Fuzzy-Based Hybrid Control Algorithm for the Stabilization of a Tri-Rotor UAV

18 Pages·2016·3.51 MB·English

sensors
Article
Fuzzy-Based Hybrid Control Algorithm for the
Stabilization of a Tri-Rotor UAV
ZainAnwarAli1,*,DaoboWang1andMuhammadAamir2
1 CollegeofAutomationEngineering,NanjingUniversityofAeronauticsandAstronautics,Nanjing210016,
China;[e-mail]
2 ElectronicEngineeringDepartment,SirSyedUniversityofEngineeringandTechnology,Karachi75300,
Pakistan;[e-mail]
* Correspondence:[e-mail];Tel.:[phone]
AcademicEditor:FelipeGonzalezToro
Received:3February2016;Accepted:28April2016;Published:9May2016
Abstract: In this paper, a new and novel mathematical fuzzy hybrid scheme is proposed for the
stabilizationofatri-rotorunmannedaerialvehicle(UAV).Thefuzzyhybridschemeconsistsofa
fuzzy logic controller, regulation pole-placement tracking (RST) controller with model reference
adaptivecontrol(MRAC),inwhichadaptivegainsoftheRSTcontrollerarebeingfine-tunedbya
fuzzylogiccontroller. Brushlessdirectcurrent(BLDC)motorsareinstalledinthetriangularframe
ofthetri-rotorUAV,whichhelpsmaintaincontrolonitsmotionanddifferentaltitudeandattitude
changes,similartorotorcrafts. MRAC-basedMITruleisproposedforsystemstability. Moreover,the
proposedhybridcontrollerwithnonlinearflightdynamicsisshowninthepresenceoftranslational
androtationalvelocitycomponents. Theperformanceoftheproposedalgorithmisdemonstratedvia
MATLABsimulations,inwhichtheproposedfuzzyhybridcontrolleriscomparedwiththeexisting
adaptiveRSTcontroller. Itshowsthatourproposedalgorithmhasbettertransientperformancewith
zerosteady-stateerror,andfastconvergencetowardsstability.
Keywords: UnmannedAerialVehicle;Tri-RotorUAV;RSTcontroller;fuzzyhybridcontroller
1. Introduction
Oneofthebestinventionsoftoday’seraisthesmallflyingmachinecommonlycalledaUAV.This
researchisdedicatedtosuchtypesofUAVs,whicharecommonlyusedinthemonitoringofdisaster
managementandmilitaryoperations,aswellassmallindooractivities[1–3]. TheresearchonUAVsis
basedonthedifferentknowledgebanksofaeronautics,signalprocessing,andcontrolautomation. For
thisresearch,multiplehardware-basedtestsareperformedtodesignthebestflyingmachineswith
precisecontrolmechanisms.
Thecurrenttrendisfocusedonthedesignofadvanced,lightweight,andperfectUAVsthatcan
beoperatedinanydisastroussituationsoverremoteareas. UAVsareclassifiedaseitherfixed-wing
orrotarywing[4]. Rotor-basedUAVsaremultipleinputandmultipleoutput(MIMO)multivariable
systems [5]. Rotorcraft have a great advantage over fixed-wing aircraft with respect to various
applications,likeverticaltakeoffandverticallanding(VTOL)capabilityandpayloads. Rotor-based
UAVsincludemanytypes,suchasbi-rotor,tri-rotor,quad-rotorandhex-rotor[6]. Moreover,atri-rotor
UAVwithVTOLabilityisconsideredinthispaper.
Real-worldapplicationofUAVsrequireintensehardwaretesting. Beforetheexperimentaltesting
ofourproposedalgorithmintherealworld,wehavetosimulatethenumericalnonlinearsimulations
fortheEulerangles,controlcommands,rotationalvelocities,andtranslationalvelocities[7]. Inthis
research,ourmainconcernistorectifytheerrorwhichoccursinayawmomentduetotheunpaired
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reactionoftherotors,therebyproducingtorque. Brushlessdirectcurrent(BLDC)motorsareinstalled
inthetriangularframeofthetri-rotorcrafttonullifythetiltanglemoment.
The dynamics of the UAV are highly nonlinear and multi-variable, with a lot of parameter
uncertainties,manyeffectstowhichapotentialcontrollerhastoberobust. Theaerodynamicsofthe
actuatorblades(flappingofbladeandpropeller),inertialtorques(angularspeedofpropellers),and
gyroscopiceffects(whichchangetheorientationoftheUAV)arefoundin[8]. Theredundancyinthe
rotorsofaUAVformulatesthemtowardsasetofpartialcollapses. Althoughthemaneuverability
and performance will probably be condensed in the case of such a collapse, it is required that a
controller stabilizes the system and tolerates reduced mode functions, such as safe arrival, steady
hover,etc.[9,10].
Previously, many control methods were used for the stabilization of UAVs, including the
conventionalproportionalintegralderivative(PID)controller,fuzzycontroller,adaptivecontroller,
andsoon[11]. ForcontrollingtheparametersofaUAVanadaptivecontrollerhasacapabilitytogive
goodperformanceinthepresenceofmodelandparametricuncertainties,whileMRACisconcerned
withthevibrantreactionofthecontrolledsystemtoasymptoticconvergence. Itfollowsthereference
systeminspiteofparametricmodeluncertaintiesinthesystem[12].
In [13] the proposed MRAC for controlling the dynamics of a quad-rotor in the presence of
actuatoruncertaintieswasconsideredtoenhanceanexistinglinearcontroller,offeringautonomous
waypointfollowing. ThestabilityoftheadaptivecontrollerwasensuredbytheLyapnauvtheorem
and,inanonlinearstructure,thealgorithmisappliedforindoorflighttest.
In [14] the hybrid control scheme to fault tolerant control (FTC) for a quad-rotor aircraft in
thepresenceoffaultsintheirrotorsduringtheflighthavebeenexploredandtestedontheMRAC
algorithmandagain-scheduledPID(GS-PID)control. MRACandGS-PIDareusedincollaboration
withalinearquadraticregulator(LQR)tocontroltheattitudeoftheUAV.MRACisbasedonMIT
rulesforcontrollingtheheightandotherparametersofaQball-X4Quad-Rotoraircraft.
Takagi-Sugenofuzzyruleswerepreviouslyusedin[15,16]tocontrolthenonlinearbehaviorof
the vehicle. On the other hand, in [17], a twin controller approach that consists of a backstepping
controllertocontrolthenonlineardynamicsofthesystemandlinguisticlogicrulesofafuzzylogic
controller(Mamdani)isusedtocontroltheattitudeoftiltofatri-rotorUAV.In[18]adualcontroller
approachwithanadaptivefuzzyslidingmodecontrollerisusedtocontroltheminiUAV,inwhich
slidingmodecontrolisutilizedtocontrolthenonlinearbehavioroftheUAV,andthenfuzzylogic
rulesareimplementedonit. Thehybridcontrollerapproachwasalsoaddressedin[19]inwhicha
fuzzy-PIDcontrollerwithaPSOalgorithmisappliedontri-rotordynamics.
Hwoever,inthispaper,weproposedafuzzyhybridcontrollerconsistingofaRSTwithMRAC,
based on MIT rules working as a main controller in the model to deal with the nonlinear system.
WecomparetheperformanceofourproposedfuzzyhybridcontrollerwiththerobustadaptiveRST
controllerof[20].
Moreover,theadaptivegainsoftheRSTcontrollerare(i.e.,regulationgain“G ”,pole-placement
R
gainG ,andtrackinggain“G ”)tunedbyafuzzylogiccontroller(Mamdanitechnique). Thismeans
S T
thatourmaincontrollerisaRSTwithMRACbasedonMITrules, andforthetuningpurposewe
use the Mamdani fuzzy logic controller. We have to implement the gains of RST by addingfuzzy
logicbetweenuniformscalesofmembershipfunctions. Itshowsthebestresultsascomparedtothe
adaptiveRSTcontroller[21,22].
Inthispaper,weareincorporatingRSTcontrollerwithourproposedsystemintwoseparateways.
First,thesystemisundergoesthroughRobustadaptiveRSTcontroller,afterthatweuseFuzzy-Hybrid
basedMITalgorithmandthenconcludetheresultsbytakingthedifferenceofrobustness.
Thecorecontributionsinthisresearchareasfollows: (1)anovelfuzzy-basedadaptiverobust
RST controller is derived by accumulating the MIT rule in the control law to remove the model
disturbanceandtoderivethesteady-stateerrortozero;(2)theproposedcontrollerusestheangular
responsesasaninputcontrolcommand,whichshowsmoreaccurateandpracticalinsightinthereal
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world;(3)inspiteofthemodeldisturbance,thecloseloopsystemerrorconvergestozero,provedin
Theorem2;and(4)lastly,thepolynomialcharacteristicsolutionisbasedontheDiophantineequation
whileleastsquareestimationisusedtochecksystemstabilityandprovedinTheorem3.
Thebreakupofthispaperisstructuredasfollows. Thesystemmodeling,dynamicrepresentation
of a tri-rotor UAV, and main engine model is discussed in Section 2. Section 3 demonstrates the
dynamiccontrolstrategiesandthecontrolalgorithmoftheUAV.Moreover,thesimulationresultsand
discussionsarediscussedinSection4. Lastly,Section5statestheconclusions.
2. SystemModelandPreliminaries
2.1. Tri-RotorModeling
The equation of motion of a rigid body is defined by Newton’s second law of motion [23,24].
Linearandangularforceschangewithrespecttothetimeframe,calledtheinitialreferenceframe,in
whichtheUAVhasasimilarvelocity,forcecomponents,andmoments,whichareusedtodevelopthe
sixdegreesoffreedomnonlinearequationsofmotion. Thenonlinearaerodynamicforces,aerodynamic
moments, rotation motion, and translational motion of a UAV are defined by using differential
Equations(1)–(4).
Equationsofaerodynamicforce:
` ˘
.
FX´mgsinθ“m u``qw´rv ˘
.
FY`mgcosθsinϕ“m`v`ru´pw˘ (1)
.
F `mgcosθcosϕ“ m w`pv´qu
Z
Equationsofaerodynamicmoments:
` ˘
. .
L“Ixp´ Ixzr `qr Iz´ Iy`´ Ixzpq˘
.
M“Iyq`rp pIx´ Izq`` Ixz p˘2´ r2 (2)
. .
N“´I p`I r `pq I ´ I ` I qr
xz z y x xz
Rotationalrates:
. .
p“ϕ´ψsinθ
. .
q“θcosϕ`ψcosθsinϕ (3)
. .
r “ψcosθcosϕ´θsinϕ
Euleranglesandbodyangularvelocities:
.
θ“ qcosϕ´rsinθ
.
ϕ“ p`qsinϕtanθ` rcosϕtanθ (4)
.
ψ“ pqsinϕ`rcosϕqsecθ
Thefourcontrolcommandsofthetri-rotorUAVare(Col,Lat,Lon,Ped),whichissimilartothe
conventionalhelicopter,inwhichColisCollective,LatisLateral,LonisLongitudinal,andPedispedal.
(Col,Lat)areusedtocontroltherollrate,(Lon)controlthepitchrateand(Ped)controlstheyawrate
ofaUAVandtiltanglebyusingtheparameter“9”[25,26]. (p,q,r)and(U,V,W)aretherotational
velocityandtranslationalvelocityofthecoordinatesystem. (L,M,N)and(φ,θ,ψ)aretheexternal
momentsandrotationalanglesofafixedbodyframe.
Atri-rotoraerialvehicleexhibitsmanyphysicaleffects,likeinertialtorque,effectsofaerodynamics,
effectsofgravity,effectsofgyroscopeandfrictionaleffects,etc. Inthepresenceofthesephysicaleffects
itisquitedifficulttodesignacontrollerwhichcaneasilyhandleallofthephysicaleffectsandstabilize
theUAVinafairamountoftime,becauseithassixdegreeoffreedom(6-DOF)withahighly-nonlinear,
multivariable,under-actuated,strongly-coupledmodelwiththerotors,asshowninFigures1and2
takenfromthedesignofMohamedMK[27].
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FiFFgiiuggruuerree1 .11T.. oTTpooppv ivveiiweewwo oofffa aat rttrri-iir--rroootttooorrrU UUAAAVVVa aalllooonnnggg wwwiiittthhh rrroootttaaatttiiiooonnnaaalll aaannnddd tttrrraaannnsssiittiiitooionnnaaall lrraarttaeetsse..s .

Figure 2. 3-D view of a tri-rotor UAV along with aerodynamic moment components.
FigFuigruer2e. 23. -3D-Dv iveiwewo offa at rtri-ir-roottoorr UUAAVV aalloonngg wwiitthh aaeerrooddyynnaammiicc mmoommeennt tcocommpponoennetnst. s.
The tri-rotor UAV, including translational and rotational subsystems, needs a vibrant strategy
The tri-rotor UAV, including translational and rotational subsystems, needs a vibrant strategy
of Tnhoenltirnie-raor tsoerqUueAnVti,ailn ccolundtrionl gint rtahnes l6a-tDioOnFa lmaonddelr.o Ttahtiiso rneasleasrucbhs yadstdermesss,ens eeerdrosra covnibtrroalnlitnsgt riant eag y
of nonlinear sequential control in the 6-DOF model. This research addresses error controlling in a
ofntroi-nrolitnoer aarirscerqaufte bnyti amlacnoangtirnogl itnhet htoer6q-uDe OpFromduocdeedl .bTy huinspreaisreeadr crhotaodr dreraecstsieosnse.r rToor ocvoenrtcroomllien tghiisn a
tri-rotor aircraft by managing the torque produced by unpaired rotor reactions. To overcome this
tri-irsostuoer, aiimrcprlaefmtbeyntmataionna goifn gditfhfeeretonrtq dueespigrnosd uhcaesd bbeyenu nmpaadiere wdirtoht othrer ehaecltpio nosf .BTLoDoCv emrcootmores,t hwishiiscshu e,
issue, implementation of different designs has been made with the help of BLDC motors, which
impacletumalelny tcaotniotrnool tfhdei nffuelrleifnytindge sainggnles. hTahsisb meeenthmoda dise uwseiftuhl tfhoer qhueilcpkeorf tBuLrnD bCy mtilotitnogr st,hwe rhoitcohr’as catxuisa.l ly
actually control the nullifying angle. This method is useful for quicker turn by tilting the rotor’s axis.
controlthenullifyingangle. Thismethodisusefulforquickerturnbytiltingtherotor’saxis.
2.2. Dynamic Representation of a Tri-Rotor UAV
2.2. Dynamic Representation of a Tri-Rotor UAV
2.2. DynamicRepresentationofaTri-RotorUAV
The orientation of the UAV is explained by Euler angles having altitude, roll ((cid:2016)), pitch ((cid:2030)), and
The orientation of the UAV is explained by Euler angles having altitude, roll ((cid:2016)), pitch ((cid:2030)), and
yyaaTwwh e(((cid:2006)(cid:2006)o))r icceoonnntattrrtooioll naaonnfddt hiitte rrUooAttaaVtteeisss aaellxoopnnlgga in((xxe,,d yyb,, yzzE)) uaalxxeeerssa,, nrreegssleppseecchttaiivvveeinllyyg.. aTTlhhtieetu ttdrraaenn,rssollaallttii(ooθnn),aapll iaatcnnhdd (rrϕoo)tt,aaattniioodnnyaalla w
(Ψ)mmcooovvneetmmroeelnnattn oodff ittthhreeo tttarriit--errsoottaoolrro nUUgAA(VVx ,iiynn,ttzoo) ttahhxeee dds,iimmreeesnnpsseiiooctnniaavlle lssypp.aaTcceeh eaanntrdda nddsyylnnaaatimmoniiccassl ooaffn dtthhreeo rrtiiaggtiiiddo nbbaoolddmyy oddveeerriimvveee nt
oftffhrrooemmtr iNN-reeowwtottoornnU’’ssA llVaawwin.. tMMoootrrheeeoodvveeimrr,, emmnoosmmioneennattlsss,, pffooarrccceeessa,,n vvdeelldooycciinttyyam ccooimmcsppooofnnteehnnettssr,,i gaainndddb aaoeedrrooydddyyennraaimmveiiccf rccooommmppNooennweenntottssn ’s
are described in Table 1.
lawa.reM doersecorvibeerd,m ino Tmaebnlets 1,.f orces,velocitycomponents,andaerodynamiccomponentsaredescribedin
Table1. Table 1. Dynamic constants of a tri-rotor UAV.
Table 1. Dynamic constants of a tri-rotor UAV.
Table1.Dynamicconstantsofatri-rotorUAV.
x, y, z Axis System Roll(φ) Pitch (θ) Yaw (ψ)
x, y, z Axis System Roll(φ) Pitch (θ) Yaw (ψ)
Aerodynamic Force Components X Y Z
x,yA,zerAodxiysnSaymstiecm Force ComRpoolnle(φnt)s XP itch(θ) Y YawZ( ψ)
Aerodynamic Moment Components L M N
Aerodynamic Moment Components L M N
AerodynaTmraincsFloartcioenal Velocity U V W
Translational Velocity X U Y V WZ
Components
Angular Rates p q r
AerodynamicMAnomguelnatr Rates p q r
ComponTTehhnrrteesee--AAxxiiss IInneerrttiiaa L IIxx M IIyy NIIzz
TranslationalVelocity U V W
TThhee oovveerraallll Assyynssgtteeummlar ccRoonnatffeiiggsuurraattiioonn iiss ddeeffiinnpeedd iinn FFiigguurree 22,, wwhhqeerree ““LL”” iiss tthhee ddiissttaarnnccee ffrroomm tthhee cceenntteerr
ooff tthhee bbooddyy ffrraaTmmheer etteoo- Aaallxll ittshhIrrneeeeer trriooattoorrss,, llaabbeelllleeddI xaass LL11,, LL22,, aanndd LL33I.. yTThhee rroottoorr ffoorrcceess aaIzrree ff11,, ff22,, ff33,, aanndd tthhee
rotor torque is defined as (cid:2028) ,(cid:2028) , and (cid:2028) , respectively. The angular velocity of the system is “(cid:2025)”.
rotor torque is defined as (cid:2028)(cid:2869),(cid:2028)(cid:2870), and (cid:2028)(cid:2871), respectively. The angular velocity of the system is “(cid:2025)”.
(cid:2869) (cid:2870) (cid:2871)
Theoverallsystemconfigurat ionisdefinedinFigure2,where“L”isthedistancefromthecenter
ofthebodyframetoallthreerotors,labelledasL1,L2,andL3. Therotorforcesaref1,f2,f3,andthe
rotortorqueisdefinedasτ , τ ,andτ ,respectively. Theangularvelocityofthesystemis“ρ”.
1 2 3
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2.3. MainEngine(ElectricMotors)
BrushlessDirectCurrent(BLDC)motorscanbeusedasamainengineoftheUAVtoachieve
requiredelectricpropulsioninthesystem. BLDCmotorsfoundtheirapplicationinthefieldofrobotics,
spacecraftsandmedicaldevices,duetohighertorquespeedfeatures,greaterperformance,minimum
repairsandvariabledegreeofspeed[28]. GenerallyBLDCismorecostlythansimpleDCmotors,
duetoitsbetterefficiencyandreliability[29]. TheelectricalandmechanicalequationsofBLDCare
givenbelow.
di
V“Ri`pL´Mq `E (5)
dt
E“K ω Fpθ q (6)
e m e
T“Ki Fpθ q (7)
t a e
d2θ dθ
T “J m `β m (8)
e dt2 dt
P
θ “ θ (9)
e m
2
dθ
ω “ m (10)
m
dt
Inwhich,Vistheappliedvoltage;Risthetotalresistance;Eisthebackelectromagneticforce;L
isthetotalinductanceofthemotor;Misthemutualinductanceofthemotor;ω istheangularspeed
m
ofthemotor;θ istherotationangleofthemotor;T istheelectricaltorqueproducedbythemotor;
e e
andK istheback-EMFconstant.
e
3. DesigningofController
3.1. Tri-RotorDynamicControlStrategies
Theflightdynamicsandcontrolstrategyofatri-rotorUAVisthesameastraditionalaircraft. The
placementororientationofflightdynamicscontrolisaproductofroll,pitch,andyaw. Thecontrol
schemeoftheUAVincludesaltitude,roll,pitch,yaw,andtiltanglecontrol,havingamajorroleforthe
displacementcontroltheparametersofthesystem.
AltitudeControlMechanism: Toachievethedesiredaltitude,thespeedofallrotorsmustbesame
ρ1=ρ2=ρ3. IncreasingthespeedofallrotorsconstantlywilleventuallyraisethealtitudeoftheUAV,
suchthattheangularvelocityofmotorsbecomesequal.
RollControlMechanism: Rollcontrolisachievedbyregulatingthefrontrotorsspeed. Decreasing
rotor1velocity,rollsthesystemtotheleftandrotor2rollsthesystemtowardsright-side. Rollcontrol
hastwoconditions.
i Whenmovingclockwiserollρ2ąρ1ąρ3.
ii Whenmovingcounter-clockwiserollρ2ăρ1ăρ3.
PitchControlMechanism: RegulatingthespeedofRotor1andrearwardrotorswillchangethe
pitch. ThesystempitchesdownwardifthespeedofRotors1and2decreaseswhileRotor3speed
keepsrising. IfwedecreasethespeedofRotor3andincreasethespeedofRotors1and2,theUAV
pitchrisesandflyflightreverses. Pitchcontrolalsohastwoconditions:
i Whennose-upρ2“ρ3ąρ1.
ii Whennose-downρ2“ρ3ăρ1.
YawControlMechanism: Theproductofreactiontorqueandtiltangle“9”ofRotor3isusedto
controlyawmovement. ThevalueofthetiltangleistoosmallandhelpstomaneuvertheUAVquickly.
Theyawcontrolconditionis: ρ1“ρ2“ρ1,with9“ 0.
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3.2. ControlAlgorithm
Inthissectiontheoverallcontrolhierarchyisdefinedtocontroltheattitudeandaltitudeofthe
tri-rotorUAV.InwhichweassumethedesiredattitudevariablesareK “ ∅(roll),K “ θ (pitch),
1 2
andK “ Ψ(yaw),respectively,andK isageneralizedtermfortherotationalanglesofthesystem.
3 T
NowthecontrolalgorithmfortheattitudecontrollingofanactualsystemoftheUAViswrittenas:
B pjq
G pjq“ “Yptq (11)
KT1 Apjq
ThedegreeofthesystemmodelnumeratorBpjqanddenominator Apjqisfoundtobe“1”and
“3”. WhereB“pB`˚B´q,suchthatB`isvariableandB´ istheconstant.
Equation(12)presentsthedesiredattituderesponseofourUAVmodel:
B pjq
m
G pjq“ “Y ptq (12)
KT2 A pjq m
m
Nowthegradienttheorywhichwasdefinedbythemodelreferenceadaptivecontrolmethod
isimplemented:
degA “2˚degApjq´1“5 (13)
c
So,theRSTcontrollerwillbesecond-ordered,andnowdegA “degApjq´degB`´1“ 1.
0
Remark 1. For the perfect system model A pjq “ Apjq and B pjq “ Bpjq. Where A and A is the
m m C Cm
characteristic polynomials of the actual and desired system models and A “ A for the constraints and
Cm C
will not affect the system to change the close loop poles of the model. Otherwise, A differes the system
Cm
modelmismatch.
FuzzyLogicController. Thevibrantperformanceofthefuzzylogiccontrollerisdescribedby
the set of linguistic procedures that was established by a knowledgeable acquaintance in [30–32],
in which the system “error” and variation in error rate are the input constraints, and RST are the
variableoutputsinourproposedcontroller. Formerly,RSTcanbeimprovedonline,usingthesetof
rules,existingerror,andvariationintheerror. Ingeneral,theerrorintheangle,combinedwiththe
mechanismoutput,increases. Furthermore,thecontrollerperformswellwhethertheerrorrisesor
therateoferrordifferencefalls. Itisimportantthat,intheminorerrorphase,theearmarkedcontrol
outputisrequiredtoinfluencethechangeinerrorassoonastheerrorfallssuddenly.
WiththehelpofEquation(13),thedegreeoftheproposedcontrollerisfoundtobe5. Thefuzzy
logic-basedadaptiveRSTcontrolleriswritteninEquations(14)–(16):
FR“q2`pr ˆqq`r (14)
0 1
FS“ps ˆq2q`ps ˆqq`s (15)
0 1 2
FT“pt ˆqˆA q`pt ˆA q (16)
0 0 1 0
PutthevaluesofFR, FS, FTintheabove:
de
U pjq“ FRpe, qG ˆepjq (17)
FRKT dt R
de
U pjq“ FSpe, qG ˆΣepijq∆ptq (18)
FSKT dt S
de ∆epjq
U pjq“ FTpe, qG ˆ (19)
FTKT dt T ∆t
whereG , G andG arethedelayedcontrolgainsofthesignalscalingfactors.
