Article Contents
Erphan A. Bhuiyan, Md. Maeenul Azad Akhand, Sajal K. Das, Md. F. Ali, Z. Tasneem, Md. R. Islam, D. K. Saha, Faisal R. Badal, Md. H. Ahamed, S. I. Moyeen. A Survey on Fault Diagnosis and Fault Tolerant Methodologies for Permanent Magnet Synchronous Machines. International Journal of Automation and Computing. doi: 10.1007/s11633-020-1250-3
Cite as: Erphan A. Bhuiyan, Md. Maeenul Azad Akhand, Sajal K. Das, Md. F. Ali, Z. Tasneem, Md. R. Islam, D. K. Saha, Faisal R. Badal, Md. H. Ahamed, S. I. Moyeen. A Survey on Fault Diagnosis and Fault Tolerant Methodologies for Permanent Magnet Synchronous Machines. International Journal of Automation and Computing. doi: 10.1007/s11633-020-1250-3

A Survey on Fault Diagnosis and Fault Tolerant Methodologies for Permanent Magnet Synchronous Machines

Author Biography:
  • Erphan A. Bhuiyan is currently pursuing the B. Sc. degree in mechatronics engineering from Rajshahi University of Engineering & Technology (RUET), Bangladesh. He is currently working on a formula student electric vehicle project. His research interests include control theory and applications, fault diagnosis, fault tolerant control, electrical machines, mechatronics systems and electric vehicle technologies. E-mail: (Corresponding author)ORCID iD: 0000-0001-6770-8300

    Md. Maeenul Azad Akhand is currently pursuing the B. Sc. degree in mechatronics engineering from Rajshahi University of Engineering and Technology, Bangladesh. He is currently working on a formula student electric vehicle project. His research interests include fault diagnosis and tolerant control of electric vehicles, automotive engineering, robust control of mechatronics system and virtual power plant. E-mail:

    Sajal K. Das received the Ph. D. degree in electrical engineering from University of New South Wales, Australia on 2014. In May 2014, he was appointed as a research engineer in National University of Singapore (NUS), Singapore. In January 2015, he joined in Department of Electrical and Electronic Engineering of American International University-Bangladesh (AIUB) as an assistant professor. He continued his work at AIUB until he joined in Department of Mechatronics Engineering of Rajshahi University of Engineering and Technology, Bangladesh as a lecturer on September, 2015. He is currently working as an assistant professor in RUET. His research interests include control theory and applications, mechatronics system control, robotics, and power system control. E-mail:

    Md. F. Ali is currently working as an assistant professor from Department of Mechatronics Engineering, Rajshahi University of Engineering and Technology, Bangladesh. His research interests include power electronics, control theory and applications, mechatronics, and artificial intelligence.E-mail:

    Z. Tasneem is currently working as an assistant professor at Department of Mechatronics Engineering, Rajshahi University of Engineering and Technology, Bangladesh from February, 2018. Her research interests include power electronics, control system, mechatronics, and aerodynamics.E-mail:

    Md. R. Islam is currently working as a lecturer in Department of Mechatronics Engineering, Rajshahi University of Engineering and Technology, Bangladesh. Previously, he was appointed as the head of Department of Mechanical Engineering, Bangladesh Army University of Science and Technology (BAUST), Bagladesh from February 2015 to August 2018. His research interests include mechatronic systems design, robotics, control system and renewable energy. E-mail:

    D. K. Saha received the B. Sc. degree in mechanical engineering from Rajshahi University of Engineering and Technology, Bangladesh in 2012. He is pursuing the M. Sc. degree in mechanical engineering in RUET. Currently, he is working as a lecturer in Department of Mechatronics Engineering, RUET. Before, he was in Walton Hi-Tech Industries Ltd. as research & development engineer in Refrigerator Cooling Design Section. His research interests include vibration-based condition monitoring, machine learning, and mechatronics.E-mail:

    Faisal R. Badal received the B. Sc. degree in mechatrans engineering from Rajshahi University of Engineering and Technology, Bangladesh in 2017. He is currently working as a lecturer in Department of Mechatronics Engineering, Rajshahi University of Engineering and Technology, Bangladesh. His research interests include smartgrid, artificial intelligence, machine learning, natural language processing, and robotics.E-mail:

    Md. H. Ahamed received the B. Sc. degree in mechatronics engineering from Rajshahi University of Engineering and Technology, Bangladesh in 2017. He is currently pursuing the M. Sc. degree in engineering in the Department of Computer Science and Engineering, Rajshahi University of Engineering and Technology, Bangladesh. He is working as a lecturer in Department of Mechatronics Engineering, Rajshahi University of Engineering & Technology. His research interests include machine vision, artificial intelligence, machine learning, robotics, and image processing.E-mail:

    S. I. Moyeen received the B. Sc. degree in computer science and engineering from Rajshahi University of Engineering and Technology, Bangladesh in 2017. In November 2019, she joined Department of Mechatronics Engineering of Rajshahi, University of Engineering and Technology, Bangladesh as a lecturer. Her research interests include data mining and big data, web development, IoT, cloud computing, image processing, and artificial intelligence.E-mail:

  • Received: 2020-04-28
  • Accepted: 2020-08-14
  • Published Online: 2020-11-11
  • This paper presents a comprehensive survey of fault diagnosis and fault tolerant approaches for permanent magnet synchronous machines (PMSM). PMSMs are prominent in the pervading usage of electric motors, for their high efficiency, great robustness, reliability and low torque inertia. In spite of their extensive appliance, they can be quite non-resilient and inadequate in operation when faults appear in motor drive apparatus such as inverters, stator windings, sensors, etc. These may lead to insulation failure, torque fluctuations, overcurrent or even system collapse. On that account, fault diagnosis and fault tolerant methods are equipped to enhance the stability and robustness in PMSMs. Progressive methodologies of PMSM fault diagnosis and tolerance are classified, discussed, reviewed and compared in this paper, beginning with mathematical modeling of PMSM and then scrutinizing various fault conditions in PMSMs. Finally, the scope of research on the topic is highlighted. The contribution of this review is to emphasize optimistic schemes and to assist researchers with the latest trends in this field for future directions.
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A Survey on Fault Diagnosis and Fault Tolerant Methodologies for Permanent Magnet Synchronous Machines

Abstract: This paper presents a comprehensive survey of fault diagnosis and fault tolerant approaches for permanent magnet synchronous machines (PMSM). PMSMs are prominent in the pervading usage of electric motors, for their high efficiency, great robustness, reliability and low torque inertia. In spite of their extensive appliance, they can be quite non-resilient and inadequate in operation when faults appear in motor drive apparatus such as inverters, stator windings, sensors, etc. These may lead to insulation failure, torque fluctuations, overcurrent or even system collapse. On that account, fault diagnosis and fault tolerant methods are equipped to enhance the stability and robustness in PMSMs. Progressive methodologies of PMSM fault diagnosis and tolerance are classified, discussed, reviewed and compared in this paper, beginning with mathematical modeling of PMSM and then scrutinizing various fault conditions in PMSMs. Finally, the scope of research on the topic is highlighted. The contribution of this review is to emphasize optimistic schemes and to assist researchers with the latest trends in this field for future directions.

Erphan A. Bhuiyan, Md. Maeenul Azad Akhand, Sajal K. Das, Md. F. Ali, Z. Tasneem, Md. R. Islam, D. K. Saha, Faisal R. Badal, Md. H. Ahamed, S. I. Moyeen. A Survey on Fault Diagnosis and Fault Tolerant Methodologies for Permanent Magnet Synchronous Machines. International Journal of Automation and Computing. doi: 10.1007/s11633-020-1250-3
Citation: Erphan A. Bhuiyan, Md. Maeenul Azad Akhand, Sajal K. Das, Md. F. Ali, Z. Tasneem, Md. R. Islam, D. K. Saha, Faisal R. Badal, Md. H. Ahamed, S. I. Moyeen. A Survey on Fault Diagnosis and Fault Tolerant Methodologies for Permanent Magnet Synchronous Machines. International Journal of Automation and Computing. doi: 10.1007/s11633-020-1250-3
    • The consistent increase in greenhouse gas emanations has led to the use of electric motors for numerous purposes since the time they were invented quite a while back[1, 2]. Around then, the safety features of the motors only had simple apparatus, such as fuse elements[3]. With the advancement of human civilization, stability and reliability have become more crucial in motor control systems[4]. However, electric motors often prompt different faults, causing degradation of performance and damage in healthy components. In complex systems, the consequences of faults can be catastrophic. It is extremely important for the systems to be stable and fault resilient, which further may not lead to any enormous situation[5].

      Over the previous decades, permanent magnet synchronous motors (PMSMs) have acquired significant ubiquity on account of high efficiency, high power to volume ratio, high torque to current ratio and higher power density[6]. They are utilized in diverse applications in the field of automotive vehicles[7], subway applications[8], aeronautics[9], turbines[10], industry applications[11] and so forth. In these applications, an unanticipated fault of the PMSM could lead to high substitution cost or even system failure. The control systems should have the capability to determine the faults and retain the performance and stability as the faults may occur by sensor errors like voltage sensor, current sensor, pressure sensor[12], or even broken plants like open phase motors[13, 14]. To increase the performance and reliability of systems, a diagnostic and monitoring approach is required that can provide maintenance and avoid system faults. These sorts of systems are frequently known as fault tolerant control system (FTCS)[15]. The basic architecture of FTCS is represented in Fig. 1.

      Figure 1.  Basic architecture of FTCS

      Faults in systems are unpermitted variations of parameters from the actual values acceptable from the systems[16]. In FTCSs, faulty conditions are observed up to a definite level of reliability and undergo a preventive maintenance. These systems are classified into two major types: active and passive[17]. The active FTCS (AFTCS) can manage various kinds of faults and results in ideal execution, though this is sensitive at the outcome which is acquired from the fault detection and isolation (FDI) module, like it can permit an incorrect information with immoderate noise. FDI is a reconfiguration system and an adjustable controller. In AFTCS, FDI performs a very essential role in detecting and isolating the parts which are faulty, in order to gain adaptability in faulty conditions[18]. For these instantaneous moves made by the controller, it is named as “active”. For nonlinear systems, it is very difficult to design because of these unpredictability. In different circumstances, passive fault tolerant control (PFTC) is faster than AFTCS, as it has no FDI unit and is computationally less complex than AFTCS. It is sufficiently quick to act instantly in any unusual condition though it can only accommodate a limited number of faults[16, 19].

      The strategy of AFTCS involves two parts: fault diagnosis and fault tolerant control (FTC). The fault diagnosis comprises of three steps, i.e., fault determination, fault isolation and fault estimation. Fault determination checks whether faults have occurred or not. It also detects the time at which the system was subject to faults. Fault isolation finds out the location of faults and fault estimation identifies the characteristics of faults. Thereafter, when the malfunctions are completely illustrated by fault diagnosis, the fault tolerance takes place. The FTC responds within an appropriate time and recovers the faulty components to a stabilized state. It examines with the interactivity between the system/plant and a redundancy controller design. The main novelty of it is to make an intelligent use of redundancies involved in the system, to enhance the system availability. Both the fault diagnosis and FTC is executed with various types of techniques. These strategies are imposed in critical systems in industries, where production loss is not granted, and persistent operation is compulsory. An embedded system approach is required for the diagnosis and controller design to accomplish the tasks on a single board computer, where the algorithms can be implemented and the measured data will be stored[20]. This board will be directly connected to the plant and act as the control unit.

      The FTCs have some distinctive type of properties that are not available in traditional controllers. The traditional controllers conventionally alter the dynamic effects of a system, where elimination of faulty situations barely occurs. Controllers designed for FTCs are intended to preserve safe working points of confinement and moderate the impacts of framework/part glitches, to ensure the security of the whole framework. It keeps minor faults from becoming serious issues[21]. Failure investigation and redundancy setup must be done before designing. The mode of failure commits to the route where potential errors may happen, especially errors that may influence the plant or administrators[22]. The controller acts as a confine control framework which does not just perform customary control capacities, but is also capable retaining the system protection in case of faults happening in the system. Any controller for FTCs can be seen as involving two fundamental parts: a controller and related redundant system elements[23]. Designing an efficacious controller for FTCs requires a definite investigation of the physical structure it will cooperate with, including the structure′s failure modes, their effect, and the various remedial control actions. Conventionally, redundancy is the key for the controller, which becomes an effective constant redundancy administration process.

      FTC of PMSMs is quite a difficult endeavour because of the diversity of various fault conditions. Fault types, fault locations, winding connections make it more challenging to conduct[24]. To achieve high efficiency and reliability, multiphase PMSMs can be used over conventional three phase PMSMs[25] for FTC. These include many advantages such as reduction of amplitude, increment of torque pulsation frequency and stabilization of stator current[26]. Research can be found for five-phase and six-phase PMSMs in many literature[13, 21, 27-30].

      Over recent decades, the increasing need for stability and reliability in real world systems have inspired noteworthy research in FTCS. A discussion about FTCS review papers are contributed by Gao et al.[31], along with descriptive explanations of model based and signal based fault diagnosis techniques. In [16], the latest advancements in FTCSs are reviewed. And in [17], a relative analysis on active and passive FTCSs are reported, while only active FTCSs are reviewed in [32]. Many review papers were written in regards of FTCS applications such as multi-agent systems[33], spacecraft altitude[34], fuel cell systems[23], safety issues[35], adjustable speed drives[36], power converters[37], switched reluctance machines[4], induction machines[25], etc. A review paper on PMSM was also presented by [38] on active FTCSs concerning doublesalient and flux switching machines, secondary windings, thermal protection, transversal flux machines. In the aim of diversity, our paper provides a detailed overview and comparison of fault diagnosis and fault tolerant methods applied for PMSMs, classified in an organized manner.

      The remainder of our paper is structured as follows: A mathematical explanation of PMSM is described in Section 2. A brief overview of fault classifications is made in Section 3. Fault diagnosis methodologies and fault tolerant methodologies are described in Sections 4 and 5 respectively. A concluding part and the future trends that can be applied in FTCS are depicted in Section 6.

    • PMSMs are brushless machines with sinusoidal flux distribution, where the stator voltage should be sinusoidal and tuned with the rotor position. The frequency and amplitude have to be aligned with the rotor speed. The most usual method in modelling PMSM is a Direct-Quadrature (d-q) axis model, which is used to mathematically rationalize three-phase circuits. It can be used to analyze transient and stable state performance of PMSMs. The model of PMSM is typically made out of equations based on voltage, flux linkage, torque and mechanical movement. The voltage equations of PMSM can be expressed as

      $ e_d = N_p \phi_d - \phi_d \omega + R_a i_d $


      $ e_q = N_p \phi_q - \phi_q \omega + R_a i_q $


      where $ e_d $ and $ e_q $ are d-q axis control voltages, $ i_d $ and $ i_q $ are d-q axis stator currents, $ N_p $ is pole pair number of PMSM, $ \omega $ is angular speed and $ R_a $ is armature resistance. The flux linkages $ \phi_d $ and $ \phi_q $ are defined as

      $ \phi_d = L_d i_d + \phi_P $


      $ \phi_q = L_q i_q $


      where $ L_d $ and $ L_q $ are inductances, $ \phi_P $ is permanent-magnet flux linkage. The equations for torque produced and mechanical movement are as follows, where $ T $ is torque produced for electromagnetism, $ M $ is moment of inertia and $ T_l $ is load torque:

      $ T = N_P (i_q \phi_d - i_d \phi_q) $


      $ \frac{M}{N_P}\times \frac{{\rm{d}}\omega}{{\rm{d}}t} = T - T_l . $


      Nevertheless, the mathematical model is an inexact depiction of the real physical model of PMSM. It cannot accurately define the variety of faults in the system.

    • On the whole, electrical machine measurements associated with torque, rotational speed, power ratings are subjected to velocity and shaft displacement (mechanical), current and voltage ratings (electrical). Digital encoders, rotor position sensors and tachometers are used to govern the angular displacement and angular velocity of the shaft respectively, sensor layers with stator winding and encoder are illustrated in Fig. 2. Shunt resistors and hall-effect sensors are accustomed to measure the current, and isolated Op-Amps, magnetic flow meters are used to measure the voltage[39]. Within all these measurements, there are constantly various noises or faults in the form of varied parameters, as a result of machine resilience. According to the format of these faults, they can be predominantly classified into mechanical faults, electrical faults and sensor faults. Fault classification of PMSM is represented in Fig. 3.

      Figure 2.  Sensor layers with PMSM stator winding

      Figure 3.  Classification of faults in PMSMs

    • Mechanical faults such as air-gap eccentricity[40], rotor misalignment[41], bearing faults[42], brush wear, etc. often occur because of rough environments. These faults alter the attributes of behavior of systems. The air-gap eccentricity is identified with a state of inconsistent air hole, which exists between the stator and rotor[3]. Due to eccentricity, the subsequent uneven radial forces can even prompt stator to rotor rub. There are two sorts of eccentricity: static and dynamic[39]. In static, the outspread air-gap length state is specified in space and is negligible. This can be brought by inaccurate situating of the rotor or stator at the amassing stage. In dynamic, the position of least air gap revolves with the rotor as the rotor core is not at the center of revolution. The reasons for this type of faults are mechanical resonance, rotor thermal bowing, bent shaft, etc. The existence of static and dynamic eccentricity can be determined by the following equations respectively, where $ P_s $ and $ P_d $ are permeance of static and dynamic eccentricity, $ k_s $ and $ k_d $ are integer numbers, $ P $ is pole pair and $ \omega $ is angular supply frequency[43]:

      $ {P_s} = \sum\limits_{{k_s} = 0}^\infty {{P_k}_s} \cos ({k_s}\theta ) $


      $ {P_d} = \sum\limits_{{k_d} = 0}^\infty {{P_k}_d} \cos [{k_d}(\frac{{\omega t}}{P} - \theta )] .$


      Broken rotor bar faults occur in PMSMs, when the bars get separated from the end rings and impede the current flow that is induced by stator field. The current flow is needed to generate a magnetic force to produce a torque. Hence, for the temperature difference, rising unbalanced force and insufficiency of magnetic field, the performance of the motor gets delimited. The root causes behind this fault are sparks, thermal overload, electromagnetic vibration, excessive shaft torque, loose coating of mechanical parts, etc. The frequency spectrum of broken rotor bar session can be represented by

      ${\psi _{{b_b}}} = [(\frac{q}{n})(1 - s) \pm s]\psi $


      where ${\psi _{{b_b}}} $ are identical broken bar frequencies, $ n $ is number of pole pairs, $ q = 1,2,3 $, slip is denoted by $ s $ and $ \psi $ is supply frequency.

      As one of the keystone element in supporting PMSM shafts, the bearings are also vulnerable and may cause serious failure[44]. Lubrication, contamination and erosion can lead to fatigue damages in bearings[45, 46].

    • According to a research, above 90% of faults are electrical faults[47]. Various electrical faults may occur in PMSMs such as open or short circuit faults of power devices, windings and inverter legs, unstable voltages, armature faults, insulation failures, etc. Open circuit faults are caused by excessive heating of components, effect of overwhelmed forces and short circuit faults are caused by overloads and high temperature[48].

      Unstable or unbalanced voltages appear when the magnitudes of phase voltages and line voltages are different. It reduces the lifespan of the motor and also decreases the performance. According to the National Electrical Manufactures Association (NEMA) standard, the temperature in the stator windings starts to rise, which in the end may prompt the winding protection to melt if the unstable voltages hit 5%. The reason for stator winding faults is short circuits between a phase winding and the ground or between two phases. Stator and armature faults are identified for protection and degeneration inside the motor. They are named as 1) turn to-turn short circuit inside a coil, 2) open circuit in one phase, 3) phase-to-phase short, 4) phase ground short and 5) short circuit between coils of a similar phase.

    • Actuators used in electrical machines employ power semiconductor devices in order to gain a variable alternating current (AC) or direct current (DC) voltage which is used to operate as well as to control the motor. AC to DC, DC to AC or DC to DC power transformation is done by these actuators as they are powered by three phase supplies. Generally, the actuator faults are also related to power supply or the semiconductor devices[49].

      Power semiconductor and DC link capacitor′s short circuit are enough to burn out the fuses which are used to protect the system. Faulty power semiconductors also shorten the ability and efficiency of the motor drives. Faults related to power supply can be modelled as[50, 51]

      $ y(t) = y_o(t) + \psi_y(t) $


      where the definite control signal is $ y $, nominal is $ y_o $, induced change due to fault is $ \psi_y (t) $.

    • The drive system could be tripped due to instant overcurrent, which is caused by rotor position sensor faults. Because of decoupling failure, this type of tripping problem happens in a high speed or heavy load conditions in d and q axes controllers. The torque linearity is not conserved in low speed or light load condition and as a result the faults steer to overcurrent trip. These faults can be detected by applying various sensorless control algorithms[52]. However, it is not easy to find out the slippage fault of the position sensor. The high frequency injection (HFI) method can be applied in a low speed condition, having a very small back electromotive force (EMF) in order to give precise rotor position[53, 54]. The seamless alteration is based on back EMF information which is estimated in the condition of high speed with an immense inertia. As the motor is capable of running, due to the inertia of the system, when the inverter is tripped by the faults of the position sensor and the back EMF also estimated by the observer, the sensorless control algorithm can be applied by using estimated back EMF without stopping the motor[55]. The sensor fault detection scheme is represented in Fig. 4.

      Figure 4.  Detection scheme for rotor position sensor fault with phase shift compensation[60]

      Faults found in voltage sensors are less difficult to detect than other sensors, as the nominal value is not approximately zero. As the power is very small itself, it is easy to detect the faults at light load and low speed operations. The nominal value of the controller replaces the measured voltage of sensor in the case when a certain threshold is exceeded by the power balance. Consequently, for fruitful activity without a voltage sensor, it is significant that the system constancy ought to be ensured with a half variety of controller gains.

      The overcurrent failure of the system occurs due to various faults of current sensor[56]. It steers to the irremediable faults of semiconductor inside the inverter if it has no protection system at the gate drive circuit. The worst performance for regulating torque happens when the offset and scaling faults are big[57]. Faults regarding offset and scaling can be detected when the machine stops, yet it is reached above a general level of overcurrent trip that would happen at heavy load operations.

    • Deviations from the actual state of plant parameters, known as faults, are determined with fault diagnosis methods concerning the determination, isolation, identification and estimation of faults. In essence, fault determination undertakes the foremost task of diagnosis, to identify the existence of faults in a system. Algorithms used in fault detection, spots the deviations of system behaviors from theoretical values of the system, using a model of the system. The time of detection is also determined during this process. When the data of the faults are determined, fault isolation and fault estimation can be accomplished. Fault isolation signifies the component where the faults are occuring. This step discerns the actual location of the faults. The final step identifies the fault characteristics and estimates the magnitudes of the faults.

      Advanced methods of fault diagnosis can be classified into model based methods, signal based methods, artificial intelligence methods and hardware based methods. The utilization of these methods means that diagnosis of various kinds of faults can be achieved effectively.

    • Model based fault diagnosis was introduced by Beard in 1971[58]. In model based methods, the availability of a certain model of the system is needed. This can be acquired by mathematical equations and physical components or data driven techniques. Algorithms monitor the input and output relation of the system, depending upon the system model. The process of model-based fault diagnosis is executed within two phases: residual generation and evaluation. Normally, a signal is produced with the input and output computations of the system. And then the signal undergoes the evaluation process, where the signal is examined for fault prospects. The basic scheme of model based diagnosis is represented in Fig. 5.

      Figure 5.  Basic scheme of model based diagnosis

      Models can be of deterministic or stochastic continuous variable systems, discrete event systems, distributed systems and hybrid systems. Continuous variable system models are sets of differential equations. When random inputs are featured, these systems are stochastic and when no randomness is involved, they are deterministic. For diagnosis of stochastic and deterministic models, the basic scheme of the model based method shown in Fig. 5 is used. In fault free continuous variable systems, the nominal value of the residual is zero, or close to zero. If a fault has occurred, the magnitude of the residual deviates from zero. The task of the evaluation part is to determine the magnitude and pattern of the residual, varying from zero to non-zero values. For discrete event systems, models are described by input and output sequences that change continuously. The fault diagnosis of these models determines whether a fault has arisen or not, analyzing the observable discrete event sequence[59]. For all of these types of models, a diversity of techniques are approached.

    • A sliding mode variable structure is able to replace the control circuit of a conventional observer. Great robustness of variable systems is maintained by sliding mode observer (SMO)[60, 61]. The dynamic performance of the entire system relies on the sliding surface when the error happens with the sliding mode and decides whether the system will fall down or keep up with good robustness. In [62], a second order SMO algorithm is used called super twisting algorithm (STA). It is suitable for systems with relative degree one, concerning the sliding variable. The sliding variable is composed of the error between real and estimated values, while the controller adopts the STA. Eventually, faults are generated by the SMO through a filter.

      The method proposed in [60], accommodates the design of an observer with an adaptive threshold for rotor position sensor fault detection. As the observer is associated with model uncertainties and variation of parameters, miscellaneous missing and false alarms take place. Thus an adaptive threshold is used to inspect the position estimation errors of the appointed observer, where the position estimator generates rotor position data by Luenberger observer (LO) for the rotor flux linkage. And the faults are detected by decision making according to input frequency and amplitude, when the residual becomes larger than the threshold. An amount of phase-shift fault also occurs, which is tolerated using a compensation algorithm.

      Another observer based algorithm is proposed in [63], based on current form factor (CFF) and adaptive-threshold, which is calculated from measured motor currents and estimated motor currents. An LO is also deployed to estimate the three-phase motor currents. Primarily, two CFFs are calculated for each phase according to the measurement and estimation. Then three residuals are generated from the CFFs for each phase and adaptive thresholds are inaugurated based on the generated residuals. From there on, the faults are detected by the comparison between each residual and the corresponding adaptive threshold. The detected open circuit faults and current sensor faults are differentiated by using current sums and isolated by current average values.

      The proposed method depicted in [64], is based on an extended dynamic model in a $ {\rm{qd}}0 $ stationary reference-frame for the detection of interturn short circuit faults in PMSM phase windings. A scheme of inter turn short circuit fault is shown in Fig. 6. It is performed with a residual current vector (RCV) signal that is found from the difference between the measurement and estimation of stator currents. A state observer was proposed to estimate the stator currents based on a nominal operation model. In qd-coordinates, the stator currents can be expressed as

      Figure 6.  Scheme of interturn short-circuit faults in PMSM phase winding[64]

      $ {I'_{{q_d}}} = {I_{{q_d}}} - \frac{2}{3}{\beta _{{q_d}}}{I_f} .$


      Then, the observer is defined as

      $ \frac{{\rm{d}}}{{{\rm{d}}t}}{\hat I'_{{q_d}}} = \frac{1}{L}({V_{{q_d}}} - {R_s}{\hat I'_{{q_d}}} - {\phi _{pm}}{\omega _e}\hat {{E_{{q_d}}}}) $


      where $ I_{q_d} $ and $ I_f $ are vectors of stator current and fault current, $ \beta_{q_d} $ is vectorial fault factor, $ L $ is stator leakage inductance, $ V_{q_d} $ is stator voltage vector, $ \phi_{p_m} $ is the amplitude of permanent magnet flux and $ \omega_e $ is electrical angular speed. $\hat{E}_{q_d}$ is the EMF, which is calculated by the electrical angular speed $ \omega_e $.

    • The idea of parameter estimation was first introduced in [65]. In this type of methods, parameters of the system are identified online by algorithms and compared to the reference parameters acquired under initial conditions. In [66], an EMF estimator is used based on open-loop physics and a current mode tracking scheme. The detection of inter turn faults are carried out on the basis of the difference between estimated back EMF and reference back EMF, where the reference back EMF is primarily calculated by a finite element method or actual measurement of the system under healthy conditions. And the EMF estimation includes the non-linearities of the system when no fault has occurred. If any fault occurs, the estimated EMF contradicts the reference EMF, which can be individually defined for each of the phase as

      $ E_{diff} = EMF_{ref} - EMF_{est} . $


      The method in [67] also incorporates back EMF estimation, with an unbalanced inductance matrix and an inverter model.

    • The extended Kalman filter (EKF) is an optimal estimator on the basis of least square approach, that estimates dynamic system states under nonlinearity, and efficiently determines the rotor position and speed of PMSMs[68], where conventional Kalman filters are not able to diagnose for nonlinear systems[69]. In [48], EKF is used for the estimation of rotor speed and phase current of an interior PMSM with corresponding signals from position, current and DC-link voltage sensors. The equations of prediction, innovation and Kalman gain calculation are represented in Table 1.

      $ \hat{S}_{k|k-1} = \hat{S}_{k-1|k-1} + Tf(S_{k-1|k-1}, \hat{V}_{k-1} ) \;\;\;\;\;\;(14) $
      $\hat{C}_{k|k-1} = \hat{C}_{k-1|k-1} + M\hat{C}_{k-1|k-1}M^{\rm{T}} + \hat{Q} \;\;\;\;\;\;(15)$
      $\hat{S}_{k|k} = \hat{S}_{k-1|k-1} + K_g[y_g - h\hat{S}_{k|k-1}] \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16)$
      $\hat{C}_{k|k} = \hat{C}_{k|k-1} - K_gH\hat{C}_{k|k-1} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(17)$
      Kalman gain calculation
      $K_g = C_{k|k-1}H^{\rm{T}}(HC_{k|k-1}H^{\rm{T}}+P)^{-1}\;\;\;\;\;\;\;\;\;\;\;\;\;(18)$

      Table 1.  Steps of EKF algorithm

      The EKF algorithm implies three major steps as shown in Table 1. A state $ \hat{S}_{k|k-1} $ is predicted from a previous state in the first step, where $ \hat{C}_{k|k-1} $ is covariance-matrix, $ \hat{V}_{k-1} $ is vector of voltage input, $ M $ is gradient-matrix and $ T $ is the period of sampling. The following step innovates the predicted step with measurement matrix $ H $. And then the Kalman gain is calculated in the final step.

      The rotor speed and phase currents from the EKF are delivered into a decision unit, where the performances of the sensors are constantly supervised to check the faults. A position sensor fault occurs, when the current value gets lower than the threshold value and the speed gets higher than the threshold value. A DC link voltage sensor fault is detected, when exactly the opposite happens. Then after the faults are isolated, the controller is reconfigured.

    • A model based linear parameter varying (LPV) fault detection and isolation method for the PMSM has been proposed in [70], which is designed based on a poly topic decomposition and a gain scheduled fault estimator in the form of linear matrix inequality (LMI). It works for isolating multiple sensor faults, both of multiplicative and additive forms.

    • The finite element analysis (FEA) is a numerical method that can be used in order to determine PMSM parameters inclusive of electromagnetic field distribution, torques, flux linkage, flux density and inductance, using geometrical proportions and properties of substances. Estimating and validating this type of parameters leads to fault detection. The proposed scheme in [15] involves the detection of interturn fault location and eccentricity fault direction, where the short turns of armature winding, eccentricity and demagnetization under varying loads are validated by FEA. In [71], magnetic field, flux linkage and d-q axis inductance are calculated via FEA. In [72], three-phase PMSM under turn-to-turn short circuit faults are evaluated by a time stepping FEA technique. A time-stepping FEA is also applied in [73], for PMSM under static eccentricity fault.

    • The signal based methods are based upon measuring signals to diagnose and extract faults, as a substitute of analyzing the input-outputs of a system. This measurement analysis can be of spectrum or phases, deviations and magnitudes by means of frequency domain and time domain respectively. Time frequency domain analysis is also performed in signal based methods, when signal evaluation becomes dynamic and transient due to torque fluctuations, unbalanced supply of power and differing loads[31]. This is capable of supervising and extracting faults precisely from non-stationary signals[74].

    • Fast Fourier transform (FFT) is commonly used to compute discrete Fourier transform of a sequence or inverse and acts with stationary signals quite proficiently. In [75], a data driven diagnosis technique based on Bayesian networks is applied to three-phase inverters. From the inverter output, two line-to-line voltages are measured by gate signals and phase currents, where the signal parameters are extracted using FFT. As it is quite hard to discern the voltages with normal computation techniques, FFT is the absolute alternative. A three-phase inverter topology for PMSMs is shown in Fig. 7.

      Figure 7.  Three-phase inverter topology for PMSMs[75]

    • The Hilbert Huang transform (HHT) analyzes signals by means of decomposition into intrinsic mode functions (IMF). In [47, 76], faults of a PMSM under demagnetization is decomposed by stator currents at various rates using HHT. First, some instantaneous frequencies of a signal are measured with empirical mode decomposition (EMD) and IMFs are produced. Then the HHT transforms the frequencies obtained from the IMFs to a full distribution of time and frequency. The signal is then filtered. The demagnetization cure of PMSMs is presented in Fig. 8.

      Figure 8.  Demagnetization curve of PMSMs[47]

    • The wavelet transform (WT) is a traditionally used time frequency domain based technique used in wide applications. In [77], incipient stator faults appearing as intermittent inter turns are diagnosed, where distortions are generated in currents and voltages. The diagnosis is done within two steps: identification of the distortions and localization of the phases that involve faulty turns. The identification is primarily done with WT and as follows compared to phase-to-phase voltages. The identification should be severally practical of its scale and good detection is needed for fault signature, thus WT is used. In [78], signal analysis is carried out by continuous and discrete WT for a PMSM under bearing damage. The WT is stabilized at $ ||\phi|| = 0 $ and centered in $ K = 0 $. Time frequency atoms are widened with a parameter $ f $ and translated by $ v $. The following equation is of continuous WT:

      $ \phi_{v,f}(K) = \frac{1}{\sqrt{f}}\phi\left(\frac{K-v}{f}\right) $


      where $ \phi_{v,f}(K) $ is the wavelet function. The discrete WT is constricted only to values that are discrete. In [79], a similar kind of operation is performed using WT.

    • Winger ville distribution (WVD) is another high resolution time frequency domain based method which works on the principle of a quadratic transformation and cross term interference. In [80], IMFs are analyzed by WVD in an operation of detecting short circuit faults in PMSM. Initially, the current is analyzed by EMD to generate IMFs and HHT is used to get the frequencies derived from IMFs. After that the WVD takes place, which is attainable for discrete signal $ f(n) $ over $ 0\leq n <N $:

      $ WVD(n,k) = \sum\limits_{N-1}^{S = -N}M[n,S]e^{\dfrac{i2\pi kS}{N}} $


      where $ M[n,S] $ is an instantaneous correlation ans $ S $ is an integer.

    • The term artificial intelligence (AI) was first announced by John McCarthy in 1956. It is a process to make machines behave intelligently and manipulated across a wide range of regions such as optimizing, neural and evolutionary computation, knowledge representation, automated reasoning, uncertainty reasoning, etc. AI is able to gain high robustness and efficiency in diagnosis of electrical machines. In [81], an overview is provided of AI applied in fault diagnosis. Several AI based techniques have been used in PMSM fault diagnosis over the years.

    • A neural network (NN) is a set of AI algorithms modeled after biological neuron distribution system, that is purposed to recognize relationships and patterns. It is based on a cluster of artificial nodes grouped into a couple of layers, that receive signals and processes to a required output, consisting of three major layers: input layer, hidden layer and output layer. Every node is connected to another layer′s node through weighted interrelations, where each node receives the value of the other node connected and multiplies it with their weight of connections. This leads to a sum of values that is passed into an activation function that determines the output of the network. The structure of NN is shown in Fig. 9. In [82], a multilayer recurrent NN (RNN) is provided to a near term current prediction, used as input to the fault diagnosis of a PMSM under load fluctuations. This is utilized with local feedback of hidden layers and global feedback of network outputs. The architecture predicts the parameters trained with a Bayesian framework of Levenberg Marquarat (LM) algorithm, where conventional techniques can be associated with many difficulties. The method in [83] uses a multilayer artificial NN (ANN) for short circuit levels classification in the operation of PMSM diagnosis.

      Figure 9.  Structure of a neural network[82]

      In [84], the diagnosis is required of extraction of current features for five non-identical PMSM states, involving two bearing fault and two demagnetization fault states. Two methods were portrayed in the paper for feature extraction. One is an unsupervised method based on 1D-convolutional NN (1D-CNN) and WT. And the other one is supervised learning incorporating a softmax layer. 1D-CNN is utilized to diagnose the faults from the current signal, where the operation is conducted between a convolution vector and a filter, which is expressed as

      $ {{x_{j:j+K} = x_{j}\oplus x_{j+1}\oplus....\oplus x_{j+K = 1}} } $


      where $ x_{j+K = 1} $ is window length, $ K $ is window size. This technique results a highly efficient performance reducing complexity in feature extraction.

      A back propagation NN (BPNN) is used in [85], to detect demagnetization faults in PMSM using acoustic noise data. It is exploited to analyze a nonlinear mapping in the acoustic index, consisting of two phases: training of indicators found from the noise and detection after the training. The BPNN uses Levenberg Marquardt and gradient descent for learning and training.

      Some recent researches on PMSM fault diagnosis based on deep learning have shown substantial results. In [84], the diagnosis method constructed with a one-dimensional CNN showed 98.8% accuracy in labelling five types of failure modes. Another method comprised of a stacked denoising auto-encoder NN (SDAE NN) in [86], is used to extract fault features with a recognition rate of 100%. In [87], another NN based method showed 99.6% accuracy in interturn short circuit fault diagnosis in PMSMs.

    • Fuzzy logic (FL) resembles human reasoning of values, which is tolerant to imprecise information with fine pliability. It is executed within three rules: fuzzification, inference and defuzzification. Fuzzification is the mapping of symptom variables using membership functions (MF). Inference establishes the correlation among input-output variables and defuzzification processes the output. The basic block diagram of fuzzy logic control is shown in Fig. 10. In [88], the proposed method applies an FL based diagnosis for voltage source inverter in a three-phase PMSM, that detects open circuit and intermittent symptom variables. Therefore, analytic symptom $ F_s $ and heuristic symptom $ Y_s $ are used as inputs to the diagnosis block. Then the symptoms can be fuzzified as

      Figure 10.  Basic block diagram of fuzzy logic control[88]

      $ F_s \in (n,z,ps,pl) $


      $ Y_s \in (n,z,p) $


      where $ z $, $ p $, $ n $, $ ps $ and $ pl $ refer to zero, positive, negative, positive small and positive large, respectively. Then the fuzzy rules are done using a Mamdani type fuzzy infernece and max-min composition based defuzzification.

      In [89], FL is applied to a PMSM under load fluctuation for stator fault detection. It uses negative sequence current and impedance, which are provided by a qualitative interpretation of MFs. The MFs are defined as

      $ A = [\beta_{low}(K), \beta_{medium}(K), \beta_{high}(K)], K {\in} A $


      $ B = [\beta_{low}(N), \beta_{medium}(N), \beta_{high}(N)], N \in B $


      where $ \beta_{s}(\lambda) $ is membership grade $[\lambda = K/N, s = low/ medium/high]$. The results derived intimates the faulty and normal conditions of the motor, depending upon MFs.

    • Random forests (RF) is a synergy of tree classifiers generated by random vectors using features selected at random, which performs regression or classification problems[90]. Output combining of trees in random forests is shown in Fig. 11. The method in [91], diagnoses broken rotor bar failure in PMSM using RF. The features from initial transient signals of PMSM currents, are used as the training inputs for RF with a view to classify the faults of the motor under different conditions and upheld an outperformed result.

      Figure 11.  Output combining of random forests trees[91]

    • Support vector machine (SVM) is used for classification problems of two groups. It uses a kernel-trick to convert data and optimize a boundary layer between the required outputs, called support vectors. In [72], it is utilized for short circuited turn detection, where support vectors, $n_j = 1,\cdots,m$ is classified into two groups. SVM resolves an optimization problem:

      $ \min_{\omega,\epsilon} = \frac{1}{2}(W^{\rm{T}} W) + P \sum\limits_{j = 1}^{N} \epsilon_j $


      where training dataset is mapped by a function, $ P $ is penalty of error parameter. The kernel function used is defined as

      $ K(n,n_j) = {\rm{exp}} \Big(-\frac{|| n-n_j ||}{2s^2}\Big) . $


      In [92], s similar type of operation is performed by SVM, where four dimensional support vectors are used for detecting inter turn short circuit faults. In [93], a fuzzified SVM estimates eccentricity severity involving degree and types of PMSM.

    • K-nearest neighbors (KNN) are classifiers based on a non-parametric method for estimating class densities. It finds out samples in the K-nearest set, by allowing a majority defined decision from the samples. In [72], KNN is used to detect short circuit faults in PMSM. With a radius hypersphere, nearest neighbors are searched from patterns to identify the actual class. In [93, 94], KNN is used for high accuracy to recognize eccentricity types, where combination of fault data is outsize in the feature space.

    • Particle swarm optimization (PSO) is a population based optimization technique, made of particle groups called swarm. It was first introduced in 1995[95]. This technique imitates the social communication process of bird or animal swarms, on how each of the animals share knowledge with each other. The particles have a velocity of their own, by which they explore within search areas to find out the best position. The searching technique is based on the equation:

      $ v_i^{k+1} = w_i v_i^k + c_1 R_1 (P_{best}-x_i^k) + c_2 R_2 (G_{best}-x_i^k) $


      $ x_i^{k+1} = x_i^k + v_i^{k+1} $


      where $ c_1 $ and $ c_2 $ are acceleration coefficients, $ R_1 $ and $ R_2 $ are random functions, $ w_i $ is inertia weight factor. PSO starts with an initialization, and the particles find their local best and global best positions with continuous calculations until the required destination is reached. The basic flowchart of PSO is depicted in Fig. 12.

      Figure 12.  PSO flowchart[96]

      In [96, 97], fault severity and location of interturn short circuit faults are identified using PSO. In [98], a dynamic estimator is used which is based on dynamic PSO (DPSO) along with an opposition based learning (OBL) operator to detect the voltage source inverter (VSI) nonlinearities and estimate the parameters of PMSM. The estimator improved the estimation of the machine by detecting the contorted voltage of VSI which helps to enhance the robustness of the PMSM. The proposed DPSO does not expand time intricacy in correlation with the fundamental PSO. As a matter of fact, it is simple to actualize to solve the enhancement issue, which helps to modify to solve the other comparative issues of PMSM. The execution of learning technique contains two angles. A dynamic OBL utilizing versatile Gaussian dissemination is proposed to survive the visual deficiency in the pursuit of $ P_{best} $ through stochastic development and empowers it to escape from particular optima. There is also a novel development update condition planned utilizing a variable investigation vector to improve the dynamic execution of PSO. In [99], PSO is implemented associated with an artificial NN, that determines short circuit faults in a PMSM stator winding. The PSO regulates the weights of the ANN, while it performs the action of detecting.

    • In PMSM fault events, finding the location, type and magnitudes of the faults can be relied on external hardwares implemented to bring out effective solutions. Many fault diagnosis techniques are based on hardware involvements, depending on different algorithms. In [42], an online order analysis (OA) method is applied for bearing fault diagnosis of PMSM as it is a key and an unfortified part. The OA method incorporates a tachometer based OA and tachometer-less OA techniques, which have been demonstrated to be viable devices for diagnosing bearing fault under variable speed conditions. The bearing signal becomes time fluctuating and the deficiency highlights shift with time, when a motor runs at variable speed, which brings trouble for fault detection. OA methods have been generally utilized in variable speed bearing deficiency finding as an order analysis strategy, which can detect this issue. It utilizes products of rotating speed rather than absolute frequencies as recurrence base. For example, wavelet denoising estimation was proposed for rolling element bearing fault diagnosis under variable speed activity through angle synchronous averaging. An OA method was also proposed for rolling element bearing fault diagnosis with time differing rotational speed which is based on time recurrence portrayal (TFR). The OA methods can be grouped into two classes. 1) The first class includes tachometer based methods which utilize a tachometer or an optical encoder to acquire rotating stage data for angular re-sampling. Outer sensors ought to be introduced on the engine to be checked, which expands estimation cost and make the structure intricate. 2) The second class includes OA methods that remove the stage data from obtained vibration or sound signal. The performance of the OA method is restricted as the vibration signal is normally ruined by substantial background noise. The algorithm flowchart of the OA method is represented in Fig. 13. FTC means fault tolerant control.

      Figure 13.  Algorithm flowchart of OA method[42]

      The method in [100] exhibits a hardware-in-the-loop (HIL) testing technique for PMSM stator fault detection. An element of hardware is subbed by a precise constant emulation in the HIL method which is accepted as a rapid checking apparatus for development of data acquisition systems[101]. It is able to precisely emulate PMSM drives in every single working mode including faulty conditions, and can give drive systems integrators a valuable and generally reasonable apparatus for the improvement and checking of fault detection and diagnosis with post-fault control activities. An ongoing PMSM model has been proposed for considering a precise variety of phase inductance because of space sounds and slotting impacts[102]. In any case, the utilization of steady inductance based models comes up short on the capacity to represent non-linearities because of immersion of the magnetic circuit. The partition of flux linkages in armature reaction and permanent magnet instigated fluxes are just legitimate under linear conditions, when current dependent inductances are utilized.

      In [103], a second-order current harmonic monitoring (CHM) method is applied in q-axis current, where faults are determined by a constant comparison of components under different conditions. Boileau et al.[104] propose a line-extraction technique of second harmonic of d-q frame voltage vector to detect interturn faults.

    • A considerable amount of research is conducted in the field of fault diagnosis for PMSMs, concerning the rotor, stator, inverter and drive systems. Within the methods surveyed, significant types of faults are diagnosed such as short circuit faults, open circuit faults, eccentricity faults, demagnetization faults, intermittent faults, etc. These faults are responsible for performance degradation, noise, vibrations, instability and failure in PMSM systems.

      A comparison of fault diagnosis methods applied are depicted in Table 2, on the subjects of embraced techniques, advantages, limitations and objective fault types. The initial four methods comprising of state observers, parameter estimation, extended Kalman filter and linear matrix inequality are of model based methods, where the models are gained by mathematical equations. Each of these methods is efficient enough in terms of accuracy and ease of implementation. The following five methods: finite element analysis, fast Fourier transform, Hilbert Huang transform, wavelet transform and winger ville distribution are of signal based methods. Except finite element analysis, the rest of the methods are based on time frequency domain. These methods are good performers when transient signals from time varying frequency spectrum and decomposition tools are required for real time signal diagnosis. The next six methods are neural network, fuzzy logic, random forests, support vector machine, K-nearest neighbor and particle swarm optimization, which are data driven AI based methods. All these techniques yield good performance in terms of accuracy and intuitiveness in classifying and detecting. The rest of the methods are hardware based which are much more feasible.

      Method Adopted techniques Advantages Limitations Fault types Refe-
      State observer Use the second order SMO with super tw-isting Algorithm (STA) Avoids chattering phenomen-on Complex design Current sensor fa-ult [62]
      Constructed on a position evaluating alg-orithm with a full order LO for rotor flux-linkage Avoids missing or false alarm, reliable two-axis linear hall ef-fect sensor used Varying deviations in resi-duals, limited to motors
      with low saliency
      Rotor position fa-ult [60]
      Based on residual generation to compute CFF and LO to estimate motor currents High robustness and effective-ness, high immunity against false alarms Results large fluctuation
      in diagnosis variables, har-monic distortions
      Open circuit fault, current sensor fa-ult [63]
      Generated RCV by measured and estima-ted stator-currents Decouples the effect of param-eter uncertainties Disturbances over RCV Interturn short ci-rcuit fault [64]
      Parameter estimation Use open-loop physics-based EMF estima-tion designed upon current mode tracking Simplicity in estimation, appl-icable for any EMF Undesired fluctuations in resistance estimation Interturn short ci-rcuit fault [66]
      Extended Kalman filter Calculate Jacobian matrix, transformati-on matrix, H-matrix and system-gradient for EKF Effective detection of three sen-sor faults, simple design Affected by parameter va-riations, decays in low spe-ed Multiple sensor fa-ults [48]
      Linear parameter varying LPV based residual generator and polyto-pic decomposition for gain scheduled dia-gnosis No external hardware needed, closed loop operation Large amount of calculati-on, influence of harmonics Multiple sensor fa-ults [70]
      Finite element analysis Calculate magnetic field by finite element method, along with flux linkage and d-q a-xis inductance for vector controlled motor Simple operation, ease of imp-lementation Only deals with less mag-netized system Demagnetization fault [71]
      Extract frequency pattern from stator cur-rent and add a white Gaussian noise Highly reliable and robust, hi-gh recognition rate Harmonics impacts, satu-ration in amplitudes Short circuit fault [72]
      Index of harmonic component amplitude with specific frequency pattern Noninvasive process, good rob-ustness More instrument needed
      in measuring torque
      Eccentricity fault [73]
      Fast Fourier transform Extract signal features using FFT for Ba-yesian diagnosis Quality classification for nor-mal and abnormal features Complex design and ope-ration Inverter faults [75]
      Hilbert Huang transform Obtain decomposition of stator currents
      for 2D FE analysis
      Good detection under differe-nt speed conditions Require high current and torque Demagnetization fault [47, 76]
      Wavelet transform Identify distortions in stator currents and reference voltages Implements in on-board diag-nosis, preventive maintenance Less turns, high short circ-uit resistance Intermittent inte-rturn fault [79]
      Winger ville distribution Analyze stator current by empirical mode decomposition to generate IMFs Obtain faults in speed variati-ons, good resolution Additional frequencies pre-sent, amplitude increase with damage Short circuit fault [80]
      Neural network Employ multilayer perceptron feed-forwa-rd NN with analytical and FE method Rigorous diagnosis, easy impl-ementation Weak training leads to err-or, lengthy procedure Short circuit fault [83]
      Deep 1D CNN including softmax layer bas-ed on current signature analysis Operates for five types of mo-tor states Require time for data coll-ection and training Demagnetization, bearing fault [84]
      Perform online fault detection using BP-NN with acoustic noise data Noninvasive: free of affect of internal parameters Complex operation, requi-re large data demagnetization fault [85]
      Fuzzy logic Calculate fuzzy approached fault symptom variables by Park′s vector method High reliability, good localiza-tion capability Limited to three phase, ne-ed extra hardwares Open circuit fault [88]
      Separate voltage and current of motor ter-minal by negative sequence component High sensitivity, avoids false alarms under fluctuation Machine reconstruction,
      complex design
      Interturn short ci-rcuit fault [89]
      Random forests Obtain transient current signal from start up of motor and extract time domain fea-tures High accuracy, 13 features us-ed Limited to line start PMS-M, extra hardwares Broken rotor bar fault [90]
      Support vec-tor machine Compose feature vectors from sparse coef-ficients of voltage and current signals Safe and rapid detection, easy to implement Only suitable for small sc-ale sample Interturn short circuit fault [72, 92]
      K-nearest neighbor Determine a correlation between a noninv-asive index and eccentricity degrees Robust on training noisy data, intuitive Sensitive, poor run time
      for large data
      Eccentricity fault [93, 94]
      Particle swarm optimization Detect VSI nonlinearities and estimate pa-rameters by PSO and opposition based learning No overlapping, no mutation calculation Undergoes partial optimi-sm, cannot handle scatter-ing problems Interturn short ci-rcuit fault [96-98]
      Hardware based Apply encoder and tachometer less fast online-order analysis Convenient and accurate, high reliability Complex design and insta-llation Bearing fault [42]
      Exhibit hardware-in-the-loop test for mo-tor stator for rapid checking Good degree of accuracy, wide range Complex design and insta-llation Stator winding fa-ult [100]

      Table 2.  Comparison of PMSM fault diagnosis methods

      The aforementioned methodologies for fault diagnosis for PMSMs, are being applied in sectors such as railway, civil aviation, subway, automotive, etc. As any kind of unwanted fault may trigger the protection unit of the system to avoid subsequent catastrophe, fault diagnostics are highly necessary for any of the applications. The improvement on the diagnosis methods is the key to getting rid of heavy economic penalties and other casualties. In accordance, all of these methods mentioned above need further improvements for faster response, precision in localization and simplicity.

    • Fault tolerant control (FTC) is the mechanism that improves reliability and availability of systems, reducing the damages by faults and risk of failure. It can be classified into two basic types: active FTC and passive FTC. The active FTCs are capable of re-configuring the controller with a fault diagnosis scheme and a reference set[35], which are not necessary in passive FTCs. Though passive FTCs have limitations when an excessive amount of faults are accommodated, it loses its flexibility[17]. The active FTCs also have drawbacks such as failure possibility, when the diagnosis associated has unexpected time delay. Various types of open-loop[105] and closed-loop techniques are implemented for active and passive approaches of PMSM fault tolerance. Often generalized control schemes and intelligent schemes are used in fault tolerant control of PMSMs. Modified and optimized hardware based techniques are also used in the approach of fault tolerance.

    • Fault tolerant techniques based on generalized control schemes are broadly utilized in PMSMs. These techniques can rely on the utilization of mathematical models and implementation of different algorithms, allowing them to be used in machine driven applications.

    • Sliding mode control (SMC) adjusts nonlinear and unpredictable system dynamics, in terms of sliding it along a cross section of the natural behavior of the system, providing signals with discontinuity. It is a non-direct current control conspire based method which effectively treats model errors and provides great dynamic execution and tracking accuracy. In [11], an SMC based current control method is used for treating model inaccuracies in a five-phase PMSM. The system equation for SMC formulation can be expressed as

      $ X' = PX + QU + d_e + d_{all} $


      where $X = [i_d\; i_q\; i_3]^{\rm{T}}$ is state vector and $U = [u_d\; u_q\; u_3]^{\rm{T}}$ is the control law. $ d_e $ and $ d_{all} $ are disturbances for back EMF and overall respectively. The sliding variable is equal to tracking error, driven by reference values of the variables:

      $ S = \begin{bmatrix} \;\;e_d \;\; \\ \;\;e_q \;\; \\ \;\;e_3 \;\; \end{bmatrix} = X_{ref} - X $


      where $ X_{ref} $ is reference current vector. The features of implementing the SMC include avoiding chattering phenomenon which is undesirable, applying space vector modulation and carrier based pulse width modulation, and eventually applying the SMC law.

      The method in [106], proposes a re-configurable higher order SMC observer (HOSM). It estimates back EMF from currents and voltages to calculate the speed and rotor position. The architecture of the HOSM method is shown in Fig. 14.

      Figure 14.  Architecture of HOSM method[106]

    • The feedback signals from the position and current sensors are essential for providing high productive drive conditions in an ordinary vector control of the PMSM. Specifically, at any rate, at minimum two current sensors and one position sensor are compulsory to ensure a moderate performance feedback control of the PMSM drives. In view of this fact, any sensor error may reduce the efficiency and durability of the PMSM. For the nonstop and dependable operations of the PMSM drive, a deficiency robust control is mandatory for the electric vehicle (EV) drive applications. EKF with fixed covariance matrix is utilized and it cannot ensure a precise speed estimation in a frequently changing condition, so an adaptive EKF (AEKF) is proposed in [107]. An FTC scheme built on the AEKF can expand the reliability and great execution of the fault tolerant control of PMSM drive for the EV applications. The proposed AEKF gives the position sensor data to accomplish the sensor fault location and reconfigure the mechanism. This AEKF does not utilize steady covariance networks like in [48, 108]. However instead they are constantly being expanded contingent upon a specific working condition to ensure an exact estimation paying little mind to the working conditions. The proposed FTC algorithm can recognize the position sensor fault immediately and reconfigure the PMSM drive, during the position sensor failure. Additionally, the reconfiguration mechanism with a precise speed estimation is constructed to guarantee an untroubled feedback signal from a faulty sensor mode to a sound (typical) sensor mode which also decreases the transient reactions in torque and speed throughout an operating session. A setup of FTC of the AEKF method is shown in Fig. 15. MTPA represents maximum torque per ampere.

      Figure 15.  FTC setup of adaptive extended Kalman filter[107]

      In [7], a two-stage EKF, an EMF based adaptive observer and a voting algorithm are merged together for position sensor tolerance of PMSM. The primary stage of the EKF is salient modeling of PMSM and the following stage is the discretization of the model. Within it, the speed and position of the PMSM is determined for further control stages.

    • Two types of control algorithms are applied to the prompt torque control for PMSM drives, vector control (VC) and direct torque control (DTC). A position sensor is not required essentially in the DTC system which builds the control calculation model in stator static coordinate legitimately[109, 110]. In [111], a fault tolerant DTC algorithm is proposed with circular adapted stator flux direction for PMSM. The adjusted stator flux, changed stator current furthermore, changed stator voltage dependent on the asymmetrical typical stator flux model are built to understand the control of circular adapted stator flux direction along with circular stator magnetomotive force (MMF) direction. The adapted stator flux and electromagnetic torque are straightforwardly constrained by the adapted stator voltage vector. DTC has a few favorable properties over other algorithms which make it meet the necessity of superiority for EV applications, e.g., rapid torque reaction, simple structure and parameter independence. By and by, numerous research on DTC algorithms for PMSM drives under sound conditions have been done. The improved DTC algorithm for low speed activity was explored in [112]. Due to the appropriation of the hysteresis comparators, DTC algorithms have variable switching recurrence, moderately enormous torque and flux ridges as well as acoustic noise. By using DTC algorithm, the torque can be estimated as

      $ T = \frac{5}{2} \psi_{pp}(\psi_a i_b - \psi_b i_a) $


      where $ \psi_{pp} $ is pole pair and $ \psi $ is stator flux linkage. In [113], a space vector modulation based DTC is used for the tolerance of voltage, current and speed sensor faults, open-switch and open-phase faults in VSI fed dual three-phase PMSM. In [114], a similar strategy is used for a five-phase interior PMSM, where the winding is fractionally slotted. It improves the current with a low amount of torque and fluctuation of flux.

    • The FTC innovation has been given significantly more consideration to keep the PMSM drive system operating under the short circuit or open circuit fault condition. In [24], a worldwide FTC algorithm is explored for a PM motor under one-phase short circuit, one-phase open circuit and two-phase open circuit shortcomings. The decoupled demonstrating for PM drives when single-phase open shortcoming was examined about in [115], in which the proposed model field oriented control (FOC) is utilized by the drive. The FTC control is proposed to be merged with actual FOC. Closed-loop speed control can be accomplished by the FOC algorithm. The proportional integral (PI) controller creates reference of quadrature current $ i_q $, or reference of torque dependent on the equation. In order to create gating signs to the inverter with the goal of producing reference of $ i_q $, space vector pulse width modulation (SVPWM) is applied. The output torque can be given by the following equation:

      $ T = \psi_f i_q $


      where $ \psi_f $ is the permanent magnet flux. On the off chance that the three currents in the PMSM have deteriorated into adapted terms and “unequal” terms, at that point the subsequent torque is the emplacement of all torque parts which are produced with FOC. At that point, FT control is successfully coordinated with existing FOC. The main outstanding issue is to identify the “unequal” currents with the goal that the yield torque is sustained[116].

    • Each control algorithm is tied by the null zero-arrangement current bond and needs to convey a sensible average torque. The current control algorithms such as those applied in [117], empower a smooth pursuit of any deficiency that is uniquely in contrast to enhancement approaches, henceforth, just the major and third time-symphonious flows are considered as well as processed systematically. Current control algorithms can be expressed by these three advance strategies: 1) The current phasors are attached to assure the bond. 2) The second-order torque harmonic is dropped following a correct decision of the current phasor edges of the sound phases. 3) The fourth-order torque harmonic is dropped including the third time harmonic current (together with a improvement of the edge of the major current phasors). In [13], a back EMF based model is used for tolerance control. In [118], a reduced order FTC based on trapezoidal back EMF is applied using a third harmonic current injection method. A current control based strategy is shown in Fig. 16.

      Figure 16.  Current control based FTC for five-phase PMSM[13]

    • As the outcome of PMSM fault events, withdrawing the defected segments, recovering the defected areas by substituting the segments and diminishing the consequence of faults are effectual provisions for fault tolerance. Influenced by these schemes, a great deal of techniques are designed for PMSM fault tolerance such as modifying inverter designs, optimizing machine infrastructures, affixing redundant switches and so forth.

    • Redundant switches are affixed with a view to the maintenance of phase currents, especially when the event of short circuit or open circuit occurs in the inverter. These switches assist as reinforcements in such circumstances. The method in [119] is based on fault switch process (FSP), where it manages a relation between two adapters, each of them connected to a distinct switch. A ten-phase PMSM is designated with dual-stator and dual-rotor in the same shaft. During the speed control of the motor, the electromagnetic torque fluctuation is taken in the FSP. The switches are connected to FPGA that triggers the faults when they occur, while metal-oxide-semiconductor field-effect transistor (MOSFETs) are used as switches. An observer based open transistor FTC scheme is introduced in [27] for VSI fed five-phase PMSM.

    • Apart from the superiorities of redundant switches, it also augments a rise of cost in installing, maintenance, continuance and usage in PMSM drives. Having considered these perspectives, many researchers have proposed modifications in inverters in different ways for fault tolerance. The method in [8] introduces a module consisting of two voltage source inverters for normal, isolated and faulty conditions providing peak velocity and torque. The inverter design is shown in Fig. 17. It is applicable for multi-machine applications. In [120], a four pole inverter is used which leads to various types of faults and generates rippleless torque. It is able to isolate and activate its poles in order to fix the faults according to the locations. It is incorporated under some remedial techniques with three-terminal motor winding and four terminal motor winding. The remedial technique disconnects one phase of the three-phase PMSM, remaining two phases produce d-q and $\alpha$-$ \beta$ current components in a stationary reference frame.

      Figure 17.  Inverter design for two PMSM drives[8]

      If a zero sequence stator current is defined as

      $ i_o = \frac{1}{3}(i_a + i_b + i_c) $


      then the transformation of the currents to α-β plane demands undergoes the following equation:

      $ \begin{bmatrix} \;\;i_a\;\; \\ \;\;i_b\;\; \\ \;\;i_c\;\; \end{bmatrix} = \begin{bmatrix} 1& 0& 1& \\ -\dfrac{1}{2}& \dfrac{\sqrt{3}}{2}& 1& \\ -\dfrac{1}{2}& -\dfrac{\sqrt{3}}{2}& 1& \end{bmatrix} \begin{bmatrix} \;\;i_\alpha\;\; \\ \;\;i_\beta\;\; \\ \;\;i_\gamma\;\; \end{bmatrix} . $


      When the faults are determined, the control scheme isolates the defected pole of the inverter and recovers the system drive. Wang et al.[121] present a neutral point clamped (NPC) three-level inverter configuration for PMSM with double stator winding, also depicted in Fig. 18. It utilizes vector space decomposition control (VSDC) achieving low copper loss and current variance.

      Figure 18.  Six-phase NPC three-level inverter[121]

      Wang et al.[122] contrast three different control schemes: normal channel direct compensation (NCDC), normal channel asymmetric current (NCAC) and equivalent current value compensation (ECVC) with paralleled voltage source inverters. These VSIs are basically used for connecting AC and DC outputs of different inverters in a parallel configuration. Errabelli and Mutschler[123] propose a three-leg two-level inverter, where during an event of fault occurrence, an auxiliary leg takes the place of the faulty leg. This is done using redundant thyristors connected back-to-back like electromagnetic relays. In [124], another four-leg inverter is designed for tolerance of open phase faults by a novel transformation matrix with current and voltage references from d-q frame.

    • As the initial objective of FTC is diminishing faults and upgrading the motor performance, this cannot only be attained by redundant switches and modified inverter designs, but also by advancing the internal structure of the motor. In [125], a modular PMSM of $ n $ modules with three-phase Y-connected winding reconstruction is proposed to deal with open circuit fault operations. The modules are shown in Fig. 19. The motor control system is presented in Fig. 20. The current equation of each module is expressed as

      Figure 19.  Modular motor of n module[125]

      Figure 20.  Modular motor control system[125]

      $ I_{A1,2,\cdots ,n} = i_{a1,2,\cdots, n} {\rm{cos}}(\omega_e t) $


      $ I_{B1,2,\cdots,n} = i_{a1,2,\cdots,n} {\rm{cos}}(\omega_e t - \frac{2}{3} \pi) $


      $ I_{C1,2,\cdots,n} = i_{a1,2,\cdots,n} {\rm{cos}}(\omega_e t + \frac{2}{3} \pi) $


      where $ i_{a1,2,\cdots,n} $ is phase current amplitude and $ \omega_e $ is angular frequency. The total torque from the motor for each module in d-q axis is

      $ T = \sum\limits_{i = 1}^n T_e = \frac{3}{2}P \big\{ \psi \sum\limits_{i = 1}^n I_q + \sum\limits_{i = 1}^n[(L_d - L_q)I_d I_q] \big\} $


      where $ P $ is the number of pole pairs.

      In [126], an interior PMSM of four pole with 15 stator slots and dual layer windings is designed, that has the advantage of minimizing high torque pulsations. In [127], a genetic algorithm (GA) based FTC is approached for dual three-phase PMSM, where external hardware modification is not required for the multi-phase structure of the motor. In [128], a strategy is proposed for phase loss fault in a flux modulated permanent magnet compact in wheel motor. It is based on the Y-$\delta$ change in the winding. The pole pairs of the rotor satisfies the following equation:

      $ P = N - P_s $


      where $ N $ is teeth number and Ps is the number of pole pairs of motor armature winding.

    • The application of intelligent algorithms in fault tolerance is to authenticate effective operations. Frequently used algorithms include neural networks, fuzzy logic and hybrid algorithms. The method in [29] proposes a recurrent fuzzy cerebellar model articular network (RFCMAN) for controlling the rotor position of six-phase PMSM. It evaluates nonlinear equations in the control law and is obtained using Lyapunov Stability. The enumerated torque of control law is

      $ U(t) = W + q(t) $


      where the nonlinear function $ W $ and auxiliary control $ q(t) $ is expressed as

      $ W = B^{-1}[\theta'(t) - A \theta(t) -h] $


      $ q(t) = B^{-1} KE. $


      As this control law is enigmatic in heuristic applications, an intelligent control law is used therefore:

      $ U_i = W_{RFNCMAN} + U_r + q(t) $


      where $ W_{RFNCMAN} $ is a RFNCMAN estimator. The RFNCMAN is built on five spaces including: input space, membership space, rule space, memory space and output space.

      In [30], a Takagi-Sugeno-Kang type fuzzy neural network with asymmetric membership function (TSKFNN-AMF) estimator is used along with complementary SMC. The structure of TSKFNN is of five layers: input layer, membership layer, rule layer, TSK type fuzzy inference, consequent layer and output layer. It is presented in Fig. 21. The output can be defined as

      Figure 21.  Structure of TSKFNN-AMF[30]

      $ H(e(t)|W) = R^{\rm{T}} WK $


      where $ W $ is adjustable weight vectors and $ K $ is output vector of rule layer. The TSKFNN is calculated from Lyapnov theorem that ensures closed loop stability.

    • The investigation regarding fault tolerant techniques is the key to reliability and failure reduction in PMSM drive systems. The tolerant techniques increase the drive availability and are applied to machines and inverters. A sufficient tolerant operation can be gained by various algorithms, additional devices or disconnecting inverters, depending upon the condition of the fault[129, 130].

      Various tolerant techniques have been used to rectify the performance of PMSMs, and are compared in Table 3. The first two methods of generalized control schemes, sliding mode control and adaptive extended Kalman filter ensure a moderate performance feedback control of PMSM drives, SMC method can control nonlinear[131] and unpredictable system dynamics where AEKF is able to provide highly productive drive condition. Later three methods consisting of direct torque control, field oriented control and current control strategy deal with changed stator current and voltage in order to adapt stator flux direction, operating PMSM drives under the short circuit and open circuit fault condition and empowering smooth pursuit of any faults[132]. The methods described in hardware enhanced are redundant switch, which maintain the phase current when open/short circuit faults occur and modified inverters which were designed for the purpose of achieving effective outcomes, depending on the configuration of the inverters. Optimized motor designs were approaches with advanced motor structures to eliminate faults, which are not attainable by redundant switches and modified inverters. Better performance is shown by these techniques. The other methods involve intelligent algorithms, that utilize the hybridization of advanced artificial intelligence and control algorithms for the computation tolerance of PMSMs, resulting in FTCs with higher accuracy[133, 134].

      Method Adopted techniques Advantages Limitations Fault types Refe-rences
      Sliding model control Reconfigure post faults by modified angular tr-ansmission for rotating frame with unchanged
      flux linkage
      Robust, treat uncertainties and nonlinearities For five-phase, affected by low frequency third harmonic disturbance Multiple type faults [11]
      Ensure finite time stabilization of observation error and demonstrate HOSM Resist perturbation, highly effective Complex design and implem-entation Position sensor fault [106]
      Adaptive EKF Estimate system states and covariance matrix
      of statistical characteristics
      Accurate speed estimation, smooth transition Needs extra hardwares, com-plex Position sensor fault [107]
      Direct torque control Construct relation between zero sequence vol-tage and current by stator flux and stator vol-tage relationship High reliability, suitable for multiphase PMSMs Large number of calculations, extra maintenance Stator winding fault [111]
      Vector space decomposition based estimation
      of current for FTC
      Controls five different faults for multiphase drives Additional costs, complex co-mputation Multiple type faults [113]
      Adopt space vector pulse width modulation by reconfiguring six equal non-zero vectors Low flux-ripple, low torque, improved current Complex stator slotting, slow switching time Multiple type faults [114]
      Field oriented control Reduce torque fluctuation by implementing fa-ult severity estimation and keeping sensible to-rque output Good post fault performan-ce, simple control with high robustness Rise of unbalance current, low accuracy Interturn short circuit fault [116]
      Current control strategy Investigate post fault strategies by an analyti-cal model Applies to any power rating, operate in ripples For five-phase PMSM, requi-re large memory Open and short circuit fault [117]
      Utilize unperturbed third harmonic air gap M-MFs with back EMF Small ripples, harmonic cur-rent injection May generate error in current injection Open circuit fa-ult [118]
      Redundant switches Characterized ten-phase PMSM by two rotors and two stators on identical shaft Uniform boundedness for lo-ad disturbance, ripple free torque Complex slot design and mot-or construction Open and short circuit fault [119]
      Modified inverters Incorporate fourth inverter pole with identical topologies of three poles High torque density and ef-ficiency Expensive hardware modifi-cation Inverter pole fa-ult [120]
      VSD modeling dual stator winding motor and develop switching strategies Minimum copper loss and current amplitude variance Additional cost for dual stat-or Open phase fa-ult, open switch fault [121]
      Three strategies are compared: NCDC, NCAC, ECVC Minimum copper loss, smo-oth torque Circulating current in parall-el channels Open circuit fa-ult [122]
      Add redundant leg to replace fault affected leg
      by thyristors
      Fault compensation, negligi-ble disturbance Additional power thyristors, complex Short circuit fa-ult, opencircuit fault [123]
      Design transformation matrix for current/vol-tage references in d-q frame Simple strategy and calcula-tion Insufficient use of controller Open phase fa-ult [124]
      Optimized motor designs Employ MMF compensation and field oriented control for reconstructed winding in different modules High feasibility and rationa-lity Complex integration, high cost Open circuit fa-ult [125]
      Provide torque producing MMFs eradicating
      the use of neutral lines and DC bus
      No additional hardware, hi-gh reliability Complex winding distributi-on Open phase fa-ult [126]
      Optimize stator current by genetic algorithm
      to control torque pulsations
      Smooth switching, induce pulses Composite demonstration Open phase fa-ult [127]
      Vector control with phase shift and winding $Y\!=\!\delta$ change for FMPMCW motor Small PMSM drive, low cost Additional switches and capa-citors Phase loss fault [128]
      Intelligent algorithms Track rotor position reference command by RF-NCMAN assisted torque controller to estimate nonlinear equations Highly effective, resilient to inference Require high amount data, hard implementation Rotor position fault [29]
      Stabilize the FTC of a six-phase PMSM drive using SMC, and TSKFNN-AMF to estimate limped uncertainty Effective under different te-st conditions Complex computational alg-orithm Open and short circuit fault [30]

      Table 3.  Comparison of PMSM fault tolerant techniques

      The techniques discussed earlier for fault tolerant control of PMSM systems need to be more enhanced with software in terms of safety issues and degraded usage[135]. Due to complications of PMSM armature winding insulation aging, electromagnetic interference, change in component features, the PMSM drive systems face unexpected errors. Hence, the emergence of more improvements is required for fault tolerance. However, all of these techniques need to be upgraded to achieve greater control, fault reduction and higher stability.

    • The appeal of PMSMs is growing in a variety of sectors as it is one of the highly efficient and reliable motors with less torque ripple and low inertia. Despite their widespread application, they can be destabilized in performance when faults occur. For that reason, fault diagnosis and fault tolerant techniques have to be equipped to ensure the stability and robustness in PMSMs. In the fields such as rail[136, 137] and aviation[138], fault diagnosis operations for PMSMs are conducted within observer based approaches, residual generation, data driven methods and so forth. Tolerant techniques based on redundancy, optimized designs, and control algorithms are also applied in PMSM systems.

      An overview of several methods regarding fault diagnosis and fault tolerance of PMSMs are presented in this paper. Miscellaneous diagnosis methods, classified into model based, signal based, artificial intelligence based and hardware based are briefed and compared concerning embraced techniques, advantages, limitations and the fault types from a number of research papers. However, there are still a lot of methodologies that are applied to induction and reluctance motors, yet not applied to PMSMs. Moreover, PMSM fault diagnosis has numerous research scopes including:

      1) Artificial intelligence based methods in the emerging field of reinforcement learning, deep learning, quantum machine learning can be employed in diagnosis for higher accuracy.

      2) The diagnosis schemes need to be less time consuming in terms of simplicity and effectiveness.

      3) Implementation of extra sensors and hardware should be lessen.

      A similar kind of review is also done for fault tolerant techniques, classified into generalized control schemes, hardware enhanced and intelligent algorithms. Future research scopes in fault tolerance are itemized as follows:

      1) Management for abrupt transience in fault tolerance should be put forward.

      2) For reducing complexity, control schemes with less components should be approached.

      3) More effective inverter topologies and motor constructions can be developed.

      4) Advanced and hybrid intelligent algorithms should be integrated in stabilization and tolerance control of PMSMs.

      To fulfill the requirements of industry and contemporary engineering, research on fault diagnosis and tolerance needs to more broadly address PMSMs for real world applications.

Reference (138)



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