Sébastien Hénaff
email: sebastien.henaff@irccyn.ec-nantes.fr

Ina Taralova
email: ina.taralova@irccyn.ec-nantes.fr

René Lozi
email: r.lozi@unice.fr

EXACT AND ASYMPTOTIC SYNCHRONIZATION OF A NEW WEAKLY COUPLED MAPS SYSTEM

Keywords: chaos, synchronization, discrete-time, observer, pseudo-random generators

The paper deals with the synchronization of a new statistically highly performant function firstly introduced by Lozi. The synchronization is reached via observers which reconstruct all the states of the original system, using only a partial information of it.

The map has been rewritten as a piece-wise affine one, which allows to design two types of observers: an exact (dead-beat) observer, and an asymptotic observer. Both observers have been analysed and compared; the dead-beat observer guarantees an exact convergence; however, synchronization can not be preserved in an autonomous regime (if the observer is switched off), because the map is highly chaotic, and the trajectories diverge due to the finite computer precision. The asymptotic observer is more robust in case of noise, but the convergence is slower, and the error converges to zero only asymptotically. In the latter case, several observers have to run in parallel, and the criteria to select the converging one are derived.

Introduction

Chaos has recently received a growing interest in various fields of science and engineering, and in particular, in secure communications. Several chaotic cryptographic schemes have been proposed since [START_REF] Kocarev | Publickey encryption based on chebyshev polynomials[END_REF], [START_REF] Fei | A secure digital signature algorithm based on elliptic curve and chaotic mappings[END_REF] and can be classified in three main categories : chaotic masking, chaotic modulation and chaotic shift keying.

In the cryptographic application, the chaotic generator must exhibit appropriate features close to those of the pseudo-random generators. These adapted properties have been studied more precisely in [START_REF] Álvarez | Some basic cryptographic requirements for chaos-based cryptosystems[END_REF], [START_REF] Gotz | Discrete-time chaotic encryption systems. i. statistical design approach[END_REF], [START_REF] Rukhin | A Statistical Test Suite for the Validation of Random Number Generators and Pseudo Random Number Generators for Cryptographic Applications[END_REF].

Further researchers have then looked for finding appropriate systems testing different architectures : traditional chaotic maps (for example, the logistic map, the Hénon map, the generalized Hénon map) [START_REF] Frey | Chaotic digital encoding: an approach to secure communication[END_REF], piece-wise linear map, cascaded map [START_REF] Tang | Chaotic communications using nonlinear transform-pairs[END_REF] or coupled map lattice. In order to evaluate the features of the system, statistical tests developed for random and pseudo-random number gen-erators (RNG and PRNG) can also be applied to chaotic maps, in order to gather evidence that the map generates "good" chaotic signals, i.e. having a considerable degree of randomness [START_REF] Álvarez | Some basic cryptographic requirements for chaos-based cryptosystems[END_REF] [START_REF] Rukhin | A Statistical Test Suite for the Validation of Random Number Generators and Pseudo Random Number Generators for Cryptographic Applications[END_REF].

It appears that most of the maps classically used for chaotic encryption do not pass successfully these tests, and don't exhibit the required features. However, most of the papers dealing with synchronization and observer synthesis consider precisely these kinds of maps, highly inefficient in the context of chaotic encryption. Unlike these models, Lozi introduced in 2008 a new ultra weakly coupled maps system [START_REF] Lozi | New enhanced chaotic number generators[END_REF] to generate pseudo-random signals which exhibits very good statistical properties.

Synchronization of chaotic systems has received a great interest since the pioneer work of Pecora and Carroll [START_REF] Carroll | Synchronizing chaotic circuits[END_REF] and its application to secure communications attracted lots of works. There are two ways of achieving synchronization: it can be done by the inverse system [START_REF] Feldmann | Communication by chaotic signals: The inverse system approach[END_REF] or by applying observers [START_REF] Angeli | Deadbeat chaos synchronization in discrete-time systems[END_REF] [START_REF] Feki | Secure digital communication using discrete-time chaos synchronization[END_REF]. Observers allows to reconstruct all the states of a system only with few components of it as shows figure 1. An overview of observers in the case of secure communications process is avaible in [START_REF] Millerioux | An observer-based approach for input-independent global chaos synchronization of discrete-time switched systems[END_REF] Figure 1: Observer autonomous system

x 1 x 2 observer x1 x2
Our previous works on the new weakly coupled map function introduced by Lozi in 2008 [START_REF] Lozi | New enhanced chaotic number generators[END_REF] demonstrated its excellent statistical and spectral properties [START_REF] Hénaff | Statistical and spectral analysis of a new coupled maps systems[END_REF]; two kinds of observers have been proposed for the synchronization of the system: an inverse lag, and an exact observer. The latter consisted of 16 sub-observers which had to run in parallel in order to guarantee the exact convergence of one of them.

In comparison with these works, the present paper deepens and completes the previous results on exact observers, in particular by taking into account and exploiting the degrees of freedom in the choice of the observation matrices. The implementation issue related to the finite machine precision and its impact on the exact synchronization in the case of hyperchaotic maps is also discussed.

In addition, this paper presents also the novel analytical and numerical results of the asymptotic synchronization of the weakly coupled map system. Indeed, the exact observers fail in the case of applications when (different kinds of) noise exist and has to be taken into account, but this problem can be successfully dealt with by the design of asymptotic observers.

This paper is organised as follows: after the presentation of the position of the problem in section two, section three analyses the observability of the system under consideration. Two types of observers are then designed in section four for exact convergence and asymptotic convergence. The problem of numerical synchronization is also discussed. A concluding section ends the paper.

Position of the problem

Lozi recently introduced a hyper chaotic system which generates signal with statistical features [START_REF] Lozi | New enhanced chaotic number generators[END_REF] competitive with the classical RNG. The aim of this work concerns the design of an observer of this particular system. Some preliminary results are available in [START_REF] Hénaff | Observer design for a new weakly coupled map function[END_REF]. A particular design of observers for the exact convergence is also available in [START_REF] Hénaff | Statistical and spectral analysis of a new coupled maps systems[END_REF]. This paper expands the previous results on exact observers analysis and deals with an additional problem: the asymptotic convergence and implementation aspects.

The system under consideration presented in [START_REF] Lozi | New enhanced chaotic number generators[END_REF] is composed by a chaotic weakly coupled map. The N -th order function f can be written as:

x(n + 1) = f (x(n)) = A Λ(x(n)) with x(n) = (x 1 (n), x 2 (n), . . . , x N (n))
where A is a N xN matrix defined by:

A = A a + A b A a =      ǫ 1 ǫ 2 . . . ǫ N      1 1 . . . 1 A b = (N -1)      ǫ 1 0 . . . 0 0 ǫ 2 . . . 0 . . . . . . . . . . . . 0 0 . . . ǫ N     
and Λ is the tent function applied to each component of

x ∈ [-1; 1] N : Λ(x(n)) =      Λ(x 1 (n)) Λ(x 2 (n)) . . . Λ(x N (n))      Λ(x) = 2x + 1 if x < 0 -2x + 1 else
ǫ i are positive parameters smaller than 1/(n -1). Since the function is piece-wise affine, it can be rewritten under a matrix form, by rewriting the tent map. For the second order, the general system f is then governed by :

x(n + 1) = 2A i x(n) + B

where matrix A i can take four possible values according to the regions of definition:

A i =        A 1 if x ∈ [0; 1] 2 A 2 if x ∈ [-1; 0] × [0; 1] A 3 if x ∈ [-1; 0] 2 A 4 if x ∈ [0; 1] × [-1; 0] A 1 = -(1 -ǫ 1 ) -ǫ 1 -ǫ 2 -(1 -ǫ 2 ) A 2 = 1 -ǫ 1 -ǫ 1 ǫ 2 -(1 -ǫ 2 ) A 3 = 1 -ǫ 1 ǫ 1 ǫ 2 1 -ǫ 2 A 4 = -(1 -ǫ 1 ) ǫ 1 -ǫ 2 1 -ǫ 2 B = 1 1
Then, the output is defined by:

x(n + 1) = f (x(n)) y(n) = Cx(n)
with C = 1 0

Observability

Observers allow the reconstruction of the evolution of a multi-component state thanks to a partial information, the output signal. But not all systems allow the realisation of this process: it must exhibit some particular features which are studied in detail by observability. It can be divided into two classes: global observability and local one. Global observability ensures that a given output describes only one internal state whereas local observability indicates that two closed states can be distinguished by the output.

Second order system

Local observability of the second order system has been studied in [START_REF] Hénaff | Statistical and spectral analysis of a new coupled maps systems[END_REF]. As we shall need it further on, the result is recalled here. An affine system can be written as :

x(n + 1) = f (x(n)) = A.x(n) + B y(n) = Cx(n)
The function is piece-wise affine, then the matrix A can be re-written in a generic form by:

A n = (1 -ǫ 1 )s 10 ǫ 1 s 20 ǫ 2 s 10 (1 -ǫ 2 )s 20
where

s ij = 2 if x i (j) < 0 -2 else
A second order affine system is observable if its observability matrix is a full-rank one :

O = C CA
Here, the system is piece-wise affine, therefore the observability matrix shall be different according to the region to which belongs the system state. It is equal to:

O = 1 0 2(1 -ǫ 1 )s 10 2ǫ 1 s 10
which is full-rank since ǫ 1 > 0 from the definition of the system. Therefore, the system is locally observable for all parameters and for all states, which is not the case of all systems. For example, the third order system is not locally observable for all configurations.

Third order system

A second order system needs the use of at least two iterations to determine weather this system is observable or not. Equivalently, at least, three iterations are necessary for a three order system. The observability matrix is:

O =   C CA CA 2  
The third order system is defined by the equation:

  x 1 (n + j + 1) x 2 (n + j + 1) x 3 (n + j + 1)   = A j   x 1 (n + j) x 2 (n + j) x 3 (n + j)   +   1 1 1   with: A j =   (1 -2ǫ 1 )s 1j ǫ 1 s 2j ǫ 1 s 3j ǫ 2 s 1j (1 -2ǫ 2 )s 2j ǫ 2 s 3j ǫ 2 s 1j ǫ 2 s 2j (1 -2ǫ 2 )s 3j  
By replacing these values, it comes:

O =   1 0 0 (1 -2ǫ 1 )s 10 ǫ 1 s 20 ǫ 1 s 30 as 11 bǫ 1 s 21 cǫ 1 s 31   with    a = (1 -2ǫ 1 ) 2 s 10 + ǫ 1 (ǫ 2 s 20 + ǫ 3 s 30 ) b = s 10 (1 -2ǫ 1 ) + s 20 (1 -2ǫ 2 ) + ǫ 3 s 30 c = s 10 (1 -2ǫ 1 ) + ǫ 2 s 20 + s 30 (1 -2ǫ 3 )
Out of singularity parameter region, that means ǫ 2 = ǫ 3 , 3ǫ 2 + 3ǫ 3 = 4ǫ 1 and 4ǫ 1 + 3ǫ 2 + 3ǫ 3 = 4, this matrix is a full rank one for all possible s ij . Then, the system is globally observable for all states in the whole phase plane if the parameter combination respects the above conditions.

Observer analysis

The aim of an observer is to recover a complete dynamics evolution thanks to a partial information only.

Since the considered system is piece-wise affine, the present section designs a piece-wise linear observer which would synchronise its dynamics with the original system and it is suitable to apply a Luenberger observer [START_REF] David | Optimization by vector space methods[END_REF]. But the application of this observer supposes that the observer exactly "knows" how the states trajectories evolve and switch from one region to another, which is not the case. For this reason, some criteria have to be defined to identify the evolution of the state trajectories. But first of all, some adapted observers would be presented in this section, before caring out numerical applications and dealing with the criteria for exact and asymptotic convergence.