R S T
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Remark2. G pjq andG pjq,arethesystemmodels. Moreover,thesystemdivergesatalowphaseangle
KT1 KT2
wSeinthsorasn 20u1n6s, t1a6b, l6e52c ontrolsignal. Theproposedcontrolalgorithmisnotbetterinthiscase. Previously,in[373 o]f, 1a7
bettercontrollerwasproposedforthistypeofcase,amodelhavingalowphaseanglelyinginthecomplexplane
wwiitthh ssoommeed damamppininggis issuseuse.sA. sAas rae sruelstu,litn, [i3n4 []3,4ze],r ozecraon ccealnlacteilolnatiinont hiens ythsete msysistesmit uisa tseidtuesaotetedr iecstohteeraicr etahwe hairceha
willcancel.
which will cancel.
Remark3. Theproposedcontrollerdesignatsamplingoftime[NT],whichwillremovethedifficultiesinthe
Remark 3. The proposed controller design at sampling of time [NT], which will remove the difficulties in the
planningstageofcontrollerimplementation.
planning stage of controller implementation.
Theorem1. Afterdesigningthecontrolsystem,thenextstepisstabilityanalysis,whichshowstherobustness
lTevheeloorfetmhe 1d. eAsifgtenre ddecsoignntrinolgl etrh.e Tcohnetsrtoal bsiylisttyemis, thhieg hnleyxtv sutlenpe risa bstleabdiuliteyt oanmaloydseisli,n wgheicrrho srhsocwasll ethdes reonbsuitsitvnietsys.
Tlehveerle foofr et,hem dodeseilgmneidsm caotncthroalnledr. mTohdee lstsaebnisliittyiv iitsy haigrehlayd dveudlninerathbele sdyustee mto dmivoedregleinngc eebrreotwrse ecnalltehde saecntusiatlivaintyd.
desiredresponsedependingupontheperformanceofthecontrolsystemanditsstabilitywhichwillbetaken
Therefore, model mismatch and model sensitivity are added in the system divergence between the actual and
from[34]Theorem5.4,andexaminethestabilityofthecontrolsystembyusingthemodeldisturbance.
desired response depending upon the performance of the control system and its stability which will be taken
Lemma. Theproposedequationillustratesthecloseloopsystemmodelhaving(NT)samplingperiod:
from [34] Theorem 5.4, and examine the stability of the control system by using the model disturbance.
” ı ” ı ” ı
Lemma. The proposedY eTquptaqtioNnT illustratGesT the pcjqloGseT looppj qsyHsTte,NmT mNoTdeGl NhaTvipnjqg (FNTTNT) sampling period:
“ ”KT1 KT2 ı KT1 F”RNT ı
F[RYN(cid:2904)T(t)](cid:2898)(cid:2904) 1`(cid:3427)GG(cid:2895)(cid:2904)TK(cid:2904)(cid:2869)T(1jp)jGq(cid:2895)(cid:2904)G(cid:2904)TK(cid:2870)(Tj2)Hpjq(cid:2904),H(cid:2898)(cid:2904)T(cid:3431),N(cid:2898)(cid:2904)TGN(cid:2895)(cid:2898)(cid:2904)(cid:2904)T(cid:2869)G(jNK)T(cid:3428)TFF1TRpj(cid:2898)(cid:2898)q(cid:2904)(cid:2904)(cid:3432)FFRSNNTT
=
FR(cid:2898)(cid:2904) FS(cid:2898)(cid:2904)
HT,NT istheconversionratean1d+w i[llGd(cid:2895)(cid:2904)(cid:2904)e(cid:2869)p(ej)nGd(cid:2895)(cid:2904)o(cid:2904)(cid:2870)n(tj)hHe(cid:2904)o,(cid:2898)r(cid:2904)ie]n(cid:2898)(cid:2904)taGt(cid:2895)(cid:2898)io(cid:2904)(cid:2904)n(cid:2869)(oj)f(cid:3428)tFhRe(cid:2898)d(cid:2904)e(cid:3432)siredsignal.
Rema(cid:1834)rk(cid:3021),(cid:3015)4(cid:3021). iTsh tehdei sctounrvbaenrsceioinn trhaetep oalnedp lwacielml denetpmenedth oodn itshoeb soorlieetentinat[i3o4n] .oSf itnhcee dtheesirreefedr esnicgenmalo. del,observer
polynomial,andreferencemodeldisturbanceactasconstraints,theyareprovedinconvergenceanalysis.
Remark 4. The disturbance in the pole placement method is obsolete in [34]. Since the reference model, observer
ProofofTheorem1. Theoutputoftheunvaryingmodelisconsideredin[35]:
polynomial, and reference model disturbance act as constraints, they are proved in convergence analysis.
Proof of Theorem 1. The output of the unv”arying mıoNdTelF iTs NcoTnsidered FinS N[3T5]: NT
YTptq“GT pjqGT pjqHT,NT GNT pjq r FRNT´ rYTptqs sT
KT1 KT2 KT1 FRNT FRNT
Y(cid:2904)(t)= G(cid:2904) (j)G(cid:2904) (j)H(cid:2904),(cid:2898)(cid:2904) (cid:3427)G(cid:2898)(cid:2904) (j)(cid:3431)(cid:2898)(cid:2904)[(cid:2890)(cid:2904)(cid:3146)(cid:3152) FR(cid:2898)(cid:2904)−(cid:2890)(cid:2903)(cid:3146)(cid:3152) [Y(cid:2904)(t)](cid:2898)(cid:2904)](cid:2904)
(cid:2895)(cid:2904)(cid:2869) (cid:2895)(cid:2904)(cid:2870) (cid:2895)(cid:2904)(cid:2869) (cid:2890)(cid:2902)(cid:3146)(cid:3152) (cid:2890)(cid:2902)(cid:3146)(cid:3152)
Remark 5. Figure 3, gives complete work flow of proposed system using model reference adaptive
Remark 5. Figure 3, gives complete work flow of proposed system using model reference adaptive control
controlalgorithm.
algorithm.

