Jie HANG, Yunhua LI, Liman YANG

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

KEYWORDS Aeroengine;Fuel metering unit;Large flowrate control;Optimization;Pressure difference compensation

Abstract To address the control accuracy of large fuel flowrate during pressure fluctuation, a novel electro-hydraulic fuel metering unit(FMU)is constructed for afterburner fuel system of military aeroengine.Different from the previous FMU,the proposed FMU can achieve the higher precision opening control by a new metering valve with double control chambers (MVDCC), and realize the lower pressure difference fluctuation regulating by a novel two-stage constant pressure difference compensated valve (CPDCV) with dynamic damping orifice and damping piston. The experimental and AMESim simulation results verify the validity and superiority of the novel FMU. Since the temperature-induced variation in fuel properties and device capabilities may degrade or even impair the properties of novel FMU, the discharge flowrate is analyzed by global sensitivity analysis to research the effect proportion of each factor, the temperature effect is explored to ensure the working reliability in long-span temperature variation.Finally,the optimization of structure parameters for novel CPDCV can further reduce pressure difference fluctuation during pressure regulation, and the overshoot, adjust time and the integral of time multiplied by absolute value of error(ITAE)can be reduced by 24%,30%and 26%,respectively.This paper provides a reference for improving the stability of large flowrate during pressure fluctuation.

1. Introduction

Afterburner fuel control system is an exclusive subsystem of military aeroengine,1which can raise the thrust-weight ratio and improve supersonic cruise and maneuverability.2Nowadays, increasing the discharge flowrate of fuel metering unit(FMU) is a mainstream method to improve thrust-weight ratio,3which may reduce the control accuracy of large flowrate during the pressure fluctuation. In addition, thermal loads from interior or exterior make it difficult to maintain fuel temperature within prescribed limits, which may degrade or even impair the property of fuel system.4Inspired by these considerations, this paper proposes a novel electro-hydraulic FMU for afterburner fuel control system, which can achieve the higher precision opening control and realize the lower pressure difference fluctuation regulation.

The existing flowrate metering methods and related pressure-difference compensation mechanisms consist of three categories: (1) Metering-based flowrate control. The discharge flowrate is controlled by metering valve,5and the pressure difference is maintained constant by constant pressure difference compensated valve(CPDCV). Despite its fast response speed,the application is limited because of high power loss.(2)Pump controlled flowrate control. The key merit is that the pressure difference can be maintained constant by CPDCV and variable displacement pump,and thereby it reduces the recirculation of excess fuel and improves thermal management.6However, the application is limited to small gas turbine due to complex structure and high cost.7(3) Motor-based flowrate control.This method replaces mechanical accessories with electro actuators,and hence reduces size and improves thermal efficiency.8However, the application in electric engine system is limited due to the low control accuracy of small flowrate.9To enlarge the applying scope, we are concerned with the FMU with meteringbased flowrate control.

Traditional metering valve can be regarded as the singlerod electro hydraulic system (EHS). However, the internal dynamic in single-rod EHS cannot be simplified as the load pressure dynamics of symmetrical hydraulic cylinder, which complicates controller design.10Numerous literature on the single-rod EHS focuses on the suppression of external disturbances under parametric uncertainties.11To address uncertain parameters, scholars have put forward some effective controllers, such as robust H∞control, parameter adaptive control,12neural adaptive control,13and other advance control algorithms.Meanwhile,numerous methods are investigated to estimate the external disturbance, such as terminal sliding mode control with disturbance observer and adaptive extended disturbance observer.14Therefore,improving the performance of single-rod EHS with nonlinearity is still a challenge.

The control accuracy of flowrate is mainly affected by the opening control of metering valve,the pressure difference fluctuation regulation of CPDCV, the variation in fuel temperature, and other key factors.3Several literature on FMU focuses on the structure simplification and performance promotion. In the early age, Carrese, et al.15proposed a new metering valve with a diaphragm valve to improve the control performance of flowrate. Georgantas, et al.16investigated a new FMU with a bypass valve to reduce the accumulation of carbon caused by excess fuel. In 1999, Mohtasebi, et al.17presented a new FMU with two independent electronic control units to control metering valve and bypass valve, respectively.Recently, rotary metering valve has attracted much attention,and related literature mainly focuses on the orifice shape optimization and cavitation.18,19Agh, et al.20presented a novel rotary FMU with Cam Nozzle. The innovations were valve shape design and the direct drive actuation. Meanwhile, the influence of back pressure21and other nonlinear factors on pressure difference have been discussed in abundant studies.Chen, et al.22presented that pressure difference fluctuation was caused by the dead zone in the zone-relay link. However,to the author’s best knowledge, these studies may be inadequate in tacking large flowrate stability during pressure fluctuation, which may lead to poor reliability and maneuverability of aeroengine.

Fuel temperature should be limited within the range from-40 °C to 120 °C to ensure the safe and efficient operation of aeroengine. However, temperature change will result in the variation in fuel properties and device capabilities, and hence degrades or even impairs the airplane characteristics.Zhang, et al.23derived a mathematical model of servo valve temperature sensitive operating force, which could explain the temperature-induced jam faults of valves. Inspired by previous results, temperature analysis should be investigated for novel FMU systematically in this work.

As a summary of above research, limitation of existing literature is absence of the precision opening control and pressure difference fluctuation regulation of FMU, which restricts the control accuracy of large flowrate. Moreover,the available literature focusing on temperature analysis of the whole FMU seems to be quite rare compared with servo valve or other components. The main contributions of this paper consist of three aspects:

(1) This paper proposes a novel electro-hydraulic FMU for large flowrate afterburner fuel control system. The high precision opening control can be achieved by a new MVDCC, and the low pressure difference fluctuation regulation can be realized by a new two-stage CPDCV with dynamic damping orifice and damping piston.The superior of the novel FMU can be verified based on AMESim.

(2) This paper is the first to conduct the systematical temperature characteristics to ensure the working reliability of novel FMU in long-span temperature variation.Meanwhile,global sensitivity analysis is used to research the effect proportion of each factor on flowrate.

(3) This paper optimizes the key structure parameters of CPDCV to further reduce pressure fluctuation.Artificial neural network (ANN) builds the nonlinear model between performance indexes and structural parameters.Non-dominated sorting genetic algorithm-II (NSGAII)searches Pareto optimal solution set,technique for order preference by similarity to an ideal solution (TOPSIS)selects the most satisfied solution.

The remainder of the paper is organized as follows: Section 2 is basic description. In Section 3, influence factors of pressure difference and temperature effect are analyzed. In Section 4, the superiority of novel FMU is verified. Meanwhile, the flowrate is analyzed by global sensitivity analysis,and fuel temperature effect is discussed. Then, multi objective optimization is employed to further reduce pressure fluctuation. In Section 5, some conclusions will be drawn.

2. Basic description

2.1. Flowrate control of FMU

The schematic of novel electro-hydraulic FMU is shown in Fig. 1. The fuel supplied by the fuel pump can be transmitted to afterburner through FMU. The fixed pressure reducing valve is used to provide constant pressure control fuel for electro-hydraulic servo valve. Meanwhile, the flow area of the metering valve is controlled by the servo valve, and pressure difference across the metering valve can be maintained constant by CPDCV. Thus, the discharge flowrate is approximately proportional to the flow area.The equation of the flowrate can be shown as follows:

The issues in previous FMU are as follows:

(1) Conventional metering valve is regulated by the servo valve with single control port.Compared with the servo valve with double control ports, it is limited by the low sensitivity, low response speed, and large zero position error.24Meanwhile, the traditional metering valve has also low efficiency and high fuel temperature rise,which may degrade thermal management of fuel control system. Moreover, the difficulty of modeling and control algorithm design for single-rod EHS are also its limitations. Using MVDCC can effectively improve opening control precision of FMU.

(2) The pressure fluctuation caused by the variation in fuel pump rotation speed directly affects the stability of discharge flowrate. Due to the flow force acting on the conventional single-stage CPDCV spool, the flowrate pressure characteristic of FMU is poor, particular in the large flowrate working condition. Meanwhile, the discharge flowrate will be fluctuation violently due to the normally open state of conventional CPDCV, when load pressure changes. Moreover, large pressure shock acting on the CPDCV exacerbates the oscillation movement during pressure regulating.3Using two-stage CPDCV with damping piston and dynamic damping orifice can reduce the pressure difference fluctuation.

Fig. 1 Schematic of novel FMU.

(3) The spools of metering valve and CPDCV are subjected to flow force,friction force,and other forces.The stability of forces is the requisite for the control accuracy of FMU. However, these forces affected by fuel temperature cannot be maintained constant.23Thus, it is significant to incorporate temperature effect into performance analysis of novel FMU to ensure working reliability in wide temperature variation condition.

2.2. Configuration and working principle

In this section, the schematic structure of the proposed novel electro-hydraulic FMU for afterburner fuel control system is shown in Fig. 2. High precision opening control can be achieved by a novel MVDCC. The design concept of symmetrical cartridge valve with double control chambers can be taken into MVDCC design, and hence improves response speed and control accuracy, and reduces the cost and size.An elongated through-hole is manufactured in the middle of MVDCC spool to make force equilibrium. Low pressure difference fluctuation regulation can be realized by a novel two-stage CPDCV. The CPDCV is composed of two essential subcomponents: a constant difference reducing valve used as the pilot valve and a slide valve used as the main valve. The pressure difference across MVDCC can be maintained constant by pilot valve and main valve, and the huge quantity of fuel can be transmitted through the main valve to combustor. Pressure difference orifice and cool orifice can reduce the oscillation of pilot valve during pressure regulation.

Meanwhile, the damping of CPDCV should be further increased to reduce the pressure shock acting on the main valve spool. Based on this theory, a damping piston is mounted in the control chamber of the main valve, of which key merit is to raise damping of FMU without increasing the mass of the main valve and reducing the response speed.Furthermore, dynamic damping orifice installed at control chamber inlet ameliorates the dynamic performance, and hence weakens the oscillation during pressure regulation.

The pressure difference regulation principle of the novel FMU is as follows: the inlet pressure of MVDCC increases as fuel pump rotation speed increases, and thereby it makes the pilot valve spool shift to the direction of increasing flow area of control chamber.Then,the increase of pressure in control chamber of main valve will make the spool shift to the direction of decreasing the flow area,thus increasing the outlet pressure of MVDCC. Then, the pressure difference can be maintained constant by the novel CPDCV, and flowrate will be linearly proportional to the MVDCC spool displacement.

3. Modeling and analysis of FMU

3.1. Pre-opening form of MVDCC

The pressure fluctuation caused by variation in rotational speed of fuel pump may cause wear and leakage of metering valve spool. In this section, the negative pre-opening form of DMMCV is adopted to compensate the leakage, and hence improves the robustness of the novel FMU. Preopening form of DMMCV is shown in Fig. 3.

Due to the imprecision in the manufacturing process, the slight clearance δ can be occurred. When the spool displacement xv1is less than overlap U, the leakage flow rate Qlcan be calculated by

Fig. 2 Schematic structure of novel FMU.

Fig. 3 Pre-opening form of metering valve.

Due to the fact U ?δ, the ratio of Kc0to Kc0′can be expressed as:

From Eqs. (2)–(5), one can conclude that the antidisturbance capability of DMMV increases as the flowpressure coefficient decreases, and hence raises valve stiffness.Therefore,the preopening form of DMMCV is adopted in this work.

3.2. Pressure-difference stability analysis

Pressure difference fluctuation control is another significant factor affecting the control accuracy of large flowrate. In this section, the key factors affecting pressure difference will be analyzed in detail. Force schematic of the pilot valve is shown in Fig. 4.

