A Novel Brushless DC Motor Speed Estimator Based on Space ...
A Novel Brushless DC Motor Speed Estimator Based on Space-Frequency Localized Wavelet Neural Networks (WNNs) Yujie Song, Ferdinanda Ponci, Antonello Monti, Lijun Gao, and Roger A. Dougal Department of Electrical Engineering University of South Carolina 301 South Main St. Columbia, SC 29208 [email protected]
Abstract---A novel high-accurate speed estimator using a recurrent Wavelet Neural Network (WNN) is proposed and validated for BLDC motor drives. The experimental results show that the WNN speed estimator yields promising results over a wide operating range including low-speed bands and transient operating conditions.
I. INTRODUCTION In brushless DC motor (BLDC) drives, there are several sensorless-based approaches to perform current commutations as well as to provide a rough speed estimation. Among these are the method using the motor back EMF [1] [2], the method detecting of the conducting state of freewheeling diodes in the unexcited phase [3], and the method using the stator third harmonic voltage components [4]. However, these methods cannot provide continual and accurate rotor speed estimation; therefore, they are not suitable for some motor drive applications, for example, motor servo drives where high accurate speed estimation is required. Moreover, the Extended Kalman Filter (EKF) based speed estimation solutions suffer from its high complexity making them hard to apply in real time and the accuracy is poor at low speed [5]. It is necessary to develop a sensorless speed estimation method which can continually provide rotor speed with high accuracy. Addressing this problem, this paper describes a novel accurate speed estimation approach using the recently developed mathematical tool – the Wavelet Neural Networks (WNNs) [6]. The network structure of the WNN-based speed estimator here presented is determined according to BLDC motor dynamic equations and the network is trained using the Levenberg-Marquardt algorithm. The proposed speed estimator is first verified through simulations. Then it is validated through hardware tests. The experimental results demonstrate that the speed estimator achieves the maximum error of ±0.25% to the rated speed over a wide operating range including low-speed bands and transient processes. II. WNN-BASED SPEED ESTIMATOR
The main challenge for motor speed estimation lies in the intrinsic non-linearity of the system. One possible approach to face the aforementioned issues is the use of Artificial Neural Networks (ANNs) that combine computational features, that may lead to high performance implementation, with the ability to represent highly non-linear behaviors [7][8]. The Wavelet Neural Networks (WNNs) are neural networks whose activation functions are wavelets and whose possible adjustable parameters include weights, dilation parameters, and translation parameters. Thanks to the capability of the wavelets to represent local non-linearities and the learning abilities of ANNs, WNNs have shown advantages on modeling non-linear systems with both rich local and global characteristics [6][9]. A. Wavelet neural network We introduce here a brief description of WNNs, starting from their activation functions and how they may approximate the behavior of a given non-linear system. Given an arbitrary
continuous function f ( x ) defined on the space L (R ) , in which the squares of all functions are integrable on real data set R, and for any positive number ε > 0 , there always exists an approximating function as described in Equation (1) 2
fˆ ( x ) =
n
∑ θ ψ (⋅) + c i =1
i
i
satisfying
the
condition
fˆ ( x ) − f ( x ) < ε for all x belonging to L2 (R ) . If the basis functions ψ i (⋅) are wavelets, then ψ i (⋅) are the dilated and translated waveletψ 0 (⋅) ,
daughter wavelets from the mother as described in Equation (2)
ψ i (⋅) = ψ 0 (α i ⋅ x + bi ) , where the network dilation parameter α i is used to rescale the independent variable x , the network translation parameter bi , is the bias added to the input, the parameter
θi
is the network weight of the activation
unit i , the integer n is the number of activation units, and the
c is a constant reflecting the non-zero mean of the function f ( x ) .
parameter
B. WNN-based speed estimator design The WNN network structure is determined according to the dynamic equations of BLDC motors. For every sextant of operation, the DC equivalent formulation can be adopted as v = Ri + L ⋅ di / dt + e , e = K Eω , Te = KT i , and
Te − Tl = J ⋅ dω / dt , where v and i are the motor voltage and current, R and L are the resistance and the inductance, e is the back electromotor force, K E and K T are the backEMF constant and the torque constant, ω is the mechanical speed, Te and Tl are the electromagnetic torque and the load torque, and J is the rotor inertia. The motor parameters R , L , K E , K T and J , are unknown in this study. The choice of load torque can be completely arbitrary; here the load torque is defined as
The constructed WNN is trained using the LevenbergMarquardt algorithm that is designed to approach secondorder training speed without having to compute the Hessian matrix. It provides a rapid converge speed from ten to one hundred times than traditional methods such as the Back Propagation algorithm [10]. III. SIMULATION VERIFICATION The constructed WNN-based speed estimator was then verified through simulations in the Virtual Test Bed (VTB) computational environment [11]. Fig. 2 illustrates the system schematic. It contains a complete BLDC drive system [12] and the designed WNN speed estimator. The speed estimator block termed “WNN” in Fig. 2 was implemented using a hierarchical model which is detailed in Fig. 3. The WNNbased speed estimator has one output of the estimated speed and three inputs including 1-step and 2-step delayed speed and the motor input voltage.
Tl = aω 2 + bω + c and the parameters
a, b, c are unknown. The motor dynamic equations are then combined
and
discretized
as
in
Equation
(3) is the system sampling step. It can be seen that, in Equation (3), the current instant motor speed not only depends on the current instant input voltage, but also non-linearly (due to the load torque expression, the possible parameter variations and intrinsic non-linearities) on the one-step time-delayed speed and two-step time-delayed speed. Therefore, the dynamic mapping between the motor input voltage and the output speed can be learned by a recurrent WNN, in which the network output has a connection with prior outputs. The corresponding structure of the BLDC speed estimator is shown in Fig. 1. Combing Equations (1), (2) and (3), the speed estimator output can be computed as follows:
ω [kh] = F { v [kh], ω [(k − 1)h], ω [(k − 2 )h]} , where h
Fig. 2 Simulation schematic in the VTB.
ωˆ [kh] = F { v [kh], ωˆ [(k − 1)h], ωˆ [(k − 2 )h]} n 3 = ∑ θ iψ 0 ∑ α i , j z j (kh ) + bi + c i =1 j =1 where z1 (kh ) = v [kh], z 2 (kh ) = ωˆ [(k − 1)h], z 3 (kh ) = ωˆ [(k − 2 )h]
v [kh]
Recurrent ωˆ [kh] WNN
ωˆ [(k − 1)h] ωˆ [(k − 2)h]
Z-1
-1
Z
Fig. 1The speed estimator using a recurrent WNN.
C. Training Method
Fig. 3 Implementation of WNN speed estimator in the VTB.
The training data, as shown in Fig. 4, were obtained by controlling the motor setting its speed reference to subsequent step values equal to 0, 30, 45, 70, 100, and 120 rad/s. The inverter duty ratio was also reordered, which was first converted to the motor input voltage and then used as the input to the WNN speed estimator. The speed estimator was then trained and the training result is illustrated in Fig. 5. Fig. 6 shows the absolute training error.
which the control loop is still closed with the actual motor speed as obtained from the model in simulation.
Fig. 4 Training data: speed (solid) and inverter duty (dash-dot). Fig. 7 Generalization property test: shaft speed (solid) and estimated speed (dashed).
IV. HARDWARE VALIDATION A. Hardware Platform The simulation-verified WNN-based speed estimator was implemented and validated through hardware tests. The experimental platform, as shown in Fig. 8, included a BLDC motor, a motor drive, and a real-time controller board (dSPACE DS1104). The motor, as shown in Fig. 9, is an axial flux BLDC motor (rated speed 1500RPM), which was built in house. The 3-phase motor drive consisted of the IRAMS10UP60A, which is an application specific intelligent power module developed by the International Rectifier. Fig. 5 Training results of the speed estimator through simulations.
Fig. 8 Experimental platform. Fig. 6 Absolute training error.
The WNN-based speed estimator was then tested by using new speed references of 0, 20, 50, 80, 110, and 120 rad/s. The estimated speed was then compared with the motor shaft speed, as shown in Fig. 7. These results refer to a system in
Fig. 9 The axial BLDC motor.
The motor control and the WNN speed estimator, trained with data acquired from hardware, were first implemented in the Simulink, and then were compiled and downloaded to the real-time controller board and executed there. The control signal was sent to the motor drive module through the 3-phase PWM generator (f = 15 kHz in this study). During experimental tests, the inverter duty ratio and the motor speed were measured by the controller board. The measured duty ratio was first converted to the motor input voltage and then was used as the input signal to the WNN speed estimator, and the measured motor speed was compared with the estimated speed to validate the estimator design. The Mexican Hat wavelet function was chosen as the activation function in this study, 5 nodes and total 26 weights were used.
Fig. 11 Relative training error.
Fig. 12 shows the implementation of the WNN speed estimator in the Simulink for the real-time controller board. Table 1 lists the corresponding well-trained network parameters.
B. WNN Training Results Figures 10 and 11 illustrate the training result of the WNN-based speed estimator and the corresponding relative error.
Fig. 12 WNN estimator implementation in Simulink for real-time controller.
TABLE I THE WELL TRAINED NETWORK PARAMETERS.
Fig. 10 Training results of the speed estimator through experiment.
W11 0.0179127991610 0.0609574414137 0.0087384632371 -0.028975862717 -0.0557237466098 W13 -0.0018223942326 0.0022190807352 -0.214194751487 0.284772268944 -0.00131611157164
W12 0.0116537849981 0.023775141521 0.00953324874617 0.00583129594087 -0.024153945634 W14 -2.25434509876 -3.28455853383 0.807212391681 -2.37379475296 3.07425306263
W21 38.7288356745 -30.4130567508 -37.1780624949 104.769335409 36.4277595722 W22 77.9373992841
G. Estimator Validation Fig. 13 shows the measured inverter duty ratio that was first converted into the motor input voltage and then used as input to the WNN speed estimator. Figures 14 and 15 illustrate the comparison between the measured motor speed and the
WNN estimated speed in motor low speed operations. Fig. 16 shows a zoom-in view of a transient process. Figures 17 and 18 show the speed estimator validation in motor high speed operations.
Fig. 16 Speed transient process validation.
Fig. 13 Measured inverter duty ratio.
Fig. 17 Validation in high speed range.
Fig. 14 Validation in low speed range.
Fig. 18 Relative error of speed estimation in high speed.
V. CONCLUSION
Fig. 15 Relative error of speed estimation in low speed.
A recurrent WNN-based speed estimator for BLDC motors has been proposed. It was verified through simulations and validated through hardware experiments. The study results
show that the WNN-based speed estimator provides accurate speed outputs over a wide speed operating range as well as in motor transient processes. Further studies are in progress for better characterization of the low-speed performance of the proposed algorithm. Besides the high accuracy, the designed WNN-based speed estimator possesses three other valuable advantages. Firstly, it is not limited by the accuracy of motor parameters. Secondly, its structure is concise since WNNs contain only one and a half activation layers. Finally, compared with a conventional speed sensor, it decreases the maintenance and increase the reliability of motor systems. The proposed approach needs further study to fully exploit its intrinsic flexibility in hybrid simulation and experimental training of the network and in higher resolution on-line training if needed. VI. ACKNOWLEDGEMENT This work is supported by the US Office of Naval Research under contracts N00014-02-1-0623 and N00014-031-0434.
REFERENCES [1] K. Iizuka, H. Uzuhashi, M. Kano, T. Endo and K. Mohri, “Microcomputer control for sensorless brushless motor,” IEEE Trans. Industry Application, Vol. IA-21, pp. 595– 601, May 1985. [2] J.Shao, D.Nolan and T.Hopkins, “A novel direct back EMF detection for sensorless Brushless DC (BLDC) motor drives,” Proc. of the IEEE Applied Power Electronic Conference, pp. 33-38, 2002. [3] S. Ogasawara and H. Akagi, “An approach to position sensorless drive for brushless dc motors,” Proc. of the
IEEE Industry Applications Society Annual Meeting, pp. 443-447, 1990. [4] J. C. Moriera, “Indirect sensing for rotor flux position of permanent magnet AC motors operating in wide speed range,” Proc. of the IEEE Industry Applications Society Annual Meeting, pp. 401-407, 1994. [5] B. Terzic and M. Jadric, “Design and implementation of the extended Kalman filter for the speed and rotor position estimation of brushless DC motor”, IEEE Trans. Industry electronics, Vol. 48, Issue 6, pp. 1065-1073, Dec. 2001. [6] Q.Zhang and A. Benviste, “Wavelet Networks”, IEEE Trans. Neural Networks, Vol.3, no. 6, pp. 889-898, Nov. 1992. [7] C. Wang, B. K. Bose, V. Oleschuk, S. Mondal and J. O. P. Pinto, “Neural-network-based space-vector PWM of a three-level inverter covering overmodulation region and performance evaluation on induction motor drive”, Proc. of the 29th Annual Conference of the IEEE Industrial Electronics Society, Vol. 1, , pp. 1-6, Nov. 2003. [8] B. K. Bose, Modern power electronics and AC drives, Prentice Hall, 2001. [9] C. L. Lin, N. C. Shieh and P. C. Tung, “Robust wavelet neuro control for linear brushless motors”, IEEE Trans. Aerospace and Electronic Systems, Vol.38, Issue 3, pp. 918-932, Jul. 2002. [10] M. T. Hagan and M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Networks, Vol. 5, no. 6, pp. 989-993, 1994. [11] The Virtual Test Bed, http://vtb.engr.sc.edu/ [12] A. Monti, D. Patterson, E. Santi, F. Ponci, etc, “An innovative low-cost drive for permanent magnet motors: a future energy challenge experience”, Proc. of the Nineteenth Annual IEEE Applied Power Electronics Conference and Exposition (APEC’2004),Vol. 2, pp.1137- 1142, 2004.
Abstract---A novel high-accurate speed estimator using a recurrent Wavelet Neural Network (WNN) is proposed and validated for BLDC motor drives. The experimental results show that the WNN speed estimator yields promising results over a wide operating range including low-speed bands and transient operating conditions.
I. INTRODUCTION In brushless DC motor (BLDC) drives, there are several sensorless-based approaches to perform current commutations as well as to provide a rough speed estimation. Among these are the method using the motor back EMF [1] [2], the method detecting of the conducting state of freewheeling diodes in the unexcited phase [3], and the method using the stator third harmonic voltage components [4]. However, these methods cannot provide continual and accurate rotor speed estimation; therefore, they are not suitable for some motor drive applications, for example, motor servo drives where high accurate speed estimation is required. Moreover, the Extended Kalman Filter (EKF) based speed estimation solutions suffer from its high complexity making them hard to apply in real time and the accuracy is poor at low speed [5]. It is necessary to develop a sensorless speed estimation method which can continually provide rotor speed with high accuracy. Addressing this problem, this paper describes a novel accurate speed estimation approach using the recently developed mathematical tool – the Wavelet Neural Networks (WNNs) [6]. The network structure of the WNN-based speed estimator here presented is determined according to BLDC motor dynamic equations and the network is trained using the Levenberg-Marquardt algorithm. The proposed speed estimator is first verified through simulations. Then it is validated through hardware tests. The experimental results demonstrate that the speed estimator achieves the maximum error of ±0.25% to the rated speed over a wide operating range including low-speed bands and transient processes. II. WNN-BASED SPEED ESTIMATOR
The main challenge for motor speed estimation lies in the intrinsic non-linearity of the system. One possible approach to face the aforementioned issues is the use of Artificial Neural Networks (ANNs) that combine computational features, that may lead to high performance implementation, with the ability to represent highly non-linear behaviors [7][8]. The Wavelet Neural Networks (WNNs) are neural networks whose activation functions are wavelets and whose possible adjustable parameters include weights, dilation parameters, and translation parameters. Thanks to the capability of the wavelets to represent local non-linearities and the learning abilities of ANNs, WNNs have shown advantages on modeling non-linear systems with both rich local and global characteristics [6][9]. A. Wavelet neural network We introduce here a brief description of WNNs, starting from their activation functions and how they may approximate the behavior of a given non-linear system. Given an arbitrary
continuous function f ( x ) defined on the space L (R ) , in which the squares of all functions are integrable on real data set R, and for any positive number ε > 0 , there always exists an approximating function as described in Equation (1) 2
fˆ ( x ) =
n
∑ θ ψ (⋅) + c i =1
i
i
satisfying
the
condition
fˆ ( x ) − f ( x ) < ε for all x belonging to L2 (R ) . If the basis functions ψ i (⋅) are wavelets, then ψ i (⋅) are the dilated and translated waveletψ 0 (⋅) ,
daughter wavelets from the mother as described in Equation (2)
ψ i (⋅) = ψ 0 (α i ⋅ x + bi ) , where the network dilation parameter α i is used to rescale the independent variable x , the network translation parameter bi , is the bias added to the input, the parameter
θi
is the network weight of the activation
unit i , the integer n is the number of activation units, and the
c is a constant reflecting the non-zero mean of the function f ( x ) .
parameter
B. WNN-based speed estimator design The WNN network structure is determined according to the dynamic equations of BLDC motors. For every sextant of operation, the DC equivalent formulation can be adopted as v = Ri + L ⋅ di / dt + e , e = K Eω , Te = KT i , and
Te − Tl = J ⋅ dω / dt , where v and i are the motor voltage and current, R and L are the resistance and the inductance, e is the back electromotor force, K E and K T are the backEMF constant and the torque constant, ω is the mechanical speed, Te and Tl are the electromagnetic torque and the load torque, and J is the rotor inertia. The motor parameters R , L , K E , K T and J , are unknown in this study. The choice of load torque can be completely arbitrary; here the load torque is defined as
The constructed WNN is trained using the LevenbergMarquardt algorithm that is designed to approach secondorder training speed without having to compute the Hessian matrix. It provides a rapid converge speed from ten to one hundred times than traditional methods such as the Back Propagation algorithm [10]. III. SIMULATION VERIFICATION The constructed WNN-based speed estimator was then verified through simulations in the Virtual Test Bed (VTB) computational environment [11]. Fig. 2 illustrates the system schematic. It contains a complete BLDC drive system [12] and the designed WNN speed estimator. The speed estimator block termed “WNN” in Fig. 2 was implemented using a hierarchical model which is detailed in Fig. 3. The WNNbased speed estimator has one output of the estimated speed and three inputs including 1-step and 2-step delayed speed and the motor input voltage.
Tl = aω 2 + bω + c and the parameters
a, b, c are unknown. The motor dynamic equations are then combined
and
discretized
as
in
Equation
(3) is the system sampling step. It can be seen that, in Equation (3), the current instant motor speed not only depends on the current instant input voltage, but also non-linearly (due to the load torque expression, the possible parameter variations and intrinsic non-linearities) on the one-step time-delayed speed and two-step time-delayed speed. Therefore, the dynamic mapping between the motor input voltage and the output speed can be learned by a recurrent WNN, in which the network output has a connection with prior outputs. The corresponding structure of the BLDC speed estimator is shown in Fig. 1. Combing Equations (1), (2) and (3), the speed estimator output can be computed as follows:
ω [kh] = F { v [kh], ω [(k − 1)h], ω [(k − 2 )h]} , where h
Fig. 2 Simulation schematic in the VTB.
ωˆ [kh] = F { v [kh], ωˆ [(k − 1)h], ωˆ [(k − 2 )h]} n 3 = ∑ θ iψ 0 ∑ α i , j z j (kh ) + bi + c i =1 j =1 where z1 (kh ) = v [kh], z 2 (kh ) = ωˆ [(k − 1)h], z 3 (kh ) = ωˆ [(k − 2 )h]
v [kh]
Recurrent ωˆ [kh] WNN
ωˆ [(k − 1)h] ωˆ [(k − 2)h]
Z-1
-1
Z
Fig. 1The speed estimator using a recurrent WNN.
C. Training Method
Fig. 3 Implementation of WNN speed estimator in the VTB.
The training data, as shown in Fig. 4, were obtained by controlling the motor setting its speed reference to subsequent step values equal to 0, 30, 45, 70, 100, and 120 rad/s. The inverter duty ratio was also reordered, which was first converted to the motor input voltage and then used as the input to the WNN speed estimator. The speed estimator was then trained and the training result is illustrated in Fig. 5. Fig. 6 shows the absolute training error.
which the control loop is still closed with the actual motor speed as obtained from the model in simulation.
Fig. 4 Training data: speed (solid) and inverter duty (dash-dot). Fig. 7 Generalization property test: shaft speed (solid) and estimated speed (dashed).
IV. HARDWARE VALIDATION A. Hardware Platform The simulation-verified WNN-based speed estimator was implemented and validated through hardware tests. The experimental platform, as shown in Fig. 8, included a BLDC motor, a motor drive, and a real-time controller board (dSPACE DS1104). The motor, as shown in Fig. 9, is an axial flux BLDC motor (rated speed 1500RPM), which was built in house. The 3-phase motor drive consisted of the IRAMS10UP60A, which is an application specific intelligent power module developed by the International Rectifier. Fig. 5 Training results of the speed estimator through simulations.
Fig. 8 Experimental platform. Fig. 6 Absolute training error.
The WNN-based speed estimator was then tested by using new speed references of 0, 20, 50, 80, 110, and 120 rad/s. The estimated speed was then compared with the motor shaft speed, as shown in Fig. 7. These results refer to a system in
Fig. 9 The axial BLDC motor.
The motor control and the WNN speed estimator, trained with data acquired from hardware, were first implemented in the Simulink, and then were compiled and downloaded to the real-time controller board and executed there. The control signal was sent to the motor drive module through the 3-phase PWM generator (f = 15 kHz in this study). During experimental tests, the inverter duty ratio and the motor speed were measured by the controller board. The measured duty ratio was first converted to the motor input voltage and then was used as the input signal to the WNN speed estimator, and the measured motor speed was compared with the estimated speed to validate the estimator design. The Mexican Hat wavelet function was chosen as the activation function in this study, 5 nodes and total 26 weights were used.
Fig. 11 Relative training error.
Fig. 12 shows the implementation of the WNN speed estimator in the Simulink for the real-time controller board. Table 1 lists the corresponding well-trained network parameters.
B. WNN Training Results Figures 10 and 11 illustrate the training result of the WNN-based speed estimator and the corresponding relative error.
Fig. 12 WNN estimator implementation in Simulink for real-time controller.
TABLE I THE WELL TRAINED NETWORK PARAMETERS.
Fig. 10 Training results of the speed estimator through experiment.
W11 0.0179127991610 0.0609574414137 0.0087384632371 -0.028975862717 -0.0557237466098 W13 -0.0018223942326 0.0022190807352 -0.214194751487 0.284772268944 -0.00131611157164
W12 0.0116537849981 0.023775141521 0.00953324874617 0.00583129594087 -0.024153945634 W14 -2.25434509876 -3.28455853383 0.807212391681 -2.37379475296 3.07425306263
W21 38.7288356745 -30.4130567508 -37.1780624949 104.769335409 36.4277595722 W22 77.9373992841
G. Estimator Validation Fig. 13 shows the measured inverter duty ratio that was first converted into the motor input voltage and then used as input to the WNN speed estimator. Figures 14 and 15 illustrate the comparison between the measured motor speed and the
WNN estimated speed in motor low speed operations. Fig. 16 shows a zoom-in view of a transient process. Figures 17 and 18 show the speed estimator validation in motor high speed operations.
Fig. 16 Speed transient process validation.
Fig. 13 Measured inverter duty ratio.
Fig. 17 Validation in high speed range.
Fig. 14 Validation in low speed range.
Fig. 18 Relative error of speed estimation in high speed.
V. CONCLUSION
Fig. 15 Relative error of speed estimation in low speed.
A recurrent WNN-based speed estimator for BLDC motors has been proposed. It was verified through simulations and validated through hardware experiments. The study results
show that the WNN-based speed estimator provides accurate speed outputs over a wide speed operating range as well as in motor transient processes. Further studies are in progress for better characterization of the low-speed performance of the proposed algorithm. Besides the high accuracy, the designed WNN-based speed estimator possesses three other valuable advantages. Firstly, it is not limited by the accuracy of motor parameters. Secondly, its structure is concise since WNNs contain only one and a half activation layers. Finally, compared with a conventional speed sensor, it decreases the maintenance and increase the reliability of motor systems. The proposed approach needs further study to fully exploit its intrinsic flexibility in hybrid simulation and experimental training of the network and in higher resolution on-line training if needed. VI. ACKNOWLEDGEMENT This work is supported by the US Office of Naval Research under contracts N00014-02-1-0623 and N00014-031-0434.
REFERENCES [1] K. Iizuka, H. Uzuhashi, M. Kano, T. Endo and K. Mohri, “Microcomputer control for sensorless brushless motor,” IEEE Trans. Industry Application, Vol. IA-21, pp. 595– 601, May 1985. [2] J.Shao, D.Nolan and T.Hopkins, “A novel direct back EMF detection for sensorless Brushless DC (BLDC) motor drives,” Proc. of the IEEE Applied Power Electronic Conference, pp. 33-38, 2002. [3] S. Ogasawara and H. Akagi, “An approach to position sensorless drive for brushless dc motors,” Proc. of the
IEEE Industry Applications Society Annual Meeting, pp. 443-447, 1990. [4] J. C. Moriera, “Indirect sensing for rotor flux position of permanent magnet AC motors operating in wide speed range,” Proc. of the IEEE Industry Applications Society Annual Meeting, pp. 401-407, 1994. [5] B. Terzic and M. Jadric, “Design and implementation of the extended Kalman filter for the speed and rotor position estimation of brushless DC motor”, IEEE Trans. Industry electronics, Vol. 48, Issue 6, pp. 1065-1073, Dec. 2001. [6] Q.Zhang and A. Benviste, “Wavelet Networks”, IEEE Trans. Neural Networks, Vol.3, no. 6, pp. 889-898, Nov. 1992. [7] C. Wang, B. K. Bose, V. Oleschuk, S. Mondal and J. O. P. Pinto, “Neural-network-based space-vector PWM of a three-level inverter covering overmodulation region and performance evaluation on induction motor drive”, Proc. of the 29th Annual Conference of the IEEE Industrial Electronics Society, Vol. 1, , pp. 1-6, Nov. 2003. [8] B. K. Bose, Modern power electronics and AC drives, Prentice Hall, 2001. [9] C. L. Lin, N. C. Shieh and P. C. Tung, “Robust wavelet neuro control for linear brushless motors”, IEEE Trans. Aerospace and Electronic Systems, Vol.38, Issue 3, pp. 918-932, Jul. 2002. [10] M. T. Hagan and M. Menhaj, “Training feedforward networks with the Marquardt algorithm,” IEEE Trans. Neural Networks, Vol. 5, no. 6, pp. 989-993, 1994. [11] The Virtual Test Bed, http://vtb.engr.sc.edu/ [12] A. Monti, D. Patterson, E. Santi, F. Ponci, etc, “An innovative low-cost drive for permanent magnet motors: a future energy challenge experience”, Proc. of the Nineteenth Annual IEEE Applied Power Electronics Conference and Exposition (APEC’2004),Vol. 2, pp.1137- 1142, 2004.
