Academic Edi o : An onio J. Ma ques
Ca doso
Recei ed: 16 Decembe 2024
Re ised: 24 Janua y 2025
Accep ed: 9 Feb ua y 2025
Published: 15 Feb ua y 2025
Ci a ion: Sa ué, M.G.; A ahal, M.R.;
O ega, M.G. Pa e o Analysis o
Elec o-Mechanical Va iables in
P edic i e Con ol o D i es. Machines
2025,13, 150. h ps://doi.o g/
10.3390/machines13020150
Copy igh : © 2025 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license
(h ps://c ea i ecommons.o g/
licenses/by/4.0/).
A icle
Pa e o Analysis o Elec o-Mechanical Va iables in P edic i e
Con ol o D i es
Manuel G. Sa ué †, Manuel R. A ahal *,† and Manuel G. O ega
Depa amen o de Ingenie ía de Sis emas y Au omá ica, Uni e sidad de Se illa, E-41092 Se ille, Spain
*Co espondence: [email p o ec ed]
†These au ho s con ibu ed equally o his wo k.
Abs ac : Va iable speed d i es a e o en con olled by a double-loop scheme in which
a p opo ional in eg al con olle akes on he speed loop. The uning o his loop is a
complex job. In mos cases jus mechanical a iables a e conside ed o uning. This pape
p esen s a new Pa e o analysis inco po a ing mechanical and elec ical a iables. A s a e o
he a ini e s a e model p edic i e con olle is used o s a o cu en con ol. The analysis
is pe o med using expe imen al da a om a i e-phase induc ion mo o and conside s
conside ing commonly ound pe o mance indica o s de i ed om expe imen al da a.
The esul s show undocumen ed connec ions be ween hose pe o mance indica o s. The
analysis no only helps in PI uning bu , mo e impo an ly, p omp s o a e ision o he
me hods usually u ilized o epo pe o mance enhancemen s o new me hods.
Keywo ds: mul iphase sys em; PI uning; p edic i e con ol; a iable speed d i e
1. In oduc ion
Va iable speed d i es o en use a double con ol loop whe e he ou e one is de o ed o
mechanical speed and he inne one o s a o cu en con ol [
1
]. The cu en loop is much
as e equi ing ac ion on he Vol age Sou ce In e e (VSI) abou e e y 50 mic oseconds [
2
].
This egula es he mo o cu en s which, in u n, p oduce a o que o manage he mechanical
speed [
3
]. Recen ly, Fini e S a e Model P edic i e Con ol (FSMPC) has been p oposed
o cu en con ol, which does no equi e modula ion such as PWM, ins ead he VSI
s a e is di ec ly se by he inne con olle [
4
]. This has been epo ed as p o iding a
high bandwid h con ol o s a o cu en s, bene i ing he d i e ope a ion [
5
,
6
]. In FSMPC,
he con ol signal is de i ed om he op imiza ion o a cos unc ion (CF). The p oblem o
uning, howe e , was s ill an issue. In pa icula , uning o he Weigh ing Fac o s (WF) o
he FSMPC needs some wo k.
In FSMPC, he in e ac ion o he in e io loop wi h he speed loop is conduc ed
ollowing he Indi ec Field O ien ed Con ol (IFOC) scheme. The me hod equi es an
app oxima e knowledge o he ime cons an o he o o ci cui o ensu e s abili y [
7
,
8
].
Following hese wo ks, some a ia ions ha e appea ed o senso less ope a ion [
9
] and
o he a ia ions [
10
]. Howe e , in mos cases, he unde lying models do no include
non-ideali ies such as nonlinea beha io o he mo o , VSI+modula ion beha io and
mechanical speed sensing ia inc emen al encode s [11,12].
In he absence o be e models, o ins ance, he inclusion o non-ideali ies, esea che s
and p ac i ione s ha e u ned o he expe imen al app oach o con olle uning. Me a-
heu is ic op imisa ion algo i hms [
13
], uzzy [
14
] and neu al app oaches [
15
] a e some o
he p oposed echniques epo ed in he li e a u e. These mos ly ely on da a bu do no
p o ide insigh ega ding ade-o s.
Machines 2025,13, 150 h ps://doi.o g/10.3390/machines13020150
Machines 2025,13, 150 2 o 12
One pa icula ype o machine ha has ecei ed pa icula a en ion is mul i-phase
mo o s. Howe e , i has been epo ed ha hei in insic ad an ages a e somehow hin-
de ed by inc eased complexi y, bu hei combina ion wi h FSMPC has made hem mo e
a ac i e. P io o his accumula i e app oach, he in e ac ion among phases made uning
mo e complex, equi ing ou p opo ional in eg al (PI) con olle s and a modula ion such
as PWM [
16
]. The decomposi ion o s a o cu en s in o a o que ela ed space (
α−β
) and
se e al ha monic spaces (
xj−yj
,
j≥
1) is a challenge o con ol design [
17
]. The numbe
o phases makes each case di e en . Howe e , o FSMPC he same simple s uc u e can be
used o any numbe o phases [
18
], and, in gene al, i s lexibili y goes beyond ha . Fo in-
s ance, some new modula ion schemes a e now possible [
19
,
20
]. In addi ion, mul i-phase
sys ems o e be e aul - ole an possibili ies [
21
]. Howe e , i s lexibili y comes a he
p ice o highe dependence on he model [
22
] and he need o uning o he weigh ing
ac o s (WF).
These new de elopmen s can po en ially a ec he uning o he ou e loop. Howe e ,
some aspec s ha e p o en di icul o model and la e ea . Fo ins ance, he delay in o-
duced by VSI modula ion [23], esul s in sa u a ion o some a iables, delay and ipple in
he speed sensing om inc emen al encode s, e c.
In addi ion o he abo e in o ma ion, some pe o mance indica o s lay somehow
be ween he mechanical and he elec ical ealms. This is he case o he o que ipple [
13
],
which plays an impo an ole in applica ions o pe o mance and main enance issues [
24
].
Howe e , i is dis ega ded by mos s udies dealing wi h uning o he ou e loop. This
is mo i a ed by he lack o equa ions o connec his igu e o me i wi h he ou e loop
con ol pa ame e s ha would allow o he uning op ions [25].
In his con ex , he ein, we p opose a new Pa e o analysis inco po a ing mechanical and
elec ical a iables. The analysis seeks links o ade-o s be ween pe o mance indica o s
as p e iously des ibed in [
26
–
28
]. Wi hou his, i can be a gued ha , he ela i e me i s
o any uning a e obscu ed. A educ ion in he se o pe o mance indica o s is p oposed
o ou e loop assessmen . This se is compu ed o di e en unings so as o ob ain an
app oxima ion o he Pa e o on by means o emo ing he domina ed solu ions. This
es ablishes a base-line agains which any p oposed enhancemen should be con as ed.
The s uc u e o he es o he pape is as ollows. The nex sec ion in oduces he
con ol s uc u e o mul iphase d i es conside ed in he analysis. The expe imen al se up
is hen p esen ed in Sec ion 3. In Sec ion 4, he discussion o he esul s is p esen ed.
Sec ion 5ends his pape .
2. Ma e ials and Me hods
Pa e o analysis uses expe imen al da a ob ained om p edic i e con ol o a a iable-
speed d i e. The ma e ials used a e p esen ed below. The main me hods equi e explana-
ion a e ela ed o he con ol s uc u e and da a-ga he ing p ocesses. These a e p esen ed
in Sec ions 2.2–2.4.
2.1. Expe imen al Se up
The me hodology o Pa e o analysis consis s o ga he ing da a om an expe imen al
se up. The se up is a es -bed o a iable-speed d i e con ol. I s main elemen s a e
lis ed below.
• A i e-phase Induc ion Machine ha is ed by a ol age sou ce in e e .
•
A i e-phase VSI ha has been cus om-made using wo SKS 22F modules (SEMIKRON
DANFOSS, Nü nbe g, Baye n, Ge many).
• Hall e ec senso s LH25-NP o s a o cu en s (LEM, Gene a, Swi ze land).
Machines 2025,13, 150 3 o 12
•
An op ical encode coupled o he o a ing sha o he machine om which eloci y
es ima es a e ob ained. These es ima es a e used in he IFOC scheme o con ol he
mechanical speed o he IM.
•
A TMS320F28335 (Texas Ins umen s, Dallas, TX, USA) digi al signal p ocesso ha
con ains he con ol p og am.
•
A di ec cu en (DC) mo o ha is coaxial wi h he induc ion mo o and capable o
p oducing an opposing o que. This allows he in oduc ion o a load o he sys em.
These elemen s a e connec ed ia he me hod shown in Figu e 1, whe e a diag am
wi h pho og aphs is shown. Also, Table 1p o ides he pa ame e s o he induc ion mo o .
These pa ame e s a e needed o he p edic i e model.
DIGITAL
SIGNAL
PROCESSOR
POWER ELECTRONIC
CONVERTERS
DC MOTOR 5-PHASE IM
i
Swi ching Signals
Posi ion Encode
DC MOTOR
DRIVE
RS232
Se ial Po s
Figu e 1. Diag am and pho og aphs o he labo a o y se up used in he expe imen s.
Table 1. Values o he i e-phase mo o pa ame e s.
Pa ame e Value Uni
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
Numbe o pai s o poles, P3 -
Di ec cu en ol age VDC 300 V
Ra ed cu en In2.5 A
Ra ed powe Pn1000 W
In addi ion o hese elemen s, he me hodology o da a ga he ing is p esen ed be-
low. In pa icula , he ollowing p ocedu es a e explained: he me hod o d i e con ol,
he me hod o s a o cu en con ol, and he igu es o me i used in Pa e o analysis.
2.2. D i e Con ol
The con ol scheme using a PI o speed con ol and FSMPC o s a o cu en con-
ol is p esen ed in Figu e 2. The ou e loop (speed) uses a PÌ as he con olle o he
mechanical speed (
ω
). The inne loop uses a p edic i e block o he con ol s a o cu en
(
is
). The s uc u e is de i ed om he IFOC scheme, p o iding independen con ol o he
o o lux and o que. The e e ence alue o he o o lux is used o de i e cu en
i∗
d
ha
magne izes he mo o . The alue o quad a u e cu en
i∗
q
di ec ly con ols he o que. I s
e e ence alue is compu ed by he speed PI as ollows:
Machines 2025,13, 150 4 o 12
i∗
q=kp·(ω∗( )−ωe( )) + kiZ∞
0(ω∗(τ)−ωe(τ))dτ(1)
whe e ω∗is he se poin o speed and ωe he measu ed speed.
PI
ω**
isq
αβ
dq
*
isd
*
isα
*
isβ
*
isy
*
isx
P edic i e
Model
αβxy
abcde
ω
is , s
is
5-ph
IM TL
i(k)
i (k+2)
min J
INVERTER
i*
s(k+2) U(k)
σ
ω
Figu e 2. Diag am o he con ol scheme o a mul i-phase mo o .
The alues in
d−q
axes a e hen ans o med in o
α−β
axes by means o he Pa k
ma ix gene a ing
I∗
α−β=Di∗
d,i∗
q⊺(2)
wi h
D= cos θasin θa
−sin θacos θa!(3)
The o o lux angle θais ound as ollows
θa=Zωed ,ωe=ωsl +Pω, (4)
whe e
ωsl =iq
id
1
ˆ
τ (5)
whe e ˆ
τ is an app oxima ion o he ime cons an o he o o ci cui
τ =L /R . (6)
The inne loop uses a e e ence cu en i∗(k)whose ampli ude is ound as
I∗=qi∗2
d+i∗2
q(7)
Finally, he α−β e e ence ajec o ies a e ound as
i∗
α( ) = I∗sin ωe ,i∗
β( ) = I∗cos ωe (8)
2.3. Cu en Loop
T acking he s a o cu en s is pe o med by FSMPC ollowing he usual MPC p ac ice.
Fi s , he mo o model is used o p edic u u e alues o he s a o cu en s. Each VSI s a e
Machines 2025,13, 150 5 o 12
p oduces a alue o s a o ol ages
s( )
. These a e linked o elec ical a iables and a e
desc ibed as ollows:
αβs( ) = Rsiαβs( ) + d
d Ψαβs( )
0=R iαβ ( ) + d
d Ψαβ ( )−jω ( )Ψαβ ( )
Ψαβs( ) = Lsiαβs( ) + Lmiαβ ( ) = Lsiαβs( ) + Lmiαβ ( )
Ψαβ ( ) = Lmiαβs( ) + L iαβ ( ) = Lmiαβs( ) + L iαβ ( )
xys( ) = Rsixys( ) + d
d Ψxys( )
Ψxys( ) = Llsixys( )(9)
whe e he a iables used a e luxes
Ψs( )
,
Ψ ( )
, cu en s
is( )
,
i ( )
, and elec ical speed
ω ( )
. The model also con ains pa ame e s iden i ied om he mo o , such as induc ances
Ls,L ,Lls,Lm, and esis ances Rs,R . These pa ame e s a e p o ided in Table 1.
The s a o ol ages depend on
VDC
(DC link ol age) and he s a e
Kj
o VSI swi ches.
Conside ing he s a e as a ec o
u= (K1
,...,
K5)∈B5
wi h
B={
0,1
}
, hen s a o ol ages
αβxys a e ound as ollows:
αβxys = ( αs, βs, xs, ys) = VDCuTM (10)
whe e TM is gi en as
TM =
abbbb
b a b b b
b b a b b
b b b a b
bbbba
·
dγc
1γc
2γc
3γc
4
0γs
1γs
2γs
3γs
4
dγc
2γc
4cϑγc
3
0γs
2γs
4γs
1γs
3
ccccc
(11)
In he abo e equa ions he ma ix en ies a e unc ions o he mo o pa ame e s and can
be ound as
c1=LsL −L2
M
,
c2=L /c1
,
c3=
1
/Lls
,
c4=LM/c1
,
a2=−Rsc2
,
a3=−Rsc3
,
a4=−LMc4ω
,
a=
4
/
5,
b=−
1
/
5,
c=−b
,
d=
2
/
5,
γc
h=cos hϑ
,
γs
h=sin hϑ
and
ϑ=2π/5.
The MPC echnique, conside ing a one-sample ime delay o he compu a ions, is
as ollows. The VSI s a e a ime
k
cons i u es he con ol ac ion
u(k+
1
)
. The op imal
alue is ound as a minimiza ion p oblem in which index
J
includes se e al objec i es.
In pa icula , de ia ions o s a o cu en s om hei e e ences is penalized in he index o
cos unc ion. This is exp essed as
J=∥i∗(k+
2
)−ˆ
i(k+
2
)∥2
. He e, symbol
ˆ.
is used o
deno e p edic ions ob ained om he model.
2.4. Figu es o Me i
The quali y o d i e con ol is ypically measu ed using some igu es o me i . He e
h ee igu es o me i a e selec ed. They a e ela ed o he beha io o he sys em a e a s ep
change in he e e ence o mechanical speed (
ω∗
). All h ee igu es o me i a e ob ained
om expe imen al esul s du ing ansien s.
The i s indica o (γ1) is he o e shoo , ha can be compu ed as ollows:
γ1=100 ·
max
k≥0ω(k)−ω∗
ω∗(12)
Machines 2025,13, 150 6 o 12
The ise ime cons i u ed he igu e o me i
γ2
. This is expe imen ally ob ained
measu ing he ime needed o he d i e o e ol e om he ini ial speed o he new
e e ence alue. Wi hou loss o gene ali y, one can suppose he s ep ime o co espond o
he disc e e ime index
k=
0, he ini ial speed o be
ω=
0 and he ini ial e e ence o be
null, hen
γ2=a gmin
k≥0
(ω∗−ω(k))(13)
Finally, he o que ipple is he oo mean squa ed (RMS) alue o he de ia ion o he
ac ual o que om i s e e ence alue T∗
γ3=
u
u
1
N
k−N
∑
j=k
(T∗(j)−T(j))2. (14)
Fo cla i y, a ec o
Γ=(γ1,γ2,γ3)
will be used o e e o all h ee igu es o me i
as an ensemble.
3. Expe imen s
The da a o he Pa e o analysis a e ob ained om expe imen al es s pe o med in he
labo a o y es bed desc ibed ea lie .
3.1. S eady S a e and T ansien Con ol Resul s
The esul s o FSMPC ha e been ea u ed in many applica ions including mul i-phase
d i es. In his case, a en ion mus be paid o o que-p oducing and ha monic subspaces.
In sinusoidal s eady s a e one seeks ha
α−β
componen s ollow hei sinusoidal e e ences
and he x−ycomponen s be egula ed a ound ze o as shown in Figu e 3.
0 0.005 0.01 0.015 0.02
−3
−2
−1
0
1
2
3
4
Time (s)
Cu en s (A)
iα
*iαiβ
*iβixiy
Figu e 3. S eady s a e e olu ion o cu en s.
T ansien condi ions a e ound o changes in e e ence speed. A se ies o s ep es s
will be used o he Pa e o analysis. In hese es s, he speed se poin is changed ollowing
a s ep as shown in Figu e 4. Fo cla i y, he case po ayed he e co esponds o a s a
es in which he mo o is no unning a he s a o he es . I is wo h poin ing ou ha ,
in mos epo s, he only igu es o me i conside ed a e de i ed om speed, including
o e shoo , ise ime, in eg al absolu e e o , and in eg al ime absolu e e o . Howe e ,
o que ipple is no isible in hese igu es o me i as seen in Figu e 4.
Machines 2025,13, 150 7 o 12
0 0.2 0.4 0.6 0.8 1 1.2
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time (s)
To que (Nm)
Te
0 0.2 0.4 0.6 0.8 1 1.2
0
10
20
30
40
50
60
Time (s)
Speed ( ad/s)
ω*,ω
Figu e 4. Elec ical o que (le column) and speed ( igh column) in a s ep es .
3.2. Pa e o Analysis
The Pa e o analysis is pe o med by ob aining expe imen al alues o
Γ
co esponding
o di e en unings o he PI. These unings a e ob ained as di e en combina ions o he
PI gains
(kp
,
ki)
. This p oduces a se o alues
γ1
,
γ2
and
γ3
o each uning. Fo he
analysis, a numbe o PI gains combina ions ha e been conside ed, whe e 0
<kp<
0.5 and
0<ki<0.02.
Some o hese combina ions a e no Pa e o op imal and mus be immed. This is
pe o med in he usual way, checking i he
Γ
alues o a pa icula
(kp
,
ki)
combina ion
A a e domina ed by some o he combina ion B. Domina ed means ha a leas one alue
γB
j<γA
j
wi h he emaining being a leas equal:
γB
h≤γA
h
. These dominan combina ions
a e excluded om he se .
The combina ions emaining, a e he imming p ocess, o m a se
P
ha can be
in e p e ed as an app oxima ion o he Pa e o on . They should lay in a lowe dimensional
su ace as will be shown la e . Poin s in se
P
a e shown in Figu e 5in conjuc ion wi h
some p ojec ions. In hose,
Γ
alues a e shown in colou . The colou is compu ed as a linea
unc ion o
γ3
. The ed hue co esponds o highe
γ3
, con e sely, he blue hue co esponds
o lowe γ3. Please no e ha colou is added jus o enhance he 3D pe cep ions.
0
0.2
0.4 0510 15
0.005
0.01
0.015
γ3
γ1
γ2
4 6 8 10 12 14
0.1
0.15
0.2
0.25
0.3
0.35
γ1
γ2
4 6 8 10 12 14
0.008
0.01
0.012
0.014
0.016
γ1
γ3
0.1 0.15 0.2 0.25 0.3 0.35
0.008
0.01
0.012
0.014
0.016
γ2
γ3
Figu e 5. P ojec ions o se P o ω∗=400 ( pm). Di e en hues co espond o di e en γ3 alues.
Poin s in se Pcan be shown o be close o a cubic Ti eica su ace cha ac e ized by
γ1·γ2·γ3=Π, (15)
Machines 2025,13, 150 8 o 12
whe e he p oduc o he
γ
alues o each poin is a cons an
Π
. In his pa icula case,
a alue Π=0.0148 is ound expe imen ally.
The Pa e o on is impo an in he con ex o con olle uning because i shows ha
he pe o mance indica o canno be imp o ed inde ini ely. In ac , he bes unings a e
hose lying in he on because hey a e he non-domina ed combina ions.
The ade-o s be ween pe o mance indica o s ollow om exp ession (15). The con-
s ain due o
γ1·γ2·γ3
being cons an means ha a educ ion in
γ1
mus p oduce a highe
alue o γ2·γ3. This means ha ei he γ2o γ3o bo h mus inc ease.
Howe e , hese ade-o s, o e lexibili y o con ol uning. One may choose which
igu e o me i o p io i ize and o wha ex en . One may be emp ed o minimize he
dis ance o he o igin as he bes uning. This case co esponds o a se o pe o mance
indica o s Γ0such ha
∥Γ0∥2=min γ2
1+γ2
2+γ2
3. (16)
The da a poin
Γ0
is expe imen ally ound as
γ0
1=
4.30,
γ0
2=
0.17,
γ0
3=
11.2.
Howe e his solu ion is, in a gene al case, jus as ele an as he o he s. This is easily
shown conside ing a change in scale in one pe o mance indica o . Then
Γ0
will mo e
al hough no hing changes in he physical sys em. The uning co esponding o
Γ0
only
makes sense i (1) he pe o mance indica o s a e held as he same impo ance and (2) hey
a e exp essed in uni s whe e ha equali y holds.
In a p ac ical applica ion one may eso o he use o a weigh ed me ic in he
γ
space.
This can be achie ed simply by applying scale ac o s o each pe o mance indica o .
Ano he aspec ha dese es some commen s is he exis ence o limi s o uning.
In some publica ions he p oposal is assessed agains a p e ious me hod esul ing in
heimp o emen o all igu es o me i a e imp o ed. This is un ealis ic unless he p e ious
me hod is e y poo ly designed. A p ope assessmen o any new p oposal should use he
Pa e o on as a way o communica e wha aspec s a e imp o ed [27].
3.3. Va ia ion wi h Speed
In a p ac ical si ua ion, he ac ual mechanical speed de ines he ope a ion egime o
he mo o . This a iable plays a ole in he beha iou o he d i e. This means, among o he
hings, ha he igu es o me i migh be di e en o di e en speeds.
This can be es ed by means o he Pa e o analysis being conduc ed o a di e en
e e ence
ω∗
. To do so, he da a ga he ing p ocess is pe o med again o a di e en alue
o
ω∗
. The esul s a e shown in Figu e 6whe e
ω∗=
120 has been used. In he plo ,
a di e en dis ibu ion o he
Γ
alues is ound. This is u he emphasized by Table 2,
whe e di e en ω∗ alues a e u ilized.
Table 2shows ha he epo ed
Γ
a e o he solu ion close o he o igin (
Γ0
). This
solu ion is p esen ed as a way o compa e he esul s o di e en speeds. The ele ance o
his solu ions can be subjec ed o he same analysis as in he p e ious case. Ne e heless, i
is in e es ing o see ha he ’op imal’ PI uning depends on he ope a ing mode. This is in
con as wi h mos applica ions whe e a single uning is used o all egimes. In pa icula ,
p ojec ions
γ1−γ3
(lowe le ) and
γ2−γ3
(lowe igh ) show a p ominen hype bola-
shaped dis ibu ion. This is in e es ing since
γ3
is no conside ed in mos pape s dealing
wi h PI uning.
A inal obse a ion ha can be made om Figu es 5and 6is ha he ade-o s a e
mo e p onounced in ela ion wi h
γ3
. This obse a ion is o impo ance as in some wo ks
he e is no conside a ion o di e en ope a ing egimes.
Machines 2025,13, 150 9 o 12
0
0.2
0.4 010 20 30
0.005
0.01
0.015
γ3
γ1
γ2
5 10 15 20 25 30
0.05
0.1
0.15
0.2
0.25
γ1
γ2
5 10 15 20 25 30
0.008
0.01
0.012
0.014
0.016
γ1
γ3
0.05 0.1 0.15 0.2 0.25
0.008
0.01
0.012
0.014
0.016
γ2
γ3
Figu e 6. P ojec ions o se P o ω∗=120 ( pm). Di e en hues co espond o di e en γ3 alues.
Table 2. Figu es o me i o se e al speed egimes.
ω∗k0
p·103k0
i·105γ0
1γ0
2γ0
3Π
( pm)
(A/( ad/s))
(A/( ad)) (%) (s) (mN·m)
400 55 103 4.30 0.17 11.2 8.19
260 96 80 7.11 0.11 12.4 9.70
120 133 58 11.2 0.09 10.3 10.4
4. Discussion
The p oposed pe o mance indica o s summa ize he beha io o he sys em.
The Pa e o-op imal unings pe ain o a su ace o he lowe dimension. In conclusion, any
PI uning mus ei he lie in he su ace o be non-op imal. This is in s a k con as o o he
app oaches, whe e jus a hand ul o con igu a ions a e conside ed and app oaches using
black-box ep esen a ions do no p o ide insigh .
Rega ding insigh , he esul s o he p e ious sec ion clea ly show he exis ence o
ade-o s be ween igu es o me i conside ing elec o-mechanical a iables. The impo -
ance o his inding esides in he ollowing poin s.
1.
The in e play be ween elec ical and mechanical a iables is made appa en . This ela-
ionship is o en dis ega ded in wo ks dealing wi h PI uning o a iable-speed d i es.
2.
The exis ence o limi a ions in PI uning is highligh ed. The limi s a e p o ided by he
Pa e o on . Wi hou knowing hese limi s, one could was e ime on
un ui ul es ing.
3.
Compa isons be ween con olle s should be made by compa ing whole Pa e o on s,
ins ead o pa icula poin s. O he wise one can always p esen a pa icula poin
whe e one con olle ou pe o ms ano he in some speci ic igu e o me i .
The indings migh seem o be nega i e in na u e. A e all, he e is no known analy ical
p ocedu e o link con olle pa ame e s o pe o mance indica o s. Such a p ocedu e would
allow o a mo e sys ema ic, and pe haps au oma ic, uning o he PI. Howe e , om
he abo e esul s, one can de i e p ac ices o help PI uning. In pa icula , one should
p epa e o accep ha he Pa e o on ie canno be pie ced jus by using uning. This has
wo consequences:
