research article multimodel predictive control approach
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Research ArticleMultimodel Predictive Control Approach forUAV Formation Flight
Chang-jian Ru1 Rui-xuan Wei1 Ying-ying Wang1 and Jun Che2
1 Air Force Engineering University Xirsquoan 710038 China2 Science and Technology on Aircraft Control Laboratory FACRI Xirsquoan 710065 China
Correspondence should be addressed to Chang-jian Ru ruchangjian1986gmailcom
Received 20 December 2013 Revised 10 March 2014 Accepted 25 March 2014 Published 6 May 2014
Academic Editor Leo Chen
Copyright copy 2014 Chang-jian Ru et al This 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
Formation flight problem is the most important and interesting problem of multiple UAVs (unmanned aerial vehicles) cooperativecontrol In this paper a novel approach for UAV formation flight based on multimodel predictive control is designed Firstly thestate equation of relative motion is obtained and then discretized By the geometrical method the characteristic points of state aredetermined Afterwards based on the linearization technique the standard linear discrete model is obtained at each characteristicstate point Then weighted model set is proposed using the idea of T-S (Takagi-Sugeno) fuzzy control and the predictive control iscarried out based on the multimodel method Finally to verify the performance of the proposed method two different simulationscenarios are performed
1 Introduction
In recent years as an advanced system with high autonomyUAVs have been widely applied in the fields of both civilianandmilitaryWhen a single UAV accomplishes tasks individ-ually it will be more likely to reduce mission success dueto its limited information accessing ability In comparisonmultiple UAVs collaborating with each other maintain acertain formation during the flight which provides themwithfull access to environmental information increases resistanceto external attack capability improves working efficiency androbustness of the system and so forth so it has attractedwide attention [1ndash3] Formation flight is an important aspectof multiple UAVs cooperative control When the forma-tion shape maintains or changes according to the missionrequirements it is necessary to control the relative positionattitude and speed between UAVs and so forth HoweverUAVrsquos control system is a nonlinear coupling system coupledwith complex operational environment constraints puttingforward higher design requirement for formation controllerSo it is essential to propose an effective control strategy tosolve those problems
Model predictive control (MPC) method is put forwardby some scholars Here several typical researches are pre-sented A hierarchical approach and a set of MPC strategiesfor the UAV formation are proposed in [4] where obstacleand collision avoidance constraints are taken into accountA distributed collision-free formation flight control law inthe framework of nonlinear model predictive control isdesigned in [5] In [6] a dual mode MPC method is used forformation control To guarantee the stability the dual modecontroller must switch from an MPC control to a terminalstate controller A simple nonlinear model predictive control(NMPC) formulation is used to adequately address theterrain avoidance problem as presented in [7] An onlinenonlinear model predictive control framework is used for thetrajectory tracking of autonomous vehicles in [8] where abicycle model is used for the prediction of future states inthe NMPC framework The validation of a formation flightcontrol techniquewith obstacle avoidance capability based onnonlinear model predictive algorithms is proposed in [9]
The nonlinear model predictive control method providesan effective means to solve the control problem of nonlinearsystems [10 11] Because the close relative distance between
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 835301 11 pageshttpdxdoiorg1011552014835301
2 Mathematical Problems in Engineering
LeaderFollower
wX
Xg
YgO
Xl
Yl
Yw
Ow119896
120593w
120593l
rarrDW rarr
DL
rarrD
Ol119896
Figure 1 The position relationship between two vehicles
the UAVs may lead to collision thus it requires highercontrol accuracy However stair-like MPC uses the way ofconstraining the variation of the future control quantitywhich restricts themaneuverability of the vehicle and is proneto causing collision between the UAVs due to the overshootproblem [12] So it is necessary to adopt a new predictivecontrol method to achieve the formation flight control Forsomemore complex systems themultimodel controlmethodhas stronger robustness and higher control accuracy undercertain conditions [13] Besides multimodel control methodcan provide the nonlinear system with transparent modeland controller facilitating the system analysis Comparedwith other nonlinear global strategies themultimodel controlmethod cannot greatly reduce computational complexity butthe model and structure of controller are more suitable foronline adjustments and learning algorithm [14] so multiplemodel-based predictive control can be used to solve UAVformation control problem
This paper is organized as follows In Section 2 thediscrete relative motion equations for UAV formation areestablished In Section 3 a multiple models-based predictivecontrol approach is used to design controller of the formationSimulation results are given in Section 4 Finally Section 5concludes the paper
2 UAV Formation Flight Control Model
21 Kinematics Model of UAV Formation Flight ControlAssume that during the formation flight an UAV is flyinghorizontally and has no sideslip In the geographic coordinatesystem the relationship between the position vectors ofleader UAV (leader) and follower UAV (follower) is shownin Figure 1
From Figure 1 it is easy to obtain the following equation
119863119871=
119863119882
+
119863 (1)
where 119863119871 119863119882are displacement vectors of two vehicles and
119863 is the relative displacement vectors between two vehicles
Differentiating (1) one can obtain
119889
119863119871
119889119905
=
119889
119863119882
119889119905
+
119889
119863
119889119905
(2)
According to the relationship between the moving coor-dinate system one can easily obtain
119889
119863119871
119889119905
=
120575
119863119882
120575119905
+ 120596119908times
119863
(3)
where 120596119908is the yaw angular rate
Since the vehicle is supposed to fly horizontally theequation of motion will be
119894= V119894cos (120593
119894)
119910119894= V119894sin (120593
119894)
119894= 120596119894 119894 = 119897 119908
(4)
where the subscript 119897 and 119908 denote leader UAV and followerUAV respectively
Combing (3) and (4) the relative motion equation of twovehicles can be obtained as
V119897[
cos120593119897
sin120593119897
] = V119908[
cos120593119908
sin120593119908
] + 119862
119871
119908([
119889
119910119889
] + [
minus120596119908sdot 119910119889
120596119908sdot 119909119889
]) (5)
where 119909119889and 119910
119889are X-axis value and Y-axis value of the dis-
tance between two vehicles in the tack coordinates of leaderUAV respectively and 119862
119871
119908is the coordinate transformation
matrixThen carry on the translational process and one can
obtain
[
119889
119910119889
] = 119862
119882
119871(V119897[
cos120593119897
sin120593119897
] minus V119908[
cos120593119908
sin120593119908
]) minus [
minus120596119908sdot 119910119889
120596119908sdot 119909119889
]
(6)
where
119862
119882
119871V119897[
cos120593119897
sin120593119897
] = V119897[
cos120593119890
sin120593119890
]
119862
119882
119871V119908[
cos120593119908
sin120593119908
] = [
V119908
0
]
(7)
So (6) can be written as follows
[
119889
119910119889
] = V119897[
cos (120593119897minus 120593119908)
sin (120593119897minus 120593119908)
] minus [
V119908
0
] minus [
minus120596119908sdot 119910119889
120596119908sdot 119909119889
] (8)
Additionally there is
119908= 120596119908 (9)
Combing (8) and (9) we can obtain
119889= V119897cos (120593
119897minus 120593119908) minus V119908+ 120596119908sdot 119910119889
119910119889= V119897sin (120593
119897minus 120593119908) minus 120596119908sdot 119909119889
119908= 120596119908
(10)
Mathematical Problems in Engineering 3
Then (10) can also be written as the state equations whichis shown as follows
[
[
119889
119910119889
119908
]
]
=[
[
minus1 119910119889
0 minus119909119889
0 1
]
]
[
V119908
120596119908
] +[
[
V119897cos (120593
119897minus 120593119908)
V119897sin (120593
119897minus 120593119908)
0
]
]
(11)
The output equation is
1199101= 119909119889
1199102= 119910119889
(12)
22 Discrete Model of UAV Formation Movement and ItsPredictive Control Analysis In the previous section the stateequation of formation control is obtained Here discretizethis equation and the following equation can be obtained
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119908 (
119896 + 1)
]
]
=[
[
119909119889 (
119896)
119910119889 (
119896)
120593119908 (
119896)
]
]
+[
[
minus1 119910119889 (
119896)
0 minus119909119889 (
119896)
0 1
]
]
[
V119908 (
119896)
120596119908 (
119896)
] Δ119879
+[
[
V119897 (119896) cos (120593119897 (119896) minus 120593
119908 (119896))
V119897 (119896) sin (120593
119897 (119896) minus 120593
119908 (119896))
0
]
]
Δ119879
(13)
Generally the sampling periodic time is short duringreceding optimization process so the velocity and yaw angleof leader UAV can be considered constant in sampling period[15] which means in a short sampling period there is
V119897 (119896 + 119894) = V
119897 (119896 + 119894 minus 1) = V
119897 (119896)
120593119897 (119896 + 119894) = 120593
119897 (119896 + 119894 minus 1) = 120593
119897 (119896)
(14)
According to these two equations the predicted value ofoutputs will be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(15)
Since at time 119896 119910119889(119896) V119897(119896) 120593
119908(119896) 120593
119897(119896) are known the
future output values of formation flight control are 119909119889119901
(119896 +
1) 119909119889119901
(119896 + 2) 119909119889119901
(119896 + 119873 minus 1) 119909119889119901
(119896 + 119873) and 119910119889119901
(119896 +
1) 119910119889119901
(119896 + 2) 119910119889119901
(119896 + 119873 minus 1) 119910119889119901
(119896 + 119873) and theseoutputs are only the function of the future control quantitiesV119908(119896) V119908(119896 + 1) V
119908(119896 + 119873 minus 1) and 120596
119908(119896) 120596119908(119896 +
1) 120596119908(119896+119873minus1) Obviously this function is amulti-input-
multioutput nonlinear control problem So these values canbe obtained using the receding optimization algorithmThenusing V
119908(119896) and 120596
119908(119896) as outputs of control quantity and
carry out the receding optimization algorithm in sequencethe input of control quantity in the next time can be obtained
According to Section 1 it can be known that for theproblem of UAV formation flight nonlinearmodel predictivecontrol and fuzzy stair-like predictive control have somelimitations but the multimodel control method has strongerrobustness and higher control accuracy the final predictivemodel of which is linear so the receding optimization
problem can be changed fromgeneral nonlinear optimizationproblem to a linear quadratic optimization problem Sincethe linear quadratic optimization has faster computation thanthe ordinary nonlinear optimization the multiple modelscan greatly improve real-time of receding optimization [16]Therefore multiple model-based predictive control approachis used to design the controller of UAV formation flight
3 Predictive Control for UAV FormationBased on Multimodel Approach
The basic principle of the multimodel control method isthat nearby the different characteristics of nonlinear systemsdifferent linear model is used to describe this nonlinearsystem and each linear model only describes a part of non-linear system dynamics The multiple linearization modelsare used to approximate the nonlinear system in its entireoperating range and the controller is designed based on eachlinearizationmodelThese controllers are combined togetherto constitute amultimodel controller in someway Finally thecontrol of entire nonlinear system can be achieved throughcoordinated control between multiple linearization modelsThe basic steps of this method can be summarized in foursteps (1) acquisition of the model set (2) local linearizationof the model set (3) establishment of the controller set (4)combination of the model set
Similarly the controller design of UAV formation flightcan also include these steps the flow chart of which isshown in Figure 2 The details of controller design willbe presented in the following parts using the multimodelprediction method
31 Determination of the State Characteristic Points of Forma-tion Control Model According to the basic principles of mul-timodel predictive control the characteristic state points ofthe nonlinear model must be obtained first before obtainingmultiple models Characteristic state points and their regionare determined using methods in [15 17] As is shown inFigure 3 the horizontal axis and vertical axis denote time andtrajectory respectively
The basic idea is as followsDetermine the first characteristic state points then com-
pute the error between reference trajectory tangent throughthe characteristic point and reference trajectory and comparethe error with the maximum permissible error If it is greaterthan the maximum permissible error redetermine the char-acteristic point to get the next characteristic state point andcalculate repeatedly until the last characteristic state pointis obtained thus the characteristic state points of nonlinearsystem and their applicable region can be determined Sothe linearization model set can be obtained through thelinearization process at the characteristic state point for eachregion
ForUAV formation control the characteristic state pointsare determined as follows
Assuming that the initial distance between two vehiclesare 119909119889(0) and 119910
119889(0) respectively when UAV formation con-
trol is carried out so the desired distances of UAV formation
4 Mathematical Problems in Engineering
Determining state characteristic points
for formation control
Generating discrete model sets of
formation control
Combining model sets of formation
control
Introduceoptimization index and solve the final
model
Figure 2 The flow chart of UAV formation flight controller design
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emaxy(t)
Figure 3 Description of tangent error
are 119909119888and 119910
119888 Assuming that the expected time arrival at the
desired value of X-axis and Y-axis is the same the referencetrajectory of UAV formation can be obtained as follows
1199101119903 (
119905) = 120585
119905119909119889 (
0) + (1 minus 120585
119905) sdot 119909119888
1199102119903 (
119905) = 120585
119905119910119889 (
0) + (1 minus 120585
119905) sdot 119910119888
(16)
Since there are two outputs generating two referencetrajectories the algorithm above cannot be applied directlyBut because the expected arrival time to the desired value isthe same the reference trajectory of an output can be used todetermine the characteristic point of state Assume that thereference trajectory of relative position on the X-axis in thetrack coordinate system is used to determine characteristicpoint of the state
At the characteristic state points there is
119889= 0 119910
119889= 0
119908= 0 (17)
Assume that ith characteristic state point is (119881119894119908 120596
119894
119908 119909
119894
119889
119910
119894
119889 120593
119894
119889) then there will be the following equation
[
[
[
[
minus1 119910
119894
119889
0 minus119909
119894
119889
0 1
]
]
]
]
[
[
119881
119894
119908
120596
119894
119908
]
]
+
[
[
[
[
119881119897cos (120593
119897minus 120593
119894
119908)
119881119897sin (120593
119897minus 120593
119894
119908)
0
]
]
]
]
=[
[
0
0
0
]
]
(18)
Meanwhile there is
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894) (19)
Solve (18) and (19) then we can obtain
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894)
(20)
Assuming the initial point is as the first characteristic statepoint then the state point value will be
120596
1
119908= 0 120593
1
119908= 120593119897 V1
119908= V119897
119909
1
119889= 1199101119903 (
0) 119910
1
119889= 1199102119903 (
0)
(21)
Then the equation of the tangent is
1199101119896 (
119905) = 1199101119903 (
119905) 119905 + 119909119889 (
0) (22)
Afterward determine the maximum permissible error119864max and the time 119905
2corresponding to the second charac-
teristic point can be obtained using the method above Thenthe state point can be obtained as follows
120596
2
119908= 0 120593
2
119908= 120593119897 V2
119908= V119897
119909
2
119889= 119909119889(1199052) 119910
2
119889= 119910119889(1199052)
(23)
So the tangent equation is
1199101119896
(119905) = 1199101119903(119905) (119905 minus 119905
2) + 119909119889(1199052) (24)
By repeating these procedures above time 119905119894correspond-
ing to the characteristic point in the region of multimodelfor formation can be obtained and at the same time thecharacteristic state point corresponding to the time 119905
119894can be
also obtained
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(25)
Calculate until the last characteristic state point isobtained and then the computation will be terminated
32 Generation of Discrete Model Sets for Formation ControlAfter obtaining the characteristic state points carry onlinearization at different discrete model sets of formationHere linearization can be realized through the followingmethods
Consider nonlinear systems as described in the form ofdiscrete-time dynamic equations
119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896))
119910 (119896) = 119892 (119909 (119896) 119906 (119896))
(26)
Mathematical Problems in Engineering 5
The system has m different characteristic state points119891(119909(119896) 119906(119896)) and 119892(119909(119896) 119906(119896)) have the first continuouspartial derivative If system is linearized at each characteristicstate point the standard discrete state-space model of mlinear models of the original system is obtained as follows
119909 (119896 + 1) = 119860119894119909 (119896) + 119861
119894119906 (119896) minus 120572
119894
119910 (119896) = 119862119894119909 (119896) + 119863
119894119906 (119896) minus 120573
119894
(27)
where
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119862119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119863119894=
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894
120572119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119910119894
(28)
Here m linearized models constitute the linearized multi-model presentation of the original system
For the UAV formation flying control the characteristicstate points are shown as
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(29)
And it has the following expression
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
minus1 119910119889(119905119894)
0 minus119909119889(119905119894)
0 1
]
]
Δ119879
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894=[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(30)
Since outputs are linear 119862119894 119863119894and 120573
119894will not be solved
Thus the linearization equation at characteristic state pointwill be
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119889 (
119896 + 1)
]
]
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
[
[
119909119889 (
119896)
119910119889 (
119896)
120593119889 (
119896)
]
]
+[
[
minus1 119910119889(119905119894)
0 119909119889(119905119894)
0 1
]
]
Δ119879[
V119908 (
119896)
120596119908 (
119896)
] minus[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(31)
So for different characteristic state points the linearmodel for UAV formation control at different horizonscan be obtained realizing the acquisition of model setsfor UAV formation And these models are denoted as119872(1)119872(2) 119872(119878)
33 Combined Method of Model Sets for Formation FlightControl Thecharacteristic state points are obtained using themethod above When the error reaches the maximum valueswitch to the new model which ensures the maximum errorvalue between the predictive trajectory and the referencetrajectory Thus the determination of model region can berealized It can be seen that the applicable range of desiredmodel is divided based on the time region so differentsampling points have different models But the predictivecontrol is based on the future time region So in this paperthe applicable model of predictive point is judged by the timeregion and then this model is used to calculate the predictivevalue The judgment rules of predictive model are describedas follows
Assuming that the time corresponding to the state char-acteristic points is 119905
1lt 1199052lt sdot sdot sdot lt 119905
119904minus1lt 119905119904and the predictive
horizon is [119905 119905 + 119873] then there will be the followingIf [119905 119905 + 119873] isin [119905
119894 119905119894+1
] then the final predictive model ofall points is 119872
119879= 119872119894 if [119905 119905 + 119873] isin [119905
119894 119905119894+119873
] one can judgethe interval [119905
ℎ 119905ℎ+1
] of all points between prediction point119905 + 1 to 119905 + 119873 in sequence and denote the model of this pointas119872119879= 119872ℎ if [119905 119905 +119873] isin [119905
119904infin) the final predictive model
of all the points will be119872119879= 119872119904
Based on this method we can obtain the predictive func-tion during the future horizon to determine the optimizationindex However the boundary point of the predictive rangemay be closer to the next linear model as shown in Figure 4Sampling point 119901
1may be closer to the linear model at
sampling point 1199054 But according to the method above the
calculation model used at the sampling point 1199011is the linear
model of the state characteristic point at sampling point1199053 Sampling point 119901
2may be closer to the linear model
of the state characteristic point at sampling point 1199055 But
according to the method above the calculation model usedby the sampling point 119901
2is the linear model of the state
characteristic point at sampling point 1199054
So the method above may decrease the performanceof the approximation capability on the boundary and eachmodel belonging to the model set cannot switch smoothly[18] However T-S fuzzy model as an intelligent controlmethod mainly uses fuzzy reasoning to approximate thenonlinear system Using this method the input space canbe divided into several fuzzy subspaces where a local linearmodel is established and then the local models are combinedsmoothly using the membership function forming a globalfuzzy model of nonlinear function which is ultimatelyidentified as a linear model [19] The predictive controlmethod based on the T-S fuzzy model belongs to multimodelpredictive control with the weighted models Comparedwith the common multimodel predictive controllers withweighted models the fuzzy weighted models have moreaccurate nonlinear approximation performance switch of
6 Mathematical Problems in Engineering
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emax
y(t)
p1 p2
Figure 4 The schematic for determining model of frontier pointsduring the predictive intervals
the model is more smooth and it is easier to understand[20 21] So in this section T-S fuzzy idea is adopted for themultimodel control of UAV formation flight as is shown inFigure 5
For each sampling point use the error between thetangent of state characteristic point and the reference tra-jectory of this sampling point to calculate the membershipdegree Assuming that the error between tangent of jthstate characteristic point and the reference trajectory at thesampling point t is 119864
119894(119905) so for the point 119905+ 119894 in the predictive
range weighted function is as follows
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(32)
This equation can ensure that the farther away from thestate characteristic point the sampling point is the lower itsweighted value is Using the weighted values the predictionmodel at the sampling point t is
119872119905=
sum
119898
119894=1(119908119894119872119894)
sum
119898
119894=1(119908119894)
(33)
For the sampling points during the predictive horizonthere is
119910119901 (
119905 + 119894) =
sum
119898
119895=1(119908119895 (119905 + 119894)119872119895
)
sum
119898
119895=1(119908119895 (119905 + 119894))
(34)
where
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(35)
Through this approach the linear prediction function forUAV formation can be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(36)
In this way predictive outputs can change from a non-linear function to a linear function This nonlinear functionincludes V
119908(119896) V119908(119896+1) V
119908(119896+119873minus1) and120596
119908(119896) 120596119908(119896+
1) 120596119908(119896 + 119873 minus 1) while this linear function includes the
control quantities mentioned above Thus the control prob-lem will become a multi-input-multioutput linear predictivecontrol problem
34 Optimization Index for Formation and Receding Opti-mization Solution During the predictive control process thegoal of receding optimization is to find a set of V
119908(119896) V119908(119896 +
1) V119908(119896+119873minus1) and 120596
119908(119896) 120596119908(119896+1) 120596
119908(119896+119873minus1)
making prediction outputs at entire optimization horizon asclose to the reference trajectory as possible
Here introduce the closed-loop
1198901 (
119905) = 119909119889 (
119905) minus 119909119889119901
(119905)
1198902 (
119905) = 119910119889 (
119905) minus 119910119889119901
(119905)
(37)
The open-loop predictive output can be directly compen-sated by the output feedback and then the predictive value ofthe closed-loop model will be
119909119889 (
119905 + 119894) = 119909119889119901
(119905 + 119894) + 1198901 (
119905)
119910119889 (
119905 + 119894) = 119910119889119901
(119905 + 119894) + 1198902 (
119905)
(38)
In this section there are two control objectives therelative distances to X-axis and Y-axis Since they have equalimportance and the same unit of quantity they are set withthe same weight when designing the performance indexThus the performance index is defined as follows
119869 =
119873
sum
119894=1
[(1199101119903 (
119905 + 119894) minus 119909119889 (
119905 + 119894))
2
+(1199102119903 (
119905 + 119894) minus 119910119889 (
119905 + 119894))
2]
+ 1205821
119873minus1
sum
119894=0
(V119908 (
119905 + 119894) minus V119908 (
119905))
2
+ 1205822
119873minus1
sum
119894=0
120596
2
119908(119905 + 119894)
(39)
Similarly the optimization constraints of control quantityare introduced as followsV119908 (
119896 + 119894 minus 1) minus ΔV lt V119908 (
119896 + 119894) lt V119908 (
119896 + 119894 minus 1) + ΔV
120596min lt 120596119908 (
119896 + 119894) lt 120596max
Vmin lt V119908 (
119896 + 119894) lt Vmax
where 119894 isin 0 1 2 119873 minus 1
(40)
Mathematical Problems in Engineering 7
Linearized prediction model 1
Linearized prediction model N
Ultimate prediction model
Prediction controller based on LQR
Referencetrajectoryof cruise
formation
Relative distance of cruise formationReference trajectory tangential error
Characteristicstatus points
LinearizeObtain characteristic point of
relative motion status for formation control
Generate relative motion model set of
formation
Tangential error of reference trajectory at sampling point and prediction point Weighted function
Relativedistance
Combinationof formation
model set
Multimodelprediction
control
Linearized prediction model 2
Weighted
Prediction Controloutputs
UAV attitude controlsystem
middot middot middot
Figure 5 The schematic of multimodel control method for UAV formation flight
After using multiple models the performance indexis linear quadratic whose constraints are linear equalityand inequality so the optimization problem is a linearquadratic programming problem The solution methods oflinear quadratic programming problem can be used to solvethe receding optimization problem The linear quadraticprogramming problem is a common programming problemand has a lot of solution methods and higher speed than theordinary nonlinear programming which increases the speedof receding optimization solution [22]
4 Simulation
In this section numerical simulations are performed todemonstrate the performance of the proposed approachHere the formation control ability can be tested in twoimportant scenarios Simulation scenarios are set as followsOne scenario is the leader UAV flying straight and the otheris the leader UAV flying with turning course Additionallythe comparison simulation between single MPC (SMPC)method and multiple MPC (MMPC) method is carried on toverify effectiveness of the method in this paper Meanwhilethe parameters used in the simulations are set as followsThe prediction horizon N is 5 and the sampling intervalis 02 s The angular velocity and velocity of two vehiclesare confined during the interval (minus01 01) and the interval(35 45) respectively All the computations and experimentshave been on a computer with Inter Core i3 CPU 330GHzand Windows XP operating systems Table 1 summarizes theinitial conditions of the formation
41 Formation Simulation of Leader UAV Flying Straight Thesimulation experiment is mainly used to verify the UAVformation control capability when the leader UAV is flyingstraight Here error exists in the position measurement of
Table 1 Initial conditions of UAV formation
Initial conditions The role of UAVLeader Follower
Initial position (0 0) (minus100 minus100)
Initial angle 0 1205872
Initial velocity 40 40Initial angular velocity 0 0
leader UAV which is plusmn05m There are two different controlgoals One is that the relative position between follower andleader ofUAV formation in the track coordinates is as follows
119883 = minus60
119884 = 30
(41)
The other is that the formation should be formed within 40 sBecause the leader UAV has its initial angle of 1205874 and
it flies straight the initial relative position in the trackcoordinates will be obtained as follows
[
119909119889
119910119889
] =
[
[
[
cos 1205874
sin 120587
4
minus sin 120587
4
cos 1205874
]
]
]
[
minus100
minus100
] = [
minus100radic2
0
] (42)
Simulation is carried out by using Matlab Simulink toolboxand the simulation results are shown from Figures 6 7 8 9and 10
According to Figures 6ndash10 it can be seen that whenleader UAV is navigating in a straight line formation controlcan be achieved through both SMPC and MMPC methodHowever the SMPC method has a larger tracking error thanthe MMPC method Meanwhile it can also be seen that ittakes a longer time for SMPCmethod thanMMPCmethod to
8 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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
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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
2 Mathematical Problems in Engineering
LeaderFollower
wX
Xg
YgO
Xl
Yl
Yw
Ow119896
120593w
120593l
rarrDW rarr
DL
rarrD
Ol119896
Figure 1 The position relationship between two vehicles
the UAVs may lead to collision thus it requires highercontrol accuracy However stair-like MPC uses the way ofconstraining the variation of the future control quantitywhich restricts themaneuverability of the vehicle and is proneto causing collision between the UAVs due to the overshootproblem [12] So it is necessary to adopt a new predictivecontrol method to achieve the formation flight control Forsomemore complex systems themultimodel controlmethodhas stronger robustness and higher control accuracy undercertain conditions [13] Besides multimodel control methodcan provide the nonlinear system with transparent modeland controller facilitating the system analysis Comparedwith other nonlinear global strategies themultimodel controlmethod cannot greatly reduce computational complexity butthe model and structure of controller are more suitable foronline adjustments and learning algorithm [14] so multiplemodel-based predictive control can be used to solve UAVformation control problem
This paper is organized as follows In Section 2 thediscrete relative motion equations for UAV formation areestablished In Section 3 a multiple models-based predictivecontrol approach is used to design controller of the formationSimulation results are given in Section 4 Finally Section 5concludes the paper
2 UAV Formation Flight Control Model
21 Kinematics Model of UAV Formation Flight ControlAssume that during the formation flight an UAV is flyinghorizontally and has no sideslip In the geographic coordinatesystem the relationship between the position vectors ofleader UAV (leader) and follower UAV (follower) is shownin Figure 1
From Figure 1 it is easy to obtain the following equation
119863119871=
119863119882
+
119863 (1)
where 119863119871 119863119882are displacement vectors of two vehicles and
119863 is the relative displacement vectors between two vehicles
Differentiating (1) one can obtain
119889
119863119871
119889119905
=
119889
119863119882
119889119905
+
119889
119863
119889119905
(2)
According to the relationship between the moving coor-dinate system one can easily obtain
119889
119863119871
119889119905
=
120575
119863119882
