Control and Flight Dynamics - Autopilot Design

Control and Flight Dynamics – Autopilot Design
Aeronautical Engineering
Wednesday, 18 February 2015
Elliot Newman
@00320195
Word Count: 4370

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Contents
Introduction.................................................................................................................................1
Objectives ................................................................................................................................ 1
Theory .....................................................................................................................................1
Longitudinal Dynamics...........................................................................................................1
Lateral Dynamics................................................................................................................... 4
Lateral Modes........................................................................................................................... 5
Roll Damping......................................................................................................................... 5
Spiral Mode .......................................................................................................................... 6
Dutch Roll Mode ................................................................................................................... 6
Aircraft Actuator Influence.....................................................................................................7
Autopilot Design........................................................................................................................... 8
Longitudinal Systems ................................................................................................................ 8
Attitude Control.................................................................................................................... 8
Proportional Control..........................................................................................................8
Proportional-plus-Derivative Control .................................................................................. 9
Trial and Error Design Process .......................................................................................... 10
Altitude Hold Control........................................................................................................... 10
Lateral Systems....................................................................................................................... 11
Roll Control......................................................................................................................... 11
Yaw Damper....................................................................................................................... 13
Heading.............................................................................................................................. 15
Heading Hold Autopilot ....................................................................................................... 16
Results....................................................................................................................................... 17
Longitudinal Results................................................................................................................ 17
.Attitude Control Results...................................................................................................... 17
Trial and Error Process Results.......................................................................................... 18
Altitude Control Results....................................................................................................... 19
Lateral Results........................................................................................................................ 20
Roll Control Results ............................................................................................................. 20
Yaw Damper Results............................................................................................................ 20
Heading Hold autopilot Results ............................................................................................ 21
Discussion & Conclusion.............................................................................................................. 23
Appendix.................................................................................................................................... 27

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Introduction
Flightcontrol manifestsitself inmanyforms, mostnotably systemsdictatingthe orientationof the
aircraft.The pilotcreatesininputintothe system, achange of headingoraltitude,encompassedin
lateral andlongitudinaldynamics, the systemthenreactsthroughaseriesof iterationstoachieve
the desiredconclusion. Thisprocesshastobe refinedinordertolineariseastable progression,
somethingthatwill be investigatedthoroughly.
Objectives
The objectivesinitiallyare tofocusonlongitudinal control,anattitude andaltitude holdautopilot,
analysinglongitudinal dynamicstodetermineif the feedbackbehaviourisacceptableanditerate
accordingly.Onthe by mathematicallymodellingthe systeminSimulink,gainanappreciationforthe
effectsof importantparameterstovisuallywitnesstheireffects.
Followingthe completionof thisand bythen applyingthe knowledge andskillsdeveloped, aheading
holdautopilotthroughinvestigatinglateraldynamics,reefingthe systemtomake itindustry
applicable bycontrollingthe maximumroll andyaw motionsthroughdamping.
Theory
The theoryfor these systemsshall firstbe examinedandappreciated,allowingforthe residual errors
to be identifiedandfactorthese intoourowndesignsata laterdate.
Longitudinal Dynamics
The control systemsof a modelled747 are immenselycomplexandwiththisinmind,primarilywe
shall focusonthe shortperiodapproximation,generatingagreaternoesisandfeel forthe subject
movingforward:
Add that,

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Usingthe data acquired,the completedpole-zeromapisillustratedinfigure 1:
Figure 1: Pole zero map for Gθδe
As with designing a system or developing a solution to a problem, interpreting the needs in the
consumer is a paramount feature, understanding what behavioural patterns a pilot wants from the
aircraft.Unfortunately,thereare legionsof empirical datathatindicate the pilotsdonotlike operating
aircraft with the flying qualities generated by this combination of frequency and damping and
therefore signalsthe needtodevelopanacceptable relationshiprange forthe natural frequencyand
the damping ratio (figure 2):
Figure 2: ‘Thumb Print’ Criterion

