adaptive observers for time-delay nonlinear systems in triangular form

8/13/2019 Adaptive Observers for Time-Delay Nonlinear Systems in Triangular Form

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Automatica 45 (2009) 23922399

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Brief paper

Adaptive observers for time-delay nonlinear systems in triangular formI

Salim Ibrir University of Trinidad and Tobago, Pt. Lisas Campus, P.O. Box. 957, Esperanza Road, Brechin Castle, Couva, Trinidad and Tobago

a r t i c l e i n f o

Article history:

Received 28 October 2008

Received in revised form

22 May 2009Accepted 21 June 2009

Available online 19 August 2009

Keywords:

Time-delay systems

Nonlinear systems

Nonlinear observers

System theory

a b s t r a c t

A simple nonlinear observer with a dynamic gain is proposed for a class of bounded-state nonlinearsystems subject to state delay. By saturating the states of the observer nonlinearities with eithersymmetric or non-symmetric saturation functions, we show that the observer exists, whatever the delayis. Furthermore, it will be highlighted that the observer design is free from any preliminary analysis ofthe time-delay system such as estimating the Lipschitz constants of nonlinearities. The proposed designencompasses a wide class of nonlinearand time-delay systems written in triangular form and generalizesprevious results on delayless nonlinear systems.

2009 Elsevier Ltd. All rights reserved.

1. Introduction

Time-delay often appears in many control systems such aschemical reactions, mechanical systems, electrical circuits (Bellen,

Guglielmi, & Ruehli, 1999), high-speed networks, and many otherprocess control systems (Dugard & Verriest, 1997;Gu, Vladimir, &Chen, 2003; Hale & Lunel, 1993; Niculescu, 2001). This delay is dueto several reasons such as communication protocols, transmission,transportation or inertia effects. Unlike systems governed byordinary differential equations, delay systems also called ashereditary or systems with aftereffects, are infinite dimensional innature and time-delay is, in many cases, a source of instability. Thestability issue and the performance of control systems with delayare, therefore, both of theoretical and practical importance. As adual problem, observer design for time-delay systems turns out tobe a stabilization issue since the observation error dynamics mustbe stabilized using only partial state measurements, which is inthe general case a quite hard problem, compared with full-state

feedback stabilization. For further results on observation of time-delay systems, we refer the reader to Boutayeb(2001),Germani,Manes, and Pepe(2002),Ibrir, Xie, and Su(2006),Wang, Goodall,and Burnham(2002) and the references therein.

Referring to many results on observation of time-delaysystems, conditions under which the observer/filter exists can be

I The material in this paper was partially presented at American Control

Conference, St. Louis, June 2009,USA. Thispaper was recommended for publication

in revised form by Associate Editor James Lam under the direction of Editor Ian R.

Petersen. Tel.: +1 868 723 2229; fax: +1 868 636 3339.E-mail address:[email protected].

classified into two main categories: delay-independent conditionsand delay-dependent ones (Mahmoud, 2000). However, delay-dependent results reveal less conservative than delay-independentconditions. Nevertheless, in most cases, a small delay is tolerable

to maintain stability by output feedbacks.In the present paper, a new adaptive observer is proposed for

bounded-state nonlinear systems written in triangular form. Ourmain task is basically focused on the development of a nonlinearobserver that converges, whatever the size of the delay, underthe assumption that large delays do not violate the boundednesscondition of the states. As a matter of fact, even the conditionof convergence of the observer is stated as a delay-independentcondition; this condition is always verified by adaptation ofthe observer gain. By the appropriate selection of a parameter-dependent Riccati equation, we show that we can update theobserver vector gain by updating only one parameter of theARE. Even though the observer design methodology does notnecessitate any preliminary analysis of the system, the knowledge

of both the system delay and the domain of variation of the systemstates remains necessary to build the nonlinear observer. Thisassumption seems to be a natural assumption in practice sincethe observation is usually made in some compact and large setcalled the observation domain. We show that the saturated-stateobserver can reproduce the behaviors of the original states byinjection of a time-varying gain. Furthermore, it is outlined thatthe observer always exists without any condition on the size of theconstant delay or the form of nonlinearities. Finally, the extensionof the obtained results to linear systems with lower-triangularnominal matrices is given.