120575119905
+ 120596119908times
119863
(3)
where 120596119908is the yaw angular rate
Since the vehicle is supposed to fly horizontally theequation of motion will be
119894= V119894cos (120593
119894)
119910119894= V119894sin (120593
119894)
119894= 120596119894 119894 = 119897 119908
(4)
where the subscript 119897 and 119908 denote leader UAV and followerUAV respectively
Combing (3) and (4) the relative motion equation of twovehicles can be obtained as
V119897[
cos120593119897
sin120593119897
] = V119908[
cos120593119908
sin120593119908
] + 119862
119871
119908([
119889
119910119889
] + [
minus120596119908sdot 119910119889
120596119908sdot 119909119889
]) (5)
where 119909119889and 119910
119889are X-axis value and Y-axis value of the dis-
tance between two vehicles in the tack coordinates of leaderUAV respectively and 119862
119871
119908is the coordinate transformation
matrixThen carry on the translational process and one can
obtain
[
119889
119910119889
] = 119862
119882
119871(V119897[
cos120593119897
sin120593119897
] minus V119908[
cos120593119908
sin120593119908
]) minus [
minus120596119908sdot 119910119889
120596119908sdot 119909119889
]
(6)
where
119862
119882
119871V119897[
cos120593119897
sin120593119897
] = V119897[
cos120593119890
sin120593119890
]
119862
119882
119871V119908[
cos120593119908
sin120593119908
] = [
V119908
0
]
(7)
So (6) can be written as follows
[
119889
119910119889
] = V119897[
cos (120593119897minus 120593119908)
sin (120593119897minus 120593119908)
] minus [
V119908
0
] minus [
minus120596119908sdot 119910119889
120596119908sdot 119909119889
] (8)
Additionally there is
119908= 120596119908 (9)
Combing (8) and (9) we can obtain
119889= V119897cos (120593
119897minus 120593119908) minus V119908+ 120596119908sdot 119910119889
119910119889= V119897sin (120593
119897minus 120593119908) minus 120596119908sdot 119909119889
119908= 120596119908
(10)
Mathematical Problems in Engineering 3
Then (10) can also be written as the state equations whichis shown as follows
[
[
119889
119910119889
119908
]
]
=[
[
minus1 119910119889
0 minus119909119889
0 1
]
]
[
V119908
120596119908
] +[
[
V119897cos (120593
119897minus 120593119908)
V119897sin (120593
119897minus 120593119908)
0
]
]
(11)
The output equation is
1199101= 119909119889
1199102= 119910119889
(12)
22 Discrete Model of UAV Formation Movement and ItsPredictive Control Analysis In the previous section the stateequation of formation control is obtained Here discretizethis equation and the following equation can be obtained
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119908 (
119896 + 1)
]
]
=[
[
119909119889 (
119896)
119910119889 (
119896)
120593119908 (
119896)
]
]
+[
[
minus1 119910119889 (
119896)
0 minus119909119889 (
119896)
0 1
]
]
[
V119908 (
119896)
120596119908 (
119896)
] Δ119879
+[
[
V119897 (119896) cos (120593119897 (119896) minus 120593
119908 (119896))
V119897 (119896) sin (120593
119897 (119896) minus 120593
119908 (119896))
0
]
]
Δ119879
(13)
Generally the sampling periodic time is short duringreceding optimization process so the velocity and yaw angleof leader UAV can be considered constant in sampling period[15] which means in a short sampling period there is
V119897 (119896 + 119894) = V
119897 (119896 + 119894 minus 1) = V
119897 (119896)
120593119897 (119896 + 119894) = 120593
119897 (119896 + 119894 minus 1) = 120593
119897 (119896)
(14)
According to these two equations the predicted value ofoutputs will be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(15)
Since at time 119896 119910119889(119896) V119897(119896) 120593
119908(119896) 120593
119897(119896) are known the
future output values of formation flight control are 119909119889119901
(119896 +
1) 119909119889119901
(119896 + 2) 119909119889119901
(119896 + 119873 minus 1) 119909119889119901
(119896 + 119873) and 119910119889119901
(119896 +
1) 119910119889119901
(119896 + 2) 119910119889119901
(119896 + 119873 minus 1) 119910119889119901
(119896 + 119873) and theseoutputs are only the function of the future control quantitiesV119908(119896) V119908(119896 + 1) V
119908(119896 + 119873 minus 1) and 120596
119908(119896) 120596119908(119896 +
1) 120596119908(119896+119873minus1) Obviously this function is amulti-input-
multioutput nonlinear control problem So these values canbe obtained using the receding optimization algorithmThenusing V
119908(119896) and 120596
119908(119896) as outputs of control quantity and
carry out the receding optimization algorithm in sequencethe input of control quantity in the next time can be obtained
According to Section 1 it can be known that for theproblem of UAV formation flight nonlinearmodel predictivecontrol and fuzzy stair-like predictive control have somelimitations but the multimodel control method has strongerrobustness and higher control accuracy the final predictivemodel of which is linear so the receding optimization
problem can be changed fromgeneral nonlinear optimizationproblem to a linear quadratic optimization problem Sincethe linear quadratic optimization has faster computation thanthe ordinary nonlinear optimization the multiple modelscan greatly improve real-time of receding optimization [16]Therefore multiple model-based predictive control approachis used to design the controller of UAV formation flight
3 Predictive Control for UAV FormationBased on Multimodel Approach
The basic principle of the multimodel control method isthat nearby the different characteristics of nonlinear systemsdifferent linear model is used to describe this nonlinearsystem and each linear model only describes a part of non-linear system dynamics The multiple linearization modelsare used to approximate the nonlinear system in its entireoperating range and the controller is designed based on eachlinearizationmodelThese controllers are combined togetherto constitute amultimodel controller in someway Finally thecontrol of entire nonlinear system can be achieved throughcoordinated control between multiple linearization modelsThe basic steps of this method can be summarized in foursteps (1) acquisition of the model set (2) local linearizationof the model set (3) establishment of the controller set (4)combination of the model set
Similarly the controller design of UAV formation flightcan also include these steps the flow chart of which isshown in Figure 2 The details of controller design willbe presented in the following parts using the multimodelprediction method
31 Determination of the State Characteristic Points of Forma-tion Control Model According to the basic principles of mul-timodel predictive control the characteristic state points ofthe nonlinear model must be obtained first before obtainingmultiple models Characteristic state points and their regionare determined using methods in [15 17] As is shown inFigure 3 the horizontal axis and vertical axis denote time andtrajectory respectively
The basic idea is as followsDetermine the first characteristic state points then com-
pute the error between reference trajectory tangent throughthe characteristic point and reference trajectory and comparethe error with the maximum permissible error If it is greaterthan the maximum permissible error redetermine the char-acteristic point to get the next characteristic state point andcalculate repeatedly until the last characteristic state pointis obtained thus the characteristic state points of nonlinearsystem and their applicable region can be determined Sothe linearization model set can be obtained through thelinearization process at the characteristic state point for eachregion
ForUAV formation control the characteristic state pointsare determined as follows
Assuming that the initial distance between two vehiclesare 119909119889(0) and 119910
119889(0) respectively when UAV formation con-
trol is carried out so the desired distances of UAV formation
4 Mathematical Problems in Engineering
Determining state characteristic points
for formation control
Generating discrete model sets of
formation control
Combining model sets of formation
control
Introduceoptimization index and solve the final
model
Figure 2 The flow chart of UAV formation flight controller design
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emaxy(t)
Figure 3 Description of tangent error
are 119909119888and 119910
119888 Assuming that the expected time arrival at the
desired value of X-axis and Y-axis is the same the referencetrajectory of UAV formation can be obtained as follows
1199101119903 (
119905) = 120585
119905119909119889 (
0) + (1 minus 120585
119905) sdot 119909119888
1199102119903 (
119905) = 120585
119905119910119889 (
0) + (1 minus 120585
119905) sdot 119910119888
(16)
Since there are two outputs generating two referencetrajectories the algorithm above cannot be applied directlyBut because the expected arrival time to the desired value isthe same the reference trajectory of an output can be used todetermine the characteristic point of state Assume that thereference trajectory of relative position on the X-axis in thetrack coordinate system is used to determine characteristicpoint of the state
At the characteristic state points there is
119889= 0 119910
119889= 0
119908= 0 (17)
Assume that ith characteristic state point is (119881119894119908 120596
119894
119908 119909
119894
119889
119910
119894
119889 120593
119894
119889) then there will be the following equation
[
[
[
[
minus1 119910
119894
119889
0 minus119909
119894
119889
0 1
]
]
]
]
[
[
119881
119894
119908
120596
119894
119908
]
]
+
[
[
[
[
119881119897cos (120593
119897minus 120593
119894
119908)
119881119897sin (120593
119897minus 120593
119894
119908)
0
]
]
]
]
=[
[
0
0
0
]
]
(18)
Meanwhile there is
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894) (19)
Solve (18) and (19) then we can obtain
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894)
(20)
Assuming the initial point is as the first characteristic statepoint then the state point value will be
120596
1
119908= 0 120593
1
119908= 120593119897 V1
119908= V119897
119909
1
119889= 1199101119903 (
0) 119910
1
119889= 1199102119903 (
0)
(21)
Then the equation of the tangent is
1199101119896 (
119905) = 1199101119903 (
119905) 119905 + 119909119889 (
0) (22)
Afterward determine the maximum permissible error119864max and the time 119905
2corresponding to the second charac-
teristic point can be obtained using the method above Thenthe state point can be obtained as follows
120596
2
119908= 0 120593
2
119908= 120593119897 V2
119908= V119897
119909
2
119889= 119909119889(1199052) 119910
2
119889= 119910119889(1199052)
(23)
So the tangent equation is
1199101119896
(119905) = 1199101119903(119905) (119905 minus 119905
2) + 119909119889(1199052) (24)
By repeating these procedures above time 119905119894correspond-
ing to the characteristic point in the region of multimodelfor formation can be obtained and at the same time thecharacteristic state point corresponding to the time 119905
119894can be
also obtained
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(25)
Calculate until the last characteristic state point isobtained and then the computation will be terminated
32 Generation of Discrete Model Sets for Formation ControlAfter obtaining the characteristic state points carry onlinearization at different discrete model sets of formationHere linearization can be realized through the followingmethods
Consider nonlinear systems as described in the form ofdiscrete-time dynamic equations
119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896))
119910 (119896) = 119892 (119909 (119896) 119906 (119896))
(26)
Mathematical Problems in Engineering 5
The system has m different characteristic state points119891(119909(119896) 119906(119896)) and 119892(119909(119896) 119906(119896)) have the first continuouspartial derivative If system is linearized at each characteristicstate point the standard discrete state-space model of mlinear models of the original system is obtained as follows
119909 (119896 + 1) = 119860119894119909 (119896) + 119861
119894119906 (119896) minus 120572
119894
119910 (119896) = 119862119894119909 (119896) + 119863
119894119906 (119896) minus 120573
119894
(27)
where
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119862119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119863119894=
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894
120572119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119910119894
(28)
Here m linearized models constitute the linearized multi-model presentation of the original system
For the UAV formation flying control the characteristicstate points are shown as
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(29)
And it has the following expression
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
minus1 119910119889(119905119894)
0 minus119909119889(119905119894)
0 1
]
]
Δ119879
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894=[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(30)
Since outputs are linear 119862119894 119863119894and 120573
119894will not be solved
Thus the linearization equation at characteristic state pointwill be
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119889 (
119896 + 1)
]
]
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
[
[
119909119889 (
119896)
119910119889 (
119896)
120593119889 (
119896)
]
]
+[
[
minus1 119910119889(119905119894)
0 119909119889(119905119894)
0 1
]
]
Δ119879[
V119908 (
119896)
120596119908 (
119896)
] minus[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(31)
So for different characteristic state points the linearmodel for UAV formation control at different horizonscan be obtained realizing the acquisition of model setsfor UAV formation And these models are denoted as119872(1)119872(2) 119872(119878)
33 Combined Method of Model Sets for Formation FlightControl Thecharacteristic state points are obtained using themethod above When the error reaches the maximum valueswitch to the new model which ensures the maximum errorvalue between the predictive trajectory and the referencetrajectory Thus the determination of model region can berealized It can be seen that the applicable range of desiredmodel is divided based on the time region so differentsampling points have different models But the predictivecontrol is based on the future time region So in this paperthe applicable model of predictive point is judged by the timeregion and then this model is used to calculate the predictivevalue The judgment rules of predictive model are describedas follows
Assuming that the time corresponding to the state char-acteristic points is 119905
1lt 1199052lt sdot sdot sdot lt 119905
119904minus1lt 119905119904and the predictive
horizon is [119905 119905 + 119873] then there will be the followingIf [119905 119905 + 119873] isin [119905
119894 119905119894+1
] then the final predictive model ofall points is 119872
119879= 119872119894 if [119905 119905 + 119873] isin [119905
119894 119905119894+119873
] one can judgethe interval [119905
ℎ 119905ℎ+1
] of all points between prediction point119905 + 1 to 119905 + 119873 in sequence and denote the model of this pointas119872119879= 119872ℎ if [119905 119905 +119873] isin [119905
119904infin) the final predictive model
of all the points will be119872119879= 119872119904
Based on this method we can obtain the predictive func-tion during the future horizon to determine the optimizationindex However the boundary point of the predictive rangemay be closer to the next linear model as shown in Figure 4Sampling point 119901
1may be closer to the linear model at
sampling point 1199054 But according to the method above the
calculation model used at the sampling point 1199011is the linear
model of the state characteristic point at sampling point1199053 Sampling point 119901
2may be closer to the linear model
of the state characteristic point at sampling point 1199055 But
according to the method above the calculation model usedby the sampling point 119901
2is the linear model of the state
characteristic point at sampling point 1199054
So the method above may decrease the performanceof the approximation capability on the boundary and eachmodel belonging to the model set cannot switch smoothly[18] However T-S fuzzy model as an intelligent controlmethod mainly uses fuzzy reasoning to approximate thenonlinear system Using this method the input space canbe divided into several fuzzy subspaces where a local linearmodel is established and then the local models are combinedsmoothly using the membership function forming a globalfuzzy model of nonlinear function which is ultimatelyidentified as a linear model [19] The predictive controlmethod based on the T-S fuzzy model belongs to multimodelpredictive control with the weighted models Comparedwith the common multimodel predictive controllers withweighted models the fuzzy weighted models have moreaccurate nonlinear approximation performance switch of
