Information Technology Reference
In-Depth Information
d
.3/
1
D f
d
1
.y; y/ d
.3/
(5.82)
D f
d
2
.y; y/
2
The state vector of the circadian oscillator is extended by including in it state
variables that describe the disturbance's variations. Thus one has
z
1
D y
1
,
z
2
Dy
1
,
z
3
Dy
1
,
z
4
D y
2
,
z
5
Dy
2
,
z
6
Dy
2
,
z
7
D d
1
,
z
8
D d
1
, and
z
9
D d
1
,
z
10
D d
2
,
z
11
D d
2
, and
z
12
D d
2
. The associated description of the system in the form of
state-space equations becomes
z
e
D A
z
e
C B
u
e
z
m
D C
e
z
e
(5.83)
where
z
e
is the extended state vector
z
e
D Œ
z
1
; ;
z
12
T
,
v
e
D Œ
v
1
;
v
2
;f
d
1
;f
d
2
T
,is
the extended control input vector, and matrices A
e
, B
e
, and C
e
are defined as
0
@
1
A
0
@
1
A
0
@
1
A
010000000000
001000000000
000000100000
000010000000
000001000000
000000000100
000000010000
000000001000
000000000000
000000000010
000000000001
000000000000
0000
0000
1000
0000
0000
0100
0000
0000
0010
0000
0000
0001
10
00
00
01
00
00
00
00
00
00
00
00
C
e
A
e
D
B
e
D
D
(5.84)
The associated disturbance observer is
z
D A
o
z
C B
o
u
C K.
z
m
z
m
/
z
m
D C
o
z
(5.85)
where A
o
D A, C
o
D C, and
001000000000
000001000000
B
o
D
(5.86)
where now
u
D Œ
u
1
;
u
2
T
. 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
,
169
]. This is
Derivative-free nonlinear Kalman
filtering
. As explained in Chap.
4
, unlike EKF, the filtering method is performed
without the need to compute Jacobian matrices and does not introduce numerical
errors due to approximative linearization with Taylor series expansion.
