Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems

HAL Id: inria-00442081

https://hal.inria.fr/inria-00442081

Submitted on 12 Jan 2010

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

class of multivalued Lur’e dynamical systems

Bernard Brogliato, Daniel Goeleven

To cite this version:

Bernard Brogliato, Daniel Goeleven. Well-posedness, stability and invariance results for a class of

multivalued Lur’e dynamical systems. [Research Report] RR-7158, INRIA. 2009. �inria-00442081�

a p p o r t

d e r e c h e r c h e

N0249-6399ISRNINRIA/RR--7158--FR+ENG

Well-posedness, stability and invariance results for a class of multivalued Lur’e dynamical systems

Bernard Brogliato — Daniel Goeleven

N° 7158

December 2009

Centre de recherche INRIA Grenoble – Rhône-Alpes

systems

Bernard Brogliato

∗

, DanielGoeleven

†

Thème: Modélisation,optimisationet ontrledesystèmesdynamiques

Équipes-ProjetsBipop

Rapportdereherhe n° 7158Deember200930pages

Abstrat: Thispaperanalyzestheexisteneand uniquenessissues in alass

ofmultivaluedLur'esystems,where themultivaluedpartisrepresentedasthe

subdierentialofsomeonvex,proper,lowersemiontinuousfuntion. Through

suitable transformations the system is reast into the framework of dynami

variationalinequalitiesandthewell-posedness(existeneanduniquenessofso-

lutions)isproved. Stabilityandinvarianeresultsarealsostudied,togetherwith

thepropertyofontinuousdependeneontheinitialonditions. Theproblemis

motivatedbypratialappliationsineletrialiruitsontainingeletronide-

vieswithnonsmoothmultivaluedvoltage/urrentharateristis,andbystate

observerdesignformultivaluedsystems.

Key-words: Lur'edynamialsystems,passivity,invariane,Kato'stheorem,

maximal monotone operators, variational inequalities, dierential inlusions,

normalones.

∗

INRIAGrenobleRhne-Alpes,Bipopteam-projet,655avenuedel'Europe,38334Sinat-

Ismier,Frane;bernard.brogliatoinrialpes.fr

†

IREMIA,UniversitédelaRéunion,97400, Saint-Denis,Frane; daniel.goelevenuniv-

reunion.fr

multivalués

Résumé : Cet artilepropose deux ritères permettant d'assurerl'existene

etl'uniité dessolutionspourdessystèmesde Lur'emultivalués,danslesquels

la partie multivaluée est représentée par la sous-diérentielle d'une fontion

onvexe, propre,semiontinue infèrieurement. Parlebiaisde transformations

adéquatesle systèmeest missouslaformed'uneinéquationvariationnelledy-

namique. La stabilité et le prinipe d'invariane sont aussi étudiés, ainsi que

la dépendane ontinue aux onditions initiales. Le problème est motivé par

l'analysedeiruitsomportantdesomposantsnon-réguliersmultivalués,ainsi

queparlasynthèsed'observateursd'état.

Mots-lés : système de Lur'e, passivité, invariane, stabilité, théorème de

Kato,opérateursmaximauxmonotones, inéquationsvariationnelles,inlusions

diérentielles,nesnormaux.

1 Introdution

Lur'esystems,whihonsistofalineartime-invariantsystemin negativefeed-

bak with a stati nonlinearity satisfying a setor ondition, have reeived a

onsiderableinterestin theapplied mathematisandontrol literature,dueto

their broadinterest (see [28℄ for a survey). More reentlythe ase where the

nonlinearityis a maximal monotone map has been studied [7℄. The maximal

monotoniityallowsonetoonsiderunboundedsetors[0,+∞]^andnonsmooth

set-valuednonlinearities. So-alledlinear omplementarity systemsanbere-

astintoLur'esystems,wherethefeedbaknonlinearitytakestheformofaset

ofomplementarityonditionsbetweentwoslakvariables[24,15, 31℄. Oneof

these slakvariables may be interpretedasaLagrange multiplier λ^{,while the}

other oneusuallytakestheform y =Cx+Dλ^{. Moregeneralpieewiselinear}

nonlinearitieshavebeenonsideredin[27,26℄. Aspointedoutin[7℄thereexists

aloserelationshipbetweensomeomplementaritysystemsanddierentialin-

lusionswith maximalmonotoneright-hand-sides,in partiularinlusions into

normalones to onvexsets (whih arein turn equivalent todynamial varia-

tional inequalitiesoftherstkind). Partiularaseshavebeeninvestigatedin

[23, 10, 11℄. All these works are howeverrestritedto the asewhere D = 0^,

exept[26℄whereaneomplementaritysystemsareonsidered. Inthispaper,

weextend the works in [23, 10℄ to the asewhere D 6= 0^{, i.e. there exists a}

feedthroughmatrix in thelinearpartof thesystem. Moreoverthenonlineari-

tieswhihweonsideraremuhmoregeneralthanomplementarityonditions

between y ^and λ ^(i.e. y ≥ 0^, λ ≥ 0^, y^Tλ = 0^{) and the onsidered systems}

maybewrittenequivalentlyasdynamialvariationalinequalitiesoftheseond

kind. Suh an extension may beimportantin pratie(for instane eletrial

iruitswith idealdiodesand transistorsusuallyyield systemswithanonzero

feedthroughmatrixD^{,possiblypossitivesemideniteandnonsymmetri). Ob-}

serversynthesisfor set-valuedsystemsisalsoanimportantappliation[8, 14℄.

Thiswork mayalsobeseenastheontinuationofpreviouseortstostudythe

relationshipsbetweenvarious types ofdierentialinlusions, omplementarity

systems,projetedsystemsin nitedimensions[11, 19,25,21℄.

The paperis organizedas follows: In setion 2 the dynamial system is pre-

sented,anditswell-posednessis studiedin setion3. Insetion4thestability

propertiesarestudied, andaninvarianeresultispresentedinsetion5. Con-

lusionsendthepaperinsetion6.

Notations: Letf : IRⁿ→IR∪{+∞}^{beaproperonvexandlowersemiontin-}

uousfuntion,wedenote bydom(f) :={x∈IRⁿ: f(x)<+∞} ^thedomainof

thefuntionf(·)^{. ReallthattheFenheltransform}f^∗(·)^off(·)^istheproper,

onvexandlowersemiontinuousfuntion denedby

(∀z∈IRⁿ) : f^∗(z) = sup

x∈dom(f){hx, zi −f(x)}.

Thesubdierential∂f(x)^off(·)^atx∈IR^{n isdened by}

∂f(x) ={ω∈IRⁿ: f(v)−f(x)≥ hω, v−xi,∀v∈IRⁿ},

where h·,·i ^{denotes the usual salar produt in} IR^{n, i.e.} hy, zi = y^Tz ^{for any}

vetorsy ^and z ^ofIR^{n. We denote by} Dom(∂f) :={x∈IRⁿ : ∂f(x)6=∅} ^the

domainof thesubdierentialoperator∂f : IRⁿ →IR^{n. Let}x0 ^beanyelement

inthedomaindom(f)^off(·)^{,thereessionfuntion}f∞(·)^off(·)^{isdened by} (∀x∈^IRn) : f∞(x) = lim

λ→+∞

1

λf(x0+λx).

Thefuntionf∞: IRⁿ →IR∪{+∞}^{isaproperonvexandlower}semiontinuous funtionwhihdesribestheasymptotibehavioroff(·)^{. Foranonemptylosed}

andonvexsetK⊂^{IRn,thedualoneof}K^{isthenonemptylosedonvexone} K^{⋆ denedby}

K^⋆:={w∈IRⁿ:hw, vi ≥0, ∀v∈K}, ⁽¹⁾

while thepolar one K^o =−K^{⋆. Let}x0 ^{be any element in} K^{, the reession}

oneofK^{isdened by}

K∞= \

λ>0

1

λ(K−x0).

ThesetK∞^{isanonemptylosedonvexonethatisdesribedintermsofthe}

diretionswhihreedefromK^{. When}K ^isaonethenK∞=K^{. Therelative}

interiorofasetK ^{isdenotedasrint}(K)^{,anditslosureas}K¯^{. Let}M ∈^IRm×n

be a given matrix, we denote by ker(M) ^{the kernel of} M ^{and by} R(M) ^the

rangeofM^. M ≥0 ^meansthat M ^ispositivesemidenite,M >0 ^meansthat

itispositivedenite.

2 The multivalued Lur'e system

Let A ∈ IR^n×n, B ∈ IR^n×p, C ∈ IR^p×n, D ∈ IR^{p×p be given matries,} f ∈ C⁰(IR+; IR) ^{suh that} f^′ ∈ L¹_loc(IR+; IRⁿ) ^and ϕi : IR → IR∪{+∞} (1 ≤ i ≤ p) ^{given proper onvex and lower} semiontinuous funtions. Let x0 ∈ IRⁿ

be some initial ondition, we onsider the problem : Find x ∈ C⁰(IR+; IRⁿ)

suh that x^′ ∈ L^∞_loc(IR+; IRⁿ) ^and x right-dierentiable on IR+^, λ ∈ C⁰(IR+; IR^p)^andy ∈C⁰(IR+; IR^p) ^{satisfyingthenonsmoothdynamial system} N SDS(A, B, C, D, f, ϕ1, ..., ϕp, x0)^:































x(0) = x0

a.e. t≥0 :

x^′(t) = Ax(t) +Bλ(t) +f(t) for all t≥0 :

y(t) = Cx(t) +Dλ(t)

λ1(t) ∈ −∂ϕ1(y1(t)) λ2(t) ∈ −∂ϕ2(y2(t))

.

.

.

λ_p(t) ∈ −∂ϕ_p(yp(t))

(2)

Thesystemis therefore in theanonialabsolute stability form sineit isthe

negativefeedbakinteronnetionofalinearinvariantsystem(A, B, C, D)^(with

input λ^{, output} y ^{and external exitation} f(·)^{) with a stati} multivalued nonlinearity (with input y ^{and output} −λ^{). In [23, 10℄ it was onsidered}

that D = 0^{. As weshall see nextthease}D 6= 0 ^{ompliatestheanalysis. It}

is noteworthy that one may have p > n^{, whih is ruial beause} λ ^{is not a}

ontrol input and p^{may in} appliations be very large. Physial examplesare given laterin the paper. Itis assumed in this paperthat the output y ^does

notdependexpliitlyontime. Ifthisistheasetheresultsofthispaperdonot

applybeauseonehastoresorttotheperturbedMoreau'ssweepingproessto

derivewell-posednessresults,see[13℄.

Let us setλ = (λ1 · · · λp)^T,Φ(·) =ϕ1(·) +· · ·+ϕp(·)^{, and}M ∈IR^{p×p is an}

invertiblematrix. OnemayonsideraslightlymoregeneralversionoftheLur'e

system(2)as:















x(0) =x0

x^′(t) =Ax(t) +Bλ(t) +f(t) λ(t)∈ −M ∂Φ(y(t))

(3)

Denoting¯λ=M⁻¹λ^,B¯ =BM^,D¯ =DM^,(3)isequivalentlyrewrittenas:















x(0) =x0

x^′(t) =Ax(t) + ¯Bλ(t) +¯ f(t) λ(t)¯ ∈ −∂Φ(Cx(t) + ¯Dλ)¯

(4)

Thereforethetransformedsystem(4)possessesthesamestrutureasthesystem

in (2). TheLur'e system(2)anberepresentedasingure1(a).

Finallyaswillappearlearlylater,alltheexisteneanduniquenessofsolu-

tionsresultswhiharederivedinthispaper(setion3)alsoholdwhenthelinear

term Ax^{isreplaed byaLipshitz ontinuousmapping}A(x)^{. Forthe sakeof}

larityofthepresentationthelinearaseAx^{iskeptallthroughthepaper,for}

thewell-posednessandthestabilityanalysis.

3 Well-posedness analysis

Inthissetiontheexisteneanduniqueness ofsolutionswill beshownrst

byusingaversionofKato'stheorem,seondviamaximalmonotoneoperators.

Examplesomingfromeletrialiruitsandstateobserverdesignareprovided

toillustrate thetheoretialdevelopments.

3.1 Well-posedness by Kato's theorem

In the remainder of this setion we shall apply some transformations to the

Lur'esystemsothat itswell-posednessanbeanalyzed.

3.1.1 System'stransformations

Letusset

(∀z∈IR) : ϕ^∗_i^,−(z) :=ϕ^∗_i(−z). ⁽⁵⁾

Assumption 1. We assume the existene of z0,i ∈ IR ^{at whih} ϕ^∗,−_i (·) ^is

ontinuous.

Assumption1is asimplequalitativeondition that isrequired to ensurethat

(see[29℄):

(∀z∈IR) : ∂ϕ^∗_i^,−(z) =−∂ϕ^∗_i(−z).

Then

λ_i∈ −∂ϕ_i(yi)⇔y_i∈∂ϕ^∗_i(−λ_i) =−[−∂ϕ^∗_i(−λ_i)] =−∂ϕ^∗_i^,−(λi).

LetusnowdenotebypI ^(andsetpII =p−pI^{)thelargestintegersuhthatthe}

matrixD ^{anbewrittenasfollows:}

D=

0pI×pI 0pII×pI

0pI×pII DII

(6)

with D_II 6= 0pII×pII^{. In using this notation, we suppose by onvention that}

p_I = 0 ^(resp. p_II = 0^{) means that the terms indexed by} I ^(resp. II^{) are}

uselessandnotonsidered. So, ifp_I = 0 ^(resp. p_II = 0^)then D≡D_II ^(resp.

D ≡ 0p×p^{). For} z ∈ IR^{p, we set also} z = z_I z_II T

with zI ∈ IR^{pI and} zII ∈IR^pII,

B = B^I B^II , C=

CI

C_II

withB^I ∈IR^n×pI, B^II ∈IR^n×pII, CI ∈IR^{pI×n and}CII ∈IR^{pII×n. Finally,we}

set

(∀y∈IR^pI) : ΦI(y) :=ϕ1(y1) +ϕ2(y2) +...+ϕpI(ypI) ⁽⁷⁾

and

(∀y∈IR^pII) : ΦII(y) :=ϕ_pI+1(y1) +ϕ_pI+2(y2) +...+ϕ_pII(ypII). ⁽⁸⁾

Wehave:

(∀z∈IR^pII) : Φ^∗_II(z) =ϕ^∗_pI+1(z1) +ϕ^∗_pI+2(z2) +...+ϕ^∗_pII(zpII).

Weset

(∀z∈IR^pII) : Φ^∗_II^,−(z) := Φ^∗_II(−z). ⁽⁹⁾

Wenote alsothatAssumption1ensuresthat:

(∀z∈IR^pII) : ∂Φ^∗_II^,−(z) =−∂Φ^∗_II(−z).

Wealsoset:

(∀x∈IR^p) : Φ(x) = ΦI(x) + ΦII(x) ⁽¹⁰⁾

and

(∀x∈IR^p) : Φ^∗,−(x) = Φ^∗(−x). ⁽¹¹⁾

Assumption1guaranteesthat

(∀x∈IR^p) : ∂Φ^∗,−(x) =−∂Φ^∗(−x).

Itfollowsthat thesystem











λ1(t) ∈ −∂ϕ1(y1(t)) λ2(t) ∈ −∂ϕ2(y2(t))

.

.

.

λ_p(t) ∈ −∂ϕ_p(yp(t))

anbewrittenequivalentlyas:

λI(t) ∈ −∂ΦI(yI(t)) λII(t) ∈ −∂ΦII(yII(t))

oras:

λ_I(t) ∈ −∂ΦI(yI(t)) yII(t) ∈ −∂Φ^∗_II^,−(λII(t))

Usingthesenotations,weseethatthesystemN SDS(A, B, C, D, f, ϕ1, ..., ϕ_p, x0)

reduesto thesystem:































x(0) = x0

a.e. t≥0 :

x^′(t) = Ax(t) +B^IλI(t) +B^IIλII(t) +f(t) for allt≥0 :

yI(t) = CIx(t) λI(t) ∈ −∂ΦI(yI(t)) y_II(t) = C_IIx(t) +D_IIλ_II(t) yII(t) ∈ −∂Φ^∗_II^,−(λII(t))

Thefeedbaknonlinearityisthereforesplittedintotwomainparts: onepart

indexedbyI ^ismultivalued,theotherpartindexed byII ^{will beshownunder}

ertainonditionstobesingle-valued.

Remark 1 The ase p_I = n ^(i.e. D = 0n×n^{) has been the objet of spei}

papers, see [23,10 ,4 ℄. The omplementarity problem (i.e. ϕi(·) = Ψ_R+(·)∀i∈ {1, ..., n}^{),has also beenthe objet ofvarious papers [24 , 15,17,16℄.}

Assumption 2. (If pI ≥ 1^{) There exists a symmetri and invertible matrix} W ∈IR^{n×n suhthat:}

W²B^I =C_I^T. ⁽¹²⁾

Weset:

V =

W if p_I ≥1

I if pI = 0 ⁽¹³⁾

and

(∀w∈IR^pI) : ΞI(w) =

ΦI(CIV⁻¹w) if p_I ≥1

0 if p_I = 0 ⁽¹⁴⁾

Notie that by [30, Exerise 1.40, Proposition 1.39℄ the funtion ΞI(·) = Φ◦ C_IV⁻¹(·)^islowersemiontinuous,proper,onvex.

Assumption 3. (IfpI ≥1^{)There exists apoint}w0 ⁱⁿ IR^{pI at whih} ΞI(·)^is

ontinuous.

Assumptions2and3ensureinasepI≥1 ^that

(∀w∈IR^pI) : ∂ΞI(w) =V^−TC_I^T∂ΦI(CIV⁻¹w) =V⁻¹C_I^T∂ΦI(CIV⁻¹w).

ThemultivaluedmappingΞI(·)^{ismaximalmonotone,beingthe}subdierential ofaonvex,proper,lowersemiontinuousfuntion. Letus nowset:

(∀x∈^IRn) : ΛII(x) :=

V B^II(DII +∂Φ^∗,−_II )⁻¹(−C_IIV⁻¹x) if p_II ≥1

0 if p_II = 0

Wesupposealsothefollowing:

Assumption 4. (If pII ≥1) ^{The operator} ΛII : ^IRn →^IRn : x7→ ΛII(x) ^is

well-dened,single-valuedandLipshitzontinuous.

Reallingthat

DIIz−q∈ −∂Φ^∗_II^,−(z)⇔q∈(DII +∂Φ^∗_II^,−)(z)⇔z∈(DII+∂Φ^∗_II^,−)⁻¹(q),

we note that Assumption 4 (in ase pII ≥ 1^{)) requires that for all} q ∈ ^IRpII,

thereexistsatleastonez(q)∈^{IRpII suhthat}

hDIIz−q, v−zi+ Φ^∗_II^,−(v)−Φ^∗_II^,−(z)≥0,∀v∈^IRpII, ⁽¹⁵⁾

and there exists a onstant K > 0 ^{suh that for all} x1, x2 ∈ ^{IRn and} z1 ∈ (DII+∂Φ^∗_II^,−)⁻¹(−CIIV⁻¹x1)^,z2∈(DII+∂Φ^∗_II^,−)⁻¹(−CIIV⁻¹x2)^:

||V B^IIz1−V B^IIz2|| ≤K||x1−x2|| ⁽¹⁶⁾

Thesolvabilityofthevariationalinequalityin(15)ensuresthat

(∀x∈^IRn) : V B^II(DII+∂Φ^∗_II^,−)⁻¹(−C_IIV⁻¹x)6=∅

while the onditionin (16)guaranteesthat ifx∈^{IRn and} z1^, z2 ∈(DII +

∂Φ^∗_II^,−)⁻¹(−CIIV⁻¹x)^then||z1−z2|| ≤0^andthusz1=z2^{. It resultsthatthe}

operatorΛII(·)^{is single-valued. TheLipshitzontinuityof}ΛII(·)^isthenalso

adiret onsequeneof(16).

Conditionson the matrix D_II ^{and on the funtion} Φ^∗_II^,−(·) ^{ensuring that As-}

sumption4holdswillbedisussedin thefollowingsetion.

The problem N SDS(A, B, C, D, f, ϕ1, ..., ϕp, x0) ^{an be redued, by setting} X(t) = V x(t) (∀t ≥ 0)^{, to the following dynamial} variational inequality problem: Find X ∈ C⁰(IR+; IRⁿ) ^{suh that} X^′ ∈ L^∞_loc(IR+; IRⁿ) ^and X ^right-

dierentiable on IR+ ^{suh that} X(0) = V x0 ^{and satisfying for a.e} t ≥ 0 ^the

variationalinequality:

hX^′(t)−V AV⁻¹X(t)−ΛII(X(t))−V f(t), v−X(t)i

+ΞI(v)−ΞI(X(t))≥0, ∀v∈IRⁿ. ⁽¹⁷⁾

whih we may name a dynamial variational inequality of the seond kind.

Indeed,letusherewritethedetailsin asepI ≥1^andpII ≥1^{. Itislearthat} x(0) =x₀⇔V x(0) =V x₀⇔X(0) =V x₀

and















x^′(t) = Ax(t) +B^Iλ_I(t) +B^IIλ_II(t) +f(t) y_I(t) = C_Ix(t)

λ_I(t) ∈ −∂ΦI(yI(t)) y_II(t) = C_IIx(t) +D_IIλ_II(t) y_II(t) ∈ −∂Φ^∗_II^,−(λII(t))

m

x^′(t)−B^II(DII+∂Φ^∗_II^,−)⁻¹(−C_IIx(t))−Ax(t)−f(t)∈ −B^I∂ΦI(CIx(t)) m

V x^′(t)−V B^II(DII +∂Φ^∗_II^,−)⁻¹(−CIIV⁻¹V x(t))

−V AV⁻¹V x(t)−V f(t)∈ −V⁻¹V²B^I∂ΦI(CIV⁻¹V x(t)) m

X^′(t)−V B^II(DII +∂Φ^∗_II^,−)⁻¹(−CIIV⁻¹X(t))

−V AV⁻¹X(t)−V f(t)∈ −V⁻¹C_I^T∂ΦI(CIV⁻¹X(t)) m

X^′(t)−V AV⁻¹X(t)−V B^II(DII+∂Φ^∗_II^,−)⁻¹(−C_IIV⁻¹X(t))−V f(t)∈ −∂ΞI(X(t))

from whih onededues(17). Theasep_I = 0 ^(resp. p_II = 0^)anbededued

fromthepreviousrelationsinremovingthetermsindexedbyI^(resp. II^{). The}

system has therefore beentransformed from (a) to (b) in gure 1, whih are

equivalentrepresentations. Aswillbemadelearinthenextsetion,thetrans-

formationonsistsofinsertingtheLipshitzontinuouspartofthemultivalued

nonlinearity,intotheontinuousdynamisofthesystem.

Ax+Bλ

λ

y=Cx+Dλ

∂ΞI(X)

−ΛII(X) V AV−1X

(a)

(b)

X

∂Φ(y)

Figure1: Lur'esystemtransformation.