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du.
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q
1.
I
N
TR
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D
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C
TI
O
N
A
ft
e
r
t
he
pi
one
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ri
n
g
w
ork
by
P
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c
ora
a
nd
C
a
rrol
l
i
n
19
90s
[
1],
Ch
a
os
s
yn
c
hro
ni
z
a
t
i
on
h
a
s
a
t
t
ra
c
t
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d
c
ons
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d
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bl
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a
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ppl
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a
t
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p
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i
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a
l
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t
e
m
s
[2]
,
bi
o
l
og
i
c
a
l
s
ys
t
e
m
s
[
3
]
,
E
nc
r
ypt
i
on
[4
,
5]
a
n
d
s
e
c
ur
e
c
o
m
m
uni
c
a
t
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o
ns
[6]
,
e
t
c
.
T
he
s
ync
hron
i
z
a
t
i
on
m
e
a
ns
m
a
k
i
ng
a
s
ys
t
e
m
c
a
l
l
e
d
a
re
s
pons
e
s
ys
t
e
m
t
o
f
ol
l
ow
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not
he
r
s
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w
h
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c
h
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s
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a
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l
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d
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ve
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m
.
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e
nor
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ous
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ync
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z
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on
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e
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ppl
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d
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ful
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/
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om
pl
e
t
e
s
ync
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i
z
a
t
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on
(CS
)
[7
-
10]
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a
nt
i
-
s
ync
h
roni
z
a
t
i
on
(A
S
)
[
11
]
,
hybr
i
d
s
ync
hroni
z
a
t
i
on
(H
S
)
[12]
,
l
a
g
s
ync
hroni
z
a
t
i
o
n,
ph
a
s
e
s
ync
h
roni
z
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t
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o
n
,
pro
j
e
c
t
i
ve
s
ync
hron
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z
a
t
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on
(P
S
)
[
13]
,
m
odi
fi
e
d
proj
e
c
t
i
v
e
s
ync
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i
z
a
t
i
on
(M
P
S
)
[14]
a
nd
g
e
ne
r
a
l
i
z
e
d
pro
j
e
c
t
i
ve
s
yn
c
hroni
z
a
t
i
on
(G
P
S
)
[
15
].
F
ul
l
s
ync
hron
i
z
a
t
i
on
a
n
d
a
nt
i
-
s
ync
h
roni
z
a
t
i
o
n
a
r
e
t
h
e
m
os
t
c
o
m
m
onl
y
us
e
d
[
9,
11
]
a
nd
p
l
a
y
a
n
i
m
por
t
a
n
t
rol
e
i
n
e
ng
i
ne
e
ri
ng
a
ppl
i
c
a
t
i
ons
[
16
,
17
].
T
he
s
e
ph
e
no
m
e
n
a
a
r
e
a
c
hi
e
ve
d
vi
a
di
ff
e
r
e
nt
va
ri
ous
t
yp
e
s
of
s
ync
hron
i
z
a
t
i
ons
s
c
he
m
e
s
i
n
c
l
ud
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ng
a
c
t
i
v
e
c
on
t
rol
[
18]
,
a
d
a
pt
i
ve
c
ont
rol
[1
6]
,
no
nl
i
ne
a
r
c
o
nt
r
ol
[
19
-
22
]
a
nd
l
i
n
e
a
r
f
e
e
dba
c
k
c
o
nt
ro
l
[
23
-
25
].
A
m
ong
t
he
a
f
ore
m
e
n
t
i
o
ne
d
s
c
h
e
m
e
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t
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a
c
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c
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ro
l
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nd
t
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non
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ve
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n
w
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us
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d
a
s
t
w
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ful
s
t
ra
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i
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s
for
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yn
c
hro
ni
z
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t
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on
o
f
di
ff
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nt
c
l
a
s
s
of
no
nl
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ne
a
r
d
yna
m
i
c
a
l
s
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m
s
.
H
ow
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v
e
r,
t
he
a
c
t
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on
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uff
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m
m
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ondi
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v
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f
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h
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m
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non
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s
uff
i
c
i
e
n
t
c
ondi
t
i
o
n
i
s
pro
pos
e
d
t
o
s
h
ow
how
m
a
ny
r
e
l
e
va
nt
v
a
ri
a
bl
e
s
of
dr
i
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e
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re
s
pons
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ys
t
e
m
s
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a
n
a
c
hi
e
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s
yn
c
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i
z
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t
i
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a
c
t
i
ve
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non
l
i
n
e
a
r
c
ont
r
ol
l
e
r
.
−
N
e
w
re
s
u
l
t
s
a
re
d
e
ri
v
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d
f
rom
a
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a
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e
s
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t
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a
n
l
e
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d
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i
z
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t
i
on
.
−
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e
w
r
e
s
ul
t
s
a
r
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de
r
i
ve
d
fr
om
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pe
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t
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on
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a
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l
e
v
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nt
va
r
i
a
bl
e
s
of
dr
i
ve
a
n
d
r
e
s
pons
e
s
ys
t
e
m
s
.
T
he
r
e
s
t
of
t
hi
s
pa
pe
r
i
s
or
ga
n
i
z
e
d
a
s
fo
l
l
ow
s
.
S
e
c
t
i
o
n
2
i
s
t
he
d
e
s
c
r
i
pt
i
on
of
hy
pe
r
c
ha
ot
i
c
Ra
bi
n
ovi
c
h
S
ys
t
e
m
.
S
e
c
t
i
o
n
3
pre
s
e
nt
s
t
he
pro
bl
e
m
of
c
om
p
l
e
t
e
s
yn
c
hron
i
z
a
t
i
on
for
t
he
hype
r
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h
a
ot
i
c
Ra
bi
n
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c
h
.
S
e
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t
i
on
4
i
s
t
he
c
onc
l
us
i
ons
t
h
i
s
p
a
p
e
r.
2.
D
ES
C
R
I
P
TI
O
N
O
F
H
Y
P
ER
C
H
A
O
TI
C
R
A
BI
N
O
V
I
C
H
S
Y
S
TEM
Ra
bi
n
ovi
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h
s
ys
t
e
m
i
s
a
f
our
-
d
i
m
e
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l
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ot
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w
h
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h
i
n
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l
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e
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m
s
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t
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o
f
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h
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m
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r
e
nonl
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ne
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ri
t
y
w
i
t
h
t
h
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e
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ra
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e
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t
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he
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[2
6
,
27
]:
{
̇
1
=
−
1
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2
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2
3
̇
2
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1
−
2
−
1
3
+
4
̇
3
=
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1
2
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−
2
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w
he
re
1
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2
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3
,
4
a
re
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h
e
s
t
a
t
e
va
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re
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os
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s
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pos
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hype
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ra
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ors
w
h
e
n
t
h
e
p
a
ra
m
e
t
e
rs
t
a
k
e
n
=
4
,
ℎ
=
6
.
75
,
=
2
as
s
how
i
n
F
i
gur
e
s
1
-
4.
F
i
gure
1
.
T
he
a
t
t
r
a
c
t
or
of
s
ys
t
e
m
(1)
i
n
1
−
3
pl
a
ne
F
i
gure
2
.
T
he
a
t
t
r
a
c
t
or
of
s
ys
t
e
m
(1)
i
n
1
,
2
,
3
s
pa
c
e
F
i
gure
3
.
T
he
a
t
t
r
a
c
t
or
of
s
ys
t
e
m
(1)
i
n
1
−
4
pl
a
ne
F
i
gure
4
.
T
he
a
t
t
r
a
c
t
or
of
s
ys
t
e
m
(1)
i
n
1
,
2
,
4
s
pa
c
e
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
1693
-
6930
T
E
L
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e
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2
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3.
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rs
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i
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l
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e
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d
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s
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m
,
a
n
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e
c
ond
s
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e
m
i
s
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a
l
l
e
d
re
s
p
ons
e
s
ys
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m
.
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he
dr
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ve
a
nd
r
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s
pons
e
s
ys
t
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m
s
for
Ra
bi
nov
i
c
h
s
ys
t
e
m
d
e
pi
c
t
i
n
(
2
)
a
nd
(
3
)
re
s
p
e
c
t
i
ve
l
y
.
{
̇
1
=
−
1
+
ℎ
1
+
1
1
̇
1
=
ℎ
1
−
1
−
1
1
+
1
̇
1
=
−
1
+
1
1
̇
1
=
−
1
(2)
{
̇
2
=
−
2
+
ℎ
2
+
2
2
+
1
̇
2
=
ℎ
2
−
2
−
2
2
+
2
+
2
̇
2
=
−
2
+
2
2
+
3
̇
2
=
−
2
+
4
(3)
W
he
r
e
=
[
1
,
2
,
3
,
4
]
i
s
t
he
c
o
nt
ro
l
l
e
r
t
o
be
de
s
i
gne
d,
T
he
s
ync
h
roni
z
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t
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rror
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4
i
s
d
e
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a
s
:
1
=
2
−
1
,
2
=
2
−
1
,
3
=
2
−
1
,
4
=
2
−
1
,
∀
=
1
,
=
1
,
2
,
3
,
4
s
o,
t
he
e
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ror
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m
i
c
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l
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ys
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m
w
h
i
c
h
i
s
g
i
ve
n
by
:
{
̇
1
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−
1
+
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2
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2
3
+
1
2
+
1
3
+
1
̇
2
=
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1
−
2
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4
−
1
3
−
1
1
−
1
3
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2
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3
=
−
3
+
1
2
+
1
1
+
1
2
+
3
̇
4
=
−
2
+
4
(4)
3.
1
.
S
yn
c
h
r
on
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z
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on
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as
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d
on
a
c
ti
ve
c
on
t
r
ol
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o
r
e
a
l
i
z
e
t
he
c
o
m
pl
e
t
e
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ync
hr
oni
z
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on
,
w
e
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e
d
t
o
de
s
i
g
n
s
ui
t
a
b
l
e
non
l
i
n
e
a
r
c
ont
r
ol
.
T
he
re
fo
re
,
t
he
c
o
nt
ro
l
f
unc
t
i
ons
a
r
e
c
hos
e
n
a
s
t
he
fo
l
l
ow
i
ng:
{
1
=
−
2
3
−
1
2
−
1
3
+
1
2
=
1
3
+
1
1
+
1
3
+
2
3
=
−
1
2
−
1
1
−
1
2
+
3
4
=
4
(5
)
i
ns
e
rt
i
ng
t
h
e
c
on
t
rol
(5)
i
n
(
4)
w
e
g
e
t
:
{
̇
1
=
−
1
+
ℎ
2
+
1
̇
2
=
ℎ
1
−
2
+
4
+
2
̇
3
=
−
3
+
3
̇
4
=
−
2
+
4
,
=
[
1
2
3
4
]
=
[
1
2
3
4
]
(6)
w
he
re
i
s
l
i
ne
a
r
c
ont
r
ol
,
i
s
a
c
o
ns
t
a
n
t
m
a
t
ri
x
,
to
m
a
ke
t
he
s
ys
t
e
m
(6
)
s
t
a
bl
e
,
t
he
m
a
t
ri
x
s
houl
d
be
s
e
l
e
c
t
e
d
by
t
he
fo
l
l
ow
i
ng:
=
[
0
−
ℎ
0
0
−
ℎ
0
0
−
1
0
0
0
0
0
0
−
1
]
(7)
he
nc
e
,
t
he
e
rr
or
dyna
m
i
c
a
l
s
ys
t
e
m
(
4)
w
i
t
h
a
bove
m
a
t
ri
x
be
c
om
e
s
:
{
̇
1
=
−
1
̇
2
=
−
2
̇
3
=
−
3
̇
4
=
−
4
(8)
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
K
O
M
N
IK
A
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e
l
e
c
om
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un
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(
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y
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997
T
he
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e
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t
he
s
t
a
b
i
l
i
t
y
of
t
he
s
ys
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(8)
.
S
o
,
t
h
e
r
e
s
po
ns
e
s
ys
t
e
m
(
3)
s
yn
c
hron
i
z
e
s
t
h
e
dr
i
ve
s
ys
t
e
m
.
H
e
nc
e
,
w
e
re
a
c
h
a
t
t
h
e
fol
l
ow
i
ng
r
e
s
ul
t
s
:
T
he
o
re
m
1
:
I
f
t
he
m
a
t
r
i
x
(7)
i
s
c
om
bi
n
e
w
i
t
h
s
ys
t
e
m
(6)
,
t
he
n
,
t
h
e
r
e
s
pons
e
s
ys
t
e
m
(3)
fol
l
ow
s
t
he
dri
ve
s
ys
t
e
m
vi
a
t
he
fo
l
l
ow
i
ng
nonl
i
n
e
a
r
a
c
t
i
ve
c
ont
ro
l
w
hi
c
h
c
ons
i
s
t
s
of
16
t
e
r
m
s
.
{
1
=
−
ℎ
2
−
2
3
−
1
2
−
1
3
2
=
−
ℎ
1
−
4
+
1
3
+
1
1
+
1
3
3
=
−
1
2
−
1
1
−
1
2
4
=
2
−
4
(
9)
P
roof:
Ba
s
e
d
o
n
t
he
L
y
a
pu
nov
s
e
c
o
nd
m
e
t
hod
,
w
e
c
on
s
t
ruc
t
a
pos
i
t
i
ve
de
fi
n
i
t
e
L
ya
p
unov
c
a
nd
i
da
t
e
func
t
i
on
a
s
:
V
(
e
)
=
=
1
2
1
2
+
1
2
2
2
+
1
2
3
2
+
1
2
4
2
(10)
w
he
re
=
[
1
2
,
1
2
,
1
2
,
1
2
]
,
t
he
de
r
i
v
a
t
i
ve
of
t
h
e
L
y
a
punov
fun
c
t
i
on
V
(
e
)
w
i
t
h
r
e
s
pe
c
t
t
o
t
i
m
e
i
s
:
̇
(
)
=
1
̇
1
+
2
̇
2
+
3
̇
3
+
4
̇
4
̇
(
)
=
1
(
−
1
)
+
2
(
−
2
)
+
3
(
−
3
)
+
4
(
−
4
)
̇
(
)
=
−
1
2
−
2
2
−
3
2
−
4
2
=
−
,
=
[
,
1
,
1
,
1
]
(11)
A
c
c
ord
i
ng
t
o
[14]
,
e
v
e
ry
d
i
a
g
ona
l
m
a
t
r
i
x
w
i
t
h
pos
i
t
i
v
e
di
a
go
na
l
e
l
e
m
e
n
t
s
i
s
pos
i
t
i
v
e
de
f
i
ni
t
e
.
S
o
>
0
.
T
he
r
e
for
e
,
̇
(
)
i
s
ne
ga
t
i
v
e
de
fi
n
i
t
e
.
A
nd
a
c
c
or
di
ng
t
o
t
he
L
y
a
punov
a
s
y
m
pt
o
t
i
c
a
l
s
t
a
bi
l
i
t
y
t
h
e
ory
,
t
he
nonl
i
n
e
a
r
a
c
t
i
ve
c
on
t
rol
l
e
r
i
s
i
m
pl
e
m
e
nt
e
d
a
nd
t
h
e
s
ync
hron
i
z
a
t
i
on
of
t
he
hyp
e
rc
h
a
o
t
i
c
s
ys
t
e
m
i
s
a
c
h
i
e
v
e
d
.
T
he
proof
i
s
n
ow
c
om
p
l
e
t
e
.
T
he
t
h
e
or
e
m
1
s
how
e
d
t
ha
t
pro
pos
e
d
c
on
t
ro
l
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c
h
c
ons
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s
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(
14)
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e
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roni
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a
s
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n
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5.
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2
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c
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r
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a
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In
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e
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z
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b
e
t
w
e
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n
s
ys
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e
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a
nd
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ys
t
e
m
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i
s
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ons
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re
d
b
y
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s
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ra
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h
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e
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l
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e
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r
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T
h
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e
m
2:
I
f
de
s
i
g
n
a
c
o
nt
ro
l
l
e
r
c
ons
i
s
t
s
of
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x
t
e
r
m
s
a
s
fo
l
l
ow
s
:
{
1
=
−
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2
2
=
4
3
=
−
2
1
1
−
1
2
4
=
−
2
−
4
(
12)
t
he
n
t
h
e
e
rr
or
dyn
a
m
i
c
a
l
s
ys
t
e
m
(4)
i
s
c
o
nve
r
ge
n
t
t
o
z
e
ro
a
s
t
i
m
e
(
4)
.
P
roof
:
W
i
t
h
t
hi
s
c
ho
i
c
e
,
t
he
e
rro
r
dyna
m
i
c
a
l
s
ys
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e
m
(4)
b
e
c
o
m
e
s
:
[
̇
1
̇
2
̇
3
̇
4
]
=
[
−
ℎ
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2
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1
0
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ℎ
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1
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1
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(
+
1
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2
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1
−
1
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1
−
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0
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1
0
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1
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[
1
2
3
4
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i
.
e
.
,
{
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3
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2
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3
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2
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1
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2
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(
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3
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1
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3
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3
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2
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1
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2
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4
=
−
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+
1
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2
−
4
(13)
t
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v
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i
s
:
̇
(
)
=
−
1
2
−
2
2
−
3
2
−
4
2
=
−
(14)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
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:
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o
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h
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he
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2
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d
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l
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nd
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t
i
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s
ync
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roni
z
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t
i
on
a
t
(
4)
t
i
m
e
(s
e
c
)
a
s
s
how
n
i
n
F
i
g
u
re
6.
F
i
gure
5
.
T
he
c
onv
e
rge
s
of
s
ys
t
e
m
(4)
w
i
t
h
c
on
t
ro
l
l
e
r
(9
)
F
i
gure
6
.
T
he
c
onv
e
rge
s
of
s
ys
t
e
m
(13)
w
i
t
h
c
on
t
ro
l
l
e
r
(1
2)
T
he
o
re
m
3
:
If
t
he
c
ont
ro
l
l
e
r
i
s
de
s
i
g
ne
d
w
i
t
h
s
i
x
t
e
r
m
s
a
s
f
ol
l
ow
s
:
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1
=
−
2
ℎ
2
−
2
1
3
−
2
3
2
=
0
3
=
0
4
=
2
−
2
−
4
(15)
t
he
n
t
h
e
s
ys
t
e
m
(4)
w
i
l
l
be
c
onv
e
rg
e
nt
t
o
z
e
ro
a
s
t
he
t
i
m
e
(
4
)
.
P
ro
of:
w
he
n
s
ubs
t
i
t
u
t
i
ng
t
h
e
c
on
t
rol
l
e
rs
(15)
i
n
t
he
s
ys
t
e
m
(4)
,
w
e
ge
t
:
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̇
1
=
−
1
−
ℎ
2
+
1
2
−
1
3
̇
2
=
ℎ
1
−
2
+
4
−
1
3
−
1
1
−
1
3
̇
3
=
−
3
+
1
2
−
1
1
+
1
2
̇
4
=
−
2
−
4
(16)
c
ons
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c
t
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t
i
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s
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r
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ore
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h
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t
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e
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h
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ons
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s
a
c
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e
ve
d
s
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z
a
t
i
o
n.
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he
t
i
m
e
s
yn
c
hron
i
z
a
t
i
on
a
t
(4)
t
i
m
e
(s
e
c
)
a
s
s
how
n
i
n
F
i
g
ure
7.
T
he
o
re
m
4:
t
h
e
s
ys
t
e
m
(4)
i
s
a
c
hi
e
v
e
d
a
nd
c
onv
e
rge
s
t
o
z
e
ro
a
t
t
he
t
i
m
e
(4
.
20)
,
i
f
t
h
e
c
ont
r
ol
l
e
r
i
s
de
s
i
gn
e
d
a
s
:
{
1
=
−
2
ℎ
2
−
2
1
3
2
=
−
1
3
+
4
3
=
0
4
=
−
2
−
4
(17)
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
K
O
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N
IK
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T
e
l
e
c
om
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or
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om
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(
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l
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999
P
roof:
U
s
i
ng
s
ys
t
e
m
(
4)
w
i
t
h
t
h
e
c
on
t
ro
l
l
e
r
(
17),
i
s
gi
ve
n
by
.
{
̇
1
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1
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2
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3
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2
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1
3
̇
2
=
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1
−
2
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(
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4
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1
3
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1
1
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3
̇
3
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3
+
1
2
+
1
1
+
1
2
̇
4
=
−
(
+
1
)
2
−
4
(18)
t
he
s
a
m
e
re
s
u
l
t
s
w
e
re
fo
und
i
n
t
h
e
or
e
m
(3)
.
T
he
t
he
or
e
m
4
s
how
e
d
t
ha
t
pr
opos
e
d
c
ont
r
ol
w
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c
h
c
ons
i
s
t
s
of
s
i
x
t
e
r
m
s
a
c
hi
e
v
e
d
s
y
nc
hr
oni
z
a
t
i
on
.
T
h
e
t
i
m
e
s
yn
c
hron
i
z
a
t
i
on
a
t
(
4.
2
0)
t
i
m
e
(s
e
c
)
i
l
l
us
t
r
a
t
e
d
i
n
F
i
g
ur
e
8.
F
i
gure
7
.
T
he
c
onv
e
rge
s
of
s
ys
t
e
m
(16)
w
i
t
h
c
on
t
ro
l
l
e
r
(1
5)
F
i
gure
8
.
T
he
c
onv
e
rge
s
of
s
ys
t
e
m
(18)
w
i
t
h
c
on
t
ro
l
l
e
rs
(17
)
4.
C
O
N
C
LU
S
I
O
N
In
t
h
e
pa
p
e
r
,
s
yn
c
hron
a
i
z
t
i
on
prob
l
e
m
f
or
4
-
D
R
a
bi
novi
c
h
h
ype
rc
ha
o
t
i
c
s
ys
t
e
m
i
s
c
ons
i
d
e
r
ed
,
ba
s
e
d
on
t
w
o
s
t
r
a
t
e
gi
e
s
:
a
c
t
i
ve
a
nd
non
-
l
i
ne
a
r
c
on
t
rol
l
e
r
.
T
h
e
s
t
a
bi
l
i
t
y
of
e
rror
dyn
i
m
c
a
l
s
ys
t
e
m
s
a
re
e
s
t
a
b
l
i
s
he
d
ba
s
e
d
on
t
he
L
ya
pu
nov
t
he
ory
a
nd
c
o
m
p
a
re
d
be
t
w
e
e
n
t
he
s
e
s
t
r
a
t
e
gi
e
s
a
nd
f
ound
t
h
a
t
bot
h
of
t
h
e
m
l
e
a
d
t
o
s
ync
hron
i
z
a
t
i
on
b
ut
t
h
e
p
e
rfor
m
a
n
c
e
of
t
h
e
no
nl
i
ne
a
r
c
ont
r
ol
l
e
r
i
s
fa
s
t
e
r
a
nd
be
t
t
e
r
t
ha
n
t
he
a
c
t
i
v
e
c
on
t
rol
.
A
l
s
o,
t
h
e
nu
m
b
e
r
of
t
e
r
m
s
of
c
on
t
rol
l
e
r
is
l
e
s
s
t
h
a
n
fi
rs
t
s
t
r
a
t
e
gy.
A
C
K
N
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WL
ED
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E
M
EN
TS
T
he
a
ut
hors
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r
e
ve
ry
g
r
a
t
e
fu
l
t
o
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n
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e
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of
M
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ul
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Col
l
e
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e
of
Com
p
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e
r
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c
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e
nc
e
s
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nd
M
a
t
h
e
m
a
t
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c
s
for
t
h
e
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prov
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d
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c
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t
i
e
s
,
w
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c
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l
pe
d
t
o
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m
prove
t
h
e
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l
i
t
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o
f
t
hi
s
w
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.
R
EF
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EN
C
ES
[
1]
S
.
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.
A
L
-
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z
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,
"
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pu
t
at
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o
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l
.
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,
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.
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-
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152,
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.
[
2]
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s
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.
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365
-
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20
19
.
[
3]
N
.
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.
A
l
-
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ha
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pu
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e
r
s
i
n
B
i
ol
ogy
and
M
e
d
i
c
i
n
e
,
vol
.
103
,
pp.
2
62
-
268
,
2018
.
[
4]
Z
.
N
.
A
l
-
ka
t
e
e
b
a
nd
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.
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.
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l
-
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,
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.
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