Comparison of Two Distributed Fault Diagnosis Approaches based on Binary Integer Linear Programming (BILP) OptimizationErdalTaskentVicenç Puigvicenc.puig@upc.edu0ReliabilitySafety Engineering SpecialistIzmirTurkey e-mail: etaskent@hotmail.comAdvanced Control System Research Group, Universitat Politècnica de Catalunya (UPC)
,
Barcelona
,
Spain
Two distributed fault diagnosis approaches were compared, by analogy, to determine which is more efficient regarding computational complexity. The first approach considered all “locally computed” global and compound sets with minimal cardinality using a heuristic optimization method while minimizing subsystems interactions (communication). The second approach aimed at obtaining minimal coupled MSOs for minimizing the number of common links between MSOs by adding constraints in already existing optimal sensor placement algorithm, which uses BILP, but not in a distributed context. As a result of comparison, complexity of both approaches is characterized.
1. Introduction
For complex systems with large-scale distribution and
communication constraints, it is appropriate to use
distributed approaches [
1
]. Distributed approaches are
more reliable than centralized approaches in case of the
failure of the centralized diagnoser (also in decentralized
schemes). Moreover, distributed approaches are preferred
because of lack of scalability and efficiency of centralized
solutions during online analysis for large-scale systems
since that can be dealt with complexity by partitioning the
system into subsystems [
11
]. Failures in communication
links and nodes, and degraded diffusion through the
affected nodes by propagation of overloads can lead to
cascading failures. Moreover, transmission delays
increasing the detection time can affect diagnostic
accuracy [
2
]. Reducing communication costs in distributed
contexts requires minimizing data transfer between local
subsystems [
1
]. Therefore, distributed algorithms should
consider the requirements of computational and
communication efficiency. To deal with computational
complexity in distributed algorithms, efficient approaches
for the sensor placement analysis and to compute feasible
MSO sets need to be developed.
The Minimal Structurally Over-determined (MSO) set
approach offers an alternate way to find all ARRs.
According to [
9
], a minimal structurally over-determined
subsystem (MSO subsystem) is a part of the
overconstrained part of a system graph from which removal of
one constraint will make the subsystem to become just
constrained, i.e., structural redundancy 1. Therefore, each
MSO set will consists of in any case one constraint that can
be used as an ARR.
The global FMSO sets are obtained from the set of local
FMSO sets, and the union of locally computed shared sets
which forms a compound FMSO set that includes at least
one shared FMSO set whose fault support is not empty,
contains equations from at least two subsystems.
In this paper, two distributed fault diagnosis approaches
were compared, by analogy, to determine which is more
efficient regarding computational complexity. The first
approach [
1
] considered generates all “locally computed”
global and compound sets with minimal cardinality using a
heuristic optimization method while minimizing
subsystems interactions (communication). The second
approach [
4
] aimed at obtaining minimal coupled MSOs,
for minimizing the number of common links between
MSOs by adding constraints in already existing optimal
sensor placement algorithm [
3
] which uses BILP, but not
in a distributed context.
The two considered approaches deal with the problem of
distributed fault diagnosis (local diagnosis with minimum
global diagnosis) that aims to obtain a set of optimal local
diagnosers that guarantee the same properties as a global
diagnoser. Both approaches target to provide the maximum
possible detectability and isolability that can be achieved
for a system given a set of measurements.
In the first approach [
1
], the Fault-Driven Minimal
Structurally Overdetermined (FMSO) Set concept is
introduced, which can be directly used to construct an
ARR (or residual generator). A heuristic optimization
method to obtain the minimal cardinality set of compound
FMSO sets is used in [
1
]. This optimization procedure can
be improved by using BILP optimization as proposed in
[
2
] utilizing MSOs (each is sensitive to a set of faults) and
the structurally equivalence to the compound FMSO sets
formation shown.
In the second approach [
4
], after applying sensor
placement algorithm proposed in [
3
], a set of minimum
coupled set of MSOs are obtained. The adaptation
presented in [
4
] aims at placing the sensors not only to
guarantee detectability and isolability properties but also to
facilitate the partition of a system into various subsystems
by reducing number of links (communication) within a
system. This algorithm also
minimizes the number of
sensors to be installed thus reducing overall cost.
For solving the BILP optimization without the need of
previous computation of the complete MSOs set, which is
a computationally complex task, some methods have been
developed [
7
]. In [
12
], an efficient method to finding all
the minimal sensors set for maximum fault detectability
and isolability from a structural model is proposed.
The structure of the paper is as follows: Section 2
presents a four tank system used as case study along the
paper. Sections 3 and 4 present the two distributed fault
diagnosis approaches. Section 5 presents the comparison
of the two approaches using the case study presented in
Section 2. Finally, Section 6 draws the main conclusions.
2. Case Study
The case study used to compare both approaches is based
on four tank system, proposed in [
2
], and is shown in Fig.
1. V1, V2, V3, V4 are the volumes of water in each tank, q12,
q23, q34, q4 represents the flows of water through each pipe
(P1, P2, P3, P4), u1 and u2 represents the water sources.
S1
u1
T1
v1
p1
3.1 Background concepts
This approach [
1
], [
6
] establishes a structure connecting
FMSOs with minimum number of shared measurements
(communication) from
neighboring subsystems by an
iterative matching procedure.
The global FMSO sets, Ф, are obtained from the set of
local FMSO sets Φ , and locally computed shared FMSO
sets Φ
and shared CMSO sets Ψ
(different subsystems)
which forms a compound FMSO set [
1
], [
6
].
The
shared
CMSO
(Clear
MinimalStructurallyOverdetermined) set, whose fault support is empty,
corresponds to the measurements (internal (subsystem i)
and from neighboring subsystems (shared variables)).
The FMSO sets including equations with shared variables
are called shared FMSO sets [
1
], [
6
].
The shared variables
don’t include, are considered as known variables [
1
],
= { 12, 2, 23, 3, 34, 4}, which
Compound FMSO set ( ′): A global FMSO set that
includes at least one shared FMSO set whose fault support
is
not empty, contains equations from at least two
subsystems [
1
], [
6
].
by a heuristic method.
The optimal compound FMSO set selection is performed
set of Σ, hence a global FMSO set [
1
], [
6
].
A local FMSO set for any subsystem Σi is also an FMSO
A local FMSO set ( ∈ Φ )'s equations include local and
shared variables of Σ
i and only involve the fault fi. To
achieve detectability of fault fi, only the equations included
in required [
1
], [
6
].
The concept of compound FMSO set allow us to establish
the relation between FMSO sets for the subsystems and
FMSO sets for the global system [
1
], [
6
].
To illustrate the previous concepts the example used in
[
1
], [
2
], [
6
] is used that considers the first tank of the
proposed case study
1 : ̇1 =
Then, the set of shared FMSO sets Φ
is { 1, 2, 3}:
equations [
4
]:
1 : ̇1 =
2 : 12 =
3 : 1 = ∫ ̇1
7 : ̇2 =
8 : 23 =
9 : 2 = ∫ ̇2
:
2 = { 1, 2, 3, 4}, ℎ
3 = { 1, 3, 4, 5}, ℎ
⊆ , ∩ ≠ ∅, and
⊆ ( ∪ )
that covers each shared variable of
FMSO set) given by Figure 4 in [
1
], [
6
].
The procedure to compute a global FMSO set , starts
by searching in the bipartite graph G(X, Г) for a matching
( is the root
According to the operational procedure of Algorithm 1 in
[
6
], [
1
], it is possible to get the set of all global FMSO sets
Φ
and shared CMSO sets Ψ .
Ф from the set of local FMSO sets Φ
found for Ф [
1
], [
5
].
Considering
all the
possible root FMSO
sets, 164
compound FMSO
sets are computed for this system.
Added to 4 = { 1, 3, 4, 5, 6} ∈
FMSO set for subsystem Σ1, the 165 global FMSO sets are
Φ1, which is a local
shared FMSO sets
3.2 Distributed diagnosis
Given a set of faults, measurements and local models for
every subsystem, we now construct local diagnosers that
together make the entire system completely diagnosable.
Using the Algorithm 2 and definitions of Chapter 2 in [
1
],
[
6
], we can develop a local full diagnosis for every
subsystem.
These results demonstrate that all considered faults can
be detected and isolated, e.g. in the considered example,
detectability is achieved
for f
1 using 4 ∈ Φ
(local
FMSO) of Table 4 in [
1
], [
6
] (no additional measurement
is needed). For f2, detectability is achieved obtaining a
compound FMSO set 9 ∈ Φ1 lumping 1 ∈ Φ1 (as root
FMSO
set)
with
Algorithm 2, using a heuristic optimization method,
produces a minimal cardinality set of compound (global)
FMSO sets while minimizing subsystems interactions.
4. Second Approach: Minimal Coupled MSOs4.1 Background concepts
In the second approach [
4
], the graph G(V, E) representing
the set of MSOs is obtained considering that
the MSOs are the graph vertices collected in a set V,
the
measured input/output variables are the graph
edges collected in a set E.
Each MSO set will consists of in any case one constraint
that can
be used
as an
ARR.
MSOs represent the
redundancies in the system and can form the basis for fault
detection and isolation. As given in [
2
], for the running
example, there
were 165
MSOs generated
using the
algorithm proposed in [
12
]. For example for the first tank,
the only MSO = { 1, 3, 4, 5, 6} as 5 is the redundant
equation. The number of ARRs generated in this way will
be larger than the set of ARRs found from a single
complete matchings (ranking algorithm), and get a set of
ARRs for each of these matchings (the number of ARRs
was 16 (C1, …, C16) as obtained in [
4
]).
The second approach by ARRs in original [
4
] performs
better concerning computational complexity, without the
need of previous computation of the complete MSOs set.
For the comparison on a common basis, in this work we
used MSOs instead of ARRs in the second approach. The
analysis is to be shown with MSOs as the same carried out
by ARRs, judging that the inference will be equivalent.
After obtaining the model from the sets of equations or set
of all MSOs, sensor placement algorithm [
3
] is applied.
A binary matrix W = [wij] of size n× k containing the set
of MSOs (the row set) and sensors (the column set) is
formed. Matrix W refers to the set of sensor faults an MSO
is sensitive to. In the same way, the binary matrix V = [vij]
of size n× l relates the set of MSOs (the row set) and
process faults (the column set). These relations are known
as fault signature matrix (FSM) [
9
]. After obtaining W and
V (the process faults not shown here) according to [
4
], the
values of W and V are used to find various constraints in
(6) [
4
], [
13
].
4.2 Minimizing the coupling between MSOs
In order to facilitate the distributed implementation of the
fault diagnosis systems, the sensors should be placed such
that the coupling between
MSOs is minimized. This is
achieved by adding additional constraints that minimizes
the number of common links between MSOs [
4
], [
13
].
First, a constraint that reduces the number of row links
coupling is written in compact form as
An analog constraint could be added to minimize the row
links coupling as follows
Wi j
Wi j
T
(0)ip
Iii
0 jy
0iu
c j1
0i1
0 j1
(4)
(5)
The MSOs (the row set (vertices)) are added.
Additional constraints
were added in the existing
optimal sensor placement algorithm using Binary Integer
The values of q, λ, rows (r), columns (c) obtained are
shown in Table 5.3 in [
13
]. The result shows that only 4
sensors are needed (minimization of sensor) which
measures the volume variable V1, V2, V3, V4 and only four
MSOs will be required to satisfy the detectability and
isolability of faults in these sensors, minimizing at the
same time the degree of coupling among the obtained
MSOs according to (4) and (5) [
4
]. The algorithm chooses
these four MSOs since it fulfills the necessary conditions,
firstly the solution obtained allows to isolate the fault
which is shown in Table 2 [
4
] how the solution fulfills the
necessary conditions of isolability (since each column is
different so it is isolable, shown in Table 2 [
4
]), secondly
the solution obtain is detectable since a unique solution is
obtained, thirdly the solution obtained gives equal number
of 1’s in respective rows and columns shown in Table 3 [
4
]
which is a necessary condition for formation of a system.
It is seen from the Table 2 that the analysis results in 4
vertices and 6 edges, whereas the ideal solution should
produce 4 edges as 1’s are only on the diagonal elements
on FSM.
The algorithm chooses MSOs also in such manner that
these MSOs form a system with minimum coupling by
choosing MSOs with minimum number of 1’s in rows and
columns (the ideal solution by algorithm for this case is
diagonal matrix, with diagonal elements are 1’s and rest of
other element are 0) but such solution is not possible in
this case. A minimum coupled or decoupled system can be
divided into various subsystems in much better way as
compared to a highly coupled system. In this case, the
system can be divided into two subsystems using the
approach presented in [
8
]. The first subsystem is formed
by MSOs, MSOi-1, MSOi (Tank 1 and Tank 2) and the
second subsystem is formed by MSOs, MSOi+1, MSOi+2
(Tank 3 and Tank 4). After dividing the system into two
subsystems, the fault signature matrices are created
following the decentralized fault diagnosis algorithm in [
8
]
(summarized in Section II) and after creating fault
signature matrices, fault detection and isolation can be
carried out. The proposed algorithm in this chapter allows
obtaining a minimum coupled system by which
partitioning of the system into various subsystems become
easy as compared to the subsystems obtained by
decentralized fault diagnosis algorithm described in [
8
]
where we get a highly coupled system using ranking
algorithm [
9
]. Tank 1 and Tank 4 have no coupling with
each other and the only coupling between two subsystems
is single coupling between MSOi and MSOi+1 (Tank 2 and
Tank 3), the system obtained is one of the least coupled
system, can be seen in Fig 5.2 in [
4
]. The other system can
also be MSOi+3, MSOi+4, MSOi+5 and MSOi+6 shown in Fig 5
in [
4
] since this system has the same type of coupling
when compared with chosen system but the cost or weight
of sensor used for MSOi-1, MSOi, MSOi+1, MSOi+2 is lower
than the cost of the sensors used for MSOi+3, MSOi+4,
MSOi+5 and MSOi+6 thus it can be seen that the algorithm
chooses the best system according to least coupling and
cost.
… , MSOi-3, MSOi-2
MSOi-1, MSOi, MSOi+1, MSOi+2
MSOi+3, …
If we cannot obtain the "best" matching property with
all the possible MSOs, the system is not structurally
monitorable and we have to place some additional sensors.
5. Comparison of Two Approaches
The matrix sizes regarding efficiency in computational
complexity for each approach are demonstrated. As the
comparison objective, the approaches are assessed under
the case of using, for both, a Binary Integer Linear
Programming (BILP) for optimization.
The first approach [
1
] considered Fault-Driven Minimal
Structurally Overdetermined Sets, used a heuristic
optimization method to obtain the minimal cardinality set
of compound FMSO sets.
To be able to apply a BILP optimization in the first
approach, the structurally equivalence of the model of
Khorasgani’s approach [
2
], which uses BILP, to the
compound FMSO sets formation and its obtaining the
equivalent results with the first approach were shown.
Hence, we can use this approach for the first approach [
1
]
for that to be comparable with the second approach [
4
] in
terms of matrix sizes.
Then, the matrix size increase for the second approach
to obtain minimal coupled MSOs, by adding constraints in
already
existing
optimal sensor
placement
algorithm
(BILP) [
3
], was demonstrated to see which approach is
more efficient.
The red colored equation relates to the shared CMSO set
(variable) from
neighboring subsystems, corresponding
minimal subsystems interactions (one or possibly more
subsystem 1(Si)
shared FMSO sets
(as root FMSO set)
and 2 ∈ Ψ2 .
The shared CMSO set, whose fault support is empty,
corresponds to the
measurements from
variables).
internal
measurements
neighboring
subsystems (shared
(Si)
From the Table 1, Φ
(i = 1..4) by a heuristic method, 5
optimal compound FMSO sets for the 4 subsystems are
obtained from 164 compound FMSO sets as given below:
from each nearest neighbor), the blue colored equation
relates to the shared CMSO set from subsystem i (Si), the
black colored equation to the shared FMSO set from S
(the root), and the yellow colored equation to the shared
i
FMSO set from a neighboring subsystem.
5.2
Applying Khorasgani’s BILP Approach [2] to
the First Approach in Optimizing the
Compound FMSO Sets by Analogy
Khorasgani [
2
] used in this approach the Binary Integer
Linear Programming (BILP) for optimization.
The
considered as structurally equivalent to the compound
FMSO set model of the first approach, and hence, we can
use this approach, which uses BILP, in the first approach
for that to be comparable with the second approach:
As given in [
2
], for the running example there were 165
MSOs generated, 3 measurements in the subsystem 1, and
8 measurements for the entire system.
Subsystem 1 has two faults of interest, and the goal is to be
able to isolate them from any of the 6 faults in the
complete system.
Therefore, to solve the optimization problem in [
2
] for
subsystem 1 (S1), matrix A has
177 rows (equal to the number of constraints):
2 constraints to guarantee the local detectability
of f1 and f2,
10 constraints to guarantee the local isolability of
f1 and f2 from the other faults, and
165
constraints
to
capture
the
between the MSOs and the measurements and
173 columns (equal to the number of binary variables):
8 constraints for the measurements 165 constraints for the MSOs and b is a vector with 177 elements (equal to the number of constraints).
Table 5 shows, in the MSOk-l form, the minimum number
of shared
measurements (from
neighboring subsystems
and possibly one from each neighbor) obtained with the
BILP Optimization. Table 5: Set of augmented measurements to each subsystem model [2]. Subsystem
neighboring subsystems as the CMSO sets (
9,10,11,12,13 (5 optimal compound FMSO sets Φ
) of
, (i =
∈ Ψ
subsystem
method [
1
].
neighboring subsystems are the same except for S2, u2 in
[
2
] is selected instead of y5 in [
1
] from the same subsystem
(S3). This difference comes from the use of shared FMSO
set from a neighboring subsystem ( 13used in 11) in this
approach [
1
] (see equation (8) and Table 5). For both
application ([
2
], Table 5 and [
1
], Table 6), subsystem 2 is
the only subsystem that shares a variable with a second
order connected subsystem, all the other subsystems only
need to communicate
with their first order (nearest)
connected subsystems. To
minimize the number of
S1
S2
S3
S4
S1
S2
S3
S4
relationship
comparison.
measurements from the other subsystems, as given in the
cost function in Eq. 9 [
2
], the cost will incur only the
external measurements from the neighboring subsystems
and not the measurements as internal in Si and as the use
of shared FMSO set from a neighboring subsystem with
the root shared set (Si).
Therefore, we could decide that both application lays on
the same basis and
Khorasgani’s approach using BILP
(i.e.,
the
is not empty and then there are two faults (f4, f5) included
in 11 corresponding two shared FMSO sets, but, since the
local diagnoser 12 will respond to its subsystem’s (S3)
fault (f5), in achieving global diagnosability, 11 can act
for only the fault (f4) of the root shared set (S2) not the one
(f5) of the shared set in the neighbor (S3).
Alternatively by another algorithm, a different compound
FMSO set for S
2 can also optimally select the same
measurements from the neighbors in Table 5 (possibly
more than one from each first order (nearest) neighbor)
which provides a practical advantage by not needing to
transfer data over long distances, which can be costly and
error-prone [
2
].
This
approach
[
2
]
now
establishes a structure
connecting FMSOs, in compound, with minimum number
of shared measurements from neighboring subsystems.
If we apply Binary Integer Linear Programming (BILP)
instead of a heuristic method in the first approach, for the
four tanks (S1, S2, S3, S4), similar to the analysis in [
2
]:
subsystem 1 (S1), matrix A has
176 rows (equal to the number of constraints):
164
constraints
to
capture
the
relationship
between the FMSOs and the measurements
In this case, 164 constraints are used for the relationship
corresponding to the number of all compound FMSO sets
computed in the first approach except for the one local
FMSO set.
Since we have 165 global FMSOs (165 constraints) in this
approach as given in [
1
], the total number of columns (c)
for each subsystem is 173.
for all the subsystems, the matrix A:
subsystem 1 (S1)
2 faults
3 measurements
subsystem 4 (S4)
1 fault
1measurement
1
5
164
170
173
for the system (S1, S2, S3, S4):
176 + 176 + 170 + 170 = 692 (r)
The analysis results in (692, 173) element matrix. This
matrix size is to be processed if we apply BILP to the first
approach for optimizing the compound FMSO sets, which
is to be compared with the one to be obtained in Section
5.3.
5.3 Second Approach: Minimal Coupled MSOs
The analysis is performed for the four tank example:
As obtained in the second approach in Section 4.1, using
the methodology in [
3
] as the first step:
n number of MSOs
The detectability constraints (14) and (17) in [
3
]
involve
The isolability constraints (20), (24), and (30) in [
3
]
involve
2 + . + 2 = 1 + 16 + 28 = 45 rows
n (λ) = 165 rows
l + k (q) = 10 rows
(10)
(11)
(12)
The matrix size by adding additional constraints, the row
set, to choose MSOs that form a system with minimum
coupling (communication) was shown below:
From Section 4.2, using equations (4) and (5) [
4
], we
checked how these additional constraints act.
For each constraint, with the values as the numbers of
MSOs (n (λ) = 165) and (candidate) sensors (q = 8) in
columns, a simple validation performed first to see that the
corresponding number of columns does not change by this
approach as 173 in total as a joint effect of the constraints,
thus maintaining the number of columns obtained with
using [
3
]:
The rows added for each constraint as 165 and 8 are given
in equation (13).
In comparison, it is shown that for optimization the
second work [
4
] performs better as (393, 173) element
matrix than the first approach [
1
] processing a (692, 173)
matrix in terms of computational complexity.
Two distributed fault diagnosis approaches were
compared, by analogy (i.e., the matrix sizes) to determine
their efficiency in the case of using, for both, a Binary
Integer Linear Programming (BILP) for optimization using
a four-tank system example.
Though, as demonstrated in the first approach [
1
], the
Fault-Driven Minimal Structurally Overdetermined
(FMSO) Sets can be directly used to construct an ARR or
residual generator, the matrix sizes to be processed for
computing the optimal sets (apart from the computation of
all global (compound) sets) were assessed.
Since the first approach used a heuristic optimization
method to obtain the minimal cardinality set of compound
FMSO sets, we applied Khorasgani's BILP optimization
method, utilizing a structurally equivalent model to the
compound FMSO sets formation, to the first approach to
decide that Khorasgani’s approach using BILP obtains the
equivalent results so as to be used in the first approach for
the purpose of comparison.
Then, we applied Binary Integer Linear Programming
(BILP) instead of a heuristic method for the four tanks to
find the matrix size to be processed in this case.
In the second approach [
4
], minimum coupled MSOs for
minimizing the number of common links (communication)
between MSOs are obtained by adding constraints (the row
set/MSOs) in already existing optimal sensor placement
algorithm [
3
], which uses BILP, but not in a distributed
context and thus uses the complete set of MSOs.
For the comparison on a common basis, in this work we
used MSOs instead of ARRs in performing the analysis of
the second approach.
In the original second approach [
4
], in applying sensor
placement algorithm, for solving the BILP optimization
without the need of previous computation of the complete
MSOs which requires a high computation time, the ARRs
(the model) were generated using ranking algorithm and
all the possible ARRs in addition to the primary ARRs
obtained from the set of system model equations.
After solving the sensor placement problem, the
algorithm ensures a set of minimum coupled (minimal
sensors) set of MSOs for maximum fault detectability and
isolability.
In comparison, it is shown that the second work
performs better in terms of the matrix sizes to handle. Then
again, it is preferential to use the second approach with
ARRs in original [
4
] concerning computational
complexity, in that case resulting as (95, 24) matrix.
In addition to this work, adding redundant sensors (the
column set) to obtain the ideal solution (best matching) in
the fault signature matrix can be shown.
As the most efficient approach from the required MSO
sets point of view causal computation approach can be
studied which needs no MSO sets.
The uncertainty in the system could be studied by using
statistical and stochastic methods for robust distributed
fault detection and isolation.
Acknowledgements
This work was partially funded by the Spanish State Research
Agency (AEI) and the European Regional Development Fund
(ERFD) through the projects DEOCS (ref.
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