Environmental Engineering Reference
In-Depth Information
Multiple observer strategies can be used to detect and isolate sensor faults on
the basis of model in Eqs.
10.33
-
10.34
. Two classical schemes can be distin-
guished, namely the Dedicated Observer Scheme (DOS) and the Generalized
Observer Scheme (GOS) [
7
,
17
]. Both schemes are presented in Fig.
10.4
.
The residual generation system per three-phase signals consists of three
observers when using the GOS or the DOS. The i-th observer, i
2f
1
;
2
;
3
g
is
designed on the basis of model dynamics described by Eq.
10.33
with the fol-
lowing output equation:
y
i
ð
k
Þ¼
C
i
x
ð
k
Þþ
v
i
ð
k
Þþ
f
i
ð
k
Þ
ð
10
:
35
Þ
For the GOS, y
i
ð
k
Þ
denotes vector y
m
ð
k
Þ
without the i-th measurement, f
i
ð
k
Þ
denotes vector f
ð
k
Þ
without the i-th component, and C
i
stands for matrix C without
the i-th row. For the DOS, y
i
ð
k
Þ
is the i-th measurement in vector y
m
ð
k
Þ
, f
i
ð
k
Þ
is
the i-th fault or the i-th component in f
ð
k
Þ
, and C
i
is the i-th row of C. v
i
ð
k
Þ
is a
zero-mean Gaussian white noise sequence with covariance matrix R
vi
. It turn out
that, for both schemes, the pair
ð
C
i
;
U
ð
x
e
ÞÞ
is observable when x
e
is a non zero
constant, or uniformly completely observable when x
e
ð
t
Þ
is varying within an
interval
½
x
min
;
x
max
with x
min
;
x
max
2
þ
. Hence, both schemes can be imple-
R
mented in the considered application.
For residual generation purposes, the observers are Kalman filters described as [
18
]
x
i
ð
k
j
k
1
Þ¼
U
ð
x
e
ð
k
1
ÞÞ
x
i
ð
k
1
Þ
ð
10
:
36
Þ
M
i
ð
k
j
k
1
Þ¼
U
ð
x
e
ð
k
1
ÞÞ
M
i
ð
k
1
Þ
U
ð
x
e
ð
k
1
ÞÞ
T
þ
R
w
ð
10
:
37
Þ
y
i
ð
k
Þ¼
C
i
x
i
ð
k
j
k
1
Þ
ð
10
:
38
Þ
1
C
i
M
i
ð
k
j
k
1
Þ
C
T
K
i
ð
k
Þ¼
M
i
ð
k
j
k
1
Þ
C
i
i
þ
R
vi
ð
10
:
39
Þ
x
i
ð
k
Þ¼
x
i
ð
k
j
k
1
Þþ
K
i
ð
k
Þð
y
i
ð
k
Þ
y
i
ð
k
ÞÞ
ð
10
:
40
Þ
M
i
ð
k
Þ¼
M
i
ð
k
j
k
1
Þ
K
i
ð
k
Þ
C
i
M
i
ð
k
j
k
1
Þ
ð
10
:
41
Þ
In Eqs.
10.36
-
10.41
, the frequency x
e
ð
k
Þ
has been replaced by its estimate
x
e
ð
k
Þ
. This estimation can be obtained, for instance, by a frequency-locked loop
(FLL) as described in [
19
].
The i-th residual can be set as the innovation of the i-th Kalman filter (KF),
namely a two-dimensional vector when using the GOS or a scalar signal when
using the DOS:
