robust observer and observer-based controller for time-delay singular systems

ROBUST OBSERVER AND OBSERVER-BASED CONTROLLERFOR TIME-DELAY SINGULAR SYSTEMS

L. Hassan, A. Zemouche, and M. Boutayeb

ABSTRACT

In this paper we address the problems of observer and observer-based controller design for a class of nonlinear time-delaysingular systems. The proposed methods use particular Lyapunov functions depending on the disturbances in order to avoid aspecific obstacle in the stability analysis. Consequently, two linear matrix inequality (LMI) conditions ensuring the H∞

convergence of the estimation error and the closed loop system were presented. These LMIs were obtained by manipulatingYoung’s inequality in order to linearize some bilinear terms.

Key Words: Time-delay systems, singular systems, observers, observer-based controller, LMI.

I. INTRODUCTION

Time-delay systems have attracted recurring interestsfrom the research community [19,22,33,20,21,9,27]. This dueto the fact that time-delay is encountered in various engineer-ing systems such as transportation, communication, mechani-cal, chemical processes, and networked control systems. Inthe case of linear time-delay systems, stability analysis aswell as observer synthesis have been widely studied [13,6,8].On the other hand, less attention has been paid toward non-linear systems with time-delay [30,31]. In general, severaldesign procedures have been proposed to design asymptoticstate observers for time-delay systems. Some of the reporteddesign methods involve the computation of the eigenvalues ofthe time-delay systems [2]. Other design methods assume theso-called matching-condition on the delayed state matrix[23]. In [11] the nonlinear system is assumed to verify theusual Lipschitz condition that permit us to transform thenonlinear system into a linear time-delay system with struc-tured uncertainties. While, as in [10], the proposed observerdesign is free from any preliminary analysis of the time-delaysystem such as estimating the Lipschitz constants of nonlin-earities, in [15] a finite-order memoryless state observer fortime-delay systems is proposed. Most of the effort has beenfocused on stability analysis and stabilization of time-delaysystems using the so-called Lyapunov–Krasovskii functionaltogether with a linear matrix inequality (LMI) approach,which provides an efficient numerical tool for handlingsystems with delays in state inputs.

Singular systems have received a lot of attentionalso because of their extensive applications. Many resultsconcerning singular systems were developed depending onthe theory of state-space. Much attention has been focused onthe problems of robust stability and robust stabilization ofsingular systems in general [26] and time delay singularsystems in particular [5,24]. Now, it is known that the robuststability problem for singular systems is much more compli-cated than that for state-space systems because it requiresconsideration of not only stability robustness, but alsoregularity and absence of impulses (for continuoussingular systems) or causality (for discrete singular systems)simultaneously.

Control and stabilization for time delay singularsystem have been treated also [28,18]. Many types of con-trollers are considered, observer-based controller [16], statefeedback controller [12,14], and dynamic output feedbackcontroller. Using the Lyapunov framework, the authors triedto overcome the conservatism stemming first from thechoice of Lyapunov–Krasovskii functional candidate whichis crucial for deriving stability criteria, and second, from thechoice of model transformation which produces cross termsin the derivative of the latter. To avoid the first reason,[17] presented a singular-type complete quadratic LKF com-bined with the discretization LKF method to obtain a newBRL. To avoid the second, [25] introduced a new finite suminequality, such that the bounding technique for cross termswas no longer needed. Other attempts have been maderegarding the H∞ control problem of a class of discrete-timeuncertain singular systems with interval time varying stateand input delays, but the resulting condition is not inthe form of an LMI. So, in order to solve the nonlinearproblem, a cone complementarity linearization algorithm isused [4].

This paper treats two problems. The first problem is anobserver design for singular systems with time-delay, as was

Manuscript received February 20, 2012; revised July 26, 2012; accepted October 3,2012.

The authors are with Centre de Recherche en Automatique de Nancy, CRANCNRS UMR 7039, Université de Lorraine, 54400 Cosnes et Romain, France.

Lama Hassan is the corresponding author (e-mail : [email protected]).The authors would like to thank ANR-EMERGENCE-PROTHERMOVERRE- for

the financial support of this work.

Asian Journal of Control, Vol. 16, No. 1, pp. 80–94, January 2014Published online 30 January 2013 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.669

© 2013 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

previously treated in [7], with known constant bounded delay,where a method was presented to design an unknown inputobserver and transform, as a result, the studied regular systeminto a singular one. The second is the design of an observer-based controller which, to the best knowledge of the authors,is rarely addressed in the literature. The design method pro-posed in this paper is capable of dealing with a class ofuncertain nonlinear time-delay systems. Like most of theproceeding work on this subject, the nonlinear functionshould be Lipschitz [18]. Contrary to the existing results, weused a Lyapunov function dependant on the disturbances,which ensures, when a certain LMI is satisfied, the H∞ con-vergence of the estimation error. In addition, the bilinearityproduced in the analysis is treated using Young’s inequalitywhich leads to less conservative results. Furthermore the con-struction of the LMI depends on a reformulation of the Lip-schitz condition [1]; this modified condition gives us morefreedom in the sense of choosing the nonlinear functions.That is, the obtained LMI allows us to choose functions withlarger Lipschitz constants. Indeed it is worth mentioning thatlarge Lipschitz constants represent the main limitation of theexisting results.

This paper is organized as follows. In Section II, ascenario that leads to the formulation of the problem isdescribed. In Section III the proposed observer synthesismethod for nonlinear time-delay singular systems is demon-strated. Finally an observer-based controller is presented inSection IV.

Notations. Throughout this paper, we will use the followingnotations:

• ||·|| is the usual Euclidean norm.• (*) is used for the blocks induced by symmetry.• AT represents the transposed matrix of A.• A� is the orthogonal matrix of A such that A�A = 0 with

A�A�T > 0• Ir represents the identity matrix of dimension r.• for a square matrix S, S > 0 (S < 0) means that this matrix

is positive definite (negative definite).

• e i ss

i

s

T s( ) ( , , , , , , ) ,= ∈ ≥0 0 1 0 0… …�

� ��

th

components

R 11 is a vector of the

canonical basis of Rs.

• The notation x x kr

k�2

2

0

1 2

= ⎛⎝⎜

⎞⎠⎟=

∞

∑ ( ) is the �2r norm of the

vector x ∈ Rr. The set �2r is defined by

� �22

r rx x r= ∈ < +∞{ : }R

• The set Co(x,y) is the convex hull of the set {x,y}, i.e..

Co x y x y( , ) { ( ) , [ , ]}.= + − ∈λ λ λ1 0 1

II. PROBLEM FORMULATION

In this paper, we treat the problems of observation andcontrol of nonlinear time delay singular system in the pres-ence of disturbances. The class of systems in which we areinterested is a discrete singular system with time-delaydescribed by:

Ex Ax A x B u Bf x x E

y Cx Dk k d k d u k k k d k

k k k

+ − −= + + + += +

⎧⎨⎩

1 ( , ) ω

ω

ωω (1)

where xk ∈ Rn is the state, uknu∈R is the input, yk ∈ Rp is the

output, and wk ∈ Rr is the disturbance vector. The matrices E,A, Ad ∈ Rm¥n; Bu

m nu∈ ×R , B ∈ Rm¥q, C ∈ Rp¥n; Ew ∈ Rm¥r,Dw ∈ Rp¥r are constant. d > 0 is a known delay. Consideran unforced linear discrete-time singular delay systemdescribed by

Ex Ax A xk k d k d+ −= +1 (2)

Definition 1 [29]. The discrete singular delay system is saidto be

• regular: if det(zd+1E - zdA - Ad) is not identically zero.• causal: if it is regular and deg(znddet(zE - A - z-dAd)) =

nd + rank(E).• stable: if it is regular and r(E,A,Ad) < 1 where

ρ λλ

( , , ) max{ | ( ) }

E A Adz det z E z A Add d

�∈ − − =+1 0

• admissible: if it is regular, causal and stable.

Definition 2. The nonlinear singular system (1) is said to beadmissible if the pair (E, A) is admissible.

Before proceeding we present the followingassumptions.

Assumption 3.

• The nonlinear function f is assumed to be Lipschitz, i.e.

f x x f x xx x

x xd d f

d d

, ,( ) − ( ) ≤ −−

⎛⎝⎜

⎞⎠⎟

ˆ ˆˆ

ˆγ (3)

Since f is differentiable with respect to its arguments, then wecan reformulate the condition (3) as follows :

af

w b wiji

kj k k ij k

sk

ri i≤ ∂∂

≤ ∀ ∈ ∀ ∈ζζ ζ( , ) , , R R (4)

af

v b vijd i

kj k k ij

dk

rk

si i≤ ∂∂

≤ ∀ ∈ ∀ ∈ζ

ζ ζ( , ) , , R R (5)

• rankE

Cn

⎡⎣⎢⎤⎦⎥

⎛⎝⎜

⎞⎠⎟= with rank(E) = m.

81L. Hassan et al.: Robust Observer and Observer-Based Controller for Time-Delay Singular Systems


Remark 4. Throughout the rest of the paper we will useinequalities (4)–(5) instead of (3), which lead to lessrestrictive synthesis conditions [32].

Remark 5. We assume, without loss of generality, that fsatisfies (4) and (5) with aij = 0 and alm

d = 0 for all i, l = 1, . . .,q, j = 1, . . . , s and m = 1, . . . , r, where s = max1�i�q(si)and r = max1�i�q(ri). Indeed, if there exist subsets S1,S qd

1 1⊂ { , , }… , S2 ⊂ {1, . . . , s} and S rd2 1⊂ { , , }… such

that aij � 0 for all (i, j) ∈ S1 ¥ S2 and almd ≠ 0 for all

( , )l m S Sd d∈ ×1 2 , we can consider the nonlinear function

f x x f x x a H x

a H

k k d k k d ij ij i

i j S S

k

lmd

l

( , ) ( , )( , )

− −∈ ×

= −⎛⎝⎜

⎞⎠⎟

−

∑ H1 2

mmd

ld

l m S S

k dd d

xH( , )∈ ×

−∑⎛

⎝⎜

⎞

⎠⎟

1 2

(6)

where

H e i e j H e l e mij q sT

lmd

q rT

i l= =( ) ( ) ( ) ( ).and

Therefore, f satisfies (4) and (5) with a aij ijd= =0 0, ,

b b aij ij ij= − and b b aijd

ijd

ijd= − , and then we rewrite (1) as

Ex Ax A x Bf x x Ek k d k d k k d k+ − −= + + +1 ( , ) ωω

with

A A B a Hij ij i

i j S S

= +∈ ×∑ H

( , ) 1 2

and

A A B a Hd d ijd

ijd

id

i j S Sd d

= +∈ ×∑ H

( , ) 1 2

2.1 Designing the proposed observer

In this section, we will present a state observer in orderto estimate robustly asymptotically the state xk in spite of thepresence of disturbances. Thus, we propose the followingstructure of the observer for uk = 0:

υ υ υk kd

k d kd

k d

q i i k id

k d

i

y y

Be i f x x

+ − −

−=

= + + +

+ ( ) ( )1 1 1 2 2Π Π Π Π

� H Hˆ , ˆ11

i q

k k kx y

=

∑= +

⎧

⎨⎪⎪

⎩⎪⎪ ˆ υ �

(7)

where

� �[ ] = ⎡⎣⎢⎤⎦⎥⎡⎣⎢⎤⎦⎥

⎛

⎝⎜⎞

⎠⎟⎡⎣⎢⎤⎦⎥

−E

C

E

C

E

C

T T1

.

Our objective is to determine the matrices P1, Π1d , P2 and Π2

d

so that the estimation error

e x xk k k= −ˆ

converges H∞ asymptotically towards zero, i.e.,

e n r� �2 2≤ λ ω

where l > 0 is the disturbance attenuation level to beminimized.

Now, by computing ek+1 we get:

e x x y xk k k k k k+ + + + + += − = + −1 1 1 1 1 1ˆ υ � (8)

Also, by construction of � and �, we have

� �E C In+ =

Hence, since yk+1 = Cxk+1 + Dwwk+1, we can write

e y E C x

Ex Dk k k k

k k k

+ + + +

+ + +

= + − += − +

1 1 1 1

1 1 1

υυ ωω

� � ��

( )(9)

By exploiting (1) and (7), we obtain

e e e Be i f

C A x

k kd

k d q i

i

i q

k

+ −=

=

= + +

+ + − −+

∑1 1 1

1

1 2 1

Π Π

Π Π ΠΠ

�

� �

( )

( ( ) )

(

δ

11 2 1

2 1

2 1

d d dd k d

k

d d

C A x

D E

D

+ − −+ − −+ −

−( ) )

(( ) )

(( ) )

Π ΠΠ ΠΠ Π

� ��

�ω ω

ω

ωωω

ωωk d

kD−

++ � 1

(10)

where

δ f f v k w k f v k w ki i i= ( ) ( )( ) − ( ) ( )( )ˆ , ˆ , . (11)

Using the differential mean value theorem (DMVT) as in [32]we deduce that there exist z Co v vi ∈ ( ), ˆ , z Co w wi

d ∈ ( ), ˆ sothat:

δ f h k e j e h k e j ei ij siT

i k

j

j si

ijd

riT

id

k d

j

j ri

= +=

=

−=

=

∑ ( ) ( ) ( ) ( )H H1 1

∑∑ (12)

where

h kf

vz k w kij

i

ji( ) ( ( ), ( ))= ∂

∂(13)

82 Asian Journal of Control, Vol. 16, No. 1, pp. 80–94, January 2014


h kf

wv k z kij

d i

jid( ) ( ( ), ( ))= ∂

∂(14)

According to (11), and using the following equations

G D n r= ×[ ]� ω 0 (15)

ωωω

= ⎡⎣⎢

⎤⎦⎥−

k

k d

(16)

and

� � � �ω ω ω ω= − − −[ ]( ) ( )Π Π Π Π2 1 2 1D E Dd d (17)

we can rewrite (10) as follows:

e e e

C A x

C A

k kd

k d

k

d d dd

+ −= ++ + − −+ + − −

1 1 1

1 2 1

1 2 1

Π ΠΠ Π ΠΠ Π Π

( ( ) )

( ( )

� �

� � ))

( )

( )

x

G

h k BH e

h k B

k d

k k

ij ij i k

j

j si

i

i q

ijd

−

+

=

=

=

=

+ +

+

+

∑∑

�

� H

�

ωω ω 1

11

HH eijd

id

k d

j

j ri

i

i q

H −=

=

=

=

∑∑11

(18)

Given the system (1) and the observer (7) then the H∞

filtering design is to determine the matrices P1, Π1d , P2 and

Π2d so that

lim ( ) , ( )k

e k for k→∞

= =0 0ω (19)

e k en r� �2 22

20 0 0≤ ∀ ≠ =λ ω ω( ) ; ( ) (20)

The problem of H∞ filtering design can be reduced to findinga Lyapunov function Vk such that

W V e k e k k kkT T= + − <Δ

2

20( ) ( ) ( ) ( )

λ ω ω (21)

where

ΔV V Vk k= −+1

III. SYNTHESIS METHOD:NEW LMI CONDITION

The content of this section consists of proposing a newobserver synthesis method for a class of nonlinear time-delaysystem.

Theorem 6. For a prescribed l > 0, the H∞ filtering designproblem corresponding to the system (1) and the observer (7)is solvable, with the H∞ performance level less than l, if theconditions below are fulfilled:

1. there exist matrices P P= >T 0, Q Q= >T 0, R andRd of adequate dimensions so that the LMI (22) holds,where

− + + −− −

− −

P Q P M � P R

� Q N � P R

� � P

I G A C

A C

G G I

nT T

dT T

d

T

0 0

0 0

2

2

( )

( ) ( )

( ) ( )λ

22 0 0

0

r

T T T

Td

T

d

D E

Dω ω

ω

R � P

R

� � � P

� � � �

−⎡⎣⎢

⎤⎦⎥

−−

( ) ( ) ( )

( ) ( ) ( ) ( )

ϒ Σϒ Σdd

T P

� � � � � P( ) ( ) ( ) ( ) ( ) −

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

< 0

(22)

M M M M H H= =[ ], [ ]1 …� �� q i i

TiT

s times

wherei

(23)

N N N N H H= =[ ], [( ) ( ) ]1 …� �� q i i

d Tid T

r times

wherei

(24)

Σ =�B H H H Hs qsq[ ]11 1 211 (25)

Σdd

rd

qrB H H H H q=� [ ]11 1 211 (26)

ϒ = diag( , , , , , )β β β β11 1 211 1 1 2I I I Is s s s qs sq q… … (27)

ϒd dr r

dr

dr qr

drI I I Iq q= diag( , , , , , )β β β β11 1 211 1 1 2… … (28)

β βijij

ijd

ijdb b

= =2 2, (29)

2. The matrices P1, Π1d , P2 and Π2

d are given by:

Π11= − −� P RA CT

Π11d

d dTA C= − −� P R

Π Π21

1= +−P R �T

Π Π21

1d

dT d= +−P R � (30)



Proof. First, notice that for any P and R , conditions (30)lead to reduce (18) as follows:

e A C e A C e

h k BH e

kT

k d dT

k d

ij ij i k

j

j s

+− −

−

=

=

= −( ) + −( )+

11 1

1

� P R � P R

� H( )ii

i

i

i q

ijd

ijd

id

k d

j

j r

i

i q

k k

h k BH e

G

∑∑

∑∑=

=

−=

=

=

=

+

+

+ +

1

11

1

( )� H

�ωω ω

(31)

where �ω becomes:

� P R � P Rω ω ω ω= −[ ]− −1 1TdTD E D . (32)

Now, it suffices to show that the matrices P , R and Rd

provided by the LMI (22) guarantee the robustness of theproposed observer. In order to do that, we use the followingLyapunov–Krasovskii function:

V e G e G e ek k kT

k k k iT

k i

i

i d

= − − + − −=

=

∑( ) ( ) .ω ωP Q

1

(33)

The term e Gk k− ω is introduced into the Lyapunov-Krasovskii-like function allows avoiding the presence of thequadratic term ω ωk

T TkG G+ +1 1P into DV. On the other hand, it

is worth mentioning, that the dual problem for continuoustime systems is to avoid deriving the disturbances, i.e. thepresence of �ω into �V .

Let us define

ζ ζij ij i k ijd

ijd

id

k dh k e h k e= = −( ) , ( ) .H H

Then from (4) and (5), we have

ζ ζijT

ij ijij

j

j s

i

i q

h b

i 1 10

11

−⎛⎝⎜

⎞⎠⎟

≥=

=

=

=

∑∑ (34)

( ) .ζ ζijd T

ijd

ijd ij

d

j

j r

i

i q

h b

i 1 10

11

−⎛⎝⎜

⎞⎠⎟

≥=

=

=

=

∑∑ (35)

The inequalities (34) and (35) become, respectively,

eb

kT

iT

ij

j

j s

i

i q

ijijT

ij

j

j s

i

i qi i

11 11

10H ζ ζ ζ

=

=

=

=

=

=

=

=

∑∑ ∑∑− ≥ (36)

eb

k dT

id T

ijd

j

j r

i

i q

ijd ij

d Tijd

j

j r

i

i i

−=

=

=

=

=

=

=∑∑ ∑−

11 1

1( ) ( )H ζ ζ ζ

11

0i q=

∑ ≥ . (37)

By calculating the difference DV = Vk+1 - Vk along the system(1) and using the aforementioned inequalities, we get

W

e

e

k

k

k d

k

k

kd

T

≤

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

−

ωζζ

Γ Γ Γ Γ ΓΓ Γ Γ

11 12 13 14 15

22 23( )� 224 25

33 34 35

44 45

55

ΓΓ Γ Γ

Γ ΓΓ

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

� �

� � �

� � � �

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

−

e

ek

k d

k

k

kd

ωζζ

(38)

where

Γ111 1= − − − + +− −( ) ( )� P R P � P R P QA C A C IT T T

n (39)

Γ121 1= − −− −( ) ( )� P R P � P RA C A CT T

d dT (40)

Γ131= − −( )� P R P�A CT T

w (41)

Γ Σ141= + − −M � P R P( ( ) )A CT T (42)

Γ Σ151= − −( ( ) )� P R PA CT T

d (43)

Γ221 1= − − −− −( ) ( )� P R P � P R QA C A Cd d

T Td d

T (44)

Γ231= − −( )� P R P�A Cd d

T Tω (45)

Γ Σ241= − −( )� P R PA Cd d

T T (46)

Γ Σ251= + − −N � P R P( )A Cd d

T Td (47)

Γ33

2

22

= − −� P� Pω ωλT T

rG G I (48)

Γ Σ34 = � PωT (49)

Γ Σ35 = � PωT

d(50)

Γ Σ Σ ϒ44 = −T P (51)

Γ Σ Σ45 = TdP (52)

Γ Σ Σ ϒ55 = −dT

ddP (53)

ζ ζ ζ ζ ζkT

sT T

qsT T

q= [ , , , , , ]11 1 211… … (54)

ζ ζ ζ ζ ζkd d T

rd T d T

qrd T T

q= [( ) , , ( ) , ( ) , , ( ) ]11 1 211… … (55)



We notice that the matrix aforementioned satisfies thefollowing inequality:

Γ Γ Γ Γ ΓΓ Γ Γ Γ

Γ Γ ΓΓ Γ

11 12 13 14 15

22 23 24 25

33 34 35

44

( )

( ) ( )

( ) ( ) ( )

�

� �

� � � 445

55

1 2 31

2 0

( ) ( ) ( ) ( )� � � �

Q QQ Q

Γ

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

= − <− T(56)

where Q1, Q2, and Q3 are matrices of appropriate dimensionssuch that

− + +−

− −

−

P Q P M

� Q N

� � P

� � �

� �

I G

G G I

n

Tr

0 0

0 0

20 0

0

2

2

( )

( ) ( )

( ) ( ) ( )

( ) ( )

λ

ϒ(( ) ( )� �

Q

−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥ϒd

1

� ��

(57)

Q

� P R

� P R

� P

P

P

2 =

−−

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

( )

( )

A C

A C

T T

dT T

d

T

T

dT

ω

ΣΣ

(58)

Q P3 0= − < (59)

Using the Schur Lemma then we deduce that equations(56)–(59) are equivalent to:

Q Q

Q Q1 2

2 3

0T

⎡⎣⎢

⎤⎦⎥< (60)

which is identical to (22). Consequently, we deduce thatunder the condition (22), the estimation error convergesrobustly asymptotically towards zero. This ends the proof ofTheorem 6.

IV. OBSERVER-BASED CONTROLLER

In this section, depending on the result of the previoussections, we propose an observer-based controller for *a thesame class of systems (1). The controller is chosen to beu Kx K xk k d k d= + −ˆ ˆ , where x̂k is the state estimate defined in(7).

Without loss of generality, we assume that f(0,0) = 0.Thus, the original system can be rewritten under the form:

E

I

x

u

e

A B

K I K

A P

k

k

k

u

nu

0 0

0 0 0

0 0

0

0 0

1

1

1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥= −

−

+

+

+ � 221

21

0 0

0

0 0

−

−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+−

⎡R

� R

T

k

k

k

d

d d

d dT

C

x

u

e

A

K K

A P C⎣⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+−

−

−

−

− −

x

u

e

E

P D E P D

k d

k d

k d

TdT

ω

ω ω

0

0 0

21

21R � R ωω

ωω

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡⎣⎢

⎤⎦⎥

+

−

=

=

=

=

∑∑

k

k d

ij ij i k

j

j s

i

i q

ij

g t BH x

h

i

( ) H11

0

(( )

( )

t BH e

g t BH

ij i k

j

j s

i

i q

ijd

ijd

i

� H

H

=

=

=

=

∑∑

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+

11

iid

k d

j

j r

i

i q

ijd

ijd

id

k d

j

j r

i

i q

x

h t BH e

i

i

−=

=

=

=

−=

=

=

=

∑∑

∑∑

11

11

0

( )� H

⎡⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

+⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡⎣⎢

⎤⎦⎥

+

+ −

0 0

0 0

0

1

1�D

k

k dω

ωω

(61)

The function f e i f v k w kq i

i

i q

==

=

∑ ( ) ( ( ), ( ))1

. Thus, by using the

DMVT [32], we deduce that there exist z Co vi ∈ ( , )0 ,z Co wi

d ∈ ( , )0 so that :

f g k e j x g k e j xi ij siT

i k

j

j s

ijd

riT

id

k d

j

j ri i

= +=

=

−=

=

∑ ∑( ) ( ) ( ) ( )H H1 1

(62)

where

g kf

vz k w kij

i

ji( ) ( ( ), ( ))= ∂

∂(63)

g kf

wv k z kij

d i

jid( ) ( ( ), ( ))= ∂

∂(64)

or more easily

E

B

B

H

H

k k d k d k k

ij

ij

ξ ξ ξ ω ωω+ − += + + +

+⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡⎣⎢

⎤⎦

1 1

0

0 0

0

0

0

A A

�

Ξ Θ

⎥⎥

+⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣⎢

⎤

⎦⎥

=

=

=

=

∑∑ ζ

ζ

ij

j

j s

i

i q

ijd

ijd ij

d

i

B

B

H

H

11

0

0 0

0

0

0�jj

j r

i

i q i

=

=

=

=

∑∑11

(65)



where

ξk

k

k

k

x

u

e

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(66)

ωωωk

k

k d

= ⎡⎣⎢

⎤⎦⎥−

(67)

A

� R

= −−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−

A B

K I K

A P C

u

nuT

0

0 0 21

(68)

A

� Rd

d

d d

d dT

A

K K

A P C

=−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−

0 0

0

0 0 21

(69)

Ξωω

ω ω ω

=−

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥− −

E

P D E P DTdT

0

0 0

21

21R � R

(70)

Θ =⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0 0

0 0

0�Dω

(71)

ζ ijij si

ij si

i

i

k

k

g I

h I

x

e= ⎡⎣⎢

⎤⎦⎥⎡⎣⎢

⎤⎦⎥⎡⎣⎢⎤⎦⎥

0

0

0

0

H

H (72)

ζ ijd ij

dri

ijd

ri

id

id

k d

k d

g I

h I

x

e=⎡

⎣⎢

⎤

⎦⎥⎡⎣⎢

⎤⎦⎥⎡⎣⎢

⎤⎦⎥

−

−

0

0

0

0

H

H(73)

So the problem to be addressed in this section is to obtainfeedback gains K,Kd for designing a robust controlleru Kx K xk k d k d= + −ˆ ˆ . But first, let us start by presenting themain theorem.

Theorem 7. System (65) in closed-loop, subject tou Kx K xk k d k d= + −ˆ ˆ is admissible with H∞ performance l, ifthere exist matrices S > 0, Q > 0, M > 0, Md > 0, L = LT, Rand Rd of adequate dimensions so that the conditions (74)and (75) are feasible.

Π1 21= − −� RA S CT

Π1 21d

d dTA S C= − −� R

Π Π2 21

1= +−S TR �

Π Π2 21

1d

dT dS= +− R � (74)

Φ

Φ Φ Φ Φ Φ

Φ Φ Φ Φ

=

−( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

11 12 13 14 15

2

22 23 24 25

2

0

0

0

0

S A C

S

T T� R

�

*

AA C

D S E

D

d dT T

T T

dT T

−( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−( )( )

⎡

⎣⎢⎢

⎤

R

R �

R* * Φ Φ Φ33 34 35

2ω ω

ω ⎦⎦⎥⎥

−( ) − −( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⊥ ⊥ ⊥ ⊥* * *

* * * *

Σ Σ ϒ Σ ΣΣ

Σ

T T T Td

d

S E LE S E LE

S

0

0

2 2

TT Td d

d

S E LE

S

S

−( ) −⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

⎡

⎣

⎢⎢

⊥ ⊥ Σ ϒΣ

0

0

2 2

2* * * * *

* * * * * *

* * * * * *

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

(75)



−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥F AT T

T

T

G

F K

F

K

0 0

0 0 0

0 0 0

12

14 ⎥⎥⎥

−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

F A

F

d dT T

d dT

d

dT

T T

G

F K

F

K

GF F

12

14

12 1

0 0

0 0 0

0 0 0

ω ωω ωΞ 22

14 14

1 12 12 1 12 12

0 0

0 0

0

0 0

0 0 0 0

0 0 0 0

F F

S L S LT T

ω ω

⎡⎣⎢

⎤⎦⎥

− −⎡

⎣

⎢Σ Σ( ) ( )

⎢⎢⎢

⎤

⎦

⎥⎥⎥

− −⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0

0 0

0 0 0 0

0 0 0 0

0 0

1 12 12 1 12 12Σ ΣdT

dTS L S L

S

( ) ( )

−− − −⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−−

⊥ ⊥E LE G G

G G

G G

M

M

T T

d

12 12

14 14

1

2

0 0

0 0

0 0 0 0

0 0 0

0 0

*

εε 00

0 01

0

0 0 01

1

2

−

−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

ε

ε

M

Md

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

< 0

where the variables are defined as follows:

Φ11 2

11 11 12

12 12 22

= − + +

++ −+ −× ×

E SE I

F A I K F B F I K

F A K F B F K

Tn

n n u n n

u

u u

Q

00 0 01

1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥+

Z

TZ

� ��

(76)

Φ12

11

12

11 11 12

12 12

0 0

0 0

0 0 0

0

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+−

F A

F A

F A F B F

F A F

d

d

d d u d

d d BB Fu d

T

−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

22 0

0 0 0

(77)

Φ Θ13

11

13

11 11 12

13

0

0

0 0

0

= +⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+−

E S

F E

F E

F A F Bu F

F A F

T

ω

ω

ω ω ω

ω ω113 14 0Bu F

T

−⎡⎣⎢

⎤⎦⎥ω

(78)

ΦΣΣ14

11 11 1

11 21 1

0

= +−−( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

M

A S L

B S S

T

uT

( )

(79)

ΦΣΣ15

11 11 1

11 21 1

0

=−−( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

A S L

B S S

Td

uT

d

( )

(80)



Φ22

11

12

0

0

0 0 02

= −

+++

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

× ×

Q

F A I K I K

F A K Kd d n n d n n d

d d d d

Z

u u

� ��

+ ZT2 (81)

Φ23

11

13

11 11 12

13 1

0

0

0 0

0

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

+−

F E

F E

F A F Bu F

F A F

d

d

d

d

ω

ω

ω ω ω

ω ω 33 14 0Bu F

T

−⎡⎣⎢

⎤⎦⎥ω

(82)

ΦΣ

24

11 11 1

0

0

=−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

A S LdT ( )

(83)

ΦΣ

25

11 11 1

0

0

= +−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

N

A S LdT

d( )

(84)

Φ Θ Θ3312

142

0

0= ⎡⎣⎢

⎤⎦⎥− −( ) −⊥ ⊥F E

F ES E LE IT T

rω ω

ω ωμ (85)

Φ Ξ Σ34 = −( )⊥ ⊥ωT TS E LE (86)

Φ Ξ Σ35 = −( )⊥ ⊥ωT T

dS E LE (87)

M M M= [ ]1 q , where

M

H

H

H

Hi

iT

iT

iT

iT

si

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0

0 0

0

0

0 0

0

� ��times

��

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

(88)

N N N= ⎡⎣ ⎤⎦1 q , where

N

H

H

H

Hi

id T

id T

id T

id T

ri

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

( )

( )

( )

( )

0

0 0

0

0

0 0

0

ttimes� ��

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

(89)

�Σ =

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

B

B

H

H

H

H

H

s

s

0

0 0

0

0

0

0

011

11

1

1

21

1

1�

,

00

0

0

0

0

21

1

2

H

H

Hqs

qs

q

q

⎡⎣⎢

⎤⎦⎥

⎡

⎣⎢

⎤

⎦⎥⎤

⎦⎥

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Σ

Σ

(90)

�Σd

d

d

sd

sd

B

B

H

H

H

H=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡0

0 0

0

0

0

0

011

11

1

1

1

1�

,⎣⎣⎢

⎡⎣⎢

⎤⎦⎥

⎡

⎣⎢

⎤

⎦⎥⎤

⎦⎥

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥

H

H

H

H

d

d

qsd

qsd

d

d

q

q

21

21

1

2

0

0

0

0

0

Σ

Σ ⎥⎥

(91)

ϒ = ⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

diagβ

ββ

ββ

11

11

1

1

21

1

1

1 1

1 1

2

0

0

0

0

I

I

I

I

I

s

s

s s

s s

s

00

0

0

021 2ββ

βI

I

Is

qs s

qs s

q q

q q

⎡⎣⎢

⎤⎦⎥

⎡

⎣⎢

⎤

⎦⎥⎞⎠⎟

(92)

ϒdd

r

dr

rd

r

rd

r

I

I

I

I=

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎛⎝⎜

diagβ

ββ

β11

11

1

1

1

1

1 1

1 1

0

0

0

0 ,

βββ

ββ

21

21

2

2

0

0

0

0

dr

dr

qrd

r

qrd

r

I

I

I

Iq q

q q

⎡⎣⎢

⎤⎦⎥

⎡

⎣⎢

⎤

⎦⎥⎞⎠⎟

(93)

β βijij

ijd

ijdb b

= =2 2, (94)

Proof. Following the method in Section 2.1, theLyapunov–Krasovskii function becomes:

V E Ek k kT

k k k iT

k i

i

i d

= − − + − −=

=

∑( ) ( )ξ ω ξ ω ξ ξΘ ΘP Q

1

(95)

with

P =

⎡⎣⎢

⎤⎦⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

P P

P

P

P

11 12

22

2

1

0

0

*� ��



By calculating DV = Vk+1 - Vk we get

Δ

Ξ Ξ

V E E IkT T T

n k

k dT

dT

d k d

kT T

= − + −+ −+

− −

ξ ξξ ξω ω ω

2

( )

( )

(

A PA P Q

A PA Q

P −−+ +++

−

Θ ΘΣ Σ Σ Σ

Tk

kT T

k kd T d T d

kd

kT T

d k d

P

P P

A PA

)

( ) ( )

ωζ ζ ζ ζξ ξ

� � � �

2

2

ξξ ωξ ζ ξ ζξ

ωkT T T

k

kT T

k kT T d

kd

k dT

dT

E

2 2

2

( )A P P

A P A P

A

Ξ ΘΣ Σ

++ ++ −

� �

PP

A P A P

P

ΞΣ Σ

Ξ Σ

ω

ω

ωξ ζ ξ ζω ζ ω

k

k dT

dT

k k dT

dT d

kd

kT T

k kT

+ ++ +

− −2 2

2 2

� �

�ΞΞ Σ

Σ Σω ζ

ζ ζ

T dkd

kT T d

kd

P

P

�

� �+

(96)

ζ ζijT ij ij

si

ij ijsi

i

g bI

h bI

1 10

01 1

−⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

jj

j

j s

i

i q i

=

=

=

=

∑∑ ≥11

0 (97)

( )ζ ijd T ij

dijd si

ijd

ijd si

g bI

h bI

1 10

01 1

−⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

≥=

=

=

=

∑∑ ζ ijd

j

j r

i

i q i

11

0 (98)

The inequalities (97) and (98) become, respectively:

ξ ζ

ζ

kT

iT

iT

ij

j

j s

i

i q

ijT ij

si

i

i

bI

b

H

H

0

0 0

0

10

01

11

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

=

=

=

=

∑∑

jjsi

ij

j

j s

i

i q

I

i

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

≥=

=

=

=

∑∑ ζ11

0

(99)

ξ ζ

ζ

k dT

id T

id T

ijd

j

j r

i

i q

ijd T

i

−=

=

=

= ⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

∑∑( )

( )

( )

H

H

0

0 0

011

110

01

011

bI

bI

ijd si

ijd si

ijd

j

j r

i

i q i

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

≥=

=

=

=

∑∑ ζ

(100)

To ensure a robust behavior of both the observer and thecontroller, we calculate

W V k k k kkT T= + − <Δ ξ ξ λ ω ω( ) ( ) ( ) ( ) .

2

20

(101)

By adding inequalities (99) and (100) to DV, we get

W

x

u

e

x

u

ek

k

k

k

k

k

kd

Tk

k

k

k

k

kd

=

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢ω

ζ

ζ

ω

ζ

ζ

Ω

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

(102)

where W is of the form

Ω

Ξ Θ

=

− + +−

⎡

⎣

⎢⎢⎢⎢⎢⎢

+

A PA P Q A PA

A PA Q

A P P A P

T Tn

Td

dT

d

T T T

E E I

E

2

*

* *

* *

* *

ω��

� ��

Σ Σ

Ξ Σ ΣΞ Ξ Θ Θ Ξ Σ Ξ

+

+− −

M A P

A P A P A P N

P P P P

Td

dT

dT

dT

d

T Tr

T TIω

ω ω ω ωμ 2

��

� �

ΣΣ Σ ϒ Σ Σ

Σ Σ ϒ

d

Td

dT

dd

*

* *

P P

P

−−

⎤

⎦

⎥⎥⎥⎥⎥⎥

(103)

Applying the Schur complement on the resulting matrix(103), we can handle the observer part separately and weobtain an equivalent matrix (104).

Ω

Ξ ΘΞ

Ξ Ξ Θ1

2

=

− + + +−

−

A PA P Q A PA A P P

A PA Q A P

P

T Tn

Td

T T

dT

d dT

T

E E I Eω

ω

ω ω

*

* * TTrIPΘ−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

μ 2

* * *

* * *

* * * (104)



T Td

T TP A C

A P M A P

� R

Σ Σ+

−( )

⎡

⎣

⎢⎢⎢⎢ 2

0

0

⎤⎤

⎦

⎥⎥⎥⎥

+

−( )

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

A P A P N

� R

P P

dT

dT

d

d dT T

T Td

P A C

Σ Σ

Ξ Σ Ξ Σ

0

0

2

ω ωRR �

R

P P

T T

dT T

Td

D P E

D

P

ω ω

ω

−( )( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

2

2 2

0

0Σ Σ ϒ Σ ΣΣ

Σ* ddT

dd

d P

P

PΣ ϒΣ

−⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

0

0

2 2

2* *

with

A = −⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

A B

K I Ku

nu

0

0 0 0(105)

Ad

d

d d

A

K K=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

0 0

0

0 0 0(106)

Ξωω

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

E 0

0 0

0 0(107)

ΣΣ

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1

0

0(108)

ΣΣ

d

d

=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1

0

0(109)

Following [34], there exist matrices S > 0 and L = LT suchthat:

P = − ⊥ ⊥S E LET

with S on the form

S S

S

S

S

11 12

22

2

1

0

0

⎡⎣⎢

⎤⎦⎥

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

� ��

In addition, according to some authors [18,3], it is possibleto treat a singular system by adding some free matrices. Inthis section, we will show how to modify this method so itcan be applied to singular time delay systems with nonlin-earities. In order to do so, notice that if W1 < 0 thus V < 0,we can always find a matrix G such that P − − <G GT 0.Consequently,

ΩΩ

=− −

⎡⎣⎢

⎤⎦⎥<1 0

00

P G GT (110)

pre-multiply by

I

I

I

I

I

I

I

T

dT

T

0 0 0 0 0

0 0 0 0

0 0 0

0 0 0

0 0

0 0

0 0 0 0 0 0

−−−

⎡ A

A*

* *

* * *

* * * *

* * * *

Ξω

⎣⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

and

post-multiply by

I

I

I

I

I

I

Id

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

0 0 0

*

* *

* * *

* * * *

* * * * *

− − −

⎡

⎣

⎢⎢

A A Ξω

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

and then

by making these change of variables:

F A A= −T TP G (111)

F A Ad dT

dTP G= − (112)

Fω ω ω= −Ξ ΞT TP G (113)



we can get the inequality (114).

− + + + + + + +− + +

E SE I E ST T Tn d

TdT T T T

d d dT

dT

FA A F Q FA A F F A F

Q F A A F F2 Ξ Θω ω

* dd dT T

T T T TrS E LE I

ΞΞ Ξ Θ Θ

ω ω

ω ω ω ω μ+

+ − −( ) −

⎡

⎣

⎢⎢⎢⎢⎢⎢

⊥ ⊥

A F

F F* *

* * *

* * *

2

AA M A

� R

F AT T T Td

T T

S E LE S E LE

S A C

−( ) + −( )−( )

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−⊥ ⊥ ⊥ ⊥

2

0

0Σ Σ TT T

dT T

dT T

d

d dT T

G

S E LE S E LE

S A C

A A N

� R

−( ) −( ) +

−( )

⎡

⎣

⎢⎢⎢

⎤⊥ ⊥ ⊥ ⊥

2

0

0Σ Σ

⎦⎦

⎥⎥⎥

−

−( ) −( ) −( )⊥ ⊥ ⊥ ⊥

F A

R �

R

d dT T

T T T Td

T T

G

S E LE S E LED S E

Ξ Σ Ξ Σω ωω ω 2

ddT T

T T

T T Td

DG

S E LE S E LE

ω

ω ω( )

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−

−( ) − −( )⊥ ⊥ ⊥ ⊥

F Ξ

Σ Σ ϒ Σ ΣΣ

2

0

0

SS

S E LE

S

S

dT T

dd

d

2

2 2

2

0

0

0 0

0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−( ) −⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

−

⊥ ⊥*

* *

Σ Σ ϒΣ

** * * S E LE G GT T−( ) − −

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

<

⊥ ⊥

0

(114)

In order to get rid of the nonlinearity, we choose F and Fd

as follows:

G

G G

G G

G

F F F

F F F=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=11 12

13 14

4

11 12 21

13 14 22

0

0

0 0

, Fωω ω ω

ω ω ω

⎡⎡⎣⎢

⎤⎦⎥

(115)

F F=⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

=× ×F I M

F M

F

F I M

F M

F

n n

d

d n n d

d d

d

u u11

13

4

11

13

0

0

0 0

0

0

0 0

,

44

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(116)

where M, Mdnu∈R are free matrices.

Ω Ψ= + +X Y Y XT T (117)

In this case, by defining K M K= −1 and K M Kd d d= −1 , thecomponents of Y become

Ψ Φij ij i j= ∈, [ , ]1 5 (118)

X

F

F

F

F

F F

F F

S L S

d

d

T T

=−

0

0

0 0

0

0

0 0

12

14

12

14

12 12

14 14

1 12 12 1 1

ω ω

ω ω

Σ Σ( ) ( 22 12

1 12 12 1 12 12

12 12

14 14

0 0

0 0

0 0

0 0

0 0

0 0

−

− −

⎡

L

S L S L

G G

G G

dT

dT

)

( ) ( )Σ Σ

⎣⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

(119)



YK K

K Kd d

T

= ⎡⎣⎢

⎤⎦⎥

0 0 0 0 0 0

0 0 0 0 0 0(120)

Applying Young’s inequality leads to

Ω Ψ≤ + +−1 1

εεX S X Y SYT T

which can be written under the form

Ω Ψ≤ + ⎡⎣⎢

⎤⎦⎥⎡

⎣

⎢⎢

⎤

⎦

⎥⎥⎡⎣⎢

⎤⎦⎥

−

X

Y S

S

S

X

SY

T

T

ε

ε

0

01

1

.

Thus, having Ω < 0 is equivalent to the following

ΨX

Y S

S

S

T

T

⎡⎣⎢

⎤⎦⎥

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

<*

ε

ε

0

01

0 (121)

By choosing SM

Md

= ⎡⎣⎢

⎤⎦⎥

0

0, we can get (75).

V. NUMERICAL EXAMPLE

In this section, we present a numerical example to showthe performances of the proposed controller. We will considera simple example of a discrete system under the form (1),where

E A Ad= ⎡⎣⎢

⎤⎦⎥

= ⎡⎣⎢

⎤⎦⎥

=−

⎡⎣⎢

⎤⎦⎥

1 0

0 0

1 6 2

1 2

0 345 0 46

0 23 0 345,

.,

. .

. .

B C Bu = −⎡⎣⎢⎤⎦⎥

= [ ] = ⎡⎣⎢

⎤⎦⎥

0

11 1

0 0

0 1, ,

E D dω ω= [ ] = =0 1 0 1 0 1 4. . , . , ,

f x xx k d

k k d( , )sin( . ( ))

− =−

⎡⎣⎢

⎤⎦⎥

0

0 5 1

. f can be rewritten under the

form f Be i f x xq i i k id

k d

i

i

= −=

=

∑ ( ) ( , )H H1

2

with H2 1 0d = [ ]. The

disturbance vector w is a Gaussian distributed random signalwith mean zero and standard deviation s = 0.1. which we willadd on the interval t = [1,2] of time, in order to showsimultaneously the robustness and the asymptoticconvergence to zero of the proposed observer, respectivelywith and without disturbances.

The bounds of the partial derivatives of f are:ad

21 0 5= − . , bd21 0 5= . . According to the remark (5) we need to

solve the LMI (75) with b b ad d d21 21 21 1= − = . Hence, we obtain

the following solutions for m = 0.7:

L =−

−⎡⎣⎢

⎤⎦⎥

101 45 0 39

0 39 1 267

. .

. .

Observer gains:

Π Π1 1

2 41 2 01

2 41 2 01

0 21 0 09

0 21 0 09=− −⎡⎣⎢

⎤⎦⎥

=− −⎡⎣⎢

⎤⎦⎥

. .

. .,

. .

. .d

Π Π2 2

2

2

0 46

0 46=−⎡⎣⎢⎤⎦⎥

=−⎡⎣⎢

⎤⎦⎥

,.

.d

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

−5

0

5

10

15

20

Rea

l sta

te V

S E

stim

ated

sta

te

Time (k)

Controlled state x1

Unforced state x1

Estimated unforced state x1^

Fig. 1. x1 and its estimate.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−15

−10

−5

0

5

10

Rea

l sta

te V

S E

stim

ated

sta

te

Time (k)

Controlled state x2

Unforced state x2

Estimated unforced state x2^

Fig. 2. x2 and its estimate.



Controller gains:

K Kd= − −[ ] = − −[ ]0 49 0 40 0 10 0 08. . , . .

The simulation results represent the behaviour of thesystem subject to the proposed control law (Figs 1–2),u Kx k K xk d d= +ˆ( ) ˆ (Figs 1 and 2) by comparison to theunforced system uk = 0. We notice a good estimation of thestate, and the robustness of the system to the presence ofdisturbances.

VI. CONCLUSION

In this paper we presented a new observer designmethod for a class of nonlinear time-delay singular systemswith disturbances. The nonlinearity of the considered systemis assumed to be Lipschitz with respect to its arguments. Anew sufficient LMI condition was proposed to ensure H∞

robustness of the proposed observer in spite of the presenceof disturbances. In addition, an observer-based controller wasproposed and the method was modified to accept the resultingtime-delay singular system. Another main contribution of thispaper lies in using a particular Lyapunov-Krasovskii func-tional disturbance-dependent and using Young’s inequality toobtain LMI stability conditions.

REFERENCES

1. Arcak, M. and P. Kokotovic, “Obserbverbased control ofsystems with slope-restricted nonlinearities,” IEEETrans. Autom. Control, Vol. 46, No. 7, pp. 1146–1150(2001).

2. Bhat, K. P. M. and H. N. Koivo, “An observer theory fortime-delay systems,” IEEE Trans. Autom. Control, Vol.21, No. 2, pp. 266–269 (1976).

3. Chadli, M. and M. Darouach, “Novel bounded reallemma for discrete-time descriptor systems: Applicationto H∞ control design,” Automatica., Vol. 48, No. 2,pp. 449–453 (2012).

4. Cui, S., H. Zhu, and X. Zhang, “ H∞ control for discrete-time uncertain singular systems with interval time-varying state and input delays,” In Proc. 30th ChineseControl Conference, Yantai, China, pp. 2374–2379,(2011).

5. Fridman, E. and U. Shaked, “An improved stabilizationmethod for linear time-delay systems,” IEEE Trans.Autom. Control, Vol. 47, No. 11, pp. 1931–1937(2002).

6. Han, Q. L. and K. Gu, “Stability of linear systems withtime-varying delay: A generalized discretized lyapunovfunctional approach,” Asian J. Control, Vol. 3, No. 3,pp. 170–180 (2001).

7. Hassan, L., A. Zemouche, and M. Boutayeb, “ H∞ designfor a class of nonlinear time-delay systems in descriptorform,” Int. J. Control, Vol. 84, No. 10, pp. 1653–1663(2011).

8. He, Y., Q. Wang, C. Lin, and M. Wu, “Delay-rangedependent stability for systems with time-varyingdelay,” Automatica, Vol. 43, pp. 371–376 (2007).

9. He, Y., M. Wu, J. H. She, and G. P. Liu, “Delaydependentrobust stability criteria for uncertain neutral systems withmixed delays,” Syst. Control Lett., Vol. 51, pp. 57–65(2004).

10. Ibrir, S., “Adaptive observers for time-delay nonlinearsystems in triangular form,” Automatica, Vol. 45, No. 10(2009).

11. Ibrir, S., W. F. Xie, and S. Chun-Yi, “Observer design fordiscrete-time systems subject to timedelay nonlineari-ties,” Int. J. Syst. Sci., Vol. 37, No. 9, pp. 629–641 (2006).

12. Ji, X. and J. Gao, “Delay-dependent robust H∞ control foruncertain discrete singular linear timedelay systems,” InProc. 30th Chinese Control Conference, Yantai, China(2011).

13. Kharitonov, V. L., “Robust stability analysis of timedelay systems: a survey,” Annu. Rev. Control, Vol. 23, pp.185–196 (1999).

14. Kim, J. H., J. H. Lee, and H. B. Park, “Robust H∞ controlof singular systems with time delays and uncertaintiesvia LMI approach,” In Proc. Amer. Control Conference,Vol. 1, Anchorage, AK, USA, pp. 620–621, (2002).

15. Leyva-Ramos, J. and A. E. Pearson, “An asymptoticmodal observer for linear autonomous time lag system,”IEEE Trans. Autom. Control, Vol. 40, No. 7, pp. 1291–1294 (1995).

16. Li, X. and R. A. DeCarlo, “Robust controller andobserver design for time-delay systems,” In Proc. Amer.Control Conference, Vol. 1, Anchorage, AK, USA, pp.2210–2215, (2002).

17. Liu, L. L., J. G. Peng, and B. W. Wu, “ H∞ control ofsingular time-delay systems via discretized Lyapunovfunctional,” J. Frankl. Inst.-Eng. Appl. Math., Vol. 348,pp. 749–762 (2011).

18. Lu, G. and D. W. C. Ho, “Generalized quadratic stabili-zation for discrete-time singular systems with time-delayand nonlinear perturbation,” Asian J. Control, Vol. 7,No. 3, pp. 211–222 (2005).

19. Raff, T. and F. Allgöwer, “An EKF-based observer fornonlinear time-delay systems,” In Proc. 2006 Amer.Control Conference, Minneapolis, MN, USA (2006).

20. Richard, J. P., “Time-delay systems: an overview of somerecent advances and open problems,” Automatica,Vol. 39, pp. 1667–1694 (2003).

21. Sename, O., “Is a mixed design of observer-controllersfor time-delay systems interesting?” Asian J. Control,Vol. 9, No. 2, pp. 180–189 (2007).



22. Trinh, H., M. Aldeen, and S. Nahavandi, “An observerdesign procedure for a class of nonlinear time-delaysystems,” Comput. Electr. Eng., Vol. 30, pp. 61–71(2004).

23. Trombè, A., “A simple observer-based control law fortime lag systems,” Int. J. Syst. Sci., Vol. 23, No. 9 (1992).

24. Wang, C., “New delay-dependent stability criteria fordescriptor systemswith interval time delay,” Asian J.Control, Vol. 14, No. 1, pp. 197–206 (2012).

25. Wang, H. J., X. D. Zhao, A. K. Xue, and R. Q. Lu,“Delay-dependent robust control for uncertain discretesingular systems with time-varying delay,” J. ZhejiangUniversity Sci. A, Vol. 9, No. 8, pp. 1034–1042 (2008).

26. Wu, H. and K. Mizukami, “Stability and robust stabili-zation of nonlinear descriptor systems with uncertain-ties,” In Proc. 3rd IEEE Conference on Dicision &Control, Lake Buena Vita, FL, pp. 2772–2777, (1994).

27. Wu, M., Y. He, and J.H. She. Stability Analysis andRobust Control of Time-Delay Systems. Springer, Berlin,Heidelberg, Germany (2010).

28. Xu, S. and J. Lam, “Robust stability and stabilization ofdiscrete singular systems: An equivalent characteriza-tion,” IEEE Trans. Autom. Control, Vol. 45, No. 4, pp.568–574 (2004).

29. Xu, S. and J. Lam, Robust Control and Filtering of Sin-gular Systems. Springer-Verlag, Berlin Heidelberg, 2006.

30. Xu, S., J. Lu, S. Zhouc, and C. Yang, “Design of observ-ers for a class of discrete-time uncertain nonlinearsystems with time delay,” J. Franklin Inst.-Eng. Appl.Math., Vol. 341, No. 4, pp. 295–308 (2004).

31. Yang, R. and Y. Wang, “Stability analysis and H∞ controldesign for a class of nonlinear time-delay systems,”Asian J. Control, Vol. 14, No. 1, pp. 153–162 (2012).

32. Zemouche, A. and M. Boutayeb, “A unified H∞ adaptiveobserver synthesis method for a class of systems withboth lipschitz and monotone nonlinearities,” Syst.Control Lett., Vol. 58, No. 4, pp. 282–288 (2009).

33. Zemouche, A. and M. Boutayeb, “Observer synthesismethod for lipschitz nonlinear discrete-time systemswith time-delay: An LMI approach,” Appl. Math.Comput., Vol. 218, No. 2, pp. 419–429 (2011).

34. Zheng, G., Y. Xia, and P. Shi, “New bounded real lemmafor discrete-time singular systems,” Automatica, Vol. 44,pp. 886–890 (2008).

Lama Hassan received her B.S. degreefrom Higher Institute for Applied Scienceand Technology (HIAST), Damascus,Syria, in 2001. She also received Master’sdegree in Automatic Control in 2007from Institut National Polytechnique deGrenoble. Currently, she is Ph.D.

student in automatic control at Université de Lorraine,France. Her research interests include time-delay systems,state estimation, and robust control.

Ali Zemouche received his B.S. degree inMathematics from Département des Sci-ences Exactes de Oued-Aissi, UniversityMouloud Mammeri, Tizi-Ouzou, Algeria,in 2000. He also received two Master’sdegrees in Mathematics. The first one is

received in 2002 from Institut National Polytechnique deGrenoble, France with specialization in operation research,combinatory and optimization, and the second one is receivedin 2003 from University Picardie Jules Verne, Amiens,France, with specialization in applied analysis and modeliza-tion. He obtained Ph.D. degree in Automatic Control in 2007,from University Louis Pasteur, Strasbourg, France, where heheld post-doctorate degree from October 2007 to August2008. Dr. Ali Zemouche joined, as an associate professor,Centre de Recherche en Automatique de Nancy (CRANUMR CNRS 7039) at University of Lorraine, since septem-ber 2008. His research activities include nonlinear systems,state estimation, observer-based control, time-delay systems,and robustness analysis.

Mohamed Boutayeb (M’97) receivedElectrical Engineer degree in ElectricalEngineering from Ecole Hassania desTravaux Publics, Casablanca, Morocco, in1988, and Ph.D. and H.D.R. degrees in

Automatic Control from University Henri Poincaré, Nancy,France, in 1992 and 2000, respectively. From 1996 to 1997,he was Invited Researcher with Alexander von HumboldtFoundation, University of Duisburg, Duisburg, Germany.From 1997 to 1999, he was Researcher with Centre Nationalde la Recherche Scientifique (CNRS), France. From 2002 to2007, he was a full Professor with University of LouisPasteur, Strasbourg, France. Since 2007, he has been a fullProfessor with Centre de Recherche en Automatique deNancy, CNRS Unité Mixte de Recherche 7039, Nancy-Université, Nancy. His research interests are identification,state estimation, and control of dynamical systems.



robust observer and observer-based controller for time-delay singular systems

Documents