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and statedependent sampling
Christophe Fiter
To cite this version:
Christophe Fiter. Contribution to the control of systems with timevarying and statedependent
sampling. Other. Ecole Centrale de Lille, 2012. English. �NNT : 2012ECLI0021�. �tel00773127�
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 , GifsurYvette
Membre: M. DanielSimon Chargé de Re her he INRIA àGrenoble
Dire teur de thèse: M. JeanPierre Ri hard Professeur àl'É ole Centrale de Lille
Codire teur de thèse: M. WilfridPerruquetti Professeur àl'É ole Centrale de Lille
Coen 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
timevarying and statedependent 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 ,GifsurYvette
Member: M.Daniel Simon INRIAResear h Asso iateinGrenoble
Thesis supervisor: M.JeanPierre Ri hard Professorat É ole Centrale de Lille
Thesis osupervisor: M.WilfridPerruquetti Professorat É ole Centrale de Lille
Thesis osupervisor: 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 JeanPierre Ri hard, Professor
WilfridPerruquetti,and CNRSResear hAsso iate LaurentiuHetel. Itwassupportedby the
"Centre National de la Re her he S ientique" (CNRS) and the "Conseil Régional de la
RégionNordPas de Calais".
Iwouldliketoexpressmysin eregratitudetomyadvisorsJeanPierreRi 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
Professor 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 GifsurYvette, 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 RhneAlpes, in Grenoble.
I would like to thank all the members of the teams "Systèmes Nonlinéaires et à
Retards" (SyNeR, LAGIS) and "NonAsymptoti Estimation for Online Systems"(NonA,
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 Sampleddata systems: an overview of re ent resear h
dire tions 23
1.1 Introdu tionto sampleddatasystems . . . 23
1.1.1 General sampleddatasystems . . . 23
1.1.2 Sampleddata lineartimeinvariantsystems . . . 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 oflineartimeinvariantsystemswith sampleddata ontrol 30 1.3 Stabilityanalysis under onstant sampling . . . 30
1.4 Stabilityanalysis undertimevarying sampling . . . 32
1.4.1 Di ulties and hallenges . . . 32
1.4.2 Timedelay approa h with Lyapunov te hniques . . . 37
1.4.2.1 LyapunovRazumikhin approa h . . . 39
1.4.2.2 LyapunovKrasovskii approa h . . . 40
1.4.3 Smallgainapproa h . . . 42
1.4.4 Convexembedding 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 ISSLyapunov fun tion approa h . . . 53
1.5.5 Upperbound onthe Lyapunov fun tion approa h . . . 55
1.5.6
L
2
stability approa h . . . 571.6 Con lusion . . . 59
Chapter 2 A polytopi approa h to dynami sampling ontrol for LTI systems: 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 statespa e . . . 66
2.3.1.2 Convex embeddinga ording to time . . . 68
2.3.2 Stability results inthe ase of statedependent sampling . . . 69
2.3.3 Stability results inthe ase of timevarying 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 systems: the perturbed ase 77 3.1 Problemstatement . . . 78
3.2 Mainstability results . . . 81
3.3 Robuststabilityanalysiswithrespe ttotimevaryingsampling Optimization of the parameters . . . 83
3.4 Eventtriggered ontrol . . . 87
3.4.1 Overapproximation based eventtriggered ontrols heme . . . 87
3.4.2 Perturbationaware eventtriggered ontrols heme . . . 88
3.4.3 Dis retetime approa h eventtriggered ontrols heme . . . 89
3.5 Selftriggered ontrol . . . 89
3.6 Statedependent sampling . . . 93
Chapter 4 A LyapunovKrasovskii approa h to dynami sampling
ontrol 101
4.1 Problem formulation . . . 103
4.2 Main
L
2
stabilityresults . . . 1054.2.1 Stability analysis of the perturbed system . . . 106
4.2.1.1 Continuity,pie ewise dierentiability,and positivity
onditions of the LyapunovKrasovskii Fun tional . . 107
4.2.1.2
L2
stability onditions . . . 1084.2.2 Stability analysis of the perturbed system with delays . . . 111
4.2.3 Algorithmtodesignthestatedependentsamplingfun tion
τ
max for
a given feedba k matrix gain
K
. . . 1184.3 Main
L2
stabilizationresults . . . 1214.3.1 Stabilization using a pie ewise onstant feedba k ontrol
u(t) =
−Kx(s
k)
. . . 1214.3.2 Stabilization using aswit hing pie ewise onstant feedba k ontrol
u(t) =
_{−K}
σ
k
x(sk)
. . . 1244.3.3 Algorithmtodesignthestatedependentsamplingfun tion
τ
max and
its asso iated feedba k matrix gain
