O F K Kalenda
email: kalenda@karlin.mff.cuni.cz

H Pfitzner
email: hermann.pfitzner@univ-orleans.fr

J Spurn
email: spurny@karlin.mff.cuni.cz

ON QUANTIFICATION OF WEAK SEQUENTIAL COMPLETENESS

Keywords: 1991 Mathematics Subject Classification. 46B20 weakly sequentially complete Banach space, L-embedded Banach space, quantitative versions of weak sequential completeness

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.

Introduction and statement of the results

If X is a Banach space, we recall that it is weakly sequentially complete if any weakly Cauchy sequence in X is weakly convergent. In the present paper we investigate quantitative versions of this property. To this end we use several quantities related to a given bounded sequence (x k ) in X.

Let clust X * * (x k ) denote the set of all weak* cluster points of (x k ) in X * * . By δ (x k ) we will denote the diameter of clust X * * (x k ) (see also (4) below). Further, if A, B are nonempty subsets of a Banach space X, then d(A, B) denotes the usual distance between A and B and the Hausdorff non-symmetrized distance from A to B is defined by d(A, B) = sup{d(a, B) : a ∈ A}.

Note that a space X is weakly sequentially complete if for each bounded sequence (x k ) in X satisfying δ (x k ) = 0 (this just means that the sequence is weakly Cauchy) we have d(clust X * * (x k ), X) = 0 (i.e., all the weak* cluster points are contained in X, which for a weakly Cauchy sequence means that it is weakly convergent). It is thus natural to ask which Banach spaces satisfy a quantitative version of weak sequential completeness, i.e., the inequality [START_REF] Behrends | New proofs of Rosenthal's l 1 -theorem and the Josefson-Nissenzweig theorem[END_REF] d(clust

X * * (x k ), X) ≤ C • δ (x k )
for all bounded sequences (x k ) in X and for some C > 0. The starting point of our investigation was the following remark made by G. Godefroy in [3, p. 829]: If X is complemented in X * * by a projection P satisfying

(2)

x * * = P x * * + x * * -P x * * , x * * ∈ X * * , Ý then X is weakly sequentially complete and

(3) d(clust X * * (x k ), X) ≤ δ (x k )
for any sequence (x k ) in X.

It can be easily seen that ( 4)

δ (x k ) = sup x * ∈B X * ( lim sup k→∞ x * (x k ) -lim inf k→∞ x * (x k ) ) = sup x * ∈B X * lim n→∞ sup{|x * (x l ) -x * (x j )| : l, j ≥ n}
The first formula of ( 4) is used in [1, Section 2.1], the second one in [3, p. 829]. Banach spaces satisfying assumption (2) above are called L-embedded, see [5, Section III.1]. The proof of (3) can be found in [START_REF] Godefroy | Operators between subspaces and quotients of L 1[END_REF]Lemma IV.7].

By what has been said above, inequality (3) is a quantitative form of weak sequential completeness. In [3, p. 829] G. Godefroy mentions that it is not clear which weakly sequentially complete spaces can be renormed to have such a quantitative form of weak sequential completeness.

The aim of our paper is twofold. On the one hand we show that the answer to G. Godefroy's question cannot be positive for all weakly sequentially complete Banach spaces, more precisely we construct a weakly sequentially complete space that cannot be renormed in such a way that (3) holds, see Example 4 below. On the other hand we put inequality (3) into context by studying some modifications and possible converses, see the following theorem. In particular, we slightly improve inequality (3) -see [START_REF] Knaust | On c 0 sequences in Banach spaces[END_REF] in the theorem -but such that now the additional factor 2 is optimal.

We will use one more quantity (cf. [START_REF] Pfitzner | Boundaries of Banach spaces determine weak compactness[END_REF]) which in some situations is more natural than the quantity δ X , namely

δ (x k ) = inf { δ ( x kj ) : (x kj ) is a subsequence of (x k ) } .
Theorem 1. Let X be a Banach space and (x k ) be a bounded sequence in X. Then

(5) δ (x k ) ≤ 2 d(clust X * * (x k ), X).
If the space X is L-embedded, then also the following inequalities hold:

2 d(clust X * * (x k ), X) ≤ δ (x k ) , (6) 2 d(clust X * * (x k ), X) ≤ δ (x k ) . ( 7
)
Since we have trivially that δ ≤ δ X and d ≤ d it is natural to ask whether one of these quantities can be replaced by a sharper one in the inequalities of the theorem. The following remark and Example 3 show that this cannot be done in any of the inequalities ( 5)- [START_REF] Pfitzner | Boundaries of Banach spaces determine weak compactness[END_REF].

Remark 2. (a) In [START_REF] Knaust | On c 0 sequences in Banach spaces[END_REF], δ X cannot be replaced by δ and in [START_REF] Pfitzner | Boundaries of Banach spaces determine weak compactness[END_REF] d cannot be replaced by d. This is witnessed by the sequence (x k ) in X = 1 such that x 2k-1 = 0 and 5) is a kind of converse of (3) and holds in all Banach spaces. We note that δ cannot be replaced by δ X in [START_REF] Harmand | M -ideals in Banach spaces and Banach algebras[END_REF], in other words, inequality (3) cannot be reversed as it is, neither in L-embedded spaces. Indeed, let X = 1 . We consider the elements x k = 1 k e k and y k = e 1 + 1 k e k , k ∈ N. Let (z k ) be the sequence

x 2k = e k for all k ∈ N. Then d(clust X * * (x k ), X) = δ (x k ) = 0, d(clust X * * (x k ), X) = 1 and δ (x k ) = 2. (b) Inequality (
x 1 , y 1 , x 2 , y 2 , . . . . Then d(clust * * 1 (z k ), 1 ) = 0 because all weak * cluster points of (z k ) are contained in 1 , but

δ (z k ) ≥ lim sup k→∞ e 1 (z k ) -lim inf k→∞ e 1 (z k ) = 1.
(c) We further remark that in all inequalities in Theorem 1 the factor 2 is optimal, as witnessed by the sequence (e k ) in X = 1 . Indeed, then

d(clust X * * (e k ), X) = d(clust X * * (e k ), X) = 1 and δ (e k ) = δ (e k ) = 2.
It is also natural to ask whether d can be replaced by d in the inequality (5), i.e., whether the inequality ( 7) can be reversed (at least for L-embedded spaces). This is not the case by the following example.

Example 3.

There is an L-embedded space X and a bounded sequence

(x k ) in X such that δ (x k ) = 2 and d(clust X * * (x k ), X) = 0.
The negative partial answer to the mentioned question of G. Godefroy is given by the following example. In fact, we obtain a slightly stronger result. Not only there is a weakly sequentially complete Banach space not satisfying (1) for all bounded sequences and some C > 0, but we get even a weakly sequentially complete space not satisfying a weaker form of (1) -with d in place of d.

Example 4.

There exists a separable Banach space X with the Schur propertyin particular, X is weakly sequentially complete -which is 1-complemented in its bidual, such that there is no constant C > 0 satisfying

d(clust X * * (x k ), X) ≤ C • δ (x k )
for every bounded sequence (x k ) in X.

Proof of Theorem 1

The proof relies on two simple properties of 1 -sequences which are formulated in the following lemma. Lemma 5. Let X be a Banach space and (x n ) be a bounded sequence in X. Suppose that c > 0 is such that

n ∑ j=1 α j x j ≥ c n ∑ j=1 |α j | whenever n ∈ N and α 1 , . . . , α n are real numbers. Then (i) δ (x n ) ≥ 2c, (ii) d(clust X * * (x k ), X) ≥ c.
Proof. (i) It is clear that the sequence (x n ) is linearly independent. Hence there is a unique linear functional defined on its linear span whose value is c at x 2k-1 and -c at x 2k for each k ∈ N. By the assumption, the norm of this functional is at most 1. Let x * ∈ B X * be its Hahn-Banach extension. Then x * witnesses that δ (x n ) ≥ 2c.

(ii) Let x * * be any weak* cluster point of the sequence (x n ) in X * * and x ∈ X be arbitrary. It follows from [6, Proposition 4.2] that there is an index

m ∈ N such that ∞ ∑ j=m α j (x j -x) ≥ c ∞ ∑ j=m |α j |
for every sequence (α j ) ∞ j=m with finitely many nonzero elements. In particular, it follows that the vectors x j -x, j ≥ m, are linearly independent. So, there is a unique linear functional on their linear span whose value at each x j -x is equal to c. By the above inequality, the norm of this functional is at most one. Let x * ∈ X * be its Hahn-Banach extension. Then we have

x * * -x ≥ (x * * -x)(x * ) ≥ lim inf j→∞ x * (x j -x) = c.
This completes the proof of the lemma. Now we are ready to prove Theorem 1: We start by proving (5): Let (x k ) be a bounded sequence in X. We assume that δ (x k ) > 0 because otherwise (5) holds trivially. Let c ∈ (0, δ (x k )) be arbitrary. By [1, Theorem 3.2] there is a subsequence (

x n k ) such that k ∑ i=1 α i x ni ≥ c 2 k ∑ i=1 |α i | whenever k ∈ N and α 1 , . . . , α k ∈ R. By Lemma 5(ii) we get d(clust X * * (x n k ), X) ≥ c 2 , hence d(clust X * * (x k ), X) ≥ c 2 . As c ∈ (0, δ (x k )
) is arbitrary, (5) follows. We continue by proving (6): We set c = d(clust X * * (x k ), X) and assume that c > 0 because otherwise (6) holds trivially. Let ε ∈ (0, c) be arbitrary and let x * * be a weak* cluster point of the sequence (x k ) in X * * such that d(x * * , X) > c -ε 2 . Set x = P x * * and x s = x * * -x where P denotes the projection on X as in [START_REF] Fabian | Functional analysis and infinite-dimensional geometry[END_REF]. Then d(x * * , X) = x s . We claim that there is a subsequence (x kn ) such that (8

) n ∑ i=1 α i (x ki -x) ≥ (c -(1 -2 -n )ε) n ∑ i=1 |α i |
for all n ∈ N and all (α i ) n i=1 in R n . This will be proved by G. Godefroy's 'ace of ♦ argument' [5, p. 170], cf. the proof of [5, Proposition IV.2.5]. Since x s is a weak* cluster point of the sequence (x k -x), there is k 1 such that x k1 -x > c -ε 2 which settles the first induction step.

Suppose we have constructed x k1 , . . . , x kn . Let (α l ) L l=1 be a finite sequence of elements of the unit sphere of n+1 1 such that α l n+1 = 0 for all l ∈ {1, . . . , L} and such that for each α in the unit sphere of n+1 1 there is an element α l such that

α -α l n+1 1 < ε 2 n+2 sup k x k .
Let l ∈ {1, . . . , L} be arbitrary. Then

∑ n i=1 α l i (x ki -x) + α l n+1 x s is a weak* cluster point of the sequence ( ∑ n i=1 α l i (x ki -x) + α l n+1 (x k -x)) ∞ k=1 and for its norm we have n ∑ i=1 α l i (x ki -x) + α l n+1 x s = n ∑ i=1 α l i (x ki -x) + α l n+1 x s ≥ (c -(1 -2 -n )ε) n ∑ i=1 |α l i | + |α l n+1 |(c - ε 2 ) > (c -(1 -2 -n )ε) n+1 ∑ i=1 |α l i | = c -(1 -2 -n )ε.
It follows that there is

k n+1 > k n such that n+1 ∑ i=1 α l i (x ki -x) > c -(1 -2 -n )ε
for all l ∈ {1, . . . , L}. By a straightforward calculation using the choice of the α l and the triangle inequality we get that inequality (8), with n + 1 instead of n, holds for all α in the unit sphere of n+1

1
and hence for all elements of R n+1 . This finishes the construction. By Lemma 5(i) we get

δ (x kn -x) ≥ 2(c -ε), hence clearly δ (x k ) ≥ δ (x kn ) = δ (x kn -x) ≥ 2(c -ε). As ε ∈ (0, c) is arbitrary, we get (6).
Finally, let us prove (7): We take any subsequence (x kn ) and observe that

2 d(clust X * * (x k ), X) ≤ 2 d(clust X * * (x kn ), X) ≤ δ (x kn )
by [START_REF] Knaust | On c 0 sequences in Banach spaces[END_REF]. Then we can pass to the infimum over all (x kn ). This finishes the proof of the theorem.

Proof of Example 3

For n ∈ N set X n = n ∞ and let X be the 1 -sum of all the spaces X n , n ∈ N. Then X is L-embedded by [5, Proposition IV.1.5].

Further, let e n 1 , . . . , e n n be the canonical basic vectors of X n and let (x k ) be the sequence in X containing subsequently these basic vectors, i.e., the sequence e 1 1 , e 2 1 , e 2 2 , e 3 1 , e 3 2 , e 3 3 , e 4 1 , . . . , e 4 4 , . . . Then we have δ (x k ) = 2 as each subsequence of (x k ) contains a further subsequence isometrically equivalent to the canonical basis of 1 .

It remains to show that d(clust X * * (x k ), X) = 0. To do so, it is enough to prove that 0 is a weak cluster point of the sequence (x k ). To verify this, we fix g 1 , . . . , g m ∈ X * and ε > 0. Let K = max{ g 1 , . . . , g m }.

The dual X * can be canonically identified with the ∞ -sum of the spaces X * n , n ∈ N. Moreover, X * n is canonically isometric to n 1 . Thus each g ∈ X * can be viewed as a bounded sequence (g n ) n∈N , where g n = (g n,j ) n j=1 ∈ n 1 for each n ∈ N. We find N ∈ N such that K N < ε and let n ∈ N be such that n > mN . Let k ∈ {1, . . . , m} be arbitrary. We have Since e n j is an element of the sequence (x k ), this completes the proof that 0 is in the weak closure of the sequence, hence 0 is a weak cluster point (as the sequence (x k ) does not contain 0).

g k n ≤ g k ≤ K. As g k n = ∑ n j=1 |g k n,

Proof of Example 4

We recall that βN is the Čech-Stone compactification of N and M (βN) is the space of all signed Radon measures on βN considered as the dual of ∞ .

Let us fix α > 0 and consider the space

Y α = ( 1 , α • 1 ) ⊕ 1 (C[1, ω], • ∞ ).
Here • 1 denotes the usual norm on 1 , ω is the first infinite ordinal, C [1, ω] stands for the space of all continuous functions on the ordinal interval [1, ω] and • ∞ is the standard supremum norm. Note that we have the following canonical identifications:

Y * α = ( ∞ , 1 α • ∞ ) ⊕ ∞ ( 1 [1, ω], • 1 )
, and

Y * * α = (M (βN), α • M (βN) ) ⊕ 1 ( ∞ [1, ω], • ∞ ). For k ∈ N, let x k = (e k , χ [k,ω] ) ∈ Y α
, where e k denotes the k-th canonical basic vector in 1 and χ [k,ω] is the characteristic function of the interval [k, ω]. Let X α be the closed linear span of the set {x k : k ∈ N}. We observe that (9)

X α = { ((η k ), f ) ∈ Y α : f (n) = n ∑ k=1 η k for all n ∈ N } .
Indeed, the set on the right-hand side is a closed linear subspace of Y α containing x k for each k ∈ N. This proves the inclusion '⊂'. To prove the converse one, let us take any point ((η k ), f ) in the set on the right-hand side. Since (η k ) ∈ 1 , we get

((η k ), f ) = ∞ ∑ k=1 η k x k ∈ X α
as the series is absolutely convergent. It follows that for each ((η k ), f ) ∈ X α we have

α (η k ) 1 ≤ ((η k ), f ) ≤ (α + 1) (η k ) 1 ,
hence X α is isomorphic to 1 . More precisely, the projection on the first coordinate is an isomorphism onto 1 . In particular, X α has the Schur property (and thus it is weakly sequentially complete). We further observe that X * * α is canonically identified with the weak* closure of

X α in Y * * α , thus (10) X * * α = {(µ, f ) ∈ M (βN) × ∞ [1, ω] : (∀n ∈ N : f (n) = µ{1, . . . , n}) and f (ω) = µ(βN)}.
Indeed, the set on the right-hand side is a weak* closed linear subspace of Y * * α containing X α , which proves the inclusion '⊂'. To prove the converse one let us fix (µ, f ) in the set on the right-hand side. Take a bounded net (u τ ) in 1 which weak* converges to µ. For each τ there is a unique f τ ∈ C [1, ω] such that (u τ , f τ ) ∈ X α . Then (f τ ) is clearly a bounded net in ∞ [1, ω]. Moreover, we will show that (f τ ) weak* converges to f . Since the weak* topology on bounded sets coincides with the topology of pointwise convergence, it suffices to show that f τ pointwise converge to f . Indeed,

f τ (n) = n ∑ k=1 u τ (k) → µ{1, . . . , n} = f (n), for each n ∈ N, f τ (ω) = ∞ ∑ k=1 u τ (k) → µ(βN) = f (ω).
It follows that X α is 1-complemented in its bidual. To show that we set

P (µ, f ) = ((µ{k}), f -µ(βN \ N) • χ {ω} ), (µ, f ) ∈ X * * α . Then P is a projection of X * *
α onto X α of norm one. Indeed, if (µ, f ) ∈ X α , then µ(βN \ N) = 0 and hence P (µ, f ) = (µ, f ). Further, by ( 9) and (10) we get that P (µ, f ) ∈ X α for each (µ, f ) ∈ X * * α . Thus P is a projection onto X α . To show it has norm one, it is enough to observe that, given (µ, f ) ∈ X * * α , we have (µ{k}) 1 ≤ µ , and that f -µ(βN \ N) • χ {ω} is a continuous function on [1, ω] coinciding on [1, ω)

with f and so f -µ(βN \ N) • χ {ω} ∞ ≤ f ∞ .
Further, for the sequence (x k ), its weak * cluster points in X * * α are equal to {(ε t , χ {ω} ) : t ∈ βN \ N}, where ε t denotes the Dirac measure at a point t ∈ βN.

We claim that, for our sequence (x k ), we have

(11) d(clust X * * α (x k ), X α ) ≥ 1 2 and δ (x k ) = 2α.
To see the first inequality, we use the fact that the distance of any weak * cluster point of (

x k ) from X α is at least d(χ {ω} , C[1, ω]) = 1 2 . On the other hand, if t, t ∈ βN \ N are distinct, then (ε t , χ {ω} ) -(ε t , χ {ω} ) X * * α = (ε t -ε t , 0) X * * α = α ε t -ε t M (βN) = 2α
. This verifies (11). Now we use the described procedure to construct the desired space X. For n ∈ N, let α n =1 n and let X 1 n be the space constructed for α n . Let

X = ( ∞ ∑ n=1 X 1 n ) 1
be the 1 -sum of the spaces X 1 n . We claim that X is the required space. First, since each X 1 n has the Schur property, X, as their 1 -sum, possesses this property as well (this follows by a straightforward modification of the proof that Further, observe that

X * = ( ∞ ∑ n=1 X * 1 n ) ∞ and X * * ⊃ ( ∞ ∑ n=1 X * * 1 n ) 1 .
Note that the latter space is not equal to X * * but it is 1-complemented in X * * (cf. the proof of [5, Proposition IV.1.5]). Now it follows that X is 1-complemented in X * * . Finally, fix n ∈ N. We consider a sequence x k = (0, . . . , 0, n-th

x k , 0, . . . ), where the elements x k ∈ X 1 n , k ∈ N, are defined above. Let y = (0, . . . , 0, n-th (ε t , χ {ω} ), 0, . . . ), where t ∈ βN \ N, be a weak * cluster point of ( x k ) in X * * . Then, for any z = (z(1), z( 2

), . . . ) ∈ X, y -z X * * ≥ (ε t , χ {ω} ) -z(n) X * * 1 n ≥ 1 2 by (11). Hence d(clust X * * ( x k ), X) ≥ 1 2 .
On the other hand,

δ ( x k ) = δ (x k ) = 2 n ,
again by (11). From this observation the conclusion follows.

Final remarks

Up to now we have tacitly assumed that we are dealing with real Banach spaces. In fact, our proofs work for real spaces but all the results can be easily transferred to complex spaces as well. Let us indicate how to see this.

Let X be a complex Banach space. Denote by X R the same space considered over the field of real numbers (i.e., we just forget multiplication by imaginary numbers). Let φ : X * → (X R ) * be defined by φ(x * )(x) = Re x * (x),

x * ∈ X * , x ∈ X.

It is well known that φ is a real-linear isometry of X * onto (X R ) * . Let us define a mapping ψ : X * * → (X R ) * * by the formula ψ(x * * )(y * ) = Re x * * (φ -1 (y * )), x * * ∈ X * * , y * ∈ (X R ) * .

It is easy to check that the mapping ψ satisfies the following properties: (i) ψ is a real-linear isometry of X * * onto (X R ) * * .

(ii) ψ is a weak*-to-weak* homeomorphism. (iii) ψ(X) = X R .

It follows that for any sequence in X all the quantities in question (i.e., δ, δ, d and d) are the same with respect to X and with respect to X R . (Recall that δ is defined as the diameter of weak* cluster points, which has good sense in a complex space as well, even though in the complex case only the second formula of (4) works.) If, moreover, we observe that X R is L-embedded whenever X is L-embedded, we conclude that Theorem 1 is valid for complex spaces as well.

As for Examples 3 and 4, it is clear that they work also in the complex setting -we can just consider complex versions of the respective spaces.

has the Schur property, see [2, Theorem 5.19]). Hence X is weakly sequentially complete.

We finish by recalling that G. Godefroy's question, for which Banach spaces (3) holds, remains open. In particular, the following question seems to be open.

Question. Let X be a Banach space which is a u-summand in its bidual, i.e., there is a projection P : X * * → X with I -2P = 1. Does (3) hold for X (at least with some multiplicative constant)?

We conjecture that the space from Example 4, although it is 1-complemented in its bidual, is not a u-summand. At least the projection we have constructed does not work.

The first and third authors were supported in part by the grant GAAV IAA 100190901 and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education.


Ryad Zemouri
email: ryad.zemouri@cnam.fr

Rafael Gouriveau
email: rgourive@ens2m.fr

Paul Ciprian

Improving the prediction accuracy of Recurrent neural network by a PID controller

Keywords: Maintenance, prognostic, error of prediction, neural network, RRBF, PID

In maintenance field, prognostic is recognized as a key feature as the prediction of the remaining useful life of a system which allows avoiding inopportune maintenance spending. Assuming that it can be difficult to provide models for that purpose, artificial neural networks appear to be well suited. In this paper, an approach combining a Recurrent Radial Basis Function network (RRBF) and a proportional integral derivative controller (PID) is proposed in order to improve the accuracy of predictions. The PID controller attempts to correct the error between the real process variable and the neural network predictions.

I. INTRODUCTION

The high costs in maintaining complex equipments make necessary to enhance maintenance support systems and traditional concepts like preventive and corrective strategies are progressively completed by new ones like predictive and proactive maintenance [START_REF] Muller | Formalisation of a new prognosis model for supporting proactive maintenance implementation on industrial system[END_REF]. Current maintenance strategies have progressed from breakdown maintenance, to preventive maintenance, then to condition-based maintenance (CBM) managed by experts, and lately towards a futuristic view of intelligent predictive maintenance systems [START_REF] Heng Aiwina | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF]. Breakdown maintenance is the earliest form of maintenance, where no actions are taken to maintain the equipment until it breaks and consequently needs a repair or replacement. To prevent catastrophic failures and emergency shutdowns, preventive maintenance was introduced in the 1950s. A typical preventive maintenance scheme includes setting periodic intervals for machine inspections and maintenance regardless of the machine's health condition. The determination of optimal maintenance interval is critical for this scheme to work effectively.

Three key elements of effective CBM are data acquisition (i.e. the collection and storage of machine health information), data processing (i.e. the conditioning and feature extraction/selection of acquired data) and decision making (i.e. the recommendation of maintenance actions through diagnosis and/or prognosis) [START_REF] Heng Aiwina | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF]. Increased automation and mechanisation have made computerised diagnostics and prognostics systems a valuable tool for maintenance personnel in making maintenance decisions, or possibly even replace maintenance experts in due time. Today's concept of machine diagnosis comprises the automated detection and classification of faults, whereas machine prognosis is the automated estimation of how soon and likely a failure will occur. Prognostics promises to significantly reduce expensive downtime, spares inventory, maintenance labour costs and hazardous conditions. However, prognostic is a relatively new research area and has yet to receive its prominence compared to the other areas of CBM.

Related reviews on prognostics have been reported in the literature [START_REF] Heng Aiwina | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF]. Pusey et al. provided a broad overview of the development in diagnostics and prognostics technologies applicable to high-performance turbo-machines up until year 1999 [START_REF] Pusey | An assessment of turbomachinery condition monitoring and failure prognosis technology[END_REF]. Jardine et al. provided an overview and a catalogue of publications on data acquisition, data processing, diagnostics and prognostics of various machines up to year 2005 [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. Vachtsevanos et al. defined and described intelligent fault diagnostics and prognostics approaches for engineering systems through examples [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF]. Ma et al. discussed the need for a new paradigm shift in condition monitoring (CM) research for engineering asset management [START_REF] Ma | Condition monitoring in engineering asset management[END_REF].

Prognostic is nowadays considered as a key feature in maintenance strategies as the estimation remaining useful life (RUL) of a system allows avoiding inopportune spending. A central problem can be pointed out: the accuracy of a prognostic system is related to its ability to approximate and predict the degradation of the equipment: starting from a "current situation", a prognostic tool must be able to forecast the "future possible situations". From the research point of view, many developments exist to support these prognostic or forecasting activities ( [START_REF] Byington | Prognostic enhancements to diagnostic systems for improved condition-based maintenance[END_REF], [START_REF] Gooijer | 25 years of time series forecasting[END_REF], [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF], [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF]). However, choosing an efficient technique depends on classical constraints that limit the applicability of the tools: available data-knowledgeexperiences, dynamic and complexity of the system, implementation requirements (precision, computation time, etc.), available monitoring devices... Also, it can be difficult to provide effective models of dynamic systems including the inherent uncertainty of prognostic. That said, developments of this paper are founded on the following two complementary assumptions. 1) On one hand, real systems increase in complexity and their behaviour is often non-linear, which makes harder a modelling step, even impossible. 2) On the other hand, in many cases, it is not too costly to equip dynamic systems with sensors, which allows gathering real data online. According to all this, artificial neural networks (ANN) appear to be very promising prognostic tools: they learn from examples and attempt to capture the subtle relationship among data. They are computationally effective techniques and are thereby well suited for practical problems, where it is easier to gather data than to formalize the behaviour of the system being studied. Actual developments confirm the interest of using ANNs in forecasting applications ( [START_REF] Yam | Intelligent predictive decision support system for condition-based maintenance[END_REF], [START_REF] Zhang | Forecasting with artificial neural networks: the state of the art[END_REF], [START_REF] Arash | Multi-step ahead forecasts for electricity prices using NARX: A new approach, a critical analysis of one-step ahead forecasts[END_REF], [START_REF] Pisoni | Forcasting peak air pollution levels using NARX models[END_REF], [START_REF] Balaguer Ballester | Forecasting of surface ozone concentrations 24 hours in advance using neural networks[END_REF], [START_REF] Daniel | Fuzzy prediction architecture using recurrent neural networks[END_REF]).

In this context, the purpose of the work is to propose an ANN as a predictive tool for prognostic purpose and to improve its prediction accuracy. More precisely, the approach combines a Recurrent Radial Basis Function network (RRBF) [START_REF] Zemouri | Recurrent Radial Basis Function network for Time-Series Prediction[END_REF] and a proportional integral derivative controller (PID) [START_REF] Zemouri | Prediction Error Feedback for Time Series Prediction: a way to improve the accuracy of predictions[END_REF]. The PID controller attempts to correct the error between the real process and the neural network predictions.

The paper is divided into four main parts. First, the concept of "prognostic" is clarified and replaced within maintenance strategies, and the relationship between prognostic and prediction is also explained; the efficiency of a prognostic system is highly dependent on its ability to perform "good" predictions. In this way, the prognostic metrics are presented and developed. Then, the use of artificial neural networks for prognostic is justified and the ways of building such models are discussed. Consequently, the combination of the RRBF and the PID controller is proposed for prognostic. In the fifth and the sixth section, data benchmark, simulation conditions, results and discussions are presented.

PROGNOSTIC FRAMEWORK AND PREDICTION

A. From maintenance to prognostic

Maintenance activity combines different methods, tools and techniques to reduce costs while increasing reliability, availability and security of equipments. Thus, one usually speaks about fault detection, failures diagnosis and response development (choice and scheduling of preventive/corrective actions). Briefly, these steps correspond to the need, firstly, of "perceiving" phenomena, secondly, of "understanding" them, and finally, of "acting" consequently. However, rather than understanding a phenomenon which has just appeared like a failure (a posteriori comprehension), it is convenient to "anticipate" its manifestation in order to take adequate actions as soon as possible. This is what can be defined as the "prognostic process", the object of this paper.

Industrials show a growing interest in prognostic which becomes a major research framework; see recent papers dedicated to condition-based maintenance [START_REF] Ciarapica | Managing the condition-based maintenance of a combined-cycle power plant: an approach using soft computing techniques[END_REF], [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. The relative positioning of detection, diagnosis, prognostic and decision / scheduling in the maintenance framework is schematized in Fig. 1.a. From the phenomenological point of view, the complementarities of detection, diagnosis and prognostic can be explained as follows (see Fig. 1.b): 1) detection aims at identifying the functioning mode of the system, i.e., its current state, 2) assuming that a failure occurred, diagnosis enables to isolate and identify the component that has ceased to operate (past propagation: from effects to causes), 3) prognostic deals with the prediction of the future(s) state(s) of the system (future propagation: from causes to effects).

B. From prognostic to prediction

Although there are some divergences in literature, prognostic can be defined as proposed by the International Organization for Standardization: "prognostic is the estimation of time to failure and risk for one or more existing and future failure modes" (ISO 13381-1, 2004) [START_REF]Condition monitoring and diagnostics of machinesprognostics -Part1: General guidelines[END_REF]. Prognostic is also a process whose objective is to predict the remaining useful life (RUL) before a failure occurs given the current machine condition and past operation profile [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. Two salient aspects of prognostic appear [START_REF] Dragomir | Framework for a distributed and hybrid prognostic system[END_REF]: [START_REF] Arash | Multi-step ahead forecasts for electricity prices using NARX: A new approach, a critical analysis of one-step ahead forecasts[END_REF] prognostic is mostly assimilated to a prediction process (a future situation must be caught), [START_REF] Barbounis | A locally recurrent fuzzy neural network with application to wind speed prediction using spatial correlation[END_REF] prognostic is based on the failure notion, which implies that it is associated with a limit of acceptability (the predicted situation must be assessed with regard to a referential).

A central problem can be pointed out: the accuracy of a prognostic system is related to its ability to approximate and predict the degradation of an equipment; the prediction phase is a critical one. A look at prognostic metrics enables to point it out.

C. Prognostic metrics

There is no general agreement as to an appropriate and acceptable set of metrics that can be employed in prognostic applications, and researchers and maintenance practitioners are still working on this [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF], [START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF], [START_REF] Saxena Abhinav | On Applying the Prognostics Performance Metrics[END_REF], [START_REF] Leao | Prognostics performance metrics and their relation to requirements, design, verification and cost-benefit[END_REF]. Various measures emerge however from literature and are presented hereafter. As for any industrial task, prognostic can be evaluated at least in two ways:

-the main objective of prognostic is to provide the efficient information that enables the underlying decision process, i.e., the choice of maintenance actions. Thus, a first set of metrics are those that quantify the risks incurred by the monitored system. This kind of metrics can be called the prognostic measures, -assuming that prognostic is in essence an uncertain process, it is useful to be able to judge from its "quality" in order to imagine more suitable actions. In this way, prognostic system performance measures can be constructed.

Prognostic measures

As mentioned earlier, the main prognostic measure pursued is the predicted time to failure (TTF), also called the remaining useful life (RUL). In addition, a confidence measure can be built to indicate the degree of certitude of the future predicted failure time. By extension, and considering that practitioners can be interested in assessing the system with regard to any performance limit, RUL and confidence can be generalized: in Fig. 2.a, TTxx refers to the remaining time to overpass the performance limit Perf/xx and Conf/xxT is the confidence with which can be taken the asset TTxx > T.

Prognostic system performance measures

The Timeliness of the predicted time to failure (TTF) is the relative position of the probability density function (pdf) of the prediction model along the time axis with respect to the occurrence of the failure event. This measure evolves as more data are available and reveals the expected time to perform preventive actions [START_REF] Vachtsevanos | Intelligent Fault Diagnosis and Prognosis for Engineering Systems[END_REF] (see Fig. 2.b). According to [START_REF] Goebel | Prognostic information fusion for constant load systems[END_REF], one needs to define two different boundaries for the maximum acceptable late and early predictions.

The Precision reveals how close predictions are grouped or clustered together and is a measure of the narrowness of the interval in which the remaining life falls. Precision follows from the variance of the predicted results for many experiments. Complementarity of accuracy and precision is illustrated in Fig. 2.c.

The Repeatability measures the ability to have the same performances if the experiments are repeated several times. This gives the robustness of the model with regard to the setup step (training process for the neural model).

The Accuracy measures the closeness of a future predicted value to a real one. It has an exponential form and is as higher as the error between the predicted value of TTF and the real one is smaller. The accuracy depends on the quality of the timeliness, precision and repeatability performances of the prediction model. This accuracy is higher: if the probability density function is closer to the real data (good timeliness performances), if the prediction values are grouped together (good precision performances) and if the model is not sensitive to the parameters setup (good repeatability).

Fig. 2. Some prognostic metrics

Prognostic neural network system performance measures

Learning is an important capability of neural networks. Learning rules are algorithms for finding suitable weights W and other network parameters. Learning of a neural network can be viewed as a nonlinear optimization problem in which the goal is to find a set of network parameters minimizing the cost function for given examples. Putting it in another way, learning is an optimization process that produces an output that is as close as possible to the desired output by adjusting network parameters. For the same learning set, the neural network structure or parameters can be different at each training run. This results from the random initialization of certain parameters of the training process.

The quality of the prediction can be completely different at each running of the training algorithm. To test the prediction performances of a neural prediction model according to the prognostic metrics cited on the previous section, several running of the "training/test" process have to be done. Suppose that M represents the number of all the "training/test" running. For every running i of the training algorithm, a new value of the mean prediction error E(i) and the standard deviation std(i) are obtained for the n data of the test set as follows :

( ) 1 1 ( ) ( ) ( ) n i j E i j j n ξ ζ = = - ¦ (1)
( )

2 1 1 ( ) ( ) ( ) n j std i E i j n ζ = = - ¦ (2)
where ( ) i j ξ is the jth output obtained by the ith neural model

( ) j ζ is the jth system output
The measures of the prognostic neural system performance are then processed on the variations of E(i) and std(i). The different training and test running steps are presented as follows:

for i=1 to M

• train the NN on the training data set • test the NN on the test data set • calculate the mean prediction error E(i) produced by the ith neural model on the test set • calculate the standard deviation std(i) produced by the ith neural model on the test set Next i

The prognostic metrics for a neural model are then processed as follows:

• The timeliness is given by the global mean of all the M values of E(i):

1 1 Timeliness = = ( ) M i E E i M = ¦ (3)
where M is the number of the training/test running, E(i) is the mean error of every running test i of the ith neural model obtained at the training step i.

For a small value of the timeliness, the probability to have a prediction close to the real value can be significant. On the contrary, if the timeliness value is important, the probability to have a wrong prediction can be significant.

• The Precision is given by the global mean of all the M

values of std(i):

1 1 Precision = = ( ) M i std std i M = ¦ (4)
where std(i) is the standard deviation of each running test i of the ith neural model obtained by the training step i.

For a small value of the precision, the probability to have predictions grouped together can be significant. On the contrary, if the precision value is important, the probability to have a big dispersion of the predictions can be significant.

• The Repeatability is given by the standard deviation of both E(i) and std(i). A simple way to calculate the repeatability parameter is :

( ) ( )

+ Repeatability = 2 std E σ σ (5)
where ( ) and ( ) std E σ σ

represent the standard deviation of the M values of respectively the std(i) and E(i) values :

( ) ( ) 2 1 1 ( ) M i std std std i M σ = = - ¦ (6) ( ) ( ) 2 1 1 ( ) M i E E E i M σ = = - ¦ (7)
Repeatability reveals how close the different values of the E(i) and the std(i) are grouped or clustered together. This parameter reveals the dispersion of E(i) and std(i) values. For small values of ( ) and ( ) std E σ σ

, it means that at each training/running time i, the neural model gives the same performances on the test set. The repeatability parameter reveals the random initialization influence of some learning parameters. The training process is completely repeatable for small values of the repeatability parameter. The structure of the neural model is always the same at each running of the training process.

• The accuracy is calculated from the third previous parameters. A simple way to calculate the accuracy is :

1 Accuracy = Repeatability + Timeliness + Precision (8)
If a neural model has a good Timeliness, Precision and is completely Repeatable, the prediction given by this neural model is very close to the real data. The prediction confidence is then very high. A big value of the accuracy parameter gives a great confidence of the prediction. Fig. 3 illustrates the three prediction metrics: the timeliness, the precision and the repeatability described by a radar graph. This Fig. reveals the quality of the prediction made by the neural network according to the value of the three metrics. Fig. 4 gives the relation between the accuracy and the three metrics. One can see that the accuracy is very high for small values of the: timeliness, precision and repeatability. On the contrary, the accuracy is very low if, at least, one of the three metrics has a great value.

D. Perform good predictions: a critical issue

All prognostic metrics derive from the notion of "prediction": prognostic measures are themselves specific prediction measures and prognostic system performance measures can be seen as a way to asses the performances of the prediction in terms that can be interpreted by practitioners. As a synthesis, one should pay a particular attention to this prediction issue when choosing and adapting a prognostic tool. This aspect is developed in the next sections.

1 1 Timeliness = = ( ) M i E Ei M = ¦ 1 1 Precision = = ( ) M i std std i M = ¦ ( ) ( ) 2 1 1 ( ) M i std std std i M σ = = - ¦ ( ) ( ) 2 1 1 ( ) M i E E E i M σ = = - ¦ ( ) ( ) + Repeatability = 2 std E σ σ 0 t 1 t n t 0 t 1 t n t 0 t 1 t n t 0 t 1 t n t 1 µ 2 µ 1 2 µ µ <<< $ S R R U UH S H D WD E LO LW \ $ J UH D W UH S H D WD E LO LW \

THE RRBF AS A PREDICTION TOOL FOR PROGNOSTIC PURPOSE

A. Prediction / forecasting approaches overview

According to some authors, the methods presented in this section are sometimes labelled as "prognostic techniques". However, most of them refer to what, in this paper, is called "prediction / forecasting". That said, the aim of this part is not to dress an exhaustive overview of prediction techniques but to explain the orientations of works that are being performed.

Various prognostic approaches have been developed ranging in fidelity from simple historical failure rate models to high-fidelity physics-based models, [START_REF] Byington | Prognostic enhancements to diagnostic systems for improved condition-based maintenance[END_REF]. These methods can be associated with one of the following two approaches, namely model-based (or Physics-based) and data-driven [START_REF] Heng Aiwina | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF].

Model-based methods assume that an accurate mathematical model for the analyzed system / phenomenon can be constructed. The main advantage of these approaches is their ability to incorporate physical understanding of monitored system. Moreover, if the understanding of the system / phenomenon improves, the model can be adapted to increase its accuracy and to address subtle performance problems. But this closed relation with a mathematical model may also be a strong weakness: it can be difficult, even impossible, to catch the system's behavior.

For most industry applications, physics-based models might not be the most practical solution since the fault type in question is often unique from component to component and is hard to be identified without interrupting operation. However, physics-based models may be the most suitable approach for cost-justified applications in which accuracy outweighs most other factors and physics models remain consistent across systems, such as in air vehicles [START_REF] Roemer | Development of diagnostic and prognostic technologies for aerospace health management applications[END_REF]. They also generally require less data than data-driven models.

Data-driven approaches attempt to derive models directly from routinely collected condition monitoring (CM) data instead of building models based on comprehensive system physics and human expertise. They are built based on historical records and produce prediction outputs directly in terms of CM data. These approaches use real data (like on-line gathered with sensors or operator measures) to approximate and track features revealing the degradation of components and to forecast the global behavior of a system. Indeed, in many applications, measured input/output data is the major source for a deeper understanding of the system degradation. The strength of data-driven techniques is their ability to capture subtle relationships among the data even if the underlying relationships are unknown or hard to describe (by a learning process).

Data-driven approaches can be divided into two categories: statistical techniques (also called conventional data-driven methods) and artificial intelligence techniques (neural networks, fuzzy systems, decision trees, etc.). The conventional data-driven methods include simple projection models, such as exponential smoothing [START_REF] Batko | Prediction method in technical diagnostics[END_REF], linear autoregressive [START_REF] Kazmierczak | Application of autoregressive prognostic techniques in diagnostics[END_REF], quadratic discriminators and partial least squares models. One major advantage of these techniques is the simplicity of their calculations, which can be carried out on a programmable calculator. However, most of these trend forecasting techniques assume that there is some underlying stability in the monitored system. They also rely on past patterns of degradation to project future degradation. This reliance could lead to inaccurate forecasts in times of change. Most of these models follow the changing pattern with a time lag of at least one observation. Cempel et al. introduced the Tribo-vibroacoustical (TVA) model, which can estimate the time to failure of a machine as well as forecasting the vibration amplitude or condition [START_REF] Cempel | Simple condition forecasting techniques in vibroacoustical diagnostics[END_REF]. The model was compared with a constant trend parabolic model, an exponential trend model and an adaptive trending model in predicting a rolling bearing's peak vibration acceleration. It was reported that none of the forecasting techniques was able to predict the sudden change in the life curve.

B. Neural Networks -a fitted forecasting technique

Real systems are complex and their behavior is often non linear, non stationary. These considerations make harder a modeling step, even impossible. Yet, a prediction computational tool must deal with it. Moreover, monitoring systems have evolved and it is now quite easy to online gathered data. According to all this, data-driven approaches have been increasingly applied to prediction problems in general and to machine prognostic in particular. More precisely, research works emphasize on the interest of using artificial neural networks for prediction ( [START_REF] Barbounis | A locally recurrent fuzzy neural network with application to wind speed prediction using spatial correlation[END_REF], [START_REF] Dai | Chained DLS-ICBP neural networks with multiple steps time series prediction[END_REF], [START_REF] Mandic | Recurrent Neural Networks for Prediction: Learning Algorithms and Architectures and Stability[END_REF], [START_REF] Yam | Intelligent predictive decision support system for condition-based maintenance[END_REF]).

Artificial neural networks (ANNs) are currently the most commonly found data-driven techniques in the prognostics literature [START_REF] Heng Aiwina | Rotating machinery prognostics: State of the art, challenges and opportunities[END_REF]. ANNs are a special case of adaptive networks that have been extensively explored in literature because of the following aspects. An ANN consists of a layer of input nodes, one or more layers of hidden nodes, one layer of output nodes and connecting weights. ANNs can perform nonlinear modelling without a priori knowledge: they are able to learn complex relationships among "inputs and outputs". The network learns the unknown function by adjusting its weights with repetitive observations of inputs and outputs. Numerous studies across various disciplines have demonstrated the merits of ANNs, including the abilities to (a) perform faster than system identification techniques in multivariate prognosis [START_REF] Lee | A systematic approach for developing and deploying advanced prognostics technologies and tools: methodology and applications[END_REF]; (b) perform at least as good as the best traditional statistical methods, without requiring untenable distributional assumptions [START_REF] Tse | Prediction of machine deterioration using vibration based fault trends and recurrent neural networks[END_REF], [START_REF] Joshi | Beyond the Cox model: artificial neural networks for survival analysis part II[END_REF]; and (c) capture complex phenomenon without a priori knowledge [START_REF] Bostwichk | Prediction of individual patient outcome in cancer[END_REF]. A widely known limitation of ANNs is the lack of transparency, or rather the lack of documentation on how decisions are reached in a trained network. However, it was argued in [START_REF] Bostwichk | Prediction of individual patient outcome in cancer[END_REF] that increase in model complexity reduces the transparency of both traditional statistical models and ANN models. It is just that ANNs are more capable in modelling complex phenomenon and consequently need a more complex structure to represent the phenomenon. Rules can actually be extracted from trained ANNs to explain how decisions are reached, see [START_REF] Setiono | Opening the neural network black box: An algorithm for extracting rules from function approximating artificial neural networks[END_REF] and [START_REF] Tickle | The truth will come to light: directions and challenges in extracting the knowledge embedded within trained artificial neural networks[END_REF].

The temporal NNs have two typical connection architectures depending on the type of time representation [START_REF] Elman | Finding Structure in Time[END_REF]: in feedforward networks (like the multi layers perceptron MLP or the radial basis function network RBF) time is represented as an external mechanism, whereas recurrent networks (like the Elman architecture or the recurrent radial basis function network RRBF) are able to treat time dimension without any external mechanism. Recurrent neural network can be described by a NARX model (Nonlinear AutoRegressive with eXogenous inputs). Both have been employed in system behavior forecasting. In order to explicitly take into account time in forecasting tools, backward networks architectures have been also developed. These NARX recurrent neural networks are fundamentally different from feedforward architectures in the sense that they not only operate on an input space but also on an internal state space. Recurrent ANNs have been compared with some of the well known methods for the prediction of non-linear time series. Results indicated that RNNs have a better forecasting performance than the classical methods and are even better than the feedforward type ANNs.

One of the first successful application of ANNs in forecasting is reported by Lapedes and Farber who designed a feedforward ANN that can accurately mimic a chaotic series [START_REF] Lapedes | Nonlinear signal processing using neural networks: prediction and system modelling[END_REF], [START_REF] Zhang | Forecasting with artificial neural networks: the state of the art[END_REF]. In general, feedforward ANNs (MLP, RBF) trained with the backpropagation algorithm have been found to perform better than classical autoregressive models for the trend prediction of non linear time series [START_REF] Yam | Intelligent predictive decision support system for condition-based maintenance[END_REF].

The most simple ANN-based machinery prognostics approach was time series prediction models [START_REF] Ekonomou L | Implementation of Artificial Intelligence in the Time Series Prediction Problem[END_REF], [START_REF] Kim | Modeling of Ion Energy Distribution Using Time-Series Neural Network[END_REF], [START_REF] Minji Kwon | Prediction of Chamber Leak Pattern Using Time-Series Neural Network[END_REF], [START_REF] Chibanga | Use of Neural Networks to Forecast Time Series: River Flow Modeling[END_REF], [START_REF] Balaguer Ballester | Forecasting of surface ozone concentrations 24 hours in advance using neural networks[END_REF]. Tse et al. and Yam et al. used recurrent neural networks (RNNs) to trend CM indices and forecast successive index value at the next time step [START_REF] Tse | Prediction of machine deterioration using vibration based fault trends and recurrent neural networks[END_REF], [START_REF] Yam | Intelligent predictive decision support system for condition-based maintenance[END_REF]. Wang et al. developed a recurrent wavelet neural network (RWNN) to predict rolling element bearing crack propagation [START_REF] Wang | Fault prognostics using dynamic wavelet neural networks[END_REF]. The network performed satisfactorily in trending an artificially seeded and manually enlarged crack, provided that sufficient data points were used and network retraining was carried out after each time step. In [START_REF] Wang | Prognosis of machine health condition using neuro-fuzzy systems[END_REF], Wang et al. used a Neuro-Fuzzy (NF) network to predict spur gear condition value one step ahead. The fuzzy interference structure is determined by experts, whereas the fuzzy membership functions are trained by the neural network [START_REF] Rivas-Echeverría | Time Series Forecasting using ARIMA, Neural Networks and Neo Fuzzy Neurons[END_REF]. The NF system performed much better than RNN when there was sufficient training data. However, it could not predict well when the train-set was small or when there were fast dynamic fluctuations, such as during the chipping of gear tooth surface material or just prior to gear failure. An adaptive training technique was later proposed by Wang et al. to improve the NF model [START_REF]An adaptive predictor for dynamic system forecasting[END_REF]. The addition of feedback links to the NF model was able to increase the forecasting accuracy. However, further work is needed to extend the prediction horizon from single step to multiple steps ahead. Feed forward neural network (FFNN) has also been used to perform single-step-ahead prediction of rolling element bearing condition by [START_REF] Shao | Prognosis of remaining bearing life using neural networks[END_REF]. Multiple-step-ahead predictions were also performed simply by feeding the predicted value back into the network input until the desired prediction horizon was reached. The authors also proposed some rules to vary the data sampling period according to the change ratio of consecutive condition index values.

C. RBF and Recurrent RBF networks

The Radial Basis Function network

In practice, Multi-layer perceptrons (MLP) have been found to perform poorly in a number of ways, slow convergence of weights and difficulty in modelling differential responses. Radial basis function (RBF) neural networks are able to surpass the MLPs as they are simpler in structure and have the ability to model any nonlinear function in a straightforward way. On one hand MLP networks are global approximators with nonlinear input-output mappings and the representation of knowledge is distributed throughout the network. On the other hand, RBF networks are local approximators with nonlinear input-output mapping. The knowledge representation in this case is localized. Thus, RBF network are able to learn faster and suffer less from interference, as compared to MLPs.

A key feature of RBF networks is that the output layer is merely a linear combination of the hidden layer signals, there being only one hidden layer. Therefore, RBF networks allow for a much simpler weight updating procedure and subsequently open up greater possibilities for stability proofs and network robustness in that the network can be described readily by a set of nonlinear equations.

The RBF network is commonly used for the purpose of modelling uncertain and nonlinear functions. Utilizing RBF networks for modelling purposes could be seen as an approximation problem in a high-dimensional space. Consider the RBF network, which can be seen as a two-layer processing structure, as shown in Fig. 5.a. The hidden layer consists of an array of computing units (i.e., 1 2 , ,... M φ φ φ ). These hidden units provide a set of basis functions of the input vectors (i.e., 1 2

, ,.... d x x x ) as they are expanded into the higher dimension hidden-unit space. The mapping from the input vectors to the outputs of the hidden units is nonlinear, whereas the mapping from the hidden units to the final output of the RBF network is linear.

The general mapping function of the RBF network can be represented by ( ) exp( ) 2

j j j φ - = - x c ı x ( 10
)
Each RBF contains a parameter vector called a centre ( j c ), and calculates a squared distance between the centre and the input vector ( x ). The result is then divided by the width ( j ı )

and then passed through an exponential function.

Training a RBF with linear outputs is very fast and is accomplished through two stages:

-The first stage is unsupervised and accomplished by obtaining cluster centres of the training set input vectors. A popular method for that purpose is the k-means clustering, -the second stage consists in solving a set of linear equations, the solution of which can be obtained by a matrix inversion technique such as singular value decomposition or least squares method.

The Recurrent Radial Basis Function network

The Recurrent RBF neural network considers time as an internal representation (Fig. 5.b). The dynamic aspect is obtained by the use of an additional self-connection to the input neurons with a sigmoid activation function. The RRBF network can thus take into account a certain past of the input signal.

Every neuron of the input layer gives a summation at the instant t between its input i x and its previous output weighted by a self-connection ii w . The output of its activation function is:

( ) ( 1) ( ) i i i i i a t w t x t ξ = -+ , ( )
( ) ( ) i i t f a t ξ = (11)
where ( ) i a t and ( ) i t ξ represent respectively the neuron activation and its output at the instant t , and f is the sigmoid activation function defined as:

( ) ( ) ( )

1 exp( ) 1 exp( ) f x k x k x = - - + - (12)
The RRBF network was described on several publications [START_REF] Zemouri | Réseaux de neurones Récurrents à Fonction de base Radiales :RRFR/ Application au pronostic[END_REF], [START_REF] Zemouri | Recurrent Radial Basis Function network for Time-Series Prediction[END_REF], [START_REF] Samantaa | Construction of a radial basis function network using an evolutionary algorithm for grade estimation in a placer gold deposit[END_REF], [START_REF] Gowrishankar; Ramesh | Neural network based BER prediction for 802.16e channel[END_REF], [START_REF] Evgueni | A novel approach for distributed application scheduling based on prediction of communication events[END_REF], [START_REF] Palluat | A neuro-fuzzy monitoring system: Application to flexible production systems[END_REF]. A complete study of the looped neuron can be found on [START_REF] Frasconi | Local Feedback Multilayered Networks[END_REF], [START_REF] Bernauer | Les réseaux de neurones et l'aide au diagnostic : un modèle de neurones bouclés pour l'apprentissage de séquences temporelles[END_REF]. The reader can find in a detailed mathematical demonstration of the dynamic behavior of the looped neuron. To have the longest memory, the self connection weight ii w and the parameter k of the sigmoid function must respect this relation: 2

ii kw = .

COMBINING THE RRBF AND THE PID CONTROLLER FOR PROGNOSTIC PURPOSE

A. Principle RRBF appears to be a good candidate for prediction in prognostic applications. Nevertheless, one can improve its prediction accuracy by combining the RRBF output with a Proportional-Integral-Derivative controller (PID) [START_REF] Zemouri | Prediction Error Feedback for Time Series Prediction: a way to improve the accuracy of predictions[END_REF]. The use of the PID as a controller to perform the neural prediction has the same objectives as all other conventional feedback control applications. Fig. 6 shows the prediction structure where the PID controller is used to minimize the neural prediction error ( ) t ε . Let call this prediction structure the RRBF PID structure.

11

w 22 w 1 φ 2 φ M φ ( 1) y t + ( ) y t ( 1) y t - τ ( ) t Ψ ( ) y t + - ( ) t ε τ ( 1) t Ψ + ( ) d t K t ε ∂ ∂ ( ) i K ε τ τ ∂ ³ ( ) p K t ε Σ Σ 11 w 12 w 1M
w Fig. 6. The RRBF PID structure for time series prediction At any time t, the prediction error of the RRBF neural network can be expressed as:

( ) ( ) ( ) t y t t ε = -Ψ (13)
where ( ) y t represents the real system output, and ( ) t Ψ the RRBF network predicted output. A one step neural prediction is:

[ ] 1 [START_REF] Arash | Multi-step ahead forecasts for electricity prices using NARX: A new approach, a critical analysis of one-step ahead forecasts[END_REF] , ( )

M kj j j t w φ = Ψ + = = ¦ g y w y ( 14
)
The aim of the PID is to apply a corrective action on this error. The output of the final one-step ahead prediction ( 1) y t + obtained by the RRBF PID structure is then defined as: 0 ( )

( 1) ( ) ( )

t p i d t y t t K t K K t ε ε ετ τ ∂ + = Ψ + + + ∂ + ∂ ³ (15) [5] 0 0 ( ) ( 1) ( 1) ( ) ( ) ( ) ( ) ( )
t p i d t p i d y t y t t K y t K y K t t K t K K t τ τ τ τ ∂ + = Ψ + + + ∂ + ∂ ∂Ψ -Ψ - Ψ ∂ - ∂ ³ ³ (16)
where, p K , i K , d K are the proportional, integral and derivate gains of the PID. In a general way, the prediction output ( 1) y t + obtained by the RRBF PID can be defined as follows:

( 1)

[ , ] y t + = F y Ȍ [START_REF] Goebel | Prognostic information fusion for constant load systems[END_REF] and

[ ] ( 1)
, t Ψ + = g y w [START_REF] Gowrishankar; Ramesh | Neural network based BER prediction for 802.16e channel[END_REF] [.] F is the whole nonlinear function for the prediction purpose. Note that if the integrator parameter is equal to zero ( 0 i K = ), the function F is similar to the well-known NAR models (Nonlinear Auto Regressive model). In a general way, a great equivalence exists between the NARX models and the recurrent neural network (RNN) [START_REF] Sum | a note on the equivalence of NARMAX anf RNN[END_REF]. The next section presents briefly the different structures of the NARMAX models, followed by a comparative test on a NARX time series prediction.

B. NARMAX and NARX model for optimal predictors

The statistical approach for forecasting involves the construction of stochastic models to predict the value of an observation t y using previous observations. A very general class of such models used for forecasting purpose is the Nonlinear AutoRegressive Moving Average with eXogenous inputs (NARMAX models) [START_REF] Gao | NARMAX time series model prediction : feedforward and recurrent fuzzy neural network approaches[END_REF], [START_REF] Sum | a note on the equivalence of NARMAX anf RNN[END_REF], [START_REF] Luh Guan-Chun | Artificial immune regulation (AIR) for model-based fault diagnosis[END_REF], [START_REF] Pisoni | Forcasting peak air pollution levels using NARX models[END_REF] given by:

( ) [ ( 1
where y, e and x are output, noise and external input of the system model respectively. y n , e n and x n are the maximum lags in the output, noise and input, respectively, and F is an unknown smooth function. It is assumed that e(t) has zero mean, is independent and identically distributed, is independent of past y and x, and has a finite variance 2 σ .

The NARMAX models are nonlinear generalization of the well-known ARX models, which constitute a standard tool in linear black-box model identification. Several special cases of the general NARMAX ( y n , e n , x n ) model which are frequently used are summarized here :

• NAR( y n ) model (Nonlinear AutoRegressive) : ( ) [ ( 1),,,, ( )] (
The NARX models can represent a wide variety of nonlinear dynamic behaviors and have been extensively used in various applications [START_REF] Pisoni | Forcasting peak air pollution levels using NARX models[END_REF], [START_REF] Arash | Multi-step ahead forecasts for electricity prices using NARX: A new approach, a critical analysis of one-step ahead forecasts[END_REF], [START_REF] Pham | A hybrid of nonlinear autoregressive model with exogenous input and autoregressive moving average model for long-term machine state forecasting[END_REF], [START_REF] José Maria | Long-term time series prediction with the NARX network: An empirical evaluation[END_REF], [START_REF] Wei | Forecasting the geomagnetic activity of the Dst index using multiscale radial basis function networks[END_REF].

C. Example on a NARX time series model

To demonstrate the validity of our proposed predictor, the RRBF PID is trained and tested on a nonlinear autoregressive process with exogenous input (NARX model) generated as follows [START_REF] Gao | NARMAX time series model prediction : feedforward and recurrent fuzzy neural network approaches[END_REF]:

2 2 ( 1) ( 2)[ ( 1) 2.5] ( ) ( 1) 1 ( 1) ( 2)
y t y t y t y t

x t y t y t

- - -+ = + - + -+ - (23)
where the input has the form ( ) sin(2 / 25)

x t t π = .
The topology of the RRBF network is estimated using the first 20 observations of the process. The predictions ^( 1) y t + of ( ) y t are then tested on the following observations. Fig. 7 presents the results obtained on the 100 last observations (from t=200 to t=300). To test the influence of the PID corrective action, three prediction models are compared. The first one is the RRBF model. On the second model, only the PID action is tested by considering the RRBF output equal to zero ( ( ) 0 t Ψ = ). On the third one, the whole model is tested (RRBF + PID).

• RRBF network prediction without PID action:

^( 1)
( 1)

y t t + = Ψ + (24)
• PID output without the RRBF network:

^0 ( ) ( 1) ( ) ( ) t p i d y t y t K y t K y K t τ τ ∂ + = + ∂ + ∂ ³ (25)
• RRBF with PID correction:

^0 0 ( ) ( 1) ( 1) ( ) ( ) ( ) ( ) ( ) t p i d t p i d y t y t t K y t K y K t t K t K K t τ τ τ τ ∂ + = Ψ + + + ∂ + ∂ ∂Ψ -Ψ - Ψ ∂ - ∂ ³ ³ ( 26
)
The RRBF is trained by the k-means algorithm to find the Gaussian parameters and the linear regression to find the output weights. 9 Gaussians nodes were used for the RRBF model and the best PID parameters were found by testing all the possibilities of ( , , p i d K K K ) from 0 to 1 by increments of 0,1. The best PID parameters are then: 0.9, 0, 0.2

p i d K K K = = = .
With this parameter values, the structure of the PID is then similar to an Auto-Regressive model (AR). The RRBF PID seems to be better than the classic RRBF structure and also better than the AR model (Fig. 7).

The prediction error obtained by the RRBF network is judiciously corrected by the PID structure (AR model). The next sections give more results and discussions of the RRBF PID obtained on three data Benchmarks.

5. DATA BENCHMARKS, SIMULATION CONDITIONS AND CROSSED VALIDATION Three experimental data sets have been used to test the prediction performances of the RRBF PID structure regards to the classical RRBF network. In the three cases, the aim of the predictions is to approximate a phenomenon by learning data gathered from the system.

For all the benchmarks, three sets were used: a training set, a validation set and a test set (Fig. 8). The first set was used for the training parameters of the neural network (the k centres of the Gaussian nodes and the output weights). 50 samples have been used for training set and 50 samples for validation set. All data have been normalized by range [-1,+1]. Predictions steps were made from "t+1" to "t+10" by increments of 1 (in order to measure the stability of results in time). Predictions have been performed with two past inputs at a step time "t" and "t-1" (Fig. 6).

In order to find the best RRBF structure with regard to the k centers, several models of neural networks have been created by varying the number of basis functions from 2 to 50 nodes. The k-means training algorithm has been used to find the best Gaussian centers for each k-structure. The basis width parameter of the Gaussian nodes was fixed to 1. The neural network trained with the first set is validated in the second one. Validation uses data different from the training set, thus the validation set is independent from the estimated model. This helps to select the best one among the different model parameters. To avoid overfitting or underfitting, the optimal model parameters should be selected so as to have the best performance measure associated with the validation set. Since this dataset is independent from the estimated model, the generalization error obtained is a fair estimated. The best model which gives the best mean prediction error in the validation set is then selected and tested in the third set (the test set). The model with the best generalization performance is then selected with this crossvalidation technique.

( ) x t ( 1 ) x t- τ Σ ˆ( 1 ) x t + ( 1 ) x t + ( ) x t ( 1 ) x t + 11 w 22 w 1 φ 2 φ M φ ˆ( 1 ) x t + ( ) x t ( 1 ) x t- τ Σ ( ) x t ( ) x t

Fig. 8. Training process and crossed validation

Once the best RRBF model found, the second step is to find the PID parameters (Kp, Ki, Kd). At this stage, theses parameters varied from 0 to a maximum of 1 by increments of 0,1. While it is appreciated that values used for this scaling variable are extremes, these values have been chosen to encapsulate all possibilities. Two sets of PID parameters have been selected: the parameters that give the best performance (mean prediction error) on the training set (RRBF PID1 ) and the parameters that give the best performance on the validation set (RRBF PID2 ). u(t)} values as inputs. The system is described by the following NARX model:

( ) [ ( ), ( 1

), ( ), ( 1)] y t n y t y t u t u t

+ = - - F (31)
6. RESULTS AND DISCUSSION To measure the prognostic system performances presented in section 2.3.3, the simulation test was running 1000 times for each benchmark (i=1 to M=1000). At each simulation running i, the mean prediction error E(i) and the standard deviation std(i) are calculated.

A. Mackey-Glass

Fig. s 10-11-12-13 illustrate the results obtained by the three neural models (RRBF, RRBF PID1 and RRBF PID2 ) for the Mackey-Glass data test for different horizons of prediction n (n=1 to 10). The results show that the prognostic performance measures obtained by the RRBF PID2 are better than those obtained by the two other models. Fig. 10 reveals the great disparity of the timeliness parameter among the three models. This Fig. shows that for the prediction horizon n=1 to 9, the RRBF PID2 has the best timeliness value. The disparity between the models decreases with the increasing of the prediction horizons.

Fig. 11 shows the precision parameter for different models for different horizons of prediction. The precision of the RRBF PID1 and RRBF PID2 are similar for the different values of the horizon n. The prediction obtained with the RRBF model is more dispersed than the two other models.

Fig. 12 reveals how the prediction can be repeatable. For the prediction horizon n=0 to 6, the RRBF PID2 is the model with the best repeatability performance.

The accuracy parameter obtained from the three metrics is shown in Fig. 13 and is calculated according to the following formula:

1 accuracy = Repeatability + timeliness + Precision (32)
The RRBF PID2 is the model with the best accuracy performance for n=1 to 6. This accuracy becomes the same for the three models for n>6.

B. Lorenz data set

The results obtained with the three prediction models for the Lorenz data set are presented by Fig.s 14-15-16-17. Prognostic performances measures obtained by the RRBF PID2 are better than those obtained by the two other models. The RRBF PID2 model obtains the best timeliness and precision parameters for all the prediction horizons n. For the repeatability parameter, we have good performances for the RRBF PID2 only for n=1 to n=5. The prediction accuracy is illustrated in Fig. 17. The RRBF PID2 model obtain the best accuracy with a great disparity for n=1, decreases until n=5 and becomes the same for the three models for n>6.

C. Box & Jenkins data set

Fig. s 18-19-20-21 show the results obtained by the three models for the Box&Jenkens data set for different horizons of prediction n (n=1 to 10). The performances of the RRBF PID2 are not as better as for the two previous data set. Nevertheless, the prediction accuracy of the RRBF PID2 is better than the two other models for n=1 to n=5. For n=6 & n=7, the prediction performance of the RRBF PID2 is very bad for the three metrics. The accuracy prediction obtained with the RRBF PID2 model is the best for n=1.

D. Discussions and future works

For all the tests, the best results of RRBF PID are obtained with the integrator parameter equal to zero 0 i K = . With this value, the PID action is reduced to a PD corrector. This correction added to the neural network output is similar to an Auto-Regressive model (AR). Fig. 9 illustrates the structure of the whole prediction model obtained after the tests. The results obtained by the RRBF PID2 are significantly better than the two other models for the Mackey-Glass and the Lorenz data benchmark. The performances obtained by the three models on Box&Jenkins benchmark are similar (there is no significant performance difference among the three structures). The Mackey-Glass and the Lorenz data benchmarks are pseudo periodic and are represented by a NAR model. These two benchmarks are described by differential equations. The Box&Jenkins data benchmark is a real system. There is no formal relation between the input and the output and there is no pseudo periodicity on the data. The relation between the input/ouput is described by a NARX model ( ( ) [ ( ), ( 1), ( ), ( 1)] y t n y t y t u t u t + = --F

). The linear PID corrector cannot perform significantly the prediction obtained by the recurrent neural network.

The experimental study presented in this paper reveals the good prediction ability of the RRBF PID and opens some interesting perspectives for our future works. The feedback error performs significantly the prediction performances of the neural network with regard to the prediction metrics: Timeliness, Precision and Repeatability. A good accuracy of the prediction is then obtained by the combination of the Neural Network and the AR model (Fig. 9). Further study is necessary to investigate the robustness of the proposed method by using various training data groups. Additional tests should be made to evaluate the performances of a NARX model to process the feedback prediction error made by the ANN (use a NARX model instead of the AR model in Fig. 9). CONCLUSIONS In maintenance field, prognostic is recognized as a key feature in estimating the remaining useful life of an equipment allows avoiding inopportune maintenance spending. However, it can be difficult to define and implement an adequate and efficient prognostic tool that includes the inherent uncertainty of the prognostic process. Indeed, an important task of prognostic is that of prediction. In this context, the purpose of the work reported in this paper is to point out an accurate prediction technique and to propose a way to improve its prediction performances.

The concept of "prognostic" has been positioned within the maintenance strategies in order to point out the importance of the prediction phase in prognostic. According to the global requirements that can be expected from a forecasting tool, the neural network RRBF has been presented as a candidate to support this activity. An improvement of this neural network has also been proposed by combining it with a proportional integral derivative controller (PID). The PID controller attempts to correct the error between the real process variable and the neural network predictions. Various simulations have been led with three benchmark problems. Results show that the proposed prediction structure enables the forecasting to be a more robust task without increasing complexity of treatments. The whole is of good omen for prognostic purpose.

Fig. 1 .

 1 Fig. 1. Prognostic within maintenance strategies

Fig. 3 .Fig. 4 .

 34 Fig. 3. Neural Network system performance measures

  the second layer of the RBF network, it functions like a summer with a set of weights (i.e.,

Fig. 5 .

 5 Fig. 5. Radial Basis Function and Recurrent Radial Basis Function networks

Fig. 7 .

 7 Fig. 7. Prediction results obtained on a NARMAX example

Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .Fig. 13 .Fig. 14 .Fig. 15 .Fig. 16 .Fig. 17 .Fig. 18 .Fig. 19 .Fig. 20 .Fig. 21 .

 9101112131415161718192021 Fig. 9. The Final prediction structure obtained after the test results.

  )

		•	NARX( y n , x n )	model		(Nonlinear
			AutoRegressive with eXogenous inputs):
	( ) y t	=	[ ( 1)... ( y t y t n x t ), ( 1)... ( y x t n ----F	x	)] ( ) e t +
						y t	=	F	y t	-	y t n -	y	+	e t	(20)
							•			NARMA( y n , e n )	model	(Nonlinear
										AutoRegressive Moving Average):
						( ) y t	=	[ ( 1)... ( y t y t n e t ), ( 1)... ( y e t n e ----F	)] ( ) e t +	(21)

A. Mackey Glass

The first data set is the chaotic Mackey-Glass time series data [START_REF] Jang | ANFIS: adaptive network-based fuzzy inference system[END_REF]. This time series is a benchmark problem extensively used: it's a non periodic and non convergent time series. Considering our final applicative objective (the prognostic of failures) able to carry out predictions on such a signal is of good omen: what makes difficult a modeling phase are real complex systems which generally have a nonstationary and non-linear behavior. The time series is governed by the following formula:

C. Box-Jenkins

The third benchmark is the Box-Jenkins furnace data [START_REF] George | Time series analysis: Forecasting and control[END_REF]. This data set was recorded from a combustion process of a methane-air mixture. During the process, the portion of methane was randomly changed, keeping constant gas flow rate. The data set consists of 296 pair of input-output measurements ({y(t), u(t)}, from t=1 to t=296). The input u(t) is the methane gas flow into the furnace and the output y(t) is the CO2 concentration in the outlet gas. Tests on this time series aim at predicting future values y(t+n) by using {y(t),


A Le Bris

L Lombez

J F Guillemoles

R Esteban

M Laroche

J J Greffet

G Boissier

P Christol

S Collin

J L Pelouard

P Aschehoug

F Pellé

HOT CARRIER SOALR CELLS : IN THE MAKING ?

Keywords: High efficiency, Devices, GaSb

Hot carrier solar cells allow potential efficiency close to the thermodynamical limit in ideal conditions. However, the feasability of such devices has not been clearly stated so far and only ideal cells were considered in previous studies. Here we develop a model with realistic energy selective contacts, carrier thermalization and absorptivity. The requirements in term of contact selectivity is investigated, showing that semi-selective contacts are not incompatible with high efficiencies. Candidates for absorbing material were synthesized and the required thermalization properties are obtained. Specific structures were designed to enhance absorption of concentrated sunlight in a small material thickness, allowing high carrier density in the absorber.

INTRODUCTION

The hot carrier solar cell (HCSC) is an attractive concept allowing a potential efficiency close to the thermodynamical limit [1], with a relatively simple structure (see figure 1). In such a cell, the carriers are kept in a thermal disequilibrium with the lattice, thanks to a reduced electron-phonon interaction [2]. They are collected through energy selective contacts, allowing only carriers having one specific energy to be collected, to prevent a heat flow upon carrier extraction towards a cold population [3]. Electron-hole pairs are photogenerated in the absorber where they are kept "hot" (at temperature TH). They are extracted through selective contacts having a small transmission range δE towards a cold distribution in the electrodes (at temperatrue TC). Models of ideal HCSC were proposed [1,3] showing a potential efficiency up to 86 % under a fully concentrated (~46000 times) black body spectrum. However, the feasibility of such a device has not been clearly stated so far. In particular, achieving the required carrier thermalization rate in the absorber may be challenging. Also, theoretical and experimental study on selective contacts have shown the difficulty to obtain good selectivity and high conductivity [4].

Here, simulations are used to determine the required carrier thermalization properties and contact selectivity for a hot carrier effect to occur. A material and device structure is proposed to achieve the targeted carrier cooling rate. Also, a design is proposed to obtain high absorption in a small material thickness to approach the high carrier density that is necessary.

THERMAL LOSSES THROUGH CONTACTS

Simulations are used to determine the effect of the contact non ideality on the cell efficiency.

Model

The model used is based on a detailed balance model [1,5], where the charge (or energy) current extracted is the difference between the absorbed photon (or energy) flux and the emitted photon (or energy) flux. Absorption and emission are described by a generalized Planck law [6]. In previous models, the contacts were assumed ideal with a discrete transmission probability. In that case, the energy collected per extracted carrier is that of the carrier, and is equal to the energy difference between the hole contact and the electron contact Eext. Minimal losses upon carrier extraction are obtained in that case.

Here, we considered a non zero transmission range δE. In that case, if the energy Eext + 2δE is extracted from the absorber, only Eext is usable to produce work, the rest being lost as heat. A thermalization process occurs at the contacts that increases when increasing the transmission range.

The charge and energy current delivered by the absorber is then defined as follows [7,8]: (2) where Jel and Pel are the electric and energy current density delivered by the cell, Emin and Emax are the lower and upper bound of the contact transmission range, τ(E) is the transmission probability of a carrier having energy E, and fC(E) and fH(E) are the carrier distribution function of the cold and hot population respectively. In this paper, τ(E) will be taken unity in the contact transmission range, zero outside.

J el ∝ ∫ E min E max τ  E  [ f H  E -f C  E  ] dE (1) P el ∝ ∫ E min E max Eτ  E  [ f H  E -f C  E  ] dE
By solving the charge and energy balance equations using equations ( 1) and ( 2) [8], the current is determined as a function of the voltage, and the efficiency is obtained for a given δE.

Results of simulations

Simulations are made considering a 1 eV band gap absorber under fully concentrated 6000K black body spectrum. Two cases are considered : an ideal case where thermalization is negligible, and a more practical case with thermalization losses in the absorber. In the latter, a thermalization factor of Q=1 W/K/cm² is considered, which is ten times lower than that of a multi quantum wells GaAs sample [START_REF] Guillemoles | 15th International Photovoltaic and Engineering Conference[END_REF][START_REF] Rosenwaks | [END_REF] (see section 3.1). Figure 2 presents the efficiency and losses of a HCSC as a function of the contact transmission range and for the two thermalization factors that are considered here. For δE<10 -2 eV, corresponding to δE small compared to kTH, the contact can be considered ideal, with an efficiency that is constant and close to the optimal value. Above 10 -2 eV, the losses in contacts increase, causing a drop of efficiency. However, for δE>>kTH ~ 0.1 eV, the efficiency becomes independant of the transmission range. In that case, the contacts are only semi-selective, allowing carrier extraction above a threshold energy Eext. One can see that in the limit of broad transmission range, the efficiency is still above the Shockley-Queisser limit. This show that semiselective contacts are compatible with efficiency enhancement which may allow a much simpler design.

REDUCTION OF CARRIER THERMALIZATION

Results of simulations

Carrier thermalization in the absorber is taken into account in the simulation by adding thermal losses in the energy balance. P el =P abs -P em -P Th (3) where Pel is the electric power, Pabs and Pem are the absorbed and radiatively emitted power, and PTh is the power lost by thermalization. The thermal losses term is expressed is considered here as a function of the carrier temperature [8,[START_REF] Guillemoles | 15th International Photovoltaic and Engineering Conference[END_REF]:

P Th =Q  T H -T C  exp  -E P k B T H  (4)
TH and TC being the temperature of the carriers and of the lattice respectively, EP is the energy of the zone center LO phonon, and Q is a thermalization factor and a material constant.

One can see on figure 2 that a HCSC having a thermalization factor of 1W/K/cm² (ten times lower than a GaAs multi quantum well sample) would give a potential efficiency around 60%. Such a value is our target for future devices.

Experimental study

We investigated antimony based compounds for their good absorption properties, and for their small band gap that is suitable for HCSC application. Also, they are expected to have a low thermalization rate.

It has been demonstrated [START_REF] Rosenwaks | [END_REF] that nanostructured materials exhibit a reduced thermalization rate (see figure 3) compared to bulk. This comes from a reduction of the phonon density of states that limits the thermalization pathways. Also, phonon confinement may prevent heat flow outside of the absorber, and could help keeping the carrier population excited. For that reason, we studied quantum wells samples.

Two types of samples have been studied: strained samples (V812) made of 5x10 nm thick InGaSb wells and 20 nm thick GaSb barriers, and lattice matched samples (V725) made of 5x10 nm thick InGaAsSb wells and 20 nm thick AlGaAsSb barriers. The samples were made at IES.

We determined the cooling rate of these samples using continuous wave photoluminescence. By looking at the carrier temperature (from the high energy side of the PL peak) as a function of the laser power, it is possible to determine the material thermalization factor [START_REF] Bris | 24th EUPVSEC[END_REF].

Results obtained at low temperature (50 K) are encouraging, with estimated values of the thermalization factor around 2 W/K/cm², close to the targeted value. However, a confirmation of this result at ambient temperature is necessary.

Light absorption enhancement

We can see on figure 3 that the thermalization time can be lengthened with increased carrier densities. With higher carrier density per volume unit, it becomes easier to saturate the thermalization pathways which is related to the number of available phonon modes.

This means that achieving a high concentration of the solar energy allows to reduce the thermalization properties requirement. It is therefore useful to be able to efficiently absorb the incident light under concentration in a reduced thickness of material.

For that purpose, resonant dielectric structure are designed to enhance the absorption of the solar spectrum over a wide spectral and angular range in a 50nm thick GaAs absorber. The proposed structure in figure 4 is made of a GaSb layer sandwiched between a dielectric 1D saw tooth grating and a silver mirror [START_REF] Esteban | APL 25th European Photovoltaic Solar Energy Conference and Exhibition / 5th World Conference on Photovoltaic Energy Conversion[END_REF]. Simulations predict 66% absorption of the total incident photon flux below 1800 nm under ~10000 suns (θ < 45°) in the GaSb layer, which represents a 76% increase compared to the same layer without the saw tooth structures.

CONCLUSION

Different technological locks in the HCSC device have been addressed. Resonant structures at the front are proposed and are expected to allow efficient absorption in a thin absorbing layer. The material and structure of the studied samples give promising thermalization properties that make it a good candidate as an absorber. Finally, simulations show that semiselective contacts enable efficiency up to 60%, higher than the Shockley-Queisser limit.

Figure 1 :

 1 Figure 1: Schematic of a hot carrier solar cell.Electron-hole pairs are photogenerated in the absorber where they are kept "hot" (at temperature TH). They are extracted through selective contacts having a small transmission range δE towards a cold distribution in the electrodes (at temperatrue TC).

Figure 2 :

 2 Figure 2: Efficiency (circles) and losses (in the absorber: squares, and in the contacts: triangles) of a HCSC as a function of the contact transmission range. Two thermalization factor are considered: negligible thermalization (filled symbols), and Q=1W/K/cm², corresponding to a thermalization rate ten times lower than MQW GaAs.

Figure 3 :

 3 Figure 3: thermalization vs carrier temperature of bulk (green and blue dots) and quantum wells (red and black dots) GaAs. Two carrier densities are considered: 5e18cm -3 and 1e19cm -3

Figure 4 :

 4 Figure 4: Resonant dielectric structure for enhanced light absorption over a wide spectral and angular range. The absorbing 50 nm thick GaSb layer is sandwiched between a SiC grating and a silver back reflector.

25th European Photovoltaic Solar Energy Conference and Exhibition / 5th World Conference on Photovoltaic Energy Conversion, 6-10 September 2010, Valencia, Spain

We thank all our partners of the THRI-PV project (IES, ILV, LCFIO, LCMCP, LGEP, LPN) and GCEP and ANR programs for financial support.