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er
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n
ce
r
tain
ty
[
1
3
]
.
T
h
e
c
o
m
b
i
n
atio
n
o
f
L
y
ap
u
n
o
v
s
tab
ilit
y
th
eo
r
y
with
f
u
zz
y
l
o
g
ic
co
n
tr
o
l
s
y
s
tem
s
ca
n
ad
ju
s
t
co
n
tr
o
l
f
o
r
ce
s
r
ea
l
tim
e,
wh
ich
ef
f
ec
tiv
e
in
m
an
ag
in
g
co
m
p
lex
n
o
n
lin
ea
r
s
y
s
tem
s
.
T
h
is
m
eth
o
d
h
as
s
h
o
wn
th
e
f
lex
ib
ilit
y
an
d
ef
f
ec
tiv
en
ess
in
co
n
tr
o
llin
g
v
ar
io
u
s
n
o
n
lin
ea
r
s
y
s
tem
s
[
1
4
]
.
I
n
m
u
lti
-
ag
en
t
s
y
s
tem
s
,
L
y
ap
u
n
o
v
-
b
ased
co
n
tr
o
l
m
eth
o
d
h
as
b
ee
n
d
o
n
e
to
r
ed
u
ce
th
e
ef
f
ec
ts
o
f
f
alse
-
d
at
a
-
in
jectio
n
attac
k
.
B
y
u
s
in
g
n
eu
r
al
n
etwo
r
k
s
f
o
r
s
tate
esti
m
atio
n
an
d
s
tab
ilit
y
an
aly
s
is
,
th
ese
co
n
tr
o
ller
s
en
s
u
r
e
r
o
b
u
s
tn
ess
u
n
d
er
attac
k
co
n
d
itio
n
s
[
1
5
]
.
R
ec
en
t
ad
v
a
n
ce
s
lev
er
ag
e
d
ee
p
n
eu
r
al
n
etwo
r
k
s
(
DNNs)
to
a
p
p
r
o
x
im
ate
L
y
a
p
u
n
o
v
f
u
n
ctio
n
s
f
o
r
s
tab
ilit
y
ass
ess
m
en
t
[
1
6
]
,
[
1
7
]
.
R
ec
tifie
d
lin
ea
r
u
n
it
(
R
eL
U
)
an
d
a
d
ap
t
iv
e
weig
h
t
-
u
p
d
ate
laws
g
u
id
e
d
b
y
L
y
a
p
u
n
o
v
th
eo
r
y
e
n
ab
le
r
o
b
u
s
t
tr
ac
k
in
g
in
u
n
ce
r
tain
n
o
n
lin
ea
r
s
y
s
tem
s
[
1
8
]
–
[
2
1
]
.
Ho
wev
er
,
m
o
s
t
m
eth
o
d
s
r
el
y
o
n
s
u
p
er
v
is
ed
lear
n
in
g
with
p
r
e
-
co
llected
d
ata
,
o
f
ten
n
eg
lect
in
g
s
y
s
tem
d
y
n
am
ics
d
u
r
in
g
tr
ai
n
in
g
[
2
2
]
,
wh
ich
ca
n
lea
d
to
s
tab
ilit
y
v
io
latio
n
s
.
W
h
ile
ad
ap
tiv
e
u
p
d
ate
laws
ex
is
t
[
2
3
]
,
th
ey
ty
p
ically
o
p
tim
ize
p
ar
am
eter
s
s
ep
ar
ately
r
ath
er
th
an
jo
in
tly
lear
n
in
g
b
o
t
h
L
y
ap
u
n
o
v
f
u
n
ctio
n
s
(
)
an
d
co
n
tr
o
l
laws
(
)
.
Mo
r
eo
v
er
,
th
e
r
eg
io
n
o
f
attr
ac
t
io
n
(
R
o
A)
is
r
ar
ely
ex
p
an
d
ed
d
u
r
in
g
tr
ai
n
in
g
,
lim
itin
g
r
o
b
u
s
tn
ess
.
Fu
r
th
er
m
o
r
e,
f
ew
s
tu
d
ies
ex
p
licitly
ad
d
r
ess
s
af
ety
-
b
o
u
n
d
co
n
s
tr
ain
ts
o
r
d
is
tu
r
b
an
ce
r
o
b
u
s
tn
ess
in
ad
ap
tiv
e
DNN
–
L
y
a
p
u
n
o
v
f
r
a
m
ewo
r
k
s
,
leav
in
g
o
p
en
c
h
allen
g
es
in
ac
h
ie
v
in
g
r
eliab
le
s
tab
ilit
y
u
n
d
e
r
r
ea
l
-
wo
r
ld
u
n
ce
r
tain
ties
.
T
o
ad
d
r
ess
th
ese
ch
allen
g
es,
th
is
p
ap
er
p
r
o
p
o
s
es
an
Ad
ap
tiv
e
L
y
ap
u
n
o
v
-
b
ased
co
n
tr
o
l
f
r
am
ewo
r
k
u
s
in
g
DNNs
f
o
r
u
n
d
e
r
ac
tu
ate
d
n
o
n
lin
ea
r
s
y
s
tem
s
.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
em
p
lo
y
s
two
n
eu
r
al
n
etwo
r
k
s
:
i
)
a
L
y
ap
u
n
o
v
n
etwo
r
k
th
at
lear
n
s
a
p
o
s
itiv
e
-
d
ef
in
ite
f
u
n
ctio
n
(
)
,
an
d
ii
)
a
n
ad
a
p
tiv
e
co
n
tr
o
l
n
etwo
r
k
th
at
en
s
u
r
es
th
e
n
eg
ativ
e
d
e
f
in
iten
ess
o
f
(
)
.
B
o
th
n
etwo
r
k
s
ar
e
tr
ain
ed
jo
in
tly
u
s
in
g
a
d
y
n
am
ics
-
g
u
id
e
d
f
ee
d
b
ac
k
p
r
o
ce
s
s
b
ased
o
n
th
e
ac
tu
al
n
o
n
lin
ea
r
m
o
d
els
o
f
th
e
Fu
r
u
ta
p
en
d
u
lu
m
an
d
wh
e
eled
p
ath
-
f
o
llo
win
g
s
y
s
tem
s
.
A
cu
s
to
m
co
m
p
o
s
ite
lo
s
s
f
u
n
ctio
n
in
co
r
p
o
r
atin
g
R
o
A
ex
p
an
s
io
n
an
d
s
af
ety
-
b
o
u
n
d
c
o
n
s
tr
ain
ts
is
d
esig
n
ed
to
e
n
h
an
ce
r
o
b
u
s
tn
ess
ag
ain
s
t
d
is
tu
r
b
an
ce
s
wh
ile
m
ain
tain
in
g
s
y
s
tem
s
tab
ilit
y
.
T
h
e
p
a
p
er
is
s
tr
u
ctu
r
ed
as
f
o
llo
ws:
s
ec
tio
n
2
r
ev
iews
s
y
s
tem
d
y
n
am
ics
u
s
ed
an
d
L
y
ap
u
n
o
v
th
eo
r
y
;
s
ec
tio
n
3
p
r
esen
ts
th
e
p
r
o
p
o
s
ed
co
n
tr
o
l
m
eth
o
d
;
s
ec
tio
n
4
d
is
cu
s
s
es
s
im
u
latio
n
r
e
s
u
lts
an
d
co
m
p
ar
is
o
n
s
with
L
QR
;
an
d
s
ec
tio
n
5
co
n
clu
d
es th
e
s
tu
d
y
.
2.
B
ACK
G
RO
UND
2
.
1
.
Sy
s
t
em
dy
na
m
ics
An
u
n
d
er
ac
tu
ated
s
y
s
tem
is
a
n
o
n
lin
ea
r
co
n
tr
o
l
s
y
s
tem
wit
h
f
ewe
r
co
n
tr
o
l
in
p
u
ts
th
an
d
eg
r
ee
s
o
f
f
r
ee
d
o
m
(
DOF)
.
Ma
th
em
atica
lly
,
it is
d
escr
ib
ed
b
y
th
e
s
tate
-
s
p
ac
e
d
y
n
am
ics:
̇
(
)
=
(
(
)
)
+
(
(
)
(
)
)
(
1
)
W
h
er
e
(
)
∈
ℝ
s
y
s
tem
s
tate
v
ec
to
r
,
(
)
∈
ℝ
co
n
tr
o
l
in
p
u
t
v
ec
t
o
r
,
:
ℝ
→
ℝ
d
r
if
t
d
y
n
am
ics,
an
d
:
ℝ
→
ℝ
×
in
p
u
t
d
is
tr
ib
u
tio
n
m
atr
ix
,
wi
th
co
n
d
itio
n
<
.
T
h
is
in
eq
u
ality
r
ef
lects
th
at
n
o
t
all
s
tate
v
ar
iab
le
is
d
ir
ec
tly
ac
tu
ated
.
T
h
eir
co
m
p
le
x
d
y
n
am
ics
p
o
s
e
ch
allen
g
es
f
o
r
t
r
ad
itio
n
al
co
n
tr
o
l
m
eth
o
d
s
,
th
u
s
m
o
tiv
atin
g
th
e
u
s
e
o
f
d
ata
-
d
r
iv
en
a
p
p
r
o
ac
h
es
s
u
ch
as
DNNs
f
o
r
lear
n
in
g
co
n
tr
o
l
l
aws
an
d
L
y
ap
u
n
o
v
f
u
n
ctio
n
s
d
ir
ec
tly
f
r
o
m
th
e
s
y
s
tem
b
eh
a
v
io
r
.
T
h
e
Fu
r
u
ta
Pen
d
u
lu
m
an
d
wh
ee
led
p
at
h
-
f
o
llo
win
g
r
o
b
o
t
a
r
e
ca
n
o
n
ical
ex
am
p
les o
f
s
u
ch
s
y
s
tem
s
.
2
.
1
.
1
.
F
uruta
pend
ulu
m
T
h
e
Fu
r
u
ta
p
e
n
d
u
lu
m
co
n
s
is
ts
o
f
a
r
o
tatin
g
ar
m
(
ac
t
u
ated
)
a
n
d
a
p
en
d
u
lu
m
(
u
n
ac
t
u
ated
)
m
o
u
n
ted
o
n
th
e
en
d
o
f
t
h
e
ar
m
.
W
e
u
s
ed
th
e
Fu
r
u
ta
p
en
d
u
lu
m
s
y
s
tem
f
r
o
m
Qu
an
s
er
1
to
p
er
f
o
r
m
s
i
m
u
latio
n
[
2
4
]
.
T
h
e
d
y
n
am
ic
e
q
u
atio
n
s
ar
e
e
x
p
r
ess
ed
as
(
2
)
,
(
3
)
:
̈
=
1
(
1
+
)
+
2
2
(
2
)
̈
=
2
(
1
+
)
+
1
2
(
3
)
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
d
a
p
tive
Lya
p
u
n
o
v
-
b
a
s
ed
c
o
n
tr
o
l fo
r
u
n
d
era
ctu
a
te
d
n
o
n
lin
ea
r
s
ystem
u
s
in
g
…
(
Tr
iya
Ha
iyu
n
n
is
a
)
719
W
ith
s
u
b
co
m
p
o
n
e
n
ts
:
=
+
2
+
2
s
in
2
(
)
(
4
)
=
−
c
os
(
)
(
5
)
=
+
2
(
6
)
1
=
−
2
,
2
=
−
−
2
(
7
)
1
=
−
s
in
(
)
−
2
̇
2
s
in
(
)
c
os
(
)
−
̇
(
8
)
2
=
−
2
2
̇
̇
s
in
(
)
c
os
(
)
−
̇
2
s
in
(
)
−
̇
(
9
)
2
.
1
.
2
.
Wheeled
pa
t
h f
o
llo
wing
T
h
e
p
ath
f
o
llo
win
g
co
n
t
r
o
l
p
r
o
b
lem
ca
n
b
e
m
o
d
eled
u
s
in
g
th
e
k
in
em
atic
b
icy
cle
m
o
d
el,
wh
ich
d
escr
ib
es
th
e
d
y
n
am
ics
o
f
a
v
eh
icle
as
it
f
o
llo
ws
a
p
ath
[
2
5
]
.
T
h
e
p
r
im
ar
y
g
o
al
in
p
a
th
f
o
llo
win
g
is
to
m
in
im
ize
th
e
d
ev
iatio
n
f
r
o
m
a
d
esire
d
tr
ajec
to
r
y
,
wh
ich
i
s
u
s
u
ally
d
ef
in
ed
as
a
f
u
n
ctio
n
o
f
th
e
v
e
h
icle'
s
p
o
s
itio
n
an
d
o
r
ien
tatio
n
er
r
o
r
s
.
T
h
e
d
y
n
am
ics o
f
t
h
e
v
eh
icle
f
o
r
p
at
h
f
o
llo
win
g
ca
n
b
e
ex
p
r
ess
ed
as
(
1
0
)
,
(
1
1
)
:
̇
=
s
in
(
)
(
1
0
)
̇
=
ta
n
(
)
−
(
)
c
os
(
)
(
1
1
)
T
h
e
g
o
al
o
f
th
e
p
ath
f
o
llo
win
g
co
n
t
r
o
l
is
to
d
esig
n
a
c
o
n
tr
o
l
law
u
s
u
c
h
th
at
th
e
later
al
e
r
r
o
r
an
d
h
ea
d
in
g
er
r
o
r
ar
e
m
in
im
ized
,
i.e
.
,
th
e
v
e
h
icle
r
em
ain
s
cl
o
s
e
to
th
e
d
esire
d
p
ath
with
m
i
n
im
al
o
r
ien
tatio
n
d
ev
iatio
n
.
T
h
e
e
q
u
ilib
r
iu
m
p
o
in
ts
r
elate
to
th
e
v
eh
icle
m
ain
tain
in
g
a
s
tead
y
tr
ajec
to
r
y
with
o
u
t
d
ev
iatio
n
f
r
o
m
th
e
p
ath
(
=
0
,
=
0
)
.
All simu
latio
n
s
wer
e
p
er
f
o
r
m
ed
u
s
in
g
th
e
n
o
m
in
al
p
ar
am
eter
s
lis
ted
in
T
ab
le
1
.
T
ab
le
1
.
Mo
d
el
p
ar
am
ete
r
o
f
F
u
r
u
ta
p
e
n
d
u
lu
m
[
2
4
]
N
o
n
l
i
n
e
a
r
s
y
st
e
m
N
o
me
n
c
l
a
t
u
r
e
V
a
l
u
e
F
u
r
u
t
a
P
e
n
d
u
l
u
m [
2
4
]
-
P
e
n
d
u
l
u
m
mass
0
.
1
2
7
k
g
-
D
i
st
a
n
c
e
f
r
o
m
p
i
v
o
t
t
o
c
e
n
t
e
r
o
f
m
a
ss
0
.
1
5
6
m
-
F
u
l
l
l
e
n
g
t
h
o
f
r
o
t
a
r
y
a
r
m
0
.
2
1
6
m
-
R
o
t
a
r
y
a
r
m m
o
me
n
t
o
f
i
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e
r
t
i
a
a
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u
t
p
i
v
o
t
9
.
9
8
3
×
10
−
4
k
g
m
2
-
P
e
n
d
u
l
u
m
mo
m
e
n
t
o
f
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t
i
a
a
b
o
u
t
p
i
v
o
t
0
.
0
0
1
2
k
g
m
2
-
V
i
sc
o
u
s
d
a
m
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i
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c
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e
f
f
i
c
i
e
n
t
o
f
a
r
m
0
.
0
0
2
4
N
m
s/
r
a
d
-
V
i
sc
o
u
s
d
a
m
p
i
n
g
c
o
e
f
f
i
c
i
e
n
t
o
f
p
e
n
d
u
l
u
m
0
.
0
0
2
4
N
m
s/
r
a
d
-
G
r
a
v
i
t
y
9
.
8
1
m/
s
2
W
h
e
e
l
e
d
P
a
t
h
F
o
l
l
o
w
i
n
g
[
2
5
]
-
V
e
h
i
c
l
e
m
a
ss
1
.
5
k
g
-
W
h
e
e
l
b
a
se
0
.
1
6
5
m
-
F
o
r
w
a
r
d
v
e
l
o
c
i
t
y
0
.
2
m
/
s
–
c
o
n
t
r
o
l
i
n
p
u
t
(
st
e
e
r
i
n
g
a
n
g
l
e
)
±
30
0
2
.
2
.
L
y
a
pu
no
v
s
t
a
bil
it
y
t
heo
ry
L
y
ap
u
n
o
v
t
h
eo
r
y
p
r
o
v
id
es
a
p
o
wer
f
u
l
m
eth
o
d
to
ce
r
tify
th
e
s
tab
ilit
y
o
f
n
o
n
lin
ea
r
s
y
s
tem
s
with
o
u
t
s
o
lv
in
g
th
eir
tr
ajec
to
r
ies.
Lemma
1
.
Glo
b
al
asy
m
p
to
tic
s
tab
ilit
y
v
ia
L
y
ap
u
n
o
v
f
u
n
ctio
n
[
2
5
]
L
et
x
=
0
b
e
a
n
e
q
u
ilib
r
iu
m
o
f
th
e
n
o
n
lin
ea
r
s
y
s
tem
ẋ
=
(
)
.
I
f
t
h
er
e
e
x
is
ts
a
co
n
tin
u
o
u
s
ly
d
if
f
er
en
tiab
le
f
u
n
ctio
n
V(
x
)
:
ℝ
ⁿ →
ℝ
s
u
ch
th
at:
(
0
)
=
0
,
(
)
>
0
f
o
r
≠
0
̇
(
)
=
∇
(
)
̇
<
0
f
o
r
≠
0
T
h
en
th
e
e
q
u
ilib
r
iu
m
x
=
0
is
g
lo
b
ally
asy
m
p
to
tically
s
tab
le.
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
.
2
,
Ap
r
il
20
2
6
:
7
1
7
-
728
720
Defin
itio
n
1
.
I
n
p
u
t to
s
tate
s
tab
ilit
y
(
I
SS
)
[
2
5
]
C
o
n
s
id
er
a
n
o
n
lin
ea
r
s
y
s
tem
i
n
[
2
4
]
,
[
2
5
]
.
T
h
e
s
y
s
tem
is
in
p
u
t
-
to
-
s
tate
s
tab
le
(
I
SS
)
if
th
e
r
e
ex
is
ts
a
class
K
ℒ
f
u
n
ctio
n
(
⋅
,
⋅
)
an
d
a
class
K
f
u
n
ctio
n
(
∙
)
s
u
ch
th
at
f
o
r
ev
er
y
in
itial
co
n
d
itio
n
(
0
)
an
d
b
o
u
n
d
e
d
in
p
u
t
(
)
,
th
e
s
o
lu
tio
n
s
atis
f
ies:
‖
(
)
‖
≤
(
‖
(
0
)
‖
,
)
+
(
s
up
0
≤
≤
‖
(
)
‖
)
(
1
2
)
W
h
er
e
(
‖
(
0
)
‖
,
)
→
0
as
→
∞
is
th
e
d
ec
ay
p
r
o
p
er
ty
an
d
(
s
up
0
≤
≤
‖
(
)
‖
)
is
b
o
u
n
d
s
th
e
in
f
l
u
en
ce
o
f
in
p
u
t d
is
tu
r
b
a
n
ce
s
.
Defin
itio
n
2
.
I
n
p
u
t
-
to
-
s
tate
s
ta
b
le
(
I
SS
)
-
L
y
a
p
u
n
o
v
f
u
n
ctio
n
[
2
5
]
A
f
u
n
ctio
n
(
)
is
an
I
SS
L
y
ap
u
n
o
v
f
u
n
ctio
n
if
th
er
e
ex
is
t
class
∞
f
u
n
ctio
n
s
1
,
2
,
3
,
an
d
,
s
u
ch
th
at:
1
(
‖
‖
)
≤
(
)
≤
2
‖
‖
,
∈
ℝ
(
1
3
)
̇
(
,
)
≤
−
3
(
‖
‖
)
+
(
‖
‖
)
,
,
(
1
4
)
T
h
is
r
ef
lects
a
d
is
s
ip
ativ
e
en
e
r
g
y
b
alan
ce
:
th
e
s
tate
d
ec
ay
s
u
n
less
th
e
in
p
u
t
is
n
o
n
ze
r
o
.
T
h
e
p
r
o
p
o
s
ed
m
eth
o
d
m
in
im
izes
a
lo
s
s
th
at
in
d
ir
ec
tly
en
f
o
r
ce
s
̇
(
,
)
<
0
ev
en
i
n
th
e
p
r
e
s
en
ce
o
f
d
is
tu
r
b
an
ce
≈
(
)
.
T
h
e
d
esig
n
en
co
u
r
ag
es
I
SS
b
eh
av
i
o
u
r
wh
er
e
th
e
s
y
s
tem
’
s
s
tab
ilit
y
d
eg
r
ad
es
u
n
d
er
p
er
t
u
r
b
atio
n
b
u
t
r
ec
o
v
er
s
wh
en
th
e
d
is
tu
r
b
an
ce
v
an
is
h
es.
Defin
itio
n
3
.
Saf
ety
b
o
u
n
d
ed
a
n
d
r
eg
i
o
n
o
f
attr
ac
tio
n
T
h
e
s
y
s
tem
s
tate
∈
ℝ
.
T
h
e
s
af
ety
b
o
u
n
d
ed
r
e
g
io
n
⊂
ℝ
is
d
ef
in
ed
as
(
1
5
)
.
=
{
∈
ℝ
|
ℎ
(
)
≤
0
,
∀
=
1
,
2
,
…
,
}
(1
5
)
wh
er
e
ea
ch
ℎ
(
)
is
a
d
if
f
er
en
tiab
le
f
u
n
ctio
n
th
at
r
ep
r
esen
ts
a
s
af
ety
co
n
s
tr
ain
t
o
n
a
s
tat
e
v
ar
iab
le
o
r
co
m
b
in
atio
n
o
f
s
tate
v
ar
ia
b
les.
T
h
e
co
n
s
tr
ain
ts
m
ay
b
e
u
p
p
er
an
d
lo
wer
b
o
u
n
d
s
o
n
ea
ch
s
ta
te:
≤
≤
,
∀
∈
{
1
,
…
,
}
.
(
1
6
)
R
eg
io
n
o
f
a
ttra
ctio
n
(
R
o
A)
,
d
en
o
ted
b
y
ℛ
is
th
e
s
u
b
s
et
o
f
s
tate
s
p
ac
e
f
r
o
m
wh
ich
all
tr
ajec
to
r
ies
co
n
v
er
g
e
to
th
e
eq
u
ilib
r
iu
m
u
n
d
er
th
e
p
r
o
p
o
s
ed
co
n
tr
o
ller
.
I
t is d
ef
in
e
d
a
s
(
1
7
)
.
ℛ: =
{x
∈
ℝ
ⁿ |
V(
x
)
<
c,
V
̇
(
x
)
<
0
}
(
1
7
)
W
h
er
e
(
)
is
p
o
s
itiv
e
d
ef
in
ite
L
y
p
u
n
o
v
f
u
n
ctio
n
a
n
d
>
0
d
en
o
tes a
co
n
s
tan
t su
b
lev
el
th
r
esh
o
ld
.
3.
P
RO
P
O
SE
D
M
E
T
H
O
D
3
.
1
.
DNN
–
ba
s
ed
co
ntr
o
ller
des
i
g
n
T
h
is
wo
r
k
u
s
es
2
DNNs,
i.e
.
,
L
y
ap
u
n
o
v
NN
th
at
le
ar
n
s
a
p
o
s
itiv
e
-
d
ef
in
ite
f
u
n
ctio
n
,
an
d
C
o
n
tr
o
lLa
wNN
to
en
s
u
r
e
th
e
n
eg
ativ
e
d
ef
in
iten
ess
o
f
̇
(
)
.
T
h
e
L
y
ap
u
n
o
v
NN
as
s
ee
n
in
Fig
u
r
e
1
,
is
a
f
u
lly
co
n
n
ec
ted
f
ee
d
-
f
o
r
war
d
n
etwo
r
k
c
o
n
s
is
tin
g
o
f
two
h
i
d
d
en
lay
e
r
s
(
6
4
an
d
1
2
8
n
eu
r
o
n
s
)
with
So
f
tPl
u
s
ac
tiv
atio
n
to
en
s
u
r
e
th
at
th
e
lear
n
ed
L
y
a
p
u
n
o
v
f
u
n
ctio
n
V
(
)
r
em
ain
s
p
o
s
itiv
e
d
ef
in
ite
f
o
r
all
n
o
n
-
ze
r
o
s
tates.
I
t
m
ap
s
th
e
f
o
u
r
-
d
im
en
s
io
n
al
s
tate
v
ec
to
r
=
[
,
̇
,
,
̇
]
in
to
a
s
in
g
le
s
ca
lar
o
u
tp
u
t
V(
x
)
,
r
ep
r
esen
tin
g
th
e
s
y
s
tem
’
s
en
er
g
y
lan
d
s
ca
p
e
.
T
h
e
C
o
n
tr
o
lLa
wNN
g
e
n
er
ates
th
e
co
n
t
r
o
l
s
ig
n
al
(
)
ac
co
r
d
in
g
t
o
th
e
n
o
n
lin
ea
r
co
n
t
r
o
l
law
(
)
=
ta
n
h
(
)
wh
er
e
th
e
i
n
itial
v
alu
e
o
f
th
e
g
ain
v
ec
to
r
is
d
er
iv
ed
f
r
o
m
th
e
o
p
tim
al
g
ai
n
m
atr
ix
o
b
tain
e
d
b
y
th
e
L
QR
m
eth
o
d
.
T
h
is
in
itiali
za
tio
n
en
s
u
r
es
th
at
th
e
lear
n
in
g
p
r
o
ce
s
s
b
eg
in
s
f
r
o
m
a
s
tab
il
izin
g
lin
ea
r
p
o
licy
an
d
th
en
ad
ap
tiv
el
y
r
ef
in
es
th
r
o
u
g
h
n
e
u
r
al
o
p
tim
izatio
n
to
s
atis
f
y
L
y
ap
u
n
o
v
s
tab
ilit
y
co
n
d
itio
n
s
.
T
h
e
C
o
n
tr
o
lLa
wNN
em
p
lo
y
s
a
tan
h
-
b
ased
ac
tiv
atio
n
to
m
ain
ta
in
s
m
o
o
th
an
d
b
o
u
n
d
e
d
co
n
tr
o
l
o
u
tp
u
ts
,
wh
ile
its
p
ar
am
eter
s
ar
e
tr
ai
n
ed
jo
in
tl
y
with
L
y
ap
u
n
o
v
NN
u
s
in
g
a
co
m
p
o
s
ite
lo
s
s
th
at
p
en
alize
s
v
io
latio
n
s
o
f
th
e
L
y
ap
u
n
o
v
co
n
d
itio
n
s
.
Un
lik
e
co
n
v
en
tio
n
al
L
y
ap
u
n
o
v
-
b
ased
co
n
tr
o
ller
s
,
th
e
two
DNNs
in
th
is
f
r
am
ewo
r
k
n
o
t
o
n
ly
lear
n
th
e
f
u
n
ctio
n
al
r
elatio
n
s
h
ip
s
o
f
V
(
)
an
d
̇
(
)
,
b
u
t
also
im
p
licitly
d
eter
m
in
e
th
e
R
o
A
u
s
in
g
s
af
ety
b
o
u
n
d
co
n
t
r
ain
ed
a
p
p
r
o
ac
h
as
s
ee
n
in
Fig
u
r
e
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
d
a
p
tive
Lya
p
u
n
o
v
-
b
a
s
ed
c
o
n
tr
o
l fo
r
u
n
d
era
ctu
a
te
d
n
o
n
lin
ea
r
s
ystem
u
s
in
g
…
(
Tr
iya
Ha
iyu
n
n
is
a
)
721
Fig
u
r
e
1
.
DNN
a
r
ch
itectu
r
e
o
f
L
y
ap
u
n
o
v
NN
Fig
u
r
e
2
.
DNN
-
b
ased
c
o
n
tr
o
ll
er
f
r
am
ewo
r
k
3
.
2
.
Sa
f
e
and
ro
bu
s
t
re
g
io
n
o
f
a
t
t
ra
ct
io
n
I
n
th
is
s
tu
d
y
,
th
e
R
o
A
was
f
u
r
th
er
en
lar
g
e
d
to
ap
p
r
o
ac
h
th
e
s
af
ety
b
o
u
n
d
c
o
n
s
tr
ain
t.
W
ith
th
e
R
o
A
clo
s
e
to
th
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
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2
0
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8
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I
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&
C
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p
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g
,
Vo
l.
1
6
,
No
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2
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Ap
r
il
20
2
6
:
7
1
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722
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s
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th
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s
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Ma
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4.
RE
SU
L
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AND
DI
SCUS
SI
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.
1
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no
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ntr
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tio
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d
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e
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f
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g
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clu
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esig
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s
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.
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s
h
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u
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3
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3
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c
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,
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b
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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p
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4
7
3
]
,
wh
ich
s
h
o
ws
th
at
ad
ap
tiv
e
co
n
tr
o
l
s
o
f
ten
s
th
e
r
esp
o
n
s
e
at
s
o
m
e
s
tates,
an
d
ad
ju
s
ts
th
e
g
ain
s
o
th
at
L
y
ap
u
n
o
v
s
tab
ilit
y
is
n
o
t
o
n
ly
g
u
ar
a
n
teed
lo
ca
lly
as
in
L
QR
,
b
u
t
also
in
a
wid
er
n
o
n
lin
ea
r
d
o
m
ain
ac
co
r
d
in
g
to
t
h
e
d
esig
n
o
b
jectiv
es.
4
.
2
.
Reg
i
o
n o
f
a
t
t
ra
c
t
io
n
(
R
o
A)
wit
h DN
N
-
ba
s
ed
co
ntr
o
ller
T
o
ev
al
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a
te
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h
e
p
er
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an
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o
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t
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r
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m
et
h
o
d
,
a
co
m
p
a
r
at
iv
e
a
n
a
ly
s
is
o
f
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A
a
r
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w
as
co
n
d
u
ct
ed
a
g
a
in
s
t c
o
n
v
e
n
t
io
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a
l
c
o
n
t
r
o
l m
et
h
o
d
s
b
as
e
d
o
n
t
h
e
L
QR
.
L
QR
was c
h
o
s
e
n
as
a
c
o
m
p
ar
is
o
n
b
ec
a
u
s
e
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is
o
n
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t
h
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m
o
s
t
wi
d
el
y
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ti
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l
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p
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h
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l
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ea
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te
m
s
ar
o
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n
d
a
n
e
q
u
ili
b
r
i
u
m
p
o
i
n
t.
T
h
e
R
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A
w
as
v
is
u
a
liz
e
d
in
2
D
s
li
ce
s
t
o
f
ac
i
lit
ate
i
n
t
e
r
p
r
etat
io
n
o
f
t
h
e
r
es
u
lts
.
T
h
e
F
u
r
u
t
a
p
en
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u
l
u
m
h
as
f
o
u
r
s
tat
e
v
a
r
i
ab
les
(
,
̇
,
,
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,
th
e
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f
u
ll
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d
es
in
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-
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i
m
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l
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w
h
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ch
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if
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i
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lt
to
v
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s
u
al
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e
d
i
r
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ctl
y
.
T
h
e
r
ef
o
r
e
,
t
h
e
R
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A
is
d
is
p
l
a
y
e
d
p
e
r
tw
o
s
t
ate
v
a
r
ia
b
les
—
f
o
r
ex
am
p
l
e,
i
n
t
h
e
p
la
n
e
s
[
θ,
θ˙
]
an
d
[
α
,
α
˙
]
.
T
h
e
R
o
A
c
o
m
p
a
r
is
o
n
r
esu
lts
i
n
Fi
g
u
r
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s
4
(
a
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a
n
d
4(
b
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s
h
o
w
a
s
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g
n
if
ica
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t
d
if
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er
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ce
b
etw
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t
io
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l
L
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b
as
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d
ap
p
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ac
h
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n
d
t
h
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p
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p
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s
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d
DNN
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tr
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le
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ase
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m
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o
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d
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g
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w
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th
e
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th
t
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co
n
t
r
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p
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ce
s
a
R
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2
ti
m
es wi
d
e
r
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c
o
m
p
ar
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t
o
t
h
e
R
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A
p
r
o
d
u
c
e
d
b
y
t
h
e
L
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m
et
h
o
d
.
A
s
im
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tr
en
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is
s
ee
n
in
th
e
wh
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led
p
at
h
f
o
llo
win
g
s
y
s
tem
in
Fig
u
r
e
4
(
c
)
.
Alth
o
u
g
h
th
e
r
esu
ltin
g
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A
ex
p
an
s
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n
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as
lar
g
e
as
in
th
e
Fu
r
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ta
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lu
m
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th
e
p
r
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p
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eth
o
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s
till
p
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o
d
u
ce
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a
lar
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er
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T
h
er
e
f
o
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e,
it
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n
b
e
c
o
n
clu
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th
at
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DN
N
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p
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ly
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th
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in
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itio
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s
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r
o
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id
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etter
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v
an
tag
es
o
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class
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L
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ap
p
r
o
a
ch
,
esp
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ially
in
n
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lin
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r
s
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s
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d
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LQ
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:
P
r
o
p
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d
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4
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9
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Fig
u
r
e
4
.
C
o
m
p
a
r
is
o
n
o
f
R
o
A:
Fu
r
u
ta
p
en
d
u
lu
m
(
a)
[
θ
–
θ
̇
]
,
(
b
)
[
α
-
α
̇
]
p
lan
e;
an
d
(
c)
wh
ee
le
d
p
ath
f
o
llo
win
g
4
.
3
.
Ro
bu
s
t
nes
s
t
est
4
.
3
.
1
.
O
nli
ne
a
da
ptiv
e
t
ra
c
k
i
ng
wit
h f
ee
db
a
ck
-
ba
s
ed
lea
rning
I
n
th
is
s
tu
d
y
,
an
o
n
lin
e
ad
a
p
tiv
e
tr
ac
k
in
g
with
f
ee
d
b
ac
k
-
b
a
s
ed
lear
n
in
g
ap
p
r
o
ac
h
is
u
s
ed
to
e
n
s
u
r
e
th
at
th
e
Fu
r
u
ta
p
en
d
u
lu
m
ca
n
f
o
llo
w
th
e
r
ef
e
r
en
ce
s
ig
n
al
(
)
with
h
ig
h
p
r
ec
is
io
n
wh
ile
m
ain
tain
in
g
s
y
s
tem
s
af
ety
.
Simu
latio
n
r
es
u
lts
s
h
o
w
th
at
th
e
p
r
o
p
o
s
ed
ad
ap
tiv
e
co
n
tr
o
ller
ca
n
tr
ac
k
th
e
an
g
le
θ
ac
co
r
d
in
g
to
th
e
g
iv
en
s
in
u
s
o
id
al
r
ef
er
e
n
ce
,
wh
ile
k
ee
p
in
g
th
e
s
y
s
tem
with
in
th
e
s
af
e
lim
its
.
B
ased
o
n
th
e
Fig
u
r
e
5
s
h
o
ws
th
at
th
e
av
er
ag
e
tr
ac
k
i
n
g
er
r
o
r
is
v
er
y
s
m
all
an
d
co
n
tr
o
lled
o
v
er
a
2
0
-
s
ec
o
n
d
tim
e
h
o
r
izo
n
,
with
th
e
er
r
o
r
co
n
tr
i
b
u
tio
n
d
ec
r
ea
s
in
g
o
v
er
tim
e
(
ev
id
e
n
ce
d
b
y
th
e
lo
w
m
ea
n
s
q
u
ar
e
d
er
r
o
r
-
MSE
v
al
u
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.
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r
th
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m
o
r
e
,
th
e
p
e
n
alty
m
ec
h
an
is
m
ap
p
lied
to
th
e
lo
s
s
f
u
n
ctio
n
is
p
r
o
v
en
ef
f
ec
tiv
e
in
k
ee
p
in
g
t
h
e
an
g
le
α
(
p
en
d
u
lu
m
p
o
s
itio
n
)
with
in
th
e
o
p
er
atin
g
r
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g
e
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f
∣
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≤
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ad
ian
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u
s
p
r
ev
e
n
tin
g
th
e
p
e
n
d
u
lu
m
f
r
o
m
m
o
v
i
n
g
o
u
ts
id
e
th
e
s
tab
le
r
eg
io
n
.
T
h
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an
g
u
lar
v
elo
city
lim
it
p
ar
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ter
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also
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o
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o
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ates saf
ely
with
o
u
t e
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in
g
±
3
r
ad
/s
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I
n
ad
d
itio
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to
th
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Fu
r
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,
t
r
ac
k
in
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f
o
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in
u
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was
also
p
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o
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m
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wh
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p
ath
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o
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win
g
.
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h
e
m
ain
o
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was
to
m
in
im
ize
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later
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r
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r
d
e
a
n
d
h
ea
d
i
n
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er
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r
θe
,
wh
ile
m
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v
eh
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tate
with
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th
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o
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n
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ed
.
T
h
e
ad
ap
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DNN
co
n
tr
o
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s
i
g
n
if
ican
tly
r
e
d
u
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d
th
e
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ter
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f
ir
s
t
f
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n
d
s
,
as
s
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n
in
Fig
u
r
e
6
,
alth
o
u
g
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th
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wer
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s
till
s
af
ety
b
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d
v
io
latio
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s
th
at
n
ee
d
ed
to
b
e
m
itig
ated
.
T
h
e
r
esu
ltin
g
MSE
was
q
u
ite
s
m
al
l
f
o
r
th
e
s
in
u
s
o
id
al
p
ath
s
ce
n
ar
io
.
T
h
is
in
d
icate
s
th
at
DNN
-
b
ased
ad
ap
tiv
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c
o
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t
r
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in
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d
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m
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ce
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tain
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in
p
ath
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f
o
ll
o
win
g
s
y
s
tem
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
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Vo
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2
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:
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724
Fig
u
r
e
5
.
T
h
eta
t
r
ac
k
in
g
in
Fu
r
u
ta
p
en
d
u
lu
m
Fig
u
r
e
6
.
Path
tr
ac
k
in
g
i
n
wh
e
eled
p
ath
f
o
llo
win
g
4
.
3
.
2
.
Dis
t
urba
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T
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ex
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test
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t
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e
c
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tr
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ller
'
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ab
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ilit
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.
Fig
u
r
e
7
s
h
o
ws
th
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r
esp
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s
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o
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th
e
Fu
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o
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ar
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e
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.
Fig
u
r
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7
(
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p
r
esen
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e
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th
e
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m
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le
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h
o
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s
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d
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im
m
ed
iately
af
ter
ea
ch
d
i
s
tu
r
b
an
ce
f
o
llo
wed
b
y
a
g
r
ad
u
al
co
n
v
er
g
en
ce
b
ac
k
to
th
e
r
ef
er
e
n
ce
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ajec
to
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y
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an
titativ
ely
,
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y
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tem
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o
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