Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3916231, 12 pages

http://dx.doi.org/10.1155/2016/3916231

## Sensorless Control of Permanent Magnet Synchronous Motors and EKF Parameter Tuning Research

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

Received 7 September 2015; Revised 22 January 2016; Accepted 31 January 2016

Academic Editor: Kalyana C. Veluvolu

Copyright © 2016 Yong Zhang and Xu-Feng Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper concerns the parameter tuning and the estimated results postprocessing of the extended Kalman filter for the sensorless control application of permanent magnet synchronous motors. At first an extended Kalman filter parameter tuning method is proposed based on the theoretical and simulation analysis of extended Kalman filter parameters. Furthermore, a sensorless control system is proposed based on the parameter tuning method and the simulation analysis of extended Kalman filter estimation results in different reference speeds and different load torques. The proposed sensorless control system consists of two parts. The first one is a module to self-regulate extended Kalman filter parameters. The second part can correct the estimated speed and the estimated rotation angle based on the reference speed and the electromagnetic torque. Finally, simulation results are presented to verify the feasibility and validity of the proposed sensorless control system.

#### 1. Introduction

The sensorless control of the PMSM (permanent magnet synchronous motor) is the technology which, respectively, uses the estimated speed and the estimated rotation angle to replace the actual speed and the actual rotation angle. Nowadays, various sensorless control methods have been proposed. These methods can be classified as follows [1, 2]: direct methods based on the motor model, model reference adaptive methods, observer based methods, signal injection methods, and intelligent control methods. Direct methods utilize the electromagnetic model of the PMSM to estimate the speed and the rotation angle directly, and the accuracy of estimated results depends on PMSM parameters [3]. Model reference adaptive methods utilize adaptive parameters identification theory to estimate the speed and the rotation angle. These methods use a closed-loop control to eliminate the dependence on PMSM parameters [4]. Observer based methods include the EKF (extended Kalman filter), the SMO (Sliding Mode Observer), and other observers [5–8]. Injection methods estimate the speed and the rotation angle by injecting high or low frequency signal to the salient PMSM [9, 10]. Intelligent control methods can be used independently to estimate the speed and the rotation angle or combined with other methods to improve the accuracy of estimated results [11, 12]. Signal injection methods can be used in low-speed situations; other methods are used in medium-speed and high-speed situations. The EKF is one of observers, which estimate the speed and the rotation angle through the input voltage and the output current of the PMSM.

The EKF parameter tuning is an important but difficult work. EKF parameters have great influence on the stability and reliability of the EKF estimated results. In many papers the EKF parameter tuning is also discussed and studied. The simple EKF parameters meaning and trend without further study have been presented in [13]. The relatively detailed studies of the EKF parameter tuning and a parameter tuning method have been given in [14]. The other EKF parameter tuning study has been presented in various papers, which concern the EKF sensorless control of the PMSM [15–18]. However, comprehensive and detailed study is lacking. Thus, it is required to do detailed and comprehensive study in the EKF parameter tuning.

Due to estimating errors it is not appropriate to use the EKF estimated results directly in sensorless control of the PMSM. And in different load torques and different reference speeds, the accuracy of EKF estimated results is different. However, few papers studied the estimated results postprocessing of the EKF and the estimated results in different load torques and different reference speeds. Thus, it is required to design a method to correct the EKF estimated results and self-regulate EKF parameters by electromagnetic torque and reference speed.

The remainder of this paper is organized as follows. The EKF sensorless control theory and the parameters theoretical analysis are presented in Section 2. In Section 3, simulation analysis of EKF estimated results in different EKF parameters, different load torques, and different reference speeds is given. The EKF parameter tuning method and the EKF sensorless control system are presented in Section 4. Simulation results are shown in Section 5 for two speed signals. Finally, the conclusion is given in Section 6.

#### 2. Theoretical Analysis of the EKF

##### 2.1. The EKF Model of the PMSM

The EKF is the Kalman filter to be applied to the continuous nonlinear system. The EKF model of the continuous nonlinear system can be obtained by linearization and discretization. The PMSM is a continuous nonlinear system. The mathematical model of the non-salient-pole PMSM in the coordinate is presented as follows. The stator voltage equation iswhere is the stator voltage, is the stator current, is the stator resistance, and is the flux linkage.

The stator flux linkage equation is where is the PMSM winding inductance, is the rotation angle, and is the rotor permanent magnet flux.

Based on (1) and (2), and the hypothesis that the speed is constant in the very short sampling time, the PMSM equation in coordinate is

State variables, control variables, and output variables are , , and , respectively. Nonlinear state equation of the PMSM is where is the input matrix, is the state noise, and is the measurement noise.

Based on the comparison between (3) and (4), the coefficient equations in (4) can be obtained:

Equation (6) is the linear equation of (4) by Taylor series method. Equation (7) is the discrete equation of (6) by sampling cycle :

The Jacobi matrix , in (6) is

Because of , the PMSM EKF equations (10) and (11) can be obtained based on Kalman filter equations:(1)Prediction is (2)Correction is where is the state covariance matrix, is the state noise covariance matrix, is the Kalman gain, and is the measurement noise covariance matrix.

The EKF model of the PMSM is built based on (10) and (11). Appropriate EKF parameters are required to obtain accurate and stable estimated results.

##### 2.2. EKF Parameters Analysis

Kalman filter parameters are the noise covariance matrix and , the PMSM initial state matrix , and the initial state covariance matrix .

represents the PMSM initial state based on (9). contains the two-phase current information, the speed information, and the rotation angle information. can be selected based on the PMSM initial state.

is measurement noise covariance of the stator current in the coordinate. It is a 2 × 2 matrix. The current noise is uncorrelated, so is actually a diagonal matrix composed of measurement noise variance of two-phase current in the coordinate. This diagonal matrix is related to the current measurement noise. can be selected by measuring the noise in practical application.

is the covariance matrix of the PMSM state . is the initial value of . Because the PMSM state is uncorrelated, is a diagonal matrix. In the matrix, and are the current covariance in the coordinate. is the covariance of the speed. is the covariance of the rotation angle. will be automatically adjusted to the appropriate value in the recursive filtering process of the Kalman filter. The selection of does not affect the steady-state output of the Kalman filter. However, the EKF equations of the PMSM are approximate equations; the inappropriate selection of may lead to the change of estimated results in both initial state and steady-state. The selection of and is based on current changes in the coordinate. They usually choose the moderate value. Speed will rapidly increase after the PMSM starts to run, so should take the large value. The change of the rotation angle is from slow to fast when the PMSM starts up, so should take the small value.

The selection of the state noise covariance matrix is the most important work in the EKF design. The state noise is assumed to be Gaussian white noise, which is independent and zero-mean; therefore, is a diagonal matrix. and are the current noise covariance in the coordinate. is the speed noise covariance. is the noise covariance of the rotation angle. The PMSM status cannot be obtained directly, so is also not directly obtained. There are many factors affecting the PMSM state noise generation. For instance, the simplification and approximation error of the PMSM equations can be considered to be noise. Therefore, cannot be selected by existing methods. Generally the optimum needs to be obtained by trial and error.

#### 3. Simulation Analysis of the EKF

In this section, the simulation analysis of the EKF will be presented. A preset model of the PMSM in MATLAB/Simulink is used to build the simulation model of the sensorless control system. The sensorless control system is based on the field-oriented control; the PMSM parameters are shown in Table 1.