Gain Scheduled Missile Control Using Robust Loop Shaping

Robust Loop Shaping

ExamensarbeteutförtiReglerteknik

vid TekniskaHögskolaniLinköping

av

HenrikJohansson

Robust Loop Shaping

ExamensarbeteutförtiReglerteknik

vid TekniskaHögskolaniLinköping

av

HenrikJohansson

Regnr: LiTH-ISY-EX-3291-2002

Supervisor: Martin Enqvist

Henrik Jonson

Division,Department Date Språk Language

2

Svenska/Swedish

2

Engelska/English

2

Rapporttyp Reportcategory

2

Licentiatavhandling

2

Examensarbete

2

C-uppsats

2

D-uppsats

2

Övrigrapport

2

URLförelektroniskversion

ISBN

ISRN

Serietitelochserienummer

Titleofseries,numbering

ISSN Titel Title Författare Author Sammanfattning Abstract Nyckelord

Robustcontrol designhasbecomeamajor researchareaduringthe

lasttwentyyearsandtherearenowadaysseveralrobustdesignmethods

available. One exampleof such amethodis therobust loopshaping

methodthatwasdevelopedbyGloverandMacFarlaneinthelate1980s.

Theideaofthismethodistousedecentralizedcontrollerdesigntogive

thesingularvaluesoftheloopgainadesiredshape. Thisstepiscalled

LoopShapingand itis followedby aRobustStabilization procedure,

whichaimstogivetheclosedloopsystemamaximumdegreeofstability

margins. In this thesis, the robust loop shaping method is used to

designagainscheduledcontrollerforamissile. Thereportconsistsof

threeparts,where therstpart introducesthe RobustLoopShaping

controllerdesigntheoryandaGainSchedulingapproach. Thesecond

part discusses the missile and itscharacteristics. In the thirdpart a

controller is designed and ashort analysis of the closed loop system

is performed. A scheduled controller is implemented in a nonlinear

environment,inwhichperformanceandrobustnessaretested. Robust

LoopShaping is easy to use and simulations show that the resulting

controllerisabletocopewithmodelperturbationswithoutconsiderable

loss in performance. The missile should to be able to operate in a

large speedinterval. There, it is shown that a single controller does

notstabilize the missileeverywhere. Thegain scheduledcontrolleris

howeverableto doso,whichhasbeenshownbymeansofsimulations

inthenonlinearenvironment. AutomaticControl,

Dept.ofElectricalEngineering

10thJanuary2003 LITH-ISY-EX-3291-2002    http://www.control.isy.liu.se

GainScheduledMissileControlUsingRobustLoopShaping Parameterstyrdmissilstyrningmedhjälpavrobustkretsformning

HenrikJohansson

×

×

Robust control design has become amajor research area during the last twenty

yearsandtherearenowadaysseveralrobustdesignmethodsavailable. Oneexample

ofsuchamethodistherobustloopshapingmethodthatwasdevelopedbyGlover

andMacFarlanein thelate1980s. Theideaofthismethodisto usedecentralized

controllerdesignto givethesingularvaluesoftheloopgainadesiredshape. This

stepiscalledLoopShapinganditisfollowedbyaRobustStabilizationprocedure,

whichaimstogivetheclosedloopsystemamaximumdegreeofstabilitymargins.

Inthis thesis,therobustloop shapingmethod isused to designagainscheduled

controller for a missile. The report consists of three parts, where the rst part

introducestheRobustLoopShapingcontrollerdesigntheoryandaGainScheduling

approach. The second part discusses the missile and its characteristics. In the

third partacontroller is designedand ashort analysis of the closedloopsystem

is performed. A scheduledcontroller is implemented in anonlinearenvironment,

in which performance androbustnessaretested. RobustLoopShaping iseasyto

useand simulationsshowthat theresultingcontroller isableto copewithmodel

perturbationswithoutconsiderablelossinperformance. Themissile should tobe

abletooperateinalargespeedinterval. There,itisshownthatasinglecontroller

doesnotstabilizethemissileeverywhere. Thegainscheduledcontrollerishowever

able to do so, which has been shown by means of simulations in the nonlinear

environment.

ThisworkhasbeencarriedoutatSaabBoforsDynamicsABinLinköping,Sweden.

IwouldliketothankmytosupervisorsHenrikJonsonandMartinEnqvistfortheir

helpandsupportduringthisproject. IwouldalsoliketothankmyexaminerTorkel

Gladforshowinginterestinmywork. Finally,Iwouldliketothankallthepeople

atSaabBoforsDynamics ABformakingmefeellikeapartofthegroup.

Linköping,December2002

Symbols

u ≡ u(t)

inputvector

δ

_a

aileron(

u

1

)

δ

_e

elevator(

u

2

)

δ

r

rudder (

u

3

)

x ≡ x(t)

statevector

˙x

thetimederivativeof

x

ˆx

theestimateof

x

y ≡ y(t)

outputvector

G

(s)

transferfunction

ω

theangularvelocityvector

p

angularvelocityaroundthe

x

axis

q

angularvelocityaroundthe

y

axis

r

angularvelocityaroundthe

z

axis

α

angleofattack

β

sideslipangle

V

themissilevelocityvector

u

velocityinthedirectionofthe

x

axis

v

velocityinthedirectionofthe

y

axis

w

velocityinthedirectionofthe

z

axis

V

theforwardspeedofthemissil(

|V |

)

I

massmomentofinertiaortheunitarymatrix

I

x

the

x

partof

I

I

y

the

y

partof

I

I

z

the

z

partof

I

Operators and functions



A

B

C

D



shorthandforthestatespace

realizationof

G

(s) = C(sI − A)

A

−1

inverse

A

†

pseudoinverse

A

∗−1

shorthandfor

(A

−1

₎

∗

A.

