Research Article LPV Control with Pole Placement

Research ArticleH

infinLPV Control with Pole Placement Constraints for

Synchronous Buck Converters with Piecewise-Constant Loads

Hwanyub Joo and Sung Hyun Kim

School of Electrical Engineering University of Ulsan (UOU) Ulsan 680-749 Republic of Korea

Correspondence should be addressed to Sung Hyun Kim shnkimulsanackr

Received 5 June 2014 Revised 12 August 2014 Accepted 26 August 2014

Academic Editor Guido Maione

Copyright copy 2015 H Joo and S H KimThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper addresses the output regulation problem of synchronous buck converters with piecewise-constant load fluctuations vialinear parameter varying (LPV) control scheme To this end an output-error state-space model is first derived in the form of LPVsystems so that it can involve amismatch error that temporally arises from the process of generating a feedforward controlThen toattenuate the mismatch error in parallel with improving the transient behavior of the converter this paper proposes an LMI-basedstabilization condition capable of achieving both H

infinand pole-placement objectives Finally the simulation and experimental

results are provided to show the validity of our approach

1 Introduction

Drawing on the development of electronic technologyswitching DC-DC converters have been widely and success-fully applied to a variety of power conversion systems such asDC power supplies DCmotor drivers and power generationsystems (see [1ndash4] and the references therein) Recently withthe growing interest in linear matrix inequalities (LMIs) [5]some advanced control techniques have been investigatedregarding to output regulation of DC-DC buck (step-down)converters that produce a lower output voltage than the inputvoltage especially for T-S fuzzy control [2 6ndash10] Indeedthe asynchronous buck converters operating in a large-signal domain are generally modeled in terms of nonlinearsystemsThus based on the T-S fuzzymodel derived from theaveraging method for one-time-scale discontinuous systems(AM-OTS-DS) [9] has successfully designed an integral T-Sfuzzy control with respect to the output regulation problemofthe asynchronous buck converter Meanwhile in the case ofsynchronous buck converters [11ndash13] the use of the low-sideFET plays an important role in eliminating the voltage dropacross the power diode of the nonsynchronous converterwhich allows buck converters to bemodeledwith linear time-varying systems

In general the operation of the DC-DC converter isusually affected by the fluctuation of output loads [3 912 13] For this reason it is of great importance to con-sider the presence of a wide load range in the problem ofregulating the output voltage and current levels of DC-DCconverters that is it has become a hot topic to maintainhigh efficiency in a great load fluctuation Moreover due tothe fact that conventional pulse-width modulation (PWM)buck converters have poor efficiency under light load [14]numerous research efforts have been invested to improvethe efficiency of the PWM converters with a wide loadrange (see [12 15ndash17] and the references therein) Howevera remarkable point is that most of references cited abovehave paid considerable attention at the hardware level tocover such problem Further [13] used a reduced systemmodel with the limits in theoretically capturing the dynamicbehavior of piecewise-constant load fluctuation in the processof implementing the robust periodic eigenvalue assignmentalgorithm [18] In other words limited work has been foundin terms of the control theory Motivated by the concern thispaper proposes a suitable approach in light of the controltheory to take the effect of load fluctuations into account

This paper addresses the output regulation problem ofsynchronous buck converters with piecewise-constant load

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 686857 8 pageshttpdxdoiorg1011552015686857

2 Mathematical Problems in Engineering

fluctuations To consider the presence of such load fluctu-ations we derive an output-error state-space model in theform of linear parameter varying (LPV) systems [19ndash21]thereby converting the underlying regulation problem intothe stabilization problem Here it is worth noticing thata mismatch error that temporally arises from the processof generating a feedforward control is clearly incorporatedinto the LPV model and it is attenuated by the H

infin-

synthesis technique [22 23] However Hinfin

design provideslittle control over the transient behavior [24 25] Hence toattenuate the mismatch error in parallel with improving thetransient behavior of the converter this paper proposes anLMI-based stabilization condition capable of achieving bothHinfin

and pole-placement objectives Finally the simulationand experimental results are provided to show the validity ofour approach

Notation The notations 119883 ge 119884 and 119883 gt 119884 mean that 119883 minus 119884

is positive semidefinite and positive definite respectively Insymmetric block matrices (lowast) is used as an ellipsis for termsinduced by symmetry For any square matrix Q He[Q] =Q + Q119879 Lebesgue spaceL

2+=L2[0infin) consists of square-

integrable functions on [0infin)

2 Modeling for DC to DC PWMBuck Converter

Theequivalent circuit for a class of synchronousDC-DCbuckconverters and the corresponding closed-loop control systemare depicted in Figure 1 where the following notations areused

(i) 119877DS denotes the static drain to source resistances ofthe high-side and low-side power MOSFETs respec-tively

(ii) V119868(119905) and V

119874(119905) denote the power input and output

voltages respectively where it is assumed that V119868(119905) =

V119868is time-invariant

(iii) 119894119871(119905) and V

119862(119905) denote the inductor current and the

capacitor voltage respectively

(iv) 119871 and 119862 denote the inductance and capacitanceselected by the given design specifications includingthe switching frequency of MOSFETs

(v) 119877DCR and 119877ESR denote the equivalent series resis-tances of the inductor and capacitor

(vi) 119877(119905) and 120591(119905) denote the piecewise-constant loadresistance subject to 119877minus le 119877(119905) le 119877

+ and the dutyratio of PWM buck converter

Then based on averaging method for one-time-scale discon-tinuous system (AM-OTS-DS) [26] the mathematical model

of the buck converter under consideration is described asfollows

119894119871(119905) = minus

1

119871

V119874(119905) minus

119877DS + 119877DCR119871

119894119871(119905) +

1

119871

V119868120591 (119905) (1)

V119862(119905) =

1

119862

119894119871(119905) minus

1

119877 (119905) 119862

V119874(119905) (2)

V119862(119905) =

1

119877ESR119862(V119874(119905) minus V

119862(119905)) (3)

Further combining (2) and (3) yields

V119874(119905) =

119877 (119905)

119877 (119905) + 119877ESR(119877ESR119894119871 (119905) + V

119862(119905)) (4)

by which (1) and (2) can be rewritten as follows

119894119871(119905) = minus

1

119871

(

119877 (119905) 119877ESR119877 (119905) + 119877ESR

+ 119877DS + 119877DCR) 119894119871 (119905)

minus

1

119871

119877 (119905)

119877 (119905) + 119877ESRV119862(119905) +

V119868

119871

120591 (119905)

(5)

V119862(119905) =

1

119862

119877 (119905)

119877 (119905) + 119877ESR119894119871(119905)

minus

1

119862

1

119877 (119905) + 119877ESRV119862(119905)

(6)

Let V119862be a unique equilibrium point of V

119862(119905) and assume that

the value of 119877(119905) is piecewise-constant that is 119877(119905) = 119877119895for

119905 isin T119895(see Figure 1) Then by (6) the equilibrium point of

119894119871(119905) that is 119894

119871 is given by

0 = 119877119895119894119871minus V119862 for 119905 isin T

119895 (7)

Further by (3) the equilibrium point of V119874(119905) is given by V

119874=

V119862 Hence from (1) the equilibrium point of 120591(119905) that is 120591 is

given by

0 = V119862+ (119877DS + 119877DCR) 119894119871 minus V

119868120591 for 119905 isin T

119895 (8)

Remark 1 This paper focuses on addressing the case wherethe variation of 119877(119905) is subject to a class of switching signalswhose finite intervalsT

119895are larger than the setting time

Remark 2 Indeed it is extremely hard to directly measurethe value of V

119862(119905) in the considered buck converter owing to

the existence of 119877ESR (see Figure 1) Thus to find the value ofV119862(119905) from the ones of the measured V

119874(119905) 119894119874(119905) and 119894

119871(119905) we

introduce a method of using the following equality derivedfrom (4)

V119862(119905) = V

119874(119905) + 119877ESR (119894119874 (119905) minus 119894119871 (119905)) (9)

Remark 3 From (9) we can see that 119894119874= 119894119871in the equilibri-

um point because V119862= V119874

Mathematical Problems in Engineering 3

+minus +

+

minus

minus

ControllerPWMgenerator

Gatedriver

120591(t)

High-sideMOSFET

Low-sideMOSFET

i

RDS

RDS RESR

C(t) C

R(t) = Rjminus1

R(t) = Rj

R(t) = Rj+1

iL(t)

Tjminus1 Tj Tj+1

R(t)

O(t)

O(t)

iO(t)

UGATE LGATE

RDCR L

Figure 1 Equivalent circuit for a class of synchronous DC-DC buck converters and piecewise-constant load fluctuations

Remark 4 To solve the regulation problem herein we needto find the desired value of 119894

119871= V119862119877119895from (7) where V

119862is a

prescribed value and 119877119895can be established as follows

119877119895=

119877minus

ifV119874(119905)

119894119874(119905)

lt 119877minus

119877+

ifV119874(119905)

119894119874(119905)

gt 119877+

V119874(119905)

119894119874(119905)

otherwise

(10)

3 LPV Control with PolePlacement Constraints

31 LPV Model Description Let 119871(119905) = 119894

119871(119905) minus 119894

119871 V119862(119905) =

V119862(119905)minusV

119862 and 120591(119905) = 119906(119905)+120595(119905) where 119906(119905) and120595(119905) indicate

the feedback and feedforward control inputs respectivelyThen for 119905 isin T

119895 the system given in (5) and (6) can be

converted into

119871(119905) = minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119871 (119905)

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862(119905) +

V119868

119871

119906 (119905)

minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119894119871

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862+

V119868

119871

120595 (119905)

V119862(119905) =

1

119862

119877119895

119877119895+ 119877ESR

119871(119905) minus

1

119862

1

119877119895+ 119877ESR

V119862(119905)

+

1

119862

1

119877119895+ 119877ESR

(119877119895119894119871minus V119862)

(11)

Proposition 5 Let us consider the feedforward control input120595(119905) of the following form

120595 (119905) =

1

V119868

((

119877119895119877119864119878119877

119877119895+ 119877119864119878119877

+ 119877119863119878+ 119877119863119862119877

) 119894119871

+

119877119895

119877119895+ 119877119864119878119877

V119862) + 119908 (119905)

=

1

V119868

(V119862+ (119877119863119878+ R119863119862119877

) 119894119871) + 119908 (119905)

= 120591 + 119908 (119905)

(12)

where 119908(119905) isin L2+

denotes the error that can temporallyoccur when finding 119877

119895and 119894119871119895

under the saturation operator(10) Here a remarkable point is that if 119908(119905) = 0 then thefeedforward control input 120595(119905) = 120591 for 119905 isin T

119895

Under (12) the system given in (11) becomes

119871(119905) = minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119871 (119905)

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862(119905) +

V119868

119871

119906 (119905) +

V119868

119871

119908 (119905)

V119862(119905) =

1

119862

119877119895

119877119895+ 119877ESR

119871(119905)

minus

1

119862

1

119877119895+ 119877ESR

V119862(119905) for 119905 isin T

119895

(13)

In what follows let us define

1198911=

119877119895

119877119895+ 119877ESR

1198912=

1

119877119895+ 119877ESR

forall119895 (14)

Then by letting 119909(119905) = col(119871(119905) V119862(119905)) we can rewrite (13) as

follows

(119905) = 119860 (1198911 1198912) 119909 (119905) + 119861

119906119906 (119905) + 119861

119908119908 (119905) (15)


where

119860 (1198911 1198912) =

[

[

[

[

minus

1

119871

(119877ESR1198911 + 119877DS + 119877DCR) minus

1

119871

1198911

1

119862

1198911

minus

1

119862

1198912

]

]

]

]

119861119906= 119861119908=[

[

V119868

119871

0

]

]

(16)

Here note that from the fact that

0 lt 11989110=

119877minus

119877minus+ 119877ESR

le 1198911le

119877+

119877++ 119877ESR

= 11989111

0 lt 11989120=

1

119877++ 119877ESR

le 1198912le

1

119877minus+ 119877ESR

= 11989121

(17)

(14) can be represented by 1198911= sum1

119894=012058811198941198911119894and 1198912= sum1

119894=01205882119894

1198912119894 where

12058810=

11989111minus 1198911

11989111minus 11989110

12058811=

1198911minus 11989110

11989111minus 11989110

12058820=

11989121minus 1198912

11989121minus 11989120

12058821=

1198912minus 11989120

11989121minus 11989120

(18)

Therefore we can derive the linear parameter varying (LPV)form of 119860(119891

1 1198912) as follows

119860 (1198911 1198912) =

4

sum

119901=1

120590119901119860119901≜ 119860 (120590)

120590119901= 1205881[119901]1

sdot 1205882[119901]2

119860119901=

[

[

[

[

minus

1

119871

(119877ESR1198911[119901]1

+ 119877DS + 119877DCR) minus

1

119871

1198911[119901]1

1

119862

1198911[119901]1

minus

1

119862

1198912[119901]2

]

]

]

]

(19)

where [119901]1and [119901]

2are decided from 119901 = 2[119901]

2+ [119901]1+

1 and [119901]119896isin 0 1 Here it should be pointed out that

the piecewise-constant parameters 120590119901satisfy the following

properties

4

sum

119901=1

120590119901=

1

sum

[119901]1=0

1

sum

[119901]2=0

1205881[119901]1

sdot 1205882[119901]2

= (

1

sum

[119901]1=0

1205881[119901]1

) sdot (

1

sum

[119901]2=0

1205882[119901]2

) = 1

0 le 120590119901le 1 forall119901

(20)

32 Control Design Now consider a state-feedback controllaw of the following form

119906 (119905) = 119865 (120590) 119909 (119905) (21)

where119865(120590) = sum4119901=1

120590119901119865119901Then the closed-loop systemunder

(15) and (21) is given by

(119905) = (119860 (120590) + 119861119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905) (22)

119911 (119905) = 119862119911119909 (119905) (23)

where 119911(119905) denotes the performance output in theHinfinsense

In order to address theD-stability problem we considerthe following theorem reported in [24] whichwill be used forthe design of LPV control with pole placement constraints

Theorem 6 (see [24]) The system matrix 119860 is D-stable (ieall the poles of119860 lie inD) if and only if there exists a symmetricmatrix119883 such that119872D(119860119883) lt 0119883 gt 0 where119872D(119860119883) isrelated with the following characteristic function 119891D(119911) on thebasis of the substitution (119883 119860119883119883119860119879) 999445999468 (1 119911 119911)

(i) left half-plane such that 119877(119911) lt minus120572 D = 119911 isin

C | 119891D(119911) = 119911 + 119911 + 2120572 lt 0

(ii) disk with center at (minus119902 0) and radius 119903

D = 119911 isin C | 119891D (119911) = [minus119903 119902 + 119911

119902 + 119911 minus119903] lt 0 (24)

(iii) conic sector centered at the origin and with inner angle2120579

D = 119911 isin C | 119891D (119911) = [sin 120579 (119911 + 119911) cos 120579 (119911 minus 119911)cos 120579 (119911 minus 119911) sin 120579 (119911 + 119911)] lt 0

(25)

The following theorem presents a set of LMIs for D-stability criterion for (22)

Theorem7 Let the required LMI regionD be given in terms of120572 119903 120579 and 119902 = 0 Then (22) isD-stable if there exist matrices119865119901119901=1234

isin R1times2 and symmetric matrix 0 lt 119883 isin R2times2 suchthat for all 119901 isin 1 2 3 4

0 gt He [119860119901119883 + 119861

119906119865119901+ 120572119883] (26)

0 gt [

minus119903119883 119860119901119883 + 119861

119906119865119901

(lowast) minus119903119883

] (27)

0 gt

[

[

[

[

[

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

cos 120579 sdot (119860119901119883 + 119861

119906119865119901minus 119883119860

119879

119901minus 119865

119879

119901119861119879

119906)

(lowast)

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

]

]

]

]

]

(28)

0 gt[

[

minus119868 119862119911119883 0

(lowast) He [119860119901119883 + 119861

119906119865119901] 119861119908

(lowast) (lowast) minus1205742

119868

]

]

(29)

Moreover the control gains 119865119901can be reconstructed by 119865

119901=

119865119901119883minus1


Proof From Theorem 6 the pole placement constraints aresatisfied if and only if there exists119883 gt 0 such that

119872119863(119860 (120590) + 119861

119906119865 (120590) 119883) lt 0 (30)

Accordingly theD-stability conditions of (22) are given by

0 gt He [Φ (120590) + 120572119883]

0 gt [

minus119903119883 Φ (120590)

(lowast) minus119903119883]

0 gt [sin 120579 sdotHe [Φ (120590)] cos 120579 sdot (Φ (120590) minus Φ(120590)119879)

(lowast) sin 120579 sdot (He [Φ (120590)]) ]

(31)

where Φ(120590) = 119860(120590)119883 + 119861119906119865(120590) and 119865(120590) = 119865(120590)119883 =

sum4

119901=1120590119901119865119901119883 By defining 119865

119901= 119865119901119883 we can express that

Φ(120590) = sum4

119901=1120590119901(119860119901119883 + 119861

119906119865119901) which leads to Φ(120590) isin

Co(119860119901119883 + 119861

119906119865119901) by (20) where Co(sdot) indicates the convex

hull Hence the D-stability conditions given in (31) can beassured by (26)ndash(28) respectively

Next we discuss the Hinfin

performance such thatsup119908(119911(119905)

2119908(119905)

2) lt 120574 for all 0 = 119908(119905) isin L

2+ where

120574 represents the disturbance attenuation capability As is wellknown theH

infinstability criterion [19] can be derived from

0 gt 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905) + (119905) (32)

Here let 119881(119905) = 119909119879

(119905)119883minus1

119909(119905) Then (32) can be naturallyrewritten as follows0 gt He [119909119879 (119905) 119883minus1 ((119860 (120590) + 119861

119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905))]

+ 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

(33)

which leads to

0 gt [

He [119883minus1 (119860 (120590) + 119861119906119865 (120590))] + 119862

119879

119911119862119911119883minus1

119861119908


119868

] (34)

As a result pre- and postmultiplying (34) by diag(119883 119868) andits transpose yield

0 gt [

He [Φ (120590)] + 119883119862119879119911119862119911119883 119861

119908


119868

] (35)

which can be converted by the Schurrsquos complement [5] into

0 gt[

[

minus119868 119862119911119883 0

(lowast) He [Φ (120590)] 119861119908


119868

]

]

(36)

Therefore condition (36) is guaranteed by (29) in light ofΦ(120590) isin Co(119860

119901119883 + 119861

119906119865119901)

Remark 8 In general when investigating the output regu-lation problem of synchronous buck converters we need toconsider the dynamic behavior of some natural phenomenasuch as piecewise-constant load fluctuations and mismatcherrors arisingwhen generating the feedforward controlThusbased on the framework of LPV control theory this papermakes an attempt to impose such natural phenomena in thecontrol design

Table 1 Parameters of system (1)ndash(3)

Parameters ValueNominal input voltage V

11986812 V

Equilibrium point of V119862(119905) 5V

Inductance 119871 47 120583HCapacitance 119862 220 120583FStatic drain to source resistance 119877DS 30mΩEquivalent series resistance of inductor 119877DCR 100mΩEquivalent series resistance of capacitor 119877ESR 105mΩLoad resistance 119877(119905) 5ndash10ΩSwitching frequency 150 kHz

4 Simulation and Experimental Results

The parameters of the considered buck converter (1)-(3) arelisted in Table 1 As shown in Table 1 the equilibrium pointof V119862(119905) is given by V

119862= 5 and the lower and upper bounds

of 119877(119905) are given by 119877minus = 3 and 119877+ = 20 which leads to11989110

= 09662 11989111

= 09948 11989120

= 000497 and 11989121

=

003221 from (17) As a result system (15) can be representedas follows

1198601= 102

times [

minus492445 minus2055710

439174 minus22609]

1198602= 102

times [


452172 minus22609]

1198603= 102

times [


439174 minus146391]

1198604= 102

times [


452172 minus146391]

119861119879

119906= 119861119879

119908= 105

times [25532 0]

119862119911= [01 01]

(37)

For three LMI regions D119894119894=123

Theorem 6 provides thecorresponding control gains and the minimized H

infinper-

formances for (37) which are listed in Table 2 Figure 2shows the behaviors of the output voltage V

119874(119905) simulated

byMATLAB (dot-line) and PSIM (solid-line) for the controlgains corresponding to the LMI region D

2 Here to verify

the effectiveness of the proposed approach we consider thepiecewise-constant load 119877

119895that changes from 5Ω to 10Ω

(see Figure 2(a)) and back to 5Ω (see Figure 2(b)) whereT2= 00025 s is set for simulation From Figure 2 it can be

found that the transient responses obtained from MATLABand PSIM simulations are approximately equal in view of theaverage mode which means that the obtained LPV model(15) is valuable in investigating the regulation problem ofsynchronous buck converters In particular from the PSIMsimulation result we can see that the proposed control offersthe short setting time 1ms (less than T

2as mentioned in

Remark 1) small overshoot 180mV and nearly zero steadystate error even though there exist piecewise-constant loadfluctuations in the synchronous buck converter


O(t)

O(t)

525

52

515

51

505

5

495

4924 26 28 3 32 34 36

times10minus3 times10minus3

T2 = 00025T1 T3

PSIMMATLAB

PSIMMATLAB

51

505

5

495

49

485

485 52 54 56 58 6 62

Time (ms)Time (ms)

(a) (b)

Figure 2 Simulation result Output voltage V119874(119905) of the buck converter using the control gains forD

2 where the load is changed (a) from

5Ω to 10Ω at 119905 = 25 times 10minus3 and (b) from 10Ω to 5Ω at 119905 = 5 times 10minus3

Table 2Hinfinperformance and control gains for each LMI region

120572 119903 120579 120574 Control gains

D1

11000 13000 1205871000 117050 [

11986511198652

11986531198654

] = [

minus00738 minus00422 minus00732 minus00357


]

D2

11000 15000 1205871000 45797 [

11986511198652

11986531198654

] = [



]

D3

11000 20000 1205871000 21914 [

11986511198652

11986531198654

] = [



]

Next an experiment is additionally carried out to confirmthe applicability of our approach verified through the simu-lation results The parameters used herein are set the sameas the ones listed in Table 1 and a dual N-channel MOSFET(FDS8949) and two current sense amplifiers (LMP8481) areused for the hardware implementation Here we need totackle several problems concerning the measurement of therequired output signals to construct the controller First ofall the ringing problem arising from the used MOSFETshould be addressed (1) by changing the Q-point factor thatinfluences the setting time of V

119874 and (2) by adding the

RC snubber circuit placed between the low-side MOSFETand inductor In what follows the residual switching noisesshould be attenuated to exactly measure the values of 119894

119874(119905)

119894119871(119905) and V

119874(119905)with respect to the common ground To do so

the power voltage V119868is thoroughly isolated from the applied

voltage used for the operation of sensor units which playsan important role in reducing such switching noises in theside of the sensor units Finally it is necessary to eliminatethe undesirable effects of electromagnetic interference (EMI)that may be caused by the wrong PCB layout In this sense all

net paths on the PCB board are designed as short as possibleand the top and bottom layers of the PCB board withoutinner layers are not assigned to draw the power and groundnets Figure 3(a) shows the construction of our experimentalbench which consists of a prototype of buck converter(see Figure 3(b)) a dSPACE board an oscilloscope and anelectronic loader Here data acquisition and real-time controlsystem are implemented on the basis of dSPACE 1104 softwareand digital processor card which have useful functionssuch as analogdigital converters (ADCs) and pulse-widthmodulation (PWM) built in TMS320F240 DSP Figure 4(a)shows the output response V

119874(119905) of the buck converter with119877

119895

changing from 5Ω to 10Ω from which we can observe thatthe maximum overshoot of V

119874(119905) is approximately 180mV

and its setting time is less than 15ms In addition Figure 4(b)shows the output response V

119874(119905) of the buck converter with

119877119895changing back to 5Ω which illustrates that the maximum

undershoot of V119874(119905) is approximately 180mV and its setting

time is less than 15ms That is by making a comparisonbetween Figures 2 and 4 we can see that this experimentachieves similar output transition performances to the ones


Control desk

dSPACE

Buck converter

DC electronic loader

Power supplyOscilloscope

(a) (b)

Figure 3 (a) Experimental bench for testing the proposed approach and (b) experimental prototype of synchronous DCDC buck converter

100mVdivminus50010V ofst

Tbase 000ms500120583sdiv

100MS 200MSs

BwLDC1MC1

(a)



100MS 200MSs

BwLDC1MC1

(b)

Figure 4 Experiment result Output voltage V119874(119905) of the buck converter using the control gains forD


5Ω to 10Ω and (b) from 10Ω to 5Ω

of the simulation results which means that our approach canbe practically applied to the output regulation problem ofsynchronous buck converters with load fluctuations

5 Concluding Remarks

In this paper we have shed some light on addressing theoutput regulation problem of synchronous buck converterswith piecewise-constant load fluctuations via linear parame-ter varying (LPV) control schemeThus based on the derivedLPV model an H

infinstabilization condition is proposed

such that (1) the mismatch error arising temporally in thefeedforward control term can be attenuated and (2) theclosed-loop poles can lie in the a prescribed LMI regionFinally the validity of the proposed approach is verifiedthrough the simulation and experimental results

Conflict of Interests

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

Acknowledgment

This work was supported by the National Research Foun-dation of Korea Grant funded by the Korean Government(NRF-2012R1A1A1013687)

References

[1] C-S Chiu ldquoT-S fuzzy maximum power point tracking controlof solar power generation systemsrdquo IEEETransactions onEnergyConversion vol 25 no 4 pp 1123ndash1132 2010

[2] D Saifia M Chadli S Labiod and H R Karimi ldquo119867infin

fuzzy control of DC-DC converters with input constraintrdquoMathematical Problems in Engineering vol 2012 Article ID973082 18 pages 2012

[3] L-F Shi and W-G Jia ldquoMode-selectable high-efficiency low-quiescent-current synchronous buck DC-DC converterrdquo IEEETransactions on Industrial Electronics vol 61 no 5 pp 2278ndash2285 2014

[4] R Silva-Ortigoza J R Garcia-Sanchez and J M Alba-Martinez ldquoTwo-stage control design of a buck converterDCmotor system without velocity measurements via a sum - Δ -modulatorrdquo Mathematical Problems in Engineering vol 2013Article ID 929316 11 pages 2013

[5] S Boyd L E Chauoi E Feron and V Balakrishan Lin-ear Matrix Inequalities in System and Control Theory SIAMPhiladelphia Pa USA 1994

[6] K Lian and C Hong ldquoCurrent-sensorless flyback convertersusing integral T-S fuzzy approachrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 66ndash74 2013

[7] H K Lam and S C Tan ldquoStability analysis of fuzzy-model-based control systems application on regulation of switchingDC-DC converterrdquo IET Control Theory amp Applications vol 3no 8 pp 1093ndash1106 2009


[8] K-Y Lian H-W Tu and C-W Hong ldquoCurrent sensorlessregulation for converters via integral fuzzy controlrdquo IEICETransactions on Electronics vol 90 no 2 pp 507ndash514 2007

[9] K-Y Lian J-J Liou and C-Y Huang ldquoLMI-based integralfuzzy control of DC-DC convertersrdquo IEEE Transactions onFuzzy Systems vol 14 no 1 pp 71ndash80 2006

[10] Y Li and Z Ji ldquoT-S modeling simulation and control ofthe Buck converterrdquo in Proceedings of the 5th InternationalConference on Fuzzy Systems and Knowledge Discovery (FSKD08) pp 663ndash667 Schandong China October 2008

[11] R Nowakowski and N Tang ldquoEfficiency of synchronous versusnonsynchronous buck convertersrdquoAnalog Applications Journalvol 4 pp 15ndash18 2009

[12] Y Qiu X Chen C Zhong and C Qi ldquoLimiting integral loopdigital control for DC-DC converters subject to changes in loadcurrent and source voltagerdquo IEEE Transactions on IndustrialInformatics vol 10 no 2 pp 1307ndash1316 2014

[13] G Cimini G Ippoliti S Longhi G Orlando and M PirroldquoSynchronous buck converter control via robust periodic poleassignmentrdquo in in Proceedings of European Control Conference(ECC rsquo14) pp 1921ndash1926 2014

[14] W-R Liou M-L Yeh and Y L Kuo ldquoA high efficiencydual-mode buck converter IC for portable applicationsrdquo IEEETransactions on Power Electronics vol 23 no 2 pp 667ndash6772008

[15] O Lucıa J M Burdio L A Barragan C Carretero and JAcero ldquoSeries resonantmultiinverter with discontinuous-modecontrol for improved light-load operationrdquo IEEE Transactionson Industrial Electronics vol 58 no 11 pp 5163ndash5171 2011

[16] X Zhang and D Maksimovic ldquoMultimode digital controllerfor synchronous buck converters operating over wide ranges ofinput voltages and load currentsrdquo IEEE Transactions on PowerElectronics vol 25 no 8 pp 1958ndash1965 2010

[17] M Qin and J Xu ldquoImproved pulse regulation control techniquefor switching DC-DC converters operating in DCMrdquo IEEETransactions on Industrial Electronics vol 60 no 5 pp 1819ndash1830 2013

[18] S Longhi and R Zulli ldquoA robust periodic pole assignmentalgorithmrdquo IEEETransactions onAutomatic Control vol 40 no5 pp 890ndash894 1995

[19] S H Kim ldquo119867infinoutput-feedback LPV control for systems with

input saturationrdquo International Journal of Control Automationand Systems vol 10 no 6 pp 1267ndash1272 2012

[20] F Wu and K M Grigoriadis ldquoLPV systems with parameter-varying time delays analysis and controlrdquo Automatica vol 37no 2 pp 221ndash229 2001

[21] X Zhang andH Zhu ldquoRobust stability and stabilization criteriafor discrete singular time-delay LPV systemsrdquo Asian Journal ofControl vol 14 no 4 pp 1084ndash1094 2012

[22] J C Doyle and G Stein ldquoMultivariable feedback designConcepts for a classicalmodern synthesisrdquo IEEE Transactionson Automatic Control vol 26 no 1 pp 4ndash16 1981

[23] B A Francis A Course in Hinfin

Control Theory Lecture Notesin Control and Information Sciences Springer New York NYUSA 1987

[24] M Chilali and P Gahinet ldquo119867infin

design with pole placementconstraints an LMI approachrdquo IEEE Transactions on AutomaticControl vol 41 no 3 pp 358ndash367 1996

[25] W Assawinchaichote and S K Nguang ldquoFuzzy 119867infin

outputfeedback control design for singularly perturbed systems withpole placement constraints an LMI approachrdquo IEEE Transac-tions on Fuzzy Systems vol 14 no 3 pp 361ndash371 2006

[26] J Sun and H Grotstollen ldquoAveraged modeling of switchingpower converters reformulation and theoretical basisrdquo in Pro-ceedings of the 23rd Annual IEEE Power Electronics SepecialistsConference (PESC rsquo92) pp 1165ndash1172 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


fluctuations To consider the presence of such load fluctu-ations we derive an output-error state-space model in theform of linear parameter varying (LPV) systems [19ndash21]thereby converting the underlying regulation problem intothe stabilization problem Here it is worth noticing thata mismatch error that temporally arises from the processof generating a feedforward control is clearly incorporatedinto the LPV model and it is attenuated by the H

infin-

synthesis technique [22 23] However Hinfin

design provideslittle control over the transient behavior [24 25] Hence toattenuate the mismatch error in parallel with improving thetransient behavior of the converter this paper proposes anLMI-based stabilization condition capable of achieving bothHinfin

and pole-placement objectives Finally the simulationand experimental results are provided to show the validity ofour approach

Notation The notations 119883 ge 119884 and 119883 gt 119884 mean that 119883 minus 119884

is positive semidefinite and positive definite respectively Insymmetric block matrices (lowast) is used as an ellipsis for termsinduced by symmetry For any square matrix Q He[Q] =Q + Q119879 Lebesgue spaceL

2+=L2[0infin) consists of square-

integrable functions on [0infin)

2 Modeling for DC to DC PWMBuck Converter

Theequivalent circuit for a class of synchronousDC-DCbuckconverters and the corresponding closed-loop control systemare depicted in Figure 1 where the following notations areused

(i) 119877DS denotes the static drain to source resistances ofthe high-side and low-side power MOSFETs respec-tively

(ii) V119868(119905) and V

119874(119905) denote the power input and output

voltages respectively where it is assumed that V119868(119905) =

V119868is time-invariant

(iii) 119894119871(119905) and V

119862(119905) denote the inductor current and the

capacitor voltage respectively

(iv) 119871 and 119862 denote the inductance and capacitanceselected by the given design specifications includingthe switching frequency of MOSFETs

(v) 119877DCR and 119877ESR denote the equivalent series resis-tances of the inductor and capacitor

(vi) 119877(119905) and 120591(119905) denote the piecewise-constant loadresistance subject to 119877minus le 119877(119905) le 119877

+ and the dutyratio of PWM buck converter

Then based on averaging method for one-time-scale discon-tinuous system (AM-OTS-DS) [26] the mathematical model

of the buck converter under consideration is described asfollows

119894119871(119905) = minus

1

119871

V119874(119905) minus

119877DS + 119877DCR119871

119894119871(119905) +

1

119871

V119868120591 (119905) (1)

V119862(119905) =

1

119862

119894119871(119905) minus

1

119877 (119905) 119862

V119874(119905) (2)

V119862(119905) =

1

119877ESR119862(V119874(119905) minus V

119862(119905)) (3)

Further combining (2) and (3) yields

V119874(119905) =

119877 (119905)

119877 (119905) + 119877ESR(119877ESR119894119871 (119905) + V

119862(119905)) (4)

by which (1) and (2) can be rewritten as follows

119894119871(119905) = minus

1

119871

(

119877 (119905) 119877ESR119877 (119905) + 119877ESR

+ 119877DS + 119877DCR) 119894119871 (119905)

minus

1

119871

119877 (119905)

119877 (119905) + 119877ESRV119862(119905) +

V119868

119871

120591 (119905)

(5)

V119862(119905) =

1

119862

119877 (119905)

119877 (119905) + 119877ESR119894119871(119905)

minus

1

119862

1

119877 (119905) + 119877ESRV119862(119905)

(6)

Let V119862be a unique equilibrium point of V

119862(119905) and assume that

the value of 119877(119905) is piecewise-constant that is 119877(119905) = 119877119895for

119905 isin T119895(see Figure 1) Then by (6) the equilibrium point of

119894119871(119905) that is 119894

119871 is given by

0 = 119877119895119894119871minus V119862 for 119905 isin T

119895 (7)

Further by (3) the equilibrium point of V119874(119905) is given by V

119874=

V119862 Hence from (1) the equilibrium point of 120591(119905) that is 120591 is

given by

0 = V119862+ (119877DS + 119877DCR) 119894119871 minus V

119868120591 for 119905 isin T

119895 (8)

Remark 1 This paper focuses on addressing the case wherethe variation of 119877(119905) is subject to a class of switching signalswhose finite intervalsT

119895are larger than the setting time

Remark 2 Indeed it is extremely hard to directly measurethe value of V

119862(119905) in the considered buck converter owing to

the existence of 119877ESR (see Figure 1) Thus to find the value ofV119862(119905) from the ones of the measured V

119874(119905) 119894119874(119905) and 119894

119871(119905) we

introduce a method of using the following equality derivedfrom (4)

V119862(119905) = V

119874(119905) + 119877ESR (119894119874 (119905) minus 119894119871 (119905)) (9)

Remark 3 From (9) we can see that 119894119874= 119894119871in the equilibri-

um point because V119862= V119874


+minus +

+

minus

minus


Gatedriver

120591(t)

High-sideMOSFET

Low-sideMOSFET

i

RDS

RDS RESR

C(t) C

R(t) = Rjminus1

R(t) = Rj

R(t) = Rj+1

iL(t)

Tjminus1 Tj Tj+1

R(t)

O(t)

O(t)

iO(t)

UGATE LGATE

RDCR L



119871= V119862119877119895from (7) where V

119862is a


119877119895=

119877minus

ifV119874(119905)

119894119874(119905)

lt 119877minus

119877+

ifV119874(119905)

119894119874(119905)

gt 119877+

V119874(119905)

119894119874(119905)

otherwise

(10)



119871(119905) minus 119894

119871 V119862(119905) =

V119862(119905)minusV




converted into

119871(119905) = minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119871 (119905)

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862(119905) +

V119868

119871

119906 (119905)

minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119894119871

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862+

V119868

119871

120595 (119905)

V119862(119905) =

1

119862

119877119895

119877119895+ 119877ESR

119871(119905) minus

1

119862

1

119877119895+ 119877ESR

V119862(119905)

+

1

119862

1

119877119895+ 119877ESR

(119877119895119894119871minus V119862)

(11)


120595 (119905) =

1

V119868

((

119877119895119877119864119878119877

119877119895+ 119877119864119878119877

+ 119877119863119878+ 119877119863119862119877

) 119894119871

+

119877119895

119877119895+ 119877119864119878119877

V119862) + 119908 (119905)

=

1

V119868

(V119862+ (119877119863119878+ R119863119862119877

) 119894119871) + 119908 (119905)

= 120591 + 119908 (119905)

(12)

where 119908(119905) isin L2+


119895and 119894119871119895


119895


119871(119905) = minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119871 (119905)

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862(119905) +

V119868

119871

119906 (119905) +

V119868

119871

119908 (119905)

V119862(119905) =

1

119862

119877119895

119877119895+ 119877ESR

119871(119905)

minus

1

119862

1

119877119895+ 119877ESR

V119862(119905) for 119905 isin T

119895

(13)


1198911=

119877119895

119877119895+ 119877ESR

1198912=

1

119877119895+ 119877ESR

forall119895 (14)


follows

(119905) = 119860 (1198911 1198912) 119909 (119905) + 119861

119906119906 (119905) + 119861

119908119908 (119905) (15)


where

119860 (1198911 1198912) =

[

[

[

[

minus

1

119871

(119877ESR1198911 + 119877DS + 119877DCR) minus

1

119871

1198911

1

119862

1198911

minus

1

119862

1198912

]

]

]

]

119861119906= 119861119908=[

[

V119868

119871

0

]

]

(16)


0 lt 11989110=

119877minus

119877minus+ 119877ESR

le 1198911le

119877+

119877++ 119877ESR

= 11989111

0 lt 11989120=

1

119877++ 119877ESR

le 1198912le

1

119877minus+ 119877ESR

= 11989121

(17)


119894=012058811198941198911119894and 1198912= sum1

119894=01205882119894

1198912119894 where

12058810=

11989111minus 1198911

11989111minus 11989110

12058811=

1198911minus 11989110

11989111minus 11989110

12058820=

11989121minus 1198912

11989121minus 11989120

12058821=

1198912minus 11989120

11989121minus 11989120

(18)



119860 (1198911 1198912) =

4

sum

119901=1

120590119901119860119901≜ 119860 (120590)

120590119901= 1205881[119901]1

sdot 1205882[119901]2

119860119901=

[

[

[

[

minus

1

119871

(119877ESR1198911[119901]1

+ 119877DS + 119877DCR) minus

1

119871

1198911[119901]1

1

119862

1198911[119901]1

minus

1

119862

1198912[119901]2

]

]

]

]

(19)

where [119901]1and [119901]

2are decided from 119901 = 2[119901]

2+ [119901]1+



properties

4

sum

119901=1

120590119901=

1

sum

[119901]1=0

1

sum

[119901]2=0

1205881[119901]1

sdot 1205882[119901]2

= (

1

sum

[119901]1=0

1205881[119901]1

) sdot (

1

sum

[119901]2=0

1205882[119901]2

) = 1

0 le 120590119901le 1 forall119901

(20)


119906 (119905) = 119865 (120590) 119909 (119905) (21)

where119865(120590) = sum4119901=1



(119905) = (119860 (120590) + 119861119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905) (22)

119911 (119905) = 119862119911119909 (119905) (23)





C | 119891D(119911) = 119911 + 119911 + 2120572 lt 0


D = 119911 isin C | 119891D (119911) = [minus119903 119902 + 119911

119902 + 119911 minus119903] lt 0 (24)



(25)




0 gt He [119860119901119883 + 119861

119906119865119901+ 120572119883] (26)

0 gt [

minus119903119883 119860119901119883 + 119861

119906119865119901


] (27)

0 gt

[

[

[

[

[

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

cos 120579 sdot (119860119901119883 + 119861

119906119865119901minus 119883119860

119879

119901minus 119865

119879

119901119861119879

119906)

(lowast)

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

]

]

]

]

]

(28)

0 gt[

[

minus119868 119862119911119883 0

(lowast) He [119860119901119883 + 119861

119906119865119901] 119861119908


119868

]

]

(29)


119901=

119865119901119883minus1



119872119863(119860 (120590) + 119861

119906119865 (120590) 119883) lt 0 (30)


0 gt He [Φ (120590) + 120572119883]

0 gt [

minus119903119883 Φ (120590)




(31)

where Φ(120590) = 119860(120590)119883 + 119861119906119865(120590) and 119865(120590) = 119865(120590)119883 =

sum4

119901=1120590119901119865119901119883 By defining 119865


Φ(120590) = sum4

119901=1120590119901(119860119901119883 + 119861


Co(119860119901119883 + 119861





2119908(119905)

2) lt 120574 for all 0 = 119908(119905) isin L

2+ where



0 gt 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905) + (119905) (32)

Here let 119881(119905) = 119909119879

(119905)119883minus1


119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905))]

+ 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

(33)

which leads to

0 gt [

He [119883minus1 (119860 (120590) + 119861119906119865 (120590))] + 119862

119879

119911119862119911119883minus1

119861119908


119868

] (34)


0 gt [

He [Φ (120590)] + 119883119862119879119911119862119911119883 119861

119908


119868

] (35)


0 gt[

[

minus119868 119862119911119883 0

(lowast) He [Φ (120590)] 119861119908


119868

]

]

(36)


119901119883 + 119861

119906119865119901)




11986812 V







= 09662 11989111

= 09948 11989120

= 000497 and 11989121

=


1198601= 102

times [


439174 minus22609]

1198602= 102

times [


452172 minus22609]

1198603= 102

times [


439174 minus146391]

1198604= 102

times [


452172 minus146391]

119861119879

119906= 119861119879

119908= 105

times [25532 0]

119862119911= [01 01]

(37)



infinper-


119874(119905) simulated


2 Here to verify





2as mentioned in



O(t)

O(t)

525

52

515

51

505

5

495

4924 26 28 3 32 34 36


T2 = 00025T1 T3

PSIMMATLAB

PSIMMATLAB

51

505

5

495

49

485

485 52 54 56 58 6 62

Time (ms)Time (ms)

(a) (b)





120572 119903 120579 120574 Control gains

D1

11000 13000 1205871000 117050 [

11986511198652

11986531198654

] = [



]

D2

11000 15000 1205871000 45797 [

11986511198652

11986531198654

] = [



]

D3

11000 20000 1205871000 21914 [

11986511198652

11986531198654

] = [



]




119874(119905)

119894119871(119905) and V






119895









Control desk

dSPACE

Buck converter



(a) (b)




100MS 200MSs

BwLDC1MC1

(a)



100MS 200MSs

BwLDC1MC1

(b)











Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+minus +

+

minus

minus


Gatedriver

120591(t)

High-sideMOSFET

Low-sideMOSFET

i

RDS

RDS RESR

C(t) C

R(t) = Rjminus1

R(t) = Rj

R(t) = Rj+1

iL(t)

Tjminus1 Tj Tj+1

R(t)

O(t)

O(t)

iO(t)

UGATE LGATE

RDCR L



119871= V119862119877119895from (7) where V

119862is a


119877119895=

119877minus

ifV119874(119905)

119894119874(119905)

lt 119877minus

119877+

ifV119874(119905)

119894119874(119905)

gt 119877+

V119874(119905)

119894119874(119905)

otherwise

(10)



119871(119905) minus 119894

119871 V119862(119905) =

V119862(119905)minusV




converted into

119871(119905) = minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119871 (119905)

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862(119905) +

V119868

119871

119906 (119905)

minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119894119871

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862+

V119868

119871

120595 (119905)

V119862(119905) =

1

119862

119877119895

119877119895+ 119877ESR

119871(119905) minus

1

119862

1

119877119895+ 119877ESR

V119862(119905)

+

1

119862

1

119877119895+ 119877ESR

(119877119895119894119871minus V119862)

(11)


120595 (119905) =

1

V119868

((

119877119895119877119864119878119877

119877119895+ 119877119864119878119877

+ 119877119863119878+ 119877119863119862119877

) 119894119871

+

119877119895

119877119895+ 119877119864119878119877

V119862) + 119908 (119905)

=

1

V119868

(V119862+ (119877119863119878+ R119863119862119877

) 119894119871) + 119908 (119905)

= 120591 + 119908 (119905)

(12)

where 119908(119905) isin L2+


119895and 119894119871119895


119895


119871(119905) = minus

1

119871

(

119877119895119877ESR

119877119895+ 119877ESR

+ 119877DS + 119877DCR) 119871 (119905)

minus

1

119871

119877119895

119877119895+ 119877ESR

V119862(119905) +

V119868

119871

119906 (119905) +

V119868

119871

119908 (119905)

V119862(119905) =

1

119862

119877119895

119877119895+ 119877ESR

119871(119905)

minus

1

119862

1

119877119895+ 119877ESR

V119862(119905) for 119905 isin T

119895

(13)


1198911=

119877119895

119877119895+ 119877ESR

1198912=

1

119877119895+ 119877ESR

forall119895 (14)


follows

(119905) = 119860 (1198911 1198912) 119909 (119905) + 119861

119906119906 (119905) + 119861

119908119908 (119905) (15)


where

119860 (1198911 1198912) =

[

[

[

[

minus

1

119871

(119877ESR1198911 + 119877DS + 119877DCR) minus

1

119871

1198911

1

119862

1198911

minus

1

119862

1198912

]

]

]

]

119861119906= 119861119908=[

[

V119868

119871

0

]

]

(16)


0 lt 11989110=

119877minus

119877minus+ 119877ESR

le 1198911le

119877+

119877++ 119877ESR

= 11989111

0 lt 11989120=

1

119877++ 119877ESR

le 1198912le

1

119877minus+ 119877ESR

= 11989121

(17)


119894=012058811198941198911119894and 1198912= sum1

119894=01205882119894

1198912119894 where

12058810=

11989111minus 1198911

11989111minus 11989110

12058811=

1198911minus 11989110

11989111minus 11989110

12058820=

11989121minus 1198912

11989121minus 11989120

12058821=

1198912minus 11989120

11989121minus 11989120

(18)



119860 (1198911 1198912) =

4

sum

119901=1

120590119901119860119901≜ 119860 (120590)

120590119901= 1205881[119901]1

sdot 1205882[119901]2

119860119901=

[

[

[

[

minus

1

119871

(119877ESR1198911[119901]1

+ 119877DS + 119877DCR) minus

1

119871

1198911[119901]1

1

119862

1198911[119901]1

minus

1

119862

1198912[119901]2

]

]

]

]

(19)

where [119901]1and [119901]

2are decided from 119901 = 2[119901]

2+ [119901]1+



properties

4

sum

119901=1

120590119901=

1

sum

[119901]1=0

1

sum

[119901]2=0

1205881[119901]1

sdot 1205882[119901]2

= (

1

sum

[119901]1=0

1205881[119901]1

) sdot (

1

sum

[119901]2=0

1205882[119901]2

) = 1

0 le 120590119901le 1 forall119901

(20)


119906 (119905) = 119865 (120590) 119909 (119905) (21)

where119865(120590) = sum4119901=1



(119905) = (119860 (120590) + 119861119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905) (22)

119911 (119905) = 119862119911119909 (119905) (23)





C | 119891D(119911) = 119911 + 119911 + 2120572 lt 0


D = 119911 isin C | 119891D (119911) = [minus119903 119902 + 119911

119902 + 119911 minus119903] lt 0 (24)



(25)




0 gt He [119860119901119883 + 119861

119906119865119901+ 120572119883] (26)

0 gt [

minus119903119883 119860119901119883 + 119861

119906119865119901


] (27)

0 gt

[

[

[

[

[

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

cos 120579 sdot (119860119901119883 + 119861

119906119865119901minus 119883119860

119879

119901minus 119865

119879

119901119861119879

119906)

(lowast)

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

]

]

]

]

]

(28)

0 gt[

[

minus119868 119862119911119883 0

(lowast) He [119860119901119883 + 119861

119906119865119901] 119861119908


119868

]

]

(29)


119901=

119865119901119883minus1



119872119863(119860 (120590) + 119861

119906119865 (120590) 119883) lt 0 (30)


0 gt He [Φ (120590) + 120572119883]

0 gt [

minus119903119883 Φ (120590)




(31)

where Φ(120590) = 119860(120590)119883 + 119861119906119865(120590) and 119865(120590) = 119865(120590)119883 =

sum4

119901=1120590119901119865119901119883 By defining 119865


Φ(120590) = sum4

119901=1120590119901(119860119901119883 + 119861


Co(119860119901119883 + 119861





2119908(119905)

2) lt 120574 for all 0 = 119908(119905) isin L

2+ where



0 gt 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905) + (119905) (32)

Here let 119881(119905) = 119909119879

(119905)119883minus1


119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905))]

+ 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

(33)

which leads to

0 gt [

He [119883minus1 (119860 (120590) + 119861119906119865 (120590))] + 119862

119879

119911119862119911119883minus1

119861119908


119868

] (34)


0 gt [

He [Φ (120590)] + 119883119862119879119911119862119911119883 119861

119908


119868

] (35)


0 gt[

[

minus119868 119862119911119883 0

(lowast) He [Φ (120590)] 119861119908


119868

]

]

(36)


119901119883 + 119861

119906119865119901)




11986812 V







= 09662 11989111

= 09948 11989120

= 000497 and 11989121

=


1198601= 102

times [


439174 minus22609]

1198602= 102

times [


452172 minus22609]

1198603= 102

times [


439174 minus146391]

1198604= 102

times [


452172 minus146391]

119861119879

119906= 119861119879

119908= 105

times [25532 0]

119862119911= [01 01]

(37)



infinper-


119874(119905) simulated


2 Here to verify





2as mentioned in



O(t)

O(t)

525

52

515

51

505

5

495

4924 26 28 3 32 34 36


T2 = 00025T1 T3

PSIMMATLAB

PSIMMATLAB

51

505

5

495

49

485

485 52 54 56 58 6 62

Time (ms)Time (ms)

(a) (b)





120572 119903 120579 120574 Control gains

D1

11000 13000 1205871000 117050 [

11986511198652

11986531198654

] = [



]

D2

11000 15000 1205871000 45797 [

11986511198652

11986531198654

] = [



]

D3

11000 20000 1205871000 21914 [

11986511198652

11986531198654

] = [



]




119874(119905)

119894119871(119905) and V






119895









Control desk

dSPACE

Buck converter



(a) (b)




100MS 200MSs

BwLDC1MC1

(a)



100MS 200MSs

BwLDC1MC1

(b)











Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119860 (1198911 1198912) =

[

[

[

[

minus

1

119871

(119877ESR1198911 + 119877DS + 119877DCR) minus

1

119871

1198911

1

119862

1198911

minus

1

119862

1198912

]

]

]

]

119861119906= 119861119908=[

[

V119868

119871

0

]

]

(16)


0 lt 11989110=

119877minus

119877minus+ 119877ESR

le 1198911le

119877+

119877++ 119877ESR

= 11989111

0 lt 11989120=

1

119877++ 119877ESR

le 1198912le

1

119877minus+ 119877ESR

= 11989121

(17)


119894=012058811198941198911119894and 1198912= sum1

119894=01205882119894

1198912119894 where

12058810=

11989111minus 1198911

11989111minus 11989110

12058811=

1198911minus 11989110

11989111minus 11989110

12058820=

11989121minus 1198912

11989121minus 11989120

12058821=

1198912minus 11989120

11989121minus 11989120

(18)



119860 (1198911 1198912) =

4

sum

119901=1

120590119901119860119901≜ 119860 (120590)

120590119901= 1205881[119901]1

sdot 1205882[119901]2

119860119901=

[

[

[

[

minus

1

119871

(119877ESR1198911[119901]1

+ 119877DS + 119877DCR) minus

1

119871

1198911[119901]1

1

119862

1198911[119901]1

minus

1

119862

1198912[119901]2

]

]

]

]

(19)

where [119901]1and [119901]

2are decided from 119901 = 2[119901]

2+ [119901]1+



properties

4

sum

119901=1

120590119901=

1

sum

[119901]1=0

1

sum

[119901]2=0

1205881[119901]1

sdot 1205882[119901]2

= (

1

sum

[119901]1=0

1205881[119901]1

) sdot (

1

sum

[119901]2=0

1205882[119901]2

) = 1

0 le 120590119901le 1 forall119901

(20)


119906 (119905) = 119865 (120590) 119909 (119905) (21)

where119865(120590) = sum4119901=1



(119905) = (119860 (120590) + 119861119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905) (22)

119911 (119905) = 119862119911119909 (119905) (23)





C | 119891D(119911) = 119911 + 119911 + 2120572 lt 0


D = 119911 isin C | 119891D (119911) = [minus119903 119902 + 119911

119902 + 119911 minus119903] lt 0 (24)



(25)




0 gt He [119860119901119883 + 119861

119906119865119901+ 120572119883] (26)

0 gt [

minus119903119883 119860119901119883 + 119861

119906119865119901


] (27)

0 gt

[

[

[

[

[

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

cos 120579 sdot (119860119901119883 + 119861

119906119865119901minus 119883119860

119879

119901minus 119865

119879

119901119861119879

119906)

(lowast)

sin 120579 sdotHe [119860119901119883 + 119861

119906119865119901]

]

]

]

]

]

(28)

0 gt[

[

minus119868 119862119911119883 0

(lowast) He [119860119901119883 + 119861

119906119865119901] 119861119908


119868

]

]

(29)


119901=

119865119901119883minus1



119872119863(119860 (120590) + 119861

119906119865 (120590) 119883) lt 0 (30)


0 gt He [Φ (120590) + 120572119883]

0 gt [

minus119903119883 Φ (120590)




(31)

where Φ(120590) = 119860(120590)119883 + 119861119906119865(120590) and 119865(120590) = 119865(120590)119883 =

sum4

119901=1120590119901119865119901119883 By defining 119865


Φ(120590) = sum4

119901=1120590119901(119860119901119883 + 119861


Co(119860119901119883 + 119861





2119908(119905)

2) lt 120574 for all 0 = 119908(119905) isin L

2+ where



0 gt 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905) + (119905) (32)

Here let 119881(119905) = 119909119879

(119905)119883minus1


119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905))]

+ 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

(33)

which leads to

0 gt [

He [119883minus1 (119860 (120590) + 119861119906119865 (120590))] + 119862

119879

119911119862119911119883minus1

119861119908


119868

] (34)


0 gt [

He [Φ (120590)] + 119883119862119879119911119862119911119883 119861

119908


119868

] (35)


0 gt[

[

minus119868 119862119911119883 0

(lowast) He [Φ (120590)] 119861119908


119868

]

]

(36)


119901119883 + 119861

119906119865119901)




11986812 V







= 09662 11989111

= 09948 11989120

= 000497 and 11989121

=


1198601= 102

times [


439174 minus22609]

1198602= 102

times [


452172 minus22609]

1198603= 102

times [


439174 minus146391]

1198604= 102

times [


452172 minus146391]

119861119879

119906= 119861119879

119908= 105

times [25532 0]

119862119911= [01 01]

(37)



infinper-


119874(119905) simulated


2 Here to verify





2as mentioned in



O(t)

O(t)

525

52

515

51

505

5

495

4924 26 28 3 32 34 36


T2 = 00025T1 T3

PSIMMATLAB

PSIMMATLAB

51

505

5

495

49

485

485 52 54 56 58 6 62

Time (ms)Time (ms)

(a) (b)





120572 119903 120579 120574 Control gains

D1

11000 13000 1205871000 117050 [

11986511198652

11986531198654

] = [



]

D2

11000 15000 1205871000 45797 [

11986511198652

11986531198654

] = [



]

D3

11000 20000 1205871000 21914 [

11986511198652

11986531198654

] = [



]




119874(119905)

119894119871(119905) and V






119895









Control desk

dSPACE

Buck converter



(a) (b)




100MS 200MSs

BwLDC1MC1

(a)



100MS 200MSs

BwLDC1MC1

(b)











Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872119863(119860 (120590) + 119861

119906119865 (120590) 119883) lt 0 (30)


0 gt He [Φ (120590) + 120572119883]

0 gt [

minus119903119883 Φ (120590)




(31)

where Φ(120590) = 119860(120590)119883 + 119861119906119865(120590) and 119865(120590) = 119865(120590)119883 =

sum4

119901=1120590119901119865119901119883 By defining 119865


Φ(120590) = sum4

119901=1120590119901(119860119901119883 + 119861


Co(119860119901119883 + 119861





2119908(119905)

2) lt 120574 for all 0 = 119908(119905) isin L

2+ where



0 gt 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905) + (119905) (32)

Here let 119881(119905) = 119909119879

(119905)119883minus1


119906119865 (120590)) 119909 (119905) + 119861

119908119908 (119905))]

+ 119911119879

(119905) 119911 (119905) minus 1205742

119908119879

(119905) 119908 (119905)

(33)

which leads to

0 gt [

He [119883minus1 (119860 (120590) + 119861119906119865 (120590))] + 119862

119879

119911119862119911119883minus1

119861119908


119868

] (34)


0 gt [

He [Φ (120590)] + 119883119862119879119911119862119911119883 119861

119908


119868

] (35)


0 gt[

[

minus119868 119862119911119883 0

(lowast) He [Φ (120590)] 119861119908


119868

]

]

(36)


119901119883 + 119861

119906119865119901)




11986812 V







= 09662 11989111

= 09948 11989120

= 000497 and 11989121

=


1198601= 102

times [


439174 minus22609]

1198602= 102

times [


452172 minus22609]

1198603= 102

times [


439174 minus146391]

1198604= 102

times [


452172 minus146391]

119861119879

119906= 119861119879

119908= 105

times [25532 0]

119862119911= [01 01]

(37)



infinper-


119874(119905) simulated


2 Here to verify





2as mentioned in



O(t)

O(t)

525

52

515

51

505

5

495

4924 26 28 3 32 34 36


T2 = 00025T1 T3

PSIMMATLAB

PSIMMATLAB

51

505

5

495

49

485

485 52 54 56 58 6 62

Time (ms)Time (ms)

(a) (b)





120572 119903 120579 120574 Control gains

D1

11000 13000 1205871000 117050 [

11986511198652

11986531198654

] = [



]

D2

11000 15000 1205871000 45797 [

11986511198652

11986531198654

] = [



]

D3

11000 20000 1205871000 21914 [

11986511198652

11986531198654

] = [



]




119874(119905)

119894119871(119905) and V






119895









Control desk

dSPACE

Buck converter



(a) (b)




100MS 200MSs

BwLDC1MC1

(a)



100MS 200MSs

BwLDC1MC1

(b)











Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










O(t)

O(t)

525

52

515

51

505

5

495

4924 26 28 3 32 34 36


T2 = 00025T1 T3

PSIMMATLAB

PSIMMATLAB

51

505

5

495

49

485

485 52 54 56 58 6 62

Time (ms)Time (ms)

(a) (b)





120572 119903 120579 120574 Control gains

D1

11000 13000 1205871000 117050 [

11986511198652

11986531198654

] = [



]

D2

11000 15000 1205871000 45797 [

11986511198652

11986531198654

] = [



]

D3

11000 20000 1205871000 21914 [

11986511198652

11986531198654

] = [



]




119874(119905)

119894119871(119905) and V






119895









Control desk

dSPACE

Buck converter



(a) (b)




100MS 200MSs

BwLDC1MC1

(a)



100MS 200MSs

BwLDC1MC1

(b)











Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Control desk

dSPACE

Buck converter



(a) (b)




100MS 200MSs

BwLDC1MC1

(a)



100MS 200MSs

BwLDC1MC1

(b)











Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article LPV Control with Pole Placement