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a
b
5
2
5
0
0
0
−5
−2
5
10
15
20
25
30
35
40
0
20
40
0
20
40
time
time
time
0.5
1
5
0
0
0
−0.5
−1
5
10
15
20
25
30
35
40
0
20
40
0
20
40
time
time
time
0.2
0.05
5
0
0
0
−0.2
−0.05
5
10
15
20
25
30
35
40
0
20
40
0
20
40
time
time
time
Fig. 5.3
Test 1: (
a
) Synchronization of state variables x
i
(i D 1;4: frq mRNA concentration,
i D 2;5: FRQ protein concentration in cytoplasm, i D 3;6: FRQ protein concentration in nucleus)
between the two circadian cells (
red continuous line
denotes concentration in cell 1 whereas the
dashed blue line
denotes concentration in cell 2) (
b
) Estimation of disturbance inputs and of their
derivatives (
blue lines
) with the use of the Derivative-free nonlinear Kalman Filter
a
b
5
5
10
0
0
5
0
−5
−5
5
10
15
20
25
30
35
40
0
10
20
30
40
0
10
20
30
40
time
ti
m
e
ti
m
e
1
1
10
0
0
5
0
−1
−1
5
10
15
20
25
30
35
40
0
10
20
30
40
0
10
20
30
40
time
time
time
0.5
0.1
10
0
0
5
0
−0.5
−0.1
5
10
15
20
25
30
35
40
0
10
20
30
40
0
10
20
30
40
time
time
time
Fig. 5.4
Test 2: (
a
) Synchronization of state variables x
i
(i D 1;4: frq mRNA concentration,
i D 2;5: FRQ protein concentration in cytoplasm, i D 3;6: FRQ protein concentration in nucleus)
between the two circadian cells (
red continuous line
denotes concentration in cell 1 whereas the
dashed blue line
denotes concentration in cell 2) (
b
) Estimation of disturbance inputs and of their
derivatives (
blue lines
) with the use of the Derivative-free nonlinear Kalman Filter
and uncertainty terms affecting the cell as well as with different setpoints for the
state vector elements of the coupled circadian cell model. The results obtained
are shown in Figs.
5.3
a,
5.4
a,
5.5
a,
5.6
a,
5.7
a, and
5.8
a. It can be observed that
the proposed control scheme that was based on linearization and decoupling of
the circadian's oscillator model with the use of differential flatness theory enabled
complete synchronization between the state variables of the first cell (x
i
;iD 1;2;3
denoted with the continuous red line) and the state variables of the second cell
(x
i
;iD 4;5;6 denoted with the dashed blue line).
