Robust Integral Backstepping Control for HEV.pdf

Received January 20, 2020, accepted February 24, 2020, date of publication March 4, 2020, date of current version March 19, 2020.
Digital Object Identifier 10.1109/ACCESS.2020.2978258
Robust Integral Backstepping Control for Unified
Model of Hybrid Electric Vehicles
SYED AHMAD SIFFAT , IFTIKHAR AHMAD , AQEEL UR RAHMAN , AND YASIR ISLAM
School of Electrical Engineering and Computer Science, National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan
Corresponding author: Iftikhar Ahmad (iftikhar.rana@seecs.edu.pk)
ABSTRACT Exponential decrease in oil and natural gas resources, increasing global warming issues and
insufficiency of fossil fuels has shifted the focus to fuel cell hybrid electric vehicles (FHEVs). FHEV model
used in this work consists of fuel cell, ultracapacitor and battery. Non-linearities present in the vehicle model
dominate because of extreme driving conditions like rough terrains, slippery roads or hilly areas. Behavior
of components like energy sources, induction motors and power processing blocks deviate significantly
from their normal behavior when driving in highly demanding situations. To tackle these shortcomings,
non-linear controllers are preferred because of their efficiency. In literature, different controllers have been
proposed for either the energy sources or the induction motor separately, whereas this research work focuses
on a unified hybrid electric vehicle (HEV) model to simultaneously control the energy sources and the
induction motor. The model used is a complete representation of electric system of FHEV and increases
the performance of the vehicle. This unified model provides improved DC bus voltage regulation along with
speed tracking when subjected to European extra urban drive cycle (EUDC). In this work, Robust Integral
Backstepping and Robust Backstepping controllers have been designed. Lyapunov based analysis ensures
the global stability of the system. Performance of proposed controllers is validated in MATLAB/Simulink
environment. A comparative analysis is also given to illustrate the importance of the unified model proposed
in this work.
INDEX TERMS Hybrid electric vehicles, unified model, non-linear control, integral backstepping control,
backstepping control, stabilizing functions.
I. INTRODUCTION
Carbon dioxide (CO2) makes 65% of the global greenhouse
gases and its majority portion is attributed to transport indus-
try [1]. The carbon footprint of renewable energy sources is
far less as compared to conventional fossil fuels or natural gas
resources. So, a lot of research is focused on the utilization
of these renewable energy sources to mitigate the global
environmental losses [2]. Rapid decline in fuel reserves and
a rise in demand of personal vehicles have increased the
necessity of eco-friendly and greener transport system.
Plugin hybrid electric vehicles (PHEV), battery electric
vehicles (BEV) and fuel cell hybrid electric vehicles (FHEV)
serve as a solution to the problems of fuel crisis and pollution
related environmental issues [3], [4]. BEVs and PHEVs both
need electricity from grid to charge the batteries. The major
downside of these two types is that the setup costs of charging
stations is too high and overall power generation is required
The associate editor coordinating the review of this manuscript and
approving it for publication was Lei Wang.
to be increased in order to facilitate the rapid charging of the
batteries [5]. On the other hand, FHEVs have no internal com-
bustion engine (ICE) and have an electric motor, hence they
are also emission free [6]. FHEVs have significant advantage
over the other two types in terms of their long driving range
and reliability [7].
ICE based vehicles have proven their stability and reli-
ability in toughest terrains and offer a long driving range.
In order to be widely accepted as a replacement of
ICE based vehicles, electric vehicles need performance and
driving range that is similar to or even better than that of
conventional vehicles. FHEVs promise longer driving range
and better performance in rough driving conditions when
hybrid of fuel cell and energy storage devices is used. Fuel
cells alone don’t have the capability to meet the require-
ments of dynamic driving range, hence FHEVs use different
combinations of fuel cells, ultracapacitor and battery. There
are still a few challenges that are required to be addressed
before these vehicles can be commercially realized on large
scale [8].
49038
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VOLUME 8, 2020

S. A. Siffat et al.: Robust Integral Backstepping Control for Unified Model of HEVs
In order to improve the performance of HEVs, it is nec-
essary that the current tracking of the energy sources and the
speed tracking of the motor are robust and accurate. A number
of linear controllers have been implemented on FHEVs in
order to enhance their performance like Model Predictive
Control, Fuzzy logic control and Proportional Integral (PI)
control [9]–[12]. The performance of these linear controllers
is acceptable but due to inherent limitations of the linear
controllers, the operating range of the vehicle is significantly
reduced.
Slippery roads, uphill/downhill drive, sudden acceleration
or braking and frequent start-stops make driving experience
very demanding and dynamic. The energy sources, power
converters and the induction/traction motor of HEVs are
dominated by non-linearities because of these high dynamic
requirements. Hence the HEV models are usually complex
and nonlinear in nature. Performance of nonlinear controllers
for a dynamic nonlinear models is better than that of linear
controllers because linear controllers fail to take into account
the system non-linearities and are only stable when the sys-
tem operates in a limited operating range [13], [14]. On the
other hand, nonlinear controllers consider all the parameter
deviations of the model. So, increased operating range of
the model and global stability of the system is ensured [15].
Parallel Hybrid Energy Storage System (HESS) modeling
and control has been done in [16]. With regards to nonlinear
controllers, Lyapunov based controller has been proposed
in [17] and sliding mode controller for FHEV has been pro-
posed in [18].
FHEV modeled with fuel cell and ultracapacitor based
HESS have been proposed in [19], [20], whereas ultraca-
pacitor and battery based HESS in [21]. An updated model
which uses all three sources: fuel cell, ultracapacitor and
battery as HESS has been proposed in [17], [22]. This model
uses fuel cell as primary source while ultracapacitor and
battery act as auxiliary sources to store energy. This model
offers increased energy storage capacity and high current and
energy density [23].
In hybrid electric vehicles, meeting the dynamic load
demands is the sole responsibility of energy sources which
operate at different voltage levels and ensure the provision
of desired DC bus voltage for a smooth ride. The electric
motor is responsible for speed tracking under a dynamic load
torque and has a wide operating range. Power converters play
a vital role in efficient voltage regulation and serve as an
interface between different energy sources and the induction
motor. Different converter topologies have been proposed
for these three sources in [24], [25]. Single bi-directional
DC-DC converter, flyback converter and multi-device inter-
leaved DC-DC converters have been proposed in [26]–[28].
In [26], [27] separate power converters for each source has
been proposed. This is known as multiple converter topology
and is more costly. Multi-input multi-output (MIMO) con-
verter topology has been proposed in [29] is a cost effec-
tive alternative but is more complex than multiple converter
topology.
FIGURE 1. Block diagram of HEV.
The aim of this work is to improve the performance and
driving experience of FHEV to the level of conventional ICE
based vehicles. FHEV under study has a 3 phase induction
motor and three energy sources; Fuel Cell, Ultracapacitor and
a Battery. In literature the energy sources part has been the
main focus for improvement of performance of vehicle and
controller has been designed separately for energy sources
and the induction motor part. Nonlinear control for FHEV
energy sources part has been proposed in [17], [19], [20], [30]
and it only improves DC bus voltage without taking into
account the motor dynamics which is the main factor for
load demand on the bus voltage. Here a complete unified
model is proposed which improves the overall performance
of the vehicle, while keeping in view the current limitations
of the energy sources and the load torque of the motor. This
model not only performs better regulation of the DC bus
voltage, but also tracks the desired European Extra Urban
Drive Cycle (EUDC) speed efficiently with perfect tracking
of energy sources’ current.
In this paper (a) backstepping, (b) robust backstep-
ping/backstepping integrated with Sliding Mode Controller
(SMC), (c) integral backstepping and (d) robust inte-
gral backstepping/integral backstepping integrated with SMC
have been proposed. A comparative analysis of the conven-
tional FHEV model and the proposed unified model, along
with a comparison of the aforementioned nonlinear con-
trollers on the proposed model using Matlab/Simulink have
also been given.
This article has been organized as; Section II gives brief
description of the system under study; sections III and IV
give mathematical modeling of unified FHEV model while
nonlinear control laws are formulated in section V; section VI
discusses performance of FHEV under proposed controllers
and the conclusion is given in section VII.
II. SYSTEM UNDER STUDY
The three important parts of FHEV are hybrid energy storage
system (HESS), power converters and traction motor.
A. HYBRID ENERGY STORAGE SYSTEM
Three energy sources of FHEV are fuel cell (FC), ultracapac-
itor and battery. FC alone can not provide a smooth driving
experience as it is unable to meet the load requirements alone.
It is always used as a primary source along with one or
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two secondary sources. Significance of all the energy sources
used in this model is discussed in this section briefly.
1) FUEL CELL
Fuel cells have higher conversion efficiency and their fuel
flexibility is more than internal combustion engines (ICE)
vehicles but they have cold start-up and poor transient
response [31]. There are many types of fuel cells used in
literature like Solid oxide FC, alkaline FC, molten carbonate
FC, phosphoric acid FC and proton exchange membrane
fuel cells (PEMFC) [32]. PEMFC has found its application
in vehicular technology because it has longer life, smaller
size, higher efficiency and its working temperature is lower
with better durability and shock resistance than other fuel
cells [33], [34].
In order to overcome the deficiencies of fuel cells, auxiliary
sources are used with them which improves the driving range
and reduces the load stress on individual sources is also
reduced. As a result, the life of the sources is also extended.
To meet the transients and start up requirements, different
combinations of FC, photovoltaic, ultracapacitor and battery
are proposed in [35]–[38].
2) ULTRACAPACITOR
Ultracapacitor’s high power density, small size, less charge
time and its ability to capture maximum regenerative energy
makes it a perfect candidate for use in FHEV [39]. The
life of ultracapacitors is also more than battery having a
range of about 1 million recharge cycles. In battery based
electric vehicles, peak current during acceleration and high
load torque is provided by ultracapacitors which relieves the
stress on battery and hence battery life is extended [40].
3) BATTERY
Fuel cells have a slow start-up and ultracapacitors have
problem of self-discharging, making battery a vital compo-
nent in FHEV for initial traction. Although batteries have
high energy storage density but their operation or recharge
cycles are quite limited in comparison to ultracapacitors. Nor-
mally battery recharge cycles range from 800 to 2000 cycles
depending upon the type of battery [41].
As each of these three energy sources complement each
other, this combination of energy sources in FHEV proves to
be a viable solution to the general shortcomings of electric
vehicles.
B. ELECTRIC MOTORS
Tough driving conditions require a motor that can oper-
ate efficiently under dynamic load requirements. Choice of
a suitable motor for electric vehicle plays a vital role in
vehicle’s performance. Electric motors have fast acceleration
and their torque is more accurate than internal combustion
engines [42]. Several types of electric motors are available
in market like DC motors, induction motors, brush-less per-
manent magnet motors and switched reluctance motors, but
the induction motors have proven to be better suited for
their use in vehicular technology. Due to cost effectiveness,
FIGURE 2. Electric circuit of power processing block.
easy availability and wide speed range, they are preferred
over other types of motors in electric and hybrid electric
vehicles [43], [44].
III. MATHEMATICAL MODEL OF ELECTRIC POWER STAGE
The circuit diagram of unified model of FHEV is shown
in figure 2. A DC-DC boost converter is used for fuel cell
whereas; a bi-directional DC-DC buck-boost converter is
used for both the ultracapacitor and the battery. A 3-phase
DC-AC converter is used for electric motor. The model of
FHEV has been derived using basic electric circuit laws on
energy sources and power converters.
The output voltage of DC-DC bus (vdc) appears across
the output capacitor Cdc. ua, ub, uc, ua0 , ub0 , uc0 are the
3-phase inverter switching control inputs. They are used to
obtain 3-phase DC-AC signal for driving the induction motor.
A. FUEL CELL - BOOST CONVERTER MODEL
Fuel cells lack the capability to utilize the regenerative brak-
ing or the excess power from any other sources, so unidirec-
tional DC-DC boost converter is sufficient for this source.
This converter consists of an insulated gate bipolar transis-
tor (IGBT) switch S1 shown in fig (2) and its value can
be 0 or 1. There are two operation modes of this converter
which depend upon the position of switch S1. Following
equations can be obtained after applying Kirchhoff’s laws to
boost converter:
difc
dt
= −(1 − u1)
vdc
L1
−
R1
L1
ifc +
vfc
L1
(1)
dvdc
dt
= (1 − u1)
ifc
Cdc
−
i1
Cdc
(2)
also
io = i1 + i2 + i3 (3)
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where
i2 = u23iuc (4)
and
i3 = u45ibat (5)
where i1, i2 and i3 represent the output currents of fuel
cell, ultracapacitor and that of battery converters respectively.
ifc, iuc and ibat are fuel cell, ultracapacitor and battery input
currents respectively and vdc is output bus voltage. u1 is the
duty cycle of switch S1.
Equation (2) can be written as
dvdc
dt
= (1 − u1)
ifc
Cdc
−
io
Cdc
+
iuc
Cdc
u23 +
ibat
Cdc
u45 (6)
B. ULTRACAPACITOR - BUCK-BOOST CONVERTER MODEL
Ultracapacitor can store the access charge of fuel cell and the
energy from regenerative braking. To use ultracapacitor as an
energy source and to charge it when required, a bidirectional
DC-DC buck-boost converter is used as shown in figure 2.
This buck-boost converter has two IGBT switches S2
and S3. The converter acts as a boost converter in order to
meet the peak power demands and as a buck converter to store
energy from regenerative braking. Boost mode refers to the
discharging of the ultracapacitor and buck mode represents
the charging. Equations (7) and (8) represent the boost and
buck modes of the converter respectively. In boost mode,
the switch S2 is on while input to S3 is zero, whereas in buck
mode the condition reverses. In this mode Kirchhoff’s laws
give
diuc
dt
=
vuc
L2
−
R2
L2
iuc − (1 − u2)
vdc
L2
(7)
diuc
dt
=
vuc
L2
−
R2
L2
iuc − u3
vdc
L2
(8)
Both of these equations can be written in a more compact
form with the introduction of a variable j and the average
control input u23 is defined as:
u23 = [j(1 − u2) + (1 − j)u3] (9)
So, equations (7) and (8) take the form
diuc
dt
=
vuc
L2
−
R2
L2
iuc − u23
vdc
L2
(10)
C. BATTERY - BUCK-BOOST CONVERTER MODEL
The battery operation is almost same as that of the ultraca-
pacitor, the equations are obtained in a similar way. Boost
and buck modes of the battery power converter are given by
equation (11) and (12) respectively as:
dibat
dt
=
vbat
L3
−
R3
L3
ibat − (1 − u4)
vdc
L3
(11)
dibat
dt
=
vbat
L3
−
R3
L3
ibat − u5
vdc
L3
(12)
Similarly, to write these equations in a compact form,
a variable k is introduced. The average control input u45 is
defined as:
u45 = [k(1 − u4) + (1 − k)u5] (13)
and equations (11) and (12) take the form
dibat
dt
= −u45
vdc
L3
−
R3
L3
ibat +
vbat
L3
(14)
D. COMBINED INVERTER-MOTOR MODEL
For tracking the speed of vehicle under changing load torque,
Field Oriented Control (FOC) strategy is used. In this strat-
egy, input power parameters are processed using a 3-phase
DC-AC inverter due to which the induction motor is con-
trolled indirectly.
The d-q axis rotated frame transformed mathemati-
cal model of the induction motor is given in [45] by
equations (15)-(19).
disd
dt
= baφrd + bpφrq − Υ isd + ωisq + m1vsd (15)
disq
dt
= baφrq − bpφrd − Υ isq − ωisd + m1vsq (16)
dφrd
dt
= −aφrd + (ωs − p)φrq + aMsr isd (17)
dφrq
dt
= −aφrq − (ωs − p)φrd + aMsr isq (18)
d
dt
= m(φrd isq − φrqisd ) − c −
Tl
J
(19)
where isd and isq are current components of stator’s d-axis and
q-axis respectively. Rotor flux d-axis and q-axis components
are represented by φrd and φrq. Load torque, stator frequency
and angular speed are represented by Tl, ωs and respec-
tively. The number of pole-pairs is denoted by p and where
a =
Rr
Lr
, b =
Msr
LsLr σ
, σ = 1 −
M2
sr
LsLr
Υ =
L2
r Rs + M2
sr Rr
σLsL2
r
, m1 =
1
Lsσ
m =
pMsr
JLr
, c =
fv
J
where Lr , Ls, Rr and Rs are rotor and stator self-inductances
and resistances respectively. Mutual inductance between sta-
tor and rotor windings is Msr . J gives inertia of the motor and
load; coefficient of viscous damping is represented by fv.
The two switching inputs of 3-phase inverter are given as
ud and uq in d-q axis rotating frame transformed mathemati-
cal model. The currents and voltages of d-axis and q-axis of
the inverter are as follows:
vdcio = isd vsd + isqvsq (20)
and the relations between inputs and outputs in term of
switching inputs are
vsd = ud vdc (21)
VOLUME 8, 2020 49041

and
vsq = uqvdc (22)
Equation (6) can be simplified using equations (20), (21)
and (22) taking the form
dvdc
dt
=(1−u1)
ifc
Cdc
+u23
iuc
Cdc
+ u45
ibat
Cdc
− ud
isd
Cdc
− uq
isq
Cdc
(23)
where ud and uq are the input signals of the inverter for the
d-q axis for motor control.
IV. UNIFIED MODEL OF FHEV
The complete mathematical model of FHEV is given in
equations (24) through (32). These equations represent the
unified model of FHEV with mathematical modeling of the
energy sources and the motor. All these controlled variables
are interlinked directly or indirectly. Renaming the following
variables as
x1 = ifc, x2 = iuc, x3 = ibat, x4 = vdc
x5 = isd , x6 = isq, x7 = φrd , x8 = φrq, x9 =
So, we get
ẋ1 = −(1 − u1)
x4
L1
−
R1
L1
x1 +
vfc
L1
(24)
ẋ2 = −u23
x4
L2
−
R2
L2
x2 +
vuc
L2
(25)
ẋ3 = −u45
x4
L3
−
R3
L3
x3 +
vbat
L3
(26)
ẋ4 = (1 − u1)
x1
Cdc
+ u23
x2
Cdc
+ u45
x3
Cdc
− ud
x5
Cdc
− uq
x6
Cdc
(27)
ẋ5 = bax7 + bpx9x8 − Υ x5 + ωsx6 + m1ud x4 (28)
ẋ6 = bax8 − bpx9x7 − Υ x6 − ωsx5 + m1uqx4 (29)
ẋ7 = −ax7 + (ωs − px9)x8 + aMsr x5 (30)
ẋ8 = −ax8 − (ωs − px6)x7 + aMsr x6 (31)
ẋ9 = m(x7x6 − x8x5) − cx9 −
Tl
J
(32)
x1 through x3 are the averaged current values of fuel cell,
ultracapacitor and battery respectively, while x4 is the DC bus
voltage. x5 and x6 represent d-axis and q-axis stator currents
respectively, whereas d-axis and q-axis rotor flux are repre-
sented by x7 and x8 respectively. The angular speed of the
induction motor is represented by x9. The difference between
the conventional FHEV model and the unified model can eas-
ily be observed. The dynamics of the energy sources depend
on DC bus, whereas the dynamics of the DC bus here also
incorporates the motor input current dynamics. The motor
current dynamics are linked with the dynamics of motor flux
and angular speed of motor as well.
V. CONTROLLER DESIGN
The proposed nonlinear controller for FHEV should be able
to fulfill the following objectives efficiently in order to be
considered effective:
• Voltage regulation of DC bus with varying load,
• Currents tracking of energy sources to their reference
values,
• Speed tracking of induction motor to its reference value,
• Global assymptotic stability of the controlled system.
As the boost converter of FC has non-minimum phase
nature, we can not directly track the reference value of
DC bus voltage. The DC bus voltage and FC current are
linked to each other such that if FC current tracks its ref-
erence value ifc, DC bus tracks reference voltage vdc. The
relationship of these reference values is given in the following
equations:
Pin = Pout (33)
Pfc + Puc + Pbat = Pout (34)
The input and output powers of the system must be equal
and the input power is the power of energy sources. Pin and
Pout are the input and output powers respectively, whereas
Pfc, Puc and Pbat are fuel cell, ultracapacitor and battery
powers respectively. Solving equation (34) for x1ref , we have
x1ref = β

vdcref io − vuciuc − vbatibat
vfc

(35)
Here, x1ref is the reference value of FC and β is the ideality
factor which must have a value greater than 1. This factor is
used to compensate the losses in power converters.
A. INTEGRAL BACKSTEPPING SLIDING MODE
CONTROLLER DESIGN
In order to track the currents of energy sources to their ref-
erence values, corresponding error signals must be defined.
The error in FC current is defined as:
e1 = x1 − x1ref (36)
Taking the time derivative of equation (36) to get its
dynamics as
ė1 = ẋ1 − ẋ1ref (37)
Actual dynamics of the error can be found using equa-
tion (24) as
ė1 = −(1 − u1)
x4
L1
−
R1
L1
x1 +
vfc
L1
− ẋ1ref (38)
Adding integral action to the error e1:
ζ1 = e1 + σ (39)
where
σ =
Z t
0
(x1 − x1ref )dt (40)
e1 given by equation (36) must approach zero for which the
Lyapunov candidate function is chosen as,
V1 =
1
2
e2
1 +
κ
2
σ2
(41)
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here κ is a positive constant. Time derivative of V1 from
equation (41) gives:
V̇1 = e1ė1 + κσσ̇ (42)
Equation (40) gives:
σ̇ = e1 (43)
Using the value of ˙
e1 and σ̇ from equations (38) and (43)
respectively, equation (42) becomes:
V̇1 = e1

−R1
L1
x1 +
1
L1
vfc −
1 − u1
L1
x4 − ẋ1ref + κσ

(44)
To ensure that V̇1 stays negative definite, let us take
−R1
L1
x1 +
1
L1
vfc −
1 − u1
L1
x4 − ẋ1ref + κσ = −k1e1 (45)
where the design parameter k1 can have any positive constant
value. Taking x4
L1
as virtual control and representing it by γ ,
we get
γ =

k1e1 + −R1
L1
x1 + 1
L1
vfc − ẋ1ref + κσ

1 − u1
(46)
For the system to be stable, x4
L1
= γ must hold true at all
times. Hence, defining e2 as:
e2 =
x4
L1
− γ (47)
which must go to zero to ensure x4
L1
= γ .
Using equation (47) and putting the value of virtual
control x4
L1
in equation (38), we get
˙
e1 =
−R1
L1
x1 +
1
L1
vfc − (1 − u1)(e2 + γ ) − ẋ1ref (48)
Putting the value of γ from equation (46) in equation (48),
we get:
˙
e1 = −κσ − k1e1 − (1 − u1)e2 (49)
Now, equation (44) can be written as:
V̇1 = −k1e2
1 − (1 − u1)e1e2 (50)
Time derivative of e2 in equation (47) gives
˙
e2 =
ẋ4
L1
− γ̇ (51)
Taking time derivative of γ from equation (46), we get:
γ̇ =

k1ė1 − R1
L1
ẋ1 + 1
L1
v̇fc − ẍ1ref + κe1
(1 − u1)

+
u̇1γ
(1 − u1)
(52)
To simplify the above expression, declaring a new variable
for numerator of first term,
ζ = k1ė1 −
R1
L1
ẋ1 +
1
L1
v̇fc − ẍ1ref + κe1 (53)
Using equation (49) for value of ė1, ζ can be expressed in
terms of e1 and e2. Now, equation (52) can be written as:
γ̇ =
ζ
(1 − u1)
+
u̇1γ
(1 − u1)
(54)
Substituting the values of ˙
x4 and γ̇ from equations (27) and
(54) respectively in equation (51), we get:
˙
e2 =
1
L1

(1 − u1)
x1
Cdc
+ u23
x2
Cdc
+ u45
x3
Cdc
− ud
x5
Cdc
−uq
x6
Cdc

−
ζ
(1 − u1)
−
u̇1γ
(1 − u1)
(55)
Combined Lyapunov function is defined as follows for
ensuring convergence of both the errors e1 and e2 to zero,
Vc1 = V1 +
1
2
e2
2 (56)
Taking time derivative of equation (56), we have
V̇c1 = V̇1 + e2ė2 (57)
Putting the value of V̇1 from equation (50), we get:
V̇c1 = −k1e2
1 − e2[(1 − u1)e1 − ė2] (58)
For making V̇c1 negative definite, taking
(1 − u1)e1 − ė2 = k2e2 (59)
where k2 is design parameter of control and k2 0 for all
times. So, equation (58) becomes
V̇c1 = −k1e2
1 − k2e2
2 (60)
Using ė2 from equation (55) in equation (59) and solving
for control law u1, we get
u̇1 =
(1 − u1)
γ

k2e2+
1
L1
(1 − u1)
x1
Cdc
+u23
x2
Cdc
+ u45
x3
Cdc
− ud
x5
Cdc
−uq
x6
Cdc

−(1−u1)e1−
ζ
(1 − u1)

(61)
This control law would ensure tracking of x1 and x4 to their
reference values x1ref and x4ref respectively.
To incorporate robustness in integral backstepping, a slid-
ing surface is defined as:
S = a1e1 + a2e2 (62)
Based on the surface S, adding a switching term in control
law u1 to formulate integral backstepping plus SMC con-
troller. Using u̇1 from equation (61), the final control law u1
becomes:
u1 =
Z t
0
u̇1dt−ks(sat(S/B)) (63)
where B ∈ [0, 1], ks 0 and sat represents saturation
function.
sgn(S) =





−1, if x 0
0, if x = 0
1, if x 0
(64)
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Being approximation of signum function (sgn), saturation
function (sat) turns out to be:
sat(S/ε) =
(
S/ε, if | S/ε |≤ 1
sgn(S/ε), otherwise
(65)
To define the control law for the remaining energy sources,
we need to find u23 for tracking x2 to its reference value x2ref .
For ultracapacitor, following error term is introduced:
e3 = x2 − x2ref (66)
Taking time derivative of e3 from equation (66) and substi-
tuting the value of ˙
x2 from equation (25), we get:
˙
e3 = −u23
x4
L2
−
R2
L2
x2 +
vuc
L2
− ẋ2ref (67)
Lyapunov candidate function of error e3 is selected as:
V2 =
1
2
e2
3 (68)
Taking time derivative of V2 gives:
V̇2 = e3 ˙
e3 (69)
To make ˙
V2 negative definite, we take
ė3 = −k3e3 (70)
so that equation (67) gives
−u23
x4
L2
−
R2
L2
x2 +
vuc
L2
− ẋ2ref = −k3e3 (71)
So, equation (69) becomes:
˙
V2 = −k3e2
3 (72)
Solving equation (71) for u23, we get following control
law:
u23 =
1
x4

L2k3e3 − R2x2 + vuc − L2ẋ2ref

(73)
The combined Lyapunov function is taken as,
Vc2 = Vc1 + V2 (74)
Taking time derivative of Vc2 and using equations (71)
and (74) we get
V̇c2 = −k1e2
1 − k2e2
2 − k3e2
3 (75)
Similarly, for battery we need to find the control law u45 so
that x3 tracks its reference value x3ref . Defining the following
error for this purpose,
e4 = x3 − x3ref (76)
Taking time derivative of e4 from equation (76)
ė4 = ẋ3 − ẋ3ref (77)
substituting the value of ˙
x3 from equation (26), we get:
˙
e4 = −u45
x4
L3
−
R3
L3
x3 +
vbat
L3
− ẋ3ref (78)
Lyapunov candidate function is defined as:
V3 =
1
2
e2
4 (79)
Time derivative of V3 gives:
V̇3 = e4 ˙
e4 (80)
For making ˙
V3 negative definite, we take
ė4 = −k4e4 (81)
where k4 0. So, equation (78) gives
−u45
x4
L3
−
R3
L3
x3 +
vbat
L3
− ẋ3ref = −k4e4 (82)
Solving equation (82) for u45 in order to get the control law
as
u45 =
1
x4

L3k4e4 − R3x3 + vbat − L3ẋ3ref

(83)
Combined Lyapunov function is taken as
Vc3 = Vc2 + V3 (84)
Its time derivative gives
V̇c3 = −k1e2
1 − k2e2
2 − k3e2
3 − k4e2
4 (85)
For tracking motor speed and d-axis rotor flux, we need to
design the control law ud and uq. Defining d-axis rotor flux
error as follows
e5 = x7 − φ∗
rd (86)
and speed error is defined as,
e6 = x9 − d (87)
where φ∗
rd and d are represent corresponding reference
values.
In order to implement field oriented control (FOC), it is
necessary that the modulus of rotor flux is equal to the mag-
nitude of d-axis component rotor flux. Following expression
must be satisfied:
q
φ2
rd + φ2
rq =k φrd k (88)
It is only possible when q-axis component of rotor flux is
made zero. Stator frequency ωs is selected in such a way that
it makes equation (31) exponentially zero, taking x8 and its
dynamics equal to zero. The effect of selection of this type of
frequency is observed in design of control law.
ωs = px6 + a
Msr
x7
x6 (89)
Taking time derivative of equation (86) gives
ė5 = ẋ7 − ˙
φ∗
rd (90)
Putting the value of ẋ7 from equation (30) in equation (90),
we get:
ė5 = −ax7 + aMsr x5 − ˙
φ∗
rd (91)
49044 VOLUME 8, 2020

Lyapunov candidate function is taken as:
V4 =
1
2
e2
5 (92)
Taking time derivative of equation (92), we get:
V̇4 = e5 ˙
e5 (93)
For making ˙
V4 negative definite we put
ė5 = −k5e5 (94)
−ax7 + aMsr x5 − ˙
φ∗
rd = −k5e5 (95)
Taking x5 in equation (95) as virtual controller x5d and
solving for it,
x5d =
1
aMsr
(−k5e5 + ax7 + ˙
φ∗
rd ) (96)
Stability is ensured only when x5d = x5, so defining
another error,
e7 = x5 − x5d (97)
from here,
x5 = e7 + x5d (98)
Putting value of x5 from equation (98) in equation (91),
we get:
ė5 = −k5e5 + aMsr e7 (99)
Substituting ė5 from equation (99) in equation (93). V̇4
becomes,
V̇4 = −k5e2
5 + e5(aMsr e7) (100)
Dynamical error of e7 is obtained by taking the time deriva-
tive of equation (97), we get:
ė7 = ẋ5 − ẋ5d (101)
Putting ẋ5 from equation (28) in above equation, we have
ė7 = bax7 − Υ x5 + ωsx6 + m1ud x4 − ẋ5d (102)
Let us define the Lyapunov candidate function as,
V5 = V4 +
1
2
e2
7 (103)
Taking time derivative of V5 and substituting value of V̇4
from equation (100), we get:
V̇5 = −k5e2
5 + e7(e5aMsr + ė7) (104)
For making equation (104) negative definite, let us take
design parameter k7 0 such that:
e5aMsr + ė7 = −k7e7 (105)
so,
V̇5 = −k5e2
5 − k7e2
7 (106)
Now, using the value of ė7 from equation (102) and solving
for ud , we get
ud =
1
m1x4
(−k7e7 − aMsr e5 − bax7 + Υ x5 − ωsx6+ẋ5d )
(107)
To find uq, taking time derivative of equation (87), we have
ė6 = ẋ9 − Ω̇d (108)
Substituting ẋ9 from equation (32) in equation (108),
we get:
ė6 = mx7x6 − cx9 −
Tl
J
− Ω̇d (109)
Taking Lyapunov candidate function as:
V6 =
1
2
e2
6 (110)
Time derivative of equation (110) gives:
V̇6 = e6 ˙
e6 (111)
In order to make ˙
V6 negative definite, we put
ė6 = −k6e6 (112)
mx7x6 − cx9 −
Tl
J
− Ω̇d = −k6e6 (113)
Replacing x6 in equation (113) as virtual controller x6d and
solving for it, we get
x6d =
1
mx7
(−k6e6 + cx9 +
Tl
J
+ ˙
d ) (114)
To ensure stability, x6d = x6 must hold true. So, defining
another error,
e8 = x6 − x6d (115)
Rearranging to get x6, as
x6 = e8 + x6d (116)
Substituting the value of x6 from equation (116) in equa-
tion (109), following expression is obtained:
ė6 = −k6e6 + mx7e8 (117)
Putting ė6 from above in equation (111). V̇6 becomes,
V̇6 = −k6e2
6 + e6(mx7e8) (118)
Taking time derivative of equation (123) gives:
ė8 = ẋ6 − ẋ6d (119)
Substituting ẋ6 from equation (29) in equation (119),
we have
ė8 = −bpx9x7 − Υ x6 − ωsx5 + m1uqx4 − ẋ6d (120)
Defining the Lyapunov candidate function as,
V7 = V6 +
1
2
e2
8 (121)
VOLUME 8, 2020 49045

Substituting value of V̇6 from equation (118) after taking
time derivative of equation (121) as,
V̇7 = −k6e2
6 + e8(e6mx7 + ė8) (122)
In order to make ˙
V7 negative definite, we put
e6mx7 + ė8 = −k8e8 (123)
where k8 0, so
V̇7 = −k6e2
6 − k8e2
8 (124)
Now, substituting ė8 from equation (120) in equation (123)
and solving for uq
uq =
1
m1x4
(−k8e8−mx7e6+bpx9x7 + Υ x6 + ωsx5 + ẋ6d )
(125)
Combined Lyapunov function becomes,
Vc = Vc3 + V5 + V7 (126)
Taking time derivative of Vc and using equations (106)
and (124), we get:
V̇c = −k1e2
1−k2e2
2−k3e2
3−k4e2
4−k5e2
5−k6e2
6−k7e2
7 − k8e2
8
(127)
V̇ =
8
X
i=1
−kie2
i (128)
From above equation, global stability of the system is
ensured.
B. BACKSTEPPING PLUS SMC CONTROLLER DESIGN
In integral backstepping, an integral term is added to error e1.
If integral action is not included in the error, equation (45)
becomes:
−R1
L1
x1 +
1
L1
vfc −
1 − u1
L1
x4 − ẋ1ref = −k1e1 (129)
Following the same steps as done in sub section V-A,
virtual control law γ for backstepping controller gets updated
as follows:
γ =

k1e1 + −R1
L1
x1 + 1
L1
vfc − ẋ1ref

1 − u1
(130)
This γ will only affect our control law u1. All other equations
remain same as for integral backstepping.
In order to incorporate robustness, equations (62) and (63)
can be used. Resulting control law u1 will constitute Back-
stepping plus SMC controller.
VI. SIMULATION RESULTS
The unified model with proposed controllers is simulated in
Matlab/Simulink environment. Tables 1, 2 and 3 represent
specifications of energy sources, converter and electric motor
respectively.
TABLE 1. Energy sources.
TABLE 2. Converter specifications.
TABLE 3. Electric motor.
A. PROPORTIONAL INTEGRAL (PI) CONTROLLER
The supremacy of the proposed unified model over the con-
ventional FHEV model is highlighted by the results of one of
the most widely used linear control techniques. The models
are subjected to same load profile as shown in figure 3.PI con-
troller is applied to both the models and the results in figure 4
clearly show that the unified model performs better voltage
regulation with similar control efforts for both the models.
Under the 10 second test, the unified model outperforms
the conventional model significantly. As it is evident from
figure 4, unified model is a better choice when it comes to
efficient control of hybrid electric vehicles.
The conventional model of FHEV is given by
equations (131 - 134). As conventional model does not
include the motor dynamics, it fails to capture the true
electrical nature of the system. Hence, the voltage regulation
is not as efficient as unified model.
ẋ1 = −(1 − u1)
x4
L1
−
R1
L1
x1 +
vfc
L1
(131)
49046 VOLUME 8, 2020

FIGURE 3. Load torque profile.
FIGURE 4. DC-DC bus voltage.
ẋ2 = −u23
x4
L2
−
R2
L2
x2 +
vuc
L2
(132)
ẋ3 = −u45
x4
L3
−
R3
L3
x3 +
vbat
L3
(133)
ẋ4 = (1 − u1)
x1
Cdc
+ u23
x2
Cdc
+ u45
x3
Cdc
−
1
Cdc
io (134)
B. INTEGRAL BACKSTEPPING PLUS SMC
The controller has been subjected to EUDC. It represents the
standard driving conditions in a city and is known for its
high speed driving mode. In this profile a vehicle can take
a maximum speed of 90 km/h and the speed of the profile
refers to the load on the motor. Equations used to compute
the power demand and corresponding load currents from DC
bus voltage and ultimately the energy sources are as follows:
Pout =
1
0.75

0.5pav2
t ACx + MgCr + M
dvt
dt

vt (135)
io =
1
(0.75)(400)

0.5pav2
t ACx +MgCr +M
dvt
dt

vt (136)
where the working efficiency of the inverter is considered to
be 75 percent. The vehicle speed is represented by vt, standard
TABLE 4. Specifications of vehicle parameters.
FIGURE 5. EUDC Cycle (a) speed reference (b) torque reference.
air density by pa, gravitational acceleration constant by g,
mass of car by M and A is the front area of the car. The table 4
gives the parameters used in the equations (135) and (136).
The EUDC speed profile is shown in figure 5(a), whereas
figure 5(b) shows the load torque of motor or the torque
reference. While driving, three modes of motor torque
are of utmost importance: acceleration, deceleration/braking
and steady speed. The load torque profile encompasses all
three modes and the controller’s functionality can be tested
effectively.
The controller gains are selected on a trial and error basis,
in which gains are chosen randomly and then checked for
error values. Those having lesser error are selected as final
gains given in table 5.
The switching control signals (duty ratio) of the control
laws of energy sources are shown in figure 6, the load current
VOLUME 8, 2020 49047

TABLE 5. Controller gains.
FIGURE 6. Converter control laws (a) u1 (b) u23 (c) u45.
profile is shown in figure 7. The currents of energy sources
are shown in figure 8 and the corresponding current tracking
errors of the energy sources are shown in figure 9. It can
be clearly observed that the current tracking errors are not
too large and are in an acceptable range because they do not
FIGURE 7. Load current profile.
FIGURE 8. Energy sources’ currents (a) Fuel Cell (b) Ultracapacitor
(c) Battery.
violate any limitation of the energy sources. Besides, this
work does not focus on energy management of the sources
due to which the peaks are observed in the errors. The errors
in figure 6(a), 6(b) and 6(c) refer to fuel cell error (e1),
ultracapacitor error (e3) and battery error (e4) respectively.
The control laws for the d-q axis motor are shown
in figure 10. The control laws ud and uq are responsible for
ensuring accurate speed tracking of the vehicle to the refer-
ence values by generating switching signals for the 3-phase
inverter. These control laws generate appropriate voltages vsd
and vsq across stator input, which in turn generates the motor
input currents isd and isq.
49048 VOLUME 8, 2020

FIGURE 9. Current tracking error (a) Fuel Cell (b) Ultracapacitor (c) Battery.
FIGURE 10. Motor control laws (a) ud (b) uq.
C. COMPARATIVE ANALYSIS
The main objectives of control design of FHEV include volt-
age regulation and accurate speed reference tracking. The
controller which achieves these objectives more efficiently
with lesser errors is considered to be better than the other
controllers in comparison. In this work four nonlinear control
controllers have been designed for the unified FHEV model:
Integral Backstepping plus SMC (IBS+SMC), Integral
FIGURE 11. DC-DC bus voltage tracking (Vdc ).
TABLE 6. Root mean square errors (RMSE).
Backstepping (IBS), Backstepping plus SMC (BS+SMC)
and Backstepping (BS). The voltage of DC-DC bus is sup-
posed to be 400 volts (V) under the dynamic load demands.
The comparison of DC-DC bus voltage (Vdc) or x4 for the
aforementioned nonlinear controllers is shown in figure 11.
The spikes in the graph correspond to the transitions or
changes of motor speed in EUDC speed profile (fig 5). At all
the transitions, the overshoots/undershoots of IBS+SMC are
smaller and they are short-lived spikes with fast settling time.
The steady state error of this controller is also lower than the
other applied nonlinear controllers and convergence is very
close to the targeted value of 400V.
The speed tracking of all the controller is very efficient
but IBS+SMC produces the better results. At each tran-
sition, the overshoot/undershoot and the settling time of
IBS+SMC is the less. A comparison of speed tracking of all
the proposed controllers is shown in figure 12. The figure 13
shows the speed tracking error for the nonlinear proposed
controllers. It can be easily observed from this figure that
IBS+SMC performs significantly better than the other three
controllers.
To further quantify the difference in performance of all the
applied controllers, Root Mean Square Errors (RMSE) of Vdc
and speed tracking is given in table 6. The table clearly high-
lights the significant improvement of the proposed controllers
over the others. The results are also supported by theoretical
concepts involving these techniques. Due to integral action,
VOLUME 8, 2020 49049

FIGURE 12. Speed tracking.
FIGURE 13. Speed error.
the steady state error of the tracking is significantly reduced;
the incorporation of switching term in equation (63) gives the
controller additional robustness, hence, making it better than
others.
The unified model performs better voltage regulation and
speed tracking than the conventional model and it is cemented
by the fact that the variants of backstepping have only been
applied to fuel cell control law u1. The remaining control laws
are based on normal backstepping, still there is a significant
FIGURE 14. Simulation control hierarchy.
difference between the tracking of all the controllers. The
effect of the different techniques in the design of FC control
law u1, for improvement of voltage regulation, has prop-
agated to the speed tracking of the motor and vice versa.
This proves irrefutably that incorporation of the motor part
improves the overall performance of vehicle by improving
the speed tracking and the voltage regulation. Figure 14 gives
the simulation control hierarchy for simultaneous control of
energy source converters and induction motor drive.
VII. CONCLUSION
In this research work, a unified model of fuel cell based
hybrid electric vehicle has been proposed along with robust
integral backstepping and robust backstepping controllers
for DC bus voltage regulation and motor speed control.
Lyapunov stability theory has been used for proving the
asymptotic stability. Performance of the proposed con-
trollers for the unified model has been analysed using
MATLAB/Simulink environment. Comparative analysis of
the unified model with the conventional model has been
done using linear controller. The results show that unified
model has significant improvements in voltage regulation and
efficiently tracks the EUDC speed profile. Also, comparative
analysis of nonlinear controllers with each other has been
given on the unified model. The results illustrate that robust
integral backstepping controller has substantially reduced the
overshoots/undershoots, convergence time and steady state
error; hence making robust it suitable for FHEV. Future works
may encompass implementation of these control methodolo-
gies on real platforms, different energy sources and converter
topologies or different controllers to further reduce speed
tracking and voltage regulation errors.
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VOLUME 8, 2020 49051

SYED AHMAD SIFFAT received the B.S. degree
from COMSATS University Islamabad, Lahore,
Pakistan, in 2018. He is currently pursuing the
M.S. degree in electrical engineering with special-
ization in power and control system at the School
of Electrical Engineering and Computer Science,
National University of Sciences and Technology
(NUST), Islamabad, Pakistan.
IFTIKHAR AHMAD received the M.S. degree
in fluid mechanical engineering from University
of Paris VI (University Pierre and Marie Curie),
Paris, and the Ph.D. degree in robotics, control,
and automation from the Universite de Versailles,
France.
He is currently an Assistant Professor with
the Department of Electrical Engineering, School
of Electrical Engineering and Computer Science,
NUST, Islamabad, Pakistan. His main research
interests include nonlinear systems, robotics, automation and control, hybrid
electrical vehicles, and maximum power point tracking.
AQEEL UR RAHMAN received the B.S. degree
from the Mirpur University of Science and Tech-
nology, in 2017, and the M.S. degree in electrical
engineering with specialization in power and con-
trol system from the School of Electrical Engineer-
ing and Computer Science, National University
of Sciences and Technology (NUST), Islamabad,
Pakistan, in 2019.
YASIR ISLAM received the B.S. and M.S. degrees
in electrical engineering with specialization in
power and control system from the School of
Electrical Engineering and Computer Science,
National University of Sciences and Technology
(NUST), Islamabad, Pakistan.
49052 VOLUME 8, 2020

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