Sensorless Vector Control of AC Induction Motor Using Sliding-Mode Observer - Diponegoro University | Institutional Repository (UNDIP-IR)
Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
ISSN: 2086-5023
Sensorless Vector Control of AC Induction Motor
Using Sliding-Mode Observer
Phuc Thinh Doan#1, Tan Tien Nguyen##2, Sang Kwun Jeong*3, Sea June Oh**4 and Sang Bong Kim#5
#
Department of Mechanical & Automotive Eng., College of Eng., Pukyong National University
Busan 608-739, Korea
5
##
Mechanical Engineering Faculty, Hochiminh City University of Technology, Viet Nam
*
**
kimsb@pknu.ac.kr
Han Sung Well Tech Co, Ltd, Busan, Korea
Division of Marine System Engineering, Korea Maritime University, Busan 606-791, Korea
Abstract— This paper develops a sensorless vector controlled
method for AC induction motor using sliding-mode observer. For
developing the control algorithm, modeling of AC induction motor is
presented. After that, a sliding mode observer is proposed to estimate
the motor speed, the rotor flux, the angular position of the rotor flux
and the motor torque from monitored stator voltages and currents.
The use of the nonlinear sliding mode observer provides very good
performance for both low and high speed motor operation.
Furthermore, the proposed system is robust in motor losses and load
variations. The convergence of the proposed observer is obtained
using the Lyapunov theory. Hardware and software for simulation
and experiment of the AC induction motor drive are introduced. The
hardware consists of a 1.5kw AC induction motor connected in series
with a torque sensor and a powder brake. A controller is developed
based on DSP TMS320F28355. The simulation and experimental
results illustrate that fast torque and speed response with small
torque ripples can be achieved. The proposed control scheme is
suitable to the application fields that require high performance of
torque response such as electric vehicles.
[Keywords— Sliding-Mode Observer;
Sensorless Vector Control]
AC Induction Motor;
I. INTRODUCTION
The Adjustable Speed Drives (ADS) are used in a wide
variety of industrial applications such as pumps, fans, elevator,
electrical vehicles, wind generation systems, etc. There are
various types of electric motors in electric drives such as DC
motors, AC motors, stepper motors. AC induction motors are
the most popularly motor type used in industrial due to their
simple construction, reliability, robustness and low cost.
Many attempts have been made to develop a sensorless
vector control of AC induction motor. Several techniques
have been proposed. Magnetic-Salience-Based methods are
very promising for standstill and low-speed operation [1,2].
Their main disadvantage is the need for high precision voltage
and current measurements. Also, these methods require a
proper machine design. Model-Reference Adaptive System
(MRAS) have a good performance over a large speed range
[3,4]. Their disadvantage is the large influence of parameter
deviation at low-speed and standstill operation. The extended
Kalman filter has been used as the most appropriate solution
for speed sensorless drives [5]. However, this stochastic
observer has some inherent disadvantages, such as the
influence of the noise characteristic, the computational burden
and the absence of design and tuning criteria. The current
harmonic spectral estimation method estimates the rotor speed
from current harmonics [6]. This method have a very good
performance at low speed, but needs a complicated hardware
and software setup for measuring and filtering the currents.
Artificial intelligence methods [7] that use artificial
intelligence techniques such as fuzzy logic and neural
networks are robust to parameter deviation and measurement
noise. However, they need long development times and an
expertise in several artificial intelligence procedures.
This paper develops a sensorless vector controlled method
for AC induction motor using sliding-mode observer. The
proposed observer estimates the motor speed, the rotor flux,
the angular position of the rotor flux and the motor torque in a
robust and efficiency manner. For developing the control
algorithm, modeling of AC induction motor is presented.
Hardware and software for simulation and experiment of the
AC induction motor drive are introduced. The simulation and
experimental results illustrate that fast torque and speed
response with small torque ripples can be achieved. The
proposed control scheme is suitable to the application fields
that require high performance of torque response such as
electric vehicles.
II. MATHEMATICAL MODEL OF AN INDUCTION MOTOR
The motor model equation in the stator-fixed coordinate
( ab coordinate) is as follows.
ì s
dψ ss
s
u
=
R
i
+
ï s
s s
dt
ï
ïï s
dψ sr
s
u
=
R
i
+
- jwr ψ rs = 0
r r
í r
dt
ï
ï ψ ss = Ls i ss + Lm i rs
ï s
s
s
ïî ψ r = Lm i s + Lr i r
where
(1)
u ss and u rs are stator and rotor voltage vectors; i ss
and i rs are stator and rotor current vectors; ψ ss and ψ rs are
stator and rotor flux vectors; Ls and Lr are stator and rotor
winding self inductances; Lm is magnetizing inductance, Rs
and Rr are stator and rotor winding resistances and wr is the
mechanical motor speed.
Depending on the viewpoint of the observer, the
unmeasurable rotor current, i rs , and the stator flux, ψ ss , shall
be eliminated from the equation system. The two flux
equations of Eq. (1) are reduced into:
39
Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
i rs =
1
( ψ rs - Lmi ss )
Lr
ψ ss = Ls i ss +
Lm
Lr
(ψ
s
r
(2)
- Lm i ss )
(3)
Now substituting i rs and ψ ss into the voltage equations of
Eq. (1) yields:
ì s
di ss Lm dψ sr
s
+
ïu s = Rs i s + s Ls
dt Lr dt
ï
í
s
ï0 = - Lm R i s + æ Rr - jw ö ψ s + dψ r
ç
÷
r
s
r
r
ï
Lr
dt
è Tr
ø
î
Lm
L2m ö
R
1 æ
, h= r
ç Rs + 2 Rr ÷ , k =
s Ls Lr
Lr
s Ls è
Lr
ø
2
s = 1 - Lm ( Ls Lr ) , the following is obtained from Eq. (4).
With
g =
ì di ss
1 s
= -g i ss + k (h - jwr ) ψ rs +
us
ï
s Ls
ï dt
í s
ï dψ r
s
s
ïî dt = h Lm i s - (h - jwr ) ψ r
A. Conventional observer
The conventional observer is designed as follows.
The estimated stator flux is calculated based on the motor
equations using a pure integrator as follows:
yˆ sa = ò ( u sa - isa Rs )dt
(9)
yˆ s b = ò ( u s b - is b Rs )dt
The estimated rotor flux is computed from estimated stator
flux as follows:
(4)
,
æ L L - L2m ö
Lr
yˆ ra = - ç s r
÷ isa + yˆ sa
L
L
m
m
è
ø
2
æ L L - Lm ö
Lr
yˆ r b = - ç s r
÷ is b + yˆ s b
L
L
m
m
è
ø
(10)
The estimated angle of rotor flux is obtained as follows.
d qˆr
wˆ r =
dt
(5)
æ yˆ r b ö
where qˆr = tan -1 ç
÷
è yˆ r a ø
(11)
B. Sliding-mode observer
First, the observer is given as follows [8]:
After separating the real and imaginary components from
Eq. (2), the followings are obtained.
1
ì disa
ï dt = -g isa + khy r a + kwry r b + s L u sa
s
ï
ï dis b
1
= -g is b - kwry r a + khy r b +
usb
ï
ï dt
s Ls
í
ï dy r a
ï dt = -hy r a - wry r b + h Lm isa
ï
ï dy r b
ïî dt = -hy r b + wry r a + h Lmis b
ISSN: 2086-5023
diˆsa
dt
diˆs b
= iˆs¢a = -g iˆsa + khyˆ r a + kwˆ ryˆ r b +
1
u sa
s Ls
(12)
= iˆs¢b = -g iˆs b - kwˆ ryˆ r a + khyˆ r b +
1
u
s Ls s b
(13)
dt
dyˆ r a
(6)
dt
dyˆ r b
dt
= yˆ r¢a = -hyˆ r a - wˆ ryˆ r b + h Lm isa
(14)
= yˆ r¢b = -hyˆ r b + wˆ ryˆ r a + h Lmis b
(15)
where iˆsa and iˆs b are the estimated stator current components,
yˆ ra and yˆ r b are the estimated rotor flux components and wˆ r
To get the complete model of the induction motor, the
electromagnetic torque and the motor angular velocity are
obtained as follows:
is the estimated value of the mechanical motor speed. The
followings are defined:
isa = iˆsa - isa , is b = iˆs b - is b , wr = wˆ r - wr , y sa = yˆ sa -y sa ,
3 p Lm
Te =
( isby sa - isay sb )
2 Lr
(7)
y s b = yˆ s b -y s b
d wr
1
= p (Te - TL )
dt
J
(8)
The definitions of Eq. (16) express the mismatch between
estimated and real components. A new system can be
developed as follows:
where Te , TL are electromagnetic torque and load torque,
respectively; p is the number of pole pairs; J is inertial
moment.