6 Mathematical Problems in Engineering
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emax
y(t)
p1 p2
Figure 4 The schematic for determining model of frontier pointsduring the predictive intervals
the model is more smooth and it is easier to understand[20 21] So in this section T-S fuzzy idea is adopted for themultimodel control of UAV formation flight as is shown inFigure 5
For each sampling point use the error between thetangent of state characteristic point and the reference tra-jectory of this sampling point to calculate the membershipdegree Assuming that the error between tangent of jthstate characteristic point and the reference trajectory at thesampling point t is 119864
119894(119905) so for the point 119905+ 119894 in the predictive
range weighted function is as follows
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(32)
This equation can ensure that the farther away from thestate characteristic point the sampling point is the lower itsweighted value is Using the weighted values the predictionmodel at the sampling point t is
119872119905=
sum
119898
119894=1(119908119894119872119894)
sum
119898
119894=1(119908119894)
(33)
For the sampling points during the predictive horizonthere is
119910119901 (
119905 + 119894) =
sum
119898
119895=1(119908119895 (119905 + 119894)119872119895
)
sum
119898
119895=1(119908119895 (119905 + 119894))
(34)
where
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(35)
Through this approach the linear prediction function forUAV formation can be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(36)
In this way predictive outputs can change from a non-linear function to a linear function This nonlinear functionincludes V
119908(119896) V119908(119896+1) V
119908(119896+119873minus1) and120596
119908(119896) 120596119908(119896+
1) 120596119908(119896 + 119873 minus 1) while this linear function includes the
control quantities mentioned above Thus the control prob-lem will become a multi-input-multioutput linear predictivecontrol problem
34 Optimization Index for Formation and Receding Opti-mization Solution During the predictive control process thegoal of receding optimization is to find a set of V
119908(119896) V119908(119896 +
1) V119908(119896+119873minus1) and 120596
119908(119896) 120596119908(119896+1) 120596
119908(119896+119873minus1)
making prediction outputs at entire optimization horizon asclose to the reference trajectory as possible
Here introduce the closed-loop
1198901 (
119905) = 119909119889 (
119905) minus 119909119889119901
(119905)
1198902 (
119905) = 119910119889 (
119905) minus 119910119889119901
(119905)
(37)
The open-loop predictive output can be directly compen-sated by the output feedback and then the predictive value ofthe closed-loop model will be
119909119889 (
119905 + 119894) = 119909119889119901
(119905 + 119894) + 1198901 (
119905)
119910119889 (
119905 + 119894) = 119910119889119901
(119905 + 119894) + 1198902 (
119905)
(38)
In this section there are two control objectives therelative distances to X-axis and Y-axis Since they have equalimportance and the same unit of quantity they are set withthe same weight when designing the performance indexThus the performance index is defined as follows
119869 =
119873
sum
119894=1
[(1199101119903 (
119905 + 119894) minus 119909119889 (
119905 + 119894))
2
+(1199102119903 (
119905 + 119894) minus 119910119889 (
119905 + 119894))
2]
+ 1205821
119873minus1
sum
119894=0
(V119908 (
119905 + 119894) minus V119908 (
119905))
2
+ 1205822
119873minus1
sum
119894=0
120596
2
119908(119905 + 119894)
(39)
Similarly the optimization constraints of control quantityare introduced as followsV119908 (
119896 + 119894 minus 1) minus ΔV lt V119908 (
119896 + 119894) lt V119908 (
119896 + 119894 minus 1) + ΔV
120596min lt 120596119908 (
119896 + 119894) lt 120596max
Vmin lt V119908 (
119896 + 119894) lt Vmax
where 119894 isin 0 1 2 119873 minus 1
(40)
Mathematical Problems in Engineering 7
Linearized prediction model 1
Linearized prediction model N
Ultimate prediction model
Prediction controller based on LQR
Referencetrajectoryof cruise
formation
Relative distance of cruise formationReference trajectory tangential error
Characteristicstatus points
LinearizeObtain characteristic point of
relative motion status for formation control
Generate relative motion model set of
formation
Tangential error of reference trajectory at sampling point and prediction point Weighted function
Relativedistance
Combinationof formation
model set
Multimodelprediction
control
Linearized prediction model 2
Weighted
Prediction Controloutputs
UAV attitude controlsystem
middot middot middot
Figure 5 The schematic of multimodel control method for UAV formation flight
After using multiple models the performance indexis linear quadratic whose constraints are linear equalityand inequality so the optimization problem is a linearquadratic programming problem The solution methods oflinear quadratic programming problem can be used to solvethe receding optimization problem The linear quadraticprogramming problem is a common programming problemand has a lot of solution methods and higher speed than theordinary nonlinear programming which increases the speedof receding optimization solution [22]
4 Simulation
In this section numerical simulations are performed todemonstrate the performance of the proposed approachHere the formation control ability can be tested in twoimportant scenarios Simulation scenarios are set as followsOne scenario is the leader UAV flying straight and the otheris the leader UAV flying with turning course Additionallythe comparison simulation between single MPC (SMPC)method and multiple MPC (MMPC) method is carried on toverify effectiveness of the method in this paper Meanwhilethe parameters used in the simulations are set as followsThe prediction horizon N is 5 and the sampling intervalis 02 s The angular velocity and velocity of two vehiclesare confined during the interval (minus01 01) and the interval(35 45) respectively All the computations and experimentshave been on a computer with Inter Core i3 CPU 330GHzand Windows XP operating systems Table 1 summarizes theinitial conditions of the formation
41 Formation Simulation of Leader UAV Flying Straight Thesimulation experiment is mainly used to verify the UAVformation control capability when the leader UAV is flyingstraight Here error exists in the position measurement of
Table 1 Initial conditions of UAV formation
Initial conditions The role of UAVLeader Follower
Initial position (0 0) (minus100 minus100)
Initial angle 0 1205872
Initial velocity 40 40Initial angular velocity 0 0
leader UAV which is plusmn05m There are two different controlgoals One is that the relative position between follower andleader ofUAV formation in the track coordinates is as follows
119883 = minus60
119884 = 30
(41)
The other is that the formation should be formed within 40 sBecause the leader UAV has its initial angle of 1205874 and
it flies straight the initial relative position in the trackcoordinates will be obtained as follows
[
119909119889
119910119889
] =
[
[
[
cos 1205874
sin 120587
4
minus sin 120587
4
cos 1205874
]
]
]
[
minus100
minus100
] = [
minus100radic2
0
] (42)
Simulation is carried out by using Matlab Simulink toolboxand the simulation results are shown from Figures 6 7 8 9and 10
According to Figures 6ndash10 it can be seen that whenleader UAV is navigating in a straight line formation controlcan be achieved through both SMPC and MMPC methodHowever the SMPC method has a larger tracking error thanthe MMPC method Meanwhile it can also be seen that ittakes a longer time for SMPCmethod thanMMPCmethod to
8 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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
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
Mathematical Problems in Engineering 3
Then (10) can also be written as the state equations whichis shown as follows
[
[
119889
119910119889
119908
]
]
=[
[
minus1 119910119889
0 minus119909119889
0 1
]
]
[
V119908
120596119908
] +[
[
V119897cos (120593
119897minus 120593119908)
V119897sin (120593
119897minus 120593119908)
0
]
]
(11)
The output equation is
1199101= 119909119889
1199102= 119910119889
(12)
22 Discrete Model of UAV Formation Movement and ItsPredictive Control Analysis In the previous section the stateequation of formation control is obtained Here discretizethis equation and the following equation can be obtained
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119908 (
119896 + 1)
]
]
=[
[
119909119889 (
119896)
119910119889 (
119896)
120593119908 (
119896)
]
]
+[
[
minus1 119910119889 (
119896)
0 minus119909119889 (
119896)
0 1
]
]
[
V119908 (
119896)
120596119908 (
119896)
] Δ119879
+[
[
V119897 (119896) cos (120593119897 (119896) minus 120593
119908 (119896))
V119897 (119896) sin (120593
119897 (119896) minus 120593
119908 (119896))
0
]
]
Δ119879
(13)
Generally the sampling periodic time is short duringreceding optimization process so the velocity and yaw angleof leader UAV can be considered constant in sampling period[15] which means in a short sampling period there is
V119897 (119896 + 119894) = V
119897 (119896 + 119894 minus 1) = V
119897 (119896)
120593119897 (119896 + 119894) = 120593
119897 (119896 + 119894 minus 1) = 120593
119897 (119896)
(14)
According to these two equations the predicted value ofoutputs will be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(15)
Since at time 119896 119910119889(119896) V119897(119896) 120593
119908(119896) 120593
119897(119896) are known the
future output values of formation flight control are 119909119889119901
(119896 +
1) 119909119889119901
(119896 + 2) 119909119889119901
(119896 + 119873 minus 1) 119909119889119901
(119896 + 119873) and 119910119889119901
(119896 +
1) 119910119889119901
(119896 + 2) 119910119889119901
(119896 + 119873 minus 1) 119910119889119901
(119896 + 119873) and theseoutputs are only the function of the future control quantitiesV119908(119896) V119908(119896 + 1) V
119908(119896 + 119873 minus 1) and 120596
119908(119896) 120596119908(119896 +
1) 120596119908(119896+119873minus1) Obviously this function is amulti-input-
multioutput nonlinear control problem So these values canbe obtained using the receding optimization algorithmThenusing V
119908(119896) and 120596
119908(119896) as outputs of control quantity and
carry out the receding optimization algorithm in sequencethe input of control quantity in the next time can be obtained
According to Section 1 it can be known that for theproblem of UAV formation flight nonlinearmodel predictivecontrol and fuzzy stair-like predictive control have somelimitations but the multimodel control method has strongerrobustness and higher control accuracy the final predictivemodel of which is linear so the receding optimization
problem can be changed fromgeneral nonlinear optimizationproblem to a linear quadratic optimization problem Sincethe linear quadratic optimization has faster computation thanthe ordinary nonlinear optimization the multiple modelscan greatly improve real-time of receding optimization [16]Therefore multiple model-based predictive control approachis used to design the controller of UAV formation flight
3 Predictive Control for UAV FormationBased on Multimodel Approach
The basic principle of the multimodel control method isthat nearby the different characteristics of nonlinear systemsdifferent linear model is used to describe this nonlinearsystem and each linear model only describes a part of non-linear system dynamics The multiple linearization modelsare used to approximate the nonlinear system in its entireoperating range and the controller is designed based on eachlinearizationmodelThese controllers are combined togetherto constitute amultimodel controller in someway Finally thecontrol of entire nonlinear system can be achieved throughcoordinated control between multiple linearization modelsThe basic steps of this method can be summarized in foursteps (1) acquisition of the model set (2) local linearizationof the model set (3) establishment of the controller set (4)combination of the model set
Similarly the controller design of UAV formation flightcan also include these steps the flow chart of which isshown in Figure 2 The details of controller design willbe presented in the following parts using the multimodelprediction method
31 Determination of the State Characteristic Points of Forma-tion Control Model According to the basic principles of mul-timodel predictive control the characteristic state points ofthe nonlinear model must be obtained first before obtainingmultiple models Characteristic state points and their regionare determined using methods in [15 17] As is shown inFigure 3 the horizontal axis and vertical axis denote time andtrajectory respectively
The basic idea is as followsDetermine the first characteristic state points then com-
pute the error between reference trajectory tangent throughthe characteristic point and reference trajectory and comparethe error with the maximum permissible error If it is greaterthan the maximum permissible error redetermine the char-acteristic point to get the next characteristic state point andcalculate repeatedly until the last characteristic state pointis obtained thus the characteristic state points of nonlinearsystem and their applicable region can be determined Sothe linearization model set can be obtained through thelinearization process at the characteristic state point for eachregion
ForUAV formation control the characteristic state pointsare determined as follows
Assuming that the initial distance between two vehiclesare 119909119889(0) and 119910
119889(0) respectively when UAV formation con-
trol is carried out so the desired distances of UAV formation
4 Mathematical Problems in Engineering
Determining state characteristic points
for formation control
Generating discrete model sets of
formation control
Combining model sets of formation
control
Introduceoptimization index and solve the final
model
Figure 2 The flow chart of UAV formation flight controller design
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emaxy(t)
Figure 3 Description of tangent error
are 119909119888and 119910
119888 Assuming that the expected time arrival at the
desired value of X-axis and Y-axis is the same the referencetrajectory of UAV formation can be obtained as follows
1199101119903 (
119905) = 120585
119905119909119889 (
0) + (1 minus 120585
119905) sdot 119909119888
1199102119903 (
119905) = 120585
119905119910119889 (
0) + (1 minus 120585
119905) sdot 119910119888
(16)
Since there are two outputs generating two referencetrajectories the algorithm above cannot be applied directlyBut because the expected arrival time to the desired value isthe same the reference trajectory of an output can be used todetermine the characteristic point of state Assume that thereference trajectory of relative position on the X-axis in thetrack coordinate system is used to determine characteristicpoint of the state
At the characteristic state points there is
119889= 0 119910
119889= 0
119908= 0 (17)
Assume that ith characteristic state point is (119881119894119908 120596
119894
119908 119909
119894
119889
119910
119894
119889 120593
119894
119889) then there will be the following equation
[
[
[
[
minus1 119910
119894
119889
0 minus119909
119894
119889
0 1
]
]
]
]
[
[
119881
119894
119908
120596
119894
119908
]
]
+
[
[
[
[
119881119897cos (120593
119897minus 120593
119894
119908)
119881119897sin (120593
119897minus 120593
119894
119908)
0
]
]
]
]
=[
[
0
0
0
]
]
(18)
Meanwhile there is
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894) (19)
Solve (18) and (19) then we can obtain
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894)
(20)
Assuming the initial point is as the first characteristic statepoint then the state point value will be
120596
1
119908= 0 120593
1
119908= 120593119897 V1
119908= V119897
119909
1
119889= 1199101119903 (
0) 119910
1
119889= 1199102119903 (
0)
(21)
Then the equation of the tangent is
1199101119896 (
119905) = 1199101119903 (
119905) 119905 + 119909119889 (
0) (22)
Afterward determine the maximum permissible error119864max and the time 119905
2corresponding to the second charac-
teristic point can be obtained using the method above Thenthe state point can be obtained as follows
120596
2
119908= 0 120593
2
119908= 120593119897 V2
119908= V119897
119909
2
119889= 119909119889(1199052) 119910
2
