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Acta Astronautica

Acta Astronautica 67 (2010) 764–773

0094-57

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/actaastro

Switching control of thrust regulation and inlet buzz protection forducted rocket

Wen Bao, Bin Li, Juntao Chang, Wenyu Niu, Daren Yu �

Harbin Institute of Technology, 150001 Heilongjiang, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 2 February 2010

Received in revised form

10 April 2010

Accepted 28 April 2010Available online 8 June 2010

Keywords:

Ducted rocket

Inlet buzz

Thrust control

Switching control

Integral limitation

65/$ - see front matter Crown Copyright & 2

016/j.actaastro.2010.04.022

responding author.

ail address: yudaren@hit.edu.cn (Daren Yu).

a b s t r a c t

The renewed interest on ducted rockets impulses their investigation. In this article,

switching control in the working process of ducted rockets is focused on, in order to

obtain optimal thrust control while avoiding phenomena like inlet buzz. Firstly multi-

objective control problems of ducted rockets during its working process are discussed.

Then the dynamic mathematical models of gas flow regulating system, thrust regulation

control loop and inlet buzz protection loop are established and analyzed. Lastly, the

switching strategy and PID controller are applied to the ducted rocket system, and the

influence of integral limitation of controllers is analyzed. In conclusion, it is useful to

introduce the multi-objective switching control to ducted rockets, and simulation

results show its validity.

Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

The solid fuel ducted rocket, also known as ramrocketor integral rocket ramjet, is a type of ramjet on whichthere has been much renewed interest lately for tacticalmissile propulsion [1,2]. It has several advantages oversolid propellant rocket motors, including increased rangeand higher speed, with only a limited increase incomplexity [3]. Particularly, its specific impulse could be4–6 times higher than that of an ordinary solid rocket.Ducted rocket was firstly applied in SAM-6 missilesdeveloped by the former Soviet Union before 1960. Thegas flow needs to be controlled over wider ranges of Machnumber (2–4) and altitude (5–25 km) to guarantee thatthe ducted rocket works with both safety and highperformance.

Control technology of aero engines has been greatlyenhanced these years because of higher demands onnewly designed aircrafts or missiles. Aero-engines, espe-cially ducted rockets, are a complex nonlinear system thatoperates in an uncertain environment of limitations:

010 Published by Elsevier

limitations in temperature, air pressure, physical accel-eration, etc. The traditional controller of an aero engine islinear-based and derived for a linearized system atmultiple trimmed flight conditions throughout the flightenvelope [4–7]. However, this method cannot performwell over the entire range of highly nonlinear aero-enginesystems. Nonlinear control methods then emerge toovercome the defects of linear design approaches, devel-oped for almost two decades [8–11]. Also, intelligentcontrol has been an important part of aero engine control,such as life extending control, adaptive control, multi-variable control, performance seeking control, modelpredictive control, etc. [12–16].

Whether a linear or nonlinear controller is applied,limitations on aero engines make the controller a familyof subsystems that switch interactively depending onvarious environmental factors. Switching systems havebeen widely investigated during the last decade [17–20]and stability of the system in the switching process is oneemphasis of this study. Since a quick switch between twocontrollers may cause large amplitude oscillations, tradi-tional control methods have to sacrifice response speedfor stability of the system. In order to achieve both fastresponse and high stability, the idea of multi-objectiveswitching control system is proposed and widely used in

Ltd. All rights reserved.

Nomenclature

a integral limiters of inlet buzz protection con-troller

Av area of valve head chamberAr force area of gasAtr throat areab integral limiters of gas regulating controllerb0 viscous friction coefficientCr characteristic velocity of gasd disturbance signalH altitudei drive current of servo valveKi integration time constant of the control sys-

temM mass of valve headM0 Mach number of free streampb backpressure of inletpr pressure in the gas generator chamberpv pressure in the valve head chamberpc pressure in the channel of the control valve

ql mass flow of leakageqie mass flow through servo valveRf friction force besides viscous friction forceRv gas constant of gas in valve head chambert timeTv temperature of gas in valve head chamberVv0 volume of valve head chamber at a operating

pointwg gas flow ratexv displacement of valve headb angle of attack of free streamx inlet buzz marginpb maximum pressure ratio of supersonic inletpu current pressure ratio of supersonic inlet

Subscripts

max maximum parametermin minimum parameterF parameter at point F(M0F, bF)

W. Bao et al. / Acta Astronautica 67 (2010) 764–773 765

all areas [21,22]. In Ref. [23], by disassembling theregulating and protecting problems of aero engine,controllers are designed separately and can easily switchto each other, which resolves the contradiction betweenfast response and security. Linear matrix inequality is aneffective method that has been applied to several practicalproblems. Refs. [24,25] derive linear matrix inequalitybased sufficient conditions to deal with switched staticoutput feedback control of systems under arbitraryswitching.

Despite much progress in aero engine control, researchon ducted rocket control is still a rarely involved field. Gasflow regulating control is one of the key technologies forducted rockets [26]. By throttling the gas flow, propellantconsumption can be optimized during missile flight alongany non-pre-known path. It provides the advantages ofexpanded no escape zone, improved high altitude operat-ing envelope and increased range. It can avoid flameoutand maintain high specific impulses. Nevertheless, due toits specificity in structure, some kinds of non-minimumphase characteristics like anti-regulation exist in gas flowcontrol of ducted rockets, which greatly affect theirdynamic performance. Anti-regulation characteristics ofgas flow control result in some particular control issues,which need to be studied in detail and considered inregulating controller design progress. Inlet buzz is aphenomenon of ducted rockets that significantly reducessystem’s performance and should be completely avoided.Therefore, inlet buzz protection control should be intro-duced in a ducted rocket system to obtain optimal thrustat safe operating conditions.

The outline of this paper is as follows. First, the multi-loop control system of ducted rockets is described inSection 2, the control problems are formulated and designcontradictions in the switching control are detailed. InSection 3, mathematical models of thrust control loopand inlet buzz protection loop in ducted rockets are

established. Section 4 shows the procedure to design themulti-loop control system of ducted rockets and simula-tion results. Finally, conclusions can be obtained inSection 5.

2. Multi-loop control system description of ductedrockets

2.1. Necessities for introducing multi-objective control

For a modern aircraft, in order to achieve high flightperformance, a highly qualified aero engine controlsystem is essentially required. But there are also manylimitations during the working process of aero engines,especially ducted rockets. Thrust is one of the mostimportant flight performance parameters of aircraft, andthrust of ducted rockets is controlled by throttling the gasflow. However, the gas regulating system works in ahighly adverse environment where gas generator dis-charges high temperature and high velocity gas carryingcondensed phase particles. Without adequate control,ducted rockets will be unable to maintain high specificimpulse or even counter phenomena like flameouts.

Inlet buzz is a phenomenon of self-sustained shockoscillations that may appear in almost every type ofsupersonic inlets, which results in high-amplitude varia-tions of inlet mass flow and pressure. It generally ariseswhen the entering mass flow is reduced below a givenvalue. Inlet buzz can lead to thrust loss and engine surge,or can even cause structural damages to the aircraft; thisviolent and dangerous phenomenon is therefore highlyundesirable. However, the maximum performance ofsupersonic inlet will be achieved when operating as closeas possible to the buzz boundary. In order to maintainhigh performance without crossing the buzz boundary, aninlet buzz protection controller is necessary. The control

W. Bao et al. / Acta Astronautica 67 (2010) 764–773766

system must be carefully constructed as a priority so as toavoid system crossing the known buzz boundary and tobe able to take some measures when approaching thebuzz boundary [27].

In the working process of ducted rockets, especiallywhen the missile accelerates, a high command in thrustoutput will require more solid fuel to be burnt to provideenough energy. However, the gas generated in gasgenerator will then pour into the combustor, whichincreases inlet backpressure. If the control system stillgoes on to satisfy the objective of thrust, inlet back-pressure would keep increasing, leading the system tocross the inlet buzz boundary. If the controller based onthe objective of inlet buzz protection takes effect at thistime, it will stop the system from approaching the inletbuzz boundary and ensure that the ducted rocket obtainsmaximum thrust at safe operating conditions.

