2.3.1. Flow Equation of Electro-Hydraulic Servo Valve

The function of the servo valve is to control the movement of the valve spool with weak current signal to achieve the control of high power hydraulic energy. There are many advantages such as small volume, high power amplification, fast response and high dynamic performance.

According to the working principle of the servo valve, when the spool displacement *xv* is used as the input and the load flow *QL* is taken as the output, the basic flow equation of the servo valve can be obtained:

$$Q\_L = f(\mathbf{x}\_{\upsilon}, p\_L) = \begin{cases} \mathbf{C}\_d \mathsf{W} \mathbf{x}\_{\upsilon} \sqrt{\frac{2(p\_s - p\_L)}{\rho}} & \mathbf{x}\_{\upsilon} \ge 0\\ \mathbf{C}\_d \mathsf{W} \mathbf{x}\_{\upsilon} \sqrt{\frac{2(p\_L - p\_t)}{\rho}} & \mathbf{x}\_{\upsilon} < 0 \end{cases} \tag{3}$$

where *Cd* is the flow coefficient of valve port, *W* is the area gradient of valve port (m), *xv* is the displacement of main spool (m), ρ is the hydraulic oil density (kg/m3), *ps* is the oil supply pressure (MPa), *pt* is the return pressure (MPa) and *pL* is the working pressure of rodless chamber of the hydraulic cylinder (MPa).

*Processes* **2019**, *7*, 766

The relationship between spool displacement of the servo valve and input current can be expressed as:

$$G\_{\upsilon}(s) = \frac{\mathbf{x}\_{\upsilon}}{I\_{c}} = \frac{K\_{\text{sv}}}{\frac{\mathbf{s}^{2}}{\omega\_{\text{sv}}} + \frac{2\mathcal{L}\_{\text{sv}}}{\omega\_{\text{sv}}}\mathbf{s} + \mathbf{1}},\tag{4}$$

where *Ic* is the input current of the servo valve (A), *Ksv* is the amplification coefficient of the spool displacement on the input current (m/A), ω*sv* is the natural angular frequency of the servo valve (rad/s) and ξ*sv* is the damping coefficient of the servo valve (N · s/m).

The servo valve also has nonlinear saturation characteristics and its input current is limited by:

$$I\_{\mathcal{E}} = \left\{ \begin{array}{c} I \\ I\_N & I \geq I\_N \end{array} \right. \tag{5}$$

where *IN* is the rated current of the servo valve (A).

#### 2.3.2. Basic Flow Equation of Hydraulic Cylinder

The flow from the servo valve into the hydraulic cylinder not only meets the flow required to push the piston, but also compensates for internal and external leakage in the cylinder, as well as the flow required to compensate for oil compression and chamber deformation.

The flow continuity equation for the rodless chamber of the hydraulic cylinder can be expressed by:

$$\dot{Q}\_{\text{L}} = A\_{\text{p}} \dot{\mathbf{x}}\_{1} + \mathbf{C}\_{\text{ip}} (p\mathbf{L} - p\_{\text{b}}) + \mathbf{C}\_{\text{cp}} p\mathbf{L} + \frac{V\_{0} + A\_{\text{p}} \mathbf{x}\_{1}}{\beta\_{\text{c}}} \dot{p}\_{\text{L}} \tag{6}$$

where *Ap* is the effective working area of the piston (m2), *x*<sup>1</sup> is the displacement of the piston rod (mm), *Cip* is internal leakage coefficient (m<sup>3</sup> · <sup>s</sup>−<sup>1</sup> · Pa<sup>−</sup>1), *Cep* is external leakage coefficient (m<sup>3</sup> · <sup>s</sup>−<sup>1</sup> · Pa<sup>−</sup>1), *pb* is the working pressure of the rod chamber (MPa), *V*<sup>0</sup> is the initial volume of the control chamber (including the oil inlet pipe and the rodless chamber) (m3) and β*<sup>e</sup>* is the bulk modulus of oil (MPa).

