Optimized Kalman filters for sensorless vector control induction motor drives

This paper presents the comparison between optimized unscented Kalman filter (UKF) and optimized extended Kalman filter (EKF) for sensorless direct field orientation control induction motor (DFOCIM) drive. The high performance of UKF and EKF depends on the accurate selection of state and noise covariance matrices. For this goal, multi objective function genetic algorithm is used to find the optimal values of state and noise covariance matrices. The main objectives of genetic algorithm to be minimized are the mean square errors (MSE) between actual and estimation of speed, current, and flux. Simulation results show the optimal state and noise covariance matrices can improve the estimation of speed, current, torque, and flux in sensorless DFOCIM drive. Furthermore, optimized UKF present higher performance of state estimation than optimized EKF under different motor operating conditions.


INTRODUCTION
Field orientation control induction motor (FOCIM) drives are widely used in high performance industrial applications when high torque and speed response are required [1], [2]. Furthermore, the main advantage of FOCIM drives is decupling control between torque and flux as separately excited direct current (DC) motor [2], [3]. In order to archive the high performance in direct field orientation control induction motor (DFOCIM) drive, accurate measurements of rotor speed and flux are required [4]. These measurements are provided by Hall sensors and sensing coils for flux measurement as well as the incremental encoder for rotor speed measurement. However, these sensors imply high cost, size, and weight as well as, lower reliability and difficult of installing [2], [5], [6]. In recent years, elimination of these sensors has been considered and the speed and flux are estimated based on voltage and current terminals to represent the senserless vector control drives [6].
In the last decade, Kalman filter algorithms have been used for the estimation of the rotor speed and flux in induction motor drives [7]. Extended Kalman filter (EKF) and unscented Kalman filter (UKF) are used to estimate the rotor speed of induction motor [2], [6]- [8]. UKF is used to estimate the speed and current FOCIM drives [9]- [12]. EKF is used to estimate the rotating speed in FOCIM drives [6], [8], [13], [14]. On the other hand, the main point in EKF and UKF is covariance matrix (Q) and measurement noise matrix (R) which are unknown matrices. These matrices were tuned manually based on trial and error method [6]- [15]. The performance of EKF and UKF highly depends on the right selection for the covariance  [15]. Recently, optimization algorithms are used to tunned the covariance and measurement noise matrices based on minimizing the mean squared error (MSE) between the actual and estimation of measuring response. Single objective optimization algorithms are used to estimate the covariance and measurement noise matrices based on specific minimization of MSE. Therefore, single objective practical swarm optimization (PSO) is used to optimize EKF for speed estimation based on minimizing MSE of the rotating speed [3], [7], [16], [17]. Also, single objective genetic algorithms are used to find the optimal tunning of the covariance and measurement noise matrices for the EKF based on minimizing MSE of the rotating speed [6], [18]. However, single objective optimization algorithms can reduce only the MSE for speed estimation and neglect the error on the other state estimation by means of flux, current, and torque. Therefore, multi objective function optimization algorithms are used to find optimal tunning of the covariance and measurement noise matrices based on minimizing the MSE in different state estimations of induction motor (IM). Multi objective genetic algorithm can reduce MSE of rotating speed and torque but MSE of current is increased [19], even though, the multi objective function of differential evolution algorithm is applied in EKF to find the optimal covariance and measurement noise matrices based on minizine the MSE of speed and current, the MSE of flux is not considered [20].
The main contribution of the present paper is to implement the multi objective genetic algorithm of the UKF and EKF to find the optimal values of state and noise covariance matrices. The optimal values of these matrices are optimized based on minimizing MSE between the actual and estimation of speed, current, and flux. Dynamic model of IM is presented and DFOCIM strategy has been included to improve torque/current capability via decoupling of stator current components. For enhancement of speed-controlled alternating current (AC) drive and increase its reliability, an accurate estimation of speed, current, and flux based on optimized UKF and optimized EKF have been included and compared. The proposed method is focused on finding the accurate state estimation for senserless DFOCIM drive. It is shown in both no-load and load conditions results that optimized UKF can find an accurate estimation of speed, current, and flux better than optimized EKF.

DYNAMIC MODEL OF INDUCTION MOTOR
The mathematical model of IM has four variables in the stationary reference frame (α, β); stator current ( , ) and flux ( , ). The induction motor model has been extended ( ) to include the rotor speed ( ). Where is state matrix, is input matrix, is output matrix, and are the voltage in (α, β) frame. , are stator and rotor resistance; , , are stator, rotor, and mutual inductance; P is Paris of pole [10], [11]. The (7) should be discretized by using Taylor series of order two to be applied to the digital implementation.

KALMAN FILTER ALGORITHMS
In this paper, the EKF and UKF algorithms are used to find the estimation of current, flux, and rotor speed to be used in sensorless DFOCIM drive. However, the accurate estimation of these state variables by using Kalman filters depends on finding the optimal values of state and noise covariance matrices. The details of the EKF and UKF algorithms can be found in the following subsections.

Extended Kalman filter
To use a nonlinear model of IM with the extended EKF, the model must be linearized about the current operating point ( ), giving a linear perturbation model represented by a Jacobian matrix [6], [8], [14]: where , are noise matrix of state and output model; is derivative of the Jacobian matrix. The equations of EKF applied in IM drive used the model in (7) can be expressed: where is error covariance matrix; is covariance matrix of system noise; is covariance matrix of measurement noise; and is Kalman filter gain. Although the EKF is straightforward, it has instability solution due to the linearization and costly calculation due to Jacobian matrices.

Unscented Kalman filter
The UKF is used with nonlinear vectors of IM without needing any derivative and Jacobian approximations. In this paper nonlinear discrete time state transition equation [9]- [12].
The UKF process is used to find the minimum mean square error (MMSE) then find the best state estimation of IM drive.
The update equations of state estimation and covariance matrix are given as (17), (18): where +1 is innovation matrix, +1 is state innovation covariance matrix. The sigma points are selected to approximate n-dimensional of state variable with ̂ and into 2n+1 weighted sample. Finally, the estimation process of set of samples can be explained in three steps: i) first is transform each sigma point ( +1), ii) second is compute the state estimation (̂ +1), and iii) third is calculate the estimation covariance matrix ( +1).

DESIGN OPTIMIZATION OF EKF AND UKF ALGORITHMS
According to the theory of Kalman filter algorithms, , , and are unknown matrices and these matrices have to be obtained based on stochastic properties of the noises [7], [13], [14]. Therefore, in most cases using tunning experimental trial-and-error to achieve the best state estimation. Finding the correct parameters of those covariance matrices can reflect on the accurate state estimation based on Kalman filter algorithms. In this paper, the objective functions to be minimized are the MSE of speed, current, and flux to find an accurate estimation for all state estimation in senserless DFOCIM drive. The four main components in the design optimization of Kalman filters are defined:

Design variables
The design variables are components of , , and matrices:

Objective function
The three main objectives to be minimized are MSE of speed ( ), MSE of current ( ), and MSE of flux .

Constraints
The constraints of the design variables are based on the minimum and maximum levels of each component in design variable.

Optimization algorithm
The multi objective function optimization algorithm is used to find the optimal design variables based on maximizing or minimizing vector of the objective functions. In this paper, the genetic algorithm is used to find state and noise covariance matrices based on minimizing MSE of rotor speed, current, and flux. Genetic algorithms are designed based on the biological process. Therefore, much of the processes are based on genetics and natural selection. The genetic algorithm has seven processes to find the optimal solution. These processes are a selection of the parameters, encoding and decoding, population, natural selection, pairing, mating, and mutations. The genetic algorithm is iterated until the chromosome gives the same value of cost. This means the genetic algorithm has been converged. Detailed information about Genetic algorithms can be found in [21], [22].

OPTIMIZATION PROCEDURE
This section describes the procedure to find the optimal state and noise covariance matrices for UKF and EKF of sensorless DFOCIM drive. The steps are: − Step 1: specify the initial value of the design variables ( , , and matrices) for UKF and EKF according to (22) Step 7: the steps (2-6) are repeated until finding the optimal Q, P, and R matrices.

SENSORLESS DFOCIM DRIVE UNDERSTUDY
The EKF and UKF are used to estimate rotor speed and flux based on voltage and current probes of sensorless DFOCIM drive as shown in Figure 1. DFOC can achieve the decoupling between torque and flux [11], [14]. Park transforms used to change from ( , ) to ( , ) reference frame.
Space vector control (SVC) pulse width modulation is used as a control for three phase IM [23]- [25]. Table 1 shows the parameter of IM under test. Table 2 shows the minimum and maximum levels of , , and matrices. Table 3 shows the optimal components of , , and matrices by using the multi objective function genetic algorithm based on minimizing MSE of speed error ( ) current error ( ), and flux ( ). In order to show the effectiveness of optimized EKF and optimized UKF in sensorless DFOCIM drive, two different operations conditions in IM have been used.

No-load conditions
In this condition, there is no load applied to IM and the speed reference is tracking response (1,000 Turn/min. from the beginning until 1.5 sec. and -1000 Turn/Min from 1.5 sec. until 3 sec.). As shown in Figures 2(a) and 2(b) (in appendix), optimized EKF and optimized UKF can estimate the speed, flux, current, and torque in sensorless DFOCIM drive. Also, the error between actual and estimation response in optimized UKF is less than optimized EKF in most estimation parameters.

Load conditions
In this section check the robustness of the system in different rotations of speed and sudden load. 50 Nm load has been applied in 0.8 sec when the speed reference is tracking response. As shown in Figures 3(a) and 3(b) (in appendix), UKF and EKF can track speed, current, flux, and torque after adding the load at a specific time. The convergence speed of the genetic algorithm takes about (344.706 min.) for optimized EKF and (232.374 min.) for optimized UKF in a powerful computer server with two Intel Xeon processors (CPU X5650) operating at 2.67 GHz (2 processors) and 64 GB of RAM. Table 4 (in appendix) shows the comparison in MSE between optimized UKF and EKF in no-load and load conditions. As shown in this table, MSE of speed, current, flux, and torque by using optimized UKF are less than optimized EKF in different operation conditions.

CONCLUSION
This paper proposed a method to optimize EKF and UKF for estimation speed, flux, torque, and current in sensorless DFOCIM drive. Multi objective genetic algorithm was used to find the optimal selection of state and noise covariance matrices in both EKF and UKF. The main objective in multi objective genetic algorithm to be minimized ware MSE of speed, current, and flux. Senserless DFOCIM drive was presented to achieve the decoupling between torque and flux based on optimized EKF and optimized UKF. The optimized EKF and UKF provided the accurate estimation of speed, flux, torque, and current in DFOCIM drive. Furthermore, optimized UKF had high accuracy of state estimations than optimized EKF. According to our expectations, the optimal control parameters of DFOC drive will be studied to find the optimal control of senserless DFOCIM drive.