IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 4, JULY/AUGUST 2005
1039
An Adaptive Sliding-Mode Observer for Induction
Motor Sensorless Speed Control
Jingchuan Li, Longya Xu, Fellow, IEEE, and Zheng Zhang, Senior Member, IEEE
Abstract—An adaptive sliding-mode flux observer is proposed
for sensorless speed control of induction motors in this paper.
Two sliding-mode current observers are used in the method to
make flux and speed estimation robust to parameter variations.
The adaptive speed estimation is derived from the stability theory
based on the current and flux observers. The system is first simulated by MATLAB and tested by hardware-in-the-loop. Then,
it is implemented based on a TMS320F2812, a 32-bit fixed-point
digital signal processor. Simulation and experimental results are
presented to verify the principles and demonstrate the practicality
of the approach.
Index Terms—Adaptive sliding-mode observer, induction motor,
sensorless speed control.
I. INTRODUCTION
T
HE approach of sensorless speed control of induction motors can reduce cost, avoid fragility of a mechanical speed
sensor, and eliminate the difficulty of installing the sensor in
many applications. Thus, the approach has been receiving more
and more attention. However, due to the high-order multiple
variables and nonlinearity of induction motor dynamics, estimation of the rotor flux and speed is still very challenging. Over the
years, many research efforts have been made and various sensorless speed control algorithms have been proposed in the literature. The model reference adaptive system (MRAS) methods
[1], [2] are based on the comparison between the outputs of
two estimates. The output errors are then used to drive a suitable adaptation mechanism that generates the estimated speed.
These schemes require integration and the system performance
is limited by parameter variations. The Kalman filter approaches
[3] are known to be able to get accurate speed information, but
have some inherent disadvantages, such as the influence of noise
and large computational burden. Adaptive observer-based approaches [4], [5] have improved performance using an adaptive mechanism with relatively simple computation. Slidingmode observers [6]–[9], due to their order reduction, disturbance rejection, and simple implementation, are recognized as
the promising control methodology for electric motors. Other
algorithms for sensorless speed control, such as artificial neural
Paper IPCSD-05-029, presented at the 2004 Industry Applications Society
Annual Meeting, Seattle, WA, October 3–7, and approved for publication in
the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial Drives
Committee of the IEEE Industry Applications Society. Manuscript submitted
for review September 25, 2004 and released for publication April 25, 2005.
J. Li and L. Xu are with the Department of Electrical and Computer Engineering. The Ohio State University, Columbus, OH 43210-1272 USA (e-mail:
li.407@osu.edu; xu.12@osu.edu).
Z. Zhang is with the R&E Center, Whirpool Corporation, Benton Harbor, MI
49022 USA (e-mail: Zheng_Zhang@whirlpool.com).
Digital Object Identifier 10.1109/TIA.2005.851585
network [10] and artificial intelligence (AI) methods [11], can
achieve high performance, but they are relatively complicated
and require large computational time.
In the sensorless speed control of induction motors with direct field orientation, the rotor flux and speed information are
dependent on the observers. However, the exact values of the
parameters that construct the observers are difficult to measure
and changeable with respect to the operating conditions. When
the motor parameters are changed and, thus, are different from
the preset values, the estimated flux and speed will deviate from
the real values. To make flux and speed estimation robust to parameter variations, a novel adaptive sliding-mode flux and speed
observer is proposed in the paper. Two sliding-mode current observers are used in the proposed method. The effects of parameter deviations on the rotor flux observer can be alleviated by
the interaction of these two current sliding-mode observers. The
stability of the method is proven by Lyapunov theory. An adaptive speed estimation is also derived from the stability theory.
II. SLIDING-MODE CURRENT AND FLUX OBSERVER DESIGN
Induction motors can be modeled in various reference frames.
Defining stator currents and rotor fluxes as the state variables,
we can express the induction motor model in the stationary
frame as
(1)
(2)
where
rotor fluxes,
are stator currents,
are stator voltages,
are
are stator and rotor resistances,
are total stator
is magnetizing inductance, is the
and rotor inductances,
,
is rotor time conleakage coefficient,
, and
is motor angular velocity.
stant,
0093-9994/$20.00 © 2005 IEEE
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Fig. 1.
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 4, JULY/AUGUST 2005
Configuration of the flux and speed observer.
