ENCODING FUZZY DIAGNOSIS RULES AS
OPTIMISATION PROBLEMS
Antonio Sala, Alicia Esparza, Carlos Ari
˜
no and Jose V. Roig
Systems Engineering and Control Dept., Univ. Polit
´
ecnica de Valencia
Cno. Vera s/n, 46022 Valencia, Spain
Keywords:
Fault detection and diagnosis, fuzzy mathematical programming, approximate reasoning, optimisation.
Abstract:
This paper discusses how to encode fuzzy knowledge bases for diagnostic tasks (i.e., list of symptoms pro-
duced by each fault, in linguistic terms described by fuzzy sets) as constrained optimisation problems. The
proposed setting allows more flexibility than some fuzzy-logic inference rulebases in the specification of the
diagnostic rules in a transparent, user-understandable way (in a first approximation, rules map to zeros and
ones in a matrix), using widely-known techniques such as linear and quadratic programming.
1 INTRODUCTION
Many industrial activities depend on the correct op-
eration of complex technological processes. A fault
changes the behaviour ofa system in such a way that it
does no longer satisfy its nominal performance objec-
tives or even the system functionality is lost. The ob-
jective of diagnosis (process monitoring) is estimating
a vector of fault state parameters, f, from measure-
ments of the outputs of a dynamic system, whose tra-
jectories depend on f and on time, initial conditions,
external input variables, physical parameters, etc. De-
tecting faults with a gradation of severity from let’s
say zero to 100% may be advantageous in practice:
detecting faults in its early stages enables corrective
actions to be taken on time, if needed.
Different approaches to the problem appear in lit-
erature: data-based, knowledge-based or based on
differential-equation analytical models. A description
of many of these techniques appear in (Chiang et al.,
2001; Blanke et al., 2003). The broadest conception
of the problem may be set up in a probabilistic set-
ting. In that situation, a comprehensive approach to
the problem would involve estimation under nonlin-
ear stochastic dynamics (Timmer, 2000; Khalil, 2002)
as well as decision-theoretic criteria (Berger, 1985)
apart, of course, of the consideration of robustness of
the resulting results when subject to possibly signifi-
cant modelling errors. The problem, as such, is quite
complex and possibly intractable. Hence, simplified
assumptions are often stated.
In quite a few practical cases, diagnostic-related
knowledge is available from experts, who express
it in linguistic terms (“fault F produces a pressure
in pipe 3 lower than normal”). The meaning of
some of those linguistic terms may be understood
as rules involving fuzzy concepts defined on the nu-
meric range of the physical variables being measured.
This fact inspires the use of fuzzy logic in diag-
nosis (Angeli, 1999; Carrasco and et. al., 2004).
Other knowledge-based frameworks extend the ba-
sic logic reasoning schemes to uncertain reasoning
(Kruse et al., 1991; Shafer and Pearl, 1990), possi-
bilistic reasoning (Dubois and Prade, 2004; Yamada,
2004), Bayesian networks (Castillo et al., 1997; Rus-
sell and Norvig, 2003) or uncertain probability (Ky-
burg, 1988), or combine the approach with neural net-
works (Ayoubi and Isermann, 1997; Jie and Morris,
1996).
This paper presents an alternative approach: the ex-
pert knowledge stemming from some (partially sim-
plified) properties of a nonlinear dynamic system
is encoded as a constrained optimisation problem,
transcoding fuzzy assertions as approximate linear
equations in the linguistic domain. The idea of trans-
forming fuzzy statements into equations also appears
in (Juuso, 1999; Sala and Albertos, 2001) in a differ-
ent context of system modelling.
The proposed approach seems to possesses signi-
ficative advantages with respect to a classical fuzzy
IF-THEN rulebase, particularly in multiple fault situ-
ations, while keeping the problem readable (reduced
34
Sala A., Esparza A., Ariño C. and V. Roig J. (2006).
ENCODING FUZZY DIAGNOSIS RULES AS OPTIMISATION PROBLEMS.
In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 34-39
DOI: 10.5220/0001203100340039
Copyright
c
SciTePress
number of “rules”) and computationally tractable (ef-
ficient off-the-shelf linear programming (LP) soft-
ware exists able to deal with hundreds, even thou-
sands, of constraints and variables, and quadratic pro-
gramming (QP) routines are also widely available).
LP is a widely known tool (Sierksma, 2001; Gass,
2003), taught in many undergraduate disciplines so
that user understanding of both the rules and the infer-
ence tools implies that the approach might be useful
in practical applications.
The structure of the paper is as follows: a prelimi-
nary section will justify some approximate additivity
properties of systems based on linearisation. Section
3 will discuss how a fuzzy rulebase may be encoded
as equations. Section 4 will discuss the available al-
gorithms to solve them and how they should be mod-
ified for the problem in consideration. A conclusion
section closes the paper.
2 PRELIMINARIES
Dynamic systems. Let us assume that a system
to be diagnosed is governed by equations ˙x =
ψ(x, u, f, θ, t) where x is the system’s dynamical
state, u is a set of known input variables, f is the fail-
ure state to be estimated and θ are a set of system pa-
rameters, also assumed to be approximately known,
and t is the time variable (Khalil, 2002).
First, it will be assumed that the initial conditions
x(t
0
) and inputs of standarised tests are known be-
forehand, as well as system-dependent parameters θ.
The result of such tests will be a finite collection of
measurements at some time instants:
y = ψ(x(t
0
), θ, u, t
0
, t, f)
where, by assumption, only f is unknown. Let us de-
note as q = (x(t
0
), θ, u, t
0
, t) the set of known vari-
ables; they will be denoted as the experiment context
variables. The diagnostic problem may then be cast
as estimating f from y(q, f), i.e., obtaining the im-
plicit function, if it exists f = γ(q, y). In general,
the analytic expression for γ cannot be obtained ex-
cept in the simplest of the cases. If enough sensors
exist,and pre-classified data from the system (y,q,f)
were available, a functional approximator such as a
neural network might be used in order to try to learn
γ by example, or at least to learn the normal behaviour
and generate some residuals in case of faults. Appli-
cations of the approach are reported in (Chow et al.,
1993; Jie and Morris, 1996). However, learning for
successful fault isolation in a complex system may
require an impractical number of data. The approach
will not be pursued further in this paper.
The diagnosis problem becomes easier if some sim-
plifications and assumptions are made. Taking a Tay-
lor series expansion of ψ on the variable f , around
f = 0, which will be defined as the “normal” situa-
tion, the result is an affine model:
y(q, f) = ψ(q, 0) +
ψ
f
(q, 0) f + o(f
2
) (1)
This Taylor series expansion justifies that at least ap-
proximately a linearity assumption holds:
y(q, f) ψ(q) + C
y
(q) f (2)
where C
y
is a linear operator (a matrix whose ele-
ments depend on the known information q). Usu-
ally, a parameterised analytical model of the system
is not available so ψ(q) and C
y
(q) cannot be calcu-
lated. However, human operators possess knowledge
that may be encoded in terms of fuzzy sets. Express-
ing such a knowledge in the form (2) is discussed in
next section.
3 FUZZY DIAGNOSTIC MODELS
In the above discussed context, a collection of fuzzy
sets mapping y to the interval [0, 1] are assumed to
defined by an expert on the system to be diagnosed,
µ : R
p
[0, 1]
k
where p is the number of measure-
ments and k is the number of fuzzy concepts. Usu-
ally, those sets are denoted by user-defined linguistic
labels.
Then, the expert also knows which effects each
fault has in the measurements. This knowledge is usu-
ally expressed in terms of rules:
If the isolated fault F
i
occurs, then abnormal
symptoms S
i
1
, . . . , S
i
p
should be observed, be-
ing the rest normal
Those rules, in a fuzzy context, may be basically un-
derstood as
If the severity of fault F
i
is f
i
, 0 f
i
1, and it
is the only fault occurring in the system, then the
intensity of symptoms S
i
1
, . . . , S
i
p
is approxi-
mately f
i
, and the intensity of the rest of them is
zero.
where f
i
= 0 denotes fault not occurring and f
i
=
1 denotes the fault occurring at a very significative
severity level requiring user attention
1
, and the inten-
sity of the symptoms is the membership function of
the suitably defined fuzzy concepts.
In a multiple-fault situation, the system is assumed
to verify an expression such as (2). If the observed
outputs are the fuzzified ones, it will be assumed that
1
It is up to the user to fix a maximum value of f
i
(usually
1) in the optimisation procedures to be discussed, or to set
f
i
= 1 as a “landmark” point but considering higher values
(more severe) possible.
ENCODING FUZZY DIAGNOSIS RULES AS OPTIMISATION PROBLEMS
35
the system verifies the following linear equation in the
domain [0, 1]
p
of logic values:
µ
q
(y) C(q)f + D(q) (3)
where C and D are a known function of the system’s
parameters, input variables and initial conditions of
the experiment. The notation µ
q
indicates that even
the definition of the membership function may de-
pend on a priori information, such as historical data,
system-dependent parameters, etc. C(q) will be sim-
ilar to
µ
y
C
y
(q). Note that a differentiable system in-
deed does verify such an equation
2
for small values
of f based in (2). The basic assumption here is that,
with suitably defined fuzzy sets, the range in which
such an equation fulfills is large enough to be useful
for diagnosis.
Fuzzy sets may be defined on the difference be-
tween the observed readings and those in a normal
situation, as an alternative to the absolute reference
frame for concepts implicit in equation (3), so that:
µ
q
(y D(q)) C(q)f (4)
In a practical situation, both types of fuzzy sets (3)
or (4) may be used. In summary, the following linear
equation must be solved in the diagnosis process:
µ(y, q) = C(q)f (5)
Other authors also pose matrix representations of the
relationship between fuzzy faults and symptoms (for
instance, (Yao and Yao, 2001) uses a fuzzy rela-
tion approach). The assumption of linearity in the
logic domain, with suitable definitions of membership
functions is also used, in a control context, in (Juuso,
1999).
3.1 Rule Encoding
The proposed expert rules are encoded in the format
required in (3) in a simple way. Let us discuss several
situations which will be clarified by examples.
First, the simplest setting would imply that, under
a standarised test (q is fixed for all diagnostic experi-
ments), a fuzzy set denoting an “abnormal” situation
is defined for every measured variable. In that case,
the knowledge:
F
i
causes abnormality in y
1
, y
3
, . . .
will define a column of matrix C, where elements at
rows 1, 3, . . . will be 1 and the rest of elements not
explicitly enumerated at the above assertion will be
set as zero. The usual fuzzy negation operator may be
used if some variables have a fuzzy set defining the
“normal” value of a variable.
2
Formally, left and right derivatives may be needed, but
details are not relevant.
Example 1 Let us have an industrial boiler where a
fault f
1
causes: no variation on a temperature mea-
surement t
1
, and increasing of temperature t
2
. An-
other fault, f
2
, causes increases in both temperatures.
They both cause an increase of pressure p. Defining
abnormally high temperatures with fuzzy sets denoted
as “T
1
abnormal”, µ
1
and “T
2
abnormal”, µ
2
, and “nor-
mal pressure” with another fuzzy set (such as a trian-
gular one), µ
3
, the basic diagnosis equation would be:
µ
1
(t
1
)
µ
2
(t
2
)
1 µ
3
(p)
=
0 1
1 1
1 1
f
1
f
2
Alternatively, if the nominal normal pressure de-
pends on a variable q, denoted as P
0
(q), (for instance,
q might be the load regime of the boiler), a fuzzy set
may be defined on the pressure increment so that the
diagnosis equation would be written as:
µ
1
(t
1
)
µ
2
(t
2
)
1 µ
3
(p P
0
(q))
=
0 1
1 1
1 1
f
1
f
2
Model errors and sensor faults. Another situation
takes into account the approximate nature of (5) al-
lowing for an instrumental “fault” variable associ-
ated to each of the measurements. That instrumen-
tal fault variable encompasses both sensor faults and
modelling errors (inaccuracy in the definition of the
membership functions).
Example 2 In the example being considered, the fault
vector may be extended as:
µ
1
(t
1
)
µ
2
(t
2
)
1 µ
3
(p)
=
0 1 1 0 0
1 1 0 1 0
1 1 0 0 1
f
1
f
2
f
1
f
2
f
3
(6)
where f
are three instrumental fault variables and f
denotes the “primary” faults.
In other situations, different faults have opposite ef-
fects on a particular variable so that its simultaneous
occurrence does not deviate its measurements from
the normal condition. The following example clari-
fies how to encode such a knowledge.
Example 3 If, in example 1 fault 1 decreases T1, be-
ing the rest of symptoms the same as previously de-
scribed, with concepts ”abnormally high”, µ
h
, and ”ab-
normally low”, µ
l
, defined for T
1
, then the rulebase
should be encoded as:
µ
h
(t
1
)
µ
l
(t
1
)
µ
2
(t
2
)
1 µ
3
(p)
=
1 1
1 1
1 1
1 1
f
1
f
2
ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION
36
In the above example, depending on the units and
membership definitions, the 1 terms may be a dif-
ferent negative number. Note that some combinations
of faults may yield negative membership values; this
is not a problem if inequality restrictions are consid-
ered, as discussed in section 4.1.
In any of the examples, a coefficient in matrix
C lower (higher) than 1 would indicate a milder
(stronger) effect of the fault on the symptom, as de-
fined by the membership function. Modifying the co-
efficients might be needed in a fine-tuning phase be-
cause not all the faults influence with the same in-
tensity a particular variable. However, taking into ac-
count “sensor” errors f
, an initial setting with mostly
0 and 1 coefficients in C may be enough in order to
achieve a reasonable output.
4 INFERENCE
A fault estimation f is consistent with the observed
symptoms if it is a solution of the basic diagnosis
equation (5). Hence, fuzzy diagnosis amounts to find-
ing the set of solutions of (5). However, some consid-
erations need to be made: indeed, if sensor faults and
modelling errors, f
, are considered, the solution set
includes any conceivable primary fault f , as sensor
faults can accommodate any reading. For instance,
in (6) a value of f
can be calculated for any sensor
reading and any value of f . However, if sensor faults
are not considered, with less faults than sensors the
set of consistent faults will be usually empty, as (5)
will have no solution due to modelling errors. So, the
above idea must be refined for practical usability.
Inference as optimisation. As above discussed,
sensor faults must be considered in practice. Then,
(5) are equality restrictions and a criterion should be
used in order to rank all the feasible solutions. Note
also that inequality restrictions f
i
> 0 implicitly ap-
ply, unless the user casts a meaning for negative fault
severities.
A possible criterion to be chosen is minimising the
norm of the “instrumental” fault component, i.e., giv-
ing as the diagnosis solution the one that achieves less
discrepancy between the measurements and the pre-
dictions C(q)f. The discrepancy µ C(q)f will be
denoted as inference error.
If the chosen norm is the Euclidean one, fuzzy in-
ference is equivalent to a least squares problem. Let
us consider an equation µ = Cf where sensor faults
are also members of f. This is a linear system of
equations with more unknowns than equations, which
can be solved in the following sense:
The feasible solution f that minimises the squared
Euclidean norm of W f, where W is a diagonal
weight matrix, is given by the pseudo-inverse formula
(Meyer, 2001):
f = W
2
C
T
(CW
2
C
T
)
1
µ (7)
In a practical setting, a high penalisation in W must
be specified for the sensor fault components f
. For
invertibility of W , small positive weights in the pri-
mary faults need to be introduced
3
.
If the chosen norm is the 1-norm of W f (sum of
absolute value of the components), then inference can
be carried out in a linear programming setting. The
LP framework needs to introduce dummy variables
for positive and negative sensor errors f
= f
+
f
,
f
+
> 0, f
> 0, to calculate the 1-norm as the sum
of f
+
+ f
. This change of variables is standard in
LP textbooks.
Note that LS algorithms produce an “intermediate”
point as a result (not a vertex of the feasible region),
sharing the error between all the equations, as small
errors are not significant (because of the squaring) so
LS tries to reduce big errors. On the contrary, LP pro-
duce a result in a vertex of the feasible solution space,
and increments from either small or big errors weight
the same.
4.1 Constrained Optimisation
Under the proposed settings, it is implicitly assumed
that a reasonable diagnostic should verify f
i
0 in
all components of the primary faults.
Also, fuzzy concepts saturate in [0, 1]. Hence, si-
multaneous faults yielding the same symptom cannot
fulfill, for instance, 1 = f
1
+ f
2
if they are fully ac-
tive. However, two easy options are available (or a
combination of them) in that case:
When a fuzzified sensor reading is saturated, re-
place the equality constraint in the diagnosis equa-
tion by an inequality (1 C
i
f, 0 C
i
f, where
C
i
denotes the i-th row of C).
Translate an ordered fuzzy partition on a domain
into numerical values (for instance, {very low, low,
normal, high, very high} into {−2, 1, 0, 1, 2}) in
the spirit of the so-called linguistic equation (Juuso,
1999; Jarvensivu et al., 2001). In this way, the basic
diagnosis equation (5) may involve sensor values
ranging more than [0, 1], but somehow keeping the
linguistic meaning.
Algorithms. LP algorithms incorporate linear in-
equality restrictions seamlessly. However, the least
squares formula (7) must then be discarded and
quadratic programming (QP) routines used instead.
3
Solving y = Cf by standard least squares, f =
(C
T
C)
1
C
T
µ, is equivalent to the proposed approach
when the primary-fault weights tend to zero and sensor ones
are equal to the same constant.
ENCODING FUZZY DIAGNOSIS RULES AS OPTIMISATION PROBLEMS
37
For instance, if the sensor reading in example (4) had
been {.2, .18, 0} the output of the LS formula would
have been f
1
= 0.197, f
2
= 0.003, out of the
constraint space so QP or non-negative least squares
would have been needed.
As the involved restrictions are linear, efficient
code exists for both LP and QP settings. For brevity,
mostly linear programming settings will be consid-
ered in the sequel, although a similar version posed as
QP would produce comparable results. For instance,
commercially available LP software is able to effi-
ciently deal with hundreds (even thousands) of vari-
ables and restrictions, allowing for large-scale imple-
mentation of the ideas in this work.
Binary faults. Some faults may be only either 0
or 1, without intermediate values (for instance, circuit
breaker ON vs. OFF). To carry out optimisation, ex-
plicit enumeration of all the involved binary (or inte-
ger) variables and solving for each of them the optimi-
sation on the remaining real variables may be an ap-
proach. Alternatively, mixed linear integer program-
ming or branch-and-bound methodologies may also
be applied (Sierksma, 2001).
Managing non-unique solutions. The optimisa-
tion routines stop at an approximately optimal point.
However, there might be other optimal points or, at
least, which may have a very similar value of the
cost index. The situation is particularly frequent in
the case of missing measurements, which amounts to
deleting the corresponding row of C.
In order to choose between possible nonunique so-
lutions, the weights of the different fault components
should penalise each fault according to its probabil-
ity: in that way, the solution would tend to be the most
likely fault consistent with the sensor measurements.
Imprecise measurements. Some sensors may be
imprecise, in the sense that a small deviation from the
expected values is frequent and acceptable to assume
when producing a diagnosis. In a sense, quadratic
cost indices naturally take that fact into account, but
LP settings need a straightforward change of variable
to do that
4
. This is easily carried out, by express-
ing each “sensor fault” by a sum of two sub-faults,
bounding the maximum value of one of them (with
a small weight in the inference cost index) and set-
ting a much larger penalisation on the deviations of
the non-limited one. If the small sensor errors have
zero weight, the setting is equivalent to interval mea-
surements:
f
= f
+
1
+ f
+
2
f
1
f
2
(8)
0 f
+
1
l
+
, 0 f
1
l
, 0 f
+
2
, 0 f
2
(9)
Heavy cost in f
+
2
and f
2
, no penalisation in f
+
1
, f
1
results in an interval sensor reading allowing, with no
4
The same change of variable may be used in QP for-
mulations to fine-tune the cost index formula, if so wished.
cost, diagnostics involving the actual reading, say σ,
plus or minus the desired bound: [σ l
, σ + l
+
].
Example 4 Let us consider a knowledge base:
Fault 1 produces S1, S2. Fault 2 produces S2,
S3
and a fuzzified sensor reading were {0.2,0.43,0.15}.
The setting for minimal squared inference error
(weighting by 0.5 the “confidence” on the accuracy of
equation 2, because the addition of individual faults
may not result in the exact addition of the results due
to possible system nonlinearity) would be carried out
by the following Matlab code:
C=[1 0;1 1;0 1];C2=[C eye(3)];
j=diag([0.01 0.01 1 .5 1]);
cw=C2
*
inv(j);
f=inv(j)
*
pinv(cw)
*
[.2 .43 .15]’
The result is f
1
= 0.21, f
2
= 0.16.
Alternatively, the Matlab code for the above problem
minimising the 1-norm of the inference error via linear
programming is:
C=[1 0;1 1;0 1];
C2=[C eye(3) -eye(3)];
j=[0.01 0.01 1 .5 1 1 .5 1];
x=linprog(j,[],[],C2,
[.2 .43.15],zeros(8,1));
which produces f
1
= 0.2, f
2
= 0.15, apportioning all
error to the less reliable sensor 2. Vector j contains
the weights for f
1
, f
2
and the positive and negative
components of the 3 possible sensor/modelling faults.
The Matlab manual explains the meaning of the argu-
ments to linprog.
5 CONCLUSIONS
This paper presents a methodology for approximately
translating expert diagnostic knowledge into mathe-
matical programming problems (constrained optimi-
sation). The knowledge is a series of statements about
the list of symptoms caused by the occurrence of a
particular fault, expressed via linguistic statements in-
volving fuzzy sets.
The approach deals naturally with multiple faults
approximately following a linear equation in the lin-
guistic domain. The diagnostic procedure operates
satisfactorily with missing measurements or a limited
number of faulty sensors.
The proposed approach can be thought of as an in-
termediate between IF-THEN rulebases and diagnosis
based on a full mathematical model. It keeps a lin-
guistic interpretation while allowing for many combi-
nations of requirements difficult to be taken into ac-
count in a pure logic framework. Marginal possible
intervals of fault severities are also easily calculated
in the case of non-unique solutions of the optimisation
problem. The problem is computationally tractable
ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION
38
even in a large-scale framework as efficient software
exists for the proposed optimisation techniques.
The presented framework has been discussed on a
theoretical level with simple academic examples. De-
tailed comparative analysis and application to realis-
tic diagnostic environments with a large number of
“rules” is under research at this moment.
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