An Enhanced Information Structure for Linear Parameter-Varying Design: Application to Reichert's Missile Benchmark

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An Enhanced Information Structure for Linear

Parameter-Varying Design: Application to Reichert’s

Missile Benchmark

Safta de Hillerin, Vincent Fromion, Gérard Scorletti, Gilles Duc, Emmanuel

Godoy

To cite this version:

Safta de Hillerin, Vincent Fromion, Gérard Scorletti, Gilles Duc, Emmanuel Godoy. An Enhanced

Information Structure for Linear Parameter-Varying Design: Application to Reichert’s Missile Bench-

mark. AIAA Guidance, Navigation, and Control Conference, Aug 2010, Toronto, ON, Canada. 18 p.,

�10.2514/6.2010-8196�. �hal-01676646�

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Parameter-Varying Design: Appli ation to Rei hert's

Missile Ben hmark

Safta de Hillerin 1

, Vin ent Fromion 2

, GérardS orletti 3

, GillesDu 4

,and EmmanuelGodoy 5

1,4,5

SUPELEC Systems S ien es (E3S), Automati ControlDepartment,Fran e

2

Institut National de la Re her he en Agronomie, Unité de Mathématique, Informatique et

Génome UR 1077, Fran e

3

E ole Centrale de Lyon, LaboratoireAmpère UMR CNRS 5005, Fran e

Thispaperis on ernedwith theappli ation oflinear parameter-varying(LPV) methods.

Itspurposeistoinvestigatetheinterestofanewinformationstru turefortheLPV ontrollers.

Theproposed improvement onsists inextending the traditionalinformation stru tureby

introdu ing,besidethesignalsusuallymeasured,spe ialsignalssupposedavailablefor ontrol.

This enhan es the designin twodire tions: rst, the performan e ofthe obtained ontroller

isimproved by amorea urate adjustmenttothe LPV systemparameter value;se ond,this

stru tureenablesthe implementationofa ontroller ofredu ed omplexity inrelation to the

LPV systemparameter.

Theadvantagesofthe proposed stru ture areillustratedonthe single-axis missile ontrol

problemproposed by Rei hertwhi h hasbeenintensivelystudiedin theexistingliterature.

I. Introdu tion

A linearparameter-varying(LPV)systemisdenedas:

z = G LP V (w)



 

 

 

 

˙x(t) = A (θ(t)) x(t) + B (θ(t)) w(t)

z(t) = C (θ(t)) x(t) + D (θ(t)) w(t)

x(t 0 ) = x 0

(1)

where

x(t) ∈ R ⁿ

^is^the^state^ve
tor,

w(t) ∈ R ⁿ ^w

^is^thedisturban einput,

z(t) ∈ R ⁿ ^z

îs^theôutputând

θ(t) ∈ R ^p

is a time-varying exogenous parameter ve tor valued in a hyper ube (ea h parameter

θ i (t)

^ranges ^between

knownextremalvalues

θ _i

^and

θ i

^).

Theinterestofthese systemsliesin thefa t thatthey anmodellineartime-varying (LTV)and nonlinear

plants. Assumingthat theparameteris measured,theideaisto usetheparametermeasurementstoimprove

thedesign omparedtoalineartime-invariant(LTI)strategy.

A ustomarymethodforndingaparameter-dependent ontroller(alsodenotedbygain-s heduled ontroller)

wasdevelopedheuristi ally byengineers from LTI methods, see referen e.

1

It onsists in designing LTI on-

trollersusinglinearizationsoftheplantasso iatedtofrozenvaluesoftheparameter. Theparameter-dependent

ontrolleristhenobtainedbyinterpolatingthese LTI ontrollersasfun tionsoftheparameter. Althoughthis

(3)

man e andeven stability, sothatin pra ti eengineersneedto a posteriori test theperforman e byintensive

dynami alsimulations.

Theseseriousdrawba ksmotivatedthesear hforasystemati approa hto onstru tagain-s heduled on-

troller, see e.g. referen es.

1,2

Important ontributions in this eld are datedfrom the beginning of the 90's

and are due to Pa kard.

3

The LPV problem wasformulatedas the problem of minimizing the

L 2

^gain ^of ^a

systemaugmentedwithweightingfun tions,knownasthe

L 2

^gain^LPVôntrol^problem,^whi
hîsânêxtension

of the

H ∞

ôntrol ^problem. Îndeed, ân ^LTI ^plantîs â^very^spe
i âseôf ^LPV ^plantând ^moreover, ^the

L 2

gainof anLTIsystem isequalto its

H ∞

^norm ^so ^that ⁱⁿ ^theâseôf ân^LTI^plant, ^the

L 2

^gain^LPV^ontrol

problem redu es to the

H ∞

ôntrol ^problem. ^The îssue ^was^then ^to ôbtain^tra
table ônditions ^to ^solve ^the

problem. The

L 2

^gain ^LPV ôntrol ^problem ^turned ôut ^to ^be ^di
ult: îndeed, ^so ^far, ⁱⁿ ^the ^general âse

only su ient onditions ould be written as a onvex LMI optimization problem, hen e tra table. These

methodsmaythereforebeover onservative. Numerousapproa heswereproposed. Thesimplest arebasedon

quadrati Lyapunovfun tions andaretherefore onservativein the asewhereparametershavebounded rates

of variation: Pa kard 3

orApkarian andGahinet 4

re asttheproblem asarobustsynthesisproblemandsolved

it using as aled versionof the small-gain theorem, leadingto onvex onditions expressed as Linear Matrix

Inequalities (LMIs). However,sin ethes aled small-gaintheoremis only on erned withsymmetri s alings,

these pro eduresare unabletotakeintoa ountthefa tthat theparametersare knowntobereal. Basedon

the exploitationof inter onne tedsystemsproperties,less onservativeresultswere obtainedby S orlettiand

El Ghaoui 5

byintrodu ingskew-symmetri s alingsandbyS herer 6

usingfull-blo ks alings.

Parameter-dependentLyapunovfun tions anfurtherredu e onservatism,howevertheyleadtoparameter-

dependentLMI optimization problemswhi h are ingeneralnottra table andmethods fortransformingthese

problems intotra tableproblemsusuallyintrodu e onservatism,seee.g. referen es.

710

Despitethesetheoreti allimitations,inpra tisethesemethodsyielden ouragingresultssin eithasbe ome

possibletoobtaina ontrollerthatguaranteesthe losed-loopstabilityandperforman e. However,somepoints

moderatethesesu esses. Indeed,itwasobservedfromthestudyoffrozenlinearizationsthattheobtained on-

troller seemsnottoadjust mu h totheparametervalue,seee.g. the on lusionsin referen e.

11

Traditionally,

thisphenomenonwas hargedonthea ountofthe onservatismintrodu edbythemethodsforsolvingthe

L 2

gainLPV ontrol problem. Anotherlimitation oftheLPVsynthesismethodsis thattheyprodu e ontrollers

of high omplexityintheparameter,thuspossiblyinvolvingheavy omputationsforimplementation.

Thepresentinvestigationsuggestsanotherexplanationforthephenomenon. Indeed,be auseLPVmethods

arisefromLTImethods(andmorepre iselyfromthe

H ∞

^method),ⁱⁿengineeringpra tisesomepro essesthat areusual andlegitimateinanLTIframeworkhavebeentransposedtotheLPV ontextad ho ,that is,some-

timeswithoutfurtherinvestigationonthevalidityoftheanalogy. Inparti ular,theadequa yofthetraditional

LTIinformationstru ture(thatis,the hoi eofsignalsavailablefor ontrol)intheLPV ontexthasnotreally

beeninvestigated.

This paper fo uses onthe issue of thesele tion of ontrol signalsand suggestsa seeminglymore suitable

hoi e: theideaistointrodu efor ontrol,besidesthe lassi almeasures,twootheravailablesignals.

Onesignal isa systemoutput that givesinformation about theoperating point. This leadsto signi ant

amelioration of thedesign, aswill be demonstrated by omparison with lassi alresults: rst, itameliorates

the performan e leveland se ond, the frozenlinearizations indi ate that the ontroller adjusts better to the

parametervariations.

The other signal introdu ed in order to further enhan e the design is the signal orresponding to the

parameter-dependentterm inthestate-spa eequations, whi h anbesupposed available withoutmakingfur-

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ther hypotheses. The ideafor this stru ture wasrst introdu ed by Wu and Lu and a ontribution of the

presentpaperisto suggestawaytoexploit that ideatoimprovethedesignof LPV ontrollers. Theresultin

referen e 12

impliesindeedthatthenewstru tureisallthemoreinterestingthatin identallyitenablestoobtain

a ontrollerofredu ed omplexityin theparameter,hen einvolvingless omputationsandthus ir umventing

amajorlimitationoftheusualsynthesismethods.

Thepaperisorganizedasfollows. Themissilemodelandthedesignobje tivesoftheben hmarkproposedby

Rei hertarepresentedinSe tionII. AnLPV ontrolleris al ulatedinSe tionIIIusinga lassi alinformation

stru ture. A new information stru ture is proposed in Se tion IV, where it is shown that the resulting LPV

ontrollerhasabetterdependen e ontheparameterandthereforea hievesbetterperforman e. InSe tion V,

it is provedthat the stru turepresentsmoreoverthepra ti al interestof permittingthe implementation of a

ontrollerofredu ed omplexityintheparameter. Con ludingremarksendthepaperin Se tionVI.

Notations

Thenotationis fairlystandard.

M ^T

^is^the^transpose^of^matrix

M

^. ^F^or^a^symmetri^real ^matrix

M

^,

M > 0

and

M < 0

^stand respe tively for positive denite and negative denite while

M ≥ 0

^and

M ≤ 0

^stand

respe tivelyfor nonnegativeandnonpositivedenite. TheLapla evariable isdenoted by

s

^and

˙x = ^dx _dt

^is ^the

timederivative.

I n

îsûsed^to^denote^theîdentity^matrixôf^size

n

^and

O m×n

^the^zero^matrix^of^dimensions

m× n

but when dimensions are obvious from ontext, only the notation

I

^and

O

^may^be ^used. ^The ^maximal ^and

minimalsingularvaluesofamatrix

M

^are^denotedrespe tivelyby

σ(M )

^and

σ(M )

^. ^Thestate-spa erealization oftransfer

G(s) = D + C(sI − A) ⁻¹ B

^is^denoted ^by

G(s) =





A B

C D





^. ^The

H ∞

^norm^of^a^stable^LTI^system

G

^with^transfer^fun
tion

G(s)

^is ^denoted^by

||G|| ∞

^and^dened^as

||G|| ∞ = sup _{ω∈[0,+∞)} σ (G(jω))

^.

Next are givensome denitions and notationsspe i to theLPV ontext. Weintrodu e theaugmented

LPVplant

P LP V

^:





z

y



 = P LP V





w

u







 



 



˙x(t) = A (θ(t)) x(t) + B w (θ(t)) w(t) + B u (θ(t)) u(t)

z(t) = C z (θ(t)) x(t) + D zw (θ(t)) w(t) + D zu (θ(t)) u(t)

y(t) = C y (θ(t)) x(t) + D yw (θ(t)) w(t)

x(t 0 ) = x 0

(2)

where

u(t) ∈ R ⁿ ^u

îs^theôntrolledînputând

y(t) ∈ R ⁿ ^y

îs ^the^measuredôutput. În^the^proposedâpproa
hes

of referen es, 3,4,13

the dependen e on the parametersof thestate-spa e matri es is supposed to berational.

The methods then require the LPV plant

P LP V

^to ^be ^written ^as ^the inter onne tion of an LTI plant

P (s)

with a so- alled parameterblo k matrix

Θ

hara terizing the parameterstru ture. This is alled the linear fra tional transform (LFT) representation. For matri es

Θ =





Θ 11 Θ 12

Θ 21 Θ 22





^and

M

^of ^ompatible ^dimen-

sions,

F l (M, Θ) = Θ 11 + Θ 12 M (I − Θ 22 M ) ⁻¹ Θ 21

^denotes ^the^lower ^LFT^of ^the inter onne tion

(M, Θ)

^and

F u (M, Θ) = Θ 22 + Θ 21 M (I − Θ 11 M ) ⁻¹ Θ 12

^theûpper^LFT.În^this^paper,^re
all^that ^the^parameter^ve
torîs

dened as

θ = [θ 1 · · · θ p ] ^T

ândîsâssumed^to^be^real. ^The^parameter^blo
kîs^then^denedâsâ^diagonal^matrix

Θ = diag(θ 1 I n 1 , ..., θ p I n p )

^where

n i

^is^the^number^of ^times

θ i

^appears ⁱⁿ ^the^LFT. ^The^dimension^(or ^size)^of

theparameterblo kisthen

n θ = n 1 + · + n p

^.

Inthe approa hes onsidered,the LPV ontroller

K LP V

îs âssumed^to ^have^the^same ^dependen
y ôn ^the

parameterastheplant, thereforeitisalsowritten inLFTform astheinter onne tion ofanLTIsystem

K(s)

and thesameparameterblo k

Θ

^as^the^plant. ^Noti
e ^that ^the losed-loopsystemfrom

w

^to

z

representedin

(5)

Figure 1denoted by

P LP V ⋆ K LP V

^readsⁱⁿ ^LFT^form:

F l (F u (P (s), θ), F l (K(s), θ))

^.

P (s)

K(s)

Θ

z w

q ₁ p ₁

y u

Θ

q ₂

p ₂

Figure1. Closed-loop LPVsysteminLFTrepresentation.

