04439774_Implementation Issues and Performance Evaluation Smpm

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 44, NO. 1, JANUARY/FEBRUARY 2008 161

Implementation Issues and Performance Evaluationof Sinusoidal, Surface-Mounted PM Machine Drives

With Hall-Effect Position Sensors and aVector-Tracking Observer

Michael C. Harke, Member, IEEE, Giulio De Donato, Student Member, IEEE, Fabio Giulii Capponi, Member, IEEE,Tod R. Tesch, Member, IEEE, and Robert D. Lorenz, Fellow, IEEE

Abstract—This paper presents the implementation and evalua-tion of a high-resolution position estimation system for sinusoidal,surface phase modulation machines based on Hall-effect sensorsand a vector-tracking observer. First, the tuning of the observeris presented and a speed-dependent gain scheduling strategy isproposed. Then various harmonic decoupling strategies are inves-tigated to improve the performance of the observer, particularly atlow speeds. Stability analysis is performed leading to the definitionof local stability limits, within which the actual position is trackedwith bounded estimation error. Both simulation and experimentaltesting illustrate the performance and limitations of the proposedobserver topology and of the drive when this observer is used forstate feedback.

Index Terms—Hall-effect sensors, phase-modulation (PM) ma-chine drive, vector-tracking observer.

I. INTRODUCTION

B INARY Hall-effect sensors provide for discrete, or quan-tized, position feedback with a typical resolution of 60◦

electrical. The idea of extracting high resolution estimates forvelocity and position in an ac brushless drive from binaryHall-effect sensors was first advanced in 1996 by Corzine andSudhoff [1], and shortly thereafter by Morimoto et al. [2]. Twodifferent solutions were proposed by these researches. In the firstpaper, a hybrid state filter based on the differential equations ofa rotating vector is used to estimate the rotor synchronous ref-erence frame transformation terms used for torque control. The

Paper IPCSD-07-058, presented at the 2006 Industry Applications SocietyAnnual Meeting Conference Record, Tampa, FL, October 8–12, and approvedfor publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS bythe Industrial Drives Committee of the IEEE Industry Applications Society.Manuscript submitted for review October 31, 2006 and released for publicationAugust 14, 2007. This work was supported in part by the Wisconsin Elec-tric Machines and Power Electronics Consortium (WEMPEC), University ofWisconsin, Madison, and in part by the University of Rome “La Sapienza,”Rome, Italy.

R. D. Lorenz is with the Departments of Mechanical and Electrical andComputer Engineering, University of Wisconsin, Madison, WI 53706-1572USA (e-mail: [email protected]).

F. G. Capponi and G. D. Donato are with the Department of ElectricalEngineering, University of Rome “La Sapienza,” Rome 00184, Italy (e-mail:[email protected]; [email protected]).

M. C. Harke is with the Hamilton Sundstrand, Rockford, IL 61125-7002USA (e-mail: [email protected]).

T. R. Tesch is with Siemens VDO Electric Drives, Inc., Dearborn, MI 48120USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIA.2007.912729

input to the state filter is the average rotor velocity between twosuccessive sensor transitions, and a change in the Hall-effectsensor state was used to limit the error in the resulting esti-mated transformation term. The second approach, also basedon average velocity, consisted in using linear extrapolation inorder to obtain a position estimate. Other researchers have fur-ther investigated this second approach, [3], [4], [7], also knownas the zeroth-order Taylor algorithm [7]. An important limita-tion of both techniques is the fact that average velocity is used.In the following sections, it will be shown that sensor mis-placement or inaccuracy causes the average velocity to be verynoisy. While this has limited repercussions on position estima-tion, closing a velocity loop on this noisy average can be verydifficult, in particular if a medium to high-performance motioncontroller is required. Other approaches to position and velocityestimation using binary Hall-effect sensors have been proposedin which the sensors are used to aid back-electromotive force(emf) estimation schemes by bounding the position estimationerror [5], [6]. These approaches require precise knowledge ofthe electrical parameters such as inductance, phase resistance,and phase modulation (PM) flux, all of which may vary in nor-mal working conditions of the machine. Moreover, back-emf isnot easily detected at low speeds.

In [7], it was also demonstrated that the sensors can be pro-cessed to form a single, spatially rotating vector, as shown inFig. 1. The vector moves around the hexagon, in quantized 60◦

increments, as each sensor output sequentially changes withthe rotation of the motor shaft. Such position vector Hαβ canbe used in a vector-tracking observer topology, as shown inFig. 2, in order to extract potentially “zero-lag” estimates ofposition and velocity. Thus, superior performances with re-spect to the zeroth-order Taylor algorithm can be achieved [7].Vector-tracking observers are a form of Luenberger-style ob-server that use the vector cross-product phase-detection meth-ods originally employed in resolver-to-digital (RTD) converters.Vector-tracking observers have been used extensively in the areaof position sensorless control of ac machines [8]–[11]. Unlikephase-locked loops (PLLs), such observers have intrinsic zero-lag tracking capability.

