Research Article Internal-Model-Principle-Based Specific...

Research ArticleInternal-Model-Principle-Based Specific Harmonics RepetitiveController for Grid-Connected PWM Inverters

Wenzhou Lu1 Jinfei Shen1 and Yunhu Yang2

1Key Laboratory of Advance Process Control for Light Industry Ministry of Education Jiangnan University Wuxi 214122 China2School of Electrical and Information Engineering Anhui University of Technology Marsquoanshan 243002 China

Correspondence should be addressed to Wenzhou Lu luwenzhou126com

Received 9 May 2016 Revised 22 July 2016 Accepted 9 August 2016

Academic Editor Alfonso Banos

Copyright copy 2016 Wenzhou Lu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper analyzes the general properties of IMP-based controller and presents an internal-model-principle-based (IMP-based)specific harmonics repetitive control (SHRC) scheme The proposed SHRC is effective for specific 119899119896 plusmn 119898 order harmonics with119899 gt 119898 ge 0 and 119896 = 0 1 2 Using the properties of exponential function SHRC can also be rewritten into the format of multipleresonant controllers in parallel where the control gain of SHRC is 1198992 multiple of that of conventional RC (CRC) Thereforeincluding SHRC in a stable closed-loop feedback control system asymptotic disturbance eliminating or reference tracking for anyperiodic signal only including these specific harmonic components at 1198992 times faster error convergence rate compared with CRCcan be achieved Application examples of SHRC controlled three-phasesingle-phase grid-connected PWM inverters demonstratethe effectiveness and advantages of the proposed SHRC scheme

1 Introduction

Repetitive control (RC) as an internal model principle (IMP)[1 2] based controller can track or eliminate periodic signalsin an effective way if it is included in a stable closed-loopsystem Conventional RC (CRC) [3] presented since the early1980 and being the most widely used RC format nowadayscan achieve zero-error tracking or disturbance rejection forany periodic signal whose fundamental period time is knownat first Through fast-Fourier transformation (FFT) analysisany known periodic signal can be decomposed into dcfundamental component and all harmonic componentsThatis CRC is effective to track or eliminate any periodic signalincluding any order harmonic Due to the time delay linebeing included in the RC structure CRC is slower thanany instantaneous feedback controller although it has bettertracking accuracy RC has been studied in several aspects[4ndash12] including its properties and structures Moreover itsapplications can be found in many fields such as disc drives[13] robots [14] satellites [15] and PWMconverters [8 9 16ndash22]

However in many applications [7 8 10] dominantharmonics only concentrate at some specific order harmonic

frequencies Especially in three-phasesingle-phase PWMinverter applications 6119896plusmn1 or 4119896plusmn1 (119896 = 1 2 ) order har-monics dominate their output voltagescurrents distortionsIn these cases CRCmight be too slow to trackeliminate thesespecific harmonics with satisfactory error convergence rate

In order to solve this problem this paper proposesan IMP-based specific harmonics RC (SHRC) for three-phasesingle-phase grid-connected PWM inverters To betterunderstand the proposed SHRC the general properties ofIMP-based controller and the performance of IMP-basedCRC are analyzed at first Then the performance of IMP-based SHRC and its error convergence rate analysis are alsogiven Finally two application cases of SHRC controlledthree-phasesingle-phase grid-connected PWM inverters areprovided to demonstrate the effectiveness and advantages ofthe proposed SHRC scheme

2 IMP-Based Controller

21 Internal Model Principle (IMP) Some specific distur-bance and reference signals can be described as the outputof a linear dynamic system with zero-input and certain

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9640403 10 pageshttpdxdoiorg10115520169640403

2 Mathematical Problems in Engineering

++

+ y

d

u

eGpGc

yref

minus

Figure 1 Closed-loop feedback control system

initial conditions Thus a general disturbance signal can bedescribed as the format of a differential equation as follows

119889119902119889 (119905)

119889119905119902

+

119902minus1

sum

119894=0

120574119894

119889119894119889 (119905)

119889119905119894

= 0 (1)

Using Laplace transform (1) can be derived into

119889 (119904) =

119873119889 (119904) 119909119889 (0)

Γ119889 (119904)

(2)

whereΓ119889(119904) is the disturbance generating polynomial (D-GP)which can be defined into

Γ119889 (119904) ≜ 119904119902+

119902minus1

sum

119894=0

120574119894119904119894 (3)

Notice that this D-GP is only related with the denomi-nator of (2) Similarly reference signal can be described inthe same way as long as replacing the symbol ldquo119889rdquo with ldquo119903rdquoReference generating polynomial (R-GP) Γ119903(119904) can be definedas follows

Γ119903 (119904) ≜ 119904119902+

119902minus1

sum

119894=0

120582119894119904119894 (4)

Then internal model principle (IMP) [2 23] can beexpressed as follows If disturbance 119889(119905) or reference input119910119903119890119891(119905) can be expressed as its D-GP or R-GP just like Γ119889(119904)

or Γ119903(119904) asymptotic error elimination or reference trackingcan be achieved for the control system using the followingcontroller

119866119888 (119904) =

119875 (119904)

Γ119889 (119904) 119871 (119904)

(5)

or

119866119888 (119904) =

119875 (119904)

Γ119903 (119904) 119871 (119904)

(6)

Therefore only D-GP or R-GP that is Γ119889(119904) or Γ119903(119904) isneeded when using IMP while the amplitude information ofdisturbance or reference input is not needed

22 The General Properties of IMP-Based Controller Figure 1shows the diagram of a closed-loop feedback control systemLet the model of plant be

119866119901 (119904) =

119861119900 (119904)

119860119900 (119904)

=

119887119899minus1119904119899minus1

+ 119887119899minus2119904119899minus2

+ sdot sdot sdot + 1198870

119904119899+ 119886119899minus1119904

119899minus1+ 119886119899minus2119904

119899minus2+ sdot sdot sdot + 1198860

(7)

and assume that Γ119889(119904) or Γ119903(119904) is not the factor of 119861119900(119904)

Then the output sensitivity function of closed-loop feed-back system shown in Figure 1 is

119878119900 (119904) =

Γ119889 (119904) 119871 (119904) 119860119900 (119904)

Γ119889 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(8)

or

119878119900 (119904) =

Γ119903 (119904) 119871 (119904) 119860119900 (119904)

Γ119903 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(9)

and the output complementary sensitivity function is

119879119900 (119904) = 1 minus 119878119900 (119904) =

119875 (119904) 119861119900 (119904)

Γ119889 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(10)

or

119879119900 (119904) = 1 minus 119878119900 (119904) =

119875 (119904) 119861119900 (119904)

Γ119903 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(11)

Assume the selected 119871(119904) and 119875(119904) can make the closed-loop characteristic equation

119860cl (119904) = Γ119889 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)(12)

or

119860cl (119904) = Γ119903 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)(13)

have the negative realistic roots Thus the output response ofdisturbance 119889(119905) can be obtained as

119910 (119904) = 119878119900 (119904) 119889 (119904) =

119871 (119904) 119860119900 (119904)119873119889 (119904) 119909119889 (0)

119860cl (119904) (14)

Because 119860cl(119904) has stable roots the inverse Laplacetransform of 119910(119904) can asymptotically converge to zero thatis 119910(119905 rarr infin) = 0

The system error response of disturbance 119889(119905) is

119890 (119904) = minus119878119900 (119904) 119889 (119904) = minus

119871 (119904) 119860119900 (119904)119873119889 (119904) 119909119889 (0)

119860cl (119904) (15)

Because 119860cl(119904) has stable roots the inverse Laplacetransform of 119890(119904) can asymptotically converge to zero that is119890(119905 rarr infin) = 0

The system error response of reference 119910ref (119905) is

119890 (119904) = 119878119900 (119904) 119910ref (119904) =

119871 (119904) 119860119900 (119904)119873119903 (119904) 119909119903 (0)

119860cl (119904) (16)


The system output response of reference 119910ref (119905) is

119910 (119904) = 119879119900 (119904) 119910ref (119904)

= 119910ref (119904) minus119871 (119904) 119860119900 (119904)119873119903 (119904) 119909119903 (0)

119860cl (119904)

(17)

Mathematical Problems in Engineering 3

+

+

++krc

cos

2120587

nm

eminussT119900neminussT119900n

eminussT119900n

minus

minus

2cos2120587

nm

e(s)

Grc(s)

c(s)

Figure 2 Specific harmonics repetitive control (SHRC)

Because 119860cl(119904) has stable roots the inverse Laplacetransform of 119910(119904) can asymptotically track its reference inputthat is 119910(119905) rarr 119910ref(119905)

Therefore if D-GP or R-GP is included in the denomi-nator of a controller asymptotic disturbance eliminating orreference tracking can be achieved for the control system

Then we will see why conventional RC (CRC) is an IMP-based controller andwhy it can trackeliminate all harmonics

23 IMP-Based Conventional Repetitive Controller (CRC)According to the properties of exponential function [24]CRC can be rewritten into

119866rc (119904) = 119896rc sdot

119890minus119904119879119900

1 minus 119890minus119904119879119900

= 119896rc sdot

119890minus1199041198791199002

2 sinh (1199041198791199002)

= 119896rc sdot

119890minus1199041198791199002

119904119879119900 sdot prodinfin

119896=1[1 + 119904

2 (11989621205962119900)]

(18)

where 119896rc is the control gain and 119879119900 = 2120587120596119900 = 1119891119900 is thefundamental period of signals with 119891119900 being the fundamentalfrequency 120596119900 being the fundamental angular frequency

If the disturbance 119889(119905) or reference input 119910ref (119905) of closed-loop feedback control system is a periodic signal whichcan be decomposed into the summation of all harmoniccomponents that is dc fundamental component and anyorder harmonic component using FFT its D-GP or R-GP thatis Γ119889(119904) or Γ119903(119904) can be derived as follows

119889 (119905) = 1198890 +

infin

sum

119896=1

119889119896 cos (119896120596119900119905 + 120593119896) 997904rArr

Γ119889 (119904) = 119904

infin

prod

119896=1

(1199042+ (119896120596119900)

2)

(19)

or

119910ref (119905) = 1199030 +

infin

sum

119896=1

119903119896 cos (119896120596119900119905 + 1205931015840

119896) 997904rArr

Γ119903 (119904) = 119904

infin

prod

119896=1

(1199042+ (119896120596119900)

2)

(20)

Then CRC can be derived into

119866119888 (119904) = 119866rc (119904) =

119896rc119890minus1199041198791199002prodinfin

119896=1(11989621205962

119900)

119879119900 sdot Γ119889 (119904)

(21)

or

119866119888 (119904) = 119866rc (119904) =


119896=1(11989621205962

119900)

119879119900 sdot Γ119903 (119904)

(22)

From (21) or (22) it can be seen that D-GP Γ119889(119904) or R-GPΓ119903(119904) is included in the denominator of CRC controllerThusasymptotic disturbance eliminating or reference tracking forany periodic signal can be achieved using IMP indicated inSection 21 if CRC is used as the controller in a closed-loop feedback control system Therefore CRC is an IMP-based controller and is effective for any periodic signal thatis composed with all harmonic components

3 SHRC

31 Specific Harmonics RC (SHRC) A specific harmonics RC(SHRC) shown in Figure 2 for efficient removingtrackingspecific 119899119896 plusmn 119898 order harmonics is proposed as follows

119866rc (119904) = 119896rc sdot

cos (2120587119898119899) 119890119904119879119900119899

minus 1

1198902119904119879119900119899

minus 2 cos (2120587119898119899) 119890119904119879119900119899

+ 1

(23)

where 119899 and 119898 are integers with 119899 gt 119898 ge 0In three-phasesingle-phase PWM inverter applications

because 6119896 plusmn 1 or 4119896 plusmn 1 (119896 = 1 2 ) order harmonicsdominate its output distortion (23) can be used for thesespecific harmonics with 119899 = 6 or 119899 = 4 and 119898 = 1

Using the similar analysis with CRC in Section 23 it canbe achieved that SHRC is also an IMP-based controller in thefollowing subsection

32 IMP-Based SHRC According to the properties of expo-nential function [24] (23) can be rewritten into


119866rc (119904) =

119896rc2

sdot

cos (2120587119898119899) minus 119890minus119904119879119900119899

cosh (119904119879119900119899) minus cos (2120587119898119899)

=

119896rc sdot [cos ((2120587119899)119898) minus 119890minus119904(119879119900119899)

]

4sin2 (119898120587119899) (1 + 1199042 (11989821205962119900)) sdot prod

infin

119896=1[1 + 119904

2 ((119899119896 + 119898)

21205962119900)] [1 + 119904

2 ((119899119896 minus 119898)

21205962119900)]

(24)

where 119898 = 0 If 119898 = 0 (23) can be rewritten into

119866rc (119904) =

119896rc119890119904119879119900119899

minus 1

=

119896rc

(119904119879119900119899) sdot 1198901199041198791199002119899

sdot prodinfin

119896=1[1 + 119904

2 (1198991198961205960)

2]

(25)

If the disturbance signal 119889(119905) of closed-loop feedbackcontrol system is a periodic signal only including specific119899119896 plusmn 119898 order harmonic components its D-GP that is Γ119889(119904)can be derived as follows

119889 (119905) =

1198890 +

infin

sum

119896=1

119889119899119896 cos (119899119896120596119900119905 + 120593119899119896) (119898 = 0)

infin

sum

119896=1

119889119899119896minus119898 cos ((119899119896 minus 119898)120596119900119905 + 120593119899119896minus119898) +

infin

sum

119896=0

119889119899119896+119898 cos ((119899119896 + 119898)120596119900119905 + 120593119899119896+119898) (119898 = 0)

997904rArr

Γ119889 (119904) =

119904

infin

prod

119896=1

(1199042+ (119899119896120596119900)

2) (119898 = 0)

(1199042+ (119898120596119900)

2)

infin

prod

119896=1

(1199042+ ((119899119896 minus 119898)120596119900)

2) sdot (1199042+ ((119899119896 + 119898)120596119900)

2) (119898 = 0)

(26)

Then SHRC can be derived into

119866119888 (119904) = 119866rc (119904) =

119896rc sdot

119899 sdot 119890minus1199041198791199002119899

prodinfin

119896=1(119899119896120596119900)

2

119879119900 sdot Γ119889 (119904)

(119898 = 0)

119896rc [cos (2120587119898119899) minus 119890minus119904119879119900119899]11989821205962

119900prodinfin

119896=1[(119899119896 + 119898)

2(119899119896 minus 119898)

21205964

119900]

4sin2 (119898120587119899) sdot Γ119889 (119904)

(119898 = 0)

(27)

Similarly if the reference input 119910ref (119905) is a periodic signalonly including specific 119899119896 plusmn 119898 order harmonic componentsits R-GP that is Γ119903(119904) and corresponding SHRC can also bederived

So it can be seen that D-GP Γ119889(119904) or R-GP Γ119903(119904) isincluded in the denominator of SHRC controller Thusasymptotic disturbance eliminating or reference tracking forany periodic signal only including these specific 119899119896plusmn119898 orderharmonic components can be achieved using IMP indicatedin Section 21 if SHRC is used as the controller in a closed-loop feedback control system Therefore SHRC is also anIMP-based controller and is effective for any periodic signalonly including these specific harmonics

33 Error Convergence Rate of SHRC According to theproperties of exponential function [24] SHRC can also berewritten into the format of multiple resonant controllers inparallel as follows

119866rc (119904) =

1

2

119896rc sdot [

119890119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890119895(2120587119899)119898

+

119890minus119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890minus119895(2120587119899)119898

] = 119896rc sdot minus

1

2

+

119899

2119879119900 (119904 minus 119895119898120596119900)

+

119899

2119879119900 (119904 + 119895119898120596119900)

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 minus 119895119898120596119900)

(119904 minus 119895119898120596119900)2+ 119899211989621205962119900

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 + 119895119898120596119900)

(119904 + 119895119898120596119900)2+ 119899211989621205962119900

(28)

From (28) it can be achieved that for specific 119899119896plusmn119898 orderharmonics the control gain of SHRC is (1198992)119896rc119879119900 From[25] the corresponding control gain of CRC is 119896rc119879119900 So


PWM

RCu

e

Switching

+

+

+

minus

DBTransform

TransformCalculation

PLL

dq

abc

minus1

L

A

B

C

+

minus

L

L0

R

R

R

+ minus+

minus + minus

+ minus

+

minus +

minus

generator

Udc

Udc

cba

ab

ab

bc

bc

ca

c

c

b

b

a

a

ic

ic

ib

ib

ia

ia

icref

ibref

iaref

ibia

ia ib icur

iqref

idref

sin (120596t)cos (120596t)

S1

S2

S3

S4

S5

S6

signals (S1ndashS6)

Pref Qref

Figure 3 Repetitive controlled three-phase grid-connected PWM inverter

the control gain of SHRC is 1198992 multiple of that of CRC Theerror convergence rate of SHRC is also 1198992 times faster thanthat of CRC Particularly for the three-phase PWM invertersapplication the error convergence rate of SHRC with 119899 = 6

and 119898 = 1 is three times (ie 1198992 = 3) faster than that ofCRC For the single-phase PWM inverters application theerror convergence rate of SHRC with 119899 = 4 and 119898 = 1 istwo times (ie 1198992 = 2) faster than that of CRC ThereforeSHRC can be used to enhance the error convergence ratefor trackingremoving specific harmonics and thus improvethe performance of three-phasesingle-phase PWM invertercontrol system

In the next two sections SHRC will be used in three-phase and single-phase grid-connected PWM inverter sys-tems to verify the effectiveness of SHRC and to improve thecontrol performance of both inverter systems

4 Case 1 Three-Phase Grid-ConnectedPWM Inverters

41 Modeling Figure 3 shows a three-phase grid-connectedPWM inverter system where 119880dc is the dc-side voltage 119871and 119877 are inductance and resistor respectively 119894119886 119894119887 and 119894119888

are the inductance currents V119886 V119887 and V119888 are the 119886 119887 and 119888

three-phase grid voltages The control objective is to achievehigh current tracking accuracy through forcing 119894119886 119894119887 and 119894119888

to exactly track their reference 119894aref 119894bref and 119894cref which aregenerated by using PQ control method and instantaneouspower theory [26] System parameter values are shown inTable 1

Table 1 Parameters for three-phase grid-connected inverter system

Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119886119887119887119888119888119886

= 25V (rms)

Reference 119875ref = 100W119876ref = 0

Switching frequency 119891119904= 6 kHz

The mathematical model [18] can be described as

(

119894119886

119894119887

119894119888

) = (

minus

119877

119871

0 0

0 minus

119877

119871

0

0 0 minus

119877

119871

)(

119894119886

119894119887

119894119888

)

+

(

(

(

(

1

119871

(V119886 minus V119860)

1

119871

(V119887 minus V119861)

1

119871

(V119888 minus V119862)

)

)

)

)

(29)


The data-sample format of (29) can be gotten

(

119894119886 (119896 + 1)

119894119887 (119896 + 1)

119894119888 (119896 + 1)

)

=(

(

1198871 minus 1198872

1198871

0 0

0

1198871 minus 1198872

1198871

0

0 0

1198871 minus 1198872

1198871

)

)

(

119894119886 (119896)

119894119887 (119896)

119894119888 (119896)

)

+

(

(

(

(

(

1

1198871

V119886 (119896) minus1

1198871

119880dc (119896)

2

119906119886 (119896)

1

1198871

V119887 (119896) minus1

1198871

119880dc (119896)

2

119906119887 (119896)

1

1198871

V119888 (119896) minus1

1198871

119880dc (119896)

2

119906119888 (119896)

)

)

)

)

)

(30)

where 119906119886(119896) 119906119887(119896) and 119906119888(119896) are the three-phase duty cycles1198871 = 119871119879 and 1198872 = 119877 Therefore three independent singlesubsystems can be decomposed into

119894119895 (119896 + 1) =

1198871 minus 1198872

1198871

119894119895 (119896) +

1

1198871

V119895 (119896)

minus

119880dc (119896)

2

1

1198871

119906119895 (119896)

(31)

where 119895 = 119886 119887 119888A dead-beat (DB) current controller is chosen as follows

119906119895 (119896) =

2

119880dc (119896)[V119895 (119896) minus 1198871119894jref (119896) + (1198871 minus 1198872) 119894119895 (119896)] (32)

where 119895 = 119886 119887 119888 Letting 119894jref(119896) = 119894119895(119896 + 1) the closed-looptransfer function without RC is 119867(119911) = 119911

minus1 Therefore thecontrol delay forDB current controller is only one step that isone sample period in theory which can achieve fast dynamicresponse

Using the parameter values in Table 1 the mathematicalmodel for the three-phase grid-connected inverter system in(31) can be rewritten into

119894119895 (119896 + 1) = 09833119894119895 (119896) + 00333V119895 (119896) minus 08333119906119895 (119896) (33)

and the DB current controller in (32) can be rewritten into

119906119895 (119896) = 004V119895 (119896) minus 12119894jref (119896) + 118119894119895 (119896) (34)

where 119895 = 119886 119887 119888

42 Experimental Results Figure 4(a) shows the experimen-tal result with single DB controller It can be seen that currentdistortion is relatively serious that is the current waveformis not a smooth sine wave

Table 2 Parameters for single-phase grid-connected inverter sys-tem

Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119904= 25V (rms)



Figures 4(b) 4(c) and 5 show the steady-state responseand the current error convergence histories with CRC con-troller and the proposed SHRC controller being pluggedinto the DB controlled three-phase grid-connected inverterrespectively For comparability both RC gains are 119896rc =

02 Figures 4(b) and 4(c) show the current distortions canbe greatly improved with both CRC and SHRC controllersMoreover it can be clearly seen from Figure 5 that thecurrent error convergence times for CRC and SHRC are032 s and 012 s respectively Therefore the current errorconvergence rate of the proposed SHRC controller is nearlythree times faster than that of CRC controller (as indicated inSection 33)

Therefore the effectiveness of SHRC for three-phase grid-connected PWM inverter system and its advantage comparedwith CRC are verified

5 Case 2 Single-Phase Grid-ConnectedPWM Inverters

51 Modeling Figure 6 shows a single-phase grid-connectedPWM inverter system where119880dc is the dc-side voltage 119871 and119877 are inductance and resistor respectively 119894119904 is the inductancecurrent and V119904 is the grid voltage The control objective is toachieve high current tracking accuracy through forcing 119894119904 toexactly track its reference 119894sref which is generated by using PQcontrolmethod and instantaneous power theory [26] Systemparameter values are shown in Table 2

The mathematical model can be described as

119894119904 = minus

119877

119871

119894119904 +

1

119871

(V119904 minus Vin) (35)


119894119904 (119896 + 1) =

1198871 minus 1198872

1198871

119894119904 (119896) +

1

1198871

V119904 (119896)

minus

1

1198871

119880dc (119896) 119906119904 (119896)

(36)

where 119906119904(119896) is the single-phase duty cycles 1198871 = 119871119879 and1198872 = 119877

A DB current controller is chosen as follows

119906119904 (119896) =

2

119880dc (119896)[V119904 (119896) minus 1198871119894sref (119896) + (1198871 minus 1198872) 119894119904 (119896)] (37)

Letting 119894jref(119896) = 119894119895(119896 + 1) the closed-loop transfer functionwithout RC is119867(119911) = 119911

minus1Therefore the control delay for DB


ab bc ca

ia

(a) DB

ab bc ca

ia

(b) DB + CRC

ab bc ca

ia

(c) DB + SHRC

Figure 4 Steady-state response for three-phase grid-connected inverter system (119909 label 25msdiv 119910 label 20Vdiv amp 2Adiv)

t (s)

minus3

minus2

minus1

0

1

2

3

29 3 31 32 33 34 35 36 3728

iaref minus ia (A)

(a) CRC

t (s)29 3 31 32 33 34 35 36 3728

minus3

minus2

minus1

0

1

2

3

iaref minus ia (A)

(b) SHRC

Figure 5 Current tracking error histories with two RCs plugged into for three-phase grid-connected inverter system

current controller is only one step that is one sample periodin theory which can achieve fast dynamic response

Using the parameter values shown in Table 2 the math-ematical model of the single-phase grid-connected invertersystem in (36) can be rewritten into

119894119904 (119896 + 1) = 09833119894119904 (119896) + 00333V119904 (119896) minus 04167119906119904 (119896) (38)

and the DB current controller can be used with the samecoefficients in (34) for a convenient implementation purposeas follows

119906119904 (119896) = 004V119904 (119896) minus 12119894sref (119896) + 118119894119904 (119896) (39)


Figures 7(b) 7(c) and 8 show the steady-state responseand the current error convergence histories with CRC con-troller and the proposed SHRC controller being pluggedinto the DB controlled single-phase grid-connected inverterrespectively For comparability both RC gains are 119896rc =

02 Figures 7(b) and 7(c) show the current distortions forboth CRC and SHRC controllers can be greatly improvedMoreover it can be clearly seen from Figure 8 that the


Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1

Figure 6 Repetitive controlled single-phase grid-connected PWM inverter

s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC

Figure 7 Steady-state response for single-phase grid-connected inverter system (119909 label 25msdiv 119910 label 20Vdiv amp 2Adiv)

current error convergence times for CRC and SHRC are032 s and 016 s respectively Therefore the current errorconvergence rate of the proposed SHRC controller is nearlytwo times faster than that of CRC controller (as indicated inSection 33)

Therefore the effectiveness of SHRC for single-phasePWM grid-connected inverter system and its advantagecompared with CRC are verified

6 Conclusions

This paper proposed an internal-model-principle-based(IMP-based) specific harmonics RC (SHRC) Using the gen-eral properties of IMP-based controller and the propertiesof exponential function it can be concluded that SHRC canachieve zero-error tracking or perfect disturbance rejectionof specific periodic signal only including these specific


26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC

Figure 8 Current tracking error histories with two RCs plugged into for single-phase grid-connected inverter system

119899119896 plusmn 119898 order harmonic components at 1198992 times faster errorconvergence rate compared with conventional RC (CRC)

Two application examples of SHRC controlled three-phasesingle-phase grid-connected PWM inverters havebeen given to show the effectives and promising advantagesof SHRCThe experimental results show that compared withCRC SHRC offers three times (for three-phase inverter)or two times (for single-phase inverter) faster current errorconvergence rate and yields nearly the same current track-ing accuracy The proposed IMP-based SHRC provides ahigh performance control solution to grid-connected PWMinverters

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the Natural ScienceFoundation of China under Grant 51407084 by the Funda-mental Research Funds for the Central Universities underGrant JUSRP11461 by the Jiangsu Province Science andTechnology Support Project under Grant BE2013025 by theNatural Science Research key Project of Anhui ProvincialEducation Department under Grant KJ2014A026 by theNational Natural Science Foundation of Anhui Provincialunder Grant 1508085ME86 and by the Jiangsu ProvinceNatural Science Foundation under Grant BK20130159

References

[1] B Francis O A Sebakhy and W M Wonham ldquoSynthesis ofmultivariable regulators the internal model principlerdquo AppliedMathematics and Optimization vol 1 no 1 pp 64ndash86 1974

[2] B A Francis andWMWonham ldquoThe internalmodel principleof control theoryrdquo Automatica vol 12 no 5 pp 457ndash465 1976

[3] S Hara Y Yamamoto T Omata and M Nakano ldquoRepetitivecontrol system a new type servo system for periodic exogenous

signalsrdquo IEEE Transactions on Automatic Control vol 33 no 7pp 659ndash668 1988

[4] M Tomizuka ldquoZero phase error tracking algorithm for digitalcontrolrdquo Journal of Dynamic Systems Measurement and Con-trol vol 109 no 1 pp 65ndash68 1987

[5] M Tomizuka T-C Tsao and K-K Chew ldquoDiscrete-timedomain analysis and synthesis of repetitive controllersrdquo inProceedings of the American Control Conference pp 860ndash866Atlanta Ga USA 1988

[6] R Grino and R Costa-Castello ldquoDigital repetitive plug-in con-troller for odd-harmonic periodic references and disturbancesrdquoAutomatica vol 41 no 1 pp 153ndash157 2005

[7] K Zhou K-S Low DWang F-L Luo B Zhang and YWangldquoZero-phase odd-harmonic repetitive controller for a single-phase PWM inverterrdquo IEEE Transactions on Power Electronicsvol 21 no 1 pp 193ndash201 2006

[8] R Costa-Castello R Grino and E Fossas ldquoOdd-harmonicdigital repetitive control of a single-phase current active filterrdquoIEEE Transactions on Power Electronics vol 19 no 4 pp 1060ndash1068 2004

[9] B Zhang D Wang K Zhou and Y Wang ldquoLinear phase leadcompensation repetitive control of a CVCF PWM inverterrdquoIEEE Transactions on Industrial Electronics vol 55 no 4 pp1595ndash1602 2008

[10] G Escobar P G Hernandez-Briones P R Martinez MHernandez-Gomez and R E Torres-Olguin ldquoA repetitive-based controller for the compensation of 6ℓplusmn 1 harmoniccomponentsrdquo IEEE Transactions on Industrial Electronics vol55 no 8 pp 3150ndash3158 2008

[11] G Escobar P R Martınez J Leyva-Ramos and P MattavellildquoA negative feedback repetitive control scheme for harmoniccompensationrdquo IEEE Transactions on Industrial Electronics vol53 no 4 pp 1383ndash1386 2006

[12] K Zhou D Wang B Zhang Y Wang J A Ferreira and SW de Haan ldquoDual-mode structure digital repetitive controlrdquoAutomatica vol 43 no 3 pp 546ndash554 2007

[13] K K Chew and M Tomizuka ldquoDigital control of repetitiveerrors in disk drive systemsrdquo IEEE Control Systems Magazinevol 10 no 1 pp 16ndash20 1990

[14] C Cosner G Anwar and M Tomizuka ldquoPlug in repetitivecontrol for industrial robotic manipulatorsrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automationvol 3 pp 1970ndash1975 Cincinnati Ohio USA May 1990


[15] H L Broberg and R G Molyet ldquoReduction of repetitiveerrors in tracking of periodic signals theory and application ofrepetitive controlrdquo in Proceedings of the 1st IEEE Conference onControl Application Dayton Ohio USA 1992

[16] K Zhou D Wang and K-S Low ldquoPeriodic errors eliminationin CVCF PWM DCAC converter systems repetitive controlapproachrdquo IEE ProceedingsmdashControl Theory and Applicationsvol 147 no 6 pp 694ndash700 2000

[17] K Zhou and D Wang ldquoDigital repetitive learning controllerfor three-phase CVCF PWM inverterrdquo IEEE Transactions onIndustrial Electronics vol 48 no 4 pp 820ndash830 2001

[18] K Zhou andDWang ldquoDigital repetitive controlled three-phasePWM rectifierrdquo IEEE Transactions on Power Electronics vol 18no 1 pp 309ndash316 2003

[19] P Mattavelli and F P Marafao ldquoRepetitive-based control forselective harmonic compensation in active power filtersrdquo IEEETransactions on Industrial Electronics vol 51 no 5 pp 1018ndash1024 2004

[20] W Lu K Zhou D Wang and M Cheng ldquoA general parallelstructure repetitive control scheme for multiphase DC-ACPWM convertersrdquo IEEE Transactions on Power Electronics vol28 no 8 pp 3980ndash3987 2013

[21] W Lu K Zhou D Wang and M Cheng ldquoA generic digital119899119896 plusmn 119898-order harmonic repetitive control scheme for PWMconvertersrdquo IEEE Transactions on Industrial Electronics vol 61no 3 pp 1516ndash1527 2014

[22] W Lu K Zhou andDWang ldquoGeneral parallel structure digitalrepetitive controlrdquo International Journal of Control vol 86 no1 pp 70ndash83 2013

[23] G C Goodwin S F Graebe and M E Salgado Control SystemDesign Prentice Hall Upper Saddle River NJ USA 2001

[24] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 7th edition2007

[25] W Lu K Zhou and Y Yang ldquoA general internal modelprinciple based control scheme for CVCF PWM convertersrdquo inProceedings of the 2nd IEEE International Symposium on PowerElectronics for Distributed Generation Systems (PEDG rsquo10) pp485ndash489 Hefei China June 2010

[26] H Akagi H E Watanabe and M Aredes InstantaneousPower Theory and Applications to Power Conditioning WileyHoboken NJ USA 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


++

+ y

d

u

eGpGc

yref

minus

Figure 1 Closed-loop feedback control system

initial conditions Thus a general disturbance signal can bedescribed as the format of a differential equation as follows

119889119902119889 (119905)

119889119905119902

+

119902minus1

sum

119894=0

120574119894

119889119894119889 (119905)

119889119905119894

= 0 (1)

Using Laplace transform (1) can be derived into

119889 (119904) =

119873119889 (119904) 119909119889 (0)

Γ119889 (119904)

(2)

whereΓ119889(119904) is the disturbance generating polynomial (D-GP)which can be defined into

Γ119889 (119904) ≜ 119904119902+

119902minus1

sum

119894=0

120574119894119904119894 (3)

Notice that this D-GP is only related with the denomi-nator of (2) Similarly reference signal can be described inthe same way as long as replacing the symbol ldquo119889rdquo with ldquo119903rdquoReference generating polynomial (R-GP) Γ119903(119904) can be definedas follows

Γ119903 (119904) ≜ 119904119902+

119902minus1

sum

119894=0

120582119894119904119894 (4)

Then internal model principle (IMP) [2 23] can beexpressed as follows If disturbance 119889(119905) or reference input119910119903119890119891(119905) can be expressed as its D-GP or R-GP just like Γ119889(119904)

or Γ119903(119904) asymptotic error elimination or reference trackingcan be achieved for the control system using the followingcontroller

119866119888 (119904) =

119875 (119904)

Γ119889 (119904) 119871 (119904)

(5)

or

119866119888 (119904) =

119875 (119904)

Γ119903 (119904) 119871 (119904)

(6)

Therefore only D-GP or R-GP that is Γ119889(119904) or Γ119903(119904) isneeded when using IMP while the amplitude information ofdisturbance or reference input is not needed

22 The General Properties of IMP-Based Controller Figure 1shows the diagram of a closed-loop feedback control systemLet the model of plant be

119866119901 (119904) =

119861119900 (119904)

119860119900 (119904)

=

119887119899minus1119904119899minus1

+ 119887119899minus2119904119899minus2

+ sdot sdot sdot + 1198870

119904119899+ 119886119899minus1119904

119899minus1+ 119886119899minus2119904

119899minus2+ sdot sdot sdot + 1198860

(7)

and assume that Γ119889(119904) or Γ119903(119904) is not the factor of 119861119900(119904)

Then the output sensitivity function of closed-loop feed-back system shown in Figure 1 is

119878119900 (119904) =

Γ119889 (119904) 119871 (119904) 119860119900 (119904)

Γ119889 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(8)

or

119878119900 (119904) =

Γ119903 (119904) 119871 (119904) 119860119900 (119904)

Γ119903 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(9)

and the output complementary sensitivity function is

119879119900 (119904) = 1 minus 119878119900 (119904) =

119875 (119904) 119861119900 (119904)

Γ119889 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(10)

or

119879119900 (119904) = 1 minus 119878119900 (119904) =

119875 (119904) 119861119900 (119904)

Γ119903 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)

(11)

Assume the selected 119871(119904) and 119875(119904) can make the closed-loop characteristic equation

119860cl (119904) = Γ119889 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)(12)

or

119860cl (119904) = Γ119903 (119904) 119871 (119904) 119860119900 (119904) + 119875 (119904) 119861119900 (119904)(13)

have the negative realistic roots Thus the output response ofdisturbance 119889(119905) can be obtained as

119910 (119904) = 119878119900 (119904) 119889 (119904) =

119871 (119904) 119860119900 (119904)119873119889 (119904) 119909119889 (0)

119860cl (119904) (14)

Because 119860cl(119904) has stable roots the inverse Laplacetransform of 119910(119904) can asymptotically converge to zero thatis 119910(119905 rarr infin) = 0

The system error response of disturbance 119889(119905) is

119890 (119904) = minus119878119900 (119904) 119889 (119904) = minus

119871 (119904) 119860119900 (119904)119873119889 (119904) 119909119889 (0)

119860cl (119904) (15)


The system error response of reference 119910ref (119905) is

119890 (119904) = 119878119900 (119904) 119910ref (119904) =

119871 (119904) 119860119900 (119904)119873119903 (119904) 119909119903 (0)

119860cl (119904) (16)


The system output response of reference 119910ref (119905) is

119910 (119904) = 119879119900 (119904) 119910ref (119904)

= 119910ref (119904) minus119871 (119904) 119860119900 (119904)119873119903 (119904) 119909119903 (0)

119860cl (119904)

(17)


+

+

++krc

cos

2120587

nm


eminussT119900n

minus

minus

2cos2120587

nm

e(s)

Grc(s)

c(s)






119866rc (119904) = 119896rc sdot

119890minus119904119879119900

1 minus 119890minus119904119879119900

= 119896rc sdot

119890minus1199041198791199002

2 sinh (1199041198791199002)

= 119896rc sdot

119890minus1199041198791199002

119904119879119900 sdot prodinfin

119896=1[1 + 119904

2 (11989621205962119900)]

(18)



119889 (119905) = 1198890 +

infin

sum

119896=1

119889119896 cos (119896120596119900119905 + 120593119896) 997904rArr

Γ119889 (119904) = 119904

infin

prod

119896=1

(1199042+ (119896120596119900)

2)

(19)

or

119910ref (119905) = 1199030 +

infin

sum

119896=1

119903119896 cos (119896120596119900119905 + 1205931015840

119896) 997904rArr

Γ119903 (119904) = 119904

infin

prod

119896=1

(1199042+ (119896120596119900)

2)

(20)


119866119888 (119904) = 119866rc (119904) =


119896=1(11989621205962

119900)

119879119900 sdot Γ119889 (119904)

(21)

or

119866119888 (119904) = 119866rc (119904) =


119896=1(11989621205962

119900)

119879119900 sdot Γ119903 (119904)

(22)


3 SHRC


119866rc (119904) = 119896rc sdot

cos (2120587119898119899) 119890119904119879119900119899

minus 1

1198902119904119879119900119899

minus 2 cos (2120587119898119899) 119890119904119879119900119899

+ 1

(23)






119866rc (119904) =

119896rc2

sdot

cos (2120587119898119899) minus 119890minus119904119879119900119899

cosh (119904119879119900119899) minus cos (2120587119898119899)

=


]

4sin2 (119898120587119899) (1 + 1199042 (11989821205962119900)) sdot prod

infin

119896=1[1 + 119904

2 ((119899119896 + 119898)

21205962119900)] [1 + 119904

2 ((119899119896 minus 119898)

21205962119900)]

(24)


119866rc (119904) =

119896rc119890119904119879119900119899

minus 1

=

119896rc

(119904119879119900119899) sdot 1198901199041198791199002119899

sdot prodinfin

119896=1[1 + 119904

2 (1198991198961205960)

2]

(25)


119889 (119905) =

1198890 +

infin

sum

119896=1

119889119899119896 cos (119899119896120596119900119905 + 120593119899119896) (119898 = 0)

infin

sum

119896=1


infin

sum

119896=0

119889119899119896+119898 cos ((119899119896 + 119898)120596119900119905 + 120593119899119896+119898) (119898 = 0)

997904rArr

Γ119889 (119904) =

119904

infin

prod

119896=1

(1199042+ (119899119896120596119900)

2) (119898 = 0)

(1199042+ (119898120596119900)

2)

infin

prod

119896=1

(1199042+ ((119899119896 minus 119898)120596119900)

2) sdot (1199042+ ((119899119896 + 119898)120596119900)

2) (119898 = 0)

(26)


119866119888 (119904) = 119866rc (119904) =

119896rc sdot

119899 sdot 119890minus1199041198791199002119899

prodinfin

119896=1(119899119896120596119900)

2

119879119900 sdot Γ119889 (119904)

(119898 = 0)


119900prodinfin

119896=1[(119899119896 + 119898)

2(119899119896 minus 119898)

21205964

119900]

4sin2 (119898120587119899) sdot Γ119889 (119904)

(119898 = 0)

(27)




119866rc (119904) =

1

2

119896rc sdot [

119890119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890119895(2120587119899)119898

+

119890minus119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890minus119895(2120587119899)119898


1

2

+

119899

2119879119900 (119904 minus 119895119898120596119900)

+

119899

2119879119900 (119904 + 119895119898120596119900)

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 minus 119895119898120596119900)

(119904 minus 119895119898120596119900)2+ 119899211989621205962119900

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 + 119895119898120596119900)

(119904 + 119895119898120596119900)2+ 119899211989621205962119900

(28)



PWM

RCu

e

Switching

+

+

+

minus

DBTransform


PLL

dq

abc

minus1

L

A

B

C

+

minus

L

L0

R

R

R

+ minus+

minus + minus

+ minus

+

minus +

minus

generator

Udc

Udc

cba

ab

ab

bc

bc

ca

c

c

b

b

a

a

ic

ic

ib

ib

ia

ia

icref

ibref

iaref

ibia

ia ib icur

iqref

idref

sin (120596t)cos (120596t)

S1

S2

S3

S4

S5

S6

signals (S1ndashS6)

Pref Qref











Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119886119887119887119888119888119886

= 25V (rms)




(

119894119886

119894119887

119894119888

) = (

minus

119877

119871

0 0

0 minus

119877

119871

0

0 0 minus

119877

119871

)(

119894119886

119894119887

119894119888

)

+

(

(

(

(

1

119871

(V119886 minus V119860)

1

119871

(V119887 minus V119861)

1

119871

(V119888 minus V119862)

)

)

)

)

(29)



(

119894119886 (119896 + 1)

119894119887 (119896 + 1)

119894119888 (119896 + 1)

)

=(

(

1198871 minus 1198872

1198871

0 0

0

1198871 minus 1198872

1198871

0

0 0

1198871 minus 1198872

1198871

)

)

(

119894119886 (119896)

119894119887 (119896)

119894119888 (119896)

)

+

(

(

(

(

(

1

1198871

V119886 (119896) minus1

1198871

119880dc (119896)

2

119906119886 (119896)

1

1198871

V119887 (119896) minus1

1198871

119880dc (119896)

2

119906119887 (119896)

1

1198871

V119888 (119896) minus1

1198871

119880dc (119896)

2

119906119888 (119896)

)

)

)

)

)

(30)


119894119895 (119896 + 1) =

1198871 minus 1198872

1198871

119894119895 (119896) +

1

1198871

V119895 (119896)

minus

119880dc (119896)

2

1

1198871

119906119895 (119896)

(31)


119906119895 (119896) =

2





119894119895 (119896 + 1) = 09833119894119895 (119896) + 00333V119895 (119896) minus 08333119906119895 (119896) (33)


119906119895 (119896) = 004V119895 (119896) minus 12119894jref (119896) + 118119894119895 (119896) (34)

where 119895 = 119886 119887 119888



Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119904= 25V (rms)









119894119904 = minus

119877

119871

119894119904 +

1

119871

(V119904 minus Vin) (35)


119894119904 (119896 + 1) =

1198871 minus 1198872

1198871

119894119904 (119896) +

1

1198871

V119904 (119896)

minus

1

1198871

119880dc (119896) 119906119904 (119896)

(36)



119906119904 (119896) =

2





ab bc ca

ia

(a) DB

ab bc ca

ia

(b) DB + CRC

ab bc ca

ia

(c) DB + SHRC


t (s)

minus3

minus2

minus1

0

1

2

3

29 3 31 32 33 34 35 36 3728

iaref minus ia (A)

(a) CRC

t (s)29 3 31 32 33 34 35 36 3728

minus3

minus2

minus1

0

1

2

3

iaref minus ia (A)

(b) SHRC




119894119904 (119896 + 1) = 09833119894119904 (119896) + 00333V119904 (119896) minus 04167119906119904 (119896) (38)


119906119904 (119896) = 004V119904 (119896) minus 12119894sref (119896) + 118119894119904 (119896) (39)





Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1


s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC




6 Conclusions



26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

+

++krc

cos

2120587

nm


eminussT119900n

minus

minus

2cos2120587

nm

e(s)

Grc(s)

c(s)






119866rc (119904) = 119896rc sdot

119890minus119904119879119900

1 minus 119890minus119904119879119900

= 119896rc sdot

119890minus1199041198791199002

2 sinh (1199041198791199002)

= 119896rc sdot

119890minus1199041198791199002

119904119879119900 sdot prodinfin

119896=1[1 + 119904

2 (11989621205962119900)]

(18)



119889 (119905) = 1198890 +

infin

sum

119896=1

119889119896 cos (119896120596119900119905 + 120593119896) 997904rArr

Γ119889 (119904) = 119904

infin

prod

119896=1

(1199042+ (119896120596119900)

2)

(19)

or

119910ref (119905) = 1199030 +

infin

sum

119896=1

119903119896 cos (119896120596119900119905 + 1205931015840

119896) 997904rArr

Γ119903 (119904) = 119904

infin

prod

119896=1

(1199042+ (119896120596119900)

2)

(20)


119866119888 (119904) = 119866rc (119904) =


119896=1(11989621205962

119900)

119879119900 sdot Γ119889 (119904)

(21)

or

119866119888 (119904) = 119866rc (119904) =


119896=1(11989621205962

119900)

119879119900 sdot Γ119903 (119904)

(22)


3 SHRC


119866rc (119904) = 119896rc sdot

cos (2120587119898119899) 119890119904119879119900119899

minus 1

1198902119904119879119900119899

minus 2 cos (2120587119898119899) 119890119904119879119900119899

+ 1

(23)






119866rc (119904) =

119896rc2

sdot

cos (2120587119898119899) minus 119890minus119904119879119900119899

cosh (119904119879119900119899) minus cos (2120587119898119899)

=


]

4sin2 (119898120587119899) (1 + 1199042 (11989821205962119900)) sdot prod

infin

119896=1[1 + 119904

2 ((119899119896 + 119898)

21205962119900)] [1 + 119904

2 ((119899119896 minus 119898)

21205962119900)]

(24)


119866rc (119904) =

119896rc119890119904119879119900119899

minus 1

=

119896rc

(119904119879119900119899) sdot 1198901199041198791199002119899

sdot prodinfin

119896=1[1 + 119904

2 (1198991198961205960)

2]

(25)


119889 (119905) =

1198890 +

infin

sum

119896=1

119889119899119896 cos (119899119896120596119900119905 + 120593119899119896) (119898 = 0)

infin

sum

119896=1


infin

sum

119896=0

119889119899119896+119898 cos ((119899119896 + 119898)120596119900119905 + 120593119899119896+119898) (119898 = 0)

997904rArr

Γ119889 (119904) =

119904

infin

prod

119896=1

(1199042+ (119899119896120596119900)

2) (119898 = 0)

(1199042+ (119898120596119900)

2)

infin

prod

119896=1

(1199042+ ((119899119896 minus 119898)120596119900)

2) sdot (1199042+ ((119899119896 + 119898)120596119900)

2) (119898 = 0)

(26)


119866119888 (119904) = 119866rc (119904) =

119896rc sdot

119899 sdot 119890minus1199041198791199002119899

prodinfin

119896=1(119899119896120596119900)

2

119879119900 sdot Γ119889 (119904)

(119898 = 0)


119900prodinfin

119896=1[(119899119896 + 119898)

2(119899119896 minus 119898)

21205964

119900]

4sin2 (119898120587119899) sdot Γ119889 (119904)

(119898 = 0)

(27)




119866rc (119904) =

1

2

119896rc sdot [

119890119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890119895(2120587119899)119898

+

119890minus119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890minus119895(2120587119899)119898


1

2

+

119899

2119879119900 (119904 minus 119895119898120596119900)

+

119899

2119879119900 (119904 + 119895119898120596119900)

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 minus 119895119898120596119900)

(119904 minus 119895119898120596119900)2+ 119899211989621205962119900

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 + 119895119898120596119900)

(119904 + 119895119898120596119900)2+ 119899211989621205962119900

(28)



PWM

RCu

e

Switching

+

+

+

minus

DBTransform


PLL

dq

abc

minus1

L

A

B

C

+

minus

L

L0

R

R

R

+ minus+

minus + minus

+ minus

+

minus +

minus

generator

Udc

Udc

cba

ab

ab

bc

bc

ca

c

c

b

b

a

a

ic

ic

ib

ib

ia

ia

icref

ibref

iaref

ibia

ia ib icur

iqref

idref

sin (120596t)cos (120596t)

S1

S2

S3

S4

S5

S6

signals (S1ndashS6)

Pref Qref











Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119886119887119887119888119888119886

= 25V (rms)




(

119894119886

119894119887

119894119888

) = (

minus

119877

119871

0 0

0 minus

119877

119871

0

0 0 minus

119877

119871

)(

119894119886

119894119887

119894119888

)

+

(

(

(

(

1

119871

(V119886 minus V119860)

1

119871

(V119887 minus V119861)

1

119871

(V119888 minus V119862)

)

)

)

)

(29)



(

119894119886 (119896 + 1)

119894119887 (119896 + 1)

119894119888 (119896 + 1)

)

=(

(

1198871 minus 1198872

1198871

0 0

0

1198871 minus 1198872

1198871

0

0 0

1198871 minus 1198872

1198871

)

)

(

119894119886 (119896)

119894119887 (119896)

119894119888 (119896)

)

+

(

(

(

(

(

1

1198871

V119886 (119896) minus1

1198871

119880dc (119896)

2

119906119886 (119896)

1

1198871

V119887 (119896) minus1

1198871

119880dc (119896)

2

119906119887 (119896)

1

1198871

V119888 (119896) minus1

1198871

119880dc (119896)

2

119906119888 (119896)

)

)

)

)

)

(30)


119894119895 (119896 + 1) =

1198871 minus 1198872

1198871

119894119895 (119896) +

1

1198871

V119895 (119896)

minus

119880dc (119896)

2

1

1198871

119906119895 (119896)

(31)


119906119895 (119896) =

2





119894119895 (119896 + 1) = 09833119894119895 (119896) + 00333V119895 (119896) minus 08333119906119895 (119896) (33)


119906119895 (119896) = 004V119895 (119896) minus 12119894jref (119896) + 118119894119895 (119896) (34)

where 119895 = 119886 119887 119888



Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119904= 25V (rms)









119894119904 = minus

119877

119871

119894119904 +

1

119871

(V119904 minus Vin) (35)


119894119904 (119896 + 1) =

1198871 minus 1198872

1198871

119894119904 (119896) +

1

1198871

V119904 (119896)

minus

1

1198871

119880dc (119896) 119906119904 (119896)

(36)



119906119904 (119896) =

2





ab bc ca

ia

(a) DB

ab bc ca

ia

(b) DB + CRC

ab bc ca

ia

(c) DB + SHRC


t (s)

minus3

minus2

minus1

0

1

2

3

29 3 31 32 33 34 35 36 3728

iaref minus ia (A)

(a) CRC

t (s)29 3 31 32 33 34 35 36 3728

minus3

minus2

minus1

0

1

2

3

iaref minus ia (A)

(b) SHRC




119894119904 (119896 + 1) = 09833119894119904 (119896) + 00333V119904 (119896) minus 04167119906119904 (119896) (38)


119906119904 (119896) = 004V119904 (119896) minus 12119894sref (119896) + 118119894119904 (119896) (39)





Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1


s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC




6 Conclusions



26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119866rc (119904) =

119896rc2

sdot

cos (2120587119898119899) minus 119890minus119904119879119900119899

cosh (119904119879119900119899) minus cos (2120587119898119899)

=


]

4sin2 (119898120587119899) (1 + 1199042 (11989821205962119900)) sdot prod

infin

119896=1[1 + 119904

2 ((119899119896 + 119898)

21205962119900)] [1 + 119904

2 ((119899119896 minus 119898)

21205962119900)]

(24)


119866rc (119904) =

119896rc119890119904119879119900119899

minus 1

=

119896rc

(119904119879119900119899) sdot 1198901199041198791199002119899

sdot prodinfin

119896=1[1 + 119904

2 (1198991198961205960)

2]

(25)


119889 (119905) =

1198890 +

infin

sum

119896=1

119889119899119896 cos (119899119896120596119900119905 + 120593119899119896) (119898 = 0)

infin

sum

119896=1


infin

sum

119896=0

119889119899119896+119898 cos ((119899119896 + 119898)120596119900119905 + 120593119899119896+119898) (119898 = 0)

997904rArr

Γ119889 (119904) =

119904

infin

prod

119896=1

(1199042+ (119899119896120596119900)

2) (119898 = 0)

(1199042+ (119898120596119900)

2)

infin

prod

119896=1

(1199042+ ((119899119896 minus 119898)120596119900)

2) sdot (1199042+ ((119899119896 + 119898)120596119900)

2) (119898 = 0)

(26)


119866119888 (119904) = 119866rc (119904) =

119896rc sdot

119899 sdot 119890minus1199041198791199002119899

prodinfin

119896=1(119899119896120596119900)

2

119879119900 sdot Γ119889 (119904)

(119898 = 0)


119900prodinfin

119896=1[(119899119896 + 119898)

2(119899119896 minus 119898)

21205964

119900]

4sin2 (119898120587119899) sdot Γ119889 (119904)

(119898 = 0)

(27)




119866rc (119904) =

1

2

119896rc sdot [

119890119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890119895(2120587119899)119898

+

119890minus119895(2120587119899)119898

119890119904(119879119900119899)

minus 119890minus119895(2120587119899)119898


1

2

+

119899

2119879119900 (119904 minus 119895119898120596119900)

+

119899

2119879119900 (119904 + 119895119898120596119900)

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 minus 119895119898120596119900)

(119904 minus 119895119898120596119900)2+ 119899211989621205962119900

+

119899

2119879119900

+infin

sum

119896=1

2 (119904 + 119895119898120596119900)

(119904 + 119895119898120596119900)2+ 119899211989621205962119900

(28)



PWM

RCu

e

Switching

+

+

+

minus

DBTransform


PLL

dq

abc

minus1

L

A

B

C

+

minus

L

L0

R

R

R

+ minus+

minus + minus

+ minus

+

minus +

minus

generator

Udc

Udc

cba

ab

ab

bc

bc

ca

c

c

b

b

a

a

ic

ic

ib

ib

ia

ia

icref

ibref

iaref

ibia

ia ib icur

iqref

idref

sin (120596t)cos (120596t)

S1

S2

S3

S4

S5

S6

signals (S1ndashS6)

Pref Qref











Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119886119887119887119888119888119886

= 25V (rms)




(

119894119886

119894119887

119894119888

) = (

minus

119877

119871

0 0

0 minus

119877

119871

0

0 0 minus

119877

119871

)(

119894119886

119894119887

119894119888

)

+

(

(

(

(

1

119871

(V119886 minus V119860)

1

119871

(V119887 minus V119861)

1

119871

(V119888 minus V119862)

)

)

)

)

(29)



(

119894119886 (119896 + 1)

119894119887 (119896 + 1)

119894119888 (119896 + 1)

)

=(

(

1198871 minus 1198872

1198871

0 0

0

1198871 minus 1198872

1198871

0

0 0

1198871 minus 1198872

1198871

)

)

(

119894119886 (119896)

119894119887 (119896)

119894119888 (119896)

)

+

(

(

(

(

(

1

1198871

V119886 (119896) minus1

1198871

119880dc (119896)

2

119906119886 (119896)

1

1198871

V119887 (119896) minus1

1198871

119880dc (119896)

2

119906119887 (119896)

1

1198871

V119888 (119896) minus1

1198871

119880dc (119896)

2

119906119888 (119896)

)

)

)

)

)

(30)


119894119895 (119896 + 1) =

1198871 minus 1198872

1198871

119894119895 (119896) +

1

1198871

V119895 (119896)

minus

119880dc (119896)

2

1

1198871

119906119895 (119896)

(31)


119906119895 (119896) =

2





119894119895 (119896 + 1) = 09833119894119895 (119896) + 00333V119895 (119896) minus 08333119906119895 (119896) (33)


119906119895 (119896) = 004V119895 (119896) minus 12119894jref (119896) + 118119894119895 (119896) (34)

where 119895 = 119886 119887 119888



Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119904= 25V (rms)









119894119904 = minus

119877

119871

119894119904 +

1

119871

(V119904 minus Vin) (35)


119894119904 (119896 + 1) =

1198871 minus 1198872

1198871

119894119904 (119896) +

1

1198871

V119904 (119896)

minus

1

1198871

119880dc (119896) 119906119904 (119896)

(36)



119906119904 (119896) =

2





ab bc ca

ia

(a) DB

ab bc ca

ia

(b) DB + CRC

ab bc ca

ia

(c) DB + SHRC


t (s)

minus3

minus2

minus1

0

1

2

3

29 3 31 32 33 34 35 36 3728

iaref minus ia (A)

(a) CRC

t (s)29 3 31 32 33 34 35 36 3728

minus3

minus2

minus1

0

1

2

3

iaref minus ia (A)

(b) SHRC




119894119904 (119896 + 1) = 09833119894119904 (119896) + 00333V119904 (119896) minus 04167119906119904 (119896) (38)


119906119904 (119896) = 004V119904 (119896) minus 12119894sref (119896) + 118119894119904 (119896) (39)





Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1


s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC




6 Conclusions



26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










PWM

RCu

e

Switching

+

+

+

minus

DBTransform


PLL

dq

abc

minus1

L

A

B

C

+

minus

L

L0

R

R

R

+ minus+

minus + minus

+ minus

+

minus +

minus

generator

Udc

Udc

cba

ab

ab

bc

bc

ca

c

c

b

b

a

a

ic

ic

ib

ib

ia

ia

icref

ibref

iaref

ibia

ia ib icur

iqref

idref

sin (120596t)cos (120596t)

S1

S2

S3

S4

S5

S6

signals (S1ndashS6)

Pref Qref











Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119886119887119887119888119888119886

= 25V (rms)




(

119894119886

119894119887

119894119888

) = (

minus

119877

119871

0 0

0 minus

119877

119871

0

0 0 minus

119877

119871

)(

119894119886

119894119887

119894119888

)

+

(

(

(

(

1

119871

(V119886 minus V119860)

1

119871

(V119887 minus V119861)

1

119871

(V119888 minus V119862)

)

)

)

)

(29)



(

119894119886 (119896 + 1)

119894119887 (119896 + 1)

119894119888 (119896 + 1)

)

=(

(

1198871 minus 1198872

1198871

0 0

0

1198871 minus 1198872

1198871

0

0 0

1198871 minus 1198872

1198871

)

)

(

119894119886 (119896)

119894119887 (119896)

119894119888 (119896)

)

+

(

(

(

(

(

1

1198871

V119886 (119896) minus1

1198871

119880dc (119896)

2

119906119886 (119896)

1

1198871

V119887 (119896) minus1

1198871

119880dc (119896)

2

119906119887 (119896)

1

1198871

V119888 (119896) minus1

1198871

119880dc (119896)

2

119906119888 (119896)

)

)

)

)

)

(30)


119894119895 (119896 + 1) =

1198871 minus 1198872

1198871

119894119895 (119896) +

1

1198871

V119895 (119896)

minus

119880dc (119896)

2

1

1198871

119906119895 (119896)

(31)


119906119895 (119896) =

2





119894119895 (119896 + 1) = 09833119894119895 (119896) + 00333V119895 (119896) minus 08333119906119895 (119896) (33)


119906119895 (119896) = 004V119895 (119896) minus 12119894jref (119896) + 118119894119895 (119896) (34)

where 119895 = 119886 119887 119888



Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119904= 25V (rms)









119894119904 = minus

119877

119871

119894119904 +

1

119871

(V119904 minus Vin) (35)


119894119904 (119896 + 1) =

1198871 minus 1198872

1198871

119894119904 (119896) +

1

1198871

V119904 (119896)

minus

1

1198871

119880dc (119896) 119906119904 (119896)

(36)



119906119904 (119896) =

2





ab bc ca

ia

(a) DB

ab bc ca

ia

(b) DB + CRC

ab bc ca

ia

(c) DB + SHRC


t (s)

minus3

minus2

minus1

0

1

2

3

29 3 31 32 33 34 35 36 3728

iaref minus ia (A)

(a) CRC

t (s)29 3 31 32 33 34 35 36 3728

minus3

minus2

minus1

0

1

2

3

iaref minus ia (A)

(b) SHRC




119894119904 (119896 + 1) = 09833119894119904 (119896) + 00333V119904 (119896) minus 04167119906119904 (119896) (38)


119906119904 (119896) = 004V119904 (119896) minus 12119894sref (119896) + 118119894119904 (119896) (39)





Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1


s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC




6 Conclusions



26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(

119894119886 (119896 + 1)

119894119887 (119896 + 1)

119894119888 (119896 + 1)

)

=(

(

1198871 minus 1198872

1198871

0 0

0

1198871 minus 1198872

1198871

0

0 0

1198871 minus 1198872

1198871

)

)

(

119894119886 (119896)

119894119887 (119896)

119894119888 (119896)

)

+

(

(

(

(

(

1

1198871

V119886 (119896) minus1

1198871

119880dc (119896)

2

119906119886 (119896)

1

1198871

V119887 (119896) minus1

1198871

119880dc (119896)

2

119906119887 (119896)

1

1198871

V119888 (119896) minus1

1198871

119880dc (119896)

2

119906119888 (119896)

)

)

)

)

)

(30)


119894119895 (119896 + 1) =

1198871 minus 1198872

1198871

119894119895 (119896) +

1

1198871

V119895 (119896)

minus

119880dc (119896)

2

1

1198871

119906119895 (119896)

(31)


119906119895 (119896) =

2





119894119895 (119896 + 1) = 09833119894119895 (119896) + 00333V119895 (119896) minus 08333119906119895 (119896) (33)


119906119895 (119896) = 004V119895 (119896) minus 12119894jref (119896) + 118119894119895 (119896) (34)

where 119895 = 119886 119887 119888



Inverter

119880dc = 50V119871 = 5mH119877 = 05Ω

V119904= 25V (rms)









119894119904 = minus

119877

119871

119894119904 +

1

119871

(V119904 minus Vin) (35)


119894119904 (119896 + 1) =

1198871 minus 1198872

1198871

119894119904 (119896) +

1

1198871

V119904 (119896)

minus

1

1198871

119880dc (119896) 119906119904 (119896)

(36)



119906119904 (119896) =

2





ab bc ca

ia

(a) DB

ab bc ca

ia

(b) DB + CRC

ab bc ca

ia

(c) DB + SHRC


t (s)

minus3

minus2

minus1

0

1

2

3

29 3 31 32 33 34 35 36 3728

iaref minus ia (A)

(a) CRC

t (s)29 3 31 32 33 34 35 36 3728

minus3

minus2

minus1

0

1

2

3

iaref minus ia (A)

(b) SHRC




119894119904 (119896 + 1) = 09833119894119904 (119896) + 00333V119904 (119896) minus 04167119906119904 (119896) (38)


119906119904 (119896) = 004V119904 (119896) minus 12119894sref (119896) + 118119894119904 (119896) (39)





Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1


s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC




6 Conclusions



26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ab bc ca

ia

(a) DB

ab bc ca

ia

(b) DB + CRC

ab bc ca

ia

(c) DB + SHRC


t (s)

minus3

minus2

minus1

0

1

2

3

29 3 31 32 33 34 35 36 3728

iaref minus ia (A)

(a) CRC

t (s)29 3 31 32 33 34 35 36 3728

minus3

minus2

minus1

0

1

2

3

iaref minus ia (A)

(b) SHRC




119894119904 (119896 + 1) = 09833119894119904 (119896) + 00333V119904 (119896) minus 04167119906119904 (119896) (38)


119906119904 (119896) = 004V119904 (119896) minus 12119894sref (119896) + 118119894119904 (119896) (39)





Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1


s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC




6 Conclusions



26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Switching

PWMgenerator

S1

S2

S3

S4

RCu

DB

Calculation

PLL

Udc

Udc

urminusminus

minus

minus

minus1is

is

is

Pref

Qref

s

s

s

Ipeak

isref sin

L R

e +

++

+

+

+

+

times

120587

divide

signals (S1ndashS4)

tanminus1


s

is

(a) DB

s

is

(b) DB + CRC

s

is

(c) DB + SHRC




6 Conclusions



26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










26 27 28 29 3 31 32 33 3425t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(a) CRC

25 26 27 28 29 3 31 32 3324t (s)

minus4

minus3

minus2

minus1

0

1

2

3

4

isref minus is (A)

(b) SHRC




Competing Interests


Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Internal-Model-Principle-Based Specific...

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