A Gain Scheduling Design Method of the Aero-Engine Fuel Servo Constant Pressure Valve with High Accuracy and Fast Response Ability

Abstract

Constant pressure valve, which has an axially symmetric structure, is an essential component to supply the servo reference pressure for aero-engine hydraulic mechanical control systems, and its performance directly impacts the performance of the control system. This paper constructs a closed-loop disturbance rejection system of the constant pressure valve based on the linear incremental description method, in which the controlled object is the controlled pressure, and the stabilization controller is constructed by the closed-loop feedback motion valve. Meanwhile, the linear models of the closed-loop system are calculated. Subsequently, based on the bode frequency domain characteristic analysis, the accurate influences of the stabilization control gain on the dynamic performance and stability of the system are given. On this basis, a gain scheduling design method of the system is proposed, and the geometry design and implementation method of the inlet orifice is proposed to complete the design work. The simulation results show that under bad conditions, which include the 1 MPa strong step disturbance of the inlet pressure and the step disturbance of the variable outlet flow area, the steady-state working range of the controlled pressure is 1.5 ± 0.01 MPa, the steady-state error is not more than 0.7%, and the regulation time is not more than 0.006 s.

1. Introduction

Constant pressure valve is the most essential and key component in the aero-engine hydraulic mechanical control system. It regulates the oil pressure after the pump within a wide working range to the required pressure to provide a servo reference pressure and ensure the stable work of the fuel metering system. Its performance directly affects the stability and dynamic performance of the control system [1,2,3]. For example, if the constant pressure valve works unstably, the output fuel pressure will fluctuate, causing engine combustion fluctuation and speed fluctuation; if it responds slowly, the servo pressure regulation will be delayed and inaccurate, and the function of the control system will be lost, resulting in engine abnormality [4,5]. Hence, the analysis and design work of the constant pressure valve with high responsiveness and high accuracy is a core issue.

In the early stage, studies on the constant pressure valve mainly focused on modeling and stability analysis. Based on the classical control theory, the transfer function block diagram of the system was established after deriving the dynamic equations, then the static and dynamic characteristics, as well as the stability, were analyzed. On this basis, some analog signal simulation works were conducted, and the simulation results showed that the regulation time of the system reached almost 40 ms [6,7,8,9,10]. However, these studies do not reveal the design theory of the system, and the analysis results are unclear. Moreover, the established linear models are complex, and they cannot clarify the decisive relationship between the design parameters and the system performance; of course, the performance of the designed system is defective. Although these works provide a research basis, the proposed methods are inefficient and cumbersome and should be improved. With the wide application of the simulation technology in recent years, research works on the constant pressure valve primarily concentrated on modeling and system performance analysis, yet theoretical exploration is still rare. Based on the nonlinear simulation, the influences of the pressure-bearing area, spring stiffness, and some other parameters on the system performance were directly studied, and some guidance measures for the performance parameters design are provided [11,12,13]. However, these works are solely implemented on the nonlinear models, and they do not involve any theoretical analysis in the research processes, which is undesirable. In addition, some studies established linear models, carried out the theoretical analysis, and improved the performance of the system, specifically, the regulation time reached 20 ms [14,15,16,17,18,19,20]. Unfortunately, these research works follow the traditional analysis methods, and the theoretical analysis results cannot explain the influence of the design parameters on the system performance clearly. Thus, the trial-and-error method is the main research method, which is inefficient. Moreover, there are some reports about the nonlinear analysis of the system stability [21,22]. The results show that the structural parameters of the system had a great influence on its nonlinear dynamic behavior, and increasing the mass and orifice diameter properly and reducing the spring stiffness were conducive to the stabilization of the system. Although these works do not involve the design methods, they provide a new perspective, and the method is instructive. Additionally, there were some studies on improving the system performance based on physical tests [23,24]. Nevertheless, these works lack adequate theoretical analysis and discussion of the results.

These research works have all contributed significantly; however, they all relied on traditional methods rather than the modern control theory. As a result, they cannot complete the accurate system design because the analysis results are inaccurate and the guidance measures are inefficient. This study first applies the state space theory to analyze the design theory of the system and proposes effective design methods and guidance measures to realize the design task. The contributions include:

• Firstly, the design theory of the system is revealed successfully by using the linear incremental analysis method. Evidently, the controlled object and the stabilization controller are the two fundamental units of the closed-loop system, as demonstrated by the study results.
• Secondly, the accurate loop transfer function models of the system are calculated. In addition, the precise influences of the design parameters on the dynamic performance and stability are analyzed by using the bode frequency domain analysis methods, and some useful guidance measures are offered.
• Finally, a gain scheduling design method for the stabilization control gain is proposed, and the proposed method is demonstrated to exactly solve the design work based on the nonlinear simulation.

This paper is organized as follows. In Section 2, the design theory of the system is analyzed, and the compositions of the system and the dynamic equations are given. In Section 3, the linear models are established. In Section 4, the precisive influences of the stabilization control gain on the dynamic performance and stability are analyzed. In Section 5, the gain scheduling design method is provided. In Section 6, a design example is established, and the nonlinear simulation is implemented. In Section 7, the conclusions are presented.

