A COMPARISON OF ENSEMBLE KALMAN FILTER AND EXTENDED KALMAN FILTER AS THE ESTIMATION SYSTEM IN SENSORLESS BLDC MOTOR

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1 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. A COMPARISON OF ENSEMBLE KALMAN FILTER AND EXTENDED KALMAN FILTER AS THE ESTIMATION SYSTEM IN SENSORLESS BLDC MOTOR M. Ri an, F. Yusivar, W. Wahab and B. Kusumoputro Computational Intelligence and Intelligent System Research Group, Department o Electrical Engineering, Faculty o Engineering, Universitas Indonesia, Kampus Baru Universitas Indonesia, Depo, West Java, Indonesia muhammad.rian@ui.ac.id ABSTRACT In this paper, a new iltering algorithm is proposed or system control o the sensorless BLDC motor based on the Ensemble Kalman ilter (EnKF). The proposed EnKF algorithm is used to estimate the speed and rotor position o the BLDC motor only using the measurements o terminal voltages and three-phase currents. The speed estimation perormance o developed EnKF was compared with the Extended Kalman Filter (EKF) under the same conditions. Results indicate that the proposed EnKF as an observer shows better perormance than that o the EKF. Keywords: speed estimation, sensorless brushless DC motor, ensemble alman ilter, extended alman ilter. INTRODUCTION Brushless Direct Current (BLDC) is a motor without brush and electronically control; and recently is one o the motor types that rapidly gaining popularity in a number o industrial applications. Because o its characteristics, such as high starting torue, high eiciency with no excitation losses, low noise (silent) operation and high durability, BLDC motors are used in various applications such as in various types o compressors, in electrical vehicles, hard disc drives and in medical applications (Liu, 26). To replace the unction o cummutators and brushes, the BLDC motor reuires an inverter and a position sensors such as Hall-eect, resolver, or absolute encoder that detects the rotor position or a proper commutation o current. The rotation o the BLDC motor is based on the eedbac o the rotor position which is obtained rom a sensor position (i.e., a hall sensors). BLDC motor usually uses three hall sensors or determining the commutation seuence, hence,the installation o these sensors poses several problems or the motor-drive system. As the conseuence, various problems occurred on the operation o the BLDC motor such as reduce reliability and system robustness, diiculty on its installation and maintenance, and increasing the size and the expense o the motor (Niapour, 214). To overcome the negative eects o the used sensor systems, the researches are doing researches on the possibility o a sensor-less systems. Iizua e al., proposed a zero-crossing point (ZCP) o the bac-emf and a speeddependent period o time delay (Iizua, 1985); however, this method has an error accumulation problem when the motor is operated at a low speed. Another sensorless method is a Flux Calculation Method (Kim, 24), but as with the previous proposed method, this method has also perormed an error accumulation problem or integration at a low speed. This method also demands a lot o computational cost and is sensitive to a parameter variation, then an expensive loating-point processor would be reuired to handle this complex algorithm. The third method is developed based on the unction o an observer (Kim, 24). Various types o observers are then used to estimate rotor position, especially the Extended Kalman Filter (EKF) (Lenine, 27). The biggest advantage o using observers is lied on that all o the states in the system can be estimated, including with the states that are hard to obtain by measurements. However, there are also some limitations on using the EKF as an observer, such as the characteristics o the EKF that can only be perormed as irst-order accuracy, a high computational complexity due to calculation o the Jacobian matrices and its covariance matrix. However, the most important problem related with the used o EKF as an observer is lied on its wea robustness characteristics against parameter detuning. Figure-1. The control diagram o sensorless BLDC motor. Figure-1 shows the bloc diagram o the system operation o a speed control sensorless BLDC motor using a amily o Kalman ilters, as an observer, or estimating the rotor speed () and the rotor position (). To turn the BLDC motor, a DC power supply is necessary to be ed through a three-phase current-controlled voltage-source 7386

2 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. inverter. And to be rotated precisely in a determined seuence o times, a control signal timed by a precisely rotor position is reuired, i.e. turning the ON/OFF o the active inverter transistor pair. As can be seen rom this igure, the input signal into the controller is the dierence between the reerence rotor speed re and the actual output, which in a sensorless system, is provided also rom the observer. Thus, the unction o the observer, in this case is the amily o Kalman Filters, is to provide the estimated o the actual rotor speed () and the rotor position (). The PID controller process this speed error signal (e) into a torue command ( re ), which in the reerence current generator, this torue command ( re ) is combined with the position signal () to provide the current signal or each phase o the current controller system. In a sensorless system, these positions signal (), which are very important in providing the best perormance curve o the BLDC motor, is provided also rom the observer. Then, the current signal o each phase o the motor is compared with the the eedbac BLDC current, then generates the ault currents o each phase o the motor or urther ed to the inverter to rotate the motor. In this paper, to provide a better estimation o the actual rotor speed () and the rotor position (), an Ensemble Kalman Filter (EnKF) as an observer is proposed. The characteristics o the EnKF are then compared with that o the EKF in an experimental procedure, with all o the programs are executed in MATLAB environment. As can be seen in our experimental results, it is proved that our proposed EnKF estimation system outperormed the EKF system, especially on its characteristics perormance at various speed reerences. This paper is organized as ollows. Section II presents description o the Brushless DC motor system, including with its mathematical representations. Section III discusses the development o the Ensemble Kalman Filter based controller in detail, including with its comparison with that o the Extended Kalman Filter. Section IV presents the experimental results and discussion, ollows by the conclusions that is presented in Section V. MATHEMATICAL MODEL OF THE BLDC As stated early, inormation regarding the rotor speed () and the rotor position () signals are necessary in order to control the BLDC motor. In this paper, a threephase BLDC motor with star connection is considered (see Figure 2) and used as the reerence. The mathematical model o the BLDC motor is derived, in order to provide the mathematical relationship between the rotor speed (), the rotor position () and the BLDC motor input current, by which the Kalman Filter could provided the estimatedinormation or the controller. The general voltage euation o BLDC motor can be written as ollows (Sheel, 212): [ ] = [ ] [ ] + [ ] [ ] + [ ] (1) which the induced bacs EMFs are trapezoidal and their pea values are eual to λ m ω The electromagnetic torue can be calculated by = + + = + + (2) where a, b, c have shapes lie e a, e b, e c, respectively, and their maximum values are one. The euation o motion or a simple system with an inertia J s, a riction coeicient B s, and a load torue T L can be written as: + = (3) Figure-2. Circuit o a BLDC motor drive. The rotor speed () and the rotor position (), can be written as: = (4) where P is number o pole on rotor. While the state space orm o such system can be deined as: = + (5) where = [ ] (6) = [ ] (7) 7387

3 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. = [ ] (8) = [ ] (9) THE SPEED CONTROLLER OF THE BLDC MOTOR The most importance parameter or controlling the BLDC motor is the speed control, and the most eective controller or the BLDC motor is the PID (Proportional, Integral, and Derivative) controller. The euation o a PID controller can be written as = + + (1) with e the error input signal, re the manipulated output signal, K p the proportional gain, T i the integral time seuence, and T d the derivative time seuence. These parameters, K p, T i, and T d are careully chosen to meet the best prescribed perormance criteria. In order to use the PID controller, the parameters related with its operation must be irstly tuned. This tuning process is utilized to synchronize the controller withthe controlled variable, thus allowing the process o the plant to be optimized accordance with the desired operating condition. Standard methods or tuning the controllers and the criteria or judging the loop tuning process have been investigated or many years. Some o them are Mathematical criteria, Cohen-coon Method, Trial and error method, Continuous cycling method, Relay eedbac method, Kappa-Tau tuning method, and Chien-Hrones- Reswic (CHR) PID tuning method. As the CHR method perormed better compare with that o Cohen-coon or Ziegler-Nichols methods (Gireesh, 214), (Xue, 27), especially or a tracing control and or a disturbance rejection problems, in this paper, we have used CHR techniue to ind the optimal values o K p, T i and T d o the PID controller or the BLDC motor speed control system. Figure-3. CHR step response tuning method. Figure-3 shows a simple approach on calculating the time constant T, the delay time L, the controller gain and determining the constant value o =L/T o the CHR method, derived rom the step time response o the system under investigation. Table-1 shows the CHR method ormula or determining those parameters constant or the P, PI, and PID controllers, more speciically using a % overshoot and 2% overshoot phenomena. Please note that the uicest periodic response o the system under step response input is labeled with % overshoot, while the uicest oscillatory process is labeled with 2% overshoot. Using those parameters value such as in the Table-1, the proportional controller gain, the integral time, and the derivative time o the P, PI, and the PID controllers using the CHR tuning rules or the determined BLDC motor are calculated and depicted in Table-2. Figure-4 shows speed response parameters o the closed loop system with the P, PI, and the PID controllers or the determined reerence speed o 15 rpm with no load condition and Table-3 shows the time domain analysis o the speed control o the BLDC motor perormance calculated rom these experiments. Results show that the PID controller perormed the best overshoot perormance, especially when using the % overshoot constants. ENSEMBLE KALMAN FILTER AS AN OBSERVER The undamental idea o an estimation system is to use a mathematical model derived rom the observed plant or a system to calculate the estimated output parameters value rom a measured input parameters. As long as there is a dierence between the estimated outputs value and the measured inputs, this error is ed bac to the 7388

4 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. estimation system or reducing this dierence by correcting the estimated output values. The best estimator rom the amily o the Kalman Filters so ar is the ExtendedKalman ilter (EKF), which hasbeen usually used to estimate the instantaneous system state variables and stator resistance o the BLDC motor byusing the actualvoltagesandcurrents derived through the mathematical model othebldc motor. To improve the perormance characteristics o the EKF estimator, we proposed in this paper the Ensemble Kalman Filter (EnKF) as an estimation system that will be described later. Extended Kalman Filter The EKF estimation system is estimator developed based on a Taylor expansion series which can handle almost every nonlinear system. Generally, estimation system by EKF can be devided into two stages: the prediction step and the correction step. In the irst stage, the predicted value o the state variables and the predicted state covariance matrix, which is expressed by P -1, can be obtained, and in the second stage, the correction step, a correction term is added to the predicted. The estimation procedures or the EKF can be listed as ollow: Selection o initial values or P, Q, R and X(). State vector prediction x = x, u (11) with x is calculated through (5) Prediction o error covariance matrix P = F P F T + Q (12) Calculation o correction actor o EKF S = H P H T + R (13) K = S H T S (14) Table-1. CHR tuning ormula or set point regulation. Controller Type With % overshoot With 2% overshoot K p T i T d K p T i T d P.3/.7/ PI.35/ 1.2T.6/ T PID.6/ T.5L.95/ 1.4T.47L Controller Type Table-2. Values o PID parameter. With % overshoot With 2% overshoot K p T i T d K p T i T d P PI PID Controller Type Time Settling Table-3. Time domain analysis. With % overshoot Overshoot Pea Time Time Settling With 2% overshoot Overshoot Pea Time P PI PID

5 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. Figure-4. Step response o the closed loop system with various parameter o PID controller. Estimation o output vector and state vector = h (15) = + (16) Estimation o error covariance matrix = (17) The most important o the EKF as an estimation system is to determine the best initial values or the three covariance matrices, namely, Q, R, and P, since these initial values are highly contributing on the ilter stability, i.e., the estimation values o the rotor speed and the rotor speed, and the convergence time. The diiculty o the EKF system to estimate the rotor speed, especially at a lower speed, maes the error o the estimation o the rotor position becomes too high (Ejlali, 212), so that the controller o the sensorless drive is no longer woring properly. Ensemble Kalman Filter The EnKF estimation system is a suboptimal estimator, where the error statistics are predicted by using a Monte Carlo or ensemble integration method to solve the Foer-Planc euation. Unlie the EKF system, the evaluation o the ilter gain in the EnKF system does not involve the approximation o the nonlinearity unctions (x, u) and h(x). Hence, the computational burden o evaluating the Jacobians o (x, u) and h(x) does no longer exist in the EnKF system. The starting point or the EnKF estimation system as a particle ilters is done by choosing a set o sample points, which is an ensemble o state estimations that captures the initial probability distribution o the state. These state estimation points are then propagated through the true nonlinear system, so that the probability density unction o the actual state can be approximated by the ensemble o the estimation system (EnKF). The EnKF estimation method consists o three stages. The irst step is called the orecast step; where to represent the error statistics o the system, assume that at time, there are an ensemble o orecasted state estimations with random sample errors. We denote these ensemble as, where X x, x,, x, (18) with the superscript i reers to the i-th orecast ensemble member. with x i is calculated through (5). Then, the ensemble mean is deined by i x x i= (19) Since the true state x is not nown, we approximate (18) by using the ensemble members. We deine the ensemble error matrix around the ensemble mean by E [x x x x ] (2) and the ensemble o the output error by E a [y y y y ] (21) 739

6 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. We then approximate the by, the by, and the by, respectively, where P E (E ) T,P E (E ) T (22) Thus, we interpret the orecast ensemble mean as the best orecast estimations o the state, and the spread o the ensemble members around the mean as the error between the best estimations and the actual state. The second step is the analysis step: To obtain the analysis estimations o the state, the EnKF perorms an ensemble o parallel data assimilation cycles, where or i = 1,..., x a i = x i + K (y i h x i ) (23) The perturbed observations are given by y i = y + v i (24) where is a zero-mean random variable with a normal distribution and covariance R. The sample error o the covariance matrix computed rom the converges to R as. We approximate the analysis o the error o the covariance matrices by, where P a E a E at (25) And is deined by with replaced by and replaced by the mean o the analysis estimate ensemble members. We use the classical Kalman ilter gain expression and the approximations o the error covariances to determine the ilter gain by by K = P (P ) (26) The last step is the prediction o error statistics in the orecast step: x i a + = i i (x, u ) + w (27) where the values are sampled rom a normal distribution with average zero and covariance. The sample error covariance matrix computed rom the converges to as. Finally, we summarize the analysis and orecast steps. Analysis Step: K = P (P ) (28) x a i = x i + K (y i h x i ) (29) x a i x a = i= (3) Forecast Step: x i a + = i i (x, u ) + w (31) = x i= + (32) x + E + = [x + i x + x + x + ] (33) E a [y y y y ] (34) P = E (E ) T,P = E (E ) T (35) RESULTS AND DISCUSSIONS A sensorless speed estimation and control system has been simulated using MATLAB. Simulation parameters o the BLDC motor are given as ollows: the stator resistance R=.7, the euivalent inductance o the stator L s = H, the maximum o each phase winding permanent magnet lux m =.5238W b, the inertia J= gm 2, the viscous riction coeicient B= Nms, the poles o the permanent magnet p=4 and simulation step length T=5 1 5 s, x =[ -1 1 ] T. For the EKF system, the x is calculated rom (5) and the Jacobian matrix F = A is calculated rom (8), while or the EnKF the same euation is used to calculate X, with the upper index, the number o the ensemble member, and in this paper is determined to be 8. In order to reach an insight distinct rom the whole system perormance o the proposed sensorless algorithm, the motor operation needs to be evaluated under dierent conditions. The eectiveness o the proposed algorithm and its comparison will be analyzed based on the perormance o the estimation systems rom a low speed operation added with the various speed increment operations. In the irst experiment, low speed operation perormance is conducted. The reerence speed is determined to be 2 rpm, and the experimental results (i.e. the perormance curves) are presented in Figure 5.As can be seen in rom this igure, the EnKF estimation system perormed a better predictive ability compare with that o the EKF estimation system. It is also clearly seen that both the rotor position estimation error and the rotor speed estimation error o the EKF estimation is increased as increasing the experimental time, while those error values or EnKF estimation system are close to constantly zero. In such conditions, the estimations error or both o the rotor position and the rotor speed o the EKF 7391

7 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. estimation system are growing to be very high, that maes the controller and the sensorless drive is no longer possible to eectively control the motor. These phenomena are in contrast with the response o the EnKF estimation system, which show nearly zero estimations error, or both the rotor position and the rotor speed. The second experiment is perormed to evaluate the response o the EKF and the EnKF estimation systems on various speed changes operation.in this experiment, the reerence speed changes rom 2 to 36 rpm within t=.25s; then the reerence speed is decreased rom 36 to 16 rpm within t=.5s, and rom 16 to 28 rpm within t=.75s, and the experimental results (i.e. the perormance curves) are depicted in Figure 6. As it was expected rom the theory, the perormance curves o the EnKF estimation system wors properly with very high reliability, with the estimated value o the actual speed could be the same as the speed changes. However, as can also be clearly seen rom this igure, the perormance curves o the EKF estimation system are showing a luctuated estimation rotor speed values, providing a high degree o estimation error rate or both rotor speed and the rotor position. speed (rpm) 1 5 sensor EKF EnKF error speed (rpm) 5-5 position (rad) error position (rad) 1 5 error EKF error EnKF time (s) sensor EKF EnKF error EKF error EnKF Figure-5. Perormance curves o low speed. 7392

8 ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Networ (ARPN). All rights reserved. speed (rpm) 4 2 sensor EKF EnKF error speed (rpm) error position (rad) -2 position (rad) Figure-6. Perormance curves o various speed changes. error EKF error EnKF 2 1 sensor EKF EnKF error EKF error EnKF -1 time (s) CONCLUSIONS This paper ocused on the estimation o BLDC motor speed or dierent speed reerences with EKF and EnKF. In order to evaluate the estimate perormance, simulation experiments are presented in the paper. It is obvious to see that, rom the simulation results, the accurate estimation perormance can be obtained and the eectiveness o EnKF algorithm can be demonstrated. Moreover, the sensorless BLDC motor can be controlled precisely according to the designed EnKF algorithm. REFERENCES [1] Y. Liu, Z. Q. Zhu, and D. Howe, "Instantaneous Torue Estimation in Sensorless Direct-Torue-Controlled Brushless DC Motors", IEEE Trans on indusial application. vol. 42, no. 5. [2] S.A.KH. Mozaari Niapour, M. Tabarraie, M.R. Feyzi, A new robust speed-sensorless control strategy or highperormance brushless DC motor drives with reduced torue ripple, Control Engineering Practice. Vol 24, March 214, Pages [3] Iizua, K., Uzuhashi, H., Kano, M., Endo, T., and Mohri, K Microcomputer control or sensorless brushless motor. IEEE Transactions on Industry Applications (IA-21), 4, [4] Tae-Hyung Kim; Hyung-Woo Lee; Ehsani, M., "State o the art and uture trends in position sensorless brushless DC motor/generator drives," Industrial Electronics Society, 25. IECON st Annual Conerence o IEEE, vol., no., pp. 8 pp. 6-1 Nov. 25. [5] Lenine, D.; Reddy, B.R.; Kumar, S.V., "Estimation o speed and rotor position o BLDC motor using Extended Kalman Filter, Inormation and Communication Technology in Electrical Sciences (ICTES 27), 27. ICTES. IET-UK International Conerence on. vol., no., pp. 433, 44, 2-22 Dec. 27. [6] Ejlali, A.; Soleimani, J., "Sensorless vector control o 3-phase BLDC motor using a novel Extended Kalman," Advances in Power Conversion and Energy Technologies (APCET), 212 International Conerence on. vol., no., pp. 1, 6, 2-4 Aug [7] Dr. Satya Sheel and Omhari Gupta. New Techniues o PID Controller Tuning o DC Motor-Development o a Toolbos. MIT IJEIE. Vol. 2. No pp [8] Gireesh. N.; Sreenivasulu. G. "Comparison o PI controller perormances or a Conical Tan process using dierent tuning methods." Advances in Electrical Engineering (ICAEE). 214 International Conerence on. vol. no. pp Jan. 214 [9] Terzic, B. and Jadric, M. 21. Design and implementation o the extended Kalman ilter or the speed and rotor position estimation o brushless DC motor. IEEE Transactions on Industrial Electronics. 48(6),