Research ArticleProportional-Type Output Voltage-TrackingController for Interleaved DCDC Boost Converter withPerformance Recovery Property

Seok-Kyoon Kim

Department of Creative Convergence Engineering Hanbat National University Daejeon 341-58 Republic of Korea

Correspondence should be addressed to Seok-Kyoon Kim lotus45krgmailcom

Received 7 August 2017 Accepted 18 February 2018 Published 19 March 2018

Academic Editor Bruno G M Robert

Copyright copy 2018 Seok-Kyoon Kim 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 proposes an offset-free proportional-type output voltage-tracking algorithm embedding the disturbance observers(DOBs) for the119873-phase interleaved DCDC boost converter through a systematical multivariable approach The contributions ofthis article fall into two partsThe first one is to design the first-order nonlinear DOBs for exponentially estimating the disturbancescaused by themodel-plantmismatchesThe second one is to prove that the proposed proportional-type controller equippedwith theDOBs guarantees the performance recovery property as well as the offset-free property The performance of the proposed methodis evaluated through simulations and experiments using a 3-kW four-phase interleaved DCDC boost converter comparing theproposed and feedback linearizing (FL) methods

1 Introduction

The DCDC converters have been widely utilized to supplya high quality DC power despite the disturbances for avariety of industrial applications such as uninterruptiblepower supply and solar photovoltaic systems [1ndash6] In theseapplications the DCDC converters are required to be con-trolled their current and output voltage with a satisfactoryclosed-loop performance for a wide range of operating regionin the presence of the converter parameter and unexpectedload variations

The cascade output voltage control strategy [7] has beencommonly adopted to adjust the output voltage of theDCDCconverters where the inner and outer-loop controllers weredesigned to control the current and output voltage in acascade manner The proportional-integral (PI) controllerequipped with nonlinearity cancellation terms has beenmainly used for implementing both inner-loop and outer-loop controllers [7 8] which can be interpreted as an feed-back linearizing (FL) method in the control theoretical pointof view The corresponding control gains were determinedfor the resulting closed-loop error dynamics to be the desired

low-pass filter (LPF) behavior using the converter parameterssuch as inductance and capacitance values However it isquestionable that their closed-loop performance would beremaining satisfactory for the various load conditions despitethe parameter uncertainties

There have been many alternative solutions based onthe novel control strategy for the inner loop to attain abetter closed-loop performance which includes the deadbeatcontrollers [9] predictive controllers [10] sliding modecontrollers [11] adaptive controllers [12] model predictivecontrollers [13] and robust controllers [14] Aside from theadaptive technique however these novel schemes requireusing the converter true parameter values so as to guaranteethe closed-loop stability and performance The adaptive andsliding mode schemes were built using the true inductancevalue that can be dramatically decreased with increasing theinductor current The disturbance observer (DOB) basedmethods were proposed for the robot manipulation [15] andthe DC motor control applications [16] Unlike [16] it isrigorously proved that the proposed proportional-type con-troller with the DOBs guarantees to not only exponentiallyrecover the target output voltage-tracking performance but

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 2536946 12 pageshttpsdoiorg10115520182536946

2 Mathematical Problems in Engineering

L

L

L

C

PWM

PWM

PWM

PWM

PWM

PWM

+

minus

Q12

iL1

iL iL2

iLN

ＣＨ

ＩＯＮ

Q11 Q21 QN1

Q22 QN2

+

minus

iC iＩ＞

Figure 1 Interleaved DCDC boost converter circuit

also remove the steady state errors which means that thetracking error integrators can be eliminated as well as thecorresponding antiwindup algorithms

This article presents a robust output voltage-trackingalgorithm based on a systematical multi-variable approachwithout tracking error integrators for the 119873-phase inter-leaved DCDC boost converter considering a planet-modelmismatch such as parameter and load variations The nov-elties of the proposed method are twofold the exponen-tially convergent DOBs are constructed for the current andoutput voltage loops and the proportional-type nonlinearcontroller is developed with the exponential performancerecovery property and the proof of the offset-free propertyThese two novelties simplify the implementations of thecontrol algorithm by eliminating the use of tracking errorintegrators with the anti-windup parts as well as the gainscheduling algorithms which is the sharp contrast to theabove mentioned existent methods such as model predictiveand adaptive methods The simulations and experimentalresults using a 3-kW four-phase interleaved DCDC boostconverter show the effectiveness of the proposed methodcomparing it with the FL method

2 Averaged Dynamics of Interleaved DCDCBoost Converter

This section briefly describes the dynamical equations ofthe synchronous-type 119873-phase interleaved DCDC boostconverter depicted in Figure 1 where the duty ratio of 119863119894 isin[0 1] 119894 = 1 2 119873 is considered as the control actionto be applied and the phase inductor current of 119894119871119894(119905) 119894 =1 2 119873 and the capacitor voltage of Vout(119905) are treatedas the state variables Unlike the traditional single-phaseboost converter a low voltage drop switch is used as theupper-side diode and is alternatively operated with a lower-side switch and this type of converter never enters thediscontinuous conduction mode (DCM) for all time [17]Note that the duty ratio of 119863119894 119894 = 1 2 119873 producesthe switching command by performing the pulse width

modulation (PWM) operation For example the duty ratioof 119863119894 isin (0 1) renders the switch to be turned on for a period119863119894119879119904 and the switch to be turned off for a period (1 minus 119863119894)119879119904with 119879119904 denoting the PWM period Due to these operationsthe averaged converter dynamics are obtained as

119871119889119894119871119894 (119905)119889119905 = minus (1 minus 119906119894 (119905)) Vdc (119905) + Vin119894 = 1 2 119873

119862119889Vdc (119905)119889119905 = 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894 (119905) minus 119894Load (119905) (1)

forall119905 ge 0 where the input source voltage is given as Vin119906119894(119905) fl 119863119894 isin (0 1) and 119894Load(119905) denotes the load currentSee [6 7 18 19] for a detailed derivation of the averageddynamical equations of (1)

Due to an abrupt load current variation and the parame-ter uncertainties such as inductance and capacitance valuesthe converter dynamics of (1) are rewritten regarding to thenominal parameters as

1198710 119889119894119871119894 (119905)119889119905 = minus (1 minus 119906119894 (119905)) Vdc (119905) + Vin0 + 119908119871119894119900 (119905) 119894 = 1 2 119873

(2)

1198620 119889Vdc (119905)119889119905 = 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894 (119905) + 119908V119900 (119905) (3)

forall119905 ge 0 with the nominal inductance and capacitance valuesof 1198710 1198620 and the initial input voltage Vin0 where 119908119871119894119900(119905)119908V119900(119905) denote the unknown lumped disturbances causedfrom the parameter mismatch unmodelled dynamics andload uncertainties

3 Output Voltage Controller Design

Let 119881lowastdc(119904) and 119881dcref(119904) be the Laplace transforms of desiredoutput voltage behavior Vlowastdc(119905) and its reference Vdcref(119905)

Mathematical Problems in Engineering 3

respectively The objective of this section is to constructthe control law for rendering the closed-loop output voltagedynamics to be

119881lowastdc (119904)119881dcref (119904) =120596vc119904 + 120596vc

forall119904 isin C (4)

with 120596vc being the assignable cut-off frequency as a designparameter while guaranteeing the current-tracking property

lim119905rarrinfin

119894119871119894 (119905) = 119894119871119894ref (119905) 119894 = 1 119873 (5)

In order to achieve the control objective of (4) considerthe time domain expression of the target dynamics of (4)given by

Vlowastdc (119905) = 120596vc (Vdcref (119905) minus Vlowastdc (119905)) forall119905 ge 0 (6)

and define the output voltage-tracking error as Vlowastdc(119905) flVlowastdc(119905)minusVdc(119905) forall119905 ge 0 which gives the output voltage-trackingerror dynamics using (3)

1198620 Vlowastdc (119905) = 1198620Vlowastdc (119905) minus 1198620Vdc (119905)= minus 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894 (119905) + 119908V (119905)

= minus 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894ref (119905)

+ 119873sum119894=1

(1 minus 119906119894 (119905)) 119871119894 (119905) + 119908V (119905) forall119905 ge 0

(7)

where 119908V(119905) fl 1198620Vlowastdc(119905) minus 119908V119900(119905) 119871119894(119905) fl 119894119871119894ref(119905) minus 119894119871119894(119905)forall119905 ge 0 An inductor current reference signal is proposedas a stabilizing solution to the output voltage-tracking errordynamics of (7)

119894119871119894ref (119905) = 1119873 1(1 minus 119906119894 (119905)) (1198620120582VVlowast

dc (119905) + 119908V (119905)) (8)

119894 = 1 119873 forall119905 ge 0 where 120582V gt 0 denotes a tuningparameter the estimated disturbance 119908V(119905) is given by

119908V (119905) fl 120577V (119905) + 119897V1198620Vlowastdc (119905) forall119905 ge 0 (9)

and 120577V(119905) stands for the state of the nonlinear observer120577V (119905) = minus119897V120577V (119905) minus 1198972V1198620Vlowastdc (119905)+ 119897V 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894 (119905) forall119905 ge 0 (10)

with 119897V gt 0 being the observer gain For the rest of this articlethe combination of two equations of (10) and (9) is called theoutput voltage disturbance observer (DOB)

The substitution of the inductance current reference of (8)to the output voltage-tracking error dynamics of (7) leads to

1198620 Vlowastdc (119905) = minus1198620120582VVlowast

dc (119905) + Δ119908V (119905)+ 119873sum119894=1

(1 minus 119906119894 (119905)) 119871119894 (119905) forall119905 ge 0 (11)

where Δ119908V(119905) fl 119908V(119905) minus 119908V(119905) forall119905 ge 0 Lemma 1 derives theclosed-loop property of (11) which is used for analyzing thewhole closed-loop stability analysis

Lemma 1 Assuming that the disturbance of119908V(119905) converges toits steady state value1199080V exponentially that is there is a positiveconstant of 119896V gt 0 such that

1199080V (119905) = minus119896V1199080V (119905) forall119905 gt 0 (12)

where 1199080V(119905) fl 1199080V minus 119908V(119905) forall119905 ge 0 then a specified designparameter of 120582V gt 0 with the DOB gains of

119897V = 341198620120582V+ 1 + 120578V 120578V gt 0 (13)

gives the strict passivity for the input-output mapping of

119873sum119894=1

(1 minus 119906119894) 119871119894 997891997888rarr Vlowastdc (14)

In [20] the strict passivity and L2 stability are formallydefined and it is proved that the strict passivity impliesthat L2 stability The result of Lemma 1 indicates thatthe output voltage of Vdc(119905) converges to the target outputvoltage trajectory of Vlowastdc(119905) exponentially as the inductorcurrent errors of 119871119894(119905) vanish exponentially For the proofof Lemma 1 a positive definite function is defined as

119881V (119905) fl 11986202 (Vlowastdc (119905))2 + 121199082V (119905) + 120574V2 (1199080V (119905))2 forall119905 ge 0

(15)

Then the proof can be completed by showing that

V (119905) le minus120572V119881V (119905) + Vlowastdc (119905) 119873sum119894=1

(1 minus 119906119894 (119905)) 119871119894 (119905) forall119905 ge 0

(16)

for some constant 120572V gt 0 where 119908V(119905) fl 1199080V minus 119908V(119905) forall119905 ge0 and 1199080V denotes a steady state value of 119908V(119905) forall119905 ge 0 SeeAppendix for details

For deriving the final control action write the inductorcurrent-tracking error dynamics using (2) as

1198710 119894119871119894 (119905) = 1198710 119894119871119894ref (119905) minus 1198710 119894119871119894 (119905)= 1198710 119894119871119894ref (119905) + (1 minus 119906119894 (119905)) Vdc (119905) minus Vin0

minus 119908119871119894119900 (119905)= (1 minus 119906119894 (119905)) Vdc (119905) minus Vin0 + 119908119871119894 (119905)= minusVlowastdc (119905) 119906119894 (119905) + Vlowastdc (119905) minus (1 minus 119906119894 (119905)) Vlowastdc (119905)minus Vin0 + 119908119871119894 (119905) forall119905 ge 0

(17)

4 Mathematical Problems in Engineering

DC voltagecontroller (8)

+

minus

DC voltageDOB(9) (10)

Targetdynamics (6)

+

minus

Currentcontroller(18)

CurrentDOB (19)-(20)

Outer-loop controller Inner-loop controller

+

minus

＞＝

＞＝Ｌ

lowast＞＝

lowast＞＝

iL1Ｌ

iLNＬ

iLN

iL1

iL1

iLN

u1

uN

lowast＞＝

iLi ui ＞＝ iLi

wL1 middot middot middot wLNw

Figure 2 Proposed controller structure

with 119908119871119894(119905) fl 1198710 119894119871119894ref(119905) minus 119908119871119894119900(119905) 119894 = 1 119873 forall119905 ge 0Then the stabilization can be accomplished by the proposedstabilizing control law for the duty ratio 119906119894(119905) as119906119894 (119905)

= 1Vlowastdc (119905) (1198710120582119871119871119894 (119905) + Vlowastdc (119905) minus Vin0 + 119908119871119894 (119905))

119894 = 1 119873 forall119905 ge 0(18)

where the inductor current references 119894119871119894ref(119905) come from(8) 120582119871 gt 0 denotes a tuning parameter and the estimateddisturbances 119908119871119894(119905) are given by

119908119871119894 (119905) fl 120577119871119894 (119905) + 1198971198711198941198710 119871119894 (119905) 119894 = 1 119873 forall119905 ge 0 (19)

and 120577119871119894(119905) stand for the state of the dynamical equations

120577119871119894 (119905) = minus119897119871119894120577119871119894 (119905) minus 11989721198711198941198710 119871119894 (119905)+ 119897119871119894 (minus (1 minus 119906119894 (119905)) Vdc (119905) + Vin0)

119894 = 1 119873 forall119905 ge 0(20)

with 119897119871119894 gt 0 being the observer gains For the rest of thisarticle the combination of two equations of (20) and (19)is called the inductor current DOB The proposed controllerstructure is visualized in Figure 2

The substitution of the proposed control law of (18) to theinductor current error dynamics of (17) yields that

1198710 119894119871119894 (119905) = minus1198710120582119871119871119894 (119905) + Δ119908119871119894 (119905)minus (1 minus 119906119894 (119905)) Vlowastdc (119905)

119894 = 1 119873 forall119905 ge 0(21)

where Δ119908119871119894(119905) fl 119908119871119894(119905) minus 119908119871119894(119905) 119894 = 1 119873 forall119905 ge 0Lemma 2 gives the closed-loop property of (21) which is usedfor analyzing the closed-loop stability

Lemma 2 Assuming that the disturbance of 119908119871119894(119905) convergesto its steady state value 1199080119871119894 exponentially that is there areconstants of 119896119871119894 gt 0 such that

1199080119871119894 (119905) = minus1198961198711198941199080119871119894 (119905) 119894 = 1 119873 forall119905 gt 0 (22)

where 1199080119871119894(119905) fl 1199080119871119894 minus 119908119871119894(119905) forall119905 ge 0 then a specified designparameter of 120582119871 gt 0 with the DOB gains of

119897119871119894 = 341198710120582119871 + 1 + 120578119871119894 120578119871119894 gt 0 119894 = 1 119873 (23)

gives the strict passivity for the input-output mapping of

minus (1 minus 119906119894 (119905)) Vlowastdc (119905) 997891997888rarr 119871119894 (119905) 119894 = 1 119873 (24)

Lemma 2 shows that the inductor current-tracking errorof 119871119894(119905) vanishes exponentially if the output voltage-trackingerror of Vlowastdc(119905) does so For the proof a positive definitefunction is defined as

119881119871 (119905) fl 11987102119873sum119894=1

2119871119894 (119905) + 119873sum119894=1

121199082119871119894 (119905)

+ 119873sum119894=1

1205741198711198942 (1199080119871119894 (119905))2 forall119905 ge 0(25)

The proof can be completed by showing that

119871 (119905) le minus120572119871119881119871 (119905) minus 119873sum119894=1

119871119894 (1 minus 119906119894 (119905)) Vlowastdc (119905) forall119905 ge 0

(26)

for some constant 120572119871 gt 0 where119908119871 (119905) fl 1199080119871 minus 119908119871 (119905) forall119905 ge 0 (27)

where 1199080119871 denotes a steady state value of 119908119871(119905) forall119905 ge 0 SeeAppendix for details

Mathematical Problems in Engineering 5

FinallyTheorem 3 asserts that the proposed control algo-rithm forces the output voltage trajectory to be exponentiallyconvergent to the target output voltage trajectory comingfrom the LPF dynamics of (6) The two resulting inequalitiesof (16) and (26) given by Lemmas 1 and 2 play the crucial roleof provingTheorem 3 and the proof is given in Appendix

Theorem 3 Assuming that the assumptions of Lemmas 1 and2 hold true then the proposed control law of (8) (18) with theDOBs of (9) (10) (19) and (20) forces the closed-loop systemto recover the target output voltage-tracking performance of (6)exponentially that is

1003816100381610038161003816Vlowastdc (119905)1003816100381610038161003816 le 120581119890minus(1205722)119905 forall119905 ge 0 (28)

for some positive constants 120581cl and 120572clNote that interestingly although the proposed propor-

tional-type control law of (8) and (18) with the DOBs of(9) (10) (19) and (20) does not include the tracking errorintegrators it ensures the offset-free property which is apractical advantage of this article See Theorem 4 for detailsand the proof is included in Appendix

Theorem 4 e closed-loop system always eliminates theoffset errors of the output voltage in the steady state at is

V0dc = V0dcref (29)

where V0dc and V0dcref represent the steady-state of Vdc(119905) andVdcref(119905) respectivelyRemark 5 The current and output voltage DOB dynamicscan be described in a first-order LPF form (for details seeAppendix)

119908V (119905) = 119897V (119908V (119905) minus 119908V (119905)) 119908119871119894 (119905) = 119897119871119894 (119908119871119894 (119905) minus 119908119871119894 (119905))

119894 = 1 119873 forall119905 ge 0(30)

The corresponding transfer functions are given by

V (119904)119882V (119904) =119897V119904 + 119897V

119871119894 (119904)119882119871119894 (119904) =119897119871119894119904 + 119897119871119894

119894 = 1 119873 forall119904 isin C(31)

which implies that the DOB gains of 119897V 119897119871119894 gt 0 119894 = 1 119873can be adjusted for a cut-off frequency (rads) of the transferfunctions of (31)

4 Simulations

This section evaluates the output voltage-tracking perfor-mance between the proposed and FL methods using the

PowerSIM (PSIM) software with the DLL block embeddingthe control algorithms in the C-language To this end a four-phase interleaved DCDC boost converter was consideredwith the parameters

119871 = 40 120583H119862 = 1650 120583F (32)

The input DC source voltage was set to be Vin = 50V andpulse-width modulation (PWM) and the control frequencieswere both chosen as 20 kHz In order to take the parameteruncertainties into account the nominal parameters of theconverter were assumed to be

1198710 = 28 120583H (=07119871) 1198620 = 2145 120583F (=13119862) (33)

which are adopted to constitute the control algorithmsFigure 3 visualizes the implementation of the closed-loopsystem

The FL controller in [7] was used for a comparison givenas

119906119894 (119905) = 1Vdc (119905) (minus119877dc119894119871119894 (119905) + 1198710120596cc 119871119894 (119905)

+ 119877dc120596cc int1199050

119871119894 (120591) 119889120591 + (Vdc (119905) minus Vin)) 119894119871119894ref (119905) = 14 (minus119877dvVdc (119905) + 1198620120596vcVdc (119905)+ 119877dv120596vc int119905

0

Vdc (120591) 119889120591) 119894 = 1 4 forall119905 ge 0

(34)

which yields the closed-loop transfer functions of theinner and outer-loops through the pole-zero cancellations119881dc(119904)119881dcref(119904) = 120596vc(119904+120596vc) 119868119871119894(119904)119868119871119894ref(119904) = 120596cc(119904+120596cc)119894 = 1 4 forall119904 isin C with the adjustable active dampingcoefficients 119877dc gt 0 119877dv gt 0 and the cut-off frequencies120596vc gt 0 120596cc gt 0 The corresponding 119891vc 119891cc were tuned to be119891vc = 15Hz 119891cc = 1000Hz so that 120596vc = 2120587119891vc = 942 rads120596cc = 2120587119891cc = 6280 rads and the active damping coefficientswere selected as 119877dc = 119877dv = 01 Note that the output voltageerror decay ratio parameter 120582V of the proposed method wasset to be the same as 120596vc and the design parameter 120582119871 wasalso let to be the same as the cut-off frequency of 120596cc for afair comparison because the inductor current error dynamicsof (21) can be approximately given as a first-order LPF form119894119871119894(119905) = minus120582119871(119894119871119894(119905) minus 119894119871119894ref(119905)) 119894 = 1 119873 forall119905 ge 0 fora slowly time varying inductor current reference signal of119894119871119894ref(119905) The DOB gains were tuned as 119897V = 119897119871119894 = 1256 rads119894 = 1 119873 for the cut-off frequency of the correspondingtransfer functions in (31) to be 200Hz which also meets theassumptions of (13)ndash(23)

In the first stage the evaluation of the output voltage-tracking performancewith the resistive load of119877119871 = 20Ωwasperformed where the output voltage reference was increasedfrom 100V to 150V and it was decreased to 120V From

6 Mathematical Problems in Engineering

Four-phaseinterleavedDCDCboost converter

AD converter

Outer-loopcontrollers

Phase shiftedcarrier signals

Comparators

PWMsignals

Inner-loopcontrollers

Implementation of the proposed control scheme Load

(times4)

(times4)

(times4)＞＝Ｌ

＞＝ ＞＝

iL1Ｌ iL4Ｌ

u1 u4

iL1 iL4 iL1 iL4

iL1 iL4u1 u4

lowast＞＝

＞＝

iＩ＞u1 u4

Figure 3 Implementation of the closed-loop system

IL

Vout Vout

IL

40

60

80

100

120

140

160

04 0602Time (s)

40

60

80

100

120

140

160

04 0602Time (s)

0

10

20

30

40

04 0602Time (s)

04 0602Time (s)

0

10

20

30

40

＞＝ (Classical Method)

iL (Classical Method)

＞＝ (Proposed Method)

iL (Proposed Method)

Figure 4 Output voltage-tracking performance comparison between proposed and FL methods with resistive load of 119877119871 = 20Ω

the simulation results depicted in Figure 4 the proposedmethod successfully maintains the closed-loop performanceto be the desired first-order LPF characteristic despite theparameter mismatches Figure 5 presents the state behaviorsof the DOBs

In the second stage the evaluation of the closed-looprobustness was carried out through investigating the outputvoltage-tracking performance variations under the sameoutput voltage reference with the different resistive loads119877119871 = 75 20 30Ω As can be seen from the results given

in Figure 6 the proposed method effectively reduces thevoltage-tracking performance changes for several differentloads This is a sharp contrast to the FL method

In this simulation setting the closed-loop tracking per-formance was quantitatively compared for the resistive loads119877119871 = 50 30 20 10Ω using the two cost functions defined as

119869int fl intinfin0

1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 119889119905119869max fl max 1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 forall119905 ge 0 (35)

Mathematical Problems in Engineering 7

Table 1 Quantitative output voltage-tracking performance comparison for resistive loads 119877119871 = 50 30 20 10ΩLoad resistance value119877119871 = 50Ω 119877119871 = 30Ω 119877119871 = 20Ω 119877119871 = 10Ω

Proposed Method119869int 23281 7700 4558 1722119869max 10 7 5 4Classical Method119869int 83654 58191 16325 29917119869max 35 35 11 8

WL_v_est WL_a_est WL_b_est WL_c_est WL_d_est

04 0602Time (s)

minus100

minus50

0

w

wL4

wL1wL2wL3

Figure 5 Disturbance estimation behaviors

which implies that the best controller would guarantee thetwo cost functions of (35) to be minimized for all operatingpointsThe resulting table is given inTable 1 which shows thatthe proposed method offers a better tracking performancethan the FLmethod at least two times for these four operatingpoints

In the last stage the evaluation of the output voltageregulation performance was conducted by using the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω at the 150Voperation mode Figure 7 depicts the output voltage regu-lation comparison results with the corresponding inductorcurrent which implies that the proposed method effectivelyreduces the overunder shoots caused by the load variationcomparing the FL method

5 Experimental Results

This section experimentally verifies the performance of theproposed method by comparing it with the FL techniqueused in the previous section In this experiment a 3-kWfour-phase interleaved DCDC converter shown in Figure 1was utilized with the input DC source voltage level of Vin =50V and the digital signal processor (DSP) of TMS320F28335was used for implementing the proposed and FL controlalgorithms with the control and PWM periods of 119879119904 = 50 120583sThe experimental setup is shown in Figure 8

Except for design parameters of 120596vc 120596cc 120582V and 120582119871the control parameters of two controllers were left the sameas that in the simulation section These four parameters

Proposed Method

Classical Method

02 03 04 0501Time (s)

50

100

150

50

100

150

02 03 04 0501Time (s)

Vout_(DOB_tracking_50_to_150_to_120_at_75 ohm) Vout_(DOB_trac

＞＝(30Ω) ＞＝(20Ω)

＞＝(75Ω)

＞＝(75 20 30Ω)

Figure 6 Output voltage-tracking performance comparisonbetween proposed and FL methods at three resistive loads119877119871 = 75 20 30Ω

were selected as 120596vc = 314 rads (119891vc = 5Hz) 120596cc =2512 rads (119891cc = 400Hz) and 120582V = 120596vc 120582119871 = 120596ccIn the first stage the evaluation of the output voltage-

tracking performance was experimentally performed with anincreasing output voltage reference from 75V to 100V underthe resistive loads of 119877119871 = 66 20Ω From Figures 9 and 10as intended the proposed controller successfully forces theoutput voltage dynamics to be the desired trajectory comingfrom (6) for the different operation modes

The second stage examines the output voltage regulationperformance regarding two step load changes the resistiveload was changed from 20Ω to 10Ω and from 10Ω to 66Ωin a sequential manner The resulting waveforms are shown

8 Mathematical Problems in Engineering

ILIL_(DOB_regulation_150 V_from_15_to_75 ohm)

Vout Vout_(DOB_regulation_150 ohm)V_from_15_to_75

04 0602Time (s)

0

20

40

60

04 0602Time (s)

0

20

40

60

04 0602Time (s)

100

120

140

160

180

＞＝ (Proposed Method)

＞＝ (Classical Method)

iL (Classical Method)iL (Proposed Method)

Figure 7 Output voltage regulation performance comparison between proposed and FL methods at the output voltage 150V with the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω

Control Board Inductors

Capacitors

MOSFETs

Figure 8 Experimental setup

in Figures 11 and 12 which demonstrates that the proposedmethod effectively enhances the output voltage regulationperformance through a reduction of the undershoot origi-nated from the abrupt load variation

From these simulation and experimental results it can beconcluded that the proposed method has the two practicalmerits

(1) The proposed method would suggest an almost sameclosed-loop performance for various operating rangeswithout any gain scheduling method

(2) The proposed method simplifies the control algo-rithms by getting rid of the use of the tracking errorintegrators with the antiwindup algorithms

6 Conclusions

This paper suggests a performance recovery output voltage-tracking controller for an unknown 119873-phase interleavedDCDC boost converter Taking the parameter and loadcurrent uncertainties into account the proposed control lawwas devised for the output voltage dynamics to convergeto the desired behavior coming from a LPF despite theparameter and load uncertainties Various simulation andexperimental evidences confirmed that the proposed tech-nique would provide a better closed-loop performance forseveral industrial applications

Appendix

This section provides the proofs of lemmas and theoremsTheproof of Lemma 1 is given as follows

Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

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Mathematical Problems in Engineering 3

respectively The objective of this section is to constructthe control law for rendering the closed-loop output voltagedynamics to be

119881lowastdc (119904)119881dcref (119904) =120596vc119904 + 120596vc

forall119904 isin C (4)

with 120596vc being the assignable cut-off frequency as a designparameter while guaranteeing the current-tracking property

lim119905rarrinfin

119894119871119894 (119905) = 119894119871119894ref (119905) 119894 = 1 119873 (5)

In order to achieve the control objective of (4) considerthe time domain expression of the target dynamics of (4)given by

Vlowastdc (119905) = 120596vc (Vdcref (119905) minus Vlowastdc (119905)) forall119905 ge 0 (6)

and define the output voltage-tracking error as Vlowastdc(119905) flVlowastdc(119905)minusVdc(119905) forall119905 ge 0 which gives the output voltage-trackingerror dynamics using (3)

1198620 Vlowastdc (119905) = 1198620Vlowastdc (119905) minus 1198620Vdc (119905)= minus 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894 (119905) + 119908V (119905)

= minus 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894ref (119905)

+ 119873sum119894=1

(1 minus 119906119894 (119905)) 119871119894 (119905) + 119908V (119905) forall119905 ge 0

(7)

where 119908V(119905) fl 1198620Vlowastdc(119905) minus 119908V119900(119905) 119871119894(119905) fl 119894119871119894ref(119905) minus 119894119871119894(119905)forall119905 ge 0 An inductor current reference signal is proposedas a stabilizing solution to the output voltage-tracking errordynamics of (7)

119894119871119894ref (119905) = 1119873 1(1 minus 119906119894 (119905)) (1198620120582VVlowast

dc (119905) + 119908V (119905)) (8)

119894 = 1 119873 forall119905 ge 0 where 120582V gt 0 denotes a tuningparameter the estimated disturbance 119908V(119905) is given by

119908V (119905) fl 120577V (119905) + 119897V1198620Vlowastdc (119905) forall119905 ge 0 (9)

and 120577V(119905) stands for the state of the nonlinear observer120577V (119905) = minus119897V120577V (119905) minus 1198972V1198620Vlowastdc (119905)+ 119897V 119873sum119894=1

(1 minus 119906119894 (119905)) 119894119871119894 (119905) forall119905 ge 0 (10)

with 119897V gt 0 being the observer gain For the rest of this articlethe combination of two equations of (10) and (9) is called theoutput voltage disturbance observer (DOB)

The substitution of the inductance current reference of (8)to the output voltage-tracking error dynamics of (7) leads to

1198620 Vlowastdc (119905) = minus1198620120582VVlowast

dc (119905) + Δ119908V (119905)+ 119873sum119894=1

(1 minus 119906119894 (119905)) 119871119894 (119905) forall119905 ge 0 (11)

where Δ119908V(119905) fl 119908V(119905) minus 119908V(119905) forall119905 ge 0 Lemma 1 derives theclosed-loop property of (11) which is used for analyzing thewhole closed-loop stability analysis

Lemma 1 Assuming that the disturbance of119908V(119905) converges toits steady state value1199080V exponentially that is there is a positiveconstant of 119896V gt 0 such that

1199080V (119905) = minus119896V1199080V (119905) forall119905 gt 0 (12)

where 1199080V(119905) fl 1199080V minus 119908V(119905) forall119905 ge 0 then a specified designparameter of 120582V gt 0 with the DOB gains of

119897V = 341198620120582V+ 1 + 120578V 120578V gt 0 (13)

gives the strict passivity for the input-output mapping of

119873sum119894=1

(1 minus 119906119894) 119871119894 997891997888rarr Vlowastdc (14)

In [20] the strict passivity and L2 stability are formallydefined and it is proved that the strict passivity impliesthat L2 stability The result of Lemma 1 indicates thatthe output voltage of Vdc(119905) converges to the target outputvoltage trajectory of Vlowastdc(119905) exponentially as the inductorcurrent errors of 119871119894(119905) vanish exponentially For the proofof Lemma 1 a positive definite function is defined as

119881V (119905) fl 11986202 (Vlowastdc (119905))2 + 121199082V (119905) + 120574V2 (1199080V (119905))2 forall119905 ge 0

(15)

Then the proof can be completed by showing that

V (119905) le minus120572V119881V (119905) + Vlowastdc (119905) 119873sum119894=1

(1 minus 119906119894 (119905)) 119871119894 (119905) forall119905 ge 0

(16)

for some constant 120572V gt 0 where 119908V(119905) fl 1199080V minus 119908V(119905) forall119905 ge0 and 1199080V denotes a steady state value of 119908V(119905) forall119905 ge 0 SeeAppendix for details

For deriving the final control action write the inductorcurrent-tracking error dynamics using (2) as

1198710 119894119871119894 (119905) = 1198710 119894119871119894ref (119905) minus 1198710 119894119871119894 (119905)= 1198710 119894119871119894ref (119905) + (1 minus 119906119894 (119905)) Vdc (119905) minus Vin0

minus 119908119871119894119900 (119905)= (1 minus 119906119894 (119905)) Vdc (119905) minus Vin0 + 119908119871119894 (119905)= minusVlowastdc (119905) 119906119894 (119905) + Vlowastdc (119905) minus (1 minus 119906119894 (119905)) Vlowastdc (119905)minus Vin0 + 119908119871119894 (119905) forall119905 ge 0

(17)

4 Mathematical Problems in Engineering

DC voltagecontroller (8)

+

minus

DC voltageDOB(9) (10)

Targetdynamics (6)

+

minus

Currentcontroller(18)

CurrentDOB (19)-(20)

Outer-loop controller Inner-loop controller

+

minus

＞＝

＞＝Ｌ

lowast＞＝

lowast＞＝

iL1Ｌ

iLNＬ

iLN

iL1

iL1

iLN

u1

uN

lowast＞＝

iLi ui ＞＝ iLi

wL1 middot middot middot wLNw

Figure 2 Proposed controller structure

with 119908119871119894(119905) fl 1198710 119894119871119894ref(119905) minus 119908119871119894119900(119905) 119894 = 1 119873 forall119905 ge 0Then the stabilization can be accomplished by the proposedstabilizing control law for the duty ratio 119906119894(119905) as119906119894 (119905)

= 1Vlowastdc (119905) (1198710120582119871119871119894 (119905) + Vlowastdc (119905) minus Vin0 + 119908119871119894 (119905))

119894 = 1 119873 forall119905 ge 0(18)

where the inductor current references 119894119871119894ref(119905) come from(8) 120582119871 gt 0 denotes a tuning parameter and the estimateddisturbances 119908119871119894(119905) are given by

119908119871119894 (119905) fl 120577119871119894 (119905) + 1198971198711198941198710 119871119894 (119905) 119894 = 1 119873 forall119905 ge 0 (19)

and 120577119871119894(119905) stand for the state of the dynamical equations

120577119871119894 (119905) = minus119897119871119894120577119871119894 (119905) minus 11989721198711198941198710 119871119894 (119905)+ 119897119871119894 (minus (1 minus 119906119894 (119905)) Vdc (119905) + Vin0)

119894 = 1 119873 forall119905 ge 0(20)

with 119897119871119894 gt 0 being the observer gains For the rest of thisarticle the combination of two equations of (20) and (19)is called the inductor current DOB The proposed controllerstructure is visualized in Figure 2

The substitution of the proposed control law of (18) to theinductor current error dynamics of (17) yields that

1198710 119894119871119894 (119905) = minus1198710120582119871119871119894 (119905) + Δ119908119871119894 (119905)minus (1 minus 119906119894 (119905)) Vlowastdc (119905)

119894 = 1 119873 forall119905 ge 0(21)

where Δ119908119871119894(119905) fl 119908119871119894(119905) minus 119908119871119894(119905) 119894 = 1 119873 forall119905 ge 0Lemma 2 gives the closed-loop property of (21) which is usedfor analyzing the closed-loop stability

Lemma 2 Assuming that the disturbance of 119908119871119894(119905) convergesto its steady state value 1199080119871119894 exponentially that is there areconstants of 119896119871119894 gt 0 such that

1199080119871119894 (119905) = minus1198961198711198941199080119871119894 (119905) 119894 = 1 119873 forall119905 gt 0 (22)

where 1199080119871119894(119905) fl 1199080119871119894 minus 119908119871119894(119905) forall119905 ge 0 then a specified designparameter of 120582119871 gt 0 with the DOB gains of

119897119871119894 = 341198710120582119871 + 1 + 120578119871119894 120578119871119894 gt 0 119894 = 1 119873 (23)

gives the strict passivity for the input-output mapping of

minus (1 minus 119906119894 (119905)) Vlowastdc (119905) 997891997888rarr 119871119894 (119905) 119894 = 1 119873 (24)

Lemma 2 shows that the inductor current-tracking errorof 119871119894(119905) vanishes exponentially if the output voltage-trackingerror of Vlowastdc(119905) does so For the proof a positive definitefunction is defined as

119881119871 (119905) fl 11987102119873sum119894=1

2119871119894 (119905) + 119873sum119894=1

121199082119871119894 (119905)

+ 119873sum119894=1

1205741198711198942 (1199080119871119894 (119905))2 forall119905 ge 0(25)

The proof can be completed by showing that

119871 (119905) le minus120572119871119881119871 (119905) minus 119873sum119894=1

119871119894 (1 minus 119906119894 (119905)) Vlowastdc (119905) forall119905 ge 0

(26)

for some constant 120572119871 gt 0 where119908119871 (119905) fl 1199080119871 minus 119908119871 (119905) forall119905 ge 0 (27)

where 1199080119871 denotes a steady state value of 119908119871(119905) forall119905 ge 0 SeeAppendix for details

Mathematical Problems in Engineering 5

FinallyTheorem 3 asserts that the proposed control algo-rithm forces the output voltage trajectory to be exponentiallyconvergent to the target output voltage trajectory comingfrom the LPF dynamics of (6) The two resulting inequalitiesof (16) and (26) given by Lemmas 1 and 2 play the crucial roleof provingTheorem 3 and the proof is given in Appendix

Theorem 3 Assuming that the assumptions of Lemmas 1 and2 hold true then the proposed control law of (8) (18) with theDOBs of (9) (10) (19) and (20) forces the closed-loop systemto recover the target output voltage-tracking performance of (6)exponentially that is

1003816100381610038161003816Vlowastdc (119905)1003816100381610038161003816 le 120581119890minus(1205722)119905 forall119905 ge 0 (28)

for some positive constants 120581cl and 120572clNote that interestingly although the proposed propor-

tional-type control law of (8) and (18) with the DOBs of(9) (10) (19) and (20) does not include the tracking errorintegrators it ensures the offset-free property which is apractical advantage of this article See Theorem 4 for detailsand the proof is included in Appendix

Theorem 4 e closed-loop system always eliminates theoffset errors of the output voltage in the steady state at is

V0dc = V0dcref (29)

where V0dc and V0dcref represent the steady-state of Vdc(119905) andVdcref(119905) respectivelyRemark 5 The current and output voltage DOB dynamicscan be described in a first-order LPF form (for details seeAppendix)

119908V (119905) = 119897V (119908V (119905) minus 119908V (119905)) 119908119871119894 (119905) = 119897119871119894 (119908119871119894 (119905) minus 119908119871119894 (119905))

119894 = 1 119873 forall119905 ge 0(30)

The corresponding transfer functions are given by

V (119904)119882V (119904) =119897V119904 + 119897V

119871119894 (119904)119882119871119894 (119904) =119897119871119894119904 + 119897119871119894

119894 = 1 119873 forall119904 isin C(31)

which implies that the DOB gains of 119897V 119897119871119894 gt 0 119894 = 1 119873can be adjusted for a cut-off frequency (rads) of the transferfunctions of (31)

4 Simulations

This section evaluates the output voltage-tracking perfor-mance between the proposed and FL methods using the

PowerSIM (PSIM) software with the DLL block embeddingthe control algorithms in the C-language To this end a four-phase interleaved DCDC boost converter was consideredwith the parameters

119871 = 40 120583H119862 = 1650 120583F (32)

The input DC source voltage was set to be Vin = 50V andpulse-width modulation (PWM) and the control frequencieswere both chosen as 20 kHz In order to take the parameteruncertainties into account the nominal parameters of theconverter were assumed to be

1198710 = 28 120583H (=07119871) 1198620 = 2145 120583F (=13119862) (33)

which are adopted to constitute the control algorithmsFigure 3 visualizes the implementation of the closed-loopsystem

The FL controller in [7] was used for a comparison givenas

119906119894 (119905) = 1Vdc (119905) (minus119877dc119894119871119894 (119905) + 1198710120596cc 119871119894 (119905)

+ 119877dc120596cc int1199050

119871119894 (120591) 119889120591 + (Vdc (119905) minus Vin)) 119894119871119894ref (119905) = 14 (minus119877dvVdc (119905) + 1198620120596vcVdc (119905)+ 119877dv120596vc int119905

0

Vdc (120591) 119889120591) 119894 = 1 4 forall119905 ge 0

(34)

which yields the closed-loop transfer functions of theinner and outer-loops through the pole-zero cancellations119881dc(119904)119881dcref(119904) = 120596vc(119904+120596vc) 119868119871119894(119904)119868119871119894ref(119904) = 120596cc(119904+120596cc)119894 = 1 4 forall119904 isin C with the adjustable active dampingcoefficients 119877dc gt 0 119877dv gt 0 and the cut-off frequencies120596vc gt 0 120596cc gt 0 The corresponding 119891vc 119891cc were tuned to be119891vc = 15Hz 119891cc = 1000Hz so that 120596vc = 2120587119891vc = 942 rads120596cc = 2120587119891cc = 6280 rads and the active damping coefficientswere selected as 119877dc = 119877dv = 01 Note that the output voltageerror decay ratio parameter 120582V of the proposed method wasset to be the same as 120596vc and the design parameter 120582119871 wasalso let to be the same as the cut-off frequency of 120596cc for afair comparison because the inductor current error dynamicsof (21) can be approximately given as a first-order LPF form119894119871119894(119905) = minus120582119871(119894119871119894(119905) minus 119894119871119894ref(119905)) 119894 = 1 119873 forall119905 ge 0 fora slowly time varying inductor current reference signal of119894119871119894ref(119905) The DOB gains were tuned as 119897V = 119897119871119894 = 1256 rads119894 = 1 119873 for the cut-off frequency of the correspondingtransfer functions in (31) to be 200Hz which also meets theassumptions of (13)ndash(23)

In the first stage the evaluation of the output voltage-tracking performancewith the resistive load of119877119871 = 20Ωwasperformed where the output voltage reference was increasedfrom 100V to 150V and it was decreased to 120V From

6 Mathematical Problems in Engineering

Four-phaseinterleavedDCDCboost converter

AD converter

Outer-loopcontrollers

Phase shiftedcarrier signals

Comparators

PWMsignals

Inner-loopcontrollers

Implementation of the proposed control scheme Load

(times4)

(times4)

(times4)＞＝Ｌ

＞＝ ＞＝

iL1Ｌ iL4Ｌ

u1 u4

iL1 iL4 iL1 iL4

iL1 iL4u1 u4

lowast＞＝

＞＝

iＩ＞u1 u4

Figure 3 Implementation of the closed-loop system

IL

Vout Vout

IL

40

60

80

100

120

140

160

04 0602Time (s)

40

60

80

100

120

140

160

04 0602Time (s)

0

10

20

30

40

04 0602Time (s)

04 0602Time (s)

0

10

20

30

40

＞＝ (Classical Method)

iL (Classical Method)

＞＝ (Proposed Method)

iL (Proposed Method)

Figure 4 Output voltage-tracking performance comparison between proposed and FL methods with resistive load of 119877119871 = 20Ω

the simulation results depicted in Figure 4 the proposedmethod successfully maintains the closed-loop performanceto be the desired first-order LPF characteristic despite theparameter mismatches Figure 5 presents the state behaviorsof the DOBs

In the second stage the evaluation of the closed-looprobustness was carried out through investigating the outputvoltage-tracking performance variations under the sameoutput voltage reference with the different resistive loads119877119871 = 75 20 30Ω As can be seen from the results given

in Figure 6 the proposed method effectively reduces thevoltage-tracking performance changes for several differentloads This is a sharp contrast to the FL method

In this simulation setting the closed-loop tracking per-formance was quantitatively compared for the resistive loads119877119871 = 50 30 20 10Ω using the two cost functions defined as

119869int fl intinfin0

1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 119889119905119869max fl max 1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 forall119905 ge 0 (35)

Mathematical Problems in Engineering 7

Table 1 Quantitative output voltage-tracking performance comparison for resistive loads 119877119871 = 50 30 20 10ΩLoad resistance value119877119871 = 50Ω 119877119871 = 30Ω 119877119871 = 20Ω 119877119871 = 10Ω

Proposed Method119869int 23281 7700 4558 1722119869max 10 7 5 4Classical Method119869int 83654 58191 16325 29917119869max 35 35 11 8

WL_v_est WL_a_est WL_b_est WL_c_est WL_d_est

04 0602Time (s)

minus100

minus50

0

w

wL4

wL1wL2wL3

Figure 5 Disturbance estimation behaviors

which implies that the best controller would guarantee thetwo cost functions of (35) to be minimized for all operatingpointsThe resulting table is given inTable 1 which shows thatthe proposed method offers a better tracking performancethan the FLmethod at least two times for these four operatingpoints

In the last stage the evaluation of the output voltageregulation performance was conducted by using the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω at the 150Voperation mode Figure 7 depicts the output voltage regu-lation comparison results with the corresponding inductorcurrent which implies that the proposed method effectivelyreduces the overunder shoots caused by the load variationcomparing the FL method

5 Experimental Results

This section experimentally verifies the performance of theproposed method by comparing it with the FL techniqueused in the previous section In this experiment a 3-kWfour-phase interleaved DCDC converter shown in Figure 1was utilized with the input DC source voltage level of Vin =50V and the digital signal processor (DSP) of TMS320F28335was used for implementing the proposed and FL controlalgorithms with the control and PWM periods of 119879119904 = 50 120583sThe experimental setup is shown in Figure 8

Except for design parameters of 120596vc 120596cc 120582V and 120582119871the control parameters of two controllers were left the sameas that in the simulation section These four parameters

Proposed Method

Classical Method

02 03 04 0501Time (s)

50

100

150

50

100

150

02 03 04 0501Time (s)

Vout_(DOB_tracking_50_to_150_to_120_at_75 ohm) Vout_(DOB_trac

＞＝(30Ω) ＞＝(20Ω)

＞＝(75Ω)

＞＝(75 20 30Ω)

Figure 6 Output voltage-tracking performance comparisonbetween proposed and FL methods at three resistive loads119877119871 = 75 20 30Ω

were selected as 120596vc = 314 rads (119891vc = 5Hz) 120596cc =2512 rads (119891cc = 400Hz) and 120582V = 120596vc 120582119871 = 120596ccIn the first stage the evaluation of the output voltage-

tracking performance was experimentally performed with anincreasing output voltage reference from 75V to 100V underthe resistive loads of 119877119871 = 66 20Ω From Figures 9 and 10as intended the proposed controller successfully forces theoutput voltage dynamics to be the desired trajectory comingfrom (6) for the different operation modes

The second stage examines the output voltage regulationperformance regarding two step load changes the resistiveload was changed from 20Ω to 10Ω and from 10Ω to 66Ωin a sequential manner The resulting waveforms are shown

8 Mathematical Problems in Engineering

ILIL_(DOB_regulation_150 V_from_15_to_75 ohm)

Vout Vout_(DOB_regulation_150 ohm)V_from_15_to_75

04 0602Time (s)

0

20

40

60

04 0602Time (s)

0

20

40

60

04 0602Time (s)

100

120

140

160

180

＞＝ (Proposed Method)

＞＝ (Classical Method)

iL (Classical Method)iL (Proposed Method)

Figure 7 Output voltage regulation performance comparison between proposed and FL methods at the output voltage 150V with the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω

Control Board Inductors

Capacitors

MOSFETs

Figure 8 Experimental setup

in Figures 11 and 12 which demonstrates that the proposedmethod effectively enhances the output voltage regulationperformance through a reduction of the undershoot origi-nated from the abrupt load variation

From these simulation and experimental results it can beconcluded that the proposed method has the two practicalmerits

(1) The proposed method would suggest an almost sameclosed-loop performance for various operating rangeswithout any gain scheduling method

(2) The proposed method simplifies the control algo-rithms by getting rid of the use of the tracking errorintegrators with the antiwindup algorithms

6 Conclusions

This paper suggests a performance recovery output voltage-tracking controller for an unknown 119873-phase interleavedDCDC boost converter Taking the parameter and loadcurrent uncertainties into account the proposed control lawwas devised for the output voltage dynamics to convergeto the desired behavior coming from a LPF despite theparameter and load uncertainties Various simulation andexperimental evidences confirmed that the proposed tech-nique would provide a better closed-loop performance forseveral industrial applications

Appendix

This section provides the proofs of lemmas and theoremsTheproof of Lemma 1 is given as follows

Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

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Submit your manuscripts atwwwhindawicom

4 Mathematical Problems in Engineering

DC voltagecontroller (8)

+

minus

DC voltageDOB(9) (10)

Targetdynamics (6)

+

minus

Currentcontroller(18)

CurrentDOB (19)-(20)

Outer-loop controller Inner-loop controller

+

minus

＞＝

＞＝Ｌ

lowast＞＝

lowast＞＝

iL1Ｌ

iLNＬ

iLN

iL1

iL1

iLN

u1

uN

lowast＞＝

iLi ui ＞＝ iLi

wL1 middot middot middot wLNw

Figure 2 Proposed controller structure

with 119908119871119894(119905) fl 1198710 119894119871119894ref(119905) minus 119908119871119894119900(119905) 119894 = 1 119873 forall119905 ge 0Then the stabilization can be accomplished by the proposedstabilizing control law for the duty ratio 119906119894(119905) as119906119894 (119905)

= 1Vlowastdc (119905) (1198710120582119871119871119894 (119905) + Vlowastdc (119905) minus Vin0 + 119908119871119894 (119905))

119894 = 1 119873 forall119905 ge 0(18)

where the inductor current references 119894119871119894ref(119905) come from(8) 120582119871 gt 0 denotes a tuning parameter and the estimateddisturbances 119908119871119894(119905) are given by

119908119871119894 (119905) fl 120577119871119894 (119905) + 1198971198711198941198710 119871119894 (119905) 119894 = 1 119873 forall119905 ge 0 (19)

and 120577119871119894(119905) stand for the state of the dynamical equations

120577119871119894 (119905) = minus119897119871119894120577119871119894 (119905) minus 11989721198711198941198710 119871119894 (119905)+ 119897119871119894 (minus (1 minus 119906119894 (119905)) Vdc (119905) + Vin0)

119894 = 1 119873 forall119905 ge 0(20)

with 119897119871119894 gt 0 being the observer gains For the rest of thisarticle the combination of two equations of (20) and (19)is called the inductor current DOB The proposed controllerstructure is visualized in Figure 2

The substitution of the proposed control law of (18) to theinductor current error dynamics of (17) yields that

1198710 119894119871119894 (119905) = minus1198710120582119871119871119894 (119905) + Δ119908119871119894 (119905)minus (1 minus 119906119894 (119905)) Vlowastdc (119905)

119894 = 1 119873 forall119905 ge 0(21)

where Δ119908119871119894(119905) fl 119908119871119894(119905) minus 119908119871119894(119905) 119894 = 1 119873 forall119905 ge 0Lemma 2 gives the closed-loop property of (21) which is usedfor analyzing the closed-loop stability

Lemma 2 Assuming that the disturbance of 119908119871119894(119905) convergesto its steady state value 1199080119871119894 exponentially that is there areconstants of 119896119871119894 gt 0 such that

1199080119871119894 (119905) = minus1198961198711198941199080119871119894 (119905) 119894 = 1 119873 forall119905 gt 0 (22)

where 1199080119871119894(119905) fl 1199080119871119894 minus 119908119871119894(119905) forall119905 ge 0 then a specified designparameter of 120582119871 gt 0 with the DOB gains of

119897119871119894 = 341198710120582119871 + 1 + 120578119871119894 120578119871119894 gt 0 119894 = 1 119873 (23)

gives the strict passivity for the input-output mapping of

minus (1 minus 119906119894 (119905)) Vlowastdc (119905) 997891997888rarr 119871119894 (119905) 119894 = 1 119873 (24)

Lemma 2 shows that the inductor current-tracking errorof 119871119894(119905) vanishes exponentially if the output voltage-trackingerror of Vlowastdc(119905) does so For the proof a positive definitefunction is defined as

119881119871 (119905) fl 11987102119873sum119894=1

2119871119894 (119905) + 119873sum119894=1

121199082119871119894 (119905)

+ 119873sum119894=1

1205741198711198942 (1199080119871119894 (119905))2 forall119905 ge 0(25)

The proof can be completed by showing that

119871 (119905) le minus120572119871119881119871 (119905) minus 119873sum119894=1

119871119894 (1 minus 119906119894 (119905)) Vlowastdc (119905) forall119905 ge 0

(26)

for some constant 120572119871 gt 0 where119908119871 (119905) fl 1199080119871 minus 119908119871 (119905) forall119905 ge 0 (27)

where 1199080119871 denotes a steady state value of 119908119871(119905) forall119905 ge 0 SeeAppendix for details

Mathematical Problems in Engineering 5

FinallyTheorem 3 asserts that the proposed control algo-rithm forces the output voltage trajectory to be exponentiallyconvergent to the target output voltage trajectory comingfrom the LPF dynamics of (6) The two resulting inequalitiesof (16) and (26) given by Lemmas 1 and 2 play the crucial roleof provingTheorem 3 and the proof is given in Appendix

Theorem 3 Assuming that the assumptions of Lemmas 1 and2 hold true then the proposed control law of (8) (18) with theDOBs of (9) (10) (19) and (20) forces the closed-loop systemto recover the target output voltage-tracking performance of (6)exponentially that is

1003816100381610038161003816Vlowastdc (119905)1003816100381610038161003816 le 120581119890minus(1205722)119905 forall119905 ge 0 (28)

for some positive constants 120581cl and 120572clNote that interestingly although the proposed propor-

tional-type control law of (8) and (18) with the DOBs of(9) (10) (19) and (20) does not include the tracking errorintegrators it ensures the offset-free property which is apractical advantage of this article See Theorem 4 for detailsand the proof is included in Appendix

Theorem 4 e closed-loop system always eliminates theoffset errors of the output voltage in the steady state at is

V0dc = V0dcref (29)

where V0dc and V0dcref represent the steady-state of Vdc(119905) andVdcref(119905) respectivelyRemark 5 The current and output voltage DOB dynamicscan be described in a first-order LPF form (for details seeAppendix)

119908V (119905) = 119897V (119908V (119905) minus 119908V (119905)) 119908119871119894 (119905) = 119897119871119894 (119908119871119894 (119905) minus 119908119871119894 (119905))

119894 = 1 119873 forall119905 ge 0(30)

The corresponding transfer functions are given by

V (119904)119882V (119904) =119897V119904 + 119897V

119871119894 (119904)119882119871119894 (119904) =119897119871119894119904 + 119897119871119894

119894 = 1 119873 forall119904 isin C(31)

which implies that the DOB gains of 119897V 119897119871119894 gt 0 119894 = 1 119873can be adjusted for a cut-off frequency (rads) of the transferfunctions of (31)

4 Simulations

This section evaluates the output voltage-tracking perfor-mance between the proposed and FL methods using the

PowerSIM (PSIM) software with the DLL block embeddingthe control algorithms in the C-language To this end a four-phase interleaved DCDC boost converter was consideredwith the parameters

119871 = 40 120583H119862 = 1650 120583F (32)

The input DC source voltage was set to be Vin = 50V andpulse-width modulation (PWM) and the control frequencieswere both chosen as 20 kHz In order to take the parameteruncertainties into account the nominal parameters of theconverter were assumed to be

1198710 = 28 120583H (=07119871) 1198620 = 2145 120583F (=13119862) (33)

which are adopted to constitute the control algorithmsFigure 3 visualizes the implementation of the closed-loopsystem

The FL controller in [7] was used for a comparison givenas

119906119894 (119905) = 1Vdc (119905) (minus119877dc119894119871119894 (119905) + 1198710120596cc 119871119894 (119905)

+ 119877dc120596cc int1199050

119871119894 (120591) 119889120591 + (Vdc (119905) minus Vin)) 119894119871119894ref (119905) = 14 (minus119877dvVdc (119905) + 1198620120596vcVdc (119905)+ 119877dv120596vc int119905

0

Vdc (120591) 119889120591) 119894 = 1 4 forall119905 ge 0

(34)

which yields the closed-loop transfer functions of theinner and outer-loops through the pole-zero cancellations119881dc(119904)119881dcref(119904) = 120596vc(119904+120596vc) 119868119871119894(119904)119868119871119894ref(119904) = 120596cc(119904+120596cc)119894 = 1 4 forall119904 isin C with the adjustable active dampingcoefficients 119877dc gt 0 119877dv gt 0 and the cut-off frequencies120596vc gt 0 120596cc gt 0 The corresponding 119891vc 119891cc were tuned to be119891vc = 15Hz 119891cc = 1000Hz so that 120596vc = 2120587119891vc = 942 rads120596cc = 2120587119891cc = 6280 rads and the active damping coefficientswere selected as 119877dc = 119877dv = 01 Note that the output voltageerror decay ratio parameter 120582V of the proposed method wasset to be the same as 120596vc and the design parameter 120582119871 wasalso let to be the same as the cut-off frequency of 120596cc for afair comparison because the inductor current error dynamicsof (21) can be approximately given as a first-order LPF form119894119871119894(119905) = minus120582119871(119894119871119894(119905) minus 119894119871119894ref(119905)) 119894 = 1 119873 forall119905 ge 0 fora slowly time varying inductor current reference signal of119894119871119894ref(119905) The DOB gains were tuned as 119897V = 119897119871119894 = 1256 rads119894 = 1 119873 for the cut-off frequency of the correspondingtransfer functions in (31) to be 200Hz which also meets theassumptions of (13)ndash(23)

In the first stage the evaluation of the output voltage-tracking performancewith the resistive load of119877119871 = 20Ωwasperformed where the output voltage reference was increasedfrom 100V to 150V and it was decreased to 120V From

6 Mathematical Problems in Engineering

Four-phaseinterleavedDCDCboost converter

AD converter

Outer-loopcontrollers

Phase shiftedcarrier signals

Comparators

PWMsignals

Inner-loopcontrollers

Implementation of the proposed control scheme Load

(times4)

(times4)

(times4)＞＝Ｌ

＞＝ ＞＝

iL1Ｌ iL4Ｌ

u1 u4

iL1 iL4 iL1 iL4

iL1 iL4u1 u4

lowast＞＝

＞＝

iＩ＞u1 u4

Figure 3 Implementation of the closed-loop system

IL

Vout Vout

IL

40

60

80

100

120

140

160

04 0602Time (s)

40

60

80

100

120

140

160

04 0602Time (s)

0

10

20

30

40

04 0602Time (s)

04 0602Time (s)

0

10

20

30

40

＞＝ (Classical Method)

iL (Classical Method)

＞＝ (Proposed Method)

iL (Proposed Method)

Figure 4 Output voltage-tracking performance comparison between proposed and FL methods with resistive load of 119877119871 = 20Ω

the simulation results depicted in Figure 4 the proposedmethod successfully maintains the closed-loop performanceto be the desired first-order LPF characteristic despite theparameter mismatches Figure 5 presents the state behaviorsof the DOBs

In the second stage the evaluation of the closed-looprobustness was carried out through investigating the outputvoltage-tracking performance variations under the sameoutput voltage reference with the different resistive loads119877119871 = 75 20 30Ω As can be seen from the results given

in Figure 6 the proposed method effectively reduces thevoltage-tracking performance changes for several differentloads This is a sharp contrast to the FL method

In this simulation setting the closed-loop tracking per-formance was quantitatively compared for the resistive loads119877119871 = 50 30 20 10Ω using the two cost functions defined as

119869int fl intinfin0

1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 119889119905119869max fl max 1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 forall119905 ge 0 (35)

Mathematical Problems in Engineering 7

Table 1 Quantitative output voltage-tracking performance comparison for resistive loads 119877119871 = 50 30 20 10ΩLoad resistance value119877119871 = 50Ω 119877119871 = 30Ω 119877119871 = 20Ω 119877119871 = 10Ω

Proposed Method119869int 23281 7700 4558 1722119869max 10 7 5 4Classical Method119869int 83654 58191 16325 29917119869max 35 35 11 8

WL_v_est WL_a_est WL_b_est WL_c_est WL_d_est

04 0602Time (s)

minus100

minus50

0

w

wL4

wL1wL2wL3

Figure 5 Disturbance estimation behaviors

which implies that the best controller would guarantee thetwo cost functions of (35) to be minimized for all operatingpointsThe resulting table is given inTable 1 which shows thatthe proposed method offers a better tracking performancethan the FLmethod at least two times for these four operatingpoints

In the last stage the evaluation of the output voltageregulation performance was conducted by using the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω at the 150Voperation mode Figure 7 depicts the output voltage regu-lation comparison results with the corresponding inductorcurrent which implies that the proposed method effectivelyreduces the overunder shoots caused by the load variationcomparing the FL method

5 Experimental Results

This section experimentally verifies the performance of theproposed method by comparing it with the FL techniqueused in the previous section In this experiment a 3-kWfour-phase interleaved DCDC converter shown in Figure 1was utilized with the input DC source voltage level of Vin =50V and the digital signal processor (DSP) of TMS320F28335was used for implementing the proposed and FL controlalgorithms with the control and PWM periods of 119879119904 = 50 120583sThe experimental setup is shown in Figure 8

Except for design parameters of 120596vc 120596cc 120582V and 120582119871the control parameters of two controllers were left the sameas that in the simulation section These four parameters

Proposed Method

Classical Method

02 03 04 0501Time (s)

50

100

150

50

100

150

02 03 04 0501Time (s)

Vout_(DOB_tracking_50_to_150_to_120_at_75 ohm) Vout_(DOB_trac

＞＝(30Ω) ＞＝(20Ω)

＞＝(75Ω)

＞＝(75 20 30Ω)

Figure 6 Output voltage-tracking performance comparisonbetween proposed and FL methods at three resistive loads119877119871 = 75 20 30Ω

were selected as 120596vc = 314 rads (119891vc = 5Hz) 120596cc =2512 rads (119891cc = 400Hz) and 120582V = 120596vc 120582119871 = 120596ccIn the first stage the evaluation of the output voltage-

tracking performance was experimentally performed with anincreasing output voltage reference from 75V to 100V underthe resistive loads of 119877119871 = 66 20Ω From Figures 9 and 10as intended the proposed controller successfully forces theoutput voltage dynamics to be the desired trajectory comingfrom (6) for the different operation modes

The second stage examines the output voltage regulationperformance regarding two step load changes the resistiveload was changed from 20Ω to 10Ω and from 10Ω to 66Ωin a sequential manner The resulting waveforms are shown

8 Mathematical Problems in Engineering

ILIL_(DOB_regulation_150 V_from_15_to_75 ohm)

Vout Vout_(DOB_regulation_150 ohm)V_from_15_to_75

04 0602Time (s)

0

20

40

60

04 0602Time (s)

0

20

40

60

04 0602Time (s)

100

120

140

160

180

＞＝ (Proposed Method)

＞＝ (Classical Method)

iL (Classical Method)iL (Proposed Method)

Figure 7 Output voltage regulation performance comparison between proposed and FL methods at the output voltage 150V with the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω

Control Board Inductors

Capacitors

MOSFETs

Figure 8 Experimental setup

in Figures 11 and 12 which demonstrates that the proposedmethod effectively enhances the output voltage regulationperformance through a reduction of the undershoot origi-nated from the abrupt load variation

From these simulation and experimental results it can beconcluded that the proposed method has the two practicalmerits

(1) The proposed method would suggest an almost sameclosed-loop performance for various operating rangeswithout any gain scheduling method

(2) The proposed method simplifies the control algo-rithms by getting rid of the use of the tracking errorintegrators with the antiwindup algorithms

6 Conclusions

This paper suggests a performance recovery output voltage-tracking controller for an unknown 119873-phase interleavedDCDC boost converter Taking the parameter and loadcurrent uncertainties into account the proposed control lawwas devised for the output voltage dynamics to convergeto the desired behavior coming from a LPF despite theparameter and load uncertainties Various simulation andexperimental evidences confirmed that the proposed tech-nique would provide a better closed-loop performance forseveral industrial applications

Appendix

This section provides the proofs of lemmas and theoremsTheproof of Lemma 1 is given as follows

Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

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Mathematical Problems in Engineering 5

FinallyTheorem 3 asserts that the proposed control algo-rithm forces the output voltage trajectory to be exponentiallyconvergent to the target output voltage trajectory comingfrom the LPF dynamics of (6) The two resulting inequalitiesof (16) and (26) given by Lemmas 1 and 2 play the crucial roleof provingTheorem 3 and the proof is given in Appendix

Theorem 3 Assuming that the assumptions of Lemmas 1 and2 hold true then the proposed control law of (8) (18) with theDOBs of (9) (10) (19) and (20) forces the closed-loop systemto recover the target output voltage-tracking performance of (6)exponentially that is

1003816100381610038161003816Vlowastdc (119905)1003816100381610038161003816 le 120581119890minus(1205722)119905 forall119905 ge 0 (28)

for some positive constants 120581cl and 120572clNote that interestingly although the proposed propor-

tional-type control law of (8) and (18) with the DOBs of(9) (10) (19) and (20) does not include the tracking errorintegrators it ensures the offset-free property which is apractical advantage of this article See Theorem 4 for detailsand the proof is included in Appendix

Theorem 4 e closed-loop system always eliminates theoffset errors of the output voltage in the steady state at is

V0dc = V0dcref (29)

where V0dc and V0dcref represent the steady-state of Vdc(119905) andVdcref(119905) respectivelyRemark 5 The current and output voltage DOB dynamicscan be described in a first-order LPF form (for details seeAppendix)

119908V (119905) = 119897V (119908V (119905) minus 119908V (119905)) 119908119871119894 (119905) = 119897119871119894 (119908119871119894 (119905) minus 119908119871119894 (119905))

119894 = 1 119873 forall119905 ge 0(30)

The corresponding transfer functions are given by

V (119904)119882V (119904) =119897V119904 + 119897V

119871119894 (119904)119882119871119894 (119904) =119897119871119894119904 + 119897119871119894

119894 = 1 119873 forall119904 isin C(31)

which implies that the DOB gains of 119897V 119897119871119894 gt 0 119894 = 1 119873can be adjusted for a cut-off frequency (rads) of the transferfunctions of (31)

4 Simulations

This section evaluates the output voltage-tracking perfor-mance between the proposed and FL methods using the

PowerSIM (PSIM) software with the DLL block embeddingthe control algorithms in the C-language To this end a four-phase interleaved DCDC boost converter was consideredwith the parameters

119871 = 40 120583H119862 = 1650 120583F (32)

The input DC source voltage was set to be Vin = 50V andpulse-width modulation (PWM) and the control frequencieswere both chosen as 20 kHz In order to take the parameteruncertainties into account the nominal parameters of theconverter were assumed to be

1198710 = 28 120583H (=07119871) 1198620 = 2145 120583F (=13119862) (33)

which are adopted to constitute the control algorithmsFigure 3 visualizes the implementation of the closed-loopsystem

The FL controller in [7] was used for a comparison givenas

119906119894 (119905) = 1Vdc (119905) (minus119877dc119894119871119894 (119905) + 1198710120596cc 119871119894 (119905)

+ 119877dc120596cc int1199050

119871119894 (120591) 119889120591 + (Vdc (119905) minus Vin)) 119894119871119894ref (119905) = 14 (minus119877dvVdc (119905) + 1198620120596vcVdc (119905)+ 119877dv120596vc int119905

0

Vdc (120591) 119889120591) 119894 = 1 4 forall119905 ge 0

(34)

which yields the closed-loop transfer functions of theinner and outer-loops through the pole-zero cancellations119881dc(119904)119881dcref(119904) = 120596vc(119904+120596vc) 119868119871119894(119904)119868119871119894ref(119904) = 120596cc(119904+120596cc)119894 = 1 4 forall119904 isin C with the adjustable active dampingcoefficients 119877dc gt 0 119877dv gt 0 and the cut-off frequencies120596vc gt 0 120596cc gt 0 The corresponding 119891vc 119891cc were tuned to be119891vc = 15Hz 119891cc = 1000Hz so that 120596vc = 2120587119891vc = 942 rads120596cc = 2120587119891cc = 6280 rads and the active damping coefficientswere selected as 119877dc = 119877dv = 01 Note that the output voltageerror decay ratio parameter 120582V of the proposed method wasset to be the same as 120596vc and the design parameter 120582119871 wasalso let to be the same as the cut-off frequency of 120596cc for afair comparison because the inductor current error dynamicsof (21) can be approximately given as a first-order LPF form119894119871119894(119905) = minus120582119871(119894119871119894(119905) minus 119894119871119894ref(119905)) 119894 = 1 119873 forall119905 ge 0 fora slowly time varying inductor current reference signal of119894119871119894ref(119905) The DOB gains were tuned as 119897V = 119897119871119894 = 1256 rads119894 = 1 119873 for the cut-off frequency of the correspondingtransfer functions in (31) to be 200Hz which also meets theassumptions of (13)ndash(23)

In the first stage the evaluation of the output voltage-tracking performancewith the resistive load of119877119871 = 20Ωwasperformed where the output voltage reference was increasedfrom 100V to 150V and it was decreased to 120V From

6 Mathematical Problems in Engineering

Four-phaseinterleavedDCDCboost converter

AD converter

Outer-loopcontrollers

Phase shiftedcarrier signals

Comparators

PWMsignals

Inner-loopcontrollers

Implementation of the proposed control scheme Load

(times4)

(times4)

(times4)＞＝Ｌ

＞＝ ＞＝

iL1Ｌ iL4Ｌ

u1 u4

iL1 iL4 iL1 iL4

iL1 iL4u1 u4

lowast＞＝

＞＝

iＩ＞u1 u4

Figure 3 Implementation of the closed-loop system

IL

Vout Vout

IL

40

60

80

100

120

140

160

04 0602Time (s)

40

60

80

100

120

140

160

04 0602Time (s)

0

10

20

30

40

04 0602Time (s)

04 0602Time (s)

0

10

20

30

40

＞＝ (Classical Method)

iL (Classical Method)

＞＝ (Proposed Method)

iL (Proposed Method)

Figure 4 Output voltage-tracking performance comparison between proposed and FL methods with resistive load of 119877119871 = 20Ω

the simulation results depicted in Figure 4 the proposedmethod successfully maintains the closed-loop performanceto be the desired first-order LPF characteristic despite theparameter mismatches Figure 5 presents the state behaviorsof the DOBs

In the second stage the evaluation of the closed-looprobustness was carried out through investigating the outputvoltage-tracking performance variations under the sameoutput voltage reference with the different resistive loads119877119871 = 75 20 30Ω As can be seen from the results given

in Figure 6 the proposed method effectively reduces thevoltage-tracking performance changes for several differentloads This is a sharp contrast to the FL method

In this simulation setting the closed-loop tracking per-formance was quantitatively compared for the resistive loads119877119871 = 50 30 20 10Ω using the two cost functions defined as

119869int fl intinfin0

1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 119889119905119869max fl max 1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 forall119905 ge 0 (35)

Mathematical Problems in Engineering 7

Table 1 Quantitative output voltage-tracking performance comparison for resistive loads 119877119871 = 50 30 20 10ΩLoad resistance value119877119871 = 50Ω 119877119871 = 30Ω 119877119871 = 20Ω 119877119871 = 10Ω

Proposed Method119869int 23281 7700 4558 1722119869max 10 7 5 4Classical Method119869int 83654 58191 16325 29917119869max 35 35 11 8

WL_v_est WL_a_est WL_b_est WL_c_est WL_d_est

04 0602Time (s)

minus100

minus50

0

w

wL4

wL1wL2wL3

Figure 5 Disturbance estimation behaviors

which implies that the best controller would guarantee thetwo cost functions of (35) to be minimized for all operatingpointsThe resulting table is given inTable 1 which shows thatthe proposed method offers a better tracking performancethan the FLmethod at least two times for these four operatingpoints

In the last stage the evaluation of the output voltageregulation performance was conducted by using the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω at the 150Voperation mode Figure 7 depicts the output voltage regu-lation comparison results with the corresponding inductorcurrent which implies that the proposed method effectivelyreduces the overunder shoots caused by the load variationcomparing the FL method

5 Experimental Results

This section experimentally verifies the performance of theproposed method by comparing it with the FL techniqueused in the previous section In this experiment a 3-kWfour-phase interleaved DCDC converter shown in Figure 1was utilized with the input DC source voltage level of Vin =50V and the digital signal processor (DSP) of TMS320F28335was used for implementing the proposed and FL controlalgorithms with the control and PWM periods of 119879119904 = 50 120583sThe experimental setup is shown in Figure 8

Except for design parameters of 120596vc 120596cc 120582V and 120582119871the control parameters of two controllers were left the sameas that in the simulation section These four parameters

Proposed Method

Classical Method

02 03 04 0501Time (s)

50

100

150

50

100

150

02 03 04 0501Time (s)

Vout_(DOB_tracking_50_to_150_to_120_at_75 ohm) Vout_(DOB_trac

＞＝(30Ω) ＞＝(20Ω)

＞＝(75Ω)

＞＝(75 20 30Ω)

Figure 6 Output voltage-tracking performance comparisonbetween proposed and FL methods at three resistive loads119877119871 = 75 20 30Ω

were selected as 120596vc = 314 rads (119891vc = 5Hz) 120596cc =2512 rads (119891cc = 400Hz) and 120582V = 120596vc 120582119871 = 120596ccIn the first stage the evaluation of the output voltage-

tracking performance was experimentally performed with anincreasing output voltage reference from 75V to 100V underthe resistive loads of 119877119871 = 66 20Ω From Figures 9 and 10as intended the proposed controller successfully forces theoutput voltage dynamics to be the desired trajectory comingfrom (6) for the different operation modes

The second stage examines the output voltage regulationperformance regarding two step load changes the resistiveload was changed from 20Ω to 10Ω and from 10Ω to 66Ωin a sequential manner The resulting waveforms are shown

8 Mathematical Problems in Engineering

ILIL_(DOB_regulation_150 V_from_15_to_75 ohm)

Vout Vout_(DOB_regulation_150 ohm)V_from_15_to_75

04 0602Time (s)

0

20

40

60

04 0602Time (s)

0

20

40

60

04 0602Time (s)

100

120

140

160

180

＞＝ (Proposed Method)

＞＝ (Classical Method)

iL (Classical Method)iL (Proposed Method)

Figure 7 Output voltage regulation performance comparison between proposed and FL methods at the output voltage 150V with the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω

Control Board Inductors

Capacitors

MOSFETs

Figure 8 Experimental setup

in Figures 11 and 12 which demonstrates that the proposedmethod effectively enhances the output voltage regulationperformance through a reduction of the undershoot origi-nated from the abrupt load variation

From these simulation and experimental results it can beconcluded that the proposed method has the two practicalmerits

(1) The proposed method would suggest an almost sameclosed-loop performance for various operating rangeswithout any gain scheduling method

(2) The proposed method simplifies the control algo-rithms by getting rid of the use of the tracking errorintegrators with the antiwindup algorithms

6 Conclusions

This paper suggests a performance recovery output voltage-tracking controller for an unknown 119873-phase interleavedDCDC boost converter Taking the parameter and loadcurrent uncertainties into account the proposed control lawwas devised for the output voltage dynamics to convergeto the desired behavior coming from a LPF despite theparameter and load uncertainties Various simulation andexperimental evidences confirmed that the proposed tech-nique would provide a better closed-loop performance forseveral industrial applications

Appendix

This section provides the proofs of lemmas and theoremsTheproof of Lemma 1 is given as follows

Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

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6 Mathematical Problems in Engineering

Four-phaseinterleavedDCDCboost converter

AD converter

Outer-loopcontrollers

Phase shiftedcarrier signals

Comparators

PWMsignals

Inner-loopcontrollers

Implementation of the proposed control scheme Load

(times4)

(times4)

(times4)＞＝Ｌ

＞＝ ＞＝

iL1Ｌ iL4Ｌ

u1 u4

iL1 iL4 iL1 iL4

iL1 iL4u1 u4

lowast＞＝

＞＝

iＩ＞u1 u4

Figure 3 Implementation of the closed-loop system

IL

Vout Vout

IL

40

60

80

100

120

140

160

04 0602Time (s)

40

60

80

100

120

140

160

04 0602Time (s)

0

10

20

30

40

04 0602Time (s)

04 0602Time (s)

0

10

20

30

40

＞＝ (Classical Method)

iL (Classical Method)

＞＝ (Proposed Method)

iL (Proposed Method)

Figure 4 Output voltage-tracking performance comparison between proposed and FL methods with resistive load of 119877119871 = 20Ω

the simulation results depicted in Figure 4 the proposedmethod successfully maintains the closed-loop performanceto be the desired first-order LPF characteristic despite theparameter mismatches Figure 5 presents the state behaviorsof the DOBs

In the second stage the evaluation of the closed-looprobustness was carried out through investigating the outputvoltage-tracking performance variations under the sameoutput voltage reference with the different resistive loads119877119871 = 75 20 30Ω As can be seen from the results given

in Figure 6 the proposed method effectively reduces thevoltage-tracking performance changes for several differentloads This is a sharp contrast to the FL method

In this simulation setting the closed-loop tracking per-formance was quantitatively compared for the resistive loads119877119871 = 50 30 20 10Ω using the two cost functions defined as

119869int fl intinfin0

1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 119889119905119869max fl max 1003816100381610038161003816Vlowastdc (119905) minus Vdc (119905)1003816100381610038161003816 forall119905 ge 0 (35)

Mathematical Problems in Engineering 7

Table 1 Quantitative output voltage-tracking performance comparison for resistive loads 119877119871 = 50 30 20 10ΩLoad resistance value119877119871 = 50Ω 119877119871 = 30Ω 119877119871 = 20Ω 119877119871 = 10Ω

Proposed Method119869int 23281 7700 4558 1722119869max 10 7 5 4Classical Method119869int 83654 58191 16325 29917119869max 35 35 11 8

WL_v_est WL_a_est WL_b_est WL_c_est WL_d_est

04 0602Time (s)

minus100

minus50

0

w

wL4

wL1wL2wL3

Figure 5 Disturbance estimation behaviors

which implies that the best controller would guarantee thetwo cost functions of (35) to be minimized for all operatingpointsThe resulting table is given inTable 1 which shows thatthe proposed method offers a better tracking performancethan the FLmethod at least two times for these four operatingpoints

In the last stage the evaluation of the output voltageregulation performance was conducted by using the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω at the 150Voperation mode Figure 7 depicts the output voltage regu-lation comparison results with the corresponding inductorcurrent which implies that the proposed method effectivelyreduces the overunder shoots caused by the load variationcomparing the FL method

5 Experimental Results

This section experimentally verifies the performance of theproposed method by comparing it with the FL techniqueused in the previous section In this experiment a 3-kWfour-phase interleaved DCDC converter shown in Figure 1was utilized with the input DC source voltage level of Vin =50V and the digital signal processor (DSP) of TMS320F28335was used for implementing the proposed and FL controlalgorithms with the control and PWM periods of 119879119904 = 50 120583sThe experimental setup is shown in Figure 8

Except for design parameters of 120596vc 120596cc 120582V and 120582119871the control parameters of two controllers were left the sameas that in the simulation section These four parameters

Proposed Method

Classical Method

02 03 04 0501Time (s)

50

100

150

50

100

150

02 03 04 0501Time (s)

Vout_(DOB_tracking_50_to_150_to_120_at_75 ohm) Vout_(DOB_trac

＞＝(30Ω) ＞＝(20Ω)

＞＝(75Ω)

＞＝(75 20 30Ω)

Figure 6 Output voltage-tracking performance comparisonbetween proposed and FL methods at three resistive loads119877119871 = 75 20 30Ω

were selected as 120596vc = 314 rads (119891vc = 5Hz) 120596cc =2512 rads (119891cc = 400Hz) and 120582V = 120596vc 120582119871 = 120596ccIn the first stage the evaluation of the output voltage-

tracking performance was experimentally performed with anincreasing output voltage reference from 75V to 100V underthe resistive loads of 119877119871 = 66 20Ω From Figures 9 and 10as intended the proposed controller successfully forces theoutput voltage dynamics to be the desired trajectory comingfrom (6) for the different operation modes

The second stage examines the output voltage regulationperformance regarding two step load changes the resistiveload was changed from 20Ω to 10Ω and from 10Ω to 66Ωin a sequential manner The resulting waveforms are shown

8 Mathematical Problems in Engineering

ILIL_(DOB_regulation_150 V_from_15_to_75 ohm)

Vout Vout_(DOB_regulation_150 ohm)V_from_15_to_75

04 0602Time (s)

0

20

40

60

04 0602Time (s)

0

20

40

60

04 0602Time (s)

100

120

140

160

180

＞＝ (Proposed Method)

＞＝ (Classical Method)

iL (Classical Method)iL (Proposed Method)

Figure 7 Output voltage regulation performance comparison between proposed and FL methods at the output voltage 150V with the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω

Control Board Inductors

Capacitors

MOSFETs

Figure 8 Experimental setup

in Figures 11 and 12 which demonstrates that the proposedmethod effectively enhances the output voltage regulationperformance through a reduction of the undershoot origi-nated from the abrupt load variation

From these simulation and experimental results it can beconcluded that the proposed method has the two practicalmerits

(1) The proposed method would suggest an almost sameclosed-loop performance for various operating rangeswithout any gain scheduling method

(2) The proposed method simplifies the control algo-rithms by getting rid of the use of the tracking errorintegrators with the antiwindup algorithms

6 Conclusions

This paper suggests a performance recovery output voltage-tracking controller for an unknown 119873-phase interleavedDCDC boost converter Taking the parameter and loadcurrent uncertainties into account the proposed control lawwas devised for the output voltage dynamics to convergeto the desired behavior coming from a LPF despite theparameter and load uncertainties Various simulation andexperimental evidences confirmed that the proposed tech-nique would provide a better closed-loop performance forseveral industrial applications

Appendix

This section provides the proofs of lemmas and theoremsTheproof of Lemma 1 is given as follows

Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

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Mathematical Problems in Engineering 7

Table 1 Quantitative output voltage-tracking performance comparison for resistive loads 119877119871 = 50 30 20 10ΩLoad resistance value119877119871 = 50Ω 119877119871 = 30Ω 119877119871 = 20Ω 119877119871 = 10Ω

Proposed Method119869int 23281 7700 4558 1722119869max 10 7 5 4Classical Method119869int 83654 58191 16325 29917119869max 35 35 11 8

WL_v_est WL_a_est WL_b_est WL_c_est WL_d_est

04 0602Time (s)

minus100

minus50

0

w

wL4

wL1wL2wL3

Figure 5 Disturbance estimation behaviors

which implies that the best controller would guarantee thetwo cost functions of (35) to be minimized for all operatingpointsThe resulting table is given inTable 1 which shows thatthe proposed method offers a better tracking performancethan the FLmethod at least two times for these four operatingpoints

In the last stage the evaluation of the output voltageregulation performance was conducted by using the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω at the 150Voperation mode Figure 7 depicts the output voltage regu-lation comparison results with the corresponding inductorcurrent which implies that the proposed method effectivelyreduces the overunder shoots caused by the load variationcomparing the FL method

5 Experimental Results

This section experimentally verifies the performance of theproposed method by comparing it with the FL techniqueused in the previous section In this experiment a 3-kWfour-phase interleaved DCDC converter shown in Figure 1was utilized with the input DC source voltage level of Vin =50V and the digital signal processor (DSP) of TMS320F28335was used for implementing the proposed and FL controlalgorithms with the control and PWM periods of 119879119904 = 50 120583sThe experimental setup is shown in Figure 8

Except for design parameters of 120596vc 120596cc 120582V and 120582119871the control parameters of two controllers were left the sameas that in the simulation section These four parameters

Proposed Method

Classical Method

02 03 04 0501Time (s)

50

100

150

50

100

150

02 03 04 0501Time (s)

Vout_(DOB_tracking_50_to_150_to_120_at_75 ohm) Vout_(DOB_trac

＞＝(30Ω) ＞＝(20Ω)

＞＝(75Ω)

＞＝(75 20 30Ω)

Figure 6 Output voltage-tracking performance comparisonbetween proposed and FL methods at three resistive loads119877119871 = 75 20 30Ω

were selected as 120596vc = 314 rads (119891vc = 5Hz) 120596cc =2512 rads (119891cc = 400Hz) and 120582V = 120596vc 120582119871 = 120596ccIn the first stage the evaluation of the output voltage-

tracking performance was experimentally performed with anincreasing output voltage reference from 75V to 100V underthe resistive loads of 119877119871 = 66 20Ω From Figures 9 and 10as intended the proposed controller successfully forces theoutput voltage dynamics to be the desired trajectory comingfrom (6) for the different operation modes

The second stage examines the output voltage regulationperformance regarding two step load changes the resistiveload was changed from 20Ω to 10Ω and from 10Ω to 66Ωin a sequential manner The resulting waveforms are shown

8 Mathematical Problems in Engineering

ILIL_(DOB_regulation_150 V_from_15_to_75 ohm)

Vout Vout_(DOB_regulation_150 ohm)V_from_15_to_75

04 0602Time (s)

0

20

40

60

04 0602Time (s)

0

20

40

60

04 0602Time (s)

100

120

140

160

180

＞＝ (Proposed Method)

＞＝ (Classical Method)

iL (Classical Method)iL (Proposed Method)

Figure 7 Output voltage regulation performance comparison between proposed and FL methods at the output voltage 150V with the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω

Control Board Inductors

Capacitors

MOSFETs

Figure 8 Experimental setup

in Figures 11 and 12 which demonstrates that the proposedmethod effectively enhances the output voltage regulationperformance through a reduction of the undershoot origi-nated from the abrupt load variation

From these simulation and experimental results it can beconcluded that the proposed method has the two practicalmerits

(1) The proposed method would suggest an almost sameclosed-loop performance for various operating rangeswithout any gain scheduling method

(2) The proposed method simplifies the control algo-rithms by getting rid of the use of the tracking errorintegrators with the antiwindup algorithms

6 Conclusions

This paper suggests a performance recovery output voltage-tracking controller for an unknown 119873-phase interleavedDCDC boost converter Taking the parameter and loadcurrent uncertainties into account the proposed control lawwas devised for the output voltage dynamics to convergeto the desired behavior coming from a LPF despite theparameter and load uncertainties Various simulation andexperimental evidences confirmed that the proposed tech-nique would provide a better closed-loop performance forseveral industrial applications

Appendix

This section provides the proofs of lemmas and theoremsTheproof of Lemma 1 is given as follows

Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

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Submit your manuscripts atwwwhindawicom

8 Mathematical Problems in Engineering

ILIL_(DOB_regulation_150 V_from_15_to_75 ohm)

Vout Vout_(DOB_regulation_150 ohm)V_from_15_to_75

04 0602Time (s)

0

20

40

60

04 0602Time (s)

0

20

40

60

04 0602Time (s)

100

120

140

160

180

＞＝ (Proposed Method)

＞＝ (Classical Method)

iL (Classical Method)iL (Proposed Method)

Figure 7 Output voltage regulation performance comparison between proposed and FL methods at the output voltage 150V with the pulseresistive load from 119877119871 = 15Ω to 119877119871 = 75Ω

Control Board Inductors

Capacitors

MOSFETs

Figure 8 Experimental setup

in Figures 11 and 12 which demonstrates that the proposedmethod effectively enhances the output voltage regulationperformance through a reduction of the undershoot origi-nated from the abrupt load variation

From these simulation and experimental results it can beconcluded that the proposed method has the two practicalmerits

(1) The proposed method would suggest an almost sameclosed-loop performance for various operating rangeswithout any gain scheduling method

(2) The proposed method simplifies the control algo-rithms by getting rid of the use of the tracking errorintegrators with the antiwindup algorithms

6 Conclusions

This paper suggests a performance recovery output voltage-tracking controller for an unknown 119873-phase interleavedDCDC boost converter Taking the parameter and loadcurrent uncertainties into account the proposed control lawwas devised for the output voltage dynamics to convergeto the desired behavior coming from a LPF despite theparameter and load uncertainties Various simulation andexperimental evidences confirmed that the proposed tech-nique would provide a better closed-loop performance forseveral industrial applications

Appendix

This section provides the proofs of lemmas and theoremsTheproof of Lemma 1 is given as follows

Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

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Mathematical Problems in Engineering 9

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)

＞＝ (20 ６＞ＣＰ)

＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

Figure 9 Experimental results of output voltage-tracking perfor-mance comparison between proposedmethod and FLmethodswithresistive load 119877119871 = 66Ω

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

Time (2 Ｍ＞ＣＰ)

Time (2 Ｍ＞ＣＰ)＞＝ = 50６

＞＝ = 50６

＞＝ = 75６

＞＝ = 75６

＞＝ = 100 ６

＞＝ = 100 ６

＞＝ (20 ６＞ＣＰ)

Figure 10 Experimental results of output voltage-tracking perfor-mance comparison between proposed andFLmethodswith resistiveload 119877119871 = 20Ω

Proof The combination of the nonlinear observer of (10) andthe output of (9) leads to

119908V minus 119897V1198620 Vlowastdc = minus119897V (119908V minus 119897V1198620Vlowastdc) minus 1198972V1198620Vlowastdc+ 119897V 119873sum119894=1

(1 minus 119906119894) 119894119871119894 forall119905 ge 0 (A1)

which gives that (see the output voltage error dynamics of (7))

119908V = 119897V (1198620 Vlowastdc + 119873sum119894=1

(1 minus 119906119894) 119894119871119894 minus 119908V)= 119897V (119908V minus 119908V) forall119905 ge 0

(A2)

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 21 iL1 = 24

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

iL1 = 21 iL1 = 24

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

C4

C4

RL = 20Ω rarr = 10ΩRL

ＩＯＮ = 100 ６( = 10Ω)RL

RL = 20Ω rarr = 10ΩRLＩＯＮ = 100 ６( = 10Ω)RLＩＯＮ = 100 ６( = 20Ω)RL

ＩＯＮ = 100 ６( = 20Ω)RL

Figure 11 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 20Ω to 119877119871 = 10Ω

C4

C4

⟨０ＬＩＪＩＭ＞ －ＮＢＩ＞⟩

⟨ＦＭＭＣ＝Ｆ －ＮＢＩ＞⟩

iL1 = 24 iL1 = 27

iL1 = 24 iL1 = 27

Time (200 ＧＭ＞ＣＰ)

Time (200 ＧＭ＞ＣＰ)

300ＧＭ

500ＧＭ

RL = 10Ω rarr = 66 Ω

RL = 10Ω rarr = 66 Ω

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

ＩＯＮ = 100 ６( = 66 Ω)ＩＯＮ = 100 ６( = 10Ω)

iL1 (50 ＞ＣＰ)

ＩＯＮ (10 ６＞ＣＰ)

RL

RLRL

RL

RL

RL

Figure 12 Experimental results of output voltage regulation perfor-mance comparison between proposed and FLmethods regarding toresistive load variation from 119877119871 = 10Ω to 119877119871 = 66Ω

The assumption of (12) results in the steady state DOBdynamics as (see (A2))

0 = 119897V (1199080V minus 1199080V) (A3)

which means that

1199080V = 1199080V (A4)

Equation (A4) implies that

119908V = minus119897V (119908V minus 1199080V) (A5)

Δ119908V = 119908V minus 1199080V forall119905 ge 0 (A6)

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Mathematical Problems in Engineering

where119908V fl 1199080Vminus119908Vforall119905 ge 0 Together with (A6) it is possibleto rewrite the tracking error dynamics of (11) as

1198620 Vlowastdc = minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894forall119905 ge 0

(A7)

Now consider the positive definite function of (15)

119881V = 11986202 (Vlowastdc)2 + 121199082V + 120574V2 (1199080V)2 forall119905 ge 0 (A8)

with 120574V denoting positive constant to be found laterThen thetime-derivative of V can be obtained by using the closed-looptrajectories of (12) (A5) and (A7)

V = Vlowastdc(minus1198620120582VVlowast

dc + 119908V minus 1199080V + 119873sum119894=1

(1 minus 119906119894) 119871119894)minus 119897V1199082V + 119897V119908V1199080V minus 120574V119896V (1199080V)2

le minus1198620120582V3 (Vlowastdc)2 minus (119897V minus 341198620120582Vminus 1)1199082V

minus (120574V119896V minus 341198620120582Vminus 1198972V4 ) (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0

(A9)

where the inequality is confirmed by Youngrsquos inequality

119909119910 le 12059821199092 + 121205981199102 forall119909 119910 isin R forall120598 gt 0 (A10)

It is easy to verify that the DOB gains of (13) with the positiveconstant of 120574V given as 120574V = (1119896V)(341198620120582V + 1198972V4 + 12)renders the inequality of (A9) as

V le minus1198620120582V3 (Vlowastdc)2 minus 120578V1199082V minus 12 (1199080V)2

+ Vlowastdc119873sum119894=1

(1 minus 119906119894) 119871119894le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 forall119905 ge 0(A11)

with 120572V fl min2120582V3 2120578V 1120574V which confirms the strictpassivity regarding to the input-output mapping of (14)

The proof of Lemma 2 is given as follows

Proof The combination of the nonlinear observer of (20) andthe output of (19) leads to

119908119871119894 minus 1198971198711198941198710 119894119871119894 = minus119897119871119894 (119908119871119894 minus 1198971198711198941198710 119871119894) minus 11989721198711198941198710 119871119894+ 119897119871119894 (minus (1 minus 119906119894) Vdc + Vin0)

forall119905 ge 0(A12)

which gives that (see the output voltage error dynamics of(17))

119908119871119894 = 119897119871119894 (1198710 119894119871119894 minus (1 minus 119906119894) Vdc + Vin0 minus 119908119871119894)= 119897119871119894 (119908119871119894 minus 119908119871119894) 119894 = 1 119873 forall119905 ge 0 (A13)

The assumption of (22) results in the steady state DOBdynamics as (see (A13)) 0 = 119897119871119894(1199080119871119894 minus 1199080119871119894) 119894 = 1 119873which means that

1199080119871119894 = 1199080119871119894 119894 = 1 119873 (A14)

The equation of (A14) implies that

119908119871119894 = minus119897119871119894 (119908119871119894 minus 1199080119871119894) (A15)

Δ119908119871119894 = 119908119871119894 minus 1199080119871119894 119894 = 1 119873 forall119905 ge 0 (A16)

where 119908119871119894 fl 1199080119871119894 minus 119908119871119894 forall119905 ge 0 Together with (A16) it ispossible to rewrite the error dynamics of (21) as

1198710 119894119871119894 = minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc119894 = 1 119873 forall119905 ge 0 (A17)

Now consider the positive definite function in (25)

119881119871 = 11987102119873sum119894=1

2119871119894 + 119873sum119894=1

121199082119871119894 +119873sum119894=1

1205741198711198942 (1199080119871119894)2 forall119905 ge 0

(A18)

with 120574119871119894 119894 = 1 119873 being positive constants to be foundlater The time-derivative of 119871 can be obtained by using theclosed-loop trajectories of (22) (A15) and (A17)

119871 = 119873sum119894=1

119871119894 (minus1198710120582119871119871119894 + 119908119871119894 minus 1199080119871119894 minus (1 minus 119906119894) Vlowastdc)

minus 119873sum119894=1

1198971198711198941199082119871119894 + 119873sum119894=1

1198971198711198941199081198711198941199080119871119894minus 119873sum119894=1

120574119871119894119896119871119894 (1199080119871119894)2

le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

(119897119871119894 minus 341198710120582119871 minus 1)1199082119871119894minus 119873sum119894=1

(120574119871119894119896119871119894 minus 341198710120582119871 minus11989721198711198944 ) (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A19)

where the inequality is confirmed by Youngrsquos inequality of(A10) It is easy to verify that the DOB gains of (23) with the

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

positive constants of 120574119871119894 given as 120574119871119894 = (1119896119871119894)(341198710120582119871 +11989721198711198944 + 12) 119894 = 1 119873 render the inequality of (A19) as

119871 le minus11987101205821198713119873sum119894=1

2119871119894 minus 119873sum119894=1

1205781198711198941199082119871119894 minus 119873sum119894=1

12 (1199080119871119894)2

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdcle minus120572119871119881 minus 119873sum

119894=1

119871119894 (1 minus 119906119894) Vlowastdc forall119905 ge 0

(A20)

with 120572119871 fl min21205821198713 21205781198711 2120578119871119873 1120574119871119894 1120574119871119873which confirms the strict passivity of (24)

The proof of Theorem 3 is given as follows

Proof The composite-type positive definite function definedas

119881cl fl 119881V + 119881119871 forall119905 ge 0 (A21)

with two positive defined functions given in (15) and (25)gives its time-derivative of cl using the closed-loop trajec-tories of (A7) and (A17) as

cl = V + 119871le minus120572V119881V + Vlowastdc

119873sum119894=1

(1 minus 119906119894) 119871119894 minus 120572119871119881

minus 119873sum119894=1

119871119894 (1 minus 119906119894) Vlowastdc le minus120572cl119881cl forall119905 ge 0(A22)

where the two inequalities of (16) and (26) give the firstinequality and the positive constant 120572cl is defined as 120572cl flmin120572V 120572119871 which confirms the inequality of (28) by theComparison principle in [20]

The proof of Theorem 4 is given as follows

Proof As can be seen in the proof ofTheorem 3 in Appendixthe inductor current and output voltage DOB dynamics canbe equivalently written as 119908V = 119897VΔ119908V 119908119871119894 = 119897119871119894Δ119908119871119894 119894 =1 119873 forall119905 ge 0 which yields the steady-state equations

Δ1199080V = 0Δ1199080119871119894 = 0

forall119897V 119897119871119894 gt 0 119894 = 1 119873(A23)

where Δ1199080V Δ1199080119871119894 denote to the steady-states of Δ119908V(119905)Δ119908119871119894(119905) 119894 = 1 119873 respectively The combination ofthe closed-loop error dynamics of (11) and (21) and thetarget output voltage dynamics of (6) gives the steady-stateequations

0 = (J minus R) x0 = minus120596vc ((Vlowastdc)0 minus V0dcref) (A24)

where

J =[[[[[[[[

0 (1 minus 11990601) sdot sdot sdot (1 minus 1199060119873)minus (1 minus 11990601) 0 sdot sdot sdot 0

d

minus (1 minus 1199060119873) 0 sdot sdot sdot 0

]]]]]]]]

R = diag 1198620120582V 1198710120582119871 1198710120582119871 x = [(Vlowastdc)0 01198711 sdot sdot sdot 0119871119873]119879

(A25)

Note that the matrix (J minus R) on the right-side of (A24) isalways nonsingular since it is negative definiteThat is x119879(JminusR)x = x119879Jx minus x119879Rx = minusx119879J119879x minus x119879Rx = minusx119879Rx lt 0 forallx = 0Equation (A24) indicates that x = 0 which shows that thefollowing chain implications hold

(Vlowastdc)0 = 00119871119894 = 0 119894 = 1 119873

dArrV0dc = (Vlowastdc)0 = V0dcref

(A26)

Therefore the verification of the result of (29) is completed

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by a grant from Transportationamp Logistics Research Program (TLRP) (17TLRP-C135446-01Development of Hybrid Electric Vehicle Conversion Kit forDiesel Delivery Trucks and its Commercialization for ParcelServices) funded by Ministry of Land Infrastructure andTransport of Korean Government

References

[1] J Moreno-Valenzuela and O Garcıa-Alarcon ldquoOn control ofa boost DC-DC power converter under constrained inputrdquoComplexity vol 2017 Article ID 4143901 2017

[2] J Huang D Xu W Yan L Ge and X Yuan ldquoNonlinearControl of Back-to-Back VSC-HVDC System via Command-Filter Backsteppingrdquo Journal of Control Science and Engineeringvol 2017 Article ID 7410392 2017

[3] N Yang C Wu R Jia and C Liu ldquoFractional-order terminalsliding-mode control for Buck DCDC converterrdquo Mathemati-cal Problems in Engineering vol 2016 pp 1ndash7 2016

[4] N N Yang C J Wu R Jia and C Liu ldquoModeling andcharacteristics analysis for a buck-boost converter in pseudo-continuous conduction mode based on fractional calculusrdquoMathematical Problems in Engineering vol 2016 Article ID6835910 11 pages 2016

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

[5] S K Pidaparthy and B Choi ldquoControl Design and Loop GainAnalysis of DC-to-DC Converters Intended for General LoadSubsystemsrdquo Mathematical Problems in Engineering vol 2015Article ID 426315 2015

[6] B Salhi H El Fadil T Ahmed Ali E Magarotto and F GirildquoAdaptive output feedback control of interleaved parallel boostconverters associated with fuel cellrdquo Electric Power Componentsand Systems vol 43 no 8-10 pp 1141ndash1158 2015

[7] R W Erickson and D Maksimovic Fundamentals of PowerElectronics Springer Berlin Germany 2001

[8] A G Perry G Feng Y-F Liu and P C Sen ldquoA design methodfor PI-like fuzzy logic controllers for DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 54 no 5 pp 2688ndash2696 2007

[9] S Bibian and H Jin ldquoHigh performance predictive dead-beatdigital controller for dc power suppliesrdquo IEEE Transactions onPower Electronics vol 17 no 3 pp 420ndash427 2002

[10] Q Zhang R Min Q Tong X Zou Z Liu and A Shen ldquoSen-sorless predictive current controlled DC-DC converter with aself-correction differential current observerrdquo IEEE Transactionson Industrial Electronics vol 61 no 12 pp 6747ndash6757 2014

[11] M Salimi J Soltani A Zakipour and V Hajbani ldquoSlidingmode control of theDC-DCflyback converter with zero steady-state errorrdquo in Proceedings of the 2013 4th Power ElectronicsDrive Systems amp Technologies Conference (PEDSTC) pp 158ndash163 Tehran Iran Feburary 2013

[12] J Linares-Flores A Hernandez Mendez C Garcia-Rodriguezand H Sira-Ramirez ldquoRobust nonlinear adaptive control of arsquoboostrsquo converter via algebraic parameter identificationrdquo IEEETransactions on Industrial Electronics vol 61 no 8 pp 4105ndash4114 2014

[13] A G Beccuti S Mariethoz S Cliquennois S Wang and MMorari ldquoExplicit model predictive control of DC-DC switched-mode power supplies with extended kalman filteringrdquo IEEETransactions on Industrial Electronics vol 56 no 6 pp 1864ndash1874 2009

[14] Y-X Wang D-H Yu and Y-B Kim ldquoRobust time-delaycontrol for the DC-DC boost converterrdquo IEEE Transactions onIndustrial Electronics vol 61 no 9 pp 4829ndash4837 2014

[15] W-H Chen D J Ballance P J Gawthrop and J OrsquoReilly ldquoAnonlinear disturbance observer for robotic manipulatorsrdquo IEEETransactions on Industrial Electronics vol 47 no 4 pp 932ndash9382000

[16] Y I Son I H Kim D S Choi and H Shim ldquoRobust cascadecontrol of electric motor drives using dual reduced-order PIobserverrdquo IEEE Transactions on Industrial Electronics vol 62no 6 pp 3672ndash3682 2015

[17] S Maniktala Switching Power Supplies A-Z Newnes 2012[18] A I Pressman ldquoSwitching Power Supply Designrdquo in 1em plus

05em minus 04em McGraw-Hill New York NY USA 1998[19] M H Rashid Power Electronics Pearson Prentice Hall 3rd

edition 2004[20] H K Khalil ldquoNonlinear Systemsrdquo in Nonlinear Systems Pren-

tice Hall 2002

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom