Benoît Beaubier
email: benoit.beaubier@lmt.ens-cachan.fr

Karine Lavernhe Taillard

Gilles Besnard

Sylvain Lavernhe

François Hild

Stéphane Roux

K Lavernhe-Taillard

CAD-based Calibration of a 3D DIC System

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

3D surface Digital Image Correlation (stereocorrelation) is a measurement technique of 3D surface displacement fields by comparison between two reference and two deformed pictures. Image registration is possible thanks to a random pattern deposited onto the observed part, here, an automotive roof. The method is based on a stereovision process, which consists in determining 3D points from a pair of pictures with two different cameras, at different fixed angles. The 3-dimensional shape reconstruction is carried out thanks to a calibration procedure, which is the subject of this paper. The calibration consists in the determination of intrinsic parameters (internal parameters of cameras) and of extrinsic parameters (angle and position of the cameras in the experimental frame).

Most of the stereovision methods use calibration targets whose geometry is known. Calibration target geometries are numerous, the most used objects are planar. To calibrate a stereocorrelation device with this kind of target, many pairs of pictures of the calibration target in various positions and angles are required. Moreover, this method is very dependent on the geometric quality of the calibration target [START_REF] Sutton | Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications[END_REF][START_REF] Garcia | A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming[END_REF]. Besnard et al. [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF], among others, have used 3D calibration targets. The chosen geometry has an open book shape. It allows the calibration to be performed with only one pair of pictures thanks to the 3D geometry. This calibration implementation is easier than the one using planar targets. However, the target size should be similar to the observed part. Furthermore, it is more difficult to ensure the geometric quality of (large) calibration target [START_REF] Garcia | A combined temporal tracking and stereo-correlation technique for accurate measurement of 3D displacements: application to sheet metal forming[END_REF]. Other studies propose "autocalibration" methods in order to calibrate stereovision systems without a priori knowledge of the observed part. However, uncertainties about dimensions of the reconstructed object are not always satisfactory [START_REF] Fusiello | Uncalibrated Euclidean reconstruction: a review[END_REF].

NEW APPROACH

This paper presents a new calibration approach. The idea is to use no more calibration target, and perform directly the calibration on the measured part. An appropriate modeling of the observed part is needed. For the final application, the part of interest is an automotive thin sheet roof for witch geometry is described with freeform surfaces (as used in Computer Aided Design (CAD) models, for instance Bézier patches [START_REF] Piegl | The NURBS Book[END_REF]). To prove the feasibility of such calibration method, a test part made of aluminum alloy was machined (Figure 1). It was created as a Bézier patch in CatiaV5 and milled on a 3 axis high speed machine (HSM). To minimize deviations induced by milling between the CAD model and the actual geometry, machining tolerance and scallop height were required to be less than 1 m. Polynomial interpolation was also used through the computer numerical control (CNC) options of the HSM to avoid chordal errors and to improve the follow-up.

ADVANTAGES

Such a calibration method has many advantages. First, as no calibration target is needed (i.e., calibration is directly performed from the pair of pictures of the object) the practical implementation of the calibration method is easier. The second advantage is the basis used to express the 3D shape. The choice of a geometry defined by a Bézier surface allows us to use a global approach to DIC [START_REF] Besnard | Finite-Element" displacement fields analysis from digital images: Application to portevin-le chatelier bands[END_REF]. In the present case, an integrated-type of approach is chosen [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF]. For the latter, the sought displacement field requires a small number of parameters, thereby regularizing the analysis. The chosen description of the geometry is suited for this kind of approach. With this point of view, the present implementation departs from that proposed by Besnard et al. [START_REF] Besnard | Characterization of necking phenomena in high speed experiments by using a single camera[END_REF], who used a global approach for the 3D matching, but it still yielded a cloud of 3D independent points.

PRINCIPLE OF CALIBRATION SYSTEM

The calibration principle is presented in Figure 3. First, an initialization is performed with the same pair of images (left and right) of the part taken by the two cameras in their chosen position. The user selects at least 6 points pairs (x i , y i ) on the right and left pictures. The three-dimensional coordinates (X i , Y i , Z i ) of these points have to be known on the CAD model. After rewriting system (2), a singular value decomposition [START_REF] Salvi | A comparative review of camera calibrating methods with accuracy evaluation[END_REF] gives a first estimation of

In order to solve Equations (2), x l,r and y l,r are divided by l l,r so that

         r l i r l i r l r l i r l i r l l X m y l X m x , , 2 , , , 1 , (3)
where

) 1 , , , ( X i Z Y X 
are the homogeneous coordinates. The problem has 24 unknowns to be determined (i.e., 24 components of matrices M r and M l ). Consequently, we have to use a kinematic basis with at most 24 components. Only 22 are considered in practice since the matrix components are determined up to a multiplicative constant. The "kinematic" basis allows us to determine matrix parameters m ij directly. In order to determine the basis the coordinate sensitivity is expressed as a function of the matrix parameters ). An image is a matrix with a gray level value for each pixel coordinate x. The conservation of gray level reads

m ij ij ij r l ij r l r l dm m d m d d     , , , ) ( x x x
) ( ) ( r r l l d g d f x x x x    (5)
Contrary to classical DIC approaches in which the reference picture does not change during the iterations of the minimization procedure, the two pictures are updated at each iteration. The linearized conservation law reads

r r l l d g g d f f x x x x        ) ( ) ( (6)
The problem thus amounts to minimizing the following residual x x x x [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF] with respect to the unknown m ij parameters. A multiscale iterative procedure is implemented to solve this problem [START_REF] Besnard | Finite-Element" displacement fields analysis from digital images: Application to portevin-le chatelier bands[END_REF].

dudv d g d f g f T r l r l          2 ) ) ( ) ( (

CALIBRATION TEST

Measurements have been carried out on the part shown in Figure 1 In the correlation algorithm, the dimension of the parametric space (u,v) is 1000 x 1000 pixels. Figure 5 shows the map of the correlation residuals (i.e. difference (f-g) in the (u,v) plane) at the end of the calibration procedure. This map accounts for both correlation and geometric errors. It is worth emphasizing that up to now, the transformation matrix was determined from the theoretical CAD geometry. No large deviation from the CAD model is observed in Figure 5. The largest difference is due to a spurious specular reflection of light onto the surface captured by the right camera, but not the left one. The dimensionless residuals were 2.1% as compared to the initial residual obtained with the manual pointing of characteristic points.

CONCLUSIONS

It has been shown that it is possible to calibrate a stereovision setup (i.e., components of the transformation matrix) using a CAD model of the observed part. A global and integrated correlation scheme was implemented. Small correlation residuals were found, thereby validating the proposed procedure. The next step is to account for optical distortions to complete the calibration of the stereovision system. The last step is the application to the analysis of an automotive roof and its subsequent deformation during weld brazing.

Figure 1 : 2 where

 12 Figure 1: Part made of aluminum alloy and machined with an HSM.

Figure 2 :

 2 Figure 2: Theoretical Bézier patch of the studied part (Figure 1). Dimensions are in millimeters. In red, the theoretical surface and in blue the corresponding control polygon of the Bézier patch.

M

  right r ~and left l M ~ transformation matrices. In a second step, the theoretical surface S(u,v) is then projected onto the right and to left pictures using the matrices previously determined. The projections are interpolated on right and left pictures. Matrix parameters m ij are then updated by using an integrated correlation algorithm. Final values of m ij are obtained after convergence of the correlation procedure.The apparent displacement field has to be expressed on a reduced judiciously chosen kinematic basis,. The mathematical formulation of the problem is presented hereafter. The relationships between three-dimensional points and their (right and left) projections are given by x l and y l are pixel coordinates on the left picture, x r and y r pixel coordinates of the right picture, l l and l r the scale factors, and X, Y, Z, 1 homogeneous coordinates of the considered point of the CAD geometry.

Figure 3 :

 3 Figure 3: Principle of the calibration procedure.

x

  denote the apparent displacements in the left and right pictures induced by a change in the component of the transformation matrices. The images shot by left and right cameras are respectively labeled as f(x l ) and g(x r

Figure 4 :

 4 , when black and white paint was sprayed onto the external surface. Cameras used herein were Canon EOS 40 D with a CMOS sensor of definition 3888 x 2592 pixels (in RGB). After transformation into gray levels, the final definition of the pictures is 1944 x 1296 pixels. The two analyzed pictures are shown in Figure 4. Left picture f(x l ) Right picture g(x r ) Pictures taken by the left and right cameras.

Figure 5 :

 5 Figure 5: Map of the final correlation residual expressed in percent of the dynamic range of the analyzed pictures (mean value η = 2.1 %)


Romain Thalineau

Sylvain Hermelin

Andreas D Wieck

Christopher Bäuerle

Laurent Saminadayar

Tristan Meunier

A few-electron quadruple quantum dot in a closed loop

Keywords:

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Controlling the path of a single electron in semiconducting nanostructures is an interesting tool in the context of spin qubits systems. In particular, it opens the route towards the transport of quantum information on a chip and represents a potential strategy to scale up the number of spin qubits in interaction [1]. In addition, in presence of spin-orbit interaction, it represents an interesting way to manipulate coherently the spin of a single electron [2][3][4]. Recently, fast and efficient single electron transport could be obtained by assisting the transport through a 1D-channel electrostatically defined with surface acoustic waves on AlGaAs heterostructures [5,6]. Nevertheless, such a technique is restricted to displacements on a straight line. To perform more complex displacements, engineering the path of the electron with series of quantum dots is a promising alternative. In this context, it has been demonstrated that topological spin manipulation can be obtained if the electron is transported adiabatically on a closed path under spin-orbit interaction [7,8].

In order to preserve spin information during transport, all other electrons of the heterostructure potentially present along the electron path have to be removed. Therefore all dots have to be emptied and one needs to control them in the so called "few-electron regime". Up to now, only three dots in series have been demonstrated to be tunable [9,10] in the few-electron regime. A triple quantum dot geometry in a star-like configuration has been demonstrated, but the geometry did not allow tunneling between all close-by dots and the few-electron regime was not reached [11].

Here we present the first step towards the transfer of a single electron spin on a closed path using a series of quantum dots in a square-like geometry. We demonstrate that we are able to set the system in the few-electron regime using charge detection techniques and therefore to remove the other electrons along the path. Moreover, the tunability of the four-dot system allows a single electron to be transported along a closed path isolated from the other electrons of the structure. The quantum dot system is fabricated using a GaAs/AlGaAs heterostructure grown by molecular beam epitaxy, with a two-dimensional electron gas (2DEG) 100nm below the surface (density 2 × 10 11 cm -2 and mobility 1 × 10 6 V -1 s -1 ). By applying negative voltages to the metal Schottky gates (Ti-Au) deposited on the surface of the heterostructure, an electrostatic potential confining electrons can be engineered. Figure 1(a) shows a Scanning Electron Microscopy (SEM) image of the sample used. The pattern of the gates allows us to define a quadruple well potential. Figure 1(b) shows a zoom image of the sample, in particular the expected position of the four quantum dots and their labels. Each quantum dot of the quadruple well potential is most strongly capacitively coupled to the closest yellow gate and is labeled accordingly. The red gates are used to create a tunnel barrier between the dots while the yellow ones are used to define tunnel barriers with the leads and to control the electrochemical potential of each dot. By modifying the voltage applied to these gates, the electrochemical potential of each dot can be changed, and consequently the electronic population can be controlled.

Quantum point contacts (QPC) are formed with the help of the green gates and used as charge detectors to probe changes in the electronic configuration of the quantum dot system [12]. Due to the screening of the gates, the coupling of each QPC to its diagonally opposite dot was too small to observe any change in its electronic population. Nevertheless, combining the signal from two QPCs was sufficient to see any charge change in the whole four-dot system. Consequently, only two of them have been used during this experiment, the ones in the upper right (I QP C2 ) and in the lower left (I QP C4 ). They were DC-biased with a voltage V R = 500µeV, defining also the bias of the four reservoirs. All measurements have been done in a dilution refrigerator with a base temperature around 20mK corresponding to an electronic temperature calibrated to 40mK from weak localization measurement realized in earlier experiments [13].

Figure 2 (a) shows the charge stability diagram with respect to gate voltages V g1 and V g3 (for V g2 and V g4 fixed) obtained from analyzing the QPCs current. We have chosen to plot

dIQP C dV = ( dIqpc2 dVg1 + dIqpc2 dVg3 ) + ( dIqpc4 dVg1 + dIqpc4 dVg3
) to emphasize any change of all electronic populations. As expected from the sample geometry, we observe four different types of charge degeneracy lines that we can identify with their slopes with respect to V g1 and V g3 . They delimit Coulomb blockaded regions where the number of electrons in each dot is fixed. Each of these lines correspond to the exchange of one electron between one of the dots and its closest reservoir and their slopes depend on their relative capacitive coupling to the two gates V g1 and V g3 . First, the dot 1 and 3 are hardly coupled to V g3 and V g1 respectively. As a consequence, the two corresponding degeneracy lines are the one with the largest (dot 1) and the smallest (dot 3) slope in figure 2(a). Second, the two remaining degeneracy lines correspond to dot 2 and 4. As expected from the sample geometry, these two dots are coupled almost symmetrically to gates V g1 and V g3 . The slope differences of the corresponding degeneracy lines observed in figure 2(a) are explained by the geometry of the yellow gates. These gates break the symmetry of the sample. For instance, V g1 is more capacitively coupled to dot 2 than to dot 4. Consequently, the degeneracy line of quantum dot 2 has to exhibit a larger slope than the one of dot 4. One can therefore assign to each charge degeneracy line a corresponding quantum dot by comparing the data of figure 2(a) with the sample geometry and four quantum dots on a closed loop have been engineered in this way.

Thanks to our identification of the four-dot positions and their corresponding charge degeneracy lines, we can infer the charge configuration of each Coulomb blockade region. The charge label used is defined from the sample geometry (see figure 1(b)). To understand the indexing, it is convenient to start from the emptied region (lower left part of figure 2(a)). By increasing voltages V g1 or V g3 , we can add electrons to the four-dot system by crossing one of these charge degeneracy lines. Depending on which charge degeneracy line is crossed, the added electron is labeled accordingly.

0 0 0 1 0 1 0 1 0 0 0 2 0 1 0 2 dI (a.u) QPC V (V) g1 g3 V (V) g2 V =-0.628V g4 V =-0.682V g1 V =-0.785V g2 V =-0.678V V (V) g4 V (V) g3 V (V) g3 V (V)
Until now, it has been assumed that the lower left part of figure 2(a) corresponds to the emptied Coulomb blockade region. The charge detection can demonstrate that the proposed labels correspond to an absolute number of charge present in the four-dot system. In figure 2(a), no degeneracy line is observed for V g1 < -0.74V and V g3 < -0.77V . Due to the strong capacitive coupling between these two gates and the two dots 1 and 3, one would have expected to observe degeneracy lines in this gate voltage region if the two dots were not emptied. This demonstrates that the dots 1 and 3 have been emptied. On the contrary, we cannot conclude from figure 2(a) on the population of the dots 2 and 4 due to the small capacitive coupling with gates V g1 and V g3 . To check whether the few-electron regime was reached for dot 2 and dot 4, stability diagrams varying gate voltages V g2 and V g4 were recorded. Figures 2(b) and (c) show the QPC response with respect to the gates V g3 and V g4 (V g2 ). For both cases, the dot chemical potentials have been set such that the dot 1 is empty (V g1 < -0.785V ). Such a large negative voltage is also applied to V g2 (V g4 ) in figure 2 (b) (figure 2 (c)) in order to empty the dot 2 (4). In both figures, we demonstrate that no degeneracy line corresponding to dots 2 and 4 are observed, confirming that no electron are in the four-dot system in the bottom left Coulomb blockade region of figure 2(a).

In addition, one can extract information from figure 2(a) about the distance between the dots. In this squarelike geometry, we expect that the distance between quantum dots to be larger if they are sitting on opposite corners rather than on adjacent corners. The closer the dots are, the larger the inter-dot capacitive coupling is. In the charge stability diagram, this coupling opens a Coulomb gap at the crossings between charge degeneracy lines. Figure 2(a) clearly shows that the gaps opened at the dot 2-dot 4 and dot 1-dot 3 degeneracy line crossings are much smaller than the other ones confirming the squarelike distribution of the four quantum dots expected from the sample geometry. From this set of data, we can therefore demonstrate that by changing the gate voltages V gi (for i ∈ {1, 2, 3, 4}, see figure 1 (b)), we are able to remove all the electrons from the quantum dot system, control the injection of a single electron within the four-dot structure as well as its transfer from one dot to the other.

The control of the four quantum dot system opens the route towards single electron transport on a closed path. In this last part, we want to give a strategy to perform such an electron displacement. The idea is to keep in close vicinity the chemical potential of dots 1 and 3 and to bring sequentially in this vicinity the chemical potential for dots 2 and 4. By modifying V g2 and V g4 at the same time, such chemical potential movements can be engineered as demonstrated in the three stability diagrams presented in figures 3(a), (b) and (c). By changing the voltage applied to one of these two gates, only one charge degeneracy line moves as expected from the sample geometry and demonstrated in figure 3. Figures 3(b) and (c) are the two gate voltage configurations used in the transport sequence. On figure 3(b), the Coulomb blockaded regions (0,0,0,0)/(1,0,0,0)/(0,0,1,0)/(0,0,0,1), where (i,j,k,l) represent the number of electrons in the dots (1,2,3,4), have been set in close vicinity. Starting from the empty region (0,0,0,0) (label 1 in figure 3(b)), an electron can be loaded into the dot 3 by lowering |V g3 | (label 2 in figure 3(b)). By changing voltages on the two gates V g1 and V g3 , this electron can then be transferred from dot 3 to dot 1 via two tunnel processes (label 3 and 4 in figure 3(b)). To complete the closed path, we need to transfer this electron into the dot 2. Consequently the region (0,1,0,0) has to be set in vicinity with (1,0,0,0) and Each QPC response step corresponds to a wait time in a stable charge configuration and fast transitions between steps appear whenever the electron transfer takes place. Movement of the gates are faster than the bandwidth of the detector (set to 800Hz). Possible exchange of the electron between dots or with the leads are too fast to be observable with our current set-up.

To be able to use such a transfer to manipulate the spin of the electron, one needs to realize the transfer on time scales faster than the spin decoherence time which is of the order of 10-100ns [14,15]. Tunneling between the close-by quantum dots needs then to be strong enough. Indeed two competing mechanisms can explain a change of charge configuration when the electron is transferred between two tunnel-coupled dots. First the electron can tunnel directly between the two dots and second it can be replaced in the dot system by an electron coming from the leads. The second scenario requires two energetically allowed tunnel processes, each of them is an electron transfer between one dot and its closest reservoir: the electron tunnels out of the dot initially occupied and an electron from the reservoir tunnels into the dot initially empty. The timescale of the second scenario is set by the coupling of the dots with their connected leads. The results of the slow QPC measurement presented in 3(d) do not permit to discriminate between these two mechanisms. Moreover, due to the restricted number of gates used to define the dots, we were not able to tune all the tunnel couplings of this system in the regime of a few tens of µeV. This coupling corresponds to a tunnel timescale inferior to 1ns, which would be at least three orders of magnitude faster than the one between the dots and the leads. Consequently, with such tunnel coupling, we would be able to transfer the electron along this closed path for several turns with a small probability of an exchange with one electron from the reservoirs. A more complete study of the tunneling in this dot configuration is needed to reach the nanosecond transfer of a single electron in a closed loop.

In conclusion, we demonstrated that a tunable four quantum dot system where each dot is located at the corner of a square in the few-electron regime can be engineered and probed with charge detection techniques. It allowed us to transfer a single electron in a closed loop free of electrons on a slow timescale. Addition of gates in the present geometry and coupling to fast electronics should permit to transfer a single electron faster than the spin decoherence and to study topological spin manipulation using spin-orbit interaction.

FIG. 1 :

 1 FIG. 1: (a) SEM image of the device : Ti-Au gates are deposited on the surface of the heterostructure allowing the definition of a quadruple quantum dot. (b) Zoom of the dotted circle of figure 1 (a). Expected position of the four quantum dots with their label.

FIG. 2 :

 2 FIG. 2: (a) Stability diagram of a quadruple quantum dot.

  g3 ) is plotted as a function of Vg1 and Vg3, while Vg2 and Vg4 are fixed. The charge label used mimic the geometry of the sample. Upper left label corresponds to the charge contained in the dot strongly coupled to Vg1, upper right to Vg2, lower left to Vg4 and lower right to Vg3.(b) and (c) Stability diagrams of double dots where the two others are set to unreachable energies. It proofs that the few-electron regime can be reached for each quantum dot. (b) Stability diagram with respect to gate Vg3 and Vg4, while Vg1 = -0.785 and Vg2 = -0.678 allowing to say that the dots 1 and 2 are emptied. (c) Stability diagram with respect to gate Vg2 and Vg3, while Vg1 = -0.785 and Vg4 = -0.8 allowing to confirm that the dots 1 and 4 are emptied.

FIG. 3 :

 3 FIG. 3: (a), (b) and (c) Stability diagram of the quadruple quantum dot for different values of Vg2 and Vg4. (b) and (c) Trajectory of the electron in the gate voltage space. d) Position of the electron during the path and the corresponding QPC signal. Each step observed in the QPC currents corresponds to a charge state.

Acknowledgments

We acknowledge technical support from the "Poles Electroniques" of the "Département Nano and MCBT" from the Institut Néel as well as Pierre Perrier for technical support. A.D.W. acknowledges expert help from PD Dr. Dirk Reuter and support of the DFG SPP1285 and the BMBF QuaHLRep 01BQ1035. T.M. acknowledges financial support from Marie-Curie ERG 224786. We are grateful to the Nanoscience Foundation of Grenoble, for partial financial support of this work. Devices were fabricated at "Plateforme Technologique Amont" de Grenoble, with the financial support of the "Nanosciences aux limites de la Nanoélectronique" Foundation and CNRS Renatech network.


Joseph Le Roux
email: leroux@univ-paris13.fr

Jennifer Foster

Joachim Wagner

Rasul Samad

Zadeh Kaljahi

Anton Bryl
email: a.bryl@systransis.ch

DCU-Paris13 Systems for the SANCL 2012 Shared Task

The DCU-Paris13 team submitted three systems to the SANCL 2012 shared task on parsing English web text. The first submission, the highest ranked constituency parsing system, uses a combination of PCFG-LA product grammar parsing and self-training. In the second submission, also a constituency parsing system, the n-best lists of various parsing models are combined using an approximate sentence-level product model. The third system, the highest ranked system in the dependency parsing track, uses voting over dependency arcs to combine the output of three constituency parsing systems which have been converted to dependency trees. All systems make use of a data-normalisation component, a parser accuracy predictor and a genre classifier.

Introduction

This paper describes the three systems we submitted to the shared task on parsing English web data organised by [START_REF] Petrov | Overview of the 2012 shared task on parsing the web[END_REF] and hosted by the 2012 NAACL-HLT Workshop on Syntactic Analysis of Non-Canonical Language (SANCL). The aim of the shared task was to encourage research groups around the world to build robust systems capable of parsing English documents belonging to the following five web genres: answers, emails, newsgroups, reviews and weblogs. To train their systems, participants were supplied with labelled Wall Street Journal (WSJ) data (annotated in the Ontonotes style) and unlabelled data from the five web genres. They were also provided with WSJ, email and weblog development data. One week before the deadline, four blind tests were released.

Our first system, DCU-Paris13-1, the top ranked constituency parsing system, employs self-training with products of PCFG-LA grammars [START_REF] Huang | Self-training with products of latent variable grammars[END_REF]. We use the Lorg parser [START_REF] Attia | Handling unknown words in statistical latent-variable parsing models for arabic, english and french[END_REF], our in-house PCFG-LA parser [START_REF] Matsuzaki | Probabilistic cfg with latent annotations[END_REF][START_REF] Petrov | Learning accurate, compact and interpretable tree annotation[END_REF]. Our second system, DCU-Paris-2, the fourth ranked constituency parsing system, uses a sentence-level product model to rerank the outputs of first-stage Brown models [START_REF] Charniak | A maximum-entropy-inspired parser[END_REF][START_REF] Charniak | Course-tofine n-best-parsing and maxent discriminative reranking[END_REF] trained using increasingly large training sets. Our third system, DCU-Paris13-Dep, the top ranked dependency parsing system, converts constituency parser output created using various versions of the DCU-Paris13-2 system into dependency trees and then combines these trees using a voting algorithm [START_REF] Surdeanu | Ensemble models for dependency parsing: Cheap and good[END_REF].

The general architecture common to all three systems is displayed in Figure 1. The training and test web sentences are normalised before parsing. The baseline parser that is used to parse the unlabelled web sentences is a Lorg product model trained on the Ontonotes-WSJ training material. The product model combines eight 5th-order PCFG-LA grammars trained using different random seeds. Five split-merge-smooth cycles are employed rather than six since there is some evidence that a 6th-order model overfits to WSJ data [START_REF] Petrov | Improved inference for unlexicalized parsing[END_REF][START_REF] Foster | cba to check the spelling" investigating parser performance on discussion forum posts[END_REF]. The parser uses the English signature list described in [START_REF] Attia | Handling unknown words in statistical latent-variable parsing models for arabic, english and french[END_REF] to assign partof-speech tags to unknown words. Parser accuracy prediction [START_REF] Ravi | Automatic prediction of parser accuracy[END_REF] is carried out on the Preliminary experiments with the email and blog data suggested that better performance could be obtained by training on sentences from the same genre and so we train genre-specific grammars (see the upper half of Table 1) and rely on a genre classifier to choose the appropriate grammar during test time.

We describe the components common to all three systems in Section 2. In Sections 3, 4 and 5, we describe in more detail our three systems. Finally, we briefly outline our plans in Section 6.

Common Components

Data Normalisation

Following our previous work on parsing the British National Corpus [START_REF] Wagner | C-structures and fstructures for the British National Corpus[END_REF] and discussion forum comments [START_REF] Foster | cba to check the spelling" investigating parser performance on discussion forum posts[END_REF], we transform the training and test data so that the sentences more closely resemble WSJ sentences. We use the following heuristics:

• Depending on its position within the input sentence, an emoticon is replaced by either a comma or full stop.

• Neutral quotes are transformed to opening or closing quotes.

• Email addresses and URLs are replaced by the generic strings EmailAddress and LinkAddress respectively.

• Sequences of uppercased words or individual uppercased words of length > 4 letters are lowercased. • Common abbreviations and spelling variants (ppl, plz) are replaced by their standard form.1 • The tokens nt and s are replaced by n't and 's respectively. • Repeated punctuations symbols, e.g. !!! are collapsed into one. • List items, e.g. # 2, are removed from the start of a sentence. • Certain "sentences" are not parsed and instead are assigned a trival parse tree rooted by the symbol X with each token tagged as NFP.

These include sequences of punctuation symbols separating the content of an email from its signature and sentences containing only a URL.

Parser Accuracy Prediction

Inspired by previous work in parser accuracy prediction [START_REF] Ravi | Automatic prediction of parser accuracy[END_REF], we parse all the unlabelled data using the baseline Ontonotes-WSJtrained parser and then sort the parsed output ac-cording to its predicted Parseval f-score. We do this in an attempt to maximize the number of highquality web trees in our training material. The fscore predictor is trained on the Ontonotes-WSJ development data using support vector regression with an RBF kernel and the following features:

• Sentence length • Number of words in the sentence that are not in the Ontonotes-WSJ training set • The f-score of the tree measured against a reference parse tree, produced using the first-stage Brown parser [START_REF] Charniak | A maximum-entropy-inspired parser[END_REF] • The presence and count of particular discriminative words, where discriminativeness is measured using information gain • The category of the root of the tree • For each Ontonotes-WSJ category, the number of nodes in the tree labelled with that category • The depth of the parse tree When we first applied this predictor to the automatically parsed trees, we observed that the highest ranked trees contained very few tokens and did not appear to represent good training material. For example, many of the highly ranked trees described sentences which were simply the addressee section of an email, e.g. John. We tried to improve the situation by filtering from the training material all those trees with a yield of less than six tokens and by introducing some randomness into the tree sorting process so that we were not relying completely on the parser accuracy predictor.

Genre Classifier

We use a 6-way classifier to choose between the 5 genre-specific grammars and the Ontonotes-WSJ baseline grammar. For simplicity, we represent documents with feature vectors listing the relative word frequency for any word that has a relative frequency higher than 0.0001 in at least one of data sets. Inspired by the cosine vector similarity used in information retrieval, we project all feature vectors to the unit sphere and then measure vector similarity with the Euclidian distance. As a classifier, we then use k-nearest neighbour (k-NN) with k = 20. To increase the number of training or reference items available to the k-NN model, we slice the training data into (overlapping) sub-documents of fixed size and number. Splitting the unlabelled genre data into training and development sets, we experiment with different values for size and number of training documents and size of test documents and find highest accuracy with 10,000 sub-documents per genre with 4,000 tokens each for training. In final application of the method to the data to be parsed, we use all unlabelled data to train the genre classifier. As this doubles the amount of training data, we cautiously increase the number of training documents to 15,000 and their size to 5,000 tokens.

Aiming for a whole document classification for the final blind test sets but also interested in classifier confidence, we also slice the test sets into overlapping sub-documents of 5,000 tokens each. The classifier achieves 100% confidence for blind A, C and D belonging to the answers, reviews and Ontonotes-WSJ genres respectively. For blind B, we have a near draw between newsgroups and weblogs. Given that the a priori probability of weblogs in the blind test sets is low as weblogs are already present in the development sets, we finally decide to use our newsgroup-specific grammars for blind B.2 3 DCU-Paris13-1

Our first system in the constituency track employs the product-of-PCFG-LA-grammars approach [START_REF] Petrov | Products of random latent variable grammars[END_REF] in conjunction with self-training [START_REF] Huang | Self-training with products of latent variable grammars[END_REF] to obtain an accurate parsing system over out-of-domain genres.

The method works as follows: each edge in the chart is scored using the product of the scores of several grammars using the max-rule algorithm of [START_REF] Petrov | Improved inference for unlexicalized parsing[END_REF]. The grammars are trained on the same dataset but, as PCFG-LA training uses the EM algorithm which does not guarantee a global likelihood optimum over the training set, different initialization setups will lead to different grammars. More precisely, the grammars have the same PCFG backbone but the weights associated with the annotated rules are different depending on the random deviation used when splitting annotations in two.

The baseline parsing system which is used to parse the unlabelled data is a product of eight grammars generated using five rounds of split/merge/smooth (5th-order grammars). For each web genre, eight 6th-order grammars are trained on two copies of the Ontonotes-WSJ training material and 2.6 million tokens of the automatically parsed genre-specific treebanks. These self-trained grammars are then combined in a product model. 3 The time it takes to train a PCFG-LA grammar 4 imposed a limit on the size of the training sets. Nonetheless, this is the best performing constituency parsing system. The Parseval f-scores for the five web genre development sets are shown in Table 2.

DCU-Paris13-2

Our previous work on parsing forum comments and tweets [START_REF] Foster | From News to Comment: Resources and benchmarks for parsing the language of Web 2.0[END_REF] suggests that the DCU-Paris-1 system does not utilise enough of the unlabelled web data. Therefore, we decided to submit a second constituency parsing system using a parser that is quicker to train.

We train the first-stage Brown parser [START_REF] Charniak | A maximum-entropy-inspired parser[END_REF][START_REF] Charniak | Course-tofine n-best-parsing and maxent discriminative reranking[END_REF] on web data parsed using the baseline parser (Ontonotes-WSJtrained Lorg product model) with 10 target sizes from 1.3 to 13.1 million tokens and two copies of the Ontonotes-WSJ training data.

We parse each test set with the 10 trained grammars (of varying size) matching the genre predicted by our genre classifier. We implement an approximated product model based on the n-best output. For each parse tree present in at least one of the nbest lists, we multiply all observed parse probabilities, substituting a constant very low probability if a parse tree is absent in an n-best list. We use n = 50. This system performs well. However, despite seeing more data, it is systematically outperformed by our first system (see the second row of Table 2).

DCU-Paris13-Dep

Our third system combines the insights of individual constituency parsing models at the sub-sentential level using the following algorithm which is applied 3 There is no hard limit on the number of grammars that our parser can use. We restricted ourselves to eight grammars mostly because of hardware resource limitations. 4 Training 5 × 8 = 40 genre-specific grammars with six rounds and 2.6 million words of training material took 4.5 days on a cluster of six mixed machines with between 24 and 48 GB of RAM and 8 to 12 CPU cores. to dependency trees: each word in the sentence is attached to a parent word and assigned a dependency label according to a majority vote amongst the individual systems [START_REF] Surdeanu | Ensemble models for dependency parsing: Cheap and good[END_REF] . Three constituency parsing systems are converted to dependencies using the Stanford converter (de [START_REF] De Marneffe | Generating typed dependency parses from phrase structure parses[END_REF] and then combined using this voting algorithm. These systems are the following: 1. Our second constituency parsing system, DCU-Paris13-2 (see Section 4) 2. A system which is the same as DCU-Paris13-2 except that instead of using the genre-specific grammars, we use the backoff grammars listed in the lower half of Table 1 3. A system which is the same as DCU-Paris13-2 except that it combines both genre-specific and backoff grammars Thus, we can view this system as a combination of combinations since the individual systems that we combine are themselves combination systems. The voting algorithm is not guaranteed to produce a wellformed tree. In the rare case that an ill-formed tree is produced , we backoff to the DCU-Paris13-2 system. The labelled attachment accuracy for this system over the five web genres are shown in the third row of Table 2.

Future Work

Our immediate next step is to perform an error analysis in order to better understand the strengths and weaknesses of the three systems, and based on this analysis, to try to improve the systems. Future work will also include the following: an analysis of the role of data normalisation, the use of the genre classifier to predict parsing models at the subdocument level, product model experiments involving larger training set sizes, sentence-level product model reranking experiments involving n-best lists > 50, and dependency voting experiments that combine both the Lorg and Brown systems.

Figure 1 :

 1 Figure 1: General Architecture

  -mixture of answers (A), emails (E), newsgroups (N), reviews (R) and weblogs (W) for genrespecific (SP) grammars and backoff (BK) grammars: All grammars are trained on 2 copies of the Ontonotes-WSJ training data and the above mixture of genre material (measured in number of tokens and rounded to the next tree). We train the genre-specific grammars with 100% of the respective genre and 2 Ontonotes copies (except for the answers due to insufficient unlabelled data). The genre-mixtures for the backoff grammars are derived from the genre classifier confusion matrices for test document lengths 500, 1k, 2k, 5k and 10k. The backoff grammars are only used in our DCU-Paris13-Dep system.

Table 1 :

 1 Genre

	BK 26.6	27.6	10.5	10.9	24.4
	E BK	5.2	50.0	19.6	0.9	24.3
	N BK	4.5	22.4	50.0	3.2	19.9
	R BK 19.9	12.1	2.2	50.0	15.8
	W BK 15.0	13.5	12.9	8.6	50.0

Table 2 :

 2 Results for our three systems on the web genre development sets

	System	A	E	N	R	W
	1 (F1)	82.19 81.04 84.33 84.03 86.17
	2 (F1)	79.62 80.79 82.65 82.52 85.38
	Dep (LAS) 81.15 80.40 85.38 83.86 87.60

The list of abbreviations is rather limited having been obtained in an ad-hoc manner by reading the annotation guidelines and manually inspecting approximately fifty sentences from each of the unlabelled training sets.

This could be formalised with the Bayesian decision rule.


Philippe Weber
email: philippe.weber@cran.uhp-nancy.fr

Christophe Simon

Didier Theilliol

Vicenç Puig

Fault-tolerant control design for over-actuated system conditioned by reliability: a drinking water network application

Keywords: Probabilistic model, Bayesian Network, actuator failures, Control allocation, Fault Tolerant System, Drinking water network

Fault-Tolerant Control Design for over-actuated System conditioned by Reliability: a Drinking Water Network Application.

P. Weber * C. Simon * D. Theilliol * V. Puig * *

INTRODUCTION

Dependability analyses are used to increase the performance, the product quality and the system reliability by improving analytic and material redundancy. In most safety critical systems, particular redundant structures have been used to design engineering systems. Actuator redundancy is often used and since these systems are overactuated therefore several solutions are available to control the system. The main problem in the dependability analysis of a system with several components is to determine the relationship between the performance level of the system and the performance levels of its components. This relationship is usually explained by a Structure Function (SF). The reliability literature develops the properties of binary systems with binary components, where only two states are supposed: "perfect functioning" and "complete failure" (see, for example, [START_REF] Barlow | Statistical theory of reliability and life testing[END_REF]). But such a binary assumption becomes an oversimplification when dealing with real situations, since it is very often found that components and systems can be in intermediate levels.

In the recent literature, as in [START_REF] Weber | Overview on bayesian network applications for dependability, risk analysis and maintenance area[END_REF], a growing interest to model the reliability of complex industrial systems by means of Bayesian Networks (BN) has appeared. This modelling method is very relevant in the context of complex systems [START_REF] Langseth | Bayesian Network in Reliability: The Good, the Bad and the Ugly[END_REF]. BN are particularly interesting because they are able to compute the reliability taking into account observations (evidences) about the state of some components. For instance, it is possible to estimate the reliability of the system and all its components knowing that a part of them are out of order.

Drinking water management in urban areas is a subject of increasing concern as cities grow [START_REF] Baumann | Urban water demand management and planning[END_REF].

Limited water supplies, conservation and sustainability policies, as well as the infrastructure complexity for meeting consumer demands with appropriate flow, pressure and quality levels make water management a challenging control problem. Fault-tolerant control (FTC) is introduced in order to address the growing demand of plant availability. The aim of FTC is to keep plant available by the ability to achieve the objectives that have been specified in faulty behaviour [START_REF] Noura | Fault tolerant control systems: Design and practical application[END_REF]. The FTC is based on material or functional redundancy. For overactuated systems, control allocation is implemented to distribute the objectives over the available solutions to control the system. Recently [START_REF] Khelassi | Reconfigurable control design for over-actuated systems based on reliability indicators[END_REF] propose to integrate the reliability of the actuators in the control allocation in order to preserve the health of the actuators and the availability of the system both in the nominal behavior and in the presence of actuator faults. Nevertheless, the SF is not used in this method, therefore the solution minimizes the actuator's failure and contributes to increase the overall system reliability but is not based its optimization. Our proposal consists mainly in formalizing a standard SF model with a BN and demonstrating the usability of this model in FTC problem of overactuated system. The paper is organized as follows: In Section 2, a solution for a reliable control of overactuated systems is presented based on the on-line actuators reliability indi-cators. Section 3 introduces BN and explains the generic structure and parameters to model reliability of an overactuated system. Section 4 is dedicated to the illustration of our approach. Finally, conclusions are given in Section 5.

PROBLEM STATEMENT WITH A SOLUTION CONDITIONED BY RELIABILITY

Various methods have been developed to control large scale systems. Few control approaches are useful to deal with multi scale systems or to coordinate a number of local controllers. Most of large scale systems require to solve control problem associated to different control configurations, also called modes (or paths). Optimizing a sequence of actions to attain or to track a general target is the main topic of control theory. Moreover, in the presence of faults/failures, the configuration structure of the system should be able to improve the robustness and fault tolerant properties of the control scheme. From a general point of view, let us consider a linear process for large scale systems with an overactuated structure described by: ẋ

(t) = Ax(t) + Bu(t) y(t) = Cx(t) z(t) = M y(t) (1)
where A ∈ IR n×n , B ∈ IR n×m and C ∈ IR p×n are respectively, the state, the control and the output matrices.

x ∈ IR n is the system state, u ∈ IR m is the control input, y ∈ IR p is the system output. M ∈ IR l×p is a known constant matrix used to select the output required to track the reference signal. z(t) ∈ IR l is the controlled output and (A, B) is stabilizable. rank(B) = k < m characterizes the overactuated property of the system. A virtual actuator is computed through a classical controller synthesis and classical control allocation optimization methods can be considered in order to distribute the load to each actuator (see [START_REF] Durham | Constrained control allocation[END_REF] or [START_REF] Harkegard | Resolving actuator redundancy-optimal control vs. control allocation[END_REF]).

As proposed recently by [START_REF] Picasso | An mpc approach to the design of two-layer hierarchical control system[END_REF], another state space representation can be considered such as:

ẋ(t) = Ax(t) + m i=1 α i (t) b i u i (t) (2)
where u i (t) is the control variable provided to the i th actuator and α i (t) is an additional input variable which fixed the active actuator or not. The control variables should be designed in order to select a subset of actuators to achieve the main target of the system. In the general case, the management level is used to provide the operational objective, and a global control level identifies one of the 2 m values taken by the vector

α (t) = [α 1 (t) . . . α m (t)] T ∈ [0 1] m .
For both representations, an optimal control input u = (u 1 , u 2 . . . u m ) is the solution of a control (re)-allocation problem. From a general point of view, a weighting matrix W u ∈ IR m×m ≻ 0 is used to give a specific priority level to the actuators (in fault-free case or failure case) according to multi-objective operational goals see for instance [START_REF] Maciejowski | Predictive Control: With Constraints[END_REF] or [START_REF] Tatjewski | Optimization of water distribution and water quality by hybrid genetic algorithm[END_REF].

In order to improve the safety of the system and to keep the set of the actuators available as long as possible, a specific choice of the weighting matrix W u has been proposed based on the actuators reliability [START_REF] Khelassi | Reconfigurable control design for over-actuated systems based on reliability indicators[END_REF]).

The weighing matrix W u is considered as a key to manage the overactuators and contributes to reliable controllers which improve the system reliability. However in order to perform simultaneously the control law synthesis and to solve the control allocation problem, we proposed to distribute the desired efforts defined by the controller proportionally to the actuator functioning probability according to the on-line observations and their reliability based on the steady-state optimization [START_REF] Tatjewski | Optimization of water distribution and water quality by hybrid genetic algorithm[END_REF]) as follows:

min{W u u(t)} (3) subject to: Qu (t) ≤ Γ (4) z d (t) = M A + Bu (t) (5) u min ≤ u (t) ≤ u max , y min ≤ y (t) ≤ y max
(6) where z d correspond to the desired outputs and w i is the actuator i th functioning probability with:

W u =     w 1 0 w 2 . . . 0 w m     ≻ 0 (7)
For simplicity reasons without any restriction, it is assumed that y(t) = x(t).

As a direct consequence, the actuators are utilized in the control strategy proportionally to their ability and necessity of functioning. Let us consider the actuator i ∈ {1 . . . m} defined as a random variable w i with two states {U p, Down} and S is the system state with also two states {U p, Down}. Its functioning probability is defined by:

P (w i = U p |S = U p ) (8)
In order to integrate the actuators degradation in the reconfigured control allocation strategy, W u is re-estimated and changed on-line according to the a priori estimation of actuator reliabilities. Therefore, if an actuator w j is unavailable, the system is still working because it is an overactuated system, and the functioning probability of each actuator is defined by: P (w i = U p |w j = Down . . . S = U p ) (9) This paper concerns the on-line synthesis of W u according to the reliability of the each component i.e. the diagonal matrix values are defined at each sample t. The following section explains how the Dynamic Bayesian Network model is used to compute (9).

BAYESIAN NETWORK MODEL

BN is a efficient solution to model the reliability of complex systems because they perform the factorization of variables joint distribution based on the conditional dependencies. The principles of this modelling tool are explained in [START_REF] Jensen | An Introduction to Bayesian Networks[END_REF].The variable handled by BN are not limited to Boolean random variables contrary to the standard reliability analysis as Fault Tree or Reliability Block Diagram.

Recall of BN characteristics

A BN is a directed acyclic graph (DAG) defined by a set of nodes and a set of directed arcs. A probability is associated to each state of the node. This probability is defined a priori for a root node and computed by inference for the others. A general inference mechanism (that permits the propagation as well as the diagnostic) is used to collect and to incorporate new information (evidences) gathered in a study. The Bayes' theorem is the heart of this mechanism and allows updating a set of events' probabilities according to the observation of some variables and the prior knowledge of the others. This theorem also is responsible of the strength of this knowledge management tool.

Generic reliability models

A BN model provides an easy solution to model the reliability of complex systems in which the nodes represent the system variables and the arcs symbolize the dependencies or the cause-effect relationships among the variables. Let us consider Q sets with δ components that represent the minimal set of components that guarantee the system functioning, these sets are the success paths. The random variables g j with j ∈ {1 . . . Q} is defined by two states {Available, N otAvailable} and represent all success paths states. Components are defined as random variables e i , i ∈ {1 . . . m} with two states {U p, Down}. The availability of a success path is computed by the intersection of the δ functioning components e i . One minimal success paths is necessary to guarantee the success of the main system mission. The system reliability modelled by variable S and its states {U p, Down}, is defined from the union of variables g j representing success paths validity. This combination respects the logical description of the system SF. Fig. 1 presents the generic BN model structures for any system.

The conditional probability table (Table 1) defines the relation between g j and components e i in the success path. This path is defined from a logical aggregation as AND gates to compute the availability of the minimal success path and the system reliability is defined as a OR gate to compute the combination of the minimal success paths (Table 2). In both tables, the parameters in the conditional probability tables (CPT) are equal to 1 or 0 because there is no uncertainty on the combination of components' events leading to the unavailability of the system.

Computation of w i

W u is estimated on-line according to the estimation of actuator reliability and the observation of actuators out of order. A Dynamic Bayesian Network (DBN) computes the reliability of each component following an exponential distribution [START_REF] Weber | Dynamic bayesian networks modelling the dependability of systems with degradations and exogenous constraints[END_REF]. Without any restrictions, we assume that the less the reliability of a component is the less it can be used in the control law and the less its weight should be. Moreover, if a component e δ is involved in only one path g j , and if this path is Not Available, the component e δ is not used and its weight is equal to zero. Of course, many actuators are involved in several paths and if a component is involved in at least one available path its weight is different from 0. The weight is thus computed by the BN from CPT and Bayes theorem. For instance, as presented in Fig. 1, the CPT of w δ depends of g 1 and g j because the component e δ is connected to the paths g 1 and g j . The marginal distribution of w δ is computed according to Table 3.

The failure of a component in a path induces the unavailability of the path. Consequently, the control law of all the actuators involved only in the failed paths is no more necessary since it as any effect on the efficiency of the paths. When several paths are in state {Available}, the control of the actuators involved in each path depends on the reliability of each path and the reliability of the component. The value associated to W u are weighted by the path availability computed from the reliability of actuators.

APPLICATION

System description

A drinking water network (DWN) is a network which considers sources (water supplies), sinks (water demand sectors) and pipelines that link sources to sinks. It also contains active elements like pumps, valves. For the study of the DWN reliability, sources, sinks, tanks and pipelines are considered perfectly reliable. Only the active elements are considered not perfectly reliable. The water transport network of Barcelona is used as the case study of this paper. This network covers a territorial extension of 425km 2 , with a total pipe length of 4,470 km. Every year, it supplies 237.7hm 3 of drinking water to a population over 2.8 millions of inhabitants. The network has a centralized tele-control system, organized in a two-level architecture.

At the upper level, a supervisory control system installed in the control centre of AGBAR is in charge of controlling the whole network by taking into account operational constraints and consumer demands.

BN Synthesis

By considering all the paths from all the sources to the sink, the combination of all success paths should follow the functional requirements. Based on this analysis, the following list of paths is detailed in table 4. Notice that all sources (AportLL1, AportLL2, ApousE13, ApousE3456, AportA) are assumed completely reliable and are conserved in the paths. The valve vAbrera is also considered perfectly reliable but is not modelled in the BN for the sake of simplification. The reliability distribution of all the components is considered exponential and the failure rates are given in table 5. The BN reliability model is given in Table 5. Failure rates

Actuator name Actuator code Failure rate iSJDSpf P 4 9.5 10

-4 h -1 iSJDSub P 3 6.3 10 -4 h -1 iSJD10 P 9 1 10 -4 h -1 iSJD50 P 8 5 10 -4 h -1 iRelieu P 5 1 10 -4 h -1 iEstrella12 P 1 1.2 10 -4 h -1 iEstrella3456 P 2 3.456 10 -4 h -1 iCornella50 P 7 5 10 -4 h -1 iCornella100 P 6 10 10 -4 h -1
Fig. 3. The nodes on the left part of the figure model the reliability of each active component as a DBN and the availability of the sources. The nodes AND i model the combination operators between components to compute the reliability of each path from source to sink defined in the Path i nodes. On the right part, the nodes AvailX define the availability of the sources X through the DWN according to the path availability. The availability of the sink is defined through the node C100CFE. Modelling the path, the nodes WiX compute the weight w i of each actuator X as explained in the previous section. The Table 6 gives the CPT to compute the weight w i of the component Cornella50. The supply of water through the network should be done by at least 2 sources from all sources. So, the combination among the sources availability through the network is a 2oo5:G system.

Results and Comments

A simulation has been done with a duration of 3600 hours.

It is done under Matlab TM for the optimization, the control and the system simulation. The BN part is computed thanks to Bayesialab TM a BN dedicated software. At the beginning, a fault-free case is considered. As illustrated in Fig. 4 during the 500 first hours, the measured output is tracking the desired output reference z d (3). The steady optimization method generates control inputs which minimize efficiently the criteria under tracking control objective constraints. A first failure is assumed to occur and to be isolated on pump P8 (iSJD50) at t=1000h, and at t=2000h, a second failure is isolated on pump P5 (iRelieu). A Fault Detection and Isolation module is assumed to carry out the failure isolation task. Based on the BN reliability model, the various w i parameters are calculated as presented in Fig. 5. Before t=1000h, the actuators P1 (iEstrella12), P2 (iEstrella3456) and P5 (iRelieu) are used to achieve the water demand z d . This solution is computed by the BN and is based on the success paths 7 and 8 (Table 4). This solution is justified by the BN that compute w 1 , w 2 and w 5 upper to the others w i because the actuators P1 (iEstrella12) and P2 (iEstrella3456) are more reliable than P3 (iSJDSub) and P4 (iSJDSpf ) and the actuator P5 (iRelieu) is more reliable than P6 (iCornella100) (table 5). Moreover, because P1 (iEstrella12) and P2 (iEstrella3456) are saturated, the actuator P3 (iSJDSub) is used to provide the desired output z d as illustrated in Fig. 5. The actuator P3 (iSJDSub) is used because the BN compute w 3 > w 4 (P3 (iSJDSub) is more reliable than P4(iSJDSpf )). This solution is based on the path 6 using P8 (iSJD50). Unfortunately after t=1000, the actuator P8 (iSJD50) is out of order, therefore the path 6 is unavailable. The path 4 {P 3, P 7, P 9} is then computed by the BN to be the solution to reconfigure the system. Between 1000h to 2000h the components used are P1, P2, P5 and P3, P7, P9 as presented in Fig. 5. After the occurrence of the failure of P5 (iRelieu) at t=2000, the success path 7 and the success path 8 become unavailable. Therefore, the BN compute w 1 = 0 and w 2 = 0 due to the fact that the actuators P1 (iEstrella12) and P2 (iEstrella3456) can not be used. The system is reconfigured using the path 1 and the path 2 with the components P3, P4, P6 and P9. The BN compute w 3 = 1, w 4 = 1, w 6 = 1, and w 9 = 1, being the only solution to provide the water to the demand sector as highlighted in Fig. 5.

CONCLUSION

In this paper, we have shown how to define a generic BN that can easily model the reliability of overactuated system. Moreover the functioning probabilities of each actuator is computed with DBN inference. These distributions of probabilities are proposed to be used to define the weighing matrix used to give a specific priority level to the actuators in the re-allocation problem of control. This method provides a control re-allocation that is based on the on-line actuators reliability estimation and the BN provide the reconfiguration solution according to the available success path in the system. The future work is to implement this method to a larger part of the DWN.

For

  Fig. 2. The considered part of the Barcelona DWN.

Fig. 3 .

 3 Fig. 3. Reliability and weights computing BN model of the Barcelona DWN.

Fig. 4 .

 4 Fig. 4. Desired and measured outputs in fault-free case.

Fig. 5 .

 5 Fig. 5. Controlled output behaviors z i (P1..P9) of flow rate and usability indicators w i with a failure on P8 (iSJD50) at 1000h and another one on P5 (iRelieu) at 2000h.

Table 1 .

 1 Conditional Probabilities Table of node g j .

					g j
	e i+δ	. . .	e i	Available Not Available
	Up	Up	Up	1	0
		. . .	. . .	0	1
			Down	0	1
	. . .	. . .	. . .	. . .	. . .
	Down		Up	0	1
		. . .	. . .	0	1
		Down	Down	0	1

Table 2 .

 2 Conditional Probabilities Table of node S.

	S

+ Fig. 1. Generic BN model computing w i according to the success paths g j .

Table

3

. Weight CPT of w δ .

Table 4 .

 4 Success paths of considered part of the Barcelona DWN

	Path	Component Set
	path 1	{AportLL2, iSJDSpf, iSJD10, iCornella100}
	path 2	{AportLL1, iSJDSub, iSJD10, iCornella100}
	path 3	{AportLL2, iSJDSpf, iSJD10, iCornella50, iRelieu}
	path 4	{AportLL1, iSJDSub, iSJD10, iCornella50, iRelieu}
	path 5	{AportLL2, iSJDSpf, iSJD50, iRelieu}
	path 6	{AportLL1, iSJDSub, iSJD50, iRelieu}
	path 7	{ApousE12, iEstrella12, iRelieu}
	path 8	{ApousE3456, iEstrella3456, iRelieu}
	path 9	{AportA, iRelieu}

Table 6 .

 6 Weight CPT of w i Cornella50

	w i Cornella50

ACKNOWLEDGEMENTS

This work was supported by the SAFFE project (Ageing Management in Fault-tolerant control system) from GIS 3SGS -France (https://www.gis-3sgs.fr/).