electrical contro and computer engineering

8/13/2019 electrical contro and computer engineering

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Abstract:A synchronous regulator that accurately tracks three-phasecommands with arbitrary balanced harmonic content is set forth. The

regulator utilizes multiple reference frames to realize both a novelharmonic estimator and an integral feedback controller. Theregulator is analyzed and demonstrated in the context of a permanent-magnet synchronous machine drive. The drive utilizes an optimizednon-sinusoidal current command to achieve superior efficiency andtorque ripple performance but which is dependent upon thecommanded waveforms being precisely realized. The multiplereference frame synchronous current regulator proposed herein

readily achieves this requirement. A computer simulation and ahardware prototype demonstrate validity of the regulator.

I. INTRODUCTION

In motor drive and power electronic applications, it isfrequently of interest to control voltage and current harmonics

precisely. This is done either to eliminate harmonics, such as

in unity-power-factor dc supplies, or to intentionally injectharmonics in certain specialty applications. Many varieties of

power converter modulation allow the designer to modify the

control signal so that the appropriate harmonics are injected.

However, this is normally an open loop procedure that does

not ensure that the exact desired operating point is achieved.

One interesting application of harmonic injection

concerns surface-mounted permanent-magnet synchronousmachine (SMPMSM) drives with non-sinusoidal back emfs.

Several authors [1-6] have shown that injection of the

appropriate current harmonics yields superior performance interms of efficiency and torque ripple. However, the control

schemes in all of these papers require that the currentharmonics injected be exactly achieved. For example, if it is

desired to minimize loss subject to achieving constant torque,

it is possible to calculate a set of corresponding Fourier series

coefficients for the stator currents that would result in the

desired performance. However, if the machine current does

not precisely possess these coefficients, then neither constanttorque nor minimum loss results. In fact, since the small high-

order currents multiply with the large fundamental back emf in

the torque equation [1-6], considerable ripple can result if the

current harmonics are in error.

These optimal control schemes are particularly relevant to

high-power Naval propulsion drive systems. For example,

maximum efficiency operation would be appropriate for

normal operation. Operating with maximum efficiency

subject to no torque ripple is appropriate in battle situations

where torsional vibrations due to torque harmonics increasedetectability. In that case, it is particularly important to

guarantee that the commanded currents reach their

commanded values exactly. Industrial applications such asrobotic positioning systems and numerically controlled

machines are two more examples where it is desirable to

mitigate torsional harmonics.In a conventional current source based drive with a

sinusoidal current command, a synchronous current regulator

is often used [7] to achieve the commanded current. This

control has many forms but the common feature is integral

feedback in a synchronous reference frame. This ensures thefundamental component of the current is precisely achieved.

Unfortunately, this control cannot be used to exactly obtain

current commands that are non-sinusoidal since it does notoperate on the individual harmonics.

In this paper, multiple reference frame theory is used to

formulate a synchronous estimator/regulator that ensures that

commanded current consisting of a fundamental component as

well as arbitrary harmonic content is exactly tracked. Whilethe regulator is presented in the context of a current regulator

herein, it may also be applied to voltage regulation by utilizing

the same architecture. It is interesting to note that althoughmultiple reference frame (MRF) theory has often been used as

a basis for analysis [8-13], it has not been widely used as a

basis for control.

The proposed control has two parts, a multiple referenceframe based estimator, which decomposes the measured

current to appropriate frames of reference, and a multiple

reference frame integral feedback controller which forces the

actual components to match their commands. The proposed

scheme is demonstrated using both computer simulation and a

hardware prototype.

II. EXAMPLE SYSTEM DESCRIPTION

Fig. 1 shows a block diagram of an example system in

which the multiple reference frame synchronous

estimator/regulator (MRFSER) is applied to a surface-

mounted permanent-magnet synchronous machine drive. In

this particular application, the input to the drive system is the

commanded torque, *eT , which is converted to an optimal qd

current command vector,*qdi . The conversion from a

A Multiple Reference Frame Synchronous Estimator/Regulator

P.L. Chapman, Member S.D. Sudhoff, Member

Purdue University

West Lafayette, IN 47907-1285


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command torque to*qdi may be accomplished by any of the

methods in [1-6], but the exact methodology is not relevant to

the work set forth herein. Instead, the objective of the paper is

to set forth a means of exactly achieving the current command.

Formal definition and notational convention for*qdi and other

vectors is set forth in the following section. Given the

reference current vector,*qdi , the vector of measured machine

currents, abci , and the rotor position, r , the multiple

reference frame synchronous estimator/regulator proposed in

this paper synthesizes a machine variable inverter command

vector, abci

~

, which is used as the current command to either adelta or hysteresis current regulator [14]. Although these

modulation strategies produce currents that are nearly equal to

the commanded current, tracking errors do exist. These

tracking errors do not scale with each harmonic

proportionally, so a different error is encountered for eachharmonic. It is the function of the MRFSER to ensure that the

resultant machine current, abci , corresponds precisely to the

reference, *qdi , in the steady state. Therefore, abci will not be

equal to the inverter command vector, abci~

. It is important to

observe that although the MRFSER is shown in the context of

a SMPMSM drive herein, it is readily applied to any situation

in which harmonic injection for voltage or current isappropriate.

III. NOMENCLATURE

This section sets forth the notation and transformationsthat are necessary for implementation of the synchronous

regulator. Phase variables, af , bf , and cf , are written in

vector form as

[ ]Tcbaabc fff=f (1)

wherefmay represent a voltage or current. The component of

these variables which is constant in a reference frame that

rotates at 'x' times the fundamental frequency is defined as

[ ]Txdxqxqd ff=f . (2)

The phase variable vector (1) may be approximately expressed

in terms of the q- and d-axis variables as

=

Nx

xqd

xabc fKf

1)( (3)

where a rotational transformation,x

K , is given by

+

+=

))(sin())(sin()sin(

))(cos())(cos()cos(

3

2

3

2

3

23

2

3

2

eee

eeex

xxx

xxxK (4)

and where the pseudo-inverse is defined as

T

2

31)()(

xxKK

= . (5)

In (3), N is the set of all reference frames considered,

where each reference frame corresponds to exactly oneharmonic present in variables of (1). The set is formally

defined as

}:{},:}3{}13{{ 0iNI += rqdrqpqpN (6)

where Iis the set of integers, Nis the set of natural numbers,

and only non-zero harmonics are considered as shown by the

set on the right-hand side of the intersection. In (4), e is the

electrical angle of a synchronous reference frame.

If the position of the synchronous reference frame is set

equal to rotor position, i.e. re = , then this transformation

is similar to the generalized Parks transformation set forth in[15] with the following exceptions. First, only referenceframes that are located at multiples of the electrical angle are

considered since these are the harmonics generally of interest.The second difference is that the multiple of the electrical

angle, x, multiplies the quantities in the cosines and sines,

)3/2( ex , rather than just the electrical angle,

3/2 ex . The harmonics of interest in a balanced,

symmetrical, power system are in the series {-5, -2, 1, 4,

7,} and since a wye-connected system is not assumed, the

triplen (zero sequence) harmonics {3,6,9,} are also of

interest. Notice that, for example, -5 is considered and 5 is

not since the 5thharmonic in a balanced, symmetrical systemexhibits a negative phase sequence. Defining (4) in this

manner has the advantage that ifxis a triplen harmonic, then it

is automatically incorporated into the same transformation as

is used for the non-triplen harmonics. In [13], the triplen

harmonics were considered under a different transformation

that only applied to zero sequence variables.

A vector of the form qdf is defined as the union of

individual vectors from each reference frame,qdf ,

qdf ,,

qdf , such that

[ ]T... = dqdqdqqd ffffff f . (7)

In (7) and throughout the work, is the first reference

frame considered and is the last frame considered. Herein,physical variables shall be depicted without a modifier as in

(1-7). In contrast, estimated values will be distinguished with

a circumflex (^), reference commands with an asterisk (*), and

inverter commands with a tilde (~), but otherwise have the

same structure as (1-7). The two types of commands

(reference and inverter) differ in that reference commands are

physically desired values. Inverter commands reflect

PMSM

Te*OptimalCurrentGenerator

MRFSERCurrent-RegulatedInverter

riabc

iabciqd* ~

Fig. 1 System Diagram


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commands to the specific inverter modulation strategy. Sincethe inverter gives rise to tracking error, the MRFSER

synthesizes the inverter commands that result in machine

currents that will exactly correspond to the reference

commands.

IV. MULTIPLE REFERENCE FRAME

SYNCHRONOUS REGULATORThe regulator portion of the MRFSER is shown in Fig. 2

in the context of a current-controlled SMPMSM drive. In this

case, fis replaced with iand the synchronous reference frame

of interest is the rotor reference frame so that re = . In Fig.

2, the vectors*

qdi ,*

qdi ,,*

qdi represent the reference qdaxis

current components in reference frames , ,, ,

respectively. Likewise, qdi ,qdi ,,

qdi represent the

estimated qd axis current components in reference frames

,...,, . Details of how the estimated currents are obtained

are in the next section. The operator 1/s denotes time

integration and cG is the associated controller loop gain. The

integrated error of each reference frame component is then

transformed into a component of the inverter command, abci~

,

abci

~,, abci

~, in each controller block. The vectors abci

~,

abci

~,, abci

~, from each block are summed in accordance with

(6) to give an aggregate inverter command, abci~

, which is

utilized by the inverter as the hysteresis or delta modulator

control signal.

The action of the MRFSER is similar to a synchronouscurrent regulator [7] that separately operates on each harmonic

component. The Rowan and Kerkman regulator [7] could beused to ensure convergence of the fundamental in the presence

of current harmonics, but would not ensure convergence of the

harmonics themselves. In essence, the integral feedback of the

MRFSER in each reference frame ensures that*qdqd ii = in

the steady state. However, there is an important difference in

that the multiple reference frame current vectorsqdi ,

qdi ,,

qdi , do not physically exist nor are readily

computed by a straightforward mathematical transformation.

A means of estimating these components is set forth in the

next section.

V. MULTIPLE REFERENCE FRAMESYNCHRONOUS ESTIMATOR

There have been several papers that discuss dynamicharmonic estimation for power system applications [16-20].

These are typically designed for use with large-scale power

systems and are not optimized for use in single converter

systems. Unlike [16-20], the estimator presented herein is

specifically designed for simultaneous three-phase

measurement for a single converter and furthermore, it is

presented in a context that makes it suitable for use with the

MRFSER presented herein.

The block diagram for the estimation system is depictedin Fig. 3. Therein, it may be observed that the estimator

consists of n branches, each of which estimates the currentvector associated with one reference frame. The branches are

interconnected in such a way that for any branch the estimated

harmonics of all the other branches are subtracted from the

measured current, abci . In this way, the harmonic of interest

to that branch becomes isolated in the steady state as the

estimated currents from the other branches converge to the

iqd*

iqd^

Gcs

(K)

-1 iabc~

iqd*

iqd^

Gcs

(K)-1 iabc

~

iqd*

iqd^

Gcs

(K)-1 iabc

~

iabc~

Fig. 2 MRFSER regulator applied to current-controlled PMSM drive

_

_

_

Fig. 3 MRFSER estimator applied to current-controlled PMSM drive


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correct values. This is necessary since applying (4) directly to

abci produces the vectorxqdi only as an average value.

Vectors from reference frames other than 'x' contribute

sinusoidally varying components to the average value and

have amplitudes proportional to the magnitudes of the other

harmonics. These sinusoidally varying components aredifficult to filter since their amplitudes may be high compared

with the average value of the signal. When the harmonic isisolated however, utilizing the transformation (4) becomes

effective since it extracts onlyxqdi and there are no

sinusoidally varying components. Each branch then has an

integral feedback loop that drives the estimated current exactly

to the actual current in the steady state.