The

L 2

^gain^of^an^LPV^system

z = G LP V (w)

^dened ^asⁱⁿ ⁽¹⁾ ^is^the^smallest

γ

^su
h ^that^for ^all

T 0 ≥ t 0

^,

wehave

Z T 0

t 0

z(t) ^T z(t)dt ≤ γ ²

Z T 0

t 0

w(t) ^T w(t)dt

forany

w

^su
h^that

R T ⁰

t 0 w(t) ^T w(t)dt < ∞

^.

Foran LPVaugmentedplant

P LP V

^dened ^asⁱⁿ ^(2), ^the

L 2

^gain^LPVôntrol ^problem ân^be^stated âs

follows: DesignanLPV ontroller

u = K LP V (y)

^su
h ^that, ^with ^the^losed^loop ^systemrepresentedFigure 1 and denedby

P LP V ⋆ K LP V = F l (F u (P (s), θ), F l (K(s), θ))

^:

• P LP V ⋆ K LP V

^isasymptoti allystable;

• P LP V ⋆ K LP V

^has^a

L 2

^gain^less^than^a^given

γ ≥ 0

^(known^as^level^ofperforman e).

Tra tablesu ient onditions forthis problem were derivedby Apkarian and Gahinet 4

asanLMI feasibility

problem. AlltheresultspresentedinthenextSe tionswereobtainedbyimplementingtheformulaeinreferen e 4

whi haregivenin detailintheappendixatSe tion VII.

II. Model of the missile and design spe i ations

A. Nonlinear modelof the missile

The onsideredsystemisthepit h-axismodelofamissile,yingatMa h

3

ândâtânâltitudeôf

20, 000

^ft,^that

wasdened byRei hert.

14

Theasso iated ontrol problemwasintensivelystudied, seee.g. referen es.

11,1519

Theideaistousethetaildee tion

δ

^to^tra
k^ana elerationmaneuver. Themissileismodeledasarigid body,seeFigure2. The ontrolinputis

δ

ând^the^measuredôutputsâre^thea eleration

η

^and^the^pit
h^rate

q

^.

Thestateofthemissileinvolvestheangleofatta k

α

^and^the^pit
h ^rate

q

^and ^thestate-spa eequationsare:







˙α = cos(α)K α M C n (α, δ, M ) + q

˙q = K q M ² C m (α, δ, M )

(3)

(6)

Thea elerationoutput

η

^is^given^by:

η = ^K _g ^z M ² C n (α, δ, M )

where

M

^is^the^Ma
h^while^the^fun
tions

C n

^and

C m

^are^dened ^by:







C n (α, δ, M ) = a n α ³ + b n |α|α + c n (2 − M/3)α + d n δ

C m (α, δ, M ) = a m α ³ + b m |α|α + c m (−7 + 8M/3)α + d m δ.

(4)

Forthisspe i model,thesetwofun tionsaredetermined. Howeverinpra ti e,the oe ientsareusually

knownonlypoorlyandsometimesnotat all. Itisinterestingtoemphasizethattheapproa hdes ribedbelow

anbeapplied eveninthese ases.

Thea tuatorismodeledasase ondordersystem:

¨ δ = −ω _a ² δ − 2ξ a ω a ˙δ + ω _a ² δ c

where

δ

îs^theâ
tual^tail^dee
tionând

δ c

^the^ommanded^tail^dee
tion.

+

x

z

Vxz

α

δ

y

G

Figure2. Denitionofthemissilevariables.

SeeTable1fortheasso iatednumeri aldataextra tedfrom Rei hert'spaper.

14

B. Designobje tives

Thedesignspe i ations onsideredfollowfrom Ferrereset al.

16

andwere usedaswellin referen es:

11,20

•

^whenâpplyingâ^stepînput^signal^to^the^referen
eînput

η c (t)

^,^the^time^onstant^must^be^less^than

0.35 s

^,

themaximalovershootlessthan

20%

^and^the^steady^state^error^less^than

5%

^;

•

^a
tuatorsaturation,bothina elerationandin speed,shouldbeavoided;

•

^due^to^the^presen
eôf^non^modeledêxible^modes,^theôntroller^bandwidth^must^be^limited^(the^transfer

from

η c

^to

η

^must^present^anattenuationof

30 dB

^at

300 rad/s

^);

•

^robustness^to un ertaintiesonaerodynami fun tions oe ientsisdemanded.

C. LPV model of themissile

Inorder to applyLPVsynthesismethodsto this system,thersttaskis toderiveanLPVmodelfrom the

nonlinear model of the missile and to write it in LFT form. Following the lines of referen e, 20

we use the

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a n 1.0286 10 ⁻⁴ deg ⁻³

b n −0.94457 10 ⁻² deg ⁻²

c n −0.1696 deg ⁻¹

d n −0.034 deg ⁻¹

a m 2.1524 10 ⁻⁴ deg ⁻³

b m −1.9546 10 ⁻² deg ⁻²

c m 0.051 deg ⁻¹

d m −0.206 deg ⁻ ¹

ω a 150

ξ a 0.7

P 0 973.3 lb/f t ²

S 0.44 f t ²

m 13.98 slugs

V 1036.4 f t/s

d 0.75 f t

I y 182.5 slug.f t ²

K α 0.7P oS/m/V

K q 0.7P oSd/I y

K z 0.7P oS/m

g 32.2

Table1. Missileparameters.

approximations

cos(α) ≈ 1

^for^the ônsidered ângles. ^Two ^polytopi ^models âreintrodu ed: one orresponds to the original nonlinearmodel written in quasi-LPV form, that is, onsidering the nonlinear dependen e as

embedded in a parameter. The other orresponds to the non stationary linearizations of the model. Sin e

a m = 2a n

^,

b m = 2b n

^, ^the êquations ôbtained ⁱⁿ ^these ^two âses âre ^the ^same, ^the ^dieren
e ^lying ⁱⁿ ^the

denition of theparameter: thus,ifwewishto onsiderthe nonlinearsystemthen

θ(t) = a m α(t) ³ + b m α(t) ²

^,

whereas ifwe onsider the non stationary linearizationsin

α 0

^we ^should ^take

θ(t) = 3a m α 0 (t) ² + 2b m |α 0 (t)|

^.

Sin e the angle

α

^varies ^between

−0.35 rad

^and

0.35 rad

^,

a m α ³ + b m α ²

^varies ^between

0

^and

−10

^while

3a m α ² + 2b m |α|

^varies^between

0

^and

−15

^. ^Hen
e^by onsideringthattheparametervariesbetween

0

^and

−15

wetakeintoa ountboththenonlinearmodelanditsnonstationarylinearizations.

