High-Gain Observer Design for Nonlinear Systems with Delayed Output Measurements using Time-Varying Gains

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High-Gain Observer Design for Nonlinear Systems with Delayed Output Measurements using Time-Varying Gains

Item Type Conference Paper

Authors Adil, Ania;Ndoye, Ibrahima;Laleg-Kirati, Taous-Meriem Citation Adil, A., N’Doye, I., & Laleg-Kirati, T.-M. (2022). High-Gain

Observer Design for Nonlinear Systems with Delayed

Output Measurements using Time-Varying Gains. 2022 IEEE 61st Conference on Decision and Control (CDC). https://

doi.org/10.1109/cdc51059.2022.9992555 Eprint version Post-print

DOI 10.1109/CDC51059.2022.9992555

Publisher IEEE

Rights This is an accepted manuscript version of a paper before final publisher editing and formatting. Archived with thanks to IEEE.

Download date 2023-11-01 19:33:47

Link to Item http://hdl.handle.net/10754/687020

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High-Gain Observer Design for Nonlinear Systems with Delayed Output Measurements using Time-Varying Gains

Ania Adil^1,2, Ibrahima N’Doye², Taous-Meriem Laleg-Kirati^2,3

Abstract— This paper proposes a high-gain observer design for nonlinear systems with delayed output measurements using time-varying gains. The proposed observer is endowed with an exponential stability guarantee and relies on the generalization of the Halanay-type inequalities. We establish that the estimated state and the adapted gain are exponentially bounded and prevent the oscillatory response of the estimates. The time- varying gain feature limits the constant high-gain values of the standard high-gain observer design to the minimum gain required to achieve stability. Furthermore, we derive an explicit relation between the maximum bound of the delay and the maximum gain parameter by using a Lyapunov-Krasovskii functional jointly with the time-varying Halanay inequality.

Finally, a comparison with the standard high-gain observer is provided through numerical simulations to demonstrate the superiority of the proposed high-gain observer in rejecting the noise and reducing the peaking phenomena.

Index Terms— High-gain observer, nonlinear triangular systems, disturbance rejection, time-varying gain, peaking atten- uation, nonlinear observer.

I. INTRODUCTION

Time-delay is widely involved in various engineering and physical applications, such as industrial processes, com- munication networks, ecological systems, and biomedical engineering [1], [2]. The presence of delay may affect the system’s performance and control design. In particular, when the system’s state is not accessible, the estimation process is challenging when there is a time delay. This problem has increased the interest of the control community [3], [4].

Various state estimation methods for nonlinear systems with time-delay have been developed in the literature. For instance, different types of observers have been extended from standard nonlinear systems to time-delay systems such as high-gain observers [5], chain of observers [6], and pre- dictors [7]. The high-gain observer is particularly interesting thanks to its easy implementation [8]. However, the high-gain observer structure relies on high gain values, which leads to numerical issues [9]. Additionally, it exhibits the peaking phenomenon during the transient and is highly sensitive to output disturbances (measurement noise, delayed outputs,

1,2Ania Adil is with Laboratoire de Math´ematiques Pures et Appliqu´ees (LMPA) Mouloud Mammeri University of Tizi-Ouzou, Tizi-Ouzou, BP No 17, RP 15000, Algeria and with Computer, Electrical and Mathematical Science&Engineering Division (CEMSE), KAUST, Saudi Arabia (email:

[email protected])

2Ibrahima N’Doye and Taous-Meriem Laleg-Kirati are with Computer, Electrical and Mathematical Science &Engineering Division (CEMSE), KAUST, Saudi Arabia. ³Taous-Meriem Laleg-Kirati is also with The National Institute for Research in Digital Science and Technology (INRIA), Paris-Saclay, France. (email: [email protected], [email protected])

sampled data, etc.). To improve the performance of the high- gain observer, several solutions have been proposed [10]–

[12]. The standard high-gain observer has been enlarged by considering time-varying gain instead of static gain. Time- varying gain-based observers have been introduced in [13], [14] for nonlinear systems. The advantages of such structures are that they allow for better convergence properties of the observers and ensure some robustness [15]–[18]. However, in the presence of a delay in the measured output, few works are developed in the literature. For instance, an exponential observer with time-varying delayed measurement and delay- dependent gain for triangular systems is derived in [19], using the Razumikhin method. Note that the Lyapunov Razumikhin conditions used in [19] to prove the convergence to zero of the observer error might provide conservative gain conditions than the Lyapunov-Krasovskii method [2].

In this work, we propose a high-gain observer design for nonlinear systems with delayed output measurements using time-varying gains. Additionally, the time-varying dynamic gain relies on the maximum allowable value of the time delay and the Lipschitz constant. Indeed, the idea is to maintain the time-varying gains at small values that offers the possibility to shape the value of the tuning high-gain parameter while ensuring exponential convergence for larger values delay through Lyapunov-Krasovskii functional and a time-varying Halanay inequality. The Halanay inequality technique introduces essential features for the analysis of time-delay systems. For instance, it overcomes differentiating the time delays and provides sufficient conditions for time- delays systems’ stability. Furthermore, the dynamic time- varying does not depend on the output injection error in the adaptive sense. Finally, the observer design is derived in a non-adaptive scheme, which eases the practical numerical implementation.

The rest of the paper is organized as follows. Section II presents the class of systems under consideration and the problem formulation. Section III is devoted to con- structing the high-gain observer with time-varying gains and preliminary results for the time-varying Halanay inequalities. Section IV provides the main results of the high- gain observer and its proof, along with the time-varying Halanay inequalities guaranteeing sufficient conditions of the observer estimation error to be exponentially stable. A numerical example comparing the proposed observer to the standard high-gain observer is provided in Section V. Finally, the paper ends with a conclusion summarizing the main contributions and some future works.

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II. PROBLEM FORMULATION

In this paper, we consider the following class of nonlinear systems with delayed output, described by:

x(t) =˙ Ax(t) +f(x(t))

y(t) =Cx(t−τ(t)), (1)

where the matricesA andC are defined by C=

1 0 . . . 0

, (A)i,j =

1 if j=i+ 1, 0 if j̸=i+ 1.

and

f(x(t)) =







f1(x1) f2(x1, x2)

...

fn−1(x1, x2, . . . , xn−1) fn(x1, . . . , xn)







where x(t) ∈ Rⁿ is the state vector of the system and y(t)∈Ris the measured output. We assume that τ(t) is a known time-varying delay satisfying

0< τ(t)⩽τM.

whereτ_M is the maximum allowable delay. The nonlineari- tiesf_i:Rⁱ→Rsatisfy the following Lipschitz property:

|fi(x1, . . . , xi)−fi(¯x1, . . . ,x¯i)|⩽γf_i i

X

j=1

|xj−x¯j|, (2) whereγf_i is the Lipschitz constant.

The main objective of this paper is to design a high- gain nonlinear observer for system (1) with time-varying gains that ensure exponential stability of the estimation error.

Indeed, the maximum bound of the delay required to ensure exponential convergence is increased. Hence, we derive an explicit relation between the maximum bound of the delay and the maximum gain parameter by using a Lyapunov- Krasovskii functional jointly with the time-varying Halanay inequality.

In the following, we recall some useful inequalities used in the proof of the main results.

Lemma 1 (Jensen’s Inequality): [20] For any constant symmetric and positive definite matrix M ∈R^n×n, scalars t₁, t₂ and vector function Υ : [t₁, t₂] → Rⁿ, then the following inequality holds

Z t₂ t₁

Υ(s)ds

⊤M Z t₂

t₁

Υ(s)ds

⩽(t2−t1) Z t2

t₁

Υ^⊤(s)MΥ(s)ds

. Lemma 2 (Young’s Inequality): [21] Let X and Y be two matrices of appropriate dimensions. Then, for every invertible matrixS and scalarη >0, we have

X^⊤Y +Y^⊤X ⩽ηX^⊤SX+1

ηY^⊤S⁻¹Y.

III. DESIGN APPROACH

In this section we present the proposed high gain observer design for system (1) and its proof of convergence. We consider the following observer:

˙ˆ

x(t) =Aˆx(t) +f(ˆx(t)) + Γ⁻¹K(y(t)−xˆ1(t−τ(t))), (3) where

Γ =diag{1/µ^1+m, . . . ,1/µ^n(1+m)},

is a scaling matrix,mis a design parameter such thatm⩾1 and,µ(t)is the time-varying function gain to be determined later.

In this approach, the dynamics of the estimation errore(t) = x(t)−x(t)ˆ is given as follows:

˙

e(t) =Ae(t) + ∆f−Γ⁻¹Ke1(t−τ(t)) (4) where

∆f :=f(x)−f(ˆx). (5) To prove the stability of system (4), the following transformation is introduced:

¯

e(t) = Γ(t)e(t). (6)

The dynamics of the transformed error are given as follows:

˙¯

e(t) =µ^1+m(t)(A−KC)¯e(t)−(1 +m)µ(t)˙ µ(t)D¯e(t)

+ Γ∆f+µ^1+m(t)KC(¯e(t)−¯e(t−τ(t))) (7) where D is a diagonal matrix defined as D = diag{1,2, . . . , n}.

First, we introduce an intermediate Halanay result in lemma 3 and its corollary that aims to improve the clarity and readability of the main theorem.

Lemma 3 (Halanay’s result): [22], [23] Consider the following differential inequality

˙

v(t)⩽−a(t)v(t) +b(t) sup

s∈[t−τ(t),t]

v(s), t⩾t0 (8) where t ∈ [0,+∞), a(.) ∈ C([0,+∞),R) and b(.) ∈ C([0,+∞),[0,+∞)) are continuous and locally bounded functions, v(.)∈ C([−τ,0]S[0,+∞),[0,+∞)), and τ⩾0 is a constant denoting the delay.

Assume that ϕa−b(τ) = sup

t∈[−τ,∞)

max

θ∈[0,τ]

Z t+θ t

(a(s)−b(s))ds

(9) is well defined. Let

Z t t0

ψ(s)ds= Z t

t0

b(s)e^ϕ^a−b^(τ)−a(s) ds <0, i.e.ψ(t)is uniformly asymptotically stable. Then the solution to the differential inequality satisfies

v(t)⩽exp Z t

t0

ψ(s)ds

v(t0),∀t⩾t0, t0∈[0,+∞), (10) hencev(t)converges exponentially to zero as t→+∞.

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Corollary 1: [24] Leta(t)andb(t)denote non-negative, continuous and bounded functions and that there exists a constantε >0 such that

a(t)−b(t)> ε, ∀t⩾t0, (11) whereε= inf a(t)−b(t)

>0. Then there exists a positive constantρ¯such that

v(t)⩽ sup

s∈[t₀−τ_M,t₀]

v(s)

!

e⁻^ρ(t−t^¯ ⁰⁾, ∀t⩾t0, (12) hencev(t)→0 as t→+∞.

IV. MAINRESULT

This section presents the main result that ensures sufficient conditions for the exponential convergence of the observer (3).

Theorem 1: Let the solutions t → (x(t),x(t), µ(t))ˆ cor- responding to systems (1), (3), and (16) and the initial pairs (ˆx(t0), µ(t0))exist and are bounded. Assume that there exist a symmetric positive definite matrix P, a matrix Y with appropriate dimension and real positive constants m,c,λ1, λ2, andε¯such that the following conditions hold:





 Hen

P A− Y^⊤Co

+λ₁I Y^⊤

Y −η1





⩽0, (13) D^TP+P D−λ2In⩾0, (14)

τ_M < 1 ρ1µ^1+m∗

, (15) with

˙

µ(t) =ν λ_max(P)

λ2(1 +m)µ(t)− λ₁

λ2(1 +m)µ^2+m(t), (16) where

ν =h2γ_fλ_max(P) +τ_M λmax(P) −ci

, µ^∗= sup(µ(t)), µ(t₀)⩾1, ρ1=γ_f²₁η1

1 +η2

1 + 1

η3

, andHen

So

:=S+S^⊤.

Then the observer error is exponentially stable, and the observer’s gain is found equal to:

K=P⁻¹Y^⊤ = [K1. . . Kn]^⊤.

Proof 1: Consider the following Lyapunov function can- didate

V(¯e(t)) = ¯e(t)^TPe(t)¯

| {z }

V₁(t)

+ Z t

t−τM

Z t ξ

(¯e₁(s))²dsdξ

| {z }

V₂(t)

, P >0

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The derivative ofV1 can be calculated as

∂V1(t)

∂t =µ^1+m(t)¯e^T(t)h

(A−KC)^TP+P(A−KC)i

¯ e(t)

−(1 +m)µ(t)˙ µ(t)¯e^T(t)h

D^TP+P Di

¯

e(t) + 2¯e^T(t)P(Γ∆f) +µ^1+m(t)¯e^T(t)P KC

¯

e(t)−¯e(t−τ(t)) +µ^1+m(t)

¯

e(t)−e(t¯ −τ(t))T

C^TK^TP¯e(t) (18)

Hence, by applying Jensen’s inequality and upper-bounding the derivative ofV₁, we obtain (19) whereη₁, η₂, η₃ andη₄ come from the application of Young inequality.

The derivative ofV2along the trajectories (7) is given by

∂V2(t)

∂t =τ_M(¯e₁(t))²− Z t

t−τ_M

(¯e₁(s))²ds. (20)

Let

(A−KC)^TP+P(A−KC) + 1 η1

Y^TY ≤ −λ1I, D^TP+P D≥λ2I,

ρ₁=γ_f²

1η₁

1 +η₂ 1 + 1

η3

, ρ2=η1

1 + 1

η2

, ρ₃=η₁

1 +η₂

1 +η₃ 1 + 1

η₄

K₁², ρ4=η1

1 +η2

1 +η3

1 +η4

,

Thus the derivative ofV becomes along the trajectories (7)

∂V(t)

∂t ⩽−λ1µ^1+m(t)¯e^T(t)¯e(t)−λ2(1 +m)µ(t)˙

µ(t)e¯^T(t)¯e(t) +τM(¯e1(s))²+ 2¯e^T(t)P(Γ∆f)

+

ρ1µ^1+m(t)τM −1 Z t

t−τ_M

¯ e1(s)2

ds +ρ2µ^1+m(t)τM

Z t t−τ_M

(µ^1+m(s)¯e2(s))²ds +ρ3µ^1+m(t)τM

Z t t−τ_M

µ^1+m(s)¯e1(s−τ(s))2

ds

+ρ4µ^1+m(t)τM

Z t t−τ_M

(1 +m)µ(s)˙ µ(s)e¯1(s)

2

ds (21)

Through the use of the Lipschitz property off, we have 2¯e^T(t)P(Γ∆f)⩽2¯e^T(t)P[max

i γf_inΓ(x−x)]¯

⩽2γfe¯^T(t)PΓ˜x⩽2γfλmax(P)¯e^T(t)¯e(t)

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∂V1(t)

∂t ⩽µ^1+m(t)¯e^T(t)

A^TP+P A−C^TY − Y^TC+ 1 η1

Y^TY

¯ e(t)

−(1 +m)µ(t)˙ µ(t)¯e^T(t)h

D^TP+P Di

¯

e(t) + 2¯e^T(t)P(Γ∆f)

+η1µ^1+m(t) 1 + 1

η2

τ_M

Z t t−τ_M

(µ^1+m(s)¯e2(s))²ds +γ²_f₁η1µ^1+m(t)

1 +η2

1 + 1

η3

τM

Zt t−τ_M

¯ e1(s)2

ds +η1µ^1+m(t)

1 +η2

1 +η3

1 + 1

η4

τMK₁²

Z t t−τ_M

µ^1+m(s)¯e1(s−τ(s))2

ds

+η1µ^1+m(t)

1 +η2

1 +η3

1 +η4

τM

Z t t−τ_M

(1 +m)µ(s)˙ µ(s)e¯1(s)

2

ds (19)

whereγ_f = max_iγ_f_in. We obtain the following inequality

∂V(t)

∂t ⩽− 1

λmax(P)

λ1µ^1+m(t) +λ2(1 +m)µ(t)˙ µ(t)

−2γfλmax(P)−τM

i V1(t)−

1

τ_M −ρ1µ^1+m(t)

V2(t) +ρ2µ^1+m(t)τM

Z t t−τM

(µ^1+m(s)¯e2(s))²ds +ρ₃µ^1+m(t)τ_M

Z t t−τM

µ^1+m(s)¯e₁(s−τ(s))² ds +ρ4µ^1+m(t)τM

Z t t−τM

(1 +m)µ(s)˙ µ(s)e¯1(s)

2

ds (22) Thus we get

d

dtV(t)⩽−α(t)V(t) +β(t) sup

[t−2τM,t]

V(s), where

α(t)≜min

α1(t);α2(t) ,

α1(t) = λ1

λ_max(P)µ^1+m(t) +λ2(1 +m) λ_max(P)

˙ µ(t) µ(t)

−2γfλmax(P) +τM

λmax(P) , α₂(t) = 1

τM

−ρ₁µ^1+m(t), and

β(t)≜(ρ₂+ρ₃)µ^1+m(t)τ_M Z t

t−τ_M

(µ^1+m(s))²ds +ρ4µ^1+m(t)τM

Z t t−τM

(1 +m)µ(s)˙ µ(s)

2

ds.

Since µ(t)⩾0 is non-increasing and bounded function, we deduce thatα₂(t)⩽α₁(t), hence by assuming that

α2(t)−β(t)>ε >¯ 0, (23) and using the fact that

λmin(P)(¯e(t))²⩽V(t)⩽λmax(P)(¯e(t))², (24)

then, there existρ¯such that

∥¯e(t)∥⩽ sup

s∈[t0−τM,t₀]

s V(s) λ_min(P)

!

e⁻^ρ(t−t^¯ ⁰⁾, (25) which is equivalent to

∥¯e(t)∥⩽ sup

s∈[t0−τM,t0]

s

λ_max(P) λmin(P)∥¯e(s)∥

!

e^−¯^ρ(t−t⁰⁾. (26) Moreover, by using the state transformation (6) and inequality (26), we have

∥e(t)∥⩽nν₁µ(t)^n(1+m) sup

s∈[t0−τM,t0]

∥¯e(s)∥

!

e^−¯^ρ(t−t⁰⁾, (27) whereν₁=

sλmax(P)

λ_min(P). This ends the proof.

Remark 1: The condition (23), which refers to the hy- pothesis related to the constant bound, has been introduced in [24]. It assumes that the time-varying features of α₂(t) and β(t) parameters are implicitly removed. Indeed, this condition is not restrictive in this work. It is due to the bound of the time-varying gain solution µ(t) and can be numerically computed along with (16). Indeed, this condition is tailored to the analysis of the high-gain observer-like design leading to essentially rejecting the measurement noise and reducing the peaking. Additionally, this condition (23) on ε¯is required to ensure the exponential convergence to zero ofe(t), according to (26).¯

Remark 2: The error trajectory has a given decay rate ρ¯ that can be chosen arbitrarily to be large enough to guarantee fast exponential convergence. One might also design a gener- ating function that encompassesε¯for the prespecified decay rate ρ¯of the time-varying Halanay inequality to provide an optimal trade-off betweenρ¯andε.¯

Remark 3: The time-varying dynamic gain µ(t)given in (16) relies on the maximum allowable value of the time delay τM and the Lipschitz constant γf. Therefore, the idea is to maintain the time-varying gain at small values that offers the possibility to shape the value of the tuning high-gainK while ensuring exponential convergence for larger values of

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TABLE I: Coupled tanks model’s parameters

Symbol Description Value Unit

V_p Pump voltage 12 V

K_p Pump flow constant 3.3 cm³/s/V

D_o1 Tank1outlet Diameter 0.635 cm

D_t1 Tank1inside diameter 4.445 cm

A_o1 Tank1outlet section area 0.3167 cm² At1 Tank1inside cross-section area 15.1579 cm²

Do2 Tank2outlet Diameter 0.45625 cm

Dt2 Tank2inside diameter 4.445 cm

Ao2 Tank2outlet section area 0.1781 cm² At2 Tank2inside cross-section area 15.1579 cm² g Gravitational constant on earth 981 cm/s²

the delay and fixed Lipschitz constant. Additionally, equation (16) does not depend on the output injection error in the adaptive sense. Hence, the observer design is derived in a non-adaptive scheme, which eases the practical numerical implementation.

V. NUMERICAL SIMULATIONS

In this section, we present a numerical example to show the performance of the proposed observer design. The simulations will be performed using MATLAB and YALMIP. The objective of this example is to demonstrate the effectiveness of the proposed observer in reconstructing the coupled tanks’

water levels with a comparison to the standard high-gain observer.

The dynamics in tanks 1 and 2 are described by the following equations [25]:





 dL1

dt =−Ao1

At1

p2gL1+ Kp

At1

Vp

dL2

dt =Ao1

At2

p2gL1−Ao2

At2

p2gL2

(28)

where the water levels in tank1and2are described byL1, L2, respectively.Vpstands for the pump voltage. The output of system (28),y, is the water level in tank2. The physical parameter’ values of system (28) are stated in Table I. To design the high-gain observer, the coupled-tanks system (28) is first transformed into an observable canonical form [25], represented by system (1) where:

f(x(t)) =

0 ϕ(x1, x2)

,

ϕ(x1, x2) =− A²_o1g At1At2

1 + KpVp

At2x2+Ao2

√2gx1

−Ao2g At2

x2

√2gx1

.

Then, the proposed observer can be established as follow x˙ˆ1(t) = ˆx2(t) +µ^1+mK1(x1(t−τ(t))−xˆ1(t−τ(t)))

˙ˆ

x2(t) =ϕ(ˆx1,xˆ2) +µ^2(1+m)K2(x1(t−τ(t))−xˆ1(t−τ(t))) wherexˆ₁ andxˆ₂ are the state estimates. The functionµ(t) is found by solving the differential equation (16) online. The curves ofµ(t)are illustrated in Fig 1. By using Matlab and YALMIP, the gainK is found equal to

K1= [2.3037 2.1376]^T. (29)

We compare the proposed observer to the standard high-gain observer [25], which can be constructed as follow

x˙ˆ1(t) = ˆx2(t) +θK1(x1(t−τ(t))−xˆ1(t−τ(t))) x˙ˆ2(t) =ϕ(ˆx1,xˆ2) +θ²K2(x1(t−τ(t))−xˆ1(t−τ(t))) The observer gain and design parameter are found equal to

K₂= [0.0430 0.0008]^⊤, θ= 15637. (30) By using Matlab, the maximum bound of delay obtained by the proposed observer is found equal toτM = 0.001s, which is considerably increased fromτM = 6.8166×10⁻¹¹swith the standard high-gain observer.

We denote by x(t)ˆ = [ˆx1,xˆ2]^⊤ and xˆSHG(t) = [ˆx_1,SHG,xˆ_2,SHG]^⊤ the state estimates for the system by using the observer design method proposed in the present paper (29) and the standard high-gain observer, respectively.

Letx(0) = [5,5]^⊤, x(0) = [4,ˆ 4]^⊤ and xˆ_SHG(0) = [4,4]^⊤. The behaviours of x_i and its estimates xˆ_i,xˆ_i,SHG,i= 1,2 are illustrated in Fig. 2. We can observe that the proposed observer converges to the original system and reduces the peaking phenomenon compared to the standard high-gain observer, thanks to the time-varying gain properties. In Fig 3,

0 0.5 1 1.5 2 2.5 3

Time [sec]

0 1 2 3 4

Fig. 1: The time-varying gainµ(t)

0 0.5 1 1.5 2

Time [sec]

0 2 4 6 8

0 0.02 0.04

4 5 6

0 0.5 1 1.5 2

Time [sec]

-100 0 100 200 300 400

0 0.01 0.02

0 100 200 300

Fig. 2: Behaviour ofx1 andx2 and their estimates we show the behavior of the proposed observer when the measured output is subject to noise. The simulations are carried out by considering an additive white Gaussian noise.

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The proposed observer states converge to the actual states despite the noise. This latter is attenuated thanks to the time-varying gain. This demonstrates the effectiveness of the proposed observer in rejecting the measurement noise.

0 0.5 1 1.5 2 2.5 3

Time [sec]

0 2 4 6 8 10

1 1.5

6.8 7 7.2

0 0.5 1 1.5 2 2.5 3

Time [sec]

0 2 4 6 8 10 12

1 1.5

1.15 1.2

Fig. 3: Behaviour ofx1andx2and their estimates with white Gaussian measurement noise chosen.

VI. CONCLUSION

In this paper, we investigated the problem of observer design for a class of nonlinear systems with time-varying delayed output measurements. We proposed a high gain observer with time-varying gains. We showed how the time- varying gains limit the constant high-gain values of the standard high-gain observer design to the minimum gains required to achieve stability. This feature eases its numerical implementation and overcomes the peaking phenomenon, one of the main drawbacks of the standard high-gain observer design. Finally, we observe through simulations that the proposed time-varying gain structure provides good performance results in terms of robustness to measurement noise.

ACKNOWLEDGMENT

This work has been supported by the King Abdullah University of Science and Technology (KAUST), Base Re- search Fund (BAS/1/1627-01-01) to Taous-Meriem Laleg- Kirati. Ania Adil thanks the Direction Générale de la Recherche Scientique et du Développement Technologique DGRSDT/MESRS-Algeria for the financial support.

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