K
(or gainsKσ
) . . . 1264.4 Numeri al examples. . . 128
4.4.1 Example 1  State dependent samplingfor systems with
perturbations and delays . . . 128
4.4.2 Example 2  Conservatism redu tion thanks tothe swit hed LKF . 129
4.4.3 Example 3  Statedependent samplingfor systems whi h are both
openloopand losedloop(witha ontinuousfeedba k ontrol)
unstable . . . 130
4.4.4 Example4Statedependentsampling ontrollerforperturbed
systems . . . 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 retetimebehaviour of the system 161
Appendix C Contru tion of a polytopi embedding based on Taylor
polynomials 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 Analogtodigital onversion . . . 17
2 Digitaltoanalog onversion . . . 18
1.1 Sampleddatasystem . . . 24
1.2 Sampleddatasystem with a onstant samplingrate . . . 31
1.3 Sampleddatasystem with atimevarying sampling . . . 33
1.4 Evolutionofthemodulus
λ
max(T )

ofthemaximumeigenvalueofthe transition matrixΛ(T )
,depending onthe sampling periodT
. . . 331.5 Constantsamplingratewith
T
1
= 0.18s
(left)andT
2
= 0.54s
(right)Stable 34 1.6 Variable sampling intervalsT
1
= 0.18s
→ T2
= 0.54s
→ T1
→ T2
→ · · ·
Unstable . . . 341.7 Stability domain (allowable sampling interval) for a periodi sampling sequen e
T1
→ T
2
→ T
1
→ T
2
→ · · ·
 rst example . . . 351.8 Stability domain (allowable sampling interval) for a periodi sampling sequen e
T1
→ T
2
→ T
1
→ T
2
→ · · ·
 se ond example . . . 361.9 Variable sampling
T
1
= 2.126s
→ T2
= 3.950s
→ T1
→ T2
→ · · ·
 Stable . 37 1.10 Samplingseen as a pie ewise ontinuous timedelay . . . 381.11 Inter onne ted system . . . 43
1.12 Sampleddatasystem with adynami sampling ontrol . . . 47
1.13 Eventtriggered ontrolfrom[Cervin 2007℄ applied onadouble integrator . 49 1.14 Lyapunovfun tionlevelsapproa htodynami sampling ontrol[Velas o2009℄ 
η
= 0.8
≥ η
∗
,stable (left) andη
= 0.65 < η
∗
, unstable (right) . . . 512.1 Covering the statespa e of dimension
2
withq
oni regionsR
s
. . . 672.2 2D representationof a onvex polytopearound thematrix fun tion
Φ
over the time intervalσ
∈ [0, τ
s]
. . . 682.3 Example 1: Stateangle dependent sampling fun tion
τ
for dierent de ay ratesβ
. . . 722.4 Example 1: Interexe ution times
τ(x(sk))
and LRFV
(x) = x
T
_{P x}
for a de ay rateβ
= 0
. . . 732.5 Example 1: Interexe ution times
τ(x(sk))
and LRFV
(x) = x
T
_{P x}
for a de ay rateβ
= 0.05
. . . 732.6 Example2: Mappingofthestatespa e(regardingtheangular oordinates)
2.7 Example 2: Interexe ution times
τ
(x(sk))
and LRFV
(x) = x
T
_{P x}
for a de ay rate
β
= 0
. . . 763.1 Illustrationof the onvex embedding design . . . 85
3.2 Illustrationof the property of the onvex embeddingdesign with
subdivisions fromAppendix C.3 aroundthe matrix fun tion
∆
. . . 913.3 Stateangledependentsamplingmap
τ
max
for dierentde ayrates(
β
)and perturbations (W
) . . . 973.4 Interexe ution times
τ
max
(x(sk))
and LRF
V
(x) = x
T
_{P x}
for ade ay rate
β
= 0.3
andW
= 0
 Statedependent sampling . . . 983.5 Interexe ution times
τ
max
(x(sk))
for a de ay rate
β
= 0.1
andW
=
0.04
(kw(t)k2
≤ 20%kx(s
k)
k2
)  First eventtriggered ontrols heme,selftriggered ontrol, and statedependent sampling . . . 98
4.1 Algorithm to design the statedependent sampling fun tion
τ
max
(x)
for a given feedba k matrix gainK
. . . 1194.2 Algorithmtodesignthe statedependentsamplingfun tion
τ
max
(x)
andits asso iatedfeedba k matrix gain
K
(or gainsKσ
). . . 1274.3 Example1: Mappingofthemaximaladmissiblesamplingintervals
τ
+
σ
withorwithout perturbations
w
and/or delaysh
. . . 1294.4 Example 1: Leftside: delayed ase (delays up to
0.1s
). Rightside:delayfree ase. In both sides, the perturbation satises
kw(t)k
2
=
1
γ
kz(t)k
2
≃
32%
kz(t)k2
. . . 1304.5 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
gainsγ
, with orwithout swit hing ontroller . . . 1324.7 Example4: State
x(t)
andsamplingintervalsτk
= τ
max
(x(sk))
for the
ontrolled 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) . . . 1331 Conversion analogiquenumérique . . . 140
2 Conversion numérique analogique . . . 140
3 Re ouvrement de l'espa ed'état de dimension
2
parq
régions oniquesR
s
1434 Système LTI é hantillonnéave perturbations etretards . . . 146
B.1 Covering the statespa e of dimension
2
withq
oni regionsR
s
. . . 160C.1 2D representation of the onvex polytope design using polytopi
 ISS = InputtoStateStability.
