Next Article in Journal
Spectral Methods in Nonlinear Optics Equations for Non-Uniform Grids Using an Accelerated NFFT Scheme
Next Article in Special Issue
Special Issue: Advances in Mechanics and Control
Previous Article in Journal
Approximation Algorithm for X-ray Imaging Optimization of High-Absorption Ratio Materials
Previous Article in Special Issue
Assessing Parameters of the Coplanar Components of Perturbing Accelerations Using the Minimal Number of Optical Observations
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

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

School of Energy and Power Engineering, Beihang University, Beijing 100191, China
*
Author to whom correspondence should be addressed.
Symmetry 2023, 15(1), 45; https://doi.org/10.3390/sym15010045
Submission received: 9 November 2022 / Revised: 3 December 2022 / Accepted: 20 December 2022 / Published: 24 December 2022
(This article belongs to the Special Issue Advances in Mechanics and Control)

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 in is the inlet pressure, P C is the controlled pressure, P 0 is the return pressure, A in is the inlet flow area, A out 1 is the fixed outlet flow area, A 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 out 2 . Generally, the regulation processes can be described as: when the controlled pressure P C changes caused by the variable inlet pressure P in or the variable outlet flow area A out 2 , the motion valve senses a change in the pressure difference ( P C P 0 ) to regulate the inlet flow area A 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 Δ P C caused by the variable inlet pressure increment Δ P in and the variable outlet flow area increment Δ A out 2 , and ensure the controlled pressure increment Δ 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 fp is the dynamic matrix of the controlled object.
  • G 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 Δ 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 Δ P in and Δ A out 2 are rejected.

2.2. Dynamic Equations

2.2.1. Controlled Object

The pressure-flow nonlinear dynamic equation of the controlled chamber is
d P C d t = B V 0 ( C q A in 2 ( P in P C ) ρ C q ( A out 2 + A out 1 ) 2 ( P C P 0 ) ρ A y x ˙ y )
where B is the oil bulk modulus, ρ is the oil density, and C q is the flow coefficient.
The pressure-flow linear dynamic differential equation can be described as
d Δ P C d t = B V 0 ( K Ain Δ A in + K Pin ( Δ P in Δ P C ) ( K Aout Δ A out 2 + K Pout Δ P C ) A y Δ x ˙ y )
where K Ain = C q 2 ( P in P C ) ρ , K Pin = C q A in 1 2 ρ ( P in P C ) , K Aout = C q 2 ( P C P 0 ) ρ , K Pout = C q ( A out 2 + A out 1 ) 1 2 ρ ( P C P 0 ) .
Then, the state space model of the controlled object is
x ˙ p = A p x p + B p u p + E p w y = C p x p + D p u p
where x p = [ Δ P C ] , u p = [ Δ A in Δ x ˙ y ] T , w = [ Δ P in Δ A out 2 ] T , y = [ Δ P C ] ,
A p = [ B V 0 ( K Pin + K Pout ) ] , B p = [ B V 0 K Ain B V 0 A y ] , E p = [ B V 0 K Pin B V 0 K Aout ] C p = [ 1 ] , D p = [ 0 0 ]

2.2.2. Stabilization Controller

By taking the steady-state working point x y 0 = 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 x ¨ y = A y P C A y P 0 K f x ˙ y K x y F L
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 Δ x ¨ y = A y ( Δ P C Δ P 0 ) K f Δ x ˙ y K Δ x y
A function A in = f in ( x u ) is used to express the geometry design of the inlet orifice. According to its linearized gain characteristics
Δ A in = d f in d x u Δ x u
and Δ x u = Δ x y , the gain control law of the stabilization controller is expressed as
Δ A in = K C Δ x y
where K C = d f in d x u , and it is the generalized stabilization control gain.
Then, the state space model of the stabilization controller is
x ˙ s = A s x s + B s y + E s r u s = C s x s + D s y
where x s = [ Δ x y Δ x ˙ y ] T , r = [ Δ P 0 ] , u s = [ Δ A in ] ,
A s = [ 0 1 K M y K f M y ] , B s = [ 0 A y M y ] , E s = [ 0 A y M y ] C s = [ K C 0 ] , D s = [ 0 ]