Thestate-spa eequationsaregivenbelow,wherethestateis

x = [α q] ^T

^:





˙x

η



 =





A + θ(t)A θ B

C + θ(t)C θ D









x

δ





⁽⁵⁾

where





A B

C D



 =







K α M c n (2 − M 3 ) 1 K α M d n

K q M ² c m (−7 + 83M) 0 K ^q M ² d m

K z

g M ² c n (2 − M 3 ) 0 K z

g M ² d n







(6)





A θ

C θ



 =







K α M 0

2K q M ² 0

K z

g M ² 0







.

⁽⁷⁾

Theequations anbefurther writteninLFTformbyisolatingtheparameter-dependentsignal. Thusin the

LFT representation, the parameter blo k input is dened as

q 1 (t) = α(t)

ând ^the ^parameter^blo
kôutput îs

(8)

p 1 (t) = θ(t) · q 1 (t)

^. ^Then^thestate-spa eequationsbe omethefollowing:



 

 

 

 







˙x

q 1

η







=







A A θ B

[1 0] 0 0

C C θ D













x

p 1

δ







p 1 (t) = θ(t) · q 1 (t)

.

⁽⁸⁾

Noti e that in this parti ularexample there isa singleparameterso that theparameter blo k

Θ

^depi
ted ⁱⁿ

Figure 1andlaterinFigure 11isinfa tas alarandthereforewillusually bedenotedby

θ

^.

III. Control with the lassi al information stru ture

Werstreviewtheresultsobtainedwithaninformationstru turethatwas lassi ally onsideredinliterature,

wherethe ontrollerinputsarethepit hrate

q

^and^the^tra
king^error

η c − η

^. ^Morespe i ally,theinformation stru ture onsideredisdire tlyinspiredfromreferen es.

11,16

A. Criterion and weighting fun tions forthe lassi al informationstru ture

We pro eed asfor an usual

H ∞

^synthesis, ^see ^e.g. ^referen
e:

21

the performan e spe i ationsare hara -

terized byLTI weightingfun tions onstrainingthe losed-looptransferfun tions. Thusfollowingthelinesof

referen e, 16

weusethe6blo s riteriondes ribedinFigure3tospe ifyperforman eandrobustness.

G

A tuator

W 4

W 3

W 1 W 2

η c − η

q

w 1

w 3

w 2

z 2

z 1

p 1 q 1

+ + +

+ +

-

δ c δ

P

θ

Figure3.

H ∞

^riterion^for^the^lassi
alinformationstru ture( ontrollerinputs:

η c − η

^and

q

^).

To hooseadequateweightingfun tions, itis in generalne essaryto doseveraltrials, seereferen e.

21

The

usualmethodgoesasfollows: rst,weightsaresoughtforonespe ialplantLTIfrozenlinearizationlikeforan

usual

H ∞

^synthesis^pro
edure. ^Next,^the^fun
tionsâre^modiedûntil^theyâre^suitable^for^the^plant^LTI^frozen

linearizations orrespondingtoallthe parametervaluesin thedenition set. A satisfyingresultwasobtained

in referen e 11

withtheweightingfun tionsgivenbelow. The orrespondingfrequen yresponsesfor

W 1 (s)

^and

(9)

W 2 (s)

^are^displayedⁱⁿ ^Figure^4.

W 1 (s) = 10 ³ s/6.93 + 1

s/3.46 · 10 ⁻³ + 1 , W 2 (s) = 10 s ² /150 ² + 0.8/150s + 1

s ² /1000 ² + 2/1000s + 1 , W 3 (s) = 0.04, W 4 (s) = 0.07.

10 ⁻⁴ 10 ⁻² 10 ⁰ 10 ² 10 ⁴

−60

−50

−40

−30

−20

−10

0

10 Singular Values

Frequency (rad/sec)

Singular Values (dB)

W ₁ ⁻¹ (s)

W ₂ ⁻¹ (s)

Figure4. Weightingfun tions

W ₁ (s)

^and

W ₂ (s)

^.

B. Resultsforthe lassi al informationstru ture

Theobtainedlevelofperforman eis

γ = 1.297

^. ^The^results^that ^were^obtained^by^performing^an^LPV^syn-

thesiswiththe lassi alinformationstru tureofFigure5areillustratedin Figure5(LTIfrozenlinearizations)

andFigure6(LPVsimulationsfortheparametertraje tory orrespondingtothenonlinear(quasi-LPV)model

θ(t) = a m α(t) ³ + b m α(t) ²

^for^several^step^inputs^of^dierentamplitudes).

TheresultsobtainedbyperforminganLPVsynthesisusingthe lassi alstru turearequitesatisfying: the

LTIandLPVplotsenableindeedto he kthatthedesignspe i ationsarerespe ted(timeresponseabout

3 s

andovershootlessthan

20%

^). ^Moreover,^ontrary^to^the^usual^heuristigain-s heduledmethods,LPVmethods guaranteethatthestabilityandperforman epropertiesarea hievedforanyparametertraje toryremainingin

thedenition set.

A questionneverthelessarises from the study of theLTI plots. Theoreti ally, weexpe t hereto obtaina

parameter-dependent ontroller. However,the frequen y responses of the ontrollerLTI frozenlinearizations

seemto indi atethat theinuen eoftheparameterisweak. Thisis onrmedbythefa t that thefrequen y

responsesofthe losed-loopLTIfrozenlinearizationsstilldependontheparameter. Asanundesirable onse-

quen e,thestepresponsesoftheLTIfrozenlinearizationsandoftheLPVsimulationsofthe ontrolledsystem

arenothomogeneousenough(e.g.

t 5% ∈ [0.3, 0.5] s

^).

Havingre alledthese lassi alresults,wenextproposeamodiedinformationstru turethatleadstobetter

results.

IV. Control with the new information stru ture

First, anewstru ture is presented and theinterestof the signalnewly introdu ed is justied empiri ally.

An LPVsynthesisisthen arriedoutwhiletheweightingfun tionsarekeptthesameaspreviously,inorderto

highlightthepotentialby omparingthelevelofperforman eobtained. Next,thestru tureisfurthermodied

byintrodu inganothersignalsupposed available,whi hpresentsanotherinterest.

Next,theweightsaremodieda uratelytoimprovetheperforman e. AnLPVsynthesisis arriedoutand

a omparativestudyoftheLTIfrozenplotsandtheLPVsimulationsispresented,emphasizingtheadvantages

of thenewstru ture.