Thus, it is necessary to introduce multi-objectivecontrol to ducted rockets to eliminate the contradictionbetween thrust control objective and inlet buzz controlobjective.

2.2. Multi-loop switching control of ducted rockets

Considering the actual working conditions of ductedrockets, the controller needs to adjust controllablevariables, such as gas flow rate and throat area, to changesof environment disturbances, which mainly manifest inMach number, altitude and attack angle of the flight. Itsobjective is to guarantee that the missile works in apredetermined schedule. Meanwhile, the controller needsto keep working parameters of the system from exceedingthe permissive limitations. Above all, control require-ments of an engine may be classified under three aspects:

(1)

stable controls, for resisting the effect of disturbance; (2) change-of-state controls, for quickly handling the

system to shift to another working state;

(3) limitation controls, for restraining the system from

exceeding the permissive margins.

For the reason that the demands of security and fastresponse should be satisfied simultaneously, traditionaldesigned control systems have to retain enough safetymargins and take conservative tuning values, which willobviously result in sacrificing system’s fast response

Fig. 1. Scheme diagram of a duc

performance for security. In order to improve theperformance of ducted rockets, it is necessary to study arational regulation/protecting control scheme to speed upsystem’s response while ensuring security according tothe characteristic of ducted rockets.

The basic idea of multi-objective control is to decom-pose demands of security and fast response to sub-controllers, and design regulation and protecting controlloop separately. The problem of multi-objective controlcan be realized by switching control. Through switchingamong relatively simple sub-controllers designed by theidea of multi-loop control, the contradiction betweensecurity and fast response can be eliminated. Consideringthe switching control applied in ducted rockets, whenducted rockets work in normal conditions, regulationcontrol loop will take effect and respond fast andaccurately to commands. If the system approaches thesafety boundary, the control system will switch to theprotecting loop, avoid system exceeding limitations andsatisfy maximum requirements of commands. Further, byseparately designing sub-controllers, protecting loopsdirectly aim at system’s limitations, which will firmlyensure system’s security, while without considering theproblem of security, regulation loop could do better tomaximize system’s performance.

A scheme diagram of multi-loop control system ofducted rockets is shown in Fig. 1. It includes sub-controlloops like thrust regulation control and inlet buzzprotection control. Though the switching control it isable to satisfy the demand of multi-objectives. Theinstability problem that may exist in the switchingprocess should be fully considered during the design ofmulti-loop switching control.

3. Mathematical modeling of ducted rockets

3.1. Modeling and characteristic analysis of gas regulation

systems

3.1.1. Modeling of gas regulation systems

Fig. 2 shows a schematic diagram of pressure-balancedpintle type gas regulating systems. The working principleof this system is as follows. The servo valve (4) iscontrolled to make the gas enter into the valve headchamber (9) from a high-pressure gas bottle (1) when theneed for gas flow increases. Pressure in the valve head

ted rocket control system.

Fig. 2. Schematic diagram of a pressure-balanced pintle-type gas regulating system.

W. Bao et al. / Acta Astronautica 67 (2010) 764–773 767

chamber (9) is increased and upsets the force balance ofthe valve head (8), which then moves forward along thevalve stem (10), reducing the throat area (7) of gasgenerator (5) and increasing pressure in the gas generator(5). Since burning rate of solid propellant in the gasgenerator is proportional to chamber pressure, moreburning gases get generated and increase flow ratethrough the control valve. In contrast, when the servovalve (4) pushes the gas leaves the valve head chamber(9), pressure in the chamber valve head (9) is reduced, andso do pressure in the gas generator (5) and flow ratethrough the gas control valve.

Investigations on models of ducted rockets these yearshave been activated by higher requirements on perfor-mance of tactical missiles. Refs. [28,29] studied modelingand control of gas regulation for ducted rockets withoutthe effect of friction.

