Robust Multiple Fault Isolation Based on Partial-orthogonality Criteria

Abstract

In this paper, a data-driven scheme for the robust Fault Isolation of multiple sensor faults is proposed. Robustness to modelling uncertainty and noise is achieved via the optimized design of the processing blocks. The main idea of the study is the introduction of a Pre-Isolation block that selects a restricted set of sensors containing (with high probability) the subset of the faulty sensors; in this phase, robustness is achieved through the datadriven design of a redundant number of Multiple Analytic Redundancy Relations (MARRs) and a voting logic for the ranking of the candidate faulty sensors. Then, robust Faults Isolation (FI) is achieved by means of another large set of specialized ARRs, whose fault signatures are specifically designed to optimize, at the same time, noise immunity while maximizing the decoupling only of the pre-isolated fault directions (Partial-Orthogonality Criteria). The proposed diagnostic system may provide an effective means for early sensor failure isolation, particularly useful for critical applications such as aerospace control systems or energy management systems. To assess the performance of the approach, we performed a comparative study with other State-of-the-Art (SoA) approaches using a well-known benchmark model that emulates faults on six sensors. Results for single and multi-contemporary faults have clearly highlighted the superiority of our method.

Full text

International Journal of Control, Automation, and Systems 20(7) (2022) 2148-2158
http://dx.doi.org/10.1007/s12555-021-0428-y
ISSN:1598-6446 eISSN:2005-4092
http://www.springer.com/12555
Robust Multiple Fault Isolation Based on Partial-orthogonality Criteria
Nicholas Cartocci* , Francesco Crocetti, Gabriele Costante, Paolo Valigi, and Mario L. Fravolini
Abstract: In this paper, a data-driven scheme for the robust Fault Isolation of multiple sensor faults is proposed.
Robustness to modelling uncertainty and noise is achieved via the optimized design of the processing blocks. The
main idea of the study is the introduction of a Pre-Isolation block that selects a restricted set of sensors containing
(with high probability) the subset of the faulty sensors; in this phase, robustness is achieved through the data-
driven design of a redundant number of Multiple Analytic Redundancy Relations (MARRs) and a voting logic
for the ranking of the candidate faulty sensors. Then, robust Faults Isolation (FI) is achieved by means of another
large set of specialized ARRs, whose fault signatures are specifically designed to optimize, at the same time, noise
immunity while maximizing the decoupling only of the pre-isolated fault directions (Partial-Orthogonality Criteria).
The proposed diagnostic system may provide an effective means for early sensor failure isolation, particularly
useful for critical applications such as aerospace control systems or energy management systems. To assess the
performance of the approach, we performed a comparative study with other State-of-the-Art (SoA) approaches
using a well-known benchmark model that emulates faults on six sensors. Results for single and multi-contemporary
faults have clearly highlighted the superiority of our method.
Keywords: Data-driven, directional residuals, multiple analytical redundancy, multiple-fault diagnosis, partial-
orthogonality robust residuals.
1. INTRODUCTION
Fault Diagnosis (FDi) plays a critical role to ensure the
safety of engineering systems. In the last decades, the sci-
entific community has indeed produced a wealth of results
in this important area. The main approaches for Fault Di-
agnosis are essentially two: Model-Based (MB) [1] and
Data-Based (DB) techniques. A clear picture of the State-
of-the-Art (SoA) of the FDi can be found in the recent
review [2]. MB and DB based methods are also known as
Analytical Redundancy (AR) techniques [3].
The present study deals with data-based FDi methods;
in this context, the most established and widespread meth-
ods are based on multivariate statistical (MVS) method-
ologies, such as: Principal Component Analysis (PCA),
Independent Component Analysis (ICA), Partial Least
Squares (PLS), Fisher Discriminant Analysis (FDA), Sub-
space Aided approach (SAP) as well as their recent ad-
vancements [4]. Among the mentioned DB techniques,
undoubtedly, PCA methods [5,6] are the most popular
because of their ease of use and because these can ef-
ficiently handle large dimensional data. In recent years,
a growing research interest is also focusing on the em-
ployment of neural networks, computational intelligence
and machine learning methods [7]. A comparison of data-
driven fault diagnosis techniques, particularly considering
the best-known machine learning-based fault diagnosis al-
gorithms, is presented in [8].
Regardless of the specific technique, FDi performance
is strictly influenced by modelling uncertainty and mea-
surement noise, whose presence implies that the residual
signals might significantly deviate from zero in fault-free
conditions. Indeed, (primary) residuals may be very sen-
sitive to noise and uncertainty and not robust enough to
detect and isolate small amplitude faults.
A general approach to build a robust FDi system is to
formulate the design as an optimization problem where
a cost function (to be optimized) quantifies desired fault
sensitivity requirements and immunity to noise and uncer-
tainty for the residual signals.
In [9,10], the sensitivity to faults and reference input
is measured using finite-frequency performance indices
so that the computed residual is sensitive to both in the
presence of faults, while it is robust only against the ref-
erence input in the fault-free cases. The paper [11] studies
the issue of the reliable asynchronous sampled-data filter-
Manuscript received May 31, 2021; revised August 24, 2021; accepted September 26, 2021. Recommended by Associate Editor Yueying
Wang under the direction of Editor Bin Jiang. This study was partially supported by the University of Perugia through the 2018, 2019 and
2020 Basic Research Funds (Projects: RICBA18MF, RICBA19MLF and RICBA20MF).
Nicholas Cartocci, Francesco Crocetti, Gabriele Costante, Paolo Valigi, and Mario L. Fravolini are with the Department of Engineering,
University of Perugia, Via G. Duranti, 67, Perugia 06125, Italy (e-mails: {nicholas.cartocci, francesco.crocetti, gabriele.costante, paolo.valigi,
mario.fravolini}@unipg.it).
* Corresponding author.
c
ICROS, KIEE and Springer 2022