Since the change of piston displacement of the hydraulic cylinder is small when the hydraulic system is working stably, that is, *Apx*<sup>1</sup> << *V*0, then the total volume of the hydraulic cylinder control chamber is approximately equal to the initial volume. In addition, with regard to the actual system, the external leakage is small and can be ignored. Therefore, the continuous flow equation of the hydraulic cylinder control chamber can be further written as:

$$Q\_L = A\_p \dot{\mathbf{x}}\_1 + \mathbb{C}\_{\dot{\mathbf{p}}p} (p\_L - p\_b) + \frac{V\_0}{\beta\_c} \dot{p}\_L. \tag{7}$$

#### *2.4. Mathematical Model of Load*

The external load of the HAGC system consists of several sets of rolls with symmetrical structure. The basic structure of the load roll system of the commonly used four-high mill is shown in Figure 3. In consideration of the load roll system of the six-high mill, the basic structure is similar, and there is a set of intermediate rolls between the support roll and the work roll.

At present, in order to facilitate the analysis, the load roll system is mainly divided according to the lumped model and distribution parameter model, into single degree of freedom (DOF) load model and multi-DOF mass distribution load model, respectively. Moreover, numerous research studies indicate that the stiffness of the upper and lower roll systems of the rolling mill is asymmetrical. The analysis for the HAGC system according to the two-DOF mass distribution load model is more consistent with the actual working conditions [44,45].

**Figure 3.** Structure diagram of four-high load roll system.

In order to get closer to the actual working conditions, the modeling method of the load roll system is studied based on the two-DOF asymmetric mass distribution model. The upper roll system is used as a mass system and the lower roll system is utilized as another mass system, then the two-DOF mechanical model of the load roll system is established, as illustrated in Figure 4.

**Figure 4.** Two degrees of freedom mechanics model of the load roll system.

According to Newton's second law, the load force balance equation of the HAGC system can be expressed as:

$$p\_L A\_p - p\_b A\_b = m\_1 \ddot{\mathbf{x}}\_1 + c\_1 \dot{\mathbf{x}}\_1 + k\_1 \mathbf{x}\_1 + F\_{L\prime} \tag{8}$$

$$F\_L = m\_2\ddot{\mathbf{x}}\_2 + c\_2\dot{\mathbf{x}}\_2 + k\_2\mathbf{x}\_{2\prime} \tag{9}$$

where *m*<sup>1</sup> is the equivalent mass of moving parts of the upper roll system (URS) (kg); *m*<sup>2</sup> is the equivalent mass of the moving parts of the lower roll system (LRS) (kg); *c*<sup>1</sup> is the linear damping coefficient of moving parts of URS (N·s/m); *c*<sup>2</sup> is the linear damping coefficient of moving parts of LRS (N·s/m); *k*<sup>1</sup> is the linear stiffness coefficient between the upper frame beam and the moving parts of URS (N/m); *k*<sup>2</sup> is the linear stiffness coefficient between the lower frame beam and the moving parts of LRS (N/m); *x*<sup>1</sup> is the displacement of URS (mm); *x*<sup>2</sup> is the displacement of LRS (mm); *Ab* is the effective working area of the rod chamber piston (m2); and *FL* is the load force acting on the roll system (N).

#### *2.5. Mathematical Model of Sensor*

The feedback component of the HAGC position closed-loop system is mainly the displacement sensor. In the actual working process, the response time of the sensor needs to be considered, so the sensor can be represented as an inertia link.

The transfer function of the displacement sensor is:

$$G\_{\mathbf{x}}(\mathbf{s}) = \frac{K\_{\mathbf{x}}}{T\_{\mathbf{x}}\mathbf{s} + 1},\tag{10}$$

where *Kx* is the amplification coefficient of the displacement sensor (V/m) and *Tx* is the time constant of the displacement sensor.

#### **3. Incremental Transfer Model of Position Closed-Loop System**

#### *3.1. Incremental Transfer Model of Hydraulic Transmission Part*

When the system is in equilibrium at the working point A, according to the mathematical model and information transfer relationship established above, the equilibrium equations of the hydraulic transmission part of the HAGC system can be derived as:

$$Q\_{LA} = f(\mathbf{x}\_{vA\prime} p\_{LA}),\tag{11}$$

$$Q\_{LA} = A\_p \dot{\mathbf{x}}\_{1A} + \mathbb{C}\_{\dot{\mathbf{p}}\mathbf{p}}(p\_{LA} - p\_{\mathbf{b}}) + \frac{V\_0}{\beta\_\varepsilon} \dot{\mathbf{p}}\_{LA'} \tag{12}$$

$$p\_{LA}A\_p - p\_bA\_b = m\_1\ddot{\mathbf{x}}\_{1A} + c\_1\dot{\mathbf{x}}\_{1A} + k\_1\mathbf{x}\_{1A} + F\_{LA\prime} \tag{13}$$

where *QLA* is the value of the load flow *QL* at the working point A; *xvA* is the value of spool displacement *xv* at the working point A; *pLA* is the value of working pressure *pL* at the working point A; and *x*1*<sup>A</sup>* is the value of piston rod displacement *x*<sup>1</sup> at the working point A.

When the system makes small disturbances near the working point A, all the variables of the system change around the equilibrium point, as follows:

$$Q\_L = Q\_{LA} + \Delta Q\_{L\prime} \tag{14}$$

$$
\Delta \mathbf{x}\_{\upsilon} = \mathbf{x}\_{\upsilon \mathcal{A}} + \Delta \mathbf{x}\_{\upsilon \star} \tag{15}
$$

$$p\_L = p\_{LA} + \Delta p\_{L\prime} \tag{16}$$

$$\mathbf{x}\_1 = \mathbf{x}\_{1A} + \Delta \mathbf{x}\_\prime \tag{17}$$

where Δ*QL* is the disturbance quantity of the load flow *QL* at the working point A; Δ*xv* is the disturbance quantity of spool displacement *xv* at the working point A; Δ*pL* is the disturbance quantity of working pressure *pL* at the working point A; and Δ*x* is the disturbance quantity of piston rod displacement *x*<sup>1</sup> at the working point A.

The load flow of the servo valve is expanded by Taylor series near the working point A, and the high-order minor terms are omitted, so:

$$Q\_L = Q\_{LA} + \frac{\partial Q\_L}{\partial \mathbf{x}\_{\upsilon}}|\_{A} \Delta \mathbf{x}\_{\upsilon} + \frac{\partial Q\_L}{\partial p\_L}|\_{A} \Delta p\_L. \tag{18}$$

Then, the approximate equation of disturbance flow can be deduced when the system makes a small disturbance motion near the working point A.

$$\begin{split} \Delta Q\_{L} = Q\_{L} - Q\_{LA} &= \frac{\partial Q\_{L}}{\partial \mathbf{x}\_{\upsilon}}|\_{A} \Delta \mathbf{x}\_{\upsilon} + \frac{\partial Q\_{L}}{\partial p\_{L}}|\_{A} \Delta p\_{L} \\ &= K\_{\mathsf{q}} \Delta \mathbf{x}\_{\upsilon} - K\_{\mathsf{c}} \Delta p\_{L} \end{split} \tag{19}$$

where *Kq* is the flow gain, *Kq* = <sup>∂</sup>*QL* <sup>∂</sup>*xv* ; and *Kc* is the flow–pressure coefficient, *Kc* <sup>=</sup> <sup>−</sup>∂*QL* <sup>∂</sup>*pL* .

When the system makes small disturbance motion near the working point A, the flow continuity equation of the hydraulic cylinder can be expressed as:

$$Q\_{LA} + \Delta Q\_L = A\_p(\dot{\mathbf{x}}\_{1A} + \Delta \dot{\mathbf{x}}) + \mathbb{C}\_{\dot{\mathbf{p}}}[(p\_{LA} + \Delta p\_L) - p\_b] + \frac{V\_0}{\beta\_t}(\dot{p}\_{LA} + \Delta \dot{p}\_L). \tag{20}$$

In combination with Equations (12) and (20), there is:

$$
\Delta Q\_L = A\_p \Delta \dot{x} + \mathbb{C}\_{ip} \Delta p\_L + \frac{V\_0}{\beta\_\varepsilon} \Delta \dot{p}\_L. \tag{21}
$$

