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A High-Speed Sliding-Mode Observer
for the Sensorless Speed Control of a
PMSM
Hongryel Kim, Jubum Son, and Jangmyung Lee, Senior Member, IEEEIEEE
TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 9,
SEPTEMBER 2011 4069-4077
教授:王明賢
學生:王沼奇
目錄
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INTRODUCTION
CONVENTIONAL SMO
 A. PMSM Modeling
 B. Conventional SMO
HIGH-SPEED SMO
 A. Sigmoid Function
 B. Stability Analysis
 C. Position and Velocity Estimation of the Rotor
IV. EXPERIMENTAL RESULTS
 A. Hardware Organization
 B. Simulation and Experimental Results
REFERENCES
INTRODUCTION
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There are two types of ac motors: the induction motor (IM) and the
permanent-magnet (PM) synchronous motor (PMSM).
The PMSM is very popular in ac motor applications since it is useful
for various speed controls. The IM has a simple structure, and it is
easy to build.
However, it is not as efficient as the PMSM considered in terms of
dynamic performance and power density.
INTRODUCTION
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Since a PMSM receives a sinusoidal magnetic flux from the PM of the
rotor, precise position data are necessary for an efficient vector control.
However, these sensors are expensive and very sensitive to
environmental constraints such as vibration and temperature.
To overcome these problems, instead of using position sensors, a
sensorless control method has been developed for control of the motor
using the estimated values of the position and velocity of the rotor.
INTRODUCTION
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This paper proposes a new sensorless control algorithm
for a PMSM based on the new SMO which uses a
sigmoid function as a switching function with variable
boundary layers.
Using this SMO, the position and velocity of the rotor can
be calculated from the estimated back EMF [25]. Also, to
overcome the sensitivity of the parameter variations in the
sensorless control and to improve the steady-state
performance, the stator resistance is estimated using an
adaptive control scheme. The superiority of the proposed
algorithm has been proved by comparison with the
conventional SMO through real experiments.
PMSM Modeling
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The PMSM consists of a rotor with a PM and a stator with a threephase Y-connected winding, which is located at every 120◦ on the
circle. The three-phase motor is an intrinsically nonlinear timevarying system.
PMSM Modeling
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The state equations, where the
stator current is a state variable
of the fixed-frame voltage
equation, can be represented as
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The electromotive force for each
phase can be represented in the
fixed frame as
Conventional SMO
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The sliding-mode control is used to restrict the state variables on the
sliding surface by changing the system structure dynamically.
It is widely used for nonlinear system control since it is robust
against system parameter variations. For the sensorless control of
the PMSM, the sliding-mode controller is adopted for use in the
observer design and so is named the SMO. However, there are the
shortcomings of chattering and time delay for the rotor position
compensation in the conventional SMO
Conventional SMO
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Fig. 1 shows the conventional sliding-mode controller where the
signum function is used as he switching function and the low-pass
filter (LPF) is used to eliminate the chattering effects from the
switching.
HIGH-SPEED SMO
Sigmoid Function

This new SMO is composed by
the PMSM current equation in
the rest frame of (1) as follows:
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The new SMO resolves the
problems of the conventional
SMO by using a sigmoid
function as the switching
function.
The sigmoid function is
represented as
where a is a positive constant used to regulate the slope of the sigmoid function. The
estimation errors of the stator current are defined as iα =ˆiα − iα and iβ =ˆiβ − iβ.
Stability Analysis
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The sliding surface sn can be defined as functions of the
errors between the actual current, i.e., iα and iβ, and the
estimated current, i.e.,ˆiα andˆiβ, for each
phase as
follows:
wheresα = iα and sβ = iβ.
Stability Analysis
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When the sliding mode is reached, i.e., when the
estimation errors are on the sliding surface, the estimation
errors become zero, i.e., ˆiα = iα and ˆiβ = iβ.
the Lyapunov function used to find the sliding condition
can be defined as
Stability Analysis
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where (1/2)(ˆRs − Rs)2 is used to estimate the stator
resistance which is a variable parameter. From the
Lyapunov stability theorem [8], the sliding-mode
condition can be derived to satisfy the condition that V˙ <
0 for V > 0. Taking the time derivative of (6), we find
Stability Analysis
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By using the current equations of (3) in the stationary
coordinates, the derivatives of the estimated phase
currents can be represented as
where ˆ A = −ˆRs/Ls and A = −Rs/Ls.
Stability Analysis
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By substituting (8) into (7), the sliding condition can be represented
as
where sα = iα, sβ = iβ, and Rs = ˆRs − Rs.
Stability Analysis
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To satisfy the condition V˙ < 0, (9)
is decomposed into two equations
as follows:
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From the condition used to
satisfy (10), the estimation of
the stator resistance can be
obtained as
Stability Analysis
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It should be noted that this
estimation of stator resistance
is directly related to the
stability of the system.
Fig. 2 shows the SMO with the stator-resistance estimation. The stator resistance is estimated by integrating (12).
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In order to keep the SMO stable, the observer gains should
satisfy the inequality condition found in (11). This condition
in (11) is described in more detail as follows:

where the observer gain can be derived to satisfy the
inequalitycondition as
Position and Velocity Estimation of the
Rotor
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The sliding mode occurs with the
suitable selection of observer gain k;
thus, the sliding surface can be
represented as

From the aforementioned
equation, the back EMF
voltages can be expressed in the
form
Position and Velocity Estimation of the
Rotor
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Fig. 4 shows the block diagram
of the new SMO. To resolve
the chattering problems, the
sigmoid function is serially
connected to the sliding-mode
control.
Position and Velocity Estimation of the
Rotor
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Using the estimated back EMF voltages, the position and velocity of
the rotor are calculated from
Position and Velocity Estimation of the
Rotor
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The proposed observer reduces the influence of the estimation error
caused by the parameter changes in the conventional adaptive control
and calculates the position and the speed of the rotor using the new
SMO without the integral operations inherent in the LPF. As a result,
the system speed performance is improved.
Hardware Organization
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Fig. 5 shows a photograph of
the sensorless speed controller
for the 1-kW PMSM.
PM300CSD060 interior PM
(IPM) modules, made by
Mitsubishi, are utilized as the
switching devices.
Hardware Organization

Fig. 6 shows the block diagram
of the sensorless speed
controller, where the SMOPOS
represents the SMO used to
estimate the rotor position.
Hardware Organization
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Table I shows the
specifications of the CSMT-10
B (1-kW) SPMSM made by
Samsung Rockwell.
Simulation and Experimental Results

Fig. 7 compares the
step responses of two
step inputs, 500 and
2000 r/min without a
load, between the
conventional sliding
mode and the proposed
SMO.
Simulation and Experimental Results
Fig. 8(a)shows the real currents
Fig. 10(a) shows the real currents
Simulation and Experimental Results
Fig. 8(b) shows the estimated currents
Fig. 10(b) shows the estimated
currents
Simulation and Experimental Results
Fig. 8(c) shows the estimated back EMF
Fig. 10(c) shows the estimated back EMF
Simulation and Experimental Results
Fig. 8(d) shows the
actual and estimated rotor position
Fig. 10(d) shows the actual and estimated
rotor position,
Simulation and Experimental Results
Fig. 8(e) shows the estimation
error of the rotor position
Fig. 10(e) shows the estimation
error of the rotor position
REFERENCES
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