Marwan Zeidan

Christophe Turpin
email: turpin@laplace.univ-tlse.fr

Frédéric Cantin

Stéphan Astier

Steady state macroscopic model of the influence of water on the performances of a micro air-breathing fuel cell

Model of the influence of water on a micro air-

Introduction

General issues

PEM fuel cells produce water coming from the cathodic O2 reduction. This water can accumulate in the porous medium of the cathode in a mixture of liquid and gaseous phases, like illustrated in [1], depending on the operating conditions. As an example, Qiangu Yan et al. show in [2] the strong impact of the RH/T of the feeding gases. In general, this water is evacuated by evaporation toward the atmosphere in the case of air-breathing fuel cells [3], or is carried by the cathodic air/O2 overflow [4]. Many studies focus on what happens in the cathode, through various twophase models [4][5][6][7], sometimes modelling the porous medium's geometry [1], taking into account the permeability, the porosity, and the wettability of the electrode [5,[8][9], and dealing with the interfacial issues that multi layers electrodes can raise [1,[10][11]. It involves complex mass transport, and energy balances, that can be carried out at different scales, from a macroscopic [9,12] to a microscopic point of view [9,[13][14]. Water can also diffuse back through the membrane toward the anode, hydrating or dehydrating the membrane [15][16]. The protonic conduction also drags water from the anode to the cathode, more or less compensating for the back diffusion. Moreover, the membrane is generally made of Nafion, which also partly constitutes the electrodes. This polymer can accumulate and restore water in a non linear way (Schroeder paradox for example), strongly depending on the activity and quantity of water at its surface, that is to say the pore water (the liquid water in the pores of the porous medium) in the membrane and in the electrodes. The hydric events located in the membrane are very complex, and are the core of a lot of dedicated studies, like [18] or [19]. Complex theories, through capillarity models, try to explain the Schroeder paradox that can occur in membranes [17]. All those phenomena, interacting with each other, depending on operating conditions, are the heart of the fuel cell's performances. Indeed, PEMFCs need to be hydrated, mostly to allow H + diffusion/conduction from the anodic active sites to the cathodic ones: a dry fuel cell will not start, or it will have very low performances; it needs to be hydrated first. Qianpu Wang et al. propose a description of the impact of water on this phenomenon in [7]. Nevertheless, exceeding water can have a clogging/flooding effect because gases, mostly O2, need to diffuse correctly toward the active Pt conglomerates that are distributed in the electrodes, in order to make the electrochemical reaction possible. The influence of water on O2 diffusion can be observed through the pressure drop measurement technique, as studied in [20][21]. Other methods, aiming at observing in situ water behaviour, can also give a qualitative idea of the impact that water can have on O2 diffusion: visual measurements on transparent fuel cells as seen in [22][23], or neutron radiology [24][25]. St-Pierre gives in [26] an exhaustive list of the experimental methods that can be used for this purpose. The influence of water on O2 diffusion is thus modelled in numerous ways [9,11,27]. However, not only the paths to those active sites need to be diffusion-friendly. The Pt conglomerates themselves also need to fulfil the triple phase boundary (TPB): it needs to be hydrated enough to allow H + arrival/departure through the neighbouring Nafion or through the pore water, but not too much, for gaseous O2 adsorption issues. This possible inerting of the active sites can be taken into account through a water content depending coverage/efficiency coefficient, cutting the active surface down in several ways, as illustrated in [4,5,9,28]. Finally, between the two extreme situations that drying out and flooding are, liquid water presence has a first order influence on every physical (diffusions and conduction) and electrochemical (activation) phenomenon. Water management thus consists in controlling the operating conditions to find the best compromise (between drying out and flooding) to get the best performances for the PEMFC. Nevertheless, all the phenomena described above are linked in a non linear and globally non monotone way: a proton-friendly hydric situation can be hostile to O2, those two species having antagonist hydric needs. Thus it is quite hard to get the best fuel cell's working point 'manually' and empirically. This becomes nearly impossible in the case of the micro air-breathing fuel cells (µABFC) we deal with in this article. N. Karst studied them and enlightened these water-linked difficulties in [29][30]. We will describe them precisely in the following, concluding that µABFC's geometrical and technological features make them very sensitive to any atmospheric conditions and operating temperature change. Moreover those parameters will vary during the final use of those µABFC; they are indeed intended for a non hermetic portable device that is supposed to operate everywhere in the world as a range extender for lithium-ion batteries of a cell phone.

In conclusion, a model is strongly needed to estimate the water content of the fuel cell first, and then to describe its influences on the PEMFC's electrical performances. The final goal is to be able to optimize online the µABFC's environment, thanks to its auxiliaries. The part 2 of this paper focuses on the first aspect of the problem: the calculation of the water content according to the atmospheric conditions and the fuel cell's temperature. The links with the physic-chemical model of the fuel cell will be proposed in the part 3, leading to a micro fuel cell model depending on atmospheric conditions. The whole study is achieved in the steady state case.

µABFC description and application

The geometric and technological features of the µABFCs are very specific and will strongly impact the choices we will make for the hydric model. The studied µABFCs are made in France at STMicroelectronics and at the CEA Liten, by successive depositions of thin layers on a pre-etched silicon substrate (for hydrogen feeding). Both anode and cathode active layers are deposited by ink-jet printing, and are respectively 10 µm and 20 µm thick. The membrane is thicker (30 µm) and is coated on the anode. To obtain the cathodic collector, a very thin (0.5 µm) gold layer is deposited on the top of the cathode. It is both thin and porous enough to allow air-breathing and water evacuation, but it is homogeneous enough to enable a good electrical contact. To get more precisions on this process, please refer to [29]. A schematic cross-section of the micro-PEMFC is shown in Fig. 2. There are no purely non-active gas diffusion layers (GDL), due to process limitations. This significant difference with most of the classically studied fuel cells will have an impact on the model: contrary to what Yun Wang et al. do in [10][11] for example, we will not have to link two different diffusion media for the H2O electrode diffusion. The geometric surface of the cell is about 1 cm², which is very small regarding to the commonly modelled fuel cells, for which a small surface is usually not less than 20 cm². Instead of using surface currents in the following, we will rather use the global current actually crossing the fuel cell, considering, for a first estimation, a homogeneous current repartition in the electrodes. This hypothesis is strengthened by the fact that both anodic and cathodic current collections are shared out between 3 gold contacts each (Fig. 1), which contributes to a better current repartition. Those small dimensions and the absence of a GDL make these fuel cells highly sensitive to atmospheric changes, as well with regard to its RH, temperature, or turbulences. Those µABFCs aim at reaching 1 W/cm². They are designed to be mounted in an active packaging, more or less opened to the ambient air and in which atmospheric conditions will be driven by the fuel cell's water production and temperature, which are themselves depending on atmospheric conditions. Micro built-in fans will be used to manage those environmental variables. In the studies we carried out, and the measures we used to develop our model, the µABFC was opened to a fully controlled atmosphere, mounted on a specific measurement equipment; we indeed wanted to model the µABFC itself, separated from its packaging's influence.