 LKF = LyapunovKrasovskii Fun tional.
 LMI = LinearMatrix Inequality.
 LTI = LinearTimeInvariant.
 LRF = LyapunovRazumikhinFun 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 realn
× m
matri es.
M
n(R)
denotes the set of realn
× n
matri es.
Sn
denotes the set of symmetri matri esinM
n(R)
.
S
+
n
(resp.S
+∗
n
) denotes the set of positive (resp. positive denite) symmetri matri esin
M
n(R)
. Co
{F
i
}
i∈I
, for given matri esFi
∈ M
n,m(R)
and a nite set of indexesI
, denotes theonvex polytope in
M
n,m(R)
formed by the verti esFi, i
∈ I
.
C
0
_{(X}
_{→ Y )}
, for two metri spa es
X
andY
,is the set of ontinuous fun tions fromX
to
Y
.
L2
isthe spa e of squareintegrablefun tions fromR
+
to
R
n
.
λX
,for a s alarλ
∈ R
and anR
ve tor spa eX
, 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 setX
.
P(X)
denotes the power set of a setX
(i.e. the set of allsubsets ofX
).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 esA, B
∈ M
n(R)
meansthatA
− 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 esAi, 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 radiusofM
∈ M
n(R)
.
.2
standsfor the operatornorm onM
n(R)
asso iatedtothe normk.k2
onR
n
: for a matrixM
∈ M
n(R)
,M2
= 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 normonR
n
: for ave torx
∈ R
n
,kxk2
=
√
x
T
_{x}
.Notations on erning s alars:

⌊x⌋
isthe oor ofx
∈ R
: the largestinteger not greaterthanx
:x
− 1 < ⌊x⌋ ≤ x
. ⌈x⌉
isthe eiling ofx
∈ R
: the smallestinteger not less thanx
:x
≤ ⌈x⌉ < x + 1
.  sgn(x)
denotes the sign ofthe s alarx
. sat
(x)
denotes a s alar that is equal to−1
if the s alarx
≤ −1
,1
ifx
≥ 1
, andx
otherwise.
Notations on erning fun tions:

xt
(resp.˙xt
)denotes thefun tioninC
0
_{([}
_{−¯h, 0] → R}
n
_{)}
,foragivenmaximaldelay
¯h
su h thatxt(θ) = x(t + θ),
∀θ ∈ [−¯h, 0]
(resp.˙xt(θ) = ˙x(t + θ),
∀θ ∈ [−¯h, 0]
).
k.k
L
2
is theL
2
norm onL
2
: fora fun tionf
∈ L
2
,kfk
L
2
=
R
∞
0
kf(t)k
2
2
dt
1
_{2}
. k.k
H
∞
is theH
∞
norm onL2
→ L2
: for an operator∆ : u
∈ L2
7→ v ∈ L2
,k∆k
H
∞
=
sup
w∈R
+
k∆(jw)k
, withk∆(jw)k =
max
kzk
2
=1, z∈C
n
k∆(jw)zk2
. It is equal to the
L2
toL2
norm:
k∆k
H
∞
=k∆kL
2
→L
2
= sup
u6=0
kvk
L
2
kukL
2
.A lass
K
fun tionisafun tionϕ
: [0, a)
→ [0, +∞)
thatisstri tly in reasing,and su h thatϕ(0) = 0
. A lass
K∞
fun tion isa lassK
fun tion su h thata
= +
∞
andlimt→+∞
ϕ(t) = +
∞
. A
C
∞
fun tion isa fun tion that isinnitely dierentiable.

