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Volume 46, Number 2, 2005 51
Speed Sensorless DTC for Induction Motor
Based on an Improved Adaptive Flux Observer
S. BELKACEM, F. NACERI and A. BETTA
LEB-Research Laboratory, Department of Electrical Engineering, University of Batna
Rue Chahid Med El Hadi Boukhlouf, Batna 05000, Algeria
E-mail: belkacem_sebti@yahoo.fr
Abstract – This paper presents a method for estimation of the rotor speed and stator flux based on the
theory of an adaptive control and a direct torque control (DTC). A linear observer for estimation of the
stator flux is synthesized, by using a Lyapunov function. The adaptive observer is associated to a direct
torque control of an induction motor. To illustrate the performances and the robustness of this observer, a
simulation results is presented.
I. INTRODUCTION performance, state estimators or observers are
usually more preferable than physical
In recent years, several studies have measurements [8]-[10].
been developed which propose alternative In DTC, the flux is conventionally
solutions to the FOC control of a PWM obtained from the stator voltage model, using
inverter-fed motor drive with two objectives: the measured stator voltages and currents.
first, achievement of an accurate and fast This method, utilizing open-loop pure
response of flux and torque, and second, integration suffers from increased noise on
reduction in the complexity of the control voltage and current and quantization errors in
system. Among the various proposals, Direct the digital system, in addition to the offset,
Torque and Flux Control (DTFC) also termed gain and conversions factors in the low speed
Direct Torque Control (DTC), has found wide operation range, even with the correct
acceptance [2]. knowledge of the stator resistance. We
Since its introduction in 1985, the direct propose to use an adaptive observer to
torque control (DTC) [1], principle was improve the estimation of the components of
widely used for IM drives with fast dynamics. the stator flux and the rotor speed.
Despite its simplicity, DTC is able to produce Adaptive speed observers seem to be
very fast torque and flux control, if the torque between the most promising methods thanks
and the flux are correctly estimated, is robust to their good performance versus computing
with respect to motor parameters and time ratio. They have the advantage to
perturbations. provide both flux and mechanical speed
As it is well known, speed sensors like estimates without problems of open-loop
tachometers or incremental encoders increase integration. Besides, the adaptive observer
the size and the cost of systems unnecessarily. has the interesting property to provide a
Similar problems arise with the addition of mechanism for on-line tuning of key model
search coils or Hall Effect sensors to the parameters such as the stator or rotor
motor for the measurement of flux, hindering resistance and then to compensate for their
functionality in terms of implementation. drift due to motor heating for example [9].
Thus, to improve the overall system
52 ACTA ELECTROTEHNICA
For the speed regulation, the saturation complex loop of return, independently of the
of the manipulated variable can involve a rotor parameters. The fact that the DTC
phenomenon of racing of the integral action control the switches directly, without passing
during the great variations (starting of the by regulators, it clearly improves its dynamic
machine), which is likely to deteriorate the performances compared to the vector control
performances of the system or even to [7]-[8].
destabilize it completely, the solution consists
in correcting the integral action [5]. A. Behavior of Stator Flux
In the reference (α, β) related to the
II. MODELLING OF THE INDUCTION stator flux, the stator flux could be obtained
MOTOR by the following equation:
In this section, the induction motor dΨ s
Vs = Rs I s + (2)
model is given by [7]: dt
⎧ x = Ax + BU By neglecting the voltage drop due to
⎨ (1) the resistance of the stator to simplify the
⎩ y = Cx study (for high speeds), we find:
where t
T Ψ s ≈ Ψ s0 + ∫ Vs dt (3)
x = ⎡isα
⎣ isβ ψrα ψ rβ ⎤
⎦ 0
T T For one period of sampling, the voltage
U = ⎡Vsα
⎣ Vsβ ⎤ , y = ⎡isα
⎦ ⎣ isβ ⎤
⎦ vector applied to the asynchronous machine
M remains constant, we can write:
C = ⎡1 0 0 0⎤ , K = , Ψ s (k +1) ≈ Ψ s (k)+VsTe (4)
⎢0 1 0 0⎥
⎣ ⎦ σLs Lr
Rs Rr M
2 Then ΔΨ s ≈ VsTe
γ= + 2 • Ψ s (k) : is the stator flux vector to the step
σLs Lr σLs
of current sampling;
⎡- γ 0
K ⎤ Kωr • Ψ s (k +1) : is the stator flux vector to the
⎢ Tr ⎥ ⎡ 1 ⎤ step of following sampling;
⎢ K ⎥ ⎢ σL 0
⎥ • ΔΨ s : is the variation of the stator vector
⎢0 -γ - Kωr ⎥
A= ⎢
Tr ⎥ ⎢ s
, B= ⎢ 0
1 ⎥ flux;
⎥
⎢M 0 -
1
- ωr ⎥ ⎢ 0 σLs ⎥ • Te : is the sampling period.
⎢ Tr Tr ⎥ ⎢ 0 0
⎢0 M 1 ⎥ ⎣ 0 ⎥⎦
For a constant period of sampling, ΔΨ s
ωr - is proportional to the voltage vector applied
⎢
⎣ Tr Tr ⎥
⎦ to the stator of the induction motor.
III. PRINCIPLE OF THE DTC B. Behavior of the Torque
The DTC Technique makes it possible The electromagnetic torque is
to precisely control the stator flux and the proportional to the vector product between
electromagnetic torque. It is based only on the the vectors of stator and rotor flux according
knowledge of the stator currents and the to the following expression:
voltages and the rotor speed in steady state Ce = k (Ψ s ×Ψ r ) = k Ψ s Ψ r sin ( δ ) (5)
operation. The stator voltage makes it
with:
possible to easily consider stator flux starting
from I S, V S and ω S. The values of the flux • Ψ s : is the stator vector flux;
and the torque are then calculated without • Ψ r : is the rotor vector flux;
Volume 46, Number 2, 2005 53
• δ: is the angle between the stator and The symbol ^ denotes estimated values
rotor flux vectors. and G is the observer gain matrix.
By using an adaptation mechanism, we
C. Development of the Commutation Strategy can estimate the rotor speed. The system
states and the parameters can also be
In order to exploit the possible
estimate. The speed adaptation mechanism is
sequences of operation of the inverter on two
deduced by using a Lyapunov theory. The
levels, the classical selection table of the DTC
estimation error of the stator current and the
is summarised in the table I, which shows the
rotor flux represents the difference between
strategy of commutation suggested by
the observer and the model of the motor. The
Takahashi to control the stator flux and the
dynamic error is given by [3].
electromagnetic torque of induction motor.
d
TABLE. I. Selection table for direct torque control. e = ( A+GC ) e+ ΔAxˆ (7)
dt
ΔΨs ΔCe S1 S2 S3 S4 S5 S6
where:
1 V2 V3 V4 V5 V6 V1
1 0 V7 V0 V7 V0 V7 V0 e= x - x ˆ (8)
-1 V6 V1 V2 V3 V4 V5 ⎡0 0 0 p kΔωr ⎤
1 V3 V4 V5 V6 V1 V2 ⎢ ⎥
0 0 V0 V7 V0 V7 V0 V7 ˆ ⎢0
ΔA= A- A= ⎢
0 -pkΔωr 0 ⎥
⎥ (9)
-1 V5 V6 V1 V2 V3 V4 ⎢0 0 0 -pΔωr ⎥
⎢ ⎥
⎣0 0 pΔωr 0 ⎦
The Fig. 1 gives the partition of the ˆ
Δω r = ω r - ω r , (10)
complex plan in six angular sectors S I = 1… 6.
we consider the following Lyapunov
β function,
V3(010) V2(110)
1
V3(DF,AC) V2 (IF, IT)
V = eT e+ ( Δωr )2 (11)
λ
3 2 V5(DF,DC) V6 (AF, DT)
V0,7(000)
Where λ is a positive constant, the derivative
V4(011)
4 7 1
V1(100)
of Lyapunov function is giving by [3]:
α d
V = eT ⎧( A ( ωr ) +GC ) + ( A ( ωr ) +GC ) ⎫ e
T
⎨ ⎬
5 6 Secteur 1 dt ⎩ ⎭
ˆ ˆ 2 ˆ
V5(001) V6(101) - 2KΔωr (e1 x4 - e2 x3 )+ Δωr ωr
λ
(13)
Fig. 1. Partition of the complex plan in six angular
With ωr is the estimated rotor speed
ˆ
sectors S I = 1… 6.
ˆ ˆ
e1 = x1 - x1 , e2 = x2 - x2
AF: Increase the Flux, DF: decrease the Flux.
IT : Increase the torque, DT: decrease the torque. From equation (12), we can deduce the
IF : Increase the flux, DF: decrease the flux.
adaptation law for the estimation of the rotor
speed by the equality between the second and
IV. ADAPTIVE FLUX AND SPEED the third term, we obtain:
OBSERVERS ωr = λ.K ( e1 x4 - e2 x3 )
ˆ ˆ ˆ (13)
A linear state observer for the stator flux With K: is a positive constant.
can then be derived as follows, by To enhance the dynamic behavior of the
considering the mechanical speed as constant speed observer, we add a proportional term.
parameter: The speed adaptation laws become [3]:
ˆ ˆ
X = AX + BU + G (iˆ - i ) (6) s s
54 ACTA ELECTROTEHNICA
t k1 : is a coefficient obtained by the pole
ωr = k p ( e1 x4 - e2 x3 ) +ki ∫ (e1 x4 - e2 x3 )dt (14)
ˆ ˆ ˆ ˆ ˆ
placement [10].
0
where: k p and ki are positive gains V. SYSTEM OF SPEED REGULATION
A. Synthesis of the adaptive observer Very often, the transfer of the
manipulated variable is affected by one or
To solve the problem related especially
more nonlinear elements (hysteresis,
to the estimation of stator flux and the rotor
saturation). The saturation of the manipulated
speed we have recourse to adaptive observer;
variable can involve a phenomenon of racing
the mechanism adaptive is a PI regulator.
of the integral action during the great
The block diagram of the observer with
variations (starting of the machine), which is
the adaptive mechanism is given in Fig. 2.
likely to deteriorate the performances of the
isα system or even to destabilize it completely.
Induction motor Vsα To overcome this phenomenon, a solution
+ isβ Vsβ consists in correcting the integral action
according to the diagram of Fig. 3.
+ ˆ
- isα Observer
kω
Model + δ1
- δ
ˆ
isβ ˆ
Φ rβ ˆ
Φ rα
+ + +
Adaptation
1/S Saturation
Mechanism ˆ
ωr ωref - - -
Integral
+
ωr
Fig. 2. Global adaptive observer structure.
1/ kω
B. Observer gain selection Fig. 3. Structure of the anti-windup system.
The observer gain matrix is chosen so
The correction of the integral action is
that to impose a fast dynamic for the
based on the difference between the values of
observer. The observer matrix is presented as
( δ ) upstream and downstream from the
follow: [3]
limiting device, balanced by the coefficient
⎡ g1 -g2 ⎤ 1
⎢ ⎥ such as [5]-[6]:
⎢ g2 g1 ⎥ kω
G= ⎢ ⎥ (15)
⎢ g3 -g4 ⎥ 1
⎢
g3 ⎥ X R (k +1) = X R (k) - ( )(δ(k) - δ1 ) (16)
⎣ g4 ⎦ kω
Where: g1, g2, g3, g4 are given by: The stator flux according to rotor flux is
⎛ R 1- σ 1 ⎞ given by:
g1 = ( 1- k1 ) ⎜ s + + ⎟
⎝ σLs σTr Tr ⎠ ⎧ M
⎪Ψ sα = σLsisα + Ψ rα
g 2 = ( k1 -1) ωr
ˆ ⎪ Lr
⎨ (17)
1- K 2 ⎛ Rs 1 - σ K ⎞ k1 - 1 ⎛ Rs 1- σ 1 ⎞ ⎪Ψ sβ = σLsisβ + M Ψ rβ
g3 = ⎜ + + ⎟+ ⎜ + + ⎟ ⎪
⎩ Lr
K ⎝ σLs σTr Tr ⎠ K ⎝ σLs σTr Tr ⎠
k -1 ˆ
g4 = - 1 ωr
K
with: k1 > 0
Volume 46, Number 2, 2005 55
Vsβ Vsα
ˆ2 ˆ2 isα
Ψ sα + Ψ sβ
Adaptive Flux α β
ˆ Observer abc
Ψs ωr
ˆ i sβ
ˆ
Ψ sβ ˆ
Ψ sα
ˆ ˆ
isα i
sβ
- Sector Select
ˆ +
Ψs ia ib ic
ˆ
θs
Flux Comparator.
ωr
+ Τe
PI Anti- Look up Inverter
Windup Table
- + -
ˆ
Τe
ωr
ˆ Torque Comparator.
ˆ
Ψ sα ˆ
Ψ sβ IM
p ⎛ Ψ sα ˆsβ - Ψ s ˆsα ⎞
⎜
ˆ i ˆ i
⎟ ˆ
isα ˆ
isβ
⎝ β ⎠
Fig. 4. Speed Sensorless DTC System Using Adaptive Flux Observer.
VI. PROPOSED SENSORLESS VII. SIMULATION RESULTS
INDUCTION MOTOR DRIVE
The proposed sensorless IM drive of
The proposed sensorless IM drive block Fig. 4 was verified using simulations. During
diagram is shown in Fig. 4. The drive the simulations, the torque set value is limited
operates at constant stator flux uses DTC to to 25 (N.m).
provide torque control. The speed controller In order to show the performances and
is a (PI anti-windup) regulator that generates the robustness of the adaptive observer, we
the reference torque. Measurements of two simulated different cases, which are presented
line currents and the average voltage over a thereafter. The static and dynamic
PWM switching period are used in the performances of the observer are analyzed
adaptive flux observer and the direct torque according to the simulation of the following
control. The IM stator flux is estimated by the transients:
adaptive observer and used in the DTC
control. The estimated speed is used in the PI A. Speed Inversion
anti-windup speed regulator and for flux To test the robustness of the System, we
weakening. The IM under study is a 4-kW vary a reference from 157 rad/sec to –157
two-pole squirrel-cage motor, the parameters rad/sec at time t = 2s. Fig. 5 presents the
of which are listed in Table II.
actual and the estimated speeds ωr , ωr ˆ
TABLE II. INDUCTION MOTOR PARAMETERS
respectively and the estimation error of ωr .
P = 4 KW f = 50 Hz
Wr = 1440 rpm Vr = 380 V
Fig. 6 presents actual flux Ψ s , the estimated
Rs = 1.2 Ω Rr = 1.8 Ω ˆ
flux Ψ s and estimation error Ψ s . The
Ls = 0.1554 H Lr = 0.1568 H estimation algorithm is robust because the
Tr = 0.0871 s M = 0.15 H variation of the speed is important and the
J = 0.07 kgm2 P=2 estimated speed follows the real speed when
the motor starts and at the moment of speed
56 ACTA ELECTROTEHNICA
er (rad/sec) Wr (rad/sec) Wr (rad/sec)
||Ψs| (Wb) |Ψs| (Wb)
^
^
er (Wb)
Fig. 5. The real, estimated speed and estimation error. Fig. 6. The real, estimated stator flux magnitude
and estimation error.
Wr (rad/sec) Wr (rad/sec)
|Ψs| (Wb)
||Ψs| (Wb)
^
^
er (rad/sec)
er (Wb)
Fig. 7. The real, estimated speed and estimation error. Fig. 8. Evolution of the stator magnitude.
inversion. Fig. 9. illustrates the response of adaptive observer, the simulation was
stator current and the electromagnetic torque. established in low speed. The Fig. 7 and
Fig. 8 illustrate the simulation results of the
B. Operation at low speed process of speed estimation with a speed
To test the speed estimation by using an reference equal to ±50 rad/sec. We can see
that the speed follow perfectly the speed
Ψsβ (Wb)
Ψsα (Wb)
Fig. 9. Responses of electromagnetic torque and
stator current. Fig. 10. Trajectory of the estimated stator flux.
Volume 46, Number 2, 2005 57
Wr (rad/sec)
Wr (rad/sec)
^
er (rad/sec)
Fig.11.Variation test of the load torque Fig.12. Response of the stator voltage
and electromagnetic torque
reference. The deviation detected in Fig. 10 is Fig.13.a, 13.b illustrates the evolution of the
caused by the instantaneous reversal of the estimated and the real magnitude of the stator
speed of the load torque at the zero crossing flux with an increase of 100 % of the stator
of the speed; however, it is important to note resistance at 0.25s. We note that according to
that the control system demonstrates a good these results that the observer corrects well
performance even under those variations. the magnitude of the stator flux and its
reference does not deviate from its reference
C. Variation of the load torques in established mode follows.
Fig. 11 and Fig. 12 illustrate the
VIII. CONCLUSION
simulation results after the introduction of
load torque between 1.5 s and 3s after a
In this paper, we use a method for the
leadless starting. We can see the insensibility
direct torque control with sensorless
of the control algorithm to load torque
induction motor using an adaptive observer to
variation.
estimate the rotor speed and the stator flux.
The adaptive observer uses an adaptation
D. Effect of the Stator Resistance Variation
mechanism for the speed estimation when the
To show the effect of stator resistance, load torques change. This approach relies on
we carried out to vary stator resistance. the improvement of an estimation of the
||Ψs| (Wb)
|Ψs| (Wb)
^
Fig. 13.a. Evolution of a real stator magnitude Fig. 13.b. Evolution of an estimate stator magnitude
developed by the induction motor. developed by the adaptive observer.
58 ACTA ELECTROTEHNICA
components of the stator flux and the rotor 4. G. Bottiglieri, G. Scelba, G. Scarcella, A. Testa,
speed. A. Consoli, "Sensorless speed estimation in
induction motor drives," IEEE Electric Machines,
A PI (anti-windup) regulator has been IEMDC'03., Vol 1, pp. 624 – 630, June.2003.
used to replace the PI controller in the speed 5. Y. Cao, Z. Lin, and D.G. Ward, "An anti-windup
control of a direct torque control. In approach to enlarging domain of attraction for
conclusion, it seems that the PI (anti-windup) linear systems subject to actuator saturation,"
controller outperforms the classical PI IEEE Trans. Automat. Control, vol. 47, pp. 140–
145, Jan. 2002.
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We have given a general vision on Trans. on Automat. Control, Vol.49, pp. 2055-
association adaptive Observer-DTC, we can 2061, November. 2004.
note that the estimation of the components of 7. C. Canudas de Wit, Modélisation contrôle
vectoriel et DTC, Edition HARMES science
the stator flux by the adaptive observer has Europe, Ltd: 2000.
well compensated the variation of the stator 8. D. Casadei, F. Profumo, G. Serra, and A. Tani
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