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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
(lowast) minus1205742
119868
] (34)
As a result pre- and postmultiplying (34) by diag(119883 119868) andits transpose yield
0 gt [
He [Φ (120590)] + 119883119862119879119911119862119911119883 119861
119908
(lowast) minus1205742
119868
] (35)
which can be converted by the Schurrsquos complement [5] into
0 gt[
[
minus119868 119862119911119883 0
(lowast) He [Φ (120590)] 119861119908
(lowast) (lowast) minus1205742
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 [
minus498833 minus2116548
452172 minus22609]
1198603= 102
times [
minus492445 minus2055710
439174 minus146391]
1198604= 102
times [
minus498833 minus2116548
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
6 Mathematical Problems in Engineering
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
minus00704 minus00216 minus00687 minus00123
]
D2
11000 15000 1205871000 45797 [
11986511198652
11986531198654
] = [
minus00817 minus00614 minus00813 minus00550
minus00773 minus00364 minus00715 minus00290
]
D3
11000 20000 1205871000 21914 [
11986511198652
11986531198654
] = [
minus01012 minus01094 minus01009 minus01018
minus00986 minus00868 minus00988 minus00813
]
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
Mathematical Problems in Engineering 7
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)
100mVdivminus50010V ofst
Tbase 000ms500120583sdiv
100MS 200MSs
BwLDC1MC1
(b)
Figure 4 Experiment 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 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 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
(lowast) minus1205742
119868
] (34)
As a result pre- and postmultiplying (34) by diag(119883 119868) andits transpose yield
0 gt [
He [Φ (120590)] + 119883119862119879119911119862119911119883 119861
119908
(lowast) minus1205742
119868
] (35)
which can be converted by the Schurrsquos complement [5] into
0 gt[
[
minus119868 119862119911119883 0
(lowast) He [Φ (120590)] 119861119908
(lowast) (lowast) minus1205742
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 [
minus498833 minus2116548
452172 minus22609]
1198603= 102
times [
minus492445 minus2055710
439174 minus146391]
1198604= 102
times [
minus498833 minus2116548
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
6 Mathematical Problems in Engineering
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
minus00704 minus00216 minus00687 minus00123
]
D2
11000 15000 1205871000 45797 [
11986511198652
11986531198654
] = [
minus00817 minus00614 minus00813 minus00550
minus00773 minus00364 minus00715 minus00290
]
D3
11000 20000 1205871000 21914 [
11986511198652
11986531198654
] = [
minus01012 minus01094 minus01009 minus01018
minus00986 minus00868 minus00988 minus00813
]
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
Mathematical Problems in Engineering 7
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)
100mVdivminus50010V ofst
Tbase 000ms500120583sdiv
100MS 200MSs
BwLDC1MC1
(b)
Figure 4 Experiment 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 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 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
(lowast) minus1205742
119868
] (34)
As a result pre- and postmultiplying (34) by diag(119883 119868) andits transpose yield
0 gt [
He [Φ (120590)] + 119883119862119879119911119862119911119883 119861
119908
(lowast) minus1205742
119868
] (35)
which can be converted by the Schurrsquos complement [5] into
0 gt[
[
minus119868 119862119911119883 0
(lowast) He [Φ (120590)] 119861119908
(lowast) (lowast) minus1205742
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 [
minus498833 minus2116548
452172 minus22609]
1198603= 102
times [
minus492445 minus2055710
439174 minus146391]
1198604= 102
times [
minus498833 minus2116548
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
6 Mathematical Problems in Engineering
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
minus00704 minus00216 minus00687 minus00123
]
D2
11000 15000 1205871000 45797 [
11986511198652
11986531198654
] = [
minus00817 minus00614 minus00813 minus00550
minus00773 minus00364 minus00715 minus00290
]
D3
11000 20000 1205871000 21914 [
11986511198652
11986531198654
] = [
minus01012 minus01094 minus01009 minus01018
minus00986 minus00868 minus00988 minus00813
]
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
Mathematical Problems in Engineering 7
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)
100mVdivminus50010V ofst
