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International Journal of Contemporary Mathematical Sciences

Vol. 11, 2016, no. 7, 343 - 358

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijcms.2016.6633

Asymptotic Regional Gradient Full-Order

Observer in Distributed Parabolic Systems

Raheam A. Al-Saphory

Department of Mathematics

College of Education for Pure Sciences Tikrit University, Tikrit, Iraq

Naseif J. Al-Jawari

Department of Mathematics College of Science

Al-Mustansriyah University, Baghdad, Iraq

Asmaa N. Al-Janabi

Department of Mathematics College of Science

Al-Mustansriyah University, Baghdad, Iraq

Copyright © 2016 Raheam A. Al-Saphory, Naseif J. Al-Jawari and Asmaa N. Al-Janabi. This

article is distributed under the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the characterization of regional asymptotic gradient

observer have been given for a parabolic system. The approach of this

characterization derived from Luenberger observer theory which is enable to

estimate asymptotically the gradient state of the original system in a subregion

of a spatial domain Ω in order that the regional asymptotic gradient observability

notion can be achieved. Furthermore, we show that the strategic sensors allows

the existence of regional asymptotic gradient observer and a sufficient condition

have been given for such regional asymptotic gradient observer in identity case.

Thus, the obtained result are applied for different types of measurements, domains

and boundary conditions.

344 Raheam A. Al-Saphory et al.

Mathematics Subject Classification: 93A30; 93B07; 93B30; 93C20

Keywords: -strategic sensors, asymptotic -detectability, asymptotic

-observers

1. Introduction

There are many situations in modern technology in which it is necessary

to estimate the state of a dynamic system using only the measured input and

output data of the system [18]. An observer is a dynamic system the purpose of

which is to estimate the state of another dynamic system using only the

measured input and output letter. If the order of is equal to the order of the

observer is called full-order state observer [3, 8, 12]. Thus, asymptotic observer

theory explored by Luenberger in [20] for finite dimensional linear systems and

extended infinite dimensional distributed parameter systems govern by strongly

continuous semi-group in Hibert space by Gressang and Lamont as in [19]. The

study of this approach via another variable like sensors and actuators developed

by El-Jai et al. as in ref.s [3, 8, 12, 14] in order to achieve asymptotic

observability. One of the most important approach in system theory is focused on

reconstruction the state of the system from knowledge of dynamic system and the

output function on a subregion of a spatial domain Ω this problem is called

regional observability problem has been received much attention as in [7, 15-17].

An extension of this notion has been given in [4, 21] to the regional gradient case.

The regional asymptotic notion has been introduced and developed by Al-Saphory

and El- Jai in [2, 11]. Thus, this notion consists in studying the asymptotic

behavior of the system in an internal subregion of a spatial domain Ω. In this

paper, we develop the results of asymptotic regional state reconstruction in [6, 10]

to the asymptotic regional gradient full order observer which allows to estimate

the state gradient of the original system. The main reason for introducing this

notion is that it provides a means to deal with some physical problems concern the

model of single room shown in (Figure 1) below.

Fig. 1: Room observation problem , workspace ω, and input-output vents.

Asymptotic regional gradient full-order observer 345

Now the object to design the room (locate vents, place sensors, etc. …) in order to

observe asymptotically the room vents near workspace (for more details see [13]).

The outline of this paper is organized as follows: Section 2 is devoted to the

problem statement and some basic concept related to the regional gradient

stability, regional asymptotic gradient detectability. Section 3 we focus on

regional asymptotic gradient observer so we introduced and characterization the

existing of identity regional asymptotic gradient observer to provide an identity

regional asymptotic gradient estimator of gradient state for the original system in

terms of sensors structure. In the last section we have been applied these result to

the two dimensional distributed parameter systems for different zone and

pointwise sensors case.

2. Problem Formulation and Preliminaries

Let Ω be a regular bounded open subset of , with smooth boundary

and be subregion of , [0, ], > 0 be a time measurement interval. We

denoted = ×]0, [, =×]0, [. We considered distributed parabolic

systems is described by the following partial deferential equations

(,)=(,)+()

(, 0)=()

(,)= 0

(1)

Augmented with the output function

(. , )=( , ) (2)

where is a second order linear differential operator, which generator a strongly

continuous semi-group ()on the Hilbert space and is self-adjoin with

compact resolvent. The operator (,) and (,), depend on the

structure of actuators and sensors [16]. The space , and be separable Hilbert

spaces where is the state space, =(0, ,) is the control space and =

(0, ,) is the observation space where and are the numbers of actuators

and sensors (see Figure 2) which is mathematical model more general spatial case

in (Figure 1).

Fig. 2: The domain of , the sub-region , various sensors locations.

346 Raheam A. Al-Saphory et al.

Under the given assumption, the system (1) has a unique solution [18]:

( , )=()()+()()

(3)

The measurements are obtained through the output function by using of zone,

point wise which may located in Ω (or ). [16]

(. , )=( , ) (4)

• We first recall a sensors is defined by any couple (D, f ), where D is its

spatial support represented by a nonempty part of

and f represents the

distribution of the sensing measurements on D.

• Depending on the nature of D and f , we could have various type of

sensors. A sensor may be pointwise if D={b} with

and =(. ),

where is the Dirac mass concentrated at b. In this case the operator C is

unbounded [10] and the output function (2) can be written in the form

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

It may be zonal when

and (). The output function (2) can be

written in the form

()=(,)()

In the case of boundary zone sensor, we consider = and

(), the output function (2) can be written as

(. , )=(. , )=(,)

()

• We define the operator by the form

:=(.)

With : is the adjoint operator of K defined by

=()()

• Consider the gradient operator :()(())

Asymptotic regional gradient full-order observer 347

• :()( )

=

where is the restriction of to and it’s adjoint is denoted by

.

• Finally, we introduced the operator = from into (( )) where

its adjoint given by .

The problem is how to build an approach which observe (estimates) regional

gradient state in subregion in of Ω asymptotically by using a dynamic system

(an observer) in identity case only may be called full-order observer in region .

The important of an observer is that to estimates all the gradient state variables,

regardless of whether some are available for direct measurements or not [18].

• The systems (1)-(2) are said to be exactly regionally gradient observable on

(exactly –observable) if

= = (()).

• The systems (1)–(2) are said to be weakly regionally gradient observable on

(weakly –observable) if

=

= (()).

It is equivalent to say that the systems (1)-(2) are weakly –observable if

== {0}.

• If The systems (1)–(2) are is weakly –observable, then ( , 0) is given by

= ()=,

where is the pseudo-inverse of the operator (see ref. [15, 17]).

• A sensor (,) is gradient strategic on (-strategic) if the observed system

is weakly -observable.

As well known the observability [8, 18] and asymptotic observability [14, 16, 18-

20] are important concepts to estimate the unknown state of the considered

dynamic system from the input and output functions. Thus, These notions are

studied and introduced to the regional distributed parameter systems analysis with

different characterizations by El-Jai , Zerrik and Al-Saphory et al. in many paper

for example [2, 4-7, 9-11, 13, 15, 17, 21] in connection with strategic sensors.

Definition 2.1: The system (1) is said to be asymptotically regionally gradient

stable (asymptotically -stable) if the operator generates a semi-group which

is asymptotically gradient stable on the (()). It is easy to see that the system

(1) is asymptotically -stable, if and only if for some positive constants

,, we have

(. )(), () ,0

348 Raheam A. Al-Saphory et al.

If (()) is -stable semi-group in (()), then for all (),

the solution of associated system satisfies

(.,)(())=

(. )(())= 0 (5)

Definition 2.2: The systems (1)-(2) are said to be asymptotically regionally

gradient detectable (asymptotically -detectable) if there exists an operator

:(()) such that generates a strongly continuous

semi-group (()) which is asymptotically -stable on (()).

Remark 2.3: In this paper, we only need the relation (5) to be true on a sub-

region of the region

lim

(.,)(())= 0.

Thus, from the previous results we can deduce the following results:

Corollary 2.4: If The systems (1)–(2) are is exactly -observable, then it is

asymptotically –detectable.

Corollary 2.5: For every (()) there exists > 0, such that

(.,)(()) = 0

Remark 2.6: For parabolic systems, the notion of asymptotic –detectability is

far less restrictive than the exact -observability.

3. Sensors and asymptotic –Observer

In this section we present the sufficient conditions which are guarantee the

existence of an asymptotic regional gradient full-order observer (asymptotic

–Observer ) which allows to construct an –estimator of the state

( , ).

3.2 Definitions and characterizations

Definition 3.1: Suppose there exists a dynamical system with state (.,)

given by

(,)=(,)+()+((,)(,))

(, 0)=()

(,)= 0

(6)

Asymptotic regional gradient full-order observer 349

In this case the operator in general case [17] is given by =

where = the identity operator. Thus the operator generator a

strongly continuous semi-group ( ()) on separable Hilbert space

which is asymptotically –stable.

Thus, ,> 0 such that

(. ) ,0.

and let (,),( , ) such that the solution of (6) similar to (3)

( , )=()()+()[()+()]

Definition 3.2: The system (6) defines asymptotic identity (full-order) –

estimator for (,)=( , )=( , )(()) where (,) is the

solution of the systems (1)-(2) if lim

(.,)( , )(())= 0, and

maps () into () where (,) is the solution of system (6).

Remark 3.3: The dynamic system (6) specifies an asymptotic –observer of

the systems given by (1) and (2) if the following holds:

1- There exists (, (())) and ((())) such that

+=.

2- =C and = B.

3- The system (6) defines an asymptotic –estimator for ( , ).

The object of an asymptotic - observer is to provide an approximation to the

original system state gradient. This approximation is given by

()=()+().

Definition 3.4:The systems (1)-(2) are asymptotically -observable, if there

exists a dynamic system which is asymptotic -observer for the original

system.

3.2 Asymptotic -Observer reconstruction

In this case, we need to consider = and =, then the operator

observer equation becomes as = where and are known. Thus,

the operator must be determined such that the operator is asymptotically

-stable. This observer is an extension of asymptotic observer as in [2, 10-11,

16]. Now Consider again system (1) together with output function (2) described

by the following form

350 Raheam A. Al-Saphory et al.

(,)=(,)+()

(, 0)=()

(,)= 0

()=(. , ) (7)

Let be a given subdomain of Ω and suppose that ((())), and

(,)=(,) there exists a system with state (,) such that

(,)=(,)=(,) with =, where is the identity

operator with respect to regional gradient state estimator. Then

(,)=(,) (8)

From equation (7) and (8) we have

=

If we assume that there exist two bounded linear operators :((())

and

: ((())((()), such that += then by

deriving (,) in (8) we have

(,)=

(,)=(,)+()

= (,)+(,) + ()

Since the operator = , then we have

(,)= (,)+()+(,)

Therefore

(,)=(,)+()+(.,)

and since =C and = B then we have

(,)=(,)+()+((,)(,))

(, 0)=()

(,)= 0

(9)

Let us consider a complete sets of eigenfunctions in (()) orthonormal to

(()) associated with the eigenvalue of multiplicity and suppose the

system (1) has unstable mode. Then, the sufficient condition of an -observer

is formulated in the following main result.

Theorem 3.5: Suppose that there are zone sensors (,) and the

spectrum of contains eigenvalues with non-negative real parts. Then the

dynamic system (9) is -observer system for the system (7), that is

lim

[(,)(,)] = 0, if :

Asymptotic regional gradient full-order observer 351

1-There exists (, (())) and ((())) such that

+ = .

2- =C , = B.

3-

4- rank = , ,= 1, … , with

= ()=

(),(.)>() for zone sensors

() for pointwise sensors

<

,(.)>() for boundary zone sensors

where sup =< and j = 1, … , .