The structure and working principle of ducted rocketsare referred to in Ref. [28]. A pintle-type gas control valveis the key component of the gas regulation system. Duringits movement, the pintle valve is subjected to severalforces. The force equation of valve head is given as

Md2xv

dt2þb0

dxv

dt¼ prAr�pvAvþðAv�ArÞpcþRf ð1Þ

As weight M of valve head is usually very small, inertialforce Mðd2xv=dt2Þ of the pintle valve can be neglected.Force (Av�Ar)pc can be neglected because pressure pc inthe valve channel is far smaller than that in the gasgenerator. Here only viscous friction force is considered,and other friction forces are neglected. Eq. (1) can besimplified as

b0dxv

dt¼ prAr�pvAv ð2Þ

Applying a dimensionless Laplace transform, thetransfer function between the pressure of gas generatorpr and the pressure of valve chamber pv can be given as

follows [28]:

D ~prðsÞ ¼K5

T42s2þ2xT4sþ1

D ~pvðsÞ ð3Þ

Also, by the movement of control valve the throat areaAtr can be adjusted to the demand of gas flow rate wg asfollows:

wg ¼prAtr

Crð4Þ

where Atr can be expressed as a function of displacementxv.Then the transfer function between flow rate wg andpressure pr can be written as

D ~wgðsÞ ¼K1

K2ð1�T2sÞD ~prðsÞ ð5Þ

The model of pneumatic servo system [28] can beexpressed as

D ~pvðsÞ ¼K6

T6sþ1D~iðsÞ ð6Þ

where

D~iðsÞ ¼DiðsÞ

imax

K6 ¼�@qie=@i

@ql=@pvþ@qie=@pv

imax

pvmax

T6 ¼�Vv0

RvTv

1

@ql=@pvþ@qie=@pv

As discussed above, the mathematical model of gasflow control for ducted rockets can be described byEqs. (3), (5) and (6), and is shown in Fig. 3.

3.1.2. Characteristic analysis

Figs. 4 and 5 show the step response of gas flow forducted rockets. As can be seen from Fig. 4, whenthe control command increases, the actual gas flowdrops first before turning to approach the demand.Vice versa in Fig. 5, when the control command

Fig. 6. Scheme diagram of thrust control loop.

Fig. 3. Mathematical model of the gas flow control for ducted rockets.

Fig. 4. Increasing step response of gas flow for ducted rockets.

Fig. 5. Decreasing step response of gas flow for ducted rockets.

W. Bao et al. / Acta Astronautica 67 (2010) 764–773768

decreases, the actual gas flow shoots up first beforeturning to approach the demand. That is to say, thegas flow regulator shows the anti-regulationcharacteristic. To account for this physics mechanism,taking the increasing step response for example, theincrease in control command makes the valve moveforward and gas flow area decreases; at the same timechamber pressure is still lower and leads the actual gasflow to reduce first. Due to the decrease of throat area,chamber pressure eventually increases, and consequentlyincreases gas flow. From the point of view of themathematical model, a positive zero point existing inthe transfer function of the gas regulation system (shownin Fig. 3) is the main reason that results in anti-regulationcharacteristics of the system.

3.2. Modeling of thrust control loop

Thrust is one of the most important parameters ofperformance of ducted rockets. However, it is difficult todetect thrust online, and inlet backpressure is detectedinstead. Ref. [30] studied the mathematical model, variedparameter characteristics analysis and model reduction,and presented a simplified mathematical model ofcontrollable flow ducted rocket engines. The transformfunction between inlet backpressure pb and gas flow ratewg is

D ~pbðsÞ ¼ K1e�as ð1þt1sÞð1þt2sÞ

ð1þt3sÞð1þt4sÞD ~wgðsÞ

þK2ð1þt5sÞ

ð1þt3sÞð1þt4sÞD ~M0ðsÞ

þK3ð1þt5sÞ

ð1þt3sÞð1þt4sÞD ~HðsÞ ð7Þ

Neglecting the change in Mach number and altitude,Eq. (7) can be simplified to

D ~pbðsÞ ¼ K1e�as ð1þt1sÞð1þt2sÞ

ð1þt3sÞð1þt4sÞD ~wgðsÞ ð8Þ

A scheme diagram of thrust control loop can be givenas shown in Fig. 6, which includes the controller, theactuator and the characteristic between backpressure andgas flow rate. This actuator is discussed in Section 3.1.1.

3.3. Modeling of inlet buzz protection control loop

3.3.1. Inlet buzz boundary and margin

For a fixed-geometry ducted rocket, inlet buzz isgenerated due to excessively large backpressure or toolow free stream pressure. In order to avoid inlet buzz,knowing the inlet buzz limit (boundary) is desirable.Refs. [31,32] investigated and proposed the method toobtain buzz boundary of supersonic and hypersonic inlet,respectively. The inlet buzz boundary refers to themaximum inlet pressure ratio pu at different free streamconditions, where pu is the critical pressure ratio abovewhich inlet buzz will occur. The result shows that theinlet buzz boundary is mainly determined by Machnumber of free stream M0 and attack angle b.

The inlet buzz boundary is shown in Fig. 7. Point F

denotes free stream, Mach number of free stream is M0F,free stream angle of attack is bF and isolator backpressureratio is pbF. The buzz boundary of a supersonic inlet at the

Fig. 8. Control system of the supersonic inlet buzz margin.

Fig. 7. Buzz boundary of a supersonic inlet.

W. Bao et al. / Acta Astronautica 67 (2010) 764–773 769

point F (M0F, bF) can be obtained by M0F and bF, namelypuF=pu(M0F, bF).

Inlet buzz margin represents distance from the currentoperation point to inlet buzz boundary. When workingcloser to inlet buzz boundary, the performance of supersonicinlet is better, while its stability gets worse. For acompromise between performance and stability of super-sonic inlet, a line 5% or 10% lower than the inlet buzzboundary is chosen as a warning line in the actual operatingprocess. Once ducted rockets cross the warning line, thecontrol system will take the corresponding measure toprevent pressure ratio from increasing and avoid inlet buzzphenomenon from taking place. In this article, the inlet buzzmargin is defined as x=(puF�pbF)/puF, where pbF is thecurrent inlet backpressure ratio.

3.3.2. Variation of inlet backpressure and gas flow

There is a complicated relationship between gas fuelflow and backpressure of the supersonic inlet. However, theemphasis of this paper is not regulation characteristics andthe law of gas fuel flow injection in the combustor. For theneeds of the present research, a simplified model ofrelationship between gas fuel flow and inlet backpressure[28] is adopted, assuming that there exists a single pointinjection in the supersonic combustion chamber and therelationship between fuel flow and backpressure is linear.That is to say, wgnor=pbnor, where wgnor=(wg�wgmin)/(wgmax�wgmin), pbnor=(pb�pbmin)/(pbmax�pbmin), wgnor isthe per-unit fuel flow, pbnor the per-unit backpressure ratioof the isolator, mf the current value of fuel flow, wgmin andwgmax are the minimum and maximum fuel flow in the flightenvelope, respectively and pb is the current value of isolatorbackpressure; pbmin and pbmax are the minimum andmaximum backpressure ratio in the flight envelope,respectively.

3.3.3. Inlet buzz protection control loop

Fig. 8 shows the control system of inlet buzz marginfor ducted rockets, which includes the controller, theactuator, the characteristic between backpressure and thefuel and the controlled plant. The actuator is discussed inSection 3.1.1 and is able to change supersonic inlet

backpressure. Consequently, inlet buzz margin can beregulated by changing inlet backpressure; zis thedesignated inlet buzz margin, u the control variable, wg

the fuel flow and d the disturbance signal.

4. Switching control of thrust regulation and inlet buzzprotection for ducted rockets

4.1. Switching control objectives

Through the whole range of working conditions,ducted rockets encounter safety limitations like inlet buzzboundary, over-temperature boundary, etc. Especiallywhen the missile is accelerating, too high thrust willincrease inlet backpressure, which can possibly lead thesystem to cross the inlet buzz boundary. These limitationsmake ducted rockets a multi-objective control system.

Switching control is a strategy widely used to dealwith problems of multi-objectives. Through switchingamong relatively simple sub-controllers, the contradictionbetween security and fast response is eliminated. Whenducted rockets work in normal conditions, regulationcontrol loop will take effect and respond fast andaccurately to commands. When the system approachesthe safety boundary, the control system will switch to theprotecting loop to prevent the system from exceedinglimitations, satisfying maximum requirements of thecommand. By designing separate sub-controllers, theprotecting loop directly aims at system’s limitations,which will firmly ensure system’s security, while withoutconsidering the problem of security, regulation loop coulddo better in maximizing system’s performance.

In this article, the aim is to improve the dynamicperformance of ducted rockets using switching controltheory. In the ducted rocket control system, different sub-controllers have different performance objectives. In thegas flow regulating control loop, the performance objec-tive is to achieve good tracking of thrust command, whichcannot be easily detected and substituted by inlet back-pressure. To perform better, regulating control loopshould guarantee certain performance indexes, such assmall overshoot, zero steady error for step command,settling time less than 0.5 s and sufficiently large phasemargin. In the inlet buzz protection control loop, it isdesired to keep the inlet buzz margin at a designated z.Therefore the main task is to compensate for disturbances,to realize which low steady error and short dynamicregulation time of disturbances are needed.

Generally the switching control system in this paperaims to achieve a fast thrust response with non-over-shoot, zero steady state error and fast settling time, whilemaintaining safe temperature limits and inlet buzzmargins during switching.

Fig. 10. Curve of controller output.

W. Bao et al. / Acta Astronautica 67 (2010) 764–773770

4.2. Switching PID controller and switching strategy

The switching system includes switching strategiesand sub-controllers. Min/max switching strategy is theone widely used in aero engine control for its simplerealization. The gas generator of ducted rockets workswithout the effect of afterburner, so that the gas injectedfrom gas generator provides a strong support of ignition,preventing ducted rockets from flameout by poor oil orrich oil. Considering the particularity of ducted rocketsabove, min switching strategy is adopted in the switchingcontrol system of ducted rockets. By selecting theminimum control signal of thrust controller and inletbuzz protection controller, the control system can beswitched to guarantee ducted rockets against crossing theinlet buzz boundary in the acceleration process ordeceleration process, and be able to work at effectiveconditions.

The selecting and tuning of sub-controller is a veryimportant part of the design process. Due to their simplestructure and robustness, PID controllers are widely usedin industrial areas and there are many approachesproposed to tune PID controllers. Some recent literatureshave works on transforming the problem of PI or PIDcontroller design to that of static output feedbackcontroller design [33,34].

Ref. [23] has applied output feedback controller tomulti-objective switching control based on linear matrixinequalities (LMIs). Consider the continuous-timeswitched system

_x ¼ AixþB1iwþB2iu,

z¼ C1ixþD11iwþD12iu,

y¼ C2ixþD21iwþD22iu,

i¼ 1, � � � ,N

8><>: ð9Þ

where the vector of state variables is x 2 Rn, w 2 Rr is anexogenous input, u 2 Rmi is the control input of subsystemi, z 2 Rp the controlled output, y 2 Rr the measured outputand Ai, B1i, B2i, C1i, C2i, D11i, D12i, D21i and D22i are thesubsystem matrices.

The aim here is to determine a stabilizing switchoutput feedback control

_xKi ¼ AKixKiþBKiyi

u¼ CKixKiþDKiyi

(ð10Þ

and the desirable controller can be obtained by themethod of H1optimal and pole assignment, which isspecifically presented in [23].

Fig. 9. Scheme diagram of switching

After the steps above, multi-objective switching con-trol system of ducted rockets can be obtained, and itsscheme diagram is shown in Fig. 9.

4.3. Simulation analysis

Figs. 10–12 show the curve of controller output, inletbuzz margin and gas flow for the acceleration process ofducted rocket, respectively. Before 1 s, inlet buzz marginstays at a constant margin larger than the given value(Fig. 11) while the output of inlet buzz protection controlkeeps rising until integral limitation (Fig. 10). Theacceleration process starts at 1 s and thrust signal makesa step rise (Fig. 10). At the same time, inlet buzz marginjumps first and then drops (Fig. 11) while gas flow dropsfirst and then shoots (Fig. 12) due to the anti-regulationcharacteristic of ducted rockets, which is discussed inSection 3. Under the action of min switching strategy,thrust control loop works before 1.46 s, while the signal ofthrust controller is larger than that of inlet buzzprotection controller later, leading the system to switchto inlet protection loop. Shortly after 1.46 s, the output ofinlet buzz protection controller varies so fast that the anti-regulation phenomenon shows up again, reducing theinlet buzz margin and increasing gas flow to a certaindegree before tending to stabilize.

controller of ducted rockets.

