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Time update
. Compute:
P
.k C 1/ D J
.x.k//P.k/J
T
.x.k// C Q.k/
x
.k C 1/ D .x.k// C L.k/
u
.k/
(4.93)
4.12.2
Design of a Disturbance Observer for the Model
of the Coupled Neurons
Using Eq. (
4.69
), the nonlinear model of the coupled neurons can be written in the
MIMO canonical form that was described in Sect.
4.8
, that is
0
@
1
A
D
0
@
1
A
0
@
1
A
C
0
@
1
A
z
1
z
2
z
3
z
4
0100
0000
0010
0000
z
1
z
2
z
3
z
4
00
10
00
01
v
1
v
2
(4.94)
where
z
1
D y
1
,
z
2
Dy
1
,
z
3
D y
2
and
z
4
Dy
2
, while
v
1
D L
f
h
1
.x/ C
L
g
1
L
f
h
1
.x/
u
1
C L
g
2
L
f
h
1
.x/
u
2
and
v
2
D L
f
h
2
.x/ C L
g
1
L
f
h
2
.x/
u
1
C
L
g
2
L
f
h
2
.x/
u
2
. Thus one has a MIMO linear model of the form
z
D A
z
C B
v
z
m
D C
z
(4.95)
where
z
D Œ
z
1
;
z
2
;
z
3
;
z
4
T
and matrices A,B,C are in the MIMO canonical form
0
1
0
1
0
1
0100
0000
0010
0000
00
10
00
01
10
00
01
00
@
A
@
A
@
A
A D
B D
C
T
D
(4.96)
Assuming now the existence of additive input disturbances in the linearized model
of the coupled neurons described in Eq. (
4.69
) one gets
L
f
h
1
.x/
L
f
h
2
.x/
!
z
2
z
2
D
(4.97)
L
g
1
L
f
h
1
.x/ L
g
2
L
f
h
1
.x/
L
g
1
L
f
h
2
.x/ L
g
2
L
f
h
2
.x/
u
1
C d
1
u
2
C d
2
C
These disturbances can be due to external perturbations affecting the coupled
neurons and can also represent parametric uncertainty for the neurons' model. It
can be assumed that the additive disturbances d
i
;iD 1;2 are described by the
