is input is displayed in Figure 13. It can be seen that the rolling force can follow the typical signal input well. This indicates that the system possesses good dynamic performance under this controller parameters combination and can be used as the initial steady state of the tests.Figure 12
Following effect of rolling force when a square wave signal is input. (a) Operation side. (b) Drive side.
(a)(b)Figure 13
Following effect of rolling force when a sine signal is input. (a) Operation side. (b) Drive side.
(a)(b)


## 6.3. Collection of Test Data

With regard to the derived instability condition expressions (48) and (49), the parameters that often change in the practical working process are the proportional factor Kp of controller and the operating pressure pL of rodless chamber. Hence, in the following signal acquisition scheme, the change of Kp is mainly reflected by the variation of the proportional coefficient of controller. The change of pL is mirrored by changing the rolling force. Then, through feature extraction and analysis for system state signals under various working conditions, the influences of the aforementioned different factors on system vibration are explored.The dynamic tests are carried out according to the various working conditions set up as below. Meanwhile, the vibration responses of the load of the HAGC system are collected by the data acquisition system. The sampling frequency is 1 kHz.The proportional coefficient of the controller is changed by adjusting the PID parameters. The setting values of the proportional coefficient of the controller are shown in Table1.Table 1
Proportional coefficient of controller.
NameProportional coefficient of controllerValue0.10.150.20.250.30.35The setting values of rolling force are represented in Table2.Table 2
Rolling force.
NameRolling force (kN)Value200280360440520600680760840920


## 6.4. Analysis of Test Results

When the proportional coefficient of controller is changed, the variation curve of vibration severity of the load of the HAGC system is displayed in Figure14. It can be seen that, with the increase of the proportional coefficient Kp of controller, the vibration severity of the load of the HAGC system gradually increases. It means that as Kp increases, the vibration intensity of the HAGC system will increase, and the system will be likely to be unstable. Therefore, if the magnitude of the proportionality coefficient Kp is appropriately controlled, it is beneficial to the stability of the system. The result reflects the effect of the gain coefficient, which is consistent with the theoretically derived law of the influence of the gain coefficient on the system instability.Figure 14
Variation curve of vibration severity under different proportion coefficient.When rolling force changes, the variation curve of vibration severity of load of HAGC system is represented in Figure15. It can be found that with the increase of rolling force, the vibration severity of the load of the HAGC system increases gradually. It indicates that the vibration strength of the HAGC system will augment with the increase of rolling force. This result reflects the effect of the operating pressure pL, which shows agreement with the influence of pL on the system stability derived from the instability condition.Figure 15
Variation curve of vibration severity under different roll force.


## 7. Conclusions

In this paper, aiming at each component of the HAGC system, the mathematical models were established. In accordance with the syntagmatic relation of each element, the incremental transfer model of the key PCL system in HAGC was deduced. Furthermore, according to the deduced information transfer relation, the transfer block diagram of the perturbation of PCL system was instituted. Moreover, in order to derive the absolute stability condition and instability condition, the Popov frequency criterion was employed. The instability conditions of the HAGC system under PCL control were obtained under two working conditions, that is, when the spool displacement of servo valve is positive and negative, respectively. Furthermore, the derived instability conditions of the HAGC system were verified by experiments under various working conditions. The main factors often change in practical working process such as the proportional coefficientKp of controller and the load pressure pL. The influences of main factors on system vibration were explored. The results indicate that the vibration strength of the HAGC system will augment with the increase of main factors, which is consistent with the derived instability condition. The research results lay a theoretical basis for the investigation on the instability mechanism of the HAGC system.The study can provide an essential foundation for further exploration of the oscillation traceability and suppression of the HAGC system. Moreover, it can provide theoretical reference for the instability condition derivation of electrohydraulic servo system in other fields.



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*Source: 6618525-2021-06-14.xml*