When the system makes small disturbance motion near the working point A, the load force balance equation can be expressed as:

$$(p\_{1A} + \Delta p\_L)A\_p - p\_b A\_b = m\_1(\ddot{\mathbf{x}}\_{1A} + \Delta \ddot{\mathbf{x}}) + c\_1(\dot{\mathbf{x}}\_{1A} + \Delta \dot{\mathbf{x}}) + k\_1(\mathbf{x}\_{1A} + \Delta \mathbf{x}) + F\_{LA}.\tag{22}$$

In combination with Equations (13) and (22), there is:

$$
\Delta p\_{\rm L} A\_p = m\_1 \Delta \ddot{\mathbf{x}} + c\_1 \Delta \dot{\mathbf{x}} + k\_1 \Delta \mathbf{x}.\tag{23}
$$

In combination with Equations (19), (21) and (23), the incremental equations of the hydraulic transmission part can be deduced when the system makes small disturbance motion near the working point A.

$$\begin{cases} \Delta Q \text{L = } K\_{\text{q}} \Delta \mathbf{x}\_{\text{v}} - K\_{\text{c}} \Delta p\_{L} \\\ \Delta Q\_{L} = A\_{p} \Delta \dot{\mathbf{x}} + C\_{\dot{\mathbf{p}}} \Delta p\_{L} + \frac{V\_{\text{0}}}{\beta\_{\text{c}}} \Delta \dot{p}\_{L} \\\ \Delta p\_{L} = \left( m\_{1} \Delta \ddot{\mathbf{x}} + c\_{1} \Delta \dot{\mathbf{x}} + k\_{1} \Delta \mathbf{x} \right) / A\_{p} \end{cases} \tag{24}$$

The incremental Equation (24) is further organized as follows:

$$\begin{array}{lcl} \mathbf{K}\_{\mathbf{q}} \Delta \mathbf{x}\_{\mathcal{V}} = & \frac{V\_{0} \mathbf{m}\_{1}}{\frac{\partial}{\partial \mathbf{r}} \mathbf{A}\_{p}} \Delta \ddot{\mathbf{x}} + \left[ (\frac{V\_{0} \mathbf{c}\_{1}}{\frac{\partial}{\partial \mathbf{r}} \mathbf{A}\_{p}} + \frac{(\mathbf{C}\_{\dot{\mathbf{w}}} + \mathbf{K}\_{\mathbf{c}}) \mathbf{m}\_{1}}{A\_{p}}) \right] \Delta \ddot{\mathbf{x}} \\ & + \left[ (\frac{V\_{0} \mathbf{k}\_{1}}{\frac{\partial}{\partial \mathbf{r}} \mathbf{A}\_{p}} + \frac{(\mathbf{C}\_{\dot{\mathbf{w}}} + \mathbf{K}\_{\mathbf{c}}) \mathbf{c}\_{1}}{A\_{p}} + A\_{p} \right) \right] \Delta \dot{\mathbf{x}} + \frac{(\mathbf{C}\_{\dot{\mathbf{w}}} + \mathbf{K}\_{\mathbf{c}}) \mathbf{k}\_{1}}{A\_{p}} \Delta \mathbf{x} \end{array} \tag{25}$$

By performing Laplace transformation on Equation (25), the relationship between the load displacement disturbance Δ*x* and the spool displacement disturbance Δ*xv* can be derived.

$$\Delta \mathbf{x} = \frac{A\_p}{s[\frac{V\_0 m\_1}{\beta\_\ell} s^2 + (K\_{c\varepsilon} m\_1 + \frac{V\_0 c\_1}{\beta\_\ell})s + (K\_{c\varepsilon} c\_1 + \frac{V\_0 k\_1}{\beta\_\ell} + A\_p^2)] + k\_1 K\_{c\varepsilon}} K\_q \Delta \mathbf{x}\_v \tag{26}$$

where *Kce* is total flow–pressure coefficient (m3 · <sup>s</sup>−<sup>1</sup> · Pa<sup>−</sup>1), *Kce* = *Cip* + *Kc*.

Suppose that:

$$G\_{1}(s) = \frac{A\_{p}}{s[\frac{V\_{0}m\_{1}}{\overline{\rho}\_{r}}s^{2} + (K\_{\rm cr}m\_{1} + \frac{V\_{0}c\_{1}}{\overline{\rho}\_{r}})s + (K\_{\rm cr}c\_{1} + \frac{V\_{0}k\_{1}}{\overline{\rho}\_{r}} + A\_{p}^{2})] + k\_{1}K\_{\rm cr}}.\tag{27}$$

In addition, according to the aforementioned theoretical formula given as Equation (3), there is:

$$K\_{\mathbb{Q}} = \frac{\partial Q\_L}{\partial \mathbf{x}\_{\upsilon}} = \begin{cases} \, \, \mathbb{C}\_d \, \mathcal{W} \sqrt{\frac{2(p\_s - p\_l)}{\rho}} & \mathbf{x}\_{\upsilon} \ge 0 \\\, \, \, \mathbb{C}\_d \, \mathcal{W} \sqrt{\frac{2(p\_l - p\_l)}{\rho}} & \mathbf{x}\_{\upsilon} < 0 \end{cases} . \tag{28}$$

From Equations (26)–(28), the information transfer relationship between the displacement disturbance Δ*x* of the load and the displacement disturbance Δ*xv* of the servo valve spool can be identified, which is transmitted by the transfer function *G*1(*s*) and the nonlinear mathematical expression *Kq*.

#### *3.2. Incremental Transfer Model of the Feedback and Control Part*

When the HAGC system adopts the position closed loop, based on the mathematical model of displacement feedback and control, the relationship between spool displacement disturbance Δ*xv* and load displacement disturbance Δ*x* can be deduced.

$$\begin{array}{rcl} \Delta \mathbf{x}\_{\upsilon} &= \mathbf{G}\_{\mathsf{c}}(s) \mathbf{K}\_{\mathsf{d}} \mathbf{G}\_{\upsilon}(s) \mathbf{G}\_{\mathsf{x}}(s) \Delta \mathbf{x} \\ &= \frac{\mathbf{K}\_{\mathsf{p}} \left( 1 + \frac{1}{T\_{\mathsf{p}}^{\mathsf{s}}} + T\_{\mathsf{d}} s \right) \mathbf{K}\_{\mathsf{a}} \mathbf{K}\_{\mathsf{x}} \mathbf{K}\_{\mathsf{o}\upsilon}}{\left( T\_{\mathsf{x}} s + 1 \right) \left( \frac{s^{2}}{\underline{\alpha} \underline{w}} + \frac{2 \underline{\epsilon} \underline{w}}{\underline{\alpha} \underline{w}} s + 1 \right)} \Delta \mathbf{x} \end{array} \tag{29}$$

Assume that:

$$\mathbf{G}\_3(s) = \frac{K\_p(1 + \frac{1}{T\_{i^\sf S}} + T\_{d^\sf S})K\_a K\_{\sf x} K\_{\sf s\vartheta}}{(T\_{\sf x}s + 1)(\frac{s^2}{\alpha\_{\sf s\vartheta}} + \frac{2\ell\_{\sf x\sf x}}{\alpha\_{\sf x\vartheta}}s + 1)}.\tag{30}$$

It can be seen from Equations (29) and (30) that the information relationship between the spool displacement disturbance Δ*xv* and the load displacement disturbance Δ*x* is transmitted by the transfer function *G*3(*s*). In addition, according to the input current limitation condition expression (Equation (5)) of the servo valve, it can be found that *G*3(*s*) possesses a nonlinear saturation characteristic and is a nonlinear transfer function.

#### **4. Absolute Stability Condition for Position Closed-Loop System**

On the basis of the aforementioned derived transfer relationship, the transfer block diagram of the disturbance of the position closed-loop system is established, as shown in Figure 5. For purpose of researching the absolute stability of system, the transfer block diagram of the disturbance is the mathematical model which uses the frequency method.

**Figure 5.** Transfer block diagram of the disturbance of the position closed-loop system.

In this work, the Popov frequency criterion is introduced to determine the absolute stability of the position closed-loop control of the HAGC system. For this, in the transfer function *G*1(*s*), suppose that *s* = iω, then the frequency characteristic is obtained:

$$G\_1(\mathrm{i}\omega) = \mathrm{Re}\_1(\omega) + \mathrm{i}\mathrm{Im}\_1(\omega). \tag{31}$$

The expression (Equation (27)) of *G*1(*s*) is substituted into Equation (31), then the real frequency and imaginary frequency characteristics can be acquired:

$$\begin{split} \text{Re}\_{1}(\omega) &= A\_{\mathbb{P}} \left[ \mathbf{k}\_{1} \mathbf{K}\_{\text{ct}} - \left( \mathbf{K}\_{\text{ct}} m\_{1} + \frac{V\_{0} \mathbf{c}\_{1}}{\overline{\beta}\_{\text{c}}} \right) \omega^{2} \right] \\ &\times \left\{ \left[ \mathbf{k}\_{1} \mathbf{K}\_{\text{ct}} - \left( \mathbf{K}\_{\text{ct}} m\_{1} + \frac{V\_{0} \mathbf{c}\_{1}}{\overline{\beta}\_{\text{c}}} \right) \omega^{2} \right]^{2} + \left[ \left( \mathbf{K}\_{\text{ct}} \mathbf{c}\_{1} + \frac{V\_{0} \mathbf{k}\_{1}}{\overline{\beta}\_{\text{c}}} + A\_{\text{p}}^{2} \right) \omega - \frac{V\_{0} m\_{1}}{\overline{\beta}\_{\text{c}}} \omega^{3} \right]^{2} \end{split} \tag{32}$$

$$\begin{split} \text{Im}\_{1}(\boldsymbol{\omega}) &= -A\_{p} \Big[ (\mathcal{K}\_{\text{c2}} \boldsymbol{c}\_{1} + \frac{V\_{0} \mathbf{k}\_{1}}{\overline{\beta\_{\text{c}}}} + A\_{p}^{2}) \boldsymbol{\omega} - \frac{V\_{0} \mathbf{m}\_{1}}{\overline{\beta\_{\text{c}}}} \boldsymbol{\alpha}^{3} \Big] \\ &\times \Big[ \left[ \mathcal{k}\_{1} \mathcal{K}\_{\text{c2}} - (\mathcal{K}\_{\text{c2}} \boldsymbol{m}\_{1} + \frac{V\_{0} \mathbf{c}\_{1}}{\overline{\beta\_{\text{c}}}}) \boldsymbol{\alpha}^{2} \right]^{2} + \left[ (\mathcal{K}\_{\text{c2}} \boldsymbol{c}\_{1} + \frac{V\_{0} \mathbf{k}\_{1}}{\overline{\beta\_{\text{c}}}} + A\_{p}^{2}) \boldsymbol{\omega} - \frac{V\_{0} \mathbf{m}\_{1}}{\overline{\beta\_{\text{c}}}} \boldsymbol{\alpha}^{3} \right]^{2} \end{split} \tag{33}$$

The expression of corrected frequency characteristic *G*∗ <sup>1</sup>(iω) is defined as:

$$G\_1^\*(\text{i}\omega) = X\_1(\omega) + \text{i}Y\_1(\omega),\tag{34}$$

$$X\_1(\omega) = \text{Re}\_1(\omega), \quad \mathcal{Y}\_1(\omega) = \omega \text{Im}\_1(\omega). \tag{35}$$