2 Calculation of the water content within the porous cathode according to the experimental conditions and the working current point

Water content model description

Global description and main hypotheses

Cathode focus:

Numerous studies have been devoted to developing mathematical models for the transport of reactants and produced water in PEMFC. They sometimes deal with global balances, taking into account the hydric interactions between the gas channels, the membrane and the two electrodes, and modelling water transport and transformation in each of these media, like Springer et al. in [31], or like in [8,37]. Nevertheless, authors often focus on modelling the electrodes hydric behaviour, and especially the cathode [5][6][7]9,22]. Indeed, this is first of all where water is produced; this is the source location for mostly every hydric issue. Secondly, the cathode is the electrode where O2 is reduced. It is commonly admitted that O2 diffusion is more problematic than the H2 one, due to the fact that O2 is a bigger molecule than H2. In fact H2 diffusion is often neglected [7]. The cathode thus is where liquid water presence will have the greater impact. Moreover, in our approach, the complex impact of the membrane's hydration will not be explicitly modelled, as well as it is done in a lot of studies. The membrane resistance will be indeed estimated (Fig. 4) starting from Electrochemical Impedance Spectroscopy (EIS) measurements that were carried out along the polarization curve. In these studies concentrated on membrane modelling, the anode is considered to be ideally hydrated; in these conditions, the H2 has no problem to reach the active sites (H2 diffusion neglected), and the produced H + have no problem to reach the membrane (no anodic drying out issues). To take into account the flow of water leaving/reaching the cathode toward/from the membrane, a net transport coefficient is often used, modifying the produced water flow [2,4]. We neglected this coefficient here, for the same reason [10] exposes. Moreover, the multi-conditions electrical measurements we used to calibrate our model were performed by Nicolas Karst at the CEA Liten. They are displayed in Fig. 5 and in Fig. 6. N. Karst proved in [30] that in the conditions the tests were carried out, the anode was not suffering from dehydration, which strengthens our cathode focus.

Model scale:

Microscopic points of view are often preferred to develop cathodic hydric models. Nevertheless, macroscopic-like models can be sometimes used, like the segmented approach of Yong-Song Chen et al. in [32], or the average approach of Cecile Gondrand in [33] also adopted by Prodip K. Das in [9]. Here, we will chose a global macroscopic scale. We will consider macroscopic parameters and variables (the water content for instance will be uniform in the whole cathode), that will not be precisely located in the fuel cell, but that will be representative of the global cathodic situation.

Classical Fickian diffusion:

Generally, cathodic water transport models deal with multicomponent diffusion, using Stefan-Maxwell equations [32,33], and Darcy transport equations, taking into account the porosity of the medium in several ways. This leads to binary (O2 and H2O) and sometimes ternary (O2, N2, and H2O) systems. Nevertheless, Cecile Gondrand showed in [33] that the generalised Fickian crossed coefficients could be neglected, leading to a classical Fickian behaviour. We propose to consider separately gaseous H2O and O2 diffusion: the water content will be calculated from the purely hydric study we will carry in the part 2, and then the effect of this water content will be included in O2 diffusion.

Phase of the water:

Almost all the hydric models in the literature only suppose gaseous water in the cathode, considering that water is produced in this phase. Regarding the conditions, and/or above a critical current value, the vapour pressure reaches a saturation value, making liquid water appear, which leads to a two-phase mode [4,5,20…]. Our µABFC operating at low temperatures (45 °C max), the produced water will be supposed liquid (generally there is always at least an equilibrated liquid drop somewhere in the porous medium). Moreover, the fuel cell itself is opened to the atmosphere: its cathode will be in contact with an air volume proportionally large compared with a classical closed fuel cell. In this case, the liquefaction of vapour will be more limited than it is in a closed confined cathode. Consequently, our model does not work with two modes (one phase/two phases). Its core is the calculation of the vapour flow that is evacuated from the fuel cell by evaporation. In steady state, there will be a balance between the evaporated and the produced water. If this balance is not assured, there will be progressively flooding or drying out.

Other hypotheses:

We will consider that the system is under steady state. Each gas is considered like a perfect gas. We neglect the influence of the diffusion of the dissolved O2. The temperature TFC is uniform within the whole µABFC.

The hydric balance

We aim at calculating the water content of the µABFC's cathode, defined by the water saturation s. The saturation is the ratio between the volume of the pores that are occupied by liquid water, and the total pore volume that is theoretically available:

tot liqu V V s (1)
Eq.2 proposes the 'purely hydric' balance that comes from the hypotheses exposed in 2.1.1:

dt t J dt t J t n dt t n evap O H prod O H O H O H . .
F I J s J FC prod O H evap O H 2 2 2
(3)

We now want to express s J evap O H 2 literally, in order to know s.

For that purpose, we consider that, in the porous medium of the cathode, only the vapour concentration difference between the inner liquid water's surface and the outside surface of the cathode leads to the evaporation of the liquid water to the atmosphere. We stated in 2.1.1 that this diffusion was Fickian:

O H evap evap eff O H evap O H C S D J 2 2 2 (4)
Where evap S is the surface through which the vapour diffusion occurs, evap is the diffusive length, and In porous media, the equilibrated vapour pressure at the interface with liquid water is not rigorously equal to the saturated value it would have in a free medium. This equilibrium pressure is theoretically modified by the capillaritylinked phenomena that are involved by the porosity of the medium. A common formalisation is given by the capillary condensation theory, through the Kelvin equation, which is for example used by Cecile Gondrand in [33]. In this case, the saturation pressure of vapour is affected by the capillary pressure Pc:

RT M s P sat v O H v O H l O H O H c e T P P , 2 2 2 2 , , (6)
Where s P c depends on s through the Levrett function [5]. A numerical application shows that in every situation, the exponential term of Eq.6 is very close to 1 (in extreme cases, between 0.8 and 1.2). That is why we assume in the following that we are in a classical situation:

FC sat O H FC sat O H in O H RT P T C C 2 2 2 (7) 2.2.2 Calculation of surf O H C 2 surf O H C 2
depends on the flow of gaseous water that gets out of the fuel cell at the interface between the cathode surface and the atmosphere, and on the atmospheric conditions (RHa, Ta) that are set "far from" the cathode surface, by the climatic chamber. Indeed,

surf v O H 2
will have to undergo a mass transfer in the atmosphere, driven by diffusion and possibly natural convection, which is due to a possible temperature gradient between the µABFC's surface and the external atmosphere temperature. We propose, for this first macroscopic approach, to describe the diffusion/convection phenomenon by a simple resistance of mass transfer DC R , representing the parallel influence of steady state diffusion and natural convection:

a conv a diff DC R R R 1 1 1 (8)
With this variable, we can calculate

surf O H C 2
, as we know the H2O flow at the interface, thanks to Eq.3:

F I R C C FC DC O H surf O H 2 2 2 (9)
Where, according to the definition of relative humidity:

a a sat a O H RT T P RH C 2 (10)
We can in the same way calculate the surface concentration of O2, using the same DC R : (11) We now need to estimate DC R , that is to say

F I R C C FC DC O surf O 4 2 2
a diff R and a conv R . Calculation of a diff R : a diff
R comes from a steady state Fickian description of diffusion in the atmosphere:

0 , 2 v O H carac carac a diff D S R (12)
In the whole study, to take into account the variations of diffusivity with temperature, we used a common correlation [11]:

5 . 1 0 , 298 T D D X X ( 13
)
Where T is the temperature of the medium the diffusion takes place in. Thus, the diffusion resistance is finally given by: There are a lot of ways to take into account the porous nature of the medium a diffusion mechanism occurs in, especially when it is partially saturated with liquid water. The most common way is to impact the effective diffusivity of the species that diffuses through the pores, through the Bruggeman formula:

X b a eff X D s D X X 1 (19)
Where X D is the "free" diffusivity of the species X through air at the operating temperature, out of the porous medium.

In this formula, the more liquid water there is in the pores, the more tortuous the path the diffusing gas will have to take will be, the weaker its diffusivity will be. The parameters X a and X b are often equal and are called tortuosity. Nevertheless, [START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF] deals with the theory of porous media diffusion, and gives another definition for tortuosity, that we will use in the following (this is just a vocabulary matter):

X eff X D D ( 20
)
The most commonly found case for parameterization is There are also other methods to make the effective diffusivity depend on s, as described in [START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF]. Regarding the diversity of parameters we can find, we will use X a and X b as adjustable parameters in our model:

O H b a eff O H D s D O H O H 2 2 2 2 1 (21)
With, according to Eq.13: Other studies are devoted to developing other approaches for evaporation in porous media. Those studies are often more focused on porous materials or powders drying processes, and are not dealing with fuel cells. For example, in [34], F. Debaste proposes various evaporation models, at an Representative Elementary Volume (REV) scale, in which he takes into account the variation of the evaporation surface, or of the vapour diffusion length, with the water content, modelling the physiognomy of water in pores. Our model is inspired by such methods, and by the Bruggeman type formulations. We will consider that a small cathodic water content s involves a small evaporating surface, which one will increase when s does. This is traduced by:

K b evap S s S O H 2 (23)
Where K S is the geometric surface of the cathode.

In the same way, we will consider that macroscopically, the more liquid water there will be in the pores, the shorter the average path the vapour will have to take will be, the complexity of this path being taken into account by the tortuosity in Eq.21:

K b evap O H s 2 1 (24)
This way, we could see

O H b 2
as being an image of the sensibility of the gaseous water diffusion phenomenon to the presence of liquid water in the porous medium where it takes place. We will see in the following that we will reproduce this definition on every diffusive species. This approach is quite original, and its relevance will be proven in the final model in the following.

Literal s calculation

Finally, according to Eq.4, 21,23 and 24 we have:

O H K b K b O H b a prod O H C s S s D s J O H O H O H O H 2 2 2 2 2 2 2 1 1 (25)
This leads, using Eq.3, to:

O H O H b K K O H a O H FC S D C F I s 2 2 2 2 1 2 (26)
Eq.26 gives a literal expression of s, the steady state water content of the cathode, depending on the operation conditions RHa, Ta and TFC (Eq.7, 9, 10) and on the operation current point IFC. To illustrate this dependency, we displayed the s curves in Fig. 3 for the experimental conditions (Tab.3) that N. Karst applied in [30] for the measurements.

This plot uses the model parameters of the Tab.4. We can notice that s values are coherent with the experimental conditions: the dryer the atmosphere is, the lower the water content is.

Steady state macroscopic modelling of the µABFC performances according to the cathodic water content

The theoretical classic fuel cell model

The classical model

The theoretical reversible fuel cell potential is given by Nernst's relation: As our water content is set by the imposed IFC (Eq.26), we reverse this relation:

Where ref X X diff eff X X diff X C D nFS I lim (31)
We also set, as a new parameter: X X (32) One can notice in Eq.30 that we considered only pure diffusion for H + , using Fick's Law, contrary to what is often done [7,35]. Indeed, we consider that the migration part of the protons' diffusive mechanism is already taken into account by the variations of the global electric resistance Relec we extracted from the EIS carried out at the end of each working point. As a matter of fact, Relec is the sum of the electronic resistance and the protonic one, which contains the information of the conductivity of the Nafion that constitutes the membrane and the electrodes. Moreover, for this model, we only took into account the pore water protonic diffusion that occurs in the cathode.

We can now express each loss separately. (33) Where I0 depends on temperature, following this relation [11,28] We added a leak current In, which is a simple and common way to take into account H2 crossover issues, like M.G. Santarelli does in [12].

Activation losses:

Diffusion losses:

X n X FC X diff I I I F RT lim 1 ln 2 ( 35
)
Ohmic losses:

n elec elec I I R (36)
Where Relec varies with IFC and is a measured data (Fig. 4).

The global steady state model is finally given by :

elec diff act rev FC V V (37)

Influence of s in the model

The parameters that are supposed to be the most dependent on water are diffusion parameters, and activation parameters.

O2 diffusion

Oxygen diffusion mechanism will undergo the same type of liquid water influence that vapour does, through

surf O O diff eff O O diff O C D FS I 2 2 2 2 2 2 lim (38) eff O H D 2 :
The more liquid water there will be in the porous medium, the more tortuous the path from the cathode surface to the active sites will be for O2, the smaller the effective diffusivity will be. : The more liquid water there will be in the pores, the smaller the average surface through which O2 diffuses will be:

K b O diff S s S O 2 2 1 (40) 2 O diff :
The more liquid water there will be in the pores, the shorter the average path the O2 will have to take will be:

K b O diff O s 2 2 1 (41)
Mixing Eq.38, 39, 40 and 41 leads to:

surf O b K O K a O C s D S F I O O 2 2 2 2 2 1 2 lim (42)

H + diffusion

H + diffusion probably occurs very locally, on very short distances. We can thus assume that the protonic effective diffusivity is not very sensitive to the water content:

H a eff H D D H . ( 43
)
H diff S : The more liquid water there will be in the pores, the higher the average surface through which H + diffuses will be:

K b H diff S s S H (44) H diff :
The more liquid water there will be in the pores, the lower the average path the H + will have to take will be:

K b H diff H s 1 (45)
Mixing Eq.31,43,44 and 45 leads to:

ref H b K H K a H C s s D S F I H H 1 lim (46)
Like described by Eq.13, we take into account the dependence of the diffusivities of H + and O2 to temperature: For each steady state measured point, an explicit water content was expressed. Laws for the dependence on this liquid saturation were proposed for VFC. In the next part, we will parameterize this model starting from several measurements. The model parameters that have to be identified are gathered in Tab.2.

Measurements and model results

Our final goal is to match our model with an unique set of the Tab.2 parameters with 8 polarization U-I curves, that were measured in 8 different experimental conditions, that we will express through this notation in the following, for readability: [RHa, Ta= th FC T ]. We will describe them in the following (Fig. 5 and Fig. 6). By "unique", we mean that ideally, one only set of optimized parameters should be able to reproduce the µABFC's behaviour for 8 different experimental conditions, independently from those conditions. This is very important, since those root parameters are supposed to be an image of the µABFC itself, whatever the environment it is in.

Measurements description

Measurements were made by N. Karst at the CEA Liten. He mounted the µABFC on a heating support, in order to control its temperature TFC, and he put it in a climatic chamber, aiming at controlling (RHa,Ta). For more details on the experimental set up, please refer to [30]. He settled 8 conditions (Tab.3), and at each condition, he plotted the steady state U-I polarization curve (Fig. 5,6). At the end of each stabilized point, he carried out an EIS, which gives Relec. As far as the contact surface between the µABFC chip and its heating support is very small, due to the presence of the H2 chamber underneath it, and as the H2 was in circulating mode, we assumed that the real temperature of the µABFC TFC could deviate from the theoretically set temperature th FC T , due to this small exchange surface, and to the thermal losses of the µABFC. We then injected a thermal resistance in the model, aiming at representing these variations:

FC FC rev th th FC FC I V V R T T (49)
In those conditions, we have a C T th FC 30 theoretical temperature, which corresponds to a good operating temperature for the µABFC (the ideal one being generally 35°C). As a benchmark, 25 °C is the lower limit for the µABFC to operate correctly. 45 °C is quite high for the µABFC: it generally does not work passed 50 °C. Moreover, a 0.1 RHa corresponds to a very dry atmosphere (leading to a high evaporation rate). RHa=0.7 corresponds to a not excessively but yet quite humid atmosphere, being able to provoke a lack of evaporation from the cathode, leading to a flooding of the µABFC. By observing Fig. 5 and Fig. 6, we can already define some points for the model to respect:

-At [0.7,30], the fuel cell voltage decreases quickly with the current, regarding the other curves: at those humid conditions, the evaporation is not favoured; we could reasonably interpret this as a flooding. This should be traduced in the model by a high O2 diffusion overpotential.

-At [0.1,45], the fuel cell voltage also decreases quickly with the current. The dryness of the conditions would point at a drying out issue. The model should traduce it with a high protonic diffusion overpotential. The [0.7,30] simulated curve in Fig. 5 shows a problem at high currents. This is due to the O2 diffusion loss, as seen in Fig. 7.a. This particular high current variation is directly due to the RDC dependence on temperature, showing that this point has to be improved. On most of the 30°C curves, the low current slope of the simulation does not match the measure. This is also a point to improve, we will propose some explanations in the following (it is probably linked to the non dependency of max 0 I on s, or from our protonic diffusion model).

Model results and discussion

The model's global behaviour

Nevertheless, the model is rather accurate. It traduces correctly the influence water should have on the fuel cell. At [0.7,30], which is the more humid condition, Fig. 7.a shows that O2 diffusion is the cause of the voltage drop, when the protonic diffusion overpotential is very low; which is logical, the experimental conditions pointing at flooding issues. At 30 °C, as RHa decreases, protonic diffusion becomes more problematic, especially at low currents (a few water is produced, s remains small), as O2 diffuses more easily (Fig. 7.a to Fig. 7.d). At 45°C (Fig. 8.a,b,c,d), where the µABFC is supposed to suffer from further dehydration, protonic diffusion is indeed globally higher than it is at 30 °C, whereas the O2 diffusion overpotential is globally smaller. This last one becomes negligible at RHa=0.1 and 0.3, conditions for which the protonic diffusion becomes very problematic and majoritary compared with the gaseous diffusion problems. Moreover, the activation losses in each experimental condition are largely predominant compared with the other losses, which is a classical result. Finally, the electric overpotential elec is also quite high. It increases with TFC, which is not what we could expect, regarding the fact that the protonic conductivity is supposed to be better when the temperature increases. This points at the fact that the evaporation process, which is stronger at higher temperatures, contributes to drying the Nafion out, leading to a lowering of its conductivity. This is confirmed by the fact that elec rises when RHa decreases. This behaviour analysis shows that the calculated overpotentials, taken individually, qualitatively illustrate the hydric behaviour we expected, and are quantitatively very plausible.

Discussion around the parameters

Every optimized parameter is in a classic range regarding the literature. Moreover, we almost succeeded in identifying a unique set of parameters valid for a large range of experimental conditions. We had to adapt only two parameters according to temperature (Tab.4): This can be a collateral effect of the problem we have on max 0 I . Nevertheless, the s dependency exposed in Eq.47 could be the cause of the problem. Further investigations, proposed in the conclusion part, could bring more precise elements on this. This problem on , as far as the equivalent exchange current should rise with a temperature elevation. Our temperature dependency, even if it could be improved, is not the source of the problem. There remain two possible explanations to this. Maybe this gap is caused by a hedge somewhere else in the model, for example on protonic diffusion, which remains uncertain. But it can also point at the need for max 0 I to depend on s, like it is often done in the literature [4,5,9,11,35]. Nevertheless, all those studies use relations of the following form: Where act b often equals 1 (nevertheless, act b =2 in [35], and act b =1,2,5,8 is explored in [9]). This traduces the fact that the active surface can only be reduced by a flooding effect. This can be explained by the fact that it is very hard to observe drying out issues on classical 'closed' fuel cells, which is absolutely not the case with µABFC. In our case, at 45°C, our simulations (Fig. 3), and the Relec measures (Fig. 4), tend to show that the µABFC is globally drier than it is at 30°C, that is to say that on average,

C s C s 45 30
. This means that a Eq.51 type relation would not be sufficient. We need another relation to traduce the fact that if the cathode is too dry, it also loses active surface (no departure/arrival possible for protons from/to the active sites). For example:

Conclusions and perspectives

We developed a full macroscopic steady state physical model for the µABFC, that takes into account the influence of atmospheric conditions on its behaviour. It is able to predict if the µABFC will suffer from drying out or from flooding, and can tell what condition would be the best for the fuel cell's performances. Nevertheless, some points need to be improved. The convection diffusion model of the atmosphere is probably too minimalist. Some other parts need further investigation, like the influence of the cathodic water content on the equivalent exchange current max 0 I , or on the protonic diffusion.

Full EIS that were carried out all along the measurements we used remain partially unanalysed. This could be a good starting point to adjust our model. Indeed, studying the spectra and their evolution with the water content could help us to refine our s depending laws, particularly for the protonic diffusion. This phenomenon clearly appears indeed in the EIS results. This analysis would also be very helpful concerning the distinction and the quantification of each phenomenon's influence.

The next step will be to connect this model to the model of the atmosphere inside the packaging the µABFC will be set in. The aim will be to know how to drive the packaging's auxiliaries (fans) in order to settle the best atmosphere for the µABFC. [1,7,8,[START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF] 1.5 [4,5,6,27,[START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF] 1 [1,38,[START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF] 2 [39] 0.71 [START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF] 4 [START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF] 3.33 [START_REF] Lefebvre | Ecoulement Multiphase en milieu poreux[END_REF] 1.67

  produced and the evaporated water molar flows. As the µABFC is supposed to be under steady state, this becomes:

,

  is the 298K reference. Eq.15 represents the average temperature of the air between the surface of the cathode and the point of space where resistance, we suppose that it depends on temperature through the following relation:

  fact this relation by analogy with thermal convection. Mixing Eq.8, 14 and 16 leads to Eq.18:

.

  Nevertheless, in some studies, X X b a , and other values than 1.5 are sometimes used. We propose to sum up some of these cases in Tab.1, focusing on the most important parameter X b .

  the current increases, the fuel cell voltage progressively drops due to physical losses.In steady state ( act I I), the activation current is described by the Butler-Volmer equation, expressed with the Tafel approximation (

  linear Fickian diffusion equations to express the species' concentrations, the voltage drop due to activation and diffusion phenomena is:

  :

Fig. 5 and

 5 Fig.5 and Fig.6 show how the model behaves. The parameters we identified are summed up in Tab.4. Fig.7.a,b,c,d and Fig.8.a,b,c,d detail the evolution with current of the overpotentials we defined in our model. The model globally predicts the µABFC's behaviour very well. Nevertheless, some particularities have to be underlined.

D

  , which can potentially traduce problems in our model.Nevertheless, we can propose some explanations.

I

  Fig.6at low current values, even if this point could also be corrected by a max 0 I dependency to s.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig.1. Photography of an non assembled µABFC

Fig. 6 .

 6 Fig.6. Comparison between the model (Tab.4 parameters) and the measures for [RHa; 45 °C]

Fig. 7 .Fig. 8 .

 78 Fig.7.a,b,c,d. Simulated (Tab.4 parameters) overpotentials (V) for 30 °C, respectively for RHa= 0.7, 0.5, 0.3, 0.1, depending on IFC(A)

  gaseous water through air, in a partially liquid-saturated porous medium.

											D	eff O 2 H	is the
	effective diffusivity of O H C 2	is the
	difference between the water concentration at the surface of the inner pore liquid water	C	in H 2	O	and the water
	concentration there is at the surface of the cathode	C	surf O H 2	:
				C	H	2	O		C	in H 2	O	C	surf O H 2	(5)
	.
	2.2 Calculation of vapour concentrations
	2.2.1 Calculation of	C	in H 2	O

Table . 1

 . . Overview of possible literature values for X b References

X b

Table . 2

 . . Definition of the tuning parameters of the model

		Table.4. The extracted parameters for the model
		Phenomenon				Parameters	30°C	45°C	units
									1.1*10 -5	m 3 .s	-1
									a diff R 0 ,	2*10 6	s.m -3
									H a	2	O	H a	2 O a	1.5	Ø
		Evaporation							b	H	2	O	2.7	Ø
									H D	2 O	,	0	5*10
	Phenomenon	Parameters	Name	units
									Mass transfer coefficient	m 3 .s	-1
			a diff R 0 ,	Diffusion resistance in the atmosphere at 298K	s.m -3
	Evaporation	H a	2	O	H a	2 O a	Porosity correction coefficient	Ø
			H b	2	O	H2O sensitivity to water presence coefficient	Ø
			D	H	, 2 O	0	Diffusivity of gaseous H2O in air at 298K	m².s	-1
									Equivalent transfer coefficient	Ø
	Activation	I	max 0	Maximal exchange current	A
			2 O D	,	0	Diffusivity of O2 in air at 298K	m².s	-1
	O2 diffusion	2 O b		O2 sensitivity water presence coefficient	Ø
				2 O		O2 corrected transfer coefficient	Ø
			H D		,	0	Diffusivity of H	+ in water at 298K	m².s	-1
	H	+ diffusion	H b				H coefficient + sensitivity to water presence	Ø
				H		H	+ corrected transfer coefficient	Ø
	Crossover	I	n					Equivalent leak current	A
	Heat transfer to the µABFC	th R		Heat transfert resistance	K.W	-1
					Table.3. Experimental conditions
			th FC T					a T	RH	a
			45 °C		0.1	0.3	0.5	0.7
			30 °C		0.1	0.3	0.5	0.7

Table . 5

 . . Some other values used for the model

	Quantity Name	Value	units
	C	ref H	Reference concentration on protons [7] 200	mol.m -3
					Reference internal diffusion length	1.5*10	5	m
	K
	K S			Reference internal diffusion surface	4*10 -5	m²
					Cathodic porosity [30]	0.4	Ø
	H P	2		H2 partial pressure	1	bar
	2 O P		O2 atmospheric partial pressure	0.2	bar
	G	0	Standard Gibbs energy	236309 J.mol -1
	R				Universal gas constant	8.314	J.mol -1 .K
	F				Faraday's constant	96485	C

Acknowledgments

This work was supported by STMicroelectronics Tours (France), and the LAPLACE Laboratory, Toulouse (France). We are grateful to Nicolas Karst, who performed the helpful measurements we used for our model.


M A Sid

S Aberkane

D Sauter

D Maquin

Fault isolation filter and sensors scheduling co-design for networked control systems

This paper addresses the problem of multiple fault detection and isolation under communication constraints. More specifically, we consider the issue of sensor scheduling and fault isolation co-design under limited bandwidth capacity. The proposed isolation filter can be viewed as special structure of the traditional Kalman filter. The sensor scheduling sequence and the proposed filter are built in order to ensure the fault isolability property and noise effect minimization.

I. INTRODUCTION

The study of networked control systems (NCS) is receiving much importance in this recent years. This is mainly due to the several advantages resulting from using a shared real time network through which sensors, actuators and controllers communicate. Compared with classic fault detection (FD) systems, diagnosis over networks can reduce the system wiring, make the system easy to supervise, maintain and increase system agility• • • etc. Nevertheless, new constraints also arise when sensor information and control information are transmitted over a network. Such constraints can be categorized in five types [START_REF] Heemels | Networked Control Systems With Communication Constraints: Tradeoffs Between Transmission Intervals, Delays and Performance[END_REF], [START_REF] Zhang | Stability of networked control systems[END_REF]:

1) Quantization errors in the signals transmitted over the network due to the finite word length of the packets; 2) Packet dropouts caused by the unreliability of the network; 3) Variable sampling/transmission intervals; 4) Variable communication delays; 5) Medium access constraints due to the limited bandwidth and sharing the network by multiple nodes and the fact that only one node is allowed to transmit its packet per transmission 6) Power consumption mainly in wireless networked control systems. The presence of these network induced effects can degrade the performance of FD systems and implies more robust algorithms to this communication constraints.

It is clear that all of these constraints can exist in a communication network, but only some of them were fully considered in the literature, mainly the induced delay effect, packet loss and sampling influences. The delay issue is considered for example in [START_REF] Wang | A New Fault Detection Scheme for Networked Control Systems Subject to Uncertain Time-Varying Delay[END_REF], [START_REF] Li | Fault detection design of networked control systems[END_REF], [START_REF] Wang | Residual generation and evaluation of networked control systems subject to random packet dropout[END_REF], [START_REF] Heemels | Networked Control Systems With Communication Constraints: Tradeoffs Between Transmission Intervals, Delays and Performance[END_REF]. In [START_REF] Mao | H ∞ fault detection filter design for networked control systems modeled by discrete Markovian jump systems[END_REF], the authors deal with the design of robust FD systems for NCS with large transfer delays, in which it is impossible to totally decouple the fault effects from unknown inputs. An adaptive Kalman filter is proposed in [START_REF] Chabir | Design of an adaptive Kalman filter for fault detection of Networked Control Systems[END_REF] to minimize the effects of the network induced delay on the residual signal. [START_REF] Li | Fault diagnosis for networked control systems with delayed measurements and inputs[END_REF] considered the problematic of FD for NCS with both delayed inputs and measurements. In [START_REF] Wang | Residual generation and evaluation of networked control systems subject to random packet dropout[END_REF], [START_REF] Zhang | Fault detection of networked control systems with missing measurements[END_REF], the FD system for NCSs with packet dropouts was designed by modeling the NCSs as a Markov jumping linear system (MJLS). The issue of FD with multiple network induced constraints has been considered for example [START_REF] Li | Fault detection design of networked control systems[END_REF], [START_REF] Song | Stabilisation of networked control systems with communication constraints and multiple distributed transmission delays[END_REF], [START_REF] Nejad | Fault detection and isolation in networked control systems with access constraints and packet dropouts[END_REF]. Note that, to the best of the author knowledge, the issue of fault isolation in networked systems has not been fully investigated nowadays.

In this paper, we will address the problem of multiple fault detection and isolation under communication constraints. More specifically, we will consider medium access constraints. In this case, the shared network can only accommodates a limited number of simultaneous communications between components. In this context, it is only meaningful to specify a fault detection and isolation module in conjunction with a communication policy which indicates the times at which the plants sensors are to be granted medium access. This communication policy is known in the literature as communication sequence. The communication sequence specifies which sensors are able to send information to the detection filter at each time step. Hence, the considered problematic leads naturally to consider a co-design problematic. That is, the design of a fault detection and isolation filter in conjunction with sensor scheduling sequence. The sensor scheduling sequence and the proposed filter are built in order to ensure the fault isolability property and noise effect minimization.

The rest of the paper is organized as follow: section II gives the problem formulation. In section III, we will give our main results. The sensor scheduling problem will be discussed in section IV. A numerical example will be given in section V to illustrate the effectiveness of the proposed codesign method. Finally, we will provide some conclusions and some future research directions in section VI.

II. PROBLEM FORMULATION

Consider the remote system depicted in Fig. 1. The state space representation of the plant under actuator and/or com-

x k+1 = Ax k + Bu k + Fϒ k + w k y k = Cx k + v k (1)
where

x k ∈ ℜ n is the state vector, u k is the control input, F = [ f 1 , f 2 , ..., f q ] ∈ ℜ n×q is the fault distribution matrix, ϒ k = [ϒ 1 k , ϒ 2 k , ..., ϒ q k ]
T ∈ ℜ q is the fault vector and y k ∈ ℜ m is the measurement signals vector. We assume that each component of the output vector y i represents the sensor i ∈ {1, 2, .., m}. The initial state vector x 0 , process noise w k and measurement noise v k are uncorrelated, zero mean white Gaussian random processes with x 0 ∼ N( x0 , P0 ), w k ∼ N(0,W ) and v k ∼ N(0, I m ) respectively, where P0 ,W and R are symmetric, positive definite matrices.

The main objective of this paper consists in the design of a fault detection and isolation filter that takes into account the communication constraints induced by the shared communication medium. More specifically, the communication constraint we deal with in this paper is referred to in the literature as a medium access constraint. In this case, the shared network can only accommodates a limited number of simultaneous communications between components. In this context, it is only meaningful to specify a filter in conjunction with a communication policy which indicates the times at which the plants sensors are to be granted medium access. This communication policy is known in the literature as communication sequence [START_REF] Hristu-Varsakelis | Short-Period Communication and the Role of Zero-Order Holding in Networked Control Systems[END_REF]. The communication sequence specifies which sensors are able to send information to the filter at each time step.

We will consider that the communication medium connecting the sensors and the residual generator has b output channels, with

1 ≤ b ≤ m (2)
At any time, only b of the m sensors can access these channels to communicate with the residual generator while others must wait.

A. Communication sequence

Suppose that there are m different sensors and that at each time step k only b < m are allowed to transmit messages. We then have σ

= C b m = m! b!(m-b)! possible configurations.
Let us introduce the application: µ k : Z → M = {1, . . ., σ }, that determine at each sample time the corresponding sensors group index. We call this application the switching pattern for the sensor side. In Fig. 1, the signal ỹk ∈ R b is related to y k by the following relation: ỹk = S(µ k )y k . The switch matrix S(µ k ) ∈ R b×m is used to select the subset of measures that will be sent to the controller at each time step k. This subset is indexed by the values of the switching pattern µ k .

Considering the band limited effect, the extended plant model is described by

x k+1 = Ax k + Bu k + Fϒ k + w k ỹk = C µ k x k + ṽk (3)
where

C µ k = S(µ k )C and ṽk = S(µ k )v k .

B. Fault detection filter

The fault detection and isolation filter proposed in this paper is a modified version of the filter proposed in [START_REF] Keller | Fault isolation filter design for linear stochastic systems[END_REF] The state space representation of this filter is given by

xk+1 = A xk + Bu k + K k ( ỹk -ŷk ) ŷk = C µ k xk (4)
where xk and ŷk denote the state and output estimation vectors, respectively.

It is important to note that in the context considered in this paper, both the filter gain K k and the switching pattern µ(k) are design parameters.

III. MAIN RESULTS

In this section, we will introduce our main results. Before doing this, we will first recall some basic definitions that will be used in the sequel.

Definition 1. [22]

The linear stochastic system (1) is said to have fault detectability indexes ρ

= {ρ 1 , ρ 2 , • • • , ρ q } if ρ i = min{ν : CA ν-1 f i = 0, ν = 1, 2, ...}.
Definition 2. The time-varying fault detectability matrix associated with the extended plant is defined as

D µ k = C µ k Ψ (5)
where

Ψ = [F 1 , AF 2 , ..., A s-1 F s ] (6)
Let

s = max{ρ i , i = 1, 2, • • • , q} be the maximum value of fault detectability indexes. We define Ῡk = [ Ῡ1T k , Ῡ2T k , ..., ῩsT k ] T , F = [ F1 , F2 , ..., Fs ]. Where Ῡi ∈ ℜ q i
represents the part of faults having detectability index ρ i and distribution matrix Fi ∈ R n,q i . The extended system can be equivalently rewritten as

x k+1 = Ax k + Bu k + F Ῡk + w k ỹk = C µ k x k + ṽk
Consider the filter given by ( 4). The estimation error e k = (x kxk ) and the output residuals q k = ( ỹkŷk ) dynamics are given by

e k+1 = (A -K k C µ k )e k + F Ῡk -K k ṽk + w k , q k = C µ k e k + ṽk (7)
From superposition principle, it follows that for an additive faults occurring at time instant r (with k > r + s), the output residuals q k can be expressed as:

q k = qk + ρk,r ηT r • • • ηT k-s • • • ηT k-2 ηT k-1 T (8) with ρk,r = C µ k G k-1,r F • • • G k-1,k-(s-1) F • • • G k-1 F F G k-1,k-j = G k-1 G k-2 • • • G k-j G k = A -K k C µ k
and qk corresponds to the output residuals for the non faulty case.

Following similar arguments as in [START_REF] Keller | Fault isolation filter design for linear stochastic systems[END_REF], the following result is derived.

Proposition 1. (Fault isolability condition)

Under the condition rank(C µ k ) = q, the solutions of the algebraic constraints:

(A -K k C µ k ) Ψ = 0 can be parameterized as K k = ωΠ µ k + Kk Σ µ k with Σ µ k = α µ k (I m -D µ k Π µ k ), Π µ k = D + µ k and ω = A Ψ (9)
where Kk ∈ ℜ n,b-q is the reduced gain describing the remaining freedom of design, D + µ k is the generalized inverse or pseudo-inverse of D µ k and α µ k is an arbitrary matrix determined so that matrix Σ µ k is of full rows rank. Under these conditions, the residual q k can then be expressed as

q k = qk + D µ k η1T k-1 η2T k-2 • • • ηsT k-s T ( 10
)
Remark 1. In the result given above, it is important to recall that the matrices C k depend on the switching pattern µ k . µ k being a design parameter, it follows that the switching patterns that contains sequences which violate the rank condition in Proposition 1 have to be excluded. Hence, let us define the set of admissible switching patterns Ξ * given by

Ξ * = {µ k : Z → M * ⊆ M } (11)
where M * contains the indices corresponding to sensor configurations (traduced by corresponding matrices C µ k ) that verify the rank condition rank(C µ k ) = q.

Based on the development above, the fault isolation filter can be designed by computing the free parameter Kk so that the trace of covariance matrix Pk = E( ēk ēT k ) is minimized.

Proposition 2. (Fault isolation filter design) For a fixed switching pattern µ k ∈ Ξ * , The proposed fault detection filter described by the following relations:

xk+1 = A xk + Bu k + ωq r k + Kµ k γ k (12) Pk+1 = ( Āµ k -Kµ k Cµ k ) Pk ( Āµ k -Kµ k Cµ k ) T + Kµ k Vµ k KT µ k + Wµ k = φ µ k ( Pk ) (13)
Kµ k = Āµ k + Pk CT µ k ( Cµ k Pk CT µ k + Vµ k ) -1 (14)
with

Āµ k = A -ωΠ µ k C µ k (15) Cµ k = Σ µ k C µ k (16) Vµ k = Σ µ k Σ T µ k , ( 17
) Wµ k = W + ωΠ µ k Π T µ k ω T ( 18
)
where

γ k = Σ µ k ( ỹk -C µ k xk ) (19) q r k = Π µ k ( ỹk -C µ k xk ) (20)
have the following properties

• γ k is decoupled from the faults • q r k satisfy the relation

q r k = Π µ k qk + η1T k-1 η2T k-2 • • • ηsT k-s T ( 21
)
Each component of the reduced output residual q r k ∈ ℜ q is sensitive to only one fault. Thus, it is used for the fault isolation.

Proof. The proof of this proposition follows similar arguments as for the proof of Theorem 3.1 in [START_REF] Keller | Fault isolation filter design for linear stochastic systems[END_REF].

One can see that the evolution of the covariance matrix given by the Riccati equation ( 13) depends on the initial covariance matrix P 0 and the switching pattern given by µ k . Hence, in addition to the isolability condition (see Remark 1), the scheduling strategy can be generated to optimize the covariance matrix evolution. This point will be further exposed in the next section.

IV. FINITE HORIZON OPTIMAL SCHEDULING

The problem addressed here is to choose which b sensors should operate at each time-step to minimize a function of the error covariance of the state estimation at each time step. Defining the scheduling strategy s N is equivalent to define the values of µ k for each k = 0,

• • • , N -1, or equivalently s N = µ 0 µ 1 • • • µ N-1 . Let S N = M N
be the set of all possible N-horizon scheduling strategies and let S * N be the set of all admissible N-horizon scheduling strategies (see Remark 1). The problem of optimal scheduling is formulated as min

s N ∈S * N J (s N ) (22)
where

J (s N ) = ∑ N i=1 tr( Pi ) = ∑ N-1 i=0 tr(φ µ i ( Pi )) and µ i = s N (i).

The search algorithm:

Search algorithms are used for solving optimization problem [START_REF] Keller | Fault isolation filter design for linear stochastic systems[END_REF]. The trivial way of solution is to perform all possible scheduling cases. This enumerating method is only tractable for relatively short time horizons. It requires much resources in memory and computational time for longer estimation horizons. To overcome this limitation, we will use in this paper a pruning technique proposed in [START_REF] Vitus | On efficient sensor scheduling for linear dynamical systems[END_REF]. As showed in [START_REF] Vitus | On efficient sensor scheduling for linear dynamical systems[END_REF], the proposed algorithm can significantly reduce the computation complexity. Before proceeding, we will first recall some definitions to ease the reading of the paper.

Definition 3. (Characteristic sets)

Let {H k } N k=0 be defined as the characteristic sets as they completely characterize the objective function. Each set is of the form ( P, γ) ∈ A × R + (A being the set of all symmetric positive semidefinite matrices) and is generated recursively by

H k+1 = π M * (H k ) from H 0 = {( P0 , tr( P0 ))} with π M * (H ) = {(φ i ( P), γ + tr(φ i ( P)) : ∀i ∈ M * , ∀( P, γ) ∈ H }
Note that the above definition differs from the original one in [START_REF] Vitus | On efficient sensor scheduling for linear dynamical systems[END_REF] by using M * instead of M . This is due to the fault isolability constraints in our context.

The sets H k , k = 1, • • • N, express the covariance of the estimate and the objective cost at every time-step under every possible sensor schedule. Let H k (i) be the i th element of H k , Pk (i) and γ k (i) be the covariance matrix and objective cost corresponding to H k (i), M * k the set of all ordered sequences of admissible (in terms of isolability constraint) sensor schedules of length k, λ ( Pk (i)) ∈ M * k be the ordered sensor schedule corresponding to the covariance matrix Pk (i) and λ * be the optimal sensor schedule for the problem. [START_REF] Vitus | On efficient sensor scheduling for linear dynamical systems[END_REF] A pair ( P, γ) ∈ H is called algebraically redundant with respect to H \{( P, γ)}, if there exist nonnegative constants

Definition 4. (Algebraic redundancy)

{α i } l-1 i=1 such that l-1 ∑ i=1 α i = 1 and P 0 0 γ l-1 ∑ i=1 α i P(i) 0 0 γ(i)
where l = card(H ) and {( P(i), γ(i))} l-1 i=1 is an enumeration of H \{( P, γ)}.

The following theorem provides a condition which characterizes the branches that can be pruned without eliminating the optimal solution of the sensor scheduling problem.

Theorem 1. [START_REF] Vitus | On efficient sensor scheduling for linear dynamical systems[END_REF] If the pair ( P, γ) ∈ H k is algebraically redundant, then the branch and all of its descendants can be pruned without eliminating the optimal solution from the search tree.

We are now in position to describe the sensor scheduling algorithm. Before doing this, let us recall the notion of equivalent subset of the search tree. This one is defined as a set that still contains the optimal sensor schedule after pruning, the pruning being realized using Theorem 1. The computation of the equivalent subsets is done via Algorithm 1 in [START_REF] Vitus | On efficient sensor scheduling for linear dynamical systems[END_REF] The sensor scheduling algorithm is given as follows Algorithm 2. Sensor scheduling for a finite horizon i)

H 0 = {( P0 , tr( P0 ))} ii) for k = 1, • • • , N, do -H k = π M * (H k-1 ) -Perform Algorithm 1 in [18] with H k end for iii) λ * = arg min j∈{1,•••,card(H N )} J (λ (H N ( j)))
Remark 2. In [START_REF] Vitus | On efficient sensor scheduling for linear dynamical systems[END_REF], the authors proposed a suboptimal solution which consists in approximating the search tree by pruning branches which are numerically redundant. To this end, they use the notion of ε-redundancy. As pointed out by the authors, the ε-redundancy concept can typically eliminate many more branches of the search tree leading to less complexity problems.

V. ILLUSTRATIVE EXAMPLE

We consider the following discrete-time system

A =     0.2 1 0 0.2 0 0.1 1 0.4 0 0 0.4 1 0 0 0 0.3     , F =     -1 1 1 0 0 -1 1 1     C =   0 0.7 0 0 0 0 0.2 0 0 0 0 5   , V = I W =     0.89 0 0 0 0 0.5 0 0 0 0 0.95 0 0 0 0 0.58    
The fault associated to the first column of the matrix F occurs at time instant r 1 = 50, with η 1 k = 10 sin(0.1k), while the second fault (associated to the second column of F) occurs at time r 2 = 120 with η 2 k = 5.

We use the suboptimal version of Algorithm 2 (see Remark 2) to compute the suboptimal sensor schedule. Figure 2 shows the reduced output residuals q r k = q r1 k q r2 k T , in the case of using the suboptimal sensor schedule sequence and an arbitrary periodic schedule, respectively . One can see that based on the proposed filter, one has the possibility to detect and isolate multiple faults.

Figure 3 shows the evolution of the trace of the covariance matrix of the estimation error for the two cases: suboptimal sensor schedule and arbitrary periodic schedule. One can see clearly the advantage and the performance of the proposed method.

VI. CONCLUSIONS

This paper addresses the problem of multiple fault detection and isolation under communication constraints. More specifically, we have considered the issue of sensor scheduling and fault isolation co-design under limited bandwidth capacity. We have proposed a detection and isolation filter in addition with an optimal (or suboptimal) sensor scheduling sequence that ensure the fault isolability property and noise effect minimization. Future directions of research will include the infinite horizon case and extension to online scheduling techniques.

Fig. 1 .

 1 Fig. 1. Remote filtering

Fig. 3 .

 3 Fig. 2. Reduced output residuals