Pneumatic pressure control based on improved NMPC and its application to aeroengine surge simulation

Xinglong ZHANG, Zhonglin LIN, Jio LI, Tinhong ZHANG,*

aCollege of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

bSchool of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350108, China

KEYWORDSAeroengine surge simulation;Asymmetric valve groups;High-speed on–off valve;Nonlinear model predictive control;Pneumatic pressure tracking control

AbstractIn the semi-physical simulation of aeroengines, using the pneumatic pressure servo control technology to provide realistic pneumatic excitation to the sensors and electronic controller can improve the confidence of the simulation and reduce the test cost and risk.However, the existing methods could not satisfy the precise simulation of large-amplitude and high-frequency pulsating pressure during aeroengine surge.In this paper,a pneumatic pressure control system with asymmetric groups of the High-Speed on–off Valve (HSV) is designed, and an Improved Nonlinear Model Predictive Control(INMPC)method is proposed.First,the volumetric flow characteristics of HSV are tested and analyzed with Pulse Width Modulation(PWM)signal input.Then,a simplified HSV model with the volume flow characteristic correction is developed.Based on these, an integrated model for the whole system is further established and used as the prediction model in INMPC.To improve the computational speed of the rolling optimization process,the mapping scheme from control signal to PWM duty cycle of HSVs and the objective function with exterior penalty function are designed.Finally,the random step,sinusoidal and real engine surge data are set as the reference pressure in multiple comparative experiments to verify the effectiveness of the pressure tracking system.

1.Introduction

Surge is a typical unstable flow pattern of aeroengines that greatly affects the performance and safety of the engine.1–3As the primary aspect of active stability control, the detection algorithm must undergo repeated reliability verification.However, the current data source for the validation of the surge detection algorithm can only rely on engine or compressor surge tests, which is costly and risky.4,5Even if the surge test or flight fault data can be replayed through electrical signals or digital signals,it also lacks the participation of the actual pressure sensor.In the semi-physical simulation for the aerospace field, pneumatic pressure servo control technology is widely used to simulate the real and effective pressure signal and provide the real excitation to sensors and electronic controllers.Therefore,applying pneumatic pressure servo control technology to simulate the large-amplitude and high-frequency pulsating pressure during aeroengine surge can ensure the confidence of the reliability verification of surge detection algorithm and avoid the high cost and high risk of surge tests,so as to provide the basis of semi-physical simulation for the research of aeroengine active stability control.

At present,the pneumatic pressure servo system adopts the proportional valve or the on–off valve as the control element.In particular, the High-Speed on–off Valve (HSV) has been more and more applied in pressure control in the aerospace field because of its low cost and strong anti-pollution ability.6–8Pneumatic servo systems using HSVs essentially control the pressure in the air chamber by controlling the flow of the HSV, or furthermore the position and force output through the actuator.Therefore, the modeling of the HSV is the basis of research into pneumatic control.Taghizadeh et al.9divided the HSV into electromagnetic, mechanical and fluid subsystems, and established a nonlinear dynamic model according to the interaction among them.However, the accuracy of the model is limited due to some unmeasured parameters needed to be identified.Based on this model, Fang et al.10considered the influence of coil heating on the electromagnetic force, and further improved the accuracy of the nonlinear dynamic model of the HSV.Due to their complexity, mechanism-based nonlinear dynamic models are mostly only suitable for simulation studies of the dynamic characteristics of HSVs and are difficult to apply to the design of real-time controllers.Therefore,most of the control methods using HSVs as control elements adopt simplified models instead of complete HSV mechanism model in order to simplify the system model and facilitate the design of the controller.Hodgson et al.11used the mass flow equation representing that gas flows through the orifice as a simplified HSV model,and designed a sliding mode controller for it.Hassan,12Le,13Wang14et al.also utilized a similar simplified HSV model to design the nonlinear controllers.

Because of the discrete switching characteristics of the HSV,Pulse Width Modulation(PWM)signals are widely used in the control of HSVs.15The average mass flow rate of the switching valve over one cycle is controlled by varying the duty cycle of the PWM signal,thus achieving approximate continuous control of HSV’s flow.In response to the inevitable effects of nonlinear and uncertain parameters in pneumatic servo systems, a variety of robust nonlinear control algorithms combined with PWM signals have been applied to HSV-based pneumatic servo control.Pipan and Herakovicˇ16,17designed a control strategy based on the volumetric flow characteristics of the HSV.The relationship among the PWM duty cycle,the pressure difference and the volumetric flow output of the HSV is measured, and then the PWM duty cycle of the HSV is deduced through bilinear interpolation according to the expected volumetric flow calculated by the PID controller.Hejrati and Najafi18 proposed a Sliding Mode Controller(SMC)combined with PWM,which uses a single HSV to control the pressure in the air chamber and realizes high-precision tracking of step and sinusoidal signals.Lin et al.19proposed a hybrid control scheme based on time interlaced modulation,SMC and seven-mode switching, which improved the control performance of the intelligent real-time pressure tracking system using HSVs.Pi et al.20proposed a combined control algorithm including a PID controller with PWM and a fuzzy controller with cooperative PWM, and realized the combined brake pressure control based on two HSVs.In addition to the widely used PID control and SMC, Li et al.21proposed a dynamic event triggered mechanism to realize the tracking control of aeroengine in finite time, which can also be applied to pneumatic pressure tracking control based on HSVs.Zhang et al.22proposed a hybrid model predictive control.The simplified model of the HSV is integrated in a mixed logical dynamical framework after linearization and discretization.In addition, a switching minimization term is added to the objective function of the optimization problem to achieve accurate pressure tracking and prolong the service life of HSVs.Although model predictive control can achieve higher accuracy pressure tracking control through online rolling optimization, it is rarely used in pneumatic pressure servo control using HSVs due to the contradiction between the prediction model’s accuracy and computational complexity.

Although the above research on pressure tracking using HSVs has made some progress, the reference pressure signals in most research are limited to step signals.Even for the sinusoidal reference pressure, its pulsation amplitude is only several tens of kPa and its pulsation frequency is lower than 0.5 Hz, which cannot satisfy the requirements of large amplitude and high frequency of pulsation pressure at the compressor outlet during engine surge.

In order to realize an accurate pressure simulation of the aeroengine surge, i.e., the accurate tracking of a large amplitude fluctuating air pressure in a short time,a pneumatic pressure simulation system with single charge HSV and double discharge HSVs is built, and an Improved Nonlinear Model Predictive Control (INMPC) method is proposed.The asymmetric HSV groups of single charge valve and double discharge valve can accelerate the bleeding speed and supplement the asymmetry of charge and discharge speed.23,24The volume flow characteristics of HSV are tested and analyzed, and it is used to modify the existing HSV simplified model which reflects the relationship between the input PWM duty cycle and the output average mass flow of HSV under different differential pressure.By combining the HSV model and chamber model,the model of the whole pneumatic simulation system is further established and used as the prediction model in INMPC.To avoid a large number of iterative calculations of multivariable constrained nonlinear optimization problems in Standard Nonlinear Model Predictive Control (SNMPC), a mapping scheme from the control signal to the duty cycle of each HSV’s PWM signal is designed, which considers the characteristics of the dead zone and saturation zone of the HSV.In addition, the constraint of the control quantity is added to the objective function as an exterior penalty function.Therefore,the multivariable constrained nonlinear optimization problem is transformed into a univariate unconstrained nonlinear optimization problem and solved.Finally,compared with the PID and the SMC in Ref.18,several groups of experiments are carried out with step signal, sinusoidal signal and real surge test data as the reference pressure to verify the steady and dynamic pressure tracking performance of the asymmetric HSV groups and INMPC method proposed.

This paper is organized as follows:Section 2 introduces the test and analysis of HSV volume flow characteristics.Section 3 introduces the mathematical model of the pneumatic simulation system,including HSV simplified model with volume flow characteristic correction and chamber.Section 4 describes the design method of INMPC.The experimental results are shown and the discussion is made in Section 5.The conclusions are drawn in Section 6.

2.Test and analysis of HSV volume flow characteristics

This section analyzes the dynamic characteristics of the HSV driven by the PWM signal, and divides five working states of HSV under different PWM duty cycle.Then, the test scheme of HSV’s volume flow characteristics is designed.According to the test results,the correlation between volume flow characteristics and dynamic characteristics of HSV is analyzed in detail.

2.1.Dynamic characteristics of HSV with PWM signal input

Fig.1 shows the structural diagram of the mac100 series highspeed on–off valve studied, which is a typical two-position three-way (3/2) valve.Port 1 of the HSV is the inlet of the air supply.Port 2 is normally closed and Port 3 is normally open so that the HSV is used as a two-position two-way valve(2/2).The main parts include manual operator,sealed solenoid enclosure,armature,push pin,fixed core,moveable pole piece,spool and valve spring.Once the solenoid is excited by an electrical signal,the magnetic field generated by the fixed iron core and the solenoid applies an electromagnetic force FEto the armature.When FEgradually increases to overcome the sum of pneumatic force FPcaused by the pressure difference between the inlet and outlet of HSV and spring preload force FS, the armature will move towards the fixed core and further drive the push pin to change the position of the spool.After the electrical signal excitation disappears, FEalso disappears.Under the action of FPand FS, the spool resets with the push pin and the HSV is then closed.

The dynamic characteristics of the HSV refer to the variation of spool displacement from giving a high-level signal to fully opening the valve and from giving a low-level signal to fully closing the valve in a PWM cycle.Fig.2 shows the standard dynamic characteristics of HSV in one PWM cycle.Assume that the period of PWM is TPWMand the duty cycle is d, and then the high-level duration is d?TPWM.Obviously,the actual movement of the spool will be affected by the delay,and there is a certain lag in the change of the PWM signal.

The opening process of the HSV can be divided into three phases:

(1)Delayed opening phase:The duration of this phase is the opening delay time Ton1.In this stage, the valve current increases slowly after the high-level electrical signal, and the magnetic field strength increases at the same time.However,FEis less than FPand FS, so the spool displacement is always 0.

Fig.1 Structure of 3/2 HSV.

Fig.2 Dynamic characteristics of HSV.

(2)Opening movement phase:The duration of this phase is the opening movement time Ton2.As the valve current continues to increase to FEgreater than the sum of FPand FS, the spool begins to displace.Meanwhile,the counter electromotive force generated by the spool motion causes the valve current to gradually decrease.

(3)Fully open phase:At this phase,the spool moves to the maximum displacement and the HSV is fully opened.The valve current stops falling,gradually increases to the maximum value and remains stable.

The closing process of HSV is also divided into three phases:

(1)Delayed closing phase:The duration of this phase is the closing delay time Toff1.When the high-level electrical signal disappears, the valve current gradually decreases due to the equivalent inductance.However, FEis still greater than the sum of FPand FS, so the valve spool is still at maximum displacement.

(2) Closing movement phase: The duration of this phase is the closing movement time Toff2.The valve current continues to decrease, causing the magnetic field strength to decrease as well.When FEon the armature is less than the sum of FPand FS,the spool displacement starts to decrease.At this point,the spool movement generates counter-electromotive force, so the valve current will gradually increase.

(3)Fully closed phase:As the spool moves,its displacement gradually decreases to 0 and the HSV is completely closed.The valve current stops rising in this phase and gradually decreases to 0.

According to the above analysis of the dynamic characteristics of the HSV, with the increase of PWM duty cycle, HSV can be divided into five states as shown in Fig.3.

(1) The HSV cannot open, which corresponds to Fig.3(a).In this state, the spool cannot move because the PWM duty cycle is small and the duration of the PWM high-level is less than Ton1,resulting in the electromagnetic force being less than the sum of the pneumatic force and the spring force.

(2) The HSV can partially open, which corresponds to Fig.3(b).In this state, the duration of the PWM high-level is between Ton1and Ton2.At this point, the electromagnetic force is greater than the sum of pneumatic force and spring force, and the spool begins to move.During movement of the spool, the high-level signal disappears, but the spool continues to open due to inertia until the valve current decreases to the point where the electromagnetic force is less than the sum of the pneumatic force and the spring force,and the valve begins to close.

(3) The HSV can open and close normally, which corresponds to Fig.3(c).In this state, the duration of the PWM high-level is greater than Tonand the duration of PWM lowlevel is greater than Toff, when the HSV can be fully opened and closed.

(4)The HSV can only partially close,which corresponds to Fig.3(d).In this state, the duration of PWM high-level is greater than Ton,and the duration of PWM low-level is greater than Toff1and less than Toff.At this point, the HSV can be fully opened but cannot be completely closed because during the closing process it goes to the next PWM cycle and the PWM signal goes high again.Due to inertia,the valve will continue to close until the valve current increases to the point where the electromagnetic force is greater than the sum of pneumatic force and spring force.Then the valve will open again and the spool will move to the maximum displacement.

(5) The HSV cannot close, which corresponds to Fig.3(e).In this state, the duration of PWM low-level is less than Toff1,and the next PWM cycle will be entered when the valve current is not reduced until the electromagnetic force is less than the sum of pneumatic force and spring force.Therefore, the HSV cannot be closed, and the spool is always at the maximum displacement.

2.2.HSV volume flow characteristic test method

As shown in Fig.4, the HSV volume flow characteristic test platform includes an air compressor, a compressed air tank,a refrigerated air dryer,pneumatic triplet consisting of the Filter,Regulator,and Lubricator(FRL),a pressure sensor,a volume flow sensor, a controller and the HSV.The model of the air compressor is DENAIR DVA-11A, and its maximum working pressure is 850 kPa.The air tank matched with the air compressor is used to store compressed air, and the refrigerated air dryer is used to reduce the temperature of compressed air and dry the discharged air.The FRL is used to stabilize the air supply, purify and filter the compressed air,and adjust the required output pressure to satisfy the needs of tests at different differential pressures.The model of the pressure sensor is MEACON MIK-P300 and the volume flow sensor model is PFMB7501-04-C.The controller adopts a high-performance embedded control unit MangoTree Robust RIO, which can be inserted into MangoTree E series cards for expansion.The test platform uses digital input and output card (MT-E750) and analog acquisition card (MT-E710), in which E750 card is configured as PWM digital output card to control HSVs, and E710 card is used as the analog acquisition card to collect sensors’signal.The HSV selects MAC 111b-611BA and is set to 2/2 mode.The pressure regulating range is from vacuum to 1 MPa.The precise movement per minute can reach more than 12000 times, with good dynamic performance.

Fig.3 Dynamic characteristics of HSV under different PWM duty cycle.

The specific test process is shown as follows:

Step 1.Set the frequency of PWM signal fPWM= 1/TPWMand find the appropriate control cycle.The HSV(model MAC 111b-611BA) used in this experiment has an opening time of 4 ms and a closing time of 6 ms.Considering that a certain processing time needs to be reserved for the control algorithm,the alternative PWM signal frequencies are set as 25, 40 and 50 Hz.

Step 2.Set the differential pressure ΔP between the inlet and outlet of the HSV.In the test, a different inlet pressure Pinis set by adjusting the regulator.Since the outlet of the HSV is connected to the atmosphere where the pressure is Patm, the differential pressure can be regarded as ΔP = Pin- Patm.Considering that the air supply pressure in the experiment is 800 kPa, set ΔP as 100, 200, 300, 400,500, 600 and 700 kPa.

Step 3.Give the duty cycle of the PWM signal in steps of 0.005 from 0 to 1 under each set differential pressure.Each duty cycle lasts for 3 s.Each duty cycle lasts for 3 s to ensure the stability of airflow, and then record the average volume flow through the HSV under the current duty cycle.

Step 4.Repeat the above steps until all frequency and differential pressure tests are completed.

2.3.Analysis of HSV volume flow characteristics

When all the tests are completed,the collected data are used to analyze the volumetric flow characteristics of the HSV.Taking the PWM signal frequency fPWM=40 Hz and the differential pressure ΔP = 0.4 MPa as an example, the relationship between the duty cycle of the PWM signal and the average volume flow of the HSV is shown in Fig.5.Obviously, there is a corresponding relationship between the measured HSV volume flow characteristics and the dynamic characteristics.With the change of PWM signal duty cycle, the volume flow through the HSV can be divided into five regions:dead zone,nonlinear region 1,linear region,nonlinear region 2 and saturation zone,which correspond to the five spool motion states shown in Figs.3(a)–3(e) respectively.

Fig.4 HSV volume flow characteristic test platform.

Fig.5 HSV volume flow characteristics with fPWM=40 Hz and ΔP = 0.4 MPa.

Fig.6 HSV volume flow characteristics with fPWM= 25, 40,50 Hz and ΔP = 0.4 MPa.

As shown in Fig.5, when the duty cycle is less than 0.175,the HSV’s volume flow Qvis 0 because the HSV cannot be opened.According to the frequency and duty cycle of the PWM signal at this point, Ton1= 4.375 ms can be calculated;when the duty cycle is 0.175–0.195, the volume flow of the HSV changes nonlinearly because it is partially open, and the delay time of valve opening movement is calculated as Ton2= 0.5 ms; when the duty cycle is between 0.195 and 0.7, the volume flow approximately changes linearly because the HSV can be opened and closed normally; when the duty cycle is 0.7–0.73, the volume flow shows a nonlinear trend of rapid increase because the HSV can only be partially closed,and the closing movement time is calculated as Toff2=0.75 ms;when the duty cycle is greater than 0.73, the volume flow reaches more than 95 % of the maximum value.It is considered that HSV is in the saturation zone.Therefore, it can be calculated that the closing delay time Toff1= 6.75 ms.

In order to further study the influence of PWM signal frequency on HSV volume flow characteristics, the volume flow test results with fPWM= 25, 40, 50 Hz and ΔP = 0.4 MPa are shown in Fig.6.Similarly, Ton1, Ton2, Toff1and Toff2in the HSV dynamic characteristics under each frequency can be calculated according to the percentage of each region in the volume flow characteristics.These results are shown in Table 1.It is obvious that Ton1, Ton2, Toff1and Toff2of the HSV with different fPWMare relatively consistent for the same differential pressure, which indicates that it is reliable to ana-lyze the dynamic characteristics of HSV based on the volumetric flow characteristics.However, with the increase of valve control frequency, that is, the increase of PWM cycle time,the percentage of controllable region of the HSV will decrease.When fPWM= 25 Hz, the linear region of volume flow accounts for 68%,while when fPWM=50 Hz,the linear region accounts for only 36.5%.The decrease of the linear region percentage will make the controllability of the system worse, but the large control cycle cannot ensure good control accuracy,so it is difficult to take both into account.Considering that the percentage of the controllable region at 40 Hz is less than that at 25 Hz and the smaller control period is conducive to improving the system’s control accuracy, 40 Hz is chosen as the control period of HSVs in this paper.

Table 1 Comparison of volume flow characteristics and dynamic characteristics with different PWM frequencies.

With fPWM= 40 Hz, the volumetric flow characteristics of the HSV under all differential pressures are shown in Fig.7.Under each differential pressure, a set of Ton1, Ton2, Toff1and Toff2can be calculated from the volume flow characteristics.Furtherly, the volume flow characteristics are normalized and the results are shown in Fig.7(b).After normalization,the variation trend of the HSV volume flow with the PWM signal duty cycle is basically the same.Another reason for normalization is that the mass flow rate of HSV is often of more interest in studies,but high-precision mass flow meters are very expensive.Considering that there is only a density conversion between the volume flow and mass flow of the air, the trend of both with the duty cycle of the PWM signal must be consistent.In other words, the normalized mass flow characteristic curves are as same as the normalized volume flow characteristic curves.The normalized mass flow characteristics of HSV also reflect its dynamic characteristics, so it can be used in the modeling of the HSV, which will be concretely described in Section 3.1.

3.System mathematical model

Fig.8 shows the schematic diagram of the pneumatic pressure control system studied.It is worth noting that, unlike the traditional pneumatic servo control system, the pneumatic pressure control system designed in this paper uses single charge HSV and double discharge HSVs to compensate for the asymmetry speed of charge and discharge.HSV1 is the charge valve and HSV2 and HSV3 are discharge valves.High-pressure air supply connects the inlet of HSV1;the chamber inlet connects the outlet of the charge valve and the chamber outlet connects to the inlet of HSV2 and HSV3;the outlet of HSV2 and HSV3 connects the atmosphere.The PWM signal is used to control the on–off of HSVs to change the mass flow in the chamber and finally control the pressure in the chamber.The models of the HSV and the chamber are introduced in each subsection first.Then the integrated simulation frame is presented at the end of this section.

Fig.7 HSV volume flow characteristics with fPWM=40 Hz and ΔP = 0.1–0.7 MPa.

3.1.Simplified model of HSV based on mass flow characteristics correction

Due to the complexity of mechanism model, it is not suitable for the design of the controller.Many previous research used the orifice mass flow equation in ISO 635825as a simplified model for the HSV:

where Qmis the mass flow; A is the cross-sectional area of the valve orifice; Cqis the flow coefficient; γ is the specific heat capacity ratio; R is the gas constant; T is the temperature; P is the pressure; d is the duty cycle of PWM signal; Pcris the critical pressure ratio; Subscript ‘‘in”represents the HSV’s inlet; Subscript ‘‘out”represents the HSV’s outlet.

This equation describes the relationship between the mass flow and the pressure at HSV’s inlet and outlet.However,there are two problems: (A) It can only accurately reflect the static mass flow characteristics of the HSV, i.e., when the HSV is fully open26; (B) Dead zone and saturation zone of the HSV cannot be reflected with PWM signal input.In order to solve the above problems, based on the analysis of the correlation between volumetric flow characteristics and dynamic characteristics in Section 2, volume flow characteristics of the HSV is used to modify Eq.(1) and establish a high-precision model that accurately reflects its dynamic characteristics.Seven volume flow characteristic lines of the HSV in Fig.7 reflect the relationship between the duty cycle of PWM signal d and the volume flow Qvunder different differential pressure.It has been explained in Section 2.3 that the conversion between the volume flow Qvand the mass flow Qmis only related to air density, and thus the normalized volume flow Q～vcan be regarded as the normalized mass flow rate Q～m.We divide Q～min Fig.7(b) into five sections for polynomial fitting, which corresponds to the dead zone, nonlinear region 1, linear region, nonlinear region 2 and saturation zone respectively:

where d is the duty cycle of PWM signal; a0, a1, a2, a3, a4are polynomial coefficients in nonlinear region 1; b0, b1are the coefficients of the first-order equation in the linear region; c0, c1, c2, c3, c4are nonlinear region 2 polynomial coefficients.

Eq.(3) is used to fit seven normalized mass flows in Fig.7(b) to obtainmfdΔPi（）, i = 1, 2,..., 7, which corresponds to the normalized mass flow under ΔP = 0.1–0.7 MP a respectively.In addition, when ΔP = 0, it is obvious that the mass flow is also 0, i.e.,fdPi（）=0.During the working process of the HSV, assuming that the current differential pressure ΔP is between ΔPiand ΔPi+1and the duty cycle is d,the correction coefficient θ can be obtained using a fit and interpolation to the dimensionless mass flow:

Fig.8 Schematic diagram of pneumatic pressure control system.

The correction coefficient θ demonstrates the nonlinear change of mass flow caused by the HSV dynamic characteristics of HSV under different ΔP and d.Combining θ with the valve orifice mass flow Eq.(1), we can obtain the HSV model based on the correction of mass flow characteristics:

3.2.Model of chamber

Assuming that air is an ideal gas,the pressure and temperature in the chamber are evenly distributed, the kinetic energy and potential energy can be ignored, the gas flow is an isentropic adiabatic process, and then the law of energy conservation can be obtained:

where Qm,inand Qm,outare the air mass flow into and out of the chamber respectively;qinand qoutare the heat flowing into and out of the chamber respectively; Tinand Toutare the air temperature into and out of the chamber respectively; cvis the specific heat capacity at constant volume; γ is the specific heat capacity ratio; ˙W is the change rate of work; ˙U is the change rate of internal energy.

Since the charge and discharge process of the chamber with its volume being V is constant volume change, which means the change rate of chamber volume ˙V=0, we know that ˙W=P ˙V=0 and ˙U in Eq.(6) satisfy

According to Meyer formula, the constant volume specific heat capacity cvand the constant pressure specific heat capacity cpsatisfies

Since the gas flow is assumed to be an isentropic adiabatic process,qin–qout=0 and Tin=Tout=T.Then substituting Eq.(7)and Eq.(9)into Eq.(6),we can obtain the relationship between the air mass flow into and out of the chamber and its internal pressure:

3.3.System modeling and verification

For the charge valve HSV1, its inlet pressure is the stable air supply pressure Psupply, so the outlet pressure is the pressure in the chamber, and the gas mass flow through it is Qm1; for the discharge valves HSV2 and HSV3, the inlet pressure is the pressure in the chamber and the outlet pressure is the atmospheric pressure Patm.Therefore, the mass flow through them is Qm2and Qm3respectively.Combining Eq.(5) and Eq.(10), we can obtain the model of the asymmetric pneumatic pressure control system in this paper:

We build the simplified model of HSV based on mass flow characteristics correction and the pneumatic pressure control system model as shown in Fig.9 in Simulink 2021b.All parameters of the pneumatic pressure control system used in the simulation framework are listed in Table 2.

Fig.9 Simulation framework of pneumatic pressure control system.

Table 2 System parameters.

In order to verify the accuracy of the whole pneumatic pressure control system model, the whole process of charge and discharge is numerically simulated and experimentally tested under different air supply pressure, and the variation curves of pressure in the chamber are compared.During the charge process, HSV1 is fully open and other HSVs are closed; conversely, during the discharge process, HSV2 and HSV3 are fully open and HSV1 is closed.The comparison between the experiment and simulation of charge and discharge characteristics is shown in Fig.10.The simulation results under different air supply pressures are in good agreement with the experimental results, which verifies the effectiveness of the proposed pneumatic pressure control system model.

4.Controller design

The core idea of model predictive control is to solve a finite time horizon open-loop optimization problem online at each sampling time according to the current measurement information,and then apply the first control signal of the optimal control sequence on the controlled object.Different from the traditional control methods such as PID and SMC, the model predictive control can realize constraints management by one controller,so as to limit the duty cycle of HSV within the normal working range and avoid the influence of dead zone and saturation zone.In addition,the online construction and solution of the optimization problem ensure good control performance, and the feedback update of the measured value at each sampling time makes it have good robustness.Based on the system model in Section 3.3, this section constructs and solves the constrained nonlinear optimization problem to minimize the pneumatic pressure tracking error, and designs the nonlinear model predictive controller for the pneumatic pressure control system.To reduce the difficulty of solving the optimization problem online,a mapping scheme from the control signal to the duty cycle of each HSV and an optimization solving algorithm based on exterior penalty function are proposed.The schematic diagram of pneumatic pressure control based on the INMPC is shown in Fig.11.

Fig.10 Charge and discharge pressure curves of system.

4.1.Design of PWM duty cycle mapping scheme

If the PWM duty cycle of each HSV is directly used as the control signal to construct the multivariable constrained nonlinear optimization problem,it is necessary to calculate three optimal PWM duty cycles of HSVs in each control cycle, which will greatly increase the difficulty and amount of calculation, and even there will be no optimal solution.Therefore,the distribution scheme from the control signal to the actual PWM duty cycle of each HSV is designed to transform the multivariable constrained nonlinear optimization problem into a univariate constrained nonlinear optimization problem, which greatly reduces the computational complexity of solving.The specific mapping scheme is shown as follows:

where dminmeans the PWM signal duty cycle corresponding to the dead zone of the HSV; dmaxmeans the PWM signal duty cycle corresponding to the saturation zone of the HSV; u means control signal of HSVs and its value range is [-1,1].

Fig.12 Mapping scheme from control signal to PWM duty cycle of each HSV.

Fig.12 graphically shows the mapping scheme from control signal to the PWM duty cycle of 3 HSVs.When the control quantity is equal to 0, all HSVs are closed.When the control quantity is greater than 0, only HSV1 is open and the system is in the charge process.With the increase of control quantity,its duty cycle of HSV1 increases accordingly.When the control quantity is less than 0,HSV1 is closed,the discharge HSVs are opened,and the system is in the discharge process.As the control quantity decreases in the range of (-0.5,0), HSV2 gradually opens.When the control quantity is less than - 0.5,HSV2 maintains the maximum opening, while HSV3 starts to work, and its duty cycle increases with the further decrease of the control quantity.It is worth noting that the dead zone and saturation zone are excluded when mapping the control quantities to the actual PWM duty cycle of the HSVs.

4.2.Construction and solution of optimization problems

In order to facilitate the construction of the optimization problem and the derivation of the solution process, the prediction model in Eq.(11) is rewritten into the form of the following nonlinear differential equation and discretized by the forward Euler method:

Fig.11 Schematic diagram of pneumatic pressure control based on INMPC.

where TPWMmeans the PWM signal cycle of HSVs, which is the same as the calculation step of the model; k represents the current control cycle; u(?) is the control quantity; P(?) is the pressure in the chamber measured by the sensor.

Let the prediction horizon be Np,the control horizon be Nc(Nc≤Np) and the control input sequence in the prediction horizon be Uk= {u(k|k), u(k + 1|k),..., u(k + Np- 1|k)}.The pressure change of the chamber in the following Npcontrol cycles can be predicted according to Eq.(16):

where ^P（?）is the pressure in the chamber predicted by the prediction model of the pneumatic pressure control system.Since model predictive control requires solving the optimization problem in real time during each control cycle, the design of the objective function is critical.In this paper, to minimize the pressure tracking error, the objective function J is designed as follows:

where q1and q2are weighting factors, where q1represents the weight of the overall error in the whole prediction time horizon and q2represents the weight of the terminal time error;Pref(?)is the reference pressure; ||?||1stands for 1-norm.

Combined with the prediction model, objective function and constraints, the optimal control sequence U*kcan be obtained by solving the following univariate constrained nonlinear optimization problem, and the first optimal control quantity u*（k）is applied to the system as the real control quantity.

Although the mapping scheme from the control quantity to the duty cycle in Section 4.1 avoids the optimal solution of three duty cycles at the same time and reduces the computational complexity to a certain extent, the solution of the constrained nonlinear optimization problem is still much more complex than the unconstrained optimization problem.Besides, the optimization problem constructed in the whole working process of the pneumatic simulation system may be without a feasible solution, resulting in the failure of the system to work normally.For example, when the change rate of the reference pressure is very large, the solution of the optimization problem will be limited by the response speed of the HSV.Regardless of what the value of the control quantity within the constraint range is,it cannot make the iteration error meet the requirements, and then the optimization problem must be a nonlinear optimization problem with no feasible solutions with constraints.However, in this case, it is obvious that the control quantity should be set to a boundary value so that the maximum charge or discharge rate is guaranteed and the rate of pressure change is as large as possible.Based on the above discussion,the exterior penalty function based method27is used in this paper to solve the nonlinear optimization problem with constraints as in Eq.(19),allowing the variables to be solved to lie outside the feasible domain and making the optimal sequence of solutions approach the optimal from outside the feasible domain.

We define the new objective function with quadratic penalty function J～:

where σ is the penalty factor.The second term at the right end of the equation represents the penalty for violating the constraint,which means that when a constraint is violated,a positive penalty value will be attached to the original objective function J according to the degree of exceeding the constraint.Therefore, when all constraints are satisfied, the two objective functions are equivalent,while if the constraint is violated,the algorithm tends to find a solution so that the amplitude beyond the constraint limit is as small as possible to minimize the objective function.

So far, the univariate constrained nonlinear optimization problem defined by Eq.(19) can be transformed into the following single variable unconstrained nonlinear optimization problem:

For the unconstrained optimization problem, a simple and efficient Davidon-Fletcher-Powell (DFP) algorithm28is used to solve it.In the kth control cycle, the optimization process of the whole model predictive control can be summarized as Algorithm 1.

Algorithm 1.Solution of optimal control quantity in the kth control cycle Input: P(k):measured pressure in the chamber,Pref(?):reference pressure.Output: u*（k ）: optimal control quantity.1.Initialize the optimal control sequence U*kand penalty factor σ 2.Calculate the predicted pressure sequence ^P（?）using the prediction model Eq.(17)3.Calculate the objective function J～using Eq.(18)and Eq.(20)4.Construct the univariate unconstrained optimization problem:min J～5.Solve the optimal control sequence U*kusing DFP algorithm 6.u*（k ）=U*k{1}

5.Experimental results and analysis

In the semi-physical simulation experiment for aerospace, it is necessary to simulate complex pressure changes to provide real pneumatic excitation to the electronic controller, which puts forward high requirements for the steady and dynamic performance of the pneumatic pressure control system.In this section, the test equipment is first introduced, and then several groups of comparative experiments are carried out to verify the effectiveness of the designed pneumatic pressure control system and the INMPC algorithm.

5.1.Experimental setup

Fig.14 Experimental results from random step reference pressure.

Fig.13 Pneumatic pressure control system test platform.

The final pneumatic pressure simulation system test platform is shown in Fig.13.On the basis of the volume flow characteristics test platform in Fig.4, it adds the chamber, additional required HSVs and another pressure sensor, meanwhile removing the volumetric flow sensor.Similarly, the air compressor, compressed air tank, refrigerated air dryer and FRL are reserved to provide a stable high-pressure air supply.The high-performance embedded controller MangoTree Robust RIO, MT-E750 and MT-E710 card are used to receive sensor signals and drive HSVs.All programs are designed in NI Lab-VIEW professional 2016 and deployed in Real-Time (RT)Module and Field Programmable Gate Array(FPGA)Module in the MangoTree Robust RIO.PWM signal generation,logic control and data interaction with the board are all carried out on the FPGA platform.For RT Module, the main task is to run the nonlinear model predictive control algorithm and communicate with FPGA.Utilizing the FPGA Module which is extremely reliable in time sequence control guarantees the highly real-time response in the pneumatic pressure control system.

Table 3 Comparison of RMSE with random step reference pressure.

Fig.15 Experimental results from sinusoidal reference pressure.

Table 4 Comparison of RMSE with sinusoidal reference pressure.

5.2.Comparison with traditional methods

In this set of comparison experiments,random step signals and multiple sinusoidal signals with different amplitudes and frequencies are used as the reference pressure to evaluate the steady and dynamic tracking performance.The INMPC proposed is compared with PID and SMC in these experiments.According to the limitation of computing power of embedded controller and the pursuit of control performance, the control parameters of INMPC are selected as Nc=2,Np=2,q1=1,q2= 2, and σ = 100.However, since the parameters of PID and SMC need to be adjusted according to different reference pressures, their control parameters are also chosen by trials with different reference pressures.To quantitatively evaluate the control effect in the pressure tracking test, the Root Mean Square Error (RMSE) between the target pressure and the actual pressure is introduced as an evaluation metric:

Fig.16 Validation of asymmetric HSV groups and INMPC.

We set Patmas the initial pressure in the chamber and Prefas a random step signal in the range of 100–800 kPa.The pressure tracking results under the three controllers are shown in Fig.14 and Table 3.Compared with traditional PID and SMC, INMPC has the best performance in aspects of overshoot, steady-state error and regulation time and its RMSE improves by 39.89 % and 33.69 % respectively.Among them,the PID controller cannot avoid the steady-state error.Although increasing the integral coefficient is conducive to quickly eliminating the steady-state error, too large integral coefficient will lead to the instability of the system.The SMC can eliminate steady-state error but its performance in overshoot and regulation time is not satisfactory because it focuses more on stability than control performance.

Further, we set the sinusoidal reference pressure with the base pressure of 450 kPa, the frequency of 0.25–2 Hz and the pulsation amplitude of 100–350 kPa to test the dynamic pressure tracking performance of the system, as shown in

where Arefis the pulsation amplitude of the sinusoidal reference pressure; frefis the frequency of the sinusoidal reference pressure.

Fig.15 shows the pressure tracking results of six sets of sinusoidal reference pressure with different pulsation frequencies frefand amplitudes Aref.The effectiveness of the controllers becomes progressively worse with increasing pulsation frequency and pulsation amplitude,and the tracking of high pulsation frequency pressure is more difficult than the tracking of high pulsation amplitude pressure.

Fig.17 Comparison of optimization problem solving time.

Fig.18 Compressor outlet pressure in surge test of turboshaft engine.

For the PID controller, when the pulsation amplitude Arefis small and the pulsation frequency fref≤1 Hz, the pressure tracking performance is good and the relative error is within 10%.However, when the pulsation frequency fref≥1 Hz,the pressure waveform gradually distorts and the relative error gradually increases,while the pulsation frequency fref=2 Hz,or even the pulsation amplitude Aref= 100 kPa, is difficult to track,and the relative error reaches 20%.For a reference pressure with fref=1 Hz and Aref=350 kPa,the relative error in the discharge process is up to 50%.It is worth noting that the tracking error of the PID controller in the discharge process is significantly larger than that in the charge process because the control quantity of the PID control output is basically the same as the tracking error without considering the working state of HSV.Even if the same error leads to the same control quantity, the charge and discharge speed of HSV are not only affected by the current PWM duty cycle but also related to the differential pressure between the inlet and outlet of HSV.Therefore, PID is difficult to achieve good control performance.

The control effect of SMC has been improved compared with the PID controller.When the pulsation amplitude is 200 kPa, the relative error is within 20% even if the pulsation frequency is 2 Hz.For the reference pressure with highfrequency large-amplitude pulsation of Aref= 350 kPa and fref= 1 Hz, the relative error reaches 25%, but the generated pressure signal waveform is basically consistent with the sinusoidal waveform, and the distortion is not obvious.There is a significant waveform hysteresis in the sliding mode controller,and the higher the frequency is, the more significant the response hysteresis is when tracking sinusoidal pressure.This is due to the fact that Ref.18used sat(s) function rather than sgn(s) function to weaken the system jitter and improve the system stability when designing the sliding mode variable structure controller, but the boundary layer in sat(s) causes the system hysteresis.When the pulsation frequency is higher,a thicker boundary layer is needed to ensure the stability of the system,which leads to a more serious system lag phenomenon.

Fig.19 Pressure tracking results during the first surge.

Fig.20 Pressure tracking results during the second surge.

The INMPC proposed in this paper introduces the prediction model including dead zone,saturation zone and nonlinear regions of the HSV, and calculates the optimal control quantity of each control cycle online by solving the optimization problem.Therefore, it has achieved excellent tracking performance with different reference pressures.For the case of low frequency (fref≤ 1 Hz) and low pulsation amplitude(Aref≤200 kPa), the relative error is less than 5%.For high-frequency pulsation pressure of Aref= 200 kPa and fref= 2 Hz, the relative error is also within 8%.Even for the extreme reference pressure with Aref=350 kPa and fref=1 Hz, the relative error of the whole inflation stage is within 10%, and the relative error is large only when the command pressure is less than 200 kPa during deflation.We can observe that the control quantity at this moment is always - 1, which indicates that all discharge HSVs have been fully opened to track the speed of reference pressure decline.

The RMSE of the above six groups of experimental results are calculated as shown in Table 4.When Arefis 200 kPa and frefis 0.25,0.5,1 and 2 Hz,the RSME of the INMPC improves by 75.13%, 69.02%, 72.74% and 81.37% respectively, compared with the PID controller; compared with SMC, it improves by 72.61%, 66.22%, 70.77% and 69.29% respectively.When Arefis 100 kPa and frefis 2 Hz, the RSME of the INMPC improves by 58.91% and 59.12% respectively compared with the PID and SMC.When Arefis 350 kPa and frefis 1 Hz, the RSME of the INMPC improves by 69.42%and 63.41% respectively, compared with the PID and SMC.In summary,the INMPC proposed in this paper is much better than the PID and the SMC for a variety of reference pressures with pulsation frequency and pulsation amplitude.

5.3.Effectiveness verification of asymmetric valve groups and INMPC

The above experimental results show that the INMPC has a great improvement over the PID and SMC control algorithm.To further illustrate the ability of the asymmetric HSV groups proposed in this paper to compensate for the asymmetry of the charge and discharge speed and the effectiveness of the INMPC algorithm in improving the real-time performance of the solution, three sets of comparison experiments were conducted with a sinusoidal reference pressure of Aref=0.2 MPa and fref= 2 Hz:

(1) Symmetry-INMPC: Use the symmetric HSV groups with a single charge HSV and a single discharge HSV and the INMPC control algorithm in this paper.

(2) Asymmetry-INMPC: Use the asymmetric HSV groups with single charge HSV and double discharge HSV and the INMPC control algorithm in this paper.

(3) Asymmetry-standard NMPC (SNMPC): Use the asymmetric HSV groups with single charge HSV and double discharge HSV and the SNMPC control algorithm without the mapping scheme from control signal to PWM duty cycle and exterior penalty function in the objective function.

In the experiments, besides the actual pressure in the gas chamber, the time cost in solving the optimization problem in each control cycle is recorded.

The pressure tracking results of the three sets of experiments are shown in Fig.16.Comparing Symmetry-INMPC and Asymmetry-INMPC, we can see that when the reference pressure drops by 250 kPa, the relative error of the symmetrical HSV groups is greater,because the discharge speed of a single HSV is not enough to reduce the system pressure rapidly.The asymmetric HSV groups with two discharge HSVs increase the discharge speed and can well track the rapidly decreasing reference pressure.Comparing Asymmetry-SNMPC and Asymmetry-INMPC, we can see that since the asymmetric HSV groups compensate for the asymmetry in the charge and discharge process,there will be no large relative error when the reference pressure drops rapidly.However, the control performance of the SNMPC algorithm is poor around 0.4 s and 0.9 s, where the control signal remains constant for several control cycles.Fig.17 shows the time consumed for solving the optimization problem.The solution time of the SNMPC algorithm has been 25 ms around 0.4 s and 0.9 s,which means that the solution fails and therefore the control signal remains the same as that of the previous cycle.In contrast,the solution time of the INMPC is around 5 ms on average, and the maximum does not exceed 10 ms, which significantly reduces the time cost of solving the optimization problem and proves the effectiveness of the INMPC proposed in this paper.

5.4.Simulation results of actual aeroengine surge pressure

Fig.18 shows the dynamic change of the pressure at the compressor outlet of a turboshaft engine in the surge test with two surges at 152 s and 394 s respectively.As the engine enters the surge from a stable working state, the pressure decreases dramatically instantaneously,followed by a high-frequency pulsation.Once the engine enters into the surge, the fuel supply is reduced in time so that the turboshaft engine exits the surge to avoid the damage.Therefore, after a short period of highfrequency pulsation, the pressure is gradually reduced to another steady state.It is clear that the surge pressure signal includes a step-like and sinusoidal-like component with large-amplitude and high-frequency variations.

Taking the compressor outlet pressure signal of the two surge processes in the surge test as the reference pressure, the experimental results are shown in Fig.19 and Fig.20.The results show that the relative error of tracking the first surge process is about 5%.The pressure during the second surge is more complex and there are many drops of the pressure, so the relative error also increases to about 8%.In general, the asymmetric HSV groups and the INMPC algorithm proposed in this paper can accurately track the complex pressure signals with substantially rapid drops and high-frequency pulsations,which verifies the effectiveness of the designed pneumatic pressure control system.

6.Conclusions

At present, the pneumatic pressure servo control technology for the aerospace semi-physical simulation cannot satisfy the demand for accurate simulation of large-value highfrequency pulsating air pressure during the aeroengine surge.Therefore, this paper builds a pneumatic pressure simulation system with asymmetric HSV groups and proposes a pressure control method based on INMPC to achieve high-accuracy simulation of the wide range of steady or dynamic pressures.The main conclusions are drawn below.

(1) The relationship between volume flow characteristics and dynamic characteristics of the HSV is tested and analyzed,and the widely used HSV simplified model is modified according to this so that it has both high precision and low computational complexity.On this basis, the prediction model of the pneumatic pressure control system is further established.

(2) The asymmetrical HSV groups of single charge valve and double discharge valve can improve the discharge speed of the pneumatic pressure control system to compensate for the asymmetry of the charge and discharge speed.

(3) We design the mapping scheme from the control signal to each HSV’s PWM duty cycle and the solution method of the optimization problem based on the exterior penalty function,thus transforming the multivariable constrained nonlinear optimization problem into a univariate unconstrained optimization problem.The results show that the average solution time of the INMPC is only 5 ms, while that of the SNMPC is 20 ms and often times out.

(4) The pneumatic pressure control system proposed can achieve high-accuracy tracking of steady and high-frequency dynamic pressure in the range of 100–800 kPa.The experimental results show that the relative error and RSME are greatly improved regardless of whether the reference pressure is a random step, sinusoidal signal, or real surge test data.

(5) This pneumatic pressure control system can be further used in the semi-physical simulation of aero-engine active stability control to provide realistic pneumatic pressure excitation for sensors and electronic controllers at very low cost,so as to improve the verification confidence of the surge detection algorithm.

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 Natural Science Foundation of China(No.51976089)and the Natural Science Foundation of Fujian Province of China (No.2021J05113).