System Analysis via Integral Quadratic Constraints Part I Megretski, Alexander; Rantzer, Anders

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LUND UNI VERSI TY PO Box 117

System Analysis via Integral Quadratic Constraints Part I

Megretski, Alexander; Rantzer, Anders

1995

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Megretski, A., & Rantzer, A. (1995). System Analysis via Integral Quadratic Constraints: Part I. (Technical Reports TFRT-7531). Department of Automatic Control, Lund Institute of Technology (LTH).

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ISRN LUTFD2/TFRT--7531--SE

System Analysis via

Integral Quadratic Constraints

Part I

Alexander Megretski

Anders Rantzer

Department of Automatic Control

Lund Institute of Technology

April 1995

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Constraints, Part I

A. Megretski

Dept. of ElectricalEngineering

IowaState University

Ames,IA 50011

USA

alexm@iastate.edu

A. Rantzer

Dept. ofAutomaticControl

Lund Instituteof Technology

Box118

S-221 00LUND

SWEDEN

rantzer@control.lth.se

Abstract

This pap er intro duces a unied approach to robustness analysis

with resp ect to nonlinearities, time-variations and uncertain param-

eters. From an original idea by Yakub ovich, the approach has b een

develop ed under a combination of inuences from the western and

russian traditions of control theory. It is shown how a complex sys-

tem can b e describ ed by using certain integral quadratic constraints

(IQC's), derived for its elementarycomp onents. A stabilitytheorem

forsystemsdescrib ed byIQC'sis presented,thatcoversclassicalpas-

sivity/dissipativityarguments,butsimpliestheuseofmultipliersand

thetreatmentofcausality.

The pap er is divided into two parts. Part I presents the basic

ideasforstabilityanalysis,referingtoasimpleexample. Asystematic

computational approach is describ ed and relations to other metho ds

of stability analysis are discussed. Last, but not least, it contains a

summarizinglistof IQC's forimp ortant typ esofsystem comp onents,

that existin various formsintheliterature.

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Itiscommonengineeringpracticetoworkwithsimplestp ossible mo d-

els fordesign of controlsystems. In particular,one often uses linear

time-invariant plant mo dels,forwhich there isa well establishedthe-

oryandcommerciallyavailablecomputerto olsthathelpinthedesign.

Toverifythatthedesignalsoworkswellin practiceoneneedsrealex-

p eriments,oftenpreceededbysimulationswithmoreaccuratemo dels.

However, there isalso a strongneed formoreformal ways toanalyse

the systems. Such analysis can help to identify critical exp erimental

circumstances or parameter combinations and estimate the p ower of

themo dels.

Inthe1960-70s,alargeb o dyofresults wasdevelop edin thisdirec-

tion,oftenreferredtoasabsolutestabilitytheory. Thebasicideawas

topartitionthesystemintoafeedbackinterconnectionoftwop ositive

op erators. See[45,78,82,75,39,17,54]andthereferencestherein. To

improvetheexibility oftheapproach,so-called multiplierswereused

to select prop ervariables for the partitioning. The absolute stability

theory is now considered as a fundamental comp onent of the theory

for nonlinear systems. However, the applicability of many of the re-

sults has b een limited by computational problems and by restrictive

causalityconditions usedin themultiplier theory.

Forcomputationofmultipliers,substantialprogresshasb eenmade

inthelastdecade,themostevidentexampleb eingalgorithmsforcom-

putation of structured singular values ( analysis) [19]. As a result,

robustnessanalysiswithresp ecttouncertainparametersand unmo d-

eleddynamics,canb ep erformedwithgreataccuracy.Aprobablyeven

morefundamentalbreakthroughinthisdirectionisthedevelopmentof

p olynomial time algorithmsfor convex optimization with constraints

dened by linear matrix inequalities [40, 7]. Such problems app ear

not only in -analysis, but in almost any analysis metho d based on

passivity-typ econcepts.

The purp ose of this pap er is to adress the second obstacle to ef-

ceint analysis, by proving that multipliers can b e intro duced in a

less restrictive manner, without causality restrictions. Not only do es

this make the theory moreaccessible by simplication of pro ofs, but

also enhances the development of computer to ols, that supp orts the

transformationof assumptionson mo del structureinto a numerically

tractableoptimizationproblem.

The term integral quadratic constraint (IQC) is used for several

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 To exploit structural informationab out a complex oruncertain

systemcomp onent.

 To characterizeprop ertiesof anexternalsignal.

 To analyze combinations ofseveral constraintson p erturbations

andsignals in a system.

Implicitly, integral quadraticconstaints have always b een present

in stability theory. For example, p ositivity of an op eratorF, can b e

expressedbytheIQC

Z

1

1 d

(Fv)(j!)



b

v (j!)d!0 8v :

In the 1960s, most of the stability theory was devoted to scalar

feedbacksystems. Thisledtoconvenientlyvisualizable stabilitycrite-

ria based on the Nyquist diagram, which was particularly imp ortant

in times whencomputerswereless accessible.

Inthe70-s,integralquadraticconstaintswereexplicitly used(and

named so) by Yakub ovich to treat the stability problem for systems

withadvancednonlinearities,including amplitudeandfrequencymo d-

ulationsystems. Somenew IQC:s,unrelatedto thepassivityorsmall

gain arguments, were intro duced, and the so-called S-pro cedure was

applied to thecase of multiple constraints[79]. Willems also gavean

energy related interpretationof the stability results, in terms of dis-

sipativity, storage functions and supply rates [75]. Later on,Safonov

interpreted the stability results geometrically, in terms of separation

of thegraphsof thetwo op eratorsinthefeedbacklo op.

Animp ortantstepinthefurtherdevelopment,wastheintro duction

ofanalysismetho dswhichessentiallyrelyontheuseofcomputers. One

exampleisthetheoryforquadraticstabilization[30,22,15],anotheris

themultilo op generalizationof thecircle criterionbasedon D-scaling,

[55,19]. BoththesearchforaLyapunovfunctionandthesearchforD-

scales canb e interpretedasoptimizationofparametersin anintegral

quadratic constaint. Another direction was the intro duction of H 1

optimization for synthesis of robust controllers [83, 61]. Again the

results can b eviewed in termsof integralquadraticconstraints,since

optimal design withresp ect toan IQC leadstoH 1

optimization.

During thelast decade, a variety of metho ds has b een develop ed

within theareaofrobustcontrol. Aswasp ointedoutin [35],manyof

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 G(s)

c c



 -

-

? 6

v f

Figure1: Perturbation inFeedbackForm

themcan b ereformulatedtofallwithintheframeworkof IQC's. This

will b efurtherdemonstratedin thecurrentpap er,whichisdivided in

twoparts.

This rst part presentssome minimal framework forthe stability

analysis of feedback interconnectionsdescrib ed in terms of IQC's. It

is intro duced by an extensive example, illustratingthe main ideas on

a feedback lo op withsaturationand an uncertaindelay. In section 3,

denitions and main theorem are stated in detail. After that follows

sections with discussions and comparisons to well known results. Fi-

nally, we givea summarizinglistof integral quadraticconstraints for

imp ortanttyp esofsystemcomp onents.

The second part of the pap er concerns analysis of robust p erfor-

mance,andgeneralizesthestabilityanalysistocaseswheretheb ound-

edness,causalityanduniquenessassumptionsofpartoneareviolated.

2 Outline of the method

Consider a feedback conguration illustrated in Figure 1, consisting

of a time-invariant linear op erator with transfer matrix G(s), inter-

connected with an op erator , that describ es the "troublemaking"

(nonlinear,time-varyingoruncertain)comp onentsofthesystem. The

notation G will in the sequel either denote a linear op erator ora ra-

tionaltransfermatrix,dep ending on thecontext.

First,wedescrib easaccuratelyasp ossible byintegralquadratic

constraints(IQC's)

Z

1

1

"

b v(j!)

d

(v)(j!)

#



(j!)

"

b v(j!)

d

(v)(j!)

#

d!0 (1)

which should holdforanysquaresummablev withFouriertransform

^

v . The class 



of all rational hermitean matrix functions  that

dene a valid IQC fora given isconvex,since thesum of two p os-

itive integrals is p ositive, and it is usually innite-dimensional. For

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is readilyavailablein thelitterature. Infact,IQC's areimplicitly

present in many results on robust/non-linear/time-varying stability.

A list of such IQC's has b een app ended to this pap er in section 7.

Whenconsistsofacombinationofseveral simpleblo cks,IQC'scan

b e generated by convex combinations of constraints for the simpler

comp onents.

Next, we search for a matrix function  2 



, that satises the

criterion



G(j!)

I





(j!)



G(j!)

I



<0 8 !2R[f1g: (2)

In combinationwith (1),this essentially provesstability of the inter-

connection. Thesearch fora suitable  canb ecarried outbynumer-

ical optimization, restricted tosome nite-dimensional subset of 

 .

Roughly sp eaking, is exp ected tob eof theform

(j!)= q =q0

X

q =1 x

q



q (j!);

wherex

q

arereal parameters.  and G areprop er rationalfunctions

withno p oleson the imaginaryaxis, sothere existsn>0,a Hurwitz

matrixAofsizenn,amatrixBofsizenm,andasetofsymmetric

real matricesM

q

of size(n+m)(n+m),such that



G(j!)

I







q (j!)



G(j!)

I



=



(j!I A) 1

B

I





M

q



(j!I A) 1

B

I



forallq. ByapplicationoftheKalman-Yakub ovich-Pop ovLemma,as

statedbyWillems[74],itfollowsthattheinequalityin(2)isequivalent

totheexistenceof asymmetricnn matrixP =P T

suchthat



PA+A T

P PB

B T

P 0



+ q =q

0

X

q =1 x

q M

q

<0: (3)

Hence the search for x

q

that pro duce a  weight satisfying (2) (i.e.

provingthestability)takestheformofaconvexoptimizationproblem

denedbyalinearmatrixinequality(LMI)inthevariablesx

q

;P. Such

problems can b e solved very eciently using the recently develop ed

numerical algorithmsbased oninterior p oint metho ds [40,7].

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k sat

P(s)

e

 s

- - - -

Figure2: Systemwith Saturation and Delay

2.1 Example with Saturation and Delay

Considerthefollowingfeedbacksystemwithcontrolsaturationandan

uncertain delay.

_

x(t) = Ax(t)+Bsat(w (t))

w (t) = k Cx(t  )

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whereris thereferencesignal, 2[0;

0

]isan unknownconstant,

P(s)=C(sI A) 1

B=

s 2

s 3

+2s 2

+2s+1

is the transfer function of the controlled plant (see the Nyquist plot

on Figure3), and

sat(w ) =



w ; jw j1;

w =jw j ; jw j1;

isthefunction thatrepresentsthesaturation. Thesetup isillustrated

in Figure2.

Letusrstconsiderstabilityanalysisforthecaseofnodelay. Then

letb ethesaturation,whileG(s)= k P(s). Applicationofthecircle

criterion

k 1

< min

!

ReP(j!) (5)

givesstabilityfor

k < k

circ

8:12

(see dashed line in Figure 3). This corresp onds to a 



containing

only thematrix



0 1

1 2



:

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−0.5 0 0.5 1

−0.5 0 0.5 1

Re P(j!)

Figure 3: Nyquist plot for P(j!) (solid line)

InthePop ovcriterion,



consistsof alllinear combinations



0 1

1 2



+



0 j!

j! 0



;

and theresultinginequality(2)givesthe minorimprovement

k 1

< max

 min

!

Re[(1+j!)P(j!)] (6)

k < k

Pop ov

8:90

A Pop ovplotis shownin Figure4.

Furthermore, b ecause the saturation is monotone and o dd, it is

p ossible toapplyamuchstrongerresult,obtainedbyZamesandFalb,

[84]. By their statement, a sucient condition for stability is the

existenceof afunction H 2RL

1

suchthat

0 < min

!

Re[(1+H(j!)



)(P(j!)+k 1

)]

H(j!) = Z

1

1 e

j! t

h(t)dt

1  Z

1

1

jh(t)jdt

This extends the class of valid IQC's further, by allowing all matrix

functionsof theform

(j!) =



0 1+H(j!)

1+H( j!) 2(1+ReH(j!))



whereH hasanimpulseresp onseofL

1

normnogreaterthanone. For

ourproblem,thechoiceH(j!)= (1+j!) 1

givesfor!2R that

Re[(1+H( j!))P(j!)] = j1+H( j!)j 2

Re



j!

! 2

+j!+1



0

(10)

−0.5 0 0.5 1

−1

−0.5 0 0.5

1

k

Re Im

Figure 4: Pop ov plot for P(j!) with stabilizing gain k

Thisshowsthatthefeedbacksystemisindeedstableforallk>0and

concludesthestabilityanalysisin theundelayedcase.

Considering also the delay uncertainty, the problem is to nd a

b ound on the maximal stabilizing feedback gain for a given delay

b ound. A crude b ound can b e received directly from the small gain

theorem, stating that, b ecause of the gain b ound ke

 s

sat()k < 1,

the feedback interconnection of e

 s

sat() and k P(s) is stable pro-

vided that

k < kPk

1 1

1:37:

Not surprisingly, this condition is conservative. For example, it do es

not utilize any b ound on the delay. In order to do that, it is useful

togeneratemoreIQC's forthedelaycomp onent. However,letusrst

step back and formulate the stability criterion more carefully. The

example will b econtinued in section6.

Notation

LetRL

1

b ethesetofprop er(b oundedatinnity)rationalfunctions

withrealco ecients. Thesubsetconsistingoffunctionswithoutp oles

intheclosedrighthalfplaneisdenotedRH

1

. Thesetofmnmatri-

ceswithelementsinRL

1 (RH

1

)willb edenotedRL mn

1

(RH mn

1 ).

L l

2

[0;1)can b e thought of as thespace of R l

-valued signals (i.e.

functionsf : [0;1)!R l

) of niteenergy

kf()k= Z

1

0

jf(t)j 2

dt:

This is a subset of the space L l

2e

[0;1), whose memb ers only need

to b e square integrable on nite intervals. By an operator we mean

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a function F : L

2e

[0;1) ! L

2e

[0;1) from one L

2e

[0;1) space to

another. The gainof an op eratorF :L a

2e

[0;1)!L b

2e

[0;1)is given

by

kFk=sup fkF(f)k=kfk: f 2L a

2

[0;1); f 6=0g

(samenotationforthegainasfortheenergy). An imp ortantexample

of an op erator is given by the past projection (truncation) P

T

, which

leavesa function unchangedon theinterval[0;T]and givesthe value

zeroon(T;1].Causalityofanop eratorF meansthatP

T F =P

T FP

T

foranyT >0.

3 A Basic Stability Theorem

The following feedback conguration, illustrated in Figure 1, is the

basic objectof thetheoreticalstudyin this pap er.



v = Gu+f

u = (v)+e;

(7)

Heref 2L l

2e

[0;1);e2L m

2e

[0;1)representtheinterconnectionnoise,

G andare thetwo causalop eratorsonL m

2e

[0;1)and L l

2e

[0;1)re-

sp ectively. It is assumed that G is a linear time-invariant op erator

withthetransferfunctionG(s)inRH lm

1

,andisb ounded(butnot

necessarilylinear ortime-invariant).

Animp ortantassumptionab outsystem(7)willb eitswell-p osedness.

Denition The feedback system (7) is said to b e well-posed, if the

op erator

I G: L l

2e

[0;1)!L l

2e [0;1);

whichmapsv tov G(v),iscausallyinvertible,i.e. ifthere existsa

causalop erator suchthatI =(I G)Æ = Æ(I G):

Inmostapplications,thisdenitionofwell-p osedness(amoregen-

eral denition will b e intro duced in the second part of the pap er) is

equivalent totheexistence,uniqueness and continuabilityof solutions

oftheunderlyingdierential equations,andisthereforeeasytoverify.

Thefollowing kindofinput/outputstabilityin theL

2

-setting,will

b econvenient.

Denition Thefeedbacksystem(7)issaidtob estableifthereexists

a C>0 such that

Z

T

0 (jvj

2

+juj 2

)dt  C Z

T

0 (jfj

2

+jej 2

)dt (8)

(12)

Indeed,awell-p osedsystem(7)isstableifand onlyif (I G) 1

is a b ounded causal op erator. In many cases, it is also desirable to

verifysomekind of exponential stability. One might exp ect that this

requires separate analysis. However, for general classes of ordinary

dierential equations,exp onential stability turnsouttob eequivalent

totheinput/output stability intro ducedab ove(compare[67],section

6.3).

Prop osition 1 Let besuch that

sup

x;t

j(x;t)j=jx(t)j<1:

Assume that for any g2L n

2

[0;1),x

0 2R

n

,t

0

0 the system

_

x(t) = (x(t);t)+g(t); tt

0

(9)

has a solution x(). Then the following two conditions areequivalent.

(i) Thereexists a c>0 such that

Z

T

0

jx(t)j 2

dt  c Z

T

0 jg(t)j

2

dt 8 T >0 (10)

for any solution of (9) withx(0)=0.

(ii) There exist;d>0 suchthat

jx(t

1 )j

2

 de

(t

0 t

1 )

jx(t

0 )j

2

+d Z

t1

t

0 jg(t)j

2

dt (11)

for any solution x of (9).

Proof. Parts, if notall, of this result can b e found in standardref-

erencesonnonlinear systems. However,foreasyreference,acomplete

pro of isgivenin section 8. 2

Next,weneed aformaldenition ofthetermIQC.

Denition Supp ose:jR!C

(l+m)(l+m)

isab oundedmeasurable

functiontakingHermiteanvalues. Let b ethequadraticformdened

on L l

2

[0;1)L m

2

[0;1)by

(v;u) = Z

1

1



b v(j!)

b u(j!)





(j!)



b v(j!)

b u(j!)



d!

(13)

A b ounded op erator :L

2e

[0;1)!L

2e

[0;1) is said to satisfy the

IQC dened by  if

(v;v)  0 8 v2L l

2

[0;1): (12)

Theorem 2 Assume that

(i) forany  2[0;1],system(7) withreplacedby is well-posed.

(ii) for any  2[0;1],the IQC dened by  issatised by .

(iii) thereexists >0such that



G(j!)

I





(j!)



G(j!)

I



 I 8 !2R: (13)

Then the feedback system (7) is stable.

Remark 1 Note that (j!) =



I 0

0 I



gives a version of the

small gaintheorem,while (j!)=



 I

I =kk 2



givesapassivity

theorem.

Remark 2 In manyapplications, (see, forexample, Remark 1), the

upp erleftcornerof(j!)isp ositivesemi-denite andthelowerright

corneris negativesemi-denite, so satisestheIQC dened by

for 2[0;1]if and onlyif do es so. This simpliesassumption(ii).

Remark 3 It isimp ortanttonotethatif with  2[0;1]satises

several IQC:s, dened by 

1

;:::;

n

, then a sucient condition for

stability is existence of x

1

;:::;x

n

0 such that (13)holds for  =

x

1



1

++x

n



n

. Hence,themoreIQC:sthatcan b everiedfor,

theb etter. Moreover,it canb e proved alongthelines of[60,36]that

if nosuchx

1

;:::;x

n

0 exist,thenthereisa b oundedop eratorthat

destabilizes (7),but satisesall theIQC:s. Inthis sense,thestability

condition ofTheorem2is non-conservative.

Proof of Theorem2.

Step 1. Show that there existsc

0

>0 such that

kvkc

0

kv G(v)k 8 v2L l

2

[0;1): (14)

Intro duce m

11

;m

12

;m

22

as the normsm

ij

= sup

! k

ij

(j!)kfor the

matrixblo cksof

(j!)=





11

(j!) 

12 (j!)



12 (j!)





22 (j!)



