Valérie Dos Santos

Yongxin Wu

Mickael Rodrigues

Valérie Dos

Santos Martins

M Wu

Ms Dos

Design of a PI Control using Operator Theory for Infinite Dimensional Hyperbolic Systems

Keywords: PDEs, de Saint-Venant Equations, Multi-Models, Semigroup Theory, IMBC. I

This paper considers the control design of a nonlinear distributed parameter system in infinite dimension, described by the hyperbolic Partial Differential Equations (PDEs) of de Saint-Venant. The nonlinear system dynamic is formulated by a Multi-Models approach over a wide operating range, where each local model is defined around a set of operating regimes. A new Proportional Integral (PI) feedback is designed and performed through Bilinear Operator Inequality (BOI) and Linear Operator Inequality (LOI) techniques for infinite dimensional systems. The new results have been simulated and also compared to previous results in finite and infinite dimension, in order to illustrate the new theoretical contribution.

The open surface channels couple transport phenomena and delay phenomena, so they have a complex nonlinear dynamics. In this case, the distributed parameter systems have a dynamic represented by hyperbolic Partial Differential Equations (PDE): the equations of de Saint-Venant, which depend on time and space [START_REF] Malaterre | Classification of canal control algorithms[END_REF], [START_REF] Papageorgiou | Flow control of a long river stretch[END_REF], [START_REF] Weyer | Decentralised PI controller of an open water channel[END_REF]. Some studies take into account the uncertainties and apply a robust control approach [START_REF] Litrico | Robust continuous-time and discrete-time flow control of a dam-river system: (i) modelling & (ii) controller design[END_REF], [START_REF] Litrico | H∞ control of an irrigation canal pool with a mixed control politics[END_REF]. Studying directly the nonlinear dynamics is also possible as in [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF], [START_REF] Dulhoste | Nonlinear control of water flow dynamics by input-output linearisation based on a collocation model[END_REF], [START_REF] Litrico | H∞ control of an irrigation canal pool with a mixed control politics[END_REF], [START_REF] Zaccarian | Anti-windup for marginally stable plants and its application to open water channel control systems[END_REF]. The Riemann approach has also been used to prove stability results for systems of two conservation laws [START_REF] Greenberg | The effect of boundary damping for the quasilinear wave equations[END_REF], and for systems of larger dimensions in [START_REF] Li | Global Classical Solutions for Quasilinear Hyperbolic Systems[END_REF].

Recently, it has been also coupled with LMI [START_REF] Amin | Exponential Stability of Switched Linear Hyperbolic Initial-Boundary Value Problems[END_REF]. The Lyapunov techniques have been used in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF], [START_REF] Santos | Boundary control with integral action for hyperbolic systems of conservation laws: Lyapunov stability analysis and experimental validation[END_REF], [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF].

In practice, industrial processes such as mining, chemical or water treatment processes are characterized by complex systems which operate in multiple operating regimes. Multi-Models methods split the operating range of a system into separate regions where local models are affected to each region [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF] for control and Fault Diagnosis purposes [START_REF] Bhagwat | Multi-linear model-based fault detection during process transitions[END_REF], [START_REF] Gatzke | Use of multiple models and qualitative knowledge for on-line moving horizon disturbance estimation and fault diagnosis[END_REF], [START_REF] Rodrigues | Sensor Fault Detection and Isolation Filter for Polytopic LPV Systems: A Winding Machine Application[END_REF]. Each local model is defined as a Linear Time Invariant (LTI) model for each operating point. The Multi-Models philosophy is based on weighting functions which ensure the transition between the different local models. Some authors speak about gain scheduling strategy for example in [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF] or Linear Parameter Varying (LPV) controllers [START_REF] Rodrigues | Fault tolerant control design for polytopic LPV systems[END_REF].

The use of Multi-Models representation for the study of the stability of a system described by nonlinear PDE has been examined in [START_REF] Amin | Exponential Stability of Switched Linear Hyperbolic Initial-Boundary Value Problems[END_REF], [START_REF] Santos Martins | A Proportional Integral Feedback for Open Channels Control Trough LMI Design[END_REF], [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF]. The nonlinear PDE stability is studied by transferring the common approaches based on finite dimension to infinite dimension. The theoretical proof has been given for the closed loop stability under a Proportional and a Proportional Integral (PI) controller with identical gains. In this paper, an analysis of the closed loop stability of the de Saint-Venant PDE is proposed with a general PI, using Multi-Models and the Internal Model Boundary Control (IMBC) structure. A variable elimination technique, as for finite-dimensional systems [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF], [START_REF] Rodrigues | LMI & BMI Technics for the Design of a PI Control for Irrigation Channels[END_REF], [START_REF] Skelton | A Unified Algebraic Approach to Linear Control Design[END_REF], has been used in order to solve a BOI (well known as Bilinear Matrix Inequality (BMI) in finite dimension) problem by the resolution of two LOI (well known as Linear Matrix Inequalities (LMI) in finite dimension).

The paper is organized as follows: firstly, the equations of de Saint-Venant are presented as well as the control problem. The Internal Model Boundary Control is explained and the physical constraints are given. In section II, the linearized models are defined around equilibrium sets as their insertion into the LOI formalism. Part III of the paper is dedicated to the design of November 20, 2013 DRAFT the feedback gains by LOI & BOI techniques which ensures the stability of the system. A Proportional Integral (PI) controller is implemented based on two propositions: these are the major contributions of this paper. The last section IV is dedicated to simulations. Comparisons between the simulations with the PI controller (for which the proportional gain is equal to the integral gain, developed in previous works) and the new PI controller (K int = K pr calculated using the LOI & BOI techniques) are realized. Simulations of a new channel are also implemented in the last section.

II. PROBLEM STATEMENT ABOUT CHANNEL REGULATION

The control problem concerns the stabilization of the water flow rate and/or the water height around an equilibrium for a reach denoted by (Z e (x), Q e (x)).

A. A model of a reach

The channel is supposed to have a sufficient length L (from x = 0 to x = L) such that one can consider that the lateral movement is uniform. Q(x, t), the water flow rate, and Z(x, t), the height of the water, are the state variables. The nonlinear PDE of de Saint-Venant which describe the flow on the channel are [START_REF] De Saint-Venant | Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit[END_REF], [START_REF] Georges | Automatique pour la Gestion des Ressources en Eau[END_REF]:

   ∂ t Z = -∂ x Q b , ∂ t Q = -∂ x ( Q 2 bZ + 1 2 gbZ 2 ) + gbZ(I -J), (1)
y(t) = C[Z(x, t) Q(x, t)] T (2)
Z 0 (x) = Z(x, 0), Q 0 (x) = Q(x, 0) (3)
∀x ∈ Ω = (x up , x do ) = (0, L), t > 0, C : (L 2 (0, L)) 2 → R. I is the slope, b is the channel width, g is the gravity constant. J is the friction slope from the formula of Manning-Strickler and R

is the hydraulic radius. The considered boundary conditions ∀x ∈ Γ = ∂Ω are two underflow gates. The controlled variable is defined as follows:

Q(x, t) = U (t)Ψ(Z(x, t)) (4)
with Ψ(Z) = κ 2g(Z up -Z do ). Z up is the water height at upstream of the gate, Z do is the water height at downstream of the gate, κ is the product of the channel width with the water flow rate coefficient of the gate. The gates opening U (t) is the control at upstream (U up = U 0 ) and at downstream (U do = U L ). The output variable is the water level at downstream, i.e. Z(L).