f(n) = O(g(n))
meansthat the growthrate of the sequen ef
(n)
,n
∈ N
, isdominatedby the sequen e
g(n)
, i.e. there existN
∈ N
andK
∈ R
∗
+
su h that for alln
≥ N
,f(n) ≤ Kg(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 iallydue 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 reusability, 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 retetime signalsand dis retetime 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 ananalogtodigital(A/D) onverter. Su h a onversion of aninput signal
x(t)
intoasampledsignal
x(sk)
, at samplinginstantssk, 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: Analogtodigital onversion
a digitaltoanalog (D/A) onverter (a zeroorderhold), so as to hold the ontrol value
that is sent to the a tuators. The onversion of a sampled input signal
u(sk)
into apie 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: Digitaltoanalog onversion
In embedded ontrol appli ations however, a dis retetime implementation may
produ eundesiredee tssu hasdelaysoraperiodi ontrolexe utions,duetotheintera tion
between ontroltasks and realtimes heduler me hanisms [HristuVarsakelis 2005℄. The
ee ts of these dis retetimedynami s brought up new hallenges regarding the stability
andstabilization,and newtheoriesandtoolshavebeen developedfor thesesampleddata
systems. In parti ular, in the last few years, two main problems have been of a great
importan efor ontroltheorists:
P1) the stability of sampleddatasystems with timevarying 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 totimevarying
samplingwillalsobein luded. The robustness aspe t with respe t toexogenous
perturbationsordelaysinthe ontrolloopwillbe onsidered,sototakeintoa ountphenomena
o uringintherealtime ontrolofphysi alsystems. Finally,a odesign 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 upperbound for timevarying 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,
hallenges, and re ent resear h dire tions in the domain of sampleddata systems in ontrol
theory. First, the notion of sampleddata systems is dened, and the main open
problems 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 sampleddatasystems with onstant or
timevaryingsampling,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,astatedependentsampling ontrolisdesignedforidealLTIsystems
with sampleddata. 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. Theproposedstatedependentsamplingfun 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 statedependent sampling law. As in the se ond hapter,
theapproa hisbasedonLyapunovRazumikhinstability onditionsandpolytopi
embeddings. Afterpresentingthemainstabilityresults,fourdierentappli ationsareaddressed.
Therstone on erns therobuststabilityanalysiswithrespe ttotimevaryingsampling.
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. Eventtriggered ontrol,
selftriggered ontrol, and the newly introdu ed statedependent sampling s hemes are
then presented.
Chapter 4
In the fourth and last hapter, an extension to the stability analysis of perturbed
timedelaylinearsystemsista kled,andthestabilizationissueis onsidered. Theobje tivehere
istodesigna ontrolleralongwiththestatedependentsamplinglaw,soastostabilizethe
onsidered perturbed LTI sampleddata system, and enlarge even further the allowable
samplingintervals. First,the aseofa lassi linearstatefeedba 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 odesign of both the ontroller and the statedependent sampling
fun tion is based on LMIs obtained thanks to the mappingof the statespa e presented
in the previous hapters, and thanks to a new lass of LyapunovKrasovskii 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
C. Fiter,L.Hetel, W.Perruquetti, andJ.PRi hardA StateDependent Sampling
for Linear State Feedba k  Automati a, Volume 48, Number 8, Pages 18601867,
August 2012. doi:10.1016/j.automati a.2012.05.063
C. Fiter, L. Hetel, W. Perruquetti, and J.P Ri hard  A Novel Stabilization
provision C.Fiter,L.Hetel,W.Perruquetti,andJ.PRi hardARobustStability Framework
forTimeVarying Sampling Automati a, submitted.
International onferen es
C.Fiter,L.Hetel,W.Perruquetti,andJ.PRi hardStateDependentSampling: an
LMIBasedMappingApproa h 18thIFACWorldCongress,Milan,Italy,September
2011.
C.Fiter,L.Hetel,W.Perruquetti, andJ.PRi hardStateDependentSamplingfor
Perturbed TimeDelay Systems  51st IEEE Conferen e on De ision and Control,
Maui, Hawaii, USA,De ember 2012.
C. Fiter, L. Hetel, W. Perruquetti, and J.P Ri hard  A Robust Polytopi
Approa h for StateDependentSampling 12th European ControlConferen e, Zuri h,
Switzerland,July 2013 submitted.
National onferen es
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.
C.Fiter,L.Hetel,W.Perruquetti,andJ.PRi hardÉ 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 123,
Pages 189203, 2011. doi:10.3166/jesa.45.189203. Best young resear her arti le
Sampleddata 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
dire tionsaboutsampleddatasystems. First,ashortintrodu tionofsampleddatasystems
willbegiven,alongwiththemainmathemati aldenitionsandproblemati s. Then,some
general on epts of stabilitywill bere alled, and the sampleddatasystems 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, timevarying sampling,
and dynami sampling ontrol.