2. Design Analysis and Dynamic Equations

The structure diagram of the constant pressure valve is shown in Figure 1.

where $P_{\mathrm{in}}$ is the inlet pressure, $P_C$ is the controlled pressure, $P_0$ is the return pressure, $A_{\mathrm{in}}$ is the inlet flow area, $A_{\mathrm{out}1}$ is the fixed outlet flow area, $A_{\mathrm{out}2}$ is the variable outlet flow area, $V_0$ is the volume of the controlled chamber, and $x_y$ is the displacement of the motion valve.

The controlled object is the controlled pressure $P_C$, and it is used to supply the servo reference pressure for other valves. Then, it returns to the oil tank through the return orifice of other valves, which is approximately represented by the variable outlet flow area $A_{\mathrm{out}2}$. Generally, the regulation processes can be described as: when the controlled pressure $P_C$ changes caused by the variable inlet pressure $P_{\mathrm{in}}$ or the variable outlet flow area $A_{\mathrm{out}2}$, the motion valve senses a change in the pressure difference $\left(P_C-P_0\right)$ to regulate the inlet flow area $A_{\mathrm{in}}$, which is the control input of the controlled object, realizing the regulation of the controlled pressure $P_C$ [25,26,27,28].

2.1. Design Analysis

The constant pressure valve is a typical closed-loop disturbance rejection system. Its design objective is to reject the disturbances on the controlled pressure increment $\Delta P_C$ caused by the variable inlet pressure increment $\Delta P_{\mathrm{in}}$ and the variable outlet flow area increment $\Delta A_{\mathrm{out}2}$, and ensure the controlled pressure increment $\Delta P_C$ is zero. Clearly, the composition of the closed-loop system is shown in Figure 2, and the derivation processes are shown in Section 2.2.

where:

• $G_{\mathrm{fp}}$ is the dynamic matrix of the controlled object.
• $G_{\mathrm{cpv}}$ is the dynamic matrix of the motion valve, $K_C$ is the generalized stabilization control gain, and they construct the stabilization controller.

Generally, the reference input $\Delta P_0$ is zero, and the main design objective is to reject disturbances. Through the feedback regulation function of the stabilization controller, the system is guaranteed to be asymptotically stable, and the disturbances caused by the disturbance inputs $\Delta P_{\mathrm{in}}$ and $\Delta A_{\mathrm{out}2}$ are rejected.

2.2. Dynamic Equations

2.2.1. Controlled Object

The pressure-flow nonlinear dynamic equation of the controlled chamber is

$$\frac{d{P}_C}{dt}=\frac{B}{V_0}\cdot (C_q{A}_{\mathrm{in}}\sqrt{\frac{2(P_{\mathrm{in}}-P_C)}{\rho }}-C_q\left(A_{\mathrm{out}2}+A_{\mathrm{out}1}\right)\sqrt{\frac{2(P_C-P_0)}{\rho }}-A_y\cdot {\dot{x}}_y)$$

(1)

where $B$ is the oil bulk modulus, $\rho$ is the oil density, and $C_q$ is the flow coefficient.

The pressure-flow linear dynamic differential equation can be described as

$$\frac{d\Delta P_C}{dt}=\frac{B}{V_0}\cdot (K_{\mathrm{Ain}}\cdot \Delta A_{\mathrm{in}}+K_{\mathrm{Pin}}\cdot (\Delta P_{\mathrm{in}}-\Delta P_C)-(K_{\mathrm{Aout}}\cdot \Delta A_{\mathrm{out}2}+K_{\mathrm{Pout}}\cdot \Delta P_C)-A_y\cdot \Delta {\dot{x}}_y)$$

(2)

where $K_{\mathrm{Ain}}=C_q\sqrt{\frac{2(P_{\mathrm{in}}-P_C)}{\rho }}$, $K_{\mathrm{Pin}}=C_q{A}_{\mathrm{in}}\sqrt{\frac{1}{2\rho ({\mathrm{P}}_{\mathrm{in}}-P_C)}}$, $K_{\mathrm{Aout}}=C_q\sqrt{\frac{2(P_C-P_0)}{\rho }}$, $K_{\mathrm{Pout}}=C_q\left(A_{\mathrm{out}2}+A_{\mathrm{out}1}\right)\sqrt{\frac{1}{2\rho (P_C-P_0)}}$.

Then, the state space model of the controlled object is

$$\begin{array}{l}{\dot{x}}_p=A_p{x}_p+B_p{u}_p+E_{p}w\\ y=C_p{x}_p+D_p{u}_p\end{array}$$

(3)

where $x_p=\left[\Delta P_C\right]$, $u_p={\left[\begin{array}{cc}\Delta A_{\mathrm{in}}& \Delta {\dot{x}}_y\end{array}\right]}^T$, $w={\left[\begin{array}{cc}\Delta P_{\mathrm{in}}& \Delta A_{\mathrm{out}2}\end{array}\right]}^T$, $y=\left[\Delta P_C\right]$,

$$\begin{array}{l}{A}_p=\left[-\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right],B_p=\left[\begin{array}{cc}\frac{B}{V_0}\cdot K_{\mathrm{Ain}}& -\frac{B}{V_0}\cdot A_y\end{array}\right],E_p=\left[\begin{array}{cc}\frac{B}{V_0}\cdot K_{\mathrm{Pin}}& -\frac{B}{V_0}\cdot K_{\mathrm{Aout}}\end{array}\right]\\ C_p=\left[1\right],D_p=\left[\begin{array}{cc}0& 0\end{array}\right]\end{array}$$

(4)

2.2.2. Stabilization Controller

By taking the steady-state working point $x_{y0}=0$ as the local coordinate origin, $x_y$ as the relative displacement, and the arrow direction as the positive direction, the local coordinates for the motion valve is established.