∗ B

elementwisemultiplicationof

A

and

B

λ

(A)

eigenvalueof

A

|β|

absolutevalueof

β

1 Introduction 1

1.1 Background . . . 1

1.2 Objectives. . . 1

1.3 Limitations . . . 2

1.4 ThesisOutline . . . 2

2 The

H

∞

Loop ShapingDesign 3 2.1 BasicRequirementsfortheClosedLoop . . . 3

2.2 LoopShaping . . . 4

2.2.1 TheScalingProcedure . . . 4

2.2.2 Pairingof theInputsandOutputs . . . 5

2.2.3 DecentralizedControllerDesign. . . 6

2.2.4 AlignmentofSingular Values . . . 6

2.2.5 SummaryoftheLoopShapingProcedure . . . 7

2.3 RobustStabilization . . . 7

2.3.1 NormalizedLeftCoprimeFactorizations . . . 8

2.3.2 TheStabilizingController

K

∞

(s)

. . . 9

2.4 GainScheduling . . . 10

2.4.1 SchedulingonEitherOneorTwoParameters . . . 11

3 The MissileModel 13 3.1 TheMissile . . . 14

3.1.1 TheGuidanceSystemoftheMissile . . . 14

3.1.2 AssumptionsandLimitations . . . 15

3.2 TheMainDynamicsoftheMissile . . . 18

3.2.1 RigidBodyDynamics . . . 18

3.2.2 TheTranslationof theMissile. . . 19

3.2.3 TheRotationoftheMissile . . . 20

3.2.4 SummaryoftheMainDynamicsoftheMissile . . . 21

3.3 ThePlant . . . 21

3.3.1 Choosing theDesignParameters . . . 23

3.3.2 TheNewton-RaphsonMethod . . . 23

4 Controller Designfor the BasicPlant 29

4.1 ThePlantUsedfortheControllerDesign . . . 29

4.2 LoopShapingandRobustStabilizationfortheBasicPlant . . . 30

4.2.1 TheScalingProcedure . . . 30

4.2.2 PairingtheInputsandOutputs. . . 31

4.2.3 DecentralizedControllerDesign. . . 33

4.2.4 AlignmentofSingular Values . . . 33

4.2.5 RobustStabilization . . . 34

5 Linear Analysis 37 5.1 TheClosed LoopSystem. . . 37

5.2 ThePerformanceoftheClosedLoopSystem . . . 38

5.3 RobustnessoftheClosedLoopSystem . . . 39

6 Nonlinear Evaluation 43 6.1 Simulations . . . 43

6.2 TheControllerDesignBasedontheBasic Model . . . 44

6.2.1 TransformationoftheReferenceSignals . . . 44

6.2.2 SimulationswithaSingleController . . . 45

6.2.3 SchedulingontheForwardSpeed. . . 47

6.3 TheControllerDesignBasedontheExtendedModel. . . 49

6.3.1 SchedulingontheAngleofAttack . . . 49

6.4 Basicvs. ExtendedModel . . . 50

7 Conclusions 51 7.1 Results. . . 51

7.2 FutureWork . . . 52

Appendices

A The SystemMatrices ofthe Nominal Plant 53

B The SystemMatrices ofthe Controller 55

C The Matlab Script Used for the ControllerDesign 57

Introduction

Thisreportisapresentationofaprojectperformedat SaabBoforsDynamicsAB

in Linköping, Sweden. For the last twenty years Saab Bofors Dynamics AB has

successfullybeendevelopingmissilesystems,wheretheSwedishdefenseisthemain

customer. Advancedautomaticcontrolisanimportantpartofthemissilesystems

of today. Hence, it is very important for Saab to stay updated in the area of

guidanceandcontrol.

1.1 Background

Robust control design has become amajor research area during the last twenty

yearsandtherearenowadaysseveralrobustdesignmethodsavailable. Oneexample

of such a method is the robust loop shaping method that was developed by K.

GloverandD. C.MacFarlaneinthelate1980s. Theideaofthismethodisto use

decentralizedcontrollerdesigntogivethesingularvaluesoftheloopgainadesired

shape. ThisstepiscalledLoopShapinganditisfollowedbyaRobustStabilization

procedure,whichaimstogivetheclosedloopsystemamaximumdegreeofstability

margins. Thismethod wasused in[8]byR. A.HydeandK.Gloverinanaircraft

application.

1.2 Objectives

The objectives of this project is to use the ideas of K. Glover and R. A. Hyde

[8] to design ascheduled

H

∞

LoopShaping controllerfor a missile application. Thecontrollerwassupposedtobeimplementedin anonlinearsimulation

environ-ment, suppliedby SaabBofors Dynamics. Furthermore, thecontroller capability

1.3 Limitations

In order to keep this projectwithin reasonablelimits there was aneed for some

restrictions. Themissileis supposedto workwellin alargeoperatingarea,which

canbedividedintospeedandaltitude. Alargeareameansthatseveralcontrollers

havetobedesigned. Inordertoreduce thenumberofdesigncases,thisprojectis

restrictedto designingandevaluatingcontrollersforaxedaltitude.

1.4 Thesis Outline

Thisreportcanbedividedintothree dierentparts. Therstpartdescribesthe

H

∞

Loop Shaping controller design and is found in Chapter 2. The controller design consists of a loop shaping procedure which species performance for the

closedloopsystem. Moreover,arobuststabilizationprocedure,whichisappliedin

orderto givetheclosed loopsystemsucientstabilitymargins, andanapproach

forgainschedulingisdescribed. Anaccuratesystemmodelisthecornerstoneofa

successfulcontroller. Moreover,sincegainschedulingis includedin thecontroller

design it is important to be able to derive several linear plant descriptions. A

methodthatmakesthiseasyisdescribedinChapter3. Thischaptercanbeviewed

asthesecondpartofthereport. Thethirdpartofthethesiscontainsanevaluation

of the designed controller. In Chapter 4a short design exampleis given and in

Chapter5theperformanceandrobustnessoftheclosedloopsystemisdiscussed.

Thebest waytoevaluateacontrollerisofcoursetoletitcontroltherealsystem.

Since it, for obvious reasons, is impractical to implement the controller in areal

missile,thenextbest thingistoevaluatethescheduledlinear

H

∞

LoopShaping controller in anonlinearsimulationenvironment. This is described in Chapter 6

The

H

∞

Loop Shaping Design

Thecontroldesignmethodthatisusedinthismasterthesisis

H

∞

LoopShaping. This particular method was proposed by D. C. MacFarlane and K. Glover, and

wasusedbyR.A.HydeandK.Gloverin[8]. Thedesignmethodconsistsofthree

parts, alldescribed in this chapter. A Loop Shaping techniqueis used as a rst

steptospecifyperformanceoftheclosedloopsystem. Theloopshapingprocedure

is followed by a Robust Stabilization procedure with the purpose of maximizing

the stability margins of the closed loop system. Finally, an approach to Gain

Scheduling suitablefor

H

∞

controllersispresented.

2.1 Basic Requirements for the Closed Loop

For a multivariable (MIMO) system there are some basic requirements for the

ClosedLoopSystem

G

c

(s)

. These canbesummarizedasfollows.

•

Thetransferfunction fromthereferenceto theoutputshouldbecloseto

I

, i.e.

G

c

(s) ≈ I

.

•

Thetransferfunctionfromreferencetoinput

G

ru

(s)

shouldnotbetoolarge.

•

The sensitivity function

S

(s)

should be small so that system disturbances andmodelperturbationshavelittleornoeect ontheoutput.

•

Formeasurementdisturbancesto havelittleornoeect on theoutput, the complementary sensitivity function

T

(s)

should be small. Furthermore, it shouldbesmallsothatmodelperturbationsdonotaectthesystemstability.

However,becausethese designobjectivesare usuallyconicting it isnotpossible

tofulllthemall. Furthermore,themissileisapoorlydecouplednonlinearsystem

controllers that give a good compromise between the desired transfer functions

mentionedearlierandthat handlepoorlydecoupledsystems.

2.2 Loop Shaping

In[8]R.A.HydeandK.GloverproposeaLoopShapingprocedureusedtospecify

performanceasarststepinthe

H

∞

LoopShapingcontrollerdesign. Thepurpose ofthisstepistogivethesingularvaluesoftheloopgainadesiredshape. Thiswill

aftertherobuststabilizationprocedureleadtosuitablebehavioroftheclosedloop

system. Accordingto [1] thedesiredshapeof thesingularvaluesoftheloopgain

ishigh gainforlowerfrequenciesand lowgainforhigherfrequencies. Duringthe

transitionbetweenhighandlowgain,theloopgainshouldhaveadecreaseof

−1

. InthismasterthesisprojecttheLoopShapingprocedureconsistsoffoursteps,

which arecarriedoutinthefollowingorder

1. TheScalingProcedure

⇒ D

u

and

D

y

. 2. PairingoftheInputsandOutputs.

3. DecentralizedControllerDesign

⇒ W

p

(s)

. 4. AlignmentofSingularValues

⇒ W

a

.

TheresultfromtheLoopShapingprocedureisashapedplant

G

s

(s)

according to Figure 2.1. The singular values of the shaped plant should havethe desired

shape.

D

1

y

−

Wa

W

p

(

s

)

Wf

Du

G

(

s

)

Figure 2.1. Theresultof theLoopShaping procedure isaloop gainthat has

charac-teristicssuitablefortherobuststabilizationprocedureandforthebehavioroftheclosed

loopsystem. Theuseofthe

W

f

matrixwillbeexplainedinChapter4.

2.2.1 The Scaling Procedure

TherststepintheLoopShapingprocedureistoscaletheinputsandtheoutputs.

Thiscanbeachieved,accordingto[1],byscalingtheinputsandtheoutputssuch

that they vary in the interval between -1and 1. A common scaling is to choose

diagonalmatrices

D

y

and

D

u

sothatthephysicaltruevariables

y

p

and

u

p

satises

y

p

_{= D}

y

y

and

u

p

_{= D}

u

u

,where

y

and

u

varybetween

−1

and

1

. Hence,therst stepoftheLoopShapingprocedure istochoosethematrices

D

y

and

D

u

in (2.1).

D

yy = G(s)Duu ⇔

y = D

−1

y

G

(s)D

u

u

Noticethatthescalingscanbealteredduringthedesignphaseandthatpreliminary

scalingsmayberough.

2.2.2 Pairing of the Inputs and Outputs

InthesecondstepoftheLoopShaping proceduretheinputsand theoutputsare

pairedto achieveaplantthatisasdiagonalaspossible. Theinteractionsbetween

inputs and outputs reect controller capabilities in the sense that strong cross

couplingsoftenleadto poorcontrol. Hence,itis importantfortheplantto beas

diagonalaspossibletomakeiteasierforthecontrollertoachievegoodperformance

androbustness. In[1]itisshownhowtheRelative GainArray can beusedas an

indication of how the inputs and outputs should be arrangedto make the plant

as diagonal as possible. The relative gain array can be calculated according to

Denition2.1.

Denition2.1 The Relative Gain Array of an arbitrary complex-valued matrix,

A

,isdenedas

RGA

(A) = A. ∗ (A

†

)

T

where

.∗

denotes elementwise multiplicationand

†

denotes the pseudoinverse. Inthedenitionabove,

A

isastaticmatrix. Hence,fortheplant

G

(s)

the

RGA

canonlybecalculatedforxedfrequencies. Thetwomostcommonfrequenciesto

considerare

ω

= 0

and thedesiredbandwidth,

ω

= ω

b

, oftheclosedloopsystem. The

RGA

ofamatrix

A

hassomeusefulcharacteristics,e.g.

•

Ifrows(columns)arerearrangedin

A

thenthecorresponding rows(columns)of

RGA

(A)

arerearranged.

• RGA

of a matrix

A

is independent of scalings, e.g.

RGA

(A) = RGA(D

−1

_y

AD

u

)

•

Thesumof theelementsin arow(column)isalways1.

For aplant evaluated in

ω

= 0

andin

ω

= ω

b

there are twoformal resultson howtoarrangetheinputsandoutputsoftheplant,namely

I. Theinputsandoutputsshouldbepairedsuch thatthediagonalelementsof

RGA

(G(iω

_b

))

areasclosetoone,inthecomplexplane,aspossible. II. Avoidpairingsthatimpliesnegativediagonalelementsfor

RGA

(G(0))

.

Thesecondstepistocomputetherelativegainarrayfor

ω

= 0

andfor

ω

= ω

b

andpairtheinputsandoutputstogethersuchthattheresultsIandIIaresatised.

2.2.3 Decentralized Controller Design

Asthethird stepof theLoopShapingprocedure adecentralizedcontroller

W

p

(s)

is designedsuch that the singularvaluesof theshapedplant (loop gain) receives

thedesiredshape. Theideaofthedecentralizedcontrolistodesignacontrolleras

ifthesystemwasdiagonal,i.e. to disregardtheinuence ofcrosscouplings. The

precompensator

W

p

(s)

chosenfor thecontroller designused in this thesisis, like in [8], supposed to addintegralaction to thescaled anddiagonalized plant. The

transferfunction matrix

W

p

(s)

canbedenedaccordingto (2.2).

W

_p

(s) =





K

1

(1 +

1

T

1

s

)

0

0

0

K

2

(1 +

_T

1

₂

_s

)

0

0

0

K

₃

(1 +

_T

1

3

s

)





(2.2)

Thecoecients,

K

1

, T

1

. . . K

3

, T

3

aredesignparametersandcanbealteredduring thedesignphase. Inthenextsection, thefourth (andoptional) stepofthe Loop

Shapingprocedurewill bedescribed. Thisstepusesacertainmethodtoalignthe

singularvaluesofthescaled, diagonalizedandprecompensatedplantat acertain

frequency.

