I
nte
rna
t
io
na
l J
o
urna
l o
f
E
lect
rica
l a
nd
Co
m
pu
t
er
E
ng
ineering
(
I
J
E
CE
)
Vo
l.
11
,
No
.
6
,
Dec
em
b
er
2
0
2
1
,
p
p
.
4
9
0
7
~
4
9
2
1
I
SS
N:
2
0
8
8
-
8
7
0
8
,
DOI
: 1
0
.
1
1
5
9
1
/
ijece
.
v
1
1
i
6
.
pp
4
9
0
7
-
4
9
2
1
4907
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ij
ec
e.
ia
esco
r
e.
co
m
Disturba
nce
obs
e
rv
er
-
ba
sed co
nt
ro
ller f
o
r i
nv
erte
d
p
endulum
with
u
ncer
taintie
s:
Linea
r ma
tr
ix
i
nequa
lity
a
ppro
a
ch
Va
n
-
P
ho
ng
Vu,
M
inh
-
T
a
m
Ng
uy
en,
Anh
-
Vu N
g
uy
en,
Vi
-
Do
T
ra
n,
T
ra
n
-
M
in
h
-
Ng
uy
et
Ng
uy
en
De
p
a
rtme
n
t
o
f
Au
t
o
m
a
ti
c
Co
n
tro
l
,
Ho
C
h
i
M
i
n
h
Cit
y
Un
i
v
e
rsity
o
f
Tec
h
n
o
l
o
g
y
a
n
d
E
d
u
c
a
ti
o
n
,
H
o
C
h
i
M
i
n
h
Cit
y
,
Vie
tn
a
m
Art
icle
I
nfo
AB
S
T
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Sep
4
,
2
0
2
0
R
ev
is
ed
Ma
y
1
9
,
2
0
2
1
Acc
ep
ted
Ma
y
3
1
,
2
0
2
1
A
n
e
w
a
p
p
ro
a
c
h
b
a
se
d
o
n
li
n
e
a
r
m
a
tri
x
in
e
q
u
a
l
it
y
(L
M
I)
tec
h
n
i
q
u
e
fo
r
sta
b
il
izin
g
t
h
e
i
n
v
e
rted
p
e
n
d
u
l
u
m
is
d
e
v
e
l
o
p
e
d
i
n
t
h
is
a
rti
c
le.
T
h
e
u
n
k
n
o
w
n
sta
tes
a
re
e
stim
a
t
e
d
a
s
we
ll
a
s
t
h
e
sy
ste
m
is
sta
b
il
ize
d
sim
u
lt
a
n
e
o
u
sly
b
y
e
m
p
lo
y
i
n
g
th
e
o
b
se
rv
e
r
-
b
a
se
d
c
o
n
tr
o
ll
e
r.
I
n
a
d
d
it
i
o
n
,
th
e
im
p
a
c
ts
o
f
th
e
u
n
c
e
rtain
ti
e
s
a
re
tak
e
n
in
t
o
c
o
n
si
d
e
ra
ti
o
n
in
th
is
p
a
p
e
r.
Un
li
k
e
t
h
e
p
re
v
io
u
s
stu
d
ies
,
t
h
e
u
n
c
e
rtain
ti
e
s
i
n
t
h
is
s
tu
d
y
a
re
u
n
n
e
c
e
ss
a
ry
to
sa
ti
sfy
t
h
e
b
o
u
n
d
e
d
c
o
n
stra
in
ts.
T
h
e
se
u
n
c
e
rtain
t
ies
will
b
e
c
o
n
v
e
rted
in
t
o
th
e
u
n
k
n
o
wn
in
p
u
t
d
istu
r
b
a
n
c
e
s,
a
n
d
t
h
e
n
a
d
istu
rb
a
n
c
e
o
b
se
rv
e
r
-
b
a
se
d
c
o
n
tro
ll
e
r
will
b
e
sy
n
th
e
siz
e
d
to
e
stim
a
te
th
e
in
f
o
rm
a
ti
o
n
o
f
t
h
e
u
n
k
n
o
wn
sta
tes
,
e
li
m
in
a
te
c
o
m
p
lete
ly
t
h
e
e
ffe
c
ts
o
f
t
h
e
u
n
c
e
rtain
ti
e
s,
a
n
d
sta
b
il
ize
in
v
e
rted
p
e
n
d
u
l
u
m
sy
ste
m
.
Wi
t
h
t
h
e
su
p
p
o
rt
o
f
l
y
a
p
u
n
o
v
m
e
th
o
d
o
lo
g
y
,
th
e
c
o
n
d
it
io
n
s
f
o
r
c
o
n
stru
c
ti
n
g
t
h
e
o
b
se
rv
e
r
a
n
d
c
o
n
tro
ll
e
r
u
n
d
e
r
th
e
fra
m
e
wo
rk
o
f
li
n
e
a
r
m
a
tri
x
in
e
q
u
a
li
ti
e
s
(
LM
Is
)
a
re
d
e
ri
v
e
d
i
n
m
a
in
t
h
e
o
re
m
s.
F
i
n
a
ll
y
,
t
h
e
sim
u
latio
n
s
fo
r
sy
ste
m
with
a
n
d
with
o
u
t
u
n
c
e
rtain
ti
e
s
a
re
e
x
h
ib
it
e
d
to
sh
o
w
t
h
e
m
e
rit
a
n
d
ef
fe
c
ti
v
e
n
e
ss
o
f
th
e
p
ro
p
o
se
d
m
e
t
h
o
d
s
.
K
ey
w
o
r
d
s
:
Dis
tu
r
b
an
ce
o
b
s
er
v
e
r
I
n
v
er
ted
p
en
d
u
lu
m
LMIs
Ob
s
er
v
er
-
b
ased
c
o
n
tr
o
ller
Un
ce
r
tain
ties
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Van
-
Ph
o
n
g
Vu
Dep
ar
tm
en
t o
f
Au
to
m
atic
C
o
n
tr
o
l
Ho
C
h
i M
in
h
C
ity
Un
iv
er
s
ity
o
f
T
ec
h
n
o
lo
g
y
an
d
E
d
u
ca
tio
n
No
.
1
Vo
Van
Ng
an
Stre
et,
T
h
u
Du
c
Dis
tr
ict,
Ho
C
h
i M
in
h
c
ity
,
Vietn
am
E
m
ail:
p
h
o
n
g
v
v
@
h
cm
u
te.
e
d
u
.
v
n
1.
I
NT
RO
D
UCT
I
O
N
I
n
v
er
ted
p
e
n
d
u
lu
m
s
y
s
tem
is
a
ty
p
ical
s
y
s
tem
wh
ich
is
u
s
ed
f
o
r
d
ev
elo
p
i
n
g
a
n
d
test
in
g
m
a
n
y
m
o
d
er
n
c
o
n
tr
o
l
th
eo
r
ies
b
ec
au
s
e
o
f
th
e
in
ter
esti
n
g
d
y
n
am
ic
ch
ar
ac
ter
is
tics
s
u
ch
as
s
tr
o
n
g
n
o
n
lin
ea
r
,
co
m
p
licated
,
m
u
lti
-
v
ar
iab
le,
a
n
d
u
n
s
tab
le
s
y
s
tem
.
T
h
e
m
o
d
el
o
f
in
v
er
ted
p
en
d
u
lu
m
is
q
u
it
e
s
im
ilar
to
th
e
p
r
ac
tical
m
o
d
els
ex
is
tin
g
in
r
e
ality
s
u
ch
as
a
m
is
s
ile,
s
elf
-
b
alan
cin
g
r
o
b
o
t,
a
n
d
h
ea
v
y
cr
an
e
liftin
g
co
n
tai
n
er
s
.
I
n
th
e
p
ast
f
ew
d
ec
a
d
es,
p
le
n
ty
o
f
p
a
p
er
s
s
tu
d
y
in
g
in
v
er
t
ed
p
en
d
u
lu
m
h
av
e
b
ee
n
p
u
b
l
is
h
ed
[
1
]
-
[
1
2
]
.
Fo
r
ex
am
p
le,
th
e
p
r
o
b
lem
s
o
f
m
o
d
elin
g
th
e
in
v
er
ted
p
en
d
u
lu
m
wer
e
in
v
esti
g
ated
in
p
ap
er
s
[
1
]
an
d
[
2
]
wh
e
r
e
th
e
m
o
d
elin
g
m
et
h
o
d
r
elied
o
n
th
e
f
u
zz
y
clu
s
ter
m
eth
o
d
was
s
tu
d
ied
in
[
1
]
an
d
th
e
D'
Alem
b
er
t'
s
p
r
in
cip
le
was
em
p
lo
y
ed
to
m
o
d
el
in
v
e
r
ted
p
en
d
u
l
u
m
in
[
2
]
.
Ad
d
itio
n
all
y
,
th
e
co
n
tr
o
ller
s
y
n
th
esis
to
s
tab
ilize
th
e
s
y
s
te
m
h
as
b
ee
n
r
ec
eiv
e
d
g
r
ea
t
atten
t
io
n
f
r
o
m
r
esear
ch
er
s
[
3
]
-
[
1
2
]
.
Fo
r
in
s
tan
ce
,
a
PI
-
s
tate
f
ee
d
b
ac
k
co
n
tr
o
ller
was
d
esig
n
ed
to
co
n
tr
o
l
th
e
in
v
er
t
ed
p
en
d
u
lu
m
s
y
s
tem
in
[
5
]
,
i
n
wh
ich
th
e
p
r
o
p
o
r
tio
n
al
an
d
in
teg
r
al
g
ain
s
wer
e
d
eter
m
in
ed
v
ia
th
e
p
o
le
p
la
ce
m
en
t
m
eth
o
d
w
h
o
s
e
in
p
u
t
co
n
tr
o
l
s
ig
n
al
was
s
am
p
led
an
d
d
id
n
o
t
h
av
e
co
n
tin
u
ity
o
f
tim
e.
An
o
th
er
m
o
d
er
n
c
o
n
tr
o
ller
,
s
lid
in
g
m
o
d
e
co
n
tr
o
l,
h
as
b
ee
n
also
ap
p
lied
to
s
tab
ilize
th
e
in
v
er
ted
p
en
d
u
lu
m
in
[
1
1
]
,
[
1
2
]
as
well.
Un
f
o
r
tu
n
at
ely
,
th
e
d
is
ad
v
an
tag
e
o
f
th
e
s
lid
in
g
-
m
o
d
e
a
p
p
r
o
ac
h
is
t
h
at
th
er
e
ex
is
t th
e
ch
atter
in
g
p
h
e
n
o
m
en
a
wh
ich
will im
p
ac
t th
e
d
ev
ices a
n
d
p
e
r
f
o
r
m
an
ce
o
f
th
e
s
y
s
tem
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
6
,
Dec
em
b
e
r
2
0
2
1
:
4
9
0
7
-
4
9
2
1
4908
I
n
r
ea
lity
,
a
lo
t
o
f
p
h
y
s
ical
s
tate
v
ar
iab
les
o
f
th
e
s
y
s
tem
ar
e
u
n
ab
le
o
r
h
ar
d
to
m
ea
s
u
r
e
b
y
u
s
in
g
th
e
s
en
s
o
r
s
.
Mo
r
eo
v
er
,
em
p
l
o
y
in
g
s
en
s
o
r
s
to
o
b
tain
th
e
in
f
o
r
m
atio
n
o
f
th
e
s
tate
v
ar
iab
les
will
ca
u
s
e
th
e
co
s
t
to
g
r
o
w
u
p
an
d
th
e
s
en
s
o
r
s
a
r
e
a
ls
o
s
en
s
itiv
e
to
th
e
n
o
is
e
th
at
lead
s
to
th
e
in
co
r
r
ec
t
m
ea
s
u
r
e
m
en
ts
.
Du
e
to
th
e
ab
o
v
e
r
ea
s
o
n
s
,
d
esig
n
in
g
an
o
b
s
er
v
er
to
r
ep
lace
th
e
s
en
s
o
r
s
is
a
p
r
es
s
in
g
is
s
u
e
th
at
attr
ac
ts
a
lo
t
o
f
r
esear
ch
er
s
.
R
ec
en
tly
,
th
er
e
a
r
e
m
an
y
p
ap
er
s
f
o
c
u
s
in
g
o
n
o
b
s
er
v
er
d
esig
n
[
1
3
]
-
[
1
6
]
.
R
eg
ar
d
in
g
o
b
s
er
v
er
d
esig
n
f
o
r
in
v
er
ted
p
en
d
u
lu
m
,
s
ev
er
al
in
ter
esti
n
g
r
esu
lts
h
av
e
b
ee
n
f
o
u
n
d
e
d
in
s
o
m
e
p
ap
er
s
[
1
7
]
-
[
1
9
]
.
Fo
r
ex
am
p
le,
a
n
ap
p
r
o
ac
h
to
d
e
s
ig
n
a
h
ig
h
-
o
r
d
er
s
lid
in
g
m
o
d
e
o
b
s
er
v
er
was
in
tr
o
d
u
ce
d
in
[
1
7
]
to
co
m
p
u
te
th
e
u
n
m
ea
s
u
r
ab
le
s
tates.
Ho
wev
er
,
th
e
d
r
awb
ac
k
o
f
th
e
s
lid
in
g
m
o
d
e
m
eth
o
d
in
p
ap
er
[
1
7
]
is
th
at
th
e
ex
is
ten
ce
o
f
th
e
ch
atter
in
g
p
h
e
n
o
m
en
o
n
will
in
f
lu
en
ce
th
e
p
er
f
o
r
m
an
ce
o
f
th
e
o
b
s
er
v
e
r
(
to
b
e
s
ee
n
i
n
[
1
7
]
)
.
I
n
p
ap
er
[
1
8
]
,
b
o
th
s
tates
an
d
f
au
lts
wer
e
esti
m
ated
b
y
d
e
s
ig
n
in
g
an
o
b
s
er
v
er
an
d
a
m
eth
o
d
b
ased
o
n
Ac
k
er
m
an
’
s
f
o
r
m
u
la
was
p
r
esen
ted
in
th
e
ar
ticle
[
1
8
]
.
I
n
th
e
p
ast
d
ec
ad
e,
a
m
ath
e
m
atica
l
tech
n
iq
u
e
ca
lled
lin
ea
r
m
atr
i
x
in
eq
u
ality
(
L
MI
)
wh
ich
ass
is
ts
to
s
o
lv
e
th
e
p
r
o
b
lem
s
o
f
t
h
e
co
n
tr
o
l
f
ield
m
o
r
e
ea
s
ily
an
d
e
f
f
icien
tly
was
in
tr
o
d
u
ce
d
in
[
2
0
]
.
Ho
wev
er
,
to
th
e
b
est
o
f
o
u
r
k
n
o
wled
g
e,
th
er
e
ex
is
t
a
f
ew
p
ap
e
r
s
em
p
lo
y
in
g
th
e
L
MI
tech
n
iq
u
e
to
s
y
n
t
h
esize
o
b
s
er
v
er
f
o
r
in
v
er
ted
p
en
d
u
lu
m
.
T
h
u
s
,
in
th
is
wo
r
k
,
we
will
p
r
o
p
o
s
e
a
m
et
h
o
d
b
ased
o
n
th
e
L
M
I
tech
n
iq
u
e
to
co
n
s
tr
u
ct
an
o
b
s
er
v
e
r
f
o
r
in
v
er
ted
p
en
d
u
lu
m
th
at
ca
n
a
v
o
id
th
e
ch
atter
in
g
is
s
u
es
in
p
ap
e
r
[
1
7
]
.
I
n
ad
d
itio
n
,
with
th
e
aid
o
f
th
e
L
MI
s
tech
n
iq
u
e,
th
e
co
n
d
itio
n
s
to
d
esig
n
o
b
s
er
v
e
r
in
th
is
ar
ticle
will
b
e
m
o
r
e
r
elax
ed
with
r
esp
ec
t
to
t
h
e
m
eth
o
d
e
m
p
lo
y
i
n
g
ac
k
er
m
an
’
s
f
o
r
m
u
la
i
n
p
ap
e
r
[
1
8
]
.
B
esid
es,
in
p
r
ac
tice,
th
e
s
y
s
t
em
s
ar
e
in
ev
itab
le
to
b
e
im
p
ac
ted
b
y
th
e
u
n
ce
r
tain
ties
wh
ich
m
ay
o
r
ig
in
at
e
f
r
o
m
th
e
m
o
d
elin
g
an
d
/o
r
p
a
r
am
eter
er
r
o
r
s
.
T
h
e
in
v
er
ted
p
e
n
d
u
l
u
m
is
n
o
t
an
ex
ce
p
tio
n
,
h
en
ce
,
s
tab
ilizin
g
th
e
in
v
er
ted
p
e
n
d
u
lu
m
with
t
h
e
im
p
ac
ts
o
f
th
e
u
n
ce
r
tain
ties
is
a
p
r
ess
in
g
a
n
d
in
ter
esti
n
g
is
s
u
e.
T
h
er
e
ar
e
m
an
y
ar
ticles p
ay
in
g
atten
tio
n
to
s
tab
iliz
in
g
th
e
u
n
ce
r
tain
in
v
er
te
d
p
en
d
u
lu
m
s
y
s
tem
in
r
ec
en
t y
ea
r
s
[
2
1
]
-
[
2
7
]
.
I
n
p
ap
er
[
2
1
]
,
a
f
u
zz
y
ty
p
e
-
2
PID
co
n
tr
o
ller
was
s
y
n
th
esized
f
o
r
t
h
e
in
v
er
ted
p
en
d
u
lu
m
to
r
ejec
t
th
e
in
f
lu
en
ce
o
f
u
n
ce
r
tain
ties
an
d
s
tab
ilize
th
e
in
v
er
ted
p
en
d
u
lu
m
s
y
s
tem
.
Ho
wev
er
,
t
h
e
u
n
ce
r
tain
ties
in
p
ap
er
[
2
1
]
m
u
s
t
b
e
s
atis
f
ied
with
t
h
e
b
o
u
n
d
e
d
co
n
s
tr
ain
ts
.
An
o
u
tp
u
t
f
ee
d
b
ac
k
c
o
n
tr
o
ller
was
p
r
o
p
o
s
ed
in
p
ap
er
[
2
2
]
w
h
er
e
th
e
u
n
k
n
o
wn
s
tates
wer
e
esti
m
ated
b
y
th
e
h
ig
h
-
g
ain
o
b
s
er
v
e
r
.
Ho
wev
e
r
,
th
er
e
ar
e
s
ev
er
al
d
r
awb
ac
k
s
in
th
is
wo
r
k
s
u
c
h
a
s
th
e
h
ig
h
-
g
ain
o
b
s
er
v
er
is
s
en
s
itiv
e
with
m
ea
s
u
r
em
en
t
n
o
is
e
s
o
r
s
o
m
etim
es
t
h
e
p
ea
k
in
g
p
h
en
o
m
en
o
n
o
cc
u
r
s
d
u
e
to
th
e
h
ig
h
g
ain
o
f
th
e
o
b
s
er
v
er
.
An
ad
a
p
tiv
e
co
n
tr
o
ller
an
d
ad
a
p
tiv
e
f
u
zz
y
s
lid
in
g
m
o
d
e
co
n
tr
o
ller
wer
e
s
y
n
th
esized
f
o
r
in
v
er
te
d
p
en
d
u
lu
m
an
d
r
o
tar
y
in
v
e
r
ted
p
en
d
u
lu
m
s
y
s
tem
with
th
e
u
n
ce
r
tain
ties
in
[
2
3
]
,
[
2
4
]
,
r
esp
ec
tiv
ely
.
A
n
ew
ap
p
r
o
ac
h
b
ased
o
n
th
e
c
o
n
tr
o
l
ly
a
p
u
n
o
v
f
u
n
ctio
n
an
d
L
MI
s
was
in
v
esti
g
ated
t
o
s
y
n
th
esize
th
e
co
n
tr
o
ller
f
o
r
in
v
e
r
ted
p
e
n
d
u
l
u
m
s
y
s
tem
with
t
h
e
ex
is
ten
ce
o
f
th
e
u
n
ce
r
tain
ties
[
2
5
]
.
R
eg
ar
d
in
g
s
y
n
th
esizin
g
f
o
r
th
e
u
n
ce
r
t
ain
in
v
er
ted
p
en
d
u
lu
m
s
y
s
tem
,
a
s
lid
in
g
m
o
d
e
tech
n
iq
u
e
was
em
p
lo
y
ed
to
d
esig
n
an
o
b
s
er
v
er
to
ca
lcu
late
th
e
u
n
k
n
o
wn
s
tates
an
d
r
ejec
t
th
e
im
p
ac
ts
o
f
th
e
u
n
ce
r
tain
ties
.
Ho
wev
e
r
,
it
is
s
ee
n
th
at
th
e
u
n
ce
r
tain
ties
o
f
th
e
p
r
e
v
io
u
s
a
r
ticles
[
2
1
]
-
[
2
7
]
h
av
e
to
b
e
b
o
u
n
d
e
d
.
I
t
m
ea
n
s
th
at
th
e
u
p
p
er
an
d
lo
wer
b
o
u
n
d
s
o
f
th
e
u
n
ce
r
tain
ties
s
h
o
u
ld
b
e
p
r
o
v
id
ed
in
ad
v
a
n
ce
,
o
th
er
wis
e
it
is
im
p
o
s
s
ib
le
to
d
esig
n
th
e
c
o
n
tr
o
ller
an
d
o
b
s
er
v
er
f
o
r
th
ese
s
y
s
tem
s
.
R
ec
en
tly
,
a
d
is
tu
r
b
a
n
ce
o
b
s
er
v
er
h
as
b
ee
n
i
n
tr
o
d
u
ce
d
t
o
esti
m
ate
th
e
d
is
tu
r
b
an
ce
i
n
[
2
8
]
.
T
h
is
d
is
tu
r
b
an
ce
o
b
s
er
v
er
allo
ws
u
s
to
o
b
tain
in
f
o
r
m
atio
n
o
f
th
e
d
is
tu
r
b
an
ce
t
h
at
n
ee
d
s
to
c
o
n
t
r
o
l
th
e
s
y
s
tem
an
d
en
h
an
ce
th
e
co
n
tr
o
l
ac
cu
r
ac
y
o
f
th
e
s
y
s
tem
.
T
h
er
e
h
av
e
b
ee
n
m
an
y
p
r
ev
io
u
s
p
ap
e
r
s
s
tu
d
y
in
g
ab
o
u
t
th
e
d
is
tu
r
b
an
ce
o
b
s
er
v
e
r
s
u
ch
as
p
ap
er
s
[
2
9
]
-
[
3
2
]
.
Un
f
o
r
tu
n
ate
ly
,
to
th
e
b
est
o
f
o
u
r
k
n
o
wled
g
e,
th
e
d
is
tu
r
b
an
ce
o
b
s
er
v
er
h
as
n
o
t
b
ee
n
em
p
lo
y
ed
to
d
ea
l
with
th
e
in
v
er
te
d
p
en
d
u
lu
m
with
th
e
p
r
esen
ce
o
f
th
e
u
n
ce
r
tain
ties
in
p
r
ev
io
u
s
p
ap
er
s
.
Owin
g
to
t
h
is
r
ea
s
o
n
,
we
p
r
o
p
o
s
ed
a
n
ew
ap
p
r
o
ac
h
r
elied
o
n
d
is
tu
r
b
an
ce
o
b
s
er
v
er
to
esti
m
ate
th
e
u
n
k
n
o
w
n
s
tates a
n
d
th
e
u
n
ce
r
tain
ties
.
W
ith
th
e
af
o
r
em
en
tio
n
e
d
an
aly
s
es,
it
m
o
tiv
ates
u
s
to
p
r
o
p
o
s
e
a
n
ew
m
eth
o
d
to
s
y
n
th
esize
th
e
o
b
s
er
v
er
a
n
d
d
is
tu
r
b
an
ce
o
b
s
er
v
er
-
b
ased
co
n
tr
o
ller
f
o
r
i
n
v
er
ted
p
e
n
d
u
lu
m
em
p
h
asizin
g
in
th
e
f
o
llo
win
g
co
n
tr
ib
u
tio
n
s
:
−
An
o
b
s
er
v
er
-
b
ased
co
n
tr
o
ller
is
s
y
n
th
esized
to
s
tab
ilize
t
h
e
in
v
e
r
ted
p
en
d
u
lu
m
.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
r
ely
in
g
o
n
th
e
L
MI
tech
n
i
q
u
e
allo
ws
u
s
ca
n
d
eter
m
in
e
th
e
o
b
s
er
v
e
r
an
d
co
n
t
r
o
ller
g
ain
s
ea
s
ily
an
d
ef
f
icien
tly
.
T
h
e
m
eth
o
d
i
n
th
i
s
p
ap
er
will
h
elp
to
av
o
id
th
e
ch
atter
in
g
p
h
en
o
m
en
o
n
in
p
a
p
er
[
1
6
]
as
well
as
th
e
co
n
d
itio
n
s
f
o
r
d
esig
n
i
n
g
o
b
s
er
v
er
-
b
ased
co
n
tr
o
ller
is
also
m
o
r
e
r
elax
ed
in
co
m
p
ar
is
o
n
with
th
e
m
eth
o
d
in
[
1
7
]
.
I
n
a
d
d
itio
n
,
s
o
m
e
s
tate
v
ar
iab
les
o
f
in
v
er
te
d
p
en
d
u
lu
m
ar
e
n
o
t
m
ea
s
u
r
e
d
b
y
s
en
s
o
r
s
,
th
u
s
,
th
e
m
eth
o
d
s
in
[
3
]
-
[
1
2
]
ar
e
f
ailed
to
s
tab
ilize
th
e
s
y
s
tem
.
T
o
d
ea
l
with
th
ese
is
s
u
e
s
,
in
th
is
ar
ticle,
a
n
o
b
s
er
v
er
is
s
y
n
th
esized
f
o
r
r
e
p
lacin
g
th
e
s
en
s
o
r
s
to
esti
m
ate
th
e
u
n
k
n
o
wn
s
tates o
f
th
e
s
y
s
tem
.
−
A
d
is
tu
r
b
an
ce
o
b
s
er
v
er
-
b
ased
co
n
tr
o
ller
is
p
r
o
p
o
s
ed
f
o
r
th
e
in
v
er
ted
p
e
n
d
u
l
u
m
s
y
s
tem
with
u
n
ce
r
tain
ties
th
at
h
as
n
o
t
b
ee
n
f
o
u
n
d
i
n
a
n
y
p
r
ev
io
u
s
p
ap
er
.
Un
lik
e
p
r
ev
io
u
s
p
ap
er
s
[
2
1
]
-
[
2
7
]
,
th
e
u
n
c
er
tain
ties
in
th
is
s
tu
d
y
d
o
n
o
t
n
ee
d
to
f
u
lf
ill
t
h
e
b
o
u
n
d
e
d
c
o
n
s
tr
ain
ts
.
T
h
e
r
ef
o
r
e,
it
is
im
p
o
s
s
ib
le
to
em
p
lo
y
th
e
m
eth
o
d
s
in
p
ap
er
[
2
1
]
-
[
2
7
]
to
d
esig
n
a
co
n
tr
o
ller
f
o
r
o
u
r
ca
s
e.
I
n
th
is
p
ap
er
,
f
i
r
s
t
s
tep
,
th
e
u
n
ce
r
tain
ties
ar
e
tr
an
s
f
o
r
m
ed
to
th
e
in
p
u
t
d
is
tu
r
b
an
ce
s
,
an
d
th
e
n
t
h
e
d
is
tu
r
b
an
ce
o
b
s
er
v
e
r
-
b
ased
co
n
t
r
o
ller
is
s
y
n
th
esized
i
n
th
e
s
ec
o
n
d
s
tep
to
esti
m
ate
u
n
k
n
o
wn
s
tates,
an
d
in
p
u
t
d
is
tu
r
b
an
ce
s
s
im
u
ltan
e
o
u
s
ly
.
T
h
is
m
eth
o
d
h
as
th
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8
7
0
8
Dis
tu
r
b
a
n
ce
o
b
s
erver
-
b
a
s
ed
c
o
n
tr
o
ller
fo
r
in
ve
r
ted
p
en
d
u
lu
m
w
ith
…
(
V
a
n
-
P
h
o
n
g
V
u
)
4909
ad
v
an
tag
es
th
at
th
e
in
f
o
r
m
atio
n
o
f
th
e
u
n
ce
r
tain
ties
is
o
b
tain
ed
b
y
o
b
s
er
v
er
,
an
d
th
en
it
is
f
ee
d
-
b
ac
k
ed
to
th
e
co
n
tr
o
ller
to
elim
in
ate
co
m
p
letely
th
e
im
p
ac
ts
o
f
th
e
u
n
ce
r
tain
ties
an
d
in
cr
ea
s
e
th
e
co
n
tr
o
l a
cc
u
r
ac
y
.
T
h
e
r
est
o
f
th
is
ar
tic
le
is
o
r
g
a
n
ized
as
f
o
ll
o
ws.
I
n
s
ec
tio
n
2
,
th
e
r
esear
c
h
m
eth
o
d
t
h
at
in
c
lu
d
es:
th
e
m
ath
em
atica
l
m
o
d
el
o
f
in
v
e
r
ted
p
e
n
d
u
l
u
m
is
d
escr
ib
ed
,
th
e
p
r
o
b
lem
s
will
b
e
s
o
lv
e
d
,
an
d
m
eth
o
d
s
to
s
y
n
th
esize
an
o
b
s
er
v
er
-
b
ased
co
n
tr
o
ller
b
ased
o
n
L
MI
te
ch
n
iq
u
e
f
o
r
in
v
e
r
ted
p
e
n
d
u
lu
m
s
y
s
tem
with
o
u
t
u
n
ce
r
tai
n
ties
in
th
is
p
a
p
er
ar
e
s
tated
as
well.
T
h
e
s
im
u
latio
n
r
esu
lts
an
d
d
is
cu
s
s
io
n
s
f
o
r
b
o
t
h
with
an
d
with
o
u
t
u
n
ce
r
tain
ties
o
f
in
v
er
ted
p
en
d
u
lu
m
s
y
s
tem
ar
e
p
r
esen
ted
in
s
ec
tio
n
3
.
Fin
ally
,
s
ev
er
al
co
n
clu
s
io
n
s
ar
e
s
u
m
m
ar
ized
in
s
ec
tio
n
4
.
No
tatio
n
s
:
I
n
th
is
p
ap
e
r
,
Θ
>
0
(
<
0
)
in
d
icate
s
th
e
m
atr
ix
is
a
p
o
s
itiv
e
(
n
eg
ativ
e)
d
ef
in
ite.
r
ep
r
esen
t
th
e
tr
an
s
p
o
s
e
o
f
a
m
atr
ix
;
−
1
d
ef
in
ed
th
e
in
v
er
s
e
o
f
;
is
d
ef
in
ed
as
an
id
en
tity
m
atr
ix
.
+
in
d
id
ca
tes
th
e
M
o
o
r
e
-
Pen
r
o
s
e
p
s
eu
d
o
-
in
v
er
s
e
o
f
with
+
=
(
)
−
1
.
T
h
e
s
y
m
b
o
l
ℜ
×
d
en
o
tes
th
e
s
et
o
f
×
m
atr
ices.
2.
RE
S
E
ARCH
M
E
T
H
O
D
2
.
1
.
Sy
s
t
em
m
o
del
C
o
n
s
id
er
th
e
in
v
er
te
d
p
en
d
u
lu
m
o
n
a
ca
r
t in
Fig
u
r
e
1
with
th
e
n
o
n
lin
ea
r
eq
u
atio
n
as
in
(1
)
:
{
̅
̈
=
+
(
)
̇
2
−
(
)
(
)
+
−
(
(
)
)
2
̈
=
(
)
−
(
+
)
(
(
)
)
+
(
(
)
(
)
)
̇
(
(
)
)
2
−
(
+
)
(
1
)
wh
er
e
th
e
p
a
r
am
eter
s
an
d
t
h
eir
v
alu
es a
r
e
d
escr
ib
e
d
in
T
a
b
le
1
[
3
3
]
.
L
et
u
s
d
ef
in
e
1
=
̅
,
2
=
̅
̇
,
3
=
,
4
=
̇
,
th
en
th
e
(
1
)
is
wr
itten
as
(
2
)
.
[
̇
1
̇
2
̇
3
̇
4
]
=
[
2
(
3
)
−
(
+
)
(
s
i
n
(
3
)
+
(
co
s
(
3
)
s
i
n
(
3
)
)
4
(
co
s
(
1
)
)
2
−
(
+
)
4
+
(
s
i
n
(
3
)
)
4
2
−
(
3
)
s
i
n
(
3
)
+
−
(
co
s
(
3
)
)
2
]
(
2
)
L
in
ea
r
izin
g
th
e
o
r
ig
in
al
n
o
n
li
n
ea
r
s
y
s
tem
(
2
)
at
t
h
e
eq
u
ilib
r
iu
m
p
o
in
t
(
0
,
0,
0,
0
)
a
n
d
s
u
b
t
itli
n
g
th
e
v
alu
es
o
f
th
e
p
ar
am
eter
s
in
T
a
b
le
1
o
b
ta
in
s
th
e
f
o
llo
win
g
s
y
s
tem
.
{
̇
(
)
=
(
)
+
(
)
(
)
=
(
)
(
3
)
W
h
er
e
=
[
1
2
3
4
]
,
̇
=
[
̇
1
̇
2
̇
3
̇
4
]
,
=
[
0
1
.
00
0
0
0
0
−
1
.
9613
0
0
0
0
1
0
0
23
.
536
0
]
,
=
[
0
1
0
−
2
]
,
an
d
=
[
1
0
0
0
0
0
0
1
]
Fig
u
r
e
1
.
I
n
v
er
te
d
p
en
d
u
lu
m
o
n
ca
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
6
,
Dec
em
b
e
r
2
0
2
1
:
4
9
0
7
-
4
9
2
1
4910
T
ab
le
1
.
Par
am
eter
s
o
f
th
e
in
v
er
ted
p
en
d
u
lu
m
[
3
3
]
P
a
r
a
me
t
e
r
s
S
y
mb
o
l
V
a
l
u
e
U
n
i
t
M
a
ss
o
f
c
a
r
t
M
1
kg
M
a
ss
o
f
p
e
n
d
u
l
u
m
m
0
.
2
kg
Le
n
g
t
h
o
f
p
o
l
e
l
0
.
5
m
G
r
a
v
i
t
a
t
i
o
n
a
l
a
c
c
e
l
e
r
a
t
i
o
n
g
9
.
8
0
5
5
6
/
2
P
o
si
t
i
o
n
o
f
C
a
r
t
̅
V
e
l
o
c
i
t
y
o
f
C
a
r
t
̅
̇
/
A
n
g
l
e
o
f
i
n
v
e
r
t
e
d
p
e
n
d
u
l
u
m
A
n
g
l
e
v
e
l
o
c
i
t
y
̇
/
2
.
2
.
P
ro
blem
des
cr
iptio
n
Su
p
p
o
s
e
th
at
o
n
ly
t
h
e
p
o
s
itio
n
o
f
th
e
ca
r
t
(
1
=
)
an
d
a
n
g
le
ac
ce
ler
a
tio
n
(
4
=
)
̇
ar
e
m
ea
s
u
r
ed
b
y
s
en
s
o
r
s
;
an
d
th
e
v
elo
city
o
f
th
e
ca
r
t
(
2
=
̇
)
an
d
an
g
le
o
f
in
v
er
ted
p
en
d
u
l
u
m
(
3
=
)
ar
e
u
n
k
n
o
wn
.
Ho
wev
er
,
th
e
in
f
o
r
m
atio
n
o
f
t
h
ese
two
s
tate
v
ar
iab
les
is
n
e
ce
s
s
ar
y
to
s
y
n
th
esize
a
co
n
tr
o
ller
to
s
tab
ilize
th
e
s
y
s
tem
(
3
)
.
Du
e
to
th
is
r
ea
s
o
n
,
th
e
o
b
jectiv
e
o
f
th
is
p
ap
er
is
to
d
esig
n
an
o
b
s
er
v
er
-
b
ased
co
n
tr
o
ller
to
esti
m
ate
th
e
u
n
k
n
o
wn
s
tate
an
d
s
tab
ilize
th
e
s
y
s
tem
(
3
)
at
th
e
eq
u
ili
b
r
iu
m
p
o
in
t
(
0
,
0
,
0,
0
)
.
T
h
er
e
ar
e
two
s
ce
n
ar
io
s
tak
en
in
to
c
o
n
s
id
er
atio
n
i
n
th
i
s
ar
ticle.
Scen
ar
io
1
:
T
h
e
o
b
s
er
v
er
-
b
as
ed
co
n
tr
o
ller
is
s
y
n
th
esized
f
o
r
th
e
in
v
e
r
ted
p
e
n
d
u
l
u
m
s
y
s
tem
(
3
)
wh
ic
h
is
n
o
t
af
f
ec
ted
b
y
th
e
u
n
ce
r
tain
ties
.
Scen
ar
io
2
:
T
h
e
d
is
tu
r
b
an
ce
o
b
s
er
v
er
-
b
ased
co
n
tr
o
ller
is
d
es
ig
n
ed
f
o
r
th
e
in
v
er
ted
p
en
d
u
lu
m
s
y
s
tem
wh
ich
is
af
f
ec
ted
b
y
th
e
u
n
ce
r
tain
ties
.
I
t
s
h
o
u
ld
b
e
n
o
ted
th
at
th
e
u
n
ce
r
tain
ty
in
th
is
ca
s
e
is
u
n
n
ec
ess
ar
y
to
s
atis
f
y
th
e
b
o
u
n
d
ed
c
o
n
s
tr
ain
ts
.
I
n
th
is
ca
s
e,
b
o
th
th
e
u
n
k
n
o
wn
s
tates
a
n
d
u
n
ce
r
tain
ties
ar
e
esti
m
ated
asy
m
p
to
tically
an
d
f
ee
d
-
b
ac
k
ed
to
t
h
e
co
n
tr
o
ller
t
o
s
tab
ilize
th
e
s
y
s
tem
.
R
em
ar
k
1
:
I
n
th
is
p
a
p
er
,
we
ass
u
m
e
th
at
th
e
v
elo
city
o
f
th
e
ca
r
t
(
2
=
̇
)
a
n
d
an
g
le
o
f
in
v
e
r
ted
p
en
d
u
lu
m
(
3
=
)
ar
e
n
o
t
m
ea
s
u
r
ed
b
y
s
en
s
o
r
s
.
I
t
m
ea
n
s
t
h
at
th
e
in
f
o
r
m
atio
n
o
f
th
ese
two
s
tate
v
a
r
iab
les
is
u
n
k
n
o
w
n
,
th
er
ef
o
r
e,
th
e
m
eth
o
d
s
to
d
esi
g
n
co
n
tr
o
ller
f
o
r
in
v
er
ted
p
en
d
u
lu
m
in
p
ap
er
s
[
3
]
-
[
1
2
]
ar
e
u
n
ab
le
to
a
p
p
ly
f
o
r
th
is
ca
s
e.
Ad
d
itio
n
ally
,
in
th
is
s
tu
d
y
,
s
en
s
o
r
s
ar
e
n
o
t
u
s
ed
f
o
r
o
b
tain
in
g
th
e
in
f
o
r
m
a
tio
n
o
f
v
elo
city
o
f
th
e
ca
r
t
an
d
an
g
le
o
f
in
v
er
ted
p
en
d
u
lu
m
lead
in
g
to
r
ed
u
ce
th
e
co
s
t f
o
r
co
n
s
tr
u
ctin
g
th
e
s
y
s
tem
.
2
.
3
.
O
bs
er
v
er
-
ba
s
ed
co
ntr
o
ller
f
o
r
inv
er
t
ed
pend
ulu
m
I
n
th
is
s
ec
tio
n
,
an
o
b
s
er
v
er
an
d
co
n
tr
o
ller
ar
e
d
esig
n
ed
f
o
r
th
e
s
y
s
tem
(
3
)
s
im
u
lta
n
eo
u
s
ly
.
T
h
e
s
tr
u
ctu
r
e
o
f
th
e
s
y
s
tem
with
th
e
o
b
s
er
v
er
-
b
as
ed
co
n
tr
o
ller
is
d
ep
icted
in
Fig
u
r
e
2
.
L
et
u
s
tak
e
co
n
s
id
er
atio
n
th
e
o
b
s
er
v
er
f
o
r
m
f
o
r
th
e
s
y
s
tem
(
3
)
as
(
4
)
.
{
̂
̇
=
̂
+
+
(
−
̂
)
̂
=
̂
(
4
)
W
h
er
e
̂
an
d
̂
ar
e
th
e
esti
m
atio
n
o
f
th
e
s
tate
an
d
o
u
tp
u
t
,
r
e
s
p
ec
tiv
ely
.
is
th
e
o
b
s
er
v
e
r
g
a
in
wh
ich
is
co
m
p
u
ted
in
n
e
x
t sectio
n
.
T
h
e
co
n
tr
o
ller
f
o
r
m
o
f
th
e
s
y
s
tem
(
3
)
is
ex
p
r
ess
ed
as
(
5
)
.
=
−
̂
(
5
)
T
h
e
esti
m
atio
n
er
r
o
r
is
d
ef
in
e
d
:
=
−
̂
(
6
)
T
h
e
d
y
n
am
ic
ex
p
r
ess
io
n
o
f
th
e
esti
m
atio
n
er
r
o
r
is
:
̇
=
̇
−
̂
̇
(
7
)
C
o
m
b
in
in
g
(
3
)
an
d
(
4
)
,
o
n
e
o
b
tain
s
:
̇
=
(
−
)
(
8
)
Su
b
s
titu
tin
g
(
5
)
in
t
o
(
3
)
,
th
e
cl
o
s
ed
-
lo
o
p
s
y
s
tem
is
o
b
tain
e
d
̇
=
(
−
)
+
(
9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8
7
0
8
Dis
tu
r
b
a
n
ce
o
b
s
erver
-
b
a
s
ed
c
o
n
tr
o
ller
fo
r
in
ve
r
ted
p
en
d
u
lu
m
w
ith
…
(
V
a
n
-
P
h
o
n
g
V
u
)
4911
Fro
m
(
8
)
an
d
(
9
)
,
it in
f
er
s
th
at
[
̇
̇
]
=
[
−
0
−
]
[
]
(
1
0
)
Den
o
te
̃
=
[
]
an
d
̃
=
[
−
0
−
]
,
th
en
(
1
0
)
b
ec
o
m
es
̃
̇
=
̃
̃
(
1
1
)
Th
eo
r
em
1
:
T
h
e
esti
m
atio
n
e
r
r
o
r
an
d
th
e
s
tate
v
ar
iab
le
o
f
th
e
s
y
s
tem
(
3
)
with
t
h
e
o
b
s
er
v
er
(
4
)
an
d
co
n
tr
o
ller
(
5
)
c
o
n
v
e
r
g
e
to
ze
r
o
asy
m
p
to
tically
,
if
th
er
e
e
x
is
t
m
atr
ices
,
,
an
d
p
o
s
itiv
e
s
y
m
m
etr
ic
m
atr
ices
1
an
d
2
s
u
ch
th
at
th
e
f
o
llo
win
g
c
o
n
d
itio
n
h
o
ld
s
[
(
−
)
1
−
1
+
1
−
1
(
−
)
1
−
1
(
−
)
(
)
1
−
1
(
−
)
2
+
2
(
−
)
]
<
0
(
1
2
)
P
r
o
o
f:
T
h
e
L
y
ap
u
n
o
v
f
u
n
ctio
n
is
ch
o
s
en
as
(
1
3
)
;
(
)
=
̃
(
)
̃
(
)
(
1
3
)
wh
er
e
=
[
1
−
1
0
0
2
]
T
ak
in
g
th
e
d
er
iv
ativ
e
o
n
b
o
th
s
id
es o
f
(
1
3
)
y
ield
s
;
̇
(
)
=
̃
̇
(
)
̃
(
)
+
̃
(
)
̃
̇
(
)
(
1
4
)
Fro
m
(
1
1
)
an
d
(
1
4
)
,
we
o
b
tain
;
̇
(
)
=
̃
(
)
[
̃
̃
]
̃
(
)
=
̃
(
)
[
(
−
)
1
−
1
+
1
−
1
(
−
)
1
−
1
(
−
)
(
)
1
−
1
(
−
)
2
+
2
(
−
)
]
̃
(
)
(
1
5
)
I
t
is
ea
s
ily
s
ee
n
th
at
if
th
e
co
n
d
itio
n
(
1
2
)
is
s
atis
f
ied
th
en
̇
(
)
<
0
,
it
in
f
er
s
th
at
an
d
co
n
v
er
g
e
to
ze
r
o
asy
m
p
to
tically
wh
en
→
∞
.
T
h
e
p
r
o
o
f
is
co
m
p
leted
.
Un
f
o
r
tu
n
atel
y
,
it
is
o
b
v
io
u
s
th
at
th
er
e
ex
is
t
two
m
atr
ix
v
ar
ia
b
les
m
u
ltip
ly
in
g
to
g
eth
er
in
o
n
e
ter
m
o
f
th
e
co
n
d
itio
n
s
(
1
2
)
,
th
er
ef
o
r
e
co
n
d
itio
n
(
1
2
)
is
a
n
o
n
-
c
o
n
v
ex
b
ilin
ea
r
m
atr
i
x
in
eq
u
ality
(
B
MI
)
th
at
is
co
m
p
licated
to
d
eter
m
in
e
m
at
r
ix
v
ar
iab
les
,
,
1
,
an
d
2
f
u
lf
ill
t
h
e
co
n
d
itio
n
(
1
2
)
.
B
ec
au
s
e
o
f
th
is
r
ea
s
o
n
,
T
h
eo
r
em
2
n
ee
d
s
to
tr
an
s
f
o
r
m
co
n
d
itio
n
(
1
2
)
to
lin
ea
r
m
a
tr
ix
in
eq
u
ality
(
L
MI
)
wh
ich
i
s
ea
s
y
to
r
eso
lv
e
b
y
u
s
in
g
th
e
L
MI
to
o
l o
f
MA
T
L
AB
.
Th
eo
r
em
2
:
T
h
e
esti
m
atio
n
e
r
r
o
r
an
d
th
e
s
tate
v
ar
iab
le
o
f
th
e
s
y
s
tem
(
3
)
with
t
h
e
o
b
s
er
v
er
(
4
)
an
d
co
n
tr
o
ller
(
5
)
ap
p
r
o
ac
h
ze
r
o
a
s
y
m
p
to
tical
ly
,
if
th
er
e
e
x
is
t
m
atr
ices
,
,
an
d
p
o
s
itiv
e
s
y
m
m
e
tr
ic
m
atr
ices
1
an
d
2
s
u
ch
th
at
th
e
f
o
llo
win
g
c
o
n
d
itio
n
s
s
atis
f
y
.
1
−
+
1
−
<
0
(
1
6
)
2
−
+
2
−
<
0
(
1
7
)
T
h
e
o
b
s
er
v
e
r
an
d
co
n
tr
o
ller
g
a
in
s
o
b
tain
=
2
−
1
(
1
8
)
=
1
−
1
(
1
9
)
P
r
o
o
f:
Acc
o
r
d
in
g
to
t
h
e
Sch
u
r
co
m
p
lem
en
t,
(
1
2
)
is
eq
u
iv
ale
n
t to
{
(
−
)
1
−
1
+
1
−
1
(
−
)
<
0
(
20
)
(
−
)
2
+
2
(
−
)
<
0
(
20
)
Pre
an
d
p
o
s
t m
u
ltip
ly
in
g
1
with
(
2
0
a)
an
d
d
en
o
tin
g
=
1
,
(
2
0
a)
b
e
co
m
es
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
6
,
Dec
em
b
e
r
2
0
2
1
:
4
9
0
7
-
4
9
2
1
4912
1
−
+
1
−
<
0
(
2
1
)
Def
in
e
=
2
th
en
(
2
0
b
)
is
wr
itten
a
s
(
2
2
)
,
2
−
+
2
−
<
0
(
2
2
)
Fro
m
(
2
1
)
an
d
(
2
2
)
,
it
is
o
b
v
i
o
u
s
th
at
(
2
1
)
an
d
(
2
2
)
ar
e
th
e
L
MI
s
an
d
th
ey
a
r
e
th
e
s
am
e
a
s
(
1
6
)
an
d
(
1
7
)
o
f
T
h
e
o
r
em
2
.
I
t
m
ea
n
s
th
at
it
is
s
u
cc
ess
f
u
l
to
c
o
n
v
e
r
t
B
MI
(
1
2
)
o
f
T
h
e
o
r
em
1
t
o
L
MI
s
(
1
6
)
an
d
(
1
7
)
o
f
T
h
eo
r
em
2
.
T
h
e
p
r
o
o
f
is
co
m
p
leted
.
T
h
e
p
r
o
ce
d
u
r
e
f
o
r
s
y
n
th
esizin
g
is
b
r
ief
ly
p
r
esen
ted
as f
o
llo
ws.
Step
1
: So
lv
in
g
th
e
L
MI
(
1
6
)
an
d
(
1
7
)
to
o
b
tain
m
atr
ices
Z
,
P
1
,
W
,
a
n
d
P
2
.
Step
2
: T
h
e
o
b
s
er
v
er
a
n
d
co
n
t
r
o
ller
g
ain
s
an
d
ar
e
co
m
p
u
te
d
b
y
u
s
in
g
(
1
8
)
a
n
d
(
1
9
)
.
2
.
4
.
Dis
t
urba
nce
o
b
s
er
v
er
-
ba
s
ed
co
ntr
o
ller
f
o
r
inv
er
t
ed
pend
ulu
m
wit
h uncer
t
a
inties
Ass
u
m
e
th
at
th
e
in
v
er
ted
p
en
d
u
lu
m
s
y
s
tem
(
1
)
is
im
p
ac
ted
b
y
th
e
tim
e
-
v
ar
y
i
n
g
u
n
c
er
tain
ties
,
th
en
it
is
r
ewr
itten
in
th
e
f
o
llo
win
g
f
r
am
ewo
r
k
(
2
3
)
:
{
̇
(
)
=
(
+
∆
(
)
)
(
)
+
(
+
∆
(
)
)
(
)
(
)
=
(
)
(
2
3
)
wh
er
e
∆
(
)
an
d
∆
(
)
ar
e
th
e
u
n
ce
r
tain
ti
es.
W
ith
th
e
ex
is
ten
ce
o
f
t
h
e
u
n
c
er
tain
ties
,
th
e
p
er
f
o
r
m
an
c
e
o
f
th
e
s
y
s
tem
(
1
)
is
d
eg
r
a
d
ed
.
Hen
ce
,
th
e
o
b
jectiv
e
o
f
th
is
s
ec
tio
n
is
to
s
y
n
th
esize
an
o
b
s
er
v
er
-
b
a
s
ed
co
n
tr
o
ller
t
o
elim
in
ate
t
h
e
ef
f
ec
ts
o
f
th
e
u
n
ce
r
tain
ties
an
d
s
tab
ilize
th
e
s
y
s
tem
.
B
ec
au
s
e
o
f
th
e
ex
is
ten
ce
o
f
th
e
u
n
ce
r
tain
ties
,
th
e
m
eth
o
d
to
s
y
n
t
h
esize
th
e
o
b
s
er
v
e
r
-
b
ased
c
o
n
tr
o
ller
i
n
s
ec
tio
n
2
.
3
ar
e
f
ailed
to
ap
p
l
y
f
o
r
th
e
s
y
s
tem
(
1
)
.
T
h
er
ef
o
r
e,
in
th
is
s
ec
tio
n
,
a
n
ew
m
eth
o
d
b
ased
o
n
th
e
d
is
tu
r
b
an
ce
-
o
b
s
er
v
e
r
b
ased
c
o
n
tr
o
ller
is
p
r
o
p
o
s
ed
to
s
tab
i
lize
t
h
e
s
y
s
tem
(
1
)
.
T
h
e
s
tr
u
ctu
r
e
o
f
t
h
e
o
b
s
er
v
er
-
b
ase
d
co
n
tr
o
ller
f
o
r
s
y
s
tem
wit
h
u
n
ce
r
tain
ties
is
s
h
o
wn
in
Fig
u
r
e
3.
Fig
u
r
e
2
.
Stru
ctu
r
e
o
f
o
b
s
er
v
e
r
-
b
ased
co
n
tr
o
ller
f
o
r
in
v
er
ted
p
e
n
d
u
l
u
m
s
y
s
tem
Fig
u
r
e
3
.
Stru
ctu
r
e
o
f
o
b
s
er
v
e
r
-
b
ased
co
n
tr
o
ller
f
o
r
in
v
er
ted
p
e
n
d
u
l
u
m
s
y
s
tem
with
u
n
ce
r
tain
ties
Ass
u
m
p
tio
n
1
:
T
h
e
u
n
c
e
r
t
a
i
n
t
i
e
s
∆
(
)
a
n
d
∆
(
)
)
a
r
e
s
u
p
p
o
s
e
d
t
o
s
a
t
i
s
f
y
t
h
e
f
o
l
l
o
w
i
n
g
m
a
t
c
h
i
n
g
co
n
d
itio
n
s
:
∆
(
)
=
(
)
a
n
d
∆
(
)
=
(
)
.
R
ema
r
k
2
:
I
t
s
h
o
u
ld
b
e
n
o
te
d
th
at
th
e
u
n
ce
r
tain
ties
∆
(
)
an
d
∆
(
)
)
in
(
2
3
)
d
o
n
o
t
n
ee
d
to
s
atis
f
y
th
e
b
o
u
n
d
ed
co
n
s
tr
ain
ts
an
d
th
e
l
o
wer
an
d
u
p
p
e
r
-
b
o
u
n
d
e
d
v
al
u
e
ar
e
u
n
k
n
o
wn
,
o
r
in
p
r
ac
tice,
it
is
d
if
f
icu
lt
to
d
eter
m
in
e
th
e
v
alu
es
o
f
th
e
lo
wer
/u
p
p
er
b
o
u
n
d
s
o
f
th
e
u
n
ce
r
tain
ties
.
T
h
er
ef
o
r
e,
it
is
im
p
o
s
s
ib
le
to
ap
p
ly
th
e
m
eth
o
d
s
in
p
ap
e
r
s
[
2
3
-
2
7
]
f
o
r
s
y
n
th
esizin
g
c
o
n
tr
o
ller
f
o
r
th
e
s
y
s
tem
in
(
2
3
)
.
B
ec
au
s
e
th
e
p
r
ev
io
u
s
s
tu
d
ies
in
p
ap
er
s
[
2
3
-
2
7
]
,
th
e
u
p
p
er
b
o
u
n
d
s
o
f
u
n
ce
r
tain
ties
m
u
s
t
b
e
in
clu
d
ed
in
th
e
co
n
d
itio
n
s
to
d
e
s
ig
n
o
b
s
er
v
er
a
n
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8
7
0
8
Dis
tu
r
b
a
n
ce
o
b
s
erver
-
b
a
s
ed
c
o
n
tr
o
ller
fo
r
in
ve
r
ted
p
en
d
u
lu
m
w
ith
…
(
V
a
n
-
P
h
o
n
g
V
u
)
4913
co
n
tr
o
ller
.
Owin
g
to
th
is
r
ea
s
o
n
,
a
n
ew
ap
p
r
o
ac
h
b
ased
o
n
d
is
tu
r
b
an
ce
o
b
s
er
v
e
r
-
b
ased
co
n
tr
o
ller
to
elim
in
ate
th
e
in
f
lu
en
ce
s
o
f
th
e
u
n
ce
r
tain
ties
an
d
s
tab
ilize
th
e
s
y
s
tem
is
in
v
esti
g
ated
in
th
is
s
ec
tio
n
.
R
ema
r
k
3
:
T
h
e
m
atch
in
g
co
n
d
itio
n
s
in
ass
u
m
p
tio
n
1
ar
e
n
ee
d
ed
to
tr
a
n
s
f
o
r
m
th
e
u
n
ce
r
tain
ties
in
to
th
e
u
n
k
n
o
wn
in
p
u
t
d
is
tu
r
b
an
ce
.
T
h
is
ass
u
m
p
tio
n
ca
n
b
e
f
o
u
n
d
in
p
r
ev
io
u
s
p
a
p
er
s
s
u
ch
as
[
1
5
]
,
[
1
6
]
a
n
d
[
3
4
]
.
Ho
wev
er
,
we
ass
u
m
e
t
h
at
th
is
ass
u
m
p
tio
n
is
s
till
co
n
s
er
v
ati
v
e
wh
en
co
m
m
o
n
m
atr
ix
B
is
u
s
ed
to
d
ec
o
m
p
o
s
e
th
e
u
n
ce
r
tain
ties
∆
(
)
an
d
∆
(
)
.
I
t is st
il
l a
n
o
p
e
n
is
s
u
e
th
at
n
ee
d
to
s
o
lv
e
in
f
u
t
u
r
e
wo
r
k
.
L
em
m
a
1
[
3
5
]
:
T
ak
e
n
in
to
ac
co
u
n
t
o
f
th
e
m
atr
ix
eq
u
atio
n
=
wh
er
e
∈
ℝ
×
,
≥
,
an
d
∈
ℝ
×
.
T
h
e
g
e
n
er
al
s
o
lu
tio
n
o
f
th
e
ab
o
v
e
m
atr
i
x
eq
u
atio
n
is
ex
p
r
ess
ed
in
t
h
e
f
o
r
m
=
+
+
(
−
+
)
in
wh
ich
∈
ℝ
×
is
an
ar
b
itra
r
y
m
atr
ix
with
ap
p
r
o
p
r
iate
d
im
en
s
io
n
an
d
+
=
(
)
−
1
is
th
e
Mo
o
r
e
-
Pen
r
o
s
e
p
s
eu
d
o
in
v
e
r
s
e
o
f
A.
Un
d
er
ass
u
m
p
tio
n
1
,
th
e
s
y
s
te
m
(
2
3
)
b
ec
o
m
es
(
2
4
)
:
{
̇
(
)
=
(
)
+
(
)
+
(
(
)
(
)
+
(
)
(
)
)
(
)
=
(
)
(
2
4
)
Den
o
te
(
)
=
(
)
(
)
+
(
)
(
)
T
h
en
th
e
s
y
s
tem
(
2
4
)
is
m
o
d
if
ied
as
(
2
5
)
:
{
̇
(
)
=
(
)
+
(
(
)
+
(
)
)
(
)
=
(
)
(
2
5
)
I
t
is
s
ee
n
th
at
u
n
d
er
ass
u
m
p
ti
o
n
1
,
th
e
in
v
er
ted
p
en
d
u
lu
m
s
y
s
tem
(
2
3
)
with
u
n
ce
r
tain
ties
h
as
b
ee
n
tr
an
s
f
o
r
m
ed
to
th
e
s
y
s
tem
(
2
5
)
with
th
e
u
n
k
n
o
wn
in
p
u
t
d
is
tu
r
b
an
ce
.
Fr
o
m
n
o
w
o
n
,
th
e
co
n
tr
o
ller
an
d
o
b
s
er
v
er
will
b
e
s
y
n
th
esized
f
o
r
th
e
s
y
s
tem
(
2
5
)
in
s
tead
o
f
(
2
3
)
.
Ad
d
itio
n
ally
,
i
n
th
is
s
ec
tio
n
,
a
d
is
tu
r
b
a
n
ce
o
b
s
er
v
er
is
s
y
n
th
esized
f
o
r
th
e
s
y
s
tem
(
2
5
)
to
esti
m
ate
th
e
u
n
k
n
o
wn
s
tates
an
d
th
e
d
is
tu
r
b
an
ce
(
)
s
im
u
ltan
eo
u
s
ly
.
Af
ter
th
at
th
i
s
in
f
o
r
m
atio
n
is
f
ee
d
-
b
ac
k
ed
to
th
e
co
n
tr
o
ller
t
o
m
ak
e
th
e
I
n
v
er
ted
Pen
d
u
lu
m
s
tab
le
at
th
e
eq
u
iv
alen
t
p
o
in
t (
0
,
0,
0,
0
)
.
C
o
n
s
id
er
th
e
d
is
tu
r
b
an
ce
o
b
s
e
r
v
er
f
o
r
m
as
(
2
6
)
:
{
̇
(
)
=
(
)
+
(
)
+
(
)
̂
(
)
=
(
)
−
(
)
̂
(
)
=
(
)
+
̇
(
)
−
̂
(
)
−
(
)
(
2
6
)
in
wh
ich
̂
(
)
an
d
̂
(
)
ar
e
th
e
esti
m
at
io
n
o
f
th
e
s
tate
(
)
,
an
d
d
is
tu
r
b
a
n
ce
(
)
,
r
esp
ec
tiv
ely
.
,
,
,
,
an
d
ar
e
th
e
o
b
s
er
v
er
g
ain
s
o
f
th
e
o
b
s
er
v
e
r
(
2
6
)
.
(
)
+
=
[
(
)
(
)
]
−
1
(
)
is
th
e
Mo
o
r
e
-
Ps
eu
d
o
in
v
er
t
o
f
(
)
.
T
h
e
co
n
tr
o
ller
f
o
r
m
is
ex
p
r
ess
ed
in
th
e
f
o
llo
win
g
f
r
am
ewo
r
k
(
2
7
)
:
(
)
=
−
̂
(
)
−
̂
(
)
(
2
7
)
L
et
u
s
d
ef
in
e
th
e
esti
m
atio
n
e
r
r
o
r
s
:
(
)
=
̂
(
)
−
(
)
(
2
8
)
Su
b
s
titu
tin
g
(
2
6
)
in
to
(
2
8
)
y
ield
s
:
(
)
=
(
)
−
(
)
−
(
)
=
(
)
−
(
)
(
2
9
)
wh
er
e
=
[
+
]
T
ak
in
g
th
e
d
er
iv
ativ
e
o
f
(
2
9
)
,
o
n
e
o
b
tai
n
s
:
̇
(
)
=
̇
(
)
−
̇
(
)
(
3
0
)
Fro
m
(
2
5
)
,
(
2
6
)
a
n
d
(
3
0
)
,
we
h
av
e
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
6
,
Dec
em
b
e
r
2
0
2
1
:
4
9
0
7
-
4
9
2
1
4914
̇
(
)
=
[
(
)
+
(
)
+
(
)
]
−
[
(
)
+
(
(
)
+
(
)
)
]
=
(
)
+
[
−
+
]
(
)
+
[
−
]
(
)
−
(
)
(
3
1
)
Den
o
te
th
e
esti
m
atio
n
er
r
o
r
o
f
th
e
d
is
tu
r
b
an
ce
:
(
)
=
̂
(
)
−
(
)
(
3
2
)
C
o
m
b
in
in
g
(
2
3
)
,
(
2
6
)
,
an
d
(
3
2
)
o
b
tain
s
(
3
3
)
:
(
)
=
(
)
+
[
+
(
+
(
)
)
]
−
̂
(
)
−
(
)
−
(
)
=
(
)
+
(
)
+
(
)
+
(
)
+
(
)
+
(
)
(
)
)
−
̂
(
)
−
(
)
−
(
)
=
−
(
)
+
[
−
(
)
+
]
(
)
(
3
3
)
Th
eo
r
em
3
:
T
h
e
s
tates
o
f
th
e
s
y
s
tem
(
2
3
)
,
th
e
esti
m
atio
n
er
r
o
r
s
o
f
s
tates
(
2
8
)
,
an
d
th
e
esti
m
atio
n
er
r
o
r
o
f
th
e
d
is
tu
r
b
an
ce
(
3
2
)
ap
p
r
o
ac
h
z
er
o
asy
m
p
to
tically
if
th
er
e
e
x
is
t
th
e
p
o
s
itiv
e
s
y
m
m
etr
ic
m
atr
ix
̅
,
an
d
t
h
e
m
atr
ices
,
,
,
,
,
an
d
s
u
ch
th
at
t
h
e
f
o
llo
win
g
co
n
d
itio
n
s
s
atis
f
y
:
−
+
=
0
(
3
4
)
−
=
0
(
3
5
)
=
0
(
3
6
)
−
(
)
+
=
0
(
3
7
)
Ξ
̅
̃
(
)
+
̅
Ξ
<
0
(
3
8
)
whe
r
e
=
[
−
1
0
0
]
,
Ξ
=
[
−
+
0
]
,
is
a
p
o
s
itiv
e
s
ca
lar
.
P
r
o
o
f:
I
f
th
e
co
n
d
itio
n
s
(
3
4
)
-
(
3
7
)
h
o
ld
th
en
th
e
d
y
n
a
m
ic
esti
m
atio
n
er
r
o
r
(
3
1
)
an
d
th
e
d
is
t
u
r
b
an
ce
esti
m
atio
n
er
r
o
r
(
3
3
)
b
ec
o
m
es
(
3
9
)
;
̇
(
)
=
(
)
(
3
9
)
an
d
(
)
=
−
(
)
(
4
0
)
Fro
m
(
4
0
)
,
it
is
s
ee
n
th
at
if
th
e
esti
m
atio
n
er
r
o
r
(
)
→
0
wh
en
→
∞
th
en
th
e
esti
m
atio
n
e
r
r
o
r
o
f
th
e
d
is
tu
r
b
an
ce
in
(
4
0
)
co
n
v
er
g
es
to
ze
r
o
as
well.
T
h
er
ef
o
r
e,
we
m
e
r
e
l
y
n
e
e
d
t
o
s
y
n
t
h
e
s
i
z
e
t
h
e
o
b
s
e
r
v
e
r
t
o
m
a
k
e
t
h
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
a
p
p
r
o
a
c
h
z
e
r
o
t
h
e
n
t
h
e
e
s
t
i
m
a
t
i
o
n
e
r
r
o
r
o
f
t
h
e
d
i
s
t
u
r
b
a
n
c
e
au
to
m
atica
lly
co
n
v
er
g
es to
ze
r
o
.
Fro
m
(
2
3
)
,
(
2
7
)
,
an
d
(
4
0
)
,
it in
f
er
s
th
at
̇
(
)
=
(
)
−
̂
(
)
−
̂
(
)
+
(
)
=
(
−
)
(
)
+
(
−
)
(
)
(
4
1
)
C
o
m
b
in
in
g
(
3
9
)
a
n
d
(
4
1
)
y
ield
s
:
[
̇
(
)
̇
(
)
]
=
[
−
−
0
]
[
(
)
(
)
]
(
4
2
)
Den
o
te;
̃
(
)
=
[
(
)
(
)
]
,
Ξ
=
[
−
−
0
]
th
en
(
4
2
)
b
ec
o
m
e
̃
̇
(
)
=
Ξ
̃
(
)
(
4
3
)
C
h
o
o
s
e
th
e
ly
ap
u
n
o
v
f
u
n
ctio
n
;
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8
7
0
8
Dis
tu
r
b
a
n
ce
o
b
s
erver
-
b
a
s
ed
c
o
n
tr
o
ller
fo
r
in
ve
r
ted
p
en
d
u
lu
m
w
ith
…
(
V
a
n
-
P
h
o
n
g
V
u
)
4915
(
̃
(
)
)
=
̃
(
)
̅
̃
(
)
(
4
4
)
in
wh
ich
̅
=
[
−
1
0
0
]
Fro
m
(
4
4
)
,
we
ca
n
h
av
e
(
4
5
)
;
̇
(
̃
(
)
)
=
̃
̇
(
)
̅
̃
(
)
+
̃
(
)
̅
̃
̇
(
)
(
4
5
)
C
o
m
b
in
in
g
(
4
3
)
a
n
d
(
4
5
)
y
ield
s
(
4
6
)
;
̇
(
̃
(
)
)
=
[
Ξ
̃
(
)
]
̅
̃
(
)
+
̃
(
)
̅
[
Ξ
̃
(
)
]
=
̃
(
)
[
Ξ
̅
̃
(
)
+
̅
Ξ
]
̃
(
)
(
4
6
)
Fro
m
(
4
6
)
,
it
is
s
ee
n
th
at
if
th
e
co
n
d
itio
n
(
3
8
)
o
f
T
h
eo
r
em
3
h
o
l
d
s
th
en
̇
(
̃
(
)
)
<
0
,
it
m
ea
n
s
th
at
̃
(
)
=
[
(
)
(
)
]
→
0
wh
en
→
∞
asy
m
p
to
tically
.
T
h
e
p
r
o
o
f
is
s
u
cc
ess
f
u
lly
co
m
p
leted
.
Un
f
o
r
tu
n
atel
y
,
th
e
co
n
d
itio
n
(
3
8
)
o
f
T
h
eo
r
em
3
is
B
MI
th
at
is
h
ar
d
to
r
eso
lv
e
in
MA
T
L
A
B
to
o
b
tain
b
o
th
o
b
s
er
v
er
a
n
d
co
n
tr
o
ller
g
ain
s
.
T
h
e
f
o
llo
win
g
T
h
e
o
r
em
i
s
n
ee
d
ed
to
tr
a
n
s
f
o
r
m
t
h
e
B
MI
(
3
8
)
i
n
to
L
MI
.
Th
eo
r
em
4
:
T
h
e
s
tates
o
f
th
e
s
y
s
tem
(
2
3
)
,
th
e
esti
m
atio
n
er
r
o
r
o
f
t
h
e
s
tate
v
ar
ia
b
les
(
2
8
)
,
an
d
th
e
esti
m
atio
n
er
r
o
r
o
f
th
e
d
is
tu
r
b
a
n
ce
(
3
2
)
c
o
n
v
er
g
e
to
ze
r
o
asy
m
p
to
tically
if
th
er
e
ex
is
t
th
e
p
o
s
itiv
e
s
y
m
m
etr
ic
m
atr
ic
,
R,
an
d
th
e
m
atr
ices
,
,
,
,
,
an
d
s
u
ch
th
at
th
e
f
o
llo
win
g
c
o
n
d
it
io
n
s
h
o
ld
:
+
+
+
+
̅
+
̅
−
−
<
0
(
4
7
)
+
−
−
<
0
(
4
8
)
ℎ
=
Γ
+
Ω
(
4
9
)
=
−
(
)
+
(
5
0
)
=
−
(
)
(
)
+
(
5
1
)
=
(
5
2
)
̅
=
(
5
3
)
=
(
5
4
)
T
h
e
o
b
s
er
v
e
r
an
d
co
n
tr
o
ller
g
a
in
s
ar
e
co
m
p
u
ted
as
(
5
5
)
-
(
5
9
)
:
=
−
(
5
5
)
=
(
+
)
−
(
5
6
)
=
(
5
7
)
=
(
)
+
(
5
8
)
=
−
1
(
5
9
)
P
r
o
o
f
:
Fro
m
(
3
8
)
,
o
n
e
o
b
tain
s
(
2
6
)
;
[
−
−
0
]
[
−
1
0
0
]
+
[
−
1
0
0
]
[
−
−
0
]
=
=
[
[
−
1
+
−
1
−
−
1
−
−
1
]
[
−
1
−
−
1
]
[
−
1
−
−
1
]
+
]
<
0
(
6
0
)
L
et
u
s
d
ef
in
e
Λ
=
−
1
+
−
1
−
−
1
−
−
1
,
Δ
=
−
1
−
−
1
,
an
d
=
+
,
th
en
s
u
b
s
titu
tin
g
in
t
o
(
6
0
)
y
i
eld
s
(
6
1
)
;
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
6
,
Dec
em
b
e
r
2
0
2
1
:
4
9
0
7
-
4
9
2
1
4916
[
Λ
Δ
]
(
∗
)
Φ
]
<
0
(
6
1
)
Acc
o
r
d
in
g
t
o
Sch
u
r
c
o
m
p
lem
e
n
t,
(
6
1
)
is
eq
u
iv
alen
t t
o
(
6
2
)
;
{
Φ
<
0
Λ
−
Δ
(
Φ
)
−
1
(
Δ
)
<
0
(
6
2
)
B
ec
au
s
e
Φ
<
0
,
Δ
(
Φ
)
−
1
(
Δ
)
≤
0
.
T
h
u
s
,
Λ
<
Δ
(
Φ
)
−
1
(
Δ
)
≤
0
.
T
h
en
(
6
2
)
i
s
eq
u
iv
alen
t to
(
6
3
)
;
{
Φ
<
0
Λ
<
0
(
6
3
)
in
wh
ich
=
+
an
d
Λ
=
−
1
+
−
1
−
−
1
−
−
1
Firstl
y
,
let
u
s
tak
e
in
to
ac
co
u
n
t o
f
th
e
m
atr
i
x
i
n
eq
u
ality
(
6
3
)
,
it in
f
er
s
th
at
(
6
4
)
;
=
+
<
0
(
6
4
)
I
t
is
ea
s
ily
s
ee
n
th
at
R
an
d
X
ar
e
b
o
th
v
ar
iab
les,
h
e
n
ce
(
6
4
)
is
a
B
MI
th
at
is
h
ar
d
to
s
o
lv
e
in
Ma
tlab
.
T
h
e
f
o
llo
win
g
s
tep
s
ar
e
to
co
n
v
er
t BMI
(
6
4
)
to
L
MI
.
Fro
m
(
3
6
)
,
we
h
av
e
(
6
5
)
,
(
6
6
)
;
[
+
]
=
0
(
6
5
)
(
)
=
−
(
6
6
)
Acc
o
r
d
in
g
t
o
L
em
m
a
1
,
th
e
g
e
n
e
r
al
s
o
lu
tio
n
o
f
(
6
6
)
is
(
6
7
)
:
=
Γ
+
Ω
(
6
7
)
in
wh
ich
=
−
(
)
+
,
=
−
(
)
(
)
+
,
is
an
ar
b
itra
r
y
m
atr
ix
with
a
co
m
p
atib
le
d
im
en
s
io
n
.
Ad
d
in
g
a
s
lack
v
ar
iab
le
L
wh
i
ch
is
d
ef
in
ed
as
(
6
8
)
.
=
+
(
6
8
)
Fro
m
(
3
4
)
an
d
(
6
8
)
,
it y
ield
s
(
6
9
)
;
=
−
(
6
9
)
C
o
m
b
in
in
g
(
6
8
)
a
n
d
(
6
9
)
y
ield
s
(
7
0
)
;
=
(
+
)
−
(
7
0
)
Su
b
s
titu
tin
g
(
6
7
)
in
to
(
6
9
)
y
ield
s
(
7
1
)
;
=
(
+
)
−
=
(
+
(
Γ
+
Ω
)
)
−
(
7
1
)
Su
b
s
titu
tin
g
(
7
1
)
in
to
(
6
4
)
r
esu
lts
in
(
7
2
)
;
Θ
=
[
(
+
(
Γ
+
Ω
)
)
−
]
+
[
(
+
(
Γ
+
Ω
)
)
−
]
=
+
+
+
+
+
−
−
<
0
(
7
2
)
Den
o
te
=
(
7
3
)
̅
=
(
7
4
)
̅
=
(
7
4
)
Evaluation Warning : The document was created with Spire.PDF for Python.