SHAO Ming-yu，WANG Zhi-gang

(School of Astronautics,Northwestern Polytechnical University,Xi’an　710072,China)

?

Study on control method of compoundadjustable ducted rocket

SHAO Ming-yu，WANG Zhi-gang

(School of Astronautics,Northwestern Polytechnical University,Xi’an710072,China)

The responses of compound-adjustable ducted-rocket to inlet/gas generator/nozzle adjustments were studied with nonlinear dynamic model.On this basis,the control strategy was proposed to improve critical total-pressure recovery by inlet adjustment and controlling thrust/afterburner pressure by gas-generator/nozzle adjustments. In design of control system,the coupling characteristic between thrust and afterburner pressure loops was weakened by INA method,while feedback variables without non-minimum phase characteristic were constructed by information fusion in frequency domain,then PI controllers were designed.The nonlinear simulation results of control system show that the controllers are effective in compound-adjustable control.

ducted rocket；compound adjustment；control method

0　Introduction

Ducted rocket is considered to be the ideal power plant for medium supersonic,medium range tactical air-to-air missile[1].The existing ducted rocket uses fixed-geometry structure generally,for taking over normally,the flow channel of inlet and nozzle were designed according to low speed requirements[1-3].When the ducted rocket works at high speed,the inflow is not compressed sufficiently by inlet;the flow capacity of nozzle is too large,while the expanding capacity is insufficient,leading to an imperfect thermal-cycle,which departs from its original purpose for being a high speed power plant[1-3].

The inlet/gas-generator/nozzle are supposed to adjust continuously to meet a wide Mach range,so that the concept of compound adjustable ducted rocket is put forward by scholars[1-3]:Improving the critical total-pressure recovery in full-design Mach range by inlet adjustment;providing the demand thrust by gas generator adjustment;regulating the flow capacity of nozzle and the afterburner operating pressure by nozzle adjustment,so as to give full play to optimum performance of inlet,and improving the expanding capacity of nozzle.

The various parts of compound adjustable ducted rocket are adjusted according to flight conditions and thrust commands.In order to satisfy thrust command with optimal performance,and to keep ducted rocket operating safely and stably,the adjustments of inlet/ gas-generator/nozzle must be controlled effectively.The primary purpose of the controller is to regulate the gas flow,so the desired thrust is obtained over full flight envelope;secondly,to regulate the afterburner operating pressure to improve the total pressure recovery of inlet,so the specific impulse of ducted rocket can be enhanced;meanwhile,the controller must ensure that no operating constraints are violated,including inlet unstart constraint,flameout constraint,rich/poor gas-air ratio constraint and afterburner temperature/structure constraint[4-5]etc.

The compound adjustable ducted rocket is a MIMO nonlinear system,which causes great troubles in dynamic modeling and controller design.Lots of studies were carried out on dynamic modeling and controller design of ramjet.Niu Wenyu[5-8]developed a linear dynamic model of controllable flow solid ducted rocket,and constructed a new feedback variable without non-minimum characteristic;then a PI controller was applied to the ducted rocket system;lastly,the switching strategy between thrust regulation and inlet buzz protection was studied.Gupta N K[4，9-10]established a nonlinear dynamic model of ramjet with regulatable fuel flow and adjustable nozzle,then derived the linear model by system identification,finally a thrust controller was designed by allocation in frequency domain.Researchers in Bayern Chemie designed a robust thrust controller of throtteable ducted rocket,and the controller was validated by Connected Pipe test and Quasi-Free Jet test[11].

The studies on multivariable control system mostly concentrate on aeroengines.Sun Jianguo[12]studied the advanced multivariable control system of aeroengines in detail.Li Huacong[3]proposed a design method for pre-compensator in frequency domain of aeroengine multivariable control system based on diagonal dominance and the method of linear matrix inequalities,effectively reducing the interaction in aeroengine multivariable system.