The configuration of the proposed flux and speed estimators is
shown in Fig. 1. The adaptive sliding-mode observer consists of
two sliding-mode current observers and one rotor flux observer.
The rotor flux observer is based on the current estimation from
the two current observers. The rotor speed observer takes the
outputs from the second current observer and the rotor flux observer as its inputs and generates the estimated rotor speed as
the output. The estimated speed is then fed back to the second
current observer for its adaptation. The estimation of the motor
speed is derived from a Lyapunov function, which guarantees
the system convergence and stability. Once the sliding functions of the current observers reach the sliding surfaces, the rotor
flux will converge to the real value asymptotically. Each sub-observer of the overall adaptive sliding-mode observer is discussed
in the following sections.
Equation (5) indicates that the equivalent control equals the
rotor flux multiplied by the matrix, which is the common part
in (2). The rotor flux can be obtained by integrating this equivalent control without speed information as discussed in [6]. The
,
flux estimation is accurate when the motor parameters
and are known. However, if the parameters in the observers
are different from the real values, there will be some errors
in the coefficients of the observers. Then, the estimated flux and speed will be incorrect. In order to compensate
for this divergence, a second sliding-mode current observer is
used for the flux estimation.
B. Current Observer II
The second sliding-mode current observer is designed differently from (3) as
A. Current Observer I
(6)
The first sliding-mode current observer is defined as [6]
(3)
where
are the first observer currents,
are the first sliding functions,
are the second observer currents,
are the observed rotor fluxes,
the second sliding function,
where
is
The sliding-mode surface is
The sliding-mode surface is defined as
According to the above formulas, the current error equation
is
By subtracting (1) from (6), the error equation becomes
(7)
(4)
where
.
large enough, the sliding mode will occur
By selecting
, and then it follows that
.
where
From an equivalent control point of view, we have
(8)
From the equivalent control concept [8], if the current trajectories reach the sliding manifold, we have
(5)
where
. The second equivalent control equals to the
negative multiplication of the estimated rotor flux error and the
matrix. It is noticed that the second current observer needs
the rotor speed as the input.
LI et al.: ADAPTIVE SLIDING-MODE OBSERVER FOR INDUCTION MOTOR SENSORLESS SPEED CONTROL
C. Rotor Flux Observer Design
Combining the results from (5) and (8), the rotor flux observer
can be constructed as
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be an arbitrary positive constant. With this
Let
assumption, the above equation becomes
(9)
where is the observer gain matrix to be decided such that the
observer is asymptotically stable.
From (3) and (6), the equivalent controls obtained individually by the two current observers will deviate from their real
values if the motor parameters are incorrect. Consequently, the
rotor flux estimation based on each individual control will also
be inaccurate. To reduce this deviation on rotor flux estimation,
the rotor flux observer is designed from the combination of two
equivalent controls, where the effects of parameter variations
are largely cancelled. From (5) and (8), the error equation for
the rotor flux is
(14)
Letting the second term be equal to the third term in (14), we
can find the following adaptive scheme for rotor speed identification:
(15)
where
In practice, the speed can be found by the following proportional and integral adaptive scheme:
(10)
(16)
where
III. ADAPTIVE SPEED ESTIMATION
In order to derive the adaptive scheme, Lyapunov’s stability
theorem is utilized. If we consider the rotor speed as a variable
parameter, the error equation of flux observer is described by
the following equation:
and
are the positive gains.
IV. STABLITY ANALYSIS
Since the second term is equal to the third term in (14), the
time derivative of becomes
(11)
where
(17)
It is apparent that (17) is negative definite. From Lyapunov
stability theory, the flux observer is asymptotically stable, guaranteeing the observed flux to converge to the real rotor flux.
is the estimated rotor speed
The candidate Lyapunov function is defined as
V. SIMULATION RESULTS
(12)
where is a positive constant. We know that
nite. The time derivative of becomes
is positive defi-
To evaluate the proposed algorithm for the rotor flux and
speed estimation, computer simulations have been conducted
using MATLAB. The block diagram of the control system is
shown in Fig. 2. To further investigate the implemental feasibility, the estimation and control algorithm are evaluated by
hardware-in-the-loop (HIL) testing. A 1-hp induction motor was
used in the simulation and also in the experiments. The motor
parameters are as follows: 1 hp, four poles, 220 V, 5 A,
,
,
mH, and
mH.
A. Simulation Results by MATLAB
(13)
Figs. 3 and 4 show the induction motor response to a step
speed command of 0.5 pu ( 900 r/min) where the motor parameters are exactly known. The actual machine model is used
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Fig. 2.
IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 4, JULY/AUGUST 2005
System configuration for simulation and implementation.
Fig. 3. Real and estimated speed at a step speed command.
Fig. 4. Real and estimated rotor flux.
to calculate the current, flux, and speed of the motor. The observer model as described above is used to estimate the rotor
flux and speed. Fig. 3 shows the speed command, real speed,
estimated speed, and the speed estimation error. Fig. 4 shows
the real and estimated rotor flux and the flux estimation error.
It can be seen that the estimated speed and flux converge to the
real values very quickly.
To study the effects of parameter variation on the speed and
flux observers, the parameters in the observers are changed
on purpose in the simulation. Fig. 5 shows the simulation
in the observers is changed by
results when the coefficient
and speed
20% from its actual value, where the flux
are estimated by the proposed method, and
and
by the previous method using only one current
LI et al.: ADAPTIVE SLIDING-MODE OBSERVER FOR INDUCTION MOTOR SENSORLESS SPEED CONTROL
Fig. 5. Coefficient k in the observer is increased by 20%. (a) : real rotor
: estimated by the proposed method; : estimated by
flux; previous method. (b) ! : real rotor speed; !
: estimated by the proposed
: estimated by previous method.
method; !
sliding-mode observer as in [6]. It is noticed that even if
is incorrect, the estimated rotor flux and speed by the new
observer still converge to the real values, but in the previous
model, there is an offset in the rotor flux estimation and
fluctuation in the rotor speed estimation. The dc offset of flux
estimation by the previous method is caused by the incorrect
. If
changes, the equivalent control
equivalent control
will detune. The integration of incorrect
causes
dc offset on the flux estimation, whereas in the new flux
observer, this dc offset is cancelled by using two current
variation on the flux
observers. The effects of coefficient
and speed estimation are shown in Fig. 6. We can also observe
obvious fluctuations in speed estimation. There is still an
error on the rotor flux estimation by the proposed method
as shown in Fig. 6(a), but the new method eliminates the
dc offset caused by the parameter variation, which can be
observed in results simulated by the previous model.
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Fig. 6. Coefficient in the observer is increased by 20%. (a) : real rotor
: estimated by the proposed method; : estimated by
flux; previous method. (b) ! : real rotor flux; !
: estimated by the proposed
: estimated by previous method.
method; !
B. HIL Evaluation Results by TI 2812 DSP
HIL evaluation is to use a computer model of the process as
the real target hardware, and on the other hand, the control and
estimation algorithm are implemented in real time. The purpose
of HIL is to make evaluation of the proposed algorithm as close
as possible to that which would be encountered in the real-time
implementation. In this paper, the evaluation is performed using
a TMS320F2812 DSP. The dynamics of the electric machine
are modeled by five differential equations. The control and estimated algorithms are implemented in 32-bit Q-math approach,
interacting with the motor model rather than the real targeted
physical system.
The main advantages of this evaluation are: 1) the control
software is implemented and evaluated in real time and can be
debugged very easily in the absence of motor and 2) the control
software can be easily transferred to the real drive system with
only minor changes.
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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 4, JULY/AUGUST 2005
0
Fig. 7. Speed step response from 0.5 pu to 0.5 pu (curve 1: speed command
! , 0.606 pu/div; curve 2: real speed ! , 0.606 pu/div; curve 3: estimated speed
!
~ , 0.606 pu/div).
6
Fig. 9. Trapezoidal speed at 0.5 pu (curve 1: phase current i , 0.6 pu/div;
curve 2: torque current i , 0.3 pu/div; curve 3: estimated speed !
~ , 1.212
pu/div).
Fig. 10.
Fig. 8. Rotor flux estimation (curve 1: real flux ~ ; curve 3: estimated flux angle ~ ).
; curve 2: estimated flux
The results evaluated by HIL are shown in Figs. 7–9. Fig. 7
shows the motor step response to a speed command at 0.5 pu
( 900 r/min). Fig. 8 shows the real and estimated rotor flux and
the estimated flux angle. Fig. 9 shows the motor response to a
trapezoidal speed command. The results show that the method
can be successfully implemented by the fixed-point DSP.
VI. EXPERIMETAL RESULTS
In order to evaluate the performance of the proposed algorithm experimentally, an induction motor drive system was set
up. The setup consists of a 1-hp induction motor, a power drive
board, and a DSP controller board. The external load is imposed
by a hysteresis dynamometer. The experimental setup is shown
in Fig. 10.
Experimental setup.
The control algorithm is implemented by a Texas Instruments
TMS320F2812 32-bit fixed-point DSP. It has the following
features:
• high-performance static CMOS technology, 150 MHz
(6.67-ns cycle time);
• high-performance 32-bit CPU;
• Flash devices: up to 128 K 16 Flash;
• 12-bit ADC, 16 channels.
The test was first performed on the motor in four-quadrant operation. Fig. 11 shows the motor response to a commanded step
change speed at 900 r/min. Fig. 12 shows the measured current and two sliding-mode observer currents. It is seen that the
sliding-mode functions enforce the two observed currents to the
measured ones very closely. Once these two observer currents
converge to the measured ones, the estimated rotor flux converges to the real rotor flux. The motor responses to a trapezoidal
speed command when the motor runs at no load are shown in
Fig. 13. To further investigate the motor transient performance
pu is applied
at load conditions, an external torque
when the motor runs at the same trapezoidal speed command
as in Fig. 13. The waveform of speed command , estimated
speed , torque current , and phase current are shown in
Fig. 14. The estimated rotor speed response to a step change of
command from 360 to 1260 r/min with a load torque of
LI et al.: ADAPTIVE SLIDING-MODE OBSERVER FOR INDUCTION MOTOR SENSORLESS SPEED CONTROL
6
Fig. 11. Transient response to speed step command 900 r/min at no load
(curve1: speed command ! , 1091 r/min/div; curve 2: estimated speed !
~ , 1091
r/min/div; curve 3: torque current i , 1.5 A/div; curve 4: phase current i , 5
A/div).
Fig. 12. Real and estimated currents (curve 1: measured current i , 3 A/div;
curve 2: observed current ^i , 3 A/div; curve 3: observed current ~i , 3 A/div).
pu is shown in Fig. 15. To investigate the speed robustness, a step
pu is applied and then removed at
disturbance torque
r/min. Fig. 16 shows the estimated rotor
motor speed
speed response and the torque current response. As evidenced
by the testing results, the induction motor drive functions very
well by the proposed algorithm.
VII. CONCLUSION
A novel adaptive sliding-mode observer for sensorless speed
control of an induction motor has been presented in this paper.
The proposed algorithm consists of two current observers and
one rotor flux observer. The two sliding-mode current observers
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6
Fig. 13. Transient response due to trapezoidal speed command ( 900 r/min)
at no load (curve 1: speed command ! , 2182 r/min/div; curve 2: estimated
speed !
~ , 2182 r/min/div; curve 3: torque current i , 1.5 A/div; curve 4: phase
current i , 5 A/div).
6
Fig. 14. Transient response due to trapezoidal speed command ( 900 r/min)
at T = 0:5 pu (curve 1: speed command ! , 1091 r/min/div; curve 2: estimated
speed !
~ , 1091 r/min/div; curve 3: torque current i , 1.5 A/div; curve 4: phase
current i , 5 A/div).
are utilized to compensate for the effects of parameter variations on the rotor flux estimation. When the motor parameters
are deviated from initial values by temperature or operation conditions, the errors of two equivalent controls from current observers will be largely cancelled, which make the flux estimation more accurate and insensitive to parameter variations. Although an additional sliding-mode current observer is used, the
complexity of the method is not increased too much. The stability and convergence of the estimated flux to real rotor flux
are proved by the Lyapunov stability theory. Digital simulation
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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 41, NO. 4, JULY/AUGUST 2005
[5] G. Yang and T. H. Chin, “Adaptive-speed identification scheme for
a vector-controlled speed sensorless inverter-induction motor drive,”
IEEE Trans. Ind. Appl., vol. 29, no. 4, pp. 820–825, Jul./Aug. 1993.
[6] A. Derdiyok, M. K. Guven, H. Rehman, N. Inanc, and L. Xu, “Design
and implementation of a new sliding-mode observer for speed-sensorless control of induction machine,” IEEE Trans. Ind. Electron., vol. 49,
no. 5, pp. 1177–1182, Oct. 2002.
[7] H. Rehman, A. Derdiyok, M. K. Guven, and L. Xu, “A new current
model flux observer for wide speed reange sensorless control of an
induction machine,” IEEE Trans. Ind. Electron., vol. 49, no. 6, pp.
1041–1048, Dec. 2002.
[8] V. I. Utkin, “Sliding mode control design principles and applications to
electric drives,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 23–36,
Feb. 1993.
[9] C. Lascu, I. Boldea, and F. Blaabjerg, “Direct torque control of sensorless induction motor drives: A sliding-mode approach,” IEEE Trans. Ind.
Appl., vol. 40, no. 2, pp. 582–590, Mar./Apr. 2004.
[10] P. Mehrotra, J. E. Quaico, and R. Venkatesan, “Speed estimation
of induction motor using artificial neural networks,” in Proc. IEEE
IECON’96, 1996, pp. 881–886.
[11] D. Schroder, C. Schaffner, and U. Lenz, “Neural-net based observers for
sensorless drives,” in Proc. IEEE IECON’94, 1994, pp. 1599–1610.
Fig. 15. Speed response due to step change command from 360 to 1260 r/min
at T = 0:5 pu (curve1: real speed, 1091 r/min/div; curve 2: estimated speed
!~ , 1091 r/min/div; curve 3: torque current i , 1.5 A/div; curve 4: phase current
i , 5 A/div).
Fig. 16. Transient response for step disturbance torque (curve1: real speed ! ,
1091 r/min/div; curve 2: estimated speed !
~ , 1091 r/min/div; curve 3: torque
current i , 1.5 A/div).
and experiments have been performed. The effectiveness of the
approach is demonstrated by the results.
REFERENCES
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30, no. 5, pp. 1234–1240, Sep./Oct. 1994.
[3] Y. R. Kim, S. K. Sul, and M.-H. Park, “Speed sensorless vector control
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Jingchuan Li received the B.S. degree from Xi’an
Jiaotong University, Xi’an, China, in 1993. He is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, The
Ohio State University, Columbus.
His research interests include power electronics,
design and control of electrical machines, and finiteelement analysis of electromagnetic devices.
Longya Xu (S’89–M’90–SM’93–F’04) received the
M.S. and Ph.D. degrees in electrical engineering from
the University of Wisconsin, Madison, in 1986 and
1990, respectively.
In 1990, he joined the Department of Electrical and
Computer Engineering, The Ohio State University,
Columbus, where he is presently a Professor. He has
served as a Consultant to several industrial companies, including Raytheon Company, US Wind Power
Company, General Motors, Ford, and Unique Mobility Inc. His research and teaching interests include
dynamic modeling and optimized design of electrical machines and power converters for variable-speed generating and drive systems, application of advanced
control theory, and digital signal processors for control of motion and distributed
power systems in super-high-speed operation.
Dr. Xu received the 1990 First Prize Paper Award from the Industrial Drives
Committee of the IEEE Industry Applications Society (IAS). In 1991, he received a Research Initiation Award from the National Science Foundation. He
was also a recipient of 1995, 1999, and 2004 Lumley Research Awards from the
College of Engineering, The Ohio State University, for his research accomplishments. He has served as Chairman of the IAS Electric Machines Committee and
as an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS.
Zheng Zhang (M’98–SM’99) received the B.S.E.E.
degree from Hefei University of Technology, Hefei,
China, in 1982, the M.S.E.E. degree from Chongqing
University, Chongqing, China, in 1985, and the Ph.D.
degree from the Politecnico di Torino, Turin, Italy in
1997, all in electrical engineering.
From 1985 to 1992, he was with the University
of Shandong, Jinan, China, where he taught several
courses and conducted research. From 1993 to 1995,
he was with the Electronics Division of the FIAT Research Center, Turin, Italy, as an Intern Researcher.
While there, he was involved in the development of motor and drive systems for
electric vehicle applications. He is currently with the R&E Center, Whirlpool
Corporation, Benton Harbor, MI. His research interests include special motors,
advanced motor controls for low-cost electronic devices, and diagnostics of appliances by use of motor drive information. He has authored more than 30 published technical papers.