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and the associated measurement equation is
0
1
z
1
z
2
z
3
z
m
D
100
@
A
(5.14)
For the linearized model of the circadian oscillator dynamics the control law is
computed using Eqs. (
5.31
) and (
5.32
), as in the case of differential flatness theory.
5.4
Protein Synthesis Control Using Differential
Flatness Theory
5.4.1
Differential Flatness of the Circadian Oscillator
Considering the model of the circadian oscillator described in Eqs. (
5.2
)-(
5.4
)the
flat output is taken to be y D x
1
.
From Eq. (
5.2
) it holds
K
i
y
K
M
Cy
)
y
D
v
s
K
i
Cx
3
v
m
D
n
.K
m
Cy/K
i
.K
m
Cy/yC
v
m
y
h
v
s
v
m
K
m
Cy
y
io
)
y
x
3
n
(5.15)
n
.K
m
Cy/K
i
.K
m
Cy/yC
v
m
y
h
v
s
v
m
K
m
Cy
y
io
1=n
y
x
3
D
Therefore, state variable x
3
is a function of the flat output and its derivatives that is
x
3
D f
1
.y; y/.
From Eq. (
5.4
) one obtains
x
3
D k
1
x
2
k
2
x
3
)
x
2
D
(5.16)
1
K
1
Œx
3
C k
2
x
3
Therefore, state variable x
2
is a function of the flat output and its derivatives that is
x
2
D f
3
.y; y/.
From Eq. (
5.3
) it holds
x
2
x
2
D K
s
x
1
v
d
K
d
Cx
2
K
1
x
2
C K
2
x
3
)
(5.17)
1
x
2
K
s
D
x
1
Œx
2
v
d
K
d
Cx
2
K
1
x
2
K
2
x
3
Taking into account that x
1
D y, x
3
D f
1
.y; y/, and x
2
D f
2
.y; y/ one obtains
that the control input K
s
is also a function of the flat output y and of the associated
derivatives, that is K
s
D f
3
.y; y/.
