aircraft control lecture 12

7/29/2019 aircraft control Lecture 12

1/27

Lecture# 12AircraftLateralAutopilots

Multi-loopclosure

HeadingControl: linearHeadingControl: nonlinear


2/27

Fall2004 16.333101LateralAutopilots

Wecanstabilize/modifythe lateraldynamicsusingavarietyofdif-ferentfeedbackarchitectures.

a - -p 1 -s

r -Glat(s)

-r 1 -

s

Lookfor

good

sensor/actuator

pairings

to

achieve

desired

behavior.

Example: Yawdamper

Can improve the damping on the Dutch-roll mode by adding afeedbackonrtotherudder: c =kr(rcr)r

3.33ServodynamicsHr = s+3.33 addedtoruddera =Hrcr rSystem:

1.618s3+0.7761s2+0.03007s+0.1883Grcr =s5+3.967s4+3.06s3+3.642s2+1.71s+0.01223

3.5 3 2.5 2 1.5 1 0.5 0

2

1.5

1

0.5

0

0.5

1

1.5

2

0.94

0.86 0.160.340.50.640.76

3.5

0.76

0.985

0.160.340.50.64

1

0.86

0.94

0.985

0.53 1.522.5

Lateral autopilot: r to rudder

Real Axis

ImaginaryAxis

c

r

Figure2: Lateralpole-zeromapG r


3/27

Fall2004 16.333102 Notethatthegainoftheplantisnegative(Kplant 0

Real Axis

ImaginaryAxis

3.5 3 2.5 2 1.5 1 0.5 02

1.5

1

0.5

0

0.5

1

1.5

2

1

0.25

0.250.5

0.5

0.75

0.25

0.5

0.75

Lateral autopilot: r to rwith k


4/27

Fall2004 16.333103YawDamper: Part2

Canavoidthisproblemtosomeextentbyfilteringthefeedbacksignal.Feedbackonlyahighpassversionofthersignal.Highpasscutsoutthelowfrequencycontentinthesignalsteadystatevalueofrwouldnotbefedbacktothecontroller.

Newyawdamper: c = kr(rcHw(s)r) whereHw(s) = s isther s+1washoutfilter.

102

101

100

101

102

101

100

Washout filter with =4.2

|Hw

(s)|

Freq (rad/sec)

Figure4: Washoutfilterwith =4.2 Newcontrolpicture

p 1a - - -s

Glat(s)c r 1rrc - - - - -Hr(s)kr6 s

Hw(s)


5/27

Fall2004 16.333104

2 1.5 1 0.5 0 0.5 11.5

1

0.5

0

0.5

1

1.5

0.250.5

0.5

0.75

0.75

0.25

1

0.5

0.25

Lateral autopilot: r to rudder WITH washout filter

Real Axis

Imag

inaryAxis

Figure5: RootLocuswiththewashoutfilter included. ZeroinHw(s)trapsapoleneartheorigin,butitisslowenoughthat

itcanbecontrolledbythepilot. Obviouslyhaschangedtheclosedlooppolelocations( ),butkr =1.6stillseemstogiveawelldampedresponse.


6/27

Fall2004 16.333105

0 5 10 15 20 25 300.5

0

0.5

1

Time

Responser

Without Washout

With Washout

0 5 10 15 20 25 30

0.5

0

0.5

1

Time

Control

r

Without Washout

With Washout

Figure 6: Impulse response of closed loop system with and without the Washoutfilter ( = 4.2). Commanded rc = 0, and both have (r)ss = 0, but without thefilter,rss = 0,whereaswith it,rss = 0.

Fordirectcomparisonwithandwithoutthefilter,appliedimpulseasrc tobothclosed-loopsystemsandthencalculatedrandr.

Bottom plot shows that control signal quickly converges to zero inbothcases,i.e.,nomorecontroleffortisbeingappliedtocorrectthemotion.

Butonlytheonewiththewashoutfilterproducesazerocontrolinputeventhoughthethere isasteadyturnthecontrollerwillnottrytofightacommandedsteadyturn.


7/27

Fall2004 16.333106HeadingAutopilotDesign

So now have the yaw damper added correctly and want to controltheheading.Needtobanktheaircrafttoaccomplishthis.Resultisacoordinatedturnwithangularrate

Aircraftbankedtoanglesothatvectorsumofmg andmU0 isalongthebodyz-axisSumminginthebodyy-axisdirectiongivesmu0 cos=mgsin

U0tan=

g Sincetypically1,wehave

U0

g

givesthedesiredbankangleforaspecifiedturnrate.


8/27

Fall2004 16.333107Problemnow isthat tendstobeanoisysignaltobaseoutbankangleon,sowegenerateasmoothersignalbyfilteringit.Assumethatthedesiredheading isknownd andwewant to

followd relativelyslowlyChoosedynamics1 + = d

1=

d 1s+ 1with1=15-20secdependingonthevehicleandthegoals.

Alowpassfilterthateliminatesthehigherfrequencynoise.

Filteredheadinganglesatisfies1

= (d)1

whichwecanusetocreatethedesiredbankangle:U0

d = = U0 (d)g 1g


9/27

7 6 5 4 3 2 1 010

8

6

4

2

0

2

4

6

8

10

0.88

0.68

0.52 0.38 0.28 0.060.120.2

0.68

0.38 0.28 0.2 0.12 0.06

8

0.88

0.52

6

2

4

6

8

2

4

Root Locus

Real Axis

ImaginaryAxis

Fall2004 16.333108RollControl

Giventhisdesiredbankangle,weneedaroll controller toensurethatthevehicletracksitaccurately.Aileronisbestactuatortouse: a = k(d)kpp

Todesignk andkp,canjustusetheapproximationoftherollmodeI

I xxp = Lpp+ Laa xxLp = Laa= pwhichgives La=

a s(IxxsLp) Forthedesign,addtheaileronservodynamics

1Ha(s) = , a = Ha(s)ca0.15s+ 1 a

PDcontroller

c= k(s

+ 1) + kd,

adds

zero

at

s= 1/a

Pick= 2/3

Figure7:

Root

Locus

for

roll

loop

closed

Loop

poles

for

Kp =20,K =30


10/27

Fall2004 16.333109

0 1 2 3 4 5 6 7 8 9 100.4

0.3

0.2

0.1

0

0.1

0.2

0.3

Reponse to initial roll of 15 degs

p

0 1 2 3 4 5 6 7 8 9 102

0

2

4

6

8

10x 10

3 Reponse to initial roll of 15 degs

r

Figure8: Rollresponsetoan initialrolloffset


11/27

Fall2004 16.3331010

0 5 10 15 20 25 300.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (sec)

Reponse to step input c=15 degs (0.26 rad)

p

0 5 10 15 20 25 300.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec)

Reponse to step input c=15 degs (0.26 rad)

r

Figure9: Rollresponsetoastepcommandforc. Notetheadverseyaweffects.


12/27

2 1.5 1 0.5 0 0.5 1 1.5 2

1

0.5

0

0.5

1

0.68 0.5 0.25

0.975

0.94

0.88

0.8

0.68 0.25

0.994

1.5

0.50.8

11.75

0.88

0.50.751.25 0.25

0.94

0.975

0.994

Heading Command Loop 1=12 sec

Real Axis

ImaginaryAxis

Fall2004 16.3331011HeadingAutopilot

PuttingthepiecestogetherwegetthefollowingautopilotcontrollerKp

?6

+ - K

Ha(s)-?

ca - -

p1s -

d Glat(s)

0

6 - Kr Hs(s)-cr - -r 1

s -

s

s+ 1U0 1g 6+ d

Laststep: analyzeeffectofclosingthed loop

Figure10: Headinglooprootlocus. Closed looppolesforgainU0/(g1). 8thordersystem,butRLfairlywellbehaved.


13/27

Fall2004 16.3331012 Measurewithaverticalgyroandwithadirectionalgyro Addalimitertod orelsewecangetsomeverylargebankangles

DesignvariablesareK,Kp,1,,andKr Multi-loopclosuremustbedonecarefully

Must choose the loop gains carefully so that each one is slowerthantheoneinsidecanleadtoslowoverallresponse

Analysisonfullycoupledsystemmightshowthatthecontrollersdesignedonsubsystemmodelsinteractwithothermodes(poles)severaliterationsmightberequired

Nowneedawaytospecifyd


14/27

Fall2004 16.3331013

0 5 10 15 20 25 30 35 40 45 500.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time (sec)

p

0 5 10 15 20 25 30 35 40 45 500.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time (sec)

r

c

Figure11: Responseto15degstepinc. Notethebankangleisapproximately30degs,which isaboutthemaximumallowed. Decreasing1 tendsto increasemax.


15/27

Fall2004 16.3331014GroundTrackControl

Considerscenarioshown- usethistodeterminedesiredheadingd

Figure12: Headingdefinitions

Areinitiallyat(0,0),movingalongXe (0 =0)Wanttobeat(0,1000)movingalongdashedlineangledatref

Separationdistanced=Yref Ya/c (d0 =1000)Desiredinertialyposition- ypositionofaircraft

d =U0sin(ref ) U0(ref ) Wanttosmoothlydecreasedtozero,useafiltersothat

d = 1d d

=30secd

1d = U0(ref )

d1 1

= = ref + d=ref + (Yref Ya/c)dU0 dU0

whichincludesafeedbackontheaircraftinertialY position. Usethisastheinputd.


16/27

Fall2004 16.3331015

-5000

pathoffset

Nwas

h(s)

Dwas

h(s)

Washou

tFIlter

ac3out

To

Workspace

Signal

Generator

Saturation

JNr(s)

JDr(s)

RudderDynamics

2*pi/180

Psi_ref

Path1

desired

ye

Mux

Mu

syslat2

Lateral

-K

-

Kp

1 sIntegrator

-K-

U0

Kphi Gai

n

emu

0

Constant

Clock

-K-

Bankingmix1

-K-

Bankingmix

JNa(s)

JDa(s)

AileronDynamics

alldata

alldata

rpp

hi

v

psi

psi

time

desiredpath

desiredpath

psi_ref

Psi_d

ye

Figure13: Simulink implementationrunac3.mfirst


17/27

Fall2004

0 100 200 300 400 500 600 700 800 900 1000

1.5

1

0.5

0

0.5

1

1.5

x 104

PathY

e

Yref

Ya/c

16.3331016

Figure14: Pathfollowingforgroundtrackcontroller

0 100 200 300 400 500 600 700 800 900 10000.8

0.6

0.4

0.2

0

0.2

0.4

0.6

andc

c

Figure15: Rollcommandfollowingforgroundtrackcontroller

0 100 200 300 400 500 600 700 800 900 10003

2

1

0

1

2

3

a

ndc

c

Figure16: Headingfollowingforgroundtrackcontroller


18/27

Fall2004 16.3331017AlternativeStrategy

Sanghyuk Park developed an alternative tracking algorithm that hepresentedthispastsummer1

Guidancelogicselectsareferencepointonthedesiredtrajectoryandusesittogeneratealateralaccelerationcommand.SelectionofReferencePoint Reference point is on the desired

pathatadistance(L1)forwardofthevehicleFigure17.LateralAccelerationCommanddeterminedbyascmd =2V

2sinL1

R

V

R

L1

desired pathreference poin

2

aScmd

aircraft

Figure17: DiagramforGuidanceLogic

Direction of acceleration depends on sign of angle between the L1linesegmentandthevehiclevelocityvector.Ifselectedreferencepointtotherightofthevehiclevelocityvec-

tor,thenthevehiclewillbecommandedtoacceleratetotheright(seeFigure17)vehiclewilltendtoalign itsvelocitydirectionwiththedirectionoftheL1 linesegment.

1SanghyukPark,JohnDeyst,andJonathanHow,ANewNonlinearGuidanceLogicforTrajectoryTracking,AIAAGNC2004.


19/27

Fall2004 16.3331018

=as

Vt

s=Vt

L1

: velocity direction change

due to the acceleration

reference point

Figure18: OneTimeStep

0 20 40 60 8010

0

10

20

30

40

50

X

Y

Desired Path

At some point, acceleration sign changes

Smooth convergence to desired path

Vehicle PositionReference PointL

1

t V L1Figure19: StepbyStep( =1, =10,and =40) Figure18showsevolutionoftheguidancelogicinonetimestepand

Figure19showsthetrajectoryofthevehicleoverseveraltimesteps.Vehicle initially starts from a location far away from the desired

path,andeventuallyconvergestoit.Ifvehiclefarfromthedesiredpath,thentheguidancelogicrotates

the vehicle so that its velocity direction approaches the desiredpathatalargeangle.

Ifvehicleclosetothedesiredpath,thentheguidancelogicrotatesthevehiclesoitsvelocitydirectionapproachesthedesiredpathatasmallangle.


20/27

Fall2004 16.3331019 Figure 20definesthenotationusedforalinearization.

d reference poin

desired flight path

1

2

L1

V

1s

cmd

a

Figure20: LinearModelforStraight-lineFollowingCaseL1 isthedistancefromthevehicletothereferencepoint.disthecross-trackerrorV isthevehiclenominalspeed.

Assuming issmallsin=1+2 andd d

1 , 2L1 V

CombiningtheabovegivesV

2ascmd =2 sin2V d + V d (1)L1 L1 L1

LinearizationyieldsaPDcontrollerforthecross-trackerror.RatioofV andseparationdistanceL1 isan important factor in

determiningtheproportionalandderivativecontrollergains.Keypoints: NL formworkssignificantlybetterthanaPDand

ismuchmoretoleranttowindsdisturbances.


21/27

Fall2004 16.3331020Implementation

Can implement this command by assuming that vehicle maintainssufficientlifttobalanceweight,eventhoughbankedatangle.Requiresthatthevehiclespeedupand/orchangetoa larger.

LiftincrementW

CL =Lmg(n1)QS QS

wherenL/W istheloadfactor.AssumethatLcos=W,thenLsin=mas,sothat

astan=

g Wecanusethistodevelopd thatweapplytotherollcontroller.

Kp

c 1a? - ? - - - -Ha(s)K6+ p sGlat(s)d

c0 r 1r- - - - -Hs(s)Kr6 s

s

s+ 1SomesimulationsshownWorkswellhardestpart isdeterminingwheretoplacethe ref-

erencepoint,whichisL1 aheadalongthepath.Recall thatL1 acts like a gain in the controller making it too

smallcandrivetheaircraftunstable.


22/27

Fall2004 16.3331021

0 1 2 3 4 5

x 104

2

1.5

1

0.5

0

0.5

1

1.5

2

x 104

Ye

xe

veh

Path

Figure21: Simulation#1: path. TurnradiusRV2/(gtan)

0 500 1000 1500 2000

30

20

10

0

10

20

30

time

andd

d

Figure22: simulation#1: bankangle. Limitedto30degshere.


23/27

Fall2004 16.3331022

2000 0 2000 4000 6000 8000 10000 12000 140000

0.5

1

1.5

2

2.5

3

3.5x 10

5

Ye

xe

veh

Path

0 200 400 600 800 1000 1200

30

20

10

0

10

20

30

time

and

d

d

Figure23: simulation#2


24/27

Fall2004 16.3331023FlightTest

Guidancealgorithm implementedandtestedwithtwoUAVs[ParentChildUnmannedAirVehicle(PCUAV)projectbyProf. Deyst]RequiredlateralaccelerationachievedusingbankanglecontrolNominalflightvelocityofabout25m/s,thechoiceofL1=150m

resultsintheassociatedcrossoverfrequencyat0.4rad/s.

Figure 24showstheflightdatafortheMinivehicleusingtheguidancelogicinthelateraldynamics.Plot shows the 2-dimensional trajectory of the Mini vehicle ()

withacommandeddesiredtrajectory(- -).Smallnumbersalongthetrajectoryaretheflighttimes recorded

intheonboardavionics.Lateral displacement between the vehicle and the desired path

within2 meters for the 75% of its flight time and within3metersfor96%oftheflighttime

.

AsimilarflighttestwasperformedfortheOHSParent(seeFigure25)After the transient period, the trajectory of the vehicle followed

thecommandedpathwithin2meters forthe78%of itsflighttimeandwithin3metersfor97%oftheflighttime.


25/27

Fall2004 16.3331024

300 200 100 0 100 200 300 400400

300

200

100

0

100

200

300

East [m]

North[m]

141

150

160

170

180

190

200

210

220

230

Desired PathPosition Trajectory

Figure24: FlightDataofMINI- TrajectoryFollowing

700 600 500 400 300 200 100100

0

100

200

300

400

500

East [m]

N

orth[m]

76

90

110

130

150

170

Desired PathPosition Trajectory

Figure25: FlightDataofOHSParent- TrajectoryFollowing


26/27

Fall2004 16.3331025 Thendemonstratedrendezvousfromanyarbitraryinitialpositionsto

aconfigurationoftightformationflight.Figures26showthepositionsoftheParentandtheMini inthe

north-east2-Dmapevery10seconds.OHSParentvehiclefollowsthecircularflightpath,withnoknowl-

edgeoftheMinivehicleslocation.Minivehicleschedulesitsflightpathandperformsformationflight

byreceivingpositioninformationfromtheOHSParent.

300 200 100 0 100 200 300 400400

300

200

100

0

100

200

300

O

M

EAST [m]

NORTH

[m]

40 [sec]

300 200 100 0 100 200 300 400400

300

200

100

0

100

200

300

O

M

EAST [m]

NORTH

[m]

120 [sec]

300 200 100 0 100 200 300 400400

300

200

100

0

100

200

300

O

M

EAST [m]

NORTH

[m]

70 [sec]

300 200 100 0 100 200 300 400400

300

200

100

0

100

200

300

OM

EAST [m]

NORTH

[m]

90 [sec]

Figure26: FlightData- RendezvousTrajectoriesofOHSandMini(O:OHS,M:Mini)


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Fall2004 16.3331026% newr.m 53 kk=1;% Analyze tracking algorithm by Park et al 54 while (~isempty(iii)) & (kk

aircraft control lecture 12

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