Chouaib Afri

Laurent Bako

Vincent Andrieu

Pascal Dufour

Adaptive identification of continuous-time MIMO state-space models

This paper studies the problem of identifying linear continuous-time state-space models from input-output measurements. An adaptive identifier is developed for the online estimation of both the state and the model parameters in a deterministic framework. Our approach is a non parametric one in the sense that it provides an arbitrary realization of the system. It relies on ideas from the subspace identification literature and adaptive observer.

I. INTRODUCTION

State-space models provide a nice and compact representation of linear multiple input multiple output (MIMO) systems. This feature probably justifies, at least partially, its tremendous and still increasing success over input-output models e.g., for multi variable control design, fault detection and isolation, performance and stability analysis. From an identification perspective, such models enjoy the advantage that their construction depends on only one structural parameter, namely the so-called McMillan degree of the system. On the other hand, estimating a black-box state-space model from input-output data is not a trivial problem due to the fact that both state and parameters are unknown.

The emergence of subspace identification methods [START_REF] Van Overschee | Subspace Identification for Linear Systems : Theory -Implementation -Applications[END_REF], [START_REF] Katayama | Subspace Methods for System Identification[END_REF], [START_REF] Verhaegen | Filtering and System Identification : A Least Squares Approach[END_REF] in the 90s has initiated a steady progress in the treatment of this problem. Since then, a number of interesting methods have been developed for efficiently recovering statespace models along with the associated state from a partial observation of the input-output behavior. Nevertheless, such techniques are mainly applied for computing discretetime models from sampled-data collected in batch mode. A key ingredient of subspace estimation techniques is a rank factorization step, the Singular Value Decomposition (SVD) algorithm. But, since it is computationally expensive, one cannot afford to run it repeatedly in online applications. Therefore, in available extensions of subspace methods to online recursive estimation, the research effort has been essentially devoted to surmounting the need for SVD [START_REF] Mercère | Propagator-based methods for recursive subspace model identification[END_REF]. Again, apart from a few exceptions [START_REF] Bergamasco | Continuous-time predictor-based subspace identification using laguerre filters[END_REF], this was done essentially for discrete-time models. The subspace approach can be viewed as a non parametric one since it relies on the estimation of an arbitrary basis for a subspace.

A complementary approach is the one of adaptive observers where a state basis is enforced by working on canonical parameterization [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF], [START_REF] Zhang | Adaptive observer for multiple-input multiple-output (MIMO) linear time-varying systems[END_REF], [START_REF] Afri | Identification of linear systems with nonlinear luenberger observers[END_REF]. While many impressive results have been achieved on this head, it is worth mentioning some drawbacks of canonical representations : (1) they are rarely minimal in the case of MIMO systems unless a great deal of prior knowledge is assumed (e.g. availability of Kronecker observability/controllability indexes [START_REF] Kailath | Linear systems[END_REF]) (2) they might be numerically badly conditioned.

The current paper proceeds from the following remark. There is a need for continuous-time models which can be viewed as more natural time-description of physical systems than discrete-time models. One can then argue that since sensing devices can only acquire sampled versions of the system data, it is easier to construct a discrete-time model first and then convert it into a continuous-time one. This is indeed the most developed path of research in system identification. Note however that accuracy of such conversion is dependent of the sampling frequency which therefore must be chosen, when possible, with great care. Moreover, in case the model is to be computed online in an adaptive fashion, the cost of the conversion might not be affordable. An alternative solution is to identify directly a continuous-time model from interpolation of sampled-data [START_REF]Identification of Continuous-time Models from Sampled Data[END_REF]. In this paper, we present an adaptive identifier for linear continuous-time MIMO state-space models. The proposed method is capable of estimating online both the state and the system matrices. The paper has two main contributions : (a) a cheap, adaptive and continuous-time identifier for MIMO state-space models which does not involve any use of the costly SVD algorithm ; (b) an exponential convergence analysis under appropriate input sufficiency of richness conditions and model minimality.

The outline of the paper is as follows. In Section I, we underline the problem to deal with and set out the assumptions. The proposed solution idea is presented in Section III. In Section IV, we show that the estimates converge exponentially fast to their respective desired values. We conclude by an application of the algorithm on a linear MIMO system in Section V. II. LINEAR SYSTEM Consider a linear system described by a state-space model of the form

S : ẋ = Ax + Bu y = Cx + Du (1)
where

x ∈ R n , u ∈ R ny , y ∈ R

III. ESTIMATION METHOD

A. Main idea

Let f > n be an integer and introduce the notations

y f = y ẏ • • • y (f -1) u f = u u • • • u (f -1) (2)
and

O f O f (A, C) = C (CA) • • • (CA f -1 ) T f T f (A, B, C, D) =     D 0 • • • 0 CB D • • • 0 . . . . . . . . . . . . CA f -2 B CA f -3 B • • • D     .
(3) Proceeding from (1), we can derive the following equation

y f = O f x + T f u f . (4)
In practice, the time derivatives of u and y are not measured in general. The only signals that can be reasonably assumed to be available are the input and output. However they are in general noisy and differentiating noisy measurements is well-known to come with some serious drawbacks. One way to avoid numerical computation of derivatives is to work with filtered data. The filter has the further advantage that when it is appropriately chosen, it is capable of attenuating the effects of noise. Let F (s) = 1/d(s) be a stable filter with d(s

) = s f + α f -1 s f -1 + • • • + α 0 a
polynomial in the Laplace operator s. The filter must be chosen to cut off potential noise. In general, the frequency spectrum of the noise is located at high frequencies. So, if we choose a linear filter with large enough eigenvalues (in absolute value) we can reduce the noise effect while preserving the dynamics of the to-beidentified-system. Note that many other choices of the filter F (s) are possible, see e.g. [START_REF]Identification of Continuous-time Models from Sampled Data[END_REF]. The method to be presented is not attached to any specific choice of the filter and so remains applicable for any valid choice1 of F (s). For this reason, we will not elaborate more on this implementation aspect here.

Posing p = F (s)y f , q = F (s)u f , x = F (s)x, we obtain p = F (s)(ξ f (∂) ⊗ I ny )y, q = F (s)(ξ f (∂) ⊗ I nu )u (5)
where

ξ f (∂) = 1 ∂ • • • ∂ f -1 with ∂ = d dt denoting the time derivative operator.
We can express ( 4) and ( 5) in state-space form as

ṗ = Λ c ⊗ I ny p + l ⊗ I ny y (6a) q = (Λ c ⊗ I nu ) q + (l ⊗ I nu ) u (6b) O f x = p -T f q (6c)
with

Λ c =        0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . • • • . . . . . . 0 0 • • • • • • 1 -α 0 -α 1 • • • • • • -α f -1        , l =        0 0 . . . 0 1        . Note that for any k, the pair (Λ c ⊗ I k , l ⊗ I k ) is control- lable.
Using the filter, p and q can be generated from the measurements using Eqs (6a)-(6b). But we are still facing the difficulty that x in Eq. ( 6c) is unknown. To overcome this we will follow the method proposed in [START_REF] Bako | A least squares approach to the subspace identification problem[END_REF].

Since the system is observable, O f has full column rank so that there exists a matrix H ∈ R n×f ny such that rank(HO f ) = n. Hence T = HO f is a square (n × n) non singular matrix. In fact if H is randomly generated, then the required rank condition is satisfied with probability one. But H can be chosen in many other ways. For example, if the system to be identified is MISO, then observability implies that the n first rows of O f are linearly independent so that H = [I n 0 n×(f -n) ] is a valid choice. In the MIMO case, if a set of rows of O f indexed by {i 1 , . . . , i n } is known to be linearly independent beforehand, then H = [e i1 • • • e in ] with e i ∈ R f ny the i-th canonical basis vector, fulfills the desired rank condition. By a change of state coordinates, let us define a new state x as x = T x.

Accordingly, the system realization becomes ( Ã, B, C, D)

with à = T AT -1 , B = T B, C = CT -1 , D = D. Multiplying (6c) by H yields x = Hp -HT f q. (7)
On the other hand, observe from (6c) that

p = O f x + T f q = Õf x + T f q where Õf = O f T -1 . Inserting (7) into the foregoing expression of p leads to p = M ϕ (8)
with

M = Õf I -Õf H T f ∈ R f ny×(n+f nu) (9) ϕ = Hp q ∈ R n+f nu . (10)
In [START_REF] Katayama | Subspace Methods for System Identification[END_REF] the vectors p and ϕ are known ; the parameter matrix to be estimated is the structured matrix M . It can be usefully remarked that (8) results from (6c) by a multiplication on the left by the projection matrix I -Õf H.

B. Adaptive identifier

From the equation ( 8) derived above, we can proceed with the estimation of the matrix M . We use the recursive least squares method. At any time t, the estimated parameter matrix is obtained by

M (t) = arg min M V t (M ) where V t (M ) = 1 2 e -αt tr (M -M 0 )P -1 0 (M -M 0 ) + 1 2 t 0 e -α(t-τ ) r(τ ) 2 p(τ ) -M ϕ(τ ) 2 2
where α > 0 is a design parameter ; r(τ ) 2 ≥ 1 is a normalizing factor which is upper-bounded, P 0 is a symmetric positive-definite matrix, M 0 = M (0) is the initial value of M (t), tr refers to matrix trace operator. Setting the gradient of V t (M ) to zero and differentiating the resulting solution with respect to time yields the continuous-time recursive least squares estimator :

Ṁ = (p -M ϕ) ϕ P r(t) 2 , M (0) = M 0 (11) Ṗ = αP - P ϕϕ P r(t) 2 , P (0) = P 0 0 (12)
Here the notation P 0 0 means that P 0 is positive-definite. Simulating ( 11)-( 12) provide an estimate of the matrix M at any time. Now the question to be addressed is how to get back to the system matrices and the state. This can be done by exploiting the structure of M as follows.

Consider partitions of M in the forms

M =   M 11 M 12 M 21 M 22 M 31 M 32   , M 11 ∈ R ny×n , M 31 ∈ R ny×n , M = M 1 M 2 , M 1 ∈ R f ny×n .
Then we can compute the system matrices by the formulas

C = M 11 , Ã = M 11 M 21 † M 21 M 31 . ( 13
)
Estimation of B and D. We are left with estimating B and D. To this end, note that the system output can be expressed as

y(t) = Ce Ãt Ψ(t) u(t) ⊗ I ny   x0 vec( B) vec(D)   ( 14
)
where the matrix Ψ(t) is defined by

Ψ(t) = t 0 u(τ ) ⊗ Ce Ã(t-τ ) dτ.
Indeed, it is easy to see that Ψ obeys the following differential equation

Ψ(t) = Ψ(t) I nu ⊗ Ã + u(t) ⊗ C. ( 15
)
Hence by using the pair ( Ã(t), C(t)) delivered by [START_REF] Verhaegen | Filtering and System Identification : A Least Squares Approach[END_REF] the matrix Ψ(t) can be generated at any time t. Writing Eq. ( 14) in the linear regression form

y(t) = Ce Ãt x0 + Φ(t)θ, (16)
with

Φ(t) = Ψ(t) u(t) ⊗ I ny (17) θ = vec( B) vec(D) , (18)
we can derive a recursive least squares algorithm for estimating θ as follows,

θ = QΦ (y -Φθ) ρ(t) 2 , θ(0) = θ 0 (19) Q = βQ - QΦ ΦQ ρ(t) 2 , Q(0) = Q 0 0 (20)
where ρ(t

) 2 = 1 + tr Φ(t)Q(t)Φ(t) .
Estimation of the state. The state is obtained as

x = H(p -T f q).
This is indeed a filtered version of the transformed state T x since x = F (s)(T x). Note that x might be more desirable in practice than the state T x for the following reason. Since the noise spectrum is generally located in a high frequency range, if the filter F (s) is properly chosen, then x is likely to contain less noise than T x. But in case one is interested in the state T x, note that it can be generated by constructing an observer based on the identified realization ( Ã, B, C, D).

IV. EXPONENTIAL CONVERGENCE

In this section, we prove exponential convergence of the recursive identifier ( 11)-( 12) and ( 19)-(20). To proceed, we need to introduce some specific definitions of persistence of excitation and sufficiency of richness.

Definition 1 (persistence of excitation): A locally integrable signal ϕ : R + → R d is said to be persistently exciting (PE) if there exist constant numbers α 1 , α 2 , T > 0 and t 0 ≥ 0 such that

α 1 I t+T t ϕ(τ )ϕ(τ ) dτ α 2 I ∀t ≥ t 0 .
Here the notation S R is used to mean that R -S is positive semi-definite.

Inspired by [START_REF] Mareels | Persistency of excitation criteria for linear, multivariable, time-varying systems[END_REF], we also define sufficiency of richness of an input signal as follows.

Definition 2 (Sufficiency of richness): A smooth signal u : R + → R nu is said to be sufficiently rich (SR) of order k if

• u (i) (0) = 0 for i = 0, . . . , k -1 and • for any γ > 0, the signal ūk ∈ R knu defined by

ūk = 1 (s + γ) k u u • • • (u (k-1) )
is PE. It can be shown that if u is SR of order k, then u is also SR of order for any ≤ k. Now we relate sufficiency of excitation of the input and persistence of excitation of the state under controllability assumption.

Lemma 1: Let x ∈ R n be a signal generated by

ẋ = Ax + Bu, x(0) = 0 (21)
where it is assumed that A is Hurwitz and (A, B) is controllable. Then the signal x is PE if u is sufficiently rich of order m where m = deg(m A (λ)), with m A (λ) denoting the minimal polynomial of A.

Proof: Let m = deg(m A (λ)) and m A (λ) = λ m + c m-1 λ m-1 + • • • + c 1 λ + c 0 . Differentiating m times the first equation in (1) yields x (m) = A m x + C m (A, B)u m ,
where u m is defined from u using the notational convention [START_REF] Bako | A least squares approach to the subspace identification problem[END_REF] and

C m (A, B) = [A m-1 B • • • AB B]. By definition of the minimal polynomial m A (A) = 0 so that A m = -c 0 I - c 1 A -• • • -c m-1 A m-1 = -(c ⊗ I m )O m (A, I), where c = [c 0 • • • c m-1 ] . It follows that x (m) = -(c ⊗ I m )O m (A, I)x + C m (A, B)u m
On the other hand, we have

x m = O m (A, I)x + T m (A, B, I, 0)u m with x m = [ x ẋ • • • x (m-1)
] and T m (A, B, I, 0) defined similarly as in [START_REF] Bergamasco | Continuous-time predictor-based subspace identification using laguerre filters[END_REF]. We get after some calculations,

x (m) + c m-1 x (m-1) + • • • + c 1 ẋ + c 0 x = C m (K ⊗ I nu )u m (22) where K =        1 c 1 1 c 2 c 1 1 . . . . . . . . . . . . c m-1 c m-2 • • • c 1 1        .
Let γ > 0 be an arbitrary number and ūm (t) = 1/(s + γ) m u m (t). By applying the Laplace transform to (22), it follows that 2

x(t) = (s + γ) m m A (s) C m (K ⊗ I nu )ū m (t) Since (A, B) is controllable, C m is full row rank. As a consequence, C m (K ⊗ I nu )
is also full row rank because K ⊗ I nu is a square non singular matrix. We can therefore apply a similar reasoning as in [10, Theorem 4.1,Theorem 4.2] to conclude.

A. Convergence of the (A, C)-estimates

To prove convergence of matrices à and C we must prove the convergence of matrix M . So, let us start with the following proposition.

Proposition 1: Assume that the matrices A and Λ c from (1) and ( 6) are Hurwitz. If the pair (A, B) of system ( 1) is controllable and the input u is SR of order f + m, then the signal ϕ defined in [START_REF] Mareels | Persistency of excitation criteria for linear, multivariable, time-varying systems[END_REF] is PE.

2. There is a slight abuse of notation here : time and Laplace variable s are simultaneously used. Now we state the exponential convergence result of M towards the true value M o .

Theorem 1 (Exponential convergence): Let Assumptions A1 and A2 hold. If the input u in system (1) is sufficiently rich of order f + m, then M defined in ( 11)-( 12) converges exponentially fast to M o .

Proof: The proof is similar to the proof of Corollary 4.3.2 in [START_REF] Ioannou | Robust Adaptive Control[END_REF].

B. Convergence of the (B, D)-estimates

To study the convergence of ( 19)-( 20), we may need some more notations. For the sake of clarity, denote with ( Ão , Bo , Co , Do ) the "true" realization of the system (with respect to the same basis as the estimated one). Indeed such a realization is obtainable from any other "true" minimal realization (A, B, C, D) by the formulas

Ão = (HO f (A, C))A(HO f (A, C)) -1 , Bo = (HO f (A, C))B, Co = C(HO f (A, C)) -1 , Do = D. (23)
It must be noted that in spite of the appearance, for a given matrix H, the expressions ( Ão , Bo , Co , Do ) in ( 23) are independent of any specific realization. Pose Φ o (t) = Φ(t, Ão , Co ), that is, Φ o (t) is the matrix arising from ( 15)-(17) if the true matrices were used for the generation of Ψ. Similarly, let Φ(t) = Φ(t, Ã(t), C(t)). By letting

θ o = vec( Bo ) vec(D o )
, define the estimation error as e(t) = θ(t) -θ o , where θ(t) is defined by ( 19)-(20). Then

ė = - QΦ Φ ρ 2 e + v (24)
with

v = QΦ ρ 2 Co e Ão t x0 + (Φ o -Φ) θ o . (25)
If we can prove that (i) the homogeneous part of (24) is exponentially stable and (ii) v(t) vanishes as t → ∞ then, we can conclude on the convergence of the estimation error e towards zero. This argument is supported by the following lemma (see, e.g., [4, chap 4]).

Lemma 2: Consider a time-varying system

ż(t) = A(t)z(t) + v(t)
with exponentially stable homogeneous part ż(t) = A(t)z(t).

Assume that v(•) is bounded and lim t→∞ v(t) = 0. Then lim t→∞ z(t) = 0. By a similar reasoning as in the proof of Theorem 1, proving assertion (i) above boils down essentially to showing that Φ is PE, which we do in Lemmas 3 and 4 below. Lemma 3: Let Assumptions A1-A3 hold for system (1) and let u be sufficiently rich of order m. Then Φ o = Φ(t, Ão , Co ) defined as in (15)-( 17) is PE.

Proof: Let F ∈ R n×nu , G ∈ R ny×nu be nonzero matrices and let θ = vec(F ) vec(G) .

We first show that for a given θ with θ 2 = 1, w(t) Φ o (t)θ is PE with lower bound α 1 (θ) > 0. The so-defined w is indeed the output of the following system

ż = Ão z + F u, z(0) = 0 w = Co z + Gu (26)
Without loss of generality, we can assume that the realization (z, Ão , F, Co , G) is minimal 3 with state dimension d ≤ n.

Note that if w is PE when d = n, then it will also be PE when d < n. It is therefore enough to study the case d = n.

Arguing as in the proof of Lemma 1, it can be shown that

w(t) = (s + γ) m m A (s) Ω(s)ū m (t)
where

Ω(s) = Co C m ( Ão , F )(K ⊗ I nu ) + m A (s)(e 1 ⊗ G) with e 1 = [1 0 • • • 0]
. The rest of the proof consists in applying [START_REF] Mareels | Persistency of excitation criteria for linear, multivariable, time-varying systems[END_REF]Theorem 4.2]. For this purpose, we must verify that Ω(s) is full row rank, that is, no γ can be found such that γ Ω(s) = 0 for all s. This follows easily from the controllability property of ( Ão , F ) and the full row rank assumption on Co .

We have hence shown that for any nonzero θ with θ 2 = 1, w(t) = Φ o (t)θ is PE, that is, there are T, t 1 and α 1 (θ) > 0 and α 2 > 0 such that

α 1 (θ)I t+T t w(τ )w(τ ) dτ α 2 I ∀t ≥ t 1 .
As a consequence, by taking the trace, we get θ R(t)θ ≥ α 1 (θ)n θ > 0 for all t ≥ t 1 , where n θ is the dimension of θ and

R(t) = t+T t Φ o (τ ) Φ o (τ )dτ.
On the other hand,

inf t≥t1 θ R(t)θ = inf t≥t1 λ min [R(t)] θ 2 2 ≥ α 1 (θ)n θ . It follows that inf t≥t1 λ min [R(t)] > n θ inf θ =0, θ 2 =1 α 1 (θ) > 0 so that Φ o is PE.
Under the conditions of Theorem 1 we know that ( Ã(t), C(t)) converges exponentially fast to ( Ão , Co ) as t → ∞. We then call rate of convergence the largest number c such that

sup k∈N C(t) Ã(t) k -Co ( Ão ) k 2 ≤ λe -ct (27)
for a certain constant λ > 0. Here, the notation • 2 refers to matrix 2-norm (largest singular value). Lemma 4: Let Assumptions A1 and A2 hold for system (1) and let u be sufficiently rich of order at least f + m. If the rate of convergence defined in (27) satisfies c > 1, then Φ = Φ(t, Ã(t), C(t)) is PE, i.e., there exist T, β 1 , β 2 > 0 such that

β 1 I t+T t Φ(τ ) Φ(τ )dτ β 2 I.
for any t larger than a certain T 1 .

3. Indeed since F = 0 one can always reduce the considered realization to a minimal one.

We turn now to checking assertion (ii), that is, we question whether v defined in (25) may vanish as t → ∞.

Lemma 5: Under the assumptions of Lemma 4, it holds that lim t→∞ v(t) = 0.

V. SIMULATION

As mentioned in Section II our goal here is to give a linear model from the knowledge of outputs and inputs. In other words, the determination of the model (1) matrices A, B, C and D in an arbitrary basis. To illustrate the performance of the proposed identification algorithm, we estimate the MIMO linear system described by the following matrices :

A =     -1.4 0.3 -0.6 -0.3 0.3 -0.9 -0.9 0.1 -0.6 -0.9 -2.6 -1.0 -0.3 0.1 -1.0 -1.8     , B =     0 0 -0.4 1.3 0.6 0.2 0 0.6     , C = 0.9 0 0 1.3 -1.1 -0.8 0.9 0.4 , D = 0.8 -1.5 0 0 .
A white noise was selected as an input signal u(t) which is sufficiently rich in frequency and the algorithm initial values are given as : M (0) = 0, θ(0) = 0, P (0) = I, ψ(0) = 0 and Q(0) = I.

We recall that the algorithm is composed of two parts. The first part is given by Eqs (6a)-(6b) and ( 11)- [START_REF] Van Overschee | Subspace Identification for Linear Systems : Theory -Implementation -Applications[END_REF]. The second part is given by Eqs (15), ( 19) and (20) ; this latter estimates ( B, D)-matrices once the first part of the algorithm has converged. The matrix H used to calculate the vector ϕ is generated in a random way.

The state basis of the identified realization is not necessarily the same as that of the real system. It follows that the estimated matrices are not numerically identical but have common characteristics such as eigenvalues that are left unchanged under a similar transformation. Fig. 1 shows the convergence of the matrix à eigenvalues to those of the true matrix A. It can be seen that estimated and true eigenvalues coincide perfectly in the noise-free scenario. Fig. 2 is the bode diagram of the transfer function between the first output and the first input ; it shows that the estimated frequency spectrum converges perfectly to the real one and the same can be seen for the phase diagram. The expression Õf B = O f B is also invariant through a similar transformation and is used to compute the error norm shown in Fig. 3 and4. The relative error norm

ε r = Õf B -O f B F O f B F
vanishes in Fig. 3 which means that the first f Markov parameters of the system are well estimated and in Fig. 4 it differ a little in the presence of a moderate amount of noise. VI. CONCLUSION In this paper, we have developed a new identification algorithm for adaptively identifying linear continuous-time MIMO systems in state-space representation. The main challenges with such a problem are (i) estimate time derivatives of input-output signals (ii) provide a cheap update process for the system matrices while the state is unknown (iii) analyze convergence of the whole algorithm. The contribution of the paper lies essentially in the last two points. Simulation results tend to show that the algorithm performs very well in a deterministic context. However further work might be needed to strengthen its robustness to noise. In particular, the H-matrix based projection step can probably be improved by choosing the matrix H to be the transpose of an off-line prior estimate of the extended observability matrix O f . Also, there is still a room for improvement concerning the filtering of the input-output signals.

Fig. 1 .

 1 Fig. 1. Ã eigenvalues convergence without output noise (α = 4.10 -1 , β = 8.10 -2 ).

Fig. 2 .Fig. 3 .

 23 Fig. 2. Bode diagram of output 1 input 1 transfer function

Fig. 4 .

 4 Fig. 4. Error norm εr with output noise of 50 dB (α = 10 -2 , β = 8.10 -3 ).

C. Afri, V. Andrieu and P. Dufour are with the Université de Lyon, F-69622, Lyon, France -Université Lyon 1, Villeurbanne, France -CNRS, UMR 5007, LAGEP, France. afri@lagep.univ-lyon1.fr

L. Bako is with Laboratoire Ampère -Ecole Centrale de Lyon -Université de Lyon, 69130 France. laurent.bako@ec-lyon.fr

For a different choice of the filter F (s), one just needs to adjust the equations (6a)-(6b) which generate p and q ; the other steps of the identification method remain unchanged.


Vincent Grelet
email: vincent.grelet@volvo.com

Pascal Dufour
email: dufour@lagep.univ-lyon1.fr

Madiha Nadri
email: nadri@lagep.univ-lyon1.fr

Vincent Lemort
email: vincent.lemort@ulg.ac.be

Thomas Reiche
email: thomas.reiche@volvo.com

Explicit multi-model predictive control of a waste heat Rankine based system for heavy duty trucks

This paper presents an explicit multi-model predictive controller (MMPC) for a waste heat recovery system (WHRS) mounted on a heavy duty truck engine. WHRS based on the Rankine cycle principle attracts a lot of interest in the heavy duty industry, over the past few years, to decrease the fuel consumption and reach the future pollutant emissions standards. Control issues have still to be faced before the integration of such a system into a vehicle. Model predictive controllers suits really well for our control purpose due to their ability to handle online optimal control problem. This paper focuses on the development of an explicit control law based on MMPC with aim to control the fluid temperature at the inlet of the expansion machine in order to increase the system performance and the reliability.

INTRODUCTION

Over the last few years, the increase in fuel price coupled to the more and more stringent pollutant emissions legislations have compelled the truck manufacturers to look into innovative solutions to decrease the fuel consumption. Waste heat recovery systems (WHRS) such as Rankine cycle could be one of those solutions [START_REF] Sprouse | Review of organic rankine cycles for internal combustion engine exhaust waste heat recovery[END_REF]. Recovering waste heat by the way of a Rankine cycle and turned it into another form of energy is not really new. This principle is used for electricity generation for years since today a large proportion of the worldwide electricity production is based on the Rankine process using a wide variety of heat sources. In the last decades, automotive industry has been placing more and more attention in recovering waste heat. The most applied system, today, is the turbocharger which turns available energy into aeraulic work. Another example is the turbo-compound which transforms wasted energy into mechanical one. In addition to those systems, energy recovery based on the Rankine cycle has attracted a lot of interest over the last years and a significant potential has been shown in heavy duty (HD) industry and especially for long haul truck applications [START_REF] Oomori | Waste heat recovery of passenger car using a combination of rankine bottoming cycle and evaporative cooling system[END_REF], [START_REF] Boretti | Recovery of exhaust and coolant heat with r245fa organic rankine cycles in a hybrid passenger car with a naturally aspirated gasoline engine[END_REF]. If working fluid (WF), heat source, heat sink and components choice are largely addressed topics, only few papers treat of control system development [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF], [START_REF] Zhang | Multivariable robust control for organic rankine cycle based waste heat recovery systems[END_REF]. To ensure safe operation and improve the operating strategy, a key variable is the working fluid temperature at the expansion machine inlet: This temperature is often turned into a superheat criterion (superheat is the temperature difference between the actual value and the saturation temperature) and needs to be accurately controlled in order to maximize the power production and increase the power generation period by feeding the expander with vapor at good conditions. Moreover, technical integration constrains that the control algorithm has to be yet simple to limit the computational complexity (attribute a large part of the computational capability of the electronic control unit to a classical optimization problem is not possible) and fast to compute (sampling time is 20ms). This paper is organized as follows: section 2 presents the principle and the studied system are explained. Section 3 approaches the controller methodology development when section 4 shows some numerical results. Finally a conclusion is drawn.

I. PRINCIPLE OF OPERATION AND STUDIED SYSTEM

A. Rankine process Rankine cycle process is well known and widely used for years. It relies on four transformations and is a variant of the Carnot cycle. First, the fluid is pumped and compressed from the condensing pressure to the evaporation pressure (1 → 2). The pressurized liquid goes through one or several heat exchangers to recover thermal power ( Qin ) from the heat source(s) to be preheated (2 → 3a), vaporized (3a → 3b) and then superheated (3b → 3c). This superheated vapor is expanded (3c → 4) through the expander which aims to convert the pressure drop of the expansion from evaporating to condensing pressure into a mechanical power on its shaft ( Ẇout ). Finally the fluid at low pressure, which can be fully vapor or partially condensed after the expansion process (depending on the technology used and the superheat level at expander inlet), is condensed (i.e. changes from vapor to liquid phase 4 → 1) releasing heat ( Qout ) into the heat sink. An usual representation of the thermodynamic cycle is done through a temperature-entropy (T-s) form: Figure 1 shows the state changes of the working fluid (or Rankine fluid) and transfer fluids (heat source and sink) of a subcritical Rankine cycle.

B. Studied system

The studied WHRS (Figure 2) recovers heat from both exhaust gas recirculation (EGR) and exhaust in series, which means that the working fluid goes first through the evaporator linked to the EGR and then through the tailpipe boiler (linked to the exhaust gases) [START_REF] Grelet | Waste heat recovery rankine cycle based system modeling for heavy duty trucks fuel saving assessment[END_REF]. This special configuration has been chosen to lower the EGR temperature downstream the boiler. The Rankine fluid is pumped in an atmospheric vented tank and pressurized by the feed pump. By controlling its speed, the working fluid mass flow rate going through the evaporators is set. The fluid then flows through the EGR and the tailpipe boiler where it is turned into vapor. It is expanded in a turbine, condensed in a water cooled condenser and goes back to the tank . The system is mounted onto an 11L 6 cylinder turbocharged engine with diesel particulate filter (DPF) and selective reduction (SCR) exhaust after treatment system. The working fluid is a mixture of water and ethanol which reduces the flammability of pure ethanol and decreases the freezing temperature of pure water.

II. RANKINE CYCLE CONTROLLER

A. Controller aim

This paper describes a temperature controller development for the system mentioned in section I-B. The aim is to reduce the standard variation around the set point. To do that, the manipulating variable (MV) is the working fluid mass flow rate, and is used to ensure that the temperature after the two evaporators (the controlled variable CV) track a desired set-point. In the following, superheat is used instead of temperature since only normal operation mode, where the goal is to keep the working fluid in vapor phase, is considered. Having an efficient control of it is important to maximize the system efficiency and to ensure that no liquid enters in the turbine which can be destroyed due to its high rotational speed. This work focuses only on the evaporators models since the system dynamic is mainly controlled by the heat exchangers (HEX) behavior. Moreover, technical integration impacts the choice of the control algorithm: it has to be simple (poor computational capability) and fast to be computed (sampling time is 20ms).

B. Nonlinear evaporator model

The input vector u contains the MV ṁ f 0 and four inputs disturbances which are: the WF pressure and enthalpy entering in the evaporator (P f 0 and h f 0 ) and the gas mass flow and

Z ẋi = f i (x, u)
where:

Z =       0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1       x T i = ṁ f i h f i T w int i T g i T w ext i u T = ṁ f 0 P f 0 h f 0 ṁg L T g L (1)
f i (x i , u) =                   ṁ f i-1 h f i-1 ρ f i-1 ∂ ρ f i-1 ∂ h f i-1 + 1 ρ f i-1 ∂ ρ f i-1 ∂ h f i-1 α f i A exch int f T f i -T w int i 1- h f i ρ f i ∂ ρ f i ∂ h f i -ṁ f i ṁ f i-1 h f i-1 -ṁ f i h f i -α f i A exch int f T f i -T w int i ρ f i V f α f i A exch int f T f i -T w int i +α g A exch intg T g i -T w int i ρ w int V w int ṁg c pg (T g i )(T g i-1 -T g i )-αg A exch intg T * g i -T w int i -A exch extg T g i -T w ext i ρ g i V g c pg (Tg i ) α amb A exch ext amb T amb -T w ext i +α g A exch extg T g i -T w ext i ρ w ext V w ext                  
(2) where i ∈ [1 : n d ] and the ambient temperature T amb is taken constant around the entire HEX.

C. Local multi linear model development

Development of a good nonlinear first principle model is a severe effort (many parameters are usually hard to get from experimental results) and may be difficult to be implemented in an online model based control strategy. Here, integration constrains lead to develop a simple and fast controller. In that case, many works report the usage of multi linear model approaches [START_REF] Banerjee | Estimation of nonlinear systems using linear multiple models[END_REF], [START_REF] Aufderheide | Extension of dynamic matrix control to multiple models[END_REF]. Let us consider the single input single output (SISO) nonlinear model of the form:

ẋ = f (x, u), (3) y = g(x, u), (4)
where the input is the considered MV (the WF mass flow rate entering in the EGR boiler) and the output is the considered CV (the WF temperature at the outlet of the tailpipe boiler). The four remaining input components in (1) are further considered as input disturbances. By plant experiments or simulation of the detailed non linear model (1-2), the relationship between the MV and the CV defined in the nonlinear SISO system (3-4) can be identified by a series of first order plus time delay (FOPTD) models (each

a subscript i ∈ [1, N]
), where the model parameters are the static gain G i , the time constant τ i and the delay L i :

F i (p) = y i u = G i 1 + τ i p e -L i p . (5)
In the sequel, the scalar MV is u and the scalar CV is y i .

To obtain a global model, theses N local models have now to be combined. The weighting scheme has to estimate the probability of one model among N to be valid by comparing the process measurements to the output of each local models.

The global model output is then calculated by weighting the local model outputs and sum them.

D. Weighting schemes

The two weighting schemes, used in the latter, estimate the validity of the i th model by calculating the modeling error:

ε i,k = y p,k -y i,k , (6)
which corresponds to the i th model error at the current time t k between the real measure y p,k and the i th model output y i,k .

1) Bayesian estimator: Most of the study on multi linear approach found in the literature use the Bayesian estimator as adaptive estimation technique to select the best model at each time step [START_REF] Rao | Multiple model predictive control of hemodynamic variables: an experimental study[END_REF], [START_REF] Aufderheide | Extension of dynamic matrix control to multiple models[END_REF]. It is based on probabilities calculation P(i), representing the probability of the i th model to be valid. This weighting scheme assigns a value between 0 and 1 to each model outputs where the weights sum is equal to 1 to ensure that the global model is always bounded by the extreme values of the N models in the bank. It also allows an exact, if it appears, model to be the only prediction model. The recursive Bayesian weighting scheme is a conditional probability of the i th model among the N models of the bank to be true given the model population in the bank and its past history of probabilities. It is written as follows:

p i,k = exp(-1 2 ε i,k K i ε i,k p i,k-1 ) N ∑ m=1 (exp(-1 2 ε m,k K m ε m,k p m,k-1 ) , (7)
w i,k =    p i,k N ∑ m=1 p m,k for p i,k > δ 0 for p i,k < δ . (8)
In the probability calculation of equations (7-8), a vector (K) containing convergence factor K i (i ∈ [1 N]) is used to improve the convergence to a single model. If more combination of the different models contained in the bank is desired, this factor should be detuned and set at a low value. To keep every models of the bank alive, an artificial probability is introduced such as all p i,k never goes below this artificial lower limit δ (which is a tuning parameter). This one is called in the following cut-off probability and is set to improve the estimator performance. However every models having this lower saturation value as probability are excluded from the weight computation as it is shown in equation [START_REF] Aufderheide | Extension of dynamic matrix control to multiple models[END_REF]. The model output is then defined as at the current time t k :

y k = N ∑ i=1 w i,k y i,k . (9)
2) New developed estimator: To simplify the weighting scheme and reduce the number of setting parameters, a weighting scheme has been developed. All values denoted by a superscript, ˜, refer to normalized values.

εi,k = ε 2 i,k N ∑ m=1 ε 2 m,k , (10)
c i,k = (1 -εi,k ) j=N ∏ j =i, j=1 ε j,k , (11) ci
,k = c i,k N ∑ m=1 c m,k . (12)
The value ci,k computed thanks to equation ( 12) is then filtered through a first order transfer function with unit static gain

w i (s) = 1 1 + τ f ilt s ci (s) (13)
to obtain the current weight of each local model w i,k . τ f ilt is a scalar which plays a similar role as the convergence matrix K used in Bayesian approach [START_REF] Banerjee | Estimation of nonlinear systems using linear multiple models[END_REF]. The number of tuning parameters is then decreased to the only parameter τ est instead of N + 1 (K and λ ) in the Bayesian recursive scheme. The global model output is then calculated in the same way than with the Bayesian method with (9).

E. MMPC strategy

At time t k (the continuous time is discretized at the constant sampling rate T s ), the model outputs y i,k of the N FOPTD models (5) allow to compute the models weights w i,k and the final model output y k with (9). This modeling (5) and ( 9) is used in a MMPC framework to ensure a tracking of the process output y p to the set-point y sp , accounting for the control move effort ∆u. Since here, a major issue for the technical integration, is to decrease the computational load, the number of optimization arguments (i.e., the control horizon) is first set to its minimum:

1 (u(t) = u(t k ) = u k ).
The control horizon is t p . Therefore, the MMPC algorithm aims to find, at each current time t k , the optimal step input u k that minimizes following the cost function J:

   min u in f ≤u k ≤u sup J(u k ) = t k +t p t k (y p (t) -y sp ) 2 + w u ∆u 2 k dt, ∆u k = u k -u k-1 , (14)
where w u is a scaling factor and a penalty weight for the control move. In this quadratic form, the ideal optimal solution leads to J(u k ) = 0, i.e.: the tracking is perfect (y p (t) = y sp ∀t) and control is at a steady state (∆u k = 0 ∀t). In the internal model control structure, the control u is both applied to the process and the model(s). Hence, it allows to compute the modeling error e at each k:

e k = y p,k -y k . ( 15
)
this error is assumed constant into the future and used to get the output prediction y p (t) in ( 14) using the model:

y p (t) = y(t) + e k . ( 16
)
In the same way, the set point y sp is taken constant on the prediction horizon and updated at each time k. The question is now to write y(t) according to u k (and also the constant values and the parameters updated at each k). Let us start from the definition of the output response y i (t) of one linear FOPTD model i (5) with any input profile u(t) from any current time t k , and starting from the current process output y p,k :

y i (t) = y p,k e -(t-t k ) τ i + t t k (e -(t-s) τ i G i τ i u(s -τ i ))ds (17)
(17) can be developed as:

y i (t) = y p,k e -(t-t k ) τ i + G i τ i e -t τ i t t k (e s τ i u(s -τ i ))ds (18)
To proceed, we need to define, based on the time delay L i :

λ i = max(a i ∈ N|a i ≤ L i T s ) ∆L i = L i -λ i T s , ∈ R + , (19)
which allows to define u(.) used in the integration interval of (18), based on past input values and the future input value of u. Then, the integration in ( 18) is done by parts, where the λ i + 2 time intervals are given in table I:

s s -L i u(s -L i ) t k → t k + ∆L i t k -L i → t k-λ i u(t k-λ i -1 ) t k + ∆L i → t k + ∆L i + T s t k-λi → t k-λi+1 u(t k-λi ) . . . . . . . . . t k + ∆L i + (λ i -j)T s → t k + ∆L i + (λ i -j + 1)T s t k-j → t k-j+1 u(t k-j ) . . . . . . . . . t k + ∆L i + (λ i -1)T s → t k + L i t k-1 → t k u(t k-1 ) t k + L i → t t k → t -L i u(t k ) = u k TABLE I INPUT SEQUENCE DEFINITION. y i (t) = y p,k e -(t-t k ) τ i + G i τ i e -t τ i u(t k-λ i -1 ) t k +∆L i t k e s τ i ds + G i τ i e -t τ i j=λ i ∑ j=1 u(t k-j ) t k +∆L i +(λ i -j+1)T s t k +∆L i +(λ i -j)T s e s τ i ds + G i τ i e -t τ i u(t k ) t t k +L i e s τ i ds. (20)
(20) can be summarized as a linear expression in the optimization argument u k :

y i (t) = y p,k f 1i (τ i ,t k ,t) + f 2i (T s , G i , τ i , ∆L i , λ i ,t k ,t, u(past)) +u k f 3i (G i , τ i , L i ,t k ,t), (21)
where the f i may be explicitly defined offline and numerically updated online at each time t k . Therefore, replacing first (21) in [START_REF] Rao | Multiple model predictive control of hemodynamic variables: an experimental study[END_REF], assuming that the weights w i,k are constant over the prediction horizon, then combining this result with (15) and ( 16) in ( 14), the cost function to minimize can be written as:

J(u k ) = t k +t p t k N ∑ i=1 w i,k y i (t) + e k -y sp k 2 + w u ∆u 2 k dt (22)
where t p = max(t pi ) ∀i is each prediction horizon has to be tuned according:

   t pi = γ p * τ i + L i ; γ p ∈ R + , e.g.: γ p = 1(63% of the dynamics is predicted)
or γ p = 3(95% of the dynamics is predicted).

(23) Based on the step response series (20) of the N linear FOPTD models, the expression ( 22) is a quadratic one in u k :

J(u k ) = β 2,k (N, G i , τ i , L i ,t p , w u , w i,k )u 2 k +β 1,k (N, T s , G i , τ i , L i ,t p , ∆L i , λ i , w u , y p,k , y sp k , e k , u(past), w i,k )u k +β 0,k (N, T s , G i , τ i , L i ,t p , ∆L i , λ i , w u , y p,k , y sp k , e k , u(past), w i,k ) (24)
The β i,k (.) can be easily explicitly defined offline (see the appendix) and are updated at each time t k .

Since J is convex in u k (due to the fact that β 2,k > 0, by definition), let us then define u min k (N, T s , G i , τ i , L i ,t p , ∆L i , λ i , w u , y p,k , y sp k , e k , u(past), w i,k ) (25) the solution of the minimization of (24) obtained with the first order optimality at each t k :

∂ J ∂ u k = 0 at u k = u min k . (26)
The calculation of u min k is then straightforward:

u min k = -β 1,k 2β 2,k (27)
which leads to the explicit formulation of the solution u k of the constrained optimization problem (14):

   if u in f ≤ u min k ≤ u sup : u k = u min k if u min k ≤ u in f : u k = u in f if u sup ≤ u min k : u k = u sup . (28)
Using (28), the solution is very fast to compute (no online optimization task, no online dynamic model resolution).

F. Simulation results

The MMPC strategy is then implemented into Simulink and coupled to a detailed experimentally validated model [START_REF] Grelet | Waste heat recovery rankine cycle based system modeling for heavy duty trucks fuel saving assessment[END_REF].

The engine operation covers a heat flow rate entering into the WHRS going from 0 to 140kW that is divided into 10kW zones, hence leading to 14 different triplets of the FOPTD parameters. As seen in figure 3 for two of them, this model structure is validated by comparing complete simulation results of the nonlinear model (1 -2) and each SISO FOPTD model ( 5) identified around an operating point. On both operating points the agreement is good. Model parameters of the FOPTD models change a lot from one operating point another as it can be seen on figure 4, hence underlying the nonlinear aspect of the model. The controller is then tested on a validated simulation model [START_REF] Grelet | Waste heat recovery rankine cycle based system modeling for heavy duty trucks fuel saving assessment[END_REF] and the two weighting schemes are compared. Perfect actuators and sensors are considered and controller tuning parameters are set to obtain the best performance. All values of the convergence matrix K are set to the same value. Table II shows the controller tuning parameters used (i ∈ [1 N]). The set point is changed during the simulation time to assess the performance as well for load changes as set point variations. Despite the large inputs disturbances (figure 5) of the representative long haul truck driving cycle used, the tracking performance (shown in figure 6) of the controller for the two implemented weighting schemes is good. The Bayesian scheme gives better performance as well in terms of accuracy as in response time. The main reason is the presence of a cut off probability (δ ) which allows here to converge to a single model faster than with the other approach. The developed weighting scheme uses more blending to compute the global model output. In addition to that, the new developed weighting scheme seems less performant to reach the set point when it moves far from the operating point where the 14 FOPTD models have been identified. For both weighting schemes, the time response is quite long compared to the use of a nonlinear model based controller [START_REF] Grelet | Model based control for waste heat recovery rankine cycle system in heavy duty trucks[END_REF], which requires meanwhile the full perfect knowledge of all nonlinear model parameters and an online nonlinear model inversion (which may be no compatible with our integration constraints).

G. Conclusion

This paper reports an explicit control law based multi-model predictive controller (MMPC). Based on FOPTD models, it is a fast control algorithm since it does not require any online optimization, nor a dynamic model resolution. A weighting scheme for multi linear models approach is also shown and compared to a classical Bayesian weighting scheme. Simulation results developed for a waste heat recovery Rankine based system used in heavy duty trucks, show similar performance to the Bayesian scheme but contrary to this method, the new proposed approach requires less effort in terms of tuning. Currently experimentally validated, the new proposed weighting scheme outperforms the Bayesian for a low number of linear models in the data bank. However, for both cases shown here, the time response is longer than with a nonlinear model based controller structure and needs to be improved. In the meantime, the explicit controller presented here is a very fast algorithm and does not require to solve online a dynamic model that might be too time consuming, contrarily to the nonlinear model based approach. The presented controller can be derived either from an available detailed model (1-2), or directly from experiments (without detailed model development). Also, a known set point trajectory into the prediction horizon could be beneficial on this aspect but needs to be designed. Indeed, the knowledge of the future operating conditions is here often hard to get and requires complex calculations. At the time of writing, the strategy proposed has been implemented on the experimental setup and first results are convincing.

H. Appendix 1) β . expression:

β 1,k = N ∑ i=1 β 1,k + β 1,k + β 1,k + β 1,k -2w u u k-1 t p , β 1,k = -G i τ i w i,

Fig. 1 .

 1 Fig. 1. Temperature-entropy diagram of the Rankine cycle

Fig. 2 .

 2 Fig. 2. Studied system schematic

Fig. 3 .

 3 Fig. 3. FOPTD and nonlinear model variations for two operating points.

Fig. 4 .

 4 Fig. 4. FOPTD model parameters

Fig. 5 .

 5 Fig. 5. Input disturbances

  k 2 y p,k e

									-	2tp τ i	e	∆L i τ i -1 . . .
						tp			L i	tp	L i +tp
		. . . e	τ i -1	e	τ i -2 e	τ i + e	τ i
	β 1,k =	λ i ∑ j=1	G i Ts 2 τ i u(t k-j ) w i,k L i -tp	2 e	∆L i -tp -j Ts +λ i Ts τ i	. . .	(29)
		. . . e	τ i -1	e	τ i	-2 . . .
		. . . -G i	2 τ i u(t k-j ) w i,k	2 e	∆L i -j Ts +λ i Ts τ i	. . .
						L i
		. . . e		τ i -2	e
									L i -tp	L i
									τ i	-τ i e	τ i
		G i	2 w i,k	2 2t p +4 τ i e	L i -tp τ i -τ i e	2 L i -2tp τ i	-4 τ i e	L i τ i +τ i e	2 L i τ i
	β 2,k = + w u t p					2

L i τ i + 2 e -tp τ ie L i -2tp τ i -2 β 1,k = -G i 2 τ i u(t k-λ i -1 ) w i,k 2 e Ts τ i -1 β 1,k = 2 G i w i,k (e kysp) t p + τ i e

NOMENCLATURE


Johan Peralez
email: johan.peralez@ifpen.fr

Madiha Nadri
email: nadri@lagep.univ-lyon1.fr

Pascal Dufour
email: dufour@lagep.univ-lyon1.fr

Paolino Tona
email: paolino.tona@ifpen.fr

Antonio Sciarretta
email: antonio.sciarretta@ifpen.fr

Control Design for an Automotive Turbine Rankine Cycle System based on Nonlinear State Estimation

This paper investigates the problem of control design for an Organic Rankine Cycle (ORC) for Waste Heat Recovering (WHR). Based on a moving boundary heat exchanger model derived from energy and mass conservation principles, an implicit model, control-oriented and experimentally validated, is presented. Then, a first controller which consists on a gain-scheduled PID combined to a dynamic feedforward is implemented on the pump speed to maintain the superheat close to the set point value. A second controller is a nonlinear feedback law which allows to adjust the evaporating pressure to time-varying demand with a good accuracy. To implement these two controllers, an observer is designed to achieve the state estimation from pressure measurement, typically available in WHR applications.

The observer and the control approach are validated on realistic operating conditions.

I. Introduction

In recent years, growing awareness of environmental and energy issues has steadily driven research interest in technologies enabling efficiency improvements for internal combustion engines. Engine waste heat recovery, in particular, has a interesting potential to increase fuel thermal efficiency, reduce fuel consumption, and decrease engine emissions. Indeed, more than one third of the energy produced by internal combustion engines is wasted through exhaust gas heat. The ORC is a thermodynamical cycle that seeks to recover the waste heat by producing expansion work. A pump circulates an organic working fluid in a closed loop where an external heat source supplies heat, via a heat exchanger. Vaporized fluid expands in a turbine or an expander to produce mechanical power. Among all the currently available technical solutions used to recover low-grade waste heat, the ORC combines the benefits of a high thermal efficiency and of a limited impact on engine operation, and has thus drawn the attention of many car and truck manufacturers [START_REF] Sprouse | Review of organic rankine cycles for internal combustion engine exhaust waste heat recovery[END_REF].

In transport applications, the main difference with conventional (stationary) ORCs lies in the highly transient behavior of the hot source, depending on driving conditions. This poses a great challenge for system control. In this context, an effective control system is essential to attain satisfactory performance over a broad range of operating conditions, acting as time-varying disturbances. This is especially true when there are few available actuators and sensors, as it often happens in the automotive industry. Despite that, literature on control design for these applications is still very scarce and most strategies for vapour pressure and temperature control are based on linear models [START_REF] Zhang | Dynamic modeling and multivariable control of organic rankine cycles in waste heat utilizing processes[END_REF]- [START_REF] Hou | Supervisory predictive control of evaporator in organic rankine cycle (ORC) system for waste heat recovery[END_REF].

The system considered here is illustrated in Fig. 1. It uses a turbine connected to an electric generator which delivers power (P L ) to the on-board electrical system. A by-pass valve controls the fraction of exhaust gas entering the evaporator. Despite the importance of this component, which is mandatory for safety reasons, only heuristic control approaches have been reported [START_REF] Feru | Experimental validation of a dynamic waste heat recovery system model for control purposes[END_REF]- [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF]. In a previous paper [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF], we have shown -with an experimental validation -that a nonlinear model-based monovariable control approach is superior to the commonly used PID approach for superheat regulation. In the present paper, we extend the approach to the multivariable case, with the use of the evaporator by-pass as an additional actuator, allowing tracking a pressure setpoint -or equivalently of a power production demand.

The paper is organized as follows. In the first section we briefly introduce the reference and the simplified models. Then the control problem is formulated. In the third section, a new multivariable control scheme is introduced. In section 4, an implicit observer is designed which allows state estimation using the pressure measurement.

The complete approach is illustrated in the last section.

II. Model and problem formulation

Heat exchangers are the main contributors to thermal inertia and therefore are the focus of attention in dynamic modelling.

A. Partial Differential Equation (PDE) reference model

A classic 1-D representation of heat exchangers (along the fluid displacement direction z) is given by mass and energy balances for the fluid and the wall [START_REF] Jensen | Dynamic modeling of thermo-fluid systems with focus on evaporators for refrigeration[END_REF]. The evolution of fluid and wall variables -namely fluid mass flow ṁ, specific enthalpy h, pressure p and wall temperature T w -is then described by

           ∂ (ρ A) ∂t + ∂ ṁ ∂z = 0 ∂ (ρAh -Ap) ∂t + ∂ ṁh ∂z = πDα(T w -T ) C w ρ w A w ∂T w ∂t = πDα(T -T w ) + πD o α o (T a -T w ) (1)
where A, D, D o , ρ w , C w are design parameters. Density ρ and temperature T of the fluid are nonlinear maps of p and h. T a is the ambient temperature. α (resp. α o ) is the heat exchange coefficient between the wall and the working fluid (resp. the ambient). Notice that the above model is a system of three coupled nonlinear PDEs. Such a representation is difficult to deal with for control purposes. Therefore, a discretization of system (1) is used in simulation for control validation. Among the methods for discretization of system (1), the use of moving boundaries (MB) results in a relatively low order model. As shown in Fig. 2, the MB model monitors the length of each fluid phase along the evaporator: the normalized zone lengths L 1 , L 2 , L 3 track the liquid, two-phase and vapour zones, respectively. Fluid and wall dynamics are then captured by seven state variables:

x = T w1 T w2 T w3 L 1 L 2 h 3 p T . A more
detailed representation on this model may be found in [START_REF] Jensen | Dynamic modeling of thermo-fluid systems with focus on evaporators for refrigeration[END_REF] and is referred as the reference model in the following.

B. Control-oriented simplified model

In literature, model-based control generally circumvents the complexity of a reference model, such the one described above, by linearization around an operating point. In [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF], we proposed a simplified model, experimentally validated over a broad range of time varying operating conditions. To achieve this, we assume a twotime-scale dynamic behaviour where wall temperatures capture the slow dynamics. Moreover, it was shown that the high and low pressure parts of the ORC could be decoupled, resulting in the simplified model described in the following. Wall energy balance and fluid equilibrium yield the following equations for each of the three zones:

dT wi dt = -β i [T wi (t) -T i (p(t), SH(t))] -η exh (t) [T wi (t) -T exh (t)] , (2)
   L i (t) = ṁ ∆h i (p(t), SH(t)) β i (T wi (t) -T i (p(t), SH(t))) 3 i=1 L i (t) = 1, ( 3
)
where SH is the superheating value, the β i s are the components of a constant vector, T i (p(t), SH(t)) and ∆h i (p(t), SH(t)) are nonlinear maps of fluid temperature and enthalpy and:

η exh (t) = ṁexh (t)β 5 1 -exp(- β 6 V o (t) ṁexh (t)
) . ( 4)

The heat exchanger model is completed by a static model of the turbine mass flow:

ṁ(t) = β 4 ρ(p(t), SH(t))p(t), ( 5
)
while a nonlinear map for power production P L (p(t), SH(t)) is considered.

Looking at expressions ( 2) and ( 3), it can be seen that the obtained control-oriented model is a nonlinear implicit model.

C. Problem formulation

The control problem is to respond to a power production demand while ensuring a safe level for pressure and superheating. From the map of power production, power demand can be converted in term of pressure set-point. So let y = SH p T , T w = T w1 T w2 T w3 T and

u = ṁ η exh T .
This two-input two-output control problem is treated in two steps, assuming a two-time scale dynamic behavior. First, the control design proposed in [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF] is used to regulate the superheating y 1 with pump mass flow u 1 . Indeed, experimental results in [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF] showed that a control based on a gain-scheduled PID with a dynamic feedforward allows a tight superheating regulation. Then, the control design for the pressure y 2 assumes the (fast) superheating dynamic as perfectly regulated to tune the evaporator by-pass u 2 . Furthermore, this control requires knowledge of all system states. To overcome this problem, the authors in [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF] used an open-loop model for the estimation of needed states.

In this work, we propose to use an observer to improve the robustness of the closed loop against the model uncertainties. Then, a second nonlinear controller for pressure will be designed. The control scheme proposed in [START_REF] Peralez | Improving the control performance of an organic rankine cycle system for waste heat recovery from a heavy-duty diesel engine using a model-based approach[END_REF] is taken up and completed as shown in Fig. 3.

Assuming that the pump mass flow is fixed by the value of SH following (5), the system of equations ( 3)-( 5) can be rewritten in the following semi-explicit form

Ṫw = f (T w , p) + u 2 g(T w , t) ϕ(T w , p) = 0, (6)
where

f (T w , p) =   β 1 (T 1 (p) -T w1 ) β 2 (T 2 (p) -T w2 ) β 3 (T 3 (p) -T w3 )   , ( 7
)
g(T w , t) =   T exh (t) -T w1 T exh (t) -T w2 T exh (t) -T w3   , ϕ(T w , p) = 1 -(L 1 + L 2 + L 3 ), (8)
with

L 1 = u 1 h l (p) -h in β 1 (T w1 -T 1 (p)) , L 2 = u 1 h v (p) -h l (p) β 2 (T w2 -T 2 (p)) (9)
L 3 = u 1 h out (p) -h v (p) β 3 (T w3 -T 3 (p))

III. Control Design

Let us consider the class of implicit systems of the following form which includes system (6):

   ẋ = F (x, z, u) Φ(x, z) = 0 y = h(x, z) (10) where y ∈ R p , u ∈ U ⊂ R m , (x, z) ∈ R n × R d , F , h and Φ = (Φ 1 , . . . , Φ d )
T are assumed to be sufficiently smooth with respect to their arguments, and

∂Φ ∂z (x,z) is of full rank ∀(x, z) ∈ M ( 11
)
where M is the set of zeros of Φ:

M = (x, z) ∈ R n × R d , s.t. Φ(x, z) = 0 . ( 12
)
Note that this condition guarantees the existence and the uniqueness of the solution z of Φ(x, z) = 0, for every x.

To transform the implicit system [START_REF] Åslund | An observer for non-linear differentialalgebraic systems[END_REF] to an Ordinary Differential Equation (ODE) system, we need the following assumption.

Assumption 1: There exists > 0 such that ∂Φ ∂z has a full rank on the tubular neighbourhood

Ω = {(x, z) ∈ R n+d ; Φ(x, z) < },
and there exist two functions σ, σ of class

K ∞ such that: i) ∀(x, z) ∈ Ω , Φ(x, z) ≥ σ( z -z(x) ) ii) ∀(x , z ) ∈ Ω , we have z(x)-z(x ) ≤ σ( z -z ).
From A 1 ), we deduce that the following system

       ẋ = F (x, z, u) ż = - ∂Φ ∂z | x,z -1 ∂Φ ∂x | x,z F (x, z, u) y = h(x, z), ( 13
)
is well defined on Ω . Moreover, the trajectories of ( 10) issued from the manifold M, coincide with those of system (13). Now, let us use this result to transform our system (6) into an explicit system. To do so, we must verify that Assumption 1 holds. Using the analytic form of ϕ(T w , p) given by ( 8), we can show that Assumption 1 is satisfied in the physical domain (for more details on the proof see the appendix). This done, system (6) can be written as

         Ṫw = f (T w , p) + u 2 g(T w , t) ṗ = - ∂ϕ ∂p | Tw,p -1 ∂ϕ ∂T w | Tw,p × (f (T w , p) + u 2 g(T w , t)) , (14)
where

∂ϕ ∂p | Tw,p = β 4 ρ(p) p        h l (p) -h in β 1 (T w1 -T 1 (p)) 2 h v (p) -h l (p) β 2 (T w2 -T 2 (p)) 2 h out (p) -h v (p) β 3 (T w3 -T 3 (p)) 2        . ( 15
)
Proposition 1: For every T i (i = 1, ..., 3), T exh bounded and for the input u(.) which takes values in some bounded set U ⊂ R + , the solution T w of system Ṫw = f (T w , p) + u 2 g(T w , t), ∀T wi (t 0 ) = x 0 (16) is uniformly bounded with respect to t in R + , i.e. the system (14) is BIBO (bounded-input bounded-output) stable.

Proof: Considering the positive definite quadratic function V (T wi ) = T 2 wi /2, the derivative of V along the trajectories of a dynamical system ( 16) is given by

V = T wi (β i (T i (t) -T wi ) + u(t) (T exh (t) -T wi )) = -T 2 wi (β i + u(t)) + T wi (β i T i (t) + u(t)T exh (t)) ≤ -2aV + b √ 2V ≤ - √ V (2a √ V -b √ 2) ( 17
)
where a = inf (β i + u(t)) and b = sup (β i T i (t) + u(t)T exh (t)) with a > 0 and b ≥ 0. From, (17), we deduce that V is negative for T wi such that {|T wi | ≤ b/a}. Furthermore, solutions starting from the set {V (T wi ) ≤ (b/a) 2 /2} at t 0 , remain there for t > t 0 since V is negative on the boundary V = (b/a) 2 /2.

Based on the same physical conditions used to show that ∂ϕ ∂p | Tw,p < 0, we can state our nonlinear controller in the following proposition. Proposition 2: Let k > 0 then, the control

1 u 2 (T w , p, p SP ) = -ϕ T Tw f (T w , p) + k(p SP -p) ϕ T Tw g(T w , t) , ( 18
)
asymptotically stabilizes p to the setpoint p SP . Moreover, the input u 2 and the states of system (6) remain bounded.

Proof: Let u 2 (T w , p, p SP ) defined as in (18). Since conditions C 1 and C 2 hold, u 2 is bounded. Then the second equation of system (14) becomes :

ṗ = -ϕ -1 p k(p SP -p). ( 19
)
Moreover, from proposition 3, we have ϕ p < 0. So, obviously, pressure dynamic in ( 19) is asymptotically attracted by p SP .

IV. Observer design

To design an observer for the implicit system [START_REF] Åslund | An observer for non-linear differentialalgebraic systems[END_REF], the authors of [START_REF] Nikoukhah | A new methodology for observer design and implementation[END_REF] gave an implicit observer described by a dynamic system together with an optimization problem permitting to solve the algebraic constraint at each time. Such observer is called an implicit observer.

Definition 1: An implicit observer for system (10) which converges for every input u ∈ U is a dynamical system of the form

   ẋ = F (x, ẑ, g, y, u) ġ = G(g, u), g ∈ O ⊂ R k Φ(x, ẑ) = 0, ( 20
)
such that for every u ∈ U we have i) ∀g(0) ∈ O, the trajectory (g(t)) t≥0 lies into a compact subset of O. 1 The following notations are used :

ϕp = ∂ϕ ∂p | Tw ,p , ϕ Tw = ∂ϕ ∂Tw | Tw ,p ii) ∀r > 0; ∀(x(0), z(0)) ∈ M; ∃r > 0; ∀(x(0), ẑ(0))) ∈ M, (x(0)-x(0), ẑ(0)-z(0)) < r implies (x(t) -x(t), ẑ(t) -z(t)) < r, for all t ≥ 0. iii) ∀(x(0), z(0)) ∈ M; ∀(x(0), ẑ(0))) ∈ M, lim t→+∞ (x(t) -x(t), ẑ(t) -z(t)) = 0,
where conditions i) and ii) ensure the stability of the observer, and condition iii) guarantees its convergence.

Based on this definition, and using our notations z = p and x = T w with (T w (t), p(t)) ∈ R 3 × R and y(t) = p, the gain G is given by a Riccati equation, and an implicit extended Kalman filter (EKF) for system ( 6) is given by (for more details, see [START_REF] Nikoukhah | A new methodology for observer design and implementation[END_REF], [START_REF] Åslund | An observer for non-linear differentialalgebraic systems[END_REF])

           Ṫw = f ( Tw , p) + u 2 g( Tw , p) -SC T R -1 (p -p) Φ( Tw , p) = 0 Ṡ = AS + SA T -SC T R -1 C T S + Q S(0) = S(0) T > 0, ( 21
)
where Q is a constant symmetric positive definite (SPD) matrix, R is a real positive constant and

A = ∂(f + u 2 g) ∂T w | ( Tw, p) , C = ϕ -1 p ϕ T Tw .
V. Simulation results To demonstrate the effectiveness of the control structure described in Fig. 3, this section presents a set of simulation results with realistic operating conditions.

A. Observer validation

Observer (21) performances in presence of measurement noise can be seen Fig. 4. System simulation is conducted under transient conditions. Observer (21) is initialized with initial condition different from the reference model. The states of the observer are found to converge faster to their target values than the reference model states, if they both start with the same initial conditions (with the same errors). Moreover, it provides an efficient filtering of pressure measurement (Fig. 5) that will be useful to the control feedback.

B. Closed-loop evaluation

Experimental data for exhaust gas conditions that are representative of a long-haul truck mission, as shown in Fig. 6, were used. Moreover, in order to assess the robustness of the control, errors were considered on two main model parameters: an error of 10% on the heat exchange coefficient between exhaust gas and evaporator wall, and of 20% on the wall volume. transients on exhaust conditions and dramatic variations of pressure. A tight pressure set-point tracking is also demonstrated as long as exhaust gas heat flow is sufficient. Indeed, in the time interval between 1400s and 1800s, pressure set-point -or equivalently, power production demand -cannot be satisfied due to insufficient exhaust gas heat flow.

Finally, to emphasize benefits of observer, a second simulation is conducted where state estimation is performed via an open-loop model. Fig. 10 compares resulting superheating regulations: it is more than 8 K in this case while the maximum error is about 5 K when the observer is used. Thus, observer was found to improve robustness of our model-based control strategy.

VI. Conclusion

This paper presents a nonlinear control strategy for an ORC system. The proposed scheme combines a superheat controller, an additional nonlinear controller -allowing pressure set-point tracking or equivalently a tracking of power production demand -and an implicit extended Kalman Filter for wall temperature estimations. The proposed approach has been illustrated with success in presence of measurement noise, model uncertainties and state initial errors. The experimental implementation of the new control scheme is in progress.

Appendix

In order to check if this assumption is satisfied, we consider normal operating conditions of an ORC. Namely, the following conditions are considered.

Condition C 1 : Pressure p in the evaporator is such that p < 25 bar and temperature of working fluid T evap,in at evaporator inlet verifies 10

• C < T evap,in < 40 • C.
Condition C 2 : T wi > T i for i = 1, 2, 3. Notice that C 1 is verified for the considered system and working fluid in normal conditions (greater pressures or inlet temperatures do not match safety conditions and/or efficiency considerations and would lead to system shutdown by the control system supervisor). C 2 means that heat is transferred from wall to fluid which might not be verified only during start-up operation. However, in practice, none of these conditions is restrictive. To proof this proposition let us show the following sufficient conditions: where L i are given by ( 9).

Fig. 1 .

 1 Fig. 1. The Organic Rankine Cycle system under investigation

Fig. 2 .

 2 Fig. 2. Moving boundaries layout for the evaporator.

Fig. 3 .

 3 Fig. 3. Proposed closed loop structure for control of SH and p. Observer feeds state estimation to the model-based controller.

Fig. 7

 7 Fig. 7 presents estimation results. The states of the reference model that are not measurable and their estimation are compared. Due to the model parameters errors, estimations do not converge exactly to their target values. But they still provide representative values, especially in fast transient periods. Fig. 8 and 9 validate control capability in regulating the process output SH to its setpoint SH SP despite fast

Fig. 4 .

 4 Fig. 4. Simulation results under varying inlets: reference model (solid line), observer (dash line) and reference model with observer initial conditions (dot line).

Fig. 5 .

 5 Fig. 5. Zoom on pressure plot of Fig. 4.

Fig. 6 .

 6 Fig. 6. Experimental data of a truck exhaust gas conditions: mass flow and temperature

Fig. 8 .Fig. 9 .

 89 Fig. 8. Pressure set-point p SP tracking and superheating regulation around SH SP .

Proposition 3 :

 3 Since conditions C 1 , C 2 hold, we get ∂Φ ∂p (Tw,p) < 0.

Fig. 10 .

 10 Fig. 10. Superheating regulation error (SH -SH SP ) with observer (solid blue line) and with state estimation performed by an openloop model (dash green line).

  ,p) > 0, 2) ∂L2 ∂p (Tw ,p) > 0, 3) ∂L1 ∂p (Tw ,p) > 0,

1 )

 1 Let L 3 = N 3 D 3, where N 3 = u 1 (h out (p) -h v (p)) andD 3 = β 3 (T w3 -T 3 (p)), according to (8). Then sign ∂L3 ∂p (.) = sign ∂N3 ∂p (.) D 3 -N 3 ∂D3 ∂p (.) .Clearly N 3 > 0 and ∂D3 ∂p (.) < 0 (the evaporation temperature T 2 increases with p). Moreover D 3 > 0 (condition C 1 ). Fig.11(a) shows that ∂N3 ∂p (.) > 0 for the considered pressure range. So ∂L3 ∂p (.) > 0. 2) Similar to 1), let L 2 = N2 D2 , and notice that N 2 > 0, ∂D2 ∂p (.) < 0 and D 2 > 0. Figure 11(a) shows that ∂N2 ∂p (.) > 0 . So ∂L2 ∂p (.) > 0. 3) Similar to 1),let L 1 = N1 D1 , and notice that N 1 > 0, ∂D1 ∂p (.) < 0 and D 1 > 0. Figure 11(b) shows that ∂N1 ∂p (.) > 0 for the considered pressure and disturbance range. So ∂L2 ∂p (.) > 0. Thus, ϕ(T w , p) decreases monotonically with p. Consequently, ∂Φ ∂p (Tw,p) has a full rank in the physical domain.

Fig. 11 .

 11 Fig. 11. N 1 (b: bottom),N 2 (a: top) and N 3 (a: top) increase with pressure (here SH SP equal 30 • C).


M ' Badi
email: miehakandambadi@yahoo.fr

Toward a culture of diversity

Keywords: Interactions, Language distortions, Representation, Cultural diversity, Reciprocal comprehension

Toward a culture of diversity

This paper deals with language practices in nurseries and pre-elementary schools of Martinique. It is focused on the account of modalities concerning everyday interactions and more precisely language distortions that may occur during interactions between adults and children in those structures. The consideration of cultural diversity generated by language distortions is suggested here to be a prerequisite for the development of a reciprocal comprehension leading to adequate mastery of language behavior and learning among young children.

This approach proposes new patterns within a didactic agreement considering adult and child differences.

INTRODUCTION

To conceive the entrance of the individual to the culture, [START_REF] Bruner | L'éducation, entrée dans la culture. Les problèmes de l'école à la lumière de la psychologie culturelle[END_REF] proposes the idea of an education susceptible to adapt a culture to the needs for its members and to adapt its members and their manners to teach the needs for the culture. According to him, "our actions are guided by values, standards which, far from being "natural", are cultural and "symbolic constructions " [START_REF] Bruner | Rencontre avec Jérôme Bruner : Pour une psychologie culturelle[END_REF]. He conceives the human development as a process of collaboration between child and adult, the adult being envisaged as mediator of the culture.

Language distortions intervene however in the interactions adults-children because of an asymmetry between their respective levels of experience in the spoken language, highlighting the disparities between the intentions of these interlocutors [START_REF] Miehakanda | Transmission et appropriation des langues en Martinique[END_REF]. These disparities are in touch with cultural, social and individual diversities occurring in the transmission of the language know-how and the reciprocal comprehension which it involves.

The present reflection suggests putting in perspective the conditions of application of such an approach in Martinique. It will be introduced by a presentation of the cultural and language context into which evolve the speakers in Martinique. The socio-cultural values conveyed by the adults who supervise the children will be taken into account in an approach of positioning collectively adopted in establishments towards the sociolinguistic standard. This standard joins in a linguistic politic which it is advisable to clarify to conceive the social orientations which it infers. The cultural diversity of the places of presence of the French language is an essential element of this linguistic politic and the elaboration of a new didactic contract is imperative upon it with the prospect of a regulation of the interactions adults-children in environment French speaker / Creole speaker.

APPROACH OF CULTURAL AND LINGUISTIC CONTEXT FROM MARTINIQUE

The Inhabitants of Martinique generally speak French and the Creole, the first idiom being the linguistic "acrolectale" shape (high language) adopted as the linguistic standard, the second corresponding to the "basilectale" shape (low language) used most of the time on the familiar exchanges and of closeness. Bernabé (1984) proposes on this matter a model which is situated in a continuum / discontinuum where every language (the Creole and French) is endowed with an "acrolecte" and with a "basilecte".

The speakers of this region are confronted at the same time with the influence of the French language established as the standard and in an underlying presence at the school of the Creole language practiced in their linguistic environment today.

This sociolinguistic situation gives rise to productions which are sometimes situated at the middle path between Creole and French: we speak then about a "creolization" of French or French-frying of the Creole, the mixing code and the switching being code besides become the current linguistic practices. [START_REF] Prudent | Pratiques martiniquaises : genèse et fonctionnement d'un système créole[END_REF] draw the attention on the notion of "interlecte" which indicates the "zone of superimposing of the Creole and French" where the speakers from Martinique deceive the grammatical standard by making use of "lectes" not ranked in their language practices. The analysis of the eco-linguistic context from Martinique is of this fact made complex by the difficulty which we feel to distinguish from French-speaking speakers and from speakers Creole speakers.

It seems convenient to wonder about the modes of transmission and appropriation of know how to say to try to understand how the children learn to speak in such conditions, and to identify the language practices in the situations of communication.

One of the objectives of this reflection is to discover the opportunity of a possible consideration of the Creole language to the contact of the French language during the transmission of know how to say in the sociolinguistic environment from Martinique marked by the selective use of two vector idioms of different socio-cultural values.

To conceive the characteristics of this context, a pedagogy based on the adaptive capacities of learners and their capacities to co-build their linguistic space inside clearly identified linguistic rites can constitute a ground favorable to a rational distribution of the functions of French and Creole, as demonstrate in experiences regrettably still too marginal in the school environment (voluntary actions Of militant teachers).

The quite small in Martinique, as their peers worldwide, will they contribute one day, through spontaneous interpretations of language practices which are brought to them by the adults, in the transfer towards bigger one balance of the eco-linguistic space in which they evolve?

LINGUISTIC ORIENTATIONS AND SOCIAL REPRESENTATIONS

The directors of the visited day-nurseries are unanimous to recognize that the requirements relative to the exclusive use of French language emanate from no legislation and are not either a part of the specifications of their organizations of supervision. This linguistic orientation nevertheless strictly applied in the whole of establishments is, according to their own statements, firstly influenced by their representations of the languages of their environment (ex: Creole = dialect) and by the education which they received. They also indicate the implication of the expectations of the families which entrust them their children, in particular those of the parents of origin from Martinique for whom the control of French would represent a main advantage in the education of the quite small. A maternal headmistress situated in residential zone shows resistances of teachers of her establishment to use supports in Creole language provided to them within the framework of poetic expressions. One of these people confirms her refractory positioning by asserting that she would agree to use these supports only if she was forced to it, what gives pause for thought on the reports which it maintains with this idiom which is nevertheless one of hers.

It gives us in any case an overview of the educational intentions and adults' linguistic representations asked to educate the children in Martinique.

The positioning of the Inhabitants of Martinique towards their languages seems at least ambivalent when it is a question of educating and of educating the children. It is difficult to them to use the Creole as language of the education and the instruction while they recognize speakers of two idioms, French and Creole about which they usually speak. It is thus advisable to stop on the representations of the languages by their speakers in Martinique, because as in quite other society, the social representations intervene actively in the transmission and the appropriation of the language practices.

The whole scientific community identifies the social representation " as a shape of knowledge, socially elaborated and shared, having an aim has a practice and contributing to the construction of a reality common to a social group " (Jodelet, on 1989, The social representations, the page 36). The representations that the speakers are made languages, their standards, their characteristics, or their statutes influence the procedures and the strategies which they develop and operate to learn them and use them, according to [START_REF] Dabène | L'image des langues et leur apprentissage[END_REF]. [START_REF] Candelier | Entre le choix et l'abandon : les langues étrangères à l'école, vues d'Allemagne et de France[END_REF] show in their approach of the motivations of choice and relinquishment of the foreign languages by the French and German students that the school contact with the language influences in a positive way the representations attached to this language. In Martinique the school contact with the Creole is daily in spite of the omnipresence of French which is the language of the education: prohibitions and censorships of which is the object the Creole are not right of its presence, even allusive, in the speeches of the adults-educators.

A consideration of the Creole at school would doubtless allow this language to benefit from the positive prejudice which mention Candelier and Hermann-Brennecke. These indeed noticed that the evolution of the representations can be distorted by the degree of relevance which the pupils assign to the language of the education, through their imagined or actual reports in their sociolinguistic context.

It is then important to raise the problem under the angle of the linguistic standard established by the school which has an incidence on the social representations, in particular in terms of language conducts, and which joins within the framework of a French linguistic politics where regional languages as Creole are not still of the domain of the official practices.

INCIDENCES OF THE CURRENT LINGUISTIC POLITICS

France is one of the countries the most reserved in front of the ratification of the European charter of the regional or minority languages which confers to these legitimacy relative to their presence on their territories of presence.

In its article 8, this charter stipulates in particular that the signatory parts make a commitment "to plan that a substantial part of the preschool education is insured in the concerned regional or minority languages; to plan a primary education insured in these languages".

In France, the Constitutional Council always opposed that it considers as a threat for the language of the Republic, the regional or minority languages which can compete to see devaluing the French language if we introduced them in the same way into the constitution. "The language of the Republic is French", it is stipulated in the article 2 of the Constitution, what establish themselves the reserves of the opponents of the ratification of the charter.

The resistances of these in front of the recent modification of this article are bound to the fears of a questioning of the national sovereignty of the other instruments representing, worth knowing the flag, the hymn, the slogan and French there.

The conclusions of the constitutional reform which took place in May, 2008, under the insistence of the President of the Republic indeed allow introducing the regional or minority languages into the French linguistic heritage. But this reform voted by the National Assembly was the object of an amendment of the Senate which refused to register the recognition of regional languages in the constitution.

On May 22nd, 2008, the French Academy more expressed its opposition to the recognition of regional languages which it considers as "an achievement on the national identity ". It has purely and simply asked the retreat of the article concerning them.

The French linguistic politics thus tends to congeal on former models concerning its national language and the heritage which it associates it, with the exception of the other spoken languages on its various places of presence. In this context, the Creole language is only partially introduced into schools from The French language remains the only language of the education, the Creole having up to here an optional implication in the school learning registered in programs where it is more a question of teaching the regional language that to teach in regional language.

Which amounts to say that a linguistic politics really directed to a consideration of the cultural diversities of regions concerned by reports French / regional language has to set up the optimal conditions of expression of this diversity by distributing in particular the fields of intervention of every language at the same time regarding communication (social aspect) and regarding pedagogy (disciplinary aspects).

In practice, all the teachers and all the professionals of the early childhood should form in an equivalent control of French and their respective regional language in their functional dimensions (oral expression and written expression) but also as didactic tools (vehicular organ of the knowledge).

It would be a question, among others, of being capable of putting in perspective an idiom with regard to an other one (the Creole with regard to French and mutually) during the exploration of the concepts and the know-how to be transmitted in all the domains appropriate to the preschool and school learning.

Such measures have a double interest: that to organize the process of reciprocal comprehension and the inter-subjective meetings which it implies around a consideration of the socio-cultural disparities bound to the presence of both idioms in the linguistic environments of the young speakers; that to direct the language practices of the pupils from Martinique to a bilingual use(custom) of French and Creole which would be in break with the monolingual's at present practiced in the French educational space.

The adoption on April 10th, 2009, within the framework of the Law for the Economic development of it Besides Sea (Loi pour le Développement Économique de l'Outre Mer : LODÉOM), of an amendment introducing Creole languages into French national heritage allows to hope that new administrative measures will know how to welcome new educational modalities adapted to the implementation of a harmonious practice of the Creole and French, in particular among quite small frequenting the preschool and school structures in Martinique.

CULTURAL DIVERSITIES AND LINGUISTIC DIVERSITIES

In most of the extra-hexagonal regions concerned by a presence of French as official language, to learn to speak in community and to the school while protecting its cultural identity requires didactic and social conditions which must be built in the contact of a common language implying the cultural and social components of the environment of every speaker in presence.

The interest of such an approach of the language practices, in Martinique for example, is bound to a diversity of the cultural expressions of a society, every language, the Creole as French, being the vector of a culture and a mode of thought different. The Creole makes a reference to a culture sending back to a stemming oral tradition largely of a real-life experience in the colonial plantations whereas

French makes a reference to a culture of the older, Western Europe, where the concepts are transcribed for a long time in written language.

It is translated by language practices today reproducing the cultural diversities of the speakers from Martinique which, following their education and the social backgrounds in which they evolve, appropriate the Creole and French through practices of their own.

Of this fact we meet not many Inhabitants of Martinique expressing himself exclusively in French or in Creole. We can so notice that these two idioms are registered, consider held (or in resentment) by conditions of diglossia still present, in the linguistic heritage of the speakers from Martinique.

According to Charaudeau (2001), the language is necessary, certainly, for the constitution of a collective identity but it is advisable to separate language and culture, and to associate rather the language practices with the culture. This distinction is locatable in the linguistic rites and in the didactic contract between teachers and pupils in Martinique where the first ones make a reference to the Creole in a more or less indirect way, but also in the disparities between the language practices of the children and those of the adults in charge of their education. There are, indeed, several manners of speaking the same language if we consider on one hand the level of maturation of the speaker, on the other hand the socio-cultural environment in which it usually evolves.

So, when it is about speakers brought to communicate in a language which is not their language of origin, we are in the presence of a meeting of different ways of thinking because every speaker is not only carrier of an identity built in the contact of the idiom which he uses but he is also an agent of the practices being connected with the culture of his language circle of acquaintances.

The respective intentions of the people implied in the interaction are not only a function of motivations in direct link with the situations of communication and their conditions of reciprocal comprehension but also the modalities of use of the idiom which they usually use on their respective social and family backgrounds.

As regards the use of the Creole and French in Martinique, we can evoke here the situation of anchoring which makes that each of these idioms holds a place favored with regard to the other one according to social situations lived by the speakers.

It is thanks to this anchoring that the speakers from Martinique appropriate on an identity mode both of these languages which constitute the cultural markings of their sociolinguistic environment.

A CULTURE OF THE DIVERSITY (DIDACTIC IMPLICATIONS)

It is not thus any more for the exclusion one of other than the French language and the Creole language must be learnt in Martinique but, quite the opposite, the one In the contact of other one, in a didactic perspective favoring the mutual consideration of the cultural disparities bound to the respective use of every idiom. Prudent proposes "to establish pedagogy of French and Creole by taking into account the "interlectale" zone which is at the heart of the mother language of the Inhabitant of Martinique" (Prudent, on 2010, page 7) to take into account the pupil from Martinique and his sociolinguistic environment. This constitutes, in the field of the didactics of the languages, a big advance which allows conceiving the variety of "lectes" used on every idiom by these young speakers in situation of learning.

The relational modalities regarding linguistic acquisition are however browsed by rites which can, thanks to a strategic mediation of the teacher, question the implication of both idioms in the various situations of learning. In day-nurseries and nursery schools, it should be translated by a posture of reception and multicultural transmission of the adult towards the children whom he supervises.

The posture of reception corresponds to an attitude of listening and observation of the spontaneous language conducts of the quite small supervised by the adults. The fruit of this approach is to be put to the advantage of a location and of a collection of the disparities of social and cultural order.

The posture of transmission corresponds to a consideration of the located disparities and to an exploitation of these during the collective construction of the know-how. The Creole and French productions are reinvested in the whole of the interactions. The didactic contract supervisorsupervised, taught teacher concentrates on the modalities implemented in situation of learning. In this context, the pupils are invited to clarify and to argue their own productions and those of their peers, the main instruction being to express himself freely in French or in Creole.

This educational plan requires, it is advisable to remind it, a common formation of the adults-referents and the teachers in regional language (oral and written) but also in didactics of the languages so as to acquire skills regarding bilingual teaching and regarding mark of the language practices. It is also essential that these actors of the early childhood are competent in the field of the observation of the language conducts.

Eventually, it is a raising awareness and a formation of the quite small to the disparities and the cultural diversities of their language environment that are put as educational principles, in particular in the situations of learning where the know-how build themselves as well in the contact of the adults as the peers, any merged language skills.

When, for example, one of the directors of day-nursery makes the presentation of the activities of its establishment, she specifies that it is seen frequently by children among whom cultural origins turn out particularly diverse notably because of its zone of presence.

Situated in a district where residents from Martinique, French, Italian, German, Cambodian and Togolese families, this day-nursery of the periphery of Fort de France welcomes children who, in the contact of cultural disparities, have to evolve together of the section of the quite small in the section of the big. This director, to take into account the intercultural differences, organizes regularly a world tour of the cultures with, in every journey, the contribution activates of a culturally involved family.

During these meetings, the children discover the cultures from each other through recreational activities (ex: typical games of the visited countries) and by means of culinary, musical, dance and linguistic animations. The adults referents intervene in mediators, by supplying to the children of new linguistic tools to communicate with their peers by referring to culturally diversified registers (ex: little songs of different parts of the country) within the framework of reports base established on an intercultural construction of the know-how very well welcomed by the children. This example shows the interest of the respect for the interpersonal differences for the development of a process of reciprocal comprehension favorable to the language self-fulfillment of young speakers.

Indeed, the quite small put in situation of discovery with their peers by an adult referent concerned about this respect for the differences succeed in building the know-how which are transmitted them by implementing their own linguistic experience (intra subjective) while growing rich of those of their small companions (inter subjective experience).

While valuing and by stimulating the expression and the reception of the cultural differences among the young speakers he is responsible for educational the adult should obtain the most successful and most adapted language conducts. Cultural diversities are to be put to the advantage of the process of reciprocal comprehension by an adult-referent who has to create the optimal conditions of exchanges among the young speakers he is responsible for. He has to for it takes into account at the same time capacities and potentialities of the children and the sociolinguistic environment where they evolve.

  Martinique, in particular through the formation of certain professors in the Vocational Certificate in the Teaching profession of the Secondary education (CAPES) Creole and other teachers in the Languages and Regional Cultures ( LCR), under the impulse among others of the Group of Study and Creole-speaking and French-speaking Research in Space (GEREC-F), become Interdisciplinary Research Center in Literature, Languages, Art and Human Sciences ( CRILLASH).