A Fuzzy Reasoning Design for Fault Detection and Diagnosis of a Computer-controlled System

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0952-1976/$ - se

doi:10.1016/j.en

Abbreviations

decision-makin

error hazard lev

reasoning and v

ship degree; MF

PNs, Petri nets;

strength of rule

total memory u�CorrespondE-mail addr

Engineering Applications of Artificial Intelligence 21 (2008) 157–170

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A fuzzy reasoning design for fault detection and diagnosis of acomputer-controlled system

Y. Ting�, W.B. Lu, C.H. Chen, G.K. Wang

Department of Mechanical Engineering, Chung Yuan Christian University, 200, Chung Pei Road, Chung Li, Taiwan 32023, ROC

Received 15 March 2005; received in revised form 23 January 2007; accepted 27 April 2007

Available online 5 July 2007

Abstract

A fuzzy reasoning and verification Petri nets (FRVPNs) model is established for an error detection and diagnosis mechanism applied

to a complex fault-tolerant PC-controlled system. The inference accuracy can be improved through the hierarchical design of a two-level

fuzzy rule decision tree and a Petri nets technique to transform the fuzzy rule into the FRVPNs model. Several simulation examples of

the assumed failure events were carried out by using the FRVPNs and the Mamdani fuzzy method with MATLAB tools. The reasoning

performance of the developed FRVPNs was verified by comparing the inference outcome to that of the Mamdani method. Both methods

result in the same conclusions. Thus, the present study demonstrates that the proposed FRVPNs model is able to achieve the purpose of

reasoning, and furthermore, determining of the failure event of the monitored application program.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Error detection; Error diagnosis; Petri nets; Fuzzy rule; Fuzzy reasoning

1. Introduction

The configuration of an error detection and diagnosismechanism (EDDM) for a fault-tolerant computer-con-trolled system has been described in the previous researches(Ting et al., 2002; Ting et al., 2004). The detectionmechanism employs the hook process to capture themessage in and between the various application programs(APs) and the operating system (OS), and detects whetherthe monitored AP is failed. Establishment of errorclassification and standardization was developed in theprevious research (Lu et al., 2003). The diagnosis mechanismidentifies the failure type and the location of error message,

e front matter r 2007 Elsevier Ltd. All rights reserved.

gappai.2007.04.007

: CPU, central processing unit; DLDM, damage level

g; EDDM, error detection and diagnosis mechanism; EHL,

el; FRDT, fuzzy rule decision tree; FRVPNs, fuzzy

erification Petri nets; HT, handling time; MD, member-

, membership function; PCPUU, process CPU usage;

ProMF, membership function of proposition; RFS, firing

; RT, response time; TCPUU, total CPU usage; TMU,

sage; WRFSwinning-rule firing strength

ing author. Tel.: +886 3 2654319; fax: +886 3 2654096.

ess: [email protected] (Y. Ting).

and makes predictable estimation on the executing APs. Asbeen known, while the AP is failed, the OS of computersends an error message with illustration of the failure event.However, most displayed error messages are difficult for theuser to understand, letting alone for them to know thedamage level, so that one can hardly deal with the failureappropriately. Also, when different APs are executed in thePC, there may be similar error message but with differenterror definition and illustration in different programminglanguage (Inprise Corporation, 2002; Richter, 1999). There-fore, it is demanding to investigate how to unify theillustration of error message and examine the closenessdegree of the same failure event but with different errordescription, so that a unified error knowledge databasecould be established, and then the error symptoms could bebetter inferred via the reasoning algorithm to acquire a finaldecision-making. The proposed EDDM, which is a fault-tolerant computer-controlled system, is designed and aimedto satisfy the above needs. Petri nets (PNs) is an idealcandidate for investigating and modeling of systems, and itcan represent the inference process as a discrete-eventdynamic system. The advantages of using PNs in rule-basedsystems include: (1) the graphical formalism, which can

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dx.doi.org/10.1016/j.engappai.2007.04.007

mailto:[email protected]

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Notation

i ith input object of an antecedent proposition,i ¼ 1, 2, 3, y

j jth fuzzy set of input object of an antecedentproposition, j ¼ 1, 2, 3, y

p, t, l, s, h, k, m a serial positive integer; p, t, l, s, h, k,

and m ¼ 1, 2, 3, y

N a set of non-negative integers

Pk kth place in the PNs

Rm mth transition in the PNs

FRk kth fuzzy ruleIPAProIO

k Input object of antecedent propositionIRAProD

k distribution of input object of antecedentproposition

IPAProij antecedent proposition of rule

IRAProMFij calculation of MD of antecedent proposition

by MFIWFij weight functionIPAProMD

ij MD of antecedent propositionIRRFS

k calculation of firing strength of a rule by MIN

composition operation

IPRFSk Firing strength of rule

RMAXð Þ MAX composition operation

PWRFSk place of winning-rule’s firing strength

IRFSCk comparison of firing strength

IPWFRk winning firing rule

IPRFS½�k� a set of ‘‘firing strength of rules’’ except the

firing strength of the winning ruleIPCProO linguistic variable of output object of a

consequent propositionIRFDR

k final decision of winning ruleIPDMI decision-making identification of winning firing

ruleIPCProM

k marking of consequent propositionIRDMI

k identification of consequent propositionIPCPro

k result of consequent proposition of a ruleIRCProMD

k calculation of MD of consequent propositionby MF

IPCProMDk MD of consequent proposition

RFDMk final decision-making computation

PFDM final decision-making result

Y. Ting et al. / Engineering Applications of Artificial Intelligence 21 (2008) 157–170158

visualize the inference states step by step; (2) the transparentmodeling, which has well-established formal mechanisms formodeling and structure inconsistency checking; (3) theanalyzing capability, which can express dynamic andstructural behaviors of a rule-based system via algebraicforms (Scarpelli et al., 1996; Tsang et al., 1999; Yang et al.,2003). The fuzzy PNs combines the graphical technique ofPNs, the fuzzy sets theory, and the fuzzy production rule, soit has the advantage of the graphical power of PNs and thecapability of fuzzy to model rule-based decision systemeffectively (Fay, 2001). Thus, fuzzy PNs would outperformthe PNs and improve the efficiency of fuzzy reasoning (Yanget al., 1997). In general, fuzzy rule-based system consists ofknowledge base, database of event facts, rule base, andinference engine. In order to deal with the input informationfrom the non-fuzzy system, the fuzzy inference mechanismwill carry on fuzzification, rule matching, and defuzzificationin advance to allow the likely incomplete and impreciseinformation to match the antecedent proposition of thefuzzy rule and then obtain an inference result. Quite a fewresearches use PNs to construct the fuzzy rule-based system(Chen et al., 1990; Koriem, 2000; Chen, 2002). The inputand output places of each transition in a PNs can be used torepresent the knowledge of the antecedent and consequentpropositions by the fuzzy production rule. Many researcheshave investigated on the extension of PNs to fuzzy PNs, andthe modeling as well as the reasoning of fuzzy rule-basedsystem. For instance, Gao et al. used the fuzzy reasoningPNs model to describe the production rule-based system andverify its performance as a diagnostic expert system on theturbine (Gao et al., 2003). Chen used fuzzy PNs to construct

knowledge description and fuzzy reasoning, and developed aweighted fuzzy PNs method (Chen et al., 1990; Chen, 2002).Looney developed an algorithm of rule-based decision-making by using fuzzy PNs (Looney, 1998).In this study, it is attempted to use the PNs, the fuzzy

sets theory, and the fuzzy production rule to establish thefuzzy reasoning and verification Petri nets (FRVPNs)model for the diagnosis mechanism. The FRVPNs isdesigned based on the fuzzy rule decision tree (FRDT) withthe merit of hierarchical structure. It provides differentlevel of abstraction, which can be used to represent the sub-model construction and the rule decision for independentlevel of the FRVPNs. Regarding the efficiency ofhierarchical design, it has been discussed in several researcharticles. For example, Delgado et al. proposed a multi-objective decision making scheme, allowing the evolution-ary parameters based on a hierarchical genetic fuzzy systemadjustable, so it not only improves the models performance(accuracy), but also guarantees the interpretability of theresulting fuzzy models (Delgado et al., 2002); Joo et al. alsodemonstrated a scheme of hierarchical fuzzy system withconstraints on the fuzzy rules approximating with highaccuracy and fewer fuzzy rules (Joo and Lee, 2005); Wangproposed a hierarchical structure in a fuzzy systemachieving good approximation accuracy (Wang, 1999).According to these works, it is therefore believed thatimprovement of inference accuracy can be achieved by useof the hierarchical design.Since different types of AP may have different definition

of the failure event, it is likely to describe the errorsymptoms in terms of linguistic variables via the fuzzy set

ARTICLE IN PRESSY. Ting et al. / Engineering Applications of Artificial Intelligence 21 (2008) 157–170 159

theory, and construct the inference contexts by use of thefuzzy production rule. The fuzzy production rule makesuse of the ‘‘IFyTHENy’’ rule to describe the conditionsof the antecedent and consequent propositions. Themembership degrees (MDs) of the propositions on thefuzzy rule are calculated via the membership function(MF). Then, the damage level decision-making is deter-mined by using the fuzzy reasoning method. In addition,the rule-checking process and the verification and mod-ification module are included in the FRVPNs. The formerone is used to confirm the correctness of the winning rule.The latter one is used to deal with the problem ofredundancy, conflict, circularity, and incompleteness whilenew fuzzy rules are added. The rule verification andmodification module will address in another article.Simulations are carried out with several sets of examplesby using the developed FRVPNs and the fuzzy logictoolbox of MATLAB. As compared with the inferredresults, both methods draw the same conclusions.

2. Fuzzy diagnosis reasoning structure of EDDM

The EDDM is proposed for a fault-tolerant computer-controlled system and is a middleware structured in andbetween the user applications of the user mode and thekernel mode for error detection and diagnosis of the AP.The EDDM consists of the retrieve process, the detectionmechanism, the diagnosis mechanism, the informationrecord database, and the knowledge base (Ting et al.,2004). Several important parameters such as the totalcentral processing unit (CPU) usage (TCPUU), processCPU usage (PCPUU), total memory usage (TMU),

Database

Reasoning Diagnosis

FuzzificationModule

Fuzzy RMo

send

User

Rule Self-LearningModule

input

input

read/write

KnowledgeAcquisition

Modulewrite

writeread

MeasurementParatemers

Rule Veand Mod

Mo

Information RecDatabase (IRB

Diagnosis Mechan

Fig. 1. Block diagram of fuzzy d

handling time (HT), and error hazard level (EHL), etc.are considered for fuzzy reasoning. Fig. 1 demonstrates theprocedure: the EDDM measures these parameters of themonitored AP, and sends them to the fuzzification module.And then the fuzzification module handles the measuredparameters of the monitored AP by fuzzification. By usingthe fuzzified parameters, the fuzzy reasoning module willsearch a corresponding fuzzy rule, read an object knowl-edge unit from the knowledge database, and then infer anoutcome of the consequent proposition according to theinput of the antecedent proposition of the fuzzy rule andthe contents of the object knowledge unit. Once theparameter fuzzification fails to acquire appropriate match-ing, the fuzzified parameters are sent to the rule self-learning module. The rule self-learning module collects thenew available information from the user or the informationrecord database, and then carries on self-learning beforeconstructing new rules. If it is impossible to acquire acorresponding fuzzy rule from the fuzzy rule base, thedomain expert will be allowed to input the domainknowledge to the knowledge acquisition module. The ruleverification and modification module will detect whetherthe new rules have problems of redundancy, conflict,circularity, and incompleteness by using the input/outputtransition function of the place and the inhibit condition.Hence, error of inference and unnecessary expense ofinference due to inaccuracy or redundancy of fuzzy rulecan be reduced; in other words, inference accuracy can thusbe improved. The fuzzy inference results will be savedto the information record database and the knowledge baseof the EDDM, and be sent to the defuzzification modulefor defuzzification and decision-making. Finally, the

s of EDDM

Processor ofMechanism

easoningdule

DefuzzificationModule

Fuzzy RuleBase

send

require

read/write

output

Results andIllustration

rificationification

duleread/write

ord)

KnowledgeBase (KB)

ism of EDDM

DomainExpert

input

iagnosis reasoning structure.

ARTICLE IN PRESSY. Ting et al. / Engineering Applications of Artificial Intelligence 21 (2008) 157–170160

defuzzification module delivers an output of damageaffection to the monitored AP and the OS.

3. Representation of fuzzy sets and rules

3.1. Definition of fuzzy sets and linguistic variables

The input variables of the antecedent propositionconsidered for the fuzzy rule include the TCPUU, thePCPUU, the TMU, the HT, and the EHL. The outputvariables of the fuzzy reasoning include the response time(RT) and the damage level decision-making (DLDM). Inreference to Negnevitsky (2002), linguistic value, notationand normalized numerical range of the linguistic variableare included in the fuzzy rule and illustrated in Table 1. Thevalues of linguistic variables are fuzzified to obtain the MDby MF. Values of all the linguistic variables are normalizedin the range of [0, 1], so the MD is bounded in the range of[0, 1]. The MDs of the concerned variables are all designedof the trapezoid shape of fuzzy sets, as listed in Table 1.

3.2. Fuzzy rules representation

The implication relationship of the antecedent andconsequent proposition is used to establish the elementsfor fuzzy rules. Assuming FR is the set of fuzzy rulesrepresented by FR ¼ {FR1, FR2, FR3, y, FRs�1, FRs},FRkAFR. The kth fuzzy rule is denoted by

FRk : IF Aa THEN CbðCFk ¼ mkÞ, (1)

where a, b, k and s are positive integer, and 1pkps; Aa isthe antecedent proposition defined in a fuzzy region of

Table 1

Linguistic values and ranges of fuzzy sets for FRVPNs

Linguistic values Notation Numerical range

(normalized)

TCPUU, PCPU, TMU

Low L 0.0–0.4

Medium M 0.2–0.8

High H 0.6–1.0

RT

Fast F 0.0–0.4

Medium M 0.2–0.8

Slow S 0.6–1.0

HT

Short S 0.0–0.4

Medium M 0.2–0.8

Long L 0.6–1.0

EHL

Low L 0.0–0.4

Normal N 0.2–0.8

High H 0.6–1.0

DLDM

Normal N 0.0–0.4

Slightly high SH 0.2–0.8

High H 0.6–1.0

input space and described by the linguistic variables such as‘‘Small’’, ‘‘Medium’’, and ‘‘Large’’; Cb is the consequentproposition defined in a fuzzy region of specified outputand described by linguistic variables such as ‘‘Low’’,‘‘Medium’’, and ‘‘High’’; CFkA[0, 1] is called the certaintyfactor (CF); mk is the value of the CF that represents thestrength of the belief in the rule FRk.In general, composite fuzzy production rule is classified

into several rule types (Gao et al., 2003). The rule type of‘‘AND’’ connection is used to describe the antecedentproposition in this study. The CF value is designed toreflect the way the experts think (Negnevitsky, 2002; Quand Shirai, 2003). In this article, assuming that the beliefstrength of a fuzzy rule assigned by human expert isCF ¼ 1, which represents that the fuzzy rule is completelybelievable, and denoted by

IF A1 AND A2 AND . . .AND Aa THEN CbðCF ¼ 1Þ, (2)

4. Structure and modeling of FRVPNs

In Figs. 2 and 3, an example of a two-level FRVPNsincluding the rule-checking process as well as the verifica-tion and modification module is presented. The proposedFRVPNs is constructed based on the FRDT with themerits of hierarchical design and PNs technique. It usesthe rule-checking transitions and places to confirm thecorrectness of the firing rule so as to protect the accuracy ofinference result of the winning rule. The verification andmodification module is another important function. It usesthe input/output transition function of the place and theinhibit condition to detect the inconsistent fuzzy rules(conflict, redundancy, incompleteness, and circularityrules) so that inference accuracy is improved.

4.1. Hierarchical structure

FRVPNs is designed based on the FRDT with thehierarchy structure. On the basis of the hierarchical design,the inference accuracy of FRDT is expected to improve(Joo and Lee, 2005; Delgado et al., 2002; Wang, 1999).Fig. 4 shows that there are three inputs considered in theLevel_I, including the TCPUU, the PCPUU, and TMU, bywhich the RT can be inferred. In Fig. 5, there are threeinputs considered in the Level_II, including the RT (theoutput of the Level_I), the HT, and the EHL, by which theDLDM can be inferred in terms of three degrees (Normal,Slightly High, High) to the AP and the OS.

4.2. Dynamic decision-making procedure

The dynamic decision-making procedures of Level_I andLevel_II are shown in Figs. 4 and 5, respectively, anddescribed as follows:

(1)
Level_I:Step 1. enter the required new object variables for thefuzzy rule in the Level_I.

ARTICLE IN PRESS

…

…

…

IWF11IWF12=1 IWF13=2 IWF21=2 IWF31

IWF32=2 IWF33=1IWF23IWF22=1

oPrA11

I P oPrA

33

I PoPrA

31

I P oPrA

32

I PoPrA

23

I PoPrA

22

I PoPrA

21

I PoPrA

13

I PoPrA

12

I P

oDPrA

1

I R oDPrA

3

I RoDPrA

2

I R

oIOPrA

1

I P oIOPrA

2

I P oIOPrA

3

I P

oMFPrA

11

I RoMFPrA

33

I RoMFPrA

32

I RoMFPrA

31

I RoMFPrA

23

I RoMFPrA

22

I RoMFPrA

21

I RoMFPrA

13

I RoMFPrA

12

I R

oMDPrA

11

I P oMDPrA

33

I PoMDPrA

32

I PoMDPrA

31

I PoMDPrA

23

I PoMDPrA

22

I PoMDPrA

21

I PoMDPrA

13

I PoMDPrA

12

I P

RFS

1

I P

RFS

1

I R

RFS

2

I P RFS

3

I P

RFS

2

I R RFS

3

I R

WRFSkP

()MAXR

WFR

1

I P

FSC

1

I R FSC

3

I RFSC

2

I R

WFR

3

I PWFR

2

I PRFS

]1[

I P RFS

]3[

I PRFS

]2[

I P

…

oOPrCI P

FDR

1

I RFDR

2

I R FDR

3

I R

DMI

1

I R

oMFPrC

1

I R

DMII PoMPrC

1

I PoMPrC

2

I P

oMPrC

3

I P

DMI

2

I R DMI

3

I R

oPrC

1

I P oPrC

2

I P oPrC

3P

oMFPrC

2

I R oMFPrC

3

I R

oMDPrC

1

I P oMDPrC

2

I P oMDPrC

3

I P

Rule-che cking

Verification andModification Algorithm

Fig. 2. The FRVPNs model of the Level_I.

Y. Ting et al. / Engineering Applications of Artificial Intelligence 21 (2008) 157–170 161

Step 2. calculate the MD of the proposition of newobject variables via the MF.Step 3. calculate the firing strength by the compositionoperator MIN.

Step 4. calculate the maximum firing strength by thecomposition operator MAX, and find the winning ruleamong the activated rules in the Level_I.

Step 5. draw a conclusion of the output sub-object inthe Level_I from the selected winning rule.

(2)
Level_II:Step 6. import the three object variables, the RT, HT,and EHL, to the Level_II.Step 7. calculate the MD of the new object variables bythe MF.

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…

…

…

…

oMDPrC

1

I P oMDPrC

2

I P oMDPrC

3

I P

oIOPrA

1

II P oIOPrA

2

II P

oPrA

11

II PoMFPrA

211R

oMDPrA

11

II P

RFS

1

II P

RFS1

II R

WRFS

wP

()MAXR

WFR

1

II P

FSC1

II R

RFS

]1[

II P

FDR

2

II R

DMIII PoMPrC

1

II P oMPrC

3

II PoMPrC

2II P

oOPrCII P

oPrA

23

II PoPrA

22

II PoPrA

21

II PoPrA

13

II PoPrA

12

II P

oDPrA1

II R oDPrA2

II R

oMFPrA

212RoMFPrA

21II RoMFPrA

213R oMFPrA

22

II R oMFPrA

23

II R

oMDPrA

12

II P oMDPrA

13

II PoMDPrA

22II PoMDPrA

21II P oMDPrA

23II P

RFS3

II RRFS2

II R

RFS

2

II P RFS

3

II P

FSC

3

II RFSC

2

II R

WFR

2

II P WFR

3

II PRFS

]2[II P RFS

3][II P

…FDR

1

II RFDR

3

II R

DMI

1

II R DMI

2

II RDMI

3

II R

oPrC

2

II P oPrC

3

II PoPrC

1

II P

FDM

1R FDM

3RFDM

2R

FDMP

IIWF23=1IIWF22

IIWF21=2

IIWF13=1IIWF12=2IIWF11

Verification and Modification Algorithm

Rule-checking

Fig. 3. The FRVPNs model of the Level_II.


Step 8. calculate the firing strength of each rule in theLevel_II by the composition operator MIN.Step 9. search for the maximum firing strength of thewinning rule among the activated rules in the Level_IIby the composition operator MAX.Step 10. draw a conclusion of the output in the Level_IIfrom the selected winning rule.

4.3. Petri nets for FRVPNs modeling

The FRVPNs takes advantage of the PNs to model thereasoning structure and represents the dynamic behaviorof the fuzzy rule. An ordinary PNs structure is a 4-tuple(P, R, I, O,) defined in Cardoso and Camargo (1999),Peterson (1981), Girault and Valk (2003), and Zhou

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Low (L)

High (H)

Medium (M)

Low (L)

High (H)

Medium (M)

Low (L)

High (H)

Medium (M)

......

RFS1

RFS2

RFS3

RFS4

RFS5

RFS6

RFS7

RFS8

RFS9

RFS26

RFS27

Inputobjects

Membership degree Calculation

)TCPUU(Lowμ

)TCPUU(Mediumμ

)TCPUU(Highμ

)PCPUU(Lowμ

)PCPUU(Mediumμ

)PCPUU(Highμ

)TMU(Lowμ

)TMU(Mediumμ

)TMU(Highμ

Step 1 Step 2 Step 3 Step 4

RT

WFR

LevLevelel_I_I

Outputsub-

object

MAXcomposition

operation

)TMU()PCPUU(

)TCPUU(RFS

LowLow

Low1

μ∧μ∧μ=

)TMU()PCPUU(

)TCPUU(RFS

HighLow

Low3

μ∧μ∧μ=

)TMU()PCPUU(

)TCPUU(RFS

MediumLow

Low2

μ∧μ∧μ=

.....

,...)RFS,

RFS,RFS(WRF

3

21MAX=

Step 5

A

Description of Mark A:The output subobjectof the FRDT Level_Ibeco mes the inputobject of the FRDTLevel_II.

TCTCPUPUU

PCPPCPUUUU

TMTMU

NoNote:

MIN composition operation

Fig. 4. Reasoning the sub-object in the Level_I.


and Venkatesh (1998), but the ordinary PNs cannotsatisfy the fuzzy reasoning design for the EDDM.Therefore, the proposed FRVPNs applied 11-tuples, aredefined by

FRVPNs ¼ (P, R, I, O, WF, M, H, Pro, ProMF, RFS,WRFS).

Related definitions are described as below.

Definition 4.1. WF: ASD(P�R)[(R�P)-+N is a weightfunction associated with the weight of the arc or the numberof the direct arcs, where AS is a set of the arcs, and the currentrelation holds only between the places and the transitions.

Definition 4.2. H: P�R-N, H(Pk) ¼ {PkAP: H(Pk, Rm)40}, is an inhibitor function. An inhibitor arc from a place Pk

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Short (S)

Low (L)

Fast (F)

Medium(M)

Normal(N)

Medium(M)

Long (L)

High (H)

Slow (S)

RFS1

RFS2

RFS3

RFS4

RFS5

RFS6

RFS7

RFS27

......

Input obj ect sMember ship degr ee

CalculationMIN composi tion

operat ion

LevelLevel_I_II

Output objec t

MAX composi tion oper ation

WRF

)EHL()HT(

)RT(RFS

LowMedium

Fast3

μ∧μ∧μ=

)EHL()HT(

)RT(RFS

NormalSlow

Fast2

μ∧μ∧μ=

)EHL()HT(

)RT(RFS

LowShort

Fast1

μ∧μ∧μ=

)HT(Shortμ

)HT(Longμ

)RT(Slowμ

)EHL(Highμ

)EHL(Normalμ

)RT(Mediumμ

)HT(Mediumμ

)RT(Fastμ

)EHL(Lowμ

.....

,...)RFS

,RFS,RFS(WRF

3

21MAX=

FRDT FRDT OutOutputput

Step 6 Step 7 Step 8 Step 9 Step 10

EHLEHL

RTRT

HTHT

HTHT

RTRT

EHLEHL

HTHT

RTRT

EHLEHL

DLDMDLDM

RFS26

RFS9

RFS8

Note:Note:

Fig. 5. Reasoning the object in the Level_II.


to a transition Rm has a ‘‘circle’’ symbol rather than anarrowhead at the transition. A transition represented by an‘‘arrow’’ symbol is enabled when tokens are in all of itsnormal inputs and no tokens are in all of its inhibitorinputs. The transition is fired by removing tokens from allof its normal inputs.

Definition 4.3. Pro ¼ {Pro11, Pro12, y, Proij} is a finite setof propositions. A proposition ProijAPro is mapped on aplace PnAP in the FRVPNs.

Definition 4.4. ProMF is a MF of a proposition. The inputobject of the proposition can be mapped to a corresponding


MD through the ProMF. The MD of a proposition of afuzzy rule FRk is represented by m(x)A[0, 1]-FR, wherexAX is the input object of the proposition.

Definition 4.5. a fuzzy composition operation represents aMD of the required proposition calculated by thecomposition operator MAX/MIN. The fuzzy compositionis defined by MAX/MIN-R.

Definition 4.6. RFSk: fuzzy composition-R, representsthe firing strength of a fuzzy rule. The larger firing strengthindicates the larger belief strength of a rule. Based on thefuzzy operators AND/OR shown in the antecedentproposition of a rule FRk, the RFSk is calculated by thecomposition operator MIN or MAX.

Definition 4.7. WRFS: a winning-rule firing strength, iscalculated by the MAX composition operator. The FRDTstructure consists of L-levels, e.g., Level_I and Level_II inthis article, and each level includes s-rules, e.g., FR1, FR2,y, FRk, y, FRs. Let RFSk be the firing strength of eachkth rule in the Level_I. Let WRFSk: MAX (RFS1, RFS2,y,RFSk, y, RFSs)-R be the firing strength of the winningrule FRk-FR. The WRFSk is used to compose the firingstrengths of the s-rules in the Level_I or Level_II and selectthe winning rule FRk with the highest confidence.

Fig. 6. (a) Fuzzy rules in the Level_I and Level_I II. (b) FR

5. FRVPNs reasoning strategy

As illustrated in Fig. 6, the examples of FR1 and FR2

are used to describe the FRVPNs modeling and thedynamic reasoning behavior. Fig. 6(a) shows the contentsof the fuzzy rule in the Level_I and Level_II. Fig. 6(b)shows part of the FRVPNs model by means of the PNstechnique. Fig. 6(c) shows the dynamic reasoning behaviorof the rules FR1 and FR2. The properties of theproposition set of places and the firing transitions aredescribed as follows:

(1)

VPNs

IPAProij and IIPAPro

ij : represents the ‘‘antecedent propo-sition’’ place of a rule FRk in the Level_I andLevel_II, respectively, where i represents the ith inputobject of a fuzzy rule FRk, and j represents the jthantecedent proposition of a fuzzy rule FRk.

(2)
IRAProMFij and IIRAProMF
ij : represents the ‘‘MF’’ transi-tion of an antecedent proposition of a fuzzy rule FRk

in the Level_I and Level_II, respectively.
(3) IPAProMD
ij and IIPAProMDij : represents the ‘‘MD’’ place

of an antecedent proposition of a fuzzy rule FRk in theLevel_I and Level_II, respectively.

(4)
IRRFSk and IIRRFS
k : represents the ‘‘firing strength’’transition of a fuzzy rule FRk in the Level_I andLevel_II, respectively.

Model. (c) Marking dynamic reasoning behavior.


(5)
IPRFSk and IIPRFS
k : represents the ‘‘firing strength’’place of a fuzzy rule FRk in the Level_I and Level_II

respectively. The place IPRFSk with a token represents

the firing strength of a fuzzy rule FRk.
(6) RMAX( ): represents the ‘‘firing strength’’ transition of
a winning rule in the Level_I or Level_II. It iscalculated by the composition operator MAX.

(7)
PWRFSk : represents the ‘‘winning fuzzy rule’’ place. It
models the firing strength FRSk of the winning ruleFRk in the Level_I or Level_II.

(8)
IRCProMFk and IIRCProMF
k : represents the ‘‘MF of aconsequent proposition’’ transition in the Level_I andLevel_II, respectively. The MD of a consequentproposition of the winning rule FRk is calculated bythe centroid of the aggregate output MF.

(9)
IPCProMDk : represents the ‘‘MD of a consequent
proposition’’ place. The place IPCProMDk with a token

represents the MD of a consequent proposition of arule FRk in the Level_I.

(10)
PFDM: is a ‘‘final decision-making’’ place. The placePFDM with a token represents the final result in theLevel_II.
5.1. FRVPNs reasoning algorithm for the Level_I

The FRVPNs model of the Level_I is illustrated inFig. 2, and described as below.

Step 1. Import the parameters of the proposition of therules by the following places and transitions:

(1)
IPAProIO ¼ IPAProIO1 ; IPAProIO
2 ; . . . ; IPAProIOk ; . . . ; IPAProIO

s

� �is a set of ‘‘input object’’ places. Each place has aninput variable.� �

(2)
IRAProD ¼ IRAProD1 ; IRAProD
2 ; . . . ; IRAProDk ; . . . ; IRAProD

s

is a set of ‘‘input object distribution’’ transitions. Itrepresents the transition distribution of linguisticvariables for each antecedent proposition of a rule.

Step 2. Establish the antecedent propositions for each
rule and calculate the MD for each antecedent propositionby the following places and transitions:
(1)
PAPro ¼ IPAPro11 ; IPAPro
12 ; IPAPro13 ; . . . ; IPAPro

ij

� �is a set of

‘‘antecedent proposition’’ places for each rule.� �
(2) RAProMF ¼ IRAProMF
11 ; IRAProMF12 ; . . . ; IRAProMF

ij is a set

of ‘‘MF of an antecedent proposition’’ transitions.I RAProMF

ij represents the MF of the jth fuzzy set

corresponding to the ith input object of the antecedent

proposition, thus the MD of the ith input object of the

antecedent proposition is computed by the jth MF.
(3) IWFij represents the weighted function corresponding
to each transition arc of the antecedent propositions.� �
(4) PAProMD ¼ IPAProMD
11 ; IPAProMD12 ; . . . ; IPAProMD

ij is a set of

‘‘MD of an antecedent proposition’’ places for each rule.

Step 3. Perform MIN composition operation andcalculate the firing strength for the activated rules by the
following places and transitions:
(1)
RRFS ¼ IRRFS1 ; IRRFS
2 ; . . . ; IRRFSk ; . . . ; IRRFS

s

� �is a set of

‘‘firing strength of rule’’ transitions for each rule in the

Level_I. IRRFSk uses the fuzzy operator ‘‘AND’’ or ‘‘OR’’

to perform MIN or MAX composition operation.

(2)
PRFS ¼ IPRFS1 ; IPRFS
2 ; . . . ; IPRFSk ; . . . ; IPRFS

s

� �is a set of

‘‘firing strength of rule’’ places. While the token exists

in the place IPRFSk , the transition IRRFS

k is disabled by

the inhibitor arc to ensure that the new result of firing

strength is transferred into the place IPRFSk after the

previous firing strength in the place have beentransferred out .

Step 4. Perform MAX composition operation on thefiring strengths of the activated rules, and select thewinning rule from the available rules in the Level_I bythe following places and transitions:

(1)
RMAXð Þ ¼MAX IPRFS1 ; IPRFS
2 ; . . . ; IPRFSk ; . . . ; IPRFS

s

� �is

a transition of MAX composition operation. Thetransition RMAX( ) is used to calculate the maximumfiring strength among the activated rules.

(2)
PWRFSk is a ‘‘WRFSk’’ place, and it represents the firing
strength RFSk of the winning rule FRk.

Step 5. Determine the winning rule for rule-checking
purpose by the following places and transitions:
(1)
RFSC ¼ IRFSC1 ; IRFSC
2 ; . . . ; IRFSCk ; . . . ; IRFSC

s

� �is a set of

‘‘firing strength comparison’’ transitions. The transitionis used to compare the firing strength of an activatedfuzzy rule with that of the WRFSk of a FRk. If thefiring strengths are not same, the reasoning process willbe terminated.� �

(2)
PWFR ¼ IPWFR1 ; IPWFR
2 ; . . . ; IPWFRk ; . . . ; IPWFR

s is a set

of ‘‘winning firing rule’’ places. While the place IPWFRk

contains a token, the place IPWFRk represents FRk is the

winning firing rule and is selected to fire.

Step 6. Determine the final decision rule for rule-
checking purpose by the following places and transitions:
(1)
IPRFS½�k� is a set of ‘‘firing strength of rule’’ places, except
the IPRFSk place.

(2)
IPCProO is a ‘‘linguistic variable of output object of aconsequent proposition’’ place.
(3)
RFDR ¼ IRFDR1 ; IRFDR
2 ; . . . ; IRFDRk ; . . . ; IRFDR

s

� �is a set

of ‘‘final decision of the winning rule’’ transitions. Thetransition is used to delete the rules with the firingstrength less than the rule with the highest firing strength.

(4)
IPDMI is a ‘‘decision-making identification’’ place. Itrepresents the decision result of the selected winningrule FRk with the highest firing strength.


Step 7. Construct the consequent propositions ofthe final decision rule by the following places and
transitions:
(1)
PCProM ¼ IPCProM1 ; IPCProM
2 ; . . . ; IPCProMk ; . . . ; IPCProM

s

� �is a set of ‘‘marking of the consequent proposition’’places. � �

(2)
RDMI ¼ IRDMI1 ; IRDMI
2 ; . . . ; IRDMIk ; . . . ; IRDMI

s is a setof ‘‘decision-making identification’’ transitions. Itrepresents the transition distribution of linguisticvariables for a consequent proposition of a fuzzy rule.

(3)
PCPro ¼ IPCPro1 ; IPCPro
2 ; . . . ; IPCProk ; . . . ; IPCPro

s

� �is a set of

‘‘consequent proposition’’ places. It describes theconclusions of the fuzzy rules.

Step 8. Calculate the MD for each consequent proposi-tion by the following places and transitions:

(1)
RCProMF ¼ IRCProMF1 ; IRCProMF
2 ; . . . IRCProMFk ; . . .

�IRCProMF

s Þ is a set of ‘‘MF of a consequent proposi-

tion’’ transitions for each consequent proposition.

When the transition IRCProMFk is fired, the MD is com-

puted by the kth MF.�
(2) PCProMD ¼ IPCProMD
1 ; IPCProMD2 ; . . . ; IPCProMD

k ; . . . ;IPCProMD

s Þ is a set of ‘‘MD of a consequent proposi-

tion’’ places. While the place IPCProMDk contains a

token, it receives an MD from the transition IRCProMFk .

5.2. FRVPNs reasoning algorithm for Level_II

The FRVPNs model of the Level_II is illustrated inFig. 4, and described as below.

Step 1. Establish the antecedent propositions of the rulesand calculate the MDs for each antecedent proposition asfollows.

(1)
The output object from Level_I is used as the inputplace to Level_II.
(2)
The new input objects of the antecedent proposition ofthe rules in the Level_II can be modeled by repetition ofthe Step1 in Section 5.1.
(3)
The MDs of the new input objects are calculated byrepetition of the Step2 in Section 5.1.
Step 2. Describe the decision behaviors of the rules asfollows.

(1)
Use Step3 in Section 5.1 to calculate the firing strengthof each activated rule.
(2)
Use Step4 in Section 5.1 to find out the maximum firingstrength from the available activated rules.
(3)
Use Step5 and Step6 in Section 5.1 to decide thewinning rule among the activated rules.
(4)
Use Step7 in Section 5.1 to determine the consequentpropositions of the Level_II.
Step 3. Determine the final decision-making by thefollowing places and transitions:

(1)
RFDM ¼ RFRDM1 ;RFRDM
2 ; . . . ;RFRDMk ; . . . ;RFRDM

s

� �is a

set of ‘‘final decision-making’’ transitions, which

determines RFDMk .

(2)
PFDM is a ‘‘final decision-making’’ place of entireFRVPNs.
6. Simulations

As shown in Figs. 2 and 3, the fuzzy rule base of thereasoning processor in the EDDM is constructed of twolevels in this study. The measurement values [TCPUU,PCPU, TMU, HT, EML] ¼ [73%, 33%, 65%, 10.5 s,72%], are assumed to be the inputs variables to theantecedent propositions of FRVPNs model, which isthe example_1 with the inference results in the Level_Iand Level_II listed in Table 2. With this example, thereasoning process is described in detail through thefollowing steps.

Step 1. Normalize the measurement inputs (e.g.[TCPUU, PCPU, TMU, HT, EHL] ¼ [73%, 33%, 65%,10.5sec, 72%]) by the fuzzy sets in Section 3.1 anddistribute the normalized inputs to the FRVPNs model.The measurement values are normalized to be

TCPUU

100%;PCPUU

100%;TMU

100%;HT

30;

EHL

100%

� �

¼ ½ð73=100Þ; ð33=100Þ; ð65=100Þ; ð10:5=30Þ; ð72=100Þ�

¼ ½0:73; 0:33; 0:65; 0:35; 0:72�,

where the maximum HT is defined to be 30 s.Step 2. Assume the Level_I of the reasoning processor

has the following rules:

FR1: IF ‘‘TCPUU is Medium’’ AND ‘‘PCPUU is Low’’AND ‘‘TMU is Low’’ THEN ‘‘RT is Fast’’.FR2: IF ‘‘TCPUU is High’’ AND ‘‘PCPUU is Low’’AND ‘‘TMU is Medium’’ THEN ‘‘RT is Medium’’.FR3: IF ‘‘TCPUU is High’’ AND ‘‘PCPUU is Medium’’AND ‘‘TMU is High’’ THEN ‘‘RT is Slow’’.

Then, TCPUU ¼ 0.73, PCPUU ¼ 0.33, and TMU ¼ 0.65are imported to the places IPAProIO

1 , IPAProIO2 , and IPAProIO

3 inthe Level_I, respectively.

Step 3. The transitions IRAProD1 , IRAProD

2 , andIRAProD3 are

concurrently fired, then the tokens are distributed to the

place IPAPro11 , IPAPro

12 , IPAPro13 , IPAPro

21 , IPAPro22 , IPAPro

23 , IPAPro31 ,

IPAPro32 , and IPAPro

33 , respectively.

Step 4. The transition set IRAProMF is fired, then the

tokens are transferred to the place IPAProMD11 , IPAProMD

12 ,IPAProMD

13 , IPAProMD21 , IPAProMD

22 , IPAProMD23 , IPAProMD

31 ,IPAProMD

32 , and IPAProMD33 , respectively. The MD of each

place is calculated by its trapezoidal MF described in

ARTICLE IN PRESS

Table 2

Simulation examples of the FRVPNs

Level_I Level_II Matlab RT Matlab DLDM

Rule No. TCPUU PCPUU TMU RT MIN HT EHL DLDM MIN

Example_1 1 M L L F 0 S N N 0.25

2 H L M M 0.35 M N SH 0.4

3 H M H S 0.25 L H H 0

Measurement values 0.73 0.33 0.65 MAX 0.35 0.35 0.72 MAX 0.4 M 0.592 SH 0.416

Example_2 1 M L L F 0.2 S L N 0

2 H L M M 0.35 M N SH 0.3

3 H M H S 0 L L N 0


Example_3 1 L M L F 0.35 S N N 0

2 H L L F 0 M L N 0.2

3 M L M M 0.4 M H SH 0.25


Example_4 1 M L H S 0 S N N 0

2 M M L M 0 M N SH 0

3 H H M S 0.6 L H H 0.7

Measurement values 0.72 0.75 0.5 MAX 0.6 0.74 0.86 MAX 0.7 S 0.83 H 0.835

Example_5 1 M L L M 0.25 S N N 0

2 M L M M 0.25 L L N 0

3 H M M S 0.6 L H H 0.75

Measurement values 0.75 0.34 0.32 MAX 0.6 0.78 0.75 MAX 0.75 S 0.71 H 0.838

Note: RT and DLDM are the crisp values calculated by the Mamdani fuzzy method of MATLAB tools.

The italic words are the winning rules.


Section 3.1. In this example, the calculated MDs are:

IPAProMD11 ¼ mLowðTCPUUÞ ¼ 0,

IPAProMD12 ¼ mMediumðTCPUUÞ ¼ 0:35,

IPAProMD13 ¼ mHighðTCPUUÞ ¼ 0:65,

IPAProMD21 ¼ mLowðPCPUUÞ ¼ 0:35,

IPAProMD22 ¼ mMediumðPCPUUÞ ¼ 0:65,

IPAProMD23 ¼ mHighðPCPUUÞ ¼ 0,

IPAProMD31 ¼ mLowðTMUÞ ¼ 0,

IPAProMD32 ¼ mMediumðTMUÞ ¼ 0:75,

IPAProMD33 ¼ mHighðTMUÞ ¼ 0:25.

Step 5. The transition set RRFS and the transition RMAX( )

are fired in order, and then the firing strength of eachactivated rule and the winning rule is calculated bythe MIN and MAX composition operator, respectively.It yields

FR1 : MINð0:35; 0:35; 0Þ ¼ 0,

FR2 : MINð0:65; 0:35; 0:75Þ ¼ 0:35,

FR3 : MINð0:65; 0:65; 0:25Þ ¼ 0:25,

MAX ðFR1;FR2;FR3Þ

¼MAX ð0; 0:35; 0:25Þ ¼ 0:35.

Step 6. According to the result of MAX compositionoperation in the above Step5, the final winning rule is

FR2: IF ‘‘TCPUU is High’’ AND ‘‘PCPUU is Low’’AND ‘‘TMU is Medium’’ THEN ‘‘RT is Medium’’.

In case that the antecedents of rules are evaluated to bethe same level, the max–min inference cannot determine thewinning rule. Hence, other defuzzification methods areconsidered to use. The center of gravity method (COG) isthe most popular method (Van Broekhoven and De Baets,2004; Negnevitsky, 2002; Yager, 1992), which is used tosolve this problem.

Step 7. Assume the Level_II of the reasoning processorhas the following rules:

FR4: IF ‘‘HT is Medium’’ AND ‘‘RT is Medium’’ AND‘‘EHL is Normal’’ THEN ‘‘DLDM is Slightly High’’.FR5: IF ‘‘HT is Short’’ AND ‘‘RT is Medium’’ AND‘‘EHL is Normal’’ THEN ‘‘DLDM is Normal’’.FR6: IF ‘‘HT is Long’’ AND ‘‘RT is Medium’’ AND‘‘EHL is High’’ THEN ‘‘DLDM is High’’.

Then, HT ¼ 0.35 and EHL ¼ 0.72 are imported to theplaces IIPAProIO

1 and IIPAProIO2 , respectively. The defuzzifica-

tion of RT is calculated as RT ¼ 0.598 by the centroid ofthe aggregate output MF in the Level_I. This RT value is

ARTICLE IN PRESS

Fig. 8. Final decision of the Level_II.


then imported to the antecedent propositions in theLevel_II.

Step 8. The transition set IIRAProDis fired to distributethe tokens to the place set IIPAPro.

Step 9. As the transition set IIRAProMF is fired, the tokensare then transferred to the place set IIPAProMD. The MDs ofantecedent propositions in the Level_II are calculated bythe trapezoidal MF and listed as below.

IIPAProMD11 ¼ mShortðHTÞ ¼ 0:25

IIPAProMD12 ¼ mMediumðHTÞ ¼ 0:75

IIPAProMD13 ¼ mLongðHTÞ ¼ 0

IIPAProMD21 ¼ mLowðEHLÞ ¼ 0

IIPAProMD22 ¼ mMediumðEHLÞ ¼ 0:4

IIPAProMD23 ¼ mHighðEHLÞ ¼ 0:6

IPCProMD23 ¼ mMediumðRTÞ ¼ 1

Step 10. By using the MIN and MAX compositionoperators, the firing strength of each activated rule and thewinning rule is calculated respectively as

FR4 : MINð0:75; 1; 0:4Þ ¼ 0:4

FR5 : MINð0:25; 1; 0:4Þ ¼ 0:25

FR6 : MINð0; 1; 0:6Þ ¼ 0

MAX ðFR4;FR5;FR6Þ ¼MAX ð0:4; 0:25; 0Þ ¼ 0:4

Step 11. According to the result of MAX compositionoperation in the above Step10, the final winning rule is:

FR4: IF ‘‘HT is Medium’’ AND ‘‘RT is Medium’’ AND‘‘EHL is Normal’’ THEN ‘‘DLDM is Slightly High’’.

Following the steps of the FRVPNs Reasoning Algo-rithm in Section 5, the final winning rule in Level_I is FR2,which indicates that the ‘‘RT is Medium’’, and in theLevel_II the final winning rule is FR4, which indicates the

Fig. 7. Final decision of the Level_I.

‘‘DLDM is Slightly High’’. The Mamdani fuzzy method ofthe MATLAB tools is also used to compare the inferenceresults under the same conditions (same inputs, samelinguistic values, same ranges). As shown in Fig. 7, rulesFR1, FR2, and FR3 are aggregated and defuzzied to have acrisp value of RT ¼ 0.592 by calculating the centroid,which indicates the rule FR2 is the winner. In reference tothe consequent proposition of FR2, ‘‘RT is Medium’’ isthus inferred for Level_I. In Fig. 8, in Level_II, the crispvalue of DLDM ¼ 0.416 is calculated, which indicates therule FR4 is the winner. In reference to the consequentproposition of FR4, ‘‘DLDM is Slightly High’’ is thusinferred for the monitored AP and the OS. In comparisonwith the inferred results, both methods have the samereasoning outcomes. This example and other four examplesby using the two methods are included and listed inTable 2, which shows that the inference outcomes of all theexamples are the same.

7. Conclusions

In this article, the fuzzy set theory and the fuzzyproduction rule method are used to establish the fuzzyrules for the failure events of the AP. The proposedFRVPNs is designed based on the FRDT in associationwith the PN technique, and constructed with the rule-checking process as well as the verification and modifica-tion module. The computational complexity (Chen et al.,1990; Gao et al., 2003; Kungas, 2005) is defined by O(ptl)and determined by the number of p, t and l, where p

and t are the numbers of places and transitions, respec-tively, and l is the maximum number of transitions in thelongest place-transition path. The FRVPNs reasoningalgorithm apparently has higher computational complexitythan a general FRDT, because it uses more places andtransitions in the rule-checking process and in theverification and modification module. Even though thecomplexity, and higher accuracy could be obtained,


problems of redundancy, conflict, circularity, and incom-pleteness could be reduced. Comparing the FRVPNs withthe fuzzy logic toolbox of MATLAB by the simulationresults of several sets of arbitrarily selected examples, bothmethods conclude the same outcomes of fuzzy reasoning. Itverifies that the damage level of the failure event for themonitored AP and the OS can be successfully reasoned bythe proposed FRVPNs. To sum up, the proposed FRVPNsmodel is capable of reasoning and determining the failureevent efficiently and accurately, and it is thus suitable to beincluded in the diagnosis mechanism of the EDDM.Regarding the verification and modification algorithm, itis addressed in another article.

Acknowledgment

This research is supported by NSC89-TPC-7-033-009 &NSC88-2212-E-033-018.

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Documents

A Fuzzy Reasoning Design for Fault Detection and Diagnosis of a Computer-controlled System