On stability and robustness of nonlinear cascaded systems - Application to mechanical systems

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systems - Application to mechanical systems

Antoine Chaillet

To cite this version:

Antoine Chaillet. On stability and robustness of nonlinear cascaded systems - Application to mechan-

ical systems. Automatic. Université Paris Sud - Paris XI, 2006. English. �tel-00480931�

THÈSE

Présentée pour obtenir

Le GRADE de DOCTEUR ÈS SCIENCE S

DE L'UNIVERSITÉ PARIS XI ORSAY

par

Antoine CHAILLET

Sujet:

On stability and robustness of nonlinear systems

Appli ations to as aded systems

Stabilité et robustesse des as ades non-linéaires

et appli ation aux systèmes mé aniques

Soutenue le 7 juillet 2006 devant la Commision d'Examen:

M.Jean-Mi helCoron Examinateur

M.Thor Fossen Rapporteur

M.Antonio Loría Examinateur

M.RodolpheSepul hre Examinateur

M.Eduardo Sontag Rapporteur

M.Andrew Teel Examinateur

A Thibaut.

Remer iements - A knowledgement

I sin erely thank Profs. Fossen and Sontag for having done me the honor of reviewing

thisdo ument,and theexaminersofmy ommittee, Profs. Coron,Sepul hre andTeel,for

their parti ipatio n to the evaluation of this work and their pre ious s ienti omments

thatsurely greatly improvedthe qualityof the thesis.

I warmly thank my supervisor, Dr. A. Loría who, with his s ienti ulture, his in-

tuition, his omprehension , his listening and his sympathy, made of these three years a

gilt-edged sour e of s ienti enri hment in a simple and friendly atmosphere. Few PhD

students,a ordingtome,areleftsu hafreedomintheirworkandgivensu hapromotion

of theirresults bytheir supervisor. Iamdeeplygratefulto him for this.

I also thank the personswith whom I had the privilege and pleasure to dire tlywork

with during the past three years: Alexei, David, Elena, Erik, Mario, Rafael and Ya ine.

These ollaborations have allbeen ofgreat interestto meandextendedmy ulture in the

eld. Animportant partofthe results presented in this do ument arefruitsof thesejoint

works.

Jetiensàexprimermaprofondere onnaissan eàmafamillesurl'amouretla onan e

de laquellej'aitoujours pu ompter.

J'adresse une pensée tout parti ulière à Hélène sans l'éternel soutien de qui je ne me

seraisprobablement paslan é dans e projet.

Enn, estroisannées n'auraient étéaussienri hissantes et animéessans les ollègues

etamisren ontrésauLSS,etplusparti ulière mentAlessandrodiRÔma,elSuperCabron,

Fernandinetta , Haïowen et Islem. Mer i enn à Vin e, mon ollo ' et ami, pour me sup-

porter même en période de réda tion!

Preliminary remark

Thisdo ument synthesizesthe resear h works ondu ted byA.Chaillet, under the super-

visionofDr. A.Loría,inordertoobtainthePhDdegreefromUniversitéParisSud. Please

notethata Fren hversionis alsoavailableon request.

For anyremark,question or omment, please feelfreeto onta tthe author at:

antoine hailletyahoo.fr.

Contents

Remer iements - A knowledgement 5

Preliminary remark 7

Preamble 11

Contribution of this thesis 15

Publi ations 17

Notation 19

1 Denitions 21

1.1 Stabilityofthe origin . . . 22

1.2 Stabilityofsets . . . 25

1.3 Semiglobal and pra ti al asymptoti stability . . . 28

1.4 Input toState Stability . . . 31

2 Semiglobal and pra ti al asymptoti stability 33 2.1 Su ient onditions . . . 36

2.1.1 Globalpra ti al stability. . . 38

2.1.2 Semiglobal pra ti al stability . . . 45

2.1.3 Semiglobal asymptoti stability . . . 51

2.2 Converse results. . . 55

2.2.1 Semiglobal pra ti al stability . . . 57

2.2.2 Semiglobal asymptoti stability . . . 63

3 Stability of nonlinear time-varying as aded systems 65 3.1 Semiglobal pra ti al asymptoti stabilityof as aded systems . . . 68

3.1.1 WithaLyapunovfun tionfor the driven subsystem . . . 68

3.1.2 Withouta Lyapunovfun tion for the driven subsystem. . . 73

3.2 Semiglobal asymptoti stabilityof as aded systems . . . 77

3.2.1 WithaLyapunovfun tionfor the driven subsystem . . . 77

3.2.2 Withouta Lyapunovfun tion for the driven subsystem. . . 78

3.3 Globalpra ti al asymptoti stabilityof as aded systems. . . 79

4 Set-stability 95

4.1 Prelimina r ydenitions andtools . . . 98

4.2 Onset-stabilityof as aded systems . . . 101

4.3 Example: ross-tra kformation ontrol ofundera tuated surfa evessels . . 106

5 Integral inputto state stability for as aded systems 113 5.1 Globalasymptoti stabilityfor as ades, Lyapunov-based . . . 116

5.2 Integral input to state stabilityfor as ades, Lyapunov-based . . . 120

5.3 Integral input to state stabilityfor as ades, traje tory-based . . . 126

6 Appli ation to me hani al systems 135 6.1 PID ontrol of robot manipulator s . . . 135

6.1.1 Robustnesswith respe tto external disturban es . . . 137

6.1.2 PID ontrol onsidering a tuators'dynami s with disturban es . . . . 141

6.1.3 Experimental results . . . 144

6.2 Spa e raft formation . . . 146

6.2.1 Problemformulation . . . 147

6.2.2 Measurements available . . . 149

6.2.3 Whenonly bounds areknown . . . 151

6.2.4 Simulation results . . . 153

6.3 Underway shipreplenishment . . . 155

6.3.1 Preliminaries . . . 156

6.3.2 Virtualvehi ledesign . . . 160

6.3.3 Follower vehi ledesign . . . 161

6.3.4 Stabilityanalysisof the overallsystem . . . 162

6.3.5 Simulation study . . . 163

Con lusion and further resear h 169 A Proof of auxiliary results 171 A.1 Proofof ofLemma 2.7 . . . 171

A.2 Proofof Proposition1.16. . . 172

A.3 Proofof Theorem 3.38 . . . 174

A.4 Proofof Corollary 3.40 . . . 176

A.5 Proofof Theorem 3.42 . . . 177

A.6 Proofof Claim 6.4 . . . 178

A.7 Proofof Claim 6.5 . . . 178

Referen es 194

Preamble

Before introdu ing, in amore detailed manner,the subje tof study ofthe present thesis,

we give, in informal terms, a on ise overview of the motivations for this work and its

ontribution s.

Stability. Roughlyspeaking,stabilityisthepropertyofadynami alsystemthatanyerror

signals an be madearbitrarilysmall providedthatthe initialerrorsaresu iently small.

It is a ru ial notion froma ontrol point of view as itensures an a eptable behavior of

the plant if its initial onguratio n is not too far from the nominal one. If, in addition,

theerrorsignalseventuallytendtozero, wesaythatthisoperatingpointisasymptoti ally

stable. The domain ofattra tion onsiststhe set ofall initial states fromwhi h solutions

gotozero. Wetalkabout globalasymptoti stabilitywhenthedomainofattra tionisthe

whole state-spa e.

Obsta les. Through intuitive examples,weexposesome ofthe reasonsthatmayprevent

the error signals from onverging to zero, as, for instan e, the presen e of an external

perturbation, measurement impre ision, fri tion, et . In the same way, we showthat the

domain of attra tion may be restri ted to a ompa t neighborhood of the origin, notably

in the ase of negle ted high order nonlineariti es. In these situations, most of existing

toolsfail at ensuringbetter than ultimate boundedness ( onvergen e ofsolutions to some

neighborhood of the origin)or to lo alstability(restri tion ofthe domain of attra tion).