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The ‘thumbprint’criterion,aconceptdevelopedduringthe 1950’s, signifiesthe acceptableregionof
whichthe valuescaninteractand isstill applicable today.The graphillustratesthatthe primary
target value toaimfor will be anatural frequency(ωn) of 3𝑟𝑎𝑑𝑠−1 anda dampingratio(ζ) of
approximately 0.6.Asbecomesapparentfromcomparingthese valueswiththose attainedfromthe
formulas,the shortperioddynamicsof the 747 are well outsidethe acceptablerange,markedon
the graph, andtherefore mustbe modifiedaccordingly.
Figure 3 clearlyillustratesthe targetpole locationsonapole-zeromapwhenunderthese
approximatedconditions:
Figure 3: Pole-zero map and target pole locations
Throughinvestigatingthe graph,the shadedarearepresentsthe regionof whichthe closedloop
dominantpolesshouldbe foundandtoaccomplishthisfeat,we willrequire some feedbackfrom
the system.There are twoprimarypracticesin achievingthiswhichare:
 Proportional Control.
 Proportional-plus-Derivative Control.
Furthermore,we canemploythe techniqueof plottingroot-loci,whichallowvaluesof ωn andζ to be
attainedaidingthe designprocess,thenusingSimulinktoassessourdesigns.

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Lateral Dynamics
Obtainingthe necessaryvaluesinthisplane canbe achievedusingaprocedure similartothatof the
longitudinalcase,where we candevelopthe equationsof motion,whichinclude;dutchroll mode,
roll mode andspiral mode, usingstate space equations:
Extractingthe numerical valuesforeachof the modescan be achieved inMatlab,performingthe
‘lat’commandon eachof the matrices,e.g.AlatandBlat:
Alat =
-0.0558 0 -235.9000 9.8100
-0.0127 -0.4351 0.4143 0
0.0036 -0.0061 -0.1458 0
0 1.0000 0 0
Blat=
0 1.7188
-0.1433 0.1146
0.0038 -0.4859
0 0

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Encompassedineachof these state spacesequationsare the behavioural patternsof all the lateral
modeswith,forthe Alatmatrix;the top tworows beingassociatedwiththe dutchroll mode,the
thirdline linkedtothe roll mode andthe final row demonstratingthe spiral mode.The Blatmatrix
allowsforan aerodevicestobe isolatedbeingeither;the aileronsforthe firstcolumnorthe rudder
by the second.
The true benefitof calculatingthese numerical valuesistosolve forthe eigenvaluesof matrix A,
relinquishingthe modesof the system:
These are stable,althoughthere isone veryslow pole.The nextstepistolinkthese toeachof the
three available modes:
Now,due to theirincreasedcomplexityfromlongitudinalmodes,before we cancontinue tothe
designof eachof the systems,we mustfirstunderstandeachof these modesandtheireffectonthe
aircraft.
Lateral Modes
Roll Damping
Thismode isheavilydamped,increasingthe ease of operationandthe severityof the destabilising
effectof the roll.Asthe plane rolls, the winggoingdownhasan increasedα,the influence of windis
effectivelyincreasedandthishasan opposite affectforthe neighbouringwing.Thisdisparity
generatesanimbalance inthe incremental liftproduced,more onthe descendingwing.Thislift
differentialcreatesamomentthattendsto restore the equilibriumof the aircraft.Aftera
disturbance,the roll rate buildsexponentiallyuntilthe restoringmomentbalancesthisdisturbance
and a stable roll isestablished,illustratedinfigure 4:
Figure 4: Roll mode illustration.

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Spiral Mode
The spiral mode isthe slowestof the lateral modesandisoftenunstable,fromlevel flightconsidera
small disturbance thatcreatesa small roll angle φ> 0. Thisresultsina small side slip 𝑣,asexpected
the tail finis nowtravelingthroughthe airat an incidence angle β, creatingextratail lift,increasing
the yawingmoment.The positiveyawingmomentfurtherincreasesthe sideslipcompoundingthe
situationwhich,if leftunchecked,wouldcause the aircrafttocommence agraduallydivergingpath
inroll,yawand altitude,‘spirallingtowardsthe ground,visuallydemonstratedinfigure5:
Figure 5: Spiral mode destabilisation.
DutchRoll Mode
The final lateral mode,the mostcomplex,involvesdampedoscillationof yaw thatcouplesintoroll.
It manifestsitself atafrequencyclose tothe shortperiodmode,althoughnotasheavilydamped
and therefore the finhaslesseffectthanthe horizontal tail plane.The termiscoinedfromskating
circles,givingreference tothe actof repeatedlyskatingfromrighttoleftonthe outeredge of their
skates,imitatingthe aircraftsmotion. Againwe shallconsideradisturbance from straightlevelflight,
where anoscillationinyawψ,withthe finprovidingthe aerodynamicstiffness.The wingsmove back
and forthwithrespecttothisyaw motionandthe resultisan oscillatorydifferentialinlift/drag,as
dependingonthe direction andmotionof the wing,the inducedliftiseitherincreasedorreduced
accordingly.Addingtothismotionisthe roll,φalsooscillatoryandisapproximatelylagsthe yaw by
90⁰ and therefore atall timesthe wingmovingforwardsduringthe cycle isthe lowest.The final
resultisan oscillatingroll withside slipinthe directionof the low wing.Whenwitnessingthe
phenomenonfirsthanditisclearto see the wingtiptraces a figure of 8 in the skyduringeach time
period(figure 6):

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Figure 6: Dutch roll mode.
AircraftActuatorInfluence
As the B matrix implicitlyexplainthe responsesof the rudderandaileroninputs,theirinfluence must
alsobe investigated.Due tothe physical placementof the rudder,beingquite high,ithasa
significantinfluence onthe aircraftsroll,whereasthe aileronsaffectthe yaw byinducingdrag
differentials.We shall viewthe impulse responseof the twoinputs:
Rudderinput
 Β showsa verylightlydampeddecay.
 𝑝, 𝑟 clearlyexcited aswell.
 𝜑 oscillatesaround 2.5⁰.
 Dutch-roll oscillationsare clear.
Aileroninput
 Large impacton.
 Causeslarge change to 𝜑.
 Verysmall change toremainingvariables.
 Influence smallerthanthe rudder.

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Autopilot Design
The main objective of thisassignmentistocomplete andtestan autopilotsystemthatcannot only
control the longitudinal modesof attitude andaltitude,butalsothe lateral effectsof aheadinghold
system,engagingroll andyawlimiterswhichshall be concludedinthissection,beginningwiththe
longitudinalsystem.
Longitudinal Systems
Attitude Control
In thisdesignthe SPOtransferfunctionmodelforaBoeing747 is considered,anactuatorforthe
elevatorsisalsoaddedwithapole at −4. UsingSimulink,the transientbehaviourof the system
underproportional control shall be modelled.
ProportionalControl
Figure 7: Simulink model for proportional controller
UsingMatlab capabilities,the aimisto obtaina root-loci plotof thissystem, aimingtoachieve a
pointon the plotwhere the natural frequencyis 3𝑟𝑎𝑑𝑠−1 anda dampingratioof 0.5for thisclosed
loopsystem.Transferringthe Simulinkmodel(figure 7) intoMatlabcode for the command window
isdemonstratedbelow:
n=4*[1.166 0.35];
a=[1 0];
b=[1 4];
c=[1 0.74 0.92];
d=conv(a,conv(b,c));
rlocus(n,d);
axis([-6 2 -3 3]);
Constrictingthe axistothe desiredregion.
From inspectingthe plot,the desiredvalues couldnotbe attained,alludingtothe shortcomingsof
the system,requiringadifferentapproach.One approachtotake isto replace the proportional
controllerwithaproportional-plus-derivativecompensator.

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Proportional-plus-DerivativeControl
Usingthis type of systemwill improve the performance of control,replacingthe proportionalcontrol
withparameters k1 and k2.There are three differenttypesof control feedbacksystemwhichare:
 Forwardsloop(figure 8).
 Feedbackloop(figure 9).
 Two loopcontrol system(figure 10).
The proportional-plus-derivative compensatorencompassesthe twoparameters k1 andk2,but there
are twodesignequationstosolve (the angleandmagnitude criteria),soanydesignisstraight
forward. Each Simulinkmodel isdepictedbelow:
Figure 8: Forwards Loop.
Figure 9: Feedback Loop.
Figure 10: Two Loop system equivalent to the P-D above.
Determiningthe valuesfork1 andk2 becomes the nextissue toaddressandthe mostpracticedform
istrial anderror, usingconvergingiterationsof the systemtoachieve the predeterminedvaluesof
ωn and ζ.

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Trialand ErrorDesignProcess
Thismethodutilisesthe ease of conductingcalculations usingMatlabandSimulink,withthe
preferredsystemtoconductthe processbeingthe forwardsloopsystem(figure 8:11).
By firstly,guessingavalue for k2,the root-loci forthisisthenplottedandsee if the roots pass
throughthe points:
𝑆 = −1.5 + 2.5981𝑖
𝑆 = −1.5 – 2.5981𝑖
The value of K2 selectedwas 0.975andthenthe code usedto obtainthe final valuesfor k1 andk2
was:
k2=0.975;
x=[1 k2];
y=4*[1.66 0.35];
n=conv(x,y);
a=[1 0];
b=[1 4];
c=[1 0.74 0.92];
d=conv(a,conv(b,c));
rlocus(n,d)
Usingthe resultingroot-locusplottodetermine theseparameters.
AltitudeHoldControl
The altitude control systemisanextenuationof the attitude system,beingfedintoanew transfer
functiontoalleviate analtitudevalue,depictedbyfigure 11below:
Figure 11: Transfer function.
By addingthisintothe attitude control system, coupledwithaproportional controlleronthe
altitude loop,the systemhasbeenconvertedtothe requiredaltitudecontrol system(figure12):
Figure 12: Altitude control system.

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UsingSimulink,obtainingavalue forthe parameter kh wouldcomplete the system.
Lateral Systems
The stabilityandmodificationof lateral dynamicscanbe controlledusingavarietyof different
feedbackarchitectures,e.g.usingintegrators(figure 13):
Figure 13: Integral control.
Usingthe integrators,valuesof roll (𝑝) andyaw (𝑟) can be convertedintoroll rate (φ) andyaw rate
(ψ),lookingforgoodsensororactuator pairingstosustainsuitable behaviourforthe pilot. The block
Glat iscomprisedof a seriesof state space equationswiththe Matlabcode depictedasappendix1
(page 27).
Roll Control
Whena desiredbankangle isselectedandinputintothe system,aroll controllerisneedtoensure
and maintainthe accuracy at whichthe vehicle tracksrequest.Inthissituation,the aileronsare the
bestactuator to use:
To obtaindesignvalue forKΦ and Kp,approximationsof the roll mode mustbe made:
Whichgives:
To fullyencompassthe design,addthe aileronservodynamics:
Thisgeneratesaroot loci plotthat istypicallydemonstrated(figure 14:14) below:

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Figure 14: Typical root loci plot.
In the case of the systemforthe 747, itis designtoachieve adampingratioof 0.667and a natural
frequencyof 3𝑟𝑎𝑑𝑠−1forthe secondordermodesof the roll damper.The roll control systemwill
deployaproportional controllerasillustrated below(figure 15):
Figure 15: Proportional roll controller.
Thenusingroot loci,the systemshall be testedtosee if the desiredperformancevaluescanbe
achieved,thenusingSimulinktosimulatethe system.
An improvementcanbe made onthe controllerbymakingita proportional plusintegralinthe form
below(figure 16):
Figure 16: Proportional plus integral roll controller.
Again,rootloci shall be usedto determine valuesfor k1 andk2,such that the systemmatchesthe
requiredperformancecharacteristics.
Upon completionthe finalcontrol systemthatwill be implementedisbelow (figure 17:13):

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Figure 17: Completed roll controller.
Yaw Damper
In the case of a headingalteration,asthe aircraftbanks,the nose also‘leans’intothe turn,knownas
yaw.In orderto avoidenteringthe pre-mentionedspiral mode (page6),a yaw damperisneed,
reducingthe amountof natural yaw that can be inducedinthe aircraft.This can be subtlycontrolled
by alteringthe feedbackinthe control system:
 Feedbackonlyahighpass versionof the 𝑟 signal.
 Highpass cuts of the lowfrequencycontentinthe signal.
 Steadystate value of 𝑟 wouldnotbe fedback intothe controller.
Newyawdamper:
Figure 18: Frequency response of the washout filter.
Thisinformationthenleadstothe completionof the final yaw damperdesign(figure19):

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Figure 19: Yaw damper control system.
The transferfunctionrelatingthe ruddermovementtothe yaw rate is:
𝐺𝑙𝑎𝑡(𝑠) 𝑑𝑟 𝑟⁄ =
1.618(𝑠 + 0.6943)(𝑠2
− 0.2146𝑠 + 0.1678)
(𝑠 + 3.33)(𝑠 + 0.5613)(𝑠 + 0.007264)(𝑠2 + 0.06629𝑠 + 0.8978)
The washoutfilterhasthe followingtransferfunctionalso:
𝐻 𝜔( 𝑠) =
𝑇𝑠
𝑇𝑠+1
Plottingacomplex rootloci of the system, finding kr andTsuch that the dominantcomplex roots
have a dampingratioof 0.83 and a natural frequencyof 0.95𝑟𝑎𝑑𝑠−1 obtainsa plotsuchas:
Figure 20: Complex root loci plot
Withthisaspect nowcompletedthe yaw dampercanbe addedto the lateral control systemand
allowsthe nextfeature tobe designed,the headingautopilot.

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Heading
Withthe yawdampercompletedandaddedtothe system, controllingthe heading(𝜓) isthe next
characteristicto design.Inordertoachieve aheadingchange,firstandforemost,the aircraftisgoing
to needtobank,resultingina ‘coordinatedturn’withandangularrate 𝜓̇.
The aircraft isbankedto a predeterminedangle 𝛷 sothatthe vectorsumof 𝑚𝑔 and 𝑚𝑈0 𝜓̇ is along
the bodyof the 𝑧 − 𝑎𝑥𝑖𝑠. Summingupthe bodyy-axisdirection,thisgives 𝑚𝑢0 𝜓̇ cos 𝜙 = 𝑚𝑔sin 𝜙
thiswill give anequationof:
tan 𝜙 =
𝑈0 𝜓̇
𝑔
Since typically ϕ<< 1, then:
𝜙 ≈
𝑈0 𝜓̇
𝑔
Whichgives the desiredbankangle foraspecifiedturnrate.
The issue withthisisthat 𝜓 tendsto be a noisysignal tobase the bankangle on,so a smoother
signal isgeneratedthroughfilteringit.Byassumingthatthe desiredheadingisknown 𝜓 𝑑 and we
want 𝜓 to follow 𝜓 𝑑 relatively slowly, then selecting the dynamics of 𝜏1 𝜓̇ + 𝜓 = 𝜓 𝑑 :

𝜓
𝜓 𝑑
=
1
𝜏1 𝑠+1
, with 𝜏1 = 15 − 20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠dependingonthe situation.
A lowpassfilterthateliminatesthe higherfrequencynoise.
The filteredheadingangle satisfiesthe equation:
𝜓̇ =
1
𝜏1
(𝜓 𝑑 − 𝜓)
Whichcan be usedto create the desiredbankangle forthe aircraft:
𝜙 𝑑 =
𝑈0
𝑔
𝜓̇ =
𝑈0
𝜏1 𝑔
(𝜓 𝑑 − 𝜓)
Nowwithall the individualaspectsof the headingautopilotdesignedandfunctional,the systemcan
be completed.

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HeadingHoldAutopilot
By compilingthe designedfeaturesintoone,all-encompassingcontrol system, the autopilot
controlleriscreated(figure 21):
Figure 21: Heading hold autopilot system.
Before the systemcanbe completedreadyforoperation,the final stepistoanalyse the effectof
closingthe 𝜓 to 𝛷 𝑑 loop.The parameterenclosedonthe loop, 𝑈0/𝑇1 𝑔,has tobe carefullyselected
due to the sensitivityof the loop,toolarge andthe systemwill gounstable.The value canbe
determinedusingthe conventional methodof rootloci,howeverthe calculationscanbe quite
complicated.Byassumingavalue of 2, the performance canbe investigated,before addingaroll
angle limiter,inthe formof a saturationblock,ontothe pathof 𝑈0/𝑇1 𝑔 yieldingafinal designof
(figure 22):
Figure 22: Final heading hold autopilot design.

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Results
Determiningcertaincharacteristics,aswell asconductingperformance testswasanecessary
practice and the concludingdatasetsare depictedinthissection.
Longitudinal Results
Attitude Control Results
Figure 23: Attitude control Simulink performance.
Figure 24: Attitude control root loci plot.

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Trialand ErrorProcessResults
Figure 25: Trial and error root loci plot.
From the root loci plot, valuesfork1 and k2 were achievedat k1 = 2 and k2 = 0.45. Applyingthese
gainsintothe three formsof P-Dcontrol systemsthe transientresponsescanbe evaluatedin
Simulink:
Figure 26: Forwards loop transient response.

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Figure 27: Two loop transient response.
AltitudeControl Results
Usingthe altitude gain kh as 2:
Figure 28: Altitude control Simulink performance.

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Lateral Results
Roll Control Results
Obtainingvaluesof k1 = −20 and k2 = 1.5, the root loci plotis illustratedas(figure 29):
Figure 29: Roll control root loci plot.
Yaw DamperResults
Employingparametersof kr = 9.26 and T = 0.4:
Figure 30: Yaw damper root loci plot.

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HeadingHoldautopilotResults
Aftermeshingall the pretestedsystems,the finalautopilotdesignwastestedtoo yielding:
Figure 31: Heading hold root loci plot.
Figure 32: Heading orientation Simulink performance.

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Figure 33: Roll angle Simulink performance.

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Discussion& Conclusion
Overall,fromthe datasetacquired,the responsesfrom the autopilotsystemswaswell withinthe
acceptable performance range, althoughrarelywill acontrol systemresonate withthe required
performance parametersperfectlyandthereforeinthe majorityof cases‘controllers’are a
necessaryfundamental.In thiscase,these characteristics couldonlybe achievedwithdampersand
limitcontrollers,hintingtotheirsignificance insystemdesign.
The yaw damperhelpsininhibitthe maximumyaw angle experiencedduringflight,making
manoeuvre conditionsmore stable andeasiertocontrol forthe pilot.Placingaceilingonthe
maximumyawincreasessafetyduringcertainflightconditions,suchasheadingalteration,where if
yawand bank conditionsincreaseabove the maximum, the circumstanceswouldbe enoughto
induce the spiral mode,whichhasclearlydangerousconnotations.Anothermethodof preventing
thisphenomenonisconstrainingthe maximumroll angle,controlledbythe additionof asaturation
block.Maintainingthisalsoenable amore pleasantflyingexperience foronboardpassengers,asthe
‘banks’experiencedwill have little effectoncabinconditions.
Pilotsrequire flightcontrol systemstoreactina stable progressivemanorafterinputs,responding
withpredictabilitybefore smoothlytracking the requestedoutput. Clearlydemonstratedwiththe
attainedresultsisthe response differencesbetweenthe three varyingP-Dcontrollers;forwards
loop,feedbackloop andtwoloop(figures8,9,10: 9).Initially,the forwardslooprisesata steep
angle,reachesthe requiredchange andbeginstooscillatebelow the requiredoutputbefore slowly
converging.Thisisinstarkcontrast to the most effective,andmostcomplex of the three,the two
loopsystem.Inthiscase there are no immediatechangesinitially,beforethe systemgradually
convergesonthe requiredyield,providingthe stable,controllable systemthe pilot’sdesire.
Continuingwithaltitude response,the systemisaspecial scenario,of whichitisapositive feedback
system,whichcausesachange in the normal practiceswhenconductingrootloci plots.Thisoccurs
whenthe flightdynamicsystemshave anon-minimumphase zerosandthe systemhastobe
modelledaspositive feedback.Certainfundamentalsmaintain;the numberof branches,the
symmetryandthe startingand endingpoints.The factors thatcharge are the fact that; on the real
axis,the root locusnowexiststothe leftof an evennumberof polesorzerosandthat the equations
to calculate the necessarycriteriahave subtle differences:
𝛷𝑙 =
360
𝑛−𝑚

𝜎 =
∑ 𝑝𝑖−∑ 𝑧 𝑖
𝑛−𝑚
Also,calculatingthe angle criteriaalters,withthe equationbecomingequal to0rather than the
conventional -180.The magnitude criteriaremainsthe same easingcalculations.

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Keyto understandingthe systemsfunctionalityishow itrespondstochangingparameters,affecting
the stabilityof the system. These testswill helpapilotdetermine the operationalperformance
boundariesof the aircraftand alsohelpthenutilisepractical efficienciesduringflight.Through
specificalterations,increasing,thendecreasinggainsanappreciationcanbe attained.
Beginningwithgainincreasesforthe value of Tau1 locatedonthe lowestfeedbackloopof the
system(figure 22:16), the Simulinkplotsrevealedhow the roll andthe yaw respondstothe
alterationwithinthe system(Tau1 = 8):
Figure 34: Roll gain change reaction.
Figure 35: Yaw gain change reaction.
From the resultsitisclearto see the unstable oscillatingnature of the performance,eachone begins
the manoeuvre inthe conventional wayuntil the incorrectgainvalue isfedbackintothe systemand
beginstodestabilise it.Continuingthe analysis, itisclearly visiblethatthe instability hasapeak
amplitude,werethe oscillations reachtheirmaximumandcontinue forthe time period. Visiblein

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the periodicgainincreasesinthe appendix (Appendices2and 3: 28,29), as the gain marginincreases
the systemsstabilitybecomesmore iritic,aswhenTau1 = 2, the systemisbarelyaffected,yetat4,
clearunstable oscillationsbegintooccuras the systemrevertstoequilibriumafterthe manoeuvre.
Furtherinvestigationyieldsthatthe peakamplitudes demonstratedare intrinsicallylinkedintothe
magnitude of the gains,illustratedbycontrastingthe increasinggainsgraphs. Interestingly,asthe
frequencyof the yawand roll are of a verysimilartime period,indicatingthe dutchrole mode is
beinginduced,increasinginseverityasthe gainincreases.
Now,investigatingthe effectsof reducingthe gainmargin, whichis a starkcontrast to the effectsof
the increase:

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The most notable difference isthere isnopronouncedinstabilityatall,the systemappearsreacts
perfectly,although,undercloserinspectionthe subtle differencesarise.The time periodof the
exercise isfargreaterthannormal,taking10 timeslongerat1600 seconds,thisalleviatesthat,the
time periodof anysystemsoscillation canbe loweredbyreducingthe gainmarginbythe required
factor,a fact reinforcedbyappendix 4(page 30), where Tau1 is0.5 and the total time take istwice
that of the designsystem.Thisfactoralsoappliestothe amplitude,increasingbythe same
magnitude asitsreduction.The onlynoticeable instabilityisatthe peakof the roll,where the
amplitude isthatat the peak,there isno stabilisationbefore beginningtoreturntolevel flight,the
change is quite sharpandcouldtherefore destabilise the aircraftduringflight.
Overall,the gainchangesexercise alludedtothe significance of awell-produced designtomaximise
the efficiencyof asystem,alterationfromthis‘sweetspot’cancause eitherlethargic manoeuvre
response ora totallyunstable,un-flyable aircraft.The factthat the performance changesaltereither
side of the designedgainillustratesthe successof the processundertakenanditsindustry
applications.Also,aswhenthe gainisoverthe desiredvalue instabilityoccurs,itmaybe helpful to
introduce asafetymarginintothe designprocesstoprotectagainstthis, by loweringthe gainby10-
20%, increasingthe time periodof modesandallowingthe pilotvaluablethinkingtime inthe event
of anyerror.
The successof thisprojectcannotbe disputedasthe resultsspeakforthemselvesandmirrorthose
of predictedplots.Also,the synergybetweenvalue alterations,the factthatno anomalousresults
are introducedduringthese changes,demonstratestheir validityandreliability. Toimprove the
data, nexttime Iwouldproduce aconsistentspreadof gainalterationstoenable agraphical
representationof the aforementionedtrendswitnessed.

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Appendix
Appendix 1– Glat State Space Matlab Code
% B747 lateral dynamics
%T= ????
Yv=-1.61e4;Yp=0;Yr=0;
Lv=-3.062e5;Lp=-1.076e7;Lr=9.925e6;
Nv=2.131e5;Np=-1.33e6;Nr=-8.934e6;
g=9.81;theta0=0;S=511;cbar=8.324;b=59.64;
U0=235.9;
m=2.83176e6/g;cbar=8.324;rho=0.3045;
Iyy=.449e8;Ixx=.247e8;Izz=.673e8;Ixz=-.212e7;
Cyda=0;Cydr=.1146;
Clda=-1.368e-2;Cldr=6.976e-3;
Cnda=-1.973e-4;Cndr=-.1257;
QdS=1/2*rho*U0^2*S;
Yda=QdS*Cyda;Ydr=QdS*Cydr;Lda=QdS*b*Clda;Ldr=QdS*b*Cldr;
Nda=QdS*b*Cnda;Ndr=QdS*b*Cndr;
Ixxp=(Ixx*Izz-Ixz^2)/Izz;
Izzp=(Ixx*Izz-Ixz^2)/Ixx;
Ixzp=Ixz/(Ixx*Izz-Ixz^2);
Alat=[Yv/m Yp/m (Yr/m-U0) g*cos(theta0);
(Lv/Ixxp + Ixzp*Nv) (Lp/Ixxp + Ixzp*Np) (Lr/Ixxp + Ixzp*Nr) 0;
(Ixzp*Lv + Nv/Izzp) (Ixzp*Lp + Np/Izzp) (Ixzp*Lr + Nr/Izzp) 0;
0 1 tan(theta0) 0];
Blat=[1/m 0 0;0 1/Ixxp Ixzp;0 Ixzp 1/Izzp;0 0 0]*[Yda Ydr;Lda Ldr;Nda Ndr];
Clat= eye(4,4);
D_lat =zeros(4,2);
c=[0 0 1 0]
b=zeros(4,1)
for n=1:1:4
b(n,1)=Blat(n,2)
end
d=0
sys1=ss(Alat,b,c,d)
zpk(sys1)
n1=[-T 0]
d1=[T 1]
sys2=tf(n1,d1)
sys3=sys1*sys2;
n3=0.333
d3=[1 0.333];
sys4=tf(n3,d3)
sys5=sys4*sys3
rlocus(sys5)
axis([-2 1 -1.5 1.5])

P a g e | 28
Appendix 2– Tau 1 = 2

P a g e | 29
Appendix 3– Tau 1 = 4

P a g e | 30
Appendix 4– Tau 1 = 0.5
Figure 42: Roll gainchange reaction.

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