A High-Speed Sliding-Mode Observer for the Sensorless Speed Control of a

PMSMHongryel Kim, Jubum Son, and Jangmyung Lee, Senior Member, IEEEIEEE TRANS

ACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 9, SEPTEMBER 2011 4069-4077

教授 :王明賢學生 :王沼奇

目錄 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

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

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

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

The PMSM consists of a rotor with a PM and a stator with a three-phase Y-connected winding, which is located at every 120◦ on the circle. The three-phase motor is an intrinsically nonlinear time-varying system.

PMSM Modeling

The state equations, where the stator current is a state variable of the fixed-frame voltage equation, can be represented as

The electromotive force for each phase can be represented in the fixed frame as

Conventional SMO

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

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:

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

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

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

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

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

By substituting (8) into (7), the sliding condition can be represented as

where sα = iα, sβ = iβ, and Rs = ˆRs − Rs.

Stability Analysis

To satisfy the condition V˙ < 0, (9) is decomposed into two equations as follows:

From the condition used to satisfy (10), the estimation of the stator resistance can be obtained as

Stability Analysis

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).

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

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

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

Using the estimated back EMF voltages, the position and velocity of the rotor are calculated from

Position and Velocity Estimation of the Rotor

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

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

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 currentsFig. 10(b) shows the estimatedcurrents

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 theactual and estimated rotor position

Fig. 10(d) shows the actual and estimated rotor position,

Simulation and Experimental Results

Fig. 8(e) shows the estimationerror of the rotor position

Fig. 10(e) shows the estimation error of the rotor position

REFERENCES

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