Semiglobal and pra ti al stability. Su h adegradation of performan e is not a ept-

ableinmany on reteappli ations,asthismayresultinatoolittleoperatingbandwidthor

atoolarge impre ision. Nevertheless, for ontrolledsystems,thedomain ofattra tion an

often be arbitrarily enlarged provided su iently large gains. We refer to this property

as semiglobal asymptoti stability. For a given system, semiglobal asymptoti stability

ensuresmu hmore interesting propertiesthan simplylo al properties, sin eitestablishes

thatnotheoreti alobsta lepreventsfromin ludinganygivennitesetofinitial onditions

to the domain ofattra tion.

Inthesameway, thesteady-stateerrors anoftenbediminishedatwillunderasimilar

tuningofthegains: we allthispropertypra ti alasymptoti stability. Again,this on ept

shouldbeseenasafarstrongerpropertythanthesimpleultimateboundednessofsolutions.

Indeed,pra ti al asymptoti stabilityimposesthatthe pre ision, afterthe transients, an

be made as ne as desired. In addition, as we will see in more details in the sequel, it

suggestsareasonablebehaviorofthetransientdynami swhi hisnotthe ase,ingeneral,

of ultimate boundedness.

Whenitfollowsfromalimitationoftheperforman esofthesystem,semiglobalpra ti-

external perturbations and model un ertainty. But, aswe will see, this stability property

doesnotariseex lusivelyfromadegradationofglobalasymptoti properties,butmayalso

be established byexistingresults in the literature, su h asaveragingte hniques.

A rst goal of the present thesis is to provide a rigorous framework for the study

of semiglobal and/or pra ti al asymptoti stability properties. To this end, we provide

su ient onditions,expressedintermsofLyapunovfun tions,thatguaranteethisstability

properties. Aswewill see, these onditions allowthatthe Lyapunov fun tion depends on

thetuningparameter. Indeed,su ha hoi eallowsto in ludeinouranalysisamu hwider

s ope of appli ations than if the Lyapunov fun tion was assumed uniform in the tuning

parameter. Forinstan e, forme hani alsystems,itis ommon to hoosethe energyofthe

system asa Lyapunov fun tion, in whi h ase the ontrol gains, then playing the role of

the tuning parameter, naturally appear in the Lyapunovfun tion.

Due to this non-uniformity, ompared to lassi al results, an additional assumption is

required that links the upper and lower bounds on the Lyapunov fun tion. We infer the

ne essityof su han additional onditionthrough anexample.

Westress thatthe stability on ept thatwe will usealongthe do ument makesuseof

two measures: the distan e to the ball for whi h we want attra tivity and the Eu lidean

norm. Here also,this hoi e is motivated by simpli ityand generality reasons. Indeed, if

wehad hosentouseonesinglemeasure,the orrespondingLyapunovfun tionwouldthen

have had to vanish on a whole neighborhood of the origin, whi h would have prevented

the use of the Lyapunov fun tion asso iated to the nominal system. On the opposite,

with this hoi e, most of the semiglobal or pra ti al stability properties that result from

a degradation of an ideal system due to perturbations an be inferred by using the

Lyapunovfun tion of the unperturbed system.

Hen e, we propose tools that allow to establish powerful stability properties, i.e.

semiglobaland/orpra ti alstability,whi husuallydonotrequiremu hmore onservative

assumptions than thoseneeded for the (weaker but ertainlymore lassi al) properties of

ultimate boundedness andlo al stability.

We also present a so- alled  onverse theorem for semiglobal pra ti al asymptoti

stability, i.e. a result that guarantees the existen e of a Lyapunov fun tion under the

assumptionofsu hastabilityproperty. Thegeneralityofthe on eptthatweuserequires

spe i pre autions ompared to the properties that would be uniform in the tuning pa-

rameter. We will see that, asthis result generatesan autonomous bound on the gradient

ofthe generated Lyapunovfun tion,itwill be of greathelpin lightening the assumptions

in our results on as ades.

Cas ades. In orderto simplify the study ofa omplex system,it is ommon, in stability

analysis, to divide it into smaller inter onne t ed subsystems. In this way, the di ulty

of the analysisis often redu ed. A parti ular type of su h inter onne tion is the as ade

stru ture. In this situation, the subsystems are inter onne ted in a unilateral way, i.e.

the output of a driving subsystem is the input of a driven subsystem. The modularity

oered by this so- alled as ade approa h gave rise to powerful results, both in analysis

and ontrol design.

However,most ofthe existingresultsin thisdomain onlytreat lo alorglobal stability

of the origin. Hen e, they do not apply to the on epts, although ommon and powerful,

of semiglobaland/orpra ti al stability.

Asa se ondobje tive,we provide su ient onditions under whi hsemiglobal and/or

ing, we show that this is the ase provided that we expli itly knowa Lyapunovfun tion

for the driven subsystem and that the solutions are uniformly bounded. In the ase of

global pra ti al asymptoti stability, we provide a stru tural riterion to ensure this uni-

form boundedness of solutions,therefore yielding an easy-to- he k ondition to guarantee

global pra ti al asymptoti stability of the as ade. An illustration of these results on-

sists in stabilizing, by a bounded output feedba k, the double integrator ae ted by a

persistently ex iting signal. Asanother appli ation, we rigorouslyshowthatsmoothing a

feedba k ontrol lawmayresult in pra ti alstability. Furthermore,weprovide a onverse

Lyapunovresult for semiglobalpra ti al asymptoti stabilitythat permits us to relaxthe

requirement ofexpli itlyknowingaLyapunovfun tion forthe drivensubsystem. Asillus-

tratedbyanexample,this latterfeature isparti ularly usefulwhenstabilityisestablished

basedon averaging te hniques.

Set-stability. Thegeneralityoeredbyset-stabilitymakesit,aswefurtherdevelopupon,

another interesting tool for the stability and robustness analysis of perturbed systems.

Indeed, this notion in ludes, as parti ular ases, the stability of a single operating point,

of a traje tory or even a more omplex domain a ording to the set that is onsidered.

Moreover, asthe latter isnot assumed to be ompa t, itis also possible to in lude to the

study the partialstability, whi h refersto the situation when the behaviorof only a part

of the state is onstrained. We will see that the latter appears very useful when dealing

with adaptive ontrol.

The thirdobje tive ofthis workis to providesu ient onditions forthe preservation

ofthe set-stabilityfor as adedsystems. Therequirement isrstgivenasaglobal bound-

edness of the solutions of the overall as ade. We establish that, in some situations, this

anberelaxedtojustforward ompletene ssprovided agrowthrestri tiononthe inter on-

ne tion term. Asan illustrative appli ation,wepropose aproof for are ently established

resultin marine ontrol.

ISS and iISS. So far, we have dis ussed Lyapunov stability for systems without inputs.

A eld of stability analysis, regrouped under the paradigm of input to state stability

(ISS), isespe ially on erned bythe impa tof external signalson the performan e ofthe

system. Without going into details, this property imposes that the norm of the urrent

statebeboundedbyafun tionoftheamplitudeoftheperturbingsignalplusafadingterm

dependingoninitial onditions. Arelaxedextensionofthispropertyis alledintegralinput

tostatestability(iISS).Insteadoftheamplitudeoftheexternalsignal,thispropertytakes

into a ount the energy that the latter feeds to the system. The iISS property is very

general in stability analysis and provides interesting information about the system. For

instan e,iftheinputenergyisnite,thenthestate onvergestozero. Inthissense,iISS(as

well asISS) therefore onstitutesanother powerfulmeasure of the robustness of asystem

to external perturbations.

A fourth part of this text is devoted to the behavior of iISS systems when pla ed in

as ades. Weprovide elementary onditions underwhi h the as ade omposedofaniISS

systemdrivenbyaglobally asymptoti allystableoneremainsglobally asymptoti allysta-

ble. These onditions areexpressed in termsof theLyapunovfun tionsasso iatedto ea h

subsystem,thus generalizing existingtraje tory-based results. Under mildly onservative

additionalassumptions, weestablishthatthe as ade oftwoiISS subsystemsisitselfiISS.

Thislatterresultisrstlyexpressed in termsofLyapunovfun tions, andthen in termsof

Appli ations. Many of the results presented in this do ument have been applied in

pra ti e, and we exposesome ofthese results in afth step.

We study the robustness of PID- ontrolled robot manipulator s to fri tions, external

disturban es, model un ertainty and taking into a ount the dynami s of a tuators. We

prove that, under these environmental onstraints, the system is semiglobally pra ti ally

asymptoti allystable. Thisis onrmed byexperimentalresults.

In an other domain, we show that the leader-follower strategy adopted for the on-

trol ofspa e raftformations yields globalasymptoti stabilitywhenall measurementsare

available. However,inpra ti e,someinformationontheleader'spositionmaynotbeavail-

able. We showthat, provided that these signalsarebounded, global pra ti al asymptoti

stability an be on luded.

Finally,in the ontextofunderwayshipreplenishment,wherethe ontrolofthesupply

vesselaimsat preservinga onstant distan efromthe mainshipduringtheoperation,the

only measurements available for the main ship areposition and heading. No information

onitsmodelisatdisposal. Underthis onstraints,weshowthatavirtualvehi leapproa h

ensuresglobal pra ti al asymptoti stabilityof the system.

We eventually stress that, although some of the results presented here impose rela-

tively heavy notations for the sake of rigor, the do ument also aims at giving intuitive

explanations of the utilized on epts. In this dire tion, we give several simple examples

to illustrate the purpose and, when possible, provide simplied orollaries that are less

general but easierto usein pra ti e.

Also, even though the results presented along this do ument on ern more stability

analysisthan stabilization,in the sense thatnoexpli itdesign of ontrol lawispresented,

theystill onstituteapres riptiveframeworkonwhi hone ouldbase ontroldesignstrate-

gies,asillustrated bythe on rete appli ationsof Chapter6.

Contribution of this thesis

We briey summarizethe main results of this thesis, hapterby hapter, and ite related

publi ations. Labels orrespond to the listof publi ationspresented in p. 17.

- Chapter2: We present new toolsfor the study of semiglobal and pra ti al stability

ofnonlinear time-varyingsystems. Somesu ient onditions, interms ofLyapunov

fun tions, are proposed. Compared to lassi al Lyapunov onditions, an additional

onditionappears,thattakesintoa ount thenon-uniformityoftheLyapunovfun -

tion in the tuning parameter. We underline the ne essity of su h a requirement

throughanexample. Conversely,weprovethatsu haLyapunovfun tion anbede-

rived providedsu ient regularityofthe right-handside ofthe ordinary dierential

equation.

This hapterformed the subje t ofthe following publi ationswith A. Loría:

[(i), (ii), (iv), (viii), (x),(xiii), (xvii)℄.

- Chapter 3: We extend the results of Chapter 2 to nonlinear time-varying systems

presenting a as ade stru ture. We prove that, under a boundedness ondition on

thesolutions ofthe overallsystem,bothsemiglobalandpra ti alstabilityproperties

arepreserved bythe as ade inter onne t ion of two subsystems. We also givesome

su ient onditionstoensurethe boundedness onditiononthesolutions,whi hare

parti ularly easytouseinthe aseofglobalpra ti al stability. Illustrativeexamples

areprovided in ea h ontext.

Theseresults wereoriginally presented in the following publi ationswith A.Loría:

[(i), (ii), (iv), (viii), (x),(xiii), (xv), (xvii)℄.

- Chapter4: Weanalyzethebehaviorofnonlinearsystemsthataregloballyasymptot-

i allystablewith respe ttoa(nonne essarily ompa t)set,whenpla edin as ade.

We provide su ient onditions under whi h set-stability, dened with respe t to

two measures,ispreserved bythe as ade inter onne tion.

Theseworks orrespond to the ollaboration [(xiv)℄ with E. Panteley. An extension

wasproposedwith the same oauthor, J.Tsønnas and T.A.Johansen in [(xix)℄.

- Chapter5: Westudythe preservationoftheintegralinputtostatestabilityproperty

of nonlinear time-invariant systems in as ade. We give some su ient onditions

for the as ade omposed of an iISS subsystem driven bya globally asymptoti ally

stable(GAS) subsystemto be GAS.These onditions are expressed in termsof the

traje tory-based results. We also provide onditions under whi h two iISS systems

pla ed in as ade remain iISS. These su ient onditions are rst expressed based

on the Lyapunovfun tion forea h of the twosubsystems, andthen on the estimate

oftheir solutions.

Theseresults wereprepared with D.Angeli in [(iii), (xii)℄.

- Chapter6: We present on rete appli ationsof our maintheoreti al ndings insta-

bilizationproblems ofme hani al systems. Weshowthat, whentakinginto a ount

external perturbation s (su h as fri tion, torque ripping, et . ) and the dynami s of

the a tuator, PID- ontrolled robot manipulator s aresemiglobally pra ti ally stable,

ba ked upwith experimentalresults.

Onthe otherhand, a ontrol for underwayfuel replenishment ofvesselsis designed,

using a virtual ship approa h, whi h requires neither a priori model knowledge nor

velo ity measurement for the ship to be replenished. Global pra ti al asymptoti

stabilityis obtained.

Athird appli ation on erns the ontrol ofa spa e raftformation, whentakinginto

a ountboundedexternaldisturban es. A ordingtotheassumedlevelofknowledge

we have on the orbital parameters of the leader, various stability properties are

derived.

Theseappli ations werethe obje tof the following joint publi ationswith R.Kelly,

E.Kyrkjebø,R.Kristiansen,A.Loría,E.Panteley,K.PettersenandP.J.Ni klasson:

[(iv), (vi) , (vii), (xvi), (xviii)℄.

Although not presented in this do ument, these three years of PhD gave rise to other

fruitful ollaborations:

- The publi ation [(ix)℄ is a joint work with J. de León Morales, A. Loría and G.

Besançon where we proposed an adaptive observer for systems that an be put in

the so- alled output feedba k form, based on a onvenient persisten y of ex itation

property.

- With A. Loría, G. Besançon and Y. Chitour, we have posed open problems for

stabilization of persistently ex ited systems, and partially solve them in the ase of

the doubleintegrator, f. [(xi)℄.

- In[(xx)℄,with M. Sigalotti,P. Mason, Y.Chitour and A.Loría, thislatter problem

was further extended and solved with a linear time invariant feedba k, with gains

uniform in the persistently ex iting signal.

Publi ations

Thefollowingisanexhaustivelistofpubli ationswrittenduringthepastthreeyears,that

areeither published, a epted for publi ation,or stillunder review. It ontains but isnot

restri tedto the ontents ofthis do ument.

Journal Papers

i/ A.ChailletandA.Loría,A onversetheoremforuniformsemiglobalpra ti al asymp-

toti stability: appli ation to as aded systems,A epted for Automati a,2006.

ii/ A.Chaillet and A. Loría,Uniform semiglobal pra ti al asymptoti stability for non-

autonomous as aded systems and appli ations, A eptedfor Autmati a, 2006.

iii/ A.ChailletandD.Angeli,Integralinputtostatestablesystemsin as ade,Submitted

to SystemsandControl Letters, 2005.

iv/ A. Chaillet and A. Loría, Uniform global pra ti al asymptoti stability for non-

autonomous as aded systems, Submitted to European Journal of Control, 2006.

A.Chaillet, A. Loríaand R.Kelly, Robustness of PID- ontrolled manipulators vis a

vis a tuator dynami sand external disturban es,Submitted to European Journal of

Control, 2006.

Book hapters

vi/ R.Kristiansen, A.Loría, A. Chaillet, and P. J. Ni klasson, Output feedba k ontrol

of relative translation in a leader-follower spa e raft formation, In Pro eedin gs of

the Workshop on Group Coordinatio n and Cooperative Control, Le ture Notes in

Control and InformationS ien es, Springer Verlag,Tromsø,Norway, 2006.

vii/ E.Kyrkjebø,E.Panteley,A.Chaillet,andK.Pettersen,Avirtualvehi leapproa hto

underwayreplenishment ,InPro eedin gsoftheWorkshoponGroupCoordinatio nand

Cooperative Control, Le ture Notes in Control and Information S ien es, Springer

Verlag,Tromsø, Norway, 2006.

viii/ A.Chaillet, Tools for semiglobal pra ti al stability analysis of as aded systems and

appli ations, InPro eedin gs oftheCTSHYCONWorkshop,International S ienti

and Te hni al En y lopedia, Paris,Fran e, July 2006.

Conferen e papers

ix/ J.de LeónMorales, A.Chaillet, A.Loría,andG. Besançon,Output feedba k ontrol

viaadaptiveobserverswithpersisten yofex itation,InIFACWorldCongress,Praha,

T he kRepubli , July 2005.

x/ A.Chaillet and A. Loría,Uniform semiglobal pra ti al asymptoti stability for non-

linear time-varying systems in as ade, In IFAC World Congress, Praha, T he k

Republi ,July 2005.

xi/ A. Loría, A. Chaillet, G. Besançon, and Y. Chitour, On the PE stabilization of

time-varying systems: openquestions andpreliminary answers, InCDC2005, pages

68476852, Sevilla, Spain,De ember 2005.

xii/ A. Chaillet and D. Angeli, Integral input to state stability for as aded systems, In

MTNS2006, Kyoto,Japan, July 2006.

xiii/ A.Chailletand A.Loría,Uniformglobal pra ti al stability for non-autonomous as-

aded systems, InMTNS2006, Kyoto,Japan, July 2006.

xiv/ A. Chaillet and E. Panteley, Stability of sets for nonlinear systems in as ade, In

MTNS2006, Kyoto,Japan, July 2006.

xv/ A.Chaillet, Estimée du domained'attra tionpourla as ade desystèmes uniformé-

ment asymptotiquement stables, In CIFA 2006,Bordeaux, Fran e, June 2006.

xvi/ A.Chaillet, A.Loría, andR.Kelly, Robustness of PID ontrolled manipulators with

respe t toexternal disturban es,InCDC 2006,San Diego,USA, De ember 2006.

xvii/ A. Chaillet and A. Loría, A onverse Lyapunov theorem for semiglobal pra ti al

asymptoti stability and appli ation to as ades-based ontrol, In CDC 2006, San

Diego,USA, De ember 2006.

xviii/ R.Kristiansen,A.Loría,A.Chaillet, and P. J.Ni klasson,Adaptive outputfeedba k

ontrol of spa e raft relative translation, In CDC 2006, San Diego, USA, De ember

2006.

xix/ J. Tsønnas, A. Chaillet, E. Panteley, and T. A.Johansen, Cas aded lemma for set-

stable systems, Submitted toCDC 2006,San Diego,USA, De ember 2006.

Other reports

xx/ M.Sigalotti, A.Chaillet,P. Mason,Y.Chitour, andA.Loría,Lineartime-invariant

stabilizationwith persisten y of ex itation, Internal report, 2006.

xxi/ A.Chaillet and A. Loría,Uniform semiglobal pra ti al asymptoti stability for non-

autonomous as aded systems and appli ations, 2006,

http://arxiv.org/PS_ a he/math/pdf/0 503/0503039.pdf.

Notation

All properties su h as positive, greater, in reasing, et . are to be understood in the

stri t sense.

N

and

R

denote the sets ofall nonnegative integers and allreal numbers respe tively.

N _≤N

ontains all the nonnegative integers lessthan or equal to

N ∈ N

^{. In the same way,}

R _≥0

is omposedof allnonnegative realnumbers.

I

^{denotes the identitymatrix of} appropriatedimension.

A ontinuous fun tion

α : R _≥0 → R ≥0

^{is of lass}

K

⁽

α ∈ K

^{), if it is in reasing and}

α(0) = 0

^{. It is said to belong to lass}

K ∞

^{if, in addition,}

α(s) → ∞

^as

s → ∞

^{. A}

ontinuous fun tion

σ : R _≥0 → R ≥0

^{is of lass}

L

⁽

σ ∈ L

^{) if it is de reasing and tends to}

zero as its argument tends to innity. A fun tion

β : R _≥0 × R ≥0 → R ≥0

^{is said to be a}

lass

KL

^{fun tion if}

β( ·, t) ∈ K

^{for any}

t ∈ R ≥0

^{, and}

β(s, ·) ∈ L

^{for any}

s ∈ R ≥0

^.

We denote by

φ( ·, t 0 , x ₀ )

^{the solutions of the} dierential equation

˙x = f (t, x)

^with

initial ondition

φ(t ₀ , t ₀ , x ₀ ) = x ₀

^.

We use

|·|

^{for the Eu lidean norm of ve tors and the indu ed}

L 2

^{norm of matri es.}

We use

k · k

^{for the essential supremum norm, i.e., for a signal}

u : R _≥0 → R ^p

^,

kuk :=

ess

sup _t≥0 |u(t)|

^.

We denote by

B δ

^{the losed ball in}

R ⁿ

of radius

δ

^{entered at the origin, i.e.}

B δ :=

{x ∈ R ⁿ : |x| ≤ δ}

^{. We use the notation}

H(δ, ∆) := {x ∈ R ⁿ : δ ≤ |x| ≤ ∆}

^{. By an}

abuseof notation,

B 0 = H(0, 0) = {0}

^and

B ∞ = H(0, ∞) = R ⁿ

^.

δ

^{being a}nonnegative onstant,we dene

|x| _δ :=

^inf

_{z∈B δ} |x − z|

^{. Moregenerally, for a}

losed set

A

^,

|·| _A

^{representsthe distan eto this set:}

|x| _A :=

^inf

_z∈A |x − z|

^.

For agiven set

E

^of

R ⁿ

,

E ◦

^{denotes itsinterior.}

Let

a ∈ {0, +∞}

^and

q ₁

^and

q ₂

^{be lass}

K

^{fun tions. Wesaythat}

q ₂ (s) = O(q 1 (s))

^as

s

^tendsto

a

^{ifthereexistsa}nonnegative onstant

k

^{su h that}

lim sup _s→a q ₂ (s)/q ₁ (s) ≤ k

^.

We say that

q ₂ (s) = o(q ₁ (s))

^{) if}

k

^{an be taken to be zero, and that}

q ₁ (s) ∼ q 2 (s)

^if

lim _s→a q ₂ (s)/q ₁ (s) = 1

^.

We say that

f : R _≥0 × R ⁿ → R ⁿ

^{satises the} Carathéodory onditions if

f ( ·, x)

^is

measurable forea h xed

x ∈ R ⁿ

^,

f (t, ·)

^{is ontinuousfor ea hxed}

t ∈ R ≥0

^{and,for ea h}

ompa t

U

^of

R _≥0 × R ⁿ

^{, there exists a integrable fun tion}

m _U : R _≥0 → R ≥0

^{su h that}

|f(t, x)| ≤ m U (t)

^{for allall}

(t, x) ∈ U

^.

Afun tion

f : R ⁿ → R ⁿ

^{issaidtobelo allyLips hitz if,forany ompa t}

U

^of

R ⁿ

,there existsanonnegative onstant

k _U

^{su h that}

|f(x) − f(y)| ≤ k U |x − y|

^{for all}

(x, y) ∈ U ²

^.

When the ontext issu iently expli it,wemayomit the arguments ofa fun tion.

Chapter 1

Denitions

Lyapunov stability. The works presented in this do ument appeal to many dierent

typesof stabilityproperties. Stabilityshould be understood in the Lyapunovsense. Gen-

erally speaking,itrefersto thepropertyofa point,a setor atraje torythatanysolution

starting su iently nearremains arbitrarily lose at all time. It onstitutes a ru ial fea-

ture in ontrol of dynami al systems, as it ensures an a eptable behavior of the plant

provided thatits initial onditions aresu iently lose to the nominal ones.

The notion of stability may easily be grasped in the ontext of me hani al systems.

Considering a ball on a non at surfa e, an equilibrium position is stable if, after any

su iently small perturbations on the position of the ball, itremains for ever arbitrarily

near to it. The equilibrium is said to be asymptoti ally stable if, in addition, the ball

approa hes itasymptoti ally. This isillustrated bythe drawings ofFigure 1.1.

Instability Asymptotic

stability

Global

asymptotic

stability

Figure1.1: Illustrationof dierent typesof stability.

In some situations, it is interesting to know how far from the asymptoti ally stable

equilibrium the ball an start and nally return to it. The region of the state spa e

that leadasymptoti onvergen e isreferred to asdomain of attra tion. Ifthe domain of

attra tion is the whole spa e, then the equilibrium under onsideration is alled globally

asymptoti allystable.

While very intuitive in the ontext of me hani al systems, Lyapunov stability is far

from being onned to this area. Generally speaking, the systems onsidered throughout

the do ument arerepresented asanite dimensionaldierential equationof the form

˙x = f (t, x) ,

^(1.1)

where

t ∈ R ≥0

^{representsthetime,}

x ∈ R ⁿ

^{isthe stateand}

f : R _≥0 × R ⁿ → R ⁿ

^isassumed

to satisfy Carathéodor y onditions ( f. p. 19) and to be lo ally Lips hitz in

x

^{. More}

pre isely, for ea h ompa t

U

^of

R _≥0 × R ⁿ

^{, we assume that there exists an integrable}

fun tion

k U : R _≥0 → R ≥0

^{su h that,for all}

(t, x) ∈ U

^andall

(t, y) ∈ U

^,

|f(t, x) − f(t, y)| ≤ k U (t) |x − y| .

By virtue of [Hal69, Theorem 5.3℄, these ombined onditions ensure both existen e and

uniqueness of the solutions of(1.1).

It is worth pointing out the wide varietyof systemsthat an be des ribed by su h an

equation. To ite a few, it overs a very large number of ontrol problems in me hani s,

ele tri alsystems,biology, ele troni s,ele tro-mag net i s,et . , f. e.g. [OLNSR98,KSV91,

Son05b℄.

Why time-varyingsystems ? Thefa tthatthe right-hand sideterm ofthe onsidered

dierential equation is time-dependent allows to in lude in the study many problems of

traje torytra king. Thisaims at designinga ontrol

u

^{in su h awaythat the solution of}

the the dynami al system

˙x = f (x, u)

^follows asymptoti ally apres ribed referen e

x _d (t)

^.

As the adopted ontrol law depends on the time-varying referen e traje tory

x _d (t)

^{, the}

systemin losed-loop,although originally time-invariant,is ofthe form

˙˜x = g(t, ˜x)

^{, where}

˜

x := x − x d

^{. Thesostated tra king ontrol problemapplies to manyphysi al systems,as}

for instan e in the area of ontrol of me hani al and ele trome hani al systems ( f. e.g.

[OLNSR98℄ andreferen es therein).

Another typi al situation in whi h expli it time-dependen e of the dynami al system

o ursis thatof regulationsproblems (thatis, stabilizationof xedoperating point)that

donotsatisfyBro kett's ondition[Bro83℄orthemore onservative onditionpresentedby

Coron in[Cor90℄. Inthis ase, the open-loop plant isnot stabilizable byany ontinuously

dierentiable time-invariant feedba k. A time-varying ontroller is then on eivable. For

instan e,itwasshownbyCoronin[Cor92℄thatany ompletely ontrollablesmoothsystem

without drift(in luding nonholonom i me hani al systems) an be stabilizedbymeans of

a smooth periodi time-varying state feedba k. Also, it was shown in [KT03℄ that, if a

system an be stabilized by a ontinuous state-feedba k, then itis stabilizable (although

possibly in anon-uniform way)byasmooth time-varyingfeedba k,whi h may onstitute

an interesting featurefor some appli ations.

However, this expli it dependen e in time of (1.1) an also be of interest from an

analysispoint ofview. Some te hniquesin the literature, seefor instan e[Kha96, Lor04℄,

onsistin simplifyinga omplexnonlinear systeminto amore simpletime-varying one by

onsidering partof the state asasimple fun tion oftime.

Although the results presented along this do ument on ern more stability analysis

thanstabilization, inthesensethatnoexpli itdesign of ontrollawispresented, theystill

onstitute a pres riptiveframeworkon whi h one ould base ontrol design strategies.

1.1 Stability of the origin

Many ontrol appli ations an be formulated asa stabilization problemof the origin of a

dynami al system. Typi ally, one requires thatthe error between the desired behavior of

the system and the a tual one onverges to zero, leading to the notion of attra tivity of

the dieren e between the desired behavior and the a tual one remains arbitrarily small

at alltime: in otherwords, stability ofthe origin is desired.

We briey re all these notions in a nonlinear time-varying ontext. In this se tion,

I

denotea losed(butnotne essarilybounded)subsetof

R ⁿ

that ontainstheorigin. Please referto the Notationpart(p. 19) fora denitionofthe mathemati al on epts usedhere.

Westart with the notion of uniform boundedness of solutions.

Denition 1.1 (UB/UGB) Let

I

^{be a losed subset of}

R ⁿ

. The solutions of (1.1) are said to be uniformlybounded on

I

^{if, for any} nonnegative onstant

r

^{, there existsa non-}

negative

c(r)

^{su h that, for all}

t 0 ∈ R ≥0

^{, they satisfy}

x ₀ ∈ I ∩ B r ⇒ |φ(t, t 0 , x ₀ ) | ≤ c , ∀t ≥ t 0 .

If

I = R ⁿ

^{, then the solutions are uniformlyglobally bounded .}

Based on this, we an introdu e a pre ise denition of the stability on ept thatwill

be usedthroughout the do ument.

Denition 1.2 (US/UGS) Let

I

^{be a losed subset of}

R ⁿ

. The origin of (1.1) is said to be uniformly stable on

I

^{if its solutions are uniformly bounded on}

I

^{and, given any}

positive onstant

ε

^{, there exists a positive}

δ(ε)

^{su h that, for all}

t ₀ ∈ R ≥0

^{, the solution of}

(1.1) satises

|x 0 | ≤ δ ⇒ |φ(t, t 0 , x ₀ ) | ≤ ε , ∀t ≥ t 0 .

