Research ArticleFault Mechanism Analysis for Manufacturing SystemBased on Catastrophe Model

Jianhua Zhu1 Chaoan Lai1 and Yanming Sun 2

1School of Business Administration South China University of Technology Guangzhou 510641 China2School of Business Administration University of Guangzhou Guangzhou 510006 China

Correspondence should be addressed to Yanming Sun sunyanminggzhueducn

Received 24 February 2019 Revised 23 May 2019 Accepted 2 June 2019 Published 24 June 2019

Academic Editor Erik Cuevas

Copyright copy 2019 Jianhua Zhu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Fault analysis is important in both research and industry Current fault analysis tasks aremainly concernedwith fault prediction andclassification and do not focus enough on fault evolution mechanisms In this paper we propose a fault analysis method based oncatastrophe theory for manufacturing system to improve the effectiveness and efficiency of real time monitoring of potential faultand causes analysisThe key advantages of our proposedmethod are (i) utilizing catastrophe theory and big data analysis to establishthe fault cusp catastrophe model of manufacturing system and create the internal fault evolution mechanism of manufacturingsystem by the cusp catastrophe model and (ii) with the established catastrophe model fulfilling fault monitoring and accuratepreventive control of the manufacturing system and ensuring the healthy operation of the manufacturing system

1 Introduction

The modern manufacturing industry is characterized byhigh quality and high production efficiency and currentmanufacturing systems are designed for good performancehigh stability and high repeatability Production equipmentis designed to be extremely precise efficient and intelligentSmall performance degradation or security risksmay bringserious consequences The global manufacturing industryloses more than 100 billion dollars every year due to qualityproblems caused by machine failures and other issues Itis vital to have a valid analysis approach to ensure safeoperation of the equipment Analyses of failure mechanismsand preventive control of manufacturing systems offer agood potential for future manufacturing such as smartmanufacturing

The current method of fault analysis is shown in Table 1Research on fault analysis started in the 1960s Early

research regarded signal processing techniques and statisticalanalysis as major tools and primarily used artificial intelli-gence to extract fault features Knowledge-driven methodsneed to establish a precise mathematical model based uponthe understanding of the physical mechanism [1] parameterestimation [2] and parity spaces [3]

However during complicated dynamic industrial pro-cesses it is very difficult to manually build a mechanismmodel according to a deep insight into the system Data-driven fault analysis rests on either an explicit mathematicalmodel derived from prior knowledge or a reasoning mecha-nism derived from experience It uses different types of datamining technology to extract and classify fault features inacquired vast operating data [4] which include signal pro-cessing [5 6] statistical analysis [7 8] and early quantitativeartificial intelligence methods [9] With increasing degrees ofautomation and intelligence of industrial equipment alongwith the development and widening of application spectrumof related advanced technologies [10 11] data began togrow exponentially It becomes more important to processand analyze manufacturing systems in order to obtain hugediagnostic valueHence in recent years value-drivenmethodhas gradually become a hot topic for researchers Amongthem deep learning is the most concerned by researchersDeep learning [12] is good at finding complex structures inhigh dimensional data It can extract fault features adaptivelyusing enough conversions and combinations and distill thephysical significance of features without manual interven-tion

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 2313581 11 pageshttpsdoiorg10115520192313581

2 Mathematical Problems in Engineering

Table 1 The research status

Methods Representative

Knowledge-drive H Ma [1]W Chen [13] G H B Foo[14] M Sepasi [15] T Jiang [2] S Zhai [16] MZ [3] H M[17]

Data-drive I Chen [5] J Yan [6] M Grbovic [7] S Yin [8] H A Talebi [9] Dong J [18] Guo W [19] WeissB A [20] Zhou Z [21]

Value-drive G E Hinton [22] J Xiong [23]W Chen [24] Zhang Y [25]Wang P [26] Zhao R [12]

In conclusion data-driven and value-driven methodsare widely employed in fault analysis which makes fastand accurate analysis possible when faults occur In recentyears scholars mainly focus on the following aspects (1)value-driven which mainly includes deep learning [22]integration and fusion [23 24] Zhang et al [25] and Wanget al [26] used deep learning in extraction and classificationof fault feature Zhao et al [12] use deep learning in machinehealth monitoring and (2) data-driven Dong et al [18] joindata-driven fault analysis integrating causality graphs withstatistical process monitoring for complex industrial pro-cesses Guo et al [19] used topological data analysis to extractcharacteristics Weiss et al [20] extracted key informationusing a combination of heuristic algorithms Zhou et al [21]used k-Nearest neighborhood in fault Isolation

However the above methods lack analysis of fault evolu-tion mechanism capability with regard to potential faults ofmanufacturing system in particular complex manufacturingsystem Therefore complex system theory was developedwhere the internal mechanism of fault operation can berevealed Unfortunately there is little research in the exist-ing literature on the application of catastrophe theory tofault analysis We found that only [27] studied and provedproperties of Catastrophe to finite-state systems and showedthat catastrophe theory can also be used in fault injec-tion

In conclusion the above research has promoted theintegration of big data and intelligentmanufacturing and pro-vided a good research idea for the research onmanufacturingsystem failures and the research results provided a scientificbasis for enterprisesrsquo production decisions However manyof the above research methods examined the identification offault types and prediction of fault occurrence time which isresult-oriented and fails to dig out the internal mechanismof fault evolution Based on the above analysis this paperproposes a method combining catastrophe theory and bigdata analysis to mine the internal fault evolution mechanismof manufacturing systems

According to the method proposed in this paper twopotential scenarios are related to the analysis of failure mech-anisms and preventive control of manufacturing systems(1) after the fault occurs the catastrophe theory is used tomodel the existing fault in the manufacturing system to findout the internal mechanism of fault evolution and (2) bymonitoring the system operation data the correspondingcatastrophe model is found using the abnormal data Andusing established catastrophe models to dig out the causesof the failure before the failure occurs will help controlthe system in terms of the fault cause As a result the

preventive maintenance of manufacturing systems will havebeen realized

The main contributions of this paper are twofold asfollows

(i) We use the catastrophe model and big data analysisto establish the cusp catastrophe model of fault about man-ufacturing system and by analyzing the catastrophe modelthe internal mechanism of fault evolution of manufacturingsystem is found

(ii) According to the established catastrophe model werealize fault monitoring and accurate preventive control ofthe manufacturing system and ensure the healthy operationof the manufacturing system

The rest of this paper is organized as follows Section 2describes an overview of catastrophe theory analysis togetherwith a concept of catastrophe theory analysis based on faultanalysis for manufacturing system A simplified proof-of-concept case study is carried out to validate the proposedfault analysis process in Section 3 The research significanceis illuminated in Section 4 Finally Section 5 concludes thepaper and outlines our future work

2 Catastrophe Theory Analysis Based Fault forManufacturing System

The catastrophe theory deals with how continuous gradualchanges in nature and human society cause catastrophesor leaps and seeks to describe predict and control thesecatastrophes and leaps with uniform mathematical mod-els

In this papermanufacturing system catastrophe ismainlyreflected in two aspects (1) the manufacturing system sud-denly went into fault from normal state and (2) the manufac-turing system suddenly returns to normal state from the faultstate The catastrophe theory researches why manufacturingsystem will occur with the above catastrophes and whatstrategies can be adopted to control the catastrophes

Before starting the model analysis we first define thethreshold value 119884 which indicates that the manufacturingsystem will fail if throughput is lower than this value

21 Model and Analysis In manufacturing systems catas-trophe theory [28] can be used to study the impact ofchanges in external conditions on industrial big data Ingeneral the catastrophe can cause irreversible damage andbring huge economic losses to the manufacturing industryTherefore it is of great practical value to use catastrophetheory to study manufacturing system fault In this paper the

Mathematical Problems in Engineering 3

operation of manufacturing system is considered from bothinternal and external aspects There are many internal andexternal factors that affect the operation of manufacturingsystems for example machine adjustment machine agingimproper product design irrational production technologyand process design improper processing methods insuf-ficient machining accuracy machine maintenance errorsand operational management errors That is very hard toanalyze the impact of these factors on the manufacturingsystem at the same timeTherefore we introduce catastrophemodels and macroscopic order parameters to reflect the realoperation situation of the manufacturing system In thispaper external macroscopic order parameters are defined interms of production throughput and the internal macro-scopic order parameters are defined in terms of productionload and duration Based on the catastrophe theory weplan to describe the behavior catastrophe of manufacturingsystem by cusp catastrophe theory The cusp mutation modelis the simplest mutation model When the cusp mutationmodel is used for fault description the critical surface iseasy to construct and there is a strong geometrical intu-ition

The external macroscopic order parameter and the inter-nal macroscopic order parameters are regarded as the statevariables 119883 and control variables 119880 and 119881 of the manufac-turing system respectively And a cusp catastrophe model isestablished to describe the abnormal behavior of the systemby using the data of each the adjacent continues data flowinterval including normal data and abnormal data Accordingto the opinion proposed by Hall [29] the basic model of cuspcatastrophe is

119881 (119909) = 11988811199094 + 11988821199061199092 + 1198883V119909 (1)

where 119909 is the state variable 119906 and V are the controlvariables In this paper 119906 is duration V is production loadand 119909 is throughput 1198881 1198882 1198883 are coefficients Thereforethe phase space is a three-dimensional space composedof state variables 119909 119906 and V The critical point of thepotential function is the solution of equation nabla119909119881(119909) =0 Therefore the equilibrium surface 119872 is also given bynabla119909119881(119909) = 0

The corresponding nonisolated singularities set 119878 is givenby

119878 = (119909 119906 V) | nabla2119909119881 (119909) = 0 nabla119909119881 (119909) = 0 (2)

Projecting 119878 onto the parametric plane u-v we can obtainthe bifurcation set equation

8119888321199063 + 27119888111988823V2 = 0 (3)

After coordinate transformation the mutant flow andbifurcation set equations are respectively as follows

Catastrophe Flow

1199093 + 119886119906119909 + 119887V = 0 (4)

Bifurcation Equation

411988631199063 + 271198872V2 = 0 (5)

where 119886 = 119888221198881 119887 = 119888341198881 We can obtain

Equilibrium Surface 119872 = (119909 119906 V) | 1199093 + 119886119906119909 + 119887V = 0

Bifurcation Set 119861 = (119906 V) | 119863 = 411988631199063 + 271198872V2 119863 = 0Where 119886 and 119887 are the coefficients For parameters 119886 and

119887 the optimal value can be obtained by solving the extremamethod with the multivariate function supposing we collect119899 sets of continuous adjoining data including normal andabnormal data (119909119894 119906119894 V119894) put them into 119872 and 119861 and getcumulative error

119890119903119900119903119903 =119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894) + (411988631199063119894 + 271198872V2119894 ) (6)

Our goal is to find the value of 119886 119887 for each group(119909119894 119906119894 V119894) we can obtain

1199093 + 119886119906119909 + 119887V = 0

411988631199063 + 271198872V2 = 0(7)

So we use the sum of squares to find the optimal 119886 119887

119890119903119900119903119903

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (8)

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (9)

According to above formula and 120597119869(119886 119887)120597119886 = 0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(10)

By solving (10) we can get all of the extremum sets

119864 = (119886119895 119887119895) | (119886119895 119887119895)

is the extreme points of 119869 (119886 119887) 119895 = 1 2 119905 (11)

where 119905 is the number of extreme points Because thegeometric surface of 119869(119886 119887) is pointing upwe canfind (119886 119887) =

4 Mathematical Problems in Engineering

M

x

v

u

v

uD=0

D=0

Dlt0

Dgt0

B

S

S

B

Upper lobe

Lower lobe

unreachable

Mode 1

Mode 2

Figure 1 Cusp catastrophe

(119886119894 119887119894) | 119869(119886119894 119887119894) lt 119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 thatis (119886 119887) being minimal points of 119869(119886 119887)

From Figure 1 it is obvious that on the control parameterplane u-v if (119906 V) isin (119906 V) | 119863(119906 V) lt 0 for each (119906 V)119909 has 3 values on the surface that is the manufacturingsystem has three kinds of mode that one of three modes istheoretically reachable but practically impossible which isunreachable When (119906 V) isin (119906 V) | 119863(119906 V) = 0 thetwo kinds of mode merge into a single one In this case themanufacturing system will change suddenly from one modeto the others When (119906 V) isin (119906 V) | 119863(119906 V) gt 0 119909has only one value which indicates that the manufacturingsystem is single mode So 119863 = 0 is the separatrix between thesinglemode and themultimode of themanufacturing systemwhere the modal catastrophe of the manufacturing systemoccurs And we found that the control parameters (119906 V) canonly go from 119863 lt 0 region to the edge 119863 = 0 as catastropheoccurs In actual operation the control parameters (119906 V)cannot be reached from the region of119863 gt 0 to the edge of119863 =0 So if the control parameters (119906 V) of the manufacturingsystem are located in (119906 V) | 119863(119906 V) lt 0 it shows thatthe manufacturing system is in a risky state In this case thecontrol parameter can easily be changed to the edge 119863 = 0and catastrophe occurs In contrast if the control parametersare located in (119906 V) | 119863(119906 V) gt 0 themanufacturing systemwill stay in a stable state So if the upper lobe represents thenormal state of manufacturing system lower lobe representsfault state and the above analysis is followed when (119906 V) isin

(119906 V) | 119863(119906 V) lt 0 the manufacturing system has twokinds of modes one of them is in a normal state while theother one is faulty With the change of control parameters(119906 V) manufacturing systems keep stable in one state until(119906 V) isin (119906 V) | 119863(119906 V) = 0 and manufacturingsystemwill suddenly jump from upper lobe to lower lobe andbreaks down or contrary Therefore in order to make themanufacturing system run healthily we should control (119906 V)and make (119909 119906 V) isin (119906 V) | 119863(119906 V) gt 0 119909 ge 119884 then thepurpose of preventive control is achieved

22 Management Research Through the analysis of themutation model shown in Figure 1 we can obtain the bound-ary between normal and abnormal in the manufacturingsystem In order to apply above research into managementwe define the following related events

(1) Event 119860 in the operation data of manufacturingsystem if the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the logical value of event A is 1 otherwise it is 0

(2) Event 119861 if the control variables (119906 V) isin (119906 V) |119863(119906 V) lt 0 119909 ge 119884 the logical value of event A is 1otherwise it is 0

(3) Event 119862 if the control variables (119906 V) isin (119906 V) |119863(119906 V) gt 0 119909 lt 119884 the logical value of event A is 1otherwise it is 0

(4) Event D in the operation data of manufacturingsystems if the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 the logical value of event A is 1 otherwise it is 0

Mathematical Problems in Engineering 5

Table 2 The related events

Event Label A B C D Implication SuggestGreen 1 0 0 0 Operating-normally - -Yellow 0 1 0 0 Warning Monitor (119906 V) to far from (119906 V) | 119863(119906 V) = 0Red 0 0 1 0 Break down RepairBlue 0 0 0 1 Break down Adjust (119906 V) to (119906 V) | 119863(119906 V) = 0

1

3

4

2

x

xo1

xo3

xo4

o

xo2

xo5

1 2 4

5

06

Internal mechanism The external display Real time data Management

A

B

C

D

Real time

asymptotic stabilityunreachable

First stepMeasure actual operation of

Second stepDetermine if the manufacturing

system will suddenly fail

Third step

Adopt management measures

evolutionary pathmappingcontrol management

manufacturing system

productiondata

Figure 2 Control process

Then according to the above definition we get thefollowing as shown in Table 2

For manufacturing enterprises the best thing is to reducethe occurrence of production failures Its frequent occurrencewill not only affect the morale of the companyrsquos employeesbut also lead to the stagnation of production which willseriously affect the companyrsquos normal operation and profitsTherefore based on the above research results we can carryout preventive control The control processes can be dividedinto three parts (Figure 2)(1) measure the actual operation ofthe manufacturing system according to the logic relationshipestablished in the table (2) judge whether the manufactur-ing system will suddenly fall into failure according to thecatastrophe model (3) take management actions to controlparameters and prevent the system from falling

In order to visually show the core ideas of this paper Letus take 119906 = constant the change of V in Figure 2 is analyzed

First map Figure 1 in the 119909-V plane and get the internalmechanism part shown in Figure 2The curve 1-3 and 3-4represent the upper branch of counter-S curve in Figure 1and indicates the normal state of the manufacturing system

where point 3 is a critical point that the system goes fromsingle mode to multimode Point 4 is in the bifurcation setand represents the critical point where the manufacturingsystem jumps from the normal state to the fault state Thecurve 4-2 represents the central part of counter-S curvein Figure 1 and represents the state that the manufacturingsystem in the actual operation cannot reach where point 2is in the bifurcation set and represents the critical point atwhich the manufacturing system jumps from the fault stateto normal state The curves 2-5 and 5-6 represent the lowerbranch of counter-S curve in Figure 1 and it indicates the faultstate of the manufacturing system The curve 2-5 indicatesthat although the manufacturing system is faulty it can berestored to the normal state by changing the values of 119906 andV but curve 5-6 indicates the manufacturing system cannotrecover itself and can only be repaired manually AnalyzingFigure 2 when V = V1 the manufacturing system statevariable is located in xo1 of upper branch and when v isincreased from v1 to v4 the state variable x changes to xo4 Atthis time if we added an infinitesimal perturbation to v4 thestate of variable x will jump from xo4 to xo5When v reduces to

6 Mathematical Problems in Engineering

Data extraction

Adjacent normal datawith abnormal data Abnormal data

Establishment of mutation model

Real-time of manufacturing system

visualization

Management

Model analysis

Figure 3 Control process

v2 the state of variable x changes to xo2 At this time as long asv2 gets a little bit smaller the state of variable xwill jump fromxo2 to xo3 in point 2 and the state of variable x enters upperbranch of counter-S curve Therefore according to the aboveanalysis we can control the failure of manufacturing systemThe specific process is shown in Figure 2 firstly the operationdata of the manufacturing system (119909 119906 V) are collected andanalyzed and then according to Table 2 the data will bedisplayed in real-time by bars in different colors By thisway we can find the operational problem in manufacturingsystems and then it is controlled according to the internalmechanism of fault in the manufacturing system to keep thesystem running in a healthy state

3 A Simplified Case Study

A simplified proof-of-concept case is illustrated to show theprocess of the proposed method Yonggu is a company thatproduces metal tools and has a complete IoT (Internet ofthings) system in its workshop We use the IoT system tocollect real-time data of the productworkshop and then selectthe adjacent data including normal data and abnormal datato establish the catastrophe model

In this section the throughput 119883 and the productionload 119881 of the manufacturing system within the duration time119880 are selected as the monitoring parameters In addition itshould be noted that according to the parameter estimationrequirements of the catastrophe model the data must be

extracted in an interval that is the data in an interval mustbe continuous in time

The specific way is shown in Figure 3

31 Data Extraction and Modeling Suppose that the dataseries of 119899 groups (119909 119906 V) can be obtained in a time period

(1199091 1199061 V1 1199051) (1199092 1199062 V2 1199052) (119909119899 119906119899 V119899 119905119899) (12)

where 119905119894 is sampling time In this paper the method of clus-tering is used to distinguish the normal and abnormal data inthe original data sequenceThismethod differentiates normaldata and abnormal data based on the similarity of databetween data (based on the distance between data points)and the effect of isolation or noise points on classification iserased using this method Therefore after the sampling databeing preprocessed the adjacent normal data and abnormaldata of the state variable are obtained Some sample data isshown in Figure 4

In each interval the data was collected continuouslyand composed of abnormal data and normal data Thecatastrophe model was established with the data of oneinterval and then the parameters of the model were revisedwith the data of other intervals to make the model moreperfect

32 Modeling and Analysis Now we use the above data to setup catastrophe model First to satisfy the above equation of

Mathematical Problems in Engineering 7

10

100

200

300

400

500

600

700

800

15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323

DurationProduction LoadThroughtput

Figure 4 Part experimental data (119909 = 119884 = 134)

catastrophe flow and bifurcation set the following equationis satisfied

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (13)

And then according to above formula and 120597119869(119886 119887)120597119886 =0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(14)

By analyzing the above model the computational com-plexity is 119874(119899) Using above data and references [26] 10solutions were obtained by using (119886 119887) = (119886119894 119887119894) | 119869(119886119894 119887119894) lt119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 And then the optimalvalues 119886 = minus87 and 119887 = 7821 can be obtained Next the otherintervals data can be used to modify the parameters of themodel finally we obtain 119886 = minus95 and 119887 = 9721

Now let us research on some of the dynamic behaviors ofcatastrophe in the manufacturing system when 119886 = minus95 and119887 = 9721

Equilibrium Surface (119909 119906 V) | 1199093 minus 95119906119909 + 8721V = 0

Bifurcation Set (119906 V) | 119863 = minus1199063 + 600V2 119863 = 0The graph of the equilibrium surface 119872 and the bifurca-

tion set 119861 is represented in Figure 5 Let the system state be

represented by (119909 119906 V) in the three-dimensional space thenthe phase point must be located on the surface and alwayson the upper lobe or lower lobe of the surface In Figure 5the upper lobe indicates the normal state of manufacturingsystem the lower lobe indicates the fault state and the foldingpart indicates an unreachable state of the manufacturingsystem

The projection of the equilibrium hypersurfaces 119872 in thecontrol plane 119862 namely u-v is a topological transformationor mapping which can be represented by 119891

119891 119872 (1198773) 997888rarr 119862 (1198772)

namely (119909 119906 V) 997888rarr (119906 V) (15)

Now we get the monitor Figure 6As shown in Figure 6 the first figure in Figure 6 cor-

responds to the path (1) in Figure 5 When the controlparameters (119906 V) are not controlled at 950 then (119906 V) willfall on the curve of 119863 = 0 at the next moment andthe manufacturing system will suddenly break down Atthis time the manufacturing system is still able to operateand though certain adjustments can be restored to normaloperation if the (119906 V) is still not controlled (119906 V) will enter119863 gt 0 and the system cannot work and manual repairis required The second figure in Figure 6 is the result ofcontrolling (119906 V) before system fault corresponding to path(2) in Figure 5 From path (2) in Figure 5 we can know thatpreventing (119906 V) from 119863 = 0 at point 1 we can change theevolutionary path of the system and prevent the system frombreaking down suddenly The third figure in Figure 6 is theresult of controlling (119906 V) after system fault corresponding topath (3) in Figure 5 From path (3) in Figure 5 we can knowthat changing the values of 119906 and V satisfying (119906 V) isin (119906 V) |119863(119906 V) = 0 at point 2 we can bring the manufacturingsystem back to normal suddenly

8 Mathematical Problems in Engineering

X

(1)

(2)

(3)

Normal

N

Control

Control

Fault

M

C1

2

u

V

UC

D=0

Dlt0Dgt0

D=0

evolution path (1)evolution path (2)evolution path (3)

Figure 5 Manufacturing system fault evolution process

t

t

t

Control Control

102010101000950

102010101000

102010101000

950

940930920910900850840830820810800

Figure 6 Logical events

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

Mathematical Problems in Engineering 3

operation of manufacturing system is considered from bothinternal and external aspects There are many internal andexternal factors that affect the operation of manufacturingsystems for example machine adjustment machine agingimproper product design irrational production technologyand process design improper processing methods insuf-ficient machining accuracy machine maintenance errorsand operational management errors That is very hard toanalyze the impact of these factors on the manufacturingsystem at the same timeTherefore we introduce catastrophemodels and macroscopic order parameters to reflect the realoperation situation of the manufacturing system In thispaper external macroscopic order parameters are defined interms of production throughput and the internal macro-scopic order parameters are defined in terms of productionload and duration Based on the catastrophe theory weplan to describe the behavior catastrophe of manufacturingsystem by cusp catastrophe theory The cusp mutation modelis the simplest mutation model When the cusp mutationmodel is used for fault description the critical surface iseasy to construct and there is a strong geometrical intu-ition

The external macroscopic order parameter and the inter-nal macroscopic order parameters are regarded as the statevariables 119883 and control variables 119880 and 119881 of the manufac-turing system respectively And a cusp catastrophe model isestablished to describe the abnormal behavior of the systemby using the data of each the adjacent continues data flowinterval including normal data and abnormal data Accordingto the opinion proposed by Hall [29] the basic model of cuspcatastrophe is

119881 (119909) = 11988811199094 + 11988821199061199092 + 1198883V119909 (1)

where 119909 is the state variable 119906 and V are the controlvariables In this paper 119906 is duration V is production loadand 119909 is throughput 1198881 1198882 1198883 are coefficients Thereforethe phase space is a three-dimensional space composedof state variables 119909 119906 and V The critical point of thepotential function is the solution of equation nabla119909119881(119909) =0 Therefore the equilibrium surface 119872 is also given bynabla119909119881(119909) = 0

The corresponding nonisolated singularities set 119878 is givenby

119878 = (119909 119906 V) | nabla2119909119881 (119909) = 0 nabla119909119881 (119909) = 0 (2)

Projecting 119878 onto the parametric plane u-v we can obtainthe bifurcation set equation

8119888321199063 + 27119888111988823V2 = 0 (3)

After coordinate transformation the mutant flow andbifurcation set equations are respectively as follows

Catastrophe Flow

1199093 + 119886119906119909 + 119887V = 0 (4)

Bifurcation Equation

411988631199063 + 271198872V2 = 0 (5)

where 119886 = 119888221198881 119887 = 119888341198881 We can obtain

Equilibrium Surface 119872 = (119909 119906 V) | 1199093 + 119886119906119909 + 119887V = 0

Bifurcation Set 119861 = (119906 V) | 119863 = 411988631199063 + 271198872V2 119863 = 0Where 119886 and 119887 are the coefficients For parameters 119886 and

119887 the optimal value can be obtained by solving the extremamethod with the multivariate function supposing we collect119899 sets of continuous adjoining data including normal andabnormal data (119909119894 119906119894 V119894) put them into 119872 and 119861 and getcumulative error

119890119903119900119903119903 =119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894) + (411988631199063119894 + 271198872V2119894 ) (6)

Our goal is to find the value of 119886 119887 for each group(119909119894 119906119894 V119894) we can obtain

1199093 + 119886119906119909 + 119887V = 0

411988631199063 + 271198872V2 = 0(7)

So we use the sum of squares to find the optimal 119886 119887

119890119903119900119903119903

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (8)

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (9)

According to above formula and 120597119869(119886 119887)120597119886 = 0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(10)

By solving (10) we can get all of the extremum sets

119864 = (119886119895 119887119895) | (119886119895 119887119895)

is the extreme points of 119869 (119886 119887) 119895 = 1 2 119905 (11)

where 119905 is the number of extreme points Because thegeometric surface of 119869(119886 119887) is pointing upwe canfind (119886 119887) =

4 Mathematical Problems in Engineering

M

x

v

u

v

uD=0

D=0

Dlt0

Dgt0

B

S

S

B

Upper lobe

Lower lobe

unreachable

Mode 1

Mode 2

Figure 1 Cusp catastrophe

(119886119894 119887119894) | 119869(119886119894 119887119894) lt 119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 thatis (119886 119887) being minimal points of 119869(119886 119887)

From Figure 1 it is obvious that on the control parameterplane u-v if (119906 V) isin (119906 V) | 119863(119906 V) lt 0 for each (119906 V)119909 has 3 values on the surface that is the manufacturingsystem has three kinds of mode that one of three modes istheoretically reachable but practically impossible which isunreachable When (119906 V) isin (119906 V) | 119863(119906 V) = 0 thetwo kinds of mode merge into a single one In this case themanufacturing system will change suddenly from one modeto the others When (119906 V) isin (119906 V) | 119863(119906 V) gt 0 119909has only one value which indicates that the manufacturingsystem is single mode So 119863 = 0 is the separatrix between thesinglemode and themultimode of themanufacturing systemwhere the modal catastrophe of the manufacturing systemoccurs And we found that the control parameters (119906 V) canonly go from 119863 lt 0 region to the edge 119863 = 0 as catastropheoccurs In actual operation the control parameters (119906 V)cannot be reached from the region of119863 gt 0 to the edge of119863 =0 So if the control parameters (119906 V) of the manufacturingsystem are located in (119906 V) | 119863(119906 V) lt 0 it shows thatthe manufacturing system is in a risky state In this case thecontrol parameter can easily be changed to the edge 119863 = 0and catastrophe occurs In contrast if the control parametersare located in (119906 V) | 119863(119906 V) gt 0 themanufacturing systemwill stay in a stable state So if the upper lobe represents thenormal state of manufacturing system lower lobe representsfault state and the above analysis is followed when (119906 V) isin

(119906 V) | 119863(119906 V) lt 0 the manufacturing system has twokinds of modes one of them is in a normal state while theother one is faulty With the change of control parameters(119906 V) manufacturing systems keep stable in one state until(119906 V) isin (119906 V) | 119863(119906 V) = 0 and manufacturingsystemwill suddenly jump from upper lobe to lower lobe andbreaks down or contrary Therefore in order to make themanufacturing system run healthily we should control (119906 V)and make (119909 119906 V) isin (119906 V) | 119863(119906 V) gt 0 119909 ge 119884 then thepurpose of preventive control is achieved

22 Management Research Through the analysis of themutation model shown in Figure 1 we can obtain the bound-ary between normal and abnormal in the manufacturingsystem In order to apply above research into managementwe define the following related events

(1) Event 119860 in the operation data of manufacturingsystem if the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the logical value of event A is 1 otherwise it is 0

(2) Event 119861 if the control variables (119906 V) isin (119906 V) |119863(119906 V) lt 0 119909 ge 119884 the logical value of event A is 1otherwise it is 0

(3) Event 119862 if the control variables (119906 V) isin (119906 V) |119863(119906 V) gt 0 119909 lt 119884 the logical value of event A is 1otherwise it is 0

(4) Event D in the operation data of manufacturingsystems if the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 the logical value of event A is 1 otherwise it is 0

Mathematical Problems in Engineering 5

Table 2 The related events

Event Label A B C D Implication SuggestGreen 1 0 0 0 Operating-normally - -Yellow 0 1 0 0 Warning Monitor (119906 V) to far from (119906 V) | 119863(119906 V) = 0Red 0 0 1 0 Break down RepairBlue 0 0 0 1 Break down Adjust (119906 V) to (119906 V) | 119863(119906 V) = 0

1

3

4

2

x

xo1

xo3

xo4

o

xo2

xo5

1 2 4

5

06

Internal mechanism The external display Real time data Management

A

B

C

D

Real time

asymptotic stabilityunreachable

First stepMeasure actual operation of

Second stepDetermine if the manufacturing

system will suddenly fail

Third step

Adopt management measures

evolutionary pathmappingcontrol management

manufacturing system

productiondata

Figure 2 Control process

Then according to the above definition we get thefollowing as shown in Table 2

For manufacturing enterprises the best thing is to reducethe occurrence of production failures Its frequent occurrencewill not only affect the morale of the companyrsquos employeesbut also lead to the stagnation of production which willseriously affect the companyrsquos normal operation and profitsTherefore based on the above research results we can carryout preventive control The control processes can be dividedinto three parts (Figure 2)(1) measure the actual operation ofthe manufacturing system according to the logic relationshipestablished in the table (2) judge whether the manufactur-ing system will suddenly fall into failure according to thecatastrophe model (3) take management actions to controlparameters and prevent the system from falling

In order to visually show the core ideas of this paper Letus take 119906 = constant the change of V in Figure 2 is analyzed

First map Figure 1 in the 119909-V plane and get the internalmechanism part shown in Figure 2The curve 1-3 and 3-4represent the upper branch of counter-S curve in Figure 1and indicates the normal state of the manufacturing system

where point 3 is a critical point that the system goes fromsingle mode to multimode Point 4 is in the bifurcation setand represents the critical point where the manufacturingsystem jumps from the normal state to the fault state Thecurve 4-2 represents the central part of counter-S curvein Figure 1 and represents the state that the manufacturingsystem in the actual operation cannot reach where point 2is in the bifurcation set and represents the critical point atwhich the manufacturing system jumps from the fault stateto normal state The curves 2-5 and 5-6 represent the lowerbranch of counter-S curve in Figure 1 and it indicates the faultstate of the manufacturing system The curve 2-5 indicatesthat although the manufacturing system is faulty it can berestored to the normal state by changing the values of 119906 andV but curve 5-6 indicates the manufacturing system cannotrecover itself and can only be repaired manually AnalyzingFigure 2 when V = V1 the manufacturing system statevariable is located in xo1 of upper branch and when v isincreased from v1 to v4 the state variable x changes to xo4 Atthis time if we added an infinitesimal perturbation to v4 thestate of variable x will jump from xo4 to xo5When v reduces to

6 Mathematical Problems in Engineering

Data extraction

Adjacent normal datawith abnormal data Abnormal data

Establishment of mutation model

Real-time of manufacturing system

visualization

Management

Model analysis

Figure 3 Control process

v2 the state of variable x changes to xo2 At this time as long asv2 gets a little bit smaller the state of variable xwill jump fromxo2 to xo3 in point 2 and the state of variable x enters upperbranch of counter-S curve Therefore according to the aboveanalysis we can control the failure of manufacturing systemThe specific process is shown in Figure 2 firstly the operationdata of the manufacturing system (119909 119906 V) are collected andanalyzed and then according to Table 2 the data will bedisplayed in real-time by bars in different colors By thisway we can find the operational problem in manufacturingsystems and then it is controlled according to the internalmechanism of fault in the manufacturing system to keep thesystem running in a healthy state

3 A Simplified Case Study

A simplified proof-of-concept case is illustrated to show theprocess of the proposed method Yonggu is a company thatproduces metal tools and has a complete IoT (Internet ofthings) system in its workshop We use the IoT system tocollect real-time data of the productworkshop and then selectthe adjacent data including normal data and abnormal datato establish the catastrophe model

In this section the throughput 119883 and the productionload 119881 of the manufacturing system within the duration time119880 are selected as the monitoring parameters In addition itshould be noted that according to the parameter estimationrequirements of the catastrophe model the data must be

extracted in an interval that is the data in an interval mustbe continuous in time

The specific way is shown in Figure 3

31 Data Extraction and Modeling Suppose that the dataseries of 119899 groups (119909 119906 V) can be obtained in a time period

(1199091 1199061 V1 1199051) (1199092 1199062 V2 1199052) (119909119899 119906119899 V119899 119905119899) (12)

where 119905119894 is sampling time In this paper the method of clus-tering is used to distinguish the normal and abnormal data inthe original data sequenceThismethod differentiates normaldata and abnormal data based on the similarity of databetween data (based on the distance between data points)and the effect of isolation or noise points on classification iserased using this method Therefore after the sampling databeing preprocessed the adjacent normal data and abnormaldata of the state variable are obtained Some sample data isshown in Figure 4

In each interval the data was collected continuouslyand composed of abnormal data and normal data Thecatastrophe model was established with the data of oneinterval and then the parameters of the model were revisedwith the data of other intervals to make the model moreperfect

32 Modeling and Analysis Now we use the above data to setup catastrophe model First to satisfy the above equation of

Mathematical Problems in Engineering 7

10

100

200

300

400

500

600

700

800

15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323

DurationProduction LoadThroughtput

Figure 4 Part experimental data (119909 = 119884 = 134)

catastrophe flow and bifurcation set the following equationis satisfied

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (13)

And then according to above formula and 120597119869(119886 119887)120597119886 =0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(14)

By analyzing the above model the computational com-plexity is 119874(119899) Using above data and references [26] 10solutions were obtained by using (119886 119887) = (119886119894 119887119894) | 119869(119886119894 119887119894) lt119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 And then the optimalvalues 119886 = minus87 and 119887 = 7821 can be obtained Next the otherintervals data can be used to modify the parameters of themodel finally we obtain 119886 = minus95 and 119887 = 9721

Now let us research on some of the dynamic behaviors ofcatastrophe in the manufacturing system when 119886 = minus95 and119887 = 9721

Equilibrium Surface (119909 119906 V) | 1199093 minus 95119906119909 + 8721V = 0

Bifurcation Set (119906 V) | 119863 = minus1199063 + 600V2 119863 = 0The graph of the equilibrium surface 119872 and the bifurca-

tion set 119861 is represented in Figure 5 Let the system state be

represented by (119909 119906 V) in the three-dimensional space thenthe phase point must be located on the surface and alwayson the upper lobe or lower lobe of the surface In Figure 5the upper lobe indicates the normal state of manufacturingsystem the lower lobe indicates the fault state and the foldingpart indicates an unreachable state of the manufacturingsystem

The projection of the equilibrium hypersurfaces 119872 in thecontrol plane 119862 namely u-v is a topological transformationor mapping which can be represented by 119891

119891 119872 (1198773) 997888rarr 119862 (1198772)

namely (119909 119906 V) 997888rarr (119906 V) (15)

Now we get the monitor Figure 6As shown in Figure 6 the first figure in Figure 6 cor-

responds to the path (1) in Figure 5 When the controlparameters (119906 V) are not controlled at 950 then (119906 V) willfall on the curve of 119863 = 0 at the next moment andthe manufacturing system will suddenly break down Atthis time the manufacturing system is still able to operateand though certain adjustments can be restored to normaloperation if the (119906 V) is still not controlled (119906 V) will enter119863 gt 0 and the system cannot work and manual repairis required The second figure in Figure 6 is the result ofcontrolling (119906 V) before system fault corresponding to path(2) in Figure 5 From path (2) in Figure 5 we can know thatpreventing (119906 V) from 119863 = 0 at point 1 we can change theevolutionary path of the system and prevent the system frombreaking down suddenly The third figure in Figure 6 is theresult of controlling (119906 V) after system fault corresponding topath (3) in Figure 5 From path (3) in Figure 5 we can knowthat changing the values of 119906 and V satisfying (119906 V) isin (119906 V) |119863(119906 V) = 0 at point 2 we can bring the manufacturingsystem back to normal suddenly

8 Mathematical Problems in Engineering

X

(1)

(2)

(3)

Normal

N

Control

Control

Fault

M

C1

2

u

V

UC

D=0

Dlt0Dgt0

D=0

evolution path (1)evolution path (2)evolution path (3)

Figure 5 Manufacturing system fault evolution process

t

t

t

Control Control

102010101000950

102010101000

102010101000

950

940930920910900850840830820810800

Figure 6 Logical events

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

4 Mathematical Problems in Engineering

M

x

v

u

v

uD=0

D=0

Dlt0

Dgt0

B

S

S

B

Upper lobe

Lower lobe

unreachable

Mode 1

Mode 2

Figure 1 Cusp catastrophe

(119886119894 119887119894) | 119869(119886119894 119887119894) lt 119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 thatis (119886 119887) being minimal points of 119869(119886 119887)

From Figure 1 it is obvious that on the control parameterplane u-v if (119906 V) isin (119906 V) | 119863(119906 V) lt 0 for each (119906 V)119909 has 3 values on the surface that is the manufacturingsystem has three kinds of mode that one of three modes istheoretically reachable but practically impossible which isunreachable When (119906 V) isin (119906 V) | 119863(119906 V) = 0 thetwo kinds of mode merge into a single one In this case themanufacturing system will change suddenly from one modeto the others When (119906 V) isin (119906 V) | 119863(119906 V) gt 0 119909has only one value which indicates that the manufacturingsystem is single mode So 119863 = 0 is the separatrix between thesinglemode and themultimode of themanufacturing systemwhere the modal catastrophe of the manufacturing systemoccurs And we found that the control parameters (119906 V) canonly go from 119863 lt 0 region to the edge 119863 = 0 as catastropheoccurs In actual operation the control parameters (119906 V)cannot be reached from the region of119863 gt 0 to the edge of119863 =0 So if the control parameters (119906 V) of the manufacturingsystem are located in (119906 V) | 119863(119906 V) lt 0 it shows thatthe manufacturing system is in a risky state In this case thecontrol parameter can easily be changed to the edge 119863 = 0and catastrophe occurs In contrast if the control parametersare located in (119906 V) | 119863(119906 V) gt 0 themanufacturing systemwill stay in a stable state So if the upper lobe represents thenormal state of manufacturing system lower lobe representsfault state and the above analysis is followed when (119906 V) isin

(119906 V) | 119863(119906 V) lt 0 the manufacturing system has twokinds of modes one of them is in a normal state while theother one is faulty With the change of control parameters(119906 V) manufacturing systems keep stable in one state until(119906 V) isin (119906 V) | 119863(119906 V) = 0 and manufacturingsystemwill suddenly jump from upper lobe to lower lobe andbreaks down or contrary Therefore in order to make themanufacturing system run healthily we should control (119906 V)and make (119909 119906 V) isin (119906 V) | 119863(119906 V) gt 0 119909 ge 119884 then thepurpose of preventive control is achieved

22 Management Research Through the analysis of themutation model shown in Figure 1 we can obtain the bound-ary between normal and abnormal in the manufacturingsystem In order to apply above research into managementwe define the following related events

(1) Event 119860 in the operation data of manufacturingsystem if the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the logical value of event A is 1 otherwise it is 0

(2) Event 119861 if the control variables (119906 V) isin (119906 V) |119863(119906 V) lt 0 119909 ge 119884 the logical value of event A is 1otherwise it is 0

(3) Event 119862 if the control variables (119906 V) isin (119906 V) |119863(119906 V) gt 0 119909 lt 119884 the logical value of event A is 1otherwise it is 0

(4) Event D in the operation data of manufacturingsystems if the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 the logical value of event A is 1 otherwise it is 0

Mathematical Problems in Engineering 5

Table 2 The related events

Event Label A B C D Implication SuggestGreen 1 0 0 0 Operating-normally - -Yellow 0 1 0 0 Warning Monitor (119906 V) to far from (119906 V) | 119863(119906 V) = 0Red 0 0 1 0 Break down RepairBlue 0 0 0 1 Break down Adjust (119906 V) to (119906 V) | 119863(119906 V) = 0

1

3

4

2

x

xo1

xo3

xo4

o

xo2

xo5

1 2 4

5

06

Internal mechanism The external display Real time data Management

A

B

C

D

Real time

asymptotic stabilityunreachable

First stepMeasure actual operation of

Second stepDetermine if the manufacturing

system will suddenly fail

Third step

Adopt management measures

evolutionary pathmappingcontrol management

manufacturing system

productiondata

Figure 2 Control process

Then according to the above definition we get thefollowing as shown in Table 2

For manufacturing enterprises the best thing is to reducethe occurrence of production failures Its frequent occurrencewill not only affect the morale of the companyrsquos employeesbut also lead to the stagnation of production which willseriously affect the companyrsquos normal operation and profitsTherefore based on the above research results we can carryout preventive control The control processes can be dividedinto three parts (Figure 2)(1) measure the actual operation ofthe manufacturing system according to the logic relationshipestablished in the table (2) judge whether the manufactur-ing system will suddenly fall into failure according to thecatastrophe model (3) take management actions to controlparameters and prevent the system from falling

In order to visually show the core ideas of this paper Letus take 119906 = constant the change of V in Figure 2 is analyzed

First map Figure 1 in the 119909-V plane and get the internalmechanism part shown in Figure 2The curve 1-3 and 3-4represent the upper branch of counter-S curve in Figure 1and indicates the normal state of the manufacturing system

where point 3 is a critical point that the system goes fromsingle mode to multimode Point 4 is in the bifurcation setand represents the critical point where the manufacturingsystem jumps from the normal state to the fault state Thecurve 4-2 represents the central part of counter-S curvein Figure 1 and represents the state that the manufacturingsystem in the actual operation cannot reach where point 2is in the bifurcation set and represents the critical point atwhich the manufacturing system jumps from the fault stateto normal state The curves 2-5 and 5-6 represent the lowerbranch of counter-S curve in Figure 1 and it indicates the faultstate of the manufacturing system The curve 2-5 indicatesthat although the manufacturing system is faulty it can berestored to the normal state by changing the values of 119906 andV but curve 5-6 indicates the manufacturing system cannotrecover itself and can only be repaired manually AnalyzingFigure 2 when V = V1 the manufacturing system statevariable is located in xo1 of upper branch and when v isincreased from v1 to v4 the state variable x changes to xo4 Atthis time if we added an infinitesimal perturbation to v4 thestate of variable x will jump from xo4 to xo5When v reduces to

6 Mathematical Problems in Engineering

Data extraction

Adjacent normal datawith abnormal data Abnormal data

Establishment of mutation model

Real-time of manufacturing system

visualization

Management

Model analysis

Figure 3 Control process

v2 the state of variable x changes to xo2 At this time as long asv2 gets a little bit smaller the state of variable xwill jump fromxo2 to xo3 in point 2 and the state of variable x enters upperbranch of counter-S curve Therefore according to the aboveanalysis we can control the failure of manufacturing systemThe specific process is shown in Figure 2 firstly the operationdata of the manufacturing system (119909 119906 V) are collected andanalyzed and then according to Table 2 the data will bedisplayed in real-time by bars in different colors By thisway we can find the operational problem in manufacturingsystems and then it is controlled according to the internalmechanism of fault in the manufacturing system to keep thesystem running in a healthy state

3 A Simplified Case Study

A simplified proof-of-concept case is illustrated to show theprocess of the proposed method Yonggu is a company thatproduces metal tools and has a complete IoT (Internet ofthings) system in its workshop We use the IoT system tocollect real-time data of the productworkshop and then selectthe adjacent data including normal data and abnormal datato establish the catastrophe model

In this section the throughput 119883 and the productionload 119881 of the manufacturing system within the duration time119880 are selected as the monitoring parameters In addition itshould be noted that according to the parameter estimationrequirements of the catastrophe model the data must be

extracted in an interval that is the data in an interval mustbe continuous in time

The specific way is shown in Figure 3

31 Data Extraction and Modeling Suppose that the dataseries of 119899 groups (119909 119906 V) can be obtained in a time period

(1199091 1199061 V1 1199051) (1199092 1199062 V2 1199052) (119909119899 119906119899 V119899 119905119899) (12)

where 119905119894 is sampling time In this paper the method of clus-tering is used to distinguish the normal and abnormal data inthe original data sequenceThismethod differentiates normaldata and abnormal data based on the similarity of databetween data (based on the distance between data points)and the effect of isolation or noise points on classification iserased using this method Therefore after the sampling databeing preprocessed the adjacent normal data and abnormaldata of the state variable are obtained Some sample data isshown in Figure 4

In each interval the data was collected continuouslyand composed of abnormal data and normal data Thecatastrophe model was established with the data of oneinterval and then the parameters of the model were revisedwith the data of other intervals to make the model moreperfect

32 Modeling and Analysis Now we use the above data to setup catastrophe model First to satisfy the above equation of

Mathematical Problems in Engineering 7

10

100

200

300

400

500

600

700

800

15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323

DurationProduction LoadThroughtput

Figure 4 Part experimental data (119909 = 119884 = 134)

catastrophe flow and bifurcation set the following equationis satisfied

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (13)

And then according to above formula and 120597119869(119886 119887)120597119886 =0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(14)

By analyzing the above model the computational com-plexity is 119874(119899) Using above data and references [26] 10solutions were obtained by using (119886 119887) = (119886119894 119887119894) | 119869(119886119894 119887119894) lt119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 And then the optimalvalues 119886 = minus87 and 119887 = 7821 can be obtained Next the otherintervals data can be used to modify the parameters of themodel finally we obtain 119886 = minus95 and 119887 = 9721

Now let us research on some of the dynamic behaviors ofcatastrophe in the manufacturing system when 119886 = minus95 and119887 = 9721

Equilibrium Surface (119909 119906 V) | 1199093 minus 95119906119909 + 8721V = 0

Bifurcation Set (119906 V) | 119863 = minus1199063 + 600V2 119863 = 0The graph of the equilibrium surface 119872 and the bifurca-

tion set 119861 is represented in Figure 5 Let the system state be

represented by (119909 119906 V) in the three-dimensional space thenthe phase point must be located on the surface and alwayson the upper lobe or lower lobe of the surface In Figure 5the upper lobe indicates the normal state of manufacturingsystem the lower lobe indicates the fault state and the foldingpart indicates an unreachable state of the manufacturingsystem

The projection of the equilibrium hypersurfaces 119872 in thecontrol plane 119862 namely u-v is a topological transformationor mapping which can be represented by 119891

119891 119872 (1198773) 997888rarr 119862 (1198772)

namely (119909 119906 V) 997888rarr (119906 V) (15)

Now we get the monitor Figure 6As shown in Figure 6 the first figure in Figure 6 cor-

responds to the path (1) in Figure 5 When the controlparameters (119906 V) are not controlled at 950 then (119906 V) willfall on the curve of 119863 = 0 at the next moment andthe manufacturing system will suddenly break down Atthis time the manufacturing system is still able to operateand though certain adjustments can be restored to normaloperation if the (119906 V) is still not controlled (119906 V) will enter119863 gt 0 and the system cannot work and manual repairis required The second figure in Figure 6 is the result ofcontrolling (119906 V) before system fault corresponding to path(2) in Figure 5 From path (2) in Figure 5 we can know thatpreventing (119906 V) from 119863 = 0 at point 1 we can change theevolutionary path of the system and prevent the system frombreaking down suddenly The third figure in Figure 6 is theresult of controlling (119906 V) after system fault corresponding topath (3) in Figure 5 From path (3) in Figure 5 we can knowthat changing the values of 119906 and V satisfying (119906 V) isin (119906 V) |119863(119906 V) = 0 at point 2 we can bring the manufacturingsystem back to normal suddenly

8 Mathematical Problems in Engineering

X

(1)

(2)

(3)

Normal

N

Control

Control

Fault

M

C1

2

u

V

UC

D=0

Dlt0Dgt0

D=0

evolution path (1)evolution path (2)evolution path (3)

Figure 5 Manufacturing system fault evolution process

t

t

t

Control Control

102010101000950

102010101000

102010101000

950

940930920910900850840830820810800

Figure 6 Logical events

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

Mathematical Problems in Engineering 5

Table 2 The related events

Event Label A B C D Implication SuggestGreen 1 0 0 0 Operating-normally - -Yellow 0 1 0 0 Warning Monitor (119906 V) to far from (119906 V) | 119863(119906 V) = 0Red 0 0 1 0 Break down RepairBlue 0 0 0 1 Break down Adjust (119906 V) to (119906 V) | 119863(119906 V) = 0

1

3

4

2

x

xo1

xo3

xo4

o

xo2

xo5

1 2 4

5

06

Internal mechanism The external display Real time data Management

A

B

C

D

Real time

asymptotic stabilityunreachable

First stepMeasure actual operation of

Second stepDetermine if the manufacturing

system will suddenly fail

Third step

Adopt management measures

evolutionary pathmappingcontrol management

manufacturing system

productiondata

Figure 2 Control process

Then according to the above definition we get thefollowing as shown in Table 2

For manufacturing enterprises the best thing is to reducethe occurrence of production failures Its frequent occurrencewill not only affect the morale of the companyrsquos employeesbut also lead to the stagnation of production which willseriously affect the companyrsquos normal operation and profitsTherefore based on the above research results we can carryout preventive control The control processes can be dividedinto three parts (Figure 2)(1) measure the actual operation ofthe manufacturing system according to the logic relationshipestablished in the table (2) judge whether the manufactur-ing system will suddenly fall into failure according to thecatastrophe model (3) take management actions to controlparameters and prevent the system from falling

In order to visually show the core ideas of this paper Letus take 119906 = constant the change of V in Figure 2 is analyzed

First map Figure 1 in the 119909-V plane and get the internalmechanism part shown in Figure 2The curve 1-3 and 3-4represent the upper branch of counter-S curve in Figure 1and indicates the normal state of the manufacturing system

where point 3 is a critical point that the system goes fromsingle mode to multimode Point 4 is in the bifurcation setand represents the critical point where the manufacturingsystem jumps from the normal state to the fault state Thecurve 4-2 represents the central part of counter-S curvein Figure 1 and represents the state that the manufacturingsystem in the actual operation cannot reach where point 2is in the bifurcation set and represents the critical point atwhich the manufacturing system jumps from the fault stateto normal state The curves 2-5 and 5-6 represent the lowerbranch of counter-S curve in Figure 1 and it indicates the faultstate of the manufacturing system The curve 2-5 indicatesthat although the manufacturing system is faulty it can berestored to the normal state by changing the values of 119906 andV but curve 5-6 indicates the manufacturing system cannotrecover itself and can only be repaired manually AnalyzingFigure 2 when V = V1 the manufacturing system statevariable is located in xo1 of upper branch and when v isincreased from v1 to v4 the state variable x changes to xo4 Atthis time if we added an infinitesimal perturbation to v4 thestate of variable x will jump from xo4 to xo5When v reduces to

6 Mathematical Problems in Engineering

Data extraction

Adjacent normal datawith abnormal data Abnormal data

Establishment of mutation model

Real-time of manufacturing system

visualization

Management

Model analysis

Figure 3 Control process

v2 the state of variable x changes to xo2 At this time as long asv2 gets a little bit smaller the state of variable xwill jump fromxo2 to xo3 in point 2 and the state of variable x enters upperbranch of counter-S curve Therefore according to the aboveanalysis we can control the failure of manufacturing systemThe specific process is shown in Figure 2 firstly the operationdata of the manufacturing system (119909 119906 V) are collected andanalyzed and then according to Table 2 the data will bedisplayed in real-time by bars in different colors By thisway we can find the operational problem in manufacturingsystems and then it is controlled according to the internalmechanism of fault in the manufacturing system to keep thesystem running in a healthy state

3 A Simplified Case Study

A simplified proof-of-concept case is illustrated to show theprocess of the proposed method Yonggu is a company thatproduces metal tools and has a complete IoT (Internet ofthings) system in its workshop We use the IoT system tocollect real-time data of the productworkshop and then selectthe adjacent data including normal data and abnormal datato establish the catastrophe model

In this section the throughput 119883 and the productionload 119881 of the manufacturing system within the duration time119880 are selected as the monitoring parameters In addition itshould be noted that according to the parameter estimationrequirements of the catastrophe model the data must be

extracted in an interval that is the data in an interval mustbe continuous in time

The specific way is shown in Figure 3

31 Data Extraction and Modeling Suppose that the dataseries of 119899 groups (119909 119906 V) can be obtained in a time period

(1199091 1199061 V1 1199051) (1199092 1199062 V2 1199052) (119909119899 119906119899 V119899 119905119899) (12)

where 119905119894 is sampling time In this paper the method of clus-tering is used to distinguish the normal and abnormal data inthe original data sequenceThismethod differentiates normaldata and abnormal data based on the similarity of databetween data (based on the distance between data points)and the effect of isolation or noise points on classification iserased using this method Therefore after the sampling databeing preprocessed the adjacent normal data and abnormaldata of the state variable are obtained Some sample data isshown in Figure 4

In each interval the data was collected continuouslyand composed of abnormal data and normal data Thecatastrophe model was established with the data of oneinterval and then the parameters of the model were revisedwith the data of other intervals to make the model moreperfect

32 Modeling and Analysis Now we use the above data to setup catastrophe model First to satisfy the above equation of

Mathematical Problems in Engineering 7

10

100

200

300

400

500

600

700

800

15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323

DurationProduction LoadThroughtput

Figure 4 Part experimental data (119909 = 119884 = 134)

catastrophe flow and bifurcation set the following equationis satisfied

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (13)

And then according to above formula and 120597119869(119886 119887)120597119886 =0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(14)

By analyzing the above model the computational com-plexity is 119874(119899) Using above data and references [26] 10solutions were obtained by using (119886 119887) = (119886119894 119887119894) | 119869(119886119894 119887119894) lt119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 And then the optimalvalues 119886 = minus87 and 119887 = 7821 can be obtained Next the otherintervals data can be used to modify the parameters of themodel finally we obtain 119886 = minus95 and 119887 = 9721

Now let us research on some of the dynamic behaviors ofcatastrophe in the manufacturing system when 119886 = minus95 and119887 = 9721

Equilibrium Surface (119909 119906 V) | 1199093 minus 95119906119909 + 8721V = 0

Bifurcation Set (119906 V) | 119863 = minus1199063 + 600V2 119863 = 0The graph of the equilibrium surface 119872 and the bifurca-

tion set 119861 is represented in Figure 5 Let the system state be

represented by (119909 119906 V) in the three-dimensional space thenthe phase point must be located on the surface and alwayson the upper lobe or lower lobe of the surface In Figure 5the upper lobe indicates the normal state of manufacturingsystem the lower lobe indicates the fault state and the foldingpart indicates an unreachable state of the manufacturingsystem

The projection of the equilibrium hypersurfaces 119872 in thecontrol plane 119862 namely u-v is a topological transformationor mapping which can be represented by 119891

119891 119872 (1198773) 997888rarr 119862 (1198772)

namely (119909 119906 V) 997888rarr (119906 V) (15)

Now we get the monitor Figure 6As shown in Figure 6 the first figure in Figure 6 cor-

responds to the path (1) in Figure 5 When the controlparameters (119906 V) are not controlled at 950 then (119906 V) willfall on the curve of 119863 = 0 at the next moment andthe manufacturing system will suddenly break down Atthis time the manufacturing system is still able to operateand though certain adjustments can be restored to normaloperation if the (119906 V) is still not controlled (119906 V) will enter119863 gt 0 and the system cannot work and manual repairis required The second figure in Figure 6 is the result ofcontrolling (119906 V) before system fault corresponding to path(2) in Figure 5 From path (2) in Figure 5 we can know thatpreventing (119906 V) from 119863 = 0 at point 1 we can change theevolutionary path of the system and prevent the system frombreaking down suddenly The third figure in Figure 6 is theresult of controlling (119906 V) after system fault corresponding topath (3) in Figure 5 From path (3) in Figure 5 we can knowthat changing the values of 119906 and V satisfying (119906 V) isin (119906 V) |119863(119906 V) = 0 at point 2 we can bring the manufacturingsystem back to normal suddenly

8 Mathematical Problems in Engineering

X

(1)

(2)

(3)

Normal

N

Control

Control

Fault

M

C1

2

u

V

UC

D=0

Dlt0Dgt0

D=0

evolution path (1)evolution path (2)evolution path (3)

Figure 5 Manufacturing system fault evolution process

t

t

t

Control Control

102010101000950

102010101000

102010101000

950

940930920910900850840830820810800

Figure 6 Logical events

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

6 Mathematical Problems in Engineering

Data extraction

Adjacent normal datawith abnormal data Abnormal data

Establishment of mutation model

Real-time of manufacturing system

visualization

Management

Model analysis

Figure 3 Control process

v2 the state of variable x changes to xo2 At this time as long asv2 gets a little bit smaller the state of variable xwill jump fromxo2 to xo3 in point 2 and the state of variable x enters upperbranch of counter-S curve Therefore according to the aboveanalysis we can control the failure of manufacturing systemThe specific process is shown in Figure 2 firstly the operationdata of the manufacturing system (119909 119906 V) are collected andanalyzed and then according to Table 2 the data will bedisplayed in real-time by bars in different colors By thisway we can find the operational problem in manufacturingsystems and then it is controlled according to the internalmechanism of fault in the manufacturing system to keep thesystem running in a healthy state

3 A Simplified Case Study

A simplified proof-of-concept case is illustrated to show theprocess of the proposed method Yonggu is a company thatproduces metal tools and has a complete IoT (Internet ofthings) system in its workshop We use the IoT system tocollect real-time data of the productworkshop and then selectthe adjacent data including normal data and abnormal datato establish the catastrophe model

In this section the throughput 119883 and the productionload 119881 of the manufacturing system within the duration time119880 are selected as the monitoring parameters In addition itshould be noted that according to the parameter estimationrequirements of the catastrophe model the data must be

extracted in an interval that is the data in an interval mustbe continuous in time

The specific way is shown in Figure 3

31 Data Extraction and Modeling Suppose that the dataseries of 119899 groups (119909 119906 V) can be obtained in a time period

(1199091 1199061 V1 1199051) (1199092 1199062 V2 1199052) (119909119899 119906119899 V119899 119905119899) (12)

where 119905119894 is sampling time In this paper the method of clus-tering is used to distinguish the normal and abnormal data inthe original data sequenceThismethod differentiates normaldata and abnormal data based on the similarity of databetween data (based on the distance between data points)and the effect of isolation or noise points on classification iserased using this method Therefore after the sampling databeing preprocessed the adjacent normal data and abnormaldata of the state variable are obtained Some sample data isshown in Figure 4

In each interval the data was collected continuouslyand composed of abnormal data and normal data Thecatastrophe model was established with the data of oneinterval and then the parameters of the model were revisedwith the data of other intervals to make the model moreperfect

32 Modeling and Analysis Now we use the above data to setup catastrophe model First to satisfy the above equation of

Mathematical Problems in Engineering 7

10

100

200

300

400

500

600

700

800

15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323

DurationProduction LoadThroughtput

Figure 4 Part experimental data (119909 = 119884 = 134)

catastrophe flow and bifurcation set the following equationis satisfied

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (13)

And then according to above formula and 120597119869(119886 119887)120597119886 =0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(14)

By analyzing the above model the computational com-plexity is 119874(119899) Using above data and references [26] 10solutions were obtained by using (119886 119887) = (119886119894 119887119894) | 119869(119886119894 119887119894) lt119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 And then the optimalvalues 119886 = minus87 and 119887 = 7821 can be obtained Next the otherintervals data can be used to modify the parameters of themodel finally we obtain 119886 = minus95 and 119887 = 9721

Now let us research on some of the dynamic behaviors ofcatastrophe in the manufacturing system when 119886 = minus95 and119887 = 9721

Equilibrium Surface (119909 119906 V) | 1199093 minus 95119906119909 + 8721V = 0

Bifurcation Set (119906 V) | 119863 = minus1199063 + 600V2 119863 = 0The graph of the equilibrium surface 119872 and the bifurca-

tion set 119861 is represented in Figure 5 Let the system state be

represented by (119909 119906 V) in the three-dimensional space thenthe phase point must be located on the surface and alwayson the upper lobe or lower lobe of the surface In Figure 5the upper lobe indicates the normal state of manufacturingsystem the lower lobe indicates the fault state and the foldingpart indicates an unreachable state of the manufacturingsystem

The projection of the equilibrium hypersurfaces 119872 in thecontrol plane 119862 namely u-v is a topological transformationor mapping which can be represented by 119891

119891 119872 (1198773) 997888rarr 119862 (1198772)

namely (119909 119906 V) 997888rarr (119906 V) (15)

Now we get the monitor Figure 6As shown in Figure 6 the first figure in Figure 6 cor-

responds to the path (1) in Figure 5 When the controlparameters (119906 V) are not controlled at 950 then (119906 V) willfall on the curve of 119863 = 0 at the next moment andthe manufacturing system will suddenly break down Atthis time the manufacturing system is still able to operateand though certain adjustments can be restored to normaloperation if the (119906 V) is still not controlled (119906 V) will enter119863 gt 0 and the system cannot work and manual repairis required The second figure in Figure 6 is the result ofcontrolling (119906 V) before system fault corresponding to path(2) in Figure 5 From path (2) in Figure 5 we can know thatpreventing (119906 V) from 119863 = 0 at point 1 we can change theevolutionary path of the system and prevent the system frombreaking down suddenly The third figure in Figure 6 is theresult of controlling (119906 V) after system fault corresponding topath (3) in Figure 5 From path (3) in Figure 5 we can knowthat changing the values of 119906 and V satisfying (119906 V) isin (119906 V) |119863(119906 V) = 0 at point 2 we can bring the manufacturingsystem back to normal suddenly

8 Mathematical Problems in Engineering

X

(1)

(2)

(3)

Normal

N

Control

Control

Fault

M

C1

2

u

V

UC

D=0

Dlt0Dgt0

D=0

evolution path (1)evolution path (2)evolution path (3)

Figure 5 Manufacturing system fault evolution process

t

t

t

Control Control

102010101000950

102010101000

102010101000

950

940930920910900850840830820810800

Figure 6 Logical events

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

Mathematical Problems in Engineering 7

10

100

200

300

400

500

600

700

800

15 29 43 57 71 85 99 113 127 141 155 169 183 197 211 225 239 253 267 281 295 309 323

DurationProduction LoadThroughtput

Figure 4 Part experimental data (119909 = 119884 = 134)

catastrophe flow and bifurcation set the following equationis satisfied

min 119869 (119886 119887)

=119899

sum119894=1

(1199093119894 + 119886119906119894119909119894 + 119887V119894)2 + (411988631199063119894 + 271198872V2119894 )

2 (13)

And then according to above formula and 120597119869(119886 119887)120597119886 =0 120597119869(119886 119887)120597119887 = 0 we can obtain119899

sum119894=1

119906119894119909119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894)

+ 1211988621199063119894 (411988631199063119894 + 271198872V2119894 ) = 0119899

sum119894=1

V119894 (1199093119894 + 119886119906119894119909119894 + 119887V119894) + 54V2119894 (411988631199063119894 + 271198872V2119894 )

= 0

(14)

By analyzing the above model the computational com-plexity is 119874(119899) Using above data and references [26] 10solutions were obtained by using (119886 119887) = (119886119894 119887119894) | 119869(119886119894 119887119894) lt119869(119886119895 119887119895) 119895 = 1 2 119894 minus 1 119894 + 1 119905 And then the optimalvalues 119886 = minus87 and 119887 = 7821 can be obtained Next the otherintervals data can be used to modify the parameters of themodel finally we obtain 119886 = minus95 and 119887 = 9721

Now let us research on some of the dynamic behaviors ofcatastrophe in the manufacturing system when 119886 = minus95 and119887 = 9721

Equilibrium Surface (119909 119906 V) | 1199093 minus 95119906119909 + 8721V = 0

Bifurcation Set (119906 V) | 119863 = minus1199063 + 600V2 119863 = 0The graph of the equilibrium surface 119872 and the bifurca-

tion set 119861 is represented in Figure 5 Let the system state be

represented by (119909 119906 V) in the three-dimensional space thenthe phase point must be located on the surface and alwayson the upper lobe or lower lobe of the surface In Figure 5the upper lobe indicates the normal state of manufacturingsystem the lower lobe indicates the fault state and the foldingpart indicates an unreachable state of the manufacturingsystem

The projection of the equilibrium hypersurfaces 119872 in thecontrol plane 119862 namely u-v is a topological transformationor mapping which can be represented by 119891

119891 119872 (1198773) 997888rarr 119862 (1198772)

namely (119909 119906 V) 997888rarr (119906 V) (15)

Now we get the monitor Figure 6As shown in Figure 6 the first figure in Figure 6 cor-

responds to the path (1) in Figure 5 When the controlparameters (119906 V) are not controlled at 950 then (119906 V) willfall on the curve of 119863 = 0 at the next moment andthe manufacturing system will suddenly break down Atthis time the manufacturing system is still able to operateand though certain adjustments can be restored to normaloperation if the (119906 V) is still not controlled (119906 V) will enter119863 gt 0 and the system cannot work and manual repairis required The second figure in Figure 6 is the result ofcontrolling (119906 V) before system fault corresponding to path(2) in Figure 5 From path (2) in Figure 5 we can know thatpreventing (119906 V) from 119863 = 0 at point 1 we can change theevolutionary path of the system and prevent the system frombreaking down suddenly The third figure in Figure 6 is theresult of controlling (119906 V) after system fault corresponding topath (3) in Figure 5 From path (3) in Figure 5 we can knowthat changing the values of 119906 and V satisfying (119906 V) isin (119906 V) |119863(119906 V) = 0 at point 2 we can bring the manufacturingsystem back to normal suddenly

8 Mathematical Problems in Engineering

X

(1)

(2)

(3)

Normal

N

Control

Control

Fault

M

C1

2

u

V

UC

D=0

Dlt0Dgt0

D=0

evolution path (1)evolution path (2)evolution path (3)

Figure 5 Manufacturing system fault evolution process

t

t

t

Control Control

102010101000950

102010101000

102010101000

950

940930920910900850840830820810800

Figure 6 Logical events

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

8 Mathematical Problems in Engineering

X

(1)

(2)

(3)

Normal

N

Control

Control

Fault

M

C1

2

u

V

UC

D=0

Dlt0Dgt0

D=0

evolution path (1)evolution path (2)evolution path (3)

Figure 5 Manufacturing system fault evolution process

t

t

t

Control Control

102010101000950

102010101000

102010101000

950

940930920910900850840830820810800

Figure 6 Logical events

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

Mathematical Problems in Engineering 9

Figure 7 Deep learning method

Best Training Performance is 00017027 at epoch 4000

Mea

n Sq

uare

d Er

ror (

mse

)

TrainBestGoal

Gradient = 00027322 at epoch 4000

0 500 1000 1500 2000 2500 3000 3500 40004000 Epochs

Validation Checks = 0 at epoch 4000

10minus6

10minus4

10minus2

100

10minus5

100

105

grad

ient

minus1

0

1

val f

ail

0 500 1000 1500 20004000 Epochs

2500 3000 3500 4000

Figure 8 Training renderings diagram

Data-driven methods such as support vector machine(SVM) PCA and spectrum analysis are based on a largenumber of real-time data for training so it has requirementson the quantity of dataHowever in this paper the fault data issmall sample so data-driven methods cannot be able to adoptin this paper In the following part the value-driven methodis used to predict and analyze manufacturing system faults

Now we establish 4 layers networks as show in Figure 7the input layers have 2 neurons the first hidden layers have12 neurons the second hidden layers have 6 neurons andthe output layer have 1 neuron The elastic conjugate gradientdescentmethodwithmomentum is used for network trainingby 368 groups of part experimental data in Figure 4 In orderto be consistent with the events in Figure 6 we predict andanalyze the next 10 events

By Figure 8 we can know that aftermore than 4000 timesof training the network model has achieved good resultsHowever by Figure 9 we can know deep learning has poor

prediction effect in such caseThe reason is that the fault datais a small sample so the deep learning method is difficult toachieve good prediction effect Moreover it is result-orientedand does not give the internal mechanism of fault evolution

The combination method of analysis model and datadriven proposed in this paper can effectively make up thedrawback that data-driven and knowledge-driven have toohigh requirements for fault data volume

The above results show that the cusp catastrophe modelestablished in this paper for manufacturing systems canexcavate the internal mechanism of fault evolution andachieve the preventive control of fault

4 Research Significance

Previous fault analysis based on data driven and value drivencan predict the type of fault and the time of fault occurrence

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

10 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8 9 10

Actual value Predicted value

Abnormal

Abnormal state

Normal state

100

150

200

250

300

Figure 9 Result comparison diagram

accurately however it is a big issue that the cause of faultand evolution mechanism cannot be found Therefore theevolution mechanism of fault is of great significance to themanagement and operation of an enterprise In this paperthe catastrophe model of manufacturing system fault isestablished and then by solving and analyzing model it isfound that

(1) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 ge 119884 the manufacturing system will stay in thenormal state

(2) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 ge 119884 the manufacturing system will stay in thenormal state However with the change of (119906 V) themanufacturing system will break down suddenly on(119906 V) isin (119906 V) | 119863(119906 V) = 0

(3) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) gt0 119909 lt 119884 the manufacturing systemwill break downand require manual repair

(4) If the control variables (119906 V) isin (119906 V) | 119863(119906 V) lt0 119909 lt 119884 by changing (119906 V) into (119906 V) | 119863(119906 V) =0 we can bring the manufacturing system back tonormal suddenly

Through the above analysis the internal mechanism offault evolution is found and the preventive control of fault isrealized

5 Conclusion

In this paper the cusp catastrophic model was proposedto describe manufacturing system fault and to explain faultevolution mechanism in a production process First theoperation state of the manufacturing system is describedwith two internal and external macro order parameters The

external macro order parameters are taken as state variables 119909and the internal macro order parameters as control variables119906 and V In the process of solving model parameters 119886 and 119887k-mean clustering algorithm is used for data preprocessingand then the extremum of multivariate functions is used forthe optimal parameters 119886 and 119887 Finally the dynamicsmethodis used to analyze the cusp catastrophe model to find out theinternal mechanism of fault evolution in the manufacturingsystem and to design the logic operation according to theinternal mechanism of evolution so as to realize the real-timemonitoring and preventive control of themanufacturingsystem Through the example verification this method istried out in the Yonggu Company for one year which canshorten 12847 hours of fault response time and reduce theloss of $33000

Data Availability

The experimental data in Figure 4 used to support thefindings of this study are belongs to Yonggu bloc in zhejiangprovince Hence we just provide part data that are includedwithin the supplementary information file(s) (available here)

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by Natural Science Foundationof China (71571072) Guangdong Provincial Natural ScienceFoundation Project (2018A030313079) and 2018 GuangzhouPhilosophy and Social Science Development ldquo13thFive-YearPlanrdquo (2018GZYB16)

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

Mathematical Problems in Engineering 11

Supplementary Materials

Some experimental data are given in the attachment threecolumns of data are represented respectively duration pro-duction load and throughput (Supplementary Materials)

References

[1] H-J Ma and G-H Yang ldquoObserver-based fault diagnosis fora class of non-linear multiple input multiple output uncertainstochastic systems using B-spline expansionsrdquo IET Controleory amp Applications vol 5 no 1 pp 173ndash187 2011

[2] T Jiang K Khorasani and S Tafazoli ldquoParameter estimation-based fault detection isolation and recovery for nonlinear satel-lite modelsrdquo IEEE Transactions on Control Systems Technologyvol 16 no 4 pp 799ndash808 2008

[3] M Zhong Y Song and S X Ding ldquoParity space-basedfault detection for linear discrete time-varying systems withunknown inputrdquo Automatica vol 59 pp 120ndash126 2015

[4] B Cai Y Zhao H Liu and M Xie ldquoA data-driven faultdiagnosismethodology in three-phase inverters for pmsmdrivesystemsrdquo IEEE Transactions on Power Electronics vol 32 no 7pp 5590ndash5600 2017

[5] J Chen Z Li J Pan et al ldquoWavelet transform based on innerproduct in fault diagnosis of rotating machinery a reviewrdquoMechanical Systems and Signal Processing vol 70ndash71 pp 1ndash352016

[6] J Yan and L Lu ldquoImproved Hilbert-Huang transform basedweak signal detectionmethodology and its application on incip-ient fault diagnosis and ECG signal analysisrdquo Signal Processingvol 98 pp 74ndash87 2014

[7] M Grbovic W Li P Xu A K Usadi L Song and S VuceticldquoDecentralized fault detection and diagnosis via sparse PCAbased decomposition and maximum entropy decision fusionrdquoJournal of Process Control vol 22 no 4 pp 738ndash750 2012

[8] S Yin X Zhu and O Kaynak ldquoImproved PLS focused on keyperformance indictor related fault diagnosisrdquo IEEE Transac-tions on Industrial Electronics vol 62 no 3 pp 1651ndash1658 2015

[9] H A Talebi and K Khorasani ldquoA neural network-basedmultiplicative actuator fault detection and isolation of nonlinearsystemsrdquo IEEE Transactions on Control Systems Technology vol21 no 3 pp 842ndash851 2013

[10] L Shu Y Chen Z Huo N Bergmann and L Wang ldquoWhenmobile crowd sensing meets traditional industryrdquo IEEE Accessvol 5 pp 15300ndash15307 2017

[11] S Wang J Wan D Li and C Zhang ldquoImplementing smartfactory of industrie 40 an outlookrdquo International Journal ofDistributed Sensor Networks vol 2016 Article ID 3159805 2016

[12] R Zhao R Yan Z Chen et al ldquoDeep learning and its applica-tions tomachine health monitoringrdquoMechanical Systems SignalProcessing vol 115 pp 213ndash237 2019

[13] W Chen W-T Chen M Saif M-F Li and H Wu ldquoSimulta-neous fault isolation and estimation of lithium-ion batteries viasynthesized design of Luenberger and learning observersrdquo IEEETransactions on Control Systems Technology vol 22 no 1 pp290ndash298 2014

[14] GH B Foo X Zhang andDMVilathgamuwa ldquoA sensor faultdetection and isolation method in interior permanent-magnetsynchronous motor drives based on an extended kalman filterrdquoIEEE Transactions on Industrial Electronics vol 60 no 8 pp3485ndash3495 2013

[15] M Sepasi and F Sassani ldquoOn-line fault diagnosis of hydraulicsystems using unscented kaiman filterrdquo International Journal ofControl Automation and Systems vol 8 no 1 pp 149ndash156 2010

[16] S Zhai W Wang and H Ye ldquoFault diagnosis based onparameter estimation in closed-loop systemsrdquo IET Controleory amp Applications vol 9 no 7 pp 1146ndash1153 2015

[17] H M Odendaal and T Jones ldquoActuator fault detection and iso-lation an optimised parity space approachrdquoControl EngineeringPractice vol 26 no 1 pp 222ndash232 2014

[18] J Dong M Wang X Zhang L Ma and K Peng ldquoJoint data-driven fault diagnosis integrating causality graphwith statisticalprocess monitoring for complex industrial processesrdquo IEEEAccess vol 5 pp 25217ndash25225 2017

[19] W Guo andA G Banerjee ldquoIdentification of key features usingtopological data analysis for accurateprediction of manufactur-ing system outputsrdquo Journal of Manufacturing Systems vol 43pp 225ndash234 2017

[20] B A Weiss M Sharp and A Klinger ldquoDeveloping a hierar-chical decomposition methodology to increase manufacturingprocess and equipment health awarenessrdquo Journal of Manufac-turing Systems vol 48 pp 96ndash107 2018

[21] Z Zhou C Wen and C Yang ldquoFault Isolation Based on 120581-Nearest neighbor rule for industrial processesrdquo IEEE Transac-tions on Industrial Electronics vol 63 no 4 pp 2578ndash25862016

[22] G E Hinton and R R Salakhutdinov ldquoReducing the dimen-sionality of data with neural networksrdquo e American Associa-tion for the Advancement of Science Science vol 313 no 5786pp 504ndash507 2006

[23] J Xiong Q Zhang G Sun X ZhuM Liu and Z Li ldquoAn infor-mation fusion fault diagnosis method based on dimensionlessindicators with static discounting factor and knnrdquo IEEE SensorsJournal vol 16 no 7 pp 2060ndash2069 2016

[24] W Chen and A M Bazzi ldquoLogic-based methods for intelligentfault diagnosis and recovery in power electronicsrdquo IEEE Trans-actions on Power Electronics vol 32 no 7 pp 5573ndash5589 2017

[25] Y Zhang X Li L Gao L Wang and L Wen ldquoImbalanceddata fault diagnosis of rotating machinery using syntheticoversampling and feature learningrdquo Journal of ManufacturingSystems vol 48 pp 34ndash50 2018

[26] P Wang Ananya R Yan and R X Gao ldquoVirtualization anddeep recognition for system fault classificationrdquo Journal ofManufacturing Systems vol 44 pp 310ndash316 2017

[27] O Kupferman W Li and S A Seshia ldquoA theory of mutationswith applications to vacuity coverage and fault tolerancerdquo inProceedings of the 2008 International Conference on FormalMethods in Computer-Aided Design FMCAD pp 1ndash9 USANovember 2008

[28] D H Vries and J B Farmer catastrophe theory 1909[29] R Lange and J Houran ldquoModeling maherrsquos attribution theory

of delusions as a cusp catastropherdquoNonlinearDynamics Psychol-ogy amp Life Sciences vol 4 no 3 pp 235ndash254 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom