International Journal of Automation and Computing 04(2), April 2007, 164168DOI: 10.1007/s1163300701647
Parity Relation Based Fault Estimation for NonlinearSystems: An LMI Approach
Sing Kiong Nguang
1
∗
Ping Zhang
2
Steven X. Ding
2
1
The Department of Electrical and Computer Engineering, the University of Auckland, Auckland, New Zealand
2
AKS, Faculty of Engineering, University of DuisburgEssen, Duisburg, Germany
Abstract:
This paper proposes a parity relation based fault estimation for a class of nonlinear systems which can be modelled byTakagiSugeno (TS) fuzzy models. The design of a parity relation based residual generator is formulated in terms of a family of linearmatrix inequalities (LMIs). A numerical example is provided to illustrate the eﬀectiveness of the proposed design techniques.
Keywords:
Fuzzy systems, nonlinear systems, fault identiﬁcation, fault detection, fault diagnosis.
1 Introduction
Many systems are subject to random variations whichmay result from component and interconnection failures,parameters shifting, tracking, sudden environmental disturbances, abrupt variations of the operating condition,
etc
.Therefore, in order to avoid production deteriorations ordamage to machines and humans, variations have to be detected as quickly as possible and decisions that stop thepropagation of their eﬀects have to be made. Over the lastthree decades various fault detection and isolation (FDI)techniques have been developed
[1
−
3]
. The parity relationapproach or analytical redundancy relation (ARR) is oneof the most commonly used techniques in FDI, see [310].This technique is ﬁrst proposed in [5] to detect sensor faultsin a ﬂight control system. Then extended in [6] to detectboth sensor and actuator faults. This technique is furthergeneralised to handle systems with unknown disturbanceand uncertainty
[8
,
11
−
13]
. Recently, an attempt has beenmade to extend the parity relation technique to the inputoutput model of a nonlinear system in [14] based on an inverse model of the system which in general is not unique.Hence, this extension is not always successful.In the past two decades, many researchers have studied a class of nonlinear systems described by a TakagiSugeno(TS) fuzzy model, see [1521]. In this TS fuzzymodel, local dynamics in diﬀerent state space regions arerepresented by local linear systems. The overall model of the system is obtained by “blending” of these linear modelsthrough nonlinear membership functions. In other words,a TS fuzzy model is essentially a multimodel approachin which simple submodels are combined to represent theglobal behavior of the system. This fuzzy model has beenshown to be able to approximate a large class of nonlinearsystems. However, to the best of our knowledge, the problem of paritybased fault estimation for TS fuzzy modelshas not been addressed in the literature.Motivated by the capability of the TS fuzzy model, thispaper generalizes the parity space approach for linear sys
Manuscript received January 10, 2006; revised November 18, 2006.This work was supported by the Alexander von Humboldt Foundation.*Corresponding author. Email address: sk.nguang@auckland.ac.nz
tems to nonlinear systems described by TS fuzzy models.The design procedure is given in terms of a family of LMIswhich can be solved eﬀectively by an algorithm given in[22]. The eﬀectiveness of the proposed design techniques isdemonstrated through a numerical example.This paper is organized as follows. The parity based faultestimation for nonlinear systems is developed in Section 2.The validity of this approach is demonstrated in Section 3.Finally, in Section 4, the conclusion is drawn.
2 Paritybased fault estimation fornonlinear systems
Consider the following nonlinear system:
x
(
k
+ 1) =
f
(
x
(
k
)) +
g
1
(
x
(
k
))
u
(
k
)+
g
2
(
x
(
k
))
w
(
k
) +
g
3
(
x
(
k
))
f
(
k
)
y
(
k
) =
h
(
x
(
k
))
x
(
k
) +
d
1
(
x
(
k
))
u
(
k
)+
d
2
(
x
(
k
))
w
(
k
) +
d
3
(
x
(
k
))
f
(
k
)
,
(1)where
x
(
k
)
∈
R
n
is the state vector,
u
(
k
)
∈
R
m
is the input,
w
(
k
)
∈
R
p
is the unknown disturbance/uncertainty,
y
(
k
)
∈
R
is the measurement,
f
(
k
)
∈
R
q
is the fault.
f
(
x
(
k
))
, g
i
(
x
(
k
))
, h
(
x
(
k
)), and
d
i
(
x
(
k
)) are function of
x
(
k
).A fuzzy dynamic model has been proposed by Takagiand Sugeno in [15] to represent local linear input/outputrelations of nonlinear systems. This fuzzy linear model isdescribed by IFTHEN rules and has been shown to be ableto approximate a large class of nonlinear systems. The
i
thrule of this fuzzy model for the nonlinear system (1) is of the following form
[15]
:Plant Rule
i:
IF
ν
1
(
k
) is
M
i
1
and
···
and
ν
ϑ
(
k
) is
M
iϑ
THEN
x
(
k
+ 1) =
A
i
x
(
k
) +
B
i
u
(
k
) +
E
i
w
(
k
) +
G
i
f
(
k
)
y
(
k
) =
C
i
x
(
k
) +
D
i
u
(
k
) +
J
i
w
(
k
) +
H
i
f
(
k
)(2)where
i
= 1
,
···
,r
,
r
is the number of IFTHENrules,
M
ij
(
j
= 1
,
2
,
···
,ϑ
) are fuzzy sets, and
ν
(
k
) =[
ν
1
(
k
)
···
ν
ϑ
(
k
)] are the premise variables.There are two major ways in constructing a TS fuzzymodel. One is the TS fuzzy model identiﬁcation
[15
,
23
,
24]
us
S. K. Nguang et al./ Parity Relation Based Fault Estimation for Nonlinear Systems: An LMI Approach
165ing inputoutput data, and the other is the TS fuzzy modelconstruction, by the idea of sector nonlinearity
[25
−
27]
.The defuzziﬁed output of the TS fuzzy system (2) is represented as follows:
x
(
k
+ 1) =
A
(
µ
(
k
))
x
(
k
) +
B
(
µ
(
k
))
u
(
k
)+
E
(
µ
(
k
)
w
(
k
) +
G
(
µ
(
k
))
f
(
k
)
y
(
k
) =
C
(
µ
(
k
))
x
(
k
) +
D
(
µ
(
k
))
u
(
k
)+
J
(
µ
(
k
))
w
(
k
) +
H
(
µ
(
k
))
f
(
k
)(3)where
A
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
A
i
, B
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
B
i
E
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
E
i
, G
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
G
i
C
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
C
i
, D
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
D
i
J
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
J
i
, H
(
µ
(
k
)) =
r
X
i
=1
µ
i
(
k
)
H
i
and
µ
i
(
k
) =
i
(
ν
(
k
))
P
ri
=1
i
(
ν
(
k
))
with
i
(
ν
(
k
)) =
Q
ϑj
=1
M
ij
(
ν
j
(
k
)) and
M
ij
(
ν
j
(
k
)) is the grade of membership of
ν
j
(
k
) in
M
ij
. Notice that
i
(
ν
(
k
))
≥
0 for
i
= 1
,
2
,...,r
, and
P
ri
=1
i
(
ν
(
k
))
>
0
,
∀
k
≥
0, hence,
µ
i
(
k
)
≥
0 for
i
= 1
,
2
,...,r
, and
P
ri
=1
µ
i
(
k
) = 1
∀
k
≥
0.Deﬁne
S
i
≡
n
x
(
k
)

µ
i
(
x
(
k
))
≥
µ
j
(
x
(
k
))
, j
∈
I
s
o
, i
∈
I
s
where
I
s
=
n
1
,
2
,
···
,r
o
is the set of indexes. Then, theglobal model of the fuzzy dynamic system can also be expressed in each subspace as
x
(
k
+ 1) =
A
i
x
(
k
+ 1) +
B
i
u
(
k
) +
E
i
w
i
(
k
) +
G
i
f
(
k
) +
¯
r
i
X
j
¯
µ
j
(
k
)(
A
j
−
A
i
)
x
(
k
) +
¯
r
i
X
j
¯
µ
j
(
k
)[(
B
j
−
B
i
)
u
(
k
) + (
E
j
−
E
i
)
w
(
k
)] +
¯
r
i
X
j
¯
µ
j
(
k
)(
G
j
−
G
i
)
f
(
k
)
y
(
k
) =
C
i
x
(
k
) +
D
i
u
(
k
) +
J
i
w
(
k
) +
H
i
f
(
k
) +
¯
r
i
X
j
¯
µ
j
(
k
)(
C
j
−
C
i
)
x
(
k
) +
¯
r
i
X
j
¯
µ
j
(
k
)[(
D
j
−
D
i
)
u
(
k
) + (
H
j
−
H
i
)
f
(
k
)]for
x
(
t
)
∈ S
i
, where ¯
µ
1
(
k
)
,
¯
µ
2
(
k
)
,
···
,
¯
µ
¯
r
i
(
k
) are the membership functions that are not equal to zero when
i
th playsa dominant role. In a more compact form:
x
(
k
+ 1) =
A
i
x
(
k
+ 1) +
B
i
u
(
k
) + ¯
E
i
¯
w
i
(
k
) +
G
i
f
(
k
)
y
(
k
) =
C
i
x
(
k
) +
D
i
u
(
k
) + ¯
J
i
¯
w
i
(
k
) +
H
i
f
(
k
)(4)for
x
(
t
)
∈ S
i
, where ¯
E
i
=
h
¯
E
1
i
¯
E
2
i
¯
E
3
i
¯
E
4
i
i
, ¯
J
i
=
h
¯
E
5
i
¯
E
6
i
¯
E
7
i
¯
E
8
i
i
, ¯
E
i
=
h
¯
E
i
1
···
¯
E
i
¯
ri
i
,
i
= 1
,
···
,
8,¯
E
1
ij
=
A
j
−
A
i
, ¯
E
2
ij
=
B
j
−
B
i
, ¯
E
3
ij
=
E
j
−
E
i
, ¯
E
4
ij
=
G
j
−
G
i
, ¯
E
5
ij
=
C
j
−
C
i
, ¯
E
6
ij
=
D
j
−
D
i
, ¯
E
7
ij
=
J
j
−
J
i
,¯
E
8
ij
=
H
j
−
H
i
, ¯
w
i
(
k
) =
h
¯
v
T1
i
(
t
) ¯
v
T2
i
(
k
) ¯
v
T3
i
(
k
) ¯
v
T4
i
(
k
)
i
T
,¯
v
1
i
(
t
) =
2664
¯
µ
1
(
k
)
x
(
k
)...¯
µ
¯
r
i
(
k
)
x
(
k
)
3775
, ¯
v
2
i
(
t
) =
2664
¯
µ
1
(
k
)
u
(
k
)...¯
µ
¯
r
i
(
k
)
u
(
k
)
3775
,¯
v
3
i
(
t
) =
2664
¯
µ
1
(
k
)
w
(
k
)...¯
µ
¯
r
i
(
k
)
w
(
k
)
3775
, and ¯
v
4
i
(
t
) =
2664
¯
µ
1
(
k
)
f
(
k
)...¯
µ
¯
r
i
(
k
)
f
(
k
)
3775
.Given the system (3), the residual signal is generated bythe temporal redundancy approach
[3
−
11]
as follows:IF
x
(
t
)
∈ S
i
THEN
r
s
(
k
) =
V
s
i
0BBBB@266664
y
(
k
−
s
)
y
(
k
−
s
+ 1)...
y
(
k
)
377775
−
H
us
i
266664
u
(
k
−
s
)
u
(
k
−
s
+ 1)...
u
(
k
)
3777751CCCCA
∈
R
q
where
µ
s
(
k
) =
n
µ
(
k
−
s
)
,µ
(
k
−
s
+1)
,
···
,µ
(
k
)
o
and
H
us
i
=
2666664
D
i
0
...
0
C
i
B
i
D
i
............ ... 0
C
i
A
s
−
1
i
B
i
... C
i
B
i
D
i
3777775
with
V
s
i
∈
R
(
s
+1)
m
×
q
isthe parity matrix for the subspace
x
(
t
)
∈ S
i
, and
s >
0 isthe order of the parity relation.Using (4),
r
s
(
k
) can be reexpressed as
r
s
(
k
) =
V
s
i
h
H
os
i
x
(
k
−
s
) +
H
ws
i
¯
w
i
s
(
k
) +
H
fs
i
f
s
(
k
)
i
where
H
os
i
=
266664
C
i
C
i
A
i
...
C
i
A
si
377775
,
¯
w
i
s
(
k
) =
266664
¯
w
i
(
k
−
s
)¯
w
i
(
k
−
s
+ 1)...¯
w
i
(
k
)
377775
f
s
(
k
) =
266664
f
(
k
−
s
)
f
(
k
−
s
+ 1)...
f
(
k
)
377775
166
International Journal of Automation and Computing 04(2), April 2007
H
ws
i
=
2666664
¯
J
i
0
...
0
C
i
¯
E
i
¯
J
i
............ ... 0
C
i
A
s
−
1
i
¯
E
i
... C
i
¯
E
i
¯
J
i
3777775
and
H
fs
i
=
2666664
H
i
0
...
0
C
i
G
i
H
i
............ ... 0
C
i
A
s
−
1
i
G
i
... C
i
G
i
H
i
3777775
The main objection of the paritybased fault estimation isto select
V
s
i
for each subspace such that the residual signal
r
s
(
k
) =
f
s
(
k
). However, in general, this objective isnot achievable. Hence, for each subspace, ﬁnd
V
s
i
suchthat
V
s
i
H
os
i
2
,
V
s
i
H
ws
i
2
,
V
s
i
H
fs
i
−
I
2
are minimised.This minimisation problem can easily be cast in terms of LMIs as follows:
LMI ParityBased Fault Estimation Problem:
For
i
= 1
,
···
,r
, ﬁnd
V
s
i
such that
−
Q
1
V
s
i
H
os
i
V
s
H
T
os
i
−
I
#
<
0
−
Q
2
V
s
i
H
ws
i
V
s
i
H
T
ws
i
−
I
#
<
0
24
−
Q
3
“
V
s
i
H
fs
i
−
I
”“
V
s
i
H
fs
i
−
I
”
T
−
I
35
<
0and traces of
Q
i
>
0 are minimised.
3 Illustrative example
Consider a nonlinear massspringdamper mechanicalsystem with a nonlinear spring:˙
x
1
(
t
) =
x
2
(
t
)˙
x
2
(
t
) =
−
x
2
(
t
)
−
2
x
1
(
t
)
−
2
x
31
(
t
) +
f
1
(
t
)+
d
1
(
t
) +
u
(
t
)
y
1
(
t
) =
x
1
(
t
)
y
2
(
t
) =
x
2
(
t
) + 0
.
1
d
2
(
t
) +
f
2
(
k
)(5)where
x
1
(
k
) is the spring’s displacement,
x
2
(
k
) = ˙
x
1
(
t
),
f
1
(
t
) and
f
2
(
k
) are, respectively, actuator and sensor faults.
d
1
(
t
) and
d
2
(
k
), respectively, represent unknown disturbances/uncertainties in the mechanical system and the sensor system. The term
−
0
.
67
x
31
is due to the nonlinearity of the spring. The spring is attached to a ﬁxed wall, thereforethe spring’s displacement
x
1
(
t
) is physically constrained bythe length of the spring and the wall. The length of thespring could be any value, in this paper,
x
1
(
t
)
∈
[
−
0
.
5
,
0
.
5]is assumed. The lower limit is the minimum length thatthe spring can be compressed. The Euler approximationmethod is employed to discretise the massspringdampersystem (5) at 0
.
1 seconds.
x
1
(
k
+ 1) =
x
1
(
k
) + 0
.
1
x
2
(
k
)
x
2
(
k
+ 1) =
x
2
(
k
)
−
0
.
1
x
2
(
k
)
−
0
.
1
x
1
(
k
)
−
0
.
2
x
31
(
k
) + 0
.
1
u
(
k
) + 0
.
1
f
1
(
k
) + 0
.
1
d
(
k
)
y
1
(
k
) =
x
1
(
k
)
y
2
(
k
) =
x
2
(
k
) + 0
.
1
d
(
k
) +
f
2
(
k
)
.
(6)The concept of sector nonlinearity
[26]
is employed toconstruct an exact TS fuzzy model for the massspringdamper system. Using the fact that
x
1
(
k
)
∈
[
−
0
.
5
,
0
.
5], this nonlinear term can be expressed as
−
0
.
2
x
31
(
k
) =
−
h
1
(
x
1
(
k
))
ˆ
0
˜
x
1
(
k
)
−
h
2
(
x
1
(
k
))
ˆ
0
.
05
˜
x
1
(
k
)where
h
1
(
x
1
(
k
)) = 1
−
x
i
(
k
)0
.
25
and
h
2
(
x
1
(
k
)) =
x
i
(
k
)0
.
25
. Using
h
1
(
x
1
(
k
)) and
h
2
(
x
1
(
k
)), we obtain the following TS fuzzymodel which exactly represents (5) under the assumptionon bounds of the state variable
x
1
(
k
)
∈
[
−
0
.
5 0
.
5]:
x
(
k
+ 1) =
P
2
i
=1
h
i
(
z
(
k
))
h
A
i
x
(
k
) +
B
i
u
(
k
)+
E
i
w
(
k
) +
G
i
f
(
k
)
i
y
(
k
) =
P
2
i
=1
h
i
(
z
(
k
))
h
C
i
x
(
k
) +
D
i
u
(
k
)+
J
i
w
(
k
) +
H
i
f
(
k
)
i
(7)where
x
(
k
) =
ˆ
x
1
(
k
)
x
2
(
k
)
˜
T
,
A
1
=
1 0
.
1
−
0
.
1 0
.
9
#
,A
2
=
1 0
.
1
−
0
.
15 0
.
9
#
B
1
=
B
2
=
00
.
1
#
C
1
=
C
2
=
1 00 1
#
D
1
=
D
2
=
0 00 0
#
,E
1
=
E
2
=
0 00 0
.
1
#
G
1
=
G
2
=
0 00
.
1 0
#
,H
1
=
H
2
=
0 00 1
#
and
J
1
=
J
2
=
0 00
.
1 0
#
. For the sake of simplicity,the order of the parity relations is selected to be
s
= 2.Applying the procedure given in Section 3, the followingresidual generator is obtainedIF
h
1
(
x
1
(
k
))
≥
h
2
(
x
1
(
k
)) THEN
r
s
(
k
) =
V
s
1
0BBBB@266664
y
(
k
−
s
)
y
(
k
−
s
+ 1)...
y
(
k
)
377775
−
H
us
1
266664
u
(
k
−
s
)
u
(
k
−
s
+ 1)...
u
(
k
)
3777751CCCCA
∈
R
q
S. K. Nguang et al./ Parity Relation Based Fault Estimation for Nonlinear Systems: An LMI Approach
167Else
r
s
(
k
) =
V
s
2
0BBBB@266664
y
(
k
−
s
)
y
(
k
−
s
+ 1)...
y
(
k
)
377775
−
H
us
2
266664
u
(
k
−
s
)
u
(
k
−
s
+ 1)...
u
(
k
)
3777751CCCCA
∈
R
q
where
V
s
1
=
89
.
87 0
−
187
.
65 0 98
.
76 09
.
99 0
.
99
−
9
.
99 0 0 0
#
and
V
s
2
=
91
.
38 0
−
189
.
76 0 99
.
87 09
.
99 1
.
00
−
9
.
99 0 0 0
#
.
In order to compare our result with the linear result, thefollowing residual generator is designed based on the linearmodel obtained by linearising the system (5) at the srcin.
r
s
(
k
) =
V
s
L
0BBBB@266664
y
(
k
−
s
)
y
(
k
−
s
+ 1)...
y
(
k
)
377775
−
H
us
L
266664
u
(
k
−
s
)
u
(
k
−
s
+ 1)...
u
(
k
)
3777751CCCCA
∈
R
q
where
V
s
L
=
45
.
50 0
−
94
.
99 0 50
.
00 09
.
90 0
.
99
−
9
.
90 0 0 0
#
.
Fig.1 Plot of the residual signal
r
s
1
(
k
) and the
f
1
(
k
)Fig.2 Plot of the residual signal
r
s
2
(
k
) and the
f
2
(
k
)
u
(
k
) = sin(0
.
1
k
), and
d
1
(
k
) and
d
2
(
k
) are white noises,respectively, with 0
.
00001 and 0
.
1 as their noise powers.Histories of the residual signals
r
s
1
(
k
) and
r
s
2
(
k
) are, respectively, depicted in Figs.1 and 2. These two ﬁgures showthat for the nonlinear case both
r
s
1
(
k
) and
r
s
2
(
k
) are able tofollow
f
1
(
k
) and
f
2
(
k
), however, for the linear case there exist a large fault estimation error for
r
s
1
(
k
). Hence, this example clearly demonstrates the superiority of the approachproposed in this paper.
4 Conclusion
In this paper, the parity space approach for linear systems has been generalized to nonlinear systems describedby the TS fuzzy models. The design procedure has beenprovided in terms of a family of linear matrix inequalities (LMIs) which can be solved eﬀectively by an algorithmgiven in [22]. A numerical example has been given to illustrate the eﬀectiveness of the proposed design technique.
References
[1] M. Basseville. Detecting Changes in Signals and Systems –A Survey.
Automatica
, vol. 24, no. 3, pp. 309326, 1988.[2] R. Isermann. Process Fault Detection Based on Modelingand Estimation Methods – A Survey.
Automatica
, vol. 20,no. 4, pp. 387404, 1984.[3] P. M. Frank. Fault Diagnosis in Dynamic Systems UsingAnalytical and Knowledge Based Redundancy – A Surveyof Some New Results.
Automatica
, vol. 26, no. 3, pp. 459474, 1990.[4] R. J. Patton, J. Chen. A Review of Parity Space Approachesto Fault Diagnosis. In
Proceedings of IFAC Symposium onSafeprocess’91
, Hull, UK, vol. 1, pp. 239255, 1991.[5] J. E. Potter, M. C. Suman. Threshold Redundancy Management with Arrays of Skewed Instruments.
Integrity inElectronic Flight Control Systems
, vol. Agardograph 224,no. 15, pp. 1125, 1977.[6] E. Y. Chow, A. S. Willsky. Analytical Redundancy and theDesign of Robust Failure Detection Systems,
IEEE Transactions on Automatic Control
, vol. 29, no. 7, pp. 603614,1984.[7] J. J. Gertler. Survey of Modelbased Failure Detection andIsolation in Complex Plants,
IEEE Control Systems Magazine
, vol. 3, no. 6, pp. 311, 1988.