R Souissi

N Bouguila
email: bouguila.nour@gmail.com

B Bouricha

C Vázquez-Vázquez

M Bendahan

A Labidi

Thickness effect on VOCs sensing properties of sprayed In2S3 films

Keywords: In2S3 film, spray pyrolysis, thickness, volatile organic compounds, gas sensor

This work reports the thickness effect on the sensing performances of In2S3 material for some Volatile Organic Compounds (VOCs). In2S3 films were deposited on glass substrates by the spray pyrolysis technique. Different samples were prepared via changing spray time in the range of 10-90 min. The film thickness varies from 0.8 m to 6.1 m. The X-ray diffraction results demonstrate that the In2S3 films are polycrystalline in nature and exhibit a cubic structure. Additionally, Scanning Electron Microscopy (SEM) and 3D profilometry examinations show that the surface roughness increases with the rising spray time. On the other side, the oxygen adsorption versus a working temperature was investigated. Sensing measurements with ethanol, methanol and acetone gases were carried out by a dynamic control of the current passing through the sensitive layers. The best sensitivity was obtained for the film matching a 70-min deposit time. An understanding detection mechanism based on the oxidation reaction between reduced vapors and chemisorbed oxygen was confirmed.

The selectivity of the sensor was analyzed for several volatile organic compounds (VOCs).

Introduction

Volatile organic compounds sensors are used in a wide range of applications such as in human breath analyzers [START_REF] Lawson | Skin alcohol perspiration measurements using MOX sensors[END_REF], food and fruit ripening monitoring devices. Many studies have recently been devoted to the development of selective and organic vapor sensitive sensors. Among these sensors developed nowadays, we can cite metal oxides including single oxide or multi-oxide structures such as LaFeO3 [START_REF] Rong | Highly selective and sensitive methanol gas sensor based on molecular imprinted silver doped LaFeO3 core-shell and cage Structures[END_REF], ZnCo2O4 [START_REF] Xiong | Multi-shelled ZnCo2O4 yolk-shell spheres for high-performance acetone gas sensor[END_REF], CuFe2O4/-Fe2O3 [START_REF] Li | Hollow CuFe2O4/-Fe2O3 composite with ultrathin porous shell for acetone detection at ppb levels[END_REF], ZnO [5][START_REF] Wongrat | Acetone gas sensors based on ZnO nanostructures dectorated with Pt and Nb[END_REF][START_REF] Zhu | Hydrothermal synthesis of hierarchical flower-like ZnO nanostructure and its enhanced ethanol gas-sensing properties[END_REF], SnO2 decorated SiO2 [START_REF] Asgari | SnO2 decorated SiO2 chemical sensors: Enhanced sensing performance toward ethanol and acetone[END_REF]. The literature review for the latest achievements has shown some very promising results. Recently, metal sulfides have been studied as a new sensitive class for vapor detection. It has been shown that cadmium sulfide (CdS) films are more sensitive to Ethanol than Cadmium Oxide (CdO) [START_REF] Gaiardo | Metal Sulfides as Sensing Materials for Chemoresistive Gas Sensors[END_REF]. Nevertheless, these materials are considered toxic. It is therefore important to develop non-toxic films such as Indium Sulfide In2S3. Various techniques such as Spray Pyrolysis [START_REF] John | CuInS2/In2S3 thin film solar cell using spray pyrolysis technique having 9.5 % efficiency[END_REF][START_REF] Kraini | Properties of In2O3 films obtained by thermal oxidation of sprayed In2S3[END_REF], Hydrothermal [START_REF] Abadi | Hydrothermal synthesis, characterization and optical properties of 3D flower like indium sulfide nanostructures[END_REF], Ultrasonic Dispersion [START_REF] Li | Preparation of In2S3 nanoparticle by ultrasonic dispersion and its tribology property[END_REF],

Chemical Bath Deposition (CBD) [START_REF] Asenjo | Indium sulfide buffer layers deposited by dry and wet methods[END_REF], Physical Vapor Deposition [START_REF] Timoumi | Fabrication and characterization of In2S3 thin films deposited by thermal evaporation technique[END_REF], etc., have been used to prepare In2S3. The Spray Pyrolysis method has been chosen in this work. Indeed, this method, which is both simple and not expensive for the deposition of large area thin films, does not require high quality targets or vacuum and is a facile way to dope material by adding a doping element to the spray solution [START_REF] Tiss | Effect of silver doping on structural and optical properties of In2S3 thin films fabricated by chemical pyrolysis[END_REF]. The physical properties of the obtained In2S3 films depend on synthesis parameters such as deposit time, substrate temperature and spray solution flow rate [START_REF] Bouguila | Thickness effect on properties of sprayed In2S3 films for Photovoltaic Applications[END_REF][START_REF] Bouguila | Substrate temperature effect on properties of sprayed In2S3 films[END_REF].

In2S3 is an n-type semiconductor of group III-VI with a wide energy band gap ranging from 2 to 3 eV [START_REF] Bouguila | Thickness effect on properties of sprayed In2S3 films for Photovoltaic Applications[END_REF][START_REF] Bouguila | Substrate temperature effect on properties of sprayed In2S3 films[END_REF][START_REF] Schulmeyer | Band offset at the CuGaSe2 ∕ In2S3 heterointerface[END_REF]. It can exist in four allotropic crystal structures as a function of temperature and pressure [START_REF] Diehl | Vapour growth of three In2S3 modifications by iodine transport[END_REF]. At room temperature and atmospheric pressure, In2S3 has been found to crystallize into the defective spinel structure -In2S3 ,which is a stable phase and presents many Sulfur vacancies VS (donor), Indium vacancies VIn (acceptor), and Indium interstitials IIn (donor) in the unit cell [START_REF] Diehl | Vapour growth of three In2S3 modifications by iodine transport[END_REF].

This material shows interesting optoelectronic and semiconductor sensitization properties which allow it to be used as a buffer layer in solar cells [START_REF] Sterner | Study on ALD In2S3/Cu(In,Ga)Se2 interface formation[END_REF][START_REF] Buecheler | Ultrasonically sprayed indium sulfide buffer layers for Cu(In,Ga)(S,Se)2 thin-film solar cells[END_REF], a heterojunction in photovoltaic electric generators [START_REF] Dalas | Primary solid-state batteries constructed from copper and indium sulphides[END_REF][START_REF] Hsiao | Characterization of photovoltaics with In2S3 nanoflakes/p-Si heterojunction[END_REF], biological imaging sensors [START_REF] Nagesha | In2S3 Nanocolloids with Excitonic Emission: In2S3 vs CdS Comparative Study of Optical and Structural Characteristics[END_REF] and a photodetector [START_REF] Kumar | Indium sulfide based metal-semiconductormetal ultraviolet-visible photodetector[END_REF][START_REF] Raid | Preparation of high-sensitivity In2S3/Si heterojunction photodetector by chemical spray pyrolysis[END_REF][START_REF] Ghosh | Excitation dependent multicolor emission and photoconductivity of Mn, Cu doped In2S3 monodisperse quantum dots[END_REF]. More recently, Cai et al. [START_REF] Cai | Hierarchical spheres In2S3-based cataluminescence sensor for ammonium sulfide[END_REF] have demonstrated that In2S3 hierarchical microspheres are good candidates for fabricating practical cataluminescence Ammonium Sulfide sensor. In our earlier paper [START_REF] Souissi | Ethanol sensing properties of sprayed -In2S3 thin films[END_REF], we reported the use of -In2S3 thin films deposited by the Spray Pyrolysis technology as an Ethanol sensor. We showed that the In2S3 layers had a fairly good sensitivity to Ethanol. The optimal operating temperature was found to be 350°C, the best response to 500 ppm Ethanol concentration was 1.7 and the response/recovery times were about 150 s. These results are encouraging to deepen the study for enhancing sensing performances by acting on the synthesis parameters.

The aim of the present study is to investigate the influence of the thickness of sprayed In2S3 films on the organic vapor-sensing properties. We discuss also in this paper how the oxygen adsorption/desorption phenomenon impacts the detection mechanism of the analyzed gas. The detection properties were generally deduced from DC measurements which provide information in the general sensor behavior.

Experimental details

Films preparation

In2S3 films were deposited by the Spray Pyrolysis technique. Glass substrates were cleaned in Ethanol and rinsed in deionised water. An aqueous solution was made by a mix of Indium Chloride (InCl3) and Thiourea (SC(NH2)2) with a ratio of S:In = 2. The solution was sprayed at a flow rate of 2 mL.min -1 onto 1 cm1 cm glass substrates at 340°C within an accuracy of ± 5°C. Nitrogen was used as a carrier gas and the flow rate was kept constant at 6 L.min -1 . The deposit time was varied in the range of 10-90 min. The In2S3 formation results from a global endothermic reaction governed by the following equation [START_REF] Asenjo | Indium sulfide buffer layers deposited by dry and wet methods[END_REF][START_REF] Bouguila | Substrate temperature effect on properties of sprayed In2S3 films[END_REF]:

2 InCl3+ 3 SC(NH2)2 + 6 H2O → In2S3 + 3 CO2 + 6 NH4Cl (1)
Finally, two rectangular Au electrodes of 5 mm long and 2 mm apart are deposited on the film surface using the Thermal Evaporation technique under high vacuum.

Films characterization

X-ray diffraction (XRD) analyses of the films were performed using monochromatic CuK1 radiation (1.54056 Å) (Bruker D8 Advance diffractometer). Topography images and surface roughness were obtained with a non-contact 3D optical profilometer (Sensofar S Neox) in a white-light vertical scanning interferometry (VSI) mode and using a CFI60-2 Nikon objective lens (10x EPI, numerical aperture 0.3 and working distance 17.5 mm). The surface morphology and the chemical analysis were studied by means of scanning electron microscopy (SEM) using a Zeiss Ultra PLUS microscope endowed with an electron dispersive spectrometer (EDS).

Experimental set-up for VOC detection

To investigate gas sensing response, the In2S3 films were tested using the experimental setup shown in Fig. 1. The global gas circuit has two sub-circuits. The first is used to test the organic vapor generated using two mass-flow controllers D1 and D2 which are used to set the dry air flow rate and the organic vapor concentration. The dry air from flow controller D1 is charged with vapor through a dilution bath containing a pure organic solvent. This latter is maintained at a fixed temperature in order to adjust the saturation vapor partial pressure. This flow is mixed with the second flow of dry air coming from the second flow controller D2. By varying d1 and d2 (d1+d2 were kept constant at 1 L.min -1 ), diverse concentrations of organic vapor can be obtained and computed using the formula [START_REF] Souissi | Ethanol sensing properties of sprayed -In2S3 thin films[END_REF][START_REF] Labidi | Ethanol and ozone sensing characteristics of WO3 based sensors activated by Au and Pd[END_REF]:

(

) 2
where x is the molar fraction of the vapor in the bottomed flask at Tvap, given by:

(

with Pvap the partial pressure of the vapor at a specified temperature Tvap, and Patm the atmospheric pressure.

The total flow charged with organic vapor has been blown on the sensor for 5 min. Then the test chamber has been purged with dry air for 15 min.

A second sub-circuit is used to check oxygen adsorption/desorption phenomenon. Since the test chamber is dynamic, the gases are controlled at 1 L.min -1 flow rate. Nitrogen (0% Oxygen atmosphere) is sent along the third arm D3. Firstly, the flow loaded with dry air (20% oxygen atmosphere) was sent along the second arm D2 and blown directly on the sensor for 5 min. Then, it was removed from the test chamber by switching the gas flow to Nitrogen for 15 min.

Samples were polarized at 0.5 V. The measurements of direct current intensity have been done in real time using a (HP 4140B) Source/Picoammeter. The electrical contact on the sensor is ensured by gold tip tungsten electrodes. Measurements were done at different operating temperatures ranging from 150°C to 425°C. The sensor is heated by a halogen lamp which is powered by a stabilized generator and controlled by an (HP 34401A) Multimeter regulator that allows to set the temperature of the semiconductor by measuring the resistance across the RTD PT100 platinum probe that will be converted into a temperature value.

In order to compare the responses of In2S3 sensors we will use the following formula of sensing response [START_REF] Souissi | Ethanol sensing properties of sprayed -In2S3 thin films[END_REF][START_REF] Labidi | Ethanol and ozone sensing characteristics of WO3 based sensors activated by Au and Pd[END_REF]:

𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑒 = 𝑅 𝑎 /𝑅 𝑔 (4)
where Ra and Rg represent respectively the sensor resistance value in air and in a sample gas.

Results and discussion

3.1. Characterization

Thickness and structure

The double weight method was used to measure In2S3 film thicknesses (d) for varying deposit time and calculated by using the relation:

𝑑 = 𝑚  𝑆 ( 5
)
where m is the deposited film mass,  is the density of the material in the bulk form ( = 4.613 g.cm -3 matching with the standard file of JCPDS 25-390) and S is the effective area on which the film was deposited.

The obtained values of thicknesses for all samples are illustrated in Table 1 and show that thickness increases from 0.8 m to 6.1 m by rising deposit time in the range 10-90 min.

The X-ray diffraction patterns of the In2S3 layers obtained for different thicknesses are shown in Fig. 2. These patterns indicate that all films were polycrystalline and can be closely indexed to pure -In2S3 phase with a simple cubic structure matching with the standard file of JCPDS 32-0456. The preferred orientation observed in the pattern for all samples is (400). The intensity of the diffraction main peak becomes more intense with the increase of spray time. Such behavior can be explained by the increase of film thickness and the increased ability of atoms to orient in proper equilibrium sites giving more preferentially oriented films.

The main peak was used to determine the crystallite size using Lorentzian fits and Scherrer's formula [START_REF] Cullity | Elements of X-ray Diffraction[END_REF]:

𝐷 = 0.9   𝑐𝑜𝑠𝜃 (6)
where λ denotes the wavelength of X-ray, θ represents the Bragg angle and  the full width at half maximum (FWHM) in radians.

Likewise, the imperfection in the films can be quantified by computing the dislocation density (δ) which is defined as the length of dislocation lines per unit volume of the crystal.

For the (400) peak, it was calculated using the Williamson-Smallman relation [START_REF] Bilgin | The effect of substrate temperature on the structural and some physical properties of ultrasonically sprayed CdS films[END_REF]:

𝛿 = 1 𝐷 2 (7)
The interplanar d-spacing (dhkl) for In2S3 nanoparticles was calculated using Bragg's equation [START_REF] Asgari | SnO2 decorated SiO2 chemical sensors: Enhanced sensing performance toward ethanol and acetone[END_REF]. The lattice parameter (a) associated with dhkl and Miller indices (h,k,l) are calculated via the theoretical equations ( 9) and ( 10) as well [START_REF] Cullity | Elements of X-ray Diffraction[END_REF]:

𝑑 ℎ𝑘𝑙 =  2 𝑠𝑖𝑛 (8) 1 𝑑 ℎ𝑘𝑙 2 = ℎ 2 +𝑘 2 +𝑙 2 𝑎 2 (9) 𝑎 = 4𝑑 400 (10)
The calculated lattice parameter reveals a constant value a = 10.7 Å, while crystallite size and dislocation density of films are presented in Table 1. It can be seen that the change in thickness affects the density of native imperfections which modify the structural quality of the films. In fact, a maximum of the crystallite size (D = 52 nm) and a minimum of the defect  ( = 3.7 10 10 lines.cm -2 ) is located around a thickness of 4.8 m reflecting the best crystallinity of the deposit in this condition.

Spray time (min)

Topography and morphology

Surface topography is an important surface property and it affects the performance of products at its application fields. Topography of prepared layers was performed by using optical profilometer. Fig. 3 presents 2D surface measurement of In2S3 samples over a scanned area of 1 mm 2 . In this case we arbitrarily amplify the z axis and we also know the altitudes using a color palette that allows appreciating the local variations of the heights. Here the variations of the relief are also amplified and the represented surfaces would appear flat. The Surface roughness of films seems to change from one layer to another with a changing thickness.

The parameters (prefix) R calculated on the roughness profile are defined in a handful of international standards Geometric Product Specifications (GPS)-Surface Texture: Profile Method (ISO 4287:1997). Some parameters calculated on segments of the profile (the base lengths) and averaged over several segments are presented in Table 1. These results are evaluated from measurements in three different areas of each sample and from five profiles for each area.

Rq, root mean squared roughness, is the standard deviation of the height distribution over the evaluation length. We can note from The morphological study by SEM shows in Fig. 4 the microscopic appearance of the surface layers for different thicknesses. The films were well covered, homogeneous, dense and continuous. We can note that the thickness effect is reflected on the film surface morphologies. SEM images reveal a granular layer for the sample corresponding to the thickness of 4.8 m, whereas the layers are smooth for the others thicknesses. In addition to this, big granules/chunks are observed at the figure for 0.8 µm, 2.3 µm, and 4.8 µm thicknesses. Such observation may be a result of a small deviation from the stoichiometric composition in starting a solution that yields a very strong influence on the microstructure and the surface morphology of the films. This behavior was the subject of previous reports on films such as In2S3 [START_REF] Revathi | Synthesis and physical behavior of In2S3 films[END_REF] and CuInS2 [START_REF] Krunks | Structural and optical properties of sprayed CuInS2 films[END_REF]. We can also note a rougher layer surface with cracks at the higher thickness of 6.1 m identifying a beginning of crystallinity degradation.

Chemical analysis

The use of EDS made chemical elements´ identification possible in the obtained thin layers.

We report in These results are evaluated from measurements in four different areas of each sample. The S/In molar ratio remains constant for all the samples with an average value of 1.27. This order of magnitude is less than the stoichiometric ratio (S/In = 1.5) of the In2S3 material. This indicates that the prepared In2S3 material contains defective sulfur positions.

VOCs sensing properties

Response

First, all the sensors elaborated with several thicknesses in the range 0.8-6.1 m were tested to 4000 ppm of Methanol, Ethanol and Acetone gas at an operating temperature of 350°C. The choice of such temperature is based on our previous publication [START_REF] Souissi | Ethanol sensing properties of sprayed -In2S3 thin films[END_REF]. This is the optimal temperature for an Ethanol detection by In2S3 films matching a thickness around 2

m. As we will see in the following section, the optimal temperature can be shifted by the film thickness or the gas nature while remaining in the vicinity of 350°C. We think that this value allows a significant response comparison. Fig. 5 illustrates dynamic responses during the alternating exposure to target gas and dry air. For each sensor, three response and recovery curves are shown. We note that for all the prepared sensors, during the exposure to target gas, the response increases quickly to an equilibrium value and then returns to the baseline when dry air is reinjected in the test chamber. This behavior is consistent with the sensing mechanism of n-type semiconductors. It is also observed that the response plot obtained is affected by the change of thickness. A possible explanation is that samples have different specific surface areas. squared roughness of 141 nm, 207 nm and 223 nm, respectively) the gas interaction is only taking place at the geometric surface. In the case of granular layer and high roughness (d = 4.8 m, Rq = 501 nm), the gas interaction can therefore take place at the surface of individual grains and at grain-grain boundaries [START_REF] Barsan | Conduction model of metal oxide gas sensors[END_REF]. So, such sensor presents a higher specific surface area involving abundant active adsorption sites which improve the gas response. This result confirms the proposal to improve the response by changing microstructure and morphology of sensors [START_REF] Dey | Semiconductor metal oxide gas sensors: A review[END_REF]. In the case of sample (d = 6.1 m, Rq = 1150 nm) cracks increase the electric potential barriers, reduce the current and diminish the sensing response. Indeed, the increase in resistance is related to the number of cracks in the semiconductor layer which depends on the applied deformation and the thickness of the film [START_REF] Cairns | Electric resistance dependent on the stress of indium oxide doped with tin on polymeric substrates[END_REF]. Despite the fact that the cracks can be the gas trapping path, it is also very possible that they cause an electrical rupture of these areas. We think that the active surface which describes the resistance decreases enormously with these cracks because they prevent the electrical contribution of a certain volume of material and therefore reduce the response of the sensor.

Rapidity

In a second stage, we investigated the time deposition effect on the sensing rapidity of In2S3 thin films. The response time res is defined as the time required for the sensor to reach 90% of the sensor response, and the recovery time rec as that needed to reach 10% of the initial resistance baseline after the analyst gas has been purged. Fig. 6 shows the comparison of samples with respect to their response time, tested by the vapors of Methanol, Ethanol and Acetone with 4000 ppm concentration and 350°C operating temperature. We note that the sensor prepared with thickness d = 4.8 m presents the faster response. Also, the shortest response time corresponds to Ethanol gas for all samples (e.g. res(ethanol) = 33 s). In addition to this, Fig. 6 illustrates the evaluation of recovery times which show a similar evolution-like response time. We have checked that the sensor matching the thickness of 4.8 m has also the shortest recovery time (e.g. rec(ethanol) = 138 s). Therefore, the performance of In2S3 sensors evaluated by response and rapidity reveals that the sample matching the thickness of 4.8 m presents the best results.

Reproducibility

Reproducibility was investigated by exposing In2S3 sensors with different thicknesses to a variety of VOCs. Fig. 7 shows the dynamic response of sensors to various concentrations of Methanol, Ethanol and Acetone ranging from 2000 ppm to 10000 ppm, 1000 ppm to 5000 ppm and 4000 ppm to 20000 ppm respectively at the operating temperature of 350°C. For each sensor, the sensor´s response has an obvious increasing relationship with VOCs concentration, demonstrating that the sensor can offer more surface active sites and adsorb more VOCs molecules. Moreover, the sensor's response is affected by the layer's thickness and the best response was performed with the thickness matching 4.8m. This fact reflects the reproducibility of the results detailed previously.

Gas sensing mechanism

In the following part, we propose to study the sensing mechanism and selectivity for some volatile organic compounds. We focused on the In2S3 sample corresponding to the 4.2 m thickness.

Oxygen adsorption/desorption

Here, the adsorption and desorption of oxygen on a polycrystalline Indium Sulfide thin layers have been experimentally demonstrated by performing alternating exposure to Nitrogen and dry air at a working temperature ranging from 150°C to 400°C. Fig. 8(a) shows that resistance increases when dry air is introduced in the test chamber and decreases after its evacuation by Nitrogen for an operating temperature higher than 200°C. In fact, under dry air an adsorption phenomenon occurs. Ho et al. [START_REF] Ho | Synthesis of In2S3 and Ga2S3 crystals for oxygen sensing and UV photo-detection[END_REF] reported that because of Sulfur vacancies and Indium interstitials defects, the -In2S3 crystal can form easily surface oxidation states of -In2S3-3xO3x in ambient air (or O2) by chemical reaction of In-O. So, we think that the interaction of atmospheric oxygen with the surface of In2S3 forms a layer of charged oxygen species, which traps electrons from the bulk of the material. The electrons trapped for this process are removed from the conduction band and are confined on the surface [START_REF] Barsan | Conduction model of metal oxide gas sensors[END_REF]. Consequently, a space charge layer is formed. The presence of negative surface charges leads to a band bending defined by potential barrier eVs at the surface. This restricts the electron flow and decreases the conductivity. On the other hand, under Nitrogen, the resistance decreases reflecting the oxygen desorption phenomenon. In this context, the study of the oxygen sensing response versus the operating temperature was illustrated in Fig. 8(b). The response increases by increasing the working temperature up to 350°C and decreases above this temperature. This fact could be interpreted as a competition between the rates of oxygen adsorption and desorption, which is shown in the Lennard-Jones [START_REF] Labidi | Impedance spectroscopy on WO3 gas sensor[END_REF] equation:

𝑑𝜃 𝑑𝑡 = 𝑘 𝑎𝑑𝑠 𝑒 -𝛥𝐸 𝐴 𝑘𝑇 -𝑘 𝑑𝑒𝑠 𝑒 -(𝛥𝐸 𝐴 +𝛥𝐻) 𝑘𝑇 (11)
where kads and kdes are respectively the rate constants for adsorption and desorption reactions,

EA the activate energy of chemisorption, H the heat of adsorption and θ is the fraction of available surface sites covered. The evolution of the response versus the working temperature follows also Lennard-Jones equation and shows an optimal temperature for each gas. It clearly shows a direct correlation between oxygen adsorption and reducing gas detection. Indeed, responses to oxygen and reducing gases present a similar evolution and a neighboring optimal temperature.

Operating temperature effect

The increase in operating temperature before the optimum value improves the oxygen adsorption and hence the VOCs response. On the other hand, desorption of all oxygen ionic species previously adsorbed occurs at high temperatures, which explains the response decrease to reducing gases beyond the optimum value.

We note that optimal temperature for Ethanol and Acetone is 365°C, whereas it is shifted to 385°C for Methanol gas. Similarly, such a trend has been studied by Bejaoui et al. [START_REF] Bejaoui | Modeling of a p-type resistive gas sensor in the presence of a reducing gas[END_REF]. They proposed a model simulating the interaction between sensors and reducing gases. They showed that the increase of the activation energy following a change of gas causes a shift of the optimal temperature towards a higher value. We also remarked that Ethanol response (16.2) is higher than the acetone one (6.3) at the same optimal temperature. This could be explained by the Ethanol and the Acetone molecules' size difference. Indeed, the surface cover rate is higher for the smaller molecules of Ethanol.

Gas concentration effect

In order to identify ionosorbed oxygen species, measurements were conducted on the In2S3 sensor at a working temperature of 350°C by varying the concentration of Methanol, Ethanol and Acetone. The transient current during cycles (air/reducing gas/air) is shown in Fig. 10(a).

The fitting of the response versus gas concentration obtained in Fig. 10(b) shows a power law:

𝑅𝑒𝑠𝑝𝑜𝑛𝑠𝑒 = 𝐶𝑡𝑒 [𝐶 𝑅𝑒𝑑𝑢𝑐𝑖𝑛𝑔 𝑔𝑎𝑠 ]  (12)
where   1/3 for the three analyzed gases. This typical value locates mainly O 2-ionosorbed oxygen on the material surface according to the metal oxide gas detection model proposed by Barsan and Weimer [36]. Thus, ionization equations of adsorbed oxygen can be written as follows:

𝑂 2(𝑔𝑎𝑠) → 𝑂 2(𝑎𝑑𝑠) (13) 𝑂 2(𝑎𝑑𝑠) + 4𝑒 -→ 2𝑂 (𝑎𝑑𝑠) 2- (14)
In the same context, when the material is exposed to reducing gases, the organic molecules are oxidized by oxygen species O 2-and simultaneously the electrons are fed back into sensing body, the potential barrier is reduced and the conductivity increases. This mechanism can be expressed as follows:

For Methanol:

𝐶𝐻 3 𝑂𝐻 (𝑎𝑑𝑠) + 𝑂 (𝑎𝑑𝑠) 2- → 𝐻 2 𝐶𝑂 (𝑔𝑎𝑠) + 𝐻 2 𝑂 + 2𝑒 - (15)
For Ethanol:

𝐶 2 𝐻 5 𝑂𝐻 (𝑎𝑑𝑠) + 𝑂 (𝑎𝑑𝑠) 2- → 𝐶𝐻 3 𝐶𝐻𝑂 (𝑔𝑎𝑠) + 𝐻 2 𝑂 + 2𝑒 - (16)
For Acetone:

𝐶𝐻 3 𝐶𝑂𝐶𝐻 3 (𝑎𝑑𝑠) + 8𝑂 (𝑎𝑑𝑠) 2- → 3𝐶𝑂 2(𝑔𝑎𝑠) + 3𝐻 2 𝑂 (𝑔𝑎𝑠) + 16𝑒 - (17)
So, oxygen plays the main element in reducing gas sensing response as detailed by the mechanism presented in Fig. 11.

Selectivity

The efficiency of a sensor to respond to a certain gas in the presence of other gases is called selectivity. Fig. 12 shows the dynamic response of an In2S3 film (d = 4.8 m) to seven VOCs at a 500 ppm concentration and a working temperature of 350°C. This figure reveals that the relatively high responses are obtained with Isopropanol (10.5) and Ethanol (9.5). The middle response is observed with Butan-1-ol (5.8) and Toluene (4.5). Lastly, weak responses correspond to Methanol (2.9), Acetone (3.3) and Chlorobenzene (1.5) vapors. Therefore, we can't expect for the time being that our sensor can be selective. However, it is possible to analyze the different responses through a mathematical technique known as principal component analysis (PCA) [START_REF] Vanden Eynde | ToF-SIMS quantification of polystyrene spectra based on principal component analysis (PCA)[END_REF]. This latter is a method of graphical data analysis which consists in finding the directions of space which best represents the correlations between the chosen variables of the measure [START_REF] Penza | Gas recognition by activated WO3 thin film sensors array[END_REF]. This study is being planned for a future publication. A comparison of the developed sensor response to those of previously published Indium Oxide sensors is reported in Table 3. The results show that the Ethanol detection response of Indium sulfide In2S3-based sensor in the present work is not high but can be improved by enhancing gas adsorption or promoting specific chemical reactions via catalytic or electronic effects using bulk dopants and by the addition of metallic clusters or Oxide catalysts [START_REF] Liu | Ultra-fast responding C2H5OH sensors based on hierarchical assembly of SnO2 nanorods on cube-like α-Fe2O3[END_REF][START_REF] Tyagi | Metal oxide catalyst assisted SnO2 thin film based SO2 gas sensor[END_REF][START_REF] Hohl | The role of noble metals in the chemistry of solid-state gas sensors[END_REF].

Material

Conclusion

In2S3 films were deposited on glass substrates by spray pyrolysis technique. Film thickness increases from 0.8 m to 6.1 m by increasing deposit time from 10 min to 90 min. XRD diffraction analysis reveals the formation of cubic -In2S3 pure phase with (400) preferred orientation. The crystallite size varies in the narrow 39-52 nm range. The EDS measurements indicate that the S/In molar ratio remains constant for all the samples with an average value of 1.27. The morphological study by SEM shows that films were well covered, homogeneous and dense. The parameter Rq calculated from the roughness profile increases eventually with increasing thickness. Thus, the In2S3 sample matching the thickness of 4.8 m combine high roughness and best crystallinity. Such results were correlated with its best sensing properties.

We have fairly improved the In2S3 sensor performances compared to that mentioned in the previous publication [START_REF] Souissi | Ethanol sensing properties of sprayed -In2S3 thin films[END_REF]. The response to 500 ppm Ethanol at 350°C is enhanced from 1.7 to Nevertheless, these results revealing the potential of β-In2S3 in gas detection are encouraging for deepening the study in order to improve sensing properties and to test other gasses.

Fig. 1 :

 1 Fig. 1: Experimental set-up for gas detection.

Fig. 2 :

 2 Fig. 2: XRD patterns of sprayed In2S3 films.

Fig. 3 :

 3 Fig. 3: Topography images of In2S3 films.

Fig. 4 :

 4 Fig. 4: SEM images of sprayed In2S3 films.

Fig. 5 :

 5 Fig. 5: Dynamic response of In2S3 films to 4000 ppm of Methanol, Ethanol and Acetone at 350°C for thickness varying from 0.8 m to 6.1 m.

  In the case of smooth layers and low roughness (d = 0.8 m, d = 2.3 m, d = 3.2 m / Root mean

Fig. 6 :

 6 Fig. 6: Response and recovery times of In2S3 films to 4000 ppm of methanol, ethanol and acetone at 350°C.

Fig. 7 :

 7 Fig. 7: Dynamic response of In2S3 sensors with different thicknesses to various concentrations of methanol, ethanol and acetone at the operating temperature of 350°C.

Fig. 8 :

 8 Fig. 8: (a) Resistance change of In2S3 film with time upon altering exposure to dry air and Nitrogen at operating temperature ranging from 150°C to 400°C, (b) response of In2S3 film to oxygen as a function of operating temperature.

Fig. 9 (

 9 Fig. 9(a) demonstrates the resistance change during an alternating exposure to dry air and

Fig. 9 :

 9 Fig. 9: (a) Resistance evolution of In2S3 sensor upon exposure to 4000 ppm concentration of various organic vapors at operating temperatures in the range: 250°C-400°C, (b) sensing response of In2S3 sensor vs. working temperature towards various organic vapors.

Fig. 10 :

 10 Fig. 10: (a) Dynamic response of In2S3 sensor to various organics vapors concentrations at 350°C, (b) experimental and power fit of sensing response of In2S3 sensor as a function of organic vapor concentration at 350°C.

Fig. 11 :

 11 Fig. 11: Detection mechanism of organic vapor by In2S3 sensor.

Fig. 12 :

 12 Fig. 12: Selectivity of In2S3 sensor (d = 4.8 m).

9. 5 .

 5 The response time and recovery time are improved from 150 s/155 s to 23 s/75 s respectively. The optimal working temperature is found to be around 385°C for Methanol and 365°C for both Ethanol and Acetone. Besides, oxygen adsorption versus working temperature were put forth and the sensing mechanism of the In2S3 semiconductor was explained on the bases of the reaction between the reducing vapors with the chemisorbed oxygen ions O 2-.

Table 1 :

 1 Thickness, data evaluated from X-ray diffraction pattern and roughness profile parameters of In2S3 films.

	Rsk

Thickness (m)

D (nm) δ (10 10 lines.cm -2 ) Rq (nm)

Table 1

 1

that Rq increases eventually with increasing thickness by rising spray time and shows high values for samples matching thicknesses of 4.8 m and 6.1 m.

Rsk, asymmetry of the profile: asymmetry of the distribution of the heights, defined on the length of evaluation. This parameter is important because it gives information on the morphology of the surface state. A positive value of Rsk (thicknesses of 4.8 m and 6.1 m) corresponds to a surface having peaks and protuberances above the surface, and thus a distribution shifts towards the lowest points, while a negative value (thicknesses of 0.8 m, 2.3 m and 3.2 m) corresponds to a plateau surface with scratches or deep pores, and a shifted distribution towards the highest points.

Table 2

 2 the counting of the atomic percentage of the main elements in the films.

	Thickness (m)	S (at.%) In (at.%)	S/In
	0.8	31.60	23.68	1.33
	2.3	37.83	30.15	1.25
	3.2	37.98	29.90	1.27
	4.8	36.92	29.07	1.27
	6.1	35.74	28.92	1.24

Table 2 :

 2 EDS results of In2S3 thin films.

Table 3 :

 3 Sensing response of the developed indium sulfide sensor compared with previously published indium oxide sensors.

Acknowledgments

This work was supported by Tunisian Ministry of Higher Education and Scientific Research, Spanish Ministry of Science and Innovation -FEDER Funds (MODENA Project CTQ2016-79461-R) and Fundación Ramón Areces (Spain, Project CIVP18A3940). NANOMAG group belongs to Galician Competitive Research Group ED431C-2017/22, programme co-funded by FEDER, and AEMAT Strategic Partnership (ED431E-2018/08,


G Pérez-Zuñiga

E Chanthery
email: elodie.chanthery@laas.fr

L Travé-Massuyès

J Sotomayor-Moriano

Near Optimal Decentralized Diagnosis via Structural Analysis

Keywords: Fault Detection and Isolation, Test selection, Decentralized diagnosis, Structural analysis. I

Health monitoring of current complex systems significantly impacts the total cost of the system. Centralized fault diagnosis architectures are sometimes prohibitive for large-scale interconnected systems such as distribution systems, telecommunication networks, water distribution networks or fluid power systems. Confidentiality constraints are also an issue. This paper presents a decentralized fault diagnosis method that only requires the knowledge of local models and limited knowledge of their neighboring subsystems. The method, implemented in the Decentralized Diagnoser Design (D 3 ) algorithm, is based on structural analysis and can advantageously be applied to high dimensional systems, linear or non-linear. Using the concept of isolation on request, a hierarchy is built according to diagnostic objectives. The resulting diagnoser is based on analytical redundancy relations generated along the hierarchy. Their number is optimized via binary integer linear programming while still guaranteeing maximal diagnosability at each level. D 3 proves of lower time complexity than its centralized equivalent. It is successfully applied to a non-linear combined cycle gas turbine power plant.

Fault Detection and Isolation (FDI) can be achieved leveraging the analytical redundancy provided by a system model. The idea is to generate tests whose outputs, termed residuals, reflect inconsistencies between a reference model prediction and the actual system behavior. Such tests are designed using analytical approaches, such as observers [START_REF] Edwards | Sliding mode observers for fault detection and isolation[END_REF], parameter estimation [START_REF] Isermann | Fault-Diagnosis applications: model-based condition monitoring: actuators, drives, machinery, plants, sensors, and fault-tolerant systems[END_REF] or parity equations [START_REF] Gertler | Structured Parity Equations in Fault Detection and Isolation[END_REF].

We adopt the parity equations approach and work with analytical redundancy relations (ARRs). The system is modeled by a set of ordinary differential equations. ARRs are generated from the model by eliminating unmeasured variables, which can be carried out in a pure analytical framework using elimination theory [START_REF] Diop | Elimination in control theory[END_REF], [START_REF] Zhang | Fault detection and isolation in nonlinear dynamic systems: A combined input-output and local approach[END_REF]. However, another way to proceed is to first use structural analysis techniques to select sets of equations that can lead to ARR generation.

Structural analysis makes use of a structural abstraction of the model. Only the links between variables and equations are modeled. Despite the apparent simplicity of the approach, it provides a set of powerful tools, based on graph theory, to analyze the system and deduce relevant information. Moreover, it applies to both linear and non-linear systems [START_REF] Armengol | Minimal structurally overdetermined sets for residual generation: A comparison of alternative approaches[END_REF]- [START_REF] Bregon | An event-based distributed diagnosis framework using structural model decomposition[END_REF].

The originality of this paper is to use structural analysis in a decentralized framework, for decentralized FDI of continuous systems. In this framework, one of the major assumptions is that the system is decomposed into a given set of subsystems. The decomposition results from taking into account functional, geographical or confidentiality constraints. This assumption arises from industrial requests aiming at protecting the data about the subsystems they develop. This is true in many application domains, like space, aeronautics, or power plants, where several suppliers are involved in the design phase.

This paper brings the following contributions:

• An original Decentralized Diagnoser Design method implemented in the D 3 algorithm : using the concept of isolation on request, this method consists in using first the local models of subsystems. If isolability of all the faults cannot be achieved, a higher level is investigated to improve isolability. A hierarchy of levels is thus built, driven by diagnostic goals. The advantages of this design are twofold: on one hand, the architecture naturally matches the system functional decomposition commonly adopted in engineering processes. On the other hand, it results in an anytime FDI algorithm. Indeed, if real time constraints are severe, the algorithm may be stopped at any level while still returning a consistent, although not necessarily precise, diagnosis. • An advantageous inter linkage of test generation and test selection thanks to a binary integer linear programming (BILP) model [START_REF] Vanderbei | Linear Programming: Foundations and Extensions[END_REF] that achieves to minimize the number of tests while maximizing fault isolability at each considered level. • A thorough theoretical and experimental analysis of the properties of the D 3 algorithm and its application to the non-linear real case study of a gas turbine combined cycle power plant. The paper is organized as follows: section II presents related work on decentralized diagnosis and structural analysis. In section III, the main concepts of the structural approach are summarized and Fault-Driven Minimal Structurally Overdetermined (FMSO) sets are introduced. The link between ARRs and FMSO sets is outlined and residual generation is explained. Section IV extends the concepts presented in the previous section to the decentralized fault diagnosis framework. Section V presents the algorithm D 3 for the design of the decentralized diagnoser that interlinks test generation and test selection and proves its properties. The computational efficiency of D 3 is proved in practice, and a Gas Turbine Combined-Cycle (GTCC) power plant is then used as a case study in section VI. Finally, section VII concludes the paper.

II. RELATED WORK

Structural analysis is a general framework that can be used to analyze complex, dynamic systems. It ignores the details of parameter values to base the analysis on the structure of the system by means of efficient graph-based tools [START_REF] Cassar | A structural approach for the design of failure detection and identification systems[END_REF].

It has been used for about two decades by the FDI community to identify redundancy in the system model, independently of their nonlinear nature [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF], [START_REF] Travé-Massuyès | Diagnosability analysis based on component-supported analytical redundancy relations[END_REF], [START_REF] Krysander | A structural algorithm for finding testable sub-models and multiple fault isolability analysis[END_REF]. If we refer to diagnosis test selection, FlexDx [START_REF] Krysander | FlexDx: A reconfigurable diagnosis framework[END_REF], can be mentioned. However, all these works frame the problem in a centralized framework.

It is a long time since decentralized control has been approached in the literature [START_REF] Travé | Large scale systems: decentralization, structure constraints and fixed modes[END_REF]. Decentralized diagnosis has also been the subject of many works for discrete event systems [START_REF] Wang | Diagnosis of discrete event systems using decentralized architectures[END_REF]- [START_REF] Wada | Decentralized diagnosis of discrete event systems subject to permanent sensor failures[END_REF] while decentralized fault diagnosis for continuous systems has only being dealt recently. The authors of [START_REF] Boem | Decentralized fault diagnosis for heterogeneous multi-agent systems[END_REF]- [START_REF] Ferdowsi | A decentralized fault detection and prediction scheme for nonlinear interconnected continuoustime systems[END_REF] or more recently [START_REF] Meynen | Robust fault detection and isolation for distributed and decentralized systems[END_REF], [START_REF] Zhou | Decentralized fault detection and faulttolerant control for nonlinear interconnected systems[END_REF] use decentralized state estimation and observers, and are therefore far from the work presented in this paper.

Like this paper, the authors of [START_REF] Gupta | Decentralized fault diagnosis using analytical redundancy relations: Application to a water distribution network[END_REF] address FDI with ARRs and make use of a graph to obtain local diagnosers for each subsystem and a global diagnoser for coordination. Although some structural properties are considered with the graph, they do not make use of structural diagnosis strictly speaking.

Among the papers that make use of structural analysis, [START_REF] Khorasgani | Minimal structurally overdetermined sets selection for distributed fault detection[END_REF] proposes two distributed diagnosis approaches. The first one computes tests in a centralized way, then uses a BILP problem to optimize the choice of tests and the decomposition into subsystems to minimize communication between the subsystems. Other works like [START_REF] Zhao | Total variable decomposition based on sparse cointegration analysis for distributed monitoring of nonstationary industrial processes[END_REF] present methods to decompose process variables into different blocks automatically to minimize communication between sub-systems but without selecting tests accordingly. In [START_REF] Pérez | Fault driven minimal structurally overdetermined set in a distributed context[END_REF], like in this paper, the decomposition structure is assumed to be dictated by functional or geographical constraints. Tests are generated locally and unknown variables shared by several subsystems are eliminated when necessary while minimizing communication between subsystems. In [START_REF] Taskent | Comparison of two distributed fault diagnosis approaches based on Binary Integer Linear Programming (BILP) optimization[END_REF], the approach of [START_REF] Khorasgani | Minimal structurally overdetermined sets selection for distributed fault detection[END_REF] is compared to the one of [START_REF] Pérez | Fault driven minimal structurally overdetermined set in a distributed context[END_REF]. The comparison metric is the size of the input matrices for the BILP problems in each approach. As [START_REF] Pérez | Fault driven minimal structurally overdetermined set in a distributed context[END_REF] proceeds directly with a distributed approach, the size of the input matrices for the BILP problems is bigger.

The second approach is a distributed algorithm that is designed to minimize communication between the subsystems. It assumes that the global model of the system is not available and proceeds incrementally by augmenting the model subsystem by subsystem until full isolability is achieved. The approach uses Minimal Structurally Overdetermined (MSO) sets that are all computed for each subsystem. In the worse case, all the MSO sets are computed and the augmented system is the whole system.

Our paper proceeds similarly but, as motivated in [START_REF] Pérez | Fault driven minimal structurally overdetermined set in a distributed context[END_REF], it does not use MSO sets but FMSO sets instead. These later necessarily involve some faults in their support, and they are much less numerous than MSO sets, which reduces significantly computation time. Although Minimal Test Equation Support (MTES) can be a solution to develop more powerful tests, FMSO sets are better suited to minimize communications between subsystems [START_REF] Pérez | Fault driven minimal structurally overdetermined set in a distributed context[END_REF].

The work presented in this paper can be seen as a continuation of [START_REF] Chanthery | Fault isolation on request based on decentralized residual generation[END_REF], driven by the idea of isolation on request. The algorithm proposed in this paper builds a diagnostic architecture that allows the diagnoser to adapt to the fault impacting the system at run time. It relies on the property that was demonstrated in [START_REF] Chanthery | Fault isolation on request based on decentralized residual generation[END_REF], i.e. the ARRs generated along a hierarchical decentralized diagnosis architecture are just the same as the ones generated in a centralized architecture when the hierarchy is pushed up. Nevertheless, the proposed algorithm brings a new perspective with respect to [START_REF] Chanthery | Fault isolation on request based on decentralized residual generation[END_REF] by taking into account inter-level restrictions and by taking benefit of BILP optimization to select only the necessary and sufficient tests (ARRs) at each level. On the other hand, although this paper resumes some ideas of [START_REF] Pérez | Faultdriven structural diagnosis approach in a distributed context[END_REF], it goes much further in the analysis of the properties of the proposed algorithm and in its application to a case study representative of a real process.

III. CONCEPTS FOR DIAGNOSIS

A. Analytical Redundancy Relations

The model Σ(z, x, f), or Σ for short, of a continuous system takes the form of a set of differential or algebraic equations e k (z, x), k = 1, . . . , n e , that involve known variables corresponding to sensored inputs and outputs represented by vector z, unknown variables represented by vector x, and variables that represent the possible faults that may act on the system gathered in vector f. Such models can be obtained from physical principles or derived from data using model identification techniques. Identification can be carried out by classical methods [START_REF] Akaike | A new look at the statistical model identification[END_REF], but it can also be achieved by methods from artificial intelligence. For a system seen as a black box, methods based on neural networks or regression can be used. Recent graph structure learning methods [START_REF] Jin | Graph structure learning for robust graph neural networks[END_REF] seem particularly promising for that purpose, as they allow to learn both the structure and the parameters of the model.

Residual generation relies on the analytical redundancy embedded in the model. It results from the number and location of sensors, hence from known variables. In the case of analytical redundancy, variable elimination can be applied to generate relations that involve measured variables only. These relations are used as residual generators.

Definition 1 (Residual Generator for Σ(z, x, f)): A relation of the form arr(z ′ , ż′ , z′ , ...) = r, with input z ′ a subvector of z and output r, a scalar signal named residual, is a residual generator for the model Σ(z, x, f) if, for all z consistent with Σ(z, x, f), it holds that lim t→∞ r(t) = 0.

The relation arr(z ′ , ż′ , z′ , ...) = r is qualified as an Analytical Redundancy Relation (ARR). It allows one to check whether the measured variables z are consistent with the system model. ARRs can be designed so that each is sensitive to a different subset of faults qualified as the fault support of the ARR. When the residual is non-zero, it means that at least one of the faults of the fault support has occurred.

There are many solutions to design residual generators based on a model or a sub-model with redundancy. Linear algebra based approaches can be used when the model is linear [START_REF] Chow | Analytical redundancy and the design of robust failure detection systems[END_REF]- [START_REF] Nyberg | Residual generation for fault diagnosis of systems described by linear differential-algebraic equations[END_REF]. [START_REF] Svärd | Residual generators for fault diagnosis using computation sequences with mixed causality applied to automotive systems[END_REF] proposes residual generation with mixed causality (integral and derivative). However, the complexity of these solutions increases rapidly with problem size and quickly becomes intractable.

The authors of [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF] propose the sequential residual generation that is very efficient and simple when the model is of low differential index. The idea is to use graph theory and search for a matching, i.e. a substitution path for eliminating unknown variables in equations of the model. A relation including only known variables with a non-empty fault support can be used as residual generator.

In [START_REF] Travé-Massuyès | Diagnosability analysis based on component-supported analytical redundancy relations[END_REF], SARR-Algorithm proposes the successive elimination of unknown variables in the equations that form the structural model with the goal of obtaining the set of minimal Structural ARRs (SARRs). Contrary to other solutions using structural analysis, SARRs account for causality and point at how the equations must be chained to generate an actual ARR.

Some considerations have to be taken into account when trying to generate residuals from redundancy in the model. If the model is perfect, all residue generators are of equivalent quality to detect defects. However, uncertainty and model noise deteriorate performance. It is indeed difficult to estimate the derivatives of known signals in a noisy environment. In [START_REF] Frisk | Lowering orders of derivatives in non-linear residual generation using realization theory[END_REF], the author shows how to reduce the need to estimate derivatives of known signals by putting the residual generator in a state-space form. More recently, [START_REF] Frisk | Residual selection for consistency based diagnosis using machine learning models[END_REF] proposes to use the Random Forest technique to select residuals properly.

B. Structural Analysis for Diagnosis

Structural analysis can be applied to large-scale systems described by numerous variables, both linear and non-linear, and even for systems under uncertainty for which the analytical model is not precisely known [START_REF] Cassar | A structural approach for the design of failure detection and identification systems[END_REF], [START_REF] Düstegör | Structural analysis of fault isolability in the damadics benchmark[END_REF].

1) Structural Model: A structural model is composed of a set of constraints, each one associated to a component of the system. The model generally represents nominal behavior, hence the violation of one constraint indicates that the system is faulty and points at the responsible component. The structural model of the system Σ(z, x, f) can be obtained by abstracting the functional relations of Σ(z, x, f). Let us define the sets Z, X, and F as the set of known (or measured) variables, unknown (or unmeasured) variables, and faults that may impact the system, respectively of cardinal n z , n x , and n f .

The structural model can be represented by a matrix qualified as the incidence matrix, whose rows are associated to equations and columns to variables. Its elements take the value "1" when the variable is involved in the equation and "0" otherwise. Equivalently, the structural model can be represented by a bipartite graph G(Σ ∪ X ∪ Z, A), where A is a set of edges linking equations of Σ and variables of X and Z. For the purpose of diagnosis and as explained later, this graph can be reduced to G(Σ ∪ X, A), where A ⊆ A and A is a set of edges such that a(i, j) ∈ A iff variable x i is involved in equation e j .

As an illustrative example, consider a system composed of 6 equations e 1 to e 6 relative to the set of known variables Z = {u, y 1 , y 2 }, the set of unknown variables X = {x 1 , x 2 , ẋ1 , ẋ2 } and the set of faults

F = {f 1 , f 2 , f 3 }. (e 1 ) ẋ1 = 0.7x 1 + 0.2x 2 (e 2 ) ẋ2 = 0.5x 2 + u + f 3 (e 3 ) y 1 = x 1 + f 1 (e 4 ) y 2 = x 2 + f 2 (e 5 ) ẋ1 = dx1 dt (e 6 ) ẋ2 = dx2 dt
Fig. 1 shows the bipartite graph relating to the illustrative example.

Fig. 1: The bipartite graphs G(Σ ∪ X ∪ Z, A) and G(Σ ∪ X, A) obtained by removing gray edges and variables for the illustrative example.

2) Diagnosis via Structural Redundancy: When used for FDI purposes, the outputs of structural analysis are in the form of subsets of equations endowed with redundancy. These can be turned into diagnosis tests, i.e. ARRs or parity relations, which are designed off-line [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF]. Diagnosis tests are then checked against observations on-line.

Redundancy in a system of the form Σ(z, x, f) can be brought to light by the well-known Dulmage-Mendelsohn (DM) canonical decomposition [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF], [START_REF] Murota | Matrices and Matroids for System Analysis[END_REF], [START_REF] Dulmage | Coverings of bipartite graphs[END_REF]. It partitions the system into three subsystems:

• Σ + has more equations than unknown variables and is named the structurally overdetermined (SO) subsystem, • Σ 0 is the structurally just determined subsystem,

• Σ -has more unknown variables than equations and is named the structurally underdetermined subsystem. In the following, the systems that are considered are welldefined systems, which means that Σ = Σ + . This is the case of the illustrative example.

Definition 2 (Structural redundancy): The structural redundancy ρ Σ ′ of a set of equations Σ ′ ⊆ Σ is defined as the difference between the number of equations and the number of unknown variables.

If a set of equations is structurally redundant (ρ Σ ′ > 0), it means that ARRs, i.e. residual generators, can be built using the equations in this set. Thus, structural analysis provides the means to find subsets of equations with structural redundancy. Once the subsets found, one of the solutions proposed in the previous section can be used to obtain residual generators.

If a set of equation Σ is such that Σ = Σ + and no proper subset of Σ is overdetermined, the concept of minimally structurally overdetermined (MSO) [START_REF] Krysander | A structural algorithm for finding testable sub-models and multiple fault isolability analysis[END_REF] is used. In the illustrative system, the set Σ 1 = {e 1 , e 2 , e 3 , e 5 , e 6 } is an MSO set and its structural redundancy is ρ Σ1 = 1. An MSO set gives naturally rise to one residual generator because it has structural redundancy 1.

However, only MSO sets impacted by faults are interesting for diagnosis. This is why we refer to the concept of fault support.

Definition 3 (Fault support): The set F Σ ′ of faults that are involved in the equations of a subset Σ ′ ⊆ Σ is defined as the fault support of Σ ′ .

In the following, we distinguish MSO sets of Σ(z, x, f) whose fault support is empty, qualified as Clear Minimal Structurally Overdetermined (CMSO) sets, and MSO sets whose fault support is not empty, qualified as Fault-Driven Minimal Structurally Overdetermined (FMSO) sets [START_REF] Pérez | Faultdriven structural diagnosis approach in a distributed context[END_REF].

In the illustrative example, the MSO set Σ 1 = {e 1 , e 2 , e 3 , e 5 , e 6 } is an FMSO set and its fault support is F Σ1 = {f 1 , f 3 }. There are no CMSO sets.

An FMSO set φ identifies a just overdeterminated subset of |φ| equations of the model, among which one is redundant. This means that all the unknown variables can be determined using |φ| -1 equations, and that an ARR can be generated by substituting in the |φ| th equation. We illustrate the process of generating an ARR from an FMSO set below [START_REF] Armengol | Minimal structurally overdetermined sets for residual generation: A comparison of alternative approaches[END_REF].

For the illustrative example, let us consider the FMSO set Σ 1 = {e 1 , e 2 , e 3 , e 5 , e 6 }. Every unknown variable can be associated to one equation that is interpreted as a mechanism to determine it. For instance:

• e 3 determines x 1 : x 1 = y 1 • e 5 determines ẋ1 : ẋ1 = dx1 dt ⇒ ẋ1 = ẏ1 • e 1 determines x 2 : x 2 = 1 0.2 ẋ1 -0.7 0.2 x 1 ⇒ x 2 = 1 0.2 ẏ1 - 0.7 0.2 y 1 • e 6 determines ẋ2 : ẋ2 = dx2 dt ⇒ ẋ2 = 1 0.2 ÿ1 -0.7 0.2 ẏ1
Finally, the ARR is obtained with the redundant equation e 2 as follows:

ẋ2 = 0.5x 2 + u ⇒ 1 0.2 ÿ1 -0.7 0.2 ẏ1 = 0.5 0.2 ẏ1 -0.5×0.7 0.2 y 1 + u ⇒ 5ÿ 1 -6 ẏ1 + 1.75y 1 -u = 0
FMSO sets are also useful in defining the concepts of detectable fault and isolable fault.

Definition 4 (Detectable fault):

A fault f ∈ F is detectable in the system Σ(z, x, f) if there exists an FMSO set φ ∈ Φ such that f ∈ F φ . Definition 5 (Isolable fault): Given two detectable faults f and f ′ of F , f ̸ = f ′ , f is isolable from f ′ if there exists an FMSO set φ ∈ Φ such that f ∈ F φ and f ′ ̸ ∈ F φ .
In the illustrative example, the faults f 1 , f 2 , and f 3 are detectable: f 1 ∈ F Σ1 and f 3 ∈ F Σ1 , where Σ 1 = {e 1 , e 2 , e 3 , e 5 , e 6 } has been shown to be an FMSO set, f 2 ∈ F Σ2 , where Σ 2 = {e 2 , e 4 , e 6 } is also an FMSO set. f 1 and f 3 are isolable from

f 2 because f 2 ̸ ∈ F Σ1 while f 1 ∈ F Σ1 and f 3 ∈ F Σ1 .
Definition 6 (Ambiguity set and Isolability degree): The ambiguity set A(Σ) of a system Σ is the set of sets of not isolable faults. The system isolability degree I(Σ) is defined as the cardinal of its ambiguity set, i.e.

I(Σ) = |A(Σ)|.
According to Definition 6, the system isolability degree I(Σ) is at most equal to n f , which is met when all the faults are isolable. The system isolability degree I(Σ) defines an upper bound for the isolability degree that can be obtained with a diagnosis algorithm.

IV. DECENTRALIZED DIAGNOSIS

A. Fault Diagnosis in a Decentralized Framework

The aim of this work is to propose a decentralized diagnoser design method. One of the assumptions is that the system is decomposable into sub-systems determined by functional, geographical or confidential constraints guided by engineering requirements. Each subsystem may represent the implementation of a primary function. The decomposition itself is not in the scope of this work but this can be a problem in its own right [START_REF] Gei | Optimal system decomposition for distributed fault detection: Insights and numerical results[END_REF], [START_REF] Kościelny | Decomposition of complex diagnostic systems[END_REF].

Fig. 2 illustrates an example of a decentralized architecture with 3 sub-systems at the lowest level and 2 levels in the supervisory hierarchy. In the decomposition, each subsystem has its own local diagnoser. If isolability of all faults cannot be achieved at some level, higher level sub-systems can be considered with their own diagnosers. This can be applied recursively, so that this type of architecture is fully scalable.

More generally, the decentralized structure of Σ(z, x, f) is defined as a hierarchical organization over several levels as illustrated in Fig. 3. Σ 1,i , i = 1, . . . , n 1 represent sub-systems of the lowest level. Π j,i sets are equation sets that are only available at level j, for confidentiality reasons or because they are too far away or costly to access. Σ j,i represents the i th subsystem at level j, with i = 1, ..., n j and j = 1, ..., m, obtained by lumping some of the equations of some subsystems of the level just beneath and the equations of Π j,i . Note that, by construction, n j < n j-1 , ∀j = 2, ..., m.

In extension, Σ j,i , j = 1, ..., m and i = 1, ..., n j , is denoted Σ j,i (z j,i , x j,i , f j,i ), where z j,i , x j,i and f j,i are the vectors of known variables, unknown variables and faults, respectively. The sets of unknown variables, known variables, and faults are denoted X j,i , Z j,i , and F j,i , respectively. These are the subsets of variables of X, Z, and F involved in the subsystem Σ j,i .

Parameter n m of the highest level m is required to be equal to 1, i.e. level m is restricted to one single subsystem. Parameter n 1 is given by the imposed decomposition. The hierarchy, i.e. the number of levels m and the number of subsystems n j , j = 1, . . . , m in each level j > 1, are constrained by Fig. 2: A decentralized diagnosis architecture with 3 subsystems.

allowed and forbidden inter-level communication as formally defined below.

Definition 7 (Inter-level communication): The inter-level communication is given by a set of m -1 bipartite graphs S j j-1 (N j j-1 , L j j-1 ), j = 2, . . . , m, such that N j j-1 = N j-1 ∪ N j , where:

• N j = {n j,i , i = 1, . . . , n j } is a set of nodes corresponding to the subsystems Σ j,i , i = 1, . . . , n j , of level j, and N j-1 is defined similarly. • L j j-1 = {l ν,ξ , ν = 1, . . . , n j-1 , ξ = 1, . . . , n j } is a set of edges such that the edge l ν,ξ between node n ν ∈ N j-1 and n ξ ∈ N j exists if communication is possible between subsystem Σ j-1,ν at level j -1 and subsystem Σ j,ξ at level j. Definition 8 (Local variable): A variable of a subsystem Σ j,i is said to be local if it is only involved in the equations of Σ j,i . The set of local variables of Σ j,i is denoted by X l j,i .

X l j,i = X j,i \( k=1,...,m l=1,...,n k (X j,i ∩ X k,l )) with (k, l) ̸ = (j, i) (1)
Let us decompose the illustrative system in two subsystems Σ 1,1 = {e 1 , e 3 , e 5 } and Σ 1,2 = {e 2 , e 4 , e 6 }, then local variables are given by X l 1,1 = {x 1 , ẋ1 } and X l 1,2 = { ẋ2 }. Definition 9 (Shared Variable): A variable of Σ j,i is said to be shared if it is involved in the equations of Σ j,i and at least in the equations of another subsystem Σ j ′ ,i ′ with (j, i) ̸ = (j ′ , i ′ ). The set of shared variables of subsystem Σ j,i is denoted X s j,i and

X s j,i = X j,i \X l j,i (2)
X s represents the set of shared variables of Σ. In the illustrative system, there is one shared variable only so that

X s 1,1 = X s 1,2 = X s = {x 2 }.

B. FMSO Sets for Decentralized Diagnosis

The concepts of FMSO sets and CMSO are extended to the decentralized context. Definition 11 (Local FMSO set): φ is a local FMSO set of Σ j,i (z j,i , x j,i , f j,i ) if φ is an FMSO set of Σ(z, x, f) and if its unknown variables can all be eliminated locally, i.e. with equations of Σ j,i . The set of local FMSO sets of Σ j,i is denoted by Φ l j,i . The set of all local FMSO sets is denoted by Φ l = j=1,...,m i=1,...,nj Φ l j,i .

Two important things have to be noticed: 1) a local FMSO set may involve shared variables if they can be eliminated locally; 2) a local FMSO set for any subsystem Σ j,i is also an FMSO set of Σ, hence, a global FMSO set.

In the illustrative system, Φ l 1,1 = {∅}, Φ l 1,2 = {φ 1 = {e 2 , e 4 , e 6 }}. As mentioned above, note that the local FMSO set φ 1 is also a global FMSO set.

If shared variables cannot be eliminated locally, they need equations of other subsystems to be substituted. This leads to the definition of shared FMSO sets for a subsystem Σ j,i .

Definition 12 (Shared FMSO set): If φ is an FMSO set of Σj,i (z j,i , xj,i , fj,i ), where zj,i is the vector of variables in Zj,i = Z j,i ∪X s j,i , xj,i is the vector of variables in Xj,i = X l j,i , and fj,i = f j,i , then φ is a shared FMSO set of subsystem Σ j,i (z j,i , x j,i , f j,i ). The set of shared FMSO sets for Σ j,i is denoted by Φ s j,i . The set of all shared FMSO sets is denoted by Φ s = j=1,...,m i=1,...,nj Φ s j,i .

From the above definition, a shared FMSO set φ for subsystem Σ j,i (z j,i , x j,i ,

f j,i ) is such that φ ⊆ Σ j,i , X φ ⊆ X l j,i , Z φ ∩ X s j,i ̸ = ∅, and Z φ ⊆ (Z j,i ∪ X s j,i ). For subsystem Σ 1,1 of system (III-B1), Z1,1 = Z 1,1 ∪ X s 1,1 = {y 1 , x 2 } and Φ s 1,1 = {φ 4 = {e 1 , e 3 , e 5 }}. For subsystem Σ 1,2 , Z1,2 = Z 1,2 ∪ X s 1,2 = {y 2 , u, x 2 } and Φ s 1,2 = {φ 5 = {e 4 }, φ 6 = {e 2 , e 6 }}.
CMSO sets can also be extended in a decentralized framework as local CMSO sets and shared CMSO sets, as done for FMSO sets. The set of local (shared) CMSO sets of Σ j,i is denoted by Ψ l j,i (Ψ s j,i ) and the set of all local (shared) CMSO sets by Ψ l (Ψ s ) [START_REF] Pérez | Faultdriven structural diagnosis approach in a distributed context[END_REF].

V. DECENTRALIZED DIAGNOSIS DESIGN ALGORITHM This section presents the Decentralized Diagnoser Design algorithm D 3 for designing a decentralized diagnoser optimizing the number of FMSO sets while providing fault isolability guarantees.

A. Algorithm for Decentralized Diagnoser Design

The D 3 algorithm runs offline. It delivers a near-optimal distribution of ARRs on a hierarchy of subsystems. The user chooses a target isolability degree as a percentage of the number of faults, i.e. %n f . Given the system isolability degree I(Σ), which may be known by the user or not, the maximal isolability degree I * (Σ) is defined as:

I * (Σ) = min(I(Σ), I t (Σ)) (3)
The D 3 Algorithm.

1: E0,1 = ∅, I0(Σ) = 0, j = 1, i = 1 ▷ Init part 2: while (j ≤ m) && (Ij-1(Σ) < I * (Σ)) do ▷ Main loop 3: while i < nj do 4:
Υj,i ← ComputeUpsilon(j, i)

5: Σj,i ← Πj,i ∪ Υj,i 6: Φ l j,i ← ComputeLocFMSOOf(Σj,i) 7: Φ l j,i * ← SelectOptLocFMSO(Φ l j,i ) 8: ARRj,i ← ComputeARR(Φ l j,i * ) 9: i = i + 1 10:
end while 11:

Ij(Σ) ← ComputeID(j) 12: j = j + 1, i = 1 13: end while
In the initialization part of the D 3 algorithm (line 1), E 0,1 and I 0 (Σ) are initial conditions for an artificial level 0 that allows us to apply the recursive operations for the following levels. Indexes j and i are initialized to 1. j represents the level in the decentralized architecture, while i numbers the subsystems in a given level. Two loops consider subsystems from i = 1 to i = n j at each level j from 1 up to possibly m.

The inter-level communication is taken into account at each level j = 2, . . . , m by Υ j,i , where:

Υ j,i = ν/lν,i∈L j j-1 E j-1,ν (4)
E j-1,ν = {e ∈ Φ s j-1,ν ∪ Ψ s j-1,ν } (5)
In other words, for levels j = 2, . . . , m, Υ j,i contains the equations involved in shared FMSO and CMSO of all subsystems of level j -1 whose connection is allowed with subsystem Σ j,i and that share variables. This is performed by function ComputeUpsilon(j,i) on line 4. Note that Υ 1,i = ∅. The subsystem Σ j,i is then formed (line 5) by unioning Υ j,i and Π j,i that brings additional information only available at level j.

Local1 FMSO sets for Σ j,i are then computed with function LocalFMSOOf() (line 6). Function SelectOptLocFMSO() selects the best local FMSO sets among the local FMSO sets by solving a BILP problem as described in Subsection V-B (line 7). The set ARR j,i (line 8) represents the set of ARRs derived from the selected FMSO sets Φ l * i,j and is computed with function ComputeARR(). Finally, function ComputeID() computes the isolability degree I j (Σ) at the current level (line 11). The condition of the loop checks whether if the isolability degree is still lower than the targeted isolability degree I * (Σ) (line 2). If this is the case, a higher level of the hierarchy is explored. D 3 stops either when the targeted isolability degree I * (Σ) is achieved or when the maximal number of levels m has been reached.

B. Optimal Selection of Local FSMO sets

For each subsystem Σ j,i , D 3 solves a BILP problem [START_REF] Artigues | Optimal test/sensor selection problems formalized as integer programs[END_REF] during the ComputeLocFMSOOf() function. This function selects the best local FMSO sets among all the computed local FMSO sets Φ l j,i of Σ j,i . The selection criterion aims at maximizing the isolability degree while minimizing the number of chosen local FMSO sets.

Let z φi be a boolean variable that is equal to 1 if the local FSMO set φ i is selected, 0 otherwise. Let f klφi be a boolean variable equal to 1 if the fault f k is isolable from the fault f l by using the local FMSO set φ i . Let e kl be a variable that indicates that the fault f k is isolable from the fault f l when it is equal to 1 and the inverse when it is equal to 0. The e kl variables are introduced in the objective function and the problem becomes a bi-objective optimization problem weighted by a parameter α. α represents a weighting between the maximization of the isolability degree and the minimization of the number of chosen local FMSO sets. The BILP model is as follows:

max( α e kl (f k ,f l )∈Fj,i -(α -1) z φi φi∈Φ l j,i ) (6)
Subject to:

φi∈Φ l j,i f klφi z φi ≥ e kl (7)
z φi ∈ {0, 1} for φ i ∈ Φ l j,i , (k, l) ∈ F j,i , k ≤ l (8) α ∈ [0, 1], 0 ≤ e kl ≤ 1 (9)
Equation ( 6) is the objective function of the BILP problem. The constraint ( 7) is used to guarantee the maximum possible isolation for subsystem Σ j,i .

Let us notice that efficiency dictates to limit the number of binary variables2 , hence the e kl 's are coded as continuous variables in the program. In practice, the fact that the left-hand side of inequality ( 7) is binary forces the value of e kl to be binary. Indeed, when equal to 0, e kl is forced to 0 and when equal to 1, e kl is forced to 1 by the maximization objective [START_REF] Armengol | Minimal structurally overdetermined sets for residual generation: A comparison of alternative approaches[END_REF].

C. Algorithm Properties

This section analyzes the properties and complexity of the D 3 algorithm.

1) Prerequisites:

• Considered systems are well-defined systems, i.e., Σ = Σ + . • Proposition 3 of [START_REF] Pérez | Faultdriven structural diagnosis approach in a distributed context[END_REF]: in a distributed architecture for which the system Σ is decomposed into a set of subsystems that can exchange information, the set of global FMSO sets is given by the union of local FMSO sets and compound FMSO sets, where compound FMSO sets are FMSO sets that include shared FMSO sets. Note that a distributed architecture can be viewed as a two levels hierarchy, in which the subsystems define level 1 and the information exchange defines level 2. 2) Properties: Theorem 1: The D 3 algorithm guarantees maximal isolability degree if: (P ) the inter-level communication is such that there exists a link l ν,ξ from any subsystem Σ j-1,ν , ν ∈ {1, . . . , n j-1 }, with shared variables of level j -1 to some subsystem Σ j,ξ , ξ ∈ {1, . . . , n j }, of level j in L j j-1 , j = 1, . . . , m.

Proof 1: If property (P ) is satisfied, any shared FMSO set φ of a given subsystem Σ j-1,ν , ν ∈ {1, . . . , n j-1 }, of level j -1 is part of some subsystem Σ j,ξ , ξ ∈ {1, . . . , n j }, of level j. There are two cases:

• Case 1: Σ j,ξ has inherited from shared FMSO sets of other sub-systems Σ j,β̸ =ξ so that φ is involved in a local FMSO set for Σ j,ξ . • Case 2: the condition of the previous item is not true and φ remains as a shared FMSO set for Σ j,ξ and by property (P ), it is part of some sub-system of level j + 1. Now, property (P ) implies that there exists a level ℓ ≤ m in the hierarchy for which the subsystems Σ m,ν , ν ∈ {1, . . . , n ℓ } do not share variables and do not hence involve shared FMSO sets. This implies that a local FMSO set that includes φ is generated at some level k ∈ {j + 1, . . . , ℓ} of the hierarchy (this is guaranteed by Σ = Σ + ). Case 1 and Case 2 imply that all shared FMSO sets are retrieved to generate local FMSO sets at some level j ∈ {2, . . . , ℓ}. In the decentralized hierarchy used by D 3 , local FMSO sets generated at a given level j can be viewed as compound FMSO sets (as defined in Proposition 3 of [START_REF] Pérez | Faultdriven structural diagnosis approach in a distributed context[END_REF]) with respect to shared FMSO sets of some lower level. Recursive application of Proposition 3 of [START_REF] Pérez | Faultdriven structural diagnosis approach in a distributed context[END_REF] to every two subsequent levels of the D 3 hierarchy hence implies that the set of global FMSO sets is given by the set of local FMSO sets generated over the ℓ first levels of the D 3 hierarchy. Consequently, all global FMSO sets can be generated by D 3 , which guarantees maximal isolability I * (Σ) when I * (Σ) = I(Σ) ≤ I t (Σ). If I * (Σ) = I t (Σ) < I(Σ), D 3 may stop before developing levels up to ℓ, guarantying however maximal isolability.■ Lemma 1: If the inter-level communication satisfies property (P ) and I t (Σ) ≥ I(Σ), the set of local FMSO sets generated by D 3 over all levels of the hierarchy is the set of global FMSO sets of system Σ, i.e., Φ l = Φ.

Proof 2: This lemma is a direct consequence of Theorem 1.■ Theorem 2: The D 3 algorithm computes a near-optimal set of analytical residual generators that guarantee maximal fault isolability if the inter-level communication satisfies property (P ).

Proof 3: By Theorem 1 and Lemma 1, the set of all local FMSO sets generated by D 3 over the levels 1 to ℓ of the hierarchy guarantee maximal fault isolability. Function SelectOptimalLocalFMSO() of D 3 makes a selection among local FMSO sets while preserving achievable isolability at the level of each subsystem. Although the solution of the BILP problem is proved optimal and so is the selected set of local FMSO sets for each subsystem, the union of these sets may overestimate the optimal solution for the whole system Σ, hence the D 3 algorithm is near-optimal.■ Theorem 3: The D 3 algorithm has lower time complexity than its centralized equivalent if and only if the highest level ℓ of the developed hierarchy has more than one subsystem, i.e., n ℓ > 1.

Proof 4: Given a system Σ with n e equations and fixed order of structural redundancy ρ, the computation of FMSO sets has been proven to have order of n ρ+1.5 e time complexity and the number of FMSO sets to be bounded by ρ.n ρ-1 e [START_REF] Krysander | An efficient algorithm for finding minimal over-constrained sub-systems for model-based diagnosis[END_REF]. BILP is known to be polynomial when the constraint matrix is totally unimodular. This is the case in the BILP formulation of FMSO set selection. The time complexity of an FMSO set selection step based on BILP is thus negligible compared to that of FMSO sets computation.

Let ρ * denote the highest structural redundancy over all subsystems of the developed hierarchy:

ρ * = max j=1,...,ℓ i=1,...,nj ρ Σj,i (10)
The computation of local FMSO sets as thus order of n ρ * +1. Let us now prove by contradiction that if n ℓ > 1, then any subsystem of the hierarchy has lower structural redundancy than the system Σ as a whole. Let Σ 1 be a subsystem of structural redundancy ρ Σ1 . Let us assume that recombining Σ 1 with a subsystem Σ 2 results in Σ = Σ 1 ∪ Σ 2 with structural redundancy ρ Σ ≤ ρ Σ1 . Σ 2 must involve at least one equation e and the constraint ρ Σ ≤ ρ Σ1 implies that e must involve at least one variable that is not shared with Σ 1 . This is impossible because Σ would then include a just determined part, which is in contradiction with the fact that the global system is overdetermined (such that Σ = Σ + ), so ρ Σ1 < ρ Σ . This is easily generalized for any proper subsystem of Σ.

The condition n ℓ > 1, i.e. there are more than one subsystem in the highest level of the hierarchy, guarantees that the whole system is never recomposed and that D 3 only handles proper subsystems of Σ. Consequently, the time complexity of the D 3 algorithm is lower than the time complexity of its centralized equivalent.■

D. Implementation of D 3 and online Decentralized Diagnoser

The implementation of the diagnoser designed by the D 3 algorithm is based on the notion of local fault signature matrix of the subsystems, defined as follows.

Definition 13 (Local Fault Signature Matrix of a subsystem): Let a set R j,i composed of n r j,i ARRs and F j,i the set of considered n f j,i faults for the subsystem Σ j,i . The signature of a fault f ∈ F j,i is a binary vector

F S j,i (f ) = [τ 1 , τ 2 , ...τ n r j,i ] T where τ k , k = 1...n r j,i , is computed from R j,i × F j,i -→ {0, 1} so that τ k = 1, if arr k ∈ R j,i
is affected by f , and τ k = 0 otherwise. The signatures of all the faults in F j,i put together constitute the Local Fault Signature Matrix, denoted S l j,i of subsystem Σ j,i , i.e. S l j,i = [F S j,i (f 1 ), . . . , F S j,i (f n f j,i )] T . The diagnoser is computed offline as a hierarchical residual generator bank based on the local FMSO sets computed for each subsystem at each level. In accordance with the isolation on request concept [START_REF] Chanthery | Fault isolation on request based on decentralized residual generation[END_REF], the computations are only carried out at a higher level if there exist faults that remain not isolable at the current level and the maximal isolability degree I * (Σ) is not achieved. This check is done with the local fault signature matrix of the subsystems. Measurements of known variables are used online to evaluate residuals up along the levels until a fault is actually detected.

VI. EXPERIMENTS A. Computational Efficiency and Scalability in Practice

In order to experimentally show the computational efficiency of D 3 and its scalability, the algorithm is applied to serially connected water tanks. Setups of 4, 6, 8 and 10 tanks are tested.

The 4 tanks system model, initially presented in [START_REF] Khorasgani | Minimal structurally overdetermined sets selection for distributed fault detection[END_REF], is composed of 20 equations with 8 known variables, 14 unknown variables and 6 faults (see [START_REF] Pérez | Faultdriven structural diagnosis approach in a distributed context[END_REF] for details). The construction of setups with more tanks is done by adding two tank patterns. Fig. 4 illustrates the 10 tanks setup, where T 1 to T 10 represent the 10 tanks, u 1 to u 5 and y 1 to y 18 are the known variables. Setups with 4, 6 or 8 tanks are a subpart of this setup. The hierarchical organization of the diagnoser proposed by D 3 for all setups is shown in Fig. 5. The D 3 algorithm and its centralized equivalent are applied to each setup. The execution time results reported in Table I leave no doubt about the gain recorded by D 3 as well as its scaling up.

B. Case Study: Gas Turbine Combined Cycle power plant

The method is applied to a gas turbine combined cycle power plant. A combined cycle power plant includes a gas turbine and a steam turbine. The remaining heat from the gas turbine is sent by a steam turbine that generates additional energy, producing up to 50% more electricity with the same amount of fuel than a conventional single-cycle plant. The nominal operation of a combined cycle power plant is more complex than than that of a conventional single cycle and therefore is more exposed to faults. First, natural gas is burned in a combustion chamber and passed through a gas turbine connected to an alternator. Hot gases are used to heat water and convert it into steam in a heat recovery unit. This steam is passed through a second turbine connected to another alternator, so that both generate electricity. One of the main components of a gas turbine combined cycle power plant is the gas turbine generator system, which in turn is composed of the following components: compressor, electric generator, combustion chamber, gas turbine and heat recovery as shown in Fig. 7.

Gas turbine installations are ubiquitous in industry so their monitoring and diagnosis meets industrial standards (cf. ISO 19860:2005 [45]). Most installations are equipped with a trend monitoring system [START_REF] De Michelis | Condition monitoring and assessment of power plant components[END_REF]. This equipment acquires operating data describing the condition of the gas-turbine installation that are used for the computation of short-term and longterm trends for selected parameters in mainly four areas: thermodynamics, vibrations, bearing lubrication, and control [START_REF] Giampaolo | Gas turbine handbook: principles and practice[END_REF]. Without calling into question the usefulness of trend monitoring systems, model-based approaches applied to gas turbines have shown their interest in many works by taking advantage of the analytical redundancy provided by the model [START_REF] Travé-Massuyès | Gas-turbine condition monitoring using qualitative model-based diagnosis[END_REF], [START_REF] Verde | Application of Structural Analysis to Improve Fault Diagnosis in a Gas Turbine[END_REF]. Beyond monitoring the trends of isolated parameters, they make it possible to take into account the underlying relationships and existing interactions, which is highly beneficial for automatic diagnostics. This is why it is interesting to illustrate the performance of D 3 on a gas turbine generator.

The non-linear mathematical model of the gas turbine generator system used in this paper is based on thermodynamics, mass and energy conservation, fluid flow and heat transfer. The assumptions of this model are: ideal gas behavior, isentropic compression and linear flow characteristics for all control valves. The whole model Σ(z, x, f) is composed of 37 equations e 1 to e 37 that relate 20 known variables Z = {z 1 , . . . , z 20 } and 29 unknown variables X = {x 1 , . . . , x 29 }. The set of target faults is F = {f 1 , . . . , f 5 } of cardinal n f =5, where f 1 is a fault in the air pressure sensor at compressor discharge, f 2 is a fault in the position of the compressor inlet guide valves, f 3 is a fault in the position of the combustion chamber gas fuel valve, f 4 is a fault in the heat recovery pressure sensor and f 5 is a fault in the position of the combustion chamber gas valve. The full description of the model is given in [START_REF] Sanchez-Parra | PID fault tolerant control for a gas turbine[END_REF]. A natural decomposition of the system provides the following five physically separated subsystems: Based on the system model and restricted information, the hierarchical organization of the diagnoser proposed for the gas turbine generator system is shown in Fig. 6. It includes 8 subsystems distributed in 3 levels, the 3rd level being the last possible level, i.e. m=3.

Compressor subsystem: Σ 1,1 Σ 1,1 = {e 1 , e 2 ,
1,1 = {f 1 , f 2 } X 1,1 = {x 1 , x 3 , x 5 , x 6 , x 8 , x 9 , x 10 , x 28 } Z 1,1 = {z 1 , z 2 , z

C. Global FMSO Sets Calculation

As a way to validate the method, we determine the system fault isolability I(Σ) by calculating the set of global FMSO sets Φ assuming that the entire model is available. TABLE II: Gas turbine generator system: Global FMSO sets

A(Σ) [f 1 ],[f 2 ],[f 3 ],[f 4 ],[f 5 ] I(Σ) 5 |Φ| 59
As shown in Table II, all faults can be detected and isolated. Hence, the system isolability degree, which bounds isolability, is I(Σ) = n f = 5.

D. Decentralized Diagnoser Design

In this section, the operational algorithm for decentralized diagnosis design D 3 is applied with a targeted isolability degree equal to the system isolability degree, i.e. I * (Σ) = I(Σ) = n f = 5.

1) Shared variables: As a previous step to apply the proposed algorithm, the set of shared variables (cf. definition 9) is determined.

X s = {x 1 , x 6 , x 8 , x 10 , x 12 , x 15 , x 28 } (11)

X s is composed of 7 variables as shown in Fig. 7. Then, X s is used to calculate shared FMSO / CMSO sets for each subsystem. 2) Local FMSO sets at level 1: These are computed for Σ 1,1 , Σ 1,2 , Σ 1,3 , Σ 1,4 and Σ 1,5 of level 1. None FMSO set is found for Σ 1,1 , Σ 1,3 and Σ 1,4 . Therefore, no fault can be detected locally for these subsystems.

For both subsystems Σ 1,2 and Σ 1,5 of level 1, only one FMSO set is found. Hence, the found FMSO sets Φ l 1,2 and Φ l 1,5 are selected without need to solve the BILP problems, allowing detectability for faults f 3 and f 4 locally in Σ 1,2 and Σ 1,5 respectively.

Σ 1,2 F 1,2 {f 3 } A(Σ 1,2 ) {{f 3 }} I(Σ 1,2 ) 1 Local FMSO sets selected Φ l * 1,2 = {φ 1 }, φ 1 = {e 16 , e 17 } Σ 1,5 F 1,5 {f 4 , f 5 } A(Σ 1,5 ) {{f 4 },{f 5 }} I(Σ 1,5 ) 2
Local FMSO sets selected Φ l * 1,5 = {φ 2 }, φ 2 = {e 29 , e 30 , e 31 , e 32 , e 33 , e 34 }

Isolability degree at level 1

A 1 (Σ) {{f 1 f 2 , f 5 },{f 3 },{f 4 }} I 1 (Σ) 3
At this stage, if f 3 or f 4 occur, they are diagnosed at level 1, without need of further analysis. However, not all system faults can be diagnosed at level 1.

The D 3 algorithm checks if the targeted isolability degree is achieved at level 1, which is not the case, since we have I 1 (Σ) = 3 < I * (Σ) = 5 (cf. Table III). The higher level 2 is thus explored as shown in Fig. 6. IV). Υ 2,1 and Υ 2,2 are then lumped together with the additional information Π 2,1 and Π 2,2 (see architecture of Fig. 6) to obtain: At this stage, the maximal isolability degree is achieved since I 2 (Σ) = I * (Σ) = 5. D 3 hence stops, having developed ℓ = 2 levels. For this system, ℓ = 2 < m = 3 and it has not been necessary to build up level 3, which is shown in transparency in Fig. 6.

Σ 2,1 = {e 1 ,

E. Online Decentralized Diagnoser

The D 3 algorithm delivers a near-optimal distribution of ARRs on a hierarchy of subsystems that guarantees maximal A faulty case has been chosen to exemplify the method. Using a simulator of the GTCC, f 3 has been injected at 600s during 25s. Fig. 8 shows the set of residuals when fault f 3 occurs. As can be seen, this fault is detected at level 1 by residue arr 1 while all other residuals remain constant 3 .

The algorithms for optimal selection of FMSO sets were implemented in Python and executed on a PC with a 1.6GHz processor. The execution times were measured in milliseconds, comparing the global diagnoser architecture that requires 4 out of 59 FMSO sets to achieve maximal isolability (cf. Section VI-C Table II) and the decentralized diagnoser architecture of D 3 that requires 5 FMSO sets in two levels (cf. Section VI-D, Tables III andV), which confirms the near-optimality result of Theorem 2. Results are presented in Table VII and show that the decentralized diagnoser architecture is faster than the global diagnoser architecture, which confirms the theoretical time complexity result of Theorem 3.

VII. CONCLUSION

Systems submitted to confidentiality constraints, geographically distributed or with limited access to some information require specific diagnosis architectures. This paper proposes a design method for decentralized diagnosis architectures named D 3 that meet this requirement. D 3 computes recursively, by developing just the necessary levels, the set of residual generators guaranteeing maximal isolability. It has been theoretically proved of lower time complexity than its centralized equivalent.

The industrial case study of a Gas Turbine Combined Cycle Power Plant has been used to demonstrate the validity and performances of the decentralized diagnoser design approach.

The main advantages of the proposed decentralized architecture are that local diagnosers are designed according to the traditional function-by-function design organization, that subsystem models do not need to be exposed and that fault isolation is guided by the concept of isolation on request.

In future work, there is interest in investigating the double problem of choosing the system decomposition and diagnosis tests while minimizing inter-subsystem communication and maximizing diagnosability. A preliminary proposal along these lines is presented in [START_REF] Chanthery | Process decomposition and test selection for distributed fault diagnosis[END_REF]. Additionally, the impact of different inter-level communication constraints on the diagnosis results should be analyzed.

Fig. 3 :

 3 Fig. 3: General decentralized architecture.

Definition 10 (

 10 Global FMSO set): An FMSO set φ of Σ(z, x, f) is called a global FMSO set. The set of global FMSO sets is denoted by Φ. Considering the illustrative system, Φ = {φ 1 , φ 2 , φ 3 }, where φ 1 = {e 2 , e 4 , e 6 }, φ 2 = {e 1 , e 3 , e 4 , e 5 } and φ 3 = {e 1 , e 2 , e 3 , e 5 , e 6 }.

5 e 5 e

 55 time complexity and the number of local FMSO sets is bounded by ρ * .n ρ * -1 e . Moreover, as shown above, the BILP algorithm time complexity is negligible compared to that of local FMSO sets computation. Thus, each while loop of the D 3 algorithm has order of n ρ * +1.5 e time complexity. The loop is calculated m.n 1 times at worst, hence, D 3 has order of n ρ * +1.time complexity.

Fig. 4 :

 4 Fig. 4: Serially connected water tanks

Fig. 6 :

 6 Fig. 6: Decentralized architecture diagnoser proposed for the gas turbine generator system.

Fig. 7 :

 7 Fig. 7: Decentralized architecture for the gas turbine generator system.

Shared FMSO sets Φ s 1 , 1 =

 11 {φ 3 , φ 4 , φ 5 }, φ 3 = {e 2 , e 5 , e 6 }, φ 4 = {e 2 , e 3 }, φ 5 = {e 7 } Shared CMSO sets Ψ s 1,1 = {ψ 1 , ψ 2 , ψ 3 } ψ 1 = {e 4 }, ψ 2 = {e 3 , e 5 , e 6 }, ψ 3 = {e 1 } Subsystem Σ 1,2 Shared FMSO sets Φ s 1,2 = {φ 6 }, φ 6 = {e 16 , e 17 } Shared CMSO sets Ψ s 1,2 = {ψ 4 , ψ 5 , ψ 6 }, ψ 4 = {e 12 , e 13 , e 14 , e 15 }, ψ 5 = {e 9 }, ψ 6 = {e 11 } Subsystem Σ 1,3 Shared CMSO sets Ψ s 1,3 = {ψ 7 , ψ 8 , ψ 9 , ψ 10 , ψ 11 , ψ 12 , ψ 13 , ψ 14 } ψ 7 = {e 18 }, ψ 8 = {e 19 }, ψ 9 = {e 20 }, ψ 10 = {e 21 }, ψ 11 = {e 22 }, ψ 12 = {e 23 }, ψ 13 = {e 24 }, ψ 14 = {e 25 } Subsystem Σ 1,4 Shared CMSO sets Ψ s 1,4 = {ψ 15 , ψ 16 , ψ 17 } ψ 15 = {e 26 }, ψ 16 = {e 27 }, ψ 17 = {e 28 } Subsystem Σ 1,5 Shared FMSO sets Φ s 1,5 = {φ 7 , φ 8 , φ 9 , φ 10 },φ 7 = {e 30 , e 31 , e 32 , e 33 , e 34 }, φ 8 = {e 29 , e 31 , e 34 }, φ 9 = {e 29 , e 30 , e 31 , e 32 , e 33 }, φ 10 = {e 35 } Shared CMSO sets Ψ s 1,5 = {ψ 8 }, ψ 18 = {e 29 , e 30 , e 32 , e 33 , e 34 } 3) Building level 2: Because faults f 1 and f 2 are not yet isolable, D 3 builds subsystem Σ 2,1 at level 2. Similarly, because fault f 5 is not yet isolable, D 3 builds subsystem Σ 2,2 . These 2 new subsystems are built on demand, following the isolation on request concept. To form Σ 2,1 and Σ 2,2 , function ComputeUpsilon() is first used to retrieve the shared FMSO / CMSO sets Υ 2,1 and Υ 2,2 of the children systems of level 1 Σ 1,1 , Σ 1,2 , Σ 1,3 , Σ 1,4 and Σ 1,5 . Note that subsystems Σ 1,3 and Σ 1,4 are not impacted by any fault, leading to CMSO sets only (cf. Table

Fig. 8 :

 8 Fig. 8: Residuals for fault f 3 showing the fault at 600s by residual arr 1 , while all other residuals remain constant 3

TABLE I :

 I Execution times for the centralized algorithm and D 3 algorithm

	Setup	4 tanks	6 tanks	8 tanks	10 tanks
	Number of tests	165	5977	33040	1296249
	Centralized case	44.00 ms	637.03 ms	4965.28 ms	3984 s
	Decentralized case	16.00 ms	28.05 ms	47.41 ms	59.81 ms
	Local diagnosers:
	Σ1,1 to Σ1,10	< 1ms	< 1ms	< 1ms	< 1ms
	Σ2,1	9.00 ms	11.02 ms	13.01 ms	12.35 ms
	Σ2,2	7.06 ms	8.03 ms	10.01 ms	11.01 ms
	Σ2,3		9.03 ms	11.26 ms	12.24 ms
	Σ2,4			13.14 ms	10.43 ms
	Σ2,5				13.78 ms

  3 , z 4 , z 5 , z 6 , z 7 , z 17 } Combustion Chamber subsystem: Σ 1,2 Σ 1,2 = {e 9 , e 11 , e 12 , e 13 , e 14 , e 15 , e 16 , e 17 }, F 1,2 = {f 3 } X 1,2 = {x 1 , x 2 , x 6 , x 7 , x 10 , x 12 , ..., x 15 , x 28 } Z 1,2 = {z 1 , z 6 , z 8 , z 18 } Gas Turbine subsystem: Σ 1,3 Σ 1,3 = {e 18 , e 19 , e 20 , e 21 , e 22 , e 23 , e 24 , e 25 } , F 1,3 = {} X 1,3 = {x 1 , x 4 , x 6 , x 8 , x 10 , x 11 , x 12 , x 15 , ..., x 19 , x 29 } Z 1,3 = {z 1 , z 2 , z 6 , z 10 , z 11 , z 12 , z 13 } Electric Generator subsystem: Σ 1,4 Σ 1,4 = {e 26 , e 27 , e 28 }, F 1,4 = {} X 1,4 = {x 20 , x 21 } Z 1,4 = {z 2 , z 13 } Heat Recovery subsystem: Σ 1,5 Σ 1,5 = {e 29 , e 30 , e 31 , e 32 , e 33 , e 34 , e 35 }, F 1,5 = {f 4 , f 5 } X 1,5 = {x 15 , x 23 , x 24 , x 25 , x 26 , x 27 } Z 1,5 = {z 10 , z 11 , z 14 , z 15 , z 16 , z 19 }Besides, the functional decomposition considers the existence of restricted information not directly available for the subsystems in the first level and therefore only available in the upper hierarchical level 2. Restricted information has different sources, e.g. owner confidentiality, large distance between subsystems or difficult access. For level 2, it is denoted by Π 2,1 and Π 2,2 . The corresponding additional sets of equations for forming subsystems Σ 2,1 and Σ 2,2 are Π 2,1 = {e 8 , e 10 } and Π 2,2 = {e 36 , e 37 }, respectively.

TABLE III :

 III Selected Local FMSO Sets and isolability degree at level 1

TABLE IV :

 IV Shared FMSO and CMSO sets at level 1

	Subsystem Σ 1,1

  e 2 , ..., e 7 , e 9 , e 11 , ..., e 17 } ∪ {e 8 , e 10 } (12) Σ 2,2 = {e 18 , e 19 , ..., e 25 , ..., e 29 , ..., e 35 } ∪ {e 36 , e 37 } (13)4)Local FMSO sets at level 2: Three local FMSO sets are found for subsystem Σ 2,1 . Solving a BILP program returns an optimal selection of two FMSO sets for this subsystem, guaranteeing isolation of faults f 1 and f 2 . On the other hand, one local FMSO set is found for subsystem Σ 2,2 , guaranteeing isolation of faults f 4 and f 5 (cf. TableV).

TABLE V :

 V Subsystems Σ 2,1 and Σ 2,2 : Selected Local FMSO Sets Σ 2,1 A(Σ 2,1 ) {{f 1 }, {f 2 }, {f 3 }} I(Σ 2,1 ) 3 Local FMSO sets selected Φ l * 2,1 = {φ 211 , φ 212 }, φ 211 = {e 7 , e 8 }, φ 212 = {e 1 , e 2 , e 5 , e 6 , e 9 , e 10 , e 11 , e 12 , e 13 , e 14 , e 15 } = {φ 221 }, φ 221 = {e 35 , e 37 } Isolability degree at level 2 A 2 (Σ) {{f 1 }, {f 2 },{f 3 },{f 4 },{f 5 }} I 2 (Σ)5 fault isolability. The analytical redundancy relations are computed from the selected FMSO sets according to Tables III and V. The decentralized diagnoser has two levels formed by residuals arr 1 for subsystem Σ 1,2 , and arr 2 for subsystem Σ 1,5 at level 1, and by residuals arr 3 and arr 4 for subsystem Σ 2,1 , and arr 5 for subsystem Σ 2,2 at level 2. Table VI reports the obtained signature matrix that indicates maximal isolability.

			Σ 2,2
	A(Σ 2,2 )	{{f 4 }, {f 5 }}	I(Σ 2,2 ) 2
	Local FMSO sets selected	Φ l * 2,2

TABLE VI :

 VI Decentralized signature matrix for the gas turbine generator Systemf 1 f 2 f 3 f 4 f 5

	Level 1	arr 1 ∈ ARR 1,2	X
		arr 2 ∈ ARR 1,5	X
	Level 2	arr 3 ∈ ARR 2,1	X
		arr 4 ∈ ARR 2,1	X
		arr 5 ∈ ARR 2,2	X

TABLE VII :

 VII Execution times for global and decentralized architectures

	Architecture	Execution Time	number of FMSO Selected
	Global diagnoser	121.088 ms	4
	Decentralized diagnoser	44.096 ms	5
	Local diagnosers:
	Σ1,1 to Σ1,5	1.021 ms	2
	Σ2,1 to Σ2,2	43.075 ms	3

Note that "local" refers to a subsystem according to Definition 11, however this subsystem can be created at any level of the hierarchy.

Note that in recent solvers, it may not be more efficient.

If there were any, constants have been ignored in the expression of the ARRs.

This project is supported by ANITI through the French "Investing for the Future -PIA3" program under the Grant agreement noANR-19-PI3A-0004.