Contributions to the Theory of Time-Delay Systems : Stability and Stabilisation

(1)

HAL Id: tel-02395141

https://tel.archives-ouvertes.fr/tel-02395141

Submitted on 5 Dec 2019

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Contributions to the Theory of Time-Delay Systems :

Stability and Stabilisation

Caetano de Brito Cardeliquio

To cite this version:

Caetano de Brito Cardeliquio. Contributions to the Theory of Time-Delay Systems : Stability and

Stabilisation. Automatic. Université Paris Saclay (COmUE); Universidade estadual de Campinas

(Brésil), 2019. English. �NNT : 2019SACLC080�. �tel-02395141�

(2)

T

h

`e

s

e

d

e

d

o

c

to

ra

t

N

T

:

2

0

1

9 S

A

C

L

C

0

8

0

50

100

150

200

250

300

350

400

450

500

0

50

100

150

200

250

300

350

400

450 Contributions to the Theory of

Time-Delay Systems: Stability and

Stabilisation

Th `ese de doctorat de l’Universit ´e Paris-Saclay

pr épar ée à CentraleSup élec

´

Ecole doctorale n

◦

_{580 Sciences et technologies de l’information et de la}

communication (STIC)

Sp ´ecialit ´e de doctorat : Automatique

Th èse pr ésent ée et soutenue à Campinas-SP, le 27 septembre 2019, par

C

AETANO DE

B

RITO

CARDELIQUIO

Composition du Jury :

Oswaldo Luiz do Valle COSTA

Professeur, USP (EPUSP)

Pr ´esident / Rapporteur

Reinaldo Martinez PALHARES

Professeur, UFMG (DELT)

Rapporteur

Sami TLIBA

Professeur Associ ´e, CENTRALE SUP ´

ELEC (L2S)

Examinateur

Matheus SOUZA

Professeur Associ ´e, UNICAMP (DCA-FEEC)

Examinateur

Catherine BONNET

Directeur de Recherche, CENTRALE SUP ´

ELEC (L2S/INRIA)

Directrice de th `ese

Andr ´e Ricardo FIORAVANTI

(3)

0

50

100

150

200

250

300

350

400

450

500

550

0

50

100

150

200

250

300

350

400

450

UNIVERSITY OF CAMPINAS

SCHOOL OF MECHANICAL ENGINEERING

DEPARTMENT OF COMPUTATIONAL MECHANICS

Caetano de Brito Cardeliquio

B. S . in Controland Automation Engineering - UNESP

Master in Ele tri alEngineering - UNICAMP

Contributions to the Theory of Time-Delay Systems:

Stability and Stabilisation

PhD Dissertation presented to the S hool of

Me hani al Engineering of UNICAMP, to the

grandé ole CENTRALESUPÉLEC and

Paris-Sa lay University as requirement for obtaining

the degree of PhD in S ien es and T

e hnolo-gies of Information and Communi ation from

Paris-Sa layUniversityand PhDinMe hani al

Engineering, Area of Appli ation:

Me hatron-i sfrom UniversidadeEstadual de Campinas.

Campinas

(4)

Le but de ette thèse est de présenter de nouveaux résultats sur l'analyse

etlasynthèse de systèmesàretard.Danslapremièrepartie,nous étendons

l'utilisation du système de dimension nie invariant dans le temps, appelé

système de omparaison, à la on eption d'un ontrleur qui dépend non

seulement de la sortie du système à l'instant

t

ainsi que du retard

maxi-mal,maiségalementd'unnombrearbitrairedevaleursentre elles- i.Cette

appro he nous permet d'augmenter le retard maximal stable sans exiger

d'informations supplémentaires. Les méthodes présentées i i on ernent la

on eptiondesystèmesde ontrleave desretardsenutilisantdesroutines

numériques lassiques basées sur la théorie

H

∞

. La deuxième partie de e

travailtraited'unenouvelleappro hepourdévelopperuneenveloppe

englo-bant tous lesples d'un système àretard. Grâ e auxLMIs (Linear Matrix

Inequalities),noussommesen mesurede déterminerlesenveloppespourles

systèmes àretarddu typeretardéetdu typeneutre. Lesenveloppes

propo-sées sontnon seulementplus étroitesque ellesde la littérature,mais,ave

notrepro édure,ellespeuventégalementêtreappliquéespourvérierla

sta-bilité du système et pour déterminer des ontrleurs par retour d'état qui

sont robustes fa e aux in ertitudes paramétriques. Les systèmes

fra tion-nairessontégalementdis utésdanslesdeux hapitresmentionnés i-dessus.

La troisième et dernière partie étudie les systèmes sto hastiques ave des

retards.Nousdis utonsd'aborddes systèmesàtemps ontinusoumisàdes

sauts de Markov. Nous dénissons la stabilité et obtenons des LMIs pour

le ontrle par retourd'état de telle sorte que la relation entre les tauxde

transition entre les modes soit ane, e qui permet don de traiter le as

dans lequel les taux sont in ertains. Nous dis utons ensuite des systèmes

positifs ave retards, tant pour le as ontinu que pour le as dis ret. Un

système linéairequimodéliseladynamiquedu premiermomentest obtenu

et lastabilitédépendantdu retardest traitée.De nombreux exemplessont

illustrés tout aulong de lathèse.

Mots lés : Systèmes à retard, Retour d'état, Retour de sortie, Norme

H

∞

, Système de omparaison, Systèmes linéairesave du saut de Markov,

(5)

The aimof this dissertation is to present new results on analysis and

on-trol design of time-delay systems. On the rst part, we extend the use of

a nite order LTI system, alled omparison system, to designa ontroller

whi h depends not only on the output at the present time and maximum

delay, but alsoon an arbitrary number of values between those. This

ap-proa h allows us to in rease the maximum stable delay without requiring

any additional information. The methods presented here onsider

time-delay systems ontrol design with lassi al numeri routines based on

H

∞

theory. These ondpart of thiswork dealswith anewapproa htodevelop

an envelope that engulfs all poles of a time-delay system. By means of

LMIs, we are able to determine envelopes for retarded and neutral

time-delay systems. The envelopes proposed are not onlytighter than the ones

inthe literaturebut, withour pro edure, they analsobeappliedtoverify

the stability of the system and design state-feedba k ontrollers whi h are

robust in fa eof parametri un ertainties. Fra tional systems are also

dis- ussed for both hapters mentioned above. The third and lastpart studies

sto hasti time-delay systems. First we dis uss ontinuous-time systems

that are subje ted to Markov jumps. We dene stability and obtain LMIs

forthe state-feedba k ontrolinsu hawaythatthe relationwiththe

tran-sition rates between the modes is ane, allowing, therefore, to treat the

ase in whi h the rates are un ertain. We then dis uss positive systems

with delays, both for the ontinuous ase as for the dis rete ase. A

lin-ear system that models the rst moment dynami s is obtained and delay

dependent stability is addressed. A fair amount of examples are presented

throughout the dissertation.