Fig. 11. Curve of inlet buzz margin.

Fig. 12. Curve of gas flow.

Fig. 13. Curve of inlet buzz margin under different integral limiters in

inlet buzz protection controller.

Fig. 14. Curve of gas flow under different integral limiters in inlet buzz

protection controller.

W. Bao et al. / Acta Astronautica 67 (2010) 764–773 771

4.4. Effect of integral limiter on switch performance

In the min/max switch strategy, control signal ischosen by a comparison of outputs of controllers, makingit strongly depend on selection of controller parameters.In order to switch smoothly and guarantee controlaccuracy, integral is a necessary link included in con-trollers, which influences the switching process consider-ably. As can be seen from Fig. 10, when the inlet buzzprotection control is not working, the output of inlet buzzprotection controller keeps increasing. If the integraloutput is not limited, it will take a long time for theswitching process, which may prevent the inlet protectingloop from acting in time to prevent the system fromcrossing the inlet buzz boundary. Therefore integrallimiter should be applied to the system to minimize theprocess time of switching, and the effect of integral limiteris discussed below.

Figs. 13 and 14 show curves of inlet buzz margin andgas flow under different integral limiters of inlet buzzprotection controller for the acceleration process ofducted rockets, where the acceleration process starts at1 s, making inlet backpressure step change. The integrallimitation of thrust control loop is 1.2. Along with adecrease of integral limiter in the inlet buzz protectioncontrol loop, settling time and peak value of inlet buzzcontrol keep reducing, consequently improving the safetyof inlet. However, a too small limiter will lead theacceleration capability to totally fail though the systemcould still switch to the inlet protecting loop. Therefore,the integral limiter of inlet buzz protection controllershould be as low as possible while at least assuring thatthe actual inlet buzz margin is lesser than the given value.

W. Bao et al. / Acta Astronautica 67 (2010) 764–773772

Figs. 15 and 16 show curves of inlet buzz margin andgas flow under different integral limiters of thrustcontroller for both acceleration and decelerationprocesses of ducted rockets, where the accelerationprocess starts at 1 s, making thrust signal step rise, andthe deceleration process starts at 3 s, making thrust signalstep down. The integral limitation of inlet buzz protectioncontrol loop is 0.4. Along with the decrease of integrallimiter in thrust control loop, the settling time ofswitching process keeps reducing. However, a too smalllimiter will lead the acceleration capability to totally failfor the reason that the system is not able to switch to theinlet protecting loop. Therefore, the integral limiter ofinlet buzz protection controller should be as low aspossible while at least assuring that the actual inlet buzzmargin is lesser than the given value.

Fig. 16. Curve of gas flow under different integral limiters in thrust

controller.

Fig. 15. Curve of inlet buzz margin under different integral limiters in

thrust controller.

5. Conclusions

Due to large demands on tactical missiles, the renewedinterest impulses the investigation of ducted rockets fortheir simple structure and high performance. In thisarticle, in order to obtain optimal thrust control in theworking process of ducted rockets while avoiding phe-nomena like inlet buzz, switching control is investigatedand applied. Firstly multi-loop control problems of ductedrockets during its working process are discussed. Then thedynamic mathematical model of gas flow regulatingsystem, thrust regulation loop and inlet buzz protectionloop is established and analyzed. Lastly, switching PIDcontroller and switching strategy are applied to theducted rocket system. There are some conclusions asfollows:

�

Anti-regulation characteristic in ducted rockets isthe maximum difference in comparison with liquidramjets. It leads to the particularity of switching controlfor ducted rockets that when the phenomenon of inletbuzz is likely to happen, the command to decrease gasflow should not be given immediately, or it willaccelerate the buzz phenomenon taking place. � Integral limiter is demanded in the switching control of

ducted rockets in order to minimize the process timeof switching but the value of limitation should beelaborately designed. Simulations show that theoptimal value is the minimum that permits the systemto switch between two sub-controllers.

� The working requirements make ducted rockets a

multi-objective system, and the introduction of switch-ing control preferably eliminates the multi-objectivecontrol problem of ducted rockets.

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