C Delebarre
email: corentin.delebarre@turbomeca.fr

V Wagner

J Y Paris

G Dessein

J Denape

J Gurt-Santanach

Toulouse Archive Ouverte HIGH SPEED INTERACTION BETWEEN AN ABRADABLE COATING AND A LABYRINTH SEAL IN TURBO-ENGINE APPLICATION

Keywords: Labyrinth seal, abradable material, high speed interaction 1-Introduction

The design of gas turbine aims to enhance the engine efficiency by developing new materials able to work at higher temperatures, or to promote new technologies, fuel management and airflow direction. One solution is the reduction of clearance between rotary parts in turbomachinery air systems. This clearance reduction causes direct interactions in the secondary air system of a turbo-engine when a rotary seal, called labyrinth seal, rubs on the turbo-engine as a result of successive starts and stops, thermal expansions and vibrations. The purpose of the present paper is to study interaction phenomena between an abradable material (Al-Si 6%) and a nickel alloy (718 alloy) during high speed contacts. A high speed test rig has been designed to simulate interactions between labyrinth seals and abradable coatings in similar operating conditions of turbo-engine in terms of geometries, rotational and linear velocities. A series of experiments has been carried out in order to get a first assessment under different turbo-engine operating conditions. Experimental results are presented using visual observations of test samples, quantitative approaches of interacting forces and micrographic observations. This work provides new basic data for a preliminary study of the interaction between a labyrinth seal teeth tips and its casing for turbo-engine applications.

During decades, turbomachinery manufacturers try to find how to save energy by increasing engine efficiency. One way to improve engine efficiency is to control the airflow direction by reducing the clearance between rotary parts in air systems [START_REF] Dorfman | Gas turbines use abradable coatings for clearance-control seals[END_REF]. In the secondary air and sealing systems, the control pressure differences and the levels of cooling between modules of the engines are crucial in order to operate the turbine. As a part of dynamic sealing systems, labyrinth seals are used to reduce the loss of high pressure gas within the engine cycle. This special type of rotary seal controls cooling air flow through the hot section of the engine and maintains pressure balance on the rotor shaft system. They are located on the rotating shafts between the compression stages and are composed of several teeth [START_REF] Dowson | Development of abradable and rub-tolerant seal materials for application in centrifugal compressors and steam turbines[END_REF].

Nevertheless, reducing the clearance might cause undesirable interactions between labyrinth teeth tips and housing [START_REF] Schmid | An overview of compressor abradable thermal sprays[END_REF]. This is also true for successive start and stop cycles, thermal expansions, vibrations and mechanical loadings occurring during engine operation [START_REF] Jacquet-Richardet | Rotor to stator contacts in turbomachines. review and application[END_REF]. In this case, labyrinth teeth rub into the housing surface and form rub-grooves on it. To avoid undesirable wear due to incursions of labyrinth teeth in the bare metal housing, the insert of a few millimeters abradable material layer was widely recognized as a robust solution. The turbo-engine housing is coated with a sacrificial abradable material (rub tolerant material) sprayed by thermal spraying [R1]. This coating can be composed of metal phase and self-lubricating non-metal phase, which provides a high porosity rate and offers a good balance between abradability and erosion resistance [START_REF] Maozhong | Erosion wear behaviour and model of abradable seal coating[END_REF]. This method is widespread in other turbine-engine applications such as blade tip and compressor housing where extensive researches are conducted to characterise blade-abradable interaction [START_REF] Collins | The degradation of abradable honeycomb labyrinth seal performance due to wear[END_REF]. The use of abradable seal to prevent rub interactions encourages aeronautical engineers to characterize the abradable behavior during labyrinth seal interactions and to validate the design choices in the various stages of the engine.

Today, research activities are mainly concentrated on the labyrinth seal behavior in their sealing performance before and after interaction. Many experimental studies (with development of specific test rig) and numerical studies help characterize the effect of interactions on the aerodynamic seal performance, leakage levels generated by the grooves left by labyrinth teeth and the effect on tooth profiles [START_REF] Delgado | Continued investigation of leakage and power loss test results for competing turbine engine seals[END_REF][START_REF] Collins | The degradation of abradable honeycomb labyrinth seal performance due to wear[END_REF][START_REF] Gamal | Labyrinth seal leakage test: tooth profile, tooth thickness, and eccentricity effects[END_REF].

Concerning interactions between labyrinth seal and abradable coating, very few studies are devoted to understanding the contact. Dowson et al [START_REF] Dowson | The investigation of suitability of abradable seal materials for application in centrifugal compressors and steam turbines[END_REF] and Whalen et al [START_REF] Whalen | Thermoplastic labyrinth seals for centrifugal compressors[END_REF] developed a full-scale facility based on a grinding machine, to study the behavior of abradable materials (mica filled tetrafluoroethylene (TFE), silicone rubber, aluminium polyester and nickel graphite) in contact with labyrinth seal at relative speed up to 25 m•s -1 and 130 m•s -1 respectively. They performed contact with incursion speeds varying from 2.54 m•s -1 to 25.4 m•s -1 which are defined as representative conditions of labyrinth thermal expansions and vibrations. They established a "good abradability" condition for tested materials by defining a special ratio. Mutazim et al [START_REF] Mutasim | Evaluation of plasma sprayed abradable coatings[END_REF] introduce an additional condition describing interaction by characterizing the rub-groove geometry of seven abradable materials (aluminium, Al-bronze, Al-Si-polyester, Al-bronzepolyester, Al-Si-polyimide, Cu-Sn-Bi, Cu-Sn-Pb) subjected to labyrinth seal interaction. A specific full-scale test rig was developed and performed under unlubricated conditions.

Later, Sulzer Innotec and Sulzer Metco companies [W1] worked closely on the development of abradable materials, by using special test rig adapted to the application of labyrinth/abradable interactions. It consists of an alloy disc that accommodates four continuous seal trips on the outer circumference, a movable specimen controlled by a programmable controller coated with abradable (step length of 0.15 µm), and a heating device (from 25°C to 700°C). The wear behavior is evaluated using the seal strip cutting-edge velocity, the temperature of the shroud segment, the wear track depth and the incursion depth and the video of test.

The present paper describes the results of an experimental investigation carried out to undertake a first assessment of high speed mechanical interactions between a labyrinth seal of 718 alloy and an abradable Al-Si 6% coating. An original test rig is specially designed to simulate these interactions in realistic operating conditions.

2-Experimental protocol

2-1 Test rig description

The experimental device presented in Fig. 1 has been specially designed and developed to simulate and to study operational wear interactions between labyrinth seals and abradable coatings in similar operating conditions as turbo-engine. For that, a 5-axis milling machine UCP600 VARIO from Mikron is used as a test rig and receives a specific experimental device. The 5-axis milling machine has the particularity to be fitted with a special magnetic bearings spindle which allows a maximum rotation speed of 40 000 rpm and maintains the spindle vertically without any mechanical contact.

A labyrinth seal, representative of an actual part of a motor shaft section is shrinking on a HSK-50 tool holder instead of a cutting tool (Fig. 2a). The contact is performed inside a special tube sample coated with abradable materials in this inner periphery. This specimen is representative of the housing facing labyrinth seals in the secondary airflow turbo-engine section (Fig. 2b). A diffuser and a base piece precisely fasten the tube sample coated with abradable on the machine tool table. The diffuser is specially designed to simulate the sealing of the labyrinth seal during operation by pressurizing the entire system. It will allow, for future tests, the measurement of their leakage level and their sealing performance after interactions. In case of sample break during the test, a protective shielding has been designed.

The rotation speed of labyrinth seal varies from 0 to 40 000 rpm, representative of typical speed reached by free turbines in turbo-engines, and corresponding to a labyrinth tip speed varying from 0 to 130 m•s -1 . Contact between labyrinth seal and abradable is controlled by using capabilities of the high speed milling machine, which gives rise to a radial incursion speed ranged from 0.001 mm•s -1 to 25 mm•s -1 . Geometric tolerances of samples and theirs assemblies require a precise control of a clearance of 150 µm between labyrinth teeth tips and abradable coating, according to real engine conditions.

2-2 Contact geometry

Interactions between labyrinth tips teeth and abradable coating are characterized by a complex contact geometry induced by the incursion of a circular tooth of trapezoidal section in a tube coated with abradable. This contact geometry evolves during the test according to the incursion depth, as shown in Fig. 3. Nevertheless, the machining precision of labyrinth seal teeth (geometric tolerances, unbalance…), the positioning precision of test pieces and the spindle stiffness (supposed infinite during contact), have an important impact on the nature of the interaction. Once the labyrinth seal precisely positioned in the abradable test piece, the concentricity error is stated to ±30 µm. In addition, for the reasons given above, the use of three teeth labyrinth seal leads to get three different contact behaviors, one for each tooth, with the same test condition. The geometric confinement of the central tooth probably leads to thermal phenomenon during the interaction.

2-3 Contact forces measurement

A special instrumentation has been developed on the rig test to record forces during interactions thought the continuous control current of the spindle magnetic bearings. This continuous control current is performed to maintain the rotation shaft axis in the centre responsive to any change of load (external force, disturbance of a solid rotor imbalance). In our case, interacting forces induced by the labyrinth seal incursion are directly transmitted in the magnetic bearings and compensated through control current signals. These compensated signals are then recorded by a special Matlab toolbox, and processed as contact force signals (Fig. 4). The force measurements are then represented using the norm defined in the following equation:

)² ( )² ( )² ( fz fy fx F + + = (1)

2-4 Test parameters

As mentioned earlier, the labyrinth seal specimens analyzed in this study was made of 718 alloy whose chemical composition is presented in Table 1. It was designed with three circular teeth of 65.8 mm in external diameter. The Al-Si 6% coating was sprayed inside the housing support on a stainless steel substrate. Spraying is realized with a thermal wire flame spray process onto a bond coat approximately ranged from 20 to 150 µm (Fig. 5). The high roughness of the bond coat promotes good cohesion of the deposit. Once coated, the housing support is machined by milling to control the abradable thickness (about 1 mm) which defined the mechanical clearance between abradable coating and labyrinth seal teeth. The main studied test parameters are labyrinth tip speed, rotation speed, incursion speed and incursion depth. A test matrix has been selected to cover various conditions encountered in turbo-engine such as engine vibrations, excessive mechanical loading or thermal expansion. Table 2 shows the test parameters studied in this work. It is important to mention that all the tests were carried out with only one labyrinth seal sample.

3-Data analysis and discussion

3-1 Analysis of test samples

A first post mortem analysis is performed by a visual observation of labyrinth seal teeth and abradable coating after test. Fig. 6 presents all the rub-grooves left by labyrinth seal teeth on Al-Si 6% abradable for each test parameter. As seen in Fig. 6, different morphologies of rub-grooves are observed according to test parameters. Some rub-grooves present a significant amount of plastically deformed material on both sides of the groove. This is the case for tests at high incursion speed (tests n°2, 4, 6) and for test n°1 at low incursion speed and labyrinth tip speed of 130 m•s -1 . Unlike others, rub-grooves from tests n°3 and 5 (low incursion speed and low rotational speed) display geometries that may be qualified as "clear-cut" appearance without positive amount of material. Visual observations of labyrinth seal teeth after test did not show any noticeable changes or severe wear except for labyrinth seal teeth from tests n°3 and 4. Al-Si 6% coating transfer was observed on the top of labyrinth seal teeth, as shown in Fig. 7. Such results suggest that incursion speed V inc is a significant parameter for rub-grooves morphology. However, results of test n°1 demonstrate the additional influence of the labyrinth rotation speed. To confirm rub-grooves visual observations, accurate profile measurements were carried out with a SOMICRONIC 2C profilometer at the maximum incursion depth area (Fig. 8). Positive heights and negative depths are identified and quantified respectively by Z 1 and Z 2 values. In most cases, rub-grooves formed on Al-Si 6% were widely deeper than the 200 µm fixed by the maximum labyrinth seal radial incursion. In addition, on each test, the deeper rubgroove corresponds to the tooth which is confined between the others two. This is consistent with the hypothesis of the presence of an additional wear phenomenon within the affected area because of thermal confinement.

Fig. 9 presents the maximum heights of the deformed coating according to the rotation and the incursion speed. As the heights bars show, the labyrinth tip speed has a significant effect on the heights of deformed coating at low incursion speed (V inc = 0.005 mm•s -1 ). In fact, at 43 m•s -1 and 17 m•s -1 , deformed coating is lower whereas rub-grooves depths are identical. This examination indicates the presence of material deformation phenomena at high rotation speed, in contrast to material removal for lower rotation speed. This difference of wear mechanism may be explained by different dissipation level of the contact temperature in the Al-Si 6% coating. This difference is affected by the rotation speed of the labyrinth seal, which impacts the ductile behavior of the coating. It is therefore interesting to observe the interaction forces during the contact. Secondly, the norm is used as a quantitative approach and allows the establishment of a maximum force value induced by the contact. Fig. 12 presents maximal forces recorded versus rotation and incursion speed. Maximum forces generated by tests at high incursion speed are significantly higher than those at low incursion speed. At low incursion speed, maximum forces are included in a range from 1 % to 9 % of the maximum forces compared to a range from 38 % to 100 % of the maximum forces for high incursion speed. Moreover, rub-grooves characterized by a low amount of deformed coating are obtained with forces lower to 9 % of the maximum forces.

Another trend is observed by comparing the maximum forces evolution for different incursion speeds. At low incursion speed, the increase of rotation speed induces a gradual increasing of the maximum interaction forces. Conversely for high incursion speed, when the rotation speed decreases, interaction forces increases.

3-3 Micrographic rub-grooves observations

Micrographic rub-grooves observations were performed on the Al-Si 6% abradable surface from test n°1. Fig. 13 shows a section view of three rub-grooves from test n°1 (V inc = 0.005 mm•s -1 , V r = 37 500 rpm, D p = 0.2 mm). This view is the place where the incursion depth of the labyrinth teeth is the greatest. Rub-grooves formed in Al-Si 6% are slightly wider than the initial tooth dimension. Considering the coiled shape of coating deformed outside the rub-groove, the SEM image demonstrates the existence of a material expulsion mechanism by plastic deformation on the both side of the rub-groove (fig. 13). Moreover, a slight Al-Si 6% densification is identified around the rub-groove.

Nevertheless, EDX analysis of the rub-grooves on the Al-Si 6% coating identified the presence of nickel and chrome, which are basic elements of 718 alloy, coming from the labyrinth seal teeth (Fig. 14a). This mater transfer from the labyrinth seal to the abradable coating is made of an accumulation of layers in the rub-groove (Fig. 14b).

4-Conclusion

To optimize turbo-engines performances by reducing the mechanical clearance between rotary parts, contact between labyrinth seals and abradable coatings must be understood. This requirement needs to study and characterize a material couple (where one material undergoes while the other imposes) by simulating interactions in operating conditions. Interactions between an Al-Si 6% abradable coating and a 718 alloy labyrinth seal were performed using a high speed test rig specially developed to achieve a maximum interaction speed of 130 m•s -1 .

A first assessment of high speed interactions has been carried out by an analysis of test samples. Visual rub-grooves observations, accurate profile measurements, maximum interaction forces analysis and micrographic examinations reveal different material deformation and wear phenomena and significant interaction parameters:

• Observation of damages by a phenomenon of deformation on the Al-Si 6% coating at high incursion speed and high rotation speed.

• Observation of a material removal wear phenomenon at low incursion speed and low rotation speed considering transfer on Al-Si 6% coating from labyrinth seal teeth, characterized by an accumulation of layers in the rub-groove.

• Influence of the labyrinth rotation speed on the rubgroove morphology and on interaction forces. However, it is necessary to complement this study by evaluating the abradable behavior about radial incursion. A test campaign should be achieved by fixing test parameters to perform successive incursions.

5-Acknowledgment

These investigations were supported by the European Commission within the FP7 E-BREAK project under Grant agreement no: 314366 and by SAFRAN-TURBOMECA, and received the financial support of the Agence Nationale de la Recherche et de la Technologie (ANRT) which are both gratefully acknowledged.

Figure 1 :

 1 Figure 1: Labyrinth-abradable interaction test rig.

Figure 2 :

 2 Figure 2: (a) Labyrinth seal specimen, (b) Abradable coating on housing support.

Figure 4 :

 4 Figure 4: Schematic diagram of forces acquisition.

Figure 3 :

 3 Figure 3: Sectional view of the evolution of the contact geometry.

Figure 5 :

 5 Figure 5: Microstructure in section view of Al-Si 6% coating on stainless steel substrate.

Figure 6 :

 6 Figure 6: Images of abradable rub-grooves after testing.

Figure 7 :

 7 Figure 7: Al-Si 6% transfer on labyrinth seal teeth after test n°3 (V r = 12 500 rpm, V inc = 0.005 m•s -1 ).

Figure 8 :

 8 Figure 8: Rub-grooves measurement depth, corresponding to the six test conditions.

Figure 9 :

 9 Figure 9: Maximum heights of deformed coating according to rotational and incursion speed. 3-2 Analysis of interaction forces The use of interaction force norm provides a global vision of the changing forces during tests. Firstly, two different sets of signals are observed and are associated to different incursion speed (25 mm•s -1 and 0.005 mm•s -1 ). Fig. 10 shows a typical signal recorded at low incursion speed. It reveals cyclic variations characterized by non-contact periods where forces are close to zero. This phenomenon may probably be associated to a sudden Al-Si 6% pick-up and non-contact time because mechanical clearance adjustment. The increase in maximum forces at each cycle can be in part explained by the increasing of the contact surfaces. Fig. 11 shows a typical signal recorded at high incursion speed. It is composed of a very fast rise time (≈ 0.005 ms), of a maximum which corresponds to the halt of the labyrinth seal incursion and of a fall of forces.

Figure 10 :

 10 Figure 10: Typical forces signal at low incursion speed (V inc = 0.005 mm•s -1 , Vr = 12 500 rpm).

Figure 11 :

 11 Figure 11: Typical forces signal at high incursion speed (V inc = 25 mm•s -1 , Vr = 37 500 rpm).

Figure 12 :

 12 Figure 12: Maximum interaction forces versus rotation speed (Vr) and incursion speed (V inc ).

Figure 14 :

 14 Figure 14: (a) SEM image in backscattered electron mode of the first rub-groove from test n°1, (b) SEM image of accumulated layers of 718 alloy.

Table 1 : Chemical composition and properties of 718 alloy.

 1

		Elements (% by mass)
		Ni	Cr	Fe	Mo	Nb	Co	Mn
	Min	50	15		2.8	4.75
	Max	55	21	Base	3.3	5.50	1.0	0.35
	Material properties :
		Tensile strength (MPa)				1310
		Yield Strength (MPa)				1110
		Elastic modulus (GPa)				206
		Hardness (HV150)					370
		Density (g•cm 3 )					8.19
		Melting point (°C)					1300
		Thermal conductivity (W•m•K -1 )			11.2

Table 2 : Test parameters.

 2

	Test n°	Rotation Speed (V r ) (rpm)	Labyrinth tip speed (V t ) (m•s -1 )	Incursion speed (V inc ) (mm•s -1 )	Incursion depth (D p ) (mm)
	1	37500	130	0.005	0.2
	2	37500	130	25	0.2
	3	12500	43	0.005	0.2
	4	12500	43	25	0.2
	5	5000	17	0.005	0.2
	6	5000	17	25	0.2


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Methodological framework for implementation of a prediction reliability model of IGBT power modules used in railway applications

Essi Ahoefa Dabla, Carmen Martin, François Pérès, Florent Andrianoelison,

Clair Fournier, Michel Piton

To cite this version: Official URL: https://doi.org/10.1109/IROS.2016.7759625

Essi

  Ahoefa Dabla, Carmen Martin, François Pérès, Florent Andrianoelison, Clair Fournier, et al.. Methodological framework for implementation of a prediction reliability model of IGBT power modules used in railway applications. 2017 4th International Conference on Control, Decision and Information Technologies (CoDIT), Apr 2017, Barcelona, Spain. pp.0639-0644, 10.1109/CoDIT.2017.8102666. hal-02487137 OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible Any correspondence concerning this service should be sent to the repository administrator: tech-oatao@listes-diff.inp-toulouse.fr This is an author's version published in: http://oatao.univ-toulouse.fr/23625 To cite this version: Dabla, Essi Ahoefa and Martin, Carmen and Pérès, François and Andrianoelison, Florent and Fournier, Clair and Piton, Michel Methodological framework for implementation of a prediction reliability model of IGBT power modules used in railway applications. (2017) In: 2017 4th International Conference on Control, Decision and Information Technologies (CoDIT), 5 April 2017 -7 April 2017 (Barcelona, Spain).


Mounir Meriem Hamdoun
email: meriemhamdoun@yahoo.fr

Frédéric Rotella
email: rotella@enit.fr

Irène Zambettakis
email: izambettakis@iut-tarbes.fr

Minimal single linear functional observers for discrete-time linear systems

 in designing minimal functional state observers is extended to deal with discrete-time systems. One of the benefits of this approach is that it does not require solving the Sylvester matrix equation that appears in other observer design procedures. Both stable observers and arbitrary fixed poles observers problems are considered for minimality. A numerical example and simulation results explain the effectiveness and the benefits of the proposed algorithm.

I. INTRODUCTION

In observer design theory, it does not usually required to observe all the states of the system so only a few number of functions of the states can be estimated. Ever since Luenberger has introduced the concept of functional observer in 1966 [Luenberger, 1966], this field of study has been a subject of intensive researches (see [O'Reilly, 1983], [START_REF] Aldeen | Reduced-order linear functional observer for linear systems[END_REF], [Darouach, 2000], [START_REF] Trinh | Functional observers for dynamical systems[END_REF]). This scheme is more general than reduced order Luenberger observers and it has been applied in several applications like observer-based feedback control, fault detection and system monitoring [START_REF] Trinh | Quasi-decentralized functional observers for the lfc of interconnected power systems[END_REF], [START_REF] Patton | Observer-based fault detection and isolation: Robustness and applications[END_REF]], [START_REF] Hou | Fault detection and isolation observers[END_REF]. Because of their ability to reduce the cost and complexity of control systems, there have been considerable attention drawn to functional observers. However, only little attention has been paid toward discrete-time systems.

In literature, we distinguish three design procedures to reconstruct a set of linear functions: the parametric approach ( [START_REF] Trinh | Design of scalar functional observers of order less than (l)[END_REF], [START_REF] Trinh | Algorithms for designing reduced-order functional observers of linear systems[END_REF]), the conventional approach ( [START_REF] Aldeen | Reduced-order linear functional observer for linear systems[END_REF], [Darouach, 2000], [Duan, 1992], and the direct approach [START_REF] Rotella | Minimal single linear functional observers for linear systems[END_REF], [START_REF] Rotella | A direct design procedure for linear state functional observers[END_REF].

The main problem faced in designing functional observers is finding a method to solve Sylvester equation that appears in the observer design approach ( [START_REF] Trinh | Design of scalar functional observers of order less than (l)[END_REF], [START_REF] Trinh | Functional observers for dynamical systems[END_REF]). These interconnected equations usually have an infinite number of solutions and each method in the solving tries to optimize some criteria and satisfy robustness and performance.

The direct procedure for designing linear state functional observers was first presented in [START_REF] Rotella | Minimal single linear functional observers for linear systems[END_REF] to estimate a single function of the states of a Linear Time-Invariant (LTI) system. This scheme is the only available one extended to a Linear Time-Varying (LTV) systems in [START_REF] Rotella | On functional observers for linear time-varying systems[END_REF]]. An extension to designing minimal multi-functional observers is proposed in the paper of [START_REF] Rotella | A direct design procedure for linear state functional observers[END_REF], [START_REF] Mohajerpoor | Minimal Multi-Functional Observers for Linear Systems Using a Direct Approach[END_REF].

Comparing to the other observer design approaches, the direct scheme is the only method that does not require solving the Sylvester equation to find the observer parameters, it's based on linear algebraic operations in a state space setting.

In the present paper, the direct approach is extended to reconstruct single linear functional of the states for large-scale linear discrete-time systems in such way that the minimum possible order of the observer is achieved.

The paper is structured as follows. The problem is illustrated in Section II, and some preliminary definitions, theorems, and remarks are given. An extension of the direct approach in designing a minimal single linear functional observers to the discrete-time framework is proposed in Section III. In Section IV, the design procedure illustrated with the same numerical example provided in [START_REF] Rotella | Minimal single linear functional observers for linear systems[END_REF], and simulation results are reported. The paper is concluded in Section V.

II. PRELIMINARIES

Lets consider the following discrete-time LTI system:

x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) (1)
where x(k) ∈ R n , u(k) ∈ R p and y(k) ∈ R m are the state vector, the control input vector and the measurement output vector, respectively. Matrices

A ∈ R n × R n , B ∈ R n × R p and C ∈ R m × R n are known constant.
Let υ(k) be a vector that is required to be reconstructed (or estimated), such that:

υ(k) = Lx(k) (2)
and L ∈ R (l×n) a constant full row rank known matrix.

To reconstruct the state function, υ(k), the following Luenberger observer structure of order q, q ≤ (np), is proposed

z (k + 1) = F z (k) + Gu (k) + Hy (k) w (k) = P z (k) + V y (k) (3)
where z(k) is the q-dimensional state vector and w (k) ∈ R l is the estimate of υ(k).

The observer matrices F ∈ R (q×q) , G ∈ R (q×p) , H ∈ R (q×m) , P ∈ R (l×q) and V ∈ R (l×m) are determined such that:

lim k→∞ (υ (k) -w (k)) = 0 (4)
This asymptotic tracking is ensured if F is a Schur matrix. Namely, if all the eigenvalues of F are inside an unit open circle.

Following [START_REF] Fortman | Design of a low order observer for linear feedback control laws[END_REF], the linear functional observer (3) exists if and only if there exists a (q×n) matrix T such that:

G = T B (5) T A -F T = HC (6)
L = P T + V C (7)
F is a Schur matrix.

Note that L can always be chosen to be a controller gain that stabilizes the closed-loop system matrix (A + BL), then the linear functional observer (3) would provide an estimate of the corresponding control signal to be directly feedback into the system. On the other hand, the designer can always chose L to represent any desired partial set of the state vector that needs to be estimated.

In Fig 1 the observer structure design is displayed. Recently, the interesting notion of functional observability of the triplet (A; C; L) defined in [START_REF] Fernando | Functional observability and the design of minimum order linear functional observers[END_REF], [START_REF] Jennings | Existence conditions for functional observability from an eigenspace perspective[END_REF] was introduced to tackle the problem of designing a minimal order functional observer. Definition 1. [START_REF] Fernando | Functional observability and the design of minimum order linear functional observers[END_REF], [START_REF] Jennings | Existence conditions for functional observability from an eigenspace perspective[END_REF] The triple (A, C, L) is functional observable if and only if

rank                           C CA . . . CA n-1 L LA . . . LA n-1                           = rank         C CA . . . CA n-1         (9)
Theorem 1. [START_REF] Fernando | Functional observability and the design of minimum order linear functional observers[END_REF], [START_REF] Jennings | Existence conditions for functional observability from an eigenspace perspective[END_REF] There exists an asymptotic functional observer with structure (3) for the system (1) if the triple (A, C, L) is functional observable.

Clearly when the condition ( 9) is satisfied, there are matrices L 0 , . . . , L n-1 , such that:

L = n-1 i=0 L i CA i (10) therefore rank             C CA . . . CA n-1 L             = rank         C CA . . . CA n-1         (11)
Conversely, suppose that L can be written as (10). Thus, it is easy to prove by induction that, for all k ∈ {0, . . . , n -1}, there exist L k,i matrices such that:

LA k = n-1 i=0 L k,i CA i
These relationships lead to (9). So, we can claim the triplet (A; C; L) is functionally observable if and only if (11) is fulfilled.

Functional observability is a sufficient but not necessary criteria for the existence of an asymptotic observer. Nevertheless, functional detectability defined in [START_REF] Fernando | Functional observability and the design of minimum order linear functional observers[END_REF] is a necessary and sufficient condition to investigate the existence of a solution to the observer design problem.

An

alternative to characterize the functional observability for linear systems has been developed in [START_REF] Rotella | A note on functional observability[END_REF]. A necessary and sufficient condition for the existence of a stable multi-functional observer of a time-invariant linear system is defined. This criterion to test functional observability of a triplet (A; C; L) leads to a constructive procedure of functional observer neither based on the use of canonical state space forms nor on the solution of a Sylvester equation.

III. DESIGN OF MINIMAL SINGLE LINEAR FUNCTIONAL OBSERVERS IN DISCRETE-TIME FRAMEWORK

Let us define q as the smallest integer such that:

rank (Σ q ) = rank Σ q LA q (12)
with:

Σq =             C L CA LA . . . CA q-1 LA q-1 CA q             (13)
After q phase-advanced of υ(k) = Lx(k), we obtain:

υ(k + q) = LA q x(k) + q-1 i=0 LA i Bu(k + q -1 -i) (14)
From ( 12), there exist Γ i for i = 0 to q and Λ i for i = 0 to q -1 such that:

LA q = q i=0 Γ i CA i + q-1 i=0 Λ i LA i (15)
Thus ( 14) can be written as:

υ (k + q) = q i=0 Γ i CA i x (t) + q-1 i=0 Λ i LA i x (k) + q-1 i=0 LA i Bu (k + q -1 -i) (16)
To eliminate the state x(k) we have the equalities :

Lx (k) = υ (k) LAx (k) = υ (k + 1) -LBu (k) . . . LA (q-1) x (k) = υ(k + q -1) - q-2 i=0 LA i Bu (k + q -2 -i) Cx (k) = y (k) CAx (k) = y (k + 1) -CBu (k) CA 2 x (k) = y (k + 2) -CABu (k) -CBu (k + 1) . . . CA q x(k) = y(k + q) - q-1 i=0 CA i Bu(k + q -1 -i)
It yields:

υ (k + q) = q-1 i=0 LA i Bu (k + q -1 -i) +Γ 0 y (k) + q i=1 Γ i y(k + i) - i-1 j=0 CA j Bu (k + i -1 -j) +Λ 0 υ (k) + q-1 i=1 Λ i υ(k + i) - i-1 j=0 LA j Bu (k + i -1 -j) = q i=0 Γ i y (k + i) + q-1 i=0 Λ i υ (k + i) + q-1 i=0 Φ i u (k + i) (17)
where for i = 0 to q -2

Φ i =   LA q-1-i - q j=i+1 Γ j CA j-i-1 - q-1 j=i+1 Λ j LA j-i-1   B
(18) and:

Φ q-1 = [L -Γ q C] B (19)
The input-output recurrence equation ( 17) can be realized as the q-order state space observable system:

z (k + 1) = F z (k) + Gu (k) + Hy (k) w (k) = P z (k) + V y (k) (20)
where

F =      0 1 . . . . . . 0 1 Λ 0 Λ 1 . . . Λ q-1      , G =     φ 0 φ 1 . . . φ q-1     H =     Γ 0 + Λ 0 Γ q Γ 1 + Λ 1 Γ q . . . Γ q-1 + Λ q-1 Γ q     , P = 0 . . . 0 1 and V = Γ q .
IV. NUMERICAL EXAMPLE In order to illustrate the proposed design approach, let us consider the same example in [START_REF] Trinh | Design of scalar functional observers of order less than (l)[END_REF] and [START_REF] Rotella | Minimal single linear functional observers for linear systems[END_REF] with the continuous time LTI system given by:

ẋ(t) = A c x(t) + B c u(t) y(t) = C c x(t) (21)
with

A c =      -1 0 0 1 -2 0 -5 3 4 0 1 1 -8 3 0 -4 0 2 -6 0 0 0 0 1 -1      , B c =      0 0 0 0 1      C c = [ 1 0 0 0 0 ] (22)
and the single functional defined by:

L = [ 1 14 42 79 2 ] ( 23
)
Discretization of linear state space models ( 22) with the sampling time T = 0.1s leads to write in the discrete-time LTI system (1) with:

A =     
0.8907 0.0002 0.0057 0.0637 -0.18 -0.0475 0.6156 0.1809 0.2569 0.0033 0.027 0.0532 0.4722 0.164 -0.0045 -0.2794 0.0053 0.1003 0.5531 0.0304 -0.0152 0.0002 0.0062 0.0714 0.9059

     , B =      -0.0093 0.0001 -0.0002 0.0011 0.0952      , C = [ 1 0 0 0 0 ]
The following steps illustrate the procedure for designing the minimum observer. Testifying condition (12), we get rank(Σ 2 ) = 5 and rank Σ 2 LA 2 = 5. It is found that q = 2. As: 1417 -0.4325 14.3742 1.3226 -22.148 ] we deduce Λ 0 = -0.4325 and Λ 1 = 1.3226, leading to:

LA 2 Σ 2 -1 = [ -1.
F = 0 -0.4325 1 1.3226 .
The eigenvalues of F are {0.5919, 0.7307}. Then F is a Schur matrix which means that a minimal second-order observer can be designed.

A. Design of the minimal second-order observer

From LA 2 Σ 2 -1 we get Γ 0 = -1.1417, Γ 1 = 14.3742 and Γ 2 = -22.148. Thus, from (20) we get:

G = -0.1534 0.0535 , H = 8.4367 -14.9178 , P = [ 0 1 ] , V = -22.1480.
For an initial condition υ (0)w (0) = 200, the simulation results are given in Fig. 2.

It is clear that the estimated function has asymptotically converged to its true value. Moreover, the observer is designed with an arbitrary asymptotic convergence speed due to the eigenvalues of F . To increase performance and obtaining a fastest convergence rate, we have to augment the order of the observer as follows.

B. Design of the minimal third-order observer

A minimal third-order observer with partially fixed poles can be designed using the following procedure.

From:

CA 3 = [ Γ 20 Λ 20 Γ 21 Λ 21 Γ 22 ] Σ 2 , LA 2 = [ Π 0 ∆ 0 Π 1 ∆ 1 Π 2 ] Σ 2 ,
we deduce:

     Γ 20 Λ 20 Γ 21 Λ 21 Γ 22      =      0.2812 -0.0008 -1.3818 0.0010 2.1149      and      Π 0 ∆ 0 Π 1 ∆ 1 Π 2      =      -1.1417 -0.4325 14.3742 1.3226 -22.1480     
On the one hand, we have:

LA 3 = LA 2 A = Π 0 CA + ∆ 0 LA + Π 1 CA 2 + ∆LA 2 + Π 2 CA 3 , = Π 0 CA + ∆ 0 LA + Π 1 CA 2 + ∆ 1 [ Π 0 ∆ 0 Π 1 ∆ 1 Π 2 ] Σ 2 + Π 2 [ Γ 20 Λ 20 Γ 21 Λ 21 Γ 22 ] Σ 2 , =      ∆ 1 Π 0 + Π 2 Γ 20 ∆ 0 ∆ 1 + Π 2 Λ 20 Π 0 + ∆ 1 Π 1 + Π 2 Γ 21 ∆ 0 + ∆ 1 ∆ 1 + Π 2 Λ 21 Π 1 + ∆ 1 Π 2 + Π 2 Γ 22      T Σ 2
and on the other hand, as:

LA 3 = [ Γ 0 Λ 0 Γ 1 Λ 1 Γ 2 Λ 2 Γ 3 ] Σ 3
and Σ 3 is not invertible, we are then led to:

LA 3 = Γ 0 C + Λ 0 L + Γ 1 CA + Λ 1 LA + Γ 2 CA 2 + Λ 2 LA 2 + Γ 3 CA 3 , = Γ 0 C + Λ 0 L + Γ 1 CA + Λ 1 LA + Γ 2 CA 2 + Λ 2 [ Π 0 ∆ 0 Π 1 ∆ 1 Π 2 ] Σ 2 + Γ 3 [ Γ 20 Λ 20 Γ 21 Λ 21 Γ 22 ] Σ 2 , =      Γ 0 + Λ 2 Π 0 + Γ 3 Γ 20 Λ 0 + Λ 2 ∆ 0 + Γ 3 Λ 20 Γ 1 + Λ 2 Π 1 + Γ 3 Γ 21 Λ 1 + Λ 2 ∆ 1 + Γ 3 Λ 21 Γ 2 + Λ 2 Π 2 + Γ 3 Γ 22      T Σ 2
where Λ 2 and Γ 3 are two design parameters. It yields

Γ 0 = ∆ 1 Π 0 + Π 2 Γ 20 -Λ 20 Π 0 -Γ 3 Γ 20 , Γ 1 = Π 0 + ∆ 1 Π 1 + Π 2 Γ 21 -Λ 2 Π 1 -Γ 3 Γ 21 , Γ 2 = Π 1 + ∆ 1 Π 2 + Π 2 Γ 22 -Λ 2 Π 2 -Γ 3 Γ 22 ,
and:

Λ 0 = ∆ 0 ∆ 1 + Π 2 Λ 20 -Λ 2 ∆ 0 -Γ 3 Λ 20 , Λ 1 = ∆ 0 + ∆ 1 ∆ 1 + Π 2 Λ 21 -Λ 2 ∆ 1 -Γ 3 Λ 21 .
When Λ 2 and Γ 3 are chosen, these five parameters are known and we can design the third-order observer (20).

The poles of the matrix F are the roots of the characteristic polynomial p F (λ) = λ 3 -Λ 2 λ 2 -Λ 1 λ -Λ 0 which depends on the parameters Γ 3 and Λ 2 .

We choose these poles {-0.0370 ± 0.1698i, 0.4346}. It yields p F (λ) = λ 3 + 0.7437λ 2 + 0.0020λ + 209.9125, we get Λ 0 = 209.9125, Λ 1 = 0.0020 and Λ 2 = 0.7437. These equalities are consistent and yield Γ 3 = 325.6895. For these values we get the Luenberger observer defined by For the same initial condition, the simulation results of the third-order observer is given in Fig. 3. It is clear that all the estimated functions converge asymptotically to their true values.

Moreover, the estimated value with the third-order observer has the fastest convergence rate than the one with the secondorder observer (Fig. 3), which was expected due to the selected observer's eigenvalues.

V. CONCLUSION

A minimal single linear functional observers design for discrete-time LTI systems using the direct approach has been addressed. The observer is designed so that an asymptotic functional observer can be obtained with arbitrary convergence speed. A numerical example and simulation illustrated the effectiveness of the proposed approach. In this example, we have points out that we can fix the observation error at any desired rate. Future works will concern the extension of the direct approach to unknown-input functional observers with minimum possible order.

Fig. 1 .

 1 Fig. 1. Linear functional observer structure design.

  Fig. 2.Simulation results for the implementation of the second-order functional observer

Fig. 3 .

 3 Fig. 3. Simulation results for the implementation of the third-order functional observer

Fig. 4 .

 4 Fig. 4. Estimation errors of the second and the third-order functional observer


Quan Liu
email: quan.liu@enit.fr

Ayeley Tchangani
email: ayeley.tchangani@iut-tarbes.fr

Bernard Kamsu-Foguem
email: bernard.kamsu-foguem@enit.fr

Franc ¸ois Pérès
email: francois.peres@enit.fr

Modelling a large scale system for risk assessment

Keywords: complex systems, interdependency, modelling, ontology, OOBN

The aim of this communication is to present an earlier study of how to structure modelling process of complex and large scale systems for risk assessment and management purpose. The approach, in a first stage, uses ontology paradigm to determine variables (or concepts) characterizing system locally and the nature of relationships relating them and, in a second stage, object oriented Bayesian network (OOBN) to characterize the strength of these relationships in terms of conditional probabilities tables given that one of main feature of complex systems is the uncertainty that affect the relationships between different variables. A case study in the domain of risk assessment of flash floods effect on the infrastructures inoperability is considered to show potential applicability of the developed approach.

I. INTRODUCTION

Capabilities of many systems (transportation infrastructures, energy and water supply infrastructures, communication infrastructures, production infrastructures, financial infrastructures, etc.) that facilitate modern life in many ways are highly interdependent, complex and evolve on a large scale. Within this context, disturbances of any entity in such formed network is likely to affect other entities by risk propagation mechanism. Integration of risk factors in decision making or risk informed decision making is receiving a great attention by researchers and decision makers in many domains such as engineering (designing technical systems that mitt some requirements in terms of safety), finance (setting up norms to monitor finance activities in order to avoid companies collapse), environment (developing sustainable agriculture and natural resources extraction actions), science and medical research (monitoring scientists activity by the society to avoid creating new threats); because national and international opinions are being more and more concerned by risk issues from all human activities as well as natural phenomena (earthquake, hurricane, tsunami, floods, etc.). Risk comes from the incapacity of human beings to correctly predict the outcomes of some events from the environment of the system under consideration or their actions on this system. Indeed, risk and uncertainty are fundamental elements of modern life so they must be addressed properly to protect people from injury. Today an ever-increasing number of professionals and managers in industry, government, and academia are devoting a larger portion of their time and resources to the task of improving their approach and understanding of, risk-based decision making. Indeed, decision making under uncertainty (risk) literally encompasses every facet, dimension, and aspect of our lives. Any decision maker needs to cope with uncertainty in order to rationally act in the sense of risks reduction. To correctly and scientifically address risk management process that is assessing, filtering risk factors, selecting and prioritizing appropriate actions, one needs to dispose of sound models able to reproduce the functional and dysfunctional aspects associated with an anthropic or natural system. These models should permit manager to take decisions of three types: pre-active decisions, these decisions consist in doing things to prepare the system under consideration to face potential adverse events (one knows that such events will occur soon or later). Actions such as transferring risk by contracting insurances, editing anti-seismic construction norms in the case of natural disasters, preparing population on how to behave in the case of an earthquake, constructing and organizing emergency facilities, etc. are proactive decisions. One may need also need reactive decisions that are real time decisions to be made when the undesirable event does really occurs and pro-active decisions that consists in doing things to avoid the occurrence of catastrophe when possible for instance [START_REF] Tchangani | A model to support risk management decisionmaking[END_REF]. This necessity of disposing of sound models is of great importance in the case of complex interdependent systems evolving on a large scale. The remainder of this communication is organized as follows: in the second section, main features of complex interdependent systems are briefly presented together with the necessity of disposing of a structured tool for the modelling process; section three presents a state of the art on ontology and object oriented Bayesian network (OOBN) approaches as structuring tools that can potentially respond to our needs; and finally section four considers applying these approaches for modelling flash floods phenomena in order to assess risk faced by infrastructures in zone where these phenomena take place.

II. COMPLEX SYSTEM

A. Main features of a complex system

A complex system is constituted by many entities, components or variables with mutual relationships. The complexity is exacerbated by the uncertainty that may affect these relationships. The behavior of the system is highly unpredictable without a sound model. The purpose of this communication is therefore to develop an approach that can be used for the modelling of complex systems with the ultimate purpose to assess the risk facing some components of the systems when one of the components is destabilized by an external event for instance. Such a generic model can be used in many socioeconomic domains to assist the assessment of indicators or the decision making. The modelling process must rely on logical and structuring tools in order to avoid incomplete, imperfect or faulty models; next paragraph reviews existing approaches and proposes those that can be used or adapted to fulfill our needs.

B. Structured modelling necessity

Classical analytic approaches that suppose many simplifying hypothesis are not suited to deal with all aspects of complex large scale interdependent systems. When it is possible to collect data driving the behavior of a complex system, learning approach offers the possibility to build a model that will be able to reproduce, by feedback mechanism, the behavior of the system as accurate as possible. Learning is an artificial intelligence approach that to relate input data (causes) of a system to the observations (consequences or output). The learning process may consist in determining the internal structure of the system that is identifying relationships (or interactions) between its different components referred to as structure learning; or in determining the strength of interactions between components of a system with known structure known in the literature as parameter learning. Within this framework, Bayesian Networks (BN) are very efficient for modelling uncertainties. Dynamic Bayesian Network (DBN) may be used when temporal dimension in the behavior of the system is to be taken into [START_REF] Murphy | Dynamic bayesian networks: representation, inference and learning[END_REF]. In DBN, each sample instant t of time horizon is constituted by a BN. For BN, there exist many learning (mainly in what concern parameters learning) sound and powerful algorithms in the literature ; this is not the case for DBN for which existing learning algorithms are so complicated that their deployment in real world applications is not easy mainly in the case of systems with a huge number of components. A possibility to reduce this complexity is to use the so called Object Oriented BN (OOBN) in order to exploit possibilities offered by this modelling technique. The idea of modelling repeatable systems by object oriented techniques has been already considered in a certain number of studies such as works undertaken in references [START_REF] Jaeger | On the complexity of inference about probabilistic relational models[END_REF], [START_REF] Weber | Complex system reliability modelling with dynamic object oriented bayesian networks (doobn)[END_REF], [START_REF] Xiang | Inference in multiply sectioned bayesian networks: Methods and performance comparison[END_REF] to mention just a few.

III. STATE OF THE ART

Existing approaches Object Oriented Bayesian Network (OOBN) standpoint

Bayesian Networks are used to formalize knowledge in the form of a causal graph associated with a probability space [START_REF] Nielsen | Bayesian networks and decision graphs[END_REF], [START_REF] Pearl | Probabilistic reasoning in intelligent systems: networks of plausible inference[END_REF]. They are directed acyclic graphs (DAGs) where knowledge is represented by variables. Each node of the graph corresponds to a variable and arcs represent the probabilistic dependencies between these variables. Formally, a Bayesian network is defined by:

• a graph-oriented without circuit, noted G = (V, ε),
with V, the set of nodes of G, and ε, the set of arcs of G,

• a finite probability space (Ω, A, P), where Ω is the universe, i.e. the set of all the elements considered in the problem, A is a σalgebra on Ω and P is a measure on Ω such that P(Ω) = 1; P(empty set) = 0;

P(A) <= P(B) if A included in B,
• a set of random variables defined on (Ω, A, P), corresponding to each node of the graph, such that the set of probabilities associated with these variables defines the distribution of probabilities attached to the network:

P(V 1 , V 2 , ..., V n ) = Π n i=1 P(V i |pa(V i ))
with pa(V i ), the parent set (also called predecessors or causes) of V i in graph G. There are two types of probability tables in Bayesian Networks [START_REF] Godichaud | Sustainable management of end-of-life systems[END_REF]. Tables of prior probabilities characterizes the chances that the variable V a without any parent is in state a i . Tables of conditional probabilities establish the chances that a variable V b is in state b j based on the state of its parents [START_REF] Matthieu | Optimising end-oflife system dismantling strategy[END_REF], [START_REF] Godichaud | Sustainable management of end-of-life systems[END_REF]. Inference in a Bayesian network consists in propagating information in the network [START_REF] Ben Hassen | Diagnosis sensor fusion for wire fault location in can bus systems[END_REF], [START_REF] Ben Hassen | Distributed sensor fusion for wire fault location using sensor clustering strategy[END_REF]. Indeed, a model using this formalism is generally not intended to be a static representation of knowledge. Beyond the a priori reasoning, evidences may be introduced to update the observed situation and to insert into the model the changes enabling the refinement of the results [START_REF] Ben Hassen | Omtdr using ber estimation for ambiguities cancellation in ramified networks diagnosis[END_REF]. This new knowledge, takes the form of a so-called elementary information, denoted J , relative to a particular node. There are two types of basic information. The deterministic information allows instantiating a variable, that is affecting it a precise value, (eg P(V a = a 1 |J ) = 1). The imprecise information modifies the distribution of probability of the variable, either by excluding a value of the universe of the variable (P(V a = a 1 |J ) = 0) or, more usually, by changing the law (P(V a = a 1 |J ) = P(V a = a 1 )).

The structured representation offered by the object oriented techniques enables to improve the performance of the Bayesian Networks in terms of complexity of specification and inference of large systems. An object-oriented Bayesian network, is a direct application of the object paradigm. The basic element is the class, fragment of a Bayesian network whose nodes are broken down into three sets: input and output interfaces together with internal nodes. The object oriented Bayesian network takes advantage of classic Bayesian networks but introduce the concept of instance nodes. An instance node is an abstraction of a part of a network into a single unit. Consequently, instance nodes can be used to represent different network classes within other networks. The notion of encapsulation allows the transmission of all properties of the net fragment. An object-oriented network can be viewed as a hierarchical description/model of a problem domain. This makes the modelling easier since the OOBN-fragments at different levels of abstraction are more readable.

An example of an OOBN is presented in figure 1. Based on [START_REF] Ambroise | La dynamique du cycle de l'eau dans un bassin versant: processus, facteurs, modèles[END_REF] the model introduces four classes, namely: rain, melting water, irrigation water and occult water. Inputs are represented by dotted line such as energy or thickness, outputs are characterized by solid lines like for instance water state and water volume. Interstructure associated with internal nodes is encapsulated in each class.

Once the structure and relationships between the nodes has been established, the main work consists in characterizing the Conditional Probability Tables (CPT). Building OOBN model may be somehow difficult when it concerns a complex system with many variables, states or relationships. The use of learning techniques might bring some help for the identification of In [START_REF] Langseth | Fusion of domain knowledge with data for structural learning in object oriented domains[END_REF], [START_REF] Wuillemin | Structured probabilistic inference[END_REF] the authors give some insight over OOBN structure learning. In [START_REF] Langseth | Parameter learning in object-oriented bayesian networks[END_REF] the author extends the parameter learning based on OO assumption and propose a parameter learning method based on the reducing of the parameter number during the learning phase.

Ontology standpoint

Historically, ontologies has a metaphysical origin by a philosophical perception, arriving in computer science with a more technical view.

On a computer view, an ontology is an explicit specification of a conceptualization [START_REF] Gruber | Toward principles for the design of ontologies used for knowledge sharing?[END_REF]. Currently, principles for the design of ontologies are means of providing a structured representation of knowledge from real world at different levels of abstraction and modelling (e.g. domain ontology or upper ontology). The purpose of employing an ontological representation is to build a common understandable model for collaborative practices and knowledge sharing to apply reusable problemsolving techniques ( [START_REF] Da Costa | Pr-owl: A bayesian ontology language for the semantic web[END_REF], [START_REF] Helsper | Building bayesian networks through ontologies[END_REF], [START_REF] Devitt | Ontology-driven automatic construction of bayesian networks for telecommunication network management[END_REF]). Here we will use the example in Figure 1. Nowadays, the fundamental meaning within artificial intelligence is a semantic model for specifying the used vocabulary that consists of a set of types of concepts, relations and their associated properties. This formal vocabulary is essential to capture and formalize the structured information needed for knowledge representation and reasoning [START_REF] Devitt | Ontology-driven automatic construction of bayesian networks for telecommunication network management[END_REF]. Hence a key benefit of using ontology is the opportunity to improve mechanisms of reasoning in the target models. So, for a special complex large real system, the generated models are semantically more informative than statistical models like Bayesian Network, since it provides a formal semantics that is a sound basis for the computerized reasoning and knowledge exploitation. In light of these characteristics, ontological models are complementary to numerical models in modelling the complex systems for risk assessment. For instance, as part of a flood forecasting system, procedural knowledge encoded in the domain-specific ontology can be triggered to monitor and display water conditions in the modelling of risk (Figure 2).

Ontology and OOBN

Ontology lacks the quantification part. That is the reason why some researchers work to combine the probabilistic and ontology language ([18], [START_REF] Ding | A probabilistic extension to ontology language owl[END_REF]) so that they can enrich each other. There are two main directions, one is to enhance ontology capabilities to support probabilistic inference, the other one is to enhance the probabilistic graphical models construction by integrating ontologies [START_REF] Ishak | Ontology-based generation of object oriented bayesian networks[END_REF].

Bayesian Network as a knowledge-based network presents quantitative as well as qualitative information. The qualitative part presents the system by a graph, and the quantitative part defines the conditional probability for each node in the graph.

As we mentioned it previously, although there exist some learning methods in the literature to building the structure of BN, the construction of the model is still a big issue for BN design and development. Ontology model can qualify the knowledge clearly and simply through modern languages like OWL (Web Ontology Language), DALM+OIL for instance. Let us note that some work has already been undertaken on how inserting probabilistic reasoning in the ontology language, and most of them have concluded that BN was the appropriate tool for that [START_REF] Da Costa | Pr-owl: A bayesian ontology language for the semantic web[END_REF], [START_REF] Helsper | Building bayesian networks through ontologies[END_REF], [START_REF] Devitt | Ontology-driven automatic construction of bayesian networks for telecommunication network management[END_REF]. An approach of combining ontology and BN is presented in [START_REF] Devitt | Ontology-driven automatic construction of bayesian networks for telecommunication network management[END_REF].

Others researches like [START_REF] Da Costa | Pr-owl: A bayesian ontology language for the semantic web[END_REF], [START_REF] Helsper | Building bayesian networks through ontologies[END_REF], [START_REF] Ding | A probabilistic extension to ontology language owl[END_REF], [START_REF] Sadeghi | Ontology driven construction of a knowledgebase for bayesian decision models based on umls[END_REF], [START_REF] Andrea | Extending ontology queries with bayesian network reasoning[END_REF] give some ways for extending an ontology to BN. Indeed, there are already some means to proceed to the transformation, but the quantitative part remains difficult.

Based on [START_REF] Ishak | Ontology-based generation of object oriented bayesian networks[END_REF], equivalence between ontologies and object oriented bayesian network has been identified.

Introduction

In this part, we propose to use the ontology model to help building the BNs. As mentioned earlier, due do some limitation of the BNs for the complex system representation we will use OOBNs which are more expressive than the standard BN in order to address an extended range of ontology [START_REF] Ishak | Ontology-based generation of object oriented bayesian networks[END_REF]. Research carried out in [START_REF] Ishak | Ontology-based generation of object oriented bayesian networks[END_REF] deals with an approach for translating an ontology model to OOBN, including a set of mapping rules allowing to generate a prior OOBN structure by morphing an ontology related to the problem under study to be used as a starting point to the global OOBN building algorithm. Research done in [START_REF] Andrea | Extending ontology queries with bayesian network reasoning[END_REF] is interesting because it shows how translating an ontology model into a two level Bayesian networks.

From Ontology to OOBN

In the following this paper assumes that all semantic relations present in the ontology are of a causal orientation. We first defines a simple concept and a normal concept in ontology. The simple concept has no property and influence other concept only by itself. The normal concept has one or more properties.

The reason of distinguishing simple concept and normal concept is mainly due to the possibility of associating directly a node to the simple concept in the future Bayesian network. It makes thus easier the modelling process.

Proposition: A simple concept correspond to a node in OOBN (see [START_REF] Helsper | Building bayesian networks through ontologies[END_REF]).

We use the generic Depth-First Search (DFS) algorithm from [START_REF] Tarjan | Depth-first search and linear graph algorithms[END_REF] for graph building. We give below a stepwise approach on how converting an ontology representation into an OOBN. We present now a brief description of each step of the approach.

Step 1: Identification of simple or normal concepts

• Check the property of each concept;

• Put the no property-concept in a 'concept-node'box;

• Put the property-concept in a 'concept-class'box.

The concept in concept-node box corresponds to a simple node in OOBN [START_REF] Helsper | Building bayesian networks through ontologies[END_REF]. Without any precision, the concept is considered to be a normal concept.

Step 2: Depth-First-Search algorithm(see [START_REF] Tarjan | Depth-first search and linear graph algorithms[END_REF]). We will illustrate this step by its application on the example provided in Figure 2. On branch A appear causal links and hierarchy links. When there is a hierarchy link "is-a", the upperconcept will appear in the subconcept placed in parentheses for underlying the difference. Applying the DFS algorithm allows the identification of the order of the concepts to be encapsulated. In this example the DFS algorithm leads to the following order: Wind(Weather), Atmosphere energy, Melting water and Net water.

Step 3: Building a Directed Acyclic Graph (DAG)

• Associate each concept CP i to a node C i in the DAG G;
• Add the link in DAG by keeping the direction of the correspond link in ontology model;

• If the link in ontology belongs to a "is-a" type, use a hierarchy link in DAG with the direction from class to subclass; for example, if CP ia "is-a" CP i , we have CP i =⇒ CP ia , where CP ia is called a subclass node of CP i ;

• Add a "cause-effect" linkage for each hierarchy link with the descendant of its subclass node in DAG. Using the example from Figure 2 and Algorithm, CP i to a DAG in class level we obtain the model in Figure 3.

Step 4: Translating a DAG into an OOBN • Find the root cause of each branch (using DFS) which is the deepest instantiation;

• Each property p ki in a concept is its own output node of class-instantiation I i : C i correspond in OOBN, add the output nodes at this class;

• The cause class-instantiation gives the input informations/nodes of the effect class, add the input node at the effect class-instantiation;

• Repeat the same processes below until meeting a hierarchy link =⇒;

• Hierarchy link =⇒ means the output nodes in cause class are the input nodes at the effect class, and the class node does not provide the information for its subclass node;

• Add a construction link "--" between the class node and its subclass node. Using this algorithm on the square part of Figure 3, we obtain results in the OOBN of Figure 4.

Ontology represents the relationships between concepts. The corresponding algorithms [START_REF] Ishak | Ontology-based generation of object oriented bayesian networks[END_REF] will have to be improved in a near future to take into account into the OOBN of the internal nodes and links.

V. CASE STUDY: MODELLING FLASH FLOOD EFFECT

Climate change has contributed to an increase in extreme weather events. Scientists predict that climate changes will increase the frequency of heavy rains, putting many communities at risk of flooding. In mountainous areas, this risk is increased by the relief and its consequences in terms of water flow kinematics.Flood torrents directly threaten human and material issues due to the intensity and suddenness of the events. The implementation of methods of protection, prevention, security and control of the risk of flooding in general requires a better knowledge of the phenomena of floods and, thus, a better ability to model these phenomena.Flood risk analyses are important insofar as they allow the assessment of the economic efficiency of the mitigation measures and optimize investments envisaged for the protection of population and infrastructure. Facing a potential, announced or proven crisis, they lead to a better design of insurance policies and actions of anticipation or remediation implemented by companies, municipalities or even citizens.

The most common approach to define flood risk requires to combine a hazard characterized by statistical aspects (frequency of occurrence) and physical aspects (flow intensity) and an impact expressed in terms of vulnerability; i.e. exposure and sensitivity of persons and goods to potential damages. Torrential floods are rapid gravity phenomena which include a share of irreducible uncertainty related to randomness events (rain, snow...) and to the knowledge of the involved processes. Risk management decisions must compose and integrate this stochastic dimension.

Bayesian networks seem a suitable tool for the implantation of such a model see [START_REF] Villeneuve | Hybridization of bayesian networks and belief functions to assess risk. application to aircraft deconstruction[END_REF]. The modelling work which is in progress has been divided into 2 parts. It consists first in identifying the influential parameters in the generation of the flood phenomenon. It endeavours to determine the cognitive structure to retrieve the corresponding information from networks of sensors, databases, expertise... Within this framework, the passage through an ontological approach is crucial and represents a natural framework to structure the problem. The second part aims to create a spatio-temporal causal model for the explanation and the probabilization of the feared events for diagnostic and prognostic purposes.

As a first step, an elementary time-independent Bayesian network will be established to characterize the influence of variables on a small geographical area homogeneous in terms of topology.

In order to characterize the spatiality of the phenomenon, the idea is then to exploit this basic model as a generic block of modelling, characteristic of the evolution of all of the variables that may be brought into play, and then to associate the different bricks instantiated with respect to the considered area and sequenced temporally according to the phenomenon timeline. Objects Oriented Bayesian Networks appear to be appropriate to characterize this chronology characterized by similar variables instantiated according to the corresponding spot. Figure 5 gives a simplified view of the modelling elementary block whilst Figure 6 represents the model of an extended geographical zone.

VI. CONCLUSION

The present work deals with the modelling and analysis of complex systems characterized, in an uncertain and evolving framework by many interactions between components. With the objective of representing large and repetitive structure in such a context , oriented object probabilistic models seem an appropriate solution. Within this framework object oriented Bayesian Networks (OOBN) appear as powerful and well suited to the representation of complex models submitted in part to random events. To facilitate the characterization of this type of model, an ontology-based approach is presented. An algorithm is then used to translate the ontological model into an object oriented Bayesian network. An illustration of the After showing the construction of the object oriented Bayesian network on a small and homogeneous geographical area with respect to topology characteristics, the extension of this basic model is proposed for the representation of a wider territory.

Fig. 1 .

 1 Fig. 1. OOBN model of water supply

Fig. 2 .

 2 Fig. 2. Ontology model of water supplyThe concepts have their own properties, shown in the

1 )

 1 Distinguish each concept CP i by separating simple concept and normal concept; 2) Use Depth-First Search in ontology model for finding every branch b 1 , b 2 , ..., b n ; 3) Build a directed acyclic graph (DAG) in class level; 4) Transfer the class level DAG to a OOBN.

Fig. 3 .

 3 Fig. 3. Ontology model of water supply to a DAG in class level

Fig. 4 .

 4 Fig. 4. OOBN at instatiation level

Fig. 6 .

 6 Fig. 6. OOBN of an extended geographical zone

Fig. 5 .

 5 Fig. 5. OOBN of modelling elementary block

table below :

 below

	Concept	Property
	Water	Water volume, water state
	Topography	Altitude, surface
	Atmosphere energy	Energy

In this research the author treats the concept as an instantiation of class, but not all the concepts in an ontology can be represented by a class level.

IV. PROPOSED APPROACH