The equilibrium equation of the pilot valve can be expressed by

where Fpmis the inertial force, Fphis the steady flow force,Fpv-cis the viscous friction and Coulomb force, and Fpsis the spring force. Fpm，F(xiàn)ph，F(xiàn)pv-cand Fpscan be described as:

Fig. 4 Schematic of load forces acting on pilot valve.

Fig. 5 Schematic of load forces acting on main valve.

where kphxp?Ap,P1-Pc=P1-P2+P2-Pc.

From Eq.(11)and Eq.(17),one can conclude that the influence factors of pressure difference are P1, P2,P3,Apand Am.These findings have significant implications to further reduce pressure difference fluctuation by decreasing Apand Amwithin prescribed limits.

3.3. Temperature effect analysis

Fig.6 shows that the fuel used as heat sink can absorb external thermal loads from airborne electromechanical systems.Meanwhile,Fuel temperature cannot be maintained within the specified range due to the throttling loss, friction loss, and other power losses in FMU.23Temperature change will result in the variation in fuel properties and device capabilities, and thus it may impact on the performance.

For the sake of simplicity, temperature variables with superscript T represent the corresponding values to temperature T. All parameters interrelated to temperature should be analyzed. Then, the discharge flowrate of metering valve can be expressed as:

The flow coefficient of metering valve is related to Reynolds number, which can expressed as:

Conclusively, the temperature-induced variation in fuel properties and device capabilities may degrade or even impair the properties of fuel control system. Thus, it is indispensable to incorporate temperature effect into the performance analysis of the novel FMU to ensure the working reliability.

4. Simulation and discussion

In this work, the related simulations will be carried out in AMESim platform, which can offer numerous facilities in terms of usability and increased model scalability.26Structure and the operating parameters of the novel FMU are shown in Table 1. Fig. 7 shows the AMESim model of novel FMU.

4.1. Validation of models

In this section,some experimental data of nozzle in afterburner FMU for a military fighter is given, which represents six typical working conditions. Thus, the related experimental fitting curves can be achieved by interpolating the typical operating points. The experimental fitting data of the discharge flow and back pressure can be taken as the input of the servo valve and nozzle for different simulation models, respectively. The validation of AMESim models can be verified by comparing the experimental fitting data of fuel flowrate and inlet pressure with simulation data.

Fig. 6 Temperature effects on stability of fuel flowrate.

Table 1 Main parameters of FMU.

Fig. 8 shows the comparison of the discharge flowrate and inlet pressure of the nozzle between experiment and simulation. The experimental discharge flowrate of original scheme is controlled by proportion integration differentiation (PID)algorithm with a certain revision for fuel temperature and quantity, while that in the AMESim simulation is only regulated by PID without revision, which results in a certain deviation between experiment curve and simulation curve.Significantly,the trends of experimental and simulation curves are similar. The mean errors of discharge flowrate of original scheme are within 0.01634 kg/s, and those of inlet pressure are within 0.1694 MPa. The two curves are identical by 98.46%and 93.59%,respectively.Meanwhile,the mean errors of discharge flowrate of optimization scheme are within 0.00645 kg/s, and those of inlet pressure are within 0.15713 MPa. The two curves are identical by 99.36% and 94.05%, respectively. Hence, AMESim model at different schemes can precisely analyze the flowrate pressure characteristic.

Meanwhile, the novel FMU is a new device of afterburner fuel control system. In this work, it is difficult for us to build the experimental conditions for the novel FMU. As AMESim model of novel scheme is built based on the original simulation model with partial quantitative modification, the following research of novel FMU can be analyzed based on the valid AMESim model.

4.2. Properties comparison at different schemes

Limited by the conventional single-rod EHS, it is a challenge to improve the control accuracy of metering valve. Using MVDCC can effectively improve the high precision opening control. In Fig. 9 and Fig. 10, the rotation speed of pump keeps constant, and the set value of input current increases from minimum to maximum,and then reduces from maximum to minimum. Fig. 9 shows the comparison of flowrate characteristic at different schemes. Due to the negative pre-opening form of metering valve, the flow gains of two schemes have a dead zone near the zero position. Meanwhile, the flowrate response of novel MVDCC is excellent in linearity, while that of the original metering valve is the opposite.The hysteresis of the original metering valve may be caused by the low response speed and large zero position error of the servo valve with single control port. Fig. 10 represents the anti-load interference ability of metering valve at different schemes.The pressure difference at both of schemes cannot be maintained constant during the whole working condition,and the pressure difference in the opening stage is slightly smaller than that in the closing stage. The range variation of pressure difference at different schemes is small, which indicates that both of schemes have strong anti-load interference ability. Fig. 11 shows dynamic step responses of flowrate at different schemes. To monitor the dynamic response of different metering valve, we apply step current ranging from 15 mA to 20 mA and from 20 mA to 15 mA at 25 s,30 s,respectively.The original metering valve has a response delay in the closing stage,while the response of novel MVDCC is stable during the whole working condition without any over-regulation,oscillation,and hysteresis.Meanwhile,the response speed of novel FMU in the opening stage is faster than that of original one. Simulation results has proved that the novel MVDCC is of high steady-state control precision, fast dynamic response and strong anti-load interference ability, which can fully meet the fuel control requirements.

Fig. 7 AMESim model of novel FMU.

Fig. 8 Comparison of inlet pressure and discharge flowrate of nozzle between experiment and simulation.

Fig. 9 Comparison of control characteristic.

Fig. 10 Flowrate-pressure difference property.

Fig. 11 Dynamic step response of fuel flowrate.

Fig. 12 Comparison of step response of pressure difference.

Fig. 13 Frequency domain analysis of flowrate fluctuation.

Fig. 14 Comparison of step response of discharge flowrate.

The pressure difference fluctuation control performance of CPDCV at different schemes can be analyzed both in time domains and frequency domain.As shown in Fig.12,the flow area of metering valve keeps constant, outlet pressure of fuel pump changes from 145 MPa to 155 MPa at 25 s. Compared with the original scheme, both dynamic damping orifice and damping piston can dramatically improve the ability of CPDCV to suppress pressure difference fluctuation. Meanwhile,simulation result indicates that design scheme 2 has better regulation performance of pressure difference fluctuation than design scheme 1. Moreover, the design scheme 3 brings the most significant enhancement effect in reducing pressure difference fluctuation, which results in 45%, 27% and 42%reduction in overshoot, adjust time and the integral of time multiplied by the absolute value of error (ITAE), respectively.margin, and hence reduce the overshoot and adjust time during pressure regulation. In the third scenario, the further increase of damping ratio and the phase margin is beneficial to further improve the transient performance. From Fig. 12 and Fig. 14, it follows that the trends of curves of discharge flowrate and pressure difference at different schemes are similar,which means flowrate stability can be effectively improved by reducing the fluctuation of pressure difference. Simulation results illustrate that the novel two- stage CPDCV with dynamic damping orifice and damping piston fully meets low pressure difference fluctuation regulation requirement of fuel control system.

4.3. Global sensitivity analysis of novel FMU

The control accuracy of discharge flowrate is mainly affected by the comprehensive influence of precise opening control of metering valve, the pressure difference fluctuation regulation of CPDCV,the change of fuel temperature,and other factors.3However, the relationship between influence factors and flowrate is complicated and cannot be accurately reconstructed by using traditional fitting methods. In this section, the back propagation(BP)27neural network can be chosen as a learning method to build the above relationship due to its strong ability of fitting arbitrary functions. Then, global sensitivity analysis can be used to analyze the effect proportion of each factor and the interactions among them on flowrate.

In this section,a simplified method28is used,and the steps are as follows: (I) Generate a (N, 2k) matrix of random numbers (N is a base sample, k is the number of technical indicators) and define two matrices of data (A and B). (II) Define a matrix Ciformed by all columns of B except the ith column,which is taken from A.(III)Compute the model output for all the input values in the sample matrices A, B, and Cito obtain three vectors of model outputs of dimension N×1:yA=f（A ),yB=f（B ), yC=f（C). First-order sensitivity index can be calculated as:Fig. 13 shows that the corresponding frequency diagram of pressure difference at different schemes can be obtained by Bode diagram. Both dynamic damping orifice and damping piston can effectively increase the damping ratio and the phase the fitting curves can be obtained by MATLAB, which are as follows:

Table 2 Performance indicators at different schemes.

In this simulation,760 sets of data would be regarded as the BP neural network sample data, of which seventy-five percent of groups could be chosen as the training samples, and other groups could be chosen as the testing samples. The set value of spool displacement increases from minimum to maximum,and the variation of temperature is supposed by the law T=40+30sin（2πt ).

Fig.15(a)shows the maximum fitting error of BP is approximately equal to 1%, which indicates the BP network has the high capacity of generalization. An area chart is presented in Fig. 15 (b) and the sensitivity indexes of each influence factor are listed. The first-order sensitivity index and total effect index of spool displacement are larger than other performance indexes,which means the spool displacement has the most primary contribution to the flow during the opening stage and followed by fuel temperature and pressure differential. Meanwhile,the effect proportion of pressure difference to flowrate is lower than that of fuel temperature to flowrate.

4.4. Temperature characteristics of novel FMU

In this section,to further explore the temperature-related characteristics of the novel FMU,several related AMESim simulations of novel FMU have been carried out and relevant results will be analyzed.

Compared with the pressure effects, temperature has more significant influence on the properties of fuel. To achieve the temperature characteristic of fuel in density and viscosity,

Fig. 16 shows that the effect of fuel temperature on viscosity is greater than that on density. In particular, the viscosity apparently is fallen by nearly 92.79%, while fuel density dips from 824.76 kg/m3to 703.81 kg/m3with the reduction of 14.66%. From Fig. 17, it follows that: (I) the discharge flowrate of the novel FMU is proportional to the spool displacement of metering valve and fractionally fluctuates with fuel temperature. (II) The temperature effect of the flowrate at large opening is higher than that at small opening due to the reduction of fuel density.More specifically,Fig.18 shows that the flowrate is fallen by 1.97% at the maximum opening, less than 2.8%, which ensures the working reliability of the novel FMU in long-span temperature variation. Fig. 19 shows that the dynamic step characteristic of flowrate at different fuel temperature.The results show that the flowrate decreases with increasing the temperature. Meanwhile, the response speed of flow at high temperature is slightly faster than that in low temperature due to the reduction in fuel viscosity. In conclusion,the variation in fuel temperature has a significant influence on the performance of the novel FMU to some extent.

4.5. Optimization of CPDCV parameters

In this section,the key structure parameters of dynamic damping orifice and damping piston (e.g. the damping piston diameter d, clearance on damping piston δl, the contact length on damping piston lc, and the dynamic damping orifice diameter dd.) have significant influence on regulation performance of pressure difference fluctuation of the novel CPDCV.

Fig. 20 shows that the trends of pressure difference fluctuations under different the damping piston diameters d seem to be similar, however, there exits steady-state error. Thus, the steady performance of novel FMU can be improved by optimizing d within reasonable bounds.

Fig. 15 Global sensitivity of each technical index.

Fig. 16 Variation in viscosity and density with temperatures.

Fig. 17 Three-dimensional variation in flow with fuel temperature and spool displacement.

Fig. 18 Static characteristic of flow varying with temperature.

Fig. 21 shows that the response speed of CPDCV can be improved by increasing the clearance on damping piston δl.However, the overshoot of pressure difference fluctuation increases with the increasing δl.

Fig. 19 Step response of flowrate at different temperatures.

Meanwhile,it follows that response speed can be effectively improved by increasing δlwithin the range from 0.005 mm to 0.015 mm. When δlexceeds 0.015 mm, the effect of δlon response speed is obviously weakened.

Fig.22 shows that the overshoot of pressure difference fluctuation can be reduced by increasing the contact length on damping piston lc. However, the response speed of CPDCV decreases rapidly with the increasing the lc, and hence the adjust time of step response will be extended.

Fig. 23 shows response speed of CPDCV will be increased by increasing the dynamic damping orifice diameter ddwith the range from 1 mm to 2 mm, when ddexceeds 2 mm, the effect of ddon response speed is obviously weakened. Meanwhile,ddhas little impact on overshoot and steady-state error.

To further improve the regulation performance of pressure difference of novel CPDCV,multi-objective optimization algorithm will be adopted. Performance indexes evaluating the effect of the key structure parameter on the suppressing pressure fluctuation of CPDCV are as follows:

(1) Overshoot of pressure difference fluctuation

Fig. 20 Comparison of pressure difference under different d.

Fig. 21 Comparison of pressure difference under different δl.

Fig. 22 Comparison of pressure difference under different lc.

Fig. 23 Comparison of pressure difference under different dr.

(3)ITAE which used to show dynamic performance of pressure difference fluctuation

The objective function is given as follows:

In this work, structural parameters of CPDCV to be optimized are as follows: the damping piston diameter d with the range from 10 mm to 30 mm,the clearance on damping piston δlwith the range from 0.005 mm to 0.035 mm, the contact length on damping piston lcwith the range from 40 mm to 70 mm, and the dynamic damping orifice diameter ddwith the range from 1 mm to 5 mm. Fig. 24 shows the flowchart of multi-objective optimization.

ANN can reconstruct the complex nonlinear relationship between performance indexes and structural parameters.Multi-objective optimization algorithm29based on NSGAII can search Pareto optimal solution set.30TOPSIS can select the most satisfied solution.

Fig. 25 shows that the maximum fitting errors of performance indexes are approximately equal to 4%,which indicates that the ANN established is high capacity of generalization and high accuracy. As seen from Fig. 26, the Pareto optimal surface is made up of a considerable number of nondominated solutions.Since only a set of optimization parameters of CPDCV is needed in the actual project, a suitable method is employed to select the optimal solution. Here, the optimal solution is measured by TOPSIS method.

Fig. 27 shows that the comparison of the performance of optimized novel FMU on pressure difference fluctuation suppression compared with the initial novel FMU.The optimized CPDCV can acquire better performance in terms of pressure difference fluctuation suppression. The overshoot of pressure difference fluctuation can be decreased by 24%,the adjust time by 30%,and ITAE by 26%.The performance improvement of low pressure difference regulation will improve the stability of large flowrate during pressure fluctuation, and hence further improve the control accuracy of large flowrate.Fig.28 verifies the above conclusions.

5. Conclusions

This paper proposed a novel fuel metering unite (FMU) to improve the control accuracy of large flowrate of aeroengine afterburner fuel control system. The key contributions of the novel FMU is to achieve the high precision opening control by metering valve with double control chambers (MVDCC),and to realize the low pressure difference fluctuation regulation by a novel two-stage constant pressure difference compensated valve (CPDCV) with dynamic damping orifice and damping piston. The theoretical analysis and simulation on this novel flowrate valve show that it can perfectly meet the performance requirements.

In addition,the global sensitivity analysis for the systematical temperature characteristics of the novel FMU illustrates the working stability of novel FMU in long-span temperature variation can be ensured. Furthermore, by virtue of multiobjective optimization algorithm, the regulation performance optimization of pressure difference fluctuation of novel CPDCV has carried out, and the performance of novel FMU can be improved: overshoot, adjust time and ITAE can be reduced by 24%, 30% and 26%, respectively.

Fig. 24 Flowchart of multi-objective optimization.

Fig. 25 Fitting error of BP.

Fig. 26 Pareto front.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was co-supported by the National Key Basic Research Program of China (No. 2014CB046403) and the National Science and Technology Major Project (2017-V-0015-0067).

Fig. 27 Comparison of the step response of flowrate.

Fig. 28 Comparison of the step response of flowrate.