An Optimized Extended Kalman Filter for Sensorless ... - IEEE Xplore

2012 IEEE International Conference on Power and Energy (PECon), 2-5 December 2012, Kota Kinabalu Sabah, Malaysia

An Optimized Extended Kalman Filter for Speed Sensorless Direct Troque Control of an Induction Motor I. M. Alsofyani, NRN Idris, T. Sutikno, Y. A. Alamri UTM-PROTON Future Drive Laboratory, Power Electronics and Drives Research Group, Faculty of Electrical Engineering, Universiti Teknologi Malaysia, 81300 Skudai, Johor, Malaysia and the effect of unmeasured disturbances. The application of EKF has been used in sensorless ac motor drives for about two decades. The EKF performance strictly depends on the right selection for the filter matrices. However, very little attempts have been made to optimize these filter’s covariances. In [9] an efficient selection of the process and the measurement noise covariances, Q and R respectively, and the weight matrices G for the EKF using a real-number code GA in the closed loop constant V/f control of an IM and a field orientation controller are reported. [10] proposed fuzzy logic using adaptive tuning of the KF for R covariance. In [8] a calculation approach for EKF matrices using a Taylor series expansion of the nonlinear equations around the nominal parameter values and by Monte Carlo simulations is addressed. In [11] by introducing Simulated Annealing for optimizing the performance of the EKF, good results over a wide speed range were achieved.This paper discusses the application of the EKF in estimating the speed for a DTC induction motor drive with closed loop control. The performance of the EKF based speed estimation is optimized using a particle swarm optimization (PSO) method.

Abstract—This paper presents a new approach of optimizing the performance of an Extended Kalman Filter (EKF) using particle swarm optimization (PSO) for speed estimation of an induction motor drive. The development of the EKF algorithm and selection of the filter covariance for the EKF based speed estimation are presented and discussed. The effectiveness of the optimization technique is verified through Matlab/Simulink simulation of a direct torque control (DTC) drive system under various operating conditions. Keywords-DTC; EKF; direct torque control; speedsstimation; induction motor; paricale pwarm optimization

I.

INTRODUCTION

Research interests on the development of sensorless techniques applied to induction machine drives (IM) has grown significantly in the last few decades. These techniques are used to estimate the induction machine state variables, such as rotor/stator flux vector, electromagnetic torque and rotor speed for control drives. The sensorless techniques are divided into two broad categories; the modelbased IM state equations methods and the signal injection techniques [1].

The paper is organized in seven sections. The following section presents the extended IM model. Section III deals with the development of the EKF algorithm. Particle swarm optimization (PSO) of the EKF and Simulation are presented in Sections IV and V, respectively. Section VI deals with the results and discussion of the EKF performance. Section VII concludes the work.

Various model-based schemes have been proposed in the literature for the speed and IM parameter estimations , such as, full order closed loop observers [2-3], the Model Reference Adaptive System (MRAS) estimators [4-5], Extend Kalman Filter (EKF) [6], and Sliding-Mode Observers [7]. It is well known that the model- based estimation methods have an inherent limit, when the stator frequency approaches zero since the rotor-induced voltage goes to zero, and consequently the induction machine system becomes unobservable [1]. Nevertheless, various speed estimation model-based techniques have been proposed, which focus on the improvement at low speed operations.

II.

Among the various techniques, the estimation that is based on EKF is one of the promising model-based techniques [8]. The EKF is useful for obtaining reliable estimates of the states from a limited number of measurements. They also can handle the model uncertainties

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EXTENDED MODEL OF INDUCTION MOTOR

In this study, the EKF is applied to a DTC induction motor drive to simultaneously estimate stator current, rotor flux, and rotor speed; the estimated speed will be used for the purpose of speed control system. However, the accurate estimation of these state variables is very much dependent on how well the filter matrices are chosen over a wide speed range. The extended model to be used in the development of the EKF algorithm can be given (as referred to the stator stationary frame) in the following general form:

319


xi (t ) = f i ( xi (t ), u(t )) + wi (t )

(1)

fi ( xi (t ),u(t )) = Ai ( xi (t )) xi (t ) + Bu(t ) Y (t ) = Hi ( xi (t )) xi (t ) + Bu(t ) + vi (t )

(2) (3)

Lσ = Ls −

III.

where xi is the ith estimated state, fi is the nonlinear function of the states and inputs, Ai is the system matrix, u is the control-input vector, B is the input matrix, wi is the process noise, H is the measurement matrix, and vi is the measurement noise. The general form of IM can be represented by (4) and (5).

X

⎡ ⎤ ⎢ ⎛ ⎥ 2 ⎞ Lm Rr ωr Lm ⎥ ⎢− ⎜ Rs + Lm Rr ⎟ 0 0 ⎢ ⎜⎝ Lσ Lσ L2r ⎟⎠ Lσ Lr ⎥ Lσ L2r ⎢ ⎥ ⎡ isd ⎤ 2 ⎛ ⎞ ⎢ ⎜ Rs + Lm Rr ⎟ − ωr Lm Lm Rr 0⎥ ⎢ isq ⎥ 0 − ⎢ ⎜ Lσ L L2 ⎟ Lσ Lr Lσ L2r ⎥ ⎢ ⎥ σ r ⎠ ⎝ ⎢ ⎥ * ⎢ψ rd ⎥ Rr Rr ⎢ ⎥ 0 − − ωr 0⎥ ⎢ψ rq ⎥ ⎢ ⎢ ⎥ Lr Lr ⎢ ⎥ ⎢ ω ⎥ R R ⎣ r ⎦ r r ⎢ 0 Lm ωr − 0⎥  ⎢ ⎥ X Lr Lr ⎢ 0 0 0 0 0⎥ ⎢  ⎥ A ⎣ ⎦ 0 ⎤ ⎡1/ Lσ ⎢ 0 1/ L ⎥ σ ⎥ v ⎢ ⎡ sd ⎤ + ⎢ 0 0 ⎥.⎢ ⎥ + w(t ) ⎢ ⎥ v sq 0 ⎥ ⎣⎦ ⎢ 0 u ⎢⎣ 0 0 ⎥ ⎦ B

L2m Lr

Stator transient inductance.

DEVELOPMENT OF THE EKF ALGORITHM

In this section, the EKF algorithm used in the IM model will be derived using the extended model in (4) and (5). For a nonlinear system, such as the one in consideration, the Kalman Filter method is not directly applicable, since linearity plays an important role in its derivation and performance as an optimal filter. The EKF technique attempts to overcome this difficulty by using a linearized approximation, where the linearization is performed about the current state estimate. This process requires the discretization of (4) and (5) as follows:

⎡ isd ⎤ ⎢  ⎥ ⎢ isq ⎥ ⎢ψ rd ⎥ = ⎢ ⎥ ⎢ψ rq ⎥ ⎢ ω ⎥ ⎣r⎦

Rotor speed.

ωr

(4)

xi (k +1) = f i ( xi (k ), u(k )) + wi (k )

(6)

f i ( xi (k ), u(k )) = Ai ( xi (k )) xi (k ) + Bu(k )

(7)

Y (k ) = H i ( xi (k )) xi (k ) + Bu(k ) + vi (k )

(8)

The linearization of (7) is performed around the current

xˆi as given as follows: ∂f ( x (k ), u(k )) Fi (k ) = i i ∂xi (k ) ˆx ( k )

estimated state vector

(9)

i

The resulting EKF algorithm can be written with the following recursive relations:

P(k ) = F (k ) P(k ) F (k ) −1 + Q

(10)

K (k + 1) = H T P(k )( HP(k + 1) H T + R) −1

(11)

xˆ (k + 1) = fˆ ( x(k ), u(k )) + K (k )(Y (k ) − Hxˆ (k ))

(12)

P(k + 1) = ( I − K (k + 1) H ) P(k )

(13)

⎡ i sd ⎤ ⎢ i ⎥ sq ⎡i sd ⎤ ⎡1 0 0 0 0⎤ ⎢ ⎥ ⎢ ⎥ . ψ ⎢i ⎥ = ⎢ rd + v (t ) 0 0 0⎥⎦ ⎢ ⎥ ⎣0 1 ⎣ sq ⎦    ⎢ψ rq ⎥ H ⎢⎣ ω r ⎥⎦ 

(5)

X

where Q is the covariance matrix of the system noise, namely, model error; R is the covariance matrix of the output noise, namely, measurement noise; and P are the covariance matrix of state estimation error. The algorithm involves two main stages: prediction and filtering. In the prediction stage, the next predicted states fˆ (⋅) and predicted state-error

where,

v s , i s , ir

ψ s ,ψ r

Ls , Lr Rs , Rr

Stator voltage and current and rotor current space vectors. Stator and rotor flux linkage space vectors. Stator and rotor self inductances.

covariance matrices, Pˆ (⋅) are processed, while in the filtering stage, the next estimated states xˆ (k + 1) is obtained as the

Stator and rotor resistances.

320


sum of the next predicted states and the correction term (second term in (12)), are calculated. The structure of the EKF algorithm is shown in Fig. 1.

va

ia vb

IM

ib

vc

ic

mux

abc dq

3.

B

abc

fˆ (k )

vdq Φ(k )

dq

I R H

idq Q

P(k + 1)

1/ z

K (k + 1)

1/ z

xˆ (k + 1)

Outputs

(c) END FOR WHILE maximum number of swarm steps has not been reached: (a) Compute the speed of each particle. (b) Compute the new positions of the particles adding the speed produced from the previous step. (c) Evaluate each of the particles in POP. (d) When the current position of the particle is better than the position contained in its memory, the particle’s position is updated. (e) Increment the loop counter (f) END WHILE DC

1/ z

ωr r

1/ z

EKF

te

Speed Cont.

+-

r

Direct Troque Control

sa sb

IM

VSI

sc

ωˆ r

Fig. 1. Structure of Extended Kalman Filter IV.

ia

ic va

OPTIMIZATION OF EKF USING PSO

Extended Kalman Filter

In this study, the EKF algorithm is implemented to a closed-loop DTC as shown in Fig. 2. PSO is applied to optimize the EKF estimator for improving the dynamic performance of DTC. An objective function is adopted aiming to minimize the mean-squared error of the estimated speed during wide speed operations:

vb

vc

Fig. 2. Closed-Loop DTC with EKF Speed Estimator V.

SIMULATION

The real coded PSO for the EKF has been implemented using Matlab/Simulink. The estimator utilizes measured voltage and stator currents from the motor. The parameters of the 3-phase 4-pole (p) squirrel cage IM with rated torque of 150 (N.m) and rated speed of 152 (rad/s) are given in Table I. For optimizing the covariance matrices Q and R of the EKF, the parameters of the PSO are set as in Table II.

2 1 n (14) ( ωai − ωˆi ) n 0 where n is sampling times, ωa is the actual speed, ωˆ is the estimated speed. Among the three matrices in EKF algorithm, the error covariance matrix (P) is initially set as a unit matrix, then it will be updated in the course of the EKF iterations as prescribed by Eq. (13).

f =

ib

∑

TABLE I Induction Motor Parameters

The PSO algorithm for optimizing the filter is described as follows: 1. Initialize the population (POP). There are two main matrices that need to be optimized. The five diagonal elements of matrix Q and two diagonal elements of matrix R are organized in seven dimensions. Each particle in the swarm consists of the seven elements called positions. The PSO begins by randomly generating an initial population of the particles. 2. FOR each particle: (a) Calculate fitness value f: The particle positions are decoded back to the corresponding diagonal elements of the two matrices, Q and R. Then, these diagonal positions from each particle are sent to the EKF speed estimator of the IM drive to obtain the fitness value. (b) If the fitness value is better than the best fitness value (pBest) in history, set current value as the new pBest.

f Hz

Rs Ω

Rr Ω

Ls H

Lr H

50

.25

.2

.0971

.0971

JL Kg. m2 .046

TABLE II Particle Swarm Optimization Parameters No. of birds Bird steps Dims. c1 20 10 7 .12

P KW

V V

15

340

c1 .9

w .9

The training set for the PSO optimization strategy composes of the stator voltages and stator currents of the closed loop DTC drive. This training set is input to the EKF speed estimator, while the actual rotor speed of the motor drive is taken to be the target set for Fitness evaluation by the PSO. The simulation time is set to 0.5 s, enough time for the estimator to complete a series of different speed conditions including, transient state and steady state, acceleration and deceleration, low speed as well as zero speed zone. The optimization procedure has gone through challenging conditions (half-rated torque, 50% increase in stator resistance, and 100% increase in rotor resistance) where the increase in these variables started at 0.1s. The PSO method

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uses Matlab and the required parameter is transferred to Simulink. The mean squared error obtained from Simulink is then sent back to Matlab for further analysis using PSO algorithm. The diagonal elements for the best (minimum) meansquared error based on PSO iterations, are given as follows: Q = Diag[2.9319*10-13 6.4079*10-13 0.4289*10-13 1.9316*10-13 3.4719*10-3], R = Diag[6.4718 *10-2 5.8305*10-2].A marked improvement in the EKF performance is therefore achieved with the PSO optimization technique. VI.

RESULTS AND DISCUSSION

Fig. 4. Results for speed estimation and error when Rs is increased by 50% and Rr is nominal

To evaluate the performance of the developed EKF approach for DTC drive, four tests are applied, with different stator and rotor resistance variations to investigate the effect of machine parameter changes (e.g., due to temperature and frequency variations) on the EKF performance. The speed variations are obtained by changing the reference speed (Ref. Sp.) of the DTC IM drive with applying half-rated load torque. Fig. 3 shows the estimated speed (Est. Sp.) of the optimized EKF, and the actual rotor speed (Act. Sp.) with the motor resistances set to their nominal values. The figure shows that the estimated and actual speeds coincide during the transient and steady state conditions, as well as zero speed zone. The accuracy of speed tracking is still good when the stator resistance is increased by 50% as shown in Fig. 4. Even when the rotor resistance is doubled (Fig. 5), the speed tracking is still satisfactory. For the worse condition when the rotor resistance is doubled and the stator resistance is increased by 50%, Fig. 6 shows that the error between the actual and estimated speed is still bounded in a satisfactory band. The extended Kalman filter shows disturbance rejection to the machine parameter variations because the latter are handled as noise in the speed estimation algorithm.

Fig. 5. Results for speed estimation and error when Rs is nominal and Rr are increased by 100%

Fig. 6. Results for speed estimation and error when Rs is increased by 50% and Rr is increased by 100% In addition to the resistance variation tests at high and low speed, the EKF is tested by reversal speed tests (changing input frequency from 50 to -50 Hz) at different resistance variations with applying half-rated load torque. By inspecting the results (Figs. 7 to 9) of these tests, it can be noted that the estimated speed tracks the actual speed through 150 (rad/s) to -150 (rad/s) within a satisfactory error band. It can be seen that the error band become wider with the increase of resistances as shown in Fig. 8 and Fig. 9. The small value of this estimation error is an important indicator for the good performance of the EKF in the high-speed range under load.

Fig. 3. Results for speed estimation and error when both Rs and Rr are nominal

322


approach. When the motor runs under various disturbances (stator resistance, rotor resistance variations and half-rated load torque), it shows good robustness in a wide speed range with excellent accuracy in the estimation of the rotor speed. The proposed scheme can be used for replacing the conventional trial and error method and offers an alternative solution for optimizing the EKE. ACKNOWLEDGMENT The authors would like to thank Universiti Teknologi Malaysia (Vot Q.J130000.2523.01H30), Ministry of Higher Education (MoHE) and Ministry of Science, Technology and Innovation (MoSTI) of the Malaysian government for providing the funding for this project.

Fig. 7. Results for reversal speed estimation and error when Rs and Rr are both nominal

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[3] (b) Fig. 8. Results for reversal speed estimation and error when Rs is increased by 50% and Rr is nominal

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[6]

Fig. 9. Results for reversal speed estimation and error when Rs is nominal and Rr is increased by 100%

[7]

Overall, the PSO optimization strategy for EKF speed estimation approach shows very good performance throughout different disturbance and speed conditions. [8]

VII. CONCLUSION The paper introduces a new method to optimize the performance of an EKF for speed estimation of an induction motor drive. The particle swarm optimization approach is employed to properly select the white noise covariance. Simulation studies on the direct toque control of induction motor have confirmed the effectiveness of the proposed

[9]

323

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