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Equivalently, it holds that
z
6
D L
f
h
2
.x/ C L
g
1
L
f
h
2
.x/
u
1
C L
g
2
L
f
h
2
.x/
u
2
(5.40)
where
L
f
h
2
.x/ D
n
v
m
K
m
2.K
m
C
x
1
/
.K
m
Cx
4
/
4
h
v
s
K
m
Cx
4
i
C
v
m
.K
m
Cx
4
/
2
o
K
i
x
4
K
m
K
m
K
i
Cx
6
v
m
.K
m
Cx
4
/
2
v
m
h
v
s
i
K
i
K
i
Cx
6
n
v
s
K
i
nx
n
1
.K
i
Cx
6
/
2
K
1
oh
K
d
Cx
5
K
1
x
5
C K
2
x
6
C K
c
.x
5
x
2
/
i
x
4
x
5
v
m
K
m
Cx
4
v
d
3
n
v
m
.K
m
Cx
4
/
2
v
s
K
i
nx
n
1
K
m
C
6
.K
i
Cx
6
/
2
v
s
K
i
n.n
1/x
n
6
.K
i
C
x
6
/
K
i
nx
n
6
2.K
i
C
x
6
/
.K
1
x
5
K
2
x
6
/
.K
i
Cx
6
/
4
.K
i
Cx
6
/
2
K
2
o
ŒK
1
x
2
K
2
x
6
C
v
s
K
i
nx
n
1
6
(5.41)
L
g
1
L
f
h
2
.x/ D 0
(5.42)
L
g
2
L
f
h
2
.x/ D
v
s
K
i
nx
.n1/
6
.K
i
C x
6
/
2
K
1
x
4
(5.43)
Therefore, for the system of the coupled circadian neurons one obtains an input
output linearized form given by
z
.3/
1
z
.3/
4
!
L
f
h
1
.x/
L
f
h
2
.x/
!
L
g
1
L
f
h
1
.x/ L
g
2
L
f
h
1
.x/
L
g
1
L
f
h
2
.x/ L
g
2
L
f
h
2
.x/
!
u
1
u
2
D
C
(5.44)
The new control inputs are defined as
v
1
D L
f
h
1
.x/ C L
g
1
L
f
h
1
.x/
u
1
C L
g
2
L
f
h
1
.x/
u
2
v
2
D L
f
h
2
.x/ C L
g
1
L
f
h
2
.x/
u
1
C L
g
2
L
f
h
2
.x/
u
2
(5.45)
Therefore, the system's dynamics becomes
x
.3/
1
D
v
1
(5.46)
x
.3/
4
D
v
2
By defining the new state variables
z
1
D x
1
,
z
2
D x
2
,
z
3
D x
3
,
z
4
D x
4
,
z
5
D x
5
, and
z
6
D x
6
one obtains a description for the system of the coupled circadian oscillators
in the linear canonical (Brunovsky) form