Robust Multiple Fault Isolation Based on Partial-orthogonality Criteria 2149
ing of Takagi–Sugeno fuzzy delayed neural networks with
stochastic intermittent faults, proposing a reliable asyn-
chronous sampled-data controller that ensures that the re-
sulting neural network is asymptotically stable. In [12],
the sensors selection process for FDi is formulated as an
optimization problem to guarantee that the monitored sen-
sors meet required performance specifications consider-
ing model uncertainties and measurement noise. Similarly,
in [13], a robust subset of residuals for FDi is computed
through the solution of a convex optimization problem,
while in [14], a FDi scheme is designed for non-linear sys-
tems using a differential evolutionary algorithm.
Another relevant aspect of FDi is the number of simul-
taneous faults handled by the diagnostic system. In sin-
gle fault schemes, the implicit assumption is that the sys-
tem is subject to a fault at a time, while, in the multi-fault
case, this restriction is not present. In the multi-fault case,
the main challenges can be summarized as [15]: smear-
ing effect, fault signatures mixing and downward/upward
estimation of the actual number of faulty sensors and par-
tial misclassifications. Although the multi-fault scenario
is more general compared to the single fault case, in the
literature, there is a limited number of complete Fault De-
tection (FD), Fault Isolation (FI) approaches developed
to handle the multiple-fault case. Among these methods,
Zhu et al. [16] proposed a PCA-based approach where
the multi-fault diagnosis is performed using the Squared
Prediction Error (SPE) index. In [17], Li et al. has devel-
oped a wavelet-Autoregressive-PCA method for the multi-
fault diagnosis of gears. Jiang et al. [18] proposed a mul-
tiwavelet based pre-filter to improve the Ensemble Em-
pirical Mode Decomposition for multi-fault diagnosis for
rotating machinery. Finally, Chen et al. [19] proposed an
approach based on PCA and Sequential Probability Ra-
tio Test (SPRT) for multifault condition monitoring of a
slurry pump.
Precisely to address the issues mentioned above in this
study, a robust data-based diagnostic scheme for the Fault
Isolation of multiple sensor faults is proposed. The main
contribution of the scheme lies in the employment of two
novel functional blocks. The first, called “pre-fault iso-
lation block” (pre-FI), has the purpose of selecting a re-
stricted subset of sensors containing (with high proba-
bility) the set of the faulty sensors; in this block, a re-
dundant group of specifically designed Multiple Ana-
lytic Redundancy Relations (MARRs), is used within a
vote-based decision-making logic to online select a sub-
set of candidate faulty sensors. Then, the second block,
named “Multi-Fault Isolation block”, performs the actual
FI based on the novel concept of Partial-Orthogonality
Criteria (POC). The POC based FI mechanism relies on
the design of a collection of specialized parity equations
(residuals) derived from the solution of a multi-objective
optimization problem that minimizes the residuals stan-
dard deviation (STD) while maximizing the decoupling
(orthogonality) of the (only) pre-isolated fault directions.
Differently from existing methods, the rationale of the
POC design is that the orthogonality condition is im-
posed on all the monitored sensors but only for the re-
stricted set of the current pre isolated faults. The reduc-
tion of the number of orthogonality constraints may sig-
nificantly decrease the STD of the residuals in the multi-
objective optimization, thus achieving less noisy residu-
als with the consequent increase of the chance of isolat-
ing small-amplitude faults. Further, the POC constraint
between the directions of the pre-isolated faults nullifies,
theoretically, the sensor fault smearing effect on the non-
faulty sensors.
A possible drawback of the POC based FI is the need
of computing an optimized set of residuals for each pos-
sible combination of pre-isolated faults. Fortunately, this
computation has not to be performed online; in fact, all
the optimizations can be solved and stored offline so that,
in the online phase, it is only requested to retrieve from
memory of the specific POC matrix.
In order to assess the performance of our approach
quantitatively, we performed a comparative study with
SoA approaches using the benchmark model proposed in
[20–26] for the analysis of single and multiple faults FDi
systems. Specifically, the following SoA algorithms were
considered.
In [22,23], Van den Kerkhof et al. performed a quanti-
tative analysis of the smearing effect comparing Contribu-
tion Analysis (CA) methods based on Complete Decom-
position Contributions (CDCs), Partial Decomposition
Contributions (PDCs) and Reconstruction-Based Contri-
butions (RBCs). The authors showed that the smearing ef-
fects increase in correlated variables and that the contribu-
tions do not guarantee, in general, correct fault isolation of
multiple sensor faults.
In [24], Zhou et al. define a new variable contribution
based on the k-Nearest Neighbors (kNN) distance. This
method, unlike CA-based, is defined in the original mea-
surement space without correlation among the defined iso-
lation indices, so it suffers less from the smearing effect.
As shown in the experiment section, our method pro-
vides superior performance compared with the mentioned
SoA methods for single and multiple sensor faults cases.
2. OVERVIEW OF THE PROPOSED FDI
APPROACH
The block diagram of the proposed FDI (Fault
Detection-Isolation) scheme is shown in Fig. 1. The sys-
tem can be conceptually divided into three functional
blocks:
•Fault Detection (FD) block. Its purpose is to detect,
online, the occurrence of single or multiple sensor
faults. The FD residual signal is derived by employ-
ing the data-based techniques described in Section 4.

2150 Nicholas Cartocci, Francesco Crocetti, Gabriele Costante, Paolo Valigi, and Mario L. Fravolini
Fig. 1. The workflow of the proposed robust fault detec-
tion and isolation scheme.
•Pre-Fault Isolation (Pre-FI) block. Its purpose is to
identify a restricted subset of candidate faulty sen-
sors (using a voting logic), thus simplifying the sub-
sequent FI task. Its design and operation are described
in Section 5.
•Multi-Fault Isolation (FI) block. Based on the set of
pre-isolated faults, its purpose is to isolate the actual
faulty sensors, providing also fault reconstruction. Its
design and operation are described in Section 6.
3. MONITORED SENSOR MODELLING
The signals associated with the monitored sensors are
concatenated in the vector x(k)∈Rnwhere kis the
discrete-time index. The proposed technique is based on
the AR concept and estimates the monitored sensors sig-
nals as a function of the other monitored signals in x(k).
We assume that x(k)is the sum of two contributions: A
linear multivariate model plus an uncertain term that char-
acterizes the residual modeling uncertainty, nonlinearities
and noise [27], that is
xi(k) =
n
∑
j=1,j6=i
wxi,jxj(k) + ∆i(k),i=1, ..., n,(1)
where wxi,jare the coefficients of the linear multivariate
model, and ∆i(k)characterizes the modelling uncertainties
associated with the i-th sensor model. Model (1) is a stan-
dard linear multivariate model used, for instance, in many
PCA and PLS process monitoring techniques; this can be
easily extended to include non-linear effects by augment-
ing x(k)with arbitrary non-linear functions of the original
measured variables [28]. Model (1) can also be extended
to take into account possible dynamic effects by including
in the regression variables the delayed version of x(k), that
is x(k−1),x(k−2), ..., x(k−d). For simplicity, models in
(1) are rearranged in compact form
xi(k) = wxix(k) + ∆i(k),i=1, ..., n,(2)
where wxi = [wxi,1, ..., wxi,i−1, 0, wxi,i+1, ..., wxi,n]∈Rn.
Putting the above nequations together, we get the follow-
ing vector expression
x(k) = W
W
Wx
x
xx(k) + ∆(k),(3)
where W
W
Wx
x
x= [wx1; ...; wxn]∈Rn×n. The linear terms in (3)
provide a linear (computable) estimation of the vector x(k)
that is defined as
ˆx(k) = W
W
Wx
x
xx(k).(4)
Equation (4) can be rearranged as parity equations [29]
(equal to zero), defining the matrix W
W
W=W
W
Wx
x
x−I
I
I∈Rn×n
and the data matrix X
X
X∈Rm×n, that is
W
W
W x(k) = 0,(5)
and in matrix form
XW
XW
XW =0
0
0.(6)
It is observed that coefficients in the diagonal of W
W
Ware
equal to −1 (by design). Matrix X
X
X∈Rm×ncontains a set
of mmeasurement samples of the signals in x(k)and it
is used to identify the unknown coefficients of W
W
Win (5).
All signals in X
X
Xare preventively normalized so that the n
columns of X
X
Xhave zero mean and unit variance.
3.1. Sensor faults modelling
In this study, we considered independent multiple addi-
tive sensor faults. These are modelled as additive signals
fi(k)that corrupt the fault-free signals xi(k).
In the presence of faults, the faulty signals replace the
fault-free signals
x(k)←x(k) + F(k),(7)
where F(k)∈Rnis the fault vector whose components
are fault modelling functions fi(k). For example, a double
fault on the i-th and j-th sensors is expressed as F(k) = [0
... 0 fi(k)0 ... 0 fj(k)0 ... 0]T.
4. THE FAULT DETECTION SUBSYSTEM
In this study, the FD is based on the approach proposed
in [30], where the FD is based on the diagnostic signal
eD(k)defined as
eD(k) = x(k)Tψ,(8)
where the vector ψ∈Rnis a design vector such that, in
faulty free conditions, results
eD(k) = 0,(9)
and eD(k)6=0 in a faulty condition. Equations (8) and (9)
are called, in the literature, parity equations. In practice,
due to noise and modelling error, eD(k)is never zero also
in fault-free condition. Therefore, in the case |eD(k)|ex-
ceeds a fixed detection threshold (T hD), the vector of mon-
itored signals are considered faulty, otherwise normal, that
is
|eD(k)| ≤ T hD=⇒normal,
|eD(k)| ≥ T hD=⇒anomaly detection.(10)

Robust Multiple Fault Isolation Based on Partial-orthogonality Criteria 2151
The vector ψis determined by performing the Singular
Value Decomposition (SVD) of the data matrix X
X
X[31],
that is
X
X
X=U
U
UΛ
Λ
ΛV
V
VT,(11)
where U
U
U∈Rm×mand V
V
V∈Rn×nare the left and right or-
thonormal singular vectors matrices respectively, and Λ
Λ
Λ∈
Rm×nis a rectangular diagonal matrix whose elements λi
are the square root of ordered eigenvalues (λ1> ... > λn)
of the matrix X
X
XTX
X
X, and the last m−nrows of Λ
Λ
Λare equal
to 0. As explained in [32], there exists a versor (in this case
ψ) belonging to the Right Null Space (RNS) of X
X
Xand de-
fined as the last column vector vnof V
V
Vsuch that X
X
Xψ≈0.
This leads to the assignment
ψ=vn.(12)
In other words, ψis the eigenvector associated to the
smallest eigenvalue of the data covariance matrix.
In this study, the threshold ThDwas derived from the
experimental Cumulative Distribution Function (CDF) of
|eD(k)|applying the procedure proposed in [33]
T hD=min(|eD(k)|):CDF(|eD(k)|)≥(1−P
F),
(13)
where P
Fis the tolerable false alarm probability, that is,
the probability that |eD(k)|exceeds the detection thresh-
olds in fault-free condition and CDF(|eD(k)|)is the ex-
perimental CDF of |eD(k)|derived in fault-free condition.
In this study, we selected P
F=5%.
Note 1: In the following experiments, two FD schemes
have been considered to be consistent with the SoA stud-
ies reported in [22,23]. In the first scheme, named “Ideal
Detection”, the FD is assumed ideal; that is, any fault is
detected regardless of the actual FD threshold crossing,
while in the second scheme, the FD operates consistently
with the FD logic in (10).
5. THE PRE-FAULT ISOLATION SUBSYSTEM
When the FD subsystem detects a fault, then the pre-
FI procedure is activated. This block aims to select a re-
stricted subset of candidate sensors that may be responsi-
ble for the FD. This goal is achieved through the voting
logic described hereafter. The pre-FI is based on a redun-
dant set of specifically designed MARRs or parity equa-
tions. In this study, MARRs were generated directly from
data computing a number (ne) of linear models having a
multivariate structure as the one in (5). A generic parity
equation is expressed in the usual form as
ePi(k) = mix(k) = 0,i=1, ..., ne,(14)
where the regressor vector mi∈Rnis designed exclud-
ing (zeroing) some ad-hoc components of x(k)in order
to generate a redundant set of parity equations with de-
sired fault sensitivities (see the procedure in Subsection
5.2). An equal number of ARRs is designed for each mon-
itored sensor resulting in a total number of neMARRs that
are grouped in the matrix M
M
M= [m1;m2;... ;mne]∈Rn×ne.
The vector of pre-Isolation residuals eP(k)∈Rneresulting
from the nepre-isolation parity equations is
eP(k) = M
M
MTx(k).(15)
5.1. Decision logic for selecting suspect faulty sen-
sors
In the time intervals when the FD block detects the occur-
rence of a fault, the pre-FI block computes a subset of po-
tential faulty sensors through the following decision logic
applied to the components of the eP(k)parity vector.
In case the i-th component ePi(k)of eP(k)exceeds pre-
defined pre-Isolation thresholds, that is
ePi(k)≤T hPL,i∨ePi (k)≥T hPU,i,(16)
then the sensor associated with the parity equation ePi(k)
gets a vote. The thresholds in (16) were derived using the
same method used for the FD thresholds in (13), that is
T hPU,i=min(ePi(k)) :CDF(ePi (k)) ≥(1−P
F),
i=1, ..., ne,(17)
T hPL,i=max(ePi(k)) :CDF (ePi (k)) ≤P
F,
i=1, ..., ne.(18)
The probability P
Fthat has been used in (17) and (18)
is equal to 5%. The pre-Isolation decision logic computes
the ratio between the number of votes received by the in-
dividual sensors and the maximum number of votes that
could be theoretically assigned to that sensor (this num-
ber is equal to the number of parity equations associated
with each sensor). In case this ratio is larger than a prede-
fined value <1 (0.5 in this study), then the corresponding
sensor is considered potentially faulty, and it is added to
the set IS(k)that contains the list of the suspect faulty sen-
sors; otherwise, the sensor is considered in normal work-
ing conditions. The number of pre isolated faults in IS(k)
at time k, is η(k), where η(k)≤n. In case the subset IS(k)
is empty, it means that no sensor reaches the quorum of
votes; in this case, the FD is interpreted as a false alarm,
and the FI procedure is not activated.
Note 2: The number of votes requested to include a
sensor in the set IS(k)is an important design parameter.
The larger this number is, the more selective is the selec-
tion; the lower this number is, the more sensitive is the
method. In the experiment section, we observed that fix-
ing the number of votes equal to 50% of the number of
MARRs assigned to a sensor implies that the actual faulty
sensors are included in the suspect sensors set with a very
high probability.

2152 Nicholas Cartocci, Francesco Crocetti, Gabriele Costante, Paolo Valigi, and Mario L. Fravolini
5.2. MARRs generation
The set of the nepre-isolation parity equations were de-
signed taking inspiration from the dropout technique used
in Neural Networks (NN) [24] training, where a fraction
of the NN inner connection is randomly zeroed to improve
the NN robustness. For each monitored sensor i, we de-
rived (n−1)parity equations where each parity equation
contains (n−2)regressors, all nmeasurements with the
exclusion of the i-th sensor and another sensor among the
remaining ones. The coefficients mi,i=1, ...,neof these
parity equations were estimated using the Least-Squares
method from the data matrix X
X
X. The resulting number of
computed parity equations is ne=n·(n−1), so that matrix
M
M
M∈Rn×n·(n−1). This implies that the maximum number of
votes that could be assigned to each sensor is (n−1).
The logic used for building matrix M
M
Mcan be extended to
increase the number of MARRs by excluding, by design,
an increasing number of sensors from the parity equation.
In this way, it is possible to generate a larger number of
additional parity equations associated with a monitored
sensor. However, it is underlined that the parity equations
based on a restricted number of regressors may result in-
accurate due to the exclusion of potentially informative
regressors. In the application study presented in Section 7,
the removal of one measurement showed to be sufficient
to generate a set on ne=n·(n−1) = 30 ARRs for n=6
sensors.
6. THE MULTI-FAULT ISOLATION SUBSYSTEM
In the time intervals, the pre-FI procedure returns a no
empty set IS(k), the FI procedure is activated. In case IS(k)
returns a single sensor, the FI isolation is immediate, and
the pre-FI isolated sensor is considered faulty. Conversely,
if the number η(k)>1, then the FI algorithm of Subsec-
tion 6.1. is applied.
It is advantageous to restart from the parity equation in
(5) to understand the rationale of the FI algorithm, tak-
ing now expressly into account the additive fault vector
F(k) = [ f1(k),f2(k),... ,fn(k)]Tand the modelling un-
certainty ∆(k). The residual that originates from the parity
equations is
r(k) = W
W
W(x(k) + F(k)) + ∆(k).(19)
Considering that, by design, W
W
W x(k) = 0, then (19) be-
comes
r(k) = W
W
W F (k) + ∆(k) =
n
∑
i=1
wifi(k) + ∆(k),(20)
where wiis the i-th column vector of the matrix W
W
W, and
fi(k)is the fault on the i-th sensor. Clearly, in case of no-
fault on the i-th sensor, then fi(k) = 0. Considering (20),
it is apparent that r(k)depends on the linear combination
of the known column vectors wi, on the fault signals fi(k)
as long as the uncertainty ∆(k). In the literature, the resid-
ual r(k)is also known as the directional residual, and the
column vectors of the matrix W
W
Was fault directions or fault
signatures [27]. Exploiting this directional property, it is
theoretically possible to isolate a faulty sensor by compar-
ing the direction of the residual vector r(k)with the known
ndirections wiand assigning the fault to the sensor with
the closest angular distance from the r(k)direction.
It is apparent that this method is effective in case the
fault contribution wifi(k)in (20) is dominant compared
to ∆i(k). However, it is emphasized that modelling uncer-
tainty and noise limit the performance of any real fault
diagnosis scheme. In the ideal case of a single fault fi(k)
and zero uncertainty ∆(k), the angular distance between
r(k)and the fault direction wiis 0, while in a realistic sce-
nario, this distance depends on the fault amplitude, fault
sensitivity and modelling uncertainty.
In multiple faults case, the FI is more involved because
the r(k)vector is the linear combination of vectors with
unknown amplitudes (see (20)). For instance, it may oc-
cur that in case of multiple faults, the resulting direction of
r(k)is parallel to the direction of a no faulty sensor, caus-
ing a wrong FI. The above issues associated with direc-
tional residuals are well-known in the FI literature [34,35].
The proposed method relies on the data-driven design
of an optimized set of parity equations W
W
WO
O
Ox(k) = 0 that
are characterized by partial-orthogonal fault signatures in
W
W
WO
O
O∈Rn×nas long as minimum modelling error ∆(k).
These requirements are formalized in the following op-
timization problem
min
W
W
WO
O
O∈Rn×nl
s.t. 




W
W
WO
O
O(i,i) = −1,i=1, ..., n,
wOiTwO j =0,∀i6=j∈IS,

X
X
XW
W
WO
O
OT
F/√nm <l,
(21)
where k...kFis the Frobenius norm and
•The constraint W
W
WO
O
O(i,i) = −1 imposes that all the el-
ements in the diagonal are set equal to −1;
•The constraint wOiTwO j =0 imposes the orthogonal-
ity between vectors wOi and wO j;
•X
X
XW
W
WO
O
OT∈Rm×nis the fault-free residual signal gener-
ated by the experimental data and kX
X
XW
W
WO
O
OTkF/√nm is
the corresponding root mean square residual error;
•The cost function to be minimized is the upper bound
lof the root mean square residual error, while the free
decision variables are the off-diagonal elements of the
matrix W
W
WO
O
O.
The novelty in (21) is that the orthogonality condition
is imposed only to the subset of the η(k)fault direc-
tions identified by the pre-FI block (the subset of sensors
in IS(k)). Since W
W
WO
O
Oorthogonalizes only a subset of di-
rections, an ad-hoc matrix W
W
WO
O
O,{IS}is requested for each

Robust Multiple Fault Isolation Based on Partial-orthogonality Criteria 2153
possible combination of the faults in IS(k),for a total of
∑n
i=2n
i=2n−n−1 matrices. Since these matrices are
computed offline, their computation is not considered a
relevant issue for online operation.
6.1. Fault isolation logic
During the online operation, in case η(k)>1, the spe-
cific matrix W
W
WO
O
O,{IS(k)}is retrieved from the memory, and
the residual r(k) = W
W
WO
O
O,{IS(k)}x(k)is calculated as long as
the angular distances between r(k)and the fault directions
given by the columns of W
W
WO
O
O,{IS(k)}.
Assuming a negligible ∆i(k)compared to the amplitude
of the faults, it results, by construction, that r(k)is orthog-
onal (90◦) to the directions of all the non-faulty sensors in
IS(k). Moreover, in the case of a single fault, r(k)is alien-
ated to the actual fault direction (0◦), while in the case of
multiple faults, r(k)has components 6=0 (angles <90◦)
along the directions of the actual faults.
As for fault isolation, to be consistent with the assump-
tion made in the SoA approaches evaluated in Section 8,
it was assumed that the number NFof faults to be iso-
lated is known. Exploiting this information, the multi-fault
isolation logic was implemented by ranking the angular
distances and considering as faulty the NFsensors with
smaller angular distance.
Note 3: In case the number of the faulty sensors NF
is larger than the cardinality η(k)of IS(k), then, it is not
possible to isolate all the faulty sensors. In this case, the
FI scheme returns only a partial subset of the faulty sen-
sors, and therefore the FI will be classified as wrong in the
experiments of Section 8 because not all the faulty sensors
are isolated correctly.
7. THE BENCHMARK MODEL AND FD, PRE-FI
AND FI BLOCKS DESIGN
In this section, the performance of our approach is eval-
uated and compared with those of SoA methods. For this
purpose, Normal Operating Conditions (NOC) data were
generated using the following linear benchmark model al-
ready used in [20–26] for the evaluation of FDi schemes
in case of n=6 monitored sensors
x=








−0.3441 0.4815 0.6637
−0.2313 −0.5936 0.3545
−0.5060 0.2495 0.0739
0.5552 −0.2405 −0.1123
0.3371 0.3822 −0.6115
−0.3877 −0.3868 −0.2045










t1
t2
t3

+noise,
(22)
where variables t1,t2, and t3are uniformly distributed ran-
dom signals in the ranges [0, 2],[0, 1.6], and [0, 1.2], re-
spectively; the noise term in (22) is white and Gaussian
with zero mean and standard deviation 0.2. The number
of generated training and validation samples was 20000
and 10000, respectively.
7.1. Design of the FD, pre-FI and FI subsystems
This section describes the design of the three blocks that
constitute the FDi scheme in Fig. 1. Specifically, the FD
vector was computed according to the method in (12), re-
sulting in ψ= [0.44 0.25 0.32 0.65 0.38 0.28]T; the cor-
responding fault detection threshold, derived through (13),
resulted equal to T hD=0.90.
The pre-FI block was designed following the approach
of Section 5, providing a total of n(n−1) = 30 ARRs.
This implies that 5 parity equations are associated to each
one of the n=6 monitored sensors. The pre-FI thresholds
vectors T hPU and T hPL defined in (17) and (18) respec-
tively, were derived from the training data applying the
same procedure used for the computation of T hD.
7.2. Computational complexity of the FI block
The Fault-Isolation block required the offline design of
∑n
i=2n
i=57 matrices W
W
WO
O
Odefined in (21). The total num-
ber of coefficients associated with the 57 matrices W
W
WO
O
Ois
of order 2nn2. Although this number grows exponentially
with n, the offline computational load and memory stor-
age requirement are not crucial, indeed, with 6 monitored
sensors, we estimated about 4.5 KB of memory occupa-
tion and about two minutes of computation time to solve
the problems (21) using an Intel Core i7-7500U CPU with
8 GB of RAM. The online computational load is deemed
not a critical factor since the computations requested in
the functional blocks of Fig. 1 involve simple matrix mul-
tiplications and threshold comparisons.
8. VALIDATION RESULTS
The first study was focused on the validation of the FDi
scheme in the case of a single fault occurring on the sen-
sor x1that is modelled as a constant additive bias. In the
second and third validation studies, random faults were in-
jected to reproduce the same test scenario used in the SoA
studies [22–24], thus allowing a quantitative performance
comparison between the techniques.
Performances were evaluated using three metrics: 1) the
True Detection Rate (TDR), which is ratio between the
number of samples when a failure is detected and the ac-
tual number of faulty samples; 2) the True Isolation Rate
(TIR), which is ratio between the number of samples when
the faults are correctly isolated and the number of the
faulty samples; 3) the True Normal Rate (TNR) that is ra-
tio between number of samples detected as non-faulty and
the number of non-faulty samples.
8.1. Step additive fault on the x1sensor
Fault free train data were generated using model (22);
this allowed the derivation of the standard deviation of the

2154 Nicholas Cartocci, Francesco Crocetti, Gabriele Costante, Paolo Valigi, and Mario L. Fravolini
Fig. 2. Fault detection signal and threshold (step fault on
x1sensor starting from sample 5001).
Fig. 3. Voting scores in pre-Isolation phase for the 6 sen-
sors (step fault on x1sensor starting from sample
5001).
signal x1that resulted σx1=0.4264. Then, a step fault of
amplitude 2.5σx1was injected between observations 5001
and 10000 to the x1signal in the testing dataset [21].
In Fig. 2, the FD signal |eD(k)|and the FD threshold
T hDare shown. It is observed that the FD signal keeps
under the threshold most of the time in the fault-free time
intervals [1, 5000], while it crosses the FD threshold im-
mediately following the fault injection starting from sam-
ple 5001.
In the sampling interval where the FD threshold is ex-
ceeded, the pre-Fault Isolation procedure is activated, and
the voting logic is applied to the 30 pre-FI parity equations
producing the results in Fig. 3 where, for each sensor, the
voting score is reported as a function of k.
Analysing the figure, in the fault-free time interval (k<
5000), 791 of the 1161 samples that exceed the FD thresh-
old do not exceed the pre-FI thresholds, and so do not
receive any votes; while it is evident that following the
fault injection (for k>5000) the x1variable receives a
large number of votes totalizing a score larger than the
pre-isolation threshold that is set to 2.5 votes (the 50% of
the maximum number of votes equal to n−1=5) and
therefore, the variable x1, most of the time, is included in
Fig. 4. The angle (degree) between the direction of the
residual r(k)and the six fault signatures (step fault
on x1sensor starting from sample 5001).
Table 1. Percent FD and FI performances in case of a sin-
gle fault (step fault on xiof the amplitude 2.5σxi).
Sensor x1x2x3x4x5x6
fi1.06 0.96 0.93 0.99 0.98 0.89
TDR [%] 80.6 51.3 64.46 95.82 72.16 55.64
TIR [%] 76.16 46.54 61.64 87.64 67.28 52.22
TNR [%] 95.12
the set IS(k)that contains the pre-Isolated faults; the same
voting procedure is applied to the other sensors.
To better understand the operation of the FI procedure,
in Fig. 4, the evolution of the angle between the direction
of the residual r(k)and the fault signatures derived from
the W
W
WO
O
O,{IS(k)}matrices is shown. Before the fault injection
(k<5000), there are few sporadic false positives, i.e., 244
samples where some sensors totalize a score larger than
the pre-isolation threshold; while after the fault injection,
it is evident that the angle between r(k)and the direction
of x1is significantly smaller than the angle between r(k)
and the other (orthogonal) directions. After the fault injec-
tion, the average angles between r(k)and the faults direc-
tions are
]=11.79
11.79
11.79 87.7 87.64 86.68 86.86 84.61
implying that the fault is correctly attributed to the sensor
x1.
The same experiment was then repeated, injecting sin-
gle faults of amplitudes fi=2.5σxi,i=1, ..., nto all the
other sensors achieving the performance reported in Table
1. It is observed that the TDR and TIR vary significantly
among the 6 sensors.
It is very important to note that the TDR and TIR differ-
ence is in the range of 3-4%. This result highlights the effi-
cacy of the proposed FI scheme that is able to isolate most
of the detected faults correctly. It seems reasonable to ar-
gue that the low TDR achieved for sensors x2,x3and x6
could be improved using less conservative FD thresholds.

Robust Multiple Fault Isolation Based on Partial-orthogonality Criteria 2155
The TNR is obviously the same for all sensors resulting
equal to 95.12%.
8.2. Random faults with increasing fault magnitude
In this study, the test scenario proposed in [22] was re-
produced to compare the performance for single and dou-
ble faults. Specifically, increasing amplitude faults in the
range [0, 2]have been generated and added (each sampling
time) to the randomly selected sensor. For each amplitude,
10000 faults have been generated.
Fig. 5 shows the TIR performance of our POC FDi
scheme compared to the technique based on PDCs of the
T2statistics proposed in [23] in case of single sensor fault;
similarly, Fig. 6 shows the comparison in case of double
simultaneous sensor faults.
As pointed out in [23], the TIR scores are significantly
Fig. 5. TIR in case of single sensor fault. PDCs-based
technique assuming Ideal Detection (red); POC
scheme considering Ideal Detection (blue); POC
scheme using the threshold-based FD approach of
Section 4 (green).
Fig. 6. TIR in case of double sensor fault: PDCs-based
technique assuming Ideal Detection (red), Partial-
Orthogonality scheme assuming Ideal Detection
(blue) and with the proposed threshold-based De-
tection (green).
influenced by the result of the FD stage; thus, to elim-
inate the effect of threshold-based FD on the FI perfor-
mance, we assumed a perfect detection as clarified in Note
1. For this reason, in Figs. 5 and 6, we used the acronyms
“PDC-ID” for the PDCs-based technique assuming Ideal
Detection and “POC-ID” for the POC technique consider-
ing Ideal Detection and “POC” for the POC technique us-
ing the actual threshold-based FD introduced in Section 4.
Analysing Fig. 5 results that the PDC approach per-
forms slightly better than our methods for single fault am-
plitude less than 0.4, while it is evident that our two meth-
ods (assuming both Ideal and realistic FD) perform sig-
nificantly better for larger amplitudes. As expected, POC
assuming Ideal Detection (POC-ID) always performs bet-
ter than the POC method based on realistic FD; instead,
it is worth noting that POC performs better than the PDC
technique for fault amplitudes larger than 0.6 even if PDC
assumes perfect fault detection.
These positive trends have also been confirmed in the
case of double contemporary faults, as clearly evident in
Fig. 6. A marked improvement in the TIR between our
methods and the PDC method is also observed. It is also
observed that the POC and POC-ID performance are sim-
ilar because, in the case of double faults, it is easier for the
FD block to detect a failure in the presence of two con-
temporary faults.
8.3. Random faults with random fault amplitude
In this experiment, multiple random amplitude faults
are randomly injected on the six sensors, and the results
are compared with those of the SoA methods in [24]
and described in Section 1. The faults are randomly (uni-
formly) selected between the six sensors; similarly, the
fault amplitudes are randomly sampled from a uniform
distribution in the range [0, 5]. This validation study was
performed using 10000 samples.
Table 2 compares, in the case of a single sensor fault,
the TDRs and TIRs performance of the POC approach and
those provided by the SoA approach [24] that is based on
the SPE, T2and kNN indices (The acronyms used refer to
those introduced in Section 1). Table 3 reports the same
indices in case of two contemporary sensor faults. Finally,
Table 4 reports the TDRs and TIRs in case of multiple
contemporary faults only for the POC technique.
Analysing the results in the tables, it is apparent that
the POC method performs better than the considered SoA
Table 2. TDRs and TIRs in case of single fault.
Index SPE T2ΦkNN POC
TDR (%) 79.9 54.35 80.20 80.50 87.69
TIR
(%)
CDC 57.6 42 79.8
80 84.96
PDC 79.15 54.3 79.65
RBC 77.1 33.6 79.9

2156 Nicholas Cartocci, Francesco Crocetti, Gabriele Costante, Paolo Valigi, and Mario L. Fravolini
Table 3. TDRs and TIRs in case of double fault.
Index SPE T2ΦkNN POC
TDR (%) 95.45 82.85 96.40 96.35 98.56
TIR
(%)
CDC 7 4.4 8.95
70.05 79.10
PDC 56.45 58.2 64.10
RBC 18 26.75 60.85
Table 4. TDRs and TIRs in case of multiple sensor faults
using POC.
Triple Quadruple Quintuple All
TDR (%) 99.84 99.99 99.99 100
TIR (%) 70.29 67.96 71.29 95.01
methods either for single, double, and multiple faults. The
TIR improvement is about 5% for a single fault, while this
is about 9% for double faults. Our TDR is nearly perfect
for multiple faults, while the TIR is close to 70%.
9. CONCLUSIONS
A data-driven Fault Diagnosis scheme based on Partial-
Orthogonality Criteria for the robust isolation of multiple
sensor faults has been presented. The diagnostic system
consists of the cascade connection of functional blocks,
all designed using data-driven approaches. Following a
generic Fault Detection, a Pre-Isolation block is employed
to select a restricted number of potentially faulty sen-
sors. In this block, the combination of redundant Multiple-
ARRs in conjunction with a voting logic has shown to
be particularly effective in selecting a subset of candi-
date faulty sensors to be processed by the successive Fault
Isolation block. The Fault isolation block contains a col-
lection of partial-orthogonality parity relations (residuals)
specifically designed to limit the so-called smearing ef-
fect, thus improving robustness. Validation experiments
based on a benchmark system have clearly highlighted,
in three different fault scenarios, that using the proposed
POC technique, it is possible to enhance both the fault
TDR and the TIR significantly compared to SoA tech-
niques.
On the other side, a drawback of the method is the need
for an offline phase to compute optimized POC parity re-
lations for all the combinations of the faulty variables. To
limit the number of POC optimizations with the increase
of the monitored variables, future research will focus on
applying hierarchical clusterization and decision-making
techniques of similar faulty patterns.
REFERENCES
[1] N. B. Hoang and H. J. Kang, “A model-based fault diagno-
sis scheme for wheeled mobile robots,” International Jour-
nal of Control, Automation, and Systems, vol. 12, no. 3, pp.
637-651, May 2014.
[2] Z. Gao, C. Cecati, and S. X. Ding, “A survey of fault diag-
nosis and fault-tolerant techniques Part I: Fault diagnosis,”
IEEE Transactions on Industrial Electronics, vol. 62, no.
6, pp. 3768-3774, 2015.
[3] E. Y. Chow and A. S. Willsky, “Analytical redundancy
and the design of robust failure detection systems,” IEEE
Transactions on Automatic Control, vol. 29, no. 7, pp. 603-
614, 1984.
[4] N. Cartocci, G. Costante, M. R. Napolitano, P. Valigi, F.
Crocetti, and M. L. Fravolini, “PCA methods and evidence
based filtering for robust aircraft sensor fault diagnosis,”
Proc. of 28th Mediterranean Conference on Control and
Automation, MED 2020, pp. 550-555, September 2020.
[5] I. Gueddi, O. Nasri, K. Benothman, and P. Dague, “Fault
detection and isolation of spacecraft thrusters using an ex-
tended principal component analysis to interval data,” In-
ternational Journal of Control, Automation, and Systems,
vol. 15, no. 2, pp. 776-789, April 2017.
[6] A. Benaicha, G. Mourot, K. Benothman, and J. Ragot, “De-
termination of principal component analysis models for
sensor fault detection and isolation,” International Journal
of Control, Automation, and Systems, vol. 11, no. 2, pp.
296-305, April 2013.
[7] Y. Lei, B. Yang, X. Jiang, F. Jia, N. Li, and A. K. Nandi,
“Applications of machine learning to machine fault diag-
nosis: A review and roadmap,” Mechanical Systems and
Signal Processing, vol. 138. p. 106587, April 2020.
[8] N. Cartocci, M. R. Napolitano, F. Crocetti, G. Costante,
P. Valigi, and M. L. Fravolini, “Data-driven fault diagno-
sis techniques: Non-linear directional residual vs. machine-
learning-based methods,” Sensors, vol. 22, no. 7, p. 2635,
2022.
[9] X. J. Li and G. H. Yang, “Fault detection in finite fre-
quency domain for Takagi-Sugeno fuzzy systems with sen-
sor faults,” IEEE Transactions on Cybernetics, vol. 44, no.
8, pp. 1446-1458, 2014.
[10] X. J. Li and X. Y. Shen, “A data-driven attack detection
approach for DC servo motor systems based on mixed op-
timization strategy,” IEEE Transactions on Industrial In-
formatics, vol. 16, no. 9, pp. 5806-5813, September 2020.
[11] K. Shi, J. Wang, Y. Tang, and S. Zhong, “Reliable asyn-
chronous sampled-data filtering of T-S fuzzy uncertain de-
layed neural networks with stochastic switched topolo-
gies,” Fuzzy Sets and Systems, vol. 381, pp. 1-25, February
2020.
[12] D. Jung, Y. Dong, E. Frisk, M. Krysander, and G. Biswas,
“Sensor selection for fault diagnosis in uncertain systems,”
International Journal of Control, vol. 93, no. 3, pp. 629-
639, March 2020.
[13] D. Jung and E. Frisk, “Residual selection for fault detection
and isolation using convex optimization,” Automatica, vol.
97, pp. 143-149, 2018.

Robust Multiple Fault Isolation Based on Partial-orthogonality Criteria 2157
[14] X. Yang, Z. Li, Q. Zhang, Q. Wu, and L. Yang, “A non-
linear adaptive observer-based differential evolution al-
gorithm to multiparameter fault diagnosis,” Mathematical
Problems in Engineering, vol. 2020, Article ID 4531075,
2020.
[15] N. Cartocci, M. R. Napolitano, G. Costante, P. Valigi, and
M. L. Fravolini, “Aircraft robust data-driven multiple sen-
sor fault diagnosis based on optimality criteria,” Mechan-
ical Systems and Signal Processing, vol. 170, p. 108668,
2022.
[16] D. Zhu, J. Bai, and S. Yang, “A multi-fault diagnosis
method for sensor systems based on principle component
analysis,” Sensors, vol. 10, no. 1, pp. 241-253, December
2009.
[17] Z. Li, X. Yan, C. Yuan, Z. Peng, and L. Li, “Virtual proto-
type and experimental research on gear multi-fault diagno-
sis using wavelet-autoregressive model and principal com-
ponent analysis method,” Mechanical Systems and Signal
Processing, vol. 25, no. 7, pp. 2589-2607, October 2011.
[18] H. Jiang, C. Li, and H. Li, “An improved EEMD with mul-
tiwavelet packet for rotating machinery multi-fault diagno-
sis,” Mechanical Systems and Signal Processing, vol. 36,
no. 2, pp. 225-239, April 2013.
[19] H. Chen, W. Huang, J. Huang, C. Cao, L. Yang, Y. He,
and L. Zeng, “Multi-fault condition monitoring of slurry
pump with principle component analysis and sequential
hypothesis test,” International Journal of Pattern Recogni-
tion and Artificial Intelligence, vol. 34, no. 7, p. 2059019,
June 2020.
[20] C. F. Alcala and S. J. Qin, “Analysis and generalization of
fault diagnosis methods for process monitoring,” Journal of
Process Control, vol. 21, no. 3, pp. 322-330, March 2011.
[21] M. Z. Sheriff, N. Basha, M. N. Karim, H. Nounou, and
M. Nounou, “Fault detection of single and interval valued
data using statistical process monitoring techniques,” Fault
Detection, Diagnosis and Prognosis, IntechOpen, 2019.
[22] P. Van den Kerkhof, J. Vanlaer, G. Gins, and J. F. M. Van
Impe, “Contribution plots for statistical process control:
Analysis of the smearing-out effect,” Proc. of European
Control Conference, ECC 2013, pp. 428-433, 2013.
[23] P. Van den Kerkhof, J. Vanlaer, G. Gins, and J. F. M. Van
Impe, “Analysis of smearing-out in contribution plot based
fault isolation for statistical process control,” Chemical En-
gineering Science, vol. 104, pp. 285-293, December 2013.
[24] Z. Zhou, C. Wen, and C. Yang, “Fault isolation based
on κ-nearest neighbor rule for industrial processes,” IEEE
Transactions on Industrial Electronics, vol. 63, no. 4, pp.
2578-2586, April 2016.
[25] J. Yang, Z. Sun, and Y. Chen, “Fault detection using the
clustering-kNN rule for gas sensor arrays,” Sensors, vol.
16, no. 12, p. 2069, December 2016.
[26] Z. Zhou, C. Yang, C. Wen, and J. Zhang, “Analysis of prin-
cipal component analysis-based reconstruction method for
fault diagnosis,” Industrial & Engineering Chemistry Re-
search, vol. 55, no. 27, pp. 7402-7410, July 2016.
[27] N. Cartocci, M. R. Napolitano, G. Costante, and M. L.
Fravolini, “A comprehensive case study of data-driven
methods for robust aircraft sensor fault isolation,” Sensors,
vol. 21, no. 5, p. 1645, February 2021.
[28] N. Cartocci, F. Crocetti, G. Costante, P. Valigi, M. R.
Napolitano, and M. L. Fravolini, “Data-driven sensor fault
diagnosis based on nonlinear additive models and local
fault sensitivity,” Proc. of 20th International Conference
on Advanced Robotics, 2021.
[29] J. Gertler, “Fault detection and isolation using parity re-
lations,” Control Engineering Practice, vol. 5, no. 5, pp.
653-661, 1997.
[30] N. Cartocci, G. Costante, M. R. Napolitano, P. Valigi, F.
Crocetti, and M. L. Fravolini, “A robust data-driven fault
diagnosis scheme based on recursive Dempster-Shafer
combination rule,” Proc. of 29th Mediterranean Confer-
ence on Control and Automation (MED), 2021.
[31] Y. Jiang, S. Yin, and O. Kaynak, “Optimized design of
parity relation-based residual generator for fault detection:
Data-driven approaches,” IEEE Transactions on Industrial
Informatics, vol. 17, no. 2, pp. 1449-1458, February 2021.
[32] B. Ochoa, “The null space of a matrix left null space,”
2015.
[33] M. L. Fravolini, M. R. Napolitano, G. Del Core, and U.
Papa, “Experimental interval models for the robust fault
detection of aircraft air data sensors,” Control Engineering
Practice, vol. 78, pp. 196-212, January 2018.
[34] Y. Hu and J. Gertler, “Design of optimal directional resid-
uals for linear dynamic systems,” IFAC Proceedings Vol-
umes, vol. 36, no. 5, pp. 245-250, 2003.
[35] Y. Hu and J. Gertler, “Design of directional residuals for
optimal testability,” IFAC Proceedings Volumes, vol. 15,
no. 1, pp. 131-136, 2002.
Nicholas Cartocci received his Master’s
degree in computer science and robotics
engineering from the University of Peru-
gia, Perugia, Italy, in 2019. From Octo-
ber 2018 to March 2019, he was a Visiting
Scholar in the Department of Mechanical
and Aerospace Engineering, West Virginia
University, Morgantown, WV, USA. He is
currently a research assistant at the Univer-
sity of Perugia.
Francesco Crocetti received his M.Sc.
degree in information and automation
engineering in 2018 from the Univer-
sity of Perugia, where he is currently a
Ph.D. student. His research interests are
mainly automation, control, fault diagno-
sis, robotics, and machine learning.

2158 Nicholas Cartocci, Francesco Crocetti, Gabriele Costante, Paolo Valigi, and Mario L. Fravolini
Gabriele Costante received his Ph.D. de-
gree in information engineering from the
Department of Engineering, University of
Perugia, Perugia, Italy, in 2016. He is
currently a Postdoctoral Researcher with
the Intelligent Systems, Automation and
Robotics Laboratory (ISARLab) and a
Lecturer of computer vision and robot per-
ception, machine learning, and data anal-
ysis with the Department of Engineering, University of Peru-
gia. His research interests include artificial intelligence, robotics,
computer vision, and machine learning.
Paolo Valigi received his Ph.D. degree
from the Tor Vergata University of Rome
in 1991. Since 2004 he is a full profes-
sor with the University of Perugia. His re-
search interests are in the field of auto-
matic control, robotics, and systems biol-
ogy.
Mario L. Fravolini received his Ph.D. de-
gree in electronic engineering from the
University of Perugia in 2000. After that,
he worked as a research assistant with the
Georgia Institute of Technology and with
West Virginia University. Currently, he is
an associate professor with the Univer-
sity of Perugia. His research interests in-
clude fault diagnosis, intelligent and adap-
tive control, and biomedical imaging.
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