Keywords: Time-delay Systems, State Feedba k, Output feedba k,

H

∞

-norm, Comparison System, Markov Jump Linear Systems, Fra tional

(6)

O objetivo desta tese éapresentar novosresultados naanálise e nasíntese

de ontroladores parasistemas omatrasos. Naprimeiraparte,estendemos

o uso de um sistema linear invariante notempo de ordem nita, hamado

sistema de omparação, para projetar um ontrolador que depende não

apenas da saída no tempo presente e do atraso máximo, mas também de

um número arbitrário de valores entre eles. Essa abordagem nos permite

aumentar o atraso estável máximo sem exigir do sistema nenhuma

infor-mação adi ional. Os métodos apresentados aqui onsideram o projeto de

ontrole de sistemas de atraso no tempo om rotinas numéri as lássi as

baseadas nateoria

H

∞

. A segunda partedestetrabalhotratade uma nova

abordagem para desenvolver um envelope que engloba todos os polos de

um sistema om atrasos. Por meio de LMIs, podemos determinar

envelo-pespara sistemas om atrasos dotiporetardo epara sistemas om atrasos

do tipo neutro. Os envelopes propostos não são somentemais estreitosdo

que os presentes na literatura, mas, além disso, om nosso pro edimento,

eles tambémpodemser apli adospara veri ara estabilidadedosistemae

empregadospara seprojetar ontroladores viarealimentaçãode estadoque

são robustos perante ain ertezas paramétri as. Sistemasfra ionários

tam-bém são dis utidos emambas as partes supra itadas. A ter eira e última

parte estuda sistemas esto ásti os om atraso. Primeiro dis utimos

siste-massujeitosasaltosmarkovianosatempo ontínuo. Denimosestabilidade

e obtemosLMIs para o ontrole por realimentação de estado de tal forma

que a relação om as taxas de transição entre modos é am, permitindo,

portanto, tratarmos o aso em que as taxas são in ertas. Dis utimos, em

seguida, sistemas positivos om atrasos, tanto para o aso ontínuo omo

para o aso dis reto. Um sistema linear que modela a dinâmi a de

pri-meiro momentoéobtidoeaestabilidadedependentedoatrasoéabordada.

Uma boaquantidadede exemplos,aolongo datese, ilustramos resultados

al ançados.

Palavras- have: Sistemas omatraso, realimentaçãode estado,

realimen-tação de saída, norma-

H

∞

,Sistema de Comparação,SistemasLineares

su-jeitos aSaltos Markovianos, SistemasFra ionários,Desigualdades

(7)

(8)

Tothegentleand kindCatherineBonnet mydeeplythankyou; fora eptingmeasyour

PhD student,forallthe helpbothonbureau rati issues asonte hni aldis ussionsand also,

foren ouragingmetospeakmoreFren handless Englishonourmeetingsand onversations.

To my advisor André R. Fioravanti, words annotexpress how mu h I amgrateful for

his patien e, help and motivation throughout the years. He always believed onmy potential

even when I did not. For his belief and onden e that I would a hieve this tough goal, my

eternal gratitude.

To the rapporteurs Oswaldo Luiz do Valle Costa and Reinaldo Martinez Palhares and

tothe members ofthe jury,Sami Tlibaand MatheusSouza, I amthankful for the a eptan e

in parti ipate on my viva vo e and also for all the remarks, suggestions and orre tions that

aggrandised this work.

To my PhD olleagues, I wish to thank, FilipePerez who gave me good tips regarding

life in Fran e; Walid Djema who was always a good friend even when I was not so friendly;

MariaBek hevaforallthe onversationsandforkeepingthegoodenvironmentatthelabwith

her optimisti way tofa e life with good humourand always with asmile on her fa e.

I thank the Erasmus-Smart

2

program whi h oered the possibility of a double degree

between universitiesfromdierent ountries. I alsothankthe Institute Nationalde Re her he

en Informatique et Automatique - INRIA, the Laboratoire de Signaux et Systèmes - L2S,

the É ole Supérieure d'Éle tri ité - SUPÉLEC, the university Paris-Sa lay, the Fa ulty of

Me hani al Engineering -FEM and the university UNICAMP.

I am thankful to the Brazilian Navy for allowing me to travel to Fran e to improve

my professional skills and be ome a better Engineer and a better person. I thank Captain

Luiz Antnio Moura and Captain Osvaldo Monteiro de Carvalho for helping me before my

departure. InFran e,Ithank,CaptainMáximoEduardoEggerandthedearSophieDesumeur

who was always sympatheti and helpful.

Finally, to Carolina, thank you for more than twelve years by my side. You bring

happiness to my life,pea e to my mind and joyto my heart. In this world full of doubtsand

(9)

1 Introdu tion 1

1.1 Obje tives . . . 1

1.2 Preliminaries . . . 1

1.2.1 Time-Delay Systems . . . 1

1.3 FinalRemarks . . . 3

2 Rational Comparison Systems 5 2.1 Introdu tion . . . 5

2.2 Comparisonsystem . . . 6

2.2.1

H

∞

Norm Cal ulation . . . 12

2.3 State-Feedba k Design . . . 13

2.4 Output-Feedba k Design . . . 19

2.4.1 FilterDesign . . . 24

2.5 Fra tionalsystem . . . 26

2.6 FinalRemarks . . . 31

3 Stability and stabilisation using envelopes 32 3.1 Introdu tion . . . 32

3.2 Retarded Systems . . . 33

3.2.1 Implementation . . . 35

3.2.2 Stability . . . 38

3.2.3 State feedba k forRetarded systems . . . 39

3.2.4 Robust ase . . . 41

3.3 Neutral Systems. . . 47

3.3.1 Implementation . . . 49

3.3.2 State feedba k forNeutral systems . . . 50

(10)

4 Sto hasti Time-Delay Systems 57

4.1 Introdu tion . . . 57

4.1.1 Un ertainrates . . . 60

4.2 Stability . . . 60

4.2.1 Stabilityof Markoviantime-delay systems . . . 61

4.3 Stabilisation . . . 62 4.3.1

H

∞

Norm . . . 62 4.3.2 State Feedba k . . . 64 4.4 Positive systems . . . 69 4.4.1 Continuous-time Case . . . 70 4.4.2 Dis rete-time Case . . . 77 4.5 FinalRemarks . . . 81 5 Con lusions 82 A Fra tional Systems 93 A.0.1 Fra tional al ulusand ontrol. . . 93

A.0.2 Stability . . . 94 B Résumé en Français 97 B.1 Introdu tion . . . 97 B.2 Système de Comparaison . . . 98 B.2.1 Cal ul de lanorme

H

∞

. . . 100 B.3 Retour d'état . . . 101 B.4 Enveloppes . . . 103 B.5 Systèmes retardés . . . 104 B.5.1 Miseen ÷uvre. . . 106

B.5.2 Retour d'étatpour systèmes retardés . . . 107

B.6 Systèmes neutres . . . 109

B.6.1 Miseen ÷uvre. . . 111

B.6.2 Retour d'étatpour systèmes neutres . . . 111

B.7 Systèmes de Markov ontinusave des retards . . . 112

B.8 Con lusions . . . 117

C Resumo em Português 120 C.1 Introdução . . . 120

(11)

C.2.1 Cál uloda norma

H

∞

. . . 123

C.3 Realimentação de estado . . . 124

C.4 Envelopes . . . 126

C.5 Sistemas om atrasos dotipo retardo . . . 127

C.5.1 Implementaçao . . . 129

C.5.2 Realimentação de estado para sistemasdo tiporetardo . . . 131

C.6 Sistemas om atrasos dotipo neutro . . . 132

C.6.1 Implementação . . . 134

C.6.2 Realimentação de estado para sistemasdo tiponeutro. . . 134

C.7 Sistemasde Markov ontínuo om atrasos . . . 135

(12)

2.1

H

∞

norm and lowerbounds as fun tionsof

τ

. . . 14

2.2 Buer ne essary to implement

u(t)

. . . 14

2.3

τ

γ

as afun tion of

N

for

γ = 0.13

.. . . 19

2.4

H

∞

performan e versus time delay for

γ = 1

. . . 24

3.1 Envelopes for dierent values of

d

and fromprevious work inthe literature . . 38

3.2

α

-stability,

α = 1

,

d = 31

. . . 40

3.3

α

-stability,

α = 1

,

d

suggested by [76℄ . . . 41

3.4 Envelope for anUn ertain Retarded Time-DelaySystem . . . 45

3.5

α

-stabilityEnvelope foran Un ertainRetarded Time-Delay System . . . 46

3.6 Envelopes for dierent values of

d

- Neutral-type . . . 50

3.7 State-feedba k -

τ

1 = τ

h

= 2

. . . 52

3.8 Stability Envelope for Fra tionalRetarded Time-Delay System . . . 54

3.9 Zoominon poleinside the envelope . . . 54

4.1 Markov hain with three states . . . 59

4.2 System norm forone un ertain parameter . . . 68

4.3 System norm fortwo un ertain parameters . . . 69

4.4 FirstExample: RealPartof Rightmost Eigenvalue of

F + G(τ )

. . . 74

4.5 Se ond Example: RealPart of Rightmost Eigenvalue of

F + G(τ )

. . . 75

4.6 Expe ted value for the state variablesfor

τ = 2

. . . 76

4.7 Expe ted value for the state variablesfor

τ = 0.1

. . . 76

4.8 Third Example: RealPart of Rightmost Eigenvalue of

F + G(τ )

. . . 77

4.9

x[k]

and

q[k]

ˆ

, for

τ = 1

. . . 81

A.1 The

ω

-stability regionfor fra tional systems . . . 96

B.1

τ

γ

en tantque fon tion de

N

pour

γ = 0.13

. . . 104

(13)

B.3 Retour d'état- Type neutre -

τ

1 = τ

h

= 2

. . . 113

B.4 1er exemple :Partie réelle de la valeur proprela plus à droitede

F + G(τ )

. . 116

B.5 2ème exemple: Partie réellede lavaleurpropre la plus àdroite de

F + G(τ )

. 116 B.6 Valeurattendue pour lesvariablesd'état -

τ = 2

. . . 117

B.7 Valeurattendue pour lesvariablesd'état -

τ = 0.1

. . . 118

C.1

τ

γ

omo função de

N

para

γ = 0.13

. . . 127

C.2 Envelopes para diferentes valores de

d

. . . 131

C.3 Realimentação de estado -Tipo Neutro -

τ

1 = τ

h

= 2

. . . 136

C.4 PrimeiroExemplo: Parte real do autovalormais a direitade

F + G(τ )

. . . . 139

C.5 Segundo Exemplo: Parte real do autovalormais adireita de

F + G(τ )

. . . 140

C.6 Valoresperado para as variáveisde estado-

τ = 2

. . . 141

(14)

N

- Set of natural numbers with zero.

N

∗

- Set of natural numbers without zero.

N

_N

- Set of the rst N+1naturalnumbers

(0, . . . , N)

.

K

- Set of the rst N numbers

∈ N

∗

_{(1, . . . , N)}

.

K

_i

-

K

-{i}.

Z

- Set of integernumbers.

R

- Set of real numbers.

R

∗

- Set of real numbers without zero.

R

₊

- Set of nonnegative real numbers.

R

₋

- Set of nonpositive real numbers.

C

- Set of omplex numbers.

ℜ(.)

- The real part of a omplex numberor a omplexmatrix.

ℑ(.)

- The imaginary partof a omplex number ora omplex matrix.

⌊x⌋

- The largest integer less than or equalto

x

,

x ∈ R.

⌈x⌉

- The least integer greater than orequalto

x

,

x ∈ R.

I

- The identity matrix of any dimension.

X

′

- The transpose of the matrix

X

.

X

∗

- The onjugated transpose of the matrix

X

.

X

−1

- The inverse of a nonsingularsquare matrix

X

.

X > 0

- The symmetri matrix

X

is positivedenite.

X ≥ 0

- The symmetri matrix

X

is positivesemi-denite.

kXk

p

- The indu ed

p

-norm of a matrix

X ∈ C

n×m

.

det(X)

- Determinant of the square matrix

X

.

Tr(X)

- Tra e of the square matrix

X

.

ker(X)

- The null spa e of the matrix

X

, i.e.,

{v ∈ V |Xv = 0}

.

diag(X , Y )

- Diagonal blo k matrix formed by the matri esX and Y.

X ⊗ Y

- Krone kerprodu t.

X ◦ Y

- Hadamardprodu t.

vec(X)

- Ve torizationof a matrix

X

, i.e.,

[x

1,1

, . . . , x

m,1

, . . . , x

1,n

, . . . , x

m,n

]

′

.

Vec(X

i

)

- Sta k matri es

X

i

,

i ∈ K

, ina olumn blo k su has

[X

′

(15)

←

- When inan algorithmindi atestoupdate a value.

λ

i

- The

i

theigenvalue of a matrix.

λ

min

- Minimum eigenvalue of asymmetri matrix.

λ

rme

- Rightmosteigenvalue of a matrix.

σ(X)

- Set of singularvalues of the matrix

X

.

σ

M

(X)

- Maximum singular value of the matrix

X

.

x

L

- Left eigenve tor, with dimension

1 × n

,of a matrix

X

, i.e.,

x

L

X = λ

L

x

L

.

0 D

α

t

- Dierintegral operator.

N

k

- Binomial oe ient.

n!

- Fa torialof n, i.e.,

n! = 1 × 2 × · · · × n

.

co{S}

- The onvex hull of a nitepointset

S

.

E[·]

- Mathemati alexpe tan e.

L

- Innitesimalgenerator.

kz(t)k

2

- Dened by

E

R

∞

0 z(t)

′

_z(t)dt

.

L

₂

- Set of allsto hasti signals

z(t) ∈ R

n

su h that

kz(t)k

2

2 < ∞

.

A

ki

-

A

k

(θ

t

)

whenever

θ

t

= i ∈ K

.

Γ(x)

- Gammafun tion, i.e.,

R

∞

0 e

−y

_y

x−1

_dy

.

1

A

(ω)

- The Dira measure overa set

A

.

1

N

- Unitve tor of order

N

, i.e.,

[1 1 . . . 1]

′

.

α

-stability -

ℜ(λ

j

) < −α

, for alleigenvalues of alinear time-invariantsystem.

o(∆)

-

f ∈ o(∆)

i

lim

∆→0

f (∆)/g(∆) = 0

.

•

- Ea hone of the Hermitianblo ksrelated tothe diagonal

ina Hermitianmatrix.

- End of proof.

(16)

[1℄ C. B. Cardeliquio,A.R.Fioravanti,C. Bonnet,S.-I Ni ules u,Stability andStabilisation

Through Envelopes for Retarded and Neutral Time-Delay Systems, IEEE - Transa tions

on Automati Control (EarlyA ess), 2019.

[2℄ C.B.Cardeliquio,A.R.Fioravanti,C.Bonnet,S.-INi ules u,StabilityandRobust

Stabil-isation Through Envelopes for Retarded Time-Delay Systems, Preprints, Joint 9th IFAC

Symposium on Robust Control Design and 2nd IFAC Workshop on Linear Parameter

Varying Systems, 2018.

[3℄ C. B.Cardeliquio,M.Souza,C.Bonnet, A.R.Fioravanti,StabilityAnalysisand

Output-Feedba k ControlDesign forTime-Delay Systems,InternationalFederationofAutomati

Control- IFAC, 2017.

[4℄ C. B. Cardeliquio, M. Souza, R. H. Korogui, A. R. Fioravanti, Stability Analysis and

State-Feedba k Control Design for Time-Delay Systems, European Control Conferen e,

2016.

[5℄ C. B. Cardeliquio, A. R. Fioravanti e A. P. C. Gonçalves,

H

2

output-feedba k ontrol of

ontinuous-time MJLS withun ertaintransitionrates, IEEE 53rdAnnualConferen e on

De ision and Control, 2014.

[6℄ C.B.Cardeliquio,A.R.FioravantieA.P.C.Gonçalves,

H

2

and

H

∞

state-feedba k ontrol

of ontinuous-time MJLS with un ertain transition rates, European ControlConferen e,

(17)

Chapter

1

Introdu tion

1.1 Obje tives

This dissertation has as its main obje tive the study of time-delay systems. Our goal is

touse Linear MatrixInequalities (LMIs) toobtainnew methods foranalysis and synthesis of

ontrollers and, also, toimproveexisting methods. The time-delay systems that we fo us on

will vary from bran hes su h as lassi alsystems, fra tional systems, sto hasti systems and

positive systems. We aim not only analysis, but state-feedba k and output-feedba k design,

as wellas delay-independent and delay-dependent stabilisation.

1.2 Preliminaries

1.2.1 Time-Delay Systems

Time-delay systems have instigated an in reasingly interest from the ontrol ommunity

[1,2, 3,4℄. This an be due to several pra ti alreasons, amongwhi h we highlight: the time

ne essary to a quire the information needed for the ontrol, the time required to transport

information, the pro essing time, the sampling period, amid many others. Moreover, due to

environmental onditions, e.g., high temperatures inside a ompartment, a ess di ulties,

su hasoshore underwater oilplatforms[5,6,7℄,unhealthyareas,amongothers,onemethod

that is being used to ommand dynami al systems is the approa h of ontrol via a network

[8,9,10℄. Controllersperformingthroughanetworkhave,intrinsi ally,delaysembeddedinits

stru ture. Eventhoughthosedelays,inall asesmentioned,areoftentimesnegle ted,they an

beresponsibleforpoorperforman eand, inworst s enarios,they mayeven leadthesystem to

instability. Forthat reason, several studies onsidering the so alled time-delay systems have

being made through the lastde ades.

Models ontainingdelays an likewiseappearinafairlyamountofpro essessu has

physi- al,biologi al[11,12℄,e onomi al[13,14℄,me hani al[15℄andsoforth. Arstextensivestudy

about delays in dierential equations, known as DDEs, is made in [16℄ while some examples

fortime-delaysystemsasmu hastheiranalysis anbeseen in[17℄. Intimedomain,ageneri

(18)

equation:

˙x(t) = A

0 x(t) + A

1 x (t − τ) + H ˙x (t − τ) + Ew(t),

z(t) = C

0 x(t) + C

1 x (t − τ) + D

z

w(t),

(1.1)

in whi h, for all

t ∈ R

+

,

x(t) ∈ R

n

is the state variable,

w(t) ∈ R

m

is the exogenous input,

z(t) ∈ R

p

is the output of interest,

τ ∈ R

+

isthe delay and

A

0

,

A

1

,

H

,

E

,

C

0

,

C

1

and

D

z

are

real matri eswith appropriate dimensions. This system is alleda neutraltime-delay system

due to the term ontainingthe derivative of the state delayed. Forthe ase where

H = 0

,we

saythatthesystemhasadelayinaretardedform. Inthis ase,thesystemis alledaretarded

time-delay system. Both retarded and neutral time-delay systems are dis ussed throughout

this work.

In frequen y domain,the transfer fun tion of (1.1) is given by

T (s, τ ) = C

0 + C

1 e

−sτ

sI − A

0 − A

1 e

−sτ

− sHe

−sτ

−1

E + D

z

.

(1.2)

The hara teristi equationisthenquasi-polynomialwith,ingeneral,innitesolutions. There

are several possible frameworks to study stability and stabilisation for time-delay systems.

Stability is dis ussed, among others, in [4℄, [18℄ and [19℄. A simple ne essary and su ient

LMI 1

ondition forthe strongdelay-independent stability of LTI systems with single delay is

thesubje tof[20℄. Thedevelopmentofe ient ontroldesignte hniquesthat opewithtime

delay has re eived mu h attention in the past de ades; see the books [21℄ and [22℄ and the

survey paper [23℄ for important theoreti al results in the area. In this ontext,

H

∞

ontrol

te hniquesplay akeyrole inthedesignof ontrollersthatattainapre-spe iedworst ase

L

2

gain for the losed-loopsystem whenever the time delay isgiven [24℄.

For the stabilisationthrough statefeedba k, delay-independent ontrollers an bedevised

using Ri ati equations [25, 26℄, whereas the delay-dependent ase is usually designed by

means of Lyapunov-Krasoviskii fun tionals [27, 28, 29℄. Similar results have been extended

to the output-feedba k framework; see [30, 31, 32,28℄. Lyapunov-Krasoviskii fun tionals are

alsoutilisedforrobust ontrolofstatedelaysystems in[33℄. Filteringandoutputfeedba kfor

time-delay systems an be seen in [34℄ and the design of observers in [35℄. State and output

feedba k stability is dealt in [36℄. A modied Ri ati equation is used in [25℄ for the design

of a memoryless

H

∞

ontroller. The

H

∞

ontrol problemfor multiple input-output delays is

also dis ussed in [37℄. A ontroller design approa h through a nite LTI omparison system

isdeveloped in[38℄and in[39℄. Anothertype ofapproa hbased onarationalapproximate to

theinnite-dimensionalsystem anbedoneusingPadé te hniquessu hasin[40℄. Criteriafor

robuststabilityand stabilisation isdealt in[41℄. Robustexponentialstabilisation forsystems

with time-varying delays an be seen in [23℄. Robust stability and stabilisation for singular

systems with parametri un ertainties are dis ussed, among others, in [42℄ and [43℄. Delay

independent stability for un ertain systems an be seen in[44℄ and delay-dependent stability

and stabilisationin [45℄, [46℄and [47℄. The dis rete ounterpartisstudied in[48℄, forpositive

systems. Guaranteed LQR ontrol is dealt in [49℄ and robust polytopi

H

∞

stati output

feedba k in[50℄. For un ertainlinear systems with multiple time-varying delays, robustlter

is design in [51℄. Additionally,

α

-stability is dis ussed in [52℄ for non ommensurate delays

and in [53℄ via LMIs. Fornon-linear time-delay systems see [54, 55, 56, 57℄.

