Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 609586, 17 pages

http://dx.doi.org/10.1155/2015/609586

## Direct Torque Control of Sensorless Induction Machine Drives: A Two-Stage Kalman Filter Approach

^{1}School of Electric Power, South China University of Technology, Guangzhou, Guangdong 510640, China^{2}Sunwoda Electronic Corporation Limited, Shenzhen 518108, China

Received 27 May 2015; Revised 27 August 2015; Accepted 27 August 2015

Academic Editor: Mohamed Djemai

Copyright © 2015 Jinliang Zhang et al. 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

Extended Kalman filter (EKF) has been widely applied for sensorless direct torque control (DTC) in induction machines (IMs). One key problem associated with EKF is that the estimator suffers from computational burden and numerical problems resulting from high order mathematical models. To reduce the computational cost, a two-stage extended Kalman filter (TEKF) based solution is presented for closed-loop stator flux, speed, and torque estimation of IM to achieve sensorless DTC-SVM operations in this paper. The novel observer can be similarly derived as the optimal two-stage Kalman filter (TKF) which has been proposed by several researchers. Compared to a straightforward implementation of a conventional EKF, the TEKF estimator can reduce the number of arithmetic operations. Simulation and experimental results verify the performance of the proposed TEKF estimator for DTC of IMs.

#### 1. Introduction

High performance control and estimation techniques for induction machines (IMs) have been finding more and more applications with Blaschke’s well-known field oriented control (FOC) method [1]. To improve the dynamic response of instantaneous electromagnetic torque and simplicity in control structure, one such technique for induction machine control is that the direct torque control (DTC) method can provide accurate fast torque control [2]. This method has become increasingly popular for industrial applications due to the simplified control strategy and lower parameter dependence, in comparison with the FOC methods [3, 4].

For DTC of IMs, the method requires information on the position and amplitude of the controlled stator flux for speed control applications. In the conventional approach, the stator flux is obtained utilizing a search coil or through Hall effect sensors, whilst speed sensors like incremental encoders or resolvers are used to monitor rotor velocity [2]. These unnecessarily increase hardware costs and the size of the control systems and degrade the reliability of the systems when encountering defective environments. So, sensorless DTC strategy has become the hot issue in research and drawn many researchers and engineers’ attention.

Conventional approaches to sensorless DTC of IMs employ the method of stator flux and rotor velocity estimation by using a stator voltage model [5, 6]. This method has a large error in rotor velocity estimation, particularly in the low-speed operation range. Some recent studies conducting simultaneous stator flux and rotor velocity estimation for sensorless DTC technology include model reference adaptive system (MRAS) [7], artificial neural networks (ANN) [8], sliding mode control (SMC) [9], extended Luenberger observer [10], and extended Kalman filter (EKF) [2, 11]. The model uncertainties and nonlinearities inherent to induction motors are well suited to the EKF’s stochastic nature [2]. Using this method, it is possible to make estimation of states whilst simultaneously performing identification of parameters in a short time [12–14], even taking measurement and system noises directly into system model. This explains why the EKF estimator is widely applicable in the sensorless DTC of IMs. However, the EKF may suffer numerical problems and computational burden due to the high order of the mathematical models. This has generally limited the applicability of the EKF to real-time signal processing problems.

In order to reduce the conventional EKF computational algorithm complexity, the main objective of this paper is to present a two-stage extended Kalman filter (TEKF) for stator flux, rotor speed, and electromagnetic torque estimation of a sensorless direct torque controlled IM drive. The proposed estimator is an effective implementation of EKF. Following the two-stage filtering technique as given in [15], the TEKF can be decomposed into two filters such as the modified bias free filter and the bias filter. Compared to the conventional EKF, the main advantage of the TEKF is the ability to reduce the computational complexity, whilst maintaining the same level of performance.

The paper is organized as follows. In Section 2, the sensorless DTC-SVM strategy of IMs is introduced briefly. In Section 3, according to the discrete model of IM, a conventional EKF algorithm for estimating stator flux, rotor speed, and position is designed. In Section 4, TEKF are developed by the two-stage filtering approach, and its stability is analyzed. In Section 5, simulation and experimental results are discussed. Finally, a conclusion wraps up the paper.

#### 2. Principle of Sensorless DTC-SVM

As elaborated in [12], a dynamic mathematical model for an IM in the stationary reference frame is obtained as follows:where , , , , , and are the stator currents, flux linkages, and voltages in the stationary reference frame. and are the stator winding resistance and inductance, respectively, is the leakage or coupling factor (where ), and are the mutual inductance and rotor inductance, is the rotor time constant (where ), and is the rotor resistance. The rotor angular velocity is measured in mechanical radians per second, is the mechanical rotor position, and is the number of pole pairs.

The behavior of an IM in DTC technique can be described in terms of space vectors by the following equations written in the stator stationary reference frame:where is known as load angle which is the angle between rotor flux and stator flux . and are amplitudes of and , respectively. From (2), it can be seen that the instantaneous electromagnetic torque control of IMs in DTC is determined by changing the values of load angle while and maintain the constant amplitude. Accelerating the stator flux, with respect to the rotor flux vector, will increase the electromagnetic torque, and decelerating the same vector will decrease the electromagnetic torque [16].

The basic idea of DTC technique of IM is to control and acquire accurate knowledge on the stator flux and electromagnetic torque to achieve high dynamic performance. DTC technique involves stator flux, electromagnetic torque estimators, hysteresis controllers, and a simple switching logic (switching tables) in order to reduce the electromagnetic torque and stator flux errors rapidly [17, 18]. Due to the fact that the universal voltage inverter has only eight available basic space vectors and only one voltage space vector is maintained for the whole duration of the control period, the conventional approach causes high ripples in stator flux, current, and electromagnetic torque, accompanied by acoustical noise. To reduce the ripples of the stator flux linkage current and electromagnetic torque in IM drives, a modified DTC using Space Vector Modulation (SVM) method called DTC-SVM is proposed in this paper. The main difference between conventional DTC and DTC-SVM is that DTC-SVM has a SVM model and two PI controllers instead of switching table and hysteresis controllers [19, 20]. The system structure of DTC-SVM can be built and shown in Figure 1. This system operates at constant stator flux (below rated speed). From Figure 1, the reference torque is generated from regulated speed proportional integral (PI); is the torque error between the reference torque and estimated torque . In order to compensate this error, the angle of stator flux vector must be increased from to as shown in Figure 2, where is the phase angle of stator flux vector that can be obtained by the flux estimator and is the increment of stator flux in the next sampling time. Therefore, the required reference stator flux in polar form is given by .