1. Fault detection and diagnosis for non-Gaussian stochastic distribution
systems with time delays via RBF neural networks
Qu Yi n
, Li Zhan-ming, Li Er-chao
College of Electrical and Information Engineering, Lanzhou University of Technology, Key Laboratory of Gansu Advanced Control for Industrial Processes, Lanzhou 730050, China
a r t i c l e i n f o
Article history:
Received 25 December 2011
Received in revised form
19 June 2012
Accepted 11 July 2012
Available online 14 August 2012
This paper was recommended for
publishing by Ricky Dubay.
Keywords:
Probability density functions
Non-Gaussian stochastic distribution
system
Radial basis functions(RBFs) neural
network
Observer-based fault detection and
diagnosis
a b s t r a c t
A new fault detection and diagnosis (FDD) problem via the output probability density functions (PDFs)
for non-gausian stochastic distribution systems (SDSs) is investigated. The PDFs can be approximated
by radial basis functions (RBFs) neural networks. Different from conventional FDD problems, the
measured information for FDD is the output stochastic distributions and the stochastic variables
involved are not confined to Gaussian ones. A (RBFs) neural network technique is proposed so that the
output PDFs can be formulated in terms of the dynamic weighings of the RBFs neural network. In this
work, a nonlinear adaptive observer-based fault detection and diagnosis algorithm is presented by
introducing the tuning parameter so that the residual is as sensitive as possible to the fault. Stability
and Convergency analysis is performed in fault detection and fault diagnosis analysis for the error
dynamic system. At last, an illustrated example is given to demonstrate the efficiency of the proposed
algorithm, and satisfactory results have been obtained.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Fault detection and diagnosis (FDD) are an important area of
research in recent years from the viewpoint of improving control
systems safety and reliability. In the past two decades, many
efficient methods have been presented for various types of system
faults [1–5]. For the stochastic system, the two kinds of
approaches including the system identification techniques [6]
and the statistic approaches based on the Likelihood methods,
Bayesian theorem, and Hypothesis test techniques [1] can be used
to deal with the related FDD problems. Besides, we have known
that observers have been extensively applied to generate the
residual signal for fault detection and diagnosis ([7], [8]), and
many significant approaches of them have been applied to
practical processes successfully ([9], [10]).
It is noted that most of the FDD methodologies for stochastic
systems only investigated Gaussian system [1–6, 8–10], and one
of the common features for these methods is performed by using
system input and output measuring values. However, in many
practical processes, non-Gaussian variables exist in many sto-
chastic systems due to nonlinearity, and these may possess
asymmetric and multiple-peak stochastic distributions, where
mean and variance are insufficient to characterize their statistical
behavior precisely. Along with the development of instruments and
image processing techniques, the measured information can be the
stochastic distribution of system output rather its instant values. As
such, there is a need to further develop FDD methods that can be
applied to the stochastic systems subjected to non-Gaussian distribu-
tion. Motivated by these factors, studies on stochastic distribution
systems and stochastic distribution control have been investigated in
Ref. [7,11–24]. Different from conventional FDD problems, the
measurement information for FDD is the output PDFs rather than
the mean or variance of the output, and the stochastic variables
involved are not confined to Gaussian ones.
Up to now, many efficient Fault detection and diagnosis
methods for non-Gaussian stochastic distribution systems have
been considered by researchers to cover various types of faulty
systems [7,10,11,18–23].
However, there are many drawbacks exists in them. For
example, the error of B-spline expansion model is omitted in
[11,12]. In [11,12,19,20], fault detection and diagnosis strategy is
presented by a conventional Lyapnuov candidate function
method, in which the threshold is not sensitive to the fault.
To overcome the obstacles and improve FDD sensitive per-
formance, in this paper, two main works have been included.
The first one, we introduce RBF neural network to approximate
the output PDFs of system, which can generate the output PDF
expansion model and overcome the problems with B-spline
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.07.003
n
Corresponding author.
E-mail address: quyi0709052@163.com (Q. Yi).
ISA Transactions 51 (2012) 786–791

2. expansion model; Secondly, By using nonlinear observer as residual
generator and tuning parameter, a more sensitive fault detection
and diagnosis method can be given.