119889= 119910119889(1199052)
(23)
So the tangent equation is
1199101119896
(119905) = 1199101119903(119905) (119905 minus 119905
2) + 119909119889(1199052) (24)
By repeating these procedures above time 119905119894correspond-
ing to the characteristic point in the region of multimodelfor formation can be obtained and at the same time thecharacteristic state point corresponding to the time 119905
119894can be
also obtained
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(25)
Calculate until the last characteristic state point isobtained and then the computation will be terminated
32 Generation of Discrete Model Sets for Formation ControlAfter obtaining the characteristic state points carry onlinearization at different discrete model sets of formationHere linearization can be realized through the followingmethods
Consider nonlinear systems as described in the form ofdiscrete-time dynamic equations
119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896))
119910 (119896) = 119892 (119909 (119896) 119906 (119896))
(26)
Mathematical Problems in Engineering 5
The system has m different characteristic state points119891(119909(119896) 119906(119896)) and 119892(119909(119896) 119906(119896)) have the first continuouspartial derivative If system is linearized at each characteristicstate point the standard discrete state-space model of mlinear models of the original system is obtained as follows
119909 (119896 + 1) = 119860119894119909 (119896) + 119861
119894119906 (119896) minus 120572
119894
119910 (119896) = 119862119894119909 (119896) + 119863
119894119906 (119896) minus 120573
119894
(27)
where
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119862119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119863119894=
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894
120572119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119910119894
(28)
Here m linearized models constitute the linearized multi-model presentation of the original system
For the UAV formation flying control the characteristicstate points are shown as
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(29)
And it has the following expression
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
minus1 119910119889(119905119894)
0 minus119909119889(119905119894)
0 1
]
]
Δ119879
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894=[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(30)
Since outputs are linear 119862119894 119863119894and 120573
119894will not be solved
Thus the linearization equation at characteristic state pointwill be
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119889 (
119896 + 1)
]
]
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
[
[
119909119889 (
119896)
119910119889 (
119896)
120593119889 (
119896)
]
]
+[
[
minus1 119910119889(119905119894)
0 119909119889(119905119894)
0 1
]
]
Δ119879[
V119908 (
119896)
120596119908 (
119896)
] minus[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(31)
So for different characteristic state points the linearmodel for UAV formation control at different horizonscan be obtained realizing the acquisition of model setsfor UAV formation And these models are denoted as119872(1)119872(2) 119872(119878)
33 Combined Method of Model Sets for Formation FlightControl Thecharacteristic state points are obtained using themethod above When the error reaches the maximum valueswitch to the new model which ensures the maximum errorvalue between the predictive trajectory and the referencetrajectory Thus the determination of model region can berealized It can be seen that the applicable range of desiredmodel is divided based on the time region so differentsampling points have different models But the predictivecontrol is based on the future time region So in this paperthe applicable model of predictive point is judged by the timeregion and then this model is used to calculate the predictivevalue The judgment rules of predictive model are describedas follows
Assuming that the time corresponding to the state char-acteristic points is 119905
1lt 1199052lt sdot sdot sdot lt 119905
119904minus1lt 119905119904and the predictive
horizon is [119905 119905 + 119873] then there will be the followingIf [119905 119905 + 119873] isin [119905
119894 119905119894+1
] then the final predictive model ofall points is 119872
119879= 119872119894 if [119905 119905 + 119873] isin [119905
119894 119905119894+119873
] one can judgethe interval [119905
ℎ 119905ℎ+1
] of all points between prediction point119905 + 1 to 119905 + 119873 in sequence and denote the model of this pointas119872119879= 119872ℎ if [119905 119905 +119873] isin [119905
119904infin) the final predictive model
of all the points will be119872119879= 119872119904
Based on this method we can obtain the predictive func-tion during the future horizon to determine the optimizationindex However the boundary point of the predictive rangemay be closer to the next linear model as shown in Figure 4Sampling point 119901
1may be closer to the linear model at
sampling point 1199054 But according to the method above the
calculation model used at the sampling point 1199011is the linear
model of the state characteristic point at sampling point1199053 Sampling point 119901
2may be closer to the linear model
of the state characteristic point at sampling point 1199055 But
according to the method above the calculation model usedby the sampling point 119901
2is the linear model of the state
characteristic point at sampling point 1199054
So the method above may decrease the performanceof the approximation capability on the boundary and eachmodel belonging to the model set cannot switch smoothly[18] However T-S fuzzy model as an intelligent controlmethod mainly uses fuzzy reasoning to approximate thenonlinear system Using this method the input space canbe divided into several fuzzy subspaces where a local linearmodel is established and then the local models are combinedsmoothly using the membership function forming a globalfuzzy model of nonlinear function which is ultimatelyidentified as a linear model [19] The predictive controlmethod based on the T-S fuzzy model belongs to multimodelpredictive control with the weighted models Comparedwith the common multimodel predictive controllers withweighted models the fuzzy weighted models have moreaccurate nonlinear approximation performance switch of
6 Mathematical Problems in Engineering
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emax
y(t)
p1 p2
Figure 4 The schematic for determining model of frontier pointsduring the predictive intervals
the model is more smooth and it is easier to understand[20 21] So in this section T-S fuzzy idea is adopted for themultimodel control of UAV formation flight as is shown inFigure 5
For each sampling point use the error between thetangent of state characteristic point and the reference tra-jectory of this sampling point to calculate the membershipdegree Assuming that the error between tangent of jthstate characteristic point and the reference trajectory at thesampling point t is 119864
119894(119905) so for the point 119905+ 119894 in the predictive
range weighted function is as follows
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(32)
This equation can ensure that the farther away from thestate characteristic point the sampling point is the lower itsweighted value is Using the weighted values the predictionmodel at the sampling point t is
119872119905=
sum
119898
119894=1(119908119894119872119894)
sum
119898
119894=1(119908119894)
(33)
For the sampling points during the predictive horizonthere is
119910119901 (
119905 + 119894) =
sum
119898
119895=1(119908119895 (119905 + 119894)119872119895
)
sum
119898
119895=1(119908119895 (119905 + 119894))
(34)
where
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(35)
Through this approach the linear prediction function forUAV formation can be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(36)
In this way predictive outputs can change from a non-linear function to a linear function This nonlinear functionincludes V
119908(119896) V119908(119896+1) V
119908(119896+119873minus1) and120596
119908(119896) 120596119908(119896+
1) 120596119908(119896 + 119873 minus 1) while this linear function includes the
control quantities mentioned above Thus the control prob-lem will become a multi-input-multioutput linear predictivecontrol problem
34 Optimization Index for Formation and Receding Opti-mization Solution During the predictive control process thegoal of receding optimization is to find a set of V
119908(119896) V119908(119896 +
1) V119908(119896+119873minus1) and 120596
119908(119896) 120596119908(119896+1) 120596
119908(119896+119873minus1)
making prediction outputs at entire optimization horizon asclose to the reference trajectory as possible
Here introduce the closed-loop
1198901 (
119905) = 119909119889 (
119905) minus 119909119889119901
(119905)
1198902 (
119905) = 119910119889 (
119905) minus 119910119889119901
(119905)
(37)
The open-loop predictive output can be directly compen-sated by the output feedback and then the predictive value ofthe closed-loop model will be
119909119889 (
119905 + 119894) = 119909119889119901
(119905 + 119894) + 1198901 (
119905)
119910119889 (
119905 + 119894) = 119910119889119901
(119905 + 119894) + 1198902 (
119905)
(38)
In this section there are two control objectives therelative distances to X-axis and Y-axis Since they have equalimportance and the same unit of quantity they are set withthe same weight when designing the performance indexThus the performance index is defined as follows
119869 =
119873
sum
119894=1
[(1199101119903 (
119905 + 119894) minus 119909119889 (
119905 + 119894))
2
+(1199102119903 (
119905 + 119894) minus 119910119889 (
119905 + 119894))
2]
+ 1205821
119873minus1
sum
119894=0
(V119908 (
119905 + 119894) minus V119908 (
119905))
2
+ 1205822
119873minus1
sum
119894=0
120596
2
119908(119905 + 119894)
(39)
Similarly the optimization constraints of control quantityare introduced as followsV119908 (
119896 + 119894 minus 1) minus ΔV lt V119908 (
119896 + 119894) lt V119908 (
119896 + 119894 minus 1) + ΔV
120596min lt 120596119908 (
119896 + 119894) lt 120596max
Vmin lt V119908 (
119896 + 119894) lt Vmax
where 119894 isin 0 1 2 119873 minus 1
(40)
Mathematical Problems in Engineering 7
Linearized prediction model 1
Linearized prediction model N
Ultimate prediction model
Prediction controller based on LQR
Referencetrajectoryof cruise
formation
Relative distance of cruise formationReference trajectory tangential error
Characteristicstatus points
LinearizeObtain characteristic point of
relative motion status for formation control
Generate relative motion model set of
formation
Tangential error of reference trajectory at sampling point and prediction point Weighted function
Relativedistance
Combinationof formation
model set
Multimodelprediction
control
Linearized prediction model 2
Weighted
Prediction Controloutputs
UAV attitude controlsystem
middot middot middot
Figure 5 The schematic of multimodel control method for UAV formation flight
After using multiple models the performance indexis linear quadratic whose constraints are linear equalityand inequality so the optimization problem is a linearquadratic programming problem The solution methods oflinear quadratic programming problem can be used to solvethe receding optimization problem The linear quadraticprogramming problem is a common programming problemand has a lot of solution methods and higher speed than theordinary nonlinear programming which increases the speedof receding optimization solution [22]
4 Simulation
In this section numerical simulations are performed todemonstrate the performance of the proposed approachHere the formation control ability can be tested in twoimportant scenarios Simulation scenarios are set as followsOne scenario is the leader UAV flying straight and the otheris the leader UAV flying with turning course Additionallythe comparison simulation between single MPC (SMPC)method and multiple MPC (MMPC) method is carried on toverify effectiveness of the method in this paper Meanwhilethe parameters used in the simulations are set as followsThe prediction horizon N is 5 and the sampling intervalis 02 s The angular velocity and velocity of two vehiclesare confined during the interval (minus01 01) and the interval(35 45) respectively All the computations and experimentshave been on a computer with Inter Core i3 CPU 330GHzand Windows XP operating systems Table 1 summarizes theinitial conditions of the formation
41 Formation Simulation of Leader UAV Flying Straight Thesimulation experiment is mainly used to verify the UAVformation control capability when the leader UAV is flyingstraight Here error exists in the position measurement of
Table 1 Initial conditions of UAV formation
Initial conditions The role of UAVLeader Follower
Initial position (0 0) (minus100 minus100)
Initial angle 0 1205872
Initial velocity 40 40Initial angular velocity 0 0
leader UAV which is plusmn05m There are two different controlgoals One is that the relative position between follower andleader ofUAV formation in the track coordinates is as follows
119883 = minus60
119884 = 30
(41)
The other is that the formation should be formed within 40 sBecause the leader UAV has its initial angle of 1205874 and
it flies straight the initial relative position in the trackcoordinates will be obtained as follows
[
119909119889
119910119889
] =
[
[
[
cos 1205874
sin 120587
4
minus sin 120587
4
cos 1205874
]
]
]
[
minus100
minus100
] = [
minus100radic2
0
] (42)
Simulation is carried out by using Matlab Simulink toolboxand the simulation results are shown from Figures 6 7 8 9and 10
According to Figures 6ndash10 it can be seen that whenleader UAV is navigating in a straight line formation controlcan be achieved through both SMPC and MMPC methodHowever the SMPC method has a larger tracking error thanthe MMPC method Meanwhile it can also be seen that ittakes a longer time for SMPCmethod thanMMPCmethod to
8 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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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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
4 Mathematical Problems in Engineering
Determining state characteristic points
for formation control
Generating discrete model sets of
formation control
Combining model sets of formation
control
Introduceoptimization index and solve the final
model
Figure 2 The flow chart of UAV formation flight controller design
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emaxy(t)
Figure 3 Description of tangent error
are 119909119888and 119910
119888 Assuming that the expected time arrival at the
desired value of X-axis and Y-axis is the same the referencetrajectory of UAV formation can be obtained as follows
1199101119903 (
119905) = 120585
119905119909119889 (
0) + (1 minus 120585
119905) sdot 119909119888
1199102119903 (
119905) = 120585
119905119910119889 (
0) + (1 minus 120585
119905) sdot 119910119888
(16)
Since there are two outputs generating two referencetrajectories the algorithm above cannot be applied directlyBut because the expected arrival time to the desired value isthe same the reference trajectory of an output can be used todetermine the characteristic point of state Assume that thereference trajectory of relative position on the X-axis in thetrack coordinate system is used to determine characteristicpoint of the state
At the characteristic state points there is
119889= 0 119910
119889= 0
119908= 0 (17)
Assume that ith characteristic state point is (119881119894119908 120596
119894
119908 119909
119894
119889
119910
119894
119889 120593
119894
119889) then there will be the following equation
[
[
[
[
minus1 119910
119894
119889
0 minus119909
119894
119889
0 1
]
]
]
]
[
[
119881
119894
119908
120596
119894
119908
]
]
+
[
[
[
[
119881119897cos (120593
119897minus 120593
119894
119908)
119881119897sin (120593
119897minus 120593
119894
119908)
0
]
]
]
]
=[
[
0
0
0
]
]
(18)
Meanwhile there is
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894) (19)
Solve (18) and (19) then we can obtain
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 1199101119903 (
119894) 119910
119894
119889= 1199102119903 (
119894)
(20)
Assuming the initial point is as the first characteristic statepoint then the state point value will be
120596
1
119908= 0 120593
1
119908= 120593119897 V1
119908= V119897
119909
1
119889= 1199101119903 (
0) 119910
1
119889= 1199102119903 (
0)
(21)
Then the equation of the tangent is
1199101119896 (
119905) = 1199101119903 (
119905) 119905 + 119909119889 (
0) (22)
Afterward determine the maximum permissible error119864max and the time 119905
2corresponding to the second charac-
teristic point can be obtained using the method above Thenthe state point can be obtained as follows
120596
2
119908= 0 120593
2
119908= 120593119897 V2
119908= V119897
119909
2
119889= 119909119889(1199052) 119910
2
119889= 119910119889(1199052)
(23)
So the tangent equation is
1199101119896
(119905) = 1199101119903(119905) (119905 minus 119905
2) + 119909119889(1199052) (24)
By repeating these procedures above time 119905119894correspond-
ing to the characteristic point in the region of multimodelfor formation can be obtained and at the same time thecharacteristic state point corresponding to the time 119905
119894can be
also obtained
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(25)
Calculate until the last characteristic state point isobtained and then the computation will be terminated
32 Generation of Discrete Model Sets for Formation ControlAfter obtaining the characteristic state points carry onlinearization at different discrete model sets of formationHere linearization can be realized through the followingmethods
Consider nonlinear systems as described in the form ofdiscrete-time dynamic equations
119909 (119896 + 1) = 119891 (119909 (119896) 119906 (119896))
119910 (119896) = 119892 (119909 (119896) 119906 (119896))
(26)
Mathematical Problems in Engineering 5
The system has m different characteristic state points119891(119909(119896) 119906(119896)) and 119892(119909(119896) 119906(119896)) have the first continuouspartial derivative If system is linearized at each characteristicstate point the standard discrete state-space model of mlinear models of the original system is obtained as follows
119909 (119896 + 1) = 119860119894119909 (119896) + 119861
119894119906 (119896) minus 120572
119894
119910 (119896) = 119862119894119909 (119896) + 119863
119894119906 (119896) minus 120573
119894
(27)
where
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119862119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119863119894=
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894
120572119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119910119894
(28)
Here m linearized models constitute the linearized multi-model presentation of the original system
For the UAV formation flying control the characteristicstate points are shown as
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(29)
And it has the following expression
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
minus1 119910119889(119905119894)
0 minus119909119889(119905119894)
0 1
]
]
Δ119879
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894=[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(30)
Since outputs are linear 119862119894 119863119894and 120573
119894will not be solved
Thus the linearization equation at characteristic state pointwill be
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119889 (
119896 + 1)
]
]
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
[
[
119909119889 (
119896)
119910119889 (
119896)
120593119889 (
119896)
]
]
+[
[
minus1 119910119889(119905119894)
0 119909119889(119905119894)
0 1
]
]
Δ119879[
V119908 (
119896)
120596119908 (
119896)
] minus[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(31)
So for different characteristic state points the linearmodel for UAV formation control at different horizonscan be obtained realizing the acquisition of model setsfor UAV formation And these models are denoted as119872(1)119872(2) 119872(119878)
33 Combined Method of Model Sets for Formation FlightControl Thecharacteristic state points are obtained using themethod above When the error reaches the maximum valueswitch to the new model which ensures the maximum errorvalue between the predictive trajectory and the referencetrajectory Thus the determination of model region can berealized It can be seen that the applicable range of desiredmodel is divided based on the time region so differentsampling points have different models But the predictivecontrol is based on the future time region So in this paperthe applicable model of predictive point is judged by the timeregion and then this model is used to calculate the predictivevalue The judgment rules of predictive model are describedas follows
Assuming that the time corresponding to the state char-acteristic points is 119905
1lt 1199052lt sdot sdot sdot lt 119905
119904minus1lt 119905119904and the predictive
horizon is [119905 119905 + 119873] then there will be the followingIf [119905 119905 + 119873] isin [119905
119894 119905119894+1
] then the final predictive model ofall points is 119872
119879= 119872119894 if [119905 119905 + 119873] isin [119905
119894 119905119894+119873
] one can judgethe interval [119905
ℎ 119905ℎ+1
] of all points between prediction point119905 + 1 to 119905 + 119873 in sequence and denote the model of this pointas119872119879= 119872ℎ if [119905 119905 +119873] isin [119905
119904infin) the final predictive model
of all the points will be119872119879= 119872119904
Based on this method we can obtain the predictive func-tion during the future horizon to determine the optimizationindex However the boundary point of the predictive rangemay be closer to the next linear model as shown in Figure 4Sampling point 119901
1may be closer to the linear model at
sampling point 1199054 But according to the method above the
calculation model used at the sampling point 1199011is the linear
model of the state characteristic point at sampling point1199053 Sampling point 119901
2may be closer to the linear model
of the state characteristic point at sampling point 1199055 But
according to the method above the calculation model usedby the sampling point 119901
2is the linear model of the state
characteristic point at sampling point 1199054
So the method above may decrease the performanceof the approximation capability on the boundary and eachmodel belonging to the model set cannot switch smoothly[18] However T-S fuzzy model as an intelligent controlmethod mainly uses fuzzy reasoning to approximate thenonlinear system Using this method the input space canbe divided into several fuzzy subspaces where a local linearmodel is established and then the local models are combinedsmoothly using the membership function forming a globalfuzzy model of nonlinear function which is ultimatelyidentified as a linear model [19] The predictive controlmethod based on the T-S fuzzy model belongs to multimodelpredictive control with the weighted models Comparedwith the common multimodel predictive controllers withweighted models the fuzzy weighted models have moreaccurate nonlinear approximation performance switch of
6 Mathematical Problems in Engineering
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emax
y(t)
p1 p2
Figure 4 The schematic for determining model of frontier pointsduring the predictive intervals
the model is more smooth and it is easier to understand[20 21] So in this section T-S fuzzy idea is adopted for themultimodel control of UAV formation flight as is shown inFigure 5
For each sampling point use the error between thetangent of state characteristic point and the reference tra-jectory of this sampling point to calculate the membershipdegree Assuming that the error between tangent of jthstate characteristic point and the reference trajectory at thesampling point t is 119864
119894(119905) so for the point 119905+ 119894 in the predictive
range weighted function is as follows
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(32)
This equation can ensure that the farther away from thestate characteristic point the sampling point is the lower itsweighted value is Using the weighted values the predictionmodel at the sampling point t is
119872119905=
sum
119898
119894=1(119908119894119872119894)
sum
119898
119894=1(119908119894)
(33)
For the sampling points during the predictive horizonthere is
119910119901 (
119905 + 119894) =
sum
119898
119895=1(119908119895 (119905 + 119894)119872119895
)
sum
119898
119895=1(119908119895 (119905 + 119894))
(34)
where
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(35)
Through this approach the linear prediction function forUAV formation can be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(36)
In this way predictive outputs can change from a non-linear function to a linear function This nonlinear functionincludes V
119908(119896) V119908(119896+1) V
119908(119896+119873minus1) and120596
119908(119896) 120596119908(119896+
1) 120596119908(119896 + 119873 minus 1) while this linear function includes the
control quantities mentioned above Thus the control prob-lem will become a multi-input-multioutput linear predictivecontrol problem
34 Optimization Index for Formation and Receding Opti-mization Solution During the predictive control process thegoal of receding optimization is to find a set of V
119908(119896) V119908(119896 +
1) V119908(119896+119873minus1) and 120596
119908(119896) 120596119908(119896+1) 120596
119908(119896+119873minus1)
making prediction outputs at entire optimization horizon asclose to the reference trajectory as possible
Here introduce the closed-loop
1198901 (
119905) = 119909119889 (
119905) minus 119909119889119901
(119905)
1198902 (
119905) = 119910119889 (
119905) minus 119910119889119901
(119905)
(37)
The open-loop predictive output can be directly compen-sated by the output feedback and then the predictive value ofthe closed-loop model will be
119909119889 (
119905 + 119894) = 119909119889119901
(119905 + 119894) + 1198901 (
119905)
119910119889 (
119905 + 119894) = 119910119889119901
(119905 + 119894) + 1198902 (
119905)
(38)
In this section there are two control objectives therelative distances to X-axis and Y-axis Since they have equalimportance and the same unit of quantity they are set withthe same weight when designing the performance indexThus the performance index is defined as follows
119869 =
119873
sum
119894=1
[(1199101119903 (
119905 + 119894) minus 119909119889 (
119905 + 119894))
2
+(1199102119903 (
119905 + 119894) minus 119910119889 (
119905 + 119894))
2]
+ 1205821
119873minus1
sum
119894=0
(V119908 (
119905 + 119894) minus V119908 (
119905))
2
+ 1205822
119873minus1
sum
119894=0
120596
2
119908(119905 + 119894)
(39)
Similarly the optimization constraints of control quantityare introduced as followsV119908 (
119896 + 119894 minus 1) minus ΔV lt V119908 (
119896 + 119894) lt V119908 (
119896 + 119894 minus 1) + ΔV
120596min lt 120596119908 (
119896 + 119894) lt 120596max
Vmin lt V119908 (
119896 + 119894) lt Vmax
where 119894 isin 0 1 2 119873 minus 1
(40)
Mathematical Problems in Engineering 7
Linearized prediction model 1
Linearized prediction model N
Ultimate prediction model
Prediction controller based on LQR
Referencetrajectoryof cruise
formation
Relative distance of cruise formationReference trajectory tangential error
Characteristicstatus points
LinearizeObtain characteristic point of
relative motion status for formation control
Generate relative motion model set of
formation
Tangential error of reference trajectory at sampling point and prediction point Weighted function
Relativedistance
Combinationof formation
model set
Multimodelprediction
control
Linearized prediction model 2
Weighted
Prediction Controloutputs
UAV attitude controlsystem
middot middot middot
Figure 5 The schematic of multimodel control method for UAV formation flight
After using multiple models the performance indexis linear quadratic whose constraints are linear equalityand inequality so the optimization problem is a linearquadratic programming problem The solution methods oflinear quadratic programming problem can be used to solvethe receding optimization problem The linear quadraticprogramming problem is a common programming problemand has a lot of solution methods and higher speed than theordinary nonlinear programming which increases the speedof receding optimization solution [22]
4 Simulation
In this section numerical simulations are performed todemonstrate the performance of the proposed approachHere the formation control ability can be tested in twoimportant scenarios Simulation scenarios are set as followsOne scenario is the leader UAV flying straight and the otheris the leader UAV flying with turning course Additionallythe comparison simulation between single MPC (SMPC)method and multiple MPC (MMPC) method is carried on toverify effectiveness of the method in this paper Meanwhilethe parameters used in the simulations are set as followsThe prediction horizon N is 5 and the sampling intervalis 02 s The angular velocity and velocity of two vehiclesare confined during the interval (minus01 01) and the interval(35 45) respectively All the computations and experimentshave been on a computer with Inter Core i3 CPU 330GHzand Windows XP operating systems Table 1 summarizes theinitial conditions of the formation
41 Formation Simulation of Leader UAV Flying Straight Thesimulation experiment is mainly used to verify the UAVformation control capability when the leader UAV is flyingstraight Here error exists in the position measurement of
Table 1 Initial conditions of UAV formation
Initial conditions The role of UAVLeader Follower
Initial position (0 0) (minus100 minus100)
Initial angle 0 1205872
Initial velocity 40 40Initial angular velocity 0 0
leader UAV which is plusmn05m There are two different controlgoals One is that the relative position between follower andleader ofUAV formation in the track coordinates is as follows
119883 = minus60
119884 = 30
(41)
The other is that the formation should be formed within 40 sBecause the leader UAV has its initial angle of 1205874 and
it flies straight the initial relative position in the trackcoordinates will be obtained as follows
[
119909119889
119910119889
] =
[
[
[
cos 1205874
sin 120587
4
minus sin 120587
4
cos 1205874
]
]
]
[
minus100
minus100
] = [
minus100radic2
0
] (42)
Simulation is carried out by using Matlab Simulink toolboxand the simulation results are shown from Figures 6 7 8 9and 10
According to Figures 6ndash10 it can be seen that whenleader UAV is navigating in a straight line formation controlcan be achieved through both SMPC and MMPC methodHowever the SMPC method has a larger tracking error thanthe MMPC method Meanwhile it can also be seen that ittakes a longer time for SMPCmethod thanMMPCmethod to
8 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
The system has m different characteristic state points119891(119909(119896) 119906(119896)) and 119892(119909(119896) 119906(119896)) have the first continuouspartial derivative If system is linearized at each characteristicstate point the standard discrete state-space model of mlinear models of the original system is obtained as follows
119909 (119896 + 1) = 119860119894119909 (119896) + 119861
119894119906 (119896) minus 120572
119894
119910 (119896) = 119862119894119909 (119896) + 119863
119894119906 (119896) minus 120573
119894
(27)
where
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119862119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119863119894=
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894
120572119894=
120597119892
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119892
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119910119894
(28)
Here m linearized models constitute the linearized multi-model presentation of the original system
For the UAV formation flying control the characteristicstate points are shown as
120596
119894
119908= 0 120593
119894
119908= 120593119897 V119894
119908= V119897
119909
119894
119889= 119909119889(119905119894) 119910
119894
119889= 119910119889(119905119894)
(29)
And it has the following expression
119860119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
119861119894=
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
=[
[
minus1 119910119889(119905119894)
0 minus119909119889(119905119894)
0 1
]
]
Δ119879
120572119894=
120597119891
120597119909
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119909119894+
120597119891
120597119906
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816(119909119894 119906119894)
119906119894minus 119909119894=[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(30)
Since outputs are linear 119862119894 119863119894and 120573
119894will not be solved
Thus the linearization equation at characteristic state pointwill be
[
[
119909119889 (
119896 + 1)
119910119889 (
119896 + 1)
120593119889 (
119896 + 1)
]
]
=[
[
1 0 0
0 1 minusV119897 (119896) Δ119879
0 0 1
]
]
[
[
119909119889 (
119896)
119910119889 (
119896)
120593119889 (
119896)
]
]
+[
[
minus1 119910119889(119905119894)
0 119909119889(119905119894)
0 1
]
]
Δ119879[
V119908 (
119896)
120596119908 (
119896)
] minus[
[
minusV119897 (119896) Δ119879
minusV119897 (119896) 120593119897 (
119896) Δ119879
0
]
]
(31)
So for different characteristic state points the linearmodel for UAV formation control at different horizonscan be obtained realizing the acquisition of model setsfor UAV formation And these models are denoted as119872(1)119872(2) 119872(119878)
33 Combined Method of Model Sets for Formation FlightControl Thecharacteristic state points are obtained using themethod above When the error reaches the maximum valueswitch to the new model which ensures the maximum errorvalue between the predictive trajectory and the referencetrajectory Thus the determination of model region can berealized It can be seen that the applicable range of desiredmodel is divided based on the time region so differentsampling points have different models But the predictivecontrol is based on the future time region So in this paperthe applicable model of predictive point is judged by the timeregion and then this model is used to calculate the predictivevalue The judgment rules of predictive model are describedas follows
Assuming that the time corresponding to the state char-acteristic points is 119905
1lt 1199052lt sdot sdot sdot lt 119905
119904minus1lt 119905119904and the predictive
horizon is [119905 119905 + 119873] then there will be the followingIf [119905 119905 + 119873] isin [119905
119894 119905119894+1
] then the final predictive model ofall points is 119872
119879= 119872119894 if [119905 119905 + 119873] isin [119905
119894 119905119894+119873
] one can judgethe interval [119905
ℎ 119905ℎ+1
] of all points between prediction point119905 + 1 to 119905 + 119873 in sequence and denote the model of this pointas119872119879= 119872ℎ if [119905 119905 +119873] isin [119905
119904infin) the final predictive model
of all the points will be119872119879= 119872119904
Based on this method we can obtain the predictive func-tion during the future horizon to determine the optimizationindex However the boundary point of the predictive rangemay be closer to the next linear model as shown in Figure 4Sampling point 119901
1may be closer to the linear model at
sampling point 1199054 But according to the method above the
calculation model used at the sampling point 1199011is the linear
model of the state characteristic point at sampling point1199053 Sampling point 119901
2may be closer to the linear model
of the state characteristic point at sampling point 1199055 But
according to the method above the calculation model usedby the sampling point 119901
2is the linear model of the state
characteristic point at sampling point 1199054
So the method above may decrease the performanceof the approximation capability on the boundary and eachmodel belonging to the model set cannot switch smoothly[18] However T-S fuzzy model as an intelligent controlmethod mainly uses fuzzy reasoning to approximate thenonlinear system Using this method the input space canbe divided into several fuzzy subspaces where a local linearmodel is established and then the local models are combinedsmoothly using the membership function forming a globalfuzzy model of nonlinear function which is ultimatelyidentified as a linear model [19] The predictive controlmethod based on the T-S fuzzy model belongs to multimodelpredictive control with the weighted models Comparedwith the common multimodel predictive controllers withweighted models the fuzzy weighted models have moreaccurate nonlinear approximation performance switch of
6 Mathematical Problems in Engineering
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emax
y(t)
p1 p2
Figure 4 The schematic for determining model of frontier pointsduring the predictive intervals
the model is more smooth and it is easier to understand[20 21] So in this section T-S fuzzy idea is adopted for themultimodel control of UAV formation flight as is shown inFigure 5
For each sampling point use the error between thetangent of state characteristic point and the reference tra-jectory of this sampling point to calculate the membershipdegree Assuming that the error between tangent of jthstate characteristic point and the reference trajectory at thesampling point t is 119864
119894(119905) so for the point 119905+ 119894 in the predictive
range weighted function is as follows
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(32)
This equation can ensure that the farther away from thestate characteristic point the sampling point is the lower itsweighted value is Using the weighted values the predictionmodel at the sampling point t is
119872119905=
sum
119898
119894=1(119908119894119872119894)
sum
119898
119894=1(119908119894)
(33)
For the sampling points during the predictive horizonthere is
119910119901 (
119905 + 119894) =
sum
119898
119895=1(119908119895 (119905 + 119894)119872119895
)
sum
119898
119895=1(119908119895 (119905 + 119894))
(34)
where
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(35)
Through this approach the linear prediction function forUAV formation can be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(36)
In this way predictive outputs can change from a non-linear function to a linear function This nonlinear functionincludes V
119908(119896) V119908(119896+1) V
119908(119896+119873minus1) and120596
119908(119896) 120596119908(119896+
1) 120596119908(119896 + 119873 minus 1) while this linear function includes the
control quantities mentioned above Thus the control prob-lem will become a multi-input-multioutput linear predictivecontrol problem
34 Optimization Index for Formation and Receding Opti-mization Solution During the predictive control process thegoal of receding optimization is to find a set of V
119908(119896) V119908(119896 +
1) V119908(119896+119873minus1) and 120596
119908(119896) 120596119908(119896+1) 120596
119908(119896+119873minus1)
making prediction outputs at entire optimization horizon asclose to the reference trajectory as possible
Here introduce the closed-loop
1198901 (
119905) = 119909119889 (
119905) minus 119909119889119901
(119905)
1198902 (
119905) = 119910119889 (
119905) minus 119910119889119901
(119905)
(37)
The open-loop predictive output can be directly compen-sated by the output feedback and then the predictive value ofthe closed-loop model will be
119909119889 (
119905 + 119894) = 119909119889119901
(119905 + 119894) + 1198901 (
119905)
119910119889 (
119905 + 119894) = 119910119889119901
(119905 + 119894) + 1198902 (
119905)
(38)
In this section there are two control objectives therelative distances to X-axis and Y-axis Since they have equalimportance and the same unit of quantity they are set withthe same weight when designing the performance indexThus the performance index is defined as follows
119869 =
119873
sum
119894=1
[(1199101119903 (
119905 + 119894) minus 119909119889 (
119905 + 119894))
2
+(1199102119903 (
119905 + 119894) minus 119910119889 (
119905 + 119894))
2]
+ 1205821
119873minus1
sum
119894=0
(V119908 (
119905 + 119894) minus V119908 (
119905))
2
+ 1205822
119873minus1
sum
119894=0
120596
2
119908(119905 + 119894)
(39)
Similarly the optimization constraints of control quantityare introduced as followsV119908 (
119896 + 119894 minus 1) minus ΔV lt V119908 (
119896 + 119894) lt V119908 (
119896 + 119894 minus 1) + ΔV
120596min lt 120596119908 (
119896 + 119894) lt 120596max
Vmin lt V119908 (
119896 + 119894) lt Vmax
where 119894 isin 0 1 2 119873 minus 1
(40)
Mathematical Problems in Engineering 7
Linearized prediction model 1
Linearized prediction model N
Ultimate prediction model
Prediction controller based on LQR
Referencetrajectoryof cruise
formation
Relative distance of cruise formationReference trajectory tangential error
Characteristicstatus points
LinearizeObtain characteristic point of
relative motion status for formation control
Generate relative motion model set of
formation
Tangential error of reference trajectory at sampling point and prediction point Weighted function
Relativedistance
Combinationof formation
model set
Multimodelprediction
control
Linearized prediction model 2
Weighted
Prediction Controloutputs
UAV attitude controlsystem
middot middot middot
Figure 5 The schematic of multimodel control method for UAV formation flight
After using multiple models the performance indexis linear quadratic whose constraints are linear equalityand inequality so the optimization problem is a linearquadratic programming problem The solution methods oflinear quadratic programming problem can be used to solvethe receding optimization problem The linear quadraticprogramming problem is a common programming problemand has a lot of solution methods and higher speed than theordinary nonlinear programming which increases the speedof receding optimization solution [22]
4 Simulation
In this section numerical simulations are performed todemonstrate the performance of the proposed approachHere the formation control ability can be tested in twoimportant scenarios Simulation scenarios are set as followsOne scenario is the leader UAV flying straight and the otheris the leader UAV flying with turning course Additionallythe comparison simulation between single MPC (SMPC)method and multiple MPC (MMPC) method is carried on toverify effectiveness of the method in this paper Meanwhilethe parameters used in the simulations are set as followsThe prediction horizon N is 5 and the sampling intervalis 02 s The angular velocity and velocity of two vehiclesare confined during the interval (minus01 01) and the interval(35 45) respectively All the computations and experimentshave been on a computer with Inter Core i3 CPU 330GHzand Windows XP operating systems Table 1 summarizes theinitial conditions of the formation
41 Formation Simulation of Leader UAV Flying Straight Thesimulation experiment is mainly used to verify the UAVformation control capability when the leader UAV is flyingstraight Here error exists in the position measurement of
Table 1 Initial conditions of UAV formation
Initial conditions The role of UAVLeader Follower
Initial position (0 0) (minus100 minus100)
Initial angle 0 1205872
Initial velocity 40 40Initial angular velocity 0 0
leader UAV which is plusmn05m There are two different controlgoals One is that the relative position between follower andleader ofUAV formation in the track coordinates is as follows
119883 = minus60
119884 = 30
(41)
The other is that the formation should be formed within 40 sBecause the leader UAV has its initial angle of 1205874 and
it flies straight the initial relative position in the trackcoordinates will be obtained as follows
[
119909119889
119910119889
] =
[
[
[
cos 1205874
sin 120587
4
minus sin 120587
4
cos 1205874
]
]
]
[
minus100
minus100
] = [
minus100radic2
0
] (42)
Simulation is carried out by using Matlab Simulink toolboxand the simulation results are shown from Figures 6 7 8 9and 10
According to Figures 6ndash10 it can be seen that whenleader UAV is navigating in a straight line formation controlcan be achieved through both SMPC and MMPC methodHowever the SMPC method has a larger tracking error thanthe MMPC method Meanwhile it can also be seen that ittakes a longer time for SMPCmethod thanMMPCmethod to
8 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 2008
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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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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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
t1t2 t3 t4 t5 t6
t
Emax
Emax
Emax
y(t)
p1 p2
Figure 4 The schematic for determining model of frontier pointsduring the predictive intervals
the model is more smooth and it is easier to understand[20 21] So in this section T-S fuzzy idea is adopted for themultimodel control of UAV formation flight as is shown inFigure 5
For each sampling point use the error between thetangent of state characteristic point and the reference tra-jectory of this sampling point to calculate the membershipdegree Assuming that the error between tangent of jthstate characteristic point and the reference trajectory at thesampling point t is 119864
119894(119905) so for the point 119905+ 119894 in the predictive
range weighted function is as follows
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(32)
This equation can ensure that the farther away from thestate characteristic point the sampling point is the lower itsweighted value is Using the weighted values the predictionmodel at the sampling point t is
119872119905=
sum
119898
119894=1(119908119894119872119894)
sum
119898
119894=1(119908119894)
(33)
For the sampling points during the predictive horizonthere is
119910119901 (
119905 + 119894) =
sum
119898
119895=1(119908119895 (119905 + 119894)119872119895
)
sum
119898
119895=1(119908119895 (119905 + 119894))
(34)
where
119908119895 (119905 + 119894) = 119890
minus(119864119894(119905+119894)119864max)2
minus119864max le 119864119895 (119905 + 119894) le 119864max
119908 (119905 + 119894) = 0 119864119895 (119905 + 119894) le minus119864max
or 119864119895 (119905 + 119894) ge 119864max
(35)
Through this approach the linear prediction function forUAV formation can be obtained as follows
119909119889119901
(119896 + 1) 119909119889119901(119896 + 2) 119909119889119901
(119896 + 119873)
119910119889119901
(119896 + 1) 119910119889119901(119896 + 2) 119910119889119901
(119896 + 119873)
(36)
In this way predictive outputs can change from a non-linear function to a linear function This nonlinear functionincludes V
119908(119896) V119908(119896+1) V
119908(119896+119873minus1) and120596
119908(119896) 120596119908(119896+
1) 120596119908(119896 + 119873 minus 1) while this linear function includes the
control quantities mentioned above Thus the control prob-lem will become a multi-input-multioutput linear predictivecontrol problem
34 Optimization Index for Formation and Receding Opti-mization Solution During the predictive control process thegoal of receding optimization is to find a set of V
119908(119896) V119908(119896 +
1) V119908(119896+119873minus1) and 120596
119908(119896) 120596119908(119896+1) 120596
119908(119896+119873minus1)
making prediction outputs at entire optimization horizon asclose to the reference trajectory as possible
Here introduce the closed-loop
1198901 (
119905) = 119909119889 (
119905) minus 119909119889119901
(119905)
1198902 (
119905) = 119910119889 (
119905) minus 119910119889119901
(119905)
(37)
The open-loop predictive output can be directly compen-sated by the output feedback and then the predictive value ofthe closed-loop model will be
119909119889 (
119905 + 119894) = 119909119889119901
(119905 + 119894) + 1198901 (
119905)
119910119889 (
119905 + 119894) = 119910119889119901
(119905 + 119894) + 1198902 (
119905)
(38)
In this section there are two control objectives therelative distances to X-axis and Y-axis Since they have equalimportance and the same unit of quantity they are set withthe same weight when designing the performance indexThus the performance index is defined as follows
119869 =
119873
sum
119894=1
[(1199101119903 (
119905 + 119894) minus 119909119889 (
119905 + 119894))
2
+(1199102119903 (
119905 + 119894) minus 119910119889 (
119905 + 119894))
2]
+ 1205821
119873minus1
sum
119894=0
(V119908 (
119905 + 119894) minus V119908 (
119905))
2
+ 1205822
119873minus1
sum
119894=0
120596
2
119908(119905 + 119894)
(39)
Similarly the optimization constraints of control quantityare introduced as followsV119908 (
119896 + 119894 minus 1) minus ΔV lt V119908 (
119896 + 119894) lt V119908 (
119896 + 119894 minus 1) + ΔV
120596min lt 120596119908 (
119896 + 119894) lt 120596max
Vmin lt V119908 (
119896 + 119894) lt Vmax
where 119894 isin 0 1 2 119873 minus 1
(40)
Mathematical Problems in Engineering 7
Linearized prediction model 1
Linearized prediction model N
Ultimate prediction model
Prediction controller based on LQR
Referencetrajectoryof cruise
formation
Relative distance of cruise formationReference trajectory tangential error
Characteristicstatus points
LinearizeObtain characteristic point of
relative motion status for formation control
Generate relative motion model set of
formation
Tangential error of reference trajectory at sampling point and prediction point Weighted function
Relativedistance
Combinationof formation
model set
Multimodelprediction
control
Linearized prediction model 2
Weighted
Prediction Controloutputs
UAV attitude controlsystem
middot middot middot
Figure 5 The schematic of multimodel control method for UAV formation flight