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
x ˙ = A open x + B open u + E open w y = C open x + D open u
where x = [ Δ P C Δ x y Δ x ˙ y ] T , u = [ e u ] ,
A open = [ B V 0 ( K Pin + K Pout ) B V 0 K Ain K C B V 0 A y 0 0 1 0 K M y K f M y ] B open = [ 0 0 A y M y ] , E open = [ B V 0 K Pin B V 0 K Aout 0 0 0 0 ] C open = [ 1 0 0 ] , D open = [ 0 ]
Then, the open-loop transfer function is
L = C open ( s I A open ) 1 B open = B V 0 A y M y A y ( s + K Ain A y K C ) ( s + B V 0 ( K Pin + K Pout ) ) ( s 2 + K f M y s + K M y )

3.2. Disturbance Input Transfer Functions

Defining
E 1 = [ B V 0 K Pin ] , E 2 = [ B V 0 K Aout ]
Then, the open-loop disturbance transfer function from Δ P in to Δ P C is
G Pin - Pc = C ( s I A ) 1 E 1 = B V 0 K Pin s + B V 0 ( K Pin + K Pout )
The open-loop disturbance transfer function from Δ A out 2 to Δ P C is
G Aout 2 - Pc = C ( s I A ) 1 E 2 = B V 0 K Aout s + B V 0 ( K Pin + K Pout )

3.3. Control Output Transfer Functions

The closed-loop sensitivity function is defined as
S = ( 1 + L ) 1 = ( s + B V 0 ( K Pin + K Pout ) ) ( s 2 + K f M y s + K M y ) ( s + B V 0 ( K Pin + K Pout ) ) ( s 2 + K f M y s + K M y ) + B V 0 A y M y A y ( s + K Ain A y K C )
Assuming Δ P 0 = 0 , the control output transfer function of the system is
Δ P C = S G Pin - Pc Δ P in + S G Aout 2 - Pc Δ A out 2
Specifically:
  • The closed-loop disturbance transfer function from Δ P in to Δ P C is
    S Pin = S G Pin - Pc = B V 0 K Pin ( s 2 + K f M y s + K M y ) ( s + B V 0 ( K Pin + K Pout ) ) ( s 2 + K f M y s + K M y ) + B V 0 A y M y A y ( s + K Ain A y K C )
Its steady-state gain is
K P = K Pin ( K Pin + K Pout ) + A y K ( K Ain K C )
2.
The closed-loop disturbance transfer function from Δ A out 2 to Δ P C is
S Aout 2 = S G Aout 2 - Pc = B V 0 K Aout ( s 2 + K f M y s + K M y ) ( s + B V 0 ( K Pin + K Pout ) ) ( s 2 + K f M y s + K M y ) + B V 0 A y M y A y ( s + K Ain A y K C )
Its steady-state gain is
K A = K Aout ( K Pin + K Pout ) + A y K ( K Ain K C )