The motion nonlinear dynamic equation of the motion valve is

$$M_y{\ddot{x}}_y=A_y{P}_C-A_y{P}_0-K_f{\dot{x}}_y-K{x}_y-F_L$$

(5)

where $M_y$ is the mass, $K_f$ is the viscous friction coefficient, $K$ is the spring stiffness, $A_y$ is the pressure-bearing area, and $F_L$ is the initial spring compression force.

The motion linear dynamic differential equation can be described as

$$M_y\cdot \Delta {\ddot{x}}_y=A_y\cdot \left(\Delta P_C-\Delta P_0\right)-K_f\cdot \Delta {\dot{x}}_y-K\cdot \Delta x_y$$

(6)

A function $A_{\mathrm{in}}=f_{\mathrm{in}}(x_u)$ is used to express the geometry design of the inlet orifice. According to its linearized gain characteristics

$$\Delta A_{\mathrm{in}}=\frac{d{f}_{\mathrm{in}}}{d{x}_u}\cdot \Delta x_u$$

(7)

and $\Delta x_u=-\Delta x_y$, the gain control law of the stabilization controller is expressed as

$$\Delta A_{\mathrm{in}}=-K_C\cdot \Delta x_y$$

(8)

where $K_C=\frac{d{f}_{\mathrm{in}}}{d{x}_u}$, and it is the generalized stabilization control gain.

Then, the state space model of the stabilization controller is

$$\begin{array}{l}{\dot{x}}_s=A_s{x}_s+B_{s}y+E_{s}r\\ u_s=C_s{x}_s+D_{s}y\end{array}$$

(9)

where $x_s={\left[\begin{array}{cc}\Delta x_y& \Delta {\dot{x}}_y\end{array}\right]}^T$, $r=\left[\Delta P_0\right]$, $u_s=\left[\Delta A_{\mathrm{in}}\right]$,

$$\begin{array}{l}{A}_s=\left[\begin{array}{cc}0& 1\\ -\frac{K}{M_y}& -\frac{K_f}{M_y}\end{array}\right],B_s=\left[\begin{array}{c}0\\ \frac{A_y}{M_y}\end{array}\right],E_s=\left[\begin{array}{c}0\\ -\frac{A_y}{M_y}\end{array}\right]\\ C_s=\left[\begin{array}{cc}-K_C& 0\end{array}\right],D_s=\left[0\right]\end{array}$$

(10)

3. Linear Models

When studied by the loop transfer functions, the relationship between the designed parameters and the system performance is explicit. The loop transfer functions of the system are then calculated and used to analyze the accurate influence of the designed parameters on the dynamic performance. The design block diagram of the closed loop is shown in Figure 3, and the calculation procedures of the loop transfer functions are as follows.

3.1. Open-Loop Transfer Function

The augmented open-loop state space model of the system is

$$\begin{array}{l}\dot{x}=A_{\mathrm{open}}x+B_{\mathrm{open}}u+E_{\mathrm{open}}w\\ y=C_{\mathrm{open}}x+D_{\mathrm{open}}u\end{array}$$

(11)

where $x={\left[\begin{array}{ccc}\Delta P_C& \Delta x_y& \Delta {\dot{x}}_y\end{array}\right]}^T$, $u=\left[e_u\right]$,

$$\begin{array}{l}{A}_{\mathrm{open}}=\left[\begin{array}{ccc}-\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})& -\frac{B}{V_0}\cdot K_{\mathrm{Ain}}\cdot K_C& -\frac{B}{V_0}\cdot A_y\\ 0& 0& 1\\ 0& -\frac{K}{M_y}& -\frac{K_f}{M_y}\end{array}\right]\\ B_{\mathrm{open}}=\left[\begin{array}{c}0\\ 0\\ -\frac{A_y}{M_y}\end{array}\right],E_{\mathrm{open}}=\left[\begin{array}{cc}\frac{B}{V_0}\cdot K_{\mathrm{Pin}}& -\frac{B}{V_0}\cdot K_{\mathrm{Aout}}\\ 0& 0\\ 0& 0\end{array}\right]\\ C_{\mathrm{open}}=\left[\begin{array}{ccc}1& 0& 0\end{array}\right],D_{\mathrm{open}}=\left[0\right]\end{array}$$

(12)

Then, the open-loop transfer function is

$$L=C_{\mathrm{open}}{(sI-A_{\mathrm{open}})}^{-1}{B}_{\mathrm{open}}=\frac{\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot A_y\cdot (s+\frac{K_{\mathrm{Ain}}}{A_y}\cdot K_C)}{\left(s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right)\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)}$$

(13)

3.2. Disturbance Input Transfer Functions

Defining

$$E_1=\left[\frac{B}{V_0}\cdot K_{\mathrm{Pin}}\right],E_2=\left[-\frac{B}{V_0}\cdot K_{\mathrm{Aout}}\right]$$

(14)

Then, the open-loop disturbance transfer function from $\Delta P_{\mathrm{in}}$ to $\Delta P_C$ is

$$G_{\mathrm{Pin}-\mathrm{Pc}}=C{(sI-A)}^{-1}{E}_1=\frac{\frac{B}{V_0}\cdot K_{\mathrm{Pin}}}{s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})}$$

(15)

The open-loop disturbance transfer function from $\Delta A_{\mathrm{out}2}$ to $\Delta P_C$ is

$$G_{\mathrm{Aout}2-\mathrm{Pc}}=C{(sI-A)}^{-1}{E}_2=\frac{-\frac{B}{V_0}\cdot K_{\mathrm{Aout}}}{s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})}$$

(16)

3.3. Control Output Transfer Functions

The closed-loop sensitivity function is defined as

$$S={\left(1+L\right)}^{-1}=\frac{\left(s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right)\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)}{\left(s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right)\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)+\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot A_y\cdot (s+\frac{K_{\mathrm{Ain}}}{A_y}\cdot K_C)}$$

(17)

Assuming $\Delta P_0=0$, the control output transfer function of the system is

$$\Delta P_C=S\cdot G_{\mathrm{Pin}-\mathrm{Pc}}\cdot \Delta P_{\mathrm{in}}+S\cdot G_{\mathrm{Aout}2-\mathrm{Pc}}\cdot \Delta A_{\mathrm{out}2}$$

(18)

Specifically:

• The closed-loop disturbance transfer function from $\Delta P_{\mathrm{in}}$ to $\Delta P_C$ is

  $$S_{\mathrm{Pin}}=S\cdot G_{\mathrm{Pin}-\mathrm{Pc}}=\frac{\frac{B}{V_0}\cdot K_{\mathrm{Pin}}\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)}{\left(s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right)\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)+\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot A_y\cdot (s+\frac{K_{\mathrm{Ain}}}{A_y}\cdot K_C)}$$

  (19)

Its steady-state gain is

$$K_P=\frac{K_{\mathrm{Pin}}}{(K_{\mathrm{Pin}}+K_{\mathrm{Pout}})+\frac{A_y}{K}\cdot (K_{\mathrm{Ain}}\cdot K_C)}$$

(20)

2.

The closed-loop disturbance transfer function from $\Delta A_{\mathrm{out}2}$ to $\Delta P_C$ is

$$S_{\mathrm{Aout}2}=S\cdot G_{\mathrm{Aout}2-\mathrm{Pc}}=\frac{-\frac{B}{V_0}\cdot K_{\mathrm{Aout}}\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)}{\left(s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right)\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)+\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot A_y\cdot (s+\frac{K_{\mathrm{Ain}}}{A_y}\cdot K_C)}$$

(21)

Its steady-state gain is

$$K_A=\frac{-K_{\mathrm{Aout}}}{(K_{\mathrm{Pin}}+K_{\mathrm{Pout}})+\frac{A_y}{K}\cdot (K_{\mathrm{Ain}}\cdot K_C)}$$

(22)

4. Bode Frequency Domain Characteristic Analysis

The corner frequencies and the open-loop gain of the bode frequency domain curve determine the system performance, and the key concept is to analyze the precise influences of the design parameters on the corner frequencies and the open-loop gain. The analysis procedures are as follows.

4.1. Calculation of the Corner Frequencies

The corner frequencies of the bode frequency domain curve are calculated according to Equation (13) as follows.

• If $K>\frac{K_f^2}{4\cdot M_y}$, the second-order section is an underdamped oscillation link, and its poles are

  $$p_{1,2}=-\frac{K_f}{2\cdot M_y}\pm j\cdot \sqrt{\frac{K}{M_y}-{\left(\frac{K_f}{2\cdot M_y}\right)}^2}$$

  (23)

The corner frequencies are defined as

$${\omega }_{1,2}=\frac{K_f}{2\cdot M_y}$$

(24)

2.

If $K<\frac{K_f^2}{4\cdot M_y}$, the second-order section is an overdamped non-oscillation link, and its poles are

$$p_{1,2}=-\frac{K_f}{2\cdot M_y}\pm \sqrt{{\left(\frac{K_f}{2\cdot M_y}\right)}^2-\frac{K}{M_y}}$$

(25)

The corner frequencies are defined as

$${\omega }_{1,2}=\frac{K_f}{2\cdot M_y}\pm \sqrt{{\left(\frac{K_f}{2\cdot M_y}\right)}^2-\frac{K}{M_y}}$$

(26)

The third pole is

$$p_3=-\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})$$

(27)

The corner frequency is defined as

$${\omega }_4=\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})$$

(28)

The zero is

$$z_1=-\frac{K_{\mathrm{Ain}}}{A_y}\cdot K_C$$

(29)

The corner frequency is defined as

$${\omega }_3=\frac{K_{\mathrm{Ain}}}{A_y}\cdot K_C$$

(30)

The open-loop gain is

$$K_{open}=\frac{K_{Ain}\cdot A_y\cdot K_C}{(K_{Pin}+K_{Pout})\cdot K}$$

(31)

The above calculation equations of the corner frequencies demonstrate that:

• The stabilization control gain $K_C$ mainly influences ${\omega }_3$ and the open-loop gain $K_{\mathrm{open}}$;
• The spring stiffness $K$ mainly influences ${\omega }_1$, ${\omega }_2$, and the open-loop gain $K_{\mathrm{open}}$;
• The volume $V_0$ of the controlled chamber mainly influences ${\omega }_4$;
• The pressure-bearing area $A_y$ mainly influences ${\omega }_3$ and the open-loop gain $K_{\mathrm{open}}$;
• The mass $M_y$ mainly influences ${\omega }_1$ and ${\omega }_2$.

