In
te
r
n
ation
a
l Jou
rn
al
o
f Po
we
r
Elec
tron
ic
s an
d
D
r
ive S
y
stem
(IJ
PED
S
)
V
o
l.
10, N
o.
3, S
ep 2019,
pp.
1
2
2
9
~1
2
4
3
ISSN: 2088-
8694,
DOI
:
10.11591
/ijpeds.
v10.
i
3.pp1229-1243
1
2
29
Jou
rn
a
l
h
o
me
pa
ge
:
ht
tp:
//i
a
e
score
.
com
/
j
o
u
r
na
l
s
/
i
n
d
e
x
.
p
hp/IJ
PED
S
A compara
t
ive study of no
n
li
near circl
e criterion based
observer and H
∞ observe
r
for induction m
otor drive
Far
i
d
Ber
rez
ze
k
1
,
W
a
f
a
B
o
u
rb
i
a
2
,
Bachir
Ben
sa
ker
3
1
F
acu
lty of. S
c
i
en
ce and
Tech
no
l
o
g
y
,
LEE
R Lab
, S
o
u
k
Ahras Univ
e
rsity,
A
l
geri
a.
2,
3
LSE
L
ab
, Bad
j
i
Mok
h
t
ar Un
i
v
e
rsi
t
y
,
A
l
g
éri
a
.
Art
i
cl
e In
fo
ABSTRACT
A
r
tic
le hist
o
r
y
:
Re
ce
i
v
e
d
Jan
7
, 2019
Re
vise
d Mar
1,
201
9
Ac
ce
p
t
ed
M
a
y
2
8
,
2
019
Th
is
p
aper
d
e
a
ls
w
it
h
a
co
m
p
arati
v
e
st
ud
y
o
f
c
ircle
crit
erion
b
ased
n
on
li
near
ob
serv
er
an
d
H
∞
o
bse
r
v
e
r
fo
r
in
du
c
t
io
n
mo
to
r
(I
M)
d
rive
.
The
a
d
v
a
nta
g
e
of
th
e
circl
e
c
riterion
a
pp
roach
f
o
r
n
o
n
l
i
near
o
bs
erver
d
e
sig
n
i
s
th
a
t
i
t
d
i
r
e
c
t
l
y
h
a
nd
le
s
the
no
nlin
e
a
r
i
t
ie
s
o
f
t
h
e
s
ystem
w
ith
l
ess
restri
c
tion
c
o
ndi
ti
on
s
in
c
o
ntra
st
o
f
the
othe
r
me
th
o
d
s
wh
ich
at
tempt
to
e
limi
n
ate
them.
Ho
w
e
ve
r
t
h
e
H
∞
o
b
s
erv
e
r
gu
aranteed
t
he
s
tab
i
lity
t
akin
g
into
accou
n
t
di
st
u
r
ba
n
ce
an
d
no
is
e
at
tenu
ati
o
n
.
L
in
ear
m
atri
x
in
equal
i
t
y
(
LMI)
opt
im
i
zatio
n
app
r
oach
i
s
us
ed
t
o
co
m
pute
th
e
g
a
ins
m
a
trices
f
o
r
t
he
t
wo
o
b
s
erv
e
rs.
The
s
im
u
l
ati
o
n
results
s
how
t
h
e
supe
ri
ority
o
f
H
∞
o
bs
e
r
ver
i
n
t
he
s
ense
t
hat
it
can
a
chi
e
ve
con
v
erg
e
n
ce
t
o
t
he
t
ru
e
state,
d
es
pi
te
t
he
nonl
in
e
a
rity
o
f
mod
e
l
and
the
pres
ence o
f
dist
u
rban
ce.
K
eyw
ord
s
:
C
i
rc
le
c
r
i
te
ri
on
H
∞
ob
se
r
v
er
Ind
u
ct
i
on m
o
to
r
Li
n
e
ar m
atri
x
ineq
ual
i
t
y
L
y
ap
un
ov
st
ab
il
ity
No
nlinear
o
b
s
erver
Co
pyri
gh
t © 2
019 In
stit
u
t
e
of Advanced
En
gi
neeri
n
g
an
d
S
c
ien
ce.
All
rights
res
e
rv
ed.
Corres
pon
d
i
n
g
Au
th
or:
Fa
rid Berr
ezz
ek,
D
e
pa
rtem
ent o
f
E
lectr
i
c
a
l E
n
gi
nee
r
in
g,
Mo
ham
e
d C
h
e
r
if Messaa
d
ia
-S
ouk
ahra
s
U
n
ivers
i
t
y
, LEE
R,
S
ouk-A
h
ras,
A
lger
ia
, 410
0
0
.
Em
ail:
f.be
rre
zze
k@
u
n
iv-
s
o
u
k
a
h
ras.dz
1.
I
N
TR
OD
U
C
TI
O
N
F
o
r
a
lon
g
t
i
m
e,
t
he
i
n
duc
t
i
o
n
m
o
t
or
i
s
co
n
s
ide
r
ed
a
s
a
pri
n
ci
p
a
l
w
or
kh
orse
i
n
t
h
e
ind
u
s
t
ry
d
ue
t
o
its
rob
u
st
n
e
ss,
h
igh
rel
i
a
b
i
l
i
t
y
,
re
lat
i
ve
ly
l
ow
c
ost
,
m
odes
t
m
a
i
nt
e
n
a
n
ce
re
qu
i
r
em
ents
a
nd
e
f
f
icie
nc
y
[1,
2]
.
H
o
w
e
ve
r,
i
nd
u
c
t
i
o
n
m
otor
i
s
als
o
k
now
n
as
a
c
om
ple
x
n
o
n
l
i
ne
ar
s
ys
tem
,
i
n
w
h
ic
h
tim
e-
varyi
ng
par
a
m
e
ters
e
n
t
a
il
a
dd
it
io
na
l
di
ff
i
c
ul
ti
e
s
f
o
r
m
a
c
h
i
n
e
c
o
n
t
r
ol
,
c
o
ndi
t
i
o
n
s
m
oni
t
o
ri
n
g
a
n
d
f
au
l
t
d
ia
gn
ost
i
c
p
u
rpo
s
es
[
1]
.
The
ma
in
p
ro
blem
w
i
t
h
i
nduc
t
i
on
mo
t
o
r
i
n
d
u
s
t
ria
l
a
p
p
l
i
c
a
t
i
o
ns
i
s
th
at
o
nly
a
f
e
w
st
a
t
e
v
a
ri
abl
e
s
o
f
t
h
e
ma
chine
are
a
v
ail
a
b
l
e
for o
n
-l
ine
me
asure
m
e
n
ts.
This is d
u
e
t
o
tech
n
i
ca
l
an
d/
or
econ
o
m
i
c
a
l c
o
n
s
t
r
ai
n
t
s.
I
n
o
rde
r
t
o
per
f
orm
adva
nce
d
c
on
t
r
ol
t
ec
h
n
i
que
s,
c
ond
it
io
n
s
m
on
i
t
o
ri
ng
a
n
d
f
a
ul
ts
d
i
a
gno
si
s
th
e
r
e
i
s
a
gre
a
t
nee
d
o
f
a
re
l
i
a
b
le
a
nd
acc
urate
e
s
tim
ati
on
of
t
he
k
e
y
u
nme
a
sura
bl
e
sta
t
e
va
ria
b
le
s
o
f
t
he
m
ac
h
i
ne
.
In
th
i
s
c
on
t
e
xt
,
t
h
e
obse
r
ver
des
i
g
n
t
he
ory
see
m
s
to
b
e
an
i
d
e
al
s
o
l
uti
on.
O
ver
t
h
e
last
t
w
o
d
ec
a
d
es,
no
n
l
i
n
e
a
r
obs
erve
r
des
i
g
n
p
r
o
b
l
em
h
as
r
e
ceive
d
muc
h
a
t
t
en
t
i
o
n
i
n
t
h
e
l
ite
ra
ture
.
S
e
ve
ral
a
t
tem
p
ts
h
ave
be
e
n
m
ad
e
fo
r
part
icu
l
ar
c
lass
es
o
f
no
n
l
i
n
ea
r
syste
m
s.
E
xist
in
g
a
pproa
c
h
es
c
a
n
be
r
o
u
g
h
l
y
clas
sifie
d
a
s
fol
l
ow
s
:
N
o
n
li
ne
ar
st
a
t
e
l
i
n
ear
i
z
a
t
i
on
ap
proa
c
h
es,
hi
g
h
-
g
a
i
n
obse
r
ver
s
,
geom
etric
al
gori
t
h
m
s,
v
a
r
i
a
bl
e
st
r
u
ctu
r
e
d
e
si
gn
proce
dures,
and
alge
bra
i
c te
c
h
n
i
q
u
e
s
[3-
6].
Th
e
p
r
ob
l
e
m
wi
th
s
en
so
rl
e
s
s
i
n
du
c
t
ion
moto
r
in
du
st
ri
a
l
a
pp
l
i
c
a
t
i
o
ns
i
s
w
h
ic
h
a
p
proac
h
t
ha
t
pro
v
i
d
e
the m
o
s
t
ac
c
ura
t
e a
nd re
lia
ble
estima
t
i
o
n
o
f
unm
ea
sur
a
ble
sta
t
e
v
a
r
i
a
b
l
es o
f
t
h
e ma
ch
i
n
e
sy
st
em.
I
n
t
his
pa
per
w
e
f
oc
us
our
a
tte
nti
on
o
n
t
h
e
p
erforma
n
ce
c
o
m
pa
ri
so
n
of
t
h
e
s
o
c
a
lle
d
c
i
rc
l
e
c
ri
ter
i
o
n
appr
oa
ch
a
n
d
H
∞
o
b
s
e
r
v
e
r
d
e
sig
n
fo
r
i
nduc
t
i
o
n
mo
to
r
(I
M
)
s
y
s
te
m.
T
h
e
a
dv
a
n
t
a
g
e
o
f
t
h
e
circ
l
e
c
riter
i
on
appr
oa
ch
i
s
th
e
dire
c
t
h
a
n
d
l
ing
t
h
e
sys
t
e
m
n
o
n
l
i
n
ear
it
ies
by
e
x
pl
o
i
t
i
ng
the
i
r
pr
opert
ies.
T
hi
s
ap
proa
c
h
i
s
les
s
restric
t
i
v
e
com
p
are
d
t
o t
h
e
o
t
her
m
e
th
ods
a
s
li
n
e
ariz
a
t
i
o
n
a
ppr
oa
ch,
h
i
g
h
g
a
i
n obser
ve
r
a
nd
va
ria
b
le
s
tr
uc
t
u
re
des
i
g
n
p
r
o
ce
d
u
res
w
itc
h
a
t
t
e
m
p
t
t
o
e
li
m
i
nate
t
hem
[5,
6].
The
H
∞
obs
erve
r
is
u
se
d
t
o
guara
nt
i
e
s
t
h
e
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
Int J
Pow
El
ec &
D
ri
S
yst
,
Vol
.
10,
N
o.
3
, Se
p
2
019 : 1229 –
1
2
43
1
230
ro
bu
stn
e
ss
ag
ai
ns
t
no
nl
in
e
a
ri
ti
es
a
ssu
me
d
to
b
e
e
q
u
i
v
a
l
e
nt
t
o
s
om
e
unc
e
r
t
a
in
tie
s
d
u
e
to
i
nter
na
l
a
n
d
e
x
t
r
e
r
nal
di
st
urba
nces
a
nd m
e
a
s
urem
ent n
o
i
ses
[7-12
]
.
To m
ake a c
o
m
p
ara
t
i
v
e s
t
ud
y be
twee
n t
h
e circ
l
e
c
ri
t
e
ri
on ba
se
d
o
b
ser
v
er
and
H
∞
observer
d
esign for
in
duc
t
i
on
m
o
tor
deri
ve,
w
e
u
se
o
ne
o
f
the
sta
n
d
a
rd
m
o
d
e
l
o
f
a
n
in
du
ct
i
o
n
mo
to
r.
T
hi
s
ty
p
e
o
f
n
onl
in
ea
r
mode
l
is
g
e
n
e
r
all
y
u
se
d
fo
r
perfor
ming
no
n
l
i
n
ear
c
ont
rol
stra
te
g
i
es,
con
d
i
t
i
ons
m
on
i
t
or
in
g
an
d
fau
lts
dia
g
no
si
s
o
f
e
le
ctric
ind
u
c
t
i
o
n m
a
c
h
ine s
y
st
em
s.
The
pa
per
i
s
organize
d
a
s
follows
:
In
t
h
e
s
eco
nd
a
nd
t
hi
rd
s
e
c
t
io
n w
e
prese
nt
t
he
t
he
o
r
y
of no
n
line
a
r
obs
erve
r
and
t
h
e
no
nl
i
n
ea
r
o
b
serve
r
b
ase
d
c
ircle
cri
t
eri
o
n
re
sp
ect
ive
l
y.
T
he
r
o
bus
t
H
∞
o
b
s
e
r
v
e
r
i
s
p
r
e
s
e
n
t
e
d
in
t
he
f
o
u
rt
h
s
e
ctio
n.
I
n
the
fift
h
se
ct
i
o
n
w
e
p
rese
n
t
t
he
c
ons
ider
ed
n
on
li
nea
r
i
nduc
tio
n
m
o
t
o
r
m
o
del.
F
inl
a
y,
w
e
p
rese
nt
s
i
m
ula
t
ion
re
su
lts
a
nd
com
m
e
n
t
s
.
A
concl
u
s
i
on
end
s
t
h
e
p
ap
er.
2.
T
H
E NONLIN
E
AR
L
U
E
N
BE
R
G
ER OBSERVER DESIGN
We
r
eca
ll
t
h
at
a
n
o
b
serve
r
i
s
a
dy
na
mica
l
sy
stem
w
h
i
c
h
u
se
s
t
h
e
a
va
i
l
a
b
l
e
i
n
put-
o
u
t
pu
t
data
t
o
re
co
n
s
t
r
u
c
t
th
e
u
n
m
ea
su
rabl
e
sy
st
e
m
s
t
a
t
e
v
ari
a
bl
es.
It
i
s
a
“s
oft
se
n
s
or
”
tha
t
p
la
ys
a
n
im
por
tan
t
r
ol
e
no
t
o
n
ly
in
s
e
n
s
o
rle
s
s
c
o
n
t
ro
l
t
e
c
h
ni
q
u
es
b
ut
a
ls
o
in
c
on
di
t
i
o
n
s
mon
i
tor
in
g,
f
a
u
l
t
di
a
g
n
o
sis,
p
re
di
c
t
i
v
e
m
a
i
n
te
nanc
e
and
fa
u
lt t
o
lera
nt
c
o
n
t
ro
l tec
h
niq
u
e
s
[
1,
2
].
In
t
hi
s
p
a
p
e
r,
w
e
con
s
i
d
e
r
a
c
l
a
ss
o
f
no
nl
i
n
ea
r
sy
st
ems
th
a
t
c
a
n
be
d
ecom
p
ose
d
i
n
line
a
r
a
n
d
nonl
inear
parts
as
:
))
(
),
(
(
)
(
)
(
t
u
t
x
F
t
Ax
t
x
(
1
)
)
(
)
(
t
Cx
t
y
(
2
)
Where
)
(
t
x
stan
d fo
r t
h
e
sys
t
em
st
a
te varia
b
l
e,
)
(
t
u
for system
in
pu
t a
n
d
)
(
t
y
f
o
r
the system
o
utput.
The
fu
nc
ti
on
))
(
),
(
(
t
u
t
x
F
r
e
p
resents
t
h
e
system
n
on
l
i
ne
ar
i
t
ies.
F
or
t
h
i
s
type
o
f
n
onl
i
n
e
a
r
s
y
s
t
e
m
,
a
g
e
n
e
r
a
l
no
n
l
i
n
ea
r
ob
se
rver
(Lue
nberg
er
) e
xpressi
o
n
is a
s
f
o
l
l
o
w
s
:
))
(
ˆ
)
(
(
)
,
ˆ
(
)
(
ˆ
)
(
ˆ
t
y
t
y
L
u
x
F
t
x
A
t
x
(
3
)
)
(
ˆ
)
(
ˆ
t
x
C
t
y
(
4
)
Where
)
(
ˆ
t
x
s
tan
d
s
for
t
h
e
es
ti
m
a
te
d
s
t
a
t
e.
W
e
assume
t
hat
t
h
e
non
li
nea
r
f
u
n
ct
ion
))
(
),
(
(
t
u
t
x
F
i
s
loca
l
l
y L
i
p
s
ch
i
t
z
wit
h
r
espec
t
t
o
t
h
e sta
t
e
vari
a
b
l
e
)
(
t
x
.
)
(
ˆ
)
(
))
(
),
(
ˆ
(
))
(
),
(
(
/
),
(
ˆ
),
(
t
x
t
x
t
u
t
x
F
t
u
t
x
F
t
x
t
x
Wi
t
h
0
as the
L
i
p
sc
hitz
co
n
st
an
t;
Th
e
o
r
e
m
1
[1
3]
:
C
o
nside
r
L
ip
sch
i
t
z
n
on
lin
ea
r
system
(
1)-
(
2)
a
long
w
i
t
h
t
he
o
bse
rver
(
3)-(
4).
The
ob
ser
v
er
err
o
r
dy
nam
i
cs
a
re
a
sym
p
t
o
t
i
call
y
s
ta
b
l
e
wi
th
m
axim
um
a
dmiss
i
b
l
e
Lip
s
ch
it
z
con
s
t
a
nt
i
f
th
ere
e
x
i
s
t
sca
l
e
r
s
0
and
0
a
n
d
m
atr
i
ce
s
0
P
a
nd
G
su
c
h t
h
at
t
he
fo
l
l
o
w
i
ng
L
MI o
pt
imi
z
a
tio
n
pro
b
lem
has a
so
l
u
tio
n
)
min(
.
0
2
)
(
)
(
P
P
P
LC
A
P
P
LC
A
T
(
5
)
0
2
1
2
1
I
P
P
I
(6
)
W
ith
1
*
)
max(
ˆ
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
A c
o
m
para
tive
st
udy o
f
non
l
i
n
e
ar c
i
rc
l
e
c
r
i
t
e
r
i
on b
a
se
d
ob
se
rver
and
H
∞
ob
s
e
r
v
er
…..
(Fa
r
id
Berrezzek)
1
231
Proof:
D
e
finin
g
t
he
o
bser
ver
error
a
s
)
(
ˆ
)
(
)
(
t
x
t
x
t
e
,
t
h
e
n
t
he
o
b
s
e
r
ver
er
ror
dy
nam
i
c
s
ar
e
give
n
b
y:
))
(
),
(
ˆ
(
))
(
),
(
(
)
(
)
(
t
u
t
x
F
t
u
t
x
F
e
LC
A
t
e
(
7
)
I
n the foll
ow
i
ng w
e
use
n
o
t
a
tio
ns
:
))
(
),
(
(
t
u
t
x
F
F
a
nd
))
(
),
(
ˆ
(
ˆ
t
u
t
x
F
F
The
tim
e
deriv
a
ti
ve
o
f the
L
y
apu
n
ov fu
nc
ti
o
n
Pe
e
V
T
can
b
e
expre
s
se
d as:
)
ˆ
(
2
)]
(
)
[(
)
(
F
F
P
e
e
LC
A
P
P
LC
A
e
t
V
T
T
T
(
8
)
By
u
sin
g
t
he
L
ip
sch
i
tz pr
ope
rty w
e
have
:
e
P
e
F
F
P
e
T
T
2
)
ˆ
(
2
(
9
)
The
n eq
ua
ti
on
(
8) be
c
om
es:
0
]
2
)
(
)
[(
)
(
e
P
e
LC
A
P
P
LC
A
e
t
V
T
T
T
(
10)
Th
ose le
ad
s to
:
0
2
)
(
)
(
1
P
PP
LC
A
P
P
LC
A
T
(
11)
By
u
sin
g
t
he
S
chur
c
om
p
l
e
m
e
n
t
t
h
e
o
rem
we
t
he
refore
o
bta
i
n
t
h
e
f
oll
o
w
i
ng l
i
near
m
atrix ine
q
ua
li
t
y
(LMI)
:
0
2
)
(
)
(
P
P
P
LC
A
P
P
LC
A
T
(
12)
The
Lipsc
h
itz
c
on
sta
n
t c
a
n b
e
com
pute
d
a
s fo
l
l
ow
s
:
F
o
r
any
Lyap
u
n
o
v
func
tio
n
Pe
e
V
T
we h
av
e:
2
max
2
min
)
(
)
(
e
P
Pe
e
e
P
T
(
13)
2
min
)
(
2
2
e
P
Pe
e
T
(
14)
Le
t:
Q
LC
A
P
P
LC
A
T
)
(
)
(
(
15)
The
n
t
he
t
ime
der
i
va
ti
ve
ca
n
be
w
r
itte
n a
s
:
0
2
)
(
2
e
P
Qe
e
t
V
T
(
16)
0
)
(
2
)
(
)
(
2
max
2
min
e
P
e
Q
t
V
(
17)
So
:
)
(
)
(
2
min
max
Q
P
)
(
)
(
2
1
max
min
P
Q
(
18)
In a
dd
iti
on,
s
i
n
c
e
P
is posi
t
i
v
e
de
fi
nite,
sup
p
o
se
I
Q
.F
ro
m t
h
e
eq
u
a
ti
on
(1
8
)
we
ca
n
w
r
i
t
e
:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
Int J
Pow
El
ec &
D
ri
S
yst
,
Vol
.
10,
N
o.
3
, Se
p
2
019 : 1229 –
1
2
43
1
232
0
)
(
2
1
max
P
(
19)
A
nd
)
(
)
(
max
P
P
S
o
f
rom
(
19):
2
1
)
(
P
(
20)
Wh
ic
h
i
s
e
q
u
i
v
alen
t t
o
:
0
)
2
1
(
2
P
P
I
T
(
21)
U
s
ing
S
c
hur c
omplem
en
t
l
e
mm
a
0
2
1
2
1
I
P
P
I
(
22)
Defi
n
i
ng
1
, (6)
i
s
ac
hi
e
v
ed.
3.
CIRCLE
CRITERION BASED NONLINE
A
R OB
S
E
R
V
E
R
D
ESIGN
I
n
c
o
n
t
r
as
t
of
t
he
line
a
riz
a
t
i
o
n-
base
d
a
n
d
h
i
g
h
-ga
i
n
a
ppr
oac
h
e
s
w
h
i
c
h
a
ttem
p
t
t
o
e
limi
n
a
t
e
t
h
e
syste
m
n
o
n
l
i
n
ear
i
t
ies
us
in
g
a
no
nl
ine
a
r
st
ate
t
r
ans
f
orm
a
ti
o
n
o
r
t
o
d
o
m
inate
t
h
e
m
by
a
hi
gh
gai
n
t
e
r
m
of
corr
ecti
on,
c
ir
cle
-
criter
i
o
n
e
xp
l
o
its
t
he
p
r
o
pert
ies
of
t
he
s
ys
tem
no
nl
i
n
e
a
r
i
t
i
es.
In
its
b
a
s
ic
f
orm
,
i
nt
r
oduc
e
d
by
A
r
cak
a
n
d
K
ok
oto
v
ic
[
14
],
t
he
a
p
p
roa
c
h
i
s
ap
p
lica
b
le
t
o
a
c
l
ass
o
f
s
ys
tem
s
t
hat
c
a
n
be
d
ec
om
p
o
se
d
i
n
line
a
r
and n
o
n
l
i
nea
r
pa
r
ts
w
i
t
h
a
cond
i
t
i
o
n t
h
at
t
he
n
on
l
i
ne
ari
t
i
es sat
isfy t
he
sec
tor
prope
r
t
y
[1
4-1
6
].
.
3.1.
Ba
s
i
c se
cto
r
pr
o
pe
rti
e
s
A
me
moryle
ss
n
o
n
l
i
ne
ar
f
un
cti
on
p
p
R
R
t
z
F
[
0
[
:
)
,
(
i
s
said
t
o
be
l
ong
t
o
the
sector
[
0
[
if
0
)
,
(
t
z
zF
.
Let
1
v
a
n
d
2
v
t
w
o
r
e
a
l
p
o
s
i
t
i
v
e
n
u
m
b
e
r
s
,
b
y
s
e
t
t
i
n
g
2
1
v
v
z
a
n
d
)]
,
(
)
,
(
[
)
,
(
2
1
t
v
F
t
v
F
t
z
F
,
the
above
s
ec
tor
pro
p
ert
y
i
s
equ
i
vale
n
t
t
o:
R
v
v
t
v
F
t
v
F
v
v
2
1
2
1
2
1
,
0
)]
,
(
)
,
(
)[
(
(
23)
Re
lat
i
o
n
(2
3)
s
ta
tes
t
h
at
t
he
n
on
li
nea
r
f
un
c
t
i
o
n
)
,
(
t
z
F
i
s
a
n
ond
ec
rea
s
in
g
func
t
i
o
n
.
O
n
th
e
o
t
h
e
r
han
d
i
f
)
,
(
t
z
F
is a
co
nt
i
n
u
ous
ly d
iff
e
re
ntiab
l
e
func
ti
o
n
the
a
b
o
v
e
rela
tio
n i
s e
q
u
i
va
l
e
n
t
t
o [7]
, [8]:
R
z
t
z
F
d
z
d
0
)
,
(
(
24)
I
f
t
he
n
o
n
l
i
ne
a
r
f
unc
t
i
o
n
)
,
(
t
z
F
do
e
s
n
o
t
s
a
t
is
fy
t
he
p
os
i
t
i
v
i
ty
c
on
d
i
t
i
on
(
24)
w
e
intr
od
uce
a
f
un
ct
i
on
)
,
(
t
z
F
su
c
h
th
a
t:
R
z
t
z
F
dz
d
z
t
z
F
t
z
F
,
)
,
(
,
)
,
(
)
,
(
(
25)
An
d:
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
A c
o
m
para
tive
st
udy o
f
non
l
i
n
e
ar c
i
rc
l
e
c
r
i
t
e
r
i
on b
a
se
d
ob
se
rver
and
H
∞
ob
s
e
r
v
er
…..
(Fa
r
id
Berrezzek)
1
233
R
z
t
z
F
d
z
d
t
z
F
d
z
d
0
)
,
(
)
,
(
(
26)
I
n
t
he
m
u
l
t
i
v
ariab
l
e
case
the
sect
or
p
r
o
pe
rt
y
ca
n
be
w
rit
t
en
a
s
:
0
)
,
(
t
z
F
z
T
.
W
h
ere
z
an
d
)
,
(
t
z
F
are
r
e
spec
tive
l
y ve
ct
ors of an
appro
p
r
i
a
t
e
d
i
m
e
nsion.
3.2.
Non
l
in
ear
Ob
serve
r
D
esign
The
c
i
r
c
le
c
rit
e
rio
n
b
ase
d
non
l
i
nea
r
o
bser
ver
desi
g
n
c
a
n
b
e
per
form
ed
f
or
a
c
lass
of
n
onl
inear
syste
m
t
ha
t
the
m
odel
can
b
e
d
e
com
p
ose
d
i
n
t
o l
i
n
ear
par
t
and
no
nl
i
n
ear
par
t as
t
he
foll
o
win
g
[1
4
-1
6]
:
)]
(
.
[
)]
(
),
(
[
)
(
)
(
t
x
H
GF
t
y
t
u
t
Ax
t
x
(
27)
)
(
)
(
t
Cx
t
y
(
28)
Wh
ere
A
,
C
a
n
d
G
a
r
e
k
n
o
w
n
c
o
n
s
t
a
n
t
m
a
t
r
i
c
e
s
w
i
t
h
a
p
p
r
o
p
r
i
a
t
e
d
i
m
e
n
s
i
o
n
s
.
T
h
e
p
air
)
,
(
C
A
i
s
assum
e
d
t
o
b
e
observa
b
l
e
.
T
he
t
erm
)]
(
),
(
[
t
u
t
y
is
a
n
a
r
bi
trary
rea
l
-value
d
ve
ct
or
t
h
a
t
de
pen
d
s
on
l
y
o
n
t
h
e
system
m
easured
c
ont
ro
l
in
pu
ts
)
(
t
u
a
n
d
ou
t
put
s
)
(
t
y
.
The
no
n
l
i
n
ea
r
pa
rt
o
f
the
system
i
s
i
n
c
l
ude
d
in
t
h
e
term
)]
(
.
[
t
x
H
F
w
hic
h
i
s
a
tim
e-va
ry
i
n
g
ve
c
t
or
f
u
n
c
t
i
on
veri
fy
i
ng
t
h
e
sec
t
or
p
r
o
pe
rty.
I
n
t
h
e
f
o
l
l
ow
i
n
g
w
e
rec
a
ll t
h
e ma
i
n
the
ore
m
and c
o
n
d
i
t
i
o
n
s
t
hat a
r
e used in th
is
w
o
r
k
to s
t
ud
y the
fea
s
i
b
i
l
i
t
y of no
n
l
i
n
ear
o
b
s
erve
r
des
i
g
n
f
or
i
nd
uct
i
on
m
o
t
o
r
sens
orless
c
o
n
t
r
o
l
w
ith
r
e
s
pe
c
t
o
f
c
irc
l
e
c
r
i
t
e
r
io
n
or
s
ec
t
o
r
pro
p
er
t
y
.
A
de
tai
l
e
d
pro
o
f
o
f
the
t
h
e
ore
m
i
s
presen
te
d i
n
re
f
er
ence
[1
7
]
.
Th
e
o
r
e
m
2
[1
4,
1
5]:
C
o
nsid
e
r
a
n
on
li
nea
r
s
y
s
tem
o
f
t
he
f
orm
(
2
7
)
-(28)
w
it
h
t
h
e
no
n
lin
e
a
r
par
t
s
at
is
fy
in
g
t
h
e
circl
e
c
riterion
r
e
lations
(23)
-(
26).
I
f
t
he
re
e
xi
st
a
s
ym
m
e
tri
c
a
n
d
po
s
itiv
e
d
e
fi
nit
e
m
a
t
ri
x
nxn
R
P
a
nd
a
se
t
of
r
ow
ve
c
t
o
rs
p
R
K
suc
h
tha
t
the
foll
ow
i
n
g li
near
m
atrix i
n
eq
ua
li
tie
s (LMI)
hol
d:
0
)
(
)
(
Q
LC
A
P
P
LC
A
T
(
29)
0
)
(
T
KC
H
PG
(
30)
The
no
n
line
a
r
ob
se
rver
d
e
s
ign
re
fe
rs
t
o
t
h
e
se
l
e
c
t
i
on
of
t
he
g
a
i
n
m
a
t
ri
ces
L
a
n
d
K
s
at
i
s
fy
i
ng
t
h
e
L
M
I
con
d
i
t
i
on
s
(29)
-(
30).
Wit
h
n
I
Q
a
s
a
define
d
p
o
s
i
tive
kn
ow
n
m
a
tri
x
,
n
I
i
s
an
n
-t
h
order
un
i
t
y
m
a
t
r
ix
a
n
d
is a
sm
all
po
siti
ve
r
ea
l
numb
e
r.
The
n
a
non
l
i
ne
ar
observe
r
can
b
e de
signe
d as:
))]
(
ˆ
)
(
(
)
(
ˆ
[
)]
(
ˆ
)
(
[
)]
(
),
(
[
)
(
ˆ
)
(
ˆ
t
y
t
y
K
t
x
H
F
G
t
y
t
y
L
t
y
t
u
t
x
A
t
x
(
31)
)
(
ˆ
)
(
ˆ
t
x
C
t
y
(
32)
A
nd
0
)
(
ˆ
)
(
)
(
lim
t
x
t
x
t
e
t
,
where
)
(
ˆ
t
x
is
t
he
e
s
tima
t
e
o
f
t
he
s
ta
te
v
ec
tor
)
(
t
x
o
f
t
h
e
n
onl
in
e
a
r
sy
st
e
m
.
On
e
c
a
n
s
ee
th
at
t
h
e
s
t
r
u
c
t
u
re
o
f
t
h
e
nonl
in
e
a
r
obse
r
ver
is
c
om
p
o
se
d
of
a
line
a
r
part
t
hat
is
si
m
i
lar
to
line
a
r
L
uen
b
e
r
ger
o
b
se
rver
a
n
d
a
n
o
n
li
nea
r
p
art
tha
t
i
s
a
n
a
d
d
it
io
na
l
term
t
ha
t
repre
s
ents
t
he
tim
e-
vary
in
g n
o
n
l
in
ear
i
t
ies
sa
t
i
sfy
i
ng t
h
e
sec
t
or
p
roper
t
y.
Proof:
The
dynam
i
cs of t
h
e s
t
ate
es
tima
t
i
on
e
rror
)
(
ˆ
)
(
)
(
t
x
t
x
t
e
a
r
e
gi
ve
n by
:
))]
(
ˆ
)
(
(
)
(
ˆ
.
(
))
(
.
(
[
)
(
)
(
)
(
t
y
t
y
K
t
x
H
F
t
x
H
f
G
t
e
LC
A
t
e
(
33)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
Int J
Pow
El
ec &
D
ri
S
yst
,
Vol
.
10,
N
o.
3
, Se
p
2
019 : 1229 –
1
2
43
1
234
Let
)
(
.
1
t
x
H
v
a
n
d
))
(
ˆ
)
(
(
)
(
ˆ
.
2
t
y
t
y
K
t
x
H
v
,
by
se
tti
n
g
)
(
)
(
2
1
t
e
KC
H
v
v
z
,
the
te
rm
b
e
t
w
e
e
n
brac
ke
ts in (
3
3)
ca
n
be
see
n
a
s
a
funct
i
on o
f
t
he
va
r
iab
l
e
z
t
hen
:
)
,
(
)]
(
)
(
[
2
1
t
z
f
v
f
v
f
.
The
ex
pre
s
si
on
)
,
(
)]
(
)
(
)[
(
2
1
2
1
t
z
zF
v
F
v
F
v
v
s
atisf
i
es the
p
ropert
y
of t
he
s
ector.
Ta
ki
ng
in
t
o
a
c
c
o
unt
t
h
e
a
bove
re
s
u
l
t
,
t
h
e
e
rro
r
d
yn
ami
c
s i
n
(3
3
)
ca
n be
r
ew
rit
t
e
n
as:
)
,
(
.
)
(
)
(
)
(
t
z
F
G
t
e
LC
A
t
e
(
34)
)
(
)
(
t
e
KC
H
z
(
35)
Re
lat
i
ons
(
3
4
)-(35)
s
how
,
on
c
e
a
ga
in,
t
h
a
t
t
he
e
r
r
or
d
y
n
a
m
ics
c
an
t
h
e
n
be
c
ons
i
d
ere
d
a
s
a
l
i
ne
ar
syste
m
c
on
t
r
o
lled
by
a
t
i
me
-varyi
ng
n
o
n
l
ine
a
ri
ty
f
u
n
c
t
i
o
n
)
,
(
t
z
F
s
a
t
i
s
fy
i
ng
t
h
e
se
ct
or
p
ro
per
t
y.
C
i
r
cle
criter
i
on
e
st
a
b
lis
hes
t
h
a
t
t
he
f
eed
ba
c
k
i
n
t
erc
onne
c
t
i
o
n
of
a
l
i
nea
r
s
yst
e
m
and
a
tim
e-va
ry
in
g
non
li
n
e
a
r
i
t
y
sa
ti
s
f
yi
ng
t
h
e
se
c
t
o
r p
r
op
erty
i
s gl
ob
al
l
y
un
i
fo
rml
y
a
sy
mpt
o
t
i
c
a
l
l
y
stab
l
e
[14,
15].
Base
d
u
p
o
n
th
e
e
rror
dyna
m
i
cs,
re
latio
n
(3
4)-
(
35),
the
no
nl
ine
a
r
o
b
serv
e
r
d
esi
g
n
pro
b
l
em
i
s
th
en
e
q
uiv
a
l
e
nt
t
o
st
ab
ili
z
a
tio
n
of
t
h
e
e
rro
r
d
yn
a
m
i
c
s
p
r
ob
l
e
m.
T
o
t
hi
s
en
d
a
c
a
nd
id
at
e
Ly
a
p
un
o
v
f
un
cti
on
Pe
e
V
T
i
s
c
o
nside
r
ed
.
I
n
o
rder
t
o
ens
u
re
a
symp
t
o
t
i
c
s
t
a
b
il
i
t
y
of
t
he
obse
r
v
e
r,
t
he
d
eri
v
a
t
ive
o
f
t
he
ca
ndi
da
te
L
ya
pu
n
ov
fu
nc
ti
o
n
m
ust
be
n
e
g
ati
v
e
.
W
i
t
h
the
he
lp
o
f
r
e
l
a
t
i
o
n
(
3
4
)
a
n
d
(
3
5
)
t
h
e
d
e
r
i
v
a
t
i
v
e
o
f
t
h
e
Lya
p
u
n
ov f
unc
tio
n
becom
e
s
:
)
,
(
)
,
(
)
(
)
(
t
z
PGF
e
Pe
G
t
z
F
e
LC
A
P
P
LC
A
e
V
T
T
T
T
T
(
36)
By
se
t
ti
n
g
:
Q
LC
A
P
P
LC
A
T
)
(
)
(
(
37)
An
d
T
KC
H
PG
)
(
(
38)
W
ith
n
I
Q
a
n
d
0
, the
d
eri
v
at
i
v
e
of t
h
e
L
yap
u
n
o
v
func
ti
on
can
b
e rew
r
itte
n
as:
)
,
(
.
.
2
t
z
F
z
Qe
e
V
T
T
(
39)
Th
us end
s
t
h
e
pro
o
f.
N
o
te
t
hat
t
h
e
e
x
i
s
te
nce
of
o
bse
r
ver
(31)-(
32)
i
s
c
o
nd
it
io
ne
d
b
y
t
he
s
o
l
u
t
i
o
n
of
L
MI
c
o
n
d
i
t
i
on
s
(2
9)
-(30
)
.
By
so
l
v
i
n
g
L
M
I
c
ons
trai
n
t
s,
o
bse
r
ver
ga
in
m
a
t
ric
e
s
L
a
n
d
K
t
ha
t
gua
ra
nte
e
o
b
s
e
r
ver
co
nver
g
e
n
ce
a
r
e
th
en
com
p
u
t
e
d
. In Ibrir [1
6
], the a
ut
hor
h
a
s
i
n
v
es
t
i
ga
te
d
the
stu
d
y
o
f
g
lo
b
a
ll
y
Li
psc
h
it
z
sy
st
ems
a
n
d
bou
nd
e
d
-st
a
t
e
no
n
l
i
n
ea
r
sy
ste
m
s.
B
ou
nde
d-s
t
ate
n
onl
i
n
ea
r
sys
t
em
s
co
nst
itu
te
a
l
ar
ge
c
l
a
ss
o
f
sys
t
em
t
hat
i
n
cl
u
d
es
e
le
ctric
mac
h
i
n
e
sy
st
ems.
E
l
e
c
t
ri
c
m
a
ch
i
n
e
m
o
d
e
l
s
i
nvo
l
v
e
th
e
ma
g
n
e
ti
c
f
l
u
x
a
s
a
k
e
y
a
n
d
bound
e
d
s
t
a
t
e
v
a
r
i
a
b
l
e
th
at
com
b
i
n
e
d
w
i
t
h
o
t
her
sta
t
e
var
i
a
b
le
o
f
t
h
e
ma
chi
n
e,
s
uc
h
a
s
r
oto
r
an
g
u
lar
v
e
loc
i
ty,
lea
d
s
t
o
t
he
n
o
n
li
nea
r
p
art
of
t
he
m
ac
hi
ne
m
ode
l.
T
his
is
d
ue
t
o
t
h
e
effe
ct
o
f
t
h
e
ma
g
n
et
ic
m
a
t
eria
l
sat
u
ra
t
i
o
n
p
ro
p
e
rty
t
h
at
i
s
s
i
milar
t
o
the se
c
t
or no
n
l
inear
i
t
y.
4.
NONL
INEAR
H
∞
OBSERVER S
Y
NT
HESIS
I
n
t
h
i
s
se
c
t
i
o
n,
w
e
propo
se
t
o
st
u
d
y
the
ro
bus
t
H
∞
o
bserver.
T
his
obser
v
er
i
s
used
i
n
the
case
o
f
syste
m
s
i
n
vo
l
v
i
n
g
u
n
cer
t
a
int
i
e
s
i
n
t
he
m
odel
m
e
a
s
ure
m
ents
[7, 9
,
10,
11,
1
2,
18].
A
l
i
ter
a
ture
r
e
v
i
e
w
show
s
th
at
m
a
ny
usef
u
l
H
∞
f
il
ter
i
n
g
a
ppr
oa
ches
h
a
v
e
be
en
d
eve
l
op
ed
f
or
s
eve
r
al
k
i
nds
o
f
systems.
O
ne
c
an
c
ite
the
H
∞
o
bse
r
vers
f
o
r
one-s
ide
d
L
ipsc
hi
tz
n
on
lin
ea
r
sys
t
em
s,
t
he
H
∞
obser
v
e
r
for
si
n
gul
a
r
L
i
p
sch
i
t
z
n
onl
in
ear
sys
t
em
s
and
the
H
∞
c
o
n
tro
l
a
nd
fi
lter
i
ng
for
uncer
ta
i
n
M
a
r
ko
v
i
a
n
j
um
p
s
y
ste
m
s
w
ith
t
i
m
e-
vary
in
g de
l
a
y [9,
10,
11,
19].
Th
e
ai
m
of
t
h
e
H
∞
a
p
proac
h
i
s
to
d
es
ig
n
a
ful
l
-
o
rder
f
il
ter
suc
h
t
ha
t
t
h
e
c
o
rre
sp
o
n
d
i
n
g
filter
i
n
g
e
r
r
or
sy
st
em
i
s
asymp
t
oti
cal
ly
s
ta
b
l
e
a
n
d
sati
sf
i
e
s
a
p
r
e
s
cribe
d
H
∞
l
e
v
e
l
d
is
t
u
r
b
a
n
ce
a
tte
nua
tio
n.
B
y
u
s
i
ng
a
Lya
p
u
n
ov f
unc
tio
n,
suffic
i
e
n
t
con
d
i
t
i
on
s a
r
e form
ula
t
e
d
i
n
te
rm
s of
line
a
r
ma
t
r
i
x
i
neq
u
a
l
i
t
ie
s
(LMIs).
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
A c
o
m
para
tive
st
udy o
f
non
l
i
n
e
ar c
i
rc
l
e
c
r
i
t
e
r
i
on b
a
se
d
ob
se
rver
and
H
∞
ob
s
e
r
v
er
…..
(Fa
r
id
Berrezzek)
1
235
To
d
o
t
h
i
s
,
we
e
xt
en
d
th
e
re
su
lt
of
t
h
e
p
re
vio
u
s
s
e
c
ti
on
t
o
n
o
n
line
a
r
r
o
b
u
s
t
H
∞
ob
se
rv
er
d
e
s
ig
n
met
h
od
.
W
e
con
s
i
d
er
t
he
sy
s
tem
(1)
-
(2) w
ith an
ad
d
iti
o
n
a
l
d
ist
u
rba
n
c
e
te
r
m
as
follow
s
:
)
(
))
(
),
(
(
)
(
)
(
t
w
B
t
u
t
x
F
t
Ax
t
x
d
(
40)
)
(
)
(
t
Cx
t
y
(
41)
Whe
r
e
)
,
0
[
2
l
w
i
s
an unk
n
o
w
n
e
x
oge
no
us d
is
tur
b
an
ce
.
The
goa
l
is
t
o
re
bu
i
l
d
the
s
t
a
t
e
)
(
t
x
o
f
t
h
e
s
y
s
t
e
m
(
4
0
)
-
(
4
1
)
w
i
t
h
s
o
m
e
a
c
c
u
r
a
c
y
d
e
s
p
i
t
e
t
h
e
p
r
e
s
e
n
c
e
o
f
di
st
urba
nce
term
)
(
t
w
.
W
e
sup
po
s
e
tha
t
:
)
(
t
He
z
, with
H
a known constant
m
a
tr
ix
s
u
c
h
t
h
a
t
w
z
,
wi
t
h
0
.
The
cor
respon
di
n
g
ob
s
erve
r for the
sys
t
e
m
(
40)-
(
41) is g
i
v
e
n
b
y:
)]
(
ˆ
)
(
[
))
(
),
(
ˆ
(
)
(
ˆ
)
(
ˆ
t
y
t
y
L
t
u
t
x
F
t
x
A
t
x
(
42)
)
(
ˆ
)
(
ˆ
t
x
C
t
y
(
43)
O
u
r
purp
o
se
i
s
to
d
es
ig
n
t
h
e
obs
erve
r
par
a
m
e
ter
L
s
uc
h
t
h
at
t
he
o
bserv
e
r
error
dynam
i
cs
a
r
e
a
sym
p
t
o
tica
lly
st
a
b
le a
n
d
t
he
f
oll
o
w
i
ng s
p
ec
i
f
ie
d n
o
r
m
H
uppe
r
b
o
u
nd
is
s
i
m
ul
t
a
neo
u
s
l
y
gu
ara
n
tee
d
.
Le
mm
a
1:
F
or
a
ny
n
R
y
x
,
a
nd
a
ny
p
o
s
i
t
i
ve
d
e
f
i
n
i
t
e
m
a
tri
x
n
n
R
P
,
w
e
h
ave:
y
P
y
Px
x
y
x
T
T
T
1
2
Th
e
o
r
e
m
3
[1
3]:
Co
nsi
d
e
r
s
t
o
cha
s
t
i
c
L
i
p
s
c
h
i
t
z
no
n
l
i
n
ear
s
yste
m
(40)-(
41
)
,
a
nd
t
h
e
c
o
rre
sp
o
n
d
i
ng
o
b
s
erve
r
(42)-
(
43).
The
obs
er
ver
err
o
r
dy
n
a
mic
s
a
re
a
sym
p
t
o
t
i
ca
ll
y
sta
b
l
e
w
i
th
t
he
m
ini
m
um
o
f
t
h
e
norm
2
l
a
n
d
,
if
there
e
x
i
s
t
sc
aler
s
s
>
1
,
0
and
0
and
ma
trice
s
0
P
an
d
G
s
uc
h
tha
t
t
he
f
o
l
lo
w
i
n
g
L
M
I
op
tim
iza
t
i
o
n prob
lem
ha
s a
sol
u
t
i
o
n
.
0
)
1
(
2
1
I
P
B
PB
I
H
H
T
d
d
s
T
(
44)
0
2
)
(
1
2
)
(
1
2
2
H
P
P
I
H
(
45)
W
ith
G
P
L
1
and
)
min(
ˆ
*
Proof:
The
observe
r e
r
ror
dynam
i
cs a
re:
)
(
)
ˆ
(
)
(
)
(
)
(
t
w
B
F
F
t
e
LC
A
t
e
d
(
46)
The
tim
e
deriv
a
ti
ve
o
f the
L
y
apu
n
ov fu
nc
ti
o
n
)
(
)
(
t
Pe
e
t
V
T
is
g
iv
e
n
b
y:
Pe
B
t
w
t
w
PB
e
F
F
P
e
e
LC
A
P
P
LC
A
e
t
V
T
T
d
T
T
T
T
d
)
(
)
(
)
ˆ
(
2
)]
(
)
[(
)
(
(
47)
)
min(
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
Int J
Pow
El
ec &
D
ri
S
yst
,
Vol
.
10,
N
o.
3
, Se
p
2
019 : 1229 –
1
2
43
1
236
U
s
ing
Lem
m
a
1
a
nd
the
Ra
y
l
eig
h
lea
ds
t
o th
e foll
ow
i
n
g
:
)
ˆ
(
)
ˆ
(
)
ˆ
(
2
1
F
F
P
PP
F
F
Pe
e
F
F
P
e
T
T
T
(
48)
)
ˆ
(
)
ˆ
(
)
ˆ
(
2
F
F
P
F
F
Pe
e
F
F
P
e
T
T
T
(
49)
The
right
term
o
f (49)
c
an be
writt
e
n
as
:
e
e
P
e
P
Pe
e
T
T
)
(
)
(
max
2
max
(
50)
2
max
2
)
(
)
ˆ
(
)
ˆ
(
e
P
F
F
P
F
F
T
(
51)
The
l
e
ft ter
m
of
(
49) can be wr
i
t
ten
as:
2
max
)
(
2
)
ˆ
(
2
e
P
F
F
P
e
T
(
52)
Re
plac
i
ng (
51)
-(52)
i
n e
qua
ti
on (
4
9),
l
e
ads to:
2
max
2
2
max
2
max
)
(
)
(
)
(
2
e
P
e
P
e
P
Assume
I
Q
,
in
t
he
d
e
t
erm
i
nis
t
ic
case
we
h
a
v
e
0
w
and
the
c
o
r
r
espon
d
i
n
g
t
i
m
e
de
riva
tive
of
t
h
e
fu
nc
ti
o
n
Lya
pu
no
v
bec
o
m
e
s:
0
)
ˆ
(
2
)
(
F
F
e
Qe
e
t
V
T
T
(
53)
0
)
(
2
)
(
2
max
e
P
Ie
e
t
V
T
(
54)
Then:
0
)
(
2
1
max
P
(
55)
2
1
)
(
P
(
56)
e
e
P
F
F
P
e
T
T
)
(
)
1
(
)
ˆ
(
2
max
2
(
57)
e
e
e
e
P
F
F
P
e
T
T
T
)
1
(
2
1
)
(
2
1
)
1
(
)
ˆ
(
2
2
(
58)
In
t
he
s
t
o
c
h
a
s
t
i
c
a
n
d
gene
ra
l
ca
se
o
f
0
w
a
n
d
I
Q
s
t
he
a
b
ove
e
q
u
a
t
i
o
n
of
L
ya
pu
n
o
v
deriva
t
i
v
e
bec
o
me
s:
Pe
B
t
w
t
w
PB
e
F
F
P
e
e
LC
A
P
P
LC
A
e
t
V
T
d
T
d
T
T
T
T
)
(
)
(
)
ˆ
(
2
)]
(
)
[(
)
(
(
59)
)
(
)
(
)
1
(
2
1
)
(
)
(
max
t
w
PB
e
Pe
B
t
w
e
e
e
e
Q
t
V
d
T
T
d
T
T
T
(
60)
Evaluation Warning : The document was created with Spire.PDF for Python.
Int J
P
o
w
E
l
e
c
&
D
ri S
yst
IS
S
N
:
2088-
86
94
A c
o
m
para
tive
st
udy o
f
non
l
i
n
e
ar c
i
rc
l
e
c
r
i
t
e
r
i
on b
a
se
d
ob
se
rver
and
H
∞
ob
s
e
r
v
er
…..
(Fa
r
id
Berrezzek)
1
237
)
(
)
(
)
1
(
2
1
)
(
t
w
PB
e
Pe
B
t
w
e
e
e
e
t
V
d
T
T
d
T
T
T
s
(
61)
)
(
)
(
)]
)
1
(
2
1
[
)
(
2
t
w
PB
e
Pe
B
t
w
e
e
t
V
d
T
T
d
T
T
(
62)
)
(
t
He
z
,
H
is a
kn
o
w
n
c
o
n
s
t
a
n
t
w
ith
w
z
N
o
w
,
w
e
de
fi
n
e
t
he
f
oll
o
w
i
ng
c
riteri
o
n
t
o
mi
nimiz
e
the
di
ff
ere
n
ce
be
tw
e
e
n
the
e
n
erg
y
o
f
t
h
e
e
s
tim
at
ion
er
ror
and
the
ene
r
g
y
in
d
u
cte
d
b
y t
h
e
di
s
t
urba
nc
es
:
dt
w
w
z
z
J
T
T
)
(
0
(
63)
Thus:
dt
t
V
w
w
z
z
J
T
T
))
(
(
0
(
64)
So
a
su
f
f
i
ci
en
t
co
nd
i
tio
n
fo
r
0
J
is that:
Th
is
m
e
a
ns tha
t
:
0
w
w
z
z
T
T
(
b
eca
use
)
(
t
V
is a
ne
gat
i
ve
fu
n
c
t
i
o
n)
h
ence
:
2
2
w
z
.
w
z
B
u
t
we
h
av
e:
w
w
Pe
B
w
w
PB
e
e
e
He
H
e
t
V
w
w
He
H
e
t
V
w
w
z
z
T
T
d
T
d
T
T
s
T
T
T
T
T
T
T
)]
)
1
(
2
1
[
)
(
)
(
2
(
65)
Th
en
, we
c
an
w
ri
t
e
:
w
e
I
P
B
PB
I
H
H
w
e
J
T
d
d
s
T
T
T
]
)
1
(
2
1
[
2
(
66)
So
a
su
f
f
i
ci
en
t
co
nd
i
tio
n
fo
r
0
J
is
t
hat t
h
e
fol
l
ow
in
g be
ne
g
a
t
ive
defin
ite
:
0
]
)
1
(
2
1
[
2
I
P
B
PB
I
H
H
T
d
d
s
T
(
67)
A
c
c
o
rdi
n
g
to t
he
S
chur
’s c
o
m
plem
en
t le
mm
a, (67)
is equiva
le
n
t
t
o:
0
I
.
0
1
)
(
2
max
P
B
PB
I
P
H
H
T
d
s
T
d
(
68)
F
o
r
equa
ti
on
(
68)
t
o
be
n
ega
t
i
v
e,
t
he
s
um
o
f
the
first
a
n
d
se
co
n
d
t
er
ms
m
ust
be
n
e
g
ati
v
e
be
ca
use
t
h
e
thir
d
term
is a
l
w
a
ys pos
i
tive
:
0
)
(
2
max
I
P
H
H
s
T
(
69)
0
)
(
,
,
0
t
V
w
w
z
z
t
T
T
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
: 208
8-
869
4
Int J
Pow
El
ec &
D
ri
S
yst
,
Vol
.
10,
N
o.
3
, Se
p
2
019 : 1229 –
1
2
43
1
238
But
as for a
ny
ot
her
sym
m
e
tric
m
a
t
r
i
x
,
for
H
H
T
, we
h
ave:
)
(
)
(
max
min
H
H
H
H
H
H
T
T
T
(
70)
I
H
H
H
I
H
T
)
(
)
(
2
min
2
max
(
71)
O
r
acc
ordi
ng t
o
the
d
e
f
in
i
tio
n
of sin
g
u
l
a
r va
lue
s
0
)
(
2
)
(
max
2
s
P
H
(
72)
Or:
2
)
(
)
(
2
max
H
P
s
(
73)
Th
is
i
s
eq
ui
va
l
e
nt t
o
(45).
5.
SIM
U
L
A
TIO
N
RESULTS
AND COMME
NTS
To
p
er
form
t
he
c
ompa
rati
ve
s
t
u
d
y
b
e
t
w
e
e
n
t
he
t
w
o
n
on
line
a
r
obse
rver
s
pr
e
v
i
ous
ly
v
i
e
w
,
w
e
use
t
h
e
mode
l
of
a
n
i
n
d
u
c
t
i
o
n
m
o
to
r.
D
esc
r
ibed
b
y
the
fo
llow
i
n
g
n
on
li
ne
a
r
d
i
ffe
ren
t
ial
e
q
ua
ti
o
n
s
w
ith,
t
h
e
stat
or
curr
ent,
r
otor flux
an
d r
o
tor
angu
lar
vel
o
c
i
t
y
a
s selec
t
e
d
sta
t
e
variab
le
s
of
t
he
m
achin
e
[1,
20,
21].
s
s
r
r
r
r
s
s
u
l
T
i
i
dt
d
1
s
s
r
r
r
r
s
s
u
l
T
i
i
dt
d
1
r
r
r
r
s
r
r
T
i
T
m
dt
d
1
(
74)
r
r
r
r
s
r
r
T
i
T
m
dt
d
1
l
l
r
f
s
r
s
r
r
T
k
k
i
i
dt
d
)
(
Where
:
p
r
r
n
,
r
p
Jl
m
n
2
,
r
s
l
l
m
m
1
1
1
,
r
s
l
l
m
2
1
,
J
f
k
r
f
,
J
n
k
p
l
,
r
r
r
R
L
T
.
The
i
n
de
xes
s
a
nd
r
r
e
f
e
r
t
o
t
h
e
sta
t
or
a
nd
t
h
e
rot
o
r
c
o
mp
o
n
en
ts
r
es
pec
t
i
v
ely.
i
,
a
n
d
u
r
e
s
p
e
ct
iv
el
y
den
o
t
e
t
h
e
sta
t
or
c
urre
n
t
,
th
e
rotor
fl
u
x
es,
t
h
e
su
ppl
ie
d
stat
o
r
vo
lt
ag
e,
R
i
s
the
r
e
si
sta
n
ce,
l
i
s
t
h
e
in
duc
ta
nce
,
m
i
s
the
m
u
t
u
a
l
i
n
duc
t
a
nce.
s
T
a
n
d
r
T
a
r
e
t
he
s
ta
tor
a
nd
t
h
e
rot
o
r
tim
e
co
ns
tan
t
r
e
s
pec
t
i
v
e
l
y.
r
i
s
the
rotor
angular
velocity,
r
f
i
s
t
h
e
fric
t
i
o
n
c
oef
f
ic
ie
n
t
,
J
i
s
t
h
e
mo
me
nt
o
f
i
n
erti
a
c
o
ef
fi
ci
e
n
t
,
p
n
i
s
the
numbe
r
o
f
p
air
po
le
s,
r
i
s
the
m
echa
n
ic
al
s
pee
d
o
f
t
h
e
rotor
and
fina
l
l
y
l
T
i
s
t
h
e
m
e
c
h
ani
c
a
l
l
o
a
d
tor
que.
Thi
s
t
yp
e
o
f
n
onl
in
ea
r
model
i
s
g
e
n
eral
ly
u
sed
fo
r
perf
o
r
mi
ng
n
onl
i
n
ea
r
co
nt
rol
,
c
on
dit
i
o
n
s
moni
t
o
ri
ng
a
n
d
fa
u
l
t
s
d
i
a
gn
o
s
is
o
f
e
l
e
c
t
ric
i
n
d
u
c
tio
n
m
a
c
h
i
n
e
s
ys
tem
s
.
P
e
rformin
g
t
h
es
e
tec
h
n
i
que
s
r
e
qu
ires
est
i
ma
t
i
n
g
un
m
ea
sured ro
tor
flu
x
lin
ka
ge
a
nd
r
o
t
or a
ng
u
l
a
r
vel
o
c
i
t
y
s
ta
te
varia
b
l
es base
d
on
t
h
e s
t
a
t
or
c
urrent
and
vo
lta
ge
m
easur
em
ents [21
].
To ca
rry ou
t
t
h
e
c
o
m
par
a
t
i
v
e st
ud
y be
t
w
e
e
n
the
tw
o
o
bser
v
e
rs pr
evi
o
us
l
y
v
i
e
w, we use t
h
e
in
d
u
c
t
io
n
motor
w
i
t
h
t
he fo
llow
i
n
g
c
hara
cteris
t
i
c
s
deta
i
l
s
on
Tab
l
e
1.
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