Before presenting the observer, the second order system should be rewritten under an affine form on each of the four domains where it is defined :

x(n + 1) = F (x(n)) = A.x(n) + B y(n) = Cx(n)    x(n + 1) = (1 -ǫ 1 )s 10 ǫ 1 s 20 ǫ 2 s 10 (1 -ǫ 2 )s 20 x(n) + B y(n) = 1 0 x(n) with B = 1 1
The Luenberger observer can be applied on a linear system. The observer is then defined as:

x(n + 1) = Âx(n) + B + K(C x(n) -y(n))
K is a predefined gain such that the error e, defined by e = x -x, tends to zero. Let consider x(n) and x(n) belonging to the same region of definition. In this case, Â = A and therefore,

e(n + 1) = (A + KC)e(n)
The value of the vector K determines the eigenvalues of the error matrix (A + KC) and its speed of convergence: zero eigenvalues would imply that the observer converges in finite number of iterations whereas singular values less than one would impose an asymptotic convergence.

Exact or dead-beat observer

Analytical convergence

Hereafter, we deal with the second order system. One can identify the values of the gain K which cancel the eigenvalues of the matrix (A + KC) as a function of the affine system model. In this case, since the matrix is of second order, (A + KC) 2 = 0 therefore if the system states x and its estimates x belong to the same region of the state space twice consecutively, the estimate shall synchronise with the original system in two steps.

This will ensure that if the dynamics falls into the same region twice consecutively, then the convergence is guaranteed. However, the observer should also synchronise when the state trajectories switch from one region of definition to another. The solution of vectors K that satisfy the synchronization has been found previously in [START_REF] Hénaff | Statistical and spectral analysis of a new coupled maps systems[END_REF]: for all (i 1 , i 2 ) ∈ 1; 4 , there exist infinite vectors K j1 and K j2 such that (A i1 + K j1 C)(A i2 + K j2 C) = 0. For example, for i 1 = i 2 = 1, a formal resolution of the equation led to the following result:

K i1 = 2(2 -ǫ 1 -ǫ 2 ) -2 1-2ǫ2+ǫ 2 2 +ǫ1ǫ2 ǫ1 K i2 = p -2+p+2ǫ1+2ǫ2-pǫ2 ǫ1 for x(n) ∈ [0; 1] 2 and x(n + 1) ∈ [0; 1] 2
where p is a number arbitrarily chosen. In this paper, we go in-depth and enhance the previous studies on exact convergence, which did not consider the degree of freedom in the choice of K. Here, this degree of freedom shall be exploited as shown in the following section.

Implementation aspects

The previous development allows to synchronise exactly the states of the autonomous system in two iterations. This can be obtained by an ideal system but it is known that all processors represent the numbers with a finite precision so the calculation of an operation does not give the exact result and this approximation would not allow to synchronise perfectly. This section deals with the implementation problems and simulates numerically the synchronization. To design the final observer, some others are simulated to approach it step by step. All the simulations are performed under the parameters: ǫ 1 = 0.2 and ǫ 2 = 0.1. Consider the autonomous piece-wise affine system:

x(n + 1) = f (x(n)) y(n) = Cx(n)
From the analytical results, it comes that the synchronization can been achieved in two iterations, after what both observer and original autonomous system have the same dynamics and they evolve identically. In this first simulation, the observer system is then designed in two steps: the Luenberger observer is used for the two first iterations before leaving its states evolving autonomously. The following system is then only used for the two first iterations.

x

(n + 1) = Âx(n) + B + K(C x(n) -y(n))
And, after being synchronised, the system runs autonomously.

x(n + 1) = f (x(n))

In practice, to synchronise, the previous section has calculated all possible gains K for all possible evolution of states and it comes that sixteen observers need to run simultaneously and generate sixteen different trajectories two iterations later. But, not all of them are necessary: indeed, the output signal y(n) = x 1 (n) is a component of the state so that it brings an information on the current belonging domain of definition of definition. On the four possible domains, only two are possible, for example, if [START_REF] Conte | Algrebraic Methods for Nonlinear Control Systems[END_REF]. Finally, the observer runs in parallel four different sub-observers and one of the four generated states converges exactly to the same dynamics as the autonomous one in two iterations. To identify it, it is possible to compare both outputs ŷ and y, the one that is identical is the sub-observer that has converged.

y(n) = x 1 (n) > 0, then x(n) ∈ [0; 1] 2 or x(n) ∈ [0; 1] × [-1; 0]. It is an input-output lineari- sation
Figure 2 shows the evolution the states, only the convergent sub-observer is represented. In two iterations, both systems have the same internal states and evolve identically. The error dynamics e = x -x is reported figure 3. In two iterations, the quadratic error does not cancel but only falls to 10 -15 due to the finite precision of the computer. The system is chaotic and therefore sensitive to initial conditions, so the trajectories of two states initialised as close as wanted to each other would diverge exponentially. That is why, the residual error increases exponentially according to the Lyapunov exponents values until stabilizing to 1, which is the size of the chaotic attractor. As a result, an initial synchronization is not sufficient to ensure its synchronization in time. This paragraph investigates the possibility of synchronising all the time. To do so, the convergence of the error during more than two iterations is considered. For example, in three iterations,

e(3) = (A i2 + K j2 C)(A i1 + K j1 C)(A i0 + K j0 C)e(0)
The value of the vectors K must be chosen such that:

(A i1 + K j1 C)(A i0 + K j0 C) = 0(A i2 + K j2 C)(A i1 + K j1 C) = 0
The choice of any particular gains combination for the resolution of the first equation fixes both K j0 and K j1 . Then, as K j1 is fixed, K j2 must be chosen such that the second equation is satisfied if it is possible and so on... But the simplest solution of the system of equations is obtained if the vector K jn depends only on the associated matrix A in . In other words, we are looking for the existence of vectors K i such that for all possible i and j, (A i + K i C)(A j + K j C) = 0. It is possible to have such a solution by loosing the degree of freedom p of the results given in the previous section. By calculating it, it comes the following gains:

K 1 = 2 (2 -ǫ 1 -ǫ 2 ) 1-2ǫ2+ǫ 2 2 +ǫ1ǫ2 ǫ1 K 2 = 2 (ǫ 1 -ǫ 2 ) 1-2ǫ2+ǫ 2 2 -ǫ1ǫ2 ǫ1 K 3 = -2 (2 -ǫ 1 -ǫ 2 ) 1-2ǫ2+ǫ 2 2 +ǫ1ǫ2 ǫ1 K 4 = -2 (ǫ 1 -ǫ 2 ) 1-2ǫ2+ǫ 2 2 -ǫ1ǫ2 ǫ1
These vectors are defined such that

∀(i, j) ∈ {1; 4} 2 , (A i + K i C)(A j + K j C) = 0
The reconstruction of the states of the original system would be composed by a simpler algorithm than the one needed by the vectors defined before. A simulation of such results has been performed and the evolution of the quadratic error is reported in figure 4. Its value is related to the precision of the computer. On the contrary to the first simulation, this one keeps both systems synchronised.

Asymptotic observer

Instead of using the exact convergence, it is also possible to converge asymptotically by positioning the singular values of the error matrix less than one to ensure its convergence to zero but greater than zero for the asymptotic aspect. This section analyses both cases of one iteration matrix and two iteration matrix.

Asymptotic convergence in one step

Let consider the following system:

   x(n + 1) = 2A i x(n) + 1 1 y(n) = 1 0 x(n)
where A i is one of the four matrices defined in section two. It is worth noting that the value of the output y can be an indicator of the value of the matrix A as for exact convergence:

y(n) ≥ 0 ⇒ A i ∈ {A 1 ; A 4 } y(n) < 0 ⇒ A i ∈ {A 2 ; A 3 }
The aim of the following is to construct an observer which converges asymptotically to the states of the original system.

The global system (original system and observer) is defined by:

               x(n + 1) = 2A i x(n) + 1 1 x(n + 1) = 2 Âi x(n) + 1 1 + K(ŷ(n) -y(n)) y(n) = 1 0 x(n) ŷ(n) = 1 0 x(n)
where the matrices A i and Âi are respectively determined by the values of the states x(n) and x(n). The error of the dynamics is then given by:

e(n + 1) = x(n + 1) -x(n + 1) e(n + 1) = 2 Âi x(n) -2A i x(n) + KCe(n) If Âi = A i , then, e(n + 1) = (2A i + KC)e(n)
The asymptotic convergence can be obtained by placing both singular values of (2A i +KC) less than 1. The norm of the error would then asymptotically decrease to zero.

The formal calculation of the minimum of the singular values gives that it is obtained for the parameter combination ǫ 1 = ǫ 2 = 0.5. Both singular values of (2A i + KC) are then equal to 0 and 2 for K = (1 1) t .

So one singular vector is compressed to zero while the second singular vector increases twice its norm under the matrix (2A i + KC), thus for any parameter combination and for any gain K, it is impossible to have both singular values less than one. The asymptotic convergence cannot be reached in this way.

Asymptotic convergence in two steps

An alternative approach consists in considering the system under two iterations. The evolution of the error is still governed by the equations:

e(n + 2) = (2A i (n + 1) + K 1 C)(2A i (n) + K 2 C)e(n)
As before, an asymptotic convergence can be obtained by placing both singular values of the matrix (2A i (n + 1) + K 1 C)(2A i (n) + K 2 C) less than 1. The smallest singular values are zero -that is also what has been tuned for the exact convergence.

It is possible to find an instance of gains K 1 and K 2 such that both singular values of (2A i (n + 1) + K 1 C)(2A i (n) + K 2 C) are less than one and also for the value of vectors K 3 and K 4 by considering the matrix (2A i (n + 2) + K 3 C)(2A i (n + 1) + K 4 C) but all gains must be adapted to each other: K 4 must be chosen such that K 3 = K 2 for guaranteeing the overall convergence of the product of all matrices to zero. A particular numerical solution has been found out for parameters {ǫ 1 , ǫ 2 } = {0.2, 0.1} such that for all possible i and j, the singular values of all matrix (2A i + K i C)(2A j + K j C) are less than one. Under these values, the singular values are {0, 0.44}. Then, in each iteration, the euclidean norm of the error is multiplied by a factor of 0.44 maximum, that means that the error would decrease to zero with the time.

                       K 1 = 3.
A simulation has been performed for these values. The evolution of both original state and observer state is reported in figure 5. The evolution of the output error is also represented figure 6. The error is converging exponentially to zero so there is an asymptotic convergence so this convergence is slower than the exact convergence case. These simulations have been performed assuming that the observer knows exactly in which domain is located the state of the original system. But this assumption is not relevant. The following simulations do not suppose anything about the value of the state, as it is the case for all observation problems. The observer should consider all possibilities of the dynamics evolution.

However, there is no need of studying all these cases. Indeed, the transmitted output brings an information and allows to locate the states, for example,

y(n) = x 1 (n) ≥ 0 ⇒    x(n) ∈ [0; 1] 2 or x(n) ∈ [0; 1] × [-1; 0]
The consideration of one of both domains lets the asymptotic synchronization be done whereas the other can increase the quadratic error. In the case of exact synchronization, the domain that puts the estimated output to the same value of the real transmitted output is identified as being the correct domain but in the case of asymptotic convergence, the decreasing of the quadratic error does not give any tendency on the evolution of the output error ŷ(n) -y(n). In other words, all possible domains of definition must be considered to be sure not to exclude the dynamics which is synchronising. Two cases have to be analysed per iteration so that the number of possible states increases exponentially.

Nevertheless, a criteria can be built to restrict the number of states to be considered. Let e be the quadratic error between the state to estimate and the estimation from the observer. For all possible states, e(n + 1) < λ e(n) e(n) < λ n e(0) where λ denotes the maximal singular value, 0.44 here. At the initialisation, the error is smaller than

√ 2 2 + 2 2 = 2 √ 2 since the state is in the space [-1; 1] 2 . Then, e(n) < λ n 2 √ 2 for all iterations. And ŷ(n) -y(n) Ce(n) < e(n) < λ n e(0) .
This last inequality can be exploited to identify the observer which has converged to the original system. So if the calculated output error does not satisfy this inequality, the considered domain is not the good one. This simulation has also been performed and the evolution of the output error is represented in figure 7. There is not only one observer state, as several crosses are plotted in the figure for a same instant. The continuous curve separates the convergence zone from the rest of the space as the discrimination is done. Figure 8 plots the same values but in the logarithmic scale. In conclusion, both asymptotic and exact observers exhibit interesting properties, and the choice between the asymptotic and the exact one has to be done according to the particular application. If it is in telecommunications (e.g. for a chaotic encryption), and the noise could be assumed to be zero (i.e. already corrected by the physical layer), than an exact observer seems to be the most suitable. On the other hand, if the application is such that the system has to be robust to (different kinds of) noise, then an asymptotic observer would be more appropriate.

