Fault diagnosis For complex systems based on dynamic evidential network and multi-attribute decision making with interval numbers

With the development of science and technology, the functional requirement and modernization level of modern equipments are increasing, which makes these systems become more and more complex and raises some challenges in fault diagnosis. These challenges are shown as follows. (1) Failure dependency of components. Modern engineering systems are becoming increasingly complex, which makes components interact with each other. So, dynamic fault behaviors should be taken into account to construct the fault model. (2) The life distributions of components are different. Modern systems include a variety of components, and they may have different life distributions. Some classical static modeling techniques, including reliability block diagram model [12], fault tree (FT) model [20], and binary decision diagrams (BDD) model [23] have been widely used to model static systems. But these models assume that all components follow the exponential distribution. However, in the practical engineering, different components may have different distributions. For complex systems, a mixed life distribution should be used to analyze these systems. (3) There are a large number of uncertain factors and uncertain information. Many complex systems have adopted a variety of fault tolerant technologies to improve their dependability. However, high reliability makes it difficult to get sufficient fault data. In the case of the small sample data, the traditional methods based on the probability theory are no longer appropriate for complex systems. Aiming at these challenges mentioned above, many efficient diagnostic methods have been proposed. In order to model the dynamic failure characteristics, DFT [6], Markov model [28] and dynamic Bayesian networks (DBN) [9, 26] have been proposed to capture the above mentioned dynamic failure behaviors. DFT is widely used to model the dynamic systems as the extensions of the traditional static fault trees with sequenceand function-dependent failure behaviors. Ge et al. present an improved sequential binary decision diagrams (SBDD) method for highly coupled DFT where different dynamic gates often coexist and interact Rongxing DUAN Longfei HU Yanni LIN


Introduction
With the development of science and technology, the functional requirement and modernization level of modern equipments are increasing, which makes these systems become more and more complex and raises some challenges in fault diagnosis.These challenges are shown as follows.(1) Failure dependency of components.Modern engineering systems are becoming increasingly complex, which makes components interact with each other.So, dynamic fault behaviors should be taken into account to construct the fault model.(2) The life distributions of components are different.Modern systems include a variety of components, and they may have different life distributions.Some classical static modeling techniques, including reliability block diagram model [12], fault tree (FT) model [20], and binary decision diagrams (BDD) model [23] have been widely used to model static systems.But these models assume that all components follow the exponential distribution.However, in the practical engineering, different components may have different distributions.For complex systems, a mixed life distribution should be used to analyze these systems.(3) There are a large number of uncertain factors and uncertain information.Many complex systems have adopted a variety of fault tolerant technologies to improve their dependability.However, high reliability makes it difficult to get sufficient fault data.In the case of the small sample data, the traditional methods based on the probability theory are no longer appropriate for complex systems.Aiming at these challenges mentioned above, many efficient diagnostic methods have been proposed.In order to model the dynamic failure characteristics, DFT [6], Markov model [28] and dynamic Bayesian networks (DBN) [9,26] have been proposed to capture the above mentioned dynamic failure behaviors.DFT is widely used to model the dynamic systems as the extensions of the traditional static fault trees with sequence-and function-dependent failure behaviors.Ge et al. present an improved sequential binary decision diagrams (SBDD) method for highly coupled DFT where different dynamic gates often coexist and interact However, these methods assume that all components obey to the same distribution and cannot handle the challenge (2).Furthermore, these methods, which are usually assumed that the failure rates of the components are considered as crisp values describing their reliability characteristics, have been found to be inadequate to deal with the challenge (3) mentioned above.Therefore, fuzzy sets theory has been introduced as a useful tool to handle the challenge (3).The fuzzy fault tree analysis model employs fuzzy sets and possibility theory, and deals with ambiguous, qualitatively incomplete and inaccurate information [8,16,18].To deal with the challenge (1) and (3), fuzzy DFT analysis has been introduced [13][14] which employs a DFT to construct the fault model and calculates the reliability results based on the continuous-time BN under fuzzy numbers.However, these approaches cannot handle the challenge (2).For this purpose, Mi et al. proposed a new reliability assessment approach which used a DFT to model the dynamic characteristics within complex systems and estimated the parameters of different life distributions using the coefficient of variation (COV) method [19].To a certain extent, this method can meet the above challenges.But it is confined to the reliability analysis and cannot be used for the fault diagnosis.Dugan introduced a diagnostic importance factor (DIF) to determine the diagnosis sequence using DFT analysis [1].However, the solution for DFT is based on Markov Chain which has an apparent state space explosion problem.In the work of [3], a hybrid fault diagnosis approach was proposed based on fault tree analysis and Bayesian network.Nevertheless, it used a static fault tree model and could not capture the dynamic failure behaviors.Furthermore, diagnosis strategies of these methods are only based on DIF and usually could not do decision making when there were many attributes for consideration.In addition, these diagnostic methods are usually assumed that the failure rates of components are regarded as crisp values and cannot deal with the challenge (3).To overcome these difficulties and limitations, Duan et al. proposed a diagnosis method based on fuzzy sets theory and DFT, which used fuzzy sets theory to estimate the failure rates of basic events and solved the DFT based on discrete-time Bayesian networks [5].However, this approach could not handle the challenge (2).In addition, all the diagnosis algorithms are based on the single attribute decision making, and usually cause minimal cut sets with a smaller DIF to be diagnosed first [24], thereby influencing the diagnosis efficiency.Motivated by the problems mentioned above, this paper presents a novel diagnosis strategy for complex systems based on DEN and an improved VIKOR algorithm shown in Fig. 1.It pays close attention to meeting above three challenges.In view of the challenge (1), it uses a DFT to capture the dynamic failure mechanisms.For the challenges (2) and ( 3), a mixed life distribution is used to analyze complex systems, and the COV method is employed to estimate the parameters of life distributions for components with interval numbers.Furthermore, relevant reliability parameters can be calculated by mapping a DFT into a DEN in order to avoid the aforementioned problems.At last, components' DIF, BIM and HIV are taken into account comprehensively to design a novel diagnosis strategy using an improved VIKOR algorithm.The proposed method takes full advantages of DFT, interval numbers for handling uncertainty, DEN for inference and VIKOR for the best fault search scheme, which is especially suitable for fault location of complex systems.
The remaining of this paper is organized as follows: In section 2, a DEN modeling is introduced and the conversion process from a DFT to a DEN is also provided; Section 3 presents a new fault diagnosis method based on an improved VIKOR algorithm; An illustrative example is provided to demonstrate the proposed method in Section 4; Finally, conclusions are made in Section 5.

DEN
D-S evidence theory has a unique ability in the expression of epistemic uncertainties.The evidence theory can be well compatible with the theory of probability.This section will describe how to compute the reliability parameters using DEN.The following simply introduces the relevant definitions and theorems in this paper, and more information can be referred to literatures [4,10,22].Evidential Network is based on graph theory and D-S theory.It is a promising graphical tool for representing and managing uncertainties.Each node represents a variable, and arcs indicate direct conditional relations between the connected nodes.DEN, an extension of evidential network, takes into account the time by defining different nodes to model variables with respect to different time slices [21].It includes the initial network and the temporal transition network.Each time slice corresponds to a static evidential network, and the time slices are a directed acyclic graph , The T V and T E are respectively nodes of time T and directed arcs.A directed arc linked two variables belonging to different time slices.
In evidence theory, is the knowledge framework of the component i and the focal elements are given by: sciENcE aNd tEchNology where { } i W and { } i F denote the working and the failure state respectively.The state of {, } i i WF corresponds to the epistemic uncertainty.Belief measure (Bel) defines the lower bound of the probabilities that the focal element exists, and plausibility measure (Pl) defines the upper bound of the probabilities that the focal element exists.The basic belief assignment in the system state expresses an epistemic uncertainty, where Bel and Pl measures are not equal and bound the system reliability.Therefore, the basic probability assignment (BPA) of component i can be computed as: Presumably, the upper and lower bounds of the component reliability [ () , () Px Px ] is equivalent to the BPA in the DEN: where

Mapping a static fault tree into a DEN
The conditional probabilities of each node in the static evidential network have been discussed in detail in [25].Fig. 2 shows an AND gate and its equivalent DEN.Equation 4 and 5 give the conditional probability of each node.

Mapping a DFT into a DEN
DFT extended the traditional fault tree by defining some dynamic gates to capture the sequential and functional dependencies.Usually, there are six types of dynamic gates defined: the Functional Dependency Gates (FDEP), the Cold Spare Gates (CSP), the Hot Spare Gates (HSP), the Warm Spare Gates (WSP), the Priority AND Gates (PAND), and the Sequence Enforcing Gates (SEQ).

Fig. 3. CSP gate and its equivalent DEN
The following section briefly discusses a CSP gate as it is used later in the example.The CSP gate includes one primary input and one or more alternate inputs.Fig. 3 shows a CSP gate and its equivalent DEN.Suppose that A and B follow the same distribution, then ( ) P x and ( ) P x denote the lower probability and upper probability of the nodes respectively.At this point, node A has the same conditional probability with the AND gate of the node A and the conditional probability of other node B can be calculated by the following equations:

Calculating reliability parameters
After a DFT model is built, The DFT is converted into an equivalent DEN using the proposed method.Once the structure of the DEN is known and the probability tables are filled, the reliability parameters of the system can be calculated using the DEN inference algorithm.These reliability parameters mainly include system unreliability, DIF, BIM and HIV, which are used for fault diagnosis in the proposed method.

System unreliability
Calculating the system unreliability is very simple using the following equation: sciENcE aNd tEchNology

DIF
DIF is defined conceptually as the probability that an event has occurred given that the top event has also occurred.DIF is the cornerstone of reliability based diagnosis methodology [1].DIF can be used to locate the faulty components in order to minimize the system checks and diagnostic cost.It is given by: where i is a component of system S; (| ) Pi S is the probability that the basic event i has occurred given the top event has occurred.
Suppose the system has failed at the mission time, we input the evidence that system has failed into DEN and get the DIF of components using the inference algorithm.

BIM
Birnbaum first introduced the concept of a components' reliability importance in 1969.This measure was defined as the probability that a component is critical to system failures.i.e. when component i fails it causes the system to move from a working to a failed state.BIM of a component i can be interpreted as the rate at which the system's reliability improves as the reliability of component i is improved [21].Analytically, Birnbaum's importance interval measure of a component i can be defined using D-S theory by the following equation: where Pl W W denote respectively the belief and plausibility measures that the system is functioning when it is known that component i is in a working state.Whereas ({ } | { }) Pl W F denote respectively the belief and plausibility measures that the system is functioning when component i is in a failed state.

HIV
The heuristic function plays an important role in the diagnostic sequence [11].Owing to the different complexity of components their test cost is different, a balance should be taken into account between the DIF and test cost.Therefore, a new heuristic function for complex systems, HIV is proposed.HIV represents the value of the heuristic information contained in each fault search path and the influence degree of the fault search on the next optimal fault search.With the combination of DIF and the test cost, HIV is defined by the following expression: The test cost of the components is usually very difficult to express as crisp values because of uncertainties.So the linguistic assessments are used for generating criteria and alternative ratings, which are transformed into interval numbers to describe test cost of the components for treatment by VIKOR.Table 1 shows the evaluation criteria and alternative ratings of the test cost.

Fault diagnosis strategy based on an improved VIKOR algorithm
The basic information provided by reliability analysis can be used to construct the diagnostic decision table for fault diagnosis.Assume that the DEN has n root nodes, each root node represents a diagnostic scheme, ( 1, 2, , ) represents the diagnostic scheme and each root node has k reliability parameters.DIF enables us to discriminate between components by their importance from a diagnostic point of view.BIM is used to quantify the contributions of components' reliability to the systems' reliability and HIV plays an important role in the diagnostic sequence.DIF, BIM, and HIV are treated as attribute v1, v2 and v3 respectively.These attributes can be considered comprehensively to obtain the best faulty search scheme using an improved VIKOR algorithm [27].

Normalizing diagnostic decision table
Fault diagnosis is a process to optimize multi-attribute decision making.After the search scheme for fault diagnosis is defined, we can construct the diagnostic decision table by the corresponding evaluation attributes.However, different evaluation attributes usually have different values and dimensions, which are not directly comparable, so we should normalize the diagnostic decision table.Evaluation attributes can be divided into two classes: benefit attributes and cost attributes.There are three attributes in the diagnostic decision table, DIF, BIM and HIV, which belong to the benefit attributes.For the different data, we use the following formula to normalize them.

When the attribute ij
x is a benefit attribute, we use the following formula to normalize them: where ij x is the j th attribute value of the i th component.

When the attribute ij
x is a cost attribute, we normalize them by using the following formula:

Determining the weights of attributes
Shannon Entropy is a measure of uncertainty of information formulated in terms of probability theory [15].It is well suited for measuring the relative contrast intensities of criteria to represent the average intrinsic information transmitted to the decision makers.Entropy weighting is a multi-attribute decision making (MADM) method used to determine the important weights of decision attributes by directly relating a criterion's importance weighting relative to the information transmitted by that criterion.However, because the elements of the sciENcE aNd tEchNology decision matrix are interval numbers, the Entropy method cannot be used directly.Therefore, before the entropy method is put into use, the decision matrix needs to be quantized.The diagnosis decision table needs to be normalized before the positive and negative ideal solutions are being calculated.The positive ideal solutions are made of all the best performance scores, and the negative solutions are made of all the worst performance scores at these measures in the diagnostic decision table.To compute the positive and negative ideals, by the relations: max , min Suppose that [ , ] a a a The larger the interval deviation degree distance (,) Dab, the greater the degree of phase separation will be.In particular, when (,) Dab= 0, then a = b, which means that a and b are equal.
The diagnostic decision table is the interval numbers, which are difficult to directly compare.In order to determine the weight of attributes, the concept of the interval deviation degree distance is used.The objective weights of attributes can be calculated based on the Entropy concept through the following steps: Step 1: Transform the normalization matrix [ , ] , where ( , ) Step 2: Normalize the evaluation criterion for the interval deviation degree distance matrix through: where Step 4: Define the value of α j through: Where α j is the divergence degree of the intrinsic information of the attributes j.The greater the value of α j , the more important the attribute is in the decision making process.
Step 5: Calculate the weights of attributes using the following equation: Suppose the optimal solutions of model ( 20) and ( 21 Similarly, the interval values [ , ], can also be computed by the linear programming method.where i R and i R are given by:

Calculating the values
[ , ] by the relations: sciENcE aNd tEchNology where S S S S R R R R max , min , max and ν is introduced as the weight for the strategy of maximum group utility, whereas 1−ν is the weight of the individual regret.Usually, ν can take any value from 0 to 1 and the value of ν is set to 0.5 in the paper.

Determining the optimal diagnosis sequence
After the value of [ , ] expressed in interval numbers is obtained, the possibility matrix should be built to rank the alternatives.The possibility matrix can be defined as: Then the corresponding possibility ( ) i p x can be obtained using the following equation.Obviously, the smaller the value ( ) i p x, the better the diagnostic scheme.Therefore, we can determine the optimal ranking order by the value ( ) i p x and choose the diagnostic scheme with the minimum value ( ) i p x.

Numerical Application
An illustrative example is given to illustrate how the proposed method can be used to perform the diagnosis strategy analysis for the braking system using multi-attribute decision making with interval numbers.Suppose all components follow the exponential distribution or two-parameter Weibull distribution.For the components with an exponential distribution, the interval failure rates of the basic events for the braking system can be calculated using the expert elicitation and the fuzzy sets theory.For the components with a two-parameter Weibull distribution, the interval failure rates are calculated using the COV method [19].DFT of the braking system is shown in Fig. 4. The interval failure rates of basic events are shown in Table 2.We can map the DFT into the equivalent DEN shown in Fig. 5.

Fig. 1 .
Fig. 1.A novel fault diagnosis framework for complex systems

Fig. 2 .
Fig. 2.An AND gate and its equivalent DEN

[
represents the failure probability of system.

Step 3 :
Obtain the entropy value of the attributes j as follows:

Fig. 4 .
Fig. 4. A DFT for service braking failure of braking syste

Table 1 .
Evaluation standards of the test cost

Table 6 .
Interval values of S, R and Q for all components