(10)

0 0.2 0.4 0.6 0.8 1 1.2

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2 Classical structure (measures η _c −η, q); γ=1.297: Step response T _η

c

→ η

_c

−η

Time (sec)

Amplitude

θ ₀ =−15

θ 0 =−12

θ ₀ =−9

θ ₀ =−6

θ 0 =−3

θ ₀ =0

0 0.2 0.4 0.6 0.8 1 1.2

−0.5

0

0.5

1

1.5

2

2.5 Step Response

Time (sec)

Amplitude

θ

0

=−15

θ

0

=−12

θ

₀

=−9

θ

0

=−6

θ

₀

=−3

θ

0

=0

10

⁻¹

10

⁰

10

¹

10

²

10

³

−70

−60

−50

−40

−30

−20

−10

0

10

20

30 Classical structure (measures η _c −η, q); γ=1.297: Controller Bode

Frequency (rad/sec)

Singular Values (dB)

T _η

c

−η → u

T q → u

10

⁻¹

10

⁰

10

¹

10

²

10

³

−80

−60

−40

−20

0

20

40

60

80 Classical structure (measures η _c −η, q); γ=1.297: Open−Loop Bode

Frequency (rad/sec)

Singular Values (dB)

θ 0 =−15

θ ₀ =−12

θ ₀ =−9

θ ₀ =−6

θ ₀ =−3

θ ₀ =0

90 135 180 225 270 315

−40

−20

0

20

40

60

80 6 dB 3 dB

1 dB

0.5 dB

0.25 dB

0 dB

Classical structure (measures η _c −η, q); γ=1.297: Open−Loop Nichols

Open−Loop Phase (deg)

Open−Loop Gain (dB)

θ ₀ =−15

θ 0 =−12

θ ₀ =−9

θ ₀ =−6

θ ₀ =−3

θ 0 =0

10

⁻¹

10

⁰

10

¹

10

²

10

³

−40

−30

−20

−10

0

10

20

30 Classical structure (measures η

c −η, q); γ=1.297: σ(T

w

_w

→ z

_e

), σ(1/W

w /W

e )

Frequency (rad/sec)

Singular Values (dB)

θ ₀ =−15

θ 0 =−12

θ ₀ =−9

θ ₀ =−6

θ 0 =−3

θ 0 =0

Figure5. FrozenLTIplotsforthe lassi alinformationstru ture( ontrollerinputs:

η c − η

^and

q

^): ^Step^response

from

η c

^to

η

^(1),Closed-loop step responseof dominantpoles from

η c

^to

η

^(2), ^Controller^Bode^(3), ^Open-loop

Bode(4),Open-loopBla k-Ni hols(5),Closed-loopBodefrom

w 3

^to

(η c − η)

^(6).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−50

−40

−30

−20

−10

0

10

20

30

40

50 LPV simulation. Classical structure (measures η _c −η, q); system output η(t); γ=1.297

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−1.5

−1

−0.5

0

0.5

1

1.5 LPV simulation. Classical structure (measures η _c −η, q); system output η(t); γ=1.297

Figure 6. LPV simulation for the lassi al information stru ture ( ontroller inputs:

η c − η

^and

q

⁾ ^with

θ(t) =

a m α(t) ³ + b m α(t) ²

^: ^Step^response

η(t)

^for^dierent^step ^sizes⁽¹⁾^and^brought^to^the^same^s
ale^(2).

(11)

G

A tuator

W 4

W 3

W 1 W 2

η c − η

q

w 1

w 3

w 2

z 2

z 1

p 1 q 1

+

+ +

+

-

δ c δ

P

θ

η

w 4

W 5

+

Figure7.

H ∞

^riterion^for^the^new informationstru ture( ontrollerinputs:

η c − η

^,

q

^and

η

^).

The new ontrol riterion onsideredis given inFigure 7. Thedieren ewith the lassi alstru ture liesin

the fa t that herebesidesthe tra king errorand the pit h rate, the a eleration

η

îs êxpli
itly^fed ^ba
k âsâ

ontrollerinput. Itis learthat byaddingan extrameasurethe levelofperforman ewill beat leastasgood

as previously, howeverthis is notthe onlyreason why we suggestadding this signal. Indeed, in the lassi al

stru ture,thefa tthatonlythetra kingerrorismeasuredimpliesthatpotentially ru ialinformationaboutthe

operatingpointisla king. Therefore,byusing

η

^not^only^betterperforman eisexpe tedbut alsoa ontroller that anadjust bettertotheparametervalue.

Using thesameweightingfun tions asfortheusual informationstru ture (seeSe tion III), weobtainwith

this newstru tureaperforman elevel

γ = 1.19

^. ^This ^suggests^that ^theperforman e anbefurther improved byanadequate hoi eoftheweightingfun tions. However,wearenotyetinterestedinthisissueatthisstage.

Rather,weseektofurtherimprovethedesignbyaddinganothersignalthatalsoprovesuseful: itistheoutput

θ(t) · q 1 (t)

^of^the^parameter^blo
kⁱⁿ^the^LFTrepresentationoftheplant.

Inthespe i aseof the onsideredmissile modelit orrespondsto theparameter-dependenttermin the

plant state-spa e equations. We do notmakea strong assumption by supposing that this signal is available

for ontrol: indeed,to designanLPV ontrollerofthemissile itisalreadyassumedthattheparameter

θ(t)

^is

availableinrealtime. However,re allthat

θ(t)

^is^a^polynomialⁱⁿ

α(t)

^and^that ⁱⁿ^the^LPV^model,^the^system

output

q 1 (t)

^is^simply

α(t)

^. ^Therefore,^supposing^that ^theôutputôf^the^parameter^blo
k^(whi
hîs^here^simply

theparameter-dependentterm)isavailablefor ontrolisarealisti hypothesis. Thenewstru ture onsidered

is depi ted in Figure 8. Performing anLPV synthesisleadsto alevelof performan e

γ = 1.18

^(whi
h ^is ^not

mu h dierentthan the levelobtained with only the measures

η c − η

^,

q

^,

η

^). În ôrder ^to ênable ânâ

urate

omparison with the lassi al stru ture, the weighting fun tion

W 1

îs âdjusted ^so ^that ^the ^time ^response îs

(12)

G A tuator

W 4

W 3

W 1 W 2

η c − η

q

w 1

w 3

w 2

z 2

z 1

p 1 q 1

+

+ +

+

-

δ c δ

P

θ

p 1

η

w 3

W 5

+

Figure8.

H ∞

^riterion^for^the^newinformationstru ture( ontrollerinputs:

η c − η

^,

q

^,

η

^and

θ · α

^).

smaller:

W 1 (s) = 5.5 · 10 ⁻¹ s + 8.35

s + 8.4 · 10 ⁻³ .

B. Resultsforthe new informationstru ture

ByperforminganLPVsynthesiswiththenewinformationstru tureofFigure8andtheadjustedweighting

fun tions,thelevelofperforman e

γ = 1.288

^is^rea
hed. IllustratingplotsaredisplayedinFigure9(LTIfrozen linearizations) and Figure 10 (LPV simulations for the parameter traje tory orresponding to the nonlinear

(quasi-LPV)model

θ(t) = a m α(t) ³ + b m α(t) ²

^for^several^step^inputs^of^dierentamplitudes).

Theplots pointout that signi ant improvementis a hieved with thenewstru ture. The performan eis

obviouslyameliorated,ashighlightedby thefa t that while themarginsand thelevel ofperforman eremain

the same,the ontrolled systemfollows thereferen e inputmu h better: from bothLTI and LPVplots, itis

learthatthestepresponsesarequi kerandmorehomogeneous(e.g. ontheLTIplots,

t 5% ∈ [0.2, 0.3] s

^). ^This

is onsistentwiththeLTIBodeplotsofthe ontrollerindi atingthat itadjustsmu hbettertotheparameter

value.

V. A pra ti al advantage of the new stru ture

Thisse tionfo usesonaninterestingpra ti aladvantageofthenewstru turefromtheimplementationpoint

of view. A tually, the fa t that theterm

θ(t) · α(t)

îs^supposed âvailable ênables ^to ônstru
tâ ôntrollerôf

redu ed omplexityintheparameter. Thisinterestingpropertyisthe onsequen eofatheoremestablishedby

Wu andLu.

12

Theinterpretationofthetheoremwhi h ismadehereisneverthelessquitedierent: referen e 12

isa tually

(13)

0 0.2 0.4 0.6 0.8 1 1.2

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2 New structure (measures η

c −η, q, η, θα), Reduced complexity; γ=1.288: Step response T _η

c

→ η

_c

−η

Time (sec)

Amplitude

θ ₀ =−15

θ ₀ =−12

θ ₀ =−9

θ ₀ =−6

θ ₀ =−3

θ ₀ =0

0 0.2 0.4 0.6 0.8 1 1.2

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8 Step Response

Time (sec)

Amplitude

θ

0

=−15

θ

₀

=−12

θ

0

=−9

θ

0

=−6

θ

₀

=−3

θ

0

=0

10

⁻¹

10

⁰

10

¹

10

²

10

³

−60

−50

−40

−30

−20

−10

0

10

20

30 New structure (measures η

c −η, q, η, θα), Reduced complexity; γ=1.2936: Controller Bode

Frequency (rad/sec)

Singular Values (dB) T _η

c

−η → u

T q → u

T _{η → u}

T _{α → u}

10

⁻¹

10

⁰

10

¹

10

²

10

³

−60

−40

−20

0

20

40

60

80 New structure (measures η

c −η, q, η, θα), Reduced complexity; γ=1.288: Open−Loop Bode

Frequency (rad/sec)

Singular Values (dB)

θ ₀ =−15

θ ₀ =−12

θ ₀ =−9

θ ₀ =−6

θ ₀ =−3

θ ₀ =0

−270 −225 −180 −135 −90 −45

−40

−20

0

20

40

60

80 6 dB 3 dB

1 dB

0.5 dB

0.25 dB

0 dB

New structure (measures η

c −η, q, η, θα), Reduced complexity; γ=1.288: Open−Loop Nichols

Open−Loop Phase (deg)

Open−Loop Gain (dB)

θ ₀ =−15

θ 0 =−12

θ ₀ =−9

θ ₀ =−6

θ ₀ =−3

θ 0 =0

10

⁻¹

10

⁰

10

¹

10

²

10

³

−40

−30

−20

−10

0

10

20

30 New structure (measures η _c −η, q, η, θα), Reduced complexity; γ=1.288: σ(T _w

w

→ z

_e

), σ(1/W _w /W _e )

Frequency (rad/sec)

Singular Values (dB)

θ ₀ =−15

θ ₀ =−12

θ ₀ =−9

θ ₀ =−6

θ ₀ =−3

θ 0 =0

Figure 9. Frozen LTI plots for the new information stru ture ( ontroller inputs:

η c − η

^,

q

^,

η

^and

θ · α

^): ^Step

response from

η c

^to

η

^(1), Closed-loop step response of dominant poles from

η c

^to

η

^(2), ^Controller ^Bode^(3),

Open-loop Bode(4),Open-loop Bla k-Ni hols(5),Closed-loopBodefrom

w 3

^to

(η c − η)

^(6).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−40

−30

−20

−10

0

10

20

30

40

50 LPV simulation. New structure (measures η

c −η, q, η, θα) ; system output η(t); γ=1.288

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−1

−0.5

0

0.5

1

1.5 LPV simulation. New structure (measures η _c −η, q, η, θα) ; system output η(t); γ=1.288

Figure 10. LPV simulation for the new information stru ture ( ontroller inputs:

η c − η

^,

q

^,

η

^and

θ · α

⁾ ^with

θ(t) = a m α(t) ³ + b m α(t) ²

^: ^Step^response

η(t)

^for^dierent^step^sizes⁽¹⁾^and^brought^to^the^same^s
ale^(2).

(14)

referen e.

22

Morepre isely,in referen e, 12

the onditions onsidered aretheonesintrodu edbyApkarianand

Gahinet.

4

Yet the theorem implies that if the spe ial signal orresponding to the output of the parameter

blo koftheplantisavailablefor ontrol,thenthe onditionsforLTIrobustsynthesisaresimplied: theyturn

intoanLMIfeasibilityproblem, thus be oming onvex. Infa t,itisprovedthatin presen eofthisparti ular

measure,the(generallynon onvex)LTI robustsynthesisproblem be omesequivalentto the( onvex)

L 2

^gain

LPV ontrolproblem.

Inour ontext,thetheoremimpliesaveryinterestingresultifitisinterpretedinthefollowingmanner: ifthe

outputoftheparameterblo koftheplantisavailablefor ontrol,thenthe onditionsofthe

L 2

^gain^LPV^ontrol

problemareequivalenttothe onditionsoftherobustLTI ontrolproblem. Thismeansthatifthe

L 2

^gain^LPV

ontrolproblem issolvable,then itispossibleto onstru ta ontrollerthat hasanLTIstru ture, asdepi ted

inFigure11. Itisimportanttonoti ethatitdoesnotmakeitanLTI ontrollerintheusualsensebe ausehere

oneinputofthe ontrollerisaparameter-varyingsignal(sin eitistheoutputoftheplantparameterblo ki.e.

thesignal

θ(t) · q 1 (t)

^). Equivalently,this anbesummarizedbysayingthatiftheparameter-dependentsignalis availablefor ontrol,then itispossibleto onstru tanLPV ontrollerwhoseLFTblo kparameterdimension

is zero. Asket hoftheproofofthistheorem isprovidedinSe tion VII.

P (s)

K(s)

z w

q ₁ p ₁

Θ

y u

Θp ₁

Figure11. Closed-loopLPVsystemwith ontrollerofredu ed omplexityintheparameter.

This property has a great advantage from a pra ti al point of view. Indeed, the ontroller omplexity is

one of the main limitations of the implementation of LPV methods. Rewriting the system equations in a

lowerfra tionalmanneroftenleadsto deningaparameterblo kofgreat dimension

n θ

^. ^While^an^usual ^LPV

synthesiswouldleadtoa ontrollerhavingthesame omplexityastheplant(thatis,havingaparameterblo k

of dimension

n θ

^), ^the^new ^stru
ture ênables^to ônstru
t âôntroller ^having â^parameter^blo
k ôf ^dimension

zero.

Here the ontroller anbe onstru tedas:

u = h

K (η c − η)→u (s) K q→u (s) K η→u (s) K (θ·α)→u (s)

i







η c − η

q

η

θ · α







.

(15)

inputthatisparameter-dependent. TheBodeplotsoftoea htransferfun tionaregivenseparatelyinFigure12

(see alsoFigure 9(3)).

10

⁻¹

10

⁰

10

¹

10

²

10

³

−60

−50

−40

−30

−20

−10

0

10

20

30 Singular Values

Frequency (rad/sec)

Singular Values (dB)

10

⁻¹

10

⁰

10

¹

10

²

10

³

−25

−20

−15

−10

−5

0

5 Singular Values

Frequency (rad/sec)

Singular Values (dB)

10

⁻¹

10

⁰

10

¹

10

²

10

³

−45

−40

−35

−30

−25

−20

−15

Singular Values

Frequency (rad/sec)

Singular Values (dB)

10

⁻¹

10

⁰

10

¹

10

²

10

³

−25

−20

−15

−10

−5

0

5

10

15

20 Singular Values

Frequency (rad/sec)

Singular Values (dB)

θ ₀ =−15

θ ₀ =−12

θ 0 =−9

θ ₀ =−6

θ ₀ =−3

Figure12. FrozenLTIplotsofthe ontroller:

K _(η _c _−η)→u

^(1),

K q→u

^(2),

K η→u

^(3),

θ · K α→u

^(4).

Asimpliedexpressionofthetransferfun tions(aftermodelredu tionbytrun ationtotheorder4)isgiven

below:



 



 



K (η c − η)→u (s) = 2.65 · 10 ⁻³ (s + 38.03)(s + 19.82)(s ² − 1.48 · 10 ² s + 2 · 10 ⁶ )

(s + 7.79 · 10 ² )(s + 1.95 · 10 ² )(s + 24.46)(s + 8.15 · 10 ⁻³ )

K q→u (s) = 0.15 (s + 27.17)(s + 8.16 · 10 ⁻³ )(s ² − 2.96 · 10 ² s + 1.17 · 10 ⁶ )

(s + 7.79 · 10 ² )(s + 1.95 · 10 ² )(s + 24.46)(s + 8.15 · 10 ⁻³ )

K η→u (s) = 2.15 · 10 ⁻ ² (s − 26.56)(s + 8.15 · 10 ⁻³ )(s ² − 3.38 · 10 ² s + 8.10 · 10 ⁶ )

(s + 7.79 · 10 ² )(s + 1.95 · 10 ² )(s + 24.46)(s + 8.15 · 10 ⁻³ )

K (θ·α)→u (s) = −7.30 · 10 ⁻² (s − 43.1)(s + 8.15 · 10 ⁻³ )(s ² − 3.35 · 10 ² s + 7.53 · 10 ⁵ )

(s + 7.79 · 10 ² )(s + 1.95 · 10 ² )(s + 24.46)(s + 8.15 · 10 ⁻ ³ )

.

VI. Con lusions

Inthispaper,theinterestofanewinformationstru tureforLPVsynthesisisinvestigatedandillustratedon

theRei hert'smissile ontrolproblem. Theproposedimprovement onsistsinaugmentingthe lassi alstru ture

bysupposingthatbesidestheusually onsideredmeasuresofthea elerationtra kingerrorandthepit hrate,

twoothersignalsareusedfor ontrol:rst,thea tuala elerationandse ond,theparameter-dependentsignal.

It isshownonthis demonstrativeexamplethatthenewstru tureyieldsimprovedperforman eandleadsto a

ontrollerthatadjustsbettertotheparametervalue. Furthermore,itenablesto onstru ta ontrollerthathas

redu ed omplexityinrelationto theparameter.

Theproposed solutionsupposesneverthelessthatthemeasure oftheparameterblo koutput

θ(t) · q 1 (t)

^is

(16)

to beestimated. Robustnessof theproposedsolutionto un ertaintiesonthe measureof theparameterblo k

output

θ(t) · q 1 (t)

^is^underinvestigation.

VII. Appendix: Draft of proof of the redu tion of the ontroller omplexity in

the parameter

This se tionproposesasket h ofanalternativeproof oftheinterestingresultestablishedby Wuand Lu.

12

The proof somehow diers from that in referen e 12

as the result is interpreted here in the ontext of LPV

ontrol. Re allindeedthat inreferen e, 12

theaim wasto showthatthepresen eoftheplantparameterblo k

outputenablesarelaxationofthenon onvexLTIrobust ontrol onditions,whi hthenbe ome onvexandin

fa t equivalenttothe onditionsofthe

L 2

^gain^LPV^ontrol ^problem.

Herethebaseisthe lassi alLPVproblem onsiderede.g. in referen es.

36

Theproofgoesasfollows: rst

are re alledtheLMI onditionsforthe

L 2

^gain^LPVôntrol^problem âs^theyâreîntrodu
ed^byÂpkarianând

Gahinet.

4

Nextitisshownthatthese onditionsaresimpliediftheplantparameterblo koutputisavailable

for ontrol,thusyieldingsomefreedominthede isionvariables. Thesevariables anthenbe hosensu hthat

the omplexityof the ontrollerin relation tothe parameteris redu ed,that is, morepre isely,su h that the

parameterblo kin theLFTrepresentationofthe ontrollerisofdimensionzero.

Denote by

P

^the ^system^formed ^by^the ôriginal^plantâugmented^with ^the^weighting^fun
tions ôf ^Figure ^3.

Theaugmentedsystemmatri esaredened inLFTform asfollows:







˙x

q 1

z

y







=







A B θ B 1 B 2

C θ D θθ D θ1 D θ2

C 1 D 1θ D 11 D 12

C 2 D 2θ D 21 0













x

p 1

w

u







.

⁽⁹⁾

Su ient onditionsfor the

L 2

^gain^LPV ôntrol ^problem âreôbtained ^by Âpkarian ând ^Gahinet⁴ âs ^the

following LMI feasibility problem: Find, if they exist, symmetri positive denite matri es

R, S ∈ R ^n×n

^,

(17)

J 3 , L 3 ∈ R ⁿ ^θ ^× ⁿ ^θ

^satisfying^the^LMIs ^(10),^(11),^(12),^(13):





N _R ^T 0

0 I











AR + RA ^T R h

C _θ ^T C ₁ ^T

i h

B θ J 3 B 1

i





C θ

C 1



 R −γ





J 3 0

0 I









D θθ J 3 D θ1

D 1θ J 3 D 11









J 3 B _θ ^T

B ^T ₁









J 3 D ^T _θθ J 3 D _1θ ^T

D ^T _θ1 D ^T ₁₁



 −γ





J 3 0

0 I















N _R ^T 0

0 I



 < 0

⁽¹⁰⁾





N _S ^T 0

0 I











A ^T S + SA S h

B θ B 1

i h

C _θ ^T L 3 C ₁ ^T

i





B _θ ^T

B ₁ ^T



 S −γ





L 3 0

0 I









D ^T _θθ L 3 D ^T _1θ

D ^T _θ1 L 3 D ^T ₁₁









L 3 C θ

C 1









L 3 D θθ L 3 D θ1

D 1θ D 11



 −γ





L 3 0

0 I















N _S ^T 0

0 I



 < 0

⁽¹¹⁾





R I

I S



 ≥ 0

⁽¹²⁾





L 3 I

I J 3



 ≥ 0

⁽¹³⁾

with

N R = Ker h

B ₂ ^T D ^T _θ2 D ^T ₁₂

i , N S = Ker h

C 2 D 2θ D 21

i

.

Re allthat therank

k

^of ^matrix

I − RS

^denes ^the^numberôf^statesôf ^theôntroller^while ^the^rank

r

^of

thematrix

I − L 3 J 3

^denes^the^dimension^of^the^parameter^blo
kⁱⁿ^the^LFTrepresentationofthe ontroller.

Inthespe ial asewheretheoutputoftheparameterblo kismeasuredasinFigure8,thematri es

C 2

^,

D 2θ

and

D 21

^an^bepartitioned sothat:

h

C 2 D 2θ D 21 0

i =





C ˆ 2 D ˆ 2θ D ˆ 21 0

0 I 0 0



 .

Consequently,one anwrite

N S =







W 1 0

0 0

W 3 0

0 I







wherethematri es

W 1

^and

W 3

^are^su
h^that

N ˆ S =





W 1

W 3



 ∈ Ker









C ˆ 2

D ˆ 21









^and





C ˆ ₂ ^T W 1

D ˆ ₂₁ ^T W 3





^is^full-rank.

After rewriting theLMI (11)and applying theelimination lemma (see referen e 23

) followed by the S hur

lemmatoeliminatethede isionvariable

L 3

^,^the^LMIfeasibilityproblem((10),(11),(12),(13))in(

R, S, J 3 , L 3

⁾

be omestheLMIfeasibilityproblem((14),(15),(16),(17))in (

R, S, J 3

^):

(18)

N _R ^T







AR + RA ^T R h

C _θ ^T C ₁ ^T

i h

B θ J 3 B 1

i





C θ

C 1



 R −γ





J 3 0

0 I









D θθ J 3 D θ1

D 1θ J 3 D 11









J 3 B ^T _θ

B ₁ ^T









J 3 D ^T _θθ J 3 D ^T _1θ

D _θ1 ^T D ₁₁ ^T



 −γ





J 3 0

0 I











N R < 0

⁽¹⁴⁾

N ˆ _S ^T







A ^T S + SA SB 1

h

C _θ ^T C ₁ ^T

i

B ₁ ^T S −γI h

D _θ1 ^T D ₁₁ ^T

i





C θ

C 1









D θ1

D 11



 −γ





J 3 0

0 I











N ˆ S < 0

⁽¹⁵⁾





R I

I S



 ≥ 0

⁽¹⁶⁾

J 3 ≥ 0

⁽¹⁷⁾

where

N R = Ker( h

B ₂ ^T D _θ2 ^T D ^T ₁₂ 0

i

), ˆ N S = Ker( h

C ˆ 2 D ˆ 21 0

i

)

^.

Theeliminationlemmaimpliesthatifthereexistsasolution

(R, S, J 3 )

^of^the^LMIfeasibilityproblem((14), (15),(16),(17)),thenthereexists

L 3

^su
h^that

(R, S, J 3 , L 3 )

îsâ^solutionôf^the^LMIfeasibilityproblem((10), (11),(12),(13)).

Thenextstepistonoti ethat

L 3 = J ₃ ⁻¹

îsâ^suitable^hoi
e. ^Re
alling^that^the^dimensionôf^the^parameter

blo kintheLFTrepresentationofthe ontrollerisdenedastherankofmatrix

I − L 3 J 3

^on
ludes^the^proof.

Referen es

1

Rugh,W.J.andShamma,J.S.,Resear honGainS heduling,Automati a,Vol.36,2000,pp.14011425.

2

Fromion,V.andS orletti,G.,Atheoreti alframeworkforgains heduling,InternationalJournalofRobustandNonlinear

Control,Vol.13,Feb.2003,pp.951982.

3

Pa kard, A.,GainS heduling viaLinearFra tional Transformations, Systemsand ControlLetters,Vol.22, No. 2,Feb.

1994,pp.7992.

4

Apkarian,P.andGahinet,P.,A onvex hara terizationofgain-s heduled

H ^∞

ontrollers,IEEETrans.Automati Control, Vol.40,No.5,May1995,pp.853864.

5

S orletti,G.andElGhaoui,L.,ImprovedLMIConditionsforGainS hedulingandRelatedProblems,InternationalJournal

of RobustandNonlinearControl,Vol.8,No.10,Aug.1998,pp.845877.

6

S herer,C.W.,LPV ontrolandfullblo kmultipliers,Automati a,Vol.37,2001,pp.361375.

7

Köse,I.E.andJabbari,F.,ControlofLPVsystemswithpartlymeasuredparameters, IEEETrans.Automati Control,

Vol.44,Mar h1999,pp.658663.

8

Iwasaki,T.and Shibata,G., LPVsystem analysisviaquadrati separator forun ertain impli itsystems, IEEE Trans.

Automati Control,Vol.46,Aug.2001,pp.11951208.

9

Bliman,P.-A.,StabilizationofLPVsystems,Pro .IEEEConf.onDe isionandControl,De .,pp.61036108.

10

M.Dinh,G.S orletti,V.F.andMagarotto,E.,Parameterdependent

H ∞

^ontrol^by^nitedimensionalLMIoptimization:

appli ationtotrade-odependent ontrol,InternationalJournalofRobustandNonlinearControl,Vol.15,2005,pp.383406.

11

S orletti,G., Appro he uniée de l'analyse et la ommande dessystèmespar formulation LMI,Ph.D.thesis,Université

d'Orsay,Paris,Fran e,1997.

12

Wu,F.andLu,B.,On onvexiedrobust ontrolsynthesis,Automati a,Vol.40,2004,pp.10031010.

An Enhanced Information Structure for Linear Parameter-Varying Design: Application to Reichert's Missile Benchmark

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