1.1 Introdu tion to sampleddata systems
1.1.1 General sampleddata systems
Sampleddatasystemsaredynami systemsthatinvolvebotha ontinuoustimedynami s
and adis retetime ontrol. They are mathemati ally asfollows:
Denition 1.1 (Sampleddata 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 timevariable,x
: R+
→ R
n
the "statetraje tory",
u
: R+
→ R
n
u
the
whi h satisfy
0 = s0
< s
1
<
· · · < s
k
<
· · ·
andlim
k→+∞
sk
= +
∞
.
The sampling law is dened as
sk+1
= sk
+ τk,
(1.2)where
τk
represents thek
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 analogtodigital onverter (a sampler) and a
digitaltoanalog onverter (a zeroorder hold) respe tively.
˙x(t) = f (t, x(t), u(t))
x(t)
u(t) = u(sk)
SYSTEM CONTROLLER A/D D/Ax(sk)
u(sk) = g(sk, x(sk))
sk+1
= sk
+ τk
Figure 1.1: Sampleddatasystem
Itisimportanttonotethatwiththesesystems,the dis retetimedynami sintrodu ed
by the (digital) ontroller implies that during the time between two sampling instants
the system is ontrolled in openloop (i.e. without updating the feedba k information).
Therefore, thesamplingperiodplaysanimportantroleinthestabilityofthe system,and
adapted tools haveto be used.
1.1.2 Sampleddata linear timeinvariant systems
The model of sampleddata systems provided in Denition 1.1 is very general. In this
thesis, we will fo us mainly on linear timeinvariant sampleddata systems with
statefeedba k, whi h are dened as follows:
Denition 1.2 (Sampleddata linear timeinvariant 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 timevariable,x
: R+
→ R
n
the "statetraje tory",
u
: R+
→ R
n
u
the
"input", or " ontrol signal", and the s alars
sk
, fork
∈ N
, are the sampling instants whi hsatisfy0 = s0
< s1
<
· · · < s
k
<
· · ·
andlim
k→+∞
sk
= +
∞
.
A
∈ M
n(R)
isthe "state matrix",B
∈ M
n,n
u
(R)
is the "input gain matrix", andK
∈ M
n
u
,n(R)
is the " ontrol gain matrix". The sampling law is dened assk+1
= sk
+ τk,
(1.4)where
τk
represents thek
th
samplinginterval.
This denition presents the ase of "ideal" sampleddata 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 ontrolloopfor 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
onse utive samplinginstants
sk
andsk+1
is given byx(t) =
eA(t−s
k
)
x(sk) +
R
t−s
k
0
eAs
_{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
, andΛ
dened onR
+
asAd(σ) =
eAσ
, Bd(σ) =
Z
σ
0
eAs
_{dsB.}
(1.6) andΛ(σ) = Ad(σ)
− B
d(σ)K =
eAσ
−
Z
σ
0
eAs
_{dsBK.}
(1.7)Usingthe notation
τk
in equation (1.4), for the sampling intervals,it is then possibletoobtain the following asso iateddis retetime modelof the linear sampleddatasystem
atinstants
sk
:xk+1
= Ad(τk)xk
+ Bd(τk)uk
= Λ(τk)xk,
_{∀k ∈ N,}
(1.8)with
xk
≡ x(s
k)
anduk
≡ u(s
k)
.Ad(τk)
andBd(τk)
are alledthe "statematrix" and the "input matrix" of the dis retetime model respe tively, andΛ(τk)
is alled the dis retetime "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, sampleddata systems bring up new hallenges. As in the more
general frameworks of delayedsystems [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 sampleddata system is stable for any onstant
samplinginterval
τk
≡ τ
with values ina bounded subsetΩ
⊆ R+
?PROBLEMB:Determineifthesampleddatasystemisstableforanytimevarying
samplinginterval
τk
with values in abounded subsetΩ
⊆ R
+
?Lately,anadditionalissuehasbeenbroughtuptolight. Withtheemergen eof
embedded 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
embedded systems for example. In the ase of networked ontrol systems, the transmission
of the sampleddatarequiresbandwidth,whi his alsolimited. Therefore, anew problem
arose:
 PROBLEM C: Design a sampling law
τk
= τ (t, sk, x(sk),
· · · )
that enlarges thesamplingintervalswhile making the sampleddatasystem 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 sampleddata
systems,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 timeinvariantand autonomous (i.e.for whi hthere isno ontrol, orfor
