Information Technology Reference
In-Depth Information
2K
Dx
2
, the initial description of the system given in
Eq. (
6.33
) is rewritten as follows:
K
Dx
2
and b D
Denoting a D
0
@
1
A
0
@
1
A
0100000 000000
b0a0000 000000
0001000 000000
a0b0a00 000000
0000010 000000
00a0b0a 000000
0000000 000100
0000000 a0b0a0
0000000 000001
0000000 00a0b0
000 00
100 00
000 00
010 00
000 00
001 00
000 00
000 10
000 00
000 01
0
@
1
A
0
@
1
A
y
1;1
y
2;1
y
1;2
y
2;2
y
1;N
1
y
2;N
1
y
1;N
y
2;N
y
1;1
y
2;1
y
1;2
y
2;2
y
1;N
1
y
2;N
1
y
1;N
y
2;N
0
@
1
A
v
1
v
2
v
3
v
N
1
v
N
D
C
(6.36)
The associated control inputs are now defined as
K
v
1
D
x
2
0
C f.y
1;1
/
v
2
D f.y
1;2
/
v
3
D f.y
1;3
/
v
N1
D f.y
1;N1
/
v
N
D
(6.37)
K
x
2
NC1
C f.y
1;N
/
By selecting measurements from a subset of points x
j
j2Œ1;2; ;m,the
associated observation (measurement) equation remains as in Eq. (
6.34
), i.e.
0
@
1
A
y
1;1
y
2;1
y
1;2
y
2;2
y
1;N
y
2;N
0
@
1
A
0
@
1
A
D
100 00
000 00
000 10
000 00
z
1
z
2
z
m
(6.38)
For the linear description of the system in the form of Eq. (
6.35
) one can perform
estimation using the standard Kalman Filter recursion. The discrete-time Kalman
filter can be decomposed into two parts: (1) time update (prediction stage), and (2)
measurement update (correction stage).
The discrete-time equivalents of matrices A, B, C in Eq. (
6.36
) and Eq. (
6.38
)
are computed using common discretization methods. These are denoted as A
d
, B
d
,
and C
d
respectively. Then the Kalman Filter recursion becomes:
