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In-Depth Information
x
2
K
d
C x
2
K
1
x
2
C K
2
x
3
C K
c
.x
2
x
5
/
x
2
D K
s
1
x
1
v
d
(5.52)
x
3
D K
1
x
2
K
2
x
3
(5.53)
K
i
K
i
C x
6
v
m
x
4
K
M
C x
4
x
4
D
v
s
(5.54)
x
5
K
d
C x
5
K
1
x
5
C K
2
x
6
C K
c
.x
5
x
2
/
x
5
D K
s
2
x
4
v
d
(5.55)
x
6
D K
1
x
5
K
2
x
6
(5.56)
It can be proven that the model of the coupled circadian neurons is differentially
flat. The flat output is defined as
(5.57)
y D Œy
1
;y
2
D Œx
1
;x
4
As it has been shown in Eq. (
5.51
) in the case of the independent circadian oscillator
it holds
n
.K
m
Cy
1
/ yC
v
m
y
1
h
v
s
v
m
K
m
Cy
1
y
1
io
1=n
.K
m
Cy
1
/K
i
y
1
(5.58)
x
3
D
therefore it holds x
3
D f
1
.y
1
; y
1
;y
2
; y
2
/. Moreover from Eq. (
5.53
) it holds
1
x
3
D K
1
x
2
K
2
x
3
)x
2
D
K
1
Œx
3
C K
2
x
3
(5.59)
therefore it holds x
2
D f
2
.y
1
; y
1
;y
2
; y
2
/. Additionally, from Eq. (
5.54
) it holds
n
.K
m
Cy
2
/ y
2
C
v
m
y
2
h
v
s
v
m
K
m
Cy
2
y
2
io
1=n
.K
m
Cy
2
/K
i
y
2
(5.60)
x
6
D
therefore it holds x
6
D f
3
.y
1
; y
1
;y
2
; y
2
/. Furthermore, from Eq. (
5.56
) one obtains
1
K
1
Œx
6
C K
2
x
6
x
5
D K
1
x
5
K
2
x
6
)x
5
D
(5.61)
therefore it holds x
5
D f
4
.y
1
; y
1
;y
2
; y
2
/. Next, using Eq. (
5.52
) one finds
x
1
h
K
d
Cx
2
C K
1
x
2
K
2
x
3
K
c
.x
2
x
5
/
i
1
x
2
K
s
1
D
x
2
v
d
(5.62)
Since, x
1
, x
2
, x
3
, and x
5
are functions of the flat output and of its derivatives it holds
that K
s
1
D f
5
.y
1
; y
1
;y
2
; y
2
/. Similarly, from Eq. (
5.55
) one obtains for the second
control input
x
4
h
K
d
Cx
5
C K
1
x
5
K
2
x
6
K
c
.x
5
x
2
/
i
1
x
5
K
s
2
D
x
5
v
d
(5.63)