This paper is organized as follows. In Section 2, the output PDFs
expansion and the nonlinear weight dynamic are established to
formulate the FDD problem. In Section 3, the observer-based
fault detection and diagnosis is investigated for the transformed
nonlinear weight model, where stability and sensitivity can be
obtained. A simple example is given in Section 4 to demonstrate
the efficiency of the proposed approach. The concluding remarks
are presented in Section 5.
2. Problem formulation
Consider a continuous-time dynamic stochastic distribution
system with time delays where u(t)ARm
is the control input,
y(t)A[a,b] represents the system output, and F is the fault to be
detected and diagnosed, a typical example of which is an actuator
fault. At any time t, the probability of output y(t) lying inside [a,b]
can be described as
PðaryðtÞobÞ ¼
Z b
a
gðy,uðtÞ,FÞdy
where g(y,u(t),F) denotes the PDFs of the stochastic variable
y(t) under the control input u(t).The control objective in the FDD
context is to use the output PDFs to design observer schemes that
can detect and diagnose the fault. As shown in [20], the well-
known RBF neural network has been used to approximate by the
following expression:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gðy,uðtÞ,FÞ
p
¼
Xn
i ¼ 1
viðuðtÞ,FÞbiðyÞþo0ðy,uðtÞ,FÞ ð1Þ
where bi(y)(i¼1,2,y,n) represents the i-th pre-specified basis
function, vi(u(t),F)(i¼1,2,y,n) are i-the weight corresponding to
the RBFs neural network used for PDFs model, and o0(y,u(t),F)
stands for either the model uncertainty or the error on the
approximation of the output PDFs, which is supposed to satisfy
9o0(y,u(t),F) 9rd0, d0 is assumed to be known positive constant.
As shown in [20,24], the RBFs basis functions are chosen as of
Gaussian shapes and expressed as
biðyÞ ¼ exp½ÀðyjÀwiÞ2
=2t2
i Š ð2Þ
wherewiandtiare the centers and widths of the RBFs, respectively.
Furthermore, we denote that
B0ðyÞ ¼ ½b1ðyÞ b2ðyÞ Á Á Á bnÀ1ðyÞŠ
VðtÞ ¼ VðuðtÞ,FÞ ¼ ½v1ðuðtÞ,FÞ v2ðuðtÞ,FÞ Á Á Á vnÀ1ðuðtÞ,FÞŠT
ð3Þ
and
L1 ¼
Z b
a
BT
ðyÞBðyÞdy, L2 ¼
Z b
a
BT
ðyÞbnðyÞdy,
L3 ¼
Z b
a
b
2
nðyÞdya0,L0 ¼ L1L3ÀLT
2L2
In this paper, similar to reference [18,22], we adopt the
following model:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gðy,uðtÞ,FÞ
p
¼ BðyÞVðtÞþhðVðtÞÞbnðyÞþoðy,uðtÞ,FÞ ð4Þ
where
BðyÞ ¼ B0ðzÞÀ
L2
L3
bnðyÞ hðVðtÞÞ ¼
1
L3
ÀL2VðtÞþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L2ðtÞÀVT
ðtÞL0VðtÞ
q
ð5Þ
From the boundedness of o0(y,u(t),F) and Ref. [7], it can be
assumed that 9o0(y,u(t),F) 9rd holds for all {y,u(t),F}, where d is
also a known positive constant.
Once the square root expansion of the output PDFs have been
made for the non-Gaussian stochastic distribution system, the
next step is to find the dynamic relationship between the control
input and weights related to the PDFs corresponds to a further
modeling. As shown in [7,24], in this paper the nonlinear dynamic
model will be considered as follows
_xðtÞ ¼ AxðtÞþAdxðkÀdÞþGgðxðtÞÞþHuðtÞþDF
VðtÞ ¼ ExðtÞ
(
ð6Þ
where x(t)ARm
is the unmeasured state vector. A, Ad, G, H, D and E
represent known coefficient matrices with compatible dimen-
sions of the weight system, these matrices can be obtained either
by physical modeling or the scaling estimation technique. d is the
known time delays. V(t) is denoted in Eq. (3). g(x(t))ARm
is a
nonlinear vector function that represents the nonlinear dynamics
of the weight model.
We can see that model (6) stands for a time-delayed nonlinear
dynamic weight system with non-zero initial conditions. Under
model (6), Eq. (4) can be rewritten as a nonlinear function of x(t)
as follows
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gðy,uðtÞ,FÞ
p
¼ BðyÞExðtÞþhðExðtÞÞbnðyÞþoðy,uðtÞ,FÞ ð7Þ
Remark 1. Compared with the models considered in [11,18,19],
there are several features as follows : first of all, A radial basis
functions(RBFs) neural network technique is proposed so that the
proposed PDFs model is more practically reasonable, in the model
adopted in [23], o(y,u(t),F) is omitted, which can lead to con-
servative result. Secondly, time delays have been considered in
our work, which can satisfy requirement of practical applications.
In the rest of this paper, the following assumptions are needed.
Assumption 1. For any x1(t) and x2(t), g(x(t)) that satisfies the
following Lipschitz condition:
:gðx1ðtÞÞÀgðx2ðtÞÞ:r:U1ðx1ðtÞÀx2ðtÞÞ:
where U1 is a known constant matrix, : Á : is denoted as the
Euclidean norm.
Assumption 2. For any V1(t) and V2(t), h(V(t)) denoted by Eq.(5)
satisfies the following Lipschitz condition:
:hðV1ðtÞÞÀhðV2ðtÞÞ:r:U2ðV1ðtÞÀV2ðtÞÞ: ¼ :U2Eðx1ðtÞÀx2ðtÞÞ:
where U2 is a known constant matrix.
3. Observer-based fault detection and diagnosis
3.1. Observer-based fault detection
Since the measured informance is the output probability
distribution, in order to detect the fault based on the changes of
output PDFs, the following full-order observer is applied to detect
the fault.
_^xðtÞ ¼ A^xðtÞþAd ^xðkÀdÞþHuðtÞþGgð^xðtÞÞþLeðkÞ
eðtÞ ¼
R b
a sðyÞ½
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gðy,uðtÞ,FÞ
p
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^gðy,uðtÞÞ
p
Šdy
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^gðy,uðtÞÞ
p
¼ BðyÞC^xðtÞþhðC^xðtÞÞbnðyÞ
8
:
ð8Þ
where
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^gðy,uðtÞ,FÞ
p
stands for the desired PDFs, ^xðtÞ is the
estimated state, ^Frepresents the estimated fault, LARm  p
repre-
sents the gain to be determined and the residual e(t) is formulated
as an integral of the difference between the measured PDFs and
the estimated ones, s(z)ARm  m
can be regarded as a pre-specified
weight vector lying [a,b] and makes the integration simple or
adjust the scale of e(t).
Q. Yi et al. / ISA Transactions 51 (2012) 786–791 787

3. Furthermore, the residual signal e(t) stands for the difference
of gðz,uðtÞ,FÞ and ^gðy,uðtÞÞ, which can be rewritten as the following
nonlinear function
eðtÞ ¼ P1eðtÞþP2
~hðtÞþDðtÞ ð9Þ
where
P1 ¼
Rb
a sðyÞBðyÞEdy, P2 ¼
Rb
a sðyÞbnðyÞdy
:DðtÞ: ¼ :
Z b
a
sðyÞoðy,uðtÞ,FÞdy: ¼ d:
Z b
a
sðyÞdy:rd
and eðtÞ is also the information for feedback in observer design.
By definingeðtÞ ¼ xðtÞÀ^xðtÞ,eT
dðtÞ ¼ xðtÀdÞÀ^xðtÀdÞ, ~gðtÞ ¼ gðxðtÞÞÀ
gð^xðtÞÞ ~hðtÞ ¼ hðCxðtÞÞÀhðC^xðtÞÞ the estimation error system can be
described by
_eðtÞ ¼ ðAÀLP1ÞeðtÞþAdedðtÞþG~gðtÞÀLP2
~hðtÞÀLDðtÞþDFðtÞ ð10Þ
Remark 2. In [17,19], for continuous-time system without the
disturbance or the model perturbation, we have provided suffi-
cient conditions for the feasible fault detection and diagnosis
observer strategy, which may lead to the conservative criteria for
fault detection and diagnosis, especially when the initial value is
relatively large.
The following result provides a criterion to detect the fault by
using output PDFs, which is an extension result of theorem 1 [18]
for the fault detection observer design.
Theorem 1:. For the parameters gi,li 40 (i¼1,2), i and B,suppose
that there matrices P40, R40 and Q satisfying (11)
O11 O12 AT
dP2 U1 U2E PT
2G QP2
n ÀBPT
2ÀBP2 BAT
dP2 0 0 BPT
2G BQP2
n n ÀR 0 0 0 0
n n n Àl2
1 0 0 0
n n n n Àl2
2 0 0
n n n n n ÀlÀ2
1 0
n n n n n n ÀlÀ2
2
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
o0 ð11Þ
where O11 ¼ RþAPT
2 þP2AT
ÀQP1ÀPT
1Q þg1I,O12 ¼ P1ÀP2 þ
BAT
P2ÀBQP1, then in the absence of F, the error system (10) with
gain L ¼ PÀT
2 Qis asymptotically stable
Proof. Define the Lyapunov candidate function as follows:
V ¼ V1 þV2 þV3 ð12Þ
Where
V1 ¼
R t
tÀd eðzÞReðzÞdz
V2 ¼ eðtÞPeðtÞ ¼
eðtÞ
_eðtÞ
#T
I 0
0 0
P1 0
P2 BP2
#
eðtÞ
_eðtÞ
#
V3 ¼
1
l2
1
Z t
0
:U1eðzÞ:
2
À:gðxðzÞÞÀgð^xðzÞÞ:
2
h i
dz
þ
1
l2
2
Z t
0
:U2eðzÞ:
2
À:hðExðzÞÞÀgðE^xðzÞÞ:
2
h i
dz
It is noted that V3 is nonnegative under Assumptions 1 and 2.
then, in the absence of F, along with the error dynamic system
(10), by using the completion-of-square method, it can be verified
that
_V 3 ¼ 2eT
ðtÞP_eðtÞ ¼ 2
eðtÞ
_eðtÞ
#T
P1 PT
2
0 BPT
2
#
_eðtÞ
0
¼ 2
eðtÞ
_eðtÞ
#T
P1 PT
2
0 BPT
2
#
_eðtÞ
À_eðtÞþðAÀLP1ÞeðtÞþAdedðtÞþG~gðtÞÀLP2
~hðtÞÀLDðtÞ
#
¼ 2eT
ðtÞP1 _eðtÞÀ2eT
ðtÞPT
2
_eðtÞÀ2B_eT
ðtÞPT
2
_eðtÞ
þ2eT
ðtÞðAPT
2ÀQP1ÞeðtÞþ2B_eT
ðtÞAPT
2eðtÞ
À2B_eT
ðtÞQP1eðtÞþ2eT
ðtÞPT
2AdedðtÞþ2B_eT
ðtÞPT
2AdedðtÞ
þ2eT
ðtÞPT
2G~gðtÞþ2B_eT
ðtÞPT
2G~gðtÞÀ2eT
ðtÞQP2
~hðtÞ
À2B_eT
ðtÞQP2
~hðtÞþ2eT
ðtÞQDðtÞþ2B_eT
ðtÞQDðtÞ
r2eT
ðtÞP1 _eðtÞÀ2eT
ðtÞPT
2
_eðtÞ
À2B_eT
ðtÞPT
2
_eðtÞþ2eT
ðtÞðAPT
2ÀQP1ÞeðtÞþ2B_eT
ðtÞAPT
2eðtÞ
À2B_eT
ðtÞQP1eðtÞþ2eT
ðtÞPT
2AdedðtÞþ2B_eT
ðtÞPT
2AdedðtÞ
þ2l2
1eT
ðtÞPT
2GGT
P2eðtÞþ
1
2l2
1
~gT
ðtÞ~gðtÞ
þ2l2
1B2 _eT
ðtÞPT
2GGT
P2 _eðtÞþ
1
2l2
1
~gT
ðtÞ~gðtÞ
þ2l2
2eT
ðtÞQP2PT
2QT
eðtÞþ
1
2l2
2
~h
T
ðtÞ ~hðtÞ
þ2l2
2B2 _eT
ðtÞQP2PT
2Q _eðtÞþ þ
1
2l2
2
~h
T
ðtÞ ~hðtÞþ2eT
ðtÞQDðtÞ
þ2B_eT
ðtÞQDðtÞ
Then, we obtain
_V reT
ðtÞReT
ðtÞÀeT
dðtÞRedðtÞþ
1
l2
1
eðtÞUT
1U1eðtÞÀ
1
l2
1
~gT
ðtÞ~gðtÞ
þ
1
l2
2
eðtÞET
UT
2U2EeðtÞÀ
1
l2
2
~h
T
ðtÞ ~hðtÞþ2eT
ðtÞP1 _eðtÞ
À2eT
ðtÞPT
2
_eðtÞÀ2B_eT
ðtÞPT
2
_eðtÞþ2eT
ðtÞðAPT
2ÀQP1ÞeðtÞ
þ2B_eT
ðtÞAPT
2eðtÞÀ2B_eT
ðtÞQP1eðtÞþ2eT
ðtÞPT
2AdedðtÞ
þ2B_eT
ðtÞPT
2AdedðtÞþ2l2
1eT
ðtÞPT
2GGT
P2eðtÞþ
1
2l2
1
~gT
ðtÞ~gðtÞ
þ2l2
1B2 _eT
ðtÞPT
2GGT
P2 _eðtÞþ
1
2l2
1
~gT
ðtÞ~gðtÞ
þ2l2
2eT
ðtÞQP2PT
2QeðtÞþ
1
2l2
2
~h
T
ðtÞ ~hðtÞ
þ2l2
2B2 _eT
ðtÞQP2PT
2Q _eðtÞþ
1
2l2
2
~h
T
ðtÞ ~hðtÞ
þ2eT
ðtÞQDðtÞþ2B_eT
ðtÞQDðtÞ
¼ eT
ðtÞReT
ðtÞÀeT
dðtÞRedðtÞþ
1
l2
1
eðtÞUT
1U1eðtÞ
þ
1
l2
2
eðtÞET
UT
2U2EeðtÞÀ2B_eT
ðtÞQP1eðtÞ
À2eT
ðtÞPT
2
_eðtÞÀ2B_eT
ðtÞPT
2
_eðtÞþ2eT
ðtÞðAPT
2ÀQP1ÞeðtÞ
þ2B_eT
ðtÞAPT
2eðtÞþ2eT
ðtÞP1 _eðtÞþ2eT
ðtÞPT
2AdedðtÞ
þ2B_eT
ðtÞPT
2AdedðtÞþ2l2
1eT
ðtÞPT
2GGT
P2eðtÞ
þ2l2
1B2 _eT
ðtÞPT
2GGT
P2 _eðtÞþ2l2
2eT
ðtÞQP2PT
2QeðtÞ
þ2l2
2B2 _eT
ðtÞQP2PT
2Q _eðtÞÀ2eT
ðtÞQDðtÞÀ2B_eT
ðtÞQDðtÞ
¼
eðtÞ
_eðtÞ
edðtÞ
2
6
4
3
7
5
T
X11 X12 AT
dP2
n X22 BAT
dP2
n n ÀR
2
6
4
3
7
5
eðtÞ
_eðtÞ
edðtÞ
2
6
4
3
7
5À2eT
ðtÞQDðtÞ
À2B_eT
ðtÞQDðtÞ
Q. Yi et al. / ISA Transactions 51 (2012) 786–791788

4. where
G1 ¼ l1PT
2G l2QP2
h i
, G2 ¼ l1BPT
2G l2BQP2
h i
,
X11 ¼ Rþ
1
l2
1
UT
1U1 þ
1
l2
2
ET
UT
2U2EþAPT
2 þP2AT
ÀQP1ÀPT
1Q þGT
1G1,
X12 ¼ P1ÀP2 þBAT
P2ÀBQP1, X22 ¼ ÀBPT
2ÀBP2 þGT
2G2
Applying (11), and assuming B:_eT
ðtÞ:rg2:eðtÞ:, it can be
shown that
_V ðtÞrÀg1:eðtÞ:
2
þ2:eðtÞ::Q:dþ2B:_eT ðtÞ::Q:d
rÀg1:eðtÞ:
2
þ2:eðtÞ::Q:dþ2g2:eðtÞ::Q:d
where Q ¼ PT
2L. We have that if :eðtÞ: ¼ mZ2gÀ1
1 :Q:dþ
2g2gÀ1
1 :Q:d occurs, then _V ðtÞo0 holds. Therefore, it can be seen
that the error satisfies :eðtÞ:rmax :eð0Þ:,2gÀ1
1 :Q:dþ2g2gÀ1
1
n
:Q:d
o
and the estimation error system is stable.
e(t) is chosen as the residual signal, and the following objective
is the evaluation of the generated residual. The well-known
approaches is to select a so-called threshold !th40. Then, when
the residual evaluation function :fðtÞ:4Gthholds, it is shown
that the fault occurs.
We know that the stability condition in theorem 1 is derived in
the absence of the fault F. In the presence of fault, the following
inequality is satisfied:
:fðtÞ:4mð:P1:þ:P2::U2::E:Þþd ð13Þ
From (13), it can be seen that the fault detection criterion in
our work can provide less conservative algorithm by tuning the
parameter B.
3.2. Observer-based fault diagnosis
After the fault is detected based on theorem 1,in order to estimate
the size of the fault, the fault diagnosis needs to be carried out. For
this purpose, we consider the following fault diagnostic observer:
_^xðtÞ ¼ A^xðtÞþAd ^xðkÀdÞþHuðtÞþGgð^xðtÞÞþLeðkÞþD^FðtÞ
eðtÞ ¼
Rb
a sðyÞ½
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gðy,uðtÞ,FÞ
p
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^gðy,uðtÞÞ
p
Šdy
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
^gðy,uðtÞÞ
p
¼ BðyÞC^xðtÞþhðC^xðtÞÞbnðyÞ
_^F ¼ ÀK1
^F þK2eðtÞ
8
:
ð14Þ
where ^F is the estimation of F(t), Ki (i¼1,2, K140) are two learing
operators to be determined by the proposed diagnostic method.
By defining eðtÞ ¼ xðtÞÀ^xðtÞ, eT
dðtÞ ¼ xðtÀdÞÀ^xðtÀdÞ, ~gðtÞ ¼ gðxðtÞÞÀ
gð^xðtÞÞ, ~hðtÞ ¼ hðCxðtÞÞÀhðC^xðtÞÞ. and ~FðtÞ ¼ FÀ^FðtÞ, the estimation
error system yields
_eðtÞ ¼ ðAÀLP1ÞeðtÞþAdedðtÞþG~gðtÞÀLP2
~hðtÞÀLDðtÞþD~FðtÞ
_~F ¼ ÀK1
~F þK1FðtÞþK2P1eðtÞþK2P2
~hðtÞþK2DðtÞ
8
:
ð15Þ
And the second equation in (15) stands for a tuning rule to
estimate unknown F.
Being similar to Ref [20,23], it is assumed that :F:rM=2,
:^FðtÞ:rM=2 and consequently :~F:rM.the following result shows
that (14) can be used as a diagnostic observer, with which the
estimated error can be small by choosing appropriate observer gain
and B. Then, an observer–Based fault diagnosis approach is pre-
sented by the following theorem.
Theorem 2:. For the parameters li40 (i¼1,2), suppose that
there exist matrices P40, Q, K1 40, K2 and parameters g3,xj4
0(j¼1y4) satisfying
Where O0
11 ¼ RþAPT
2 þP2AT
ÀQP1ÀPT
1Q þg3Iþ þx2
1QQT
þxÀ2
4 ET
UT
2U2E, O12 ¼ P1ÀP2 þBAT
P2ÀBQP1then under the diagnostic
observer (14) with gain L ¼ PÀT
2 Q, the error system (15) are
asymptotically stable in the presence of F.
Proof. For this purpose, define the Lyapunov candidate function
as follows:
VF ðtÞ ¼ V þ ~F
T
ðtÞ~FðtÞ ð17Þ
where V is defined by (12). It is noted that _V F Z0. Furthermore,
along the trajectories of (15), it can be verified that
_V r
eðtÞ
_eðtÞ
edðtÞ
2
6
4
3
7
5
T
X11 X12 AT
dP2
n X22 BAT
dP2
n n ÀR
2
6
4
3
7
5
eðtÞ
_eðtÞ
edðtÞ
2
6
4
3
7
5
À2eT
ðtÞQDðtÞÀ2B_eT
ðtÞQDðtÞþ2eT
ðtÞDP2
~F þ2B_eT
ðtÞDP2
~F
r
eðtÞ
_eðtÞ
edðtÞ
2
6
4
3
7
5
T
X11 þx2
1QQT
X12 AT
dP2
n X22 þx2
2QQT
BAT
dP2
n n ÀR
2
6
6
4
3
7
7
5
eðtÞ
_eðtÞ
edðtÞ
2
6
4
3
7
5
þxÀ2
1 DT
ðtÞDðtÞþxÀ2
2 DT
ðtÞDðtÞþ2eT
ðtÞDP2
~F þ2B_eT
ðtÞDP2
~F
¼ yT
ðtÞX1yðtÞþxÀ2
1 DT
ðtÞDðtÞþxÀ2
2 DT
ðtÞDðtÞþ2eT
ðtÞDP2
~F
þ2B_eT
ðtÞDP2
~F
where X11, X12, X22 is denoted by theorem 1,
yðtÞ ¼ ½eT
ðtÞ _eT
ðtÞ eT
dðtÞŠT X1 ¼
X11 þx2
1QQT
X12 AT
dP2
n X22 þx2
2QQT
BAT
dP2
n n ÀR
2
6
6
4
3
7
7
5
O0
11 O12 AT
dP2 U1 U2E PT
2G QP2
n ÀBPT
2ÀBP2 BAT
dP2 0 0 BPT
2G BQP2
n n ÀR 0 0 0 0
n n n Àl2
1 0 0 0
n n n n Àl2
2 0 0
n n n n n ÀlÀ2
1 0
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
n
ÀlÀ2
2
n
n
n
n
K2P2
0
0
0
0
0
0
xÀ2
4
n
n
n
K2
0
0
0
0
0
0
0
xÀ2
3
n
n
Q
0
0
0
0
0
0
0
0
xÀ2
2
n
DT
P2ÀK2P1
BDT
P2
0
0
0
0
0
0
0
0
À2KT
1
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
o0 ð16Þ
Q. Yi et al. / ISA Transactions 51 (2012) 786–791 789

5. According to (13), it can be seen that
_V F ryT
ðtÞX1yðtÞþxÀ2
1 DT
ðtÞDðtÞþxÀ2
2 B2
DT
ðtÞDðtÞþ2eT
ðtÞDP2
~F
þ2B_eT
ðtÞDP2
~F þ2~F
T _~F
¼ yT
ðtÞX1yðtÞþxÀ2
1 DT
ðtÞDðtÞþxÀ2
2 DT
ðtÞDðtÞþ2eT
ðtÞDP2
~F
þ2B_eT
ðtÞDP2
~F þ2~F
T
ðtÞKT
1
^FðtÞÀ2~F
T
ðtÞK2DðtÞÀ2~F
T
ðtÞK2½P1eðtÞ
þP2
~hðtÞŠ
ryT
ðtÞX1yðtÞþxÀ2
1 DT
ðtÞDðtÞþxÀ2
2 DT
ðtÞDðtÞþ2eT
ðtÞDP2
~F
þ2B_eT
ðtÞDP2
~F þ2~F
T
ðtÞKT
1
^FðtÞþx2
3
~F
T
ðtÞK2KT
2
~FðtÞþxÀ2
3 DT
ðtÞDðtÞ
À2~F
T
ðtÞK2P1eðtÞþx2
4
~F
T
ðtÞK2P2PT
2KT
2
~FðtÞ
þxÀ2
4 yeT
ðtÞET
UT
2U2EeðtÞ
¼ yT
1ðtÞX2y1ðtÞþxÀ2
1 DT
ðtÞDðtÞþxÀ2
2 DT
ðtÞDðtÞþ þxÀ2
3 DT
ðtÞDðtÞ
þ2FT
KT
1
~FðtÞ
where
It can be shown that (16) leads to X2 odiag Àg3I 0 0 0
n o
by means of the schur complement formula. Furthermore, it can
be verified that
_V F oÀg3:eðtÞ:
2
þxÀ2
1 DT
ðtÞDðtÞþxÀ2
2 DT
ðtÞDðtÞ
þxÀ2
3 DT
ðtÞDðtÞþ2FT
KT
1
~FðtÞrÀg3:eðtÞ:
2
þðxÀ2
1 þxÀ2
2
þxÀ2
3 Þd
2
þ:K1:M2
Thus, _V F o0 if ðxÀ2
1 þxÀ2
2 þxÀ2
3 Þd
2
þ:K1:M2
og3:eðtÞ:
2
holds,
and the estimation error system (15) withL ¼ PÀT
2 Qis stable.
Therefore, it can be seen that the estimation errors satisfies.
:eðtÞ:
2
rmin :eð0Þ:
2
,gÀ1
3 ððxÀ2
1 þxÀ2
2 þxÀ2
3 Þd
2
þ:K1:M2
Þ
n o
It is shown that the diagnostic error can be retained within a
satisfactory range by tuning the parameters xi(i¼1,y,4),g3 and
learing operators K1, K2. The design approach resulting from
theorem 2 can be transferred to LMI-based convex optimization.
4. Simulation
To further illustrate the above-mentioned approach, an appli-
cation of paper-making process is given to demonstrate the
applicability of the proposed approach. It is supposed that the
output PDF can be approximated using there-layer neural net-
work with three radial basis activation functions with the
following initial condition.
yA½0,1Š, w1 ¼ 0:2, w2 ¼ 0:5, w3 ¼ 0:8, t1 ¼ t2 ¼ t3 ¼ 0:05
It is assumed that the weighting system is formulated by
(6) with the following coefficient matrices:
A ¼
À1 0
0 À1
, Ad ¼
À0:1 0
0 À0:1
,G ¼
1 0
0 0:5
, H ¼
1 0
0 À3
,
D ¼
0:5
0:5
,E ¼
1 0
0 2
, gðxÞ ¼ sinðxðtÞÞ
Let that the model error exists and satisfies :oðz,uðtÞ,FÞ:r
0:001.
Parameters can be chosen as l1 ¼ l2 ¼ 1; x1 ¼ x2 ¼ x3 ¼ x4 ¼ 1.
The fault is supposed as follows:
FðtÞ ¼
1, tZ20
0, to20
(
Response of the residual under the diagnosis observer and the
step fault is shown in Fig.1, it is demonstrated that the response
of the residual signal converges to zero asymptotically. In Fig.2,
the fault and its estimated value are given, it is shown the fault
can be well estimated through the fault diagnostic observer after
its occurrence. Furthermore, we consider the time-varying fault
described as follows
FðtÞ ¼
1þ0:02sinð0:2tÞ, tZ20
0, to20
(
Fig. 3 shows the fault and its estimated value using the
diagnostic observer. From these three figures it can be concluded
y1ðtÞ ¼ ½eT
ðtÞ _eT
ðtÞ eT
dðtÞ ~F
T
ðtÞŠT
X2 ¼
X11 þx2
1QQT
þxÀ2
4 ET
UT
2U2E X12 AT
dP2 DT
P2ÀK2P1
n X22 þx2
2QQT
BAT
dP2 BDT
P2
n n ÀR 0
n n n À2KT
1 þx2
3K2KT
2 þx2
4K2P2PT
2KT
2
2
6
6
6
6
6
4
3
7
7
7
7
7
5
Fig.1. Response of the residual under the diagnosis observer and the step fault. Fig.2. Response of diagnosis observer and step fault.
Q. Yi et al. / ISA Transactions 51 (2012) 786–791790

6. that the post-fault PDF can still follow the given distribution,
leading to a good fault tolerant control result.
5. Conclusion
In this paper, a new FDD algorithm is presented for non-
Gaussian stochastic distribution system using (RBFs) neural net-
work. Different from the conventional FDD methods, the mea-
sured informance is the output PDFs rather than its output values,
where the RBFs neural network technique is introduced so that
the output PDFs can be formulated by the dynamic weightings,
then based on LMI techniques, a new fault detection and diag-
nosis observer design scheme is proposed, and by introducing the
tuning parameter, the detection and diagnosis sensitivity is
improved. The simulation of the illustrated example demon-
strates the efficiency of the proposed approach.
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