Research ArticleControl Allocation Design of Reaction Control System forReusable Launch Vehicle

Rongjun Mu and Xin Zhang

School of Astronautics Harbin Institute of Technology Harbin 150001 China

Correspondence should be addressed to Rongjun Mu murjun163com

Received 31 January 2014 Revised 13 April 2014 Accepted 13 April 2014 Published 12 June 2014

Academic Editor Hui Zhang

Copyright copy 2014 R Mu and X ZhangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

During the early stage of reusable launch vehicle (RLV) reentry flight reaction control system (RCS) is the major attitude controldevice RCS which is much different from the atmospheric steerrsquos control requires a well designed control allocation system to fitthe attitude control in high altitude In this paper an indexed control method was proposed for RCS preallocation a 0-1 integerprogramming algorithm was designed for RCS allocation controller and then this RCS schemersquos effect was analyzed Based on thespecified flight mission simulation the results show that the control system is satisfied Moreover several comparisons between theattitude control effect and RCS relevant parameters were studied

1 Introduction

In the reentry flight phase the aerodynamic characteristicsof RLV vary rapidly when serious uncertainties and nonlin-earity exist Therefore RLV needs high robust and accurateattitude control algorithm Different from normal vehiclesRLV usually relies on aerosurfaces and reaction controlsystem for attitude control [1] In its on-orbit operationand the initial reentry RCS is the major attitude controldevice Because the dynamic pressure is not strong enough foraerosurfaces to meet the needs of spacecraft attitude control

Many control methods had already been presented tosolve the attitude control problem of RLV with hypersonicspeed In [2] a variable structure control was used to controlthe attitude of RLV in ascent and reentry phases in case ofsmall disturbance In [3] a neural network algorithm wasadopted to learn 7 level fuzzy rules online This methodhas the problems of a large amount of computation andlow reliability Currently the conventional gain-schedulingtechnique called operating point-based linearization can beapplied for the nonlinear controller designing

In this study the attitude control algorithm is onlydesigned for RCS [4ndash7] The control moment from controllaw is handled by RCS command logic and operation logicto yield the final commands of thrusters [8] Usually RCS

consists of several thrusters Naturally the actual controlmoments [9] are produced by thruster ejecting the propellantfrom the jet In consequence the control allocation [10ndash14]between all the thrusters of RCS should be the key in theattitude control algorithm designing [15] (see Figure 1)

Therefore the rest of the paper is organized as followsThe nonlinearmodel and control model of RLV are presentedin Section 2 In Section 3 RCS is extensively studied andmodeled An indexed control method is proposed for RCSpreallocation and a 0-1 integer programming algorithm isdesigned for RCS control allocation in Section 4 Sections 5and 6 respectively give the RCS schemersquos simulation resultsand the concluding remarks

2 Mathematical Model of RLV

21 Nonlinear Model In this study a high-fidelity RLVmodel is used to demonstrate the proposed reliable controlapproach The body configuration of RLV provides theinertial coupling in the lateraldirectional channel Hence thedynamicmodel of RLV is a complicated and highly nonlineartime-variation uncertain system [16]

In modeling the English-American coordinates systemis used [17] To simplify the complex model [18] some

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 541627 13 pageshttpdxdoiorg1011552014541627

2 Abstract and Applied Analysis

Controller Valve

Propellant

Jet RLV

Attitudesensor

120579c c

u120576

Tj

TdRCS =+

n lowast thruster120579 120579120579

minus

Figure 1 Block diagram of RCS

installation errors such as RCS thruster location error areignored and several assumptions are made as follows

(a) take RLV as an ideal rigid body and ignore any elasticvibration

(b) its sideslip angle is small enough to make little angleassumption thus the lateral moments are providedmainly by RCS

(c) the RLV principal axes of inertia and the body axesare superposed regardless of the influence of inertiaproduct

(d) the dynamic characteristics of the navigation deviceare ignored considering the vehiclersquos state can beobtained in time

Then the dynamic model of the reusable launch vehiclecan be expressed as

119889119909

119889119905= V119909

119889119910

119889119905= V119910

119889119911

119889119905= V119911 (1)

The attitude motion model of RLV is represented by

[

[

120574

]

]

=[[[

[

1

cos120595(1205961199101sin 120574 + 120596

1199111cos 120574)

1205961199101cos 120574 minus 120596

1199111sin 120574

1205961199091

+ tan120595 (1205961199101sin 120574 + 120596

1199111cos 120574)

]]]

]

(120595 = plusmn 90∘

)

(2)

During the initial reentry process the translational equa-tion of RLV can be described as

119898 = 119866 + 119877 + 119865co + 119865rel + 119865119896+ 119865119908 (3)

where 119866 is referred to as gravity 119877 corresponds to theaerodynamic force 119865co is the control force which is providedby RCS 119865rel and 119865

119896are respectively convected force and

Coriolis force which are small enough to be neglected theremaining 119865

119908covers all the disturbing force

The equations of rotation around the centroid can bedescribed as

1198681199091199091

+ (119868119911minus 119868119910) 1205961199101

1205961199111

= 119872119909

1198681199101199101

+ (119868119909minus 119868119911) 1205961199091

1205961199111

= 119872119910

1198681199111199111

+ (119868119910minus 119868119909) 1205961199091

1205961199101

= 119872119911

(4)

where the variables 119868119909 119868119910 119868119911are moments of inertia [19]

the variables 119872119909 119872119910 119872119911represent all the external moment

around the body axis 119909 119910 119911 of RLV

u

Channel BaffleHeater

Jetnozzle

CatalystArmature

Electromagnet Controlsignal

Figure 2 Thruster structure diagram

22 Control Model Respectively denote attack angle 120572 yawangle 120595 roll angle 120574 and the angular velocity of rotation 120596

119909

120596119910120596119911as state variables of the control system So we adopt the

control model [20]

= 120596119911minus

119902119904

119898119881(119862120572

119910120572 + 119862

120575119911

119910120575119911)

120573 = 120596119909sin120572 + 120596

119910cos120572 +

119902119904

119898119881(119862120573

119911120573 + 119862

120575119910

119911120575119910)

120574 = 120596119909

119909=

119872119909

119868119909

+(119868119910minus 119868119911) 120596119910120596119911

119868119909

119910=

119872119910

119868119910

+(119868119911minus 119868119909) 120596119911120596119909

119868119910

119911=

119872119911

119868119911

+(119868119909minus 119868119910) 120596119909120596119910

119868119911

(5)

In this paper during the RCS-only working stage thecontrol efficiency of the aerosurfaces is not strong enoughcompared with RCS So the aerodynamic moment generatedby the rudders can be ignored

3 Mathematical Models for RCS

31 RCS Characteristics Before illuminating the characteris-tics of the reaction control system this paper needs to clarifyits working process which is simply outlined by Figure 2

When the thruster does not work the baffle under thetension of the spring blocks the propellant pipeline So

Abstract and Applied Analysis 3

uc

F

1

OTn

Td

Fj

ΔTD

ΔTRO

t

t

Thru

stC

ontro

lsig

nal

Figure 3 Thruster mathematical model

the propellant cannot flow to the catalyst Therefore thereis no thrust When a positive control signal is generatedthe electromagnet will produce an attraction force to thearmature When the suction is larger than the tension ofthe spring the armature will pull the baffle away Then afterthe propellant meets the catalyst layer chemical reaction willoccur which can generate heat and thrust Conversely whenthe suction is less than the spring force armature will comeback to the original position and the reaction will be cut off

By understanding the working process of RCS the studyconcludes the following advantages [21]

(1) the thruster canwork at any orbital position thus RCShas already been widely used in spacecraft attitudecontrol

(2) control moment provided by RCS along the RLVbody axis is far greater than the coupling torquewhich makes control logic more simple and flexible

(3) compared to the external and internal disturbancetorques RCS controlmoment is larger andhas shortertransition time so the disturbance torques can beneglected in the primary design stage of the controlsystem

(4) RCS is applicable to nonperiodic disturbance torqueoccasions Due to the consumption of propellant itsservice life is short which makes it suitable for SpaceShuttle RLV and recoverable satellite

(5) thrusters usually adopt the stationary force [22 23]and the ONOFF switch working mode

Certainly RCS has several following disadvantages [24]

(1) RCS needs propellant to produce control momentbut the propellant is limited

(2) lateral jet has the complex aerodynamic interferenceto the rudders especially in the phase of hypersonicspeed

(3) it is difficult to maintain the continuous and uniformjet flux which results inmore difficulty for the controlscheme designing

(4) as one thruster can only produce one direction offorce and moment the control system may requiremore than one of them to produce various directionsof attitude control moment Usually 6 thrusters cancomplete the attitude control task [25] But consider-ing the reliability the actual system always needs thenecessary redundancy For example the RCS in thispaper has 12 thrusters

32 Thruster Model Considering electromagnetic hysteresisand nonlinear characteristics of the thruster the relation-ship between the control signal and the thrust output iscomplicated and nonlinear Therefore according to specificrequirements some practical mathematical models [26] forthe thruster are given as follows

(1) Ideal Mathematic Model This model is relatively simplewithout considering the thrustersrsquo ONOFF switching delayand assuming that the thrust output is constant

This ideal thruster model is usually used for the theoryvalidation preliminary design and mathematical analysis

(2) Mathematical Model Considering Switching Delay Thismodel is more accurate than the ideal one It considersthe thrustersrsquo ONOFF switching delay but still ignoresthe nonlinear dynamic characteristics during the ONOFFswitching process Figure 3 gives the curve of thrust outputversus time

Usually thismodel is used at the design and analysis stageof attitude control system

(3) The Index Mathematical Model This model considersboth the time delay and the dynamic characteristics duringthe ONOFF switching (the thrust rises or falls like theexponential function) So obviously this model usually usedat the simulation stage is the most precise among the three

Taking account of time delay and the dynamic character-istics during the switching process the thrustermodel is builtby using MatlabSimulink (Figure 4)

In Figure 4 the module of ldquoTransport Delayrdquo representsthe switching time delay of the thruster the variable ldquodelta1rdquoin the module of ldquoTransfer Fcnrdquo represents the switching onacceleration the variable ldquodelta2rdquo in the module of ldquoTransferFcn1rdquo represents the switching off deceleration Figure 5 givesthe simulation curve of thruster model responding to thecontrol command signal

33 Force and Moment of RCS Figures 6 and 7 present thethrusters layout map which shows the jet directions andpositions of the 12 thrusters

The output of the 119894th thruster is

119865119877119894

= [

[

0

119865119894cos 120576119894

119865119894sin 120576119894

]

]

119872119877119894

= 119865119877119894

times [

[

119909119894minus 119909119888

119910119894minus 119910119888

119911119894minus 119911119888

]

]

(6)

where 119865119894is the resultant force of thruster 119894 120576

119894is the deflection

angle of corresponding jet control moment 119872119877119894

is the crossproduct of 119865

119877119894

and its corresponding force arm relative to thecenter of mass

4 Abstract and Applied Analysis

1

In1Transport

Delay Transfer Fcn Transfer Fcn1

Tj0

Gain Out11

1 1

delta110 middot s + 1 delta110 middot s + 1

Figure 4 The thruster model in Simulink

0 05 1 15

0

05

1

15

t (s)

CommandActual thrust

minus05

Figure 5 The response curve of thruster model to control com-mand

Figure 6 Thrusters longitudinal layout map

49924

1000141

1640

90∘45∘

0

Figure 7 Thrusters lateral layout map

Table 1 Control moments of thrusters

No Roll moment(kNlowastm)

Yaw moment(kNlowastm)

Pitch moment(kNlowastm)

1 minus082 0 +14222 +082 0 +14223 +017 +1422 04 minus017 minus1422 05 +017 +1377 06 minus017 minus1377 07 +017 +1331 08 minus017 minus1331 09 minus142 +1006 +100610 +142 minus1006 +100611 +059 +1006 minus100612 minus059 minus1006 minus1006

Based on (6) the control moments of all the thrusters canbe obtained in Table 1

According to Table 1 a thruster alone can produce atleast two axes control moment which makes specific controlunsuccessful with the proper selection of thrusters we canachieve a variety of control moment levels to ensure thehardware redundancy of RCS

4 Design of Control Allocation

The designed control allocation system consists of two partspreallocation unit and allocation controller unit Accordingto the configuration of RCS the preallocation unit uses theindexed control method to select all the reasonable sets ofthrusters and process the moment data of the entire thrusterselection For the allocation controller unit the paper appliesthe 0-1 integer programming algorithm to select the optimalselection of thrusters to meet the expected control demandsBy changing some parameters of the algorithm this unit canensure that the system achieves the optimal effect of the fuelconsumption control precision and so on

Figure 8 gives the block diagram of the RCS controlallocation system In the figure the expected controlmomentobtained by the gain-scheduling controller is the input of theallocation system Hence considering the expected controldemands and the entire available selections of thrusters givenby the preallocation unit the allocation controller can obtainthe optimal selection and give the corresponding ignitingcommand to the current actuator namely RCS To the bestof the authorsrsquo knowledge this control allocation system canensure RCS work normally even with some thrusters failure

Abstract and Applied Analysis 5

Preallocation data processing

Controllerjet allocation

Multiplethrusters

Expected control moment

Ignitingcommand

Output control moment

Indexedcontrol

0-1 integerprogramming

Allocation controller

RCS

Figure 8 RCS control allocation system diagram

Priority tabulation of

Multiplethrusters

Expectedcontrol moment

Ignitingcommand

Output control moment

Modifiedindexed control

method

Get models of force and moment

from RCS arrangement

Optimalselection of

thrusters

Simplified0-1 integer

programmingInterval dt

single)thrusters (includingpossible selection

Figure 9 RCS control allocation flow diagram

1165 117 1175 118 1185 119 1195 12 12055

55

6

65

7

75

8

85

9

95

X (m)

Y (m

)

Actual trajectoryExpected trajectory

times105

times104

Figure 10 Simulation result of trajectory in 2D

300 305 310 315 320 325 330 335 340 345t (s)

200

0

minus200

minus400

minus600

minus800

minus1000

minus1200

Vx (ms)

Vz (ms)Vy (ms)

Figure 11 Simulation result of flight speed

As we can see the designed control allocation systemhas a lot of differences with the rudder control system Alsothe control model of RLV needs some appropriate deductionand simplification Generally we need to consider the timedelay characteristics the dynamic characteristics of thrustersrsquoONOFF switching the minimum pulse limit (define thevariable 119889119905 as the minimum interval of two thruster selec-tions ie the shortest interval between the ignition andshutdown of each thruster) the maximum ignition durationthe total ONOFF switching times and the fuel consumption[27]

41 RCS Preallocation Method According to Figure 8 thefirst task of preallocation is to decide how to get all thereasonable sets of thrustersWhen choosing the RCS thrusterselection we should follow the following principles [28]

(1) during the thruster selection try not to bring addi-tional coupling moment

(2) choose the selection with the larger control momentarm since it is good to reduce the consumption ofRCS

6 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 345210

2105

211

2115

212

2125

213

2135

214

Actual pitch angleExpected pitch angle

t (s)

120601(d

eg)

Figure 12 Simulation result of pitch angle

300 305 310 315 320 325 330 335 340 345

Actual yaw angleExpected yaw angle

t (s)

002

0015

0005

0

minus0005

minus001

001

minus0015

minus002

minus0025

120595(d

eg)

Figure 13 Simulation result of yaw angle

(3) in order to improve the redundancy of the systemnormally choose the selection with the higher prior-ity

Based on the above principles we can get all the possibleand reasonable thruster selections which is essential for thenext data process

This preallocation unit is to process data where all thepreparation for the control allocation is accomplished In[24] there are three methods for data processing dependentcontrol method graded control method and indexed controlmethod

(1)TheDependent ControlMethodAccording to thismethodthe resultant moment of each selection is parallel with one ofthe body axes so that there is no coupling problem And if

1798

180

1802

1804

1806

1808

181

1812

1814

300 305 310 315 320 325 330 335 340 345t (s)

Actual roll angleExpected roll angle

120574(d

eg)

Figure 14 Simulation result of roll angle

300 305 310 315 320 325 330 335 340 34551

515

52

525

53

535

Actual attack angleExpected attack angle

t (s)

120572(d

eg)

Figure 15 Simulation result of attack angle

a thruster is in this selection it cannot be in any otherselection Although this method does not fully utilize allthe thrusters it is logically simple and really simplifies theprocess However when one thruster fails RCS may lose thecontrol ability which means the systemrsquos redundancy is zero

(2) The Graded Control Method This method is an improve-ment on the first one in terms of the thruster utilization Thedifference between the twomethods is whether a thruster canbe used in more than one selection Therefore in a way thismethod relatively increases the redundancy of the system

(3) The Indexed Control Method This method takes allthe possible selections of thrusters (considering the singlethruster situation) into consideration and ranks them by

Abstract and Applied Analysis 7

300 305 310 315 320 325 330 335 340 345

0

05

1

15

2

Actual slide angleExpected slide angle

minus05

minus15

minus1

minus2

120573(d

eg)

t (s)

Figure 16 Simulation result of slide angle

300 305 310 315 320 325 330 335 340 345263

2635

264

2645

265

2655

266

t (s)

Actual angleExpected angle

120579(d

eg)

Figure 17 Simulation result of flight path angle

using the expected control moment as the index or criteriafor evaluation So if the moment of the selection is closer tothe expected moment its priority is higher

The indexed control method has the largest redundancyamong the three methods and guarantees that the systemworks normally even when some thrusters fail Howevercompared to the other two methods this one needs toestablish large tables of data covering all reasonable andavailable thruster selection But after modifying the methodby adding the selection principals into the method as a filterthe amount of calculation can be reduced greatly

In this paper RCS can have only 12 thrusters and someof them have serious coupling moment between pitching

300 305 310 315 320 325 330 335 340 345

Actual angleExpected angle

2

15

1

05

0

minus05

minus1

minus15

minus2

120595(d

eg)

t (s)

Figure 18 Simulation result of trajectory deflection angle

300 305 310 315 320 325 330 335 340 34523

24

25

26

27

28

29

3

31

32

t (s)

Actual machExpected mach

Ma

Figure 19 Simulation result of Mach number

and rolling channelsTherefore themodified indexed controlmethod is the best solution

42 RCS Control Allocation Method Denote

Δ119872 =1003816100381610038161003816119872119862 minus 119872

1003816100381610038161003816 (7)

where the variable119872119862is the expected controlmoment which

is obtained by the gain-scheduling controller the variable119872 is the actual control moment provided by RCS As we allknow the closer the resultant controlmoment of the selectionis to the expected moment the better the selection is That isto say the less Δ119872 is the better the selection is Neverthelesswhen the values of Δ119872 in two different selections are closeto each other the variables such as the fuel consumption

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

Controller Valve

Propellant

Jet RLV

Attitudesensor

120579c c

u120576

Tj

TdRCS =+

n lowast thruster120579 120579120579

minus

Figure 1 Block diagram of RCS

installation errors such as RCS thruster location error areignored and several assumptions are made as follows

(a) take RLV as an ideal rigid body and ignore any elasticvibration

(b) its sideslip angle is small enough to make little angleassumption thus the lateral moments are providedmainly by RCS

(c) the RLV principal axes of inertia and the body axesare superposed regardless of the influence of inertiaproduct

(d) the dynamic characteristics of the navigation deviceare ignored considering the vehiclersquos state can beobtained in time

Then the dynamic model of the reusable launch vehiclecan be expressed as

119889119909

119889119905= V119909

119889119910

119889119905= V119910

119889119911

119889119905= V119911 (1)

The attitude motion model of RLV is represented by

[

[

120574

]

]

=[[[

[

1

cos120595(1205961199101sin 120574 + 120596

1199111cos 120574)

1205961199101cos 120574 minus 120596

1199111sin 120574

1205961199091

+ tan120595 (1205961199101sin 120574 + 120596

1199111cos 120574)

]]]

]

(120595 = plusmn 90∘

)

(2)

During the initial reentry process the translational equa-tion of RLV can be described as

119898 = 119866 + 119877 + 119865co + 119865rel + 119865119896+ 119865119908 (3)

where 119866 is referred to as gravity 119877 corresponds to theaerodynamic force 119865co is the control force which is providedby RCS 119865rel and 119865

119896are respectively convected force and

Coriolis force which are small enough to be neglected theremaining 119865

119908covers all the disturbing force

The equations of rotation around the centroid can bedescribed as

1198681199091199091

+ (119868119911minus 119868119910) 1205961199101

1205961199111

= 119872119909

1198681199101199101

+ (119868119909minus 119868119911) 1205961199091

1205961199111

= 119872119910

1198681199111199111

+ (119868119910minus 119868119909) 1205961199091

1205961199101

= 119872119911

(4)

where the variables 119868119909 119868119910 119868119911are moments of inertia [19]

the variables 119872119909 119872119910 119872119911represent all the external moment

around the body axis 119909 119910 119911 of RLV

u

Channel BaffleHeater

Jetnozzle

CatalystArmature

Electromagnet Controlsignal

Figure 2 Thruster structure diagram

22 Control Model Respectively denote attack angle 120572 yawangle 120595 roll angle 120574 and the angular velocity of rotation 120596

