ESTIMATION OF ROTOR OPERATIONAL CHARACTERISTICS FOR A WIND TURBINE

Abstract

Provided is a method of estimation of rotor operational characteristics, in particular rotor speed, rotor azimuth and rotation direction, of a rotating rotor of a wind turbine, the method including: measuring pulse rising edge time and pulse falling edge time of pulses generated by each of multiple proximity sensors originating from multiple detection targets arranged on the rotor; estimating values of parameters associated with the sensors and/or targets, in particular parameters associated with the positioning and/or detection range of at least one sensor and/or the parameters associated with the positioning and/or size of at least one target, based on the measured pulse rising edge times and pulse falling edge times; estimating rotor operational characteristics, in particular a rotor speed and/or a rotor azimuth and/or a rotation direction, based on the measured pulse rising and/or falling edge times and/or the estimated values of parameters associated with the sensors and/or targets.

Claims

1-15. (canceled)

16. A method of estimation of rotor operational characteristics including rotor speed, rotor azimuth and rotation direction, of a rotating rotor of a wind turbine, the method comprising: measuring pulse rising edge time and pulse falling edge time of pulses generated bv each of multiple proximity sensors originating from each of multiple detection targets arranged on the rotor; estimating values of parameters associated with the positioning and detection range of at least one sensor based on the measured pulse rising edge times and pulse falling edge times, and estimating rotor operational characteristics, including a rotor speed and a rotor azimuth and a rotation direction, based on the measured pulse rising edge times and pulse falling edge times and the estimated values of parameters associated with the sensors.

17. The method according to claim 16, further comprising: estimating values of parameters associated with the positioning and/or size of at least one target, based on the measured pulse rising edge times and pulse falling edge times: estimating the rotor operational characteristics further based on the estimated values of parameters associated with the targets, and wherein estimating values of parameters associated with the sensors and targets includes estimating target diameters and/or target angular size of at least one target, and target relative angular spacing of at least one pair of targets, and/or sensor-target radial distance of at least one pair of sensor anti target, and a sensor detection angular range of at least one sensor, and sensor relative angular spacing of at least one pair of sensors.

18. The method according to claim 16, wherein estimating rotor operational characteristics and estimating values of parameters associated with the sensors and/or targets is based on a measurement model, including a mathematical model relating the measured pulse rising edge times and pulse falling edge times to the rotor operational characteristics and/or parameters associated with the sensors and/or targets.

19. The method according to claim 18, wherein estimation of the rotor speed and the rotor azimuth and the rotation direction and/or tire values of parameters associated with the sensors and/or targets is performed by applying an adaptive filter to the measurement model.

20. The method according to claim 16, wherein estimation of the rotor speed and the rotor azimuth and the rotation direction and parameters associated with the sensors and targets is based on at least one of: measured times of pulses being high and times of pulses being low as detected by at least one sensor: and measured times between pulses as detected by at least one pair of sensors.

21. The method according to claim 16, wherein the targets comprise: a reference target: and at least one relative target.

22. The method according tp claim 16, wherein the sensors comprise: at least one reference sensor, and at least one relative sensor, wherein pulses generated by the at least one reference sensor originate from detections of the reference target and pulses generated bv the at least one relative sensor originate from detections of a relative target, wherein the at least one reference sensor, the at least one relative sensor, the reference target and the at least one relative target are positioned, in particular regarding a respective radial position, such that: each of the at least one relative target is out of the sensor detection range of the at least one reference sensor: and each of the at least one reference target is out of the sensor detection range of each of the at least one relative sensor.

23. The method according to claim 21, wherein the reference target, a reference sensor, and a relative sensor are configured and positioned such that pulses originating from the reference target, as detected by the reference sensor, overtap in time with pulses originating from a relative target as detected bv the relative sensor.

24. The method according to claim 21. wherein estimating the direction of rotation includes: detecting whether a reference sensor or its associated relative sensor first detects its corresponding target.

25. The method according to claim 21, wherein a reference azimuth position is detected when a reference sensor detects a pulse originating from the reference target, wherein a mounting position of the reference sensor is determined by calibration using external azimuth measurement.

26. The method according to claim 21. further comprising: determining a difference between subsequent pulse rising edge and/or pulse failing edge times, the first and second pulse edge times detected by any of the at least one relative sensor and originating from any of the at least one relative target, and wherein estimating the rotor azimuth is based on a previsoly estimated rotor azimuth increased or decreased, depending on estimated rotation direction, by an integral of the estimated rotor speed over the difference.

27. The method according iq claim 16, further comprising: validating of at least one of the sensors by making a consistency check of measurement results of this sensor and the estimated rotor speed and/or estimated rotor azimuth; and disqualifying of one sensor, if it does not pass the consistency check.

28. The method according to claim 27, further comprising, if only one sensor passes the consistency check. estimating the rotor speed based on pulse high and pulse low time measurements of only the sensor having passed the consistency check.

29. An arrangement for estimation of rotor operational characteristics, in particular rotor speed, rotor azimuth and rotation direction, of a rotating rotor of a wind turbine, the arrangement comprising: multiple proximity sensors; wherein the arrangement is configured; to measure pulse rising edge time and pulse falling edge time of pulses generated by each of multiple proximity sensors originating from each of multiple detection targets arranged on the rotor: to estimate values of parameters associated with the positioning and detection range of at least one senior based on the measured pulse rising edge times and pulse tailing edge times, and to estimate rotor operational characteristics, including a rotor speed and a rotor azimuth and a rotation direction, based on the measured pulse rising edge times and pulse falling edge times and the estimated values of parameters associated with the sensors

30. A wind turbine, comprising: a rotor in which plural rotor blades are mounted according to the arrangement according to claim 29.

Description

BRIEF DESCRIPTION

[0049] Some of the embodiments will be described in detail, with reference to the following figures, wherein like designations denote like members, wherein:

[0050] FIG. 1 schematically illustrates a portion of a wind turbine according to an embodiment of the present invention comprising an arrangement for estimation of rotor operational characteristics according to an embodiment of the present invention;

[0051] FIG. 2 schematically illustrates a conceptional circuit diagram of an arrangement for estimation of rotor operational characteristics of a rotating rotor of a wind turbine according to an embodiment of the present invention;

[0052] FIG. 3 illustrates the modelled angular distances; and

[0053] FIG. 4 illustrates the corresponding measured time differences associated with sensor pulse traces of several sensors as considered in embodiments of the present invention.

DETAILED DESCRIPTION

[0054] The portion 3 of a wind turbine according to an embodiment of the present invention schematically illustrated in FIG. 1 in a sectional view along an axial direction 5 (direction of a rotation axis of a rotor) comprises an arrangement 1 for estimation of rotor operational characteristics of a rotating rotor 7 according to an embodiment of the present invention. A schematic circuit diagram of the arrangement 1 is also depicted in FIG. 2 which will be described below.

[0055] The arrangement 1 comprises multiple inductive proximity sensors S.sub.Al, S.sub.R1, S.sub.R2 which are mounted (in particular close to a frontal face of a rotor lock disk) at a not in detail illustrated stationary portion near to the main bearing of the wind turbine 3. The absolute inductive proximity sensor SA1 is mounted at a different radial position, in particular further outwards, from the centre of rotation than the relative sensors S.sub.R1, S.sub.R2. The arrangement 1 is configured to measure pulse rising edge time and pulse falling edge time of pulses (see for example FIG. 4) P1, P2, P3, P4, P5, generated by each of the multiple inductive proximity sensors S.sub.A1, S.sub.R1, S.sub.R2 originating from an absolute target T.sub.A and two of multiple relative targets T.sub.R1, T.sub.R2, . . . , T.sub.Rn which are arranged at the rotor 7. The absolute target T.sub.A is mounted at a different radial position than the relative targets T.sub.R1, T.sub.R2, . . . , T.sub.Rn. The absolute target T.sub.A can only be detected by the absolute sensor (but not any of the relative sensors) and the relative targets T.sub.R1, T.sub.R2, . . . , T.sub.Rn can only be detected by one of the relative sensors (but not the absolute sensor).

[0056] The arrangement 1 is further configured to estimate the rotor operational characteristics based on the measured pulse rising edge times and pulse falling edge times.

[0057] Rather than utilize and mount only one proximity sensor to detect targets on a dedicated, precision-machined rotating part, embodiments of the present invention propose to utilize and mount multiple (inductive) proximity sensors to detect targets already pre-existing at a rotating part, and in particular at the rotor 7. In particular, the rotor lock disk near the main bearing of the wind turbine may be a part including the detection targets TA, T.sub.R1, T.sub.R2, . . . , T.sub.Rn which are monitored by the sensors S.sub.A1, S.sub.R1, S.sub.R2. Although only three sensors are depicted in FIG. 1, other embodiments of the present invention utilize more than three sensors, such as four, five, six or in-between three and 50 or even more.

[0058] In particular, the absolute target ‘T.sub.A’ (also referred to as reference target) may be realized by a (iron) ferrous piece to be detected by the absolute sensor S.sub.A1, while the relative targets T.sub.R1, TR2 , . . . , T.sub.Rn may each be a previously-machined hole around the circumference of the ferrous disk acting as the relative target to be detected by the relative sensors S.sub.R1, S.sub.R2. However, any combination of holes, exposed bolt heads, gear teeth, mounted pieces, etc. may be utilized as targets (in particular relative targets) according to embodiments of the present invention.

[0059] By using multiple sensors S.sub.A1, S.sub.R1, S.sub.R2 embodiments of the present invention may have the benefit of being able to simultaneously estimate the rotor speed, rotor azimuth and also determine the direction of the rotation. Further advantages may include measurement redundancy that may allow for improved sensor validation, robustness to vibration or to faults with individual sensors, and the ability to estimate and compensate for variations in sensor sensitivity, sensor placement, target size and target spacing. In addition, the total cost of the inductive proximity sensors may be less than alternative solutions, including the cost of precision-machined dedicated parts used by conventional inductive proximity sensor methods.

[0060] In FIG. 1 also relevant sensor positioning parameters, for example designed (or nominal) azimuthal distance between adjacent sensors, labelled as Δϕ and misalignment ε.sub.1,2 between sensors S.sub.R1 and S.sub.R2 are depicted in FIG. 1. Furthermore, target positioning parameters, such as the radial and/or circumferential positioning of the absolute target T.sub.A, the radial and/or azimuthal positioning of the relative targets T.sub.R1, T.sub.R2 are indicated in FIG. 1. Furthermore, target size parameters, such as the relative target diameter d.sub.rel, the azimuthal relative target size Θ.sub.rel are indicated in FIG. 1. Furthermore, target relative positioning parameters, such as relative target spacing Φ.sub.rel is indicated in FIG. 1. These positioning and/or size parameters of the sensors and/or targets are utilized in the algorithm to determine the direction of rotation, rotational speed and rotor azimuth.

[0061] Although the embodiment illustrated in the figures demonstrates the use of three sensors, any number of sensors more than two may be used. While q≥1 sensors (referred to as sensors S.sub.A1, . . . , S.sub.Aq, or the “absolute sensors”) each detect an absolute target T.sub.A once per revolution of the rotor, the remaining m≥1 sensors (referred to as sensors S.sub.R1, S.sub.R2, . . . , S.sub.Rm, or the “relative sensors”) each detect multiple “relative targets” arranged around the circumference of the part n times per revolution, where n≥1 is the number of relative targets used. In the illustrated embodiment the mounted ferrous piece ‘T.sub.A’ assumes the role of the absolute target while the previous-machined holes T.sub.R1, T.sub.R2, . . . , T.sub.Rn assume the role of the relative targets.

[0062] The arrangement 1 comprises a processing section 9 which receives measurement results 11 of all the proximity sensors S.sub.A1, S.sub.R1, S.sub.R2 and processes these measurement results 11 in order to estimate the rotor operational characteristics based thereon.

[0063] FIG. 1 further indicates the sensor sensitivity p (also referred to as azimuthal viewing range of the sensor or angular viewing range of the sensor).

[0064] The arrangement 1 is depicted in more detail in FIG. 2 as a schematic circuit diagram including the processing section 9 and the sensor section 10. The conceptional diagram illustrated in FIG. 2 may be implemented in software and/or hardware. The arrangement 1 includes the absolute sensor S.sub.A1, a first relative sensor S.sub.R1 and a second relative sensor S.sub.R2. In the block 13 the pulse times of pulses as detected by the sensors S.sub.R1 and S.sub.R2 are measured and determined. The measurement results 11 of the absolute sensor S.sub.A1 and the first relative sensor S.sub.R1 are supplied to a block 15 which determines the direction of rotation of the rotor.

[0065] In a module 17 a pre-validation of the proximity sensor is carried out, wherein the measurement results for the different sensors are assessed for consistency.

[0066] If it is determined that multiple sensors are valid, it is branched into the branch 19 which leads to a multiple sensor estimator 21. Therein, the measurement results of multiple sensors are utilized. In a calibration step 23, the relevant parameters, such as sensor positioning parameters and/or target positioning parameters and target size parameters are calibrated based on the pulse measurements 11. In a method block 25 the rotor speed and the change in the azimuth is estimated based on the measurement results after calibrating the parameters.

[0067] When the proximity sensor pre-validation (module 17) assesses only one sensor as valid, it is branched into the branch 27 which leads to a single sensor estimation module 29. Also herein, the relevant positioning and/or sizing parameters of the sensors and/or targets are calibrated in a module or method step 31. The calibration of parameters is not always being performed, but only in the case the decision element 33 assesses that the parameters are not yet calibrated. After calibration of the parameters, it is proceeded to the method step 34 where the rotational speed and the change in azimuth is estimated based on the measurement results after calibration or using the calibrated parameters.

[0068] The multi-sensor estimator 21 as well as the single-sensor estimator 29 provide their output to a proximity sensor post-validation block 35 in which a post-validation of the proximity sensors is carried out including one or more consistency checks. The block 35 outputs the status 37 of the sensors which is fed-back to the proximity sensor pre-validation module 17.

[0069] Furthermore, the proximity sensor post-validation outputs the estimate 39 of the rotational speed. Furthermore, the proximity sensor post-validation outputs the estimation 41 of the change in the azimuth.

[0070] The estimate 39 of the rotational speed as well as the estimate 41 of the change in azimuth are both provided to a proximity-based azimuth estimator 43, namely to an integrator 45 and a relative azimuth update module 47, respectively. The proximity-based azimuth estimator 43 receives for calibration external azimuth data 49 which is supplied to an azimuth sensor calibration module 51. To this module 51 also the measurement data 53 as determined by the absolute proximity sensor SA1 are provided.

[0071] If the decision block 55 assesses that the azimuth is not calibrated, the azimuth will be calibrated in the calibration module 51. If it is assessed that the azimuth is calibrated, an absolute azimuth update is performed in module 57 and the output is provided to the relative azimuth update module 47. This relative azimuth update module 47 further receives the signal 59 indicating the direction of rotating from the rotational direction determination module 15. The estimate 61 of the rotor azimuth is finally output by the integration module 45 and the estimation 59 of the direction rotation is also output by the arrangement 1.

[0072] As shown in FIG. 2, the algorithm not only estimates the rotor speed, azimuth, and direction of rotation, but also takes steps to calibrate the azimuth and unknown parameters. Moreover, steps are taken to validate both the input and output signals of the rotor speed estimation. This includes ensuring that there is no chatter in the sensor pulses and that the pulse time measurements and rotor speed estimates appear consistent over time. These checks on the inputs and outputs of the speed estimation are referred to as the “Pre-Validation” (block 17) and “Post-Validation” (block 35) steps, respectively. Depending on the severity and duration of the detected fault, it can either be ignored, corrected, or used to disqualify the sensor from the rotor speed estimation. Each of these responses are made within the “Pre-Validation” step, using the measurements from the sensors and the previous status reported by “Post-Validation” as inputs. In the case when the azimuth sensor is disqualified, the azimuth can continue to be estimated by using the incremental azimuth fixes at each relative target ( T.sub.R1, T.sub.R2, T.sub.R1) and integrating (block 45) the speed estimate between these targets. An absolute azimuth fix can still be performed by determining the relative target associated with the fix. This can be done by a combination of counting through the known number of relative targets and finding the edge of the target that is closest to the calibrated absolute azimuth value.

[0073] After disqualification of a speed sensor, the rotor speed and azimuth estimations can continue depending on the number of sensors that remain valid. If more than one sensor are valid, each of their pulse times as well as the times between their pulses can be used to estimate rotor speed by the multi-sensor estimator described above.

[0074] However, if only one sensor is valid, its pulse times can still be used to estimate rotor speed by a variant of the original algorithm referred to as the single-sensor estimator (block 29). This estimator performs the same function as the multi-sensor estimator (block 21) but using only the high and low time measurements associated with the one valid sensor. Thus, since the number and spacing of sensors S.sub.R1, SR2, . . . , S.sub.Rm are chosen to optimize the update rate of the estimates, improving performance, the performance begins to degrade as the number of relative sensors m reduces to 1. Finally, if no valid sensors remain to be used in the estimation, an alarm must be reported by the functionality to indicate that the rotor speed and rotor azimuth cannot be estimated.

[0075] As is evident from the conceptual diagram of the proximity-based speed, azimuth, and direction estimation algorithm illustrated in FIG. 2, the algorithm first determines the direction of rotation and validates the proximity sensors before updating the rotor speed and rotor azimuth estimates using the pulses and pulse times acquired. The algorithm provides the direction of rotation, estimates of the speed and azimuth, as well as the status of the proximity sensors.

[0076] FIG. 3 schematically illustrates pulses as detected by the proximity sensors S.sub.A1, S.sub.R1, S.sub.R2 and the combination of sensor S.sub.R1 and sensor S.sub.R2. Furthermore, the relative positioning parameters of the relative targets Φ.sub.rel, the relative target size (azimuthal target extent Θ.sub.rel and the sensor azimuthal viewing range ρ are indicated as defining for example the angular differences between a falling edge and a rising edge between subsequent pulses or a rising edge and a falling edge of one pulse.

[0077] FIG. 4 illustrates these same pulses, wherein however the time differences as determined by the arrangement 1 are indicated. From the time differences the respective rotor operational characteristics are determined and values of parameters associated with the sensors and/or targets are calibrated according to embodiments of the present invention.

[0078] As the rotor rotates, the proximity sensors S.sub.A1, S.sub.R1, S.sub.R2 output binary digital signals 11, 53 indicating whether or not they detect a target. As a function of time the sensor signals resemble pulses (e.g. P1, . . . , P5 in FIGS. 3, 4 for the case with an absolute target, at least two relative targets (n>1), one absolute sensor (q=1) and two relative sensors (m=2)), whose high and low times depend on the speed of the rotor as well as on the sensitivity and alignment of the sensors and on the size and spacing of the targets (see FIG. 1). An embodiment of the present invention uses computational hardware to measure the time (e.g., Δt1, Δt2, Δt3, Δt4 in FIG. 4) that each sensor's pulse is high and low, as well as the times between the pulses produced by each pair of successive sensors (e.g. Δt5, Δt6 in FIG. 4). By mapping the angular distances (see FIG. 3) associated with the sensor pulses to each of their corresponding time measurements (see FIG. 4), the rotor speed can be estimated along with the values for the unknown parameters that influence the time measurements.

[0079] For example, if any of the sensors are misaligned by various amounts around the circumference of the rotating part with respect to the other sensors, if these sensors have varied sensitivities to ferrous metal, or if the targets vary in size and placement (e.g., due to being poorly machined), an adaptive filter automatically estimates these uncertainties and compensates for them in its estimation of the rotor speed. Depending on the degree of variation between the targets, an additional step may be introduced in which compensation is applied as a function of the targets. In other words, the parameter values may be estimated separately for each individual target and the appropriate set of values would be applied when the corresponding target is encountered.

[0080] By integrating multiple proximity sensors in the solution and utilizing existing rotating machined features, embodiments of the invention remove several limitations confronting prior solutions. Since embodiments of the invention use multiple proximity sensors in the detection of targets, each target is able to be detected more than once. This measurement redundancy allows the uncertainty caused by variations in the sensors and the targets described above to be estimated and prevented from corrupting the rotor speed and azimuth estimates. Thus, the need to precisely machine the targets, and the associated cost of this process, is mitigated. By spacing the multiple proximity sensors out such that they span the distance between two consecutive targets, a sufficiently fast update rate can be achieved with significantly less targets per revolution. Misalignments in the sensors and variations in their sensitivity are automatically estimated and compensated for by the algorithm. Moreover, by attaching a unique target to be detected once per revolution by one of the sensors (i.e., the “absolute target”), accurate determination of the rotor azimuth and the direction of rotation can also be achieved.

[0081] At least one reference (a.k.a. absolute sensor) may be needed and at least one relative sensor may be needed to estimate rotor speed, azimuth, and rotation direction. Reference sensors are only able to detect the reference target and the relative sensors are only able to detect the relative targets. If the reference target was visible to the relative sensors, there would be no need for a reference sensor: direction of rotation and absolute azimuth updates could come from the relative sensors by identifying which pulse is distinct from the others and associating it with the reference target. However, this can introduce more complexity to the solution or be more inconvenient (it can be more convenient or appropriate to introduce a reference target of opposite type as the relative targets) than introducing a reference sensor. However, the reference sensor and target may be moved out of range from the relative sensors/targets, otherwise the reference and relative sensors will become functionally equivalent.

[0082] There may be two types of targets, e.g., metal and absence of metal (i.e., holes). Often it may be convenient to attach a reference target that is the opposite type as the relative targets. In this case a reference sensor is required to detect the reference target type, since the relative sensors have been mounted and positioned to trigger upon detection of the relative target type. In other cases, it is more convenient to attach a reference target that is the same type, however as mentioned above, it must be out of range of the relative sensors.

[0083] To improve aspects of the solution such as accuracy, robustness to individual sensor faults, increased update rate, it may become beneficial to introduce additional reference and/or relative sensors. With more reference sensors, there is simply more robustness to faults with one or more reference sensors. However, with more relative sensors, there is an increase in the update rate of the estimated rotor characteristics, increased robustness to individual sensor faults, but also the solution has increased abilities, including the ability to estimate the size of and spacing between the relative targets.

[0084] One alternative to using reference targets and reference sensors is to leverage the extra ability of estimating the sizes of and spacings between the relative targets: since each real target would be expected to have a slightly different size and spacing, each relative target could be directly identified according to the unique values it has for these parameters. However, this would require that the target fabrication be imprecise enough to be detected by the sensitivity and resolution of the hardware/software used in the solution. Again, it may be more feasible to simply use a reference target and reference sensor(s).

[0085] The reference target and reference sensor may be placed such that the pulse generated by the reference sensor when detecting the reference target will overlap with a pulse generated by the relative sensor when detecting a relative target.

[0086] The model according to one embodiment is described in detail below:

[0087] When relating the modelled angular distances and measured times associated with the pulses originating from the detection of a target by two relative sensors (m=2), the following equations are obtained:

[00001]$\begin{array}{cc}{\theta }_{\mathrm{rel}}+2{\rho }_1=\left(\omega \times \Delta t_{X2S1F}\right){\varphi }_{\mathrm{rel}}-\left({\theta }_{\mathrm{rel}}+2{\rho }_1\right)=\left(\omega \times \Delta t_{X2S1R}\right){\theta }_{\mathrm{rel}}+2{\rho }_2=\left(\omega \times \Delta t_{X2S2F}\right){\varphi }_{\mathrm{rel}}-\left({\theta }_{\mathrm{rel}}+2{\rho }_2\right)=\left(\omega \times \Delta t_{X2S2R}\right)\left({\varphi }_{\mathrm{rel}}/2\right)+{ϵ}_{1,2}-\left({\theta }_{\mathrm{rel}}+{\rho }_1+{\rho }_2\right)=\left(\omega \times \Delta t_{X4S2R}\right)\left({\varphi }_{\mathrm{rel}}/2\right)-{ϵ}_{1,2}-\left({\theta }_{\mathrm{rel}}+{\rho }_1+{\rho }_2\right)=\left(\omega \times \Delta t_{X4S1R}\right).& \left(1\right)\end{array}$

[0088] , where ω is the average rotor angular speed as exhibited by the target. In these equations, some variables are known constants, some variables are unknown and must be estimated, while some variables are measurements of low and high times obtained by counter modules at the rising edge and falling edge of sensor pulses, respectively. The terms on the left hand side of eq. (1) correspond to the angular distances travelled by each target as it passes sensor S.sub.R1 and sensor S.sub.R2 to produce the pulse traces given in FIG. 3. The difference times on the right hand side of eq. (1) correspond to the time differences measured by sensor S.sub.R1 and sensor S.sub.R2 in FIG. 4.

[0089] The variables ϕ.sub.rel and θ.sub.rel are the angular spacing (in degrees) between the relative targets and the angular width (in degrees) of the relative targets, respectively. Moreover, o, and are angular quantities that represent the error in the width of the pulses (affecting the high times measured) produced by S.sub.R1 and S.sub.R2, respectively, which is caused by uncertainties in the sensitivities of the sensors compared to the specifications given by the sensor manufacturer (e.g., due to variations in temperature, input voltage, wear, etc.). The initial values assumed for ρ.sub.1 and can be determined using the specifications found in the datasheets for the sensors. Note that the sign convention taken for ρ.sub.1 and ρ.sub.2 is as follows: positive values will tend to expand the pulse widths from the sensors while negative values will tend to reduce the pulse widths. Thus, sensors configured to detect metal targets (e.g., bolt heads, gear teeth, attached pieces, etc.) will tend to have positive values for these parameters while sensors configured to “detect” holes or gaps will tend to have negative values. In addition to these parameters, ϵ.sub.1,2 is an angular quantity that represents the error in the spacing between S.sub.R1 and S.sub.R2 (i.e., the deviation from the ideal spacing of ϕ.sub.rel/.sup.2 affecting the low times measured). Since the true value of this misalignment should be relatively small in either direction, an initial value of zero is assumed for ϵ.sub.1,2. Note that the sign convention taken for ϵ.sub.1,2 is as follows: positive values for this parameter will tend to expand the distance and measured time from the falling edge of S.sub.R1 to the rising edge of S.sub.R2 while reducing the distance from the falling edge of S.sub.R2 to the rising edge of S.sub.R1, with negative values having the reverse effect. In the case of more than two speed sensors, positive values of ϵ.sub.k−1.k can be thought of as expanding the distance on the left side of the pulse from Sensor k while reducing the distance on the right side of this pulse.

[0090] The measurements of high and low times have subscripts that adhere to a particular convention. Measurements whose subscripts begin with ‘X2’ are obtained by monitoring the pulse trains associated with only one sensor and thus consist of 2 edges (i.e., one rising and one falling) per period. Meanwhile, measurements whose subscripts begin with ‘X4’ are obtained by monitoring the pulse trains from all of the available relative sensors together: in the case where m=2, these are the pulse trains that consist of 4 edges (i.e., two per sensor) per target; thus, the subscripts ‘X4’ are used. While the measurements that end in ‘R’ are obtained at the rising edge of S.sub.R1 or S.sub.R2, measurements ending in ‘F’ are obtained at the falling edge of that sensor. Although six equations are listed in (1), only four of them are independent. With four unknowns to be estimated (i.e., ω, ρ.sub.1, ρ.sub.2, and ϵ.sub.1,2), this is enough information to complete the estimation. Fortunately, it can be shown that, by dividing both sides of (1) by ω the equations can be made linear in the unknowns:

[00002]$\begin{array}{cc}{\theta }_{\mathrm{rel}}\frac{1}{\omega }+2{\tilde{\rho }}_1=\Delta t_{X2S1F}\left({\varphi }_{\mathrm{rel}}-{\theta }_{\mathrm{rel}}\right)\frac{1}{\omega }-2{\tilde{\rho }}_1=\Delta t_{X2S1R}{\theta }_{\mathrm{rel}}\frac{1}{\omega }+2{\tilde{\rho }}_2=\Delta t_{X2S2F}\left({\varphi }_{\mathrm{rel}}-{\theta }_{\mathrm{rel}}\right)\frac{1}{\omega }-2{\tilde{\rho }}_2=\Delta t_{X2S2R}\left({\varphi }_{\mathrm{rel}}/2-{\theta }_{\mathrm{rel}}\right)\frac{1}{\omega }-{\tilde{\rho }}_1-{\tilde{\rho }}_2+{\widetilde{ϵ}}_{1,2}=\Delta t_{X4S2R}\left({\varphi }_{\mathrm{rel}}/2-{\theta }_{\mathrm{rel}}\right)\frac{1}{\omega }-{\tilde{\rho }}_1-{\tilde{\rho }}_2-{\widetilde{ϵ}}_{1,2}=\Delta t_{X4S1R}.& \left(2\right)\end{array}$

[0091] Note that formulating (2) involved a transformation of variables: rather than being linear in the original unknowns (i.e., ω, ρ.sub.1, ρ.sub.2, and ϵ.sub.1,2), (2) is linear in

[00003]$\frac{1}{\omega },$

{tilde over (ρ)}.sub.1, {tilde over (ρ)}.sub.2, and {tilde over (ϵ)}.sub.1,2, where {tilde over (ρ)}.sub.1=ρ.sub.1/ω, {tilde over (ρ)}.sub.2=ρ.sub.2/ω, and {tilde over (ϵ)}.sub.1,2=ϵ.sub.1,2/ω. Therefore, the procedure to estimate all four original parameters involves first estimating the values of

[00004]$\frac{1}{\omega },$

{tilde over (ρ)}.sub.1, {tilde over (ρ)}.sub.2, and {tilde over (ϵ)}.sub.1,2, calculating the reciprocal of to obtain the speed estimate, and then multiplying the speed estimate by {tilde over (ρ)}.sub.1, {tilde over (ρ)}.sub.2, and {tilde over (ϵ)}.sub.1,2 to yield ρ.sub.1, ρ.sub.2, and ϵ.sub.1,2, respectively. To estimate

[00005]$\frac{1}{\omega },$

{tilde over (ρ)}.sub.1, {tilde over (ρ)}.sub.2, and {tilde over (ϵ)}.sub.1,2, a linear least mean squares (LMS) filter is applied that recursively adjusts their values until their mean-square error is minimized. However, note that only 4 of the 6 measurements are used to perform the estimation. If both sensors are deemed to be performing well, the measurements Δt.sub.X2S1F, Δt.sub.X4S2R, Δt.sub.X2S2F, and Δt.sub.X4S1R are related to the unknowns in the matrix equation Hx=y:

[00006]$\begin{array}{cc}\left[\begin{array}{cccc}{\theta }_{\mathrm{rel}}& 2& 0& 0\\ \left({\varphi }_{\mathrm{rel}}/2-{\theta }_{\mathrm{rel}}\right)& -1& -1& 1\\ {\theta }_{\mathrm{rel}}& 0& 2& 0\\ \left({\varphi }_{\mathrm{rel}}/2-{\theta }_{\mathrm{rel}}\right)& -1& -1& -1\end{array}\right]\left[\begin{array}{c}1/\omega \\ {\tilde{\rho }}_1\\ {\tilde{\rho }}_2\\ {\widetilde{ϵ}}_{1,2}\end{array}\right]=\left[\begin{array}{c}\Delta t_{X2S1F}\\ \Delta t_{X4S2R}\\ \Delta t_{X2S2F}\\ \Delta t_{X4S1R}\end{array}\right]& \left(3\right)\end{array}$

[0092] Thus, x is the vector of transformed unknowns, y is the vector of measurements, and H is the matrix of coefficients that map the transformed unknowns onto the measurements. Then, an LMS filter can be used to recursively arrive at an estimated vector of transformed unknowns, {circumflex over (x)};

[00007]$\begin{array}{cc}e\left(k\right)=y\left(k\right)-H.Math.\hat{x}\left(k\right)\hat{x}\left(k+1\right)=\hat{x}\left(k\right)+\mu .Math.H.Math.e\left(k\right).& \left(4\right)\end{array}$

[0093] Where e(k) is the error between the actual time measurements obtained, y(k), and the predicted measurements H.Math.{tilde over (x)}(k), and μ is a constant scale parameter whose value is chosen to balance speed of responsiveness with stability. Note that the reciprocal of the first element of is the rotor speed estimate that we are after and is expected to change significantly over time. The remaining three elements of {tilde over (x)}, which can be used to peer into the amount of uncertainty (e.g., sensor misalignment, machining error, etc.) affecting the measurement system, are expected to approximately constant over time. Note that, the lower the value of μ, the more time required for the rotor speed estimates to converge. Therefore, a method is introduced to evaluate whether the three estimated parameters have sufficiently converged so that, when convergence has occurred, the LMS filter is replaced with direct calculation of rotor speed using an alternative reformulation of the equations in (1):

[00008]$\begin{array}{cc}\frac{{\theta }_{\mathrm{rel}}+2{\rho }_1}{\Delta t_{X2S1F}}=\omega \frac{\left({\varphi }_{\mathrm{rel}}/2\right)+{ϵ}_{1,2}-\left({\theta }_{\mathrm{rel}}+{\rho }_1+{\rho }_2\right)}{\Delta t_{X4S2R}}=\omega \frac{{\theta }_{\mathrm{rel}}+2{\rho }_2}{\Delta t_{X2S2F}}=\omega \frac{\left({\varphi }_{\mathrm{rel}}/2\right)-{ϵ}_{1,2}-\left({\theta }_{\mathrm{rel}}+{\rho }_1+{\rho }_2\right)}{\Delta t_{X4S1R}}=\omega .& \left(5\right)\end{array}$

[0094] Each of the equations in (5) represents a calculation of the rotor speed from the converged estimated values of the parameters and the time measurements obtained from the sensor pulses. Depending on which edge of which sensor pulse has arrived (i.e., the falling or rising edge of S.sub.R1 or S.sub.R2), the corresponding equation in (5) is used to directly update the rotor speed estimate. However, under certain conditions (e.g., significant changes in temperature, sensor wear, input voltage, etc.), the sensor parameters may need to be re-calibrated using the LMS filter. Therefore, a routine parameter check can be scheduled once each day or so by applying the LMS filter, provided that conditions allow for it; otherwise, the solution should run the direct calculations in (5) until conditions change.

[0095] If only one sensor is deemed to be performing well, then the above procedure can be applied to the one working sensor by ignoring the pulses and parameters associated with the faulty sensor and applying the relevant equations in (1) and (2). For example, if S.sub.R2 is determined to be faulty, the measurements that depend on the pulses from S.sub.R2 (i.e., Δt.sub.X2S2F, Δt.sub.X4S2R, and Δt.sub.X4S1R) as well as the associated parameters (i.e., {tilde over (ρ)}.sub.2 and {tilde over (ϵ)}.sub.1,2) are ignored. The remaining measurements and their associated equations in (2) are used to form the following matrix equation:

[00009]$\begin{array}{cc}\left[\begin{array}{cc}{\theta }_{\mathrm{rel}}& 2\\ \left({\varphi }_{\mathrm{rel}}-{\theta }_{\mathrm{rel}}\right)& -2\end{array}\right]\left[\begin{array}{c}1/\omega \\ {\tilde{\rho }}_1\end{array}\right]=\left[\begin{array}{c}\Delta t_{X2S1F}\\ \Delta t_{X2S1R}\end{array}\right]& \left(6\right)\end{array}$

[0096] This equation is used with the LMS filter to estimate ω and ρ.sub.1 and, upon convergence of ρ.sub.1, the equations are reformulated to directly calculate ω at each rising and falling edge of the pulses from S.sub.R1. The same can be done using only the measurements from S.sub.R2 (i.e., Δt.sub.X2S2F and Δt.sub.X2S2R). However, note that as the number of sensors used to estimate the rotor speed reduces, so will the update rate with which the rotor speed estimates can be provided.

[0097] In general, more than two sensors can be used to obtain a speed estimate. This would be necessary as the spacing between the targets increases sufficiently relative to the sensitivities of the sensors, and thus can be covered by more sensors in order to maximize the update rate of the speed estimates. With m sensors, the equations used as the basis of the solution are as follows:

[00010]$\begin{array}{cc}{\theta }_{\mathrm{rel}}+2{\rho }_1=\left(\omega \times \Delta t_{X2S1F}\right){\varphi }_{\mathrm{rel}}-\left({\theta }_{\mathrm{rel}}+2{\rho }_1\right)=\left(\omega \times \Delta t_{X2S1R}\right){\theta }_{\mathrm{rel}}+2{\rho }_2=\left(\omega \times \Delta t_{X2S2F}\right){\varphi }_{\mathrm{rel}}-\left({\theta }_{\mathrm{rel}}+2{\rho }_2\right)=\left(\omega \times \Delta t_{X2S2R}\right).Math.{\theta }_{\mathrm{rel}}+2{\rho }_m=\left(\omega \times \Delta t_{X2SmF}\right){\varphi }_{\mathrm{rel}}-\left({\theta }_{\mathrm{rel}}+2{\rho }_m\right)=\left(\omega \times \Delta t_{X2S\mathrm{mR}}\right)\left({\varphi }_{\mathrm{rel}}/m\right)+\left({ϵ}_{1,2}-{ϵ}_{m,1}\right)-\left({\theta }_{\mathrm{rel}}+{\rho }_1+{\rho }_2\right)=\left(\omega \times \Delta t_{X4S2R}\right)\left({\varphi }_{\mathrm{rel}}/m\right)+\left({ϵ}_{2,3}-{ϵ}_{1,2}\right)-\left({\theta }_{\mathrm{rel}}+{\rho }_2+{\rho }_3\right)=\left(\omega \times \Delta t_{X4S3R}\right).Math.\left({\varphi }_{\mathrm{rel}}/m\right)+\left({ϵ}_{m,1}-{ϵ}_{m-1,m}\right)-\left({\theta }_{\mathrm{rel}}+{\rho }_m+{\rho }_1\right)=\left(\omega \times \Delta t_{X4S\mathrm{mR}}\right),& \left(7\right)\end{array}$

which yields the following matrix equation of the form Hx=y to be used with the LMS filter:

[00011]$\begin{array}{cc}\left[\begin{array}{ccccccccccc}{\theta }_{\mathrm{rel}}& 2& 0& 0& & 0& 0& 0& & 0& 0\\ \left({\varphi }_{\mathrm{rel}}/m-{\theta }_{\mathrm{rel}}\right)& -1& -1& 0& .Math.& 0& 1& 0& .Math.& 0& -1\\ {\theta }_{\mathrm{rel}}& 0& 2& 0& .Math.& 0& 0& 0& .Math.& 0& 0\\ \left({\varphi }_{\mathrm{rel}}/m-{\theta }_{\mathrm{rel}}\right)& 0& -1& -1& & 0& -1& 1& & 0& 0\\ & .Math.& .Math.& & & & .Math.& & & .Math.& .Math.\\ {\theta }_{\mathrm{rel}}& 0& 0& 0& .Math.& 2& 0& 0& .Math.& 0& 0\\ \left({\varphi }_{\mathrm{rel}}/m-{\theta }_{\mathrm{rel}}\right)& -1& 0& 0& .Math.& -1& 0& 0& .Math.& -1& 1\end{array}\right]\left[\begin{array}{c}1/\omega \\ {\tilde{\rho }}_1\\ {\tilde{\rho }}_2\\ {\tilde{\rho }}_3\\ .Math.\\ {\tilde{\rho }}_m\\ {\widetilde{ϵ}}_{1,2}\\ {\widetilde{ϵ}}_{2,3}\\ .Math.\\ {\widetilde{ϵ}}_{m-1,m}\\ {\widetilde{ϵ}}_{m,1}\end{array}\right]=\left[\begin{array}{c}\Delta t_{X2S1F}\\ \Delta t_{X4S2R}\\ \Delta t_{X2S2F}\\ \Delta t_{X4S3R}\\ .Math.\\ \Delta t_{X2SmF}\\ \Delta t_{X4S1R}\end{array}\right]& \left(8\right)\end{array}$

[0098] As sensors are determined to be faulty, the relevant rows and columns of H, x, and y can be removed above and the process can still be used. When two or one healthy sensors remain, the equations given for m=2 or m=1 above are used.

[0099] It is known that the recursive least squares (RLS) filter often converges faster than the LMS filter in general but is more computationally intensive than the LMS. Therefore, LMS is being desired over the RLS for this application.

[0100] Below the relationship between number of sensors/targets, update rate, and critical rotor speed is discussed:

[0101] The primary performance limitation of the proposed solution may be that the update rate of the solution is proportional to the magnitude of the rotor speed. Hence at low rpm, the solution is forced to wait some nontrivial amount of time between target detections, introducing a time lag in the speed and azimuth updates. However, by introducing multiple relative sensors at the optimal spacing (i.e.,

[00012]$\frac{{\varphi }_{\mathrm{rel}}}{m},$

where ϕ.sub.rel is the spacing between the targets in degrees and m is the number of relative sensors), the update rate is reduced by a factor of m. Therefore, there is a direct relationship between the number of sensors, number of targets, rotor speed, and amount of time the solution is forced to wait between estimation updates. The critical rotor speed, ω.sub.cr can be defined as the rotor speed below which the update rate falls below the requirement (i.e., the amount of time that we are willing to wait for an update), and can be directly calculated as follows

[00013]$\begin{array}{cc}{\omega }_{\mathrm{cr}}=\frac{{\varphi }_{\mathrm{rel}}/m}{T_{\mathrm{req}}}.Math.\left(\frac{1\mathrm{rpm}}{6\frac{\mathrm{deg}}{s}}\right)=\frac{{\varphi }_{\mathrm{rel}}}{6.Math.m.Math.T_{\mathrm{req}}}=\frac{360/n_{\mathrm{rel}}}{6.Math.m.Math.T_{\mathrm{req}}}=\frac{60}{m.Math.T_{\mathrm{req}}.Math.n_{\mathrm{rel}}}& \left(9\right)\end{array}$

[0102] where T.sub.req is the required update period from the sensor in seconds, and n.sub.rel is the number of relative targets. If this critical rotor speed is reasonable, then the number of sensors and targets are appropriate. If the number of targets, update rate requirement, and desired critical rotor speed are known, then the formula can be manipulated to calculate the required number of sensors:

[00014]$\begin{array}{cc}{n}_{\mathrm{rel}}\ge \frac{60}{m.Math.T_{\mathrm{req}}.Math.{\omega }_{\mathrm{cr}}}& \left(10\right)\end{array}$

[0103] Although the present invention has been disclosed in the form of preferred embodiments and variations thereon, it will be understood that numerous additional modifications and variations could be made thereto without departing from the scope of the invention.

[0104] For the sake of clarity, it is to be understood that the use of “a” or “an” throughout this application does not exclude a plurality, and “comprising” does not exclude other steps or elements.