Information Technology Reference
In-Depth Information
4.12.3
Disturbances Compensation for the Model
of the Coupled Neurons
For the aforementioned model, and after carrying out discretization of matrices A
o
,
B
o
, and C
o
with common discretization methods one can perform Kalman filtering
[
157
-
159
,
166
,
169
,
170
]. This is
Derivative-free nonlinear Kalman filtering
which,
unlike EKF, is performed without the need to compute Jacobian matrices and does
not introduce numerical errors due to approximative linearization with Taylor series
expansion.
In the design of the associated disturbances' estimator one has the dynamics
defined in Eq. (
5.85
), where K2R
82
is the state estimator's gain and matrices A
o
,
B
o
, and C
o
have been defined in Eqs. (
4.100
) and (
4.101
). The discrete-time equiv-
alents of matrices A
o
, B
o
, and C
o
are denoted as A
d
, B
d
, and C
d
, respectively, and
are computed with the use of common discretization methods. Next, a Derivative-
free nonlinear Kalman Filter can be designed for the aforementioned representation
of the system dynamics [
157
,
158
]. The associated Kalman Filter-based disturbance
estimator is given by the recursion [
159
,
166
,
169
].
Measurement update
:
K.k/ D P
.k/ C
d
ΠC
d
P
.k/ C
d
C R
1
z
.k/ D
z
.k/ C K.k/ΠC
d
z
.k/ C
d
z
.k/
P.k/D P
.k/ K.k/ C
d
P
.k/
(4.104)
Time update
:
A
d
.k/P.k/ A
d
.k/ C Q.k/
P
.k C 1/ D
(4.105)
A
d
.k/
z
.k/ C B
d
.k/
v
.k/
z
.k C 1/ D
To compensate for the effects of the disturbance inputs it suffices to use in the control
loop the estimates of the disturbance terms that is
z
5
D d
1
and
z
7
D d
2
and the
modified control input vector
v
1
!
or
v
D
v
1
z
5
v
2
z
7
f
a
v
D
(4.106)
f
b
v
2
4.13
Simulation Tests
Results on the synchronization between the master and the slave FitzHugh-Nagumo
neuron are provided for three different cases, each one associated with difference
disturbance and uncertainty terms affecting the neuron model as well as with
different setpoints for the state vector elements of the coupled neurons' model.
