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 [1214], 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 .

Define the stator flux deviations between and as ; thenwhere and are the stationary axis components of stator flux and and are the stator flux components estimation. In order to make up for stator flux deviations and , the reference stator voltages and should be applied on the IM which can be expressed by

Substituting (3) into (4), (5) can be acquired:

Based on the reference stator voltage components and , the drive signal for inverter IGBTs can be obtained through SVM module. Then, both the electromagnetic torque and the magnitude of stator flux are under control, thereby generating the reference stator voltage components.

3. Conventional EKF Theory

By choosing the system state vector and estimated parameter vector as and , respectively, as the input vector, and as the output vector, the IM model is described by the general nonlinear state space model:with

Remark 1. Matrices and are not affected by uncertainties.

Remark 2. Matrix is time-varying because it depends on the rotor speed .

For digital implementation of estimator on a microcontroller, a discrete time mathematical model of IMs is required. These equations can be obtained from (6):

The solution of nonhomogenous state equations (6) satisfying the initial condition isIntegrating from to , we can obtain thatThe above equations lead toIn the same way,

Tolerating a small discretization error, a first-order Taylor series expansion of the matrix exponential is used:with

Based on discretized IM model, a conventional EKF estimator is designed for estimation of stator flux, current, electromagnetic torque, and rotor speed of IM for sensorless DTC-SVM operations. Treating as the full order state and as the augmented system state, the state vector is chosen to be . and are chosen as input and output vectors because these quantities can be easily obtained from measurements of stator currents and voltage construction using DC link voltage and switching status. Considering the parameter errors and noise of system, the discrete time state space model of IMs in the stationary reference frame is described bywith

The system noise and measurement noise are white Gaussian sequence with zero-mean and following covariance matrices:where , , , and is the Kronecker delta. The initial states and are assumed to be uncorrelated with the zero-mean noises , , and . The initial conditions are assumed to be Gaussian random variables and that are defined as follows:

The overall structure of the EKF is well-known by employing a two-step prediction and correction algorithm [13]. Hence, the application of EKF filter to the state space model of IM (15) is described bywith

4. The Two-Stage Extended Kalman Filter

4.1. The TEKF Algorithm

As mentioned in conventional EKF estimator previously, the memory and computational costs increase with the augmented state dimension. Considering sampling time is very small, only high performance microcontroller can qualify for this work. Hence, the conventional EKF algorithm may be impractical to implement. The extra computation of terms leads to this computational complexity. Therefore we can reduce the computational complexity from application point of view if the terms can be eliminated. In this section, a two-stage extended Kalman filter without explicitly calculating terms is discussed.

Following the same approach as given in [15], the TEKF is decomposed into two filters such as the modified bias free filter and the bias filter by applying the following two-stage - transformation:where

The main advantage of using the transformation is that the inverse transformation involves only a change of sign. Two blending matrices and are defined by and , respectively. Using characteristic of , (25) become And the following relationships are obtained from (25):

Based on two-step iterative substitution method of [15], the transformed filter expressed by (27) can be recursively calculated as follows:Using (35), (37), and the block diagonal structure of , the following relations can be obtained:where and are defined asThe above equations lead toDefine the following notation:The equations of the modified bias free filter and the bias filter are acquired by the next steps.

Expanding (34), we have whereExpanding (35), we haveThen using (40), (43), and (47), (49) can be written aswhereExpanding (38) and using (41) and (44) we haveThenwhereExpanding (36) and using (41) we have ThenExpanding (37), we haveThen, using (41) and (44),Finally, using (25), the estimated value of original state can be obtained by sum of the state with the augmented state :Moreover, the unknown parameter is defined as

Based on the above analysis, the TEKF can be decoupled into two filters such as the modified bias free filter and bias filter. The modified bias filter gives the state estimation , and the bias filter gives the bias estimate . The corrected state estimate of the TEKF is obtained from the estimates of the two filters and coupling equations and [21]. The modified bias free filter is expressed as follows:and the bias filter iswith the coupling equationsThe initial conditions of TEKF algorithm are established with the initial conditions of a classical EKF , so thatAccording to variables of full order filter , the stator flux and torque estimators for DTC-SVM of Figure 1 are then given by where is the pole pairs of IM. The estimated speed and electromagnetic torque obtained from the TEKF observer are used to close the speed and torque loop to achieve sensorless operations.

4.2. The Stability and Parameter Sensitivity Analysis of the TEKF

Theorem 3. The discrete time conventional extended Kalman filter (19)–(23) is equivalent to the two-stage extern Kalman filter (see (61)~(83)).

Proof. Before proving the theorem, the following five relationships are needed:(1)Using (72) and (78),(2)Using (67) and (73),where(3)Using (20), we have(4)Using (21), (5)Using (22), By inductive reasoning, suppose that, at time , the unknown parameter and estimated state are equal to the parameter and state of the control system, respectively; we show that TEKF is equivalent to the conventional EKF because these properties are still true at time .
Assume that at time where and represent the variance-covariance matrices of the system and estimated variables, respectively.
From (19), we have Then using (98), (41), (62), (79), (81), (71), and (61),Using (19), (71), (98), (63), and (64), we haveUsing (91), (98), (78), (66), (79), (82), (86), and (72), we obtain Using (90), (98), (72), (32), (71), (29), and (97), we obtainUsing (92), (98), (33), (32), (79), (86), and (91),Using (93), (101), (55), (73), (67), (80), and (87),Using (94), (30), and (88), we obtainNext we will show that (98) holds at time . From (23) we haveThen using (61) and (105), the above equation can be written asUsing (95), (105), (102), and (77),Then using (80), (68), (74), and (31), we obtainUsing (96), (30), (28), (105), and (80),Using (97), (106), (95), (29), and (30), we obtainFinally, we show that (98) holds at time . This can be verified by the initial conditions of TEKF algorithm.

4.3. Numerical Complexity of the Algorithm

Tables 1 and 2 show the computational effort at each sample time by the conventional EKF algorithm and TEKF (where rough matrix-based implementation is used) in which, as defined above, is the dimension of the state vector , is the dimension of the measurement , is the input vector , and is the dimension of the parameter . The total number of arithmetic operations (additions and multiplications) per sample time of the TEKF is 1314 compared with 1778 for a rough implementation of a conventional EKF, which means the operation cost can reduce by 26%.

5. Simulation and Experimental Results

5.1. Simulation Results

To test the feasibility and performance of the TEKF method, the sensorless DTC-SVM technique for IM drives described in Section 2 is implemented in MATLAB/SIMULINK environment. The values of the initial state covariance matrices , , and have a great influence on the performance of the estimation method. The diagonal initial state covariance matrix represents variances or mean-squared errors in the knowledge of the initial conditions. Matrix gives the statistical description of the drive system. Matrix is related to measured noise. They can be obtained by considering the stochastic properties of the corresponding noises. However, a fine evaluation of the covariance matrices is very difficult because they are usually not known. In this paper, tuning the initial values of covariance matrices , , and is using particular criteria [22] to achieve steady-state behaviors of the relative estimated states, as given by

In the simulation, a comparison is made to verify the equivalence of EKF and TEKF. Real-time parameters estimated by TEKF are used to formulate the closed loop, such as rotor speed, stator flux, and electromagnetic torque. The estimations obtained by EKF algorithm are not included in the sensorless DTC-SVM strategy and only evaluated in open loop. A step reference speed was applied to the simulation.

The machine is accelerated from 0 rpm to 1000 rpm at 0 s and the torque load is set to 4 N. The simulation results of parameter estimation are shown in Figure 3. Figures 3(a) and 3(c) represent the performance of the speed and rotor position tracking capabilities of the control system with TEKF and EKF, respectively. Figures 3(e), 3(f), and 3(g) represent the estimated stator current and flux; they show that ripples are significantly suppressed due to the SVM modulation scheme. Figures 3(b), 3(d), and 3(h) show that, for variations of speed reference, the rotor speed, rotor position, and stator flux errors between the two observers are very little. It is verified that the two observers are equivalent. The difference between two estimators is caused by accuracy loss in TEKF, which uses more calculation steps.

In order to further verify the performance of TEKF against model-plant parameter mismatches and the equivalence of two observers, the change in rotor resistance is considered. Rotor resistance will increase due to temperature rise while the motor is running. To simulate this condition, the rotor resistance in TEKF and EKF is increased to 200% compared with the normal value, which is equivalent to a 50% decrease in the actual rotor resistance. The machine is still accelerated from 0 rpm to 1000 rpm at 0 s and the torque is set to 4 N. Figure 4 shows that, for variation of rotor resistance, the steady-state speed and rotor position errors are negligible, and the difference of the speed and rotor position estimations between the two observers is rather null.

5.2. Experimental Results

The overall experimental setup is shown in Figure 5 and the specifications and rated parameters of the IM, controller, and inverter are listed in Table 3. In the experimental hardware, an Expert3 control system from Myway company and a three-phase, two-pole 1.5 kW IM are applied. The IM is mechanically coupled to a magnetic clutch (MC), which provides rated torque, even at very low speed. The main processor in Expert3 control system is a floating point processor TMS320C6713 with a max clock speed of 225 MHz. All the algorithms including TEKF, EKF, DTC algorithm, and some transformation modules are implemented in TMS320C6713 with 100 μs sampling time and data acquisition of the parameter estimations, measured variables, and their visualization are realized on the cockpit provided by PEView9 software. Insulated Gate Bipolar Transistor (IGBT) module is driven by the PWM signal with a switching frequency of 10 kHz and 2 μs dead time. The stator currents are measured via two Hall effect current sensors. The rotor angle and speed of IM are measured from an incremental encoder with 2048 pulses per revolution.

This experiment test is here to testify the performance of TEKF and demonstrate that the two estimators are mathematically equivalent. The machine is accelerated from 600 rpm to 1000 rpm and 4 N torque load is set. The experimental results of parameter estimation based on two observers are given in Figures 6 and 7. Figures 6(a) and 6(c) show that the TEKF still has a good tracking performance of the speed and rotor position in experiment. Figures 6(d), 6(e), and 6(f) illustrate stator flux and stator current estimation robustness. Figures 6(b), 6(g), and 6(f) referring to the difference in speed and stator current estimations given by the two observers are still small. These experiment results prove that the two estimators are mathematically equivalent. Figure 7 shows the speed and rotor position estimations based on TEKF and EKF for a 50% decrease of rotor resistance (the same as the simulation). As expected, the steady error of the TEKF and the difference in speed and rotor position estimations are still tiny. Robustness of TEKF is verified.