For sto hasti systems, one of the rst works inthe literature dealing with Markov jump

(19)

linearsystems,hen eforth alledMJLS,withoutdelays,is[58℄fordis retesystemsand[59℄for

ontinuoussystems. Fordis rete-timesystems,alargeamountoftheoryanddesignpro edures

hasbeendeveloped toextendthe on eptsof deterministi systemstothisparti ular lass. In

parti ular, the on epts of stability and the onditions for testing them, whi h are dis ussed

in [60℄, [61℄ and [62℄. Considering MJLS in ontinuous-time there are also several results in

the literature. In [63℄ the

H

2

ontrol is treated through state feedba k via onvex analysis.

Controllability and stability on epts are studied in [64℄ and the optimal quadrati ontrol

withsolutionviatheSeparationTheorem ispresentedin[65℄. TheMJLS ontroland ltering

proje ts assume, for the most part, that transition rates between modes are known a priori.

However, inpra ti e,onlyestimatedvalues oftheseratesare availableandtheseun ertainties

an generate instabilities or at least degrade the system performan e in the same way that

o urs when there are un ertainties in the matrix of the state spa e representation of the

plant. For this ase in whi h transition rates between the modes are not fully known, there

are works inthe literaturethat showstability onditions, as an beseen, for example,in[66℄,

where the robust ase isdis ussed. The state feedba k an be seen in [67℄ and [68℄. A major

referen eforMJLSwithdelaysisthebook[24℄. Otherimportantworksare[69℄, [70℄and[71℄.

1.3 Final Remarks

This brief hapter introdu es the subje t and some basi denitions. The three major

hapters that follows are independent and an be readin any preferable order by the reader.

The stru ture of this dissertation isthe following:

Chapter 2: In this hapter an extended Rekasius substitution [72℄ is applied to repla e

the delay operator by a rational transfer fun tion; in [73℄, a useful te hnique for stability

analysisoftime-delaysystemsthat ombinestheRekasiussubstitutionandtheRouth-Hurwitz

riterionisproposed. Animportant onsequen eof theRekasius substitution,aswearegoing

to present, is the denition of a nite order linear time invariant system, alled omparison

system, whi h provides a tight lower bound to the

H

∞

norm of the time-delay system and

allowsthe developmentofsimpleande ientsynthesis algorithms;seealso[74,75℄. Applying

thisequivalen ywemay opewithstatefeedba kandoutputfeedba k fortime-delaysystems.

Filters an be designed asaparti ular ase ofthe output-feedba k problem. The te hnique is

then adapted for fra tional systems. The obje tive is divided into two ategories: rstly, to

in rease the maximum delay allowed in time-delay linear systems for a given

H

∞

level

γ

and

se ondly, when the delay is given, to minimise

γ

.

Chapter 3: This hapter deals with stability and stabilisation of time-delay systems

through thedesign ofanenvelope thatengulfs allpolesof the system. The use ofanenvelope

thatensuresthatallpolesare ontainedinsideitisdis ussedin[4℄. Dierenttypesofenvelopes

are alsodis ussed in[76℄ and[77℄. Inany ase,nomethodsutilisingenvelopeswere developed

to test stability nor to design ontrollers. In fa t, in general, the envelope extends to the

right half-plane and therefore, it only provides a region where the poles are allowed to be

without any guarantee about the stability of the system. In this work we provide a dierent

analysis for the use of envelopes. Instead of using a singular value approa h, su h as in [4℄,

our method is based on LMIs. We are able to provide a new pro edure to test stability for

both retarded and neutral time-delay systems. Furthermore, it allows to ope with some

(20)

extend the analysis result to fra tionalsystems.

Chapter 4: On this hapter we leave the deterministi domain and we handle with

sto hasti time-delay systems. Markov Jump Linear Systems, orMJLS for short, withdelays

arethemaintarget. Stabilityforsto hasti systemsisdened,see[78℄. State-feedba k ontrol

is then designed through LMIs with the novelty of a hieving an ane relation with respe t

to the transition rate between modes, allowingpolytopi un ertainty to be treated. We then

obtain a linear system that models the dynami s of the rst moment for positive-Markovian

systems and propose a method to analyse delay-dependent stability for both the ontinuous

and the dis rete-time ase.

Chapter 5: Thisnal hapterendswithasummarisationofallthatisdealtinthepresent

dissertation. The on lusion of the work is presented as same as the perspe tives for future

works.

Appendix A: An introdu tiononFra tional Systems.

Appendix B: A summary of the dissertation inFren h.

(21)

Chapter

2

Rational Comparison Systems

This hapter dealswith the

H

∞

ontrolsynthesisfor time-delaylinear systemsusing both

state-feedba k and output-feedba k approa hes. The ltering problemis alsopresented.

2.1 Introdu tion

Our goalistoin rease themaximumdelayallowedintime-delaylinearsystems foragiven

H

∞

level

γ

through state-feedba k and through output-feedba k ontrol design. A se ond

problem that is also addressed is to minimise

γ

whenever the time delay is given. In [79℄,

whi h is the main work that this hapter relies on, the Rekasius substitution [80℄ for

k = 1

was su essfullyapplied toobtaina nite orderLTIsystem, alled omparisonsystem, whi h

was used to al ulate a lower bound for the

H

∞

norm of the time-delay system. Here, we

extend this approa h nding the linear dependen e on the matri es of the system with its

omparisonsystemforasubstitutionof order

N

. We an thenusethis newsystemtodesigna

ontrollerwhi hdepends notonlyon themeasured outputatthe present timeand maximum

delay,butalsoonanarbitrarynumberof intermediatevaluesinbetween,for bothminimising

H

∞

normormaximisingthealloweddelay. Hen e,weareabletoin reasethemaximumstable

delay using information that is already in the buer. Illustrative examples are presented to

reinfor e the theoreti al results.

The main ideais basedonthe followingobservation. Shouldthe ontrollawtobedevised

be of the form

u(t) = K

0 x(t) + K

1 x(t − τ),

(2.1) forsome

τ > 0

, thenapossiblegeneralisationofsu hsignal onsistsinalsousingthe

interme-diate values

x(t − kτ/N)

,

k ∈ {1, . . . , N − 1}

, for feedba k. Note that the reasoning for this

approa h is based on the fa t that, aslong as the delayed state

x(t − τ)

must be stored in a

buer, su hbuer would also ontain the intermediate values ofinterestand, thus, ompared

tothe originalapproa h, noadditionalinformationis required.

Compared to [75℄ and [79℄, the main novelties are:

•

The ommensurate delay problem demands a new parametrisationfor the omparison

system, whi h, to the best of the author's knowledge, has not been presented in the

literature. Moreover, the design pro edure is simple to be implemented and, when

(22)

•

Most feedba k design pro edures demand the knowledge of the delayed state. This

re-quires a memory buer in order to store information from allsensors during this time.

Nevertheless,formostpartofpro edures,in luding[75℄and[79℄,theonlyuseful

informa-tionisthe onethatmat hes thetime delay. In ourpro edure, werelaxedthis onstraint

and showed that intermediary state information an be used for both minimising

H

∞

norm ormaximising the allowed delay.

The same idea isthen used foroutput-feedba k ontrollersand to solvethe ltering

prob-lem.

2.2 Comparison system

Consider the time-delay linear system with

M

ommensurate delays, whose realisation is

given by

˙x(t) = A

0 x(t) +

M

P

k=1

¯

A

k

x (t − ¯τ

k

) + E

0 w(t),

z(t) = C

z0

x(t) +

M

P

k=1

¯

C

zk

x (t − ¯τ

k

) ,

(2.2)

in whi h, for all

t ∈ R

+

,

x(t) ∈ R

n

is the state,

w(t) ∈ R

m

is the exogenous input,

z(t) ∈ R

p

is the output of interest and

τ

¯

k

= τ (M − k + 1)/M

,

k ∈ {1 · · · M}

, for a given onstant time

delay

τ ≥ 0

.

We address the ase of the ommensurate delayed system (2.2) by applying the following

substitution tothe time delay operator on erning the largest delay:

e

−τ s

=

λ − s

λ + s

N

,

(2.3)

whi his anexa t relationfor

s = ω

, whenever

τ, λ, ω ∈ R

+

and

N ∈ N

∗

are su h that

ωτ = 2N arctan

ω

λ

.

(2.4)

When

N = 1

this is known as Rekasius substitution [80℄. We extend this result allowing

N = hM

,

h ∈ N

∗

. For the following developments, regarding the analysis of this system, it

will be ne essary that the number of delays be the same as the order of the approximation

(2.3). Note however, that whenever

N = hM

for some

h ∈ {1, 2, . . .}

, system (2.2) an be

equivalently restated as

˙x(t) = A

0 x(t) +

N

P

k=1

A

k

x (t − τ

k

) + E

0 w(t),

z(t) = C

z0

x(t) +

N

P

k=1

C

zk

x (t − τ

k

) ,

(2.5) where

A

k

← ¯

A

j

,

τ

k

← ¯τ

j

whenever

N − k + 1

N

=

M − j + 1

M

,

(2.6)

(23)

for all

k ∈ {1 · · · N}

,

j ∈ {1 · · · M}

and

A

k

← 0

otherwise. Thus, without loss of generality,

hereafter we are going to work with the rearranged system (2.5) whi h satises

N = hM

for

some

h ∈ {1, 2, . . .}

.

Remark 2.1. The analysis will be done using the number of delays as the order of the

substitution. The hange of variables (2.6) is used to ir umvent this, restating the system

(2.2) as(2.5), a hievinga stronger resultfor the synthesis. We an nowhave

M

delays and

N = hM

for the order of the substitution. Let us illustrate this with an example. Let us

assume asystem with two delays and letus use

N = 4

.

˙x(t) = A

0 x(t) + ¯

A

1 x (t − τ) + ¯

A

2 t −

τ

₂

.

(2.7)

Applying (2.6) we an restate (2.7) as

˙x(t) = A

0 x(t) + A

1 x (t − τ) + A

2 t −

τ

4 + A

3 x

t −

2τ

4 + A

4 t −

3τ

4 ,

(2.8)

inwhi h

A

1 = ¯

A

1

,

A

2 = A

4 = 0

and

A

3 = ¯

A

2

. On thenew variablesthe orderofthe system

is the same as the order hosen for the Rekasius substitution allowing us to use an order

higher than the amount of delays fromthe original system.

One of our goals is to determine the maximal time delay

τ

⋆

_{> 0}

whi h ensures that the

system isglobally asymptoti allystablefor any

τ ∈ [0, τ

⋆

₎

. To a hievethis, one must analyse

the non-rational transfer fun tionof (2.5), whi h isgiven by

T (s, τ ) =

C

z0

+

N

X

k=1

C

zk

e

−τ

k

s

!

sI − A

0 −

N

X

k=1

A

k

e

−τ

k

s

!

−1

E

0 .

(2.9)

Applying the substitution (2.3) tothe transfer fun tion

T (s, τ )

in(2.9), we an dene the

omparison system with transfer fun tion

H(s, λ)

su h that

H(jω, λ) = T (jω, τ )

, whenever

(2.4) holds. In this ase, the following lemma willhelp us dene the omparison system and

(24)

Lemma 2.1. Forany nite

s ∈ C

and matri es

C

k

∈ R

p×n

,

A

k

∈ R

n×n

and

E

0 ∈ R

n×m

N

X

k=0

C

k

s

k

!

s

N +1

_{I −}

N

X

k=0

A

k

s

k

!

−1

E

0 =







C

′

0 C

′

1

. . .

C

′

N







′







sI −







0 I

0 _{· · ·}

0

0 I

_{· · ·}

0

. . . . . . . . . . . . . . . . . .

0

0 _{· · ·}

0 I

A

0 A

1 A

2 · · · A

N −1

A

N













−1







0

. . .

0 E

0 





.

(2.10)

Proof. First of all,we adopt the followingpartitionof the

Nn × Nn

matrix appearingin the

inverse of the se ond lineof (2.10):

X

Y

Z

W

=







sI

_−I

0 _{· · ·}

0

0 sI

_−I

_{· · ·}

0

. . . . . . . . . . . . . . . . . .

0

0 _{· · ·}

sI

_−I

−A

0 −A

1 −A

2 · · · −A

N −1

sI − A

N







.

(2.11)

In order to al ulateitsinverse, we onsider the following identity.

X Y

Z W

−1

=

I −X

−1

_Y

0 I

X

−1

₀

0 Λ

I

0 −ZX

−1

_I

,

(2.12) with

Λ = (W − ZX

−1

_{Y )}

−1

.

On e

X

in (2.11)is triangularsuperior and

s

is nite,

X

is non singularand wehave

X

−1

=







s

−1

_{I s}

−2

_{I s}

−3

_{I · · ·}

_s

−N

_I

0 s

−1

_{I s}

−2

_{I · · · s}

−(N −1)

_I

0

0 s

−1

_{I · · · s}

−(N −2)

_I

. . . . . . . . . . . . . . .

0

0 _{· · ·}

s

−1

_I







,

(2.13) whi hleads to

X

−1

_{Y = −}

s

−N

_{I s}

−(N −1)

_{I s}

−(N −2)

_{I · · · s}

−1

_I

′

(2.14)

(25)

and then to

Λ

−1

_{= sI − A}

N

−

N −1

X

k=0

A

k

s

−(N −k)

= s

−N

s

N +1

_{I −}

N

X

k=0

A

k

s

k

!

.

(2.15)

Finally, makingall the multipli ations involved in the se ond lineof (2.10),we obtain







C

′

0 C

′

1

. . .

C

′

N







′

−X

−1

_{Y Λ}

Λ

E

0 =

=







C

′

0 C

′

1

. . .

C

′

N







′

I s

1 _{I s}

2 _{I · · · s}

N

_I

′

s

N +1

_{I −}

N

X

k=0

A

k

s

k

!

−1

E

0 =

N

X

k=0

C

k

s

k

!

s

N +1

_{I −}

N

X

k=0

A

k

s

k

!

−1

E

0 ,

(2.16)

whi his the proposed equality.

Lemma 2.2. For a given pair

(τ, λ) ∈ R

+

, using (2.3) and applying Lemma 2.1, one an

put (2.9)in anequivalentform as

H(s, λ) =

A

λ

E

C

z

0 =







0 λI

0 N

P

k=0

α

k

(0)A

k

N

P

k=0

A

k

Γ

k

− λΓ

λ

E

0 N

P

k=0

α

k

(0)C

zk

N

P

k=0

C

zk

Γ

k

0 





,

(2.17) in whi h

Γ

k

, Γ

λ

∈ R

n×N n

are given by

Γ

k

=

α

k

(1) α

k

(2) α

k

(3) · · · α

k

(N − 1) α

k

(N)

⊗ I,

(2.18)

Γ

λ

=

α

0 (0) α

0 (1) α

0 (2) · · · α

0 (N − 2) α

0 (N − 1) ⊗ I,

(2.19)

(26)

and

α

0 (i)

,

α

k

(i)

, for

k = 0

and

k ≥ 1

, respe tively, are given by

α

0 (i) =

N

i

,

(2.20)

α

k

(i) =

k−1

X

ℓ=0

N − k + 1

i − ℓ

k − 1

ℓ

(−1)

i−ℓ

.

(2.21)

Proof. Substituting the Rekasius expression (2.3) in (2.9) we get

H(s, λ) =

C

z0

+

N

X

k=1

C

zk

λ − s

λ + s

N −k+1

!

sI − A

0 −

N

X

k=1

A

k

λ − s

λ + s

N −k+1

!

−1

E

0 .

(2.22)

Then, we an multiply

H(s, λ)

by

(λ+s)

N

(λ+s)

N

to obtain

H(s, λ) =

C

z0

(λ + s)

N

+

N

X

k=1

C

zk

(λ − s)

N −k+1

(λ + s)

k−1

!

×

(sI − A

0 ) (λ + s)

N

−

N

X

k=1

A

k

(λ − s)

N −k+1

(λ + s)

k−1

!

−1

E

0 .

(2.23)

Expanding the binomialsthe previous expression be omes

H(s, λ) = C

z

(s, λ)(A(s, λ))

−1

E

0 ,

(2.24) in whi h

C

z

(s, λ) = C

z0

N

X

i=0

N

i

λ

N −i

s

i

+ C

z1

N

X

i=0

N

i

λ

N −i

_(−s)

i

+

+ C

z2

N −1

X

i=0

N − 1

i

λ

N −1−i

_(−s)

i

1 X

ℓ=0

1 ℓ

λ

1−ℓ

s

ℓ

_{+ · · ·}

+ C

zN

1 X

i=0

1 i

λ

1−i

_(−s)

i

N −1

X

ℓ=0

N − 1

ℓ

λ

N −1−ℓ

s

ℓ

(2.25)

= C

z0

N

X

i=0

N

i

λ

N −i

s

i

+

N

X

k=1

C

zk

N −k+1

X

i=0

k−1

X

ℓ=0

N − k + 1

i

k − 1

ℓ

λ

N −i−ℓ

s

i+ℓ

₍₋₁₎

i

(27)

and

A(s, λ) = sI

N

X

i=0

N

i

λ

N −i

s

i

_{− A}

0 N

X

i=0

N

i

λ

N −i

s

i

_{− A}

1 N

X

i=0

N

i

λ

N −i

_(−s)

i

₋

− A

2 N −1

X

i=0

N − 1

i

λ

N −1−i

_(−s)

i

1 X

ℓ=0

1 ℓ

λ

1−ℓ

s

ℓ

_{− · · ·}

(2.26)

− A

N

1 X

i=0

1 i

λ

1−i

_(−s)

i

N −1

X

ℓ=0

N − 1

ℓ

λ

N −1−ℓ

s

ℓ

,

whi h an bewritten ina more ompa t way:

A(s, λ) = (sI − A

0 )

N

X

i=0

N

i

λ

N −i

s

i

₋

N

X

k=1

A

k

N −k+1

X

i=0

k−1

X

ℓ=0

N − k + 1

i

k − 1

ℓ

λ

N −i−ℓ

s

i+ℓ

₍₋₁₎

i

.

(2.27)

One an immediately see that the powers of

s

are in the interval

[0 N]

and that the

powerof

s

and the power of

λ

alwaysadd to

N

. Hen e, itis possible togroupthe terms that

multiplythe same power of

s

as

H(s, λ) =

N

X

i=0

˜

C

zi

λ

N −i

s

i

!

s

N +1

_{I −}

N

X

i=0

˜

A

i

λ

N −i

s

i

!

−1

E

0 ,

(2.28) in whi h

˜

C

zi

=

N

X

k=0

C

zk

α

k

(i),

(2.29)

˜

A

i

=

N

X

k=0

A

k

α

k

(i) − λα

0 (i − 1)I,

(2.30)

and

α

k

(i)

is given by (2.20) when

k = 0

and by (2.21) when

k ≥ 1

. Finally,for being able to

apply Lemma2.1allweneedtodoisasimilaritytransformationon(2.10)using thefollowing

matrix

M

.

M = diag(λ

−N

I, λ

−N +1

_{I, · · · , λ}

−1

I, I),

(2.31) whi hresults in







C

′

0 λ

−N

C

′

1 λ

−(N −1)

. . .

C

′

N







′







sI

_−λI

0 _{· · ·}

0

0 sI

_−λI

_{· · ·}

0

. . . . . . . . . . . . . . . . . .

0

0 _{· · ·}

sI

_−λI

−A

0 λ

−N

−A

1 λ

−(N −1)

−A

2 λ

−(N −2)

· · · −A

N −1

λ

−1

sI − A

N







−1







0

. . .

0 E

0 





.

(2.32)

(28)

an elled and we obtain

H(s, λ) =







˜

C

′

z0

˜

C

′

z1

. . .

˜

C

′

zN







′







sI

_−λI

0 _{· · ·}

0

0 sI

_{−λI · · ·}

0

. . . . . . . . . . . . . . . . . .

0

0 _{· · ·}

sI

_−λI

− ˜

A

0 − ˜

A

1 − ˜

A

2 · · · − ˜

A

N −1

sI − ˜

A

N







−1







0

. . .

0 E

0 





.

(2.33)

Using(2.18)-(2.21),(2.29) and(2.30) we nallya hieve(2.17) whi h on ludes the proof.

2.2.1

H

∞

Norm Cal ulation

We willnowshow howto approximate

kT (s, τ)k

∞

= sup

ω∈R

σ

M

(T (jω, τ ))

(2.34)

for agiven

τ ∈ [0, τ

⋆

₎

. The purpose isto show that the rationaltransfer fun tion

H(s, λ)

an

be su essfully used for

H

∞

norm al ulation of the time-delay system.

In the light of the results presented in[75℄, we extra t animportantproperty relatingthe

H

∞

norm for both the omparison system and the original time-delay one. To this end, we

need to dene the s alar

λ

o

= inf{λ | A

λ

is Hurwitz}

and for ea h

λ ∈ (λ

o

, ∞)

, we dene an

α ≥ 0

su h that,

α ∈ arg sup

ω∈R

σ

M

(H(jω, λ)).

(2.35)

Finally,determining the time delay

τ (λ, α)

that satises

α/λ = tan(ατ /2N),

(2.36)

allows us tostate the following theorem, extendingTheorem

1

of [75℄.

Theorem 2.1. Consider the system (2.5) with noexogenous inputs. Assume that

N

P

i=0

A

i

is

Hurwitz and let

α

begiven by (2.35). If

τ (λ, α) ∈ [0, τ

⋆

₎

su h that

λ

satises (2.36)then,

kH(s, λ)k

∞

≤ kT (s, τ(λ, α))k

∞

.

(2.37)

Proof. The proof follow dire tly fromthe denition of the

H

∞

norm. We havethat

kH(s, λ)k

∞

= σ

M

(H(jα, λ)).

(2.38)

Sin e

α

is given by (2.35) and re allingthat (2.4) makes (2.3) anexa t relation, wehave

(29)

and thus,

kH(s, λ)k

∞

= σ

M

(H(jα, λ))

= σ

M

(T (jα, τ (λ, α)))

≤ kT (s, τ(λ, α))k

∞

.

(2.40)

The following example illustratesthe result presented in Theorem 2.1 and pointsout the

behaviourof the bound (2.37) with respe t tothe Rekasius order

N ∈ N

∗

.

Example 2.1. Let us onsider the time-delaysystem (2.5), whose realisationisdened by







A

0 A

1 C

z0

C

z1

E

′

0 





=







−1.65

0.34

0.91 0.31 −2.21

0.63

0.18 _{0.51 −2.32}

−2.03

0.43

0.49 0.05 −1.42

0.89

0.50 _{0.81 −2.28}

0.71

0.62

0.34

0.94

0.12

0.73

0.39

0.93

0.92 





.

We analyse the behaviour of both the

H

∞

norm

kT (s, τ)k

∞

as well as its lower bounds

kH(s, λ)k

∞

given by the omparison system, yielded by the Rekasius substitution of order

N = 1

and

N = 3

. Theseresults are shown in Figure2.1; the solid line orresponds to the

real norm of the time-delay system, whilst the dashed ones represent the result obtained

fromthe omparisonsystem. Notethattheuseofahigherorder omparisonsystemprovides

a tighter bound.

This exampleindu esus to onje ture that, onsideringtwointegers

N

1 < N

2

,and letting

H

1 (s

1 , λ

1 )

and

H

2 (s

2 , λ

2 )

bethe omparisonsystems asso iated with

N

1

and

N

2

respe tively,

thentheysatisfy

kH

1 (s

1 , λ

1 )k

∞

≤ kH

2 (s

2 , λ

2 )k

∞

whenever

[0, τ

⋆

_{) ∋ τ = 2N}

1 /ω

1 arctan (ω

1 /λ

1 ) =

2N

2 /ω

2 arctan (ω

2 /λ

2 )

.

2.3 State-Feedba k Design

In this se tion, let us add some ontrol to the rearranged time-delay system (2.5), whi h

be omes

˙x(t) = A

0 x(t) +

N

P

k=1

A

k

x (t − τ

k

) + B

0 u(t) + E

0 w(t),

z(t) = C

z0

x(t) +

N

P

k=1

C

zk

x (t − τ

k

) + D

zu

u(t).

(2.41)

(30)

0

0.5

1

1.5

2

0

2

4

6

8

10 N=1

N=3

PSfrag repla ements

k

H

(s

,λ

)k

∞

,

k

T

(s

,τ

)k

∞

τ

[s]

Figure 2.1:

H

∞

norm and lowerbounds as fun tionsof

τ

.

PSfrag repla ements

x(t)

x

_{(t −}

τ

6 )

x

(t −

τ

3 )

x

(t −

τ

2 )

x

(t −

2τ

3 )

x

(t −

5τ

6 )

x

(t − τ)

Input Output Output

Figure 2.2: Buer ne essary to implement

u(t)

.

Our goal isto designa stabilising ontrolrule of the form

u(t) = K

0 x(t) +

N

X

k=1

K

k

x

t −

N − k + 1

_N

τ

,

(2.42)

in whi h the order

N

forthe feedba k law is hosen apriori and the orresponding gains

K

k

,

for

1 ≤ k ≤ N

, must be properly designed. The reasoning for this approa h is based on the

fa tthat,aslong asthe state

x(t − τ)

anbeheld, ifthe hoi eof asamplingperiodof

τ /N

is

feasible, itispossible tohandlethe states

x(t − τ/N), x(t − 2τ/N), · · · , x(t − τ)

inabuer to

beused toimplement(2.42). Figure2.2illustratesthis buer fora designer hoi eof

N = 6

.

The unknown gains

K

k

together with the s alars

α

k

(i)

, for

(k, i) ∈ {0, · · · , N}

2

(31)

multipliedas

K =







K

′

0 K

′

1

. . .

K

′

N







′







α

0 (0)

α

0 (1) · · · α

0 (N)

α

1 (0)

α

1 (1) · · · α

1 (N)

. . . . . . . . . . . .

α

N

(0) α

N

(1) · · · α

N

(N)







⊗ I,

(2.43)

to obtain again matrix

K

whi his exa tly the gain that appears when we lose the loopfor

the omparison system as an be seen onthe following realisation

H(s, λ) =

A

λ

+ BK

E

C

z

+ D

zu

K

0 ,

(2.44)

in whi h the indi ated matri esin the state-feedba k framework are dened as

A

λ

=





0 λI

N

P

k=0

α

k

(0)A

k

N

P

k=0

A

k

Γ

k

− λΓ

λ





,

B =

0 B

0 ,

C

z

=

_N

P

k=0

α

k

(0)C

zk

N

P

k=0

C

zk

Γ

k

,

E =

0 E

0 .

(2.45)

The previousrelationsallowustostatethe followinglemma,whi hprovidesanimportant

result that shall be exploited to yield design onditions for the state-feedba k ontrol law