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which can be also written in the matrix canonical form
z
1
z
2
01
00
z
1
z
2
0
1
v
D
C
(4.40)
Next, the relative degree of the system is computed. It holds that L
g
h
1
.x/ D 0,
whereas
@f
1
@x
2
g
2
)
L
g
L
f
h
1
.x/ D0 C 1 D ¤0
@f
1
@x
1
g
1
C
L
g
L
f
h
1
.x/ D
(4.41)
and L
g
L
n1
f
h
1
.x/¤0 for n D 2. Therefore, the relative degree of the system is
n D 2.
4.9.2
Linearization of the FitzHugh-Nagumo Model Using
Differential Flatness Theory
Next, the FitzHugh-Nagumo model is linearized with the use of the differential
flatness theory. The flat output of the system is taken to be y D h.x/ D x
1
. It holds
that y Dx
1
. From the first row of the state space equation of the neuron given in
Eq. (
4.33
) one gets
x
1
C
x
1
y
C
y
x
2
D
)x
2
D
(4.42)
Moreover, from the second row of Eq. (
4.33
) one gets
(4.43)
u
Dx
2
x
2
.x
2
˛/.1 x
2
/ C x
1
and since x
1
, x
2
are functions of the flat output and its derivatives one has that the
control input
u
is also a function of the flat output and its derivatives. Therefore, the
considered neuron model stands for a differentially flat dynamical system.
From the computation of the derivatives of the flat output one obtains
y
1
Dx
1
D f
1
(4.44)
f
1
D
@
@
@x
1
.x
1
C x
2
/x
2
)
y
1
D x
1
C x
2
)y
1
D.x
1
C x
2
/CŒx
2
.x
2
a/.1 x
2
/ x
1
C
u
:
(4.45)
y
1
D
@x
1
.x
1
C x
2
/x
1
C