Then, according to Equations (32), (33) and (35), the corrected real frequency and imaginary frequency characteristics can be obtained:

$$\begin{split} \left[X\_{1}(\omega)\right] &= A\_{p} \left[k\_{1}K\_{\rm cr} - (K\_{\rm cr}m\_{1} + \frac{V\_{0\rm c}}{\beta\_{\rm r}})\omega^{2}\right] \\ &\times \left\{\left[k\_{1}K\_{\rm cr} - (K\_{\rm cr}m\_{1} + \frac{V\_{0\rm c}}{\beta\_{\rm r}})\omega^{2}\right]^{2} + \left[(K\_{\rm cr}c\_{1} + \frac{V\_{0\rm c}}{\beta\_{\rm r}} + A\_{p}^{2})\omega - \frac{V\_{0\rm m}}{\beta\_{\rm r}}\omega^{3}\right]^{2}\right\}^{-1} \end{split} \tag{36}$$

$$\begin{split} Y\_{1}(\omega) &= -A\_{p}\omega \big[ (K\_{\rm c2}\mathbf{c}\_{1} + \frac{V\_{0}k\_{1}}{\frac{\beta\_{\rm r}}{\beta\_{\rm r}}} + A\_{p}^{2})\omega - \frac{V\_{0}m\_{1}}{\frac{\beta\_{\rm r}}{\beta\_{\rm r}}}\omega^{3} \big] \\ &\times \Big\{ \big[ \mathbf{k}\_{1}K\_{\rm c2} - \left(K\_{\rm c2}m\_{1} + \frac{V\_{0}\mathbf{c}\_{1}}{\frac{\beta\_{\rm r}}{\beta\_{\rm r}}}\right)\omega^{2} \big]^{2} + \Big[ (K\_{\rm c2}\mathbf{c}\_{1} + \frac{V\_{0}k\_{1}}{\frac{\beta\_{\rm r}}{\beta\_{\rm r}}} + A\_{p}^{2})\omega - \frac{V\_{0}m\_{1}}{\frac{\beta\_{\rm r}}{\beta\_{\rm r}}}\omega^{3} \big]^{-1} \end{split} \tag{37}$$

The intersection between *G*∗ <sup>1</sup>(iω) and the real axis is the critical point of the Popov frequency criterion. The coordinate is defined as (−*P*−<sup>1</sup> <sup>1</sup> , 0). The abscissa value of the critical point can be obtained by using Equations (36) and (37):

$$X\_1(\omega^\*) = -\frac{A\_p V\_0 m\_1 \beta\_\varepsilon}{\beta\_\varepsilon (K\_{\rm cr} m\_1 \beta\_\varepsilon + V\_0 c\_1)(K\_{\rm cr} c\_1 + A\_p^2) + V\_0^2 k\_1 c\_1}.\tag{38}$$

Then by the definition of Popov line, we can know that:

$$P\_1 = -\frac{1}{X\_1(\omega^\*)} = \frac{\beta\_\varepsilon (K\_{\varepsilon\varepsilon} m\_1 \beta\_\varepsilon + V\_0 c\_1)(K\_{\varepsilon\varepsilon} c\_1 + A\_p^2) + V\_0^2 k\_1 c\_1}{A\_p V\_0 m\_1 \beta\_\varepsilon}.\tag{39}$$

According to Popov's theorem [46,47], if the nonlinear characteristic function *f*1(Δ*e*) = *G*3(*s*)*Kq*Δ*e* of the position closed-loop system satisfies Equation (40), the equilibrium point of the system is absolutely stable, that is:

$$f(0) = 0, \ 0 < \frac{f\_1(\Delta \varepsilon)}{\Delta \varepsilon} \le P\_1. \tag{40}$$

From Equation (40), it can be concluded that if the characteristic curve of the nonlinear transfer function *G*3(*s*)*Kq* is located in the sector region, the position closed-loop system is globally asymptotically stable. The sector region is composed of the horizontal axis and the Popov line *l*<sup>1</sup> which passes through the origin with a slope *P*1, as shown in Figure 6a. Conversely, if the characteristic curve of *G*3(*s*)*Kq* exceeds the sector region (as illustrated in Figure 6b), the position closed-loop system is unstable. At this time, complex nonlinear dynamic behavior is likely to occur when the system parameters change.