Conclusion

The majority of the papers devoted to synchronization and observer synthesis dealt with maps with poor statistical ans spectral properties. Unlike these authors, in this work we have proposed the observers design for a new weakly coupled and highly performant chaotic map, which beats most of the classical random number generators. Two kinds of observers have been synthesised: and exact -or dead-beat observer, and an asymptotic one. The exact convergence of the dead-beat observer has been theoretically demonstrated (e.g. two iterations for a second order system). In addition, it has been shown that the exact convergence is not guaranteed anymore if the system starts to work autonomously (when the observer is switched off), since the observation error increases up to the size of the chaotic attractor itself. The asymptotic observer is more robust in the case of noise, but the error convergence to zero is slower, and only in an asymptotic way. In the latter case, it has been shown that several observers have to run in parallel, and the criteria to identify the one which has synchronised have been proposed. In conclusion, both observers are performant but in a different way; therefore, the most appropriate one has to be chosen according to the particular application which will be envisaged.

Figure 2 :

 2 Figure 2: evolution of the output y and ŷ -exact synchronisation

Figure 3 :

 3 Figure 3: evolution of the quadratic error -synchronization only in the two first iterations (observer switched off after)

Figure 4 :

 4 Figure 4: evolution of the reconstruction error -perpetual synchronisation

Figure 5 :Figure 6 :

 56 Figure 5: evolution of the outputs y and ŷ in the case of asymptotic synchronization

Figure 7 :

 7 Figure 7: evolution of the output error in the case of asymptotic synchronization and discrimination criteria for convergence

Figure 8 :

 8 Figure 8: evolution of the output error in the case of asymptotic synchronization and discrimination criteria for convergence


O Dezellus

R Voytovych

A P H Li

G Constantin

F Bosselet

J C Viala

Wettability of Ti 3 SiC 2 by Ag-Cu and Ag-Cu-Ti melts

Keywords: Wettability, Reactivity, Ti 3 SiC 2, Braze alloy, Silver, Copper

Recently, the ternary carbide Ti 3 SiC 2 has gained much attention due to its unique characteristics combining the properties of metals and ceramics (i.e., a low density, decent thermal and electrical conductivities, an excellent thermal shock resistance, a good machinability, damage tolerance, low friction and so on). The present study describes an investigation of the wettability in high vacuum of bulk Ti 3 SiC 2 by a classical braze alloy based on the Ag-Cu-Ti system. Two techniques, i.e., the sessile drop and dispensed drop methods, were utilized. The results indicated that spreading kinetics is controlled by deoxidation kinetics of Ti 3 SiC 2 surface under vacuum. The final contact angle on clean Ti 3 SiC 2 is very small (~ 10°), testifying the development of strong, metallic interactions across the liquid-solid interface. The reactivity between the ternary carbide and the liquid phase during isothermal heating at 800°C was also considered.

Introduction

Recently, nanolaminate ternary ceramics M n+1 AX n (where M corresponds to an early transition metal, A is an A-group element, and X represents either carbon or nitrogen) have attracted extensive attention due to their unique combination of metals and ceramic properties [START_REF] Barsoum | Synthesis and characterization of a remarkable ceramic: Ti 3 SiC 2[END_REF]. Among these compounds, titanium silicon carbide (Ti 3 SiC 2 ) has gained much interest as a promising candidate for numerous applications such as interphases in ceramic matrix composites [START_REF] Lan | Joining of carbon/carbon composites for nuclear applications[END_REF][START_REF] Rocher | Composite material with a ceramic matrix with lamellar interphase between refractory reinforcing fibres and matrix, and process for its manufacture[END_REF], electrical [START_REF] Tuller | Stable electrical contact for silicon carbide devices[END_REF][START_REF] Wenzel | Bulk diffusion studies of metallizations on titanium-based contacts at 600°C[END_REF] and tribological [START_REF] Myhra | Ti 3 SiC 2 -a layered ceramic exhibiting ultra-low friction[END_REF] contact layers or high-temperature structural applications [START_REF] Barnes | Corrosion of ternary carbides by molten lead (2008)[END_REF][START_REF] Nappé | Damages induced by heavy ions in titanium silicon carbide: Effects of nuclear and electronic interactions at room temperature[END_REF]. However, as for other ceramics, the synthesis of bulk Ti 3 SiC 2 on a larger scale has proven difficult in practice, and for this reason, studies devoted to joining Ti 3 SiC 2 with metals or ceramics are significant for promoting its applications [START_REF] Wenzel | Bulk diffusion studies of metallizations on titanium-based contacts at 600°C[END_REF].

Among the various joining technologies that exist, brazing is one of the most suitable for ceramic materials, especially for metal-to-ceramic joints [START_REF] Eustathopoulos | Wettability at High Temperatures[END_REF]. It is thus important to investigate the wettability of Ti 3 SiC 2 by liquid metallic alloys. Although some studies concerning the joining of Ti 3 SiC 2 with metals have already been conducted [START_REF] Gao | Joining of Ti 3 SiC 2 with Ti-6Al-4V alloy (2002)[END_REF][START_REF] Gu | Reactions between Ti and Ti 3 SiC 2 in temperature range of 1273-1573 K[END_REF], there exist, to the best of the authors' knowledge, no results on fundamental wettability characteristics.

The present work describes an investigation of the wetting of Ti 3 SiC 2 by Ag-40at.%Cu eutectic alloy (containing Ti or not) at 800 °C by way of the sessile and dispensed drop techniques under high vacuum. The reactivity at the liquid/solid interface was also considered in order to sustain the analysis of the wetting results.

Experimental section

The classical sessile drop technique was used to study the wetting in the case of a Ti-containing alloy, and the dispensed drop method was employed for Ag-40at.%Cu eutectic alloys. The experiments were carried out in a metallic furnace under a vacuum of about 5×10 -5 Pa obtained with a rotary vane pump and an oil diffusion pump, each equipped with a liquid nitrogen trap. In the classical sessile drop experiment, copper-silver eutectic alloy droplets of 50 to 100 mg were prepared prior to the wetting experiment by: (i) melting of pure Cu (99.999 wt.%) and Ag (99.999 wt.%) in an alumina crucible, and (ii) their deposition on vitreous carbon substrates, because of the high contact angle and low work of adhesion in this system, under high vacuum.

Finally the CuAg-Ti alloy was processed in situ during the sessile drop experiment by directly melting a piece of Ti (purity 99.7 wt.%) over the CuAg on the Ti 3 SiC 2 substrate. This procedure was used to avoid any direct contact between Ti and the ceramic before melting the alloy.

In the "dispensed drop" technique, the experiment consisted in heating Cu and Ag in an alumina crucible placed above the Ti 3 SiC 2 substrate [START_REF] Dezellus | Spreading of Cu-Si alloys on oxidized SiC in vacuum: experimental results and modelling[END_REF]. When employing this method, the melting and spreading processes could be separated, thus permitting fully isothermal tests to be performed. In both experiments, the drop images produced with an optical system fitted with a zoom (30× magnification) were recorded with a video camera connected to a computer, permitting automatic image analysis. This device enabled the linear dimensions (i.e., the drop base radius, R, and height) and the contact angle, θ, of the drop to be measured with an accuracy of ±1% and ±2°, respectively.

All the experiments were performed using Ti 3 SiC 2 substrates provided by 3-ONE-2 LLC [13]. The substrates were manufactured by pressureless sintering of a Ti 3 SiC 2 powder, which resulted in a dense polycrystalline monolithic ceramic. The Ti 3 SiC 2 powder was synthesized by milling of Ti, C and SiC powders followed by cold-pressing and reactive hot-pressing [START_REF] Barsoum | Synthesis and characterization of a remarkable ceramic: Ti 3 SiC 2[END_REF]. It should be noted that a significant amount of TiC (approximately 15 wt.%) was detected by XRD analysis. This was due to the unavoidable deviation to homogeneity when large batches of powders are processed. The substrate surface was polished down with a 1-µm diamond paste, thus giving rise to an average roughness value, R a , of approximately a few hundred nanometers.

The chemistry, morphology and microstructure of the interfacial reaction products were determined by means of SEM (equipped with energy dispersive X-ray spectroscopy -EDXS) and electron probe microanalysis (EPMA).

Results and discussion

Wetting Ag-40at.%Cu eutectic alloy Time is presented by solid line.

In the first case (filled square), the substrate was heated to a temperature of 900°C for 15 min, after which the temperature was decreased to 800°C. Once the latter temperature was attained, the classical dispended drop procedure (i.e., extrusion and deposition of the droplet) was applied.

Under such conditions, an instantaneous and excellent wetting could be observed and the first measurable contact angle after 40 ms of contact was already equal to 12°.

As calculated with a liquid-vapor surface energy for AgCu eutectic alloys of 950 mJ.m -2 [14], the work of adhesion at the liquid-solid interface has been found to be close to 1880 mJ.m -2 . This value represents 99% of the cohesion energy of the liquid, which is a sign of strong chemical interactions between the AgCu alloy and Ti 3 SiC 2 across the interface. Moreover, the triple line velocity is higher than 70 mm.s -1 (spreading over 5.4 mm for a contact time lower than 40 ms) -a feature characteristic of non-reactive metal/solid (ceramic or metallic) systems. The obtained value is similar to those of low viscosity liquid metals on solid metals [START_REF] Eustathopoulos | Wettability at High Temperatures[END_REF][START_REF] Naidich | Temperature effect on the kinetics of spreading and wetting in systems with different character of interaction of contacting phases[END_REF][START_REF] Saiz | Atomic dynamics and marangoni films during liquid-metal spreading[END_REF]. The results obtained regarding the work of adhesion and triple line velocity thus indicate that strong metal-like interactions has been established with the surface of Ti 3 SiC 2 .

It is already known that Ti 3 SiC 2 has various metal-like properties: at room temperature it demonstrates a thermal conductivity similar to pure Ti (37 W.m -1 .K -1 as opposed to 30 W.m -1 .K -1 for pure Ti) and an electrical conductivity higher than that of pure Ti or TiC (4.5×10 6 V -1 .m -1 as opposed to 2.3×10 6 V -1 .m -1 for Ti and 1.6×106 V -1 .m -1 for TiC) [START_REF] Barsoum | Synthesis and characterization of a remarkable ceramic: Ti 3 SiC 2[END_REF][START_REF] Barsoum | Thermal properties of Ti 3 SiC 2 (1999)[END_REF]. The apparent combination of ceramic and metallic properties has been explained by the existence of two types of Ti bonding in the crystal structure [START_REF] Zhou | Electronic structure and bonding properties in layered ternary carbide Ti 3 SiC 2[END_REF]. Indeed, the Ti 3 SiC 2 phase consists of a layered structure with a double Ti-C block, each block made up of two edge-sharing CTi 6 octahedra, separated by a square planar Si sheet. It should be noted that the titanium carbide structure can also be described as a stacking of such octahedra. The stacking sequence of Ti 3 SiC 2 (Ti II -C-Ti I -C-Ti II -Si) contains two unique sites for Ti, and there consequently also exist two unique chemical states for the Ti atom.

The charge density of Ti in the first C-Ti I -C layer has been shown to be similar to that of Ti in Ti-C but with a lower average spacing (0.213 nm instead of 0.216) [START_REF] Kisi | Structure and crystal chemistry of Ti 3 SiC 2 (1998)[END_REF][START_REF] Medvedeva | Electronic properties of Ti 3 SiC 2 -based solid solutions[END_REF][START_REF] Riley | Comparative analysis of Ti 3 SiC 2 and associated compounds using x-ray diffraction and x-ray photoelectron spectroscopy[END_REF]. In contrast, the charge density of the second type of Ti site (i.e. C-Ti II -Si sequence) contains a higher proportion of metallic bonds and hence contributes more to the metallic properties of Ti 3 SiC 2 . This is due to a slight distortion of the CTi 6 octahedra and a Ti II -Si bond distance that is somewhat larger than that of a normal covalent bond [START_REF] Kisi | Structure and crystal chemistry of Ti 3 SiC 2 (1998)[END_REF][START_REF] Riley | Comparative analysis of Ti 3 SiC 2 and associated compounds using x-ray diffraction and x-ray photoelectron spectroscopy[END_REF].

In the second case (open circles in Fig. 1), the liquid droplet was extruded from the crucible and dispensed onto the Ti 3 SiC 2 substrate once the experimental temperature reached 800°C. The spreading proceeded from the first measurable contact angle value of 126° with a triple line velocity of about 1 µm.s -1 . After 1250 s, the temperature was raised to 900°C with a heating rate of 20°C.min -1 , which was followed by an increase in the spreading velocity up to a maximum of about 20 µm.s -1 . Finally, the capillary equilibrium was rapidly obtained, with a contact angle value of 20°, before the temperature of 900°C was reached. One should note that this final contact angle value could not be compared to the one from the first experiments. Indeed, during spreading, the surface chemistry of the substrate was modified by the reactivity between the liquid and Ti 3 SiC 2 .

This is described further in the section concerning the reactivity.

The low spreading rate and the high activation energy of the spreading process are typical for reactive metal/ceramic systems when the reaction at the solid/liquid/vapor triple line controls the spreading rate [START_REF] Dezellus | Spreading of Cu-Si alloys on oxidized SiC in vacuum: experimental results and modelling[END_REF]. In the Ag-40at.%Cu/Ti 3 SiC 2 system, the long spreading time is attributed to the deoxidation of the Ti 3 SiC 2 surface under very low oxygen partial pressure (approximately 10 -20 mbar). Indeed, it has been established that the formation of the native oxide layer on a pure polycrystalline Ti 3 SiC 2 sample is the result of a chemical interaction in two systems, i.e., in Ti-C-O for the C-Ti I -C bonds and in Si-Ti-C-O for the C-Ti II -Si bonds [START_REF] Riley | Comparative analysis of Ti 3 SiC 2 and associated compounds using x-ray diffraction and x-ray photoelectron spectroscopy[END_REF].

The obtained experimental results, especially the high spreading rate observed after the heat treatment at 900°C (70 mm.s -1 ), are attributed to the fact that the Ti 3 SiC 2 surface has been deoxidized at 900°C, without any contact with the liquid, and thus has acquired a metallic character (see Fig. 1). It is interesting to note that Kozlova et al. observed a similar behavior for Ti containing stainless steel substrates [START_REF] Kozlova | Wetting and brazing of stainless steels by copper-silver eutectic[END_REF]. According to their results the deoxidation of the solid surface is mainly due to reduction by residual carbon present in solid steel to form carbon monoxide (CO). In the present study, the same atmosphere is used and the C activity in the solid should also be approximately the same because some TiC is present both in stainless steels and in Ti 3 SiC 2 substrates. Therefore, the same mechanism, reduction of oxide film by C leading to CO formation, can be responsible of deoxidation of the surface under vacuum. At 800°C, this process can remain efficient but its kinetics is significantly reduced, and instantaneous wetting cannot be observed.

Because of structural similarities (i.e., the presence of Ti 6 C edge-sharing octahedra), it is interesting at this point to compare the present results with contact angle values reported for TiC x in the literature [START_REF] Riley | Comparative analysis of Ti 3 SiC 2 and associated compounds using x-ray diffraction and x-ray photoelectron spectroscopy[END_REF], even if deoxidation at low temperatures of that carbide is difficult [START_REF] Li | Influence of oxygen partial pressure on the wetting behaviour of titanium carbide by molten copper and other metals[END_REF]. For TiC x with the same ratio between C and Ti as in Ti 3 SiC 2 (i.e., x = 0.67), contact angles are much higher than on Ti 3 SiC 2 (from 60° to 80° for pure Cu and Ag as opposed to 10° in the present study for Ag-40at%Cu) [START_REF] Frage | Wetting of TiC by non-reactive liquid metals[END_REF]. Therefore, the metallic character of the MAX phase Ti 3 SiC 2 is more pronounced than for TiC x , even when sub-stoechiometric.

Ti-containing Ag-40at.%Cu eutectic alloys

Ti is commonly added to the Ag-40at.%Cu eutectic alloy to produce the most widely used active braze family. Moreover, the dissolution of Ti from a Ti 3 SiC 2 substrate in the liquid alloy can be expected during the spreading process. For this reason, various experiments have been performed

on AgCu-2,5wt.%Ti by the classical sessile drop technique. It is worth noting that pre-heating to higher temperatures is not allowed with this procedure. Subsequently, spreading occurred with a triple line velocity of about 1 µm.s -1 , which was similar to the value observed for the AgCu eutectic alloy without Ti (cf. filled square in Fig. 1). After 1300s, the temperature was raised to 900°C with a heating rate of 20°C.min -1 , which led to an increase in the spreading velocity by one order of magnitude and, finally, to capillary equilibrium with a contact angle value of approximately 10°.

It is worth noting that the addition of Ti has no effect on the initial contact angle value, nor on the triple line velocity. Concerning the final contact angle values (12° against 8°), they can be considered as nearly the same.

It can be concluded that Ti has no significant effect on the process of deoxidation of the Ti 3 SiC 2 substrate. This conclusion is counter-intuitive since Ti is recognized as an element promoting wetting on numerous solids such as oxides [START_REF] Voytovych | The relation between wetting and interfacial chemistry in the CuAgTi/alumina system[END_REF], nitrides [START_REF] Loehman | Reactions of Ti and Zr with AlN and Al2O3[END_REF][START_REF] Nicholas | Some observations on the wetting and bonding of nitride ceramics[END_REF] or carbides [START_REF] Dezellus | Diffusion-limited reactive wetting: spreading of Cu-Sn-Ti alloys on vitreous carbon[END_REF]29]. This effect of Ti is usually attributed to the formation by reaction with the solid substrate of a better wetted compound with a metallic character, e.g., TiC 1-x or TiO 1-x [START_REF] Voytovych | The role of adsorption and reaction in wetting in the CuAg-Ti/alumina system[END_REF]. With Ti 3 SiC 2 , a possible reaction would be the transformation of the iono-covalent native oxide film to a thin continuous layer of the wettable oxide TiO 1-x .

In the present case, the thermodynamic activity of Ti is too low to promote the formation of TiO 1-x with decent wetting. Indeed, during temperature rise to 900°C the kinetics of dissolution of Ti in Ag-40at.%Cu eutectic is significantly reduced by the formation at solid Ti/liquid AgCu interface of different Cu x Ti y compounds such as Cu 4 Ti, Cu 3 Ti 4 or CuTi [START_REF] Andrieux | Low-temperature interface reaction between titanium and the eutectic silver-copper brazing alloy (2009)[END_REF]. As a consequence the effective activity of Ti in the drop is lower than the nominal one.

Reactivity

Ag-40at.%Cu eutectic alloy

Figure 3 shows an SEM micrograph representing a cross-section of the interface between an AgCu eutectic and a Ti 3 SiC 2 substrate after a prior heat treatment of the substrate at 900°C and a subsequent isothermal spreading at 800°C (filled square in Figure 1). No interfacial reaction layer was formed at the substrate surface. However the spreading was followed by a deep penetration of the liquid into the grain boundaries of the substrate, as revealed by the presence of bright solidified silver channels in Fig. 3. This penetration phenomenon was associated with solid state diffusion of Cu into the grains of Ti 3 SiC 2 . As a consequence the substrate was transformed up to 40 micrometers below the liquid/solid interface after 420 s contact at 800°C. Quantitative electron probe micro analyses have pointed out a composition of the transformed light grey grains that is likely Ti 47 Cu 8 Si 13 C 32 . At the present time, it is not possible to determine whether this composition is that of a quaternary phase or of an intimate mixture of fine-grained binary or ternary phases.

Note that Si was barely detectable by EPMA in the solidified liquid with a mean amount lower than 1 wt%.

Fig. 3: SEM micrograph representing a cross-section of the interface between an AgCu eutectic and a Ti 3 SiC 2 substrate after a prior heat treatment of the substrate at 900°C and a subsequent contact time of 420 s at 800°C (filled square in Figure 1).

It is worth noting that the existence of reactivity between Cu and Ti 3 SiC 2 has already been reported on by Guo et al. in the case of Cu magnetron sputtering layers heat-treated at temperatures ranging from 900°C to 1100°C [START_REF] Guo | Surface strengthening of Ti 3 SiC 2 through magnetron sputtering cu and subsequent annealing[END_REF]. The authors identified Cu 3 Si and substoechiometric TiC 1-x . as the reaction products. Moreover, other researchers have already reported on topotactic transformation of Ti 3 SiC 2 by outward diffusion of Si along the layers of the structure that can be partially balanced by inward diffusion of certain other elements [START_REF] Barsoum | Oxidation of Ti 3 SiC 2 in air (1997)[END_REF][START_REF] Barsoum | Comment on "reaction layers around SiC particles in Ti: an electron microscopy study[END_REF][START_REF] El-Raghy | Reaction of al with Ti 3 SiC 2 in the 800-1000°C temperature range[END_REF]. In the present case, the partial exchange of Si atoms for Cu in the structure between two layers of CTi 6 octahedra could be considered. A complete reactivity study of Ag-Cu(Ti) melts on Ti 3 SiC 2 is currently in progress.

Ti-containing AgCu eutectic alloys

As demonstrated in Figure 4, an AgCu eutectic to which Ti is added differs from a pure AgCu eutectic in that a secondary phase has appeared in the liquid. Crystals with hexagonal shape were identified by EDXS analysis as Ti 5 Si 3 C x , signifying that some limited dissolution of Ti 3 SiC 2 had occurred. This, in turn, led to an increase in the amount of Si in the liquid and finally to the precipitation of Ti 5 Si 3 C x . It was also observed that, after 20 min at 800°C and 10 min at 900°C, the Ti 3 SiC 2 substrate was penetrated and transformed up to a depth of 100 µm.

Conclusions

Wetting experiments performed by the dispensed drop technique have demonstrated that a Ti 3 SiC 2 surface can be deoxidized under high vacuum above 850°C leading to a strong metallic character.

The results indicated that spreading kinetics is controlled by deoxidation kinetics of Ti 3 SiC 2 surface under vacuum. The final contact angle on clean Ti 3 SiC 2 is very small (~ 10°), testifying the development of strong, metallic interactions across the liquid-solid interface. The wetting experiments also confirmed the existence of a complex reactivity between Cu and Ti 3 SiC 2 and a study dedicated to this reactivity is on-going.

  hal-00423015, version 1 -9 Oct 2009 Author manuscript, published in "Journal of Materials Science (2009) xxx" DOI : 10.1007/s10853-009-3941-6

Fig. 1 Fig. 1 :

 11 Fig. 1 shows the variations in contact angle, θ, and temperature as functions of time for the spreading of two AgCu eutectic alloys over Ti 3 SiC 2 in the experiments performed by the dispensed drop technique.

Fig. 2 :Figure 2

 22 Fig. 2: Contact angle measured by the classical sessile drop technique as a function of time forAgCu-2,5wt.%Ti alloy on Ti 3 SiC 2 at 800°C during about 1300 s and then after heating up to

Fig. 4 :

 4 Fig. 4: SEM micrograph representing a cross-section of the interface between an AgCu-2,5wt.%Ti melt and a Ti 3 SiC 2 substrate after 7 min of contact at 800°C and 10 min at 900°C (experiment presented by open triangles in Figure 2)