119909

120596119910120596119911as state variables of the control system So we adopt the

control model [20]

= 120596119911minus

119902119904

119898119881(119862120572

119910120572 + 119862

120575119911

119910120575119911)

120573 = 120596119909sin120572 + 120596

119910cos120572 +

119902119904

119898119881(119862120573

119911120573 + 119862

120575119910

119911120575119910)

120574 = 120596119909

119909=

119872119909

119868119909

+(119868119910minus 119868119911) 120596119910120596119911

119868119909

119910=

119872119910

119868119910

+(119868119911minus 119868119909) 120596119911120596119909

119868119910

119911=

119872119911

119868119911

+(119868119909minus 119868119910) 120596119909120596119910

119868119911

(5)

In this paper during the RCS-only working stage thecontrol efficiency of the aerosurfaces is not strong enoughcompared with RCS So the aerodynamic moment generatedby the rudders can be ignored

3 Mathematical Models for RCS

31 RCS Characteristics Before illuminating the characteris-tics of the reaction control system this paper needs to clarifyits working process which is simply outlined by Figure 2

When the thruster does not work the baffle under thetension of the spring blocks the propellant pipeline So

Abstract and Applied Analysis 3

uc

F

1

OTn

Td

Fj

ΔTD

ΔTRO

t

t

Thru

stC

ontro

lsig

nal

Figure 3 Thruster mathematical model

the propellant cannot flow to the catalyst Therefore thereis no thrust When a positive control signal is generatedthe electromagnet will produce an attraction force to thearmature When the suction is larger than the tension ofthe spring the armature will pull the baffle away Then afterthe propellant meets the catalyst layer chemical reaction willoccur which can generate heat and thrust Conversely whenthe suction is less than the spring force armature will comeback to the original position and the reaction will be cut off

By understanding the working process of RCS the studyconcludes the following advantages [21]

(1) the thruster canwork at any orbital position thus RCShas already been widely used in spacecraft attitudecontrol

(2) control moment provided by RCS along the RLVbody axis is far greater than the coupling torquewhich makes control logic more simple and flexible

(3) compared to the external and internal disturbancetorques RCS controlmoment is larger andhas shortertransition time so the disturbance torques can beneglected in the primary design stage of the controlsystem

(4) RCS is applicable to nonperiodic disturbance torqueoccasions Due to the consumption of propellant itsservice life is short which makes it suitable for SpaceShuttle RLV and recoverable satellite

(5) thrusters usually adopt the stationary force [22 23]and the ONOFF switch working mode

Certainly RCS has several following disadvantages [24]

(1) RCS needs propellant to produce control momentbut the propellant is limited

(2) lateral jet has the complex aerodynamic interferenceto the rudders especially in the phase of hypersonicspeed

(3) it is difficult to maintain the continuous and uniformjet flux which results inmore difficulty for the controlscheme designing

(4) as one thruster can only produce one direction offorce and moment the control system may requiremore than one of them to produce various directionsof attitude control moment Usually 6 thrusters cancomplete the attitude control task [25] But consider-ing the reliability the actual system always needs thenecessary redundancy For example the RCS in thispaper has 12 thrusters

32 Thruster Model Considering electromagnetic hysteresisand nonlinear characteristics of the thruster the relation-ship between the control signal and the thrust output iscomplicated and nonlinear Therefore according to specificrequirements some practical mathematical models [26] forthe thruster are given as follows

(1) Ideal Mathematic Model This model is relatively simplewithout considering the thrustersrsquo ONOFF switching delayand assuming that the thrust output is constant

This ideal thruster model is usually used for the theoryvalidation preliminary design and mathematical analysis

(2) Mathematical Model Considering Switching Delay Thismodel is more accurate than the ideal one It considersthe thrustersrsquo ONOFF switching delay but still ignoresthe nonlinear dynamic characteristics during the ONOFFswitching process Figure 3 gives the curve of thrust outputversus time

Usually thismodel is used at the design and analysis stageof attitude control system

(3) The Index Mathematical Model This model considersboth the time delay and the dynamic characteristics duringthe ONOFF switching (the thrust rises or falls like theexponential function) So obviously this model usually usedat the simulation stage is the most precise among the three

Taking account of time delay and the dynamic character-istics during the switching process the thrustermodel is builtby using MatlabSimulink (Figure 4)

In Figure 4 the module of ldquoTransport Delayrdquo representsthe switching time delay of the thruster the variable ldquodelta1rdquoin the module of ldquoTransfer Fcnrdquo represents the switching onacceleration the variable ldquodelta2rdquo in the module of ldquoTransferFcn1rdquo represents the switching off deceleration Figure 5 givesthe simulation curve of thruster model responding to thecontrol command signal

33 Force and Moment of RCS Figures 6 and 7 present thethrusters layout map which shows the jet directions andpositions of the 12 thrusters

The output of the 119894th thruster is

119865119877119894

= [

[

0

119865119894cos 120576119894

119865119894sin 120576119894

]

]

119872119877119894

= 119865119877119894

times [

[

119909119894minus 119909119888

119910119894minus 119910119888

119911119894minus 119911119888

]

]

(6)

where 119865119894is the resultant force of thruster 119894 120576

119894is the deflection

angle of corresponding jet control moment 119872119877119894

is the crossproduct of 119865

119877119894

and its corresponding force arm relative to thecenter of mass

4 Abstract and Applied Analysis

1

In1Transport

Delay Transfer Fcn Transfer Fcn1

Tj0

Gain Out11

1 1

delta110 middot s + 1 delta110 middot s + 1

Figure 4 The thruster model in Simulink

0 05 1 15

0

05

1

15

t (s)

CommandActual thrust

minus05

Figure 5 The response curve of thruster model to control com-mand

Figure 6 Thrusters longitudinal layout map

49924

1000141

1640

90∘45∘

0

Figure 7 Thrusters lateral layout map

Table 1 Control moments of thrusters

No Roll moment(kNlowastm)

Yaw moment(kNlowastm)

Pitch moment(kNlowastm)

1 minus082 0 +14222 +082 0 +14223 +017 +1422 04 minus017 minus1422 05 +017 +1377 06 minus017 minus1377 07 +017 +1331 08 minus017 minus1331 09 minus142 +1006 +100610 +142 minus1006 +100611 +059 +1006 minus100612 minus059 minus1006 minus1006

Based on (6) the control moments of all the thrusters canbe obtained in Table 1

According to Table 1 a thruster alone can produce atleast two axes control moment which makes specific controlunsuccessful with the proper selection of thrusters we canachieve a variety of control moment levels to ensure thehardware redundancy of RCS

4 Design of Control Allocation

The designed control allocation system consists of two partspreallocation unit and allocation controller unit Accordingto the configuration of RCS the preallocation unit uses theindexed control method to select all the reasonable sets ofthrusters and process the moment data of the entire thrusterselection For the allocation controller unit the paper appliesthe 0-1 integer programming algorithm to select the optimalselection of thrusters to meet the expected control demandsBy changing some parameters of the algorithm this unit canensure that the system achieves the optimal effect of the fuelconsumption control precision and so on

Figure 8 gives the block diagram of the RCS controlallocation system In the figure the expected controlmomentobtained by the gain-scheduling controller is the input of theallocation system Hence considering the expected controldemands and the entire available selections of thrusters givenby the preallocation unit the allocation controller can obtainthe optimal selection and give the corresponding ignitingcommand to the current actuator namely RCS To the bestof the authorsrsquo knowledge this control allocation system canensure RCS work normally even with some thrusters failure

Abstract and Applied Analysis 5

Preallocation data processing

Controllerjet allocation

Multiplethrusters

Expected control moment

Ignitingcommand

Output control moment

Indexedcontrol

0-1 integerprogramming

Allocation controller

RCS

Figure 8 RCS control allocation system diagram

Priority tabulation of

Multiplethrusters

Expectedcontrol moment

Ignitingcommand

Output control moment

Modifiedindexed control

method

Get models of force and moment

from RCS arrangement

Optimalselection of

thrusters

Simplified0-1 integer

programmingInterval dt

single)thrusters (includingpossible selection

Figure 9 RCS control allocation flow diagram

1165 117 1175 118 1185 119 1195 12 12055

55

6

65

7

75

8

85

9

95

X (m)

Y (m

)

Actual trajectoryExpected trajectory

times105

times104

Figure 10 Simulation result of trajectory in 2D

300 305 310 315 320 325 330 335 340 345t (s)

200

0

minus200

minus400

minus600

minus800

minus1000

minus1200

Vx (ms)

Vz (ms)Vy (ms)

Figure 11 Simulation result of flight speed

As we can see the designed control allocation systemhas a lot of differences with the rudder control system Alsothe control model of RLV needs some appropriate deductionand simplification Generally we need to consider the timedelay characteristics the dynamic characteristics of thrustersrsquoONOFF switching the minimum pulse limit (define thevariable 119889119905 as the minimum interval of two thruster selec-tions ie the shortest interval between the ignition andshutdown of each thruster) the maximum ignition durationthe total ONOFF switching times and the fuel consumption[27]

41 RCS Preallocation Method According to Figure 8 thefirst task of preallocation is to decide how to get all thereasonable sets of thrustersWhen choosing the RCS thrusterselection we should follow the following principles [28]

(1) during the thruster selection try not to bring addi-tional coupling moment

(2) choose the selection with the larger control momentarm since it is good to reduce the consumption ofRCS

6 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 345210

2105

211

2115

212

2125

213

2135

214

Actual pitch angleExpected pitch angle

t (s)

120601(d

eg)

Figure 12 Simulation result of pitch angle

300 305 310 315 320 325 330 335 340 345

Actual yaw angleExpected yaw angle

t (s)

002

0015

0005

0

minus0005

minus001

001

minus0015

minus002

minus0025

120595(d

eg)

Figure 13 Simulation result of yaw angle

(3) in order to improve the redundancy of the systemnormally choose the selection with the higher prior-ity

Based on the above principles we can get all the possibleand reasonable thruster selections which is essential for thenext data process

This preallocation unit is to process data where all thepreparation for the control allocation is accomplished In[24] there are three methods for data processing dependentcontrol method graded control method and indexed controlmethod

(1)TheDependent ControlMethodAccording to thismethodthe resultant moment of each selection is parallel with one ofthe body axes so that there is no coupling problem And if

1798

180

1802

1804

1806

1808

181

1812

1814

300 305 310 315 320 325 330 335 340 345t (s)

Actual roll angleExpected roll angle

120574(d

eg)

Figure 14 Simulation result of roll angle

300 305 310 315 320 325 330 335 340 34551

515

52

525

53

535

Actual attack angleExpected attack angle

t (s)

120572(d

eg)

Figure 15 Simulation result of attack angle

a thruster is in this selection it cannot be in any otherselection Although this method does not fully utilize allthe thrusters it is logically simple and really simplifies theprocess However when one thruster fails RCS may lose thecontrol ability which means the systemrsquos redundancy is zero

(2) The Graded Control Method This method is an improve-ment on the first one in terms of the thruster utilization Thedifference between the twomethods is whether a thruster canbe used in more than one selection Therefore in a way thismethod relatively increases the redundancy of the system

(3) The Indexed Control Method This method takes allthe possible selections of thrusters (considering the singlethruster situation) into consideration and ranks them by

Abstract and Applied Analysis 7

300 305 310 315 320 325 330 335 340 345

0

05

1

15

2

Actual slide angleExpected slide angle

minus05

minus15

minus1

minus2

120573(d

eg)

t (s)

Figure 16 Simulation result of slide angle

300 305 310 315 320 325 330 335 340 345263

2635

264

2645

265

2655

266

t (s)

Actual angleExpected angle

120579(d

eg)

Figure 17 Simulation result of flight path angle

using the expected control moment as the index or criteriafor evaluation So if the moment of the selection is closer tothe expected moment its priority is higher

The indexed control method has the largest redundancyamong the three methods and guarantees that the systemworks normally even when some thrusters fail Howevercompared to the other two methods this one needs toestablish large tables of data covering all reasonable andavailable thruster selection But after modifying the methodby adding the selection principals into the method as a filterthe amount of calculation can be reduced greatly

In this paper RCS can have only 12 thrusters and someof them have serious coupling moment between pitching

300 305 310 315 320 325 330 335 340 345

Actual angleExpected angle

2

15

1

05

0

minus05

minus1

minus15

minus2

120595(d

eg)

t (s)

Figure 18 Simulation result of trajectory deflection angle

300 305 310 315 320 325 330 335 340 34523

24

25

26

27

28

29

3

31

32

t (s)

Actual machExpected mach

Ma

Figure 19 Simulation result of Mach number

and rolling channelsTherefore themodified indexed controlmethod is the best solution

42 RCS Control Allocation Method Denote

Δ119872 =1003816100381610038161003816119872119862 minus 119872

1003816100381610038161003816 (7)

where the variable119872119862is the expected controlmoment which

is obtained by the gain-scheduling controller the variable119872 is the actual control moment provided by RCS As we allknow the closer the resultant controlmoment of the selectionis to the expected moment the better the selection is That isto say the less Δ119872 is the better the selection is Neverthelesswhen the values of Δ119872 in two different selections are closeto each other the variables such as the fuel consumption

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

uc

F

1

OTn

Td

Fj

ΔTD

ΔTRO

t

t

Thru

stC

ontro

lsig

nal

Figure 3 Thruster mathematical model

the propellant cannot flow to the catalyst Therefore thereis no thrust When a positive control signal is generatedthe electromagnet will produce an attraction force to thearmature When the suction is larger than the tension ofthe spring the armature will pull the baffle away Then afterthe propellant meets the catalyst layer chemical reaction willoccur which can generate heat and thrust Conversely whenthe suction is less than the spring force armature will comeback to the original position and the reaction will be cut off

By understanding the working process of RCS the studyconcludes the following advantages [21]

(1) the thruster canwork at any orbital position thus RCShas already been widely used in spacecraft attitudecontrol

(2) control moment provided by RCS along the RLVbody axis is far greater than the coupling torquewhich makes control logic more simple and flexible

(3) compared to the external and internal disturbancetorques RCS controlmoment is larger andhas shortertransition time so the disturbance torques can beneglected in the primary design stage of the controlsystem

(4) RCS is applicable to nonperiodic disturbance torqueoccasions Due to the consumption of propellant itsservice life is short which makes it suitable for SpaceShuttle RLV and recoverable satellite

(5) thrusters usually adopt the stationary force [22 23]and the ONOFF switch working mode

Certainly RCS has several following disadvantages [24]

(1) RCS needs propellant to produce control momentbut the propellant is limited

(2) lateral jet has the complex aerodynamic interferenceto the rudders especially in the phase of hypersonicspeed

(3) it is difficult to maintain the continuous and uniformjet flux which results inmore difficulty for the controlscheme designing

(4) as one thruster can only produce one direction offorce and moment the control system may requiremore than one of them to produce various directionsof attitude control moment Usually 6 thrusters cancomplete the attitude control task [25] But consider-ing the reliability the actual system always needs thenecessary redundancy For example the RCS in thispaper has 12 thrusters

32 Thruster Model Considering electromagnetic hysteresisand nonlinear characteristics of the thruster the relation-ship between the control signal and the thrust output iscomplicated and nonlinear Therefore according to specificrequirements some practical mathematical models [26] forthe thruster are given as follows

(1) Ideal Mathematic Model This model is relatively simplewithout considering the thrustersrsquo ONOFF switching delayand assuming that the thrust output is constant

This ideal thruster model is usually used for the theoryvalidation preliminary design and mathematical analysis

(2) Mathematical Model Considering Switching Delay Thismodel is more accurate than the ideal one It considersthe thrustersrsquo ONOFF switching delay but still ignoresthe nonlinear dynamic characteristics during the ONOFFswitching process Figure 3 gives the curve of thrust outputversus time

Usually thismodel is used at the design and analysis stageof attitude control system

(3) The Index Mathematical Model This model considersboth the time delay and the dynamic characteristics duringthe ONOFF switching (the thrust rises or falls like theexponential function) So obviously this model usually usedat the simulation stage is the most precise among the three

Taking account of time delay and the dynamic character-istics during the switching process the thrustermodel is builtby using MatlabSimulink (Figure 4)

In Figure 4 the module of ldquoTransport Delayrdquo representsthe switching time delay of the thruster the variable ldquodelta1rdquoin the module of ldquoTransfer Fcnrdquo represents the switching onacceleration the variable ldquodelta2rdquo in the module of ldquoTransferFcn1rdquo represents the switching off deceleration Figure 5 givesthe simulation curve of thruster model responding to thecontrol command signal

33 Force and Moment of RCS Figures 6 and 7 present thethrusters layout map which shows the jet directions andpositions of the 12 thrusters

The output of the 119894th thruster is

119865119877119894

= [

[

0

119865119894cos 120576119894

119865119894sin 120576119894

]

]

119872119877119894

= 119865119877119894

times [

[

119909119894minus 119909119888

119910119894minus 119910119888

119911119894minus 119911119888

]

]

(6)

where 119865119894is the resultant force of thruster 119894 120576

119894is the deflection

angle of corresponding jet control moment 119872119877119894

is the crossproduct of 119865

119877119894

and its corresponding force arm relative to thecenter of mass

4 Abstract and Applied Analysis

1

In1Transport

Delay Transfer Fcn Transfer Fcn1

Tj0

Gain Out11

1 1

delta110 middot s + 1 delta110 middot s + 1

Figure 4 The thruster model in Simulink

0 05 1 15

0

05

1

15

t (s)

CommandActual thrust

minus05

Figure 5 The response curve of thruster model to control com-mand

Figure 6 Thrusters longitudinal layout map

49924

1000141

1640

90∘45∘

0

Figure 7 Thrusters lateral layout map

Table 1 Control moments of thrusters

No Roll moment(kNlowastm)

Yaw moment(kNlowastm)

Pitch moment(kNlowastm)

1 minus082 0 +14222 +082 0 +14223 +017 +1422 04 minus017 minus1422 05 +017 +1377 06 minus017 minus1377 07 +017 +1331 08 minus017 minus1331 09 minus142 +1006 +100610 +142 minus1006 +100611 +059 +1006 minus100612 minus059 minus1006 minus1006

Based on (6) the control moments of all the thrusters canbe obtained in Table 1

According to Table 1 a thruster alone can produce atleast two axes control moment which makes specific controlunsuccessful with the proper selection of thrusters we canachieve a variety of control moment levels to ensure thehardware redundancy of RCS

4 Design of Control Allocation

The designed control allocation system consists of two partspreallocation unit and allocation controller unit Accordingto the configuration of RCS the preallocation unit uses theindexed control method to select all the reasonable sets ofthrusters and process the moment data of the entire thrusterselection For the allocation controller unit the paper appliesthe 0-1 integer programming algorithm to select the optimalselection of thrusters to meet the expected control demandsBy changing some parameters of the algorithm this unit canensure that the system achieves the optimal effect of the fuelconsumption control precision and so on

Figure 8 gives the block diagram of the RCS controlallocation system In the figure the expected controlmomentobtained by the gain-scheduling controller is the input of theallocation system Hence considering the expected controldemands and the entire available selections of thrusters givenby the preallocation unit the allocation controller can obtainthe optimal selection and give the corresponding ignitingcommand to the current actuator namely RCS To the bestof the authorsrsquo knowledge this control allocation system canensure RCS work normally even with some thrusters failure

Abstract and Applied Analysis 5

Preallocation data processing

Controllerjet allocation

Multiplethrusters

Expected control moment

Ignitingcommand

Output control moment

Indexedcontrol

0-1 integerprogramming

Allocation controller

RCS

Figure 8 RCS control allocation system diagram

Priority tabulation of

Multiplethrusters

Expectedcontrol moment

Ignitingcommand

Output control moment

Modifiedindexed control

method

Get models of force and moment

from RCS arrangement

Optimalselection of

thrusters

Simplified0-1 integer

programmingInterval dt

single)thrusters (includingpossible selection

Figure 9 RCS control allocation flow diagram

1165 117 1175 118 1185 119 1195 12 12055

55

6

65

7

75

8

85

9

95

X (m)

Y (m

)

Actual trajectoryExpected trajectory

times105

times104

Figure 10 Simulation result of trajectory in 2D

300 305 310 315 320 325 330 335 340 345t (s)

200

0

minus200

minus400

minus600

minus800

minus1000

minus1200

Vx (ms)

Vz (ms)Vy (ms)

Figure 11 Simulation result of flight speed

As we can see the designed control allocation systemhas a lot of differences with the rudder control system Alsothe control model of RLV needs some appropriate deductionand simplification Generally we need to consider the timedelay characteristics the dynamic characteristics of thrustersrsquoONOFF switching the minimum pulse limit (define thevariable 119889119905 as the minimum interval of two thruster selec-tions ie the shortest interval between the ignition andshutdown of each thruster) the maximum ignition durationthe total ONOFF switching times and the fuel consumption[27]

41 RCS Preallocation Method According to Figure 8 thefirst task of preallocation is to decide how to get all thereasonable sets of thrustersWhen choosing the RCS thrusterselection we should follow the following principles [28]

(1) during the thruster selection try not to bring addi-tional coupling moment

(2) choose the selection with the larger control momentarm since it is good to reduce the consumption ofRCS

6 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 345210

2105

211

2115

212

2125

213

2135

214

Actual pitch angleExpected pitch angle

t (s)

120601(d

eg)

Figure 12 Simulation result of pitch angle

300 305 310 315 320 325 330 335 340 345

Actual yaw angleExpected yaw angle

t (s)

002

0015

0005

0

minus0005

minus001

001

minus0015

minus002

minus0025

120595(d

eg)

Figure 13 Simulation result of yaw angle

(3) in order to improve the redundancy of the systemnormally choose the selection with the higher prior-ity

Based on the above principles we can get all the possibleand reasonable thruster selections which is essential for thenext data process

This preallocation unit is to process data where all thepreparation for the control allocation is accomplished In[24] there are three methods for data processing dependentcontrol method graded control method and indexed controlmethod

(1)TheDependent ControlMethodAccording to thismethodthe resultant moment of each selection is parallel with one ofthe body axes so that there is no coupling problem And if

1798

180

1802

1804

1806

1808

181

1812

1814

300 305 310 315 320 325 330 335 340 345t (s)

Actual roll angleExpected roll angle

120574(d

eg)

Figure 14 Simulation result of roll angle

300 305 310 315 320 325 330 335 340 34551

515

52

525

53

535

Actual attack angleExpected attack angle

t (s)

120572(d

eg)

Figure 15 Simulation result of attack angle

a thruster is in this selection it cannot be in any otherselection Although this method does not fully utilize allthe thrusters it is logically simple and really simplifies theprocess However when one thruster fails RCS may lose thecontrol ability which means the systemrsquos redundancy is zero

(2) The Graded Control Method This method is an improve-ment on the first one in terms of the thruster utilization Thedifference between the twomethods is whether a thruster canbe used in more than one selection Therefore in a way thismethod relatively increases the redundancy of the system

(3) The Indexed Control Method This method takes allthe possible selections of thrusters (considering the singlethruster situation) into consideration and ranks them by

Abstract and Applied Analysis 7

300 305 310 315 320 325 330 335 340 345

0

05

1

15

2

Actual slide angleExpected slide angle

minus05

minus15

minus1

minus2

120573(d

eg)

t (s)

Figure 16 Simulation result of slide angle

300 305 310 315 320 325 330 335 340 345263

2635

264

2645

265

2655

266

t (s)

Actual angleExpected angle

120579(d

eg)

Figure 17 Simulation result of flight path angle

using the expected control moment as the index or criteriafor evaluation So if the moment of the selection is closer tothe expected moment its priority is higher

The indexed control method has the largest redundancyamong the three methods and guarantees that the systemworks normally even when some thrusters fail Howevercompared to the other two methods this one needs toestablish large tables of data covering all reasonable andavailable thruster selection But after modifying the methodby adding the selection principals into the method as a filterthe amount of calculation can be reduced greatly

In this paper RCS can have only 12 thrusters and someof them have serious coupling moment between pitching

300 305 310 315 320 325 330 335 340 345

Actual angleExpected angle

2

15

1

05

0

minus05

minus1

minus15

minus2

120595(d

eg)

t (s)

Figure 18 Simulation result of trajectory deflection angle

300 305 310 315 320 325 330 335 340 34523

24

25

26

27

28

29

3

31

32

t (s)

Actual machExpected mach

Ma

Figure 19 Simulation result of Mach number

and rolling channelsTherefore themodified indexed controlmethod is the best solution

42 RCS Control Allocation Method Denote

Δ119872 =1003816100381610038161003816119872119862 minus 119872

1003816100381610038161003816 (7)

where the variable119872119862is the expected controlmoment which

is obtained by the gain-scheduling controller the variable119872 is the actual control moment provided by RCS As we allknow the closer the resultant controlmoment of the selectionis to the expected moment the better the selection is That isto say the less Δ119872 is the better the selection is Neverthelesswhen the values of Δ119872 in two different selections are closeto each other the variables such as the fuel consumption

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

1

In1Transport

Delay Transfer Fcn Transfer Fcn1

Tj0

Gain Out11

1 1

delta110 middot s + 1 delta110 middot s + 1

Figure 4 The thruster model in Simulink

0 05 1 15

0

05

1

15

t (s)

CommandActual thrust

minus05

Figure 5 The response curve of thruster model to control com-mand

Figure 6 Thrusters longitudinal layout map

49924

1000141

1640

90∘45∘

0

Figure 7 Thrusters lateral layout map

Table 1 Control moments of thrusters

No Roll moment(kNlowastm)

Yaw moment(kNlowastm)

Pitch moment(kNlowastm)

1 minus082 0 +14222 +082 0 +14223 +017 +1422 04 minus017 minus1422 05 +017 +1377 06 minus017 minus1377 07 +017 +1331 08 minus017 minus1331 09 minus142 +1006 +100610 +142 minus1006 +100611 +059 +1006 minus100612 minus059 minus1006 minus1006

Based on (6) the control moments of all the thrusters canbe obtained in Table 1

According to Table 1 a thruster alone can produce atleast two axes control moment which makes specific controlunsuccessful with the proper selection of thrusters we canachieve a variety of control moment levels to ensure thehardware redundancy of RCS

4 Design of Control Allocation

The designed control allocation system consists of two partspreallocation unit and allocation controller unit Accordingto the configuration of RCS the preallocation unit uses theindexed control method to select all the reasonable sets ofthrusters and process the moment data of the entire thrusterselection For the allocation controller unit the paper appliesthe 0-1 integer programming algorithm to select the optimalselection of thrusters to meet the expected control demandsBy changing some parameters of the algorithm this unit canensure that the system achieves the optimal effect of the fuelconsumption control precision and so on

Figure 8 gives the block diagram of the RCS controlallocation system In the figure the expected controlmomentobtained by the gain-scheduling controller is the input of theallocation system Hence considering the expected controldemands and the entire available selections of thrusters givenby the preallocation unit the allocation controller can obtainthe optimal selection and give the corresponding ignitingcommand to the current actuator namely RCS To the bestof the authorsrsquo knowledge this control allocation system canensure RCS work normally even with some thrusters failure

Abstract and Applied Analysis 5

Preallocation data processing

Controllerjet allocation

Multiplethrusters

Expected control moment

Ignitingcommand

Output control moment

Indexedcontrol

0-1 integerprogramming

Allocation controller

RCS

Figure 8 RCS control allocation system diagram

Priority tabulation of

Multiplethrusters

Expectedcontrol moment

Ignitingcommand

Output control moment

Modifiedindexed control

method

Get models of force and moment

from RCS arrangement

Optimalselection of

thrusters

Simplified0-1 integer

programmingInterval dt

single)thrusters (includingpossible selection

Figure 9 RCS control allocation flow diagram

1165 117 1175 118 1185 119 1195 12 12055

55

6

65

7

75

8

85

9

95

X (m)

Y (m

)

Actual trajectoryExpected trajectory

times105

times104

Figure 10 Simulation result of trajectory in 2D

300 305 310 315 320 325 330 335 340 345t (s)

200

0

minus200

minus400

minus600

minus800

minus1000

minus1200

Vx (ms)

Vz (ms)Vy (ms)

Figure 11 Simulation result of flight speed

As we can see the designed control allocation systemhas a lot of differences with the rudder control system Alsothe control model of RLV needs some appropriate deductionand simplification Generally we need to consider the timedelay characteristics the dynamic characteristics of thrustersrsquoONOFF switching the minimum pulse limit (define thevariable 119889119905 as the minimum interval of two thruster selec-tions ie the shortest interval between the ignition andshutdown of each thruster) the maximum ignition durationthe total ONOFF switching times and the fuel consumption[27]

41 RCS Preallocation Method According to Figure 8 thefirst task of preallocation is to decide how to get all thereasonable sets of thrustersWhen choosing the RCS thrusterselection we should follow the following principles [28]

(1) during the thruster selection try not to bring addi-tional coupling moment

(2) choose the selection with the larger control momentarm since it is good to reduce the consumption ofRCS

6 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 345210

2105

211

2115

212

2125

213

2135

214

Actual pitch angleExpected pitch angle

t (s)

120601(d

eg)

Figure 12 Simulation result of pitch angle

300 305 310 315 320 325 330 335 340 345

Actual yaw angleExpected yaw angle

t (s)

002

0015

0005

0

minus0005

minus001

001

minus0015

minus002

minus0025

120595(d

eg)

Figure 13 Simulation result of yaw angle

(3) in order to improve the redundancy of the systemnormally choose the selection with the higher prior-ity

Based on the above principles we can get all the possibleand reasonable thruster selections which is essential for thenext data process

This preallocation unit is to process data where all thepreparation for the control allocation is accomplished In[24] there are three methods for data processing dependentcontrol method graded control method and indexed controlmethod

(1)TheDependent ControlMethodAccording to thismethodthe resultant moment of each selection is parallel with one ofthe body axes so that there is no coupling problem And if

1798

180

1802

1804

1806

1808

181

1812

1814

300 305 310 315 320 325 330 335 340 345t (s)

Actual roll angleExpected roll angle

120574(d

eg)

Figure 14 Simulation result of roll angle

300 305 310 315 320 325 330 335 340 34551

515

52

525

53

535

Actual attack angleExpected attack angle

t (s)

120572(d

eg)

Figure 15 Simulation result of attack angle

a thruster is in this selection it cannot be in any otherselection Although this method does not fully utilize allthe thrusters it is logically simple and really simplifies theprocess However when one thruster fails RCS may lose thecontrol ability which means the systemrsquos redundancy is zero

(2) The Graded Control Method This method is an improve-ment on the first one in terms of the thruster utilization Thedifference between the twomethods is whether a thruster canbe used in more than one selection Therefore in a way thismethod relatively increases the redundancy of the system

(3) The Indexed Control Method This method takes allthe possible selections of thrusters (considering the singlethruster situation) into consideration and ranks them by

Abstract and Applied Analysis 7

300 305 310 315 320 325 330 335 340 345

0

05

1

15

2

Actual slide angleExpected slide angle

minus05

minus15

minus1

minus2

120573(d

eg)

t (s)

Figure 16 Simulation result of slide angle

300 305 310 315 320 325 330 335 340 345263

2635

264

2645

265

2655

266

t (s)

Actual angleExpected angle

120579(d

eg)

Figure 17 Simulation result of flight path angle

using the expected control moment as the index or criteriafor evaluation So if the moment of the selection is closer tothe expected moment its priority is higher

The indexed control method has the largest redundancyamong the three methods and guarantees that the systemworks normally even when some thrusters fail Howevercompared to the other two methods this one needs toestablish large tables of data covering all reasonable andavailable thruster selection But after modifying the methodby adding the selection principals into the method as a filterthe amount of calculation can be reduced greatly

In this paper RCS can have only 12 thrusters and someof them have serious coupling moment between pitching

300 305 310 315 320 325 330 335 340 345

Actual angleExpected angle

2

15

1

05

0

minus05

minus1

minus15

minus2

120595(d

eg)

t (s)

Figure 18 Simulation result of trajectory deflection angle

300 305 310 315 320 325 330 335 340 34523

24

25

26

27

28

29

3

31

32

t (s)

Actual machExpected mach

Ma

Figure 19 Simulation result of Mach number

and rolling channelsTherefore themodified indexed controlmethod is the best solution

42 RCS Control Allocation Method Denote

Δ119872 =1003816100381610038161003816119872119862 minus 119872

1003816100381610038161003816 (7)

where the variable119872119862is the expected controlmoment which

is obtained by the gain-scheduling controller the variable119872 is the actual control moment provided by RCS As we allknow the closer the resultant controlmoment of the selectionis to the expected moment the better the selection is That isto say the less Δ119872 is the better the selection is Neverthelesswhen the values of Δ119872 in two different selections are closeto each other the variables such as the fuel consumption

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Preallocation data processing

Controllerjet allocation

Multiplethrusters

Expected control moment

Ignitingcommand

Output control moment

Indexedcontrol

0-1 integerprogramming

Allocation controller

RCS

Figure 8 RCS control allocation system diagram

Priority tabulation of

Multiplethrusters

Expectedcontrol moment

Ignitingcommand

Output control moment

Modifiedindexed control

method

Get models of force and moment

from RCS arrangement

Optimalselection of

thrusters

Simplified0-1 integer

programmingInterval dt

single)thrusters (includingpossible selection

Figure 9 RCS control allocation flow diagram

1165 117 1175 118 1185 119 1195 12 12055

55

6

65

7

75

8

85

9

95

X (m)

Y (m

)

Actual trajectoryExpected trajectory

times105

times104

Figure 10 Simulation result of trajectory in 2D

300 305 310 315 320 325 330 335 340 345t (s)

200

0

minus200

minus400

minus600

minus800

minus1000

minus1200

Vx (ms)

Vz (ms)Vy (ms)

Figure 11 Simulation result of flight speed

As we can see the designed control allocation systemhas a lot of differences with the rudder control system Alsothe control model of RLV needs some appropriate deductionand simplification Generally we need to consider the timedelay characteristics the dynamic characteristics of thrustersrsquoONOFF switching the minimum pulse limit (define thevariable 119889119905 as the minimum interval of two thruster selec-tions ie the shortest interval between the ignition andshutdown of each thruster) the maximum ignition durationthe total ONOFF switching times and the fuel consumption[27]

41 RCS Preallocation Method According to Figure 8 thefirst task of preallocation is to decide how to get all thereasonable sets of thrustersWhen choosing the RCS thrusterselection we should follow the following principles [28]

(1) during the thruster selection try not to bring addi-tional coupling moment

(2) choose the selection with the larger control momentarm since it is good to reduce the consumption ofRCS

6 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 345210

2105

211

2115

212

2125

213

2135

214

Actual pitch angleExpected pitch angle

t (s)

120601(d

eg)

Figure 12 Simulation result of pitch angle

300 305 310 315 320 325 330 335 340 345

Actual yaw angleExpected yaw angle

t (s)

002

0015

0005

0

minus0005

minus001

001

minus0015

minus002

minus0025

120595(d

eg)

Figure 13 Simulation result of yaw angle

(3) in order to improve the redundancy of the systemnormally choose the selection with the higher prior-ity

Based on the above principles we can get all the possibleand reasonable thruster selections which is essential for thenext data process

This preallocation unit is to process data where all thepreparation for the control allocation is accomplished In[24] there are three methods for data processing dependentcontrol method graded control method and indexed controlmethod

(1)TheDependent ControlMethodAccording to thismethodthe resultant moment of each selection is parallel with one ofthe body axes so that there is no coupling problem And if

1798

180

1802

1804

1806

1808

181

1812

1814

300 305 310 315 320 325 330 335 340 345t (s)

Actual roll angleExpected roll angle

120574(d

eg)

Figure 14 Simulation result of roll angle

300 305 310 315 320 325 330 335 340 34551

515

52

525

53

535

Actual attack angleExpected attack angle

t (s)

120572(d

eg)

Figure 15 Simulation result of attack angle

a thruster is in this selection it cannot be in any otherselection Although this method does not fully utilize allthe thrusters it is logically simple and really simplifies theprocess However when one thruster fails RCS may lose thecontrol ability which means the systemrsquos redundancy is zero

(2) The Graded Control Method This method is an improve-ment on the first one in terms of the thruster utilization Thedifference between the twomethods is whether a thruster canbe used in more than one selection Therefore in a way thismethod relatively increases the redundancy of the system

(3) The Indexed Control Method This method takes allthe possible selections of thrusters (considering the singlethruster situation) into consideration and ranks them by

Abstract and Applied Analysis 7

300 305 310 315 320 325 330 335 340 345

0

05

1

15

2

Actual slide angleExpected slide angle

minus05

minus15

minus1

minus2

120573(d

eg)

t (s)

Figure 16 Simulation result of slide angle

300 305 310 315 320 325 330 335 340 345263

2635

264

2645

265

2655

266

t (s)

Actual angleExpected angle

120579(d

eg)

Figure 17 Simulation result of flight path angle

using the expected control moment as the index or criteriafor evaluation So if the moment of the selection is closer tothe expected moment its priority is higher

The indexed control method has the largest redundancyamong the three methods and guarantees that the systemworks normally even when some thrusters fail Howevercompared to the other two methods this one needs toestablish large tables of data covering all reasonable andavailable thruster selection But after modifying the methodby adding the selection principals into the method as a filterthe amount of calculation can be reduced greatly

In this paper RCS can have only 12 thrusters and someof them have serious coupling moment between pitching

300 305 310 315 320 325 330 335 340 345

Actual angleExpected angle

2

15

1

05

0

minus05

minus1

minus15

minus2

120595(d

eg)

t (s)

Figure 18 Simulation result of trajectory deflection angle

300 305 310 315 320 325 330 335 340 34523

24

25

26

27

28

29

3

31

32

t (s)

Actual machExpected mach

Ma

Figure 19 Simulation result of Mach number

and rolling channelsTherefore themodified indexed controlmethod is the best solution

42 RCS Control Allocation Method Denote

Δ119872 =1003816100381610038161003816119872119862 minus 119872

1003816100381610038161003816 (7)

where the variable119872119862is the expected controlmoment which

is obtained by the gain-scheduling controller the variable119872 is the actual control moment provided by RCS As we allknow the closer the resultant controlmoment of the selectionis to the expected moment the better the selection is That isto say the less Δ119872 is the better the selection is Neverthelesswhen the values of Δ119872 in two different selections are closeto each other the variables such as the fuel consumption

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 345210

2105

211

2115

212

2125

213

2135

214

Actual pitch angleExpected pitch angle

t (s)

120601(d

eg)

Figure 12 Simulation result of pitch angle

300 305 310 315 320 325 330 335 340 345

Actual yaw angleExpected yaw angle

t (s)

002

0015

0005

0

minus0005

minus001

001

minus0015

minus002

minus0025

120595(d

eg)

Figure 13 Simulation result of yaw angle

(3) in order to improve the redundancy of the systemnormally choose the selection with the higher prior-ity

Based on the above principles we can get all the possibleand reasonable thruster selections which is essential for thenext data process

This preallocation unit is to process data where all thepreparation for the control allocation is accomplished In[24] there are three methods for data processing dependentcontrol method graded control method and indexed controlmethod

(1)TheDependent ControlMethodAccording to thismethodthe resultant moment of each selection is parallel with one ofthe body axes so that there is no coupling problem And if

1798

180

1802

1804

1806

1808

181

1812

1814

300 305 310 315 320 325 330 335 340 345t (s)

Actual roll angleExpected roll angle

120574(d

eg)

Figure 14 Simulation result of roll angle

300 305 310 315 320 325 330 335 340 34551

515

52

525

53

535

Actual attack angleExpected attack angle

t (s)

120572(d

eg)

Figure 15 Simulation result of attack angle

a thruster is in this selection it cannot be in any otherselection Although this method does not fully utilize allthe thrusters it is logically simple and really simplifies theprocess However when one thruster fails RCS may lose thecontrol ability which means the systemrsquos redundancy is zero

(2) The Graded Control Method This method is an improve-ment on the first one in terms of the thruster utilization Thedifference between the twomethods is whether a thruster canbe used in more than one selection Therefore in a way thismethod relatively increases the redundancy of the system

(3) The Indexed Control Method This method takes allthe possible selections of thrusters (considering the singlethruster situation) into consideration and ranks them by

Abstract and Applied Analysis 7

300 305 310 315 320 325 330 335 340 345

0

05

1

15

2

Actual slide angleExpected slide angle

minus05

minus15

minus1

minus2

120573(d

eg)

t (s)

Figure 16 Simulation result of slide angle

300 305 310 315 320 325 330 335 340 345263

2635

264

2645

265

2655

266

t (s)

Actual angleExpected angle

120579(d

eg)

Figure 17 Simulation result of flight path angle

using the expected control moment as the index or criteriafor evaluation So if the moment of the selection is closer tothe expected moment its priority is higher

The indexed control method has the largest redundancyamong the three methods and guarantees that the systemworks normally even when some thrusters fail Howevercompared to the other two methods this one needs toestablish large tables of data covering all reasonable andavailable thruster selection But after modifying the methodby adding the selection principals into the method as a filterthe amount of calculation can be reduced greatly

In this paper RCS can have only 12 thrusters and someof them have serious coupling moment between pitching

300 305 310 315 320 325 330 335 340 345

Actual angleExpected angle

2

15

1

05

0

minus05

minus1

minus15

minus2

120595(d

eg)

t (s)

Figure 18 Simulation result of trajectory deflection angle

300 305 310 315 320 325 330 335 340 34523

24

25

26

27

28

29

3

31

32

t (s)

Actual machExpected mach

Ma

Figure 19 Simulation result of Mach number

and rolling channelsTherefore themodified indexed controlmethod is the best solution

42 RCS Control Allocation Method Denote

Δ119872 =1003816100381610038161003816119872119862 minus 119872

1003816100381610038161003816 (7)

where the variable119872119862is the expected controlmoment which

is obtained by the gain-scheduling controller the variable119872 is the actual control moment provided by RCS As we allknow the closer the resultant controlmoment of the selectionis to the expected moment the better the selection is That isto say the less Δ119872 is the better the selection is Neverthelesswhen the values of Δ119872 in two different selections are closeto each other the variables such as the fuel consumption

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

300 305 310 315 320 325 330 335 340 345

0

05

1

15

2

Actual slide angleExpected slide angle

minus05

minus15

minus1

minus2

120573(d

eg)

t (s)

Figure 16 Simulation result of slide angle

300 305 310 315 320 325 330 335 340 345263

2635

264

2645

265

2655

266

t (s)

Actual angleExpected angle

120579(d

eg)

Figure 17 Simulation result of flight path angle

using the expected control moment as the index or criteriafor evaluation So if the moment of the selection is closer tothe expected moment its priority is higher

The indexed control method has the largest redundancyamong the three methods and guarantees that the systemworks normally even when some thrusters fail Howevercompared to the other two methods this one needs toestablish large tables of data covering all reasonable andavailable thruster selection But after modifying the methodby adding the selection principals into the method as a filterthe amount of calculation can be reduced greatly

In this paper RCS can have only 12 thrusters and someof them have serious coupling moment between pitching

300 305 310 315 320 325 330 335 340 345

Actual angleExpected angle

2

15

1

05

0

minus05

minus1

minus15

minus2

120595(d

eg)

t (s)

Figure 18 Simulation result of trajectory deflection angle

300 305 310 315 320 325 330 335 340 34523

24

25

26

27

28

29

3

31

32

t (s)

Actual machExpected mach

Ma

Figure 19 Simulation result of Mach number

and rolling channelsTherefore themodified indexed controlmethod is the best solution

42 RCS Control Allocation Method Denote

Δ119872 =1003816100381610038161003816119872119862 minus 119872

1003816100381610038161003816 (7)

where the variable119872119862is the expected controlmoment which

is obtained by the gain-scheduling controller the variable119872 is the actual control moment provided by RCS As we allknow the closer the resultant controlmoment of the selectionis to the expected moment the better the selection is That isto say the less Δ119872 is the better the selection is Neverthelesswhen the values of Δ119872 in two different selections are closeto each other the variables such as the fuel consumption

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

300 305 310 315 320 325 330 335 340 3450

50

100

150

200

250

300

350

t (s)

q (P

a)

Actual dynamic pressureExpected dynamic pressure

Figure 20 Simulation result of dynamic pressure

300 305 310 315 320 325 330 335 340 345t (s)

2000

1500

1000

500

0

minus500

minus1000

minus1500

minus2000

(Nlowast

m)

Mrc

s x

Figure 21 Simulation result of RCS roll moment

and the number of selected thrusters should be taken intoaccount

Thus define the target function 119869rcs as [29 30]

min 119869rcs = 119908th119906119872 + 119908119872

1003816100381610038161003816119872119862 minus 1198721003816100381610038161003816

st 119872 =

119899

sum

119894=1

119872119894119906119872119894

119872 gt 119872119862

(8)

where the variable 119899 is the total number of the thrustersTaking the RCS in this paper as an example 119899 = 12 Thevariable119908

119872is the priority of the corresponding selection Or

it can be regarded as the weight of this selection Based on (8)the bigger the variable is the worse the selection is

300 305 310 315 320 325 330 335 340 345t (s)

times104

1

05

15

0

minus05

minus1

minus15

(Nlowast

m)

Mrc

s y

Figure 22 Simulation result of RCS yaw moment

300 305 310 315 320 325 330 335 340 345

0

1

2

3

4

5times104

minus1

minus2

minus3

t (s)

(Nlowast

m)

Mrc

s z

Figure 23 Simulation result of RCS pitch moment

0

1

2

3

4

5

6

RCS

fuel

cons

umpt

ion

(kg)

300 305 310 315 320 325 330 335 340 345t (s)

Figure 24 Simulation result of RCS consumption

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

Uoff = 05

0

minus5

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

Figure 25 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 002

0

minus002

120575120595(d

eg) 005

minus005

120575120595(d

eg) 005

minus005

Figure 26 Simulation result of yaw angle deviation

Define the vector 119908th as

119908th = [1199081198721

1199081198722

119908119872119899

] (9)

where the variable 119908119872119894

(119894 = 1 2 119899) is the weightor the priority of the thruster 119894 in this current selectionThe bigger 119908

119872119894

is the lower the priority is The remain-ing variable 119906

119872is the working condition of thruster 119894

For example if only the 9th and 10th thrusters work119906119872

= [0 0 0 0 0 0 0 0 1 1 0 0]119879 In general terms

the closer the real moment is to the expected momentthe smaller the value of the target function isThe smaller thenumber of the selected thrusters is the better the selection is

Meanwhile to reduce the consumption of propellantdefine the range of the dead zone as [minus119880off 119880off ] [31] In this

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000Nlowastm

Nlowastm

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 8000

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

5

minus5120575120574(d

eg)

120575120574(d

eg)

1

minus1

120575120574(d

eg)

Figure 27 Simulation result of roll angle deviation

range the deviation of the attitude angles is too small to startthe RCS thruster So the larger 119880off is the lower the controlprecision is

So the control allocation problem can be described as thefollowing 0-1 integer linear programming problem

min119906119872Δ119872

[1199081

1199082

sdot sdot sdot 119908119899| 119908119872] [

119906119872

Δ119872] forall119872

119862ge 119880off

st119899

sum

119894=1

10038161003816100381610038161003816119872119894119906119872119894

10038161003816100381610038161003816gt 119872119862

(10)

According to (10) how to choose the values of theweight is the key of success or failure for this allocationmethod as these values vary in different situations So bythe simplification of the weight the programming model ismodified to the following one

min 1198690= 119906119879

119872sdot 11987112times119897

+ 119896 sdot 119880off minus 119872119862

st 119896 = sum (119906119872)

12 le 119897 le

12

sum

119894=1

119862119894

12

(11)

where the variable 119871 is a 12-by-119897 matrix including theresultant force and moment of all the selections given bythe preallocation unit the variable 119897 is the sum of all thereasonable and available selections the variable 119896 representsthe sum of selected thrusters

Based on the above analysis the control allocation systemis built by considering the current thruster configuration thepriority of selection the deadweight of RCS and so on Thedesigning process is described in Figure 9

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

5000

0

minus5000

2000

0

minus2000

2000

0

minus2000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 28 Simulation result of roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

NlowastmUoff = 8000

times104

2

0

minus2

times104

2

0

minus2

times104

2

0

minus2

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 29 Simulation result of RCS yaw moment

5 Mathematical Simulation

In the initial stage of reentry reusable launch vehicle main-tains the large attack angle (the boundary dynamic pressureis 300 Pa betweenRCS control stage andRCSrudders [32 33]compound control stage) [34] At this time the aerodynamicefficiency of the rudders is far not enough tomeet the attitudecontrol need So the RCS control is the only way to achievelarge angle attitude maintaining Here by the mathematicalsimulation we verify the control allocation strategies ofRCS and make the comparison between the attitude controlperformance and RCS relevant parameters [35]

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 30 Simulation result of RCS pitch moment

Uoff = 0

Uoff = 4000 Nlowastm

Uoff = 8000 Nlowastm

300 305 310 315 320 325 330 335 340 3450

5

10

15

20

25

30RC

S fu

el co

nsum

ptio

n (k

g)

t (s)

Figure 31 Simulation result of RCS consumption

Initial conditions are as follows [36]

[

[

119909

119910

119911

]

]

= [

[

102150

90308

4025

]

]

m

119881 = [

[

minus718

minus6054

205

]

]

ms

[

[

120593

120595

120574

]

]

= [

[

212∘

0∘

180∘

]

]

(12)

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg)

dt = 0025

0

minus5

Uoff = 01

s

s

Uoff =

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

300 305 310 315 320 325 330 335 340 345t (s)

120575120572(d

eg) 5

0

minus5

04 s

Figure 32 Simulation result of attack angle deviation

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

120575120595(d

eg) 5

0

minus5

120575120595(d

eg) 005

minus005

120575120595(d

eg) 02

minus02

times10minus3 dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 33 Simulation result of yaw angle deviation

Simulation results of the RLV results are listed as follows

(1)Weight119880119900119891119891

= 4000119873lowast119898 the Selection Interval 119889119905 = 01 119904Based on Figures 10 11 12 16 17 18 19 20 21 22 23 and 24we find that the control precision declines as time goes onand the consumption rate of RCS fuel at 340 s is far greaterthan ever The reason is that along with the increase of thedynamic pressure pneumatic restoring torque also increaseswhich decreases the efficiency of RCS control Therefore atthe end of the simulation only using RCS for attitude controlbecomes more and more difficult

In the 0-1 integer programming algorithm the allocationsof pitch yaw and roll moments are interactional So consid-ering that the expected rolling moment is relatively smaller

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

0

0

2

minus2

0

10

minus10120575120574(d

eg)

120575120574(d

eg)

2

minus2120575120574(d

eg)

dt = 002

Uoff = 01

s

s

Uoff = 04 s

Figure 34 Simulation result of roll angle deviation

than the others it has less effect on the control allocationThiswill cause the rolling angle deviation to be the biggest in thethree channels as we can see by comparing Figures 13 and 14with Figure 15This characteristic is decided by the algorithmIt does not mean the controller of the roll channel failed

(2) Comparisons between Different Selection Weights 119880119900119891119891

Inorder to analyze the influence of the different weights 119880offon the allocation method and the control precision of RCSrespectively set 119880off as 0 4000Nlowastm and 8000Nlowastm andkeep the selection interval 119889119905 as 01 second The simulationresults are shown in Figures 28 29 and 30

By the comparisons between different selection weights119880off we conclude that the fuel consumption of RCS willbe reduced to a certain extent when the 119880off increase InFigure 31 the consumption of 119880off = 4000 (N lowast m) is muchless than that of 119880off = 0 but the consumption of 119880off =

8000 (N lowastm) is approximates to that of 119880off = 4000 (N lowastm)Also based on Figures 25 26 and 27 the increase of 119880off

will decrease the control precision especially for the yaw androll channels The reason is that the selection result of theRCS allocation method is mainly influenced by the biggestexpected moment demand In this case the expected controlmoment of the pitch channel is much bigger than the othersSo the pitch angle control precision is the highest and rollingangle control accuracy is the worst

(3) Comparisons betweenDifferentThruster Selection Intervals119889119905 In order to analyze the influence of the different thrusterselection intervals 119889119905 on the allocation method and thecontrol precision of RCS respectively set 119889119905 as 002 second01 second and 04 second and keep weight 119880off as 4000N lowast

mThe simulation results are shown in Figures 32 33 34 and38

From the simulation comparisons of different intervalswe learn that the shorter the single selection interval isthe higher the control precision is But at the same time

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

1000

0

minus1000

2000

0

minus2000

5000

0

minus5000300 305 310 315 320 325 330 335 340 345

t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

dt = 002 s

Uoff = 04 s

Uoff = 01 s

(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x(Nlowast

m)

Mrc

s x

Figure 35 Simulation result of RCS roll moment

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

5

0

minus5

times104

times103

2

0

minus2

times105

1

0

minus1

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y(Nlowast

m)

Mrc

s y

Figure 36 Simulation result of RCS yaw moment

according to Figures 35 36 and 37 we can see that the fre-quency of thrusterrsquos switchingONOFF has greatly increasedwhich will reduce the service life of RCS Hence the intervalshould be properly decided according to actual demand

6 Conclusion

This paper designs the control allocation method of reactioncontrol system for reusable launch vehicle A modifiedindexed control method has been presented to solve thedata processing at the preallocation stage and a simplified0-1 integer programming algorithm has been proposed todesign the allocation controller To demonstrate the schemea series of mathematical simulations in different controlallocation strategies of RCS has been made The results

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

300 305 310 315 320 325 330 335 340 345t (s)

times104

5

0

minus5

times104

5

0

minus5

times104

5

0

minus5

dt = 002

Uoff = 01

s

s

Uoff = 04 s

(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z(Nlowast

m)

Mrc

s z

Figure 37 Simulation result of RCS pitch moment

300 305 310 315 320 325 330 335 340 3450

2

4

6

8

10

12

14

t (s)

RCS

fuel

cons

umpt

ion

(kg)

dt = 04 sdt = 01 s

sdt = 002

Figure 38 Simulation result of RCS consumption

show that the designed RCS control allocation method caneffectively handle tracking accuracy and robustness in theinitial reentry phase Moreover the method proposed inthis paper can readily be applied to the design of reliableallocation controllers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 13

References

[1] Z Jun J Pengfei andHWeijun ldquoAttitude control algorithm forreusable launch vehicle in reentry flight phaserdquo Journal of ChinaOrdnance vol 5 no 1 pp 15ndash19 2009

[2] Y Shtessel and J McDuffie ldquoSliding mode control of the X-33 vehicle in launch and re-entry modesrdquo in Proceedings of theGuidance Navigation and Control Conference and Exhibit pp1352ndash1362 AIAA Boston Mass USA 1998

[3] Y Fengxian L Hua and Z Li-Jun ldquoThe application of thefuzzy-neural network controller of the flighter pose controlrdquoAerospace Control vol 17 no 1 pp 43ndash49 1999 (Chinese)

[4] C A TobinW Bong and C Stanley ldquoPulse-modulated controlsynthesis for a flexible spacecraftrdquo Journal of Guidance Controland Dynamics vol 13 no 6 pp 1014ndash1022 1990

[5] M Piero and C LePome Robert ldquoDesign of a model predictivecontrol flight control system for a reusable launch vehiclerdquo inProceedings of the Guidance Navigation and Control Conferenceand Exhibit pp 5360ndash5370 AIAA Austin Tex USA 2003

[6] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the kistler KI orbital vehiclerdquo AIAA vol4420 pp 1410ndash1422 1998

[7] T Ieko Y Ochi and K Kanai ldquoNew design method for pulse-width modulation control systems via digital redesignrdquo Journalof Guidance Control and Dynamics vol 22 no 1 pp 123ndash1281999

[8] Z Jun J Pengfei and H Weijun ldquoDesign of neural networkvariable structure reentry control system for reusable launchvehiclerdquo Journal of China Ordnance vol 4 no 3 pp 191ndash1972008

[9] N Guodong ldquoResearch on reaction control system for space-craft re-entry flightrdquo Flight Dynamics vol 23 no 3 pp 16ndash192005 (Chinese)

[10] T Ting and N Flanagan ldquoSpace shuttle transition controllerOMS-TVC and RCS jet thruster stability analysisrdquo Tech RepAIAA-91-2220 1991

[11] V E Haloulakos ldquoThrust and impulse requirements for jetattitude-control systemsrdquo Tech Rep AIAA-63-237 1963

[12] J D Gamble K M Spratlin and L M Skalecki ldquoLateraldirectional requirements for a low LD aero-maneuveringorbital transfer vehiclerdquo Tech Rep AIAA-84-2123 1984

[13] R Taylor ldquoAdaptive attitude control for long-life space vehiclesrdquoTech Rep AIAA-69-945 1969

[14] F Bernelli-Zazzera and P Mantegazza ldquoPulse-width equivalenttopulse-amplitude discrete control of linear systemsrdquo Journal ofGuidance Control and Dynamics vol 15 no 2 pp 461ndash4671992

[15] A S Hodel and R Callahan ldquoAutonomous ReconfigurableControl Allocation (ARCA) for reusable launch vehiclesrdquo TechRep 2002-4777 AIAA 2002

[16] F Liao J L Wang E K Poh and D Li ldquoFault-tolerant robustautomatic landing control designrdquo Journal of Guidance Controland Dynamics vol 28 no 5 pp 854ndash871 2005

[17] W Sentang and F Yuhua Flight Control System Beijing Univer-sity of Aeronautics and Astronautics Press 2006 (Chinese)

[18] P P Khargonekar K Poolla and A Tannenbaum ldquoRobustcontrol of linear time-invariant plants using periodic compen-sationrdquo IEEE Transactions on Automatic Control vol 30 no 11pp 1088ndash1096 1985

[19] B N Pamadi and G J Brauckmann ldquoAerodynamic character-istics database development and flight simulation of the X234

vehiclerdquo in Proceedings of the 38th Aerospace Sciences Meetingand Exhibit pp 900ndash910 AIAA Reno Nev USA 2000

[20] W R Van Soest Q P Chu and J A Mulder ldquoCombined feed-back linearization and constrainedmodel predictive control forentry flightrdquo Journal of Guidance Control and Dynamics vol29 no 2 pp 427ndash434 2006

[21] W Wenzheng Research on Composite Control Using AeroControl Surface and RCS China Aero Dynamic Research andDevelop Center Mianyang China 2005 (Chinese)

[22] D Zimpfer ldquoOn-orbit flight control design for the Kistler K-1reusable launch vehicle [R]rdquo AIAA Paper 99-4210 1999

[23] C Zimmerman and G Duckeman ldquoAn automated method tocompute orbital re-entry trajectories with heating constraintsrdquoAIAA Paper A0237761 2002

[24] Y Fang ldquoReaction control system control method for reusablelaunch vehiclerdquoActa Aeronautica et Astronautica Sinica vol 29pp 97ndash101 2008 (Chinese)

[25] P A Servidia and R S Sanchez Pena ldquoThruster design forpositionattitude control of spacecraftrdquo IEEE Transactions onAerospace and Electronic Systems vol 38 no 4 pp 1172ndash11802002

[26] C S Qian Q X Wu C S Jiang and L Zhou ldquoFlight controlfor an aerospace vehicles reentry attitude based on thrust ofreaction jetsrdquo Journal of Aerospace Power vol 23 no 8 pp1546ndash1552 2008 (Chinese)

[27] H Chenglong C Xin and W Liaoni ldquoResearch on attitudecontrol of reaction control system for reusable launch vehiclerdquoJournal of Projectiles Rockets Missiles and Guidance vol 30 no1 pp 51ndash57 2010 (Chinese)

[28] H Weijun and Z Jun ldquoResearch on revised pulse widemodulation of reaction control system for reusable launchvehiclerdquoMissiles and Guidance vol 26 no 4 pp 313ndash316 2006(Chinese)

[29] W C Durham ldquoConstrained control allocation three-momentproblemrdquo Journal of Guidance Control and Dynamics vol 17no 2 pp 330ndash336 1994

[30] J A Paradiso ldquoA highly adaptable method of managing jetsand aerosurfaces for control of aerospace vehiclesrdquo Journal ofGuidance Control and Dynamics vol 14 no 1 pp 44ndash50 1991

[31] H Chenglong Research of Guidance and Control Technologiesfor Reusable Launch Vehicle Sub-Orbital Ascent Flight 2010(Chinese)

[32] P Calhoun ldquoAn entry flight controls analysis for a reusablelaunch vehiclerdquo Tech Rep AIAA-2000-1046 2000

[33] R R Costa Q P Chu and J A Mulder ldquoRe-entry flightcontroller design using nonlinear dynamic inversionrdquo TechRep AIAA-2001-37110 2001

[34] D S Rubenstein and D W Carter ldquoAttitude control systemdesign for return of the Kistler K1 Orbital Vehiclerdquo Journal ofSpacecraft and Rockets vol 37 no 2 pp 273ndash282 2000

[35] H Weijun and Z Jun ldquoA design method of controller for reac-tion control system in high atmosphererdquo Computer Simulationvol 27 no 3 pp 89ndash93 2010 (Chinese)

[36] L Wu Y Huang and C He ldquoReusable launch vehicle lateralcontrol design on suborbital reentryrdquo in Proceedings of the2008 2nd International Symposium on Systems and Control inAerospace and Astronautics (ISSCAA rsquo08) Shenzhen ChinaDecember 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of