Figure3.Themodelreferenceadaptivecontrolsystem.
Figure 3. The model reference adaptive control system.
CCoonnvveerrggeennccee:: TTaakkininggt htheei nipnuptuct ocnosntsratrianitnstosf oofu orucro ncotrnotlrloerllesirg sniaglnsaglsiv geisvtehse tchleo sceloloseo ploeorpro errorofrth oef
styhset esmystmemod meloadnedl uannsdt aubnlestpaabrlte opfatrhte odf itshtuer dbaisntucer.bIanncoeth. eInr wotohredrs ,woourrdpsr, oopuors pedroaplgooserdit hamlgoisriatbhlme tios
satbalbei ltioz esttahbeiluiznes ttahbel eupnasrtatbolfet hpearnt ooifs ethoer ndoisistue robra ndcisetumrobdanelc.eT mheodcoenl. vTehrge ecnocnevaetrgtheencdee saitr ethdev daelusiereodf
pvaarlaume oetfe prasriasmdeotneersb iys danonoep btiym aanl ocopntitmroallm coenthtroodl mbaestehdodo nbathseedM oInT thruel MetIoTi rduelnet tiofy idtheneteifryro trhse. errors.
Remark 6. For the proposed controller the adaptive gain is in the range of 0.15 to 5 and above this range the
controller performance deteriorates.
e(j)=Y(cid:2911)(cid:2913)(cid:2930)(cid:2931)(cid:2911)(cid:2922)(t)−Y(cid:2923)((cid:2914)(cid:2915)(cid:2929)(cid:2919)(cid:2928)(cid:2915)(cid:2914)) (20)
The sensitivity derivative is presented in Equation (21):
Sensors2016,16, 652 8of18
Remark6. Fortheproposedcontrollertheadaptivegainisintherangeof0.15to5andabovethisrangethe
controllerperformancedeteriorates.
epjq“Y ptq´Y (20)
actual mpdesiredq
ThesensitivityderivativeispresentedinEquation(21):
BpjqT
Yptq“ (21)
pApjqR`BpjqSq
ApjqR`BpjqS
U ptq“ ˚Yptq (22)
c
BpjqT
Theconvergenceproofcontractsdistinctlywiththeconstraints,firstlyidentifyingthesensitivity
derivativeofalloftheparameters(t , t ,r , r ,s , s , s )ofthecontroller.
0 1 0 1 0 1 2
TheDiophantineequationrepresentsasA “ApjqR`BpjqSandA “A A pjq;therefore:
C C 0 m
ApjqR`BpjqS“ A A pjq (23)
0 m
TheMITrule-basedsensitivityderivativeofEquation(20)is:
BpjqT
epjq“ ˚U ptq´Y (24)
c m
ApjqR`BpjqS
Theorem2. ThemodelinEquation(11)withcontrollerEquation(14–16)onthebasisofsystem(Apjq,Bpjq),
havingUAVmodeldisturbance,theerrorofcloseloopoutputEquation(24)goestozeroasymptoticallyifand
onlyif A = A .
Cm C
ProofofTheorem2. ThecloseloopoutputerrorEquation(24)issettledby A andcomparedwith
C
Equation(23).
1. If,andonlyif,Apjq“A pjqandB pjq“ B pjq;therefore,thecloseloopoutputerrorresponds
m m
toepjqandgoestozeroasymptoticallybecauseA isstableinEquation(13)andA issupposedto
C Cm
bestable.
2. Contradiction: if Apjq ‰ A pjq and B pjq ‰ B pjq, the value is not cancelled by the close
m m
loopoutputerrorinEquation(24). Theinstabilityisinthedenominatorofepjqif A isunstable.
Cm
Hereafter,ifA ‰A orA isunstable,thenepjqcloseloopoutputerrorraises,unbounded,andit
C Cm Cm
isnecessaryforA tobestabletomakethecloseloopoutputerrorzero.
Cm
NowputinthevalueofT“pt q`t )A inEquation(24):
0 1 0
Bpjqpt q`t qA
epjq“ 0 1 0˚U ptq´pY q (25)
c m
ApjqR`BpjqS
Therefore,w.r.t“t ”,thepartialderivativeofEquation(25)is:
0
$
’’’’& δδ´petp0jqq “ pApBjqpRjq`q¯ABp0jqSq ˚Ucptq
δepjq BpjqqA ApjqR`BpjqS
“ 0 ˚ ˚Yptq (26)
’’’’% δpt0q pApδjqeRp`jqB“pjqSqA0q ˚YpBtpqjqT
δpt0q T
Sensors2016,16, 652 9of18
ThepartialderivativeofEquation(25)w.r.t“t ”is:
1
$ ´ ¯
’’’’& δδpet´p1jqq “ pApjqRB`pj¯qBpjqSq ˚Ucptq
δepjq Bpjq ApjqR`BpjqS
“ ˚ ˚Yptq (27)
’’’’% δpt1q pApjqδRe`pjqBp“jqSqA0 ˚YpBtqpjqT
δpt1q T
FromEquation(24),replacethevalueofR:(R=pqq2`r q`r )
0 1
BpjqT
` ˘
epjq“ ˚U ptq´pY q (28)
Apjq q2`r q`r `BpjqS c m
0 1
PartialdifferentiateEquation(25)w.r.t“r ”:
0
$ ˆ ˙
’’’’’’& δδperp0jqqˆ“´ pApjBqRpj`qTB˙ApSjqSq2 ˚Ucptq
δepjq BpjqTAS ApjqR`BpjqS (29)
’’’’’’% δpr0q “´ δpeApjpqjqR“`´BpjqSqA2S ˚˚YBpptjqqT ˚Yptq
δpr0q A0Ampjq
PartialdifferentiateEquation(25)w.r.t“r ”:
1
δepjq Apqq
“´ ˚Yptq (30)
δpr q A A pjq
1 0 m
PutthevalueofS,S“ps ˚q2`s ˚q+s qinEquation(24):
0 1 2
BpjqT
` ˘
epjq“ ˚U ptq´pY q (31)
ApjqR`Bpjq s q2`s q`s c m
0 1 2
NowthePartialderivativeofEquation(31)w.r.t“s ”:
0
$ ´ ¯
’’’’’’’& δδpesp0jqq “´ ´pApBjpqjRq¯2`˚Bpqjq2Sq2˚Ucptq
’’’’’’’% δδpesp0jqq “´δeppAjqpBjpqjRq2`˚BpBqjqp2Sjqq2˚Aqp2jqBRp`jqTBpjqS ˚Yptq (32)
“ ´ ˚Yptq
δps0q A0Ampjq
NowtakingPartialderivativeofEquation(31)w.r.t“s ”
1
δepjq Bpjqq
“´ ˚Yptq (33)
δps q pA pjqA q
1 m 0
Likewise,PartialderivativeoftheEquation(31)w.r.t“s ”gives
2
δepjq Bpjq
“´ ˚Yptq (34)
δps q pA pjqA q
2 m 0
ByapplyingtheMITalgorithminthedesiredmodelofthesystem,whereEquations(35)and(36)
denotethecostfunctionwhichisbasedontheMITrule. Whitakerdemonstratesthedifferencein
systemboundsasafunctionofthesystemerrorandthegradientofthesystemerrorwithrespecttothe
systemconstraintsandtakesthepartialderivativeofthegradienterrorwithrespecttoitsconstraints.
