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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 6, DECEMBER 2001

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Design and Implementation of the Extended Kalman Filter for the Speed and Rotor Position

Estimation of Brushless DC Motor

Bozo Terzic, Member, IEEE, and Martin Jadric, Member, IEEE

Abstract—A method for speed and rotor position estimation of a brushless dc motor (BLDCM) is presented in this paper. An extended Kalman filter (EKF) is employed to estimate the motor state variables by only using measurements of the stator line voltages and currents. When applying the EKF, it was necessary to solve some specific problems related to the voltage and current waveforms of the BLDCM. During the estimation procedure, the voltage- and current-measuring signals are not filtered, which is otherwise usually done when applying similar methods. The voltage average value during the sampling interval is obtained by combining measurements and calculations, owing to the application of the predictive current controller which is based on the mathematical model of motor. Two variants of the estimation algorithm are considered: 1) speed and rotor position are estimated with constant motor parameters and 2) the stator resistance is estimated simultaneously with motor state variables. In order to verify the estimation results, the laboratory setup has been constructed using a motor with ratings of 1.5 kW, 2000 r/min, fed by an insulated gate bipolar transistor inverter. The speed and current controls, as well as the estimation algorithm, have been implemented by a digital signal processor (TMS320C50). The experimental results show that is possible to estimate the speed and rotor position of the BLDCM with sufficient accuracy in both steady-state and dynamic operation. Introducing the estimation of the stator resistance, the speed estimation accuracy is increased, particularly at low speeds. At the end of the paper, the characteristics of the sensorless drive are analyzed. A sensorless speed control system has been achieved with maximum steady-state error between reference and actual motor speed of 1% at speeds above 5% of the rated value. Index Terms—Brushless dc motor, digital signal processor, extended Kalman filter, predictive current controller, speed and rotor position estimation.

I. INTRODUCTION HE brushless dc motor (BLDCM) has trapezoidal electromotive force (EMF) and quasi-rectangular current waveforms. Three Hall sensors are usually used as position sensors to perform current commutations every 60 electrical degrees. In addition, for servo drive applications with high stationary accuracy of the speed and rotor position, the BLDCM requires a rotor position sensor, such as resolver or absolute encoder. All the sensors mentioned increase the cost and size of the motor and reduce its sturdiness. Because of these reasons, the BLDCM

Manuscript received March 10, 2000; revised June 1, 2001. Abstract published on the Internet October 24, 2001. The authors are with the Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, 21000 Split, Croatia (e-mail: bterzic@fesb.hr). Publisher Item Identifier S 0278-0046(01)10269-8.

T

without position and speed sensors has attracted wide attention and many papers have reported on it. In most existing methods, the rotor position is detected every 60 electrical degrees, which is necessary to perform current commutations. These methods are based on: 1) using the back EMF of the motor [1], [2]; 2) detection of the conducting state of freewheeling diodes in the unexcited phase [3]; and 3) the stator third harmonic voltage components [4]. Since these methods cannot provide continual rotor position estimation, they are not applicable for the sensorless drives in which high estimation accuracy of the speed and rotor position is required. In that case, it is necessary to estimate rotor position continually, not only every 60 electrical degrees. In [5], the rotor position of the BLDCM is estimated continually using measured motor voltages and currents with the aim of estimating flux linkage. At each time step, the motor current is estimated in two stages to correct the predicted rotor position and the estimated flux linkage. The estimation results have been obtained using offline-measured voltages and currents with a 10- s sampling time. The accuracy of the rotor position estimation depends significantly on the motor parameter variation and accuracy of measured voltages and currents. In [6] and [7], the rotor position and speed of the permanent-magnet (PM) motor have been estimated by the extended Kalman filter (EKF). This method is applied to the motor with trapezoidal EMF and sinusoidal waveform currents, and is not directly applicable to the motor with rectangular currents. In this paper, a method is presented by means of which the speed and rotor position of the BLDCM are continually estimated. This method is based on the application of the EKF, which is an optimal recursive estimation algorithm for nonlinear systems that are disturbed by random noise. The EKF approach appears to be a viable and computationally efficient candidate for the online estimation of the speed and rotor position of the PM motors [8]. This is possible since mathematical models of motors are sufficiently well known. As is different from most of the similar methods dealing with estimation of the electric machine variables, in which the measuring voltages and currents are filtered in order to eliminate high harmonic components [6], [8], with this method, voltages and currents are measured without previous filtering. A special procedure is applied to obtain the line voltages average value combining measurements and calculations, which is made possible owing to the application of the predictive current controller. The experimental results of the speed and rotor position estimation are obtained using two variants of the estimation algorithm. In the first of

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 6, DECEMBER 2001

Fig. 1. System configuration for speed and rotor position estimation of BLDCM.

them, the speed and rotor position are estimated with constant motor parameters and, in the second variant, the motor variables and stator resistance are estimated simultaneously. At the end of this paper, the characteristics of the sensorless drive are analyzed. II. DESCRIPTION OF THE ESTIMATION ALGORITHM A. System Configuration Fig. 1 shows the block diagram of the system for speed and rotor position estimation of a BLDCM. The system can be functionally divided in two basic parts: the speed control system and the estimation system. The first one consists of a power circuit (dc supply, inverter, and motor) and control circuits which perform three functions: current commutation, current control, and phase currents and speed control. The measured speed , as well as the estimated rotor position , are used as feedback signals. Although application of the estimated variables in the control circuits is not necessary for the estimation procedure, it turned out that using the estimated rotor position as a feedback signal improves the estimation results. The main blocks of the estimation algorithm are the EKF and the block for calculating average motor line voltages during sampling time. The average line voltages vector, defined on the basis of average line voltages in the -sampling time , is calculated at the beginning of the sampling time by means of terminal voltages to neutral-point , the inverter transistors duty vector , the inverter dc voltage , the estimated speed cycle , and measured currents vector and rotor position . Vector forms, along with the measured , a set of the EKF input variables, currents vector by means of which speed and rotor position are estimated. The algorithm for determining line voltages (without filtering), based on measurements and calculations, represents a peculiarity differing this estimation method from similar methods dealing with estimation of the electric machine variables [6], [8]. For this purpose, it was necessary to apply a discrete-time , indispensable current controller calculating duty cycle

for average line voltage determination. The predictive current controller has been chosen [9]. In addition to the reference and actual currents, its input variables are measured speed and estimated rotor position. The control signal , turning on/off the active pair of the inverter transistors, is generated on the . The active pair basis of the duty cycle and sampling time of inverter transistors is determined by commutation logic, using the estimated rotor position. B. Predictive Current Controller The current controller algorithm is based on the voltage equation describing the two-phases interval, i.e., the mode in which the current flows through two phases and equals zero in the third phase. By means of this equation, the average value of the line would voltage is calculated so that the feedback current at the end of the sampling be equal to the reference one interval. The feedback current is different at intervals of 60 , because the phase current which in the respective interval increases from zero absolutely is chosen. In this way, the fastest commutation is achieved. The average line voltage is (1) where , , and are motor parameters, and is sampling to be constant, time. Assuming the inverter input voltage the duty cycle can be calculated according to (2) In addition, the task of the current controller is to generate a for controlling the active pair of the continuous-time signal transistors. In order to insure that the average value of the motor current is equal to the reference one, the turn-on interval has been distributed in two equal parts, one of which is at the beginning and the other at the end of the sampling interval [Fig. 2(a)]. Fig. 2(b) illustrates the case when the complete turn-on interval is at the beginning of the sampling interval, due to which a difference arises between the average value of the motor current and the reference current .

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as state variables. As the stator winding neutral point is not accessible, which makes it impossible to directly measure phase voltages, it is necessary to define a BLDCM model with line . On the basis of the voltages as input variables BLDCM model with phase voltages, the following model has been derived:

Fig. 2. Disposition of the turn-on/off intervals within the sampling time. (a) ). Correct ( = ). (b) Incorrect (

I

i

I >i

C. Line Voltages Calculation are average values The input vector elements of the EKF of the motor line voltages during the sampling interval and are calculated by means of (3) (4) (5) , , and are average values of the motor where terminal voltage to neutral point ( in Fig. 1) in the interval . The , , and are calcu, , and lated using the duty cycle and the voltages measured at the beginning of the sampling interval. Various relations are obtained because the waveform of these voltages varies in each one-sixth of the electric period, depending on whether or not the current flows through the respective phase. These relations are given in Table I for all six intervals, for the chosen direction of rotation. In the two-phases mode, the voltages of the phases through which the current flows change on impulse with period and duty cycle . The corresponding relations in Table I have been obtained on the assumption that the forward voltage drops of the inverter transistors and freewheeling diodes are equal. The voltage of the phase through which the current does not flow changes almost linearly and its average value can be assumed to be equal to the measured value at the beginning of the sampling interval. Analyzing the commutation process, the conclusion can be arrived at, that the relations for the average values of the line voltages, obtained for the two-phases mode and shown in Table I, are valid during commutation, too. A special problem is the determination of the voltage average value in the interval within which the commutation and two-phases modes interchange. For this interval, it is not possible to determine with sufficient accuracy the voltage average value of the corresponding phase, so that, in order to achieve a satisfactory estimation accuracy, it is necessary to increase the corresponding elements of the state noise covariance matrix in the respective interval. D. Motor Discrete-Time Model reference frame The BLDCM is modeled in the stationary , speed , and rotor position using phase currents

(6)

where is the inertia moment and is the number of pole pairs. – and the angles of commutation start The coefficients , by means of which the trapezoidal back electromotive force is described, vary in dependence of the rotor position and are given in Table II. Taking into consideration the fact that the electromagnetic torque is a function of the state variables, system (6) can be written as a matrix equation (7) The Kalman filter being based on the discrete-time system, it is necessary that, from model (7), representing a continuous statespace model, the following equivalent discrete-time system be deduced: (8) If the continuous system is sampled with interval , which is at least ten times shorter than the stator time constant , then the matrices of the discrete-time system (8) can be obtained by Euler’s approximation (9) The sampling interval being much shorter than the motor inertia constant, the speed can be supposed to be constant during one sampling interval, which results in the mechanical equation of motion within system (8) assuming the following form [6]: (10) In order to estimate the stator resistance at the same time with the state variables, it is necessary to extend the state vector with andintroduce anewsystemequationwhichassumes thatthestator resistance does not change during one sampling period [10] (11) For the EKF application, it is necessary to extend the machine discrete model (8) with the equation connecting output with state variables (12)

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 6, DECEMBER 2001

TABLE I RELATIONS FOR THE CALCULATION OF THE AVERAGE VALUES OF THE MOTOR TERMINAL VOLTAGES TO NEUTRAL POINT

TABLE II COMMUTATION START ANGLES AND COEFFICIENTS OF THE SYSTEM (6) AS A FUNCTION OF THE ROTOR POSITION

At the instant , the predicted state variables and their covariance are corrected using the following [11]: (18) (19) (20) To start the calculation, it is necessary to define the state variables vector and the estimation error covariance matrix at . Assuming all the currents to have the initial instant the same measurement error, the measurement noise covariance matrix has the following diagonal structure:

Because of measurement errors and inaccuracy of the machine model, stochastic variables are introduced, i.e., measurement and system noise vector . These varinoise vector ables are supposed to be Gaussian white noises. They have a zero expectation and are characterized by their covariance ma[11]. trix The BLDCM stochastic discrete-time model can finally be described as (13) (14) where definitions are given at the bottom of the next page. E. EKF The EKF is an efficient state estimator for nonlinear systems. The EKF consists of the prediction and correction equations. By means of the estimated state variables and input variables at , the state variables and the estimation the instant at the instant are error covariance matrix predicted. The prediction equations are [11] (15) (16) where (17)

(21) is the measurement and A/D converting error of the where current. The state noise covariance matrix also has a diagonal structure (22) , , and represent the estimation incerThe elements titude of the machine currents, being equal for all the phases, and are defined differently in the commutation interval and out of it. Estimation incertitudes of the speed and rotor position are determined by the change of the rotor speed during one sampling time, so that different values are set in steady-state and dynamic conditions [6]. The element represents the estimation incertitude of the stator resistance. Since the change of the resistance during a sampling interval is should be very low or zero [10]. negligible, III. EXPERIMENTAL RESULTS A. Laboratory Setup For experimental verification of the proposed estimation method, a laboratory setup has been designed and made. Its configuration is shown in Fig. 3. The motor rating is: 1.5 kW, 118 V, 15.9 A, 2000 r/min, and the rating of the insulated gate bipolar transistor (IGBT) inverter is: 300 V, 20 A. The three-phase supply voltage is rectified by a three-pulse diode

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a digital signal processor (DSP) (TMS320C50) [11]. The EKF sampling period equals the sampling time of the current control and is 200 s. Within this interval, the EKF algorithm (140 s) as well as the control algorithm (30 s) are performed. B. Results Obtained Assuming Constant Motor Parameters All vectors and matrices used in matrix equations (13)–(20) are defined for the estimation algorithm, which make possible simultaneous estimation of the motor state variables and stator resistance. In the variant of the estimation algorithm with conand stant stator resistance, it is necessary to reduce vector some matrices so that their elements relating to the stator resistance estimation need to be cancelled. These elements are the in the state vector , the sixth rows and stator resistance , , and , the sixth row in matrix columns in matrices , and the sixth column in matrix . The experimental results were obtained with following motor , parameters in the estimation algorithm: mH, and V s/rad. The elements of the measurement noise covariance matrix , (21), were chosen according to the measurement error of the current LEM sensors (0.7%) and the coding effects of the 12-bit A/D converters. is The evaluation of the system noise covariance matrix more complicated. It accounts for the model inaccuracy, the

Fig. 3. Configuration of the experimental system.

rectifier, so that the dc-link minus pole is connected to the neutral point of supply. Thus, the measurement of terminal voltages to the neutral point by means of the voltage divider without galvanic isolation of the power circuit from the control circuits is rendered possible. The currents are measured with Hall-effect sensors. The voltage and current signals are adjusted and sampled simultaneously with 12-bit A/D converters. The control and estimation algorithm has been implemented using

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 6, DECEMBER 2001

Fig. 4.

n

Phase currents (i

T

= 1000 r/min;

; i ) and line voltage (u = 3 N1m.

) at the operating point.

system disturbances, and the measurement errors of the voltages (sensors and A/D converting errors). The computation errors due to the fixed word length of the processor are considered as additional sources of system noise. In this work, the following selection procedure for the matrix has been applied. 1) First is the calculation of the matrix elements on the basis of known inaccuracies of the discrete-time motor model (quantization errors, assumption of the constant speed during sampling interval) and of the voltage measurements (sensor noise, A/D converters quantization). 2) Then, is the simulation of the BLDCM drive together with the estimation algorithm, including random noises which have been added to the currents and voltages. The results for the matrix elements calculated in the previous item have been used as initial values. The simulations have and been used to investigate the influences of the matrices on the EKF performance in transient and steadystate conditions. 3) Using the laboratory setup, by variation of the elements and matrices, their final values have been of the determined. Using the presented procedure, the following elements of the and matrices have been obtained: A ; ? measurement noise covariance: ? system noise covariances: A , rad /s , and (electrical degree) . have been The current noise covariances changed adaptively in dependence of the operation mode during a sampling interval. The above-mentioned ones have been used in the two-phases mode, while in the commutation their values have been ten times greater and, in the sampling interval where these two modes interchange, they have been even 100 times greater. It has been necessary because of the increased system noise caused by the problem of the voltage determination in the two intervals mentioned. By increasing – , the EKF becomes slower and, thus, the influence of the voltage measurement error on the estimation results is

Fig. 5. N1m.

Estimation results at the operating point.

n

= 1000 r/min;

T

= 4

Fig. 6. Estimation results at the operating point. n = 100 r/min; T N1m.

= 0:8

decreased. Since the commutation mode is shorter than the two-phases mode, these adaptive increases of the current noise covariances practically do not slow down the convergence. and line voltage Fig. 4 shows the motor phase currents at the stationary operating point: r/min, N m, measured on the laboratory setup shown in Fig. 3. The phase currents are of an approximately trapezoidal waveform with pulsations of the constant frequency and typical commutation ripples. This is practically the closest possible approximation to the ideal rectangular current waveform. In other words, the current control by means of the estimated rotor position has not affected its waveform.

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Fig. 7. Estimation results with the stator resistance estimation at various operating points. (a) n = 1000 r/min; T N1m. (c) n = 100 r/min; T = 4 N1m.

= 4 N1m. (b) n = 100 r/min; T = 0:8

Fig. 5 shows the estimated and the measured speed, as well at the stationary opas the rotor position estimation error r/min, N m. The speed estierating point mation error is less than 10 r/min, the rotor position estimation error being less than 1 electrical degree. As expected, worse estimation results have been obtained at lower speeds and loads, where the proportion of noise to measured signals is significantly larger than one at higher speeds and loads. At 100 r/min and no load, the speed estimation error is about 30 r/min, the maximum rotor position estimation error being about 18 electrical degrees (Fig. 6). C. Results With Stator Resistance as Estimated Variable All parameters used in the estimation algorithm with constant motor parameters have also been applied in the variant of the estimation algorithm with stator resistance as the estimated variable. In addition, it is necessary to define the sixth diagonal element of matrix , i.e., the covariance factor of the stator resis. tance. This factor is Fig. 7(a) shows the estimation results at the operating point r/min, N m. In comparison to the results with constant stator resistance (Fig. 5), it can be seen that introducing

the stator resistance estimation, the speed estimation error is decreased from 10 to 5 r/min. Better still results are obtained at the lower speeds, as one can see in Fig. 7(b), which shows the estimation results at 100 r/min and no load. Comparing Fig. 7(b) with Fig. 6, it can be seen that the speed estimation error is decreased from 30 to 12 r/min, and the position estimation error is slightly decreased, too. It is interesting to note that, after the initial error has converged, the average values of the estimated resistance in both considered operating points are significantly different. As both experiments were made at the same temperature of the stator winding, it is obvious that differences obtained in the estimated stator resistance do not actually exist. Therefore, introducing the stator resistance as the state variable, the speed estimation accuracy is increasing, while the resistance is becoming stationary on some fictitious value which can be greatly distinguished from the actual one. However, it should be noted that there is no difference between the estimated and r/min, measured resistance at operating point N m [Fig. 7(c)]. Generally, the uncertainty in the stator resistance estimation can be explained if a low influence of the stator resistance on the BLDCM characteristics is taken into consideration, which

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 48, NO. 6, DECEMBER 2001

is also inherent for the other electrical machines. For instance, an error of voltage average value can be balanced in the voltage equations only with a corresponding error of the estimated speed and stator resistance . Namely, the fact that currents are state variables and also measured variables has been used in the EKF algorithm implementation. It makes it possible that in (20) the current calculation can be omitted, and because of that the calculation time is shortened and the stability of the estimation procedure is improved. It means that the currents are not estimated and, because of that, they have not impacted on balancing of the voltage average value error. Since the voltage has a low influence in the voltage equation, it is obdrop on vious that the estimation error of the stator resistance is greater than the speed estimation error. Therefore, a big error in the stator resistance estimation at a low load [Fig. 7(b)] can be explained in such way. One can say the stator resistance as the estimated variable has low correlation with the other variables in the motor mathematical model, especially at no load. This is conr/min, firmed by the results at operating point N m [Fig. 7(c)] in which the stator resistance is estimated practically without error. That is in conformity with a relatively great in the voltage equation at influence of the voltage drop on this operating point. D. Sensorless Drive Performance Above results are obtained without the estimated speed as feedback signal in control circuits (Fig. 1). In order to realize the BLDCM sensorless drive the estimated speed is used as feedback signal in the speed and current controller. In that case it is necessary to filter the estimated speed and take EKF delay into consideration at synthesis of speed control circuit. Since the presented method does not solve problem of starting, the characteristics of the sensorless drive has been investigated so that the motor accelerates to a certain speed using the rotor position from the incremental encoder. The estimation procedure is started thereupon, and the estimated speed and rotor positions are used in the control circuits, the encoder being used only to check the estimation accuracy. Fig. 8 shows the estimation results of the sensorless drive during a speed reference step and load torque step. The speed estimation error in a dynamic condition and in a steady-state one is less than 10 r/min, i.e., 0.5% of the rated speed, the rotor position estimation error being less than 1 electrical degree. Fig. 9 shows the estimation results at the speed reference step from 100 to 1000 r/min. It can be seen that the estimated speed is tracking the measured one with satisfactory accuracy over the whole speed range. As expected, the rotor position estimation error is increased at lower speeds, and below a certain speed that error is so great that the correct commutation is not possible, i.e., sensorless operation is impossible. The minimum speed reference at which sensorless operation of the laboratory setup is feasible is 50 r/min. It is evident that the described method does not make possible the rotor position estimation at standstill, so that for the motor starting, one of the already known procedures would have to be applied. Most of these starting strategies are based on arbitrarily energizing the two or three windings and expecting the rotor to align to a certain definite position [1], [12].

Fig. 8. Estimation results at the speed reference step from 900 to 1000 r/min, and at the load torque step from 0 to 3 N1m.

Fig. 9. Estimation results at the speed reference step from 100 to 1000 r/min, T = 0.

IV. CONCLUSION Using the presented method, which applies the EKF, it is possible to estimate the speed and rotor position of the BLDCM with sufficient accuracy in both the steady-state and dynamic modes of operation. By simultaneous estimation of state variables and stator resistance, the speed estimation accuracy is increased, particularly at low speeds. Applying estimated speed as the feedback signal in the control circuits, a sensorless speed control system has been achieved. The maximum steady-state error between reference and actual motor speed is 1% over a speed range from 5% to 100% of the rated value. At lower speeds, the estimation accuracy decreases so that at the speed of about 50 r/min the rotor position estimation error becomes too big and the sensorless drive is no longer possible. REFERENCES

[1] K. Iizuka, H. Uzuhashi, M. Kano, T. Endo, and K. Mohri, “Microcomputer control for sensorless brushless motor,” IEEE Trans. Ind. Applicat., vol. IA-21, pp. 595–601, May/June 1985.

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[2] R. C. Beccerra, T. M. Jahns, and M. Ehsani, “Four quadrant sensorless brushless motor,” in Proc. IEEE APEC’91, 1991, pp. 202–209. [3] S. Ogasawara and H. Akagi, “An approach to position sensorless drive for brushless dc motors,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1990, pp. 443–447. [4] J. C. Moriera, “Indirect sensing for rotor flux position of permanent magnet AC motors operating in wide speed range,” in Conf. Rec. IEEE-IAS Annu. Meeting, 1994, pp. 401–407. [5] N. Ertugrul and P. P. Acarnley, “A new algorithm for sensorless operation of permanent magnet motors,” IEEE Trans. Ind. Applicat., vol. 30, pp. 126–133, Jan./Feb. 1994. [6] B. J. Brunsbach and G. Henneberger, “Field-oriented control of synchronous and asynchronous drives without mechanical sensors using a Kalman filter,” in Proc. EPE’91, vol. 3, Florence, Italy, 1991, pp. 3.664–3.671. [7] H. Brunsbach, G. Henneberger, and T. Klepsch, “Position controlled permanent excited synchronous motor without mechanical sensors,” in Proc. EPE’93, Brighton, U.K., 1993, pp. 38–43. [8] R. Dhaouadi, N. Mohan, and L. Norum, “Design and implementation of an extended Kalman filter for the state estimation of a permanent magnet synchronous motor,” IEEE Trans. Power Electron., vol. 6, pp. 491–497, July 1991. [9] D. M. Brod and D. W. Nowotny, “Current control of VSI-PWM inverters,” IEEE Trans. Ind. Applicat., vol. IA-21, pp. 562–570, May/June 1985. [10] L. Loron and G. Laliberte, “Application of the extended Kalman filter to parameters estimation of induction motors,” in Proc. EPE’93, Brighton, U.K., 1993, pp. 85–90. [11] B. Anderson and J. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [12] R. Krishnan and R. Ghosh, “Starting algorithm and performance of a PM DC brushless motor drive system with no position sensor,” in Proc. IEEE PESC’89, 1989, pp. 815–821. [13] L. Cardolleti and A. Cassat, “Sensorless position and speed control of a brushless DC motor from start-up to nominal speed,” EPE J., vol. 2, pp. 25–34, Mar. 1992. [14] B. Terzic, “Speed and rotor position estimation of brushless dc motor,” Ph.D. dissertation (in Croatian), Univ. Split, Split, Croatia, 1998. [15] B. Terzic and M. Jadric, “Brushless dc motor without position and speed sensors,” in Proc. PEMC’98, vol. 4, Prague, Czech Republic, 1998, pp. 4.60–4.65. , “Sensorless brushless dc motor with improved speed estimation [16] accuracy using stator resistance estimation,” in Proc. EPE’99, Lausanne, Switzerland, 1999, CD-ROM.

[17] “Digital control applications with the TMS320 family,” Texas Instruments Incorporated, Dallas, TX, 1991.

Bozo Terzic (M’00) was born in Grab, Croatia, in 1962. He received the Dipl.Ing. and Ph.D. degrees in electrical engineering from the Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia, in 1986 and 1998, respectively, and the M.S. degree from the Faculty of Electrical Engineering, University of Zagreb, Zagreb, Croatia, in 1993. In 1986, he became an Assistant in the Department of Electrical Power Engineering, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, where, since 2000, he has been an Assistant Professor. His research interests include ac electrical machines and drives, in particular, sensorless permanent-magnet motor drives.

Martin Jadric (M’95) received the B.S. degree in electrical engineering from the University of Split, Split, Croatia, and the M.S. and Ph.D. degrees from the University of Zagreb, Zagreb, Croatia, in 1964, 1970, and 1976, respectively. After completing his graduate studies, he joined Koncar Company, Zagreb, Croatia. Since 1996, he has been with the Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, first as an Assistant (through 1978), then as an Associate Professor (1978–1984), and currently as a Full Professor. During his career with the university, he has been responsible for the electric machines and drives curriculum, was the Head of the Department of Electric Power Engineering and, from 1980 to 1983, he was the Dean of the Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture. He has been involved in numerous government- and industry-sponsored projects in the areas of electrical machines and drives, and has authored more than 50 published technical papers and one book, Dynamics of Electrical Machines (Zagreb, Croatia: Graphis, 1997) (in Croatian).

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- 跟踪物体空间运动轨迹
*of**the*object position, moving*speed*,*the*region... is very suitable*for*hardware*implementation*. ...Based*Kalman**Filter**for*Micro-Machined Strapdown 6...

- 英文文献fang
*the**speed**of**the*controller*and*reduce ... used it in*implementation**of*a*Kalman**filter*. ...*design**and*explore*the*effects*of*fixed-point ...

- MVHI 2010 Content
*Extended**Kalman**Filter*in Unknown Environment Ming ...*speed*Data Acquisition Card Based on USB Bus TAN...Aimin Sha 354*Design**and**Implementation**of*...

- applied optimal estiation
*for**the*frequency domain*design**of*statistically optimal*filters*(Refs. 3, ...as*the**Kalman*fdter, is ideally suited*for*digital computer*implementation*. ...

- ...experiments validated the strengths of the basic...
*The*appropriate technology theory*extended*with non...With adequate accuracy,*Kalman**filter*(KF) is ...Process*of**implementation**of*Open Innovation 2.3....

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