S Tarbouriech

I Queinnec

C Prieur

Stability Analysis and Stabilization of Systems with Input Backlash

Keywords: Stability analysis, stabilization, backlash, uniform ultimate boundedness stability

This paper deals with the stability analysis and stabilization of linear systems with backlash in the input. Uniform ultimate boundedness stability and stabilization problems are tackled allowing to characterize suitable regions of the state space in which the closedloop trajectories can be captured. In the state feedback control design, computational oriented solutions are derived to solve suboptimal convex optimization problems able to give a constructive solution.

I. INTRODUCTION

Backlash operators are nonlinear elements with memory. They are usual phenomena in many physical systems, such as electrical inductors, piezo-actuators, gear trains and mechanical friction systems (see e.g. the survey [START_REF] Macki | Mathematical models for hysteresis[END_REF] on such memory elements and also on other operators as Krasnolsel'skii-Pokrovskii hysteresis). Other closely related infinite dimensional operators are also considered in the literature such as the Preisach models [START_REF] Brokate | Hysteresis and phase transitions[END_REF], the Coleman-Hodgdon operator [START_REF] Coleman | A constitutive relation for rateindependent hysteresis in ferromagnetically soft materials[END_REF] and the counterclockwise input-output dynamics [START_REF] Angeli | Systems with counterclockwise input-output dynamics[END_REF]. For many applications, such nonlinear operators affect the inputs of control systems (see e.g. [START_REF] Logemann | Discrete-time and sampled-data lowgain control of infinite-dimensional linear systems in the presence of input hysteresis[END_REF], [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF]). The negligence of such nonlinearities during the control design or the stability analysis can lead to an important degradation of the closed-loop performance or even to the loss of stability (see, in particular, [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF]).

Here we focus on the stability analysis and the control design for a linear system with a backlash operator in the input. This induces to study an infinite dimensional system, since the backlash operator is a memory-based relation. Several approaches have been developed in the context of this type of non-smooth nonlinearity: see, for example [START_REF] Corradini | Robust stabilization of nonlinear uncertain plants with backlash or dead zone in the actuator[END_REF] and references therein. Moreover, some solutions consisting in applying inverse nonlinearity have been proposed [START_REF] Taware | Control of sandwich nonlinear systems[END_REF], [START_REF] Taware | Design and analysis of a hybrid control scheme for sandwich nonsmooth nonlinear systems[END_REF]. However, although there is a large literature dealing with the stability analysis and the estimation of input/output properties (see e.g. [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF], [START_REF] Ouyang | Stability analysis and controller design for a system with hysteresis[END_REF] for recent results), there are not many papers suggesting design methods for nonlinear systems with memory operators, except [START_REF] Jayawardhana | Stability of systems with Duhem hysteresis operator: Dissipativity approach[END_REF] where a dissipativity property is used, and [START_REF] Su | Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis[END_REF] where the dynamics of the differential equation modeling a one-dimensional hysteresis is controlled.

In the present paper, sufficient conditions are provided to characterize a compact set for the nonlinear system as a finite-time attractor for all admissible initial conditions. In this set, the system trajectories are uniformly ultimately bounded. The method developed does not impose any assumption on the system matrix at s = 0, where s is the Laplace variable, as required in [START_REF] Paré | A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities[END_REF] and [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF]. Furthermore, the approach pursued does not need to consider that the static gain of the system has to be either null or nonsingular contrary to [START_REF] Haddad | Linear controller analysis and design for systems with input hystereses nonlinearities[END_REF] and [START_REF] Haddad | Absolute stability criteria for multiple slope-restricted monotonic nonlinearities[END_REF]. More precisely, generalized sector conditions using properties of the backlash operator and Lyapunov arguments allow us to analyze the stability, i.e., we exhibit, in our first main result S. Tarbouriech and I. Queinnec are with CNRS, LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France and Univ de Toulouse, LAAS, F-31400 Toulouse, France. tarbour@laas.fr, queinnec@laas.fr C. Prieur is with Gipsa-lab, Department of Automatic Control, Grenoble Campus, 11 rue des Mathématiques, BP 46, 38402 Saint Martin d'Hères, France christophe.prieur@gipsa-lab.fr (see Theorem 2 below), some conditions to prove that a compact set is a finite-time attractor, for all admissible initial conditions. These conditions are given in terms of Linear Matrix Inequalities (LMI), which can be solved using standard numerical algorithms, within a time of computation which is polynomial with respect to the dimension of the data. Even if the spirit of the approach proposed is quite similar to the SISO statement in Corollary 16 in [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF], it allows to deal with MIMO systems. Moreover, while both [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF] and our paper address the characterization of the finite-time attractor, our approach, differently from [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF], uses the knowledge of the Lyapunov function. In this sense, our approach can be considered as complementary with the method proposed in [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF]. Furthermore, at the opposite of conditions developed in [START_REF] Tarbouriech | Stability analysis for linear systems with input backlash through sufficient LMI conditions[END_REF], we do not need to use the time-derivative version of the system. An implicit objective in the characterization of the finite-time attractor is to make it as small as possible. Then, the second contribution of the paper (see Theorem 3) resides in proposing a technique to design both the state feedback gain not only stabilizing the linear part of the system but also allowing to address the minimization of the associated finite-time attractor. In this sense, the conditions obtained are complementary to those resulting for the stability analysis. This design result is again based on Lyapunov techniques. The computational issues and several examples illustrating the interests and the drawbacks of the conditions are also presented.

Notation. For two vectors x, y of ℜ n , the notation x y means that x (i) -y (i) ≥ 0, ∀i = 1, . . . , n. 1 and 0 denote the identity matrix and the null matrix of appropriate dimensions, respectively. x ∈ ℜ n + means that x 0. The Euclidian norm is denoted • . A ′ and trace(A) denote the transpose and the trace of A, respectively. He{A} = A + A ′ . diag(A; B) denotes the diagonal matrix which diagonal blocks are formed by squared matrices A and B. For two symmetric matrices, A and B, A > B means that A -B is positive definite. In partitioned symmetric matrices, the symbol ⋆ stands for symmetric blocks. λmax(A) (respectively, λ min (A)) denotes the maximal (respectively, minimal) eigenvalue of the matrix A.

II. PROBLEM FORMULATION

The class of systems under consideration is described by:

ẋ = Ax + BΦ[w] w = Kx (1)
where x ∈ ℜ n is the state and w ∈ ℜ m is the input of the nonlinearity Φ. A, B and K are matrices of appropriate dimensions. The pair (A, B) is supposed to be controllable. K is the state feedback control gain to be designed or supposed known. Φ is a componentwise backlash operator (see, for example, [START_REF] Macki | Mathematical models for hysteresis[END_REF], [START_REF] Paré | A KYP lemma and invariance principle for systems with multiple hysteresis non-linearities[END_REF], [START_REF] Taware | Control of sandwich nonlinear systems[END_REF], [START_REF] Brokate | Hysteresis and phase transitions[END_REF]). We denote the set of continuous, piecewise differentiable functions f : [0, +∞) → ℜ m by C 1 pw ([0, +∞); ℜ m ), that is the set of continuous functions w being, for some unbounded sequence (tj) ∞ j=0 in [0, +∞) with t0 = 0, continuously differentiable on (tj-1, tj) for all j ∈ N. Given the vector ρ in ℜ m + and L = diag(ℓ (i) ), with positive values ℓ (i) , i = 1, . . . , m, the operator Φ is defined as follows, for all f ∈ C 1 pw ([0, +∞); ℜ m ), for all j ∈ N, for all t ∈ (tj-1, tj) and for all i ∈ {1, . . . , m}:

( ˙ Φ[f ](t)) (i) =            ℓ (i) ḟ(i) (t) if ḟ(i) (t) ≥ 0 and (Φ[f ](t)) (i) = ℓ (i) (f (i) (t) -ρ (i) ) ℓ (i) ḟ(i) (t) if ḟ(i) (t) ≤ 0 and (Φ[f ](t)) (i) = ℓ (i) (f (i) (t) + ρ (i) ) 0 otherwise (2)
where 0 = t0 < t1 < . . . is a partition of [0, +∞) such that f is continuously differentiable on each of the intervals (tj-1, tj), j ∈ N. Thus, Φ is a time-invariant nonlinearity with slope restriction, as in [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF]. Note however that it is a memory-based operator, since to compute it, we need to have information about the past values of w (this is not the case in [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF]). Such an operator can be depicted in Figure 1. Since it is possible to stack several backlash operators into one backlash operator (with the dimension equal to the sum of all dimensions), without loss of generality, it is suitable to assume that only one backlash is present in [START_REF] Angeli | Systems with counterclockwise input-output dynamics[END_REF].

Throughout the paper, we define the set of admissible initial conditions (w(t = 0)), from which we want to guarantee the stability, as follows

L(w(0) + ρ) Φ[w](0) L(w(0) -ρ) (3)
According to [START_REF] Nordin | Controlling mechanical systems with backlash: a survey[END_REF], [START_REF] Corradini | Robust stabilization of nonlinear uncertain plants with backlash or dead zone in the actuator[END_REF], that means that the nonlinearity Φ is active. Then, with (2), one gets

L(w(t) + ρ) Φ[w](t) L(w(t) -ρ), ∀t ≥ 0 (4)
In a neighborhood of the origin, system [START_REF] Angeli | Systems with counterclockwise input-output dynamics[END_REF] operates in open loop and the presence of the backlash Φ[w](t) may induce the existence of multiple equilibrium points or a limit cycle. Thus, one aims at characterizing compact sets S0 that are globally attractive in finite-time, or equivalently, such that, for any initial condition x(0) belonging to ℜ n \S0, the resulting trajectory of the closed-loop system (1) is uniformly ultimately bounded in the set S0 (see [START_REF] Khalil | Nonlinear Systems[END_REF]Section 4.8]). This is stated in the following problem:

Problem 1 (Stability analysis): Given a gain K such that A + BLK is Hurwitz, characterize the region of the state space in which the solutions of system (1) are uniformly ultimately bounded when initialized as in [START_REF] Brokate | Hysteresis and phase transitions[END_REF].

Imposing A + BLK Hurwitz is classical since it corresponds to system (1) when Φ[w] is neglected and replaced by Lw. Let us emphasize that, unlike what is generally considered in the literature, in the current paper, the Hurwitz assumption on matrix A has been relaxed, i.e. the matrix A can be exponentially unstable.

The second problem we intend to solve is complementary to Problem 1. Actually, we want to make the domain S0 as small as possible, through the design of the state feedback gain K, as explicated below.

Problem 2 (Synthesis problem): Determine the gain K such that A +BLK is Hurwitz and which minimizes the size of the associated region of the state space in which the solutions of the system (1) are uniformly ultimately bounded when initialized as in (3).

III. MAIN RESULTS

A. Preliminary results

For conciseness, throughout the paper, we denote Φ instead of ˙ Φ[w], and Φ instead of Φ[w]. Recall that the nonlinear operator Φ verifies the properties stated in Lemma 3.1 in [START_REF] Tarbouriech | Stability analysis for linear systems with input backlash through sufficient LMI conditions[END_REF]. Let us define the nonlinearity

Ψ Ψ = Φ -Lw = Φ -LKx (5)
Hence, from Lemma 3.1 in [START_REF] Tarbouriech | Stability analysis for linear systems with input backlash through sufficient LMI conditions[END_REF] and (4), we can formulate the following properties with respect to Ψ. Lemma 1: For any diagonal positive definite matrices N1, N2, N3 in ℜ m×m , with N3 ≥ 1, we have, for all w ∈ C 1 pw ([0, +∞); ℜ m ), for all t ∈ (ti-1, ti)

( Ψ + L ẇ) ′ N1Ψ ≤ 0, (6)
( Ψ + L ẇ) ′ N2( Ψ + (1 -N3)L ẇ) ≤ 0, (7) -Lρ Ψ Lρ (8)
where 0 = t0 < t1 < . . . is a partition of [0, +∞) such that w is continuously differentiable on each of the intervals (tj-1, tj), j ∈ N.

Remark 1: Property ( 6) can be seen as a property of passivity with respect to the output Φ and input N1Lw, and with the storage function 1 2 Φ ′ N1Φ [START_REF] Khalil | Nonlinear Systems[END_REF].

B. Stability analysis

The following result relative to Problem 1 can be stated. Theorem 2: Given K such that A0 = A + BLK is Hurwitz. A positive scalar τ1 being given, if there exist a symmetric positive definite matrix P ∈ ℜ n×n , three diagonal positive definite matrices N1 ∈ ℜ m×m , N2 ∈ ℜ m×m and T3 ∈ ℜ m×m satisfying the following LMI conditions  

He{A ′ 0 P } + τ1P ⋆ ⋆ B ′ P -N1LKA0 -T3 -He{N1LKB} ⋆ -N2LKA0 -N1 -N2LKB -2N2   < 0 (9) ρ ′ LT3Lρ -τ1 ≤ 0 (10)
then, for any initial admissible conditions (x(0), Ψ(0)), the resulting trajectories of the closed-loop system (1) are uniformly ultimately bounded in the set S0(P ) defined as follows:

S0(P ) = {x ∈ ℜ n ; x ′ P x ≤ 1} (11)
Proof. Consider a quadratic Lyapunov function candidate V defined by V (x) = x ′ P x, P = P ′ > 0, for all x in ℜ n . We want to verify that there exists a class K function α such that V (x) ≤ -α(V (x)), for all x such that x ′ P x ≥ 1 (i.e. for any x ∈ ℜ n \S0), and for all nonlinearities Ψ satisfying Lemma 1 (i.e. satisfying relations ( 6), ( 7), ( 8)). By using the S-procedure, it corresponds to verify that L < 0, where

L = V (x) -τ1(1 -x ′ P x) - m i=1 τ3i(Ψ ′ (i) Ψ (i) -ρ 2 (i) ℓ 2 (i) ) -2( Ψ + L ẇ) ′ N1Ψ -2( Ψ + L ẇ) ′ N2( Ψ + (1 -N3)L ẇ) (12)
with τ1 a positive scalar and T3 = diag(τ3i) a positive diagonal matrix. Choosing N3 = 1 and noting that ẇ = KA0x + KBΨ and V (x) = x ′ (A ′ P + P A)x + 2x ′ P BΦ, or equivalently, from the

definition of Ψ in (5), V (x) = x ′ (A ′ 0 P + P A0)x + 2x ′ P BΨ, it follows that L = L0 + ρ ′ LT3Lρ -τ1 with L0 =   x Ψ Ψ   ′ M0   x Ψ Ψ  
where M0 is the left-hand-side matrix in [START_REF] Corradini | Variable structure control of systems with sandwiched backlash[END_REF]. The satisfaction of relations ( 9) and ( 10) implies both L0 < 0 and ρ ′ LT3Lρ -τ1 ≤ 0, and then L < 0. Therefore, there exists ε > 0, such that L ≤ -ε (x ′ Ψ ′ Ψ′ ) ′ 2 ≤ -εx ′ x. Hence, since by definition one gets V (x) ≤ V (x) -τ1(1 -x ′ P x) ≤ L one can also verify V (x) -τ1(1 -x ′ P x) ≤ -εx ′ x, for any x ∈ ℜ n \S0, or equivalently,

V (x) ≤ -εx ′ x , ∀x such that x ′ P x ≥ 1 . ( 13
)
Consider now a solution of (1) starting from an admissible initial condition at t0 = 0 such that x(t0) ′ P x(t0) ≥ 1. According to [START_REF] Jayawardhana | Stability of systems with Duhem hysteresis operator: Dissipativity approach[END_REF], there exists a time T ≥ t0 + (x(t0) ′ P x(t0) -1)λmax(P )/ǫ such that x(t) ∈ S0(P ), ∀t ≥ T . Furthermore, S0(P ) is an invariant set for system [START_REF] Angeli | Systems with counterclockwise input-output dynamics[END_REF]. Hence, in accordance with [START_REF] Khalil | Nonlinear Systems[END_REF], it follows that the trajectories are uniformly ultimately bounded in S0(P ) for any x(0). That concludes the proof of Theorem 2. Theorem 2 ensures the finite-time convergence of the solutions of system (1) inside S0(P ) for any initial admissible condition. Hence, the set S0(P ) contains the possible equilibrium points or limit cycles potentially induced by the backlash operator. The equilibrium points of system (1), when they exist, satisfy: ẋe = 0 = (A + BLK)xe + BΨe, or equivalently, since A0 is Hurwitz, xe = -A -1 0 BΨe, x ′ e P xe ≤ 1 and Lρ Ψe -Lρ.

C. Control design

To address Problem 2 and to remove the products between A0 and K appearing in condition [START_REF] Corradini | Variable structure control of systems with sandwiched backlash[END_REF], the control design conditions are developed by using the Finsler lemma [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Such an approach is stated in the following result.

Theorem 3: A positive scalar τ1 being given, if there exist a symmetric positive definite matrix P ∈ ℜ n×n , five matrices Q1 ∈ ℜ n×n , Q2 ∈ ℜ n×n , F3 ∈ ℜ m×n , F4 ∈ ℜ m×n and K ∈ ℜ m×n , three diagonal positive definite matrices N1 ∈ ℜ m×m , N2 ∈ ℜ m×m and T3 ∈ ℜ m×m satisfying the matrix condition [START_REF] Jayawarshana | Sufficient conditions for dissipativity on Duhem hysteresis model[END_REF] (see top of the next page) and:

ρ ′ LT3Lρ -τ1 ≤ 0 ( 15
)
then the state feedback gain K is such that A + BLK is Hurwitz and, for any initial admissible conditions (x(0), Ψ(0)), the resulting trajectories of the closed-loop system (1) are uniformly ultimately bounded in the set S0(P ) as defined in [START_REF] Haddad | Absolute stability criteria for multiple slope-restricted monotonic nonlinearities[END_REF].

Proof. Consider the quadratic Lyapunov function defined by V (x) = x ′ P x, P = P ′ > 0, for all x in ℜ n . The time-derivative of V (x) along the trajectories of system (1) reads V (x) = ẋ′ P x + x ′ P ẋ. Moreover, by noting ξ = x ′ ẋ′ Ψ ′ Ψ′ ′ , system (1) can be also written as follows:

A + BLK -1 B 0 ξ = 0 (16)
We then want to verify that there exists a K function α such that V (x) ≤ -α(V (x)), for all x such that x ′ P x ≥ 1, for all nonlinearity Ψ satisfying Lemma 1 and for x, Ψ satisfying [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF]. In other words, by using the S-procedure, we can express L defined in [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF] as L = ξ ′ L0ξ -τ1 + ρ ′ LT3Lρ, i.e., we want to verify:

ξ ′ L0ξ = ξ ′     τ1P P 0 0 P 0 -K ′ LN1 -K ′ LN2 0 -N1LK -T3 -N1 0 -N2LK -N1 -2N2     ξ < 0 and -τ1 + ρ ′ LT3Lρ ≤ 0 for ξ such that A + BLK -1 B 0 ξ = 0 (17)
with τ1 a positive scalar and T3 a positive diagonal matrix. By using the Finsler lemma, the satisfaction of ( 17) consists of finding some multipliers

F1 ∈ ℜ n×n , F2 ∈ ℜ n×n , F3 ∈ ℜ m×n , F4 ∈ ℜ m×n such that L0 + He{F A + BLK -1 B 0 } < 0, with F = F ′ 1 F ′ 2 F ′ 3 F ′ 4 ′ , which reads:     He{F1A + F1BLK} + τ1P ⋆ P + F2A + F2BLK -F ′ 1 -F2 -F ′ 2 F3A + F3BLK + B ′ F ′ 1 -N1LK + B ′ F ′ 2 -F3 F4A + F4BLK -F4 -N2LK ⋆ ⋆ ⋆ ⋆ He{F3B} -T3 ⋆ -N1 + F4B -2N2     < 0
A necessary condition for this inequality to be strict is the nonsingularity of matrix F2. Hence, by considering that F1 is also nonsingular and by multiplying the previous inequality at left by diag(Q1; Q2; 1; 1) and at right by

diag(Q ′ 1 ; Q ′ 2 ; 1; 1) with Q1 = F -1 1 and Q2 = F -1
2 , one obtains relation [START_REF] Jayawarshana | Sufficient conditions for dissipativity on Duhem hysteresis model[END_REF]. The satisfaction of relations ( 14) and ( 15) implies the satisfaction of condition [START_REF] Logemann | Discrete-time and sampled-data lowgain control of infinite-dimensional linear systems in the presence of input hysteresis[END_REF]. Furthermore, due to the block (1, 1) of the matrix in relation ( 14), the satisfaction of relation ( 14) means that the gain K is such that A + BLK is Hurwitz. Similarly to the proof of Theorem 2, by definition, S0(P ) is an invariant set for the trajectories of system (1). Hence, in accordance with [START_REF] Khalil | Nonlinear Systems[END_REF], it follows that the trajectories are uniformly ultimately bounded in S0(P ) for any x(0). That concludes the proof.

D. Discussion on systems with backlash in the output

The solution to the analysis problem (Theorem 2) may be directly extended to the case of backlash in the output, considering the following system:

ẋ = Ax + B1u y = Cx + DΦ[w] with w = Ex u = Ky (18)
which can also be written as:

ẋ = (A + B1K(C + DLE))x + B1KDψ (19)
with ψ = Φ -Lw defined as in equation ( 5). The static output feedback gain K is supposed chosen such that (A+B1K(C+DLE)) is Hurwitz. Then, by denoting A0 = (A + B1K(C + DLE)) and B = B1KD, Theorem 2 directly applies. It also applies to the case of a dynamic output feedback with backlash in the measured output.

On the other hand, the extension of Theorem 3 to the design problem with backlash in the sensor is not direct.

IV. NUMERICAL IMPLEMENTATION

A. Computational issues

As far as τ1 is fixed, conditions stated in Theorem 2 are linear in the decision variables. The problem may then be solved as an optimization problem to evaluate the smallest set S0(P ), typically described by its volume, proportional to det(P -1 ) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], in which the trajectories of the closed-loop system are uniformly ultimately

    He{AQ ′ 1 + BLKQ ′ 1 } + τ1Q1P Q ′ 1 ⋆ ⋆ ⋆ Q2P Q ′ 1 + AQ ′ 1 + BLKQ ′ 1 -Q2 -Q2 -Q ′ 2 ⋆ ⋆ F3AQ ′ 1 + F3BLKQ ′ 1 + B ′ -N1LKQ ′ 2 + B ′ -F3Q ′ 2 F3B + B ′ F3 -T3 ⋆ F4AQ ′ 1 + F4BLKQ ′ 1 -F4Q ′ 2 -N2LKQ ′ 2 -N1 + F4B -2N2     < 0 (14)
bounded. In the numerical examples which follow, thanks to the use of Yalmip [START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF] to build the LMI optimization problem, we use the convex minimization problem to solve Problem 1:

min -geomean(P ) under conditions ( 9), ( 10) [START_REF] Ouyang | Stability analysis and controller design for a system with hysteresis[END_REF] where geomean(P ) is the geometric mean of the eigenvalues of the matrix P .

To solve the design problem expressed through Theorem 3, we exploit the fact that matrices Q1, Q2, F3 and F4 are Finsler multipliers which may be considered as degrees of freedom in the design conditions. Actually, one selects Q2 = Q1, F3 = 0, F4 = 0, and introduces the variables

Y = KQ ′ 1 , R = Q1P Q ′ 1 , S1 = N -1 1 and S2 = N -1 2 .
With this change of variables and by pre-and post-mulitplying the matrix in condition ( 14) by diag(1; 1; S1; S2), condition ( 14) is then updated as follows:

    He{AQ ′ 1 + BLY } + τ1R ⋆ ⋆ ⋆ R + AQ ′ 1 + BLY -Q1 -Q1 -Q ′ 1 ⋆ ⋆ B ′ LY + S1B ′ -S1T3S1 ⋆ 0 -LY -S2 -2S2     < 0 (
R Q1 Q ′ 1 αP -1 g > 0 ( 22
)
where

• Pg is selected as the solution to the analysis problem [START_REF] Ouyang | Stability analysis and controller design for a system with hysteresis[END_REF] or as directions of interest for the minimization of the set

S0(P ) ⊆ {x ∈ ℜ n , x ′ Pgx ≤ α}, with P = (Q1) -1 R(Q ′ 1 ) -1
, in which the trajectories of the closed-loop system are uniformly ultimately bounded;

• τ1 is selected, for example on a grid; • S1 is selected as the inverse of the solution N1 to the analysis problem. A grid search may alternatively be performed to select S1.

Additional constraints may be added to set the performance of the closed-loop system, such as a limit on the size of the gain, or a pole-placement requirement [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]. This can be done by prescribing additionally the following LMI condition with extra variable Rpp, issued from the use of Finsler Lemma, which allows to place the poles of (A + BLK) in the disk centered in σ with radius r:

-Rpp Q1 (A ′ -σ1) r + Y ′ L ′ B ′ r (A-σ1) r Q ′ 1 + BL r Y Rpp -Q1 -Q ′ 1 > 0 (23)

B. Illustrative Examples 1) Example 1:

The first example is of academic nature and illustrates the conservatism of the approach. Consider the unstable system (1) defined by the following data:

A = 0 1 0 0 ; B = 0 1 ; K = Kan = -2 -3 L = 1 ; ρ = 0.5
Considering τ1 = 0.9 and applying Theorem 2, the optimization problem [START_REF] Ouyang | Stability analysis and controller design for a system with hysteresis[END_REF] gives the solution to the stability analysis problem for which the indicator of the volume is v(Pan) = det(P -1 an ) = 0.0716. The influence of the parameter τ1 on the size of S0(P ) is illustrated in Figure 2 (the problem becomes unfeasible for τ1 ≥ 2 = 2|λmax(A0)|). An iterative search on τ1 allows to quickly select a convenient value for this parameter. For the synthesis problem, let us consider the optimization problem [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF]. τ1 is selected unchanged (equal to 0.9), Pg = Pan, S1 = N -1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0
1an . An additional pole-placement requirement ( 23) is used with σ = -6, r = 5. A solution to the optimization problem [START_REF] Park | The asymptotic stability of nonlinear (Lur'e) systems with multiple slope restrictions[END_REF] is given by Ksyn = -10.3327 -11.2293

It is recommended to proceed to the analysis of this new control gain Ksyn, which results, for τ1 unchanged, to the solution v(Psynan) = det(P -1 synan) = 0.0057, with Psynan = 422.7379 33.4682 33.4682 74.3510

The phase portraits of some trajectories of the system in closed loop with the control gain Kan (solid line) or Ksyn (dashed line), issued from the initial states x(0), equal to 0 1 ′ and 1 1 ′ , are plotted in Figure 3 (Φ[w](0) = LKx(0) which satisfies (3)). They show that the trajectories of the closed-loop system with Kan converge to a limit cycle included in the ellipsoid S0(Pan). The trajectories of the closed-loop system with Ksyn are uniformly ultimately bounded in the set S0(Psynan) included in the set S0(Pan). They may converge either to a limit cycle or to some equilibrium point (trajectory issued from 1 1 ′ ), which are included in S0(Psynan).

To evaluate the influence of the choice of S1 on the solution to the design problem, a grid search in the sets defined by S1 = 10 3 , 10 4 , 10 5 , 10 6 , 10 7 , τ1 = {0.1, 0.2, • • • , 1.8, 1.9} has been performed. It gives as the best solution, after an additional analysis step v(P best ) = det(P -1 best ) = 0.0033 (to be compared to the value 0.0057 obtained with a priori given values for τ1 and S1 = N -1 an solutions to the analysis problem). 2) Example 2: As a second example, we consider an unstable F-8 aircraft MIMO example borrowed from [START_REF] Castelan | L 2 -stabilization of continuous-time systems with saturating actuators[END_REF], given by: 4, issued from the initial state x(0) = 1 1 1 1 ′ and initial nonlinearity Φ[w](0) = LKx(0). It brings out the reduction of the size of the limit cycle with the gain Ksyn (plots in dashed line) in accordance with the objective which was actually to reduce the set S0(P ). Note however that this induces larger amplitude of the system input Φ[w], as it may also be seen on the form of the backlash operator characteristic in Figure 5. To evaluate the degree of conservatism of the sets S0(Pan) and S0(Psynan) in which the trajectories are uniformly ultimately bounded, the time-evolution of x ′ Panx (solid line, associated to the control gain Kan) and x ′ Psynanx (dashed line, associated to the control gain Ksyn) are plotted in Figure 6. They show that, once the trajectory enters the ellipsoidal set S0(P ), i.e. x ′ P x ≤ 1, P = Pan or P = Psynan, it evolves over the time and comes more or less close to the boundary but never exits.

Ap =     -0.
3) Example 3: A third example is proposed to illustrate how the analysis conditions stated in Theorem 2 may be used in the case of a feedback between the system output and the backlash input involving a dynamic system. This is typically the case when considering a dynamic feedback controller or an actuator device system such as proposed in [START_REF] Corradini | Variable structure control of systems with sandwiched backlash[END_REF], defined by:

ẋ1 = x2 ẋ2 = c1x1 + c2x2 + Φ[w] with ẋA = c3xA + v w = xA
and a backlash operator with L = 1, ρ = 1.5. For numerical evaluation, values slightly different from [START_REF] Corradini | Variable structure control of systems with sandwiched backlash[END_REF] are used to manipulate an unstable open-loop system. We consider the coefficients c1 = 1, c2 = -1, c3 = -5 and a state feedback control v = -9x1 -3x2.

V. CONCLUDING REMARKS

This paper considered systems with backlash operator in the input, aiming at analyzing the stability of such control systems. Uniform ultimate boundedness stability and stabilization problems were tackled allowing to characterize suitable regions of the state space in which the closed-loop trajectories can be captured. Future research directions are open. One idea would be to extend the technique to more general models of nonlinearities as studied, for example, in [START_REF] Cerone | Bounding the parameters of linear systems with input backlash[END_REF], [START_REF] Jayawarshana | Sufficient conditions for dissipativity on Duhem hysteresis model[END_REF]. Furthermore, as recalled in [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF]Page 38], any Prandtl operator may be derived from an infinite weighted sum of backlash operators. It may be fruitful to adapt the present Lyapunov technic to such nonlinear systems. It may also be interesting to consider set-valued nonlinear operators encompassing Prandtl operators (as in [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF]Page 37]). Finally, the case of a feedback between the system output and the backlash input involving an actuator device system subject to other nonlinearities (like, for example, saturation) should be considered.

Fig. 1 .

 1 Fig. 1. An illustration of a backlash operator.

21 ) 2 α

 212 Then the control design Problem 2 is solved as the optimization problem: min R,Y,Q 1 ,T 3 ,S under conditions (15), (21),

Fig. 2 .

 2 Fig. 2. Example 1 -Analysis problem. Influence of the parameter τ 1 on the size of S 0 (P ).

Fig. 3 .

 3 Fig. 3. Example 1. Ellipsoidal sets S 0 (Pan) and S 0 (Psynan). Trajectories issued from (0 1) ′ and (1 1) ′ for the system closed with control gains Kan (solid line) and Ksyn (dashed line).

  interconnects the system with a state-feedback control, which values are given by Considering τ1 = 1.5 selected from an iterative procedure, the optimization problem[START_REF] Ouyang | Stability analysis and controller design for a system with hysteresis[END_REF] gives the solution v(Pan) = det(P -1 an )

Fig. 4 . 2 Fig. 5 .

 425 Fig. 4. Example 2 -Time-evolution of the state x. System in closed loop with Kan (solid line) or Ksyn (dashed line) + zoom of the steady-state behavior of x 1 between t = 20 and t = 50.

Fig. 6 .

 6 Fig. 6. Example 2 -Time-evolution of x ′ Panx (solid line) associated to the control gain Kan and of x ′ Psynanx (dashed line) associated to the control gain Ksyn.

Fig. 7 .

 7 Fig. 7. Example 3 -a. Ellipsoidal set S 0 (Pan) and trajectory issued from (5 5 5) ′ -b. Backlash characteristic Φ[w].

  For the synthesis case, one considers Pg = Pan and a poleplacement requirement with σ = -10, r = 8.5. A grid search in the sets defined by S1 = 10 2 1, 10 3 1, 10 4 1, 10 5 1, 10 6 1 ,

		1381	0.0035	-0.2112	0.2818	
		0.0035 -0.2112 -0.6522 0.0266	-0.6522 16.7322	0.4999 -13.0185	  
		0.2818	0.4999	-13.0185	11.5846
	τ1 = {1.1, 1.2, • • • , 2.1, 2.2}, followed by an additional analysis
	step, allows to determine the control gain
	Ksyn =	0.9533 0.1246 -2.0493 37.6241 -24.2286 0.2757 -6.6111 7.9905
	with associated set defined from Psynan with indication of the
	volume v(Psynan) = det(P -1 synan) = 0.0831. The time-evolution
	of the state x is plotted in Figure


Josette Cadusseau
email: josette.cadusseau@inserm.fr

Nilusha Ragunathan-Thangarajah
email: nilusha.thangarajah@hmn.aphp.fr

Mathieu Surenaud
email: mathieu.surenaud@inserm.fr

Sophie Hue
email: sophie.hue@inserm.fr

Francois-Jérôme Authier
email: authier@u-pec.fr

Romain K Gherardi
email: romain.gherardi@hmn.aphp.fr

Selective elevation of circulating CCL2/MCP1 levels in patients with longstanding post-vaccinal macrophagic myofasciitis and ASIA

Keywords:

Several medical conditions sharing similar signs and symptoms may be related to immune adjuvants. These conditions described as ASIA (Autoimmune/inflammatory Syndrome Induced by Adjuvants), include a condition characterized by macrophagic myofasciitis (MMF) assessing long-term persistence of vaccine derived-alum adjuvants into macrophages at sites of previous immunization. Despite increasing data describing clinical manifestations of ASIA have been reported, biological markers are particularly lacking for their characterization and follow up. We report an extensive cytokine screening performed in serum from 44 MMF patients compared both to sex and agematched healthy controls and to patients with various types of inflammatory neuromuscular diseases. Thirty cytokines were quantified using combination of LuminexR technology and ELISA. There was significant mean increase of serum levels of the monocyte-chemoattractant protein 1 (CCL2/MCP-1) in MMF patients compared to healthy subjects. MMF patients showed no elevation of other cytokines. This contrasted with inflammatory patients in whom CCL2/MCP-1 serum levels were unchanged, whereas several other inflammatory cytokines were elevated (IL1!, IL5 and CCL3/MIP1"). These results suggest that CCL2 may represent a biological marker relevant to the pathophysiology of MMF rather than a non specific inflammary marker and that it should be checked in the other syndromes constitutive of ASIA.

INTRODUCTION

During the last century, billions of people have received a variety of vaccines leading to virtual eradication of several infectious diseases. The extending scope of possible immunization strategies has promoted vaccine as a priviledged weapon against a variety of diseases. As a general rule, vaccine safety has been considered excellent in the overall population [START_REF] Moxon | The next decade of vaccines: societal and scientific challenges[END_REF] but adverse reactions to vaccines were long suspected to occur in a very small minority of cases [START_REF] Agmon-Levin | Vaccines and autoimmunity[END_REF]. Their mechanism usually remains unclear. Vaccine adverse reactions could represent autoimmune phenomena similar to those following natural infections [START_REF] Molina | Infection, vaccines and other environmental triggers of autoimmunity[END_REF]. For example, autoimmune diseases may result from: (a) molecular mimicry between an infectious agent and host antigenic components; (b) epitope spreading, with progressive appearance of immune responses to different epitopes on the same or on another antigen; (c) release of cryptic epitopes or previously hidden auto-antigens; (d) bystander activation of T cells and polyclonal activation of B-cells; (e) superantigenic T cell activation by infectious products cross-linking the T-cell receptor and MHC molecule independently of specific antigen recognition; and (f) anti-apoptotic effects on autoreactive cells. Besides classical explanations centered on vaccine antigens [START_REF] Stübgen | Neuromuscular disorders associated with Hepatitis B vaccination[END_REF], attention has progressively focussed on possible implication of some adjuvants used to boost the immune response, including aluminium oxyhydroxide or alum [START_REF] Gherardi | Macrophagic myofasciitis lesions assess long-term persistence of vaccinederived aluminium hydroxide in muscle[END_REF][START_REF] Petrik | Aluminum adjuvant linked to Gulf War illness induces motor neuron death in mice[END_REF] and squalene [START_REF] Asa | Antibodies to squalene in Gulf War syndrome[END_REF][START_REF] Nohynek | AS03 adjuvanted AH1N1 vaccine associated with an abrupt increase in the incidence of childhood narcolepsy in Finland[END_REF]. It is likely that a fine balance exists between the efficacy of vaccine adjuvants and their potential toxicity, and these may be one and the same effect [START_REF] Exley | The immunobiology of aluminium adjuvants: how do they really work?[END_REF]. Adjuvants may exert their immunostimulatory effect by many different actions, including: (a) protection of antigens resulting in prolonged delivery; (b) induction of prompt vaccine particle phagocytosis by dendritic cells and macrophages with up-regulation of their antigen-presenting function;

(c) translocation of antigens to lymphoid organs where the primary activation of naïve T cells takes place; (d) amplification of the inflammatory reaction in the injection site and its draining lymph nodes through interaction with pattern recognition receptors and release of inflammatory cytokines; and (e) priming of B cells in spleen [START_REF] Exley | The immunobiology of aluminium adjuvants: how do they really work?[END_REF][START_REF] Marrack | Towards an understanding of the adjuvant action of aluminium[END_REF][START_REF] Hornung | Silica crystals and aluminum salts activate the NALP3 inflammasome through phagosomal destabilization[END_REF].

Vaccine adverse effects include specific autoimmune diseases that are easily identified using well established criteria but also ill-defined conditions usually manifesting by symptoms such as myalgia, arthralgia, chronic fatigue and the appearance of autoantibodies [START_REF] Rosenblum | The common immunogenic etiology of chronic fatigue syndrome: from infections to vaccines via adjuvants to the ASIA syndrome[END_REF]. Yehuda Shoenfeld had the great merit to tentatively delineate a syndrome entitled "Autoimmune (Autoinflammatory) syndrome Induced by Adjuvants" (ASIA) in order to alert physicians when the above-mentioned symptoms do appear following vaccination [START_REF] Shoenfeld | %ASIA! -autoimmune/inflammatory syndrome induced by adjuvants[END_REF]. As noted previously by others [START_REF] Gherardi | Lessons from macrophagic myofasciitis: towards definition of a vaccine adjuvant-related syndrome[END_REF] he pointed out remarkable similarities between different syndromes linked to vaccine adjuvants, including the one associated with macrophagic myofasciitis, a lesion assessing longterm persistence of vaccine-derived alum into macrophages at site of previous immunization [START_REF] Gherardi | Aluminum inclusion macrophagic myofasciitis: a recently identified condition[END_REF] and the so-called Gulf war syndrome that developed in soldiers exposed to intense immunization protocols [START_REF] Hotopf | Role of vaccinations as risk factors for ill health in veterans of the Gulf war: cross sectional study[END_REF] with vaccines containing alum [START_REF] Shaw | Aluminum hydroxide injections lead to motor deficits and motor neuron degeneration[END_REF] and possibly squalene [START_REF] Asa | Antibodies to squalene in Gulf War syndrome[END_REF]. Yehuda Shoenfeld also noted similarities with siliconosis, a disease complex observed in patients with leaky breast silicone implants which was long attributed to deleterious adjuvanticity of silicone particles [START_REF] Miyoshi | Hypergammaglobulinemia by prolonged adjuvanticity in man disorders developed after augmentation mammoplasty[END_REF][START_REF] Shoaib | Human adjuvant disease: presentation as a 414 multiple slcerosis-like syndrome[END_REF]. He unified, under the heading of ASIA, the 3 syndromes with other post-vaccinal events, and proposed a set of major and minor diagnostic criteria as a guide for characterization and follow-up of these conditions [START_REF] Shoenfeld | %ASIA! -autoimmune/inflammatory syndrome induced by adjuvants[END_REF]. These criteria still need validation but, notably, ASIA grab the attention of the international medical community rapidly after its delineation [START_REF] Meroni | Autoimmune or auto-inflammatory syndrome induced by adjuvants (ASIA): old truths and a new syndrome?[END_REF][START_REF] Cervera | ASIA': a new systemic autoimmune syndrome?[END_REF][START_REF] Hughes | Foreword-Special issue of Lupus on ASIA[END_REF] pointing to a need in the field [START_REF] Gherardi | Aluminum inclusion macrophagic myofasciitis: a recently identified condition[END_REF][START_REF] Vasey | Where there!s smoke there!s fire: the silicone breast implant controversy continues to flicker: a new disease that needs to be defined[END_REF]. In addition to clinical manifestations, it could prove very useful to identify biomarkers that may either help unify the ASIA concept in the future or, alternatively, point out subtle differences among its constitutive syndromes. In the present study we performed extensive cytokine screening in serum of patients with alum-related MMF. This allowed us to detect selective increase of the monocyte chemotactic protein 1/ C-C motif ligand 2 (MCP-1/CCL2), pointing out this chemokine as a first candidate biomarker to be tested in other ASIA subsets.

