Guidelines o Weigh ing Fac o s Adjus men in
Fini e S a e Model P edic i e Con ol o Powe
Con e e s and D i es
Pa icio Co és, Sami Kou o, B uno La Rocca,
René Va gas and José Rod íguez
Elec onics Enginee ing Depa men
Uni e sidad Técnica Fede ico San a Ma ía
Valpa aiso, Chile
Email: sami [email p o ec ed]g
José I. León, Se gio Vazquez
and Leopoldo G. F anquelo
Elec onics Enginee ing Depa men
Uni e si y o Se ille
Se ille, Spain
Email:[email p o ec ed]
Abs ac —Fini e S a e Model P edic i e Con ol (FS-MPC) has
eme ged as a p omising con ol ool o powe con e e s and
d i es. One o he majo ad an ages is he possibili y o con ol
se e al sys em a iables wi h a single con ol law, by including
hem wi h app op ia e weigh ing ac o s. Howe e , a he p esen
s a e o he a , hese coe icien s a e de e mined empi ically.
The e is no analy ical o nume ical me hod p oposed ye o ob ain
an op imal solu ion. In addi ion, he empi ical me hod is no
always s aigh o wa d, and no p ocedu es ha e been epo ed.
This pape p esen s a i s app oach o a se o guidelines
ha educe he unce ain y o his p ocess. Fi s a classi ica ion
o di e en ypes o cos unc ions and weigh ing ac o s is
p esen ed. Then he di e en s eps o he empi ical p ocess a e
explained. Finally, esul s o se e al powe con e e s and d i es
applica ions a e p esen ed, which show he e ec i eness o he
p oposed guidelines o each app op ia e weigh ing ac o s.
I. INTRODUCTION.
The con inuous e olu ion and g owing capabili ies o mod-
e n mic op ocesso s and signal p ocessing echnologies, has
enabled he implemen a ion o mo e sophis ica ed con ol
me hods de ised o ul ill he indus y’s inc easing demand o
highe pe o mance. P edic i e con ol is one o hese me hods,
and has gained ecen ly mo e a en ion specially o powe
con e e and d i e applica ions [1]. In essence p edic i e
con ol is a g oup o di e en con ol me hods ha sha e
one common cha ac e is ic, which is, he use o ma hema ical
models o he sys em o p edic u u e beha io s and selec
app op ia e con ol ac ions. Se e al p edic i e con ol me hods
ha e been applied o powe con e e and d i e sys ems,
among hem: Dead Bea Con ol [2]–[8], Model P edic i e
Con ol (MPC) [9], [10], Gene alized P edic i e Con ol [11],
and Fini e S a e Model P edic i e Con ol (FS-MPC) [12].
FS-MPC can be desc ibed as a pa icula case o MPC
which akes in o accoun he inhe en disc e e na u e o he
powe con e e swi ching s a es and he digi al implemen a-
ion. Since powe con e e s ha e a ini e numbe o swi ching
s a es, he MPC op imiza ion p oblem can be simpli ied and
educed o he p edic ion o he sys em beha io only o
hose possible swi ching s a es. The ini e numbe o sys em
p edic ions a e used o e alua e a cos unc ion (also known
as quali y o decision unc ion), which usually is composed
by he e o s o he con olled a iables. Hence, he swi ching
s a e associa ed wi h he p edic ion ha minimizes he cos
unc ion is selec ed and gene a ed by he con e e . Wi h
his app oach he numbe o calcula ions is g ea ly educed,
making eal ime implemen a ions easible wi h cu en mic o-
p ocesso echnology. FS-MPC has been success ully applied
o a wide ange o powe con e e s and d i es applica ions
[12]–[28].
One o he majo ad an ages o FS-MPC is ha se e al
con ol a ge s, a iables and cons ain s can be included in
a single cos unc ion and simul aneously be con olled. In
his way adi ional a iables such as cu en , ol age, o que
o lux can be con olled while achie ing addi ional con ol
equi emen s like swi ching equency educ ion, common
mode ol age educ ion and eac i e powe con ol, o name
a ew. This can be accomplished simply by in oducing he
addi ional con ol a ge s in he cos unc ion o be e alua ed
o he di e en swi ching s a es. Howe e , he combina ion o
a iables ha mos likely a e o di e en na u e (di e en uni s
and di e en o de s o magni ude) in a single cos unc ion
is no a s aigh o wa d ask. Each addi ional e m in he cos
unc ion has a co esponding weigh ing ac o , which is used o
une he impo ance o cos o ha e m in ela ion o he o he s
con ol a ge s. These pa ame e s ha e o be p ope ly designed
in o de o achie e he desi ed pe o mance. Un o una ely,
he e a e no analy ical o nume ical me hods o con ol design
heo ies o adjus hese pa ame e s, and cu en ly hey a e
de e mined based on empi ical p ocedu es. Al hough his chal-
lenge has no kep back FS-MPC o be applied success ully o
se e al powe con e e s, i is highly desi able o es ablish
a p ocedu e o de ine some basic guidelines o educe he
unce ain y and imp o e he e ec i eness o he uning s age.
This pape p esen s a i s app oach o add ess his
challenge. Fi s some ep esen a i e examples o FS-MPC
cos unc ions a e classi ied acco ding o he na u e o hei
e ms, in o de o g oup ypes o weigh ing ac o s ha
could be uned simila ly. Then a se o simple guidelines
a e analyzed and es ed o e alua e he e olu ion o he
sys em pe o mance in ela ion o changes in he weigh ing
ac o s. Se e al con e e and d i e con ol applica ions will
be s udied o co e a wide a ie y o cos unc ions and
weigh ing ac o s. In addi ion, esul s o h ee di e en
weigh ing ac o s a e p esen ed o compa e esul s and
alida e he me hodology.
II. FINITE STATE MODEL PREDICTIVE CONTROL
OVERVIEW.
Conside he gene ic and simpli ied block diag am o FS-
MPC illus a ed in Fig. 1, ha con ols a sys em a iable x
h ough a con ol ac ion S, usually he ga ing signals o a
con e e . The measu ed a iable x( k)is ed back and used o
e alua e a disc e e p edic i e model o unc ion o he sys em
p, o ob ain he p edic ed u u e alues o he sys em xp
i( k+1)
o each possible con ol ac ion Si
xp
i( k+1) = p{x( k), Si} ∀i= 1, . . . , n. (1)
No e ha nco esponds o a ini e numbe o con ol ac ions
o swi ching s a es. Then he np edic ions oge he wi h he
e e ence a e e alua ed in a cos unc ion g, leading o n
di e en cos s g
gi= g{x∗, xp
i} ∀i= 1, . . . , n. (2)
Since he a ge is o con ol a iable x, usually he cos
unc ion gis de ined by a measu e o he e o wi h espec
o he e e ence. Some example o gene ic cos unc ions a e
he absolu e e o , quad a ic e o and mean alue o he e o
gi=|x∗−xp
i(Si)|,
gi= [x∗−xp
i(Si)]2,(3)
gi=1
TsZTS
[x∗( )−xp
i( , Si)]d .
No e ha he nex con ol ac ion S( k+1)will be he
swi ching s a e ha minimizes he cos unc ion g
S( k+1) = min
Si
g{x∗, xp
i(Si)} ∀i= 1, . . . , n. (4)
I is clea ha FS-MPC akes ad an age o he disc e e
na u e o powe con e e s by ela ing he swi ching s a e
Fig. 1. FS-MPC gene ic simpli ied con ol diag am.
di ec ly o he con ol e o . In addi ion, since he swi ching
s a e is di ec ly chosen om he cos unc ion minimiza ion,
no linea con olle s and modula o s a e necessa y. This con-
ol p inciple has been success ully applied o se e al powe
con e e and d i e con ol sys ems, including: ol age sou ce
in e e s, mul ile el in e e s, ma ix con e e s, egene a i e
ec i ie s and o que con ol o ac mo o s, o name a ew
[11]–[28].
III. COST FUNCTION CLASSIFICATION.
Al hough he cos unc ion’s main objec i e is o keep ack
o a pa icula a iable and con ol he sys em, i is no limi ed
o only do so. In ac one o he main ad an ages o FS-MPC
is ha he cos unc ion admi s any necessa y e m ha could
ep esen a p edic ion o ano he sys em a iable, sys em
cons ain o sys em equi emen . This lexibili y enables FS-
MPC o achie e easie mo e con ol a ge s ha can ansla e o
inc eased sys em pe o mance, e iciency, powe quali y, sa e y
and o he possible igu es o me i . Since hese e ms mos
likely can be o di e en physical na u e (cu en , ol age,
eac i e powe , swi ching losses, o que, lux, e c.) i can lead
o coupling e ec s be ween a iables, o o o e es ima e he
impo ance o one e m espec he o he s in he cos unc ion,
making hei p esence no wo h, hence no con ollable.
As men ioned be o e his issue has been commonly deal
wi h in MPC by including weigh ing coe icien s o weigh ing
ac o s λ, o each e m o he cos unc ion
g=λx|x∗−xp|+λy|y∗−yp|+...+λz|z∗−zp|.(5)
Depending on he na u e o he di e en e ms in ol ed in
he o mula ion o he cos unc ion, hey can be classi ied
in di e en g oups. This classi ica ion is necessa y in o de
o acili a e he de ini ion o a weigh ing ac o adjus men
p ocedu e ha could be applied o simila ypes o cos
unc ions o alike e ms.
A. Cos unc ions wi hou weigh ing ac o s.
In hese kind o cos unc ions, only one, o he componen s
o one a iable, a e con olled. This is he simples case, and
since only one ype o a iable is con olled, no weigh ing
ac o s a e necessa y. Some ep esen a i e examples o his
ype o cos unc ions a e ob ained o : p edic i e cu en
con ol o a ol age sou ce in e e [14], p edic i e powe
con ol o a back o back ac/dc/ac con e e [16], p edic i e
ol age con ol o an UPS sys em [24] and p edic i e cu en
con ol wi h imposed swi ching equency [18], among o he s.
The co esponding cos unc ions a e summa ized in Table I.
TABLE I
COST FUNCTIONS WITHOUT WEIGHTING FACTORS.
Applica ion Cos unc ion
Cu en con ol o a VSI |i∗
α−ip
α|+|i∗
β−ip
β|
Powe con ol o ac/dc/ac con e e |Qp|+|P∗−Pp|
Vol age con ol o UPS ( ∗
cα − p
cα)2+ ( ∗
cβ − p
cβ)2
Imposed swi ching equency in a VSI |F(i∗
α−ip
α)|+|F(i∗
β−ip
β)|
No e ha all he e ms in a cos unc ions a e composed
o a iables o he same na u e (same uni and o de o
magni ude). Mo eo e , some a e a decomposi ion o a single
a iable in o wo componen s. The e o e, no weigh ing ac o s
and hei co esponding uning a e necessa y.
B. Cos unc ions wi h seconda y e ms.
Some sys ems ha e a p ima y goal o a mo e impo an
con ol objec i e ha mus be achie ed in o de o p o ide a
p ope sys em beha io , and addi ional seconda y cons ain s
o equi emen s ha should also be accomplished o imp o e
sys em pe o mance, e iciency o powe quali y. In his cases
he cos unc ion p esen s a p ima y and seconda y e ms,
whe e he impo ance o he seconda y e m can a y wi hin
a wide ange, depending on he applica ion and i s speci ic
needs. Some examples a e: p edic i e cu en con ol wi h
educ ion o he swi ching equency o imp o e e iciency
[26], p edic i e cu en con ol wi h educ ion o common
mode ol ages o p e en mo o damage [28], and p edic i e
cu en con ol wi h eac i e powe educ ion o imp o e
powe quali y [15], [27]. The co esponding cos unc ions a e
lis ed in Table II.
The impo ance o he second e m, i.e. how much he
swi ching equency, he common mode ol age o he eac i e
powe a e educed, will depend on he speci ic needs o
he applica ion and will impose a adeo wi h he p ima y
con ol objec i e, in his case cu en con ol. No e ha in
each cos unc ion a weigh ing ac o λis included wi h he
co esponding seconda y e m. Hence, sol ing he adeo can
be seen as he weigh ing ac o adjus men in he cos unc ion.
C. Cos unc ions wi h equally impo an e ms.
Unlike he p e ious case, he e a e sys ems in which se e al
a iables need o be con olled simul aneously wi h equal
impo ance in o de o con ol he sys em. He e he cos
unc ion can include se e al e ms wi h equal impo ance,
and i is he job o he weigh ing ac o s o compensa e he
di e ence in na u e o he a iables. Such is he case o que
and lux con ol o an induc ion machine, whe e bo h a iables
need o be con olled accu a ely in o de o ha e p ope sys em
pe o mance [17]. An o he example is cu en con ol o an
neu al poin clamped in e e , in which he dc-link capaci o
ol age balance is a mus in o de o educe ol age dis o ion
and a oid sys em damage (exceed he pe mi ed ol age le el
o he capaci o s, o he wise o e a ed capaci o s should be
used) [26]. Bo h cos unc ions a e included in Table III.
TABLE II
COST FUNCTIONS WITH SECONDARY TERMS.
Applica ion Cos unc ion
Swi ching equency educ ion |i∗
α−ip
α|+|i∗
β−ip
β|+λswnp
sw
Common mode ol age educ ion |i∗
α−ip
α|+|i∗
β−ip
β|+λcmVp
cm
Reac i e powe educ ion |i∗
α−ip
α|+|i∗
β−ip
β|+λQ|Qp|
TABLE III
COST FUNCTIONS WITH EQUALLY IMPORTANT TERMS.
Applica ion Cos unc ion
To que and lux con ol 1
T2
en (T∗
e−Tp
e)2+λψ
ψ2
sn (|ψs|∗− |ψp
s|)2
Capaci o ol age balance 1
isn h|i∗
α−ip
α|+|i∗
β−ip
β|i+λ∆V
Vcn |∆Vp
c|
IV. WEIGHTING FACTOR ADJUSTMENT.
The weigh ing ac o uning p ocedu e will a y depending
on which ype o e ms a e p esen in he cos unc ion as
classi ied in he p e ious sec ion.
A. Fo cos unc ions wi h seconda y e ms.
This is he easies case o weigh ing ac o adjus men ,
since he sys em can be i s con olled using only he p ima y
con ol objec i e o e m. This can be e y simply achie ed be
neglec ing he seconda y e ms o cing he weigh ing ac o o
ze o λ= 0. Hence he i s s ep o he p ocedu e is o con e
he cos unc ion wi h seconda y e ms in o a cos unc ion
wi hou weigh ing ac o s. This will se he s a ing poin o
he measu emen o he beha io o he p ima y a iable.
The second s ep is o es ablish measu emen s o igu es o
me i ha will be used o e alua e he quali y achie ed by
he weigh ing ac o . Fo all he examples gi en in Table II a
s aigh o wa d quan i y should be one ela ed o he p ima y
a iable, which is cu en e o . Se e al e o measu es o
cu en can be de ined, in his wo k he oo mean squa e
(RMS) alue o he e o in s eady s a e has been used. A
leas one addi ional measu e is necessa y o es ablish he
adeo wi h he seconda y e m. Fo he h ee cos unc ions
o Table II he co esponding measu es ha we e selec ed a e:
he de ice a e age swi ching equency sw, he RMS common
mode ol age and he s eady s a e inpu eac i e powe .
Once he measu es a e de ined, e alua e he sys em beha io
wi h simula ions s a ing wi h λ= 0 and inc ease he alue
g adually. Reco d he co esponding measu es o each alue
o λ. S op he inc emen s o λonce he measu ed alue o he
seconda y e m has eached he desi ed alue o he speci ic
applica ion, o keep inc easing λun il he p ima y a iable is
no con olled p ope ly. Then plo he esul s and selec a alue
o λ ha ul ills he sys em equi emen s o bo h a iables.
1) Resul s o swi ching equency educ ion: The esul s o
he p e ious p ocedu e o he i s cos unc ion o Table II
a e gi en in Fig. 2(a). He e he seconda y e m is aimed
o educe he swi ching equency in a cu en con ol o
a NPC con e e applica ion [26]. The seconda y e m np
sw
co esponds o he p edic ed numbe o swi chings in ol ed
when changing om he p esen o he u u e swi ching s a e.
Thus by inc easing he associa ed weigh ing ac o λsw i is
expec ed ha his e m gains mo e impo ance in he cos
unc ion and o ces a educ ion in he swi ching equency,
e ec ha can be clea ly obse ed in Fig. 2(a). Howe e , a
educ ion in he swi ching equency in oduces highe dis-
o ion a ec ing he quali y o he load cu en . This adeo
is e y clea in Fig. 2(a) since he cu es ep esen ing each
0 0.02 0.04 0.06 0.08 0.1
0
0.1
0.2
0.3
0.4
0.5
RMS cu en e o [A]
Weigh ing ac o λsw
0
200
400
600
800
1000
De ice sw [Hz]
(a)
0 0.01 0.02 0.03 0.04 0.06
-20
-10
0
10
20
Load cu en [A]
0.05
0 0.01 0.02 0.03 0.04 0.05 0.06
-400
-200
0
200
400
Time [s]
Load ol age [V]
λ =0.1
sw
λ =0
sw λ =0.05
sw
λ =0.1
sw
λ =0
sw λ =0.05
sw
Time [s]
(b)
Fig. 2. a) Weigh ing ac o in luence o e he cu en e o and he
de ice a e age swi ching equency sw. b) Resul s compa ison o di e en
weigh ing ac o s (load cu en and load ol age).
measu e ha e opposi e e olu ions o he di e en alues
o λsw. A sui able selec ion o λsw would be any alue
0.04 ≤λsw ≤0.06 since he cu en e o is s ill below 10%
o he nominal cu en (15[A] in his example) and a educ ion
om 1000[Hz] o 500[Hz] is achie ed o he a e age de ice
swi ching equency. Finally λsw = 0.05 has been selec ed.
Figu e 2(b) shows compa a i e esul s o he sys em wo king
wi h h ee di e en alues o λsw, one o hem he selec ed
alue. No e how he load cu en p esen s highe dis o ion
o he la ge alue o λsw due o he s ong educ ion o he
numbe o commu a ions. On he o he hand o λsw = 0 he
cu en con ol wo ks a i s bes , howe e a expense o highe
swi ching losses. Since he NPC is aimed o medium ol age
high powe applica ions whe e losses become impo an , he
selec ion o λsw = 0.05 me ges e iciency wi h pe o mance.
2) Resul s o common mode ol age educ ion: The guide-
lines ha e been used o une he weigh ing ac o o he second
equa ion o Table II, which co esponds o p edic i e cu en
con ol o a ma ix con e e [28]. He e he addi ional e m
Vp
cm is he p edic ed common mode ol age o he di e en
swi ching s a es and i will be conside ed an addi ional cos
by uning p ope ly he weigh ing ac o λcm. The measu es
0 0.1 0.2 0.3 0.4 0.5
0
0.01
0.02
0.03
0.04
0.05
Weigh ing ac o λcm
RMS cu en e o [A]
0
25
50
75
100
125
RMS common
mode ol age [V]
(a)
0 0.01 0.02 0.03 0.04
-200
-100
0
100
200
Time [s]
CM ol age [V]
λ =0.05
cm
λ =0
cm λ =0.15
cm λ =0.5
cm
0 0.02 0.04 0.06 0.08 0.1
-20
-10
0
10
20
Time[s]
Load cu en [A]
λ =0.05
cm
λ =0
cm λ =0.15
cm λ =0.5
cm
(b)
Fig. 3. a) Weigh ing ac o in luence o e he cu en e o and he common
mode ol age. b) Resul s compa ison o di e en weigh ing ac o s (load
cu en and common mode ol age).
ha will be used o e alua e he di e en λcm a e he RMS
cu en e o and he RMS common mode ol age. Figu e 3(a)
shows he esul s ob ained ollowing he p oposed p ocedu e.
No e ha , like o he p e ious case, simila e olu ions o bo h
measu es a e ob ained, i.e, o highe alues o λcm smalle
CM ol age a e ob ained, while he cu en con ol becomes
less impo an and looses some pe o mance. The esul shows
also ha CM ol ages is a a iable mo e decoupled o he load
cu en compa ed o he swi ching equency since he cu en
e o emains e y low h oughou he wide ange o λcm.
Hence he selec ion o a app op ia e alue is easie , and alues
o λcm ≥0.05 will pe o m well. This can be obse ed o he
esul s shown in Fig. 3, whe e clea ly a no o ious educ ion
o he CM ol ages is achie ed wi hou a ec ing he cu en
con ol.
3) Resul s o inpu eac i e powe educ ion: The las cos
unc ion o Table II co esponds o a cu en con ol o a
ma ix con e e [27] wi h inpu powe ac o co ec ion. The
addi ional e m in he cos unc ion is di ec ly he p edic ed
inpu eac i e powe Qpwi h i s co esponding weigh ing
ac o λQ. The measu es used o une λQa e he RMS cu en
e o and he inpu eac i e powe .
0 0.05 0.1 0.15 0.2 0.25 0.3
0
0.02
0.04
0.06
0.08
Weigh ing ac o λQ
RMS cu en e o [A]
0
300
600
900
1.200
Reac i e powe [W]
(a)
0 0.02 0.04 0.06 0.08
-20
-10
0
10
20
Time [s]
Load cu en [A]
λ =0.05
Q
λ =0
Qλ =0.3
Q
0 0.02 0.04 0.06 0.08
-4000
-2000
0
2000
4000
Time [s]
Reac i e powe [W]
λ =0.05
Q
λ =0
Qλ =0.3
Q
(b)
Fig. 4. a) Weigh ing ac o in luence o e he cu en e o and he inpu
eac i e powe . b) Resul s compa ison o di e en weigh ing ac o s (load
cu en and inpu eac i e powe ).
The esul s o he uning p ocedu e a e depic ed in Fig. 4(a).
Since his cos unc ion belongs o he same classi ica ion as
he p e ious wo, i is expec ed o p esen simila measu e-
men s e olu ion when inc easing λQ. As wi h he p e ious
case, he inpu eac i e powe seems o be e y decoupled
o he load cu en , hence he cu en e o emains e y low
o a wide ange o λQ. I becomes easy o ob ain a sui able
alue conside ing λQ≥0.05. This can be co obo a ed wi h
he esul s gi en in Fig. 4(b), showing an impo an educ ion
o he inpu eac i e powe o λQ= 0.05.
The p oposed p ocedu e can be p og ammed by au oma ing
and epea ing he simula ion in oducing an inc emen in he
weigh ing ac o a e each simula ion. An o he way is o
educe he numbe o epe i ions, by applying a b anch and
bound algo i hm. Fo his app oach i s selec a couple o
ini ial alues o λ, usually wi h di e en o de s o magni ude
o co e a e y wide ange λ=0, 0.1, 1 and 10, o example.
A quali a i e example o his algo i hm is illus a ed in Fig. 5.
Then simula e o hese weigh ing ac o s and ob ain he
measu es o bo h e ms, M1and M2 o he p ima y and
seconda y e ms espec i ely. Then compa e hese esul s wi h
he desi ed maximum e o s admi ed by he applica ion and
i hem in o an in e al o wo weigh ing ac o s (0.1≤λ≤1
< λ <
S a
M1(λ0), M2(λ0)M1(λ0.1), M2(λ0.1)M1(λ1), M2(λ1)M1(λ10), M2(λ10)
λ = 0 λ = 0.1 λ = 1 λ = 10
M1(λ0.1), M2(λ0.1)M1(λ1), M2(λ1)M1(λ0.5), M2(λ0.5)
λ = 0.5
M1(λ0.25), M2(λ0.25)
λ = 0.25
M1(λ0.1), M2(λ0.1)M1(λ0.5), M2(λ0.5)
0.1 0.25
Fig. 5. B anch and bound algo i hm o educe simula ions o ob ain sui able
weigh ing ac o s.
in he example). Then compu e he measu es o he λin
he hal o he new in e al (λ= 0.5in he example) and
con inue so on un il you achie e a sui able lambda. No e in
Fig. 5 ha each ha d line co esponds o a simula ion and
dashed lines co esponds o alues al eady simula ed. This
me hod educes he numbe o simula ions necessa y o ob ain
a wo king weigh ing ac o .
The quali a i e example o Fig. 5 can be ma ched wi h he
esul s o he common mode educ ion case o Fig. 3(a). No e
ha wi h only 7 simula ions he sea ch o λcm would ha e
na owed o an in e al 0.1≤λcm ≤0.25 whe e any λcm
would wo k p ope ly.
B. Fo cos unc ions wi h equally impo an e ms
Fo cos unc ions like hose lis ed in Table III, he p oce-
du e needs some mino adjus men s since λis no allowed
o be ze o. Ano he di icul y is he di e en na u e o he
a iables. Fo example, when con olling o que and lux in
an adjus able speed d i e applica ion wi h a nominal o que
and lux o 25[Nm] and 1[Wb] espec i ely, he o que e o
can ha e di e en o de s o magni udes making bo h a iable
no equally impo an in he cos unc ion, a ec ing he sys em
pe o mance. Thus he is s ep is o no malize he cos unc-
ion. Once no malized, all he e ms will be equally impo an
and now λ= 1 can be conside ed as s a ing poin . Usually a
sui able λis closely loca ed o 1. No e ha he cos unc ions
in Table III ha e al eady included his no maliza ion (nominal
alues a e deno ed by subindex n).
The second s ep is he same as wi h he p e ious p ocedu e,
i.e., measu emen s o igu es o me i ha e o be de ined ha
will be used o e alua e he quali y achie ed by he weigh ing
ac o .
The las s ep is o pe o m he b anch and bound algo i hm
o Fig. 5 conside ing a couple o s a ing poin s. Na u ally
λ= 1 has o be conside ed, and λ= 0 has o be a oided.
When a small in e al o weigh ing ac o s has been eached,
meaning by small in e al, ha he e a e no big di e ences
in he measu es be ween he uppe and lowe bounds o he
in e al, hen he weigh ing ac o has been ob ained.

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
RMS lux e o [Wb]
Weigh ing ac o log10(λψ)
0
2
4
6
8
10
RMS o que e o [Nm]
(a)
0
10
20
30
To que [Nm]
0.8
0.9
1
1.1
Flux [Wb]
0 0.01 0.02 0.03 0.04 0.05 0.06
λ =10
ψ
λ =0.85
ψ
λ =0.1
ψ
λ =10
ψ
λ =0.85
ψ
λ =0.1
ψ
0 1.5 3 0 1.5 3 0 1.5 3
-5
0
5
Load cu en [A]
0 0.01 0.02 0.03 0.04 0.05 0.06
λ =10
ψ
λ =0.85
ψ
λ =0.1
ψ
Time [s]
Time [s]
Time [s]
(b)
Fig. 6. a) Weigh ing ac o in luence o e he lux and o que e o s. b)
Resul s compa ison o di e en weigh ing ac o s ( o que s ep esponse, lux
magni ude in s eady s a e and load cu en s).
1) Resul s o o que and lux con ol: The p edic i e o que
and lux con ol o an induc ion mo o speed d i e [17], can
be implemen ed using he i s cos unc ion in Table III. No e
ha he e ms appea al eady no malized. The wo e ms in
he cos unc ion a e he o que and lux e o . Hence he
measu es ha will be sui able o selec he p ope λψwill
be he espec i e RMS e o s. A b anch and bound algo i hm
s a ing wi h λψ=0.01,0.1,1,10 and 100 i s ga e he in e al
0.1≤λψ≤1, and hen 0.5≤λψ≤1wi h e y small
di e ences. Finally λψ= 0.85 was chosen. Figu e 6(a) shows
ex ensi e esul s conside ing much mo e alues o λψ(no e
ha he alues a e ep esen ed in log10() scale), o show ha
he b anch and bound me hod eally ound a sui able solu ion.
Resul s o di e en λψ, including λψ= 0.85 a e gi en in
Fig. 6(b) o show he pe o mance achie ed by he FS-MPC.
Weigh ing ac o log
-16 -14 -12 -10 -8 -6 -4 -2 0 2
0
0.05
0.1
0.15
0.2
RMS cu en e o [A]
0
5
10
15
20
RMS Vol age
unbalance [V]
10(λ∆c)
(a)
0 0.05 0.1 0.15 0.2 0.25 0.3
0
200
400
600
Time [s]
Cap. ol ages [V]
λ =0
∆c
λ =0.001
∆c
0 0.01 0.02 0.03 0.04 0.05 0.06
-400
-200
0
200
400
Time [s]
Load ol age [V]
λ =100
∆c
λ =0
∆cλ =0.001
∆cλ =100
∆c
(b)
Fig. 7. a) Weigh ing ac o in luence o e he cu en e o and he dc-link
capaci o s unbalance. b) Resul s compa ison o di e en weigh ing ac o s
(load cu en and dc-link capaci o ol ages dynamic beha io ).
No e ha λψ= 0.85 p esen s he bes combina ion o o que
s ep esponse and s eady s a e, lux con ol and load cu en
wa e o ms.
2) Resul s o ol age balancing: The p edic i e cu en
con ol o a NPC con e e [26], can be implemen ed using he
second cos unc ion in Table III. The addi ional e m ∆Vp
c
co esponds o he p edic ed ol age unbalance o he dc-link
capaci o s o he con e e . I his unbalance is no con olled,
he dc-link ol ages will d i and in oduce conside able ou -
pu ol age dis o ion, no o men ion ha he dc-link capaci o s
could ge damaged by o e ol age, unless hey a e o e a ed.
No e he e ms appea al eady no malized in he cos unc ion,
as indica ed in he i s s ep o he p ocedu e. The measu es
ha will be used o e alua e he weigh ing ac o λ∆Va e
he RMS cu en e o and he peak ampli ude o he ol age
unbalance.
A b anch and bound algo i hm s a ing wi h λ∆V=10−4,
10−2, 1, 102and 104 i s ga e he in e al 10−4≤λ∆V≤
10−2a e his i s e alua ion e y small di e ences we e
ob ained. Finally λ∆V= 10−3was e alua ed leading o he
same measu es. Hence his alue was chosen. Figu e 7(a)
shows ex ensi e esul s conside ing much mo e alues (no e
ha he alues a e ep esen ed in log10() scale), o show ha
he b anch and bound me hod eally ound a sui able solu ion.
Resul s o di e en λψ, including λ∆V= 0.001 a e gi en
in Fig. 7(b) o show he pe o mance achie ed by he FS-MPC.
No e ha o λ∆V= 0, which no mally is no allowed since
i s does no con ol he unbalance p oducing he maximum
d i o he dc-link capaci o s, he load ol age only p esen s
5 di e en ol age le els, while 9 le els should appea in
he load phase-neu al ol age (since he NPC has 3 le els
in he con e e phase-neu al ol age). Only 5 appea since
he NPC is no gene a ing 3 ou pu le els, due o he ol age
d i o i s capaci o s i is only gene a ing 2 le els. On he
o he hand, λ∆V= 100 con ols he ol age unbalance e y
accu a ely, i e en makes ol age unbalance so impo an in he
cos unc ion ha i disables he gene a ion o hose swi ching
s a es ha p oduce unbalance elimina ing ol age le els a he
ou pu and inc eases he swi ching equency as can be seen
in he load ol age o Fig. 7(b). On he con a y, he selec ed
λ∆V= 0.001 p esen s he 9 load ol age le els, con ols he
load cu en and keeps he capaci o ol ages balanced.
V. CONCLUSION
In his pape he design o he weigh ing ac o s used in cos
unc ions o Fini e S a e Model P edic i e Con ol has been
analyzed. A i s app oach based on an empi ical p ocedu e o
ob ain sui able weigh ing ac o s has been p esen ed.
Fo cos unc ions wi h a p ima y con ol objec i e and
seconda y e ms, he s a ing poin is λ=0, hen es inc emen s
o λun il he desi ed beha io is ob ained (b anch and bound
can also be used). Fo cos unc ions wi h equally impo an
e ms, i s no malize he cos unc ion and se λ= 1, wi h his
alue he sys em will be con olled, o ine uning use b anch
and bound o mo e sligh ly λa ound 1. A leas wo di e en
igu es o me i o sys em pa ame e s ha e o be conside ed,
depending on he applica ion, o se le he adeo p esen in
he designing choice o he weigh ing ac o s.
This con ibu ion is a i s design app oach o educe he
unce ain y i he co s unc ion design in FS-MPC o sys ems
wi h mo e han one con ol objec i e. The examples s udied in
his pape show he po en ial and lexibili y o FS-MPC and
how easy i is o include addi ional con ol objec i es in one
single con olle compa ed o classic con ol schemes.
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