Information Technology Reference
In-Depth Information
one has the dynamics of the circadian oscillator in the form
x
.3/
D f.x/C g.x/
u
(5.26)
or equivalently
y
.3/
D f.y; y/C g.y; y/
u
(5.27)
where function f.y; y/ comprises all terms in the right part of Eq. (
5.23
) that do not
multiply the control input K
s
, whereas function g.y; y/ comprises those terms in
the right part of Eq. (
5.23
) that multiply the control input K
s
.
After transforming the circadian oscillator in the form y
.3/
D f.y; y/Cg.y; y/
u
and by defining the new control input
v
D f.y; y/Cg.y; y/
u
one has the dynamics
y
.3/
D
v
(5.28)
Next, by defining the new state variables
z
1
D y,
z
2
Dy, and
z
3
Dy one obtains
the following state-space description for the circadian oscillator
0
1
0
1
0
1
0
1
z
1
z
2
z
3
010
001
000
z
1
z
2
z
3
0
0
1
@
A
D
@
A
@
A
C
@
A
v
(5.29)
and the associated measurement equation is
0
@
1
A
z
1
z
2
z
3
z
m
D
100
(5.30)
z
3
D y
.3/
Knowing that
D
v
the control law for the synthesis of the FRQ protein
and consequently the control law of the circadian rhythms is
D
v
D y
.3/
d
y
.3/
(5.31)
K
1
. y y
d
/ K
2
. y y
d
/ K
3
.y y
d
/
Since
v
D f.y; y/ C g.y; y/
u
, the control input that is finally applied to the model
of the circadian oscillator is
u
D g
1
.y; y/Œ
v
f.y; y/
(5.32)
The above control law assures that lim
t!1
y.t/ D y
d
.t/ which implies
lim
t!1
x
1
.t/ D x
1;d
.t/ (concentration of frq mRNA), lim
t!1
x
2
.t/ D x
2;d
.t/
(concentration
of
FRQ
protein
in
cytoplasm),
and
lim
t!1
x
3
.t/
D
x
3;d
.t/
(concentration of FRQ protein in nucleus).
