control allocation design of reaction control system for reusable
TRANSCRIPT
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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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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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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
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Decision SciencesAdvances in
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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
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Mathematical Problems in Engineering
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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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Decision SciencesAdvances in
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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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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Differential EquationsInternational Journal of
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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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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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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
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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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Decision SciencesAdvances in
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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
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Complex AnalysisJournal of
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OptimizationJournal 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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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