Ad ances in Elec ical and Elec onic Enginee ing
114
EXPERIMENTAL EVALUATION OF PI TUNING TECHNIQUES FOR FIELD
ORIENTED CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTORS
B. Zigmund
3
, A. Te lizzi
1
, X. T. Ga cia
2
, R. Pa lanin
3
, L. Sal a o e
1
1Poli ecnico di Ba i, Dipa imen o di Ele o ecnica ed Ele onica, V. O abona 4, 70125 Ba i, I aly, sal a o [email protected]
2Uni e si y o Glamo gan, School o Elec onics, CF37 1DL Pon yp idd, Wales, UK, [email p o ec ed].uk
3Uni e si y o Zilina,
Facul y o Elec ical Enginee ing, Depa men o Mecha onics and
Elec onics, Uni e zi na 1,
010 26 Zilina, SK, b [email protected], pa lani[email p o ec ed].sk
Summa y
This pape p esen s he expe imen al e alua ion o a commonly used me hodology o une he PI egula o s o he Field
O ien ed Con ol (FOC) s a egy o Pe manen Magne Synch onous Mo o s (PMSMs). The me hodology used is based on he
Absolu e Value Op imum (AVO) and Symme ic Op imum (SO) c i e ions. These me hods employ a simpli ied model o he plan o
be con olled. Due o he complexi y and non-linea i ies p esen in he FOC PMSM d i e some di e gence be ween he ideal
esponse and he eal esul s will occu . In his pape a compa ison be ween simula ed and expe imen al esul s is ca ied ou o
e alua e he goodness o he solu ion ob ained. The di e gences obse ed be ween simula ions and he esul s ob ained in he
expe imen al se up a e shown and i is a emp ed o jus i y he causes o hem. O e all i is concluded ha he me hod p o ides a
sa is ac o y ini ial commissioning o he PI egula o s o he d i e sys em unde s udy.
1. INTRODUCTION
PID egula o s a e widely employed in indus y due o
hei sa is ac o y beha iou in mos o he con ol
applica ions. One o he mos impo an enginee ing
asks du ing he commissioning o con ol sys em is he
pa ame ic op imiza ion o he egula o s o ob ain he
desi ed con ol esponse [1].
Di e en me hods can be employed o pe o m he
uning o he PID egula o s. A possible classi ica ion o
hese me hods can be as ollows:
• Expe imen al me hods based on he iden i ica ion o
ce ain esponse cha ac e is ics o he sys em. The
Ziegle -Nichols open and closed loop me hods a e an
example o hese me hods [2].
• Ma hema ical model based me hods. These me hods
employ a ma hema ical model ha app oxima es he
beha iou o he sys em [1, 3, 4].
• Op imiza ion echniques. By means o a me i
unc ion ha can be e alua ed in a es , a numbe o
solu ions consis ing on di e en se s o pa ame e s is
e alua ed. A he end he bes -pe o ming solu ion is
ound. Di e en echniques can be used o pe o m
he sea ch o he bes solu ion [5, 6].
In he ield o elec ical d i es PI egula o s a e also
employed o mo o con ol. The s uc u e and an
equi alen ans e unc ion o his con olle a e shown
in Fig. 1. The a iables o be con olled a e gene ally
posi ion, speed, o que, cu en o ol age. The ac ha
he measu emen o hese signals can con ain
conside able noise makes he PI s uc u e wi hou he
de i a i e pa mo e sui able. One example o
applica ion whe e PI egula o s a e employed is he
FOC s a egy o PMSM d i es.
PI
+
+
1/s
k
p i
s
1
i
s
= =
k
i
k
p
Fig. 1. PI con olle s uc u e and ans e unc ion
This pape p esen s he applica ion o wo popula
me hods o uning he PI egula o s o he FOC PMSM
d i e: he AVO and SO c i e ions [3, 7]. The model o
he PMSM is p esen ed oge he wi h he FOC con ol
scheme. The AVO and SO c i e ions a e explained and
applied o he sys em unde s udy. Finally some
simula ed and expe imen al es s a e pe o med in o de
o e alua e he solu ion ob ained.
2. PMSM FOC
The model o he PMSM in a o a ing d-q ame ixed o
he o o is gi en by he ollowing equa ions:
sd sd sd
L i
ψ
= + Ψ
(1)
sq sq sq
L i
ψ
= (2)
sd
sd s sd sd sq sq
di
R i L L i
d
ω
= + − (3)
( )
sq
sq s sq sq sd sd
di
R i L L i
d
ω
= + + + Ψ
(4)
( )
3
2
e sd sq sq sd
P i i
ψ ψ
Γ = − (5)
whe e
sd
ψ
,
sq
ψ
,
sd
,
sq
,
sd
i
and
sq
i
a e espec i ely
he mo o luxes, ol ages and cu en s in d-q axes;
ω
is he elec ical angula speed,
e
Γ
is he elec omagne ic
o que,
Ψ
is he lux o he pe manen magne and P is
he numbe o pole pai s.
s
R
is he s a o esis ance and
he s a o induc ance can be di ided in o wo di e en
componen s
sd
L
and
sq
L
due o he pa icula i ies o
he PMSM. The model is comple ed by he mechanical
equa ion, which is de ined as:
m
e l m
d
J B
d
ω
ω
= Γ − Γ − (6)
m
P
ω ω
= (7)
whe e J is he ine ia o he mo o and coupled load,
l
Γ
is he load o que, B is he ic ion coe icien and 
m
is
Expe imen al e alua ion o PI uning echniques…
115
he mechanical angula speed.
Simila ly o induc ion mo o s, in PMSMs a decoupled
con ol o he o que and lux magni udes can be
achie ed, emula ing a DC mo o , by means o he FOC
s a egy. This is done using he d-q ans o ma ion ha
sepa a es he componen s d and q o he s a o cu en
esponsible o lux and o que p oduc ion espec i ely
[8]. Due o he p esence o he cons an lux o he
pe manen magne , he e is no need o gene a e lux by
means o he i
sd
cu en , and his cu en can be kep
equal o ze o alue, which in u ns dec eases he s a o
cu en and inc eases he e iciency o he d i e. The
con ol scheme o he FOC s a egy is shown in Fig. 2.
PMSM
PI
PI
d,q
a,b,c
d,q
a,b,c
PI
+
-
+
-
+
-
s
+
-
+
L
sq
i
sq
+
L
sd
i
sd
i
sd
i
sd
i
sq
i
sa
i
sb
*
*
d
d
Fig. 2. FOC con ol scheme o PMSM
The con ol sys em is di ided in o h ee di e en loops:
he d loop, which con ols he lux, he q loop, which
con ols he o que and he speed con ol loop. The d
loop pe o ms he con ol o i
sd
wi h a cu en PI
egula o . The e e ence alue is 0. The q-axis con ol
sys em con ains wo con ol loops in cascade. The inne
loop con ols he o que by means o con olling i
sq
wi h
a cu en PI egula o . The ac ha he o que can be
con olled by means o i
sq
comes om he ollowing
simpli ica ion o (5), alid o Su ace Moun ed (SM)
PMSM:
3
2
e sq
P i
Γ = Ψ
(8)
The e e ence o his inne loop is gi en by he ou e
loop, which con ains a speed PI egula o .
F om he ol age equa ions o he PMSM model (3) and
(4) i can be seen ha d and q axes a e no comple ely
independen and he e a e coupling e ms which depend
on he cu en om he o he axis. To achie e
comple ely independen egula ion i is necessa y o
cancel he e ec o hese coupling e ms a he ou pu o
he cu en PI egula o . Wi h he use o decoupling i is
achie ed he linea iza ion o he con ol sys em as well
as highe dynamics. This decoupling ac ion can be seen
in Fig. 2 [9, 10].
3. PI TUNING WITH THE AVO AND SO
CRITERIONS
In o de o ob ain a sa is ac o y con ol pe o mance i is
necessa y o adjus he pa ame e s o he PI egula o s
included in he FOC scheme. In he ield o elec ical
d i es wo me hods a e equen ly employed in his
pa ame ic op imiza ion: he AVO and SO c i e ions
[3, 7]. The AVO c i e ion can be applied o design bo h
cu en egula o s, while he SO can be employed o
design he speed egula o shown in Fig. 2 [3]. These
wo me hods employ an app oxima e ans e unc ion
o he sys em o be con olled. Some modelling and
pa ame e iden i ica ion o he sys em a e he e o e
needed.
The AVO me hod assumes he sys em’s ans e
unc ion in open loop o he ollowing o m [1]:
1
( )
2 (1 )
G s
s s
τ τ
Σ Σ
=+ (9)
On he o he hand, he SO me hod conside s he
sys em’s open loop ans e unc ion as ollows:
2 2
1 4
( )
8 (1 )
s
G s
s s
τ
τ τ
Σ
Σ Σ
+
=+ (10)
whe e
τ
Σ
is sum o all small delays in loop. Bo h
me hods can be applied when he
τ
Σ
is much smalle
han he ime cons an o he sys em. The s ep esponse
o bo h ans e unc ions in closed loop inco po a ing
he PI egula o designed using he AVO and SO
c i e ions is as shown in Fig. 3. Table 1 p esen s he
main ea u es o he s ep esponses ob ained wi h bo h
me hods.
0 1 2 3 4 5
0
0.5
1
1.5
Time (s)
Ampli ude
AVO
SO
Fig. 3. AVO and SO cha ac e is ic s ep esponses
Tab. 1. S ep esponses o AVO and SO me hods
AVO SO
Rise Time 4.7
τ
Σ
3.1
τ
Σ
Se ling ime (2%) 8.4
τ
Σ
16.5
τ
Σ
O e shoo 4.3% 43.4%
Phase Ma gin 65.5º 37º
I bo h me hods a e compa ed i can be said ha he SO
is as e ega ding dis u bance ejec ion, which makes i
sui able o he speed loop. On he o he hand AVO has
a smalle se ling ime and lowe o e shoo , which
makes i app op ia e o ha e quicke and mo e accu a e
inne loops [3].
Ad ances in Elec ical and Elec onic Enginee ing
116
a) Cu en loop PI (i
sd
and i
sq
con ol)
In o de o design he egula o an app oxima ed ans e
unc ion has o be de ined acco ding o he AVO
c i e ion. Fi s o all, he exis ing delays in he sys em
need o be aken in o accoun . These delays in he case
o a mo o d i e a e due o he digi al implemen a ion o
he con ol (which implies he sampling o signals), he
use o il e s, he p ocessing o he con ol algo i hm
and he use o Pulse Wid h Modula o s (PWM). Fig. 4
shows he block diag am o he cu en con ol in closed
loop wi h all he delays conside ed.
+
-
k
p
i
sq
i
i
sq
s
1
i
i
sq
s
1
1
s
s
1
1
s
2
s
1
R
s
1
L
sq
R
s
s
1
1
i
sq
s
1
1
s
2
s
i
sq
i
sq
*
Mo o VSIP ocessing delayPI egula o
Sampling
Cu en il e
Fig. 4. Cu en loop block diag am
To make i possible o apply he AVO c i e ion, he
eedback delays mus be ans e ed o he o wa d pa h.
The esul ing block diag am is shown in Fig. 5.
+
-
k
p
i
s q
i
i
sq
s
1
i
i
sq
s
1
1 2
s
i
sq
s
1
R
s
1
L
sq
R
s
s
i
sq
i
sq
*
1
1
s
2
i
sq
s
Fig. 5. Simpli ied block diag am o he cu en loop
The ollowing app oxima ion can be made due o ac
ha
sq
i
s
τ τ
= :
1
( )
(1 )(1 ( /2) )(1 ( /2) )(1 )
sq
i
s s s
G s
s s s s
τ τ τ τ
=
+ + + +
(11)
'
1
( ) ( )
1 (2 )
sq
i
s
G s G s
s
τ τ
≈ =
+ +
(12)
In o de o adap he ans e unc ion o he de ini ion
o he AVO me hod, he PI egula o ime cons an has
o be equal o ime cons an o he plan . This in u ns
will dec ease o de o open loop ans e unc ion.
sq
i
sq
i
s
L
R
τ
= (13)
1 1
( )
2 (1 )
(1 (2 ) )
sq
sq sq
sq
sq
i
open i i
i
sq
s
i
p
G s L
s s
s s
k
τ τ
τ τ
Σ Σ
= =
+
+ +
(14)
Finally he alues o
sq
i
p
k
and
sq
i
i
k
can be calcula ed as
ollows:
2 ; ;
2
sq
sq sq sqsq
sq sq
i
i i i
i
sq p
s p i
i i
i
L k
k k
τ τ τ
τ τ
Σ
Σ
= + = = (15)
The esul ing closed loop ans e unc ion is o he
second o de ype. Because second o de ans e
unc ions can be app oxima ed by a i s o de ans e
unc ion wi h he same se ling ime, he inne loop can
be app oxima ed as ollows:
1
( ) 1 2
sq
sq
i
close i
G s
s
τ
Σ
=
+
(16)
In o de o une he PI cu en egula o in d axis i can
be ollowed he same p ocedu e desc ibed, bu in his
case using he induc ance in d axis
sd
L
.
b) Speed loop PI
The speed loop is designed by means o he SO me hod.
As in he cu en loop, he speed loop con ains some
delays ha need o be de ined. Fo his loop, i is
common o use a sampling ime en imes highe han
he sampling ime o he cu en loop. Fig. 6 shows he
ans e unc ion o he speed loop wi h all he delays
conside ed and inco po a ing he inne q axis cu en
loop ans e unc ion de ined in (16).
+
-
k
p
i
s
1
i
s
1
1
s
s
3
2
P
*
G
close
i
sq
+
-
P
Js
T
l
1
1
s
2
s
1
1
s
2
i
sq
s
PI egula o P ocessing delay
Cu en con ol and mo o
Sampling and speed il e
Fig. 6. Speed loop block diag am
A simila design p ocess as in he AVO me hod is
ollowed o adap he ans e unc ion o he SO
c i e ion in he speed con ol loop. Fo simplici y he
load o que and ic ion e ms a e neglec ed. The
simpli ied block diag am passing he eedback delays o
he o wa d loop and g ouping all he delay is shown in
Fig. 7.
+
-
k
p
i
s
1
i
s
1
1
s
3
P
2
2Js
1
1
s
2
s
*
Fig. 7. Simpli ied speed loop block diag am
The sum o all delays in he o wa d loop
ω
τ
Σ
is de ined
as:
32
2 2
sq sq
i i
s
s
ω ω
ω
τ
τ τ τ τ τ
Σ Σ
= + + − − (17)
Expe imen al e alua ion o PI uning echniques…
117
and he esul ing open loop ans e unc ion has he
app op ia e o m o apply he SO me hod:
2 2
2
2
1 1 4
( )
8( ) (1 )
2 (1 )
3
i
open
i
p
s s
G s
s s
J s s
P k
ω ω
ω
ω ω ω
ω
ω
τ τ
τ τ τ
τ
Σ
Σ Σ
Σ
+ +
= = +
+
Ψ
(18)
Finally he alues o he egula o can be ob ained as
ollows:
2
4 ; ;
3
p
i p i
i
k
J
k k
P
ω
ω ω ω
ω
ω ω
τ τ
τ τ
Σ
Σ
= = =
Ψ (19)
4. SIMULATED AND EXPERIMENTAL
RESULTS
The me hods desc ibed in he p e ious sec ion ha e
been employed o une he i
sd
and i
sq
cu en egula o s
(AVO c i e ion) and he speed egula o (SO c i e ion).
The esul ing PI egula o s ha e been es ed using a
simula ion model and also an expe imen al se up. The
expe imen al se up consis s on a Siemens 1KF7 PMSM
wi h he cha ac e is ics shown in Table 2, a Dan oss
VLT5006 powe con e e and a DSPACE DS1103
con ol boa d.
Tab. 2. PMSM cha ac e is ics (Siemens 1KF7)
Nominal Ou pu Powe (P
n
) 2135W
Nominal Speed (
n
) 3000 pm
Nominal To que (M
n
) 6.8N·m
Nominal Cu en (I
n
) 4.4A
Numbe o pole pai s (P) 4
S a o esis ance (R
s
) 1.09
S a o induc ance (L
sd
and L
sq
) 0.0124H
Ine ia (J) 4.15e
-4
Kg·m
2
Pe manen Magne Flux (
Ψ
) 0.1821Wb
The esul ing se o con ol pa ame e s calcula ed a e
shown in Table 3.
Tab. 3. Calcula ed PI pa ame e s
PI egula o
p
k
i
k
Ou pu
Sa u a ion
sd
i
8.86 778.6
2 2
n
I
sq
i
8.86 778.6
2 2
n
I
ω
0.0934 3.18
3
DC
V
Some addi ional in o ma ion ega ding he con ol is
shown in Table 4.
Tab. 4. Addi ional con ol pa ame e s
Sampling ime o cu en (
s
τ
) 100s
Sampling ime o speed (
s
ω
τ
) 1ms
DC-link Vol age (V
DC
)
380 2
Cu en il e ime cons an 500s
Speed il e ime cons an 5ms
The i s es pe o med consis s on he speed s ep
esponse. A compa ison be ween he ideal esponse
ob ained employing he AVO me hod and he simula ed
and expe imen al esul s is shown in Fig. 8. I can be
seen ha simula ed and expe imen al esul s a e e y
simila . In bo h cases he o e shoo is smalle han he
ideal esponse. This is due o he ic ion e m o he
mechanical equa ion (6) ha inc eases he damping o
he sys em. I can be also app ecia ed he noise p esen
in he expe imen al esponse and a highe se ling ime.
1 1.05 1.1 1.15 1.2 1.25 1.3
-10
0
10
20
30
40
50
60
70
80
ime [s]
speed [ ad/s]
Ideal
Expe imen al
Simula ed
Fig. 8. Compa a i e s ep esponses
The nex es implemen ed is a speed p o ile which
pe o ms a speed e e sal. Fig. 9 and Fig. 10 show he
simula ed and expe imen al esul s. The a iables
shown a e he mo o elec ical speed (
), he cu en s
(i
sd
and i
sq
), and wo phase cu en s (i
sa
and i
sb
).
The simula ed and expe imen al esul s a e also e y
simila o he speed p o ile es . These esul s illus a e
a sa is ac o y acking o he speed p o ile. The load
o que in his es is only due o he ic ion. I can be
seen how he i
sd
is kep o ze o excep o he ansien s
whe e some cu en peaks a e p oduced.
Figu es 10 and 11 show he simula ed and expe imen al
esul s espec i ely o he acking o an i
sq
p o ile. Fo
his es he speed PI is elimina ed and he e e ence o
i
sq
is gi en by a p o ile which includes a o que e e sal.
I can be seen ha he expe imen al esul s p esen mo e
oscilla ion and dis o ion caused by he noise o he eal
sys em and he ope a ion o he powe con e e .
Ad ances in Elec ical and Elec onic Enginee ing
118
1 1.5 2 2.5 3
-100
0
100
w [ ad/s]
1 1.5 2 2.5 3
-10
0
10
isq [A]
1 1.5 2 2.5 3
-10
0
10
ime [s]
isa,isb [A]
1 1.5 2 2.5 3
-0.5
0
0.5
isd [A]
1.5 2 2.5 3
-100
0
100
1.5 2 2.5 3
-10
0
10
1.5 2 2.5 3
-5
0
5
1.5 2 2.5 3
-10
0
10
ime [s]
Fig. 9. Simula ed (le ) and expe imen al ( igh ) esul o he speed p o ile es
0 0.2 0.4 0.6 0.8 1 1.2
-4
-3
-2
-1
0
1
2
3
4
ime [s]
isq [A]
Fig. 10. Simula ed esul s o he i
sq
p o ile es
0 0.2 0.4 0.6 0.8 1 1.2
-4
-3
-2
-1
0
1
2
3
4
ime [s]
isq [A]
Fig. 11. Expe imen al esul s o he i
sq
p o ile es

Expe imen al e alua ion o PI uning echniques…
119
The inal es p esen ed has been ca ied ou only in
simula ion o illus a e he e ec o he load o que on
he speed esponse. I can be seen ha inc easing he
load o que esul s on a bigge ise ime due o he
sa u a ion o he speed PI egula o ou pu s. The le el o
he sa u a ion he e o e in luences he esponse ime o
he speed loop. Fig. 12 shows he speed esponse wi h 3
di e en le els o load o que.
1 1.05 1.1 1.15
0
50
100
150
200
ime [s]
speed [ ad/s]
100% Tload
50% Tload
0% Tload
Fig. 12. Compa a i e s ep esponses o di e en load o que
condi ions
5. CONCLUSION
This pape p esen s he expe imen al and simula ed
pe o mance o he PI uning AVO and SO me hods
applied o a FOC PMSM d i e. The p ocedu e o ob ain
he pa ame e s o he PI egula o s has been desc ibed
o his applica ion. The esul s o he di e en dynamic
es s p esen ed show a sa is ac o y con ol esponse o
he speed and cu en loops. O e all he uning me hods
employed seem o p o ide a alid solu ion o he ini ial
commissioning o he d i e sys em unde s udy.
In addi ion he simula ion model employed has p o ed
o be e y accu a e and he simila i y wi h he
expe imen al esul s is e y high. Any di e gences
obse ed be ween simula ed and expe imen al esul s
a e due o: noise in he eal se up, inaccu acy o he
mo o and load pa ame e s and he ope a ion o he
con e e .
Acknowledgemen s
This esea ch p ojec has been suppo ed by a Ma ie
Cu ie Ea ly S age Resea ch T aining Fellowship o he
Eu opean Communi y’s Six h F amewo k P og amme
unde con ac numbe MEST-CT-2004-504243
Elec ical Ene gy Con e sion and Condi ion.
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