Tbase 000ms500120583sdiv
100MS 200MSs
BwLDC1MC1
(b)
Figure 4 Experiment 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 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 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
(lowast) minus1205742
119868
] (34)
As a result pre- and postmultiplying (34) by diag(119883 119868) andits transpose yield
0 gt [
He [Φ (120590)] + 119883119862119879119911119862119911119883 119861
119908
(lowast) minus1205742
119868
] (35)
which can be converted by the Schurrsquos complement [5] into
0 gt[
[
minus119868 119862119911119883 0
(lowast) He [Φ (120590)] 119861119908
(lowast) (lowast) minus1205742
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 [
minus498833 minus2116548
452172 minus22609]
1198603= 102
times [
minus492445 minus2055710
439174 minus146391]
1198604= 102
times [
minus498833 minus2116548
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
6 Mathematical Problems in Engineering
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
minus00704 minus00216 minus00687 minus00123
]
D2
11000 15000 1205871000 45797 [
11986511198652
11986531198654
] = [
minus00817 minus00614 minus00813 minus00550
minus00773 minus00364 minus00715 minus00290
]
D3
11000 20000 1205871000 21914 [
11986511198652
11986531198654
] = [
minus01012 minus01094 minus01009 minus01018
minus00986 minus00868 minus00988 minus00813
]
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
Mathematical Problems in Engineering 7
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)
100mVdivminus50010V ofst
Tbase 000ms500120583sdiv
100MS 200MSs
BwLDC1MC1
(b)
Figure 4 Experiment 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 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 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
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
(lowast) minus1205742
119868
] (34)
As a result pre- and postmultiplying (34) by diag(119883 119868) andits transpose yield
0 gt [
He [Φ (120590)] + 119883119862119879119911119862119911119883 119861
119908
(lowast) minus1205742
119868
] (35)
which can be converted by the Schurrsquos complement [5] into
0 gt[
[
minus119868 119862119911119883 0
(lowast) He [Φ (120590)] 119861119908
(lowast) (lowast) minus1205742
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 [
minus498833 minus2116548
452172 minus22609]
1198603= 102
times [
minus492445 minus2055710
439174 minus146391]
1198604= 102
times [
minus498833 minus2116548
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
6 Mathematical Problems in Engineering
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
minus00704 minus00216 minus00687 minus00123
]
D2
11000 15000 1205871000 45797 [
11986511198652
11986531198654
] = [
minus00817 minus00614 minus00813 minus00550
minus00773 minus00364 minus00715 minus00290
]
D3
11000 20000 1205871000 21914 [
11986511198652
11986531198654
] = [
minus01012 minus01094 minus01009 minus01018
minus00986 minus00868 minus00988 minus00813
]
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
Mathematical Problems in Engineering 7
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)
100mVdivminus50010V ofst
Tbase 000ms500120583sdiv
100MS 200MSs
BwLDC1MC1
(b)
Figure 4 Experiment 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 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 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
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
minus00704 minus00216 minus00687 minus00123
]
D2
11000 15000 1205871000 45797 [
11986511198652
11986531198654
] = [
minus00817 minus00614 minus00813 minus00550
minus00773 minus00364 minus00715 minus00290
]
D3
11000 20000 1205871000 21914 [
11986511198652
11986531198654
] = [
minus01012 minus01094 minus01009 minus01018
minus00986 minus00868 minus00988 minus00813
]
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
Mathematical Problems in Engineering 7
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)
100mVdivminus50010V ofst
Tbase 000ms500120583sdiv
100MS 200MSs
BwLDC1MC1
(b)
Figure 4 Experiment 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 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 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
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)
100mVdivminus50010V ofst
Tbase 000ms500120583sdiv
100MS 200MSs
BwLDC1MC1
(b)
Figure 4 Experiment 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 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 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of