Academic Edi o : And eas Sumpe
Recei ed: 25 Ma ch 2025
Re ised: 3 May 2025
Accep ed: 15 May 2025
Published: 16 May 2025
Ci a ion: Ba e o, F.; Be múdez, M.;
A ahal, M.R.; González-P ie o, I.
P edic i e Cu en Con ol o a
Fi e-Phase D i e Using a Lead-Pu sui
S a egy and Vi ual Vol age Vec o s.
Appl. Sci. 2025,15, 5604. h ps://
doi.o g/10.3390/app15105604
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Licensee MDPI, Basel, Swi ze land.
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A icle
P edic i e Cu en Con ol o a Fi e-Phase D i e Using a
Lead-Pu sui S a egy and Vi ual Vol age Vec o s
Fede ico Ba e o 1,* , Ma io Be múdez 2, Manuel R. A ahal 3and Ignacio González-P ie o 4
1Elec onic Enginee ing Depa men , Uni e si y o Se ille, 41008 Se ille, Spain
2Elec ical Enginee ing Depa men , Uni e si y o Se ille, 41008 Se ille, Spain; [email p o ec ed]
3
Depa men o Sys ems Enginee ing and Au oma ion, Uni e si y o Se ille, 41008 Se ille, Spain; [email p o ec ed]
4Elec ical Enginee ing Depa men , Uni e si y o Malaga, 29016 Málaga, Spain; [email p o ec ed]
*Co espondence: ba e [email p o ec ed]
Abs ac : Mode n elec ic machines a e a ac ing he g ea es in e es om he esea ch
communi y due o hei cu en inc easing numbe o applica ions, including elec ic
ehicles and wind powe gene a o s. Thei use equi es he de elopmen o complex
egula o s, whe e p edic i e con olle s appea as in e es ing and iable al e na i es in
ecen esea ch wo ks. Al hough hese con olle s ha e an easy o mula ion and high
lexibili y o inco po a e di e en con ol objec i es in mul idimensional sys ems, hey
ha e limi a ions ha equi e a en ion and limi hei applica ion: a high compu a ional cos
and cu en ha monic con en . This wo k p esen s a no el con olle ha ocuses on hese
limi a ions, whe e he addi ional deg ee o eedom in oduced in he p edic i e con olle
h ough he lead-pu sui guidance law concep is combined wi h he use o i ual ol age
ec o s o educe he ha monic con en in a con olled d i e. The e ec i eness o he
p oposed con olle is explo ed using a i e-phase d i e and se e al igu es o me i , such
as he oo mean squa e e o in cu en acking, he o al ha monic dis o ion in he
s a o cu en s, and he numbe o swi ching commu a ions pe cycle. Di e en p edic i e
con olle s a e compa ed wi h he p oposal in e ms o speed egula ion, s a o cu en
con ol, and s eady-s a e pe o mance, whe e he esul s ob ained a e analyzed o show he
in e es , imp o emen s, and limi a ions o he p oposal.
Keywo ds: elec ical d i es; i e-phase d i e; i ual ol age ec o s; s a o cu en p edic i e
con ol
1. In oduc ion
The con ol o elec ic machines using elec onic con e e s has an impo an ole in
mode n socie ies and is he basis o many indus ial applica ions such as elec ic ehicles,
ele a o s, wind gene a o s, and obo s [
1
,
2
]. They equi e accu a e dynamic egula ions ha
can be achie ed by he adequa e con ol o he s a ic cu en . T adi ional con ol me hods
ha use con inuous con ol a iables mus use speci ic modula ion phases, such as pulse
wid h modula ion (PWM) o i s a ia ions [
3
]. Fo mul iphase d i es, he ex ension o such
echniques poses p oblems due o he exis ence o mul iple planes in s a ic cu en space:
α
-
β
,x
1
-y
1
, e c. Fo example, in dis ibu ed mul iphase winding machines, he only cu en
p oducing o que is he
α
-
β
componen , while o he s only con ibu e o losses [
4
]. In his
case, model p edic i e con ol (MPC) has ecen ly appea ed as an in e es ing and bene icial
con ol me hod compa ed o adi ional con olle s [
5
]. In he 1960s, he concep o MPC
was de eloped, and o a long ime, i s main applica ion ields we e limi ed o sys ems wi h
slow dynamic esponses, such as he pe ochemical indus y, which allows he pe o mance
Appl. Sci. 2025,15, 5604 h ps://doi.o g/10.3390/app15105604
Appl. Sci. 2025,15, 5604 2 o 15
o he equi ed calcula ions and he high compu a ional cos o he con olle [
6
]. This use
was ecen ly ex ended o mode n elec ic de ices, such as mul iphase d i es [
7
], whe e
he easons o his in e es a e he g ea e complexi y and numbe o a iables in he
mul iphase d i e and he in ui i e and lexible o mula ion o he MPC echnique. MPC
can now be conside ed a p omising con ol me hodology in mul iphase d i es, whe e
i p esen s an easy o mula ion and high lexibili y o inco po a ing di e en con ol
objec i es [8].
Howe e , he applica ion o MPC in mul iphase d i es aces some limi a ions ha
equi e a en ion [
9
–
11
]. One limi a ion is he implemen a ion o he con olle . MPC is
in ensi e in compu a ion, and a limi on he sampling equency is ound o any gi en
compu ing ha dwa e [
12
]. Many a emp s ha e been epo ed o dec ease his compu a-
ional load, wi h mos o hem based on a educ ion in a ailable con ol ec o s [
13
], he
de ini ion o sea ch echniques based on he physical conside a ions o he sys em o a oid
he usual exhaus i e sea ch o inding he con ol ac ion [
14
], o he simpli ica ion o he
combina o ial sea ch [
15
]. Fo example, he compu a ional cos o a i e-phase d i e is
educed using a subse o ol age ec o s a ailable in he p edic ion algo i hm in [
16
] o by
elimina ing he need o i e a ion in [
17
]. Bu he dependence o MPC on ‘unce ain ies’ in
he eal sys em ha a ec he p edic i e model (i.e., non-measu able sys em a iables o
he disc epancy be ween ac ual and es ima ed o measu ed alues o elec ical pa ame e s
du ing no mal d i e ope a ion) is pa icula ly c i ical in hese lowe complexi y MPC
schemes [
18
], and he inco po a ion o mo e obus con olle s, such as deadbea s a egies,
is manda o y o achie e sa is ac o y d i e pe o mance [18,19].
A second impo an limi a ion is caused by he absence o an explici modula ion
s age due o he ixed- ime disc e iza ion na u e o MPC, which p oduces no only high-
ha monic componen s bu also in e ha monics and elec ical noise in he d i e, pa icula ly
o low alues o he s a o impedance [
20
]. While ha monics e e o cu en and ol age
wa e o ms wi h equencies ha a e mul iples o he undamen al powe equency and
appea due o he powe con e e s used in a iable- equency d i es, an impo an
po ion o wa e o m dis o ion is due o he in e ha monics (no in ege ha monics) ha
include subha monics below he undamen al equency when p edic i e con olle s a e
u ilized. All ha monic componen s can cause a ious p oblems in he elec ical d i e,
including o e hea ing he equipmen , e a ic beha io , and educed e iciency. Howe e ,
subha monic cu en s and he hea gene a ed by hei losses can damage he elec ical
machine as e han o he causes, such as ol age imbalance [
21
,
22
]. Again, a emp s
ha e been made o sol e his issue, and ecen esea ch wo ks mi iga e he si ua ion by
applying wo o mo e swi ching ec o s du ing he same sampling pe iod. Fo example,
ze o and ac i e ol age ec o combina ions a e applied a each sampling ime in [
23
]. The
ac i e ol age ec o is selec ed in esponse o con en ional MPC op imiza ion p oblems,
and he ime o applica ion is calcula ed online using a linea and educed-o de model
ha depends on he p edic ed, measu ed, and e e ence alues o he con olled a iable.
Simila app oaches can be ound in [
24
], using app op ia e PWM me hods o ensu e ixed
swi ching equencies. An al e na i e s a egy is o use o line compu a ion and de ini ion
o a ailable ol age ec o s. These ol age ec o s a e called i ual ol age ec o s (VVVs),
which a e basically p ede ined ol age models ha ensu e low a e age ol ages in he
x
i
-y
i
plane [
25
]. The ob ained esul s ha e demons a ed an imp o emen in con olle
e iciency by educing he ha monic con en in he x
i
-y
i
planes a he cos o highe swi ching
equencies and lowe a ailable ol age limi s [26].
In his con ex , a p edic i e con ol ac ion is de ined in [
27
] whe e a iable con ol
imes a e u ilized. The p oposal is pa icula ly ele an because i adds an ex a deg ee o
eedom o he con ol ac ion wi h nonuni o m con ol equencies, which allows highe
Appl. Sci. 2025,15, 5604 3 o 15
empo al esolu ions and mi iga es he compu a ional cos o he p edic i e con olle .
The mo i a ion o ou wo k is o ex end he idea in [
27
], in oducing VVV o educe he
ha monic con en o he con olled sys em and simpli y he con ol algo i hm, whe e he x-y
plane is no longe conside ed. The es o he pape is o ganized as ollows. The nex sec ion
de ails he sys em unde s udy, which is based on a i e-phase induc ion mo o (5pIM)
d i e. Then, he p oposed con olle , which combines he lead-pu sui and VVV concep s,
is shown in Sec ion 3. The p oposed analysis and he esul s ob ained a e p esen ed in
Sec ion 4, while he conclusions a e inally indica ed in Sec ion 5.
2. The Case S udy: Fi e-Phase Dis ibu ed Winding Induc ion Mo o
D i e Using a Con en ional Two-Le el VSI
The analyzed sys em is based on a 5pIM, a i e-phase wo-le el ol age sou ce in e e
(5p2L-VSI) cons uc ed om wo SEMIKRON SKS 22F modules, and a DC powe supply
ha ac s as a DC link (KDC 300-50 om Cali o nia Ins umen s Co po a ion, San Diego,
CA, USA). The co e o he con ol uni is an MSK28335 boa d wi h a TMS320F28335
digi al signal p ocesso model om Texas Ins umen s (Dallas, TX, USA). The eedback
o he mechanical speed signal is p o ided by he GHM510296R/2500 encode , while
ou LH25-NP Hall e ec senso s om LEM a e used o measu e he s a o phase-cu en
signals. No e ha he 5pIM has an isola ed neu al poin , a balanced load is assumed
(
sa
+
sb
+
sc
+
sd
+
se
= 0 and i
sa
+i
sb
+i
sc
+i
sd
+i
se
= 0), and a educed numbe o cu en
senso s ( ou ins ead o i e) is used o ob ain he s a o phase cu en s. A DC mo o is also
used o gene a e load o que. The sys em is shown in Figu es 1and 2, whe e some pic u es
o he ha dwa e and he schema ic diag am o he 5pIM d i e a e p o ided.
The 5pIM was designed using equally dis ibu ed windings,
ϑ
= 2
π
/5, and he 5p2L-
VSI gene a es 32 ol age ec o s (30 ac i e and 2 ze o- ol age ec o s), which is cha ac e -
ized by he swi ching ec o [SaSbScSdSe] as ollows:
sa
sb
sc
sd
se
=Vdc
5·
4
−1
−1
−1
−1
−1
4
−1
−1
−1
−1
−1
4
−1
−1
−1
−1
−1
4
−1
−1
−1
−1
−1
4
·
Sa
Sb
Sc
Sd
Se
(1)
whe e he midpoin o he ex e nal powe supply (V
dc
) is conside ed he g ound o he
elec ical sys em.
Cu en
Senso s
DC MOTOR 5-PHASE IM
a
b
DC MOTOR
DRIVE
RS232
Se ial
Po s
5-PHASE IM DRIVE
Swi ching
Signals
Phase
Cu en s
CONTROL
BOARD
Speed Encode (ω
m
)
DC-LINK
V
dc
abde
c
POWER ELECTRONIC
CONVERTERS (VSIs)
Figu e 1. Illus a ion diag am o he sys em used in he case s udy.
Appl. Sci. 2025,15, 5604 4 o 15
axis
Rs
R
Lls
Ll
M
axis
as
axis
a
axis
es
axis
ds
axis
bs
axis
cs
axis
as
bs
cs
ds
es
a
b
c
d
e
Figu e 2. Elec ical scheme o he 5pIM d i e.
The elec omagne ic pe o mance o he machine can be ep esen ed wi h a se o
phase- ol age equilib ium equa ions o he s a o and he o o in he s a iona y e e ence
ame (a,b,c,d,e), ob ained by assuming sinusoidal magne omo i e o ce dis ibu ions and
uni o m ai gaps and neglec ing co e and magne ic losses:
[ s]=[Rs]·[is]+[Lss]d
d [is]+d
d [Ls (θ)] ·[i ]
[ ]=[R ]·[i ]+[L ]d
d [i ]+d
d [L s(θ)] ·[is]
[ s]=h sa sb sc sd seiT
[ ]=h00000iT
[is]=hisa isb isc isd iseiT
[i ]=hi a i b i c i d i eiT
[Rs]=Rs·[I5]
[R ]=R ·[I5]
[Lss]=Lls ·[I5]+M·[Λ(ϑ)]
[L ]=Ll ·[I5]+M·[Λ(ϑ)]
[Ls (θ)]=[L s(θ)]T=M·[∆(θ)]
θ=R
0ω ·d
[Λ(ϑ)] ==
1 cos(ϑ)cos(2ϑ)cos(3ϑ)cos(4ϑ)
cos(4ϑ)1 cos(ϑ)cos(2ϑ)cos(3ϑ)
cos(3ϑ)cos(4ϑ)1 cos(ϑ)cos(2ϑ)
cos(2ϑ)cos(3ϑ)cos(4ϑ)1 cos(ϑ)
cos(ϑ)cos(2ϑ)cos(3ϑ)cos(4ϑ)1
[∆(θ)] ==
cos(∆1)cos(∆2)cos(∆3)cos(∆4)cos(∆5)
cos(∆5)cos(∆1)cos(∆2)cos(∆3)cos(∆4)
cos(∆4)cos(∆5)cos(∆1)cos(∆2)cos(∆3)
cos(∆3)cos(∆4)cos(∆5)cos(∆1)cos(∆2)
cos(∆2)cos(∆3)cos(∆4)cos(∆5)cos(∆1)
(2)
whe e and ideno e he ol age and cu en , espec i ely, and he indices sand ep esen
he s a o and o o a iables. Subsc ip s a,b,c,d, and eiden i y he phases conside ed,
supe sc ip T designa es he anspose ope a o , and
ω
is he elec ical speed o he o o .
[I
5
] is he iden i y ma ix o o de 5; R
s
and R
a e he esis ances o he s a o and he
o o , espec i ely; M is he mu ual induc ance pa ame e ; L
ls
and L
l
a e he leakage
induc ance pa ame e s o he s a o and he o o . Finally,
∆
k deno es he angles de ined as
∆k = θ+ (k −1) ϑ
, wi h k = {1, 2, 3, 4, 5}, whe e
θ
ep esen s he ins an aneous azimu h o
he o o wi h espec o he
α
-axis o he s a iona y e e ence ame. An o line calib a ion
es o he sys em is pe o med a s ands ill o minimize he in luence o he senso s on he
con olled sys em. The u ilized me hod is desc ibed in de ail in [28].
Appl. Sci. 2025,15, 5604 5 o 15
I is in e es ing o men ion ha he 5pIM d i e is gene ally ep esen ed o cla i y and
simplici y using wo o hogonal planes
αβ
and xy, plus he homopola componen z. This
is ca ied ou o g oup he undamen al equency oge he wi h he ha monics o o de
10k
±
1 (k = 0, 1, 2, e c.) in he
αβ
plane and ha monics o o de 10k
±
3 in he xy plane and
p ojec ha monics o o de 5k on he zaxis. No e ha while
αβ
componen s a e ela ed o
he p oduc ion o lux/ o que in he machine, componen s xy a e ela ed o he losses in he
conside ed i e-phase IM, and he homopola componen is neglec ed due o he isola ed
neu al poin . To his end, he Cla ke ans o ma ion ma ix T is used o mul iply he s a o
and o o ol age/ lux/cu en ec o s in he (a,b,c,d,e) ame, leading o an in a ian
ampli ude ans o ma ion o magni udes in he (
α
,
β
,x,y,z) ame. The machine model in
he (α,β,x,y,z) ame is he ollowing [29]:
[T]=2
5
1 cos ϑcos(2ϑ)cos(3ϑ)cos(4ϑ)
0 sin ϑsin(2ϑ)sin(3ϑ)sin(4ϑ)
1 cos(2ϑ)cos(4ϑ)cos(6ϑ)cos(8ϑ)
0
1
2
sin(2ϑ)
1
2
sin(4ϑ)
1
2
sin(6ϑ)
1
2
sin(8ϑ)
1
2
sα=Rsisα+Lsdisα
d +Lmdi α
d
sβ=Rsisβ+Ls
disβ
d +Lm
di β
d
sx =Rsisx +Lls disx
d
sα=Rsisy +Lls
disy
d
α=0=R i α+L di α
d +Lmdisα
d +ω L i β+Lmisβ
β=0=R i β+L
di β
d +Lm
disβ
d −ω (L i α+Lmisα)
(3)
whe e a ailable s a o ol age ec o s (VVs) a e now assigned o he
αβ
and xy planes, as
shown in Figu e 3, and h ee ca ego ies depending on hei size in he
αβ
plane a e shown:
la ge ( ed), medium (g een) and small (blue) ec o s, wi h leng hs o 0.6472V
dc
, 0.4V
dc
, and
0.272V
dc
, espec i ely. The da a and main pa ame e s o he 5pIM used on he es pla o m
a e de ailed in Table 1, complemen ed by a simula ion en i onmen ha includes Gaussian
noise o c ea e a ealis ic scena io.
31
2
8
10
1
3
9
11
4
6
12
14
5
7
13
15 16
18
24
26
17
19
25
27
20
22
28
30
21
23
29
0
α
β
31
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15 16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0
x
y
Figu e 3. A ailable ol age ec o s in he
αβ
and xy subspaces, iden i ied by he decimal numbe
de i ed om he swi ching ec o .
Appl. Sci. 2025,15, 5604 6 o 15
Table 1. Induc ion machine used in ou s udy.
Pa ame e Value/Desc ip ion
Ra ed cu en (A ms) 2.5
Pai s o poles 3
Conduc o Coppe
Diame e (mm) 0.7
Numbe o slo s 30
Numbe o u ns 165
Type o windings Single laye
Winding pi ch 5 slo s ( ull pi ch)
S a o esis ance, Rs12.85 Ω
Ro o esis ance, R 4.80 Ω
S a o leakage induc ance, Lls 79.93 mH
Ro o leakage induc ance, Ll 79.93 mH
Mu ual induc ance, Lm681.7 mH
Ro a ional ine ia, Jm0.02 kg-m2
3. Fini e Con ol Se MPC in 5PIM D i es
Di e en p edic i e s a o cu en con ol s a egies can be applied o he 5pIM d i e,
which di e mainly in he sys em a iables ha a e egula ed and in he way con ol ac ions
a e applied. The ini e con ol se MPC s uc u e (FCS-MPC) selec s he op imal con ol
ac ion ha commands he 5p2L-VSI. An ou e speed/ o que PI-based con olle , which
is usually based on he con en ional ield-o ien ed con ol (FOC) s a egy, needs o be
added o p o ide cu en e e ences o he FCS-MPC algo i hm. These e e ences a e gi en
in a o a ing e e ence ame, called dq, o be e egula ion, whe e he e e ence o he
q-cu en s a o (i
sq
*) is he ou pu o a PI egula o , while he e e ence o he d-cu en (i
sd
*)
is se o i s nominal alue. Once he objec i e cu en s a e de ined, he FCS MPC selec s
one swi ching s a e om he 32 possibili ies o ollow hose e e ences. This selec ion is
made by minimizing a cos unc ion ha compa es he cu en e e ences wi h a p edic ion
made based on a ma hema ical model o he sys em: he p edic ion model. The basis o
he p edic i e con olle is hen he op imal selec ion o he swi ching s a e o he powe
con e e using a p edic i e model o he con olled sys em (a i e-phase induc ion d i e
in ou case).
An al e na i e way o implemen ing an MPC algo i hm is he a iable sampling ime
lead-pu sui con olle o VSTLPC echnique in wha ollows. The basis o his echnique
is he op imal selec ion o bo h he swi ching s a e o he powe con e e and i s ime
o applica ion be ween all possibili ies wi hou he necessi y o a cos unc ion. VSTLPC
begins inding he op imal swi ching s a e om some measu emen s ( o o speed and s a o
cu en s) and applying he lead-pu sui concep : Hi ing a mo ing a ge equi es some
an icipa ion, since i akes some ime o he con ol ac ion o p oduce an e ec on he
sys em, and du ing such ime, he a ge changes i s posi ion. In his way, he con olle
poin s o an ad anced s a o cu en e e ence, and he swi ching ec o ha p oduces he
ajec o y o he s a o cu en s o he e e ence is selec ed.
This wo k combines VSTLPC and he VVV concep s in he cu en con ol o he 5pIM
d i e o educe ha monic dis o ion in machine cu en s and o imp o e he e iciency o
he sys em (see Figu e 4). Then, he ou e speed loop is main ained while he con ol ac ion
is implemen ed using VSTLPC and conside ing di e en se s o VVV.
Appl. Sci. 2025,15, 5604 7 o 15
u
op
abcde
αβxy
i
*
sq
i
*
sd
i
sαβ
(k)
5φ
IM
VSI i
a
i
c
i
b
i
d
i
e
αβ
dq
ω
m
PI
ω
*
m
i
*
sαβ
(k+2)
u
i
P edic i e Model
Vol age
Vec o se
J compu a ion
u
op
abcde
αβxy
i
*
sq
i
*
sd
i
sαβ
(
0
)
5φ
IM
VSI i
a
i
c
i
b
i
d
i
e
αβ
dq
T
a
ω
m
PI
ω
*
m
i
*
sαβ
(
0
+
L
)
u
i
u
op
selec ion
T
a
compu a ion
u
op
i (k+2)
p
minimize J
FCS-MPC cu en con ol algo i hm:
Read measu ed cu en s i
sαβ
(k) and e e ences i
*
sαβ
(k+2).
Ini ia e J
0
=∞.
o i=1 o 32
Take u
i
Compu e a p edic ion o he cu en s i
p
sαβ
(k+2).
Compu e he cos unc ion J.
i J<J
0
hen
J
0
=J, u
op
= u
i
end i
end o
VVVs
VSTLPC cu en con ol algo i hm:
Read measu ed cu en s i
sαβ
(
0
)
Compu e ad anced e e ences i
*
sαβ
(
0
+
L
).
Ini ia e cos(δ)
0
=0
o i=1 o 10
Take u
i
Compu e cos(δ)
i cos(δ)>cos(δ)
0
hen
cos(δ)
0
=cos(δ), u
op
= u
i
end i
end o
Compu e he nex sampling pe iod T
a
.
FCS-MPC VSTLPC
Figu e 4. Con ol schemes o a 5pIM d i e using FCS-MPC (le side) and VSTLPC wi h VVV
( igh side) echniques.
3.1. Con en ional VSTLPC Algo i hm
The VSTLPC algo i hm can be desc ibed as an inno a i e p edic i e con ol echnique
wi h he basic p inciple o gi ing eedom o he sampling ime alue o he p edic i e
con olle , i.e., applying a iable sampling imes. Then, he con en ional VSTLPC con olle
de ia es om FCS-MPC echniques, including a kind o modula ion s age in he con ol
s age [
30
]. The sampling ime is selec ed om a limi ed numbe o possibili ies oge he
wi h he swi ching ec o h ough a ime-consuming op imiza ion algo i hm. He e, he
basic idea p esen ed in [
30
] is ex ended o he mul iphase case and complemen ed by
he lead-pu sui concep [
31
] o selec he mos adequa e con ol ac ion and i s ime o
applica ion wi h a ine esolu ion in he commu ing imes.
The inpu s o he con ol algo i hm a e he measu ed s a o cu en s and speeds and
he cu en e e ences imposed by he ou e speed con ol loop. To an icipa e he e olu ion
o he cu en s, he e e ences a e ad anced o a ime
L
,i
s
*(
0
+
L
). In he nex s ep, 1 o
he 32 possible VVs o he 5p2L-VSI, u
op
= [
sα
,
sβ
,
sx
,
sy
], is selec ed by looking o he
closes ajec o y be ween he measu ed s a o cu en s and he ad anced e e ences. Fi s ,
he op imal ajec o y is calcula ed as x
s
*(
0
+
L
)
−
x
s
(
0
), whe e x
s
= [i
sα
i
sβ
i
sx
i
sy
]. Then,
he ollowing maximiza ion p oblem is sol ed:
uop =max
ui
{cos(δ)}=a gmax
ui
(x∗
s( 0+ L)−xs( 0)) · (x( 0),ui)
∥x∗
s( 0+ L)−xs( 0)∥ · ∥ (x( 0),ui)∥(4)
whe e x= [i
sα
i
sβ
i
sx
i
sy
i
α
i
β
], (x,u) is a unc ion ha cha ac e izes he e olu ion o he
s a o cu en s: (x,u)=dx
s
/d .
δ
is he angle be ween (x(
0
), u
i
) and he dis ance (x*(
0
+
L
)
−
x
s
(
0
)). This unc ion is de ined om Equa ion (3) and has ou componen s, o
which hei alues depend on he measu ed s a o and o o cu en s in he
αβ
plane and
he measu ed o o speed. Finally, he ime du ing which he chosen VV mus be applied,
T
a
, is calcula ed, as indica ed in Equa ion (5). The o mula is de i ed om an op imiza ion
p oblem whe e he de ia ion be ween he s a o e e ences and a p edic ion based on he
sys em model is minimized.
Appl. Sci. 2025,15, 5604 8 o 15
Ta=(x∗
s( 0+ L)−xs( 0))T x( 0),uop
∥ x( 0),uop 2∥(5)
3.2. P oposed VSTLPC Conside ing VVV as Con ol Ac ions
The inco po a ion o VVV as con ol ac ions in he FCS-MPC has pe mi ed he im-
p o emen o cu en quali y in mul iphase d i es, hanks o he simul aneous egula ion
o he a ailable subspaces [
29
]. These ec o s a e c ea ed h ough a combina ion o wo o
mo e ol age ec o s in such a way ha ze o o educed a e age ol age ampli udes a e
p oduced in he xy plane. Consequen ly, he egula ion o seconda y cu en s is di ec ly
ca ied ou wi h he VVV applica ion, wi hou including a speci ic con ol loop in he
algo i hm. This cha ac e is ic esul s in he simpli ica ion o he p edic ion model and
he p edic i e con ol algo i hm. In his wo k, di e se se s o 10 i ual VVVs a e used in
he op imiza ion p oblem o he VSTLPC algo i hm ins ead o he 32 o iginal VVs o he
5p2L-VSI.
Fo he case unde s udy, h ee se s o VVVs will be independen ly conside ed as
con ol ac ions, wi h he objec i e o compa ing he p os and cons o each one (Table 2
summa izes he se s o VVVs ha will be conside ed in his wo k):
Table 2. Se o VVVs conside ed.
VVV G oup Name Vol age Vec o s Used Dwell Ra ios
ML-VVV One medium + one la ge λ1= 0.382
λ2= 0.618
LL-VVV Two adjacen la ge λ1= 0.5
λ2= 0.5
LLL-VVV Th ee adjacen la ge
λ1= 0.382
λ2= 0.236
λ3= 0.382
Medium–la ge VV (ML-VVV): A la ge ec o is combined wi h a medium ec o wi h
dwell a ios equal o
λ1
= 0.382 and
λ2
= 0.618, espec i ely. As seen in Figu e 3, hey a e
aligned in he
αβ
plane, bu hey a e in he opposi e di ec ion in he xy plane ( ec o s 12
and 30, o example), and hei combina ion leads o a VVV wi h ze o xy ol age on a e age.
This is he common s a egy in mos esea ch, whe e a o al o 10 VVVs can be cons uc ed
conside ing all possible combina ions. As a disad an age, he ins an aneous alue o he
xy ol age componen s is no ze o, hus p oducing he ci cula ion o xy cu en s in he
machine. Fu he mo e, he use o he DC link is educed compa ed o he o iginal ol age
ec o s, since he size o he VVV is 0.5528V
dc
. The la es has an impo an in luence on he
ope a ing ange o he d i e, mainly in high-speed/ o que ope a ion.
Duo o la ge VVs (LL-VVV): Two adjacen la ge ol ages a e combined wi h dwell
imes a ios equal o
λ1
=
λ2
= 0.5. In his case, he combina ion will no p o ide ze o xy
ol age ampli udes (see ec o s 12 and 28, o example), bu i s magni ude is limi ed by
applying he indica ed dwell a ios. The e o e, seconda y cu en s will no e ec i ely be
egula ed o ze o, bu hei ins an aneous alues will be lowe han hose wi h con en ional
VVVs. Addi ionally, he esul ing VVV has a size o 0.6155V
dc
in he p ima y plane,
which ex ends he use o he DC link. As in he p e ious case, 10 di e en VVVs can
be cons uc ed.
T io o la ge VVs (LLL-VVV): Th ee adjacen la ge ec o s a e combined wi h
λ1= 0.382
,
λ2
= 0.236, and
λ3
= 0.382 as dwell a ios. In his case, ze o-a e age xy ol ages
a e ob ained wi h low ins an aneous xy injec ion, hus esul ing in he expec a ion o im-
p o ed cu en quali y in he machine compa ed o he applica ion o he medium–la ge
Appl. Sci. 2025,15, 5604 9 o 15
combina ion. Howe e , he magni ude o he new VVV is 0.5528V
dc
, and he use o he DC
link is poo e compa ed o he se o VVVs o med by only wo adjacen ol age ec o s.
Following he same algo i hm desc ibed abo e, he VSTLPC selec s he op imal VVV
o be applied and i s applica ion ime based on he measu ed cu en s and he es ablished
e e ences. Only a se o VVVs and a ze o ec o a e conside ed as possible con ol ac-
ions. In his case, he op imiza ion p oblem used o choose he op imal con ol ac ion,
uop = [ sα, sβ], is educed o he ollowing:
uop =max
ui
{cos(δ)}=a gmax
uix∗
sαβ( 0+ L)−xsαβ( 0)· αβxαβ( 0),ui
∥x∗
sαβ( 0+ L)−xsαβ( 0)∥ · ∥ αβxαβ( 0),ui∥(6)
A his poin , he xy plane is no conside ed in he machine’s model due o he
inco po a ion o VVV, so x
αβ
= [i
sα
,i
sβ
,i
α
,i
β
] and x
sαβ
= [i
sα
,i
sβ
];
αβ
(x
αβ
,u) is a ec o o
wo componen s o which hei alues depend on he measu ed s a o cu en s in he
αβ
subspace, he
αβ
o o cu en s, and he measu ed o o speed. A e wa ds, he ime o
applica ion o he selec ed VVV is ob ained using Equa ion (7). No e ha he dwell a ios
o he co esponding VVVs mus be used o de ine he 5p2L-VSI swi ching sequence, and
he desc ibed algo i hm is alid o any VVV g oup.
Ta=x∗
sαβ( 0+ L)−xsαβ( 0)T αβxαβ( 0),uop
∥ αβxαβ( 0),uop ∥2(7)
4. Resul s and Discussion
The e ec i eness and e iciency o he p oposed con olle a e explo ed, and he se s
o VVVs a e de ailed in Table 2. No e ha he selec ion o he pa ame e s used in he expe -
imen s is based on heu is ic adjus men and empi ical obse a ion. In all es s, maximum
and minimum alues a e imposed o he sampling ime T
a
o conside limi a ions in he
e iciency o he con ol algo i hm and in he 5p2L-VSI swi ching equency. In ou case,
he maximum alue is se a 150
µ
s, while he minimum alue is o ced o be 32.5
µ
s using
he con en ional VSTLPC me hod and he p oposed con olle , and he ad anced ime
L
is
se o T
amin
. A g oup o es s was conduc ed o compa e he pe o mance in he s eady s a e
o he o iginal VSTLPC and he p oposed one once he machine was close o he achie able
limi s. Figu es 5–8summa ize he esul s ob ained.
In he i s expe imen (see Figu e 5), a e e ence speed o 300 pm (
ωm
) is imposed, and
a load o que (T
L
) equal o 50% o he nominal is applied. The egula ion o speed and he
con ol o he s a o cu en in he
αβ
plane a e achie ed e ec i ely in all cases. Howe e ,
in e es ing commen s ega ding he s a o cu en ’s pe o mance in he xy subspace mus
be made. Conside ing VSTLPC wi h i ual ML and LLL ec o s, a educ ion in he xy
cu en componen s is obse ed ela i e o he o iginal VSTLPC echnique. This leads o
a educ ion in he ha monic con en o he s a o cu en s and, consequen ly, o a highe
sys em e iciency, hanks o he use o i ual ol age ec o s hemsel es, which p oduce an
e ec i e open-loop egula ion o he xy plane. Howe e , a di e en si ua ion is obse ed
wi h he LL i ual ec o s, whe e a non-ze o-a e age xy ol age is imposed and highe
xy s a o cu en componen s appea , as expec ed. A e e ence speed o 500 pm (
ωm
) is
imposed, and a load o que (T
L
) equal o 85% o he nominal is applied in a second ba ch o
es s (see Figu e 6). Again, he egula ion o speed and con ol o he s a o cu en in he
αβ
plane a e achie ed e ec i ely in all cases, and he main conclusions achie ed in he i s
expe imen a e con i med.
