γ = 1 -β 2 -1

lwys t onstnt speed nd for onstnt emitter urrent i 1 exp (jω 0 t)F e hve supposed onstnt ftor X

WFI teps in the method F F F F F F F F F F F F F F F F F F F F F F F IPQ WFP glosing frontier edges F F F F F F F F F F F F F F F F F F F F F F F IPR WFQ edding ords for lines F F F F F F F F F F F F F F F F F F F F F F F IPT WFR qrouping lines in hrnesses F F F F F F F F F F F F F F F F F F F IPU WFRFI qrouping lines in low frequenies F F F F F F F F F F F F IPU WFRFP qrouping lines in high frequenies F F F F F F F F F F F F IPV WFS smmersing the hrness in strutureD in low frequenies F F F F IQP WFSFI wking system in low frequeny ssumption F F F F F IQW WFT smmersing the hrness in strutureD in high frequenies F F F IRH WFTFI esontor pproh F F F F F F F F F F F F F F F F F F F F IRH WFU gonlusion for iex F F F F F F F F F F F F F F F F F F F F F IRT 10 BISE approach for system conception 147

IHFI ystem onstrution mehnism F F F F F F F F F F F F F F F F F IRW IHFP qme theory F F F F F F F F F F F F F F F F F F F F F F F F F F F IRW IHFQ es F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ISV 11 Systemic concepts applied to BISE 165 IIFI riniples F F F F F F F F F F F F F F F F F F F F F F F F F F F F F ITS IIFP sntrodution F F F F F F F F F F F F F F F F F F F F F F F F F F F F ITS IIFQ ex history F F F F F F F F F F F F F F F F F F F F F F F F F F F F ITT IIFR e de(nition for systems F F F F F F F F F F F F F F F F F F F F F ITT IIFS o properties F F F F F F F F F F F F F F F F F F F F F F F F F F F ITU
IIFT prom o to yerEphysil o @goA F F F F F F F F F F F F ITV IIFTFI ome more detils on the numeri side F F F F F F F F F ITW IIFTFP glssil themtis onsidered for g F F F F F F F F F IUH IIFU he pyEo' mtrix F F F F F F F F F F F F F F F F F F F F F F F F IUI IIFV fsi X onlusion for this (rst mnul F F F F F F F F F F F F IUP IIFVFI eside X nd wht out quntum mehnis nd sfety cIUP IIFVFP e simple irle for fsi F F F F F F F F F F F F F F F F F IUS Prologue fsi is for fio snspired ystem ingineeringF eg is for enfold enlyE sis of gomplex ystemsF he priniple is to develop rti(il systems eing inspired y nturl systemsD with ene(t of the eg formlismF his ook is user9s guide for fsi through egF iex is for ystems nder ixtended ensoril enlysis of xetworksF st9s softwre tht my e developed to help in system designF his ook is set of proposls grouping vrious tehniques in reltion with system engineering under the tensoril nlysis of networks formlismF ith ene(t of ll previous worksD the ide is to present trks in order to develop system engineering method le to integrte future tehnologies nd in prtiulr rti(il intelligene nd eletronilly inresed mterilsF his little ook must e seen s unorgnized set of idesD the whole eing tools helping for omplex system modelingF ystem of systemsD yerEphysil systemsD etFD ll re omplex systems in generl nd elong to the wide themti of systemiF U Chapitre 1

From classical to Bio Inspired System Engineering glssil pproh to mke system mens to strt from known prts @they were developed during previous projetA nd ssemling them in order to relize the ustomer needsF xturl systems think di'erentlyF sn order to survive in given onditions nd ontextD si prts evolve to reh the est nswer to this surviving ojetiveF he iggest di'erene omes from the pity of nturl systems to evolve ut y remining in dequy with the rest of the worldF ith yerE physil oneptD this kind of pity n e pprohed with rti(il sysE temsF ht9s the strting ide of fsiF goming form ll my previous worksD the tensoril nlysis of networks hs evolved to the tenfold nlysis of omplex systemsF he ojetive nd reson to live of the eg formlism is to filitte the estishment of the reltions etween the rel ehviours of systems nd their mthemtil expressionsF 1.1 Natural processing ht is the most remrkle with nturl systems is tht they re ll mde sed on the sme elementry riksF rumns hve ommon elements with treesF he fore of xture is de(ned in the nturl seletionF xturl seletion tests proilistilly new solutions nd retins the est onesF sn ny sesD the si olletion of elements used to mke the ojets is reltively smllF he numer of funtions n e listed X W IHCHAPITRE 1. FROM CLASSICAL TO BIO INSPIRED SYSTEM ENGINEERING " pereption Y " tion Y " resoning nd deidingF gommon ojetive is to surviveF ut depending on the environmentsD surviE ving mens vrious things nd vrious hoiesD vrious strtegiesF yften in lssil pprohesD engineers think solution efore to think innovtionF st mens tht they re minly inspired y previous experienesD of themselves or olleguesD strting from known solutions to reh n ojetiveF xture mkes the sme for prtF he strts from existing niml or plnt to mke them evolving nd rehing new versionD perhps etter thn the previous oneD perhps notF sn humn proessesD we lwys try to reh immeditely etter versionF hoing thtD we proess y the sme mnner thn 4y fthers judgement4 X we hve mjor hne to reprodue the previous system thn to imgine new oneF here is without disussion some new domins where the pprohes is rdilly new X sy @internet of thingsA systemsF sn tht seD people imgine new onepts tht never existed eforeF fut they priniplly onern softwreD not so hrdwreF sf we wnt to follow the nturl proessD we my respet two steps X IF uppress older systems when new nd etter systems pper Y PF test stohsti solutions without looking only to improve previous ones Y ou my sy tht this is not so fr from wht we do tody c efter llD the humn nd eonomi world is similr to the nturl one y the wy tht through ompetitivenessD it suppresses not optimized solutionsF st9s prtly true if we onsider the riterion of surviving s eing purely the eonomil oneF fut new riterion ppers this lst yersD imgined from long time ut tking its ple with fore from short time X the eologil riterionF o todyD the gol is not only to (nd the est eonomil solution ut to (nd the est eonomil nd eologil evolvingF ht9s lso the purpose of fsiF 1.2 An ADN for the BISE ehx is (rst of ll the key to mnufture ny living eenF st9s progrm tht gives sense nd ojetives to n ssemly of elementry prtsF sn n rti(il systemD elementry piees my e some generi funtions nd the 1.2. AN ADN FOR THE BISE II softwre my llow to use them for prtiulr ojetiveF ivolving is possile thnks to the internet networkF he softwre n hnge in order to etter use the hrdwre or for orreting some mlfuntionsF prom this point of viewD modern systems re similr to nturl onesF he vrious moleules implied in the ehx n e ssoited with the vrious prts involved in the systemF fut the sme physil susystem n e tivted di'erently thnks to the softwreF oD the ehx inludes lso the softwre omponentF end this softwre prt gives the opportunity to the element to evolveF vet9s tke the exmple of sensorF e n dispose of generi sensorD whih needs power supply nd us onnetionF his sensor n mesure vrious tegories of pressure depending on progrm emedded in the sensor miroomputerF fut we understnd tht (rst ssumption is to dispose of generi sensorF e sensor eomes n stndrd element of the olletion of the possile elements llle y ny ehxF e disern in these 0rmtions si priniples for fsiF 1.2.1

Global principles for BISE e omplex system n lwys e seen s system of systemsF st9s ll the more true for nturl systems s the smllest ell lredy emeds intelligene nd omplexityF e just hve to look to the omplexity of teri to undestnd tht from this point of viewD nturl systems re fr from rti(il systems in omplexity nd pitiesF he more impressive pity is perhps the pity of teri nd some smll living to sleep during yersD witing for dequte environment inviting them to ome k to lifeF nderstnding the omplexity of these smll elementsD we understnd lso tht systems mde of these elements re intrinsilly system of systemsF pinllyD eyond the elementry entities opertorD eh of these entities disE siptes prtly energy nd onsums prtly energyF es prt of the dissiption n e used y other entities for souresD the whole dissiption or onsumpE tion t the system level is not the simple summtion of the dissiptions or onsumptions t the entities levelF xowD tking the de(nition of irdo qF nfelie ITD yerEphysil system is system tht omines physil nd yer omponentsF he phyE sil omponents ome from nture or n e rti(il mdeEmn systemsF he yer side resends to softwres emedded in hrdwresF he question of dissiption nd soures for the yer omponents is rrely sidF IPCHAPITRE 1. FROM CLASSICAL TO BIO INSPIRED SYSTEM ENGINEERING he lssil representtion of physil omponents omes from utomE tiins nd is in simple wy written like X ẋ = F (x, u)

@IFIA

x is often lled stte vrileD u n input nd F funtion giving the stte vrile evolving depending on the input u nd the opertor F F he yer omponent is written s mhine giving outputs O depending of some inputs νD sttes η nd logil mhine k X O = k (η, ν) @IFPA goupling oth physil nd yer systems mens to (nd some ridge to exhnge informtion etween the stte vriles of the physil system x nd the inputs nd outputs of the yer systemF fut using the lssil utomtis9 writingD this ridge nnot e inluded in ommon opertorD while if we note the physil prt s X y = ζ (u, x) @IFQA y eing dul stte vrile pointing out the stte vrile x vlue through some projetionF e n note lso the yer system like X Ok (η, ν) = 0 or k (O, η, ν) = 0F hen reting oupling opertor χ de(ned y morphisms linking x nd ν X x → νD nd nother one linking O nd x X O → xF nder these ssumptions we otin two equtions ssoite in single system X

   y = ζ (u, x) + χ (O) 0 = χ (x) + O -k (η, ν)
@IFRA e ioEinspired system engineering works with entities le to hve ports onnetle through the opertor χF sn other wordsD they re yerEphysil systems nd moreD the yer side is (nely di'used in ll the system prtsF xow we rememer tht ny system is omposed of systems in fsi pprohF fut s these systems lso enlose yerEphysil systemsD fsi onept system is yerEphysil system of systems @goAF 1.3 Conclusion e hve presented in this hpter fundmentl onepts proposed for go modellingF go seem to e the more omplex kind of system tht 1.3. CONCLUSION IQ n e imgined todyD exept living onesF qlol orgns re present in these systems s well s in the living systems X nervous @intelligeneA networksD pereption networks et tutors networksF he prtiulrity of go is the pity to exhnge dt through very long distne while living systems seem to e limited to some hundred of kilometres @for our urrent knowledgeAF fut on nother side their evolving remin very wek nd they re not tody ple of reprodutionF enywyD one of their priniple interest is giving opportunity to rete formlism le to model this systemsD (rst step to elorte living systems modelsF his hpter sumits (rst presenttion of this pprohD sed on the xex method [12,13,14] invented y the uthorF e understnd esily tht the notions disussed leds to the ioEinspired engineering tehniqueF IRCHAPITRE 1. FROM CLASSICAL TO BIO INSPIRED SYSTEM ENGINEERING Chapitre 2 Dynamic and perception he purpose of this hpter is to introdue some ses or minding on system dynmi nd pereptionF e wnt to present vrious prolemsD not relly orderedD ut to show how the tensoril nlysis n e used in eh sitution efore to see its implition in more omplex prolemsF iveryody think see the worldF fut everyody feels only his own universeF st needs n inommensurle e'ort for imging wht the other feels E the uthorF 2.1 Classical dynamic e n look t xewton9s dynmi y prgmti wyF till reentlyD the equivlene etween the inerti mss nd the weighted mss ws demonstrE ted with remrkle uryF his result implies tht msse is no more n intrinsi property of n ojetD ut the ÷0ient of metri whih gives the reltion etween fore nd impulseF xewton9s eqution n e written in lssil mehnis like X

F k = d dt (m kq v q )
@PFIA sf the msse depends on timeD this reltion eomes X

F k = dm kq dt v q + m kq dv q dt
@PFPA e fore is opposited to the displementF e n mke the ssumption tht this fore n e given y -K kq v q nd tht it is ompleted y soure of IS IT CHAPITRE 2. DYNAMIC AND PERCEPTION movement f k D we otin X f k = dm kq dt v q + m kq dv q dt + K kq v q @PFQA xow if our ody identi(ed y the speeds v q wers little ondutive loopF his loop in mgneti (eld will respet the eqution X

e ν = L νµ di µ dt + R νµ i µ @PFRA
he eletromotive fore @emfA e ν is reted y the loop movement in onstnt mgneti (eld with speed v q F st omes from the mgneti )ux φ knowing X

e = - dφ dt = - d dt (xyB z )
@PFSA por onstnt (eld nd movement in the only diretion xD the emf is given y X e = -yB z v x @PFTA fy generlizingD our system is (nlly desried y the eqution X

f α = ζ αβ ω β @PFUA with X f α = f k 0 @PFVA ζ αβ = dm αβ dt + m αβ d dt + K αβ 0 αγηβ l γ B η L αβ d dt + R αβ @PFWA nd ω β = v β i β @PFIHA
hese terms n e hnged depending on the studied sesD ut the pproh remins the sme with this ojetive in (nl to dispose of tensoril eqution in hosen on(gurtion speF he hoie must mke pper the interesting vrilesD here for exmple the speed nd the urrentF fsi ssumption of the lssil dynmis is to set the time s invrintF ime remins onstnt nd thisD whtever the referentil from whih it is oservedF ht9s why we n ompute hnging tking time s refereneF st ehves like ommon rhythm to mnge the vlues of vrious vriles 2.2. RELATIVISTIC DYNAMIC IU during physil proessF row to do if this referene is no more referene c st mens tht time itself is vlue tht hnge depending on the referentil from whih it is seenF ime is no more n invrintF he unique solution is to (nd nother invrintD nother ommon refereneF 2.2 Relativistic dynamic sfD in given on(gurtion speD time is lolly invrintD the equtions in this spe re de(ned y lssil dynmiF igorouslyD suh spe perhps doesn9t existF fut if we re urte on the domins where the ssumption is vlidD this spe n exist in these limitsF ometimesD these ssumptions n work t sle level whih is no more vlid t higher sleF vet9s tke the exmple of simple trnsformerF et its mromodel levelD the system is stti nd the mutul indutne is su0ient to desrie with ury its ehviourF fut if we look with detils inside the trnsformerD we see urrents nd mgneti intertions oming from reltivisti intertions etween these urrents whih re prtiles in movementF e understnd tht reltivisti reltions n e studied one the sle of the nlysed system is well de(nedF eltivisti dynmi sks for de(ning the referentilD the sle nd its invrintF 2.2.1 Scale notion e system of eqution in one physis must e homogeneous in sleF st mens tht it must tret of vriles tht elong to the sme sleF wo vriles elong to two di'erent sles if it is neessry to integrte to go from one vrile to the otherF por exmple going from quntum mehnis to lssil mehnis needs n integrtion of the wve funtionF his doesn9t men tht it is impossile to study simultneously equtions of vrious slesF st mens only tht these equtions must e seprtedF he oupling funtions etween the vrious systems of equtions ssoited with eh sle will inlude integrtion opertorsF e onsider tht if we n write for vrile AD reltion with vrile B like X

A = ˆxk dx k B @PFIIA
these two vriles re of the sme kindD ut written in two di'erent slesF IV CHAPITRE 2. DYNAMIC AND PERCEPTION sn this seD system of equtions tht my grouped these two vriles will not e homogeneous in sleF 2.2.2 Invariant rving set the onditions for good on(gurtion speD we n try to understnd the mening of the invrintF hen we derive urrentD for exmple in the se of indutne metriF e hve X e 2 = -M di 1 dt @PFIPA his opertion is tully omplitedF he urrent i 1 elongs to nother networkD (rst repository tht we suppose to e fr wyF he derivtion opertion is performed in the reeiving networkD the seond repositoryF M is funtion tht rries the urrent nd its intertion from the (rst network to the seondF st is ler tht we re oliged to speify the nture of this funtionF es it stndsD it links two di'erent repositories without providing the slightest detil on this linkF he simplest wy to detil this funtion is to detil the intertion t the level of the eletromgneti (eldF e ould hve sked ourselves the question of how we ould pereive the urrent sine the repository P c fut this pereption itself uses n exhnge osonD prtile ple of trnsporting informtion from repository I to repository PF sn ftD the (eld is unvoidleF he urrent i 1 ompnies n eletromgneti (eld A 1 rdited in the surrounding immedite speF por the two referene systems to e di'erentD they must e trvelling t di'erent speedsF sf we hve the expression tht the repository P pereives vetor elonging to the repository ID we n trnsform the (eld emitted in the repository I to know how it is pereived in the repository PF o pprehend this prolem we n tke gin the simple prolem of the mirror nd the vehileF e onsider vehile moving in the diretion x to the speed v x F his vehile rries n xis perpendiulr to its h height seF et the top of this xis is mirror tht re)ets light emitted t the se of the xisD prllel to the ltterF he light moves in the diretion yF qive us n invrint X the speed of light or speedF he distne trvelled y the vehile nd mesured from the repository P for time t 2 is x = v x t 2 F he distne trvelled y the light in the vehile9s referene frme is hF he (rst thing is to write the expression of the invrint in oth frmes IW of refereneF sn the referene I of the vehile we hve X c = h t 1 @PFIQA sn repository PD the distne trvelled y the light is

√ h 2 + x 2 F pt X c = √ h 2 + x 2 t 2 @PFIRA
prom where X

t 2 = 1 + x 2 h 2 t 1 @PFISA
fy repling x with v x t 2 we get quite esily X

t 2 = 1 - v 2 x c 2 t 1 @PFITA
he time intervl is shorter in the oserver frme PD sttiD thn in the reE ferene frme I in motionF hih mens tht time psses more quikly in the seond repositoryF gonverselyD the distnes in the repository I in moE tion pper shorter views of the repository PF e see tht the proess for estlishing pereptions from one repository or nother is s follows X " de(ning n invrint Y " writing this invrint in di'erent referentil Y " then otining the trnsformtion reltions etween referentilF 2.2.3 An example for an electrical circuit gonsider iruit mde up of two rnhes onneted nd trversed y the sme urrent of mesh J 1 F een in eh rnh the spe of the urrents is of dimension P nd the two urrents of the two onneted rnhes re i 1 nd i 2 F en invrint to oth spes of rnhes nd meshes is powerF he power of the (rst rnh in rnh spe is P 1 = R 11 (i 1 ) 2 F his power is written P 1 = R 11 (J 1 ) 2 in the spe of the meshesF he developed power in the seond rnh is P 2 = R 22 (i 2 ) 2 or P 2 = R 22 (J 1 ) 2 F e immeditely dedue the onnetion X Ω = 1 1 @PFIUA PH CHAPITRE 2. DYNAMIC AND PERCEPTION yne the onnetion E or space change matrix E is foundD it eomes possile to rewrite the equtions of the prolem seen from the oserver9s referene frme @on this sujet we ould disuss the ft tht the spe of the meshes is for eletril iruitsD the oservtion speAF he vorentz trnsformtion thus otined trnsforms the oordintes t 1 , x 1 , y 1 , z 1 of moving origin spe into the oordintes t 2 , x 2 , y 2 , z 2 of the oservtion spe X

Λ =     γ γ c 2 v x γ c 2 v y γ c 2 v z γv x 1 + α(v x ) 2 αv x v y αv x v z γv y αv y v x 1 + α(v y ) 2 αv y v z γv z αv z v x αv y v y 1 + α(v z ) 2     @PFIVA with α = (γ -1)/v 2 D v is the vetoril speed etween the two spes nd γ = (1 -v 2 /c 2 ) -1/2 F
yne the pssing mtrix is knownD we n put n eqution in the referene I nd trnsform it into n eqution in the referene PF prom this point of viewD the trnsformtion of vorentz gives the pssge for the oordintesD ut not diretly for the speedsF sf we imgine two repositories in motion relting to the speed v long n xis xD the mtrix of trnsformtion of veloities etween the two frmes of referene is X

Λ =   γβ(u x -v) 0 0 0 βu y 0 0 0 βu z   @PFIWA knowing tht X β = γ 1 - u x v c 2 -1
he reltion etween the fores f k nd the veloities v q printed on the moE iles is diretly determined y the metri m ij F his reltion X f k = m kq v q is determined in repository I y n oserver who elongs to this repositoryF vikewiseD seond lw of the form e a = ζ ab i b governs the eletromgneti intertionsF por exmpleD it is emissions rdited y equipment emedded in vehileF his rdition is similr to tht mesured in nehoi hmer s prt of equipment quli(tion testingF xow this rditionD in prtieD we wnt to oserve it y ordinry reeivers tht re sttiD when the vehile PI tht rries them is movingF e must perform the trnsformtions X

   m µν = Λ k µ Λ q ν m kq , f µ = Λ k µ f k ζ αη = Ω a α Ω b η ζ ab , e α = Ω a α e a @PFPHA
he trnsformtions etween the strems Ω η q i q = i η nd Λ µ k v k = v µ eing the origin of the determintion of their mtriesD they re impliitly knownF sn summryD the knowledge of n invrint nd the endowment of on(E gurtion spe led to the estlishment of trnsition mtrix etween the )ow vriles of two spes tthed to di'erent referene frmesF Chapitre 3

Dening metrics

3.1 Origins he onept of metris is the entrl element of di'erentil geometryF sn generl wy we estlish system of equtions whih one n represent y the expression X

e k = ψ k (m kq • f q )
@QFIA where ψ k is funtion dependent on m kq opertors ting on f q stremsF e then look for how the numer e k evolves s funtion of the )ow f q F iither to lulte the terms X ∂ψ k ∂f q @QFPA hese lultions generte sequene of terms for eh vlue tken y k for vlue of qF his suite is used to de(ne vetorD se of vetor spe b q F pollowing the m kq opertorD the terms of b q re simple or re themselves funtionsF sf m kq is slrD then the orresponding term is m kq itselfF por exmple if we onsider vetor of funtions of dimension Q @k vries from I to QA nd three )ows exist @q lso vries from I to QAF e hve X

b 1 = ∂ψ 1 ∂f 1 ∂ψ 2 ∂f 1 ∂ψ 3 ∂f 1 @QFQA sf ψ 1 = m 11 f 1 D it is ler tht b 1 (1) = m 11 F uppose prtiulr funtion ψ suh tht ψ k = m kq • f q = m kk f k F sn tht se X b q = δ k q m kk @QFRA PQ PR CHAPITRE 3. DEFINING METRICS
he G metri of the spe de(ned y the si vetors b q is then given y X

G kq = b k | b q
@QFSA sn the prtiulr spe de(ned y the eqution QFRD the omponents of the G metri re ll equl to (m kk ) 2 nd redued to the omponents G kk D the omponents G kq , k = q eing equl to zeroF sf the elements of the metri m kk re terms of dissiptionD the term ω developed in the studied system nd given y X

ω = k (m kk f k ) 2 = G kq f q @QFTA the mgnitude of potentil di'erene V D or pulse pF ke n eletril iruitD we get X ω = (R 1 i 1 ) 2 + (R 2 i 2 ) 2 + (R 3 i 3 ) 2 @QFUA whih hs the dimension of di'erene of potentilsF sn mehnis X ω = (k 1 v 1 ) 2 + (k 2 v 2 ) 2 + (k 3 v 3 ) 2
@QFVA whih hs the dimension of pressure pulseD etF sn ny seD if we integrte ω with generlized stte vrile tht we will write x nd derive from timeD we get the power X P = d dt ˆx dxω @QFWA the powerD whih is n invrintF sn the se of eletromgnetismD the generE lized vrile is the lodD nd P = iω → iV F sn mehnis it is the integrted fore on speed X P = vω → vF = v ṗF he eqution QFW shows lerly tht given isoE)uxD the power depends diretly on ω whih ppers s wy to mesure this powerF st9s ω tht turns our vrile into power nd ω is de(ned s qudrti vlue of the G produt y the feedF pinlly our metri element is diretly given y GF

Inertia

sf the m opertor is derivtorD wht hppens to the omponents of the si vetors c e re getting X

∂ ∂f a m kq df q dt = ∂ ∂f a m kq df q dt + m kq ∂ ∂f a df q dt @QFIHA 3.2. INERTIA PS sing oisson9s rket X ∂ ∂f a d dt - d dt ∂ ∂f a = 0
@QFIIA he seond term of the eqution QFIH eomes nullF xoting ψ the toin mtrix ssoited with the funtion ψD this mtrix hs s olumn the omE ponents of the si vetorsF ixept for the existene of lossesD or euse the term ∂ f a m kq is nonEzeroD these vetors themselves eome null vetorsF pinlly the time derivtion opertors @or following hosen invrint ωA onstitute the kernel of the geometry of the funtion ψF hese re soure terms just like the e k pointersF he eqution QFI n e rewritten X e k -m kq d dω f q = ψ m kq f q @QFIPA he term m kq ∂ ω f q whih opposes the soure energy pulse of the system is lled the inerti termF st still exists in dynmi systems 1 F por exmpleD in eletronis where it orresponds to indutive termsD n eletrodynmi iruit nnot exist without indutorsF he (rst term of the eqution QFIH uses the derivtive with respet to the )ow of the root of the omponent of the metriF e hve X

∂ω ∂f a = (m aa ) 2 f a ω -1 @QFIQA nd so with digonl metri X ∂ ∂f q m ka df a dω = (m aa ) 2 f a ω ∂ ∂f a m kq @QFIRA
he term etween hook hs the dimension of omponent of ψF e will now nme this dimension impedance or impedance operatorF he dependene of the impedne @whih n e therefore mssD n eletril impedneD etFA with the )ux impliesD if it must exist generlly so s not to hve seond term null in the sene of losses in QFIP tht the spe n not e 4)t4F vet9s see the onsequenes of this oservtionF 1. We will write T k = e k -m kq d dω f q 3.3 Cartesian space and diagonality of G por G to e purely digonl nd ording to its de(nitionD the si vetors b q must hve no omponents in ommon diretionF sn other wordsD in the slr produt b k | b q only the produt exists X

G kq = δ q k b k | b q @QFISA sn this se the invrint is equl to X ω = (G aa f a ) 2 @QFITA
he preeding ondition requires tht none of the si vetors hve olliE nerity with notherF hey only hve one omponent 2 F he orresponding spe is lled 4Cartesian4F end we hve seen tht if the studied system is devoid of losses nd without dependene of the )ux impedneD this spe is the kernel of the morphism p suh tht X

F : ψ ∂ f a --→ b k @QFIUA
his kernel de(nes the soure term T k F fut the physil systems omplete this kernel y dissiption or urvtureD whih is form of dissiptionF he grtesin spe is therefore neessrily redued vision of relityD vision tht is only lol pproximtion of this relityF gonsider eletronisF he omponents E mgnetiD resistiveD pitive E hve models tht lwys deE pend on the intensity of the urrent or the voltge t their terminlsF enother ddition tht is interesting is tht depending on the movementF vet9s put ourselves in n oservtion frme of vriles x , y , z , t F e look t n eletromotive fore generted in this frme of referene X e = d dt (Li y ) @QFIVA efter hoosing geometri xes for the two referentilsD prllel nd reltive speed u y F e diret urrent i y )ows in driver portion in the frme of refeE rene oserved long the diretion yF his is n interesting prolem euse

for example

b 1 =   a 0 0   b 2 =   0 d 0   b 3 =   0 0 f  

CARTESIAN SPACE AND DIAGONALITY OF G

PU

normlly the eletromotive fore @femA generted is zero euse the urrent is onstntF he eletromotive fore is expressed in the referene frme of the oserver X e = d dt µ x y z i y @QFIWA sing the previous reltions of pssge we otin y repling the indutne y nonil expression X

e = d dγt µ γ t + u y c 2 y α(u y ) 2 y + z i y -i y u 1 -u y v y c 2 @QFPHA i y u est le ournt qu y F e otin if u y is onstnt X e = µ 1 γ γ (α(u y ) 2 y + z) i y -i y u 1 -u y v y c 2 @QFPIA
por n oserver in the sme frmeD t restD there is no emf generted y the )ow of diret urrent in the wireF yn the other hndD for n externl oserver moving t the speed u y with respet to the referentil oservedD emf exists in this moving iruitF his emf is not reted y the derivtion of the oserved urrentD whih remins zeroD ut y the derivtion of the expression of the indutneF sf we study the e funtions nd try to extrt the metriD we get onstnt vlue I e omprle to losses tht is expressed y X

I e = µ 1 (α(u y ) 2 y + z) 1 -u y v y c 2 @QFPPA
st is ler tht dispersions re ove ll eigenvetors of the xes of speF fut the eletromgneti omptiility hs singulrity from this point of view X the ommon impedne ouplingD whih is prtly resistive oupling termF sf we onsider three iruits inluding two oupledD one trnsmitter nd the other reeiverD the third eing independent suh s X

           e 1 -L 1 d dt i 1 = R 1 i 1 + R 12 i 2 -L 2 d dt i 2 = R 12 i 1 + R 2 i 2 e 3 -L 3 d dt i 3 = R 3 i 3 @QFPQA sf we ompute ∂ i x T k we (nd the se X b 1 =   R 1 R 12 0   b 2 =   R 12 R 2 0   b 3 =   0 0 R 3   @QFPRA nd the metri X G =   (R 1 ) 2 + (R 12 ) 2 R 12 (R 1 + R 2 ) 0 R 12 (R 1 + R 2 ) (R 12 ) 2 + (R 2 ) 2 0 0 0 (R 3 ) 2   @QFPSA
he losses here generte nonEdigonl metriF e sw tht the si vetors were found in the toin ψF e n write the system QFPQ in the form 3 X

e k -L kx d dt i x = b kx i x @QFPTA
wultiply eh term y the trnsposed mtrix of toin ψT X

ψT k µ e k -L kx d dt i x = ψT k µ b kx i x @QFPUA ut ψT k µ b kx = G µx @QFPVA nd T µ = ψT k µ e k -L kx d dt i x @QFPWA pinlly X T µ = G µx i x
@QFQHA he spe s desried y G is not grtesinF roweverD there is no moving repository here nd ll urrents elong to the sme nd unique referentil of the prolemF e need to e le to (nd proper repository tht mkes G digonlF vet9s note G in the form X

G =   g 1 g 2 0 g 3 g 4 0 0 0 g 5   @QFQIA 3. see O.
                           ag 1 + bg 3 = α ag 2 + bg 4 = 0 cg 5 = 0 dg 1 + eg 3 = 0 dg 2 + eg 4 = β f g 5 = 0 gg 1 + hg 2 = 0 gg 3 + hg 4 = 0 ig 5 = γ @QFQQA prom where we dedue X                        c = 0 f = 0 g = h = 0 a = g 4 α ∆ b = -g 2 α ∆ d = -g 3 β ∆ e = g 1 β ∆ i = γ g 5 @QFQRA
with ∆ = g 1 g 4 -g 2 g 3 F e n impose tht the metri e tht of the eigenE vlues to postpone the ouplings in soure termsF o X

G =   (R 1 ) 2 0 0 0 (R 2 ) 2 0 0 0 (R 3 ) 2   D = 1 ∆   (R 1 ) 2 ((R 12 ) 2 + (R 2 ) 2 ) -(R 1 ) 2 (R 12 (R 1 + R 2 )) -(R 2 ) 2 (R 12 (R 1 + R 2 )) (R 2 ) 2 ((R 1 ) 2 + (R 12 ) 2 ) 0 0   @QFQSA he eqution QFQH n e written X T µ = G µx i x ⇒ D µ σ T µ = D µ σ G µx i x = G σx i x @QFQTA tht to sy X T σ = G σx i x
@QFQUA e n look t the prtiulr se of solutions otined without inertiD ie without derivtive terms with respet to the invrint X without accelerationF hese solutions onstitute the geodesic of speF es X

ψT k µ =   R 1 R 12 0 R 12 R 2 0 0 0 R 3   @QFQVA
we (nd tht the soure term of the eqution QFPW is redued to X

T σ = D µ σ ψT k µ e k = . . .   [(R 1 ) 3 ((R 12 ) 2 + (R 2 ) 2 ) -(R 1 ) 2 (R 12 ) 2 (R 1 + R 2 )] e 1 [(R 2 ) 2 R 12 ((R 1 ) 2 + (R 12 ) 2 ) -R 1 (R 2 ) 2 R 12 (R 1 + R 2 )] e 1 R 3 e 3   @QFQWA
he given metri eqution QFQS eing purely digonl nd the solutions given y the eqution QFQU Y T eing de(ned y the previous reltionD the sis of the spe is X

b 1 =   R 1 0 0   b 2 =   0 R 2 0   b 3 =   0 0 R 3   @QFRHA
the geodesis following the three elementry inrements of mesures R 1 D R 2 nd R 3 nd the three urviliner diretions i 1 D i 2 nd i 3 F he spe is tully )t euse wherever one is on the xesD in other words whtever the urrents vlues i 1 to i 3 D the omponents of the se keep the sme vluesF sn dditionD the three diretions re perpendiulr two y twoF xote tht ny vetor

i = i k b k is )uxD like the distne ωF 3.4. THE MAGNETIC SPACE QI 3.4
The magnetic space guriouslyD it is the ustom tht engineers use the eletri (eld more often thn the mgneti (eld to model eletroni systems in generlF et when there re urrents @so )owsAD there is mgneti (eldF he eletri (eld is primrily ssoited with the lodsD nd therefore with the stte vrilesF fut (nllyD is not there trik euse the (eld is not the most relevnt c sf we tke the vetor potentil in the goulom gugeD the question of the relevne of the eletri or mgneti (eld is no longer relevntF he )ux of the mgneti (eld tht elongs to the ellulr spe of dimension P T 2 n lso e written X φ = S.B = S.∇ × A @QFRIA his (eld hs ll its legitimy sine it is y its use tht we demonstrte xeumnn9s formul of mutul indutneF st is lso through it tht one n otin the orrespondene etween the photon nd the free eletromgneti energyF fut in wht wy n this (eld onstitute the frmeworkD the geodesi mp of the eletromgneti spe c qeodesis re the strt lines tht follow prtile of mss null nd not sujeted to foreF he prtile following the geodesis is in free fllF wxwell9s (eld is otined y writing X

F ik = ∂A k ∂x i - ∂A i ∂x k @QFRPA
st is lso legitimte to hoose the potentil ovrint vetorD euse its exE pression s funtion of the urrent involves n impedneD twie ovrintF sf we strt from the REvetor 4 (c, v) Y we otin the vetor potentil y mulE tiplying the veloity v y µqGα -1 Y where G is funtion of qreen nd α funtion of the distne involved in GF he produt y the (rst term c gives the slr potentil divided y the elerityF he REpotentil vetor is therefore

( ψ c , A)F fut the REvetor is vetor v k F he
REpotentil otined is therefore of ontrvrint sis A k whih genertes F ik F he produt y its dul must led to the invrint tht is the powerF sn the se of REvetors the mehnis re simpleF sf A 0 is the soElled 4temporl4 omponent of the RA vetor @its (rst termA nd if A k re the three sptil omponentsD we hve X

A 0 = A 0 , A 1 = -A 1 , A 2 = -A 2 , A 3 = -A 3 @QFRQA 4.
See, "Theory of elds" of L.Landau and E.Lifchitz, editions MIR nd so X

F ik ← ∂ [x i ,x k ] ψ c , -A @QFRRA fy de(nitionD F ii = 0 nd for exmple F 12 = ∂ x 1 A 2 -∂ x 2 A 1 F he eletromE gneti (eld is therefore of n inherently tensor ntureF e n write X F ik =     0 B z B y E x /c -B z 0 B x E y /c B y -B x 0 E z /c -E x /c -E y /c -E z /c 0     @QFRSA
he lines of the F (eld de(ne )ow tues to whih we n ssoite n imE pedne opertor X the relutneF fetween the urrents or displements of lods nd the lines of (eld we hve reltion prtly dissiptive X

∇ βσ α F βσ = (µ αβ + σ αβ ) J β
@QFRTA nd seond eqution whih gives the emf X βσ α F βσ = 0 @QFRUA he ∇ βσ α opertor omines the prtil derivtions to give the wxwellE empère eqution while the βσ α opertor genertes the rottionl of the (eld for the (rst wxwellEprdy equtionF his seond eqution inludes the indution of hrge displement under goulom foreF roweverD oisson9s eqution is not expliitly given ut is not useful in the dynmi expression of the (eldF wore omplete expressions re villeF he seond eqution interests us more prtiulrlyD the (rst giving the link etween the prtiles nd the (eld vi the urrent density JF his seond eqution is expressed lssilly y reltions of the type X

∂E x ∂y = - ∂B z ∂t @QFRVA fut these reltions re expressed muh etter E nd in more syntheti wy E in n integrl form X ˛l(S) dS • E l = - d dt φ η @QFRWA
η is norml to the surfe SF he integrl of the eletri (eld on losed ontour results in the produt of n impedne opertor on mesh y

THE MAGNETIC SPACE

QQ urrent of meshF emong these impednes is inevitlyD s we hve seenD n indutne L whih re)ets the inerti of the iruit with respet to externl indutionsF vet Z e the rest of the impedne of the iruitD we write X

Zi + d dt (Li) = 0
@QFSHA ine there n e no lossless iruitsD Z inludes resistive portion RF pinllyD the externl (eld soures indue emf e in this iruitF pirst of ll our fundmentl eqution QFRU in the presene of these intertions with the environment eomes X βσ α F βσ = e α @QFSIA then the eqution QFSH is now written in the presene of these soures nd with only losses X

R αβ i β + d dt L αβ i β = e α @QFSPA sn sene of 4acceleration4D ie for X L αβ d dt i β = 0
@QFSQA @this is my de(nitionAD remins e α = R αβ i β F he si vetors of the dissiE ptive spe re the geodesis of spe here grtesinF his is our previous disussionF smgine generlized vrition @whtever the size of the spe onsideredA of the indutne s funtion of the urrent X

d dt L αβ i β = ∂ ∂i σ di σ dt L αβ i β @QFSRA riting X Ω αβ,σ = ∂ ∂i σ L αβ
@QFSSA term is dded to the usul expression of the indutne in the funtion of the eqution QFI whih is X

Ω αβ,σ di σ dt i β
@QFSTA hen we derive this term in serh of si vetorsD we otin X

∂ ∂i η Ω αβ,σ di σ dt i β = ∂ ∂i η Ω αβ,σ di σ dt i β + 0 + Ω αβ,σ di σ dt δ β η @QFSUA
sf the dependene of the indution s funtion of the urrents is of the (rst orderD it remins s omponent of the si vetors X

∂ ∂i β Ω αβ,σ di σ dt i β = Ω αβ,σ di σ dt @QFSVA
whih hs the dimension of n impedneD sine it is twie ovrintF sn this se we hve geodesisD even in the sene of resistive lossesF smgine purely mgneti spe @iruitA in medium where only indutnes susistD with lossless eletron plsmsF he eqution QFPT would eome of the form X

e α -L αβ d dt i β = Ω αβ,σ di σ dt i β
@QFSWA e must speify the term L of this equtionF sf there re networks of induE tors mde of mgneti mterilsD there re neessrily mutul indutnes etween these elementsD however wek they my eF sf these intertions tke ple in vuumD the mutul indutnes re expressed y X

L αβ = μ0 ˆxα ˆxβ dx α dx β d(α, β) G αβ @QFTHA
with μ0 = µ 0 4π et G qreen9s funtionF yn the other hndD the derivtion of these mutuls s funtion of the mesh urrents is zeroF sn ft the metri is redued to terms (Ω αα,α ) 2 sine we suppose tht the intertions re in vuumD so priori without intensity dependene of the urrent @we will review this hypothesisAF he spe eing rtesinD

ψ T = √ G nd T σ = ψ T α σ (e α -L ασ i σ ) ≡ Ω α σ,σ di σ dt (e α -L ασ i σ ) @QFTIA
his grtesin spe is then ompletely de(ned y the eqution X

T σ = G σα i α
@QFTPA fut wht re the geodesis c eording to the de(nition tht s proposeD we n not determine them y the eqution QFTP euse the nelltion of ll the terms in time derivtive leds to the nullity of the two memersF et just invert this eqution nd write X

i α = y ασ T σ ↔ i α = R ασ φ σ @QFTQA

INDEPENDENCE OF MUTUAL INDUCTANCES IN VACUUM WITH RESPECT TO CURREN

with y ασ = [G σα ] -1 et R the medium relutneF feuse the soure term T neessrily inludes the derivtions of the mgneti )ux φF he eqution QFTP n e rewritten s funtion of the mgneti )uxF e n rewrite this eqution in its ovrint formF e only need to use the inverse of the relutne nmed 4permeance4 U nd X φ σ = U σα i α @QFTRA he si vetors re then de(ned y permene nd the metri is de(ned y digonl mtrix ontining the squres of permenesF fut in this new representtionD it is no longer the emf tht re geometrilD ut the mgneti )uxesF es the reltion etween the )ux nd the emf is derivtion opertorD working on the )ux llows to extrt it nd t the sme time to de(ne nonE zero metriD even in spe without lossesF roweverD it is lwys prtiles tht generte the )owF st is therefore under the ondition tht these prtiles do not use losses in their displements tht this dynmi n e envisgedD whih onstitutes very prtiulr seF 3.5 Independence of mutual inductances in vacuum with respect to currents e mutul indutne in the lssil sense is given y xeumnn9s reltion nd is independent of urrents X

M = µ 4π ¨s1 s 2 ds 1 • ds 2 r 12
@QFTSA vooking t ny urrent i on length x s lod q t the speed vbD we onsider urrent qvb in iruitD itself in reltive speed u with respet to seond oserver iruitF he speed of the v c hrge pereived y the oserver is given y X

v c = v + u 1 + v • uc -2
@QFTTA he prtile rdites (eld µqv c G(r)D G(r) eing the funtion of qreen for distne r etween the soure nd the oserverF fut we n lso diretly lulte the (eld in the oserver9s frme of refereneF sf Λ is the repository QT CHAPITRE 3. DEFINING METRICS hnge mtrixD with for exmple X

Λ =     γ 0 0 -βγ 0 1 0 0 0 0 1 0 βγ 0 0 γ     @QFTUA @β = u/c et γ = 1 -β 2 -1
AF fy de(nitionD we hve the F (eld in the given oservtion frme s funtion of the F (eld in the oserved repository given y X

F µν = Λ σ µ Λ η ν F ση
@QFTVA e n de(ne A with X F ση = A σv x η i v F sf the emf e in reeption re de(ned y X

e α = d ds S ση α A σv x η i v @QFTWA he mutul indutnes re de(ned y X e α = M αv d ds i v ⇒ M αv = S ση α A σv x η
@QFUHA where the mutul indutne M αv depends on the speed of the referene of the hrgesD ut not on the speed of the hrges themselvesF es long s the u speed is onstntD the prolem is speil reltivity prolem nd the mutul indutne retins its intrinsi propertiesF yn the other hndD if the speed u is vrileD the derivtion of the A (eld is nonEzero nd we hve X

e α = S ση α A σv x η d ds i v + S ση α x η d dx i dx i ds A σv i v @QFUIA ut X d dx i dx i ds A σv = 1 s Γ σv,i x i
@QFUPA he oupling n then e de(ned more rodly y the opertor X

M αv = M αv d ds + S ση α x η 1 s Γ σv,i x i
@QFUQA the term dded in the generlized reltivisti frmework depends on the vrition of speed in the expression of the (eldF e ould ll 4Coriolis' Mutual4 5 this seond termF his term is used s the term of the metri omponent in the oservtion frmeF 5. This term was used for the rst time by Gabriel Kron.

Chapitre 4

Exploration of various cases eny modern system of systems is mde of eletronisF he gol of this hpter is to give some illustrtion of how the tensoril nlysis of networks n solve eletroni iruitsD even with quntum propertiesF e nnot over ll the possile intertionsD ut the exmples studied should give ll the mterils nd ides to solve ny other one in the sme (eldsF

EMC & functional EMC

st is ustomry to ll funtionl iwg @funtionl eletromgneti omE ptiilityAD the eletromgneti omptiility etween the funtions emedE ded in the sme systemF es the system is ommonD we n dmit tht in the vst mjority of sesD the eletronis elonging to this system re in the sme frme of refereneF he prolem of the eletromgneti omptiility of eletronis in di'erent referene systems ove ll points to eletronis in seprte systemsF 4.2 Fixed transmitter and mobile receiver e imgine (xed emitterD with f 0 rrierF his emitter is positioned t the oordintes (x e , y e , 0) in referentil RF e vehile with wired link is t the oordinte point (x v , y v , 0) in R referentil in uniform retiliner motion t the speed v x with respet to RF he (eld rdited y the ntenn is pereived modi(ed y the reeiverF e must (rst lulte the (eld in the R referentil t the ling levelF vet9s sty here in lssi settingF sf we QU n identify in the soure referentil smll urrent segment long the three diretions of RD let9s look t how the mgneti (eld from these segments is trnsformedD nd seen from R F ht isD speed etween the two frmes ssumed in the only diretion xF he trnsformtion of the (elds is given y

1 X      B x = B x B y = γ B y + β Ez c B z = γ B z + β Ey c @RFIA
sf our urrent segment in R is long the z xisD the mgneti (eld hs two omponents following x nd yD hene B z = 0F fut if we re in frE(eld intertionD

E z = cB y nd    B x = B x B y = γ (1 + β) B y B z = 0
@RFPA he oupling funtion for loop loted in R with norml in the y diretion is expressed here y X

e 2 = -χ v z v d dt B y = -χ v z v d dt γ (1 + β) B y @RFQA
sf the speed is onstntD only the derivtion of the (eld reminsD with X

B y = µ i 1 2πd(t) δ d(t) c @RFRA
where d(t) is the meitterEreeptor distne depending on time X d(t) = (v x t) 2 + ∆ 2 y D ∆ y = y v -y e F pinlly the oupling funtion e 2 (i 1 ) is given y X

e 2 = -χ v z v d dt γ (1 + β) µ 2πd(t) exp -jω 0 d(t) c i 1 (t)
QW e (nd (rst term from the derivtive X

T = -jω 0 v x t 2 cd 2 (t) e -jω 0 d(t) c @RFUA seond one X U = - v x t 2 d 3 (t) e -jω 0 d(t) c @RFVA nd third one X V = 1 d(t) e -jω 0 d(t) c @RFWA pinlly X e 2 = -α (T + U ) i 1 + V d dt i 1
@RFIHA o ompre with wht this oupling funtion would hve een in the sme repository X

M 0 21 = -χ v z y µ 2πd 0 exp -jω 0 d 0 c d dt @RFIIA sf β is smll enoughD γ → 1F hen X M 0 21 = - α d 0 exp -jω 0 d 0 c d dt @RFIPA if v x = 0D U = 0 nd T = 0 Y y hoosing the speil vlue of d(t)D d 0 we get M 21 = e 2 /i 1 = M 0
21 F his equlity in symptoti limit tends to reinfore the reltivisti resultF xow let9s nlyze the resultF e wnt to ompre X

   M 0 21 = -αG(d 0 ) d dt (•) M 21 = -α (T + U ) (•) -αG(d(t)) d dt (•)
@RFIQA he funtion vetor relted to the trnsmitter E reeiver intertion tht we onsider is given y X

           e 1 = r + L∂ t + 1 C ´t dt i 1 -´t dt M 0 21 M 21 ∂ t i 2 0 = -´t dt M 0 21 M 21 ∂ t i 1 + (r + L ∂ t ) i 2 @RFIRA
(r, L, C, r , L ) re the omponents of the equivlent digrms of trnsmit nd reeive ntennsF prom the system RFIR we dedue the metri tht deE pends on the dissiptions nd ouplings y mgneti (eld in the only moile se X

G = r + t C 2 + α 2 (T + U ) 2 -α (T + U ) r + r + t C -α (T + U ) r + r + t C (r ) 2 + α 2 (T + U ) 2 @RFISA he metri G o in the stti se eing X G o = r + t C 2 0 0 (r ) 2
@RFITA he di'erene etween the moile se nd the stti se pper lerlyF he moving metri hs extrEdigonl terms while the stti metri is puE rely digonlF he moving spe is urviliner where the stti spe is )tF snertiD on the other hndD is not fundmentlly di'erentF sn the stti se X

L o = [ψ] T L -αG(d 0 ) -αG(d 0 ) L @RFIUA nd in the dynmi se X L = [ψ] T L -αG(d(t)) -αG(d(t)) L @RFIVA
ividently the evolution of the distne in the intertion hnges the inertiD ut its expression remins identilF his mens tht under the onditions we hve hosenD the response to eletromgneti indution is intrinsi nd does not depend on motionD whih seems logilF row do we interpret the hnge in metris nd wht re the onsequenes c 4.2.1 Metric of a dynamic system e n not lulte the metri of system mde of two odies in reltive motionF yn the other hndD we n projet the ojet in motion into the oservtion frme to trnslte its trnsformtion nd to e le to reson with single frme of refereneF he given metri eqution RFIS hs extrE digonl termsF sf the metri is purely digonlD the distne s etween two points in the ψ plne is given in the stti se y X

s = G o 11 i 1 i 1 + G o 22 i 2 i 2 + 2G
i 1 i 1 + G o
22 i 2 i 2 F his distne hs the dimension of potentilF his mens tht the inrese in ny one of the diretions i 1 or i 2 mkes it possile to trverse the pln in ontinuous nd regulr wyF fy snning ll the possile vlues of i 1 for eh vlue of i 2 we over the ψ plnF xow onsider the seond metriF e distne in this new pln is given y X

s = G 11 i 1 i 1 + G 22 i 2 i 2 + 2G 12 i 1 i 2
@RFPHA roweverD the expressions of the previous metris do not predit possile urvtureF he (gure RFI shows the surfe desried y the stti metri G o nd the (gure RFP tht desried y the dynmi metri GF yn the digonlD 2D so smller mplitude on the digonl thn on the edges for the sme vlues of x nd yF sn the seond seD for the sme oordintesD the digonl distne is identil nd is equl to HFUF his mens thtD in generlD dding 2xy to the seond metri rises the digonl t the edges nd the surfe desried is )tF sn the (rst se it is onveF fut the urvture of the urviliner surfe is not expressed y its ppE rent geometri urvtureF sn the two previous sesD wherever we re on the ψ o nd ψ surfesD the metri remins the smeF he lol rte of inrese of the potentil is onstnt on the whole surfeF sn the (rst seD if we shre the ville energy in oth diretions @mening etween the two unoupled iruitsAD the resulting potentil will e deresedF sn the seond seD it is mintinedD the prtition eing redistriuted y the ouplingF row to exploit this informtion c he urrents i x n e generted y soures pplied to the meshes tht rry themF fy studying the surfe ψ we n detet whether exittions on ertin xes generte signi(nt inreses in potentilsF e indiretly see the oEspe e k F hese onditionsD in the se where these soures would e undesired souresD re oviously to e voided nd these mxim on the surfe ψ re regions to void exept tht the system supports these mxim of onstrintsF 4.3. EXPLORING RISK AGAINST AN EXTERNAL FIELD IN A SYSTEMRQ 4.3 Exploring risk against an external eld in a system e reurring prolem in iwg is to de(ne the levels reported on the eletroE nis y the presene of (eld in the environment of the studied systemF yne solution is to provide us with ontrol line elementD whih we will ple t di'erent points of the system nd with di'erent lodsD under the ssumption tht the line is not signi(ntly di'erent from ple to the otherF enother prolem tht is not simple is the ft tht the links n e di'erentil or ommon modeF hese re di'erent strtegies for hrdening the systemF 4.4 Analysis of a transistor oscillator en osilltor is relxtion if single lterntion tends towrds stle stte nd the osilltion omes from the threshold ontrol of stge reE ting the dul lterntionD itself then ontrolling the stge reting the (rst lterntionD etF pinlly the osilltion results from opposite ommnds of non osillting struturesF gonverselyD we n mintin nturl osilltor with losses like ny rel osilltorD the mintenne tking re to ompenE ste these losses to mintin the osilltion to in(nityF hen n osilltor is servied nd strtedD the osilltion mplitude initilly inreses until nonE liner proess is involved whih will stilize this mplitudeF st is minimum of lossesD or the presene of level nonEliner mterilD etF e wnt to study here prtiulr struture inspired y nother one lled 4oyer4 nmed fter its inventorF his is the si shem shown in (gure RFQF e immeditely understnd from the view of this digrm tht the osilltion is relxtion sine eh iruit onneted to swith @n xx trnsistorA n not e intrinsilly osilltingF wodify this struture y dding pitor to rete nturl osilltor (gure RFRF st is this struture tht we wnt to studyF he (rst step for this gol is to develop the model for xx trnsistorsF he priniple is to tivte the urrent in the oil I for exmpleD the trnsistor P eing lokedF I pssingD the trnsplnting y the trnsformer indues on the ses of the trnsistors ddp whih loks I nd mkes P pssingF he urrent is reversed in the windingsD etF hen we dd pitorsD the purpose of the swithes is to mintin the osilltion of the vg iruits formed y the trnsformer windings nd the pitorsF First model of transistor en xx trnsistor is hrterized y olletion of urves I c = f (V ce ) whih depend on the urrent )owing in the seEemitter juntion of the trnsistorF he lw giving the urrent in the olletor s funtion of the voltge ross the trnsistor V ce is of the form X

I c = -I cs + αV ce + T cb 1 -e -Vce/V O V be D >0 -I cs V be D <0 @RFPQA he oe0ient I cb is funtion of i b si urrent X I cb = γi b Y nd α is side tht lso depends on i b X α = σi b F
ζ = L 1 p + R 1 + R 2 0 0 L 2 p + R 3 @RFPRA
fy de(ning the ondition α = 0 @hrteristi urve RFUAD the olletor urrent n e written X

i c = -I cs + γi b @RFPSA with γ = β 1 -e -Vce/V 0 V be D >0.7 F he system of equtions to solve is given y X    (L 1 p + R 1 + R 2 ) i b = E b (L 2 p + R 3 ) i c = V ce @RFPTA
sn opertion we n initite si urrent to strtF efter whih the olletor urrent ppersD depending on the se urrent nd the olletor E emitter voltgeF his voltge is then updted ording to the new olletor urrent vlueD etF he olletor urrent is onsidered soure pir urrent soureF his orgniztion hs the disdvntge of hving to estimte the voltge V ce in two steps to hek if the juntion is in sturtion nd to djust the vlue of the olletor urrent whih depends on the voltge V ce F e n look t the funtion of the xx juntion in notherD less usul wyF he (gure RFV shows the returned urve of the previous i c (V ce ) hrteristiF 4.4.2

Another macromodel attempt hen the trnsistor is reverse isedD the olletor E emitter juntion is equivlent to very strong resistorF he preeding urves hve for eqution X

V ce i b , i c = -V 0 ln 1 - [i c + I cs ] βi b @RFPUA
he reltionship etween the voltge developed ross the trnsistor nd the urrent in its olletorEemitter juntion is n opertor z t prmetrized y the se urrent X

V ce = z t (i c )| i b F
vet9s lulte iruit equipped with trnsistorF iither simple pir of meshes of equtions X

         e b -V be0 V be D >0.7 = (R b + R e ) V be D >0.7 + R bb V be D <0.7 i b E DC = [z t (i c )| i b + R c i c ] V be D >0.7 + [R cb + R c ] V be D <0.7 i c
@RFPVA E DC is ontinuous dietF row to solve suh system numerilly c st is rther pinful euse it is neessry to ppel to methods y pproximtions nd onvergenesF sn dditionD the logrithm funtion does not drift wellF st is etter to (nd mromodel tht llows to ondut the lultion diretly if it is possileF por tht we re inspired y the grph presented (gure RFWF

z =   R b 0 0 0 R j 0 0 0 R c   @RFPWA
eording to the shemeD the onnetion etween rnh urrents nd the

K 1 mesh or imposed J c is X C =   1 0 1 0 0 1   @RFQHA
fy lulting the trnsformtion C T zC we (nd the tensor ζ suh tht its mtrix is X

ζ = R b + R j 0 0 R c @RFQIA
nd for the soures X

C T e = e b e c + V ce @RFQPA he system of equtions to e solved is then X

           e b = (R b + R j ) K 1 V ce = e c + R c J c J c = βK 1
@RFQQA tht we n reorder this system to ring up the resolution lgorithm nd introdue the se juntion threshold voltge V beo s well s the olletor ondution ondition X

             K 1 = (e b -V beo ) (R b + R j ) -1 J c = βK 1 K 1 D >0 V ce = e c + R c J c
@RFQRA e progrm the sequene of equtions inluding the threshold voltge V beo of the juntion for smple index t nd domin of omputtion of N smples for N/T periods X

" e b (t) = 1 + 1 2 Sin 2π t.dt T e -t N " K 1 (t) = e b (t)-V beo R b +R j " J c (t) = βK 1 (t) K 1 D >0 " V ce (t) = e c + R c J c (t)
he result otined with this progrm is given in (gure RFIHF Figure RFIH ! veforms otined vet9s tlk out these resultsF et the eginning of the wveform for e b the level is lwys higher thn the diode voltge V beo F he se urrent follows the pplied ontrol voltge with pek vlue of pproximtely PFS meF he olletor urrent follows the si ontrol with pek mplitude of PPFS meD whih is urrent mpli(tion of out IHF hen the ommnd dereses nd does not systemtilly exeed the si thresholdD the ommnd is nE eled nd the output urrent is neledF he resulting shpe then eomes deformedD pping t H e minimum vlueF he grph given in RFW thus represents the opertion of n xx trnsisE torD exept tht it does not reprodue the sturtion t the olletor if the SI olletor urrent is greter thn the ville urrentF e must dd domin to trnslte this sturtionF e write X

J c (t) = βK 1 (t) K 1 D >0 K 1 D < ec βRc + e c R c K 1 D >0 K 1 D > ec βRc @RFQSA
his 4empiril4 model fils to ount for the vlue of the si resisE tneF e see tht the development of model from urves without further knowledge of the detil of the funtioning of the omponent t the underlying sle is di0ultF vet us tke miroeletroni pproh ut try to dpt the formlism to the desription onstrint of the omponentF 4.4.3

Ebers-Moll's model he iersEwoll model is sed on the desription of the trnsistor s set of diodesF his pproh desries the urrent s funtion of the voltge developed ross the diodeD knowing its sturtion urrent nd this voltgeD supplemented y the gin of the trnsistorF ememering the importnt prmeter X

kT q = α = 26mV
@RFQTA he emitter urrent for n xx trnsistor is de(ned y X

i E = I es e V be α -1 @RFQUA unowing i E we dedue i C X i C = β β + 1 i E @RFQVA nd the se urrent y X i b = i C β @RFQWA
hepending on the polriztion of the trnsistorD we dedue if the trnsistor is sturted or linerF he oupling of the xx trnsistor is expressed in dE mittneF e must therefore ouple equtions in dmittne nd impedneF his work hs een studied in generl wy to keep oherene nd form

i C = β 1 + β I es e V in α + 1 @RFRHA
his single eqution my su0e to express the osilltor iruit with urrent soure determined y the seEemitter voltgeF ypilly I es = 10 -14 F st remins to express the sturtion of the trnsistor if the emitter urrent lled y the se voltge eomes greter thn the ville urrent of the iruit X

i C max = V c -V cesat R L @RFRIA with V cesat ≈ 0, 2F e omplete this model X i C = 0, 99 I es e V in α + 1 1 -e -i C i C max @RFRPA
e n now resume the digrm of the osilltor on the previous sis whose limits nd dvntges we msterF

X RFRFRF z =                 L 2 s 0 0 0 0 0 0 -M 18 s 0 0 0 R b + R j 0 0 0 0 0 0 0 0 0 0 R 0 0 0 0 0 0 0 0 0 0 0 L 1 s 0 -M 46 s 0 0 0 0 0 0 0 0 R c 0 0 0 0 0 0 0 0 -M 64 s 0 L 2 s 0 0 0 0 0 0 0 0 0 0 R b + R j 0 0 0 -M 81 s 0 0 0 0 0 0 L 1 s 0 0 0 0 0 0 0 0 0 0 R c 0 0 0 0 0 0 0 0 0 0 1 sC                 @RFRQA
here re two ouplings through the trnsformers X one etween the rnhes I nd VD nd one etween the rnhes R nd TF o otin the expression of ζD the impedne opertor in the mesh speD we ompute the reltion C T .z.C whih gives the mtrix shown eqution RFRFRF

ANALYSIS OF A TRANSISTOR OSCILLATOR

SU ζ =       R j + R b + L 2 s 0 M 18 s 0 -M 18 s 0 R j + R b + L 2 s -M 46 s -M 46 s 0 M 81 s -M 46 s 2L 1 s + 1 sC L 1 s -L 1 s 0 -M 46 s L 1 s R 0 + R c + L 1 s R 0 -M 81 s 0 -L 1 s R 0 R 0 + R c + L 1 s       @RFRRA SV CHAPITRE 4. EXPLORATION OF VARIOUS CASES he tensoril eqution e k = ζ km K m is ompleted y the equtions X        J 1 = βK 1 K 1 D K 1 >0 J 2 = βK 2 K 2 D K 2 >0 @RFRSA he soure vetor e k is given y X e k = -V beo K 1 D K 1 >0 -V beo K 2 D K 2 >0 0 V 1 ce V 2 ce @RFRTA nd R j = R jp K 1 D K 1 >0 + R jb K 1 D K 1 <0
@RFRUA @the sme for R j on K 2 AF he prolem is ompletely solved using the eqution e q = ζ qm K m F his is not relly di0ulty in time domin (nite di'erenesF ht is more interesting is to study the in)uene of the lod on the osilltor ehviourF st mens to omputeD knowing

e b = ζ ba K a X ∂ζ ba ∂R c @RFRVA ∂ζ ba ∂R c =       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1       @RFRWA nd so in tht se X ∂ζ ba ∂R c K a = ∂ ∂R c e b ⇒ ∂ ∂R c e b = δ bc J c @RFSHA
δ eing the uroneker symolF st mens here tht the iruit depends on R c only for the reltion with the imposed urrent souresF R c doesn9t 'et here other omponents of the prolemF Chapitre 5

Light, information & nervous transmission system eny informtion network n e modelled using the tehnique of the gamma matrix 1 F vet9s tke n exmple to illustrte this tehniqueF e imE gine hnnel mde of (rst frontier etween soure nd some propgtion mediumF his medium is losed t its end y mirrorF e work in the optiE l dominF yn the left nd on the right of the frontier we n nme optil wves on the left @index gA nd on the right @index dAF e hve propgtion wves identi(ed with symol p nd kwrd wves identi(ed with σF he inident wve on the frontier I is p g 1 on the left sideF e wve n e reE )eted from this inident power σ g 1 D nd nother wve n e trnsmitted on the other side of the frontier p d 1 F his trnsmitted wve propgtes nd reh the left side of the extremityD p g 2 F e prt n e re)eted on this limit σ g 2 F enother prt n e sored y the limit ondition p 2 F pinlly the prolem hs six oservlesF e n de(ne wve vetor v tht hs for omponents the vrious oservles X

v = p g 1 σ g 1 p d 1 σ d 1 p g 2 σ g 2 @SFIA the (
γ =         0 0 0 0 0 0 r 1 gg 0 0 t 1 gd 0 0 t 1 dg 0 0 0 0 0 0 0 0 0 0 e -x 0 0 e -x 0 0 0 0 0 0 0 r 2 gg 0        
@SFPA e n now look t wht ppends if we pply γ to v X γvF et the eginningD the vetor v hs for only not zero omponent the soure p g 1 F o X v = 1 0 0 0 0 0 @SFQA sf we ompute γv we otin X v = 0 r 1 gg t 1 dg 0 0 0 @SFRA es the soure hs normlized mplitudeD the wve mplitude tht n e mesured on port P @r 1 gg A is the re)eted prt of the inident wveF he rest of the power is trnsmitted through the frontier in the propgtion mediumF xow let9s tke look to wht ppends if we pply two times γ X γγv = 0 0 0 0 t 1 dg e -x 0 @SFSA here e -x is delyF he trnsmitted wve on the (rst frontier is propgted until the seond frontierD with dely x @x is something like -x/csAF e pply now three times γ X γγγv = 0 0 0 0 0 r 2 gg t 1 dg e -x @SFTA 5.1. FROM CLASSICAL TO QUANTUM DYNAMICS TI the wve n go k depending on the re)etion ÷0ient r 2 gg D to propgte until the input of the hnnel mking γγγγv X γγγγv = 0 0 0 r 2 gg t 1 dg e -2x 0 0 @SFUA he previous wve is now loted on the right of the (rst frontierF vet9s multiply y γ one more time X

γ 5 v = 0 t 1 gd r 2 gg t 1 dg e -2x 0 0 0 0
@SFVA he wve is trnsmitted to the mesurement on port PF fut it lks someE thing X prt my e re)eted to the mediumF st lks ÷0ient r 1 dd in our mtrixF e hve now X

γ =         0 0 0 0 0 0 r 1 gg 0 0 t 1 gd 0 0 t 1 dg 0 0 r 1 dd 0 0 0 0 0 0 0 e -x 0 0 e -x 0 0 0 0 0 0 0 r 2 gg 0         @SFWA
nd the wve vetor eomes X

γ 5 v = 0 t 1 gd r 2 gg t 1 dg e -2x r 1 dd r 2 gg t 1 dg e -2x 0 0 0
@SFIHA e prt is mesured on port PD nother prt strts gin to propgte in the mediumD etF 5.1 From classical to quantum dynamics he wve vetor gives intensities ssoited with the numer of photons in the lightF fut if the soure sends only one photonD the previous intensities nd ÷0ients eomes no more )oting vlues ut should eomes one or zeroF st mens tht the ÷0ients of the gmm mtrix must eome proilitiesF the results of these proilities trnsform the ÷0ients in integer vlues I or zeroF row it works c sf omponent r hs for mplitude HFHID it mens tht the photon hs I hne over IHH to go k from this frontierF e n ompute rndom funtion P (100) tht extrts one numer etween IHHF he funtion a = E [P (100)/100] hs one hne over IHH to e equl to IF r is so repled y this proility aF he trnsmitted proility is utomtilly given y the onservtion lwF es r + t = 1D if t orresponds to the proility bD b = 1 -aF he sme replement is operted for ll the omponents of the gmm mtrixF epplying in this se the gmm mtrix to the wve vetor will propgte photon over ll the network studiedF efter eh pplition of γ on wve vetor vD v must e projeted from the proilisti world of the quntum mehnis in the deterministi world y giving vlues to these proilitiesF e toss is done t eh time step to reple the omponents of v y the integers I or HF efter wht the pplition of γ to the vetor wve v n e redF vet9s tke n exmpleF sf omponent r hs for mplitude HFHT for exmpleD it mens tht we hve T hnes over IHH to selet ID nd WR hnes over IHH to selet HF o relize this kind of proilityD we n onstrut list mde of six 4I4 nd WR 4H4F efter whtD we mke rndom seletion in this listD rndom seletion tht gives the wnted mplitudeF e onsider soure of light of normlized mplitudeF his soure illumiE ntes (rst equipmentD (rst frontier I tht gives re)etion oe0ient of r 1 gg = 0.02F he re)eted prt is direted to mesurement unit tht ount the photonsF hen ehind the frontierD medium of length x propgtes the trnsmitted photons @with proility p d 1 = 1 -r 1 gg p 1 g A until seond fronE tier with dely of P nsF et this seond frontierD the mplitude of proility to re)et the photon is r 2 gg = 0.85F e wnt to model this optil hnnelF pirst we must de(ne the γEmtrixF he (gure SFP shows the hnnel we onsiderF he γ mtrix is given y X

γ =           0 0 0 0 0 0 0 r 1 gg 0 0 t 1 gd 0 0 0 t 1 dg 0 0 r 1 gd 0 0 0 0 0 0 0 0 e -θx 0 0 0 e -θx 0 0 0 0 0 0 0 0 r 2 gg 0 0 0 0 0 0 t 2 dg 0 0          
@SFIIA e strt the proessF he (rst pplition of γ t given time t to v gives X v(t) = γv(0) = 0 r 1 gg t 1 dg 0 0 0 0 @SFIPA yne omponent of the wve vetor n e written in generl X

ψ (k) = Ae -θs @SFIQA 5.1. FROM CLASSICAL TO QUANTUM DYNAMICS TQ Figure SFP ! ghnnel studied
s is vple9s opertor 2 F sf we wnt to extrt the phse property of this omponentD we n operte X

- 1 A ∂ψ (k) ∂s = θ @SFIRA if n is the power of IH whih is de(ned y X ∀(x, y), r x yy = R x yy .10 -n = R x yy 10 n @SFISA
st mens tht we n mke distriution D x yy of numersD only onstituted with H nd ID hving 4I4 nd (10 n -R) 4H4F por exmple if r x yy = 0.2D D x yy = [1, 1, 0, 0, 0, 0, 0, 0, 0, 0]D with R = 2 4I4 nd 10 -2 = 8 4H4F e meE surement pplied to the vetor v mens to de(ne rndom vetor vrile ũ whih hs for dimension the sme dimension s D @equl to 10 n A nd for whih only one omponent is equl to I @rndomlyAF he mesurement m is so de(ned y X

m = ˆt dtũ x • D x yy (t)
@SFITA 2. Remember that when the exponent is between parenthesis, it means that it points out a particular abstract component and not the whole vector TRCHAPITRE 5. LIGHT, INFORMATION & NERVOUS TRANSMISSION SYSTEM sf we spot prtile on port r 1 yy y the ouple (t, m)D the vlues of the ouple omponent re given y X

- 1 A ∂ψ (k) ∂s , ˆt dtũ x • D x yy (t)
@SFIUA row to proeed in order to extrt one mesurement on one prtiulr omponent (k) of v c trting from the stte of v t given time SFIVD this vetor n e written X

v(t = 1) = γ 1 v(0) = 0 ψ 2 ψ 3 0 0 0 0 @SFIVA he omponent ψ (x) is omplex numerD ∀xF e hve for exmple r 1 gg = 2.10 -2 on port PF o X ψ 2 = 2.10 -2 ⇒ R = 2, n = 2 @SFIWA e pply the qunti(tion X D 2 → ψ 2 nd D 2 = [1] 2 + [0] 100-2 where [1] 2 is list of two 4I4 nd [0] 98 is list of WV 4H4F es onsequene t 2 dg = 1 -r 1 gg = 98.10 -2 F he(ning ũ = [1] x + [0] 99 D x eing ny lotion in ũD we ompute m 2 = ũ • D 2 F m 2
is the numer of photon mesured on port PF ntil the order SD nothing ppers on port PF efter wht X

γ 5 v = 0 t 1 gd t 1 dg r 2 gg e -2θx
r 1 gd r 2 gg t 1 dg e -2θx 0 0 0 0 @SFPHA yn port P we quire t 1 gd t 1 dg r 2 gg e -2θx F fut these numers re omplexD nd it is not esy to extrt the delys from the mplitudes in this expressionF e n de(ne (rst mtrix γ where the delys re repled y ÷0ients

I X γ =           0 0 0 0 0 0 0 r 1 gg 0 0 t 1 gd 0 0 0 t 1 dg 0 0 r 1 gd 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 r 2 gg 0 0 0 0 0 0 t 2 dg 0 0           @SFPIA
5.1. FROM CLASSICAL TO QUANTUM DYNAMICS TS nd seond mtrix γ where the mplitude ÷0ients re repled y I nd delys pper X 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 e -θs 0 0 0 e -θs 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0

γ =          
          @SFPPA with these new de(nitions X v(t) = (γ + γ) n v(0)
@SFPQA he mesurement on port x eomes esier to de(ne X

- 1 s log [(γ n v(0)) x ] , ˆt dtũ(t)D x (t)
@SFPRA e n ompute the evolving of the mplitude on port PD y delineting this mplitude on urveF ht9s wht ws done nd shown (gure SFQF he urve shows tht the proility to pture prtile on port P is (rst round I7 while it eomes to e ner of WV7 in (nlF xote tht this urve is otined for ontinuous exittionD iFeF when the γ 11 omponent is equl to IF he impulse response is given y the time derivtive of this resultF xote tht in ft ll the time step re knownF sf we n de(ne the length of the propgtion struturesD let sy x 1 , x 2 , x 3 , . . .F een from given portD the wves must mke round trip eh time they ome k to this portF o the times to mke these trvels is twie the summtion of the vrious trvelling times mde in the vrious medi whih n e rossed oneD twoD three timesD etF e prtiulr trvel time is given y X Mechanics es muh eletronis ould ene(t of ll the developments round sheE mtis nd grphsD s muh it9s not so evident for mehnisF sn mehnisD initil positions pper s initil onditionsF hen we wnt to ompute the displements either internl or externl to the system depending on pplied foresF glerly the )uxes re speedsF he (rst property involved in mehnis is elstiity or plstiityF hile the two fundmentl xewton9s lws sys tht oth impulse nd inerti moment re invrintsF e write X

τ 1 = 2 n x 1 v 1 + m x 2 v 2 + p x 3 v 3 + . . .
f = d dt p τ = d dt L @TFIA
wjor property for strutures is the elstiityF ht9s why we tke look (rst to this propertyF 6.1 Elasticity ilstiity omes from rook9s ide tht ody nswers to n pplied foreD in some de(ned limitsD proportionlly to the foreF iFeF X

f = ku ⇒ f = ˆt dtkv @TFPA k is lled the elstiity onstntD u is displement nd v speedF
he energy stored in the mteril oming from the elstiity is given y X

E = 1 2 ku 2 @TFQA TU he pressure if de(ned y σ = F/SD nd oung9s module y E = σ/ F is the deformtion given y X = ∆u u 0 @TFRA nd (nlly X f = σS = ES = ES u 0 ∆u = k∆u @TFSA
leding to the reltion etween rook9s onstnt nd oung9s one X k = ES u 0 @TFTA imilr de(nitions n e used for sher or ompression onstrintsF vet9s detil little more these oneptsF 6.1.1

Tensor of tensions e hve sid tht the hnge in length of the solid n e written X

δl l = σ E
@TFUA E is oung9s module nd l the solid lengthF σ is the pressureF por segment perpendiulr to the foreD the hnge in its length R is given y

X δR R = - ν E σ @TFVA ν is lled oisson9s ÷0ientF his gives oisson9s lw X δR R = -ν δl l
@TFWA rooke sid tht the vrition of volume is proportionl to the pressureD iFeF X δV V = -χσ @TFIHA he three elsti onstnts (E, ν, χ) n e ssoited using superposition prinipleF 6.2. GENERAL APPROACH FOR THE STRUCTURE TW e n mke ting the fores in the three diretions of the spe writing X

δV V = ∆x 1 x 1 + ∆x 2 x 2 + ∆x 3 x 3 = 3σ E (1 -ν -ν) @TFIIA whih gives X χ = 3 E (1 -2ν)
@TFIPA he vlue ν = 1/2 hrterize inompressile solidF por sheringD the oservle is the ngle of deformtionD ut the priniple remins the smeD we ssoite n ngle to the tension for smll deformtionsF Deformation tensor e n de(ne the displement y(x + δx) of point D neighour of under the form X

y i x k + δx k = y i x k + D ik δx k @TFIQA veding to D ik = ∂ ∂x k y i x k @TFIRA
D n e seprted in its symmetri nd symmetri prts with X

D ik = R ik + ik @TFISA nd X    R ik = 1 2 (D ik -D ki ) = -R ki ik = 1 2 (D ik + D ki ) = + ki @TFITA
R is the rottion tensor while is the deformtion tensorF 6.2 General approach for the structure e generl eqution for the strutures my e X

m ik dv k dt + F ik dx k dt + (R ik + ik ) x k = f i (t)
@TFIUA m is the inerti opertorD F the dissiption opertorF he eqution n e rewritten X

m ik dv k dt + F ik v k + (R ik + ik ) ˆt dtv k = f i (t)
@TFIVA UH CHAPITRE 6. MECHANICS his eqution nswer to the (rst form of TFIF e similr reltion nswers to the seond form X

d dt L i + Θ ik v k + Ω ik ˆt dtv k = τ i (t)
@TFIWA Θ re dissiptions in rottionsD Ω ik x k re the torques pply on strutureF vet9s mke some exmple to see how to use these reltionsF pirst we study simple struture mde of timers 1 D shown (gure TFIF with Ω 00 = 25.10 3 D this leds to P = 62, 5.10 3 xF sf we wnt now to determine the fore on the xis N R knowing the fore P we n seprte the domin in two using the aa xis whih muh ut 1. This example is extracted from Rea's problem solvers : Mechanics.

DYNAMIC

UI not more thn three xisF ghoosing s previously y xis upD the simple summtion of the fores i f i = 0 gives X -F N R -5.10 3 -5.10 3 -20.10 3 + 62, 5.10 3 = 0 ⇒ F N R = 32, 5.10 3 @TFPPA prom the seprte prt on the right of the segment bb we n dd two moE ments desried (gure TFPF ih mss M x is node where e'orts re ppliedF ih spring hs its own lossesD represented y the frition prt k x F ih spring hs een tested efore in order to hrterize its properties k nd k nd M F eny kind of mehnil piee n e hrterized suh wyF sf s nd s re the siss ssoited with eh prtD we n write the system using TFIV X

   M d 2 dt 2 s + k 1 d dt s + k 1 s = 0 M d 2 dt 2 s + k 2 d dt s + k 2 s = 0
@TFPRA sf we exite the (rst mss using soure of virtion F m Sin (ωt)D the previous system eomes X

   M d 2 dt 2 s + k 1 d dt s + k 1 s = F m Sin (ωt) M d 2 dt 2 s + k 2 d dt s + k 2 s = 0 @TFPSA
his n e written de(ning mehnil impedne opertor z tthed with the system of msses X

z = M d 2 dt 2 (•) + k 1 d dt (•) + k 1 0 0 M d 2 dt 2 (•) + k 2 d dt (•) + k 2 @TFPTA
6.3. DYNAMIC UQ xow we wnt to ouple these two piees for relizing the system shown (gure TFQF he (rst tion is to seprte in z the intrinsi properties tthed with the rnhes from the properties oming from the environmentF xoting

z = m + σ + we de(ne X m = M d 2 dt 2 (•) 0 0 M d 2 dt 2 (•) @TFPUA σ = k 1 d dt (•) 0 0 k 2 d dt (•) @TFPVA = k 1 0 0 k 2 @TFPWA
he opertors of mss m nd frition σ elong to the rnhes spe B while the onstrints opertor is tthed with the environmentF e reple from now s nd s y v nd v D k y σD nd we use indiesF vooE king t the (gure TFR we see the proesses under topologil representtion etween the seprte piees hrteriztion nd the system onstrution @we mke the ssumption tht the two msses re sujet to the spring restoring fores loneAF Figure TFR ! wo oupled msses nd the system onstrution e must determine the reltions etween the externl onstrints nd the msses in oth sesF sn the seprte seD these reltions led to the onstrints tensor given eqution TFPWF hen we onnet the two systemsD externl onstrints eome involved following the reltions X

k 1 k 2 w 1 -1 w9 0 1 hese reltions led to the onnetion mtrix A with X A = 1 -1 0 1
@TFQHA he onstrints tensor in the rnh spe for the new system is de(ned y X

= A.k.A T nd f = A.f D whih gives @k ↔ A = k 1 + k 2 -k 2 -k 2 k 2 = k 11 k 12 k 21 k 22 f = F 0 @TFQIA he system TFPS n now e written X    m 11 d 2 dt 2 s 1 + σ 11 d dt s 1 + k 11 s 1 -k 12 s 2 = F m Sin (ωt) -k 21 s 1 + m 22 d 2 dt 2 s 2 + σ 22 d dt s 2 + k 22 s 2 = 0
@TFQPA hen we use X

s 1 = ´t dtv 1 s 2 = ´t dtv 2 @TFQQA we (nlly otin X    m 11 d dt v 1 + σ 11 v 1 + k 11 ´t dtv 1 -k 12 ´t dtv 2 = F m Sin (ωt) -k 21 ´t dtv 1 + m 22 d dt v 2 + σ 22 v 2 + k 22 ´t dtv 2 = 0 @TFQRA yrD in generl in tensoril writing X m ij d dt v j + σ ij v j + ij ˆt dtv j = f i @TFQSA
whih is our eqution TFIVF e hve shown here how to link the uron9s spirit with mehnisF ih prt must e hrterized in preliminry work to know its properties m , σ , k F e system is onstrut reporting the externl onstrints on eh prtD whih de(nes the onnetion mtrix AF prom the originl eqution X

m ij d dt v j + σ ij v j + ij ˆt dtv j = f i @TFQTA 6.3. DYNAMIC US
we otin the eqution for the system @U is the unity mtrixA X

U i α m ij U i β d dt v β + U i α σ ij U i β v β + A i α ij A j β ˆt dtv β = A i α f i @TFQUA
U is pplied on the intrinsi properties nd A on ll externl onstrintsF me proess is used for eqution TFIWF xote tht in more generl writingD we should employ X

d dt U i α m ij U i β v β + U i α σ ij U i β v β + ˆt dt A i α ij A j β v β = A i α f i @TFQVA
feuse the tensors mD nd the onnetion A n hnge with timeF es in eletromgnetismD eh impedne opertor in mehnis is governed y domins setting their lws depending on tempertureD pressureD etF 6.3.1 Connection he onnetion A is the mjor onept in mehnis when mking systemF e strt from olletion of piees (rstly seprtedF hen we hve to identify the intrinsi e'orts nd the externl onesF orking on the externl e'ortsD we estlish the reltions etween the onstrints nd eh prtD iFeF etween the msses nd the elstiitiesF his implies tht ny solid n e seen s olletion of mssesF eny solid n e identi(ed mehnilly looking t its mss enter @lso lled grvity enterAF sf we ut the solid in n prtsD eh prt eing of mss m n nd of lotion P n D the mss enter is de(ned y X

G = n m n P n n m n @TFQWA
foth vetors n e projeted in ny referentilF unowing G we n ompute the inerti moment L of eqution TFIW X

L = α Gx α × m αα (x α ) d dt x α @TFRHA
st mens lso tht we n deompose ny solid s set of suEsolidsF trting from the mss enter of one solid SD we n seprte its volume in n solid nglesD then in m suEprts utting the rdius in m lengthsF he mss of eh prt n e esily omputed X for suEvolume V nd mss density ρD the mss m i of the suEsolid is

m i = ˚V dV ρ
@TFRIA hepending on the use we wnt to do with solidD we will seprte it in vrious prts where some opertions will e mde to onnet it with other prtsF efter whtD if we rete hole in solid in order to put srewD this llows to onstrut system ut implies wekened of the solidF por ny system of linkgeD the tensor sys how the onstrints re trnsmitted in the three diretions nd if some link reking down oursF he tensor must inlude domins to trnslte this kind of eventF vet9s tke n exmpleF e onsider simple wekenD lled to e sumitted to entrl weightF st mens tht in (nlD the weken is tthed y three pointsD two t the extremities giving fore tht ompenste the weken weightD nd one t the enter pplying n externl foreF he (gure TFS shows the se imgined nd the projetion of the fore vetor on oth xes intrinsi to the wekenF he fores n e projeted following two diretions X horizontl @xA one nd vertil @yA oneF

ω = ω 1 0 0 ω 2 @TFRRA
hen we ouple oth hlf weken we use the two onnetions @for k nd ωA X

k 1 k 2 ω 1 ω 2 m 1 I EI I I m 2 EI I I I whih gives for the onnetion mtrix on kD A nd on ωD B X A = 1 -1 -1 1 B = 1 1 1 1 @TFRSA
unowing the whole weken length χ we otin the system equtions fter linking the two hlf weken sumitted to the fore

f 0 @τ 0 = f 0 χA    d dt U i α m ij U j β v β + A i β ij A j β ´t dtv β = A i α [f 0 Cos (ξ)] i d dt U i α L i + B i β ω ij B j β ´t dt θβ χ 2 = B i α 1 4 f 0 Sin (ξ) i @TFRTA ξ = arctg (y/x)F
he ehviour of the mteril remins elsti until given vlue of the deformtionF feyond this vlueD the mteril eomes plstiF row n we model this c sntrinsi properties n e 'et if degrdtion ours on mterilF sf the mteril is roken in mny prtsD its mss will e modi(ed for sure 3 sf the deformtion eomes plstiD n o'set in its lotion trnsltes this modi(tionF he lw kx eomes k x + x 0 F st mens tht for exmple

X ij = y D 1 ij (•) + y D 2 ij (•) + x j 0 @TFRUA
his pity should e hrterized during the hrteriztion phse of the primitive mehnil elementF en importnt spet is the temperE ture dependeneF st9s ler tht the elstiity of ny mteril depends on tempertureF 6.4 Vibrations dynamic gonsidering spring supporting mss M D of rigidity kD for urviliner oordinte u we write X

-ku = M d 2 u dt 2
@TFRVA o the rigidity orresponds rell ouple Γ = -Cα linked with the ngle devition αF por n inerti moment J we hve X

-Cα = J d 2 α dt 2 @TFRWA sf p is vple9s opertorD we n write X M p 2 u + ku = 0 Jp 2 α + Cα = 0
@TFSHA for sinusoidl solutionsF he (rst eqution leds to u (M p 2 + k) = 0 whih hs for root

k -M ω 2 0 = 0 ⇒ ω 0 = k M
@TFSIA nd for the devition ngle X

ω 0 = C J @TFSPA
ω 0 is the eigenvlue of the osilltor in free ondition of osilltionF he ÷0ients C nd k n e determined in stti phseF he deformtion of the solids sumitted to known stti fore F gives their rells k or CF 6.4.1 Springs in parallel sf we onsider mss m 1 of vlue M/2 tthed to spring of rigidity k 1 nd the sme for seond mss

m 2 of vlue M/2 nd rigidity k 2 Y we hve X m = m 1 0 0 m 2 @TFSQA nd X = k 1 0 0 k 2
@TFSRA xow if we tth oth msses in order to mke single oneD the onnetion eomes X 6.4. VIBRATIONS DYNAMIC UW k 1 k 2 m 1 I I m 2 I I e understnd tht if rell k x ts on hlf mssD it ts lso on the seond hlf mss tht prtiiptes to the whole mss M F sf g = d 2 u/dt 2 D we otin X

m 1 g = -(k 1 + k 2 )u m 2 g = -(k 1 + k 2 )u
@TFSSA he result is when two springs re in prllelD the equivlent single spring is equl to the sum of the two seprte springsF xow if the two springs re in seriesD the totl length x overed y the springs is x = x 1 + x 2 F his length is in reltion with fore F y F = kxF he fore is pplied on mss tthed t the extremity of the two springs in seriesD nd is reported on eh springF o we hve X

x = F k 1 + F k 2 ⇒ k = k 1 k 2 k 1 + k 2 @TFSTA
sf we look t two free mssesD s desried (gure TFTF Figure TFT ! wo free msses with spring fefore to link the mssesD they oey to the lredy seen reltion X

-k i x i = m i g i ⇔ m i g i + k i x i = 0
@TFSUA efter linking themD eh mss see the rell of the otherF pollowing our meE thodD it leds to oth equtions X

   m 1 g 1 + k 1 x 1 -k 2 x 2 = 0 -k 1 x 1 + m 2 g 2 + k 2 x 2 = 0 @TFSVA xoting x = x 1 -x 2 nd g = g 1 -g 2 D we otin of the previous system X -kx 1 m 1 + 1 m 2 = g
@TFSWA he(ning the redued mss X

1 µ = 1 m 1 + 1 m 2 @TFTHA
his osilltor de(ned y -kx = µg ehves like single osilltor mde with one mss nd its eigenvlue is X

ω 0 = k µ @TFTIA 6.4.2 Moment expression
he reltions in se of rottion n e lwys otined from the trnsE ltion reltionsF por exmpleD strting from

F = m d dt v
@TFTPA multiplying y the distne r we hve X

Fr = mr d dt v @TFTQA ut v = θr nd so X Fr = mr d dt θr ⇒ Γ = J d dt θ
@TFTRA whih imply the inerti moment de(nition X J = mr2 F 6.5 Finite element approach pigure TFU shows the prolem we tret here 2 F yn the (gureD the edge re numered with red letters nd the nodes with lk lettersF he weight (xed on the lst weken implies fore F on the onsoleF es the nodes I nd P re tthed to the wllD only the nodes QDR nd S n moveF @TFTSA e must e reful tht this ssumption doesn9t imply tht the sene of elongtion mens smll displementsF por exmple weken n rotte without hnging in lengthF sn ftD the ssumptions given when we delre the prolem with n initil geometryD we (x the mehnil ehviour of the prolemF xow to onnet the wekens in order to mke the onsoleD we need to ssure two properties X IF the sme displements reported on the shred nodes X continuity relation Y PF the opposition of the fores @tion retion xewton9s lw X equilibrium relationA fut s the onsole elongs to the sme referentilD the properties of eh weken de(ned in their referentil must e projeted in ommon referentil nmed 4glol referentil4F o we n write for eh weken involved in the onsole repired y its extremities I nd PD the omponents of displement u 1 , v 1 nd u 2 , v 2 s the reported fores f 1 , g 1 nd f 2 , g 2 F pigure TFW shows this onvention used for the whole wekensF

∆ = (Lcos(α) = u 2 -u 1 ) 2 + (Lsin(α) + v 2 -v 1 ) 2 -L @TFTTA
his expression n e redued using the smll displements ssumption to reh X

∆ ≈ cos(α) (u 2 -u 1 ) + sin(α) (v 2 -v 1 )
@TFTUA ih weken sti'ness mtrix links the fores nd the displements folE lowing their xes diretionF he eqution is X

F • t = kU • t @TFTVA
sing the previous reltionsD we otin for one element nd its displement t eh node X

U 1 • t U 2 • t = cos(α) sin(α) 0 0 0 0 cos(α) sin(α)     u 1 v 1 u 2 v 2     @TFTWA
his reltion n e written s a = C a b u b where s re the oordintes tthed with eh weken nd u b → (u, v) the oordintes tthed with the ommon referentil (x, y)F yn nother side we hve X

    f 1 g 1 f 2 g 2     =     cos(α) 0 sin(α) 0 0 cos(α) 0 sin(α)     F 1 • t F 2 • t @TFUHA or f k = C q k F q where f k → (f, g
) re the fores pplied t eh node nd for eh diretion in the ommon referentil oming from the fore soure f projeted on eh xes of the lol referentil tthed with the weken xesF e n onsider eqution TFTV to write it under the form X

F • t = kU • t ⇔ F α = k αβ s β @TFUIA e n reple in this seond eqution s β y C β b u b D whih gives X F α = k αβ C β b u b
@TFUPA now we multiply on the left oth memers y C α k to otin X

C α k F α = C α k k αβ C β b u b @TFUQA he(ning kb = C α k k αβ C β b D this leds to X f k = kb u b
@TFURA his reltion gives the link etween the fores pplied on eh weken t their extremities nd for ll the diretions of the ommon referentil @or ommon speA nd the displements of the nodes of these extremities in the sme ommon speF he solution with null fores kb u b = 0 gives the kernel of the sti'ness mtrixF yne we hve the expressions of the weken properties in the ommon speD ut for the seprte prolemD it remins to onnet the weken etween eh other to mke the systemF he (gure TFII shows the struture with the numers of the nodesD wekensD nd weken extremitiesF sf ω σ re the displements in the ommon spe one the weken re onnetedD we n (nd reltions etween ω σ nd u σ writing X

u α = C α σ ω σ
@TFUSA por exmpleD seeing (gure TFII we n estlish X u 1 (1) = ω 1 nd u 2 (2) = ω 3 D etF st must e ler tht is digonl mtrix of suEmtrix kD eh suE mtrix eing of dimension PxP or QxQD depending on the geometril spe where the prolem is studied @rememer tht in our se we work in two dimensions nd lol oordintes of the ommon spe ws u 1 , v 1 AF e re now fmilir with this kind of developmentF sf h re the fores one the weken onnetedD we n verify h α = C η α f η F end strting from eqution TFUR we otin X

f k = kb u b ⇒ f k = kb C b σ ω σ ⇒ C k α f k = C k α kb C b σ ω σ @TFUTA xoting ζασ = C k α kb C b σ D we (nlly otin X h α = ζασ ω σ @TFUUA
hih gives the solution of the onstruted onsoleF ζ is the sti'ness tensor of the onneted system in the ommon speF 6.6 Surfaces description eny surfe n e desried y urviliner oordintesF he trnsforE mtion of these oordintes into rtesin spe is the key to de(ne the system struture we wnt to useF es usul eh prt of struture is (rstly de(ned in lol spe tthed with the prt nd giving hrteriztion of the prtF ih prt n e seen s weken s previouslyD the di'erene eing tht prt is surfe nd not only monoEdiretionl ojetF he urviliner oordintes s α of eh prt is lolly trnsformed into ommon spe of oordintes u β F hen fter onnetion to mke the system through CD the omplete struture is desriedF pinlly omponent of the struture is tenfold mde of two mjor ojets X IF the prt geometril desription with u η Y PF the sti'ness mtrix of the prt k Y QF the eletril properties of the prt σ, , µ wo tensoril equtions over oth mehnil nd eletroni properties of the system X

   h α = ζασ ω σ e α = ζ αβ k β @TFUVA
fut oupling must e dded etween the mehnil nd eletroni networksF he struture properties impts the line propgtion properties while the eletroni dds weight on the struture prtF he lol environment @presE sureD tempertureA hnges the struture nd the eletronis propertiesF he pproh mens to study nd ouple the strutures nd the eletronis one their own glol networks re de(nedD then to ompute the lol environE mentF es the impedne opertors re de(ned on dominsD the environment is utomtilly tken into ount in the system workingF hynmi ehviours follow extly the sme equtionsF he only di'eE rene omes from the ft tht the fores nd displements hnges with timeF Chapitre 7

Temperature emperture is perhps one of the mjor onept of physisF invironE ments re desried in term of eletromgneti (eldsD pressureD grvittion nd tempertureF e wnt to present here mjor ides out tempertureD in order to inlude this environment prmeter in our studiesF 7.1 Thermal dilatation c÷cient nder some temperture mineD piee of mter hs its length hnE gingF he reltion etween the temperture vrition nd the length vrition is liner nd we write X ∆L = αL 0 ∆T @UFIA L 0 is the nominl length of the pieeF por nylon for exmpleD the ÷0ient is equl to 81.10 -6 F he sme ide n e pplied to volumeF sf V 0 is the nominl volume of piee of mterD the vrition of volume ∆V is de(ned y X ∆V = βV 0 ∆T @UFPA β is the ÷0ient of volume diltionF sn given volume V D the pressure P on the volume wlls is ssoited with the numer of moleules N nd their movement @giving temperture T AF his evidene is de(ned y the perfet gs reltion X

P V = N k B T @UFQA k B is
Q = mc ˆt2 t 1 dT @UFRA
he ftor c is lled the mssive pity of the mteril of the odyF yn nother sideD we tke look to the het trnsferF por plte of surfe S nd length LD the het trnsfer n e de(ned y X

dQ dt = k T S ∆T L @UFSA or in generl X dQ dt = k T S dT dx @UFTA
k T is lled the therml ondutivityD nd trnslte the pity of ody to trnsfer the het through its surfe nd ll long its lengthF es Q is energyD we n de(ne the power P y X

P = dQ dt @UFUA xoting C th = mc nd y th = k T S/L nd X θ = ˆT dT @UFVA we otin X ˆt dtP = C th θ ⇒ θ = 1 C th ˆt dtP @UFWA he seond reltion eomes X P = y th θ ⇒ θ = R th P @UFIHA
with R th = 1/y th F sf we hve piee of mtter heted y eletril wireD prt P s of the energy goes out through the dmittne y th X P s = y th θF enother prt of the 7.2. TEMPERATURE TOPOLOGY VW energy P b is used to het the ody of therml pity C th X P b = C th dθ/dtF pinllyD P e eing the totl energy trnsmitted to the wire X

P e = C th dθ dt + y th θ @UFIIA
qenerlizing this reltionD we rete nturl spe of nodesEpir soures or urrent of power P k D the temperture hnging θ u giving X

P k = C ku d dt θ u + y ku θ u @UFIPA
he soure of het is urrent driven in some eletroni iruitF he temE perture θ hnges the resistnes vlues or the solid length vi the therml dilttion ÷0ientF he three modes of het trnsfer re X " ondution Y " onvetion Y " rditionF he rdition modelling follows the sme priniple s for the onvetion modellingF e de(ne rdition ondutivity y X y R = h R S @UFITA whih gives the reltion etween the power exhnged y rdition P R nd the temperture evolving X P R = y R θ @UFIUA 7.3 Thermal environment yne identi(ed the vrious soures of hetD grph must e onstruted to model the het trnsfers etween the system nd the environmentD nd etween the vrious orgns inside the systemF hese exhnges must led to grph representing the het )uxes with their vrious properties of onduE tivities or pitnesF he soures of het rete nodesEpir of urrent in this grphF he temperture vritions θ u eomes inputs s prmeters for the domins of the impedne opertorsD or to set the length dilttion of solidsF por mny systemsD we n onlude tht X " two grphs represents the systemD one for eletronisD one for therml Y " one grph represents the environmentD one for eletromgnetismD one for thermlF yther grphs n e usedD for pressure for exmpleF hese grphs re sE soited with tensoril equtions inluding lgrngin opertorsD etF hese ojets re grouped in tenfoldsF st mens tht we hve one tenfold for the systemD nd one tenfold for the environmentF Chapitre 8

General modelling of a cyber-physical system (CPS) vet9s rell wht is yerEphysil system X it9s system mde with physil mteril nd emedding yerneti tehniques X omputersD numeri networksD loudD etF he mteril is modelled using the tensoril nlysis of networksF he vrious networks re grouped in whole system of systems under some eqution like X

T a = g ab ψ b ⇔    h α = ζασ ω σ e α = ζ αβ k β @VFIA
ome of the emf e a re mde with known urrents @urrent souresA multiplied y impednesD like R ab .J b F hrough the mehnil proess of gmm mE trixD this soure ssoited with (rst network is trnsported eoming the emf of nother networkFhis proess n trnslte the exhnge of ommuE nition etween the networksD while the eletromgneti intertions @like through ntenns for exmpleA re tken into ount y lssil qreen funtionF foth funtions n represent somewhere the sme physil seD ut the gmm mtrix eomes mny more esier to use when numeri mesE sges re involved @other ses n e interestingAF st n e used for quntum proessF he (nl eqution for the g hs the form X " pereption network E e π Y " n tion network E k a Y " nervous system nd omputer entres E n πa Y " n energy network E e E Y " wste mngement network E R uv Y " ommunition network E γ a b Y " skeleton E s uv F o de(ne ompletely the onept of g we now detil ll the notions ssoited with the onept of systemD usully delined in the siene of system engineering 1 F 8.1 Concept of system here re mny kind of de(nitions for systemF fsillyD system is setF fetween othersD it exists de(nition tht is more in ordne with our formlism X to sy tht system is ox with inputs nd outputsF snside nd outside the ox we n disern )uxes @)uxes of energyD ommunitionD etFAF e system n e seen s losed if no )uxes go from inside to outsideD or open if )uxes n go from inside to outsideF he )uxes k a re the soures for ords under the xex formlism @see referenesA or for ny kind of ommunition hnnel to other systemsF es mtter of ftD ll the system ehviour n e oserved looking t these )uxesF he system orgniztion is hrterized y its fundmentl opertor ζF his opertor n hnge with time @dynmi systemA or not @stti systemAF e system yle of life goes through six steps X " gonept nlysis " fesiility " development " mking " strt of servie " use " out of servie e ell of system n e seen s the ox presented (gure VFIF e will see tht ll the lssil notions of system engineering n e delined using mthemtil representtion of the systemF his representE tion must inlude ll the omponents of the systemD the intertions etween 1. See Luzeaux's reference. these omponents nd the humn ftorF he di'erene etween yerE physil system nd ioEinspired system holds in tking into ount the humn ftorF sntertions involving models oming from the gme theory will e used in the ioEinspired system engineeringF ht9s the purpose of the next hpterF por the momentD we will explore the vrious themtis tE thed with the notion nd development of systemsF pirst of llD we rell the mthemtil representtion of system under the xex method @exE tended tensoril nlysis of networksD developed y the uthorAD leding to the onept of mnifoldF sully system is desried y set of stte vriles xF hese vriles evolve depending on ommnds u nd externl unontrolled prmeters θF he eqution of the system life n e written X

dx i dt = g i (x, u, θ)
@VFQA his form fouses on the stte vrile hnges without giving ny prE tiulr mening to eh kind of vrileF hisD while their roles re very di'erentF he previous eqution n e written X

u i = ψ i (x, θ) @VFRA
he funtion ψ i inludes the vrile x nd its time derivtiveF st remins thtD exept when studying the stti stteD the vrile ẋ is more interesting thn xF ẋ trnsltes diretly )ux whih is our inditors for systemsF here re no prolems to work with ẋ rther thn with xF x is simply otined through time integrlF e generlize the previous eqution y X

u i = ψ i (x, ẋ, ẍ, θ) @VFSA
ψ is ny funtion pplied to x nd θF fut ψ hnges with θF he externl prE meters θ j intervene through domins DD hnging the funtions ψ i depending on the environment in)ueneF he reder will reognize in this desription the tehnique previously used to model the osilltorF his tehnique llows to model ny mteril systemF henD to go deeper in the system nlysisD we mke some opertion with the ojetive of seeing the system s omplex geometril ojetF 8.2. TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESWS 8.2 Tenfold analysis of complex systems (TACS) preliminaries xow we must give us n lger in order to identify ny system nd to model its evolutionF eny system n e represented y list of mthemtil ojets giving the time evolve of their )uxesF he nture of these )uxes point out utomtilly the physis onerned y the modelF st9s ler tht n engineer using of these ojets needs drw orresponding to the rel ojetsD to help understnding the grphs nd equtions ssoited in the modelF o the system is identi(ed with for one side set of primitive ojets used to mke the system @O {. . .}A nd for seond side list of mthemtil ojets giving the system models (C, T, G)F ih primitive ojet hs orrespondene with our ell VFIF T is the energy soures of primitive prtD G its metri nd C onnetivity tht onstruts the primitive prtF he simplest wy to understnd the pproh is to pply it on rel systemF e tke s exmple pinEsoket oupleD under externl virtions nd trnsmitting high voltge signlF he system is mde of S ojets t lest X " e pin nd soket @O p , O s A Y " resistor nd power supply @O R , O u A Y " virtion fility @O v AF o for the moment X O {. . .} = {O p , O s , O R , O u , O V }F ss this set my e lled system c xoF sn the qreek9s mening yesD ut it limit the of the word to its si mthemtil meningF pinllyD de(nition tht pleses to me is given y vuzeux X 4 system n e seen s lk ox where inputs nd outputs mterilize some numer of )uxes4F por the momentD our set eing mde of seprte funtions is not soure of )uxesF trting from these primitive ojets O i we hve to onnet them to hope mking systemF henD the )uxes nlysis will give us the se to nlyse the systemF gonneting the primitive ojets mens to rete onnetions etween themF here re vrious kinds of onnetions por the mjority of prolemsD it exists si solutionsF here is group of fundmentl solutionsD ut glollyD there re often knownF snnovtion elongs more to the hoie of tehnologiesD to reh the sme needF he story strts with the desription of needF sn our seD the need is to onnet high voltge genertor with lod using pinEsoket oupleF he ontt must e gurnteed for one thousnd opertions of onnetionD disonnetionF e n imgine tht there re lot of possiilities for this kind of onttF he o'er is ontined in set of ll ville ontts c {}F ih kind of ontt in the set c {} n e modelled y rnh in ellulr topologyF ht does it men c sf we power supplied the ontt with soure of energy eD urrent i @ )uxA is reted through the onttF his implies tht potentil u is developed long the onttD ssoite with the work developed y the )ux movingF e n write for this primitive element X e = u + z.i @VFTA z is the lw guiding the )ux vlue depending on eF his lw is in)uened y the environment onditions X tempertureD pressureD humidityD etF he work u n e written somewhere s X

u = ˆb a dx • J σ @VFUA
tht9s uirhho'9s lw nd fundmentl reltionF J is the )uxD σ ontt property nd x the urviliner oordinte long the onttF ine we use sptil integrtionD we n ssoite n strt rnh whih orders re the nodes a nd bD de(ning the limits of the integrlF his ellulr ojet represents strtly the onttD whtever it isD with its limits nd propertiesF ell the properties giving the reltions etween e nd i must e ontin in the opertor z @here ontining 1/σAF xote tht more thn one rnh n e neessry to reh this modelling qulityF et lest in generlD one rnh is needed for eh physis X one for eletronisD one for mehnisD etF pinlly the primitive ojet is group of rnh in multiEphysiD see little networksD tht will e prts in wider network representing the systemF 8.2. TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESWU e understnd tht we n strt from sets of ontts @pinEsoket oupleA c {}D high voltges h {}D wires w {} nd resistnes r {}F he system onepE tion onsists in tking one element from eh of these sets to onstrut the systemF fsilly we mus tke one element to eh of these setsF e (rst pproh in oneption n e to test rndomly the qudruplet otined y tking one element to eh setD then to verify if it is in ordne with the requirementsF por the onstrution onlyD this strtegy n e writtenD if e is rndom funtion sying if the system is in ordne with the requirements X

P (A| c i ∈ c {} , h i ∈ h {} , w i ∈ w {} , r i ∈ r {}) > 1 -
@VFVA st is ler tht this proility is wekF sf we look t the wiresD etween ll the wires ville in the mrketD few re le to support high voltgeF oD the proility to tke good wire etween ll the possile smple is lredy smllF he dvntge of this strtegy is tht it n e rih for innovtionF he ft to tke rndomly the element n led to qudruplet tht noody my hve imgineD nd perhps the est e0ient oneF fut s sid eforeD the hne to selet this omintion is very thinF e long time my e spend efore to (nd the good hoieF e more lssil strtegy my e to hoie the est prt oming from eh set @higher e0ienyD lower ostA nd to mke the system with these seletionsF por eh prtD this ojetive n e writtenD if f is n oservtion funtion sed on de(ned riteriD returning the performne of the element X

∃c i ∈ c {} / f (c i ) > f (c j ) , ∀c j ∈ c {} with j = i
@VFWA his pprohD unfortuntely the more used euse the simplest oneD doesn9t led neessrily to the est systemF his omes form the ft tht the system results emerge from ouplings etween its omponentsD in ddition to the omponents themselvesF he lk of onsidertion for the system imE pt ppers fstlyF iven if wire nswer to the hight voltge nd virtion y itselfD its performne in virtions n e indequte when the ontt will e onneted to the wireF rere ppers ompletely the gp etween the seprte omponent pities nd the system onstrintsF enother prolem oming from the lssil pproh is thtD s they tend to forget the systemD they tend to forget the ustomer needsF his is euse the ustomer needs doesn9t pper in the deomposition of the trget performnes into eh system omponent performnesF fy the ftD the ustomer needs re memoE rized only in the originl requirements nd during the system onstrutionF fut it my e too lte to reh the optimum solutionD s seen in the previous exmpleF wny round trip will e neessry to otin the wited nswer for the ustomerF e new method reently developed tries to reinfore the ustomer oriented tehniques X the viexF sn this pprohD the ide is to fous ll the deveE lopment on the ustomer needsF fut it doesn9t give proposls to improve the whole tehnique nd leds to the sme fundmentl prolem thn the lsE sil pprohF st ws imgined (rst for the mkingD then extended to 8hF s personlly think tht this tehnique is not the nswerF he pproh onsidering the system prolem is the systemi pprohF por this oneptD we ept the ft tht the system properties nnot e otined strting from the properties of its omponentsF he systemi pE proh is somewhere mix of the two previous pprohesF he mthemtil formultion of this strtegy my e given t the omponent level y X

∃c i ∈ O i {c i , h i , w i , r i } / f (O i ) > f (O j ) , ∀O j , with j = i
@VFIHA he di0ulty is to (nd the omponent whih nswers dequtely to the system needs nd to its own performne ojetivesF his mens to ntiiE pte the impt of ouplingsF his seems to e impossile efore the system existsF he key of suess elongs to the pityD one the glol system rhiteture hosenD to deline omplete environment de(nition to ll posE sile omponents implied in the system onstrutionF his exist sometimes under the form of design rulesF fut the design rules hve the defet of giving onstrintsD indeed eginning of solutions rther thn ojetivesF omeone ould sy tht the ft to tke into ount the whole system onstrints n e mde through the lssil pproh lso c st9s true ut the ig di'erene is tht in the lssil strtegyD the system knowledge is mde a posterioriD while in the systemi pproh the system ojetives is tken into ount a prioriF ht we write for the lssil pproh X

P (A(t + ∆t)| f (c i (t)) > f (c j ) , ∀c j ∈ c {} with j = i) @VFIIA nd for the systemi pproh X P (f (O i (t + ∆t)) > f (O j (t + ∆t)) , ∀O j , with j = i| A(t)) @VFIPA

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESWW

e see here tht lerlyD we onverge to mix of the previous oneptsF sf we resume oth nottions y P (A| c i ) nd P (O i | A)D we hve y one side X

P (A| c i ) P (c i ) = P (c i | A) P (A)
@VFIQA while on the other side X

P (O i | A) P (A) = P (A| O i ) P (O i ) @VFIRA sf we ept P (A) > P (A) nd P (c i ) > P (O i ) then it implies for P (O i | A) ≈ P (c i | A) X P (A| O i ) > P (A| c i )
@VFISA whih tends to give more weight to the systemi pprohF he mthemtil trnsription of systemi uses opertors de(ned on dominsF e see tht in the system engineeringD two steps re preliminries to the development X

" the onept nlysis Y " the fesiilityF st mens tht in this top-down pprohD the time llotted to the 4up4 phse is signi(ntF nfortuntelyD this time is often negleted nd redued to 4the prt of the poor4F gonept nlysis mens to hoie glol system onept etween vrious villeF ome keyword for onepts re X " endoskeleton or exoskeleton Y " distriuted system or lol system Y " system with entrl rti(il intelligene or distriuted rti(il intelE ligene Y " system with emedded tutors or remote tutors Y " etF he question is X why nswering one system kind rther thn nother c yften the history shows tht humns follow the nture in this hoieF fut not lwysF eout speD it9s ler tht nture shows tht exoskeletons seems to e the unique wy to protet the living things emed in sperft to protet them from the spe environmentF sf we spek of trnsportD rs re exoskeleton systems while humns (rst use nimls like horses whose re endoskeleton systemsF en exeption re the motoryle X (nlly they re more inspired y nturl moilities thn rs or plnesF hoes it men tht it is impossile for trnsport system emedding more thn one or two persons to e endoskeleton one c por plnesD the prolem eomes similr to sperftF por this resonD we n think tht exoskeleton strutures re neessryF por rsD usD etFD we see tht more nd more strutures eomes opened struturesD showing tht endoskeleton re possile hoiesF fut s they re nerest to nturl systemsD they ll surely for higher pities of oneption eoming just now to e imginleF he dvntge of exoskeleton re ler X this struture protets the orgns versus the mehnil @nd in some prtiulr se the hemilA onstrints of the externl environmentD inluding ll the living eing emed in the sysE temF hile in endoskeleton struturesD the orgns nd emedded living eing re exposed to these mehnil @nd sometimes hemilA onstrints oE ming from the externl environmentF he (rst disdvntge of the exoskeleE ton systems re their weight nd some smller performnes in some dominsF st seems tht some of the (rst 4lrge4 nimls ws exoskeletonF st ws loE gil evolve for nimls tht were eoming more omplex nd must mke fe to the externl onstrint of wter then tmosphereF st ws n evolve from skin to exoskeletonF fut inresing the dimensionsD nture ws oliged to evolve into endoskeletonsF pigure VFP shows lerly the gin in mss of the endoskeleton strutureF sf we evlute the weight of the skeleton in oth se of exoskeleton E or shell nd endoskeleton E or spine X for given volume V of orgnsD the weight m e of the shell mde with mteril of density ρ is given for n thikness δ nd volume surfe S y X m e = ρδS @VFITA he thikness δ is determined y the mehnil needs to support the whole weight nd dynmi of the systemF sn se of endoskeletonD we hve the skin weight ρ s δ s SD nd the spine weight determined y the whole weight to e le to supportF fut s the surfe is redued y ftor ten typillyD for the sme mterilD the skeleton weight is lso redued y tenF sn nother sideD the skin weight n e drstilly redued y the skin mteril nd its thiknessF st mens tht this dded orgn is fr from weighting the di'erene ρδS -ρ s δ s SF sn glolD the endoskeleton is lighter thn the exoskeletonF pinllyD wht helps to deide is the externl environmentF e will now spek of environment nd selfEonsisteny for the onstrints oming form the system itselfF When it is impossible to nd a skin that protect the organs from the environment, a exoskeleton is necessary. If not, endoskeleton por rti(il humn mhinesD extrterrestril environment is generlly the reson to hoie exoskeletonsF fut the sptil environment in ft doesn9t inlude plnesF roley we my (nd plnes in the future without exoskeE letonF ome irds )ight t the sme ltitude tht stndrd plnes @round IH HHH mAF sn the deep oenD (shes @nd some mmml like the perm whleA n liveF he sme in some prtiulr wter zone where the wter is very hetF st mens tht even in these hrd environmentsD endoskeleton re possile solutionsF e my sy tht mny lssil systems re tody thinking under exosE keleton strtegyD euse it ws simplest to mke them eforeD nd it9s hrd to mke something ompletely di'erent todyF fut we will see tht under ioEinspired engineeringD this kind of pproh ould e hngedD with eneE (t of lighter systems nd oth eletromehnil nd intelligent skinsF pine nd other ones will e there to give ommon rigid struture le to ept the vrious tutors nd orgnsF e n onlude the onept nlysis sying tht humn eing n ene(t of ll the reserh onduted y the xture depending on ll the mission lloted to the living eingF por eh kind of needsD there ws vrious nswers nd kind of systemsD very well dpted to their mission with timeF yur previous ides n e prtilly enfored using lssesF sf we (rst spek of the systemF st is mde of strutureD of energy nd informtion netE work @les in generlA nd of orgns @equipmentsAF e n rete lsses for eh of these elementsF sn one lssD we put the properties E prmeters E of the ojet nd methods modelling the ojet ehviourF e lss 4rE hiteture4 groups one of eh lss of strutureD etF in order to mke n rhitetureF ih omponent is hosen in set of mny of themF he result is lrge proposl of rhiteture hving vrious performnes nd ostsF fsed on this prinipleD mny 4test4 n e try like do the xtureF e don9t know in dvne if suh n rhiteture is interesting or notF st9s only one the rE hiteture is mde tht we n perform its propertiesF e similr de(nition of lss n e mde for the environmentF he di'erent lsses n e instntiE ted in progrm written to imgine the system through vrious ssumptions of hoiesF sn prtieD prtiulr point merits to e itedF he methods must e delred s stringD inluding prmeters delred in the properties (eldFefter whtD when we hve to relize the omputtions for evluting the performnesD these strings must eomes rehle equtionsF sn python for exmpleD very interesting possiility exists X the funtion eval trnsforms string in )oting point equtionF ih lss of strutureD leD eletroni hs in its methodsD one method tht de(nes the impedne opertor @in generl meningAF he rhitet lss tkes one lss of eh kind of omponent nd tries vrious solutions of rhitetureF he proess is explined (gure VFQF yne n rhiteture is mdeD it remins to verify if it is in dequy with the requirementsF ht9s the purpose of the fesiility phseF he two phses of onept nd fesiility n e explored in loop efore to (nd the est omintion for the given ojetiveF rowever the system onstrution n e guided y preliminry study mking simultion of the system missionF he system oneption goes through the ll of lsses ssoited with the vrious system omponents X strutureD lesD eletronisD etF row this mehnism n work c he ojetive is to trnslte s soon s possile the system desription using lsses in modelsD to ontinue the development strE ting from these models rther thn remining in some textul desriptionsF

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIHI

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIHQ

Figure VFQ ! glss rhiteture sf we onsider funtion y depending on tD ut lso on two other prE meters g nd v 0 F vet9s de(ned X y(t) = v 0 t -gt 2 → y(t; v 0 , g) @VFIUA enother funtion ould e g(t, A, a) = Aexp (-ax)F e my de(ned these funtions using lssil funtions in pythonD suh s X def y@ t D vo A X gaWFVI return vo * t -HFS * g * t * * PF def g @x D DeA X return e * exp@- * xA he mjor prolem of this progrmming is tht it is stritly reserved for funtions tht hs extly the sme numer nd type of prmeters nd vrilesF sn lssesD vriles nd prmeters re visile y nyodyD they ehve s glol vrilesF he previous funtion y my e de(ned s the next lss X l s s X def init@ s e l f D vo A X s e l f F voavo s e l f F gaWFVI def vlue @ s e l f D t A X return s e l f F vo * t -HFS * s e l f F g * t * * PF e n then use the lss y reting n instne of the lss X y = Y (3.)F his instrution sets v O to QF o otin the vlue y(t = 0.1; v 0 ) we just hve to omplete the previous instrution y v = y.value(0.1) whih gives the result for t = 0.1F he keyword self return to the instne of the lssF wking y = Y (3.) we impliitly reple self y yF glling v = y.value(0.1) we otin the result for v 0 = 3 nd t = 0.1F he syntxe vay F vlue @ H F I A is trnslted y python in X vaF vlue @y D H F I A eny suEsystem is hrterized y tenfold inluding the tensor ζD evenE tully soures ED nd some other propertiesF por exmple we wnt to rete struture 4ground plne4F st9s generi lsses desriing plte of thikness dD length lD width wD ondutivity sF e my rete X l s s plte X def init@ s e l f D d D l DwD s A X s e l f F dad s e l f F lal s e l f Fwaw s e l f F sas s e l f F uoaRF * pi * Ii-U s e l f Fhxa s e l f F d D s e l f F l D s e l f Fw def @ s e l f D p A X return IFG s e l f F s * s e l f F l G s e l f Fw * @IFG s e l f F dAC sqrt @@ pi * p * s e l f F uo * s e l f F s AG@I t * PF * pi AA es this lsse lls funtions tht elong to the pkge numpyD the use of this lss must e preeded y numpy lling X from numpy import *

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIHS

he dimensions of the plte re inluded in one of its propertiesD self.DF efter n instne of the lss plteD we n ompre the requirements nd the plte propertiesD ompring

x Q i nd R.D if R
is the instne of plteF xow we n imgine two eletronisF hey elong to the sme lss elec-tronicF l s s e l e t r o n i X def init@ s e l f D r D i A X s e l f F rar s e l f F i zai s e l f F hia-IiT DIiT s e l f F hraIiT DIiT def t@ s e l f D p A X return s e l f F rCs e l f F i z * p his lss llows to rete the extremities of lineF he lss line is given y X l s s wire X def init@ s e l f D le D di D d A X s e l f F l eal e s e l f F diadi s e l f F dad s e l f F uoaRF * pi * Ii-U s e l f F hsaHF DIiQ * s e l f F di def @ s e l f D p A X return s e l f F l e G@ s e l f F d * pi * s e l f F di A * @RFG s e l f F diCsqrt @@ pi * p * s e l f F uo * s e l f F d AG@I t * PF * pi A A A ith plteD two lods nd wire we n onstrut simple system mde of the wire onneting two lods nd instlled over the plteF enyody my sy X ut how do we know tht the hosen prts re in dequy with the required system performnes c rere the frontier etween system onept nd fesiility is not so lerF e prt of experiene oming from the engineers intervenes in this proessF fut we my sy tht the mjor engine to mke the (rst hoies omes from evidenesD themselves oming from nture oservtionsF sn the humn eing historyD we see eh time tht humns tke exmple on the nturl systems to rete their own rti(il onesF lnes re good exmple for tht X in his (rst tenttiveD the humn tries to opy irdsF he plnes were reted step y step to ondut to the IHTCHAPITRE 8. GENERAL MODELLING OF A CYBER-PHYSICAL SYSTEM (CPS) rti(il irds we known todyF 8.2.2 Feasibility fetween ll the ojets possily used nd oming from the olletion O i D we must verify their ompline with the system requirements in its missionF his mens to verify etween othersD the pity to ept the environE ment onditionsF he fesiility is this ompline etween the omponent pilities nd the system needsF his informtion is inluded in the domin vlues nd the ssoited lws in the de(nition of the opertor zF he domin intersetions led to the system pilityF he fesiility of solution @ soE lution eing set O i AF fy exploring the vrious omintions of omponents nd dding the ouplings etween these omponentsD we n highlight the lks of the system prototypedF st is ler tht to reh suh performne in the nlyseD the models ssoited with eh omponentD their domins de(nitions depending of the environment prmeters must e urte nd pertinentF wthemtillyD we onnet the omponents of O i using onnetivity C whih retes losed )uxesF he reltion is simply given y X

b k = C k a Q a @VFIVA
unowing this reltionD the opertor z is trnsformed in the losed )uxes spe mking C T zC = ζF e then dd the ouplings to otin the mjor opertor representing the system nd its dominsD the whole mking mnifold whih is the system modelF ζ ontins ll the mehnil nd eletril informtionF herml exhnges re emedded in the domins settingsF st mens tht the omponent of ζ seems like X

ζ ij = t D I1 L 1 (•) + t D I2 L 2 (•) + . . .
@VFIWA yne kind of system is hosen @exoskeletonD endoskeletonA nd the misE sion known the mthemtil de(nition of fesiility n e detiled strting from the previous exposed oneptF he mission is delined y the tions mdeD the pity of ommuniE tionD the pity of hving soures of energy nd the energy onsumptionD nd the environment where the mission is relizedF yne kind of system hosen to nswer to the ustomer needs ensure the possiility to over the 8.2. TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIHU tions spei(edD the mission nd ustomer needs re priniplly delined in onstrints nd environment de(nition during the missionF sn our exmpleD it mens tht we hve hosen to mke system with plte Y mounting on this plte there is onnetor to mke the pinEsoket onttD nd some mehniE l prts to relize the rest of the mokEupF he environment onstrints re omponent or the soure vetor T @see eqution VFIAF he trget of roustness opposite the environment is desried in dominsF por temperture @tAD presE sure @pAD intervls re de(ned nd we know t Q i , p Q j @other onstrints my e de(ned s grvittionD etFAF por eh elements hosen etween others in the olletions O i D their tenfolds 2 should e de(ned on domins tht over the domins required y the ustomer needsF xoting x D i these domins we wnt to ompre x D i nd x Q i for ll the system spei(tionsF fut it is ler tht ll the system orgns won9t see these levels of onstrintF pirst exerise when mking new system is to propgte the onstrints from the environment to eh system prtF he system must e represented s set of suEprts interonnetedF ih suEprt is seen s zone nd reeives the onstrints nd on nother side trnsmits the onstrints to enlosed zonesF his study hs vrious unertintiesD ut t this phse of the projetD we don9t need to e very urteF he (gure VFR shows how the proess worksF sn the fesiE ility phseD the mjor work is onentrted in this tsk of trnslting the environment onstrints to the vrious prts of the system depending on its glol rhitetureF o determine this onstrints propgtion for the pressure we n use the sme pproh s for mehnisF e grph representing the vrious zones s nodes nd their links s edgesF imilr tehnique n e used for temperture s seen previouslyF ht9s wht we will see lterF he omputtion of the dequy etween the virtul system nd the future one tht will e used is relized with ene(t of the lsses informtionsF sn ftD the fesiility step is the trnsition step etween the oneption of the system nd the eginning of its developmentF vet9s tke n exmpleF e imgine system mde of one resistneF 2. A tenfold is a set of mathematical objects usable in the tensorial analysis of networks and leading to the manifold that represents a system or a part of the system. The tenfold concept was rstly developed in a thesis whose the author is the redactor : Maurice, O. (2013). Introduction d'une théorie des jeux dans des topologies dynamiques (Doctoral dissertation, Limoges).

Figure VFR ! gonstrints propgtion

Architecture side he lss rhiteture will ll vrious suElsses in order to mke the systemF feing very simple to illustrte this mehnismD we n rete struE ture 4resist4 de(ned for (rst option y X l s s r e s i s t I @ ojet A X def f i x r @ s e l f A X s e l f FaIHF return s e l f F hen the rhiteture is relized lling for eh suElss for struturesD lesD eletronisD etF sn our simple illustrtion it gives X l s s r h i t e t u r e @ e l e tr o n i q ue A X Par e s i s t I @A QaPF f i x r @A sf the temperture is the environment prmeterD we (rst dd the perforE mne of the struture temperture s lss method X l s s r e s i s t I @ ojet A X

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIHW

def h@ s e l f A X return I D P l s s rhi @ r e s i s t I A X ar e s i s t I @A hen reting n instne of the lss to de(ne n rhiteture X g=archi()D we n fter ll for the rhiteture performnes in temperture writing X g.R.D_T() whih return [1, 2]F he domin n e ompred with the needs x Q i F his seond domin is insried in n environment lss whih my e X l s s envir @ ojet A X def @ s e l f A X return I D Q rere we would hve got Q > D nd the struture resist1 would not e omE plint with the requirementsF he struture lss s the le lss or the eletroni lss hve for method n impedne opertor mtrix @illustrted y R in the lss resistIA onstruted on funtions delred in the property (eldF ith olletion of instntited lssesD the rhitet lss relizes the diret sum leding to the system de(nition opertor ζF his llows to evlute the system response nd its dequy with the requirementsF he intervls omprison indites the system pility to work in the mission environment while the glol opertor ζ sys if the system relizes the requiE red missionF trting from our previous exmpleD we (rst tke look to the dequy of the three omponents X strutureD le nd eletronis with the requirementsF he(ning " the struture length tolerne

x Q 1 = [0.9, 1.1] Y " the le urrent strength requirements

I Q 1 = [0, 0.1] Y " the eletronis lods r Q 1 = [100, 10 3 ]F
pirst step of fesiility mens to ompre the required intervls j Q i nd the informtions of pities inlude in the lssesF st mens to do there the omprisons X " plteFhx with

x Q 1

" wireFhs with

I Q 1 " eletroniFhr nd r Q 1 IIHCHAPITRE 8

. GENERAL MODELLING OF A CYBER-PHYSICAL SYSTEM (CPS)

sn the se of the wireD we see tht the model depends on prmeter tht n e set for the le X the dimeterF he rhitet hs to hoie the dequte dimeter to e omplint with this requirementF sn (nlD this olletion of orgns n e possile rhiteture for the needsF xow we hve to onstrut the model ssoited with this system to verify tht it n e solution for the requirementsD even for vrious mission pro(les or environment onstrintsF sn generl wyD the previous exerise n neessitte to mke delintion of onstrints to evlute the omprisonsF Declination of constraints he prolem is to lulte the evolving of the intervls α Q β from outside to insideF he exerise is very di0ult whtever the physis onsideredF he prdox is tht the onstrints n e delined only one the system is mdeD nd we wnt to deline the onstrints to determine the system 3 hepending on the system rhitetureD the onstrints re propgted more or less from the environment into the system odyF his propgtions determined y the oupling tensor µF his tensor is the elsti tensor for mehnisD the eletromgneti oupling tensor for eletronisD etF here re no other wys thn to mke ssumptions on these propgtionsF o mke the system we use three mjor opertions X IF seletion of the system omponents Y PF onnetion of the omponents to mke the system frme Y QF dding the oupling mehnisms reting the system omplete rhiE teture nd workingF st is ler tht the (rst step of system onept llows only to rejet options tht re out of the requirements y n evident wyF pesiility needs to go furtherF o do thisD the system is ssemled nd onstrints re pplied in orE der to mke (rst evlution of the system response versus these onstrintsF ht9s the only solution tking into ount tht some oupling mehnisms in ner (elds nnot e evluted without ssemling the systemF oD let9s mke the system 3 e omplete the previous lsses with some dded funtionlities tht ws neessry to onstrut system nd determine its dequy with the requirementsF 5--------------------------------------------------------------------8.2. TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIII l s s plte X def init@ s e l f D d D l DwD s A X s e l f F dad s e l f F lal s e l f Fwaw s e l f F sas s e l f F uoaRF * pi * Ii-U s e l f Fhxa s e l f F d D s e l f F l D s e l f Fw def @ s e l f D p A X return IFG s e l f F s * s e l f F l G s e l f Fw * @IFG s e l f F dAC sqrt @@ pi * p * s e l f F uo * s e l f F s AG@I t * PF * pi AA l s s e l e t r o n i X def init@ s e l f D r D iv D ig A X s e l f F rar s e l f F izvaiv s e l f F izgaig s e l f F hia-IiT DIiT s e l f F hraIiT DIiT def t@ s e l f D p A X i f @ s e l f F izg aaHFAX return s e l f F rCs e l f F izv * p e l s e X return s e l f F rCs e l f F izv * p CIFG@ s e l f F izg * pA l s s wire X def init@ s e l f D le D di D d A X s e l f F l eal e s e l f F diadi s e l f F dad s e l f F uoaRF * pi * Ii-U s e l f F hsaHF DIiQ * s e l f F di s e l f FhaHF DPHHi-Q def @ s e l f D p A X return s e l f F l e G@ s e l f F d * pi * s e l f F di A * @RFG s e l f F diCsqrt @@ pi * p * s e l f F uo * s e l f F d AG@I t * PF * pi A A A IIPCHAPITRE 8. GENERAL MODELLING OF A CYBER-PHYSICAL SYSTEM (CPS) 5d e f i n i g n r h i t e t u r e struaplte @Ii-Q DIF DHFP DIi-UA leawire @HFS DIi-QDIiUA lodIae l e t r o n i @IHHF DIi-W DHFA lodPae l e t r o n i @IiQ DIi-WDIHi-WA 5 omponent s e l e t i o n paIt * PF * pi * IiT 5 t e s t f o r I wrz frequeny

a stru F @p A D H F D H F D H F D H F D le F@p A D H F D H F D H F D H F D lodI F t@p A D H F D H F D H F D H F D lodP F t@p
A 5onnetions g a I F D I F D I F D I F miadot @ trnspose @gA DA madot @mi DgA 5dding oupling mehnism X here i n e r t i tensor u haSi-P 5 i n s t l l t i o n option f o r the le over the groundplne aTHF * log @RF * hG le F di A aQiV 5 seond i n s t l l t i o n option X no d i e l e t r i etween 5the wire nd the groundplne vIIaG * le F l e uavII * p mam H H C u foaIi-Q 5 mgneti f i e l d of the environment ea-le F l e * h * p * fo 5 eletrmotive f o r e uaeGm 5 urrent indued in the le ale F@pA * s @ sqrt @ dot @uD onj @uA A A A 5 i s i t in ompline with the wire pity c ompreamx@ le FhAG i f @ ompre bIFAX print 4his le n e used in the defined environment4 pirstD olletion of orgnsD inluding the lesD re villeF sn this olE letion we test n rhiteture ssumptionD mde of one plteD one wire nd two lodsF he (gure VFS shows the system we hve onstrutF efter mking the seletionD we de(ne the impedne opertor tensor from the digonl orgniztion of the seleted orgnsF his gives n ojet Z b de(ned in the edge speF

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIIQ

Figure VFS ! erhiteture relized hen we must onnet these orgns etween themF ht9s the role of the onnetivity mtrix CF yur system is mde of one mesh @its genus is IAF ih element oming from the lsses instnes prtiiptes to the ommon mesh of the systemD whih is shown in the t X yrgn snstne gommon mesh stru plte I le wire I lodI eletroni I lodP eletroni I his t leds to the onnetivity mtrix C whih gives the impedne opeE rtor tensor in the mesh spe Z m = C T Z b CF fut this opertion gives the vrious meshes mking the system glol strutureD without de(ning the ouE plings etween the elementsD inluding the inerti tensor 3 F he(ning here for the unique msh involved L = L 11 D we ompute Z m = Z m + L 11 pF L 11 is the indutne etween the wire nd the plte given y X

L 11 = Zc c Z c = 60ln 4 h φ @VFPHA
φ is the wire dimeterD h the height of the wire over the plteD c the propgE tion speed in the line mde y the wire over the plte nd Z c the hrteristi impedne of this lineF yne the system onstrutedD we n lulte the power dissipted y the urrent indued in the line y n externl mgneti 3. See Elements of theory for electromagnetic compatibility and systems, O.Maurice, Bookelis editor 2017.

(eld oming from the environment spei(tionF hen we ompre this vE lue to the mximum power eptle y the wireD whih is the omprison etween Q nd DF o do thtD we hve omplete the lss wire with the intervl wire.DP de(ned in powerF e hve relized the onstrints deliE ntion through the glol omputtion of the systemF his opertion n e mde for ll set of possile rhiteturesF fut if it gives the ompline of the imgined system with the environment requirementsD it remins to verify the ompline with the funtionl requirementsF his is done y exiting the system through soures to provoke its tions in movementD pereptionsD exhnge of informtionD etF hese system response must e ompred with the funtionl requirementsF sn ftD it hs een done qusi impliitly during the system onept elortionD ut this gte must e veri(ed one detiled system desription is villeF 8.2.3 Development et the strtD the system is simple ideD oneptF his onept n e seen s onstrution mde with elements oming from strt setsF st9s the sme s previouslyD ut the set h {} ontins no rel omponentF st points out the kind of susystem impliitly identi(ed y the nme of the setF his steps llows to de(ne the lrge lines of the rhiteture efore to detil the tehnologies involved in its reliztionF his reliztion goes through studiesD theoretil nlysisD simultions to end y some prototype not so fr from the (nl systemF his is the development phseF he vrious ojets oming from the seleted set O i {c i , h i , w i , r i } re identi(ed with their impedne opertor ζ onstruted from ll the seprte element in the ojet nd from the onnetivity C k a F he si system is electromechanical nd the local environment is thermo-acoustical while the far environment is electromagnetic and gravitationalF yften the (rst tion mde in lssil systems onerns the strutureF st is mde of prts ζ mv oupled eh otherF hen eletril prts ζ ev re dded nd oupled eh other nd with the mehnil prtsF pirst tion mens to diretly summing oth opertors X

ζ mv ⊕ ζ ev @VFPIA
then to dd the ouplings α me nd α em F he sme opertions n e pplied to the grph of eh ojetF sn ft the more usul tehnique strts from

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIIS

the grph representtion of the ojets nd guide the opertions to e mde on the orresponding ellulr omponentsF he ouplings re emedded y extrEdigonl omponents euse even in hrd oupling se @when mteE ril is shredAD rnh is shred etween two meshesF his is trnslted y oupling funtion whih is the shred rnh lwF sn other sesD for ll fr (eld intertionsD ords wer the oupling funtionsF he industril mngement of the development is not so trivilF hiE tionries must e used to rhive the knowledge nd hrteristis of the primitive ojets seletedF sn up phsesD eh omponent is repired y its mjor propertiesF hese properties will e ompred to the ville ones in some list of prts p {}F he previous proess is followed s desried eforeF sf the oupling mehnism ppers nd is lssilly used in eletromE gneti for uron9s formlismD he exerise hd to e done for mehnisF sn the development phseD ll detils re explored nd should e urE tely de(ned t the end of this phseF gompred to the fesiility phseD dded opertions desrie t lower sle the di'erent system prts in order to e nerer to the relityF yne the hoies re de(nitively writtenD some simulE tions using meshed tehniques @testing virtully the systemA re doneF hey on(rm the wited ehviours nd n e ompleted y some experiments on prototypesF por exmpleD when mking the system priniplesD we developed some links for usesD power supplyD FFFD ut not for the whole signlsF he demonstrtion of fesiility do not need to onsider ll the eletronis nd their urte de(nitionF ynly some representtive signls re su0ient to demonstrte the pertinene of the rhitetureF fut in the vlidtion phseD the whole hrnesses re tken into ount in the @virtul or relA protoE type nd its ehviour in vrious environment is omputedF e prt of this detiled development is devoted to the suppliersF por thtD the delintion of onstrints must e lerly written s spei(tion requirements given to the suppliersF por exmpleD sed on (rst ssumption of impednesD the eletromgneti onstrints re mesured ross the eletronis inputD using the input lods electronic.ZJF king some mrginD this vlue is trnsmitE ted to the equipment supplier s requirement to e overF he equipment supplier will develop his eletroni equipment tking into ount this trget onstrintF iven if the system is not desried for ll its signlsD eh kind of links is lredy desried s funtionl linkF end nother very importnt tehnique intervene in order to mke the system glolly modelled X zoningF por vrious reson nd (rst for seurityD it is importnt to seprte the system in vrious zones su0iently segregted etween eh otherF ith this proE IITCHAPITRE 8. GENERAL MODELLING OF A CYBER-PHYSICAL SYSTEM (CPS) pertyD the system n e studied y prtsF e n try to give this de(nitions under mthemtil onesF he (rst step is to relize the diret summtion of ll the orgns opertors z i X z = ⊕ i z i F efter wht we dd the ouplingsF sing zoning mens tht the oupling mtrix presents wek oupling etween some system prtsF sf z a is (rst suEmtrix of z nd z b seond suEmtrix of zD wek oupling etween z a nd z b mens tht single oupling funtion [b] exists nd this oupling ours without modifying the soure hrteristisF nder this ssumptionD the prtil mtrix tthed to the ouple z a nd z b seems like X

z ij , i ∈ [a], j ∈
z a z ij z ij z b @VFPPA he seond eqution leds to Z ij k j = Z b k b he
two prts a nd b n e seprted reporting the oupling s n externl onstrint on the seond prtF @the sme n e mde with the (rst prtAF his priniple llows to study the system y prtsF his is key tehnique to gurntee the system sfetyD its whole onstitution eing too muh omplex to e nlysleF pinlly two mjor ssumption re mde in order to ondut the system de(nition X IF he system n e studied y prt Y PF typil impednes for the eletronis re knownF his seond point is ritil oneF hen mking the (rst lultion on the system to see its fesiilityD it9s neessry to know the input impednes of the eletronisF nder this ssumptionD delintion of onstrintsD spei(E tionsD etF n e mdeF ss this ssumption resonle c wore nd more it is 3 wodern usesD power suppliesD numeri iruits present similr impednesF he diversion in their vlues n e overed y mrgins or prmetri stuE diesF vet9s illustrte this pproh with our previous exmple of wire over ground plneF he mesh impedne is simply

L f p + R f + Z G + (Z 1 + Z 2 )F L f p nd R f p
re the impedne of the wireF Z G the impedne of the ground plneF Z 1 nd Z 2 re the impednes on the equipment inputsF hepending of the kind of eletroni involved in the iruitD these two impednes tke vrious omplex vluesF he onstrint reported on one equipment is de(ned y X

P = 1 4 {ui * + u * i} @VFPQA
u nd i eing the voltge ross the lod nd the urrent )owing in the lodF

TENFOLD ANALYSIS OF COMPLEX SYSTEMS (TACS) PRELIMINARIESIIU

sn our seD this gives X

P = 1 4 Z x k 0 (k 0 ) * + Z * x (k 0 ) * k 0 = 1 2 k 0 2 (Z x )
@VFPRA he urrent is given y X

k 0 = e 0 L f p + R f p + Z G + (Z 1 + Z 2 ) @VFPSA or X k 0 2 = e 0 [ Z G + R f + (Z 1 + Z 2 )] 2 + ω 2 [L f + (Z G + Z 1 + Z 2 )] 2 @VFPTA yften Z G ≈ R f → 0F ith Z G << L f we otin X k 0 2 = e 0 [ (Z 1 + Z 2 )] 2 + ω 2 [L f + (Z 1 + Z 2 )] 2 @VFPUA por Z x = Z 1 = Z 2 this leds to X P = 1 2 e 0 Z x [2 Z x ] 2 + ω 2 [L f + 2 Z x ] 2 @VFPVA
por the mjority of eletronis @exept eletril mhinesD etFAD 2 Z x << L f nd X

P ≈ 1 2 e 0 Z x [2 Z x ] 2 + ω 2 L 2 f @VFPWA
e see tht under these ssumptionsD the result depends only of the rel prt of the lodsF he prolem is redued y studying the order of vlue for this rel prtF end they re often known X for nlogilD numeril signls it9s typilly I M ΩF por hyperEfrequeniesD SH ΩD etF fut if we onsider the voltge s riterionD the onlusion is di'erent X

|u| = |Z x | k 0 = e 1/2 0 Z 2 x + Z 2 x [2 Z x ] 2 + ω 2 L 2 f @VFQHA
sn tht seD the imginry prt in)uenes the resultF sn tht se we my onsider the worst se given y the higher redile impedneD s muh for the rel prt s for the imginry prtF he equipment input geometry gives mny informtion for evluting the indutneF he frequeny nd of the IIVCHAPITRE 8. GENERAL MODELLING OF A CYBER-PHYSICAL SYSTEM (CPS)

iruit is nother soure of informtion to evlute the pitne of the impedneD nd the resistne is sometimes given y the iruit onsumption or the mth ondition in hyperEfrequeniesF snformtions on the impednes n lso e otin reding dtEsheetsF he min prolem omes from the impedne knowledge nd iruit ehviour in outEnd stressF he threshold lso must e evlutedD s it is not the sme for outEnd or inEndF por outEndD the threshold level is in generl unknownF fut mny expeE riments mde in the yer VH9 hve shown the next results 4 IF outEnd riterion is de(ned in trnsmitted power Y PF the trnsmitted ontinuous level @or in generl 4inEnd4 levelA is shre of the totl powerF he prolem eomes to evlute the ÷0ient to e pplied to the pek vlue of the trnsmitted power to lulte the e'etive inEnd energyF his ÷0ient is lled ÷0ient of detetion e0ienyF ithout ny more inE formtionD we n tke HDI for vlue of this ÷0ientF unowing the level trnsmitted to the omponent through its impedne input shemtiD the proportion of this level is ompred with the funtionl inEnd threshold in order to onlude on the disturne riskF 8.2.4

On the risk of disturbances costing he proility to provoke the disturne of omponent is ssoiE ted with the distne etween two funtions 5 F sf we test omponent to eletril stressD the numer of omponents of extly the sme referene depending on the stress level seems like qussin urveF fy the ftD the proility tht the omponent eomes distured for stress superior to given level u is urve hving the form shown (gure VFTF his urve sys tht under give levelD no omponent re disturedF yver given levelD no omponent exists eing le to support the onstrintD nd n verge of the tested omponents re distured for the level t mid rise of the urveF he onstrint n e hrterized y the sme wyF feyond given vlueD no onstrint existsF felow given vlueD it exists lwys hen the onstrint funtion ounded y v c hs the form X

P (u > v c ) = 1 -1 + βexp - u v c -1 @VFQPA
nd the proility P df to distur the omponent is given y X

P df = 1 -1 + βexp - u v c -1 1 + αexp - u s l -1 @VFQQA
8.3 Application to system of systems (SoS) ystem of systems re group of systems in intertionsF hese interE tions n e ommunitionsD eletromgneti intertionsD mehnil inE tertions @we n think in meteorite swrmAD etF gompre to systemsD the di'erene is tht there re no physil ontts etween the systemsF o the intertions re only through fr (eldF it mens tht the o dimension is the opertor ζ dimensionF he system onstrution is relized y oupling vrious independent systems y wvesF xo onnetion y rnhes or ner (eld intertions re usedF he ide is to sy tht the system of systems doesn9t hnge the intrinsi hrteristis of eh systemF he est wy to sy tht is the de(nition X Denition : Giving a set of systems of operators {ζ i }, a system of systems is made by the direct summation of the systems ⊕ i ζ i enriched by an interaction tensor only made of weak interactions cords. he pproh n e perfetly similr to the one of systemsF ih system n e seen s lss with its methods @pities of tionsA nd propertiesF he rhiteture eomes the tion of o oneptionF edding the interE tions like ommunition network trnsforms the set of systems in system of systemsF e shred informtion is the si nd indispensle key to mke oF wore thn elsewhereD the o uses prtiulr intertion X the deE ision intertionF his kind of exhnge lls for ords elorted from the gme theoryF here re three kinds of informtions in o X IF informtions oming from outside E the environment Y PF informtions exhnge etween the systems in the o Y QF informtions exhnge inside eh systemF

APPLICATION TO SYSTEM OF SYSTEMS (SOS)

IPI here is one level more thn in systemsD oming from the ft to group systems in order to mke systemF st9s true tht this kind of seprtion n e mde t lower slesD for exmple onsidering miroproessors in systemF ith this point of viewD ny system is oF he distintion omes from the de(nition of system itselfF sf we onsider miroproessorD it9s not system in ft euse it is not le to relize ny mission y itselfF sf we ept in system de(nition in the sense of rti(il system tht it leds to some mission in the soietyD our levels of intertions seems to e su0ientF enother wy to disern o from systems is to sy tht in ny se nd whtever the riteriD the o hs one more intertion lyer thn systemsF he need of gme theory to model ll intertions for o is the (rst step to go from rti(il systems to nturl onesD or to e inspired y these nturl systemsF fut efore llD we must relize some softwre implementing this pproh nd thenD hving demonstrte the possile use of the formlismD it is possile to inrese its omplexity tking into ount humn ftorF Chapitre 9 SUETAN iex ojet is to implement previous tehniques in softwre for eletromgneti omptiilityF he sme pproh n e used for other josF 9.1 Steps in the method iquipments must e de(ned y mtrix z giving their struture @note tht the pproh we sumit for systems n e eforehnd used for equipE mentsD onsidering them s systems 3AF sn these mtrixD some rnhes elongs to the equipment frontier nd n e identi(ed for hving this propertyF fE silly it9s lwys possile to dd rnh of high impedne @round I qΩA without disturing t ll the nominl working of the equipmentF hen this impedne n e repled y nother impedne in order to trnslte onnetions with lines etween two or more equipmentsF ht9s the seond step when mking systemF vst step onsists in dding the intertions in reltion with the struture @intertions etween equipments or lesAF ditive ports n e esily reted like proes oupled with the eletronis inside the equipmentsF sn these equipments there re lso (eld mines imposed y the strutures nd leding to oupling mehnisms etween the eletronis nd the strutures openingsF sn ll ses the opertors must over the whole frequeny ndF st mens tht domins re used to point out di'erent lws dpted to these frequeny ndsF enother point is tht the opertor re written under vple9s formlismF fsillyD it overs oth time nd frequeny dominsF hepending on eh irE IPQ uitD vrious solutions will e used to ompute non liner omponentsF st n use nonil vple9s funtion properties nd non liner omponent resE ponse reltively to these propertiesF he non liner omponents n inlude genertors reting the hrmonis nd setted y the soure wveform ross themF hese re here the si priniples for iexF 9.2 Closing frontier edges efter relizing the instntitions of the equipments lssesD we need to lose their ports y opertors in reltion with their intertions kind with other equipmentsF sn generl this is limited to wire intertionsF sn lss @whih is the softwre expression of tenfoldA we (nd X IF the equipment opertor Y PF the equipment soures Y QF the ports de(nitions Y RF the intrinsi signls ntures nd wveforms Y SF the onstrints @threshold levels proilityD soure level proilityD prmeter limitsD etFA Y TF the equipment dimensionsF por tenfold @or lssA t 1 D we hve the iGy ports numering IF ememer tht the opertor in tenfold re desried in the meshes speF sf this tenfold is simple sinus genertor of soure e 1 D its opertor my e given y R 1 + P Q 1 where P Q is the port defult nme nd vlueF e n imgine seond equipment t 2 with n opertor R 2 + P Q 2 F xow we deide to mke system with these two equipmentsF ht is line model to onnet them c sn low frequeniesD line is simply telegrph ellF e hve one onstrint X our model must de(ne opertor for the frontiers then ord to onnet the iGyF o our line model must respet these rulesF sf we onsider one telegrph ellD it is mde of two indutne nd one pitor @eventullyD it inludes resistnes nd one ondutneD ut this is detilsAF gonneting equipments t 1 nd t 2 y ports P Q 1 nd P Q 2 mens to group the two equipments mking 9.2. CLOSING FRONTIER EDGES IPS

t 1 ⊕ t 2 then to reple X    P Q 1 → Lx 2 p + 1 Cxp P Q 2 → Lx 2 p + 1 Cxp @WFIA
L nd C eing the line indutne per meter nd pitor per meterD x the line length nd p vple9s opertorF he ord etween the two meshes @eh equipment is meshA is the shred rnhD iFeF the pitorF hen the whole opertor for the onneted two equipments is @we note

L/2 = L X z -   R 1 + L xp + 1 Cxp -1 Cxp -1 Cxp L xp + 1 Cxp + R 2   @WFPA
his model n e used in low frequeniesD iFeF until the line length remins short ompred to the wvelengthF he limit frequeny f l n e de(ned y

f l = c 4x
@WFQA he previous model must e limited to this frequeny dominF o indite this onditionD we multiply the orresponding lws y the domin restrition

ν D BF with BF = [0, c/4x[ X z =     R 1 + L xp + 1 Cxp ν D BF - ν D BF 1 Cxp - ν D BF 1 Cxp L xp + 1 Cxp ν D BF + R 2     @WFRA
et higher frequeniesD frnin9s model must e usedF sn tht se P Q 1 = P Q 2 = z c the hrteristi impedne of the line nd the two de(ned ords re X

   α 12 = (R 2 -z c ) e -τ p α 21 = (z c -R 0 ) e -τ p
@WFSA e seond soure e 1 e -τ p eing dded to the t 2 souresF sf HF = [c/4x, f T EM [ @f T EM eing the limit frequeny of the line in iw modeD fter whtD modl propgtion should e modelledA nd if X

     β 12 = -1 Cxp z bf = L xp + 1 Cxp @WFTA he whole frequeny nd opertor is X z =    R 1 + ν D BF z bf + ν D HF z c ν D BF β 12 + ν D HF α 12 ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF z c + R 2    @WFUA
por the (rst step tht interests usD we (nlly relize the losing of the frontiers y X

     P Q 1 → ν D BF z bf + ν D HF z c P Q 2 → ν D BF z bf + ν D HF z c @WFVA
xext step is to de(ne the ordsD thing tht we hve lredy omputed for linesF 9.3 Adding cords for lines pirst step of losing frontiers leds to the opertor de(ned y X

z =    R 1 + ν D BF z bf + ν D HF z c 0 0 ν D BF z bf + ν D HF z c + R 2    @WFWA
fter wht we n dd the ords oupling the iGy of the equipments in reltion through the lineF e otin the opertor desried eqution WFUF he opertion to rete the linesD fter the losing step is nothing more ut nothing less thn thisF xote tht there is prdox in de(ning lines efore to de(ne the strutureF sf the line is simple wire using the struture s seond ondutorD the theoretil expression of the hrteristi impedne must e setted one the struture is de(nedF he tehnique is the sme tht the one used for the losing stepF en strt vlue is written in ple of the de(nitive vlue depending on the strutureF e understnd tht for the theoretil nlysis of our systemD everything n e expressed in the order we sumitF por two wires lineD the hrteristi impedne is (xed y the line itselfF he struture n only distur this vlueF 9.4. GROUPING LINES IN HARNESSES IPU 9.4 Grouping lines in harnesses sf we onsider two lines l 1 nd l 2 of opertors z 1 nd z 2 similr to the previous oneF hese two lines re grouped in shred hrnessD whih mens to mke

z = z 1 ⊕ z 2 F o X z =              R 1 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 0 0 ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 2 0 0 0 0 R 3 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 0 0 ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 4              (9.10)
es previouslyD we hve to dd the ouplings etween the two linesF hese ouplings re seprte like the lines in two domins X low nd high frequeny ndsF 9.4.1 Grouping lines in low frequencies fetween eh ouple of lines we ompute the mutul indutneD de(ned y xeumnn9s reltion X

M = µ 0 4π ‹ x,y dx • dy r xy @WFIIA
he mutul indutne is reported on eh extremity of the opposite lineF his dd the oupling terms X

z =              R 1 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 -M p 0 ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 2 0 -M p -M p 0 R 3 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 0 -M p ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 4              (9.12)
he lod umulted on (rst pitor of (rst line q genertes n eletri (eld E given y X E = q 4π r 2 @WFIQA he eletromotive fore indued on the seond line is e = hE if h is the line heightF xoting X

C 12 = h 4π r 2 @WFIRA we (nd e = q/C 12 F fut q = C 1 V with V = 1/C 1 piF pinlly X e i = 1 C 12 C 1 1 C 1 p = 1 C 12 p @WFISA nd (nlly X z =                R 1 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 -M p - 1 C 12 p 0 ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 2 0 -M p - 1 C 12 p -M p - 1 C 12 p 0 R 3 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 0 -M p - 1 C 12 p ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 4                (9.16)
his result is due prtly to the ft tht there is no propgtion in the modelF 9.4.2

Grouping lines in high frequencies hue to the propgtionD the sitution is more omplited ut ws studied y tenEul re @see refereneAF e de(ne (rst two ÷0ients α nd β with X

α = γ C+γ K = M (C+γ)
Lγ @WFIUA nder some ssumptionsD the nerEend rosstlk is given y X

e n = α K + 1 2 z c i opp @WFIVA
where i opp is the urrent of the opposite line nd z c its hrteristi impeE dneF he frEend rosstlk is given y X

e f = -α(K -1)τ pz c i opp @WFIWA
τ is the eletril length of the lineF es the line is represented y ouple of meshesD the ords reted y the two previous reltions ssoites one mesh with two othersF pigure WFI shows these ordsF hese two ords exist etween ll the ouple of meshes of the two linesF he (gure WFP shows the four ords tht intervene in the oupling proessF pinlly for the nerEend ross tlk we hve X

z 31 = - ν D BF M p + 1 C 12 p + ν D HF α K + 1 2 z c @WFPHA
z 41 = - ν D HF (α(K -1)τ pz c ) @WFPIA z 42 = z 31 D z 32 = z 41 D z 13 = z 31 D z 24 = z 42 D z 14 = z 41 nd z 23 = z 32 F he whole opertor z is now X 9.4. GROUPING LINES IN HARNESSES IQI R 1 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 - ν D BF M p + 1 C 12 p + ν D HF α K+1 2 zc - ν D HF (α(K -1)τ pzc) ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 2 - ν D HF (α(K -1)τ pzc) - ν D BF M p + 1 C 12 p + ν D HF α K+1 2 zc - ν D BF M p + 1 C 12 p + ν D HF α K+1 2 zc - ν D HF (α(K -1)τ pzc) R 3 + ν D BF z bf + ν D HF zc ν D BF β 12 + ν D HF α 12 - ν D HF (α(K -1)τ pzc) - ν D BF M p + 1 C 12 p + ν D HF α K+1 2 zc ν D BF β 21 + ν D HF α 21 ν D BF z bf + ν D HF zc + R 4 (9.22)
IQP CHAPITRE 9. SUETAN ith this opertor the two lines re oupled in ll the frequeny ndF 9.5 Immersing the harness in a structure, in low frequencies he (eld ehvior t low frequenies is ompletely di'erent from the one t high frequeniesF he in)uene of propgtion nd uslity whih doesn9t exist in low frequenies hnges the lwsF es for linesD the low frequeny (eld is well modeled using lumped elements indutnes nd pitorsF enother golden rule is to rememer tht the (eld nnot e stopped 3 he (eld n e omputed nd exists everywhereF ht9s the omintion of inident (eld nd refrted one whih leds sometimes to null (eld in some spe regionF et low frequeniesD the equivlent shemti of vity is given (gure WFQF Figure WFQ ! iquivlent shemti of vity in low frequenies he inident mgneti (eld retes n eletromotive fore given y -pS.BF his emf retes urrent i tht retes mgneti (eld in opposition with the inident mgneti (eldF he glol mgneti (eld t low frequenies is ner to the inident oneD ut t higher frequeniesD nd more prtiulrly eyond the skin e'etD the reted (eld is equl to the inident one nd the totl mgneti (eld eomes equl to zeroF wthemtillyD we hve X

IMMERSING THE HARNESS IN A STRUCTURE, IN LOW FREQUENCIESIQQ

e LF = -pS.B ⇒   e LF e LF   =   R + 1 Cp + Lp -1 Cp -M p -1 Cp -M p R + Lp + 1 Cp     k 1 k 2   @WFPQA ut p → 0 so X ∆ = R + 1 Cp 2 - 1 C 2 p 2 = R 2 + 2 R Cp @WFPRA nd k x = 1 ∆   R + 1 Cp 1 Cp 1 Cp R + 1 Cp     e LF e LF   @WFPSA nd for exmple X k 1 = 1 R R + 1/Cp R + 2/Cp e LF → e LF 2R
@WFPTA whih is the hg solutionF he retion (eld is so wek tht it doesn9t interE vene in the urrent vlueF he (gure WFR shows the mehnisms involved in low frequenies for the mgneti (eldF @WFPUA sf the soure of B i is frD the totl (eld t point y is X

B i + ˆx B d [y, i(x)]
@WFPVA sf we onsider the prtiulr exmple of ylindril vityD (xy) = r nd in the enter X

B d = ˆθ dθµi r.1 4πr 2 = µ i 2r
@WFPWA end the totl (eld n e omputed X

B = B i + B d = B i 1 - µpπr 4R → B i @WFQHA
e see tht t very low frequenies @p → 0AD the inident mgneti (eld is not ttenuted t llF fut fortuntelyD the indued emf is lso lowF ht this emf whih is indued on the lines enlosed in the vityF he equivlent shemti for the lines re lso the low frequeny oneF ih meshes of the opertor WFIT will reeived n emf -pS.B i reting noise dded to the funtionl souresF he prolem of low frequeny (elds inside vity leds to n impedne of the potentil V inside the vity whih is very highF es soon s we inrese little the frequenyD the mjor shemti eomes RL oneF vet9s show thtF e n hoie the meshes s desiredF he rnh urrents re not 'eted y this hoieF st mens tht whtever the hoieD the onnetivity leds to the sme results in the rnh speF his omes from the property of invrine ssoited with the tensoril nlysis of the networkF e n hoie meshes leding for the sme prolem s efore to the impedne opertor X

z =   R + Lp + R + Lp R R R + Lp + 1 Cp   @WFQIA
nder this ssumption we (nd X

k 1 ≈ e LF 2R+2Lp k 2 ≈ e LF R+1/Cp → 0 @WFQPA

IMMERSING THE HARNESS IN A STRUCTURE, IN LOW FREQUENCIESIQS

he urrent is determined y the losses nd mgneti dimensionsF the seond uto' frequeny is given y X

f 0 = 1 2π √ LC @WFQQA
sn low frequeniesD the ehvior is similr to v iruitF hen the pitor intervenes ner the resonnt frequenyF he indued urrent is driven y the resistne then the indutneF his indutne is de(nedD for vity of depth w y X

L = µ S w

@WFQRA

S is the )ux setionD the sme seen in the emf -pS.BF R is de(ned y ρP/(w.d)F P is the setion perimeter Y d the wll depth nd ρ the vity ondutivityF he urrent frequeny evolving is X

k x = -pS.B i 2R + 2Lp
@WFQSA hen the frequeny @(rst uto' R/LA is high enough this tends to X

k x = - B i µ w @WFQTA
or written in nother wy X ˛w dw.B i = µk x @WFQUA whih is empere9s equtionF his eqution is the sme for the retion (eld -B d F his implies tht B d = -B i F o fter this (rst uto' frequenyD the totl mgneti (eld tends to zeroF he (eld deresing go fster nd fster with frequeny due to the skin e'etF pigure WFS gives the urve of the indued urrent nd of the totl mgneti (eld inside the vityF e see tht tking into ount the skin e'et in the resistneD the (eld mplitude dereses very fst to led zero up to round I krzF he totl mgneti (eld mplitude is evluted y X

|B T | = α µ w B i -k x @WFQVA
he onstnt β = w/(2πµ) n e lled the mgneti struture onstntF st gives the frequeny eyond whih the totl (eld disppersF he inident eletri (eld is cB i while the eletri (eld inside the vity is otined from the voltge ross the pitne X

E = 1 h (Rk)
@WFQWA sf there ws not the skin e'etD the eletri (eld my exist t higher frequenE iesD respeting the reltion

E = R h k 1 + 1 hCp k 1 -k 2 @WFRHA
fut due to the ft tht the skin e'et uto' frequeny rrives sooner thn the RC uto' frequenyD the urve @given (gure WFTA is driven y the mgneti (eld nd the single reltion X ˛w dwB T = µσS.E T @WFRIA ithout ny openers in the vityD the ojets inside the vity re omE pletely isolted one the skin e'et uto' frequeny is exeededF fefore this frequenyD the emf indued in ny ojet enlosed in the vity n e omE putedD knowing the totl (eld vlueF his gives the intertion etween the vity nd the ojetF his intertion is symetriD iFeF it is the sme for the intertion etween the ojet nd the vity in the system impedne opertorF yne the skin e'et frequeny exeededD only (eld oming from openers n e trnsmitted inside the frequenyD s the totl (eld resulting from the intertion of the inident (eld nd the indued urrents is equl to zeroF xote like in shielded lesD the ommon impedne oupling proess injets voltge in ojets onneted to the vity nd this for frequenies elow the skin e'et uto' frequenyF ih domin delimiting the low nd high frequeny domin n evolve di'erentlyD depending on the ojets dimensionsF ht9s why one onstrint of this modeling is to keep the mesh numering nd ssoitionF fut it is sometimes very di0ultF enother solutionD simplest when prtied onsists in pplying the frequeny domins to the whole opertors rther thn on their omponentsF st mens tht the whole system is seen for known sle in time or frequeny dominF homins re kept t the omponent level when they re intrinsi to the devie de(nitionsF por exmpleD for diodeD it n e the non liner voltge nd urrent thresholdsF he est method is proly to study the system strting from low frequenies nd inresing fter the frequeniesD using opertor dpted with eh dominF Making a system in low frequency assumption trting from WFIT we n diretly ene(t of the expression z eq of ouple of lines onneted to two equipments used in low frequenies X

z eq = ν D BF            R 1 + z bf β 12 -M p -1 C 12 p 0 β 21 z bf + R 2 0 -M p -1 C 12 p -M p -1 C 12 p 0 R 3 + z bf β 12 0 -M p -1 C 12 p β 21 z bf + R 4           
@WFRPA yn nother side we hve losed vity of opertor X

z ca = R + Lp @WFRQA e mke system grouping z ca nd z eq X z s = z ca ⊕ z eq X z s =                R + Lp 0 0 0 0 0 R 1 + z bf β 12 -M p -1 C 12 p 0 0 β 21 z bf + R 2 0 -M p -1 C 12 p 0 -M p -1 C 12 p 0 R 3 + z bf β 12 0 0 -M p -1 C 12 p β 21 z bf + R 4               
@WFRRA e write the intertion etween the vity nd the ouple of equipments X

e k = p S l 2 α = z eqca @WFRSA
knowing tht B T = α k -k x nd S l the whole setion etween the lines nd the vity @ hlf vlue is ssoited with eh hlf line under the model with two meshesAF o this intertion is dded genertor e B = -pS l α k/2F enother oupling mehnism omes from the shred prt of the vity used y the lines enlosed in the vityF sf R l /2 is this ommon impedne oupling vlueD we hve X

z s =                R + Lp R l 2 + z eqacav R l 2 + z eqacav R l 2 + z eqacav R l 2 + z eqacav R l 2 + z eqacav R 1 + z bf β 12 -M p -1 C 12 p 0 R l 2 + z eqacav β 21 z bf + R 2 0 -M p -1 C 12 p R l 2 + z eqacav -M p -1 C 12 p 0 R 3 + z bf β 12 R l 2 + z eqacav 0 -M p -1 C 12 p β 21 z bf + R 4               
@WFRTA ith the soure vetor T = e i e B e B e B e B D with e i = -pSB i D the tensoril eqution T a = ζ ab k b solve ll the prolem for the low frequeny dominF 9.6 Immersing the harness in a structure, in high frequencies here re two wys to model vitiesF e (rst wy uses resontors nd seond wy uses guidewves struturesF he (rst is the more physil one nd the esier to useF fut it requiers mny dimensions when we wnt to over wide rnge of frequeniesF et the ontrry the seond needs more e'ort to e understood ut overs impliitly lrge domin of frequeniesF 9.6.1

Resonator approach eny struture n e seen s topologil surfeF ves hrterize these struturesF here re two kinds of lesF ves going round n empty volume nd les going in the diretion of volume xeF pigure WFU illustrtes this mehnismF eross the les surrounding the volume we n tre two lines perpendiulr eh otherF he longitudinl line is kept s it isF pinlly the three lines wer diretions of (eld modesF qiving nme to eh xe @for exmple DDA we n onstrut fter tht n pproximtion of the (eld mode A abc F unowing the (eld distriutionD we n ompute the intertion etween this (eld nd vrious reeivers insert inside the volumeF he modl (eld is @WFRUA while for the eletri (eld we hve X

W E = 2 ˚V dvE 2 (a, b, c) @WFRVA nd we hve X 1/2CV 2 = W E nd 1/2Li 2 = W B F ytherwise X V = ´a da • E i = 1 µ ¸c dc • B
@WFRWA he mjor di0ulty in generl omes from the estimtion of the resisE tne vlueF he simplest wy to determine its vlue omes from the qulity ÷0ientF he qulity ÷0ient Q is the rtio etween the losses nd the stored (eld energyF vosses n ome from openersD losses in wllsD soring mterils inside the vity volumeD etF st n e seen lso s the rtio etE ween the energy tht )ows outside the vity @tht9s the se of lossesA on the energy stying inside the vityF et low frequenies we hve seen tht the losses re diretly ssoited with the resistnes of the wll inluding the skin e'etF ht9s hppen t the (rst mode for exmple c he urrents in the wll follow the eletri (eld limit ondition through yhm9s reltion J = σ •EF sn ft rther thn working with (elds we my work with urrents nd lods nd de(ne urrent modesF fut to ompute the ouplings etween elements inside the vity we need nywhere the (eldsF peynmn9s hd tried to omE pute intertions without (eldsF fut in (nl it9s not ompletely stisfying nd (elds must e kept 3 por the (rst mode in ui vity

E xyz = E 011 D the eletri (eld is given y X E(x, y, z) = E 0x Sin π y Y Sin π z Z
@WFSHA he urrents follow the wllF yn the two plnes (y, z) for x = 0 nd x = XD the urrent onverge to the lod orresponding to the mximum mplitude of the eletri (eldF o X

x = 0, x = X, (y, z) ∈ [0, Y ], [0, Z] ⇒ k = w B(x, y, z) µ u r @WFSIA
r eing the rdius of the irulr referentil ssoited with the plnF he power of losses P R is given for these plns y X

P R =    ˆ2π 0 ˆR 0 drρ 1 δrdθ w B(x, y, z) µ 2 -1    -1
@WFSPA he urrent mplitude is given y the mgneti (eld mplitude on the limit onditionsF yn the other wllsD the distriution of the urrent is esily desriedF por exmple on the (rst wll X

P R =    ˆY 0 ρ δdy ˆX 0 dx w B(x, y, z) µ 2 -1    -1 @WFSQA
in oth ses δ is the skin depthF he resistne vlue n e otined knoE wing the urrent trjetoriesF por the seond se this is X R -1 = ˆy dy ˆx dx ρ δdy ¯1 @WFSRA wking these omputtionsD one resontor is ssoited with one modeF por eh omintion (x, y, z) of mode numer we hve vg iruitF unowing 9.6. IMMERSING THE HARNESS IN A STRUCTURE, IN HIGH FREQUENCIESIRQ the (eld distriution ssoited with this mode through the ssumption of les xesD we n ompute the emf indued y this modl (eld in vrious ojets enlosed in the vityF he (eld n e ssoited with the urrent in the vg iruitF vet9s tke n exmpleF sf in retngulr vityD the eletri (eld is given y X

E x = E 0 Sin nπ y Y Sin mπ z Z → T E n0m @WFSSA wxwell9s eqution ∇ × E leds to X    ∂E ∂y = -pB z -∂E ∂z = -pB y @WFSTA
es on nother side ∇ × B = µJ we otin X

µJ x = 1 p ∂ 2 E x ∂z 2 + ∂ 2 E x ∂y 2 @WFSUA
por polristion following xe xD the reltion etween the urrent nd the (eld is given y the potentil developed ross the pitor X

1 Cp k x = V = ˆx dx • E x @WFSVA
sf we ple little loop inside the vityD perpendiulr to the xes y t oordintes (x l , y l , z l )F he setion of the loop is S l nd the emf e l indued in the loop is X e l = -S l • pB y (x l , y l , z l ) = -S l ∂E x (x l , y l , z l ) ∂z @WFSWA epling the eletri (eld pr its derivtive nd for the oordintes where the loop is lotedD this gives X IF from the les of the empty volume of the vitiesD we determine for one polristion the modes xes Y PF for this (eld distriution nd polristion qD we otin from relmotz9s eqution X

∂ 2 E q ∂x 2 + ∂ 2 E q ∂y 2 + ∂ 2 E q ∂z 2 + ω c 2 E q = 0
F his leds to the dispersion K nd the resonnt frequenies f nm,q Y QF knowing the resonnesD we n ompute for eh of them the pE itor omputing X

1 2 Cχ 2 E 2 0 = 1 2 ˚v dvE q (n, m) • E q (n, m)
Y RF he indutne is otined from ω 0 = 1/ √ LC Y SF the resistne is otined omputing the resistne of the urrent linesF his lines re perpendiulr to the mgneti (eld ner the wlls nd de(ned y the eqution ∇ × B = µJ Y TF if n ojet is insert in the vityD we ompute the emf indued on this ojet nd express it depending on the urrent in the resontorF he rtio of the emf on the urrent de(nes the intertion opertor etween the ojet nd the vity ζ oc F he system is solved mking the diret sum of the ojets nd the resontors nd dding the oupling opertors ζ oc F vet9s pply the oupling proess to line inserted in the vityF

IMMERSING THE HARNESS IN A STRUCTURE, IN HIGH FREQUENCIESIRS

he oupled hrness is de(ned y the opertor X

z = ν D HF           R 1 + z c α 12 α K+1 2 z c -(α(K -1)τ pz c ) α 21 z c + R 2 -(α(K -1)τ pz c ) α K+1 2 z c α K+1 2 z c -(α(K -1)τ pz c ) R 3 + z c α 12 -(α(K -1)τ pz c ) α K+1 2 z c α 21 z c + R 4          
@WFTPA sn (rst we mke the diret summtion of this opertor with the opertor of one mode X the resontor RLCF he we must ompute the intertion opertor etween the vity on this mode nd the hrnessF en emf e t indued in the line t the extremity tx integrtes the mgneti (eld perpendiulr to the line setion on its lengthF sf we lote the line t the enter of the vity etween siss z a nd z b D we hve if the line is of height x l nd using WFST X (9.65) sn this se the soure vetor T ontins only the soure for the vity @soure oming from n externl (eld nd some element inserted in the vity to rete the (eldAF he eqution T a = ζ ab k b solve entirely the prolemF 9.7 Conclusion for SUETAN ell iex @it my e softwreD ut for the moment it9s only method while egRfsi is formlismA is sed on this prinipleF he ojet lled to prtiipte to the system onstrution re grouped on tleF e diret summtion rete (rst opertor without oupling etween them @it mens tht the system remins virtulAF hen onnetivities nd oupling pplied on this tensor retes the (nl systemF teps of losingsD onneting y wiresD inserting in struture etF re prts of this opertionF yne the system is retedD the soure n e moved from omponents to others in the soure vetor T to study ssumptions of self energies or onstrints oming from outsideF yften the question ours to know if the numeril pplitionD if there re someD my e relized in the time domin or in the frequeny dominF st is ler tht the frequeny domin is esier for ll the eletromgneti intertionsF fut even with non liner omponentD nother solution existsD often forgottenF mll signl nlysis onsiders tht non linerities hve known stte for known polrityF his polrity omes from the zero omE ponent of the spetrumF snside time domin loop where the non linerities re (xed in term of impedne opertorD frequeny loop n e omputed using these (xed impednes for the non lineritiesF he spirit of this method mens to ut the whole soures signl in suprts of onstnt vlues leding to onstnt lol hg omponents nd to onstnt polritiesF huring these stepsD the spetrum is omputed nd the solution ppers like sequene of instntneous spetrumF his pproh is very e0ient when the utting is in reltion with the system mission nd sort of prtition of his sttes where eh note is n oservle system iGy @see yFwurie 4ilements of theory for eletromgneti omptiility nd systems4D PHIUD fookelis editorAF Chapitre 10 BISE approach for system conception o e inspired y nturl systems mens two essentil things X " to think in evolution E pity to evolve Y " to see the system tions s results from system deisionsF hese two properties in generl do not elong to rti(il systemsF sn prE tiulrD rti(il systems re not le to reprodue themselvesF fut rti(il systems n evolve thnks to eletronis nd informtis nd with the inE tervention of rti(il intelligene @esAF e gme involve hoies oming from vrious systems seen s gmersF ih gmer mkes hoies depending on the gme results nd on the other gmer hoiesF trting from the tions of one systemD enlosed in some vriles k x D nd strting from the tions of seond system q y D we onstrut gme through the onstrution of pyEo' mtrixF his mtrix sys for eh ouple (k x , q y ) wht re the gins of eh gmerF hen eh gmer hs its own es nlysing the gins nd reting next ommnd e x or a y whose determine next ehviour of the gmersF he proess is represented (gure IHFIF he mjor di'erene with lssil ords is tht this ord implies group of system nd not only two of themF omewhere these ords onerns the whole system in one shot for eh events step timeF enother prtiulrityD is tht this ord lls for psyhology nd ognitive proessesF o the ompuE ttion is not so deterministi s lssil ords lling for stndrd physil lwsF fy the ftD results n vry from one sientist to notherF fut simple resoning llow to led to similr resultsF vet9s tke n exmpleF e wnt to model n engineer tht must tke deision on some reportF sf he9s very IRU IRV CHAPITRE 10. BISE APPROACH FOR SYSTEM CONCEPTION por gme theoryD no dout tht ier9s ook is perhps the est referene to egin disovering this themtiF ome of next exmples re tking o' from this ookF 10.2 Game theory e n explore some ses strting from hoies lredy mde y two gmers @it mens tht we don9t re of how the systems rrive to these possiilities of hoiesAF he gme is presented through its pyEo' mtrixF his mtrix is orgnized in wy tht the ouple (i, j) t the oordintes a, b gives the gins of the plyer i nd plyer j for the hoies a nd b respetivelyF vet9s onsider the pyEo' mtrix X

ζ =                  Rmn + Lmnp + 1 Cmn p 2ζpx 2ζ tx 2ζpx 2ζ tx 2ζpx R 1 + zc α 12 α K+1 2 zc -(α(K -1)τ pzc) 2ζ tx α 21 zc + R 2 -(α(K -1)τ pzc) α K+1 2 zc 2ζpx α K+1 2 zc -(α(K -1)τ pzc) R 3 + zc α 12 2ζpx -(α(K -1)τ pzc) α K+1 2 zc α 21 zc + R 4                 
Ξ = 1 -1 -1 1 g1 + -1 1 1 -1 g2 @IHFPA
Figure IHFQ ! yEo' mtrix s vetors X seond se epplying the previous ruleD we don9t see ny xsh9s equilirium in this seF sf ny of the two gmers hoies the vlue to win ID the other loses ID nd the inverseF sf A is the (rst hoie of gin CID the (rst plyer hs the proility p 1 to ply e nd the seond plyer hs the proility p 2 to ply eF henD the proility for the (rst plyer to win plying e is p 1 multiplied y the gin equl to ID plus (1 -p 2 ) tht the seond plyer plys B @the seond hoieA multiplied y his gin in this seD to know EIF pinlly the (rst plyer hs the hope of erning plying e h A given y X

h A = 1 • p 2 -1 • (1 -p 2 ) = 2p 2 -1
@IHFQA sf the (rst plyer plys fD y the sme wy he wins X

h B = -1 • p 2 + 1 • (1 -p 2 ) = 1 -2p 2 @IHFRA
he optimized solution for the (rst plyer @if it existsA is otined writing h A = h B F his leds to p 2 = 0, 5F he gme desription is de(ned y X e f e IDEI EIDI f EIDI IDEI unowing p 2 D the seond plyer hs hne p 1 to win given y X p 1 = 0, 5F nder this new mixed formD xsh9s equilirium ppers giving oth gmers the est strtegyF he strtegy here is de(ned for the ouple of gmer X plying SH7F xsh9s equilirium is determined here looking t the hope of erningsF rere we hve only one pointD so xsh9s equilirium is evidentF fut how n we interpret this xsh9s equilirium sed on mixed strtegy @with hope of erning proilityA c here re two interprettions desried y xsh himselfF he (rst interprettion tht s my ll 4the quntum interprettion4 sys tht one gmer plys e or f with equl hneF o he don9t hve ny preferene etween the two optionsF he seond interprettion llle the 4mss interprettion4 sys tht if mny persons ply the gmeD there will e SH 7 of them plying e nd SH 7 plying fF sn this new gmeD plyer egins lone nd n ply e or gF sf he plys eD he wins I piee nd the seond plyer wins S pieesF sf he plys gD the seond gmer plysF sf the seond gmer plys fD eh gmer wins H pieeD if the seond gmer plys hD eh gmer wins P pieesF he pyEo' mtrix n e de(ned y X Ξ = 1 0 x 2 g1 + 5 0

x 2 g2 @IHFSA he symol x showing tht this possiility doesn9t existF ime in this representtion disppersD nd we will see tht this n modify the undersE tnding of the situtionF in ft the numer x is known X if the (rst plyer hoies AD he wins I nd the seond plyer wins notingD iFeF HD even if its without plying 3 o (nlly X @IHFVA lyer I should ply option 4x4F ss this result ppers in vetoril repreE senttion c pigure IHFU shows the ouples otined for eh omintion of erning nd for the hope of erning in se of the two types of plyer PF es the plyer I doesn9t know the type of the seond plyerD he nnot hoie etween the two longer vetors for N, N or R, RF sf he plys nd if plyer P is of type f @ouple R, N AD he looses the gmeF o this (rst deE terministi representtion doesn9t give solutionF xow if we tke look to the hope of erningsD it eomes ler tht the (rst plyer must hoie N F sn tht seD oth plyers keep good hope of erning equl to (2.5, 1) @type eA or (2, 1) @type fAF hese proilities re fyesin onesF e spek of the proility for plyer I to hoose xD knowing tht plyer P is of type eD etF X P (N | A)F he vetor representtion n e seen under two wysF e n serh for the est ommon gin tht ene(t t the mximum to oth plyersF fut we n lso onsider wht is the est interest of the opponentD inditing wht he9s prepred to plyF e n tke look to very well known gme X the prisoner9s dilemmF sts pyEo' mtrix is X sn ft the mximum erning is rehed for omintion tht needs trust etween the plyersF e understnd tht if the plyer greed on ontrt @impliit or expliitAD they n inrese the theoretil equiliriumF sf the plyers plys individullyD xsh9s equilirium is the est solutionF nder the individul ehviourD the plyer look t the minimum risk whtever plys the (rst plyer knowing tht they ply simultneouslyF 10.3 AI he purpose here is not to expose lssil es ut more to develop some ides round this oneptF ht mens es c ht mens intelligene c urely not something in reltion with living eing intelligeneF st9s more the notion of some eletronis le to nlyse pereption with pity to lernD iFeF to inrese its performne in nlysis with the inresing numer of ses studiedF he inputs of es re signls oming from vrious sensorsF ih kind of signl is treted di'erentlyF ht is lled lssilly neurl network is for me iruitF por suession of signls k α t given time t 0 D the iruits generte vetor of signture s β identifying the informtion reeivedF sf we imgine vetor a giving pereived tions nd vetor r giving the orresponding nswersD the ehvior is de(ned y some funtion r(a)F wking the ssumption tht this funtion follows known lw L of unknown prmetersD we n use the lest squres method to determine these prE metersF sf we onsider for exmple simple lw of the form αa i + βD for n smples we ompute X

S = n 1 (r i -(αa i + β)) 2 @IHFWA in order to minimize SF his mens tht X ∂S ∂α = 0 ∂S ∂β = 0 @IHFIHA st leds to the system of equtions X    α n 1 a 2 i + β n 1 a i -n 1 r i a i = 0 nβ + α n 1 a i -n 1 r i = 0
@IHFIIA 10.3. AI ISW yne the lw determinesD ny soliittion a i will result in n nswer r i X the system is modelledF por vrious outputs a i of systemD the options tht n e plyed r i hve to e de(nedF hey re given y the funtion y i = αa i + βF e lerning phse must e followed to onstrut the lw representing the ehviourF his tehnique @or similr onesA n e pplied with more thn only one funtion r(a)F king a i = t i D nd set of oservles o j D the ÷0ients α ji elongs to the funtions y j = α ji t i F hey re determined y minimizing S = n 1 (o j -y j ) 2 F he set of polynomils of ÷0ients α ji nd the ssoited lws @or polynomil ordersA re the neuronl network @xxAF st is ler tht if the ÷0ients were determined with n inomplete set of dt a i or with flsi(ed responses r i D the xx will hve defets nd might e unle to show stle ehviourF he lerning phse is key phse for the xx pertinene nd e0ienyF xowD omining xx nd gme theory @qA leds to n es strutureF he priniple is to insert xx etween some outputs of system nd the possile set of hoies presented in qF hen nother xx is insert etween the hope of erning vetor of the q nd the set of ommnds @inputsA tthed with the systemF his triptyh onstituted the es in the wy s see itF wny personsD nd for (rst speilists my sy tht this is not esF hey will spek of perceptronsD etF hese onepts ws developed thinking tht the mind working ould e similr to simple eletril deviesF fut with timeD iologists lern us tht neurons re of omplexity fr wy from ny omplex logil iruitF he previous iruit n e omplited s desired dding ouplings etE ween eh referenes nd lwsF ih polynomils n e seen s n opertor pplied to given )ux f i F e otin in (rst step digonl mtrix with ll the r i depending through lws of the f i F hen we n dd dependenies etween deisions nd )uxesF ih neuron is onstruted seprtelyF he polynomil gives the lw etE ween the pereption nd the deision whih leds fter to some tionF fut the lw n e hnged depending on nother deision tken y nother neuE ronF fsed on these priniple we otin omplex network whih is the esF es sid eforeD the es inludes n tionD neuronl network giving posE sile hoies to gme theory kernelF his kernel @pyEo' mtrix nd hope of erningsA gives deision nd linked with this deisionD new tion is opertedF e typil pplition onerns n utonomous vehileF e pedestrin heE sittes on rossing the rodF re nlyzes the vehile movement using its neuronl networkF his xx tkes into ount the distneD the pprent veE hile speedD the rod widthD etF hen this xx trnsmits two possile hoies to pyEo' mtrixF he sme exerise is mde y the vehileF he pedestrin omputes its mximum hope of erning nd tkes deision to ross the rod or notF his deision uses trnsltor to trnsmit the desired tion to the orresponding mhinesF sn this desriptionD you note tht we mke no disE tintion etween rti(il nd rel networks or mhinesF he xx represents the minding of the humn nd the q tkes into ount the pity to deideF ou my sy 4it9s not gme theory ut deision helping theory4F sn ft not euseD in mny ses it is surely deision helping proessD ut in mny other ses it9s gmeD gme to survive in the generl sense of lifeF s men tht the deision is tken (nlly to gurntee survivingF st9s typilly the se for the utonomous vehile prolemF pigure IHFW shows the proesses involved in this exmpleF we n sy in our ontext tht the (rst di'erene etween es nd 4real I (RI)4 is tht the omplexity dimension of polynomils for s re in(nite ompre to those of esD nd we re very very fr from pprohing this dimensionF he gmm mtries ply n importnt role in this sequeneF hey trnsE mit the soure vetor from eh network to the other nd gives the system rhythmF o illustrte this mehnism we imgine system mde of two systemsF ih of them re de(ned y n opertor ζ given y X

ζ n =           1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1           @IHFIPA
hese opertors desrie the pity of the systems to follow some lotion in (x, y) pln @)uxes f 1 nd f 2 AD nd the ft tht the systems trnsmit their oordintes to the other one @)uxes f 3 nd f 4 AF et initilD the soure vetor of the system is T n = e x , e y , 0, 0 F eeiving the oordintes of the seond systemD how the (rst system nlyzes this nd uses this for its next opertions c he system reeives the opposite oordintes b x , b y F sing n esD the system trnsltes this informtion in hoies for gmeF he possile hoies re xorthD outhD ist nd est @xDDiDAF ther thn polynomilsD the xx here is suite of inequlities like X ∃c ∈ [N, S, E, W ] / if ∆x = 0 and ∆y > 0 ⇒ c = N @IHFIQA he seond system movingD mkes hoie cF he (rst system hs the sme possiilities c 0 F e pyEo' mtrix explores the gins ville for the IT posE sile ouples of hoies oming from the two systems @xDx Y xD Y etFAF he ernings ea ij re de(ned y lw X ea ij = dc (i, j, t) + d θ (i, j) -1 @IHFIRA where dc (i, j, t) is the verge distne etween trget tht must e rehed y the system of systems @oA nd d θ (i, j) the distne etween the two systemsF he other system plying the hoie cD the (rst system nlyzes this looking t the di'erent hopes of ernings ssoited with this hoie c nd its own possile hoie c 0 in (N, S, E, W ) @the proilities n e silly (xed to IGR for ll hoiesAF sts possile hoie c 0 is guided y mximistion of the hope of erning ea cj otined when the seond system plys c nd the (rst one plys c 0 F he hoie c 0 ∈ [N, S, E, W ] must fter e trnslted in ommnd for moving the (rst systemF he sme resoning is pplied seen from the seond systemF he trnsltor hs the form X

                   if c 0 = N ⇒ e y = A (1 -e -τ p ) if c 0 = S ⇒ e y = -A (1 -e -τ p ) if c 0 = E ⇒ e x = A (1 -e -τ p ) if c 0 = W ⇒ e x = -A (1 -e -τ p )
@IHFISA τ is pulse durtion ontrolling motor to rete the system displement on ∆x or ∆y stepF e n now drw the omplete mehnism etween the vlues of the seond system displement nd the (rst system movingF his hin is represented (gure IHFIHF

ζ =             1 0 A xx A xy 0 0 0 0 0 1 A yx A yy 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 A xx A xy 0 0 0 0 0 1 A yx A yy 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1            
@IHFITA he opertor is not symmetri nd n e esily extends to x systems mking the oF e n remrk tht the es whih n e seen s ehviorl opertor is inluded in ordD iFeF n extrEdigonl omponent rther thn in digonl omponentF hen using ehviorl opertionsD it is often esier to use ord to inlude this proessesF higonl omponent model the limit onditionD or frontier opertors ssoited with these modelingsF o hve prt of them s digonl omponent my men to detil their hrdwreF his is possileD nd n e neessry sometimes in se of studies foused on their strutureF fut t the o levelD we look for mromodels without keeping ll the mirodetils ssoited with these systemsF e understnd tht in this seD ords re the simplest wy to inlude es in systemsF e system @n equipmentD vehileA n e modeled using eight fundamental kinds of components (8-fkc) X a les or ntennsD wveguides X tehniques to trnsmit informtions Y b opertors on the frontiersD interfes etween vrious environments @inE ternlD externlA nd system hert Y c ehviorl ords tking in hrge omplex system resonings inluding es Y d sensors or elements for pereption Y e motors or elements s tutors Y f skeleton to ssure the struture le to wer ll these omponents Y g some omponents used to evute the system het nd other wste Y ITR CHAPITRE 10. BISE APPROACH FOR SYSTEM CONCEPTION h soure of energy oming from outsideF Chapitre 11 Systemic concepts applied to BISE 11.1 Principles ystems modeling ws widely ddressed y qriel uron nd 0lite uthors using tensoril nlysis of networks @exAF fsed on dikopti prinE iplesD the method leds to the lgrngin of the prolemD in multiEphysique writingF e expose here the mjor hrteristis of this pprohD fousing on multiEsle prolems nd how to tke into ount these inputs in proE lem of system of systemsF 11.2 Introduction ystem of systems mens mny things like group of nimls or group of roots exhnging dts through ntennsD etF e rell the history of ex used for systems modeling nd the systemi de(nition for systemsF efterD we rell the evolving of systems in order to detil wht we onsider s system of systems @oA in this pperD other onsidertions eing possileF e re interest in yerEphysil systems seen s o X yerEphysil systems of systems @goAF he yer side refers to dtsD numeri ommunitions nd intelligenes while the physil side refers to the mteril @even living oneA tht support the intelligene nd ommunitionsF ITS 11.3 TAN history qriel uyx hd studied the systems strting from eletril mhines whih re omplex systemsF ris personl pproh for networks seeing them through the mesh speD hs leded to prtiulrly e0ient formlismF wE jority of hmiltonin pprohes work onto rnhes nd nodes spesF uron hs extended pplitions in ellulr topology to the mesh speD developing method tht gives the lgrngin of the prolemF e (rst ook written in IWUQ gives synthesis of his rtwork on systemsIF here ws fter the works of the prenh rmy diretion X engotD pinD uu'mn write vrious mthemtil ooks explining uron9s tehnique in order to study omplex eletroni sysE temsF fut until thereD intertions ws limited to mgneti intertions nd eletroni pplitionsPQF wurie 8 goF sumitted generlized onept of ordD desriing ny intertion etween two onnex networksRSF his new pity leds to the opportunity of pplying the ex formlism to ny kind of prolem inluding iologilD eonomil onesD etFTF his is with these lst developments tht we study here yerEphysil systems of systemsF 11.4 A denition for systems fy listening to the systemi ommunityD omplex system is hrterized y three properties X " lot of intertions Y " stohsti ehvior even prtilly Y " n emergene propertyF his lst property is often unknownF st mens tht the system hs proE perties tht nnot e oserved t lower sleF st lerly pose the onept of slesF st mkes the ssumption tht edges exist etween sles nd tht going from one sle level to nother ontinuously is impossileF st doesn9t men tht oth sles nnot exist simultneouslyF fut the existene of one higher sle n e otined from lower one through n integrtion proess whih is not ontinuousUF eny system hs n ssoite grph nd n ssoite mnifoldF e pose s preliminry postulte tht system of systems @oA is olletion of onnet grphs nd diret summtion of mnifoldsF ih of these mniE folds is system of the oF he systems intert etween them through ords 11.5. SOS PROPERTIES ITU in the ex formlism of the oF ih system is onstruted from rnhes @T 1 elements of the ellulr topologyA onneted through the onnetivity CF ih system is hrterized y n opertor ζ giving ll the prtile speeds v k inside the system for given onstrints vetor e m F he mnifold ssoites with system is de(ned y the eqution e m = ζ mk v k F he intertions re de(ned y the extrEdigonl omponents of ζ X ζ mn , m = nF he numer of these intertions is n imge of the system omplexityF he stohsti property is intrinsi of the eletroni omponents like miroproessorsF ht out the emergene property c es non ontinuous hnge of sles we onsiE der n integrtion from one oservle tht elongs t lower sleF sf we work t the sme sle to de(ne the systemD it is possile tht it doesn9t present ny emergene spetF his property my pper t the o levelF 11.5 SoS properties e o S is de(ned y the diret summtion of ll the mnifolds of group of systems Q X S = ⊕ k Q k @IIFIA he o y de(nition inludes the properties of the set of systems QF wy it hve new property ssoites with the union of the systems c ypillyD the new property tht n ppers when the systems Q re grouped nd n ommunite is ssoited with this pity of ommunition tht doesn9t exist when the systems re seprtedF et lest we n postulte tht new property of the o is the indued y this pityF his pity n guide group ehviour of the systemsF he o mnifols S n e ompleted y set of intertions G whih re the qreen9s funtions of rdition ptterns for ntennsF fut more generllyD the mnifold must e ompleted y two ojets X " the ommunition network etween ll systems Q k Y " the tions lde y ll the systems nd interpreted y ll the systemsF he (rst network n e modelled y speil tensor G dded to the mnifold S or y seprte mehnism lling tehnique nmed 4gmm mtrix4VF he seond mehnism is more omplex to expressF sn one system referentil of prtiulr systemD it strts from set of )ux K x tht re responsile for the tions of ll the other systemsF his )ux is (rstly perceived nd interpreted through omplex mtrix ΛF en opertor of attention A deides if these tions re onsidered y the reeiverF hen fyesin proility P c sys wht is the answer @analysis tivityA of the reeiver in front of these pereptionsF his nswer depends on pay-o matrix ω whih leds to the hope of erning ω µc F hen lst opertor δ of deision trnsltes these informtions in decision @ ommndA e v tken y the reeiverWF e trnslte these reltions writing X

e v = δ z v ω zc P c (e c | A z y Λ y x K x @IIFPA
xote tht this ognitive hin n e pplied in ft to ny reltion etween to systemsD even for exmple eletromgneti onesF e n synthesize the whole proess through omplex ut unique opertor h de(ned y X

h vx (K x ) = δ z v ω zc P c (e c | A z y Λ y x K x @IIFQA
he glol retion of system inside the o eomes priniplly represenE ted y the intertion h it hs with ll the other systemsF 11.6 From SoS to cyber-physical SoS (CPSoS) he ommunition using loud or nything else is inluded in the tensor GD dded to ζ for SF his tensor hs for soures some )uxes KF o the intertion h inludes lso the impt of the ommunitions etween the systems Q inside STF fsillyD the formlism ontins ll the elements for modelling the yerEphysil sideF he di'erene omes from the ft tht yerEphysil o n ontin systems tht re very fr etween eh otherF fut mthemtilly this property doesn9t hnge the expression of the ognitive ord hD it only gives to the pereption λ whih inludes the hnge of referentil spei( writingF he di0ult prt for modelling suh go is not the ommunition network ut the presene of multi physil lyers nd of vrious slesF he sle of desription uses opertors pplied on )uxes whih re the oordintes of the speF he )ux vetor emeds ll the )uxes of ll the physis onern y the oF here re urrentsD speedsD therml powersD etF sf some reltions need to use oservle of lower sleD for exmple quntum oservlesD integrtions re the key to go from this sle to higher one X they re the hnge of dimension in ellulr topologyF e olletion of equtions n ompny the system of eqution of the go mnifold to give the de(nitions of some lol opertor used in its own de(nitionF he go mnifold is ompletely de(ned y the tensoril 11.6. FROM SOS TO CYBER-PHYSICAL SOS (CPSOS) ITW eqution e q = ζ qm K m plus the domin de(nitions of eh prmeter pD p D tht determines the lws L involved in ζF e men here tht the opertors ζ ij re de(ned on vrious domins depending on prmeters @tempertureD etFA X

ζ ij = p D 1 L 1 ij + p D 2 L 2 ij + . . . @IIFRA
he system of equtions e q = ζ qm K m n e seen s vetor of funtions e q F e se n e de(ned through the join ∂e q /∂K m F fut vrious ses re de(ned for the vrious domins DF e must ensure tht the ontinuity etween two lol spe tthed with two lol ses existsD iFeF X

det ∂K m [D i ] ∂K q [D j ] = 0
@IIFSA his onstrint ensure tht the mnifold is ontinuousD iFeF tht the go n e de(ned nywhereD for ny vlues of the prmeters nd oservlesF unowing the lol sesD we n ompute the metris ssoited with these lol spesF st mens tht we n ompute the distne etween the enter of domin nd ny points in the sme speF st mens tht we n ompute the neighorhood of ny pointF e hve de(ned topology nd the mnifold de(ning the goIHF 11.6.1 Some more details on the numeric side he ord n wer ny kind of funtionF sn prtiulr it n wer numeri funtions like sigmoid onesF iven if to model whole systemD it9s etter to use mroEmodels X it is lwys possile to synthesize susystem mde of wide numer of omponents y low dimension mroEmodel hving the sme trnsfer funtionF ther thn detiling omplete iruit mde of digitl gtes we n rete for exmple funtion tht outputs 4I4 if the input word is the wited oneF st is possile to study theoretilly the ommunition networkD whtever it isD y this kind of modelsF fut the ehviour of the hnnel in front of piry needs to e studied using the rel nd omplete iruitF ht9s n exmple of mehnism sed on vrious slesF 11.6.2

Classical thematics considered for CPS wore thn ommunitionD the g9 involve lso the notions of ontrolD moilityD rhiteture nd seurityF hey re omposed of intelligene netE workD sensor network nd tutors networkIIF e detil here eh of these oneptsF CPSoS control go ontrol goes through the deisions δD ut lso lling for the tE tention A nd the kind of nswers P of eh systemF he deision opertor n e prtly in)uened y n exogen prmeterF st n e n order given to the systems or set of systems in nswer to other systems ehviourF Mobility he moility is trnslte y the vlues tken y some )uxes K v in relE tion with the tutor in hrge of the systems moilityF fut the moility in)uenes lso the pereption of the systems etween them through the opeE rtor ΛF Architecture he rhiteture is the desription of the networks of intelligeneD sensors nd tutorsF hree suEgrphs re ssoited to these networksF Security eurity is tody mjor tsk for the g9F iletromgnetism nd eleE tromgneti omptiility re the jos ddressed y this themti s well s the iwi nd ryptogrphyIPF he pereption of the systems tions n e fked due to the eletromgneti disturnes in the systems enviE ronmentF st mens tht the opertor Λ is polluted y omponent of noise ΛF his pollution n led to some erroneous deision δ nd n inompreE hensile ehviour of goF fut we n even imgine tht seurity psses through the ttention opertor AF st9s ler tht strtegy of go ttk my e to fous the ttention of one system in diretion opposite to the diretion of ttkF yne this system infetedD the whole go n e inE feted step y stepF enother strtegy my e to deform the informtion of 11.7. THE PAY-OFF MATRIX IUI deisionF sn tht seD the opertor ∆ is orrupted nd leds to norml deisionsF here re mny wys for mliious system to distur the whole goF here re generl rules to void this kind of riskF pirst tion must ts on ΛF Λ must e eventully orreted in order to trnsmit the good informtionF hen if n ttk is identi(edD A must llow to lo the ess of this ttk efore it my infet the system furtherF sf this protetion filsD the only wy to derese the risk is to derese the proility of nswer to the fked informtionF his mkes the ssumption tht the yesin proE ility inludes prmeters in)uening its vlues on the se of some trust inditorsF he hope of erning n e modulted y the detetion of risk tthed with some deisionF hen the lst rrier is linked with the deision opertor whih n lso deide to exlude the hope of erning in reltion with some soureF 11.7 The pay-o matrix he pyEo' mtrix sys how mny wins system y mking suh hoieF st n e very ftul in se of simple gmes with de(ned erningsD or very omplex in se of go where the systems gins depends on mny prmeters t short nd long time rngeF his mtrix my e the key point of the systemD nd then of the o ehviourF fsilly we n mke the ssumption tht the systems ehve depending on their hope of erningsF he yesin proility P is (rst driven y the pyEo' mtrix ωF hen the deision is generlly driven y the hope of erningD ut n lso e deided y the deision opertorD de(ned itself y the psyhologil pro(le of the systemF sn our previous illustrtionD we reognize the input (e x , e y ) s the input dt )ux K x D the trnsformtion Λ relize the pssge from (e x , e y ) vlues to the orresponding hoie cF he ttention A mkes the orrespondenes etween the hoie nd the diretions N, S, E, W F he fyesin proility involving the input informtion AΛK nd the hoie mde y the (rst system cD so P c (e c | A z y Λ y x K x ws tken here lwys equl to IGRF he pyEo' mtrix ω zc is here the tensor ea zc F end the opertor δ z v mkes the orrespondene with the ommnds e x , e y @it9s the trnsltorAF 11.8 BISE : a conclusion for this rst manual fsi mens to onsider ny system s mde of skeletonD skinD muslesD nervous systemF wde of n orgn le to trnsform energy oming from the environment nd orgns to evute wste nd hetF e system eomes so o where ll orgns n e onepts with these notions in mindF he struture tkes in hrge prt of the nervous networkD the skin tkes in hrge prt of the pereptionD etF intities inluding miroproessors re le to ondut vrious opertions of signl proessingD memory mngementD omputtionsD etF wore nd more these kind of entities eomes generi @think in rduinoD etFA nd system eomes mde of mny of these entities with entrl proessor mnging the whole nd tking in hrge the mjor memoryF his entrl unit hs nother importnt role X it deides when some peripherl proessors re out of use to report some tions to other orgnsF his is simplexity X ility to relize mission even when some tors re no more villeF his is strong pity of nturl systems X the pity to survive even in degrded modeF his pity is rehed seeing the system s group of suEsystems hving the VEfkF his rhitetures give lso pities to evolveF st remins two domins where nture is prtiulr X " reprodution nd evolving through genertions Y " n intelligene in the wy tht eyond simple omputtions nd omE plex lgorithmsD n strt thinking exists in ll nimlsF e must rememer tht uring9s riteri ws never rehedD though some syF xture is our inspirtion ut we re her hildren nd we will not e le to remke herF 11.8.1

Aside : and what about quantum mechanics and safety ?

he hnge of sle etween quntum nd lssil mehnis remin difE (ultD eyond ll the reltions ville todyF hen the energy eomes very smllD ontinuous ehviors eomes quntum ehviorsF hen we study hin of trnsmission or re)exion of informtionD we n look t the vrious wves involved with high levelsD in order to ssoite with this informtion mplitudes oming from re)exion nd trnsmission ÷0ientsF ht9s wht we disuss hpter SF untum ehviors re similr to rre eventsF st gives models for prtiles of smll energy or in other word for very smll numer 11.8. BISE : A CONCLUSION FOR THIS FIRST MANUAL IUQ of prtilesF hen there is very high numer of prtilesD they ehves lssillyD following thermodynmisF untum mehnis is the mnul for rre eventsF fety is prtiulrly interested in this kind of prolemF ss it possile tht rre event ours c por exmpleD if we onsider proility for n event a equl to 10 -9 F sf we mke one jetF ss it possile tht a rrives c heoretilly yesD a n pper for one hne over 10 9 F st9s poor hneD ut it doesn9t men tht this event nnot rrives on the (rst shootF st9s prdox euse rehing suh trget in sfety mens tht the risk a should never existF sn ft wht we wnt to reh is sturtion phenomenonF yur (rst ssumption is tht we hve n oserver eing le to oserve one event t timeF sf it9s tivted on one eventD it nnot her next eventsF st9s typilly the prolems of ritil events for sfety on one missionF sing this detetor @oserverAD if proess results from sequene of opertions modeled using gmm mtrix nd if a n pper during the (rst step for n input vetor of informtion vD it mens tht γv gives aF xext steps n give high proilitiesD rehing 1 -a in (nlF oD γγv → 1 -aF sn tht seD the event a n exist on (rst shoot s γv = a nd the detetor nnot see next eventsF xow if the proess implies (rst 1 -a then aF pirst step eing de(ned y γv = 1 -aD the hne to mesure the event is tht time ner to e IF o s the detetor is one shoot mesurementD it will lwys give the sme response X 1 -aF he event a ssoited with γγv will never een seen X tht9s the sturtion phenomenonF por sfetyD this seond proess leds to the onlusion tht the ritil event a will never rriveF ht9s not the se in the previous sequeneF e understnd tht in proess resulting from sequene of funtions tht n tivte or not eventsD with known proilityD the ft to guE rntee tht prtiulr event nnot e relized depends on this sequeneF nder some ssumption nd γEmtrixD rre events eome event tht never rriveF por other γEmtrixD they n pper on (rst test with their own proilityD ut ppering on the (rst experimentF heir smll proility mening tht the next experiments will not mke pper these rre eventsF sf we imgine g of llsD if we tke ll t eh rndom seletion ut without tking new gD the proility to selet rre event is 1/N on the (rst seletionD then 1/(N -1)D then 1/(N -2)D etF until 1F st mens tht the more seletion we doD the more hne we hve to selet wht ws t the eginning rre eventF ht my seems to e disdvntge n e in ft n dvntgeD euse we (nlly know wht is the rre eventF ht9s lwys the di0ulty in ll jos X if we mke system in order to void some event nd if our oneption is goodD we will never seen this eventD ut somewhere we never demonstrte tht it n exist nd tht it is wht we thinkF he dvntge with prtiles is tht we n mnipulte enormous numers of prtiles while it is more di0ult to hve n enormous numer of lls in gF hen experiments re onduted on prtilesD two wys re ville X " sending one prtile t eh rndom seletion Y " sending set of prtiles t eh seletionF sf we send one prtile on eh seletion nd if this prtile is mde eh time extly y the sme proessD we re in the se of plying with the sme g of lls without dding lls efore the seletionF yne given numer of seletion mde @sy N AD we re sure tht lw proilities should pper @ner 1/N AF ime goingD we must see ll the ses given y the theoryF et the ontrry if we mke set of prtiles eh time @N prtiles sent eh timeA nd if we rete proess in order to mke pper the rre events through γ 2 nd not γD the sturtion proess n e oserved nd the rre event my never pperF his kind of resoningD written di'erentlyD is mde ll the dys y the sfety workersF fut it remins under this resoning one strong ssumption X we suppose tht the rre event is knownF end one spei( property of rre events is sometimes to e unknownF row is tht possile c e should perE fetly known our system nd its environment c st is lso known @qruyere prinipleA tht the more ritil rre events ome from the oupling of sysE temsD eh of them eing well ontrolledF wultiEphysi nd humn ftor inE tervene to rete rre events tht do not exist efore to onstrut the whole systemF heir grvity is ll the more importnt tht their onstrution is not preditedF oD ll the proess inlude to give some solution in se of rises re not e0ient to fe these eventsF hese rre events re emergenies in the systemi thinking of systemsF he proposed tehniques in this ook n llow rehing solutions in the system oneption in order to (nd prries in front of these risksF γ 1 systems responses n e omputed tking into ount humn ftor ords nd multiEphysil onditions tht led to the system ritil filureF 11.8. BISE : A CONCLUSION FOR THIS FIRST MANUAL IUS 11.8.2

A simple circle for BISE pinllyD system engineering n e simply writtenF sn generlD the projet egins y the ustomer needsF fut in xtureD there re no ustomers 3 ih system is its own ustomerF his di'erene mkes tht systems for humns re (rst tools efore to e utonomous systemsF yne this di'erene pointed outD we n onsider tht in oth seD it exists ustomer X in xtureD the ustomer is the system itselfF eepting this ssumptionD the fsi sequene n e simply represented y the irle shown (gure IIFIF Figure IIFI ! fsi sequene he needs step desries wht is wited from the systemF ht the system is supposed to doF sn se of ioEsystemD this need is priniplly to survive in some eologil environmentF he requirements sy wht re the funtions tht must e grnteed nf with whih sfety trgetF he simplexityD whih is nturl propertyD onstruts strtegies in order to reh these requirements y the most optiE mized rhitetureF wission nd environment de(ne the senes tht the system will enounterD the exhnges it will e lled to ssureD the lotions where it will e le to (nd energyD system wste nd dissiptionsD etF yrgns nd rhiteture de(ne the whole system strutureF st detils the hoies like n exoEskeleton for support or other optionsD etF his irle is explored until hving the est solution for the prolem onsideredDinluding post knowledge of the persons in hrge of the system oneptionD mteril ostsD etF wthemtillyD needs re oservles k x de(ned in intervls ID k x ∈ IF equirements re proilities P (k x ) ssigned to these oservle ojetivesF wission nd environment re pities in ommunitionsD speedD etF hey re prmeters nd stndrds giving riteri to judge of the system perforE mne nd of its deqution with the needs nd requirementsF yrgns nd rhiteture re (rts lsses in order to test hrdwre nd softwre solutions to onstrut the systemF his onstrution go hed s nd when tht the irle is rowsedF his fourth step lls for the more personsF he rhitet on the se of one systemD distriutes the orgns oneption to vrious projet ledersF xervous nd power supply networks re suontrted y the sme wyF gommon meetings llow to evlute the system performne otined with (rst set of orgns return k y the projet ledersF djustments re operted to tke into ount the oupling e'ets in multiEphysisF hen new irle is egun until onverging to n optiml solutionF he rhitet is in ft tem enlosing system rhitetD ommerilD (nnilD etF rious tions re X Architect strts ζ µν for the needs k ν nd environment T µ F gonstrut ∈ P (I)D return k new ojetives for ζF e see here tht the ft to work s soon s possile with mthemtiE l models llows to urtely de(ne the tehnil tions tht need to e leded nd void miguities nd the sene of tking into ount multiE physi e'etsD indeed humn ftorF his lst point implies to ontrol the es ordD ut this lst intertion modeling tht s propose needs yet more developmentsF s hope tht this ompiltion of trks of solutions will inspire engineersF s think tht the mjor xis of progress for system engineering omes from the ft to introdue more rigor nd mthemtis s soon s possile in its proessF yften engineers exhnge ides rther thn modelsF his leds to miguitiesD fke solutions nd unontroled hrdwre nd softwresF his without forgetting tht models llow to keep nd trnsmit knowledge uE rtelyD nd tht proilities must e inluded in ll the proess )owF Bibliographie I rppD rF rF @idFAF @IWUQAF qriel uron nd ystems heoryF nion gollege ressF P engotD eF @IWSUAF gompléments de mthémtiquesF Q henisEpinD wFD 8 uufmnD eF xF @IWTTAF gours de lul tensoriel ppliqué X géométrie di'érentielle solueF R wurieD yFD eineixD eFD hurndD FD 8 huoisD pF @PHIPD tulyAF yn mthemtil de(nition of hords etween networksF sn iuropen eleE tromgnetisD iuroiw PHIP @pF snonnuAF S wurieD yFD eineixD eFD hurndD FD 8 huoisD pF @PHIRAF uron9s meE thod nd ell omplexes for mgnetomotive nd eletromotive foresF riv preprint riv XIRIPFHIVUF T wurie yF 4ilements of theory for eletromgneti omptiility nd systems4F fookelisD sfx WUWEIHEPPUESUUTEVD prne PHIUF U hitneyD rF @PHIPAF qeometri integrtion theoryF gourier gorportionF V isxisD elin et wesgiD ylivierF wodèle de gmm mE tries pour l prise en ompte d9intertions ryonnées entre réseuxF sn X ITème golloque snterntionl et ixposition sur l gomptiilité ÉletromgnétiqueEgiw PHIPF PHIPF pF ession ReF W efeD tF gFD ve woigneD tF vFD exhiiuD iFD eviD fFD lléeD FD vvsiwiD tF vFD FFF 8 fygryxEwixsiD fF @IWWIAF ystémique 8 ognitionF hunodF IH hourovineD fFD pomenkoD eFD 8 xovikovD F @IWVPAF qéométrie ontemE porine X méthodes et pplitionsF II wtD hF fFD odriguesD tF tFD 8 tojmenoviD sF @idsFAF @PHISAF gyerE physil systems X from theory to prtieF gg ressF IVI IP fokD hF qF @IWUQAF e history of ommunitions seurity X he hvid qF fok leturesD tulyF IQ ylivier wurieD hilippe hurndF wodélistion des systèmes omE plexesF X résenttion des grnds prinipes de l méthode xexF réE senttion e'etuée lors de l réunion du groupe gis de l9epgi à l9isqivigD le IW déemre PHISF hlEHIPRTURH IR ylivier wurieD elin eineixF roposition d9une méthode xex pour modéliser les proessus omplexesFPHIPF hlEHHTWSRSVvI IS wurieD yF @PHIQAF sntrodution d9une théorie des jeux dns des topoE logies dynmiques @hotorl disserttionD vimogesAF IT yhsqiD toel tose gD ytwixysgD svnD 8 eeD hnd fF gyerEphysil systems X from theory to prtieF gg ressD PHISF Other references " rppD rF rF @idFAF @IWUQAF qriel uron nd ystems heoryF nion gollege ressF " engotD eF @IWSUAF gompléments de mthémtiquesF " henisEpinD wFD 8 uufmnD eF xF @IWTTAF gours de lul tensoriel ppliqué X géométrie di'érentielle solueF " wurieD yFD eineixD eFD hurndD FD 8 huoisD pF @PHIPD tulyAF yn mthemtil de(nition of ords etween networksF sn iuropen eleE tromgnetisD iuroiw PHIP @pF snonnuAF " wurieD yFD eineixD eFD hurndD FD 8 huoisD pF @PHIRAF uron9s meE thod nd ell omplexes for mgnetomotive nd eletromotive foresF riv preprint riv XIRIPFHIVUF " wurie yF 4ilements of theory for eletromgneti omptiility nd systems4F fookelisD sfx WUWEIHEPPUESUUTEVD prne PHIUF " hitneyD rF @PHIPAF qeometri integrtion theoryF gourier gorporE tionF " efeD tF gFD ve woigneD tF vFD exhiiuD iFD eviD fFD lléeD FD vvsiwiD tF vFD FFF 8 fygryxEwixsiD fF @IWWIAF ystémique 8 ognitionF hunodF " hourovineD fFD pomenkoD eFD 8 xovikovD F @IWVPAF qéométrie ontemE porine X méthodes et pplitionsF " wtD hF fFD odriguesD tF tFD 8 tojmenoviD sF @idsFAF @PHISAF gyerEphysil systems X from theory to prtieF gg ressF " fokD hF qF @IWUQAF e history of ommunitions seurity X he hvid qF fok leturesD tulyF " ylivier wurieD hilippe hurndF wodélistion des systèmes omE plexesF X résenttion des grnds prinipes de l méthode xexF réE senttion e'etuée lors de l réunion du groupe gis de l9epgi à l9isqivigD le IW déemre PHISF hlEHIPRTURH " ylivier wurieD elin eineixF roposition d9une méthode xex pour modéliser les proessus omplexesFPHIPF hlEHHTWSRSVvI " wurieD yF @PHIQAF sntrodution d9une théorie des jeux dns des toE pologies dynmiques @hotorl disserttionD vimogesAF " eineixD eFD 8 wurieD yF @PHIPD eprilAF wodèle de gmm mE tries pour l prise en ompte d9intertions ryonnées entre réseuxF sn ITème golloque snterntionl et ixposition sur l gomptiilité iletromgnétiqueEgiw PHIP @ppF essionEReAF " hFvuzeuxD tEFuultD 4v9ingénierie système4F epxy editorD PHIQF " xiols ierD 4héoris des jeux4D hxyh editorF " efiD tenEulF Életronique des impulsionsF ome s vignes ouE plées en régime trnsitoireF psiule I gouplges et prsitges entre lignesF wssonD IWUPF " tenEul reD 4wonogrphie sur les lignes ouplées4F ennles des téléommunitionsD numer QHD n IIEIPD IWUSF

  Maurice, "Elements of theory for electromagnetic compatibility and systems" at Bookelis, 2017 3.3. CARTESIAN SPACE AND DIAGONALITY OF G hek DG = G D whih llows us to (nd new digonl G metriF olving the previous eqution we (nd X

Figure

  Figure RFI ! urfe for stti metri the distne overed s for i 1 = x nd i 2 = y per G o per unit is given y s = x 2 + y 2 2 Cos π 4

Figure

  Figure RFQ ! rnsistor osilltor shemti

Figure

  Figure RFR ! wodi(ed osilltor

  Figure RFS ! xx trnsistor hrteristi sntrinsillyD the xx trnsistor is ontrolled urrent soure nd pE rmeterizedF o ontrol the model of the trnsistorD we will (rst lulte simple iruitD hlf the previous osilltor y tking vg iruit powered y genertor nd the xx trnsistor n or not swith this iruit to the power supplyF pigure RFT shows the shemeF fy ssoiting the si iruit i b with two rnhesD nd y nming i c the olletorEemitter urrent soure we (nd the next impedne opertor in

Figure

  Figure RFU ! gonsidered hrteristis

Figure

  Figure RFV ! everse hrteristis

Figure

  Figure RFW ! rnsistor mromodel he olletion of the three rnhes leds to the mtrix impedne z

Figure

  Figure RFII ! iersEwoll in dequy with the desription of networks in ellulr topology 2 F gonsider the digrm presented (gure RFIIF e hve for this mokEup X

  digrm of the osilltor given (gure RFIPF 2.Maurice, O., & Reineix, A. (2010, April). Analyse tensorielle des réseaux hybridée comme outil théorique pour la CEM et l'électromagnétisme. In CEM2010 (pp. A-Reineix).

  Figure RFIP ! ysilltor shemti

  Figure SFI ! predsheet for the gmm mtrix PF etween wves tht elong to two di'erent frontiersD the reltions re those of dispersion Y QF in se of mintined soureD ÷0ient I is put for the (rst propE gtion wve ellF epplying the previous rules to onstrut the tD we n de(ne the γ mtrix X

  Figure SFQ ! gurve of the funtion e@tA on port P

Figure

  Figure TFI ! e (rst struture es we don9t hve ny soure or lossesD our reltion is redued to X Ω ik x k = 0

Figure

  Figure TFP ! eprte prt in the struture he summtion leds to X 10F lk + 4 × 62, 5.10 3 = 0 ⇒ F lk = -25.10 3 @TFPQA he negtive sign sys tht the fore hs n opposite sense to the diretion indited on the (gureF

Figure

  Figure TFS ! e weken sumitted to one fore epplying the sme method s previously without looking t losses y fritionD we de(ne (rst mss tensor X m = m 11 0 0 m 22

  Figure TFU ! he onsole onsidered

Figure

  Figure TFW ! eported vetors

Figure

  Figure TFII ! eken onnetion

  GENERAL MODELLING OF A CYBER-PHYSICAL SYSTEM (CPS)

  Figure VFI ! epresenttion of system ell

Figure

  Figure VFP ! ixo to endoskeleton evolve

Figure

  Figure VFT ! eprtition funtion of disturne

  Figure WFI ! gouplings etween two lines

Figure

  Figure WFR ! pield proess of vity in low frequenies

Figure

  Figure WFS ! wgneti (eld ttenution inside losed vity

Figure

  Figure WFT ! iletri (eld ttenution inside losed vity

  Figure WFU ! olume topology

Figure

  Figure IHFI ! se proess s ord

  Figure IHFP ! yEo' mtrix s vetors eeing this (gureD we understnd tht it exists se for whih oth gE mers wins the mximum ginF sf ny of the two gmers mkes nother hoieD their gin would e lowerF his vetor points out the est hoie for oth

  Figure IHFR ! etor projetion of the gme

  Figure IHFU ! etor representtion of the gme

Figure

  Figure IHFW ! roess involving n es emining modest on wht we my understnd on the humn thinkingD

Figure

  Figure IHFIH ! ghin for n es ord he whole proess represents n es ordD tking the opposite system lotion e x , e y nd giving the new displement vlues e x , e y F sf we note A xx

  ⊕ µ ζ µν Y Project leaders look for ζ D onsidering mission 8 environment T µ with ∃k ν / |y νµ T µ -k ν | → ∩ P ( > r| T µ , requirements) → 0 Y Common meeting verifying is T µ = ⊕ µ ζ µν k ν ⇒ P (k ν ) ∈ P (I)D I11.8. BISE : A CONCLUSION FOR THIS FIRST MANUALIUU nd P (I) eing de(ned y the requirementsF wking djustments for oservles P (k ν ) /

  foltzmnn9s onstntD one of the fundmentl onstnt of ntureF VU 7.2 Temperature topology he het Q umulted when inresing the temperture of ody of mss m etween two times t 1 nd t 2 is given y X

  is indued t the other extremity px of the lineD with n opposite sign X ζ px = -ζ tx F o report the openiruit emfD the expression omputed previously is multiplied y twoF pinlly X

	e t = -p	ˆzb za	dzx l	E 0 p	Sin nπ	y 0 Y	mπ	z Z	Cos mπ	z Z	@WFTQA
	ut y 0 = Y /2 nd E 0 = k x /χCp we otin X
	ζ tx =	e t k x =	ˆzb za	dz	x l χCp	mπ	z Z	Cos mπ	z Z	@WFTRA
	e similr emf

References

All the chapter follows the explanations given by Pierre Thomas in his book "Éléments nis pour l'ingénieur" at Lavoisier editor, 2006.