Synchronization of Circadian Neurons and Protein Synthesis Control - Advanced Models of Neural Networks - page 115

Consequently, all state variables and the control input of the system are written
as functions of the flat output and its derivatives and this proves that the model of
the circadian oscillator is a differentially flat one.

5.4.2 Transformation into a Canonical Form

From Eq. ( 5.2 ) it holds that

K i
K i Cx 3 v m

x 1
K M Cx 1

x 1 D v s (5.18)

Deriving the aforementioned relation with respect to time one has

x 1 D v s K i nx n 1

K M

.K i Cx 3 / 2 x 3 v m

.K M Cx 1 / 2 x 1

(5.19)

3

Substituting x 2 from Eq. ( 5.3 ) and x 3 from Eqs. ( 5.4 )into( 5.19 ) one has

.K M Cx 1 / 2 v s

K m Cx 1 (5.20)

x 1 D v s K i nx n 1

K i

K M x 1

.K i Cx 3 / 2 .K 1 K 2 k 2 x 3 / v m

K i Cx 3 v m

3

which can be also written as

x 1 D v s K i nx n1

.K i C x 3 / 2 K 1 x 2 v s K i nx n1

3 3

.K i C x 3 / 2 K 2 x 3

K i

K M
.K M C x 1 / 2 v s

K M
.K M C x 1 / 2 v m

x 1
K m C x 1

v m

K i C x 3 C v m

(5.21)

By deriving once more with respect to time one obtains

D v s ŒK i n.n 1/x n 3 .K i C x 3 / 2

C K i nx n 3 2.K i C x 3 /
.K i C x 3 / 4

x .3/
1

x 3 K 1 x 2

C v s K i nx n1

3

.K i C x 3 / 2 K 1 x 2
v s ŒK i n.n 1/x n2

.K i C x 3 / 2 C K i nx n1 2.K i C x 3 /

3 3

x 3 K 2 x 3

.K i C x 3 / 4

v s K i nx n1

3
.K i C x 3 / 2 K 2 x 3
v m ΠK m 2.K m C x 1 /
.K m C x 1 / 4

K i
K i C x 3

x 1 v s