chapter 14 design of large engineering...

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Chapter14DesignofLargeEngineeringSystems Gyung-JinPark,AmroM.Farid

Summary

Onedefiningcharacteristicoftwenty-firstcenturyengineeringchallengesisthebreadthoftheirscope. TheNationalAcademyofEngineering(NAE)hasidentified14“game-changinggoals”. Atfirstglance,eachoftheseaspirationalengineeringgoalsissolargeandcomplexinitsownrightthat it might seem entirely intractable. Fortunately, design science provides a continuallyadvancingperspectivebuiltuponameta-problem-solvingskillset. ThischapterintroducesthedesignengineertotheworldoflargecomplexsystemsfromanAxiomaticDesignperspective. Inparticular, the chapter focuses on two critical “systems-thinking” design skills that help thedesignermanage the inherent and abstract complexity of large systems. They are1.) systemdecomposition and 2.) the allocation of function to form. The chapter also practicallydemonstratesthesedesignskillsinseveraldesignstoriesandcasestudies. Thechapterdiscusseswhyandhowthesedesignskillsareuseddifferentlywhenthesystemhasafixedversusflexiblestructure. Finally,thechapterconcludeswithseveralavenuesforfurtherinvestigation.

14.1 Introduction

14.1.1 Motivation

14.1.2 Contribution

14.1.3 ChapterOutline

14.2 AxiomaticDesignofLargeFixedEngineeringSystems

14.2.1 WhatareLargeFixedEngineeringSystems?

14.2.2 DivideandConquer: DecompositionofSystemHierarchy

14.2.3 AllocationofFunctiontoForm–theZigzaggingProcess

14.3 LargeFixedEngineeringSystemDesignCaseStudies

14.3.1 AirConditioner

14.3.2 AutomobileCoolingSystem

14.3.3 AutomobileSuspensionSystem

14.3.4 MobileHarbor

14.3.5 On-lineElectricalVehicle

14.3.6 DesignChallenge: ControlLoopsinCyber-PhysicalSystems

14.4 AxiomaticDesignofLargeFlexibleEngineeringSystems

14.5 Conclusion

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14.1 Introduction

Therearetwokindsoflargesystems.Thefirstkindisatree-likelargesystem.ItstartswithalimitednumberofFRsandDPsatthehighest-levelbutrequiresmanylayersofdecomposition foractual implementation, involvinga largenumberoflower-levelFRstosatisfythelimitednumberofthehighestlevelFRs.Forexample,theoriginal idea formakingadispersion-strengthenedcopperalloyhadonlyalimitednumberofFRs.Itconsistedofapurecoppermatrixandaplethoraofnano-scale titanium di-boride particles as dispersoids. The process devisedwas thehigh-speed,isothermalimpingementmixingofCu/TisolutionwithCu/Bsolution,followedbyrapidcoolingandshreddingofsolidribbonof thealloy,whichwascompactedthroughhydrostaticcompaction/extrusionofthealloy.ThenumberofFRsat thehighest levelwasrelativelysmall.However,whentheprocessdesignwascompletedtomanufacturethealloycommercially,themanufacturingsystembecame quite complicated because many lower-level FRs were introduced toimplementthehighest-levelFRs.

ThesecondkindoflargesystemhasalargenumberofFRsatthehighestlevel.Forexample,ifwearedesigninganairportforamajorhub,therearemanyhighest-levelFRs.TheseFRs,inturn,generatemanylower-levelFRs,makingthesystemverylarge.Thischapterdealswiththissecondkindoflargesystems,althoughthemethodologyandtheoryareequallyapplicabletothefirstkindoflargesystem.

14.1.1 Motivation

Onedefiningcharacteristicoftwenty-firstcenturyengineeringchallengesisthebreadthoftheirscope. TheNationalAcademyofEngineering(NAE)hasidentified14“game-changinggoals”:

1. Advancepersonalizedlearning

2. Makesolarenergyeconomical

3. Enhancevirtualreality

4. Reverse-engineerthebrain

5. Engineerbettermedicines

6. Advancehealthinformatics

7. Restoreandimproveurbaninfrastructure

8. Securecyber-space

9. Provideaccesstocleanwater

10. Provideenergyfromfusion

11. Preventnuclearterror

12. Managethenitrogencycle

13. Developcarbonsequestrationmethods

14. Engineerthetoolsofscientificdiscovery

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Atfirstglance,eachoftheseaspirationalengineeringgoalsissolargeandcomplexinitsownrightthateachmightseementirelyintractable. Furthermore,eachgoalmightappearsodifferentfromthe next that an aspiring engineer might easily conclude that the skills needed to solve onechallengeareentirelydistinct fromthoseofanother. Consequently,ourengineeringeducationsystemwouldhavetoturn“onadime”,orientitselftowardseachofthese14challenges,andaskour first-year engineering students to commit themselves to one of these challenges; never tochangedirectionagain. Andintheunlikelyeventthatwearesuccessfulonsuchacourse, theengineeringeducationsystemwouldhavetopivotagainyearslatertoaddressthenewlycropped-upgrandchallenges.

Fortunately,designscienceprovidesanalternativeandcontinuallyadvancingperspective. WhileeachoftheseaspirationalNAEgoalsmightseementirelydifferent, inreality, theyexhibitmanycommoncharacteristicswhichcanbeintegratedintoaconsistentdesignframework. Intime,thisdesign framework increasingly spans individual engineering disciplines and real-life problemdomains. Italsoincreasinglysetsasidelimitingparadigmsandassumptionsinitsquesttowardsarefinedmeta-problem-solvingskillset.

Figure 14.1. Four Domains in the Engineering Design of Systems – An Axiomatic Design Perspective

ADesignStory: (Note:Weshouldputchapternumberinfrontofthefigureandtablenumberstominimizeconfusion.)

Thedesignengineerreconsidersthe14NAEchallengesinthecontextofthefourAxiomaticDesignengineeringdomainsshowninFigure14.1.

• First, they recognize that the “design solution” to each of the grand challenges can beviewedasanewlydesignedlargecomplexsystemthatisverymucharefinedversionofthelargecomplexsystemthatexistinitsplacetoday.

• Second, they realize thatunlike “traditionalproducts”, these systemshavenot justone“customer” in the stakeholder requirements (SR) domain but rather a diversity ofinternalandexternalstakeholders. Thesestakeholdersimposeawidevarietyofhardandsoftrequirementswhichmustultimatelyberesolvedintothefunctionalrequirements(FRs)andconstraints(Cs)ofthefunctionalarchitecturedomain.

• There,theaspiringdesignengineerfindsthatlargecomplexsystemsarecharacterizedbya large number of FRs. Managing such a large number is a psychological andorganizationalchallengeinitsownright.

DP PVSR FR

StakeholderRequirements

Domain

FunctionalArchitecture

Domain

PhysicalArchitecture

Domain

Process Architecture

Domain

synthesis synthesis synthesis

analysis analysis analysis

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• Beyondjustnumber,thedesignsolutioniscomplicatedbytheheterogeneity(ordiversity)of the FRs. Functions are closely tied to siloed engineering disciplines. At theirsimplest, an electrical engineer knows electricity, a chemical engineer knows chemicalreactions,andamechanicalengineerknowsclassicalmechanics. Adiversityoffunctionmeansthatthedesignengineermusteitherabsorbthefunctionsassociatedwithotherdisciplinesthemselvesorworkeffectivelywithotherdesignengineerswhohave.

• Itisatthispoint,whenthedesignengineercrossesthesynthesispathfromthefunctionalarchitecture domain to the physical architecture domain and its associated designparametersthattheyfacetheirgreatestchallenge: creativity. Thewordstrikesfearinthe hearts of many young designers. As young designers ourselves, we recall beingperpetuallyinaweofhowourdesignprofessorsseemedtomagicallycomeupwithdesignsolutionsfrom“thinair”. Andyet,creativityisnotabinarycharacteristicencodedinDNA. Rather, it is cultivatedby awillingness to loosen and expandone’s establishedmentalconstructs; whether by picking up new books or immersing oneself amongst andappreciatingtheperspectivesofabroaddiversityofpeople. Eachperson,theirfield,andpersonalsetofexperiencescanbecharacterizedbyasetof“mentalconstructs”thatserveas“meta-design-parameters”thatthedesignengineerdeployscreativelyinthemomentof synthesis. Thedesignengineersoonrecognizes that the14NAEgoalsare indeedchallengesforthespecificreasonthattheyrequireabroadnumberanddiversityofsuch“meta-design-parameters”. No single design engineer will have them all, but designteams can produce maximally effective large complex systems by fostering anenvironmentwherenewideasareencouragedandeasilycommunicated.

• As the design engineer implements their solution and crosses over to the processarchitecture domain, they find that the large complex system has already beenimplemented–-atacostofmillions,billions,oreventrillionsofdollars--poorlynoless-- in the form of the legacy system! The likelihood of implementing their “forward-engineeringdesign”islow. Despaircansetin. Butwithnewfoundenergy,thedesignengineer sets off in the reverse direction; now along the “analytical path”, “reverse-engineering”thelegacysystem. TheytaskthemselveswithmodelingthePVs,DPs,FRs,andSRsofthelegacysystem;knowingfullwellthattheywillhavetomakeasequenceofmeaningfulpiecewisetransformationstowardsan“ideal”designsolutionthatreconcilestheirforwarddesignwiththeexistingsystemallwhileitremainsoperational.

14.1.2 ChapterContribution

This chapter introduces the design engineer to the world of large complex systems from anAxiomaticDesignperspective. Itwouldbeentirelyintractabletotrytoaddressallofthespecificchallenges mentioned in the large complex system design story relayed above. Rather, thischapterfocusesontwocritical“systems-thinking”designskillsthathelpthedesignermanagetheinherentandabstractcomplexityoflargesystems. Theyare1.)systemdecompositionand2.)theallocationof functiontoform. Thechapteralsopracticallydemonstratesthesedesignskills inseveraltractabledesigncasestudies. Thechapteralsodiscusseswhyandhowthesedesignskillsare used differently when the system has a fixed versus a flexible structure. The chapterconcludeswithadiscussionofseveralopenchallengesinthedesignoflargecomplexsystems.

14.1.3 ChapterOutline

Theremainderofthechapterisorganizedasfollows. Section14.2treatstheAxiomaticDesignoflarge fixed systems. Section 14.3 then provides several design stories of large fixed systems. Finally, Section 14.4 discusses large flexible systems and contrasts them with the large fixedsystemsdiscussedearlierinthechapter.

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14.2 AxiomaticDesignofLargeFixedEngineeringSystems

14.2.1 WhatareLargeFixedEngineeringSystems?

Theprevioussectionusedtheexamplesofthe14NAEchallengestomotivatethetopicof largecomplex systems. This section addresses an important subset of these called “large fixedengineeringsystems”. Togaininsight,thetermmustbedeconstructedintoitsconstituentwords.

Thereisnoshortageofdefinitionsintheliteraturefortheterm“system”. Thischapteradoptsthedefinitionbelow:

Definition14.1.System: Asetofcomponents(subsystems,segments)actingtogethertoachieveasetofcommonobjectivesviatheaccomplishmentofasetoftasks.

Notethatthemeredefinitionofthetermrequiresthedefinitionofthreemoreabstractsystemsthinkingconcepts:

Definition14.2.SystemBoundary:Thedelineationbetweenthesystemanditsenvironmentorcontext.

Definition14.3.SystemForm: Whatasystem“is”.Itincludesadescriptionof:

1. Allthesystem’scomponents(ordesignparameters). 2. Howthecomponentsareinterconnected. 3. Whatportionofthetotalsystembehavior/functioniscarriedoutbyeachcomponent.

Definition14.4.SystemFunction/Behavior:Whatasystem“does”.Itisitsreasonforexistence.Asetofsubfunctions(orfunctionalrequirements)thatmustbeperformedtoachieveaspecificobjective.

ReturningtotheAxiomaticDesignframeworkshowninFigure1,thesystemformconstitutesthephysical architecture domain and the system function constitutes the functional architecturedomain.

“Largesystems”arereferredtoassuchbecausetheyhavea“large”numberofsystemelements;betheyFRsorDPs. Howlargeisa“largenumber”? Forallpracticalpurposes,theanswerisdrivenbythepsychologicallimitationsofthehumanmindasitattemptstodesignthesystem. In1956,Miller recognized thathumanbeingscan typically recall7+/-2numericaldigits inshort termmemory. Consequently,asaruleofthumb,theliteraturereferstosystemswithapproximately7elementsas“small-sized”,72»50elementsas“medium-sized”,and73»300elementsas“large-size”. Interestingly,at74»2500elementsorgreater,thesystemcannolongerbedesignedpracticallybyasingledesigner(orsystemarchitect). Instead,multipledesignersordesignteamswiththeirrespectiveresponsibilitiesmustcooperatetoproduceawell-functioninglargesystem.

Themereexistenceofmultipledesignersimpliesthattheymustmanagetheinterdependenciesbetweenthesystems’elements. Theterm“complexsystem”isoftenusedtorefertosystemswithalargenumberofinterdependenciesbetweenitsconstituentelements. Forclarity,itisusefultodistinguishbetweentwotypesofsuchinterdependencies:

Definition14.5.Interactions: Aninterdependencycausedbythesequenceofonefunctionalrequirementfollowedbyanother.

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Definition 14.6. Interfaces: An interdependency caused by the relationship between onedesign parameter and another. For example, theremay exist a flow ofmatter between twocomponents.

One can imagine that a system with N elements can have up to N2 interdependencies. Consequently,alargesystembyvirtueofitsnumberofelementsislikelytobecomplexaswellbyvirtueoftheinterdependenciesbetweentheseelements.

Asthelargecomplexsystemgrowsinsize,potentiallyovermanyyears,itislikelythatitselementsandtheirinterdependencieswillchange. Consequently,AxiomaticDesigndistinguishesbetween“largefixedengineeringsystems”and“largeflexibleengineeringsystems”.

Definition14.7.Largefixedengineeringsystem: Anengineeringsystemwithalargesetoffunctionalrequirementswhichdonotevolveovertimeandwhosecomponents(DPs)alsodonotchangeovertime.

Definition 14.8. Large flexible engineering system: An engineering system with manyfunctionalrequirementsthatnotonlyevolveovertime,butalsocanbefulfilledbyoneormoredesignparameters.

Finally,thisbookisdevotedto“engineeringsystems”ratherthan“systems”broadly; inthatthelatterincludes(natural)systems(e.g.thesolarsystem,thehumanbody)thatarenotengineeredbutstilladheretothethreeminimalrequirementsofbeingasystemstatedabove.

14.2.2 DivideandConquer: DecompositionofSystemHierarchy

Thefirstcritical“systems-thinking”designskillisdecompositionofthesystemhierarchy. Here,theengineermustusea“divideandconquer”mentalitytomanagethedesignofthelargesystem. Inbrief, the system’selements;be they functional requirementsordesignparametersmustbedecomposedintotheirconstituentparts.

Forthemoment,let’sassumealargefixedengineeringsystem. ByDefinition14.7,ithasmanyfunctional requirements. Following Figure 14.1, these were identified in a requirementengineering process where the requirements of the stakeholder requirements domain weretransformedintothehigh-levelfunctionalrequirementsofthefunctionalarchitecturedomain.

Torecall,thefunctionalrequirements(FRs)describewhatthesystemmustdoandsometimestheyaresimplyreferredtoasthesystems’functionstodescribewhatthesystemdoes. Saiddifferently,theyareamutuallyexclusiveandcollectivelyexhaustivesetthatdescribethesystemfunctionally. EachFRmustbedefinedinasolution-neutralwaythatdoesn’tpresupposethetechnologiesofthedesignsolution. Byconvention,eachfunctionisdefinedasatransitiveverbstatedinthethirdpersonsingularfollowedbyitsassociatedobject/operand. Forexample,inahomedesign,onefunctionalrequirementmaybe:

FR1: ProtectInternalClimate.

Thereisalotpackedintosucha“high-level”functionalrequirement. Forthisreason,itisoftennecessarytodecomposeeachfunctionintoanumberoflower-leveloffunctionsthataggregatetoachieve the same end. In decomposing a given functional requirement, the designermust becarefultokeepthelower-levelFRsmutuallyexclusiveandcollectivelyexhaustive. Forexample,FR1canbedecomposedinto:

FR1.1KeepoutMoisture FR1.2DampoutHot/ColdFluctuationsintheExternalEnvironmentFR1.3HeatandCoolInteriorAreatodesiredtemperature

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FR1.4RedirectfallingrainawayfromthehouseFR1.5AdmitnaturallightFR1.6ProtectfromInsectsFR1.7Allowentry/exitofinhabitantsFR1.8ProtectfromIntruders

ThedesignercanfurtherdecomposeeachoftheseFRsintoevenlower-levelFRsuntilthelargesystemhassufficientfunctionaldetailtobefullyrealized. Inanabstractandgeneralsense,thisiterativeprocessgeneratesafunctionalhierarchyasshowninFigure14.2. Furthermore,inmanycases,itisusefultoexplicitlyidentifythefunctionalinteractionsbetweeneachsystemfunctionateach level of functional decomposition. At such a point, the functional hierarchy becomes acompletefunctionalarchitecture.

Figure 14.2. A functional architecture that has been decomposed two levels. Parallel and serial interactions are shown between each function.

In an analogous fashion, a large system also has a physical hierarchy composed of multipledecompositionsofdesignparameters(DPs);betheysystems,subsystems,componentsorsinglenumerical parameters. At the highest level of hierarchy, the large system is a single designparameter. Forexample,inhomedesign,

DP1: House’sExternalBarrier

Thishigh-leveldesignparametercanbedecomposedintoseverallower-leveldesignparameters(assubsystems):

DP1.1WaterproofShellDP1.2InsulationlayerDP1.3Air-sourceheatpumpDP1.4GuttersystemDP1.5WindowDP1.6WindowscreensDP1.7DoorsDP1.8Doorlocks

SystemFunction

PeopleMoney

EnergyMaterialSignals

Aggregation

Decom

position

A0

PeopleMoney

EnergyMaterialSignals

SystemFunctionA2

SystemFunctionA1

SystemFunctionA3

SystemFunctionA4

SystemFunctionA2.2

SystemFunctionA2.1

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Finally,whenthephysicalhierarchyisdepictedwiththephysicalinteractionsbetweeneachdesignparameter, it becomes the physical architecture. At such a point, it resembles the functionalarchitectureinFigure14.2;butwiththefunctionsreplacedwithcomponents(orDPs).

FunctionalandphysicaldecompositioncreatetheAxiomaticDesigndualhierarchyshowninFigure14.3. Itservesastheprimarymeansbywhichadesignertacklestheinherentlylargenumberofelements(i.e.FRsandDPs)inthesystem. Ratherthanviewthelargesystemasaloosecollectionoffunctionsandcomponents,thesesetsareorganizedmethodicallyinatree-likestructure. Theprimaryadvantageistoreducethe“mental-load”ofthedesignersothattheyareonlythinkingoftheelementsthataredirectlyrelatedwithintheassociatedhierarchy.

Onequestionthatoftenarises ishowtoorganizethefunctionalandphysicalhierarchies. OnemayimaginethatagivenFRorDPcanbedecomposedallatonceintomany(potentiallyhundreds)ofelements. Alternatively,onemayimaginethatthedesignerorganizestheseelementsintomanydecomposition layers. Intheory, thechoiceofthenumberofdecompositionsteps isarbitrary. Inpractice,itmatterstremendously. Forexample,thephysicalhierarchymaybeorganizedintoassemblies and subassemblies that have real manufacturing significance. Similarly, designteams may have certain functional expertise that drives how we may think of the functionalrequirements. Finally,the7±2rulementionedaboveprovidesapractical“designruleofthumb”fromwhichdesignersshouldnotdeparttoofar.

Figure 14.3. The Axiomatic Design Dual Hierarchy: Functional decomposition, Physical Decomposition and the Allocation of Function to Form in a “Zigzagging” Process.

14.2.3 AllocationofFunctiontoForm–theZigzaggingProcess

Thesecondcritical“systems-thinking”designskillistheallocationoffunctiontoforminwhatiscalledthe“zigzaggingprocess”. Intheprevioussection,functionalandphysicaldecompositionwere presented independently. In reality, a given functional requirement can rarely bedecomposedwithoutfirstassumingsometechnologicalsolutionthatfulfillsit. Inthemeantime,decomposingagivendesignparameterhaslittlemeaningwithoutunderstandingthefunctionalrequirementsthat thedecomposeddesignparameters fulfill. Consequently, in forward-design,thedesignerconsidersagivenfunctionalrequirement,conceivesadesignparametertofulfillit,and then allocates the function to this new element of form. At that point, the designerdecomposesthe functionalrequirementwhileassumingthenewlyconceiveddesignparameter. Insuchaway,thetwocriticaldesignskillsofdecompositionandfunction-to-formallocationareused in an alternating fashion. A single designer can handle several zig-zigging processes toaccommodateseveralhundreddesignparameters. Beyondthat,adesignteamcanworktogethertoaccommodateasystemofpotentiallyarbitrarysize.

2.3 Elements of System Performance Measurement

Figure 2.10: Simultaneous Hierarchical Physical and Functional Decomposition

where the design matrix should be ideally diagonal and square. Very low in the functional/physical

dual hierarchy, the functional requirements and physical requirements are represented by real num-

bers. A standard matrix multiplication · is used between the design matrix and the design param-

eters vector. At higher levels of the dual-hierarchy, the FR’s and DP’s cannot be represented by a

single real number. Instead, the name of the FR or DP is used, and the design matrix elements is

represented by an X to denote the existence of a relationship or 0 otherwise. Suh[197] continues to

use the · operation in this case, although its formal mathematical definition is no longer clear.

As it is fundamental to measurement, later chapters in this thesis make a clear distinction

between (empirical) physical objects and the (formal) mathematical objects that represent them.

At high levels of the dual-hierarchy, the design equation is represented in terms of physical objects

rather than mathematical ones and the design matrix elements only signify the existence of a

relationship. To maintain mathematical formality in this case, the design equation is reinterpreted

in the context of this thesis as:

FR = DM � DP (2.4)

where the � operation acts between the boolean design matrix and the design parameters set. This

noncommutative, non-associative operation is defined as:

Definition 2.14. Aggregation Operator �: Given boolean matrix A and sets B and C, C = A�B

is equivalent to:

C(i) =�

j

a(i, j)⌥ b(j) (2.5)

Intuitively, the � operator acts to aggregate chosen elements of a set.

Axiomatic design theory uses the design matrix when one or more design parameters are re-

quired to realize a given functional requirement. It also uses a knowledge base for “large flexible

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Aspartofthezigzaggingprocess,AxiomaticDesignkeepstrackoftheallocationoffunctiontoformusingadesignequation:

FR$f(DP)whereFRisthesetoffunctionalrequirementsonagivenlevelofdecomposition,DPisthesetofdesignparametersonthesamelevelofdecomposition,andf()isafunctionrepresentingthelawsofphysicsthatgovernthedesignandtherelativelynewsymbol$means“satisfies”whenreadfromrighttoleft. Whenthe$symbolisreplacedwithan=andthefirstderivativeistakenityieldsalinearequation:

DFR=BDDPwhereBisthedesignmatrix. TheIndependenceAxiominstructsdesignerstoensurethatthisdesignmatrix issquareanddiagonal toreachanuncoupleddesign,and ifnot thensquareandlowertriangulartoreachadecoupleddesign. Whenthedesignmatrixisanythingelse,thedesignissaidtobecoupledanditviolatestheIndependenceAxiom. Forexample,thedecomposedFRsandDPsmentionedintheprevioussectionyieldthedesignmatrixbelow.

Figure 14.4. Axiomatic Design Matrix of a House’s External Barrier

OneimportantconsequenceoftheIndependenceAxiomisthatitaffectstheprojectmanagementofthedesign,orthedesignworkflow. Adesigner’staskistocompletetheselectionofalltheDPsthatfulfilltheFRs. ThedesignmatrixshowstheselectionofDPsmustbetakeningroups. ThegroupofDPsthatfulfillagivenFRiscalledamodule. Thedesignmatrixalsoshowsthatthereisaspecificorderinwhichtodesigneachmodule. Inthecaseofanuncoupleddesign,themodulescanbedesignedentirely independently inparallel. As shown inFigure14.5, in adesign flowdiagram,suchacaseisdefinedbyasummation(○S )junctionoftwoormodules. Inthecaseof

decoupleddesign,Figure5showsacontrol(○C )junctiontoindicatethedesignofonemodulemustfollowanothersequentially. Finally,inthecaseofacoupleddesign,Figure5showsafeedback(○F )junctiontoindicatethatdesignofonemoduleisinaniterativefeedbackloopwithanothermodule. Theproblemwithsuchfeedbackloopsisthatit isnotalwaysclearhowmanydesigniterationswillberequiredtoescapethedesignfeedback! Inthedesignof largesystems,suchdesign feedback loops can be entirely debilitating and ultimately bring design progress to ascreechinghalt! Inshort,theIndependenceAxiomdoesnotjustfacilitategooddesigns,italsofacilitatesefficientdesignworkflows.

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.81.1 X1.2 X X1.3 X X X1.4 X X1.5 X X1.6 X X1.7 X X1.8 X X

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Figure 14.5. Elements and junctions in a design workflow diagram. Summing, control and feedback junctions correspond to uncoupled, decoupled, and coupled designs respectively.

Tosummarize,theforward-designzigzaggingprocessfollowsseveralrules. Thedesignermust:

1. ConceivetheDPsassociatedwiththeFRsinthesamelevel.

2. RespecttheIndependenceAxiomwhenallocatingfunctiontoform. Thedesignisideallyuncoupled,andotherwisedecoupled. Ifnecessary,drawadesignworkflowdiagramtooperationalizethesequenceofdesigntaskamongstthedesignteam.

3. Decompose theFRsof the subsequent levelbasedupon the choiceofDPs in the levelimmediatelyabove.

4. Ensure that theFRsandDPs ina lower level areamutually exclusiveandcollectivelyrepresentationoftheFRsandDPsinthelevelimmediatelyabove.

5. Continue the zigzagging process until the Axiomatic Design dual hierarchy includessufficientdetailtoimplementthedesignsolution.

Zigzagging in reverse-engineering simply runs in reverse. The designer must start from theindividualcomponents(orDPs)andthendeducetheassociatedFRs. Fromthere,thedesignerproceedstoahigherlevelofaggregationintheDPsandthendeducestheassociatedFRsonthatlevel. Suchareverse-engineeringapproachisnecessarywhenapartorawholeofthesystemhasalreadybeenbuiltandthedevelopmentofthefunctionalarchitectureisrequiredtodeterminehowto“evolve”thesystemforwardstoamoreadvancedstageofdevelopment. Insomecases,thedeductionoftheassociatedfunctionalrequirementisstraightforward. Manytried-and-truephysicalsolutionshavewell-knownfunctions.Forexample,I-beamsinbuildingssupportweight,andrailwaystransporttrains.

14.3 Examples

The twocritical “systems-thinking”designskillsof systemdecompositionand theallocationoffunctiontoformaresurprisinglypowerfultoolsintheengineer’sarmamenttowardsthedesignoflarge complex systems. This sectionpresents case studieswhere these two skills are appliedpractically. Theprovidedexampleshavebeenchosenfortheirmanageablesizesoastofacilitatelearning.

14.3.1 AxiomaticDesignoftheMountTypeAirConditioningSystem

Themounttypeairconditioningsystemisatypeofheating,ventilatingandairconditioning(HVAC)

○S

(a)Summingjunction (uncoupledcase)

Left ○C Right

(b)Controljunction (decoupledcase)

Left Right○F

(c)Feedbackjunction(coupledcase)

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controlsystemthatisinstalledbetweenaceilingandtheceilingboardsofaroomtocontrolroomtemperature. SuchaproductisinvestigatedfromAxiomaticDesignpointofview. Theoveralldecomposition of the system hierarchy is established down to the “atomic” level components. Somecoupledaspectsarefound;butultimatelykeptattherequestofthedesignsponsor. Thesponsorwantstokeepsomeofexistingcomponentlevelparts,andsothecoupledaspectsofthedesignarereportedaswarningsinthedesignprocess.

Thebasicoperatingprincipleofthemounttypeairconditioningsystemistoabsorbheatinaroomandemitittotheexterior. Figure6illustratesthecomponentsofthemounttypeairconditioningsystemwhich consists of indoor and outdoor machines. Most of the air conditioning systemsemploytherefrigerationcycleusingrefrigeranttogeneratecoolairinaroom.Figure14.7presentstherefrigerationcycleusingtherefrigerant.Intheoutdoorpart,acompressorgeneratestheflowofrefrigerantandacondenserreleasesheat.Theindoorpartconsistsofacapillarytubetocontrolthe flow of the refrigerant and an evaporator to absorb the heat in the room. The refrigerantperiodically circulatesbetween the two sides so that the indoorheat is absorbedand releasedoutside.Currently,thedesigniscarriedoutbasedontheconventionalprocessillustratedinFigure14.8. Since themachine is not a new one,most of the design components are already known.Therefore, the designer selects appropriate components and sets the values of the associateddesign parameters. Performance tests are then conducted to meet the requirements. Thedesigneriterativelyredesignsthecomponentsuntiltheperformancerequirementsaresatisfied. TheassociatedworkflowisillustratedinFigure14.9.

Figure 14.6. Indoor and Outdoor Machines

Figure 14.7. Composition of an Air Conditioning System

(a)Indoormachine (b)Outdoormachine

Condenser

Compressor

Evaporator

Capillary

tube

Outdoor

Indoor

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Figure 14.8. The Conventional Design Process for an Overall Air Conditioning System

Figure 14.9. The Conventional Design Process for the Design of a Part of an Air Conditioning System

Eachcustomerpresentsanewsetofperformancerequirements.Insuchacase,theproductshouldbe re-designedand re-manufacturedeach time to satisfy thenewconditions.Theconventionaldesignmethodreliesonatrialanderrorapproachtomeetthenewsetofrequirementsbecauseitisnoteasytoidentifytherelationshipsbetweenthecomponentsandtheircharacteristicdesignparameters. Furthermore, thedesignermayfinditdifficulttofollowarationaldesignprocessthat converges towards satisfied functional requirements. Along the way, the designer mayexercise engineering judgement that relies on tacit knowledge. Such knowledge is neitherwrittennorsystematicandconsequentlyisdifficulttotransfertojuniordesigners. Allofthesechallengesmaketheconventionaldesignprocesscostlyandslow.

Incontrast,anewAxiomaticDesignapproachisdemonstrated. Customerneeds(CNs)–asubsetofthestakeholderrequirementsdescribedinSection14.1--aregatheredfirstbasedoncustomersurveysandcustomerinterviewswithdesignengineers. TheCNsareshowninTable14.1. FRsarethenidentifiedbasedontheCNs,andDPsareselectedtosatisfytheindependenceoftheFRs.TheFR-DPrelationshipismadebythedesignmatrixandthedual-hierarchyofthedesignprocessisestablished. Again,thedecompositionofthefunctionalandphysicalhierarchiesismadebythezigzagging approach. CNs are defined from the customer survey and interviewswith design

Decisionofpositioningforinstallation

Designofcomponentsforthesystem

Endthedesignprocess

NoConfirmationoftheperformancetests

Yes

DesignofEvaporator/Condenser

Designofairflow

Designofcontrolsystem

Designofdrainage

<DesignofHVACsystem>

Designofexterior/assemblyparts

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engineers.

Table 14.1. Customer Needs for a Mount Type HVAC Control System

CustomerNeedsfromGeneralUsers CustomerNeedsfromDesignEngineersPeopleshouldfeelcool. DustorbadsmellshouldbeeliminatedTheroomshouldbecooledassoonaspossible. The maintenance and management of the

systemshouldbeeasyThecoolairmustsufficientlycirculate. The system should need a small space and

haveanaestheticshape. Uniformtemperatureisrequired. Regulationsmustbesatisfied.Eachvaneshouldbecontrolledseparately. Productioncostsshouldbeminimized.Temperaturecontrolshouldbeeasy. Controlfromadistanceisrequired. Themachineshouldoperatequietly. Energyconsumptionshouldbeminimized BasedontheCNsinTable1,theFRsandconstraints(Cs)aredefinedatthetoplevel.

FR1=Minimizeapossessedspaceofthemounttypeairconditioningsystem. FR2=Generateappropriateaircurrentintheroom. FR3=Makeenoughcoldair. FR4=Minimizethevibration/noiseofthemounttypeairconditioningsystem. FR5=Maintainpurityoftheairqualityintheroom. FR6=Controlthetemperatureunderuser’sdirections.

NotethattheFRsarestatedinasolution-neutrallanguage. AllofthenounsintheFRseither1.)refertothehigher-levelDPofthe“mount-typeairconditioningsystem”or2.)refertonounsinthedesigncontext(e.g.airspace,aircurrent,coldair,noise,airpurity,temperature). Constraintsaredefinedaswell.TheyprovideboundsonacceptabledesignsolutionsanddifferfromtheFRsinthattheydonothavetobeindependent.

C1=Satisfythefirst-gradeforenergyefficiency.C2=Satisfyrelatedstandards.C3=Satisfytheproductsizetosufficientlyinserttheairconditioningsystembetweenthe

ceilingandceilingboard.C4=Minimizeproductioncost.C5=Makemaintenanceandrepairofthesystemeasy.

TomeettheFRs,anappropriatesetofDPsaredefined.

DP1=Ceilingtypestructure DP2=Aircurrentformationsystem DP3=Mutualassistancesystem DP4=Vibration/noisereductionsystem DP5=Aircleanersystem DP6=Temperaturecontrolsystem

Thentheallocationoffunctiontoformiscapturedinthedesignmatrix.

DP1 DP2 DP3 DP4 DP5 DP6FR1 X 0 0 0 0 0FR2 X X 0 0 0 0FR3 0 X X 0 0 0FR4 0 X X X 0 0FR5 0 0 0 0 X 0

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FR6 0 X X 0 X X

whereXrepresentswheretheFRisbeingfulfilledbyoneormoreDPsandOrepresentsotherwise. At the top level of decomposition, the design is a decoupled, and the design matrix is lowertriangular. Therefore,theindependenceofFRs(intheIndependenceAxiom)isguaranteediftheDPsaredeterminedinincreasingnumericalorderfromFR1toFR6.

TheFRsandDPsat the top levelare then iterativelydecomposedusing thezigzaggingprocessdescribedintheprevioussectionuntilthebottomor“atomic”levelcomponentsarereached.Forthesakeofbrevity,afullexplanationofthesystemdecompositionisomittedhere. Thecompletedesignmatrix,however,ispresentedinFigure10.TheleftcolumnofthetablerepresentstheFRsandtheupperrowincludesthecorrespondingDPs.ThemounttypeairconditioningsystemhasadecoupleddesignatthetoplevelasshowninEquation(14.1). Theentiredesignmatrixat itslowestlevelofdecompositionis,however,non-squareandtheresultingproductdesignisclassifiedasredundantandcoupled. Thissituationisthedirectresultofthesponsor’srequesttokeeptheexistingcomponent-levelparts.

Figure 14.10. Complete Design Matrix for Mount Type Air Conditioning System

Thecoupledaspectsareexplained.TheentirematrixinFigure11isanon-squareandconsequentlysomeFR-DP relationshipsare coupledat somepointsof thehierarchy. For example, consider

15

FR222.

FR2.2.2=Generatetheair-flowintheperpendiculardirectionusingarotarymotion.

TheexistingsystemhasfivecorrespondingDPsthatmakeupthecomponentsoftheturbofan.

DP2.2.6=Thenumberofblades DP2.2.7=Ratiooftheinside/externaldiameters DP2.2.8=Theangleamongblades DP2.2.9=Shapeoftheblade DP2.2.10=Ratioofpitch/chord

Consequently,therelationshipbetweenFR222anditsDPsis

DP2.2.6 DP2.2.7 DP2.2.8 DP2.2.9 DP2.2.10FR2.2.2 X X X X X

Obviously,itisaredundantdesignandalotoffeedbacksarerequired.Theredundancyisalsofoundinothercomponentssuchasthemotor,theorifice,thefinoftheevaporator,andthecondenser.

Severalsubsystemswithinthisdetaileddesignwarrantfurtherdiscussion. TheFRsandDPsforthecompressorsystemaredefinedas:

FR3.1=Generatethepressuretochangethestateoftherefrigerant.FR3.2=Generate theair-flowfor theabilityofairconditioningtosufficientlycool the

room.DP3.1=Compressorsystem

Theassociateportionofthedesignmatrixis DP3.1FR3.1 XFR3.2 X

Inthiscase,thecompressorsystemhasaninsufficientnumberofDPs,anditsdesignmatrixisfulland therefore the compressor subsystem constitutes a coupled design. The capillary tubesubsystemsuffersfromthesameproblem.

TheheatexchangersystemhasanequalnumberofFRsandDPsbuttheyarefullycoupled.

FR3.3.2=Changerefrigerantfromgastoliquid. FR3.3.3=Changerefrigerantfromliquidtogas. DP3.2.2=Evaporatorsystem DP3.2.3=Condensersystem

DP3.2.2 DP3.2.3FR3.3.2 X XFR3.3.3 X X

Becausetherefrigeratorcycleconsistsofthecompressorsystem,thecapillarytube,andtheheatexchangersystem,therefrigeratorcycleasawholealsoconstitutesacoupleddesign. Notethatonlyonecoupledsubsystemisrequiredforthesystemasawholetobeclassifiedascoupled. Inanidealsituation,theDPsofthecoupleddesignshouldbemodifiedorreplacedwithasetthatyieldseitheranuncoupledordecoupleddesign. Inthiscase,thefirmdecidedtoflagthecoupledcharacteristicstoalertdesignerstothepotentialforiterativefeedbackloopsinthedesignprocess.

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Figure 14.11. Design Flow of a Mount Type Air Conditioning System

Returningbackuptothetoplevelofdesigndecomposition,theassociateddesignflowdiagramisillustratedinFigure11.Becauseitisadecoupleddesignatthetoplevelofhierarchy,thedesignflow links themodules sequentially. That said, if the designworkflowwere detailed further,feedbackloopswouldappearwithineachofthesystem’smodules.

Insummary,thisexamplehasservedtoapplyAxiomaticDesignforlargefixedsystemstoamounttype air conditioning system. Axiomatic Design demonstrates a rational and effective designprocessthatdecomposestheFRsandDPsateachlevelofhierarchyandthenallocatesfunctiontoformusingthedesignmatrix. Thesetwosystemsthinkingtechniquesallowthedesignteamtomanage the complexity of the design. Finally, the design flowdiagramorganizes the design’sprojectmanagementandisestablishedfromthedesignmatrix.

TheAxiomaticDesignapproachhighlightedsomedifferencesbetweentheexistingconventionalapproach. In theconventionalapproach,eachmodulewasdesigned inparalleland then laterintegratedintoalargersystem;resolvinginconsistenciesinthedesignallatonce. Incontrast,AxiomaticDesignhighlightedtheneedforasequentialapproachtothedesignofthehigh-levelmodules;thusavoidingtheneedtoresolvedownstreamdesigninconsistencies. Theimproveddesignprocessismoreeffectivefordesignerstonotonlyunderstandtheoveralldesignprocessbut also tomeet the diversity of customer demands. Finally, the Axiomatic Design approachhighlightsthepresenceofcoupleddesignfeedbackinlowerlevelsofthedual-hierarchy. Thesearefairlyinefficientandtime-consumingandhavethepotentialtoderailthedesignofthesystemasawhole. Thesecoupledaspectsareidentifiedandflaggedformanagementattentionwithouteliminatingthemfromthedesignasawhole.

14.3.2AutomobileCoolingSystem We consider the design of a cooling system for a hybrid vehicle as illustrated in Figure 14.12. Coolantmaterialiscirculatedinacoolingsystemtoexpelthegeneratedheatfromtheengine.ThecirculationisillustratedinFigure14.13.Someoftheheatisreusedforheatingtheinsideofthecar.

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Figure 14.12. An Automobile Cooling System

Figure 14.13. Circuit of Coolant and Air in the Cooling System

Thecustomerneeds(CNs)ofthesystemaredefinedas:

CN1:Theinsidetemperatureshouldbekeptwithinacertainrange.CN2:Environmentalpollutionisundesired.CN3:Thevehiclefueleconomyshouldbehigh.

Thetop-levelFRisdefinedbasedontheCNs: Increasetheenergyefficiencyofahybridcarwhilemaintainingtheinsidetemperature.Fromthetop-levelFR, thenextlevelFRs,DPs,anddesignmatrixare

FR1=Maintainthetemperatureoftheairinsidethevehicle.FR2=Reducetheemissionofhazardoussubstancesduringoperation.FR3=Controlfuelconsumptionwithinacceptablebounds.

DP1=Airconditioningsystem

Radiator

Engine

Exhaustgasrecirculation (EGR)

Airconditioningsystem(Blower,HeatCare,Flap)

Energyrecycling system

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DP2=HazardousmaterialreductionsystemDP3=Energymanagementsystem

DP1 DP2 DP3FR1 X x xFR2 0 X 0FR3 x 0 X

Note thatFR1andFR3havea coupleddesignwith respect toDP1andDP3. This coupling isdiscussedingreaterdetailatlowerlevelsofdecomposition. DP1andFR1aredecomposedto:

FR1.1=Decreaseroomtemperature.FR1.2=Increaseroomtemperature.DP1.1=AircoolingsystemDP1.2=Airheatingsystem

DP1.1 DP1.2FR1.1 X 0FR1.2 0 X

The air cooling system is not considered further in this example. FR1.2 and DP1.2 are thendecomposedto:

FR1.2.1=Generateaheatsource.FR1.2.2=Generateadditionalheat.FR1.2.3=Absorbthegeneratedheat.FR1.2.4=Delivertheheat.FR1.2.5=Heattheair.FR1.2.6=Movetheheatedair.FR1.2.7=Adjusttheairdirection.DP1.2.1=EngineoperationrequestlogicDP1.2.2=Positivetemperaturecoefficient(PTC)systemDP1.2.3=CoolantDP1.2.4=CoolantcircuitDP1.2.5=HeatcoreDP1.2.6=BlowerDP1.2.7=Flap

Theassociateddesignmatrixis:

DP1.2.1 DP1.2.2 DP1.2.3 DP1.2.4 DP1.2.5 DP1.2.6 DP1.2.7FR1.2.1 X 0 0 0 0 0 0FR1.2.2 0 X 0 0 0 0 0FR1.2.3 0 0 X 0 0 0 0FR1.2.4 0 0 0 X 0 0 0FR1.2.5 0 0 x 0 X 0 0FR1.2.6 0 0 0 0 x X xFR1.2.7 0 0 0 0 0 x X

FR2andDP2canbedecomposedto:

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FR21=Reducethehazardousmaterialbeforethecombustion.FR22=Reducethehazardousmaterialafterthecombustion.FR23=Reducethehazardousmaterialduringthecombustion.DP21=EvaporativegasreductionsystemDP22=EngineemissionreductionsystemDP23=Catalystsystem

Theassociateddesignmatrixis:

DP21 DP22 DP23FR21 X 0 0FR22 0 X 0FR23 0 0 X

FR3andDP3aredecomposedaswell.Figure14.14presentstheentiredesignmatrixafterseveralmoredecompositionandzigzaggingsteps.

Figure 14.14. Complete Design Matrix of an Automobile Cooling System Prior to Resorting FRs and DPs.

Atfirstglance,thedesignoftheautomobilecoolingsystemappearstobehighlycoupled. TherearefilledXelementsinboththeupperandlowertrianglesofthedesignmatrix. RecallthatFR1andFR3haveacoupleddesignbyvirtueofDP1andDP3atthehighestlevelofdecomposition. Thesecoupledelementsmanifestthemselvesinlowerlevelsdecompositionaswell. Thesituation,however,isnotasbleakasonemightthink. Thehigh-levelcouplingbetweenFR1,FR3,DP1,andDP3doesnotmeanthatallofthelow-levelFRsandDPsarecoupled. Quiteabitofsparsityisintroducedintothedesignmatrixwitheachsubsequentdecomposition. Aftercarefulinspection,thedesignerfindsthatmanyofthefilledelementsintheuppertriangleofthedesignmatrixdonothavetheirassociatedfilledelementsinthelowertriangle. Indeed,aminimumconditionofatruly

1 2 1 2 1 2

X O O O O O O O O O O O O O O O O O O O O O

O X O O O O O O O O O O O O O O O O O O O O

O O X O O O O O O O O O O O O O O O O O O x

O O O X O O O O O O O O O O O O O O O O O O

O O O O X O O O O O O O x O O O O O O O O x

O O O x O X O O O O O O O O O O O O O O O O

O O O O O x X x O O O O O O O O O O O O O O

O O O O O O x X O O O O O O O O O O O O O O

O O O O O O O O X O O O O O O O O O O O O O

O O O O O O O O O X O O O O O O O O O O O O

O O O O O O O O O O X O O O O O O O O O O O

1 O O O O O O O O O O O X x O O O O O O O O O

2 O O O O O O O O O O O O X O O O O O O O O O

O O O O O O O O O O O O O X O O O O O O O O

O O O O O O O O O O O O O O X O O O O O O O

O O O O O O O O O O O O O O O X O O O O O O

O O O O O O O O O O O O O O O O X O O O O x

O O O O O O O O O O O O O O O O O X O O O x

1 O O O O O O O O O O O O O O O O O O X O x O

2 O x O O O O O O O O O O O O O O O O O X O O

1 O O O O O O O O O O O O O O O O O O O O X O

2 O O O O O O O O O O O O O O O O O O O O O X

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1

2

3

4

1

7

6

5

2

1

1

22

3

2

1

2

3

1

3

11

2

2

1

2

1 2 3

1

1

1 2 3 4 5 6 7

2

3

1 2

1 21 2

1 2 12

1 2

2 3

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coupleddesignisthatthereexiststwoDPsthatarecoupledtotwoFRs. Ormathematically,thereexistsatleastonepairofelementsinthedesignmatrixDMsuchthatDM(i,j)=DM(j,i)wherei¹j. Suchaconditionoccurs inexactlyoneplaceabove: FR1.2.6andFR1.2.7arecoupledwithDP1.2.6 and DP1.2.7. All of the other filled elements in the designmatrix are asymmetric andindicateadecoupleddesignfortheremainderoftheautomobilecoolingsystem. Toemphasizethispoint,therowsandthecolumnsofthedesignmatrixcanbesortedinsuchawayastobringalloftheoff-diagonaltermstoeitherthematrix’supperorlowertriangle. Figure15showsthecompletedesignmatrixofthesameautomobilecoolingsystemaftertheFRsandDPshavebeensorted. ItrevealsclearlytheoriginalconclusionthatonlyFR1.2.6andFR1.2.7arecoupled. Thisexample illustrates that a high-level coupled design can be “rescued” with respect to theIndependenceAxiomtobecomeadecoupleddesigninlowerlevelsofdecompositionifthenatureofthecouplingisreflectedinanasymmetricdesignmatrix! Fortunately,acreativedesignercanoftenfindcleverwaystointroducesuchasymmetryintothedesign.

Figure14.15.CompleteDesignMatrixofAutomobileCoolingSystemafterResortingFRs&DPs.

14.3.3AutomobileSuspensionSystem

1.1 1.2.2 1.2.4 1.2.7 1.2.6 1.2.5 1.2.3 2.1.1 2.1.2 2.2.1 2.2.2.1 2.2.2.2 2.3.1 2.3.2 2.3.3 3.1.1 3.1.2 3.2.1.1 3.2.1.2 1.2.1 3.2.2.1 3.2.2.21.1 X

1.2.2 X X1.2.4 X X X1.2.7 X X1.2.6 X X X1.2.5 X X1.2.3 X2.1.1 X2.1.2 X2.2.1 X

2.2.2.1 X X2.2.2.2 X2.3.1 X2.3.2 X2.3.3 X3.1.1 X X3.1.2 X X

3.2.1.1 X X3.2.1.2 X X1.2.1 X

3.2.2.1 X3.2.2.2 X

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TheIndependenceAxiomisusedtodesigntheautomobilesuspensionsystemshowninFigure16. It connects thewheels and the body tomake the automobilemove. There are two suspensionsubsystems;oneforthefrontwheelsandtheotherfortherealwheels.

Figure14.16.AnAutomobileSuspensionSystem

Thecustomerneeds(CNs)are:

CN1:Thevehiclemusthaveadrivingcapability.CN2:Safetymustbeprovidedtothevehiclewhendriving.

FromtheCNs,thetop-levelFRis:

FR:Suspendthevehiclewithstabilitywhiledriving(CN1+CN2)

whichisinturndecomposedintosixFRs:

FR1=Generateforwardthrustfromtransmissionsystemtorque. FR2=Permitwheelrotation. FR3=Securethecarbody.FR4=SuspendthevehiclewithstabilitywhiledrivingonruggedroadsFR5=SuspendthevehiclewithstabilitywhileturningFR6=Provideturningability.

TomeetFRs,anappropriatesetofDPsaredefinedasfollows:

DP1=WheelsystemDP2=Knuckle&bearingDP3=MountingsystemDP4=IndependentlinksystemDP5=Stabilizerbar&shockabsorberDP6=Tierod&balljoint

Theassociateddesignmatrixshowsadecoupleddesignatthehighestlevelofdecomposition.

DP1 DP2 DP3 DP4 DP5 DP6FR1 X 0 0 0 0 0FR2 x X 0 0 0 0FR3 0 0 X X 0 0

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FR4 0 x x X x 0FR5 x 0 x x X 0FR6 0 x 0 0 x X

FR1andDP1arethendecomposed.

FR11=Generatefrictionforcefromtransmissionsystemtorque.FR12=Deliverforwardthrustfromfrictionforce.DP11=TireDP12=Wheel

Theassociateddesignmatrixshowsadecoupledsubsystem.

DP11 DP12FR11 X 0FR12 x X

FR2andDP2arethendecomposedandthecorrespondingdesignmatrixisdefined.

FR21=Impedethemotionofthewheelsystemthroughfivecoordinates(3translationand2rotation)FR22=Permittherotationofthewheelsystemaboutitsaxis.DP21=KnuckleDP22=Bearing

DP21 DP22FR21 X xFR22 x X


FR31=Absorbvibrationinmounting.FR32=Connectthecarbody.DP31=MountingbushDP32=Mountingbolt



FR41=Absorbvibrationandnoiseatthejunctions.FR42=Connectthemountingandwheelsystem.FR43=Allowthewheelsystemtomoveindependently.DP41=A,GbushDP42=Sub-frameDP43=Lowerarm

DP41 DP42 DP43FR41 X 0 0FR42 x X x

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FR43 x x X


FR51=Providerigidityforrollmode. FR52=Providerigidityfortheridemode.FR53=Decreaseinverticalenergy.FR54=Limittheverticaldistanceofmotion.DP51=StabilizerbarDP52=SpringDP53=DamperDP54=Bumpstopper

DP51 DP52 DP53 DP54FR51 X x 0 0FR52 x X 0 0FR53 0 x X 0FR54 0 x x X


FR61=Restrainthewheelsystemtoturn.FR62=Provideaturninginput.DP61=BalljointDP62=Tierod DP61 DP62FR61 X 0FR62 x X

1.1 1.2 2.1 2.2 3.1 3.2 4.1 4.2 4.3 5.1 5.2 5.3 5.4 6.1 6.21.1 X1.2 X X2.1 X X X2.2 X X3.1 X3.2 X X X X4.1 X X X4.2 X X X4.3 X X X X5.1 X X X5.2 X X X X X5.3 X X5.4 X X X6.1 X X X6.2 X X

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Figure14.17.CompleteDesignMatrixofanAutomobileSuspensionSystem

The entire designmatrix is illustrated in Figure 14.17. Although the suspension system hasreceivedmanydesigniterationsoveralongperiodoftime,therearemanyandmoreimportantlynon-negligiblepointsof couplingbetween theFRsand theDPs. In the early stageofproductdevelopment,thesuspensionsystemhadrelativelyfewandsimpleFRs. Overtime,moreFRswereadded,andthedesignbecamequitecomplexwithmanynon-zerooff-diagonaltermsinthedesignmatrix. Despitethis largenumber,onlythreepairsofDPs(showninredabove)havecoupleddesigns. TheyareDP2.1andDP2.2,DP4.2and4.3,andDP5.1and5.2. AlthoughthesepairsofDPs formallyviolate the IndependenceAxiom, the situation isquitemanageable. These threepairs of DPs can be treated as three DPs; with each pair being treated as a single entity. Furthermore,theDPsappearatthebottomoftheAxiomaticDesigndualhierarchy,andsotheircouplednaturedoesnot“ripple”acrossthedesignoftherestofthesuspensionsystem. Finally,productdesigncompaniescanfinditbeneficialtohavehighlycoupledcomponentsatthebottomofthedualhierarchybecausetheyeffectivelyembodyproprietaryandpotentiallysecretdesignknow-how that is not easily reproduced. That said, this special condition should not beinterpretedasalicensetoignoretheIndependenceAxiom. Theproduct’soveralldual-hierarchyshouldremainuncoupledordecoupled(asabove)andonlyifnecessaryintroducecoupledDPsattheverybottomofthehierarchy.

14.3.4MobileHarbor

Astheworld-widevolumecontainershipmentsincreasesandverylargecontainershipsemergeas a dominant player in the maritime cargo transport market, the functional capabilities ofcontainerportsneedtobegreatlyenhanced.Forexample,largecontainershipswithstreamlinehullslackmaneuverabilityintightandhighlysuggestedmaritimeportsandoftenrequiretugboatswhichmay in and of themselves be otherwise occupied. To address this problem, the KoreaAdvancedInstituteofScienceandTechnology(KAIST)isundertakingaprojecttodesignanovelcontainer transport system calledMobileHarbor (MH).A conceptual illustrationof themobileharborispresentedinFigures14.18and14.19. MobileHarborreferstoasystemthatcangoouttoalargecontainership,anchorintheopensea,loadandunloadcontainersbetweenitselfandthecontainership,and transport themto theirdestination. Ithasa flatbottomdesignso that it ismaneuverableinnarrowshippinglanesandstableenoughtohandleawidevarietyofocean-goingcargo. Likemanyotherlarge-scaleengineeringprojects,thedesignofMobileHarborpresentsanumberofchallengesatthebeginningstagesoftheproject.Intheconceptualdesignphaseoftheproject, thedesign teammustproperlydefineanddisseminate functional requirements, clarifyinterfacerequirementsbetweenitssubsystems, identifyandreconcilepotentialdesignconflictslikefunctionalcoupling.

Figure14.18.IllustrationofaRevisedMobileHarborConcept

(approvedforfundingbyanationalR&Dprogram)

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Figure14.19.TheMobileHarborasConceptualizedbyAxiomaticDesign

TobegintheAxiomaticDesign,thetop-levelFRs,DPs,andconstraintsaredefined.

FR:Transfercontainersfromshipsanchoringintheseatoaharbor.FR1=Load/unloadthecontainers.FR2=KeepthecontainersinMH(MobileHarbor).FR3=LetMHfloatonthesea.FR4=LetMHnavigateonthesea.FR5=DockMHagainstthecontainershipintheopensea.DP:Mobileharbor(MH)DP1:CranesystemDP2:DecksystemforloadingthecontainersDP3:Floatingbodystructure DP4:Drivingsystem<propulsion+navigation> DP5:Mooringsystem+Dockingsystem

Constraints C1:Thecapacityofthepowergeneratorshouldbelargeenoughtocovertheneeded energy.(Generateacertainpowerforoperating.) C2:Thetotalweightshouldbelessthanaspecifiedvalue. C3:TheMHcarriesthecontainersupto250TEU. C4:Safety,reliability,efficiency,cost-effective,environmentallyacceptable,automation,

thesatisfactionofstandardandshippingregistrationorotherrules.

AccordingtotherelationshipbetweenFRsandDPs,thedesignmatrixisdefinedasfollows:

DP1 DP2 DP3 DP4 DP5FR1 X x 0 0 0FR2 x X X 0 0FR3 x x X 0 0FR4 x 0 x X 0FR5 x 0 x x X

For illustrativepurposes, theAxiomaticDesignofDP1ispresentedindetailandthecompleteddesignmatrixoftheMobileHarborisshowninFigure19.

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DecompositionofFR1(DP1:Cranesystem)FR11:TranslocatethecontainersbetweenMHandacontainershipintheopenseaora harbor.FR12:LoadandunloadthecontainerswhileMHisonthesea.DP11:ContainercraneDP12:DynamiccontrolsystemofMHmovement


DecompositionofFR11(DP11:Containercrane)FR111:LetthecontainercraneofMHapproachtothecontainersinacontainershipor opensea.FR112:Holdthecontainer.FR113:Movethecontainer.DP111:ThesizeandshapeofthecontainercraneandmovingsystemofthecontainerDP112:Anappropriateconfigurationofthecranestructuretohavesufficientstrength DP113:Thehoistdeviceincludingatrolley

DecompositionofFR12(DP12:DynamiccontrolsystemofMHmovement)FR121:Minimizethemovementofthecontainerduetothevibrationoftheboomwhileworkingonthesea.FR122:Minimizethemovementofthecontainerduetothevibrationofthehoistwhileworkingonthesea.DP121:Zeromomentpoint(ZMP)systemforstabilitycontrolDP122:DynamiccontrolsystemofthetrolleyCs:Theconditionofseastate3(awavelevel)shouldbesatisfied.

DP121 DP122FR121 X 0FR122 x X

Themobileharborconstitutesanentirelynovellargecomplexproduct. Itisabletogoouttoashipanchoredintheopenseatoloadandunloadcargofromitwithoutoccupyingapierofaland-basedharbor. Suchafunctionalityrelievesthecargoshipfromhavingtocomeintoastationaryharbortounloadandloaditscargo. Furthermore,theincreasingsizeofcontainershipsmeansthatfewerandfewerharborshaveasufficientlylargepierandaresufficientlydeep. TheMobileHarborisalsocapableofloadingandunloadingcargotoandfromalargeshiptothestationaryland-basedharborwithoutacontainerquaybypre-loadingthecargoinfreightcarsplacedonthemobileharborwhichthemselvescanbemovedontoandremovedfromtheMobileHarbor. TheMobileHarborwasdesignedsothatitcanapproachbothsidesofacargoshipandattachitselfsecurely;furtherreducingthetimetoloadandunloadcargo. Inadditiontotheseusescases,theMobileHarborcanbeusedinlandwithinacanaltoallowthecrossingoftransportcontainersandgoods. Oneintegralpartofthedesignispositioning. NotonlymusttheMobileHarborcontrolhorizontal translation on the sea and vertical height alignment; it also uses a large gyroscopicapparatustostabilizerotationeveninthepresenceoflargewaves.

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Figure14.20.CompleteDesignMatrixoftheMobileHarborThe finaldesignmatrix is illustrated inFigure14.20. It is almostentirelydecoupled. FR2 isweaklycoupledwithFR1andFR3. Keepingthecontainersinthemobileharborhasthepotentialtoimpacttheirloading/unloadingaswellastheoverallfloatingfunctionalityofthemobileharbor. Over most conditions, these coupling are quite weak. They only represent a few designparametersandcanbegivensufficientdesignattentionupfrontsoastoallowtheremainderofthedesigntoproceed. Nevertheless,onemustrecognizethatasthedesignerspursueever-more“aggressive”mobile-harbordesignswithanever-largernumberofincreasinglyheavycontainers,thecouplingwillstrengthenandeventuallyleadtodesigninfeasibility. Indeed,thepresenceofsuchcouplingisentirelyinherenttothemobileapplicationandispredictedbyAxiomaticDesignTheory. The“DesignRangeandCoupling”theoreminAxiomaticDesignTheorystates:

Theorem20.DesignRangeandCoupling:Ifthedesignrangesofuncoupledordecoupleddesignsaretightened,theymaybecomecoupleddesigns.Conversely,ifthedesignrangesofsomecoupleddesignsarerelaxed,thedesignsmaybecomeeitheruncoupledordecoupled.

Many mobile applications ultimately suffer from such coupling because the mobility of the

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application effectively places hard constraints on the design range for space, energy, andfunctionality. Theenergyandpowercapabilitiesofmanyrobotsanddronesarelimitedbytheirsizeandweight. Sea-faringvesselsmustalsoensureflotation,andaerospaceapplicationsmustensure flight. Finally, the most recent smart city research demonstrates that as an urbanpopulationexpands,andthedemandsforwater,energy,mobilityandotherinfrastructureservicesgrow, the design and planning of the city’s infrastructure systems becomes increasinglyinterdependentwithinthecity’sconfinedgeography. Whereassuchcouplingscanprovidetheintegrateddeliveryofinfrastructureservices,theyarealsosusceptibletofailurespropagatingfromonesubsystemtoanother. Alternatively,thecity’sconfinedgeographycanbeloosenedtoavoidsuchasituation;atwhichpointurbansprawlbecomestheconcern. 14.3.5TheOn-LineElectricalVehicle

TheOn-LineElectricalVehicle(OLEV)isatypeofelectricvehiclethatwasdevelopedatKAISTtoovercomemanyoftheproblemsposedbyconventionalelectricvehicles: limiteddrivingrangeona single charge, thebattery’sheavyweight, and itshigh cost. Theconceptual configuration isillustrated in Figure 14.21. TheOLEV design team is composed ofmany designers frommanyengineeringdisciplinesincludingautomobileengineeringandelectro-magnetics.

Figure14.21.ConceptualOrganizationoftheOLEV

Theintegrationofthesemultiplefieldspresentsasignificantdesignchallenge. Consequently,thedesignwasconductedbya“designteam”andan“integrationteam”. TheirrolesandworkflowsareillustratedinFigure14.22. TheintegrationteamdefinestheFRs,atwhichpoint,thedesignteamdefinestheDPs,atwhichpointtheintegrationteamdefinesthedesignmatrixanddefinesthenew set of decomposed FRs. This alternation between the two teams realizes the zigzaggingprocessontheflyandpreventstheinadvertentintroductionofdesigncoupling.

29

Figure14.21.DesignProcessoftheOLEV

TheFRsandDPsatthetoplevelare

FR1=ImportelectricpowerFR2=Transmitelectricpowertothevehicle.FR3=Operatethevehicleusingelectricpower.FR4=Protectsystemsfromexternalloads.

DP1=ThreephaseAC370-440VDP2=InductionCouplingSystemDP3=ElectricVehicleDP4=Systemprotectiondevices

FromtheFR-DPrelationship,thedesignmatrixisdefined.

DP1 DP2 DP3 DP4FR1 X 0 0 0FR2 X X 0 XFR3 0 X X 0FR4 0 X 0 X

Throughout the design process, much effort was exerted to maintained a decoupled design. Nevertheless, thedesignmatrixaboveshowsahigh-level couplingofbyvirtueof theFR2-DP4

:Design

:SystemIntegration

Sendresultsandadviceswhicharegeneratedduringtheevaluationprocess

Definelower-levelFRsbasedonthehigherlevelDPs

Generate/Selectlower-levelDPs Investigaterelationshipsbetweenlower-levelFRsandDPs

Doesthedesigncandidatesatisfythe

Independence

Start

End

N

Refinetop-leveldeignparameters(DPs)

Refinelower-leveldesignparameters(DPs)

Sendresultsandadviceswhicharegeneratedduringtheevaluationprocess

N

Ye

Ye

30

coupling. Investigating further, this couplingarisesbecause thesystemprotectiondevicesarecoupledtothetransmissionofelectricpowertothevehicle. Thiscouplingisultimatelyinevitablebecausethesafetyconsiderationsoftransferringmanykilowattsofelectricityinductivelythroughanairgaprequiresahighlyintegrateddesignbetweentheinductioncouplingandtheprotectiondevices. Fromthishigh-leveldesignmatrix,thezigzaggingprocessiscarriedoutuntilthebottomlevelisreached.ThefinaldesignmatrixispresentedinFigure14.22. TheFR2-DP4couplingishighlightedtoalertthedesignandintegrationteamstothepotentialfordesigniterations.

Figure14.22.CompleteDesignMatrixoftheOLEV

Thisdesignexample,much like theMobileHarborexamplebefore it, serve todemonstrate theutilityofAxiomaticDesign in large systemsofunprecedented function. Theengineerson theprojectcannotrelyonotherproductsorlargesystemsforinspiration. Rather,theymustdeeplyreflectonthewhatthesystemmustachieveandconsequentlysynthesizetheassociatedFRsandDPs. Fromthere,thetwocriticalsystemsthinkingskillsdetailedatthebeginningofthechapterbecometheguidingprinciplesofforward-design: decompositionofthefunctionalandphysicalhierarchyandtheallocationoffunctiontoformthroughthezigzaggingprocess. Whenusedinconcert,theyserveastheprimarymeansbywhichadesignerordesignteamcantacklethedesignoflargefixedengineeringsystems. 14.4AxiomaticDesignofLargeFlexibleEngineeringSystems

Thusfar,thechapterhasmotivatedtheroleofdesignscienceasameansbywhichengineerscantacklethe14grandchallenges identifiedbytheNationalAcademyofEngineers. Itrecognizedthatdesignscienceprovidesengineerswithameta-problem-solvingskillsetcomposedofseveralsystems-thinkingdesign skills. The chapter focusedon twoof these: decompositionof thesystemfunctionandformasameansto“divide-and-conquer”andtheallocationoffunctiontoformin a manner consistent with the Independence Axiom. These two design skills weredemonstratedonahalf-dozendesignexamplesandinsodoingdemonstratedtheirsaliencetoawidevarietyoflargefixedengineeringsystemsinmanyapplicationdomains.

31

A curious engineeringdesign studentmaynaturally ask: “What ifwewished to design a largeflexibleengineeringsystem?” (SeeDefinition14.8atthebeginningofthechapter.).Indeed,manyof the NAE grand challenges identified at the beginning of the chapter will require enduringengineeringsolutionsthatrespondtothechangingneedsofsociety. Furthermore,theyarelikelytorequiresuchlargeinvestmentsthatitwouldbeentirelycost-prohibitivetodesignsystemsfromscratch. Thesedesignsolutionswillneedtobuilduponexistinglegacysystemsandwillneedtocontinually evolve their functionality over and over again. Unfortunately, the answer to thecurious student’s question cannot be answered comprehensively in a single book chapter, orperhapsevenbook,becauseinmanywaysitpresentsarelativelyopenfrontierforthedevelopmentofdesignscienceasanengineeringdiscipline. TheliteratureontheAxiomaticDesignofLargeFlexibleEngineeringSystemshasexpandedover theyearsandhasdevelopedtonowbecalledHetero-functional Graph Theory; given its heavy reliance on graph theoretic concepts. Nevertheless,thischaptercanservetoopenthedoortothisfascinatingrealmofdesignscienceandsystemsengineering.

TheAxiomaticDesignoflargeflexibleengineeringsystemswasfirstmentionedbySuhinhis2001text. Hepresented the followingabstractexampleofa large flexibleengineeringsystemknowledgebase:

FR1$(DPa1,DPb1,…,DPr1)FR2$(DPa2,DPb2,…,DPq2)FR3$(DPa3,DPb3,…,DPw3)

…FRm$(DPam,DPbm,…,DPsm)

wherethefirstlinemeansthatFR1(asindicatedbythe$)canbesatisfiedbyDPa1orDPb1orDPr1etc. Thesefewlinesareremarkablyprofound. First,notethattheknowledgebasedoesnotsaythatDPa1andDPb1and…andDPr1satisfyFR1. Ifitdid,itwouldbecomethedesignmatrixusedthroughouttheearlierpartsofthischapter,andDPa1andDPb1and…andDPr1wouldformeitheradecoupledor coupledengineering system. Instead, the focus is on theword “or”. Inotherwords,atagivenmomentintime,DPa1alonesatisfiesFR1. Suchastatementissimplyanotherway of stating an adherence to the Independence Axiom. Consequently, the knowledge baseabove(aspresentedintheSuh2001text)didnotallowfordecoupledanduncoupleddesigns. Insodoing, it implicitlyreachedaprofoundconclusion: large flexibleengineeringsystemsmustadheretotheIndependenceAxiomiftheyaretomaintaintheirabilitytoevolvetheirFRsandDPsovertime. Weoffer a logical proof by contradiction. In the case of an ideal uncoupleddesign (with anidentitydesignmatrix),theengineerwouldremoveaDPfromafunctioningsystem,andanentireFRwouldberemovedatthesametime. Thesystemwouldthencontinuetofunctionwithreducedfunctionality. In contrast, if a givenFRdemonstrated adecoupleddesign andwas coupled tomultipleDPs,oriftheFRdemonstratedacoupledesignandwascoupledtootherFRs,thenwhenone of the associatedDPswere removed, the systemwould demonstrate broken functionality! Furthermore,removingadditionalDPstoremoveanentiremoduleanditsassociatedFRwouldlead to downstream effects on other FRs. By Definition 14.8, such a condition contradicts afunctioning large flexible engineering system. Indeed, large flexible engineering systemsdemonstrate a “plug-and-play” functionality similar to that found inmodern computerswherefunctionalandphysicalelementscanbeaddedorremovedatwill. Theknowledgebaseabovealsoserves thedesignof large flexibleengineeringsystems for twoother reasons. First, by Definition 14.8, large flexible engineering systems have functionalrequirementsthatcanbefulfilledbypotentiallymanydesignparameters. An identitydesignmatrixdoesnotshowthis. Therefore,inordertorevealthisfunctionalredundancy1,thesetof

1 Note that functional redundancy refers to having the same functional requirement repeated. For example, FR1=generate electric power and FR2=generate (backup) electric power.

32

functional requirements instances FR must be distinguished from the set of functionalrequirementclasses 2. Second, theknowledgebaseallowsa singleDP to fulfillmore thanfunctionalrequirementclass . Theimportancefordoingsoisdiscussedlaterinthecontextofhierarchyinlargeflexibleengineeringsystems. Inbrief,theknowledgebasebecomesthesinglemostimportantconceptinthedesignoflargeflexibleengineeringsystems,inmuchthesamewaythatthedesignmatrixisimportanttothedesignoflargefixedengineeringsystems. As theAxiomaticDesignof large flexibleengineering systemsdeveloped intohetero-functionalgraphtheory,theknowledgebasewasgivenanexplicitlyquantitativedefinition. Definition14.9SystemKnowledgeBase:AbinarymatrixJofsizes( )xs(DP)whoseelementJ(w,v)Î {0,1} is equal to one when an action ewv (in the SysML sense) exists as a functionalrequirementclass wbeingexecutedbyadesignparameterDPv. Theseactionsrepresentthe“capabilities”intheengineeringsystem. Consequently,thedesignequationofthelargeflexibleengineeringsystemcanbewrittenintermsofthesystemknowledgebase.

where representsmatrixBooleanmultiplication. Definition14.10MatrixBooleanMultiplication :GivensetsorBooleanmatricesBandCandBooleanmatrixA,C=A Bisequivalentto:

where^referstothescalar“AND”operationand isan“OR”operationoverjelementsmuchlikethewell-knownsigmasumS. ThedesignequationofthelargeflexibleengineeringsystemstatedaboveinnowayreplacesthedesignmatrixB. Infact,thedesignequationstatedatthebeginningthechaptercanbewritteninaset-theoreticexpressionaswell.

wheretheaggregationoperation isdefinedas: Definition14.11AggregationOperator : GivenBooleanmatrixAandsetsBandC,C=A Bisequivalentto:

Wherethe operationisaunionoperationofjelementsmuchlikethewell-knownsigmasumS.

Functionalredundancyshouldnotbeconfusedwith“redundantdesigns”intheAxiomaticDesignofLargeFixedSystemswerethenumberofdesignparametersexceedsthenumberoffunctionalrequirements. 2 Thetermsclassandinstancearedrawnfromthefieldsofsoftware/systemsengineering. NotethatmanyworksonAxiomaticDesigndonotmakethisdistinctionbetweenfunctionalrequirementinstancesand functional requirementclassesbecause it is rarelyneededwithinasingledesignwork. Here,thedistinctionismadeinordertomaintaintheconceptuallinkbetweenlargefixedand large flexibleengineeringsystemsand theuniversalityof the IndependenceAxiom inbothcases.

Chapter 1 An Engineering Systems Introduction to Axiomatic Design 31

be analyzed [72].DP = Ap ~ DP (16)

where, again, strict mutually exclusive aggregation requires

1�(DP)T Af = 1�(DP)T (17)

Consequently,B ⇤ Ap = AfB (18)

where B is the higher level design matrix and B is the lower level designmatrix. Equation 18 may be solved when the left and right inverses of Band B respectively exist. Furthermore, when they are identity matrices, (e.g.the Independence Axiom is fulfilled), the aggregation in the functional andphysical architectures becomes the same. Af = Ap.

5.2 Large Flexible Engineering Systems

The Axiomatic Design of large flexible engineering systems was first men-tioned by Suh in his 2001 text [3] and has since been significantly devel-oped [27, 54, 55, 72–74]. Large flexible engineering systems typically requireattention at higher levels of abstraction but are otherwise similar to largefixed systems. Equation 4 describes design synthesis and Equation 11 de-scribes design analysis. Recalling Definition 27, the distinguishing featureof flexibility is achieved by a strict adherence to the Independence Axiom.Therefore, B = In, where n equivalently represents the number of designparameters or functional requirements. Conceptually, this is because a non-identity design matrix would imply that either a single design parametera↵ects more than one functional requirement or vice versa. Consequently,when it comes time to reconfigure the engineering system with an additionor removal of a functional or physical element, other changes would need tobe made as well. In contrast, adherence to the Independence Axiom, enablesa “plug & play” engineering system where functional and physical elementscan be added or removed at will [56, 71].

By Definition 27, large flexible engineering systems have functional re-quirements that can be fulfilled by potentially many design parameters. Anidentity design matrix does not show this. Therefore, in order to reveal thisfunctional redundancy, the set of functional requirement instances FR mustbe distinguished from the set of functional requirement classes FR3. A new

3 Note that many works on Axiomatic Design do not make this distinction between func-tional requirement instances and functional requirement classes because it is rarely neededwithin a single design work. Here, the distinction is made in order to maintain the concep-tual link between large fixed and large flexible engineering systems and the universality ofthe Independence Axiom in both cases.




1�(DP)T Af = 1�(DP)T (17)










1�(DP)T Af = 1�(DP)T (17)










1�(DP)T Af = 1�(DP)T (17)







32 Amro M. Farid

design equation can then be written to relate FR to DP.

FR = J�DP (19)

where J represents the system knowledge base and � represents matrixboolean multiplication.

Definition 30. System Knowledge Base [27, 54, 55, 72–74]: A binary matrixJ of size �(FR)⇥�(DP) whose element J(w, v) 2 {0, 1} is equal to one whenaction ewv (in the SysML sense)4 exists as a functional requirement FRw

being executed by a design parameter DPv.

Definition 31. Matrix Boolean Multiplication � [27, 54, 55, 72–74]: Givensets or boolean matrices B and C and boolean matrix A, C = A � B isequivalent to:

C(i, k) =_

j

A(i, j) ^B(j, k) (20)

Interestingly, it is equally valid to replace the set of design parameters in-stances DP with the set of design parameter classes DP. In such a case,Axiomatic Design addresses the design of generic or reference architectures

rather than specific, instantiated or system architectures [75–77].By Definition 27, large flexible engineering systems have functional re-

quirements that can evolve over time. To that e↵ect, the Axiomatic Designliterature introduces a system constraints matrix.

Definition 32. System Constraints Matrix [27,54,55,73,74]: A binary matrixK of size �(FR)⇥�(DP) whose element K(w, v) 2 {0, 1} is equal to one whena constraint eliminates action ewv from the action set.

A reconfiguration process is said to change the value of the system constraintsmatrix [78]. Therefore, the system knowledge base contains information onthe existence of capabilities in the engineering system. Meanwhile, the con-straints matrix contains information of their availability [77, 79]. Quantita-tively keeping track of these capabilities is done via the system’s structuraldegrees of freedom as a measure [27, 54,55,72–74].

Definition 33. LFES Sequence-Independent Structural Degrees of Freedom[27,54,55,72–74]: The set of independent actions ES that completely definesthe available processes in a LFES. Their number is given by:

DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)

4 The word “action” is meant in the technical sense of allocated functional elements inSysML’s activity diagram. See Figure 9 for details. These actions represent capabilities inthe engineering system.

32 Amro M. Farid


FR = J�DP (19)





C(i, k) =_

j

A(i, j) ^B(j, k) (20)







DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)


32 Amro M. Farid


FR = J�DP (19)





C(i, k) =_

j

A(i, j) ^B(j, k) (20)







DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)


32 Amro M. Farid


FR = J�DP (19)





C(i, k) =_

j

A(i, j) ^B(j, k) (20)







DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)


32 Amro M. Farid


FR = J�DP (19)





C(i, k) =_

j

A(i, j) ^B(j, k) (20)







DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)


32 Amro M. Farid


FR = J�DP (19)





C(i, k) =_

j

A(i, j) ^B(j, k) (20)







DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)



conceptually at a corresponding level of abstraction. For example, the designparameters can now be whole subsystems such as whole drive trains, build-ings, or organizations. This brings about the second practical issue which isin the nature of FR and DP in Equation 4. FR and DP are no longer realnumbers and so fa() is no longer well-defined as an algebraic or di↵erentialequation. In practice, designers may not know the exact impact of a givendesign parameter on a given functional requirement, and yet they must con-tinue to synthesize engineering systems in spite of this. Inevitably, this causesa profound intellectual conflict between the mathematical rigor of engineeringanalysis and the creativity of engineering synthesis. It appears most vividlyearly on in the conceptualization of an engineering system where interestinglyengineering design decisions have the greatest impact.

Axiomatic Design resolves this conflict by allowing design analysis to oc-cur, albeit with a less precise form of mathematics. At higher levels of ab-straction, early on in the conceptualization of an engineering system, FRand DP represent elements not numbers. Therefore, Equation 3 must berepresented using graph and set theory. In large fixed engineering systems,Equation 3 becomes

FR$(B ~ DP) (9)

where the aggregation operation ~ is defined as:

Definition 28. Aggregation Operator ~ [54, 71]: Given boolean matrix Aand sets B and C, C = A ~ B is equivalent to:

C(i) =[

j

a(i, j) ^ b(j) (10)

The $ in Equation 9 is often replaced with a simple = as a matter of notationalconvenience without change in the underlying meaning.

FR = B ~ DP (11)

Note that, B now comes to represent an (undirected) incidence matrix be-tween the sets FR and DP.

Definition 29. Incidence matrix [47]: M of size �(B) ⇥ �(E) is given by:

M(i, j) =

8<

:

�1 if bi is the head of arc ej

1 if bi is the tail of arc ej

0 otherwise(12)

As mentioned previously, the presence of nonzero elements in the designmatrix B, is graphically represented in SysML as an allocation of a functionalelement to a physical element as shown in Figure 9. Axiomatic Design collatesthese graphical interactions to highlight their underlying mathematical form.Furthermore, Equation 11 also implies that Equation 6 remains true andconsequently the Independence Axiom can still be applied without change.




FR$(B ~ DP) (9)



C(i) =[

j

a(i, j) ^ b(j) (10)


FR = B ~ DP (11)



M(i, j) =

8<

:



0 otherwise(12)





FR$(B ~ DP) (9)



C(i) =[

j

a(i, j) ^ b(j) (10)


FR = B ~ DP (11)



M(i, j) =

8<

:



0 otherwise(12)





FR$(B ~ DP) (9)



C(i) =[

j

a(i, j) ^ b(j) (10)


FR = B ~ DP (11)



M(i, j) =

8<

:



0 otherwise(12)





FR$(B ~ DP) (9)



C(i) =[

j

a(i, j) ^ b(j) (10)


FR = B ~ DP (11)



M(i, j) =

8<

:



0 otherwise(12)





FR$(B ~ DP) (9)



C(i) =[

j

a(i, j) ^ b(j) (10)


FR = B ~ DP (11)



M(i, j) =

8<

:



0 otherwise(12)


33

Theengineeringdesignstudentmustrecognizethatthesetwostatementsofthedesignequationareequivalentinmeaning,butmustabsolutelybedistinguishedfromeachother. TheysimplyexpresstheallocationoffunctiontoformintermsofthesystemknowledgebaseJorthedesignmatrixB. Thefinemathematicaldistinctionsbetweentheaggregationoperator andmatrixBoolean multiplication and the set of functional requirement instances FR and the set offunctionalrequirementclasses facilitatestheanalysisoflargeflexibleengineeringsystemsincomplementaryways. Example: Tosolidifytheselargeflexibleengineeringsystems,considertheexampleofthesimplemanufacturing systemdepicted in Figure 23. It consists of a drill press andmillingmachine. Theformerisabletodrillahole,andthelatterisabletodothesameandmillsurfaces. Eachcontains its respective fixture. The manufacturing system also has two one-way conveyorsbetweenthem.

Figure14.23. Asimplemanufacturingsystemthatconsistsofonedrillpress,

onemillingmachineandtwoconveyors.

Aquickanalysisofthesystemyields: FR=drillhole,drillhole,millsurface,storethepartatpointA,transportpartfrompointAtopointB,transportpartfrompointBtopointA,storethepartatB.DP={drill press,millingmachinedrill,millingmachineendmill, drill press fixture, conveyor1,conveyor2,millingmachinefixture}. Consequently, the design matrix B=I7x7. As expected, the system is uncoupled, and theIndependenceAxiomissatisfied. Tocontinuetheanalysis,thefunctionalrequirementclassesareviewedinsteadoftheirinstances. =drillhole,millsurface,storethepartatA,transportpartfrompointAtopointB,transport

partfrompointBtopointA,storethepartatB}. Ratherthanviewingthedesignparametersatthe very lowest level of aggregation, it is oftenuseful to aggregate theDPs to ahigher level ofabstraction. Inthiscase,alogicalaggregationyields: 𝐷𝑃 ={drillpress,millingmachine,conveyorsystem} Atthislevelofphysicalaggregation,thesystemknowledgebaseJisdefined.

Notethatthefirstcolumnoftheknowledgedenotesthatthedrillpressiscableofbothdrillingholesaswellasstoringthepart. Thesecondcolumnshowsthatthemillingmachineiscableofdoingbothofthosefunctionsaswellasalsomillingsurfaces. Thethirdcolumnshowsthattheconveyorsystemiscapableoftransportingpartsbackandforthbetweenthetwomachines. Inall




FR$(B ~ DP) (9)



C(i) =[

j

a(i, j) ^ b(j) (10)


FR = B ~ DP (11)



M(i, j) =

8<

:



0 otherwise(12)


32 Amro M. Farid


FR = J�DP (19)





C(i, k) =_

j

A(i, j) ^B(j, k) (20)







DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)





1�(DP)T Af = 1�(DP)T (17)







Drill Press Milling Machine

Conveyor 1

Conveyor 2




1�(DP)T Af = 1�(DP)T (17)








A large fixed engineering system analysis yields:

FR = {drill hole, drill hole, mill surface, store the part at point A, transport

part from point A to point B, transport part from point B to point A, store

the part at B}.DP = {drill press, milling machine drill, milling machine end mill, drill

press fixture, conveyor 1, conveyor 2, milling machine fixture}.The design matrix B = I7

. The Independence Axiom is satisfied.

For a large flexible engineering system analysis, the functional requirement

classes are viewed instead of their instances.

FR = {drill hole, mill surface, store the part at A, transport part from point

A to point B, transport part from point B to point A, store the part at B}.

An aggregation matrix is applied so that the drill press, milling machine and

conveyor system appear as single resources.

DP = {drill press, milling machine, conveyor system}.

J =

2

6666664

1 1 00 1 01 0 00 0 10 0 10 1 0

3

7777775=

JM | 0

JH

�(25)

That partitioning of the system knowledge base into JM and JH comes from

generalization. JM represents structural degrees of freedom that have a trans-

formational function. JH represents structural degrees of freedom that have a

transportational function.

Resource flexibility: The three resources have flexibilities of 2, 3 and 2

structural degrees of freedom respectively.

Functional redundancy: All the functional requirements have a functional

redundancy of 1 except “drill hole”.

The failure of the conveyor system would appear as two constraints in the

constraints matrix

K =

2

6666664

0 0 00 0 00 0 00 0 10 0 10 0 0

3

7777775(26)

After the failure of the conveyor system, DOF = 5.

34

three cases, the columns assigned to each of these aggregated DPs, (often called resources inhetero-functionalgraphtheory),wasgreaterthanonebecauseeachwascapableofmorethanonefunction. Conversely,thefirstrowhasasumgreaterthanonetoreflectthatthefunction“drillhole”hastwoinstancesinthesystem. Inshort,thedesignmatrixandthesystemknowledgebase give complementary insights into how function is allocated to form in large flexibleengineeringsystems. Returningtotheoriginalexpositionoflargeflexibleengineeringsystemsinthe2001Suhtext,theflexiblenaturewasemphasizedbyasetoffunctionalrequirementsthatchangedintime. @t=0 {FR}0={FR1,FR5,FR7,FRm} @t=T1 {FR}1={FR3,FR5,FR8,FRz} @t=T2 {FR}2={FR3,FR9,FR10,FRm}As theAxiomaticDesignof large flexibleengineering systemsdeveloped intohetero-functionalgraphtheory,thistime-dependentsystemfunctionalitywasquantifiedusingasystemconstraintsmatrix. Definition14.12SystemConstraintsMatrix: AbinarymatrixKofsize xDPwhoseelementB(w,v)Î{0,1}isequaltoonewhenaconstrainteliminatesactionewvfromtheactionset. Areconfigurationprocessissaidtochangethevalueofthesystemconstraintsmatrix. Therefore,thesystemknowledgebasecontainsinformationontheexistenceofcapabilitiesintheengineeringsystem. Meanwhile, the constraints matrix contains information of their availability. Quantitativelykeeping trackof these capabilities isdonevia the system's structuraldegreesoffreedomasaquantitativemeasure. Definition 14.13 Structural Degrees of Freedom3 : The set of independent actions ES thatcompletelydefinestheavailablecapabilitiesinalargeflexibleengineeringsystem. Theirnumberisgivenby:

Perhaps the last necessary topic in this introductory discussion of large flexible engineeringsystemsisthatoftheAxiomaticDesigndualhierarchy. Whenafunctionalorphysicalelementisaddedorremoved,ithasthepotentialtodisrupttheirrespectivehierarchiesaswell. Simplyaspeaking,aplaneceasestobeoneifitweretoloseawing. Anditshigh-levelfunctionofflightwouldbeimpossibleifitlosespropulsion. Nevertheless,thedesignercanproceedcautiously.

Developing the Axiomatic Design dual hierarchy for large flexible engineering systems,downwards in the direction of design synthesis, proceeds in the same way as for large fixedengineeringsystems. Thesystemisviewedintermsoffunctionalrequirementinstancesratherthan classes. Because the Independence Axiom has been strictlymaintained, each structuraldegreeof freedomcanbedesignedaspreviouslydescribedas if itwere itsownsystem. Theengineering design problem is separable. Therefore, the addition or removal of a structuraldegreeoffreedomaddsorremovesalloftheassociatedlowerbranchesinthedualhierarchy.

Itisalsousefultoconsiderthedual-hierarchyofalargeflexibleengineeringsystemupwardsinthedirection of design analysis. Here, it is no longer required to aggregate the physical andfunctional hierarchies simultaneously. It is particularly common in bottom-up design toaggregate only the physical hierarchy into higher-level design parameters (or resources). A

3 The termstructuraldegreesof freedom isappropriatelynamed. Previousworksonhetero-functionalgraphtheoryhaveshownthatitisageneralizationofthewell-knownconceptofdegreesoffreedominmechanicalsystems.




1�(DP)T Af = 1�(DP)T (17)







32 Amro M. Farid


FR = J�DP (19)





C(i, k) =_

j

A(i, j) ^B(j, k) (20)







DOF = �(E) =

�(FR)X

w

�(DP)X

v

[J K] (w, v) (21)


35

corresponding functional aggregation does not need to occur. Such was the case of themanufacturing system example above. This is because, in bottom-up design synthesis andanalysis,physicalaggregationandfunctionalaggregationdonothavethesamemeaninganddonotnecessarilyimplyeachother. Consider,forexample,fivetasksasfunctionalrequirementsandfiveindividuals as design parameters; each of whom completes one task. This a large flexibleengineering system that fulfills the Independence Axiom. The five individuals may beaggregatedintoaresourcecalledateamwithoutmakinganystatementaboutthefivetasks. Theymaynotberelatedinanyway(.i.e.shareanyfunctionalinteraction). Similarly,thefivetasksmaybeaggregatedintoaprojectwithoutmakinganystatementaboutthefiveindividualswhocompletethem. Theymay have nevermet (i.e. share any physical interface). Physical aggregation isparticularly interesting because it yields resources with many capabilities. An addition orremovalofadesignparameteryieldsthecorrespondingchangeinaresource'scapabilities. Incontrast,thefunctionalaggregationofalargeflexibleengineeringsystemmayresultinaarigidtop-down structure. Any time the set of functional requirements changes, the functionalhierarchywouldneedtochangeaswell. Inaproject,theeliminationofasingletaskcausestheeliminationoftheprojectasawhole. 14.5 Conclusion

ThischapterhasintroducedthedesignengineertotheworldoflargecomplexsystemsfromanAxiomaticDesignperspectiveandmotivatedthepressingneedforsuchdesignskillsintermsofthe14 grand challengesof theNationalAcademyofEngineers. The chapter focusedon twocritical systems-thinkingdesign skills thathelp thedesignermanage the inherent andabstractcomplexity of large systems. They are 1.) system decomposition as a means to “divide-and-conquer”,andtheallocationoffunctiontoforminwhatisoftencalledthe“zigzagging”process. These two skills were demonstrated in several tractable design case studies of large fixedengineering systems. The chapter also discusses why and how these design skills are useddifferentlywhenthesystemhasafixedversusaflexiblestructure. In distinguishing between large fixed and flexible engineering systems, this chapter has alsoopenedthedoortoafascinatingrealofdesignscienceandsystemsengineering;onethatremainsvery open frontier for methodological development. Three broad directions of enquiry areworthyofnotehere.

§ AQuantitativeUnderstandingofLifeCycleProperties§ TheTreatmentofCyber-PhysicalSystems§ Hetero-functionalNetworksinLargeFlexibleEngineeringSystems

First, the fieldof systemsengineering is increasingly concernedwithdevelopingaquantitativeunderstandingoflifecyclepropertieswhichcanbedeployedwithinengineeringdesign. Manylifecyclepropertiesarecalled“ilities”becausetheyendwiththatsuffix. Flexibility,sustainability,reconfigurability,maintainability,andinteroperabilityarebutafewthatfollowthisgrammaticalpattern. Inthemeantime,researchintosafety,quality,andevenstabilityarewell-establishedinmanyengineeringcurriculumsasindependentdesignmethods. Stillotherlifecyclepropertieslikeresiliencearerelativelynewandgarneractiveresearchinterest. These“ilities”areusuallyclassifiedasnon-functionalrequirements;whicharefurtherclassifiedasconstraintsinAxiomaticDesignTheory. Nevertheless,mostoftheselife-cyclepropertieshaveahighlyintegrativenatureand consequently their integration into formal engineering design approaches like AxiomaticDesignTheorypresentsasignificantintellectualchallenge. Second,mostlargecomplexsystemshaveacyber-physicalnature. Inadditiontotheunderlyingengineering physics, the system as a whole deploys control, automation, and decision-makingcomponentsthatmakethesystemoperateefficiently,reliably,andstably. ThesecomponentscanbeentirelyautomatedasinthecaseofPIDcontrollers,orentirelymanualasinthecaseofaircraftpilots. Furthermore, thecontrol,automation,ordecision-makingsystemcanhavecentralized,distributed, or decentralized algorithms. Figure 23 contrasts four types of cyber-physical

36

systems: a.) open-loop physical systems b.) closed-loop cyber-physical systems c.) closed-loopcyber-physicalsystemsacentralizedcontrollerandd.)closed-loopcyber-physicalsystemswithadistributed control architecture. Each of these can be modeled as a SysML block diagram,analyzedforitssystembehavior,orinspectedintermsofitsunderlyingsystemstructure. Thechallenge,here,isthatmanyclosed-loopcontrolsystemsendupcreatingcyber-physicalsystemswithcoupleddesigns. Consequently,thedesignworkflowsdevelopfeedbackloopsthatarenoteasilymanagedandpronetodesignerror;particularlyforlargecomplexsystemswithdemandingengineeringphysics. OnecanperhapsspeculatewhethertherecentnewsabouttheBoeing737MAXairplanesistheresultofthefeedbackdesignloopscausedbytheplane’sunderlyingcyber-physicalnature. Inanycase,wemustrecognizethatasthemodernworldcontinuestoautomateitstechnologies,itdesignscyber-physicalsystemsthatareincreasinglysafetycritical. Developingengineering design approaches that target the closed loop nature of cyber-physical systems isimperative.

Figure14.23. Cyber-physicalsystemsfromthePerspectivesofSysML,TransferFunctions,

andAxiomaticDesignFinally,thestudyoflargeflexibleengineeringsystemsmustcontinuetodevelopthroughhetero-functionalgraphtheory. ThreebroadcommunitiesareactivelyworkingtodevelopthemethodstotacklethetypesofsystemsthatsitattheheartoftheNAE’sgrandchallenges. Thenetworksciencecommunityhasdeployedgraphtheoryasameans toanalyze the formof large flexibleengineering systems. These works, however, are relatively divorced from the establishedunderstandingoffunctionwithintheengineeringdesignfield. Furthermore,manyofthelarge-scalearchitecturaltransformationsofthe21stcenturyappearwithinthefunctionalarchitectureorinthesystemknowledgebase,butnotinthephysicalarchitecturealone. Inthemeantime,themodel-basedsystemsengineeringhasestablishedgraphicalmodeling techniques likeSysML toapproachsystemsofarbitrarysizeandcomplexity. However,thequantitativeunderstandingoftheunderlyingsystemstructurehasremainedelusive. Inthemeantime,muchoftheAxiomaticDesigncommunityfocusesonthequantificationofsuchasystemstructure,butmostinvestigationshavebeenlimitedtolargefixedengineeringsystems. Tothenetworksciencecommunity,hetero-functionalnetworkspresentanewviewastohowtoconstructgraphswithfundamentallydifferent

Cyber-Physical System Activity/Block Diagram System Behavior Axiomatic Design

a.) Open-LoopPhysical System C0P1

1-Resource

b.) Closed-LoopCyber-Physical System C1P1

1-Resource1-Controller

c.) Closed-LoopCyber-Physical System C1PN

n-Resources1-Controller

d.) Closed-LoopCyber-Physical System CNPN

n-Resourcesn-Controller

<<allocated>>Cyber System: K1

K1

<<allocated>>Cyber System: G1

G1


K2


G1

<<allocated>>Cyber System: Kn

Kn

<<allocated>>Cyber System: Gn

G1

+-

+-

+-

act [Activity] Transform U into Y

YU


K1


G1


G1

<<allocated>>Cyber System: Gn

G1

+-

+-


YU

<<allocated>>Physical System: G1

G1


K1+-


YU

<<allocated>>Physical System: G1

G1


YU

AXIOMATIC DESIGN IN LARGE SYSTEMS: COMPLEX PRODUCTS, BUILDINGS & MANUFACTURING SYSTEMS (WORKING VERSION) 19

D. Closed-Loop Cyber-Physical System CNPN – n-Resources,n-Controller

Y =GcpU (39)

Y ⇡

2

6664

K1G1

K2G2

. . .

KnGn

3

7775

I +

2

6664

K1G1

K2G2

. . .

KnGn

3

7775

U (40)

Y = [1] ~ Gcp (41)

Y =

2

6664

1 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

3

7775~

2

6664

Gcp1

Gcp2...

Gcpn

3

7775(42)


D. Closed-Loop Cyber-Physical System CNPN – n-Resources,n-Controller

Y =GcpU (39)

Y ⇡

2

6664

K1G1

K2G2

. . .

KnGn

3

7775

I +

2

6664

K1G1

K2G2

. . .

KnGn

3

7775

U (40)

Y = [1] ~ Gcp (41)

Y =

2

6664

1 · · · 00 1 · · · 0...

.... . .

...0 0 · · · 1

3

7775~

2

6664

Gcp1

Gcp2...

Gcpn

3

7775(42)


[49] J. E. Bartolomei, “Qualitative knowledge construction for engineeringsystems: extending the design structure matrix methodology in scope andprocedure,” Massachuseets Institute of Technology Engineering SystemsDivision, Tech. Rep., 2007.

[50] J. E. Bartolomei, D. E. Hastings, R. de Neufville, and D. H. Rhodes,“Engineering systems multiple-domain matrix: An organizing frame-work for modeling large-scale complex systems,” Systems Engineering,vol. 15, no. 1, pp. 41–61, 2012.

[51] S. Ullmann, “{Semantics: An Introduction to the Science of Meaning},”1979.

[52] C. K. Ogden, I. A. Richards, B. Malinowski, and F. G. Crookshank, Themeaning of meaning. Kegan Paul London, 1923.

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[54] G. Guizzardi, Ontological foundations for structural conceptual models.CTIT, Centre for Telematics and Information Technology, 2005.

[55] J. E. Shigley, C. R. Mischke, and T. H. Brown, Standard handbook ofmachine design, 3rd ed. New York: McGraw-Hill, 2004.

[56] J. D. Sterman, Business dynamics: systems thinking and modeling for acomplex world. Irwin/McGraw-Hill Boston, 2000, vol. 19.

[57] G. Guizzardi, “On ontology, ontologies, conceptualizations, modelinglanguages, and (meta) models,” Frontiers in artificial intelligence andapplications, vol. 155, p. 18, 2007.

[58] H. P. Grice, “Logic and conversation,” in Syntax and Semantics. NewYork: Academic Press, 1970, vol. 3, pp. 43–58.

[59] M. Blaha, J. Rumbaugh, and M. BlBlaha, “Object-oriented modelingand design with UML,” pp. xvii, 477 p., 2005.

[60] Anonymous, “Synthesis,” Dictionary.com, Tech. Rep., 2015. [Online].Available: http://dictionary.reference.com/browse/synthesis?s=t

[61] A. Kossiakoff, W. N. Sweet, and Knovel (Firm), Systems engineeringprinciples and practice. Hoboken, N.J.: Wiley-Interscience, 2003.[Online]. Available: http://www.knovel.com/knovel2/Toc.jsp?BookID=1430

[62] G. A. Miller, “The magical number seven, plus or minus two: Somelimits on our capacity for processing information.” Psychological review,vol. 63, no. 2, p. 81, 1956.

[63] A. M. Farid, “Reconfigurability Measurement in Automated Manufactur-ing Systems,” Ph.D. Dissertation, University of Cambridge EngineeringDepartment Institute for Manufacturing, 2007. [Online]. Available:http://amfarid.scripts.mit.edu/resources/Theses/IEM-TP00.pdf

[64] I. S. Khayal and A. M. Farid, “Axiomatic Design Based VolatilityAssessment of the Abu Dhabi Healthcare Labor Market,” Journal ofEnterprise Transformation (in press), vol. 1, no. 1, pp. 1–20, 2015.

[65] A. M. Farid, “An Axiomatic Design Approach to Non-AssembledProduction Path Enumeration in Reconfigurable ManufacturingSystems,” in 2013 IEEE International Conference on SystemsMan and Cybernetics, Manchester, UK, 2013, pp. 1–8. [Online].Available: http://dx.doi.org/10.1109/SMC.2013.659

[66] A. Viswanath, E. E. S. Baca, and A. M. Farid, “An Axiomatic DesignApproach to Passenger Itinerary Enumeration in ReconfigurableTransportation Systems,” IEEE Transactions on IntelligentTransportation Systems, vol. 15, no. 3, pp. 915 – 924, 2014. [Online].Available: http://amfarid.scripts.mit.edu/resources/Journals/TES-J08.pdf

Amro M. Farid received his Sc.B and Sc.M degreesfrom MIT and completed his Ph.D. degree at theInstitute for Manufacturing within the Universityof Cambridge Engineering Department in 2007. Heis currently an assistant professor of EngineeringSystems and Management and leads the Laboratoryfor Intelligent Integrated Networks of EngineeringSystems (LIINES) at Masdar Institute of Scienceand Technology, Abu Dhabi, U.A.E. He is also avisiting scientist at the MIT Mechanical EngineeringDepartment. His research interests address the sys-

tems engineering of intelligent energy systems including smart power grids,energy-water nexus, transportation electrification, and industrial production.He is a senior member of the IEEE and is actively involved in the ControlSystems Society, the Systems, Man & Cybernetics Society, and the IndustrialElectronics Society.

APPENDIX

A. Open-Loop Physical System C0P1 – 1-Resource

Y = G1U (29)

Y = [1] ~ G1 (30)

B. Closed-Loop Cyber-Physical System C1P1, 1-Resource, 1-Controller

Y =GcpU (31)

Y =K1G1

I + K1G1U (32)

Y = [1] ~ Gcp (33)

Y =⇥

1 1⇤~

K1

G1

�(34)

C. Closed-Loop Cyber-Physical System C1PN – n-Resources,1-Controller

Y =GcpU (35)

Y =

K1

2

6664

G1

G2

. . .

Gn

3

7775

I + K1

2

6664

G1

G2

. . .

Gn

3

7775

U (36)

Y = [1] ~ Gcp (37)

Y =

2

6664

1 1 0 · · · 01 0 1 · · · 0...

......

. . ....

1 0 0 · · · 1

3

7775~

2

666664

K1

G1

G2...

Gn

3

777775(38)






















APPENDIX


Y = G1U (29)

Y = [1] ~ G1 (30)


Y =GcpU (31)

Y =K1G1

I + K1G1U (32)

Y = [1] ~ Gcp (33)

Y =⇥

1 1⇤~

K1

G1

�(34)


Y =GcpU (35)

Y =

K1

2

6664

G1

G2

. . .

Gn

3

7775

I + K1

2

6664

G1

G2

. . .

Gn

3

7775

U (36)

Y = [1] ~ Gcp (37)

Y =

2

6664

1 1 0 · · · 01 0 1 · · · 0...

......

. . ....

1 0 0 · · · 1

3

7775~

2

666664

K1

G1

G2...

Gn

3

777775(38)






















APPENDIX


Y = G1U (29)

Y = [1] ~ G1 (30)


Y =GcpU (31)

Y =K1G1

I + K1G1U (32)

Y = [1] ~ Gcp (33)

Y =⇥

1 1⇤~

K1

G1

�(34)


Y =GcpU (35)

Y =

K1

2

6664

G1

G2

. . .

Gn

3

7775

I + K1

2

6664

G1

G2

. . .

Gn

3

7775

U (36)

Y = [1] ~ Gcp (37)

Y =

2

6664

1 1 0 · · · 01 0 1 · · · 0...

......

. . ....

1 0 0 · · · 1

3

7775~

2

666664

K1

G1

G2...

Gn

3

777775(38)






















APPENDIX


Y = G1U (29)

Y = [1] ~ G1 (30)


Y =GcpU (31)

Y =K1G1

I + K1G1U (32)

Y = [1] ~ Gcp (33)

Y =⇥

1 1⇤~

K1

G1

�(34)


Y =GcpU (35)

Y =

K1

2

6664

G1

G2

. . .

Gn

3

7775

I + K1

2

6664

G1

G2

. . .

Gn

3

7775

U (36)

Y = [1] ~ Gcp (37)

Y =

2

6664

1 1 0 · · · 01 0 1 · · · 0...

......

. . ....

1 0 0 · · · 1

3

7775~

2

666664

K1

G1

G2...

Gn

3

777775(38)






















APPENDIX


Y = G1U (29)

Y = [1] ~ G1 (30)


Y =GcpU (31)

Y =K1G1

I + K1G1U (32)

Y = [1] ~ Gcp (33)

Y =⇥

1 1⇤~

K1

G1

�(34)


Y =GcpU (35)

Y =

K1

2

6664

G1

G2

. . .

Gn

3

7775

I + K1

2

6664

G1

G2

. . .

Gn

3

7775

U (36)

Y = [1] ~ Gcp (37)

Y =

2

6664

1 1 0 · · · 01 0 1 · · · 0...

......

. . ....

1 0 0 · · · 1

3

7775~

2

666664

K1

G1

G2...

Gn

3

777775(38)






















APPENDIX


Y = G1U (29)

Y = [1] ~ G1 (30)


Y =GcpU (31)

Y =K1G1

I + K1G1U (32)

Y = [1] ~ Gcp (33)

Y =⇥

1 1⇤~

K1

G1

�(34)


Y =GcpU (35)

Y =

K1

2

6664

G1

G2

. . .

Gn

3

7775

I + K1

2

6664

G1

G2

. . .

Gn

3

7775

U (36)

Y = [1] ~ Gcp (37)

Y =

2

6664

1 1 0 · · · 01 0 1 · · · 0...

......

. . ....

1 0 0 · · · 1

3

7775~

2

666664

K1

G1

G2...

Gn

3

777775(38)

37

meaning and insight. To the systems engineering community, hetero-functional graph theorymay come to be viewed as a quantification ofmanyof the structural concepts inmodel-basedsystems engineering. Finally, to the engineering design student interested in large flexibleengineering systems, hetero-functional graph theory presents itself as a natural extension ofAxiomaticDesignTheorywheretheIndependenceAxiomisappliedtostudysystemsofflexiblestructure. FurtherReading

14.1 National Academy of Engineering, “NAE Grand Challenges for Engineering,”

http://www.engineeringchallenges.org/challenges.aspx, National Academy ofEngineering,2019.

14.2 N.P.Suh,AxiomaticDesign:AdvancesandApplications.OxfordUniversityPress,2001.

14.3 A.M.FaridandN.P.Suh,AxiomaticDesigninLargeSystems:ComplexProducts,BuildingsandManufacturingSystems.Berlin,Heidelberg:Springer,2016.14.4 W.C. Schoonenberg, I. S.Khayal, andA.M. Farid,AHetero-functionalGraphTheory for Modeling Interdependent Smart City Infrastructure. Berlin, Heidelberg:Springer,2018.14.5 D.M.Buede,Theengineeringdesignofsystems:modelsandmethods.Hoboken,N.J.:JohnWiley&Sons,2nded.,2009.14..6 E.Crawley,B.Cameron,andD.Selva,SystemArchitecture:StrategyandProductDevelopmentforComplexSystems.UpperSaddleRiver,N.J.:PrenticeHallPress,2015.

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