Theconstraintsoftheparticularmodelwiththeinitialestimate JpKTq,andtherateofchangeofspeed
Sensors 2016, 16, 652  9 of 17 
δe(cid:4666)j(cid:4667) B(cid:4666)j(cid:4667)TAS
(cid:1747) (cid:3404)(cid:3398)(cid:4678) (cid:4679)∗U (cid:4666)t(cid:4667)
δ(cid:4666)r (cid:4667) (cid:4666)A(cid:4666)j(cid:4667)R(cid:3397)B(cid:4666)j(cid:4667)S(cid:4667)(cid:2870) (cid:2913)
(cid:1750) (cid:2868)
(cid:1750)
δe(cid:4666)j(cid:4667) B(cid:4666)j(cid:4667)TAS A(cid:4666)j(cid:4667)R(cid:3397)B(cid:4666)j(cid:4667)S
(cid:3404)(cid:3398)(cid:4678) (cid:4679)∗ ∗Y(cid:4666)t(cid:4667)  (29)
(cid:1748)δ(cid:4666)r (cid:4667) (cid:4666)A(cid:4666)j(cid:4667)R(cid:3397)B(cid:4666)j(cid:4667)S(cid:4667)(cid:2870) B(cid:4666)j(cid:4667)T
(cid:2868)
(cid:1750)(cid:1750) δe(cid:4666)j(cid:4667) (cid:1827)(cid:1845)
(cid:3404)(cid:3398) ∗Y(cid:4666)t(cid:4667)
(cid:1749) δ(cid:4666)r (cid:4667) A A (cid:4666)j(cid:4667)
(cid:2868) (cid:2868) (cid:2923)
Partial differentiate Equation (25) w.r.t “r ”: 
(cid:2869)
δe(cid:4666)j(cid:4667) A(cid:4666)q(cid:4667)
(cid:3404)(cid:3398) ∗Y(cid:4666)t(cid:4667)  (30)
δ(cid:4666)r (cid:4667) A A (cid:4666)j(cid:4667)
(cid:2869) (cid:2868) (cid:2923)
Put the value of S, S(cid:3404)(cid:4666)s ∗q(cid:2870)(cid:3397)s ∗q+s (cid:4667) in Equation (24):  
(cid:2868) (cid:2869) (cid:2870)
B(cid:4666)j(cid:4667)T
e(cid:4666)j(cid:4667)(cid:3404) ∗U (cid:4666)t(cid:4667)(cid:3398)(cid:4666)Y (cid:4667)  (31)
A(cid:4666)j(cid:4667)R(cid:3397)B(cid:4666)j(cid:4667)(cid:4666)s q(cid:2870)(cid:3397)s q(cid:3397)s (cid:4667) (cid:2913) (cid:2923)
(cid:2868) (cid:2869) (cid:2870)
Now the Partial derivative of Equation (31) w.r.t "s ": 
(cid:2868)
δe(cid:4666)j(cid:4667) (cid:4666)B(cid:4666)j(cid:4667)(cid:2870)∗q(cid:2870)(cid:4667)
(cid:1747) (cid:3404)(cid:3398) ∗U (cid:4666)t(cid:4667)
δ(cid:4666)s (cid:4667) (cid:4666)A(cid:4666)j(cid:4667)R(cid:3397)B(cid:4666)j(cid:4667)S(cid:4667)(cid:2870) (cid:2913)
(cid:1750)(cid:1750) (cid:2868)
δe(cid:4666)j(cid:4667) (cid:4666)B(cid:4666)j(cid:4667)(cid:2870)∗q(cid:2870)(cid:4667) A(cid:4666)j(cid:4667)R(cid:3397)B(cid:4666)j(cid:4667)S
(cid:3404)(cid:3398) ∗Y(cid:4666)t(cid:4667)  (32)
(cid:1748)δ(cid:4666)s(cid:2868)(cid:4667) (cid:4666)A(cid:4666)j(cid:4667)R(cid:3397)B(cid:4666)j(cid:4667)S(cid:4667)(cid:2870) B(cid:4666)j(cid:4667)T
(cid:1750)(cid:1750) δe(cid:4666)j(cid:4667) B(cid:4666)j(cid:4667)∗q(cid:2870)
(cid:3404) (cid:3398) ∗Y(cid:4666)t(cid:4667)
(cid:1749) δ(cid:4666)s (cid:4667) A A (cid:4666)j(cid:4667)
(cid:2868) (cid:2868) (cid:2923)
Now taking Partial derivative of Equation (31) w.r.t "s " 
(cid:2869)
δe(cid:4666)j(cid:4667) B(cid:4666)j(cid:4667)q
(cid:3404)(cid:3398) ∗Y(cid:4666)t(cid:4667)  (33)
δ(cid:4666)s (cid:4667) (cid:4666)A (cid:4666)j(cid:4667)A (cid:4667)
(cid:2869) (cid:2923) (cid:2868)
Likewise, Partial derivative of the Equation (31) w.r.t "s " gives 
(cid:2870)
δe(cid:4666)j(cid:4667) B(cid:4666)j(cid:4667)
(cid:3404)(cid:3398) ∗Y(cid:4666)t(cid:4667)  (34)
δ(cid:4666)s (cid:4667) (cid:4666)A (cid:4666)j(cid:4667)A (cid:4667)
(cid:2870) (cid:2923) (cid:2868)
By applying the MIT algorithm in the desired model of the system, where Equations (35) and 
(36) denote the cost function which is based on the MIT rule. Whitaker demonstrates the difference 
in system bounds as a function of the system error and the gradient of the system error with respect 
to the system constraints and takes the partial derivative of the gradient error with respect to its 
Sensors2016,16, 652 10of18
constraints. The constraints of the particular model with the initial estimate (cid:1836)(cid:4666)(cid:1837)(cid:1846)(cid:4667), and the rate of 
change of speed among the desired and actual model is taken from [36]. To minimize the error with 
respeacmt oton gtitmhee dseos itrheadt atnhde adcetusiarlemd ordeseploisntsaek eisn afrcohmiev[3e6d] .rTeoqumiriensim ainz eotphteimerurmor wcointhtrroels pweicttht ocotismt eso
functtihoant. thedesiredresponseisachievedrequiresanoptimumcontrolwithcostfunction.
` ˘
TheoTrhemeo r3e. mIn3 [.2I0n] [t2h0e ]lethasetl esaqsutasrqeu easrteimesattiimona t(cid:1836)io(cid:4666)(cid:1837)(cid:3364)n(cid:3364)(cid:3364)(cid:1846)(cid:3364)J(cid:4667) KanTd tahneidr tphoeliyrnpoomlyinalo mchiaarlacchtaerriascttiecr sisotliuctisoonlu dtieopnenddesp ends
uponu tphoe nDtihoephDainotpihnaen etqinueateiqouna atniodn satnabdilsittayb oilfi t(cid:1827)yof. ACm.
(cid:3004)(cid:3040)
ProofofTheorem3. ThesolutionalsodependsupontheDiophantineequationaswellasthestability
Proof of Theorem 3. The solution also depends upon the Diophantine equation as well as the stability 
of A . ByusingTheorem2:
of (cid:1827) . ByCm using Theorem 2: 
(cid:3004)(cid:3040)
` ˘
1. 1I.f (cid:1857)(cid:4666)(cid:1862)I(cid:4667)f →epj0q Ñas 0(cid:1864) →as∞l Ñ an8d laenaddilnegad (cid:1836)i(cid:4666)n(cid:1837)(cid:3364)g(cid:3364)(cid:3364)(cid:1846)(cid:3364)J(cid:4667)(cid:3404)KT0, t“he 0s,otlhuetisoonl ugtiivoensg tihvee ssmthaellsemsta plloesstitpivoes ivtiavleuev aolfu eof
cost fcuonstctfiuonnc. tion. ` ˘
2. 2 (cid:1827). (cid:3004)(cid:3040) AisC ma icsoantcroandtircatdioicnt,i oifn ,iti fisit nisont osttasbtaleb lferofrmom ThTehoeroermem 22; ;(cid:1857)e(cid:4666)p(cid:1862)j(cid:4667)q⇸00 aassl (cid:1864)Ñ→8∞,, tthhuussJ (cid:1836)(cid:4666)K(cid:1837)(cid:3364)(cid:3364)T(cid:3364)(cid:1846)(cid:3364)(cid:4667)‰`(cid:3405)00..˘ This
This iiss nnootto opptitmimalabl ebceacuasues,ea,s asse esneeinn sinte pst1epo f1t hoef pthroeo pf,rtohoefr, ethexeirset seaxissotslu ati osnoltuhtaiotnm tahkaets mJ aKkTes “0.
(cid:1836)(cid:4666)(cid:1837)(cid:3364)(cid:3364)(cid:3364)(cid:1846)(cid:3364)(cid:4667)(cid:3404)0.
1
JpK1Tq“ e2pKTq (35)
J(cid:4666)KT(cid:4667)(cid:3404) e(cid:2870)(cid:4666)KT2(cid:4667)  (35)
2
dKT l
“ ´γ1 y e“ ´γ˚y ˚epjq (36)
pdtq l pmq pmq
o

Afterthat,applytheMITruletoderivatecontrolvariables(s , s , s , r , r , t , andt qandsetin
0 1 2 0 1 0 1
thecontrollergives
dps q δe dps q γ q2BpjqYptq
0 “ ´γ 0 “ e (37)
dt eδs dt A pjqA
0 m 0
dps q δe dps q γ BpjqqYptq
1 “ ´γ 1 “ e (38)
dt eδs dt A pjqA
1 m 0
dps q δe dps q γ BpjqYptq
2 “ ´γ 2 “ e (39)
dt eδs dt A pjqA
2 m 0
dpr q δe dpr q γ ApjqqYptq
0 “ ´γ 0 “ e (40)
dt eδr dt A pjqA
0 m 0
dpr q δe dpr q γ ApjqYptq
1 “ ´γ 1 “ e (41)
dt eδr dt A pjqA
1 m 0
dpt q δe dpt q γ Bpjqq
0 “ ´γ 0 “´ e (42)
dt eδt dt A
0 0
dpt q δe dpt q γ
1 “ ´γ 1 “´ e (43)
dt eδt dt pA pjqq
1 m
Now,themaincontrollerequationbecomes:
ˆ ˙ ˜ ¸
pFpT q`T qA q FpS q2`S q`S q
U “ 0 1 0 ˚pUcptqq´ `0 1 ˘2 Yptq
F´Hybrid∅,θ,Ψ Fpq2`r q`r q F q2`r q`r
0 1 0 1
ThechangeinUAVorientationdependsupontherateofchangeofthecontrolcommands,which
makesthesystemrespondquitebettertowardsstabilitybyusingourproposedalgorithm.
Thelinguisticslevelsofthefuzzyhybridcontrollerareassignedas(BN)belownegative,(SN)
smallnegative,(ZR)zero,(SP)smallpositive,and(BP)bigpositive. Thefuzzylogiccontrollerif-then
rulesaredefinedinTables2–4suchthaterror“e”istherotorspeedhavingrangein´10to+10,the
derivativeerrorrangeis´5to+5,andtheoutputrangeofthefuzzyhybridcontrolleris0to1with
R=0.667,S=0.5,andT=0.5,respectively.