Elodie Péronnet

Marie-Laetitia Pastor
email: marie-laetitia.pastor@iut-tarbes.fr

Richard Huillery

Olivier Dalverny

Sébastien Mistou

Hélène Welemane

NON DESTRUCTIVE INVESTIGATION OF DEFECTS IN COMPOSITE STRUCTURES BY THREE INFRARED THERMOGRAPHIC TECHNIQUES

Keywords: Non Destructive Testing, non-contact technique, full-flied measurements, infrared thermography, sandwich and monolithic composite structure

This paper investigates full-field measurement techniques based on Infrared Thermography for Non Destructive Testing (NDT) applications on composite materials. Three methods have been implemented and the paper intends to characterize and compare their defect detection limit and related specific application fields. Various composites have been considered in this study, namely laminates and sandwich structures, in order to address many important issues of performance assessment for the aviation industry.

INTRODUCTION

In order to achieve outstanding performance, it now seeks to optimize more and more the design and process of composite structures. Many applications require specific technical inspections at various steps of the product lifetime to assess their structural health. In such a context, NDT offer an interesting and appropriate tool for the analysis of structural parts, from manufacturing to service conditions.

For aviation industry, the AITM standards are precisely based on the NDT ultrasonic testing for the validation of composite structures. However, such technique is very restrictive in terms of used transducers and inspected part shape (local measurement technique). Accordingly, Infrared (IR) Thermography is more widely used in recent years for structural investigation: full-field measurements (global inspection), non-contact technique, fast execution and analysis (2D mapping in one shot) are significant benefits for this technique [START_REF] Meola | Non-destructive evaluation of aerospace materials with lock-in thermography[END_REF]. This work intends to characterize and to compare three different Infrared Thermographic techniques (IR, Lock-in IR and Pulse IR) on laminated and sandwich composite specimens with artificial defects. Such investigation allows to demonstrate the abilities of these NDT methods. Precisely, a specific attention is given to the following points:

 the defect detection limit, in order to know the equipment limits in relation with the critical defect size determined by aviation rules (namely 6 mm diameter),  the specific material structures for each technique,  their advantages and limitations for industrial applications.

INFRARED THERMOGRAPHIC TECHNIQUES

IR Thermography

The principle of the IR thermography is to send a heat flow on the surface of an inspected composite part during a certain time and then to capture the resulting thermal response. The sample is stimulated by a heat flow generated by halogen lamp with distribution that can be considered as square pulse stimulation and during a time depending on the material, its thickness and emissivity [START_REF] Péronnet | Caractérisation et comparaison des limites de détection de techniques de contrôle non destructif : méthodes ultrasonores et méthodes optiques[END_REF]. The full-filed measurement is carried out on the sample surface with an infrared camera leading in one shot to the film of the surface thermal mapping.

This technique can be used with two experimental configurations [START_REF] Dattoma | Thermographic investigation of sandwich structure made of composite material[END_REF][START_REF] Péronnet | Caractérisation et comparaison des limites de détection de techniques de contrôle non destructif : méthodes ultrasonores et méthodes optiques[END_REF]). On the one hand, the transmission setup consists in having the heat source on the opposite side of the camera; the measure obtained is then the heat flow which has passed through the material and which is perturbed by internal defects (Fig. 1). For the reflection device on the other hand, the heat source and the camera stand on the same side; the thermal mapping given by the camera corresponds therefore to the flow reflected by internal heterogeneities and the material surface which increases the measurement noise (Fig. 2). In both cases, the analysis of thermal mapping enables to highlight defects within the composite material (Fig. 3). In our study, the following equipment has been used: 2 halogen lamps 120V 1000W, PULSAR signal amplifier, a CEDIP Infrared camera JADE retrofitted in FLIR Titanium (20 mK resolution, lens : MW 50 mm 2.0 Jade), ALTAIR software (for film capture and results visualization/analysis).

Lock-in IR Thermography

Lock-in IR thermography relies on the same methodology but is based on a different kind of thermal stimulation. Precisely, the heat source signal exhibits a sinusoidal form, whose wave frequency depends on the specimen thickness [START_REF] Choi | Quantitative determination of a subsurface defect of reference specimen by lock-in infrared thermography[END_REF].

The electrical connection diagram of the Lock-in IR thermography is detailed on Fig. 4. The heat source signal parameters, namely the frequency and amplitude of the sinus function, are defined by means of the signal generator. At the same time, these data are provided to the IR camera in order to capture the module and the phase of the signal response. In addition to the previous ones, specific equipment has been needed for such methodology: HAMEG signal generator, ALTAIR LI software (for film capture).

Fig. 5 shows the experimental device used in the laboratory. To combine a full-field detector (IR camera) with a pulse heat excitation allows a quick control of large composites. For example, a multilayered composite of 2 mm can be controlled in 2 seconds.

Here the following equipment has been employed (Fig. 6):

2 flash lamps 230V 1500J, A FLIR Infrared camera SC7500 (20 mK resolution), ALTAIR software, THERMO-CND software (supplier : THERMOCONCEPT). For all these specimens, the three full-field measurement methods based on Infrared Thermography have been applied. For each technique, the choice between transmission or reflection set-up has been done according to the considered material in order to get the best representation of heterogeneities. The study of A sample demonstrates the significant impact of the material thickness for infrared thermography. Indeed, the specimen exhibits a variable thickness (from 4 mm to 8 mm) and we can note a clear limitation of IR thermography and Pulse IR thermography over 4 mm thick (even with reflection setup). For Lock-in IR thermography, the use of a sinusoidal signal with low frequency (0.05Hz) allows to optimize the wave spread in the specimen thickness and provides better results with the detection of five defects with transmission setup. Concerning the 4 mm thick part, all the methods applied in transmission appear as powerful for aviation inspection (only the 2 mm diameter defects are not identified, which is much less than the limit of 6 mm required for reliable assessment). In this case, the Lock-in IR and the Pulse IR thermography seem more efficient for quantitative investigations since the defects outline can be more clearly distinguished. For such kind of material, the most accurate and powerful method is then the Lock-in IR thermography.

Results are quite similar between the three techniques in term of defect detection limit and precision for B specimen. Again, transmission methodology is used for such thin laminates (thickness less than 3 mm) and results clearly show the better identification of defects located closer to the camera. Between all the techniques, the Lock-in IR thermography leads to the best mapping quality since the choice of a low frequency signal allows reducing the measurement noise. Moreover, this technique is the only one able to capture the porosity on the specimen surface induced during the manufacturing phase.

Concerning foam-core composites (C and D samples), the use of infrared thermography is quite difficult since the foam is a thermal insulator. Precisely, one should rely in such case on the reflection methodology to control the two skins of the sandwich specimens. Each skin is inspected each in turn. Accordingly, the different defects are not as well detected as for A or B samples due to the large part of the noise measurement. We note that results for the D sample are better than for the C specimen, precisely concerning IR thermography and Lock-in IR thermography whereas the pulse IR thermography method have similar results for these two foam core composite structures and is the only one able to detect the majority of defects. In order to improve the result evaluation, we can make image processing which can delete the heat spot but may induce the loss of some important information such as the presence of small size defect. For such materials, the most accurate and efficient technique seems then to be the Pulse IR thermography thanks to a more powerful thermal source.

CONCLUSION

Thermography has allowed considerable progress for Non Destructive Testing in composite materials. This study intended to compare three methods based on such principle (IR, Lock-in IR and Pulse IR) on different kind of composite structures representative of aviation industry.

The three IR thermography methods presented in this work lead to quite similar results in terms of accuracy and efficiency concerning aviation industry monolithic composite part (2 mm or 3 mm of thickness). Above all, the Lock-in IR thermography with low frequency signal provides least noisy results.

For monolithic composite structure with bigger thickness (the range investigated is from 4 mm to 8 mm), the most suitable method is also the Lock-in IR since the optimization of the wave propagation allows a better crossing through the specimen.

Concerning foam core composite structure, the most accurate and efficient technique still remains the pulse IR thermography whose high power heat source manages to capture the material heterogeneities.

Soon, we will test more powerful halogen lamps in order to improve IR and Lock-in IR thermography results. The aim is to increase the heat flow to reduce the significant impact of the material thickness for infrared thermography and to obtain more clearly defects outline. These results allow us to apply image processing to size identified defects distinguishing clearly the image background, the measurement noise and small size defects.

  Fig. 1 : Transmission setup

Fig. 4 :

 4 Fig. 4 : Connection diagram of Lock-in IR thermography

Fig. 5 :

 5 Fig. 5 : Lock-in IR thermography experimental device3. Pulse IR ThermographyAs before, Pulse IR thermography differs from the others by the type of excitation. A pulse excitation is sending to the inspected surface of the sample, the main advantage of this heating is to stimulate all the frequencies of the material during a very short time[START_REF] Ibarra Castanedo | Quantitative subsurface defect evaluation by pulsed phase thermography : depth retrieval with the phase[END_REF]. The evolution of the surface temperature is recorded by a camera (around 500 images) and a specific signal treatment, developed by Thermicar and I2M-TREFLE department, is

Fig. 6 :

 6 Fig. 6: Pulsed IR thermography experimental device

Fig. 7 :

 7 Fig. 7 : Laminated and sandwich composite specimens

  Fig. 9 : Manufacturing drawing of Type B samples

Table 1 .

 1 Table 1 provides a complete summary of the gray-scale results obtained during this investigation study. Thermal mappings according to NDT Infrared technique employed and material investigated

	Specimens	IR thermography	Lock-in IR thermography	Pulse IR thermography


Tomas Menard
email: tomas.menard@irccyn.ec-nantes.fr

Emmanuel Moulay
email: emmanuel.moulay@irccyn.ec-nantes.fr

Tomas Ménard

Wilfrid Perruquetti
email: wilfrid.perruquetti@ec-lille.fr

Global finite-time observers for non linear systems

come

Introduction

The problem of observation has been studied by a number of researchers these last years. The linear case has been solved by Kalman and Luenberger, but the nonlinear case is still an active domain of research. Several ways of investigation have been borrowed. Linearization of nonlinear system with algebraic methods have been investigated in [START_REF] Conte | Algrebraic Methods for Nonlinear Control Systems[END_REF] and [START_REF] Glumineau | New algebrogeometric conditions for the linearization by inputoutput injection[END_REF]. The high-gain observer approach which is closely related to triangular structure has been developped by Gauthier et al. (see [START_REF] Gauthier | A simple observer for nonlinear systems. applications to bioreactors[END_REF] and [START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]) and is derived from the uniform observability of nonlinear systems. Other methods have been developped • Kazantzis and Kravaris observer which uses the Lyapunov auxiliary theorem and a direct coordinate transformation in [START_REF] Kazantzis | Nonlinear observer design using lyapunov's auxiliary theorem[END_REF],

• backstepping design in [START_REF] Li | Global finite-time stabilization by dynamic output feedback for a class of continuous nonlinear systems[END_REF],

• adaptive observer in [START_REF] Young | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF],

and many other ones. Almost all of these approaches give us asymptotic convergence. In some applicative fields, finite time convergence is needed for example like in secure communication, synchronisation or robot walks (see for instance [START_REF] Azemi | Sliding-mode adaptive observer approach to chaotic synchronization[END_REF][START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF][START_REF] Plestan | Stable walking of a 7-dof biped robot[END_REF]).

The problem of finite time observers has been less considered because it requires nonsmooth techniques. Nevertheless new conditions of stability and stabilization have been developped in the continuous finite-time domain, involving the settling time function, associated with the Lyapunov theory and homogeneous systems by Bhat and Bernstein in [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF]Finite-time stability of continuous autonomous systems[END_REF], and by Moulay and Perruquetti in [START_REF] Moulay | Finite-time stability of nonlinear systems[END_REF][START_REF]Finite time stability and stabilization of a class of continuous systems[END_REF].

There exists several methods achieving finite-time convergence, e.g. sliding mode observers (see [START_REF] Drakunov | Sliding mode observers. tutorial[END_REF][START_REF] Haskara | On sliding mode observers via equivalent control approach[END_REF]), moving horizon observer (see [START_REF] Michalska | Moving horizon observers and observer-based control[END_REF]), but some of these are not continuous like sliding mode observers.

We consider here continuous finite-time observer. Such one has been designed for linear systems in [START_REF] Engel | A continuous-time observer which converges in finite-time[END_REF] extended to linear time-varying system in [START_REF] Menold | finite-time convergent observers for linear time varying systems[END_REF]. A global finite time observer for a linearizable system via input output injection is constructed in [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF] and extended to uniformly observable systems in [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] in a semi-global way. Let us note that such output injection structure : a linear part plus a homogeneous part can be derived from the results obtained by Andrieu-Praly-Astolfi ( [START_REF] Andrieu | Homogeneous approximation, recursive observer design and output feedback[END_REF][START_REF]High gain observers with updated high-gain and homogeneous correction terms[END_REF] using the concept of homogeneity in the bi-limit : finite time convergence being obtained if the homogeneity degree in the 0-limit is negative. However, the construction and proof are recursive one. This paper aim at designing a global observer for systems which are uniformly observable. For this class of systems, there exists a change of coordinates which transforms the system into an observability normal form. This paper is organized as follows. The class of considered systems, the definitions and the properties of finite time stable systems are given in section 2. Section 3 presents a global finite-time observer followed by the proof of convergence and an illustrative example.

Preliminaries Notations

• We denote by t → x(t, x t 0 ) a solution of the system:

ẋ = f (x), x ∈ R n x(t 0 ) = x t 0 (1)
where f is a continuous vector field,

• x α = sign(x).|x| α , with α > 0 and x ∈ R,

• . i,k denotes the i-norm on R k ,

• if x ∈ R n , x i denotes the truncated vector in R i with the i th first components of x (1 ≤ i ≤ n),

• B . (ε) is the ball centered at the origin and of radius ε, w.r.t. (with respect to) the norm . ,

• L f h(x) = ∂ ∂ x h(x). f (x)
denotes the Lie derivative of h along the vector field f , and

L n f h = L f (L n-1 f h
) is the n-th Lie derivative of h along the vector field f .

Context

Let us consider a multi-input single-output nonlin-

ear system on R n      ẋ = F(x) + m ∑ i=1 G i (x)u i , y = h(x), (2)
where F and G i are analytic functions. If system (2) is uniformly observable (see [START_REF] Gauthier | A simple observer for nonlinear systems. applications to bioreactors[END_REF]). Then a coordinate change can be found to transform system (2) into the form

                     ẋ1 = x 2 + ∑ m j=1 g 1, j (x 1 )u j ẋ2 = x 3 + ∑ m j=1 g 2, j (x 1 , x 2 )u j . . . ẋn-1 = x n + ∑ m j=1 g n-1, j (x)u j ẋn = ϕ(x) + ∑ m j=1 g n, j (x)u j y = x 1 = Cx (3)
where C = (1 0 . . . 0).

Finite-time stability

In this paper, we are interested in observer, but especially in finite-time observer. So we recall here the main definitions and properties for finite-time stability.

In system (1), f is a continuous but not Lipschitzian function, so it may happen that any solution of the system converges to zero in finite time.

Example 1. For x ∈ R, the solutions of ẋ = -sign(x) |x| 1 3 , are given by:

x(t, x 0 ) = sign(x 0 ) |x 0 | 1 3 - t 3 3 , if 0 < t < 3 |x 0 | 1 3 x(t, x 0 ) = 0, if t ≥ 3 |x 0 | 1 3 ,
and thus reaches zero in finite time.

It is aimed here to exploit this property of such dynamical nonlinear systems to design a finite time observer (FTO). Due to the non Lipschitz condition on the right hand side of (1) backward uniqueness may be lost, and thus we only consider forward uniqueness which is recalled here. Definition 1. The system (1) is said to have unique solutions in forward time on a neighbourhood U ⊂ R n if for any x 0 ∈ U and two right maximally defined solutions of (1), x(t, x 0 )

: [0, T x [ → R n and y(t, x 0 ) : [0, T y [ → R n , there exists 0 < T x 0 ≤ min T x , T y such that x(t, x 0 ) = y(t, x 0 ) for all t ∈ 0, T x 0 .
It can be assumed that for each x 0 ∈ U , T x 0 is chosen to be the largest in R + ∪ {+∞}. Various sufficient conditions for forward uniqueness can be found in [START_REF] Kawski | Stabilization of nonlinear systems in the plane[END_REF]. Let us consider the system (1) where f is continuous on R n and where f has unique solutions in forward time. We recall the definition of finite-time stability.

Definition 2. The origin of the system (1) is said finite time stable if:

1. there exists a function T : V \ {0} → R + defined on a neighbourhood V of the origin, such that for all x 0 ∈ V \ {0}, x(t, x 0 ) is defined (and unique) on

[0, T (x 0 )), x(t, x 0 ) ∈ V \ {0} for all t ∈ [0, T (x 0 )) and lim t→T (x 0 )
x(t, x 0 ) = 0.

T is called the settling-time function of the system (1).

2. for all ε > 0, there exists δ (ε) > 0 such that for every

x 0 ∈ B . 2,n (δ (ε)) \ {0} ∩ V , x(t, x 0 ) ∈ B . 2,n (ε) for all t ∈ [0, T (x 0 )).
The following result gives a sufficient condition for system (1) to be finite time stable (see [START_REF] Moulay | Finite-time stability conditions for non-autonomous continuous systems[END_REF][START_REF] Perruquetti | Finite time stability and stabilisation[END_REF] for ordinary differential equations, and [START_REF] Moulay | Finite time stability of differential inclusions[END_REF] for Differential inclusions): Theorem 1. [START_REF] Moulay | Finite-time stability conditions for non-autonomous continuous systems[END_REF][START_REF] Perruquetti | Finite time stability and stabilisation[END_REF] Let the origin be an equilibrium point for the system (1), and let f be continuous on an open neighborhood V of the origin. If there exists a Lyapunov function1 V : V → R + and a function r : R

+ → R + such that d dt V (x(t)) ≤ -r(V (x)), (4)
along the solutions of (1) and ε > 0 such that

ε 0 dz r(z) < +∞, (5)
then the origin is finite time stable.

In particular, assuming forward uniqueness of the solution and the continuity of the settling time function, Bhat and Bernstein (see [START_REF]Finite-time stability of continuous autonomous systems[END_REF]Definition 2.2]) showed that finite time stability of the origin is equivalent to the existence of a Lyapunov function satisfying (4) where r(x) = cx a , with a ∈]0, 1[, c > 0. In order to circumvent the classical Lyapunov function art of design, one can use homogeneity conditions recalled hereafter.

Homogeneity Definition 3. A function V : R n → R is homogeneous of degree d w.r.t. the weights (r 1 , . . . , r n ) ∈ R n >0 if V (λ r 1 x 1 , . . . , λ r n x n ) = λ d V (x 1 , . . . , x n ), ∀λ > 0. ( 6
)
Definition 4. A vector field f is homogeneous of degree d w.r.t. the weights (r 1 , . . . , r n ) ∈ R n >0 if for all 1 ≤ i ≤ n, the i-th component f i is a homogeneous function of degree r i + d. The system (1) is homogeneous of degree d if the vector field f is homogeneous of degree d.

Theorem 2. [27, Theorem 5.8 and Corollary 5.4] Let g be defined on R n and be a continuous vector field homogeneous of degree d < 0 with respect to dilation Λ r . If the origin of ( 1) is locally asymptotically stable, then it is globally finite time stable.

Previous observers

Perruquetti et al. first construct a finite time observer for a canonical observable form in [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], i.e. for a linear system in the state. The proof of finite-time stability is based on homogeneity property (specifically on Theorem 2). Then, a semi-global observer was constructed by Shen and Xia in [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] based on the Perruquetti et al. observer. This result is recall here: Theorem 3. System (3) admits a semi-global observer of the form:

           ẋ1 = x2 + k 1 y -x1 α 1 + ∑ m j=1 g 1, j ( x1 )u j ẋ2 = x2 + k 3 y -x1 α 2 + ∑ m j=1 g 2, j ( x1 , x2 )u j . . . ẋn = ϕ( x) + k n y -x1 α n + ∑ m j=1 g n, j ( x)u j (7)
where the powers α i are defined by

α i = iα + (i -1), α ∈ 1 - 1 n , 1 (8)
and

[k 1 , . . . , k n ] T = S -1 ∞ (θ )C T , (9)
where S ∞ (θ ) is the unique solution of the Riccati equation:

θ S ∞ (θ ) + A T S ∞ (θ ) + S ∞ (θ )A -C T C = 0 S ∞ (θ ) = S T ∞ (θ ) (10)
where (A) i j = δ i, j-1 and C = (1 0 . . . 0).

Global Observers

In this section, we propose a global finite-time observer for system (3) based on the semi-global finitetime observer [START_REF] Young | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF]. In order to prove our result, we need the following lemma (issued from [START_REF] Hong | Finite-time control for robot manipulators[END_REF], remark 1): Lemma 1. Assume system (1) is globally asymptotically stable and finite-time attractive on a neighborhood of the origin. Then system (1) is globally finitetime stable.

The proof of the following technical lemma is not given here, but an explicit computation (easy but long) gives the first equality and the second easily follows from the first. Lemma 2. The matrix S ∞ (θ ) and S -1 ∞ (θ ) verify the following properties:

S ∞ (θ ) i, j = S ∞ (1) i, j 1
θ i+ j-1 (11)
S -1 ∞ (θ ) i, j = S -1 ∞ (1) i, j θ i+ j-1 (12)
for any θ > 0 and 1 ≤ i, j ≤ n.

Theorem 4. Let us consider system (3) with a bounded input u = (u 1 , . . . , u m ) and assume functions (g i, j ) 1≤i, j≤n , and ϕ are globally Lipschitz. Then there exists θ * > 0 and ε > 0 such that for all θ > θ * and α ∈]1ε, 1[, system (3) admits the following global finite-time high-gain observer:

           ẋ1 = x2 + k 1 ( e 1 α 1 + ρe 1 ) + ∑ m j=1 g 1, j ( x1 )u j ẋ2 = x3 + k 2 ( e 1 α 2 + ρe 1 ) + ∑ m j=1 g 2, j ( x1 , x2 )u j . . . ẋn = ϕ( x) + k n ( e 1 α n + ρe 1 ) + ∑ m j=1 g n, j ( x)u j ( 13
)
where the powers α i are defined by [START_REF] Azemi | Sliding-mode adaptive observer approach to chaotic synchronization[END_REF], the gains k i are defined by [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], and ρ

= n 2 θ 2 3 S 1 +1 2 , where S 1 = max 1≤i, j≤n |S ∞ (1) i, j |.|S -1 ∞ (1) j,1 |. Proof.
We denote e = xx. By using

D(x, x, u) =      0 . . . 0 ϕ(x) -ϕ( x)      + m ∑ j=1 (g j (x)-g j ( x))u j (t), (14)
where g j = (g 1, j , . . . , g n, j ), and

F(K, e) =    k 1 e 1 α 1 . . . k n e 1 α n    , (15)
the error system is given by: ė = Ae -F(K, e) -ρS -1 ∞ (θ )C T Ce + D(x, x, u). ( 16) In order to prove the global convergence of the observer, we prove first the existence of a "Lyapunov" function V for the error system ( 16) which is positive definite on R n and such that d dt V is negative definite on R n -B S ∞ (θ ) (r). Then we will prove that the error system is finite time stable at the origin on B S ∞ (θ ) (2r). The negativeness of d dt V on R n -B S ∞ (θ ) (r) and the finite time stability on B S ∞ (θ ) (2r) prove that the error equation is globally asymptotic stable and finite-time stable at the origin. We apply then the Lemma 1 which gives the result. Let us consider:

V (e) = e T S ∞ (θ )e. (17)
By using ( 10) and ( 16), the derivative of V is given by:

d dt (e T S ∞ (θ )e) = -θ e T S ∞ (θ )e -(2ρ -1)(Ce) 2 -2e T S ∞ (θ )F(K, e) +2e T S ∞ (θ )D(x, x, u). (18)
It leads to:

d dt (e T S ∞ (θ )e) ≤ -θ e 2 S ∞ (θ ) -(2ρ -1)(Ce) 2 -2e T S ∞ (θ )F(K, e) +2 e S ∞ (θ ) D(x, x, u) S ∞ (θ ) . (19)
Since ϕ and g i, j (i = 1, . . . , n, j = 1, . . . , m) are globally Lipschitzian with a constant l and u is bounded by u 0 , it follows that:

D(x, x, u) S ∞ (θ ) = ∑ 1≤i, j≤n S ∞ (θ ) i, j D i (x i , xi , u)D j (x j , x j , u) 1 2 ≤ (lu 0 ) 2 ∑ 1≤i, j≤n |S ∞ (1) i, j |. e i 1,i e j 1, j 1
θ i+ j-1 1 2 ≤ (lu 0 ) 2 Sθ ∑ 1≤i, j≤n 1
θ i e i 1,i 1 θ j e j 1, j 1 2
(20) where S = max 1≤i, j≤n |S ∞ (1) i, j |. Set ξ i = e i θ i , clearly for θ ≥ 1:

1

θ i e i 1,i ≤ ξ i 1,i ≤ ξ 1,n . (21)
By norm equivalence, there exists C 1 > 0 such that:

ξ 1,n ≤ C 1 ξ S ∞ (1) , (22)
ξ 2 S ∞ (1) = 1 θ e 2 S ∞ (θ ) , (23)
it leads to

D(x, x, u) S ∞ (θ ) ≤ nlu 0 C 1 √ S e S ∞ (θ ) . (24)
Finally, we have

d dt V (e) ≤ (-θ + M)V (e) -(2ρ -1)(Ce) 2 -2e T S ∞ (θ )F(K, e), (25)
where

M = 2nlu 0 C 1 √ S.
According to [START_REF] Perruquetti | Finite time stability and stabilisation[END_REF] in order to prove the negative definitness of d dt V on R n -B S ∞ (θ ) (r), we use an overvaluation of e T S ∞ (θ )F(K, e). According to lemma 2, we have the following equalities:

e T S ∞ (θ )F(K, e) = ∑ 1≤i, j≤n e i (S ∞ (1)) i, j θ i+ j-1 S -1 ∞ (1) j,1 θ j e 1 α j , = n ∑ j=1 S -1 ∞ (1) j,1 e 1 α j n ∑ i=1 e i θ i-1 (S ∞ (1)) i, j . (26)
We overvalue [START_REF] Moulay | Finite time stability of differential inclusions[END_REF] in two times. We cut the subset P r = R n -B S ∞ (θ ) (r) in two complementary parts:

P r <1 = {e ∈ R n : |e 1 | < 1} ∩ P r , ( 27
) P r ≥1 = {e ∈ R n : |e 1 | ≥ 1} ∩ P r . ( 28
)
On P r <1 , we have:

|e 1 | α i < 1, i = 1, . . . , n. (29)
Hence

|e T S ∞ (θ )F(K, e)| ≤ nS 1 θ n ∑ i=1 e i θ i , (30)
where

S 1 = max 1≤i, j≤n S -1 ∞ (1) j,1 × S ∞ (1) i, j . Let ξ i = e i θ i for i = 1, . . . , n, it follows: |e T S ∞ (θ )F(K, e)| ≤ nS 1 θ ξ 1,n . (31)
As

ξ 1,n ≤ C 1 ξ S ∞ (1) , (32)
ξ 2 S ∞ (1) = 1 θ e 2 S ∞ (θ ) , (33)
we have

|e T S ∞ (θ )F(K, e)| ≤ nS.C 1 √ θ e S ∞ (θ ) . ( 34
) Let C 2 = nSC 1 . Taking r > 1, then e S ∞ (θ ) ≤ e 2 S ∞ (θ )
for e ∈ P r , so:

|e T S ∞ (θ )F(K, e)| ≤ C 2 √ θ e 2 S ∞ (θ ) . (35)
Thus it leads to:

d dt V (e) ≤ (-θ + M +C 2 √ θ )V (e) (36)
-(2ρ -1)(Ce) 2 ≤ (-θ + M +C 2 √ θ )V (e). (37)
So there exists θ 1 > 0 such that for all θ ≥ θ 1 , d dt V (e) < 0 for all e ∈ P r <1 .

On P r ≥1 , we have

|e 1 | ≥ 1 so |e 1 | α i ≤ |e 1 | for i = 1, . . . , n. |e T S ∞ (θ )F(K, e)| ≤ nSθ ∑ n i=1 e i θ i .|e 1 |, ≤ nS ∑ n i=1 θ 2 3 e i θ i θ 1 3 |e 1 | , ≤ nS 2 θ 4 3 ∑ n i=1 e i θ i 2 + nθ 2 3 |e 1 | 2 , ≤ nSθ 4 3 2 ξ 2 2,n + n 2 θ 2 3 S 2 |e 1 | 2 . ( 38
) But ξ 2 2,n ≤ C 3 ξ 2 S ∞ (1) , (39)
ξ 2 S ∞ (1) = 1 θ e 2 S ∞ (θ ) , ( 40
) hence |e T S ∞ (θ )F(K, e)| ≤ C 4 θ 1 3 e 2 S ∞ (θ ) + n 2 θ 2 3 S 2 |e 1 | 2 , (41)
where C 4 = nSC 3 . Thus we have:

d dt V (e) ≤ -θ + M +C 4 θ 1 3 V (e), (42)
and there exists θ 2 > 0 such that for all θ ≥ θ 2 , d dt V (e) < 0 for all e ∈ P r ≥1 . Combining the two previous parts, for all θ ≥ max{θ 1 , θ 2 }, we have:

d dt V (e) < 0, e ∈ P r , r > 1. ( 43
)
Let us prove the finite-time stability of the error system (16) on B . S∞(θ ) (2r). We consider the following Lyapunov function:

Ṽα (e) = ẽT S ∞ (θ ) ẽ, (44)
where ẽ = e 1 1 q e 2 1 α 1 q . . . e n

1 α n-1 q , q = n-1 ∏ i=1 [(i - 1)α -(i -2)
] is the product of the weights. We have:

d dt Ṽα (e) = A + B +C (45)
where

A = 2 ẽT S ∞ (θ )     1 q |e 1 | 1 q -1 ( 1 2 e 2 -k 1 e 1 α 1 ) . . . 1 α n q |e n | 1 αnq -1 (-k n e 1 α n )     , (46)
and

B = 2 ẽT S ∞ (θ )     1 q |e 1 | 1 q -1 ( 1 2 e 2 -ρk 1 e 1 ) . . . 1 α n q |e n | 1 αnq -1 (-ρk n e 1 )     , (47)
and

C = 2 ẽT S ∞ (θ )      1 q |e 1 | 1 q -1 D 1 . . . 1 α n-1 q |e n | 1 α n-1 q -1 D n      . (48)
We prove now that the second term B is negative. We use the same technique as in [START_REF] Perruquetti | Finite-time observers: application to secure communication[END_REF], i.e. we use the tube lemma. As V is proper, B .

According to the proof of the main theorem of [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF], it follows that for every compact set U containing the origin, there exists θ 3 > 0 and ε 2 > 0 such that for θ ≥ θ 3 and α ∈ (1ε 2 , 1), U ⊂ Ω, where Ω is the domain of attraction of the observer. But according to [START_REF] Gauthier | A simple observer for nonlinear systems. applications to bioreactors[END_REF], there exists δ 0 > 0 such that:

S ∞ (θ ) ≥ δ 0 I, ∀θ ≥ 0. (52)

Hence, there exists θ 4 > 0 such that for all θ ≥ θ 4 :

B S ∞ (θ ) (2r) ⊂ δ 0 B . 2 (2r) ⊂ U . ( 53
)
Finally we take θ * = max{θ 1 , . . . , θ 4 } and ε = min{ε 1 , ε 2 }.

  S∞(θ ) (2r) is a compact set of R n . Define the function ϕ : R >0 × B . S∞(θ ) (2r) → R (α, e) → B.(49)Using the same technique as in Gauthier et al.[START_REF] Gauthier | A simple observer for nonlinear systems. applications to bioreactors[END_REF], we easily prove that ϕ(1, e) < 0 for e ∈ R n . Since ϕ is continuous,ϕ -1 (R <0 ) is an open subset of R >0 × B . S∞(θ ) (2r) containing the slice {1} × B . S∞(θ ) (2r). Since B . S∞(θ ) (2r) is compact, it follows from the tube lemma that ϕ -1 (R <0 ) contains some tube (1µ 1 , 1 + µ 2 ) × B . S∞(θ ) (2r) about {1} × B . S∞(θ ) (2r). For all (α, e) ∈ (1µ 1 , 1 + µ 2 ) × B . S∞(θ ) (2r) ϕ(α, e) < 0. (50)Thus there exists ε 1 > 0 such that for α ∈ (1ε 1 , 1):d dtṼα (e) ≤ A +C.

V is a continuously differentiable function defined on V such that V is positive definite and dV dt is negative definite.