To demonstrate the intuitive explanation of the estimator,

formal mathematical justification is necessary. Beginning

from analysis of the block diagram in Fig. 3,

])()([1 11

+=

Nm

mqd

mxqd

xabc

xxqd

e dt

d

GiKiKiKi (8)

may be written for any given Nx . Substituting (6) andsimplifying yields

=Nm

mqd

mqd

mxe

xqd G

dt

d][)(

1iiKKi (9)

from which it will be shown thatxqd

xqd ii for all Nx .

First let us assume that each xqdi is a constant, since the

objective here is to prove stability and convergence of the

isolated estimator. As will be discussed later, this assumptiondoes not restrict the use of the regulator to conditions in which

eachxqdi is constant in practice it simply means the

estimator should be faster than the control. Therefore,

0=xqddtd

i may be subtracted from the left side of (9)

=Nm

mqd

mqd

mxe

xqd

xqd G

dt

d][)(][

1iiKKii . (10)

It is convenient to define error vectors associated with each

reference frame as

xqd

xqd

xqd ii

= . (11)

Substitution of (11) into (10) and pre-multiplying by the1

)(

x

K yields

mqd

Nm

me

xqd

x Gdt

d=

11 )()( KK . (12)

Since this equation was written for arbitrary x, it may bewritten for each member of the setN. If this is done, and all of

the nequations are added, then

mqd

Nm

me

mqd

Nm

mnG

dt

d=

11)()( KK . (13)

Note that each 1)( xK is a sinusoidal function of r . By

considering multiple values of r it can be seen that a

solution of (13) must satisfy

NxnGdt

d xqde

xqd = . (14)

The solution to this differential equation is a decayingexponential since enG is negative. Therefore, the error

vectors,xqd , approach zero exponentially as time increases

which implies

Nxxqdxqd

t=

iilim (15)

as desired. The conclusion is that with a constant input, the

estimator output will converge to the correct value. Although

constant current was assumed in this derivation, the estimator

does not require that the physical variable be constant in orderto converge, it merely requires that signal change slowly with

respect to the speed of the estimator. Such behavior is typicalof any estimator. This does not prove that the overall system

will be stable; that is a function of machine parameters and

both the estimator and control gains. This problem, however,

is no different than the design of any other control that

involves both an estimator and controller and typically

involves making the estimator faster than the control. For thepurpose of selecting gain, it is appropriate to use a nonlinear

average value model of the system and control [21]. The

framework for such a model for this particular system is set

forth in [13]. A convenient way to do this is to linearize themodel at the fully loaded condition and use linear systems

techniques to select eG and cG . Then, the plant is linearized

versus operating point to make sure pole locations are globallyacceptable. Finally, it should be mentioned that it is possible

to select different values of eG and cG for each reference

frame, but this is not explored herein.

VI. SIMULATED PERFORMANCE

In this section, a computer simulation is used to explore

the operation of a MRFSER. Here, the motor drive considered

is described in [13] and utilizes an optimal control strategy set

forth in [6]. The significant harmonics are the first, third, and

fifth, and therefore, the set Nis {1 3 5} and n=3. Since it isdesirable for the estimator to converge faster than the

controller, gains of 1s100 =eG and-1s20=cG are used.

For this study, a constant torque command of 1.4 Nm is issued

and the dc bus voltage is 100 V. A load torque proportional to

speed is assumed. Current regulated delta modulation [14] (as

opposed to voltage source delta modulation [22]) with a 10

kHz switching frequency is used. A current regulated inverter,

such as delta-modulated, is appropriate for this comparison

since [1-5] suggest that it is sufficient to achieve optimal

control. Since this application requires triplen harmoniccurrents, it is necessary to use an inverter topology that allowszero sequence current (such as H-bridges); otherwise, the


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inverter topology used is irrelevant to the discussion herein.Furthermore, the inverter drives a SMPMSM that is equipped

with suitable rotor position feedback but may also use a

suitable estimator [23].Figs. 4-5 depict traces from an example simulation where

the SMPMSM drive is started from rest. A step torque

command is given, which represents a worst-case scenario

wherein the transients are likely to be most severe. Fig. 4shows the estimated q-axis currents on start-up of the machine

where the dashed lines indicate the commanded values. It is

apparent that the estimated currents converge to the

commanded values. A similar plot may be constructed for thed-axis quantities, however, it is omitted for brevity. Fig. 5

depicts the same study, except that abcvariables are shown.

For reference, the electrical rotor speed, r , is shown in the

first trace. The reference a-phase current, *ai , is shown in the

second trace, with the estimated current, ai , below. The

estimated current is reconstructed from the vector of estimated

q- and d-axis currents, qdi , using (6). The actual a-phase

current, ai , is depicted in the final trace. While the bottom

trace of Fig. 4 shows an overshoot in the estimated 5th

harmonic. This overshoot is small in comparison with the

1.5

0

-1.5

ia (A)5 ms

Fig. 6 Measured a-phase current

1.0

-1.0

0

1.0

-1.0

0

ia(A)

ia(A)

(a)

(b)

5 ms

Fig. 7 Experimentally reconstructed (dashed) and commanded (solid) a-

phase currents for 5 kHz switching frequency

(a) with MRFSER (b) without MRFSER

1.2

0

-0.2

0.2

-0.3

0

iq1(A)^

iq3(A)^

iq-5

(A)^

80 ms

Fig. 4 Simulated estimated q-axis currents on start-up (dashed lines

depict the commanded values)

1

-1

0

1

-1

0

1

-1

0

250

0

r (rad/s)

i *a (A)

ia(A)^

ia(A)

50 ms

Fig. 5 Simulated electrical speed, and a-phase quantities on start-up


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fundamental, and so ai tracks ai very closely except for the

high frequency switching noise, which is of limited interest in

terms of torque control. The actual current gradually

converges to the reference current as the synchronousregulator operates.

VII. EXPERIMENTAL PERFORMANCE

The test set-up is the same as that simulated in theprevious section except a dynamometer is used to load the

machine and regulate the speed at 600 rpm.

It is interesting to observe the measured currents in the

time domain both with and without the MRFSER. First, a

measured a-phase current waveform is shown in Fig. 6.Therein, both the low frequency harmonic content and

switching noise are evident. As stated, the switching induced

pulsation is of little interest here. Therefore, in Figs. 7-8 tofollow, the switching noise has been subtracted out of the

measured signals. This is accomplished by capturing the

waveform with a digital storage oscilloscope, extracting the

Fourier series coefficients of the harmonics of interest, and

then reconstructing the waveform from these coefficientsusing (6). Fig. 7 (a) shows the measured (dashed) and

reference (solid) current with the MRFSER in place with a 5

kHz switching frequency. Here, good agreement is apparent

between the two. Fig. 7 (b) shows a similar plot, exceptwithout the MRFSER. It is evident that the measured and

reference current do not agree. Fig. 8 depicts the same studyas shown in Fig. 7, except that the switching frequency is now

15 kHz. Less error is evident in Fig. 8 (b) than in Fig. 7 (b),

but there is still significant difference between the measured

and commanded current.

Another interesting experimental study is depicted in Fig.

9. Therein, the dynamometer is used to measure the

mechanical torque (electromagnetic torque minus friction andwindage) at a fixed speed for various switching frequencies ofthe delta-modulated inverter. The mechanical torque

measured differs slightly from the commanded

electromagnetic torque (1.4 Nm) due to frictional and windage

torque. This difference is the same for each point in Fig. 9

since the speed is held constant, and therefore only amounts to

a uniform offset. As may be observed, the resultant torquewhen the MRFSER is used is independent of switching

frequency. However, when the MRFSER is not used,

significant dependence on switching frequency is evident.

Since increased switching frequency leads to larger

semiconductor losses requires and higher edge rates that lead

to electromagnetic compatibility problems, it is advantageousin some applications to utilize the MRFSER in conjunction

with lower switching frequency.

VIII. CONCLUSIONS

A multiple reference frame based synchronous regulator

for power converter applications has been presented and the

operation has been demonstrated both by computer simulation

and experimentally. The synchronous regulator ensures that

each harmonic of a commanded current is reached exactly inthe steady state. This is useful in a variety of power converter

applications, but is particularly useful in optimal currentcontrol schemes for permanent-magnet synchronous machine

drives.

IX. ACKNOWLEDGEMENTS

The authors wish to gratefully acknowledge the Office of

Naval Research for financial support, and Todd Walls of

Emerson Electric for donating the test motor.

1.30

1.20

1.10

1.00

5 7 9 11 13 15fsw(kHz)

Tm(Nm)

o- with controlx- without control

Fig. 9 Measured mechanical torque vs. switching frequency: with and

without the MRFSER

1.0

-1.0

0

ia(A)

1.0

-1.0

0

ia(A)

(a)

(b)

5 ms

Fig. 8 Experimentally reconstructed (dashed) and commanded (solid) a-

phase currents for 15 kHz switching frequency

(a) with MRFSER (b) without MRFSER


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X. REFERENCES

[1] H. Le-Huy, R. Perret, R. Feuillet, Minimization of Torque Ripple in

Brushless DC Motor Drives, IEEE Transactions on Industry

Applications, Vol. IA-22, No. 4, 1986, pp. 748-755.[2] J.Y. Hung, Z. Ding, Minimization of Torque Ripple in Permanent-

Magnet Motors: A Closed Form Solution, Proceedings of the 18thIEEE

Industrial Electronics Conference, 1992, pp. 459-463.[3] D.C. Hanselman, Minimum Torque Ripple, Maximum Efficiency

Excitation of Brushless Permanent Magnet Motors, IEEE Transactionson Industrial Electronics, Vol. 41, No. 3, 1994, pp. 292-300.

[4] E. Favre, L. Cardoletti, M. Jufer, Permanent-Magnet SynchronousMotors: A Comprehensive Approach to Cogging Torque Suppression,

IEEE Transactions on Industry Applications, Vol. 29, No. 6, 1993, pp.1141-1149.

[5] C. Kang, I. Ha, An Efficient Torque Control Algorithm for BLDCM

with a General Shape Back EMF, PESC Record- Power ElectronicsSpecialists Conference, 1993, pp. 451-457.

[6] P.L. Chapman, S.D. Sudhoff, C.A. Whitcomb, Optimal Control

Strategies for Permanent-Magnet Synchronous Machine Drives,Accepted for publication in IEEE Transactions on Energy Conversion,1998.

[7] T.M. Rowan, R.J. Kerkman, "A New Synchronous Current Regulatorand an Analysis of Current-Regulated Inverters," IEEE Transactions on

Industry Applications, v IA-22, n 4, 1986, pp. 678-690.

[8] S.D. Sudhoff, Multiple Reference Frame Analysis of an UnsymmetricalInduction Machine, IEEE Transactions on Energy Conversion, Vol. 8,No. 3, Sept. 1993, pp. 425-432.

[9] S.D. Sudhoff, Multiple Reference Frame Analysis of a MultistackVariable Reluctance Stepper Motor, IEEE Transactions on EnergyConversion, Vol. 8, No. 3, Sept. 1993, pp. 418-424.

[10] T.A. Walls, S.D. Sudhoff, Analysis of a Single-Phase InductionMachine with a Shifted Auxiliary Winding, IEEE Transactions on

Energy Conversion, Vol. 11, No. 4, Dec. 1996, pp. 681-686.

[11] J.L. Tichenor, P.L. Chapman, S.D. Sudhoff, R. Budzynski, Analysis ofGenerically Configured PSC Induction Machines, Accepted for

publication inIEEE Transactions on Energy Conversion, 1997.

[12] P.C. Krause, Method of Multiple Reference Frames Applied to theAnalysis of Symmetrical Induction Machinery, IEEE Transactions on

Power Apparatus and Systems, Vol. PAS-87, Jan. 1968, pp. 218-227.

[13] P.L. Chapman, S.D. Sudhoff, C.A. Whitcomb, Multiple Reference

Frame Analysis of Non-sinusoidal Brushless DC Drives, accepted for

publication inIEEE Transactions on Energy Conversion, 1998.[14] B.K. Bose (ed.), Power Electronics and Variable Frequency Drives,

IEEE Press, 1997.[15] P.C. Krause, O. Wasynczuk, S.D. Sudhoff, Analysis of Electric

Machinery, IEEE Press, 1995.[16] A. Azemi, E.Yaz, K. Olejniczak, Reduced-Order Estimation of Power

System Harmonics, Proceedings of the IEEE Conference on Control

Applications, 1995, pp. 631-636.

[17] A.A. Girgis, W.B. Chang, E.B. Makram, A Digital RecursiveMeasurement Scheme for On-Line Tracking of Power System

Harmonics,IEEE Transactions on Power Delivery, Vol. 6, No. 3, July1991, pp. 1153-1160.

[18] H.M. Beides, G.T. Heydt, Dynamic State Estimation of Power System

Harmonics Using Kalman Filtering Methodology, IEEE Transactionson Power Delivery, Vol. 6, No. 4, Oct. 1991, pp. 1663-1669.

[19] P.K. Dash, D.P. Swain, A.C. Liew, S. Rahman, An Adaptive Linear

Combiner for On-Line Tracking of Power System Harmonics, IEEETransactions on Power Systems, Vol. 11, No. 4, 1996, pp. 1730-1735.

[20] M. Najjar, G.T. Heydt, A Hybrid Nonlinear-Least Squares Estimation

of Harmonic Signal Levels in Power Systems, IEEE Transactions onPower Delivery, Vol. 6, No. 1, Jan. 1991, pp. 282-288.

[21] S.D. Sudhoff, S.F. Glover, "Modeling Techniques, Stability Analysis,

and Design Criteria for DC Power Systems with Experimental

Verification," Proceedings of SAE Aerospace and Power SystemsConference, 1998, pp. 55-69.

[22] M.A. Rahman, J.E. Quaicoe, M.A. Choudhury, Performance Analysisof Delta Modulated Inverters, IEEE Transactions on Power

Electronics, Vol. PE-2, No. 3, July 1987, pp. 227-233.

[23] K.A. Corzine, S.D. Sudhoff, "Hybrid observer for high performance

brushless DC motor drives,"IEEE Transactions on Energy Conversion,v 11 n 2 Jun 1996. p 318-323.

Patrick L. Chapman1974- (S 94, M 96) is native to Centralia, Missouri.

He received the degrees of Bachelor of Science and Master of Science inElectrical Engineering in 1996 and 1997, respectively, from the University ofMissouri-Rolla. Currently, he is pursuing a Ph.D. in Electrical Engineering at

Purdue University. During his education, he has conducted research in theareas of power electronics, electric machinery, and solid-state power systems.

Scott D. Sudhoff (M 88) received the BSEE, MSEE, and Ph.D. degreesfrom Purdue University in 1988, 1989, and 1991, respectively. From 1991-

1993 he served as a half-time visiting faculty and half-time consultant for P.C.

Krause and Associates. From 1993-1997 he served as an assistant professor atthe University of Missouri-Rolla and became an associate professor at UMRin 1997. Later in 1997, he joined the faculty at Purdue University as an

associate professor. His interests include the analysis, simulation, and designof electric machinery, drive systems, and finite inertia power systems. He hasauthored or co-authored over twenty journal papers in these areas.

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