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 > K f 2 4 M y , the second-order section is an underdamped oscillation link, and its poles are
    p 1 , 2 = K f 2 M y ± j K M y ( K f 2 M y ) 2
The corner frequencies are defined as
ω 1 , 2 = K f 2 M y
2.
If K < K f 2 4 M y , the second-order section is an overdamped non-oscillation link, and its poles are
p 1 , 2 = K f 2 M y ± ( K f 2 M y ) 2 K M y
The corner frequencies are defined as
ω 1 , 2 = K f 2 M y ± ( K f 2 M y ) 2 K M y
The third pole is
p 3 = B V 0 ( K Pin + K Pout )
The corner frequency is defined as
ω 4 = B V 0 ( K Pin + K Pout )
The zero is
z 1 = K Ain A y K C
The corner frequency is defined as
ω 3 = K Ain A y K C
The open-loop gain is
K o p e n = K A i n A y K C ( K P i n + K P o u t ) K
The above calculation equations of the corner frequencies demonstrate that:
  • The stabilization control gain K C mainly influences ω   3 and the open-loop gain K open ;
  • The spring stiffness K mainly influences ω   1 , ω   2 , and the open-loop gain K open ;
  • The volume V 0 of the controlled chamber mainly influences ω 4 ;
  • The pressure-bearing area A y mainly influences ω   3 and the open-loop gain K open ;
  • The mass M y mainly influences ω   1 and ω   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 ω   1 and ω   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 ω   3 or the open-loop gain K 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 < K f 2 4 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 ω   3 increases, and the open-loop gain K 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 ω 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 ) = ( s + B V 0 ( K Pin + K Pout ) ) ( s 2 + K f M y s + K M y ) + B V 0 A y M y A y ( s + K Ain A y K C )
The equation is expressed as
d ( s ) = a 0 s 3 + a 1 s 2 + a 2 s + a 3
where
a 0 = 1
a 1 = [ K f M y + B V 0 ( K Pin + K Pout ) ]
a 2 = [ K M y + B V 0 ( K Pin + K Pout ) K f M y + B V 0 A y M y A y ]
a 3 = [ B V 0 ( K Pin + K Pout ) K M y + B V 0 A y M y K Ain K C ]
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 = a 1 a 2 B V 0 ( K Pin + K Pout ) K M y B V 0 A y M y K Ain

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 in 2 ( P in P C ) ρ = C q ( A out 2 + A out 1 ) 2 ( P C P 0 ) ρ
Then, the steady-state flow area of the inlet orifice is calculated as
A in = ( A out 2 + A out 1 ) ( P C P 0 ) ( P in P C )
Thereby, K Ain , K Pin , K Aout , and K Pout can be obtained. Then:
(1) According to the stability constraint, the control gain should meet
K C < K C max
(2) According to the dynamic performance requirements, the control gain should meet
K C [ K C , l , K C , h ]
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 = A y K ( P C , 0 P 0 )
where P C , 0 is the designed controlled pressure.
According to the steady-state performance requirements P C [ P C , l , P C , h ] , the variation range of the spring compression is
x s [ x s s A y K ( P C , 0 P C , l ) , x s s + A y K ( P C , h P C , 0 ) ]
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 Δ x u = Δ x y = Δ x s , it has
| Δ x u | | Δ x s s |
where Δ x s s = A y K ( 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 Δ x s s can be calculated.
Step 2. n × m steady-state working points are selected within the working range of the disturbance inputs P in and A out 2 , and the parameters K Ain , K Pin , K Aout , and K 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 , , n × m
Step 3. According to Equations (41) and (42), it is determined that K C , i , i = 1 , , n × m .
Step 4. According to the geometric characteristics of the inlet orifice Δ A in = K C Δ x u , the iterative equations of the underlap x u , i , the inlet flow area A in , i , and the control gain K C , i is
x u , i = x u , i 1 + 2 ( A in , i A in , i 1 ) ( K C , i + K C , i 1 ) , i = 1 , , n × m
Step 5. If the variation range of inlet orifice underlap increment Δ 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 s 0 , according to the steady-state spring compression x s s , the displacement range of the motion valve Δ x y is
Δ x y = x s s x s 0
Then the initial underlap of the inlet orifice is
x u 0 = x u + Δ x y

6. Design and Simulation

The given structural parameters of the constant pressure valve are shown in Table 1.
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 in is [2,8] MPa;
  • The working range of the variable outlet flow area A 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 in = f 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 Δ x s s is 0.22340214 mm.
Step 2. We select 4 × 3 steady-state working points within the working range of the disturbance inputs P in and A out 2 , and calculate the values [ A in , i , K C , i ] at each steady-state point using the method proposed in Section 5.2, as shown in Table 2.
For instance, assuming the inlet pressure of the first steady-state working point P in , 1 is 4 MPa, and the variable outlet flow area A out 2 , 1 is 0.7854 m2, then
A in , 1 = ( A out 2 , 1 + A out 1 ) ( P C P 0 ) ( P in , 1 P C ) = 1.0882796   mm 2
Then, the values of the parameters K AJ , 1 , K PJ , 1 , K AZ , 1 , K PZ , 1 , and K PT , 1 are obtained.
Step 3. (1) According to stability Equation (41), it has
K C < K C max , 1 = 0.007491835
(2) According to the dynamic performance requirements, the design range of the control gain is K C [ 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 [ Δ x u , i , A i n , i ] at each steady-state point are calculated, as shown in Table 3.
Step 5. Because of Δ x u > Δ 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 Δ x s s is 1.3404129 mm. The above design steps are repeated to obtain Table 4.
Because of Δ x u = 1.2335868 < Δ x s 0 = 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 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 s 0 as 80 mm, since the steady-state spring compression x s s is 80.424772 mm, then
Δ x y = x s s x s 0 = 0.424772   mm
x u 0 = x u + Δ x y = 1 . 2866166   mm

