Contribution to the control of systems with time-varying and state-dependent sampling

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and state-dependent sampling

Christophe Fiter

To cite this version:

Christophe Fiter. Contribution to the control of systems with time-varying and state-dependent

sampling. Other. Ecole Centrale de Lille, 2012. English. �NNT : 2012ECLI0021�. �tel-00773127�

N d'ordre :

1 9 7

ÉCOLE CENTRALE DE LILLE

THÈSE

présentée en vued'obtenir legrade de

DOCTEUR

Spé ialité : Automatique, Génie Informatique,Traitement du Signal etImage

par

Christophe Fiter

Ingénieur diplmé de l'É ole Centrale de Lille

Do torat délivré parl'É ole Centrale deLille

Contribution à la ommande robuste des

systèmes à é hantillonnage variable ou ontrlé

Soutenue le25 septembre2012 devantle jury omposé de:

Président: M. Jamal Daafouz Professeur àl'INPL, Nan y

Rapporteur: Mme. EmiliaFridman Professeur àl'Université de TelAviv

Rapporteur: Mme. SophieTarbourie h Dire teur deRe her he CNRS auLAAS

Rapporteur: M. Hugues Mounier Professeur au LSS,Supéle , Gif-sur-Yvette

Membre: M. DanielSimon Chargé de Re her he INRIA àGrenoble

Dire teur de thèse: M. Jean-Pierre Ri hard Professeur àl'É ole Centrale de Lille

Co-dire teur de thèse: M. WilfridPerruquetti Professeur àl'É ole Centrale de Lille

Co-en adrant dethèse: M. Laurentiu Hetel Chargé de Re her he CNRS auLAGIS

Thèse préparée auLaboratoired'Automatique, Génie Informatique etSignal

L.A.G.I.S, UMR CNRS 8219 - É ole Centrale de Lille

É ole Do toraleSPI 072 (LilleI, Lille III, Artois, ULCO, UVHC,EC LILLE)

Serial N :

1 9 7

ECOLE CENTRALE DE LILLE

THESIS

Presented to obtainthe degreeof

DOCTOR

Spe iality : Controltheory, omputer s ien e, signal pro essing and image

by

Christophe Fiter

Engineer from É ole Centrale de Lille

PhD awardedbyÉ ole Centrale de Lille

Contribution to the ontrol of systems with

time-varying and state-dependent sampling

Defendedon September25th, 2012 inpresen eof the ommittee:

Chairman: M.Jamal Daafouz Professorat INPL,Nan y

Examiner: Mme. Emilia Fridman Professorat Tel AvivUniversity

Examiner: Mme. SophieTarbourie h CNRSResear h Dire tor at LAAS

Examiner: M.Hugues Mounier Professorat LSS,Supéle ,Gif-sur-Yvette

Member: M.Daniel Simon INRIAResear h Asso iateinGrenoble

Thesis supervisor: M.Jean-Pierre Ri hard Professorat É ole Centrale de Lille

Thesis o-supervisor: M.WilfridPerruquetti Professorat É ole Centrale de Lille

Thesis o-supervisor: M.Laurentiu Hetel CNRSResear h Asso iateat LAGIS

Thesis prepared at Laboratoired'Automatique, Génie Informatiqueet Signal

L.A.G.I.S, UMR CNRS 8219 - É ole Centrale de Lille

É ole Do toraleSPI 072 (LilleI, Lille III, Artois, ULCO, UVHC,EC LILLE)

ThePhDworkpresentedinthisthesishasbeen ondu tedat"Laboratoired'Automatique,

GénieInformatique etSignal" (LAGIS)at É ole Centrale de Lille, fromO tober 2009to

September 2012, under the supervision of Professor Jean-Pierre Ri hard, Professor

Wil-fridPerruquetti,and CNRSResear hAsso iate LaurentiuHetel. Itwassupportedby the

"Centre National de la Re her he S ientique" (CNRS) and the "Conseil Régional de la

RégionNord-Pas de Calais".

Iwouldliketoexpressmysin eregratitudetomyadvisorsJean-PierreRi hard,Wilfrid

Perruquetti, and Laurentiu Hetel for everything they did for me in the last three years.

Firstofall,Iwouldliketothankthemforgivingmetheopportunitytoworkwiththemas

partoftheirteam. Iwouldalsoliketothankthemforthetrusttheyhavepla edinmeand

uponmyworkduringtheseyears, fortheirpatien e, andfortheir disponibility. Finally,I

would like tothank them fortheir help and advi e,their experien eand knowledgethey

shared,and for allthe dis ussions that havebroughtlife tothis work.

Likewise,Iwouldliketoexpressmysin eregratitudetoea handeverymemberofmy

PhD ommittee, forhavinga epted to readand examine the presentwork, namely

Pro-fessor EmiliaFridman, fromthe University of Tel Aviv, CNRS Resear h Dire tor Sophie

Tarbourie h, from the "Laboratoire d'Analyse et d'Ar hite ture des Systèmes" (LAAS)

inToulouse,Professor HuguesMounier, fromthe "Laboratoiredes SignauxetSystèmes"

at Supéle in Gif-sur-Yvette, Professor Jamal Daafouz, from the "Centre de Re her he

en Automatique de Nan y" at the "Institut National Polyte hnique de Lorraine", and

INRIAResear hAsso iate Daniel Simon,from INRIA Rhne-Alpes, in Grenoble.

I would like to thank all the members of the teams "Systèmes Nonlinéaires et à

Re-tards" (SyNeR, LAGIS) and "Non-Asymptoti Estimation for Online Systems"(Non-A,

INRIA) for the dis ussions we have had and for the great atmosphere. I would like to

mention espe ially the olleagues and friends from my o e: my two fellow sportsmen

Hassan Omran and Bo Zhang, Romain Delpoux, Emmanuel Bernuau, Yang Tian, as

wellasAsso iateProfessor AlexandreKruszewski andCNRS Resear hAsso iate Thierry

Floquet,who have taught a melotpreviously to and during the PhD.

Iwould alsoliketothankProfessorPhilippeVanheeghe, dire torofLAGIS,aswellas

the se retary sta, my fellow CNRS olleague Christine Yvoz, Brigitte Fon ez, Virginie

Le ler qandVanessaFleury,whorelievedmeofanenormousamountofadministrational

work,andRégineDuploui h,fromthedo umentation enter. Additionally,Iwouldliketo

and the great atmosphere they bring to the laboratory.

Finally, I would like to thank my family for their ontinuous support, with spe ial

A ronyms 13

Notations 15

General introdu tion 17

Chapter 1 Sampled-data systems: an overview of re ent resear h

dire -tions 23

1.1 Introdu tionto sampled-datasystems . . . 23

1.1.1 General sampled-datasystems . . . 23

1.1.2 Sampled-data lineartime-invariantsystems . . . 24

1.1.3 Problemati s . . . 26

1.2 Classi al stability on epts . . . 26

1.2.1 Some stability denitions. . . 27

1.2.2 Se ond Lyapunov method . . . 28

1.2.3 Properties oflineartime-invariantsystemswith sampled-data ontrol 30 1.3 Stabilityanalysis under onstant sampling . . . 30

1.4 Stabilityanalysis undertime-varying sampling . . . 32

1.4.1 Di ulties and hallenges . . . 32

1.4.2 Time-delay approa h with Lyapunov te hniques . . . 37

1.4.2.1 Lyapunov-Razumikhin approa h . . . 39

1.4.2.2 Lyapunov-Krasovskii approa h . . . 40

1.4.3 Small-gainapproa h . . . 42

1.4.4 Convex-embedding approa h . . . 44

1.5 Dynami ontrol of the sampling: a short survey . . . 46

1.5.1 Deadband ontrolapproa h . . . 48

1.5.3 Perturbation reje tion approa h . . . 51

1.5.4 ISS-Lyapunov fun tion approa h . . . 53

1.5.5 Upper-bound onthe Lyapunov fun tion approa h . . . 55

1.5.6

L

2

-stability approa h . . . 57

1.6 Con lusion . . . 59

Chapter 2 A polytopi approa h to dynami sampling ontrol for LTI sys-tems: the unperturbed ase 61 2.1 Problemstatement . . . 62

2.2 A generi stability property . . . 64

2.3 Mainstability results . . . 66

2.3.1 Te hni al tools . . . 66

2.3.1.1 Coni overing of the state-spa e . . . 66

2.3.1.2 Convex embeddinga ording to time . . . 68

2.3.2 Stability results inthe ase of state-dependent sampling . . . 69

2.3.3 Stability results inthe ase of time-varying sampling . . . 69

2.4 Generalalgorithmto designthe samplingfun tion . . . 70

2.5 Numeri alexamples. . . 71

2.5.1 Example 1 . . . 71

2.5.2 Example 2 . . . 74

2.6 Con lusion . . . 76

Chapter 3 A polytopi approa h to dynami sampling ontrol for LTI sys-tems: the perturbed ase 77 3.1 Problemstatement . . . 78

3.2 Mainstability results . . . 81

3.3 Robuststabilityanalysiswithrespe ttotime-varyingsampling- Optimiza-tion of the parameters . . . 83

3.4 Event-triggered ontrol . . . 87

3.4.1 Over-approximation based event-triggered ontrols heme . . . 87

3.4.2 Perturbation-aware event-triggered ontrols heme . . . 88

3.4.3 Dis rete-time approa h event-triggered ontrols heme . . . 89

3.5 Self-triggered ontrol . . . 89

3.6 State-dependent sampling . . . 93

Chapter 4 A Lyapunov-Krasovskii approa h to dynami sampling

on-trol 101

4.1 Problem formulation . . . 103

4.2 Main

L

2

-stabilityresults . . . 105

4.2.1 Stability analysis of the perturbed system . . . 106

4.2.1.1 Continuity,pie ewise dierentiability,and positivity

onditions of the Lyapunov-Krasovskii Fun tional . . 107

4.2.1.2

L2

-stability onditions . . . 108

4.2.2 Stability analysis of the perturbed system with delays . . . 111

4.2.3 Algorithmtodesignthestate-dependentsamplingfun tion

τ

max for

a given feedba k matrix gain

K

. . . 118

4.3 Main

L2

-stabilizationresults . . . 121

4.3.1 Stabilization using a pie ewise- onstant feedba k ontrol

u(t) =

−Kx(s

k)

. . . 121

4.3.2 Stabilization using aswit hing pie ewise- onstant feedba k ontrol

u(t) =

_−K

σ

k

x(sk)

. . . 124

4.3.3 Algorithmtodesignthestate-dependentsamplingfun tion

τ

max and

its asso iated feedba k matrix gain

K

(or gains

Kσ

) . . . 126

4.4 Numeri al examples. . . 128

4.4.1 Example 1 - State dependent samplingfor systems with

perturba-tions and delays . . . 128

4.4.2 Example 2 - Conservatism redu tion thanks tothe swit hed LKF . 129

4.4.3 Example 3 - State-dependent samplingfor systems whi h are both

open-loopand losed-loop(witha ontinuousfeedba k ontrol)

un-stable . . . 130

4.4.4 Example4-State-dependentsampling ontrollerforperturbed

sys-tems . . . 131

4.5 Con lusion . . . 132

General on lusion 135

Appendix A Proofs 151

A.1 Proofs fromChapter 2 . . . 151

A.2 Proofs fromChapter 3 . . . 152

Appendix B Constru tion of the oni regions overing 159

B.1 Isotropi state- overing: using the spheri al oordinates of the state . . . . 159

B.2 Anisotropi state- overing: usingthe dis rete-timebehaviour of the system 161

Appendix C Contru tion of a polytopi embedding based on Taylor

poly-nomials 163

C.1 General ontru tionfor polynomial matrix fun tions. . . 163

C.2 Case of unperturbed LTI systems (Chapter 2) . . . 164

C.3 Case of perturbed LTI systems (Chapter 3). . . 167

Appendix D Some useful matrix properties 173

1 Analog-to-digital onversion . . . 17

2 Digital-to-analog onversion . . . 18

1.1 Sampled-datasystem . . . 24

1.2 Sampled-datasystem with a onstant samplingrate . . . 31

1.3 Sampled-datasystem with atime-varying sampling . . . 33

1.4 Evolutionofthemodulus

|λ

max

(T )

|

ofthemaximumeigenvalueofthe tran-sition matrix

Λ(T )

,depending onthe sampling period

T

. . . 33

1.5 Constantsamplingratewith

T

1

= 0.18s

(left)and

T

2

= 0.54s

(right)-Stable 34 1.6 Variable sampling intervals

T

1

= 0.18s

→ T2

= 0.54s

→ T1

→ T2

→ · · ·

-Unstable . . . 34

1.7 Stability domain (allowable sampling interval) for a periodi sampling se-quen e

T1

→ T

2

→ T

1

→ T

2

→ · · ·

- rst example . . . 35

1.8 Stability domain (allowable sampling interval) for a periodi sampling se-quen e

T1

→ T

2

→ T

1

→ T

2

→ · · ·

- se ond example . . . 36

1.9 Variable sampling

T

1

= 2.126s

→ T2

= 3.950s

→ T1

→ T2

→ · · ·

- Stable . 37 1.10 Samplingseen as a pie ewise- ontinuous time-delay . . . 38

1.11 Inter onne ted system . . . 43

1.12 Sampled-datasystem with adynami sampling ontrol . . . 47

1.13 Event-triggered ontrolfrom[Cervin 2007℄ applied onadouble integrator . 49 1.14 Lyapunovfun tionlevelsapproa htodynami sampling ontrol[Velas o2009℄ -

η

= 0.8

≥ η

∗

,stable (left) and

η

= 0.65 < η

∗

, unstable (right) . . . 51

2.1 Covering the state-spa e of dimension

2

with

q

oni regions

R

s

. . . 67

2.2 2D representationof a onvex polytopearound thematrix fun tion

Φ

over the time interval

σ

∈ [0, τ

s]

. . . 68

2.3 Example 1: State-angle dependent sampling fun tion

τ

for dierent de ay rates

β

. . . 72

2.4 Example 1: Inter-exe ution times

τ(x(sk))

and LRF

V

(x) = x

T

_{P x}

for a de ay rate

β

= 0

. . . 73

2.5 Example 1: Inter-exe ution times

τ(x(sk))

and LRF

V

(x) = x

T

_{P x}

for a de ay rate

β

= 0.05

. . . 73

2.6 Example2: Mappingofthestate-spa e(regardingtheangular oordinates)

2.7 Example 2: Inter-exe ution times

τ

(x(sk))

and LRF

V

(x) = x

T

_{P x}

for a de ay rate

β

= 0

. . . 76

3.1 Illustrationof the onvex embedding design . . . 85

3.2 Illustrationof the property of the onvex embeddingdesign with

subdivi-sions fromAppendix C.3 aroundthe matrix fun tion

∆

. . . 91

3.3 State-angledependentsamplingmap

τ

max

for dierentde ay-rates(

β

)and perturbations (

W

) . . . 97

3.4 Inter-exe ution times

τ

max

(x(sk))

and LRF

V

(x) = x

T

_{P x}

for ade ay rate

β

= 0.3

and

W

= 0

- State-dependent sampling . . . 98

3.5 Inter-exe ution times

τ

max

(x(sk))

for a de ay rate

β

= 0.1

and

W

=

0.04

(

kw(t)k2

≤ 20%kx(s

k)

k2

) - First event-triggered ontrols heme,

self-triggered ontrol, and state-dependent sampling . . . 98

4.1 Algorithm to design the state-dependent sampling fun tion

τ

max

(x)

for a given feedba k matrix gain

K

. . . 119

4.2 Algorithmtodesignthe state-dependentsamplingfun tion

τ

max

(x)

andits asso iatedfeedba k matrix gain

K

(or gains

Kσ

). . . 127

4.3 Example1: Mappingofthemaximaladmissiblesamplingintervals

τ

+

σ

with

orwithout perturbations

w

and/or delays

h

. . . 129

4.4 Example 1: Leftside: delayed ase (delays up to

0.1s

). Rightside:

delay-free ase. In both sides, the perturbation satises

kw(t)k

2

=

1

γ

kz(t)k

2

≃

32%

kz(t)k2

. . . 130

4.5 Example 3: Mapping of the maximal admissible sampling intervals for

dierentminimalsamplingintervals

τ

−

(onthe left)andsimulationresults

using the samplingfun tion obtained with

τ

−

_{= 0.25}

(on the right) . . . . 131

4.6 Example 4: Mapping of the maximal admissible sampling intervals for

dierent

L

2

gains

γ

, with orwithout swit hing ontroller . . . 132

4.7 Example4: State

x(t)

andsamplingintervals

τk

= τ

max

(x(sk))

for the

on-trolled system without perturbation (on the left) and with a perturbation

satisfying

kw(t)k

2

=

1

γ

kz(t)k

,

γ

= 2

(on the right) . . . 133

1 Conversion analogiquenumérique . . . 140

2 Conversion numérique analogique . . . 140

3 Re ouvrement de l'espa ed'état de dimension

2

par

q

régions oniques

R

s

143

4 Système LTI é hantillonnéave perturbations etretards . . . 146

B.1 Covering the state-spa e of dimension

2

with

q

oni regions

R

s

. . . 160

C.1 2D representation of the onvex polytope design using polytopi

- ISS = Input-to-StateStability.

- LKF = Lyapunov-Krasovskii Fun tional.

- LMI = LinearMatrix Inequality.

- LTI = LinearTime-Invariant.

- LRF = Lyapunov-RazumikhinFun tion.

Notations on erning sets:

-

R

+

isthe set

{λ ∈ R, λ ≥ 0}

. -

R

∗

is the set

{λ ∈ R, λ 6= 0}

.

-

M

n,m(R)

denotes the set of real

n

× m

matri es.

-

M

n(R)

denotes the set of real

n

× n

matri es.

-

Sn

denotes the set of symmetri matri esin

M

n(R)

.

-

S

+

n

(resp.

S

+∗

n

) denotes the set of positive (resp. positive denite) symmetri matri es

in

M

n(R)

.

- Co

{F

i

}

i∈I

, for given matri es

Fi

∈ M

n,m(R)

and a nite set of indexes

I

, denotes the

onvex polytope in

M

n,m(R)

formed by the verti es

Fi, i

∈ I

.

-

C

0

_(X

_{→ Y )}

, for two metri spa es

X

and

Y

,is the set of ontinuous fun tions from

X

to

Y

.

-

L2

isthe spa e of square-integrablefun tions from

R

+

to

R

n

.

-

λX

,for a s alar

λ

∈ R

and an

R

ve tor spa e

X

, represents the set

{λx, x ∈ X}

.

-

R

∗

_x

, with

x

∈ R

n

, is the set dened as

{y ∈ R

n

_,

_{∃λ 6= 0, y = λx}}

.

-

|X|

, is the ardinality of the nite set

X

.

-

P(X)

denotes the power set of a set

X

(i.e. the set of allsubsets of

X

).

Notations on erning matri es:

-

M

T

stands for the transpose of

M

∈ M

n,m(R)

.

-

M

+

isthe pseudoinverse of

M

∈ M

n,m(R)

.

-

A

 B

(resp.

A

≻ B

)formatri es

A, B

∈ M

n(R)

meansthat

A

− B

isapositive(resp.

denitepositive) matrix.

-

I

is the identity matrix (of appropriate dimension).

-

∗

, ina matrix, denotes the symmetri elements of a symmetri matrix.

- diag

(A1

,

· · · , A

m)

is the blo k diagonalmatrix designed by the square matri es

Ai, i

∈

{1, · · · , m}

, of any dimension.

-

λ

max

(M)

(resp.

λ

min

(M)

) denotes the largest (resp. lowest) eigenvalue of a symmetri

matrix

M

∈ M

n(R)

.

-

ρ(M)

denotes spe tral radiusof

M

∈ M

n(R)

.

-

|||.|||2

standsfor the operatornorm on

M

n(R)

asso iatedtothe norm

k.k2

on

R

n

: for a matrix

M

∈ M

n(R)

,

|||M|||2

= sup

kxk

2

=1

kMxk2

=

pρ(M

T

_M

₎

.

Notations on erning ve tors:

-

x

T

stands for the transpose of

x

∈ R

n

.

-

k.k2

stands for the Eu lidean normon

R

n

: for ave tor

x

∈ R

n

,

kxk2

=

√

x

T

_x

.

Notations on erning s alars:

-

⌊x⌋

isthe oor of

x

∈ R

: the largestinteger not greaterthan

x

:

x

− 1 < ⌊x⌋ ≤ x

. -

⌈x⌉

isthe eiling of

x

∈ R

: the smallestinteger not less than

x

:

x

≤ ⌈x⌉ < x + 1

. - sgn

(x)

denotes the sign ofthe s alar

x

.

- sat

(x)

denotes a s alar that is equal to

−1

if the s alar

x

≤ −1

,

1

if

x

≥ 1

, and

x

otherwise.

Notations on erning fun tions:

-

xt

(resp.

˙xt

)denotes thefun tionin

C

0

_([

_{−¯h, 0] → R}

n

₎

,foragivenmaximaldelay

¯h

su h that

xt(θ) = x(t + θ),

∀θ ∈ [−¯h, 0]

(resp.

˙xt(θ) = ˙x(t + θ),

∀θ ∈ [−¯h, 0]

).

-

k.k

L

2

is the

L

2

-norm on

L

2

: fora fun tion

f

∈ L

2

,

kfk

L

2

=

R

∞

0

kf(t)k

2

2

dt

1

₂

. -

k.k

H

∞

is the

H

∞

-norm on

L2

→ L2

: for an operator

∆ : u

∈ L2

7→ v ∈ L2

,

k∆k

H

∞

=

sup

w∈R

+

k∆(jw)k

, with

k∆(jw)k =

max

kzk

2

=1, z∈C

n

k∆(jw)zk2

. It is equal to the

L2

-to-

L2

norm:

k∆k

H

∞

=

k∆kL

2

→L

2

= sup

u6=0

kvk

L

2

kukL

2

.

-A lass

K

fun tionisafun tion

ϕ

: [0, a)

→ [0, +∞)

thatisstri tly in reasing,and su h that

ϕ(0) = 0

.

- A lass

K∞

fun tion isa lass

K

fun tion su h that

a

= +

∞

and

limt→+∞

ϕ(t) = +

∞

.

- A

C

∞

fun tion isa fun tion that isinnitely dierentiable.

-

f(n) = O(g(n))

meansthat the growth-rate of the sequen e

f

(n)

,

n

∈ N

, isdominated

by the sequen e

g(n)

, i.e. there exist

N

∈ N

and

K

∈ R

∗

+

su h that for all

n

≥ N

,

|f(n)| ≤ K|g(n)|

.

Notations on erning logi :

-

∧

denes the "AND" logi gate. -

∨

denes the "OR"logi gate.

Other notations:

Untilthe

50s

,mostsystemswere ontrolledusinganalogi al ontrollers. However, thefast developmentof omputersledtoanin reasinguse ofdigital ontrollers. Thisisespe ially

due to their omputational power and exibility. Nowadays, digital ontrollers have

be- omeomnipresent,andenabledtheexplosionofembeddedsystemsandnetworked ontrol

systems. They oerseveral advantages: low ost installationand maintenan e, in reased

exibility and re-usability, redu ed wiring ost, and ease of programming. Furthermore,

they oer the possibilityto ontrolmore than one pro ess ata time.

Unlikeanalogi al ontrollers,digital ontrollers,duetotheirnature,introdu e

dis rete-time signalsand dis rete-time dynami s,via sample and hold devi es [Aström 1996℄.

First, the information sent from the sensors to the ontroller is sampled, by means

of ananalog-to-digital(A/D) onverter. Su h a onversion of aninput signal

x(t)

intoa

sampledsignal

x(sk)

, at samplinginstants

sk, k

∈ N

isshown inFigure 1.

s

₀

s

₁

s

₂

s

₃

s

₄

s

₅

s

₆

s

₇

s

₈

s

₉

s

₁₀

s

₁₁

s

₁₂

s

₁₃

s

₁₄

s

₁₅

s

₁₆

x

t

Continuous signal x(t)

Sampled−data signal x(s

_k

)

Figure 1: Analog-to-digital onversion

a digital-to-analog (D/A) onverter (a zero-order-hold), so as to hold the ontrol value

that is sent to the a tuators. The onversion of a sampled input signal

u(sk)

into a

pie ewise- onstant signal

u(t)

, isshown in Figure2.

s

0

s

1

s

2

s

3

s

4

s

5

s

6

s

7

s

8

s

9

s

10

s

11

s

12

s

13

s

14

s

15

s

16

s

17

u

t

Sampled−data signal u(s

k

)

Piecewise−constant signal u(t)

Figure 2: Digital-to-analog onversion

In embedded ontrol appli ations however, a dis rete-time implementation may

pro-du eundesiredee tssu hasdelaysoraperiodi ontrolexe utions,duetotheintera tion

between ontroltasks and real-times heduler me hanisms [Hristu-Varsakelis 2005℄. The

ee ts of these dis rete-timedynami s brought up new hallenges regarding the stability

andstabilization,and newtheoriesandtoolshavebeen developedfor thesesampled-data

systems. In parti ular, in the last few years, two main problems have been of a great

importan efor ontroltheorists:

P1) the stability of sampled-datasystems with time-varying sampling;

P2) the dynami ontrolof the samplingevents.

The new trend is to ontrol dynami ally the sampling so as to enlarge the sampling

intervalsand redu ethe omputationaland energeti osts.

Goals

The work presented in this thesis is on erned with these two problems P1) and P2).

system stability.

In order avoidpossible s hedulingissues, the robustness with respe t totime-varying

samplingwillalsobein luded. The robustness aspe t with respe t toexogenous

pertur-bationsordelaysinthe ontrolloopwillbe onsidered,sototakeintoa ountphenomena

o uringinthereal-time ontrolofphysi alsystems. Finally,a o-design ofthe ontroller

and sampling law is proposed. Here, in order to redu e the onservatism, the ontrol

gainsand the samplinginstants willbe omputed jointly.

Throughout the thesis, dierent designs of sampling ontrol laws will be presented.

They an be used to ompute a simple upper-bound for time-varying samplings, or to

dynami ally ontrolthe sampling intervals, using onlineor oinealgorithms.

Stru ture of the thesis

The thesis isorganized as follows:

Chapter 1

The rst hapter is a literature survey whi h presents an overview of problems,

hal-lenges, and re ent resear h dire tions in the domain of sampled-data systems in ontrol

theory. First, the notion of sampled-data systems is dened, and the main open

prob-lems in the literature are presented. Then, some general stability on epts ne essary to

the omprehension are re alled. Finally, several resear h dire tions, theories, and results

are presented on erning the stability analysis of sampled-datasystems with onstant or

time-varyingsampling,or on erningthe dynami ontrolofthesampling. Thestrengths

andweaknesses ofthedierentapproa hesareanalyzed,soastohighlightwhi hproblems

have already been solved, and what stillremains tobe done orimproved.

Chapter 2

Inthese ond hapter,astate-dependentsampling ontrolisdesignedforidealLTIsystems

with sampled-data. The goalis to design a sampling lawthat willtake intoa ount the

system's state, soas toenlarge the samplingintervals,or inother terms, togenerate the

samplingevents assparselyaspossible. Theproposedstate-dependentsamplingfun tion

takes advantage of anoine design based onLMIs obtained thanks toa mappingof the

Chapter 3

In the third hapter, the robustness aspe t with respe t to exogenous disturban es is

onsidered for the design of a state-dependent sampling law. As in the se ond hapter,

theapproa hisbasedonLyapunov-Razumikhinstability onditionsandpolytopi

embed-dings. Afterpresentingthemainstabilityresults,fourdierentappli ationsareaddressed.

Therstone on erns therobuststabilityanalysiswithrespe ttotime-varyingsampling.

The other three appli ations propose dierent approa hes to the dynami ontrol of the

sampling with the obje tive to enlarge the sampling interval. Event-triggered ontrol,

self-triggered ontrol, and the newly introdu ed state-dependent sampling s hemes are

then presented.

Chapter 4

In the fourth and last hapter, an extension to the stability analysis of perturbed

time-delaylinearsystemsista kled,andthestabilizationissueis onsidered. Theobje tivehere

istodesigna ontrolleralongwiththestate-dependentsamplinglaw,soastostabilizethe

onsidered perturbed LTI sampled-data system, and enlarge even further the allowable

samplingintervals. First,the aseofa lassi linearstate-feedba k ontrolleris onsidered.

Then, a new ontroller is proposed, the gains of whi h are swit hing a ording to the

system's state. The o-design of both the ontroller and the state-dependent sampling

fun tion is based on LMIs obtained thanks to the mappingof the state-spa e presented

in the previous hapters, and thanks to a new lass of Lyapunov-Krasovskii fun tionals

with matri esswit hing with respe t to the system's state.

Personal publi ations

The resear h exposed in this thesis an befound in the followingpubli ations:

Journals

ˆ C. Fiter,L.Hetel, W.Perruquetti, andJ.-PRi hard-A StateDependent Sampling

for Linear State Feedba k - Automati a, Volume 48, Number 8, Pages 1860-1867,

August 2012. doi:10.1016/j.automati a.2012.05.063

ˆ C. Fiter, L. Hetel, W. Perruquetti, and J.-P Ri hard - A Novel Stabilization

provision-ˆ C.Fiter,L.Hetel,W.Perruquetti,andJ.-PRi hard-ARobustStability Framework

forTime-Varying Sampling -Automati a, submitted.

International onferen es

ˆ C.Fiter,L.Hetel,W.Perruquetti,andJ.-PRi hard-StateDependentSampling: an

LMIBasedMappingApproa h -18thIFACWorldCongress,Milan,Italy,September

2011.

ˆ C.Fiter,L.Hetel,W.Perruquetti, andJ.-PRi hard-State-DependentSamplingfor

Perturbed Time-Delay Systems - 51st IEEE Conferen e on De ision and Control,

Maui, Hawaii, USA,De ember 2012.

ˆ C. Fiter, L. Hetel, W. Perruquetti, and J.-P Ri hard - A Robust Polytopi

Ap-proa h for State-DependentSampling -12th European ControlConferen e, Zuri h,

Switzerland,July 2013 -submitted.

National onferen es

ˆ C. Fiter - E hantillonnage Dépendant de l'Etat: une Appro he par Cartographie

Basée sur des LMIs - 4èmes Journées Do torales MACS, Marseille, Fran e, June

2011.

ˆ C.Fiter,L.Hetel,W.Perruquetti,andJ.-PRi hard-É hantillonnageDépendantde

l'ÉtatpourlesSystèmes ave PerturbationsetRetards -8èmeColloqueFran ophone

sur la Modélisation des Systèmes Réa tifs, Villeneuve d'As q, Fran e, November

2011. Journal Européen des Systèmes Automatisés, Volume 45, Number 1-2-3,

Pages 189-203, 2011. doi:10.3166/jesa.45.189-203. Best young resear her arti le

Sampled-data systems: an overview of

re ent resear h dire tions

In this hapter, we intend to present several basi on epts and some re ent resear h

di-re tionsaboutsampled-datasystems. First,ashortintrodu tionofsampled-datasystems

willbegiven,alongwiththemainmathemati aldenitionsandproblemati s. Then,some

general on epts of stabilitywill bere alled, and the sampled-datasystems stability and

stabilizability problems will be formulated. Finally, the main re ent resear h dire tions

and results from the literature willbe presented. They will be lassied into three main

ategories a ording to their sampling type: onstant sampling, time-varying sampling,

and dynami sampling ontrol.

1.1 Introdu tion to sampled-data systems

1.1.1 General sampled-data systems

Sampled-datasystemsaredynami systemsthatinvolvebotha ontinuous-timedynami s

and adis rete-time ontrol. They are mathemati ally asfollows:

Denition 1.1 (Sampled-data system)

˙x(t) = f (t, x(t), u(t)),

∀t ≥ 0,

u(t) = g(x(sk), sk),

_{∀t ∈ [s}

k, sk+1), k

_{∈ N,}

(1.1)

where

t

is the time-variable,

x

: R+

→ R

n

the "state-traje tory",

u

: R+

→ R

n

u

the

whi h satisfy

0 = s0

< s

1

<

· · · < s

k

<

· · ·

and

lim

k→+∞

sk

= +

∞

.

The sampling law is dened as

sk+1

= sk

+ τk,

(1.2)

where

τk

represents the

k

th

sampling interval.

Su h systems an be represented by the blo k diagram in Figure 1.1, in whi h the

blo ks A/D and D/A orrespond to an analog-to-digital onverter (a sampler) and a

digital-to-analog onverter (a zero-order hold) respe tively.

˙x(t) = f (t, x(t), u(t))

x(t)

u(t) = u(sk)

SYSTEM CONTROLLER A/D D/A

x(sk)

u(sk) = g(sk, x(sk))

sk+1

= sk

+ τk

Figure 1.1: Sampled-datasystem

Itisimportanttonotethatwiththesesystems,the dis rete-timedynami sintrodu ed

by the (digital) ontroller implies that during the time between two sampling instants

the system is ontrolled in open-loop (i.e. without updating the feedba k information).

Therefore, thesamplingperiodplaysanimportantroleinthestabilityofthe system,and

adapted tools haveto be used.

1.1.2 Sampled-data linear time-invariant systems

The model of sampled-data systems provided in Denition 1.1 is very general. In this

thesis, we will fo us mainly on linear time-invariant sampled-data systems with

state-feedba k, whi h are dened as follows:

Denition 1.2 (Sampled-data linear time-invariant system)

˙x(t) = Ax(t) + Bu(t),

_{∀t ≥ 0,}

u(t) =

−Kx(s

k),

∀t ∈ [s

k, sk+1), k

∈ N,

where

t

is the time-variable,

x

: R+

→ R

n

the "state-traje tory",

u

: R+

→ R

n

u

the

"input", or " ontrol signal", and the s alars

sk

, for

k

∈ N

, are the sampling instants whi hsatisfy

0 = s0

< s1

<

· · · < s

k

<

· · ·

and

lim

k→+∞

sk

= +

∞

.

A

∈ M

n(R)

isthe "state matrix",

B

∈ M

n,n

u

(R)

is the "input gain matrix", and

K

∈ M

n

u

,n(R)

is the " ontrol gain matrix". The sampling law is dened as

sk+1

= sk

+ τk,

(1.4)

where

τk

represents the

k

th

samplinginterval.

This denition presents the ase of "ideal" sampled-data LTI systems, in whi h no

disturban e nor any other phenomenon is taken into a ount. Throughout this thesis

however, additionalphenomenawillbe onsidered like exogenous perturbationsordelays

inthe feedba k ontrol-loopfor example. In that ase, when these lasses of systems are

onsidered,the asso iated system equations willbe provided.

In the absen e of perturbations,the evolution of the system's state between two

on-se utive samplinginstants

sk

and

sk+1

is given by

x(t) =

e

A(t−s

k

)

x(sk) +

R

t−s

k

0

e

As

_dsBu(sk)

= Ad(t

_{− s}

k)x(sk) + Bd(t

_{− s}

k)u(sk)

= [Ad(t

− s

k)

− B

d(t

− s

k)K] x(sk)

= Λ(t

− s

k)x(sk),

∀t ∈ [s

k, sk+1], k

∈ N,

(1.5)

with the matrix fun tions

Ad

,

Bd

, and

Λ

dened on

R

+

as

Ad(σ) =

e

Aσ

, Bd(σ) =

Z

σ

0

e

As

_dsB.

(1.6) and

Λ(σ) = Ad(σ)

− B

d(σ)K =

e

Aσ

−

Z

σ

0

e

As

_dsBK.

(1.7)

Usingthe notation

τk

in equation (1.4), for the sampling intervals,it is then possible

toobtain the following asso iateddis rete-time modelof the linear sampled-datasystem

atinstants

sk

:

xk+1

= Ad(τk)xk

+ Bd(τk)uk

= Λ(τk)xk,

_{∀k ∈ N,}

(1.8)

with

xk

≡ x(s

k)

and

uk

≡ u(s

k)

.

Ad(τk)

and

Bd(τk)

are alledthe "statematrix" and the "input matrix" of the dis rete-time model respe tively, and

Λ(τk)

is alled the dis rete-time "transitionmatrix".

1.1.3 Problemati s

From the ontrol theory point of view, due to the existen e of both a ontinuous and

a dis rete dynami s, sampled-data systems bring up new hallenges. As in the more

general frameworks of delayed-systems [Ri hard 2003℄, [Gu 2003℄, hybrid systems [der

S haft 2000℄, [Zaytoon 2001℄,[Goebel2009℄, [Prieur 2011℄, orreset systems [Nesi 2008℄,

[Beker2004℄, some problems are raised.

- PROBLEM A: Determine if a sampled-data system is stable for any onstant

samplinginterval

τk

≡ τ

with values ina bounded subset

Ω

⊆ R+

?

-PROBLEMB:Determineifthesampled-datasystemisstableforanytime-varying

samplinginterval

τk

with values in abounded subset

Ω

⊆ R

+

?

Lately,anadditionalissuehasbeenbroughtuptolight. Withtheemergen eof

embed-ded and networked systems parti ularly[Zhang 2001 ℄,[Hespanha 2007℄, [Ri hard 2007℄,

[Chen 2011℄, ontrols ientists realisedthat omputing the next ontrolat ea h sampling

time has a ost [Buttazzo 2002℄, [Cervin 2002℄, [Bro kett 2000℄,[Nair 2000℄. Indeed, the

omputationsfor anew ontrolredu es thelimitedpro essorresour es,inthe ase of

em-bedded systems for example. In the ase of networked ontrol systems, the transmission

of the sampled-datarequiresbandwidth,whi his alsolimited. Therefore, anew problem

arose:

- PROBLEM C: Design a sampling law

τk

= τ (t, sk, x(sk),

· · · )

that enlarges the

samplingintervalswhile making the sampled-datasystem stable?

In this thesis,wewillmainlyfo us onndingsolutionstothis lastparti ular problem

whi h on erns the redu tion of the number of sampling instants (i.e. for parti ular

systems with periodi sampling, the redu tion of the sampling frequen y). We will also

adapt the proposed tools in order to further derive solutions tothe other two problems.

During this study, some stability performan es will be taken into a ount, su h as the

speed of onvergen e of the system's state, or the robustness with respe t to possible

exogenous perturbationsor delays.

1.2 Classi al stability on epts

Before providinganoverview of some works fromthe literature about sampled-data

sys-tems,were allsomefundamental on eptsaboutstability,andsome lassi stabilitytools

1.2.1 Some stability denitions

Intuitively, stability is a property that orresponds to staying lose to an equilibrium

position, when the state is pun tually disturbed. Originally, stability is analyzed for

systemsthat are time-invariantand autonomous (i.e.for whi hthere isno ontrol, orfor

a losed-loop system with agiven ontrol). Su h systems are dened as follows:

Denition 1.3 (Autonomous system) Theordinary dierential equation:

˙x(t) = f (x(t)),

∀t ≥ 0,

(1.9)

with

f

: R

n

_{→ R}

n

Lips hitz ontinuous

1

, is said to be autonomous if

f

(x(t))

does not

depend expli itely on the free variable

t

(often regarded as time).

An "equilibriumpoint"

xe

represents a real solutionof the equation

f

(x) = 0

.

Denition 1.4 ( [Khalil 2002℄) An equilibrium point

xe

of the system (1.9) is

ˆ stable(in the sense of Lyapunov) if

∀ǫ > 0, ∃δ = δ(ǫ) > 0

su h that

kx(0) − x

e

k < δ ⇒ kx(t) − x

e

k < ǫ, ∀t ≥ 0;

ˆ attra tive if

∃ρ > 0

su h that

kx(0) − x

e

k < ρ ⇒ lim

t→+∞

kx(t) − x

e

k = 0;

ˆ asymptoti ally stable if it isstable and attra tive;

ˆ exponentially stable if there exist three s alars

α, β, δ >

0

su hthat

kx(0) − x

e

k < δ ⇒ kx(t) − x

e

k ≤ αkx(0) − x

e

k

e

−βt

_.

For su h a s alar

β

, alled (exponential) "de ay-rate", the equilibrium point is also saidto be "

β

-stable";

ˆ globally asymptoti ally stable if it is stableand

∀x(0) ∈ R

n

,

lim

t→+∞

kx(t) − x

e

k = 0

1

Giventwometri spa es

(X, d

X

)

and

(Y, d

Y

)

, where

d

X

denotesthemetri ontheset

X

and

d

Y

is themetri onset

Y

, afun tion

f

: X

→ Y

is alled Lips hitz ontinuous(or simplyLips hitz)if there existsareal onstant

K

≥ 0

su hthat forall

x1, x2

∈ X

,

d

Y

(f (x1), f (x2))

≤ Kd

X

(x1, x2)

.

Note that by using atranslation of the origin,it isalways possibletoreformulate the

problem as a stability analysis around

xe

= 0

. Therefore, all the results and stability

properties willnowbewritten whiletaking

xe

= 0

as the studied equilibriumpoint.

1.2.2 Se ond Lyapunov method

The most ommon stability tool is the Lyapunov stability approa h. It is based on the

fa tthatasystemwhi htraje toryapproa hes theorigin,losesitsenergy. TheLyapunov

stability approa h makes use of a fun tion

V

: R

n

_{→ R+}

, alled " andidate Lyapunov

fun tion", whi h depends on the system's state, and symbolizes some sort of potential

energy of the system, with respe t to the origin. Very often, this fun tion is hosen as a

norm ora distan e. The Lyapunov stability theory is des ribed as follows [Khalil 2002℄.

Theorem 1.5 Consider the autonomous system (1.9) with an isolated equilibrium point

(

xe

= 0

∈ Ω ⊆ R

n

, with

Ω

a neighborhoodof

xe

). If thereexista lo allyLips hitz fun tion

V

: R

n

_{→ R+}

with ontinuouspartialderivativesandtwo lass

K

fun tions

2

α

and

β

su h that

α(

_{kxk) ≤ V (x) ≤ β(kxk), ∀x ∈ Ω,}

then the origin

x

= 0

of the system is

ˆ stable (inthe senseof Lyapunov) if

dV

(x)

dt

≤ 0, ∀x ∈ Ω, x 6= 0;

ˆ asymptoti ally stable if there exists a lass

K

fun tion

ϕ

su h that

dV

(x)

dt

≤ −ϕ(kxk), ∀x ∈ Ω, x 6= 0;

ˆ exponentially stable if, moreover, there exist four s alars

α, ¯

¯

β, γ, p >

0

su h that

α(

kxk) = ¯

α

_kxk

p

, β(

kxk) = ¯

β

_kxk

p

, ϕ(

kxk) = γkxk.

In su h a ase, the equilibrium point

xe

allows a de ay-rate equal to

γ

p

.

There alsoexists adis rete-time version of the Lyapunov stability theory.

2

Theorem 1.6 Considerthe dis rete-time autonomous system

xk+1

= f (xk),

(1.10)

with an isolated equilibrium point (

xe

= 0

∈ Ω ⊆ R

n

, with

Ω

a neighborhood of

xe

). If

there exist a lo ally Lips hitz fun tion

V

: R

n

_{→ R+}

with ontinuous partial derivatives

and two lass

K

fun tions

α

and

β

su h that

α(

_{kxk) ≤ V (x) ≤ β(kxk), ∀x ∈ Ω,}

then the origin

x

= 0

of the system is

ˆ stable(in the sense of Lyapunov) if

∆V (xk)

_{≤ 0, ∀x}

k

∈ Ω, x

k

6= 0

where

∆V (xk) = V (xk+1)

_{− V (x}

k)

= V (f (xk))

− V (x

k);

ˆ asymptoti ally stable if there exists a lass

K

fun tion

ϕ

su h that

∆V (xk)

_{≤ −ϕ(kx}

k

k), ∀x

k

∈ Ω, x

k

6= 0;

ˆ exponentially stable if there exist four s alars

α, ¯

¯

β, γ, p >

0

su h that

α(

kxk) = ¯

α

_kxk

p

, β(

kxk) = ¯

β

_kxk

p

, ϕ(

kxk) = γkxk.

Remark 1.7 Thelo aldenitionsof theabovetwotheoremsaregloballyvalid ifthegiven

fun tions are lass

K

∞

fun tions

3 and

Ω = R

n

. The fun tion

V

: R

n

_{→ R+}

that veries the properties in the previous theorems is

alled a "Lyapunov fun tion". By abuse of language, espe ially in the ase of linear

systems, a system with a stable and unique equilibrium point is often alled a "stable

system". Furthermore,if a system is not stable,wewillsay that itis "unstable".

3

1.2.3 Properties of linear time-invariant systems with

sampled-data ontrol

Very interesting properties arise inthe ontext of sampled-data LTI systems, on erning

ontinuous and dis rete-time analysis approa hes. One of the rst on erns the

equilib-rium's attra tivity,and is formulated asfollows:

Theorem 1.8 (From [Fujioka 2009b℄) Foragivensampled-dataLTIsystem(1.3)with

bounded sampling intervals and a given initial state

x(0)

, the following onditions are

equivalent:

(i)

limt→+∞

x(t) = 0

,

(ii)

limk→+∞

x(sk) = 0

.

Thispropertymeansthattheattra tivityofthe ontinuous-timesystem(1.3)is

equiv-alent tothe attra tivity of the dis rete-time system (1.8).

Further analysis [Hetel 2011a℄ allows for proving that the ontinuous-time system's

(asymptoti ) stability is equivalent to the dis rete-time system's (asymptoti )

stabil-ity, in the more general ase of reset ontrol systems ( [Nesi 2008℄, [Beker 2004℄

[Tar-bourie h2011℄, [Za arian 2005℄).

Therefore, it ispossible touse both a ontinuous ora dis rete-timeapproa hinorder

to study the stabilityof sampled-data systems.

In the following, we will present an overview of some results from the litterature

regarding the three main studies on erning sampled-datasystems:

ˆ the stability analysis regarding a onstant sampling(Problem A);

ˆ the stability analysis regarding time-varying sampling(ProblemB);

ˆ the dynami ontrolof the sampling(Problem C).

1.3 Stability analysis under onstant sampling

The rst andeasiest way tostudy sampled-datasystems isto onsider the ase whenthe

samplinginterval is onstant, for agiven value

T

(see Figure 1.2).

In this ase, the system's stability is usually analysed using the dis rete-time LTI

modelof the system:

˙x(t) = Ax(t) + Bu(t)

x(t)

u(t) = u(sk)

SYSTEM CONTROLLER A/D D/A

x(sk)

u(sk) =

_−Kx(s

k)

sk+1

= sk

+ T

Figure1.2: Sampled-datasystem with a onstant samplingrate

Foragivensamplingperiod

T

,themost ommonapproa htoanalysethestability(the

so- alled "S hur method") onsists in studying the eigenvalues of the transition matrix

Λ(T )

. We all

λ

max

(T )

the eigenvalue of

Λ(T )

with the largest modulus. We then have

the following properties [Aström 1996℄.

Theorem 1.9 Theequilibrium

xe

= 0

of (1.11) is

ˆ S hur-stable (globally asymptoti ally stable) if and only if

|λ

max

(T )

| < 1

. In that

ase,

Λ(T )

is alled a S hur matrix;

ˆ exponentially stable (globally) with a de ay-rate

α >

0

if and only if

|λ

max

(T )

| ≤

e

−αT

.

Equivalent Linear Matrix Inequality (LMI) stability onditions an also be obtained

using the Lyapunov stabilitytheory for dis rete-time systems.

Theorem 1.10 The onsidered system (1.11) is

ˆ stable(globally) if and only if there exists a matrix

P

∈ S

+∗

n

su h that

Λ(T )

T

_P

_{Λ(T )}

− P  0;

ˆ S hur-stable(globally asymptoti ally stable) ifand only if there exists a matrix

P

∈

S

n

+∗

su hthat

ˆ exponentially stable (globally) with a de ay-rate

α >

0

if and only if there exists a matrix

P

∈ S

+∗

n

su h that

Λ(T )

T

_P

_{Λ(T )}

−

e

−αT

_P

 0.

The dis rete-time analysis of sampled-data systems with a given onstant sampling

has sin elongbeensolved. However, someproblemsstillremainopen, sin etheproposed

solutions remain onservative regarding the ontinuous-time analysis of su h systems, or

regarding the robustness with respe t to exogenous perturbations. For more results

re-gardingrobuststabilityandoptimal ontrolof sampled-datasystems both in

ontinuous-time and dis rete-time, we point to the handbooks [Chen 1991℄ and [Aström 1996℄. In

the followingse tion,we will onsider the robustness aspe t with respe t tovariationsin

the sampling interval.

1.4 Stability analysis under time-varying sampling

In the literature, there exist numerous studies about sampled-data systems with a

on-stant sampling interval. In pra ti e however, it may a tually be impossible to maintain

a onstant sampling rate during the real-time ontrol of physi al systems. Embedded

and networked systems for example are oftenrequiredto sharea limitedamountof

om-putational and transmission resour es between dierent appli ations. This may lead to

u tuations of the sampling interval, be ause of delays that ould appear during the

omputation of the ontrol, during the transmission of the information, or be ause of

s hedulingissues [Zhang 2001 ℄,[Bushnell 2001℄,[Mounier2003a℄. Su h systems are

rep-resented by the blo k diagramin Figure1.3.

1.4.1 Di ulties and hallenges

From the ontroltheory pointof view, these variationsinthe sampling intervalbring up

new hallenges sin e they may have a destabilizing ee t if they are not properly taken

into a ount [Wittenmark1995℄, [Zhang2001b℄, [Li2010℄.

Consider forexample the system [Zhang2001b℄:

˙x(t) =

"

1 3

2 1

#

x(t) +

"

1

0.6

#

u(t),

∀t ≥ 0,

u(t) =

₋

h

1 6

i

x(sk),

_{∀t ∈ [s}

k, sk+1

), k

_{∈ N.}

(1.12)

˙x(t) = Ax(t) + Bu(t)

x(t)

u(t) = u(sk)

SYSTEM CONTROLLER A/D D/A

x(sk)

u(sk) =

_−Kx(s

k)

sk+1

= sk

+ τk

Figure1.3: Sampled-datasystem with a time-varying sampling

In the ase of a onstant samplingrate, one an use a gridding onthe sampling step

T

and the stability onditions from Theorem 1.9, as shown in Figure 1.4, to nd that

the origin of the system is S hur-stable if

T

∈ [0s, T

max

onst

= 0.5937s]

, and unstable for

T

_{∈ [T}

max onst

,

0.9s]

(as wellas for highervalues).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

1

2

3

4

5

6

7

8

T

|

λ

max

(T)|

Figure1.4: Evolutionof the modulus

|λ

max

(T )

|

of themaximum eigenvalueof the

transi-tion matrix

Λ(T )

, dependingon the samplingperiod

T

Therefore, for onstant sampling intervals

T

1

= 0.18s

or

T

2

= 0.54s

for example, the

system isS hur-stable, asillustrated by Figure1.5.

However, if we sample using a sequen e of samplingintervals

T1

→ T

2

→ T

1

→ T

2

→

· · ·

, the system be omesunstable, aswe an see inFigure1.6.

0

2

4

6

8

10

−0.5

0

0.5

1

t

x(t)

0

2

4

6

8

10

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t

x(t)

Figure1.5: Constant samplingratewith

T1

= 0.18s

(left)and

T2

= 0.54s

(right)-Stable

0

2

4

6

8

10

−3

−2

−1

0

1

2

3

t

x(t)

Figure 1.6: Variable sampling intervals

T

1

= 0.18s

→ T2

= 0.54s

→ T1

→ T2

→ · · ·

-Unstable

matrix produ t(i.e. the produ tof twoS hurmatri esisnot ne essarilyS hur). Indeed,

inthis ase,the dis rete-timeequivalentsystemovertwosamplinginstants anbewritten

as

xk+2

= Λ(T2)Λ(T1

)xk,

_{∀k ∈ 2N,}

whi h an also be writtenas

with

h

representing the

2k

th

sampling,and the transitionmatrix

Λ(T1

, T

2)

≡ Λ(T2)Λ(T1) =

"

0.8069

_−3.2721

0.6133

−2.1125

#

overtwo sampling intervals

T1

and

T2

,whi his not S hurin this example.

In the ase of sampled-data systems with a periodi sequen e of sampling intervals,

it is possible to design a stability domain that depends on the sampling sequen e. For

instan e, Figure 1.7presents the stability domain(in blue) obtained by using agridding

onthe values of

T

1

and

T

2

, in the ase of a periodi sequen e of two sampling intervals,

forthe sampled-datasystem (1.43).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

T

const

max

T

const

max

T

₁

T

2

Figure 1.7: Stability domain (allowable sampling interval) for a periodi sampling

se-quen e

T

1

→ T2

→ T1

→ T2

→ · · ·

-rst example

Inthis gure,one an see thatthereexistunstable samplingsequen es made ofstable

samplingintervals 4

,whi h onrms our earlierremark. Also,one an see that thereexist

stable sampling sequen es made of both stable and unstable sampling intervals (with

T1

= 0.46s

and

T2

= 0.8s

forexample). 4

by"stablesamplinginterval",wemeanthatthetransitionmatrixoftheasso iatedsamplinginterval isS hur.

Consider nowthe example

˙x(t) =

"

0

1

−2 0.1

#

x(t) +

"

0

1

#

u(t),

∀t ≥ 0,

u(t) =

−

h

−1 0

i

x(sk),

∀t ∈ [s

k, sk+1), k

∈ N,

(1.13)

and its asso iated stability domain (see Figure 1.8). Here, one an see that there also

existstablesamplingsequen es whi hare omposedsolelyofunstable samplingintervals.

0

1

2

3

4

5

6

7

8

0

1

2

3

4

5

6

7

8

T

1

T

2

Figure 1.8: Stability domain (allowable sampling interval) for a periodi sampling

se-quen e

T1

→ T

2

→ T

1

→ T

2

→ · · ·

- se ondexample

Let us look at the sampling values

T

1

= 2.126s

and

T

2

= 3.950s

for example. The

sampled-datasystem(1.13)isunstablewithboth onstantsamplings

T1

and

T2

. However,

asitisshown inFigure1.9,the system'stransitionmatrix

Λ(T1

, T

2)

isS hur-stableunder

the periodi sampling

T

1

→ T2

→ T1

→ T2

→ · · ·

.

A ording to the previous observations, itis lear that the existing stability toolsfor

sampled-datasystemswitha onstantsamplingwillnotprovideanyguaranteeofstability

for sampled-data systems with unknown time-varying sampling that arises in real-time

ontrol onditions. Forthisreason, onsideringthedi ultyoftheproblem,severalworks

0

20

40

60

80

100

−4

−3

−2

−1

0

1

2

3

4

t

x(t)

Figure1.9: Variablesampling

T

1

= 2.126s

→ T2

= 3.950s

→ T1

→ T2

→ · · ·

- Stable

with time-varying samplings with bounded values [Mirkin 2007℄, [Naghshtabrizi 2008℄,

[Hetel 2007℄, [Fujioka 2009b℄, [Seuret 2009℄, [Fridman 2010℄, and [Hetel 2011b℄. Very

often, the sampling intervals that are onsidered an take any value in a bounded set

[τ , ¯

τ]

. In the rest ofthis se tion, wepropose a shortoverview of various notablemethods regarding this issue.

1.4.2 Time-delay approa h with Lyapunov te hniques

Oneoftheapproa hestodealwithtime-varyingsamplingwasinitiatedin[Mikheev1988℄,

and onsists in onsidering the dis rete-time dynami s indu ed by the digital ontroller

asa pie ewise ontinuous delay (see Figure1.10):

sk

= t

_{− (t − s}

k) = t

_{− h(t), ∀t ∈ [s}

k, sk+1), k

_{∈ N,}

where

h(t)

≡ t − s

k

istheindu ed delay. The LTIsystem with sampled-data(1.3) isthen

re-modeled asan LTI system with time-varying delay

˙x(t) = Ax(t) + Bu(t),

_{∀t ≥ 0,}

u(t) =

−Kx(t − h(t)), ∀t ≥ 0,

(1.14)

and is studied with lassi al tools designed for time-delay systems [Ri hard 2003℄,

[Frid-man 2003℄, [Zhong 2006℄, [Mounier 2003b℄ whi h are dened by retarded fun tional

Denition 1.11 (Time-delay system) Atime-delay system isdes ribed bythe

follow-ing fun tional dierential equation:

˙x(t) = f (t, xt),

∀t ≥ 0,

xs

₀

(θ) = φ(s

0

+ θ),

∀θ ∈ [s0

− ¯h, s0

]

(1.15) where

f

: R+

× C

0

_([

_{−¯h, 0] → R}

n

₎

→ R

n

,

φ

∈ C

0

_([

_{−¯h, 0] → R}

n

₎

, with

¯h ≥ 0

the maximal

delay, and

xt

∈ C

0

_([

_{−¯h, 0] → R}

n

₎

, whi h represents the state fun tion

5

and is dened by:

xt(θ) = x(t + θ),

_{∀θ ∈ [−¯h, 0].}

(1.16)

0

2

4

6

8

10

12

14

16

18

20

0

1

2

3

t

h(t)=t−s

k

Figure1.10: Samplingseen as apie ewise- ontinuous time-delay

It is assumed that there exists a unique solution to the above dierential equation

(some Lips hitz onditions for the existen e and uni ity of solutionsfor su h systems are

provided in [Gu 2003℄), and that there is a unique equilibrium point

6

:

xe

= 0

(as in the delay-free ase,iftheequilibriumpointisnot

0

,we an omedowntoitbyusingasimple

hange of oordinates).

Inthegeneral aseoftime-delaysystems,itisdi ulttoapplythe lassi alLyapunov

stability theory from Theorem 1.5, be ause the derivative

dV

(x)

dt

will depend on the past

values of the state:

xt

. To over ome this issue, two dierent stability approa hes, better

suited totime-delay systems, have been developped. Both of them make use of a wider

lass of fun tions or fun tionals as Lyapunov andidates. The rst approa h is alled

Lyapunov-Razumikhin [Gu 2003℄, and makes use of atime-dependent "energy"fun tion

V

_{≡ V (t, x(t))}

. The se ond approa h, alled Lyapunov-Krasovskii [Gu 2003℄, makes use of a fun tional

V

≡ V (t, x

t)

instead.

5

Notethat

x(t)

isthevalueofthestateat

θ

= 0

:

x(t) = x

t

(0)

. 6

Under existen eand uni ityof thesolution, it anbeshown[Dambrine 1994℄ that theequilibrium statedened by

˙x(t) = 0

isa onstant fun tion

x

t

(θ)

≡ x

e

, thus the expression"equilibrium point" is justied.

1.4.2.1 Lyapunov-Razumikhin approa h

In this approa h, it is onsidered a fun tion

V

≡ V (t, x(t))

. The originality is to show

that it is not ne essary to he k the ondition

V

˙

(t, x(t))

≤ 0

along all the traje tories of the system. Indeed, it is possible to limit this test to solutions whi h tend to leave a

neighbourhood

V

(t, x(t))

≤ c

of the equilibrium point. The approa h is formulated as

follows.

Theorem 1.12 (Lyapunov-Razumikhin (from [Gu 2003℄)) Consider three

ontin-uous non-de reasing fun tions

α, β, γ

: R+

→ R

+

,

β

stri tly in reasing, su h that

α(θ)

and

β(θ)

are stri tly positive for all

θ >

0

, and

α(0) = β(0) = 0

. Assume that the ve tor

eld

f

from (1.15)is bounded for bounded values of its arguments.

If there exists a ontinuously dierentiable fun tion

V

: R+

× R

n

→ R

+

su h that:

α(

kxk) ≤ V (t, x) ≤ β(kxk), ∀t ∈ R+

,

_{∀x ∈ R}

n

,

(1.17)

with

k.k

any normon

R

n

, and ifthe derivative of

V

alongthe solutions of (1.15)satises

˙

V

(t, x(t))

_{≤ −γ(kx(t)k)}

whenever

V

(t + θ, x(t + θ))

≤ V (t, x(t)), ∀θ ∈ [−¯h, 0],

(1.18)

thenthe origin of system (1.15) is uniformly stable.

If, in addition,

γ(θ) > 0

for all

θ >

0

, and if there exists a ontinuous non-de reasing fun tion

p

: R+

→ R

+

satisfying

p(θ) > θ

for all

γ >

0

, and su h that ondition (1.19)is strengthened to

˙

V

(t, x(t))

≤ −γ(kx(t)k)

whenever

V

(t + θ, x(t + θ))

≤ p(V (t, x(t))), ∀θ ∈ [−¯h, 0],

(1.19)

then the fun tion

V

is alled a Lyapunov-Razumikhin fun tion, and the origin of system

(1.15)is uniformly asymptoti ally stable.

If in addition

lims→+∞

α(s) = +

∞

, then the origin of system (1.15) is globally

uni-formly asymptoti ally stable.

In pra ti e, for simpli ity, most existing works about Lyapunov-Razumikhinstability

usealinearfun tion:

p(θ) = qθ

,withas alar

q >

1

. Moreover, theLyapunov-Razumikhin

andidates are very often taken as quadrati and time-invariant:

V

(x) = x

T

_{P x}

, where

P

_{∈ S}

_n

+∗

. Some works about the Lyapunov-Razumikhin approa h for delayed systems

One ofthe advantagesof the Lyapunov-Razumikhinstabilitytheory isthat itredu es

the onservatism with respe t to the lassi Lyapunov stability theory, and it makes

it possible to work with simple Lyapunov(-Razumikhin) fun tions. Its main drawba k

is that it may be di ult to obtain he kable delay (or sampling interval)-dependent

stability onditions, sin ethe delay (or samplinginterval)isnot expli itelyintrodu edin

the equations. This will be a motivation for employing Lyapunov-Krasovskii te hniques

to bepresented now.

1.4.2.2 Lyapunov-Krasovskii approa h

The Lyapunov-Krasovskii approa h isanextension of the Lyapunov theory tofun tional

dierential equations. Here, we are sear hing for positive fun tionals

V

≡ V (t, x

t)

whi h are de reasing along the traje tories of (1.15).

Theorem 1.13 (Lyapunov-Krasovskii (from [Gu 2003℄)) Considerthree ontinuous

non-de reasingfun tions

α, β, γ

: R+

→ R+

, su hthat

α(θ)

and

β(θ)

are stri tlypositive

for all

θ >

0

, and

α(0) = β(0) = 0

. Assume that theve tor eld

f

from (1.15) isbounded

for bounded values of its arguments.

If there exists a ontinuousdierentiable fun tional

V

: R+

× C

0

_([

_{−¯h, 0] → R}

n

₎

→ R

+

su h that

α(

_{kφ(0)k) ≤ V (t, φ) ≤ β(kφkC}

),

(1.20)

with

k.k

any norm on

R

n

,

k.kC

its asso iated norm on

C

0

_([

_{−¯h, 0] → R}

n

₎

dened by

kφk

C

= max

θ∈[−¯

h,0]

kφ(θ)k

, and if

˙

V

(t, φ)

_{≤ −γ(kφ(0)k),}

(1.21)

then the origin of system (1.15) is uniformly stable.

If in addition

γ(θ) > 0

for all

θ >

0

, then the fun tional

V

is alled a

Lyapunov-Krasovskii fun tional, and the origin of system (1.15) is uniformlyasymptoti ally stable.

If in addition

lims→+∞

α(s) = +

∞

, then the origin of system (1.15) is globally

uni-formly asymptoti ally stable.

The fun tionalsthat are being onsidered usuallyhave theform [Kolmanovskii 1996℄:

V

(t, φ) = φ

T

_{(0)P (t)φ(0) + φ}

T

₍₀₎



R

0

−¯

h

Q(t, s)φ(s)ds



+



R

0

−¯

h

φ

T

_(s)Q

T

_{(t, s)ds}



_φ(0)

+

R

0

−¯

h

R

0

−¯

h

φ

T

_{(s)R(t, s, p)φ(p)dsdp +}

R

0

−¯

h

φ

T

_{(s)S(s)φ(s)ds,}

(1.22)