November 20, 2013 DRAFT

B. A regulation model

The fluvial case, i.e. the subcritical case [START_REF] Malaterre | Classification of canal control algorithms[END_REF], is considered. Let ξ(t) = (z(t) q(t)) T be the linearized state variable, then the model around the equilibrium state (Z e (x) Q e (x)) T is:

∂ t ξ(x, t) = Aξ(x, t) = A 1 (x)∂ x ξ(x, t) + A 2 (x)ξ(x, t), (5)
F b ξ(t) = B b u(t) and ξ(0, t) = ξ 0 (t), (6)
where A 1 and A 2 are matrices of the space variable x. The linearized boundary conditions [START_REF] Curtain | An introduction to Infinite Dimensional Linear Systems[END_REF] are equivalent to:

q(x up , t) = U up,e ∂ z Ψ(Z e (x up , t))z(x up , t) + u up (t)Ψ(Z e (x up , t)), (7)
q(x do , t) = U do,e ∂ z Ψ(Z e (x do , t))z(x do , t) + u do (t)Ψ(Z e (x do , t)), (8)
where U up,e and U do,e are the opening gates for the upstream and downstream (respectively) at the equilibrium and u up (t), u do (t) are the variations of these opening gates to be controlled.

The control problem is to find the variations of u up (t) at extremity x = x up and u do (t) at extremity x = x do of the reach such that the downstream water level, Z(x do , t) = Z(L, t)

(measured variable) tracks a reference signal r(t). The reference signal r(t) is chosen for all cases either constant or non-persistent (a stable step answer of a non-oscillatory system).

In this paper, the control scheme based on the Internal Model Boundary Control (IMBC) is adopted [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF]. This control strategy integrates the process model in real time and allows to regulate the water height in all the points of the channel by taking into account the error between the linearized model and the real system (or the nonlinear model for the simulations).

C. Open-loop system stability

Equation ( 5) describes the open loop system dynamics. In this representation, the state vector ξ(x, t) is not explicitly linked with the boundary control. In order to design an output feedback and to study the closed-loop stability, a distribution operator D of control at the boundary is

introduced [15] 1 , D : C k ([0, ∞], R n ) → (L 2 (0, L)) 2 .
It is a bounded operator such that Im(D) = Ker(A) and Du ∈ D(A) and [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF], [START_REF] Fattorini | Boundary Control Systems[END_REF], [START_REF] Touré | Controller design for distributed parameter systems, Encyclopedia of LIFE Support on Control Systems[END_REF]:

ξ(x, t) = ϕ(x, t) + Du(t). ( 9
)
This operator is naturally null in the domain of A(x) as it is active only on the boundary of the domain. This change of variables allows to get a Kalman representation of the system [START_REF] Moghadam | LQR control of an infinite dimensional time-varying cstr-pfr system[END_REF], [START_REF] Fattorini | Boundary Control Systems[END_REF], [START_REF] Touré | Controller design for distributed parameter systems, Encyclopedia of LIFE Support on Control Systems[END_REF]:

∂ t ϕ(x, t) = A(x)ϕ(x, t) -D u, (10)
ϕ(x, 0) = ϕ 0 (x) = ξ 0 (x) -Du(0), (11)
y(t) = Cϕ(x, t) + CDu(t). (12)
It has been proved that the open loop system (10-12) is exponentially stable [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF], as the operator of the linearized system in infinite dimension generates an exponentially stable C 0 -semigroup.

Moreover, under a PI-control

u(t) = α i K i ε(s)ds + α p K p ε(t) ∈ U = R n , u ∈ C k ([0, ∞], U ),
the stability of the closed-loop nonlinear system is ensured under some specific conditions on the gain synthesis. They are deduced from the properties of the IMBC structure and from the stability of the closed-loop linearized system.

For example, for the tuned gains of the PI-control the stability conditions are ensured if:

0 α i < α i,max = min λ∈Γ (a R(λ; A e ) + 1) -1 , (13)
0 α p < α p,max = (sup λ∈Γ a R(λ; A) ) -1 , (14)
where A e is a part of the series development of the closed loop operator [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF], R(λ; K) is the resolvent operator of K, and a is a constant which depends on A e .

These theoretical results have been corroborated by simulations and experimentations [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF]. Those experimentations have shown the limitations due to the linearization around an equilibrium state.

A first attempt with a Multi-Models approach has been realized with success. However, it was not optimal and no theoretical proof has been given. The first approach by an integral control [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF] had been extended to a PI control in [START_REF] Santos Martins | A Proportional Integral Feedback for Open Channels Control Trough LMI Design[END_REF], but the proportional and the integral gains were equal. The aim of this paper is to extend the previous results in infinite dimension with the proportional gain different from the integral gain. The theoretical proof realized in finite dimension with [START_REF] Rodrigues | LMI & BMI Technics for the Design of a PI Control for Irrigation Channels[END_REF] is developed to infinite dimensional systems. In order to control the water level over a wide operating range, a set of models is considered around judicious operating regimes: a control is synthesized and activated on the intervals when the system gets through the intervals.

K int = K pr
November 20, 2013 DRAFT

D. A Multi-Models representation of de Saint-Venant's Equation

The Multi-Models representation [START_REF] Rodrigues | LMI & BMI Technics for the Design of a PI Control for Irrigation Channels[END_REF], [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF] of de Saint-Venant's PDE around N operating points is defined by the following equations:

∂ t ξ(x, t) = N i=1 µ i (ζ(t))A i (x)ξ(x, t) with A i (x) = A 1,i (x)∂ x + A 2,i (x), (15)
ξ 0 (x) = ξ(x,
N i µ i (ζ(t)) = 1 and µ i (ζ(t)) ≥ 0.
In the following section, the control law synthesis by LOI techniques is considered.

III. STUDY OF THE CLOSED-LOOP SYSTEM STABILITY BY LOI

In this part, the closed loop structure is studied under a proportional integral feedback. Recall that the aim is to control the water height over all the operating range, so the output y(t) is not the variations around an equilibrium but the total water height. To this end, the output y(t) is modified:

y(t) = Cξ(x, t) + Eq(x, t),
where Eq(x, t)

= N i=1 µ i (ζ(t))(Z e,i (x, t) Q e,i
) T is the equilibrium state and for this paper

CEq(x, t) = N i=1 µ i (ζ(t))Z e,i (
L) as the aim is to regulate the water level at x = L.

A. Closed-loop structure for a proportional integral feedback

Let K int and K pr be the integral and proportional gains respectively. It follows that [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF]:

u(t) = K int [r(τ ) -y(τ )]dτ + K pr [r(t) -y(t)], (16)
where r(t) is the physical water level wanted (not the variations). So by using (9) one gets:

y(t) = Cϕ(x, t) + CEq(x, t) + CDu(t), (17)
November 20, 2013 DRAFT and by replacing y(t) into the equation ( 16), it becomes:

u(t) = K int [r(τ ) -(Cϕ(x, τ ) + CEq(x, τ ) + CDu(τ ))]dτ +K pr [r(t) -(Cϕ(x, t) + CEq(x, t) + CDu(t))] (18)
In each local model, Eq(x, t) is a piecewise function ( Ėq(x, t) = 0)2 . This is also the case of r(t). So, u can be simplified to:

u(t) = K int [r(t) -Cϕ(x, t) -CEq(x, t) -CDu(t)] -K pr CA i (x)ϕ(x, t) (19)
By replacing u into the equation [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF] and with Kint = D K int , Kpr = D K pr , the expression of the closed-loop system can be expressed as follows:

∂ t ϕ(x, t) = N i=1 µ i (ζ(t)) (A i (x) + Kint C + Kpr CA i (x))ϕ(x, t) + Kint CDu(t) + CEq(x, t) -r(t)) = N i=1 M i (x, t). (20)
The stability conditions are ensured by using a quadratic Lyapunov function [START_REF] Rodrigues | Fault tolerant control design for polytopic LPV systems[END_REF] in order to guarantee the convergence of the water height to the reference r(t) over the widest operating range.