Proof:

First step: The proof is limited to the case of pointwise sensors. Under the

assumptions of section 2, the system (1) can be decomposed by the projections

and on two parts, unstable and stable. The state vector may be given by

(,)= [(,),(,)] where (,) is the state component of the

unstable part of the system (1) may be written in the form

(,)=(,)+()

(, 0)=()

(,)= 0

(10)

and (,) is the component state of the stable part of the system (1) given by

(,)=(,)+ ()()

(, 0)=()

(,)= 0

(11)

The operator is represented by a matrix of order ( ,

)

given

by =diag[,…,,, … , , … , ,…,] and = [,, … , ]. The

condition (4) of this theorem, allows that the suit (,) of sensors is -

strategic for the unstable part of the system (1), the subsystem (10) is weakly -

observable [4] and since it is finite dimensional, then it is exactly -observable

[5]. Therefor it is asymptotically -detectable, and hence there exists an operator

such that (

) which is satisfied the following:

,

,

> 0 such that (

)(())

,

and then we have

(. , )(())

,

(.)(()) .

352 Raheam A. Al-Saphory et al.

Since the semi-group generated by the operator is stable on (()) , then

there exist

,

> 0 [20] such that

(. , )(())

,

()()(.)(())

+

,

()(

)(.)(()) ()

and therefor (,)0 when . Finally, the system (9) are asymptotically

-detectable.

Second step: From equation (8), we have (,)=(,) with the observer error

is given by the following form

(,)=(,) (,)

Where (,) is a solution of the dynamic system (9). Derive the above equation,

and by using equation (8) and condition 2, we can get the following forms

(,)=

(,)

(,)=

(,)

(,)

=(,)+() (,)()(,)

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

=((,)(,))

= (,)

Thus, from the first part of this proof we obtain (,)=(0, ) is

asymptotically -stable with (0, )=()().

(,)(()))()()(()))

therefore lim

(,)= 0. Now, let the approximate solution to the gradient state of

the original system is

(,)=(. , )+(,) with = and =, then we

have

(,)=(,)

Now, we can calculate the error of gradient state estimator

(,)=(,)(,)=(,)(,)+(,)(,)

= (,)(,)=(,)=(0, )

is asymptotically -stable with (0, )=()(). Consequently we get

lim

(. , )( , )(())= lim

(.,)( , )(())= 0.

Then, the dynamical system (9) is -0bserver to the system (7).

Asymptotic regional gradient full-order observer 353

Corollary 4.4 From the previous results, we can deduce that:

1. Theorem 3.5 gives the sufficient conditions which guarantee the dynamic

system (9) is a -observer for the system (7).

2. If a system which is an -observer, then it is -observer for system (7).

3. If a system is -observer, then it is

-observer for every subset of

, but the converse is not true [6].

4. Application to asymptotic -observer in diffusion system

In this section we consider the distributed diffusion systems defined on

;

k

Ω

where

21 ≤≤ k

. Various results related to different types of sensor have

been extended. In the case of two-dimensional, we take

] [ ] [

21

2

,0 ,0 aa ×=Ω

and

] [

∞×Ω= ,0

22

Q

,

] [

∞×Ω=Σ ,0

2

2

, with boundaries

] [

∞

×Ω∂

=Θ ,

0

22

.

4.1. Two-dimensional system with rectangular domain

4.1.1. Case of zone sensors

Consider a two dimensional system defined in =]0, [×]0, [ by parabolic

equation

(,,)=

(,,)+

(,,)

(,, 0)=(,)

(,,)= 0

(12)

Augmented with output function measured by internal or boundary zone sensors

(. , )=(

,,)(,) or ( (. , )=

(

,,)(,)) (13)

Where and , , see (Figure 3).

Fig. 3: shows the domain

2

Ω

region

2

ω

and locations of internal (boundary) zone

sensors

354 Raheam A. Al-Saphory et al.

Let =],[×],[ be a subregion of . The eigenfunctions of the operator

(

+

) are defined by

(,)=

()()

Associated with the eigenvalues

=(

()+

())

Now, consider the dynamical system

(,,)=

(,,)+

(,,) (,,)()

(,, 0)=(,)

(,,)= 0

(14)

And suppose that the sensors is -strategic for unstable subsystem part of the

system (8), then we have the following results:

Proposition 4.1:

1- Internal zone case: Suppose that is symmetric about = and is

symmetric about =, then the dynamic system (14) is -observer systems

(12)-(13) if ()

and ()

N for every ,= 1, … , .

2- One side boundary zone case: Suppose that

Ω∂⊂Γ

and f is symmetric with

respect to

1

01

ηη

=

, then the dynamic system (14) is -observer for the

systems (12)-(13) if

( ) ( )

Ni ∉−−

1111

/

αβαη

for every, = 1, , . . . , .

3- Two side boundary zone case:

Let

( )

{ } { }

[ ]

Ω∂⊂+××−=Γ×Γ=Γ

20110

21

21

,00, laal

ηη

and

1

Γ

f

is symmetric

with respect to

1

01

ηη

=

, and the function

2

Γ

f

is symmetric with respect to

2

02

η

η

=

, then the dynamic system (14) is --observer for the system (12)-

(13), if

( )

( )

( )

( )

Nji ∉−−−− 22201

110 /and

/21

αβαηαβαη

for every i, j =1 ,...,

J.

4.1.2. Case of pointwise sensors

Consider again the systems (12)-(13) augmented with output function

measured by internal or boundary pointwise sensors (Figure 4).

Asymptotic regional gradient full-order observer 355

Fig. 4: Rectangular domain and locations b, σ of pointwise sensors

(. , )=(

,,)(,)((.,)=

(

,,)(,) (15)

Proposition 4.2:

1- The internal pointwise case:

If

( ) ( ) ( ) ( )

Nbjbi ∉−−

−−

22221111

/

and/

αβαα

βα

, for every , = 1, … , , then

the dynamic system (14) is -observer for the systems (12)-(15).

2- Filament pointwise case: Suppose that the observation is given by the filament

sensor

( )

σ

δσ

,

, where

)

Im(

γσ

=

is symmetric with respect to the line

( )

21

,bbb =

.

The dynamic system (14) is -observer for the system (12)-(15), if

( ) ( )

1111

/

α

βα

−−bi

and

( ) ( )

Nbj ∉−− 2222 /

αβα

, for every

.1,. . . , Ji, j=

3- The boundary pointwise case:

If

( ) ( )

and/111

1

αβα

−−bi

( ) ( )

Nbj ∉−−

2222

/

αβα

, for , =

1, .. . ,, then the dynamic system (14) is -observer for the system (12)-(15).

4.2. Two-dimensional systems with circular domain

Remark 4.3: We can extend these results to the case of two dimensional systems

with circular domain in different sensor structures as in [2, 10].

4.3. One-dimensional systems domain case

Remark 4.4: We can extend the above results of the two dimensional systems

(12)-(13) to case of one dimensional systems case if we take

] [

a,0

1

=Ω

. We

denote

] [

∞×Ω= ,0

11

Q

,

] [

∞×Ω=Σ ,0

1

1

with boundaries

] [

∞×Ω∂=Θ ,0

11

as in ref.s [2, 4-7, 9-11, 13, 15, 17, 21].

Remark 4.5: We know that the previous results have been developed with

Dirichlet boundary conditions, then we can extend with Neumann or mixed

boundary conditions as in [1, 17].

356 Raheam A. Al-Saphory et al.

5. Conclusion

The concept studied in this paper is related to the -observer in

connection with sensors structure for a class of distributed parameter systems.

More precisely, we have been given a sufficient condition for existing an -

observer which allows to estimate the gradient state in a subregion . For future

work, one can be extension these result to the problem of regional boundary

gradient observer in connection with the sensors structures as in [5].

Acknowledgments. Our thanks in advance to the editors and experts for

considering this paper to publish in this estimated journal and for their efforts.

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Received: July 2, 2016; Published: August 10, 2016