a losedloop 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 notdepend expli itely on the free variable
t
(often regarded as time).An "equilibriumpoint"
xe
represents a real solutionof the equationf
(x) = 0
.Denition 1.4 ( [Khalil 2002℄) An equilibrium point
xe
of the system (1.9) is stable(in the sense of Lyapunov) if
∀ǫ > 0, ∃δ = δ(ǫ) > 0
su h thatkx(0) − x
e
k < δ ⇒ kx(t) − x
e
k < ǫ, ∀t ≥ 0;
attra tive if
∃ρ > 0
su h thatkx(0) − x
e
k < ρ ⇒ lim
t→+∞
kx(t) − x
e
k = 0;
asymptoti ally stable if it isstable and attra tive;
exponentially stable if there exist three s alars
α, β, δ >
0
su hthatkx(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 ayrate", the equilibrium point is also saidto be "β
stable"; globally asymptoti ally stable if it is stableand
∀x(0) ∈ R
n
,
lim
t→+∞
kx(t) − x
e
k = 0
1Giventwometri spa es
(X, d
X
)
and(Y, d
Y
)
, whered
X
denotesthemetri onthesetX
andd
Y
is themetri onsetY
, afun tionf
: X
→ Y
is alled Lips hitz ontinuous(or simplyLips hitz)if there existsareal onstantK
≥ 0
su hthat forallx1, 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 stabilityproperties 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 neighborhoodofxe
). If thereexista lo allyLips hitz fun tionV
: R
n
_{→ R+}
with ontinuouspartialderivativesandtwo lass
K
fun tions2
α
andβ
su h thatα(
_{kxk) ≤ V (x) ≤ β(kxk), ∀x ∈ Ω,}
then the origin
x
= 0
of the system is stable (inthe senseof Lyapunov) if
dV
(x)
dt
≤ 0, ∀x ∈ Ω, x 6= 0;
asymptoti ally stable if there exists a lass
K
fun tionϕ
su h thatdV
(x)
dt
≤ −ϕ(kxk), ∀x ∈ Ω, x 6= 0;
exponentially stable if, moreover, there exist four s alars
α, ¯
¯
β, γ, p >
0
su h thatα(
kxk) = ¯
α
_{kxk}
p
, β(
kxk) = ¯
β
_{kxk}
p
, ϕ(
kxk) = γkxk.
In su h a ase, the equilibrium point
xe
allows a de ayrate equal toγ
p
.There alsoexists adis retetime version of the Lyapunov stability theory.
2
Theorem 1.6 Considerthe dis retetime autonomous system
xk+1
= f (xk),
(1.10)with an isolated equilibrium point (
xe
= 0
∈ Ω ⊆ R
n
, with
Ω
a neighborhood ofxe
). Ifthere exist a lo ally Lips hitz fun tion
V
: R
n
_{→ R+}
with ontinuous partial derivatives
and two lass
K
fun tionsα
andβ
su h thatα(
_{kxk) ≤ V (x) ≤ β(kxk), ∀x ∈ Ω,}
then the origin
x
= 0
of the system is 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);
asymptoti ally stable if there exists a lass
K
fun tionϕ
su h that∆V (xk)
_{≤ −ϕ(kx}
k
k), ∀x
k
∈ Ω, x
k
6= 0;
exponentially stable if there exist four s alars
α, ¯
¯
β, γ, p >
0
su h thatα(
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 tions3 and
Ω = R
n
. The fun tionV
: 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 timeinvariant systems with
sampleddata ontrol
Very interesting properties arise inthe ontext of sampleddata LTI systems, on erning
ontinuous and dis retetime analysis approa hes. One of the rst on erns the
equilibrium's attra tivity,and is formulated asfollows:
Theorem 1.8 (From [Fujioka 2009b℄) ForagivensampleddataLTIsystem(1.3)with
bounded sampling intervals and a given initial state
x(0)
, the following onditions areequivalent:
(i)
limt→+∞
x(t) = 0
,(ii)
limk→+∞
x(sk) = 0
.Thispropertymeansthattheattra tivityofthe ontinuoustimesystem(1.3)is
equivalent tothe attra tivity of the dis retetime system (1.8).
Further analysis [Hetel 2011a℄ allows for proving that the ontinuoustime system's
(asymptoti ) stability is equivalent to the dis retetime system's (asymptoti )
stability, in the more general ase of reset ontrol systems ( [Nesi 2008℄, [Beker 2004℄
[Tarbourie h2011℄, [Za arian 2005℄).
Therefore, it ispossible touse both a ontinuous ora dis retetimeapproa hinorder
to study the stabilityof sampleddata systems.
In the following, we will present an overview of some results from the litterature
regarding the three main studies on erning sampleddatasystems:
the stability analysis regarding a onstant sampling(Problem A);
the stability analysis regarding timevarying sampling(ProblemB);
the dynami ontrolof the sampling(Problem C).
1.3 Stability analysis under onstant sampling
The rst andeasiest way tostudy sampleddatasystems 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 retetime LTI
modelof the system:
˙x(t) = Ax(t) + Bu(t)
x(t)
u(t) = u(sk)
SYSTEM CONTROLLER A/D D/Ax(sk)
u(sk) =
_{−Kx(s}
k)
sk+1
= sk
+ T
Figure1.2: Sampleddatasystem with a onstant samplingrate
Foragivensamplingperiod
T
,themost ommonapproa htoanalysethestability(theso alled "S hur method") onsists in studying the eigenvalues of the transition matrix
Λ(T )
. We allλ
max
(T )
the eigenvalue of
Λ(T )
with the largest modulus. We then havethe following properties [Aström 1996℄.
Theorem 1.9 Theequilibrium
xe
= 0
of (1.11) is S hurstable (globally asymptoti ally stable) if and only if
λ
max
(T )
 < 1
. In that
ase,
Λ(T )
is alled a S hur matrix; exponentially stable (globally) with a de ayrate
α >
0
if and only ifλ
max
(T )
 ≤
e−αT
.Equivalent Linear Matrix Inequality (LMI) stability onditions an also be obtained
using the Lyapunov stabilitytheory for dis retetime systems.
Theorem 1.10 The onsidered system (1.11) is
stable(globally) if and only if there exists a matrix
P
∈ S
+∗
n
su h thatΛ(T )
T
_{P}
_{Λ(T )}
− P 0;
S hurstable(globally asymptoti ally stable) ifand only if there exists a matrix
P
∈
S
n
+∗
su hthat exponentially stable (globally) with a de ayrate
α >
0
if and only if there exists a matrixP
∈ S
+∗
n
su h thatΛ(T )
T
_{P}
_{Λ(T )}
−
e−αT
_{P}
0.
The dis retetime analysis of sampleddata systems with a given onstant sampling
has sin elongbeensolved. However, someproblemsstillremainopen, sin etheproposed
solutions remain onservative regarding the ontinuoustime analysis of su h systems, or
regarding the robustness with respe t to exogenous perturbations. For more results
regardingrobuststabilityandoptimal ontrolof sampleddatasystems both in
ontinuoustime and dis retetime, 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 timevarying sampling
In the literature, there exist numerous studies about sampleddata systems with a
onstant sampling interval. In pra ti e however, it may a tually be impossible to maintain
a onstant sampling rate during the realtime ontrol of physi al systems. Embedded
and networked systems for example are oftenrequiredto sharea limitedamountof
omputational 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
represented 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/Ax(sk)
u(sk) =
_{−Kx(s}
k)
sk+1
= sk
+ τk
Figure1.3: Sampleddatasystem with a timevarying 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 thatthe origin of the system is S hurstable 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
transition matrix
Λ(T )
, dependingon the samplingperiodT
Therefore, for onstant sampling intervals
T
1
= 0.18s
orT
2
= 0.54s
for example, thesystem isS hurstable, 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)andT2
= 0.54s
(right)Stable0
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 retetimeequivalentsystemovertwosamplinginstants anbewritten
as
xk+2
= Λ(T2)Λ(T1
)xk,
_{∀k ∈ 2N,}
whi h an also be writtenas
with
h
representing the2k
thsampling,and the transitionmatrix
Λ(T1
, T
2)
≡ Λ(T2)Λ(T1) =
"
0.8069
_{−3.2721}
0.6133
−2.1125
#
overtwo sampling intervals
T1
andT2
,whi his not S hurin this example.In the ase of sampleddata 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
andT
2
, in the ase of a periodi sequen e of two sampling intervals,forthe sampleddatasystem (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
sequen e
T
1
→ T2
→ T1
→ T2
→ · · ·
rst exampleInthis 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
andT2
= 0.8s
forexample). 4by"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
sequen e
T1
→ T
2
→ T
1
→ T
2
→ · · ·
 se ondexampleLet us look at the sampling values
T
1
= 2.126s
andT
2
= 3.950s
for example. Thesampleddatasystem(1.13)isunstablewithboth onstantsamplings
T1
andT2
. However,asitisshown inFigure1.9,the system'stransitionmatrix
Λ(T1
, T
2)
isS hurstableunderthe periodi sampling
T
1
→ T2
→ T1
→ T2
→ · · ·
.A ording to the previous observations, itis lear that the existing stability toolsfor
sampleddatasystemswitha onstantsamplingwillnotprovideanyguaranteeofstability
for sampleddata systems with unknown timevarying sampling that arises in realtime
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
→ · · ·
 Stablewith timevarying 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 Timedelay approa h with Lyapunov te hniques
Oneoftheapproa hestodealwithtimevaryingsamplingwasinitiatedin[Mikheev1988℄,
and onsists in onsidering the dis retetime 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 sampleddata(1.3) isthenremodeled asan LTI system with timevarying 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 timedelay systems [Ri hard 2003℄,
[Fridman 2003℄, [Zhong 2006℄, [Mounier 2003b℄ whi h are dened by retarded fun tional
Denition 1.11 (Timedelay system) Atimedelay system isdes ribed bythe
following fun tional dierential equation:
˙x(t) = f (t, xt),
∀t ≥ 0,
xs
_{0}
(θ) = φ(s
0
+ θ),
∀θ ∈ [s0
− ¯h, s0
]
(1.15) wheref
: R+
× C
0
_{([}
_{−¯h, 0] → R}
n
_{)}
→ R
n
,φ
∈ C
0
_{([}
_{−¯h, 0] → R}
n
_{)}
, with
¯h ≥ 0
the maximaldelay, 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 timedelay
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 delayfree ase,iftheequilibriumpointisnot0
,we an omedowntoitbyusingasimplehange of oordinates).
Inthegeneral aseoftimedelaysystems,itisdi ulttoapplythe lassi alLyapunov
stability theory from Theorem 1.5, be ause the derivative
dV
(x)
dt
will depend on the pastvalues of the state:
xt
. To over ome this issue, two dierent stability approa hes, bettersuited totimedelay 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
LyapunovRazumikhin [Gu 2003℄, and makes use of atimedependent "energy"fun tion
V
_{≡ V (t, x(t))}
. The se ond approa h, alled LyapunovKrasovskii [Gu 2003℄, makes use of a fun tionalV
≡ V (t, x
t)
instead.5
Notethat
x(t)
isthevalueofthestateatθ
= 0
:x(t) = x
t
(0)
. 6Under existen eand uni ityof thesolution, it anbeshown[Dambrine 1994℄ that theequilibrium statedened by
˙x(t) = 0
isa onstant fun tionx
t
(θ)
≡ x
e
, thus the expression"equilibrium point" is justied.1.4.2.1 LyapunovRazumikhin approa h
In this approa h, it is onsidered a fun tion
V
≡ V (t, x(t))
. The originality is to showthat 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 aneighbourhood
V
(t, x(t))
≤ c
of the equilibrium point. The approa h is formulated asfollows.
Theorem 1.12 (LyapunovRazumikhin (from [Gu 2003℄)) Consider three
ontinuous nonde reasing fun tions
α, β, γ
: R+
→ R
+
,β
stri tly in reasing, su h thatα(θ)
and
β(θ)
are stri tly positive for allθ >
0
, andα(0) = β(0) = 0
. Assume that the ve toreld
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 normonR
n
, and ifthe derivative of
V
alongthe solutions of (1.15)satises˙
V
(t, x(t))
_{≤ −γ(kx(t)k)}
wheneverV
(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 allθ >
0
, and if there exists a ontinuous nonde reasing fun tionp
: R+
→ R
+
satisfyingp(θ) > θ
for allγ >
0
, and su h that ondition (1.19)is strengthened to˙
V
(t, x(t))
≤ −γ(kx(t)k)
wheneverV
(t + θ, x(t + θ))
≤ p(V (t, x(t))), ∀θ ∈ [−¯h, 0],
(1.19)then the fun tion
V
is alled a LyapunovRazumikhin 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 globallyuniformly asymptoti ally stable.
In pra ti e, for simpli ity, most existing works about LyapunovRazumikhinstability
usealinearfun tion:
p(θ) = qθ
,withas alarq >
1
. Moreover, theLyapunovRazumikhinandidates are very often taken as quadrati and timeinvariant:
V
(x) = x
T
_{P x}
, where
P
_{∈ S}
_{n}
+∗
. Some works about the LyapunovRazumikhin approa h for delayed systemsOne ofthe advantagesof the LyapunovRazumikhinstabilitytheory 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 LyapunovKrasovskii te hniques
to bepresented now.
1.4.2.2 LyapunovKrasovskii approa h
The LyapunovKrasovskii 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 (LyapunovKrasovskii (from [Gu 2003℄)) Considerthree ontinuous
nonde reasingfun tions
α, β, γ
: R+
→ R+
, su hthatα(θ)
andβ(θ)
are stri tlypositivefor all
θ >
0
, andα(0) = β(0) = 0
. Assume that theve tor eldf
from (1.15) isboundedfor 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 onR
n
,
k.kC
its asso iated norm onC
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 allθ >
0
, then the fun tionalV
is alled aLyapunovKrasovskii 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 globallyuniformly asymptoti ally stable.
The fun tionalsthat are being onsidered usuallyhave theform [Kolmanovskii 1996℄: