Sensor Fault Diagnosis Observer for an Electric Vehicle Modeled as a Takagi-Sugeno System

Abstract

A sensor fault diagnosis of an electric vehicle (EV) modeled as a Takagi-Sugeno (TS) system is proposed. The proposed TS model considers the nonlinearity of the longitudinal velocity of the vehicle and parametric variation induced by the slope of the road; these considerations allow to obtain a mathematical model that represents the vehicle for a wide range of speeds and different terrain conditions. First, a virtual sensor represented by a TS state observer is developed. Sufficient conditions are given by a set of linear matrix inequalities (LMIs) that guarantee asymptotic convergence of the TS observer. Second, the work is extended to perform fault detection and isolation based on a generalized observer scheme (GOS). Numerical simulations are presented to show the performance and applicability of the proposed method.

Full text

Research Article
Sensor Fault Diagnosis Observer for an Electric Vehicle
Modeled as a Takagi-Sugeno System
S. Gómez-Peñate,
1
F. R. López-Estrada ,
1
G. Valencia-Palomo ,
2
R. Osornio-Ríos ,
3
J. A. Zepeda-Hernández,
1
C. Ríos-Rojas,
1
and J. Camas-Anzueto
1
1
Tecnológico Nacional de México (TecNM)/Instituto Tecnológico de Tuxtla Gutiérrez, TURIX Dynamics-Diagnosis and Control
Group, Carretera Panam, km 1080, Tuxtla Gutiérrez, CHIS, Mexico
2
Tecnológico Nacional de México (TecNM)/Instituto Tecnológico de Hermosillo, TURIX-Hermosillo, Av. Tecnológico y Periférico
Poniente S/N, 83170 Hermosillo, SON, Mexico
3
HSPdigital-CA Mecatrónica, Facultad de Ingeniería Campus San Juan del Río, Universidad Autónoma de Querétaro, Río
Moctezuma 249, San Cayetano, 76807 San Juan del Río, QRO, Mexico
Correspondence should be addressed to F. R. López-Estrada; frlopez@ittg.edu.mx
Received 25 August 2017; Revised 28 November 2017; Accepted 4 December 2017; Published 28 March 2018
Academic Editor: Jing Xu
Copyright © 2018 S. Gómez-Peñate et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
A sensor fault diagnosis of an electric vehicle (EV) modeled as a Takagi-Sugeno (TS) system is proposed. The proposed TS model
considers the nonlinearity of the longitudinal velocity of the vehicle and parametric variation induced by the slope of the road; these
considerations allow to obtain a mathematical model that represents the vehicle for a wide range of speeds and different terrain
conditions. First, a virtual sensor represented by a TS state observer is developed. Sufficient conditions are given by a set of
linear matrix inequalities (LMIs) that guarantee asymptotic convergence of the TS observer. Second, the work is extended to
perform fault detection and isolation based on a generalized observer scheme (GOS). Numerical simulations are presented to
show the performance and applicability of the proposed method.
1. Introduction
In recent years, there has been a substantial increase in the
number of electric vehicles (EV), due to the increase of
pollution by CO2emissions to the environment. Recent
studies show that currently there are more than 800 million
cars circulating every day, which represent a distribution
of 400 to 800 vehicles per 1000 inhabitants [1]. As a result,
vehicles are responsible for a high percentage of global energy
consumption and greenhouse gas emissions. This tendency
shows an accelerated growth of vehicles per inhabitants,
and, while the energy consumption in other sectors
decreases, the consumption due to the continuous use of
transport vehicles grows [2]. On the other hand, new sensor
systems and actuators on EV are increasing in complexity,
and the probability for a fault taking place is high [3].
For example, recently, a Tesla driver died in a crash while
using the autopilot mode because the car’s sensor system
failed to distinguish a large white 18-wheel truck and trailer
crossing the highway. This accident caused a severe crisis in
the EV industry. Therefore, safety, reliability, and energy-
saving optimization systems are a demand of the new grow-
ing industry. In line with this demand, this work is dedicated
to propose a method to detect and isolate sensor faults in an
electric vehicle.
An important stage in the design of the diagnosis system
is the mathematical model that represents the dynamic char-
acteristics of the EV, which is expressed by a set of nonlinear
differential equations depending on exogenous nonstation-
ary parameters [4], for example, slopes or poor conditions
of the road. However, typical models of EV consider a simpli-
fied representation given by linear models, in which it is not
Hindawi
Journal of Sensors
Volume 2018, Article ID 3291639, 9 pages
https://doi.org/10.1155/2018/3291639

possible to consider these nonstationary parameters [5–7].
Nonetheless, it is possible to obtain better representations
when nonstationary exogenous parameters, such as the road
slope, could be measured online, such that the desired diag-
nosis system, which also depends on these measurable
parameters, is better and less conservative [8]. In typical lin-
ear time-invariant systems, it is not possible to consider these
variations. However, a viable alternative is Takagi-Sugeno
models that consider nonlinearities and varying parameters
as part of the mathematical modeling [9, 10], which increases
the physical representativity of a real physical system.
The main advantage of a TS model is its capability of
describing nonlinear dynamics through a collection of
local linear models that are interpolated by nonlinear
functions [11]. These functions are known as weighting
functions and depend on exogenous variables that can be
measurable (e.g., system inputs, outputs, exogenous non-
stationary parameters, velocity, and the slope of the ter-
rain) or unmeasurable (e.g., state variables, magnetic flux,
and slip angle of the tires) [12, 13]. In this work, weight-
ing functions are considered measurable. Additionally, an
important property of this type of models is that the
weighting functions are convex, which allows to extend
some of the tools and methods developed for linear systems
to TS systems. In particular, it has been shown that a TS
dynamic model obtained through the nonlinear sector trans-
formation approach can describe the overall behavior of a
highly complex nonlinear system with a high degree of accu-
racy [11, 14]. As a result, its applicability in designing con-
trollers, diagnosis systems, and observers, among others,
has become of high importance; see, for instance, [15–17]
and references therein. Applications on vehicles can be found
in the literature; for example, in [4], a predictive control strat-
egy using a TS model is presented to control the velocity in an
EV, and the authors in [18] proposed a linear parameter
varying (LPV) controller in order to control the tracking of
the longitudinal velocity and the yaw velocity of the EV. In
[19], a TS fuzzy model is used to represent the nonlinear
behavior of an electric power steering (EPS) system, and sta-
bilization conditions for nonlinear EPS system with both
constrained and saturated control input cases are proposed
in terms of linear matrix inequalities. Some works related to
fault diagnosis can be consulted in the following references:
in [20], an observer design strategy is presented to estimate
the lateral dynamics of a vehicle and the curvature of the
road. The nonlinear model of vehicle dynamics is trans-
formed into an exact TS model with weighting functions
depending on unmeasured states. In [21], a TS observer is
designed to detect faults and estimate states in an induc-
tion motor, in order to implement a fault-tolerant control.
Recently, in [22], an observer was designed to estimate the
lateral dynamics of a motorcycle represented as a quasi-
LPV system. It is important to note that, in the works
reported in [4, 18, 21], the slope of the road is considered
constant or close to zero; nevertheless, in real driving con-
ditions, this parameter is not constant and has a great
impact on the vehicle performance and battery consump-
tion. Unlike previous papers, this work considers the slope
of the road in order to obtain an improved sensor fault
diagnosis system.
In this paper, we propose the design of an observer-based
fault diagnosis for an electric vehicle. The main contributions
of this paper are listed as follows: (i) a Takagi-Sugeno model
is developed, whose weighting functions depend on the
longitudinal velocity and the slope of the terrain in order
to increase the operation range of the diagnostic system;
(ii) sufficient conditions are proposed in order to guaran-
tee the asymptotic convergence of the observer that is
deduced through a quadratic Lyapunov function and a set
of linear matrix inequalities (LMIs); finally, (iii) a bank of
observers based on a generalized observer scheme is pro-
posed to detect and isolate sensor faults. The combination
of both techniques results in a scheme for detecting faults
in the traveled-distance and speed sensors at different operat-
ing and slope conditions.
The paper is organized as follows: in Section 2, the longi-
tudinal model of the electric vehicle is presented; in Section 3,
a TS model that uses the velocity and the slope as premise
variables is developed; the conditions of the designed
observer for the TS model of the EV are formulated in
Section 4; the numerical simulation results are presented
in Section 5. Finally, conclusions and perspectives of this
work are presented in Section 6.
2. Nonlinear Model of the Electric Vehicle
Figure 1 shows the general scheme of an EV, which is consti-
tuted mainly by an energy source (battery bank), a power
inverter, an electric motor, and a transmission system
coupled to the wheels. Considering Newton’s second law
and the translational equilibrium principle, the longitudinal
dynamics can be represented by (see Figure 2) [2]
mv
dx2t
dt =Ftt−Fat−Frt−Fgt, 1
Battery CD/CA
inverter
Electric
motor
Transmission
system
Wheel
Wheel
Figure 1: Scheme of an electric vehicle.
2 Journal of Sensors

where mv(kg) is the total mass of the vehicle, x2t(m/s)
is the speed of the vehicle, Ftt(N) is the traction force
on the wheels, Fat(N) is the aerodynamic force, Frt
(N) is the rolling resistance, Fgt(N) is the slope resistance
due to the vehicle’s weight and the road slope, and t(s) is
the time.
The traction force is given by
Ftt=ηktgr
rw
Ibat t, 2
where η=Im/Ibat is the efficiency of the power converter,
kt=Tm/Im(N·m/s
2
) is the motor constant, Im(A) is the
motor current, Tm(N·m) is the motor torque, Ibat t(A) is
the battery current, gr(m) is the transmission gear ratio,
and rw(m) the radius of the wheel.
It is important to note that, for this case, the motor
current Imis not available and the battery provides the
electric current to generate all the traction power, so that
the battery current Ibat tis considered as the system input.
The aerodynamic force is given by
Fat=1
2ρaAfCdx2
2t, 3
where ρa(kg/m
3
) is the air density (ρa=1225 k g/m
3
under
standard conditions), Af(m
2
) is the frontal area, and Cd
(—) is the aerodynamic drag coefficient.
The rolling resistance is given by
Frt=Crmvgcos αt, 4
where Cr(—) is the rolling resistance coefficient and αt
(rad) is the slope of the road that depends on the position
of the EV. However, in this work, it is considered time depen-
dent without losing generality. The coefficient of rolling fric-
tion is complex and depends on the speed of the vehicle and
the conditions of the tires and the terrain.
Finally, the gravitational force due to the slope of the road
and to the weight of the vehicle is
Fgt=mvgsin αt5
It is important to note that most of the reported work
[4, 18, 21] consider that the vehicle is traveling on a road
with a slope equal to zero or constant, which reduces the
complexity of the mathematical model due to the fact that
Frtand Fgtbecomes constant. Nonetheless, in real driv-
ing conditions, the vehicles have smooth and abrupt slope
changes. These changes can be measured with appropriate
sensors, for example, using inertial measurement units
(IMUs). By considering typical linear models, it is not
always possible to include the slope dynamics in the model.
However, variation of this parameter makes the nonlinear
system a candidate to be represented by a TS model. In
this case, the slope dynamics can be included as an exoge-
nous nonstationary parameter, which increases accuracy of
the model representation with respect to the real physical
system [23].
Substituting (2)–(5) into (1), the differential equation,
which represents the longitudinal dynamics, is given by
mv
dx2t
dt =ηktgr
rw
Ibat t−1
2ρaAfCdxt2
2
−Crmvgcos αt−mvgsin αt
6
For this mathematical model, the internal frictions, rota-
tional inertias of the power train, and the inertia of the elec-
tric motor are neglected because they are small compared to
the mass of the vehicle. The vehicle displacement x1tcan be
computed as
dx1t
dt =x2t7
Finally, the nonlinear model affine to the input is
given by
xt =
x2t
−ρaAfCdx2
2t
2mv
+
0
ηktgr
mvrw
Ibat t
+
0
−Crgcos αt−gsin αt
,
yt =Cx
1t,x2tT,
8
where y t is the system output. In the following section, the
TS model for the nonlinear model (8) is developed.
3. Takagi-Sugeno Model of the Electric Vehicle
A TS model describes a nonlinear system using a collection of
local linear time invariant (LTI) models, which are interpo-
lated by nonlinear functions known as weighting functions.
The general form of a TS model is given by
xt =〠
k
i=1
μiξt Aixt +Biut +Δi,
yt =Cx t ,
9
Fr
Fg
rw
mv · g
Ft
Fa

Figure 2: Longitudinal forces acting on an electric vehicle.
3Journal of Sensors

where x t ∈Rn,u t ∈Rm, and y t ∈Rpare the state, input,
and output vectors, respectively; Ai∈Rn×n,Bi∈Rn×m,Δi∈
Rn, and C∈Rp×nare known matrices of appropriated
dimensions; and kis the number of local models.
The weighting functions μiξtdepend on ξt, which
can represent the system states, measured inputs, or an exog-
enous measured signal, for example, the slope of the road.
The weighting functions satisfy the convex sum:
〠
k
i=1
μiξt=1, ∀i∈1, 2, …,k,μiξt≥0, ∀t10
There are three methods for obtaining a TS model. When
it is not possible to obtain analytical models by physical prin-
ciples, the most appropriate method is system identification
[24]. This method considers a structure of the TS model
and its weighting functions. Then, the problem is reduced
to identify the parameters of the subsystems defined as local
linear models and the parameters of the activation functions
using numerical optimization algorithms. However, when
the analytical equations of the nonlinear system is available,
a TS model can be obtained by linearizing around different
selected operation points [25]. Nonetheless, the main draw-
back of this method is that is not easy to find such operating
points, which are usually computed by heuristic methods.
Finally, the most appropriate method is called the nonlinear
sector transformation [26], which is based on the following
idea: consider a simple nonlinear system x=f x t , with
f0=0, where the goal is to find a global sector such that
x=f x t ∈a1,a2x t ; see Figure 3(a). This approach
guarantees an accurate approximation of the nonlinear
system; however, it is difficult to find a global sector to
enclose the nonlinear terms, and generally, a local sector is
considered. This is reasonable since the variables of the phys-
ical system are always bounded. Figure 3(b) shows the local
nonlinear sector, where two lines define a local sector
between ξ<x t <ξ. The model TS represents the nonlinear
system in the local region, given by ξ<x t <ξ.
In order to derive a TS model, the nonlinear model of the
EV in (8) is represented in matrix format, as follows:
xt =
01
0−1
2mv
ρaAfCdx2t
x1t
x2t
+
0
ηktgr
mvrw
Ibat t
+
0
−Crgcos αt−gsin αt
11
Two premise variables are considered, the nonlinear term
x2tand the slope of the road αt. For x2t, a minimum
velocity of 0 m/s and a maximum of 16.66 m/s (approxi-
mately 60 km/h) are considered; and for αt, the minimum
and maximum slopes of −π/9 and π/9 (−20 and 20, resp.)
are considered, such that
0≤x2t≤16 66,
−π
9≤at ≤π
9
12
The premise variables are selected as
ξ1t=−1
2mv
ρaAfCdx2t,
ξ2x=−Crgcos at −gsin at
13
The premise variables are replaced in (11),
xt =
01
0ξ1t
x1t
x2t
+
0
ηktgr
mvrw
Ibat t+
0
ξ2t
,
14
with the defined dimensions, and the premise variables are
limited as follows:
ξ1≤ξ1t≤ξ1,
ξ2≤ξ2t≤ξ2
15
These premise variables comply with (10), such that
f (x(t))
x (t)
2x (t)
1x (t)
(a)
f (x(t))
x (t)


(b)
Figure 3: Nonlinear sector transformation approach.
4 Journal of Sensors

ζ1t=ξ1t−ξ1
ξ1−ξ1
,
ζ2t=1−ζ1t,
ζ3x=ξ2t−ξ2
ξ2−ξ2
,
ζ4t=1−ζ3t
16
The nonlinear model of the EV represented by the TS
approach is expressed as
xt =〠
4
i=1
μiξt Aixt +Bu t +Δi, 17
where
μ1t=ζ1tζ3t,
μ2t=ζ1tζ4t,
μ3t=ζ2tζ3t,
μ4t=ζ2tζ4t,
18
with
A1=0
0
1
ξ1
,
B=
0
ηktgr
mvrw
,
Δ1=0
ξ2
,
A2=0
0
1
ξ1
,
Δ2=0
ξ2
,
A3=0
0
1
ξ1
,
Δ3=0
ξ2
,
A4=0
0
1
ξ1
,
Δ4=0
ξ2
19
3.1. Validation of the Takagi-Sugeno Model. The model
parameters considered in this paper were presented in
[27], whose values are displayed in Table 1. Initial conditions
x0 = 0, 6 5 and an input current Ibat tas shown in
Figure 4 are considered to validate the TS model. The systems
responses are displayed in Figure 4. As can be observed, the
nonlinear system and the proposed TS model have the same
behavior, and the mean square error between the two
responses is 123 × 10−21%, which demonstrates that the TS
model represents the dynamic characteristics of the nonlin-
ear system. In the next section, a fault diagnosis method
based on a generalized observer scheme is proposed to detect
and isolate sensor faults.
4. Sensor Fault Diagnosis
The fault diagnosis approach proposed in this paper is based
on the generalized observer scheme. This method considers a
reduced order TS observer for each of the measured outputs
as shown in Figure 5. For the particular case of the EV, two
TS observers are considered. Both observers consider the
same input (the current Ibat t), but each observer is dedi-
cated to estimate only one output. That is, the first observer
Table 1: Parameters of the electric vehicle [27].
Parameter Value Unit
m150 kg
η0.97 —
kt0 0604 N·m/A
gr8 5 m
rw0 24 m
ρa1 225 kg/m
3
CdAf0 1031 m
2
Cr0 00081549 —
g981 m/s
2
x2
x2TS
6.4
6.6
6.8
7.0
7.2
7.4
7.6
7.8
Speed (m/s)
90 130100 110 12080 140 15070
Time (s)
(a)
−2
0
2
4
6
8
10
Ibat (A)
100 14080 110 120 13090 15070
Time (s)
(b)
Figure 4: Comparison between the general nonlinear model and the
TS model in response to variations of the input current.
5Journal of Sensors

is dedicated to estimate the vehicle displacement y1t, and
the second observer the EV velocity y2t. The main idea is
to perform fault diagnosis by the evaluation of two residual
signals, which are the difference between the measured and
estimated outputs given by the observers.
Each observer has the following form:
̂xt =〠
4
i=1
μiξt Aîxt +Bu t +Δi+Giyt −̂yt ,
̂yt =Ĉxt,
20
where ̂x t is the estimated state vector, ̂y t is the esti-
mated output vector, and Gi∈Rn×pare the observer gain
matrices to be computed. The design problem is then to
find Gisuch that limt→0e t = limt→0x t −̂x t ≈0. Fur-
thermore, the observer (20) needs to achieve the well-
known observability conditions.
The estimation error between (20) and (17) is com-
puted as
et =xt −̂xt 21
The error dynamics is given by
et =xt −̂xt 22
Replacing (17) and (20) in (22), the following is obtained:
et =〠
4
i=1
μiξt Ai−GiCet 23
From the above expression, it follows that if the state
estimation (23) error converges asymptotically to zero, the
estimated state vector converges asymptotically to the real
state vector; then, the problem is reduced to find the matrices
Githat satisfies condition (23). To achieve this goal, the
following theorem is proposed.
Theorem 1. The observer TS (20) asymptotically converges
to (17) if there exist matrices P=PT>0,Mi, satisfying the
following matrix inequality:
AT
iP−MT
iC+PAi−MiC<0, ∀i∈1, …,4 24
The gains of the observer are determined as Gi=P−1Mi.
Proof. The stability conditions for the derivative of the error
(23) can be obtained using the quadratic Lyapunov func-
tion Vx t =eTtPe tand V t x <0, where P=PT>0,
whose derivative is
Vxt ≔eTt Pet +eTtPe t <0, 25
and replacing (23) into (25) gives
Vxt ≔〠
4
i=1
μiξt eTt
AT
iP−PTGT
iCT+PAi−PGiCet <0
26
As can be observed from (26), it is sufficient that the
following expression has to be negative to satisfy the
Lyapunov condition:
AT
iP−PTGT
iCT+PAi−PGiC<0 27
Nonetheless, expression (27) contains nonlinear terms
PG and PTG. In order to eliminate these terms, it is consid-
ered that Mi=PGi. Then by substituting this term in (27),
the LMI (24) is obtained. This completes the proof.
4.1. Fault Diagnosis Scheme. Note that in order to perform
fault diagnosis, two observers need to be designed according
to Theorem 1. Then, each observer takes the following form:
̂xt =〠
4
i=1
μiξt Aîxt +Bu t +Δi+Gi,pypt−̂ypt,
̂ypt=Cp̂xt, p∈1, 2
28
Two normalized residual signals are computed and
evaluated as follows:
r1t=y1t−̂y1t,
r2t=y2t−̂y2t,29
such that if rptis less than a predefined threshold, then the
system is working under nominal conditions; but if rp>th,
then the system is working on faulty conditions. For example,
if the fault occurs in sensor 1, the residual r1tis insensitive,
but r2tchanges its value. On the other hand, for a fault
occurring in sensor 2, the residual r2tremains insensitive,
but r1tchanges its values (see Figure 5). This scheme
produces a decoupled sensor fault detection and isolation.
Nevertheless, it should be noted that it is not possible to
detect simultaneous faults, which will be addressed in future
contributions by considering a different diagnosis scheme.
Ibat (t)Electric
vehicle
y1 (t) = x1 (t)
y2 (t) = x2 (t)
TS observer
1
TS observer
2
ˆy1 (t)
ˆy2 (t)
r1 (t)
r2 (t)
Figure 5: General scheme of the observer bank to detect sensor
faults.
6 Journal of Sensors

5. Simulation Results
Numerical simulation results are presented in this section in
order to illustrate the applicability of the proposed method.
The EV parameters are the same as the ones presented in
[27], which are shown in Table 1.
Two reduced observers are considered as discussed in the
previous sections, whose gains (28) were calculated by solv-
ing Theorem 1 in Python with the solver MOSEK [28].
The gains for observer 1 are
G1,1=G2,1=1
06596 ,
G3,1=G4,1=1
06666
30
The gains for observer 2 are
G2,1=G2,2=13061
16082 ,
G2,3=G2,4=13042
16041
31
In order to verify the convergence of both observers,
the considered initial conditions are x0 = 0, 6 5 for the
system and ̂x0 = 0, 0 for the observers. For simulation
purposes, variations on the slope of the road are considered
as displayed in Figure 6(a). Furthermore, when the vehicle
is climbing a slope, the motor requires more power, which
is provided by the bank of batteries. This can be measured
as an increment on the electrical current, which is displayed
in Figure 6(b).
In addition, to illustrate the fault detection and isolation
method, different faults are considered to affect the two sen-
sors in different time intervals. Sensor can fail for different
reasons. Usually this behavior can be observed as additive
bias, for example, offsets or calibration problems. These mal-
functions can be described by ramp or step functions in order
to represent abrupt or slow variation faults, as described in
[29, 30]. By considering the previous remark, the fault
induced to the displacement sensor, denoted as f1,isdefined
in (32), and the fault induced to the velocity sensor, denoted
as f2,isdefined in (33). Both faults are displayed in Figure 7.
In this simulation, two types of faults are considered: an
abrupt fault in sensor 1 and an incipient fault in sensor 2,
defined in (32) and (33), respectively.
f1t=
0 for 0 ≤t< 20,
5 for 50 ≤t< 60,
15 for 60 ≤t< 70,
10 for 70 ≤t< 80,
0 for 80 ≤t≤200,
32
f2t=0 for 0 ≤t< 120,
008t−9 6 for 120 ≤t≤200 33
The normalized residual signals are also shown in
Figure 7. In both cases, the fault detection turns out to be suc-
cessful. The fault identification is done by comparing the
fault signature with the incident matrix given in Table 2.
For example, for the fault affecting the speed sensor, the
residual r2presents some changes at t= 120 s, while the resi-
due r1remains at zero. This particular signature allows to iso-
late the fault in the speed sensor. A similar analysis can be
50 100 150 2000
Time (s)
−5
0
5
10
15
(t) (degrees)
(a)
0
10
20
30
40
50
60
70
80
90
Ibat (t) (A)
2000 15050 100
Time (s)
(b)
Figure 6: Variation on the slope of the road (α) and the input
current (Ibat).
−50
0
50
100
150
200
250
(Amplitude)
[r2 (t)]
[r1 (t)]
f1 (t)
f2 (t)
50 100 150 2000
2000 15050 100
2000 15050 100
Time (s)
0
2
4
6
8
(Amplitude)
0
4
8
12
16
(Amplitude)
Figure 7: Normalized residuals vectors r1t,r2tand induced
fault signals f1t,f2t.
7Journal of Sensors

done for the traveled-distance sensor. The proposed fault
diagnosis scheme is effective to detect faults in both sensors
even in the presence of variations on the slope of the road.
6. Conclusions
In this work, an observer-based fault detection system was
designed for an electric vehicle. The EV is represented by a
TS model whose weighting functions depend on the velocity
and the slope of the road. This representation is more general
compared with that of typical models, which consider a con-
stant slope. Sufficient conditions, which guarantee the con-
vergence of the TS observer, and the observer bank were
proposed by a set of linear matrix inequalities. Finally, the
detection of fault on the velocity and traveled-distance sensor
was demonstrated through simulation by the variation of the
residues compared to an incidence matrix. A variable slope
was considered in order to increase the range of representa-
tivity of the nonlinear dynamics. The result shows that the
TS fault diagnosis observer can detect and isolate sensor
faults for different driving conditions and different types of
faults. However, the model can be improved by measuring
the mass of the vehicle in order to be considered as a premise
variable. Future work will be done in order to consider simul-
taneous faults and inexact premise variables.
Conflicts of Interest
The authors declare that there is no conflicts of interest
regarding the publication of this paper.
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