**Figure 6.** Relation between the nonlinear characteristic curve of the position closed-loop system and *l*1.

From the above analysis, the absolute stability conditions of the position closed-loop system can be derived:

$$\mathcal{G}\_3(s)\mathcal{K}\_\emptyset \le \frac{\beta\_\varepsilon (\mathcal{K}\_{\varepsilon\varepsilon} m\_1 \beta\_\varepsilon + V\_0 c\_1)(\mathcal{K}\_{\varepsilon\varepsilon} c\_1 + A\_p^2) + V\_0^2 k\_1 c\_1}{A\_p V\_0 m\_1 \beta\_\varepsilon}. \tag{41}$$

The expression of *G*3(*s*) and *Kq* are substituted into Equation (41), then the absolute stability condition of the position closed-loop system when the spool displacement is positive (*xv* ≥ 0) can be obtained as:

$$\frac{\beta\_{\varepsilon}(K\_{\rm cr}m\_{1}\beta\_{\varepsilon} + V\_{0}c\_{1})(K\_{\rm cr}c\_{1} + A\_{p}^{2}) + V\_{0}^{2}k\_{1}c\_{1}}{A\_{p}V\_{0}m\_{1}\beta\_{\varepsilon}} \geq \frac{K\_{\rm p}(1 + \frac{1}{T\_{\rm sp}} + T\_{d}s)K\_{\rm d}K\_{\rm cr}}{(T\_{\rm rS} + 1)(\frac{s^{2}}{\omega\_{\rm sv}} + \frac{2L\_{\rm pr}}{\omega\_{\rm sv}}s + 1)}C\_{d} \mathcal{W} \sqrt{\frac{2(p\_{\rm s} - p\_{L})}{\rho}}.\tag{42}$$

When the spool displacement is negative (*xv* < 0), the absolute stability condition of the position closed-loop system can be acquired as:

$$\frac{\beta\_{\varepsilon}(K\_{\rm cr}m\_{1}\beta\_{\varepsilon} + V\_{0}c\_{1})(K\_{\rm cr}c\_{1} + A\_{p}^{2}) + V\_{0}^{2}k\_{1}c\_{1}}{A\_{p}V\_{0}m\_{1}\beta\_{\varepsilon}} \geq \frac{K\_{\rm pr}(1 + \frac{1}{T\_{\rm r}^{\rm s}} + T\_{d}\rm s)K\_{d}K\_{\rm cr}}{(T\_{\rm r}s + 1)(\frac{s^{2}}{a\lambda\_{\rm br}} + \frac{2\underline{\ell\_{\rm cr}}}{a\lambda\_{\rm br}}s + 1)} C\_{d}W\sqrt{\frac{2(p\_{L} - p\_{l})}{\rho}}.\tag{43}$$

#### **5. Conclusions**

In this paper, the function of key position closed-loop system in HAGC was introduced in detail. Based on the theoretical analysis, the mathematical model of each component was established. According to the connection relationship of each component element, the incremental transfer model of the position closed-loop system was derived. Moreover, according to the derived information transfer relationship, the transfer block diagram of the disturbance of the system was established. Furthermore, the Popov frequency criterion method was introduced to derive the absolute stability condition. The absolute stability conditions of the system are acquired in the following two conditions: when the spool displacement of the servo valve is positive or negative.

The obtained results lay a theoretical foundation for the study of the instability mechanism of the HAGC system. This research can provide a significant basis for the further investigation on the vibration traceability and control of the HAGC system.

**Author Contributions:** Conceptualization, Y.Z. and W.J.; Methodology, S.T.; Investigation, Y.Z. and S.T.; Writing-Original Draft Preparation, Y.Z.; Writing-Review & Editing, J.Z. and G.L.; Supervision, C.W.

**Funding:** This research was funded by National Natural Science Foundation of China (No. 51805214, 51875498), China Postdoctoral Science Foundation (No. 2019M651722), Natural Science Foundation of Hebei Province (No. E2018203339), Nature Science Foundation for Excellent Young Scholars of Jiangsu Province (No. BK20190101), Open Foundation of National Research Center of Pumps, Jiangsu University (No. NRCP201604) and Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems (No. GZKF-201714).

**Conflicts of Interest:** The authors declare no conflict of interest.
