Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
V
o
l.4
,
No
.1
, Mar
c
h 201
4,
p
p
.
1
2
~
23
I
S
SN
: 208
8-8
6
9
4
12
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJPEDS
/
FPGA-Based Implementation Non
linear Backstepping Control
of a P
M
SM Dri
ve
Badre B
o
ss
oufi*, Mohamm
ed Karim**,
Ahmed Lagri
o
ui**, Mohammed T
a
ouss
i**
* Labor
ator
y
of
Electrical Eng
i
n
eering
and
Main
te
nan
c
e,
Higher School
of Techn
o
log
y
, EST-Oujda,
University
of
Mohammed I, Mor
o
cco
** S
T
IC T
eam
,
F
acult
y
of S
c
i
e
n
ces
Dhar
El
M
a
h
r
az,
Si
di
M
o
ha
m
e
d B
e
n
A
bde
l
l
a
h U
n
i
v
e
r
si
t
y
, F
e
z
,
M
o
ro
cco
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 7, 2013
Rev
i
sed
D
ec 18
, 20
13
Accepte
d Ja
n
9, 2014
In this p
a
per
,
we present a new
cont
ribution
of F
P
GAs (Field-Pro
grammable
Gate Arra
y) for control of ele
c
tr
i
cal m
achin
es
. T
h
e adapt
a
tiv
e Ba
cks
t
epping
control appro
a
ch for a permanent magnet s
y
nchronous motor drive is
dis
c
us
s
e
d and anal
yz
ed. W
e
pres
en
t a Matlab
&
Sim
u
link simulation an
d
experimental res
u
lts from a benchmar
k based on FPGA. The Backsteppin
g
techn
i
que provides a s
y
stematic method to
address this ty
pe of
problem. It
combines the notion of Ly
apu
nov function and a controller
procedure
recurs
ive
l
y. F
i
rs
t, th
e ad
apta
tiv
e and no
adapt
a
tiv
e Backs
t
epp
i
ng contro
l
approach
is utilized to ob
tain
th
e robustness for mismatched parameter
uncertainties. The over
a
ll stability
of
the s
y
stem is shown using Ly
apunov
techn
i
que. Th
e simulation results
clear
ly
show that the proposed scheme can
track
the speed refer
e
nce. S
econdly
,
some exper
i
ment
al results ar
e
demonstrated to
validate
the pr
oposed controllers. The experim
e
ntal results
carri
ed from
a p
r
otot
yping p
l
atfo
rm
are given
to
illustra
te
the
effi
cien
c
y
an
d
the benef
its of the proposed appr
oach an
d the various stages of
implementation
of this stru
ctur
e in FPGA.
Keyword:
Ada
p
t
i
v
e bac
k
st
eppi
ng
co
nt
r
o
l
B
ackst
ep
pi
n
g
desi
g
n
t
e
c
hni
q
u
e
FPGA
Lyap
uno
v stabilit
y
Perm
anent
m
a
gnet
sy
nc
hr
on
o
u
s
machine (PMSM)
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Bad
r
e B
O
SSOUFI,
Depa
rtem
ent of Electrical a
nd Co
m
p
u
t
er
Engin
eer
ing
,
Mohamm
ed I
Uni
v
ersity,
45
1,
A
d
a
r
i
ssa,
Fès, M
o
r
o
cc
o
Em
a
il: b
a
d
r
e_
isai@ho
t
m
a
i
l
.co
m
1.
INTRODUCTION
Th
r
e
e-p
h
a
se Per
m
an
en
t Magn
et Syn
c
hr
onou
s M
o
tor
(PM
S
Ms)
is
str
ong
ly u
s
ed
i
n
ind
u
s
t
r
y and
consum
es
m
o
re than 70%
of
industrial
electricity. Th
is is
wh
y con
s
id
erab
le
ef
fo
rts a
n
d
diffe
re
nt searc
h
es a
r
e
bei
n
g d
o
n
e t
o
i
m
prove t
h
ei
r
p
e
rf
orm
a
nces and t
h
ei
r ef
fi
ci
ency. T
h
e efficiency of el
ectrical
m
achine drives i
s
g
r
eatly redu
ced
at lig
h
t
lo
ads, wh
ere th
e
flu
x
m
a
g
n
itu
d
e
referen
ce is h
e
ld
on
its in
itial v
a
lu
e. The lo
ss
min
i
mizatio
n
is realized
u
s
ing
h
i
g
h
-q
u
a
lity materials an
d
ex
cellen
t
d
e
si
gn
pro
cedu
r
es in
th
e m
a
n
u
f
act
u
r
i
n
g
pr
ocess
.
M
o
re
ove
r, e
x
pert
c
o
nt
r
o
l
al
g
o
ri
t
h
m
s
are em
pl
oy
ed
i
n
or
der t
o
i
m
pr
o
v
e m
achine perform
ance.
In t
h
is
pape
r we a
r
e
i
n
t
e
rest
ed i
n
t
w
o m
ode cont
rol
s
fo
r
PMSM d
r
iv
e, th
e n
o
t
ad
ap
t
a
tiv
e an
d
ad
ap
tativ
e
backstepping.
The n
o
t
ada
p
t
i
ve bac
k
st
ep
pi
ng a
p
pr
oach
o
ffe
rs a ch
oi
ce of d
e
si
g
n
t
o
o
l
s for acc
om
m
odat
i
on of
u
n
c
ertain
ties
no
n
lin
earities.
And
can
avo
i
d wastefu
l
ca
n
c
ellatio
n
s
.
Howev
e
r, th
e no
t ad
ap
tiv
e back
st
ep
p
i
n
g
approach is ca
pable
of
kee
p
ing alm
o
st
all
the robustness
properties of
the m
i
s
m
a
t
ched uncer
tain
ties. Th
e no
t
adapt
i
v
e
bac
k
s
t
eppi
n
g
i
s
a ri
go
r
ous a
nd
pr
oced
u
r
e desi
g
n
m
e
t
hodol
ogy
fo
r n
onl
i
n
ea
r f
eedbac
k
c
ont
r
o
l
.
Th
e
p
r
i
n
cip
a
l id
ea o
f
th
is appro
a
ch
is to
recursiv
ely d
e
si
gn c
ont
rol
l
e
rs f
o
r m
achi
n
e t
o
r
q
u
e
const
a
nt
unc
ert
a
i
n
t
y
su
bsystem
s
in
t
h
e stru
ctur
e and
‘‘
step b
a
ck
’’
th
e f
e
ed
b
a
ck
sig
n
a
ls t
o
w
a
rd
s t
h
e co
n
t
ro
l input. Th
is app
r
o
a
ch
is
di
ffe
re
nt
f
r
om
t
h
e a
p
p
r
o
ach
of t
h
e c
o
n
v
e
n
t
i
onal
feed
ba
ck lin
earizatio
n in
th
at it can
av
o
i
d
can
cellatio
n of
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
PED
S
Vo
l. 4,
No
.
1,
Mar
c
h
2
014
:
12
–
2
3
13
u
s
efu
l
n
o
n
lin
earities in
pu
rsu
i
ng
th
e obj
ectiv
es of
stab
i
lizatio
n
and
track
ing
.
A n
o
n
lin
ear b
a
ck
st
ep
p
i
n
g
cont
rol
de
si
g
n
schem
e
i
s
devel
ope
d f
o
r t
h
e s
p
eed t
r
ac
ki
ng c
ont
rol
o
f
PM
S
M
t
h
at
has exa
c
t
m
odel
kn
ow
l
e
dg
e
.
The asym
ptoti
c
stability of t
h
e resulting cl
ose
d
loop
syste
m
is guarante
e
d accord
i
ng t
o
Lyapunov stability
th
eorem
.
The spee
d va
ri
at
i
on of t
h
e P
M
SM
i
s
wi
del
y
used
in high-perform
ance app
lication
s
. The PMSM h
a
s
very
l
a
r
g
e
po
wer
de
nsi
t
y
, hi
gh
p
o
w
er
fact
or a
n
d
hi
g
h
ef
fi
ci
ency
. I
n
a
hi
g
h
-
p
er
f
o
rm
ance c
ont
r
o
l
of
PM
S
M
,
t
h
e i
n
f
o
rm
at
i
on of
rot
o
r p
o
si
t
i
on an
d s
p
eed i
s
very
im
port
a
nt
. I
n
t
h
e spee
d co
nt
r
o
l
l
o
o
p
,
for t
h
e
fi
el
d o
r
i
e
nt
ed
co
n
t
ro
l, t
h
e coo
r
d
i
n
a
te tran
sfo
r
m
a
tio
n
h
a
s
n
eeds preci
se
ro
t
o
r
po
sitio
n
.
Ro
tor po
sition
an
d
sp
eed
can
b
e
m
easured
by
a shaft
enc
ode
r or ot
her t
y
pe o
f
sens
ors
,
i
n
ot
her case t
h
e sp
eed i
s
m
easure
d
wi
t
h
an E
n
c
ode
r
resol
v
er c
o
nne
cted to the
PMSM m
ach
ine drive.
Howe
ver, the
presence of suc
h
se
ns
ors
is not acce
ptable for
co
st, m
a
in
ten
a
n
ce an
d reliab
i
lity reaso
n
s
. Th
e co
n
c
ep
t
o
f
sen
s
o
r
less con
t
ro
l
was
propo
sed
in th
e 197
0s and
has bee
n
co
nt
i
nual
l
y
devel
o
ped f
o
r PM
S
M
rot
o
r
po
si
t
i
on a
nd s
p
eed
est
i
m
a
t
i
on. The basi
c p
r
i
n
c
i
pl
e o
f
sens
orl
e
ss co
nt
rol
i
s
t
o
ded
u
c
e
t
h
e rot
o
r spe
e
d an
d p
o
si
t
i
on usi
n
g va
ri
o
u
s
i
n
fo
rm
at
i
on
and m
eans, i
n
cl
udi
n
g
d
i
rect calcu
latio
n, p
a
ram
e
ter id
en
tificatio
n,
co
nd
itio
n es
timatio
n
,
ind
i
rect
m
easu
r
ing
an
d
so
on
. Th
e stato
r
cu
rren
ts and
vo
ltag
e
s are
g
e
nerally u
s
ed to calcu
l
ate th
e info
rm
atio
n
of speed
an
d ro
tor
po
sitio
n.
The
FPGA technology is
now used by
a
n
increas
i
n
g
nu
m
b
er
of
d
e
si
g
n
e
r
s
i
n
v
a
r
i
ou
s f
i
eld
s
of
application s
u
ch as
signal
processi
n
g
,
t
e
l
ecom
m
uni
cat
i
on,
vi
de
o
,
em
bed
d
ed
c
ont
r
o
l
sy
st
em
s, an
d el
e
c
t
r
i
cal
cont
rol system
s. This last dom
ain, i.e.
the s
t
udies
of c
ont
rol of electrical
m
ach
in
es, wil
l
b
e
p
r
esen
ted
in
th
is
pape
r [1].
Inde
ed,
t
h
ese com
pone
nts ha
ve already
been us
e
d
with s
u
ccess
in
m
a
ny diffe
rent a
pplications suc
h
as Pul
s
e
W
i
dt
h M
o
dul
at
i
o
n (P
W
M
), c
ont
r
o
l
of i
n
d
u
ct
i
o
n
m
achi
n
e dri
v
e
s
and m
u
l
t
i
m
a
chi
n
e sy
st
em
cont
ro
l
.
This is becaus
e
the FPGA-based im
ple
m
e
n
tation of cont
rollers can e
f
ficiently an
swer curre
n
t and future
challenges
of t
h
is field.
C
onsi
d
eri
ng t
h
e com
p
l
e
xi
t
y
of t
h
e
di
ve
rsi
t
y
of t
h
e electric control de
vices of the m
achines
, it is
di
ffi
c
u
l
t
t
o
defi
ne
wi
t
h
uni
ver
s
al
m
a
nner
a
g
e
neral
st
ruct
ur
e fo
r s
u
c
h
sy
st
em
s. Ho
we
ver
,
by
h
a
vi
ng a
re
fl
exi
o
n
com
p
ared t
o
t
h
e elem
ents
most c
o
mm
only enc
o
untere
d i
n
t
h
e
s
e sy
st
em
s, i
t
i
s
p
o
ssi
bl
e t
o
defi
ne a
gene
ral
structure of
an electric
cont
rol
de
vice of m
achines
which is
show i
n
Fi
gure
1:
Fi
gu
re 1.
A
r
chi
t
ect
ure of
t
h
e C
ont
r
o
l
Thi
s
pape
r pr
e
s
ent
s
t
h
e
real
i
zat
i
on of
a pl
at
fo
rm
for
not
ada
p
tative a
n
d a
d
aptative
Backstepping
cont
rol
of P
M
SM
usi
n
g FPG
A based
cont
rol
l
e
r.
Thi
s
realization i
s
especially aim
e
d for future hi
gh
p
e
rform
a
n
ce ap
p
lication
s
.
In
th
is app
r
o
a
ch
,
n
o
t
on
ly th
e arch
itectu
r
e co
rrespo
n
d
i
ng
to th
e co
n
t
ro
l algorith
m
is
studie
d
,
but also a
r
chitecture
and
the
ADC i
n
terface
, E
n
c
o
der interface
a
n
d RS232
UART arc
h
itecture [2].
2.
PMS
M
MO
DEL
SYSTE
M
In t
h
i
s
paper, we appl
y
t
h
e different
al
gori
t
h
m
s
cont
rol
on a
m
a
chi
n
e ty
pe
PM
SM
(Per
m
a
nent
M
a
gnet
Sy
nchro
n
o
u
s
M
o
t
o
r) [
3
]
,
wh
i
c
h consi
s
t
s
of t
h
ree st
at
or wi
ndi
n
g
s and a r
o
t
o
r m
a
gnet
.
Thi
s
m
o
t
o
r i
s
descri
bed
b
y
th
e fo
llo
wing
equ
a
tio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
FPGA
-
B
a
sed
I
m
pl
e
m
ent
a
t
i
o
n
N
o
nl
i
n
ear
B
a
c
k
st
eppi
ng
co
nt
rol
of
a
PM
SM
Dri
ve (
B
adre
BO
SS
OU
FI)
14
.
.
.
.
.
.
.
.
.
p
C
f
dt
d
J
C
i
L
i
L
dt
d
i
r
V
dt
d
i
r
V
r
e
sq
sq
sq
f
sd
sd
sd
sd
sq
sq
s
sq
sq
sd
sd
s
sd
(1
)
Whe
r
e
Ω
is the ro
tation
'
s sp
eed
,
p
t
h
e N
u
m
b
er of pai
r
s of
pol
es
,
J
th
e
m
o
m
e
n
t
o
f
inertia,
f
th
e
C
o
ef
fi
ci
ent
of
vi
sco
u
s f
r
i
c
t
i
on,
C
r
th
e resistiv
e to
rq
ue,
Φ
f
t
h
e fl
u
x
p
r
o
d
u
c
e
d by
t
h
e perm
anent
m
a
gnet
,
L
sd
and
L
sq
th
e d-
q ax
i
s
st
ator induct
ance,
V
sd
and
V
sq
the d-q
axi
s
stator
volta
ge,
r
s
the stator
winding re
sistance a
nd
C
e
the electromagnetic torque.
3.
NO
NLINE
A
R
BAC
K
STEP
PING
A
PPR
OA
CH
The Backstepping a
p
proach
algorithm
is c
ont
rol t
echniques that can li
near
ize a
nonl
inear syste
m
suc
h
as the PMSM
m
achine drive
i
n
th
e p
r
esen
ce of
u
n
c
ertain
ties.
Un
lik
e
o
t
h
e
r feed
b
a
ck
lin
earizatio
n
tech
n
i
qu
es, adap
tiv
e Back
step
p
i
n
g
h
a
s the flex
i
b
ility o
f
k
e
ep
ing
u
s
efu
l
n
o
n
lin
earity’s in
tact du
ri
n
g
stabilization. T
h
e esse
nce
of
Backsteppi
ng
i
s
th
e
stab
ilizati
o
n of a v
i
rtu
a
l
co
n
t
ro
l state.
Hen
c
e, it
g
e
n
e
rates
a
co
rr
esp
ond
ing er
ror
v
a
r
i
ab
le wh
ich
can be
stabilized
by
care
f
ully sele
ct
i
ng
pr
o
p
er
c
ont
rol
i
n
p
u
t
s
.
These
inputs
ca
n be determ
ined from
Lyapunov st
ability analysis [
4
]
.
It
i
s
obvi
ous t
h
at
t
h
e dy
nam
i
c
m
odel
of PM
SM
i
s
hi
ghl
y
nonl
i
n
ea
r beca
u
s
e of t
h
e co
upl
i
ng bet
w
e
e
n
t
h
e spee
d an
d
t
h
e st
at
or c
u
r
r
e
nt
s (e
quat
i
o
n
(1
)).
Acc
o
r
d
i
ng t
o
t
h
e
vect
or c
ont
rol
pri
n
ci
pl
e, t
h
e di
re
ct
axi
s
cu
rren
t id
is always fo
rced
to
b
e
zero
in
o
r
d
e
r to
orien
t
all
th
e lin
k
a
ge flu
x
in
th
e d
ax
is and
ach
ieve
m
a
xim
u
m
t
o
rq
ue
per
am
pere.
J
C
J
f
i
i
L
L
i
J
p
dt
d
L
V
p
L
i
p
L
L
i
L
r
dt
di
L
V
i
p
L
L
i
L
r
dt
di
r
sq
sd
sq
sd
sq
f
sq
sq
sq
f
sd
sq
sd
sq
sq
s
sq
sd
sd
sq
sd
sq
sd
sd
s
sd
)
)
(
(
2
3
.
.
(2
)
The vect
or
T
sq
sd
i
i
x
choice as state vector is justified by
the fact that curre
n
ts and spee
d
are m
easurable
and that the
c
ont
rol
of t
h
e i
n
stanta
ne
o
u
s t
o
r
q
ue can
be
d
one
com
f
orta
b
l
e via the c
u
r
r
e
nts
i
sd
and/
or
i
sq
.
A
n
d stator volta
ges as
co
ntr
o
l
va
ri
ables
T
sq
sd
V
V
u
.
The p
r
inci
pal ob
jective
of th
e backste
ppi
ng
contr
o
ller
is to re
gulate the spee
d of
the PMSM drive to its
refe
rence value
Ω
ref
whateve
r
external dist
urba
nces.
W
e
assum
e
that t
h
e engine
para
m
e
ters are known a
nd
inva
riant.
3.
1.
Backstepping Speed Contr
o
l
l
er
The
first ste
p
i
s
de
fine
d the
tracking e
r
rors:
ref
e
(3
)
The deri
vative of
(3)
is:
r
sq
sd
sq
sd
sq
f
ref
ref
C
f
i
i
L
L
i
p
J
dt
de
e
)
)
(
(
2
3
1
(4
)
We defi
ne
the
following quadratic
function:
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I
S
SN:
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08
I
J
PEDS Vo
l.
4,
No
.
1,
Mar
c
h
2
014
:
12
–
2
3
15
2
1
2
1
e
V
(5
)
Its de
rivative
a
l
on
g the
s
o
lution
o
f
(5
),
is gi
ven
by
:
r
sq
sd
sq
sd
sq
f
ref
C
f
i
i
L
L
i
p
J
e
e
e
V
)
)
(
(
2
3
1
1
(6
)
Usin
g the B
a
c
k
step
pin
g
design m
e
thod
, we
con
s
ider t
h
e
d
-
q a
x
es c
u
r
r
ent
s
com
pone
nts
i
sd
and
i
sq
as
ou
r virt
ual co
ntr
o
l elem
ents and s
p
ecify
its desire
d
be
ha
vio
r
, w
h
ich a
r
e called stabilizing f
u
nction
in the
back
step
pin
g
d
e
sign
term
inology
a
s
f
o
llo
ws:
)
.
.
(
3
2
0
e
k
J
C
f
p
i
i
r
f
sqref
sdref
(7
)
Wi
t
h
k
Ω
is a
positive constant
Substituting (7) in (6) the
deri
vative
of
V
1
:
0
2
1
e
k
V
(8
)
3.
2.
Backstepping Current Controller
We ha
ve the asym
ptotic sta
b
ility of the origin
of the syste
m
(1).
W
e
defi
ned current following
errors:
sq
sqref
q
sdref
sd
sdref
d
i
i
e
i
with
i
i
e
0
(9
)
Their
dynam
i
cs can be
written:
sd
sd
sq
f
sd
sq
sd
sq
sq
s
r
f
sq
sqref
q
sd
sd
sq
sd
sq
sd
sd
s
sd
sdref
d
L
V
L
p
i
p
L
L
i
L
r
e
k
J
C
f
p
i
i
e
L
V
i
p
L
L
i
L
r
i
i
e
.
)
.
.
(
3
2
.
(1
0)
To analyze the
stability of
this syste
m
we
propose the following Lyapunov
f
u
nction:
)
(
2
1
2
2
2
2
q
d
e
e
e
V
(1
1)
Its de
rivative
a
l
on
g the
tra
j
ectories
(
8
),
(
9
)
a
n
d
(
1
0) is:
]
2
3
)
)
(
2
3
2
3
.(
3
)
(
2
[
]
)
(
2
3
[
2
2
2
2
sq
f
sd
sq
sd
sq
sq
s
sq
sq
f
sq
d
sq
sd
q
f
f
q
q
q
sq
sq
sd
sq
sd
sq
sd
s
sd
sd
d
d
d
q
q
d
d
q
q
d
d
L
i
L
L
i
L
r
L
V
e
J
p
e
k
i
e
L
L
J
p
e
J
p
p
f
J
k
e
k
e
i
e
L
L
J
p
i
L
L
L
r
L
V
e
k
e
e
k
e
k
e
k
e
e
e
e
e
e
V
(1
2)
Th
e exp
r
essio
n
(1
2)
f
oun
d abo
v
e
r
e
quires t
h
e followi
ng control laws:
q
sq
q
f
sd
sd
sq
s
sq
f
sq
d
sq
sd
q
f
f
sq
sq
sq
sq
sd
sd
sq
sq
sd
s
d
sd
d
sd
e
L
k
i
L
i
r
e
J
L
p
e
k
i
e
L
L
J
p
e
J
p
p
f
J
k
L
V
i
e
L
L
J
pL
i
L
i
r
e
L
k
V
2
3
)
(
2
3
2
3
3
)
(
2
)
(
2
3
(1
3)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
FPGA
-
B
a
sed
I
m
ple
m
ent
a
tion
N
o
nlinear
B
a
c
k
steppi
ng
co
nt
rol
of a
PM
SM
Drive (
B
adre
BO
SS
OU
FI)
16
Wit
h
this
ch
oic
e
the
deri
vativ
es o
f
(1
3)
bec
o
m
e
:
0
2
q
q
d
d
e
k
e
k
e
k
V
(1
4)
4.
NO
NLINE
A
R
A
DAPT
A
TI
VE BA
CK
STEPPING
A
P
P
R
O
A
C
H
CO
NTROL
4.
1.
Principle
In t
h
e pre
v
ious section, the
cont
rol laws a
r
e
de
velo
pe
d
un
de
r the ass
u
m
p
tion that th
e
m
achine
param
e
ters are kn
ow
n an
d in
varia
n
ts. This
assum
p
tion is
not always true. In fact
, the
flow created by the
m
a
gnet va
ries with increasing tem
p
erature
and
with the
fiel
ds created
by
the stator currents. Stator resistance
also va
ries
with tem
p
erature.
Also,
the c
h
a
n
ge in
operating conditions is
i
m
pl
icitl
y load
torque a
n
d
hence the
coefficient of
friction and ine
r
tia. Ad
a
p
tive
B
ackstep
pin
g
versi
on take
s in
to account the variations of these
param
e
ters.
In
e
q
uation
(
7
)
,
we d
o
not kn
ow ex
actly the
value
of the l
o
ad torque
C
r
, it will be re
placed
by its
esti
m
a
te
r
C
ˆ
.
)
.
.
ˆ
.
(
3
2
ˆ
e
k
J
C
f
p
i
r
f
sqref
(1
5)
From
(1
3
)
a
n
d
(1
5)
,
we
ded
u
c
e
the
dy
nam
i
cs o
f
the
spee
d e
r
ro
r as
f
o
llow
s
:
e
k
J
i
e
L
L
p
e
p
C
J
dt
de
sq
d
sq
sd
q
f
r
.
.
)
(
2
3
2
3
~
1
(1
6)
Wi
t
h
r
r
r
C
C
C
ˆ
~
is the e
r
ror of the
estim
ated loa
d
t
o
rque.
The
Dynam
i
c errors and
dire
ct
currents qua
d
ratic
write:
sq
sd
sq
sd
sd
s
sd
sd
sd
i
L
L
i
L
R
L
V
dt
di
dt
de
(1
7)
r
f
sq
f
sd
sq
sd
sd
sq
s
sq
sd
sq
d
sq
sd
q
f
f
sq
sqref
q
C
k
J
f
p
L
i
L
L
i
L
R
L
V
e
k
i
e
L
L
J
p
e
J
p
p
f
J
k
dt
di
dt
di
dt
de
~
)
(
3
2
.
)
(
2
3
2
3
3
)
(
2
(1
8)
To analyze the
stability of
this syste
m
we
propose the following Lyapunov
f
u
nction:
3
2
2
2
1
2
2
2
2
2
~
~
~
2
1
f
s
r
q
d
R
C
e
e
e
V
(1
9)
Its de
rivative
a
l
on
g the
tra
j
ectories
(
1
6
)
,
(
1
7
)
an
d
(1
8)
is:
q
sq
q
f
q
f
f
sq
q
sq
sd
d
sd
s
s
f
q
f
q
r
r
sq
f
sd
sq
sd
sq
sq
s
sq
sq
f
sq
d
sq
sd
q
f
f
q
q
q
sq
sq
sd
sq
sd
sq
sd
s
sd
sd
d
d
d
q
q
d
d
f
f
s
s
r
r
q
q
d
d
e
L
e
J
f
J
k
e
e
J
p
i
e
L
i
e
L
R
R
J
e
pJ
fe
p
e
k
C
C
L
i
L
L
i
L
R
L
V
e
J
p
e
k
i
e
L
L
J
p
e
J
p
p
f
J
k
e
k
e
i
e
L
L
J
p
i
L
L
L
R
L
V
e
k
e
e
k
e
k
e
k
R
R
C
C
e
e
e
e
e
e
V
1
ˆ
2
3
~
1
~
1
1
~
1
~
ˆ
3
2
ˆ
3
2
~
1
~
]
ˆ
2
ˆ
3
)
)
(
2
3
2
ˆ
3
.(
ˆ
3
)
(
2
[
)
(
2
3
~
~
1
~
~
1
~
~
1
2
3
2
1
2
2
2
3
2
1
2
(2
0)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
PEDS Vo
l.
4,
No
.
1,
Mar
c
h
2
014
:
12
–
2
3
17
Th
e exp
r
essio
n
(1
6)
f
oun
d abo
v
e
r
e
quires t
h
e followi
ng control laws:
q
sq
q
f
sd
sd
sq
s
sq
f
sq
d
sq
sd
q
f
f
sq
sq
sq
sq
sd
sd
sq
sq
sd
s
d
sd
d
sd
e
L
k
i
L
i
R
e
J
L
p
e
k
i
e
L
L
J
p
e
J
p
p
f
J
k
L
V
i
e
L
L
J
pL
i
L
i
R
e
L
k
V
ˆ
ˆ
2
ˆ
3
)
(
2
3
2
ˆ
3
3
)
(
2
)
(
2
3
ˆ
(2
1)
There
f
ore t
h
e
dynam
i
cs of the Lyapunov
function can be si
m
p
lified as fol
l
ows:
q
sq
q
f
q
f
f
sq
q
sq
sd
d
sd
s
s
f
q
f
q
r
r
q
q
d
d
e
L
e
J
f
J
k
e
e
J
p
i
e
L
i
e
L
R
R
J
e
pJ
fe
p
e
k
C
C
e
k
e
k
e
k
V
1
ˆ
2
3
~
1
~
1
1
~
1
~
ˆ
3
2
ˆ
3
2
~
1
~
2
3
2
1
2
2
2
2
(2
2)
Hence
the a
d
a
p
tation la
ws as follows:
J
e
pJ
fe
p
e
k
C
f
q
f
q
r
ˆ
3
2
ˆ
3
2
~
1
(2
3)
sq
q
sq
sd
d
sd
s
i
e
L
i
e
L
R
1
1
~
2
(2
4)
q
sq
q
f
q
f
e
L
e
J
f
J
k
e
e
J
p
1
ˆ
2
3
~
2
3
(2
5)
Wit
h
this
ch
oic
e
, the e
x
pres
sion
(
1
9)
bec
o
m
e
s:
0
2
2
2
2
q
q
d
d
e
k
e
k
e
k
V
(2
6)
So t
h
e system
i
s
gl
obally asym
ptotical
ly sta
b
le
in t
h
e
pre
s
e
n
ce
of pa
ram
e
t
r
ic uncertai
n
ties.
4.
2.
Simulati
on and
Test Perfor
mance
Figu
re
2.
Sy
stem
config
uratio
n
of
ada
p
tive B
ackstep
pin
g
C
ont
rol
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
FPGA
-
B
a
sed
I
m
ple
m
ent
a
tion
N
o
nlinear
B
a
c
k
steppi
ng
co
nt
rol
of a
PM
SM
Drive (
B
adre
BO
SS
OU
FI)
18
The follo
win
g
results
a
r
e obta
i
ned by
c
h
o
o
si
ng
the f
o
llowi
n
g
values:
Gains of the
control law:
15
.
0
k
,
01
.
0
d
k
,
01
.
0
q
k
.
Ada
p
tation gains:
15
.
0
1
,
01
.
0
2
,
015
.
0
3
.
Fol
l
o
w
of the
t
r
ajec
tor
y
(a)
(b
)
(c)
(d
)
Figu
re
3.
Test
per
f
o
r
m
a
nce of
the a
d
apti
ve c
ont
roller
for
t
r
aj
ectory
trac
king, (a) Spee
d response
tra
j
ectory (b)
Error S
p
eed re
sponse
(c
)
d-q
axis c
u
rrent
wi
thout unce
rtainties (d) a
b
c a
x
is curre
nt
Distur
bance r
e
jection
(a)
(b
)
(c)
Figu
re
4.
Test
per
f
o
r
m
a
nce of
the a
d
apti
ve c
ont
roller
f
o
r
re
jecting
dist
ur
b
a
nce to
r
que
loa
d
a
pplied
at t =
0.
3s.
(a)
Sp
eed
r
e
spon
se tr
aj
ecto
r
y
(
b
) d-
q ax
is cur
r
en
t
without unc
e
rtainties (c
) E
l
ectrom
a
gnetic Torque
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
PEDS Vo
l.
4,
No
.
1,
Mar
c
h
2
014
:
12
–
2
3
19
Par
a
metric uncertainties
(a)
(b
)
(c)
(d
)
Figu
re
5.
Test
per
f
o
r
m
a
nce of
the a
d
apti
ve c
ont
roller
f
o
llo
win
g
a c
h
a
nge
in R
s
(a)
Spee
d
response tra
j
ectory (b)
d-q axis c
u
rre
n
t
without uncertainties
(c) Electrom
a
gnet
i
c Torque
(d)
current i
sa
(a)
(b
)
Figu
re
6.
Test
per
f
o
r
m
a
nce of
the a
d
apti
ve c
ont
roller
f
o
llo
win
g
a c
h
a
nge
in
Φ
f
(
a
)
Sp
eed
r
e
spo
n
s
e
tr
aj
ector
y
(b
)
d-
q
axis c
u
rre
n
t without uncertainties
(a)
(b
)
Figu
re
7.
Test
per
f
o
r
m
a
nce of
the a
d
apti
ve c
ont
roller
f
o
llo
win
g
a c
h
a
nge
in L
sd
a
n
d
Ls
q,
(a)
S
p
eed
res
p
ons
e
tr
aj
ector
y (b)
d-
q ax
is cu
rre
n
t without unce
r
t
a
inties
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
FPGA
-
B
a
sed
I
m
ple
m
ent
a
tion
N
o
nlinear
B
a
c
k
steppi
ng
co
nt
rol
of a
PM
SM
Drive (
B
adre
BO
SS
OU
FI)
20
5.
FPGA-BASE
D IMPLEME
N
TATION
OF
AN
R
O
BU
ST
BA
CKSTEPPING C
O
NTR
O
L
SY
STEM
5.
1.
Devel
o
pme
n
t
of
the
Implem
ent
a
tio
n
There
are
seve
ral m
a
nufacturers
of
FP
GA
c
o
m
pone
nts s
u
c
h
: Actel,
Xilinx a
n
d Altera
…etc.
T
h
ese
m
a
nufact
ure
r
s use dif
f
ere
n
t techn
o
lo
gies f
o
r
the im
plem
entation of
FP
GA
s
.
T
h
ese technologies are attractive
because they provide rec
o
nfi
g
ura
b
le st
ructure that is the
m
o
st interestin
g because they allow great flexibilit
y
in desi
gn
. N
o
waday
s
,
FP
GA
s
offer the
possibility to use dedicate
d
bl
oc
ks such as
RA
Ms
, m
u
ltipliers wire
d
interfaces
PCI
and
CPU
c
o
res
.
T
h
e a
r
chitecture
desi
gni
n
g
was
d
one
usi
n
g
with
CAD
t
o
ols.
T
h
e
desc
ri
ption
is
m
a
de g
r
ap
hica
lly
or via a
ha
r
d
wa
re
desc
ri
pt
ion la
ng
ua
ge
h
i
gh le
vel, als
o
called
HD
L
(Hardware
Desc
ri
ption
Lan
gua
ge)
.
Is
com
m
only
u
s
ed
la
ng
ua
ge
VH
D
L
a
n
d Ve
rilog.
These
t
w
o langua
ges are sta
n
dardized a
n
d
provide the de
scription with
diffe
r
e
n
t levels, and es
pecially
the advanta
g
e of bei
ng
po
rtable and c
o
m
p
atible
with all
FP
GA
technologies previo
usly introduced [
7
]
.
The sim
u
lation proce
d
ure
be
gins
by ve
rifying the
fu
nctiona
lity of the c
o
ntrol al
gorithm
by trailding
a functional m
odel using Sim
u
link’s Sy
stem
Gene
rator
fo
r
Xilinx bl
ock
s
.
Fo
r t
h
is applic
ation, the functional
m
odel consists in a Si
m
u
li
nk tim
eis dis
c
retired m
odel of the
N
o
a
d
a
p
tative B
a
c
k
steppi
ng
algor
ith
m
associated
with a voltage i
n
ve
rter a
n
d PMS
M
m
odel.
The Fi
gure
8
sum
m
a
rizes the differe
n
t step
s
of
p
r
o
g
r
am
m
i
ng
an
FP
GA
.
The sy
nthesize
r ge
ne
rate
d
with
CAD
tool
s first one
Net
list which describes the c
o
nn
ectivity of the architecture
.
The
n
the place
m
e
nt-
routing optim
a
lly place co
m
p
one
nts and pe
rform
s
a
ll
the
routing bet
w
ee
n differe
n
t logi
c.
T
h
ese two st
eps are
use
d
to gene
ra
te a configurat
ion file to
be
downloa
d
ed into the
m
e
m
o
ry of the
FP
GA
.
This file is ca
lled
bitstream
. It can
be
directly loaded into
FP
GA
f
r
om
a h
o
st c
o
m
puter.
Figu
re
8.
Pr
o
g
r
a
m
m
i
ng F
P
G
A
de
visees
In this
work an
FPGA XC
3S500E
Spa
r
tan
3
E
fr
om
Xilinx is
used
. T
h
is
FP
GA
con
t
ain
s
40
0,
000
log
i
c
gates and includes an internal oscillator whic
h issue
r
a 50M
Hz fre
q
uency c
l
ock. T
h
e m
a
p
is com
posed from
a
m
a
trix of
5
3
7
6
slices linke
d t
oget
h
er
by
pr
o
g
ram
m
able connectio
ns
.
5.
2.
Simulati
on Pr
ocedure
Figu
re
9.
F
unct
i
onal M
odel
fo
r N
o
ada
p
tativ
e B
ackstep
pi
ng
C
o
ntroller
f
r
o
m
SYSTEM
G
E
NER
A
TOR
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-87
08
I
J
PEDS Vo
l.
4,
No
.
1,
Mar
c
h
2
014
:
12
–
2
3
21
The sim
u
lation proce
d
ure
be
gins
by ve
rifying the
fu
nctiona
lity of the c
o
ntrol al
gorithm
by trailding
a functional m
odel using Sim
u
link’s Sy
stem
Gene
rator
fo
r
Xilinx bl
ock
s
.
Fo
r t
h
is applic
ation, the functional
m
odel consists in a Si
m
u
link ti
m
e
discretired m
odel of
the
No ada
p
tative
Backste
ppi
ng algorithm
associated
with a volta
ge
inverter a
n
d PM
SM
m
odel.
The Fig
u
re
8
sho
w
s in det
a
il the pro
g
ra
m
m
i
ng of the cont
rol
shown in Figure 9 in the SYSTEM GENER
A
TOR e
nvi
ronm
ent from
Xili
nx, we
will i
m
plem
ent i
t
later
in the
m
e
m
o
ry
of t
h
e
FPG
A
fo
r t
h
e s
i
m
u
lation o
f
P
M
SM
.
The second step of the si
m
u
lation is the determ
ination o
f
the suitable sam
p
ling perio
d
an
d fix
e
d
poi
nt f
o
rm
at.
5.
3.
Pro
t
ot
yping pla
t
fo
rm
To test the
FP
GA
base
d c
ont
roller,
a
pr
otot
y
p
ing
platf
o
rm
fo
r th
e co
ntr
o
l of a
Perm
ane
n
t m
a
gnet
Syn
c
hro
nou
s Mach
in
e was
assem
b
led
(
F
igur
e 1
0
)
.
Figu
re 1
0
. Pr
ototy
p
in
g platfo
r
m
control
6.
E
X
PERI
MEN
T
AL RES
U
L
T
S
The im
plem
entation of the in
direct co
ntrol
by
sliding m
ode on
FP
GA
de
vices is charac
terized by a
red
u
ce
d ope
rat
i
on
tim
e.
Th
e Figur
e 11
sho
w
n the ex
perim
e
ntal results of I
n
di
rec
t
Sliding Mode
PM
SM
with the FPG
A
platfo
rm
are sh
ow
n.
U
p
d
a
te fr
eque
ncy
f
o
r thi
s
im
plem
enta
ti
on is 20
kHz
.
All results
were extracte
d
from
the
FPGA
by the
ChipSc
ope tool of Xilinx.
(a)
(b
)
(c)
F
i
g
u
r
e
11
.
(
a
)
S
t
a
t
o
r
cu
rr
en
t
l
o
c
u
s
for
IS
MC,
(b
) a
b
c-
ax
i
s
cu
rr
e
n
t
in
t
h
e PMS
M
,
(
c
)
d
-
axi
s
a
n
d
q
-
ax
i
s
cu
rr
e
n
t
in the
PMSM
In
Fig
u
re
1
1
.a
the e
x
p
e
rim
e
ntal res
u
lts
No Ada
p
tative Backstepping Control of
PMS
M
with
t
h
e
FPGA
platform
are shows t
h
e e
vol
ution of the
stator curre
n
t
i
sd
whic
h shows that t
h
e output follows the
refe
rence
i
sdref
and
i
sq
. The
Figu
re 1
1
.
b
shows the stator c
u
rrent
i
sa
and
i
sb
. U
pdat
e
freq
ue
ncy
f
o
r this
i
m
ple
m
entatio
n
is 20 kHz
.
LOAD
PMSM
F
P
GA
In
verte
r
(
IGBT
)
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