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 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 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.

Author Contributions

Methodology, W.Z.; formal analysis, W.Z.; writing—original draft preparation, W.Z.; writing—review and editing, X.W. and Z.J.; validation, Y.L. All authors have read and agreed to the published version of the manuscript.

Funding

This study was supported by the National Science and Technology Major Project (J2019-V-0010-0104), and AECC Sichuan Gas Turbine Establishment Stable Support Project (GJCZ-0011-19).

Data Availability Statement

The data used to support the findings of this paper are contained in the text.

Conflicts of Interest

There is no conflict of interest.

References

  1. Wang, X.; Yang, S.B.; Zhu, M.Y.; Kong, X.X. Aeroengine Control Principles; Science Press: Beijing, China, 2021; pp. 59–106. [Google Scholar]
  2. Fan, S.Q.; Li, H.C.; Fan, D. Aeroengine control: Volume II; Northwest Polytechnic University Press: Xi’an, China, 2008; pp. 186–294. [Google Scholar]
  3. McCloy, D.; Martin, H.R. Control of Fluid Power, 2nd ed.; Ellis Horwood Limited: New York, NY, USA, 1980; pp. 1–150. [Google Scholar]
  4. Xu, C.S.; Jia, S.F.; Li, G. Analysis of the thrust pulsation of the aeroengine. Aviat. Maint. Eng. 2002, 222–224. [Google Scholar]
  5. Wang, H.W.; Wang, X.; Li, Z.P.; Dang, W. Quantitative analysis on constant pressure valve stability. J. Aerosp. Power 2015, 30, 754–761. [Google Scholar] [CrossRef]
  6. Liu, Q.H.; Jia, M.X. Study on linear and nonlinear characteristics of pilot pressure relief valve (Part 1). Modul. Mach. Tool Autom. Manuf. Tech. 1985, 19–23. Available online: http://www.cnki.com.cn/Article/CJFDTotal-ZHJC198509002.htm (accessed on 28 September 1985).
  7. Liu, Q.H.; Jia, M.X. Study on linear and nonlinear characteristics of pilot pressure relief valve (Part 2). Modul. Mach. Tool Autom. Manuf. Tech. 1985, 17–29. Available online: http://www.cnki.com.cn/Article/CJFDTOTAL-ZHJC198510003.htm (accessed on 28 October 1985).
  8. Zeng, X.R.; Song, T.T. Steady state hydraulic compensation single stage overflow valve and its experimental study. Mach. Tool Hydraul. 1984, 12–19. Available online: http://www.cnki.com.cn/Article/CJFDTotal-JCYY198406001.htm (accessed on 26 December 1984).
  9. Li, S.N.; Ge, S.H. Development and research of new single stage overflow valve. Mach. Tool Hydraul. 1987, 24–32. Available online: http://www.cnki.com.cn/Article/CJFDTotal-JCYY198704004.htm (accessed on 29 August 1987).
  10. Ray, A. Dynamic modeling and simulation of a relief valve. Simulation 1978, 31, 167–172. [Google Scholar] [CrossRef]
  11. Yu, X.; Chen, H.H.; Zheng, Y.; Shi, H.Y. Modeling and simulation analysis of pressure relief valve based on AMESim. Hoisting Conveying Mach. 2011, 32–35. [Google Scholar] [CrossRef]
  12. Li, X.J.; Pan, Y.T.; Yue, X.K.; Ma, L.Q.; Lin, Y. Simulated analysis on the dynamic behavior influence caused by pilot valve parameters upon pilot operated pressure reducing valve. J. Taiyuan Univ. Technol. 2013, 44, 594–597, 603. [Google Scholar] [CrossRef]
  13. Hu, C.X. Dynamic modeling and simulation for converse unloading pressure reducing valve. Rocket. Propuls. 2014, 40, 60–64. [Google Scholar] [CrossRef]
  14. Wu, R.; Tang, W.; Lin, W.X. Dynamic performance simulation of pressure relief valve and test. J. Xiamen Univ. 2011, 50, 847–851. [Google Scholar]
  15. Zhang, L.; Li, W.F.; Zhao, Y.B.; Chao, C.H. Modeling and simulation of uniform-pressure-drop valve based on AMESim. Mech. Eng. Autom. 2013, 58–60. [Google Scholar] [CrossRef]
  16. Dong, J.W.; Ma, W.Q.; Guan, G.F. Simulation and dynamic characteristics analysis of a pressure reducing valve based on AMESim. Hydraul. Pneum. Seals 2015, 35, 46–49. [Google Scholar] [CrossRef]
  17. Gu, C.H.; Mao, H.P.; Wang, Q.; Shi, Y.C. Simulation analysis of direct-acting pressure reducing valves dynamic characteristics based on the AMESim. Mach. Des. Manuf. 2017, 234–237. [Google Scholar] [CrossRef]
  18. Zhou, F.; Gu, L.Y.; Chen, Z.H. Model linearization and stability analysis of thruster system controlled by proportional pressure reducing valves. J. Mech. Eng. 2017, 53, 187–194. [Google Scholar] [CrossRef]
  19. Teng, H.; Shi, Y.P.; Zhang, L.; Zang, H. Simulative and experimental study of dynamic characteristics of pressure relief valve based on AMESim. Aerospace 2015, 32, 48–53, 67. [Google Scholar] [CrossRef]
  20. Dasgupta, K.; Karmakar, R. Modelling and dynamics of single-stage pressure relief valve with directional damping. Simul. Model. Pract. Theory 2002, 10, 51–67. [Google Scholar] [CrossRef]
  21. Gabor, L.; Alan, C.; Csaba, H. Nonlinear analysis of a single stage pressure relief valve. IAENG Int. J. Appl. Math. 2009, 39. [Google Scholar] [CrossRef]
  22. Jiang, W.L.; Zhu, Y.; Yang, C. Study on nonlinear dynamic behavior of a hydraulic relief valve. China Mech. Eng. 2013, 24, 2705–2709. [Google Scholar] [CrossRef]
  23. He, X.F.; Zhao, D.X.; Sun, X.; Zhu, B.H. Theoretical and experimental research on a three-way water hydraulic pressure reducing valve. J. Press. Vessel. Technol. 2017, 139, 041601. [Google Scholar] [CrossRef]
  24. Shi, R.; Wang, C.; He, T.; Xie, T. Analysis of dynamic characteristics of pressure-regulating and pressure-limiting combined relief valve. Math. Probl. Eng. 2021, 1–13. [Google Scholar] [CrossRef]
  25. Merritt, H.E. Hydraulic Control System; John Wiley: New York, NY, USA, 1976. [Google Scholar]
  26. Sullivan, J.A. Fluid Power: Theory and Application, 4th ed.; Prentice-Hall: Hoboken, NJ, USA, 1998. [Google Scholar]
  27. Manring, N.D. Hydraulic Control System; John Wiley and Sons: New York, NY, USA, 2005. [Google Scholar]
  28. Zhang, Y. Hydraulic and Pneumatic Transmission; Electronic Industry Press: Beijing, China, 2011; pp. 113–114. [Google Scholar]
Figure 1. Structure diagram of the constant pressure valve.
Figure 1. Structure diagram of the constant pressure valve.
Symmetry 15 00045 g001
Figure 2. Design block diagram of the closed-loop system.
Figure 2. Design block diagram of the closed-loop system.
Symmetry 15 00045 g002
Figure 3. Loop design block diagram of the closed-loop system.
Figure 3. Loop design block diagram of the closed-loop system.
Symmetry 15 00045 g003
Figure 4. Bode diagram curves when changing the control gain.
Figure 4. Bode diagram curves when changing the control gain.
Symmetry 15 00045 g004
Figure 5. Bode curves of the open-loop system under different stabilization control gains.
Figure 5. Bode curves of the open-loop system under different stabilization control gains.
Symmetry 15 00045 g005
Figure 6. (a) Step curves of the controlled pressure under different stabilization control gains when the inlet pressure step disturbs; (b) Step curves of the controlled pressure under different stabilization control gains when the variable outlet flow area step disturbs.
Figure 6. (a) Step curves of the controlled pressure under different stabilization control gains when the inlet pressure step disturbs; (b) Step curves of the controlled pressure under different stabilization control gains when the variable outlet flow area step disturbs.
Symmetry 15 00045 g006
Figure 7. Relational curve of the inlet orifice flow area and the designed stabilization control gain.
Figure 7. Relational curve of the inlet orifice flow area and the designed stabilization control gain.
Symmetry 15 00045 g007
Figure 8. Geometry design curve of the inlet orifice.
Figure 8. Geometry design curve of the inlet orifice.
Symmetry 15 00045 g008
Figure 9. The nonlinear model of the constant pressure valve.
Figure 9. The nonlinear model of the constant pressure valve.
Symmetry 15 00045 g009
Figure 10. Disturbance input curve of the inlet pressure.
Figure 10. Disturbance input curve of the inlet pressure.
Symmetry 15 00045 g010
Figure 11. Step response curves of the inlet pressure disturbance.
Figure 11. Step response curves of the inlet pressure disturbance.
Symmetry 15 00045 g011
Figure 12. Disturbance input curve of the variable outlet flow area.
Figure 12. Disturbance input curve of the variable outlet flow area.
Symmetry 15 00045 g012
Figure 13. Step response curves of the variable outlet flow area disturbance.
Figure 13. Step response curves of the variable outlet flow area disturbance.
Symmetry 15 00045 g013
Table 1. Structural parameters of the constant pressure valve.
Table 1. Structural parameters of the constant pressure valve.
Parameter/UnitValueParameter/UnitValue
M y /Kg0.0115 V 0 /m310−5
K f /(N·m/s)10 ρ /(Kg/m3)800
d /m0.008 B /bar17,000
P C /MPa1.5 d o u t 1 /m0.001
P 0 /MPa0.3 C q 0.7
Table 2. Values of the steady-state inlet flow area and the designed stabilization control gain.
Table 2. Values of the steady-state inlet flow area and the designed stabilization control gain.
P i n /MPa d out 2 /mm A in /mm2 K C max [ K C l , K C h ] K C
226.083668000.139094570[0.002, 0.007]0.0068
12.433467200.041787194[0.002, 0.015]0.0048
01.216733600.019299005[0.002, 0.009]0.0028
323.512407400.034057185[0.001, 0.006]0.0058
11.404962900.011826533[0.001, 0.0055]0.0028
00.702481470.006026016[0.001, 0.0025]0.0018
422.720699000.020693472[0.0010, 0.005]0.0048
11.088279600.007491835[0.0005, 0.003]0.0028
00.544139810.003934234[0.0005, 0.001]0.0008
522.299410400.015556697[0.0008, 0.0045]0.0043
10.919764150.005747635[0.0007, 0.0027]0.0025
00.459882070.003064452[0.0005, 0.0009]0.0007
622.027889300.012802030[0.0005, 0.004]0.0038
10.811155740.004787408[0.0005, 0.002]0.0018
00.405577870.002576085[0.0005, 0.001]0.0008
721.834294900.011061443[0.0005, 0.0035]0.0033
10.733717970.004169889[0.0005, 0.0020]0.0018
00.366858980.002257730[0.00055, 0.00065]0.0004
821.687305900.009849004[0.0005, 0.0030]0.0028
10.674922370.003734197[0.0005, 0.0017]0.0015
00.337461180.002030850[0.0004, 0.0005]0.0003
Table 3. The values [ Δ x u , i , A i n , i ] at each steady-state point when k = 4500 N/m.
Table 3. The values [ Δ x u , i , A i n , i ] at each steady-state point when k = 4500 N/m.
A i n mm2 K C Δ x u mm
0.337461180.00030
0.366858980.00040.097992667
0.405577870.00080.19478989
0.459882070.00070.26267014
0.544139810.00080.38303834
0.674922370.00150.54651654
0.702481470.00180.56488927
0.733717970.00180.58224289
0.811155740.00180.62526387
0.919764150.00250.68560187
1.08827960.00280.75300805
1.21673360.00280.79888448
1.40496290.00280.86610923
1.68730590.00280.96694602
1.83429490.00331.01944210
2.02788930.00381.07810710
2.29941040.00431.14956000
2.43346720.00481.18073600
2.72069900.00481.24057590
3.51240740.00581.40551520
6.08366800.00681.84883600
Table 4. The values [ Δ x u , i , A i n , i ] at each steady-state point when k = 750 N/m.
Table 4. The values [ Δ x u , i , A i n , i ] at each steady-state point when k = 750 N/m.
A i n mm2 K C Δ x u mm
0.337461180.00060
0.366858980.00070.045227385
0.405577870.00080.096852571
0.459882070.00080.16473282
0.544139810.00150.23800042
0.674922370.00170.31973952
0.702481470.00270.33226638
0.733717970.00180.34614927
0.811155740.00250.38216684
0.919764150.00270.42393931
1.08827960.00390.47500459
1.21673360.01000.49348718
1.40496290.00490.51875286
1.68730590.00400.58220073
1.83429490.00380.61989021
2.02788930.00550.66152342
2.29941040.00490.71373902
2.43346720.01600.72656742
2.72069900.00800.75050340
3.51240740.00750.85265932
6.08366800.00601.23358680
Table 5. The designed values of the underlap and the flow area of the inlet orifice.
Table 5. The designed values of the underlap and the flow area of the inlet orifice.
x u mm A i n mm2
00
0.20.33746118
0.245227380.36685898
0.296852570.40557787
0.364732820.45988207
0.438000420.54413981
0.519739520.67492237
0.532266380.70248147
0.546149270.73371797
0.582166840.81115574
0.623939310.91976415
0.675004591.0882796
0.693487181.2167336
0.718752861.4049629
0.782200731.6873059
0.819890211.8342949
0.861523422.0278893
0.913739022.2994104
0.926567422.4334672
0.950503402.7206990
1.052659303.5124074
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Zhao, W.; Wang, X.; Jiang, Z.; Long, Y. A Gain Scheduling Design Method of the Aero-Engine Fuel Servo Constant Pressure Valve with High Accuracy and Fast Response Ability. Symmetry 2023, 15, 45. https://doi.org/10.3390/sym15010045

AMA Style

Zhao W, Wang X, Jiang Z, Long Y. A Gain Scheduling Design Method of the Aero-Engine Fuel Servo Constant Pressure Valve with High Accuracy and Fast Response Ability. Symmetry. 2023; 15(1):45. https://doi.org/10.3390/sym15010045

Chicago/Turabian Style

Zhao, Wenshuai, Xi Wang, Zhen Jiang, and Yifu Long. 2023. "A Gain Scheduling Design Method of the Aero-Engine Fuel Servo Constant Pressure Valve with High Accuracy and Fast Response Ability" Symmetry 15, no. 1: 45. https://doi.org/10.3390/sym15010045

APA Style

Zhao, W., Wang, X., Jiang, Z., & Long, Y. (2023). A Gain Scheduling Design Method of the Aero-Engine Fuel Servo Constant Pressure Valve with High Accuracy and Fast Response Ability. Symmetry, 15(1), 45. https://doi.org/10.3390/sym15010045

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop