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1159
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An
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CC B
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Facu
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Un
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M
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Djir
3
1
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5
9
@
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co
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1.
I
NT
RO
D
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A
Ham
ilto
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s
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s
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Ham
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[
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,
[
3
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L
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eq
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(
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ar
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m
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s
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l
in
c
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t
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s
y
s
tem
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esis
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p
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ly
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∞
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Ham
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well
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ate
d
b
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[
4
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,
[
5
]
.
T
h
e
n
u
m
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ical
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lu
tio
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s
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ted
in
[
6
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an
d
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v
elo
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in
[
7
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.
T
h
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R
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q
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atio
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,
ed
ited
b
y
Ab
o
u
-
Kan
d
il
et
a
l.
[
8
]
,
is
a
s
u
cc
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ct
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R
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s
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m
er
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s
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lu
tio
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m
eth
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d
s
.
Og
ata
[
9
]
p
r
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th
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class
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q
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L
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No
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o
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I
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1
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I
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3
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20
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4
1160
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esire
d
o
s
cillato
r
y
b
eh
av
io
r
.
I
n
o
r
d
er
to
o
v
e
r
co
m
e
ce
r
tain
d
if
f
icu
lties
r
elate
d
t
o
th
e
s
tr
at
eg
y
a
n
d
co
n
tr
o
l
wea
k
n
ess
es,
L
o
n
g
[
1
2
]
p
r
o
p
o
s
es
a
h
y
b
r
i
d
co
n
t
r
o
l
s
tr
at
eg
y
f
o
r
r
o
b
o
tic
m
a
n
ip
u
lato
r
s
t
h
at
co
m
b
in
es
R
icca
ti
eq
u
atio
n
–
b
ased
g
ain
d
esig
n
(
L
QR
/AR
E
)
,
s
lid
in
g
m
o
d
e
co
n
tr
o
l
(
SMC
)
an
d
a
d
ap
tiv
e
o
b
s
er
v
er
f
o
r
s
tate
esti
m
atio
n
.
Fro
m
d
if
f
ic
u
lties
d
u
e
to
th
e
co
m
p
lex
ity
o
f
t
h
e
co
n
tr
o
l
ler
m
ath
em
atica
lly
an
d
co
m
p
u
tatio
n
ally
,
ad
d
itio
n
ally
m
u
ltip
le
p
ar
am
eter
s
o
f
o
b
s
er
v
er
a
n
d
SMC
g
ain
s
m
u
s
t
b
e
ca
r
e
f
u
lly
t
u
n
ed
.
R
o
v
ed
a
an
d
Pig
a
[
1
3
]
s
h
o
ws
th
e
c
o
m
p
u
t
atio
n
al
d
em
an
d
s
o
f
s
o
lv
in
g
th
e
SDR
E
o
n
lin
e
p
o
s
e
a
ch
allen
g
e,
as
r
ea
l
-
tim
e
u
p
d
atin
g
o
f
co
n
tr
o
l
g
ain
s
is
r
eq
u
ir
ed
f
o
r
e
f
f
ec
tiv
e
f
o
r
ce
co
n
tr
o
l.
Ov
er
all,
t
h
e
p
a
p
er
d
em
o
n
s
tr
ates
h
o
w
SDR
E
-
b
ased
v
ar
iab
le
im
p
ed
an
ce
c
o
n
tr
o
l
ca
n
a
d
d
r
ess
th
ese
d
if
f
icu
lties
,
en
a
b
lin
g
r
o
b
u
s
t,
s
en
s
o
r
less
f
o
r
ce
-
tr
ac
k
in
g
in
d
y
n
am
ic
r
o
b
o
tic
m
an
i
p
u
latio
n
task
s
.
Ç
im
en
[
1
4
]
E
x
p
lain
s
h
o
w
s
tate
-
d
ep
e
n
d
en
t
R
icca
ti
e
q
u
atio
n
(
SDR
E
)
co
n
v
er
ts
n
o
n
lin
ea
r
s
y
s
tem
s
i
n
to
a
p
s
eu
d
o
-
lin
ea
r
f
o
r
m
,
allo
win
g
r
ea
l
-
tim
e
n
o
n
lin
ea
r
o
p
tim
al
co
n
tr
o
l.
Ho
wev
er
,
th
e
d
if
f
ic
u
lties
ar
is
e
in
p
ar
am
ete
r
izatio
n
d
ep
en
d
e
n
ce
;
s
tab
ilit
y
is
n
o
t
g
u
ar
an
teed
g
lo
b
ally
;
with
co
m
p
lex
ity
in
tu
n
in
g
th
e
weig
h
tin
g
m
atr
ices.
Xin
an
d
B
alak
r
is
h
n
an
[
1
5
]
d
ev
elo
p
a
n
SDR
E
-
b
ased
co
n
tr
o
l
ap
p
r
o
a
ch
th
at
in
co
r
p
o
r
ates
r
o
b
u
s
tn
e
s
s
to
h
an
d
le
m
o
d
el
u
n
ce
r
tain
ties
an
d
d
is
tu
r
b
an
c
es,
im
p
r
o
v
in
g
s
tab
ilit
y
an
d
tr
ac
k
in
g
p
er
f
o
r
m
an
ce
in
n
o
n
lin
ea
r
r
o
b
o
tic
m
an
ip
u
lato
r
s
.
Ne
k
o
o
[
1
6
]
p
r
o
p
o
s
es
a
m
o
d
el
-
r
ef
e
r
en
ce
a
d
ap
tiv
e
SDR
E
co
n
tr
o
ller
f
o
r
n
o
n
lin
ea
r
u
n
ce
r
tain
s
y
s
tem
s
an
d
ap
p
lies
it sp
ec
if
ically
to
r
eg
u
latio
n
an
d
t
r
ac
k
in
g
o
f
f
r
ee
-
f
lo
atin
g
s
p
ac
e
m
an
ip
u
lato
r
s
.
On
th
e
o
th
er
h
an
d
,
th
e
m
o
d
el
u
n
ce
r
tain
ties
ar
e
n
o
t
ea
s
y
to
h
a
n
d
le;
a
n
d
t
h
e
co
m
p
u
tatio
n
al
co
m
p
lex
ity
ass
o
ciate
d
with
th
e
o
n
lin
e
SDR
E
s
o
lu
tio
n
is
a
s
ig
n
if
ican
t
ch
allen
g
e.
Sh
awk
y
et
a
l
.
[
1
7
]
,
[
1
8
]
p
r
esen
ts
an
SD
R
E
-
b
ased
n
o
n
lin
ea
r
H∞
co
n
tr
o
l
s
ch
em
e
f
o
r
f
lex
i
b
le
m
an
ip
u
lato
r
s
,
en
h
an
cin
g
r
o
b
u
s
tn
ess
ag
ain
s
t
d
is
tu
r
b
a
n
c
es
wh
ile
r
ed
u
cin
g
v
ib
r
atio
n
s
an
d
im
p
r
o
v
in
g
tr
ac
k
in
g
p
er
f
o
r
m
a
n
ce
.
T
h
e
p
r
o
b
le
m
ar
is
es
in
h
an
d
lin
g
f
lex
ib
le
-
lin
k
v
ib
r
atio
n
s
an
d
th
e
co
m
p
u
tatio
n
al
e
f
f
o
r
t
f
o
r
n
o
n
lin
e
ar
SDR
E
,
ef
f
ec
tiv
el
y
h
an
d
lin
g
v
ib
r
atio
n
s
u
p
p
r
es
s
io
n
an
d
p
ar
am
ete
r
s
en
s
itiv
ity
r
em
ain
s
a
d
if
f
ic
u
lt
task
.
Ko
r
ay
e
m
an
d
Nek
o
o
[
1
9
]
d
ev
elo
p
s
a
n
SDR
E
-
b
ased
co
n
tr
o
l
m
eth
o
d
f
o
r
tim
e
-
v
ar
y
in
g
n
o
n
li
n
ea
r
m
an
ip
u
lato
r
s
.
T
h
e
ch
allen
g
e
lies
in
m
ain
tain
in
g
s
tab
ilit
y
u
n
d
e
r
n
o
n
lin
ea
r
v
ar
iatio
n
s
;
co
m
p
u
tatio
n
al
c
o
m
p
lex
ity
an
d
s
en
s
itiv
ity
to
p
ar
am
eter
u
n
ce
r
tain
ties
.
Mo
r
eo
v
e
r
,
Ho
an
g
an
d
Kh
a
n
g
[
2
0
]
p
r
esen
ts
an
a
d
ap
tiv
e
R
icca
ti
-
b
ased
co
n
tr
o
l
m
eth
o
d
f
o
r
r
o
b
o
tic
m
an
ip
u
lato
r
s
,
ad
d
r
ess
in
g
n
o
n
lin
e
ar
ities
an
d
p
ar
am
eter
u
n
ce
r
tain
ties
to
en
s
u
r
e
ac
cu
r
ate
tr
ajec
to
r
y
tr
ac
k
in
g
an
d
r
o
b
u
s
t
s
y
s
tem
p
er
f
o
r
m
a
n
ce
.
T
h
e
wo
r
k
b
y
Saleem
et
a
l
.
[
2
1
]
h
as
s
ev
er
al
ch
allen
g
es.
First,
u
n
d
e
r
-
ac
tu
a
ted
s
y
s
tem
s
ar
e
h
ar
d
to
co
n
tr
o
l
b
ec
au
s
e
th
er
e
ar
e
f
ewe
r
in
p
u
ts
th
an
m
o
v
em
e
n
ts
.
T
h
e
r
o
b
o
t’
s
n
o
n
lin
ea
r
b
e
h
av
i
o
r
also
m
ak
es c
o
n
t
r
o
l m
o
r
e
d
if
f
icu
lt.
T
h
e
ad
a
p
tiv
e
weig
h
t
ad
ju
s
tm
en
t
n
ee
d
s
ca
r
e
f
u
l
tu
n
i
n
g
;
in
ad
d
itio
n
,
th
e
m
eth
o
d
r
eq
u
ir
es
h
ig
h
co
m
p
u
tati
o
n
,
wh
ic
h
ca
n
lim
it
r
ea
l
-
tim
e
u
s
e.
I
t c
an
also
b
e
s
en
s
itiv
e
to
n
o
is
e
an
d
d
is
tu
r
b
an
c
es.
Ho
wev
er
,
wh
en
th
e
s
tate
n
u
m
b
er
is
im
p
o
r
tan
t
it
is
n
o
t
ea
s
y
to
f
in
d
th
e
an
aly
tic
AR
E
s
o
l
u
tio
n
f
o
r
a
s
tate
d
ep
en
d
en
t
co
ef
f
icien
t,
s
i
n
ce
it
is
d
if
f
icu
lt
to
f
in
d
eig
e
n
v
alu
es
-
eig
en
v
ec
to
r
s
v
al
u
es,
esp
ec
ially
wh
en
th
e
s
tate
n
u
m
b
er
is
h
ig
h
er
.
A
n
an
aly
tical
m
eth
o
d
f
o
r
ca
lc
u
latin
g
eig
en
v
al
u
es
-
eig
en
v
ec
to
r
s
o
f
th
e
d
if
f
u
s
io
n
ten
s
o
r
d
ir
ec
tly
f
r
o
m
th
e
d
i
f
f
u
s
io
n
te
n
s
o
r
elem
en
ts
h
as b
ee
n
e
x
am
in
ed
b
y
[
2
2
]
.
Sin
ce
th
e
eig
en
v
alu
es
o
f
an
y
m
atr
ix
ar
e
th
e
s
am
e
as
th
o
s
e
o
f
its
tr
an
s
p
o
s
e
an
d
th
e
eig
en
v
alu
es
o
f
a
m
atr
ix
ar
e
th
e
r
ec
ip
r
o
ca
ls
o
f
t
h
o
s
e
o
f
its
in
v
er
s
e,
it
co
u
ld
b
e
co
n
clu
d
ed
th
at
th
e
eig
e
n
v
al
u
es
o
f
Ham
ilto
n
ian
m
atr
ix
H
ca
n
b
e
wr
itten
as
s
tab
le
p
ar
t
an
d
u
n
s
tab
le
p
ar
t
wi
th
o
n
ly
s
ig
n
c
h
an
g
i
n
g
.
T
h
e
s
tab
le
eig
en
v
alu
es
o
f
th
e
m
atr
ix
ar
e
r
elate
d
to
th
e
d
y
n
am
ics
o
f
th
e
cl
o
s
ed
-
lo
o
p
o
p
tim
al
co
n
t
r
o
l
s
y
s
tem
.
T
h
er
ef
o
r
e,
th
e
u
n
iq
u
e
s
tab
ilizin
g
s
o
lu
tio
n
ca
n
b
e
o
b
tain
ed
b
y
c
o
n
s
tr
u
ctin
g
an
in
v
ar
ian
t
s
u
b
s
p
ac
e
ass
o
ciate
d
with
th
e
s
tab
le
eig
en
v
alu
es
o
f
th
e
Ham
ilto
n
i
an
m
atr
ix
H
[
2
3
]
.
Hen
ce
th
e
Ham
ilto
n
ian
m
atr
ix
h
as
b
ee
n
in
tr
o
d
u
ce
d
in
th
is
co
n
tex
t
to
an
aly
ze
t
h
e
s
tab
ilized
s
o
lu
tio
n
.
Fo
r
tu
n
ately
f
o
r
t
wo
d
eg
r
ee
s
(
s
tate
n
u
m
b
e
r
n
=
2
)
an
d
th
r
ee
d
eg
r
ee
s
(
n
=
3
)
o
f
f
r
ee
d
o
m
,
t
h
er
e
ar
e
alwa
y
s
p
o
s
s
ib
ilit
ie
s
to
f
in
d
an
aly
tic
eig
en
v
alu
es
an
d
eig
en
v
ec
to
r
s
,
h
en
ce
th
e
an
aly
tic
s
o
lu
tio
n
o
f
AR
E
ca
n
b
e
co
m
p
u
ted
.
T
h
e
d
er
iv
ed
c
o
n
tr
o
ller
th
en
co
m
b
in
es
th
e
attr
ac
tiv
e
f
ea
tu
r
es
o
f
H∞
o
p
tim
al
co
n
tr
o
ller
an
d
th
e
ad
v
a
n
tag
es
o
f
th
e
b
ac
k
s
tep
p
in
g
tec
h
n
iq
u
e
.
T
h
e
b
ac
k
s
tep
p
in
g
tec
h
n
iq
u
e
u
s
ed
with
H∞
th
eo
r
y
h
as
b
ee
n
d
esig
n
ed
b
y
b
r
ea
k
in
g
d
o
w
n
co
m
p
lex
n
o
n
lin
ea
r
s
y
s
tem
s
in
to
s
m
aller
s
u
b
s
y
s
tem
s
o
f
two
o
r
th
r
ee
s
tates.
Per
f
o
r
m
a
n
ce
is
s
u
es
o
f
th
e
co
n
tr
o
ller
a
r
e
illu
s
tr
ated
in
a
s
im
u
latio
n
s
tu
d
y
m
ad
e
f
o
r
a
2
-
DOF
s
y
s
tem
with
s
tate
d
ep
en
d
en
t c
o
ef
f
icien
ts
.
T
h
e
p
ap
e
r
is
o
r
g
an
ized
as
f
o
llo
ws
: in
tr
o
d
u
ctio
n
illu
s
tr
ated
in
s
ec
tio
n
1
.
Ham
ilto
n
ian
m
atr
ix
f
o
r
m
alis
m
in
s
ec
tio
n
2
.
E
ig
e
n
v
alu
es
an
d
eig
e
n
v
ec
to
r
s
illu
s
tr
ated
in
s
ec
tio
n
3
.
Gr
am
–
Sch
m
id
t
Flo
w
ch
ar
t
in
s
ec
tio
n
4
.
T
h
e
ap
p
licatio
n
to
t
wo
r
o
b
o
t
a
r
m
s
is
illu
s
tr
ated
i
n
s
ec
tio
n
5
.
Fin
ally
,
co
m
m
en
ts
with
a
co
n
cl
u
s
io
n
a
r
e
p
r
esen
ted
.
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:
2088
-
8
7
0
8
A
n
a
lytic
a
lg
eb
r
a
ic
R
icc
a
ti so
lu
tio
n
fo
r
a
r
o
b
u
s
t c
o
n
tr
o
l sys
tem:
a
p
p
lica
tio
n
to
…
(
Men
a
d
Mer
iem
)
1161
2.
H
AM
I
L
T
O
N
I
AN
M
AT
RIX
2
.
1
.
Def
ini
t
io
n
1
L
et
th
e
s
q
u
a
r
e
m
atr
i
x
∈
2
×
2
d
ef
in
e
d
b
y
=
[
0
−
0
]
,
with
∈
×
is
a
ze
r
o
m
atr
ix
,
∈
×
is
th
e
id
e
n
tity
m
atr
i
x
,
th
e
n
a
m
atr
ix
∈
2
×
2
is
ca
lled
Ham
ilto
n
ian
if
is
s
y
m
m
etr
ic,
s
o
:
+
=
0
.
No
te
th
at:
=
−
.
2
.
2
.
P
r
o
po
s
it
io
n
1
L
et
a
H
am
ilto
n
ian
an
d
(
)
is
th
e
ch
ar
ac
ter
is
tic
p
o
ly
n
o
m
ial
o
f
th
e
m
atr
ix
,
th
en
:
(
)
=
(
−
)
.
2
.
3
.
Def
ini
t
io
n
2
L
et
th
e
d
y
n
a
m
ic
s
y
s
tem
with
a
s
tate
d
ep
en
d
en
t c
o
ef
f
icien
t
b
e
:
̇
=
(
)
+
(
)
+
(
)
an
d
th
e
2
×
2
h
am
ilto
n
ian
m
at
r
ix
b
e
p
r
esen
ted
as:
=
[
−
]
with
=
−
1
,
is
a
s
y
m
m
etr
ic
m
at
r
ix
(
=
)
an
d
is
a
d
iag
o
n
al
m
at
r
ix
.
L
et
th
e
co
lu
m
n
s
o
f
[
11
,
21
]
,
11
,
21
∈
×
s
p
an
a
-
in
v
ar
ian
t,
-
d
im
en
s
io
n
a
l,
th
en
th
e
f
o
llo
win
g
e
q
u
atio
n
h
o
ld
s
:
[
−
]
[
11
21
]
=
[
11
21
]
,
∈
×
(
1
)
with
11
is
ass
u
m
ed
n
o
n
s
in
g
u
lar
,
we
o
b
tain
f
r
o
m
(
1
)
:
11
+
21
=
11
11
−
1
11
+
11
−
1
21
=
a
nd
11
−
21
=
21
=
21
(
11
−
1
11
+
11
−
1
21
)
21
11
−
1
11
+
21
11
−
1
21
−
11
+
21
=
0
−
21
11
−
1
+
21
11
−
1
21
11
−
1
−
−
21
11
−
1
=
0
s
ettin
g
−
21
11
−
1
=
we
g
et:
+
−
−
1
+
=
0
(
2
)
T
h
e
s
o
lu
tio
n
o
f
th
e
AR
E
is
th
en
o
b
tain
ed
,
an
d
is
s
y
m
m
etr
ic
an
d
s
tab
ilizin
g
(
2
)
.
Sin
ce
th
e
m
atr
ix
is
r
ea
l,
it
ca
n
b
e
s
h
o
wn
th
at
th
e
s
o
lu
tio
n
=
−
21
11
−
1
is
also
r
ea
l.
Hen
ce
th
e
f
o
llo
wi
n
g
th
eo
r
em
h
o
ld
:
Th
eo
r
em
2
.
1
:
Su
p
p
o
s
e
th
e
p
ai
r
(
,
)
is
co
n
tr
o
llab
le
an
d
th
e
p
air
(
,
)
is
o
b
s
er
v
ab
le.
W
e
as
s
u
m
e
th
at
is
p
o
s
itiv
e
s
em
id
ef
in
ite
an
d
=
−
1
,
wh
er
e
is
p
o
s
itiv
e
d
ef
in
ite.
1
.
T
h
en
th
e
2
×
2
Ham
ilto
n
ian
m
atr
ix
[
−
]
h
as
n
o
p
u
r
e
im
a
g
in
ar
y
eig
en
v
alu
es.
I
f
is
an
eig
en
v
alu
e
o
f
H,
th
en
−
is
al
s
o
an
eig
en
v
alu
e
o
f
.
T
h
u
s
h
as
n
eig
en
v
alu
es
in
th
e
o
p
en
lef
t
h
al
f
p
lan
e
an
d
eig
en
v
alu
es in
th
e
o
p
en
r
i
g
h
t h
alf
p
lan
e.
2.
I
f
th
e
2
×
m
atr
ix
[
11
21
]
h
as
co
lu
m
n
s
th
at
co
m
p
r
is
e
a
b
asis
f
o
r
th
e
in
v
ar
ian
t
s
u
b
s
p
ac
e
o
f
ass
o
ciate
d
with
th
e
eig
en
v
alu
es o
f
in
th
e
lef
t
h
alf
p
la
n
e
(
th
e
s
tab
le
in
v
a
r
ian
t
s
u
b
s
p
ac
e
)
,
th
e
n
11
is
in
v
er
tib
le
an
d
=
−
21
11
−
1
is
a
s
o
lu
tio
n
to
th
e
alg
eb
r
ai
c
R
icca
ti
eq
u
atio
n
.
m
o
r
e
o
v
er
is
s
y
m
m
etr
ic
an
d
p
o
s
itiv
e
d
ef
in
ite
an
d
t
h
e
in
p
u
t
.
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.
1
6
,
No
.
3
,
J
u
n
e
20
2
6
:
1
1
5
9
-
1
1
7
4
1162
(
,
)
=
−
−
1
(
,
)
(
,
)
(
)
m
in
im
izes th
e
co
s
t f
u
n
ctio
n
:
=
∫
∞
0
(
(
)
(
)
+
(
)
(
)
)
Fo
r
p
r
o
o
f
o
n
e
ca
n
r
ef
er
to
[
2
4
]
.
3.
E
I
G
E
NVA
L
UE
S
-
E
I
G
E
NV
E
CT
O
RS F
O
R
M
UL
AT
I
O
N
3
.
1
.
E
ig
env
a
lues
C
ase1
:
L
et
H
is
a
4
×
4
m
atr
ix
i.
e
A
is
a
2
×
2
m
atr
ix
.
T
h
en
its
ch
ar
ac
ter
is
tic
eq
u
atio
n
ca
n
b
e
ex
p
r
ess
ed
as:
(
)
=
de
t
(
−
)
wh
er
e
is
th
e
d
eter
m
in
an
t
an
d
is
th
e
eig
en
v
alu
es
o
f
.
Fo
r
=
4
.
T
h
e
C
ay
ley
-
Ham
ilto
n
th
e
o
r
e
m
[
2
5
]
s
tate
th
at:
(
)
=
4
+
2
2
+
0
=
0
T
h
e
co
ef
f
icien
ts
o
f
ca
n
b
e
d
ir
ec
tly
wr
itten
in
ter
m
s
o
f
co
m
p
lete
B
ell
p
o
ly
n
o
m
ials
b
y
co
m
p
ar
i
n
g
th
is
ex
p
r
ess
io
n
with
th
e
g
e
n
er
ati
n
g
f
u
n
ctio
n
o
f
th
e
B
ell
p
o
l
y
n
o
m
ial.
No
te
th
at
B
ell
p
o
l
y
n
o
m
ials
p
r
o
v
id
e
a
p
o
wer
f
u
l
t
o
o
l
in
co
m
b
i
n
ato
r
y
an
d
an
al
y
s
is
,
p
ar
ticu
lar
ly
f
o
r
r
ep
r
esen
tin
g
s
et
p
ar
titi
o
n
s
an
d
f
o
r
s
im
p
lify
i
n
g
th
e
co
m
p
u
tatio
n
o
f
h
ig
h
er
-
o
r
d
er
d
er
iv
ativ
es in
n
o
n
lin
ea
r
s
y
s
tem
s
[
2
6
]
.
Dif
f
er
en
tiatio
n
o
f
th
is
ex
p
r
e
s
s
io
n
with
r
esp
ec
t
to
allo
ws
th
e
d
eter
m
in
atio
n
o
f
th
e
g
en
er
ic
co
ef
f
icien
ts
o
f
th
e
ch
ar
ac
ter
is
t
ic
p
o
ly
n
o
m
ial
f
o
r
g
en
er
al
,
as
d
eter
m
in
an
ts
o
f
×
m
atr
ices,
th
e
n
:
−
=
(
−
1
)
!
de
t
(
)
(
3
)
=
|
|
(
)
−
1
0
…
0
(
2
)
(
)
−
2
…
⋮
⋮
⋮
⋮
⋮
⋮
(
(
−
1
)
)
(
(
−
2
)
)
…
…
1
(
)
(
(
−
1
)
)
…
⋯
(
)
|
|
with
=
an
d
(
)
=
0
=
h
en
ce
:
2
=
−
1
2
(
2
)
0
=
de
t
(
)
=
(
1
8
2
(
2
)
−
1
4
(
4
)
)
if
is
a
lin
ea
r
tr
an
s
f
o
r
m
atio
n
f
r
o
m
a
v
ec
to
r
s
p
ac
e
o
v
er
a
f
ield
in
to
its
elf
an
d
is
a
v
ec
to
r
in
th
at
is
n
o
t
th
e
ze
r
o
v
ec
to
r
,
t
h
en
is
an
ei
g
en
v
ec
to
r
o
f
if
(
)
is
a
s
ca
lar
m
u
ltip
le
o
f
.
T
h
is
co
n
d
itio
n
ca
n
b
e
w
r
itten
as th
e
eq
u
atio
n
:
(
)
=
wh
er
e
is
a
s
ca
lar
in
th
e
f
ield
,
k
n
o
wn
as
th
e
eig
en
v
alu
e,
ch
ar
ac
ter
is
tic
v
alu
e,
o
r
ch
ar
ac
ter
is
tic
r
o
o
t
ass
o
ciate
d
with
th
e
eig
en
v
ec
to
r
.
T
h
e
s
o
lu
tio
n
o
f
th
e
ca
r
ac
ter
is
tic
eq
u
atio
n
ca
n
b
e
r
ep
r
esen
t
ed
as:
1
,
2
,
3
,
4
=
±
(
1
4
(
2
)
±
1
2
√
(
4
)
−
1
4
2
(
2
)
)
1
2
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:
2088
-
8
7
0
8
A
n
a
lytic
a
lg
eb
r
a
ic
R
icc
a
ti so
lu
tio
n
fo
r
a
r
o
b
u
s
t c
o
n
tr
o
l sys
tem:
a
p
p
lica
tio
n
to
…
(
Men
a
d
Mer
iem
)
1163
C
ase2
:
L
et
H
i
s
a
6
×
6
m
atr
ix
i.
e
.,
A
is
a
3
×
3
m
atr
ix
.
T
h
en
its
c
h
ar
ac
ter
is
tic
eq
u
atio
n
ca
n
b
e
ex
p
r
ess
ed
as:
de
t
(
−
)
=
6
+
4
4
+
2
2
+
0
=
0
co
ef
f
icien
ts
ar
e
d
e
f
in
ed
b
y
th
e
d
eter
m
in
an
t
o
f
th
e
(
3
)
d
ef
in
e
d
ab
o
v
e:
0
=
−
1
48
3
(
2
)
+
1
8
(
2
)
(
4
)
−
1
6
(
6
)
2
=
1
8
2
(
2
)
−
1
4
(
4
)
4
=
−
1
2
(
2
)
L
et
Λ
=
2
,
th
en
th
e
c
h
ar
ac
ter
is
tic
eq
u
atio
n
ca
n
b
e
r
ep
lace
d
b
y
:
Λ
3
+
4
Λ
2
+
2
Λ
+
0
=
0
T
h
e
s
o
lu
tio
n
o
f
th
e
eig
e
n
v
alu
e
s
eq
u
atio
n
is
as f
o
llo
w
[
2
2
]
:
=
c
os
(
3
2
)
3
=
(
−
4
3
)
2
−
(
2
3
)
=
−
1
72
2
(
2
)
+
1
12
(
4
)
=
(
−
4
3
)
3
+
4
2
6
−
0
2
=
1
216
3
(
2
)
−
1
24
(
2
)
(
4
)
+
1
12
(
6
)
T
h
e
s
o
r
ted
eig
e
n
v
alu
es:(
1
=
−
6
)
>
(
2
=
−
5
)
>
(
3
=
−
4
)
ar
e
th
en
r
ep
r
esen
ted
as f
o
llo
w
[
2
7
]
:
1
,
6
=
±
√
−
4
3
+
2
1
2
c
os
(
)
2
,
5
=
±
√
−
4
3
−
2
1
2
c
os
(
3
+
)
(
4
)
3
,
4
=
±
√
−
4
3
−
2
1
2
c
os
(
3
−
)
3
.
2
.
E
ig
env
ec
t
o
rs
Giv
en
a
Ham
ilto
n
ian
2
×
2
s
q
u
ar
e
m
atr
ix
o
f
r
ea
l
n
u
m
b
er
s
,
an
ei
g
en
v
alu
e
an
d
its
ass
o
ciate
d
g
en
er
alize
d
eig
e
n
v
ec
to
r
ar
e
a
p
air
o
b
e
y
in
g
th
e
r
elatio
n
:
(
−
)
=
0
Giv
en
th
e
s
q
u
ar
e
m
atr
ix
,
b
y
m
in
o
r
o
f
a
n
elem
e
n
t
,
we
m
e
an
th
e
v
alu
e
o
f
t
h
e
d
ete
r
m
in
a
n
t
o
b
tain
e
d
b
y
d
eletin
g
th
e
ℎ
r
o
w
an
d
ℎ
co
lu
m
n
o
f
m
atr
ix
.
I
t
is
d
en
o
ted
b
y
.
I
n
o
r
d
er
to
f
in
d
th
e
ℎ
eig
en
v
ec
to
r
co
m
p
u
ted
f
o
r
,
we
co
m
p
u
te
t
h
e
d
eter
m
in
a
n
ts
o
f
t
h
e
m
in
o
r
s
r
elate
d
to
t
h
e
ℎ
r
o
w
o
f
th
e
s
q
u
ar
e
m
atr
ix
,
s
o
we
h
av
e
to
er
ase
o
u
t
a
r
o
w
a
n
d
a
c
o
lu
m
n
o
n
e
b
y
o
n
e
at
th
e
tim
e.
T
h
e
f
o
llo
win
g
s
tep
s
ar
e
u
s
ed
to
co
m
p
u
te
m
in
o
r
s
f
r
o
m
a
m
atr
ix
:
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.
1
6
,
No
.
3
,
J
u
n
e
20
2
6
:
1
1
5
9
-
1
1
7
4
1164
L
et
th
e
f
u
n
ctio
n
:
−
=
[
−
−
1
−
−
]
T
h
en
th
e
eig
e
n
v
ec
to
r
will b
e
r
ep
r
esen
ted
as:
=
[
1
2
…
]
with
=
2
,
=
1
.
.
2
,
=
1
.
.
4
f
o
r
∈
ℝ
4
×
4
an
d
=
1
.
.
3
,
=
1
.
.
6
f
o
r
∈
ℝ
6
×
6
.
4.
O
RT
H
O
G
O
NALI
T
Y
AND
M
O
DIFI
E
D
G
RA
M
–
SCH
M
I
DT
O
RG
AN
I
G
RA
M
T
h
e
Gr
am
–
Sch
m
id
t
p
r
o
ce
s
s
is
a
m
eth
o
d
f
o
r
o
r
th
o
g
o
n
alizi
n
g
a
s
et
o
f
v
ec
to
r
s
in
an
in
n
er
p
r
o
d
u
ct
s
p
ac
e,
m
o
s
t c
o
m
m
o
n
ly
th
e
E
u
clid
ea
n
s
p
ac
e
eq
u
ip
p
ed
with
t
h
e
s
tan
d
ar
d
in
n
er
p
r
o
d
u
ct.
T
h
e
Gr
am
–
Sch
m
id
t
p
r
o
ce
s
s
tak
es
a
f
in
ite,
li
n
ea
r
ly
in
d
ep
e
n
d
en
t
s
et
=
{
1
,
.
.
.
,
}
f
o
r
≤
an
d
g
e
n
er
ates
an
o
r
th
o
g
o
n
al
s
et
′
=
{
1
,
.
.
.
,
}
th
at
s
p
an
s
th
e
s
am
e
-
d
im
en
s
io
n
al
s
u
b
s
p
ac
e
o
f
as
.
Defin
itio
n
4
.
1
A
s
et
o
f
v
ec
to
r
s
{
,
1
≤
≤
}
is
o
r
th
o
g
o
n
al
if
⋅
=
0
wh
en
ev
e
r
≠
.
Hen
ce
1
=
1
=
−
∑
−
1
=
1
(
)
w
ith
(
)
=
=
tak
in
g
th
e
m
atr
ix
r
e
p
r
esen
ted
b
y
:
=
[
1
2
…
]
No
te
th
e
eig
en
v
alu
es
h
a
v
e
b
ee
n
s
o
r
ted
in
an
ascen
d
i
n
g
way
,
s
o
th
e
eig
en
v
ec
to
r
s
will
also
b
e.
T
h
en
ca
n
b
e
r
ep
r
esen
ted
in
a
m
atr
ix
f
o
r
m
a
s
:
=
[
1
1
×
1
2
×
2
1
×
2
2
×
]
Fin
ally
,
th
e
p
o
s
itiv
e
d
ef
i
n
ite
s
o
lu
tio
n
f
o
r
R
icca
ti
eq
u
atio
n
wi
ll b
e
p
r
esen
ted
as:
(
)
=
−
21
(
)
11
−
1
(
)
5.
AP
P
L
I
CA
T
I
O
N
W
e
co
n
s
id
er
a
two
-
d
e
g
r
ee
-
of
-
f
r
ee
d
o
m
(
2
-
DOF)
p
lan
ar
r
o
b
o
tic
ar
m
co
n
s
is
tin
g
o
f
two
r
i
g
id
lin
k
s
:
1
an
d
2
with
m
ass
es
1
,
an
d
2
r
esp
ec
tiv
ely
.
T
h
e
two
r
ev
o
l
u
te
jo
in
t
s
ar
e
1
an
d
2
with
a
m
o
m
en
t
o
f
i
n
er
tia
1
,
2
r
esp
ec
tiv
ely
.
T
h
e
e
n
d
-
ef
f
ec
to
r
m
o
v
es
in
a
2
p
lan
e
(
p
lan
e)
F
ig
u
r
e
1
,
with
t
h
e
g
o
al
o
f
d
esi
g
n
in
g
a
r
o
b
u
s
t
co
n
t
r
o
ller
to
tr
ac
k
d
e
s
ir
ed
tr
ajec
to
r
ies.
T
o
s
im
p
lif
y
th
e
d
y
n
am
ic
m
o
d
el
wh
ile
r
etain
in
g
ess
en
tial
n
o
n
lin
ea
r
ities
,
th
e
f
o
llo
win
g
a
s
s
u
m
p
tio
n
s
ar
e
m
ad
e:
a.
Ma
s
s
d
is
tr
ib
u
tio
n
:
T
h
e
m
ass
o
f
ea
ch
lin
k
is
co
n
ce
n
tr
ated
at
its
tip
,
s
o
th
e
ce
n
ter
-
of
-
m
ass
d
i
s
tan
ce
s
ar
e
s
e
t
to
th
e
lin
k
len
g
th
s
:
1
=
1
an
d
2
=
2
b.
Neg
lect
r
o
tatio
n
al
in
er
tia:
T
h
e
m
o
m
en
ts
o
f
in
e
r
tia
o
f
th
e
li
n
k
s
ab
o
u
t
th
eir
ce
n
ter
s
o
f
m
a
s
s
ar
e
ass
u
m
ed
n
eg
lig
ib
le.
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:
2088
-
8
7
0
8
A
n
a
lytic
a
lg
eb
r
a
ic
R
icc
a
ti so
lu
tio
n
fo
r
a
r
o
b
u
s
t c
o
n
tr
o
l sys
tem:
a
p
p
lica
tio
n
to
…
(
Men
a
d
Mer
iem
)
1165
Un
d
er
t
h
ese
ass
u
m
p
tio
n
s
,
t
h
e
m
an
ip
u
lato
r
’
s
e
q
u
atio
n
o
f
m
o
tio
n
is
e
x
p
r
ess
ed
i
n
s
tan
d
ar
d
r
o
b
o
tic
d
y
n
am
ic
f
o
r
m
:
(
)
̈
+
(
,
̇
)
̇
+
(
)
=
(
5
)
W
ith
M
ϵ
R
2
×
2
,
C
ϵ
R
2
×
2
,
G
ϵ
R
1
×
2
T
h
e
v
elo
city
o
f
lin
k
1
’
s
ce
n
te
r
o
f
m
ass
is
1
2
=
̇
1
2
+
̇
1
2
with
1
=
1
c
os
(
1
)
,
1
=
1
s
in
(
1
)
,
an
d
̇
1
=
−
1
̇
1
s
in
(
1
)
,
̇
1
=
1
̇
1
c
os
(
1
)
,
h
en
ce
1
2
=
1
2
̇
1
2
.
T
h
e
k
in
etic
en
er
g
y
o
f
lin
k
1
will b
e:
1
=
1
2
1
1
2
=
1
2
1
1
2
̇
1
2
.
T
h
e
v
elo
city
o
f
lin
k
2
’
s
ce
n
t
er
o
f
m
ass
is
2
2
=
̇
2
2
+
̇
2
2
with
2
=
1
c
os
(
1
)
+
2
c
os
(
1
+
2
)
,
2
=
1
s
in
(
1
)
+
2
s
in
(
1
+
2
)
,
an
d
̇
1
=
−
1
̇
1
s
in
(
1
)
−
2
(
̇
1
+
̇
2
)
s
in
(
1
+
2
)
,
̇
1
=
1
̇
1
c
os
(
1
)
+
2
(
̇
1
+
̇
2
)
c
os
(
1
+
2
)
,
h
en
ce
2
2
=
1
2
̇
1
2
+
2
(
̇
1
+
̇
2
)
2
+
2
1
2
̇
1
(
̇
1
+
̇
2
)
c
os
(
2
)
.
T
h
e
k
in
etic
en
er
g
y
o
f
lin
k
2
will b
e:
2
=
1
2
2
2
2
=
1
2
2
(
1
2
̇
1
2
+
2
(
̇
1
+
̇
2
)
2
+
2
1
2
̇
1
(
̇
1
+
̇
2
)
c
os
(
2
)
)
.
T
h
e
to
tal
en
er
g
y
will b
e:
=
1
+
2
wh
ich
ca
n
b
e
p
r
esen
ted
as
[
̇
1
̇
2
]
[
̇
1
̇
2
]
.
W
ith
(
1
,
1
)
=
(
1
+
2
)
1
2
+
2
2
2
+
2
2
1
2
c
os
(
2
)
(
2
,
1
)
=
2
2
2
+
2
1
2
c
os
(
2
)
(
1
,
2
)
=
2
2
2
+
2
1
2
c
os
(
2
)
(
2
,
2
)
=
2
2
2
T
h
e
C
h
r
is
to
f
f
el
f
o
r
m
u
la
g
i
v
es
th
e
co
ef
f
icien
t
o
f
C
o
r
io
lis
=
∑
1
2
2
=
1
(
+
−
)
̇
h
en
ce
:
(
1
,
2
)
=
−
2
2
1
2
s
in
(
2
)
̇
1
̇
2
−
2
1
2
s
in
(
2
)
̇
2
2
(
2
,
1
)
=
2
1
2
s
in
(
2
)
̇
1
2
T
h
e
g
r
a
v
ity
to
r
q
u
e
is
g
iv
en
b
y
:
(
)
=
[
1
2
]
,
U
is
th
e
p
o
ten
tial e
n
er
g
y
.
T
h
en
:
(
1
,
1
)
=
(
1
+
2
)
1
s
in
(
1
)
+
2
2
s
in
(
1
+
2
)
(
1
,
2
)
=
2
2
s
in
(
1
+
2
)
Sin
ce
is
in
v
er
tib
le,
th
e
(
5
)
ca
n
b
e
r
ewr
itten
as:
̈
=
−
−
1
(
,
̇
)
̇
−
−
1
(
)
+
−
1
(
6
)
T
ak
in
g
1
=
1
;
2
=
2
;
̇
1
=
3
;
̇
2
=
4
;
let
=
[
1
,
2
,
3
,
4
]
th
en
th
e
m
o
d
el
ca
n
b
e
r
ep
r
esen
ted
as:
̇
=
(
)
+
(
)
+
(
)
(
7
)
W
ith
(
)
=
[
0
2
×
2
2
×
2
0
2
×
2
−
−
1
]
;
(
)
=
[
0
2
×
2
−
1
]
;
(
)
=
[
0
2
×
2
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.
1
6
,
No
.
3
,
J
u
n
e
20
2
6
:
1
1
5
9
-
1
1
7
4
1166
1
(
)
=
[
(
1
+
2
)
1
2
2
0
2
2
]
;
=
[
s
in
(
1
)
s
in
(
1
+
2
)
]
L
et
1
=
[
1
2
]
T
h
en
th
e
c
o
n
ca
ten
atio
n
o
f
t
h
e
m
o
d
el
in
to
two
p
ar
ts
ca
n
b
e
r
e
p
r
esen
ted
as:
̇
1
=
2
(
8
)
̇
2
=
2
+
2
+
1
(
9
)
T
h
e
p
r
o
ce
d
u
r
e
o
f
b
ac
k
s
tep
p
in
g
tech
n
iq
u
e
ca
n
b
e
in
v
esti
g
ate
d
in
s
ec
tio
n
5
.
1
to
5
.
3
.
Fig
u
r
e
1
.
2
-
DOF
r
o
b
o
t
ar
m
5
.
1
.
S
t
ep
1
L
et
̇
1
=
2
u
s
in
g
b
ac
k
s
tep
p
in
g
tech
n
i
q
u
e
[
2
8
]
we
g
et:
̇
1
=
1
1
=
1
−
1
̇
1
=
̇
1
−
̇
1
=
̇
1
−
1
L
et
̇
1
=
1
1
+
1
1
+
11
1
w
ith
1
=
0
;
1
=
;
11
=
0
;
1
=
̇
1
−
1
T
h
e
co
n
tr
o
l la
w
lead
s
to
1
=
−
1
−
1
1
=
−
1
−
1
1
1
1
f
o
r
1
=
0
;
1
=
an
d
11
=
0
;
in
th
at
ca
s
e
th
e
r
icca
t
i e
q
u
atio
n
b
ec
o
m
es
−
̇
1
=
−
1
1
−
1
+
1
(
1
0
)
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:
2088
-
8
7
0
8
A
n
a
lytic
a
lg
eb
r
a
ic
R
icc
a
ti so
lu
tio
n
fo
r
a
r
o
b
u
s
t c
o
n
tr
o
l sys
tem:
a
p
p
lica
tio
n
to
…
(
Men
a
d
Mer
iem
)
1167
with
1
is
a
s
y
m
m
etr
ic
m
atr
ix
an
d
1
is
a
d
iag
o
n
al
m
atr
ix
; th
e
n
th
e
co
n
tr
o
l la
w
is
ca
lcu
lated
as:
1
=
−
1
−
1
1
=
̇
1
−
1
T
h
en
:
1
=
(
1
−
1
1
+
̇
1
)
(
1
1
)
T
h
e
Ham
ilto
n
ian
m
atr
i
x
is
1
=
[
1
1
1
−
1
1
1
−
1
]
with
=
[
1
0
0
2
]
;
=
[
1
0
0
2
]
,
s
o
:
=
1
−
=
[
−
0
1
1
0
0
−
0
1
2
1
0
−
0
0
2
0
−
]
C
o
m
p
u
tin
g
th
e
d
eter
m
in
a
n
t o
f
will g
iv
e:
de
t
(
)
=
(
2
2
−
2
)
(
2
1
−
1
)
(
1
2
)
H
en
ce
th
e
eig
en
v
alu
es a
r
e:
1
=
−
√
1
1
;
2
=
−
√
2
2
;
3
=
√
1
1
;
4
=
√
2
2
T
h
e
eig
en
v
ec
to
r
s
ca
n
b
e
d
eter
m
in
ed
th
r
o
u
g
h
t
h
e
d
eter
m
in
a
n
t
o
f
m
in
o
r
s
o
f
.
I
t
is
d
en
o
ted
b
y
.
L
et
11
b
e
th
e
m
in
o
r
o
f
th
e
h
am
ilto
n
ian
m
atr
ix
b
y
d
eletin
g
th
e
1
r
o
w
an
d
1
co
lu
m
n
.
Hen
ce
f
o
r
1
1
=
1
.
.
4
ar
e
co
m
p
u
ted
an
d
we
g
et:
1
=
[
−
1
3
+
2
2
1
0
−
1
1
2
+
1
2
2
0
]
=
[
√
1
1
(
1
1
+
2
2
)
0
−
1
2
1
+
1
2
2
0
]
an
d
2
2
=
1
.
.
4
ar
e
co
m
p
u
ted
an
d
we
g
et
:
2
=
[
0
2
3
−
2
2
2
0
2
2
2
−
1
2
2
]
=
[
0
√
2
2
(
2
2
−
1
1
)
0
2
2
2
−
1
2
1
]
5
.
1
.
1
O
rt
ho
g
o
na
liza
t
io
n
L
et
1
=
1
u
s
in
g
Gr
am
–
Sch
m
id
t o
r
g
an
ig
r
am
lead
s
to
:
2
=
2
−
1
2
1
1
1
=
2
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.
1
6
,
No
.
3
,
J
u
n
e
20
2
6
:
1
1
5
9
-
1
1
7
4
1168
C
o
m
p
u
tin
g
th
e
two
eig
en
v
ec
t
o
r
s
1
an
d
2
is
s
u
f
f
icien
t to
f
in
d
th
e
m
atr
ix
.
So
:
[
1
2
]
4
×
2
=
[
11
(
2
×
2
)
21
(
2
×
2
)
]
W
ith
11
=
[
√
1
1
(
1
1
−
2
2
)
0
0
√
2
2
(
2
2
−
1
1
)
]
21
=
[
−
1
2
1
+
1
2
2
0
0
2
2
2
−
1
2
1
]
Hen
ce
=
−
21
11
−
1
Fin
ally
=
[
√
1
1
0
0
√
2
2
]
No
te
th
at
to
av
o
i
d
s
in
g
u
lar
ity
f
o
r
th
is
ca
s
e
we
ch
o
o
s
e
1
1
≠
2
2
5
.
2
.
S
t
ep
2
L
et
th
e
s
y
s
tem
b
e:
̇
2
=
2
+
2
+
1
(
1
2
)
T
h
e
ap
p
licatio
n
f
o
r
(
9
)
with
1
=
5
;
2
=
2
;
1
=
2
=
0
.
34
;
=
9
.
81
−
2
th
e
m
o
d
el
is
r
ep
r
esen
ted
as:
=
−
1011
.
5
+
289
2
(
2
)
;
11
=
(
−
289
(
1
+
(
2
)
)
(
2
)
3
2
)
/
12
=
(
−
578
(
2
)
4
(
3
+
.
5
4
)
)
/
21
=
(
−
144
.
5
(
−
9
−
4
(
2
)
)
(
2
)
3
2
)
/
22
=
(
578
(
1
+
(
2
)
)
(
2
)
4
(
3
+
.
5
4
)
)
/
11
=
1250
×
23
.
3478
12
=
1250
×
6
.
6708
−
8338
.
5
(
1
+
(
2
)
)
21
=
1250
×
23
.
3478
(
1
+
(
2
)
)
22
=
1250
×
6
.
6708
−
4169
.
25
(
−
9
−
4
(
2
)
)
T
h
en
Evaluation Warning : The document was created with Spire.PDF for Python.