Speed Sensorless Indirect Field-Oriented of Induction
Motor Using Two Type of Adaptive Observer
Chiheb Ben Regaya1, Abderrahmen Zaafouri2 and Abdelkader Chaari3
University of Tunis
Unit C3S, Higher School of Sciences and Techniques of Tunis (ESSTT),
5 Av. Taha Hussein, BP 56, 1008 Tunis, Tunisia.
2
1
E-mail: chiheb_ben_regaya@yahoo.fr; E-mail: abderrahmen.zaafouri@isetr.rnu.tn;
3
E-mail: assil.chaari@esstt.rnu.tn
Abstract— In this paper, the indirect vector control speed
sensorless is presented, two adaptive mechanisms have been
proposed to estimate the rotor speed. The first adaptive observer
based on sliding mode, a study was made to present the steps
needed to design a sliding mode observer. The second has been
developed from the backstepping technique to design an observer
for the rotor speed. Finally, tests show the robust performance of
the control law obtained by these two types of adaptive observers.
Key words—Sliding mode observer, Backstepping observer,
Induction motor, Speed sensorless, Indirect vector control.
I. INTRODUCTION
An asynchronous machine controlled by a vector
controller behaves as a DC machine with separate excitation,
in which the torque and flux are decoupled and independently
controlled, to obtain a good control accuracy and high
dynamic performance. However, vector control has the
disadvantage of requiring the use of a speed sensor, which
imposes additional costs and increases the complexity of the
arrangements.
Variable speed drives incorporate new industrial constraints
such as the speed control without encoder. It is then necessary
to use observation techniques in the sense of automatic speed
estimate based on information collected by the measurement
of stator currents and voltages.
Speed estimation methods for sensorless induction motor
drives can be divided into two groups. The first is based on
high-frequency components injection in stator voltages or
currents [1], [2]. the second is based on fundamental
components in stator voltages and currents [3-6], [8-10]. The
first method may operate stably under zero-frequency
condition, which occurs in regenerating mode at low speeds.
The high-frequency components injection, however, causes
loss increasing and torque ripple.
The present article is organized as follows: in section 2,
the model of induction motor is defined. The sliding mode
observer of speed is established in Section 3. In section 4, we
developed another type of rotor speed observer using the
backstepping technique. We close our article with a
conclusion.
II. INDUCTION MOTOR MODELING
The induction motor model can be described by the
following state equation in the stationary reference frame:
[4-6].
⎡iαs ⎤ ⎡⎡a1
⎢i ⎥ ⎢⎢
d ⎢ βs ⎥ ⎢⎣0
=
dt ⎢φαr ⎥ ⎢⎡a4
⎢ ⎥ ⎢⎢
⎢⎣φβr ⎥⎦ ⎢⎣⎣0
⎡ 1
⎤
0 ⎥
a3ωr ⎤⎤ ⎡iαs ⎤ ⎢σ Ls
0 ⎤ ⎡ a2
⎥
⎥ ⎢ ⎥ ⎢
a1 ⎥⎦ ⎣⎢−a3ωr a2 ⎦⎥⎥ ⎢iβs ⎥ ⎢
1 ⎥ ⎡vαs ⎤
⋅
+ ⎢0
⎥ ⋅ ⎢ ⎥ (1)
−ωr ⎤⎥ ⎢φαr ⎥ ⎢ σ Ls ⎥ ⎣vβs ⎦
0 ⎤ ⎡a5
⎥ ⎢ ⎥
⎥ ⎢ω
0⎥
a4 ⎦⎣
a5 ⎦⎥⎥⎦ ⎢⎣φβr ⎥⎦ ⎢0
r
⎢
⎥
0⎦
⎣0
X = A(ω) ⋅ X + B⋅ u
Y
T
= ⎡⎣iαs iβs ⎤⎦
Where:
1 ⎛ Rs 1 − σ ⎞
M
+
⎜
⎟; a =
τ r ⎠ 2 σ L s L rτ r
1
M
M
;a =
; a5 = −
a3 =
σ Ls Lr 4 τ r
τr
a1 = −
σ ⎝ Ls
R s , Rr : Resistance of Stator and Rotor.
L s , Lr : Stator and Rotor inductance.
M
: Mutual inductance.
(
σ
: Total leakage factor, σ = 1 − M 2 L s L r
τr
ωr
: Rotor time constant, τ r = L r R r .
)
: Angular rotor speed.
III. SLIDING MODE OBSERVER OF ROTOR SPEED
The role of the observer is to estimate the rotor flux and
stator currents knowing the current measurement and stator
voltages without the rotational speed. Consider now the
system of induction motor taking into account the variables of
U.S. Government work not protected by U.S. copyright
currents and flux. Variables to observe are iˆα s , iˆβ s , φˆα r and φˆβ r .
In most cases, we prefer a system fast without overshoot.
Therefore the choice of the right commutation or sliding
surface defined by:
(2)
S( x ) = υ x1 + x1
Where υ is a positive constant, which determines the band
width of the system.
The graphical interpretation of equations is shown in Fig. 1.
e
e
P2 ( e2 , e2 )
S =0
Fig. 1.Graphical interpretation of equation
The system of equation (1) can be described as follows:
The observer of currents and flux have no coupling between
them, and we can see that the flux estimation is a simple
integration of sliding functions and estimated currents without
the need to know the speed when the estimated currents
converges to the measured currents.
With τ is the constant time of the low-pass filter.
Where:
T
is = ⎡⎣iα s iβ s ⎤⎦ , φr = ⎡⎣φα r φβ r ⎤⎦ , vs = ⎡⎣vα s vβ s ⎤⎦
0⎤
0⎤
⎡a
⎡a
M
, A21 = ⎢ 4
A11 = ⎢ 1
⎥ , A12 =
⎥
0
0
a
a
σ Ls Lr
1⎦
4⎦
⎣
⎣
− ωr ⎤
⎡a5
⎡−a5 ωr ⎤
A22 = ⎢
⎥, ζ = ⎢
⎥
ω
a
5⎦
⎣ r
⎣−ωr − a5 ⎦
The rotor flux can be calculated from equation (5) and (6)
as follows:
T
φˆα r = ∫ A21iˆsα −ψ1eq dt
(
)
φˆ = ∫ ( A iˆ −ψ )dt
βr
We note that the matrix ζ r appears simultaneously in the
equations of flux and currents. This implies that the design of
the observer for current and flux can be based on the
replacement of the common term, which is the matrix ζ r by
the same sliding function ψ:
⎡−a5
⎡ψ 1 ⎤
⎢ψ ⎥ = ψˆ = ⎢−ωˆ
⎣ 2⎦
⎣ r
⎧⎪ 1 if iα s > 0
with sign ( iα s ) = ⎨
⎪⎩ − 1 if iα s < 0
⎧⎪ 1 if iβ s > 0
sign ( iα s ) = ⎨
⎪⎩ − 1 if iβ s < 0
(3)
d
φr = A21is + A22φr = A21is − ζφr
dt
T
iβ s = iˆβ s − iβ s
This type of sliding mode function depends on motor
parameters and is difficult to implement because of the
chattering phenomenon, so we assume that the equivalent
control is only the slow component of real control and we can
obtained it by using a low-pass filter of the discontinuous
control [7].
⎡ 1
⎤
0 ⎥
⎡ψ 1eq ⎤ ⎢ τ s + 1
⎡ψ 1 ⎤
(6)
⎢
⎥=⎢
⎥⎢ ⎥
eq
1 ⎥ ⎣ψ 2 ⎦
⎣⎢ψ 2 ⎦⎥ ⎢ 0
⎢⎣
τ s + 1 ⎥⎦
P1 ( e1 , e1 )
d
is = A11is + A12ζφr + Bvs
dt
and iα s = iˆα s − iα s ,
ωˆ r ⎤ ⎡φˆα r ⎤
⎢ ⎥
− a5 ⎥⎦ ⎢φˆβ r ⎥
⎣ ⎦
(4)
21 sβ
(7)
eq
2
Using equation (4) and (5), we can calculate the estimate
rotor speed and the angle rotor flux as:
ωˆ r =
1 ˆ eq ˆ eq
φβ rψ1 − φα rψ 2
φˆ
(
)
(8)
r
(
θˆ = tan −1 φˆβ r φˆα r
)
(9)
Taking account of the above, the current and flux observer
using the sliding function can be written as:
IV. BACKSTEPPING OBSERVER OF ROTOR SPEED
d ˆ
is = A11iˆs + A12ψ + Bvs
dt
d ˆ
φr = A21iˆs −ψ
dt
In this section we will propose a type of adaptive control
using a backstepping speed observer. The variables measured
are the stator currents and the rotor speed of the asynchronous
machine. The behavior of the rotor flux and rotor speed will
generate by our backstepping observer[8-9].
(5)
Where:
iˆs = ⎡⎣iˆα s
T
iˆβ s ⎤⎦ , φˆr = ⎡φˆα r
⎣
φˆβ r ⎤⎦
T
ψ 1 = −u0 sign ( iα s ) , ψ 2 = −u0 sign ( iβ s )
The model of the backstepping observer can be described in
a reference connected to the stator by the following equations
from the system of equation (1):
d ˆ
iα s = a1iˆα s + a2φˆα r + a3ωˆ rφˆβ r + BVα s + V1
dt
d ˆ
iβ s = a1iˆβ s − a3ωˆ r φˆα r + a2φˆβ r + BVβ s + V2
dt
d ˆ
φα r = a4iˆds + a5φˆα r − ωˆ r φˆβ r
dt
d ˆ
φβ r = a4iˆβ s + ωˆ rφˆα r + a5φˆβ r
dt
(10)
(
(
(
(
)
(17)
epβ = a1ei β s − a3 ωr eφαr − Δωrφˆα r + a2eφβ r − V2 + γ 2ei β s
Finally we can design the observer of the rotor speed using
the following Lyapunov function [8], [10]:
L=
(11)
Δω 2 ⎞
1⎛ 2
2
2
2
2
2
+
+
+
+
+
+
+ ⎟ (18)
e
e
e
e
e
e
⎜ xα
xβ
pα
pβ
φα r
φβ r
2⎝
kω ⎠
After the derivation of this function and knowing that its
derivative must be defined negative we can write the
mechanism of adaptation as follows [8]:
eφqr = φqr − φˆqr
⎛
d Δω
M ⎞
= δω ⎜ n p
⎟ (φα r − φβ r )
dt
⎝ σ Ls Lr ⎠
Using equations (10) and (11), the dynamical equations
errors are giving by the following system:
(
)
eiα s = a1eiα s + a2eφαr + a3 ωr eφβ r − Δωφˆβ r − V1
(
)
epα = a1eiα s + a2eφαr + a3 ωr eφβ r − Δωrφˆβ r − V1 + γ 1eiα s
All the parameters of induction motor are considered
constant, The stator error can be defined as follows:
eφdr = φdr − φˆdr
)
)
We can get the dynamics of tracking errors epα and epβ as
follows:
Where V1 and V2 are the control input to be designed.
eids = ids − iˆds
eiqs = iqs − iˆqs
γ1 and γ2 are positive constant to guarantee the exponential
convergence of the observer. Using equations (12), (13), (14),
(15) and we choose the control law as follows:
V1 = a3 ωr eφβ r − Δωr φˆβ r + γ 1eiα s + γ 2 e pα + exα
(16)
V2 = a3 Δωr φˆα r − ωr eφα r + γ 1eiβ s + γ 2 e pβ + ex β
V. SIMULATION RESULTS AND INTERPRETATION
)
eiβ s = a1ei β s − a3 ωr eφαr − Δωφˆα r + a2eφβ r − V2
eφαr = a4eiα s + a5eφαr + Δωφˆβ r − ωr eφβ r
(19)
(12)
eφβ r = a4ei β s − ωr eφβ r + Δωφˆβ r + a5eφβ r
The block diagram of the proposed control using a sliding
mode observer of the currents, flux and speed or backstepping
observer is given in fig. 2. The model of the tree phase
induction motor and adaptive observer are emulated by
Matlab-Simulink with a sample time 10µs. The motor
parameters are shown in Table 1.
With Δω = ω − ωˆ
E
ids *
+
Vds *
ω +
iqs * +
−
Vqs *
−
−
*
ω̂
(13)
The tracking error epα and epβ of the estimation errors of
the stator currents eiαs and eiβs can be defined as follows:
e pα = eiα s − f s1
e pβ = eiβ s − f s2
(14)
Where:
f s1 = −γ 1exα
f s2 = −γ 2 exβ
(15)
iαβ
Iabc
ids
iqs
The first step of our observer design is to define the
functions stabilizing fs1 and fs2 of the virtual variables eiαs and
eiβs.
Assume that :
eiα s = exα
eiβ s = exβ
θ dq
Sensors
To solve the problem of pursuit and the convergence of
estimation errors, a control algorithm will be proposed using a
backstepping observer [8-10].
Vαβ
Fig. 2. Block diagram of speed sensorless control system based on sliding
mode observer.
TABLE I. PARAMETERS OF INDUCTION MOTOR
Designation
Notations
Rating values
Resistance statorique
Resistance rotorique
Stator self-inductance
Rotor self-inductance
Mutual inductance
Moment of inertia
Friction coefficient
Number of poles
Rated power
Rated voltage
Rs
Rr
Ls
Lr
M
j
f
np
Pn
Vsn
2.3Ω
1.83Ω
261µH
261µH
245µH
0.03kgm2
0.002Nm
2
1.5Kw
220V
ωˆr( SM )
ωˆr( BS )
ωˆr( SM )
ωˆr( BS )
ωr*
Speed [p.u]
ωr*
In the second and the third tests we decrease the speed
reference to (150r/min) then to (50r/min) to see the robustness
of the observer for low speed.
In the figures 6, 7, 8 and 9, we can see that the two
observers can estimate the rotor speed for a low step (150 or
50r/min), and the estimation error does not exceed 0.5% for
the sliding mode observer and 0.2% for the another observer.
Speed [p.u]
Figure 3, 4 and 5 shows the simulation results of the first
tests to a step speed (1500r/min) under a load torque of 10 Nm.
The estimated speed by a sliding mode observer or
backstepping observer reaches steady state after 0.35s (Fig. 3).
The error between the actual and estimated speed for the
two type of observer is given by Fig. 4. We can see the effect
of the phenomenon of chattering on the speed estimated and
the estimation error by the sliding mode observer(Fig. 3, 4),
this is due to the nature of the sliding functions ψ1 and ψ2.
Figure 5 shows the electromagnetic torque and the effect of
chattering phenomenon developed during the starting and the
stationary phase.
Zoom
Time [s]
Fig. 6.Estimated speed to a step (150r/min) using the proposed adaptive
observer (SMO and BSO).
Time [s]
Fig. 3.Estimated speed using the proposed adaptive observer (SMO and BSO).
eω( SM )
eω( BS )
Error [p.u]
eω( BS )
Error [p.u]
eω( SM )
Time [s]
Fig. 7. Error between real and estimated speed to a step (150r/min).
(SMO and BSO).
Time [s]
Fig. 4. Error between real and estimated speed (SMO and BSO).
ωˆr( BS )
Cem( SM)
ωr*
Zoom
Speed [p.u]
Torque [p.u]
ωˆr( SM )
Cem( BS)
Time [s]
Time [s]
Fig. 5. Torque response for step varying of load torque (SMO and BSO).
Fig. 8.Estimated speed to a step (50r/min) using the proposed adaptive
observer (SMO and BSO).
[9]
eω( SM )
eω( BS )
Speed [p.u]
[10]
Time [s]
Fig. 9. Error between real and estimated speed to a step (50r/min).
(SMO and BSO).
VI. CONCLUSION
Two type of observer were proposed in this paper. The
first is the sliding mode speed observer which presents good
estimation results even at low speeds, but the second observer
which is based on backstepping technique is better because the
estimation error for low speed does not exceed 0.2%.
The backstepping observer presents no corrugations at the
estimated speed and torque unlike the sliding mode observer.
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