Assia Benabdallah

Yves Dermenjian
email: dermenji@cmi.univ-mrs.fr

Jérôme Le Rousseau

On the controllability of linear parabolic equations with an arbitrary control location for stratified media

We prove a null controllability result with an arbitrary control location in dimension greater than or equal to two for a class of linear parabolic operators with non-smooth coefficients. The coefficients are assumed to be smooth in all but one directions.

Résumé

De la contrôlabilité des équations paraboliques linéaires avec une localisation arbitraire du contrôle pour des milieux stratifiés. Nous prouvons un résultat de contrôlabilité à zéro avec une localisation arbitraire de la zone de contrôle en dimension plus grande que deux pour une classe d'opérateurs paraboliques avec des coefficients non réguliers. Les coefficients sont supposés singuliers dans une seule direction.

Version franc ¸aise abrégée

. On ne traite donc que le cas linéaire.

Soit Ω un ouvert borné de R n , avec Ω = Ω ′ × (0, H), où Ω ′ est un ouvert non vide de R n-1 de frontière C 2 . On pose x = (x ′ , x n ) ∈ Ω ′ × (0, H). Soit B(x), une matrice n × n qui possède la forme diagonale par blocs

B(x ′ , x n ) = diag(c 1 (x n )C 1 (x ′ ), c 2 (x n )) où c 1 ∈ L ∞ (0, H), c 2 ∈ BV(0, H) et C 1 ∈ C 1 (Ω ′ , M n-1 (R))
. La matrice C 1 (x ′ ) est symétrique. On suppose 0 < c min ≤ c i (x n ) ≤ c max , x n ∈ (0, H), i = 1, 2, et 0 < c min I n-1 ≤ C 1 (x ′ ) ≤ c max I n-1 , x ′ ∈ Ω ′ , où I k est la matrice identité d'ordre k, ce qui implique une ellipticité uniforme. Soit l'opérateur autoadjoint A = -∇ x • (B∇ x ) dans L 2 (Ω) de domaine D(A) = {u ∈ H 1 0 (Ω); ∇ x • (B∇ x u) ∈ L 2 (Ω)}. Soit T > 0 et Q T = (0, T ) × Ω. On considère le système (2) où q 0 ∈ L 2 (Ω) et ω est un ouvert non vide de Ω tel que ω ⋐ Ω. On choisit ω ′ un ouvert non vide de Ω ′ et ω n un ouvert non vide de (0, H) tels que ω ′ × ω n ⊂ ω. On suppose de plus que le coefficient c 2 est de classe C 1 dans un ouvert non vide de ω n . Notre résultat principal de contrôlabilité à zéro est le suivant. Théorème 1 Sous les hypothèses précédentes, pour tout T > 0 et tout q 0 ∈ L 2 (Ω), il existe u ∈ L 2 ((0, T ) × Ω) tel que la solution associée, q, du système [START_REF] Fursikov | Controllability of evolution equations[END_REF] vérifie q(T ) = 0 p.p. dans Ω.

Un conséquence immédiate est l'observabilité pour le système homogène adjoint du système [START_REF] Fursikov | Controllability of evolution equations[END_REF]. Nous donnons maintenant un esquisse de la démonstration du Théorème 1.

On introduit les espaces H k = vect{ϕ k,p ; p ≥ 1} = {φ k ⊗ f ; f ∈ L 2 (0, H)} où les φ k , k ∈ N * , sont les fonctions propres de l'opérateur

A ′ = -∇ x ′ • (C 1 ∇ x ′ ), de domaine D(A ′ ) = {u ∈ H 1 0 (Ω ′ ); ∇ x ′ • (C 1 ∇ x ′ u) ∈ L 2 (Ω ′
)}, associées aux valeurs propres µ 1 ≤ µ 2 ≤ . . . , que l'on choisit formant une base orthonormée de L 2 (Ω ′ ). On définit les espaces

E j = ⊕ k≤2 j H k pour j ∈ N * , et on vérifie que ∪ j∈N E j = ⊕ k∈N * H k = L 2 (Ω).
On décompose l'intervalle de temps (0, T ), (0,

T ) = ∪ j∈N [a j , a j+1 ], a j+1 -a j = 2T j , avec T j = Kσ -ρ j , σ j = 2 j , avec K choisi tel que 2 ∞ j=0 T j = T et ρ ∈ (0, 2 3(n-1)
). Comme dans [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], on construit une suite de contrôles u j de L 2 ((a j , a j + T j ) × Ω), j ≥ j 0 pour un certain j 0 ∈ N, chargés de ramener à 0 la composante suivant E j de la solution de (2) au temps a j + T j . Le contrôle est en revanche nul dans l'intervalle de temps [a j + T j , a j+1 ], afin de profiter de la décroissance exponentielle en temps de la solution (qui se trouve dans E ⊥ j à l'instant t = a j + T j ). L'inégalité de Carleman pour l'opérateur [START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF] permet d'estimer en fonction de µ σ j le coût d'un contrôle agissant sur Ω ′ × ω n pendant l'intervalle de temps [a j , a j + T j ]. Puis, en utilisant l'inégalité [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF] relative aux fonctions propres de l'opérateur A ′ [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], on déduit une estimation du coût d'un contrôle, u j , agissant sur ω ′ × ω n pendant l'intervalle de temps [a j , a j + T j ]. Ce coût est de la forme :

∂ t + A k , où A k = -∂ x n (c 2 (x n )∂ x n ) + c 1 (x n )µ k de domaine D(A k ) = {u ∈ H 1 0 (0, H); c 2 ∂ x n u ∈ H 1 (0, H)}, prouvée dans
u j L 2 ((a j ,a j +T j )×Ω) ≤ Ce Cµ 2/3 σ j q j L 2 (Ω) , (1)
pour j suffisamment grand, où q j est la valeur de la solution de (2) au temps t = a j . On peut alors conclure comme dans [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. Remarque 2 Comme dans [START_REF] Fursikov | Controllability of evolution equations[END_REF], le résultat obtenu pour un contrôle distribué permet d'obtenir un résultat pour un contrôle frontière (ici sur une partie de la frontière Ω ′ × {0} ou Ω ′ × {H}).

A = c 1 (x n )A ′ -∂ x n c 2 (x n )∂ x n avec c 1 ∈ L ∞ (0, H), c 2 ∈ BV(0, H) (vérifiant

Introduction

The question of the null controllability of linear parabolic partial differential equations with smooth coefficients was solved in the 1990's [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Fursikov | Controllability of evolution equations[END_REF]. In the case of discontinuous coefficients in the principal part of the parabolic operator, the controllability issue and its dual counterpart, observability, are not fully solved yet. A result of controllability for a semilinear parabolic equation with a discontinuous coefficient was proven in [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] by means of a Carleman observability estimate. Roughly speaking, as in the case of hyperbolic systems (see e.g. [7, page 357]), the authors of [START_REF] Doubova | Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients[END_REF] proved their controllability result in the case where the control is supported in the region where the diffusion coefficient is the 'lowest'. In both cases, however, the approximate controllability, and its dual counterpart, unique-ness, are true without any restriction on the monotonicity of the coefficients. It is then natural to question whether or not an observability estimate holds in the case of non-smooth coefficients and arbitrary observation location.