B. Stability study with a Lyapunov function

Let us consider:

V (ϕ(x, t), t) = ϕ(x, t), P ϕ(x, t) , (21)
where ., . is the considered inner product. The Multi-Models representation of the linearized PDE of de Saint-Venant defined by equation ( 20) is asymptotically stable if there exists an operator P > 0, such that:

φ, P ϕ + ϕ, P φ = -ϕ, ϕ . (22)
The main difference here between the stability result in finite and infinite dimension [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF], lies in the inequality of the Lyapunov function for finite dimensional systems and equality for infinite ones [START_REF] Litrico | Robust continuous-time and discrete-time flow control of a dam-river system: (i) modelling & (ii) controller design[END_REF]. This equality complexity can be removed in some cases; as for example for operators with compact resolvent [START_REF] Curtain | An introduction to Infinite Dimensional Linear Systems[END_REF], [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF], [START_REF] Triggiani | On the stability problem in banach space[END_REF] or [START_REF] Sasane | Optimum Hankel Norm Approximation for the Pritchard-Salamon class of infinitedimensional systems[END_REF]. In this case, the same inequality from finite dimension is a sufficient and necessary condition for the infinite dimensional case; it needs to satisfy the spectral growth assumption [START_REF] Triggiani | On the stability problem in banach space[END_REF], [START_REF] Fattorini | Boundary Control Systems[END_REF]. Moreover, for the equations of de Saint-Venant, it has been shown that the operator has a compact resolvent [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF] so it satisfies the spectral growth assumption. Then, by taking into account ( 20)-( 22), one has to prove the following inequality:

M i , P ϕ + ϕ, P M i < 0, (23)
where M i is defined in [START_REF] Li | Global Classical Solutions for Quasilinear Hyperbolic Systems[END_REF].

The development of the inequality (23) leads us to consider an inequality for each local system of index i such that 3 :

[A i + Kint C + Kpr CA i ]ϕ(.,
+ ϕ(., t), P [A i + Kint C + Kpr CA i ]ϕ(., t) + ϕ(., t), P Kint [CDu(t) + CEq(., t) -r(t)] < 0.

In the inequality [START_REF] Mareels | Systems engineering for irrigation systems: Successes and challenges[END_REF], which defines the stability condition of the system ∀i, the control parameter u appears; this is a difficulty for the gain synthesis: Kint , Kpr .

A first approach was made in [START_REF] Santos Martins | A Proportional Integral Feedback for Open Channels Control Trough LMI Design[END_REF] with K pr = K int . In this paper, the previous results are improved as K int is considered different from K pr . It has been proved that a good choice of K int and K pr based on semigroup theory is K int = -α i [CD] † and K pr = α p [CD] † (where † stands for the right pseudo-inverse) in [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF]. α i and α p are defined in ( 13)-( 14). So, one can assume that ∃β ∈ R such that K pr = βK int , i.e. Kpr = β Kint . Then, the equation (24) becomes: Note that the open-loop system ( 5)-( 8) is exponentially stable as the closed loop one under a PI-control, with gains correctly tuned [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF] for a time t well chosen. So, one can assume that ∃k > 0, such that:

[A i + Kint C + β Kint CA i ]ϕ(.,
| Cξ(x, t) + CEq(x, t) -r(t) |≤ k | Cϕ(x, t) | (26)
and with α = (k + 1)ε (ϕ T P Kint Cϕ) [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF]:

ϕ, P Kint (CDu(x, t) + CEq(x, t) -r(t)) ≤ ϕ, αP Kint Cϕ ( 27
)
3 Due to the lack of space, (x, t) is replaced by (., t).

November 20, 2013 DRAFT For finite dimensional systems, a stability study has been given in our paper [START_REF] Rodrigues | LMI & BMI Technics for the Design of a PI Control for Irrigation Channels[END_REF], by using well-known linear techniques, but not developed for infinite dimensional systems. Here, the main contribution consists in a tool developed using the semigroup theory.

Proposition 1: Let Z be a Hilbert space and let G, U , V , X four linear operators on Z such that G : D(G) ⊂ Z → D(G). The domains of U , V and X are densely defined on D(G), the domain of G. X is a self-adjoint operator such that X ≤ 2σ.

If ∃σ ∈ R\{0} which satisfy, ∀φ, ϕ ∈ D(G) Gϕ + U * φ, ϕ + U ϕ + σ -1 φ, φ < 0 (28a) Gϕ + V * φ, ϕ + V ϕ + σ -1 φ, φ < 0 (28b)
then, the following inequality is also satisfied:

Gϕ, ϕ + U * XV ϕ, ϕ + V * XU ϕ, ϕ < 0 ( 40
)
Proof: See appendix.

The following proposition extends our results [START_REF] Rodrigues | LMI & BMI Technics for the Design of a PI Control for Irrigation Channels[END_REF] to infinite dimensional systems.

Proposition 2: If there exists a self-adjoint operator P , matrices W int and W pr , scalars σ, γ ∈ R, such that the inequalities (28) are satisfied with G = A * i P + P A i + γW int C + γC * W * int , U * = W int , V = CA i , then one gets the following inequalities:

(A * i P + P A i + γW int C + γC * W * int ) ϕ + W int φ, ϕ + W * int ϕ + σ -1 φ, φ < 0 (41a) (A * i P + P A i + γW int C + γC * W * int ) ϕ + (CA i ) * φ, ϕ + CA i ϕ + σ -1 φ, φ < 0 (41b)
and the closed-loop system (20) under the PI control law ( 16) is stable.

Proof: Let us consider [START_REF] Litrico | H∞ control of an irrigation canal pool with a mixed control politics[END_REF] with the closed-loop system (20) under a PI control law [START_REF] Gatzke | Use of multiple models and qualitative knowledge for on-line moving horizon disturbance estimation and fault diagnosis[END_REF].

By considering the inequality ( 23), we can obtain [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] with K pr = βK int . Now, let us assume that the inequality ( 27) holds true, then (25) becomes:

P [A i + γ Kint C + β Kint CA i ] ϕ, ϕ + ϕ, P [A i + γ Kint C + β Kint CA i ] ϕ < 0 (42)
with γ = 1+α, Kint = P -1 W int , Kpr = β Kint . The inequality (42) has two variables: β and W int that lead to a BOI problem. By using proposition 1, the BOI problem can be solved as two LOI problem. The inequality (42) is equivalent to (40

) with G = A * i P + P A i + γW int C + γC * W * int , U * = W int , V = CA i , X = βId.
Proposition 1 allows to conclude that if the inequalities (41) November 20, 2013 DRAFT are satisfied, the Lyapunov inequality is also true. Thus the system (15) under PI control is stable.

Remark 3: The inequalities (41) seem to be linked with the stability of each submodel in infinite dimension, with a Lyapunov Input-to-state stability (ISS) function [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], [START_REF] Dashkovskiy | Input-to-state stability of infinite-dimensional control systems[END_REF].

The aim of section IV is to compare the simulated curves obtained with this method and the ones obtained on the experimental benchmark [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF] and the simulations with the gains Kint = Kpr [START_REF] Santos Martins | A Proportional Integral Feedback for Open Channels Control Trough LMI Design[END_REF], [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF].

IV. SIMULATION RESULTS

Two benchmarks are used for the simulations: the micro-channel of Valence (France) and the channel of Gignac (France). The simulations are based on a Chang and Cooper scheme, for more details see [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF], [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF].

For both applications, the weighting function µ i (ζ(t)) is equal to 1 if the output's height is included into the validity domain of the model and 0 in the other case for each operating state.

The output of the system is the decision variable. The parameter ζ(t) is a function of it. Both coefficients β and σ of the proposition 1 are negative in both cases simulated, so the condition X ≤ -2σ is always satisfied whatever X = βId. Here, the inequalities (41) are solved after discretization by LMI as in [START_REF] Rodrigues | LMI & BMI Technics for the Design of a PI Control for Irrigation Channels[END_REF].

A. The micro-channel of Valence

The equilibrium profiles have been chosen such that the calculated control law from the local models can be efficient over all the operating range of the water height [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF]. Notice that it has been experimentally verified that a local model is valid around ±20% of an equilibrium profile.

In order to assign references which are included between 0.06 m and 0.2 m, the operating points at x = 0 are given in the Table I.

The following simulation (Figure (1.a)) compares PI controllers: one with gains K int = K pr [START_REF] Santos Martins | A Proportional Integral Feedback for Open Channels Control Trough LMI Design[END_REF] and the new one with gains K int = K pr . The better than the experimental PI which has been implemented in [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF].

B. The channel of Gignac

In this paper, we have also studied a channel which is located in Gignac (France). The

V. CONCLUSION

First attempts of a Multi-Models approach on irrigation channels control, through an IMBC structure, have been realized some years ago [START_REF] Santos | Multivariable Boundary Control approach by internal model, applied to irrigations canals regulation[END_REF]. Good experimental results were obtained in order to design the feedback gain through LMI have been realized in the case of an Integral controller in [START_REF] Santos Martins | A Multi-Models Approach of Saint-Venant's Equations: A Stability study LMI[END_REF]. Preliminary results of a PI controller in a particular case (K int = K pr ) have been published in [START_REF] Santos Martins | A Proportional Integral Feedback for Open Channels Control Trough LMI Design[END_REF] for infinite dimensional systems and with K int = K pr in [START_REF] Rodrigues | LMI & BMI Technics for the Design of a PI Control for Irrigation Channels[END_REF] for finite dimensional systems. In the present paper, the authors take into account the more general case of PI controller with K int = K pr for infinite dimensional systems.

G : D(G) ⊂ Z → D(G). U , V and X are densely defined on D(G), the domain of G. X is a self-adjoint operator such that X ≤ 2σ. If ∃σ ∈ R which satisfy, ∀φ, ϕ ∈ D(G) Gϕ + U * φ, ϕ + U ϕ + σ -1 φ, φ < 0, (43a)
Gϕ + V * φ, ϕ + V ϕ + σ -1 φ, φ < 0, (43b)
November 20, 2013 DRAFT then, the following inequality is also satisfied:

Gϕ, ϕ + U * XV ϕ, ϕ + V * XU ϕ, ϕ < 0. (44)
Proof: The objective is to prove this equation (44):

Gϕ, ϕ + U * XV ϕ, ϕ + V * XU ϕ, ϕ < 0 ⇒ Gϕ, ϕ + XV ϕ, U ϕ + XU ϕ, V ϕ < 0 (45) Gϕ, ϕ + V ϕ, XU ϕ + U ϕ, XV ϕ < 0 (46)
Developing (43), one can obtain the following inequalities:

     Gϕ, ϕ + U * φ, ϕ + U ϕ, φ + σ -1 φ 2 < 0, Gϕ, ϕ + V * φ, ϕ + V ϕ, φ + σ -1 φ 2 < 0. ⇔      Gϕ, ϕ + φ, U ϕ + U ϕ, φ + σ -1 φ 2 < 0 Gϕ, ϕ + φ, V ϕ + V ϕ, φ + σ -1 φ 2 < 0 (47)
Let consider firstly that σ is positive, φ = XV ϕ and φ = XU ϕ in equations (47), one gets:

     Gϕ, ϕ + XV ϕ, U ϕ + U ϕ, XV ϕ + σ -1 XV ϕ 2 < 0, Gϕ, ϕ + XU ϕ, V ϕ + V ϕ, XU ϕ + σ -1 XU ϕ 2 < 0. (48)
Summing those both inequalities of (48), one obtains:

2 { Gϕ, ϕ + XV φ, U ϕ + U ϕ, XV φ } + σ -1 XV ϕ 2 + XU ϕ 2 < 0. (49)
With σ -1 > 0, it is equivalent to:

⇔ Gϕ, ϕ + V ϕ, XU ϕ + U ϕ, XV ϕ < 0. (50)
So the inequality (46) and also the inequality (44) have been proved, the proposition is verified.

Let consider now that σ -1 < 0, four constants α, β, γ, δ ∈ R, and the inequalities (47), with this time φ = αU ϕ + βXV ϕ and φ = γV ϕ + δXU ϕ. One can obtained the following November 20, 2013 DRAFT inequalities:

    
Gϕ, ϕ + φ, U ϕ + U ϕ, φ + σ -1 φ 2 < 0, Gϕ, ϕ + φ, V ϕ + V ϕ, φ + σ -1 φ 2 < 0 (51)

⇒                   
Gϕ, ϕ + βXV ϕ, U ϕ + U ϕ, βXV ϕ + αU ϕ, U ϕ + U ϕ, αU ϕ + σ -1 αU ϕ + βXV ϕ 2 < 0, Gϕ, ϕ + δXU ϕ, V ϕ + V ϕ, δXU ϕ + γV ϕ, V ϕ + V ϕ, γV ϕ + σ -1 γV ϕ + δXU ϕ 2 < 0.

Let sum both equations, then one finds: ⇒ 2 Gϕ, ϕ + (β + δ + σ -1 (αβ + γδ)) [ XV ϕ, U ϕ + U ϕ, XV ϕ ] + σ -1 β 2 XV ϕ, XV ϕ

+(2α + σ -1 α 2 ) U ϕ, U ϕ + (2γ + σ -1 γ 2 ) V ϕ, V ϕ + σ -1 δ 2 XU ϕ, XU ϕ ≤ 2 Gϕ, ϕ + (β + δ + σ -1 (αβ + γδ)) [ XV ϕ, U ϕ + U ϕ, XV ϕ ] + (2α + σ -1 [α 2 + X 2 δ 2 ]) U ϕ, U ϕ + (2γ + σ -1 [γ 2 + X 2 β 2 ]) V ϕ, V ϕ < 0. ( 52
)
The aim is to defined the constants α, β, γ, δ ∈ R such that {(52)⇒ (44)} is satisfied ∀ σ -1 < 0. Three constraints appear:

1) β + δ + σ -1 (αβ + γδ) = 2,

2) (2α + σ -1 [α 2 + δ 2 X 2 ]) > 0,

3) (2γ + σ -1 [γ 2 + β 2 X 2 ]) > 0.

If they are satisfied then Gϕ, ϕ + XV ϕ, U ϕ + U ϕ, XV ϕ < 0.

Consider the first constraint and let define α = γ = -σ m, where m ∈ R, then the first equality implies that β + δ = 2 1-m m = 1. As β and δ have the same role, we define for m = 1 β = δ = 1 1-m then

(2α + σ -1 [α 2 + δ 2 X 2 ]) = -2σ * m + σm 2 + σ -1 X 2 (1 -m) 2 , (53)
(2α + σ -1 [α 2 + δ 2 X 2 ]) > 0 ⇔ σ 2 m(2 -m)(1 -m) 2 > X 2 iff 0 < m < 2 (54)
Idem for the third inequality, as the coefficients are equals.

So, for every σ < 0, ∃0 < m < 2, m ∈ R\{1} and an operator X self-adjoint such that

Figure ( 1 Fig. 1 .

 11 Fig. 1. Valence channel simulation

  following set of parameters of this channel is considered: L = 2272 m is the length of the channel, b = 3 m is the width of the channel, N = 40 is the number of the discretizated points, Z L is the water height to regulate such that 1.7 m < Z < 2.5 m.

Figure ( 2

 2 Figure (2.a) shows that the output converges to the reference over a wide operating range. In Fig. (2.b), the water flows at the upstream and downstream stay in physical proportion, which is an important practical point.

  of the downstream water level b) Water flows

Fig. 2 .

 2 Fig. 2. Gignac channel simulation

σ m( 2 2 / 2 ,Remark 5 :

 2225 -m)(1 -m) < -X ,November 20, 2013 DRAFT and the three constraints above are satisfied with α = γ = -σ m and β = δ = 1 1-m . As it is true for all m, let take m = 1 -√ then m(2 -m)(1 -m) = 0.5 and one gets σ < -2 X . Then, the inequality (52) is realized and (46)⇔(44) is too. The proposition is proved. The choice of m is arbitrary, and one can consider another one (m = 1, 0 < m < 2). Here we have chosen the critical case i.e. the minimum value of the inverse function of m(2 -m)(1 -m) 2 .

  0),where A i (x) is the operator which corresponds to the i th equilibrium state. ζ(t) is a function which depends on some decision variables directly linked with the measurable state variables and eventually to the input. µ i (ζ(t)) are the weighting functions which activate the control law in function of the output of the process Z L . They belong to a convex set such that

  Proposition 4: Let Z be a Hilbert space and let G, U , V , X four linear operators on Z such that

	APPENDIX A
	PROOF OF THE PROPOSITION 1

They synthesize the new PI controller feedback gains by solving a BOI problem. Simulations show a better performance than the previous results, through this new PI feedback controller designed by BMI & LMI [13]. The forthcoming experimentation should emphasize the improvements of the new PI controller with weighting functions, non-piecewise constant ( µ i (ζ(t)) ∈ [0, 1]).

Regularity coefficient is generally taken as k =

 November 20, 2013 DRAFT

The following notation is considered: ∂tφ = φ whatever the function φ.November 20, 2013 DRAFT

November 20, 2013 DRAFT


Mohamed Anis Dhuieb

Florent Laroche

Alain Bernard

Toward a cognitive based approach for knowledge structuring

Keywords: Knowledge structuring, Knowledge Extraction, Cognitive InfoCommunication, Man-Machine Interactions, Working situation. I

Competitiveness led factories, today, to be much more careful with their product development process, more accurate with their human resources and involve a diversity of actors such as operators, technicians, engineers, etc. creating heterogeneous knowledge domains. In this paper, we aim to present a comprehensive explanation of an approach regarding knowledge extraction and structuring in the factory. We begin our paper with an overview about the context of our research proposal. Then, we explain different constraints to take into account during the process of building our model.

INTRODUCTION

Regarding the increasing of products complexity and customers desires in product development field, the need for knowledge and expertise in production lines become an important key factor to handle quality and efficiency requirements. Production environment requires assistance tools that enable retrieval, reuse, structuring and representation of the entire product and process related knowledge.

Visions about the factory of the future affirm that future production lines will be characterized by learning capability and decisions support in all stages of the process chains involved [START_REF] Westkämper | EMO Hannover[END_REF]. The aim of such propositions is to achieve higher manufacturing outputs by integrating knowledge modules into the engineering systems concerned in the factory mainly thanks to digital tools. Despite the evolution of digital tools in the factory, the role of the human being in the product lifecycle development is not insignificant. The main challenge in today's production systems is to provide factory actors with assistant tools that help them to work more efficiently and to react quickly in front of critical working situations. In some practical situations, factory actors need to get information and to acquire more knowledge to perform better in front of difficult tasks.

The most common example of digital assistance tools are Decision Support Systems (DSS). DSS evolved in a significant way thinks to the parallel evolution made on information and communication technologies. This evolution results the appearance of new types of DSS such as web-based DSS [START_REF] Boreisha | Web-based decision support systems as knowledge repositories for knowledge management systems[END_REF], Mobile DSS [START_REF] Wen | A mobile knowledge management decision support system for automatically conducting an electronic business[END_REF], Knowledge Based DSS [START_REF] Klein | Knowledge-based decision support systems with applications in business: a decision support approach[END_REF], and ubiquitous DSS [START_REF] Kwon | UbiDSS: a proactive intelligent decision support system as an expert system deploying ubiquitous computing technologies[END_REF]. Despite the research progress made on DSS architectures, these systems are still not responding to the dynamic relation between the system and the user. There is a need for a non-static decision support system: a decision support system should be in a dynamic relation with the user. This dynamic relation can be ensured by capturing user's environment, understanding the context and providing decision support according to his/her situation. Based on representationbridging communication technics as defined on cognitive infocommunication research field [START_REF] Baranyi | Definition of Cognitive Infocommunications and an Architectural Implementation of Cognitive Infocommunications Systems[END_REF] [21], the set of knowledge have to be presented in a structured way that a simple decision support system cannot provide.

In this paper, an approach for knowledge structuring will be discussed. The aim of this approach is to respond to every actor needs in the factory according to the current working situation and depending on cognitive parameters. The proposed concept will be a part of research issue in the proposition of "Digital Factory Assistant" [START_REF] Dhuieb | Digital Factory Assistant: Conceptual Framework and Research Propositions PLM[END_REF] [START_REF] Laroche | Towards the factory of future An integrated approach of material-processesinformation-human being[END_REF].

At first, we will discuss some related works. Then, we explain our research proposal and its context. The third part will expose different points token into account through the cognitive based approach. Before the conclusion, we made an overview of the use cases that will be used to validate our approaches.

II. RELATED WORKS

Many propositions related to factory actors assistance in the factory have been encountered in the literature. In a maintenance context, CARMMI model [START_REF] Espíndola | A model-based approach for data integration to improve maintenance management by mixed reality[END_REF] provide support to operators during maintenance tasks through mixed reality providing information access from different sources. This model allows the acquisition and presentation of different data: CAx data, maintenance, and virtual data. From a decision making point of view, the big quantity of the information presented in the interface can reduce the efficiency of the operator to make the right decision. In fact, the interface contains augmented reality, visualization system, information and historical information data. The human machine interactions are not studied in this project. A second work in the same context was presented by Toro [START_REF] Toro | Knowledge based industrial maintenance using portable devices and augmented reality[END_REF]. A framework and a system implementation were developed for the exploitation of embedded knowledge in the domain of industrial maintenance. This framework uses a shared ontology designed to model and support pervasive computing applications called SOUPA. A knowledge acquisition system is also developed in this project called Experience Knowledge Structure. This system extends the functionality of SOUPA in a way that formal decision events can be suggested to the user during the working process. Both of these two technologies were used to develop the UDKE system (User, Device, Knowledge, and Experience). The UDKE provide a conceptual model of a maintenance system that combines knowledge, user experience and Augmented Reality techniques. The context awareness notion was absent from this project and therefore, the possible decisions and information provided to the user were not relied to the current task and did not consider user's knowledge. Kwon [START_REF] Kwon | UbiDSS: a proactive intelligent decision support system as an expert system deploying ubiquitous computing technologies[END_REF] [START_REF] Kwon | ubiDSS: a framework of multi-agent based proactive decision support system with context-awareness[END_REF] has developed a framework called ubiDSS. ubiDSS is a framework of multi-agent and context-aware based proactive decision support system. The architecture of this framework is composed by a number of subsystems: context subsystem, dialogue subsystem, knowledge subsystem, model management subsystem, and a database management subsystem. Even that the application field of the ubiDSS system is far from industry, it was very important to study the interactions between the components of its framework and the models behind. Many other propositions were encountered in the literature relied to the human being assistance in the different product lifecycle stages, especially, in design process. However, the application of such propositions in manufacturing context is very rare. In fact, despite the diversity of these projects, few of them deal with human assistance in production systems at a "machine level" where information is less available comparing to other departments in the factory.

III. RESEARCH PROPOSAL

A. ARTUR project

Related to previous cited works, the ARTUR project aims to offer factory actors with a so called "Digital Factory Assistant". Through this assistant the operator will benefit from an omnipresent support along the manufacturing process.

The digital assistant restitutes knowledge from distributed knowledge sources and relies in more than one way of interaction with the user in order to act upon the task that the operator is doing by providing relevant information and decision support.

It consists of an advanced knowledge based system which allows knowledge access from different sources in the factory such as Product Lifecycle management (PLM) CAx models (Computer aided Design/ Manufacturing/engineering CAD/CAE/CAM…), material flows, process chains, and simulation data. The basic idea is to allow each person to be able to respond on any given situation, mainly thanks to high speed simulation and performance evaluation models and methods. The vision about the "Digital Factory Assistant" is based on the concept of ubiquitous environments. Fig. 1 outlines the first conceptual architecture proposition of the digital assistant in a high level. The global architecture is based on four main units:

B. Scientific propositions

• The working situation module is an abstraction of the working situation. This module is related to the human-machine level, where contextual information related to user's working situation can be acquired. • The second unit is related to the issue of coupling knowledge models with the information system in the factory. This coupling aims to ensure a better reuse of the entire knowledge resides on the factory through multiple sources. Cognitive infocommunication is defined as the communication capability between two actors where the cognitive processes can co-evolve with infocommunication devices [START_REF] Baranyi | Definition and Synergies of Cognitive Infocommunications[END_REF]. Based on an inter-cognitive infocommunication, the factory actor will interact with the system through a cognitive infocommunication interface.

• The advantage behind implementing advanced numerical simulation of the manufacturing process is to guarantee a better understanding and control of the process. Thanks to this simulation, the user can manipulate some parameters related to the process and anticipate the outcome of such manipulation on the simulated process before committing on the real manufacturing process.

•

The last part deal with the problem of information representation. Virtual technologies will be used to propose a better restitution of the information. We can imagine a process of relevant information projection on the product itself or in the machine. Combining all these aspects to work together is the real issue of the project. Dealing with the particular problem of linking knowledge parts with the whole information system in the factory, and to build our "Digital Factory Assistant" model, we have to consider many points of view and to deal with technological and modeling constraints. In the following section we will try to detail some of them.

IV. BUILDING THE ASSISTANT MODEL

A. Context awareness

Situation awareness has been defined in the cognitive sciences as an intermediate state in the decision-making process of dynamic systems where the system should be able to understand the situation in order to make an appropriate decision for future development [START_REF] Schmidt | Context-Aware Computing: Context-Awareness, Context-Aware User Interfaces, and Implicit Interaction[END_REF]. The situation-awareness [START_REF] Belkadi | A situation model to support awareness in collaborative design[END_REF] is a particular aspect of context awareness information systems. Context-aware systems have the particularity of anticipating the user's needs in a particular situation and act proactively to provide appropriate assistance. These systems relied on the notion of context. Dey [START_REF] Dey | Understanding and Using Context, Personal and Ubiquious Computing[END_REF] defined this notion as any information that can be used to characterize the situation of an entity. An entity is a person, place, or object that considered relevant to the interaction between the user and the application, including the user and the applications themselves.

The human being has a central role in the working situation. Therefore, some works relied on the concept of user profiling [START_REF] Silvia | Intelligent user profiling[END_REF] in order to achieve a comprehensive abstraction of contextual information about the user.

Thus, we can affirm that a context-aware system is a system that uses the contextual information to provide relevant information/service. Based on context definitions the assistant model should respond to three main features: understand the user's context, extract the relevant set of information and provide the user with the right information. To validate the ubiquity [START_REF] Weiser | The computer for the 21st century[END_REF] in the desired model, the assistant should execute automatic work on background. The automatic execution of some tasks is based on already known contextual information, for example, user experience or current executed task. To do so, algorithms will be implemented to act like sensors in the manufacturing environment and detect contextual information.

B. Human-Computer interactions

The human computer interactions issue is decomposed on two parts. The first part is related on ergonomic issue that will be studied by other members of ARTUR project described in the second section. The second part is related to the way that information will be presented to the user.

As we described previously, contextual information will be extracted from manufacturing environment. These information will be used to extract the appropriate information related to the decision support process. The set of relevant information should be presented in a structured way and should not be the same for every user. The infocommunication interface that we aim to build will interact with the user in an intelligent way. This cognitive infocommunication interface will take into account the information "quantity" to be presented. This quantity will depend on already known contextual information and precisely user experience. According to this cognitive approach, the information quantity will be different from a beginner operator to an expert operator. In a context of uncertainty, the system will transfer a small amount of knowledge to a beginner user and led him, as his/her needs, to a larger amount of knowledge and do the opposite for a more expert user. This intelligent information exploitation system will be deployed in mobile device to make the information available in any context, even in complex working situations.

The knowledge visualization model that we aim to design should take the psychological aspects into account and reflects the fact that the interactions between factory actors assistance tools and their working environment have a significant impact on knowledge acquisition and consequently their role is vital in order to improve efficiency in production systems.

C. Cognitive-based Knowledge structuring approach issues

Foundations of our research work relate to a new proposition for modeling and structuring information, and more generally knowledge [START_REF] Dhuieb | Digital Factory Assistant: Conceptual Framework and Research Propositions PLM[END_REF]. We believe that the set of knowledge in the factory should be structured in a multi-scale way. Where every scale is corresponding to a set of knowledge that should be given to a user according to cognitive related parameters such as user level of expertise in a domain or in a task. Like in video games, the user will not just "play" in his/her level but, s/he can "jump" to another one according to his/her needs and if s/he feels that s/he is well mastering the existent set of knowledge in his/her level.

This cognitive approach is related on some concepts mainly the granularity of knowledge [START_REF] Jakubczyc | Contextual Knowledge Granularity[END_REF] [START_REF] Mach | Knowledge Granularity and Representation of Knowledge: Towards Knowledge Grid[END_REF]. The concept of knowledge granularity is based on dividing the set of knowledge into pieces. Generally speaking, granularity can be defined as the possibility to represent and operate on different levels of detail in data structures.

By proposing to apply this concept on the knowledge set, we aim to structure, in a granular way, the whole knowledge sources and match it with different knowledge domains combined in the manufacturing field by factory actors by granular reasoning. This process of knowledge granularity structuring is presented in Fig. 2. Our assistant will be based on a dynamic knowledge representation: in a process of decision making, the actor will combine different types of data and knowledge available in various forms. This knowledge set can be decomposed into hierarchical structure and represented in a multi-level way. According to his/her needs, the user will look for more granular set of knowledge that satisfies his/her query.

At first, we will define knowledge acquisition mechanisms that will be coupled with other mechanisms for context interpretation. The output of this level will be used to construct the knowledge structure and accordingly the multiscale representation of the knowledge set. Fig. 2 shows the different subsystems handling the knowledge structuring process. The role of user request and user context interpretation subsystems is to understand the user's need and get relevant contextual information respectively. After knowing the user request and the context of the current working situation, these subsystems will be the input for the knowledge extraction subsystem. The output will be unstructured relevant knowledge. Structuring the set of knowledge is the role of the next subsystem. The knowledge visualization mechanism will extract the already structured set of knowledge and present it to knowledge consumers in a multi-scale way.

Contextual information will be extracted from user profile and the working environment. These contextual information will be used as cognitive parameters to be injected to the assistant model. The most important contextual parameters can be:

• Experience Level: the factory actor will receive the information according to his/her experience level. This parameter can be extracted from information system like Enterprise Resource Planning (ERP) or human resource systems used by some industries to manage their human capitals. • Activity: this parameter determines the task that the user is doing. By knowing what is the current activity exercised by the user, we can distinguish between activities that demand a decision to take or not and providing information according to the current task. • Location: determine the location of the user, in which department he is working or in which part of the production line. • Time: the time can correspond to the needed time for performing the task or to take the decision. This parameter determines if the user does need a quick response or not to make his/her decision? • Relations: according to the user's task, the system can search and propose "knowers" that have more experience than the current user. These contextual information will be extracted by the context interpretation subsystem and consumed by the knowledge extraction mechanism in order to extract relevant knowledge according to them. 3 summarizes foundations of our research work. This illustration outlines the difficulties and issues related to the proposition of the digital factory assistant. In fact, the user will deal with only the end-user dedicated infocommunication interface and virtual reality technology, compared as the top part of an iceberg. "Back-end" problems are represented by the lower part of the iceberg. Actually, based on knowledge extraction mechanisms, the assistant will interact with the existing information system in the factory which creates interoperability problems between the desired architecture and the existing one. Another problem is structuring the set of extracted information in order to fit with the cognitive based knowledge structuring assumption and the desired interface for intelligent knowledge visualization. The last problem is related to the context handling. This issue is decomposed on two parts: contextual information detection and contextual information interpretation.

These scientific constraints will be joined with industrial constraints. Through a use case we will try to validate our approach in the real world and combine real issues. In the next section we present an overview of the industrial use case.

V. CURRENT WORK

To validate our proposition, we will be based on two use cases from aeronautic industry. The tasks during the study of the use cases are as following:

• Process observation: This task helps us to know what different actions that the factory actors are doing and to analyze how the actor resolve different problems in front of his/her machine from a cognitive point of view by answering to the questions: what are the different token decisions along the process execution? What are the used tools?

• Technical documentation analyzing: Besides, the tacit knowledge hold by factory actors, the explicit knowledge is stored in different technical documentation related to process and the developed product. In most of cases, this knowledge is just stored in some place in the factory and they are not well exploited.

• Information system audit: the aim of this task is to identify the different sources of knowledge stored in factory information system especially in PLM software and other applications directly related to the process.

These tasks allow us to set up cartography of different knowledge domains and to build a map of different knowledge sources in the factory. For confidential reasons we cannot describe the use case. Based on the study of our use case, the next step of our work is to build the information model and set up knowledge extraction mechanisms from knowledge sources in the information system. The main issue of our work is to find which information to give to the operator, where, when, and how.

A more detailed version of the global proposition architecture will be established including the different Man-Machine and Machine-Machine interactions.

VI. CONCLUSION

In a production system, the user may be faced with different working situations that require different decisions to take. Based on a cognitive approach, we aim to design and implement a framework that interacts with the information system in the factory in order to extract relevant information and present them in a multi-scale structure through a cognitive infocommunication interface. In this paper, we presented an overview about the proposed approach and the different scientific locks.

Fig. 1 .

 1 Fig. 1. ARTUR project

Fig. 2 .

 2 Fig. 2. Process of knowlegde granularity structuring

Fig. 3 .

 3 Fig. 3. Context-aware knowledge structuring process

Fig. 4 .

 4 Fig.[START_REF] Klein | Knowledge-based decision support systems with applications in business: a decision support approach[END_REF]. Global illustration of the cognitive knowledge structuring approach Fig.3summarizes foundations of our research work. This illustration outlines the difficulties and issues related to the proposition of the digital factory assistant. In fact, the user will deal with only the end-user dedicated infocommunication interface and virtual reality technology, compared as the top part of an iceberg. "Back-end" problems are represented by the lower part of the iceberg. Actually, based on knowledge extraction mechanisms, the assistant will interact with the existing information system in the factory which creates interoperability problems between the desired architecture and the existing one. Another problem is structuring the set of extracted information in order to fit with the cognitive based knowledge structuring assumption and the desired interface for intelligent knowledge visualization. The last problem is related to the context handling. This issue is decomposed on two parts: contextual information detection and contextual information interpretation.These scientific constraints will be joined with industrial constraints. Through a use case we will try to validate our approach in the real world and combine real issues. In the next section we present an overview of the industrial use case.


Mohamed Anis Dhuieb

Alain Bernard

Florent Laroche

Multi-scale structuring of enterprise's knowledge Towards an ubiquitous virtual assistant

Keywords: Decision Support System, Factory of the future, Virtual assistant, Knowledge Management, Virtual Reality. I

When the question about the "factory of the future" is discussed, it is often mentioned the use of innovative approach for knowledge acquisition and exploitation interfering in the different phases of the product lifecycle. Consequently, the use of innovative Knowledge Management Systems based on Information and Communications Technology (ICT) in the factory environment has become a must. Our research work will focus on the development of a "digital factory assistant" to help factory's actors in their daily tasks and in particular tasks that demand to make a decision.

INTRODUCTION

Despite of the "Digitalization" of today's factory, either by the emergence of ICT, robots colonization, or by the use of advanced simulation that implements the new wave of virtual and augmented reality technologies, the human factor in today's factory is still vital. In the level of the production system, and when the employee have to act upon a task that demand decision making, he\she still in an uncertain context where the expertise level does not allow him/her to properly accomplish a task and make the right decision: what effort should he/she make? What is the right place to apply this effort? What are the consequences if he/she applies a choice?

In this context and to answer these questions, we introduce the concept of the "digital factory assistant". It consists of a smart agent based on an interaction between knowledge models and the entire factory's information system in order to provide the right information in the right time and place and for the right user. In addition, this assistant will extend the dimensions of interaction models by implementing virtual engineering technologies in order to properly represent the information and for a better reuse of the knowledge [START_REF] Laroche | Towards the factory of future An integrated approach of materialprocesses-information-human being[END_REF].

The first part of this paper presents the evolution made on decision support systems. In the second part we expose a state of the art in Decision Support Systems and data, information, and knowledge retrieval. After that, we expose the ARTUR project and present the idea about multi-scale knowledge structuring before we conclude.

II. EVOLUTION OF DECISON SUPPORT SYSTEMS

A. Definition

Basically, Decision Support Systems (DSS) are information systems with a high capability of reasoning in order to provide their users with a set of appropriate information. Shim [START_REF] Shim | Past, present, and future of decision support technology[END_REF] defines this concept as a set of computer technology solutions that can be used to support complex decision making and problem solving. Many other definitions of those systems were found in the literature. Definitions are highly related to the application field of decision support systems but they share a common aspect: DSS do not make decisions by themselves but attempt to automate several tasks of the decision making process. [START_REF] De Kock | Decentralising the codification of rules in a decision support expert knowledge base[END_REF]

B. Evolution of DSS

Since the appearance of DSS, those systems evolved in a significant way, thinks to the parallel evolution made in information and communication technologies. From the technology implemented point of view, this evolution lead to the appearance of a new era of DSS like web-based DSS [START_REF] Boreisha | Web-based decision support systems as knowledge repositories for knowledge management systems[END_REF], Mobile DSS [START_REF] Wen | A mobile knowledge management decision support system for automatically conducting an electronic business[END_REF], Knowledge Based DSS [START_REF] Klein | Knowledge-based decision support systems with applications in business: a decision support approach[END_REF], and ubiquitous DSS [START_REF] Kwon | UbiDSS: a proactive intelligent decision support system as an expert system deploying ubiquitous computing technologies[END_REF].

As showed in figure 1, types of decision support systems had moved from individual DSS to a new category of DSS called ubiquitous decision support systems (ubiDSS). The ubiquitous technology is a vision introduced by Weiser [START_REF] Weiser | The computer for the 21st century[END_REF] in the end of 80s. It consists of a new model of interaction where the information processing is integrated in our daily life. The main idea of ubiDSS concept is the use of new ICT technologies that assure a high level of information and knowledge availability. Many works argue the use of this concept to build a DSS and the development of frameworks based on ubiDSS [START_REF] Kwon | UbiDSS: a proactive intelligent decision support system as an expert system deploying ubiquitous computing technologies[END_REF] [9] [START_REF] Lee | A Study on Building Web 2.0-based Decision Support System: The extended concept from ubiDSS[END_REF]. Even that many applications are built to validate those frameworks; few of them deal with industrial production domain. Our proposition is to move the application field of DSS to a "machine level" in production systems and combine it with knowledge based models. We believe that this field of application of decision support systems will lead to efficient product development by improving the effectiveness of employer's decision in the different phases of product lifecycle.

III. DATA, INFORMATION, AND KNOWLEDGE RETRIEVAL

Using advanced data management is a critical point to develop an efficient decision support system. In fact, due to the big evolution of digital tools in the industry, the output in terms of information is consequently the quantity of data and information manipulated in today's factory is growing. Consequently, forwarding the right set of knowledge to the right person who needs it becomes a crucial issue. In this section, we highlight the difference between Data, Information, and Knowledge and the retrieval systems related to each of them.

A. From Data to Wisdom: The DIKW hierarchy

In the literature review regarding knowledge Management, it was often highlighted the relation between data, information, and knowledge. The DIKW (Data-Information-Knowledge-Wisdom) hierarchy reflects this relation. It was first introduced by Ackoff [START_REF] Ackoff | From data to wisdom[END_REF] in 1988; he gave a hierarchical representation of the terms through the DIKW pyramid (figure 2). Since that, many authors have tried to describe more and more the relation between the different concepts in both Knowledge Management and Computer Science research field. [START_REF] Xu | Proposition d'un système intégrée de référence de connaissances pour l'évaluation de la connaissance dans le développement de produits[END_REF][13] We will not discuss those prepositions in this paper but we will emphasize the difference between Data, Information, Knowledge, and Wisdom.  Data: data can exist in any form and assumed to be simple isolated facts. [START_REF] Ahsan | Data, information, knowledge, wisdom: A doubly linked chain[END_REF][12]  Information: information is data with a meaning.  Knowledge: knowledge is information connected with relations.  Wisdom: wisdom can be defined as putting into action accumulated knowledge B. Data, Information, and Knowledge Retrieval Many computer systems have been developed to manage data, information, and knowledge as has been seen in the previous section. Data Retrieval Systems like database management systems are well designed to manage and structure data. Information Retrieval Systems are designed to get relevant information and acquire more knowledge (web search engines). Also, Knowledge Retrieval Systems that support knowledge discovery, organization, and retrieval with the aim of providing knowledge to users in structured way.

Yao [START_REF] Yao | Knowledge retrieval (kr)[END_REF] defines in his work a conceptual framework of a knowledge retrieval system based on three main components: discovery of knowledge, construction of knowledge structures, and the inference of required knowledge. Table 1 represents a comparison between data, information, and knowledge retrieval. In some practical situations like manufacturing systems, factory actors need to get information and to acquire knowledge to perform better in front of difficult tasks. The set of knowledge have to be presented in a structured way that a simple search engine, for example, cannot provide.

Région Pays de la Loire

The next section describe ARTUR project. Among the difficulties that this project aims to resolve is the difficulty of information/knowledge structuring in a Decision Support System for the purpose of a better reuse.

IV. ARTUR: TOWARD AN UBIQITIOUS DECSION SYSTEM

Related to visions of the factory of the future, the proposition of the project ARTUR is to offer the operator in the factory with a so called "Digital Factory Assistant". Through this assistant the operator can have information from distributed knowledge resources relying in different ways of interaction in order to act upon the task that he is doing. So, the main research issue of our work is to find which information to give to the operator, where, when, and how. The basic idea is to allow each person to be able to react in given situation, mainly thanks to high speed simulation and performance evaluation models and methods. [1][20] Figure 3. The different elements in ARTUR project Project ARTUR deals with some particular aspects that other DSS seen in the literature does not treat:  Modeling the interactions in the working situation:

What are the different elements that the operator deals with in front of the machine and how they will be presented in the assistant? A particular interest will be given to the evaluation of human-machine interactions.  Creating the ubiquitous environment: the ubiquitous environment will be created by the data flow coming from different resources of the information system in the factory including PLM systems, ERP, Cax models…  Using Virtual Engineering technologies: this aspect is used in other decision support systems but the difficulty that we try to resolve is how to define new ways of interaction dedicated to non-specialized employers by extending the dimensions of Virtual Technologies models with further data sets.

 Coupling the information system in the factory with other knowledge models so the decision support aspect according to our proposition will be not disconnected from the information system in the factory.

Other particular aspect from those cited previously is the multi-scale knowledge structuring. This aspect will be more detailed in the next section.

V. MULTI-SCALE KNOWLEDGE STRUCTURING

As mentioned above, one of the main research propositions in the project ARTUR is the way of structuring the knowledge and information to be given to the operator. Figure 3 illustrates a high conceptual level of our vision of multi-scale knowledge structuring. Taking into account decision support models, we aim to introduce a new way of definition, representation, and exploration of knowledge. We believe that the unit of knowledge should be presented to an operator in a multi-level structure where each level will be based on contextual set of information. Contextual information will be based on user profile: maturity level, knowledge already known, previous experience... The relevant set of knowledge can be decomposed into hierarchical structure and represented in a multi-level way. According to his/her needs, the user will look for more granular set of knowledge that satisfies his/her query. Some authors discuss the granularity of knowledge. [16][17]. March [START_REF] Mach | Knowledge Granularity and Representation of Knowledge: Towards Knowledge Grid[END_REF] [19] link the origins of knowledge granularity to three reasons: the multiple sources of knowledge, existence of different ways of knowledge representation, and the diversity of knowledge application domains.

Foundations of our research proposition relate to a new proposition for modeling and structuring information, and more generally knowledge. Consequently, dealing with the global data, information, knowledge and knowhow, we will be able to contextualize reusing this knowledge in any given situation that would need this knowledge.

Our assistant will be based on a dynamic knowledge representation: In a process of decision making, the actor will combine different types of data and knowledge available in various forms [START_REF] Dhuieb | Digital Factory Assistant : Conceptual Framework and research propositions[END_REF]. CONCLUSION In a production system, the user may be faced with working situations that require him/her to interact with the machine in front of him/her by choosing the right decisions. The decisions can be a force to be applied, a position of a tool in the machine… So, to make his/her decision the user will need to have such propositions of what he/she can do or what measure he/she will take and, using advanced simulation techniques, he/she can anticipate the influence of his/her choice.

To validate all those aspects in our project ARTUR, we will be based on two use cases in aeronautic industry. A survey will give us more detail on what elements in the factory's information system should interact with our assistant model and to identify the different knowledge sources and domains related to the executed process.

Figure 1 .

 1 Figure1. Evolution of Decision Supoort Systems[START_REF] Kwon | UbiDSS: a proactive intelligent decision support system as an expert system deploying ubiquitous computing technologies[END_REF]

Figure 2 .

 2 Figure2. The DIKW hierarchy[START_REF] Ackoff | From data to wisdom[END_REF]

Figure 4 .

 4 Figure 4. Conceptual vision of multi-scale structuring approach

TABLE I

 I

	.	COMPARISON OF DATA, INFORMATION, AND KNOWLEDGE
		RETRIEVAL [15]
		Data retrieval	Information Retrieval	Knowledge Retrieval
				Knowledge
	Query	Artificial language	Natural language	structure + natural
				language
				Knowledge unit
	Organization	Table, index	Table, index	and knowledge
				structure
	Storage	Database	Document collections	Knowledge base
	Retrieved results	Data set	Sections or documents	A set of knowledge unit