PATIENTS AND METHODS

1-The MMF Cohort. Analysis of our files yielded 583 patients with stereotypical MMF

detected by muscle biopsy from 1994 to 2012. Patients were either investigated and biopsied in our Reference Centre at Créteil (n=352), or referred for follow-up or investigation after MMF detection by the other myopathologic centers located in Bordeaux, Marseille, and Paris (n= 231) that described the entity with us [START_REF] Gherardi | Macrophagic myofasciitis: an emerging entity[END_REF]. For the present study, prevalence of core clinical symptoms were determined in 105 unselected consecutive patients, 25 males and 80 females (age range 12-64 years), who benefited from systematic evaluation of their brain function in the Cognition Center of Créteil, using a comprehensive battery of neuropsychological tests detailed elsewhere [START_REF] Couette | Long-term persistence of vaccine-derived aluminum hydroxide is associated with chronic cognitive dysfunction[END_REF][START_REF] Passeri | Long-term follow-up of cognitive dysfunction in patients with aluminium hydroxide-induced macrophagic myofasciitis (MMF)[END_REF]. The persistent MMF lesion consisted in stereotyped accumulations of large particle-loaded macrophages without multinucleated giant cell formation, associated with T and B cells. Granular inclusions were hematein-positive in cryostat sections, grey after formalin fixation, PAS positive, and green-fluorescent on Morin stain for aluminium (Figure 1). MMF was always detected in the deltoid muscle with a median delay of 6.5 years elapsed from the last alum-containing vaccine administration and the biopsy. Patients had received a combination of various vaccines within 10 years before biopsy: anti-hepatitis B virus vaccine administered in 3 shots was implicated in a vast majority of patients (89.7% of patients), followed by tetanus toxoid (67.9%), and anti-hepatitis A virus (14.1%) vaccines. Main symptoms declared by patients were arthromyalgias (96/105: 91.4%), fatigue (88/105: 83.8%), and cognitive alterations affecting attention and memory (102/105: 97.1%). Neuropsychological evaluation confirmed abnormal cognitive status in 93/105 (88.6%) of patients, with a pattern suggestive of subcortico-frontal alterations in 89/105 (85%). Onset of these manifestations was typically delayed from immunization. Thus 3 Major Criteria proposed for ASIA [START_REF] Shoenfeld | %ASIA! -autoimmune/inflammatory syndrome induced by adjuvants[END_REF] were met, including (1) "exposure to adjuvant prior to clinical manifestations" (i.e. aluminium oxyhydroxide), with a previously reported median time of 7 months (range 0.5-84) elapsed from last vaccine administration to initial systemic symptoms, and 11 months (range 0-72) for first myalgia [START_REF] Gherardi | Aluminum inclusion macrophagic myofasciitis: a recently identified condition[END_REF]; (2) "the appearance of typical clinical manifestations" including: myalgia; arthralgias; chronic disabling fatigue; urefreshing sleep or sleep disturbances; cognitive impairment and memory loss [START_REF] Authier | Chronic fatigue syndrome in patients with macrophagic myofasciitis[END_REF]; and, less often, overt neurological manifestations associated with demyelination [START_REF] Authier | Central nervous system disease in patients with macrophagic myofasciitis[END_REF] ; and (3) "typical biopsy of involved organs" , i.e. MMF [START_REF] Gherardi | Macrophagic myofasciitis: an emerging entity[END_REF] . Notably, in contrast to the leaky silicone breast implant-associated ASIA, the criterion "removal of inciting agent induces improvement" could not be met in our patients subjected to multiple vaccine shots. A variable proportion of MMF patients also met minor criteria for ASIA, including (1) "the appearance of autoantibodies", (2) "other clinical manifestation" such as dyspnea, (3) "specific HLA", i.e. HLADRB1; and (4) "evolvement of an autoimmune disease", i.e. mainly multiple sclerosis thyroiditis, and autoimmune myopathies [START_REF] Gherardi | Aluminum inclusion macrophagic myofasciitis: a recently identified condition[END_REF].

RESULTS

Multiplex cytokine serum analysis showed significantly increased mean CCL2/MCP-1 levels in MMF patients with ASIA compared with healthy subjects (median = 57.7 vs.

37.9 pg/mL; mean = 64.5+/-32.9 vs. 38.2+/-11.5 pg/ml, p<0.0001) (Fig. 2). Increase was unique to CCL2/MCP-1, as the other tested inflammatory cytokines, chemokines, and growth factors were either similar (IFN#, TNF", IL5, IL6, IL7, IL13, G-CSF, PDGFbb, VEGF, CCL3, CCL4, CCL5, CCL7, CCL11, CXCL8, CXCL1, CXCL10, and CX3CL1, p= NS), or even decreased (IL1!, IL1ra, IL4, IL10, IL12p70, IL17, FGFb, all p <0.05) in MMF patients compared to healthy controls. The CCL2/MCP1 variation could not be attributed to smoking habits (p=NS for all tested molecules). Inflammatory controls showed no difference for serum CCL2/MCP-1 levels (median 47.9; mean 52.8+/-25.9pg/mL) compared with both MMF and healthy control patients (p=NS). In contrast, inflammatory controls had significantly increased mean levels of several inflammatory cytokines compared to MMF patients (IL1b, IL5 and CCL3/MIP1a, all p< 0.04), and to healthy controls (CCL3/MIP1a, p<0.02). Similarly to MMF patients, inflammatory controls had decreased mean IL17 and FGFb levels compared to healthy controls (Fig. 3).

DISCUSSION

The salient finding of this crossectional screening of 30 cytokines was the selective increase of CCL2/MCP-1 serum levels in MMF patients with ASIA compared to ageand sex-matched healthy individuals. Notably, similar increase of circulating CCL2/MCP1 levels was not detected in inflammatory controls, whereas other inflammatory molecules, including IL1!, IL5 and CCL3/MIP1a levels, were higher in this group. Furthermore, MMF patients had lower mean levels of IL1b and IL1ra compared to healthy controls, pointing to CCL2/MCP-1 as a marker relevant to the pathophysiology of MMF rather than a non-specific inflammatory marker. They also had lower mean levels of IL 4 and IL10 of unknown significance.

MMF assesses long-term persistence of alum crystal agglomerates within macrophages at the site of previous i.m. immunization [START_REF] Gherardi | Macrophagic myofasciitis lesions assess long-term persistence of vaccinederived aluminium hydroxide in muscle[END_REF]. Such a biopersistence of alum is detected in the deltoid muscle a small minority of alum-adjuvanted vaccine receivers, strongly suggesting the influence of genetic factors, as previously documented in rats in which the lesion size conspicuously varies according to the genetic background [START_REF] Authier | AlOH3-adjuvanted vaccine induced macrophagic myofasciitis in rats is influenced by the genetic background[END_REF], The rarity of ASIA in the general population may also reflect specific gene-environment interactions [START_REF] Shoenfeld | %ASIA! -autoimmune/inflammatory syndrome induced by adjuvants[END_REF]. Adjuvant effects of alum remain incompletely understood [START_REF] Exley | The immunobiology of aluminium adjuvants: how do they really work?[END_REF]. In vitro studies have shown that alum acts mainly on macrophages and monocytes, skewing monocyte differentiation toward inflammatory DCs (iDCs) [START_REF] Seubert | The adjuvants aluminium hydroxide and MF59 induce monocyte and granulocyte chemoattractants and enhance monocyte differentiation toward dendritic cells[END_REF]. Whereas classical iDCs do not secrete CCL2, iDCs differentiating in the presence of alum produce high amounts of CCL2 from day 4 onward, and this does not appear to simply reflect particle uptake [START_REF] Seubert | The adjuvants aluminium hydroxide and MF59 induce monocyte and granulocyte chemoattractants and enhance monocyte differentiation toward dendritic cells[END_REF]. Unlike LPS which induces a wide proinflammatory response, alum does not upregulate other cytokines, if one excepts CCL4, and, therefore, likely acts through a mechanism distinct from that of danger signals [START_REF] Seubert | The adjuvants aluminium hydroxide and MF59 induce monocyte and granulocyte chemoattractants and enhance monocyte differentiation toward dendritic cells[END_REF]. In addition to monocyte/macrophage lineage cells, CCL2 is produced by many cell types, including for example endothelial, fibroblasts, smooth muscle, astrocytic, and microglial cells [START_REF] Deshmane | Monocyte chemoattractant protein-1 (MCP-1): an overview[END_REF]. CCL2 acts through its receptor CCR2, which expression is restricted to certain type of cells, such as monocytes, T and NK cells [START_REF] Deshmane | Monocyte chemoattractant protein-1 (MCP-1): an overview[END_REF].

CCL2 is the main chemokine responsible for recruiting inflammatory monocytes to tissues, as previously documented in both injured muscle [START_REF] Brigitte | Muscle resident macrophages control the immune cell reaction in a mouse model of notexin-induced myoinjury[END_REF] and brain [START_REF] Izikson | Resistance to experimental autoimmune encephalomyelitis in mice lacking the CC chemokine receptor (CCR)2[END_REF][START_REF] Mildner | Microglia in the adult brain arise from Ly-6ChiCCR2+ monocytes only under defined host conditions[END_REF].

Certain individuals are genetically programmed to produce copious amounts of MCP-1, and these individuals were reported to be at higher risk of developing severe forms of diseases implicating macrophages in their pathophysiology, including atherosclerosis [START_REF] Brenner | Cytokine polymorphisms associated with carotid intima-media thickness in stroke patients[END_REF], tuberculosis [START_REF] Flores-Villanueva | A functional promoter polymorphism in monocyte chemoattractant protein-1 is associated with increased susceptibility to pulmonary tuberculosis[END_REF], sarcoidosis [START_REF] Navratilova | The MCP-1-2518 (A to G) single nucleotide polymorphism in Czech patients with pulmonary sarcoidosis: association with Löfgren's syndrome[END_REF], pneumoconiosis [START_REF] Nadif | Polymorphisms in chemokine and chemokine receptor genes and the development of coal workers' pneumoconiosis[END_REF] , lupus erythematosus [START_REF] Tucci | Strong association of a functional polymorphism in the monocyte chemoattractant protein 1 promoter gene with lupus nephritis[END_REF][START_REF] Malafronte | Association of the MCP-1 -2518 A/G polymorphism and no association of its receptor CCR2 -64 V/I polymorphism with lupus nephritis[END_REF], and HIV-associated dementia [START_REF] Gonzalez | HIV-1 infection and AIDS dementia are influenced by a mutant MCP-1 allele linked to increased monocyte infiltration of tissues and MCP-1 levels[END_REF]. In light of these reports, selective increase of CCL2/MCP1 serum levels in MMF patients suggests that functional polymorphisms in the promoter of the MCP-1 gene should be screened as possible genetic susceptibility factors to develop the condition after alum administration. For example, CCL2 expression is associated with the development of polarized Th2 immune responses [START_REF] Gu | Control of TH2 polarization by the chemokine monocyte chemoattractant protein-1[END_REF], and both CCL2/MCP1 and Th2 cells increase in blood during normal ageing [START_REF] Mansfield | Normal ageing is associated with an increase in Th2 cells, MCP-1 (CCL1) and RANTES (CCL5), with differences in sCD40L and PDGF-AA between sexes[END_REF]. Th2 cytokines may disturb the xeno/autophagy process [START_REF] Harris | T helper 2 cytokines inhibit autophagic control of intracellular Mycobacterium tuberculosis[END_REF] used by macrophages to handle both infectious and non-biodegradable particles [START_REF] Eidi | Drug delivery by polymeric nanoparticles induces autophagy in macrophages International[END_REF]. Here again, genetic susceptibility factors further impacting the xeno/autophagic machinery should be searched in MMF patients in light of recent reports linking polymorphisms in certain genes with inability of cells to clear out infectious particles in Crohn disease [START_REF] Rioux | Genome-wide association study identifies new susceptibility loci for Crohn disease and implicates autophagy in disease pathogenesis[END_REF], a chronic inflammatory gastrointestinal disease in which aluminium may also play a role [START_REF] Lerner | Aluminum as an adjuvant in Crohn's disease induction[END_REF].

In a prelimary series of mouse experiments, we observed that nanomaterials injected into muscle, including latex beads and fluorescent alum surrogates, are quickly captured by monocyte lineage cells and disseminate in the body in a CCL2-dependent manner, as demonstrated by gain and loss of CCL2 function experiments [START_REF] Khan | Nanosized particles injected into muscle slowly disseminate within phagocytes and accumulate in brain through MCP1-driven mechanism[END_REF].

Particles are transported by MO-lineage cells to the draining lymph nodes, then reach blood and spleen, and, after a long delay and at a much lower rate, gain the brain where they accumulate and persist mainly within microglial cells. Alum adjuvant injected twice into mice at human equivalent doses to mimick vaccine exposure of individuals with Gulf war syndrome results in conspicuous neurotoxic effects, as assessed by behavioural and motor testings, and by neuropathologic evidence of microglial activation and motoneuronal death [START_REF] Shaw | Aluminum hydroxide injections lead to motor deficits and motor neuron degeneration[END_REF]. CCL2 is constitutively present in the central nervous system where it acts both as chemoattractant and neuromodulator [START_REF] Mélik-Parsadaniantz | Chemokines and neuromodulation[END_REF].

CCL2 is an important mediator of the neuroinflammatory responses in chronic neurologic disorders such as multiple sclerosis [START_REF] Mahad | The role of MCP-1 (CCL2) and CCR2 in multiple sclerosis and experimental autoimmune encephalomyelitis (EAE)[END_REF], Alzheimer!s disease [START_REF] Sokolova | Monocyte chemoattractant protein-1 plays a dominant role in the chronic inflammation observed in Alzheimer's disease[END_REF] and human HIV-1-associated dementia [START_REF] Dhillon | Roles of MCP-1 in development of HIV-dementia[END_REF]. Several CCL2 neuromodulatory effects have been documented. For example CCL2 protects against toxicity of trace metals such as methylmercury at low dose, protection ceasing and marked toxicity ensuing when exposure becomes chronic or when too much CCL2 is released by glial cells [START_REF] Godefroy | The chemokine CCL2 protects against methylmercury neurotoxicity[END_REF].

Consistently, chronic CCL2 overexpression is associated with delayed encephalopathy and impaired microglial function [START_REF] Huang | Chronic expression of monocyte chemoattractant protein-1 in the central nervous system causes delayed encephalopathy and impaired microglial function in mice[END_REF]. Moreover, CCL2 is crucially involved in the development of pain [START_REF] Menetski | Mice overexpressing chemokine ligand 2 (CCL2) in astrocytes display enhanced nociceptive responses[END_REF][START_REF] Jung | Monocyte chemoattractant protein-1 functions as a neuromodulator in dorsal root ganglia neurons[END_REF][START_REF] Thacker | CCL2 is a key mediator of microglia activation in neuropathic pain states[END_REF], CCL2-induced central sensitization to pain being mediated by increasing the activity of NMDA receptors in dorsal horn neurons [START_REF] Gao | neuronal-glial interactions, and central processing of neuropathic pain[END_REF]. In keeping with these effects, circulating CCL2/MCP1 elevation was previously reported in patients with fibromyalgia, in combination with IL8 elevation [START_REF] Ang | MCP1 and IL8 as pain biomarkers in fibromyalgia: a Pilot Study[END_REF], which was not detected in the present study, and with mild cognitive impairment [START_REF] Galimberti | Serum MCP-1 levels are increased in mild cognitive impairment and mild Alzheimer's disease[END_REF]. Consistently, impairment of the multi-domain type [START_REF] Passeri | Long-term follow-up of cognitive dysfunction in patients with aluminium hydroxide-induced macrophagic myofasciitis (MMF)[END_REF]. We suggest that circulating MCP-1 levels should be tested as a possible biomarker in the other ASIA subsets.

2 - 3 -

 23 The cytokine study subjects and controls. Chemokine serum levels were determined in 44 adult MMF patients (M/F sex ratio 16/28, mean age 46.2+1.6yrs) and in 44 normal age and sex-matched controls (M/F sex ratio 16/28, mean age 48.5+0.7yrs, p=NS). According to the principles expressed in the Helsinki Declaration, all patients gave written individual informed consent to participate to the study and approval was obtained from French institutional review boards (registration number 2010-A01338-31 at AFSSAPS, and 11-042 at CPP-Ile-de-France IX). MMF had received 1 to 17 i.m. alum-containing vaccine administrations (mean 5.2) within the past 10 years. Inclusion criteria were as follows: (1) onset of clinical manifestations posterior to alum-containing vaccine administration; (2) ASIA symptoms, including diffuse arthro-myalgias lasting >6 months and/or profound fatigue lasting >6 months and/or disabling cognitive deficiency affecting attention and memory; (3) histologic MMF in deltoid muscle detected by biopsy at least 18 months after the last vaccine; (4) no other disease that could explain the manifestations. Similarly to MMF patients, healthy controls included both smokers (n=16, 1 paquet per day) and non-smokers (n=28). Ten unmatched patients with various types of inflammatory myopathies and neuropathies served as inflammatory controls. Cytokine screening. Quantitation of 30 cytokines was performed on serum kept frozen at -80°C until analysis by the immunomonitoring platform of the Vaccine Research Institute of Créteil. Analysis used in combination the LuminexR technology and conventional ELISA. The Bio-Plex Pro Human Cytokine 27-plex Assay from Bio-Rad (M50-0KCAF0Y) was used following the procedure recommanded by the manufacturer. This assay simultaneously detects IL-1", IL-1ra, IL-2, IL-4, IL-5, IL-6, IL-7, IL-8 (CXCL8), IL-9, IL-10, IL-12 (p70), IL-13, IL-15, IL-17, basic FGF, G-CSF, GM-CSF, IFN-#, IP-10 (CXCL10), MCP-1 (CCL2), MIP-1$ (CCL3), MIP-1" (CCL4), RANTES (CCL5), eotaxin (CCL11), PDGF-BB, TNF-$, and VEGF. The 27plex was enriched with two simplex sets for MCP-3 (CCL7) and KC/Gro-" (CXCL1), and an ELISA kit for the detection of human Fractalkine (CX3CL1) purchased from R&D (quantikine DCX-310). Controled intraindividual variations were <5%. Statistical analyses used Mann-Whitney U-test. p <0.05 was considered significant.

Figure 1 :

 1 Figure 1 : Morin stained section from muscle biopsy with macrophagic myofasciitis showing rounded alum agglomerates within cells. (magnification x250)

Figure 2 :

 2 Figure 2 : Serum CCL2-MCP1 levels in MMF patients compared to control nonsmoking and smoking healthy subjects and inflammatory patients. CCL2-MCP1 is selectively increased in MMF patients and unchanged in inflammatory patients.

Figure 3 :

 3 Figure 3 : Serum cytokine levels in MMF patients, control healthy subjects and inflammatory patients. MIP-1", 1L-1! and IL-5 are significantly increased in inflammatory patients but similar to healthy controls in MMF patients.

Figure 1 ! 2

 12 Figure 1!

Acknowlegments : This work has benefited from research funding from Région Ile-de-France through a programme PICRI (Partenariat Institutions-Citoyens pour la Recherche et l!Innovation) "Recherche de polymorphismes dans les genes codant pour des facteurs inflammatoires (chimiokines) dans la myofasciite à macrophages". We thank Yasmine Baba-Amer (Technician), and Dr Bijan Ghaleh (Plateforme de Ressources Biologiques de Créteil) and for collecting sera.


P Diener

P Rodière

J D Fletcher

A Carrington

J P Brison

R Prozorov

T Olheiser

R W Giannetta

Anisotropic superconductivity in NbSe 2 probed by magnetic penetration depth

Keywords: NbSe 2, penetration depth, superfluid density PACS:

NbSe2 shows coexistence of a charge density wave (TCDW ∼32K) with a superconducting state below T =7.2K. Recent ARPES measurements revealed different values of the superconducting gap on the main sheets of the Fermi Surface. These results suggest a multigap superconductivity such as in MgB2. The temperature dependence of the magnetic penetration depth (λ(T ) down to Tc/16 has been measured on high quality single crystals in the Meissner state. A strong increase of the in-plane penetration depth is observed, signaling the presence of low lying excitations. Given the relative contributions of each Fermi surface sheet, these measurements indicate that a reduced gap is not necessarily only found on the small Se sheet as suggested by the ARPES measurements. These results are discussed in a framework of multigap superconductivity.

The hexagonal dichalcogenides NbSe 2 has attracted a large interest in the last few decades for the coexistence of an incommensurate charge density wave with a superconducting state below T c =7.1K [1]. Furthermore, the superconducting state shows unusual properties, which can not be explained by an isotropic BCS weak coupling model. For example, the electronic specific heat has already shown the presence of a reduced energy gap [2]. Recently, NbSe 2 has been revisited in the light of multigap superconductivity. ARPES measurement suggested that the low energy excitation gap is due to the Se p-band which has a small electron-phonon coupling constant [3]. A directional probe combined to the particular anisotropy of the Fermi Surface of NbSe 2 allows us to test the excitation gap on the different sheets.

In this paper, we present high sensitivity measurements of the change of the in plane and c-axis temperature dependence of the magnetic penetration depth in the Meissner state (respectively ∆λ a and ∆λ c ). The detailed results are published elsewhere [4]. Tsukuba, Bell Lab) have been measured in three different laboratories (Grenoble, Bristol, Urabana Champain respectively). Crystals of thickness t were grown with large flat layers perpendicular to the c-axis . Each side of the samples were cut. Samples from different batches have a RRR between 33 and 70 (see inset fig. 1). No drastic change with sample quality has been observed.

Experimental method

When the magnetic field is applied along the caxis, the supercurrents are flowing only in the basal plane. However, for a magnetic field applied perpendicular to the basal plane, both, a and c-axis directions are probed. For a sample of rectangular shape with a section (⊥ H) of width w and thickness t, the frequency shift is proportional to ∆λ a + t w ∆λ c . To extract the out-of plane penetration depth the aspect ratio of a sample is changed by cutting.

Results

In fig. 1 the low temperature dependence of the in-plane penetration depth is shown. All the curves are fitted with the approximated expression (valid for kT < T c /3): where ∆ 0 is the superconducting gap at T = 0K. We find ∆ 0 =1.1±0.1kT c , less than the value expected for a weak coupling BCS gap (∼1.76kT c ). In fig. 2 the low temperature dependence of the c-axis penetration depth is fitted for T < 2K with the same expression. A gap of ∆ 0 = 1.3 ± 0.1kT c is measured.

∆λ i (T ) ≃ λ i (0) π∆ 0 2T exp - ∆ 0 T (1)
The experimental results have to be compared with the calculated Fermi surface. The Fermi surface of NbSe 2 is formed of 2 cylinders along the c-axis from the 4d-electrons of the Nb and also a small flat pancake around the center of the Brillouin zone from the p-electron of the Se. This sheet contributes only to 2% to the total in-plane superfluidity but 85% to the out-plane superfluidity [5]. So the reduced energy gap measured by in-plane penetration depth is on one or more of the quasi-2D Nb sheets. Moreover, ∆λ c shows that the superconducting gap associated to the Se sheet is not smaller than the smallest gap of the quasi-2D Nb band. These results are in strong contrast with previous measurements [3].

2 Fig. 1 .

 21 Fig. 1. Temperature dependence of the in-plane penetration depth for samples from different sources ( measured at Bristol ✷, Grenoble •, Urbana ⋄). The field configuration is H c for Bristol's samples with an aspect ratio of 40. The contribution from the c-axis is negligible. For the other sample H is perpendicular to the c-axis. For clarity the data are offset. The lines are a fit with a superconducting gap of ∆=1.1±kTc. Inset, temperature dependence of the in plane resistivity for a single crystal from the same batch as Grenoble's samples.

Fig. 2 .

 2 Fig. 2. Temperature dependence of the out-plane penetration depth calculated from a change of the aspect ratio (see text) (B2 measured at Bristol, G2 data from Grenoble is offset).


Mehdi Zaamoune
email: mehdi.zaamoune@irstea.fr

Sandro Bimonte
email: sandro.bimonte@irstea.fr

François Pinet
email: francois.pinet@irstea.fr

Philippe Beaune
email: pbeaune@agaetis.fr

A New Relational Spatial OLAP Approach For Multi-resolution and Spatio-multidimensional Analysis of Incomplete Field Data

Keywords: Spatial OLAP, incomplete continuous field, multi-resolution geographic data

Integrating continuous spatial data into SOLAP systems is a new research challenge. Moreover, representation of field data at different scales or resolutions is often mandatory for an effective analysis. Thus, in this paper, we propose a logical model to integrate spatial dimensions representing incomplete field data at different resolutions in a classical SOLAP architecture.

INTRODUCTION

Spatial Data Warehouse (SDW) and Spatial OLAP (SOLAP) systems play an important role in helping decision-makers obtain the maximum benefits of these large amounts of geographic data [START_REF] Bédard | Fundamentals of Spatial Data Warehousing for Geographic Knowledge Discovery[END_REF]). These technologies extend Data Warehouse (DW) and OLAP systems to integrate spatial data with warehoused classical data to achieve the on-line analysis of large georeferenced data sets. SOLAP systems integrate advanced OLAP and Geographic Information Systems (GIS) in a unique framework usually based on the relational storage (i.e. Oracle, etc.) of spatial data according to the vector model, and their analysis through SOLAP operators (Spatial Roll-Up, Spatial Slice, etc.) implemented by the SOLAP server (e.g. Map4Decision, etc.) and visualized by means of tabular, graphical and cartographic displays [START_REF] Gomez | A Generic Data Model and Query Language for Spatiotemporal OLAP Cube Analysis[END_REF]. SDW are modeled according to the spatio-multidimensional model that extends the traditional multidimensional model to define spatial dimensions (i.e. analysis axes with spatial attributes) and spatial measures (i.e. analysis subjects) that integrate geographic information using the vector model [START_REF] Bédard | Spatial On-Line Analytical Processing (SOLAP): Concepts, Architectures and Solutions from a Geomatics Engineering Perspective[END_REF]. SOLAP technology can be applied in different domains (e.g. archeology, public health, etc.).

Geographic information can be represented by two models, depending on the nature of data: discrete (vector) and continuous field [START_REF] Mennis | Cubic Map Algebra functions for spatio-temporal analysis[END_REF]. Continuous fields (also called continuous spatial data) represent physical phenomena that continuously change in space [START_REF] Paolino | Integrating Discrete and Continuous Data in an OpenGeospatial-Compliant Specification[END_REF], for example the temperature, population, etc. Two representations of field data have been proposed: incomplete and complete [START_REF] Paolino | Integrating Discrete and Continuous Data in an OpenGeospatial-Compliant Specification[END_REF]. Incomplete representations store a sample of points and need additional functions to calculate the field in nonsampled areas (e.g. grid of points, TIN, etc.) (e.g. Figure 2). Complete representations associate estimated values to regions and assume that these values are valid for each point in the regions (e.g. raster). For those representations some ad-hoc analysis operators have been defined that allow a point by point analysis (i.e. map algebra [START_REF] Mennis | Cubic Map Algebra functions for spatio-temporal analysis[END_REF]). Representation of geographic data at different scales or resolutions (e.g. Figure 2b) is mandatory for an effective analysis of spatial complex phenomena since it represents a geovisualization method [START_REF] Camossi | Adaptive Management of Multigranular Spatio-Temporal Object Attributes[END_REF]. Consequently, these resolutions or scales represent decision-makers analysis needs that should be explicitly represented in any data and query model. Indeed, in the context of Geographic Information Systems and Spatial Databases Management Systems (SDBSM), several works addresses this issue by proposing conceptual, logical and physical data models and analysis techniques [START_REF] Parent | Conceptual Modelling for Traditional and Spatio-temporal Applications[END_REF].

Motivated by the important analysis capabilities offered by the continuous field representation of geographic data when integrated in SOLAP systems (visualization, querying, etc.) recently some works investigated the extension of the spatiomultidimensional model and SOLAP operators with complete and incomplete field data (cf. Section 6). In the same way, handling multi-resolutions of spatial data into spatial multidimensional models has been proposed in few works [START_REF] Yvan | Modeling multirepresentations into spatial data warehouses : A UML-Based approach[END_REF] [START_REF] Gascueña | A MULTIDIMENSIONAL METHODOLOGY WITH SUPPORT FOR SPATIO-TEMPORAL MULTIGRANULARITY IN THE CONCEPTUAL AND LOGICAL PHASES[END_REF] that propose conceptual models to represent SDW with several representations (scales, resolutions, etc.) of spatial dimensions and measures.

However, to best of our knowledge existing works concerning field data and multi-resolutions lack of a complete implementation in a full-featured SOLAP architecture, or in other terms they do not propose a coupled relational and SOLAP server model for a generic SOLAP architecture allowing (i) the map algebra operators, (ii) the multi-resolution, and (iii) a continuous view of the field.

In order to handle the spatio-multidimensional analysis of incomplete regular grid field data at different resolutions, we propose in this paper: (i) a specific logical model, extending the well-know relational star schema; (ii) and some new MDXbased defined functions. We validate our proposal using a real case study concerning the odor monitoring, and we provide some experiments showing the feasibility also in terms of storage and computation performances.

MODELING AND ANALYSIS REQUIREMENTS

In order to show our proposal, we present a case study based on data issued from the monitoring of urban odor. proportion the value of a neighbour impact on the value of the point to be estimated (Figure 1). Finally, as stated in the previous section, since visualization of spatial data at different resolutions is mandatory for the exploration/analysis process, decision-makers should be able querying spatial warehoused data at different resolutions. It is very important to note that for each spatial phenomenon a set of useful known resolutions exist, so they could be predefined according to data and users needs. Moreover, in order to calculate values at finer resolutions spatial interpolation functions as previously described can be used.

To summarize, spatio-multidimensional analysis of field data implies: supporting (i) OLAP classical operators as Map Algebra, (ii) continuous view of spatial data, (iii) spatial slice operators using field data, and (iv) visualizing and querying data at different predefined resolutions.

Spatio-multidimensional model for incomplete field data

In this section we describe our spatiomultidimensional model for handling incomplete fields at different resolutions. Our model extends the classical spatio-multidimensional models to generate the continuity of the phenomena over the studied area, and represents pre-defined levels of resolution.

In particular, a "Cube" is composed of "Facts" and "Dimensions". A "Dimension" is composed of "Hierarchies", which are composed of "Levels". A "Level" can be spatial or conventional. This means that it can contain "Spatial attributes" (e.g. points, etc.), or contain only alphanumerical attributes respectively. "Facts" is composed of "Spatial Measures" or "Conventional Measures". Moreover, our extension defines a "Field level" as a special type of spatial level where each member has a geometric attribute (e.g. point), a "neighbourhood relationship" association, and a re which it belongs.

(a)

(b) (c) (d)
Figure 2: a) local map algebra incomplete field. b) multiresolution r incomplete field c) querying an un querying an unsampled zone Indeed, as described in the pr spatial interpolation function uses which depend on the function (e.g. uses a 2*2 grid), to estimate the Thus, in our approach values are measures, the "Neighbourhood rela detailed "Field level" to its neighb detailed level. As soon as the v resolution "Field level" is required i found through the "Neighbour However, this type of relatio implemented in different ways, de intended purpose as shown in the our case, we want to estimate the v (x, y) of a Field level, so neighbou the fly by the relation "Neighbourh the case of a change in the level o members of a high resolution (e.g predefined and therefore, th neighbours, which belongs to 100*100, can be pre-stored as attrib

As described in the model, contain several field levels r phenomena at different resolutions.

Incomplete field

In order to implement field levels we have defined a GeoMDX user-defined function that represents a spatial interpolation as:

NumericType InterpolatePoint(Geometry)

This function takes as input a geometry (point) and returns a numerical value, which is a derived measure in the OLAP model, representing an estimated value calculated using the neighbourhood values of the point given in input. Thus, let us suppose that we want to retrieve a value of the field in a location whose geometric property is set to the geometric coordinates POINT (-72.1235 42.3521). Then in order to answer to that need using the Bilinear interpolation function, decision-makers have to simply use a GeoMDX function in the following way: InterpolatePoint(POINT(-72.1235 42.3521)).

Thus, the function will look for the neighbours of the point given as a parameter, in the field level ([Field].[res100]), on the basis of the distance, and then find neighbours' respective values in the fact table, evaluate the value of the point to estimate using these values, and then return an estimated derived measure. Here is an example query that uses the "InterpolatePoint" function:

Query 2: select a field member's value at coordinates (721148 3140020) for the year 2012.

With member [Measures].[value] as' InterpolatePoint(ST_GeomFromText(" POINT (721148 3140020)"))' SELECT [Measures].[value]ON ROWS, [time].[2012] ON COLUMNS FROM [odorCube]

Figure 4: Classical star schema integrating regular grid of points Note that generally MDX allows defining userdefined functions in several programming languages (i.e. Java, .NET, etc.) depending of the OLAP Server used. In this work we have used a Java-based implementation in GeoMondrian (see Sec. 5). In particular, the interpolation is done using an existing interpolation Java API "javax.media.jai api" (JAI).

In this way we achieve the continuous view of field data using incomplete fields as stated in Section 2.

Multiresolution

Theoretically, we can measure a value of a field at every position inside a geographic space. However, not all resolutions are necessarily relevant. Indeed, according to the type of analysis performed by the user, a more or less detailed resolution can be requested. The multiresolution is an approach that consists in defining resolution levels likely to improve the rendering of the requests made by the user. To model an incomplete field at several resolutions in a multi-dimensional model, we propose two Approaches based on the "Classical Star Schema": The "field aggregation star-schema" approach and the "field interpolation star schema" approach.

Field Aggregation Star-Schema Approach

Based on the star schema model previously described, we propose a logical schema where the spatial dimension presents different field levels at different resolutions (fig. 5-a). This model extends the spatial dimension of figure 4 Each level of the field dimension is composed of an identifier and a geometry representing a point. The fact table is associated, classically, to the most detailed level of the field dimension.

In this way, decision-maker can explore warehoused field data at different resolutions during the same analysis MDX-based session. Only need to change the level of resolution in the query to change the level of details of the result. Using this approach, we use in an MDX query, the appropriate level of resolution of the field dimension as in the following where the Query 1 becomes:

Query 3: select average odor for each field member at the 400*400 resolution during 2012

SELECT [Field].[res400]. Members ON ROWS, {[time].[2012]} ON COLUMNSFROM [odorCube] WHERE [Measures].[value]

Field Interpolation Star-Schema Approach

As stated in Section 2, in order to provide field data at finer resolutions, spatial interpolation methods can be used. Then, here we propose a variation of the previously proposed schema for handling multiple field resolution levels, by associating the fact table to the field at less detailed resolution as shown on figure 5-b. In our approach moving from fact table values to finer spatial members' values implies applying spatial interpolation functions. Note that this approach is possible only when dealing with spatial data, because according to the Tobler law geographical position of data can be used for estimating missing values. We have implemented a GeoMDX function in the same way of the function defined in Section 4.1:

Numeric-type InterpolateBilinear (Field Member)

However, this function, named "InterpolateBilinear" is prepared to receive as input a field level member instead of geometry and return an interpolated value of this member. We can also see that in this case, the neighbors of each member of a higher resolution than the original one are also stored in the "Field" (Neighbours2, Neighbours3), since members of each resolution are pre-defined in advance, but their values are not since they depend on other dimensions. Calling this function as follows:

InterpolateBilinear ([Field].[res400].CurrentMember) in the formula of a derived measure, allows to find the values of all the members of the level "res400" (incomplete field at a 400*400 resolution) using their neighbors "Neighbors3" . Thus, the query 3 can be performed as follows:

SELECT {[Field].[res400].Members} ON ROWS, {[time].[2012]} ON COLUMNS FROM [odorCube] Where [Measures].[EstimatedValue]
While in the multidimensional SOLAP schema, the "InterpolateBilinear" function is called in the "EstimatedValue" calculated measure formula as:

formula="InterpolateBilinear([Field].[ res400].CurrentMember)"

As we can see in the previous query, the call of the calculated measure enables to find the values at a given scale transparently to the decision maker as a classical aggregation (SQL). This approach is motivated by performance issues as described in the next section.

Experimentations

In this section we detail the performances of the two approaches proposed in Section 4.2 (FASS and FISS) in terms of storage and time computation. The computer used for the following tests has the following configuration: processor Intel® core ™ i3 2,20 GHz, RAM 4 Go, Operating system Windows 7 professional, System OS 64 bits.

In particular, spatial data is stored in PostGIS Spatial DBMS. PostGIS is an open source software that adds support for geographic objects to the PostgreSQL object-relational database. PostGIS follows the Simple Features for SQL specification from the Open Geospatial Consortium (OGC); we use GeoMondrian as a SOLAP server; and JPivot as a client. GeoMondrian is an Open Source Spatial Online Analytical Processing Server.

In order to test our proposal we define different cases where the spatial dimension presents: one field level at the 100*100 resolution; two levels at the resolutions 100*100 and 200*200; and finally three levels at the resolutions 100*100, 200*200 and 400*400. We also vary the size of the temporal dimension in order to understand impact of the spatial and non spatial dimension on performances.

Figure 6-a shows the size of the fact table measured in function of the number of spatial and temporal members (spatial finest resolution / temporal finest granularity) using the two approaches. We can easily see two important differences: i) the field aggregation approach is expensive in terms of storage than the field interpolation one since the latter stores only facts values at a less detailed spatial granularity, ii) in the field interpolation approach the size of the fact table only varies depending on the size of the non spatial dimensions. Thus, even increasing the size of the spatial dimension, the fact table does not change since it contains only measures related to the first level of resolution.

In order to evaluate computation performance we execute the queries previously cited, where we combine roll-up operation on non spatial dimensions, and spatial slice operators over different field resolutions. Figure 6-b represents the execution time of the query 3, which consists in generating values of the members at different resolutions taking into account different sizes of the time dimension. This figure shows a certain degree of approximation in execution time between the two approaches to a certain level. Beyond this level, we note that the gap widens considerably. Thus, minimizing storage and relations has allowed the field interpolation approach we propose to have better execution time than the field aggregation approach at all resolution levels (100*100, 200*200 and 400*400). Figure 9 shows that the execution time in the "field aggregation approach" increases depending on the number of spatial and temporal members, whereas in the "field interpolation approach", it increases mainly depending on the number of temporal members. Indeed the size of the spatial dimension does not influence much on performance, since there is no relationship between the fact table and the members who belong to high resolutions. a) Figure 6: a) Fact table size with time for Query 3 for the Field aggre 6

Related work

In order to integrate fields da model, (Ahmed and Miquel 20 multidimensional model for hand discrete fields, storing a sample of members, to create a discrete interpolated in the client-side continuity. [START_REF] Mchugh | Intégration de la structure matricielle dans les cubes spatiaux[END_REF] define dimensions handling fields as a squares (raster): "hybrid dimension dimension", "mixed matrix d "geometric matrix dimension". She "matrix cube" where facts are cel grid. The "field aggregation approa section 4.2.1 is based on this wor "field interpolation approach" we p it gives the same result, it is more e of storage and execution time. (Gom 2012) presents a discrete data mode continuous fields and an algebra th OLAP operators (e.g. Dice, Slice down …). However, the discrete m propose does not support the cont the field, which consists to retrieve point with coordinates x and y [START_REF] Gomez | Physical Design and Implementation of Spatial Data Warehouses Supporting Continuous Fields[END_REF], the a multidimensional model handlin define two types of fields, "field" (spatial field and temporal field), a the operators associated to these d include the notion of field dimen measures. They define the "field dimension containing at least one le (temperature, precipitation…), the as a measure represented by a fiel Finally, (McGuire, Gangopadhyay et al. 2008)define a snowflake schema for an environmental application where three dimensions represent the same spatial members at different resolutions.

Conclusion and future work

In this paper we present a multidimensional model for incomplete fields at several resolutions and its implementation in a SOLAP architecture based on standards (e.g. SQL and MDX). We are working on using spatial data mining to speed-up map algebra operations and implement a SOLAP visualization client. We also work in integrating other interpolation functions to generalize the proposed approach.

  Figure 3: Odor SDW multi-dimensional model Query 1: select average odor for each field member during 2012 SELECT [Field].[res100]. Members ON ROWS, {[time].[2012]} ON COLUMNS FROM [odorCube] WHERE [Measures].[value]

  with 2 other levels each representing a different level of resolution ([Field].[res200] and [Field].[res400]).

Figure 5 :

 5 Figure 5: (a) Field Aggregation Star Schema (FASS), (b) Field Interpolation Star Schema (FISS)

  They also propose a physical schema, but any implementation into a classical ROLAP architecture is presented.

		b)
	ith the field aggregation and the field interpolation app approaches b) Execution
	regation Star Schema and the Field Interpolation Star Sc Schema approaches
		hierarchy" as a set of relate ted field levels, which
		allows a field to be seen a at different levels of
		granularity . They also propose ose a physical model for
	data in a SOLAP	data warehouses with continu inuous fields. However,
	2005) propose a	no implementation has been een proposed and the
	andling continuous	hierarchical relationship betwe ween field levels has not
	of points as spatial	been brought to light. (Bimon onte and Myoung 2011)
	e cube which is	provide a multidimensional m l model that integrates
	e to simulate a	field data independently from m their implementation,
	ines new types of	as measures and dimensions. ns. They also present a
	a regular grid of	formal representation of the sp spatio-multidimensional
	on", "mixed hybrid	model schema where they de define the concepts of
	dimension" and	field dimensions, field measure ures, and field views. To
	e also defines the	our knowledge, no implemen entation including the
	cells of the matrix	continuous appearance of inc incomplete field or the
	roach" presented in	multiresolution over incompl plete fields has been
	ork. However, the	proposed.
	e propose, although	Representation of multidim idimensional data under
	re efficient in terms	different resolution levels s or scales may be
	omez, Gomez et al.	considered as multirepresentat tation. (Bernier, Bédard
	del for representing	et al. 2005) proposes an app pproach to provide on-
	that makes use of	Demand multi-scale maps. Al Although this approach
	ice, Roll-up, Drill-	models maps features at diffe ifferent scales by using
	model the authors	spatial hierarchies, but it does es not contain measures.
	ntinuous aspect of	(Yvan, Proulx et al. 2002) d defines a UML-based
	ve a value for each	conceptual model that integrat rates multiple geometric
	y in the map. In	and semantic representations ns properties of spatial
	e authors propose a	levels. However, this work rk does not present a
	ling fields. They	complete multidimensional m model with facts and
	d" and "tempfield"	hierarchies.
	, and semantics for	Moreover, (Bédard et al., 200 2002) suggests (without
	e data types. They	providing details) using a d different spatial data
	ensions and field	warehouse for each repres resentation. Therefore,
	ld dimension" as a	changing the representation co corresponds to move to
	level that is a field	another spatial data wareho house. (Gascueña and
	he "field measure"	Guadalupe 2009) propose a co conceptual model with a
	field and the "field	multi-representation of spatial ial members.

Acknowledgements

The authors wish to thank the FEDER and the region for funding this project and project partners who provide us the data.