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Process decomposition and test selection for distributed
fault diagnosis
Elodie Chanthery, Anna Sztyber, Louise Travé-Massuyès, Carlos Gustavo
Pérez-Zuñiga
To cite this version:
Elodie Chanthery, Anna Sztyber, Louise Travé-Massuyès, Carlos Gustavo Pérez-Zuñiga. Process
decomposition and test selection for distributed fault diagnosis. 33th International Conference on
Industrial, Engineering & Other Applications of Applied Intelligent Systems (IEA/AIE 2020), Sep
2020, Kitakyushu, Japan. �hal-02875561�
distributed fault diagnosis
E. Chanthery1, A. Sztyber2, L. Trav´e-Massuy`es1, and C. G. P´erez-Zu˜niga3
1
LAAS-CNRS, University of Toulouse, CNRS, INSA, Toulouse, France {elodie.chanthery,louise}@laas.fr
2 Institute of Automatic Control and Robotics
Warsaw University of Technology, Warsaw, Poland anna.sztyber@pw.edu.pl
3
Engineering Department, Pontifical Catholic University of Peru, PUCP, Peru gustavo.perez@pucp.pe
Abstract. Decomposing is one way to gain efficiency when dealing with large scale systems. In addition, the breakdown into subsystems may be mandatory to reflect some geographic or confidentiality constraints. In this context, the selection of diagnostic tests must comply with decom-position and it is desired to minimize the number of subsystem intercon-nections while still guaranteeing maximal diagnosability. On the other hand, it should be noticed that there is often some flexibility in the way to decompose a system. By placing itself in the context of structural analysis, this paper provides a solution to the double overlinked prob-lem of choosing the decomposition of the system by leveraging existing flexibility and of selecting the set of diagnostic tests so as to minimize subsystem interconnections while maximizing diagnosability .
Keywords: structural analysis · diagnosis test selection · system de-composition.
1
Introduction
One way to manage complexity is to use the precept of ”divide and conquer”. This is all the more true as the current systems are complex and large-scale. However, when constructing health monitoring processes such as diagnosers for such systems, decomposition influences feasible diagnostic tests. Indeed, these must be evaluable by measurements local to each subsystem or, if this is im-possible to reach the desired level of diagnosability, minimize the interactions required between subsystems.
In the literature, there are two points of view. Either the breakdown of the system is fixed by strong constraints, whether geographic, functional, or con-fidential, or it is completely flexible (cf. the related work section). In the first case, diagnostic tests must adapt to the decomposition while in the later, it is
This project is supported by ANITI through the French ”Investing for the Future – PIA3” program under the Grant agreement noANR-19-PI3A-0004.
the decomposition which adapts to the chosen diagnostic test. However, these two extreme cases do not reflect the reality of most systems. In practice, the constraints only define a partial decomposition and there remains a certain flex-ibility. By placing itself in the context of structural analysis, this paper provides a solution to the double overlinked problem of choosing the decomposition of the system by leveraging existing flexibility and of selecting the set of diag-nostic tests so as to minimize the required subsystem interconnections while maximizing diagnosability. This is performed thanks to an iterative algorithm that guarantees to improve the solution at each iteration, providing a nice and efficient distributed fault diagnosis architecture.
The paper is organized as follow. Section 2 presents the works that are related to the proposal of this paper. Section 3 presents the formulation of the problem. Section 4 recalls the prerequisites concerning structural analysis for fault diag-nosis and the graph partitioning problem that is used for system decomposition. Section 5 explains our iterative DecSel algorithm that decomposes the system and selects optimal tests, improving the solution at each iterative step. Finally, Section 6 concludes the paper.
2
Related work
Diagnosis and test selection are crucial problems to maintain the health status of a system, and consequently, a large body of research is devoted to these problems. Part of these works position themselves in a distributed framework in which the system is decomposed in subsystems. Another subset formulates the problem in a structural analysis framework like we do. It is by these two aspects that these works are related to this paper. Two main tracks exist: either the breakdown of the system is fixed by strong constraints, whether geographic, functional, or confidential, which limit the feasible diagnostic tests or it is completely flexible and based on the selected diagnostic tests. To the best of our knowledge, no prior work attempted to consider the situation in which constraints only define a partial decomposition and there remains a certain flexibility.
In the first track, we find [3] which presents an approach for decentralized fault-focused residual generation based on structural analysis. The work in [13] is a direct continuation of [3], the diagnostic tests calculated with the decentral-ized architecture are the same as those computed in a centraldecentral-ized architecture. However, a more efficient algorithm (BILP) is used for test selection.
In [11, 8] distributed diagnosis approaches are proposed, where only local models and limited information of neighboring systems are required. [14] shows a design of communication network for distributed diagnosis, where subsystems are given and contain subsets of variables, communication - transferring values of known variables using BILP formulation.
The work of [15] proposes a fuzzy diagnostic inference in the two-level de-centralized structure where subsystems contain faults and residuals and infor-mation is transferred to second level if needed. Decomposition of the diagnostic system relies on association of diagnostic subsystems with the separate
techno-logical nodes. Nevertheless, all these methods consider pre-existing constraints of confidentiality, distance, or information access limitations through inter-level communications as mandatory and therefore considers totally predefined sub-systems.
In the second track, we find the possible conflicts approach proposed in [2] where possible conflicts are presented as a way to decompose the system. Possible conflicts can be used to generate diagnostic tests, hence the decomposition is such that no interconnection is needed between subsystems. As a drawback, no practical constraint is taken into account by the decomposition.
Concerning the decomposition process, the authors of [4] present an opti-mal decomposition for distributed fault detection applied to linear models with known parameters. It is based on a cost function to measure the detection per-formance of the fault diagnosis decomposition method. The authors argue that minimization of coupling is not necessarily the best (based on numerical results) and minimize detection time.
An approach for diagnostic system decomposition minimizing interconnec-tions between subsystems was proposed in [9]. The method is based on quali-tative models in the form of graphs of a process (GP), introduced in [16], and diagnosis system. Variables and faults are divided into disjoint subsystems. De-composition is based on causal links and residuals are assigned to subsystems after decomposition in a way minimizing interconnections.
As it was shown, the development of fault diagnosis systems applying dis-tributed and decentralized architectures has become an active research area, however, most of these works consider predefined subsystems, with few ap-proaches proposing solutions to the problem of decomposition.
3
Problem formulation
Let us consider a continuous system described by a set of neequations Σ(z, x, f)
including known (or measured) variables given by a vector z, unknown variables given by a vector x and possible faults that may occur on the system given in the vector f. The possibility to generate diagnosis tests from Σ(z, x, f) relies on the analytical redundancy embedded in the model. Diagnosis tests, or residual generators, can be implemented by relations that only involved measured vari-ables and their derivatives of the form arr(z0, ˙z0, ¨z0, ...) = r, where r is a scalar
signal named residual.
Definition 1 (Diagnosis test for Σ(z, x, f)). A relation of the form arr(z0, ˙z0,
¨
z0, ...) = r, with input z0 a subvector of z and output r, a scalar signal named
residual, is a residual generator for the model Σ(z, x, f) if, for all z consistent with Σ(z, x, f), it holds that lim
t→∞r(t) = 0.
The relation arr(z0, ˙z0, ¨z0, ...) = r is qualified as an Analytical Redundancy
Relation (ARR). The resulting diagnosis test can be used to check whether the measured variables z are consistent with the system model.
As detailed later in Section 4.1, each diagnosis test arises from a subset of equations of Σ(z, x, f), also called its equation support, and it can be designed so as to be sensitive to a specific subset of faults qualified as its fault support. When the residual is non zero, it means that at least one of the faults of the fault support has occurred.
Without loss of generality, a decomposition D of the system leads to subsys-tems denoted Σi(zi, xi, fi), with i = 1, ..., ns, where zi is the vector of known
variables, xithe vector of unknown variables, and firefers to the vector of faults.
The corresponding sets of unknown variables, known variables, and faults are defined as Xi∈ X, Zi ∈ Z, Fi∈ F of Σi.
Definition 2 (Shared variables in D). xs is said to be a shared variable in
D if there exist two subsystems Σi and Σj such that xs ∈ Xi∩ Xj. The set of
shared variables of the whole system Σ in a given decomposition D is denoted by Xs
D.
Note that a variable that is involved in two equations that belong to the same subsystem is not a shared variable and it is qualified as a joint variable.
Knowing the decomposition D and the set of diagnosis tests Φ for Σ, it is possible to evaluate the total number of interconnections between subsystems required to evaluate diagnosis tests.
Definition 3 (Number of interconnections between two equations in D). Let eiand ejbe two different equations of Σ. The number of interconnections
between ei and ej in D, denoted ID(ei, ej) is the number of shared variables
between ei and ej.
Definition 4 (Number of interconnections of ϕ in D). Let ϕ be a subset of equations of Σ. The number of interconnections of ϕ in D is:
ID(ϕ) =
X
ei,ej ∈ϕ i6=j
ID(ei, ej) (1)
Let S ⊂ Φ be a selection of diagnosis tests {ϕ}, then the total number of interconnections for S in D is defined as:
ID(S) =
X
ϕ∈S
ID(ϕ) (2)
Knowing D, the goal is to select the subset S∗ of diagnosis tests that
min-imizes the number of interconnections and maxmin-imizes diagnosability, i.e. the number isolable fault pairs (cf. Definition 8), also called the isolability degree.
In several works like [13], it is hypothesized that the decomposition in sub-systems is guided by functional constraints. This paper aims at removing this hypothesis and searching for a decomposition that improves the efficiency of the architecture by decreasing the number of interconnections while keeping maxi-mal diagnosability.
Furthermore, some additional constraints are added to the problem formula-tion. The first constraint is that the number of subsystems in the decomposition is fixed at the beginning of the process. This is an important hypothesis that could be studied in future works. The second constraint is that the number of equations in the subsystems defined by the decomposition D has to be as much balanced as possible. This constraint aims at avoiding the algorithm to put all the equations in the same subsystem, hence trivially minimizing the number of interconnections.
As far as the criterion is concerned, the objective is to minimize the number of interconnections, but we have added a second term to avoid adding ”free” structurally independent diagnosis tests, i.e. without any interconnections.
The formal definition of the criterion can be written as follows. Once a de-composition D of ns subsystems is chosen, the goal is to select a subset S of
diagnosis tests that minimizes C(S) defined by:
C(S) = ID(S) + α|S| (3)
where α ∈ R+ is such that the term α|S| is lower than 1. For example, it is always possible to take α = |Φ|1 .
The goal of this paper is to present a method for finding simultaneously the decomposition D and the subset S of diagnosis tests to minimize the total number of interconnections while still guaranteeing maximal diagnosability. The selection of D and S is a challenging task, because the optimal D depends on the optimal S and vice versa. The problem of simultaneously selecting decom-position and diagnosis tests is computationally intensive. Therefore we propose an iterative algorithm, named DecSel, which computes D and S in alternating steps, improving the solution at each step.
The solution that we propose relies on graph theory to formalize the decom-position search and on structural analysis to formalize the diagnosis tests search. These two frameworks are perfectly consistent given that the structural model of a system Σ can be represented by a graph and that the generation of diagnosis tests has a graphical interpretation.
4
Prerequisites
4.1 Structural analysis for fault diagnosis
Structural analysis consists in abstracting a model by keeping only the links between equations and variables. The main advantages of structural analysis are that the approach can be applied to large scale systems, linear or non linear, even under uncertainty. The structural model of Σ(z, x, f) can be represented by a bipartite graph G(Σ ∪ X ∪ Z, E ), where E is a set of edges linking equations of Σ and variables of X and Z. An edge exists if the variable is involved in the equation.
When used for fault diagnosis purposes, structural analysis can be used to to find subsets of equations endowed with redundancy. Structural redundancy is defined as follow.
Definition 5 (Structural redundancy). The structural redundancy ρ
Σ0 of
a set of equations Σ0 ⊆ Σ is defined as the difference between the number of equations and the number of unknown variables.
Actually, subsets of equations endowed with structural redundancy have been proved to be the equation support of diagnostic tests that may take the form of ARRs. ARRs are designed offline [1] and then the corresponding diagnosis tests are used on-line to check the consistency of the observations with respect to the model.
4.2 Fault-Driven Minimal Structurally Overdetermined (FMSO) sets
Previous work [10] defined the concept of Fault-Driven Minimal Structurally Overdetermined (FMSO) sets that we recall briefly here.
Let us define Fϕ as the set of faults that are involved in a set of equations
ϕ ⊆ Σ(z, x, f).
Definition 6 (FMSO set). A subset of equations ϕ ⊆ Σ(z, x, f) is an FMSO set of Σ(z, x, f) if (1) Fϕ 6= ∅ and ρϕ = 1 that means |ϕ| = |Xϕ| + 1, (2) no
subset of ϕ is overdetermined, i.e. with more equations than unknown variables. The set of FMSO sets of Σ is denoted Φ.
FMSO sets are proved to be of special interest for diagnosis purpose because they can be turned into ARR. Indeed, by definition, all the unknown variables of Xϕcan be determined using |ϕ| − 1 equations and can then be substituted in the
|ϕ|th equation to generate an ARR off-line. They are used on-line as diagnosis
tests.
The concept of FMSO set is also relevant to define detectable fault, isolable faults and ambiguity set. We recall here these definitions.
Definition 7 (Detectable fault). A fault f ∈ F is detectable in the system Σ(z, x, f) if there exists an FMSO set ϕ ∈ Φ such that f ∈ Fϕ.
Definition 8 (Isolable faults). Given two detectable faults f and f0 of F , f 6= f0, f is isolable from f0 if there exists an FMSO set ϕ ∈ Φ such that f ∈ Fϕ
and f06∈ Fϕ.
Definition 9 (Ambiguity set). An ambiguity set is a set of faults not isolable two by two. The set of ambiguity sets is denoted A.
4.3 Graph partitioning
Given the structural model of the system Σ(z, x, f), one can represent the in-terconnections existing among equations by a graph G(Σ, A), where the set of nodes Σ is the set of equations and A is the set of edges linking equations of Σ. An edge (ei, ej) exists if there is a shared variable between equation ei and
ej. The graph G(Σ, A) can be used to support system decomposition thanks to
graph partitioning algorithms.
The graph partitioning problem consists in dividing nodes of a graph into ns
parts, as equal as possible, in a way that minimizes interconnections between the parts. If graph partitioning is applied to G(Σ, A), it hence breaks down the system into subsystems that are connected as loosely as possible.
The edges of the graph can be weighted according to some property of the corresponding links. Formally, the graph partitioning algorithm partitions Σ into nssubsets Σ1, . . . , Σns, where Σi∩ Σj= ∅ for i 6= j,
S
i
Σi= Σ, minimizing:
ncuts= Σw(ei, ej) : ei∈ Σk, ej∈ Σl, l 6= k (4)
where w(ei, ej) is the weight of the edge between ei and ej.
Graph partitioning is an NP-complete problem, but there exist algorithms for finding approximate solutions [7, 6].
5
Algorithms
5.1 Main loop algorithm
As said before, the problem of simultaneously selecting decomposition and di-agnosis tests via FMSO sets is computationally intensive. In the following we propose the algorithm DecSel (Decompose and Select) given by Algorithm 1 that is guaranteed to improve the solution at each iteration k.
Algorithm 1: DecSel Algorithm
/* Initialization */
1 S−1= ∅, k = 0 ;
/* Section 5.2 */
2 D0 ← Compute initial decomposition taking into account existing constraints ;
/* Section 5.3 */
3 S0← Select FMSO sets minimizing interconnections from the decomposition
D0;
/* Main loop */
4 while (Sk6= Sk−1) do
/* Section 5.2 */
5 Gk+1(V, Ak+1) ← Build a new graph from the selected FMSO sets Sk;
6 Dk+1 ← Compute decomposition on Gk+1;
/* Section 5.3 */
7 Sk+1← Select FMSO sets minimizing interconnections from the
decomposition Dk+1;
Section 5.2 presents the way to obtain the initial decomposition of the system, named D0. It also presents how to obtain, at each iteration k, a new
decompo-sition of the system Dk+1 given the current selection of FMSO sets Sk. Then,
Section 5.3 presents how to obtain a new selection Sk+1of FMSO sets minimizing
interconnections between the subsystems given the decomposition Dk+1.
5.2 System decomposition
Initial system decomposition We build a graph G0(Σ, A0) such that Σ is
the set of equations in the system model and A0the set of edges of the graph. An
edge (ei, ej) exists if there is a joint variable to equation ei and ej. The weight
of edge (ei, ej) is the number of joint variables to ei and ej.
Complex process decompositions have to take into account constraints related to physical components, geographical location and confidentiality. In this work we interpret this requirement by assuming that some equations must be assigned to some specific subsystem a priory, leaving flexibility on the assignment of the remaining equations. The graph partitioning algorithm accounts for these constraints by adding some artificially big weights to the edges between equations which should be in the same subsystem.
System decomposition given selected FMSO sets To account for the selected FMSO sets, we use a graph with edges representing only active connec-tions, i.e. edges between equations which are both in the support of one of the selected FMSO sets.
From the selected FMSO sets Sk, we build a new graph Gk+1(V, Ak+1) such
that V is the set of equations in the system model and Ak+1the set of edges of
the graph. One edge exists if there is a joint variable to equation ei and ej and
if ei and ej are in the support of the same selected FMSO set. The weight of the
edge (ei, ej) is the number of joint variables to eiand ej. The graph partitioning
algorithm is applied to Gk+1(V, Ak+1) and a new system decomposition Dk+1is
computed.
5.3 FMSO set selection
This subsection explains the FMSO sets selection given the decomposition Dk.
The selection is performed by an A∗ (A-star) algorithm. This idea has been initially proposed in [11]. The A∗ algorithm [5] is an algorithm that is widely used in path-finding to find a shortest path in a graph between a start node n0
and one end node included in a set of goal nodes denoted Vf. We recall here the
main concepts used in A∗, but interested readers should refer to [5] for further
information.
Considering a search weighted graph G∗(V, E) with nodes V and edges E weighted by a cost, the A∗ algorithm is a best-first search that searches a path associating to each node v a value f (v) = g(v) + h(v), where:
– h(v) is a heuristic estimate of the cost of a path from node v to a node included in Vf. This value is a mere estimation rather than an exact value.
The more accurate the heuristic the faster the goal state is reached and the higher the accuracy.
– f (v) = g(v) + h(v) is the current approximation of the shortest path to a goal state. f is called the evaluation function. f (v) is calculated for any node v to determine which node should be expanded next.
At each iteration of the algorithm, the node with lowest evaluation function value is chosen for expansion.
In our FMSO set selection problem, the weighted graph G∗= (V, E) is built from the current decomposition Dkand the set Φ of system’s FMSO sets. A path
identifies a selection S ⊆ Φ of FMSO sets. The originality of the work is that each node vi∈ V has a state that corresponds to a set of ambiguity sets Ai and edges
correspond to the assignment of one FMSO set ϕi to S. The initial node v0has
state A0= F = {{f0, f1, . . . , fnf}}, which is reduced to one single ambiguity set
representing the whole set of faults. A node vi of the search graph is identified
by the ambiguity state Ai resulting from the FMSO sets that have been selected
on the path from the start node v0 to node vi. In the case where all the faults
are isolable, a goal node vf ∈ Vf has state Af = {{f0}, {f1}, . . . , {fn}}.
The neighbors of a node vi are all the nodes that can be reached by selecting
one additional FMSO set to be added to S that increases the cardinal of the ambiguity set Ai.
The weight of the edge representing the selection of the FMSO set ϕiis equal
to:
c(ϕi) = IDk(ϕi) + α, (5)
with α defined as in Eq. 3.
g(vi) is the sum of weights in the path from v0 to vi. For example, let S =
{ϕi1, ϕi2, . . . , ϕik} be a path from the initial node v0 to the current node vi,
then: g(vi) = k X l=1 c(ϕil). (6)
A heuristic estimate for a node has been proposed in [11] assuming that the cost of each test is 1. It uses the notion of dichotomic cut that has been proved to be the most efficient manner to increase the cardinal of the set of isolability sets. The same heuristic is used in this work, adapted to the cost given in (5). The heuristic uses the minimum number of FMSO sets that are necessary to disambiguate all the sets Aji of the ambiguity set Ai and is given by:
h(vi) = M axj & ln(|Aji|) ln(2) ' ( min ϕ∈φ\S IDk(ϕ) + α) (7)
where the Aji are the different ambiguity sets of the set Ai at step i.
Running the A∗ algorithm on a decomposition Dk, results in a selection of
5.4 Application
We use a four tank system, illustrated in Fig. 1, to test the proposed algorithm. The model for this system is composed of 20 equations e1 to e20 given in [8].
Fig. 1: Four Tank System
All decompositions in the paper were calculated using the approximate graph partitioning algorithm proposed in [6]. Computations are done on Ubuntu run in VirtualBox on Windows with Intel i7-5500U CPU 2.40 GHz.
We assume that the number of subsystems is equal to 4. We also assume that measurements and faults are assigned to the same subsystems as in [8] and [12]. The sets of equations specified by constr should belong to the same subsystem, the rest of the equations is left free and will be assigned to subsystems by the successive decompositions in a way minimizing the number of interconnections. Fig. 2 illustrates all steps of DecSel algorithm. The initial decomposition D0 with constraints constr is shown in Fig. 2(a). The equations in each of the
subsets indicated by constr must be assigned to the same subsystem by the de-composition whereas the other equations are free. The obtained subsystems are depicted by colors. The cost of decomposition D0, given by Eq. 4, is ncuts=19,
which provides the sum of weights between nodes in different subsystems, i.e. the number of shared variables in total. The FMSO sets selection S0 for
decomposi-tion D0 and graph G1 are shown in Fig. 2(b). There are four FMSO sets, each
indicated by a different color by bold edges between equations. The number of interconnections for decomposition D0given by ID0(S0) is equal to 10.
Decom-position D1 (Fig. 2(c)) is a graph partitioning based on the FMSO selection S0.
Note that D1differs from D0in that equation e19 has been reassigned. This new
decomposition decreases the number of interconnections (i.e. number of shared variables, cf. Definition 3) to ID1(S0)=7. Then, the FMSO sets of S1are selected
based on decomposition D1 (Fig. 2(d)); because the set of selected FMSO sets
remains the same as in the previous step, i.e. S0= S1, the algorithm stops.
The running time of the DecSel algorithm applied to the four tanks example is 14.1s ± 78 ms per loop (mean ± standard deviation of 7 runs, one loop each), which would be 2 or 3 times faster if run directly on Intel i7-5500U CPU 2.40 GHz instead of in VirtualBox.
(a) D0 with constraints constr = {{e1, e2, e3, e4, e6}, {e7, e8, e10, e11}, {e13, e15, e16}, {e17, e20}} ncuts= 19 (b)
S0: First FMSO sets selection
ϕ00= {e16, e17, e18, e19, e20} ϕ01= {e8, e10, e11, e13, e16, e20} ϕ02= {e1, e3, e4, e5, e7, e9, e10, e12, e13, e14, e15, e16, e20} ϕ03= {e1, e2, e3, e4, e6, e10} ID0(S0) = 10 (c) D1: New decomposition ID1(S0) = 7 (d)
S1: FMSO sets selection
ϕ10= {e16, e17, e18, e19, e20}
ϕ11= {e8, e10, e11, e13, e16, e20}
ϕ12= {e1, e3, e4, e5, e7, e9, e10, e12, e13, e14, e15, e16, e20}
ϕ13= {e1, e2, e3, e4, e6, e10}
ID1(S1) = 7 Fig. 2: Results for four tanks system example
6
Conclusion
In the paper we consider the double overlinked problem of choosing the decom-position of a system by leveraging existing flexibility and of selecting a set of diagnostic tests so as to minimize their number and the communication between subsystems. The algorithm DecSel that we propose articulates diagnosis test selection solved with a specific A* algorithm and system decomposition solved with a graph partitioning algorithm. These two methods are used iteratively to improve the solution at each step. When most of the works proposed in the lit-erature about distributed diagnosis assume that the decomposition is given or that it is totally flexible, in this paper we assume a more realistic situation in which there are some constraints on decomposition, but there is some flexibility. We propose a solution to design subsystems taking into account the existing constraints on decomposition, fault isolability and communication.
However the formulated optimization problem guarantees to improve the so-lution at each iteration but it does not guarantee to provide the globally optimal solution and the algorithm requires the computation of all FMSO sets. These limitations should be considered in future work.
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