After using multiple models the performance indexis linear quadratic whose constraints are linear equalityand inequality so the optimization problem is a linearquadratic programming problem The solution methods oflinear quadratic programming problem can be used to solvethe receding optimization problem The linear quadraticprogramming problem is a common programming problemand has a lot of solution methods and higher speed than theordinary nonlinear programming which increases the speedof receding optimization solution [22]
4 Simulation
In this section numerical simulations are performed todemonstrate the performance of the proposed approachHere the formation control ability can be tested in twoimportant scenarios Simulation scenarios are set as followsOne scenario is the leader UAV flying straight and the otheris the leader UAV flying with turning course Additionallythe comparison simulation between single MPC (SMPC)method and multiple MPC (MMPC) method is carried on toverify effectiveness of the method in this paper Meanwhilethe parameters used in the simulations are set as followsThe prediction horizon N is 5 and the sampling intervalis 02 s The angular velocity and velocity of two vehiclesare confined during the interval (minus01 01) and the interval(35 45) respectively All the computations and experimentshave been on a computer with Inter Core i3 CPU 330GHzand Windows XP operating systems Table 1 summarizes theinitial conditions of the formation
41 Formation Simulation of Leader UAV Flying Straight Thesimulation experiment is mainly used to verify the UAVformation control capability when the leader UAV is flyingstraight Here error exists in the position measurement of
Table 1 Initial conditions of UAV formation
Initial conditions The role of UAVLeader Follower
Initial position (0 0) (minus100 minus100)
Initial angle 0 1205872
Initial velocity 40 40Initial angular velocity 0 0
leader UAV which is plusmn05m There are two different controlgoals One is that the relative position between follower andleader ofUAV formation in the track coordinates is as follows
119883 = minus60
119884 = 30
(41)
The other is that the formation should be formed within 40 sBecause the leader UAV has its initial angle of 1205874 and
it flies straight the initial relative position in the trackcoordinates will be obtained as follows
[
119909119889
119910119889
] =
[
[
[
cos 1205874
sin 120587
4
minus sin 120587
4
cos 1205874
]
]
]
[
minus100
minus100
] = [
minus100radic2
0
] (42)
Simulation is carried out by using Matlab Simulink toolboxand the simulation results are shown from Figures 6 7 8 9and 10
According to Figures 6ndash10 it can be seen that whenleader UAV is navigating in a straight line formation controlcan be achieved through both SMPC and MMPC methodHowever the SMPC method has a larger tracking error thanthe MMPC method Meanwhile it can also be seen that ittakes a longer time for SMPCmethod thanMMPCmethod to
8 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
Mathematical Problems in Engineering 7
Linearized prediction model 1
Linearized prediction model N
Ultimate prediction model
Prediction controller based on LQR
Referencetrajectoryof cruise
formation
Relative distance of cruise formationReference trajectory tangential error
Characteristicstatus points
LinearizeObtain characteristic point of
relative motion status for formation control
Generate relative motion model set of
formation
Tangential error of reference trajectory at sampling point and prediction point Weighted function
Relativedistance
Combinationof formation
model set
Multimodelprediction
control
Linearized prediction model 2
Weighted
Prediction Controloutputs
UAV attitude controlsystem
middot middot middot
Figure 5 The schematic of multimodel control method for UAV formation flight
After using multiple models the performance indexis linear quadratic whose constraints are linear equalityand inequality so the optimization problem is a linearquadratic programming problem The solution methods oflinear quadratic programming problem can be used to solvethe receding optimization problem The linear quadraticprogramming problem is a common programming problemand has a lot of solution methods and higher speed than theordinary nonlinear programming which increases the speedof receding optimization solution [22]
4 Simulation
In this section numerical simulations are performed todemonstrate the performance of the proposed approachHere the formation control ability can be tested in twoimportant scenarios Simulation scenarios are set as followsOne scenario is the leader UAV flying straight and the otheris the leader UAV flying with turning course Additionallythe comparison simulation between single MPC (SMPC)method and multiple MPC (MMPC) method is carried on toverify effectiveness of the method in this paper Meanwhilethe parameters used in the simulations are set as followsThe prediction horizon N is 5 and the sampling intervalis 02 s The angular velocity and velocity of two vehiclesare confined during the interval (minus01 01) and the interval(35 45) respectively All the computations and experimentshave been on a computer with Inter Core i3 CPU 330GHzand Windows XP operating systems Table 1 summarizes theinitial conditions of the formation
41 Formation Simulation of Leader UAV Flying Straight Thesimulation experiment is mainly used to verify the UAVformation control capability when the leader UAV is flyingstraight Here error exists in the position measurement of
Table 1 Initial conditions of UAV formation
Initial conditions The role of UAVLeader Follower
Initial position (0 0) (minus100 minus100)
Initial angle 0 1205872
Initial velocity 40 40Initial angular velocity 0 0
leader UAV which is plusmn05m There are two different controlgoals One is that the relative position between follower andleader ofUAV formation in the track coordinates is as follows
119883 = minus60
119884 = 30
(41)
The other is that the formation should be formed within 40 sBecause the leader UAV has its initial angle of 1205874 and
it flies straight the initial relative position in the trackcoordinates will be obtained as follows
[
119909119889
119910119889
] =
[
[
[
cos 1205874
sin 120587
4
minus sin 120587
4
cos 1205874
]
]
]
[
minus100
minus100
] = [
minus100radic2
0
] (42)
Simulation is carried out by using Matlab Simulink toolboxand the simulation results are shown from Figures 6 7 8 9and 10
According to Figures 6ndash10 it can be seen that whenleader UAV is navigating in a straight line formation controlcan be achieved through both SMPC and MMPC methodHowever the SMPC method has a larger tracking error thanthe MMPC method Meanwhile it can also be seen that ittakes a longer time for SMPCmethod thanMMPCmethod to
8 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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 Mathematical Problems in Engineering
minus500 0 500 1000 1500 2000 2500 3000minus500
0
500
1000
1500
2000
2500
3000
Relative position of Y axis (m)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 6 Flight trajectories of two UAVs
SMPCMMPC
0 20 40 60 80 100minus200
minus175
minus150
minus125
minus100
minus75
minus50
minus25
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
Figure 7 Relative position of X-axis
form a steady formation The UAV formation can be realizedin 40 seconds by the MMPCmethod which meets the actualdesign demand
42 Formation Simulation for LeaderUAVwith Turning FlightThe UAV formation control capability is proved in thissection when the leader UAV flies with a turning flight pathThe UAV flies 20 s with an initial angle of 0∘ between theleader UAV and X-axis and then the UAV flies with angularvelocity of 120587200 for 100 seconds and then it moves straight
0 20 40 60 80 1000
20
40
60
80
100
120
140
160
Time (s)
Rela
tive p
ositi
on o
f Y ax
is (m
)
MMPCSMPC
Figure 8 Relative position of Y-axis
0 20 40 60 80 1000
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 9 Change curve of follower UAVrsquos velocity
in Y-axis directionThere are also two different control goalsOne is that the relative position between follower and leaderof UAV formation in the track coordinates is as follows
[
119909dref119910dref
] = [
minus50
minus50
] (43)
The other is the formation should form within 40 sFrom Table 1 the relative position in the track coordinate
system between two vehicles is obtained as follows
[
119909119889
119910119889
] = [
cos 0 sin 0
minus sin 0 cos 0] [
minus100
minus100
] = [
minus100
minus100
] (44)
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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
Mathematical Problems in Engineering 9
0 20 40 60 80 100minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 10 Change curve of follower UAVrsquos yaw rate
minus1000 0 1000 2000 3000 4000 5000minus500
0
500
1000
1500
2000
2500
3000
Rela
tive p
ositi
on o
f X ax
is (m
)
Relative position of Y axis (m)
MMPCSMPC
Figure 11 Flight trajectories of two UAVs
Simulation is carried out by using Matlab Simulink toolboxand the results are shown in Figures 11 12 13 14 and 15
According to Figures 10ndash14 when leader UAV flies witha turning flight path using method proposed in the paperformation control can be achieved better than the SMPCmethod whenever the UAV flies straight or flies with aturning path The SMPC method has a larger tracking errorthan the MMPC method Meanwhile it can also be seenthat it takes a longer time for SMPC method than MMPCmethod to form a steady formation The UAV formation canbe realized in 40 seconds by theMMPCmethod whichmeetsthe actual design demand
According to the Matlab simulation process of UAVformation in those two scenarios above when the sampling
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ositi
on o
f X ax
is (m
)
MMPCSMPC
Figure 12 Relative position of X-axis
0 50 100 150minus100
minus80
minus60
minus40
minus20
0
Time (s)
Rela
tive p
ostio
n of
Y ax
is (m
)
MMPCSMPC
Figure 13 Relative position of Y-axis
interval is 02 s the simulation time of the receding optimiza-tion program on the PC is less than 02 s each time and thetime will be shorter if the simulation is done on a dedicatedchip So it meets the real-time needs It can be seen fromthe relative position on the X-axis and Y-axis of two vehiclesin the track coordinate system that the UAV formation isrealized within 40 s All in all the simulation shows thatthe control requirements and real-time requirements can besatisfied by using multimodel predictive control method forUAV formation control
10 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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 Mathematical Problems in Engineering
0 50 100 1500
10
20
30
40
50
60
Time (s)
Velo
city
(ms
)
MMPCSMPC
Figure 14 Change curve of follower UAVrsquos velocity
0 50 100 150minus02
minus015
minus01
minus005
0
005
01
015
02
Time (s)
Yaw
rate
(rad
s)
MMPCSMPC
Figure 15 Change curve of follower UAVrsquos yaw rate
5 Conclusion
In this paper the main work can be concluded as follows tosolve the problem of UAV formation control
(1) Discrete relative motion equations are established forUAV formation by using the leader-follower method
(2) Multimodel sets for UAV formation are establishedand the weighted model sets method is proposed
(3) The formation controller based on multimodel pre-dictive control is designed
(4) Simulation in two scenarios is carried out and theeffectiveness of controller designed and control strat-egy is verified
The multimodel predictive control method can be usedfor UAV formation control This method can meet controlrequirements and real-time requirements well The result ofthis paper is the basis of further research on formation recon-figuration control problem In the future we will introducethe approach proposed in this paper to the controller designof actual UAV formation flight
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Research is supported by the National Science Foundationof China (NSFC) under Grants no 51201182 and 61105012andNationalAviation Science Foundation ofChina (NASFC)under Grant no 20135896027 Among these foundations theNASFC is a cooperation program of our research group andFACRI and this foundation requires both sides to publish anarticle
References
[1] X-Y Wang X-M Wang and C-C Yao ldquoDesign of UAVsformation flight controller based on neural network adaptiveinversionrdquo Control and Decision vol 28 no 6 pp 837ndash8432013
[2] C-J Ru R-X Wei J Dai D Shen and L-P ZhangldquoAutonomous reconfiguration controlmethod forUAVrsquos forma-tion based onNash bargainrdquoAutaAutomatica Sinica vol 39 no8 pp 1349ndash1359 2013
[3] L Jieun S K Hyeong and K Youdan ldquoFormation geometrycenter based formation controller design using Lyapunov sta-bility theoryrdquoKSAS International Journal no 2 pp 71ndash76 2008
[4] A Bemporad and C Rocchi ldquoDecentralized hybrid modelpredictive control of a formation of unmanned aerial vehiclesrdquoin Proceedings of the 18th IFAC Word Congress Milanno Italy2011
[5] Z Chao S-L Zhou L Ming and W-G Zhang ldquoUAV for-mation flight based on nonlinear model predictive controlrdquoMathematical Problems in Engineering vol 2012 Article ID261367 15 pages 2012
[6] K Wesselowski and R Fierro ldquoA dual-mode model predictivecontroller for robot formationsrdquo in Proceedings of the 42ndIEEE Conference on Decision and Control pp 3615ndash3620 MauiHawaii USA December 2003
[7] B J N Guerreiro C Silvestre and R Cunha ldquoTerrainavoidance nonlinear model predictive control for autonomousrotorcraftrdquo Journal of Intelligent amp Robotic Systems Theory andApplications vol 68 no 9 pp 69ndash85 2012
[8] M A Abbas J M Eklund and R Milman ldquoReal-time analysisfor nonlinearmodel predictive control of autonomous vehiclesrdquoin Proceedings of the 25th IEEE Canadian Conference on Electri-cal amp Computer Engineering (CCECE rsquo12) pp 1ndash4 2012
[9] J Shin and H J Kim ldquoNonlinear model predictive formationflightrdquo IEEE Transactions on Systems Man and Cybernetics ASystems and Humans vol 39 no 5 pp 1116ndash1125 2009
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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
Mathematical Problems in Engineering 11
[10] F Alessandro L Sauro and M Andrea ldquoNonlinear decen-tralized model predictive control strategy for a formation ofunmanned aerial vehiclesrdquo in Proceedings of the 2nd IFACWorkshop on Multivehicle System vol 2 pp 49ndash54 2012
[11] C Gorman and N Slegers ldquoPredictive control of generalnonlinear systems using series approximationsrdquo in Proceedingsof the AIAA Guidance Navigation and Control Conference andExhibit AIAA 2009-5994 Chicago Ill USA August 2009
[12] W Dhouib M Djemel and M Chtourou ldquoFuzzy predictivecontrol of nonlinear systemsrdquo in Proceedings of the 8th Inter-national Multi-Conference on Systems Signals and Devices (SSDrsquo11) pp 1ndash8 Sousse Tunisia March 2011
[13] T Keviczky F Borrelli and G J Balas ldquoDecentralized recedinghorizon control for large scale dynamically decoupled systemsrdquoAutomatica vol 42 no 12 pp 2105ndash2115 2006
[14] Q Chen L Gao R A Dougal and S Quan ldquoMultiple modelpredictive control for a hybrid proton exchange membrane fuelcell systemrdquo Journal of Power Sources vol 191 no 2 pp 473ndash482 2009
[15] N N Nandola and S Bhartiya ldquoA multiple model approachfor predictive control of nonlinear hybrid systemsrdquo Journal ofProcess Control vol 18 no 2 pp 131ndash148 2008
[16] D Dougherty and D Cooper ldquoA practical multiple modeladaptive strategy for single-loop MPCrdquo Control EngineeringPractice vol 11 no 2 pp 141ndash159 2003
[17] K S Narendra and C Xiang ldquoAdaptive control of discrete-time systems using multiple modelsrdquo IEEE Transactions onAutomatic Control vol 45 no 9 pp 1669ndash1686 2000
[18] L-L Liu L-F Zhou T Ji and Y-H Zhao ldquoResearch onmodel switchingmethod ofmulti-hierarchicalmodel predictivecontrol systemsrdquoActa Automatica Sinica vol 39 no 5 pp 626ndash630 2013
[19] R J Spiegel M W Turner and V E McCormick ldquoFuzzy-logic-based controllers for efficiency optimization of inverter-fed inductionmotor drivesrdquo Fuzzy Sets and Systems vol 137 no3 pp 387ndash401 2003
[20] Z-Q Chen and H-M Jiang ldquoT-S fuzzy model predictivecontrol simulation based on intelligent optimization algorithmrdquoJournal of System Simulation vol 2 pp 79ndash85 2009
[21] Y Gu H O Wang K Tanaka and L G Bushnell ldquoFuzzycontrol of nonlinear time-delay systems stability and designissuesrdquo in Proceedings of the American Control Conference (ACCrsquo01) pp 4771ndash4776 Arlington Calif USA June 2001
[22] T Keviczky F Borrelli K Fregene D Godbole and G J BalasldquoDecentralized receding horizon control and coordination ofautonomous vehicle formationsrdquo IEEE Transactions on ControlSystems Technology vol 16 no 1 pp 19ndash33 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