:

(14)

For >0,letc()=m

11 +m

11

=+m

12

=. Then

j(v;u) (v+Æ;u)j  m

11 kÆk

2

+2kÆk(m

11

kvk+m

12 kuk)

 c()kÆk 2

+(kuk 2

+kvk 2

)

forall v;Æ2L l

2

[0;1),u2L m

2

[0;1). Notethat(13)implies that

(Gu;u)  kuk 2

8u2L m

2 [0;1)

Let 2[0;1],u=(v),v2L l

2

[0;1),

1

==(2+2kk 2

). Since 

satisestheIQC dened by,we have

0  (v;u)=(Gu;u)+(v;u) (Gu;u)

 kuk 2

+c(

1

)kv Guk 2

+

1 (kuk

2

+kvk 2

)





2 kuk

2

+c(

1

)kv Guk 2

Hencekuk p

2c=kv Gukand

kvk  kGuk+kv Guk

 (1+kGk p

2c=)kv G(v)k:

Step 2. Show that if(I G) 1

isbounded for some 2[0;1]then

(I G) 1

isboundedforany 2[0;1]withj j<(c

0

kGkkk) 1

.

By the well-p osedness assumption, the inverse (I G) 1

is well

dened on L l

2e

[0;1). Boundednessof theinverse meansthat

kP

T

vkconstkP

T

(v G(v))k 8v2L l

2e [0;1)

Furthermore, whenthis inequalityholds for someconstant,it follows

from (14)thatitholds withtheconstant c

0 . Then

kP

T

vk  c

0 kP

T

(v G(v))k

 c

0 kP

T

(v G(v))+( )P

T

G(v)k

 c

0 kP

T

(v G(v))k+c

0

kGkkkj jkP

T vk:

Boundednessof (I G) 1

follows,since c

0

kGkkkj j<1.

Step 3. Now,since (I G) 1

isb ounded for =0, step2 shows

that (I G) 1

is b ounded for < (c

0

kGkkk) 1

, then for <

2(c

0

kGkkk) 1

, etc. Byinduction, (I G) 1

isb ounded aswell.

2

(15)

-

-

- -

-

? e

6

0 -



C(s)

jj 2

dt

R

jj 2

dt v

v u

+

Figure 5: Testing ablo ck for an IQC.

4 Hard and soft IQC's

Asa rule, an integralquadraticconstraintis an inequality describing

correlation b etweentheinput andoutputsignals of acausalblo ck.

VerifyinganIQCcanb eviewedasavirtualexp erimentwiththesetup

shownon Fig.5,whereistheblo cktestedforanIQC,f isthetest

signalof niteenergyand C(s)is astable lineartransfermatrixwith

twovectorinputs,twovectoroutputsandzeroinitialdata. Theblo cks

with R

jj 2

dt indicate calculationof theenergyintegral ofthe signal.

We say that  satises the IQC describ ed by the test setup, if the

energy of the second output of C is always at least as large as the

energy of the rst output. Then the IQC can b e representedin the

form (1),where

(j!)=C(j!)





I 0

0 I



C(j!): (15)

The most commonly used IQC is the one that expresses a gain

b ound on the op erator. For example C(s)= I corresp onds to the

b oundkk1. Theenergyb oundshavetheparticularprop ertythat

theenergydierenceuntiltimeT willb enon-negativeatanymoment

T,notjustT =1. SuchIQC'sarecalledhardIQC's,incontrasttothe

moregeneralsoftIQC's,which need nothold fornite timeintervals.

Some of the most simple IQC's are hard, but the generic ones are

not. In the theory of absolute stability, the use of soft IQC's was

oftenreferredtoasallowing non-causalmultipliers. While forscalar

systems thiswasusuallynota seriousproblem,theknownconditions

for applicability of non-causal multipliers were far to o restrictive for

multivariablesystems. TheformulationofTheorem2makesitp ossible

(andeasy)tousesoftIQC'sin a verygeneralsituation. Forexample,

consider thefollowingcorollary.

Corollary 3 (Non-causal multipliers) Assume that condition (i)

of Theorem 2 is satised. If thereexist some M 2RL lm

1

and  >0

(16)

R

1

1 Re(bv



M c

v)d!  0 for v2L

l

2 [0;1)

M



G+G



M  G



G on jR

then the system isinput/output stable.

Proof. This isTheorem2 with

(j!) =



 M(j!)

M(j!)



=kk 2



2

For multivariable systems, the ab ove conditions on M are much

weakerthanfactorizabilityasM =M M

+

,withM

+

;M

+ 1

;M



;(M



) 1

all b eing stable, which is required forexample in [84] and [17]. The

price paidforthis in Theorem2isthe verymild assumptionthat the

feedbacklo op is well-p osed notonly for =1,but forall  2[0;1].

Anotherexample isprovidedbytheclassicalPop ovcriterion.

Corollary 4 (Pop ov criterion) Assume that  : R ! R is such

that 0  ()  const 2

for  2 R. Let H(s)= C(sI A) 1

B,

whereA is Hurwitz. Assume that the system

_

x(t)=Ax(t)+B(Cx(t))+f(t) (16)

has unique solution on [0;1) for any  2 [0;1] and for any square

summable f. If for someq2R

inf

! >0

Re[(1+j!q)H(j!)] > 0 (17)

then the system (16)with =1 is exponentially stable.

Remark5 Infact,theconditionofexistenceanduniqueness,usedto

deneasanop erator,isnotreallyimp ortantinthestabilityanalysis.

In the second part of this pap er, a stronger version of Theorem 2 is

given,which allowsus todroptheuniqueness assumption.

Proof. For q2Rand adierentiable w2L l

2

[0;1),wehavethesoft

IQC

Z

1

0

(w+qw)(w )dt_ q

"

Z

w (t)

0

()d

#

1

0

=0

(17)

-

-

-

?

i

6

0 - f(t)

C(s)

jj 2

dt

R

jj 2

dt

+

Figure 6: Testing a signal f for an IQC.

Application of Corollary3with

G(s) = (s+1)H(s)

(v)(t) = 

Z

t

0 e



v()d



M(s) = (1+qs)=(s+1)

shows that the conditions of Prop osition 1 hold, which ensures the

exp onentialstability. 2

Integral quadraticconstraints can b eusedto describ e an external

signal (noise ora reference) entering thesystem. The virtual exp er-

iment setup for a signal f is shown on Fig. 6. The setup clearly

shows the sp ectral analysis nature of IQC's describing the signals.

Mathematically,theresultingIQC hastheform

Z

1

1t

^

f(j!)



(j!)

^

f(j!)d!0;

whereisgivenby(15). Inthesecondpartofthispap er,p erformance

analysis of systemswith b oth interior blo cks and externalsignals de-

scrib ed in termsof IQC'sis considered.