Throughout this paper, we note by Rthe set of real numbers.The notation A > 0 (resp. A < 0) means that the matrix A is

0005-1098/$ see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.06.027
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S. Ibrir / Automatica 45 (2009) 23922399 2393

positive definite (resp. negative definite). A0 is the transpose ofmatrix A. ? is used to notify an element which is induced bytransposition. , stands for an equality by definition. k(A)standsfor thekth eigenvalue of the matrix A. k k is the Euclidean norm,k k is the infinity norm, and Spec(A) stands for the set ofeigenvalues of the matrixA.Ckn is the binomial coefficient.

2. System description

Consider the time-delay nonlinear system given in lower-triangular form:

x1(t)= x2(t)+f1(x1(t),x1(t ), u(t)),x2(t)= x3(t)+f2(x1(t),x2(t),x1(t ),x2(t ), u(t)),...

xi(t) = xi+1(t)+fi(x1(t), . . . ,xi(t),x1(t ), . . . ,xi(t ), u(t)),

...

xn(t)= fn(x1(t), . . . ,xn(t),x1(t), . . . ,xn(t

), u(t)),

y(t)= x1(t),

(1)

wherex(t) Rn is the state vector,u(t) U Rm is a boundedcontrol input andy(t) R is the system output. We assume thatthe delay is constant and x(t) = (t) for t . In matrixnotation, system(1)takes the form

x(t)= Ax(t)+f(x(t),x(t ), u(t)),y(t)= Cx(t),

(2)

where

A ,

0 1 0 00 0 1 0...

..

....

. . . 0

0 0 0 10 0 0 0

Rnn, C,

100

...

0

0

R

n,

f(x(t),x(t ), u(t))

,

f1(x1(t),x1(t ), u(t))

f2(x1(t),x2(t),x1(t ),x2(t ), u(t))...

fn(x(t),x(t ), u(t))

Rn.(3)

To complete the system description, the following assumptions areconsidered.

Assumptions 1. The nonlinearityf(x(t),x(t ), u(t))is smoothand well-defined for allx(t) Rn withf(0, 0, 0)= 0.

Assumptions 2. For allt 0, the system inputu(t) U Rm isglobally bounded.

Assumptions 3. For given constant delay , u U, and initialconditionx0 M Rn, the state vector x(t)is well-defined andglobally bounded for allt 0.

Assumptions 4. For allt 0, the delay is known and constant.In the following, for simplicity of notations,u(t),x(t)andx(t

)will be noted byu,xandx, respectively.

3. Observer design

Before giving the main result of this paper, let us present somepreliminary results.

3.1. Preliminary results

First, let us recall the result of the following lemma.

Lemma 1 (Derese & Noldus, 1980).Let A, Q2, and W2be given n nmatrices such that Q2 is symmetric and W2 is positive definite and

symmetric. Furthermore, assume that S is positive definite symmetric

matrix satisfying

SW2S+ AS+ SA0+ Q2 =0 (4)

and W1 is a positive definite symmetric matrix such that W1 < W2.

Then there exists a positive definite symmetric matrix S+ such that

S 0; = diag(1, 2, 3, . . . , n), (7)

and

Q()= 2D()Q D(),

D()= diag

1, , 2, . . . , n1

.

(8)

Then, P()is positive definite for > 0 and has the following prop-erties:

(i) for any >0, the Cholesky decomposition of P() is given by:

P()= R()R0(), R()=D()eR, (9)

whereP=eReR0is the solution of the ARE:PA0+ APPC0CP+ Q =0. (10)

(ii) For any >0,

d

dP() >0;

d

dP1()


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2394 S. Ibrir / Automatica 45 (2009) 23922399

Remark 2. The properties (i)(iii) ofLemma 2do not require thematrixQ() to be diagonal as it has been required in the designof the ARE-based differentiation observers, see Chitour (2002)andIbrir (2001,2006).Therefore, more freedom in assigning theeigenvalues ofAP()C0Cis tolerated.Explicit formulae thatgivesQ()in terms of the coefficients of the characteristic polynomial(not on the roots of this polynomial) is given in Ibrir(2006).

The analysis of the observer is given in the following section.

3.2. Adaptive observer design

The observer design methodology is basically founded onestimating the bounds of the system states and saturating thestates of the observer nonlinearities according to the estimatedlevels of variation of the true states. The idea of this new designis to adapt the observer gain without any care of the maximumlevel that can reach the Jacobian norm of the system nonlinearityinside the observation domain. The analysis of the observer issummarized in the following statement.

Theorem 1. Consider the time-delay nonlinear system (2) under

Assumptions 14. Assume that there exist positive saturation levels1,. . ., nsuch that for x0 M Rn

t 0, |xi(t)| i, 1 i n. (14)Let

x = Ax+f((x), (x), u) P()C0(Cx Cx), =

x1 y)2, (0) >1, >0,

P()A0+ AP() P()C0CP()+Q()= 0,PA0+ APPC0CP+ Q =0,

(15)

be the adaptive observer where Q + Q > 0, = diag(1, 2,

. . . , n), Q() = 2D()Q D()and (x) = [1(x1), . . . , n(xn)]0

is a vector of saturation functions defined as

i(xi) ,

xi if,|xi| i, i >0,isign(xi) otherwise,

1 i n. (16)

Then, the observation error e= x x that results from (2) and (15)is asymptotically stable for all x(0) Rn, x(t) = (t) for t ,

x0 M Rn.Proof. See theAppendixsection.

Remark 3. The design of the observer remains valid when thesaturation functions employed in the observer dynamics are non-symmetric. This technique limits the variation of the observernonlinearity vector in the same range of variation of the system

nonlinearity. We refer the reader to the referenceBenzaouia andBurgat (1988) to see the features of controlling linear systems withnon-symmetrical constrained control.

As a comparison with previous results, the differential Lyapunov-like equation(17)proposed first in the reference Ibrir and Diop(1999) todealwiththe problem of the peaking has a fixed terminalvalue which is fixed by the coefficient. This means that the finalvalue of the matrixP()is a priori known even the gainP1()C0

that results from the differential Lyapunov-like equation:

P(t)= P(t) A0P(t) P(t)A+C0C (17)is time-varying. In our proposed design the final value of theobserver gain is not a priori known. However, the observer gain

is updated through the difference (y Cx)2

. The only criteria thatdetermines the final value ofis the conditiony = Cxthat makes

the derivative ofconstant. Other recent results that concern thedynamic-output stabilizability of controllable linear systems withsaturated inputs are given inZhou, Duan, and Lin(2008).

In the following statement, we show that we can apply theprevious results forlinear systems whose statesare notnecessarilybounded. This result is given by the following statement.

Corollary 1. Consider the time-delay linear system

x(t)= (A+A)x(t)+Adx(t )+(u)y(t)= Cx(t),

x(t)= (t), 0 t ,(18)

where A Rnn and C Rn are defined as in (3) and x(t) isnot necessarily bounded. LetA and Ad be known, real, and lower-

triangular matrices and(u) is an input-injection vector of dimensionn. Then the following system

x(t)= (A+A)x(t)+Ad x(t ) P()C0(Cx y)+(u), = (x1 y)2, (0) >1, >0

(19)

is an asymptotic observer for system(18)where P()is defined as inEq.(6).

Proof. The matricesA andAdcan be seen as the matrix Jacobian.Therefore, the proof becomes straightforward as it was developedbefore.

4. Extension to multiple and unknown time-delay systems

4.1. Known multiple delays systems

It is important to outline that the strategy of the observerremains the same if the system nonlinearities involve multipletime delays. The only condition to preserve the same design ismaintaining the triangular structure of the system nonlinearitywith respect to each state of different time-delay. In other words,if we consider ` different known and constant delays: 1, 2,. . ., , the vector nonlinearity should have the following structure:

f1 = f1

x1(t),x1(t 1),x1(t 2), . . . ,x1(t ), u(t)

,

f2 = f2

x1(t),x2(t),x1(t 1), . . . ,x1(t ),

x2(t 1), . . . ,x2(t ), u(t)

,

..

.

fn = fn

x(t),x(t 1),x(t 2), . . . ,x(t ), u(t)

.

(20)

Consequently, an observer is readily constructed as

x(t) = Ax(t)+f

(x(t)), (x(t 1)), (x(t 2)), . . . ,

(x(t )), u(t)

P1()C0(Cx y),

= (Cx(t) y(t))2, (0) 1, >0.

(21)

The proof of this result is deduced by following the same steps ofthe proof of the main statement. It is sufficient to expand then the

Jacobian to a sum of Jacobians as done in(45).


4/8


4.2. Unknown time-varying delay nonlinear systems

If an unknown time-varying delay = (t) is present in thesystem nonlinearity, the whole nonlinearity vector becomes un-known. Therefore, constructing the observer as a copy of the dy-namics of the true system is consequently not possible. Therefore,the design of a delayless observer seems, for instance, the only

strategic solution to handle this complex problem as it hasbeen done for linear time-delay systems under some conditions,seeFairman and Kumar(1986).

The only possible situation where we can design a delaylessobserver for our system is the case where the system admits aninputoutput representation of the form:

y(n) =

Y(t), Y(t (t)), u(t)

, (22)

whereY(t) = [y(t),y(t),y(t), . . . ,y(n1)(t)]. If we put z1 = yasa measured output and zi(t) = y

(i1)(t), i = 2, . . . , n then, wecan always construct a model-free algebraic observer as discussedin the referenceIbrir(2003). In this situation, the boundedness ofthe state is not a necessary condition for the construction of an

asymptotic algebraic observer.

5. Example

Let us consider the time-delay system

x1(t)= x2(t),

x2(t)= x1(t) x2(t ) cos(x2(t )),y(t)= x1(t).

(23)

The system nonlinearity is not globally Lipschitz, but the systemstates exhibit bounded behaviors for the initial condition x0 =[1 1]0 with = 3.5, = 1.2, = 3. By tacking 1 = 4 and

2=8, the states of the following saturated-state observer

x1(t)= x2(t)+`1()(y(t) x1(t)),x2(t) =1(x1(t)) 2(x2(t )) cos(2(x2(t ))),

+ `2()(y(t) x1(t)), =(y(t) x1(t))2

(24)

converge asymptotically to the true ones as shown inFig. 1whereP()C0 = [ `1() `2()]0. The evolution of the parameter is given in the last subplot ofFig. 1. InFig. 2, we show that fordifferent and large initial conditions of the nonlinear observer,the saturated-state observer with dynamic gain guarantees thestability of the observation error and avoids the divergence of theestimates for a time delay equal to =3.

6. Conclusion

The results given in this paper can been seen as an extensionof the previous results on observation of uniformly-observablenonlinear systems written is lower-triangular form Gauthier,Hammouri, and Othman (1992). The proposed design is validwhatever the size of the delay that preserves the boundedness ofthe states. We showed that by state saturation and adaptation ofthe observer gain we could eliminate preliminary analysis stepsand avoid excessive high-gain design. The proposed design isdedicated to a large class of bounded-state nonlinear systems thatare not necessarily globally Lipschitz. In the meantime, even the

computation of theobservergain does notdepend upon thesystemdelay, the knowledge of this constant delay remains necessary to

x2anditsestimate

Time in (sec)

(t)

10

5

0

5

x1anditsestimate

10

0

10

20

30

0

2

4

6

8

0 5 10 15 20 25 30

0 5 10 15 20 25 30

0 5 10 15 20 25 30

Fig. 1. The performance of the saturated nonlinear observer. The observer states

are represented in dashed line.

e2

e1

40

30

20

10

0

10

20

30

15 10 5 0 5 10

Fig. 2. The phase plane of the observation error.

set up the observer. This particular issue is under investigation andshall be the topic of our next contributions.

Appendix

Proof of Lemma 2. SinceQ()is symmetric and positive definitefor all > 0 then, the matrix P()is the only solution of the ARE(6)which is always symmetric and positive definite for >0.

To prove (i) it is sufficient to prove that P() = D()PD().Pre- and post-multiplying the ARE(10)by D()then, we have:

2D()PA0D()+2D()APD()

2D()PC0CPD()+2D()Q D()= 0. (25)


5/8


where Q = Q()|=1. Using the following properties:

D()A0 =A0D(),D()A= AD(),

CD()= C, D()C0 =C0,

2D()Q D()= Q(),

(26)

then, the first ARE in(6)can be rewritten asD()PD()

A0+ A

D()PD()

D()PD()

C0C

D()PD()

+Q()= 0. (27)

By comparing the last ARE with the first ARE of(6),we conclude

that

P()= D()PD(). (28)

Since P=

eR

eR0, this immediately implies that

P()= D()eReR0D() (29)and consequently,

R()=D()eR. (30)

This ends the proof of item (i).

(ii) In fact, the derivative property of P() is a direct conse-

quence of the DereseNoldus Lemma. Starting from our ARE de-fined in(6),we have

P1()Q()P1()+A0P1()+P1()A C0C=0. (31)By replacing in(4)the matrixAbyA0, which is the transpose of theanti-shift matrix given in (3), Q2by

C0C, W2by Q(), and then us-

ing the DereseNoldus Lemma, we can write that for any 2 >1,P1(2) 0. It remains to

prove here that dd

Q() >0. We have,

d

dQ() =

d

d

2D()Q D()

=

d

d

D()

QD()

+D()

Q

d

d

D()

.

(34)

Since

d

d(D())= diag

1, 2, . . . , nn1

= D(). (35)

This gives

dd

Q()= D()Q D()+D()Q D(). (36)

SinceD()and commute then,D() = D(), which makesthe derivative ofequal to

d

dQ() =D()

Q+ Q

D(). (37)

By assumption ofLemma 2,Q+ Q > 0 then, dd

Q() > 0 for

all >0. Since the derivative of the matrix P()can be explicitly

written as

d

dP()= P() d

dP1()P(). (38)

Using the proved result, we conclude that dd

P() >0.

(iii) Since each(i,j)-entry of the matrixD1()LD()is in theform i,j+

i,j

then, we conclude that

sup1

kD1()LD()k n sup1

kD1()LD()k

c1+ c2

.

(39)

Proof of Theorem 1. The dynamic equation of the observationerror is given by:

e= (A P()C0C)e+f((x), (x), u) f((x), (x), u).(40)Let us assign the functional

V(e)= e0P1()e+1

2

Z tt

e0(s)P1()Q()P1()e(s)ds (41)

to the dynamics(40).Then, we have

V = e0P1()e+e0P1()e+1

2e0P1()Q()P1()e

1

2

e0P1()Q()P1()e + e0dP1()

d

e. (42)

Since > 0 for allt 0 and dP1()d

< 0 for all > 0, see the

result ofLemma 2then,

V(e) e0P1()e+e0P1()e+ 12

e0P1()Q()P1()e

12

e0P1()Q()P1()e

= e0

A0P1()+P1()A 2C0C

+1

2P1()Q()P1()

e

+ 2e0P1()f((x), (x), u) f((x), (x), u) 1

2e0P1()Q()P1()e. (43)

Since for anyx1,x2 Rn and any differentiable vector(x) Rn ,we have

(x1) 2(x2)=Z 1

0

(s)

s

s=x1(x1x2)

(x1 x2)d. (44)

Then, by decomposition of the Jacobian, we can write that

f((x), (x), u) f((x), (x), u)

= Z 10

f(,, u)

=()=()

(x) (x)d


6/8


+

Z 10

f(,, u)

=()=()

(x) (x)

d (45)

where

() = (x) ((x) (x)),= (x) ()((x) (x)),

() = (x) ((x) (x))= (x) ()((x) (x)),

(46)

where

()=

if, 0 1,1 otherwise.

(47)

Let us note

f(,, u)

=()=()

= L(),

f(,, u)

=()=()

= L().

(48)

This immediately gives

V(e)Z 1

0

e0

C0C 12

P1()Q()P1()

ed

+ 2

Z 10

e0P1()L()(x) (x)

d

+ 2

Z 10

e0P1()L()(x) (x)

d

12

Z 10

e0

P1()Q()P1()

ed. (49)

Using the fact that for given vectors 1

Rn, 2

R

n the following

inequality 2012 01X01+ 02X12 holds for anyX Rnn > 0.Then,

2

Z 10

e0P1()L()(x) (x)

d

Z 10

e0P1()ed

+

Z 10

(x) (x)

0L

0()P

1()L()(x) (x)

d, (50)

and

2

Z 10

e0P1()L()(x) (x)

d

Z 10

e0P1()ed

+ Z 1

0 (x)

(x)

0L

0()P

1()L()

(x) (x)d. (51)Consequently,

2

Z 10

e0P1()L()(x) (x)

d

+2

Z 10

e0P1()L()(x) (x)

d

2Z 1

0

e0P1()ed

+

Z 1

0 (x) (x)

0L

0()P

1()L()

(x) (x)0d

+

Z 10

(x) (x)

0L

0()P

1()L()

(x) (x)

d. (52)

This implies that

V(e)

1

2 Z 1

0

e0P1()Q()P1()ed

+2

Z 10

e0P1()ed

+

Z 10

(x) (x)

0L

0()P

1()L()

(x) (x)

0d

+

Z 10

(x) (x)

0L

0()P

1()L()

(x) (x)

d

12 Z

1

0

e0

P1()Q()P1()

ed. (53)

Let us define = D1()e, = D1()e. Using the result ofLemma 2,we have

P1()= 1

D1()P1D1()

Q()= 2D()Q D().

(54)

Note thatZ 10

(x) (x)

0L

0()P

1()L()(x) (x)

0d

+

Z 1

0 (x) (x)

0L

0()P

1()L()

(x) (x)dZ 1

0

kD()L0()P1()L()D()k

kD1()(x) (x)

k2d

+

Z 10

kD()L0()P1()L()D()k

kD1()(x) (x)

k2d. (55)

SinceD()is a diagonal matrix then,

kD1()

(x) (x)

k2

x x

0D1()D1()

x x

= 0,kD1()(x) (x)k2

x x0

D1()D1()

x x

= 0. (56)

From inequalities(53)(55),we get

V(e) 12

Z 10

0P1QP1d+ 2

Z 10

0P1d

+ 1

Z 10

kD()L0()D1()P1

D1()L()D()kkk2d

+ 1

Z 1

0

kD()L0()D1()P1

D1()L()D()kkk2d


7/8


12

Z 10

0P1QP1d. (57)

This implies that

V(e)

Z 1

0

1

2min

P1QP1

+max(P1)

2+ kD1()L()D()k2

kk2d+

Z 10

1

2min

P1QP1

+max(P

1)

kD1()L()D()k2

kk2d. (58)

As the matricesL() andL() are lower-triangular andall their

entries are saturated,thenby the use of(iii) ofLemma 2, and taking

into account that for 0 1, we have

kD1()L()D()

k2

nmax1in

nXj=1

D1()L()D()

i,j

!2

c1()+ c2()

2,

kD1()L()D()k2

nmax

1in

nXj=1

D1()L()D()

i,j

!2

c1 ()+

c2 ()

2

,

(59)

where c1(), c2(), c1 (), and c

2 () are bounded positive

constants that depend on . Finally, we can write that

V(e)Z 1

0

"1

2min

P1QP1

+max(P

1)

2+

c1()+

c2()

2#kk2d

+

Z 10

" 1

2min

P1QP1

+ max(P

1

)

c1 ()+ c2 () 2#kk2d. (60)

Since is increasing whenever x1 6= y then, there exist a finite-timeTwhereby

12min

P1QP1

+max(P

1)

2+

c1()+

c2()

2= 2 0,P()A0+ AP() P()C0CP()+Q()= 0.

(63)

The vectorf((x), (x

), u)can be seen as a bounded vector thatperturbs the stable system

x = (A P()C0C)x. (64)Then, it is easy to prove that the observer state trajectories arebounded for 0 t T wheneveru(t)is bounded. It remains toprove that(t)converges to some constant when x1 y = 0. Toprove the boundedness of(t)assume first that

limt

(t)= +. (65)

Starting from inequalities(60)(62), we can write that V(e) 2kk2 2(Cx y)2 fort T whereT is the first instantthat verifies the inequalities(60)(62).This immediately implies

thatZ tT

V(e())dZ t

T

2(Cx y)2d, (66)

or equivalently,Z tT

2(Cx y)2d = 2

Z tT

()d

V(e(T)) V(e(t)) V(e(T)). (67)

Based on(67),fort T, (t)is bounded as

(t)

2V(e(T))+(T)= Constant (68)

which is in contradiction with(65).By extending the interval oftime[T, t]to [T, [, we conclude that limt(t) = ?


8/8


Ibrir, S., & Diop, S. (1999). On continuous-time differentiation observers. InProceedings of the European control conference.

Ibrir, S., Xie, W. F., & Su, C.-Y. (2006). Observer design for discrete-time systemssubject to time-delay nonlinearities. International Journal of Systems Science,37(9), 629641.

Mahmoud, M. S. (2000). Robust control and filtering for time-delay systems. New-York: Marcel Dekker.

Niculescu, S.-I. (2001).Delay effect on stability: A robust control approach. Springer-Verlag.

Wang, Z., Goodall, D. P., & Burnham, K. J. (2002). On designing observers for time-

delay systems with nonlinear disturbances. International Journal of Control,75(11), 803811.

Zhou, B., Duan, G., & Lin, Z. (2008). A parametric Lyapunov equation approach tothe design of low gain feedback.IEEE Transactions on Automatic Control, 53(6),15481554.

Salim Ibrir received hisB. Eng. degreefrom Blida Instituteof Aeronautics, Algeria, in 1991, his Masters degree fromInstitut des Sciences Appliques, Lyon, France, in 1994and his Ph.D. degree from Paris-11 University, in 2000.From 1999 to 2000, he was a research associate (ATER)in the department of Physics of Paris-11 University. Dr.Ibrirspent manyyears as a postdoctoral fellow andvisitingAssistantProfessor in diverse NorthAmericanUniversitiesbefore joining theUniversity of Trinidadand Tobagowherehe is an Associate Professor. His current research interests

arein theareas of adaptive control of non-smoothsystems,nonlinear observers, robust system theory and applications, time delay systems,fuel-cell powered systems,hybrid systems,signal processing, ill-posed problems inestimation, singular hybrid systemsand Aero-Servo-Elasticity. Dr. Ibriris theauthorof more than 70 technical research papers and one text book.

adaptive observers for time-delay nonlinear systems in triangular form

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