Optimal Sensor Placement for Model-based
Fault Detection and Isolation
Ramon Sarrate, Vicenc¸ Puig, Teresa Escobet and Albert Rosich
Abstract— The problem of optimal sensor placement for FDI
consists in determining the set of sensors that minimizes a
pre-defined cost function satisfying at the same time a pre-
established set of FDI specifications for a given set of faults.
The main contribution of this paper is to propose an algorithm
for model-based FDI sensor placement based on formulating
a mixed integer optimization problem. FDI specifications are
translated into constraints of the optimization problem consid-
ering that the whole set of ARRs has been generated, under
the assumption that all candidate sensors are installed. To show
the effectiveness of this approach, an application based on a
two-tanks system is proposed.
I. INTRODUCTION
The sensor placement problem consists in determining
the optimal set of sensors to install in a process such
that several goals are fulfilled. For instance, observability
is a key process property, seek in the design of a process
control algorithm. Other desirable properties are reliability,
precision, robustness, etc.
There are several articles devoted to the study of the
design of sensor networks using goals corresponding to
normal monitoring operations. Aside from cost, different
other objective functions such as precision [1], reliability [2],
or simply observability [1] were used. Different techniques
were also used, such as graph theory [2], mathematical
programming [3], genetic algorithms [4] and multiobjective
optimization [4], among others. The problem has also been
extended to incorporate upgrade considerations [5] and main-
tenance costs [6]. In [7][8], Bagajewicz reviews all these
methods and also discusses the applications to bilinear and
fully nonlinear systems.
Process disturbances or faults, if undetected, have a serious
impact on process economy, product quality, safety, produc-
tivity, and pollution level. In order to detect, isolate and
correct these abnormal process behaviors, efficient and ad-
vanced automated diagnostic systems are of great importance
to modern industries. Considerable research has gone into
the development of such diagnostic systems [9][10][11]. All
model-based approaches for fault detection and isolation in
some sense involve the comparison of the observed behavior
of the process to a reference model. Process behavior is
inferred using sensors measuring the important variables in
the process. Hence, the efficiency of the diagnostic approach
This work was supported in part by the Research Comission of the
Generalitat of Catalunya (Grup SAC, ref. 2005SGR00537) and by CICYT
(ref. DPI-2005-05415) of Spanish Ministry of Education
All authors are with the Automatic Control Department, Universitat
Polit
`
ecnica de Catalunya, Rambla de Sant Nebridi, 10, 08222 Terrassa, Spain
ramon.sarrate@upc.edu
critically depends on the location of sensors monitoring
process variables. The emphasis of most of the work on
model-based fault diagnosis has been more on procedures
to perform diagnosis given a set of sensors and less on the
actual location of sensors for efficient identification of faults.
This paper focuses in the design of a sensor network for
model-based Fault Detection and Isolation (FDI) such that
faults are detected and eventually isolated. Some contribu-
tions have already been done in this direction [12][13][14].
In model-based FDI, faults are modeled as deviations of
parameter values or unknown signals and diagnostic models
are often brought back to a residual form. Residual quantities
are zero in the absence of faults and each residual acts as an
alarm that is expected to trigger to a non-zero value upon the
occurrence of some faults, in which case the residual is said
to be sensitive to these faults. The expected triggering pat-
tern(s) of a set of residuals under some fault is interpreted as
the fault signature. Fault isolation is performed by checking
the observed residual pattern against different fault signatures
[15]. The main approaches to construct residuals are based
on using Analytical Redundancy Relations (ARRs) generated
either using the parity space [16] or observer approaches
[17].
As noticed in [7], the problem of sensor placement in the
model-based FDI community is still an open problem. In [13]
the sensor placement problem is solved by the analysis of a
set of possible ARRs using algorithms of cycle generation
in graphs. More recent approaches consist in finding the set
of all possible ARRs under the assumption that all possible
sensors are installed [14]. Just recently, several exhaustive
methods have been developed that claim to generate the
complete set of ARRs [18][19][20]. For sensor placement,
it is required to use an ARR generation algorithm that is
complete. Otherwise, the sensor placement could exclude
from consideration some sensor configurations just because
some ARRs were not generated. Excluded configurations
could provide better FDI results that the ones that were
generated. Or, even in some dramatic cases, the sensor
placement could not find a solution because of this lack of
completeness, whereas, in fact, if all ARRs were generated
a solution would have been found.
The main contribution of this paper is to propose an
algorithm for model-based FDI sensor placement based on
formulating a mixed integer optimization problem. FDI spec-
ifications are translated into constraints of the optimization
problem considering that the whole set of ARRs has been
generated, under the assumption that all candidate sensors
are installed. It has been inspired until some extent in [21].
Proceedings of the
46th IEEE Conference on Decision and Control
New Orleans, LA, USA, Dec. 12-14, 2007
ThPI22.3
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There, a mixed-integer linear programming (MILP) formula-
tion for the design of sensor networks for simultaneous pro-
cess monitoring and fault detection/isolation was presented.
The objective was to find a cost-optimal sensor set for a
chemical process that provided a good estimate of the state of
the system and detected as well as isolated a preestablished
set of faults. The optimization problem was casted as an
MILP formulation inspired in [3]. The idea was to define a
cost function based on a binary optimization vector which
stated whether a sensor was installed (1) or not (0). However,
constraints were formulated as linear inequalities based on
a digraph description of the fault propagation behavior of
the process in presence of faults [22]. In the present paper,
as already noticed, constraints are formulated on the set of
all ARRs generated from the system model considering that
the whole set of candidate sensors has been installed. For an
alternative MILP formulation of this approach see [23].
The structure of the paper is the following. Section II
introduces some model-based FDI basics and states the
sensor placement problem for FDI. In Section III, the sensor
placement problem is formulated as an optimization prob-
lem. Next, Section IV applies this optimal sensor place-
ment approach to a two-tanks system. In Section V, some
computation complexity issues are analyzed. Finally, some
conclusions and extensions are suggested in Section VI.
II. MODEL-BASED FDI
A. The ARR Table
In model-based FDI, the behavior of a plant is usually
modeled by a set of equations, E, which in general depend
not only on known variables (i.e., measured input and
output process variables) but also on unknown variables
(i.e., unmeasured internal process variables). In order to
evaluate the consistency between the model and measure-
ments taken from available sensors in the process, Analytical
Redundancy Relations (ARRs) that only depend on known
variables should be generated. ARRs can be obtained by
eliminating unknown variables through the convenient ma-
nipulation of process equations. For that purpose, structural
analysis theory has been extensively used in model-based
FDI [9][18][19][20]. A structural model is an abstraction of
the equations model, E, in which only appears the variables
involved in the relations. The structural model can be rep-
resented by a binary Incidence Matrix, IM, which crosses
model relations in rows and model variables in columns: an
entry im
ij
of the matrix is 1 when variable j appears in
relation i, and 0 otherwise.
According to the structural analysis theory, the binary ARR
Table, A, crosses measured variables or sensors in columns
and all possible ARRs in rows, denoted by R: a
ij
= 1 means
that ARR r
i
∈ R depends on sensor s
j
, a
ij
= 0 otherwise.
For instance, according to Table I, r
1
only depends on the
variables measured by sensors s
5
, s
7
and s
8
.
B. The Fault Signature Matrix
According to the structural analysis approach to FDI [18],
each ARR is expected or not to be sensitive to a fault,
TABLE I
EXAMPLE OF AN ARR TABLE
s
5
s
6
s
7
s
8
r
1
1 0 1 1
r
2
1 1 0 0
r
3
0 1 1 1
r
4
0 0 0 1
r
5
1 0 1 0
TABLE II
EXAMPLE OF A FAULT SIGNATURE MATRIX
f
1
f
2
f
3
f
4
f
5
f
6
f
7
f
8
r
1
0 0 1 0 1 0 1 1
r
2
1 0 0 0 1 1 0 0
r
3
0 1 1 1 0 1 1 1
r
4
1 0 0 0 0 0 0 1
r
5
1 0 0 1 1 0 1 0
characterizing the binary Fault Signature Matrix, M . In
this matrix, columns represent faults and rows represent all
possible ARRs R: m
ik
= 1 means that whenever fault f
k
occurs, the ARR r
i
∈ R is violated.
Assume that Table II shows the Fault Signature Matrix
that corresponds to Table I. According to this table, whenever
fault f
3
is present, ARRs r
1
and r
3
are violated.
On a given system, fault detection and isolation properties
can be stated based on the information stored by this matrix.
Possible properties are:
• Detectability: A set of faults are detectable if their
effects on the system can be observed on the available
set of ARRs. A fault f
k
is detectable if at least there is
a 1 present in the k
th
-column of M .
• Isolability
1
: A set of faults are (fully) isolable if their
effects can be discriminated one of each other consid-
ering the available set of ARRs. Two faults f
k
and f
l
are isolable if the k
th
-column and the l
th
-column of M
are different.
For instance, in Table II all faults are detectable and
isolable.
C. Sensor Placement for Model-Based FDI
Let P be the set of fixed process components. A tank,
a valve, a level sensor and a controller are examples of
process components. This set contains the components that
are needed for the proper operation of the process, so that
the predesigned functional specifications are met. The term
’fixed’ denotes that these components are present in any
sensor placement configuration. Fixed process components
can be affected by faults. Assume that F
P
is the set of all
fixed process components faults.
Let S be the set of candidate sensors. This set contains all
possible sensors that can be installed in the system, so that
the fault detection and isolation specifications are fulfilled.
The term ’candidate’ means that the chosen sensor placement
1
Under single-fault isolability assumption
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 ThPI22.3
2585
configuration will state which sensors will be present in the
process and which not. Let S
∗
⊆ S denote such set of
installed sensors. Assume that every sensor s
j
∈ S can be
affected by a fault f
j
∈ F
S
, where F
S
is the set of candidate
sensors faults. Then, F
S
∗
will denote the set of installed
sensors faults.
As sensors are a kind of process component, let F =
F
S
∪F
P
be the set of all possible process components faults.
Then, the sensor placement problem for model-based FDI
can be stated as follows:
Sensor placement problem for model-based FDI:
GIVEN a set of candidate sensors, S, a structural
model, IM (obtained from the set of model equa-
tions, E), a Target Fault Set, denoted by F
D
⊆
F , and a set of model-based FDI specifications,
denoted by T , FIND a set of installed sensors,
S
∗
⊆ S, such that F
D
fulfils T .
Possible model-based FDI specificacions are fault de-
tectability and fault isolability, as stated in the preceding
section. The assessment of these specifications for every
possible subset of candidate sensors requires the generation
of the ARR Table and the Fault Signature Matrix, which must
be obtained from IM , since both tables are different for each
subset.
Given a set of installed sensors, S
∗
⊆ S. Let A(S
∗
) and
M(S
∗
) denote the ARR Table and the Fault Signature Matrix,
with ARRs that just depend on any subset of S
∗
. Let R(S
∗
)
be this set of ARRs. A particular case is
b
A = A(S) and
c
M = M (S), denoting the Full ARR Table and the Full Fault
Signature Matrix, when all candidate sensors are installed.
In this particular case, let
b
R = R(S) denote the Full ARR
Set.
Given
c
M and
b
A, it is easy to obtain any possible M (S
∗
).
It suffices to eliminate in
c
M the ARRs which depend on
sensors s ∈ S \ S
∗
, according to
b
A. Assume that in Table
II F
P
= {f
1
, f
2
, f
3
, f
4
} and F
S
= {f
5
, f
6
, f
7
, f
8
}. Note
that for candidate sensor faults the Fault Signature Matrix
coincides with the ARR Table. The reason for this is that if
an ARR r
i
depends on a sensor s
j
, then r
i
is sensible to
faults affecting s
j
. If the set of installed sensors is S
∗
=
{s
5
, s
7
, s
8
}, then A(S
∗
) just comprises ARRs belonging to
the set R(S
∗
) = {r
1
, r
4
, r
5
}. ARRs r
2
and r
3
has been
discarded since they depend on s
6
, which is not available
according to the current configuration, S
∗
. Consequently,
M(S
∗
) just comprises ARRs belonging to this set R(S
∗
).
Then, assuming that F
D
= {f
1
, f
2
, f
3
, f
4
} and according to
the resulting Fault Signature Matrix, faults f
1
, f
3
and f
4
are
detectable and isolable, whereas fault f
2
is not detectable.
Consequently, a possible approach to solve the sensor
placement problem for model-based FDI involves that the
Full ARR Table and the Full Fault Signature Matrix has
already been generated using any of the available complete
algorithms [18][19][20]. From these tables, and introducing a
cost for each candidate sensor, the sensor placement problem
can be translated to an optimization problem, as presented
in next section.
III. OPTIMAL SENSOR PLACEMENT PROBLEM
FORMULATION
A. Optimization Problem Statement
Let q be a vector of binary elements that denotes which
candidate sensors are installed or not. q
j
= 1 means that
sensor s
j
∈ S is installed, whereas q
j
= 0 means that s
j
is not. Then, the optimal sensor placement problem can be
formulated as the following optimization problem:
min : J(q) =
m
X
j=1
w
j
q
j
subject to
F
D
is detectable
F
D
is isolable, (1)
where m is the total number of candidate sensors and w
j
is the cost of sensor s
j
comprising purchase, maintenance,
installation and reliability costs.
Problem (1) will be solved for two general cases:
• CASE I: F
I
D
= F
P
.
• CASE II: F
II
D
= F
P
∪ F
S
∗
In CASE I , the Target Fault Set is known a priori, before
solving the optimization problem. In CASE II, this is not true,
since F
S
∗
will be known a posteriori, after the optimization
problem is solved.
To solve (1), fault detection and isolation specifications
must be stated as a set of optimization constraints. Next
sections describe how the Full ARR Table and the Full Fault
Signature Matrix will serve that purpose.
B. The ARR Selector
Given a set of installed sensors S
∗
⊆ S, let ρ
i
be the
binary ARR selector denoting whether ARR r
i
is valid (ρ
i
=
1) or not (ρ
i
= 0), according to S
∗
.
The ARR selector can be expressed as in (2), where set S
and table
b
A are given, whereas q is the optimization vector.
ρ
i
=
Y
s
j
∈S
h
b
A
ij
q
j
+ (1 −
b
A
ij
)
i
(2)
For each candidate sensor s
j
, if r
i
depends on s
j
, this
sensor is required to be installed. If r
i
does not depend on
s
j
, it is not a requirement. Then, r
i
is valid as long as all
required sensors are installed ( i.e., they belong to the current
sensor placement configuration).
For instance, according to Table I, ρ
5
= q
5
q
7
, which
means that r
5
is valid as long as sensors s
5
and s
7
are
installed.
C. Fault Detectability Constraint Formulation
First, CASE I will be considered. The fault detectability
requirement can be expressed as (3), where sets
b
R and F
P
and matrix
c
M are given, and ρ
i
corresponds to (2).
F
I
D
is detectable ↔
X
r
i
∈
b
R
c
M
ik
ρ
i
≥ 1, ∀f
k
∈ F
P
(3)
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 ThPI22.3
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Constraint (3) assures that the column of
c
M which cor-
responds to fault f
k
contains at least one 1 associated to a
valid ARR.
For instance, given Tables I and II, the fault detectability
constraint associated to fault f
1
is q
5
q
6
+ q
8
+ q
5
q
7
≥ 1 and
for fault f
2
is q
6
q
7
q
8
≥ 1. So, if the set of installed sensors
is S
∗
= {s
5
, s
7
}, then f
1
is detectable, whereas f
2
is not.
In CASE II, faults affecting fixed process components
as well as candidate sensors are considered. The constraint
formulation (see (4)) depends on the type of fault considered.
F
II
D
is detectable ↔
X
r
i
∈
b
R
c
M
ik
ρ
i
≥
(
1 if f
k
∈ F
P
,
q
k
if f
k
∈ F
S
.
∀f
k
∈ F (4)
According to (4), (3) is applicable whenever a fixed
process component fault or a candidate sensor fault is
considered, as long as this candidate sensor is installed
(i.e., q
k
= 1). For non-installed candidate sensors, the right
hand side of the inequality becomes 0, meaning that no
detectability property is expected for them.
Given Tables I and II, the fault detectability constraint
associated to fault f
1
is q
5
q
6
+ q
8
+ q
5
q
7
≥ 1 and for fault
f
5
is q
5
q
7
q
8
+ q
5
q
6
+ q
5
q
7
≥ q
5
. So, if the set of installed
sensors is S
∗
= {s
5
, s
7
}, then f
1
and f
5
are detectable.
D. Fault Isolability Constraint Formulation
First, CASE I will be considered. The fault isolability
requirement can be expressed as (5), where sets
b
R and F
P
and matrix
c
M are given, and ρ
i
corresponds to (2).
F
I
D
is isolable ↔
X
r
i
∈
b
R
¯
¯
¯
c
M
ik
−
c
M
il
¯
¯
¯
ρ
i
≥ 1, ∀f
k
, f
l
∈ F
P
, f
k
6= f
l
(5)
Constraint (5) assures that every two columns of
c
M are
different at least in one row associated to a valid ARR.
For instance, given Tables I and II, the fault isolability
constraint associated to faults f
3
and f
4
is q
5
q
7
q
8
+q
5
q
7
≥ 1.
So, if the set of installed sensors is S
∗
= {s
5
, s
7
}, then f
3
and f
4
are isolable.
Again, the constraint formulation for CASE II (see (6))
depends on the type of fault considered.
F
II
D
is isolable ↔
X
r
i
∈
b
R
¯
¯
¯
c
M
ik
−
c
M
il
¯
¯
¯
ρ
i
≥
1 if f
k
, f
l
∈ F
P
,
q
k
if f
l
∈ F
P
and f
k
∈ F
S
,
q
l
if f
k
∈ F
P
and f
l
∈ F
S
,
q
k
q
l
if f
k
, f
l
∈ F
S
.
∀f
k
, f
l
∈ F, f
k
6= f
l
(6)
According to (6), (5) is applicable whenever fixed process
component faults or candidate sensor faults are considered,
as long as the candidate sensors are installed (i.e., q
k
= 1
and q
l
= 1). For non-installed candidate sensors (either s
k
q
p
(t)
q
v
(t)
h
u
(t)
h
l
(t)
u
v
(t)
u
p
(t)
Fig. 1. Two-tanks system
TABLE III
VARI ABLES OF THE TWO-TANKS SYSTEM
Variable Description
h
u
upper tank level
h
l
lower tank level
q
v
valve flow
q
p
pump flow
u
v
valve control input
u
p
pump control input
or s
l
, or both), the right hand side of the inequality becomes
0, meaning that no isolability property is expected for them.
For instance, given Tables I and II, the fault isolability
constraint associated to faults f
4
and f
8
is q
5
q
7
q
8
+ q
8
+
q
5
q
7
≥ q
8
. So, if the set of installed sensors is S
∗
= {s
5
, s
7
},
then f
4
and f
8
are isolable.
IV. APPLICATION TO A TWO-TANKS SYSTEM
A. Process Description
The system is made up of two tanks interconnected by
a pump and a valve (see Fig. 1). In all, there are four
internal variables and two input variables in the system,
as summarized in Table III. So the candidate sensor set
comprises up to six sensors S = {h
u
, h
l
, q
v
, q
p
, u
v
, u
p
}.
Eight hypothetical faults are considered in the system (see
Table IV): leaks in the upper and lower tanks, and wrong
readings of each candidate sensor. So the fault sets are F =
F
P
∪ F
S
= {f
u
, f
l
} ∪ {f
h
u
, f
h
l
, f
q
v
, f
q
p
, f
u
v
, f
u
p
}.
TABLE IV
HYPOTHETICAL FAULTS OF THE TWO-TANKS SYSTEM
Fault Description
f
u
upper tank leak
f
l
lower tank leak
f
h
u
wrong upper tank level sensor reading
f
h
l
wrong lower tank level sensor reading
f
q
v
wrong valve flow sensor reading
f
q
p
wrong pump flow sensor reading
f
u
v
wrong valve control input sensor reading
f
u
p
wrong pump control input sensor reading
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 ThPI22.3
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TABLE V
RESULTS FOR CASE I
Cost distribution 1 Cost distr. 2
Sensor Cost q alternative Cost q
h
u
10 X X X X 100
h
l
100 X X X 10 X
q
v
10 X X X X 10 X
q
p
10 X X X X X X 10 X
u
v
10 X X X X 100
u
p
100 X X X 10 X
TABLE VI
RESULTS FOR CASE II
Cost distribution 1 Cost distribution 2
Sensor Cost q alternative Cost q alternative
h
u
10 X X 100 X
h
l
100 X X X X 10 X X X
q
v
10 X X 10 X X
q
p
10 X 10 X
u
v
10 X X X 100 X X
u
p
100 X X X X 10 X X X
B. Optimization results
Applying the exhaustive ARR generation algorithm de-
scribed in [19] a Full ARR Table and a Full Fault Signature
Matrix was created, comprising up to 35 ARRs.
Problem (1) along with constraints (3) and (5) corresponds
to CASE I. This optimization problem was implemented
in ILOG OPL Studio [24], involving 3 constraints (i.e.,
2 regarding the detectability specification and 1 for the
isolability specification). Results are given in Table V for
different candidate sensors cost distributions. A ’X’ in the
table indicates that the corresponding sensor is installed.
Cost distribution 1 suggested six alternative sensor config-
urations with the same minimum total cost of 130, whereas
cost distribution 2 produced just a unique solution, with a
minimum global cost of 40. Other cost distributions were
tested and the optimization algorithm always suggested a
sensor configuration of 4 sensors.
Problem (1) along with constraints (4) and (6) corresponds
to CASE II. This optimization problem was also implemented
in ILOG OPL Studio, now involving 36 constraints (i.e., 8
regarding the detectability specification and 28 the isolability
specification). Results are given in Table VI for different
candidate sensors cost distributions.
In this case, cost distribution 1 suggested just four alterna-
tive sensor configurations with a minimum total cost of 220.
Despite its higher cost, all these alternatives included sensors
h
l
and u
p
. Thus, these sensors will be part of all solutions,
no matter what cost is assigned to them. In cost distribution
2, three alternatives where given; all included both sensors.
Again, other cost distributions were tested and the opti-
mization algorithm always suggested a sensor configuration
of 4 sensors. So, for the two-tank system, the cardinality of
S
∗
will always be 4.
Notice that for CASE I the optimization algorithm pro-
duced cheaper solutions than for CASE II. This was expected,
TABLE VII
SIMPLIFIED FAULT SIGNATURE MATRIX FOR S
∗
= {h
l
, q
v
, u
p
, u
v
}
f
u
f
l
f
h
l
f
q
v
f
u
p
f
u
v
r
9
0 1 1 1 1 1
r
18
1 0 1 1 1 1
r
23
1 1 1 1 0 1
r
27
1 1 1 1 1 0
r
32
1 1 0 1 1 1
r
34
1 1 1 0 1 1
since constraints in CASE I are more relaxed (i.e., a solution
to CASE I is more easily attainable than for CASE II).
In order to verify that the optimization algorithm is sug-
gesting feasible solutions to the FDI specifications, Table VII
shows the simplified fault signature matrix corresponding
to the first sensor configuration alternative given in Table
VI for cost distribution 1. This simplified fault signature
matrix has been obtained by eliminating, in the Full Fault
Signature Matrix, the columns that correspond to faults f
h
u
and f
q
p
, that affect sensors not installed, and the rows
that correspond to ARRs that depend on these sensors not
installed. Every column in Table VII contains at least one
1, and all columns are different. Thus, sensor configuration
S
∗
= {h
l
, q
v
, u
p
, u
v
} satisfies the detectability and isolabil-
ity specifications.
V. COMPUTATIONAL COMPLEXITY ISSUES
Finding a solution to problem (1) is not trivial because of
its combinatorial nature. As it is known, combinatorial prob-
lems fall in the NP category with a complexity that depends
exponentially with the number of optimization variables. In
particular, solving time for problem (1) clearly depends on
the number and complexity of the optimization constraints,
which in turn, according to section III, depend on the sizes
of the Full ARR Table and the Full Fault Signature Matrix.
On the one hand, for CASE II the number of constraints,
n
C
, depends on the number of faults, card(F) (see (7)).
There is a fault detectability constraint for each fault in F
(see (4)) and a fault isolability constraint for every possible
combination of two faults out of F (see (6)).
n
C
= n
C
¯
¯
detectability
+ n
C
¯
¯
isolability
=
card
(
F
) +
µ
card(F)
2
¶
=
card(F)(card(F ) + 1)
2
(7)
On the other hand, the complexity of the constraints
depends on the number of candidate sensors, card(S) (see
(2)), and the number of all possible ARRs, card(
b
R) (see
(3)-(6)).
In the previous section, an application to a simple two-
tanks system has been presented. In this case, computational
complexity was not a real concern. In order to see the
limitations of the proposed sensor placement method, a more
demanding application was used (see [25]), involving 17
faults and 8 candidate sensors, which, according to (7), posed
an optimization problem with up to 153 constraints.
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 ThPI22.3
2588