5 IQC's and Quadratic Stability

Thereis a close relationship b etween quadraticstabilityand stability

analysisbasedon IQC's. Asarule, ifa systemisquadraticallystable

thenitsstabilitycanalsob eprovedbyusingasimpleIQC.Conversely,

insomegeneralizedsense,asystemthatcanb eprovedtob establevia

IQC'salwayshasaquadraticLyapunovfunction. However,toactually

present this Lyapunovfunction, one has toextend thestate space of

the system(by addingthe statesof C(s)from Figure 5). Even then,

(18)

sign-denite, and may not decrease monotonically along the system

trajectories. Inanycase,useofIQC'sreplacestheblind searchfora

quadraticLyapunovfunction,whichistypicalforthequadraticstabil-

ity,byamoreintelligentsearch. Ingeneral,forexampleinthecaseof

so-called parameter-dep endent Lyapunovfunctions, therelationship

withtheIQC typ eanalysis isyettob eclaried.

Belowweformulateandprovearesultontherelationshipb etweena

simpleversionofquadraticstabilityandIQC's. LetDb eap olytop eof

mlmatrices,containingthezeromatrix=0. Let

1

;:::;

N b e

theextremalp ointsofD . Considerthesystemofdierential equations

_

x(t)=(A+B(t)C)x(t); (t)2D ; (18)

whereA;B;Caregivenmatricesofappropriatesize,AisaHurwitzn 

nmatrix. (Themostoftenconsidered caseof system(18)is obtained

when m =l and D is the set of all diagonal matrices withthe norm

notexceeding1. ThenN =2 m

,and

i

arethediagonalmatriceswith

1 on the diagonal). The system iscalled stable if x(t)!0 for any

solutionof(18)where()isameasureablefunctionand(t)2Dfor

allt. Therearenoecientgeneralconditions,thatareb othnecessary

andsucientforstabilityofsystem(18). Instead,wewillb econcerned

withstabilityconditionsthat areonlysucient.

Thesystem(18)iscalledquadraticallystableifthereexistsamatrix

P =P T

such that

P(A+B

i

C)+(A+B

i C)

T

P <0 8i: (19)

Note that, since 0 2 D and A is a Hurwitz matrix, this condition

implies that P > 0. It follows that V(x) = x T

Px is a Lyapunov

function forthe system (18), in thesense that V is p ositive denite,

and dV(x(t))=dt is negative denite on the trajectories. Quadratic

stability is a sucient condition for stability of the system and (19)

can b esolved eciently withresp ect toP =P T

as asystemoflinear

matrixineqalities.

An IQC-based approach to stability analysis of system (18) can

b e formulated as follows. Note that stabilityof (18)is equivalent to

stability of the feedback interconnection (7), where G is the linear

timeinvariantop eratorwithtransferfunction G(s)=C(sI A) 1

B,

and  is theop eratorof multiplication by (t)2D . Onecan apply

Theorem 2, using the fact that  satises the IQC's given by the

(19)

(j!)=



Q S

S T

R



;

whereQ=Q T

;R=R T

;S are realmatricessuchthat

Q+S+ T

S T

+ T

R>0 82D : (20)

Fora xed matrixsatisfying (20),a sucient condition ofstability

given byTheorem2is



G(j!)

I









G(j!)

I



<0 8 !2R[f1g;

whichisequivalent(bytheKalman-Yakub ovich-Pop ovLemma)tothe

existenceof amatrixP =P T

such that



PA+A T

P +C T

QC PB+C T

S

B T

P +S T

C R



<0: (21)

For an indenite matrix R, condition (20)may b e dicult toverify.

However(21)yields R<0. In thatcase, itis sucient tocheck(20)

atthevertices =

i

ofD only, i.e. (20)can b ereplacedby

Q+S

i +

T

i S

T

+ T

i R

i

>0 8i: (22)

ItiseasytoseethattheexistenceofthematricesP =P T

,Q=Q T

,S,

R=R T

, suchthat (21),(22)hold,is a sucient condition ofstability

of system(18).

Nowwe have thetwo seemingly dierent conditions of stabilityof

system (18), b oth expressed in terms of systemsof LMI's: quadratic

stability(19), and IQC-stability (21),(22). Condition (19)hasn(n+

1)=2 free variables (the comp onents of the matrix P = P T

), while

conditions (21),(22) have n(n +1)=2+(n+m)(n+m +1)=2 free

variables. However,theadvantageofusing(21),(22)isthattheoverall

size ofthe corresp ondingLMIisn+m+Nlwhilethesize of(19)

is Nn. IfN is alargenumb er and nissignicantly largerthanl and

m, mo dest (ab out 2 times) increase of the numb er of free variables

in (21),(22)results in a signicant (ab outn=l times) decrease in the

sizeofthecorresp ondingLMI.Thefollowingresultshowsthatthetwo

sucientconditionsofstability(21),(22)and(19)areequivalentfrom

thetheoretical p oint of view.

(20)

to the convex hull of matrices 

1

;:::;

N

. Then a given symmetric

matrix P solves the system of LMI's (19), if and only if P together

with the matrices Q=Q T

, R=R T

, S solves(21) and(22).

Apro of isgiven in section8.

6 Example Revisited

Inordertoapplytheresultstosystem(4),werewriteitasafeedback

interconnectionon Fig. 1,with

G(s) =



k P(s) k

P(s) 0



;

and

(v)(t) =



sat(v

1 (t))

v

2

(t  ) v

2 (t)



The equationsfortheinterconnection arethen

_

x(t) = Ax(t)+Bu

1 (t)

u

1

(t) = sat[v

1

(t)]+e

1 (t)

u

2

(t) = v

2

(t  ) v

2 (t)+e

2 (t)

v

1

(t) = k Cx(t) k u

2 (t)+f

1 (t)

v

2

(t) = Cx(t)+f

2 (t)

(23)

where x(0)=0 and v

2

(t  ) =0for t< . One can see that (23)is

equivalenttotheequationsfrom(4),disturb edbytheinterconnection

noise e;f.

For the uncertain time delay, several typ es of IQC's are given in

the list. Here we shall use a simple (and notcomplete) set of IQC's

fortheuncertain delay

^ u

2

(j!)=(e j !

1)^v

2

(j!); 2[0;

0 ];

based ontheb ounds

j^v

2 (j!)j

2

j^v

2

(j!) u^

2 (j!)j

2

 0

0 (

0

!)j^v

2 (j!)j

2

j^u

2 (j!)j

2

 0

(24)

where

0 (!)=

! 2

+0:08!

4

1+0:13!

2

+0:02!

4

(21)

0 5 10 15 20 0

1 2 3 4 5



0 0

(!)

 (j!)

Figure 7: Comparison of

0

(!) and

 (j!)

ischosen asa rational upp erb ound(see Figure7)of



(j!)= max

 2[0;

0 ]

je j!  =

0

1j 2

=



4sin 2

(!=2); !<

4 !

By integratingthep ointwiseinequalities(24)withsomenonnegative

rational functions, one can obtain a huge set of IQC's valid for the

uncertain delay. Using these in combination with some set of IQC's

forthesaturationnonlinearity,onecanestimatetheregionofstability

forthesystemgiven in(4). InFigure8,we haveplottedtheresulting

stabilityb oundforthe casewhen onlyone IQC

Z

1

1



^ v

1

^ u

1







0 1+H

1+H



2(1+Re H)



^ v

1

^ u

1



d!0;

with H(s)= (s+1) 1

,describ es the saturation,while (24)utilizes

the information ab out the delay. The guaranteed instability region

wasobtainedanalyticallybyconsideringtheb ehaviorofthesystemin

thelinearunsaturated regionaroundtheorigin.

7 A List of IQC's

The collection of IQC:s presented in this section is far from b eing

complete. However, the authors hop e it will supp ort the idea that

many imp ortant prop erties of basic system interconnections used in

stabilityanalysiscan b echaracterized byIQC:s.

(22)

0 0.5 1 0

5 10 15 20 25



0 stable

unstable

Figure 8: Bound on stabilizing gain k versus delay uncertainty

0

7.1 Uncertain LTI Dynamics

Letb eanylineartime-invariantop eratorwithgain(H

1

norm)less

thanone. Then satisesallIQC's of theform



x(j!)I 0

0 x(j!)I



wherex(j!)0is ab oundedmeasurablefunction.

7.2 Constant Real Scalar

If isdened bymultiplication withareal numb erofabsolute value

1,thenitsatisesall IQC:sdened bymatrixfunctionsoftheform



X(j!) Y(j!)

Y(j!)



X(j!)



(25)

whereX(j!)=X(j!)



0 and Y(j!)= Y(j!)



areb ounded and

measurablematrixfunctions.

This IQC and the previous one are the basis for standard upp er

b ounds forstructuredsingular values[20,80].

7.3 Time-varying Real Scalar

Let  b e dened by multiplication in the time-domainwith a scalar

function Æ 2L

1

withkÆk

1

1. Then satisesIQC:s dened bya

matrixof theform



X Y

Y T

X



whereX=X T

0and Y = Y T

arereal matrices.

(23)

Letb edenedbymultiplication inthetime-domainwithameasur-

ablematrix(),suchthat(t)2Dforanyt,whereDisap olytop e

of matriceswiththeextremal p oints(vertices)

1

;:::;

N

. satises

theIQC's givenbytheconstant weight matrices

(j!)=



Q F

F T

R



;

whereQ=Q T

;F ;R=R T

are realmatricessuchthatR0,and

Q+F

i +

T

i F

T

+ T

i R

i

>0 8 i:

This IQC corresp onds to quadratic stability and wasstudied in sec-

tion 5.

7.5 Periodic Real Scalar

Letb edened bymultiplicationin thetime-domainwithap erio dic

scalarfunctionÆ 2L

1

withkÆk

1

1andp erio dT. Thensatises

IQC:sdenedby(25)whereXandY areb ounded,measurablematrix

functionssatisfying

X(j!) = X(j(!+2=T))=X(j!)



0

Y(j!) = Y(j(!+2=T))= Y(j!)



:

This set of IQC:s gives theresult by Willems on stability of systems

withuncertain p erio dicgains [74].

7.6 Multiplication by a Harmonic Oscillation

If(v)(t)=v(t)cos(!

0

t) thensatisestheIQC's dened by

(j!)=



X(j! j!

0

)+X(j!+j!

0

) 0

0 2X(j!)



;

where X(j!) = X(j!)



 0 is any b ounded matrix-valued rational

function. Multiplicationbyamorecomplicated(almostp erio dic)func-

tion can b erepresentedas a sum of several multiplications bya har-

monic oscillation,with theIQC'sderivedforeach of themseparately.

Forexample,

v(t)fa

1 cos(!

1 t)+a

2 cos(!

2

t)g=a

1 (

1

v)(t)+a

2 (

2 v)(t);

where(

1

v)(t)=v(t)cos(!

1

t),and (

2

v)(t)=v(t)cos(!

2 t).

(24)

Here  is the op erator of multiplication by a slowly time-varying

scalar, v = Æ(t)v(t), where jÆ(t)j  1, j _

Æ (t)j  d. Since the 60-s,

variousIQC:shaveb eendiscoveredthatholdforsuchtime-variations.

See, forexample,[21, 34,24].

Here we describ e a simple but representative family of IQC:s de-

scribingtheredistributionofenergyamongfrequencies,caused bythe

multiplication by a slowly time-varying co ecient. For any transfer

matrix

H(s)=H

0 +

Z

+1

1 e

ts

h(t)dt;

where h() 2 L nm

1

( 1;+1) and H

0

is a constant, let (H ;d) b e

an upp er b ound of the norm of the commutatorÆH HÆ, for

example

(H ;d)= Z

+1

1

kh(t)kminf2;djtjgdt:

Thefollowing weightingmatricesthendene valid IQC:s:

=

"

(1+)fH



H+

(H ;d) 2

 I

m

g 0

0 H



H

#

(26)

where>0isaparameter,andHisacausaltransferfunction(h(t)=

0 fort<0). Anotherset ofIQC:s isgiven by

=



(H ;d)I H

H



0



(27)

where H is skew-Hermitean along the imaginary axis (i.e. H(j!) =

H(j!)



),but notnecessarilycausal. Since

(H ;d)=O (d) as d!0

whenever kh(t)k = O (t 2 

), the constraints used in the  case

(multiplication by aconstantgain Æ 2 [ 1;1])can b e recoveredfrom

(26)and(27)asd!0. Similarly,thetime-varyingrealscalar IQC's

willb erecoveredasd!1byusingconstanttransfermatricesH(s)=

H

0 .

In [49, 50], IQC's are instead derived for uncertain time-varying

parameters with b ounds on the supp ort of the Fourier transform

^

Æ.

Slow variation then means that

^

Æ(j!) is zero except for ! in some

small interval[ a;a].

(25)

The uncertain b ounded delay op erator (v)(t) = v(t  ) = u(t),

where2[0;

0

], satisesthep ointwise quadraticconstraintsin the

frequency domain:

j^u(j!)j 2

=j^v (j!)j 2

; (28)

1 (!

 )(jj!



^

u(j!)+v(j!)j^ 2

(1+! 2



)j^v (j!)j 2

)

2 (j!



)j^v (j!) u(j!)j^ 2

;

(29)

where!



=!

0

=2,and

1;2

arethefunctionsdened by

1 (!)=



sin!

!

;j!j

0 ;j!j>:

;

2 (!)=



cos! ;j!j

0 ;j!j>:

:

Note that (29) is just a sector inequality for the relation b etween

^

v (j!) u(j!)^ andj(v (j!)^ +u(j!)):^

j(^v (j!)+u(j!))^ =

cos(! =2)

sin(! =2)

(^v (j!) u(j!)):^

Multiplying (28) by any rational function and integrating over the

imaginary axis yields a set of IQC's for the delay. Unfortunately,

theseIQC'sdonotutilizetheb oundonthedelay. ToimprovetheIQC-

description,onecanmultiply(29)byanynon-negativeweightfunction

and integrate overtheimaginary axis. The resultingIQC's, however,

will have non-rational weight matrices (). To x the problem, one

shouldusearationalupp erb ound

1+

of

1

andrationallowerb ounds

1 and

2 of

1 and

2

resp ectively. Forexample,areasonablygo o d

approximationisgiven by

1+

=

(1 0:0646!

2

) 2

1+0:038!

2

+0:0001!

4

+0:00085!

6

;

1

=

1 !

2

=

2

1+(1=6 1=

2

)!

2

+(2=

4

1=6

2

)!

4

;

2

=

1 0:4073!

2

1+0:0927!

2

+0:0085!

4 :

(26)

2 2

and with

1

replaced by

1

(the upp er b ound for the jj!



^ u+w j^

2

multiplier, the lower b ound for the j^v j 2

multiplier) resp ectively, and

can b eintegrated witha non-negative rational weightfunction to get

rational IQC's utilizing theupp erb oundon thedelay.

Asimpler,butlessinformative,setofIQC:sisdenedfor(v)(t)=

v(t  ) v(t), 

0 ,by



(j!)

0 (!

0

=2) 0

0 (j!)



;

where()is anynon-negativerationalweightingfunction, and

0 (!)

isanyrational upp erb ound of



(j!)= max

 2[0;

0 ]

je j!  =

0

1j 2

=



4sin 2

(!=2); !<

4 !

;

forexample,

0 (!)=

! 2

+0:08!

4

1+0:13!

2

+0:02!

4 :

7.9 Memoryless nonlinearity in a sector

If(v)(t)=(v(t);t),where :RR!R isa function,suchthat

v 2

(v;t)v v 2

8v2R;t0;

thenobviouslytheIQC with

(j!)=



2 +

+ 2



holds.

7.10 The Popov IQC

Ifu(t)=(v)(t)=(v(t)),where:R!Risacontinuousfunction,

v(0)=0, andb oth u()andv()_ aresquaresummable,then

Z

1

0 _

v(t)u(t)dt=0:

Inthefrequency domain, thislo oks like an IQCwith

(j!)=



0 j!

j! 0



:

(27)

theimaginaryaxis. Toxtheproblem,consider

1

=Æ 1

s+1

instead

of , i.e. u(t) = (

1

f)(t) = (v(t)), where v(t)_ = v(t)+f(t),

v(0)=0. Now,

1

satisestheIQC with

(j!)=

"

0

j!

1+j!

j!

1 j!

0

#

:

TogetherwiththeIQC foramemoryless nonlinearityin asector,this

IQC yields thewell-knownPop ovcriterion.

7.11 Monotonic Odd Nonlinearity

Supp ose  op erateson scalarsignals accordingto thenonlinear map

(v)(t)=Æ(v(t)),where Æ is an o dd function on R such that _

Æ(x)2

[0;k ]forsomeconstant k . Then satisestheIQC:sdened by



0 1+H(j!)

1+H( j!) (2+2ReH(j!))=k



;

where H 2RL

1

is arbitrary exceptthat the L

1

-normof its impulse

resp onse isno largerthanone [84].

7.12 IQC:s for Signals

Performanceof a linear controlsystem is oftenmeasured in terms of

disturbance attenuation. An imp ortant issue is then the denition

of theset ofexp ected externalsignals. Hereagain, integral quadratic

constraintscanb eusedasaexibleto ol,forexampletosp ecifyb ounds

on autocorrelation, frequency distribution, or even tocharacterize a

given nite set of signals. Then, the informationgiven by the IQC:s

can b e usedin the p erformanceanalysis, along thelines discussed in

[33,44]and furtherin thesecond partof this pap er.

7.13 IQC's from robust performance

Oneofthemostapp ealingfeaturesofIQC'sistheirabilitytowidenthe

eldofapplication ofalreadyexistingresults. Thismeansthatalmost

anyrobustnessresult derived by somemetho d (p ossibly unrelated to

the IQC techniques) for a sp ecial class of systems can b e translated

intoan integralquadraticconstraint.

(28)

connection of a particular linear time-invariant system G

0

= G

0 (s)

withan "uncertain"blo ck:

v=G

0

u+f; u=(v); (30)

wheref isthe externaldisturbance. Assumethat stabilityof this in-

terconnection(i.e. theinvertibilityoftheop eratorI G

0

)isalready

proved,and,moreover,anupp erb oundontheinducedL

2

gain"from

f tov ("robustp erformance")isknown: kvk 2

dkfk 2

foranysquare

summable f, v satisfying (30). Then, since foranysquare summable

v thereexistsa squaresummablef =v G

0

(v)satisfying (30),the

blo cksatisestheIQC givenby

(j!)=



d 1 dG

0 (j!)

dG

0

( j!) djG

0 (j!)j

2



: (31)

ThisIQCimplies stabilityof system(30)viaTheorem2,butcan also

b eusedin theanalysis ofsystemswith additionalfeedbackblo cks,as

wellaswithdierent nominal transferfunctions.

Forexample,considertheuncertainblo ckwhichrepresentsmul-

tiplication of a scalar input by a scalar time-varying co ecient k =

k (t),suchthatk (t)2[ 1;1]. ThereisoneobviousIQCforthisblo ck,

stating that the L

2

-induced norm of  is not greater than 1. Let

us show how additional non-trivial IQC's can b e derived based on a

particularrobustp erformanceresult. Considerthefeedbackintercon-

nection of  with a given LTI blo ck with a stable transfer function

G

0

(s)=C(sI A) 1

B. Thisis thecase ofa systemwithone uncer-

tain fasttime-varyingparameterk=k (t), k (t)2[ 1;1]:

_

x(t)=Ax(t)+Bk (t)(Cx(t)+f(t)); (32)

where A;B;C are given constant matrices, A is a Hurwitz matrix,

f() is the external disturbance. It is known that, for this system,

the norm b ound kvk 2

 k(v)k 2

, yields the circle stability criterion

jG

0

(j!)j<1,whichgivesonlysucientcontitionsofstability. Never-

theless,foralargeclassoftrasnsferfunctionsG

0

(s),notsatisfyingthe

circlecriterion,system(32)isrobustlystable. Apro ofofsuchstability

usually involvesusing anon-quadratic Lyapunovfunction V =V(x),

andprovidesanupp erb ounddoftheworst-caseL

2

-inducedgain"from

v to y = Cx+v". Thisupp er b ound, in turn, yields theIQC given

by (31), describing the uncertain blo ck . The fact that stability of

(29)

norm b oundkvk 2

k(v)k 2

,shows thatthenew IQC indeed carries

additional informationab out .

8 Proofs

Proof of Proposition1

(i))(ii): Fort

0

1and c

0

>c,dene theLyapunovfunction

V(x

0

;t

0

) = sup

g 2L

2

;x(t

0 )=x

0 Z

1

t

0

fjx(t)j 2

c

0 jg(t)j

2

gdt

wherex;g satisfy(9).

Ourrst objectivesaretoshowconvergenceoftheintegralforany

g2L

2 [t

0

;1)and existenceof ; >0 such that

jx

0 j

2

V(x

0

;t

0

)  jx

0 j

2

:

Anysolutionx;gof(9)on[t

0

;1)withx(t

0 )=x

0

canb eextendedto

[0;1)with x(0)=0, bysetting



g(t) = x

0

=t

0

(x

0 t=t

0

;t)

x(t) = x

0 t=t

0

0tt

0 :

Letkk=sup

x;t

j(x;t)j=jx(t)jand notethat

Z

t0

0 jgj

2

dt  2jx

0

=t

0 j

2

+2 Z

t0

0

(x

0 t=t

0

;t) 2

dt

 2jx

0

=t

0 j

2

(1+kk 2

t

0 3

=3)

= c

2 jx

0 j

2

forsomec

2

>0. Theinequality (10)implies that

Z

1

t

0

jx(t)j 2

dt  kxk 2

ckgk 2

c Z

1

t

0 jgj

2

dt+cc

2 jx

0 j

2

Thisprovesconvergenceoftheintegralin thedenitionofV andwith

=c

2

,itshowsthat V(x

0

;t

0

) jx

0 j

2

. To provetheexistenceof ,

let g0 andnotethat

jxj_  kkjxj

d

dt lnjxj

 kk

jx(t)j  jx

0 je

(t0 t)kk

; tt

0

V(x

0

;t

0

)  jx

0 j

2

:

(30)

Nowconsideraxed solutionx;g of (9). By denitionofV

V(x(t

0 );t

0

)  V(x(t

1 );t

1 )+

Z

t

1

t

0

fjx(t)j 2

c

0 jg(t)j

2

gdt

for anyt

1

 t

0

1. Hence, with k (t)= V(x(t);t) 0, the measure

dk (t) isabsolutely continuous, andsatisestheinequalities

dk (t)  [c

0 jg(t)j

2

jx(t)j 2

]dt[c

0 jg(t)j

2

k (t)= ]dt

d[e t=

k (t)]  c

0 e

t=

jg(t)j 2

dt

k (t

1

)  e (t

0 t

1 )=

k (t

0 )+c

0 Z

t

1

t

0 jg(t)j

2

dt

 e (t

0 t

1 )=

jx(t

0 )j

2

+c

0 Z

t

1

t0 jg(t)j

2

dt

This implies (11) for t

1

 t

0

 1. The result follows for arbitrary

t

1

t

0

0,since

jxj_  kkjxj+jgj

jx(1)j 2

 c

3 jx(t

0 )j

2

+c

3 Z

1

t0 jg(t)j

2

dt;

forsomec

3

>0.

(ii))(i): Let T > 0 b e such that d

1 := de

T

< 1 where d is the

constantfrom(11). Then,by(11)

jx(k T +T)j 2

d

1

jx(k T)j 2

+d Z

k T+T

k T

jg(t)j 2

dt

fork=0;1;2;:::Hence

1

X

k =0

jx(k T)j 2

 d

2 1

X

k =0 Z

k T+T

k T

jg(t)j 2

dt=d

2 kgk

2

for some d

2

> 0 if x(0) = 0. Also, the inequality (11) applied for

t

0

=k T,t

1

2[k T;k T+T]yields

jx(t)j 2

 d



jx(k T)j 2

+ Z

k T+T

k T

jg(t)j 2

dt



; t2[k T;k T +T]

Z

k T+T

k T

jx(t)j 2

dt  dT



jx(k T)j 2

+ Z

k T+T

k T

jg(t)j 2

dt



kxk 2

 (d

2

+1)dTkgk 2

;

(31)

Proof of Theorem 5 The suciency is straightforward: multiplying

(21) by [I C T

 T

i

] from the left, and by [I C T

 T

i ]

T

from the right

yields

P(A + B

i

C) + (A + B

i C)

T

P+ C T

(Q + S

i + 

T

i S

T

+  T

i R

i

)C<0;

which implies (19)b ecauseof theinequalityin (22).

To prove the necessity, let P = P T

satisfy (19). Let 

0 : R

n



R m

!Rb ethequadraticform



0

(x;)= (jxj 2

+jj 2

) 2x T

P(Ax+B);

where>0 is asmall parameter. Dene :R l

R m

!Rby

(y;)=inff

0

(x;): Cx=yg; (33)

wherethe inmum is takenoverall x2R n

such thatCx =y. Since

the zero matrix b elongs to the convex hull of D , (19) implies that

PA+A T

P < 0. Hence, for a suciently small  > 0,  is strictly

convex in the rst argument, and a nite minimum in (33) exists.

Moreover,since 

0

is aquadraticform,thesameistruefor and the

matricesQ;R;S canb e intro ducedby

(y;)=y T

Qy+2y T

S+ T

R:

Letusshowthattheinequalities(21),(22)aresatised. First,by(19),

foranyy we have

y T

(Q+S

i +

T

i S

T

+ T

i R

i )y

=(y;

i y)

=inff

0 (x;

i

Cx): Cx=yg

=inff x T

(P(A+B

i

C)+(A+B

i C)

T

P)x

(jxj 2

+j

i Cxj

2

): Cx=yg



1 jyj

2

;

(providedthatand 

1

aresucientlysmall). Hence(22)holds. Sim-

ilarly,foranyx; we have

x T

P(Ax+B)+(Cx;)

=x T

P(Ax+B)+inff

0 (x

1

;): Cx

1

=Cxg

x T

P(Ax+B) (jxj 2

+jj 2

) x T

P(Ax+B)

 (jxj 2

+jj 2

);

(32)

quadraticformx T

P(Ax+B)+(Cx;). 2

Acknowledgement

The authors are greatful to many p eople, in particular to K.J. As-

tröm,J.C.Doyle,U.JönssonandV.A.Yakub ovich forcommentsand

suggestions ab out this work. The work has b een supp orted by the

NationalScienceFoundation,grantECS9410531,andtheSwedishRe-

searchCouncilforEngineering Sciences, grant94-716.

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