Robust LQR Design Method for the Aero-Engine Integral Constant Pressure Drop Control Valve with High Precision

Abstract

The closed-loop constant pressure drop control valve is widely used in aero-engine fuel servo metering systems. However, the available constant pressure drop control valve cannot realize servo tracking without static error and, often, a high proportional gain is used to reduce the static error and improve the servo tracking performance, which reduces the stability margin. In this paper, an integral constant pressure drop control valve is designed, which consists of an integral controller and a stabilizing controller. Moreover, a robust LQR design method is proposed to complete the design task. Firstly, the controlled plant’s state–space model is derived, and the augmented model with tracking error is established based on the robust servo system design theory. Secondly, a servo controller with dual functions of integral control and stabilization control is constructed and decoupled, in which the stabilizing controller guarantees the asymptotic stability as well as the anti-disturbance performance, and the integral controller realizes the servo tracking without static error. Finally, based on the robust LQR design method, two key design parameters, including the integral control gain and the stabilization control gain, are designed to complete the design task. The simulation results indicate that, even when suffering 50 mm² metered flow area step disturbance and 1 MPa inlet pressure step change, the designed integral constant pressure drop control valve can realize the function of servo tracking without static error. The static error is almost 0, the settling time is within 0.01 s, the overshoot is within 10%, and the phase margin is more than 55°.

1. Introduction

The aero-engine fuel servo metering system basically adopts the constant pressure drop control principle to complete fuel metering, except for very few applications of the variable pressure drop control structure, such as the fuel regulator in the Spey MK202 turbofan engine in the Rolls-Royce [1]. However, the available constant pressure drop control valve cannot realize servo tracking without static error and, often, a high proportional gain is used to reduce the static error and improve the servo tracking performance. The control theory explains that if there is no integral part in the closed-loop system, but only a proportional part, then the closed-loop system exhibits static error under any circumstances [2,3]. Although the high proportional gain can reduce the static error, it reduces the system stability and, even, causes aero-engine instability in the acceleration and deceleration process, such as thrust and speed swing [4]. In addition, as the high precision, high stability, and high robustness requirements in the modern aero-engine control system are proposed, it is crucial to improve the servo tracking ability and robustness of the constant pressure drop control valve.

Early studies about the constant pressure drop control valve primarily concentrated on frequency domain modeling and analysis. Specifically, based on the derived frequency domain models, the steady-state and dynamic characteristics of the whole system, as well as the stability variation under different structure parameters, were analyzed [5,6,7]. These studies offer some basic methods, however, due to the limitations to classical control theory and simulation technology, the established models are complex, the proposed methods cannot explain the design theory, and the conclusions cannot be verified because of lacking simulation works. In recent years, simulation technology has developed rapidly, and many studies relying on simulation technology have appeared, mainly concentrating on nonlinear modeling and simulation. For example, when the structure parameter values change, such as the hole diameter and the spring stiffness, the system performance variations are analyzed and simulated, and some guidance on the design structure parameters is offered [8,9,10,11,12]. However, these studies lack theoretical analysis and only rely on nonlinear simulation, causing difficulty in guiding the design process. Moreover, relying on the transfer function models, the system’s stability conditions are analyzed [13]. Indeed, the analysis results should be confirmed, because the model is oversimplified, and the study lacks a simulation comparison. Besides, some detailed studies, concentrating on the system’s characteristic changes with the return orifice profile structure, explain the correlation between the profile structure and the system control gain [14,15,16]. Nevertheless, these studies have limitations, because they only offer guidance for orifice profile design, rather than system design. Furthermore, there are some unique studies. For example, an ideal variable orifice is constructed and tested physically, and the test results indicate that the variable structure can enhance system performance [17]. Despite lacking design theory analysis, the design idea for the variable orifice structure in this study is helpful. In addition, through CFD simulation, the valve balance characteristic related to the flow force are explored [18,19,20,21]. The conclusions indicate that the flow force will affect the system performance, but it is not the most crucial aspect. Moreover, some physical tests have been executed to explore the influence of hysteresis on the speed fluctuation [22,23]. However, the theoretical analysis and results discussion in these studies are unclear, and they do not involve the design process. A new remarkable study, using the linear incremental analysis method, reveals the design theory for a constant pressure drop control valve, and proposes the frequency domain analysis and design methods, realizing a pretty good performance [24]. The study proposes efficient guidance measures and design methods; however, it is still limited to classical control design methods, and cannot realize servo tracking without static error.

These studies concentrate on classical control theory, rather than modern control theory, to study the high proportional gain constant pressure drop control valve, causing complex analysis and design processes. With the design object of servo tracking without static error, this paper uses the modern servo control theory to complete the design of the constant pressure drop control valve with static error-free tracking capability and provides a technical approach that can be used in engineering for high precision fuel metering. This paper makes the following contributions:

• Firstly, based on the modern servo model design theory, an integral design structure for the constant pressure drop control valve is proposed, and a servo controller with dual functions of integral control and stabilization control is constructed;
• Secondly, based on the decoupling design theory, the servo controller is decoupled. Where the stabilizing controller guarantees the asymptotic stability, as well as the disturbance rejection performance, and the integral controller realizes the command servo tracking without static error;
• Finally, a robust LQR design method is proposed to design the control gains of the system, and it completes the design task well. The method is proven to guarantee fine performance and stability, as well as strong robust performance.

The chapters are arranged as follows. In Section 2, an integral design structure is constructed and the relevant design theory is derived. In Section 3, the servo controller is decoupled and realized. In Section 4, a robust LQR design method is proposed, and the design and implementation processes for the system are given. In Section 5, a design example is provided. In Section 6, the conclusions are presented.

2. Theoretical Design

A typical fuel metering system structure is shown in Figure 1. In general, the fuel flow metering equation is represented as $Q=C_{q}A\sqrt{2\mathsf{\Delta }P/\rho }$, in which the pressure drop $\mathsf{\Delta }P$ is designed as a constant, and the required fuel flow $Q$ is metered by the metered flow area $A$ [13,24]. Specifically, the pressure drop control valve is used to guarantee the constant pressure drop $\mathsf{\Delta }P$, and the fuel metering valve is used to realize the control of the metered flow area $A$.

This paper concerns the design problem of the pressure drop control valve. The designed integral constant pressure drop control valve is shown in Figure 2, which is improved from the general pressure drop control valve [12,24].

The parameters are defined as follows: $P_S$ is the inlet pressure, $P_T$ is the return pressure, $P_C$ is the controlled pressure, $P_Z$ is the adjusting pressure, $P_O$ is the ejection pressure; $A_J$ is the metered flow area, $A_{in}$ is the servo inlet flow area, $A_{out}$ is the servo outlet flow area, $A_Z$ is the adjusting flow area, $A_1$ is the fixed inlet flow area, $A_2$ is the fixed ejection flow area; $V_C$ is the controlled chamber volume, $V_Z$ is the adjusting chamber volume, $V_O$ is the ejection chamber volume; $x_z$ is the adjusting valve displacement, $x_y$ is the servo valve displacement; $A_y$ is the servo valve pressure bearing area, and $A_{zx}$ is the adjusting valve pressure bearing area [24].

2.1. Working Principle

For the integral constant pressure drop control valve, the fuel flow metering equation is correspondingly represented as $Q=C_q{A}_J\sqrt{2(P_S-P_C)/\rho }$, in which the pressure drop is $(P_S-P_C)$, and the metered flow area is $A_J$.

Generally, for the pressure drop control valve, the metered flow area $A_J$ is considered as the disturbance input and the inlet pressure $P_S$ is considered as the reference input, and the working principle of the system is described as: when the metered flow area $A_J$ disturbs or the inlet pressure $P_S$ changes, the controlled pressure $P_C$ changes, and the pressure drop $(P_S-P_C)$ deviates from the designed value. Instantaneously, because the pressure drop $(P_Z-P_C)$ changes, the adjusting valve moves and the adjusting flow area $A_Z$ changes, realizing a rapid regulation of the $P_C$. Simultaneously, because the pressure drop $(P_S-P_C)$ changes, the servo valve also moves and the servo inlet and outlet flow area $A_{in}$ and $A_{out}$ change, realizing a precise regulation of the $P_C$. The dual function restores the pressure drop $(P_S-P_C)$ to the design value [24,25,26,27,28,29,30].

Hence, the design objective is to ensure the controlled pressure $P_C$ servo tracks the inlet pressure $P_S$ without static error, and rejects the disturbance related to the metered flow area $A_J$.

2.2. State–Space Model of the Controlled Plant

The controlled plant is composed of the controlled pressure and the ejection pressure, and their pressure–flow nonlinear dynamic equations are:

$$\frac{d{P}_C}{dt}=\frac{B}{V_C}\cdot \left(\left(C_{q1}{A}_1+C_{qj}{A}_J\right)\sqrt{\frac{2(P_S-P_C)}{\rho }}-C_{qz}{A}_Z\sqrt{\frac{2(P_C-P_O)}{\rho }}+A_y{\dot{x}}_y+A_{zx}{\dot{x}}_z\right)$$

(1)

$$\frac{d{P}_O}{dt}=\frac{B}{V_O}\cdot \left(C_{qz}{A}_Z\sqrt{\frac{2(P_C-P_O)}{\rho }}-C_{qo}{A}_2\sqrt{\frac{2(P_O-P_T)}{\rho }}\right)$$

(2)

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

The calculation formula for $C_q$ is:

$$C_q=C_{q\max }\cdot \tanh (\sqrt{\frac{8\left|\mathsf{\Delta }P\right|}{\rho }}\cdot \frac{\rho \cdot d_h}{Nu\cdot lamc})$$

(3)

where $C_{q\max }$ is the maximum flow coefficient, $d_h$ is the hydraulic diameter, $Nu$ is the absolute viscosity, and $lamc$ is the critical flow number [24].

The relevant linear dynamic differential equations are:

$$\mathsf{\Delta }{\dot{P}}_C=\frac{B}{V_C}\cdot \left(-(K_{PJ}+K_{PZ})\cdot \mathsf{\Delta }{P}_C+K_{PZ}\cdot \mathsf{\Delta }{P}_O+K_{PJ}\cdot \mathsf{\Delta }{P}_S+K_{AJ}\cdot \mathsf{\Delta }{A}_J-K_{AZ}\cdot \mathsf{\Delta }{A}_Z+A_{zx}\cdot \mathsf{\Delta }{\dot{x}}_z+A_y\cdot \mathsf{\Delta }{\dot{x}}_y\right)$$

(4)

$$\mathsf{\Delta }{\dot{P}}_O=\frac{B}{V_O}\cdot \left(-(K_{PZ}+K_{PT})\cdot \mathsf{\Delta }{P}_O+K_{PZ}\cdot \mathsf{\Delta }{P}_C+K_{AZ}\cdot \mathsf{\Delta }{A}_Z\right)$$

(5)

where $K_{AJ}=C_{qj}\sqrt{\frac{2(P_S-P_C)}{\rho }}$, $K_{AZ}=C_{qz}\sqrt{\frac{2(P_C-P_O)}{\rho }}$, $K_{PZ}=C_{qz}{A}_Z\sqrt{\frac{1}{2\rho (P_C-P_O)}}$, $K_{PT}=C_{qo}{A}_2\sqrt{\frac{1}{2\rho (P_O-P_T)}}$, $K_{PJ}=\left(C_{qj}{A}_J+C_{q1}{A}_1\right)\sqrt{\frac{1}{2\rho (P_S-P_C)}}$.

Finally, the state–space model of the controlled plant is:

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

(6)

where $x_p={\left[\begin{array}{cc}\mathsf{\Delta }{P}_C& \mathsf{\Delta }{P}_O\end{array}\right]}^T$, $u_p={\left[\begin{array}{ccc}\mathsf{\Delta }{A}_Z& \mathsf{\Delta }{\dot{x}}_y& \mathsf{\Delta }{\dot{x}}_z\end{array}\right]}^T$, $w={\left[\begin{array}{cc}\mathsf{\Delta }{A}_J& \mathsf{\Delta }{P}_S\end{array}\right]}^T$, $y_p={\left[\begin{array}{cc}\mathsf{\Delta }{P}_C& \mathsf{\Delta }{P}_O\end{array}\right]}^T$,

$$\begin{array}{l}{A}_p=\left[\begin{array}{cc}-\frac{B}{V_C}\cdot (K_{PJ}+K_{PZ})& \frac{B}{V_C}\cdot K_{PZ}\\ \frac{B}{V_O}\cdot K_{PZ}& -\frac{B}{V_O}\cdot (K_{PZ}+K_{PT})\end{array}\right]\\ B_p=\left[\begin{array}{ccc}-\frac{B}{V_C}\cdot K_{AZ}& \frac{B}{V_C}\cdot A_y& \frac{B}{V_C}\cdot A_{zx}\\ \frac{B}{V_O}\cdot K_{AZ}& 0& 0\end{array}\right],E_p=\left[\begin{array}{cc}\frac{B}{V_C}\cdot K_{AJ}& \frac{B}{V_C}\cdot K_{PJ}\\ 0& 0\end{array}\right]\\ C_p=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],D_p=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]\end{array}$$

(7)

2.3. Structure Design of the Servo Controller

As mentioned previously, the adjusting flow area $A_Z$ is regarded as the main control input. Considering the controlled pressure $P_C$ as the output feedback variable, according to the robust servo system design theory, as well as the steady-state deviation design method [2,3], the theoretical design diagram of the control architecture is shown in Figure 3.

Correspondingly, the design objective is to ensure the controlled pressure increment $\mathsf{\Delta }{P}_C$ servo tracks the inlet pressure increment $\mathsf{\Delta }{P}_S$ without static error, and rejects the disturbance related to the metered flow area increment $\mathsf{\Delta }{A}_J$. The detailed design processes are detailed below.

The tracked input signal $\mathsf{\Delta }{P}_S$ and the rejected disturbance signal $\mathsf{\Delta }{A}_J$ are both constant value signals, and their first order differential equations are expressed as $\mathsf{\Delta }{\dot{A}}_J=0$ and $\mathsf{\Delta }{\dot{P}}_S=0$, respectively. Since the input signals are constant, an integrator needs to be added to realize servo tracking without static error.

The tracking error signal is defined as:

$$e=\mathsf{\Delta }{P}_C-\mathsf{\Delta }{P}_S$$

(8)

Then, the first order differential equation of the tracking error signal is:

$$\dot{e}=\mathsf{\Delta }{\dot{P}}_C$$

(9)

A new state vector is defined as:

$$z={\left[\begin{array}{cc}e& \xi \end{array}\right]}^T$$

(10)

The augmented servo system design model is:

$$\dot{z}=\tilde{A}z+\tilde{B}\mu$$

(11)

where $\xi ={\dot{x}}_p={\left[\begin{array}{cc}\mathsf{\Delta }{\dot{P}}_C& \mathsf{\Delta }{\dot{P}}_O\end{array}\right]}^T$, $\mu ={\dot{u}}_p={\left[\begin{array}{ccc}\mathsf{\Delta }{\dot{A}}_Z& \mathsf{\Delta }{\ddot{x}}_y& \mathsf{\Delta }{\ddot{x}}_z\end{array}\right]}^T$, and

$$\tilde{A}=\left[\begin{array}{ccc}0& 1& 0\\ 0& -\frac{B}{V_C}\cdot (K_{PJ}+K_{PZ})& \frac{B}{V_C}\cdot K_{PZ}\\ 0& \frac{B}{V_O}\cdot K_{PZ}& -\frac{B}{V_O}\cdot (K_{PZ}+K_{PT})\end{array}\right],\tilde{B}=\left[\begin{array}{ccc}0& 0& 0\\ -\frac{B}{V_C}\cdot K_{AZ}& \frac{B}{V_C}\cdot A_y& -\frac{B}{V_C}\cdot A_{zx}\\ \frac{B}{V_O}\cdot K_{AZ}& 0& 0\end{array}\right]$$

(12)

The output feedback vector is defined as $y_c={\left[\begin{array}{cc}e& \mathsf{\Delta }{\dot{P}}_C\end{array}\right]}^T$, and the output model is:

$$y_c=\tilde{C}z+\tilde{D}\mu$$

(13)

where $\tilde{C}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$, $\tilde{D}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]$.

Then, the output feedback control law of the servo model is:

$$\mu =-K_c\cdot y_c=-\left[\begin{array}{cc}{k}_1& k_2\end{array}\right]\left[\begin{array}{c}e\\ \mathsf{\Delta }{\dot{P}}_C\end{array}\right]=-\left(k_{1}e+k_2\mathsf{\Delta }{\dot{P}}_C\right)$$

(14)

where $k_1$ and $k_2$ are the elements of the output feedback control gain vector $K_c$.

Then, the control input of the controlled plant can be expressed as:

$$\begin{array}{l}u={\displaystyle \int \mu } dt\\ =-{\displaystyle \int \left(k_1\cdot e+k_2\cdot \mathsf{\Delta }{\dot{P}}_C\right) }dt\\ =-k_1\cdot {\displaystyle \int e }dt-k_2\cdot \mathsf{\Delta }{P}_C\end{array}$$

(15)

Represented by the physical control input $\mathsf{\Delta }{A}_Z$, it has:

$$\begin{array}{l}\mathsf{\Delta }{A}_Z=-k_1\cdot {\displaystyle \int (\mathsf{\Delta }{P}_C-\mathsf{\Delta }{P}_S) }dt-k_2\cdot \mathsf{\Delta }{P}_C\\ =k_2\cdot \left(\frac{k_1}{k_2}\cdot {\displaystyle \int (\mathsf{\Delta }{P}_S-\mathsf{\Delta }{P}_C) }dt+\mathsf{\Delta }{P}_C\right)\end{array}$$

(16)

The specific structure design diagram of the servo controller is shown in Figure 4.

Since the above servo controller is coupled, it cannot be directly realized in a hydraulic control system. The decoupling problem of the servo controller needs to be considered, which is completed in the following chapters.

3. Decoupling Design of the Servo Controller

As mentioned in Section 2.3, the servo controller consists of two parts: an integral link and a stabilizing link. For hydraulic control systems, these two links can only be realized through hydraulic components. The design processes are detailed below.

3.1. Integral Controller

3.1.1. Characteristics Analysis of the Servo Valve

The nonlinear dynamic equation for the servo valve is:

$$A_y{P}_S-A_y{P}_C-F_L-M_y{\ddot{x}}_y-K_{f1}{\dot{x}}_y-K_1{x}_y=0$$

(17)

where $M_y$ is the mass, $K_1$ is the spring stiffness, $K_{f1}$ is the viscous friction coefficient, and $F_L$ is the initial spring force.

The relevant linear dynamic differential equation is:

$$A_y\cdot \left(\mathsf{\Delta }{P}_S-\mathsf{\Delta }{P}_C\right)-M_y\cdot \mathsf{\Delta }{\ddot{x}}_y-K_{f1}\cdot \mathsf{\Delta }{\dot{x}}_y-K_1\cdot \mathsf{\Delta }{x}_y=0$$

(18)

Its steady-state characteristic is:

$$\mathsf{\Delta }{x}_y=-\frac{A_y}{K_1}\left(\mathsf{\Delta }{P}_C-\mathsf{\Delta }{P}_S\right)$$

(19)

Obviously, the steady-state value of $\mathsf{\Delta }{x}_y$ reflects the tracking error.

3.1.2. Characteristics Analysis of the Adjusting Chamber

The pressure–flow nonlinear dynamic equation of the adjusting pressure is:

$$\frac{d{P}_Z}{dt}=\frac{B}{V_Z}\cdot \left(C_{qin}{A}_{in}\sqrt{\frac{2(P_S-P_Z)}{\rho }}-C_{qout}{A}_{out}\sqrt{\frac{2(P_{\mathrm{Z}}-P_T)}{\rho }}-A_{zx}{\dot{x}}_z\right)$$

(20)

The relevant linear dynamic differential equation is:

$$\mathsf{\Delta }{\dot{P}}_Z=\frac{B}{V_Z}\cdot \left(-(K_{PY}+K_{PT2})\cdot \mathsf{\Delta }{P}_Z+K_{AY}\cdot \mathsf{\Delta }{A}_{in}-K_{AT2}\cdot \mathsf{\Delta }{A}_{out}+K_{PY}\cdot \mathsf{\Delta }{P}_S-A_{zx}\cdot \mathsf{\Delta }{\dot{x}}_z\right)$$

(21)

where $K_{AY}=C_{qin}\sqrt{\frac{2(P_S-P_Z)}{\rho }}$, $K_{PY}=C_{qin}{A}_{in}\sqrt{\frac{1}{2\rho (P_S-P_Z)}}$, $K_{AT2}=C_{qout}\sqrt{\frac{2(P_Z-P_T)}{\rho }}$, $K_{PT2}=C_{qout}{A}_{out}\sqrt{\frac{1}{2\rho (P_Z-P_T)}}$.

The functions $A_{in}=f_{in}(x_{uin})$ and $A_{out}=f_{out}(x_{uout})$ are adopted to represent the geometry relationship of the servo valve inlet orifice and outlet orifice, respectively. According to their linearized gain characteristics:

$$\mathsf{\Delta }{A}_{in}=\frac{d{f}_{in}}{d{x}_{uin}}\cdot \mathsf{\Delta }{x}_{uin}, \mathsf{\Delta }{A}_{out}=\frac{d{f}_{out}}{d{x}_{uout}}\cdot \mathsf{\Delta }{x}_{uout}$$

(22)

Since $\mathsf{\Delta }{x}_{uin}=-\mathsf{\Delta }{x}_{uout}=\mathsf{\Delta }{x}_y$, then

$$\mathsf{\Delta }{A}_{in}=\frac{d{f}_{in}}{d{x}_{uin}}\cdot \mathsf{\Delta }{x}_y, \mathsf{\Delta }{A}_{out}=-\frac{d{f}_{out}}{d{x}_{uout}}\cdot \mathsf{\Delta }{x}_y$$

(23)

If the steady-state servo inlet and outlet flow area are both designed as 0, that is $A_{in,0}=A_{out,0}=0$, there are:

$$K_{PY}=0,K_{PT2}=0$$

(24)

Then, it has:

$$\mathsf{\Delta }{\dot{P}}_Z=\frac{B}{V_Z}\cdot \left(\left(K_{AY}\cdot \frac{d{f}_{in}}{d{x}_{uin}}+K_{AT2}\cdot \frac{d{f}_{out}}{d{x}_{uout}}\right)\cdot \mathsf{\Delta }{x}_y-A_{zx}\cdot \mathsf{\Delta }{\dot{x}}_z\right)$$

(25)

where the generalized integral control gain is defined as:

$$K_C=\frac{B}{V_Z}\cdot \left(K_{AY}\cdot \frac{d{f}_{in}}{d{x}_{uin}}+K_{AT2}\cdot \frac{d{f}_{out}}{d{x}_{uout}}\right)$$

(26)

Then, it has:

$$\mathsf{\Delta }{P}_Z={\displaystyle \int \mathsf{\Delta }{\dot{P}}_Z}dt=K_C{\displaystyle \int \mathsf{\Delta }{x}_y}dt-\frac{B}{V_Z}\cdot A_{zx}\cdot \mathsf{\Delta }{x}_z$$

(27)

Because $\mathsf{\Delta }{x}_y$ reflects the tracking error $(\mathsf{\Delta }{P}_C-\mathsf{\Delta }{P}_S)$, the integrator for the tracking error is embedded through Equation (27). Obviously, the integral function can be realized, and it ensures the performance of servo tracking without static error.

Finally, the state–space model of the integral controller is:

$$\begin{array}{l}{\dot{x}}_i=A_i{x}_i+B_i{u}_i\\ y_i=C_i{x}_i+D_i{u}_i\end{array}$$

(28)

where $x_i={\left[\begin{array}{ccc}\mathsf{\Delta }{x}_y& \mathsf{\Delta }{\dot{x}}_y& \mathsf{\Delta }{P}_Z\end{array}\right]}^T$, $u_i={\left[\begin{array}{cc}e& \mathsf{\Delta }{\dot{x}}_z\end{array}\right]}^T$, $y_i=\left[\mathsf{\Delta }{P}_Z\right]$,

$$\begin{array}{l}{A}_i=\left[\begin{array}{ccc}0& 1& 0\\ -\frac{K_1}{M_y}& -\frac{K_{f1}}{M_y}& 0\\ K_C& 0& 0\end{array}\right],B_i=\left[\begin{array}{cc}0& 0\\ -\frac{A_y}{M_y}& 0\\ 0& -\frac{B}{V_Z}\cdot A_{zx}\end{array}\right]\\ C_i=\left[\begin{array}{ccc}0& 0& 1\end{array}\right],D_i=\left[\begin{array}{cc}0& 0\end{array}\right]\end{array}$$

(29)

3.2. Stabilizing Controller

The nonlinear dynamic equation of the adjusting valve is:

$$A_{zx}{P}_Z-A_{zx}{P}_C-M_z{\ddot{x}}_z-K_{f2}{\dot{x}}_z-K_2{x}_z-F_{L2}=0$$

(30)

where $M_z$ is the mass, $K_2$ is the spring stiffness, $K_{f2}$ is the viscous friction coefficient, and $F_{L2}$ is the initial spring force.

The relevant linear dynamic differential equation is:

$$A_{zx}\cdot \left(\mathsf{\Delta }{P}_Z-\mathsf{\Delta }{P}_C\right)-M_z\cdot \mathsf{\Delta }{\ddot{x}}_z-K_{f2}\cdot \mathsf{\Delta }{\dot{x}}_z-K_2\cdot \mathsf{\Delta }{x}_z=0$$

(31)

Its steady-state characteristic is:

$$\mathsf{\Delta }{x}_z=\frac{A_{zx}}{K_2}\left(\mathsf{\Delta }{P}_Z-\mathsf{\Delta }{P}_C\right)$$

(32)

Obviously, the steady-state value of $\mathsf{\Delta }{x}_z$ reflects the error $e_u$.

The function $A_Z=f_Z(x_{uz})$ is adopted to represent the geometry relationship of the adjusting valve orifice. Its linearized gain characteristic is:

$$\mathsf{\Delta }{A}_Z=\frac{d{f}_Z}{d{x}_{uz}}\cdot \mathsf{\Delta }{x}_{uz}$$

(33)

Since $\mathsf{\Delta }{x}_{uz}=-\mathsf{\Delta }{x}_z$, then:

$$\mathsf{\Delta }{A}_Z=-\frac{d{f}_Z}{d{x}_{uz}}\cdot \mathsf{\Delta }{x}_z$$

(34)

Because $\mathsf{\Delta }{x}_z$ reflects the error $(\mathsf{\Delta }{P}_Z-\mathsf{\Delta }{P}_C)$, the negative feedback function is embedded through Equation (34), and it ensures the robust asymptotic stability, as well as the disturbance rejection performance.

The generalized stabilizing control gain is defined as:

$$K_Z=\frac{d{f}_Z}{d{x}_{uz}}$$

(35)

Finally, the state–space model of the stabilizing controller is:

$$\begin{array}{l}{\dot{x}}_s=A_s{x}_s+B_s{u}_s\\ y_s=C_s{x}_s+D_s{u}_s\end{array}$$

(36)

where $x_s={\left[\begin{array}{cc}\mathsf{\Delta }{x}_z& \mathsf{\Delta }{\dot{x}}_z\end{array}\right]}^T$, $u_s=\left[e_u\right]$, $y_s=\left[\mathsf{\Delta }{A}_z\right]$,

$$\begin{array}{l}{A}_s=\left[\begin{array}{cc}0& 1\\ -\frac{K_2}{M_z}& -\frac{K_{f2}}{M_z}\end{array}\right],B_s=\left[\begin{array}{c}0\\ \frac{A_{zx}}{M_z}\end{array}\right]\\ C_s=\left[\begin{array}{cc}-K_Z& 0\end{array}\right],D_s=\left[0\right]\end{array}$$

(37)

3.3. State–Space Model of the Servo Controller

According to the aforementioned content, the structure of the decoupling controllers is shown in Figure 5.

Combining the integral link and the stabilization link, the state–space model of the servo controller can be expressed as:

$$\begin{array}{l}{\dot{x}}_c=A_c{x}_c+B_{c1}y+B_{c2}r\\ u=C_c{x}_c+D_{c1}y+D_{c2}r\end{array}$$

(38)

where $x_c={\left[\begin{array}{ccccc}\mathsf{\Delta }{x}_y& \mathsf{\Delta }{\dot{x}}_y& \mathsf{\Delta }{P}_Z& \mathsf{\Delta }{x}_z& \mathsf{\Delta }{\dot{x}}_z\end{array}\right]}^T$, $y=\left[\mathsf{\Delta }{P}_C\right]$, $r=\left[\mathsf{\Delta }{P}_S\right]$, $u=\left[\mathsf{\Delta }{A}_Z\right]$,

$$\begin{array}{l}{A}_c=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ -\frac{K_1}{M_y}& -\frac{K_{f1}}{M_y}& 0& 0& 0\\ K_C& 0& 0& 0& -\frac{B}{V_Z}\cdot A_{zx}\\ 0& 0& 0& 0& 1\\ 0& 0& \frac{A_{zx}}{M_z}& -\frac{K_2}{M_z}& -\frac{K_{f2}}{M_z}\end{array}\right],B_{c1}=\left[\begin{array}{c}0\\ -\frac{A_y}{M_y}\\ 0\\ 0\\ -\frac{A_{zx}}{M_z}\end{array}\right],B_{c2}=\left[\begin{array}{c}0\\ \frac{A_y}{M_y}\\ 0\\ 0\\ 0\end{array}\right]\\ C_c=\left[\begin{array}{ccccc}0& 0& 0& -K_Z& 0\end{array}\right],D_{c1}=\left[0\right],D_{c2}=\left[0\right]\end{array}$$

(39)

Then, the integrated structure of the servo controller is shown in Figure 6.

4. Robust LQR Design Method

As mentioned in Section 3.1.2, the steady-state flow area of the servo inlet and outlet orifices are both designed as 0. It follows that once the flow area variation gradient of the orifice is determined, the generalized integral control gain $K_C$ is a constant value at any steady-state working point and cannot be designed arbitrarily.

When combining the servo controller and the controlled plant, embedding the generalized integral control gain $K_C$ in the state–space model, and considering the generalized stabilizing control gain $K_Z$ as the static output feedback gain, the open-loop augmented state–space model of the whole system can be expressed as:

$$\begin{array}{l}\dot{x}=Ax+Bu+Ew\\ y=Cx+Du\end{array}$$

(40)

where $x={\left[\begin{array}{ccccccc}\mathsf{\Delta }{x}_y& \mathsf{\Delta }{\dot{x}}_y& \mathsf{\Delta }{P}_Z& \mathsf{\Delta }{x}_z& \mathsf{\Delta }{\dot{x}}_z& \mathsf{\Delta }{P}_C& \mathsf{\Delta }{P}_O\end{array}\right]}^T$, $u=\left[\mathsf{\Delta }{A}_Z\right]$, $y=\left[\mathsf{\Delta }{x}_z\right]$,

$$\begin{gathered} A=\left[\begin{array}{ccccccc}0& 1& 0& 0& 0& 0& 0\\ -\frac{K_1}{M_y}& -\frac{K_{f1}}{M_y}& 0& 0& 0& -\frac{A_y}{M_y}& 0\\ K_C& 0& 0& 0& -\frac{B}{V_Z}\cdot A_{zx}& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& \frac{A_{zx}}{M_z}& -\frac{K_2}{M_z}& -\frac{K_{f2}}{M_z}& -\frac{A_{zx}}{M_z}& 0\\ 0& \frac{B}{V_C}\cdot A_y& 0& 0& \frac{B}{V_C}\cdot A_{zx}& -\frac{B}{V_C}\cdot (K_{PJ}+K_{PZ})& \frac{B}{V_C}\cdot K_{PZ}\\ 0& 0& 0& 0& 0& \frac{B}{V_O}\cdot K_{PZ}& -\frac{B}{V_O}\cdot (K_{PZ}+K_{PT})\end{array}\right] \\ \begin{array}{l}B=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ -\frac{B}{V_C}\cdot K_{AZ}\\ \frac{B}{V_O}\cdot K_{AZ}\end{array}\right],E=\left[\begin{array}{cc}0& 0\\ 0& \frac{A_y}{M_y}\\ 0& 0\\ 0& 0\\ 0& 0\\ \frac{B}{V_C}\cdot K_{AJ}& \frac{B}{V_C}\cdot K_{PJ}\\ 0& 0\end{array}\right]\\ C=\left[\begin{array}{ccccccc}0& 0& 0& 1& 0& 0& 0\end{array}\right],D=\left[0\right]\end{array} \end{gathered}$$

(41)

This is a typical static output feedback design problem, and its output feedback control law is:

$$u=-K_Z\cdot y$$

(42)

The topology of the output feedback control system is shown in Figure 7.

4.1. Output Feedback Control Gains Design

4.1.1. Integral Control Gain

Defined as:

$$\frac{d{f}_{in}}{d{x}_{uin}}\in \left[k_{in,\min },k_{in,\max }\right],\frac{d{f}_{out}}{d{x}_{uout}}\in \left[k_{out,\min },k_{out,\max }\right]$$

(43)

where $k_{in,\min }$, $k_{in,\max }$, $k_{out,\min }$ and $k_{out,\max }$ is the extremum values of the flow area variation gradient of the servo inlet orifice and outlet orifice, respectively.

According to Equation (26), it has:

$$K_C\in \left[\frac{B}{V_Z}\cdot \left(K_{AY}\cdot k_{in,\min }+K_{AT2}\cdot k_{out,\min }\right),\frac{B}{V_Z}\cdot \left(K_{AY}\cdot k_{in,\max }+K_{AT2}\cdot k_{out,\max }\right)\right]$$

(44)

According to the structural parameter limitations, appropriate values of the flow area variation gradient can be designed, then the integral control gain can be determined.

4.1.2. Stabilization Control Gain

The quadratic optimization objective for Equation (40) is defined as:

$$J=\frac{1}{2}{\displaystyle \underset{0}{\overset{\infty }{\int }}\left[y^T\cdot Q\cdot y+u^T\cdot R\cdot u\right]dt}$$

(45)

where $Q=Q^T\ge 0$, $R=R^T>0$.

Then, the LQR state feedback control law is:

$$u=-K_x\cdot x=-R^{-1}{B}^{T}P\cdot x$$

(46)

where $K_x$ is the state feedback control gain vector, and $P$ is the solution of an ARE algebraic equation, expressed as:

$$PA+A^{T}P+C^{T}QC-PB{R}^{-1}{B}^{T}P=0$$

(47)

By designing the weight matrices $Q$ and $R$ to optimize the quadratic objective, a positive definite solution $P$ of the ARE equation can be obtained. Then, the above system yields satisfactory closed-loop reference dynamics, described by:

$$\dot{x}=\left(A-B{K}_x\right)x=Fx$$

(48)

Using the output feedback control law expressed by Equation (42), the dominant eigenvalue ${\mathsf{\Lambda }}_1$ and its associated eigenvector $X_1$ of the state feedback design can be retained, and the static output feedback gain is given by [3]:

$$K_Z=K_x{X}_1{\left(C{X}_1\right)}^{-1}$$

(49)

where the eigenvector $X_1$ and the eigenvalue ${\mathsf{\Lambda }}_1$ satisfy the eigen equation for the state feedback system, described as:

$$F{X}_1=X_1{\mathsf{\Lambda }}_1$$

(50)

Finally, the closed-loop system is calculated as:

$$\dot{x}=\left(A-B{K}_{Z}C\right)x=A_{cl}x$$

(51)

4.2. Design and Implementation Method of the Servo Controller

Assuming that the performance requirements are:

• Steady-state requirement: The steady-state pressure drop is designed as $P_e$, and the phase margin should be more than N°;
• Dynamic requirement: The settling time should be within the $t_s$, and the overshoot should be within the $\sigma$.

4.2.1. Control Gains Design

The steady-state flow balance equation of the controlled plant is:

$$\left(C_{qj}{A}_J+C_{q1}{A}_1\right)\sqrt{\frac{2\left(P_S-P_C\right)}{\rho }}=C_{qz}{A}_Z\sqrt{\frac{2\left(P_C-P_O\right)}{\rho }}=C_{qo}{A}_2\sqrt{\frac{2\left(P_O-P_T\right)}{\rho }}$$

(52)

At a steady-state working point, if the inlet pressure is represented as $P_{S,i}$, and the metered flow area is represented as $A_{J,i}$, then:

$$P_{C,i}=P_{S,i}-P_e$$

(53)

$$P_{O,i}={\left(\frac{C_{qj,i}{A}_{J,i}+C_{q1,i}{A}_1}{C_{qo,i}{A}_2}\right)}^2\cdot \left(P_{S,i}-P_{C,i}\right)+P_T$$

(54)

$$A_{Z,i}=\left(\frac{C_{qj,i}{A}_{J,i}+C_{q1,i}{A}_1}{C_{qz,i}}\right)\sqrt{\frac{\left(P_{S,i}-P_{C,i}\right)}{\left(P_{C,i}-P_{O,i}\right)}}$$

(55)

Subsequently, the parameters $K_{AJ,i}$, $K_{AZ,i}$, $K_{PZ,i}$, $K_{PJ,i}$ and $K_{PT,i}$ can be calculated.

The steady-state balance equation of the adjusting valve is:

$$0=A_{zx}{P}_{C,i}-A_{zx}{P}_{Z,i}-K_2{x}_{szd,i}$$

(56)

Then:

$$P_{Z,i}=P_{C,i}+\frac{K_2}{A_{zx}}\cdot x_{szd,i}$$

(57)

Subsequently, the parameters $K_{AY,i}$ and $K_{AT2,i}$ can be calculated.

When designing appropriate values of the flow area variation gradient $\frac{d{f}_{in}}{d{x}_{uin}}$ and $\frac{d{f}_{out}}{d{x}_{uout}}$, then the integral control gain $K_C$ can be determined according to Equation (26).

When designing optimized weight matrices $Q$ and $R$ that meet the above performance requirements, a positive definite solution $P_i$ of the ARE equation can be obtained, and the state feedback control gain vector $K_{x,i}$ can be obtained, then the stabilization control gain is expressed as:

$$K_{Z,i}=K_{x,i}{X}_1{\left(C{X}_1\right)}^{-1}$$

(58)

4.2.2. Orifice Geometry Relationship Design

A design formula is provided to deal with the geometry relationship design problem of the flow area increment $(A_{Z,i}-A_{Z,i-1})$ and the orifice underlap increment $\mathsf{\Delta }{x}_{uz,i}$, expressed by:

$$\mathsf{\Delta }{x}_{uz,i}=\frac{\left(A_{Z,i}-A_{Z,i-1}\right)}{K_{Z,i-1}},i=2,3,\dots ,N$$

(59)

where $N$ is the number of the designed steady-state working points [24].

Assuming that the adjusting valve steady-state spring compression at the (i − 1)-th steady-state working point is $x_{szd,i-1}$, then the calculation formula of the steady-state spring compression at the i-th steady-state working point is provided as:

$$x_{szd,i}=x_{szd,i-1}+\mathsf{\Delta }{x}_{uz,i}$$

(60)

4.2.3. Valve Initial Parameters Design

(1)

Adjusting valve

The initial underlap and the underlap at the first steady-state working point are designed as $x_{uz,0}$ and $x_{uz,1}$, respectively. Since $\mathsf{\Delta }{x}_{uz}=\mathsf{\Delta }{x}_{szd}$, the initial spring compression $x_{szd,0}$ is calculated as:

$$x_{szd,0}=x_{szd,1}-\left(x_{uz,1}-x_{uz,0}\right)$$

(61)

(2)

Servo valve

The steady-state balance equation of the servo valve is:

$$0=A_y{P}_{S,i}-A_y{P}_{C,i}-K{x}_{scd,i}$$

(62)

Then, at the first steady-state working point, the steady-state spring compression is:

$$x_{scd,1}=\frac{A_y}{K}{P}_e$$

(63)

The initial underlap and the underlap at the first steady-state working point is designed as $x_{uin,0}$ and $x_{uin,1}$, respectively. Since $\mathsf{\Delta }{x}_{uin}=\mathsf{\Delta }{x}_{scd}$, the initial spring compression $x_{scd,0}$ is calculated as:

$$x_{scd,0}=x_{scd,1}-\left(x_{uin,1}-x_{uin,0}\right)$$

(64)

5. Design Example

The structural parameters of the pressure drop control valve are shown in

Table 1. Structural parameters of the pressure drop control valve.

Parameter/Unit	Value	Parameter/Unit	Value
$M_y$/Kg	0.08	$K_1$/(N/m)	4 × 104
$M_z$/Kg	0.05	$K_{f1}$/(N/(m/s))	200
$d_y$/m	0.036	$K_2$/(N/m)	1.5 × 104
$d_z$/m	0.036	$K_{f2}$/(N/(m/s))	200
$V_C$/m3	2 × 10−6	$\rho$/(Kg/m3)	780
$V_Z$/m3	2 × 10−6	$B$/MPa	1.7 × 103
$V_O$/m3	4.9087 × 10−4	$C_{q\max }$	0.7
$A_1$/m2	2.8274 × 10−7	$Nu$/Pas	0.051
$A_2$/m2	1.9007 × 10−4	$lamc$	1 × 103
$P_T$/MPa	0.2

.

The input conditions include:

• The inlet pressure $P_S$ is within [3, 9] the MPa;
• The metered flow area $A_J$ is within [10, 240] × 10⁻⁶ m².

The design objectives include:

• Geometry design of the adjusting orifice $A_Z=f_Z(x_{uz})$;
• Geometry design of the servo orifices $A_{in}=f_{in}(x_{uin})$ and $A_{out}=f_{out}(x_{uout})$.

The performance requirements include:

• The rated pressure drop $P_e$ is 0.92 MPa, and the variation range is within 0.01 MPa;
• The settling time is within 0.01 s, and the overshoot is within 10%;
• The phase margin should be more than 50°.

5.1. Dynamic Design

5.1.1. First Steady-State Working Point

The input conditions of the first steady-state working point are: the metered flow area $A_{J,1}$ is 10 × 10⁻⁶ m², and the inlet pressure $P_{S,1}$ is 9 MPa. According to Equations (53)–(55), then:

$$P_{C,1}=P_{S,1}-\mathsf{\Delta }{P}_e=8.08 \mathrm{MPa}$$

(65)

$$P_{O,1}={\left(\frac{C_{qj,1}{A}_{J,1}+C_{q1,1}{A}_1}{C_{qo,1}{A}_2}\right)}^2\cdot \left(P_{S,1}-P_{C,1}\right)+P_T=0.20265 \mathrm{MPa}$$

(66)

$$A_{Z,1}=\left(\frac{C_{qj,1}{A}_{J,1}+C_{q1,1}{A}_1}{C_{qz,1}}\right)\sqrt{\frac{\left(P_{S,1}-P_{C,1}\right)}{\left(P_{C,1}-P_{O,1}\right)}}=3.48627\cdot {10}^{-6}{ \mathrm{m}}^2$$

(67)

Then, the parameters $K_{AJ,1}$, $K_{AZ,1}$, $K_{PZ,1}$, $K_{PJ,1}$ and $K_{PT,1}$ are calculated as:

$$K_{AJ,1}=C_{qj,1}\sqrt{\frac{2(P_{S,1}-P_{C,1})}{\rho }}=33.99849$$

(68)

$$K_{AZ,1}=C_{qz,1}\sqrt{\frac{2(P_{C,1}-P_{O,1})}{\rho }}=99.48459$$

(69)

$$K_{PZ,1}=C_{qz,1}{A}_{Z,1}\sqrt{\frac{1}{2\rho (P_{C,1}-P_{O,1})}}=2.20144\cdot {10}^{-11}$$

(70)

$$K_{PJ,1}=\left(C_{qj,1}{A}_{J,1}+C_{q1,1}{A}_1\right)\sqrt{\frac{1}{2\rho (P_{S,1}-P_{C,1})}}=1.88494\cdot {10}^{-10}$$

(71)

$$K_{PT,1}=C_{qo,1}{A}_2\sqrt{\frac{1}{2\rho (P_{O,1}-P_T)}}=6.54328\cdot {10}^{-8}$$

(72)

Designing the spring compression $x_{szd,1}$ as 10 mm. According to Equation (57), it has:

$$P_{Z,1}=P_{C,1}+\frac{K_2}{A_{zx}}\cdot x_{szd,1}=8.22737 \mathrm{MPa}$$

(73)

Then, the parameters $K_{AY,1}$ and $K_{AT2,1}$ are calculated as:

$$K_{AY,1}=C_{qin,1}\sqrt{\frac{2(P_{S,1}-P_{Z,1})}{\rho }}=31.15679$$

(74)

$$K_{AT2,1}=C_{qout,1}\sqrt{\frac{2(P_{Z,1}-P_T)}{\rho }}=100.42741$$

(75)

When designing the weight matrices $Q$ as 2.041 and $R$ as 2000, the different control gain design schemes are shown in

Table 2. Different control gain design schemes at the first steady-state working point.

$\frac{\mathit{d}{\mathit{f}}_{\mathit{i}\mathit{n}}}{\mathit{d}{\mathit{x}}_{\mathit{u}\mathit{i}\mathit{n}}}$$/\frac{\mathit{d}{\mathit{f}}_{\mathit{o}\mathit{u}\mathit{t}}}{\mathit{d}{\mathit{x}}_{\mathit{u}\mathit{o}\mathit{u}\mathit{t}}}$	${\mathit{K}}_{\mathit{Z},1}$	Phase/°
0.005	0.0213	60.4
0.010	0.0307	59.8
0.015	0.0303	59.0
0.020	0.0302	58.1
0.025	0.0289	57.2
0.030	0.0311	56.4
0.035	0.0318	55.6
0.040	0.0385	55.0

.

5.1.2. Other Steady-State Working Points

In this paper, the flow area variation gradient of the servo inlet orifice and outlet orifice $\frac{d{f}_{in}}{d{x}_{uin}}$ and $\frac{d{f}_{out}}{d{x}_{uout}}$ are both designed as 0.03, and the 4 × 5 steady-state working points are selected as the designed points. Subsequently, when executing the design processes shown in Section 5.1.1, the stabilization control gains that meet the above performance requirements can be obtained, as shown in

Table 3. Stabilization control gains at each steady-state working point.

${\mathit{P}}_{\mathit{S}}$ MPa	${\mathit{A}}_{\mathit{J}}$ mm2	${\mathit{A}}_{\mathit{Z}}$ mm2	${\mathit{K}}_{\mathit{Z}}$	Phase/°
3	10	7.1413178	0.0305	69.2
	30	21.258905	0.0251	87.6
	80	58.720918	0.0161	89.4
	160	138.75330	0.0299	83.1
	240	359.53286	0.0305	89.8
5	10	4.9691655	0.0302	61.5
	30	14.750547	0.0288	84.5
	80	39.904931	0.0151	84.5
	160	85.546194	0.0201	76.0
	240	148.38949	0.0259	70.4
7	10	4.0360839	0.0298	58.2
	30	11.969912	0.0301	82.6
	80	32.175269	0.0152	81.1
	160	67.213819	0.0141	73.7
	240	109.70300	0.0159	68.5
9	10	3.4862689	0.0311	56.4
	30	10.334703	0.0281	81.3
	80	27.693258	0.0221	76.6
	160	57.160884	0.0121	70.7
	240	90.995231	0.0149	62.8

.

5.1.3. Valve Initial Parameters Design

(1)

Adjusting valve

The initial underlap $x_{uz,0}$ and the underlap at the first steady-state working point $x_{uz,1}$ are both designed as 0.1 mm. According to Equation (61), it has:

$$x_{szd,0}=x_{szd,1}-\left(x_{uz,1}-x_{uz,0}\right)=10 \mathrm{mm}$$

(76)

(2)

Servo valve

According to Equation (63), at the first steady-state working point, the steady-state spring compression is:

$$x_{scd,1}=\frac{A_y}{K}{P}_e=23.39155 \mathrm{mm}$$

(77)

The initial underlap $x_{uin,0}$ and the underlap at the first steady-state working point $x_{uin,1}$ are both designed as 0 mm. According to Equation (64), it has:

$$x_{scd,0}=x_{scd,1}-\left(x_{uin,1}-x_{uin,0}\right)=23.39155 \mathrm{mm}$$

(78)

Finally, the summary of the designed initial parameters is shown in

Table 4. Summary table of the designed initial parameters.

Parameter/Unit	Value
$x_{szd,0}$/mm	10
$x_{scd,0}$/mm	23.39155
$x_{uz,0}$/mm	0.1
$x_{uin,0}$/mm	0
$x_{uout,0}$/mm	0

.

5.1.4. Orifice Geometry Relationship Design

According to Equations (59) and (60), the geometry relationship value pair of the adjusting orifice is shown in

Table 5. Geometry relationship value pair of the adjusting orifice.

${\mathit{x}}_{\mathit{u}\mathit{z}}$ mm	${\mathit{A}}_{\mathit{Z}}$ mm2
0	0
0.1	3.4862689
0.117678939	4.0360839
0.148990402	4.9691655
0.220915975	7.1413178
0.325617129	10.334703
0.38380962	11.969912
0.47618952	14.750547
0.702174173	21.258905
0.958522898	27.693258
1.161328826	32.175269
1.669859221	39.904931
2.812637565	57.160884
2.941565995	58.720918
3.469075374	67.213819
4.769243813	85.546194
5.040340181	90.995231
6.295895148	109.70300
8.122958041	138.75330
8.445238643	148.38949
16.59749231	359.53286

.

Moreover, the geometry relationship diagram of the adjusting orifice is shown in Figure 8.

In addition, the geometry relationship diagrams of the servo inlet orifice and outlet orifice are shown in Figure 9a,b.

5.2. Simulation and Discussion

A nonlinear model is established based on the AMESim software, and the structure diagram is shown in Figure 10.

According to Table 1, the relevant structure parameters are set, and according to Table 4, the relevant initial parameters are set. In addition, the remaining structure parameters and the simulation parameters are set to the default values of AMESim.

5.2.1. Simulation

Executing the following simulation tasks, the performance of the designed system can be verified. The input conditions and the simulation results are detailed below.

• The step signal of the inlet pressure $P_S$ is shown in Figure 11. Besides, the steady-state working points of the metered flow area $A_J$ are designed as 10 mm², 50 mm², 100 mm², 150 mm², 200 mm², and 240 mm², respectively. The simulation results are shown in Figure 12.
• The step signal of the metered flow area $A_J$ is shown in Figure 13. Besides, the steady-state working points of the inlet pressure $P_S$ are designed as 3 MPa, 4 MPa, 5 MPa, 6 MPa, 7 MPa, 8 MPa, and 9 MPa, respectively. The simulation results are shown in Figure 14.

5.2.2. Discussion

The simulation results shown in Figure 12 and Figure 14 indicate that:

• Despite suffering from strong step inputs, the controlled pressure drop is always 0.92 MPa, and the static error is almost 0. Evidently, the designed control device has the ability to perform servo tracking without static error. It follows that the theoretical design architecture is relevant and the LQR design method is effective;
• During each transient process, the settling time is within 0.01s and the dynamic overshoot is within 10%. Obviously, the dynamic performances match the design requirements. It follows that the derived models are precise and the designed weight matrices, Q and R, are reasonable.

In addition, although the proposed method is based on the linear model, the designed system still has good performance and strong robustness when faced with nonlinear effects, such as leakage. The verification processes are detailed below.

According to the aforementioned content, the adjusting chamber is almost dead at each steady-state working point, hence it is most likely to leak. Assuming that the adjusting chamber leaks, a nonlinear model can be established, and the structure diagram is shown in Figure 15.

The leakage area is represented as $A_L$, and it is designed as 0 mm², 0.5 mm², 1.0 mm², 1.5 mm², 2.0 mm², and 2.5 mm², respectively. Selecting some steady-state working points for verification and executing the following simulation tasks:

• The step signal of the inlet pressure $P_S$ is still shown in Figure 11 and the metered flow area $A_J$ is designed as 150 mm². The simulation results are shown in Figure 16;
• The step signal of the metered flow area $A_J$ is still shown in Figure 13 and the inlet pressure $P_S$ is designed as 7 MPa. The simulation results are shown in Figure 17.

The simulation results shown in Figure 16 and Figure 17 indicate that:

• When the adjusting chamber leaks, the pressure drop in the system deviates from the design value, and as the leakage area increases, the deviation increases; however, even if the leakage area reaches 2.5 mm², the deviation is still within 0.01 MPa. Thus, the steady-state performance is acceptable;
• Besides, during each transient process, the settling time is still within 0.01 s and the dynamic overshoot is still within 10%. Obviously, the dynamic performances match the design requirements and the leakage has a small impact on the dynamic performances.

To enhance the reliability of the simulation results, the same input conditions as mentioned in Section 5.2.1 are used and the leakage area is set as 2.5 mm². When executing corresponding simulation tasks, the simulation results are shown in Figure 18 and Figure 19.

The simulation results indicate that the pressure drop in the system deviates from the design value, but the deviation is all within 0.01 MPa, and the designed system still has good steady-state and dynamic performance.

These simulation results show that, when faced with nonlinear effects, the designed system has strong robustness, and the proposed method is robust and effective.

6. Conclusions

In this paper, with the design object of achieving servo tracking without static error, an integral constant pressure drop control valve is designed, and a robust LQR design method is proposed. The conclusions are as follows:

• Based on the servo system design theory, a servo control architecture for the pressure drop control valve is constructed and implemented, which can clearly explain the design theory of the system. In addition, the control architecture clearly displays the key structural design parameters, including the generalized stabilization control gain and the generalized servo control gain. Compared with classic design methods, the proposed design architecture and design method are more illustrative;
• Based on the output feedback design theory, the robust LQR design method can realize the design of the key structural design parameters effectively and obtain optimized structural parameters to guarantee high precision and high robustness in performance. The proposed method provides more accurate guidance for the design of the structural parameters and improves the design efficiency;
• The simulation results show that the designed integral constant pressure drop control valve has dual control functions of integral control and stabilization control and can realize tracking without static error and pretty good dynamic performance. Besides, when faced with nonlinear effects, the designed system still has good performance and strong robustness. Evidently, the proposed design method is robust and effective, and can also be used in the design process of other fuel system components.

However, nonlinearities also play an essential role in the design process, and it is meaningful to study how nonlinear characteristics affect system performance in a future study. Besides, if executing physical tests, the proposed design methods may not perform as well as the simulation results, which should also be tested in future work.