Modeling and model predictive control for hybrid electric vehicles

Abstract

This paper builds up a typical model of a parallel hybrid electric vehicle and develops model predictive controllers for this model to control the speeds and torques for fast clutch engagement with high driving comfort and low jerk. Some modified algorithms for model predictive controllers are studied to improve their ability to track the desired speed setpoints, subject to input and output constraints.

Full text

International Journal of Automotive Technology, Vol. 13, No. 3, pp. 477−485 (2012)
DOI 10.1007/s12239−012−0045−0
Copyright © 2012 KSAE/064−14
pISSN 1229−9138/eISSN 1976−3832
477
MODELING AND MODEL PREDICTIVE CONTROL
FOR HYBRID ELECTRIC VEHICLES
V. T. M I N H * and A. A. RASHID
Mechanical Engineering Department, Universiti Teknologi PETRONAS, Tronoh 31750, Malaysia
(Received 1 February 2011; Revised 3 May 2011; Accepted 12 August 2011)
ABSTRACT−This paper builds up a typical model of a parallel hybrid electric vehicle and develops model predictive
controllers for this model to control the speeds and torques for fast clutch engagement with high driving comfort and low jerk.
Some modified algorithms for model predictive controllers are studied to improve their ability to track the desired speed
setpoints, subject to input and output constraints.
KEY WORDS : Vehicle modeling, Model predictive control, Speed tracking setpoints
1. INTRODUCTION
Hybrid electric vehicles (HEVs) are combined hybrid and
electric vehicles that combine the best features of internal
combustion engines (ICEs) and electric motors (EMs).
HEVs can be divided into two main groups: serial and
parallel hybrids. In the serial hybrid, the primary power
source (ICE) is not mechanically linked to the powertrain,
but it is used to provide electrical power to a battery. The
secondary power source (EM) can then draw the energy
from the battery to run the vehicle. In the parallel hybrid,
both power sources are independently installed so that they
can run the vehicle either individually or together. Most
commonly, the ICE and EM and gearbox are coupled by
automatically controlled clutches. For electric driving at
low speeds, the clutch between the ICE and the EM is
open, and only the EM propels the vehicle. At high speeds,
the clutch is closed, and the ICE is activated and runs the
vehicle while the EM is turned off. At very high speeds or
under very heavy loads, the EM can also be automatically
turned on to join with the ICE to propel the vehicle.
The motivation behind using the Model Predictive
Control (MPC) method in this work is the ability of MPC
to solve optimization problems online with both linear and
nonlinear systems. MPC refers to a class of algorithms that
compute a sequence of input variables to optimize the
future behavior of the output variables. One of the
superiorities of MPC is its ability to deal with constraints
within open-loop optimal control problems. Finding the
solution for general constrained nonlinear models over an
infinite prediction horizon is impractical because numerical
methods to solve these problems can only be formulated in
a finite horizon length to find a real-time numerical
solution. For this reason, only a finite moving horizon
regulator is considered, in which the optimization problem
is performed over a finite prediction horizon and the cost of
prediction after the end of the horizon is approximated by a
terminal penalty. To ensure the stability within a finite
prediction horizon, most Nonlinear Model Predictive
Control (NMPC) schemes use a terminal region constraint
at the end of the prediction horizon in (Rawlings, 2000;
Morari and Lee, 1999; Mayne, 1993). A particular NMPC
scheme using a terminal region constraint, namely, a quasi-
infinite horizon that guarantees asymptotic closed-loop
stability with input constraints, was presented in (Minh and
Afzulpurkar, 2006). However, the nonlinear systems then
have both input and output constraints, and difficulties
arise from the failure to satisfy the output constraints due to
constraints on input. Therefore, a new NMPC scheme
without a terminal region constraint is developed in (Minh
and Afzulpurkar, 2006) using softened output constraints.
A Robust Model Predictive Control (RMPC) that
guarantees stability in the present of model uncertainty
using Linear Matrix Inequalities (LMIs) subject to input
and output saturated constraints was presented in (Minh
and Afzulpurkar, 2005a). In this novel RMPC, the
controller softens the output constraints as penalty terms
are added into the objective function. These terms maintain
the output violation at low values until a constrained
solution is returned.
When there are too many input and output constraints,
the control system may not be able to meet all of the
desired outputs. Because the MPC regulator is designed for
an online implementation, any infeasible solution of the
optimization problem cannot be tolerated. To guarantee
system stability, the traditional MPC methods delete one or
*Corresponding author. e-mail: vutrieuminh@petronas.com.
my

478 V. T. MINH and A. A. RASHID
more of the output variables from the control objective
function. If some output setpoints are deleted, the system
becomes looser, and the probability that MPC can find a
solution will increase. The robustness of MPC can also
increase if some output setpoints can be relaxed. A new,
modified MPC algorithm that changes from the setpoints to
regions was proposed in (Minh and Afzulpurkar, 2005b).
In this research, the robustness of the MPC controller is
also tested by both deleting and changing the output
setpoints into regions with outputs that have ranges of
desired values instead of specific values. The output
violation can be regulated by changing the values of the
weighting factor.
Next, we review characteristics of some HEV
configurations, and then a typical HEV configuration is
selected and dynamic equations for this model are built.
MPC algorithms are constructed to control the speeds of
the combustion engines and electric motors for each part of
the HEV. Examples and simulations are also given after
each section to illustrate the main ideas of the section.
Finally, conclusions are drawn, and some directions for
future research are discussed.
2. HEV MODELING
Figure 1 shows the configuration of a very common parallel
HEV system developed by Daimler Chrysler called the
P12-Configuration, which appeared in the Detroit Motor
Show in 2004. It consists of a conventional ICE and two
electric motors, EM1 and EM2. An automatically controllable
friction clutch separates the drivetrain into two parts: Part 1:
ICE with EM1; and Part 2: EM2 and the rest of the
transmission. EM1 serves as a starter and a generator. There
is no torque converter in this configuration. The rear wheels
are the driven wheels, and they are connected with a
standard automated transferred gearbox and a differential
gearbox.
During pure electrical driving at low speeds (below 50 km/
h), the clutch is opened, and a series hybrid configuration is
achieved. In this operating range, the second DC Motor,
EM2, propels the vehicle. The transition from series mode
to parallel mode takes place at high speeds (above 50 km/h)
by closing the clutch. The first DC Motor, EM1, activates
the ICE to run the vehicle. Then EM2 turns off. EM1 turns
on as a generator to charge the battery. In critical times of
an urgent load demanded by the driver, both EM1 and EM2
can join together with the ICE to propel the vehicle.
This research focuses on the development of a reliable
controller system that can control the speeds of the two
drivetrain parts and synchronize these parts with a
controllable clutch, which can reduce jerk and increase
driving comfort.
A simplified dynamic model for this hybrid drivetrain is
shown in Figure 2. The first part of the powertrain can be
approximated by a lumped inertia J1, where the shaft
holding the ICE, EM1 and one clutch plate is modeled as a
rigid body. As proposed in (Fredrikssong and Egbert, 2000)
and (Powell et al., 1998), the second part of the powertrain
is modeled by two inertias J2 and J3, which are connected
by a mechanical spring. The automated transmission
gearbox and the differential gearbox are modeled as a
simple gain transforming torque with a variable factor
depending on the selected gear ratios.
J1: lumped inertia of the ICE and EM1, J2: inertia of
EM2, and J3: lumped inertia of the rest of the powertrain.
The powertrain can be considered as a connection of a
spring with the torsional rigidity kθ, the velocity damping
coefficient kβ and the acceleration damping coefficient kα.
The torques MICE, MEM1, and MEM2 are from the ICE, EM1,
and EM2, respectively. Rotating angles θ1, θ2, and θ3 and
angular velocities ω1, ω2, ω3 and are from shaft1, shaft2,
and shaft3, respectively. r is the vehicle dynamic wheel
radius.
There are three separate contributions to the vehicle
resistance model: the air drag; the rolling resistance and the
combined mechanical losses in the gearbox—the differential
and the shaft bearings due to friction. They can be
approximated into the vehicle velocity resistance torque, Mv,
as:
(1)
where ρ: air density, cw: drag coefficient, A: vehicle frontal
area, r: wheel dynamic radius, fr: resistance coefficient, m:
vehicle mass, g: natural gravity, and ai: polynomial
coefficients. The influence of the road inclination and the
road dynamics can be considered as disturbances to the
system. These disturbances can lead to some additional
accelerating or decelerating torques in the rolling resistances.
However, the changes of these loads and the change of the
vehicle mass during the vehicle dynamics are not
Mv
ρ
2
---cwArω3
()
2frmg+
⎝⎠
⎛⎞
ra
0a1ω3a2ω3
2
++ +=
Figure 1. Configuration of the parallel hybrid powertrain.
Figure 2. Simplified drivetrain structure.

MODELING AND MODEL PREDICTIVE CONTROL FOR HYBRID ELECTRIC VEHICLES 479
considered in this paper.
Because the development of this powertrain model is
built for the clutch engagement at low vehicle speeds
(below 50 km/h), the contribution of the exponential term
of to Mv is small and the linearization of the resistance
torque Mv in equation (31) can be considered as:
(2)
where Mv0 is the initial constant for air drag torque and the
kv is the linear air drag coefficient. Because of the
difference in gear ratios, the coefficients M0 and kv can vary
for each gear ratio.
The following torque equations are applied for the first
part:
(3)
The driving torque is computed as:
(4)
The friction torque MC transmitted by the clutch can be
divided into two engagement modes:
the clocked mode, when the friction torque (MC) exceeds
the static friction capacity ( ):
(5)
where FNC: normal force exerted on the clutch, rC: clutch
corresponding radius, and µs: clutch static friction
coefficient and the slipping mode:
(6)
where µK: clutch slipping kinetic friction coefficient.
The following torque equations are applied for the
second part:
(7)
or:
(8)
and
(9)
The torque M2o is computed as:
(10)
with η: efficiency of gearbox and differential.
The above torque equations can be transformed to the
following dynamic equations:
(11)
(12)
where kβ1 is the friction coefficient in shaft 1.
(13)
(14)
where kβ2 is the friction coefficient in shaft 2.
(15)
(16)
where kβ3 is friction coefficient in shaft 3.
(17)
Replace the torque generated by a DC Motor in the
following formula:
(18)
where MDC_MOTER : torque by DC Motor; kT : motor torque
constant, where (Nm/A); kE: motor electromotive
force (EMF) constant (V-sec/rad), where kE=kT
; RI : terminal
resistance (Ohm); V: power supply (Volts); and ω: angular
velocity (rev/min - RPM).
Then a new set of vehicle dynamics is installed as:
(19)
(20)
(21)
(22)
(23)
ω3
2
MvMv0kvω3
+=
M1oJ1ω
·
1
=
M1oMICE MM1MC
–+=
Mfmax
Staitc
MC
2
3
---rCFNCµs when MCMf max
Static
=()=
MCrCFNCsign ω1ω2
–()µK when MCMf max
Static
<()=
M2okθθ2
kθ
i
---- θ3kvω3
++=
M2oJ2ω
·2iJ
3ω
·3kvω3
++=
M
··
2oJ2ω
·· 2ik
α
ω
·2
i
----- ω
·3
–
⎝⎠
⎛⎞
kβ
ω2
i
----- ω3
–
⎝⎠
⎛⎞
++=
M2oMEM2Mc
+()ηiM
v0
–=
θ
·1ω2
=
ω
·2
kβω1
J1
---------- MICE
J1
---------- MM1
J1
--------- M–C
J1
----------
+++=
θ
·
2ω2
=
ω
·2
kβ2ω3
J2i
------------J3ω3
J2i
---------- ηMM2
J2
-------------
–– ηMC
J2
-----------M–vC
J2i
------------
–+=
θ
·
3ω3
=
ω
·3
kβ3ω3
J3
------------MV0
+=
kT
MTorque
ICurrent
---------------
=

480 V. T. MINH and A. A. RASHID
(24)
(25)
If we define the state variables as
for the positions, angular velocities, and
acceleration on vehicle shafts 1, 2, and 3, respectively, and
the input variables as for the
torque for the ICE, voltage for EM1, voltage for EM2,
torque on clutch, and the initial air-drag torque load, then a
space state form of the vehicle dynamics is set up:
(26)
The new dynamic modeling in (26) makes it possible to
gain a deeper understanding of the acceleration and jerk
of HEVs. This is also one of the main contributions of
this study.
When the vehicle travels at low speeds (below 50 km/h),
only EM2 is running. Then, for the inputs MICE =0, V1=0,
and MC= 0 and the state variables θ1=0 and ω1=0, the
system becomes:
(27)
where xp=[θ2 ω2 θ3 ω3]’ and , and the
outputs, , are the vehicle velocity
(measured) ω3 and the vehicle torque (unmeasured) TTorque3
generated on shaft 3. In this case, the torsional rigidity
(torque/angle) is , where ϕ is the angle of
twist (radians). G is the shear modulus or
modulus of rigidity of mild carbon steel, G= 81500*106(N/
mm²). l is the length of the shaft where the torque is being
applied. In this case, we assumed l= 1.5(m) for the vehicle
drivetrain length. J is the moment of inertia: J=J2+J3(m4).
At a high speed (more than 50 km/h), EM1 starts the
ICE while the clutch is still open, and the dynamic
equations in the first part become:
(28)
(29)
where ζ is a new coefficient added to EM1 as a compensated
load to the starting period.
Then the system becomes:
; (30)
where , and .
TTor que 1 is the output torque (unmeasured) on shaft 1.
Using the comprehensive HEV modeling equations from
(29) to (30), we can develop MPC controllers for this HEV
in the next section.
3. MODEL PREDICTIVE CONTROLLER
DESIGN
The tasks of MPC are to control the output torques among
the components and the speeds of each shafts to ensure
smooth clutch engagements for driving comfort. The
RMPC algorithms for uncertain systems subject to input-
and output-saturated constraints are referred to in (Minh
and Afzulpurkar, 2005a). MPC with softened output
constraints is referred to in (Minh and Afzulpurkar, 2005b),
where a new MPC controller with output regions is
developed to improve the robustness of the controller for
handling input and output constraints and rejecting
disturbances. The conditions for NMPC stability using
x0[θ2 ω1 θ2 ω2
=
θ2 ω2 ω
·3]'
u0MICE V1 V2 MC Mv0
[]'=
ω
·3
ω
·· 3
upV2 Mv0
[]'=
ypω3 TTorque3
[]'=
kθ
MTorque
ϕ
---------------GJ
l
-------
==
ϕθ2
θ3
i
----
–=
θ
·10 ω1
+[]0 0+[]+=
xeθ1 ω1
[]'= ueV1 MICE
[]'= yeω1 Torque1
[]'=

MODELING AND MODEL PREDICTIVE CONTROL FOR HYBRID ELECTRIC VEHICLES 481
softened output constraints are referred to in (Minh and
Afzulpurkar, 2006). Some fault detection and control
systems with MPC are referred to in (Minh et al., 2007);
MPC can be reconfigured automatically on-line upon
detecting a fault to always maintain an error-free offset.
The system in (27) can be discretized into the following
form:
,(31)
where xt, ut and yt are states, inputs and outputs and A, B, C,
and C are fixed matrices.
Subject to the input and output constraints:
(32)
Then the optimization problem for an MPC controller
for tracking setpoints can be presented in the following
objective function:
,
Subject to:
(33)
is solved at each time t, where denotes the predicted
state vector at time t+k, obtained by applying the input
increment sequence and the new
inputs to the model equation (31),
starting from the state xt=x(t). and r are the predicted
outputs and the output setpoints, respectively. The output
setpoints can be changed dynamically to the desired speeds
by the driver or r=r(t). and are the
weighting matrices for the predicted outputs and the input
increments, respectively.
In the tracking setpoints MPC regulator, the steady-state
values are equal to the target tracking setpoints if there are
no constraints and disturbances. The formula in (33) is the
one we consider for the remainder of this research to verify
the ability of MPC to control the HEV speeds.
For the limits of this research, the authors assume that
the prediction horizon is set equal to the control horizon,
i.e., Nu=Ny=Np (the predictive lengths). The quadratic
objective junction J(U, x(t)) in equation (33) is minimized
over a vector Np future prediction inputs starting from the
state x(t).
For the MPC with hard constraints, by substituting
, equation (33) can be
rewritten as a function of the current state x(t) and the
current setpoints r(t):
(34)
subject to the hard combined constraints of ,
where the column vector is the
prediction optimization vector; , and H, F, Y G, W
and E are matrices obtained from Q, R, x(t), and r(t) in (34).
Because only the optimizer U is needed, the term involving
Y is usually removed from (34). The optimization problem in
(34) is a quadratic program and depends on the current state
x(t) and the current setpoints r(t). The implementation of
MPC requires the on-line solution of this quadratic program
at each time step.
In reality, the system might have both input and output
constraints, and the difficulty arises because of the inability
to satisfy all output constraints due to the input constraints.
Because MPC is designed for on-line implementation, no
infeasible solution of the optimization problems can be
allowed. Normally, the input constraints are based on the
physical limits to the system and can normally be
considered hard constraints. For this MPC controller, the
outputs constraints are the measured speeds, and the
unmeasured torques are not strictly imposed and can be
violated somewhat during the evolution of the system. To
guarantee the system stability once the outputs violate the
constraints, we can modify the optimization in (34) to some
softened constraints, such as:
, (35)
Subject to:
.
New items are added into the new softened objective
function: (usually a small value) is a new weighting
factor, and εi(t) represents the violation penalty terms
( ) added to the MPC objective function. These
terms keep the output violations at low values until the
constrained solution is returned. A comprehensive study of
MPC’s robustness subject to the softened state constraints
can be seen in (Minh and Afzulpurkar, 2005a).
To increase the ability of the controller to obtain
solutions at certain transitional critical times, we can also
delete some output setpoints. If certain output setpoints are
deleted, the system becomes looser, and the possibility that
the MPC controller can find a solution increases. The
deletion of some output setpoints can be done by choosing
some zeros in the weighting matrix Q in (35). The
robustness of MPC can also increase if we can relax certain
xt1+ AxtBut
+=
ytCxtDut
+=
⎩
⎨
⎧
utuu
t
∆utut1– u and yty∈,∆∈–=,∈
xtkt+
U∆ut
∆…utN
n1–+
∆,,{}
xtkt+utk1t–+ utkt+
∆+=
ytkt+
QQ'0≥=RR'0>=
xtN
pt+ANpxt() AkButN
p1k––+
k0=
Np1–
∑
+=
GU W Ex t()+≤
U∆ut
∆…utN
p1–+
∆,,[]'u∆∈
−
HH'0>=
Λ0>
εit() 0≥
−
−

482 V. T. MINH and A. A. RASHID
setpoints into regions rather than into certain specific
values. When the system turns to output regions, the MPC
formulation needs to be changed slightly because the
setpoints r(t) in (35) now become regions. An output
region is defined by the minimum and maximum values of
a desired range. The minimum value is the lower limit, and
the maximum value is the upper limit, .
Then a modified objective function of MPC with the output
regions is:
,(36)
.
As long as the outputs still lie inside the desired regions,
no control actions are taken because none of the control
objectives have been violated ( ). However, when
the outputs violate the desired regions, the control objective
in the MPC regulator will activate and push them back to
the desired regions. Studies on MPC with output constraints
deletion and regions can be seen in (Minh and Afzulpurkar,
2005b). We illustrate the robustness improvements of the
modified MPC controller in the following simulations.
3.1. MPC Simulation for EM2
EM2 alone propels the HEVs at speeds below 50 km/h. By
that time, the clutch is open, and both the ICE and EM1
stop. The system dynamic equation is indicated in (27).
The MPC regulator applied for this system is indicated in
(33), (35), and (36). The discrete time interval for all
simulations is set at 0.01 sec.
The following parameters are used for EM2: torsional
rigidity, kθ= 1158; motor constants, kE2=kT2=10; motor
inertia, J2=1; load inertia, J3=1; gear ratio, i=2.34; motor
damping, kβ2=0.5; load damping, kβ3=12; and armature
resistance, RI2=5.
The control constraints are set as follows: the input
constraints for the DC voltage applied for the vehicle are
and , or we do not set the limit
constraint on ∆u(t). The output constraints are imposed on
the motor shaft having a finite shear strength (carbon steel),
τ=25 (MPa or N/mm²). The output limit for the torque on
shaft2 is , where d=0.05m is the output motor shaft
diameter. Then the torque constraint is .
For the MPC parameters, we select a horizon prediction
Nu=Ny=Np=5 and the weighting matrices and
R= [1]. Figure 3 shows the MPC performance with the
input voltage, the output speed and torque.
The MPC performance is dependent on choosing the
weighting matrices Q and R. If we select the condition that
Q is much greater than R ( ), then the input increment
∆u(t) is much greater than the output penalty value of (y(t)−
r) as per the objective function (33). MPC will track the
setpoints very quickly, but in this case, greater energy for
the input is needed, as shown in Figure 4.
For this simulation, we select a horizon prediction
Nu=Ny=Np=5 and the weighting matrices and
R[1].
The horizon prediction of the MPC controller is also
affected by its performance. The MPC controller becomes
looser when we select the longer prediction horizon, and
the system will have better performance because it becomes
ylower ytkt+yupper
≤≤
ztkt+0=
V2300V≤ut()∆inf≤
Tτπd3
16
------
=
T2455Nm≤
Q10 0
0 10
=
Q R»
Q10 0
0 10
=
Figure 3. MPC performance with Np= 5 and Q=10R.
Figure 4. MPC performance with Np=5 and Q=100R.
Figure 5. MPC performance Np = 2 with and Q=50R.

MODELING AND MODEL PREDICTIVE CONTROL FOR HYBRID ELECTRIC VEHICLES 483
more flexible and easier to find a better solution. However,
with longer prediction, the computer calculation burden
will exponentially increase, and the on-line optimization is
limited by the ability of the CPU and its communication
cables. In the next simulation, we run the MPC with a
shorter horizon length of Nu=Ny=Np=2 and medium values
of and R[1], as shown in Figure 5.
In Figure 5, we can see that with a shorter horizon
prediction, Np=2, the MPC performance gets worse
because the system cannot correctly track the output speed
setpoints. Of course, compared with the next simulation
with a longer horizon prediction, we can see a much better
performance of the MPC controller.
The MPC horizon length is now selected to be longer,
Nu=Ny=Np=10. For the same values of and
R[1], the MPC performance is shown in Figure 6.
The MPC controller with a longer prediction horizon in
Figure 6 shows better performance compared with Figure
5. However, we can see the greater energy needed for the
input voltages.
It is not realistic to set the input constraint for
because the DC motor normally runs with only positive
voltage, , and the brake and the recharge
system will be activated when we want to reduce the
vehicle speed. In the next simulation, we set the input
voltage for this DC motor to only positive values,
. This simulation opens up the possibility of
regenerative braking when we use the brakes to reduce the
speed. The simulation is shown in Figure 7 with
Nu=Ny=Np=5 and high values of and a small
value of R[1].
The MPC performance for EM2 is determined by
changing the parameters in the weighting matrices, the
horizon length, and the input constraints. In the next part,
we will consider the MPC simulations for the ICE and
EM1 to track the desired speed setpoints at high speeds
(above 50 km/h) and show how to synchronize the speeds
of these two parts.
3.2. MPC Simulation for the ICE and EM1
When the vehicle speed exceeds 50 km/h, EM1 activates
the ICE to propel the vehicle. Depending on the speeds and
the torque requirements, the parallel hybrid vehicle can run
only on the ICE or on the ICE, EM1 and EM2. At the
starting time, the clutch is still open, and the dynamic
equations for the ICE and EM1 are as shown in Figure 1.
The parameters used for EM1 are as follows: motor
constants, kE2=kT2= 15; motor inertia, J1=1; motor
damping, kβ1=0.5; armature resistance, RI1=7; and
compensation factor, ζ= 0.5. The system is discretized at a
time interval of 0.01 sec. The air drag resistance torque at
ω3= 2000 rpm is selected as Nv0=30 N m.
The constraints are set as follows: the input constraints
for the DC voltage applied for the vehicle are
and /sec. The output torque constraint for
shaft1 is set at .
For the MPC parameters, we also select a horizon
prediction Nu=Ny= Np=5 and the weighting matrices
and . Figure 8 shows the MPC performance
at the starting point.
In Figure 6, after a delay of 1 second, the ICE is fully
ignited, and after 2.4 seconds, the engine speed has reached
the setpoint of 2000 rpm and runs stably at 6 kW with an
output torque of 30 N m (the clutch is still open).
Next, we run the simulations with EM1 and the ICE for
tracking setpoints with the clutch engaged. We assume that
when EM2 runs in excess of 1500 rpm, EM1 will cause the
ICE to engage with the system. For driving comfort and
jerk reduction, the engagement will take place only when
or ω1=1.05*ω2, or EM1 and the ICE must track
EM2 at 5% positive offset. The MPC objective function in
Figure 7 is now changed from setpoint r(t) to ω2(t) track
with 5% positive offset. The results of the simulation are
shown in Figure 9. The system reaches the setpoint and is
ready for the clutch engagement after 2.5 seconds.
In the simulation in Figure 9, we run both EM1 and the
ICE to track EM2, and we can see that after 2 seconds, the
Q50 0
0 50
=
Q50 0
0 50
=
V2300V≤
0V2300V≤≤
0V2300V≤≤
Q1000 0
0 1000
=
V148V≤
u∆t() 180V≤
T1628Nm≤
Q10 0
0 10
=R1 0
0 1
=
ω1ω2
≥
Figure 6. MPC performance Np = 10 with and Q=50R.
Figure 7. MPC performance with , Np=5 an
d
Q=1000R.
0V2300V≤≤

484 V. T. MINH and A. A. RASHID
speed of the left-hand clutch has exceeded 5% of the speed
on the right-hand side, and the system is ready for the
clutch engagement. However, EM1 is only used for the
start-up of the ICE, and once the ICE is fully ignited, EM1
will be switched to a generator to charge the battery. In the
next simulation, we turn off EM1 and let only the ICE track
EM2. The MPC objective function is now similar to the
one we used in Figure 9. The results of the simulation are
shown in Figure 10. The system reaches the speed setpoint
and is ready for clutch engagement after 4.4 seconds.
How to control the ω1 while rapidly tracking ω2+5%
with an MPC controller is still a considerable challenge.
Next, we test a new MPC controller using the softened
output constraints when we consider the output tracking
setpoints as in Figure 8 with some additional penalty terms
added into the MPC objective function, .
The results of the simulation are shown in Figure 11.
In figure 11, the penalty terms εi and the new weighting
factor can be changed independently with Q and R to have
a good softened-constraints MPC performance. The system
JSoften JHard Λεi
2t()+=
Figure 8. MPC at the starting point.
Figure 9. MPC controller for tracking setpoints with both EM1 and the ICE.
Figure 10. MPC controller for tracking setpoints with only the ICE.

MODELING AND MODEL PREDICTIVE CONTROL FOR HYBRID ELECTRIC VEHICLES 485
becomes looser and more flexible with more regulated
parameters. The new system reaches the setpoint faster and
is ready for the clutch engagement after only 3.4 seconds.
4. CONCLUSIONS
In this paper, the mathematical model and the design of a
controller for tracking the speeds of a parallel hybrid
electric vehicle have been developed. The new model
enables better studies on how to achieve fast engagement
as well as how to reduce jerking during the transitional
period. Simulations show that MPC controllers can control
the speeds of this HEV very well for a fast clutch
engagement with low jerk and high driving comfort. The
MPC performance can also be considerably improved if we
modify and select appropriate prediction horizon lengths
and values for the weighting parameters. Modified MPC
algorithms can help increase the robustness of the system if
we soften certain output constraints or turn them into
constrained regions.
ACKNOWLEDGMENT−This work was supported by Universiti
Teknologi Petronas (UTP) and funded by Project FRGS 2/2010/
TK/UTP/02/28 from Ministry of Higher Education, Malaysia
(MOHE).
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Figure 11. Softened Constraints of the MPC controller for tracking setpoints with only the ICE.