Es a egias de con ol obus o
pa a p ocesos de soldadu a po
a co eléc ico
Au o :
Manuel Masenlle Núñez
Di ec o es:
J. Xabie Os olaza Zamo a
Jo ge Elso To al a
2025
(cc) 2025 Manuel Masenlle Núñez (cc by 4.0)
iii
EUSKAL HERRIKO UNIBERTSITATEA
Abs ac
Facul y o Enginee ing, Gipuzkoa
Sys ems Enginee ing and Con ol Depa men
Doc o o Philosophy
Es a egias de con ol obus o pa a p ocesos de soldadu a po
a co eléc ico
by Manuel Masenlle Núñez
i
Welding, a undamen al p ocess in mode n manu ac u ing,
joins me al componen s by mel ing hem and allowing hem o
solidi y, o en wi h a ille ma e ial. I s widesp ead use is e i-
den in he ac ha o e hal o manu ac u ed p oduc s a e es-
ima ed o con ain welded join s. Howe e , achie ing p ecise
con ol o e he welding p ocess o ensu e desi ed mechanical
p ope ies, such as s ong, dis o ion- ee join s wi h minimal
esidual s ess, emains a signi ican challenge. These p ope -
ies a e hea ily dependen on he hea ing and cooling cycles
o he mol en zone, making hei p ecise manipula ion essen-
ial. Despi e welding’s impo ance, he applica ion o sophis i-
ca ed con ol echniques has no kep pace wi h o he manu ac-
u ing domains. This esea ch aims o b idge his gap by de-
eloping models ha link con ollable pa ame e s (powe and
a el speed) o empe a u e, and by designing con ol sys ems
o main ain he p ocess wi hin speci ied pa ame e s.
This s udy in es iga es wo dis inc welding applica ions.
Fi s , a long-du a ion welding p ocess, common in la ge s uc-
u e ab ica ion, is examined. A simpli ied 1D model se es as
he basis o a decen alized con ol sys em ha manipula es
powe and a el speed o achie e a a ge cooling cu e. Sec-
ond, he complexi ies o addi i e manu ac u ing a e add essed.
The laye -by-laye na u e o his p ocess and he associa ed hea
accumula ion demand a mo e ad anced con ol s a egy. A mul-
i a iable con olle , designed using sequen ial Quan i a i e Feed-
back Theo y (QFT), egula es he su ace empe a u e p o ile by
adjus ing powe and a el speed, wi h pa icula emphasis on
he challenges posed by di ec ion changes du ing laye c ea ion.
A c i ical aspec o hea ans e modeling in welding is he
ep esen a ion o mo ing hea sou ces. Building on he ounda-
ional wo k o Rosen hal, which es ablished he heo y o hea
low om a mo ing sou ce unde quasi-s a iona y condi ions,
his esea ch aims o de i e equa ions ha desc ibe he dynamic
ela ionship be ween empe a u e and he con ol pa ame e s.
Speci ically, ans e unc ions a e de i ed ha ela e empe a-
u e a ia ions o changes in bo h hea sou ce powe ( o 1D,
2D, and 3D cases) and a el speed ( o 1D and 2D cases).
These ans e unc ions a e hen used o c ea e a 1D welding
con ol model. Fo he mo e complex 3D case, a pu ely analy i-
cal app oach p o es di icul , especially in de i ing he ans e
unc ion o a el speed. The e o e, a nume ical app oach is
employed, linea izing an FEM model o c ea e a s a e-space sys-
em. While sac i icing some analy ical p ecision, his me hod
o e s g ea e geome ic lexibili y. This 3D FEM-based model is
hen used o design a con ol sys em aimed a imp o ing pe -
o mance du ing di ec ion changes, a c i ical aspec o addi i e
manu ac u ing.
This model is u he enhanced by inco po a ing a dis u -
bance inpu ha cap u es he dynamics o di ec ion lips. A
modi ied 2-Deg ee-o -F eedom (2DOF) MIMO QFT me hodol-
ogy is in oduced o dis u bance ejec ion, whe e speci ica ions
i
a e de ined as de ia ions om a desi ed esponse. The ma he-
ma ical unc ions needed o gene a e QFT bounds o his ype
o p oblem a e de i ed, expanding he applicabili y o he QFT
echnique. An algo i hm o au oma ically op imizing he coe -
icien s used o balance he ole ance o he bounds is also de-
eloped, u he enhancing he MIMO QFT oolbox.
The enhanced QFT me hod is applied o he de ined bench-
ma k p oblem, yielding sa is ac o y esul s in bo h he ime and
equency domains. Compa ison wi h he uncon olled p ocess
demons a es signi ican imp o emen s, pa icula ly in main-
aining lowe empe a u es a he poin s ep esen ing he iso he -
mal zone. This is expec ed o imp o e he quali y o deposi ed
laye s in addi i e manu ac u ing by minimizing he need o
heigh co ec ions o cooling pe iods be ween laye s. In sum-
ma y, his wo k p o ides aluable con ibu ions o bo h he mod-
eling and con ol o welding p ocesses, including he de elop-
men o no el modeling echniques, he ex ension o he QFT
con ol me hodology, and he demons a ion o i s e ec i eness
in challenging welding scena ios.
ii
Con en s
Abs ac i
1 In oduc ion 1
1.1 Au oma iza ion o a welding p ocess ........ 5
1.2 Impo ance o welding in he local indus y .... 7
1.3 Challenges ....................... 8
1.4 Objec i es ....................... 12
1.5 S uc u e ........................ 14
2 S a e o he A 17
2.1 Mo ing hea sou ce model .............. 17
2.2 Moni o ing o welding ................ 19
2.3 Modelling and con ol o welding .......... 23
2.4 Summa y ........................ 28
3 Modelling o a c welding 31
3.1 G een’s unc ions ................... 34
3.2 Ob aining ans e unc ions ............. 38
3.2.1 T ans e unc ion o welding o he 1-D case 39
3.2.2 T ans e unc ion o welding o he 2-D case 41
3.2.3 T ans e unc ion o welding o he 3-D case 42
iii
3.3 S eady-s a e model alida ion ............ 43
3.4 Modelling he welding a el speed as he inpu
o he sys em ...................... 44
3.4.1 T ans e unc ion o he 1-D case ...... 45
3.4.2 T ans e unc ion o he 2-D case ...... 47
3.5 FEM welding model and simula ion ........ 48
3.5.1 Simula ion Model ............... 51
3.5.2 Linea Model ................. 55
3.5.3 Valida ion o 1-D analy ical model in he
equency domain .............. 58
4 Decen alized MIMO con ol o he 1-D case 63
4.1 Con ol p ope ies ................... 64
4.1.1 Ou pu s selec ion ............... 64
4.1.2 RGA analysis ................. 66
4.2 Decen alised mul i a iable con ol ......... 68
4.2.1 Benchma k p oblem s a emen ....... 68
4.2.2 Con olle design ............... 69
4.2.3 Resul s ..................... 71
5 Mul i a iable QFT con ol o he 3D case 75
5.1 The lip o di ec ion p oblem ............ 78
5.1.1 Di ec ion lip modelled as an impulse dis-
u bance .................... 79
5.2 Con ol app oach ................... 81
5.2.1 Benchma k sys em .............. 82
5.2.2 Analysis .................... 83
ix
5.2.3 Con ol s a egy ................ 83
5.3 The QFT me hod ................... 84
5.3.1 Dis u bance ejec ion in QFT MIMO . . . . 89
5.3.2 Measu ed dis u bance ejec ion ....... 94
5.3.3 Measu ed dis u bance ejec ion wi h model
ma ching .................... 96
5.3.4 Ob aining an op imal di ision o ole ances 99
5.4 Design .........................100
5.4.1 Speci ica ions .................101
5.4.2 Con olle design ...............104
5.5 Resul s .........................108
5.5.1 Speci ica ion compliance ...........108
5.5.2 Time esponse .................108
5.5.3 Nonlinea simula ion .............110
5.5.4 Tempe a u e p o iles .............112
6 Conclusion 115
6.1 Con ibu ions .....................120
6.2 Fu u e wo k ......................121
7 Resumen 123
7.1 In oducción ......................123
7.2 Es ado del a e .....................127
7.3 Modelado .......................131
7.3.1 Ob ención de unciones de ans e encia . 132
7.3.2 Ob ención de modelo numé ico ......135
7.4 Con ol MIMO descen alizado pa a el caso 1-D . 138
x ii
Lis o Abb e ia ions
AM Addi i e Manu ac u ing
BJ Binde Je ing
DED Di ec ed Ene gy Deposi ion
DOF Deg ee O F eedom
FEM Fini e Elemen s Me hod
HAZ Hea A ec ed Zone
MAG Me al Ac i e Gas
MIG Me al Ine Gas
MIMO Mul iple Inpu Mul iple Ou pu
PBF Powde Bed Fusion
QFT Quan i a i e Feedback Theo y
SISO Single Inpu Single Ou pu
TIG Tungs en Ine Gas
WAAM Wi e A c Addi i e Manu ac u ing
xix
Lis o Symbols
AC oss-sec ional a ea m2
CpSpeci ic hea capaci y J/(KgK)
hCon ec ion coe icien W/(m2K)
KThe mal conduc i i y J/(msK)
PC oss-sec ional pe ime e m
QHea inpu powe W
TTempe a u e ◦C
T a el speed mm/s
XDimension in X axis mm
YDimension in Y axis mm
ZDimension in Z axis mm
αThe mal di usi i y m2/s
ρMa e ial densi y kg/m3
ωangula equency ad
1
Chap e 1
In oduc ion
Welding is a p ecise, sa e and cos -e ec i e me hod o joining
ma e ials in manu ac u ing indus ies. I is es ima ed ha mo e
han i y pe cen o manu ac u ed p oduc s con ain welded
join s, his echnology being he p edominan me hod o joining
me allic ma e ials.
To mel he me als o be welded, usually s eel, a conside ably
powe ul hea sou ce mus be employed. The e a e h ee main
ypes o hea sou ces ha can be applied:
1. Gas Welding: I is ypically used in manual welding and
uses he combina ion o oxygen and ace ylene gases o c e-
a e a high empe a u e lame. I can also be used o cu -
ing and shaping.
2. Elec ic A c: I is he mos used me hod. I consis s in
c ea ing an elec ic a c by applying a ol age be ween an
elec ode and he ma e ials. The elec ode can be ungi-
ble o non- ungible, wi h he ungible me hod being he
mos widesp ead because i is as e . The non- ungible
2Chap e 1. In oduc ion
me hod, known as TIG (Tungs en Ine Gas), uses a ung-
s en elec ode and is p edominan ly used when p ecision
o quali y is mo e impo an han speed. In bo h cases,
an ine gas is used o s abilise he a c. Depending on he
gas used, u he dis inc ions a ise be ween a c me hods
such as MIG (Me al Ine Gas) and MAG (Me al Ac i e
A c). This echnology can be applied manually o h ough
a obo ic o au oma ic me hod.
3. Lase : Th ough a lase ene gy sou ce, highe p ecision and
smoo he hea inpu can be achie ed. This me hod can
only be used in au oma ed welding and equi es mo e so-
phis ica ed equipmen .
Apa om he main me hods abo e, he e a e speci ic niche
echnologies ha a e used in ce ain cases:
1. F ic ion Welding: I is no mally used o weld cylind ical
pa s wi hou ille ma e ial. I consis s o o a ing one o
he pieces o e he o he pa un il he pieces each he
mel ing empe a u e h ough ic ion. Then he o a ing
pa is s opped and p essu e is applied. I can be used o
join pieces o di e en me als.
2. Resis ance Welding: This me hod consis s o applying a cu -
en h ough he pa s ha a e in con ac whe e he e is an
elec ical esis ance ha will p oduce hea ha will mel
he a ea. I can be used o join pa s using a achmen
poin s.
Chap e 1. In oduc ion 3
In he Fi s Wo ld Wa , welding was used mainly o make e-
pai s, bu la e , he manu ac u ing me hods o many pa s made
by cas ing using moulds, o example ank olle s, we e eplaced
by manu ac u ing me hods using welded s eel pa s. [1]. The
e olu ion o manu ac u ing me hods con inues oday, whe e cas
aluminium pa s a e con inually being eplaced by welded pa s,
which a e somewha ligh e and much mo e economical [2]. Ad-
di i e manu ac u ing (AM), also known as 3D p in ing, is he
na u al pa h o his e olu ion end o manu ac u ing me hods.
I consis s in he p ocess o c ea ing h ee-dimensional objec s by
building hem up laye by laye . This is opposi e o sub ac i e
manu ac u ing echniques, like milling, which emo e ma e ial
om a solid block o c ea e he desi ed shape.
Me al 3D p in ing is an e ol ing manu ac u ing p ocess, wi h
new me hods con inually eme ging. The echnologies can be
classi ied as Powde Bed Fusion (PBF), Di ec ed Ene gy Depo-
si ion (DED) and Binde Je ing (BJ), mainly. The echnology
ha sha es he mos wi h adi ional welding is DED. Wi hin
DED echnology, ine classi ica ions can be made depending
on he hea sou ce and how he ma e ial is p o ided. The hea
sou ce can be an elec ic a c o a lase , while he added ma e-
ial can come in he o m o powde o wi e. Each echnology
has i s ad an ages and disad an ages in e ms o cos o equip-
men , speed o ope a ions, quali y o he esul s, limi a ions in
he shapes o he pa s and limi a ions in he size o he pa s.
When he hea sou ce is an elec ic a c and he ille ma e ial
4Chap e 1. In oduc ion
is wi e, he ad an ages a e educed equipmen cos , apid p o-
cessing, and he abili y o p oduce la ge pa s. Disad an ages,
on he o he hand, a e limi a ions in pa shapes and inal qual-
i y, which ypically equi e some addi ional machining.
Wi e A c Addi i e Manu ac u ing (WAAM) has a su p is-
ingly long his o y, e en hough i ’s ecei ing enewed a en-
ion ecen ly. The oo s o WAAM can be aced back o he
1920s. This is when he co e concep o using an elec ic a c o
mel and deposi ma e ial laye -by-laye eme ged [3]. Despi e
he longe i y o he undamen al idea, WAAM is becoming in-
c easingly a ac i e oday due o i s ad an ages. These include
cos -e ec i eness o la ge pa s, as e build imes compa ed o
o he AM echniques, and e icien use o ma e ials [4]. O e -
all, WAAM ep esen s an in e es ing b idge be ween adi ional
echniques like welding and mode n ad ancemen s in addi i e
manu ac u ing. I s long his o y and ongoing de elopmen posi-
ion i as a po en ially aluable ool o a ious indus ial appli-
ca ions.
A he same ime, pa s manu ac u e s a e inc easingly see-
ing AM as a iable al e na i e o adi ional machining. How-
e e , while he echnology’s po en ial o minimize ma e ial was e
has long been acknowledged, de ec s such as de o ma ion, po os-
i y, and c acking ha e impeded i s b oade indus ial adop ion.
Fo una ely, exis ing manu ac u ing wo kshop echnologies
and con ol sys ems can epu pose a c welding obo s in o e -
sa ile 3D me al p in e s. This ans o ma ion, enabled by Wi e
1.1. Au oma iza ion o a welding p ocess 5
A c Addi i e Manu ac u ing (WAAM), o e s he bene i s o ad-
di i e manu ac u ing, including educed p oduc ion cos s, wi h-
ou equi ing signi ican in es men s in new machine y. WAAM
democ a ises addi i e manu ac u ing, making i accessible o
medium-sized welding wo kshops.
1.1 Au oma iza ion o a welding p ocess
The e a e di e en le els o de elopmen in ela ion o he p o-
cess o obo iza ion and digi aliza ion o elec ic a c welding
p ocesses. S a ing om he adi ional si ua ion in which he
welde pe o ms he wo k manually, his de elopmen goes
h ough he ollowing s ages:
1. The inco po a ion o obo s o he manual ask. In addi ion
o he sa e y and economical easons, he epea abili y and
accu acy o he p ocess is imp o ed.
2. The e olu ion owa ds he applica ion o senso s and p o-
cess moni o ing. Wi h he ga he ing and egis e ing o
da a, knowledge o he p ocess is acqui ed.
3. The adjus men s o he p ocess inpu s h ough eedback
con ol o imp o e quali y and educe de ec s.
4. Inco po a ion o a i icial in elligence o make decisions
and ac .
12 Chap e 1. In oduc ion
o es ima e pene a ion and cooling a e.
4. Maximum empe a u e: I is he maximum empe a u e ha
he ma e ial eaches a each poin .
All in all, welding con ol aims o achie e consis en , high-
quali y welds by egula ing he hea ing and cooling p ocesses.
Howe e , welding p esen s signi ican con ol challenges: non-
linea i y, mul i a iable na u e, and he di icul y o measu ing
key a iables. A con ol sys em mus cope wi h hese challenges
o deli e eliable and high-quali y esul s.
1.4 Objec i es
Wi h he p e ious challenges in mind, he objec i e o his wo k
is o de elop models ha ela e con ollable pa ame e s o em-
pe a u e, and o design con ol sys ems o main ain he p ocess
in he desi ed s a e.
Two use cases a e conside ed. Fi s , a long welded join p o-
cess (case o a ship, b idge, la ge con aine , e c.), in which he
objec i e is o adjus he powe and ad ance speed o main ain
a desi ed cooling cu e. Using a one-dimensional model, he
objec i e is o design a decen alised mul i a iable con ol sys-
em. Figu e 1.1 shows a la ge gan y used o weld ship pa s ha
ma ch he use case conside ed o he one-dimensional model.
The second use case ocuses on an addi i e manu ac u ing
p ocess using elec ic a c welding. This case has been selec ed
1.4. Objec i es 13
FIGURE 1.1: One o he la ges lase welding
ins alla ion in he wo ld, ope a ing a Meye
We , Papenbu g since 2010, Sou ce: Meye
We
because i is a longe p ocess and can bene i mo e han a simple
welding bead o a ew cen ime es. In addi i e manu ac u ing,
pa s a e buil in laye s and hea accumula es, especially when
changing he di ec ion o a el o he welding o ch. The ob-
jec i e o his case is, using obus mul i a iable con ol (mo e
speci ically, Quan i a i e Feedback Theo y, o QFT), o adjus
he powe and o wa d speed o egula e he geome ic empe -
a u e p o ile. Figu e 1.2 shows a pic u e o a WAAM p ocess
whe e he o ch needs o pe o m cons an lips o di ec ion o
o m he pa .
In summa y, he objec i es o his wo k a e:
1. De elop con ol models o aid in designing con ol sys-
ems ha imp o e weld quali y in manu ac u ing.
14 Chap e 1. In oduc ion
FIGURE 1.2: Indus ial gas u bine blade buil
by WAAM echnology, Sou ce: WAAM3D Lim-
i ed
2. Design a mul i a iable decen alised con ol sys em o eg-
ula e he cooling cu e o a long welding p ocess.
3. Design a obus mul i a iable con ol sys em using an ad-
anced echnique o compensa e o hea accumula ion dis-
u bances in an addi i e manu ac u ing welding p ocess.
1.5 S uc u e
Chap e 2 p esen s an o e iew o he cu en ad ancemen s
and exis ing esea ch on welding p ocess con ol. I p o ides a
comp ehensi e o e iew o he ield, ou lining es ablished ech-
niques and highligh ing a eas o po en ial imp o emen .
Chap e 3 del es in o he ela ionship be ween G een’s unc-
ions and ans e unc ions in he con ex o hea conduc ion
du ing welding p ocesses. The chap e p esen s analy ical solu-
ions o calcula ing ans e unc ions in bo h one-dimensional
1.5. S uc u e 15
(1-D) and wo-dimensional (2-D) welding con igu a ions. Fu -
he mo e, i de elops a 3-D con ol model based on a nume i-
cal model de i ed using he Fini e Elemen Me hod. Addi ion-
ally, a non-linea simula ion model based on he Fini e Elemen
Me hod o alida e con ol models and con ol designs is p e-
sen ed.
In chap e 4 a decen alised con ol s a egy is designed upon
he 1-D model de eloped in he p e ious chap e wi h he pu -
pose o egula ing he cooling a e o he welded pa . A bench-
ma k p oblem o he 1-D welding con ol scena io is p esen ed,
ollowed by he p oposi ion o a p ac ical decen alised con ol
design app oach.
Chap e 5 aims o design a MIMO con olle using he QFT
sequen ial echnique. The ocus he e is on main aining he de-
si ed shape o he Hea A ec ed Zone (HAZ) du ing wi e a c
addi i e manu ac u ing. The con ol s a egy is designed o ad-
d ess dis u bances in oduced by laye ansi ions, whe e he
welding di ec ion lips be ween laye s. These ansi ions dis-
up he he mal equilib ium o he weld pool. To coun e ac
his, he con ol s a egy u ilises eed o wa d con ol o expedi e
he sys em’s e u n o a s able s a e.
Chap e 6 summa ises he key indings and con ibu ions o
he esea ch wo k.
17
Chap e 2
S a e o he A
The e is an abundance o published esea ch on a c welding.
Howe e , mos o i is ocused on he sea ch o op imum pa-
ame e s o speci ic p ocesses, he sea ch o new ma e ials o
addi i es o acili a e he join s o imp o e hei mechanical p op-
e ies. Many wo ks also seek o imp o e he inal quali y in
e ms o educ ion o po es, c acks o p ojec ions wi h pa am-
e e adjus men s, sys ems ela ed o he p o ec i e a mosphe e
o addi i es in he ille ma e ials.
The bibliog aphy used in his hesis is ocused on he con-
ol o he mal aspec s in elec ic a c welding, al hough in some
cases wo ks using lase s a e also p esen ed due o hei ele-
ance o he modelling o his wo k.
2.1 Mo ing hea sou ce model
In hea ans e , a c i ical challenge o enginee s, pa icula ly
in welding, is modelling mo ing hea sou ces. The ea ly 20 h
18 Chap e 2. S a e o he A
cen u y saw he dawn o esea ch on his opic by welding engi-
nee s, who employed bo h empi ical and heo e ical app oaches
o unde s and how hea beha es du ing welding [12].
Theo e ical solu ions o hese ca ego ies can be used o p e-
dic empe a u e dis ibu ion and cooling a es wi hin he weld.
This knowledge empowe s enginee s o g asp he impac o hea
sou ces on a weld’s quali y and he inal p oduc ’s pe o mance.
The solu ion o his p oblem depends on welding pa am-
e e s, ma e ial p ope ies and geome y o he pa . Un il he
mid-1930s, he s udy o he heo y o hea ans e om a mo -
ing sou ce was neglec ed, and empe a u e dis ibu ion due o
mo ing hea sou ces could only be calcula ed app oxima ely
[13]. Bu , in 1935, Rosen hal [14] published he heo y o hea
low om a mo ing sou ce o a c welding, whe e he hea dis-
ibu ion o welding in he quasi-s a iona y condi ions is gi en
by
T(x,y,z)=Q
2πKe−λ x e−λ √x2+y2+z2
px2+y2+z2(1)
whe e Tis he empe a u e, Qis he hea inpu , he speed o
welding, K he he mal conduc i i y and 1
2λ he he mal di u-
si i y. This model assumes ha he ma e ial p ope ies a e con-
s an h oughou he wo k piece and ha he hea sou ce is a
poin sou ce. The assump ion ha he hea sou ce is a poin
leads o inaccu acy in he icini y o he usion zone, and i is no
good o p edic ing he empe a u e wi hin he weld pool, bu
i can be conside ed accu a e enough o empe a u es wi hin
2.2. Moni o ing o welding 19
abou 20% o he mel ing empe a u e [15].
As i is conside ed ha he quali y o a weld is in luenced
by he dynamics o he empe a u e be ween he anges below
he mel poin — om 800 o 1200ºC in he case o s eel — due
o he cooling a e o he hea a ec ed zones [16], he same as-
sump ions a e conside ed in his wo k. Fo he same eason, he
dynamics o he s a e change o he ma e ial and adia ion a e
no conside ed as hey ha e a negligible e ec a lowe empe -
a u es compa ed o hea conduc ion [14].
2.2 Moni o ing o welding
The i s a emp s o apply eedback con ol in TIG welding ook
place in he 1980s [17]. F om he beginning, i was clea ha he
key ac o o he de elopmen o his echnology lay in he de-
elopmen o sui able senso s o he p ocess [18,19]. Thus, he
ype o signal ed back la gely de e mined he con ol echnique
o be employed.
The wo mos widely used measu emen me hods in weld-
ing ha e been he measu emen o a c pa ame e s and he use
o imaging. In he i s case, he non-linea ela ionship be ween
a c leng h, a c cu en and a c ol age is exploi ed. Since hese
las wo pa ame e s a e easily measu able, i is possible o de-
sign ac ua ion s a egies ha main ain a cons an ela ionship
be ween hem, achie ing a highe quali y weld han in he non-
eedback case [20–22]. O he mo e sophis ica ed schemes use
20 Chap e 2. S a e o he A
a c measu emen s o es ima e he weld pene a ion dep h, y-
ing o con ol i by adap i e s a egies [23,24], o e en o weld
join acking [25].
Howe e , he mos widesp ead op ion in he li e a u e is he
use o images cap u ed by came as. Since his is an en i onmen
cha ac e ised by he high b igh ness caused by he a c and he
high e lec i i y o many o he ma e ials o be welded, a obus
imaging sys em is equi ed. The mos widesp ead op ion in he
li e a u e is he one ha uses he lase iangula ion p inciple
o ex ac he p o ile o he pa o be welded, [26–35]. In gen-
e al, hese echniques include algo i hms o join ype ecogni-
ion and join acking, de e mining, among o he a iables, he
amoun o ma e ial o be deposi ed.
In his ega d, se e al welding p oduc manu ac u e s o e
"welding senso s" (see Figu e 2.1) whose iangula ion p inciple
p o ides da a o he p ocess. These welding senso s a e eady
o connec wi h indus ial obo s h ough d i e s o he main
manu ac u ing i ms and a e mainly used o ollow he weld
seam and o p ecompu e he welding gap.
Lase -scanning echnology has also been used o con ol weld
pene a ion based on a mul i a iable model ha ela es welding
cu en and a c leng h o wo geome ical pa ame e s ha se e
o es ima e weld pene a ion [36,37].
In ecen yea s, he high cos o lase echnology and im-
p o emen s in machine ision echniques ha e led some esea ch-
e s o op o di ec image acquisi ion o he welding p ocess as a
2.2. Moni o ing o welding 21
FIGURE 2.1: Welding Senso
sensing me hod [6,38–48]. E iden ly, hese images a e subjec ed
o a il e ing p ocess by bo h physical de ices and so wa e.
A hi d g oup o con ol me hods a e based on he use o
in a ed he mog aphy. Schemes ha e been implemen ed ha
adjus he pene a ion and wid h o he weld, as well as i s posi-
ion, based on he empe a u e p o iles de ec ed by his ype o
came a [49]. O he wo k ocuses on weld dep h, bu educing
he need o pos -p ocessing by using s a egically placed poin
senso s [50]. In a ed echnology has also been equen ly used
in he ield o GMAW welding [51,52].
Finally, he use o ul asonic senso s has been p oposed as
a low-cos al e na i e o ision sys ems. Thei use allows weld
join localiza ion e en on highly e lec i e su aces [53–55].
In any case i mus be conside ed se e al ac o s o de e -
mine he possibili y o being used in indus ial applica ions by
pa manu ac u ing companies. The ollowing lis ga he s some
ac o s ha may de e mine he sui abili y o hei use:
1. A ailabili y o comme cial p oduc s o he echnology.
28 Chap e 2. S a e o he A
app oaches o he mal managemen o hin-walled s uc u es.
They conclude ha simila geome ies can be achie ed using ei-
he ac i e o passi e cooling echniques, bu passi e cooling
esul s in longe deposi ion imes.
A mono a iable PID con olle is p oposed in [98] o adjus
he hea powe using he mel pool wid h ins ead o using he
empe a u e o he su ace. [99] implemen s also a PID con-
olle o add ess he esul ing geome y, using he heigh o he
deposi ed laye as inpu and he wi e eed speed o he WAAM
sys em as con ol ou pu .
The issue o laye di ec ion lip in WAAM has been explo ed
in [100] and [101]. These s udies p opose open-loop a ia ions
in speed a he beginning and end o hin pa s o mi iga e hea
accumula ion. To da e, no esea ch has add essed he con ol o
su ace empe a u e du ing laye di ec ion lips in WAAM using
a mul i a iable, closed-loop app oach based on a obus con ol
echnique.
2.4 Summa y
In he case o moni o ing, p ac ically all he wo k ca ied ou
op s o a speci ic echnology, which allows hem o deal wi h a
pa icula p oblem wi hin he complex welding p ocess. The e
is s ill a lo o wo k o be done in he in eg a ion o di e en
ypes o senso s and he c ea ion o con ol algo i hms o man-
age all his in o ma ion in a obus way, gene a ing an op imal
2.4. Summa y 29
welding p ocess in all i s aspec s.
Despi e i s impo ance as a manu ac u ing p ocess, he use
o con ol loops o imp o e welding esul s is no gene alised
in indus y. The e seems o be a need o a amewo k whe e
con ol enginee s can wo k com o ably wi hou depending on
low-o de iden i ied models o black boxes.
The lack o con ol-o ien ed models is he e o e pe cei ed
as one o he limi ing ac o s o he gene alisa ion o eedback
loops in he welding indus y. In pa icula , ans e unc ion
models, which a e s ill he p e e ed choice o he p ac ising
con ol enginee , a e shockingly ou o he pic u e.
This wo k aims o be pa o he pa h o ill ha oid, p e-
sen ing ans e unc ion models o 1D and 2D welding p o-
cesses, and a p elimina y con ol design o se e as a bench-
ma k o he con ol communi y. Inspi ed by Rosen hal’s quasi-
s a iona y model, dynamical models o de e mine how he em-
pe a u e p o ile changes when he hea inpu and a el speed
changes will be ob ained. The ans e unc ions a e de i ed us-
ing G een’s unc ions, as hey p o ide a e y con enien me hod
in his ega d. Fo he 3D case, he app oach ollowed is a nu-
me ical model based on he Fini e Elemen Me hod, whe e a
s a e-space con ol model is de i ed.
In his way, by being able o c ea e con ol loops o indus-
ial sys ems, his wo k b idges he gap be ween pu e heo e ical
models and he p ac ical applica ions. The model can be applied
30 Chap e 2. S a e o he A
o a ious hea ea men s o p ocesses in which he e is mo e-
men like welding whe e hea is applied h ough lase , a c o
plasma [102]. Also o cu ing p ocesses wi h hea [103], cu ing
ma e ials [104] o con olled cooling o me als [105] o example.
31
Chap e 3
Modelling o a c welding
The quali y o a weld depends on many ac o s. Some can be
di ec ly con olled in eal ime, such as he hea powe and i s
posi ioning and he amoun o ille ma e ial. O he s, howe e ,
a e ac o s ha canno be con olled, such as impu i ies in he
ma e ials o geome ic impe ec ions. Du ing he welding p o-
cess, each spa ial poin o he ma e ials in ol ed expe iences a
empe a u e a ia ion o e ime. A quali y weld equi es each
poin o each a su icien ly high empe a u e and a su icien ly
slow cooling a e. The e o e, i is conside ed ha egula ing he
empe a u e by adjus ing he con ollable pa ame e s can im-
p o e he quali y o he welds.
This chap e p esen s models o empe a u e e olu ion o
a weld, sui able o use in he design o con ol sys ems. The
weldmen can be modelled in one-dimensional, wo-dimensional
o h ee-dimensional space, depending on he shape o he pa .
Fo he one-dimensional and wo-dimensional cases, a pu ely
analy ical model is p esen ed. Fo he h ee-dimensional case,
32 Chap e 3. Modelling o a c welding
he app oach ollowed is a linea isa ion o a nume ical model
based on he ini e elemen me hod. In bo h cases he s a ing
poin is he hea ans e equa ion. Hea ans e means he en-
e gy anspo be ween ma e ial bodies due o a empe a u e
di e ence. The e a e h ee modes o hea ans e : conduc ion,
con ec ion and adia ion.
Fo hea conduc ion, he a e equa ion is known as Fou ie ’s
law, which is exp essed as
q=−k∇T(2)
whe e qis he local hea lux densi y (W/m2), kis he he mal
conduc i i y o he ma e ial (W/mK) and ∇Tis he empe a u e
g adien (K/m). Fo con ec i e hea ans e , he a e equa ion
is gi en by New on’s law o cooling as
q=h(Tw−Ta)(3)
whe e qis he con ec i e hea lux (W/m2), (Tw−Ta)is he em-
pe a u e di e ence be ween he wall and he ambien and his
he con ec ion hea ans e coe icien (W/m2K). The lux emi -
ed by adia ion om a su ace is gi en by he S e an-Bol zmann
Law
q=ϵσT4
w(4)
Chap e 3. Modelling o a c welding 33
whe e qis he adia i e hea lux (W/m2), ϵis he adia i e p op-
e y o he su ace e e ed o as he emissi i y, σis he S e an-
Bol zmann cons an (5.669x10−8) in W/m2K4and Twis he su -
ace empe a u e (K).
Fo he pu pose o de eloping a con ol model o welding o
a pa , o he 3-D case, only he mode o conduc ion is aken
in o conside a ion as i s con ibu ion o hea ans e is much
g ea e han he ans e in he su ace. Fo he 1-D and 2-D
models, he con ec ion mode is also conside ed bu no he a-
dia ion mode.
When conside ing also an ex e nal hea inpu sou ce, he
a ia ion o he empe a u e in a welding pa in he 3-D space,
is gi en by
ρCp
∂T
∂ =k∇2T+Q(5)
whe e ρis he densi y o he ma e ial (kg/m3), Cpis he speci ic
hea capaci y o he ma e ial (J/(KgK)) and Qis he hea inpu
om an ex e nal sou ce.
In a welding o simila hea inpu ea men p ocess, he hea
inpu comes om a o ch de ice ha mo es along some su ace
o he pa . I i is conside ed a poin inpu hea sou ce a he
o igin o coo dina es, and a mo emen o speed o he coo di-
na es, he ansien hea ans e equa ion o his case becomes
ρCp
∂T
∂ =k∇2T+δQ− ∇T. (6)
34 Chap e 3. Modelling o a c welding
I should be no ed ha al hough he empe a u e T(x,y,z, )
is a unc ion in R4, in p ac ice, he empe a u e ha can be mea-
su ed is only he su ace empe a u e, and ha in a con inuous
p ocess i would no make sense o measu e each poin o e
ime. Bu i con inuous mo emen is conside ed, and he sys-
em is in a quasi-s a iona y s a e, a geome ic p o ile o su ace
empe a u e akes in o accoun he empo al e ec .
3.1 G een’s unc ions
A G een’s unc ion is an in eg al ke nel used o sol e a ious
di e en ial equa ions, anging om simple o dina y di e en-
ial equa ions wi h ini ial o bounda y alue condi ions o mo e
complex inhomogeneous pa ial di e en ial equa ions (PDEs)
wi h bounda y condi ions. They p o ide a gene al way o de-
sc ibe he esponse o a di e en ial-equa ion solu ion o an a bi-
a y sou ce e m. They ep esen he impulse esponse a poin
x′o an inhomogeneous linea di e en ial ope a o (L=L(x))
de ined on a domain (x∈X⊂R3),
LGx,x′=δ(x−x′), (7)
whe e G(x,x′)is he G een’s unc ion and δ he del a o Di ac’s
unc ion. By mul iplying he abo e iden i y by a unc ion (x′)
3.1. G een’s unc ions 35
and in eg a ing wi h espec o x′yields
ZLGx,x′ (x′)dx′=Zδ(x−x′) (x′)dx′. (8)
The igh -hand side educes me ely o (x)due o p ope ies o
he del a unc ion, and because Lis a linea ope a o ac ing only
on xand no on x′, he le -hand side can be ew i en as
LZGx,x′ (x′)dx′. (9)
This educ ion is pa icula ly use ul when sol ing o u=u(x)
in di e en ial equa ions o he o m
Lu(x) = (x), (10)
because hey hold ha
Lu(x) = LZGx,x′ (x′)dx′, (11)
and hus, u(x)has he speci ic in eg al o m
u(x) = ZGx,x′ (x′)dx′. (12)
The unc ions o igina ed in he 1820s wi h he wo k o B i ish
ma hema ician Geo ge G een. He was pa icula ly in e es ed
in sol ing bounda y alue p oblems, speci ically o elec ici y
36 Chap e 3. Modelling o a c welding
and magne ism. I s applicabili y o sol ing ODEs is e y con-
enien when he unc ion o he speci ic di e en ial ope a o
and he bounda y and ini ial condi ions is known, since h ough
he supe posi ion p inciple, gi en a linea o dina y di e en ial
equa ion, one can i s sol e (7) o each poin and hen sum he
esul s.
The e a e se e al me hods o inding G een’s unc ions, in-
cluding eigen alue expansions, sepa a ion o a iables, and La-
place ans o ms. Due o he esea ch in di e en ields, gi en
he di e en ial ope a o and he domain, he speci ic G een’s
unc ion o some p oblems a e known and a e ga he ed in he
li e a u e. They ha e been ex ensi ely analysed o he p oblem
o hea conduc ion [106], and hey ha e also been used o he
de elopmen o analy ical models in welding [58].
Fo he case o he hea equa ion (3) in s eady s a e, he dis-
ibu ion o empe a u e along he space x∈X⊂R3is gi en
by
k∇2T(x) = Q(x), (13)
in which he linea di e en ial ope a o Lis he Laplacian, ∇2,
and in he absence o bounda y condi ions he alue o i s G een’s
unc ion is known,
Gx,x′=1
4π|x−x′|. (14)
3.1. G een’s unc ions 37
Using (7), T(x)can be calcula ed by adding all he impulse con-
ibu ions o Qas ollows. I Qis exp essed as
Q(x) = Zδ(x−x′)Q(x′)dx′, (15)
he dis ibu ion o empe a u e T(x)can be ob ained by
T(x) = ZV
Q(x′)
4πk|x−x′|d3x′. (16)
When ansien alues a e conside ed, G een’s unc ion
G( , | ′,τ) ep esen he empe a u e a he loca ion , a ime
, due o an ins an aneous poin sou ce o uni s eng h, loca ed
a a poin ′in a egion R, eleasing i s ene gy spon aneously
a ime =τ. The main ad an age o G een’s unc ions is ha
once G( , | ′,τ)is known o a speci ic p oblem [107], he em-
pe a u e dis ibu ion T( , )in he medium is ob ained [108] as
T( , )=
Ini ial Condi ions, T0( )
z }| {
ZRG , | ′,0T0 ′d
+
Bounda y condi ions, i( , )
z }| {
αZ
0dτ
N
∑
i=1ZSiG , | ′,τ1
ki
i ′,τdsi
+
Hea Inpu s, g( , )
z }| {
α
kZ
0dτZRG , | ′,τg ′,τd . (17)
whe e α(m2/s) is he he mal di usi i y and is calcula ed om
he he mal conduc i i y and capaci y by he ela ion α=k
ρCp.
44 Chap e 3. Modelling o a c welding
be apidly e alua ed by means o Laplace’s inal- alue heo em.
Thus, o he 1-D case, he s eady-s a e alue is
T0(x)=lim
s→0sT (x,s)=αe−1
2αh x+√x2(4αb+ 2)i
k√4αb+ 2·q0, (32)
which ully ag ees wi h he 1-D Rosen hal solu ion [109].
In a simila manne , o he 2-D and 3-D p oblems he inal
alue heo em leads o
T0(x,y)=e− x
2α
2πkK01
2αq(x2+y2) (4αb+ 2)·q0, (33)
and
T0(x,y,z)=e− (x+√x2+y2+z2)
2α
2πkpx2+y2+z2·q0(34)
which a e in ull ag eemen wi h he exp essions gi en by Rosen-
hal [14].
3.4 Modelling he welding a el speed as he
inpu o he sys em
As s a ed be o e, equa ions (28), (31) and (25) ep esen he ma h-
ema ical models o he welding p ocess o he 1-D, 2-D and 3-D
espec i ely, whe e he ob ained empe a u e p o ile is he ou -
pu , he hea sou ce powe is he inpu and he o ch a elling
speed ac s as a pa ame e .
3.4. Modelling he welding a el speed as he inpu o he
sys em 45
Bu , inasmuch as o ch speed is an ope a ional pa ame e
ha is easily modi iable, i seems easonable o include his pa-
ame e as a di ec con ol ac ion in he welding con ol model.
This can be achie ed by means o linea iza ion echniques.
3.4.1 T ans e unc ion o he 1-D case
When di e en ial equa ion (23) is exp essed abou an equilib-
ium poin , assuming he e a e small pe u ba ions among pa-
ame e s and a iables —[∆Q( ),∆ ( ),∆T(x, )]— we ge
∂∆T
∂ − 0
∆T
∂x−∆ ∂T0
∂x=α∂2∆T
∂x2+δ(x)∆Q
ρCp−b∆T. (35)
As T0=T0(x)is known om (32), he di e en ial equa ion (35)
is he same as (23), bu conside ing ha inpu s e ms come om
wo sou ces, i.e.
g∆(x, )=δ(x)
ρCp
∆Q( )+∂T0(x)
∂x∆ ( ), (36)
whe e he i s one is loca ed a he o igin and is due o inc e-
men s in he hea inpu , and welding o ch speed inc emen s ac
in a dis ibu ed manne .
As he e ec o ∆Q( )is exac ly he same as in eq. (23), he
i s ans e unc ion is s aigh o wa d om eq. (25):
∆Tq(x,s)
∆Q(s)=T(x,s)
Q(s) = 0
=1
ρCp
e−
0x+ x2(4αs+4αb+ 2
0)
2α
q4αs+4αb+ 2
0
. (37)
46 Chap e 3. Modelling o a c welding
On he o he hand, he e ec o a elling speed inc emen s
on empe a u e p o iles can be ob ained ollowing (17), bu he
G een’s unc ion o he p oblem (24) is modi ied [107] o con-
side dis ibu ed hea inpu s as ollows:
Gx, |x′=e− b+ 2
4α
2√πα exp "− (x−x′)
2α−(x−x′)2
4α #. (38)
Then, i is possible o w i e
∆T (x, )=α
kZ
0dτZ∞
−∞Gx, −τ|x′∂T0(x′)
∂x′∆ (τ)dx′, (39)
and, applying Laplace ans o m p ope ies,
∆T (x,s)=α
kZ∞
−∞Gx,s|x′∂T0(x′)
∂x′dx′∆V(s). (40)
Once he space in eg al is analy ically calcula ed, we ob ain
he ans e unc ion ha desc ibes he e ec o changes in o ch
a elling speeds on he empe a u e p o ile:
∆T (x,s)
∆V(s)=q0e− 0x
2α
2sk hχ(x,s)−χ(x,0)i, (41)
whe e
χ(x,s)=
sgn(x)+ 0
q4αs+4αb+ 2
0
e− x2(4αs+4αb+ 2
0)
2α. (42)
3.4. Modelling he welding a el speed as he inpu o he
sys em 47
3.4.2 T ans e unc ion o he 2-D case
Fo he 2-D case, he same p ocess is ollowed, by which he
ans e unc ion o he empe a u e p o ile wi h espec o he
a el speed is ob ained:
∆T (x,y,s)
∆V(s)=q0e− x
2a
4πaks [χ(x,y,s)−χ(x,y,0)], (43)
whe e
χ(x,y,s) = K0 px2+y2√4ab +4as + 2
2a!
+x
px2+y22| |− 2
√4ab +4as + 2
K1 px2+y2√4ab +4as + 2
2a!, (44)
whe e K0deno es he Bessel unc ion o second kind and ze o
o de and K1 he Bessel unc ion o second kind and i s o de .
Fo 3-D case, he ollowing in eg al unc ion mus be sol ed
o ob ain he ans e unc ion:
T(x,y,z) = Z∞
−∞Z∞
−∞Z∞
0
1
(x2+y2+z2)3
2
e−(z+z′)2
b+e−(−z+z′)2
b
0x2− 0xqx2+y2+z2+ 0y2+ 0z2−x
e 0(−x+x′)−(−x+x′)2+(−y+y′)2
bdz′dy′dx′(45)
48 Chap e 3. Modelling o a c welding
bu o he ime being i has no been achie ed due o i s com-
plexi y. In subsec ion 3.5.2, a nume ical app oach is employed
o ob ain he ans e unc ion o he empe a u e change wi h
espec o a el speed o he 3-D case. The nume ical model
will also enable simula ion and alida ion o con olle s.
3.5 FEM welding model and simula ion
Due o he di icul y o sol ing (45) and ob aining an analy ical
ans e unc ion as a con ol model o he 3-D case, a nume -
ical model is used o de i e a s a e space sys em om a Fini e
Elemen Model.
The Fini e Elemen Me hod (FEM) ollows a gene al s uc-
u e, bu he e a e a ia ions wi hin ha amewo k. The a i-
a ional app oaches a e he mos popula a ia ion, being he
Gale kin me hod he mos widely used. The Gale kin me hod
aims o con e a con inuous PDE in o a sys em o algeb aic
equa ions ha can be sol ed nume ically on a compu e . I a-
chie es his by:
1. Di iding he domain: The physical domain o he p oblem
(e.g., welding pa ) is i s disc e ized in o smalle subdo-
mains called elemen s. These elemen s can be iangles,
squa es, e ahed ons, o o he shapes depending on he
p oblem’s dimensionali y.
3.5. FEM welding model and simula ion 49
2. Choosing basis unc ions: Simple unc ions wi h speci ic p op-
e ies (o en polynomials) a e chosen wi hin each elemen .
These unc ions a e called basis unc ions and a e used o
app oxima e he unknown solu ion o he PDE wi hin ha
elemen .
3. P ojec ing he esidual: The go e ning PDE is e o mula ed
in o a weak o m, which o en in ol es in eg a ing he
equa ion o e he en i e domain. The Gale kin me hod
hen en o ces his weak o m by equi ing he esidual ( he
di e ence be ween he le and igh sides o he equa ion)
o be o hogonal (ze o a e age) o a se o weigh ing unc-
ions.
4. Sol ing he sys em: By applying he o hogonali y condi-
ion o e each elemen and using he chosen basis unc-
ions, a sys em o algeb aic equa ions is ob ained. This
sys em ela es he unknown coe icien s o he basis unc-
ions, essen ially ep esen ing he app oxima e solu ion a
speci ic poin s (nodes) wi hin he elemen s.
The sys em unde s udy is he h ee-dimensional e sion o
he sys em p esen ed in he p e ious sec ion. I consis s in a dy-
namic model ha simula es p ocesses in which he e is a mo ing
hea inpu . The inpu s o he sys em a e he powe o he hea
sou ce and he a el speed. I s ou pu s a e he empe a u e a
wo gi en poin s on he su ace o he piece.
50 Chap e 3. Modelling o a c welding
To de elop he con ol sys em, a model o he welding p o-
cess ( he plan ) is c ea ed. Fo his pu pose, he sys em is asumed
o be in s eady s a e, whe e he base me al has in ini e leng h,
he ene gy inpu and eloci y a e cons an , and he empe a u e
p o ile o e he en i e pa wi h espec o he poin o he hea
sou ce is in equilib ium. Conside ing he geome ic coo dina es
ela i e o he hea sou ce, his allows wo king wi h a bounded
coo dina e sys em e en i he o ch is in con inuous mo ion. In
his coo dina e sys em, unde s eady s a e condi ions, empe -
a u es a e cons an in space, whose s a e is said o be quasis a-
iona y. Figu e 3.4 shows his condi ion.
Conside ing Rosen hal equa ions [14], he con ol challenge
o ob aining a good empe a u e p o ile could be ansla ed o
he empe a u e con ol o wo ep esen a i e poin s: P1behind
he hea inpu , and P2on i s side; bo h belonging o he same
iso he m. Once again, inpu s a e he o ch speed and he inpu
powe Q.
The nex sec ion de elops in de ail a FEM simula ion model
ha will allow o he es ing and e i ica ion o he con ol sys-
ems o be de eloped.
3.5. FEM welding model and simula ion 51
FIGURE 3.4: Con ol inpu s and posi ioning o
poin s o empe a u e measu emen .
3.5.1 Simula ion Model
Recall ha he sys em can be modelled wi h he hea equa ion,
gi en by
∂T(x,y,z, )
∂ −k
ρCp∇2T(x,y,z, )
+ ( )∂T(x,y,z, )
∂x=δ(x,y,z)Q( )
ρCp
, (46)
whe e T(x,y,z, )is he empe a u e a any poin in he space
o he domain, is he ime, ( )is he a el speed, kis he
he mal conduc i i y cons an o he ma e ial, ρis he mass den-
si y o he ma e ial, Cpis he speci ic hea capaci y and Q( )
he hea inpu . Con ec ion and adia ion a e no conside ed be-
cause hea exchanges (losses) h ough he su ace o su ound-
ing a mosphe e can be neglec ed in ega d o he hea low in
he piece i sel . This assump ion is expe imen ally suppo ed
and is explained by he ac ha he hea conduc i i y o me als
is much g ea e han hei hea ansmission h ough he su ace
[14].
52 Chap e 3. Modelling o a c welding
The me hod o he Fini e Elemen s uses he weak o a ia-
ional o mula ion o es ablish he nume ical app oxima ion in
he domain o some space disc e iza ion. To con e a pa ial
de i a i e equa ion in o a a ia ional p oblem he equa ion is
mul iplied by a unc ion wand in eg a e he esul ing equa ion
o e he domain Ω,
ZΩ
∂T(x,y,z, )
∂ wdΩ−k
ρCpZΩ∇2T(x,y,z, )wdΩ
+ ( )ZΩ
∂T(x,y,z, )
∂xwdΩ=ZΩ
δ(x,y,z)Q( )
ρCp
wdΩ, (47)
whe e he unc ion wwhich mul iplies he PDE is called a es
unc ion and he unknown unc ion T o be app oxima ed is e-
e ed o as he ial unc ion. Then in eg a ion by pa s o e ms
wi h second-o de de i a i es is hen pe o med,
ZΩ∇2TwdΩ=−ZΩ(∇T·∇w)dΩ+Z∂Ω
∂T
∂nwds (48)
whe e ∂T
∂n=∇T·nis he de i a i e o Tin he ou wa d no mal
di ec ion non he bounda y. In he FEM me hod, he es unc-
ion is made o be ze o in he bounda y o he p oblem esul ing
in a a ia ional o m o he equa ion (46), o
ρCpZΩ
δT
δ wdΩ+kZΩ(∇T·∇w)dΩ
+ρCpZΩ( ( )·∇T)wdΩ=δ(xq,yq,zq)Q( ). (49)
whe e wis he es unc ion and Ω he domain o he p oblem.
3.5. FEM welding model and simula ion 53
Using one o he se e al FEM me hods (see o example he
Gale kin me hod in [110]) a sys em o equa ions is buil ha can
be exp essed in ma ix o m as
L˙
T( ) + K( ( ))T( ) = Q( ), (50)
whe e Land K( ( )) a e known as he elas ici y and s i ness
ma ices espec i ely, and as he load ec o . Ob aining he
alues o he ma ices in (50) gi es he dynamic FEM model o
he hea equa ion o he domain. To pe o m simula ions and o
de i e he p oblem ma ices, he FEniCS so wa e [111] is used.
The FEniCS so wa e acili a es he assembly o ma ices o di -
e en ial equa ions exp essed in he weak o mula ion. The spe-
ci ic pa ame e s used in he spa ial disc e iza ion a e shown in
Table 3.1. To imp o e he p ecision o he nume ical model, an
i egula mesh (A.2) is used whe e g ea e g anula i y is applied
in he icini y o he hea sou ce poin , whe e g ea e empe a-
u e g adien s a e expec ed.
TABLE 3.1: Disc e iza ion pa ame e s.
Pa ame e Value Desc ip ion
X120 Dimension o he X axis
Y40 Dimension o he Y axis
Z10 Dimension o he Z axis
N1484 Numbe o poin s
M5899 Numbe o cells
DOF 1 Deg ee o eedom
FS Lag ange Func ion space
60 Chap e 3. Modelling o a c welding
FIGURE 3.7: FEM Simula ion o dynamic ali-
da ion
and he simula ion is un om 0 o 1000 seconds wi h d =0.01
seconds.
Once empe a u e da a a e ob ained, he Fas Fou ie ans-
o m o he inpu s and ou pu s a e ob ained as U(ω)and Y(ω)
and om hose he ans e unc ion is calcula ed as
H(ω) = U(ω)Y(ω)
U(ω)2. (58)
Then, gains and phases a e compu ed o ep esen and compa e
wi h he analy ical unc ions in he Bode plo diag am.
This way, Figu e 3.8 ep esen s he compa ison be ween he
ob ained simula ion esul s and he equency esponse
∆Tq(xi,jω)/∆Q(jω) o he same se o poin s {xi}wi h e y
good ag eemen among hem.
The model has also been alida ed conside ing a iable e-
quency o ch a elling speeds and, as Figu e 3.9 shows, ob-
ained beha iou ma ches ha o ∆T (xi,jω)/∆V(jω)wi h g ea
accu acy.
3.5. FEM welding model and simula ion 61
a)
b)
xi=−0.04
xi=−0.03
xi=−0.02
xi=−0.01
Magni ude (dB)
F equency ( ad/s)
Phase (deg)
10-3 10-2 10-1 100101
-400
-300
-200
-100
0
10-3 10-2 10-1 100101
-2000
-1500
-1000
-500
0
FIGURE 3.8: Compa ison o equency e-
sponses o FEM and ∆Tq(xi,jω)
∆Q(jω): a) Magni ude,
b) Phase
a)
b)
xi=−0.04
xi=−0.03
xi=−0.02
xi=−0.01
Magni ude (dB)
F equency ( ad/s)
Phase (deg)
10-3 10-2 10-1 100101
60
70
80
90
100
110
10-3 10-2 10-1 100101
-135
-90
-45
0
45
90
135
180
FIGURE 3.9: Compa ison o equency e-
sponses o FEM and ∆T (xi,jω)
∆V(jω): a) Magni ude,
b) Phase
63
Chap e 4
Decen alized MIMO
con ol o he 1-D case
Now ha we ha e models o he welding p ocess we can ace
con ol p oblems. In his chap e , a con ol sys em is designed
o he use case o a long welded join p ocess (a ship, b idge,
la ge con aine , e c.) in which, using a one-dimensional model,
he powe and ad ance speed a e adjus ed o main ain a desi ed
cooling cu e, using a decen alised con ol.
The model p edic s he empe a u e dis ibu ion wi hin a
ma e ial based on i s powe inpu and a el speed. In he one-
dimensional case, empe a u es along he ma e ial’s x-axis, and
in he wo and h ee-dimensional cases, he empe a u es o some
supe icial poin s can be measu ed using a non-con ac senso
like an in a ed py ome e . The ollowing sec ion discusses he
speci ic empe a u e measu emen poin s equi ed o imple-
men ing a decen alised mul i a iable con ol s a egy o he
1-D case.
64 Chap e 4. Decen alized MIMO con ol o he 1-D case
4.1 Con ol p ope ies
4.1.1 Ou pu s selec ion
A key objec i e o he mal sys ems is o con ol he cooling cu e
o he ma e ial [115–117]. In his wo k, his is achie ed by go -
e ning he empe a u e o wo poin s along he slab —see x1,x2
on Figu e 3.6—. The choice o wo poin s has o do wi h he ac
ha he e a e wo manipula ed a iables: Hea sou ce powe
and o ch a elling speed. This way, a 2 ×2 mul i a iable sys-
em is con igu ed. Howe e , i s p ope ies depend e y much
upon he pa icula wo poin s whose empe a u e is measu ed.
Based on classical mul i a iable con ol ools, his sec ion de-
ines a p ocedu e o make such choice.
To begin wi h, solid lines in Figu es 3.8 and 3.9 show he
Bode plo s de ined by he ans e unc ions (37) and (41) a
poin s xi=−0.01, −0.02, −0.03, −0.04m. Simila ly, Figu e
4.1 shows he esponses o he sys em, ob ained wi h FEniCS,
when each o he wo con ol inpu s su e s a s ep de ia ion
om i s equilib ium alue.
a) b)
xi=−0.04
xi=−0.03
xi=−0.02
xi=−0.01
Time (s)
∆Tq(xi, )
Time (s)
∆T (xi, )
0 10 20 30 40
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40
-2.4
-1.8
-1.2
-0.6
0
0.6
1.2
FIGURE 4.1: No malised s ep esponses a
poin s xi=−0.01, −0.02, −0.03, −0.04: a)
Hea sou ce powe , b) To ch a elling speeds
4.1. Con ol p ope ies 65
F om a con ol poin o iew, Figu e 3.8 shows ha low e-
quency gain be ween powe and empe a u e is almos he same
o all xipoin s. Howe e , as he dis ance be ween he sou ce
and he senso inc eases, he Bode plo o he i a ional ans-
e unc ion educes i s bandwid h and accumula es phase lag,
which jus i y he slowe esponse obse ed in he ime domain
—Figu e 4.1a—. To minimise his ha m ul delay, he i s poin
should be placed in he closes posi ion o he sou ce ha s ill
p o ides a good eading o he empe a u e. In he example,
his poin is assumed o be x1=−0.01 m.
The plo s de ining he ela ion be ween o ch a elling speeds
and empe a u e —Figu e 3.9— a e mo e complex. Speeding
up he ad ance educes he inal empe a u e on poin s nea he
sou ce, bu inc eases i as he dis ance g ows. In o he wo ds,
he gain goes om nega i e alues o posi i e ones. A he
same ime, s ep esponses —Figu e 4.1b— show ha he plan
is clea ly non-minimum phase on poin s nea he sou ce. Wi h
all his in mind, i is ha d o decide whe e o place he second
senso : oo close o he i s one implies nega i e gain and un-
de shoo , and oo a om i ensu es limi ed bandwid h and a
slow esponse.
To ge ou o his conund um, and also o de ine he bes
inpu -ou pu pai ing, a Rela i e Gain A ay (RGA) analysis o
he plan is p esen ed in he nex sec ion.
66 Chap e 4. Decen alized MIMO con ol o he 1-D case
4.1.2 RGA analysis
In mul i a iable p ocess con ol sys ems, RGA helps de e mine
he bes pai ings be ween manipula ed a iables and measu ed
a iables [118]. I conside s how much o a change in one in-
pu a iable a ec s a pa icula ou pu a iable, compa ed o
how much i a ec s o he ou pu s. Selec ing op imal pai ings
h ough RGA analysis leads o imp o ed con ol sys em pe o -
mance and s abili y and educed coupling e ec s.
To calcula e i , i s he s eady-s a e ans e unc ion ma ix
o he MIMO sys em is ob ained, which ep esen s he gain a io
be ween each inpu and ou pu . Nex , i is mul iplied elemen -
wise by he in e se o he anspose o i sel . A good pai ing is
achie ed when he ob ained ma ix esembles he iden i y ma-
ix by ea anging he o de o he iles o columns.
In wha ollows, pai ing ules sugges ed by [119] a e ol-
lowed:
Pai ing ule 1. P e e pai ings such ha he ea -
anged sys em, wi h he selec ed pai ings along he
diagonal, has an RGA ma ix close o iden i y a e-
quencies a ound he closed-loop bandwid h.
Pai ing ule 2. A oid (i possible) pai ing on neg-
a i e s eady-s a e RGA elemen s.
The e o e, he i s ask is o de e mine he desi ed closed-
loop bandwid h. Looking a he Bode plo s in Figu e 3.9, his
4.1. Con ol p ope ies 67
bandwid h is es ima ed o be 0.4 ad/s, as beyond ha poin ,
oo much phase lag is accumula ed in he powe - o- empe a u e
ans e unc ion. Nex , Figu e 4.2 shows he magni ude o e-
quency dependen RGA elemen s o di e en choices o he sec-
ond measu ing poin , in pa icula x2=−0.02, −0.03, −0.04,
−0.05m. I can be seen ha he op ion p o iding he bes al-
ues a he bandwid h equency, ep esen ed in Figu e 4.2 wi h
a dashed e ical line, is x2=−0.04 m. In his case, he RGA
ma ix becomes
Λ(j0.4)=
0.7784 +0.1708j0.2216 −0.1708j
0.2216 −0.1708j0.7784 +0.1708j
, (59)
which sugges s ha he empe a u e a x1=−0.01 mshould be
pai ed wi h he powe inpu , whe eas he empe a u e a x2=
−0.04 mshould be pai ed wi h he eloci y inpu .
To check he second pai ing ule, he s eady s a e RGA is
calcula ed o x2=−0.04 m, and i s alues a e ound o be
non-nega i e. I is in e es ing o no ice how he s a ic RGA ge s
close o he iden i y as x2mo es away. Howe e , he p ice
o his coupling educ ion a e la ge delays in he open-loop e-
sponse, which in u n implies a e y limi ed closed-loop band-
wid h.
A e selec ing he poin s in he ma e ial whe e he empe a-
u e is going o be ead —see Figu e 3.6—, he sys em o con ol
68 Chap e 4. Decen alized MIMO con ol o he 1-D case
a)
b)
x2=−0.05
x2=−0.04
x2=−0.03
x2=−0.02
Rela i e Gain
F equency ( ad/s)
Rela i e Gain
10-3 10-2 10-1 100101
0
0.5
1
1.5
2
10-3 10-2 10-1 100101
0
0.2
0.4
0.6
0.8
1
FIGURE 4.2: RGA plo s o poin s x2=−
0.02, −0.03, −0.04, −0.05, when x1=−0.01:
a) Diagonal Rela i e Gain, b) O -diagonal Rel-
a i e Gain
is de ined by he ans e ma ix
G(s) =
g11(s)g12(s)
g21(s)g22(s)
=
∆Tq(x1,s)
∆Q(s)
∆T (x1,s)
∆V(s)
∆Tq(x2,s)
∆Q(s)
∆T (x2,s)
∆V(s)
, (60)
whe e x1=−0.01 mand x2=−0.04 m.
4.2 Decen alised mul i a iable con ol
4.2.1 Benchma k p oblem s a emen
As explained be o e, he objec i e is o each and main ain ce -
ain empe a u es in wo poin s o he mo ing piece, he e o e
4.2. Decen alised mul i a iable con ol 69
con olling he cooling cu e and ensu ing good mechanical p op-
e ies in he ma e ial. Mo e speci ically, he p oposed bench-
ma k consis s in a acking con ol p oblem: beginning a he
equilib ium condi ions de ined by Table 3.2, i.e. T(x1) = 1397
and T(x2) = 389, manipula e he inpu powe and speed in o -
de o each a new cooling cu e in which T(x1) = 1600 and
T(x2) = 500.
4.2.2 Con olle design
This chap e ocuses on es ing he model a he han on con-
olle syn hesis. The e o e, he simples app oach o mul i-
a iable con ol design is adop ed: independen decen alised
con ol. As a consequence, he sys em is go e ned by a diag-
onal con olle whose elemen s a e uned aking in o accoun
only he diagonal elemen s o he ans e ma ix (60). The ob-
jec i e is o ob ain a wo king con ol scheme whose esponse
can be compa ed wi h o he s p o ided by mo e sophis ica ed
echniques.
In spi e o he ac ha he ans e unc ions (37) and (41) a e
i a ional, hei equency esponse can be used o adjus a con-
olle jus like o a ional unc ions. This is a g ea ad an age o
he equency domain app oach. In his case, he loop shaping
in he Bode plo seeks o maximise he sys em bandwid h while
keeping a phase ma gin o 40◦in bo h loops. The non-minimum
phase na u e o he plan makes hese goals easily a ainable o
76 Chap e 5. Mul i a iable QFT con ol o he 3D case
he welding pool by adequa ely manipula ing he powe and
speed o he hea sou ce. A benchma k sys em is es ablished o
e alua e he e ec i eness o he p oposed con ol sys em. The
esul s demons a e signi ican imp o emen in empe a u e con-
ol, leading o enhanced laye cons uc ion quali y and educed
need o heigh co ec ions o cooling pauses.
Op imising he pa h o new laye s in addi i e manu ac u -
ing (AM) is c ucial o se e al easons. I can:
1. Imp o e quali y: A well-planned pa h minimises de ec s
like wa ping, c acking, and poo su ace inish.
2. Reduce ime: By minimising a el dis ances and op imis-
ing ool mo emen s, p in ing ime can be signi ican ly e-
duced.
3. Sa e ma e ial: Op imised pa hs can minimise was ed ma e-
ial and imp o e ma e ial u ilisa ion.
Pa h planning s a egies a e key aspec s o op imising he
pa h o new laye s in AM. Speci ic so wa e ools a e used in
o de o ansla e a 3D model in o laye in o ma ion and gene -
a e oolpa hs based on use -de ined pa ame e s and chosen pa h
planning s a egies. Among he pa ame e s, he mal conside -
a ions a e impo an : Hea dis ibu ion is c i ical in AM. Op i-
mising he pa h helps manage hea buildup o p e en wa p-
ing and ensu e consis en ma e ial p ope ies. Bu his can im-
pac he ime needed o c ea e he pa because back-and- o h
Chap e 5. Mul i a iable QFT con ol o he 3D case 77
FIGURE 5.1: Illus a ion showing a pa c ea ed
by laye s ollowing a pa h in a single di ec ion
and wi h back-and- o h mo emen s.
mo emen s a e a oided. I he hea buildup can be con olled
by o he means and di ec ion lips a e allowed, he ime o c e-
a e he pa s is educed.
Figu e 5.1 illus a es an addi i ely manu ac u ed pa wi h 3
laye s. In one case, he laye s a e deposi ed in he same o wa d
di ec ion (a ow wi h he solid line) whe e he ools a e eposi-
ioned wi hou deposi ing ollowing he pa h o he a ow wi h
he dashed line. In he second case, he i s laye is deposi ed in
he o wa d di ec ion, he second is deposi ed in he backwa d
di ec ion and he hi d again in he o wa d di ec ion. A lip
di ec ion occu s each ime he ool eaches he end o he laye .
The second app oach is ob iously much as e han he i s , bu
i o he measu es a e no aken in o accoun , i can c ea e hea
build-up p oblems a he edges.
78 Chap e 5. Mul i a iable QFT con ol o he 3D case
5.1 The lip o di ec ion p oblem
Con ol in addi i e manu ac u ing o me als is mo e challenging
han one pass welding, as empe a u e p o iles a y o e ime,
due o epea ed hea inpu om he o ch. Howe e , compen-
sa ing o his cons an hea build-up is no a di icul challenge
in con olle design as he dynamics a e e y slow.
On he o he hand, compensa ing o speci ic i egula i ies
in he base me al, such as small holes o c acks h ough which
he o ch passes, is ex emely di icul , since senso s only de ec
such si ua ions oo la e o any co ec ed ac ion.
In his way, i a si ua ion ha may cause a sudden dis u -
bance in he sys em is known in ad ance, eed o wa d con ol
can be applied, as in he case o a di ec ion lip manoeu e. In
his ype o p oblem, whe e he e is a sudden change o he s a e
o he sys em, pulling i ou o i s equilib ium, he use o eed-
o wa d is o g ea help, and he QFT echnique is known and
s ands ou o i s use.
A case ha occu s qui e equen ly is he con ibu ion o se -
e al co ds o when a wall is being buil by such addi i e manu-
ac u ing. When he a el di ec ion changes, ma e ial and en-
e gy begin o be added o he newly hea ed a ea, so i con ol is
no applied, he empe a u e ises apidly p oducing aniso opy
[120]. Unlike welding sys ems unde s eady s a e condi ions,
his case does p esen a challenge o con ol design. I is also
a case ha , al hough speci ic, occu s ela i ely equen ly in he
5.1. The lip o di ec ion p oblem 79
pa hs ollowed by addi i e manu ac u ing sys ems [121].
When di ec ion lip occu s, he poin s a e mo ed o hei
symme ic posi ions as in he case whe e a obo ool e e ses
wi h he senso s a ached. This p oduces a sudden dis u bance
in he sys em, as he s a e o he poin s ahead o he o ch is e-
placed by ha o he poin s behind i .
The p ima y goal o he con ol sys em is o es o e he em-
pe a u e a he designa ed poin s o hei p ede ined alues by
manipula ing he a el speed and powe Q. This s able em-
pe a u e p o ile is c ucial o main aining he desi ed mechani-
cal p ope ies h oughou he p ocess, as shown in [11].
This chap e add esses p ecisely he egula ion o he HAZ
a e he di ec ion lip o he mo emen in he ab ica ion o a
piece using addi i e manu ac u ing. To his aim, con ol is used
o e u n as quickly as possible o he p e ious equilib ium s a e.
5.1.1 Di ec ion lip modelled as an impulse dis u bance
As men ioned in 3.5.2, he linea s a e space model ob ained
om he nonlinea Fini e Elemen model will be used o he de-
sign. Howe e , his model needs o be enhanced o adequa ely
ep esen he lip o di ec ion manoeu e. In he linea ized model,
his mo ion is equi alen o a sudden change o s a e o he sys-
em, in which empe a u es a e ins an ly eplaced by ha o he
poin s symme ically loca ed wi h espec o he x axis.
80 Chap e 5. Mul i a iable QFT con ol o he 3D case
0 5 10 15
0
500
1000
1500
2000
0 5 10 15
1000
1500
2000
2500
3000
FIGURE 5.2: Open loop ime esponse o he
impulse dis u bance in he linea and nonlin-
ea models.
Conside ing ha in s a e space ini ial condi ions can be ep-
esen ed by impulse inpu s eaching di ec ly each s a e wi h he
co esponding alue, a change o s a e can be modelled by a di -
e ence be ween he new and he p e ious s a e. This allows he
lip o di ec ion o be modelled by an impulse inpu whose am-
pli ude is he di e ence be ween he equilib ium in he o wa d
di ec ion (+) and he equilib ium in he backwa d di ec ion (−).
The esul ing s a e space sys em is he e o e
˙
x=Ax+[BQB x0−−x0+]∆Q( )
∆ ( )
δ( ). (66)
No ice ha he inpu ma ix Bis decomposed in o i s cons i uen
ec o s BQ,B , and now includes a hi d column x0−−x0+ ha
e lec s he di ec ion lip.
5.2. Con ol app oach 81
To he bes o he au ho ’s knowledge, his me hod o de-
i ing a linea model di ec ly om he FEM me hod, as well as
he ep esen a ion o a sudden manoeu e as an impulse dis u -
bance, has no been used o documen ed be o e.
Once he linea con ol model has been ob ained, we can
p oceed o design a con ol sys em ha a enua es he e ec s
o he di ec ion lip impulse dis u bance. Howe e , he ac ha
such dis u bance is known in ad ance allows he in oduc ion
o eed o wa d compensa ion. The nex sec ion p esen s he ol-
lowed app oach in he design o he con olle . The design is
unde aken in sec ion 5.4.
5.2 Con ol app oach
The sys em de i ed in sec ion 3.5.2 is o he same o de as he
numbe o poin s o he FEM model. To p oceed wi h he con-
ol design his sys em is educed o he minimum o de model
keeping he same equency esponse. Figu e 5.2 shows he ime
esponse compa ison be ween he linea sys em ob ained wi h
he explained p ocedu e and educed o he 40 h o de , and he
nonlinea FEM model. A MATLAB sc ip ha pe o ms his
educ ion using he ma ices ob ained be o e is shown in Ap-
pendix A.3.
82 Chap e 5. Mul i a iable QFT con ol o he 3D case
5.2.1 Benchma k sys em
To es di e en con ol s a egies, a benchma k sys em is de-
ined using he de eloped model. The piece used o he bench-
ma k is he one desc ibed in subsec ion 3.5. The sys em wo ks
wi h a nominal powe o Q=7500 Wa s and a a el speed o
=5 mm/s. The measu emen poin s ha e been placed in he
1200 deg ee iso he m when he sys em is in a s a iona y si ua-
ion [11].
Table 5.1 shows he pa ame e s used in he p oposed con-
ol p oblem. I mus be aken in o accoun ha he dimensions
o he piece ha e been chosen in such a way ha he model
is equi alen o a semi-in ini e space, ha is, X∈(−∞,∞),
Y∈(−∞,∞)and Z∈[0, ∞), o a oid edge e ec s.
TABLE 5.1: Pa ame e s o he piece used in he
benchma k.
Symbol Uni Value Desc ip ion
Xmm 120 Dimension in X axis
Ymm 40 Dimension in Y axis
Zmm 10 Dimension in Z axis
ρkg/m37870 Ma e ial densi y
CpJ/(Kg K) 2719 Speci ic hea capaci y
KJ/(m s K) 60 The mal conduc i i y
hW/(m2K)0 Con ec ion coe icien
Q0W 7500 S eady s a e hea inpu
0mm/s (-5,0,0) S eady s a e a el speed
Pimm (0,0,0) Hea applica ion poin
P1mm (14.53,0,0) Measu emen poin 1
P2mm (0,2.35,0) Measu emen poin 2
T1 e ◦C 1200 P1 empe a u e e e ence
T2 e ◦C 1200 P2 empe a u e e e ence
5.2. Con ol app oach 83
5.2.2 Analysis
Following [11], he measu emen poin s selec ed a e on he igh
o he hea inpu and behind i , in he poin s c ossing he same
iso he m. Se e al iso he ms ha e been conside ed, ying o
s ike a balance be ween esponsi eness and he measu emen
di icul y nea he hea sou ce. A highe iso he m makes he sys-
em esponse as e bu p esen s he p oblem o ha ing o mea-
su e he empe a u e e y nea he hea sou ce, whe e in a ed
senso s a e a ec ed by adia ion. Looking o his comp omise,
he 1200 deg ee iso he m was selec ed. Taking in o accoun he
selec ed benchma k pa ame e s, he i s measu ed empe a u e
T1belongs o he poin in he mo emen axis P1, and empe a-
u e T2co esponds o a poin P2in he pe pendicula axis, as
indica ed in Table 5.1.
Fo he inpu -ou pu pai ing, an RGA analysis has been pe -
o med, yielding simila esul s as in he 1D case [122] i.e. o
ecommend pai ing T1wi h Qand T2wi h . In he 3D case as
in he 1D case, he poin whose empe a u e is used o con ol
he speed inpu is he one closes o he hea sou ce.
5.2.3 Con ol s a egy
To add ess he dis u bance ejec ion p oblem, a con ol s uc-
u e wi h wo deg ees o eedom is employed, by inco po a -
ing a p e il e a he inpu o he known dis u bance. Figu e 5.3
84 Chap e 5. Mul i a iable QFT con ol o he 3D case
−
E(s)G(s)ΔQ(s)
Δ (s)P(s)
M(s)F(s)
δ(s)
D(s)
ΔT1(s)
ΔT2(s)
FIGURE 5.3: Con ol s uc u e.
depic s he con olle a chi ec u e, whe e he elemen s o be de-
signed a e he 2 ×2 diagonal ma ix G(s)and he 2 ×1 ec o
F(s).P(s)and D(s) ep esen , espec i ely, he modelled plan
and dis u bance dynamics. Finally, δ(s)se es as he inpu o
his con ol con igu a ion, gene a ing an impulse when a dis-
u bance occu s and emaining ze o o he wise.
The ou pu s a e he wo empe a u es ha he con olle aims
o main ain a he equilib ium alues. M(s)is he ec o o
ans e unc ions ha encapsula es he desi ed esponse o he
sys em o an impulsi e dis u bance. Employing a model o
ma ching he dis u bance esponse demands an enhancemen
in exis ing MIMO QFT echniques. This de elopmen is elabo-
a ed on in Subsec ion 5.3.3. The desi ed sys em dynamics M(s)
a e speci ied in subsec ion 5.4.1.
5.3 The QFT me hod
The selec ed con olle design me hod is Quan i a i e Feedback
Theo y (QFT) [123]. The a ionale behind he QFT echnique is
5.3. The QFT me hod 85
o minimise he use o eedback, shi ing he esponsibili y o
he eed o wa d con olle . The undamen al p inciple is ha
eedback should be limi ed o compensa ing o sys em unce -
ain ies, including unknown dis u bances.
I is a obus con ol design me hod ha ocuses on ensu ing
sys em pe o mance o e a speci ied ange o plan unce ain-
ies. The echnique equi es he de ini ion o a desi ed esul
and a maximum allowable ole ance, known as a speci ica ion.
Then a se o plan s a e de ined o which his speci ica ion mus
be me . A disc e e se o ele an equencies is selec ed o ans-
la e he speci ica ions in o equency-domain cons ain s, which
a e ep esen ed as QFT bounds in he Nichols cha . The con-
olle is designed by manipula ing i s pa ame e s o shape he
nominal open-loop esponse in he Nichols cha , a oiding he
QFT bounds a all design equencies. Con olle s and il e s
a e designed o mee he speci ica ions in his equency lis , bu
i he equencies a e wisely chosen, ul ilmen o he speci ica-
ions can be expec ed in he whole spec um and by he whole
se o plan s.
The inal design using his me hod usually in ol es se e al
i e a ions o he men ioned s eps and can be de ined h ough he
ollowing algo i hm ha he designe mus ollow:
1. Model he unce ain plan and he nominal plan P(s),P0(s)
2. Un il a sa is ac o y design is eached
(a) De ine/ une speci ica ions
92 Chap e 5. Mul i a iable QFT con ol o he 3D case
ˆ
p21d1+ˆ
p22d2
ˆ
p22 +g22 ≤x2b2, (79)
ˆ
p21
ˆ
p22 +g22 b1≤(1−x2)b2, (80)
whe e x1∈(0,1)and x2∈(0, 1)a e design pa ame e s ha
mus be adjus ed o ind he poin ha a ou s less bandwid h
in g11 and g22.
Unlike he sequen ial me hod, using he non-sequen ial p o-
cedu e, he loops can be designed in any o de , as he designed
con olle is no applied in he design o subsequen loops. Ne -
e heless, an i e a i e edesign o loops gene ally akes place, as
educing he ole ances b1and b2, when he design allows, can
bene i he pe o mance o he o he loops. This can pe mi u -
he educ ion o ole ances in o he loops, and so on.
In o de o use he sequen ial me hod, i s , bo h sides o (68)
a e mul iplyed on he le by he ma ix
h1 0
−ˆ
p21
ˆ
p11+g11 1i, (81)
gi ing hˆ
p11+g11 ˆ
p12
0ˆ
p2
22+g22 iy1
y2=hˆ
p11 ˆ
p12
ˆ
p2
21 ˆ
p2
22 ihd1
d2i(82)
whe e
ˆ
p2
21 =ˆ
p21 −ˆ
p21 ˆ
p11
ˆ
p11 +g11 =ˆ
p21g11
ˆ
p11 +g11 (83)
and
ˆ
p2
22 =ˆ
p22 −ˆ
p21 ˆ
p12
ˆ
p11 +g11 . (84)
5.3. The QFT me hod 93
F om he p e ious equa ions, y1and y2can be exp essed, e-
spec i ely, as
y1=ˆ
p11d1+ˆ
p12d2−ˆ
p12y2
ˆ
p11 +g11 , (85)
y2=ˆ
p2
21d1+ˆ
p2
22d2
ˆ
p2
22 +g22 . (86)
Due o he Cauchy-Schwa z inequali y we know ha y1and y2
a e uppe bounded by
y1≤|ˆ
p11d1+ˆ
p12d2|+|ˆ
p12|y2
|ˆ
p11 +g11|, (87)
y2≤
ˆ
p2
21d1+ˆ
p2
22d2
ˆ
p2
22 +g22 . (88)
Taking ad an age o his, we can design he MIMO 1DOF diag-
onal con olle as wo SISO con olle s. Fi s , we de ine uppe
bound speci ica ions, b1and b2, o each loop. Then, we can use
hese speci ica ion o calcula e g11 and g22, espec i ely, ensu -
ing hey comply wi h he ollowing inequali ies:
|ˆ
p11d1+ˆ
p12d2|
|ˆ
p11 +g11|≤x1b1, (89)
|ˆ
p12|
|ˆ
p11 +g11|b2≤(1−x1)b1, (90)
and
ˆ
p2
21d1+ˆ
p2
22d2
ˆ
p2
22 +g22 ≤b2, (91)
whe e a di ision o ole ances h ough pa ame e x1is applied
in (87), in he same manne as in he non-sequen ial me hod.
94 Chap e 5. Mul i a iable QFT con ol o he 3D case
u(s)
F(s)
−
G(s)P(s)
y(s)
−
e(s)G(s)P(s)
w(s)
F(s)D(s)
y(s)
——–
——-
———
−
e(s)G(s)u(s)P(s)
d(s)
y(s)
1
FIGURE 5.5: Block ep esen a ion o a dis u -
bance ejec ion 1DOF MIMO p oblem.
u(s)
F(s)
−
G(s)P(s)
y(s)
−
e(s)G(s)P(s)
w(s)
F(s)D(s)
y(s)
——–
——-
———
−
e(s)G(s)u(s)P(s)
d(s)
y(s)
1
FIGURE 5.6: Block ep esen a ion o a mea-
su ed dis u bance ejec ion 2DOF MIMO p ob-
lem.
F om hese equa ions he decoupling QFT bounds a e c e-
a ed and he design p ocess can p oceed wi h he me hod as
desc ibed in he p e ious subsec ion.
5.3.2 Measu ed dis u bance ejec ion
When he dis u bance can be measu ed, he solu ion can be g ea ly
imp o ed using an 2DOF s uc u e wi h a p e- il e F(s)as illus-
a ed in Figu e 5.6. This me hod was de eloped in [128]. The
ans e ma ix om he dis u bance o he e o Tew is gi en by
Tew =[1+PG]−1[−D−PF], (92)
5.3. The QFT me hod 95
ha can be ea anged as
hP−1+GiTew =−P−1D−F, (93)
and using again he no a ion P−1= [ ˆ
pij], he elemen -wise ex-
p ession becomes
ew
ij =1
ˆ
pii +gii "−
n
∑
k=1
ˆ
pikdkj − ij −∑
k=1
ˆ
pik ew
kj #. (94)
Due o o e bounding, he dis u bance ejec ion speci ica ion
ew
ij ≤bij
is gua an eed i a solu ion gij, ij is ound o he SISO-equi alen
p oblems
− ij/ˆ
pii −∑n
k=1ˆ
pikdkj/ˆ
pii
1+gii/ˆ
pii ≤bij(1−xij)
and
∑k=1|ˆ
pik/ˆ
pii|bkj
|1+gii/ˆ
pii|≤bijxij.
wi h xij ∈(0,1)being used o di ide he ole ance bij be ween
bo h exp essions. This di ision is usually done manually o
each design equency so ha he bound equi emen a e he
dis ibu ion has been applied is balanced. In his wo k, a me hod
has been c ea ed o au oma e he choice o his dis ibu ion, which
is desc ibed in Subsec ion 5.3.4.
96 Chap e 5. Mul i a iable QFT con ol o he 3D case
The sequen ial me hod is de eloped applying Gauss elimi-
na ion in (5.3.2) wi h gi es ise o he elemen -wise exp ession
ew
ij =1
ˆ
pi
ii +gii "−
n
∑
k=1
ˆ
pikdkj +∑
k<1
ˆ
pk
ik k
kj
ˆ
pk
kk +gkk − ij −∑
k>1
ˆ
pi
ik ew
kj #,
whe e ˆ
p
ij and
ij inco po a e he p e iously designed elemen s
o Gand F, acco ding o he ecu si e o mulas
ˆ
p +1
ij =ˆ
p
ij −ˆ
p
i ˆ
p
j
ˆ
p
+g
, (95)
+1
ij =
ij −ˆ
p
i
j
ˆ
p
+g
, (96)
whe e ≥1, ˆ
p1
ij =ˆ
pij and 1
ij = ij. Wi h his a angemen o
he sequen ial me hod, he SISO-equi alen p oblems a e
− i
ij/ˆ
pi
ii −∑n
k=1ˆ
pi
ikdkj/ˆ
pi
ii +∑k<1ˆ
pk
ik k
kj/(ˆ
pk
kk +gkk)
1+gii/ˆ
pi
ii ≤bij(1−xij)
and
∑k>1|ˆ
pi
ik/ˆ
pi
ii|bkj
|1+gii/ˆ
pi
ii|≤bijxij.
5.3.3 Measu ed dis u bance ejec ion wi h model ma ch-
ing
To add ess he di ec ion lip p oblem, we employ a wo-deg ee-
o - eedom con ol s uc u e (as desc ibed in Subsec ion 5.2.3).
This s uc u e includes a p e il e a he inpu o he known dis-
u bance o imp o e dis u bance ejec ion. Figu e 5.3 illus a es
he con olle a chi ec u e, whe e he design pa ame e s a e he
5.3. The QFT me hod 97
2×2 diagonal ma ix G(s)and he 2 ×1 ec o F(s). The ec o
o ans e unc ions M(s)encapsula es he desi ed sys em e-
sponse o an impulsi e dis u bance, i.e., he speci ica ion. This
is a e inemen o he p oblem ha was no p esen in he o ig-
inal me hodology. To ma ch he dis u bance esponse using a
model, exis ing MIMO QFT echniques a e enhanced.
To ob ain he unc ions de ining he bounds o he cu en
p oblem, a ma hema ical de i a ion has been ca ied ou based
on he wo k o [128]. In he ci ed wo k, he QFT me hodology is
ex ended o he p oblems o model ma ching and dis u bance
ejec ion o n×nmul i a iable plan s. Following he same
p ocedu e, a de i a ion o dis u bance ejec ion wi h an ideal
esponse o ma ch will be shown. Acco ding o Figu e 5.3, he
closed loop esponse be ween he dis u bance and he e o Te i
is gi en by
[1+PG]Te i =[M−PF −D], (97)
and mul iplying by ˆ
P=P−1, we ge
ˆ
P+GTe i =ˆ
PM −F−ˆ
PD. (98)
Conside ing a 2 ×2 case, i his exp ession is p emul iplied by
he ma ix
Mg=1 0
−b
p1
21
b
p1
11+g11 1, (99)
i is possible o pe o m Gaussian elimina ion and ge he uppe
diagonal ma ix cha ac e is ic o he sequen ial me hod, whose
98 Chap e 5. Mul i a iable QFT con ol o he 3D case
elemen -wise exp ession o a 2 ×2 sys em is
eδ
11 =1
b
p1
11 +g11
b
p1
11m11 +b
p1
12m21 − 1
11 −b
p1
11d11 −b
p1
12d21 −b
p1
12 eδ
21
and
eδ
21 =1
b
p2
22 +g22
b
p2
21m11 +b
p2
22m21 − 2
21 −b
p2
21d11 −b
p2
22d21, (100)
whe e
b
p2
21 =b
p1
21 −b
p1
21 b
p1
11
b
p1
11 +g11 ,
b
p2
22 =b
p1
22 −b
p1
21 b
p1
12
b
p1
11 +g11 ,
and
2
21 = 1
21 −b
p1
21 1
11
b
p1
11 +g11 . (101)
Thanks o o e bounding, he dis u bance ejec ion speci ica ion
eδ
ij ≤bij
is gua an eed i a solu ion gij, ij is ound o he SISO-equi alen
p oblems
1
1+g11
ˆ
p1
11 m11 +ˆ
p1
12
ˆ
p1
11
m21 −1
ˆ
p1
11
11 −d11 −ˆ
p1
12
ˆ
p1
11
d21
≤b11 (1−x11), (102)
5.3. The QFT me hod 99
1
1+g11
ˆ
p1
11
ˆ
p1
12
ˆ
p1
11
b21≤b11x11, (103)
and
1
1+g22
ˆ
p2
22 ˆ
p2
21
ˆ
p2
22
m11 +m21 1
ˆ
p2
22
21 −1
ˆ
p2
22
2
11 −ˆ
p2
21
ˆ
p2
22
d11 −d21
≤b21,(104)
whe e x11 exp esses he di ision o ole ances be ween he pe -
o mance equa ion (102) and he coupling equa ion (103) in he
i s loop. The pe o mance equa ions (102) and (104) co e-
spond o 2 DOF SISO p oblems o which a nonconse a i e
solu ion and a bound-building p ocedu e is p o ided in [128].
Once again, he coe icien x11 balances he equi emen s o
decoupling and pe o mance in he i s loop, and i s alue is
adjus ed o make he bounds a ising om each condi ion lay
oge he . Thanks o he sequen ial me hod, he di ision o ole -
ances is unnecessa y in he second loop.
5.3.4 Ob aining an op imal di ision o ole ances
To de e mine he op imal ec o o coe icien s x11, an op imiza-
ion algo i hm is employed. The op imum alue o he coe -
icien s is achie ed when he bounds o pe o mance and cou-
pling a e as close as possible. P e iously o his wo k, op imiza-
ion o his pa ame e was pe o med manually, equi ing he
enginee o isually inspec plo s o he bounds and adjus he
100 Chap e 5. Mul i a iable QFT con ol o he 3D case
coe icien by hand un il he bounds aligned o each design e-
quency. This manual p ocess was ime-consuming and ine i-
cien , especially when i e a ing h ough mul iple speci ica ions
and design modi ica ions. To add ess his, he cu en wo k em-
ploys an op imiza ion algo i hm implemen ed in MATLAB. The
implemen a ion u ilises a cos unc ion ha calcula es he di e -
ence be ween he bounds and employs
minbnd
o minimise his
di e ence. The algo i hm employed is as ollows:
1. Fo ωi=ω1..ωn
(a) =@(x)cos _ unc ion(calcula e_bounds(ωi,...))
(b) x11(i) = minbnd( ,0,1)
The cos unc ion (cos _ unc ion) compu es he di e ence be-
ween he bounds, being ze o when he bounds coincide pe -
ec ly. Figu e 5.7 shows he bounds ob ained by he algo i hm
o he op imal x11 ha esul ed in he minimal alue o he cos
unc ion. Wi hou he algo i hm, he designe has o y di e -
en x11 alues and decide when he bounds a e mos alike in he
Nichols plo .
5.4 Design
Be o e he con olle syn hesis, he linea ized plan is scaled o
a oid nume ical calcula ion p oblems. The ope a ion is pe -
o med ollowing Skoges ad [119] ecommenda ions wi h e o
and inpu scaling ma ices (Deand Du), esul ing in he scaled
5.4. Design 101
-260 -240 -220 -200 -180 -160 -140 -120 -100
-8
-6
-4
-2
0
2
4
FIGURE 5.7: Bounds b11 and b21 o equency
0.5 ad/s a he op imal x11 alue.
plan
P=De−1¯
PDu. (105)
whe e Pis he scaled plan , Deis he e o scaling ma ix, Duis
he inpu scaling ma ix and ¯
Pis he o iginal pla .
5.4.1 Speci ica ions
Dis u bance ejec ion con ol ypically aims o e u n o he equi-
lib ium s a e exis ing be o e he dis u bance. In his case, he
dis u bance implies a sudden change in he s a e o he sys-
em due o he ins an aneous eposi ioning o he measu emen
poin s. The ideal ou pu would be an immedia e e u n o p e-
ious alues, bu i is no easonable no p ac ical o demand an
a bi a ily quick e u n o he equilib ium. Ins ead, a ajec o y
108 Chap e 5. Mul i a iable QFT con ol o he 3D case
-360 -315 -270 -225 -180 -135 -90 -45 0
-80
-60
-40
-20
0
20
40
FIGURE 5.12: Loop shaping o 21.
5.5 Resul s
5.5.1 Speci ica ion compliance
Figu e 5.13 p esen s he magni ude o he Bode diag am ep-
esen ing he e o be ween he unce ain plan dis u bance e-
sponse and he desi ed esponse. The e o emains in all cases
below he speci ied ole ance, indica ed in ed. This e i ies ha
he QFT me hodology, which employs a disc e e numbe o e-
quencies, also complies wi h he speci ied ole ance along he
whole equency spec um.
5.5.2 Time esponse
Figu e 5.15 displays he ime esponse o he sys em. The plo
shows he con olled esponse o he linea sys em plan s used
in he design (in blue) and he esponse o he simula ion using
5.5. Resul s 109
10-2 10-1 100101102
-80
-40
0
40
80
10-2 10-1 100101102
-80
-40
0
40
80
FIGURE 5.13: Magni ude o he Bode diag am
o he e o o plan s wi h espec o he speci-
ica ion.
he nonlinea ini e elemen s model (in g een) along wi h he
limi s o he desi ed esponses es ablished in he speci ica ion
(in ed). The imp o emen wi h espec o he open loop sys em
(Figu e 5.2) is no iceable o all he plan s because he e u n o
he s a iona y s a e is p oduced mo e han 10 seconds ea lie .
The nonlinea esponse simula ed wi h he FEM model is also
complian wi h he speci ica ion de ined in subsec ion 5.4.1.
The con ol ac ions a e checked o ensu e ha hei alues
all wi hin accep able bounds. Figu e 5.16 shows he powe
(∆Q) and speed (∆ ) changes equi ed by he sys em o achie e
he esul s shown in Figu e 5.15. The designed con ol ac ions
o he plan s a e shown in blue, while he esul s o he con ol
ac ions ob ained om he simula ion using he nonlinea FEM
model a e p esen ed in g een.
110 Chap e 5. Mul i a iable QFT con ol o he 3D case
In bo h cases, he nonlinea simula ion alls wi hin he e-
sul s o he di e en plan s used in he design, demons a ing
he achie emen o obus pe o mance.
5.5.3 Nonlinea simula ion
In o de o check he alidi y o he designed con ol, a closed
loop nonlinea simula ion is pe o med. The simula ion loop
ollows he algo i hm depic ed in subsec ion 3.5.1, whe e he
speed and powe a e upda ed in each i e a ion. To do ha , he
il e s and con olle s o he con ol s uc u e M,Fand G(see
Figu e 5.3) a e disc e ized using Ma lab’s unc ion c2dwi h a
ime pe iod o 0.02 seconds. Equa ion (112) shows he Z- ans o m
o he g11 il e , and Figu e 5.14 shows a Bode diag am o he
con inuous and disc e e il e e sions.
Gd11(z) = 0.0001818z2+0.0003636z+0.0001818
z2−1.818z+0.8182 . (112)
The coe icien s o he esul ed Z- ans o med unc ions a e
used o implemen IIR il e s in he simula ion loop and eed
he alues needed o pe o m he nex i e a ion. The inpu s and
ou pu s a e s o ed in each i e a ion and sa ed as a Ma lab ma ix
a he end o he simula ion o be isualised in he plo diag ams
shown (Figu es 5.15 and 5.16).
5.5. Resul s 111
-100
-80
-60
-40
-20
0
Magni ude (dB)
10-1 100101102
-180
-135
-90
Phase (deg)
F equency ( ad/s)
FIGURE 5.14: Bode diag ams o he con inuous
and disc e e e sions o G11.
0 5 10 15 20 25 30
-1500
-1000
-500
0
500
0 5 10 15 20 25 30
0
500
1000
1500
2000
FIGURE 5.15: Time esponse o he linea plan s
and he nonlinea nominal plan .
112 Chap e 5. Mul i a iable QFT con ol o he 3D case
0 5 10 15 20 25 30
-4000
-3000
-2000
-1000
0
0 5 10 15 20 25 30
-1
0
1
2
3
FIGURE 5.16: Con ol ac ions o he linea de-
sign and nonlinea simula ion.
5.5.4 Tempe a u e p o iles
As s a ed in he in oduc ion, he me allog aphic cha ac e is ics
o he esul ing pa s a e la gely de e mined by he hea ing-
cooling p o iles o he mol en zone [131–133]. In his subsec-
ion, he empe a u e p o iles ob ained wi h he de eloped sys-
em a e conside ed.
To e alua e he imp o emen s ob ained wi h he con ol sys-
em, he su ace empe a u e o he pa in he case o a 180 de-
g ee di ec ion change in he absence o any con ol sys em is
compa ed agains he designed solu ion. Figu e 5.17 shows he
a ea be ween he empe a u es o poin s P1and P2 o wo simu-
la ions. The solid blue line ep esen s he empe a u e o e ime
a e he di ec ion lip occu s in he absence o any con ol. The
5.5. Resul s 113
dashed ed line ep esen s he empe a u e in he case o a sim-
ula ion whe e he designed con ol sys em is ac i e. The a ea o
he di e ence o empe a u es o he bo h cases is ep esen ed
in yellow.
I can be seen ha in he case o he sys em unde con ol
he empe a u es emain close o he e e ence alue o 1200
◦C. Tha means ha he hea a ec ed zone shape e u ns as e
o his nominal alue de ined by he iso he ms (see Figu e 3.4).
The s abilisa ion ime o he sys em unde con ol is 2 seconds
whe eas he uncon olled sys em needs 12 seconds o e u n o
equilib ium. This imp o emen is u he ein o ced by a sub-
s an ial educ ion in he squa ed e o o empe a u es calcu-
la ed by
SE =Z20
0(T( )−T e )2d . (113)
The i s poin exhibi s a 5- old educ ion, while he second poin
achie es a 4- old imp o emen . These esul s a e p omising, bu
expe imen al es s a e necessa y o quan i y he ac ual quali y
bene i s.
114 Chap e 5. Mul i a iable QFT con ol o he 3D case
0 2 4 6 8 10 12 14 16 18 20
1000
1500
2000
2500
3000
0 2 4 6 8 10 12 14 16 18 20
1000
1500
2000
2500
3000
FIGURE 5.17: Di e ence in empe a u e p o-
iles wi h and wi hou con ol.
115
Chap e 6
Conclusion
Welding is a echnique used o pe manen ly join pieces o me al.
By applying hea , hese pieces a e mel ed and hen allowed o
solidi y, o ming a s ong bond. In some ins ances, a ille ma-
e ial is in oduced o c ea e a bead o me al ha connec s he
pieces. Reco ds indica e ha he i s expe imen ela ed o wha
we now know as a c welding da es back o 1880. A cen u y
and a hal a e i s in en ion, he signi icance o welding p o-
cesses in mode n manu ac u ing is ha d o o e s a e. Welding
is a p ecise, sa e, and cos -e ec i e me hod o joining ma e i-
als in manu ac u ing indus ies. I ’s es ima ed ha o e i y
pe cen o manu ac u ed p oduc s con ain welded join s, mak-
ing his echnology he p edominan me hod o joining me allic
ma e ials.
Welding p ocesses mus adap o di e se ope a ional condi-
ions such as he ma e ials o be joined, hickness anges, join
116 Chap e 6. Conclusion
ypes, e c., o mee inc easingly s ingen pe o mance and di-
mensional equi emen s. As a esul , p ope weld con ol e-
mains an unsol ed echnological challenge. The pu pose o weld
con ol is o ob ain a weld wi h adequa e mechanical p ope ies
in any scena io. S ong join s wi h low esidual s esses and ee
o dis o ions a e sough . These me allu gical cha ac e is ics a e
la gely de e mined by he hea ing and cooling p o iles o he
mol en zone, so i makes sense o y o con ol hese p o iles.
Despi e i s impo ance as a manu ac u ing p ocess, he use
o con ol loops o imp o e me al joining esul s has no been
he subjec o as much esea ch by he con ol communi y as
o he a eas. The e seems o be a need o a amewo k in which
con ol enginee s can wo k mo e com o ably. This wo k aimed
o de elop models ha ela e con ollable pa ame e s o em-
pe a u e, as well as con ol sys ems o main ain he p ocess in a
desi ed s a e.
To achie e his goal, wo applica ions we e conside ed. Fi s ,
a long-du a ion welding p ocess, ypically ound in la ge s uc-
u es, is analyzed. A simple decen alized con ol sys em ad-
jus s powe and a el speed based on a 1D model o main ain a
desi ed cooling cu e. Second, an addi i e manu ac u ing p o-
cess is in es iga ed. Due o he laye ed na u e o addi i e manu-
ac u ing and hea accumula ion, a mul i a iable con olle de-
signed using he sequen ial QFT echnique is employed o eg-
ula e he su ace empe a u e p o ile by adjus ing powe and
a el speed. Special conside a ion is gi en o he changes in
Chap e 6. Conclusion 117
di ec ion ha he o ch has o make o s a new laye s.
In hea ans e , a c i ical challenge o enginee s, pa icu-
la ly in welding, is o model mo ing hea sou ces. In he ea ly
20 h cen u y, welding enginee s began o in es iga e his opic,
employing bo h empi ical and heo e ical app oaches o unde -
s and how hea beha es du ing welding and Rosen hal pub-
lished he heo y o hea low om a mo ing sou ce o a c weld-
ing, whe e he hea dis ibu ion o he weld unde quasi-s a iona y
condi ions was exp essed in an equa ion whe e powe an a el
speed a e pa ame e s.
In o de o design con olle s, we need no only he s eady-
s a e exp essions bu also equa ions ha indica e how empe -
a u e alues change wi h changes in he powe and speed pa-
ame e s. One o he objec i es o his hesis has been o ind
hese equa ions. T ans e unc ions ha indica e how empe a-
u e changes a any poin o he pa when he e a e changes in
he powe o he hea sou ce ha e been p esen ed o he 1-D,
2-D, and 3-D cases. In he case o changes in empe a u e when
he a el speed changes, he ans e unc ions a e p esen ed
o he 1-D and 2-D cases.
The ans e unc ions we e applied o c ea e a 1-D welding
con ol 2 ×2 model. The model included hea powe and o ch
a eling speed as con ol inpu s, leading o a mul i a iable sys-
em. The ou pu a iables being he empe a u es o wo posi-
ions along he piece. Thei loca ions we e de e mined using
RGA analysis. A simple decen alized PI con ol was p oposed
124 Chap e 7. Resumen
his o ia so p enden emen e la ga, aunque ecien emen e es á
ganando más a ención. Las aíces de la WAAM se emon an
a la década de 1920, cuando su gió la idea de u iliza un a co
eléc ico pa a undi y deposi a ma e ial capa po capa. Si bien
la idea exis e desde hace iempo, la WAAM se es á ol iendo
cada ez más a ac i a hoy en día debido a sus en ajas, que
incluyen la en abilidad pa a piezas g andes, iempos de con-
s ucción más ápidos en compa ación con o as écnicas de ab-
icación adi i a y un uso e icien e de los ma e iales. En gene al,
la WAAM ep esen a un puen e in e esan e en e las écnicas
adicionales, como la soldadu a, y los a ances mode nos en la
ab icación adi i a. Su la ga his o ia y su desa ollo con inuo la
posicionan como una he amien a po encialmen e aliosa pa a
di e sas aplicaciones indus iales.
La ab icación adi i a de me ales se es á con i iendo en una
al e na i a compe i i a a la ab icación po mecanizado adi-
cional. Aunque el po encial pa a aho a ma e ial se ha econo-
cido desde los inicios, la apa ición de de ec os de de o mación,
po osidad y ag ie amien o ha obs aculizado su adopción gen-
e alizada en la indus ia. A o unadamen e, la combinación de
las ecnologías de soldadu a ac uales y los sis emas de con ol
pueden ans o ma los obo s de soldadu a po a co en imp e-
so as 3D pa a piezas me álicas. Con ello se ob ienen las en a-
jas de la ab icación adi i a, sin la necesidad de g andes in e -
siones.
7.1. In oducción 125
La au oma ización de los p ocesos de soldadu a en la in-
dus ia local es dispa y depende en g an medida del sec o
indus ial en el que se cen e el análisis. Así, sec o es como el
na al siguen con ando hoy en día con una ele ada p esencia
de soldado es que ealizan abajos manuales muy a iados, en
si uaciones complejas desde el pun o de is a de la segu idad, la
higiene y la e gonomía labo al. En es a línea adicional, no sólo
se iden i ican sec o es, sino ambién ope aciones, des acando el
ma cado ca ác e manual del p oceso de epa ación de compo-
nen es, siendo en la mayo ía de los casos p ocesos de epa ación
manuales. En el o o ex emo del desa ollo hacia la obo i-
zación y digi alización del p oceso de soldadu a al a co se en-
con a ían indus ias con un cla o en oque ecnológico, como el
sec o de la au omoción o el ae onáu ico.
Los p ocesos de soldadu a deben adap a se a di e sas condi-
ciones ope a i as como los ma e iales a uni , angos de espe-
so es, ipos de unión, e c. pa a cumpli con equisi os de endi-
mien o y dimensionales cada ez más es ic os, po lo que el
con ol adecuado de la soldadu a sigue siendo un e o ecnológico
sin esol e . El p opósi o del con ol de la soldadu a es ob ene
una soldadu a con p opiedades mecánicas adecuadas en cualquie
escena io. Se buscan uniones ue es, con pocas ensiones esid-
uales y lib es de dis o siones. Es as ca ac e ís icas me alog á i-
cas es án de e minadas en g an medida po los pe iles de calen-
amien o-en iamien o de la zona undida, po lo que iene sen-
ido a a de con ola es os pe iles.
126 Chap e 7. Resumen
En e las a iables de soldadu a des acan el mo imien o del
obo y en consecuencia el mo imien o de la an o cha de sol-
dadu a (ubicación de la uen e de ene gía), los pa áme os de
soldadu a (pa áme os eléc icos pa a el caso de soldadu a po
a co) o los de ec os ocasionados en las uniones soldadas. Es-
os pa áme os dinámicos, como la in ensidad de co ien e, el
apo e é mico o la elocidad de a ance de la soldadu a, en e
o os, es án in e elacionados in luyendo en la calidad inal de
la soldadu a y haciendo de és a un p oceso complejo. En conse-
cuencia, con el obje i o de maximiza la calidad de las uniones,
mejo a la es abilidad del p oceso, educi cos es y aumen a la
p oduc i idad, se han impulsado p ocesos de au oma ización
del p oceso de soldadu a y sis emas de moni o ización y con-
ol en lazo ce ado en iempo eal.
A pesa de su impo ancia como p oceso de ab icación, el
uso de lazos de con ol en la soldadu a no ha sido obje o de an-
os abajos de in es igación po pa e de la comunidad de con-
ol como o os ámbi os. Pa ece exis i la necesidad de un ma co
en el que los ingenie os de con ol puedan abaja más cómoda-
men e. Es e abajo iene como obje i o desa olla modelos que
elacionen pa áme os con olables con la empe a u a, así como
sis emas de con ol pa a man ene el p oceso en el es ado de-
seado.
Con ese obje i o se conside an dos casos de uso. En p ime
luga , un p oceso de unión soldada de la ga du ación (caso de
un ba co, puen e, con enedo de g andes dimensiones, e c.) en
7.2. Es ado del a e 127
el que, u ilizando un modelo unidimensional, se ajus a la po en-
cia y la elocidad de a ance pa a man ene una cu a de en i-
amien o deseada, u ilizando un con ol simple descen alizado.
El segundo caso de uso se cen a en un p oceso de ab icación
adi i a. Se ha seleccionado es e caso po que es un p oceso más
la go y se puede bene icia más que un simple co dón de sol-
dadu a de unos pocos cen íme os. En la ab icación adi i a, las
piezas se cons uyen en capas y el calo se acumula, especial-
men e al cambia la di ección de desplazamien o de la an o cha
de soldadu a. En ese caso, u ilizando la écnica secuencial QFT
pa a diseña un con olado mul i a iable, se ajus a la po encia
y la elocidad de a ance pa a egula el pe il de empe a u a
de la supe icie de la pieza.
7.2 Es ado del a e
La mayo ía de los abajos de in es igación publicados sob e
soldadu a se cen an en la búsqueda de pa áme os óp imos
pa a p ocesos conc e os y nue os ma e iales pa a acili a las
uniones o mejo a sus p opiedades mecánicas. Muchos abajos
ambién buscan la educción de po osidad, g ie as o de ec os
median e ajus e de pa áme os, adi i os en el gas de p o ección
o adi i os en los ma e iales de apo ación.
En las úl imas décadas se ha hecho un g an es ue zo pa a
comp ende mejo las écnicas de soldadu a, lo que ha lle ado
128 Chap e 7. Resumen
al desa ollo de modelos ma emá icos pa a el p oceso de sol-
dadu a, que se di iden en dos en oques p incipales.
El p ime o consis e en modelos analí icos que esuel en la
ecuación de calo bajo supues os simpli icados sob e o mas de
en ada de calo , pé didas é micas y con igu aciones geomé i-
cas. A pesa de su simplicidad, es os modelos p opo cionan
una comp ensión más gene al de los p ocesos de soldadu a. La
con ibución seminal en es e campo la da Rosen hal [14], quien
esuel e el caso de es ado es aciona io cuando el me al base
se mue e a elocidad cons an e, las p opiedades é micas pe -
manecen cons an es y la uen e de calo es una uen e pun ual.
Pos e io men e se han in en ado supe a es as limi aciones. La
usión del me al base ha sido in oducida po [56], se han con-
side ado modelos de uen e de calo dis ibuida, incluyendo dis-
ibuciones gaussianas [57,58], gaussianas doble-elipsoidales [59]
y pseudo-gaussianas [11]. Incluso se ha es imado el amaño del
baño de soldadu a y la pene ación [60,61]. El modelo analí ico
se esuel e u ilizando las unciones de G een en [62].
Po o o lado, los mé odos numé icos, como FEM o FVM
[63], pe mi en ealiza simulaciones complejas de soldadu a, p e-
s ando a ención a los de alles especí icos de cada aplicación, es o
es, modelo de uen e de calo , geome ía y ijación de la pieza,
ma e iales y sus ans o maciones de ase, e c. E iden emen e,
es a lexibilidad iene un cos e, ya que cada aspec o de la apli-
cación debe conside a se en el modelo [64,65], lo que a menudo
implica esol e p oblemas de ansmisión in e sa de calo [66],
7.2. Es ado del a e 129
o incluso ensayo y e o [67], pa a ob ene esul ados de simu-
lación p ecisos. Lindg en en su lib o "Compu a ional Welding
Mechanics" [68] con el mismo í ulo que su men o ([64]) desa -
olla aún más el abajo haciendo hincapié en los mé odos FEM
que incluían dinámica de luidos (Figu a 2.3). Además, la dis-
c e ización espacial simpli ica la esolución, pe o la con e gen-
cia a la solución exac a solo es á ga an izada pa a modelos de
o den supe io , y se han p opues o nue os mé odos adap a i os
[69] pa a educi los cos es compu acionales.
En [78] se u iliza el mé odo de mig ación de iso e mas pa a
modela la dinámica del baño de usión. Es e mé odo consis e
en modi ica la ecuación de calo de al mane a que la a i-
able independien e sea el espacio en luga de la empe a u a.
U ilizando la disc e ización y el mé odo de di e encias ini as,
se ob iene un modelo en espacio de es ados del sis ema cuasi-
es aciona io dada la uen e de calo pun ual y la pa e mó il.
U ilizando el modelo desa ollado, en [79] se diseña un con o-
lado PI que ajus a la po encia pa a egula la posición de la
iso e ma de cambio de ase en una posición especí ica. Con es e
en oque, es posible ajus a el amaño de la zona a ec ada po el
calo po encima de la empe a u a especí ica, pe o no la o ma
de la zona.
Más ecien emen e, se han desa ollado écnicas de con ol
pa a p ocesos de ab icación adi i a basados en a co eléc ico.
Algunos abajos se han cen ado en el desa ollo de modelos
de los p ocesos ísicos [80–82]. En e los abajos que u ilizan el
130 Chap e 7. Resumen
con ol, la mayo ía se cen an en la pa e geomé ica de la capa
esul an e [83–86]. En algunas o as uen es [87,88], el con ol
de empe a u a se ealiza median e en iamien o o zado, pe o
no ac úan sob e la po encia de la uen e. También han su gido
écnicas de con ol que emplean in eligencia a i icial [73–77],
que o ecen soluciones adap a i as e independien es del modelo
pa a el con ol de la soldadu a.
Es udios ecien es se han cen ado en mi iga el p oblema
de los cambios de di ección en la cons ucción de WAAM de
pa edes delgadas. En [93], los au o es in es igan la iabilidad
de u iliza mediciones de empe a u a en e capas pa a moni-
o iza y con ola las p opiedades geomé icas y me alú gicas
de pa edes delgadas du an e WAAM. Si bien es a in es igación
p opo ciona in o mación aliosa, no p opone una es a egia de
con ol especí ica.
Va ios es udios, en e ellos [94–96], se han cen ado en de-
e mina los pa áme os WAAM óp imos pa a ab ica piezas
especí icas. En [97], los au o es compa an la e icacia de dos en-
oques de en iamien o pa a la ges ión é mica de es uc u as de
pa edes delgadas. Concluyen que se pueden log a geome ías
simila es u ilizando écnicas de en iamien o ac i o o pasi o,
pe o el en iamien o pasi o da como esul ado iempos de de-
posición más p olongados.
En [98] se p opone un con olado PID mono a iable pa a
ajus a la po encia é mica u ilizando el ancho del baño de usión
7.3. Modelado 131
en luga de u iliza la empe a u a de la supe icie. [99] am-
bién implemen a un con olado PID pa a abo da la geome ía
esul an e, u ilizando la al u a de la capa deposi ada como en-
ada y la elocidad de alimen ación del hilo del sis ema WAAM
como salida de con ol.
La cues ión de la in e sión de la di ección de las capas en
WAAM se ha explo ado en [100]y[101]. Es os es udios p o-
ponen a iaciones de lazo abie o en la elocidad al p incipio y
al inal de las piezas delgadas pa a mi iga la acumulación de
calo . Has a la echa, ningún abajo de in es igación ha abo -
dado el con ol de la empe a u a de la supe icie du an e las
in e siones de la di ección de las capas en WAAM u ilizando
un en oque mul i a iable de lazo ce ado basado en una écnica
de con ol obus a.
7.3 Modelado
La calidad de una soldadu a depende de muchos ac o es. Al-
gunos se pueden con ola di ec amen e en iempo eal, al como
la can idad de ma e ial de apo ación, la po encia calo í ica y su
pun o de aplicación. O os, en cambio, son ac o es que no se
pueden con ola , como las impu ezas en los ma e iales o las
impe ecciones geomé icas. Du an e el p oceso de soldadu a,
cada pun o espacial de los ma e iales implicados expe imen a
132 Chap e 7. Resumen
una a iación de empe a u a a lo la go del iempo. Una sol-
dadu a de calidad equie e que cada pun o alcance una em-
pe a u a su icien emen e al a y una elocidad de en iamien o
su icien emen e len a. Po ello, se conside a que egula la em-
pe a u a ajus ando los pa áme os con olables puede mejo a
la calidad de las soldadu as.
En es a sección se p esen an modelos de e olución de la em-
pe a u a de una pieza soldada, adecuados pa a su uso en el dis-
eño de sis emas de con ol. La pieza soldada se puede mod-
ela en un espacio unidimensional, bidimensional o idimen-
sional, dependiendo de la o ma de la pieza. Pa a los casos uni-
dimensional y bidimensional, se p esen a un modelo pu amen e
analí ico. Pa a el caso idimensional, el en oque seguido es una
linealización de un modelo numé ico basado en el mé odo de
elemen os ini os. En ambos casos el pun o de pa ida es la
ecuación de ans e encia de calo .
7.3.1 Ob ención de unciones de ans e encia
Las unciones de ans e encia se u ilizan ampliamen e en inge-
nie ía de con ol como modelos de sis emas, ya que pe mi en
la ex acción de p opiedades en el dominio de la ecuencia. Se
ob ienen a pa i de ecuaciones lineales dinámicas u ilizando la
ans o mada de Laplace. En el caso de la ecuación di e encial
7.3. Modelado 133
pa cial en (6), la unción de ans e encia se calcula esol iendo
T(s) = 1
ρCpZ∞
0e−s (k∇2T( ) + δQ( )− ( )∇T( ))d .
Las unciones de G een G(x,x′)p opo cionan una o ma
gene al de desc ibi la espues a de una solución de ecuación
di e encial a un é mino uen e a bi a io. Rep esen an la e-
spues a impulsional en el pun o x′de un ope ado di e encial
lineal no homogéneo (L) de inido en un dominio (x∈X⊂R3)
con condiciones iniciales y de con o no especi icas:
LGx,x′=δ(x−x′).
La exp esión (17) es adecuada pa a modela una an o cha
de soldadu a como una uen e de calo pun ual en el o igen,
g( , )=δ( )Q( ), donde δ(·) ep esen a la unción del a de
Di ac, y además se supone que las condiciones iniciales y de
con o no son ce o. En es e caso la ecuación (17) se educe a (18).
Las unciones de ans e encia dinámicas de los p ocesos de
soldadu a se pueden ob ene en gene al como
T( ,s)
Q(s)=α
kG( ,s|0,0),
donde G( ,s|0,0)oG( ,s)es di ec amen e la ans o mada de
Laplace de la unción de G een G( , |0,0), y el ac o α
k=1
ρCp
in oduce las p opiedades ísicas del ma e ial base.
Es e esul ado no depende del plan eamien o del p oblema
