Asymptotic Regional Boundary Observer in Distributed Parameter Systems via Sensors Structures

Abstract

The purpose of this paper is to study the concept of asymptotic regional observer in connection with the characterizations of sensors. We give various results related to different types of measurements, of domains and boundary conditions. Furthermore, we show that the measurements structures allow the existence of regional observer and we give a sufficient condition for such observer. We also show that, there exists a dynamical system for the considered system is not observer in the usual sense, but it may be regional boundary observer.

1. Introduction

The observer theory was introduced by Luenberger in [1] and is generalized to infinite dimensional control systems described by semi-group operators in [2]. The analysis of distributed parameter systems has received much attention in the literatures [3, 4, 5]. The study of this notion, through the structure of the sensors and the actuators, was developed by El Jai and Pritchard [6, 7].

The notion of regional analysis was extended by El Jai and Zerrik et al. [8, 9]. This study motivated by certain concrete-real problems, in thermic, mechanic, environment [10, 11]. If a system defined on a domain Ω is represented as in the (Fig. 1), then we are interested in the regional state on ω or Г of the domain Ω.

The concept of asymptotic regional analysis was introduced recently by Al-Saphory and El Jai in [12, 13, 14], consists in studying the behavior of systems not in all the domain Ω but only on particular regions ω or Г of the domain. In this paper, we are only interested in the asymptotic regional boundary state on a region Г of the boundary ∂Ω. The purpose of this paper is to give some results related to the link between the Г-observer and the number of sensors, their locations, the geometrical domains and boundary conditions.

Figure 1. Domain Ω, regions ω, Г and locations of measurements.

Figure 1. Domain Ω, regions ω, Г and locations of measurements.

The paper is organized as follows. Section 2 concerns the formulation problem and preliminaries. Section 3, devotes to the introduction of Г-detectability problem. In section 4, we give some definitions and characterizations concerns Г-observer and strategic sensors. We show that there exists a counter-example of the case Г-observer which is not observer, in the whole domaine. In the last section, we illustrate applications with various situations of sensor locations and we characterize such a regional boundary observer to a diffusion system.

2. Problem Statement

Let Ω be a regular, bounded and open set of Rⁿ, with smooth boundary ∂Ω and Г be a nonempty given subregion of ∂Ω, with positive measurement. We denote Q = Ω × ]0,∞[ and Θ = ∂Ω × ]0,∞[. We consider the system described by the following parabolic equation

(2.1)

with the output function

y(., t) = Cz(., t)

(2.2)

and we assume that Z, U, O be separable Hilbert spaces where Z is the state space, U the control space and O the observation space. Usually we consider Z = H¹ (Ω), U = L² (0,∞, R^p) and O = L² (0,∞, R^q) where p and q hold for the number of actuators and sensors. A is a second order linear differential operator which generates a strongly continuous semi-group (S_A (t)) _t≥0 on Z and is selfadjoint with compact resolvent. The operators B∈L (R^p, Z) and C ∈L (Z, R^q) depend on the structure of actuators and sensors [7]. Under the given assumptions, the system (2.1) has an unique solution given by

(2.3)

The measurements can be obtained by the use of zone, pointwise or lines sensors which may be located in Ω (or ∂Ω) [7].

• Let us recall that a sensor by any couple (D, f) where D denote closed subsets of $\overline{\Omega }$, which is spatial support of sensor and f ∈ L² (D) defines the spatial distribution of measurement on D. According to the choice of the parameters D and f, we have various types of sensors. A sensors may be a zone types when D ⊂ Ω. The output function (2.2) can be written in the form

  y (t) = Cz(., t) = ∫_D z(ξ, t) f (ξ)dξ

  (2.4)

A sensor may also be a point when D = {b} and f = δ(. - b) where δ is the Dirac mass concentrated in b. Then, the output function (2.2) may be given by the form

(2.5)

In the case of boundary zone sensor, we consider D = Γ with Γ ⊂ ∂Ω and f ∈L ² (Γ). The output function (2.2) can then be written in the form

(2.6)

The operator C is unbounded and some precautions must be taken in [7, 15].

• The operator K defined by

  and in the case of internal zone sensors is, linear and bounded with an adjoint
• The trace operator of order zero

  is linear, surjective and continuous with adjoint denoted by γ₀^*
• Consider a subdomain Γ of ∂Ω and let χ_Γ be the function defined by

  where z_|Γ is the restriction of the state z to Γ. We denote by ${\chi }_{\Gamma }^{*}$ the adjoint of χ_Γ.
• Let χ_ω be the function defined by

  where z_|Γ is the restriction of the state z to ω.
• The operator T_Γ : H¹(Ω) → H^1/2(Γ) is given by

Recalling some definitions concern the notion of Γ-observability as in [9, 16] :

• The autonomous system associated to (2.1)-(2.2) is said to be exactly (respectively weakly) Γ-observable if :
• The suite (D_i, f_i)_1≤i≤q of sensors is said to be Γ-strategic if the system (2.1) together with the output function (2.2) is weakly Γ-observable. For the dual results concerning the actuators structures [17].

3. Regional boundary detectability

The main reason for introducing Γ-detectability is, the possibility to construct an Γ-estimator for the current state of the original system. This concept is extended by Al-Saphory and El Jai in [13]. For this objective we recall some definitions concern this concept :

• The system (2.1) is said to be ∂Ω-stable, if the operator A generates a semi-group which is stable on the space H^1/2 (∂Ω). It is easy to see that the system (2.1) is ∂Ω-stable, if and only if, for some positive constants M and α, we have

If (S_A(t))_t≥0 is stable semi-group H ^1/2(∂Ω), then for all z₀ ∈H¹ (Ω), the solution of the associated autonomous system satisfies

(3.1)

• The system (2.1) together with the output (2.2) is said to be ∂Ω-detectable, if there exists an operator H_∂Ω: R^q → H^1/2(∂Ω) such that (A – H_∂ΩC) generates a strongly continuous semi-group (S_H∂Ω (t))_t≥0 which is stable on H^1/2 (∂Ω).
• If a system is ∂Ω-detectable, then it is possible to construct an asymptotic ∂Ω-observer for the original system. If we consider the system

  (3.2)

then x(ξ,t) estimates asymptotically the state z(ξ,t) because the error e(ξ,t) = z(ξ,t)-x(ξ,t) satisfies $\frac{\partial e}{\partial t}(\xi ,t)=(A-H_{\partial \Omega }C)e(\xi ,t)$ with e(ξ,t) = z(ξ,t)-x(ξ,t). Then, if the system is ∂Ω-detectable, it is possible to choose H_∂Ω which realizes $\underset{t\to \infty }{\lim }{\Vert e(.,t)\Vert }_{H^{1/2}(\partial \Omega )}=0$.

Remark 3.1. 

In this paper, we only need the relation (3.1) to be true on a given subregion Γ of the region ∂Ω

(3.3)

we may refer to this as Γ-stability.

• The system (2.1) is said to be Γ-stable, if the operator A generates a semi-group which is stable on H^1/2 (Γ).
• The system (2.1)-(2.2) is said to be Γ-detectable if there exists an operator H_Γ: R^q → H^1/2(Γ) such that (A − H_ΓC) generates a strongly continuous semi-group (S_HΓ(t))_t≥0 which is stable on H^1/2 (Γ).

For more detail [18]. However, one can easily have the following results :

Corollary 3.2. 

If the system (2.1) together with output function (2.2) is exactly Γ-observable, then it is Γ-detectable.

This result leads to:

(3.4)

Thus, the notion of Γ-detectability is a weaker property than the exact Γ-observability as in (Ref. [16].

4. Asymptotic Γ-observer and strategic sensors

In this section, we propose an approach which allows to determinate a regional asymptotic estimator of z(ξ,t) on Γ, based on the internal asymptotic Γ-observer. This method need some definitions concern regional observer as in [11].

4.1. Definitions and characterizations

• Consider the system (2.1) - (2.2) together with the dynamical system

  (4.1)

  where F_ω generates a strongly continuous semi-group (S_Fω(t))_t≥0 which is stable on X = H¹ (ω), i.e. :

  (4.2)

  and G_ω∈L(R^p, H¹ (ω)) and H_ω ∈L(R^q, H¹ (ω)). The system (4.1) defines an asymptotic ω-estimator for χ_ωT(ξ,t) where z(ξ,t) is a solution of the system (2.1) - (2.2) if $\underset{t\to \infty }{\lim }{\Vert x(.,t)-{\chi }_{\omega }T(.,t)\Vert }_{H^1(\omega )}=0$ and χ_ωT maps D(A) into D(F_ω) where x(ξ, t) is the solution of the system (4.1).
• The system (4.1) specifies an ω-observer for the system (2.1) - (2.2) if the following conditions hold:

  • There exists M_ω ∈ L(R^q, L² (ω)) and N_ω ∈ L(L² (ω)) such that M_ωC + N_ωχ_ωT = I_ω.
  • χ_ωTA + F_ωχ_ωT = G_ωC and H_ω=χ_ωTB.
  • The system (4.1) defines an asymptotic ω-estimator for χ_ωT (ξ,t).
• The system (4.1) is said to be an identity ω-observer for the system (2.1) - (2.2) if and Z = X.
• The system (4.1) is said to be a reduced-order ω-observer for the system (2.1) - (2.2) if Z = O ⊕X.

  Figure 2. The considered domain Ω and the subregion ω_r.

  Figure 2. The considered domain Ω and the subregion ω_r.
• The boundary regional observer in Γ may be seen as internal regional observer in ω if we consider the following transformations. Let ℜ be the continuous linear extension operator [19] ℜ: H^1/2 (∂ Ω) → H¹ (Ω) such that

  (4.3)
• Let r>0 is an arbitrary and sufficiently small real and let the sets

  where B(z, r) is the ball of radius r centered in z(ξ,t) and where Γ is a part of ${\overline{\omega }}_r$(Fig. 2).

Proposition 4.1. 

If the system (2.1) - (2.2) is exactly (respectively weakly) ${\overline{\omega }}_r$-observable, then it is exactly (respectively weakly) Γ-observable (see [9]).

From this result, we can deduce the following proposition :

Proposition 4.2. 

The dynamical system (4.1) is Luenberger ${\overline{\omega }}_r$-observer for the system (2.1) - (2.2), then it is Luenberger Γ-observer.

Proof: 

Let x(ξ, t) ∈H^1/2 (Γ) and $\overline{x}$(ξ, t) its extension to H^1/2 (∂Ω) using (4.3) and trace theorem, there exists ℜ$\overline{x}$(ξ, t)∈ H¹ (Ω) with a bounded support such that

(4.4)

Since the system (4.1) is regional observer on ${\overline{\omega }}_r$ so we can deduce that :

The system (4.1) is regional observer on ω_r, there exists a dynamical system with x(ξ,t) ∈ X such that

then we have

(4.5)

The equations (2.2) and (4.5) allow

and there exist two linear bounded operators R and S satisfy the relation

There exists an operator $F_{{\overline{\omega }}_r}$ is regionally stable on ${\overline{\omega }}_r$, then it is regionally stable on Γ [18]. Finally the system (4.1) is a regional boundary observer on Γ. ■

4.2 Sufficient condition for Γ-observer

As in (Refs. [7,18]), we develop a characterized result that links the Γ-observer and strategic sensors and we give a sufficient condition for Γ-observer. For that purpose, we assume that there exists a complete set of eigenfunctions ϕ_n of A in H¹ (Ω), associated to the eigenvalues λ_n with a multiplicity s_n and suppose that the functions ψ_n defined by ψ_n = χ_Γγ₀ϕ_n(ξ), is a complete set in H^1/2 (Γ). If the system (2.1) has J unstable modes, then we have the following theorem.

Theorem 4.3. 

Suppose that there are q zone sensors (D_i, f_i)_1≤i≤q and the spectrum of A contains J eigenvalues with non-negative real parts. If the following conditions are satisfied :

• q > s
• rank G_n = r_n, ∀n, n = 1,..., J with

  where sup s_n = s < ∞ and j = 1,..., s_n. Then the dynamical system

  (4.6)

  is Γ-observer for the system (2.1) - (2.2), i.e. $\underset{t\to \infty }{\lim }{\Vert x(.,t)-T_{\Gamma }z(.,t)\Vert }_{H^{1/2}(\Gamma )}=0$.

Proof : 

The proof is limited to the case of zone sensors in the following stapes :

Step 1. Under the assumptions of section 2, the system (2.1) can be decomposed by the projections P and I - P on two parts, unstable and stable. The state vector may be given by where z₁(ξ,t) is the state component of the unstable part of the system (2.1), may be written in the form

(4.7)

and z₂ (ξ,t) is the component state of the stable part of the system (2.1) given by

(4.8)

The operator A₁ is represented by a matrix of order (${\displaystyle \sum _{n=1}^J{s}_n},{\displaystyle \sum _{n=1}^J{s}_n}$) given

Step 2. The condition (2) of this theorem, allows that the suite (D_i, f_i)_1≤i≤q of sensors is Γ-strategic, the subsystem (4.7) is weakly Γ-observable [16] and since it is finite dimensional, then it is exactly Γ-observable. Therefore it is Γ-detectable, and hence there exists an operator $H_{\Gamma }^1$ such that (A₁ −$H_{\Gamma }^1$c) which is satisfied the following :

and then we have

Since the semi-group generated by the operator A₂ is stable on H^1/2 (Γ), then there exist ${\overline{M}}_{\Gamma }$, ${\overline{\alpha }}_{\Gamma }$ > 0 such that

and therefore ║z(ξ, t)║_H^1/2(Γ) → 0 when t → ∞. Finally, the system (2.1) - (2.2) is Γ-detectable.

Step 3. Let e(ξ,t) = z(ξ,t) − x(ξ,t) where x(ξ,t) is the solution of the system (4.6). Deriving the above equation and using the equations (2.1) and (4.6), we obtain

Since the system (2.1) - (2.2) is Γ-detectable, there exists an operator H_Γ ∈ L(R^q, H^1/2 (Γ)), such that the operator (A − H_ΓC), generates a stable, strongly continuous semi-group (S_HΓ(t))_t≥0 on the space H^1/2 (Γ) which is satisfied the following relations :

Finally, we have

with e₀ (ξ,t) =z₀ (ξ,t) − x₀ (ξ,t) and therefore

Thus, the dynamical system (4.6) a Γ-observer for the system (2.1) - (2.2). ■

Remark 4.4. 

From theorem 4.3., we can deduce the following statements :

• A dynamical system which is an ∂Ω -observer is Γ-observer.
• If a system is Γ-observer, then it is Γ₁ -observer in every subset Γ₁ of Γ, but the converse is not true. This may be proven in the following example:

Example 4.5. 

Consider a two-dimensional system described by the diffusion equation

(4.9)

where Ω=]0,1[×]0,1[. The operator A = Δ generates a strongly continuous semi-group (S_A(t))_t≥0 on the Hilbert space H¹ (Ω) given by

where λ_nm = −(n² + m²)π², ϕ_nm(ξ₁, ξ₂) = 2a_nm cos(nπξ₁) cos(mπξ₂) and 2a_nm = (1 −λ_nm)^−1/2.

Consider the dynamical system

(4.10)

where H ∈ L(R^q, Z), Z is a Hilbert space and C :H¹ ($\overline{\Omega }$) → R^q is a linear operator. Consider the boundary sensor (Γ₀, f) defined by Γ₀ ={0} × ]0,1[ and f (η₁,η₂) = cosπη₂. Thus, the output function can be written by

(4.11)

If the state z₀ is defined in Ω by z₀ (ξ₁, ξ₂) = cos(πξ₁)cos(2πξ₂), then the system (4.9) - (4.11) is not weakly observable in Ω, i.e. the sensor (Γ₀, f) is not strategic and therefore the system (4.9) - (4.11) is not detectable in Ω. Thus, the dynamical system (4.10) is not observer for the system (4.9) - (4.11) (see [6]). Here, we consider the region Γ = ]0,1[×{0} ⊂∂Ω (Fig. 3) and the dynamical system

Figure 3. Domain Ω, region Γ and location Γ₀ of zone sensor.

Figure 3. Domain Ω, region Γ and location Γ₀ of zone sensor.

(4.12)

where H ∈L(R^q, H^1/2 (Γ)) . In this case, the system (4.9) - (4.11) is weakly observable in Γ and the sensor (Γ₀, f) is Γ-strategic [9]. Thus, the system (4.9) - (4.11) is Γ-detectable [13]. Finally the dynamical system(4.9) is Γ-observer for the system (4.9) - (4.11) [18].

5. Application to sensors structures

In this section, we consider the distributed diffusion systems defined on Ω =]0,a₁[×]0,a₂[. We explore various results related to different types of measurements, domains and boundary conditions.

5.1 Case of a zone sensor

We study the following cases :

5.1.1 Rectangular domain

In this case, the system (2.1) - (2.2) is given by

(5.1)

where $\Sigma =\overline{\Omega }\times ]0,\infty [,\mathrm{D}\subset \Omega$ is the location of zone sensor and above system represent the heat-conduction problem (see Ref. [20]). Let Γ = {a₁}×]0, a₂[ be a region of on ]0,a₁[×]0,a₂[ The eigenfunctions and the eigenvalues are given by

(5.2)

and

(5.3)

If $a_1^2/a_2^2\notin Q$, then the multiplicity of s_nm = 1 and one sensor can be guaranteed Γ -strategic sensor (see [21]). The dynamical system (4.6) may be given by

(5.4)

Let the measurement support is rectangular with D = [ξ₁ − l₁, ξ₁ + l₁] × [ξ₂ − L², ξ₂ + L²] ∈ Ω.

If f₁ is symmetric about ξ₁ = ξ₀₁ and f₂ is symmetric with respect to ξ₂ = ξ₀₂, then we have the following result :

Corollary 5.1. 

The dynamical system (5.4) is Γ-observer for the system (5.1) if nξ₀₁ / a₁ and mξ₀₂ / a₂ ∉ N for every n, m =1,..., J.

In the case where Γ ⊂ ∂Ω and f ⊂ L² (Γ), the sensor (D, f) may be located on the boundary in Γ₀ = [η₀₁ − l, η₀₁ + l]×{ a₂ } then we have :

Figure 4. Domain Ω, region Γ and locations D, Γ₀, $\overline{\Gamma }$ of zone sensor.

Figure 4. Domain Ω, region Γ and locations D, Γ₀, $\overline{\Gamma }$ of zone sensor.

Corollary 5.1. 

• One side case : Suppose that the sensor (D, f) is located on Γ₀ =[η₀₁ − l, η₀₁ + l] × { a₂ } ⊂ ∂Ω and f is symmetric with respect to η₁ = η₀₁, then the dynamical system (5.4) is Γ-observer for the system (5.1) if n η₀₁ / a₁ ∉N for every n, m =1,..., J.
• Two side case : Suppose that the sensor (D, f) is located on $\overline{\Gamma }=[0,{\overline{\eta }}_{0_1}+l_1]\times \text{{0}}\cup \left\{0\right\}\times [0,{\overline{\eta }}_{0_2}+l_2]\times \subset \partial \Omega$ and $f_{|{\Gamma }_1}$ is symmetric with respect to η₁ = η₀₁ and the function $f_{|{\Gamma }_2}$ is symmetric with respect to η₂ = η₀₂, then the dynamical system (5.4) is Γ-observer for the system (5.1) if n η₀₁ / a₁ and mη₀₂ / a₂ ∉N for every n, m = 1,..., J.

5.1.2 Disk domain

The system (5.1) may be given by the following form

(5.5)

where 0 < θ_i< 2π, Ω = D(0, a), r = a > 0 and Θ = θ ∈[0, 2π], t > 0 are defined as in (Fig. 5).

Let the eigenfunctions and eigenvalues concerning the region Γ = D(a, θ_i)_2≤i≤q of ∂Ω with θ_i ∈ [0, 2π] are defined by

(5.6)

where β_nm are the zeros of the Bessel functions J_n and

(5.7)

Figure 5. Domain Ω, region Γ and locations p₁, p₂ of internal (boundary) zone sensors.

Figure 5. Domain Ω, region Γ and locations p₁, p₂ of internal (boundary) zone sensors.

with multiplicity s_nm = 2 for all nm ≠ 0 and s_nm = 1 for all nm = 0. In this case, the Γ–strategic sensor is required at least two zone sensors (D_i,f_i)_2≤i≤q where D_i =(r_i,θ_i)_2≤i≤q (see [6]). The dynamical system (5.4) can be written by

(5.8)

If f_i and D_i are symmetric with respect to θ = θ_i, for all 2 ≤i ≤ q, then we have :

Corollary 5.3. 

The dynamical system (5.8) is Γ-observer for the system (5.5) if n(θ₁ − θ₂)/π ∉ N for every n, n = 1,..., J.

When the sensors (Γ_i,f_i)_2≤i≤q are located on ∂Ω and the function f_|Γi is symmetric with respect to θ = θ_i, 2 ≤ i ≤ q as in (Fig. 5). So, we have.

Corollary 5.4. 

The dynamical system (5.8) is Γ-observer for the system (5.5) if n(θ₁ − θ₂) /π ∉ N for every n, n =1,..., J.

5.2 Case a pointwise sensor

In this subsection, we consider the following cases :

5.2.1 The domain Ω =]0,a₁[×]0,a₂[

Now the system (5.5) is given by the form

(5.9)

If b = (b₁,b₂) ∈∂Ω then, we have :

Figure 6. Rectangular domain, region Γ and locations b, σ of pointwise sensors.

Figure 6. Rectangular domain, region Γ and locations b, σ of pointwise sensors.

Corollary 5.5. 

• Internal case : If nb₁ / a₁ and mb₂ / a₂ ∉N for every n, m = 1,..., J, then the dynamical system (5.4) is Γ-observer for the system (5.9).
• Filament case : Suppose that the observation is given by the filament sensor where σ = Im(γ) is symmetric with respect to the line b = (b₁,b₂), if nb₁ / a₁ and mb₂ / a₂ ∉N for every n, m = 1,..., J, the dynamical system (5.4) is Γ-observer for the system (5.9).
• Boundary case : If mb₂ / a₂ ∉N for every m = 1,..., J, then the dynamical system (5.4) is Γ- observer for the system (5.9).

5.2.2 The domain Ω = D(0,a)

The system (5.5) may be given by the following form

(5.10)

Where 0 ≤ θ_i ≤ 2π. The sensors may be located in p₁ =(r₁,θ₁) and p₂ =(r₂,θ₂) ∈ Ω (or in p_i = (a,θ_i) ∈Ω) (Fig. 7).

Figure 7. Disc domain, region Γ and locations p₁, p₂ of internal (boundary) pointwise sensors.

Figure 7. Disc domain, region Γ and locations p₁, p₂ of internal (boundary) pointwise sensors.

Corollary 5.6. 

• If the sensors (p_i,δ_pi)_2≤i≤q are located in p_i =(r_i,θ_i) and n(θ₁ − θ₂)/π ∉ N for every n, n =1,..., J, then the dynamical system (5.8) is Γ-observer for the system (5.10).
• If the sensors (p_i,δ_pi) are located in p_i =(a,θ_i) _2≤i≤q and n(θ₁ − θ₂)/π ∉ N for every n, n =1,..., J, then the dynamical system (5.8) is Γ-observer for the system (5.10).

6. Conclusion

The concept studied in this paper is related to the case of identity Γ-observer in connection with the sensors structure. For this class of parabolic distributed parameter systems, many interesting results concerning the choice of the sensor are explored and illustrated in specific situations of the domain. In similar way, we can characterize the general case (reduced-order) Γ-observer by sensors structure. An important extension of these results, related to the problem of regional boundary gradient observer, in connection with the structure of sensors; is under consideration.