Information Technology Reference
In-Depth Information
The design of a disturbance observer based on the Derivative-free nonlinear
Kalman Filter enables simultaneous estimation of the nonmeasurable elements of
the state vector, that is y
2
, y
3
and y
5
, y
6
and estimation of the disturbance term
d
1
D
z
7
and d
2
D
z
10
. For the compensation of the disturbance term the control
input that is applied to the circadian oscillator model is modified as follows:
v
1
.k/ D
v
.k/ d
1
.k/ or
v
1
.k/ D
v
.k/
z
7
.k/
(5.87)
v
2
.k/ D
v
.k/ d
2
.k/ or
v
2
.k/ D
v
.k/
z
10
.k/
In the design of the associated disturbances' estimator one has the dynamics defined
in Eq. (
5.85
), where K2R
121
is the state estimator's gain and matrices A
o
, B
o
, and
C
o
have been given in Eqs. (
5.84
)-(
5.86
). The discrete-time equivalents 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 [
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/
(5.88)
Time update
:
A
d
.k/P.k/ A
d
.k/ C Q.k/
P
.k C 1/ D
(5.89)
A
d
.k/
z
.k/ C B
d
.k/
v
.k/
z
.k C 1/ D
5.8
Simulation Tests
The nonlinear dynamical model of the coupled circadian oscillators has been
described in Eqs. (
5.51
)-(
5.56
). Using the canonical form model control law for the
FRQ protein synthesis was computed according to the stages described in Sect.
5.6
.
The Derivative-free nonlinear Kalman Filter used for model uncertainty, measure-
ment noise, and external disturbances compensation was designed according to
Sect.
5.7
.
Results on the synchronization between a primary and a secondary circadian cell
are provided for six different cases, each one associated with different disturbance
