System and method for extracting motor fault signatures using sparsity-driven joint blind deconvolution and demodulation

Abstract

A fault detection system is provided for extracting fault signatures from time-domain stator current signals of a motor under varying load operation by solving a joint blind deconvolution demodulation problem. The stator currents of the motor under varying operating conditions are modeled as stator currents under steady-state operating conditions affected by a system response vector and a load modulation vector. A proximal alternating linearization minimization method is used to solve the joint blind deconvolution demodulation problem, assuming that the spectrum of the desired signal is sparse. Motor fault detection is then performed using the recovered stator currents under steady-state operating conditions.

Description

[Technical Field]

[0001] The present invention relates to a system for extracting fault signatures of a motor operating under varying speed or varying load conditions based on motor current signature analysis. [Background technology]

[0002] Motor faults such as bearing faults, eccentricity faults and broken bar faults generate asymmetric rotating magnetic flux in the air gap between the stator and rotor, resulting in extra frequency components in the stator current. Therefore, when there is a motor fault, the frequency spectrum of the motor current will contain not only the operating frequency components but also fault signature frequency components. Depending on the type of fault, different faults will have different signatures. For example, in the case of an eccentricity fault, the frequency components will be f s ±mf r where f s is the frequency of the power supply, and f r is the rotor frequency in terms of rotational speed, m=0, 1, 2, …

[0003] Motor current signature analysis (MCSA) aims to extract fault signatures in the frequency domain and has been widely used to detect motor faults. Over the past few decades, various MCSA-based methods have been developed to detect various types of faults in motors. MCSA-based methods generally work well for motors operating in a steady state at a constant load and constant rotational speed. However, when the motor under test operates under varying conditions, such as a fluctuating load, the magnitude of the motor current will adaptively change to provide sufficient torque to drive the load. In such situations, fault signatures, which are typically much smaller in magnitude than the operating frequency components, may be corrupted by the distorted frequency spectrum of the motor current due to the operating fluctuations or may be hidden by noise and other systematic perturbations. This problem becomes more apparent in the early stages of fault development. Therefore, it is desirable to develop advanced methods for recovering fault signatures from fluctuating motor currents. Recent research has considered specific models of load fluctuations, such as phase modulation due to sinusoidally varying loads. In situations where the load varies in a specific pattern, time-frequency analysis can be used by analyzing multiple time-domain segments. However, time-frequency analysis inherently creates a trade-off between time and frequency resolution. Recently, minimum variance (MV) beamforming-based denoising techniques have been introduced into the field of motor fault detection to extract small fault signatures under varying load conditions. While MV beamforming-based denoising is effective at removing random noise, it cannot remove structural load fluctuations. Therefore, it is necessary to develop a new method and system that can extract motor fault signatures while removing structural load fluctuations. Summary of the Invention

[0004] In this work, we propose a sparsity-driven method for extracting fault signatures from time-domain stator current signals of motors under varying load operating conditions by solving a joint blind deconvolution and demodulation problem. Our contributions to the literature mainly focus on three aspects: First, we construct a physical model of the stator currents and combine the signal modulation with an unknown load signal and convolution with an unknown filter. Second, we cast the problem of recovering steady-state stator signals with fault signatures as a joint deconvolution and demodulation optimization problem with appropriate assumptions. Third, we develop a proximal alternating linearization minimization method to solve the problem, assuming that the spectrum of the sought signal is sparse. We demonstrate the utility of our method on signals collected from a real two-pole induction motor under varying load conditions.

[0005] Some embodiments of the present invention are based on the recognition that a fault detection system is provided for detecting faults in induction motors operating under varying operating conditions exhibiting varying load conditions, varying speed conditions, or a combination thereof.The fault detection system includes a sensor device connected to a power cable of an induction motor, the sensor device measuring stator currents of the induction motor under the varying operating conditions via the power cable, and the fault detection system further includes a memory storing a computer-implemented fault detection method and a signal processor executing steps of the computer-implemented fault detection method, the steps including forming a stator current vector by sampling the measured stator currents using a sampling frequency according to a time series; reducing noise in the stator current vector using a minimum variance (MV) beamforming method; formulating a joint blind deconvolution demodulation optimization problem based on the stator current vector under steady operating conditions, a response vector of the induction motor, a modulation vector under varying operating conditions, and a noise vector to satisfy the measured stator current vector with reduced noise; and The method includes the steps of: estimating the stator current vector for the steady operating condition, the response vector of the induction motor, the modulation vector for the fluctuating operating condition, and the noise vector by solving a lined deconvolution demodulation optimization problem; and extracting a fault signature from the clean stator current estimation vector, wherein the clean stator current estimation vector is generated by using the estimated response vector and the estimated stator current vector, and further including the steps of generating a control command to a controller of the induction motor and controlling the induction motor by sending the control command to the controller to reduce an operating speed of the induction motor to a safe level if at least one of the fault signatures is greater than a threshold; and generating and sending a normal command to the controller indicating normal operation of the induction motor if at least one of the fault signatures is not greater than the threshold.

[0006] According to some embodiments of the present invention, there is provided a computer-implemented method for detecting faults in an induction motor operating under varying operating conditions exhibiting varying load conditions, varying speed conditions, or a combination thereof, the method comprising the steps of measuring stator currents of the induction motor under the varying operating conditions via current sensors connected to a power cable of the induction motor, forming a stator current vector by sampling the measured stator currents using a sampling frequency according to a time series, reducing noise in the stator current vector using a minimum variance (MV) beamforming method, formulating a joint blind deconvolution optimization problem based on the stator current vector for a steady operating condition, a response vector of the induction motor, a modulation vector for a varying operating condition, and a noise vector to satisfy the measured stator current vector with reduced noise, and solving the joint blind deconvolution optimization problem to obtain the stator current vector for the steady operating condition, the response vector of the induction motor, and a modulation vector for a varying operating condition. and extracting a fault signature from a clean stator current estimation vector, wherein the clean stator current estimation vector is generated by using the estimated response vector and the estimated stator current vector, and the method further includes generating a control command to a controller of the induction motor and controlling the induction motor by sending the control command to the controller to cause the controller to reduce an operating speed of the induction motor to a safe level if at least one of the fault signatures is greater than a threshold, and generating and sending a normal command to the controller indicating normal operation of the induction motor if at least one of the fault signatures is not greater than the threshold.

[0007] The accompanying drawings are included to provide a further understanding of the invention, illustrate embodiments of the invention, and together with the description, serve to explain the principles of the invention. The drawings shown are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of embodiments of the present disclosure. [Brief explanation of the drawings]

[0008] [Figure 1A] FIG. 1 is a schematic diagram illustrating a system for in-situ operating and monitoring an induction motor according to one embodiment of the invention. [Figure 1B] FIG. 1 is a schematic diagram illustrating a system for operating and monitoring an induction motor online (on the cloud) according to one embodiment of the invention. [Figure 2] FIG. 1 is a block diagram illustrating a fault detection process for an induction machine operating under varying operating conditions in accordance with an embodiment of the present invention. [Figure 3] FIG. 10 illustrates a detailed illustration of a spectral analysis of stator currents measured from an induction machine operating under varying conditions using a blind deconvolution decomposition method, in accordance with an embodiment of the present invention. [Figure 4] FIG. 2 illustrates an example plot of stator current in the time domain of an induction machine operating under varying operating conditions in accordance with an embodiment of the present invention. [Figure 5] FIG. 2 is a schematic diagram illustrating estimation of an induction machine system response vector and modulation vector of a motor system according to an embodiment of the present invention. [Figure 6A] FIG. 10 shows an example plot of the stator current spectrum under one varying load condition (sinusoidal modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method. [Figure 6B] FIG. 10 shows an example plot of the stator current spectrum under one varying load condition (sinusoidal modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method. [Figure 7A]10A-10C show exemplary plots of stator current spectra under different varying load conditions (rectangular modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method. [Figure 7B] 10A-10C show exemplary plots of stator current spectra under different varying load conditions (rectangular modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method. [Figure 8A] 10A-10C show exemplary plots of stator current spectra under different varying load conditions (random modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method. [Figure 8B] 10A-10C show exemplary plots of stator current spectra under different varying load conditions (random modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method. DETAILED DESCRIPTION OF THE INVENTION

[0009] Various embodiments of the present invention will now be described with reference to the figures. It should be noted that the figures are not drawn to scale, and that elements of similar structure or function are represented by like reference numerals throughout the figures. It should also be noted that the figures are intended only to facilitate the description of particular embodiments of the present invention. They are not intended as an exhaustive description of the invention or as limitations on the scope of the invention. In addition, aspects described in connection with a particular embodiment of the present invention are not necessarily limited to that embodiment, but may also be practiced in any other embodiment of the present invention.

[0010] FIG. 1A is a schematic diagram of a fault detection system 100 for controlling and monitoring an induction motor, according to one embodiment of the invention.

[0011] Induction motor (system) 200 includes rotor assembly 102, stator assembly 104, main shaft 106, and two main bearings 108. In this example, induction motor 200 is a squirrel-cage induction motor. Induction motor 200 is connected to a load device 210, such as an electric fan or a transmission belt, attached to the rotor shaft.

[0012] Controller 220 is powered by power source 230 and can be used to monitor and control the operation of induction motor 200 in response to various inputs in accordance with embodiments of the present invention. For example, controller 220 connected to induction motor 200 can control the speed of the induction motor based on input received from fault detection system 100, which is configured to obtain data regarding the operating conditions of induction motor 200 from current sensor 150. According to certain embodiments, the electrical signal of current sensor 150 can be current from one or more of the phases of the induction motor. More specifically, if the induction motor is a three-phase induction motor, the current sensor senses current data from three phases of the three-phase induction motor. While certain embodiments of the present invention are described with respect to a polyphase induction motor, other embodiments of the present invention are applicable to other polyphase electric machines.

[0013] Some embodiments of the present invention describe a system for fault detection in an electric machine, such as an induction motor 200. The system configured for detection includes a fault detection system 100 for detecting the presence of a rotor 102 fault condition, including an eccentricity fault, in an induction motor assembly. In one embodiment, the fault detection system 100 is implemented as a subsystem of a controller 220. In an alternative embodiment, the fault detection system 100 is implemented using a separate processor. The fault detection system 100 may be a hardware circuit module operably connected to the controller 220. In some implementations, the fault detection system 100 and the controller 220 may share information. For example, the fault detection system 100 may reuse sensor data used by the controller to control the operation of the induction motor.

[0014] The fault detection system 100 further includes a processor 110, a memory 120, and a fault detection program 121, which is stored in the memory 120 when instructions of the program are executed by the processor 110. The fault detection system 100 also includes a sensor interface 130 configured to acquire signals from a current sensor 150. The interface 130 includes an analog / digital (A / D) and an A / D converter for data communication with the processor 110, the memory 120, the fault detection program, the user interface 140, and the current sensor 150. The processor 110 may be multiple processors, and the memory 120 may be a memory module including multiple memories. The processor 110 is configured to execute the fault signature detection program 121 by receiving, using the sensor interface 130, signals of stator current and rotor current of the induction motor 200 via the current sensor 150 connected to the fault detection system 100.

[0015] The user interface 140 is configured to connect to a keyboard and a display unit configured to show normal / fault status information of the induction motor 200 in response to the output of the fault detection module 200 .

[0016] 1B is a schematic diagram of a fault detection system operated by a user (operator) via a network 250 to control and monitor an induction motor, according to one embodiment of the invention. In this case, fault detection system 100 may be included in the operating system of the induction motor, which may be located at the user / operator's location or at a location other than the operator's location. If fault detection system 100 determines that a serious fault has occurred in induction machine (induction motor) 200 based on the sensor signal of current sensor 150 via network 250 during operation of induction machine (induction motor), fault detection system 100 can transmit a control signal indicative of the level of the fault occurring in induction machine 200 to controller 220 via network 250, causing controller 220 to reduce the operating speed of the induction machine to a safe level or even stop the induction machine. For example, controller 220 slows or stops the operation of induction machine 200 in accordance with the control signal indicative of the fault level determined by fault detection system 100, based on a pre-programmed algorithm (not shown) stored in the memory of controller 200. The control signal may indicate an operation status of the induction machine 200. For example, the control signal may indicate normal operation of the induction machine 200 or the level of a fault level that has occurred in the induction machine 200. The controller 220 includes a display monitor system (not shown) and a user interface (not shown) configured to input control commands to the induction machine 200, thereby allowing an operator to control the operation of the induction machine 200 by inputting control commands that cause the induction machine 200 to slow down or stop the operation of the induction machine 200. Furthermore, when the controller 220 receives the control signal from the fault detection system 100, the display monitor system, which has a monitor and a graphical user interface (GUI), may display the operation status of the induction machine 200 on a monitor of the display monitor system, where the operation status indicates normal operation of the induction machine 200 or the level of a fault detected by the fault detection system 100.

[0017] In some cases, calibration of current sensor 150 may be performed based on a sensor calibration program (not shown) stored in a memory of fault detection system 100. In some cases, network 250 may be an optical fiber network, a wireless network, an Internet network, or a data communication network consisting of a combination of at least two of an optical fiber network, a wireless network, and an Internet network.

[0018] This configuration may be a maintenance system managed by a user / client that operates induction machine system 20 installed separately from the induction machine location. For example, this system configuration may be used in a user-operated power generation system and a train system for controlling induction motors that drive trains. In some cases, when fault detection system 100 detects that the induction machine location is far from the fault detection system location, data communication between the induction machine location and the fault detection system location is performed over a network. Network 250 may be an optical fiber network, a wireless network, an Internet network, or a data communication network consisting of a combination of at least two of an optical fiber network, a wireless network, and an Internet network.

[0019] Additionally, the fault detection system is included as part of the user's maintenance system. If the fault level (e.g., eccentricity) determined for the induction machine is equal to or greater than a critical threshold level, the fault detection system generates a control command and sends the control signal of the control command via network 250 to controller 220 of induction machine 200 using sensor interface / control interface 130 to slow down or stop operation of induction machine 200.

[0020] 2 is a block diagram of a fault detection process for an induction machine operating under varying operating conditions, according to an embodiment of the present invention. When motor 200 is operating with a varying load 210, sensor interface 130 monitors the motor current by continuously measuring the stator current using current sensor 150. Through a monitoring motor current process 410, time-domain current 415 is measured and saved at regular time intervals for spectral analysis 420 using the proposed blind deconvolution decomposition spectral analysis method, resulting in a frequency-domain spectrum 425 corresponding to the constant operating conditions, with the effects of the varying load removed. Based on the frequency-domain spectrum, a fault signature at a certain frequency is extracted, e.g., frequency components at approximately 30 Hz and approximately 90 Hz for an eccentricity fault when the operating frequency is 60 Hz. The magnitude of the fault signature is then used to determine the presence or absence of a fault (440). If the fault component of the fault signature magnitude is greater than a certain value (threshold amplitude), e.g., approximately 1 / 100 (-40 dB) of the operating frequency component at 60 Hz, the fault detection system determines that a fault has been detected, and then a control command (control signal) 450 is sent to the control circuit of the induction motor's controller to slow down or stop the motor for safety purposes. Furthermore, the control command can indicate various fault types, various fault levels, and fault locations based on the detected frequency component and the magnitude of the fault signature signal. For example, if the current intensity at the bearing's fault signature frequency increases, a bearing fault is detected. Otherwise, the fault detection system continues to monitor the stator current. In some cases, if the fault detection system does not detect a fault based on the threshold, the fault detection system generates a normal command indicating normal operation of the induction motor and sends it to the controller's display, causing the display's GUI to display a sign / text indicating normal operation status.

[0021]

number

[0022]

number

[0023]

number

[0024]

number

[0025]

number

[0026]

number

[0027]

number

[0028]

number

[0029]

number

[0030]

number

[0031]

number

[0032] To make our method robust to noise, we preprocess the measurements using minimum variance (MV) beamforming noise reduction. In the minimum variance beamforming method, the time-domain stator current under test is segmented into multiple overlapping time series with a fixed time shift. Each sequence is treated as an independent measurement of a virtual current sensor. Because the time difference between adjacent sequences is identical, these virtual sensors form a linear virtual sensor array. The spectral analysis problem is then transformed into a linear array beamforming problem in the frequency domain. The minimum variance beamforming method minimizes the spectral noise variance for each frequency with respect to the weights for the frequency components of the separate sequences. Therefore, although it is effective in reducing the noise in the measured current signal, it cannot remove the structured artifacts caused by the fluctuating load.

[0033]

number

[0034]

number

[0035]

number

[0036]

number

[0037]

number

[0038] Using the obtained optimization variable initialization, estimates of the variables are obtained by the following optimization problem, which represents joint blind deconvolution and demodulation: A similar problem has been solved using alternating minimization techniques from the perspective of blind deconvolution.

[0039]

number

[0040] 6A and 6B are example plots of the stator current spectrum under one varying load condition (sinusoidal modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method, respectively.

[0041] These plots show that MV beamforming is successful in denoising the signal, as expected, because it effectively destroys the noise by averaging the random phase component of the noise while actively adding structured periodic signal sections. However, as expected, MV beamforming is unable to remove artifacts caused by load variations because these artifacts are structured. The proposed technique is effective in removing the effects of load variations. The eccentricity fault signature is obtained by subtracting the corresponding rotor frequency f r = 30Hz, it can be clearly seen around 30Hz and 90Hz, which is consistent with the MSCA-based fault detection model.

[0042] 7A and 7B are example plots of the stator current spectrum under another varying load condition (rectangular modulation) using Fourier transform and the proposed blind deconvolution decomposition spectral analysis method, respectively. Similarly, the proposed approach significantly reduced the modulation effect due to the varying load.

[0043] 8A and 8B are example plots of stator current spectra under different varying load conditions (random modulation) using Fourier transform and the proposed blind deconvolution decomposition spectrum analysis method, respectively. The proposed algorithm reduces the effects of modulation from the time-domain plot. In the frequency-domain plot, the proposed approach is effective in reducing the effects of random convolution and modulation as well as noise, and can retain the eccentricity fault signature at approximately 30 Hz and 90 Hz, respectively.

[0044] The above-described embodiments of the present invention can be implemented in any of numerous ways. For example, embodiments may be implemented using hardware, software, or a combination thereof. If implemented in software, the software code may execute on any suitable processor or collection of processors, whether provided on a single computer or distributed among multiple computers. Such a processor may be implemented as an integrated circuit, with one or more processors within an integrated circuit component. However, a processor may be implemented using circuitry in any suitable format.

[0045] Additionally, embodiments of the present invention may be implemented as a method, an example of which has been provided above. The operations performed as part of the method may be ordered in any suitable manner. Thus, embodiments may be constructed in which operations are performed in an order different from that illustrated, which may include performing some operations simultaneously although shown as sequential operations in the exemplary embodiment.

[0046] The use of ordinal terms such as "first" and "second" to modify claim elements in the claims does not, in itself, imply a priority, precedence, or order of one claim element relative to another, nor does it imply a chronological order in which method actions are performed; it is merely used as a label to distinguish one claim element having a particular name from another element having the same name (but using ordinal terms) to distinguish between claim elements.

[0047] Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention.

[0048] Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.

Claims

1. 1. A fault detection system for detecting faults in an induction motor operating under varying operating conditions exhibiting varying load conditions, varying speed conditions, or a combination thereof, comprising: a sensor device connected to a power cable of the induction motor, the sensor device measuring a stator current of the induction motor under the varying operating conditions via the power cable; and the fault detection system further comprising: a memory storing a computer-implemented fault detection method; a signal processor for executing the steps of the computer-implemented fault detection method, the steps comprising: forming a stator current vector by sampling the measured stator currents using a sampling frequency according to a time sequence; reducing noise in the stator current vectors using minimum variance (MV) beamforming; Formulating a joint blind deconvolution demodulation optimization problem based on the stator current vector under steady-state operating conditions, a response vector of the induction motor, a modulation vector under fluctuating operating conditions, and a noise vector to satisfy the measured stator current vector with reduced noise; estimating the stator current vector for the steady-state operating condition, the response vector of the induction motor, the modulation vector for the varying operating condition, and the noise vector by solving the joint blind deconvolution demodulation optimization problem; and extracting a fault signature from a clean stator current estimate vector, wherein the clean stator current estimate vector is generated by using the estimated response vector and the estimated stator current vector, said step further comprising: A fault detection system comprising the steps of: generating a control command to a controller of the induction motor when at least one of the fault signatures is greater than a threshold value; and controlling the induction motor by sending the control command to the controller to cause the controller to reduce the operating speed of the induction motor to a safe level; and generating and sending a normal command to the controller when at least one of the fault signatures is not greater than the threshold value, indicating normal operation of the induction motor.

2. The fault detection system of claim 1 , wherein the signal processor uses minimum variance (MV) beamforming to denoise the stator currents before calculation.

3. 2. The fault detection system of claim 1, wherein the controller monitor displays an operational signature of the fault in response to receiving the control signal.

4. The fault detection system of claim 1 , wherein the control signal indicates a fault level and location occurring in the induction motor.

5. The fault detection system of claim 1 , wherein electromagnetic torque is approximated by the stator or rotor current of the induction motor during the varying operating conditions.

6. 6. The fault detection system according to claim 5, wherein a half-width of the modulation signal relating to the electromagnetic torque is set to approximately half the operating frequency of the induction motor.

7. The fault detection system of claim 1 , wherein the stator current vector for the steady-state operating condition is initialized as a linear combination of harmonics of the operating frequency of the induction motor.

8. The fault detection system of claim 1 , wherein the fault signature is determined by running a spectrum analysis program on the cleaned stator current estimate vector.

9. The solution is initializing the response vector and the modulation vector; 2. The fault detection system of claim 1, further comprising the step of iteratively updating the stator current vector, the response vector, and the modulation vector until the updates converge.

10. The fault detection system of claim 1 , wherein electromagnetic torque is approximated by the stator and rotor currents of the induction motor during the varying operating conditions.

11. 1. A computer-implemented method for detecting faults in an induction motor operating under varying operating conditions exhibiting varying load conditions, varying speed conditions, or a combination thereof, comprising: measuring a stator current of the induction motor under the varying operating conditions via a current sensor connected to a power cable of the induction motor; forming a stator current vector by sampling the measured stator currents using a sampling frequency according to a time sequence; reducing noise in the stator current vectors using minimum variance (MV) beamforming; Formulating a joint blind deconvolution demodulation optimization problem based on the stator current vector under steady-state operating conditions, a response vector of the induction motor, a modulation vector under fluctuating operating conditions, and a noise vector to satisfy the measured stator current vector with reduced noise; estimating the stator current vector for the steady-state operating condition, the response vector of the induction motor, the modulation vector for the varying operating condition, and the noise vector by solving the joint blind deconvolution demodulation optimization problem; and extracting a fault signature from a clean stator current estimate vector, wherein the clean stator current estimate vector is generated by using the estimated response vector and the estimated stator current vector, the method further comprising: If at least one of the fault signatures is greater than a threshold, generating a control command to a controller of the induction motor and controlling the induction motor by sending the control command to cause the controller to reduce the operating speed of the induction motor to a safe level, and if at least one of the fault signatures is not greater than the threshold, generating and sending a normal command to the controller indicating normal operation of the induction motor.

12. The method of claim 11 further comprising denoising the stator currents using minimum variance (MV) beamforming prior to calculation.

13. The method of claim 11 , wherein the controller monitor displays a faulty operating signature on a display of the controller in response to receiving the control command.

14. The method of claim 11 , wherein the control command indicates a fault level and a fault location occurring in the induction motor.

15. The method of claim 11 , wherein electromagnetic torque is approximated by the stator or rotor current of the induction motor during the varying operating conditions.

16. 16. The method of claim 15, wherein a half-width of the modulation signal for the electromagnetic torque is set to approximately half the operating frequency of the induction motor.

17. The method of claim 11 , wherein the stator current vector for the steady-state operating condition is initialized as a linear combination of harmonics of an operating frequency of the induction motor.

18. The method of claim 11 , wherein the fault signature is determined by running a spectrum analysis program on the cleaned stator current estimate vector.

19. The solution is initializing the response vector and the modulation vector; 12. The method of claim 11, further comprising the step of iteratively updating the stator current vector, the response vector, and the modulation vector until the updates converge.

20. The method of claim 11 , wherein electromagnetic torque is approximated by the stator and rotor currents of the induction motor during the varying operating conditions.

Images