^(1.2)

If

I = R ⁿ

^{, then the origin is uniformly globally stable.}

Stri tly speaking,stabilityofthe origin is apurely lo al on ept whi his summarized

by(1.2). Inmanyappli ations,itisalsointerestingtoknowadomainfromwhi hsolutions

remainbounded,whi h explains why aboundedness requirement is imposed in the above

denition.

Next, we re allthe notion ofattra tivity ofthe origin.

Denition 1.3 (UA/UGA) Let

I

^{be a losed subset of}

R ⁿ

. The origin of (1.1) is said to be uniformlyattra tive on

I

^{if, for all positive numbers}

r

^and

ε

^{, there exists a positive}

time

T (r, ε)

^{su h that, for all}

x ₀ ∈ B r ∩ I

^andall

t ₀ ∈ R ≥0

^{, the solutionof (1.1)satises}

|φ(t, t 0 , x 0 ) | < ε , ∀t ≥ t 0 + T .

If

I = R ⁿ

^{, then the origin is uniformly globally} attra tive.

When the two latterproperties are ombined, the resulting propertyis alled uniform

asymptoti stability.

Denition 1.4 (UAS/UGAS) Theoriginof(1.1)issaidtobe uniformlygloballyasymp-

toti allystableon

I

^{ifitisbothuniformlystableanduniformlyattra tive on}

I

^{. If}

I = R ⁿ

^,

then the origin is uniformlyglobally asymptoti allystable.

The uniformity requirement in the above denitions refers to the initial time. It

orrespondstothe independen e of

δ

^and

T

ⁱⁿ

t ₀

^{. Inotherwords: nomatteratwhattime}

the system'straje tories start, onvergen e-ra te to zeroandovershoot remainun hanged.

The importan e of uniformity. Uniformity is a ru ial property of time-varying sys-

tems, as it provides a ertain robustness with respe t to external disturban es. More

pre isely, as more detailed in [LLLP 05, Se tion 2.1℄, it an be shown that the uniform

asymptoti stability of the origin of a zero-input system

˙x = f (t, x, 0)

^{ensures stability}

with respe t to onstantly a ting disturban es of

˙x = f (t, x, u)

^{, where}

u

^{denotes an ex-}

ternal signal 1

, provided that the

f (t, x, u)

^{is lo ally Lips hitz in}

x

^{uniformly in}

t

^{. This}

on ept, alsoknownastotalstability,wasintrodu edbyMalkin, f. e.g. [Mal58℄. Itstates

that the traje tories remain arbitrarily small at all time ifthe initial state and the input

signalaresu iently small. Morepre isely, it isdened asfollows.

Denition 1.5 (Total stability) The origin of

˙x = f (t, x, 0)

^{is said tobe totally stable}

if, for ea h

ε > 0

^{, there exists}

δ(ε) > 0

^{su h that, for all}

t 0 ∈ R ≥0

^{, the solution of}

˙x = f (t, x, u)

^satises

max {|x 0 | , kuk} ≤ δ ⇒ |φ(t, t 0 , x ₀ , u) | ≤ ε , ∀t ≥ t 0 .

In a nutshell, byestablishing uniform asymptoti stability, we guarantee thatthe be-

haviorofthe systemis not toomu h alteredbythe presen eofsu iently smallexternal

disturban es. This robustness property does not hold for non-uniform properties, as il-

lustrated by [LLLP 05, Example 2.1, p. 28℄ in whi h a simple s alar time-varying system

is exhibited with the following properties:

˙x = f (t, x, 0)

^isglobally asymptoti ally stable (but not uniformly), nevertheless one an design an arbitrarily small perturbation

u

ⁱⁿ

su h awaythat

˙x = f (t, x, u)

^{generatesunbounded solutions.}

Larger perturbations. While uniform asymptoti stability thus ensures a natural ro-

bustnessto small externaldisturban es, itprovidesno information onthe behaviorofthe

systemsubje t to larger perturbations. Instabilityanalysis, itis lassi al to observe that

the presen e ofa bounded non-vanishing disturban e impedes asymptoti stability, yield-

inginsteadthe onvergen etoa(possiblylarge)neighborhoodoftheoperatingpoint. This

propertyis referredto asultimateboundedness, f. e.g. [Kha01, Yos66℄.

Denition 1.6 (Ultimate boundedness) The solutions of

˙x = f (t, x)

^{are said to be}

uniformly ultimately bounded if there exist positive onstants

∆ ₀

^and

c

^{su h that, for}

every

∆ ∈ (0; ∆ 0 )

^{, there exists apositive onstant}

T (∆)

^{su h that, for all}

x ₀ ∈ B ∆

^andall

t 0 ∈ R ≥0

^{, theysatisfy}

|φ(t, t 0 , x ₀ ) | ≤ c , ∀t ≥ t 0 + T .

If this holds for arbitrarily large

∆

^{, then the solutions are globally uniformly ultimately}

bounded .

Inmanysituations,thispropertyisnotenoughtoensure orre tperforman es. Indeed,

weseethatuniformultimateboundednessisonly on ernedwiththebehaviorofthesystem

afterasu ientlylongtimeand,hen e,doesnottakeintoa ountthetransientdynami s.

Inaddition,the domain towhi hsolutions onvergemaybe large,then preventingagood

pre ision.

The aim of the following se tions is to introdu e stability properties that may help

guaranteeing stronger featuresto perturbed systems.

1

u : R ≥0 → R ^m

^{may onsistinanymeasurablelo ally}essentiallyboundedfun tion.

1.2 Stability of sets

In the above denition of ultimate boundedness, solutions are required to onverge to

someball,ofradius

c

^{, enteredat theorigin. Itisnaturalto extend thispropertyto more}

generalsets. Inaddition,itisinterestingto onstrainthebehaviorofthesystemduringthe

transients, in orderto avoiddisproportioned overshoots. Thismotivatedthe introdu tion

of set-stability[Zub57, HP73℄.

A general on ept. The analysis of set-stability is very general and onsequently very

ommon in ontrol pra ti e. This ensues from the fa t that the set under onsideration

may onsistinasingleoperatingpoint(then orrespondingtoDenitions1.2,1.3and1.4),

a path,or a more omplex,possiblyunbounded, regionof the state-spa e.

As it will appear more learly in the following denitions, stability (and, similarly,

attra tivity)of asingleoperating point

x ^∗

^isobtainedby onsideringthe set

{x ^∗ }

^{. Inthis}

respe t, we always onsiderthatthereferen epoint

x ^∗

^{isthe origin. This an beassumed}

without lossof generality, sin e, if

x ^∗

^{is an} equilibriumfor (1.1),then

0

^{is an} equilibrium for

˙z = g(t, z) := f (t, z + x ^∗ )

^{with the oordinate hange}

z := x − x ^∗

^.

In the same way, stabilityof a path maybe onsidered by hoosing the set ontaining

all the points ofthis path.

Inthe asewhenthe appli ationdoesnotrequire onvergen eto theorigin but justto

a small neighborhood of it, it is appropriate to onsider the set as a ball of small radius

entered at zero. This allows to dene a rigorousformulation ofthe problems for whi h a

steady-state error is tolerated, and isalso at the basisof pra ti al stabilityas we will see

in the next se tion.

The set may also be de omposed as

R ^{n ′} × {0}

^{, with}

n ^′ ∈ N <n

^{, whenonly partof the}

state is required to be stable. We refer to this property as partial stability, f. [Vor98℄.

Many appli ations indeed require the onvergen e of a redu ed number of variables to

operate orre tly. This on epthasalsoproved usefulin presen eof superuous states,or

whenthe plantisinherently unstablewith respe ttopartofthe states. SeeChapter4for

details.

Thefollowing stabilitydenitionsshouldthereforebe seenasgeneral statements, from

whi h allthese parti ular ases maybederived.

When dealing with set stability, spe ial attention has to be paid to the existen e of

solutions forallpositivetime.

A

^and

I

^{denoting two losed (butnot} ne essarilybounded) sets of

R ⁿ

that ontain the origin

2

, we therefore start by re alling the notion of forward

ompleteness. Please see[AS99℄for a Lyapunov hara terizat ion of thisproperty.

Denition 1.7 (Forward ompleteness) The system (1.1) is said to be forward om-

plete on

I

^{if, for all}

x ₀ ∈ I

^{and all}

t ₀ ∈ R ≥0

^{, its solution}

φ(t, t ₀ , x ₀ )

^{est dénie pour tout}

t ≥ t 0

^.

Based onthis,we anextendDenition 1.1to the asewhenwearenotinterestedin a

boundedness of the distan eof thesolutions from the origin,but froma given losed (not

ne essarily ompa t) set

A

^.

2

This assumption, whi h an be made withoutloss of generality, is imposed inorder to ensurethat

|·| _A ≤ |·|

^.

Denition 1.8 (UB/UGB with respe t toa set) The solutions of (1.1) are said to

be uniformly bounded on

I

^{with respe t to}

A

^{if (1.1) is forward omplete on}

I

^{and, for}

any nonnegative onstant

r

^{, there exists a} nonnegative

c(r)

^{su h that, for all}

t ₀ ∈ R ≥0

^,

they satisfy

x ₀ ∈ I ∩ B r ⇒ |φ(t, t 0 , x ₀ ) | _A ≤ c , ∀t ≥ t 0 .

If

I = R ⁿ

^{, then the solutions are uniformly globally bounded with respe t to}

A

^{. Fur-}

thermore, for the ase that

A = {0}

^and

I = R ⁿ

^{we simply say, with a slight abuse of}

terminology, that the solutions of (1.1)are uniformlyglobally bounded .

Inthe asewhen

A = {0}

^{,were overuniform} boundednessasintrodu edin Denition 1.1. We see that an additional requirement, namely forward ompletene ss, is imposed in

the above denition. As the set

A

^{may be unbounded,} traje tories mayexplode in nite time while the quantity

|φ(t, t 0 , x ₀ ) | _A

^{remains bounded at all time. Assuming forward}

ompleteness ex ludes this possibility. It should be stressed that, in the ase when

A

^{is a}

ompa t set, this additional requirement is not needed anymore. These remarks hold as

well for the nextthree denitions.

Denition 1.9 (US/UGS of a set) Assume that (1.1) is forward omplete on

I

^{. The}

set

A

^{issaidtobe uniformlystableon}

I

^{for(1.1)ifthesolutionsofthelatterare uniformly}

bounded on

I

^{with respe t to}

A

^{and, given any positive onstant}

ε

^{, there exists a positive}

δ(ε)

^{su h that, for all}

t ₀ ∈ R ≥0

^{, the solutionof (1.1)satises}

|x 0 | ≤ δ ⇒ |φ(t, t 0 , x ₀ ) | _A ≤ ε , ∀t ≥ t 0 .

If

I = R ⁿ

^{, then the set}

A

^{is uniformlyglobally stable.}

Denition 1.10 (UA/UGA of a set) Assumethat(1.1)isforward ompleteon

I

^{. The}

set

A

^{is said to be uniformly attra tive on}

I

^{for (1.1) if, for all positive numbers}

r

^and

ε

^{, there exists a positive time}

T (r, ε)

^{su h that, for all}

x ₀ ∈ B r ∩ I

^{and all}

t ₀ ∈ R ≥0

^{, the}

solutionof (1.1)satises

|φ(t, t 0 , x 0 ) | _A < ε , ∀t ≥ t 0 + T .

If

I = R ⁿ

^{, then the set}

A

^{is uniformlyglobally} attra tive.

Denition 1.11 (UAS/UGAS of a set) Assume that (1.1) is forward omplete on

I

^.

The set

A

^{issaid tobe uniformlyglobally}asymptoti allystableon

I

^{for (1.1)ifit isboth}

uniformly stable and uniformly attra tive on

I

^{. If}

I = R ⁿ

^{, then the set}

A

^{is Uniformly}

Globally Asymptoti allyStable .

Twomeasures. Itisworthpointingoutthatthesedenitions arespe ial asesofstability

withrespe ttotwo measures, f. [Mov60,LL93℄. This on eptisverygeneral andin ludes,

aswe have seen,stabilityofa singlepoint,of a ompa t set, ofa pres ribed traje toryas

well as partial stability [Vor98, Vor02℄. It was used in e.g. [LS76, TP00, Lee04℄. Here,

the rst measure is the distan e to the set under onsideration

|·| _A

^{, while the se ond is}

the Eu lidean norm

|·|

^{. As we will see later (see Se tion 2.1), for perturbed systems or}

whendealingwithadaptive ontrol,this hoi eallows,in manysituations,tousethesame

Lyapunovfun tionasthenominalsystem,whi hmakesthisstabilitypropertymu heasier

In thisrespe t, we stressthatthe term uniform usedin the above denitions on erns

only the dependen e in the initial time. More pre isely, the onstants

c

^,

δ

^and

T

ⁱⁿ

Denitions1.8,1.9and1.10areallrequiredtobeindependentof

t ₀

^{. Otherexistingresults}

in the literature( e.g. [Yos66,LSW96,TPL02℄)usethisterminology to underlinethatthe

set-stabilityisdenedwiththesamemeasure,notablyimplyingthattheset

A

^ispositively

invariant, whi his not the asehere.

As in the spirit of Hahn's formulations [Hah63℄ of stability in terms of

K

^and

KL

estimates (seealso [Son98a℄), the properties dened above an be written in the following

pre iseway.

Proposition 1.12 (

K

hara terization of UB/UGB) Assumethat(1.1)isforward om- plete on

I

^{. The solutions of (1.1) are uniformly bounded on}

I

^{(resp. uniformly globally}

bounded)withrespe tto

A

^{ifandonlyifthereexistsa lass}

K

^{fun tion}

η

^andanonnegative onstant

µ

^{su h that, for any}

x ₀ ∈ I

^(resp.

x ₀ ∈ R ⁿ

^{) and any}

t ₀ ∈ R ≥0

^{, the solution of}

(1.1) satises

|φ(t, t 0 , x ₀ ) | _A ≤ η(|x 0 |) + µ , ∀t ≥ t 0 .

Proposition 1.13 (

K

hara terization of US/UGS) Assumethat(1.1)isforward om- plete on

I

^{. A losed set}

A

^{is uniformly stable on}

I

^{(resp. uniformly globally stable) for}

(1.1) if and only if there exists a lass

K

^{fun tion}

γ

^{su h that, for any}

x 0 ∈ I

^(resp.

x ₀ ∈ R ⁿ

^{)and any}

t ₀ ∈ R _≥0

^{,the solution of (1.1) satises}

|φ(t, t 0 , x ₀ ) | _A ≤ γ(|x 0 |) , ∀t ≥ t 0 .

