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Sensorless torque control scheme of
induction motor for hybrid electric vehicle
Yan LIU 1,2, Cheng SHAO1
(1.Research Institute of Advanced Control Technology, Dalian University of Technology, Dalian Liaoning 116024, China;
2.School of Information Engineering of Dalian University, Dalian Liaoning 116622, China)
Abstract: In this paper, the sensorless torque robust tracking problem of the induction motor for hybrid electric vehicle
(HEV) applications is addressed. Because motor parameter variations in HEV applications are larger than in industrial
drive system, the conventional field-oriented control (FOC) provides poor performance. Therefore, a new robust PI-based
extension of the FOC controller and a speed-flux observer based on sliding mode and Lyapunov theory are developed in
order to improve the overall performance. Simulation results show that the proposed sensorless torque control scheme is
robust with respect to motor parameter variations and loading disturbances. In addition, the operating flux of the motor is
chosen optimally to minimize the consumption of electric energy, which results in a significant reduction in energy losses
shown by simulations.
Keywords: Hybrid electric vehicle; Induction motor; Torque tracking; Sliding mode
1 Introduction
Being confronted by the lack of energy and the increasingly
serious pollution, the automobile industry is seeking
cleaner and more energy-efficient vehicles.A Hybrid Electric
Vehicle (HEV) is one of the solutions. A HEV comprises
both a Combustion Engine (CE) and an Electric Motor
(EM). The coupling of these two components can be in
parallel or in series. The most common type of HEV is the
parallel type, in which both CE and EM contribute to the
traction force that moves the vehicle. Fig1 presents a diagram
of the propulsion system of a parallel HEV [1].
Fig. 1 Parallel HEV automobile propulsion system.
In order to have lower energy consumption and lower pollutant
emissions, in a parallel HEV the CE is commonly
employed at the state (n > 40 km/h or an emergency speed
up), while the electric motor is operated at various operating
conditions and transient to supply the difference in torque
between the torque command and the torque supplied by
the CE. Therefore fast and precise torque tracking of an EM
over a wide range of speed is crucial for the overall performance
of a HEV.
The induction motor is well suited for the HEV application
because of its robustness, low maintenance and low
price. However, the development of a drive system based
on the induction motor is not straightforward because of the
complexity of the control problem involved in the IM. Furthermore,
motor parameter variations in HEV applications
are larger than in industrial drive system during operation
[2]. The conventional control technique ranging from the
inexpensive constant voltage/frequency ratio strategy to the
sophisticated sensorless control schemes are mostly ineffective
where accurate torque tracking is required due to their
drawbacks, which are sensitive to change of the parameters
of the motors.
In general, a HEV operation can be continuing smoothly
for the case of sensor failure, it is of significant to develop
sensorless control algorithms. In this paper, the development
of a sensorless robust torque control system for HEV
applications is proposed. The field oriented control of the induction
motor is commonly employed in HEV applications
due to its relative good dynamic response. However the classical
(PI-based) field oriented control (CFOC) is sensitive to
parameter variations and needs tuning of at least six control
parameters (a minimum of 3 PI controller gains). An improved
robust PI-based controller is designed in this paper,
Received 5 January 2005; revised 20 September 2006.
This work was supported in part by State Science and Technology Pursuing Project of China (No. 2001BA204B01).
Y. LIU et al. / Journal of Control Theory and Applications 2007 5 (1) 42–46 43
which has less controller parameters to be tuned, and is robust
to parameter variation.The variable parameters model
of the motor is considered and its parameters are continuously
updated while the motor is operating. Speed and
flux observers are needed for the schemes. In this paper,
the speed-flux observer is based on the sliding mode technique
due to its superior robustness properties. The sliding
mode observer structure allows for the simultaneous observation
of rotor fluxes and rotor speed. Minimization of the
consumed energy is also considered by optimizing operating
flux of the IM.
2 The control problem in a HEV case
The performance of electric drive system is one of the
key problems in a HEV application. Although the requirements
of various HEV drive system are different, all these
drive systems are kinds of torque control systems. For an
ideal HEV, the torque requested by the supervisor controller
must be accurate and efficient. Another requirement is to
make the rotor flux track a certain reference λref . The reference
is commonly set to a value that generates maximum
torque and avoids magnetic saturation, and is weakened to
limit stator currents and voltages as rotor speed increases.
In HEV applications, however, the flux reference is selected
to minimize the consumption of electrical energy as it is one
of the primary objectives in HEV applications. The control
problem can therefore be stated as the following torque and
flux tracking problems:
min
ids,iqs,we Te(t) − Teref (t), (1)
min
ids,iqs,we λdr(t) − λref (t), (2)
min
ids,iqs,we λqr(t), (3)
where λref is selected to minimize the consumption of electrical
energy. Teref is the torque command issued by the
supervisory controller while Te is the actual motor torque.
Equation (3) reflects the constraint of field orientation commonly
encountered in the literature. In addition, for a HEV
application the operating conditions will vary continuously.
The changes of parameters of the IM model need to be accounted
for in control due to they will considerably change
as the motor changes operating conditions.
3 A variable parameters model of induction
motor for HEV applications
To reduce the elements of storage (inductances), the induction
motor model used in this research in stationary reference
frame is the Γ-model. Fig. 2 shows its q-axis (d-axis
are similar). As noted in [3], the model is identical (without
any loss of information) to the more common T-model in
which the leakage inductance is separated in stator and rotor
leakage [3]. With respect to the classical model, the new
parameters are:
Lm = L2
m
Lr
= γLm, Ll = Lls + γLlr,
Rr = γ2Rr.
Fig. 2 Induction motor model in stationary reference frame (q-axis).
The following basic w−λr−is equations in synchronously
rotating reference frame (d - q) can be derived from the
above model.
⎧⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
dλdr
dt
= −ηλdr + (we − wr)λqr + ηLmids,
dλqr
dt
= −(we − wr)λdr − ηλqr + ηLmiqs,
dids
dt
= ηβλdr+βwrλqr−γids+weiqs+
1
σLs
Vds,
diqs
dt
=−βwrλdr+ηβλqr−weids−γiqs+
1
σLs
Vqs,
dwr
dt
= μ(λdriqs − λqrids) −
TL
J
,
dθ
dt
= wr + ηLm
iqs
λdr
= we,
Te = μ(λdriqs − λqrids)
(4)
with constants defined as follows:
μ = np
J
, η = Rr
Lm
, σ = 1−
Lm
Ls
, β =
1
Ll
,
γ = Rs + Rr
Ll
, Ls = Ll + Lm,
where np is the number of poles pairs, J is the inertia of the
rotor. The motor parameters Lm, Ll, Rs, Rr were estimated
offline [4]. Equation (5) shows the mappings between the
parameters of the motor and the operating conditions (ids,
iqs).
Lm = a1i2
ds + a2ids + a3, Ll = b1Is + b2,
Rr = c1iqs + c2.
(5)
4 Sensorless torque control system design
A simplified block diagram of the control diagram is
shown in Fig. 3.
44 Y. LIU et al. / Journal of Control Theory and Applications 2007 5 (1) 42–46
Fig. 3 Control structure.
4.1 PI controller based FOC design
The PI controller is based on the Field Oriented Controller
(FOC) scheme. When Te = Teref, λdr = λref , and
λqr = 0 in synchronously rotating reference frame (d − q),
the following FOC equations can be derived from the equations
(4).
⎧⎪
⎪⎪⎪⎪⎪⎨⎪
⎪⎪⎪⎪⎪⎩
ids = λref
Lm
+ λref
Rr
,
iqs = Teref
npλref
,
we = wr + ηLm
iqs
λref
.
(6)
From the Equation (6), the FOC controller has lower performance
in the presence of parameter uncertainties, especially
in a HEV application due to its inherent open loop
design. Since the rotor flux dynamics in synchronous reference
frame (λq = 0) are linear and only dependent on the
d-current input, the controller can be improved by adding
two PI regulators on error signals λref − λdr and λqr − 0 as
follow
ids = λref
Lm
+ λref
Rr
+ KPd(λref − λdr)
+KId (λref − λdr)dt, (7)
iqs = Teref
npλref
, (8)
we = wr + ηLm
iqs
λref
+ KPqλqr + KIq λqrdt. (9)
The Equation (7) and (9) show that current (ids) can control
the rotor flux magnitude and the speed of the d − q rotating
reference frame (we) can control its orientation correctly
with less sensitivity to motor parameter variations because
of the two PI regulators.
4.2 Stator voltage decoupling design
Based on scalar decoupling theory [5], the stator voltages
commands are given in the form:
⎧⎪
⎪⎪⎨⎪⎪⎪⎩
Uds = Rsids − weσLsiqs = Rsids − weLliqs,
Uqs = Rsiqs + weσLsids + Lm
Lr
weλref
= Rsiqs + weσLsids + weλref .
(10)
Because of fast and good flux tracking, poor dynamics decoupling
performance exerts less effect on the control system.
4.3 Speed-flux observer design
Based on the theory of negative feedback, the design of
speed-flux observer must be robust to motor parameter variations.
The speed-flux observer here is based on the sliding
mode technique described in [6∼8]. The observer equations
are based on the induction motor current and flux equations
in stationary reference frame.
⎧⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
d˜ids
dt
= ηβ˜λdr + β ˜ wr˜λqr − γ˜ids +
1
Ll
Vds,
d˜iqs
dt
= −β ˜ wr˜λdr + ηβ˜λqr − γ˜iqs +
1
Ll
Vqs,
d˜λdr
dt
= −η˜λdr − ˜ wr˜λqr + ηLm
˜i
ds,
d˜λqr
dt
= ˜wr˜λ dr − η˜λqr + ηLm
˜i
qs.
(11)
Define a sliding surface as:
s = (˜iqs − iqs)˜λdr − (˜ids − ids)˜λqr. (12)
Let a Lyapunov function be
V = 0.5s2. (13)
After some algebraic derivation, it can be found that when
˜ wr = w0sgn(s) with w0 chosen large enough at all time,
then ˙V = ˙s · s 0. This shows that s will converge to
zero in a finite time, implying the stator current estimates
and rotor flux estimates will converge to their real values
in a finite time [8]. To find the equivalent value of estimate
wr (the smoothed estimate of speed, since estimate wr is a
switching function), the equation must be solved [8]. This
yields:
˜ weq = wr
˜λ
qrλqr + λdr˜λdr
˜λ
2q
r +˜λ2
dr −
η
np
˜λ
qrλdr − λqr˜λdr
˜λ
2q
r +˜λ2
dr
. (14)
The equation implies that if the flux estimates converge to
their real values, the equivalent speed will be equal to the
real speed. But the Equation (14) for equivalent speed cannot
be used as given in the observer since it contains unknown
terms. A low pass filter is used instead,
˜ weq =
1
1 + s · τ
˜ wr. (15)
Y. LIU et al. / Journal of Control Theory and Applications 2007 5 (1) 42–46 45
The same low pass filter is also introduced to the system
input,which guarantees that the input matches the feedback
in time.
The selection of the speed gain w0 has two major constraints:
1) The gain has to be large enough to insure that sliding
mode can be enforced.
2) A very large gain can yield to instability of the observer.
Through simulations, an adaptive gain of the sliding
mode observer to the equivalent speed is proposed.
w0 = k1 ˜ weq + k2. (16)
From Equation (11), the sliding mode observer structure
allows for the simultaneous observation of rotor fluxes.
4.4 Flux reference optimal design
The flux reference can either be left constant or modified
to accomplish certain requirements (minimum current,
maximum efficiency, field weakening) [9,10]. In this paper,
the flux reference is chosen to maximum efficiency at steady
state and is weaken for speeds above rated. The optimal efficiency
flux can be calculated as a function of the torque
reference [9].
λdr−opt = |Teref| · 4Rs · L2r
/L2
m + Rr. (17)
Equation (17) states that if the torque request Teref is
zero, Equation (8) presents a singularity. Moreover, the
analysis of Equation (17) does not consider the flux saturation.
In fact, for speeds above rated, it is necessary to
weaken the flux so that the supply voltage limits are not exceeded.
The improved optimum flux reference is then calculated
as:
⎧⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
λref = λdr-opt,
if λmin λdr-opt λdr-rated ·
wrated
wr-actual
,
λref = λmin, if λdr-opt λmin,
λref = λdr-rated ·
wrated
wr-actual
,
if λdr-opt λdr-rated ·
wrated
wr-actual
.
(18)
where λmin is a minimum value to avoid the division by
zero.
4.5 Simulations
The rated parameters of the motor used in the simulations
are given by
Rs = 0.014 Ω, Rr = 0.009 Ω, Lls = 75 H,
Llr = 105 H, Lm = 2.2 mH, Ls = Lls + Lm,
Lr = Llr + Lm, P = 4, Jmot = 0.045 kgm2,
J = Jmot +MR2
tire/Rf, ρair = 1.29, Cd = 0.446,
Af = 3.169 m2, Rf = 8.32, Cr = 0.015,
Rtire = 0.3683 m, M = 3000 kg, wbase = 5400 rpm,
λdr−rated = 0.47 Wb.
Fig.4 shows the torque reference curve that represents
typical operating behaviors in a hybrid electric vehicle.
Fig. 4 The torque reference curve.
Load torque is modeled by considering the aerodynamic,
rolling resistance and road grade forces. Its expression is
given by
TL = Rtire
Rf
(
1
2ρairCdAfv2 +MCr cos αg +M sin αg).
Figures in [5∼8] show the simulation results of the
system of Fig.3 (considering variable motor parameters).
Though a small estimation error can be noticed on the observed
fluxes and speed, the torque tracking is still achieved
at an acceptable level as shown in Figs. [5, 6, 8]. The torque
control over a wide range of speed presents less sensitivity
to motor parameters uncertainty.
Fig.5 presents the d and q components of the rotor flux.
Rotor flux λr is precisely orientated to d-axis because of the
improved PI controllers.
Fig.8 shows clearly the real and observed speed in the
different phases of acceleration, constant and deceleration
speed with the motor control torque of Fig.4. The variable
model parameters exert less influence on speed estimation.
Fig.7 shows the power loss when the rotor flux keeps constant
or optimal state. A significant improvement in power
losses is noticed due to reducing the flux reference during
the periods of low torque requests.
Fig. 5 Motor rotor flux λr.
46 Y. LIU et al. / Journal of Control Theory and Applications 2007 5 (1) 42–46
Fig. 6 Motor torque.
Fig. 7 Power Losses.
Fig. 8 Motor speed.
5 Conclusions
This paper has described a sensorless torque control system
for a high-performance induction motor drive for a
HEV case. The system allows for fast and good torque
tracking over a wide range of speed even in the presence of
motor parameters uncertainty. In this paper, the improved
PI-based FOC controllers show a good performance in the
rotor flux λdr magnitude and its orientation tracking. The
speed-flux observer described here is based on the sliding
mode technique, making it independent of the motor parameters.
Gain adaptation of the speed -flux observer is used to
stabilize the observer when integration errors are present.
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