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| SDG-Based Fault Isolation for Large-Scale Complex Systems |
| 作者：佚名 来源：本站整理 发布时间：2008-3-29 9:00:29 发布人：guo8130 |
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Signed directed graph (SDG) is an important qualitative model that is used to describe large-scale complex
systems and the cause-effect relationships among variables. It has been successfully applied in fault diagnosis,
hazard assessment and other areas. In the fault isolation problem, the task is to find the fault origin that causes
the abnormal phenomenon. However, as the basis of analysis, the inference method based on SDG, is simply
a traversal search or a rule-based expert system. Because of the redundant or disordered information, the
efficiency of these algorithms is quite low. Rough set theory provides an idea of handling vague information
and can be used to data reduction, thus it can be introduced to the fault isolation problem (a kind of decision
problems) to optimize the decision rules. The decision algorithm is proposed in this paper, in which the
generation and reduction methods of the rules are related to the structure of the SDG model. We combine the
algebraic and logical expression ways to achieve the purpose. Moreover, due to the convenience of expressing
granularity, the decision algorithm is still applicable when the types of the faults we concerned are changed or
reformed. Finally, an example of a 65t/h boiler system is carried out to illustrate and validate the proposed
method, and some future trends of this method are also discussed.
Keywords：Signed Directed Graph；Rough Set；Large-Scale Complex System
1 Introduction
Signed directed graph (SDG) is a modelling method for complex systems to exhibit the process behaviours.
A SDG uses nodes to represent process variables and uses branches between nodes to represent the
cause-effect relations between variables (Iri et al., 1979). The sign on the node represents the direction of
the variable deviation. The sign on the branch represents the direction of influence and takes the value of
“+”, “-” or “0”. The sign “+” implies that a positive (negative) deviation leads to a positive (negative)
deviation. When the sign on the edge is “-”, an increase (decrease) leads to decrease (increase). Up to now,
SDG has been broadly used in many areas, especially for large-scale systems such as enterprises. The most
typical application covers fault diagnosis and hazard assessment based on the fault inference along
consistent paths (Yang et al., 2005).
The SDG-based fault isolation is actually a traversal search (usually deep-first) along the consistent paths in
the graph to find the fault origin (Iri et al., 1979). This approach uses the deep-level knowledge of the
system and shows the propagation path of each fault. Kramer et al. (1987) proposed to use rules and expert
systems to make inference. Yang et al. (2007a) introduced structural residual to simplify the rules. These
approaches, however, have many disadvantages – each fault is located on a single node and the type is only
the value deviation caused by device malfunction or misoperation; when multiple faults or complex faults
occur, it is hard to identify them, so we have to add extra nodes to denote complex faults (Yang et al.,
2006b); the algorithm is fixed and not easy to be adjusted if the actual demands of users or the choices of
fault types are changed. Yang et al. (2006c) proposed a hierarchical description of SDG, which reveals
1 The work was sponsored by the National High Technology Research and Development Programme of China (No.
2003AA412310) and National Natural Science Foundation of China (No. 60736026). We gratefully acknowledge the
financial aid for this research.
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some granularity, but the decomposition and concentration methods have not been well established. The
rough set theory provides a new idea of this problem.
The rough set theory was presented by Pawlak (1982, 1991) and becomes an effective new mathematical
approach to uncertain and vague data analysis. Rough sets describe a kind of roughness in knowledge
representation by the idea of indiscernibility between elements (more formally, indiscernibility relations).
Rough set theory can be used in many areas, among which data reduction is an important application in
data mining. In an information system or decision system, mass decision rules can be reduced and
simplified by data reduction. Moreover, the operations among rules can be used to change the granularity or
roughness. Therefore, by regarding the fault isolation problem as a decision problem, we can use rough set
theory to improve the inference efficiency.
This paper is organized as follows: In Section 2, the basic concepts of rough set theory and decision
algorithm are introduced. Combined with the background of the fault isolation problem and the method of
SDG, the fault isolation algorithm is proposed in Section 3. Section 4 gives an example of a boiler system
to illustrate the proposed method. In Section 5, the conclusions and prospective applications are discussed.
2 Decision System and Decision Algorithm
2.1 Basic Concepts
In order to describe a decision problem, we should first give some definitions, which are the fundamental
concepts of rough set theory (Pawlak, 1982, 1991; Liu, 2001).
Definition 1 (Decision system). An information system is a formal structure viewed as a four-tuple of the
form
S = <X, Q, V, f> (1)
where, X is a finite universe of discourse including all elements (objects) we are interested in some problem
description; Q is a finite set of attributes used in the description of elements of X; V describes values of all
attributes; f is called a decision function
f: X×Q→V (2)
indicating the attribute of the element. If the attribute set V is divided into two disjoint sets called condition
attributes (C) and decision attributes (D), then such information systems are referred to as decision tables
S = <X, Q, C∪D, f> (3)
Definition 2 (Indiscernibility). With the same information system S in mind, denote by A a subset of
attributes, A∈Q. We say that two objects (x and y) are indiscernible by the set of attributes A in S iff f(x,a)
= f(y,a) for every a in A.
Indiscernibility forms an equivalence relation in X,
IND(A) = {(x,y)∈X×X | f(x,a) = f(y,a) for all a∈A} (4)
Definition 3 (Partition and class). Given the information system S and its condition attributes C and
decision attributes D, X|IND(C) and X|IND(D) are the partitions of the universe X on the attribute sets C
and D respectively. The elements in the sets X|IND(C) and X|IND(D) are called condition classes and
decision classes respectively.
Definition 4 (Consistency). Given the information system S and its condition attributes C, if each condition
class E∈X|IND(C) has the same decision value, then we call E is consistent, otherwise we call E is
inconsistent. For a decision table S, if all the condition classes are consistent, then S is consistent; otherwise
S is inconsistent.
Definition 5 (Core and reduct). For an information system S with a subset A∈Q, attribute a∈A is
dispensable if IND(A)= IND(A\{a}). Otherwise we call a to be indispensable. The set of all indispensable
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attributes of A is called a core of A, denoted by CORE(A). A minimal set of attributes that discerns all
objects in S that are discernable by A and cannot be further reduced is called a reduct of A, denoted by
RED(A). The intersection of all reducts of A is a core of A
CORE(A) = all reducts ∩
RED(A) (5)
CORE(A) is composed of such attributes that cannot be removed from A without causing any loss in the
quality of classification.
As to the expressions of the decision algorithm, besides the table form, we can also express it in logic form.
Definition 6 (Rough logic). Rough logic language (RLL) is composed of attribute set Q, value set
q∈ q = ∩ Q V V ( q V is the value set of the attribute q), logic connectives, and well formed formulas (wffs):
(1) (q,v) is an atomic formula, where q∈Q, v∈Vq, and atomic formulas are all wffs;
(2) If ϕ and ψ are formulas, then ∼ ϕ , ϕ ∧ψ , ϕ ∨ψ , (ϕ), and ϕ →ψ are all wffs;
(3) The result of operating the formulas defined in (1) and (2) by logic connectives for limited times, is a
wff.
Definition 7 (Granule). Function f-1(ϕ ) denotes the object set in which the element satisfies the wff ϕ . A
granule is defined as Gr = (ϕ , f-1(ϕ )).
By the operation of wffs, granules are also changed and describe things from different levels. For example,
if two wffs are operated by ‘ ∨ ’, the corresponding granules combines into a bigger one. So the granularity
is lower because we cannot distinguish the initial two granules.
2.2 2.2 Decision Algorithm
By the above definitions, we can describe a decision algorithm by a decision table or rough logics. A
decision table can be regarded as a set of formulas. We should deal with all the possible decision rules and
obtain a concise and self-contained decision algorithm. The way of inference is as follows (Liu, 2001):
(1) List all the possible rules as Table A (as Table 1), with each row denoting a rule ϕ →ψ , where ϕ
denotes the values of the condition attributes are assumed and ψ denotes the decision to be obtained. For
convenience, we can give each attribute value a notion.
Table 1: The framework of a decision table
Attributes
Objects Q
X
Condition
attributes
C
Decision
attributes
D
(2) Try to delete each condition attribute in turn and test the consistency of the formula and obtain the
reducts and the core. Delete all the elements except the cores and get Table B. There are several methods to
test the consistency. For example,
– Each condition class E∈X|IND(C) has the same decision value.
– For each object x, the condition class covering x is contained in the decision class covering x.
– For every two decision rules ϕ →ψ and ϕ'→ψ', we have ϕ =ϕ'→ψ =ψ'.
(3) Calculate the reducts of each rule by use of Table B, and get Table C.
(4) Delete redundant rules and thus get Table D.
(5) Educe the rules and the decision algorithm according to Table D.
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The decision algorithm derived here assures the minimization of the resulted condition attribute set.
Theoretically it is an NP-hard problem (Pal et al., 2001).
3 Fault Isolation Algorithm Based on the SDG Model
Fault isolation problem is an instance of decision problem. The purpose is to determine the system state,
that is, normal or abnormal? Where does the fault occur? What kind of fault?
In expert systems, the decision is realized by rules. It is similar here that in order to express the decision
problem in the framework of rough set theory, we should transform the problem expression into decision
table or rough logics at first.
The basic method is to take the variable set as the condition attribute set, and to take all the possible
samples (combinations of all the variable values) as the objects, and to take the system states as the
decision values. In SDG, the values of the variables are “+”, “-”, or “0”, so the condition attribute set is
composed of these three signs. The decision attribute is the system state including all the kinds of faults and
a normal state.
According to the structure of the SDG, we can get some observations:
Observation 1. If two variable sets have no intersection and no branches linking them, then the rules
concerning these two sets are independent. For example, in Fig. 1, variable a and {b, c, d} are separate, so
in Table 2, b, c, and d can be reduced in the 1st row, and a can be reduced in the 2nd and 3rd rows.
Figure 1: An example of SDG
Table 2: An example of the decision table
A b c d State
1 + / / / F1
2 / + / / F2
3 / / + / F3
4 0 0 0 0 Normal
Observation 2. If two or more nodes in SDG have the same downstream node, then the core is null, because
the conjunct node and its upstream nodes can all be reduced. In Table 2, the 2nd and 3rd rows are reducts
and there are other reducts omitted here.
Considering the different granularity, the rules can be combined or disjoined. In the SDG in Fig. 1, if we
only pay attention to the fault of two clusters, a and {b, c, d}, then the 2nd and 3rd rows can be combined
together.
The advantages of this method are: Firstly, it can figure out the complex fault problem that is hard to handle
by the search on SDG. Secondly, faults can be classified into different types and combined into different
granules according to our actual needs, and are not limited to the value deviation of the variables. Thirdly,
the samples are possible decision rules that can be created by the actual measurement or by inference on
SDG. So this method takes SDG method into account to the sample generation, but for real-time inference,
it also uses the expert system to improve the efficiency.
a b
c
d
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4 Examples
A 65t/h steam boiler system, which is widely applied in large-scale petrochemical enterprise, is taken as our
example system. We implement it by simulation software PS (Wu, 2002) with its flow chart shown as Fig. 2.
Some key variables in the process are controlled by single loops, and valves can be manipulated manually
or automatically. There are 16 controlled variables in the model: inlet flow rate of the boiler, FR-01; outlet
flow rate of the overheated steam, FR-02; flow rate of the cooling water, FI-03; flow rate of the soften
water, FR-04; flow rate of the smoke, FI-06; flow rate of the fuel oil, FR-07; flow rate of the deoxidizing
water to be catalyzed, FI-08; pressure of the smoke at the exit, PI-05; oxygen percentage of the smoke,
AI-01; pressure of the main steam, PIC-01; pressure of the high pressure gas, PIC-02; pressure of the liquid
hydrocarbon, PIC-03; pressure of the deaerator, PIC-04; water level of the top steam drum, LIC-01; water
level of the deaerator, LIC-02; and temperature of the overheated steam, TIC-01. Besides, key variables in
the process include the temperature of the hearth, TI-07; the flow rate of the inlet air, FA; and the flow rate
of the high-, medium- and low-pressure gas denoted by FH, FM and FL individually. The SDG of this system
is established by Yang et al. (2006a), shown as Fig. 3, which illustrates the cause-effect relationships among
the key variables.
Figure 2: Flow chart of the boiler system Figure 3: SDG of the boiler system
The most typical faults are operational malfuctions, because each controlled variable is involved in a
control loop and is controlled by a valve, and the open of the loop or the wrong settings may lead to the
corresponding fault, that is, the deviation of the controlled variable. Besides, there are other kinds of faults
caused by complicated or compositive reasons. These faults are listed in Table 3 together with their
consequences.
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Table 3: Typical faults of the boiler system
Notation Name Consequence
F2 Full of water in steam drum Inlet reduces heavily
F3 Lack of water in steam drum Water level decreases gradually
F4 Fire extinguished All the gas muzzles are extinguished; pressure and temperature
of the stream decrease
F5 Power off Several phenomena
F6 Failure in the cooler Temperature of overheated steam reduces; cooling water reduces
abnormally, etc.
First, we consider three operational malfunctions C1, C8, and C7, which correspond to the deviation of
PIC-04, PIC-02, and PIC-03 individually. According to SDG, we can search along the consistent paths to
find out the consequence of each operational malfunction, the affected variables of which are listed in Table
4.
Table 4. The affected variables of three operational malfunctions
PIC-04 TIC-01 PIC-02 PIC-01 PIC-03 AI-01
C1 + 0 0 0 0 0
C8 0 + 0 + + -
C7 0 + + + 0 -
According to the above decision algorithm, this table can be regarded as a decision table (Table 5) by
adding a row denoting normal state, whose condition attributes are all the variables and whose decision
attribute is the state of the system. Obviously this table is consistent. And because of the redundancy, the
column of PIC-01 and AI-01 can be deleted. In the first row, which determines the fault C1, the columns
except PIC-04 are reducible and thus PIC-04 is the core. In the next two rows, all the attributes are
reducible, so there are no cores. But we can reduce PIC-04 and choose one or two attributes to combine a
reduction. In its SDG, PIC-04 is a separate node, so it becomes the only attribute after reduction. PIC-02
and PIC-03 have the same downriver nodes, so at least one of these two attributes must be chosen to
identify the two faults. Here we have validated the two observations in Section 3.
Table 5: Decision table when considering three operational malfunctions
PIC-04 TIC-01 PIC-02 PIC-01 State
+ 0 0 0 C1
0 + 0 + C8
0 + + + C7
0 0 0 0 Normal
Finally we obtain the decision table by collecting all the reductions, shown in Table 6.
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Table 6: Reduced decision table when considering three operational malfunctions
PIC-04 TIC-01 PIC-02 PIC-03 State
+ / / / C1
/ / / + C8
/ + 0 / C8
/ / + / C7
/ + / 0 C7
0 / 0 0 Normal
The decision rules are written with rough logic formulas as follows:
(PIC-04,+)→C1
(PIC-03,+)→C8
(TIC-01,+) ∧ (PIC-02,0)→C8
(PIC-02,+)→C7
(TIC-01,+) ∧ (PIC-03,0)→C7
(PIC-04,0) ∧ (PIC-02,0) ∧ (PIC-03,0)→Normal
Or we can combine them as
(PIC-04,+)→C1
(PIC-03,+) ∨ ((TIC-01,+) ∧ (PIC-02,0))→C8
(PIC-02,+) ∨ ((TIC-01,+) ∧ (PIC-03,0))→C7
(PIC-04,0) ∧ (PIC-02,0) ∧ (PIC-03,0)→Normal
Obviously, the so-called normal state is not a real normal state, because there are many other faults that
have not been considered here. Moreover, the combinational faults, such as the simultaneous fault of C8
and C7, are also not considered here. We note that these two faults compose a fault on the superior level or
lower granularity, which means the flow rate of the medium-pressure gas FM is deviated and ignore its
original cause.
Up to now, we find that the search method and the method proposed in this paper can both meet the
demands of the fault isolation problem in large-scale complex systems. SDG method demonstrates the
propagation process of the faults, so it is more applicable in the system analysis. However the method in
this paper reduces the variables to be considered, so it is more applicable in the real-time diagnosis.
When a fault occurs, the affected variables are probably not located in a concentrated area, but in several
parts that are not connective in the graph. For example, when power off (F5), many variables will be
abnormal and they consist of several origins. Table 7 is a sample set of the typical faults.
Table 7: Sample set of the typical faults
C1 C8 C7 F2 F3 F4 F5 F6
PIC-04 + 0 0 0 0 0 0 0
FR-04 0 0 0 0 0 - - 0
LIC-02 0 0 0 0 0 0 + 0
FR-01 0 0 0 + - - - 0
FI-08 0 0 0 0 0 0 - 0
LIC-01 0 0 0 + - 0 - 0
FI-03 0 0 0 0 + - - -
FR-02 0 0 0 0 - - - 0
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TIC-01 0 + + 0 0 0 - -
PIC-02 0 0 + 0 0 0 0 0
PIC-01 0 + + 0 0 - - 0
PIC-03 0 + 0 0 0 0 0 0
FR-07 0 0 0 0 0 0 - 0
FI-06 0 0 0 0 0 0 0 0
PI-05 0 0 0 0 0 0 + 0
AI-01 0 - - 0 0 + 0 0
PI-03 0 0 0 0 0 - 0 0
By the above algorithm, we can execute the data reduction to get the decision table (Table 8) and decision
rules.
Table 8: Decision table without reduction
PIC-04 LIC-01 FI-03 FR-04 PIC-02 PIC-03 State
+ 0 0 0 0 0 C1
0 0 0 0 0 + C8
0 0 0 0 + 0 C7
0 + 0 0 0 0 F2
0 - + 0 0 0 F3
0 0 - - 0 0 F4
0 - - - 0 0 F5
0 0 - 0 0 0 F6
0 0 0 0 0 0 Normal
5 Conclusion and future works
This paper combines the theories of SDG and rough sets to improve the inference efficiency of decision
problem and uses them to the fault isolation problem. It shows the flexibility of building the rules and
classifying the faults. The decision algorithm proposed here implemented is to obtain the minimum set to
achieve the purpose.
Also, this method can be used in other problems, such as sensor location problem in system design. In order
to monitor the state or performance of the system, many sensors are placed on the devices to measure the
variables. Theoretically, more sensors located in more places are better for fault detection, but because of
economic and technical limitations, we cannot use too many sensors. The basic principles are set to be able
to detect all the faults (i.e. detectability) and to distinguish between different faults (i.e. identifiability)
(Yang et al., 2007b). These two principles are consistent with the decision problem. Thus we can choose the
minimum attribute set as the sensor location set. And we can expand, reduce, or reform the table according
to the demands of the fault isolation task.
References
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Kramer, M.A. and B.L. Palowitch (1987). A rule-based approach to fault diagnosis using the signed directed graph,
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unpublished.
Author Brief Introduction：
Fan Yang received his BS degree from Tsinghua University in 2002. Currently he is a PhD candidate
in Department of Automation, Tsinghua University. His research interests include hazard assessment
and fault diagnosis.
Deyun Xiao graduated from Tsinghua University in 1970. Currently he is a professor in Department of
Automation, Tsinghua University. His research interests include fault diagnosis, system identification,
data fusion, etc.
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