An Integrated Model-Based and Data-Driven Gap Metric Method for Fault Detection and Isolation

Abstract

This article proposes an integrated approach of model-based and data-driven gap metric fault detection and isolation in a stochastic framework. For actuator and sensor faults, an adaptive Kalman filter combining with the generalized likelihood ratio method is suggested. For component faults, especially incipient faults, the model-based scheme maybe not a good choice due to the existence of disturbances or noises. Hence, a novel data-driven gap metric strategy is presented. The design of the appropriate fault cluster center model and radius via the gap metric technique is put forward to enhance the isolability of the incipient faults. Numerical simulation results are given to demonstrate the effectiveness of the proposed fault detection and isolation algorithm.

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IEEE TRANSACTIONS ON CYBERNETICS 1
An Integrated Model-Based and Data-Driven Gap
Metric Method for Fault Detection and Isolation
Hailang Jin, Zhiqiang Zuo , Senior Member, IEEE, Yijing Wang, Lei Cui , and Linlin Li ,Member, IEEE
Abstract—This article proposes an integrated approach of
model-based and data-driven gap metric fault detection and iso-
lation in a stochastic framework. For actuator and sensor faults,
an adaptive Kalman filter combining with the generalized likeli-
hood ratio method is suggested. For component faults, especially
incipient faults, the model-based scheme maybe not a good choice
due to the existence of disturbances or noises. Hence, a novel
data-driven gap metric strategy is presented. The design of the
appropriate fault cluster center model and radius via the gap
metric technique is put forward to enhance the isolability of
the incipient faults. Numerical simulation results are given to
demonstrate the effectiveness of the proposed fault detection and
isolation algorithm.
Index Terms—Data driven, fault detection and isolation (FDI),
gap metric.
I. INTRODUCTION
FAULT diagnosis technology has become a significant
research topic in various industrial systems due to the
increasing requirements for higher reliability and safety [1].
In recent years, the main direction in terms of system relia-
bility is concentrated on the large amplitude of faults via the
model-based method. In general, the model-based fault detec-
tion and isolation (FDI) is an effective tool for an accurate
system model, especially the actuator and sensor faults [2].
However, there may exist incipient fault, which is easily cov-
ered by modeling error, external disturbances, or measurement
noises. It is difficult to guarantee the tradeoff between the
robustness against the disturbances and the sensibility for the
faults. Hence, the model-based method has certain limitations
for incipient fault. Though the degree of a system deviation
from its normal state is relatively inconspicuous when an incip-
ient fault occurs at the early stage, it may still pose a significant
security risk to the system over time. If the incipient faults
Manuscript received December 24, 2020; revised April 22, 2021;
accepted May 31, 2021. This work was supported by the National Natural
Science Foundation of China under Grant 61933014, Grant 61773281, and
Grant 62073029. This article was recommended by Associate Editor B. Jiang.
(Corresponding author: Zhiqiang Zuo.)
Hailang Jin, Zhiqiang Zuo, Yijing Wang, and Lei Cui are with the School
of Electrical and Information Engineering, Tianjin University, Tianjin 300072,
China (e-mail: hljin@tju.edu.cn; zqzuo@tju.edu.cn; yjwang@tju.edu.cn;
cui_lei@tju.edu.cn).
Linlin Li is with the School of Automation and Electrical Engineering,
University of Science and Technology Beijing, Beijing 100083, China (e-mail:
lilinlin216@126.com).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TCYB.2021.3086193.
Digital Object Identifier 10.1109/TCYB.2021.3086193
can be accurately detected and isolated earlier by an FDI tech-
nique, the system will be regularly maintained and overhauled.
By doing so, people can reduce or avoid the occurrence of
catastrophic faults. Therefore, a data-driven gap metric FDI
(DDGMFDI) method is expected to detect and isolate incipient
faults.
Residual generator plays a key role for fault diagnosis
in terms of the model-based method, which mainly consists
of two categories: 1) deterministic model framework and 2)
stochastic model framework [6]. For the former, a lot of
work has been done in this field, for example, [4], [5], [7]–
[10]. In parallel with the development of the deterministic
model method, the stochastic framework has also captured
wide attention since the 1970s. A common fault detection
(FD) approach was first presented based on the Kalman fil-
ter to generate residual signals and then the faults can be
detected via the changes of the mean for the residual sig-
nals [12]. A slice of statistical tools, such as χ2testing [13],
the cumulative sum algorithm [14], multiple hypothesis test-
ing [15], and generalized likelihood ratio test (GLRT) [16],
were further applied to check the possibility of fault occur-
rence [6]. Meanwhile, the parameter estimation method via
a system identification technique [e.g., recursive least square
estimation (RLSE)] is also powerful for stochastic fault diag-
nosis [17], [18]. In recent years, many researchers have done
a lot of work in a stochastic framework. To detect and iden-
tify incipient faults for high-speed trains, Chen et al. [40]
and Jafari et al. [46] developed a novel robust FD and diag-
nosis approach. Similarly, a data-driven robust FDI scheme
was proposed to apply in a three-phase induction motor with
small magnitude faults [41]. Jiang et al. [42] came up with
a stochastic representation of maximized mutual information
analysis approach to achieve quality monitoring and faults
detection. Xue et al. [44] put forward an optimal data-driven
approach to address the issue of distribution independent
FD for stochastic linear discrete-time systems. Meanwhile,
Raptis and Noursadeghi [45] developed a distributed FDI filter
for stochastic nonlinear systems subjected to multiple fail-
ure modes. A new bearing fault diagnosis algorithm based
on the approach of moments for stochastic resonant systems
was investigated in [47].
With the rapid development of information and commu-
nication technology, a huge amount of input/output data
for the industrial process can be stored [20]–[22]. Thus,
the data-driven FDI approach has attracted wide attention.
See [23]–[25] and [43] for details. Roughly speaking, the gap
metric is used to measure the distance between two closed
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2IEEE TRANSACTIONS ON CYBERNETICS
subspaces in the Hilbert space [26], [27]. In other words, it
can determine the variation of a system’s closed-loop char-
acteristics when a component fault occurs. As we know, the
component faults may be caused by certain parameter vari-
ations. They will have a certain impact on the zero-pole
distribution of the closed-loop system. Furthermore, it may
affect the closed-loop characteristics. In order to analyze
directly the influence of parameter variation on the system,
a performance index that quantifies this effect is essential.
Fortunately, the gap metric can exactly measure the impact of
the variations caused by certain component faults. Meanwhile,
the gap metric technique is also critical for stability analy-
sis in robust control [28]–[32]. There has been much progress
of the gap metric for fault detectability, fault isolability, and
threshold computation [3], [11], [33], [34]. More recently, the
data-driven calculation of the gap metric between two linear
time-invariant (LTI) systems was suggested in [35] through
the data-driven realization of the stable image representa-
tion (SIR) [36]–[38]. With the above preparations, a novel
DDGMFDI approach is proposed for incipient component
faults in this article.
The contributions of this article are summarized as follows.
1) An integrated FDI scheme is proposed. The proposed
approach could deal with most of the faults that may
occur in practical applications, for example, sensor
faults, actuator faults, or component faults. In contrast,
the existing results only focus on one aspect, that is,
the traditional model-based method for actuator and sen-
sor fault or the data-driven approach. To the best of our
knowledge, it is not easy for the model-based method to
balance the tradeoff between the robustness against the
disturbances and the sensibility for the incipient faults,
while the data-driven approach shall cost huge com-
putation. Meanwhile, there are few reports concerning
the component faults, but it has an important impact on
system stability.
2) A novel DDGM approach is developed to realize FDI of
the incipient faults. The incipient component faults are
easily covered by external disturbances in a traditional
model-based FDI method. Our scheme shows the effec-
tiveness for incipient component faults from a system
closed-loop characteristic viewpoint with a gap metric
technique.
3) A design method for determining the appropriate fault
cluster center model and radius is suggested. To enhance
the fault isolability, a design method that aims to deter-
mine the appropriate fault cluster center model and
radius via the gap metric technique is presented.
The remainder of this article is organized as follows. In
Section II, some preliminaries and the problem formulation are
introduced. Section III develops the model-based fault diagno-
sis strategy combining the adaptive Kalman filter (AKF) and
the GLRT. In Section IV, the DDGMFDI approach is formu-
lated. The last section concludes the results and presents the
future research direction.
The following notations will be adopted throughout this arti-
cle. H2denotes the subspace of all signals that are bounded
energy and zero for t<0. H∞represents the set of all stable
transfer functions. RH∞stands for the set of all real rational
transfer functions of stable systems. ·
∞is the H∞-norm.
¯σdenotes the maximum singular value of a matrix. (·)∗is the
complex conjugate transpose of a matrix.
II. PRELIMINARIES AND PROBLEM FORMULATION
In this section, we briefly introduce the process description
and recall the definitions of SIR, gap metric.
A. Process Description
Consider a discrete-time plant Gwith its model of the form
x(k+1)=Ax(k)+Bu(k)+Eff(k)+w(k)
y(k)=Cx(k)+Fff(k)+v(k)(1)
where x(k)∈Rn,u(k)∈Rku,y(k)∈Rky,w(k)∈Rn,
and v(k)∈Rkydenote the state, control input, measurement
output, process noise, and measurement noise of the system,
respectively. w(k)and v(k)are normally distributed and statis-
tically independent of u(k)and x(0)with Qw(k)∈Rn×nand
Rv(k)∈Rky×kybeing the corresponding covariance matrices.
A,B, and Care parameter matrices with compatible dimen-
sions. f(k)=[fA,fS]Tdenotes the fault of the plant. More
specifically, fA=θrepresents the actuator gain loss fault,
where θ∈Rkudenotes the actuator gain loss coefficient
belonging to [0, 1]. fS=ςstands for the sensor offset fault,
which can be described by a constant. Ef=[(k), 0], where
(k)is a known matrix sequence. Ff=[0,C].
The term (k)θ indicates the loss of actuator gain. Once
the actuator fault occurs, the control term Bu(k)becomes
BIku−diag(θ)u(k)=Bu(k)−Bdiag(u(k))θ
where Ikuis an identity matrix, and (k)∈Rn×kuis
defined as
(k)=−Bdiag(u(k)).
When a component fault occurs, Aand Bof system G
change to
A−→ Af(k), B−→ Bf(k)
where Af(k)and Bf(k)are the parameter matrices with
component fault.
This article mainly focuses on incipient component faults.
A crucial issue is the decoupling among different faults in the
system. For instance, the occurrence of the component fault
has a weak effect on the fault diagnosis of fA. This problem
can be addressed by resorting to an AKF.
Remark 1: Incipient faults generally refer to slight changes
relative to the normal system state, which have a strong hidden
at the early stage. Meanwhile, they may lead to faults or even
damage over time [48], [49]. Generally speaking, incipient
faults are normally caused by the aged components or some
abnormal operation [50]. Unfortunately, there are no uniform
and standard definitions for incipient faults yet, however, a lot
of researchers had come up with the key features [51]–[53].
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JIN et al.: INTEGRATED MODEL-BASED AND DATA-DRIVEN GAP METRIC METHOD FOR FDI 3
B. SIR and Gap Metric
In this section, the definition of SIR will be given. We first
introduce the transfer function representation of a nominal LTI
system G0
y(z)=G0(z)u(z).
As shown in [28], G0(z)has the following right coprime
factorizations (RCFs) form:
G0(z)=N(z)M−1(z)(2)
where (M,N)is the right coprime (RC) pair over RH∞.If
M∗(z)M(z)+N∗(z)N(z)=Iku×ku, the RCF is said to be nor-
malized. The corresponding system model form of the SIR
representation of G0can be described by
u(z)
y(z)=ϕι(z), ϕ =M(z)
N(z),ι(z)∈H2(3)
where ϕdenotes the SIR of system G0. Similarly, if RCF (2)
is normalized, (3) is called the normalized SIR. It is clear that
for ι(z)∈H2, all pairs (u,y)are from a closed H2subspace.
This subspace can be represented as
G=z=u
y=M(z)
N(z)ι, ι(z)∈H2.(4)
According to (3) and (4), let G1=¯
N1¯
M−1
1and G2=¯
N2¯
M−1
2
be the normalized RCF of G1and G2, respectively. The
directed gap metric 
δ(G1,G2)for two graphs G1and G2can
be defined as

δ(G1,G2)=sup
z1∈SIS(G1)
inf
z2∈SIS(G2)z1−z22
z12
(5)
where SIS(·)denotes stable image space [35]. It is easy to
find that 0 ≤
δ(G1,G2)≤1. The calculation scheme of gap
metric is introduced in the sequel.
Lemma 1 [27]: Formula (5) can be calculated by solving
the following model matching problem:

δ(G1,G2)=inf
Q∈H∞


¯
M1
¯
N1−¯
M2
¯
N2Q


∞
.
If δ(G1,G2)<1, one has
δ(G1,G2)=
δ(G1,G2)=
δ(G2,G1).
III. AKF AND GLTR
In this part, we will introduce the results of AKF and GLRT.
For actuator fault, the actuator gain θcan be estimated via
AKF. For sensor fault, the residual signals can be obtained
by a residual generator that is designed through AKF. Then,
the mean variation of residual signals can be detected via the
GLRT method.
A. Fault Diagnosis for Actuator Fault
It is noted that the AKF consists of two parts: 1) the classical
Kalman filter for state estimation and 2) the RLS algorithm
for parameter estimation. First, an assumption is needed.
Assumption 1 [19]: The pair [AF(k), Q(1/2)
w] is uniformly
completely controllable, and [AF(k), C] is uniformly com-
pletely observable, in the sense of the uniform positive
definiteness of the corresponding Gramian matrices.
Remark 2: Compared to the LTI system for the require-
ment of conventional observability condition, we can regard
system (1) as a linear time-varying system when a fault occurs.
Actually, the uniformly completely observable condition is
necessary for ensuring the convergency of the adaptive system
parameters.
