Discrete Constraints in Control : Discontinuous feedback under Sampled-data and Switching implementations

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under Sampled-data and Switching implementations

Laurentiu Hetel

To cite this version:

Laurentiu Hetel. Discrete Constraints in Control : Discontinuous feedback under Sampled-data and

Switching implementations. Automatic. Université de Lille 1 - Sciences et Technologies, 2017.

�tel-01692115�

Universit´

e Lille 1 - Science et Technologie

M´

emoire de recherche

soumis en vue de l’obtention de

l’HABILITATION `

A DIRIGER DES RECHERCHES

Sp´ecialit´e : Automatique et Productique

par

Laurentiu HETEL

Titre :

Discrete Constraints in Control :

Discontinuous feedback under Sampled-data and Switching

implementations

Soutenue le 14 Juin 2017 `

a Centrale Lille

Garant :

J.P. Richard,

Professeur `

a Centrale Lille

Rapporteurs :

B. Brogliato

Directeur de Recherche INRIA au Centre INRIA Rhones-Alpes

D. Liberzon

Professeur `

a l’Universit´e d’Illinois `a Urbana-Champaign

L. Zaccarian

Directeur de Recherche CNRS au LAAS

Examinateurs :

O. Colot

Professeur `

a l’Universit´e de Lille 1

W. Michiels

Professeur `

a la KU Leuven

D. Peaucelle

Directeur de Recherche CNRS au LAAS

´

Thisreport presentsasele tionof theresultsthatIhavedeveloped sin e myre ruitmentasan

Asso iateResear her(ChargédeRer her hes-CR)withCNRS(CentreNationaldelaRe her he

S ientique 1

), inO tober, 2008. The resear h a tivitiesare arriedon inthegroup CO2

(Con-trol and S ienti Computing), team SYNER (Systèmes hybrides, non-linéaires et à retard 2

)

of CRIStAL (Centre de Re her he en Informatique, Signal et Automatique de Lille 3

- UMR

CNRS 9189). Ijoined the team SYNER inO tober 2008 as a2nd lass Asso iate Resear her

(CR2). ThisteamissupervisedbyProf. LotBelkoura. UntilDe ember2014,SYNERhasbeen

part ofLAGIS (Laboratoired'Automatique, Génie Informatiqueet Signal 4

) UMRCNRS8219.

On January 1st, 2015, LAGIS merged with LIFL (Laboratoire d'Informatique Fondamentale

de Lille 5

- UMRCNRS 8022), reating CRIStAL.In the ontext of the reation of CRIStAL,

SYNER is oordinating its resear h a tivities with the teams CFHP (Cal ul Formel et Haute

Performan e 6

)andDEFROST (DEFormable ROboti SofTware)inthegroupCO2-supervised

byProf. Jean-PierreRi hard.

The team SYNER addresses a large panel of problems related to the study of time-delay,

hybrid dynami al systemsandnonlinear systems. The a tivities ofthe team anbestru tured

a ordingtotwomainaxes: ononesidethemembersofSYNERdevelopestimationtoolsbased

ontheuseofdierentialalgebraandoperational al ulationinthe ontextoftheINRIAproje t

NON-A (Non-Asymptoti estimationfor onlinesystems). Onthe otherside,theteam proposes

Lyapunov based methods for analysis and ontrol design. My resear h a tivities are mainly

on ernedwiththisse ondaxisofSYNER.Atthenationallevel,mya tivities ontributetothe

workinggroupsonHybridDynami alSystemsandTimeDelaySystemofGDRMACS(Groupe

de Re her hedu CNRSen Modélisation,AnalyseetConduite desSystèmesdynamiques 7

), and

theregionalresear hgroupGRAISYHM(GroupementdeRe her heenAutomatisationIntégrée

1

NationalCenterforS ienti Resear h,apubli resear horganizationundertheresponsibilityoftheFren h

MinistryofEdu ationandResear h.

2

Hybrid,nonlinearandtime-delaysystems. 3

CenterofResear honComputerS ien es,SignalPro essingandAutomati Control. 4

LaboratoryofAutomati ontrol,ComputerEngineeringandSignalpro essing. 5

Atheoreti ComputerS ien elaboratory. 6

ComputerAlgebraandHighPerforman eComputing. 7

Anationalresear hgrouponmodellinganalysisand ontrolofdynami alsystemsundertheresponsibilityof

et Systèmes Homme-Ma hine 8

) of Région Hauts-de-Fran e. At the international level, they

have ontributedtotheHYCONNetworksofEx ellen e(Highly- omplexandnetworked ontrol

systems-FP6HYCON andFP7HYCON2).

Thisdo ument presentsseveral ontributions thathavebeenobtainedin ollaborationwith

Emmanuel BERNUAU (Ass. Prof. Agro Parite h), Mi hael DEFOORT (Ass. Prof. UVHC,

LAMIH),MohamedDJEMAI(Prof. UVHC,LAMIH),ThierryFLOQUET(DRCNRS,CRIStAL),

EmiliaFRIDMAN(Prof. Univ. Tel-Aviv),HisayaFUJIOKA(Ass. Prof. Univ. Kyoto),

Alexan-dreKRUSZEWSKI(Ass. Prof. CentraleLille,CRIStAL),Fran oiseLAMNABHI-LAGARRIGUE

(DRCNRS,L2S),Silviu-IulianNICULESCU(DRCNRS,L2S),WilfridPERRUQUETTI(Prof.

CentraleLille,CRIStAL),MihalyPETRECZKY(CRCNRS,CRIStAL),Jean-PierreRICHARD

(Prof. Centrale Lille, CRIStAL), Alexandre SEURET (CR, CNRS, LAAS), and young

re-sear hers, PhDs and post-do toral students, supervised at LAGIS and CRIStAL: Christophe

FITER (PhD Centrale Lille, defended in November 2012, now Ass. Prof., Univ. Lille),

Has-san OMRAN (PhD Centrale Lille, defended in Mar h 2014, now Ass. Prof., TP Strasbourg),

Srinath GOVINDASWAMY (post-do Centrale Lille, 2012-2013), Romain DELPOUX (ATER

Univ. Lille 1, 2013, now Ass. Prof., INSA Lyon). Other results, not mentioned in this

do -ument, have been obtained in ollaboration with Denis EFIMOV (CR INRIANon-A), Jamal

DAAFOUZ(Prof. Univ. Lorraine, CRAN),Marieke CLOOSTERMAN (PhD,TUEindhoven),

TijsDONKERS (Ass. Prof. TUEindhoven), Mauri e HEEMELS(Prof. TUEindhoven), Mar

JUNGERS(CRCNRS,CRAN),IvanMALLOCI(PhD,CRAN),SorinOLARU(Prof. Centrale

SUPELEC Paris, L2S), Worody LOMBARDI (PhD, L2S), Andrey POLYAKOV (CR INRIA

Non-A), ChristophePRIEUR(DRCNRS,GIPSA-lab),Patri k SZCZEPANSKI(Ar elor

Mit-tal), Sophie TARBOURIECH (DR CNRS, LAAS), Nathan van de WOUW (Ass. Prof. TU

Eindhoven). I would like tothank them all for their fruitful ollaboration, dynamism and

pa-tien e.

I am extremely grateful to Bernard BROGLIATO, Daniel LIBERZON and Lu a

ZACCA-RIANfor giving me the honourof reviewingthis do ument, tothe members ofthe ommittee,

OlivierCOLOT,WimMICHIELSandDimitriPEAUCELLE,forhavinga eptedtoparti ipate

in the evaluation of my resear h a tivity, and to Jean-Pierre RICHARD, for his guidan e and

support.

Iwould alsolike tothankall my olleagues from CRIStAL,INRIAand Centrale Lille who

dire tly orindire tlyinuen edthiswork.

Finally,Iwishtothankmy familyfortheirtremendous support.

8

the INTERREG IV program SYSIASS and the H2020 program UCOCOS, from the National

Resear hAgen y(ANR)through theyoungresear herproje tROCC-SYS(agreement

ANR-14-CE27-0008), fromtheRégion Hauts-de-Fran ethrough theARCIRproje t ESTIREZandfrom

Prefa e i

A ronyms ix

Notations xi

General introdu tion 1

Part I Contributions to aperiodi sampled-data ontrol 7

Chapter 1 Generalities 11

1.1 System onguration . . . 11

1.2 Classi al designmethods . . . 12

1.3 Complex phenomenainaperiodi sampling . . . 14

1.4 Problem set-ups . . . 16

Chapter 2 State of theart onaperiodi sampled-data systems 17 2.1 Stability analysisunder arbitrarytime-varyingsampling . . . 17

2.1.1 Time-delayapproa h . . . 18

2.1.2 Hybridsystemapproa h. . . 22

2.1.3 Dis rete-time approa h and onvex-embeddings. . . 29

2.1.4 Input/Outputstabilityapproa h . . . 35

2.2 Sampling asa ontrolparameter . . . 40

2.2.1 Event-Triggered (ET)Control . . . 41

2.2.2 Self-Triggered (ST) Control . . . 42

2.3 Con lusion . . . 43

Chapter 3 Main ontributions 45 3.1 LinearTimeInvariantsampled-datasystem . . . 46

3.1.1 Quasi-quadrati Lyapunovfun tions . . . 46

3.1.2 Continuous-time analysis basedon onvexembeddings. . . 49

3.1.3 Extension tothe sampling ontrolproblem . . . 54

3.2 Sampled-data ontrolofbilinear systems . . . 60

3.2.1 Hybridsystemapproa h. . . 61

3.2.2 Input /Output approa h . . . 65

3.3 Sampled-data ontrolofinputanenonlinearsystems. . . 71

3.4 Swit hing ontrollersundersampled-data implementations . . . 77

3.4.1 Swit hedanesystems . . . 77

3.4.2 Relay ontrol . . . 81

Con lusion 85 PartII Designofswit hing ontrollers-anemergingresear hdire tion 87 Chapter 4 Linear systems 91 4.1 Simplied problemformulation . . . 91

4.2 Basi idea. . . 92

4.3 Slidingdynami s androbustnesstoperturbations . . . 94

4.4 LPV aseandParameter DependentRelay Control. . . 96

Chapter 5 Swit hed ane systems 103 5.1 System des ription . . . 104

5.2 Main results . . . 104

5.3 Numeri al issues . . . 109

Chapter 6 Appli ations 111 6.1 Controlof aPermanent Magnet Syn hronousMotor . . . 111

6.2 Controlof amulti-level power onverter . . . 116

6.2.1 LMI designforageneri bilinearmodel . . . 120

6.2.2 Experimentalresults. . . 121

Perspe tives 131

ˆ IQCIntegralQuadrati Constraint

ˆ LKFLyapunov-KrasovskiiFun tional

ˆ LMILinearMatrix Iequalities

ˆ LPVLinearParameter-V arying

ˆ LTILinearTime-Invariant

ˆ MSIMaximum Sampling Interval

ˆ NCSNetworked Control System

ˆ PDRParameter DependentRelay

ˆ PWMPulse-W idth Modulation

ˆ SOSSumOf Squares

ˆ

R

+

denotesthe set

{λ ∈ R, λ ≥ 0}

.

ˆ

|c|

denotesthe absolutevalueofas alar

c

∈ R.

ˆ

kxk

representsanynormofthe ve tor

x

.

ˆ

kxk

p

, p

∈ N

,denotes the

p

normofave tor

x

.

ˆ Foramatrix

M

,

M

T

denotesthetranspose of

M

and

M

⋆

,its onjugatetranspose.

ˆ For square symmetri matri es

M, N

,

M

 N

(resp.

M

≻ N

) means that

M

− N

is a positivesemi-denite (resp. denite positive)matrix.

M

 N

(resp.

M

≺ N

)meansthat

M

_{− N}

is anegativesemi-denite(resp. negativedenite)matrix. ˆ Foramatrix

M

∈ R

n×n

,wedenotetheHermitian of

M

by

He{M} = M + M

T

_.

ˆ

∗

inasymmetri matrixrepresentselementsthatmaybeindu edbysymmetry.

ˆ

kMk

p

, p

∈ N

denotestheindu ed

p

-normof amatrix

M

.

ˆ

σ (M )

¯

denotes themaximumsingularvalueof

M

.

ˆ

C

0

_{(X, Y )}

,fortwometri spa es

X

and

Y

, isthesetof ontinuousfun tionsfrom

X

to

Y

.

ˆ

L

n

p

(a, b), p

∈ N

denotes the spa e of fun tions

φ : (a, b)

→ R

n

with norm

kφk

L

p

=

hR

b

a

kφ(s)k

p

_ds

i

1

p

. ˆ

L

n

2e

[0,

∞)

is the spa eof fun tions

φ : [0,

∞) → R

n

whi h are squareintegrable on nite

intervals.

ˆ Givenaset

S ⊂ R

n

, onv

{S}

denotesits losed onvexhullandInt

{S}

itsinterior.

ˆ For a onvex polytope

S ⊂ R

n

and a s alar

α > 0

, we denote

α

S := {αx, x ∈ S}

and vert

{S}

thesetofverti esof

S

.

ˆ The

n

dimensional open ball in

R

n

entred on

x

∈ R

n

with radius

c > 0

is denoted

B(x, c) := {y ∈ R

n

_:

_{kx − yk}

2

< c

} .

and nitesampling frequen y, et . This dissertation is on erned witha fundamental problem

in modern ontrol systems: the o urren e of dis rete onstraints in ontrol loops. Two main

aspe ts will be onsidered. On one side, we will dis uss the o urren e of dis rete- onstraints

in the time domain, related to sampled-data ontrol implementations and fa t that in pra ti e

the ontrola tionis omputedsporadi ally,ataperiodi samplinginstants. Inthis ontext,the

main hallenges are todeterminethe maximumsampling interval whi h preserves stability and

tos hedule thesampling instants soastoensuredesiredperforman es. Thistopi ismotivated

bytheuprisinginterestinnetworkedandembedded ontrolelementswherereal-times heduling

algorithms intera twith ontroltasksandwhere ommuni ation andenergeti onstraintshave

to be taken into a ount. On the other side, we will present results on erning the design of

feedba k laws subje t to dis rete onstraints inthe setsof possible ontrol values: the ontrol

signal is allowed totake onlyanite number ofvalues. Su h onstraintsare typi al in systems

with swit hes, relays or binary (on-o) a tuators. The main hallenge here is to design the

swit hing surfa es while guaranteeing desired safety onstraints in terms of (lo al) stability.

Bothof thesetopi sbringup open problemsinthe domainof hybriddynami al systems. They

involvethestudyofdierentialequationswithdis ontinuousright-handsideandofsystemswith

impulsivedynami s.

With respe t tothe resear h a tivity arried inthe team SYNER, overthe last eight years

we have investigated the ee t of aperiodi sampling on several lasses of dynami al systems

intera ting withsampled-dataimplementationsofboth ontinuousandswit hingfeedba klaws.

We have tried to address the main hallenges in aperiodi sampled-data ontrol using several

dierent approa hes. One of the main purposes of our work is to propose numeri al tools for

addressing the onsideredproblems. We havededi ated some eortto expresssolutions tothe

analysisand ontroldesignproblemsinaformthatis onvenienttothe derivationof

omputer-aidedtools. Aparti ularattentionisgiventotheformulationofanalysisandsynthesis riteriaas

simple onvex optimizationproblems whi h anbe easilyaddressednumeri ally usingpowerful

numeri alalgorithms.

First,themain ontributions inthe ontextofsampled-datasystemsarebrieypresentedas

follows:

ˆ New onditions for the stability oflinear time invariant (LTI) sampled-data systems with

arbitrarytime-varyingsamplingintervals[Hetel2011b℄. Themainideaistousea

dis rete-timesystemmodelandquasi-quadrati Lyapunovfun tionspreviouslyen ounteredinthe

ontext of polytopi dieren e in lusions in order to provide stability onditions. The

existen eofaquasi-quadrati Lyapunovfun tionde reasingatsampling instantsisshown

to be a ne essary and su ient ondition for stability. Using approximations based on

onvexpolytopesleadstosu ientstability riteria. Thisapproa h allowsaverya urate

numeri al implementation of algorithms for evaluating the maximum allowable sampling

intervalwhi h ensuresstability.

ˆ A new framework for the analysis of sampled-data systems inspired by the Dissipativity

Theory [Omran2014b,Omran2014a,Omran2016a℄. Theideais to hara terize theee t

ofsamplingusing"supply"fun tions. Themethodgeneralizestothe aseofnonlinearane

systemsseveralfrequen ydomain riteria initiallyusedforLTIsystems. Theadvantageof

thisapproa h is its exibility: the approa h anbe easily extended inorder totakeinto

either ontinuous-time ordis rete-time models. We have proposed a ontinuous-time

ap-proa h based on onvex embeddings that is able to ombine the advantages of the

time-delay system modelling (inter-sampling behaviour, robustnessto perturbations) with the

onesof dis rete-time models(a ura y of analysis). This approa h has been usedfor the

design of even-/self-triggered ontrol algorithms. We have provided tools for optimizing

the sampling mapsso asto enlarge the minimum inter-event timebetween two sampling

instantswhileensuring desiredperforman eandrobustnessproperties.

Inordertotransferourexperien eoverthisdomain,wehavegathereda olle tionofmainresults

onaperiodi sampled-datasystemsinanoverviewofstabilityanalysisapproa heswhi hhasbeen

presented asa tutorial paperat ECC [Fiter 2014a℄. A detailed survey arti le [Hetel 2017℄ has

beena eptedfor publi ationinAutomati a.

Se ond,thedo umentwillpresentamorere enteldofoura tivity: thedesignofswit hing

surfa es under dis rete onstraints. While the study of systems with aperiodi sampling has

now rea hed an advan ed phase of development, the se ond main topi of resear h, the design

ofswit hingsurfa es forsystemssubje ttodis rete onstraints,representsanemergingresear h

dire tion in the team SYNER. The design of swit hing ontrollers (relays, sliding mode

on-trollers, variable stru ture systems, et .) is an old problem in the ontrol theory. However,

veryfew numeri al tools exist for optimizing the design of swit hing surfa es while optimizing

the systems performan es (domain of attra tion, robustness toperturbations and delay, de ay

rate, et .). We are urrently investigating a re ent resear h dire tion by addressing this topi

from a hybrid systemperspe tive. The main idea of our work is to use a simple onvex

opti-mization approa h for the design of swit hing ontrollers based on Linear Matrix Inequalities

(LMIs). We have addressed this problem for LTI, polytopi approximations of nonlinear

sys-tems,bilinearsystemsandswit hedanesystems. Thisnewmethodhasleadtoseveraljournal

publi ations [Hetel 2015 ,Hetel 2015a,Delpoux 2015,Hetel 2016℄. For the ase of linear

sys-tems it is shown that the robustness requirements of lassi al sliding mode ontrollers an be

in orporated inthenew designmethodologywhileoptimizingthe domainof attra tionandthe

robustness withrespe t toperturbations [Hetel 2015 ℄. Forswit hedane systemswe provide

a new point of view in the design of stabilizing state feedba k laws: we show that the design

of swit hing ontrollers an be re-stated asa lassi al design problem for nonlinear ane

sys-tems[Hetel2015a℄. Themethodallowstotakeintoa ountsome lassesofswit hedanesystem

that an bestabilizedonly lo ally, onwhi h the existingmethodsdo notapply. Simple ontrol

design riteriaareproposedforswit hedanesystemsthatdonotsatisfythe lassi onstraints

relatedtotheexisten eofHurwitz onvex ombinations. Thenewmethodologyhaspotentialin

appli ation toele tro-magneti systems ( ontrol of steppermotors [Delpoux 2015℄)andenergy

management problems (DC/DC power onverters [Hetel 2016℄). The analysis of sampled-data

implementationsof swit hing ontrollershasequallybeenaddressed[Hetel 2013b℄.

Afterthisgeneralintrodu tion, therestofthisdissertationisorganizedintotwomajorparts

anda on lusion.

Part Ideals with dis rete onstraints inthe time domain. Itis mainly on erned with the

stabilityproblemforsampled-datasystemswithaperiodi sampling. Afterpresentingsome

gen-eralities on erningsystemswith time-varyingsampling inChapter 1,the se ond hapter gives

a overview of the literature on the eld. Chapter 3 presents our main ontributions to this

topi . Our resear h eort hasbeen dedi ated tothe analysis of various lassesof systems

sultsisgiven. Forlineartimeinvariantsystems,weshowinChapter3.1hownumeri allye ient

onditions anbederivedusingtheexa tsystemdis retizationand onvexembeddings.

Numer-i altoolsfortheoptimizationof(event/self-triggered)samplingmapsareproposed,basedonthe

usedofLinearMatrixInequalities(LMIs). Inamoregeneral ontextofbilinear(Chapter3.2)and

nonlinear anesystems (Chapter 3.3),we propose anew stability analysis frameworkinspired

by DissipativityTheory. Control designtoolsare presentedfor LTIsystemswithdis ontinuous

ontrollersusingatime-delayapproa h inChapter 3.4.

Part II presents new results for systems with inputs onstrained to a nite set of values.

Chapter 4 deals with the design of swit hing ontrollers for linear systems and some

approxi-mations of nonlinear systems as linear polytopi systems. The ase of swit hed ane systems

is dis ussed in Chapter 5, while Chapter 6 presents results on erning bilinear systems. The

potentialoftheapproa h isillustratedattheendofthispartthroughexperimentalappli ations

on erning the ontrol ofsteppermotorsandDC/DCpower onverters.

A on lusionsummarizesthemainresultspresentedinthisdo ument. Finally,several

Contributions to aperiodi

is mainly due to the ubiquitous presen e of embedded ontrollers in relevant appli ation

do-mainsandthegrowingdemandinindustryonsystemati methodstomodel,analyseanddesign

systems where sensor and ontrol data are transmitted over a digital ommuni ation hannel.

The study of systems with aperiodi sampling emerged as a modelling abstra tion whi h

al-lows tounderstand the behaviour of Networked Control Systems (NCS) with sampling jitters,

pa ketdrop-outsoru tuationsduetotheinter-a tionbetween ontrolalgorithmsandreal-time

s hedulingproto ols [Zhang 2001 ,Antsaklis 2007,Astol 2008℄. With the emergen e of

event-based andself-triggered ontrolte hniques[Heemels 2012℄,thestudyof aperiodi sampled-data

systems onstitutes nowadaysaverypopularresear h topi in ontrol.

Inthispart,wefo usonquestionsarisinginthe ontrolofsystemswithtime-varyingsampling

intervals. Importantpra ti alquestionssu h asthe hoi eof theminimal samplingbandwidth,

theevaluationofne essary omputationalandenergeti resour esortherobust ontrolsynthesis

are mainly related tostability issues. These issuesoftenlead tothe problem ofestimating the

Maximum SamplingInterval(MSI)forwhi hthestability ofa losed-loopsampleddatasystem

is ensured.

Thestudyofaperiodi sampled-datasystemshasbeenaddressedinseveralareasofresear h

in Control Theory. Systems with aperiodi sampling an be seen asparti ular time-delay

sys-tems. Sampled-and-holdin ontrolandsensorsignals anbemodelledusinghybridsystemswith

impulsivedynami s. Aperiodi sampled-datasystemshavealsobeenstudiedinthedis rete-time

domain. In parti ular, LinearTime Invariant (LTI)sampled-data systemswith aperiodi

sam-plinghavebeenanalysedusingdis rete-timeLinearParameterVarying(LPV)models,typi ally

used in gain s heduling ontrol. The ee t of sampling an be modelled using operators and

the stability problem an be addressed in the framework of Input/Output inter onne tions as

typi allydone inmodernRobust Control. While signi ant advan eson thissubje t havebeen

intheliterature, problems relatedtoboththe fundamentalsof su h systemsandthederivation

of onstru tivemethodsfor stabilityanalysis remainopen,evenforthe aseof linearsystem.

Therestofthispartisstru tured: Chapter1isdedi atedtogeneralities on erningaperiodi

sampled-data ontrol. A state of the art on aperiodi sampled-data ontrol will be given in

Generalities

1.1 System onguration

Asfollows we willstudy the propertiesof sampled-data systems onsistingof aplant, a digital

ontroller, andappropriate interfa e elements. A general onguration of su h a sampled-data

systemisillustratedbytheblo kdiagramofFigure1.1. Inthis onguration,

y(t)

isa ontinuous-timesignalrepresentingtheplantoutput(theplantvariablesthat anbemeasured). Thissignal

is represented asafun tionof time

t

,

y :

R

+

→ R

p

.

The digital ontrollerisusually implemented asanalgorithm onanembedded omputer. It

operateswithasampledversionoftheplantoutputsignal,

{y

k

}

k∈N

,obtainedupontherequestof asamplingtriggersignalatdis retesamplinginstants

t

k

andusingananalog-to-digital onverter (thesamplerblo k,

S

,inFigure1.1). Thistriggermayrepresentasimple lo k,asinthe lassi al periodi samplingparadigm,oramore omplexs hedulingproto olwhi hmaytakeintoa ount

thesensorsignal,amemoryofitslastsampled values,et . Thesampling instantsare des ribed

byamonotone in reasingsequen e ofpositiverealnumbers

σ =

{t

k

}

k

∈N

where

t

0

= 0, t

k+1

− t

k

> 0, lim

k→∞

t

k

=

∞.

(1.1)

u(t) = u

k

PLANT

y(t)

H

u

k

CONTROL

y

k

= y(t

k

)

S TRIGGER

Thedieren e betweentwo onse utivesampling times

h

k

= t

k+1

− t

k

is alledthe

k

th

sampling

interval. Assuming that the ee t of quantizers may be negle ted, the sampled version of the

plant outputisthe sequen e

{y

k

}

k∈N

where

y

k

= y(t

k

).

In asampled-data ontrolloop, the digital ontroller produ esa sequen e of ontrol values

{u

k

}

k

∈N

usingthesampledversionoftheplantoutputsignal

{y

k

}

k

∈N

. Thissequen eis onverted intoa ontinuous-time signal

u(t)

,where

u :

R

+

→ R

m

( orresponding tothe plantinput) viaa

digital-to-analog interfa e. We onsider thatthe digital-to-analog interfa e isa zero-orderhold

(the hold blo k,

H

, in Figure 1.1). Furthermore, we assume that there is no delay between thesampling instant

t

k

andthemomentthe ontrol

u

k

(obtainedbasedonthe

k

th

plantoutput

sample,

y

k

)isee tivelyimplementedattheplantinput. Thentheinputsignal

u(t)

isapie ewise onstantsignal

u(t) = u(t

k

) = u

k

,

∀t ∈ [t

k

, t

k+1

).

Overthe hapter,wewill onsiderthattheplantismodelledbyanitedimensionalordinary

dierential equationof theform



˙x = F (t, x, u) ,

y = H (t, x, u) ,

(1.2)

where

x

∈ R

n

representstheplantstate-variable. Here

F :

R

+

×R

n

_×R

m

_{→ R}

n

with

F (t, 0, 0) =

0,

_{∀t ≥ 0,}

and

H :

R

+

× R

n

_{× R}

m

_{→ R}

p

. It is assumed that for ea h onstant ontrol and

ea hinitial ondition

(t

0

, x

0

)

∈ R

+

× R

n

thefun tion

F

des ribing theplantmodel(1.2) issu h that auniquesolution exists foran interval

[t

0

, t

0

+ ǫ)

with

ǫ

large enough withrespe t tothe maximum sampling interval. The dis rete-time ontroller is onsidered to be des ribed by an

ordinary dieren eequation ofthe form



x

c

_k+1

= F

_d

c

(k, x

c

_k

, y

k

) ,

u

k

= H

d

c

(k, x

c

k

, y

k

) ,

(1.3) where

x

c

k

∈ R

n

c

is the ontrollerstate. Here,

F

c

d

:

N × R

n

c

× R

p

→ R

n

c

and

H

c

d

:

N × R

n

c

× R

p

→

R

m

. We will use the denomination sampled-data system for the inter onne tion between the

ontinuous-time plant(1.2) withthedis rete-time ontroller (1.3)viathe relations

y

k

= y(t

k

), u(t) = u

k

,

∀t ∈ [t

k

, t

k+1

),

∀k ∈ N,

(1.4) under asequen e ofsampling instants

σ =

{t

k

}

k

∈N

satisfying(1.1) .

Thedierent on eptsandresultswillbemostlyillustratedonLinearTimeInvariant(LTI)

models

˙x = Ax + Bu,

(1.5)

under astati linear statefeedba k,

u

k

= Kx

k

, k

∈ N,

(1.6)

with

x

k

= x(t

k

)

. However,whenpossible,wewillpresenttheextensionstomoregeneral nonlin-ear systems.

1.2 Classi al design methods

There are various approa hes for the designof a sampled-data ontroller (1.3) (see the

lassi- al textbooks [Åström 1996,Chen 1993℄ and the tutorial papers [Mona o 2007,Mona o 2001,

Ne²i¢ 2001,Laila 2006℄).

using lassi almethods[Khalil2002,Isidori1995,Krsti 1995,Sastry1999℄. Next,adis rete-time

ontrollerof the form(1.3) is obtainedby integrating the ontroller solutions over the interval

[t

k

, t

k+1

)

.This approa h is usually alled emulation. Generally, it is di ult to ompute in a formal mannerthe exa t dis rete-time model and approximations must beused[Mona o 2007,

Laila 2006℄. In the LTI ase (1.5) with state feedba k (1.6) , the emulation simplymeans that

the gain

K

is setsu h that thematrix

A + BK

is Hurwitzandthat theplantis driven by the ontrol

u(t) = Kx(t

k

),

∀t ∈ [t

k

, t

k+1

), k

∈ N.

While the intuition seems to indi ate that for su iently small sampling intervalsthe obtained sampled-data ontrolgivesan approximation

of the ontinuous-time ontrolproblem, no guarantee anbe given when the sampling interval

in reases, even for onstant sampling intervals. In orderto ompensatethe ee t of ontroller

dis retisation,re-designmethods maybeused[Grüne2008,Ne²i¢2005℄.

Dis rete-timedesign. Inthisframework,adis rete-timemodeloftheplant(1.2)isderivedby

integration. The obtainedmodel represents the evolutionof the plantstate

x(t

k

) = x

k

at sam-pling times

9

. Then,adis rete-time ontroller(1.3) isdesignedusing theobtaineddis rete-time

model. Inthe simplestLTI ase (1.5) ,(1.6) ,theevolutionofthe statebetweentwo onse utive

sampling instants

t

k

and

t

k+1

is given by

x(t) = Λ(t

− t

k

)x(t

k

),

∀t ∈ [t

k

, t

k+1

], k

∈ N,

(1.7) withamatrixfun tion

Λ

dened on

R

as

Λ(θ) = A

d

(θ) + B

d

(θ)K =

e

Aθ

₊

Z

θ

0

e

As

_dsBK.

(1.8)

Evaluatingthe losed-loopsystem'sevolutionat

t = t

k+1

andusingthenotation

h

k

= t

k+1

− t

k

leads tothe lineardieren e equation

x

k+1

= Λ(h

k

)x

k

,

∀k ∈ N

(1.9)

representingthe losed-loopsystematsamplinginstants. Whenthesamplingintervalis onstant,

h

k

= T,

∀k ∈ N

,alarge varietyofdis rete-time ontroldesignmethodologiesisavailableinthe literature(see[Åström1996,Chen1993℄andthereferen eswithin). Itiswellknownforthis ase

thatsystem(1.9)isasymptoti allystableifandonlyifthematrix

Λ(T )

isS hur. Inotherwords, todesignastabilizing ontrollaw(1.6),thematrix

K

must besetsu h asall theeigenvaluesof

Λ(T )

lay stri tlyintheunit ir le.

Fornonlinearsystemswith onstantsamplingintervals,anoverviewof ontroldesign

method-ologies and relatedissues an befound in[Mona o 2007,Mona o 2001,Ne²i¢ 2001,Laila 2006℄.

Notethatthedis rete-timemodelssu has(1.9)donottakeinto onsiderationtheinter-sampling

behaviour of the system. Relations between the performan es of the dis rete-time model and

the performan es of the sampled-data loop, an be dedu ed using the methodology proposed

in[Ne²i¢1999℄.

Sampled-data design . Innitedimensionaldis rete-time modelswhi h takeintoa ount the

inter-sampling systembehaviour usingsignal lifting [Bamieh1992,Bamieh 1991,Tadmor 1992,

Toivonen 1992a,Yamamoto 1994℄ have been proposed in the literature for the ase of linear

systems. Spe i design methodologies, that are able totake in onsideration ontinuous-time

9

Notethatgenerallyapproximationsof thesystemmodelmustbeused sin ethe dis retizedplantmodelis

di ultto omputeformally[Mona o1985,Veliov1997℄. Evenforthe aseofLTIsystemswith onstantsampling

intervals, the numeri al omputation of the matrix exponential (or itsintegral) is subje t to approximations

system performan es, inter-sample ripples and robustness spe i ations, an be found in the

textbook[Chen1993℄forthe aseof lineartimeinvariantsystemswithperiodi sampling.

1.3 Complex phenomena in aperiodi sampling

Whileinthelastftyyearsanintensiveresear hhasbeendedi atedtotheanalysisanddesignof

sampled-datasystemsunderperiodi sampling,thestudyofsystemswithtime-varyingsampling

intervals is quiteunderdeveloped ompared tothe periodi onterpart. Thefollowing examples

illustrate theri h omplexityof phenomenathat may o urunder aperiodi sampling.

Example 1.1 [Zhang 2001a℄ Consider an LTI sampled-data system of the form (1.5),(1.6)

where

A =



1 3

2 1



,

B =



1

0.6



,

K =

−



1 6



.

(1.10)

For this example, system's (1.9) transition matrix

Λ(T )

is a S hur matrix for any onstant sampling interval in

T

∈ T = {T

1

, T

2

}

, with

T

1

= 0.18

, and

T

2

= 0.54

. Then, in the ase of periodi sampling, thesampled-datasystemisstablefor onstantsamplingintervalstakingvalues

in

T .

Anillustrationofthesystem'sevolution for onstantsamplingintervals

T

1

,

T

2

,is givenin Figure 1.2. Clearly,whenthesamplinginterval

h

k

isarbitrarilyvaryingin

T

,theS hurproperty of

Λ(T ),

∀ T ∈ T ,

represents a ne essary ondition for stability of the sampled-data system (1.1) ,(1.5) ,(1.6). However, it is not a su ient one. For example, the sampled-data system

with a sequen e of periodi ally time-varying sampling intervals

{h

k

}

k∈N

=

{T

1

, T

2

, T

1

, T

2

, . . .

}

is unstable, as it an be seen in Figure 1.3. This is due to the fa t that the S hur property of

matri es is not preserved under matrix produ t (i.e. the produ t of two S hur matri es is not

ne essarily S hur). Indeed, the dis rete-time system representation over two sampling instants

an be written as

x

k+2

= Λ(T

2

)Λ(T

1

)x

k

,

∀k ∈ 2N,

and thetransitionmatrix

Λ(T

2

)Λ(T

1

) =



0.8069

−3.2721

0.6133

−2.1125



over two sampling intervals

T

1

and

T

2

, is not S hur. This example shows the importan e of takinginto onsiderationtheevolutionofthesamplinginterval

h

k

whenanalysingthestabilityof sampled-datasystemssin earbitraryvariationsofthesamplinginterval

h

k

mayindu einstability.

Example 1.2 [Gu 2003a℄Considernowan LTIsystemwith

A =



0

1

−2 0.1



, B =



0

1



K =



1 0



(1.11)

Assume thatthe samplinginterval

h

k

isrestri ted tothe set

T = {T

1

, T

2

}

with

T

1

= 2.126

and

T

2

= 3.950

. Thesystemisunstableforboth onstantsamplingintervals

T

1

and

T

2

sin eforthese valuessystem's(1.9)transitionmatrix

Λ(T ), T

∈ T

isnotaS hurmatrix. However,theprodu t of transitionmatri es

Λ(T

1

)Λ(T

2

)

has the S hur property. Therefore, the sampled data system is stable under a periodi evolution ofthe samplinginterval

{h

k

}

k

∈N

=

{T

1

, T

2

, T

1

, T

2

, . . .

}

. An example ofsystemevolutionwiththisparti ular samplingsequen eis providedin Figure 1.4. In

0

2

4

6

8

10

−30

−20

−10

0

10

t

Constant sampling T

1

= 0.18

x

₁

(t)

x

₂

(t)

u(t)

0

2

4

6

8

10

−100

−50

0

50

100

t

Constant sampling T

2

= 0.54

x

1

(t)

x

2

(t)

u(t)

Figure1.2: Stabilityofthe systeminExample 1.1 withperiodi sampling intervals.

0

2

4

6

8

10

−100

−50

0

50

100

t

x

₁

(t)

x

2

(t)

u(t)

Figure 1.3: Instability of the systemin Example 1.1 with aperiodi sampling sequen e

T

1

→

whi h ensures the system's stability while the possible onstant sampling ongurations are not

ableto guarantee thisproperty.

0

10

20

30

40

50

−1

−0.5

0

0.5

1

t

x

₁

(t)

x

₂

(t)

u(t)

Figure1.4: Periodi sampling sequen ewithastablebehaviour.

1.4 Problem set-ups

The oreofPartIisdedi atedtotherobustanalysis ofsampled-datasystemswithsampling

se-quen esoftheform(1.1)wherethesamplinginterval

h

k

= t

k+1

−t

k

takesarbitraryvaluesinsome interval

T = [h, h] ⊂ R

+

.

This rst problem set-up may orrespond, for example, tothe sam-plingtriggeringme hanismfromFigure1.1witha lo ksubmittedtojitter[Wittenmark1995℄,or

withsomes hedulingproto olwhi h istoo omplextobemodelledexpli itly[Zhang2001 ,

Hes-panha 2007℄. Basi ally, for the ase of LTI models (1.5) withlinear state feedba k (1.6) under

a sampling sequen e (1.1) we will address the robust stability of the losed-loop system(1.12)

givenbelow







˙x(t) = Ax(t) + BKx(t

k

),

∀t ∈ [t

k

, t

k+1

),

∀k ∈ N,

t

k+1

= t

k

+ h

k

,

∀k ∈ N,

t

0

= 0, x(t

0

) = x

0

∈ R

n

(1.12)

asif

h

k

is atime-varying "perturbation" takingvaluesinabounded set

T .

We will also indi ate some ideas on erning a re ently emerging resear h topi where the

samplinginterval

h

k

playstheroleofa ontrolparameterthatmaybe hangeda ordingtothe plantstateoroutput. Thisproblemset-up orrespondstothedesignofas hedulingme hanism.

Forthe ase of system(1.12) ,

h

k

is onsideredasan additionalinputwhi h, byan appropriate open/ losed-loop hoi e, anensurethesystemstability. Inthefollowing hapter,wewillpresent

State of the art on aperiodi

sampled-data systems

This hapterpresentsbasi on eptsandre entresear hdire tionsaboutthestabilityof

sampled-datasystems withaperiodi sampling 10

. Wefo usmainly on the stability problem for systems

witharbitrary time-varying sampling intervalswhi h hasbeen addressedinseveralareasof

re-sear h inControlTheory. Systems withaperiodi sampling anbeseen astime-delay systems,

hybridsystems,Input/Outputinter onne tions,dis rete-timesystemswithtime-varying

param-eters, et . The goal is to provide a stru tural overview of the progress made on the stability

analysisof systemswithaperiodi sampling. Withoutbeingexhaustive,whi hwouldbeneither

possiblenoruseful,wetrytobringtogetherresultsfromdiverse ommunitiesandpresentthem

inauniedmanner. Forea h oftheexisting approa hesthebasi on epts,fundamentalresults

and relations withthe other approa hes are dis ussed in detail. Results on erning extensions

of Lyapunov and frequen y domain methodsfor systemswith aperiodi sampling are re alled,

as they allow to derive onstru tive stability onditions. Furthermore, numeri al riteria are

presented whileindi ating the sour esof onservatism,the problems thatremain open andthe

possibledire tionsofimprovement. Atlast,someemergingresear hdire tions,su hasthedesign

of stabilizingsampling sequen es,are brieydis ussed.

2.1 Stability analysis under arbitrary time-varying sampling

Inthefollowing,wereviewsomeresultswhi h provideaqualitativeestimationofthemaximum

sampling interval ensuring stability for sampled-data systems with sampling intervals that are

arbitraryvarying. Moreformally,overthese tion, wepresentresultsthataddressthe following

problem:

ˆ ProblemA(Arbitrarysamplingproblem): Considerthesampled-datasystem(1.1) ,(1.2) ,

(1.3) ,(1.4)andaboundedsubset

T ⊂ R

+

. Determineifthesampled-datasystemisstable (insomesense)foranyarbitrarytime-varyingsampling interval

h

k

= t

k+1

− t

k

withvalues in

T

.

Often the set

T

is onsideredof the form

T = (0, h]

where

h

is some positive s alar. The largestvalueof

h

forwhi h thestability ofthe losedloopsystemisensuredis alledMaximum SamplingInterval(MSI).

10

The material presented inthis hapter ispart of a surveypaper a epted for publi ation inAutomati a

t

t

τ

(t

)

=

t−

t

k

Figure 2.1: Samplingseen asapie ewise- ontinuoustime-delay

Severalperspe tivesforaddressingProblemAexist. First,wepresentresultsthatarebased

on a time-delay modelling of the sampled-data system (1.1) ,(1.2) ,(1.3) ,(1.4) . Next, we show

howthe problem anbeaddressedfrom the point ofview of hybrid systems. We ontinue with

approa hesthatusetheexpli itsystemintegrationin-betweensu essivesamplinginstants,su h

asthe ones lassi allyusedinthe dis rete-time framework. Last, resultsaddressingProblem A

fromthe robust ontrol theorypointofview arepresented.

2.1.1 Time-delay approa h

Tothebestof ourknowledge, thiste hniquewasinitiatedin[Mikheev1988,Åström1989℄,and

further developed in [Fridman 1992,Teel 1998b,Louisell 2001℄ andin several other works. For

the aseofan LTIsystemwithsampled-data statefeedba k (1.12) ,wemayre-write

u(t) = Kx(t

k

) = Kx(t

− τ(t)),

τ (t) = t

_{− t}

k

,

∀t ∈ [t

k

, t

k+1

),

(2.1)

where the delay is pie ewise-linear, satisfying

˙τ (t) = 1

for

t

6= t

k

, and

τ (t

k

) = 0

. This delay indi ates the timethat has passed sin e the last sampling instant. Anillustration of atypi al

delayevolutionisgiveninFig.2.1. TheLTIsystemwithsampled-data(1.12)isthenre-modeled

asanLTIsystemwithatime-varyingdelay

˙x(t) = Ax(t) + BKx(t

_{− τ(t)), ∀t ≥ 0.}

(2.2) Thispermitstoadaptthe toolsforstability ofsystemswithfast varyingdelays [Fridman 2003,

Gu 2003b,Ri hard 2003,Ni ules u 2004℄. Thismodel isequivalenttotheoriginal sampled-data

systemwhen onsideringthat the sampling indu eddelay hasa known derivative

˙τ (t) = 1

, for all

t

∈ [t

k

, t

k+1

), k

∈ N

.

2.1.1.1 Basi results

Forsystem(2.2)itisnaturalto onsider,asastatevariable,thefun tional

x

t

(θ) = x(t+θ),

∀θ ∈

[

_{−¯h, 0]}

,and,asstatespa e,theset

C

0



_{−h, 0}



_,

_R

n



−h, 0



into

R

n

[Fridman2014,Ni ules u2001,Ni ules u1998℄. Inthegeneral aseoftime-delay

systems, it is di ult to apply the lassi al Lyapunov stability theory, be ause the The most

popular generalization ofthedire t Lyapunovmethod fortime-delaysystemhasbeenproposed

by Krasovskii[Krasovski 1963℄. Ituses the existen eof fun tionals

V (t, x

t

)

depending on the state ve tor

x

t

.

In the sampled-data ase [Fridman 2004,Fridman 2010,Liu 2012a℄ fun tionals

V (t, x

t

, ˙x

t

)

dependingbothon

x

t

and

˙x

t

(see[Kolmanovskii1992℄, p.337)are useful.

Denote by

W [

−h, 0]

the Bana h spa e of absolutely ontinuous fun tions

φ : [

−h, 0] → R

n

with

φ

˙

∈ L

n

2

(

−h, 0)

(thespa eof squareintegrable fun tions)withthenorm

kφk

W

= max

s

∈[−h,0]

kφ(s)k +

Z

0

−h

˙φ(s)

2

ds



1

2

.

Theorem 2.1(Lyapunov-Krasovskii Theorem) [Kolmanovskii 1992℄ Consider

f :

R

+

×

C

0

_[

_{−h, 0] → R}

n

ontinuous in both arguments and lo ally Lips hitz in the se ond argument.

Assume that

f (t, 0) = 0

for all

t

∈ R

+

and that

f

maps

R×

(bounded sets in

C

0

_[

_{−h, 0]}

) into

bounded sets of

R

n

. Suppose that

α, v, w :

R

+

→ R

+

are ontinuous nonde reasing fun tions,

α(s)

,

β(s)

and

γ(s)

are positive for

s > 0

,

lim

s→∞

α(s) =

∞

and

α(0) = β(0) = 0

. The trivial solution of

˙x(t) = f (t, x

t

)

is Globally Uniformly Asymptoti ally Stable if there exists a ontinuous fun tional

V :

R ×

W [

−h, 0] × L

n

2

(

−h, 0) → R

+

,

whi h is positive-denite, i.e.

α(

_{kφ(0)k) ≤ V (t, φ, ˙φ) ≤ β(kφk}

W

)

for all

φ

∈ W [−h, 0], t ∈ R

+

,

and su h that its derivative along the system's solutionsis non-positive

˙

V (t, x

t

, ˙x

t

)

≤ −γ(kx

t

(0)

k).

(2.3)

Thefun tional

V

satisfyingthe onditionsofTheorem2.1is alledaLyapunov-Krasovskii Fun -tional (LKF).Inthegeneral aseofsampled-datanonlinearsystemstheunderlyingdelaysystem

˙x = f (t, x

t

)

usedinTheorem2.1from[Kolmanovskii 1992℄isdes ribedbyafun tion

f

whi h is pie ewise ontinuous withrespe t to

t

. However, theproofof the resultin [Kolmanovskii1992℄ anbeadapted to overthis ase.

2.1.1.2 Constru tive stability onditions

VariousgeneralisationsoftheLyapunov-Krasovskiitheoremhavebeenproposedintheliterature.

Forthe ase ofsampled-datasystems,in[Fridman 2004℄theLyapunov-KrasovskiiTheoremwas

extended to linear systems with a dis ontinuous sawtooth delay by use of Barbalat lemma.

Another extension tolinear sampled-data systems hasbeen provided in[Fridman 2010℄, where

the LKF is allowed to have dis ontinuities at sampling times. Itleads toan LKF of the form

[Fridman 2010℄:

V (t, x(t), ˙x

t

) = x

T

(t)P x(t) + (h

k

− τ(t))

R

_t

t

_−τ(t)

˙x

T

(s)R ˙x(s)ds

(2.4) whi himprovestheresultsfrom[Fridman2004℄,astheinformation

˙τ = 1

anbeexpli itlytaken intoa ount whenevaluatingits derivative.

Theorem 2.2 [Fridman2010℄ Let there exist

P

≻ 0

,

R

≻ 0

,

P

2

and

P

3

su h thattheLMI



Φ

s

P

− P

2

T

+ (A + BK)

T

P

3

∗

−P

3

− P

3

T

+ hR



≺ 0

(2.5)





Φ

s

P

− P

T

2

+ (A + BK)

T

P

3

−hP

2

T

A

∗

−P

3

− P

3

T

−hP

3

T

A

∗

∗

−hR



 ≺ 0

(2.6) with

Φ

s

= P

T

2

(A + BK) + (A + BK)

T

P

2

,

are feasible. Then system (1.12) is Exponentially Stable for allsamplingsequen es

σ =

{t

k

}

k

∈N

with

h

k

= t

k+1

− t

k

≤ ¯h

.

The result takes into a ount information about the sawtooth shape of the delay, whi h is

thespe i ityofthetime-delay model (2.2)whenrepresenting exa tlythesampled-datasystem

(1.12) . It anensurethestabilityfortime-varyingdelays

τ (t)

whi harelongerthanany onstant delay that preservesstability, provided that

˙τ (t) = 1.

See also[Seuret 2009℄ for an alternative LMIformulation.

2.1.1.3 Anextension tononlinear systems

Con erning nonlinearsystems,[Mazen 2013a℄hasextendedtheideas in[Fridman 2004℄forthe

aseof ontrolanenon-autonomoussystemswithsampled-data ontrol. Considerthenonlinear

system:

˙x(t) = f (t, x(t)) + g(t, x(t))u(t),

(2.7) with the state

x(t)

∈ R

n

and the input

u(t)

∈ R

m

, and with fun tions

f

,

g

that are lo ally Lips hitzwithrespe t to

x

andpie ewise ontinuousin

t

. Assumethatthe

C

1

ontroller

u(t) =

K(t, x)

isdesignedinordertomakethesystem(2.7)GloballyUniformlyAsymptoti allyStable. Moreover,assumethatthereexista

C

1

positivedeniteandradiallyunboundedfun tion

V

,and a ontinuous positivedenite fun tion

W

su h that:

−

h∂V

_∂t

(t, x) +

∂V

∂x

(f (t, x) + g(t, x)K(t, x))

i

≥ W (x)

(2.8)

forall

t

≥ t

0

and

x

∈ R

n

. Also, onsider

K(t, 0) = 0

forall

t

∈ R

. Hen e,

V

isastri tLyapunov fun tionfor

˙x = f (t, x) + g(t, x)K(t, x)

andone anx lass

K

∞

fun tions

α

1

and

α

2

su h that

α

1

(

kxk

2

)

≤ V (t, x) ≤ α

2

(

kxk

2

)

,for all

t

≥ t

0

and

x

∈ R

n

. Dene thefun tion

ρ(t, x) =

∂K

∂t

(t, x) +

∂K

∂x



f (t, x) + g(t, x)K(t, x)



.

(2.9)

Theorem 2.3 (adaptedfrom [Mazen 2013a℄)Suppose thatthere exist onstants

c

1

,

c

2

,

c

3

and

c

4

su h that:

∂K

_∂x

(t, x)g(t, x)

2

2

≤ c

1

,

∂V

_∂x

(t, x)g(t, x)

2

2

≤ c

2

,

kρ(t, x)k

2

2

≤ c

3

W (x),

∂V

∂x

(t, x)g(t, x)K(t, x)

₂

≤ c

4

(V (t, x) + 1),

holdforall

t

≥ t

0

and

x

∈ R

n

. Considerthesystem(2.7)in losed-loopwith:

u(t) = K(t

k

, x(t

k

)),

t

_{∈ [t}

k

, t

k+1

)

,

σ =

{t

k

}

k∈N

as dened in (1.1) and

h

k

= t

k+1

− t

k

∈ [h, h]

,

∀k ∈ N

. Then, the losed-loop systemisGlobally UniformlyAsymptoti allyStable if

h

≤ (4c

1

+ 8c

2

c

3

)

−1/2

_.

The stabilityis provenby meansofaLyapunov fun tionalof theform

U (t, x

t

) = V (t, x(t)) +

ǫ

h

Z

0

−h

Z

t

t+θ

kΨ(s, x

s

)

_k

2

₂

dsdθ,

where

Ψ(t, x

t

) =

∂K

∂t

(t, x

s

(0)) +

∂K

∂x

(t, x

t

(0)) ˙x

t

(0).

This fun tional is reminis ent of the form (2.4) used in [Fridman 2004℄ to study LTI systems.

However,dierentlyfromtheLTI ase,itisfarmore omplextodeterminehow onservativethe

result is.

2.1.1.4 Further reading

The resear h on LKFs for sampled-data system is stilla wide-open domain. Currently, an

im-portanteort isdedi atedtondingbetterLKFs andbetter over-approximationsof thederiv

a-tives. Note that the derivation of onstru tive stability onditions may be quite an elaborate

analyti al pro ess and it is not always very intuitive. However, a notable advantage of this

methodology is the fa t that for linear systems the approa h an be easily extended to

on-trol design [Fridman 2004,Suplin 2007,Liu 2012a℄ and to the ase of systems with

parame-ter un ertainties [Fridman 2010,Seuret 2012,Orihuela 2010,Gao 2010,Peng 2011℄, delays

[Su-plin 2009,Mazen 2012,Gao 2008,Mazen 2013b,Seuret 2011,de Wouw 2010℄ and s hedulling

proto ols[Liu2012b,Liu2015b,Liu2015a℄. Seealso[Fridman2012,Fridman2013℄forthe ontrol

of semilinear1-D heatequations.

AsidefromtheLyapunov-Krasovskiimethod,thestability ofsampled-datasystems analso

be analysed using the method proposed by Razumikhin [Razumikhin 1956℄. Conne tions

be-tweenRazumikhin'smethod andthe ISS nonlinearsmallgaintheorem [Sontag1998℄havebeen

established in [Teel 1998a℄. This relation has been used in [Teel 1998b℄ in order to show the

preservation of ISS properties under su iently fast sampling for nonlinear systems with an

emulated sampled-data ontroller. Razumikhin'smethod hasbeen usedin[Fiter 2012a℄for the

aseofLTIsampled-datasystems. In[Karafyllis2009b℄, theRazumikhin methodisexploredfor

nonlinear sampled-data systemon the basisof ve tor Lyapunov-Razumikhin Fun tions (LRF).

For more general extensions to the ontrol design problem, see [Karafyllis 2012a℄, on erning

the ase of nonlinear feed-forward systems and [Karafyllis 2012b℄, for nonlinear sampled-data

systemwithinputdelays. Atlast,wewouldliketomention the Input/Output approa hfor the

analysisoftime-delaysystems[Fu1998,Gu2003a,Kao2004℄,whi hmakesuseof lassi alrobust

ontroltools[Zhou1996,Megretski1997℄. Theappli ationoftheInput/Outputapproa hforthe

ase of sampled-datasystemshas beendis ussed in[Mirkin 2007,Liu2010,Mi hiels2009℄. The

approa hwasfurtherdevelopedby[Fujioka2009 ,Omran2012a,Omran2014a,Omran

2014b,Om-ran2013,Chen2014℄withoutpassingthrough thetime-delaysystemmodel. Itwillbepresented

2.1.2 Hybrid system approa h

Due to the existen e of both ontinuous and dis rete dynami s, it is quite natural to model

sampled-data systems as hybrid dynami al systems [Goebel 2009,Goebel 2012,Haddad 2014,

Brogliato1996,Brogliato2016℄. Therstmentionstosampled-datasystemsashybriddynami al

systemsdateba ktothemiddleofthe'80s[Mousa1986℄. Lateron,inthe'90s,theuseofhybrid

modelshasbeendevelopedforlinearsampled-datasystemswithuniformandmulti-ratesampling

as an interesting approa h for the

H

∞

and

H

2

ontrol problems [Kabamba 1993,Sun 1993, Toivonen 1992b℄. The approa h has also been developed for nonlinear sampled-data systems

in [Hou1997,Ye1998℄. Forsystems withaperiodi sampling, impulsive modelshadbeen used

startingwith[Toivonen1992b,Dullerud1999,Mi hel1999℄. Re ently,moregeneralhybridmodels

havebeenproposedinthe ontextofNetworkedControlledSystemsby[Ne²i¢2004b,Ne²i¢2009℄.

Asolidtheoreti foundationhasbeenestablished forhybridsystemsintheframeworkproposed

by [Goebel 2009,Goebel 2012℄ and it proves to be very useful inthe analysis of sampled-data

systems.

In this se tion we will present some basi hybrid models en ountered in the analysis of

sampled-data systems. The extensionsof theLyapunov stability theoryfor hybridsystemswill

beintrodu edtogether with onstru tivenumeri alandanalyti stability analysis riteria.

2.1.2.1 Impulsive modelsforsampled-data systems

Considerthe ase of LTIsampled-data systems withlinear state feedba k, asin system(1.12) .

Let

x

ˆ

denote apie ewise onstant signalrepresentingthe mostre entstatemeasurement ofthe plant available atthe ontroller,

x(t) = x(t

ˆ

k

),

for all

t

∈ [t

k

, t

k+1

), k

∈ N.

Using the augmented systemstate

χ(t) = [x

T

_{(t), ˆ}

_x

T

_(t)]

T

_{∈ R}

n

χ

with

n

χ

= 2n

,the dynami sof theLTIsampled-data system(1.12) anbewritten undertheform



˙

χ(t)

= F χ(t),

t

6= t

k

, k

∈ N,

χ(t

k

) = Jχ(t

−

k

),

k

∈ N,

(2.10) with

χ(t

−

) = lim

θ

↑t

χ(θ), F =



A BK

0

0



, J =



I 0

I 0



.

(2.11)

Similar models an be determined by onsidering an augmented state ve tor

χ

in luding the most re ent ontrol value implemented at the plant

u(t) = u(t

ˆ

k

),

the sampling error

e(t) =

x(t)

_{− ˆx(t)}

, the a tuation error

e

u

(t) = u(t)

− ˆu(t)

, et . Models of the form (2.10) ,(2.11) t into the framework of impulsive dynami al systems [Milman 1960,Haddad 2014,

Lakshmikan-tham ,Bainov 1993℄ (sometimes also alled dis ontinuous dynami al systems or simply jump

systems). More general nonlinear sampled-data systemslead to impulsive systemsof the form

[Naghshtabrizi2008,Ne²i¢ 2004b℄

˙χ(t) = F

k

(t, χ(t)),

t

6= t

k

, k

∈ N,

(2.12a)

χ(t

k

) = J

k

(t

k

, χ(t

−

_k

)),

k

∈ N

(2.12b) where the augmented state mayalsoin lude the ontrollerstate and someof its sampled

om-ponents (state,output, et .). Generally, for animpulsive system,(2.12a) is alledthe system's

2.1.2.2 Lyapunov methodsfor impulsive systems

The stability of equilibria for the impulsive systems of the form (2.12) an be ensured by

the existen e of andidate Lyapunov fun tions that depend both on the system state and on

time, and evolve in a dis ontinuous manner at impulse instants [Bainov 1993,Haddad 2014,

Naghshtabrizi2008℄.

Theorem 2.4 [Naghshtabrizi 2008℄ Consider system (2.12) and denote

τ (t) = t

− t

k

,

∀t ∈

[t

k

, t

k+1

)

. Assume that

F

k

and

J

k

are lo ally Lips hitz fun tions from

R

+

× R

n

χ

to

R

n

χ

su h

that

F

k

(t, 0) = 0, J

k

(t, 0) = 0,

for all

t

≥ 0.

Let there exist positive s alars

c

1

,

c

2

,

c

3

,

b

and a Lyapunov fun tion

V :

R

n

χ

_{× R → R}

, su hthat

c

1

kχk

b

≤ V (χ, τ) ≤ c

2

kχk

b

,

(2.13) for all

χ

∈ R

n

χ

_{, τ}

_{∈ [0, h].}

Suppose that for any impulse sequen e

σ =

{t

k

}

k∈N

su h that

h

≤

t

k+1

− t

k

≤ h, k ∈ N,

the orrespondingsolution

χ(

·)

to (2.12) satises:

dV (χ(t), τ (t))

dt

≤ −c

3

V (χ(t), τ (t)) ,

∀t 6= t

k

,

∀k ∈ N,

and

V (χ(t

k

), 0)

≤ lim

t

→t

−

k

V (χ(t), τ (t)) ,

∀k ∈ N.

Then, theequilibrium point

χ = 0

ofsystem (2.12) is Globally Uniformly Exponentially Stable over the lass of sampling impulse instants,

i.e. there exist

c, λ > 0

su h thatforany sequen e

σ =

{t

k

}

k∈N

thatsatises

h

≤ t

k+1

− t

k

≤ h,

k

_{∈ N,}

kχ(t)k ≤ ckχ(t

0

)

ke

−λ(t−t

0

)

,

∀t ≥ t

0

.

The previous stability theorem requires in (2.13) the andidate Lyapunov fun tion to be

positive at all times. For the ase of system (2.12) with globally Lips hitz

F

k

, k

∈ N,

the ondition an be relaxed by requiring the Lyapunov fun tion to be positive only at impulse

times [Naghshtabrizi 2008℄, i.e.

c

1

kχ(t

k

)

k

b

_{≤ V (χ(t}

k

), 0)

≤ c

2

kχ(t

k

)

k

b

,

∀k ∈ N,

instead of (2.13) .

In the ase of impulsive systems (2.10) , with linear ow and jump dynami s, andidate

Lyapunov fun tionsof the form

V (χ, τ ) = χ

T

_{P (τ )χ,}

with

P : [0, ¯

h]

→ R

n

χ

×n

χ

adierentiable

matrix fun tion, have been used [Toivonen 1992a,Sun 1993,Briat 2013,Naghshtabrizi 2008℄.

Su ient stability onditions an be obtained from Theorem 2.4 in terms of existen e of a

dierentiable matrixfun tion

P : [0, h]

→ R

n

χ

×n

χ

_{, c}

1

I

≺ P (τ) ≺ c

2

I

, satisfyingthe parametri setof LMIs

F

T

P (θ

1

) + P (θ

1

)F + c

3

P (θ

1

) +

∂P

∂τ

(θ

1

)

≺ 0,

∀ θ

1

∈ [0, h],

(2.14a)

J

T

P (0)J

− P (θ

2

)

≺ 0, ∀ θ

2

∈ [h, h],

(2.14b) withpositives alars

c

1

, c

2

, c

3

. Thisformulationisreminis ent ofthe Ri atiequationapproa h usedforrobustsampled-data ontrolin[Toivonen 1992b,Sun 1993℄.

2.1.2.3 Numeri allytra table riteria

In pra ti e,the di ulty of he king the existen eof andidate Lyapunov fun tions usingLMI

in

[0, ¯

h]

or

[h, h]

, whi h leads to an innite number of LMIs. As follows we will dis uss the derivationof anitenumberofLMIs from(2.14) .

Con erning the parametri set of LMIs (2.14) , a nite number of LMI onditions an be

derived by onsidering parti ular forms for the matrix fun tion

P (τ )

. Forexample, onsider a matrix

P (τ )

linear withrespe tto

τ

P (τ ) = P

1

+ (P

2

− P

1

)

τ

h

,

(2.15)

for some positive denite matri es

P

1

, P

2

, asin [Hu 2003,Allerhand2011℄. There, su h a Lya-punov matrix has been usedfor sampled-data systems with multi-rate sampling and swit hed

linear systems. Fora andidate Lyapunov fun tion

V (χ, τ ) = χ

T

_{P (τ )χ}

, with

P (τ )

as dened in(2.15) , anitesetof LMIs that aresu ient for stability an beobtainedfrom (2.14)using

simple onvexity arguments:

F

T

P

1

+ P

1

F + c

3

P

1

+

P

2

− P

1

h

≺ 0,

(2.16a)

F

T

P

2

+ P

2

F + c

3

P

2

+

P

2

− P

1

h

≺ 0,

(2.16b)

J

T

P

1

J

≺ P

2

,

(2.16 )

J

T

P

1

J

≺ P

1

+ (P

2

− P

1

) h/h.

(2.16d)

Fortheparti ular aseofLTIsampled-data systemsrepresented by (2.10) ,(2.11), Lyapunov

fun tions of the form

V (χ, τ ) = χ

T

_{P (τ )χ}

are proposed in the literature by summing various

terms su has:

V

1

(χ, τ ) = x

T

P

0

x

(2.17)

V

2

(χ, τ ) = (x

− ˆx)

T

Q (x

− ˆx) (h − τ)

(2.18)

V

3

(χ, τ ) = (x

− ˆx)

T

R (x

− ˆx) e

−λτ

(2.19)

V

4

(χ, τ ) = χ

T

 Z

0

−τ

(s + h)(F e

F s

)

T

U (F e

˜

F s

)ds



χ,

(2.20) where

U :=

˜



U

0

0

0



, λ > 0

and

P

0

,

R

,

U

are symmetri positive denite matri es. Using su h parti ular formsof Lyapunov fun tions,LMI stability onditionshave beenderived inthe

literature[Hu2003,Naghshtabrizi2008,Ne²i¢2009,Omran2012b,Goebel2012℄. Wepointin

par-ti ulartotheterm(2.20)usedin[Naghshtabrizi2008℄whi hprovidedasigni antimprovement

in what on erns the onservatism redu tion. This term is inspired by Lyapunov-Krasovskii

fun tionals fromthe input-delay approa h, likethe onein [Fridman 2004℄. Notethat the term

(2.20) an alsobe written as

R

t

t

−τ

(s + h

− t) ˙x

T

(s)U ˙x(s)ds.

Ithas been motivated by the term

R

0

−h

R

t

t+θ

˙x

T

(s)U ˙x(s)dsdθ

usedinthetime-delayapproa h (see[Fridman 2004℄). Vi eversa,the hybrid systemapproa h hasalsoinspired the useof dis ontinuous Lyapunov fun tionals inthe

time-delay approa h (see for example the fun tional (2.4) whi h is dis ontinuous at sampling

times). Notethatthe term

(h

k

− τ)

R

t

t

−τ

˙x

T

(s)R ˙x(s)ds

inthe fun tional(2.4) anbere-written as

(h

k

− τ)χ

T

 R

0

−τ

(F e

F s

)

T

R(F e

˜

F s

)ds



χ,

with

˜

R =



R 0

0

0