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

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

Hindawi Publishing Corporationhttpwwwhindawicom

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

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 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

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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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

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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

Volume 2014

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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

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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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

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 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