on-line diagnosis and uncertainty management using evidence theory––experimental illustration to...

Journal of Process Control 14 (2004) 747–763

www.elsevier.com/locate/jprocont

On-line diagnosis and uncertainty management usingevidence theory––experimental illustration to anaerobic

digestion processes

Laurent Lardon, Ana Punal, Jean-Philippe Steyer *

Laboratoire de Biotechnologie de l’Environnement, INRA, Avenue des �etangs, 11100 Narbonne, France

Available online

Abstract

The on-line diagnosis is a key requirement in biological processes. This is particularly true in the case of wastewater treatment

processes due to the composition of media, the requirements of operating conditions and the wide variety of possible disturbances

that necessitate careful and constant monitoring of the processes. Moreover, because only partial information is available in an on-

line context and because of the technical and biological complexities of the involved processes, specific characteristics are required

for diagnosis purposes. Several approaches like quantitative model based, qualitative model based and process history based

methods were applied over the years. This paper present a methodological framework based on evidence theory to manage the fault

signals generated by conventional approaches (i.e., residuals from hardware and software redundancies, fuzzy logic based modules

for process state assessment) and to account for uncertainty. The advantages of using evidence theory like modularity, detection of

conflict and doubt in the information sources are illustrated with experimental results from a 1 m3 fixed bed anaerobic digestion

process used for the treatment of industrial distillery wastewater.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Fault detection and isolation; Diagnosis; Evidence theory; Biological process; Anaerobic digestion

1. Introduction

More than their control, the diagnosis (i.e., detection,

isolation and analysis) of faults occurring in biological

processes has become a challenging research area. In-deed, several types of disturbances like influence of the

inoculate, contamination of the media, presence of toxic

in the feeding line, fouling of sensors, can be present even

in normal operational conditions. These disturbances

can largely affect the process operation and damage the

quality of the end products. Moreover, these distur-

bances can be either sudden or slow and they can be

related to normal or faulty process operation provokingreal or apparent deviations from the normal operation.

Hence, there is a clear need for advanced supervisory

control (i.e., gathering on-line control and diagnosis) in

*Corresponding author. Tel.: +33-468-425-151; fax: +33-468-425-

160.

E-mail addresses: [email protected] (L. Lardon), punal@

ensam.inra.fr (A. Punal), [email protected] (J.-P. Steyer).

0959-1524/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2003.12.007

order to keep the system performance as close as pos-

sible to optimal. This is particularly true for biological

processes with environmental purposes like WasteWater

Treatment Plants (WWTPs) where the state of the

‘‘living’’ part of the system is to be closely monitoredtogether with large possible disturbances occurring on

any part of the systems.

In the present study, anaerobic digestion has been

chosen as an illustrative example of biological WWTPs.

Anaerobic digestion (AD) is a serie of biological

processes that take place in the absence of oxygen and

by which organic matter is decomposed and converted

into biogas, a mixture of mainly carbon dioxide andmethane, microbial biomass and residual organic mat-

ter.

Several advantages are recognised to AD processes

when used in WWTPs: high capacity to treat slowly

degradable substrates at high concentrations, very low

sludge production, potentiality for production of valu-

able intermediate metabolites, low energy requirements

and possibility for energy recovery through methanecombustion. AD is indeed one of the most promising

mail to: [email protected]

748 L. Lardon et al. / Journal of Process Control 14 (2004) 747–763

options for delivery of alternative renewable energy

carriers, such as hydrogen, through conversion of

methane, direct production of hydrogen, or conversion

of by-product streams.However, despite of these large interests and more

than 1400 commercial installations refereed world-wide

in 1999 [1], many industries are still reluctant to use AD

processes, probably because of the counterpart of their

efficiency: they can become unstable under some cir-

cumstances. Hence, actual research aims not only to

extend the potentialities of anaerobic digestion [2], but

also to optimise AD processes and increase theirrobustness towards disturbances [3].

Throughout our studies, it has been experimentally

demonstrated that AD processes could be very effi-

ciently controlled despite large changes in the influent

concentrations. Important organic loading rates (more

than 100 kgCOD/m3 d) were indeed achieved while keep-

ing the carbon removal above 75% [4]. Nearly perfect

control of intermediate products (e.g., volatile fattyacids and alkalinity) or end products (e.g., residual

pollution, total biogas produced or methane flowrate)

was also possible over a long period of time while

minimising the rejected pollution (expressed in terms of

chemical oxygen demand at the output of the process).

Several control approaches were used, each of them

demonstrating several interesting characteristics for the

control of AD processes [5]:

• PID and fuzzy logic [6–8],

• artificial neural networks [9],

• non-parametric adaptive control [10],

• adaptive control [11],

• disturbance accommodating control [12],

• interval based approaches [13],

• robust output feedback [14,15].

However, these studies were only devoted to lab or

pilot scale digesters. In addition, it is important to note

that all these control laws were to meet the specific

objective they have been designed for. As a consequence,

it is not to be expected that, for example, a control law

will manage a technical problem in the feeding circuit

(e.g., a clogging of a pipe) if its goal is to control volatilefatty acids in the output of the reactor. In addition––and

since there does not exist an ‘‘universal’’ control law that

could manage all the disturbances occurring on a pro-

cess––it is mandatory to couple control laws with ad-

vanced diagnosis scheme. It is felt that it is the only

possible way for achieving successful optimisation of

AD processes at industrial scale.

In the past, we tackled these objectives using quan-titative [16,17] and qualitative [18,19] model based

diagnosis approaches, sometimes combining them to-

gether [20] or with process history based methods [21].

Nevertheless, it is our strong belief that a unified ap-

proach––based on evidence theory––could be of great

help for overall optimisation of AD processes.

This paper will present this approach and is organised

as follows. Next section details the diagnosis problemstatement with specific emphasis on WWTPs in general

and AD processes in particular. Then, our motivations

are presented and the scientific basis of evidence theory

is briefly underlined. Before concluding, experimental

results obtained on a pilot scale fixed bed AD process

are described and discussed.

2. Problem statement

From a very general prospect, timely detection,

diagnosis and correction of abnormal conditions of

faults in a process are the central components ofabnormal event management.

There is an abundant literature on process fault

diagnosis ranging from analytical methods to artificial

intelligence and statistical approaches. From a model-

ling perspective, accurate quantitative, semi-quantitative

or qualitative models can be required. At the other end

of the spectrum, there are methods that do not assume

any form of model information and rely only on historicprocess data. It is however out of the scope of the

present paper to review all these methods but the reader

interested in could refer to a serie of three very inter-

esting review papers by Venkatasubramanian et al. [22–

24].

Several desirable characteristics can be identified for a

diagnostic system:

• quick detection and diagnosis,

• isolability,

• robustness,

• novelty identifiability,

• classification error estimate,

• adaptability,

• explanation facility,

• modelling requirements,• storage and computational requirements,

• multiple fault identifiability.

These characteristics were introduced by [22] to

evaluate and to compare the various diagnosis ap-

proaches (i.e., quantitative model based, qualitative

model based and process history based) for any possible

application. In the following, these requirements arespecifically explicated for bioprocesses in general and

AD systems in particular.

When dealing with biological processes, some of these

characteristics are indeed more important than other

ones. For example, since these processes can be con-

sidered as slow systems (typical time constants range

from minutes to months in WWTPs), storage and

L. Lardon et al. / Journal of Process Control 14 (2004) 747–763 749

computational requirements are not very high and for

sure, they are not the limiting steps.

In the same way, isolability, even though being a

non-trivial task, can be solved implementing appropri-ate on-line sensors that provide the diagnosis scheme

with relevant information. Moreover, when hardware

sensors are not available (due to technological or eco-

nomical reasons), they can be replaced by software

sensors, using mathematical models. This (i.e., imple-

mentation of hard and/or soft sensors) is the most effi-

cient––and may be the only––way to improve fault

isolability.On the other side, WWTPs are continuous processes

working 24 hours a day and 365 days a year; thus their

robustness requirements are very high. These require-

ments are further reinforced by the length of the start-up

phase of a WWTP. For example, at the industrial scale,

it takes several weeks to several months to start-up an

AD process. AD processes indeed produce 5–10 times

less sludge than aerobic activated sludge processes. Thismeans that biomass growth is much slower in AD pro-

cesses, which is an advantage regarding the sludge pro-

duction but a drawback regarding the length of the

start-up phase. It is thus much more important to avoid

breakdowns of AD processes than to achieve high per-

formances in normal operating conditions. Quick

detection and diagnosis together with robustness are

clearly two key issues. However, quick response tofailure and tolerable performance during normal oper-

ation are two conflicting goals especially in model-based

diagnosis approaches [25]. A trade-off between the

robustness and quick detection is thus needed and

robustness is often prioritised for AD processes.

Another important requirement is adaptability. In-

deed, natural inoculate used when starting up an AD

process may have completely different dynamic behav-iours fewmonths later. This is an advantage when dealing

with complex compounds that cannot be degraded in

aerobic conditions [26] but lag periods of few weeks to

few months are sometimes needed prior quantifiable

degradation can be noticed (see for example [27] for

anthropogenic phtalic acid isomers). As a consequence,

any control or diagnosis system must present large

adaptability to handle process changes as they develop

Fig. 1. Pictures of carrier (Cloisonyle�) of the pilot-scale fixed bed AD proce

volume was 984 l in 1997 and only about 220 l in 2002.

with time. Another reason for this requirement is the

changes in wastewater composition that are very likely to

occur at the industrial scale. To date, AD is indeed the

dominant treatment method for brewery, distillery andnumerous food-processing wastewaters but these waste-

waters have very dynamic daily or seasonal compositions

that affect the process dynamics accordingly.

The need for adaptability is even more pronounced

for fixed biomass reactors like upflow anaerobic sludge

blanket, fixed bed or fluidised bed reactors. In these

configurations, larger biomass concentrations can be

achieved compared to the classical continuous stirredtank reactors, thus increasing volumetric treatment

capacity and lowering hydraulic retention time. How-

ever, along with biofilm development is the substrate

limitation within the biofilm, thus limiting substrate

availability for microorganisms [28,29] and calling for

adaptivity in the supervisory of these processes. As an

illustration, the working volume of the AD fixed bed

reactor that will be described in later sections wasoriginally 984 l (i.e., in 1997) whereas five years later,

only about 220 l were available due to biofilm formation

(see Fig. 1).

It is thus clear that adaptability is a key requirement

for any WWTP in general and AD processes in partic-

ular. One way to deal with this aspect are intrinsically

adaptive schemes that have been demonstrated as a very

powerful approach for AD processes [30–40]. Anotherway is to directly account for the uncertainty in an

interval-based approach, either in the case of observers

[41], controllers [42,43] or diagnosis [44].

However, to be applicable at the industrial scale, it is

felt that modelling efforts should be as low as possible in

any of the approaches chosen to tackle either control or

diagnosis objectives. Building up a model for a biopro-

cess is indeed a tedious task that requires long term ef-forts. When specifically concerned with AD processes,

several studies have been performed since the early

seventies [45–47] to obtained mathematical models able

to handle dynamic behaviour in transient phases and to

provide useful insights about the internal functioning of

the processes [48–66]. However, the obtained models can

be very complex (i.e., up to 26 dynamic state concen-

tration variables and 19 biochemical kinetic processes as

ss used in the present study and associated biofilm. The working liquid


in [67]), making their use in on-line control and/or

diagnosis scheme very difficult, not to say impossible.

Moreover, despite the very large number of models

specifically developed for AD processes, none of themwere developed up to now to handle abnormal situa-

tions, which is basically the objective of diagnosis.

Related to this aspect is novelty identifiability. In-

deed, when an unknown fault occurs, one would like the

diagnosis system to be able to recognise the occurrence

of novel faults and not misclassify them as one of the

other known malfunctions or as normal operation. This

is even more required when concerned with remotemonitoring aspects [68]. Indeed, very often, small and

medium enterprises (SMEs) cannot afford the cost of a

human expert available on a daily basis to run their AD

processes. In such a case, it is important that the diag-

nosis system is able to indicate when it is impossible to

take a decision when an unknown fault occurs. In other

words, an appropriate fault detection output for

WWTPs should be a triple indicating that ‘‘everything isok’’, ‘‘there is a fault’’ or ‘‘I don’t know’’ (i.e., the fault is

unknown). In this last case, safety procedures could be

temporarily activated and/or a remote expert could be

called to handle these abnormal situations.

Another key but difficult requirement for diagnosis

systems is multiple fault identifiability. In cases where

different faults occur at the same time, it is indeed

desirable for the diagnosis system to handle each faultseparately while not forgetting any. One way to handle

these situations is to separate faults in different catego-

ries (e.g., sensors and actuators faults, subprocess faults

and bioprocess faults) and to manage them in a hierar-

chical structure as it is the case in [18]. However, in this

approach, deep knowledge of the process is required and

one would prefer a more systematic and universal ap-

proach.

X

Y

Z Operating conditions

Area where fault #ican be detected

fault #i

a) Non optimised diagnosis system

Fig. 2. Comparison between non-optimised and optimised diagnosis system.

difficult in case (a) if the area where another fault is detected overlaps the a

Last but not least, human dimension should be ac-

counted for. This is particularly true for WWTPs (and

AD processes in particular) since human operators in

charge of their daily functioning have usually lowqualification. This implies that an appropriated diag-

nosis system should propose explanation facility to

justify why certain hypotheses are proposed but also to

explain why certain other hypotheses are not. Also,

classification error estimates should be described to

build up user’s confidence while indicating them the

urgency of the fault.

3. Key requirements for design of AD fault detection anddiagnosis systems

To summarise, two key requirements are of prime

importance for the diagnosis of AD processes: uncer-

tainty management and modularity in the design of the

diagnosis system. This allows one (i) to handle the poor

knowledge available on-line about the internal func-

tioning of the process (due to the lack of on-line sensors,to the simplicity of the models usable in a real-time

context, to the large unmeasured disturbances occurring

at the input, etc.) and (ii) to account for adaptativity,

novelty and multiple fault identifiability requirements as

previously explained.

These motivations are further explained in the fol-

lowing. In fact, assuming a fault space defined by

three variables (X , Y and Z), our objective is to build adiagnosis system able to detect a fault from partial

information about the operating conditions (cf. Fig. 2).

This means that the precise location of the fault is

impossible to achieve (i.e., only a volume surrounding

the fault will be detected in the 3D dimension fX ; Y ; Zg).Moreover, without optimisation of the diagnosis system,

X

Y

Z Operating conditions

Area where fault #ican be detected

fault #i

b) Optimised diagnosis system

In both cases, fault #i is correctly detected but isolation could be more

rea associated to detection of fault #i.


this could lead to bad isolation of the precise fault (e.g.,

if the area where another fault is detected overlaps the

area associated to the detection of fault #i).To achieve this optimisation, one natural way is to

account for all the information available in a single

diagnosis system (cf. Fig. 3). This is of course possible

but this would also lead to a high complexity of the

diagnosis system in cases of high dimension of the fault

space (as it is the case in AD processes). Moreover,

adding or removing one dimension (e.g., one sensor)

could break down the overall structure of the diagnosis

system and lead to false alarms or wrong diagnosis.This is why a modular diagnosis system is preferable

as it is shown in Fig. 4. In this case, separate fault

detection systems are built, each of them handling only

partial information on the process. This is in fact similar

to different persons analysing the same situation but

with different point-of-views and/or different sources of

X

Y

Z

Fault signal = f(X,Y,Z)Diagnosis system

analysingall variables together

Fig. 3. Structure of an overall diagnosis system accounting for all the

available sources of information at the same time.

X

Y

Fault signal = f1(X,Y)Fault detectionsystem

analysingonly X and Y

Y

Z

Fault signal = f2(Y,Z)Fault detectionsystem

analysingonly Y and Z

X

Z

Fault signal = f3(X,Z)Fault detectionsystem

analysingonly X and Z

Fig. 4. Structure of a modu

X

Z

X

Y

Z

(a) (b)

Fig. 5. Outputs of the fault detection system in a modular diagnosis schem

(b) Y and Z, (c) X and Z.

information. Of course, these fault detection systems are

to be further combined within an overall system called

here a ‘‘state manager’’.

The role of the state manager is to cross-check thefault signals to gain more insights about the faulty sit-

uation (i.e., in Fig. 4, if f1ðX ; Y Þ says that the fault is

either A or B while f2ðY ; ZÞ says it is either B or C, then it

can be reasonably assumed that fault B is present).

But the role of the state manager is also to detect

conflict and doubt about the fault signals. For example,

in case where f1ðX ; Y Þ ¼ fA;Bg and f2ðY ; ZÞ ¼ fC;Dg,then it could be a conflict (i.e., f ðX ; Y ; ZÞ ¼ ;) oruncertainty about the current situation (i.e., f ðX ; Y ; ZÞ ¼fA;B;C;Dg). Conflict is present for example in the case

where one or more sensors are miscalibreated or present

fouling whereas doubt appears when not enough on-line

sensors are used, thus calling for off-line manual

measurements to distinguish among the possible situa-

tions.

Fig. 5 illustrates this modular diagnosis scheme onthe previous 3D example. In fact, each fault detection

system analyses the situation in a 2D dimension (i.e., the

grey parts in Fig. 5) and projects the determined surface

in the third dimension without any assumption on the

non-analysed variable. The role of the state manager is

then simply to manage the intersection of these volumes.

State manager

Fault signal = f(X,Y,Z)

lar diagnosis system.

Y

X

Y

Z

(c)

e. Fault detection output when analyzing only variables (a) X and Y ,


As it is shown in the following, the evidence theory

offers a theoretical framework particularly suited to

tackle the objectives of the state manager. In addition,

the evidence theory is independent of the method usedto determine the fault signals. As illustrated in the

experimental section, the evidence theory can indeed

manage (i) measurements analysed by different fuzzy

logic based modules used to assess the state of the

process or (ii) residuals (i.e., fault signals) generated

from the difference between two redundant hardware

sensors or between the measurement provided by a

sensor and the estimated variable obtained from amathematical model. As a consequence, evidence theory

should not be seen as a fault signal generation method

but as a fault signal management methodology, the

main advantages being its modularity, its robustness

and its capability to detect conflict and doubt in the

information sources.

1 1 1

4. The evidence theory

Evidence theory, first introduced by Dempster and

later formalised by Shafer, allows one to manipulate

non-necessarily exclusive events and thus to represent

explicitly process uncertainty [69]. This theory assumes

the definition of (i) a frame of discernment X consisting

of the exhaustive and exclusive hypothesis and (ii) thereference set 2X of all the disjunctions of the elements

of X.When used for diagnosis purposes, the frame of dis-

cernment includes singletons of all the possible faults

that could occur on a process while the reference set

contains all the possible combinations of faults.

In the evidence theory, a basic belief assignment (bba)

is an elementary mass function m : 2X ! ½0; 1� verifying:XfA22X=A�X and A6¼;g

mðAÞ6 1: ð1Þ

The elements of 2X whose mass is non-zero are called

focal elements of m and constitute the core Nm of thebelief assignment.

In a diagnosis context, a mass function will be for

example a fault signal. In the more general framework

of evidence theory, a bba is a distribution of a unit mass

of evidence among all the elements of 2X. In the evidence

theory, a piece of evidence is indeed distributed between

all subsets of X rather than between the elements of X as

it is the case in probability theory. Moreover, in theabsence of disjunction as focal element, the mass of a

singleton can be understood as its probability. As a

consequence, a bba represents the exact knowledge of an

information source and there is no hypothesis to make

when two situations or events cannot be distinguished.

A bba is also characterised by two functions: the

belief function denoted Bel and the plausibility function

denoted Pl. The belief in a subset A is the sum of all

pieces of evidence that support A and the plausibility of

A the sum of pieces of evidence not supporting :A:

8A 2 2XBelmðAÞ ¼

PfB22X=B�Ag

mðBÞ;

PlmðAÞ ¼P

fB2ð2XÞ=B 6� :AgmðBÞ ¼ 1� BelðAÞ;

8><>:

ð2ÞThe interval ½BelðAÞ;PlðAÞ� can thus be considered as

bounds of the unknown probability of A.Based on these definitions, combination operators

can be defined. For example, it is possible to build a

unique elementary mass function m from two elemen-

tary mass functions m1 and m2, arising from two distinct

and independent sources but defined on the same set,

such that m ¼ m1 � m2 where � denotes the combina-tion operator. The so-called Dempster’s rule consists of

calculating:

8C 2 2X=f;g

mXðCÞ ¼ 1

1� K

XfðA;BÞ2ð2XÞ2=A\B¼Cg

mX1 ðAÞmX

2 ðBÞ;

K ¼X

fðA;BÞ2ð2XÞ2=A\B¼;g

mX1 ðAÞmX

2 ðBÞ:

ð3Þ

Let us here take a simple example that, even though

being not related to biological processes, illustrates thesenotions and can help to understand the evidence theory.

Let us suppose that a policeman analyses a crime

scene. Two persons (i.e., W1 and W2) witnessed the scene

while three persons are suspected to have committed the

crime (i.e., S1, S2 and S3). The suspects can be differen-

tiated using physical characteristics: S1 is tall with brown

hair, S2 is small with brown hair while S3 is tall and

blond.In this case, the objective is to find the criminal. As

a consequence, X ¼ fS1; S2; S3g and 2X ¼ fS1; S2; S3;S1[ S2; S1 [ S3; S2 [ S3; S1 [ S2 [ S3g.

Let us also assume that the two witnesses give the

following description of the scene: W1 says that the guilty

person was tall with a probability of 90% and W2 says

that the guilty person had brown hair with a probability

of 80%.These two independent sources of information can be

described by the following belief functions:

mW1ðS1 [ S3Þ ¼ 0:9 and mW1

ðS1 [ S2 [ S3Þ ¼ 0:1;


ðS1 [ S2 [ S3Þ ¼ 0:2:

In other words, W1 says that the guilty person is most

probably S1 or S3 while W2 says that it is either S1 or S2.If we compute the belief and plausibility functions, we

get:

BelW1ðS1Þ ¼ 0; BelW1

ðS2Þ ¼ 0; BelW1ðS3Þ ¼ 0;

PlW ðS1Þ ¼ 1; PlW ðS2Þ ¼ 0:1; PlW ðS3Þ ¼ 1

Fig. 6. Illustration of information combination with the example of

the crime scene analysis.


and

BelW2ðS1Þ ¼ 0; BelW2

ðS2Þ ¼ 0; BelW2ðS3Þ ¼ 0;

PlW2ðS1Þ ¼ 1; PlW2

ðS2Þ ¼ 1; PlW2ðS3Þ ¼ 0:2:

Combination of the information provided by the two

witnesses is illustrated in Fig. 6.

The combination of mW1and mW2

gives the following

new belief function:

mW1;W2¼ mW1

� mW2

with

mW1;W2ðS1Þ ¼ 0:72;

mW1;W2ðS1 [ S3Þ ¼ 0:18;

mW1;W2ðS1 [ S2Þ ¼ 0:08;

and

mW1;W2ðS1 [ S2 [ S3Þ ¼ 0:02:

In this case, there is no conflict between both sources

of information and the combination leads to:

BelW1;W2ðS1Þ ¼ 0:72; PlW1;W2

ðS1Þ ¼ 1;

BelW1;W2ðS2Þ ¼ 0; PlW1;W2

ðS2Þ ¼ 0:1;

BelW1;W2ðS3Þ ¼ 0; PlW1;W2

ðS3Þ ¼ 0:2:

This example is an ideal case since there is no conflict

between the different sources of information (i.e., the

two witnesses) and there are no plausibility functions

higher than the highest belief function. In this case, the

decision making process is very simple and S1 can be

assessed to be guilty.

However, this is not always the case and situations canarise where empty conjunctions of focal elements are

present (i.e., Ai \ Bj ¼ ; in the general framework). In-

deed, by definition, the mass assigned to the empty set

must be zero and the sum of the masses of the focal

elements must be equal to one. To comply with these two

conditions, a renormalisation is carried over the com-

plete total mass not assigned to the empty set. This

operation (known as the Dempster’s rule) is open to

criticism since it entails two drawbacks. On the one hand,

it has the effect of masking the aspect of conflict of the

sources in question. Hence, there is a loss of information.

On the other hand, when the conflict is great, renor-malisation may lead to counter-intuitive results [70].

To solve this problem, several other combination

rules have been defined and they often differ by the way

the mass of evidence of an empty intersection is allo-

cated. The choice of the combination rule reflects the

interpretation of the mass allocated to the empty set.

For example, the Smets’ combination rule assumes that

the sources are reliable and that the conflict betweenthem can stem only from one or more hypotheses not

having been taken into account in the frame of dis-

cernment [71,72]. In other words, this combination rule

consists to assign the conflicting mass to the empty set,

which is interpreted as a reject class:

8C 2 2X mXðCÞ ¼X

fðA;BÞ2ð2XÞ2=A\B¼Cg

mX1 ðAÞmX

2 ðBÞ: ð4Þ

Another view is to see the frame of discernment as

certain, so the conflict can stem only from a wrong

information source. This is the case of the Yager’s

method. This combination rule performs a partial dis-

junctive combination where the conflicting mass is

allocated to X [73]:

8C 2 2X mðCÞ ¼P

A\B¼Cm1ðAÞ � m2ðBÞ;

mðfXgÞ ¼ m1ðXÞ � m2ðXÞ þP

A\B¼;m1ðAÞ � m2ðBÞ:

8<: ð5Þ

Compared to the Dempster’s rule, there is no risk of

non-linear comportment caused by the normalisation

factor and the conflict is explicitly represented by theresulting mass of X.

Moreover, in practice, it is possible to have sources of

information whose frames of discernment X and X0 are

different but compatible. To combine and merge these

sources, relationships between the frames of discernment

have to be defined. To this end, two operations––

refinement and coarsening––express the correspondences

in the form of compatibility rules. In fact, a refinementassociates a set of compatible elements of X0 to an ele-

ment of X, and a coarsening is the antagonist relation.

For more details about the theory of evidence, the

reader can refer to [74,75]. A comparison with boolean

and fuzzy logic based reasoning in a diagnosis frame-

work can also be found in [76].

Finally, the last step is the decision making process

which is supported by the results provided by the com-bination rules. Indeed, as previously highlighted, the

combination of the available sources of information

provides us with a new belief function which represents

the most reliable and complete information. However, if

the choice of the most likely hypothesis is straightfor-

ward in the probabilistic framework, it can become quite

complex in the evidence theory.

Table 1

Composition of the raw wine distillery wastewater used in this study


The decision can be based on the maximum of belief

or on the maximum of plausibility, which are respec-

tively pessimistic and optimistic decision rules. Another

method proposed by [75] consists to share equally themass of a proposition between all singletons of which it

is composed. The new function is a probability reparti-

tion called pignistic probability and defined as:

8A 2 X PigðAÞ ¼X

fB22X=A�Bg

mðBÞcardðBÞ :

In this case, the decision will be based on the singleton

with the maximal pignistic probability.If we go back to the simple example previously pre-

sented and if we assume now that the first witness says

‘‘I am sure he had brown hair and, with a probability of

70%, that he was tall’’ while the second witness says

‘‘with a probability of 80%, he had brown hair and, with

a probability of 20%, he had blond hair’’, we obtain:


ðS1Þ ¼ 0:7;


ðS3Þ ¼ 0:2:

And the combination rules provides us with the fol-

lowing results:

• using the Dempser’s rule:

mW1;W2ðS1Þ ¼ 0:7, mW1;W2

ðS1 [ S2Þ ¼ 0:3,BelW1;W2

ðS1Þ ¼ 0:7, PlW1;W2ðS1Þ ¼ 1, BelW1;W2

ðS2Þ ¼ 0,PlW1;W2

ðS2Þ ¼ 0:3,PigðS1Þ ¼ 0:85, PigðS2Þ ¼ 0:15;

• using the Smets’ rule:

mW1;W2ðS1Þ ¼ 0:56, mW1;W2

ðS1 [ S2Þ ¼ 0:24, mW1;W2ð;Þ ¼

0:2,BelW1;W2

ðS1Þ¼0:56, PlW1;W2ðS1Þ¼0:8, BelW1;W2

ðS2Þ¼0,

PlW1;W2ðS2Þ¼0:24, PigðS1Þ¼0:68, PigðS2Þ¼0:12;

• using the Yager’s rule:mW1;W2

ðS1Þ ¼ 0:56, mW1;W2ðS1 [ S2Þ ¼ 0:24, mW1;W2

ðXÞ ¼0:2,BelW1;W2

ðS1Þ ¼ 0:56, PlW1;W2ðS1Þ ¼ 1, BelW1;W2

ðS2Þ ¼ 0,

PlW1;W2ðS2Þ ¼ 0:44, PigðS1Þ ¼ 0:78, PigðS2Þ ¼ 0:22.

In this example, the three methods give different re-

sults but a decision based on the maximum of pignistic

probability is the same in all cases. However, it is to benoticed that for more complex cases and/or with higher

conflict levels, this is not always the case and the final

result can be sensitive to the chosen combinationmethod.

Concentrations Units

mean min max

Total COD 38.8 35.5 42.1 gO2/l

Soluble COD 36.2 32.5 39.8 gO2/l

TOC 11.5 11.2 11.8 g/l

Total VFAs 11.2 9.3 13 g/l

NTK 484.5 421 548 mgN-NTK/l

NHþ4 137 89.6 184 mgN-NHþ

4/l

pH 4.3 3.5 5.0

5. Results and discussion

The following illustrates the management of residual

generation from fuzzy fault detection modules using the

evidence theory. This is applied to the monitoring of a

pilot scale AD process under small (i.e., hydraulic

overload, low pH in the influent, low toxicant added)

and large disturbances (i.e., inhibition of the biomass

activity due to volatile fatty acids accumulation)

occurring separately and sometimes simultaneously.

The next section presents the wastewater used anddetails the process of configuration and the on-line

instrumentation available. Then, experimental data are

described before analysing them with the evidence the-

ory diagnosis approach previously presented.

Note: The Matlab� code source developed to imple-

ment the evidence theory in real time applications is

freely available from the authors.

5.1. Wastewater and process used in this study

Raw industrial distillery wastewaters obtained from

local wineries in the area of Narbonne, France, wereused in this study. Neither sterile nor homogeneous,

they have changing characteristics (see Table 1)

according to the wineries where the wastewater is taken

from. However, the main changes were done by diluting

the raw wastewater with water throughout the experi-

ments (dilution factor between 1 and 4).

The process is a pilot-scale up-flow anaerobic fixed

bed reactor made of a circular column of 3.5-m height,0.6-m diameter and an originally useful volume of 0.984

m3 (cf. Fig. 7). This process has a classical on-line

instrumentation gathering measurements of liquid flow

rates (at the input of the reactor and in the recirculation

loop), temperature and pH in the reactor and biogas

flowrate and composition (i.e., CO2, CH4 and H2 con-

tent in the biogas) every 2 min. In addition, a FT-IR

spectrometer provides us with the following on-linemeasurements in the liquid phase every 30 min: soluble

chemical oxygen demand (COD), total organic carbon

(TOC), total volatile fatty acids (VFA), acetate (Ac),

dissolved CO2 (CO2d). More details about the process

and its instrumentation can be found in [77,78].

5.2. Experimental illustration of overall AD process

supervision using evidence theory

During the 12 days presented in Fig. 8, several faulty

situations were encountered:

Fig. 7. Schematic view of the process and the associated on-line instrumentation.


• From t ¼ 0 to 10 h, the situation is normal (i.e., Qin,

the input liquid flow rate is around 20 l/h, VFA, the

volatile fatty acids concentration in the reactor, is

low (less than 0.5 g/l), pH in the process is around

neutrality and CH4 in the biogas is between 60%

and 70%).• From 10 to 25 h, the input flow rate shows a step in-

crease and the process response is normal for this

small hydraulic overload (even though pH in the

influent is lower than usual, i.e., 4 instead of 6).

• Starting at t ¼ 40 h, a large increase of the input flow

rate is applied, inducing a high accumulation of VFA

in the reactor (more than 6 g/l), thus lowering the pH

0 100 2000

20

40

60

80

Time ( hours )

Qin

(L/h

)

0 1000

2

4

6

8

Time (

VF

A (

g/L

)

0 100 20040

50

60

70

80

90

100

Time ( hours )

CH

4 in

biog

as(%

)

0 1002

3

4

5

6

7

8

Time (

pH in

the

infl

uent

(U

pH)

0 100 2000

20

40

60

80

Time ( hours )

Qin

(L/h

)

0 1000

2

4

6

8

Time (

VF

A (

g/L

)

0 100 20040

50

60

70

80

90

100

Time ( hours )

CH

4 in

biog

as(%

)

0 1002

3

4

5

6

7

8

Time (

pH in

the

infl

uent

(U

pH)

Fig. 8. A 12 day experiment showing inhibition of the overall AD pr

in the process (down to almost 5). This leads to a

large inhibition of the biological reaction scheme

(the gas flow rate decreases sharply while it should

have increased if no inhibition was present).

• To solve this problem, a first solution was applied

through the increase of the pH in the influent at t ¼60 h.

• However, this was not enough and thus, the input

feed flow was decreased to 10 l/h after 120 h and

the process was underloaded.

• In addition, at about t ¼ 150 h, a toxicant was added to

the process (cf. the spike in the feed flow) inducing an

increase of the pH in the reactor. Fortunately, the

200hours )

0 100 2000

100

200

300

400

Time ( hours )

Qga

s(L

/h)

200hours )

0 100 2004

5

6

7

8

9

10

Time ( hours )

pH in

the

proc

ess

(UpH

)

200hours )

0 100 2000

100

200

300

400

Time ( hours )

Qga

s(L

/h)

200hours )

0 100 2004

5

6

7

8

9

10

Time ( hours )

pH in

the

proc

ess

(UpH

)

ocess due to accumulation of undissociated volatile fatty acids.


problem lasted for only few hours and the process

could slowly recover until t ¼ 300 h (the fast step

change ofQin at t ¼ 200 hwas done onpurpose to eval-

uate the process performances during the recovery).

A global diagnosis system (i.e., accounting for all the

available information sources in a single box) could

have been developed to detect, isolate and analyse the

above mentioned faults. However, as already stated, this

would have lead to a very complex diagnosis system

almost impossible to maintain.

This is why, as presented in Fig. 4, the diagnosissystem was chosen to be in two parts: the fault signals

are generated by several fuzzy logic based modules, each

of them manipulating three process variables at the most

for simplicity reason, and then sent to the state manager

where they are analysed using the evidence theory.

Fig. 9 details all the fuzzy modules developed. In

addition to the fault signal, they also provide the asso-

ciated fault intensity, this combined information beingsent to the state manager. Two main reasons are behind

these choices: easiness of maintenance of the overall

diagnosis structure and better understanding of the

explanations provided to the human operators.

As an illustration of the different modules, Fig. 10

describes the rules associated to Qin, Qgas and CH4 (i.e.,

an advanced––but focused on the gas phase––instru-

mentation at the industrial scale). In this figure, thefollowing nomenclature was adopted:

• HO: hydraulic overload,

• OO: organic overload,

• UL: underload,

• Tox: presence of a toxic,

while ‘‘))’’ to ‘‘+ +’’ is used for the fuzzy qualificationof the variables from ‘‘very low’’ to ‘‘very high’’. It is to

Qin

VFA

CH4

H2

Qgas

pH

COD

Measurements

Dedicated Fuzzy Diagnosis modules

VFA

CH4

COD

H2

Qgas

pH

Qin

Validated values

Qin

VFA

CH4

H2

Qgas

pH

COD

Measurements


VFA

CH4

COD

H2

Qgas

pH

Qin

Validated values


VFA

CH4

COD

H2

Qgas

pH

Qin

Validated values

VFAVFA

CH4CH4

CODCOD

H2H2

QgasQgas

pHpH

QinQin

Validated values

Fig. 9. Modular fault detection system

be noticed that in some occasions, this module cannot

differentiate between an organic overload and the pres-

ence of a toxic. This doubt is normal and is due to the

low number of sensors managed by this module to assessthe state of the AD process. The same impossibility to

distinguish between two (or more) situations is present

in all the modules which makes relevant the choice of

using the evidence theory within the state manager.

Of course, neither all the modules of Fig. 9––nor all

the sensors in Fig. 7––are needed to supervise the AD

process and once again, modularity is central in this

architecture. As already noticed, one main advantage ofhaving multiple modules associated to different sensors

is indeed that if one of the sensors is not working due to

fouling or is not connected due to economical reasons,

the overall structure is still operational and it can to

provide the human operator with relevant information.

As an illustration of this statement, Figs. 11–13 de-

tails the faults detected by fuzzy module #1 (i.e., Qin

and pH), module #6 (i.e., Qin, Qgas and pH) andmodule #9 (i.e., Qin, Qgas and CH4) when analysing the

experimental data described in Fig. 8. These different

sensors are chosen because they are the most widely

used at the industrial scale.

In these different plots, a signal close to 0 means that

the associated process state is not present whereas when

the signal reaches 1, the associated state is present.

As it can be seen, the fuzzy module #1 detects well thehydraulic overloads (i.e., HO) from 10 to 25 and from 40

to 120 h (cf. Fig. 11c). Then, it cannot differentiate

among organic overload and toxicity (i.e., OO [ Tox)

between 120 and 150 h (cf. Fig. 11e). Addition of the

toxic is well detected between 150 and 155 h (cf. Fig.

11d) while after that, underload (i.e., UL) is the main

fault detected (cf. Fig. 11b).

Module #6, despite managing more sensors thanmodule #1 (i.e., Qgas in addition to Qin and pH),

Sent to theState

Manager

pH / Qin

H2 / Qin

Qgas / Qin

VFA / Qin

COD / Qin

Qgas / pH / Qin

H2 / pH / Qin

VFA / pH / Qin

CH4 / Qgas / Qin

VFA / Qgas / Qin

F1I1

F2I2

F3I3

F4I4

F5I5

F6I6

F7I7

F8I8

F9I9

F10I10

Fault and Intensity

Sent to theState

Manager

pH / Qin

H2 / Qin

Qgas / Qin

VFA / Qin

COD / Qin

Qgas / pH / Qin

H2 / pH / Qin

VFA / pH / Qin

CH4 / Qgas / Qin

VFA /Qgas / Qin

F1I1

F2I2

F3I3

F4I4

F5I5

F6I6

F7I7

F8I8

F9I9

F10I10

Fault and Intensity

previous to the state manager.

CH4 Qin Qgas State Intensity

++

0, -,--

+

Abnormal

Abnormal

--

UL alarm1+

+

0-

+++

UL alarm2

Abnormal

HO alarm3HO alarm2

0

+

0

-

++,+0, -,--

++

0-

HO alarm1

OO alarm2OO alarm1

NormalUL alarm1UL alarm2

+

--

++

0-

OO alarm2OO alarm1UL alarm1UL alarm2UL alarm3

+

0, -

+

--

--

-

+

0

-

++,+,0-,--

++

0-

HO alarm2

OO alarm3OO alarm2

++,+

---

OO alarm3OO alarm20

HO alarm3

OO Tox alarm1OO Tox alarm2OO Tox alarm3

OO Tox alarm2OO Tox alarm3

HO alarm3

OO Tox alarm3

CH4 Qin Qgas State Intensity

++

0, -,--

+

Abnormal

Abnormal

--

UL alarm1+

+

0-

+++

UL alarm2

Abnormal

HO alarm3HO alarm2

0

+

0

-

++,+0, -,--

++

0-

HO alarm1

OO alarm2OO alarm1

NormalUL alarm1UL alarm2

+

--

++

0-

OO alarm2OO alarm1UL alarm1UL alarm2UL alarm3

+

0, -

+

--

--

-

+

0

-

++,+,0-,--

++

0-

HO alarm2

OO alarm3OO alarm2

++,+

---

OO alarm3OO alarm20

HO alarm3

OO Tox alarm1OO Tox alarm2OO Tox alarm3

OO Tox alarm2OO Tox alarm3

HO alarm3

OO Tox alarm3

∪∪∪

∪∪

∪

Fig. 10. Rules associated to the fault detection module #9 analysing Qin, Qgas and CH4.


provides also more uncertainty. Indeed, as can be seen in

Fig. 12f, some unknown faults are detected after 155 h

while it cannot distinguish between a hydraulic overload

and the presence of a toxic from 50 to 120 h and from

200 and 210 h (cf. Fig. 12g). A similar doubt can be

noticed between an organic overload and the presence of

a toxic from 120 to 150 h, from 165 to 190 h and from210 h until the end (cf. Fig. 12h). Other process states

are correctly assessed by this fuzzy module, even though

in some cases, uncertainty is still present (see for

example Fig. 12b where the fault signal never reaches 1).

This uncertainty is lowered when using the compo-

sition of methane in the biogas instead of pH as it is the

case in module #9. However, despite very good and

accurate process state assessments for underload (cf.Fig. 13b), organic overload (cf. Fig. 13c), hydraulic

overload (cf. Fig. 13d), unknown situations are still

present after 155 h (cf. Fig. 13e) and distinction cannot

be made in some cases between organic overload and

toxicity (cf. Fig. 13f).

As illustrated in Figs. 11–13, each of these three

modules, when taken separately, cannot provide clear

and definitive process state assessment and they need to

be combined within the state manager to reinforce the

diagnosis. To show the benefits of such a combination

using the evidence theory, results obtained from the

application of the Yager’s rule for the combination of

these three modules (i.e., modules #1, #6 and #9) are

presented in Fig. 14. Clearly, the different faults are wellanalysed until 155 h, more belief being put afterwards

on the underload of the process while unknown situa-

tions are detected at a low level.

In fact, these results are in very good agreement with

our expectations in such situations. Indeed, during these

12 days, none of the sensors provided erratic signals and

the fuzzy fault detection modules were developed using a

deep expertise on the process. As a consequence, eachmodule reinforced the others ones in their belief about

the encountered events.

However, the picture would have been different in

case of sensor(s) fouling. Nevertheless, in such situation,

the conflict could be again detected with the same

methodological approach. Indeed, as illustrated in [79],

assumptions could be easily made on the faulty sensor(s)

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

a) Normal

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

b) UL

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

c) HO

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

d) Tox

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

e) OO Tox

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

a) Normal

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

b) UL

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

c) HO

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

d) Tox

0 100 2000

0.2

0.4

0.6

0.8

1

Time ( hours )

e) OO ∪ Tox

Fig. 11. Fault detection when analysing only Qin and pH (i.e., fuzzy module #1 alone).

0 100 2000

0.5

1

Time (hours) Time (hours) Time (hours)

Time (hours)Time (hours) Time (hours)Time (hours)

Time (hours) Time (hours)

Time (hours) Time (hours)

Time (hours)

Time (hours)

Time (hours) Time (hours)Time (hours)

a) Normal

0 100 2000

0.5

1b) UL

0 100 2000

0.5

1c) OO

0 100 2000

0.5

1d) HO

0 100 2000

0.5

1e) Tox

0 100 2000

0.5

1f) Unknown

0 100 2000

0.5

1g) HO Tox

0 100 2000

0.5

1h) OO Tox

0 100 2000

0.5

1a) Normal

0 100 2000

0.5

1b) UL

0 100 2000

0.5

1c) OO

0 100 2000

0.5

1d) HO

0 100 2000

0.5

1e) Tox

0 100 2000

0.5

1f) Unknown

0 100 2000

0.5

1g) HO ∪ Tox

0 100 2000

0.5

1h) OO ∪ Tox

Fig. 12. Fault detection when analysing only Qin, Qgas and pH (i.e., fuzzy module #6 alone).


0 100 2000

0.2

0.4

0.6

0.8

1a) Normal

0 100 2000

0.2

0.4

0.6

0.8

1b) UL

0 100 2000

0.2

0.4

0.6

0.8

1c) OO

0 100 2000

0.2

0.4

0.6

0.8

1d) HO

0 100 2000

0.2

0.4

0.6

0.8

1e) Unknown

0 100 2000

0.2

0.4

0.6

0.8

1f) OO Tox

0 100 2000

0.2

0.4

0.6

0.8

1a) Normal

0 100 2000

0.2

0.4

0.6

0.8

1b) UL

0 100 2000

0.2

0.4

0.6

0.8

1c) OO

0 100 2000

0.2

0.4

0.6

0.8

1d) HO

0 100 2000

0.2

0.4

0.6

0.8

1e) Unknown

0 100 2000

0.2

0.4

0.6

0.8

1f) OO ∪ Tox



Fig. 13. Fault detection when analysing only Qin, Qgas and CH4 (i.e., fuzzy module #9 alone).

0 100 2000

0.5

1a) Normal

0 100 2000

0.5

1b) UL

0 100 2000

0.5

1c) OO

0 100 2000

0.5

1d) HO

0 100 2000

0.5

1e) Tox

0 100 2000

0.5

1f) OO Tox

0 100 2000

0.5

1g) Unknown

0 100 2000

0.5

1a) Normal

0 100 2000

0.5

1b) UL

0 100 2000

0.5

1c) OO

0 100 2000

0.5

1d) HO

0 100 2000

0.5

1e) Tox

0 100 2000

0.5

1f) OO ∪ Tox

0 100 2000

0.5

1g) Unknown

Time (hours)Time (hours)

Time (hours)Time (hours) Time (hours)Time (hours) Time (hours)Time (hours)



Fig. 14. Results from the state manager when receiving faults detection signals from module #1 (i.e., Qin and pH), #6 (i.e., Qin, Qgas and pH) and

#9 (i.e., Qin, Qgas and CH4). The Yager’ rule is used for the information combination.


0 100 2000

0.2

0.4

0.6

0.8

1a) Normal

0 100 2000

0.2

0.4

0.6

0.8

1b) UL

0 100 2000

0.2

0.4

0.6

0.8

1c) OO

0 100 2000

0.2

0.4

0.6

0.8

1d) HO

0 100 2000

0.2

0.4

0.6

0.8

1e) Unknown

0 100 2000

0.2

0.4

0.6

0.8

1f) OO Tox

0 100 2000

0.2

0.4

0.6

0.8

1a) Normal

0 100 2000

0.2

0.4

0.6

0.8

1b) UL

0 100 2000

0.2

0.4

0.6

0.8

1c) OO

0 100 2000

0.2

0.4

0.6

0.8

1d) HO

0 100 2000

0.2

0.4

0.6

0.8

1e) Unknown

0 100 2000

0.2

0.4

0.6

0.8

1f) OO ∪ Tox


Time (hours)Time (hours) Time (hours)Time (hours) Time (hours)Time (hours)


Fig. 15. Fault detection when analysing only Qin, Qgas and VFA (i.e., fuzzy module #10 alone).


by using hardware and/or software sensors redundancy.

It is here to be recall that not all the sensors available on

our process are needed to develop the presented diag-

nosis approach and that hardware sensors can be re-

placed without any problems by software sensors tocreate a sensors network easily managed by the evidence

theory. In this case, X is simply fOKi;KOig for each

signal (i.e., hard/soft sensor #i is working fine or not)

and the basic belief assignment is calculated from the

distance between two hard or soft measurements (i.e.,

there is no need to use the fuzzy modules).

Another main advantage of the evidence theory is

that it can also be used to detect conflicts in the expertrules associated to each module. Indeed, a badly tuned

rule or an inconsistency in the rule base could be de-

tected following again the same approach. This aspect

is of particular interest during the development of

the overall diagnosis system and speed up the tuning

stage.

Last but not least, the benefits of this diagnosis

structure also lie in the minimality of the needed mod-ules. For example, fault detection results only using the

fuzzy module #10 are presented in Fig. 15 and they are

very similar to those presented in Fig. 14. One could

thus think that module #10 alone could manage the

overall AD process and, as a consequence, she/he could

choose to implement only a gas flow meter and a VFA

on-line sensor. However, in this situation, robustness of

the diagnosis would be very low. Indeed, if the VFAsensor delivers wrong measurements (which is likely to

appear in practice due to interference of chemical spe-

cies, to sensor fouling, etc.), the whole supervisory

scheme breaks down. In critical safety industries (e.g.,

nuclear and aeronautic), the solution would be to

implement two VFA sensors. However, in WWTPs, thissolution is not viable economically. And maybe more

importantly, it is felt that this would not be an appro-

priate solution. Indeed, if pH and gas composition

sensors are implemented in addition to only one VFA

sensors (instead of one gas flow meter and two VFA

sensors), many more faulty situations can be handled

(e.g., presence of different toxic compounds, technical

problems, etc.) while validating the obtained measure-ments by cross-checking each of them using the evidence

theory.

6. Conclusion

This paper presented the development of a modular

diagnosis system using evidence theory. This approach

gathers many advantages for the advanced supervision

of biological wastewater treatment processes: robust-ness, novelty identifiability, adaptability, low modelling

requirements, multiple fault identifiability.

Application has been made to a pilot scale fixed bed

AD reactor and illustrated experimentally how different

faults could be managed in a simple but efficient way.

Moreover, several others fields of applications (i.e.,

integration of software sensors in the diagnosis scheme,


automatic activation and tuning of control loops, vali-

dation of different simple models developed to handle

specific situations) could be managed using the

same approach and our current studies go in thesedirections.

Acknowledgements

This work has been carried out with the support of

the European commission, Information Society Tech-

nologies programme (contract TELEMAC number IST-2000-28256). This information is provided under the

sole responsibility of the authors and does not neces-

sarily represent the opinion of the European Commis-

sion, which is not responsible for any use that might be

made of it.

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