Optimized sensor selection for control and fault tolerance of electromagnetic suspension systems

Research Article
Optimised sensor selection for control and fault tolerance of
electromagnetic suspension systems: A robust loop shaping approach
Konstantinos Michail a,n
, Argyrios C. Zolotas b
, Roger M. Goodall c
a
Department of Mechanical Engineering and Materials Science and Engineering, Cyprus University of Technology, Limassol, Cyprus
b
School of Engineering and Informatics, University of Sussex, United Kingdom
c
Control Systems Group, School of Electronic, Electrical and Systems Engineering, Loughborough University, United Kingdom
a r t i c l e i n f o
Article history:
Received 27 September 2012
Received in revised form
14 May 2013
Accepted 4 August 2013
Available online 14 September 2013
This paper was recommended for publica-
tion by Dr. Didier Theilliol.
Keywords:
Optimised sensor selection
Robust control
Fault tolerant control
Magnetic levitation
Multiobjective optimisation
Electromagnetic suspension
a b s t r a c t
This paper presents a systematic design framework for selecting the sensors in an optimised manner,
simultaneously satisfying a set of given complex system control requirements, i.e. optimum and robust
performance as well as fault tolerant control for high integrity systems. It is worth noting that optimum
sensor selection in control system design is often a non-trivial task. Among all candidate sensor sets, the
algorithm explores and separately optimises system performance with all the feasible sensor sets in
order to identify fallback options under single or multiple sensor faults. The proposed approach
combines modern robust control design, fault tolerant control, multiobjective optimisation and Monte
Carlo techniques. Without loss of generality, it's efficacy is tested on an electromagnetic suspension
system via appropriate realistic simulations.
& 2013 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Optimum sensor selection in practical control system design
can be a complex process especially if the selection is done with
respect to a number of properties in order to achieve robust
optimum performance and reliability properties.
A typical closed-loop control system is shown in Fig. 1. Typi-
cally, a system to be controlled has a number of candidate control
inputs (actuators) and outputs (sensors) that could be used to
control it by proper controller design using one of the existing
modern control methods. Moreover the system suffers from input
disturbances and uncertainties or model inaccuracies.
Additionally, faults highly affect the closed-loop performance of
a control system. Particularly, actuator and sensor faults can cause
performance degradation or even instability. This problem has
been extensively considered by the scientific community in the
last few years. Some typical work with applications to industrial
systems includes [1–4].
The problem of sensor/actuator selection has been addressed
before in the literature [5] but none of the methods considers
simultaneous satisfaction of the aforementioned properties except
in [6] where the authors have considered both optimum perfor-
mance and sensor fault tolerance using Linear Quadratic Gaussian
(LQG) control. Therefore the problem is to find the ‘best’ set of
sensors, Yo, subject to the aforementioned control properties i.e.
optimum performance, robustness, fault tolerance and minimum
number of sensors.
The novelty in this paper relies on the fact that optimum robust
performance with sensor fault tolerance is achieved by combining
robust control methods, Fault Tolerant Control (FTC), Multi-Objec-
tive OPtimisation (MOOP) and Monte Carlo (MC) method as
illustrated in Fig. 2.
Robust control design in a practical control system has a vital
role because real systems have uncertainties, disturbance inputs
and other effects that affect the nominal performance of the
closed-loop control system. In that context robust control theory
has been developed in the last few years including H1 robust
control methods [7]. Among the existing robust control methods
the H1 Loop Shaping Design Procedure (LSDP) is merged into the
framework for the design of robust nominal controller [8].
Control system design for safety-critical systems [9,10] is vital
for the integrity of such systems when sub-system faults occur.
Therefore the scientific community developed an area where the
faults can be accommodated. Fault tolerant control systems are
divided into two categories, the Active FTC (AFTC) and the Passive
Contents lists available at ScienceDirect
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ISA Transactions
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
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n
Corresponding author. Tel.: þ357 25002133.
E-mail addresses: kon_michael@ieee.org (K. Michail),
a.zolotas@sussex.ac.uk (A.C. Zolotas), r.m.goodall@lboro.ac.uk (R.M. Goodall).
ISA Transactions 53 (2014) 97–109

FTC (PFTC) systems. In this work, the AFTC concept is introduced in
order to accommodate multiple sensor faults [11–13].
Multiobjective constrained optimisation using heuristic approaches
is very popular and has gained a lot of intention in the last few years
[14,15]. Among the heuristic methods, Genetic Algorithms (GAs) are
favoured in control optimisation [16–18]. Since the beginning of GAs
by Goldberg [19] many versions of GAs have been published all
summarised in [20] with the latest version called Dynamical Multi-
objective Evolutionary Algorithm (DMOEA) been described in [21]. In
this paper the Non-dominated Sorting Genetic Algorithm II (NSGAII)
[22] is used for the optimisation part of the framework and shows
to be a very strong optimisation tool in sensor selection for control
design.
The Monte Carlo method has gained a lot of attention after the
initial introduction by Metropolis [23,24]. Until today many
methods have been introduced for random generation of numbers
with this method [25,26]. Moreover the MC method can be used in
control systems to assess the robustness in a probabilistic way
[27]. In this paper the robustness against model parametric
uncertainties is assessed using a combination of the MC and
constraint handling functions as used in MOOP. Particularly, the
MC is used to produce a number of models for the uncertainties
and those are tested in closed-loop simulations using the nominal
controller.
The proposed framework is assessed on an Electro-Magnetic
Suspension (EMS) system. The EMS systems are being used on the
MAGnetic LEVitated (MAGLEV) trains that have a number of
advantages against the conventional wheel-on-rail trains [28]. As
indicated in [29] the EMS is Non-Linear (NL), safety-critical and
inherently unstable system with non-trial requirements. Such a
system can easily serve as a good example for testing the efficacy
of the proposed optimum sensor selection framework.
Summarizing, the novelty in this paper relies on the fact that
H1 Robust Control, FTC, MC method and optimised tuning via GA
concepts are combined to form a systematic framework in an
attempt to simplify the selection of the best sensor set defined as,
Yo, for the EMS system subject to optimum closed-loop perfor-
mance and ensuring integrity and robustness of the system under
possible sensor faults. In this context, the algorithm explores and
separately optimises the performance of the EMS using all feasible
sensor sets in order to identify fallback options under single or
multiple sensor faults, i.e. instants of one or multiple sensors
failing stability constraints but with optimum performance main-
tained by controller reconfiguration using the remaining healthy
sensors.
The rest of the paper is separated into six sections: Section 2
explains the problem under consideration and describes the
details of the proposed algorithm that leads to the optimum
sensor selection for the control system design. Section 3 describes
the rigorous modelling issues of the EMS system along with the
disturbance inputs and multiple control objectives and constraint
requirements. In Section 4 the multiobjective constraint optimisa-
tion concept as used in the algorithm is given emphasizing its
usefulness and importance. Further Section 5 describes the sensor
fault tolerance concept for the EMS system with the robustness
assessment of the optimally tuned controller using the Monte
Carlo method in combination with constraint handling technique
as used in the previous section. In Section 6 data analysis is done
from the realistic simulations done from the proposed framework.
Finally, the conclusions of this work are given in Section 7.
2. The problem statement and description of the
proposed framework
2.1. Problem statement
The plant shown in Fig. 1 has a set of control inputs (actuators)
U ¼ fu1; u2; …; unu
g, where nu is the total number of actuators, a set
of input disturbances D ¼ fd1; d2; …; dnd
g, where nd is the total
number of input disturbances, and a set of possible outputs
(sensors), fY ¼ y1; y2; …; yns
g, where ns is the total number of
sensors, and a set of sensor sub-sets of Y, Y ¼ fY1; Y2…; YNss
g to
choose from, where Nss is the total number of sensor sub-sets in Y.
The formal problem is defined as to determine the set of sensors,
Yo, in Y (i.e. select Yo & YðiÞ), for which the system
1. satisfies a set of closed-loop performance criteria,
2. satisfies a set of fault tolerance criteria,
3. the sensor set has minimum redundancy i.e. the number of
elements in Yo is minimal,
4. has sufficient robustness against parametric uncertainties and
5. has low cost (although this property is not considered in the
paper, its part of this problem and left for future work).
The following section gives a rigorous description of the proposed
algorithm which attempts to solve this problem.
2.2. The proposed framework
The proposed framework can be summarised in the flow chart
of Fig. 3. The particular points include the use of H1 loop-shaping
design and the heuristic optimisation (evolutionary algorithms)
method for tuning the controller subject to strict requirements
(objectives and constraints) for each feasible sensor set of the EMS
system. Prior to running the algorithm (initialization phase), some
parameters are assigned including:
Formulate the model of the system: Prior to algorithm execution
formulate the model of the system to be examined, i.e. non-
linearities, uncertainties, linearization, etc.
Generate the sensor sets: A set, Y, which contains all sensor sets
is generated at this stage.
Define the control objective functions ðϕiÞ and constraints ðf h; f sÞ:
Usually, in a system's optimisation the control objectives are
conflicting to each other therefore a trade-off exists between
them. Also there is a number of control constraints that have to
be satisfied in order for the system to have proper control
l
ll
l
i i
Fig. 1. A typical closed-loop control system.
l
i
l l
l
l
l i
i
l i i
i i i
l
Fig. 2. The simplified diagram of the proposed framework for optimum sensor
selection with robust control and fault tolerance.
K. Michail et al. / ISA Transactions 53 (2014) 97–10998

properties. Both can be time of frequency related characteris-
tics and will effectively deﬁne the optimum performance of the
closed-loop.
Enter the performance weights: Since the LSDP is used for the
design of the controller the structure of the Performance Weights
(PW) has to be decided. This will effectively have one PW for each
measurement for the loop-shaping and it has to be optimally
tuned for each feasible sensor set using the NSGAII.
Robustness assessment parameters: Each sensor set is tested for
robustness against parametric uncertainties using MC and the
Weighted Overall Constraint Violation Function (WOCVF). Some
parameters have to be assigned for the robustness assessment
like the number of uncertain samples to be tested, Nmc and the
Constraint Violation Weights (CVWs), wi with the latter been
rigorously described in Section 4.
Enter the optimisation parameters: As it has been mentioned the
NSGAII will be used for the optimisation part of the proposed
framework. As all GAs this also requires some parameter
assignment which effectively affects the successful recovery
of the optimum Pareto front between the objective functions.
This parameter assignment includes: the mutation probability
pm, crossover probability pc, population number Np, maximum
generation number Ng, penalty parameters, number of vari-
ables nu and the search space for each variable.
Enter the controller selection criteria: After completion of the
optimisation for each sensor set there is a number of
- Formulate the uncertain model
- Set the control objective functions and constraints
- Enter the number of uncertain model samples to be tested,
- Enter constraint violation weights,
- Enter the NSGAII parameters
- Set the controller selection criteria,
Find the Pareto-Optimality between
the control objective functions subject
to the control constraints with the
current sensor set,
- Enter the LSDP performance weights structure
- Generate the feasible sensor sets
Select the controller/s,
that satisfy all constraints.
Select the controller that satisfy
user's selection criteron,
Save the results for the
current sensor set
Select the controller/s that satisfy
controller selection criteria,
Robustness assessment of using
Monte Carlo and
Sensor fault accomodation
ratio evaluation of using
Key points of the algorithm
Increase the counter
Initialisation
Select the ith sensor set
Optimisation with NSGA-II
False
Falseis
is all
Sensor Fault Tolerance
Assessment
l ll
i i i
i ii ll
l
l
ii
l
l
Fig. 3. Flowchart of the proposed sensor optimisation framework with robust H1 loop-shaping design.
K. Michail et al. / ISA Transactions 53 (2014) 97–109 99

controllers that result in a closed-loop performance equal to
the number of population of the NSGAII. Among those the best
one which suites to the user has to be selected therefore there
are two selection criteria used and they are explained latter in
this section: (i) controller selection criteria ðf ci
Þ and (ii) user's
controller selection criterion ðf kc
Þ.
Starting the optimisation procedure, the first sensor set, ðY1Þ, is
selected and the evolutionary algorithm seeks the Pareto-optimality
of the objective functions ϕi subject to the control constraints ðf h; f sÞ.
At the end of this stage there is a set of optimally tuned controller C1
and a set of Overall Constraint Violation Functions (OCVFs), Ω1
consisting of the Ωki
(see (18) in Section 4) one for each controller
both sets lengths equal to the number of population Np.
In the sequence, the algorithm seeks to select the ’best’
controller. At this point there are two paths to follow:
(i) If there is no controller to satisfy the performance (this can be
easily verified by checking the OCVF, Ωki
in Ω1 for each
individual response) then the controller, kc, which gives the
minimum Ωki
is selected and saved in KðiÞ. Ωn
wÀc is not
evaluated in this case.
(ii) Those controllers satisfying Ω are collected in C2. The next
step is to collect in C3 the controllers that satisfy the controller
selection criteria f ci
. Finally, the user's controller selection
criteria, f kc
is used to select the controller, kc which results in
the desired closed-loop response. If no controller exists to
satisfy f ci
then the algorithm directly selects a controller based
only on f kc
.
The particular sensor set in combination with the selected
controller provides a nominal performance that is assessed for
parametric uncertainties by combining the MC method using the
WOCVF as explained in Section 4 in the following way:
(i) For every qth among the Nmc samples of the uncertain EMS
system calculate the WOCVF, Ωn
kc;q.
(ii) From the closed-loop responses of the Nmc model samples that
result in a set of WOCVF, i.e. Ωn
kc
, the maximum value of Ωn
kc
is
assigned to the current sensor set as the assessment of the
worst-case response of the uncertain model. In case the
closed-loop response with any sampled model is unstable,
Ωn
kc
is quantified as infinity.
In this way the robustness of the optimally tuned nominal
controller, kc, is assessed.
The next step for the ith sensor set optimisation is for the
proposed algorithm to save the Yi in YðiÞ, kc in KðiÞ, Ωn
wÀc in Ω2ðiÞ
and Ωkc
in Ω3ðiÞ.
Following that, the evaluation of the selection criterion for
optimum sensor fault tolerance is done. At this stage the Sensor
Fault Accomodation Ratio (SFAR) is evaluated which indicates at
which percentage the faults can be recovered in the sense of the
number of fault conditions that could be accommodated by using
the remaining healthy sensors in a given sensor set Yi. Assuming
that the sensor set Yi has the Ωkc
a0 then the SFAR¼0 otherwise
SFAR is calculated using the following formula:
SFARðYi;Ωkc ¼ 0Þffi
NYh;Ωkc
¼ 0
NYf
100 ð%Þ ð1Þ
where Yh is the healthy sensor sub-sets of Yi that have Ωkc
¼ 0
and Yf are the possible sensor sub-sets that could happen in Yi
(assuming that it is not possible to lose all sensors). NYh
and NYf
are the number of the sensor sets in Yh and Yf respectively. Then,
Yh are the healthy sensor sets in Yi that can be successfully used
for maintaining the performance under any sensor fault combina-
tion that could occur in Yi. A typical detailed example on the SFAR
calculation is found in Appendix A. SFAR metric can be used for
optimum sensor set selection that offers the highest degree of
fault tolerance against sensor failures.
Next, the algorithm increases the counter i and moves to the
next feasible sensor set until i ¼ Nss. At the final end a detailed
report is given by the framework that is analysed, in this paper
from the simulations of an EMS system.
3. The single-stage EMS system
3.1. The non-linear model and linearization
The single-stage, one degree-of-freedom model of the EMS
system represents the quarter of a typical MAGLEV vehicle. In
order to design a control system with optimum performance using
linearized controllers, a careful and rigorous modelling analysis is
required and given in this section.
Single-stage electro-magnetic suspension as the one used on
the Birmingham Airport MAGLEV and which operated for 12 years
in the 1980s and 1990s is suitable for low speed vehicles [30].
The EMS is a non-linear, inherently unstable system with non-trivial
control requirements.
The basic quarter car diagram of the MAGLEV vehicle is shown
in Fig. 4. The suspension consists of an electromagnet with a
ferromagnetic core and a coil of Nc turns which is attracted to the
track that is made of ferromagnetic material. The carriage mass
(Mc) is supported on the electromagnet, with zt being the track's
position and z the electromagnet's position (both are given as
the small variations around the operating point). The distance
between them is the airgap, ðztÀzÞ. The airgap is to be kept as close
as possible to the operating point at values that will not exceed the
maximum allowed as given later in this section. There are four
important variables in an electromagnet named as force F, flux
density B, airgap G and the coil's current I that give non-linear
characteristics to the suspension as depicted in Fig. 5 [29]. The
straight lines show the theoretical relationships and the dashed
l
il
l
il
l
i
ii l
i
Fig. 4. Single-stage suspension for MAGLEV vehicles.
fluxdensity
fluxdensity
force
flux densityairgap
I I
Fig. 5. Relationship between the key variables describing the magnet.

lines indicate the effects of magnetic saturation in the magnet
core. Assuming that the vertical downwards motion is taken as
positive the non-linear model of the EMS system is described by (i)
Newton's equation of motion given in (2), (ii) the coil's voltage Vc
given in (3) across the electromagnet's coil from Kirchoff's law and
(iii) the equations in (4), (5) and (6) that give the force, the flux
density and the airgap velocity respectively
Mc
d
2
Z
dt
2
¼ McgÀF ð2Þ
Vc ¼ IRc þLc
dI
dt
þNcAp
dB
dt
ð3Þ
B ¼ Kb
I
G
; ð4Þ
F ¼ Kf B2
ð5Þ
dG
dt
¼
dzt
dt
À
dZ
dt
ð6Þ
where Rc and Lc are the coil's resistance and inductance respec-
tively, Ap is the pole face area and constants Kb, Kf and g are the
flux, force and gravity constants with values equal to 0.0015,
0.0221 and 9.81 m/s2
respectively.
The linearisation of the EMS is based on small variations
around the operating point. The following definitions are used
with lower case letters defining the small variations and subscript
‘o’ referring to the operating point
B ¼ Bo þb; F ¼ Fo þf ð7Þ
I ¼ Io þi; G ¼ Go þðztÀzÞ ð8Þ
Vc ¼ Vo þuc; Z ¼ Zo þz ð9Þ
Following the linearization procedure described by Michail[29]
the following linearised equations describe the linear model of the
EMS:
Mc
d
2
z
dt
2
¼ À2Kf
Io
G2
o
iþ2Kf
I2
o
G3
o
ðztÀzÞ ð10Þ
di
dt
¼ À
Rci
Lc þ
KbNcAp
Go
þ
KbNcApIo
G2
o Lc þ
KbNcAp
Go

dzt
dt
À
KbNcApIo
G2
o Lc þ
KbNcAp
Go

dz
dt
þ
uc
Lc þ
KbNcAp
Go
ð11Þ
dðztÀzÞ
dt
¼
dzt
dt
À
dz
dt
ð12Þ
Given the states as x ¼ ½i _z ðztÀzÞŠT
, the state space description
of the EMS system can be expressed as in (13). The output
equation, y, corresponds to i the coil's current, b the flux density,
ðztÀzÞ the airgap, _z the vertical velocity and €z the vertical
acceleration
_x ¼ AxþBuc uc þB_zt
_zt
y ¼ Cx ð13Þ
The matrices A, Buc
, B_zt
and C are given by
A ¼
À
Rc
Lc þ
KbNcAp
Go
À
KbNcApIo
G2
o Lc þ
KbNcAp
Go
0
À2Kf
Io
McG2
o
0 2Kf
I2
o
McG3
o
0 À1 0
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
ð14Þ
Buc
¼
1
Lc þ
KbNcAp
Go
0
0
2
6
6
6
6
6
4
3
7
7
7
7
7
5
; B_zt
¼
KbNcApIo
G2
o Lc þ
KbNcAp
Go

0
1
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð15Þ
C ¼
1 0 0
Kb
Go
0 À
KbIo
G2
o
0 0 1
0 1 0
À2Kf
Io
McG2
o
0 2Kf
I2
o
McG3
o
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
ð16Þ
Using the output matrix, C different combination of sensors can
be obtained for feedback to the controller defined as the ‘sensor
set’.
The electromagnet design of MAGLEV vehicles is described in
more detail by Goodall [31]. In this paper, a typical quarter car
vehicle of 1000 kg is studied. It requires an operating force of
Fo ¼ Mc Â g and operating airgap (Go) of 15 mm in order to
accommodate effects from the track roughness. According to these
requirements the rest of the electromagnet's parameters listed in
Table 1 can be calculated.
Additionally, the EMS system is characterised by uncertainties
that can be caused due to various reasons i.e. temperature change,
mass change, etc. The maximum boundaries of the uncertainties
that could possibly occur are listed in Table 1. The subscript ‘o’
indicates that the variable is at the value of the operating point.
3.1.1. Total number of sensor sets
The total number of sensor sets, Nss, in Y, is calculated from
Nss ¼ 2ns
À1. Given that the EMS has 5 sensor outputs, 31 candidate
sensor sets, Yi, can be obtained using combination of the rows of
the output matrix, C. However, since the H1 LSDP controller
design technique is used in this paper, the airgap measurement is
used as a standard measurement and therefore the number of
candidate sensor sets , Yi, reduces to 16 with the full sensor set
given as i.e. Y16 ¼ fi; b; ðztÀzÞ; _z; €zg.
3.2. Disturbance inputs to the EMS
3.2.1. Stochastic input
The stochastic inputs are random variations of the track vertical
position as the vehicle moves along the track. This is caused by the
track installation discrepancies due to track-laying inaccuracies
and unevenness. Considering the vertical direction, the velocity
variations can be approximated by a double-sided power spectrum
density expressed as S _zt
¼ πArVv [32]. Vv is the vehicle speed
Table 1
Parameters of the EMS system.
Parameter Value Uncertainty (%)
Operating point
Airgap, Go 0.015 m 710
Flux density, Bo 1 T 710
Current, Io 10 A 710
Input voltage, Vo 100 V 0
Force, Fo 9810 N 710
Electromagnets' parameters
Carriage mass, Mc 1000 kg 710
Coil's resistance, Rc 10 Ω 750
Coil's inductance, Lc 0.1 H 750
Number of turns, Nc 2000 0
Pole face area, Ap 0.01 m2
0

(taken as 15 m/s) and Ar represents the track roughness that is
given a realistic value for high quality track 1 Â 10À7
m. The
corresponding autocorrelation function is given as RðτÞ ¼ 2π2
Ar
VvδðτÞ where δðτÞ is a dirac delta function.
3.2.2. Deterministic input
Another disturbance input to the EMS system is the position
change of the track, zt, while the vehicle is running caused by the
intended changes of the track's inclination. This intended input is
due to the track gradients due to the pre-designed infrastructure
of the maglev vehicle and is considered the only deterministic
input to the suspension in the vertical direction. This transition
onto the track's gradient is simulated by the signal depicted in
Fig. 6. In this work, a deterministic input with a track gradient of
5% at a vehicle speed of 15 m/s, an acceleration of 0.5 m/s2
and a
jerk of 1 m/s3
is used.
3.3. Performance requirements of the EMS control system
The control design requirements of an EMS system depend on
the type and speed of the train [33,30]. Typically, the EMS should
be able to follow the gradient onto the track (deterministic input)
and reject the random variations of the track. Fundamentally,
there is a trade-off between the deterministic and stochastic
closed-loop responses of the EMS system. The optimisation of
the performance is taken as minimisation of the stochastic
characteristics subject to control constraints that include the
deterministic and stochastic maximum allowable bounds. The
control objectives to be minimized are the RMS vertical accelera-
tion €zrms and the RMS current variations irms subject to the control
constraints listed in Table 2. This means that the trade-off between
€zrms and irms will give the trade-off between the ride quality and
input power while the deterministic response will keep the
bounds within safe bounds.
The robust stability margin, ϵ (robustness margin), calculated
from the LSDP is set to be maximized for maximum robustness
against the uncertainties (note that γ ¼ 1=ϵ).
Another objective function taken into consideration is the level
of the noise unrms
that appears on the input voltage uc. The level of
the noise at the output of a sensor depends on the sensor
characteristics itself and the picked-up interference from the
surrounding environment. Although the sensors are designed to
have noise immunity they still produce noise at the output. Thus
an amount of this noise will appear on uc which effectively affects
the ðztÀzÞ. The dynamical behaviour of the EMS has low pass filter
characteristics so most of this noise is filtered but it still affects the
performance of the suspension if it is not kept at low levels [34].
Hence, the level of the noise at the input voltage is added as the
4th objective to be minimized i.e. ϕ4 ¼ ucrms
. This helps the GA to
take it into account during the controller optimised design.
Additionally, ucrms
is used in the controller selection criteria f ci
in
order to constrain its value to a maximum of 10 Vrms. In order to
measure ucrms
an extra simulation is executed with idle track
profile i.e. zt ¼0 and since there is no exact information about
the sensor noise, the noise covariance is typically taken as 1% of
the maximum value of the deterministic response of the suspen-
sion for the corresponding measured signal.
Summarizing, the objective functions ϕi to be minimized are
formally written as
ϕ1 ¼ irms; ϕ2 ¼ γ; ϕ3 ¼ €zrms; ϕ4 ¼ unrms
ð17Þ
4. Multiobjective constrained optimization
NSGA-II is an evolutionary process that requires some para-
meters to be assigned in order to ensure proper population
convergence towards the optimum Pareto front between the
objective functions. These are mainly selected from experience
rather than from a priori knowledge of the optimisation problem.
The crossover probability is generally selected to be large in order
to have a good mix of genetic material. The crossover probability,
pc, is set to 90% and the mutation probability, pm, is defined as 1/nu,
where nu is the number of variables. The population consists of 50
chromosomes (Np ¼50) and the stopping criterion is the maximum
generation number Ng. Ng has a significant role in the Pareto-
optimality and the computational time therefore proper selection
of Ng is vital. It depends among other factors on the number of
variables to be tuned because the larger the nu the larger the Ng is
required with the expense of having longer computational time.
The number of variables varies according to the number of sensors
hence the Ng varies dynamically. It is set at 250 for sensor sets
with up to 2 sensors and for the rest including the full sensor set is
set at 300 generations.
In order to achieve the control constraints as described in
Section 3.3 a constraint handling technique is necessary. Different
constraint handling methods have been developed as described in
[35]. The dynamically updated penalty function approach is used
in this work. A rigorous description of this method is given in [36].
A function is used in order to ‘guide’ the objective functions in (17)
towards the Pareto-optimality while the control constraints listed
Table 2
Control constraints on the EMS system.
EMS limitations Value
Stochastic track profile
f s1
RMS acceleration, €zrms r1 m sÀ2
f s2
RMS airgap variation, ðzt ÀzÞrms r5 mm
f s3
RMS control effort, ucrms r300 V ð3I0RcÞ
Deterministic track profile
f s4
Maximum airgap deviation, ðzt ÀzÞp r7:5 mm
f s5
Control effort, ucp r300 V ð3I0RcÞ
f s6
Settling time, ts r3 s
f h1
Airgap steady state error, ðzt ÀzÞess
¼0
Other constraints
f s7
Robust stability margin, ϵ Z0:15
f si
, soft constraint; f hi
, hard constraint.
Time (s)
lilii
Position
Velocity
Acceleration
Fig. 6. Deterministic input to the suspension with a vehicle speed of 15 m sÀ1
and
5% track's gradient.

in Table 2 are satisfied. The OCVF is given as
Ωki
ðf
ðjÞ
s ; f
ðqÞ
h Þ ¼ ∑
J
j ¼ 1
ωjðf
ðjÞ
s Þþ ∑
Q
q ¼ 1
ψqðf
ðqÞ
h Þ ð18Þ
where ωj is the jth soft constraint violation for the corresponding
jth quantity to be constrained ðf sÞ and J is the total number of soft
constraints. Similarly, ψq is the hard constraint violation for the
qth quantity to be constrained ðf hÞ and Q is the total number of
hard constraints.
The OCVF not only serves as a constraint handling technique but
also serves other purposes as described in Section 2.2: (i) as a nominal
performance indicator, (ii) the level of sensor fault tolerance and (iii)
robustness assessment with MC. The first two properties are straight-
forward but the third is described in the following.
For the robustness assessment (18) is modified as follows:
Ωn
kc
ðf
ðjÞ
s ; f
ðqÞ
h Þ ¼ ∑
J
j ¼ 1
Jwjωjðf
ðjÞ
s Þþ ∑
Q
q ¼ 1
Q ~wqψqðf
ðqÞ
h Þ ð19Þ
where wj and ~wq are the CVWs, for the soft and hard constraints
respectively. Depending on the design engineer, the two CVWs can
be treated as a set of unified weights or separately. There is no
single systematic method to do that and it depends on the design
engineer. However, in this work they are unified and the authors
will propose a simple and effective analytical method to calculate
them.
The reason for using CVWs is because not all constraints have
the same significance i.e. effect on the closed-loop performance.
For example if there is a small violation on the airgap deflection
ðztÀzÞ it is not as important as having the same violation on the
input voltage, uc. Therefore weights are assigned to emphasise the
difference in the constraint violation. During the MC simulation
tests, the WOCVF is evaluated with infinity ð1Þ if the closed-loop
response of the EMS system becomes unstable.
The control constraints are separated into bands of importance
as depicted in Fig. 7. The lower band contains the least significance
constraints while the top band contains the most significant ones.
There are 1, 2, 3 …, P bands, λp are the set of the assigned control
constraints in the pth band, np is the number of control constraints
in the λp and wp and αp are the weight and the weighting
parameter for pth band with the lower having the value of one
ðα1 ¼ 1Þ.
The weighting, wi, for each band is calculated as shown in
Fig. 7. Their values are then given with respect to the weight of the
lowest band that have the smallest weighting parameter:
w2 ¼ α2w1
w3 ¼ α3w1
w4 ¼ α4w1
⋮ ð20Þ
wP ¼ αPw1 ð21Þ
weights and weighting parameters can be written in a series of
summation to be equal to one
∑
P
p ¼ 1
npαpw1 ¼ 1 ð22Þ
from where the first CVW, w1, can be calculated as
w1 ¼
1
∑P
p ¼ 1npαp
ð23Þ
Then, the ith CVW wi can be calculated as
wi ¼
αi
∑P
p ¼ 1npαp
; i ¼ 1; …; P ð24Þ
For the EMS system the control constraints are divided into
three bands, i.e. P¼3. The control constraints listed in Table 2 are
assigned to each band as follows: λ1 ¼ fts; ðztÀzÞess
g, λ2 ¼ fucp
; ucrms
;
ϵ; €zrms; g and λ3 ¼ fðztÀzÞp; ðztÀzÞrmsg therefore n1¼2, n2 ¼4 and
n3 ¼2. The weighing parameters can be assigned taking into
account that λ1 is of low importance, λ2 is of medium importance
and λ3 is of the highest importance. Hence, since α1 ¼ 1 the α2 and
α3 weighting parameters are selected to be α2 ¼ 5; α3 ¼ 25. This
means that the set of constraints in λ2 are 5 times and λ3 are
25 times ‘more important’ than the ones in λ1. These values are
generated ad hoc but a formula can be used if many bands exist i.e.
the weighting parameter for the ith band can be calculated as
αi ¼ βiÀ1
where β is a free selected integer greater than one. From
the data above the CVW are calculated as w1 ¼0.0139, w2 ¼0.0694
and w3 ¼0.347.
I
I
Fig. 7. Control constraints weighting assignment.
i
i
i
i
il
l l i
i ll
l l
li
i i
i l
ii
l ll
iii
li
Fig. 8. AFTC diagram for multiple sensor failures using a bank of H1 LSDP controllers.

5. Sensor fault tolerance via LSDP design and
robustness assessment
Sensor failures occur often in engineering systems reducing the
reliability of a control system and thread the integrity of safety-
critical systems. In this paper, the AFTC concept is used to
accommodate sensor failures in combination with the proposed
framework.
The AFTC scheme composes of a bank of H1 LSDP designed
controllers, the Fault Detection and Isolation unit and reconfigura-
tion mechanism. The overall mechanism is applied on the non-
linear model of the EMS system with the operating point defined
in Table 1 [37]. The diagram which describes the sensor fault
tolerant control concept is depicted in Fig. 8.
When multiple sensor faults occur the faulty sensors are
detected and isolated. After that, controller reconfiguration follows
in order to switch to another controller that is tuned for the
remaining healthy sensors (sub-set of the selected sensor set).
Note that for simplicity the switching delay is assumed to be
negligible. In order to detect multiple faults, the FD mechanism
monitors the residual from each healthy sensor. The residual for
each healthy sensor is typically produced by means of dedicated
observers [38]. Therefore, for each sensor set there are one
controller and one observer. Hence a bank of dedicated observers
(i.e. Ko1
; Ko2
…; Kon
) is used for fault detection. Each time that
faults happen, the reconfiguration mechanism gives a signal to
reconfigure the controller, the dedicated observers, as well as
isolate the faulty sensors. Isolating the faulty sensors is done by
taking the sensor out of the loop in such a way that the faulty
signal is not fed into the new controller (i.e. can be done using
switches).
The nominal design of the H1 controller is based on the
normalised coprime-factor plant description as proposed by [8],
which incorporates the simple performance/robustness tradeoff
obtained in loop-shaping, with the normalised Left Coprime
Factorisation (LCF) robust stabilization method as a means of
guaranteeing the closed-loop stability.
Controller design using the so-called loop-shaping design
procedure (LSDP) requires the linearized model of the plant to
be controlled (for the EMS system, the linear time-invariant state
space model in (13) is used in the design) and is done in two steps.
The first is shaping the open-loop characteristics of the plant using
pre- and post-weighting functions, W1 and W2 as shown in Fig. 9
(a). The plant is temporarily redefined as ^GðsÞ ¼ W2GW1 and the
H1 optimal controller ^KðsÞ is calculated. In the second step the
weighting functions are merged with the controller by defining
the overall controller KðsÞ ¼ W1
^KW2 as shown in Fig. 9(b). The size
of model uncertainty is quantified by the stability radius ϵ i.e. the
stability margin [8,7]. For values of ϵZ0:25, 25% coprime factor
uncertainty is allowable. However, in this paper a relaxed stability
margin is used set at ϵZ0:15.
The vital part of the LSDP is the performance weights whose
structures and bounds to be varied are selected by the user of the
framework. In order to achieve the desired optimum performance
those weights are optimally tuned using the NSGAII. This is an
iterative process where the bounds of the weights are randomly
varied until the optimum performance is achieved.
In typical design the structure of the performance weights and
thus the controller are to be kept as simple as possible. Thus, the
W1 pre-compensator is chosen as a single scalar performance
weight set to unity. For the W2 post-compensators there can be
five weighting functions that are used depending on the selected
sensor set. The airgap ðztÀzÞ measurement is a compulsory
measurement required for proper control of the magnet distance
from the rail and thus a low pass filter ðWðzt ÀzÞÞ is chosen with
integral action allowing zero steady state airgap error. The perfor-
mance weights used in this paper are given as
W1 ¼ 1; W2 ¼ diagðWi; Wb; Wðzt ÀzÞ; W _z ; W €z Þ ð25Þ
with
Wðzt ÀzÞ ¼
s
M1=nw
w
þωw
sþωwA1=nw
w
0
B
B
@
1
C
C
A
nw
ð26Þ
The low pass filter results in a minimum phase and stable
performance weight Wðzt ÀzÞ, with roll-off rate nw. Mw is the high
frequency gain, Aw the low frequency gain and ωw the crossover
frequency. Details for weighting function selection for the loop-
shaping design can be found in [18,7].
Taking advantage of the fact that any changes in the closed-
loop response (both stability and performance) will be reflected on
the WOCVF, Ωn
kc
in (19), robustness against parametric uncertain-
ties can be tested when it is combined with the MC technique.
Monte Carlo is a probabilistic method that is used to randomly
sample an uncertain variable with a given probability distribution.
There are many distribution functions that can be used to sample
an uncertain variable. The structured uncertainties of the EMS
Linear controller,
Linearised EMS model
Fig. 9. H1 loop-shaping controller design. (a) Shaped plant and (b) final controller.
Probability density function
Fig. 10. Uniform probability distribution function.
model
Uncertainties
Nominal Controller
Fig. 11. Monte Carlo test of the nominal optimally tuned LSDP designed controller.

system are listed in Table 1 and they are assumed to be uniformly
sampled. The probability density function used for each uncertain
parameters is illustrated in Fig. 10 and expressed in (27). Θo, Θmin
and Θmax are the mean, the minimum and the maximum values of
the uncertain parameter respectively and UðΘÞ is the probability of
each variable
UðΘÞ ¼
Um ¼ 1=ðΘmaxÀΘminÞ if Θmin rΘrΘmax
0 if Θmin 4Θ4Θmax
(
ð27Þ
As illustrated in Fig. 11 the nominal controller is tested under Nmc
multiple samples of the EMS model producing a vector of WOCVFs,
Ωi; i ¼ 1; …; Nmc from where the worst-case WOCVF is selected, i.e.
Ωn
wÀc ¼ maxðΩn
kc
Þ. The Ωn
wÀc is the worst case performance that can
happen among the Nmc model samples.
6. Framework assessment and data analysis
The framework is tested in MATLAB R2009b simulation envir-
onment without Java function due to large computational need
(simulation based). The computer used is the powerful DELL T610
with 2.93 GHz IntelsXeonsX5570 processor and 8 GB RAM. The
average simulation time per sensor set is about 5.5 h while
completion of the framework takes around 87 h.
The controller selection criteria ðf ci
; f kc
Þ for the desired closed-
loop response are given as follows:
f c1
€zrms r0:5 m=s2
; f c2
unrms
r10 V; ð28Þ
f kc
maxðϵÞ ð29Þ
The first set of closed-loop desired characteristics in (28) ensures
that the controllers to be selected are within the limits indicated.
f kc
in (29) ensures that the selected controller has the maximum
robust stability margin (maximum robustness).
Table 3 lists the detailed results from the framework execution.
The second column lists the sensor sets and the first the corre-
sponding identification number. The next four columns are the
variables from the closed-loop response with the stochastic track
profile and the further four lists the variable values from the
deterministic track profile. The next is the rms value of the noise
on the driving signal from idle track profile and the following
column is the resulting stability margin from the H1 loop-shaping
design. The 13th and 14th columns show whether the controller
selection criteria in (28) and (29) are satisfied or not. The 15th
column shows the SFAR value of each sensor set and 16th column
shows whether the OCVF, Ωkc
¼ 0 (marked as ‘✓’) or not Ωkc
a0
(marked as ‘ Â ’). As explained if Ωkc
¼ 0 then all control con-
straints are within the pre-set limits otherwise there is one or
more control constraint violations marked with bold fonts. The
last column lists the WOCVF, Ωn
kc
. Recall that the WOCVF is
evaluated by infinity during the MC test if there is instability in
the closed-loop response. The table gives a full picture of the EMS
optimum response under multiple circumstances that are rigor-
ously described next.
The proposed systematic framework is able to identify stabiliz-
ing controllers that satisfy the OCVF for 15 out of 16 sensor sets.
Inspecting columns fc and f kc
there are 6 sensor sets found that
satisfy (28) and (29) i.e. id:3, id:8 and id:13–16. Comparing the
corresponding sensor set rows with the control constraints listed
in Table 2 it can be seen that the optimum nominal performance
has been fully satisfied. The f c1
is not satisfied with many sensor
sets (i.e. the €zrms exceeds the given criterion) but those could be
still used for fault tolerance under multiple sensor faults.
If the user is to consider optimum sensor selection for optimum
control of the EMS then the id:3 which satisfies fc, f kc
and Ωkc
with
the minimum number of sensors can be used. The parallel coordinate
plot represents the Pareto-optimality of the objective functions with
the id:3 sensor set fb; ðztÀzÞg that is found using the NSGA-II. This
represents the response of a pool of optimally tuned controllers, C1,
that minimize the functions in (17) subject to the control constants
tabulated in Table 2. At the last Nth
g generation of the optimisation the
population of the optimised solutions is depicted in Fig. 12(a). The
trade-off of the objective functions is illustrated through a parallel
coordinate plot. All solutions of each ϕi are placed on the vertical axis
and connected with Np straight lines (equal to the number of
population) creating the parallel coordinate system which illustrates
the trade-off of the minimized objectives. Note that all control
constraints are satisfied i.e. all elements Ωki
AΩ1 are zero. The
objective functions are normalised around one for easy graphical
illustration using their maximum values shown on top of the graph.
From this last generation the controller, kc, is selected according
to (28) and (29). The response of kc with the nominal EMS model
to deterministic input disturbance (described in Section 3.2) is
depicted in Fig. 12(b). The maximum airgap deflection is less than
Table 3
Optimised sensor configurations for control and sensor fault tolerance of the EMS system.
id Sensor set Yi Stochastic input Deterministic input Idle input
grms ucrms
€zrms irms gp ucp
ts ess unrms
ϵ fc f kc
SFAR Ωkc
Ωn
kc
mm V m sÀ2
A mm V s V
1 ðzt ÀzÞ 1.41 37.19 0.77 1.30 1.24 11.92 2.29 ✓ 0.20 0.15 Â ✓ 0 ✓ 1
2 i; ðzt ÀzÞ 1.41 37.19 0.77 1.30 1.24 11.92 2.29 ✓ 0.20 0.15 Â ✓ 100 ✓ 1
3 b; ðzt ÀzÞ 1.24 24.88 0.47 1.02 3.67 27.50 2.10 ✓ 0.37 0.26 ✓ ✓ 100 ✓ 28.45
4 ðzt ÀzÞ; _z 1.41 37.19 0.77 1.30 1.24 11.93 2.29 ✓ 0.20 0.15 Â ✓ 100 ✓ 1
5 ðzt ÀzÞ; €z 1.41 37.19 0.77 1.30 1.24 11.92 2.29 ✓ 0.20 0.15 Â ✓ 100 ✓ 1
6 i; b; ðzt ÀzÞ 1.40 37.35 0.77 1.30 1.23 11.87 2.29 ✓ 0.20 0.15 Â ✓ 100 ✓ 1
7 i; ðzt ÀzÞ; _z 1.40 43.19 0.88 1.35 1.06 10.74 2.27 ✓ 0.12 0.15 Â ✓ 100 ✓ 1
8 i; ðzt ÀzÞ; €z 1.01 40.60 0.48 0.87 3.36 25.53 2.15 ✓ 2.04 0.27 ✓ ✓ 100 ✓ 1.01
9 b; ðzt ÀzÞ; _z 1.41 37.19 0.77 1.30 1.24 11.92 2.29 ✓ 0.20 0.15 Â ✓ 100 ✓ 1
10 b; ðzt ÀzÞ; €z 1.41 37.19 0.77 1.30 1.24 11.92 2.29 ✓ 0.20 0.15 Â ✓ 100 ✓ 1
11 ðzt ÀzÞ; _z; €z 1.28 35.28 0.75 1.18 2.87 22.17 2.08 Â 15.98 0.12 – – – Â –
12 i; b; ðzt ÀzÞ; _z 1.41 37.18 0.77 1.30 1.24 11.92 2.29 ✓ 0.20 0.15 Â ✓ 100 ✓ 1
13 i; b; ðzt ÀzÞ; €z 1.16 25.90 0.45 0.96 3.78 28.40 2.13 ✓ 0.90 0.21 ✓ ✓ 100 ✓ 0.76
14 i; ðzt ÀzÞ; _z; €z 1.16 25.21 0.42 0.95 4.13 30.78 2.15 ✓ 0.98 0.24 ✓ ✓ 85.71 ✓ 0.26
15 b; ðzt ÀzÞ; _z; €z 1.43 20.76 0.45 1.15 4.34 31.86 2.18 ✓ 0.26 0.18 ✓ ✓ 85.71 ✓ 12.37
16 i; b; ðzt ÀzÞ; _z; €z 1.00 42.27 0.49 0.86 3.28 25.00 2.15 ✓ 2.13 0.27 ✓ ✓ 93.33 ✓ 1.25
gp ðzt ÀzÞp, grms ðzt ÀzÞrms, ess ðzt ÀzÞess
.

7.5 mm and it settles back to the operating point (15 mm) within
less than 3 s with the steady state error being almost zero. The x-
axis (zero line) of the figure represents the operating airgap of the
suspension. The same graphical representation is used with the
rest of suspension's variables in this paper.
Looking at the results from the robust control point of
view, optimum sensor selection includes the WOCVF
Ωn
kc
tabulated in Table 4. This means that the id:3 with
Ωn
kcÀid:3 ¼ 28:45 cannot be used to accommodate the parametric
uncertainties of the EMS since there are very important control
Table 4
Robustness assessment results with Monte Carlo for the EMS system.
id Stochastic input Deterministic input
wi⟶ grms ucrms
€zrms gp ucp
ts ess ϵ Ωn
kc
w3 w2 w2 w3 w2 w1 w1 w2
Sensor set mm V m sÀ2
mm V s mm
3 b; ðzt ÀzÞ 12.0 141 0.42 20.0 232 2.1 172 0.26 28.45
8 i; ðzt ÀzÞ; €z 1.5 35.10 0.44 5.0 847 2.14 5 Â 10À5 0.27 1.01
13 i; b; ðzt ÀzÞ; €z 1.3 23.28 0.36 5.5 344 2.14 6.19 0.21 0.76
14 i; ðzt ÀzÞ; _z; €z 1.7 24.39 0.37 6.3 440 2.15 0.0041 0.24 0.26
15 b; ðzt ÀzÞ; _z; €z 4.6 57.21 0.37 19.0 219 3.17 69.68 0.18 12.37
16 i; b; ðzt ÀzÞ; _z; €z 1.4 40.31 0.45 4.8 976 2.14 0.090 0.27 1.25
wi ¼ fw1 ¼ 0:0139; w2 ¼ 0:0694; w3 ¼ 0:347g.
gp ðzt ÀzÞp, grms ðzt ÀzÞrms, ess ðzt ÀzÞess
.
Normilizedobjectivefunctions
i
Objective function Time seconds
li
Fig. 12. Parallel coordinates plot showing the trade-off between the objective functions in (17) and airgap response to deterministic input both with id:3. (a) Trade-off
between the object functions and (b) airgap response.
i
i
Nominal airgap response Nominal airgap response
Fig. 13. Airgap deflections (with deterministic track profile) for 100 samples of the non-linear uncertain model. (a) Airgap deflections with id : 8 ¼ fi; ðzt ÀzÞ; €zg and (b) airgap
deflections with id : 14 ¼ fi; ðzt ÀzÞ; _z; €zg.

constraint violations from both the stochastic and deterministic
track profiles.
More useful information from the results is extracted using the
WOCVF in the last column. Note that for the sensor sets that do not
satisfy the OCVF, the WOCVF is not evaluated. Typically, the lower
the value of the Ωn
kc
is the higher the robustness against the
uncertainties is. Looking at the value of Ωn
kc
for each sensor set it
can be seen that it varies from values between 0.26 and 28. The
detailed results of the candidate sensor sets (id:3, id:8, id:13–16)
from the corresponding worse case closed-loop performances are
listed in the same table. The table shows the measurements from
both the corresponding responses with stochastic and determi-
nistic inputs to the EMS system. The measurements with bold
numbers show that its maximum value is exceeded. id:3 has
2 constraint violations (i.e. ðztÀzÞrms; ðztÀzÞp) with the highest
importance (i.e. w3). Following that option, the id:15 with 2 con-
straint violations, the ðztÀzÞp and ts where the first is of highest
importance (w3) while the second is of lowest importance ðw1Þ.
This explains why both id:3 and id:15 have the highest Ωn
kc
. The
rest of the constraint violations are from the peak values of the
input voltage ðucp
Þ with medium importance (w2) and from the
ðztÀzÞess
with lowest importance (w1). This results in the lowest
Ωn
kc
for sensor sets with id:8, id:13, id:14 and id:16. The figures in
Fig. 13(a) and (b) show the deterministic closed-loop responses of
the EMS with the id:8 and id:14. The responses from both sensor
sets fully agree with the control constraints as given except with
the steady state error which is around 1 mm but this does not
threaten the integrity of the system. Looking at the results in
Table 3 from the sensor fault tolerance point of view the SFAR and
the OCVF are used as a metric selecting the sensor set which
results the optimum performance and have a desired level of fault
tolerance. The SFAR is evaluated for most of the sensor sets except
the id:1 because there is only one sensor and the id:11 because the
OCVF is not satisfied. Most sensor sets (id:2–10, id:12–13) have a
SFAR of 100%. The last three sensor sets with larger number of
sensors have lower level of sensor fault recovery ratio. Never-
theless, they still offer fault tolerance up to 93%.
The final step is to select the best sensor set starting with
elimination of the sensor sets that do not satisfy Ωkc
, fc, f kc
and the
sensor sets with Ωn
kc
¼ 1. After that there are 6 candidate sensor
sets: (id:3, id:8, id:13–16). From these candidates, it is easy to
identify that the id:3 and id:15 have Ωn
kcÀid:3 ¼ 28:45 and
Ωn
kcÀid:15 ¼ 12:37 respectively both excluded although they have
SFAR¼100%. Finally, from the four left candidates it is easy to
identify that the id:8 has both the higher SFAR and sufficient
robustness with the minimum number of sensors. If better robust-
ness is required the other option is the id:14 with Ωn
kc
¼ 0:26 but
with smaller SFAR.
At this stage, the optimum sensor set selection comes to the point
where the selection is relative to the system's requirement and at
what level of fault tolerance and robustness is required by the
engineer. For the work in this paper, the id:8 is selected to be the
optimum set of sensors that can offer 100% sensor fault recovery and
robustness with the minimum number of sensors i.e. Yo ¼
fi; ðztÀzÞ; €zg. Additionally, the optimum nominal performance is
achieved even under multiple sensor faults. At this point it is
mentioned that robustness under sensor faults is not quarantined
in this version of the framework but is left as future work.
Sensor fault tolerance is a vital issue in control system design
because the stability of the system depends on it. The sensor fault
modelling can be done in three ways [38]: (i) abrupt fault
(stepwise), (ii) incipient fault (drift-like) and (iii) intermittent
fault. For the simulation tests the first case is assumed in this
paper. Since the Yo ¼ fi; ðztÀzÞ; €zg and the airgap measurement is a
standard measurement it is assumed that faults can affect the
current and the acceleration sensors. Fig. 14 illustrates the abrupt-
like fault on the accelerometer during the deterministic and
stochastic track profiles at 1 s. The output signal is normal until
the first second but afterwards the fault is injected having low
frequency random characteristics. Similar profile appears on the
current sensor after the fault is injected.
The sensor fault tolerance based on the diagram in Fig. 8
involves all possible sensor fault conditions that could possibly
happen with the i and €z and the EMS performance after the
controller reconfiguration. The results are listed in Table 5. The
Fault
Fault time
Fig. 14. Abrupt fault profiles of the accelerometer sensor in Yo. (a) Fault profile of the acceleration sensor for the deterministic response. (b) Fault profile of the acceleration
sensor for the stochastic response.
Table 5
Possible sensor fault conditions with Yo ¼ id : 8 and nominal performance recovery.
Healthy sensor
sets, Yoh
Faulty sensor
sets, Yof
LSDP
controller
EMS performance
Deterministic
input
Stochastic
input
i; ðzt ÀzÞ; €z – Ki;ðzt ÀzÞ;€z
H11
✓ ✓
i; ðzt ÀzÞ €z Ki;ðzt ÀzÞ
H12
✓ ✓
ðzt ÀzÞ; €z i Kðzt ÀzÞ;€z
H13
✓ ✓
ðzt ÀzÞ i; €z Kðzt ÀzÞ
H14
✓ ✓

first row is the optimum sensor set id:8 and the rest are the
healthy sensor sets after faults with the corresponding controllers
and EMS performance. The results show that the nominal perfor-
mance of the EMS system is maintained in both stochastic and
deterministic track profiles.
Fig. 15 shows the closed-loop response of the EMS system
under the worst case scenario where both i and €z fail at 1 s. The
stability of the system is successfully maintained in both cases.
Additionally, the figures show the error of the airgap with fault-
free condition and with faults on the aforementioned sensors. The
peak-error with the deterministic input is at 2.5 mm and then it
goes to zero while the stochastic response has a small continuous
error without any significant effect on the performance of the EMS.
7. Conclusions
The paper presented an approach for selecting sensor sets for
control and fault tolerance on EMS systems, aiming to offer a level
of simplicity and flexibility in such a demanding design problem.
The computational expensive, off-line framework successfully
identifies fallback options in case of multiple sensor faults with
minimum sensor redundancy. Advanced software programming,
i.e. parallel programming, can alleviate some of the time consuming
nature of the framework (which arises due to the complex nature of
the problem and related constraints and parameter dependence) by
reducing computational effort. It is worth noting that user experi-
ence can also relax the issue of parameter dependence. With no loss
of generality the framework can be applied to a wider range of
engineering systems with multiple conflicting and complex control
requirements.
Acknowledgements
Parts of this research were supported by Engineering and
Physical Sciences Research Council in United Kingdom, BAE
Systems (Systems Engineering Innovation Center) under project
grand ref. EP/D063965/1 and by NEW-ACE EPSRC network under
project grand ref. EP/E055877/1. The authors would also like to
thank the anonymous reviewers for their rigorous review of this
paper that significantly improved the quality of it.
Appendix A. SFAR calculation example
Assume that there are three sensors at the output of a plant to
be used for control, i.e. y1; y2; y3. The total number of sensor sets in
Yi is given by Nss ¼ 2ns
À1 ¼ 7 hence Yi ¼ fY1; Y2; Y3; Y4; Y5;
Y6; Y7g. After the performance optimization of the plant with each
sensor set in Yi, the SFAR to be calculated for the full sensor set,
Y7 ¼ fy1; y2; y3g. If there are no control constraint violations with
Y7 (i.e. Ωkc
= 0) the SFAR is calculated, otherwise the algorithm
sets SFAR = 0. Until this stage, Ωkc
in Ω3 for each sensor sub-set (i.
e. Yð1–6Þ) is calculated as shown in Table A1. Note that Ωkc
for Y3
and Y6 is not equal to zero indicated by ‘ Â ’ mark. This means that
there are one or more control constraint violation using the
corresponding sensor set. If Ωkc
¼ 0 means that all control con-
straints are with in the pre-set limits and is indicated by ‘✓’. By
definition the SFAR ðYi;Ωkc ¼ 0Þ is given by
SFARðY7;Ωkc ¼ 0Þffi
NYh;Ωkc ¼ 0
NYf
100 ð%Þ ðA:1Þ
Fault
Fault
Fig. 15. Airgap error ðzt ÀzÞe between the airgap with fault-free condition ðzt ÀzÞ and the airgap under the sensor faults condition ðzt ÀzÞf . Both i and €z in the Yo ðid : 8Þ
simultaneously fail at 1 s. (a) Response with deterministic input and (b) response with stochastic input.
Table A1
Sensor subsets of Y7 with the corresponding Ωkc
.
Yi Sensor sub-sets of Y7 Ωkc
Y1 y1 ✓
Y2 y2 ✓
Y3 y3 x
Y4 y1,y2 ✓
Y5 y2,y3 ✓
Y6 y1,y3 x
Table A2
SFAR calculation algorithm.
if Ωkc
of Y7 equal to zero
Set NYh;Ωkc
¼ 0
¼ 0, NYf
¼ 2nY7 À2
for i¼1 to NYf
if Ωkc
of Yi equal to zero
NYh;Ωkc
¼ 0
¼ NYh;Ωkc
¼ 0
þ1
end
end
SFARðY7;Ωkc ¼ 0Þ ¼ ðNYh;Ωkc
¼ 0
=NYf
Þ100n%
else SFARðY7;Ωkc ¼ 0Þ ¼ 0
end

assuming that not all sensors can fail in Y7, NYf
¼ 2nY7 À2 ¼ 6, nY7
is the number of sensors in Y7. Then, NYh;Ωkc ¼ 0 is calculated using
a simple iterative algorithm given in Table A2.
In this case it is found that SFAR of Y7 is found to be
SFARðY7;Ωkc ¼ 0Þ ¼ ð4=6Þn100 ¼ 66:66% which means that 4 sub-
sets of Y7 can be used to restore the performance of the plant
with various sensor faults.
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