A limitation was found in [7] and [12] when using only asimple vector model ejθel , in the feedback path. In fact, theobserver tracks—within its bandwidth—the discrete quantizedposition vector Hαβ , thus inducing harmonic errors in the

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162 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 44, NO. 1, JANUARY/FEBRUARY 2008

Fig. 1. Quantized position vectors with Hαβ set at 60◦.

estimated position. Therefore, a varying gain was proposedin [7]. A strategy for gain variation is a topic of this paperand is detailed in the following sections.

It was realized by Tesch [12] that the quantized rotating posi-tion vector can be modeled as a continuous rotating fundamen-tal vector with a set of positive and negative rotating harmonicvectors. This same principle can be applied to the Hall-effectposition vector Hαβ , as shown in Fig. 3. The spatial harmonicmodel for Hαβ can be expressed as in (1)

Hαβ = ej(θel− π6 ) +

+∞∑k=1

[− 1

6k − 1

× e−j((6k−1)θe l + π6 )+ 1

6k − 1ej((6k+1)θe l + π

6 )]

. (1)

Degner [9] demonstrated that harmonic vectors can be decou-pled in vector-tracking observer topologies. A general solutionfor harmonic-decoupling tracking observers using quantized po-sition vectors was given in [12]. Although not recognized asspatial harmonic vectors, [13] implemented a hardware PLL,which interpolated for distorted position feedback with lookuptables.

In this research, the primary decoupling observer topologyused is shown in Fig. 4. With respect to this observer structure,the paper focuses on the implementation issues and discusses itsperformance and limitations when used in an ac brushless drivefor sinusoidal, surface-mounted PM machines.

This paper demonstrates the benefits of harmonic decouplingwith respect to position estimation. It also demonstrates thatthe disturbance rejection of any discretized Hall-effect sensor-based drive is reduced at low speeds because the position isnot sampled often enough. Both simulation and experimentaltesting are used to illustrate the performances of the proposedobserver topology.

II. IMPLEMENTATION ISSUES IN POSITION ESTIMATION USING

A VECTOR-TRACKING OBSERVER

Three issues, analyzed in the following subparagraphs, areparticularly relevant for a sinusoidal, surface-mounted PM ma-chine drive that uses Hall-effect position sensors and a vector-tracking observer. The first issue is that position samplingdepends on the state change of the Hall sensors, and is thusposition (and speed) dependent. This variable sampling affectsperformance and must therefore be considered when tuning theobserver. The second is that as previously illustrated, the addi-tional harmonics present in the quantized signal result in har-monic errors in position estimation. These spatially determinanterrors can be mitigated by decoupling the additional harmon-ics. Finally, disturbance rejection properties are not the sameas those of traditional encoder-based drives and, must there-fore, be analyzed to better understand the limits of the proposeddrive.

A. Tuning the Vector-Tracking Observer

The frequency of the Hall vector Hαβ transitions betweenthe six quantized states is speed-dependent and equivalent to theposition sample rate. As a result, the bandwidth of the positionestimator should not be too high, causing the estimated positionto track the quantized nature of the input and be erroneous. Onthe other hand, relatively high bandwidth is usually desired inorder to obtain good dynamic estimation of disturbances. Thus,to maximize performance while limiting errors, the bandwidthof the vector-tracking observer can be made speed-dependent.However, estimator bandwidths significantly higher than themotion controller will add quantization noise to the positionestimate, leading to additional losses in the machine with im-measurable improvement in disturbance rejection. Thus, the es-timator bandwidth should limited to the drive motion controllerbandwidth.

In order to achieve this desired bandwidth variation, one so-lution is to make the observer poles linearly speed-dependentbetween zero speed and an upper speed limit ωlim , beyond whichthe poles remain constant at their nominal values. This results ina nonlinear observer gain variation. Another solution is to varythe observer gains linearly with speed. This results in a simplerimplementation without drastically affecting the desired prop-erties of the observer.

This gain variation can be tuned as follows. First, the nominalgains of the observer are determined given the desired observerpole locations at rated speed. The gains are determined usingan operating point model of the vector-tracking observer. Then,the appropriate limit velocity must be determined based on thesample ratio SR between the sample rate of motor positionand the desired limit bandwidth BW, in hertz. The sample rateof motor position depends on the number of discrete statesper electrical period (NDS ) and on electrical pulsation ωel , inradians per second. For Hall-effect position sensors spaced 120◦

electrically, NDS = 6

SR =NDSωEL

2πBW. (2)

HARKE et al.: IMPLEMENTATION ISSUES AND PERFORMANCE EVALUATION OF SINUSOIDAL, SURFACE-MOUNTED PM MACHINE DRIVES 163

Fig. 2. Discrete-time vector-tracking observer.

Fig. 3. Quantized position vector Hαβ its harmonic decomposition.

The sample ratio SR is typically selected to be around 8–10,and is theoretically limited by the Nyquist frequency whereSR = 2. More aggressive tuning results with a lower SR. Thelimit velocity for the linear interpolation gain scheduling is de-termined by rearranging (2) into (3)

ωlim =2πBWSR

NDS. (3)

One caveat of implementing the speed-dependent tuning isthat at zero speed, the observer gains would be zero and thetracking observer would rely only on the feedforward pathto estimate the position, using an initial position based onthe Hall-sensor state. Theoretically, this will work providedthat the physical system model (i.e., estimated inertia J) usedon the observer is correct. In practice, the physical systemmodel is not well known, so that some gain is needed at lowspeeds. As a rule of thumb, the minimum value of the gainscan be set at 10% of their nominal values. Fig. 5 shows thegain scheduling strategy for a generic observer gain; the up-per and lower velocity limits are the same for all three gains,kI , kP , and kD .

B. Decoupling Additional Harmonics in the Hall Vector Hαβ

To improve the performance of the vector-tracking observer,the additional harmonics present in the Hall vector Hαβ shouldbe decoupled, resulting in a rotating vector with strongly re-duced quantization harmonic content. From (1), the spatial har-monic content other than the fundamental rotating vector can be

expressed as

Hαβ harmonics

= Hαβ − ej (θe l + π6 ) =

+∞∑k=1

[− 1

6k − 1e−j ((6k−1)θe l + π

6 )+

× 16k + 1

ej ((6k+1)θe l + π6 )

]. (4)

The ideal, full order, decoupling waveform includes all ofthe harmonics of the fundamental equation (4), and is shownin Fig. 6 (continuous line). However, this results in a discon-tinuous decoupling waveform, making the decoupling sensitiveto disturbances and position errors: a small error in positionnear one of these discontinuities results in a large change in thedecoupling waveform.

Another option, truncated harmonic decoupling, is to onlydecouple a subset of the additional harmonics. For exam-ple, the decoupling waveform for the 5th and 7th harmonicsis

Hαβ dec 5th and 7th =15

e−j(5θe l + π6 ) − 1

7e−j(7θe l + π

6 ). (5)

If harmonics up to the 13th are decoupled, then the waveformis

Hαβ dec 5th,7th,11th, and 13th =15

e−j(5θe l + π6 )

− 17

e−j(7θe l + π6 ) +

111

e−j(11θe l + π6 ) − 1

13e−j(13θe l + π

6 ).

(6)

Truncated harmonic decoupling results in continuous decou-pling waveforms, but with significant deviations from the fullorder decoupling, as can be seen in Fig. 6 (dashed and dash-dotlines).

Another option is to create a filtered decoupling waveformby forward and reverse filtering the full order (ideal) decouplingwaveform. This results in a continuous decoupling waveformwith minimal ripple, and is also shown in Fig. 6 (dotted line).

One advantage of decoupling the additional harmonics isthat observer bandwidth can be increased without adding


Fig. 4. Vector-tracking observer with decoupling of additional harmonic terms.

Fig. 5. Vector-tracking observer gain scheduling as a function of speed.

Fig. 6. Decoupling waveforms. (a) Hα signals. (b) Hβ signals. Ideal—continuous line; 5th and 7th—dashed line, 5th, 7th, 11th, and 13th—dash-dotline, filtered—dotted line.

quantization noise to the position estimate. Another advantage isthat the enhanced velocity estimate (Fig. 4) has less steady-stateerror. These characteristics improve the overall performance ofthe drive, particularly at low speeds.

Fig. 7. Simulated dynamic stiffness of the proposed drive. Note: the observergains were constant for all cases, BW = 0.5, 5, 50 Hz, (ωe = 40400 and800 rad/s).

TABLE IRATED VALUES OF THE MOTOR PROTOTYPE

C. Disturbance Rejection of Hall Sensor-Based Drives

Fig. 7 is a plot of the simulated dynamic stiffness frequencyresponse for three commanded velocity cases (10, 100, and200 rad/s). In the simulation, the observer gains were thesame for each case (not speed-dependent). The motor pa-rameters used for the simulation are listed in Table I. Whenthe velocity results in a sufficient position sample rate, thedynamic stiffness is governed by the observer and motion con-troller bandwidths and remains unchanged at different speeds:100 and 200 rad/s. In these cases, the position sample rates are382 and 764 Hz, respectively. When the velocity is reduced to


Fig. 8. Prototype axial flux PM machine and dc load machine testbed.

10 rad/s, the dynamic stiffness is reduced in the intermediatefrequency range (∼ 3–50 Hz). In this case, the effective posi-tion sample rate is 38 Hz that is not sufficient to achieve thespecified bandwidths. However, if the disturbance frequency islow enough (< 3 Hz), it is sampled often enough to be rejected.Above 50 Hz, the dynamic stiffness is governed by the iner-tia. This speed-dependent disturbance rejection property doesnot depend on the position/velocity estimation algorithm; it isinherent when using low-resolution sensors.

In the 10 rad/s plot, there are resonant peaks at 38 Hz and itsharmonics. This is a result of imperfect harmonic decouplingof the quantization noise and does not actually increase thedynamic stiffness. For the remaining cases, the resonant peaksexist, but occur at higher frequencies than those plotted.

III. PERFORMANCE EVALUATION OF THE OBSERVER

In the following paragraphs, performance of a PM machinedrive with Hall-effect sensors and a vector-tracking observerwill be evaluated at two levels:

1) observer performance;2) drive performance.This evaluation will take place using both simulation and

experimental results.The drive testbed consisted of a prototype 500 W axial-flux

PM machine, a full bridge IGBT inverter, and a 16-bit fixedpoint DSP. The inverter switching frequency was 15 kHz witha dead time of 2.5 µs. A dc machine is coupled to the motorshaft and its armature is connected to a resistive load, so thatat 100 rad/s, the motor is operating at nominal load conditions.A 1024 pulse incremental encoder was mounted on the drive’sshaft to allow a comparison between the estimated angle anda high-resolution measurement. The main motor characteristicsare shown in Table I, and Fig. 8 shows a photograph of the motorand load at the test bench.

The observer eigenvalues (at the upper speed limit) are setto 80, 8, and 0.8 Hz. The various harmonic decoupling wave-forms are calculated offline and implemented in stored, 256word lookup tables.

A. Stability

The stability of the proposed observer is primarily due to theheterodyning structure and secondarily due to the decouplingharmonics. These are functions of the position estimation error.

Fig. 9. Vector cross-product output versus estimation position error.

The Z-transform model of the physical system with a latchedtorque input is

θ(z)T (z)

= (1 − z−1)Z{

1s

1Js2

}

=1J

Tsz−1

(1 − z−1)(1 + z−1)

2Ts

(1 − z−1). (7)

This implies that the feedback position response to a stepdisturbance in torque (acceleration scaled by the inertia) willcreate a quadratic response in the position of the rotor. Since thestability of the observer is a function of its ability to estimatedisturbances, through tracking of the position feedback, it isinsightful to focus on the effects of position errors in order todetermine stability limits.

For global stability, the heterodyning tracking observer hasthe same stability criteria as any software-based phase-lockedloop. It is subject to the same issues of “pull-in” and “pull-out.”Decoupling harmonics play only a minor role in determiningthe global stability [14].

Conversely, local stability is fundamentally determined bythe harmonic decoupling. It was shown in [9] that the localstability of spatial harmonic decoupling tracking observers canbe determined by investigating the output ε of the vector cross-product portion of the observer. Fig. 9 shows the output ε for adecoupling observer using full order harmonic decoupling as afunction of the position estimation error ∆θerr , when the actualposition is θel = 30◦.

When the position estimation error ∆θerr is positive, i.e., theestimated position is lagging the actual position, the observerwill track well because the polarity of the output of the vectorcross-product produces a negative feedback action. The stabilitycrossover point for positive estimation errors occurs at 150◦.Beyond this point, the ε reverses polarity, thus causing localinstability, i.e., positive feedback.

When ∆θerr is negative, i.e., the estimated position is leadingthe actual position, ε is zero until ∆θerr is equal to −60◦. Insidethis interval, the “zero ε zone,” there is an unobservable error dueto quantization of the feedback, and therefore, the observer doesnot react. For larger errors, the vector cross product produces anegative feedback action whereby the stability crossover pointoccurs at −210◦.


Fig. 10. Vector cross-product output as a function of ∆θerr and θel .

Fig. 10 illustrates the dependency of the vector cross-productoutput ε with both ∆θerr and θel . For increasing values of theactual position θel , the projection on the ε, ∆θerr plane hasthe same waveform, but it shifts progressively toward positive∆θerr . After 60◦, when a transition occurs (θel = 90◦ in Fig. 9),the waveform resumes the same appearance as at θel = 30◦. Inparticular, the zero ε zone shifts linearly by 60◦ degrees through-out each sector so that the observer becomes progressively moresensitive to leading estimation errors and less sensitive to lag-ging estimation errors.

Just before the end of each sector, the zero ε zone will be en-tirely contained in the positive ∆θerr part of the ε, ∆θerr plane.Furthermore, the stability crossover points shift linearly by 60◦

throughout each sector. For example, the stability crossoverpoint for positive estimation errors shifts from 150◦ at the be-ginning of the sector to 210◦ immediately before the followingtransition. This shifting property is present even if no harmonicdecoupling is used, i.e., when ε is a sine wave, because it isdirectly related to the quantized nature of Hαβ .

Fig. 11 shows a simulation of the estimated and actual po-sition waveforms following an impulsive decelerating torquedisturbance equal to 6 p.u. Position estimation error and thevector cross-product output are also plotted. It can be seen thatε is initially zero because of the zero ε zone. It becomes nonzeroand negative as the actual position advances through the sectorand the zero ε zone shifts toward positive ∆θerr .

Fig. 12 shows the same plots for an impulsive acceleratingdisturbance of 6 p.u. Here, the position estimation error is alwayspositive, i.e., lagging estimate, and ε is nonzero in the first partof each sector, when the zero ε zone is mostly in the negative∆θerr region. As the actual position progresses into each sector,the zero ε zone shifts, causing ε to go to zero.

As a final comment, the use of different harmonic decou-pling strategies will cause deviations from Figs. 9–10 that areprogressively greater as the number of decoupled harmonics isreduced, but the main characteristics will remain. In particular,the shifting of the ε waveform throughout each sector and thepresence of a region with reduced ε (the zero ε zone for the fullorder decoupling) will appear in any case. The observer will,however, remain stable.

Fig. 11. Simulated estimated (—) and actual (–) position, position estimationerror (—), and vector cross-product output (—) following a 6 p.u. deceleratingtorque disturbance.

Fig. 12. Simulated estimated (—) and actual (–) position, position estimationerror (—), and vector cross-product output (—) following a 6 p.u. acceleratingtorque disturbance.

B. Position Estimation With Harmonic Decoupling

Fig. 13 shows the locus of the decoupled Hall vector (Hαβ −Hαβ dec) when the filtered waveform decoupling is used. Thelocus is practically circular with some error close to the sensors’transitions.


Fig. 13. Simulated vector locus with harmonic decoupling (filteredwaveform).

Fig. 14. Simulated steady-state position estimation error without decouplingat the rate of 100 rad/s.

Fig. 14 shows a simulation of the steady-state position estima-tion error (at the rate of 100 rad/s) when decoupling is not used:the induced harmonic error caused by tracking of the quantizedinput (Hαβ ) is clearly visible.

Fig. 15 shows the steady-state position estimation error withwavelet decoupling. In this case, the induced error is absent.Behavior at a lower speed (10 rad/s) is investigated in Figs. 16and 17. The lower the speed, the lower the position sample rate,and the higher the quantization noise that is fed into the observervia the Hall sensors’ vector.

The harmonic decoupling has an increasing impact on steady-state performance at low speed as shown in Fig. 17. However,there is a lower speed limit beneath which performance startsto degrade. This lower speed limit depends, among others, onthe precision of the feedforward signal. At very low speeds, theclosed-loop observer exhibits nearly open-loop behavior sincethe gains are kept very low, thus increased estimation errorscannot be avoided even with harmonic decoupling techniques.

Fig. 18 shows the experimental XY plots relative to the vectorlocus without decoupling, while Fig. 19 shows the case withfiltered waveform decoupling.

Fig. 15. Simulated steady-state position estimation error with decoupling (fil-tered waveform) at the rate of 100 rad/s.

Fig. 16. Simulated steady-state position estimation error without decouplingat the rate of 10 rad/s.

Fig. 17. Simulated steady-state position estimation error with decoupling (fil-tered waveform) at the rate of 10 rad/s.

The experimental position estimation error for various veloc-ity and decoupling cases are plotted in Figs. 20–25.

The experimental results correlate with the simulations pre-viously presented. In particular, experimental results confirmthe increased beneficial effect of harmonic decoupling at lowspeeds, when the effective position sample rate becomes very


Fig. 18. Experimental XY graph of Hall vector locus without decoupling.

Fig. 19. Experimental XY graph of Hall vector locus with decoupling (filteredwaveform).

Fig. 20. Experimental steady-state position estimation error without decou-pling at the rate of 100 rad/s.

Fig. 21. Experimental steady-state position estimation error with decoupling(filtered waveform) at the rate of 100 rad/s.

Fig. 22. Experimental steady-state position estimation error with decoupling(5th, 7th, 11th, and 13th) at the rate of 100 rad/s.

Fig. 23. Experimental steady-state position estimation error without decou-pling at the rate of 10 rad/s.

low. The position estimation error in any case is not zero, butthere is a certain amount of lag between the estimated and actualposition due to armature reaction [14]. This effect is common toall drive implementations that use Hall-effect sensors, but caneasily be compensated by using a lookup table to take into ac-count the phase shift due to armature reaction at various loads.Furthermore, the error plot is not as smooth as it is in the steady-state simulations. This can be attributed to sensors misplacementand inaccuracy [7].


Fig. 24. Experimental steady-state position estimation error with decoupling(filtered waveform) at the rate of 10 rad/s.

Fig. 25. Experimental steady-state position estimation error with decoupling(5th, 7th, 11th, and 13th) at the rate of 10 rad/s.

IV. PERFORMANCE EVALUATION OF THE PM MACHINE DRIVE

Various tests have been performed to evaluate the perfor-mance of the drive when the observer estimates are used as statefeedback. In order to better explain improvements that can beachieved using this technique, results are compared with thezero-order Taylor algorithm [7] used for state feedback.

A. Current Controller

A synchronous reference frame cross-coupling decouplingPI controller was used [15]. The controller gains were tuned toachieve a 400 Hz bandwidth. It is of interest to try to evalu-ate the influence of position estimation errors on the bandwidthof the current controller. Whenever there are errors in posi-tion estimation, the Park transformations will not be correct,thus causing a reduction in achievable bandwidth. In particular,startup is the most critical situation since position is unobserv-able in this condition. Plots of the estimated and actual q-axiscurrent can be obtained, allowing a graphical representation ofthe reduction in settling time that occurs due to error in positionestimation. Fig. 26 shows the resulting waveforms when using

Fig. 26. Q-axis current at startup: commanded value i∗q , measured valuewith observer feedback iq (ω, θ), and measured value with encoder feedbackiq (ω, θ).

Fig. 27. Q-axis current at startup: commanded value i∗q , measured valuewith zero-order feedback iq (ω, θ), and measured value with encoder feedbackiq (ω, θ).

the observer, while Fig. 27 shows the same transient responsewhen the zero-order algorithm is used for position feedback.Neither exhibits the desired performance; however, the vectortracking observer quickly reduces the position estimation erroronce the shaft rotates, allowing the steady-state value to be at-tained more quickly than with the zero-order Taylor algorithm.Similar tests carried out at various nonzero speeds demonstratedno significant difference from the desired step response: this isbecause the position estimation error remains small.

B. Speed Controller

A classical PI speed controller was used. It was tuned toachieve a 13 Hz bandwidth. The unenhanced signal was used asthe speed state feedback instead of the enhanced estimate (whichwould theoretically guarantee a higher dynamic stiffness). Thisis because with the enhanced signal, the loop was unstable due to


Fig. 28. Steady-state measured current waveform without harmonic decou-pling at the rate of 10 rad/s.

Fig. 29. Steady-state measured current waveform with harmonic decoupling(filtered waveform) at the rate of 10 rad/s.

the increased quantization noise. This indicates that the tuningof the observer and drive was too aggressive.

An investigation into the effects of harmonic decoupling onthe steady-state performance of the speed controller at lowspeeds was also carried out and the results were insightful.Fig. 28 shows the steady-state current waveform when thespeed command is set to 10 rad/s and harmonic decouplingis not present. In this case, the quantization noise present inthe velocity estimate causes the controller to command unnec-essary harmonic torque, resulting in deteriorating performanceand causing additional losses in the motor.

Figs 29 and 30 show the steady-state waveforms for thecase when harmonic decoupling is present (respectively, filteredwaveform and 5th, 7th, 11th, and 13th). A great improvementin the current waveform can be seen, due to the reduced quan-tization noise present in the speed estimate. Both waveforms,however, present some residual discontinuities due to imperfectdecoupling. It is worth noting that a similar test on the drivewhen the speed loop was closed around the zero-order Taylor

Fig. 30. Steady-state measured current waveform with harmonic decoupling(5th, 7th, 11th and 13th) at the rate of 10 rad/s (0.5 A/div 500 ms/div.).

Fig. 31. Response to a step speed input using observer-based position andunenhanced velocity estimates for state feedback.

estimate (i.e., average speed between sensor transitions) was notcarried out due to stability problems at low speeds caused bythe noise in the speed estimate (due to sensor inaccuracy andmisplacement).

A transient response test to a speed step input was also carriedout both with the observer and the zero-order Taylor algorithm.Fig. 31 shows the performance of the speed controller whenusing the observer estimate. The steady-state error with respectto the encoder speed measurement (clock pulse counting tech-nique) was about 0.5 rad/s. Fig. 32 shows the performance ofthe controller when the zero-order algorithm is used. For thezero-order algorithm, the speed bandwidth had to be lowered to1 Hz to avoid unstable operation due to the quantization noisein the average speed measurement.

A disturbance response test was carried out at 60 rad/s and theperformance of the observer-based drive was compared to theperformance of the encoder-based drive. The disturbance (about30% of the nominal torque) was created by short circuiting partof the dc machine resistive load. Fig. 33 shows the behavior of


Fig. 32. Response to a step speed input using the zero-order average velocityand position estimates for state feedback.

Fig. 33. Disturbance response of the speed loop when the encoder is used forstate feedback.

the drive when the encoder is used. It can be seen that the driveresponds very quickly to the disturbance and no speed variationcan be seen. Fig. 34 shows the performance of the drive whenthe decoupling observer is used (wavelet) and significant dif-ferences can be seen with the previous case: due to the speedestimation error, the controller keeps the estimated speed prac-tically constant, thus causing the actual speed to be reduced. Ittakes a significant length of time for the actual speed to recover(i.e., for the observer to reduce the speed estimation error), thusdemonstrating the reduced disturbance rejection properties thatthis drive has compared to the classical encoder-based drive.

Finally, low-speed performance of the drive while using boththe velocity and position estimates for state feedback is investi-gated in Figs. 35–37. Fig. 35 shows a step response from 0 to25 rad/s (4 Hz). Fig. 36 shows a speed reversal from +25 to−25 rad/s. It can be seen that the vector-tracking observer main-tains proper tracking during the zero speed crossing.

Fig. 37 shows the behavior of the drive during a forced stop-page of the mechanical shaft while the velocity command is

Fig. 34. Disturbance response of the speed loop when the decoupling observer(filtered waveform) is used for state feedback.

Fig. 35. Response to a 0–25 rad/s (4 Hz) step speed input using the observerestimates as the speed and position state feedback.

Fig. 36. Velocity reversal from +25 to −25 rad/s with decoupling (filteredwaveform).


Fig. 37. Position estimation error at 25 rad/s velocity command and forcedstoppage (and release) of the mechanical shaft with decoupling (filteredwaveform).

kept constant and equal to 25 rad/s. The quantized rotating vec-tor stops rotating as the shaft velocity goes to zero and theestimated position converges to a constant value that dependson the sector and on the type of decoupling used. In any case,the position estimation error is bounded to ±60◦ electrical sothe observer can rapidly recover the correct position and speedestimates once the shaft is released, as demonstrated by Fig. 37.

V. CONCLUSION

Key conclusions can be summarized as follows.1) Disturbance rejection properties of any quantized sensor

drive (such as a Hall-effect sensor-based drive) are speeddependent, resulting in reduced disturbance rejection atlow speeds.

2) The vector-tracking observer gains should be speed depen-dent with both minimum and nominal values. The nominalvalue limits the quantization noise on the position esti-mate. The minimum value ensures closed-loop start-up ofthe drive.

3) Decoupling of the additional harmonics results in im-proved tracking performance, both dynamic and steadystate, and allows for increased bandwidth in the vector-tracking observer; a higher bandwidth leads to improveddisturbance rejection.

4) Stability of the observer has been explored and maximumtransient position estimation errors have been found thatguarantee local stability.

5) The vector-tracking observer is based on a mechanicalmodel of the physical system, so the only parameter thatit is sensitive to is the moment of inertia J. An erroneousknowledge of inertia only causes a variation in the band-width of the observer, but does not affect its steady-statetracking properties. This is an important difference com-pared to the back-emf estimation schemes proposed in [5]and [6] that are sensitive to electrical parameter variations,even at steady state.

6) Performance of the drive is improved with respect to al-gorithms based on average velocity calculation, particu-larly at low speeds. However, at sustained zero speed, i.e.,forced mechanical shaft stoppage, the position error canbe as high as 60◦ (electrical), due to the limited resolutionof the sensors’ interface. It is, therefore, not possible touse this drive for static positioning applications requiringbetter than 60◦ (electrical) precision.

ACKNOWLEDGMENT

The authors would like to thank the Wisconsin Electric Ma-chines and Power Electronics Consortium (WEMPEC) of theUniversity of Wisconsin, Madison, and the University of Rome“La Sapienza,” Rome, Italy, for technical motivation.

REFERENCES

[1] K. A. Corzine and S. D. Sudhoff, “A hybrid observer for high performancebrushless DC motor drives,” IEEE Trans. Energy Conv., vol. 11, no. 2,pp. 318–323, Jun. 1996.

[2] S. Morimoto, M. Sanada, and Y. Takeda, “Sinusoidal current drive systemof permanent magnet synchronous motor with low resolution positionsensors,” in Proc. IEEE IAS Annu. Meeting, San Diego, CA, Oct. 1996,pp. 9–13.

[3] J. Bu, L. Xu, T. Sebastian, and B. Liu, “Near-zero speed performanceenhancement of PM synchronous machines assisted by low cost hall-effectsensors,” in Proc. IEEE Appl. Power Electron. Conf. (APEC), Anaheim,CA, Feb. 1998, pp. 68–74.

[4] J. X. Shen, Z. Q. Zhu, and D. Howe, “PM brushless drives with lowcostand low-resolution position sensors,” in Proc. Int. Power Electron. Mo-tion Control Conf. (IPEMC), Xian, China, vol. 2, Aug. 2004, pp. 1033–1038.

[5] T. D. Batzel and K. Y. Lee, “Slotless permanent magnet synchronousmotor operation without a high resolution rotor angle sensor,” IEEETrans. Energy Conv., vol. 15, no. 4, pp. 366–371, Dec. 2000.

[6] A. Lidozzi, L. Solero, F. Crescimbini, and A. Di Napoli, “SVM PMSMdrive with low resolution hall effect sensors,” IEEE Trans. Power Elec-tron., vol. 22, no. 1, pp. 282–290, Jan. 2007.

[7] F. G. Capponi, G. D. Donato, L. D. Ferraro, O. Honorati, M. C. Harke, andR. D. Lorenz, “AC brushless drive with low-resolution hall-effect sensorsfor surface-mounted PM machines,” IEEE Trans. Ind. Appl., vol. 42,no. 2, pp. 526–535.

[8] P. L. Jansen and R. D. Lorenz, “Transducerless field orientation conceptsemploying saturation-induced saliencies in induction machines,” IEEETrans. Ind. Appl., vol. 32, no. 6, pp. 1380–1393.

[9] M. W. Degner “Flux, position and velocity estimation in AC machinesusing carrier signal injection,” Ph.D. Dissertation, Univ. Wisconsin-Madison, WI, 1998.

[10] H. Kim, M. C. Harke, and R. D. Lorenz, “Sensorless control of interiorpermanent-magnet machine drives with zero-phase lag position estima-tion,” IEEE Trans. Ind. Appl., vol. 39, no. 6, pp. 1726–1733.

[11] M. C. Harke “Fundamental sensing issues in motor control,” Ph.D. Dis-sertation, Univ. Wisconsin-Madison, WI, 2006.

[12] T. R. Tesch “Dynamic torque estimation in a sensor limited environment,”Ph.D. Dissertation, Univ. Wisconsin-Madison, WI, 2005.

[13] L. Wang and T. Emura, “High-precision interpolation using two-phasetype PLL for encoders that have distorted waveforms,” in Proc. IEEE Int.Conf. Intell. Process. Syst., Beijing, China, 1997, pp. 82–87.

[14] G. De Donato, “Implementation and performance evaluation of alow cost surface-mounted PM machine drive using binary hall-effectsensors,” Ph.D. Dissertation, Univ. Rome-La Sapienza, Rome, Italy,2006.

[15] F. Briz, M. W. Degner, and R. D. Lorenz, “Analysis and design of currentregulators using complex vectors,” IEEE Trans. Ind. Appl., vol. 36, no. 3,pp. 817–825.


Michael C. Harke (S’98–M’07) received the B.S.,M.S., and Ph.D. degrees in mechanical engineeringfrom the University of Wisconsin, Madison, in 1997,1999, and 2006, respectively.

In 2006, he joined the Applied Research Depart-ment, Hamilton Sundstrand, Rockford, IL, where hewas engaged in research on the application of electricmachines and power electronics in the aerospace in-dustry. His current research interests include controlsystems, machine drives, and power electronics.

Giulio De Donato (S’05) was born in Cork, Ireland,in 1978. He received the M.S. and Ph.D. degrees inelectrical engineering from the University of Rome“La Sapienza,” Rome, Italy, in 2003 and 2007,respectively.

He is currently a Research Associate in the Depart-ment of Electrical Engineering, University of Rome“La Sapienza”. His current research interests includeadvanced digital control of electrical drives, powerelectronics, and digital modulation techniques.

Dr. De Donato is a Registered Professional Engi-neer in Italy, and is a member of the IEEE Industry Applications Society andthe IEEE Control Systems Society.

Fabio Giulii Capponi (M’98) received the M.S. andPh.D. degrees in electrical engineering from the Uni-versity of Rome “La Sapienza,” Rome, Italy, in 1994and 1998, respectively.

In 1996, he joined the Department of Electrical En-gineering, University of Rome “La Sapienza,” wherehe is currently an Assistant Professor. During 2003and 2004, he was a Visiting Scholar at the WisconsinElectrical Machines and Power Electronics Con-sortium (WEMPEC), University of Wisconsin,Madison. His current research interests include per-

manent magnet motor drives and digital control systems design for unconven-tional power converter topologies.

Dr. Giulii Capponi is a member of the IEEE Industry Applications, the IEEEIndustrial Electronics, the IEEE Power Electronics, and the IEEE Control Sys-tems Societies.

Tod R. Tesch (S’93–M’95) received the B.S. andM.S. degrees in electrical engineering and the Ph.D.degree in mechanical engineering from the Univer-sity of Wisconsin, Madison, in 1989, 1996, and 2005,respectively.

In 2001, he joined Ford Motor Company–EcostarElectric Drives, LLC, which was acquired by Bal-lard Power Systems in 2001, and in 2007 by SiemensVDO, Dearborn, MI, where he is currently a PrincipalEngineer in the Controls and Software Department,Powertrain HEV Division, and is responsible for mo-

tor control development and power electronic and system simulation for fuelcell and hybrid vehicles, where he is engaged in research and development onautomotive applications of electric drives and power electronics, and their con-trol. His current research interests include control systems, electric machines,power electronics, and mechatronics.

Dr. Tesch is a member of the IEEE Industry Applications and the IEEEPower Electronics Societies.

Robert D. Lorenz (S’83–M’84–SM–91–F’98) re-ceived the B.S., M.S., and Ph.D. degrees from theUniversity of Wisconsin, Madison, and the M.B.A.degree from the University of Rochester, Rochester,NY.

Since 1984, he has been with the University ofWisconsin, where he is the Mead Witter FoundationConsolidated Papers Professor of Controls Engineer-ing in both the Department of Mechanical Engineer-ing and the Department of Electrical and ComputerEngineering. He is also the Research Leader for con-

trol and sensor integration and for integrated modular motor drives in the Centerfor Power Electronic Systems, a National Science Foundation (NSF) Engi-neering Research Center. From 1972 to 1982, he was with the Gleason Works,Rochester. He has also been a Visiting Research Professor in the Electrical DrivesGroup of the Catholic University of Leuven, Leuven, Belgium, and in the PowerElectronics and Electrical Drives Institute of the Technical University of Aachen,Aachen, Germany. He was the SEW Eurodrive Guest Professor. His current re-search interests include sensorless electromagnetic motor/actuator technologies,power electronic device junction temperature estimation and real-time control,fast signal processing and estimation techniques, precision multiaxis motioncontrol, and ac/dc drive, and high-precision machine control technologies. Heis the author or coauthor of more than 180 technical published papers and is theholder of 23 patents with three more pending.

Dr. Lorenz was the IEEE Division II Director during 2005–2006, the IEEEIndustry Applications Society (IAS) President during 2001, a DistinguishedLecturer of the IEEE IAS during 2000–2001, the Chair of the IAS AwardsDepartment, and the Chairman of the IAS Industrial Drives Committee. Heis currently a member of the IAS Industrial Drives Committee, the ElectricalMachines Committee, the Industrial Power Converter Committee, and the In-dustrial Automation and Control Committee. He is an immediate past Chair ofthe Periodical Committee and current Chair of Periodicals Review Committeefor the IEEE Technical Activities Board. He is a member of the IEEE SensorCouncil AdCom. He is the recipient of 21 Prize Paper Awards. He is also therecipient of the 2003 IEEE IAS Outstanding Achievement Award. He is a Reg-istered Professional Engineer in the States of New York and Wisconsin. He isalso a member of the American Society of Mechanical Engineers (ASME), theInstrument Society of America (ISA), and the International Society of OpticalEngineers (SPIE). He is the Co-Director of the Wisconsin Electric Machinesand Power Electronics Consortium.

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