Above all,though there are lots of methodologies applied in control of ducted rocket and multivariable aeroengines,the study on control system design methodology of multivariable ducted rocket is rarely involved.This paper focuses on the multivariable control methodology of compound adjustable ducted rocket.Firstly,the nonlinear dynamic model of compound adjustable ducted rocket was established,and the response to inlet/gas generator/ nozzle step inputs was studied.On this basis,this paper further established the control strategy of compound adjustments.Then the coupling characteristic between thrust and afterburner operating pressure loops and the non-minimum phase characteristic of ducted rocket were analyzed.The INA technique and information fusion were employed to solve these problems,and PI controllers were designed to control the thrust and the afterburner operating pressure.At last,the controllers are validated by nonlinear simulations.

1　Dynamic modeling and simulation

1.1Control variables

The primary purpose of the controller is to satisfy the thrust command,then to regulate the terminal shock close to the inlet throat,so that high specific impulse can be obtained.The thrust can be difficult to be measured directly,but is estimated by observing the related parameters.The shock location can be difficult to quantify from direct measurement,but can be effectively regulated by controlling the inlet backpressure(afterburner operating pressure).

The extreme locations of the shock appear when it lies on the inlet throat and exit;the corresponding afterburner operating pressures are denoted bypmaxandpmin.The pressure margin is defined as[4]:

(1)

pmtherefore directly correlates with the shock location in the duct[4].According to the requirement ofpmat different flight stages and inlet throat parameters,the control command of afterburner operating pressure can be obtained.

So the thrust and afterburner operating pressure are chosen as control variables.

1.2Dynamic modeling

The sketch of typical ducted-rocket is shown in Fig.1.A typical ducted rocket consists mainly of four parts:inlet,gas generator,afterburner and nozzle.The dynamic model of compound-adjustable ducted-rocket can be obtained by combining the dynamic model of all parts and the model of disturbance propagation.

Fig.1　Sketch of ducted rocket with station number

(1)Inlet:The critical performance of the inlet depends on the parameter values of inlet throat.It can be obtained from interpolation of CFD results or theoretical calculation.The subsonic diffusion is separated by the terminal shock,the upstream flow is adiabatic and isentropic;while the downstream flow is adiabatic,and the factorσsubis employed to reflect the total pressure loss in subsonic diffusion.The velocity of terminal shock moving in the duct is expressed as:

(2)

The downstream parameters of the shock can be obtained from the mass conservation law and the energy conservation law,and the parameters of inlet exit can be obtained according to one-dimensional duct flow relations.

(2)Gas-Generator:According to the mass conservation law,the change in operating pressure of gas generator can be expressed as[14]:

(3)

The gas flow through gas generator nozzle throat can be expressed as:

(4)

(3)Afterburner:The temperature dynamic model and pressure dynamic model of afterburner represent the energy storage effect and the mass storage effect respectively[15].

The temperature dynamics is caused by the mixing and combustion of rich-fuel gas and air.The temperature change caused by the chemical reaction is very rapid,and volume of afterburner is pretty small,so the dynamic process of temperature change can be ignored.0-dimensiodal model is applied to calculate the parameter values of secondary gas.The ideal temperature rise of the combustion is the function of fuel gas-air ratio,but the actual temperature rise depends on the combustion efficiency.

(5)

The steady mass flow value of secondary flow can be obtained according to the mass conservation law,and the total pressure recovery of afterburner,σab,is estimated from the engineering experience.

The zone between the terminal shock and the nozzle throat is chosen as control volume,and divided into two zones:cold zone and hot zone,by inlet exit section.According to mass conservation law,and ignoring the volume change of hot zone due to the gas generator/nozzle adjustment,the change in afterburner operating pressure can be computed by:

(6)

where,

A(x)andT(x)are the section area and the temperature at positionxrespectively.

(4)Nozzle:The nozzle of ducted rocket consists of a contraction section and a divergent section.The flow in contraction section is adiabatic,while the flow in divergent is adiabatic and isentropic.The nozzle efficiency,nN,is employed to reflect the total pressure loss in the nozzle[16].The flow capacity of nozzle can be expressed as:

(7)

(5)Disturbances propagation:The disturbances propagate upstream and downstream in the form of acoustic wave and entropy wave[17].The variables are classified into two kinds,nonacoustic variables and acoustic variables,for simplification.Time constants for nonacoustic variable traveling downstream with mean flow velocity are computed as follows[4，9-10]:

(8)

Time constants for the propagation of acoustic variables are obtained by considering acoustic waves traveling against/with the main flow as follows[4，9-10]:

(9)

The delay in updating the system properties at various stations due to disturbances propagation,upstream and downstream,is modeled by a first order dynamical system as follows[4，9-10]:

(10)

whereyis the physical parameter,yssis its steady-state value.