4.2. Analysis of the Frequency Domain Characteristic

Designing the four corner frequencies and the open-loop gain so that the system can meet the required steady-state and dynamic and robust performance is the key part of the work. For instance:

By increasing the corner frequencies ${\omega }_1$ and ${\omega }_2$ or decreasing the open-loop gain to make the phase curve move up or the magnitude curve move down, the stability margin is improved. The measures include:

• Reducing the stabilization control gain $K_C$;
• Reducing the mass of the motion valve $M_y$;
• Increasing the viscous friction coefficient $K_f$.

By increasing the corner frequency ${\omega }_3$ or the open-loop gain $K_{\mathrm{open}}$ to make the magnitude curve move right or up, the response performance is improved. The measures include:

• Increasing the stabilization control gain $K_C$;
• Reducing the spring stiffness $K$;
• Reducing the pressure-bearing area $A_y$ of the motion valve.

4.3. Influence of the Stabilization Control Gain on the Frequency Domain Characteristic

Assuming $K<\frac{K_f^2}{4\cdot M_y}$ and only changing $K_C$, the corresponding bode curves are shown in Figure 4.

According to the bode diagram curves, when the control gain $K_C$ increases, the corner frequency ${\omega }_3$ increases, and the open-loop gain $K_{\mathrm{open}}$ increases, then:

• According to Equations (20) and (22), the steady-state gain $K_P$ and $K_A$ decrease, thus the steady error caused by the disturbance inputs is reduced, resulting in better disturbance rejection performance;
• The crossover frequency ${\omega }_c$ increases and the regulating time is reduced, resulting in faster response performance;
• The phase margin decreases, resulting in worse robustness performance.

4.4. Stability Analysis

The characteristic polynomial of the closed-loop system is

$$d(s)=\left(s+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right)\cdot \left(s^2+\frac{K_f}{M_y}s+\frac{K}{M_y}\right)+\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot A_y\cdot (s+\frac{K_{\mathrm{Ain}}}{A_y}\cdot K_C)$$

(32)

The equation is expressed as

$$d(s)=a_0\cdot s^3+a_1\cdot s^2+a_2\cdot s+a_3$$

(33)

where

$$a_0=1$$

(34)

$$a_1=\left[\frac{K_f}{M_y}+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\right]$$

(35)

$$a_2=\left[\frac{K}{M_y}+\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\cdot \frac{K_f}{M_y}+\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot A_y\right]$$

(36)

$$a_3=\left[\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\cdot \frac{K}{M_y}+\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot K_{\mathrm{Ain}}\cdot K_C\right]$$

(37)

According to the stability conditions of the third-order system: All the coefficients of the characteristic polynomial should be positive and $a_1{a}_2>a_0{a}_3$. It has

$$K_C<K_{C\max }=\frac{a_1{a}_2-\frac{B}{V_0}\cdot (K_{\mathrm{Pin}}+K_{\mathrm{Pout}})\cdot \frac{K}{M_y}}{\frac{B}{V_0}\cdot \frac{A_y}{M_y}\cdot K_{\mathrm{Ain}}}$$

(38)

5. Gain Scheduling Design Method

The design objectives for the steady-state and dynamic performance are as follows:

• Steady-state performance: The designed controlled pressure is $P_C$, and the working range is $[P_{C,l},P_{C,h}]$.
• Dynamic performance: The regulation time is not greater than $t_s$.

5.1. Dynamic Design

The system can be regarded as a fixed system with multiple stable working points, each of which has a fixed stabilization control gain. Therefore, the gain scheduling design method is suitable. The fundamental concept is to first determine the steady-state parameters and then calculate the control gain.

5.1.1. Calculation of the Stabilization Control Gain

The steady-state flow balance equation of the controlled object is described as

$$C_q{A}_{\mathrm{in}}\sqrt{\frac{2(P_{\mathrm{in}}-P_C)}{\rho }}=C_q\left(A_{\mathrm{out}2}+A_{\mathrm{out}1}\right)\sqrt{\frac{2(P_C-P_0)}{\rho }}$$

(39)

Then, the steady-state flow area of the inlet orifice is calculated as

$$A_{\mathrm{in}}=\left(A_{\mathrm{out}2}+A_{\mathrm{out}1}\right)\sqrt{\frac{(P_C-P_0)}{(P_{\mathrm{in}}-P_C)}}$$

(40)

Thereby, $K_{\mathrm{Ain}}$, $K_{\mathrm{Pin}}$, $K_{\mathrm{Aout}}$, and $K_{\mathrm{Pout}}$ can be obtained. Then:

(1) According to the stability constraint, the control gain should meet

$$K_C<K_{C\max }$$

(41)

(2) According to the dynamic performance requirements, the control gain should meet

$$K_C\in [K_{C,l},K_{C,h}]$$

(42)

where $K_{C,l}$ and $K_{C,h}$ are the minimum and maximum values meeting the dynamic performance, respectively.

5.1.2. Constraint Condition

According to the steady-state balance condition of the motion valve, the steady-state spring compression is

$$x_{s-s}=\frac{A_y}{K}\cdot (P_{C,0}-P_0)$$

(43)

where $P_{C,0}$ is the designed controlled pressure.

According to the steady-state performance requirements $P_C\in [P_{C,l},P_{C,h}]$, the variation range of the spring compression is

$$x_s\in [x_{s-s}-\frac{A_y}{K}\cdot (P_{C,0}-P_{C,l}),x_{s-s}+\frac{A_y}{K}\cdot (P_{C,h}-P_{C,0})]$$

(44)

The steady-state performance can only be satisfied if the displacement variation range of the motion valve is less than the spring compression variation range. Because of $\Delta x_u=-\Delta x_y=-\Delta x_s$, it has

$$\left|\Delta x_u\right|\le \left|\Delta x_{s-s}\right|$$

(45)

where $\Delta x_{s-s}=\frac{A_y}{K}\cdot (P_{C,h}-P_{C,l})$.

5.2. Calculation processes

The calculation processes of the gain scheduling design method is given as follows:

Step 1. The structural design scheme of the constant pressure valve is selected, then the variation range of the spring compression $\Delta x_{s-s}$ can be calculated.

Step 2. $n\times m$ steady-state working points are selected within the working range of the disturbance inputs $P_{\mathrm{in}}$ and $A_{\mathrm{out}2}$, and the parameters$K_{\mathrm{Ain}}$, $K_{\mathrm{Pin}}$, $K_{\mathrm{Aout}}$, and $K_{\mathrm{Pout}}$ at each steady-state point are calculated. Then, the open-loop transfer function can be obtained, and it has

$$L_i=f(K_{C,i}),i=1,\cdots ,n\times m$$

(46)

Step 3. According to Equations (41) and (42), it is determined that $K_{C,i}$, $i=1,\cdots ,n\times m$.

Step 4. According to the geometric characteristics of the inlet orifice $\Delta A_{\mathrm{in}}=K_C\cdot \Delta x_u$, the iterative equations of the underlap $x_{u,i}$, the inlet flow area $A_{\mathrm{in},i}$, and the control gain $K_{C,i}$ is

$$x_{u,i}=x_{u,i-1}+\frac{2\cdot (A_{\mathrm{in},i}-A_{\mathrm{in},i-1})}{(K_{C,i}+K_{C,i-1})},i=1,\cdots ,n\times m$$

(47)

Step 5. If the variation range of inlet orifice underlap increment $\Delta x_u$ satisfies Equation (45), proceed to Step 6; Otherwise, return to Step 1 and repeat the above steps.

Step 6. Determine the initial parameters: After designing the initial spring compression $x_{s0}$, according to the steady-state spring compression $x_{s-s}$, the displacement range of the motion valve $\Delta x_y$ is

$$\Delta x_y=x_{s-s}-x_{s0}$$

(48)

Then the initial underlap of the inlet orifice is

$$x_{u0}=x_u+\Delta x_y$$

(49)

6. Design and Simulation

The given structural parameters of the constant pressure valve are shown in

Table 1. Structural parameters of the constant pressure valve.

Parameter/Unit	Value	Parameter/Unit	Value
$M_y$/Kg	0.0115	$V_0$/m3	10−5
$K_f$/(N·m/s)	10	$\rho$/(Kg/m3)	800
$d$/m	0.008	$B$/bar	17,000
$P_C$/MPa	1.5	$d_{out1}$/m	0.001
$P_0$/MPa	0.3	$C_q$	0.7

.

In addition, there are four available spring stiffness values of 750, 1500, 3000, and 4500 N/m, and the working range of the disturbance inputs are:

• The working range of the inlet pressure $P_{\mathrm{in}}$ is [2,8] MPa;
• The working range of the variable outlet flow area $A_{\mathrm{out}2}$ is [0, 3.1416] × 10⁻⁶ m².

The design objects are:

• The designed controlled pressure $P_C$ is 1.5 ± 0.01 MPa;
• The designed regulating time $t_s$ is not more than 0.006 s.

The design task is:

• Designing the stabilization control law as $A_{\mathrm{in}}=f_{\mathrm{in}}(x_u)$.

6.1. Design

Step 1. The selected spring stiffness value is 4500 N/m, then:

• Since the designed pressure $P_C$ is 1.5 MPa, according to Equation (43), the steady-state spring compression $x_{s-s}$ is 13.404129 mm;
• Since the design range of $P_C$ is [1.49, 1.51] MPa, according to Equation (44), the variation of the spring compression $\Delta x_{s-s}$ is 0.22340214 mm.

