Solving Engineering and Science Problems Using Complex Bioinspired Computation Approaches
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Zhu Huang, Tao Wang, Wei Liu, Luis ValenciaCabrera, Mario J. PérezJiménez, Pengpeng Li, "A Fault Analysis Method for ThreePhase Induction Motors Based on Spiking Neural P Systems", Complexity, vol. 2021, Article ID 2087027, 19 pages, 2021. https://doi.org/10.1155/2021/2087027
A Fault Analysis Method for ThreePhase Induction Motors Based on Spiking Neural P Systems
Abstract
The fault prediction and abductive fault diagnosis of threephase induction motors are of great importance for improving their working safety, reliability, and economy; however, it is difficult to succeed in solving these issues. This paper proposes a fault analysis method of motors based on modified fuzzy reasoning spiking neural P systems with real numbers (rMFRSNPSs) for fault prediction and abductive fault diagnosis. To achieve this goal, fault fuzzy production rules of threephase induction motors are first proposed. Then, the rMFRSNPS is presented to model the rules, which provides an intuitive way for modelling the motors. Moreover, to realize the parallel data computing and information reasoning in the fault prediction and diagnosis process, three reasoning algorithms for the rMFRSNPS are proposed: the pulse value reasoning algorithm, the forward fault prediction reasoning algorithm, and the backward abductive fault diagnosis reasoning algorithm. Finally, some case studies are given, in order to verify the feasibility and effectiveness of the proposed method.
1. Introduction
As an important part of industrial and agricultural productions, the normal operation of threephase induction motors plays a pivotal role in economic benefits and security risks. For a motor, any potential failure that cannot be predicted or detected in time may produce damage on it, resulting in downtime with potentially huge economic losses [1–4]. In addition, when a motor has faults and is shut down, the first task is to perform abductive fault diagnosis to find its failure causes, which can effectively help the operation and maintenance personnels to locate faulty parts quickly. Therefore, fault prediction and abductive fault diagnosis are of great significance for improving the working reliability and stability of motors [5].
The fault prediction of a motor is usually realized based on an online monitoring system to detect the early failure symptoms and trend parameters that can reflect hidden troubles. Then, these symptoms and parameters are processed by prediction algorithms to obtain earlywarning information and integrated decision making [6] to prevent motor failures. For example, [7] diagnosed mechanical faults of motors by vibration analysis, which was carried out through a noncontact approach based on an optical computer mouse and a digital signal processing device. Reference [8] proposed a twostage machine learning analysis architecture, where a recurrent neural networkbased variational autoencoder was proposed in the first stage, and principal components analysis and linear discriminant analysis techniques were applied in the second stage. This architecture was useful to accurately predict the motor fault modes only by using motor vibration timedomain signals. In [9], a hybrid technique for bearing prognostics was proposed, which utilized a regressionbased adaptive prediction model to find the evolution trend of bearing health indices. However, so far, most fault prediction methods require a huge number of historical data to perform the training and learning processes of their predictive models.
The abductive fault diagnosis of a motor consists in finding failure causes from its fault phenomena and operation data, so that a motor can be effectively repaired, thus reducing economic losses [10]. In [11], an instantaneous frequency analysis method based on abnormal sounds was proposed. However, when the acoustic signals of a motor were mixed by other acoustic signals (such as reﬂected signals and overlapped signals), it was difficult to extract the features of bearing fault information. In [12], a new current signature analysisbased fault detector for motors based on a matched subspace technique was proposed. However, it was only effective for detecting eccentricity faults, bearing faults, and broken rotor bars. Reference [13] proposed a technique based on vibration information to identify and classify different bearing failure conditions. The setting and testing of parameters was strict and difficult; for example, the accelerometer needs to be very close to the motor, and the setting of accelerometer and data logger should be the same. However, this method needed much historical data with a complex computing process. In [14], an intelligent fault diagnosis of threephase induction motors using a signalbased method was proposed and tested in different situations, in order to verify its availability in diagnosing failures, even when the operating mode data were limited. However, the experimental results showed that it was only suitable for the diagnosis of broken bars and bearing failure.
The aforementioned methods have their own advantages with the same disadvantages implying that they mainly focus on the diagnosis of a single fault, such as the rotor bar breaking or the stator short circuit. Thus, they cannot effectively diagnose multiple faults, not achieving the requirement of performing an overall fault analysis of the whole machine.
Therefore, how to improve the abovementioned fault prediction and abductive fault diagnosis methods or put forward new ones is the main issue in the corresponding engineering domain for the motors. On the other hand, with the rapid development of artificial intelligence technology, intelligent analysis and diagnosis methods are gradually developed, such as expert systems (ESs) [15], artificial neural networks (ANNs) [16–20], Petri nets (PNs) [21–23], tissue P systems (TPSs) [24–26], and spiking neural P systems (SNPSs) [27–34]. Specifically, SNPS is a novel highperformance bioinspired distributed parallel computing model with powerful information processing ability. It is a special kind of neurallike P system [29] inspired by the topological structure of biological neural networks and the way that biological neurons store, transmit, and exchange messages, i.e., by sending electrical impulses (spikes) along axons in a distributed and parallel manner [30–32].
The SNPSbased fault diagnosis methods (for example, the ones for power systems) are derived from the similarities between the pulse transmission between neurons via synapses and the fault propagation in power systems. Accordingly, the basic mechanism to address fault diagnosis based on SNPSs is to find faulty sections by dealing with the uncertainty [35] of fault alarm information. In general, the input neurons of an SNPS correspond to protective devices (including protective relays and circuit breakers), and the output neurons are associated with suspicious fault sections. Thus, the pulse values of input neurons represent the action information of protective devices, that is, the actual tripping information from the supervisory control and data acquisition system or the action confidence levels represented by fuzzy numbers [36]. On the other hand, the pulse values of output neurons express the trip information o fault confidence levels of the suspicious sections. When the fault reasoning is finished, faulty sections are finally determined based on the fault confidence levels according to criterion rules.
Because of the high requirement of fault diagnosis methods for processing fault information, the SNPSbased diagnosis methods have become a hot research topic with rich research results [27–29, 33, 34]. However, up to now, the relevant research work is mainly focused on the fault diagnosis of power systems. Besides, the existing work mainly studies the postevent diagnosis problems. Therefore, to give full play to the excellent information processing ability and computing power of SNPSs, it is of great importance to expand their scope to different application fields, as well as extend the applications from the postante ones to new exante analysis and prediction frameworks.
Therefore, this paper moves forward in this widening of the scope of SNPSs. More specifically, the work proposes a fault analysis method based on modified fuzzy reasoning spiking neural P systems with real numbers (rMFRSNPSs) for threephase induction motors. As an important part of this new method presented here, the forward fault prediction reasoning algorithm (FFPRA) and the backward abductive fault diagnosis reasoning algorithm (BAFDRA) are proposed. The main contributions of this paper are described as follows:(1)Based on the existing variants of SNPSs, we propose a modified fuzzy reasoning spiking neural P system with real numbers by simplifying previously existing ones. In order to enable the rMFRSNPSs to achieve fault prediction and abductive diagnosis, three algorithms are proposed, i.e., the pulse value reasoning algorithm (PVRA), the FFPRA, and the BAFDRA, respectively.(2)Fault fuzzy production rules for motors are presented to obtain the relationships between failure symptoms and different faults. Moreover, the rMFRSNPSbased model for a motor is built via modelling the production rules, which is the basis for the fault analysis from the point of view of a whole machine.(3)Firstly, the SNPS is introduced to solve the fault diagnosis of motors, including forward fault prediction and backward abductive fault diagnosis. In addition, we also extend its application from the postante diagnosis to a new exante prediction framework. The new framework not only can take full advantages of the SNPS for the fault prediction with potential fault paths and their occurrence probabilities in an exante prediction problem but also can effectively find failure causes with abductive reasoning paths and their probabilities in a postante fault diagnosis problem.
2. The rMFRSNPSBased Fault Model for Motors
In this section, we first present the rMFRSNPS and then propose fault fuzzy production rules of motors. Finally, the rMFRSNPS is employed to model the rules to propose a universal rMFRSNPSbased fault analysis model.
2.1. Modified Fuzzy Reasoning Spiking Neural P Systems with Real Numbers
Definition 1. A modified fuzzy reasoning spiking neural P system with real numbers (rMFRSNPS, for short) of degree is a tuplewhere(1) is a singleton alphabet ( is called a spike, is a set of spikes).(2) is a neuron set, where is the proposition neuron set and is the rule neuron set, being . Each proposition neuron is of the form , , where(a) is a real number in [0, 1] representing the potential value of spikes (i.e., value of electrical impulses) contained in .(b) is a real number in (0, 1) representing the firing threshold of .(c) represents a firing (spiking) rule of proposition neuron with the form , where and are real numbers in [0, 1], and is the firing condition. The firing rule of can be applied if and only if receives, at least, spikes and the potential value of spikes satisfies that . By applying rule , will consume (remove) a spike with pulse value and then not only produce (emit) a new spike with pulse value but also transmit it to its postsynaptic neurons. Each rule neuron is of the form , , where(a) is a real number in [0, 1] representing the potential value of spikes (i.e., value of electrical impulses) contained in .(b) is a real number in [0, 1] representing the truth value of .(c) is a real number in (0, 1) representing the firing threshold of .(d) represents a firing (spiking) rule of with the form , where and are real numbers in [0, 1], and is the firing condition. The firing rule of can be applied if and only if receives, at least, spikes and the potential value of spikes satisfies that . By applying rule , will consume (remove) a spike with pulse value and then not only produce (emit) a new spike with pulse value but also transmit it to its postsynaptic neurons.(3) with for , is a directed graph of synapses between linked neurons.(4) and represent the sets of input and output neurons of , respectively.Fuzzy production rules can be modelled in the framework of rMFRSNPSs. Let us recall that there are, basically, three types of fuzzy production rules [33].(a)GENERAL rule, whose format is : IF THEN , where is an antecedent proposition and is a consequent proposition(b)Compound AND rule, whose format is : IF AND…AND THEN where are antecedent propositions, is a consequent proposition, and (c)Compound OR rule, whose format is : IF OR…OR THEN where are antecedent propositions, is a consequent proposition, and In fact, there exists another type of rule whose format is : IF THEN AND…AND where is an antecedent proposition and are consequent propositions, with . However, this kind of rules can be considered as a particular case of a composition of GENERAL rules.
In order to model fuzzy production rules by means of rMFRSNPSs, a proposition neuron in an rMFRSNPS is associated with a proposition in the fuzzy production rules. Such neurons will be represented by a circle. If a proposition neuron is an input neuron, then its initial potential value represents the information that has received from the environment.
A general rule neuron in an rMFRSNPS consists of only one presynaptic proposition neuron and one or more postsynaptic proposition neurons. Therefore, in a natural manner, a general rule neuron can be associated with a general rule, that is, with a fuzzy production rule which has only one proposition on its antecedent part. An and rule neuron in an rMFRSNPS consists of, at least, two presynaptic proposition neurons with an AND relationship among them and only one postsynaptic proposition neuron. Thus, in a straightforward way, an and rule neuron can be associated with each compound AND fuzzy production rule. Finally, an or rule neuron in an rMFRSNPS consists of, at least, two presynaptic proposition neurons with an OR relationship among them and only one postsynaptic proposition neuron. According to the previous comments, an or rule neuron can be associated with each compound OR fuzzy production rule. These rule neurons are represented by a rectangle, and they are graphically illustrated in Figure 1.
(a)
(b)
(c)
2.2. Fault Fuzzy Production Rules for Motors
In this paper, the possible failures in a motor include electrical faults and mechanical ones. The first class includes failures such as the excessive current in a phase, the excessive excitation current, a phase voltage loss, the phaseabsent operation, the threephase current asymmetry, and the insulation winding burned down. The second class contains failures such as the bearing expansion by heat, the excessive wear of bearing, the excessive vibration of motor in operation, the abnormal noise, the rotor stuck or stopped rotating, and the motor sweeping. According to the principle of motor failures [23, 37–41] and the fault simulation model in Figure 2, fault fuzzy production rules of motors are obtained as follows, where events corresponding to the propositions in the rules are shown in Table 1. Rule 1 (c_{1}): IF p_{1} OR p_{2} occurs, THEN p_{17} occurs Rule 2 (c_{2}): IF p_{2} AND p_{3} occur, THEN p_{18} occurs Rule 3 (c_{3}): IF p_{3} occurs, THEN p_{19} occurs Rule 4 (c_{4}): IF p_{4} occurs, THEN p_{20} occurs Rule 5 (c_{5}): IF p_{5} occurs, THEN p_{21} occurs Rule 6 (c_{6}): IF p_{6} OR p_{7} occurs, THEN p_{22} occurs Rule 7 (c_{7}): IF p8 occurs, THEN p_{23} occurs Rule 8 (c_{8}): IF p_{8} occurs, THEN p_{24} occurs Rule 9 (c_{9}): IF p_{9} OR p_{10} occurs, THEN p_{25} occurs Rule 10 (c_{10}): IF p_{10} OR p_{11} occurs, THEN p_{26} occurs Rule 11 (c_{11}): IF p_{12} OR p_{13} occurs, THEN p_{27} occurs Rule 12 (c_{12}): IF p_{14} occurs, THEN p_{28} occurs Rule 13 (c_{13}): IF p_{15} AND p_{16} occur, THEN p_{29} occurs Rule 14 (c_{14}): IF p_{17} OR p_{18} OR p_{19} occurs, THEN p_{30} occurs Rule 15 (c_{15}): IF p_{20} occurs, THEN p_{31} occurs Rule 16 (c_{16}): IF p_{21} occurs, THEN p_{32} occurs Rule 17 (c_{17}): IF p_{22} occurs, THEN p_{33} occurs Rule 18 (c_{18}): IF p_{22} OR p_{23} occurs, THEN p_{34} occurs Rule 19 (c_{19}): IF p_{24} occurs, THEN p_{35} occurs Rule 20 (c_{20}): IF p_{25} occurs, THEN p_{36} occurs Rule 21 (c_{21}): IF p_{26} occurs, THEN p_{37} occurs Rule 22 (c_{22}): IF p_{27} occurs, THEN p_{38} occurs Rule 23 (c_{23}): IF p_{28} occurs, THEN p_{39} occurs Rule 24 (c_{24}): IF p_{29} occurs, THEN p_{40} occurs Rule 25 (c_{25}): IF p_{30} OR p_{31} OR p_{32} OR σ_{33} occurs, THEN p_{41} occurs Rule 26 (c_{26}): IF p_{34} OR p_{35} occurs, THEN p_{42} occurs Rule 27 (c_{27}): IF p_{36} occurs, THEN p_{43} occurs Rule 28 (c_{28}): IF p_{37} OR p_{38} occurs, THEN p_{44} occurs Rule 29 (c_{29}): IF p_{39} OR p_{40} occurs, THEN p_{45} occurs Rule 30 (c_{30}): IF p_{41} occurs, THEN p_{46} occurs Rule 31 (c_{31}): IF p_{42} OR p_{43} occurs, THEN p_{47} occurs Rule 32 (c_{32}): IF p_{44} occurs, THEN p_{48} occurs Rule 33 (c_{33}): IF p_{45} occurs, THEN p_{49} occurs Rule 34 (c_{34}): IF p_{46} OR p_{47} OR p_{48} OR p_{49} occurs, THEN p_{50} occurs

2.3. The rMFRSNPSBased Model for a Motor
This section models the fault fuzzy production rules proposed in Section 2.2 and builds a universal rMFRSNPSbased fault analysis model for threephase induction motors, as shown in Figure 3. The designed rMFRSNPS is of degree and specifically contains proposition neurons and rule neurons.
3. Fault Analysis Method Based on rMFRSNPSs
This section proposes a fault analysis method based on rMFRSNPSs for threephase induction motors, whose flowchart is shown in Figure 4, where 0. The proposed method includes two parts, one is for fault prediction before fault occurrence while the other one is for abductive diagnosis reasoning after failures. Moreover, a diagrammatic sketch of the application scenario for the proposed method is shown in Figure 5, where red circles represent the already happened events while blue circles express the not occurred ones. The status of a motor is monitored in real time. When the motor has fault symptoms or faults, relevant state data will be transmitted to the fault analysis center, where our method will be used to handle the events.
Specifically, in this proposed method, the PVRA (Algorithm 1) is proposed to get the potential value of spikes in neurons using historical data and expertise. When a motor has no faults, but is accompanied by fault symptoms, the FFPRA (Algorithm 2) is employed to predict propagation paths with occurrence probabilities. When a motor fails, the fault positions (corresponding to neurons with fault pulses) are found according to failure phenomena, and then, failure causes with probabilities are obtained according to the BAFDRA (Algorithm 3). Thus, the maintenance efficiency can be improved accordingly to check the motor on the basis of results got by the prediction reasoning or abductive reasoning. Note that the historical data include fault probabilities of fault events (Algorithms 1–3), certainty factors of fault production rules (Algorithms 1 and 2), and the tightness degree between related fault events (Algorithm 3).



Next, we describe Algorithms 1–3 in detail as follows.
3.1. Pulse Value Reasoning Algorithm
To explain this algorithm, we introduce its vectors, matrices, and operators as follows (PN denotes proposition neuron and RN denotes rule neuron):(1) is a pulse value vector of PNs, where represents the pulse value of the ith PN . If a PN has not any pulse, then its pulse value is 0.(2) is a pulse value vector of RNs, where is the pulse value of the jth RN . If an RN has not any pulse, then its pulse value is 0.(3) is a firing threshold vector of PNs.(4) is a firing threshold vector of RNs.(5) is a diagonal matrix of truth values of RNs, where is the truth value of the jth RN .(6) is a synaptic matrix, which represents the directed synaptic connections from PNs to general RNs. If there is a synapse from the PN to the general RN , then d_{ij} = 1; otherwise, d_{ij} = 0.(7) is a synaptic matrix, which represents the directed synaptic connections from PNs to and RNs. If there is a synapse from the PN to the and RN , then d_{ij} = 1; otherwise, d_{ij} = 0.(8) is a synaptic matrix, which represents the directed synaptic connections from PNs to or RNs. If there is a synapse from the PN to the or RN , then d_{ij} = 1; otherwise, d_{ij} = 0.(9) is a synaptic matrix, which represents the directed synaptic connections from RNs to PNs. If there is a synapse from the RN to the PN , then d_{ji} = 1; otherwise, d_{ji} = 0.(10) is null vector.(11), where .(12), where .(13), where .
3.2. Forward Fault Prediction Reasoning Algorithm
To explain this algorithm, we introduce its vectors, matrices, and operators as follows:(1) is the number vector of PNs where pulses are located. If a PN contains a pulse, then the number of the neuron in which the pulse occurs is numbered as 1; otherwise, it is 0.(2) is the number vector of RNs where pulses are located. If an RN contains a pulse, the number of the neuron in which the pulse occurs is numbered as 1; otherwise, it is 0.(3) where and .(4) where . If then otherwise .(5) where and .(6) where and .(7) where . If , then , otherwise, , .
Note that the vectors , and , the matrices , and , and the operators and in Algorithm 2 are the same as the ones in Algorithm 1.
3.3. Backward Abductive Fault Diagnosis Reasoning Algorithm
To improve the accuracy of backward abductive reasoning, this paper integrates a fault screening mechanism of the precise minimum cut set (please see Definition 2) into the parallel reasoning ability of SNPSs to propose the BAFDRA for the rMFRSNPS, i.e., Algorithm 3. The precise minimum cut set effectively combines the abductive principle of top events in minimum cut sets [42] with the screening mechanism, where, in two adjacent fault events, a bottom event corresponds to a fault or a fault symptom and a top event corresponds to a fault. Moreover, the screening mechanism is used to improve the abductive reasoning accuracy by eliminating pulses contained in the minimum cut set whose danger degree is lower than the dangerous threshold, where the danger degree is used to access the fault risk of motors [43].
Definition 2. A precise minimum cut set (PMCS) is a tuplewhere(1) are l minimum cut sets (MCSs), where(a)A general rule neuron has one presynaptic proposition neuron (corresponding to a bottom event) and one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can be triggered only by the bottom event; thus, the MCS of the top event consists of the presynaptic neuron.(b)An or rule neuron has more than one presynaptic proposition neurons (each of them corresponds to a bottom event) and only one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can be triggered by any bottom event; thus, the MCS of the top event consists of any presynaptic neuron. That is, if there are bottom events that can trigger the top event, then the top event will have MCSs with each MCS consisting of one presynaptic neuron corresponding to one of the bottom events.(c)An and rule neuron has more than one presynaptic proposition neuron (each of them corresponds to a bottom event) and only one postsynaptic proposition neuron (corresponding to a top event). For this kind of rule neurons, the top event can only be triggered by all bottom events at the same time, so that the MCS of the top event consists of all the presynaptic neurons. That is, if there are bottom events that can trigger the top event, then the top event will have only one MCS and it should consist of all the presynaptic neurons corresponding to the bottom events.(2) is the th MCS, where(a) is the danger degree of the ith PN , defined as where is a weighted value in [0, 1] representing the tightness degree between PN and its postsynaptic neurons.(b) is a danger degree of the th minimum cut set, i.e., , which is defined as(c) is a number in (0, 1) representing the danger degree threshold of an MCS. When the danger degree of an MCS is greater than , then the MCS is called a PMCS and the PNs with pulse in the PMCS form the fault paths, with the first PN in each path being called the fault source.Algorithm 3 is shown as follows:
To explain the algorithm, we introduce its vectors, matrices, and operators as follows:(1) is the number vector of PNs where fault pulses are located. If a PN contains a fault pulse, then the number of the neuron is numbered as 1; otherwise, it is 0.(2) is the number vector of RNs where fault pulses are located. If a rule neuron contains a fault pulse, then the number of the neuron is numbered as 1; otherwise, it is 0.(3) is a fault pulse value vector of PNs, where represents the pulse value of the ith PN . If a PN has not any pulse, then its pulse value is 0.(4) is a fault pulse value vector of RNs, where represents the pulse value of the jth RN . If a RN has not any pulse, then its pulse value is 0.(5) is a dangerous threshold of an MCS.(6) is a weight matrix, where the matrix elements represent the tightness degree between adjacent PNs. If the PNs and are connected, then is a weighted value in [0, 1] representing the tightness degree between and ; otherwise, , .(7) is a danger degree matrix of PNs, where is a number in [0, 1] representing the danger degree of the event corresponding to the PN triggers the one associated with the PN , . If can emit a spike to , then is obtained via (3); otherwise, .Note that the vectors , and , the matrices , and , and the operators , and in Algorithm 3 are the same as the ones in the Algorithms 1 and 2.
4. Case Studies
In this section, several cases about possible faults on a motor are considered, in order to show the feasibility and validity of our proposed method. Note that the initial pulse values of input neurons in Algorithms 1 and 2 are the occurrence probabilities of fault symptoms obtained based on historical data and expertise. Since Algorithm 3 is used to find fault causes and fault sources after a motor fails, its initial pulse values are the event probabilities obtained by Algorithm 1, including the occurrence probabilities of both the fault symptoms and failures.
4.1. Pulse Value Reasoning of Neurons
The initial pulse value of input neurons and truth value of rule neurons are obtained via historical data and expert experience [23].
Here, we take the “insulation winding burned down” as an example. Then, we can get that the initial pulse value vectors of proposition neurons and rule neurons are respectively.
The truth value diagonal matrix of RNs is .
The firing threshold vectors for PNs and RNs are and , respectively.
The synaptic matrices , and are obtained via the rMFRSNPSbased fault analysis model, as shown in Figure 3.where is a null matrix, and is an identity matrix.
The pulse value reasoning process is described as follows: When k = 1, , When k = 2, , When k = 3, , When k = 4, , When k = 5,
Thus, the termination condition is satisfied and the reasoning stops. We obtain the reasoning results, i.e., the pulse value of all neurons, shown as follows:
4.2. Forward Fault Prediction Reasoning
Let us assume that the following fault symptoms of a motor are monitored online: overload (p1), resistance value of a phase winding decreases (p3), damage of shaft seal ring structure (p5), and excessive roughness value of seal surface shaft (p7). Accordingly, the initial number vector of the PNs with fault pulses is obtained: .
The synaptic matrices , and are the same as the ones in Section 4.1.
The initial pulse value vectors of PNs and RNs are respectively.
The truth value diagonal matrix of RNs is .
The fault prediction reasoning process is described as follows: When k = 1, , . , . When k = 2, , . , . When k = 3, , ; , . When k = 4, , . , . When k = 5, , .
Thus, the termination condition is satisfied and the reasoning stops. We find that the neurons with fault pulses are shown in Figure 6. Therefore, the potential fault paths are obtained; that is, , , , , and , where , and are potential faults. The occurrence probability of each fault path is , , , , and , respectively. Thus, the checking order of the fault paths is . Note that the probability of a path is obtained by multiplying the pulse value of the neurons in each path.
4.3. Backward Abductive Fault Diagnosis Reasoning
This section assumes that the motor has failures. Let us take insulation winding burned down (p46) as an example. Accordingly, the number vector of PNs with fault pulse is .
The pulse value vectors of PNs and RNs are and , respectively, given as follows:
The synaptic matrices , and are the same as those in Section 4.1.
The tightness degree between PNs is shown in Figure 7, from which the weight matrix can be obtained.
The abductive fault diagnosis reasoning process is described as follows: When k = 1, To start with the process, the pulse of PNs in the MCS whose danger degree is less than is deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., and , respectively. When k = 2, Similarly, the pulse of PNs in the MCS whose danger degree is less than must also be deleted. Accordingly, the number vectors of PNs and their corresponding pulse values of fault pulses are updated, i.e., , . When k = 3,