2.2.4 Alignment of Singular Values

As thefourth and last stepofthe LoopShaping procedure,thealignmentof the

singular values of the shaped plant is carried out. The goal of the alignment

procedureistochooseamatrix

W

a

suchthatthesingularvaluesofthetheloopgain aregatherednearoneatthedesiredbandwidthoftheclosedloopsystem. Theidea

istousethe

W

a

matrixfrom Figure2.1to minimizethefrequencyintervalwhere boththesensitivityfunction

S

(s)

andthecomplementarysensitivityfunction

T

(s)

arelarge. Themethodisdescribedin[1]andusesthesingularvaluedecomposition

(SVD).

Denition2.2 Thesingularvalue decompositionof a

n

× m

matrix

A

is

A

= UΣV

∗

where

∗

denotesthe complexconjugatetransposeofamatrix.

U

isa

n

× n

unitary (

U U

∗

_{= I}

) matrix,

Σ

isa

n

× m

matrixwhich has the singular values of

A

along the diagonalandzeroselsewhere and

V

isaunitary

m

× m

matrix.

InthealignmentstepitishereonlynecessarytoconsiderSVD:sforquadratic

3 × 3

matrices. LettheSVDforthematrix

A

be

U

= u1

u

2

u

3

Σ =





σ

_{0 σ}

1

0

₂

0

₀

0

0 σ

3





V

= v1

v

2

v

3

where

u

i

= (u

1i

, u

2i

, u

3i

)

T

and

v

i

= (v

1i

, v

2i

, v

3i

)

T

. Based on the denitions

aboveitispossibleto write

A

as

A

= UΣV

∗

=

3

X

i=1

u

i

σ

i

v

i

∗

Astheequality

Av

j

v

j ∗

= u

j

σ

j

v

j ∗

holds,

A

(I + αv

j

v

j ∗

)

hasthesamesingular valuesas

A

apartfromthe

j

:thwhichhasbeenalteredto

σ

j

(1 + α)

. It ispossible toalterthesingularvaluesofthe

3 × 3

matrix

G

(iω)

atthefrequency

ω

withthe followingmatrix

W

a

= I + α

1

v

1

v

1

∗

+ α

2

v

2

v

2

∗

+ α

3

v

3

v

3

∗

where the coecients,

α

1

to

α

3

, can be chosen such that the singular values of

G

(iω)W

a

willbealignednearone.

2.2.5 Summary of the Loop Shaping Procedure

Themain stepsoftheLoopShapingprocedureare

1. Scale the inputs and outputs by means of matrices

D

−1

y

and

D

u

so that a scaledplant

D

−1

y

G

(s)D

u

isobtained.

2. ComputetheRGAforthefrequencies0andthedesiredbandwidth. Pairthe

inputsandtheoutputsaccordingtothemainresultsinSection2.2.2.

3. Designadecentralizedcontroller

W

p

(s)

sothat thesingularvaluesof

D

−1

_y

G

(s)D

u

W

f

W

p

(s)

havethedesiredshape.

4. Alignthesingularvaluesatthedesiredbandwidthbymeansofthe

W

a

matrix. Thisstepisoptional.

The result of the design steps in the Loop Shaping procedure is the shaped

plant

G

s

(s)

accordingto

G

s

(s) = D

y

−1

G

(s)D

u

W

f

W

p

(s)W

a

The shapedplant

G

s

(s)

can be foundin Figure 2.1and it willbethe subject oftherobuststabilizationproceduredescribedin thenextsection.

2.3 Robust Stabilization

The second part of the controller design phase is to stabilize the shaped plant

G

s

(s)

. This isdonein orderto achievemaximumstabilitymarginsforthe closed loopsystem

G

c

(s)

,i.e. makingtheresultingclosedloopsystemasrobusttomodel perturbations and disturbances as possible. The shape of the singular values of

2.3.1 Normalized Left Coprime Factorizations

TheRobustStabilizationprocedureproduceacontroller

K

∞

(s)

thatwillstabilize a certain class of systems. These systems are those who can be described by

normalizedleft coprime factorizations. A normalizedleftcoprimefactorization of

asystem

G

(s)

isgivenby

G

(s) = ˜

M

−1

(s) ˜

N

(s)

(2.3) where

M

˜

(s) ˜

M

∗

_{(s) + ˜}

_N

_{(s) ˜}

_N

∗

_{(s) = I}

.

The normalizedcoprime factorizationplant descriptionof theplant

G

(s)

can bederivedaccordingtothefollowingtheorem.

Theorem2.1 (Normalizedleft coprimefactorization) Let

G

(s)

begivenby

G

(s) =



A

B

C

D



anddene

˜

R

= I + DD

∗

>

0

Suppose

(C, A)

is detectable and

(A, B)

controllable. Then there is a normalized leftcoprimefactorization

G

(s) = ˜

M

(s)

−1

_N

_˜

_(s)

,with

˜

M

(s) =



A

+ LC

L

˜

R

1/2

C

R

˜

−1/2



and

˜

N

(s) =



A

+ LC B + LD

˜

R

1/2

C

R

˜

−1/2

D



where

L

= −(BD

∗

+ ZC

∗

) ˜

R

−1

Z

isthe positive semidenite solutiontothe algebraicriccati equation

(A − BD

∗

_R

_˜

−1

_C

_{)Z + Z(A − BD}

∗

_R

_˜

−1

_C

₎

∗

_{− ZC}

∗

_R

_˜

−1

_CZ

_{+ B ˜}

_R

−1

_B

∗

_{= 0.}

Proof. SeeTheorem 13.37,in[10].

2

Theclassofperturbationsthat

K

∞

(s)

stabilizesisgivenby(2.4).

G

_∆

(s) = ( ˜

M

(s) + ∆

M

(s))

−1

( ˜

N

(s) + ∆

N

(s))

(2.4) where

∆

M

(s), ∆

N

(s)

arestable transferfunctions that representthe uncertainty in thenominalplant. Noticethat theuncertainties canintroducebothnewpoles

and zerosinto the plant. This meansthat theperturbed plantmight havemore

unstablepoles andzerosthanthenominal plant. Thecontroller

K

∞

(s)

resulting fromthe

H

∞

LoopShapingcontrollerdesignstabilizesperturbedsystems

G

∆

(s)

with

k [∆

N

(s) ∆

M

(s)] k

∞

<

1

2.3.2 The Stabilizing Controller

K

∞

(s)

This section is a review of the almost fully automatic procedure that, given a

shaped plant

G

s

(s)

, producesa stabilizingcontroller

K

∞

(s)

. For ashaped plant givenby(2.6)

G

s

(s) =



A

s

B

s

C

_s

0



(2.6)

thefollowingstepssummarizestheRobustStabilization procedure.