(6)Thrust:The nominal thrust is introduced to study the influence of operating condition of ducted rocket on the thrust.It is defined as the difference of impulse between the entrance and the exit of ducted rocket[14]:

(11)

The nonlinear dynamic model of the compound adjustable ducted rocket can be obtained by combining the dynamic models of all parts according to their positions.The structures of the model and the parameters transmission between different modules are shown in Fig.2.

Fig.2　Sketch of compound adjustable ducted rocket dynamic model

1.3Open-loop simulation

Simulations are carried out to study the responses to inlet/gas-generator/nozzle step inputs with cruising condition chosen as the typical operating condition.The trim values of inputs and other important system variables are given in Table 1,and the parameters of step inputs are given in Table 2.

Table 1　Trim values at cruising condition

Table 2　Parameters of step input

The results of open loop simulations are shown in Fig.3.The steady-state values of the thrust and the afterburner operating pressure don’t change due to the inlet adjustment,and the dynamic process is very small,so that it can be ignored.But the steady-state value ofpmchanges due to the change of the critical performance of the inlet.Both the nozzle adjustment and the gas generator adjustment have great influences on the thrust and the afterburner operating pressure (pressure margin).The anti-regulation characteristic exists in the thrust and the afterburner operating pressure (pressure margin)due to the gas generator adjustment,but only exists in the thrust due to the nozzle adjustment.The gas generator adjustment has a larger influence on the thrust but a less influence on the afterburner operating pressure (pressure margin)than the nozzle adjustment.Besides,the ducted rocket responds to the nozzle adjustment more rapidly than that to the gas-generator adjustment.

Fig.3　Responses to step inputs

2　Control system design

The results of the simulations show that,the inlet adjustment only changes its critical performance but not the downstream system properties when the capture flow of the inlet doesn’t change due to the adjustment;and the gas generator/nozzle adjustment has no influence on the upstream system properties because the existence of the terminal shock.In another word,the responses of the ducted rocket due to the inlet adjustment and the gas-generator /nozzle adjustment are decoupled,so the controllers can be designed separately.

2.1Control of the inlet adjustment

The purpose of the inlet adjustment is to improve its critical total pressure recovery without causing any instability.The critical performance of inlet depends on the compression angle.The responses of the thrust and the afterburner operating pressure due to the inlet adjustment can be ignored,and the dynamic process of oblique shock system is very rapid,so the control on the inlet adjustment can be simplified to the control on the compression angle.Considering the margin of incidence angle,the optimal compression angle of the inlet can be computed by Oswatitsch’s Theory.

The actuator of inlet adjustment typically has a high bandwidth and its design is beyond the scope of current study,so the study on the control system of inlet adjustment won't be carried out in detail.

2.2Control of gas generator/nozzle adjustment

2.2.1Controller design procedure

The purpose of gas generator/nozzle adjustment is to control the thrust and the afterburner operating pressure,so that the ducted rocket can meet the thrust command with optimal performance.According to the simulation results,the thrust responds mainly to the gas generator adjustment,while the afterburner operating pressure responds mainly to the nozzle adjustment;and the ducted rocket responds more rapidly to the nozzle adjustment than to the gas generator adjustment.These allow us to control the thrust by the gas-generator adjustment,and to control the afterburner pressure by the nozzle adjustment.

Fig.4　Sketch of typical multivariable system design

For the coupling characteristic between the thrust loop and the afterburner pressure loop,the decoupling control is the main method for the multivariable control applied to process control in the engineering practice.Among them,the INA (Inverse Nyquist Array)method,which has a simple structure,is easy to realize,as shown in Fig.4[18].WhereKc(s)is the pre compensation matrix,which allowsG(s)Kc(s) to have a diagonal dominance.By weakening the coupling between different loops,controllers of each loop can be designed as single variable system[18].

The mathematical model of ducted rockets is a non-minimum phase system[7].A contraction between the rapid response and stability induced by the anti-regulation characteristic renders the control system of ducted rockets difficult to design[5，7].The information fusion technique in frequency domain[5，7]is introduced to construct new feedback variables without non-minimum phase characteristics to make the design of the controller more feasible,and then the PI controller can be designed for each loop.

The INA method,the information fusion and the PI controller are carried out for a linear,constant parameter system.However the compound adjustable ducted rocket is a nonlinear system with parameters varying during flight.So we can choose a set of representative operating points over the flight envelop to develop linearized models and to design controllers at each operating point,and then schedule the gains as a function of parameters such as dynamic pressure so that appropriate values of gains are obtained at all points of the flight envelop by interpolation between the representative operating points.But these beyond the scope of present work which focus on the control methodology of the compound adjustment,so cruising condition is chosen to describe the design procedure of the control system as follows:

(1)Linearize the nonlinear model to obtain the Linearized model;

(2)Diagonal dominance analysis,design the pre compensation matrix by the INA technique;

(3)Construct new feedback variables without non-minimum phase characteristics;

(4)Design the PI controllers for the thrust loop and the afterburner operating pressure loop.

2.2.2Controller design

(1)Model linearization

Linearize the nonlinear model and reduce the order of results[5，7],the linearized model of the ducted rocket with the throtteable gas flow and adjustable nozzle can be expressed as:

(12)

The gains and time constants at cruising condition are given in Table 3.

Table 3　Gains and time constants at cruising condition

(2)Pre compensation

The INA plot with the Gershgorin zone of the ducted rocket system is shown is Fig.5(a).The plot shows that the original system has a diagonal dominance,but the Gershgorin zone is wide in low frequency,so the controller can’t be designed directly.The pre compensation matrixKcis obtained by pseudo dragonizing design.The INA plot with the Gershgorin zone of compensated system is shown in Fig.5(b).With pre compensation,system has a lager diagonal dominance and stability in low frequency,so the controller can be designed as single variable system.

(a)Original system　　　(a)Compensated system

(3)Feedback variables construction

Both the thrust control and the afterburner operating pressure control cause the response of the gas generator operating pressure,in which no anti regulation characteristic exists.Therefore,though the design of complementary filter in frequency domain,the steady state information of the thrust and the afterburner operating pressure is obtained by low-pass filters,while the dynamic information of the gas generator operating pressure is obtained by high-pass filters,and fused into new feedback variables without non-minimum phase characteristics[5,7].

The new feedback variables constructed by the information fusion can be expressed as follows:

(13)

(14)

The values ofT，T1，T2should make the new system meet these requirements:

① No anti-regulation characteristic exits in the thrust loop and the afterburner loop;

② The amplitude of high frequency oscillation should be small;

③ The settling time is closed to the original system

(4)PI controllers design

The control on the thrust changes the velocity of missiles,which belongs to long period motion.While the operating pressure of the afterburner influences the operating status directly,so a rapid response and small overshoot are required.Above all,the performance specifications are summarized as follows:

① The steady state error of the thrust and the afterburner operating pressure during step response is zero;

② The rise times of the thrust and the afterburner operating pressure during step responsetrare smaller than 0.5 s and 0.1 s respectively;

③ The overshoots of the thrust and the afterburner operating pressure during step response are smaller than 5% and 2% respectively;

④ Phase marginγof both loops is larger than 50°.

PI controllers are applied to both the thrust loop and the afterburner operating pressure loop.However the PI controller design is not the focus of this paper,the parameters of the controllers and the performance of the close-loop system are directly given in Table 4.The results show that the PI controllers satisfy the performance requirements.

Table 4　Parameters and performance of PI controller

3　Nonlinear simulation validation

The pre compensation matrix,the information fusion feedback variable,and the PI controllers are all designed based on the linearized model derived from the nonlinear model.So the effectiveness of the controllers must be validated by nonlinear simulation.

3.1Thrust loop

The step response of the close loop system to the thrust command is shown in Fig.6.The appropriate gas flow is regulated by the controller,so the thrust can meet the control command.At the same time,the controller regulates the nozzle to keep the steady state value of the afterburner operating pressure unchanged.Due to the rapid response to the nozzle adjustment,the thrust surges up first,and then shows the anti regulation characteristic caused by the gas generator.The rise time of the thrust is 0.31s,and the overshoot is 4.8%,so the controller is effective.

Fig.6　Step response of thrust loop

3.2Afterburner pressure loop

The step response of the close loop system to the afterburner operating pressure margin command is shown in Fig.7.The controller regulates the pressure by nozzle adjustment;moreover the steady state value of the thrust is kept constant by the gas-generator adjustment.The response of the thrust in the afterburner operating pressure loop shows the same characteristic as that in the thrust loop.The rise time of the pressure margin is 0.061s,and the overshoot is 0.3%,which means the controller satisfies the performance requirements.

Fig.7　Step response of afterburner pressure loop

3.3Inlet adjustment

The response of the close loop system to the inlet step input is shown in Fig.8.The critical total-pressure recovery changes because of the inlet adjustment.Thenpmchanges when the downstream parameters keep constants.The controller regulates the pressure to keep the steady state value ofpmunchanged.

Fig.8　Response of close-loop to inlet adjustment

The results of nonlinear simulations show that,the controllers designed in this paper can weaken the coupling between the thrust and the afterburner pressure operating loop,while the rapid response and stability are well taken care of.The controllers are effective in the thrust and the afterburner operating pressure (pressure margin)control.

4　Conclusion

Based on the nonlinear model of the compound adjustment ducted rocket,the responses to the inlet/gas generator/nozzle adjustments were studied in this paper.On this basis,the control strategy of inlet/gas generator/nozzle adjustments was established.For the coupling and non-minimum phase characteristics,the INA technique and information fusion in frequency domain were employed in PI controller design.Finally,the controllers designed are validated by nonlinear simulation.Some conclusions derived from the studies are shown as below:

(1)The critical total-pressure recovery can be improved by the inlet adjustment,while the thrust/afterburner operating pressure can be regulated by the gas-generator/nozzle adjustment.The controllers of the inlet adjustment and the gas-generator/nozzle adjustments can be designed separately.

(2)The coupling characteristic between the thrust loop and the afterburner operating pressure loop was weakened by the INA method,so the controller of each loop can be designed as single variable system.With new feedback variables without non-minimum phase characteristic constructed by the information fusion of the gas-generator operating pressure,the afterburner operating pressure and the thrust in frequency domain,the rapid response and stability in controllers design can be well taken care of.

(3)The results of nonlinear simulations show that,the controllers designed in this paper are effective in the thrust and the afterburner operating pressure (pressure margin)control,and can keep the steady state values of the thrust and the afterburner pressure margin unchanged during the inlet adjustment.

(4)The compound adjustment strategy and the control method proposed in this paper emphasize particularly on theoretical study,and may cause a complex system.Further research should be taken out to study its feasibility in engineering application.

[1]XU Dong-lai,CHEN Feng-ming,CAI Fei-chao,et al.Assessment of design techniques of ducted rockets[J].Journal of Solid Rocket Technology,2010,33(2):142-147.

[2]CAI Fei-chao,CHEN Feng-ming,XU Dong-lai,et al.Study on fixed-geometry supersonic inlet design for wide Mach number range application[J].Journal of Solid Rocket Technology,2010,33(2):163-166.

[3]XU Dong-lai,Chen Feng-ming,Cai Fei-chao,et al.Study on fixed-geometry supersonic inlet design for wide Mach number range application[J].Journal of Solid Rocket Technology,2010,33(1):45-48.

[4]Kumar P,Bharani Chandra,Nitin K,et al.Modeling,simulation and controller design for an air-breathing combustion system[J].Journal of Propulsion and Power,2010,26(3):562-574.

[5]NIU Wen-yu.Study on control methods for variable flow ducted rockets[D].Dissertation for the Doctoral Degree in Engineering,Harbin Institute of Technology,2009.

[6]Niu Wen-yu,BAO wen,CUI Tao,et al.Dynamic modeling and model reduction order of controllable flow solid ducted rockets[J].Journal of Solid Rocket Technology,2008,31(4):325-330.

[7]Chang Jun-tao,LI Bin,Bao Wen,et al.Thrust control system design of ducted rockets[J].Acta Astronautica 2011,69:86-95.

[8]Bao Wen,Bao Lin,Chang Jun-tao,et al.Switching control of thrust regulation and inlet buzz protection for ducted rocket[J].Acta Astronautica,2010,67:764-773.

[9]Nitin K Gupta,Basant K Gupta,Ananthkrishnan N,et al.Integrated modeling and simulation of an air-breathing combustion system dynamics[C]//AIAA Modeling and Simulation Technologies Conference.August 2007,Hilton Head,South Carolina.

[10]Park Ik-soo,Kim Sun-kyoung,Yeom Hyo-won,et al.Control-oriented model for intake shock position dynamics in ramjet engine[J].Journal of Propulsion and Power,2011,27(2):499-503.

[11]Pedro C Pinto,Guido Kurth.Robust propulsion control in all flight stages of a throtteable ducted rocket[C]//47th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit 31 July -03 August 2011,San Diego,California.

[12]Sun Jian-guo.Advanced multivariable control systems of aeroengines[M].Beijing:Beihang University Press,2005.

[13]Li Hua-cong,Wu Zhi-kun,HAN Xiao-bao.A multivariable control based on diagonal dominance for aeroengine[J].Journal of Propulsion Technology,2007,28(6):665-668.[14]Bao Fu-ting,Huang Xi-jun,Zhang Zhen-peng.Integral solid propellant ramjet rocket motor[M].Beijing:China Astronautic Publishing House,2006.

[15]Link C Jaw,Jack D Mattingly.Aircraft engine controls design,system analysis,and monitoring[M].AIAA Education Series,2012.

[16]John J Mahoney.Inlets for supersonic missile[M].AIAA Education Series,2004.

[17]Douglas G MacMartin.Dynamics and control of shock motion in a near-isentropic inlet[J].Journal of Aircraft,2004,41(4):846-853.

[18]Xue Ding-yu.Computer aided control systems design using MATLAB language[M].Beijing:Tsinghua University Press,2006.

(編輯：薛永利)

復(fù)合調(diào)節(jié)固沖發(fā)動(dòng)機(jī)控制方法研究

邵明玉，王志剛

(西北工業(yè)大學(xué) 航天學(xué)院，西安710072)

基于復(fù)合調(diào)節(jié)固沖發(fā)動(dòng)機(jī)非線性動(dòng)力學(xué)模型，對(duì)發(fā)動(dòng)機(jī)在氣道/燃?xì)獍l(fā)生器/噴管調(diào)節(jié)下的響應(yīng)進(jìn)行了分析。在此基礎(chǔ)上，提出采用進(jìn)氣道調(diào)節(jié)提高其所保有的最佳性能，利用燃?xì)獍l(fā)生器/噴管調(diào)節(jié)控制發(fā)動(dòng)機(jī)推力和補(bǔ)燃室工作壓力?？刂葡到y(tǒng)設(shè)計(jì)中，采用INA方法削弱推力回路和補(bǔ)燃室工作壓力回路間的耦合，采用頻域信息融合方法構(gòu)造沒有非最小相位特性的反饋?zhàn)兞浚⒃O(shè)計(jì)PI控制器。非線性仿真結(jié)果表明，文中所設(shè)計(jì)的控制器能對(duì)復(fù)合調(diào)節(jié)固沖發(fā)動(dòng)機(jī)實(shí)現(xiàn)有效控制。

固沖發(fā)動(dòng)機(jī)；復(fù)合調(diào)節(jié)；控制方法

date：2015-08-05；Revised date：2015-11-06。

National Natural Science Foundation of China(61104195)。

Biography：SHAO Ming-yu(1987—)，male，doctor，speciality：design and research of ramjet.E-mail：mingyupiaoxue@126.com

V438Document code：AArticle ID：1006-2793(2016)01-0001-08

10.7673/j.issn.1006-2793.2016.01.001