dead beat control

8/11/2019 dead beat control

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October 1999 1Simone Buso - Universit di Padova Lesson 4

LessonLesson 44

DigitalDigital Control of ThreeControl of Three --PhasePhase DC/ACDC/ACConvertersConverters :: Current Control TechniquesCurrent Control Techniques

DeadDead --Beat Current ControlBeat Current Control


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Typical control setTypical control set --upup

LL FF

abc

i* i*

SVM

EE

+

-+

-

PowerPowerConverterConverter

V*

V*

ii LLFF

ZZ LL VVLL

LoadLoad

Digital ControlDigital Control

DeadBeatControl

uu SSuu avav


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Dead-Beat Current Control EquationDead-Beat Current Control Equation

current referencecurrent reference

inverter currentinverter current

average inverter average inverter voltagevoltageload voltageload voltage

ii**LLFFiiLLFF

uu ss

[[ ]] )k(u)k(u2)k(i)k(iT

L)1k(u avsL

*L

sw

Fmav

FF

++==++ [[ ]] )k(u)k(u2)k(i)k(iT

L)1k(u avsL

*L

sw

Fmav FF ++==++

u av


kTkT (k+1)T(k+1)T (k+2)T(k+2)T

ii**LLFF iiLLFF

uu

TTswsw


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TheThe control equationcontrol equation assumes aassumes a very simplevery simple loadloadmodelmodel ::

+LLFmFm

uu SS

TheThe parameterparameter LLFmFm andandvoltagevoltage uu SS mustmust bebe

knownknown to compute theto compute therequired voltagerequired voltage uu avav (k+1) .(k+1) .

WhileWhile measuringmeasuring LLFmFm

is quiteis quite easy,easy, some problemssome problemsmay arise withmay arise with voltagevoltage uu SS ..

NoteNote also thatalso that the equationthe equation isis independentindependent ofof E.E.


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The controlThe control equationequation assumesassumes that thethat the voltagevoltage atatthe load connection pointthe load connection point uu SS is knownis known .. ThisThisvoltage can either bevoltage can either be measuredmeasured oror estimatedestimated ..

InIn anyany case, for case, for the control systemthe control system toto workworkproperlyproperly ,, the load must exhibit voltage sourcethe load must exhibit voltage sourcecharacteristicscharacteristics ,, withwith lowlow seriesseries impedance withinimpedance withinthe current controller bandwidththe current controller bandwidth ..

If this is not theIf this is not the case, acase, a model mismatchmodel mismatch isispresent andpresent and significant deviationssignificant deviations fromfrom thetheexpectedexpected closed loopclosed loop performanceperformance can take placecan take place((instabilities or lightly damped oscillationsinstabilities or lightly damped oscillations ).).


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[[ ]])k(i)1k(i

T

L)1k(u)1k(e

FF LLsw

Fm

avs ++==

[[ ]])k(i)1k(i

T

L)1k(u)1k(e

FF LLsw

Fm

avs ++==

The controlThe control

equation can be easilyequation can be easily

rearrangedrearranged

toto

create ancreate an estimation algorithmestimation algorithm for for the loadthe loadvoltagevoltage uu SS ,, asas the followingthe following ::

wherewhere ee SS standsstands for for thethe estimationestimation ofof uu SS .. ActuallyActually ,, the value of uthe value of u SS (k)(k) is needed in theis needed in the

control equationcontrol equation ;; thereforetherefore somesome interpolation isinterpolation isneededneeded toto generategenerate ee SS (k)(k) fromfrom ee SS (k-1).(k-1).

,,,


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The simplestThe simplest

wayway

toto

generate egenerate e

SS (k)(k)

fromfrom

ee

SS (k-1)(k-1)

isis

by means of by means of linear linear interpolationinterpolation .. Applying thisApplying this totoee SS (k-1)(k-1) we getwe get to:to:

10),2k(e)1k(e)1()k(e s ++== 10),2k(e)1k(e)1()k(e s ++==

By varyingBy varying ,, it is possibleit is possible toto vary the dynamics of vary the dynamics of the estimator the estimator .. ThisThis affects the system stabilityaffects the system stability ininthe presencethe presence of of mismatchesmismatches .. InIn thethe ideal case,ideal case,therethere areare nono effectseffects on the stabilityon the stability if if is changedis changed ..

It isIt is importantimportant toto noticenotice thatthat control and estimationcontrol and estimationequationequation areare dynamically not independentdynamically not independent ..


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DeadDead --Beat Control Behavior Beat Control Behavior

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-50

-40

-30

-20

-10

0

10

20

Current reference tracking (no PWM)Current reference tracking (no PWM)

The controller is ableto replicate thecurrent referencewith a two-cycle

delay. Note theabsence of anytransient at the exitof saturated mode of operation.

The controller is ableThe controller is abletoto replicate thereplicate thecurrent referencecurrent referencewith awith a two-cycletwo-cycle

delay.delay. Note theNote theabsence ofabsence of anyanytransienttransient at the exitat the exitofof saturated modesaturated mode of of operation.operation.


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3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

x 10 -3

7

7.5

8

8.5

9

9.5

10

2T sw2T2T swsw


Current reference tracking (no PWM)Current reference tracking (no PWM)

This generates aThis generates atracking error.tracking error.

TheThe reference valuereference valueisis correctlycorrectlyreproducedreproduced onlyonlyafterafter two controltwo controlperiods.periods.


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Load voltage estimation in ideal conditionsLoad voltage estimation in ideal conditions

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

In ideal conditions,i.e. when noparameter or modelmismatches are

present, the error inthe estimation isvery small. In thiscase, we have = 1.

InIn ideal conditions,ideal conditions,i.e. wheni.e. when nonoparameter or modelparameter or modelmismatchesmismatches areare

present,present, the error the error ininthe estimationthe estimation isisvery small.very small. In thisIn thiscase, we havecase, we have = 1.= 1.


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Load voltage estimation in ideal conditionsLoad voltage estimation in ideal conditions

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Estimation error as afunction of time andparameter :

= 0.0 = 0.5 = 0.75 = 1.0

A minimum is almostreached for = 0.75.

Estimation error Estimation error as aas afunction offunction of timetime andandparameterparameter ::

= 0.0= 0.0 = 0.5= 0.5 = 0.75= 0.75 = 1.0= 1.0

AA minimumminimum is almostis almostreached forreached for = 0.75.= 0.75.


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DeadDead --Beat Control StabilityBeat Control Stability

InIn principleprinciple ,, if thereif there areare nono parameter andparameter and / / or or model errorsmodel errors ,, the closed loop current controlthe closed loop current control isisdynamicallydynamically equivalentequivalent to ato a double unitydouble unity --delaydelay ,,i.e. :i.e. :

iiLLFF(k) = i(k) = i**LLFF

(k-2)(k-2)

FromFrom aa stability analysisstability analysis standpoint this meansstandpoint this meansthat thethat the closed loop systemclosed loop system exhibitsexhibits two polestwo poleslocatedlocated in the origin of the complexin the origin of the complex Z-planeZ-plane andandnono zeroeszeroes .. InIn other wordsother words ,, it is equivalentit is equivalent to ato asecond order allsecond order all --passpass FIRFIR filter filter ..


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UnfortunatelyUnfortunately ,, parameter mismatchesparameter mismatches arearealways encountered in practicealways encountered in practice .. TheThe sensitivitysensitivityof the controls stabilityof the controls stability toto errorserrors ,, at least in theat least in theidentification ofidentification of parameter parameter LLFF ,, needsneeds toto bebeinvestigatedinvestigated ..

In general,In general, it is possibleit is possible toto calculate thecalculate the closedclosedloop polesloop poles of the systemof the system as aas a function of thefunction of therelative identificationrelative identification errorerror LL%% ,, defineddefined asasfollowsfollows ::

LL%%=1 -=1 - LLFmFm /L /LFF


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%2,1 Lp == %2,1 Lp ==


It is possible toIt is possible to verifyverify that, in the case ofthat, in the case of loadloadvoltage measurement,voltage measurement, the closed loop systemsthe closed loop systemshashas two simple polestwo simple poles given by:given by:

As can be seen,As can be seen, in the ideal casein the ideal case the poles arethe poles arelocated in the origin,located in the origin, butbut move awaymove away from it asfrom it as

soon assoon as there isthere is an identification error.an identification error.


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DeadDead --Beat Control StabilityBeat Control Stability Anyway,Anyway, the stability robustnessthe stability robustness is veryis very good.good. InIn

principle, the relative error can be as high aprinciple, the relative error can be as high a100%,100%, beforebefore instability occurs.instability occurs.

Practically, as soon as the error goes overPractically, as soon as the error goes over 60%60%the controller behaviorthe controller behavior becomes unacceptable,becomes unacceptable,i.e. lightly damped oscillationsi.e. lightly damped oscillations appear.appear.

The situation isThe situation is totally differenttotally different if, instead of if, instead of measuring the load voltage, anmeasuring the load voltage, an estimationestimationalgorithmalgorithm is adopted.is adopted.

This modifies theThis modifies the system dynamics,system dynamics, negativelynegativelyaffecting theaffecting the stability robustness.stability robustness.


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0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-10

0

10

20

30

40

50


Example of control instability.Example of control instability.

The system showspersistentoscillations, at a half of the samplingfrequency (Nyquistfrequency). Note thatthe simulation doesnot include PWM, sothe high frequency is

not current ripple.

The system showsThe system showspersistentpersistentoscillations,oscillations, at a half at a half of the samplingof the samplingfrequencyfrequency ((NyquistNyquistfrequency).frequency). Note thatNote thatthe simulationthe simulation doesdoesnot include PWM,not include PWM, sosothethe high frequency ishigh frequency is

notnot currentcurrent ripple.ripple.


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DeadDead --Beat Control StabilityBeat Control Stability ToTo analyze the control stabilityanalyze the control stability when voltagewhen voltage

estimationestimation is employed,is employed, it is necessary toit is necessary toinclude in the systems dynamic equations alsoinclude in the systems dynamic equations alsothethe estimator equation.estimator equation.

TheThe system order is increased by one,system order is increased by one, but thebut theanalytic calculationanalytic calculation of the closed loop poles isof the closed loop poles isstill possible.still possible. It can be found that:It can be found that:

%%3 L2zL3z)z( ++== %%3 L2zL3z)z( ++==

isis the systems characteristic polynomial,the systems characteristic polynomial, whosewhoseroots are the systemsroots are the systems eigenvalueseigenvalues ..


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1

-1

1

(a)

(b)(c)

(c)(b)

(c) (b)

Re(z)

Im(z)


SystemSystem eigenvalueseigenvalues plot in Z-planeplot in Z-plane

Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop poles(solutions of(solutions of (z)=0)(z)=0)for different valuesfor different values

ofof LL%% parameter:parameter:a)a) LL%% = 0%,= 0%,b)b) LL%% = 20%,= 20%,

c)c) LL%% = 30%.= 30%.Instability!Instability!Instability!


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0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.5

0

0.5

1

1.5

2

Example of control instability (voltage estimation)Example of control instability (voltage estimation)


Estimation error Estimation error Estimation error Estimated voltageEstimated voltageEstimated voltageLoad voltageLoad voltageLoad voltage

As can be seen, with

only a 25% error onthe value of parameter L F thesystem goes clearlyunstable.

As can be seen, withAs can be seen, with

only aonly a 25%25% error onerror onthe value of the value of parameter Lparameter L FF thethesystem goessystem goes clearlyclearlyunstable.unstable.


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AA different type of instabilitydifferent type of instability may affect themay affect thedead-beat control in case of adead-beat control in case of a model mismatch.model mismatch.

AA model mismatchmodel mismatch arises any timearises any time the actualthe actualload of the power converter differs from theload of the power converter differs from theinternal modelinternal model the controller uses to computethe controller uses to computethe control variable.the control variable.

AA typical casetypical case is the presence ofis the presence of capacitivecapacitivefiltersfilters at the converters output,at the converters output, needed inneeded incertain applications to reduce the current ripple.certain applications to reduce the current ripple.This is the case ofThis is the case of PWM rectifiersPWM rectifiers oror UPSsUPSs ..

These cases need to be analyzed.These cases need to be analyzed.


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====

++++==

0C1

C1

L1

LR

LR

L1

LR

LR

A,uii

x

u

0L10

u00

L1

xAdt

xd

ss

c

s

cFF

c

F

c

c

s

L

ss

avF

Fr

r

r

====

++++==

0C1

C1

L1

LR

LR

L1

LR

LR

A,uii

x

u

0L10

u00

L1

xAdt

xd

ss

c

s

cFF

c

F

c

c

s

L

ss

avF

Fr

r

r


The dynamicequations of theinput PWM filter are not included inthe converter

internal model.Stability analysiscan be performedby computing the

closed loopeigenvalues of thewhole system.

TheThe dynamicdynamicequations of theequations of theinput PWM filter input PWM filter areare not included innot included inthe converter the converter

internal model.internal model.Stability analysisStability analysiscan be performedcan be performedbyby computing thecomputing the

closed loopclosed loopeigenvalueseigenvalues of theof thewhole system.whole system.


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The control equation, together with aThe control equation, together with a discretizeddiscretizedversion of the load equationsversion of the load equations can be used tocan be used toderive aderive a complete state space representationcomplete state space representation of of the controlled system.the controlled system.

TheThe eigenvalueseigenvalues of the A matrixof the A matrix can then becan then becomputed.computed. Assuming theAssuming the output voltage is measuredoutput voltage is measured andand

there isthere is no identification error in parameter Lno identification error in parameter L FF ititis possible to evaluate the effect of the inputis possible to evaluate the effect of the inputcapacitivecapacitive filter alone.filter alone.


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Real and imaginarycomponents of theclosed loop polesfor different valuesof capacitor C:

a) C = 0.005 pub) C = 0.01 pu

c) C = 0.015 pu

d) C = 0.5 pu

Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop polesfor different valuesfor different valuesofof capacitor C:capacitor C:

a) C = 0.005a) C = 0.005 pupub) C = 0.01b) C = 0.01 pupu

c) C = 0.015c) C = 0.015 pupu

d) C = 0.5d) C = 0.5 pupu

1

(a)

(c)

(b)

Re(z)

Im(z)

(d)

(a)

(a) (b)

(c)

(d)

10

(a)

(d)

(d)instability!instability!instability!



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(a)

(c)

(b)

(d)

Re(z)

Im(z)

1

(d)

(d)(d)

(d)(a)

(a)

(a)

(b)(a)

(c)

Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop poles for for different values of different values of capacitor C:capacitor C:

a) C = 0.005a) C = 0.005 pupu ,,

b) C = 0.01b) C = 0.01 pupu ,,

c) C = 0.015c) C = 0.015 pupu ,,

d) C = 0.5d) C = 0.5 pupu ..



instability!instability!instability!


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DeadDead --Beat Control StabilityBeat Control Stability The last analysis shows thatThe last analysis shows that when the outputwhen the output

voltage is estimated the stability robustness isvoltage is estimated the stability robustness isworsened.worsened.

TheThe stability limitstability limit moves frommoves from less than 0.01less than 0.01 pupucapacitivecapacitive filterfilter to about 0.015to about 0.015 pupu ..

In fact,In fact, the critical capacitor valuesthe critical capacitor values are a littleare a littlesmaller than the ones typically adoptedsmaller than the ones typically adopted to filter to filter the current ripple.the current ripple.

As a consequence,As a consequence, in general only a littlein general only a littleoversizingoversizing of theof the capacitivecapacitive filter filter may bemay benecessary tonecessary to guarantee the systems stability.guarantee the systems stability.


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WithWith a little increasea little increase in the algorithm complexityin the algorithm complexityit is anywayit is anyway possible to improve the stabilitypossible to improve the stabilityrobustness.robustness.

The oscillationsThe oscillations that take place in the controlledthat take place in the controlled

system when it goes unstablesystem when it goes unstable are due to theare due to theinteraction between the estimation algorithminteraction between the estimation algorithmand the current control.and the current control.

TheThe propagation of such oscillationspropagation of such oscillations can becan be

avoided remembering that theavoided remembering that the estimated outputestimated outputvoltage waveform is known to be sinusoidal.voltage waveform is known to be sinusoidal.

DeadDead --Beat Control Stability ImprovementBeat Control Stability Improvement


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The control algorithm robustness can be greatlyThe control algorithm robustness can be greatlyimproved with aimproved with a pass-band filter centered at thepass-band filter centered at thesupply frequencysupply frequency applied to the estimated voltage.applied to the estimated voltage.

LLFmFmTT

swswzz-1-1

ReferenceReferenceGenerationGeneration

SelectiveSelectiveFilter Filter

Estimator Estimator

GGeqeq

++--

+ +

+

+

-

iiLLFF(k)(k)ii** (k) (k)LLFF

-

u (k)u (k)avavu (k)u (k)ss

TTswsw

LL FF

11

z - 1z - 1..

221

2211

zmzcosm21

z)1m(z)m1(cos2)z(W

++

++==221

2211

zmzcosm21

z)1m(z)m1(cos2)z(W

++++==Pass-band filter Pass-band filter



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TheThe selective band pass filter selective band pass filter eliminates theeliminates theoscillationsoscillations from the estimated voltagefrom the estimated voltagewaveform and sowaveform and so avoids their propagation inavoids their propagation inthe current loop.the current loop.

The idea is toThe idea is to exploit the informationexploit the information about theabout theinput voltage waveform, which is known to beinput voltage waveform, which is known to besinusoidal at the line frequency.sinusoidal at the line frequency.

TheThe stability analysis can now be repeatedstability analysis can now be repeatedtaking into account thetaking into account the presence of thepresence of theselective filter selective filter in the control system.in the control system.



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(a)

(a)

-1

1

(a)

(a)

(a)(c)

(c)

(c)

(c)

Re(z)

Im(z)

(b)

(b) (a)

(c)

0


Real and imaginaryReal and imaginarycomponents of thecomponents of theclosed loop polesclosed loop polesfor different valuesfor different values

ofof capacitor C, withcapacitor C, withLL%% = 20%.= 20%.

Note how the polesNote how the polesnownow remain insideremain inside

the unity circle.the unity circle.Filter dynamicsFilter dynamicsFilter dynamics


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PWM rectifier behavior with the conventionalestimation techniqueand L% = 20%.Upper trace: linevoltage (u S ).Middle trace: converter output current (- i LF).

Lower trace: estimatedvoltage (e S ).

PWM rectifier behavior PWM rectifier behavior with thewith the conventionalconventionalestimation techniqueestimation techniqueandand LL%% = 20%.= 20%.Upper trace:Upper trace: linelinevoltage (voltage ( uu SS ).).Middle trace:Middle trace: converter converter output current (-output current (- iiLLFF ).).

Lower trace:Lower trace: estimatedestimatedvoltage (voltage ( ee SS ).).


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Modified estimationModified estimationtechnique andtechnique andLL%% = 20%.= 20%.Upper trace:Upper trace: linelinevoltage (voltage ( uu SS ).).

Middle trace:Middle trace: converter converter output current (-output current (- iiLLFF).).

Lower trace:Lower trace: estimatedestimatedvoltage (voltage ( ee SS ).).



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Experimental ResultsExperimental ResultsPWM Rectifier PWM Rectifier

Conventional estimationLL 12%12%

Test conditions:Test conditions:no filter no filter on the estimated lineon the estimated linevoltage;voltage;VVdcdc = 120V;= 120V;LL%% = 0.12;= 0.12;time-scale: 2ms/div;time-scale: 2ms/div;Upper trace:Upper trace: estimated lineestimated linevoltage (200mV/div)voltage (200mV/div)Middle trace:Middle trace: line to neutralline to neutralmeasured voltage (40V/div)measured voltage (40V/div)

Lower trace:Lower trace: phase currentphase current(1A/div)(1A/div)


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Experimental ResultsExperimental ResultsPWM Rectifier PWM Rectifier

LL 30%30%Modified estimation

Test conditions:Test conditions:no filter no filter on the estimated lineon the estimated linevoltage;voltage;VVdcdc = 120V;= 120V;LL%% = 0.30;= 0.30;time-scale: 2ms/div;time-scale: 2ms/div;Upper trace:Upper trace: estimated lineestimated linevoltage (200mV/div)voltage (200mV/div)Middle trace:Middle trace: line to neutralline to neutralmeasured voltage (40V/div)measured voltage (40V/div)

Lower trace:Lower trace: phase currentphase current(1A/div)(1A/div)


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ReferencesReferences

[1][1] L.L. MalesaniMalesani , P., P. MattavelliMattavelli , S., S. BusoBuso :: Robust Dead-Beat CurrentRobust Dead-Beat CurrentControl for PWM Rectifiers and Active Filters, IEEE TransactionsControl for PWM Rectifiers and Active Filters, IEEE Transactionson Industry Applications,on Industry Applications, VolVol . 35, No. 3, May/June 1999, pp. 613-. 35, No. 3, May/June 1999, pp. 613-620.620.

[2][2] L.L. MalesaniMalesani , P., P. MattavelliMattavelli , S., S. BusoBuso :: "Dead-Beat Current Control for "Dead-Beat Current Control for

Active Filters"Active Filters" 24th Annual Conference of the IEEE Industrial24th Annual Conference of the IEEE IndustrialElectronics SocietyElectronics Society (IECON),(IECON), AachenAachen , Germany, August 31 -, Germany, August 31 -September 4, 1998, pp. 1859-1864.September 4, 1998, pp. 1859-1864.

[3][3] L.L. MalesaniMalesani , P., P. MattavelliMattavelli , S., S. BusoBuso :: On the Applications of ActiveOn the Applications of ActiveFilters to Generic Loads,Filters to Generic Loads, 88 thth International Conference onInternational Conference on

Harmonics and the Quality of PowerHarmonics and the Quality of Power ICHQP 98,ICHQP 98, Athens, Greece,Athens, Greece,October 14-16, 1998, pp. 310-319.October 14-16, 1998, pp. 310-319.

dead beat control

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