The AKF can be formulated as
P(k|k−1)=AF(k)P(k−1|k−1)AT
F(k)+Qw(k)
(k)=CP(k|k−1)CT+Rv(k)
K(k)=P(k|k−1)CT−1(k)
P(k|k)=[In−K(k)C]P(k|k−1)
(k)=[In−K(k)C]AF(k)(k−1)
−[In−K(k)C](k)
(k)=CAF(k)(k−1)+C(k)
(k)=γ +(k)(k−1)T(k)−1
ϒ(k)=(k−1)T(k)(k)
(k)=1
γ(k−1)
−1
γ(k−1)T(k)(k)(k)(k−1)
¯y(k)=y(k)−CAF(k)ˆx(k−1|k−1)
+BF(k)u(k)+(k)ˆ
θ(k−1)
ˆ
θ(k)=ˆ
θ(k−1)+ϒ(k)¯y(k)
ˆx(k|k)=AF(k)ˆx(k−1|k−1)+BF(k)u(k)
+(k)ˆ
θ(k−1)+K(k)¯y(k)
+(k)ˆ
θ(k)−ˆ
θ(k−1)
ˆy(k)=Cˆx(k|k)
where (k)∈Rn×ku,(k)∈Rky×ku,ϒ(k)∈Rku×ky, and
(k)∈Rku×kuare the recursively auxiliary matrices, respec-
tively. γ∈(0,1)is the forgetting factor. ˆ
θstands for the
parameter estimation for θ.ˆx(k|k)denotes the state estimation
for x(k).ˆy(k)represents the output estimation for y(k). The ini-
tial state x(0)satisfies x(0)∼N(x0,P0), where ˆx(0|0)=x0
and P(0|0)=P0. By estimating the parameter θ, fault diagno-
sis of the actuator can be performed. Next, we will put forward
the GLRT method for the sensor fault.
B. FD for Sensor Fault
Let ζ(k)∈Rabe an arbitrary data vector. Introduce the
following augmented vector:
ζs(k)=⎡
⎢
⎣
ζ(k−s)
.
.
.
ζ(k)
⎤
⎥
⎦∈R(s+1)a(6)
where s∈Z+denotes the number of the stacked data vectors.
According to AKF, a residual generator is designed with the
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4IEEE TRANSACTIONS ON CYBERNETICS
white Gaussian process
r(k)=y(k)−ˆy(k)=e(k), r(k)∼N(0,I).
This is the case of fault-free. Meanwhile, r(k)is a zero-mean
white Gaussian process. Note that there will be a change in the
mean value if a fault occurs, and the change can be determined
via the GLRT approach.
Suppose that the residual signals are available. Then
rs(k)=⎡
⎢
⎣
r(k−s)
.
.
.
r(k)
⎤
⎥
⎦∈R(s+1)ky.
According to [5], we can directly obtain the following decision
rule:
≤Jth,fault-free
>Jth,fault (7)
where =rT
s(k)rs(k)denotes the testing statistic. To reduce
the fault false-alarm rate, we need to choose an appropri-
ate threshold Jth. To this end, we can suggest the following
relation:
probχ2s,J2
r>Jth=α(8)
where sand J2
rdenote the degree of freedom and the non-
centrality parameter of the noncentral χ2distribution, respec-
tively. αstands for the fault false alarm rate. Therefore, the
sensor fault can be detected by the GLRT method. However,
for the incipient component faults, we need to propose a novel
FDI method in the next section.
IV. DDGMFDI METHOD
Here, we propose the DDGMFDI method for component
fault in system (1). To achieve it, the main work consists of
two parts.
1) The data-driven realization of the gap metric via the SIR
identification technique.
2) The construction of an appropriate fault cluster center
model.
A. Date-Driven Gap Metric Matrices
According to (6), the Hankel matrix can be formulated as
Uk,s=[us(k−N+1)··· us(k)]∈R(s+1)·ku×N
Yk,s=ys(k−N+1)··· ys(k)∈R(s+1)·ky×N
where Nis a positive integer. The input/output dataset can be
divided into past and current ones, which are described by the
subscript pand f, respectively. Hence, we define the following
notations:
zp(k)=us(k−s)
ys(k−s),zf(k)=us(k)
ys(k).
This yields
Zp,k=Up
Yp=Uk−sp−1,sp
Yk−sp−1,sp.
Without loss of generality, we assume that sp=s.
Based on the above discussions, the definition of the
realization of a data-driven SIR is introduced.
Definition 1: Assume that system (1) with w(k)=0 and
v(k)=0 is sufficiently excited by the input signal u(k).Ifan
integer ssatisfies: ∃β(k)∈Rrsuch that
us(k)
ys(k)=Id,szp(k)
β(k)=[Id,pId,f]zp(k)
β(k).
Id,sis called a data-driven realization of the SIR.
Definition 2: The data-driven SIR is normalized if
IT
d,fId,f=I. To distinguish the normalized and general forms
of SIR, the former will be denoted by ¯
Id,s.
To calculate the data-driven SIR, an LQ decomposition of
the input/output dataset is suggested as
⎡
⎣
Zp,k
Uk,s
Yk,s⎤
⎦=⎡
⎣
L11 00
L21 L22 0
L31 L32 L33 ⎤
⎦⎡
⎣
Q1
Q2
Q3⎤
⎦.(9)
Some related lemmas are given as follows.
Lemma 2 [35]: From (9), the data-driven SIR is
calculated by
IG0
d,s=IG0
d,p¯
IG0
d,f
with
IG0
d,p=0
K,IG0
d,f=I
L
where
K=L31L†
11 −L32L†
22L21L†
11,L=L32L†
22
and (·)†denotes the pseudo-inverse. Moreover, ¯
IG0
d,sadmits the
form
¯
IG0
d,s=IG0
d,p¯
IG0
d,f
where ¯
IG0
d,f=UG0
1is obtained according to the following
singular value decomposition (SVD):
IG0
d,f=UG0G0VG0T
=UG0
1UG0
2G0
10
00
⎡
⎢
⎣VG0
1T
VG0
2T⎤
⎥
⎦.
Lemma 3 [35]: Suppose that the normalized SIR of system
G1and G2are represented by ¯
IG1
d,sand ¯
IG2
d,s, respectively.
The data-driven realization of the gap metric δd,scan be
calculated by
δd,s(G1,G2)=max
i
j=1
2,2
1
δd,sG(i),G(j)
with
δd,sG(1),G(2)=¯σ¯
IG1
d,f−¯
IG2
d,f¯
IG2
d,fT¯
IG1
d,f.
In addition, as sapproaches to infinity, we have
δd,s(G1,G2)−→ δ(G1,G2).
Based on the above preparations, the DDGMFDI method can
be proposed in the next section.
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JIN et al.: INTEGRATED MODEL-BASED AND DATA-DRIVEN GAP METRIC METHOD FOR FDI 5
(a)
(b)
Fig. 1. Schematic diagram of FI. (a) Fault Gfican be isolated. (b) Fault Gfi
cannot be isolated.
B. DDGMFDI for Component Fault
As mentioned in Section I, the occurrence of a component
fault will affect the performance of the closed-loop system.
Hence, the FDI of component fault is important for enhancing
system reliability. The gap metric method is an appropriate
technique to detect component faults. We can identify the SIR
of the nominal system G0and the operating system Gwith
measurement noises through the off line way. Then, the maxi-
mal gap metric δd,s(G,G0)is available as a threshold δth.The
FD logic is described by
δd,s(G,G0)≤δth,fault-free
δd,s(G,G0)>δ
th,fault.(10)
To isolate the component faults, we first need to introduce the
following definition.
Definition 3: The faults Gfi,i=1,...,L, are isolable, if
∀Gfi∈Sfi,Gfi/∈Sfj,i= j,j=1,...,L
with
Gfi⊆Gf:δGf,GFi≤δi,0<δ
i<1
where Gfi[shown in Fig. 1(a)] denotes the graph of a system
in the case of faults. Sfiis the set of all graphs for the faulty
systems, namely, a fault cluster. GFidenotes a center to the
fault cluster. δiis a radius of the fault cluster.
Fig. 1(a) characterizes the schematic diagram of FI via
the gap metric technique. Similar to [3], we can present the
condition of FI.
In Fig. 1(a), the fault Gfiis isolable, if ∀i,j,i= j,i,j=
1,...,L
δGfi,Gfj>δ
i+δj.(11)
Remark 3: Assume that δ(Gfi,Gfj)≤δi+δj. In such a case,
points Gf1and Gf2are depicted in Fig. 1(b), which means
∃Gfi∈Sfi∩Sfj.
This contradicts with Definition 3. Thus, the fault is isolable
under condition (11).
We emphasize that condition (11) is critical for investigat-
ing the FI problem via the gap metric method no matter it
is actuator fault, sensor fault, or component fault. However,
condition (11) is associated with the choice of the fault clus-
ter center model GFiand the fault cluster radius δi. If these
parameters are not properly selected, the case of Fig. 1(b)
may occur. To deal with this problem, two related lemmas are
needed.
Lemma 4 [31]: Given two real rational transfer functions
G1=k1/(s+1)and G2=k2/(s+1)for any k1and k2, one
has
δ(G1,G2)=⎧
⎨
⎩
|k1−k2|
|k1|+|k2|,|k1k2|>1
|k1−k2|
1+k2
11+k2
2,|k1k2|≤1.
Lemma 5 [39]: Let G1and G2be any two closed operators
in C.IfG1and G2are single-input single-output and have
dense ranges, it follows that:
δ(G1,G2)=sup
ω∈R
|G1(jω)−G2(jω)|
1+|G1(jω)|21+|G2(jω)|2.
Remark 4: In practice, Lemma 5 is the so-called ν-gap
metric in the robust control theory [27]. Without taking into
account the so-called winding number condition [3], [28],
the ν-gap metric is identical with the gap metric under a
considerable framework.
With Lemmas 4 and 5, an appropriate fault cluster center
model can be developed in the following theorem.
Theorem 1: Given the fault model Gfi(s)=βfim/(s+1),
0<β
1<β
fi<β
2,m= 0, i=1,...,L, where βfistands for
the amplitude of a fault. β1and β2denote its lower and upper
bounds. The corresponding fault cluster can be represented
by Sfiand the appropriate fault cluster center model gain β01
satisfies
β01=⎧
⎪
⎨
⎪
⎩
√β1β2, β1β2m2 >1
β11+β2
2m2+β21+β2
1m2
1+β2
1m2+1+β2
2m2, β1β2m2 ≤1.
Proof: Given a fault plant
Gf(s)=βfm
s+1
our aim is to find an appropriate fault cluster center model
Gf01(s)=β01m
s+1
in the sense of
inf
β01∈[β1,β2]sup
βf∈[β1,β2]
δGf,Gf01.
According to Lemma 4, it yields
δGfβ1,Gfβ2=⎧
⎨
⎩
|β1−β2|
|β1|+|β2|, β1β2m2 >1
|β1−β2|
1+β2
1m21+β2
2m2, β1β2m2 ≤1.
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6IEEE TRANSACTIONS ON CYBERNETICS
Fig. 2. Single schematic diagram of Gf01under gap metric.
If |β1β2m2|>1, one has
β01−β1
β01+β1=β2−β01
β2+β01
which means
β01=!β1β2.
If |β1β2m2|≤1, a suitable β01admits
β01−β1
1+β2
1m2=β2−β01
1+β2
2m2.
This leads to
β01=β11+β2
2m2+β21+β2
1m2
1+β2
1m2+1+β2
2m2.
The proof is completed.
Remark 5: The single schematic diagram of the appropri-
ate fault cluster center model Gf01under |β1β2m2|>1 can be
found in Fig. 2. The β-axis denotes the variation of the fault
amplitude, while the δ-axis stands for the corresponding posi-
tion of this fault model under gap metric. Generally speaking,
for the variation range (β1,β
2)of the fault amplitude, a suit-
able fault cluster center parameter may be set to (β1+β2)/2in
subconscious. However, Theorem 1 presents an explicit value
of the fault cluster center parameter √β1β2, which is more
appropriate for the considered problem.
Theorem 2: Given fault model Gfj(s)=m/(s+βfk),0<
β1<β
fj<β
2,m= 0, k= 0, and j=1,...,L.The
corresponding fault cluster can be represented by Sfj.The
appropriate fault cluster center model gain β02satisfies
β02=β1a+β2b
a+b
with
a=β2
1k2+m2,b=β2
2k2+m2
and
β1+β2
2<β
02<β
2.
Proof: Given a fault plant
Gf(s)=m
s+βfk.
The purpose is to find a favorable fault cluster center model
Gf02(s)=m
s+β02k
Fig. 3. Single schematic diagram of Gf02under gap metric.
such that
inf
β02∈[β1,β2]sup
βf∈[β1,β2]
δGf,Gf02.
Using Lemma 5, one obtains
δGfβ1,Gfβ2=mk|β1−β2|
β2
1k2+m2β2
2k2+m2
.(12)
Then, an appropriate β02for the proposed question satisfies
β02−β1
a=β2−β02
b(13)
with
a=β2
1k2+m2,b=β2
2k2+m2.
This further implies
β02=β1a+β2b
a+b.(14)
It is known that b>a>0, let b=κa,κ>1. According
to (12)–(14), we have
β02=β1+β2
1+κ.
Then
β1+β2
2<β
02<β
2.
This ends the proof.
Remark 6: Compared to Theorem 1, Theorem 2 considers
a fault that has any influence on the system pole-zero location
and stability. Similarly, the single schematic diagram of the
appropriate fault cluster center model Gf02under the gap metric
is plotted in Fig. 3. It is interesting noting that the appropriate
fault cluster center model is close to G2in Theorem 2, which
is different from Theorem 1.
Therefore, the FI logic via the DDGM method is
formulated as
⎧
⎨
⎩
δd,sG,Gfβ0,i<δ
i
δd,sG,Gfβ0,j≥δj=⇒ fault in cluster Sfi(15)
with
∀i,j,i= j,i,j=1,...,L
where Gdenotes the process plant. Gfβ0,iis the fault cluster
center model with suitable parameter β0.δistands for the fault
cluster radius.
With all these preparations, we propose Algorithm 1 for our
integrated FDI approach. The proposed DDGMFDI method
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JIN et al.: INTEGRATED MODEL-BASED AND DATA-DRIVEN GAP METRIC METHOD FOR FDI 7
Algorithm 1 Integrated FDI Approach
Offline:
1: Set the initial states and related parameters of system G.
2: Collect sufficient data of the operating system G. Identify
and save the normalized and data-driven SIR ¯
IG
d,sby
Lemma 2.
3: Solve (14) for each appropriate fault cluster center param-
eter β0,i,i=1,...,M. Obtain fault cluster center model
Gfi,i=1,...,M.
4: Collect sufficient data from each fault cluster center
model. Identify and save each normalized and data-driven
SIR ¯
IGfi
d,s.
Online:
5: Run system Gand AKF. Calculate the testing statistic
=rT
s(k)rs(k). Implement the FD decision logic (7).
6: Calculate δd,s(G,G0)online by Lemma 3. Conduct the
decision logic (10). If a fault is detected, execute the
isolation logic (15).
Fig. 4. Overall structure diagram of the FDI system.
Fig. 5. Boost converter electrical topology.
can be used to detect and isolate incipient component faults.
Fig. 4 is the overall structure diagram of the FDI system.
Remark 7: Our proposed method matches the framework
of [3], [11], and [54], and the main difference may be just the
system model. As [54] shows, the operation mechanism and
the structure of the wind turbine system are more complicated,
and it is easy to occur all kinds of faults in harsh environ-
ments, especially offshore wind turbines. If varying faults can
be detected and isolated at their early stages (we call them
incipient faults in our research), some damages and danger-
ous situations may be avoided in wind turbine systems. On the
other hand, different systems have their own particularities,
TAB L E I
PARAMETERS OF THE BOOST CONVERTER
Fig. 6. Dynamic process of the system with noise in the fault-free case.
including the dynamics, the description of faults, the sensi-
bility for external disturbances, and so on. In other words,
circumstances alter cases. Therefore, we should make a con-
crete analysis of the considered problem. To sum up, the main
result of our research provides an integrated model-based and
data-driven gap metric approach for different faults. We are
trying our best to apply it to other industrial systems and the
research advance will be reported in the near future.
Remark 8: According to Algorithm 1, the proposed data-
driven approach can be implemented in two steps: 1) offline
identification SIR and 2) online calculation gap metric. For
the former, it has no effect on the real-time performance of
the considerable plant. For the latter, a lag will appear for the
real-time response and its value depends on the size of the
data in the data-driven approach.
V. NUMERICAL EXAMPLE
In this section, we will verify the validity of the developed
approach through an example.
A. Plant Description
Consider an ideal boost dc–dc converter with its electrical
topology depicted in Fig. 5. The state-space average model is
formulated as
⎡
⎢
⎣
diL1
dt
dVo
dt
⎤
⎥
⎦=⎡
⎢
⎣
0−1−D
L1
1−D
C1−1
C1R
⎤
⎥
⎦⎡
⎣
iL1
Vo⎤
⎦+⎡
⎣
1
L1
0⎤
⎦Vi.(16)
The relevant parameters are listed in Table I.
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8IEEE TRANSACTIONS ON CYBERNETICS
(a)
(b)
Fig. 7. Fault diagnosis of the actuator gain loss fault via AKF. (a) θ=0.4:
Vigain loss 40% at k=1000. (b) θ=0.7: Vigain loss 70% at k=1000.
Fig. 8. ς=10: The FD of output Vovoltage sensor offset 10Vfault.
(a)
(b)
(c)
Fig. 9. Coupling between component fault and actuator fault. (a) Resistance
Rfault: R→10R. (b) Inductor L1fault: L1→0.5L1. (c) Capacitor C1fault:
C1→0.001C1.
The discrete-time state-space average model of the boost
converter system can be described by (1) with
A=0.9867 −0.0979
0.1957 0.5953 ,B=0.2488
0.0266 ,C=[0,1].
x=iL1,VoT,u=Vi,y=Vo.
Qw=0.01I2and Rv=0.01I1. The initial conditions of AKF
are: γ=0.97, θ(0)=0, (0)=0, and (0)=Iku.The
length of evaluation window is s0=15. In addition, α=5%.
According to (8), we can calculate Jth =27.9175. To verify
the proposed DDGMFDI approach, the parameters associated
with the identification of SIR are: s=sp=30 and N=
(a)
(b)
(c)
Fig. 10. FD of the plant parameter variations at k=1000. (a) Resistance R
fault: R→10R. (b) Inductor L1fault: L1→0.5L1. (c) Capacitor C1fault:
C1→0.001C1.
(a)
(b)
(c)
Fig. 11. DDGM FD method for the component fault at k=1000.
(a) Resistance Rfault: R→10R. (b) Inductor L1fault: L1→0.5L1.
(c) Capacitor C1fault: C1→0.001C1.
1000. With sufficient excitation of the input signal, 20 000
pairs of input/output data for both cases are collected. For the
fault-free case, the dynamic process of the system is shown in
Fig. 6. The related thresholds can be found as ˆ
θth =0.1210
and δth =0.0220.
B. Fault Diagnosis of Actuator/Sensor Faults
In this case, Algorithm 1 is applied. The simulation results
of the faults consisting of actuator and sensor are plotted in
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JIN et al.: INTEGRATED MODEL-BASED AND DATA-DRIVEN GAP METRIC METHOD FOR FDI 9
(a) (b) (c)
Fig. 12. DDGM FI method for the component fault. (a) FI of Rfault: R→18R. (b) FI of L1fault: L1→0.3L1.(c)FIofC1fault: C1→0.01C1.
Figs. 7 and 8. We can see that the AKF combined with the
GLRT scheme is effective for our purpose.
C. FDI for Component Faults
According to (16), three types of component faults are
considered. First, we know that component faults are always
decoupled with actuator faults (see Fig. 9). The FD of compo-
nent faults has also been investigated via the GLTR approach
as in Fig. 10. It is clear that the change of resistance param-
eter [see Fig. 10(a)] can be successfully detected. However,
as shown in Fig. 10(b) and (c), the GLRT approach does not
seem to be suitable for the inductor and capacitor cases. To
illustrate it, we plot the output error e(k). It is obvious that the
change of e(k)induced by the resistance parameter is much
larger than those by the inductor and capacitor. In other words,
the parameter variations of inductor and capacitor have a weak
effect on residual signal r(k).
We perform the simulation results of the same cases in
Fig. 11. It is found that the incipient component faults for
inductor and capacitor can be effectively detected via the
DDGM technique.
For FI of incipient faults, an appropriate fault cluster cen-
ter and radius are crucial to enhance the isolability of a fault.
From Table I, these component parameters in faulty case sat-
isfy 1.5<β
A1<20, 0 <β
A2<0.8, and 0 <β
A3<0.8. In
terms of (14), we can capture the appropriate fault cluster cen-
ter model parameters βR0=11, βL0=0.45, and βC0=0.41.
Applying the calculation scheme of the gap metric results in
δGfR0,GfL0=0.6467,δ
GfR0,GfC0=0.6886
δGfL0,GfC0=0.4204.
From Theorem 2, the corresponding fault cluster radii can be
obtained with
δR=0.3465,δ
L=0.3001,δ
C=0.1202.
Fig. 12 gives the simulation, where Gdenotes the plant
with component fault and noise. GfR0,GfL0, and GfC0stand
for the corresponding fault cluster center model. It is obvi-
ous that δ(G,GfR0)<δ
R,δ(G,GfL0)>δ
L, and δ(G,GfC0)>
δCin Fig. 12(a) all the time. Then, we can conclude that
the location of fault occurrence is resistance R. Meanwhile,
Fig. 12(b) and (c) demonstrates the locations of fault occur-
rence for inductor L1and capacitor C1, respectively. It is
evident that the proposed DDGM fault isolation method is
effective for the incipient component faults.
VI. CONCLUSION
In this article, we have developed an integrated model-based
and data-driven gap metric approach for actuator fault, sensor
fault, and component fault. For actuator fault and sensor fault,
an algorithm has been put forward to address them. For an
incipient fault, the model-based FDI has inherent flaws due to
the existence of external disturbances or measurement noises.
Hence, a novel DDGMFDI algorithm has been presented. To
enhance the faulty isolability, we have given a design algo-
rithm of an appropriate fault cluster center model and the
corresponding fault cluster radius. Simulation results for the
incipient component faults have indicated the effectiveness of
the proposed approach. This article provides a scheme for
the fusion of the different fault diagnosis approaches. Future
work will focus on the fault-tolerant control via the gap metric
technique.
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Hailang Jin received the B.E. degree from the
Shenyang University of Technology, Liaoyang,
China, in 2016, and the M.E. degree from Guangxi
University, Nanning, China, in 2018. He is cur-
rently pursuing the Ph.D. degree with the School
of Electrical and Information Engineering, Tianjin
University, Tianjin, China.
His research interests include fault diagnosis and
fault-tolerant control, robust control, and adaptive
control.
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JIN et al.: INTEGRATED MODEL-BASED AND DATA-DRIVEN GAP METRIC METHOD FOR FDI 11
Zhiqiang Zuo (Senior Member, IEEE) received the
M.S. degree in control theory and control engineer-
ing from Yanshan University, Qinhuangdao, China,
in 2001, and the Ph.D. degree in control theory from
Peking University, Beijing, China, in 2004.
In 2004, he joined the School of Electrical and
Information Engineering, Tianjin University, Tianjin,
China, where he is a Full Professor. From 2008 to
2010, he was a Research Fellow with the Department
of Mathematics, City University of Hong Kong,
Hong Kong. From 2013 to 2014, he was a Visiting
Scholar with the University of California at Riverside, Riverside, CA, USA.
His research interests include nonlinear control, robust control, and multiagent
systems.
Prof. Zuo is an Associate Editor of the Journal of the Franklin Institute.
Yijing Wang received the M.S. degree in con-
trol theory and control engineering from Yanshan
University, Qinhuangdao, China, in 2000, and
the Ph.D. degree in control theory from Peking
University, Beijing, China, in 2004.
In 2004, she joined the School of Electrical
and Information Engineering, Tianjin University,
Tianjin, China, where she is a Full Professor. Her
research interests include intelligent vehicles, anal-
ysis and control of switched/hybrid systems, and
robust control.
Lei Cui received the Ph.D. degree in mechanics
system and control from Peking University, Beijing,
China, in 2012.
He was a Senior Engineer with the Beijing
Institute of Control and Electronic Technology,
Beijing, from 2013 to 2017. Since 2018, he has been
an Associate Professor with the School of Electrical
and Information Engineering, Tianjin University,
Tianjin, China. His research interests include robust
control, adaptive control, fault detection and fault-
tolerant control, and their applications to flight
control system.
Linlin Li (Member, IEEE) received the B.E. degree
from Xi’an Jiaotong University, Xi’an, China, in
2008, the M.E. degree from Peking University,
Beijing, China, in 2011, and the Ph.D. degree from
the Institute for Automatic Control and Complex
Systems, University of Duisburg-Essen, Duisburg,
Germany, in 2015.
Since 2016, she has been a Researcher with the
School of Automation and Electrical Engineering,
University of Science and Technology Beijing,
Beijing. Her current research interests include fault
diagnosis and fault-tolerant control, fuzzy control, and estimation for nonlinear
systems.
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