•

Firstandforemostitisnecessarytondthesymmetricandpositivedenite solutions

X

and

Z

to thecontrol algebraic riccati equation andtheltering algebraicriccati equation

A

_s

∗

X

+ XA

_s

− XB

_s

B

_s

∗

X

+ C

_s

∗

C

_s

=0

A

s

Z

+ ZA

∗

s

− ZC

s

∗

C

s

Z

+ B

s

B

∗

s

=0

(2.7)

•

Theoptimal(smallest)

γ

opt

maximizing(2.5)isgivenby

γ

opt

=

p

1 + λ

m

(XZ)

(2.8)

where

λ

m

(XZ)

isthelargesteigenvalueof

XZ

. However,itis,accordingto [8],provenbyexperiencethatbetterresultsareoftenachievedwithaslightly

larger

γ

. With this in mind it is usually better to set

γ

to approximately

1.1γ

opt

.

•

Denetwomatrices

F

and

H

as

F

=γ

2

B

_s

∗

X

((1 − γ

2

)I + XZ)

∗−1

H

= − ZC

_s

∗

(2.9)

Thenthestabilizingcontroller

K

∞

(s)

isgivenby

K

_∞

(s) =



A

s

+ HC

s

+ B

s

F

−H

F

0



(2.10)

Thecontroller

K

∞

(s)

can,forexample,beimplementedaccordingtotheblock diagraminFigure2.2. Thebenetofhavingthecontroller

K

∞

(s)

inthefeedback comparedtoaunityfeedbackisthatanabruptchangeintheinputsdoesnotexcite

y

s

r

G

s

(

s

)

K

∞

(

s

)

K

∞

(

0

)

Σ

Figure2.2. Theclosedloopsystemfortheshapedplant

G

s

(s)

andthecontroller

K

∞

(s)

. Noticethestaticgain

K

∞

(0)

.

The implementation in Figure 2.2 is however not suitable due to the gain

schedulingapproachin this controllerdesign method. An alternative

implement-ation will bedescribed in the nextsection. There the controller is written asan

exactplantobserverplus statefeedback and this structure turns outto bemore

suitableforgainscheduling.

2.4 Gain Scheduling

Linearcontrollerdesignisoftenusedtostabilize anonlinearsystem,especiallyin

ightapplications. Sincealinearcontrollerdesignonlyisbasedonalinearmodel

one controller is usually insucient. The third step of the

H

∞

Loop Shaping controllerdesign is theGain Scheduling approach. The ideais to implement the

controller asa plantobserverplus state feedback and to use linear interpolation

betweencontrollersofadjacentdesignpoints. In[2]itisshownhowthecontroller

K

_∞

givenby(2.10)canbewrittenasanexactplantobserverplusstatefeedback. Theresultis

(

˙ˆx = A

s

ˆx + B

s

u

s

+ H(C

s

ˆx − y

s

)

u

s

= F ˆx + P r

(2.11)

All matrices in (2.11) apart from

P

can be recognized from (2.10). The

P

matrixisintroducedinordertogivethetransferfunctionfromreferencesignalsto

outputsignalsthestaticgain

I

. InSection5.1 itwill beshownthat

P

shouldbe chosenas

Theobserverimplementationisshownintheblockdiagramin Figure2.3. For

thismasterthesisprojectthecontrollerisscheduledoneitheroneortwo

paramet-ers,thespeedofthemissile

V

andtheangleofattack

α

.

Σ

Observer

F

P

u

_s

y

_s

r

G

s

(

s

)

Figure 2.3. Theobserverimplementationofthe

H

∞

LoopShapingcontroller.

2.4.1 Scheduling on Either One or Two Parameters

Forthismaster thesisprojectitisonlynecessarytodesign againscheduled

con-trollerthat useseither the forward speed

V

, or

V

andthe angle of attack

α

, for theinterpolation.

SchedulingontheForward Speed

Assume,forexample,thattheforwardspeedvariesintheinterval

[V

1

, V

2

[

andthat twocontrollerswith

H

matrices

H

1

and

H

2

areavailable. These controllershave beendesigned using the linearizationsaround

V

= V

1

and

V

= V

2

, respectively. Let

η

beafunction of

V

η

=

V

− V

1

V

2

− V

1

(2.12)

Aninterpolated

H

matrix

H

(η)

canthenbedened as

H

(η) = (1 − η)H

1

+ ηH

2

(2.13)

SchedulingontheForward Speedandthe Angleof Attack

Assume, in analogywith the previous section, that the missile forward speed

V

varies in the interval

[V

1

, V

2

[

, that the angle of attack

α

varies in the interval

[α

1

, α

2

[

andthatfouradjacentcontrollersareavailable.Thesecontrollershavebeen designedaroundfourlinearizations.Forexample,the

H

matrixforthelinearization around

V

= V

1

and

α

= α

1

iscalled

H

11

. Let

ε

beafunction of

α

ε

=

α

− α

1

Aninterpolated

H

matrix

H

(η, ε)

canbedenedas

H

(η, ε) = (1 − η)[(1 − ε)H

₁₁

+ εH

₁₂

]+

η

[(1 − ε)H

₂₁

+ εH

₂₂

]

(2.15)

Itisstraightforwardto extendtheschedulingtothreeormorevariables.

Thetopicofthenextchapteristhemissileanditscharacteristics. Thechapter

willdiscussthemissile ingeneralanditwillalsoshowhowalineardescriptionof

The Missile Model

An accurate mathematical descriptionof the control objectis the cornerstone of

asuccessfulcontroller. Theplantis heretheair-to-air missile in Figure 3.1.

Al-though this missile does not exist yet, it has several features in common with

existingones. The

H

∞

LoopShapingcontrollerdesignmethod isbasedonlinear controltheory. Thus,thetopicofthischapteristodescribehowalinearplant

G

(s)

, which describesthemaindynamics ofthemissilein Figure3.1,can bederived.

3.1 The Missile

This sectionis asummary of themissile characteristicsand theguidance system

of themissile. Themissile studied in this master thesis projectis a bank-to-turn

missile,which meansthatwhenthemissile isgoinginto aturnitshould rstroll

inthedirectionofitsvelocityvector

V

andthenaccelerateinthedirectionofthe

z

axis.

3.1.1 The Guidance System of the Missile

Thissectionwillgiveanintroductiontotheguidancesystemofamissile. InFigure

3.2asimplied descriptionoftheguidancesystemisgiven.

Figure 3.2.AsimplieddescriptionoftheguidancesystemoftheMissile.

AccordingtoFigure3.2,theguidancesystemconsistsofthreeparts,whereeach

Strap Down Navigation. Whenthe missileis released thestrap down

naviga-tion system receivesinitial position, attitude, rotationand speed from the

aircraft navigation system. The strap down navigation system uses

meas-urementsfromrategyrosandaccelerometersincombinationwiththeinitial

conditionstocalculateposition,attitude, rotationandspeed. Theresultsof

thesecalculationscanbeusedbybothGuidanceandAutopilot.

Guidance. The core of the guidance system is the block Guidance, which by

predened laws decides what the missile should do. Guidance receives

in-formationconcerningthetargetpartlyfrom theaircraftandpartlyfromthe

seeker. This information, combined with the missile state variables

calcu-latedbythestrapdownnavigationsystemandthepredenedlaws,produces

demanded referencesignalstotheautopilot.

Autopilot. The autopilot receivesreference signalsas demanded output signals

in termsof desiredangular velocity aroundthe

x

axis, desiredacceleration in thedirectionofthe

y

axisand desiredaccelerationinthedirection ofthe

z

axis. These reference signals are

p

d

,

A

yd

and

A

zd

and the main task of theautopilot istoproducecommanded ndeectionssuchthat thedesired

referencesareachieved.

The

H

∞

LoopShapingcontrollerofthisprojectisimplementedintheshaded blockAutopilotinFigure3.2. Theautopilotreceivesdemandsin

p

d

,

A

yd

and

A

zd

fromtheguidancesystem. DuetotheBank-to-Turnsteeringprinciplethedemands

in

p

d

,

A

yd

and

A

zd

aresuitableas referencesignals. 3.1.2 Assumptions and Limitations

Thepurposeofthischapteristoproducelinearmodelsofthemaindynamicsofthe

missile that canbeused forthe

H

∞

Loop Shapingcontroller design. Thelinear modelof themissile iscalled aplantanditisdesirablethattheplantdescription

has aslow degree aspossible. Hence, it is necessary to make some assumptions

duringthemodelingofthemaindynamicsofthemissile. First,thereisaneedfor

anexplanationofthenotationinFigure 3.1.

• (x, y, z)

isthexedbodyframeofthemissile.

• ω = (p, q, r)

T

is theangularvelocityof themissile

[rad/s]

.

• V = (u, v, w)

T

isthemissilevelocityexpressed

in thebodyframeofthemissileand

V

= |V | [m/s]

.

• α

istheangleof attack

[rad]

.

• β

isthesideslipangle

[rad]

.

The missile also has a mass moment of inertia matrix

I

, which due to the symmetryofthemissile containnomixed inertiatermsandisgivenby

I

=





I

_{0 I}

x

0

_y

0

₀

0

0 I

z



 [kgm

2

_]

(3.1)

Thenexttwosectionscontaindescriptionsofthelimitationsandtheassumptions

madeinthisproject.

TheLimitations ofthe Modeling Procedure

Inoperating conditionsthere areloads acting onthemissile, e.g. gravity, engine

thrust, engine torque aerodynamic contributions etc. Some of the loads are not

goingtobeconsideredforthemodelingof themain dynamicsofthemissile. The

gravityisneglectedduetoitssmallcontributionandtomaintainmodelsimplicity.

The contribution from the engineis neglectedbecause the rate of change of the

enginethrustismuch slowerthantheotherdynamics. Theonlyforceand torque

contributionthataregoingtobeconsideredaretheforceandtorquethatdescend

fromtheaerodynamicsofthemissile,see[5]. Theaerodynamicforceactingonthe

missileisaccordingto [6]givenby

F

a

= −q

d

S





C

C

T

C

C

N





(3.2) where

C

_T

=1

C

C

=C

Cβ

β

+ C

Cδ

r

δ

r

C

N

=C

N α

α

+ C

N δ

e

δ

e

andtheaerodynamictorqueactingonthemissileisaccordingto[6]givenby

M

a

= −q

d

Sd





C

C

m

l

C

n





(3.3) where

C

l

= C

lβ

β

+ C

lp

d

2V

p

+ C

lδ

a

δ

a

C

m

= C

mα

α

+ C

m|β|

|β| + C

mq

d

2V

q

+ C

mδ

e

δ

e

C

_n

= C

_nβ

β

+ C

_nαβ

αβ

+ C

_nr

d

2V

r

+ C

nδ

a

δ

a

+ C

nδ

r

δ

r

In Equations(3.2) and (3.3)

q

d

is the dynamic pressure,

d

areference length and

S

= πd

2

_/

₄

Mach 1.5 2.0 2.5 3.0 Mach 1.5 2.0 2.5 3.0

C

_Cβ

20 20 20 20

C

Cδ

r

-6 -6 -6 -6

C

N α

33 31 29 27

C

N δ

e

8 7 6 5

C

lβ

1.5 2.5 3.0 3.5

C

lp

-30 -27 -24 -21

C

lδ

a

-6 -6 -6 -6

C

mα

-16 -14 -12 -10

C

m|β|

-14 -11 -8 -5

C

mq

-1900 -1600 -1300 -1000

C

mδ

e

-49 -45 -41 -37

C

nβ

24 22 19 16

C

nαβ

-200 -175 -150 -125

C

nr

-2000 -1700 -1400 -1100

C

_nδ

_a

11 9 7 5

C

nδ

r

-50 -45 -40 -35

Table3.1. Theaerodynamiccoecients.

The missile ight envelopeis rather large, with a largespan in both altitude

andmissileforwardspeed. Thealtitudeandspeedintervalsthatthemissileshould

beabletooperateinare

•

Altitude:

0 ≤ h ≤ 20000 m

.

•

Forwardspeed:

450 ≤ V ≤ 1200 m/s

.

Due to thelimitations mentionedin Chapter 1 thecontrollersstudied in this

thesisarederivedfortheforwardspeedintervalmentionedearlierandforthexed

altitudeof

h

= 1000 m

.

TheAssumptionsofthe Modeling Procedure

Forthemodeltobesucientlysimpleitisnecessarytomakesomeapproximations.

Besidesneglectingsomeforceandtorquecontributionsfouradditionalassumptions

aremade,namely

1. Assume that theangles

α

and

β

in Figure 3.1are small, which forexample meansthat

sin β ≈ β

and

tan α ≈ α

.

2. Assume smallvariationsin themissile forwardspeed,i.e.

˙V ≈ 0

.

3. Assume that themassofthemissile isconstant, i.e. neglectdecrease ofthe

massofthemissilecausedbythecombustionoffuel.

3.2 The Main Dynamics of the Missile

Themaindynamicsofthemissileisdescribedwithbasicrigidbodydynamicsand

suitableapproximations.

3.2.1 Rigid Body Dynamics

When a rigid body is inuenced by forces it will go into translation and when

inuencedbytorqueitwillstartrotating. Hence,itisnecessarytounderstandthe

dynamicsof therigid body. Thetoolsused to create amathematicaldescription

areold but powerful. IsaacNewton rst postulated therelationshipbetweenthe

force actingon arigid body and itsaccelerationand this relationis hencecalled

Newton'slaw of motion. Thefactthattheappliedtorqueequalstherateofchange

of angularmomentum was rstpostulated byLeonardEuler and this equalityis

usuallyknownas Euler'sequation. Theyaretwowellknownlawsofnature buta

remindermightbeappropriate.

Denition3.1 (Newton'slaw of motion) Theforceonarigidbodyequalsthe

massof the rigidbodytimesitsinertial acceleration:

F = ma.

Denition3.2 (Euler'sequation) Theappliedtorqueequalstheinertialrateof

changeof the angularmomentumofarigid body:

M = ˙

H

Theangularmomentum canbeexpressedin termsofmassmomentofinertiaand

therotationoftherigidbody,i.e.

H = Iω

. Noticetheuseofinertial in thetwo denitionsabove. Inordercalculate theinertialtimederivativeofarigid body it

is necessaryto pay attention to its rotation. The theorem below shows how the

inertialtimederivativeofarigidbodycanbecalculated.

Theorem3.1 (Inertial derivative) The time derivative of a vector

v

with re-spect to the inertial (

i

) frameis related tothe time derivative with respectto the bodyframe(

b

) by

d

dt

i

v =

d

dt

b

v + ω

bi

_{× v}

where

ω

bi

isthe angularvelocity ofthe bodyframeand

v

isthe vectorinthe body frame.

Proof. Theproofisgivenin[9].

2

The main purposeof derivinga mathematical descriptionof the missile is to

use it for the controller design. Theearlier mentioned simplications havebeen

madeinordertomaketheresultingmodeleasiertowork with,hopefullywithout

changingthemaindynamicsofthemissiletoomuch. Themodelingapproachisto

3.2.2 The Translation of the Missile

Forthemissile translationitissensibletoexpresstheacceleration

a

inDenition 3.1asthetimederivativeofthemissilevelocity

V

. Thistimederivativeisobtained byapplyingTheorem3.1onthemissile velocityaccordingto(3.4).

a = ˙V + ω × V

(3.4)

Sinceonlytheaerodynamicforce

F

a

isconsidered,thetranslationofthemissile isobtainedbycombining(3.4)andDenition3.1. Theresultis





F

F

ax

ay

F

az



 = m





˙u

_˙v

˙w



 + m





p

q

r



 ×





u

v

w





(3.5)

Equation(3.5)containsthetimederivativesof thesuitablestatevariables

u, v

and

w

. The mathematicaldescriptionwhich isthepurposeofthis sectionshould expressthetimederivativesofthestatesasfunctionsof thestatesandtheinputs

anditisthereforesensibleto rewrite(3.5)accordingto(3.6).

˙u =

F

ax

m

+ rv − qw

˙v =

F

ay

m

+ pw − ru

˙w =

F

az

m

+ qu − pv

(3.6)

Itispossibletoinclude thetwoangles

α

and

β

asstatesinsteadof

u, v

and

w

. Todothatthereisaneedforsomefurtherassumptions. AsseeninFigure3.1the

following relationbetweenthe angles,

α

and

β

, and the velocities,

u, v

and

w

, is obvious.

sin β =

v

V

tan α =

w

u

(3.7)

The assumption mentioned earlier considering small angles is useful and the

approximations

sin β ≈ β

and

tan α ≈ α

are in this application sucient up to

|α| ≤ 30

◦

and

|β| ≤ 30

◦

. Hence,therelationshipbetween

α

and

β

and

u

and

w

is

β

=

v

V

α

=

w

u

(3.8)

Forsmallanglesitispossibletoassumethat

V ≈ V x

,whichmeansthat

u

≈ V

, andifthevariationsin thespeed

V

aresmallthen

˙V ≈ 0

and sois

˙u

. Thisleads,

combinedwith Equations(3.6)and (3.8),to tworst orderdierential equations in

α

and

β

˙α =

F

az

mV

+ q − pβ

˙β =

F

ay

mV

+ pα − r

(3.9)

The forces

F

az

and

F

ay

areobtained from

F

a

in (3.2)and this results in the descriptionofthe

α

and

β

dynamicsaccordingto (3.10).

˙α = − pβ + q −

q

d

S

mV

(C

N α

α

+ C

N δ

e

δ

e

)

˙β =pα − r −

q

d

S

mV

(C

Cβ

β

+ C

Cδ

r

δ

r

)

(3.10)

3.2.3 The Rotation of the Missile

The total torque acting on the missile equals the rate of change of the angular

momentumofthemissile. Thisis,duetothefactthat

H = Iω

,thedescriptionof therotationof themissile. Theinertialtimederivativeof

H

isthen givenbythe followingequation.

˙

H = I ˙ω + ω × Iω

(3.11)

The Euler equation from Denition 3.2 combined with (3.11), leads to the

followingequation.





M

M

ax

_ay

M

az



 =





I

I

x

_y

˙p

˙q

I

z

˙r



 +





p

q

r



 ×





I

I

x

_y

p

q

I

z

r





(3.12)

Suitable statevariablesin (3.12)are

p, q

and

r

. Byextractingthetime deriv-ativesofthesevariables from(3.12)thefollowingexpressionisobtained.

˙p =

M

x

I

x

+

I

_y

− I

_z

I

x

qr

˙q =

M

y

I

_y

+

I

z

− I

x

I

_y

pr

˙r =

M

z

I

_z

+

I

x

− I

y

I

_z

pq

(3.13)

Equation(3.13)incombinationwith(3.3)leadstothreerstorderdierential

equationsin

p

,

q

and

r

accordingto (3.14).

˙p =qr

I

y

− I

z

I

_x

+

q

d

Sd

I

_x

(C

lβ

β

+ C

lp

d

2V

p

+ C

lδ

a

δ

a

)

˙q =pr

I

z

− I

x

I

_y

+

q

d

Sd

I

_y

(C

mα

α

+ C

m|β|

|β| + C

mq

d

2V

q

+

+ C

mδ

e

δ

e

)

˙r =pq

I

x

− I

y

I

_z

+

q

d

Sd

I

_z

(C

nβ

β

+ C

nαβ

αβ

+ C

nr

d

2V

r

+

+ C

nδ

a

δ

a

+ C

nδ

r

δ

r

)

(3.14)

3.2.4 Summary of the Main Dynamics of the Missile

The main dynamics of the missile, described by equations (3.10) and (3.14), are

summarizedhere.

˙p =qr

I

y

− I

z

I

_x

+

q

d

Sd

I

_x

(C

lβ

β

+ C

lp

d

2V

p

+ C

lδ

a

δ

a

)

˙q =pr

I

z

− I

x

I

_y

+

q

d

Sd

I

_y

(C

mα

α

+ C

m|β|

|β| + C

mq

d

2V

q

+

+ C

mδ

e

δ

e

)

˙r =pq

I

x

− I

y

I

z

+

q

d

Sd

I

z

(C

nβ

β

+ C

nαβ

αβ

+ C

nr

d

2V

r

+

+ C

nδ

a

δ

a

+ C

nδ

r

δ

r

)

˙α = − pβ + q −

q

d

S

mV

(C

N α

α

+ C

N δ

e

δ

e

)

˙β =pα − r −

q

d

S

mV

(C

Cβ

β

+ C

Cδ

r

δ

r

)

(3.15)

There is need to be able to use several linearized models for the

H

∞

Loop Shapingcontrollerdesign. Theselinearizationscanbederivedfrom (3.15).

3.3 The Plant

Theplantusedforthe

H

∞

LoopShapingcontrollerdesignisalineartransfer func-tion

G

(s)

mappingtheinput

u

totheoutput

y

. Thisplantisalineardescription ofthemaindynamicsofthemissile inFigure3.1anditisdesirablethattheplant

isasimplebut yet accurate descriptionof themissile. The resultof theprevious

sectionwasEquation(3.15),whichisanonlineardescriptionofthemaindynamics

ofthemissile, andfrom thisequation aplantisderived. Sincegainschedulingis

includedinthecontrollerdesignitisnecessarytobeabletoproduceseverallinear

Theresultfromthissectionwillbetheplant

G

(s)

mappingtheinput

u

tothe output

y

anditcan eitherbewritten asastatespacerealizationaccordingto

(

˙x = Ax + Bu

y = Cx + Du

(3.16)

orasatransferfunction matrixaccordingto

G

(s) = C(sI − A)

−1

B

+ D

(3.17)

In(3.16)

x

isthestatesdened as

x =













p

q

r

α

β













(3.18)

and the input

u

is the n deections of theaileron, the elevator and therudder accordingto

u =





δ

δ

a

_e

δ

_r





(3.19)

Inboth(3.16)and (3.17) thematrices

A

,

B

,

C

and

D

are includedandthese matricesarecalledsystemmatrices.

Themaindynamicsofthemissiledescribedbyequation(3.15)willinthefuture

bereferredto as

˙x = f(x, u, h, V )

(3.20)

andthelineardescriptionof (3.20)includesthetwosystemmatrices

A

and

B

and hasthefollowingappearance

˙x = Ax + Bu

(3.21)

The followingtheorem, which can be found in [1], is used to calculate the

A

and

B

matrices.

Theorem3.2 (Linearization) If

f(x, u, h, V )

isdierentiableinaneighborhood of the stationary point

x

0

,

u

0

,

h

0

and

V

0

it is possible to approximate equation (3.20)with

˙z = Az + Bv + g(z, v)

(3.22)

where

z = x − x

0

,

v = u − u

0

and

g(z,v)

|z|+|v|

→ 0

when

|z| + |v| → 0

. The two matrices

A

and

B

aregiven by

A

=

∂f

(x, u)

∂x









x=x

0

,u=u

0

,h=h

0

,V =V

0

(3.23) and

B

=

∂f

(x, u)

∂u









u=u

0

,x=x

0

,h=h

0

,V =V

0

(3.24)

Proof. SeeTheorem

11.1

in [1].

2

A plant description of the missile can be obtainedby choosing the

C

and

D

matricesof (3.16)suchthat thedesiredoutputs areobtained. Accordingto

The-orem3.2 thelinearizationis onlyvalid in aneighborhood of thestationary point

x

₀

and

u

0

of (3.20)andthispointsatises

f(x

0

, u

0

, h

0

, V

0

) = 0

(3.25) The Newton-Raphson method is a techniquethat makes it possible to nd a

numericalsolution

x

0

,

u

0

,

h

0

and

V

0

to(3.25). Thismethodisrestrictedtosystems with

n

equationsand

n

unknowns. Thisis notthecasefor(3.25), which hasve equations and tenunknowns. The unknowns are theve states, thethree input

signals,the speed of themissile

V

and thedynamic pressurewhich is afunction of

V

and thealtitude

h

. Beforeitispossibletocalculate thestationarypointsof (3.25)usingtheNewton-Raphsonmethoditisthusnecessarytochooseveofthe

tenunknownsasdesignparameters.

3.3.1 Choosing the Design Parameters

Theightenvelopeisdividedintothespeedofthemissile

V

andthealtitude

h

ac-cordingtoSection3.1.2. Thisimpliesthat

V

and

h

aresuitabledesignparameters. Themain focus forthis masterthesisis to evaluate controllersfor xedaltitudes

andvariationsin speed. Theangleofattack

α

isagainschedulingparameterand shouldalsobechosenasadesignparameter. Therearetworemainingdesign

para-metersto choose. Theangular velocity

p

aroundthe

x

axisisasensiblechoiceas thefourthdesignparameterduetothefact that

p

d

isareferencesignal. Another referencesignalistheaccelerationin thedirectionof the

y

axis,which makesthe sideslipangle

β

agood choiceasthefthdesignparameter.

3.3.2 The Newton-Raphson Method

Asseenin theprevioussectionthevedesignparametersarechosenas

•

The angular velocity aroundthe

x

axis,

p

.

•

Theangleofattack,

α

.

•

Thesideslipangle,

β

.

•

The forward speed of the missile,

V

.

•

Thealtitude,

h

.

Thisleavesasystemofveequationsandveunknownswhichaccordingto[7]

canbesolved by using theNewton-Raphson method. With the designpointsas

xedvalues(3.20) canberewrittenas