Recently, in the one-dimensional case, the controllability result for parabolic equations was proven for general piecewise C 1 coefficients in [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF], and for coefficients with bounded variations (BV) in [START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF]. The proof relies on global Carleman estimates, which moreover allows to treat semilinear equations. Such global Carleman estimates are also of interest to prove stability results for some inverse problems. A controllability result for parabolic equations with general bounded coefficients was independently proven in [START_REF] Alessandrini | Null-controllability of one-dimensional parabolic equations[END_REF]. The method used there to achieve null controllability is that of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], which limits the field of applications to linear equations.

In the n-dimensional case, n ≥ 2, the controllability with an arbitrary control location is still open. In particular, an extension based on the proof of the Carleman estimate in the one dimensional case, leads to uncontrolled tangential terms at the interfaces of discontinuities of the coefficient. This work provides a positive answer to the controllability question for a class of discontinuous coefficients: the main assumption we make is that the coefficients are smooth w.r.t. to all but one variables, which includes the case of stratified media. The proof relies both on the Carleman estimates of [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF][START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF] in the one-dimensional case and the method of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. We thus only treat linear equations.

We let Ω be an open subset in R n , with Ω = Ω ′ × (0, H), where Ω ′ is an nonempty regular bounded open subset of R n-1 with C 2 boundary. We shall use the notation x = (x ′ , x n ) ∈ Ω ′ × (0, H). For a real Hilbert space X, • X (resp. (., .) X ) will denote the norm (resp. the real scalar product) in X. Let B(x), x ∈ Ω, be with values in M n (R), the space of square matrices with real coefficients of order n. We make the following assumption. Assumption 1.1 The matrix diffusion coefficient B(x ′ , x n ) has the following block-diagonal form

B(x ′ , x n ) =           c 1 (x n )C 1 (x ′ ) 0 0 c 2 (x n )           , where c 1 ∈ L ∞ (0, H), c 2 ∈ BV(0, H) and C 1 ∈ C 1 (Ω ′ , M n-1 (R)). The matrix C 1 (x ′ ) is symmetric. We further assume 0 < c min ≤ c i (x n ) ≤ c max , x n ∈ (0, H), i = 1, 2, and 0 < c min I n-1 ≤ C 1 (x ′ ) ≤ c max I n-1 , x ′ ∈ Ω ′
, where I k is the identity matrix of order k, which implies uniform ellipticity.

We consider the selfadjoint operator

A = -∇ x • (B∇ x ) in L 2 (Ω) with domain D(A) = {u ∈ H 1 0 (Ω); ∇ x • (B∇ x u) ∈ L 2 (Ω)}. Let T > 0.
We shall use the notation Q T = (0, T ) × Ω. We consider the following parabolic system

             ∂ t q -∇ x • (B∇ x q) = 1 ω u in Q T , q(t, x) = 0 on (0, T ) × ∂Ω, q(0, x) = q 0 (x) in Ω, (2)
(coefficients and solutions are real valued) where q 0 ∈ L 2 (Ω) and ω is a nonempty open subset of Ω such that ω ⋐ Ω.

We choose ω ′ a nonempty open subset of Ω ′ and ω n a nonempty open subset of (0,

H) such that ω ′ × ω n ⊂ ω. Assumption 1.2 The coefficient c 2 is of class C 1 in some nonempty open subset of ω n .
We analyze the null controllability of System (2), or equivalently its exact controllability to the trajectories, when a distributed control u ∈ L 2 ((0, T ) × Ω) acts on the system. The main result is the following theorem. Theorem 1.3 Under Assumption 1.1, for arbitrary time T > 0 and initial condition q 0 ∈ L 2 (Ω), there exists u ∈ L 2 ((0, T ) × Ω) such that the corresponding solution q of System (2) satisfies q(T ) = 0 a.e. in Ω. Corollary 1.4 There exists an observability inequality for the homogeneous adjoint system of System [START_REF] Fursikov | Controllability of evolution equations[END_REF].

The proof makes use of the technique introduced by G. Lebeau and L. Robbiano [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], as well as the onedimensional Carleman estimates of [START_REF] Benabdallah | Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem[END_REF][START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF].

In this article, when the constants C or C ′ , etc, are used, their values may change from one line to the other. If we want to keep track of the value of a constant we shall use another letter.

Spectral properties

Similarly to A, we define the selfadjoint operator

A ′ = -∇ x ′ • (C 1 ∇ x ′ ), in L 2 (Ω ′ ), with domain D(A ′ ) = {u ∈ H 1 0 (Ω ′ ); ∇ x ′ • (C 1 ∇ x ′ u) ∈ L 2 (Ω ′ )}.
With orthonormal eigenfunctions (φ k ) k≥1 , associated to the eigenvalues, with finite multiplicities, µ 1 ≤ µ 2 ≤ ... ≤ µ k ≤ µ k+1 ≤ ..., we construct a Hilbert basis of L 2 (Ω ′ ).

We also define the selfadjoint operators

A k , k ∈ N * , on L 2 (0, H) by A k = -∂ x n (c 2 (x n )∂ x n ) + c 1 (x n )µ k with domain D(A k ) = {u ∈ H 1 0 (0, H); c 2 ∂ x n u ∈ H 1 (0, H)}. We denote by ψ k,p (x n ), p ∈ N * , orthonormal eigenfunctions with associated eigenvalues λ k,1 ≤ λ k,2 ≤ • • • ≤ λ k,p ≤ λ k,p+1 ≤ . . . . Note that we have λ k,p > c min µ k .
From the separation of variables in the coefficients of the matrix B, we have the following proposition.

Proposition 2.1 The eigenfunctions of the operator A given by ϕ

k,p (x ′ , x n ) = (φ k ⊗ ψ k,p )(x ′ , x n ) = φ k (x ′ )ψ k,p (x n ) with associated eigenvalue λ k,p , k, p ∈ N * , form a Hilbert basis of L 2 (Ω).
Let us denote by H k the following closed infinite dimensional subspace of L 2 (Ω):

H k = span{ϕ k,p ; p ≥ 1} = {φ k ⊗ f ; f ∈ L 2 (0, H)}
and let us set E j = ⊕ k≤2 j H k for j ∈ N * . In the sequel we shall denote by Π E j the orthogonal projection onto E j in L 2 (Ω). We have the following proposition. Proposition 2.2 The following properties hold:

(i) E j ⊂ E j+1 , j ∈ N, and

∪ j∈N E j = ⊕ k∈N * H k = L 2 (Ω);
(ii) The operator (-A, D(A)) generates a C 0 -semigroup of contraction, S (t) = e -tA for t ≥ 0, and for all f ∈ L 2 (Ω) we have S (t) f = k,p≥1 e -tλ k,p ( f, ϕ k,p )ϕ k,p ; (iii) For all k ≥ 1, S (t) is reduced by the space H k . See for instance [START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF] or [START_REF] Reed | Methods of modern Mathematical Physics[END_REF].

Existence and estimation of a control acting on E j

Following [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], for ρ ∈ (0, 2 3(n-1) ), we set T j = Kσ -ρ j , with σ j = 2 j , for all j ∈ N. The constant K is adjusted so that 2 ∞ j=0 T j = T . Then, we set a 0 = 0, a j+1 = a j + 2T j , for j ≥ 0. We show that for all q j ∈ L 2 (Ω), j ≥ j 0 , for some j 0 ∈ N, there exists u j ∈ L 2 (a j , a j + T j ; L 2 (Ω)) such that the solution q to              ∂ t q -∇ x • (B∇ x q) = 1 ω u j in (a j , a j + T j ) × Ω, q(t, x) = 0 on (a j , a j + T j ) × ∂Ω, q(a j , x) = q j (x)

in Ω,

satisfies Π E j q(a j + T j , x) = 0. Since S (t) and Π E j commute, this is equivalent to the observability inequality [START_REF] Fernández-Cara | Global Carleman inequalities for parabolic systems and application to controllability[END_REF] y(a j , .)

2 L 2 (Ω) ≤ C 2 T j a j +T j a j ω |y(t)| 2 dt dx, ( 4
)
for the solution y ∈ C ([a j , a j + T j ]; E j ) of the adjoint system

-∂ t y -∇ x • (B∇ x y) = 0 in (a j , a j + T j ) × Ω, and y(a j + T j ) = y 0 in Ω, y 0 ∈ E j , (5)
which moreover yields the existence of a control u j ∈ L 2 ((a j , a j + T j ); L 2 (Ω)) such that

u j L 2 ((a j ,a j +T j )×Ω) ≤ C T j Π E j q j E j ≤ C T j q j L 2 (Ω) . (6)
We now prove (4). We first recall a Carleman estimate for a parabolic operator with a BV diffusion coefficient proven in [START_REF] Rousseau | Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients[END_REF]. For a positive function β, we introduce β = β + K with K = m β ∞ and m > 1. For λ > 0 and t ∈ (a j , a j + T j ), we define the following weight functions [START_REF] Fursikov | Controllability of evolution equations[END_REF]

ϕ(t, x) = e λβ(x) (t -a j )(a j + T j -t) , η(t, x) = e λβ -e λβ(x) (t -a j )(a j + T j -t)
, with β = 2m β ∞ .

= λ 0 (H, O, c min , c max ) > 0, s 0 = s0 (H, O, c min , c max )(T j + T 2 j ) > 0 and a positive constant C = C(H, O, c min , c max ) so that s -1 a j +T j a j H 0 e -2sη ϕ -1 (|∂ t z| 2 + |∂ x (γ∂ x z)| 2 ) dxdt + sλ 2 a j +T j a j H 0 e -2sη ϕ |∂ x z| 2 dxdt + s 3 λ 4 a j +T j a j H 0 e -2sη ϕ 3 |z| 2 dxdt ≤ C s 3 λ 4 a j +T j a j O e -2sη ϕ 3 |z| 2 dxdt + a j +T j a j H 0 e -2sη | f | 2 dxdt ,
for s ≥ s 0 , λ ≥ λ 0 and for all z (weak) solution of

∂ t z + ∂ x (γ∂ x z) = f in (a j , a j + T j ) × (0, H), z(t, 0) = z(t, H) = 0, and z(a j + T j , x) = z 0 (x) in (0, H), with z 0 ∈ L 2 (0, H) and f ∈ L 2 ((a j , a j + T j ) × (0, H)).
With the previous Carleman estimate we can prove an observability inequality for the parabolic operator -∂ t +A k . Proposition 3.2 There exist positive constants C = C(H, ω n , c min , c max ), and C ′ = C ′ (H, ω n , c min , c max ) such that, for all k ∈ N * and for s k = max(C ′ T 2 j µ 2/3 k , s0 (T j + T 2 j )), the solutions to

       -∂ t z + A k z = 0 in (a j , a j + T ) × Ω, z(t) = 0 on (a j , a j + T j ) × ∂Ω, (7)
satisfy z(a j , •) 2 L 2 (0,H) ≤ 1 T j Ce Cs k T -2 j a j +T j a j
ω n |z(t, x)| 2 dx dt. Proof. We apply Theorem 3.1, with f = -c 1 µ k z, γ = c 2 , λ = λ 0 and s ≥ s0 (T j + T 2 J ), and obtain

a j +T j a j H 0 e -2sη ϕ 3 (s 3 -Cϕ -3 µ 2 k )|z| 2 dx n dt ≤ C ′′ s 3 a j +T j a j ω n e -2sη ϕ 3 |z| 2 dx n dt.
Noting that ϕ -1 ≤ CT 2 j , the coefficient s 3 -Cϕ -3 µ 2 k will be positive for s

≥ C ′ T 2 j µ 2/3 k . Setting s = s k = max(C ′ T 2 j µ 2/3 k , s0 (T j + T 2 j ))
, we obtain

a j +3T j /4 a j +T j /4 H 0 |z| 2 dx n dt ≤ Ce Cs k T -2 j a j +T j a j
ω n |z| 2 dx n dt. Making use of the parabolic "dissipation effect", i.e. d dt |z(t)| 2 ≥ 0 here, we obtain the desired inequality. We can now obtain the observability in the space E j . Proposition 3.3 Let ω ′ be a nonempty open subset of Ω ′ with ω ′ ⋐ Ω ′ . There exists j 0 ∈ N such that, if j ≥ j 0 , solutions of ( 5) satisfy ( 4) with C T j = Ce Cµ 2/3 σ j .

To prove Proposition 3.3, we shall need the following result which was first proven in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF].

Theorem 3.4 There exists C = C(Ω ′ , ω ′ , c min , c max ) > 0 such that for all l ∈ N * , k≤l |b k | 2 ≤ Ce C √ µ l ω ′ k≤l b k φ k (x ′ ) 2 dx ′ , (b 1 , ..., b l ) ∈ R l . (8)
Proof of Proposition 3.3. The definition of E j implies that y(t, x ′ , x n ) = k≤σ j φ k (x ′ )y k (t, x n ) and one sees that y is solution in C ([a j , a j + T j ]; E j ) of ( 5), if and only if each function y k , 1 ≤ k ≤ σ j , is solution in C ([a j , a j + T j ]; L 2 (0, H)) of ( 7) and y k (a j + T j ) = y 0,k , where

y 0 (x ′ , x n ) = k≤σ j φ k (x ′ )y 0,k (x n ). We then have y(t, .) 2 L 2 (Ω) = k≤σ j y k (t, .) 2 L 2 (0,H) and similarly y(t, .) 2 L 2 (Ω ′ ×ω n ) = k≤σ j y k (t, .) 2 L 2 (ω n ) for t ∈ [a j , a j + T ].
According to the Weyl formula [12, Theorem 14.6, p. 250], µ σ j ∼ C(Ω ′ )(σ j ) 2 n-1 , we see that, for j sufficiently large, we have s

σ j = CT 2 j µ 2/3 σ j . We then have T -2 j max 1≤k≤σ j s k = T -2 j s σ j = Cµ 2/3 σ j . From Proposition 3.2 we obtain y(a j , .) 2 L 2 (Ω) ≤ Ce Cµ 2/3 σ j a j +T j a j ω n k≤σ j |y k (t, x n )| 2 dtdx n . (9)
If we use y k (t, x n ) in place of b k in (8), we deduce

y(a j , .) 2 L 2 (Ω) ≤ Ce Cµ 2/3 σ j a j +T j a j ω n C ′ e C ′ √ µ σ j ω ′ k≤σ j φ k (x ′ )y k (t, x n ) 2 dx ′ dx n dt. ( 10
)
Following the method of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], with the value of C T j we have obtained and the choice made for T j , the result of Theorem 1.3 follows.

Remark 1 For the sake of presentation, we chose to take the open set Ω of the form Ω ′ × (0, H) with Ω ′ a bounded open subset of R n-1 . There are other situations that can be handled by the method we have presented. We could for instance consider an uniformly elliptic operator in cylindrical coordinates and address the case of a ring with variations of the medium in the radial direction. In this case, the interfaces that locate the jumps of the coefficients of the diffusion matrix (in the case of piecewise continuous coefficients) do not reach the boundary of the domain Ω. As before we can address the case of a BV-type regularity in the radial direction.

One other natural extension would be the case of a domain of the form M × (0, H) where M is a smooth (n -1)-dimensional Riemannian compact manifold with or without a boundary. The parabolic operators under consideration would then be of the form ∂ t + A, where A = c 1 (x n )A ′ -∂ x n c 2 (x n )∂ x n with c 1 ∈ L ∞ (0, H), c 2 ∈ BV(0, H) (satisfying the same assumptions as those given above), and say A ′ = -∆, with ∆ the Laplace operator on M. In such a case, estimate (8) can be found in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]. Remark 2 As usual, as in [START_REF] Fursikov | Controllability of evolution equations[END_REF], the result obtained on distributed controls yields a boundary control result. Here the control function could act in a nonempty open region of Ω ′ × {0} or Ω ′ × {H} as a boundary condition for the parabolic system under consideration. This is of particular interest for geometrical situations like that described in the previous remark.

  les mêmes hypothèses que celles données plus haut), et par exemple A ′ = -∆, avec ∆ le Laplacien sur M.

Theorem 3 . 1

 31 Let O ⋐ (0, H) be a nonempty open set and γ ∈ BV(0, H) with 0 < c min ≤ γ ≤ c max and γ of class C 1 in O. There exist a positive continuous function β, and λ 0

Assia Benabdallah, Yves Dermenjian, Jérôme Le Rousseau. On the controllability of linear parabolic equations with an arbitrary control location for stratified media. Comptes rendus hebdomadaires des séances de l'Académie des sciences, 2007, 344 (6), pp.357-362. 10.1016/j.crma.2007.01.012. hal-


Khaled Laib

Ahmed Redha Meghnous

Minh Tu Pham

Xuefang Lin-Shi

Averaged state model based design of nonlinear observer for the on/off solenoid valve pneumatic actuators

This paper presents an averaged model and nonlinear observer for an on/off pneumatic actuator. The actuator is composed of two chambers and four on/off solenoid valves. The averaged model is elaborated which has the advantage of using only one continuous input instead of four binary inputs. Based on this new model, a sliding mode observer is designed using the piston's position and the pressure measurements in one of the chambers to estimate the piston's velocity and the pressure in the other chamber. The finite time convergence of the observer is proven and its performance is verified in closed loop on an experimental benchmark.

Introduction

Pneumatic actuators are widely used in the automation of different industrial production lines, robotics and also in medical applications. These systems present many advantages: reliability, velocity, low cost and effort/input energy ratio, which allow to build light and powerful actuators. Unfortunately, the major difficulty concerning pneumatic actuators is their nonlinear dynamics due to air compressibility, friction forces and mainly the nonlinear behavior of air flow rate through the power modulators. Hence, the design of a control strategy is not simple. In order to have high performance in terms of positioning, it is possible to use proportional servo-valves instead of the traditional on/off solenoid valves. These servo-valves deliver an air flow rate, depending on the control voltage and the upstream pressure, ranging from 0% to 100% contrary to the traditional on/off valves which deliver either 0% or 100% of the available mass flow rate depending on the binary input voltage of the valves. Many control strategies and techniques have been elaborated for this type of actuators: PID [START_REF] Rahmat | Non linear modeling et cascade control of an industrial pneumatic actuator system[END_REF], fuzzy control [START_REF] Schulte | Fuzzy state feedback gain scheduling control of servo pneumatic actuators[END_REF], sliding modes [START_REF] Smaoui | Robust position control of an electropneumatic system using second order sliding mode[END_REF], backstepping [START_REF] Smaoui | Systematic control of an electropneumatic system: Integrator backstepping and sliding mode control[END_REF], adaptive control [START_REF] Tsai | Fat-based adaptive control for pneumatic servo systems with mismatched uncertainties[END_REF],

neural networks [START_REF] Yamazaki | An intelligent control for state-dependent nonlinear actuator and its application to pneumatic servo system[END_REF]. However, servo valves are still far less common than solenoid on/off valves because of their price and the high expertise needed to fully exploit them.

To overcome these drawbacks, recently improved solenoid valves could be used instead of the servo-valves. However, precise control is difficult due to the inputs of these on/off solenoid valves. From an operating point of view, the pneumatic actuators equipped with these kind of components belong to the class of the switching systems with nonlinear dynamics.

Traditionally, there is no general theory able to characterize the stability of such systems.

Fortunately, with the recent development of high frequency switching on/off valves, it is possible to handle new perspectives in terms of modeling and control.

A significant amount of high quality research has been addressed in this direction using averaging techniques. In power electronics, the concept of switching systems with discrete inputs is well known. Averaging techniques are used to create one unique model with one single continuous input. In fluid power systems, solenoid valves are being controlled with a Pulse Width Modulation (PWM) controller to determine its input: 1 or 0. One can describe the equivalent continuous dynamics of PWM controlled nonlinear systems by transforming the system, originally discontinuous and possibly non-affine in the input, into an equivalent system that is both continuous and affine in the control input, i.e. transforms the system to nonlinear control canonical form [START_REF] Shen | Nonlinear averaging applied to the control of pulse width modulated PWM pneumatic systems[END_REF].

Many works show the interest of using on/off valves: [START_REF] Shen | Nonlinear averaging applied to the control of pulse width modulated PWM pneumatic systems[END_REF][START_REF] Shih | Position control of a pneumatic cylinder using fuzzy PWM control method[END_REF][START_REF] Zhang | Sliding mode approach to pwm-controlled pneumatic systems[END_REF][START_REF] Messina | Experimenting and modelling the dynamics of pneumatic actuators controlled by the pulse width modulation PWM technique[END_REF][START_REF] Ahn | Intelligent switching control of pneumatic actuator using on/off solenoid valves[END_REF][START_REF] Taghizadeh | Improving dynamic performances of PWMdriven servo-pneumatic systems via a novel pneumatic circuit[END_REF][START_REF] Hodgson | Sliding mode control of nonlinear discrete-input pneumatic actuators[END_REF][START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF][START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] to control the position of the pneumatic actuators. In order to design these control strategies, one needs to have access to all the state variables of the actuator: position, velocity and the pressures of the two chambers. However, in order to reduce the price, decreasing number of sensors is required. Therefore, the estimation of the non measured variables is essential. In the literature, there are some works related to the observation of pneumatic systems with servovalves: [START_REF] Gulati | A globally stable, load-independent pressure observer for the servo control of pneumatic actuators[END_REF] presents the synthesis of pressure observers based on Lyapunov theory. The authors of [START_REF] Girin | High gain and sliding mode observers for the control of an electropneumatic actuator[END_REF] propose two observers using the position of the piston and the pressure in one chamber. Nevertheless, the observer design for pneumatic systems with on/off valves is a standard solution in automotive applications and particularly in the control of clutch systems. In [START_REF] Prabel | Model-based control of an electro-pneumatic clutch using a sliding-mode approach[END_REF], the authors propose a reduced order observer to estimate the pressure to have an internal pressure controller in order to control the clutch position. In [START_REF] Langjord | Nonlinear observer and parameter estimation for electropneumatic clutch actuator[END_REF], the authors consider the design of a full sate observer for electro pneumatic clutch actuator with position sensor. This observer is used in [START_REF] Langjord | Adaptive observer-based switched control for electropneumatic clutch actuator with position sensor[END_REF] to obtain a dual-mode switched controller for the clutch actuator position. Nevertheless, this observer uses the input of the on/off solenoid valve. As it is mentioned earlier, there is no general theory to guarantee the observer stability of such systems since they use switched models in their design. Therefore, it's important to establish a model for which the stability can be guaranteed. In this context, averaging techniques will be used to obtain such a model. The authors of [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF][START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] presented an output averaged model for an on/off pneumatic actuator equipped with four solenoid valves which is the same as the experimental benchmark considered in this paper. Unfortunately, the elaborated output averaged model can only be used for position control strategies and it can not be used for observer design. One of the main contribution of this paper is the extension of the model elaborated in [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF][START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] to a full state averaged model to describe the behavior of the on/off pneumatic systems. A nonlinear sliding mode observer is designed [START_REF] De Wit | Sliding observers for robot manipulators[END_REF] using only two outputs. The convergence of the observation error towards zero is proven using the averaged model and Lyapunov theory. Furthermore, the performance of this observer is verified experimentally to control the position of the actuator using sliding mode strategy [START_REF] Utkine | Variable structure systems with sliding modes[END_REF][START_REF] Utkin | Sliding Model Control in Electromechanical Systems[END_REF]. This paper is organized as follows: Section 2 presents the modeling of a pneumatic actuator with four on/off solenoid valves. The result is a switching model with four binary inputs. In Section 3, a full state averaged model is elaborated using techniques from power electronics. Section 4 shows the design of sliding mode observer with a finite time convergence proof. This observer is used to calculate a control input to track a desired position trajectory.

The experimental results are shown in Section 5.

Modeling

The pneumatic system (Fig. 1) is a one degree of freedom actuator composed of two pneumatic chambers. Every chamber is connected to a pressure source and to an exhaust pressure using two on/off solenoid valves. Each valve i is controlled with a discrete binary input U i .

Let us assume that [START_REF] Hodgson | Sliding mode control of nonlinear discrete-input pneumatic actuators[END_REF]:

• The only considered friction forces are viscous;

• The air is a perfect gas and its kinetic energy is negligible in each chamber;

• The pressure, the temperature and the density of the air are homogeneous in each chamber;

• The evolution of the air inside each chamber is polytropic;

• The supply and the exhaust pressure are constant and there is no leakage.

The complete modeling could be divided into three stages: Piston's dynamics, the air evolution in each chamber and the valve dynamics. Using Newton's second law and the Figure 1: Pneumatic system with four on/off valves first principle of thermodynamics, the piston's dynamics and the air evolution inside each pneumatic chamber are given by [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF]:

           mÿ = (A P P P -A N P N ) -b v ẏ ṖP = k (l/2 + y)A P (rT Q P -A P P P ẏ) ṖN = k (l/2 -y)A N (rT Q N + A N P N ẏ) (1)
where l is the total length, y is the piston's position and m is its mass. b v is the viscosity coefficient. P P , A P and Q P (respectively P N , A N and Q N ) are the air pressure, the cylinder's area and the input mass flow rate in chamber P (respectively chamber N ). Please note that A P and A N are explicitly made equal. r is the perfect gas constant and k is the polytropic constant. T is the air temperature. Q P and Q N are the input mass flow rate in each chamber.

They are given by:

Q P = U 1 Q(P S , P P ) -U 2 Q(P P , P E ) Q N = U 3 Q(P S , P N ) -U 4 Q(P N , P E ) (2)
where U i with i = {1, 2, 3, 4} is the binary input voltage of valve i. P S and P E are the supply and the exhaust pressures respectively. The air flow inside each valve depends on the upstream pressure P u and temperature T u and the downstream pressure P d and temperature

T d .
It depends also on the valve's characteristics like the mass flow rate constant C val and the critical pressure ratio b crit [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF]:

Q(P u , P d ) = C val P u T /T u                  1 -     P d P u -b crit 1 -b crit     2 if P d P u > b crit 1 if P d P u ≤ b crit (3)
The global model of (1) becomes:

             ÿ = 1 m (A P P P -A N P N -b v ẏ) ṖP = -k P P l/2 + y ẏ + krT U 1 Q(P S , P P ) -U 2 Q(P P , P E ) (l/2 + y)A P ṖN = k P N l/2 -y ẏ + krT U 3 Q(P S , P N ) -U 4 Q(P N , P E ) (l/2 -y)A N (4)
Each binary input voltage U i can be in two situations either "on" or "off". Hence, every chamber can be in one of the following three states:

• Pressurizing: connected to the supply pressure (for the chamber P : U 1 = 1, U 2 = 0);

• Venting: connected to the exhaust pressure (U 1 = 0, U 2 = 1);

• Close: both of the valves are closed (U 1 = 0, U 2 = 0).

From these three states, nine modes can be obtained for the two chambers. It has been proved in [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF] and [START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] that only seven modes can be used depending on the input vector

U = (U 1 U 2 U 3 U 4 ) T .
For each mode j, the input vector U (j) is associated such that:

Mode 1: U (1) = (0 0 0 0) T Mode 2: U (2) = (1 0 0 0) T Mode 3: U (3) = (0 1 0 0) T Mode 4: U (4) = (0 0 0 1) T Mode 5: U (5) = (0 0 1 0) T Mode 6: U (6) = (1 0 0 1) T Mode 7: U (7) = (0 1 1 0) T (5)
3 Averaged model

For power converters, the averaged model can be obtained using the different dynamics of the system undergoing pulse-width modulation of the switched inputs. The PWM duty cycle will act as a continuous input for the averaged model. The average modeling consists of evaluating all the different modes of the switching system in order to obtain one equivalent model for all the p modes:

F = (F (1) . . . F (p) ) T
Due to the variation of the input injected into the system, it can switch between several modes within a PWM period. For each mode F (j) , a duty ratio d (j) is associated:

D = (d (1) . . . d (p) ) T
The total duration of all the duty ratios d (j) is equal to the PWM period i.e. D can be normalized: D 1 = 1. Then, the averaged model will represent all the modes F (j) within a switching period PWM:

Ẋa = p j=1 d (j) F (j) (6)
where X a represents the averaged state vector.

The averaged model presented in [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF][START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] uses three mode-model (mode 1 is for no actuation, mode 6 corresponds to the movement in positive direction while mode 7 is for the negative direction) to control the position of the piston. In these works, there is no need to average all the state variables, only the output y is averaged and replaced by y a . However, this model is not suitable for state observer design. There is a need to elaborate a state averaged model.

To obtain a state averaged model, all the states dynamics must be explicitly given with the binary discrete input U i . Model presented in (4) becomes:

                             ... y = - k m A P P P l/2 + y + A N P N l/2 -y ẏ - b v m ÿ + krT m U 1 Q(P S , P P ) -U 2 Q(P P , P E ) (l/2 + y) - krT m U 3 Q(P S , P N ) -U 4 Q(P N , P E ) (l/2 -y) ṖP = -k P P l/2 + y ẏ + krT U 1 Q(P S , P P ) -U 2 Q(P P , P E ) (l/2 + y)A P with: P N = A P P P -ÿ -b v ẏ A N (7)
As it can be seen in ( 7), the dynamics ... y and ṖP change according to the input U (j) (Mode j) of ( 5). Hence, let us introduce the following notations:

x 1 = y f 1 = x 2 x 2 = ẏ f 2 = x 3 x 3 = ÿ f (j) 3 = ... y | U (j)
x 4 = P P f

(j) 4 = ṖP | U (j)
where the dynamics f

(j) 3
and f

(j) 4

correspond to the dynamics ... y and ṖP evaluated for the input U (j) . The model ( 4) can be written as:

Ẋ(t) = F (j) (X(t), U (j) ) j = 1, . . . , 7 (8)
where

X = (x 1 x 2 x 3 x 4 ) T and F (j) = (f 1 f 2 f (j) 3 f (j) 4 ) T .
Averaging techniques consists of mixing all the p modes of the on/off actuator by introducing one single continuous input to create an equivalent continuous input similar to the one in servo-valves. The use of the averaged model is justified by the high switching frequency of the on/off valves used in this paper, i.e. their dynamics are much faster than the other dynamics. Then, the state averaged model using only three modes (mode 1, 6 and 7) can be given by: Ẋa = d (1) F (1) + d (6) F (6) + d (7) F (7) (9

)
where the vector X a = (x 1a x 2a x 3a x 4a ) T corresponding to the vector (y a ẏa ÿa P Pa ) T is the averaged state variable vector.

Remark 3.1 Unlike the Takagi-Sugeno fuzzy modeling or the T-S modeling [START_REF] Takagi | Fuzzy identification of systems and its applications to modeling and control[END_REF][START_REF] Tanaka | Fuzzy Control Systems Design and Analysis[END_REF], averaging techniques consist in combining the different existing discontinuous modes of a system.

T-S modeling approximates a nonlinear system by an interpolation of several local linear models and by combining them using activation functions. Therefore, T-S modeling can be seen as creating and combing artificial modes to approximate the nonlinear behavior of system. However, the relevance of the fuzzy model depends on its ability to represent the actual system. The averaging technique used in this paper is another alternative to represent a switched system with different discontinuous modes and combine them using the duty ratios to represent the real switched system as in [START_REF] Zhang | Sliding mode approach to pwm-controlled pneumatic systems[END_REF]. Therefore, unlike T-S methods, averaging techniques do not create any artificial modes.

Since modes 1 and 6 correspond to the actuation in the positive direction, from the chamber P to the chamber N (see Fig. 1), the modes 1 and 6 can be modulated together while mode 7 is not used. For the negative direction, from the chamber N to the chamber P, one can modulate the modes 1 and 7 while mode 6 is not used. A new and a continuous input u is introduced to modulate modes 1 and 6 together and modes 1 and 7 together. For the positive direction, u is positive and its value corresponds to the duty ratio of mode 6.

For the negative direction, u is negative and its absolute value corresponds to the duty ratio of mode 7. Hence the following modulation (switching) scheme between the coefficients d (j) and the input u is proposed:

• For the positive direction: d (1) and d (6) are mixed d (1) = 1 -|u| and d (6) = |u| (while d (7) = 0)

• For the negative direction: d (1) and d (7) are mixed d (1) = |u| and d (7) = 1 -|u|

(while d (6) = 0)
The averaged model of these three modes can be given by:

Ẋa = F (1) (X a ) + (F (6) (X a ) -F (1) (X a ))u if u ≥ 0 (F (1) (X a ) -F (7) (X a ))u if u < 0
Hence, the state averaged model is given by:

       ẋ1a = x 2a ẋ2a = x 3a ẋ3a = ϕ 1 (X a ) + g 1 (X a )u ẋ4a = ϕ 2 (X a ) + g 2 (X a )u (10)
with:

ϕ 1 (X a ) = - k m A P x 4a l/2 + x 1a + A N ψ(X a ) l/2 -x 1a x 2a - b v m x 3a g 1 (X a ) = krT m        Q(P S , x 4a ) (l/2 + x 1a ) + Q(ψ(X a ), P E ) (l/2 -x 1a ) if u ≥ 0 Q(x 4a , P E ) (l/2 + x 1a ) + Q(P S , ψ(X a )) (l/2 -x 1a ) if u < 0 ϕ 2 (X a ) = -k x 4a l/2 + x 1a x 2a g 2 (X a ) = krT        krT Q(x 4a , P E ) (l/2 + x 1a )A P if u ≥ 0 krT Q(P S , x 4a ) (l/2 + x 1a )A P if u < 0 and: ψ(X a ) = (A p x 4a -x 3a -b v x 2a )/A N .
Remark 3.2 System (10) describes the equivalent nonlinear dynamics of the on/off solenoid valve pneumatic system (8) where the initial state vector X is averaged. Similarly to the servo valve pneumatic actuators, a part of the dynamics of (10) depends on the sign of the input u; if u ≥ 0, chamber P is charging while chamber N is discharging and vice versa. Unlike power converters as SEPIC, which has two modes and a unique averaged model, pneumatic actuators have seven modes, which makes the elaboration of a unique averaged model similar to those of power converters impossible.

Remark 3.3 Unlike the switched model ( 4) with four binary discrete inputs the averaged model [START_REF] Messina | Experimenting and modelling the dynamics of pneumatic actuators controlled by the pulse width modulation PWM technique[END_REF] has the advantage of using only one continuous input u ∈ [-1, 1]. The dynamics g 1 (X a ) and g 2 (X a ) correspond to the variation of actuation between the modes F (6) and F (7) , which performed gradually and smoothly unlike the switched model ( 4) where the switch are instantaneous.

The state averaged model of [START_REF] Messina | Experimenting and modelling the dynamics of pneumatic actuators controlled by the pulse width modulation PWM technique[END_REF] belongs to the class of dynamical systems given by:

Ẋa = AX a + Φ(X a ) + G(X a )u u ∈ [-1, 1] (11)
with:

A = A 1 0 0 A 2 A 1 =   0 1 0 0 0 1 0 0 0   A 2 = 0 Φ(X a ) =     x 2 x 3 ϕ 1 (X a ) ϕ 2 (X a )     G(X a ) =     0 0 g 1 (X a ) g 2 (X a )    
Compared to the output averaged model in [START_REF] Hodgson | Sliding mode control of nonlinear discrete-input pneumatic actuators[END_REF][START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF] the new model ( 11) is a full state averaged model. It can be used to satisfy control objectives but also for designing observers.

In the next section, the averaged model [START_REF] Ahn | Intelligent switching control of pneumatic actuator using on/off solenoid valves[END_REF] will be used to design a sliding mode observer.

4 Sliding mode Observer

Observability

The observability of a system expresses the possibility to reconstruct its initial conditions, with nothing else than the measure of its inputs and outputs. The observability analysis allows to determine the minimum number of ouputs to have the observability of a system [START_REF] Hammouri | Nonlinear observers for locally uniformly observable systems[END_REF].

Let us consider the following general form of nonlinear system:

(Σ) : Ẋ = f (X, u) Y = h(X) (12)
Where

X(t) ∈ V (an open set of R n ), the input u(t) ∈ U where U is measurable set of R m .
Definition 4.1 The system ( 12) is said to be observable if for any initial conditions X 0 and X0 , there exists a control input u such that the corresponding outputs trajectories h(X 0 (t))

and h( X0 (t)) are not equal for all time t ∈ [t 0 ; t 0 + T h ] with t 0 an initial instant and T h > 0 a time horizon [START_REF] Besançon | On uniform observation of nonuniformly observable systems[END_REF].

Thus, the concept of observability is based on the possibility to distinguish two distinct initial conditions using two different outputs. It can easily proved that system (11) could be observable using only two outputs y 1a = x 1a and y 2a = x 4a [START_REF] Girin | High gain and sliding mode observers for the control of an electropneumatic actuator[END_REF]. Hence, system (11) can be rewritten as:

Ẋa = AX a + Φ(X a ) + G(X a )u Y a = CX a (13)
with:

C = C 1 0 0 C 2 , C 1 = (1 0 0) and C 2 = 1

Sliding Mode Observer Design

Given a switched pneumatic model that can be averaged and rewritten as the model ( 13), consider the following standard sliding mode observer [START_REF] De Wit | Sliding observers for robot manipulators[END_REF]:

     ˙ X a = A X a + Φ( X a ) + G( X a )u -K( Y a -Y a ) -Lsign( Y a -Y a ) Y a = C X a (14)
where:

K =     K 1 0 K 2 0 K 3 0 0 K 4     , L =     L 1 0 L 2 0 L 3 0 0 L 4     K is chosen in order to make A -KC Hurwitz.
System ( 14) is an observer for system [START_REF] Hodgson | Sliding mode control of nonlinear discrete-input pneumatic actuators[END_REF] where the observation errors converge toward zero in finite time if L 1 , L 2 , L 3 and L 4 satisfy the following conditions:

           L 1 > max |e 2 | L 2 > 0 - ϑ -|ζ| P 3 L 1 < L 3 < - ϑ + |ζ| P 3 L 1 L 4 > max |δ 2 | (15)
with: where P 2 and P 3 are strictly positive scalars. and e 2 = x 2a -x 2a , e 3 = x 3a -x 3a . δ 1 and δ 2 are given by:

ζ = - 4L 2 P 2 P 3 L 1 δ 1 e 2 2 +
δ 1 = ϕ 1 ( X a ) -ϕ 1 (X a ) + (g 1 ( X a ) -g 1 (X a ))u δ 2 = ϕ 2 ( X a ) -ϕ 2 (X a ) + (g 2 ( X a ) -g 2 (X a ))u
Proof The proof is based on Lyapunov theory. System (13) is observable using two outputs y 1a and y 2a which give two accessible observation errors e 1 = x 1a -x 1a and e 4 = x 4a -x 4a .

The other errors, e 2 and e 3 , are given by: e 2 = x 2a -x 2a and e 3 = x 3a -x 3a . The error dynamics are given by

       ė1 = e 2 -K 1 e 1 -L 1 sign(e 1 ) ė2 = e 3 -K 2 e 1 -L 2 sign(e 1 ) ė3 = δ 1 -K 3 e 1 -L 3 sign(e 1 ) ė4 = δ 2 -K 4 e 4 -L 4 sign(e 4 ) (16)
In order to make e 1 converge toward zero, one needs to define a sliding surface s 1 = e 1 .

Then, a Lyapunov function V 1 = 1 2 e 2 1 can be chosen. The convergence of the estimation error to zero is obtained if V1 < 0. Hence, L 1 has to satisfy:

L 1 > e 2 -1 e 1 if e 1 > 0 L 1 > -e 2 + K 1 e 1 if e 1 < 0
Both conditions can be satisfied by estimating the worst case scenario [START_REF] Hammouri | Nonlinear observers for locally uniformly observable systems[END_REF] i.e. considering the maximum value of e 2 that can occur. Hence, both condition are satisfied if:

L 1 > max |e 2 |
Then, the sliding surface s 1 is attractive and an ideal sliding motion takes place on this surface, e 1 = 0 and ė1 = 0. Using ( 16 Then, the dynamics of e 2 and e 3 can be written as:

ė2 ė3 ė23 = -L 2 /L 1 1 -L 3 /L 1 0 A e 2 e 3 e 23 + 0 δ 1 Υ
The convergence of e 23 toward zero is ensured if V2 < 0, where V 23 = e T 23 P e 23 and P is a matrix given by P = P 2 0 0 P 3 . Recall that P 2 and P 3 are strictly positive scalars. Then, the derivative of V 2 with respect to time is given by: V23 = -e T 23 -P A -A T P - To make V23 < 0, one needs to guarantee that M is strictly positive. M is given by: of the averaged model [START_REF] Messina | Experimenting and modelling the dynamics of pneumatic actuators controlled by the pulse width modulation PWM technique[END_REF].

M =    2 L 2 P 2 L 1 L 3 L 1 P 3 -P 2 - e 2 δ 1 P 3 e 2
e p dτ
where λ is a positive tunning parameter and e p is the position tracking error: e p = y d -x1a .

The equivalent control u is given by: u = ... The constraints [START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] give sufficient conditions for the system (14) to be a finite convergence time observer for system [START_REF] Hodgson | Sliding mode control of nonlinear discrete-input pneumatic actuators[END_REF]. Please note that they do not offer a direct method to calculate L. The constraints [START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] include the observation errors e 2 and e 3 which are not available. However, in practice, one can use experimental data to estimate the worst case scenario i.e. the worst estimation of e 2 and e 3 in order to satisfy the constraints.

y d + k m A P x4a l/2 + x1a + A N ψ( Xa ) l/2 -x1a x2a + b v m x3a -3λë p -3λ 2 ėp -λ 3 e p 0
The parameters K and L of the observer [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF] to the velocity calculated using a robust differentiation algorithm [START_REF] Smaoui | A robust differentiator-controller design for an electropneumatic system[END_REF]. For the pressure P N , there is a slight error in the estimation. and it can reach 0.5 bar. Hence, the sliding mode observer can achieve good performance with small estimation errors. Furthermore, it is clear that the control objective which is to track y d is satisfied as it is shown in Fig. 2 with small oscillations of the measured position around y d . However, this is due the dry frictions which were not considered in the modeling step in the design of the control strategy.

Conclusion

In this paper, an averaged state model is proposed for a pneumatic actuator with four on/off solenoid valves. The advantage of this averaged model is to have one continuous input instead of having four binary inputs. This model can be easily used to design control

e 3 -

 3 ), one can write: sign(e 1 ) = e 2 /L 1 . The error dynamics becomes L 2 e 2 /L 1 ė3 = δ 1 -L 3 e 2 /L 1

e 23 Υ T P e T 23 e 23 -

 23

Remark 4 . 1

 41 The proof above gives conditions on L in order to make the observation errors converge towards zero. Please note that those conditions hold even with the switching in G( X a ) by considering the worst case scenario in both cases: u ≥ 0 or u < 0 i.e. considering the worst estimation of δ 1 and δ 2 in both cases. The proof thus holds for any input u ∈ [-1, 1]

Figure 2 :

 2 Figure 2: Real and estimated position and velocity

Fig. 2 and

 2 Fig.2and Fig.3. show the estimation of the averaged state variable corresponding to y, ẏ, P P and P N . y and P P are measured directly and used in the sliding mode observer: their

Fig. 4 Figure 3 :

 43 Fig.4shows the estimation errors. The y estimation error does not exceed 0.1 mm. The

  Hence, the conditions on L 2 and L 3 given in[START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] are obtained and the errors e 2 and e 3 converge toward zero. Finally for the remaining error e 4 , similarly to e 1 , a Lyapunov Both conditions can be satisfied if L 4 > max |δ 2 |.

	using Schur's Lemma [28], M is positive definite if and only if:
			      	2 -4 L 2 P 2 L 1 P 3 δ 1 e 3 > 0 e 2 2 + e 2 3 L 2 P 2 L 1	-	L 3 L 1	P 3 -P 2 -	e 2 δ 1 P 3 e 2 3 2 + e 2	2	> 0
	function V 4 =	1 2	e 2 4 could be defined. In order to make V4 < 0, L 4 must satisfy:
					L 4 > δ 2 -K 4 e 4	if	e 4 > 0
					L 4 > -δ 2 + K 4 e 4	if	e 4 < 0
									
				P 3	L 3 L 1	-P 2 -	P 3 δ 1 e 2 e 2 2 + e 2 3	-2	2 + e 2 3 3 2 + e 2 e 2 P 3 δ 1 e 3	 

τ

Experimental Results

The observer of ( 14) is used in closed loop i.e. its variables Xa are used in the control strategy. Then, the control input u is used to calculate the different coefficients d (1) , d (6) and d (7) . Finally, using a PWM technique on each valve i, the different inputs U i are generated. With such fast switching times, the on/off valves are appropriate for the purposes of control and observation. The controller and the observer are implemented using a dSPACE board (DS1104), running at a sampling rate of 500 Hz. This sampling rate has been chosen according to the open/close bandwidth of the valves opening and closing time smaller than 1.3 ms and to enable an acceptable tracking response.

The benchmark used to validate the observer has three sensors: a position sensor (y) and two pressure sensors (P P and P N ). However, the observer ( 14) only uses the measures of y and P P to estimate the velocity ẏ and the pressure P N . The measurement the pressure P N is used neither in the observer nor in the control strategy. It is only used to validate the estimation of the pressure P N given by the observer. The benchmark parameters are summarized in TABLE 1.

The authors of [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF][START_REF] Hodgson | Nonlinear discontinuous dynamics averaging and PWM-based sliding control of solenoid-valve pneumatic actuators[END_REF] used the output averaged model to control the position of the ac- tuator y with a sliding mode control strategy [START_REF] Utkine | Variable structure systems with sliding modes[END_REF][START_REF] Utkin | Sliding Model Control in Electromechanical Systems[END_REF]. The objective is to make the piston's position y track a desired sinusoidal trajectory y d (t). Using the sliding mode control strategy, the control input u is given by:

Where: u is the equivalent control, K c is a time variant gain to ensure robustness (See [START_REF] Hodgson | Dynamical model averaging and PWM based control for pneumatic actuators[END_REF] for more details), s p is the sliding surface, it is given by: