Improved Sensorless Control of Multiphase Synchronous Reluctance Machine Under Position Sensor Fault

This article presents an investigation on the self-sensing capability of a dual three-phase synchronous reluctance motor. Self-sensing capability refers to the ability of the motor to properly operate in a sensorless drive. The multiphase machine is decomposed into two different three-phase systems according to the multistator approach. Several supply scenarios are studied where the two three-phase windings are controlled at different operating points along a reference trajectory. The analysis is carried out both with finite element analysis (FEA) simulations and experimental tests. In the first part of this article, the rotor is locked to derive the observer trajectories and find the regions in which the motor can operate without position sensor. A comparison between simulated and experimental results is given. Finally, a sensorless control strategy that allows exploiting the motor self-sensing capability under position sensor fault is developed and validated through etests.

Abstract-This article presents an investigation on the selfsensing capability of a dual three-phase synchronous reluctance motor.Self-sensing capability refers to the ability of the motor to properly operate in a sensorless drive.The multiphase machine is decomposed into two different three-phase systems according to the multistator approach.Several supply scenarios are studied where the two three-phase windings are controlled at different operating points along a reference trajectory.The analysis is carried out both with finite element analysis (FEA) simulations and experimental tests.In the first part of this article, the rotor is locked to derive the observer trajectories and find the regions in which the motor can operate without position sensor.A comparison between simulated and experimental results is given.Finally, a sensorless control strategy that allows exploiting the motor self-sensing capability under position sensor fault is developed and validated through etests.

I. INTRODUCTION
I N THE last years, three-phase synchronous reluctance mo- tors [1], [2] have seen increasing interest thanks to the absence of permanent magnets, no rotor losses, easy manufacturing, and their relatively low cost compared with the other types of permanent magnets synchronous motors (PMSMs) [3], [4].Moreover, due to the enhanced attention on the fault-tolerant capability [5], [6], redundancy [7], and high power density, multiphase motors have attracted great attention.One of the main multiphase topology for a six-phase motor is a dual threephase (DT) machine with two identical three-phase windings as it exhibits high reliability and employs commercial standard three-phase inverters.
An effective motor control relies on an accurate knowledge of the rotor position to achieve high performance.The position is obtained by means of an encoder or resolver, which increase the overall price and decrease the reliability of the system, indeed one of main common fault in a sensored drive is a fault in the position sensor.To remove the mechanical sensor, several estimation algorithms have been developed.These methods allow the removal of the position sensor by reducing the motor frame size, lowering the price, and increasing the reliability of the drive.According to the operating speed of the motor, these techniques can be bundled into two groups.In medium-high speed range, position observers based on fundamental frequency signals are implemented, such as those based on the back electromotive force or the active flux [8].At standstill and lowspeed region, additional high-frequency (HF) signal injected on the fundamental components [9] or unconventional pulsewidth modulation (PWM) patterns are exploited in order to depict the rotor anisotropy [10] and retrieve the rotor position.However, low-speed signal-injection sensorless controls are affected by an estimation error due to the motor cross-differential inductances.Stability issues and the estimator convergence could be affected due to the estimation error.
Low-speed sensorless algorithms and compensation techniques have been applied to multiphase machine in [11], [12], [13], [14], and [15].In [12], a method taking advantage of the additional degree of freedom of DT motor is proposed, and it allowed for reducing both the torque and the dc-link current ripple.The method was applied to PMSM and a constant steadystate error is shown.A different approach is used in [13] where the HF signals are injected in only one three-phase electrical winding.The method is applied on a low-voltage PMSM and good performance is achieved.The zero-sequence voltage is exploited in [14] to estimate the rotor position and the undesired harmonic was suppressed by applying an optimal phase shift between the two independent injected signals.However, a quite large constant estimation error is reported.A compensation method based on current pulse injection is investigated in [15].The additional degree of freedom of a PM DT motor is exploited to estimate motor parameters and, in turn, improve the sensorless accuracy.This article proposes a low-speed sensorless control strategy based on the DT motor self-sensing capability.The self-sensing capabilities are evaluated for different operating conditions, as the three-phase configuration, the half-control (HC) one, and six-phase one, and an in-depth discussion is reported.In the case of mechanical sensor fault, a three-phase winding is controlled at low current and HF signals are injected to retrieve the rotor position.The reduced magnetic saturation due to the working condition allows for achieving an accurate rotor position estimation, which is used by both windings of the DT machine.The second three-phase winding can work until its nominal current and overloaded.The proposed control algorithm reduces the motor performance but it allows operation in the case of a mechanical sensor fault.Moreover, the estimated position is accurate and not affected by the estimation error due to cross-magnetic saturation that is the main flaw of low-speed algorithm and no motor parameters are required.
In [16], an experimental investigation of the self-sensing capability is carried out.In this article, finite element analysis (FEA) is included to study and compare experimental and simulated results.Flux linkage maps and theoretical observer trajectories for different supply scenarios are included.In addition, an improved experimental study is proposed with the online calculation of the estimation error and the observer trajectories for the three-phase configuration of the considered motor.
The rest of this article is organized as follows.The description and the analysis of the adopted motor are described in Section II.The sensorless drive and the proposed control technique are thoroughly described in Section III.The results are reported in Section IV.Finally, Section V concludes this article.

II. MOTOR DESCRIPTION AND MODELING
A DT synchronous reluctance machine is adopted in this article.The winding arrangement is shown in Fig. 1.It is composed by two identical three-phase windings labeled as abc and xyz where each electrical system is supplied by a dedicated inverter.The former one is distributed at the top of the stator, whereas the latter one at the bottom.The layout employed reduces the mutual inductances between the two windings, improving the faulttolerant capability [17], [18], as one winding can be supplied, namely, HC mode, even if the other one is short-circuited.Table I gives the main data of the motor.Further details of the motor design, the thermal analysis under fault operating conditions and its flux weakening capabilities can be found in [19] and [20].
In the following, both FEA analysis and experimental tests on the same motor to verify the accuracy of the proposed control strategy are reported.Flux linkage and torque maps have been measured and compared with FEA simulations for different supply configurations.Magnetic maps have been measured with a constant speed method and both windings have been fed with the current vector imposed in the dq plane.In Section II-A the three-phase configuration is analyzed, whereas in Section II-B, the HC mode is studied.

A. Three-Phase Configuration Analysis
In three-phase configuration, both windings are connected in series and supplied with the same current.Flux linkage maps are obtained both through FEA simulations and experimental measurements.The results are depicted in Fig. 2 and the flux density plot is shown in Fig. 5(a).Simulated results are akin to the measured one along both d-and q-axes in the rotating reference frame.A slight difference occurs when the motor is heavily saturated.
By means of the flux linkage maps, the simulated torque of the motor is calculated.Fig. 3 shows the comparison between simulated and measured torques.
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B. FEA Analysis in HC Condition
Only the abc winding is supplied during HC-mode.The motor is controlled as a single three-phase system and supplied by only one inverter.The xyz winding is completely open, hence no voltages and currents are applied to it.This is a particular scenario in which the motor can operate, for instance during a fault in the second three-phase winding.Fig. 4 shows the flux linkage maps λ abc d (i d , i q ) and λ abc q (i d , i q ) calculated through FEA simulations.The superscript on a variable means that the quantity refers to the indicated three-phase system.Same results can be obtained by supplying only the xyz winding.
The maps show that the magnetization of the motor is not homogeneous and symmetric.Flux density plot for the HC-mode The torque map during the HC-condition is depicted in Fig. 6.The zero-value torque level is rotated with respect to the dqcurrent plane axes since the magnetic flux produced by the supplied three-phase winding is not bounded in the active region.In HC condition, the d-and q-axes flux linkage are not null even if the d-axis or the q-axis currents are zero, respectively, namely, λ d (0, i q ) = 0 and λ q (i d , 0) = 0.It is worth reminding that the chosen winding configuration minimizes the mutual coupling between the two electrical systems and, in turn, the induced current in the case of a short-circuit in one of three-phase system [18], but exhibits a nonconventional torque map in the case of an open circuit fault.

III. SENSORLESS CONTROL
Sensorless algorithms estimate the rotor position by injecting HF signals on the fundamental one.Rotating injection is a position estimation method used at low speed, where two sinusoidal HF voltage signals are injected in the stationary reference frame and U h and ω h are magnitude and pulsation frequency of the injected sinusoidal waves.The injected HF voltage signals induce HF currents, which draw an ellipse in the αβ reference frame.Its tilt contains the information related to the electrical rotor position ϑ me obtained by multiplying the mechanical one ϑ m for the pole pairs p.The elliptical trajectory due to the induced currents is centered on the fundamental current vector and can be fitted by using the implicit equation of a generic ellipse which describes the measured stator currents in the stationary reference frame i α and i β .The ellipse fitting (EF) algorithm processes the measured currents by fitting current samples on the mathematical ellipse equation by exploiting the least square algorithm and avoiding any filters [21].The estimated rotor position ϑ me can be retrieved from its sine and cosine components by means of a quadrature Q-PLL.The EF algorithm together with the Q-PLL forms the position estimator (PE).It is worth remembering that the selected estimation algorithm requires no Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.motor parameters knowledge for the position estimation, so it is inherently robust against parameter mismatches.To close the current control loop, the HF currents in the rotating reference frame are filtered by means of a low-pass filter (LPF).The filtered currents i d and i q are then used as feedback and compared with the reference one i * d and i * q .Proportionalintegrator (PI) controllers are designed to drive the error to zero and calculate the fundamental reference voltages.Fig. 7 shows the adopted scheme for the motor control and position estimation for a single three-phase system.The aforementioned blocks together with the HF injection and the inverter pulse width modulation constitute the control (CTRL) loop of the motor.

A. Self-Sensing Capability
Self-sensing capability refers to the ability of a motor to be properly controlled in low-speed sensorless mode.Magnetic saturation and cross-coupling negatively affect estimator's performance.The former reduces the detectable anisotropy, whereas the latter one induces a position estimation error, which degrades the electric drive performance and it could lead to stability issues.The estimated rotor position ϑ me (i d , i q ) differs from the actual one by the estimation error as ϑ me is the actual electromechanical rotor position and (i d , i q ) is the estimation error.It can be computed as where l dq (i d , i q ) is the cross-differential inductance obtained as ∂λ d (i d , i q )/∂i q = ∂λ q (i d , i q )/∂i d and l Δ is the semidifference inductance defined as the difference between the self-differential inductances l Δ (i d , i q ) = (l dd (i d , i q ) − l qq (i d , i q ))/2.It is worth noting that the estimation error depends on the operating point since saturation and crosssaturation inductance vary according to the current operating point.At low current cross-saturation inductance is usually negligible and, in turn, the estimation error.As the current increases becomes more relevant.
In a sensored electric drive, the current control loops work in the actual dq reference frame and the operating point coincides where i x d and i x q represent the current operating point rotated by the estimation error, so the trajectory t 1 is obtained.The trajectory t 1 exists for any current level, since the drive is not affected by the observer and, in turn, by its estimation error.Finally, it is worth noting that the electric drive does not work along the sensored trajectory 1 since it is only a graphical expedient show the observer's estimation error for any load level.
In a sensorless electric drive, the estimated position is used to perform the Park transformation and to compute the speed feedback, so the estimator performance and its accuracy affect the overall electric drive behavior.The motor operates along the trajectory t 2 , which differs from t 1 , and an effective method to compute it is derived in [22].The trajectory t 2 represents the electric drive operating locus in sensorless mode computed from a reference trajectory.The trajectory t 2 does not exist for any current level, since if the estimation error is too large, no stable points exist for the overall system, leading the electric drive in unstable condition.Stable points around t 2 can be found by computing the intersection between the condition where the HF currents are zero and their derivate with respect the current angle has negative slope [23].

IV. RESULTS
The proposed sensorless control strategy and the self-sensing capability of the motor under test (MUT) are verified throughout an extensive experimental stage.The MUT is a DT synchronous reluctance motor, whose parameters are given in Table II.The injection quantities are listed in Table III.The sampling frequency is equal to the switching one at 10 kHz.The test bench is shown in Fig. 8.The MUT is coupled to a 4.5-kW PMSM, which is supplied by its own inverter.The PMSM, namely, master Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.motor, enables to drag the MUT at a fixed speed.The MUT is controlled along a reference trajectory and each three-phase winding is supplied by its own inverter.The control algorithm is implemented on a dSpace MicroLabBox platform connected to the host PC.Current controllers are designed to achieve a bandwidth of 100 Hz.
Five different tests are carried out to study the self-sensing capabilities of the motor and to verify the effectiveness of the proposed method.The first test is aimed to study the self-sensing capability of the motor when the two three-phase windings are connected in series.In this scenario the motor is supplied by only one inverter, namely, in three-phase configuration.This test is reported in Section IV-A.The second test investigates the HC-mode, hence only the one winding is supplied and the other one disconnected.This scenario is reported in Section IV-B.The interaction between the two three-phase systems and its effect on the self-sensing capability is investigated in Section IV-C by controlling the two three-phase windings in different operating points.The condition where both three-phase windings are supplied with different control strategies by two separate inverter is hereafter called six-phase control.In Section IV-D, the control loops of both windings are closed on the estimated position obtained from the electrical system operating at low current along REF.These tests are implemented at locked rotor ϑ m = 0 rad, namely, at standstill condition.In Section IV-E, the MUT is coupled to a PMSM and dragged at 50 r/min to test the effectiveness of the control strategy developed in Section IV-D at low speed.

A. Three-Phase Configuration
Both three-phase windings are connected in series to operate the motor in three-phase configuration.The result is a single three-phase machine; therefore, a unique estimator is used to retrieve the rotor position.The layout of the experiment is shown in Fig. 9.When the switch is in position 1, the current control operates with the actual measured position and the estimated one is exploited to compute the sensored trajectory t 1 .With the switch in position 2, the CTRL is closed on the estimated Fig. 9. Block scheme of the experiment during the three-phase configuration.When the switch is closed on position 1, the motor operates in sensored mode; otherwise, it operates sensorless.The MUT follows a rampwise reference.Fig. 10.Observer trajectories comparison between measured (t m 1 , t m 2 ) and simulated (t f 1 , t f 2 ) ones when the motor operates as a three-phase system.Unstable points around the trajectory t 2 are highlighted in red line.
position and the electric drive operates in the estimated rotating reference frame, so the sensorless trajectory t 2 can be obtained.Fig. 10 shows the comparison between the trajectories measured during the experimental test (t m 1 , t m 2 ) and computed by FEA magnetic maps (t f 1 , t f 2 ).Experimental and simulated results look very similar.For current magnitude lower than 4 A, the estimation error is almost negligible being t 1 almost overlapped to the reference.The sensorless trajectory t 2 is overlapped as well, since estimation error does not affect the electric drive.For higher current values, an estimation error appears, and it affects the sensorless operation.The trajectory t 2 starts to diverge, according to the analysis carried out in Section III-A.For current values higher than 5.5 A, the system becomes unstable since the open loop estimation error is quite large (see t 1 ) and a convergence point does not exist for the observer.It is worth highlighting that not all the points around t 2 are stable.Indeed the points both in red line both for FEA and experimental measurement represents the unstable points around the sensorless trajectory as discussed in Section III-A.

B. Half Control
In HC-mode, only one three-phase system is supplied.A schematic of the experiment is shown in Fig. 11.The observer trajectories are calculated as in the previous test.The experimental trajectories are compared with the theoretical one obtained by the Apollo software.The comparison is shown in Fig. 12.The measured and the simulated observer trajectories are comparable.A slightly difference can be noted at low current where the simulated trajectories are overlapped but not with the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.reference and the measured one.The current value at which the trajectories are overlapped is the same both for experimental and simulated results and it is equal to 3 A. At higher current values, the sensored trajectories t m 1 and t f 1 are almost the same.The simulated sensorless trajectory t f 2 exhibits convergence issue for current values close to 4.5 A. While the measured sensorless trajectory t m 2 is slightly wider than the simulated one, showing that real operational limit for sensorless operation is slightly higher.
In the following tests, the xyz winding is controlled at constant operating point (COP), whereas the abc one follows a rampwise current reference, as in previous tests.In addition HF signals are injected in both windings to estimate the rotor position.The estimated rotor position with the abc winding is used to study the self-sensing capability of the motor and trace the trajectory t 1 and t 2 , whereas the xyz estimated rotor position is exploited to compute the Park transformation in the closed-loop control.For this reason, the abc winding is called load one and the xyz one is referred as estimation winding.

C. Six-Phase Control: Part 1
This test investigates the effects of the operating point of the estimation winding on the self-sensing capability of the load one.t 1 and t 2 are calculated by implementing the PE in the load winding as in the previous tests but the estimation winding is supplied and it operated at COP.Moreover, the estimation winding control loop is closed on the measured position.A block scheme of the experiment is depicted in Fig. 13.The results of the experiments are shown in Fig. 14 where two Fig. 13.Six-phase control: part 1 block scheme of the experiment.Estimation winding is supplied and operate at COP along REF.Its control loop is closed on the measured position during both the load winding observer trajectories calculation.Fig. 14.Six-phase control: part 1. Load winding observer trajectories when the estimation winding operates at COP.Its control loop is closed on the measured position for both the load winding observer trajectories calculation.operating points are evaluated, namely, (i d , i q ) = (0.5, 0.5) A and (5, 5) A. The self-sensing capability of the load winding is improved in the second case indeed both trajectories t 1 and t 2 follow the reference line for a larger segment and the sensorless trajectory becomes unstable at a higher current level.It is worth remembering that when the sensored and sensorless trajectories overlap with the reference, no estimation error is detected.
FEA analysis is carried out to investigate the obtained results.The open source package Dolomites [24] is used to calculate the estimation error of the load winding in the same scenario experimentally analyzed and results are shown in Fig. 15.A plateau can be recognized from 4 to 6 A when the estimation winding operates at 5 A. In this range, the estimation error of the load winding is lower than the 0.5 A load winding operating condition.It is worth noting that the plateau is located where the self-sensing capability of the machine increases in accordance with the experimental evidences.

D. Six-Phase Control: Part 2
In the test described in Section IV-C, a negligible estimation error is observed with the estimation winding partially loaded indeed both sensored and sensorless trajectories overlap the reference one.This test exploits previous results and verifies the self-sensing capability of the motor in complete sensorless operating mode.The measured rotor position is used only for checking the estimation accuracy.The estimation winding is Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.controlled to a low current COP as a negligible estimation error is obtained.The estimated rotor position ϑ xyz me is used by the control of the estimation winding as well as the load one during the tracing of the trajectory t 1 .The estimated position ϑ abc me is used to measure the sensorless trajectory t 2 , as in previous tests.Fig. 16 depicts the block scheme of the experiment.Fig. 17 shows the experimental results obtained with the estimating three-phase system working in two different COP.The sensorless trajectory t 2 is computed as follows: where the measured position is replaced by the estimated ϑ xyz me .It can be noted that the self-sensing capability of the load winding is superior when the estimation winding has a higher current, according to the analysis carried out in Section IV-C.Fig. 18 compares both sensored and sensorless trajectories obtained by closing the xyz winding control with the measured position or the estimated one.Both trajectory perfectly overlaps, confirming the developed control strategy.Measured rotor position can be replaced without self-sensing capability degradation by the estimated one obtained with a three-phase electrical system operating at low current level.
It is worth remembering that the complete sensorless mode just analyzed refers to the condition where the rotor of the motor is locked.It is equivalent to the motor operating at zero or standstill condition.The estimation error between the measured position and ϑ xyz me is shown in Fig. 19, and it is negligible as smaller than 0.02 rad.This is further proof of good performance of the proposed control strategy under zero or standstill conditions.

E. Six-Phase Control: Part 3
The aim of this section is further validate the proposed postfault sensorless control.The motor is coupled to a PMSM motor and dragged at 50 r/min.Both control loops are closed on the estimated position obtained by the estimating winding operating at low current, as in test carried out in Section IV-D.The load winding is controlled to follow a current reference till its nominal value.Fig. 20 depicts the position estimation error during the test, which amplitude is comparable with the one shown in Fig. 19.It is worth remembering that both tests are carried out in the same conditions, except for the operating motor speed.Fig. 21 depicts the current trajectory of the load winding in the rotating current plane obtained by using the measured or the estimated rotor position.The current reference line is depicted, as well.For sake of comparison, both standstill and steady-state condition tests are reported.The load winding exhibits a stable behavior since current follows the desired reference.It is worth noting that a 40% overload current is applied in both tests without any stability issue.In such a condition, the motor is expecting to produce almost 70% of the rated nominal torque from FEA simulation analysis.The estimating winding provided a reliable and error-free rotor position estimation since it works at low current values where the cross-coupling effect is negligible.In turn, the load winding is able to work in the desired working condition even in overload.Finally, it is worth noting that the sensorless trajectory t 2 diverged around 5 A in the HC test (see Section IV-B) where the load winding is closed on the estimated position obtained by its estimator (the xyz electrical system is disconnected), whereas the proposed scheme exhibits a stable behavior over the whole current range.

V. CONCLUSION
A postfault sensorless control strategy based on the study of the self-sensing capability of a DT synchronous reluctance motor is investigated in this article.Different supply scenarios, namely, three-phase and HC conditions, has been analyzed, and FEA analysis and experimental results are compared.In both configurations, the motor is not able to operate in sensorless mode at full load since the system is unstable due to the large estimation error induced by the cross-saturation inductance.The peculiar self-sensing capability and the additional degree of freedom of a DT motor are exploited by the proposed low-speed sensorless strategy.The former three-phase electrical winding operates at low speed and estimates the rotor position by injecting a HF rotating signal in the stator reference frame.The EF estimation algorithm is exploited to retrieve the rotor position since no motor parameters knowledge are required for its design and tuning.The negligible magnetic saturation at low-load condition allows an error-free rotor position estimation, which is used by both three-phase electric drives.The latter electric system can operate and accurately follow its current reference, even in overload condition.An extended experimental campaign proved the proposed control strategy.The proposed sensorless algorithm allows for controlling a dual-three synchronous reluctance motor in mechanical sensor postfault condition.No motor parameters are required for the estimation of the rotor position as well as compensation algorithm.The motor performance is reduced as a three-phase system that must work at low current level, but the service continuity is guaranteed and the motor is able to exploit up to 70% of the nominal torque in the analyzed condition.

Fig. 1 .
Fig.1.Adopted layout of the stator windings.One three-phase system is distributed at the top of the stator, the latter one at the bottom.

Fig. 3 .
Fig.3.Torque map comparison between FEA and experimental measurements when the motor operates as three-phase system.

Fig. 4 .
Fig. 4. Flux linkages maps obtained through FEA simulations in halfcondition mode (only the abc three-phase system is supplied).(a) λ abc d(i d , i q ).(b) λ abc q (i d , i q ).

Fig. 7 .
Fig.7.Control (CTRL) loop and PE scheme with rotating injection in αβ and EF technique.The scheme is implemented for both windings.System operates in sensored mode when the switch is in position 1; otherwise, it operates in sensorless mode.

Fig. 11 .
Fig. 11.Block scheme of the experiment during the HC-condition.The abc winding is supplied and the xyz one disconnected.

Fig. 15 .
Fig. 15.Six-phase control: part 1. Open loop estimation error of the load winding.The estimation one is supplied at COP and its control loop closed on the measured position.(a) (i xyz d , i xyz q ) = (0.5, 0.5) A. (b) (i xyz d , i xyz q ) = (5, 5) A.

Fig. 16 .
Fig. 16.Six-phase control: part 2 block scheme of the experiment.Estimation winding is supplied and operate at COP along REF. its estimated position ϑ xyz me replaces the measured one.When the switch is in position 1, the motor operates in complete sensorless mode.

Fig. 18 .
Fig.18.Comparison of the load winding observer trajectories when the estimation one operates at 3 A in complete sensorless and sensored mode.

Fig. 19 .
Fig. 19.Six-phase control: part 2. Estimation error between the measured position and the estimated one from the estimation winding ϑ xyz me working at (i xyz d , i xyz q ) = (3, 3) A. Standstill operation in complete sensorless mode.

Fig. 20 .
Fig. 20.Six-phase control: part 3. Estimation error between the measured position and ϑ xyz me from the estimation winding working at (i xyz d , i xyz q ) = (3, 3) A. Complete sensorless operation when the motor rotates at 50 r/min.

Fig. 21 .
Fig. 21.Six-phase control part 3. Load winding current in the rotating reference frame obtained both with the measured and estimated rotor position.Both standstill and steady state at 50 r/min tests are reported.(a) Standstill condition.(b) Steady-state condition.
Improved Sensorless Control of Multiphase Synchronous Reluctance Machine Under Position Sensor Fault Giuseppe Galati , Student Member, IEEE, Ludovico Ortombina , Member, IEEE, Luigi Alberti , Senior Member, IEEE, and Matteo Berto, Student Member, IEEE REF) one.The estimator works in openloop mode indeed the estimated position ϑ me does not affect the control.The estimation error can be computed for any current level along a predetermined reference trajectory.It can be depicted in the dq current plane by