III. SLIDING MODE OBSERVER DESIGN
For sensorless control, it is necessary to make a
simultaneous observation of the rotor flux and rotor speed.
The proposed observer is compared with a conventional
observer.
(16)
is¢a = -g isa + khy r a + kwry r b + kwryˆ r b
(17)
is¢b = -g is b - kwry r a + khy r b - kwryˆ r a
(18)
y r¢a = -hy ra - wryˆ r b - wry r b
y r¢b = -hy r b + wryˆ ra + wry r a
(19)
(20)
Switching functions are defined as
40
Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
sˆa = is byˆ r a - isayˆ r b
sa = is by r a - isay r b
(21)
sb = is by r a - isay r b
ISSN: 2086-5023
Substituting the equivalent speed value wˆ req into Eqs. (19,
20), the following system of equations exist:
y r¢a =
The following positive Lyapunov function is defined.
V=
(yˆ
2
ra
+
1 2
( isa + isb2 ) ³ 0
2
+yˆ r2b )
(yˆ
(yˆ
2
ra
+ yˆ r2b )
+ yˆ r2b )
2g
V + h sb
k
y ra
2
ra
+yˆ r2b )
ìl1 + l2 = -h
ìl = -h
Ûí 1
í
îl1l2 = 0
îl2 = 0
(30)
Therefore, under feedback wˆ r = K sgn ( sˆa ) , the proposed
The following equation is selected as
wˆ r = K sgn ( sˆa )
(25)
From Eq. (19), the following is satisfied V& < 0 :
K sˆa > wr sa + d ³ wr sa + d
(26)
2g
V + h sb
k
If K is big enough with Eq. (26), the estimated current
values, iˆsa and iˆs b , converge to the real values, isa , is b
where d = -
( isa ® 0, is b ® 0 , sˆa ® 0 ).
According to sliding-mode theory, under the discontinuous
control wˆ r = K sgn ( sˆa ) , the equivalent motion can be
described by the set of equations sˆa = 0, sˆa¢ = 0 where sˆa¢
denotes derivatives of sˆa .
isa = 0, is b = 0 , sˆa = 0 , the
condition sˆa¢ = 0 becomes:
observer is forced to slide on sˆa = 0 with equivalent value
wˆ r = wˆ req , and the system of Eq. (26) is neutrally stable
( y r¢a ® 0 , y r¢b ® 0 , y r a = ca , y r b = cb where ca and cb
are constants). At steady state, Eqs. (19, 20) become:
-h ca - wr cb = wryˆ r b
(31)
-wr ca - h cb = -wryˆ r a
(32)
Because yˆ r a and yˆ r b are time-variant function, the only
solution for the above system of equations is ca = cb = 0 and
wr = 0 , which means that y ra = 0 , y r b = 0 and wˆ r = wˆ req .
Fig. 1 shows the block diagram of the proposed observer.
yˆ r b
yˆ ra
isa
is b
wˆ r
å
iˆsa
iˆs b
u sa
u sb
)
+ kw (y
(
r
yˆ r a -y r byˆ r b
ra
y rb
The roots of the characteristic polynomial of the above
system are the following:
(24)
sˆa¢ = kh y r byˆ r a -y r ayˆ r b - kwˆ r yˆ r2a + yˆ r2b
(29)
-hyˆ r2b - wryˆ r ayˆ r b
(yˆ
y rb
(23)
The condition V& < 0 can be written as follows:
Using the conditions,
2
ra
wryˆ r2b - hyˆ rayˆ r b
+
V& = is¢a isa + is¢b is b
V& = -2g V - kwr ( sˆa - sa ) - kwr sˆa + kh sb
wr ( sˆa - sa ) + wr sˆa > -
y r¢b =
y ra
-wryˆ r2a - hyˆ r ayˆ r b
(22)
Its derivative after substitution from Eq. (17) and (18) can
be written as follows:
(
-hyˆ r2a + wryˆ r ayˆ r b
)
)=0
å
å
wˆ r
Sˆa
wˆ r
(27)
wˆ r
wˆ req
From Eq. (22), the equivalent rotor speed is obtained.
Fig. 1 Block diagram of the sliding mode observer
wˆ r = wˆ req = wr
(y
yˆ ra -y r byˆ r b )
ra
(yˆ
2
ra
+ yˆ r2b )
(y
+h
ˆ
ˆ
r by r a -y r ay r b )
(yˆ
2
ra
+ yˆ r2b )
(28)
IV. SIMULATION AND EXPERIMENTAL RESULTS
The proposed sliding mode observer has been tested in
order to verify its behaviour and robustness using testing
system as shown in Fig. 2. The testing system consists of an
AC induction motor (1.5kw), a torque sensor, a powder brake.
The motor parameters are given in Table 1.
41
Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
ISSN: 2086-5023
(a) Stator current
5
0
-5
0
0.5
1
1.5
2
2.5
(b) Rotor speed
600
Reference
Real
400
200
0
0
0.5
1
1.5
2
2.5
(c) Torque
Fig. 2 Testing system in experiment
100
Simulation has been done with the conventional observer
and the proposed observer in condition of load and speed
variations. Fig. 3 and Fig. 4 show the simulation result using
the conventional observer and the proposed observer,
respectively. The simulation results consist of stator current,
rotor speed, electromagnetic torque and DC bus voltage. From
the results in Fig. 3 and Fig. 4, we can conclude that: The
implementation of pure integrator to estimate the stator flux is
difficult because of initial value problem. The dynamic
performance is lower and torque oscillation is bigger than the
proposed observer.
Fig. 5 and Fig. 6 show the experimental results using the
conventional observer and the proposed observer, respectively.
In experimental results, the upper pictures are phase voltages
and the lower pictures are phase current. From the
experimental results, the phase current in conventional
observer has a larger ripple than the one in proposed observer.
Tl
Te
0
-100
0
0.5
1
1.5
2
2.5
2
2.5
(d) DC voltage
240
220
200
180
0
0.5
1
1.5
Time (s)
Fig. 3 Simulation results using conventional observer
TABLE I
PARAMETER'S VALUES OF INDUCTION MOTOR
Symbol
Value
Unit
2
-
J
Rs
Description
Number of pole
pairs
Inertial moment
Stator resistance
0.007
0.01
Kgm2
W
Rr
Rotor resistance
0.01
W
Lr
Rotor inductance
3.1
H
Ls
Stator inductance
3.1
H
Lm
Mutual inductance
3
H
p
Fig. 4 Simulation results using proposed observer
42
Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
ISSN: 2086-5023
V. CONCLUSIONS
This paper developed a sliding-mode observer for
sensorless vector control of AC induction motor. The
modeling was also presented. Hardware for the simulation and
experiment is described. The proposed observer was
compared with the conventional observer. The simulation and
experimental results showed that the proposed observer has
smaller torque ripple than the conventional observer. The
proposed observer was robust in high speed and low speed
operation and load variation.
ACKNOWLEDGEMENT
This research was supported by a grant from Construction
Technology Innovation Program (CTIP) funded by Ministry
of Land, Transportation and Maritime Affairs (MLTM) of
Korean government.
REFERENCES
Fig. 5 Phase voltage and phase current using conventional observer in
experiment
[1] M. Schroedl, “Sensorless control of AC machines at low speed and
standstill based on the INFORM method,” Proc. IEEE-IAS Annual
meeting (1996), pp. 270-277.
[2] P.L. Jansen, R.D. Lorentz, “Transducer-less position and velocity
estimation in induction and salient AC machines,” IEEE Trans. Ind.
Appl. , vol. 31, pp 240-247, Mar. 1995.
[3] C. Schauder, “Adaptive Speed Identification for Vector Control without
Rotational Transducers,” IEEE Trans. Ind. Appl. Vol.28, pp.1054-1061,
Sep. 1992.
[4] J. Maes, J.A. Melkebeek, “Speed Sensorless Direct Torque Control of
Induction Motors Using an Adaptive Flux Observer,” IEEE Trans. Ind.
Appl. Vol. 38, pp. 778-785, May 2000.
[5] R. Kim, S.K. Sul, M.H. Park, “Speed Sensorless Vector Control of
Induction Motor using extended Kalman Filter,” IEEE Trans. Ind. Appl.
Vol. 30, pp. 1225-1233, Sep. 1994.
[6] K.D Hurst, T.G. Habetler, “Sensorless Speed Measurement Using Current
Harmonic Spectral Estimation in Induction Machines Drives,” IEEE
Trans. Power Electronics, vol. 11, pp.66-73, Jan. 1996.
[7] P. Vas, Artificial-Intelligence-Based Electrical Machines and Drives
Application of Fuzzy, Neural, Fuzzy-Neural and Genertic-Algorithm
Based Techniques, Oxford Univ. Press, UK, 1999.
[8] V. Utkin, J. Gulder and J. Shi, Sliding Mode Control in
Electromechanical Systems, Taylor & Francis, 1999.
Fig. 6 Phase voltage and phase current using sliding-mode observer in
experiment
43
ISSN: 2086-5023
Sensorless Vector Control of AC Induction Motor
Using Sliding-Mode Observer
Phuc Thinh Doan#1, Tan Tien Nguyen##2, Sang Kwun Jeong*3, Sea June Oh**4 and Sang Bong Kim#5
#
Department of Mechanical & Automotive Eng., College of Eng., Pukyong National University
Busan 608-739, Korea
5
##
Mechanical Engineering Faculty, Hochiminh City University of Technology, Viet Nam
*
**
kimsb@pknu.ac.kr
Han Sung Well Tech Co, Ltd, Busan, Korea
Division of Marine System Engineering, Korea Maritime University, Busan 606-791, Korea
Abstract— This paper develops a sensorless vector controlled
method for AC induction motor using sliding-mode observer. For
developing the control algorithm, modeling of AC induction motor is
presented. After that, a sliding mode observer is proposed to estimate
the motor speed, the rotor flux, the angular position of the rotor flux
and the motor torque from monitored stator voltages and currents.
The use of the nonlinear sliding mode observer provides very good
performance for both low and high speed motor operation.
Furthermore, the proposed system is robust in motor losses and load
variations. The convergence of the proposed observer is obtained
using the Lyapunov theory. Hardware and software for simulation
and experiment of the AC induction motor drive are introduced. The
hardware consists of a 1.5kw AC induction motor connected in series
with a torque sensor and a powder brake. A controller is developed
based on DSP TMS320F28355. The simulation and experimental
results illustrate that fast torque and speed response with small
torque ripples can be achieved. The proposed control scheme is
suitable to the application fields that require high performance of
torque response such as electric vehicles.
[Keywords— Sliding-Mode Observer;
Sensorless Vector Control]
AC Induction Motor;
I. INTRODUCTION
The Adjustable Speed Drives (ADS) are used in a wide
variety of industrial applications such as pumps, fans, elevator,
electrical vehicles, wind generation systems, etc. There are
various types of electric motors in electric drives such as DC
motors, AC motors, stepper motors. AC induction motors are
the most popularly motor type used in industrial due to their
simple construction, reliability, robustness and low cost.
Many attempts have been made to develop a sensorless
vector control of AC induction motor. Several techniques
have been proposed. Magnetic-Salience-Based methods are
very promising for standstill and low-speed operation [1,2].
Their main disadvantage is the need for high precision voltage
and current measurements. Also, these methods require a
proper machine design. Model-Reference Adaptive System
(MRAS) have a good performance over a large speed range
[3,4]. Their disadvantage is the large influence of parameter
deviation at low-speed and standstill operation. The extended
Kalman filter has been used as the most appropriate solution
for speed sensorless drives [5]. However, this stochastic
observer has some inherent disadvantages, such as the
influence of the noise characteristic, the computational burden
and the absence of design and tuning criteria. The current
harmonic spectral estimation method estimates the rotor speed
from current harmonics [6]. This method have a very good
performance at low speed, but needs a complicated hardware
and software setup for measuring and filtering the currents.
Artificial intelligence methods [7] that use artificial
intelligence techniques such as fuzzy logic and neural
networks are robust to parameter deviation and measurement
noise. However, they need long development times and an
expertise in several artificial intelligence procedures.
This paper develops a sensorless vector controlled method
for AC induction motor using sliding-mode observer. The
proposed observer estimates the motor speed, the rotor flux,
the angular position of the rotor flux and the motor torque in a
robust and efficiency manner. For developing the control
algorithm, modeling of AC induction motor is presented.
Hardware and software for simulation and experiment of the
AC induction motor drive are introduced. The simulation and
experimental results illustrate that fast torque and speed
response with small torque ripples can be achieved. The
proposed control scheme is suitable to the application fields
that require high performance of torque response such as
electric vehicles.
II. MATHEMATICAL MODEL OF AN INDUCTION MOTOR
The motor model equation in the stator-fixed coordinate
( ab coordinate) is as follows.
ì s
dψ ss
s
u
=
R
i
+
ï s
s s
dt
ï
ïï s
dψ sr
s
u
=
R
i
+
- jwr ψ rs = 0
r r
í r
dt
ï
ï ψ ss = Ls i ss + Lm i rs
ï s
s
s
ïî ψ r = Lm i s + Lr i r
where
(1)
u ss and u rs are stator and rotor voltage vectors; i ss
and i rs are stator and rotor current vectors; ψ ss and ψ rs are
stator and rotor flux vectors; Ls and Lr are stator and rotor
winding self inductances; Lm is magnetizing inductance, Rs
and Rr are stator and rotor winding resistances and wr is the
mechanical motor speed.
Depending on the viewpoint of the observer, the
unmeasurable rotor current, i rs , and the stator flux, ψ ss , shall
be eliminated from the equation system. The two flux
equations of Eq. (1) are reduced into:
39
Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
i rs =
1
( ψ rs - Lmi ss )
Lr
ψ ss = Ls i ss +
Lm
Lr
(ψ
s
r
(2)
- Lm i ss )
(3)
Now substituting i rs and ψ ss into the voltage equations of
Eq. (1) yields:
ì s
di ss Lm dψ sr
s
+
ïu s = Rs i s + s Ls
dt Lr dt
ï
í
s
ï0 = - Lm R i s + æ Rr - jw ö ψ s + dψ r
ç
÷
r
s
r
r
ï
Lr
dt
è Tr
ø
î
Lm
L2m ö
R
1 æ
, h= r
ç Rs + 2 Rr ÷ , k =
s Ls Lr
Lr
s Ls è
Lr
ø
2
s = 1 - Lm ( Ls Lr ) , the following is obtained from Eq. (4).
With
g =
ì di ss
1 s
= -g i ss + k (h - jwr ) ψ rs +
us
ï
s Ls
ï dt
í s
ï dψ r
s
s
ïî dt = h Lm i s - (h - jwr ) ψ r
A. Conventional observer
The conventional observer is designed as follows.
The estimated stator flux is calculated based on the motor
equations using a pure integrator as follows:
yˆ sa = ò ( u sa - isa Rs )dt
(9)
yˆ s b = ò ( u s b - is b Rs )dt
The estimated rotor flux is computed from estimated stator
flux as follows:
(4)
,
æ L L - L2m ö
Lr
yˆ ra = - ç s r
÷ isa + yˆ sa
L
L
m
m
è
ø
2
æ L L - Lm ö
Lr
yˆ r b = - ç s r
÷ is b + yˆ s b
L
L
m
m
è
ø
(10)
The estimated angle of rotor flux is obtained as follows.
d qˆr
wˆ r =
dt
(5)
æ yˆ r b ö
where qˆr = tan -1 ç
÷
è yˆ r a ø
(11)
B. Sliding-mode observer
First, the observer is given as follows [8]:
After separating the real and imaginary components from
Eq. (2), the followings are obtained.
1
ì disa
ï dt = -g isa + khy r a + kwry r b + s L u sa
s
ï
ï dis b
1
= -g is b - kwry r a + khy r b +
usb
ï
ï dt
s Ls
í
ï dy r a
ï dt = -hy r a - wry r b + h Lm isa
ï
ï dy r b
ïî dt = -hy r b + wry r a + h Lmis b
ISSN: 2086-5023
diˆsa
dt
diˆs b
= iˆs¢a = -g iˆsa + khyˆ r a + kwˆ ryˆ r b +
1
u sa
s Ls
(12)
= iˆs¢b = -g iˆs b - kwˆ ryˆ r a + khyˆ r b +
1
u
s Ls s b
(13)
dt
dyˆ r a
(6)
dt
dyˆ r b
dt
= yˆ r¢a = -hyˆ r a - wˆ ryˆ r b + h Lm isa
(14)
= yˆ r¢b = -hyˆ r b + wˆ ryˆ r a + h Lmis b
(15)
where iˆsa and iˆs b are the estimated stator current components,
yˆ ra and yˆ r b are the estimated rotor flux components and wˆ r
To get the complete model of the induction motor, the
electromagnetic torque and the motor angular velocity are
obtained as follows:
is the estimated value of the mechanical motor speed. The
followings are defined:
isa = iˆsa - isa , is b = iˆs b - is b , wr = wˆ r - wr , y sa = yˆ sa -y sa ,
3 p Lm
Te =
( isby sa - isay sb )
2 Lr
(7)
y s b = yˆ s b -y s b
d wr
1
= p (Te - TL )
dt
J
(8)
The definitions of Eq. (16) express the mismatch between
estimated and real components. A new system can be
developed as follows:
where Te , TL are electromagnetic torque and load torque,
respectively; p is the number of pole pairs; J is inertial
moment.
III. SLIDING MODE OBSERVER DESIGN
For sensorless control, it is necessary to make a
simultaneous observation of the rotor flux and rotor speed.
The proposed observer is compared with a conventional
observer.
(16)
is¢a = -g isa + khy r a + kwry r b + kwryˆ r b
(17)
is¢b = -g is b - kwry r a + khy r b - kwryˆ r a
(18)
y r¢a = -hy ra - wryˆ r b - wry r b
y r¢b = -hy r b + wryˆ ra + wry r a
(19)
(20)
Switching functions are defined as
40
Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
sˆa = is byˆ r a - isayˆ r b
sa = is by r a - isay r b
(21)
sb = is by r a - isay r b
ISSN: 2086-5023
Substituting the equivalent speed value wˆ req into Eqs. (19,
20), the following system of equations exist:
y r¢a =
The following positive Lyapunov function is defined.
V=
(yˆ
2
ra
+
1 2
( isa + isb2 ) ³ 0
2
+yˆ r2b )
(yˆ
(yˆ
2
ra
+ yˆ r2b )
+ yˆ r2b )
2g
V + h sb
k
y ra
2
ra
+yˆ r2b )
ìl1 + l2 = -h
ìl = -h
Ûí 1
í
îl1l2 = 0
îl2 = 0
(30)
Therefore, under feedback wˆ r = K sgn ( sˆa ) , the proposed
The following equation is selected as
wˆ r = K sgn ( sˆa )
(25)
From Eq. (19), the following is satisfied V& < 0 :
K sˆa > wr sa + d ³ wr sa + d
(26)
2g
V + h sb
k
If K is big enough with Eq. (26), the estimated current
values, iˆsa and iˆs b , converge to the real values, isa , is b
where d = -
( isa ® 0, is b ® 0 , sˆa ® 0 ).
According to sliding-mode theory, under the discontinuous
control wˆ r = K sgn ( sˆa ) , the equivalent motion can be
described by the set of equations sˆa = 0, sˆa¢ = 0 where sˆa¢
denotes derivatives of sˆa .
isa = 0, is b = 0 , sˆa = 0 , the
condition sˆa¢ = 0 becomes:
observer is forced to slide on sˆa = 0 with equivalent value
wˆ r = wˆ req , and the system of Eq. (26) is neutrally stable
( y r¢a ® 0 , y r¢b ® 0 , y r a = ca , y r b = cb where ca and cb
are constants). At steady state, Eqs. (19, 20) become:
-h ca - wr cb = wryˆ r b
(31)
-wr ca - h cb = -wryˆ r a
(32)
Because yˆ r a and yˆ r b are time-variant function, the only
solution for the above system of equations is ca = cb = 0 and
wr = 0 , which means that y ra = 0 , y r b = 0 and wˆ r = wˆ req .
Fig. 1 shows the block diagram of the proposed observer.
yˆ r b
yˆ ra
isa
is b
wˆ r
å
iˆsa
iˆs b
u sa
u sb
)
+ kw (y
(
r
yˆ r a -y r byˆ r b
ra
y rb
The roots of the characteristic polynomial of the above
system are the following:
(24)
sˆa¢ = kh y r byˆ r a -y r ayˆ r b - kwˆ r yˆ r2a + yˆ r2b
(29)
-hyˆ r2b - wryˆ r ayˆ r b
(yˆ
y rb
(23)
The condition V& < 0 can be written as follows:
Using the conditions,
2
ra
wryˆ r2b - hyˆ rayˆ r b
+
V& = is¢a isa + is¢b is b
V& = -2g V - kwr ( sˆa - sa ) - kwr sˆa + kh sb
wr ( sˆa - sa ) + wr sˆa > -
y r¢b =
y ra
-wryˆ r2a - hyˆ r ayˆ r b
(22)
Its derivative after substitution from Eq. (17) and (18) can
be written as follows:
(
-hyˆ r2a + wryˆ r ayˆ r b
)
)=0
å
å
wˆ r
Sˆa
wˆ r
(27)
wˆ r
wˆ req
From Eq. (22), the equivalent rotor speed is obtained.
Fig. 1 Block diagram of the sliding mode observer
wˆ r = wˆ req = wr
(y
yˆ ra -y r byˆ r b )
ra
(yˆ
2
ra
+ yˆ r2b )
(y
+h
ˆ
ˆ
r by r a -y r ay r b )
(yˆ
2
ra
+ yˆ r2b )
(28)
IV. SIMULATION AND EXPERIMENTAL RESULTS
The proposed sliding mode observer has been tested in
order to verify its behaviour and robustness using testing
system as shown in Fig. 2. The testing system consists of an
AC induction motor (1.5kw), a torque sensor, a powder brake.
The motor parameters are given in Table 1.
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Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
ISSN: 2086-5023
(a) Stator current
5
0
-5
0
0.5
1
1.5
2
2.5
(b) Rotor speed
600
Reference
Real
400
200
0
0
0.5
1
1.5
2
2.5
(c) Torque
Fig. 2 Testing system in experiment
100
Simulation has been done with the conventional observer
and the proposed observer in condition of load and speed
variations. Fig. 3 and Fig. 4 show the simulation result using
the conventional observer and the proposed observer,
respectively. The simulation results consist of stator current,
rotor speed, electromagnetic torque and DC bus voltage. From
the results in Fig. 3 and Fig. 4, we can conclude that: The
implementation of pure integrator to estimate the stator flux is
difficult because of initial value problem. The dynamic
performance is lower and torque oscillation is bigger than the
proposed observer.
Fig. 5 and Fig. 6 show the experimental results using the
conventional observer and the proposed observer, respectively.
In experimental results, the upper pictures are phase voltages
and the lower pictures are phase current. From the
experimental results, the phase current in conventional
observer has a larger ripple than the one in proposed observer.
Tl
Te
0
-100
0
0.5
1
1.5
2
2.5
2
2.5
(d) DC voltage
240
220
200
180
0
0.5
1
1.5
Time (s)
Fig. 3 Simulation results using conventional observer
TABLE I
PARAMETER'S VALUES OF INDUCTION MOTOR
Symbol
Value
Unit
2
-
J
Rs
Description
Number of pole
pairs
Inertial moment
Stator resistance
0.007
0.01
Kgm2
W
Rr
Rotor resistance
0.01
W
Lr
Rotor inductance
3.1
H
Ls
Stator inductance
3.1
H
Lm
Mutual inductance
3
H
p
Fig. 4 Simulation results using proposed observer
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Internat. J. of Sci. and Eng., Vol. 4(2)2013:39-43, April 2013, Phuc Thinh Doan et al.
ISSN: 2086-5023
V. CONCLUSIONS
This paper developed a sliding-mode observer for
sensorless vector control of AC induction motor. The
modeling was also presented. Hardware for the simulation and
experiment is described. The proposed observer was
compared with the conventional observer. The simulation and
experimental results showed that the proposed observer has
smaller torque ripple than the conventional observer. The
proposed observer was robust in high speed and low speed
operation and load variation.
ACKNOWLEDGEMENT
This research was supported by a grant from Construction
Technology Innovation Program (CTIP) funded by Ministry
of Land, Transportation and Maritime Affairs (MLTM) of
Korean government.
REFERENCES
Fig. 5 Phase voltage and phase current using conventional observer in
experiment
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Fig. 6 Phase voltage and phase current using sliding-mode observer in
experiment
43