Proposition 1.14 (

KL

hara terization of UAS/UGAS) Assume that (1.1) is for- ward omplete on

I

^{. A losed set}

A

^{is uniformly} asymptoti ally stable on

I

^{(resp. uni-}

formlyglobally asymptoti ally stable)if andonlyifthere existsa lass

KL

^{fun tion}

β

^{su h}

that, for all

x 0 ∈ I

^(resp.

x 0 ∈ R ⁿ

^{)and all}

t 0 ∈ R ≥0

^{, the solutionof (1.1) satises}

|φ(t, t 0 , x ₀ ) | _A ≤ β(|x 0 | , t − t 0 ) , ∀t ≥ t 0 .

The proof ofthese hara terizations follows along the same linesas[Vid93, Theorems

53 and61℄, we thereforedo not re allthem here.

Whenthe onvergen eratetotheset

A

^isexponentialandthedependen eintheinitial state islinear, the stabilityis saidto beexponential.

Denition 1.15 (UES/UGES of a set) If, in Proposition1.14 , the lass

KL

^{fun tion}

an be pi ked as

β(s, t) = k ₁ se ^{−k 2 t} , ∀s, t ∈ R ≥0

forsome positive onstants

k ₁

^and

k ₂

^{, thenthe set}

A

^{issaid tobe uniformly}exponentially stableon

I

^{(resp. uniformlyglobally} exponentially stable) withparameters

(k ₁ , k ₂ )

^.

Forthestudy ofthealteration ofastabilitypropertyunder theinuen edisturban es,

anoteworthyparti ular aseof the above denitionsis whenthe sets under onsideration

are losed balls. It is indeed at the basisof all the denitions ofsemiglobal and pra ti al

stability properties introdu ed next. The following proposition, thatfollows from Propo-

sitions 1.12 and 1.14, establishes a strong linkof this on ept with the

(σ → ρ)

^-stability

Proposition 1.16 (UAS and

σ → ρ

^{stability) Let}

∆ > δ > 0

^{. Then the following}

impli ations hold:

- If

B δ

^{is UAS on}

B ∆

^{, then (1.1) is}

(∆ → δ)

^-stable;

- If (1.1) is

(∆ → δ)

^{-stable, then}

B δ

^{is UAS on}

B ∆ ^′

^{, for all}

∆ ^′ ∈ (δ, ∆)

^.

Theproofofthispropositionisdetailedin Se tionA.2 . We annoti e thatno forward

ompleteness assumption is needed anymore as the set under onsideration, namely

B δ

^,

is ompa t. In this ase, uniform asymptoti stability naturally ensures the existen e of

solutions for allforward time.

1.3 Semiglobal and pra ti al asymptoti stability

The need of a ner analysis. As already pointed out by Hahn in [Hah63℄ and byLa

Salle and Lefs hetz in [SL61℄, pra ti al onsiderations should be taken into a ount when

studyingtheasymptoti stabilityoftheequilibriumofagivenplant. Toquoteanexample

of thelatter referen e, the asymptoti stability ofan ele tri alsystemoperating at 110

V

ensuresthatsmallvariationswill be an elledout. However,iftheamplitude ofthesetol-

eratedvariationsistoolsmall,sayofsomemillivolts,thesystemmaynotoperate orre tly.

Onthe opposite, the operating point of a given system maybe mathemati ally unstable,

thusgeneratingsmallos illationsaround it,butstillguaranteeasu ientpre isionfor an

a eptable behavior. Using the intuitive illustration, already used in Figure 1.1, of a ball

on anon-at surfa e,these would orrespond to the following situations:

Asymptotic stability with a

small domain of attraction

Instability with a small

steady-state error

Figure1.2: Pra ti al onsiderations about stability.

A tighter analysisisthen apital.

Steady-state errors and restri ted domain of attra tion. As already noted, non-

vanishingperturbations a ting onthe plant ormeasurement impre isionsmayimpede the

onvergen e to theorigin byyielding asteady-state error. Inthe same way, itisoftenthe

ase thatsome negle ted high-order nonlinearityin the dynami s prevent global stability,

generating instead an unbounded basin of attra tion. In ea h of these situations, an we

In the stability analysis of losed-loop systems, but also in some ontexts that are

developed later (su h as averaging te hniques or output feedba k ontrol; see Chapter

2), the tuning of some free parameters (typi ally ontrol gains) often allow to arbitrarily

enlarge the domain ofattra tion, or to diminish atwill the magnitude ofthe steady-state

errors. Theseproperties arerespe tively referredto assemiglobal and pra ti al stability.

In moreformalterms, semiglobaland pra ti alstabilityproperties pertain to parame-

terizednonlinear time-varying systemsof the form

˙x = f (t, x, θ) ,

^(1.3)

where

x ∈ R ⁿ

^,

t ∈ R _≥0

^,

θ ∈ R ^m

^{is a onstant parameter and}

f : R _≥0 × R ⁿ × R ^m → R ⁿ

is lo ally Lips hitz in

x

^{and satises}Carathéodory onditions for any parameter

θ

^under

onsideration.

Denition 1.17 (USAS) Let

Θ ⊂ R ^m

^{be a set of} parameters. The system (1.3) is said to be uniformly semiglobally asymptoti allystable on

Θ

^{if, given any}

∆ > 0

^{, there exists}

θ ^⋆ (∆) ∈ Θ

^{su h that the origin is uniformly} asymptoti ally stable on

B ∆

^{for the system}

˙x = f (t, x, θ ^⋆ )

^.

Denition 1.18 (UGPAS) Let

Θ ⊂ R ^m

^{be asetof}parameters. Thesystem(1.3)issaid to be uniformly globally pra ti ally asymptoti ally stableon

Θ

^{if, given any}

δ > 0

^{, there}

exists

θ ^⋆ (δ) ∈ Θ

^{su h that the ball}

B δ

^{is uniformly globally} asymptoti ally stable for the system

˙x = f (t, x, θ ^⋆ )

^.

Denition 1.19 (USPAS) Let

Θ ⊂ R ^m

^{be aset of} parameters. The system(1.3)issaid tobe uniformlysemigloballypra ti allyasymptoti allystableon

Θ

^if,givenany

∆ > δ > 0

^,

there exists

θ ^⋆ (δ, ∆) ∈ Θ

^{su hthat the ball}

B δ

^isuniformly asymptoti ally stableon

B ∆

^for

the system

˙x = f (t, x, θ ^⋆ )

^.

In the above denitions,

θ

^{represents the tuning parameter, e.g. ontrol gains or any}

freedesign parameter.

Θ

^{is the set of allowed tuning}parameters, whi h maybe bounded due to physi al onstraints su h as limitation of the output of a tuators.

∆

^{an be seen}

as the radius of the estimate of the domain of attra tion; in most appli ations, a larger

∆

^{indu es better} performan e sin e the operating bandwidth is enlarged. In ontrast,

δ

^{represents the radius of the ball to whi h solutions ultimately onverge; therefore it is}

typi allyrequiredtobesmall,inordertoredu ethesteady-stateerrorasmu haspossible.

Pra ti al stability and ultimate boundedness. As it is further dis ussed in the

sequel (see Chapter 2), pra ti al stability shares similarities with the lassi al ultimate

boundedness property ( f. Denition 1.6), in the sense that solutions eventually rea h a

neighborhood of the operating point. It should however be lear to the reader that the

above Denitions 1.18 and 1.19 are usually more interesting in pra ti e, as they require

the size of this neighborhood to be redu ible at will by an adequate tuning and as they

requirethe ball

B δ

^{notonlyto beattra tivebutalso stable(inthe senseofDenition??).}

We also stress that Denitions 1.18 and 1.19 do not require the origin to be an equi-

librium for the system(1.3). This indeed failsfor many pra ti ally stable systemsas, for

instan e, Examples 2.2and 2.8givenbelow.

In viewof Proposition1.14, USPAS an be expressed in termsof

KL

^estimates.