Step 2. We select $4\times 3$ steady-state working points within the working range of the disturbance inputs $P_{\mathrm{in}}$ and $A_{\mathrm{out}2}$, and calculate the values $[A_{\mathrm{in},i},K_{C,i}]$ at each steady-state point using the method proposed in Section 5.2, as shown in

Table 2. Values of the steady-state inlet flow area and the designed stabilization control gain.

${\mathit{P}}_{\mathit{i}\mathit{n}}$/MPa	${\mathit{d}}_{\mathit{out}2}$/mm	${\mathit{A}}_{\mathrm{in}}$/mm2	${\mathit{K}}_{\mathit{C}\max }$	$[{\mathit{K}}_{\mathit{C}\mathit{l}},{\mathit{K}}_{\mathit{C}\mathit{h}}]$	${\mathit{K}}_{\mathit{C}}$
2	2	6.08366800	0.139094570	[0.002, 0.007]	0.0068
	1	2.43346720	0.041787194	[0.002, 0.015]	0.0048
	0	1.21673360	0.019299005	[0.002, 0.009]	0.0028
3	2	3.51240740	0.034057185	[0.001, 0.006]	0.0058
	1	1.40496290	0.011826533	[0.001, 0.0055]	0.0028
	0	0.70248147	0.006026016	[0.001, 0.0025]	0.0018
4	2	2.72069900	0.020693472	[0.0010, 0.005]	0.0048
	1	1.08827960	0.007491835	[0.0005, 0.003]	0.0028
	0	0.54413981	0.003934234	[0.0005, 0.001]	0.0008
5	2	2.29941040	0.015556697	[0.0008, 0.0045]	0.0043
	1	0.91976415	0.005747635	[0.0007, 0.0027]	0.0025
	0	0.45988207	0.003064452	[0.0005, 0.0009]	0.0007
6	2	2.02788930	0.012802030	[0.0005, 0.004]	0.0038
	1	0.81115574	0.004787408	[0.0005, 0.002]	0.0018
	0	0.40557787	0.002576085	[0.0005, 0.001]	0.0008
7	2	1.83429490	0.011061443	[0.0005, 0.0035]	0.0033
	1	0.73371797	0.004169889	[0.0005, 0.0020]	0.0018
	0	0.36685898	0.002257730	[0.00055, 0.00065]	0.0004
8	2	1.68730590	0.009849004	[0.0005, 0.0030]	0.0028
	1	0.67492237	0.003734197	[0.0005, 0.0017]	0.0015
	0	0.33746118	0.002030850	[0.0004, 0.0005]	0.0003

.

For instance, assuming the inlet pressure of the first steady-state working point $P_{\mathrm{in},1}$ is 4 MPa, and the variable outlet flow area $A_{\mathrm{out}2,1}$ is 0.7854 m², then

$$A_{\mathrm{in},1}=\left(A_{\mathrm{out}2,1}+A_{\mathrm{out}1}\right)\sqrt{\frac{(P_C-P_0)}{(P_{\mathrm{in},1}-P_C)}}=1.0882796 {\mathrm{mm}}^2$$

(50)

Then, the values of the parameters $K_{\mathrm{AJ},1}$, $K_{\mathrm{PJ},1}$, $K_{\mathrm{AZ},1}$, $K_{\mathrm{PZ},1}$, and $K_{\mathrm{PT},1}$ are obtained.

Step 3. (1) According to stability Equation (41), it has

$$K_C<K_{C\max ,1}=0.007491835$$

(51)

(2) According to the dynamic performance requirements, the design range of the control gain is $K_C\in [0.0005,0.003]$, and the calculation process is as follows.

When the stabilization control gain changes, the bode curves of the open-loop transfer function are shown in Figure 5, and the step curves of the controlled pressure are shown in Figure 6a,b. In addition, the relational curve of the inlet orifice flow area and the designed stabilization control gain is shown in Figure 7.

Step 4. The values $[\Delta x_{u,i},A_{in,i}]$ at each steady-state point are calculated, as shown in

Table 3. The values $[\Delta x_{u,i},A_{in,i}]$ at each steady-state point when k = 4500 N/m.

${\mathit{A}}_{\mathit{i}\mathit{n}}$ mm2	${\mathit{K}}_{\mathit{C}}$	$\mathit{\Delta }{\mathit{x}}_{\mathit{u}}$ mm
0.33746118	0.0003	0
0.36685898	0.0004	0.097992667
0.40557787	0.0008	0.19478989
0.45988207	0.0007	0.26267014
0.54413981	0.0008	0.38303834
0.67492237	0.0015	0.54651654
0.70248147	0.0018	0.56488927
0.73371797	0.0018	0.58224289
0.81115574	0.0018	0.62526387
0.91976415	0.0025	0.68560187
1.0882796	0.0028	0.75300805
1.2167336	0.0028	0.79888448
1.4049629	0.0028	0.86610923
1.6873059	0.0028	0.96694602
1.8342949	0.0033	1.01944210
2.0278893	0.0038	1.07810710
2.2994104	0.0043	1.14956000
2.4334672	0.0048	1.18073600
2.7206990	0.0048	1.24057590
3.5124074	0.0058	1.40551520
6.0836680	0.0068	1.84883600

.

Step 5. Because of $\Delta x_u>\Delta x_{s-s}$, the selected spring stiffness value is unreasonable. Therefore, the selected spring stiffness value is 750 N/m, then the corresponding steady-state spring compression $x_{s-s}$ is 80.424772 mm, and the variation of the spring compression $\Delta x_{s-s}$ is 1.3404129 mm. The above design steps are repeated to obtain

Table 4. The values $[\Delta x_{u,i},A_{in,i}]$ at each steady-state point when k = 750 N/m.

${\mathit{A}}_{\mathit{i}\mathit{n}}$ mm2	${\mathit{K}}_{\mathit{C}}$	$\mathit{\Delta }{\mathit{x}}_{\mathit{u}}$ mm
0.33746118	0.0006	0
0.36685898	0.0007	0.045227385
0.40557787	0.0008	0.096852571
0.45988207	0.0008	0.16473282
0.54413981	0.0015	0.23800042
0.67492237	0.0017	0.31973952
0.70248147	0.0027	0.33226638
0.73371797	0.0018	0.34614927
0.81115574	0.0025	0.38216684
0.91976415	0.0027	0.42393931
1.0882796	0.0039	0.47500459
1.2167336	0.0100	0.49348718
1.4049629	0.0049	0.51875286
1.6873059	0.0040	0.58220073
1.8342949	0.0038	0.61989021
2.0278893	0.0055	0.66152342
2.2994104	0.0049	0.71373902
2.4334672	0.0160	0.72656742
2.7206990	0.0080	0.75050340
3.5124074	0.0075	0.85265932
6.0836680	0.0060	1.23358680

.

Because of $\Delta x_u=1.2335868<\Delta x_{s0}=1.3404129$, the selected spring stiffness value is reasonable.

The designed values of the underlap and the flow area are shown in

Table 5. The designed values of the underlap and the flow area of the inlet orifice.

${\mathit{x}}_{\mathit{u}}$ mm	${\mathit{A}}_{\mathit{i}\mathit{n}}$ mm2
0	0
0.2	0.33746118
0.24522738	0.36685898
0.29685257	0.40557787
0.36473282	0.45988207
0.43800042	0.54413981
0.51973952	0.67492237
0.53226638	0.70248147
0.54614927	0.73371797
0.58216684	0.81115574
0.62393931	0.91976415
0.67500459	1.0882796
0.69348718	1.2167336
0.71875286	1.4049629
0.78220073	1.6873059
0.81989021	1.8342949
0.86152342	2.0278893
0.91373902	2.2994104
0.92656742	2.4334672
0.95050340	2.7206990
1.05265930	3.5124074

.

The geometry design diagram of the inlet orifice is shown in Figure 8.

Step 6. The underlap of the inlet orifice at the design working point $x_u$ is designed as 0.86184456 mm. Designing the initial spring compression $x_{s0}$ as 80 mm, since the steady-state spring compression $x_{s-s}$ is 80.424772 mm, then

$$\Delta x_y=x_{s-s}-x_{s0}=0.424772\mathrm{mm}$$

(52)

$$x_{u0}=x_u+\Delta x_y=1.2866166\mathrm{mm}$$

(53)

6.2. Simulation

Using the constant pressure valve’s nonlinear model, as shown in Figure 9, which can be established using AMESim, the simulation works are implemented after setting the structural parameters and the design parameters.

• The variable outlet flow area input $A_{\mathrm{out}2}$ is set as 3.1416, 0.7854, and 0 mm², respectively, within the working range of the disturbance inputs, and the inlet pressure step input signal is provided, as illustrated in Figure 10. The simulation results are displayed in Figure 11.
• The inlet pressure input $P_{\mathrm{in}}$ is set as 2, 4, 6, and 8 MPa, respectively, within the working range of the disturbance inputs, and the variable outlet flow area step input signal is provided, as illustrated in Figure 12. The simulation results are displayed in Figure 13.

According to the simulation results, both the steady-state and dynamic performance satisfy the design specifications. In particular, the steady-state working range of the controlled pressure is 1.5 ± 0.01 MPa, the steady-state error is not more than 0.7%, and the dynamic regulating time is not more than 0.006 s under different inlet pressure step disturbances and different variable outlet flow area step disturbances.

7. Conclusions

Using the linear incremental analysis method, this study explains the design theory of the constant pressure valve and presents the gain scheduling design method for the system. The nonlinear simulation results demonstrate the precision of the suggested methods, and the following conclusions are drawn:

• The linear incremental analysis method vividly displays the design theory of the constant pressure valve when compared to the conventional direct transfer function transformation analysis method. In addition, the key design parameter, the stabilization control gain, is clarified, as well as the system’s input-output characteristics, which were not covered in earlier studies.
• The analysis results of the bode frequency domain characteristics explain quantitatively how the design parameters affect the system performance and give correct guidance to design the performance parameters, preventing trial and error in the research process.
• Evidently, when using the gain scheduling design method, both the steady-state and dynamic performance of the designed system satisfies the design requirements. The predesign and analysis of other components of the fuel metering system, including the position control system, can be realized using the proposed design methods.

However, when executing actual physical tests, the suggested methods might not perform as well as the simulation tests, which should be confirmed in future research. Moreover, the flow coefficient $C_q$ is a constant, but in fact, it is a variable value during nonlinear simulation; whether it causes the differences between the simulation results and the theoretical design results will be explored in future works.