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where x D Œx
1
; ;x
n
T
is the state vector of the transformed system (according to
the differential flatness formulation),
u
D Œ
u
1
; ;
u
p
T
is the set of control inputs,
y D Œy
1
; ;y
p
T
is the output vector, f
i
are the drift functions, and g
i;j
;i;jD
1;2; ;pare smooth functions corresponding to the control input gains, while d
j
is a variable associated with external disturbances. It holds that r
1
Cr
2
CCr
p
D n.
Having written the initial nonlinear system into the canonical (Brunovsky) form it
holds
D f
i
.x/ C
P
jD1
g
ij
.x/
u
j
C d
j
y
.r
i
/
i
(4.28)
Next the following vectors and matrices can be defined f.x/DŒf
1
.x/; ;f
n
.x/
T
,
g.x/
Œg
1
.x/; ;g
n
.x/
T
Œg
1i
.x/; ;g
pi
.x/
T
, A
D
with g
i
.x/
D
D
diag
ŒB
1
; ;B
p
, C
T
diag
ŒA
1
; ;A
p
;B
D
diag
ŒC
1
; ;C
p
;d D
Œd
1
; ;d
p
T
, where matrix A has the MIMO canonical form, i.e. with block-
diagonal elements
D
0
1
010
000
:
:
:
:
:
:
@
A
:
:
:
001
000
A
i
D
(4.29)
r
i
r
i
B
i
D
0001
1r
i
C
i
D
1000
1r
i
Thus, Eq. (
4.28
) can be written in state-space form
x D
Ax
C
Bv
C B d
y D Cx
(4.30)
where the control input is written as
v
D f.x/C g.x/
u
.
4.9
Linearization of the FitzHugh-Nagumo Neuron
4.9.1
Linearization of the FitzHugh-Nagumo Model Using
a Differential Geometric Approach
The model of the dynamics of the FitzhHugh Nagumo neuron is considered
dV
dt
D V.V a/.1 V/
w
C I
dw
dt
(4.31)
D .V
w
/
