POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
PID, 2-DOF PID and Mixed Sensi i i y
Loop-Shaping Based Robus Vol age Con ol o
Quad a ic Buck DC-DC Con e e
Fa eh OUNIS, Nou eddine GOLEA
Elec ical Enginee ing Depa men , Sciences and Applied Sciences Facul y, La bi Ben M’hidi Uni e si y,
04000 Oum El Bouaghi, Alge ia
ounis a eh_01@yahoo. , nou _golea@yahoo.
DOI: 10.15598/aeee. 14i5.1821
Abs ac . DC-DC Quad a ic Buck Con e e (QBC)
is la gely used in applica ions whe e he high s ep-down
con e sion a io is equi ed. In he p ac ical imple-
men a ion, QBC is subjec o unce ain ies, dis u -
bance, and senso noise. To add ess he QBC con-
ol p oblems, a wo-deg ee o eedom PID (2-DOF
PID) is designed in he obus con ol amewo k. Fu -
he , o compa ison pu pose, a one-deg ee o ee-
dom PID (1-DOF PID) and mixed sensi i i y loop-
shaping (MS-LS) con olle a e also p oposed. Con-
side ing QBC pa asi ic componen s, he QBC small-
signal ans e unc ion is de i ed based on a p ac ical
app oach. Sensi i i y unc ions a e used o speci y he
desi ed design equi emen s, and non-smoo h op imiza-
ion is used o une bo h PID’s pa ame e s. The h ee
con ol s uc u es a e implemen ed and es ed in he
Ma lab/Simulink en i onmen . As a es ed by simula-
ion esul s, he 2-DOF PID exhibi s a be e egula ion
accu acy wi h enhanced obus s abili y and obus pe -
o mance o a wide ange o supply ol age/load a i-
a ion and senso noise e ec .
Keywo ds
DC-DC con e e s, mixed sensi i i y, loop-
shaping, PID con ol, quad a ic buck con e e ,
obus con ol.
1. In oduc ion
DC-DC con e e s a e key elemen s in powe ene gy
modula ion and con e sion. Basic DC-DC con e e s,
such as buck, boos and buck-boos , a e widely used
in a ious ields o echnology [1]. Recen ly, new appli-
ca ions, such as LED lamps, mic op ocesso s, po able
de ices and GPS, equi e e y low dc ol ages and hey
ope a e a e y high cu en s. Such applica ions e-
qui e con e e s wi h low ipples in he ol age and
cu en and high e iciency in o de o achie e p ecise
ou pu ol age egula ion agains pa ame e , line and
load dis u bances. Basic s ep-down con e e s a e no
sui able o high s ep-down ol age con e sion since
ope a ing a a small du y a io a ec s he con e e
dynamic pe o mance and cause asymme y in he on
and o imes o he swi ches. Mo eo e , e y small
du y a io limi s he con e e s swi ching equency
and inc eases peak swi ch cu en ha leads o mo e
swi ching losses, se e e e e se- eco e y p oblems and
con e e ’s e iciency deg ada ion [2]. Some cascade
in e connec ed powe con e e s s uc u es we e de el-
oped o ace his p oblem [3]. Howe e , a no able dis-
ad an age o cascaded con e e s is ha he o e all
e iciency is educed by losses in swi ching de ices. To
imp o e o e all e iciency, Quad a ic Buck Con e e s
(QBC) we e de eloped in [4], [5], [6] and [7]]. QBC is
designed based on cascade connec ion o wo buck con-
e e s and has only one ac i e swi ching de ice. The
DC con e sion a io is he p oduc o he con e sion
a ios o he wo single buck con e e s. QBC ope a es
a highe swi ching equencies wi h wide load ange
and achie es an imp o ed s ep-down con e sion a io.
The e iciency is also enhanced since only one ac i e
swi ch is used [8] and [9].
As QBC exhibi s complex nonlinea dynamics sub-
jec o pa ame e s unce ain ies and inpu /load a i-
a ions, con ol loops mus be in oduced o gua an-
ee s abili y and ope a ing pe o mance. Se e al QBC
con ol echniques such as linea s a e eedback, eed-
back linea iza ion, sliding mode con ol and passi i y
based con ol, we e p esen ed in [10]. In [11], QBC
nonlinea con ol scheme is p oposed. QBC wi h LC
inpu il e and damping con ol is de eloped in [12].
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 551
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
Robus QBC con ol based on iden i ied Hamme s ein
model is p oposed in [13]. A e age cu en -mode con-
ol o he QBC is p oposed in [14], whe e he ou e
ol age con ol loop bandwid h is limi ed by inne cu -
en con ol loop bandwid h. In [15], obus con ol
o QBC is designed based on Kha i ono ’s heo em
and D-s abili y concep . To ensu e obus ou pu eg-
ula ion, obus s a e eedback s abilize wi h sa u a ed
in e nal model, is p oposed in [16]. In [17], inne cu -
en loop PI pa ame e s a e selec ed om QBC la ge-
signal model, and ou e ol age loop is con olled us-
ing a con en ional PI egula o . To ensu e obus ness
in he p esence o dis u bances and unce ain ies, H∞
based con ol is in es iga ed in [18].
In his pape , 2-DOF PID is de eloped o sol e he
QBC obus con ol p oblem. Fo he compa ison,
a 1-DOF PID and MS-LS con ol a e also p oposed.
Taking he pa asi ic componen s in o accoun , he
QBC small signal ans e unc ion om ou pu ol age
o con ol signal is de i ed. Robus pe o mance e-
qui emen s a e de ined using he same weigh ing unc-
ions o bo h PID and by ano he se o weigh ing
unc ions o MS-LS con ol. Con a y o MS-LS con-
ol and 1-DOF PID, he 2-DOF PID p o ides a wide
ange o he c osso e equency o speci y esponse
ime-pe o mance comp omise. The non-smoo h ap-
p oach p esen ed in [19] is used o une bo h PID’s
pa ame e s. Based on he model educ ion me hods,
MS-LS con olle is educed om 8 o 5, and he e-
duced e sion is p esen ed and used in simula ions.
Simula ion esul s illus a e he 2-DOF PID in e m
o accu acy and s abili y obus ness.
The emainde o his pape is o ganized as ollows.
Sec ion 2. p esen s he QBC nominal ans e unc-
ion compu a ion. The MS-LS con ol is de eloped in
Sec ion 3. PID con olle ’s design is p o ided in Sec-
ion 4. The h ee con olle s’ obus ness analysis
is es ablished in Sec ion 5. Simula ion esul s a e
shown in Sec ion 6. Concluding ema ks a e gi en in
Sec ion 7.
2. QBC Nominal Model
As a i s s ep o he con ol design, he QBC open
loop small-signal con ol- o-ou pu ol age ans e
unc ion should be es ablished. The objec i e can be
eached using analy ical modeling and a e aging ech-
niques [20] and [21]. In his wo k, a p ac ical app oach
is adop ed. Based on QBC pa ame e s and ope a ing
poin gi en in App. A, a Simulink implemen a ion o
he open loop exci a ion is ealized (Fig. 1). The PWM
con ol signal du y cycle is adjus ed o ge he desi ed
ou pu ol age le el. Taking in o accoun he pa asi ic
componen s, he QBC disc e e- ime ans e unc ion
G(z) = B(z)/A(z)is assumed o 6 h o de . Hence,
Mos e
C1
c1
l1
L1
E
C2
c2
l2
L2
R
45 kHz PWM
Mean
ZOH To Wo kspace
ua
Mean
Mean 2
ZOH
To Wo kspace
To Wo kspace
ua
Fig. 1: QBC open loop exci a ion se up.
applying he S eigli z-McB ide ecu si e iden i ica ion
me hod [22] on con e e a e aged inpu /ou pu sig-
nals o 5 i e a ions, yields he es ima ed ans e unc-
ion:
B(z)=0.7108 −1.6654z−1+ 1.9263z−2
−1.6483z−3+ 1.2154z−4−0.5151z−5,
A(z)=1−4.829z−1+ 10.1746z−2−11.9259z−3
+8.1753z−4−3.1028z−5+ 0.5094z−6.
(1)
Fu he , using he nume ical algo i hm p oposed in
[23], he equi alen con inuous- ime ans e unc ion
G(s) = N(s)/D(s)is gi en by:
N(s) = 544300s5+ 1.618 ·1011s4+
+2.184 ·1018s3+ 1.524 ·1023s2+
+7.774 ·1029s+ 3.487 ·1034,
D(s) = s6+ 674400s5+ 7.804 ·1011s4+
+3.354 ·1017s3+ 1.502 ·1023s2+
+2.677 ·1028s+ 2.342 ·1033.
(2)
The G(s)Bode plo is shown in Fig. 2. I is clea ha
QBC has wo second-o de il e s wi h high quali y-
ac o Q, which depends on he selec ed ci cui alues.
All poles and ze os a e loca ed on he igh hal o he
s-plane as shown he Fig. 3. The igh hal plane ze os
a e esponsible o he excessi e phase lag in he ideal
case. The Equi alen Se ies Resis ances (ESR) p o ide
some damping in o he sys em, which is bene icen as
i will ease eedback con ol design.
3. MS-LS Con ol Design
A diag am o he con ol design is shown in Fig. 4,
whe e Gis he quad a ic buck con e e ans e unc-
ion. W1,W2and W3a e he pe o mance, con ol
and noise weigh ing unc ions, espec i ely. Fu he ,
wdeno e inpu signals, zou pu ec o ha includes
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 552
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
40
20
0
-20
-40
-60
90
0
-90
-180
-270
103104105106107108
F equency ( ad )·s-1
Bode diag am
Magni ude (dB)
Phase (deg)
Q
Fig. 2: F equency open loop QBC esponse.
-2 -1.5 -1 -0.5 0
·105
·106
-2
-1
0
1
2
Real axis ( )s-1
Imagina y axis ( )s-1
Pole-Ze o Map
Fig. 3: T ans e unc ion pole-ze o map.
bo h pe o mance and obus ness measu es, is he
ec o o measu emen s a ailable o he con olle K
and u he con ol signal. In he con ex , he necessa y
de ini ions a e gi en by:
S(s) = I+G(s)K(s)−1,(3)
T(s) = G(s)K(s)I+G(s)K(s)−1,(4)
whe e S(s)is he sensi i i y unc ion and T(s)is he
complemen a y sensi i i y unc ion. The gene alized
closed loop ans e unc ion is gi en by:
Tzw =

W1S
W2RS
W3T

,(5)
whe e R(s) = K(s)I+G(s)K(s)−1. In his mixed
p oblem, he con ol objec i e is o design a s able
con olle ha minimizes he no m o he gene alized
K
G
y
uu
w=
PW1z1
z2
z3
W2
W3
z
Fig. 4: MS-LS con ol design.
ans e unc ion Tzw such ha :
||Tzw||∞<1.(6)
3.1. Weigh ing Func ions Selec ion
The closed loop pe o mance o he sys em is la gely
dependen on he shape o he weigh ing unc ion. The
weigh unc ion W1speci ies he con ol pe o mance
and W1is selec ed acco ding o me hodology sugges ed
by Zhou [24],
W1=s/Ms+ws
s+wses
,(7)
whe e esis he maximum allowed s eady-s a e o se
ixed o es= 0.001,wsis he desi ed bandwid h ixed o
6·103 ad·s−1and Msis he sensi i i y peak ( ypically
M= 1.6). The e o e,
W1=0.625(s+ 9600)
s+ 6 .(8)
In o de o a oid impulsi e inpu e ec on he con-
e e , W2is chosen as:
W2(s)=0.01.(9)
W3is used o shape he complemen a y sensi i i y
unc ion T, and hus i mus be la ge a high equen-
cies. Hence W3is chosen as:
W3=s+wb/Mb
ebs+wb
.(10)
To keep he sys em s able, he complemen a y sensi-
i i y Tmus be small o high equencies. Thus, he
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 553
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
alue 0.001 is selec ed o he pa ame e eb. In o de
o limi he closed loop bandwid h, he pa ame e Mb
is ixed as 1.6and wbis ixed o 104 ad·s−1. Then:
W3=1000(s+ 6250)
s+ 107.(11)
In o de o adop a uni ied solu ion p ocedu e, he
abo e ma ix inequali y Eq. (6) can be ecas in o a
s anda d con igu a ion as in Fig. 4. This can be ob-
ained by using he Linea F ac ional T ans o ma ion
(LFT), and he gene alized plan Pis ob ained by
g ouping signals in o se s o ex e nal inpu s, ou pu s,
inpu o he con olle and ou pu om he con olle ,
which yields:
hz
i=




W1−W1G
0W2G
0W3G
I−G




| {z }
P
h
ui,(12)
whe e =wis he e e ence ol age,
z=z1z2z3Tis he ou pu signals ec o , is
he con ol signal and is he con olled QBC ou pu
ol age. W1,W2and W3a e he weigh ing unc ions
desc ibed by Eq. (8), Eq. (9) and Eq. (11), espec i ely.
Based on he abo e con igu a ion, he gene alized
plan can be buil up, and consequen ly he con olle
can be calcula ed using Ma lab obus con ol oolbox.
Hence, he ob ained 8 h o de con olle is:
K(s) = NK(s)
DK(s),(13)
wi h
NK(s) = 5.504 ·106s7+ 5.875 ·1013s6+ 4.141 ·1019s5
+4.48 ·1025s4+ 1.929 ·1031s3+ 8.414 ·1036s2
+1.486 ·1042s+ 1.289 ·1047,
and
DK(s) = s8+ 5.444 ·109s7+ 5.525 ·1015s6
+2.3·1022s5+ 1.718 ·1028s4+ 8.868 ·1033s3
+5.922 ·1039s2+ 2.5·1044s+ 1.5·1045.
The ob ained con olle Eq. (13) has high o de ,
and can be u he , educed by examining K(s)Han-
kel singula alues σi. Hankel singula alues based
model educ ion ou ines a e g ouped by he ypes o
e o bound. In Balanced T unca ion (BT) and ela ed
me hods, an e o bound is a measu e o how close he
educed o de con olle K (s)is o he o iginal sys-
em and is compu ed based on he in ini y no m o he
addi i e e o ,
||K(s)−K (s)||∞=
n
X
+1
(σi),(14)
wi h
K =NK (s)
DK (s).(15)
The basic idea o BT elies on balancing he wo
con olle s’ con ollabili y G amian and ope abili y
G amian [25]. The Hankel singula alues plo ed in
Fig. 5 a e used o decide which s a es o he con olle
can be sa ely disca ded. To achie e a leas 1 % ela-
i e accu acy, he lowes -o de con olle K (s)should
be compa ible wi h he desi ed le el o accu acy chosen
o be 5.
10-3
10-2
10-1
100
101
102
12345678
O de
Hankel singula alues o K
Fig. 5: Hankel singula alues o K(s).
The unc ion " educe" is he ga eway o all model e-
duc ion ou ines a ailable in he oolbox MATLAB. We
use he de aul , squa e- oo balance unca ion (’bal-
ancm ’) op ion o " educe" as he i s s ep. This
me hod uses an "addi i e" e o bound o he abo e
desc ibed educ ion me hod, meaning ha i ies o
keep he absolu e app oxima ion e o uni o mly small
o all equencies.
The e o bound o addi i e-e o algo i hms is de-
ined as:
||K(s)−K (s)||∞= 2(σ6+σ7+σ8) = 0.0088,(16)
which yields:
NK (s)=1.045 ·104s4−3.36 ·109s3+ 5.937 ·1015s2
−1.393 ·1021s+ 6.808 ·1026,
DK (s) = s5+ 2.848 ·105s4+ 4.018 ·1012s3
+1.128 ·1017s2+ 1.43 ·1024s+ 7.464 ·1024.
Acco ding o condi ion Eq. (6), i is necessa y ha
he magni ude esponse o Slies bellow he magni ude
esponse o W−1
1in he whole equency ange, and
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 554
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
he magni ude esponse o Tshould lie bellow he e-
sponse o W−1
3. Figu e 6 shows ha hese condi ions
a e e i ied using he educed con olle .
-10
-20
-30
-40
-50
10
20
30
40
0
102103104105106
F equency ( ad )·s-1
Singula Values (dB)
Singula Values
Fig. 6: MS-LS con ol design esul s.
4. 1-DOF PID and 2-DOF
PID Con olle s Design
In he ollowing sec ion, he con ol sys em s uc u e o
Fig. 7 is adop ed, whe e C(s)is he s anda d con olle ,
CF(s) he inpu il e , and Gis he con e e ans e
unc ion.
Cs
F() Cs() Gs()
d
y
u
Fig. 7: 2-DOF PID s uc u e.
The s anda d PID con olle is used wi h he ans e
unc ion:
C(s) = KP+KI
s+KDs
T s+ 1,(17)
wi h he p opo ional gain KP, he in eg a o gain KI,
he de i a i e gain KD, and he de i a i e il e ime
cons an T .
4.1. 2-DOF PID Con olle
The ou pu signal o a 2-DOF egula o is de ined as:
u(s) = KPep+KIeI+KDeD,(18)
whe e 








ep=b (s)−y(s)
eI(s) = 1
s (s)−y(s)
eD(s) = c (s)−y(s)
,(19)
whe e b,ca e weigh ing pa ame e s o p opo ional
e m and de i a i e e m, espec i ely. The 2-DOF
con olle can be ans o med in o a 1-DOF con olle ,
i band ca e selec ed o be equal o 1. To o mula e he
closed loop ans e unc ion, he ou pu o con olle
Eq. (19) is ew i en as:
u(s) = CF(s) (s)−y(s)C(s).(20)
The closed loop con ol sys em ou pu o he pe u -
ba ion is gi en by:
y(s) = CF(s)G(s)
1 + C(s)G(s) (s) + G(s)
1 + C(s)G(s)d(s).(21)
The sys em closed loop ans e unc ion is de ined as:
Ty (s) = CF(s)C(s)G(s)
1 + C(s)G(s).(22)
The pa ame e s {KP, TI, TD, b, c}a e ob ained con-
side ing he a ge ed speci ica ions.
4.2. F equency Speci ica ions
To ensu e ha he ou pu ol age acks he e e ence
wi h a desi ed esponse ime and acking e o , ans-
e unc ion is used o speci y he maximum equency-
domain acking e o :
emax =Aes+ωcDe
s+ωc
,(23)
whe e ωc= 2/ s( sis he se ling ime) is he acking
bandwid h, Deis he maximum ela i e s eady-s a e
e o and Aeis he peak ela i e e o ac oss all e-
quencies. Fo he QBC, we se De= 0.001,Ae= 1
and ωc= 103 ad·s−1, since he open loop-gain should
be high wi hin he con ol bandwid h. To ensu e good
dis u bance ejec ion, he minimum loop gain p o ile is
chosen as:
Ws=0.03wc
s.(24)
To ensu e insensi i i y o measu emen noise, he
open loop gain should be less han 1 ou side he con ol
bandwid h, so he maximum loop gain p o ile is chosen
as:
WT=0.3wc
s.(25)
Using so wa e p o ided by Ma lab, he abo e e-
qui emen s a e con e ed in o no malized scala s unc-
ions (x)and g(x)such as:
g(x) = 






1
emax T(s, x)





∞
.(26)
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 555

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
(x) = 









WsSa
W−1
TTa









∞
,(27)
whe e T(s, x) = L
1 + Lis he ou pu complemen-
a y sensi i i y unc ion, L(s, x)is he open-loop e-
sponse being shaped, Ta=D−1TD is he scaled
ou pu complemen a y sensi i i y unc ion, Sa=
D−11 + L(s, x)−1Dis he scaled ou pu sensi i -
i y unc ion, xis he ec o o ee ( unable) pa ame e s
KP,KIand KD.
Then, de e mining he PID pa ame e s is equi alen
o sol ing he op imiza ion p oblem:
min
xmaxa (x), g(x),(28)
whe e a > 0is a pa ame e weigh ing he subp oblems
impo ance in o de o ge he mos op imal solu ion
o he op imiza ion p oblem. Nonsmoo h op imiza-
ion algo i hms [26] and [27] a e used o sol e he QBC
con e e con ol p oblem. Acco ding o he equi ed
desi able pe o mance, he ul ima e PIDs pa ame e s
a e achie ed as ollows:
1-DOF:Kp= 0.00865,KI= 230,KD=−8.03 ·10−5,
2-DOF:Kp= 0.00824,KI= 298,KD= 1.68 ·10−5,
b= 0.00015,c= 0.226.
5. Robus Analysis
We can es he obus ness p ope ies o he h ee con-
olle s by execu ing he p ope µ es s o he QBC
unce ain eedback sys em shown in Fig. 8, whe e he
dashed box ep esen s he QBC eal ans e unc ion
Gunc. The ans e unc ions Wdel and ∆Gpa ame e -
ize he mul iplica i e unce ain y a he con e e in-
pu . The ans e unc ion Wdel is assumed known, and
he ans e unc ion ∆Gis assumed o be s able and
unknown, excep o he no m condi ion ||∆G||∞<1.
The unce ain y weigh Wdel is desc ibed as:
Wdel(s) = 100s+ 7.035 ·107
s+ 7.035 ·108.(29)
con olle G
u
e
Wdel
ΔG
Gunc
y
d
Fig. 8: Unce ain eedback sys em.
The unce ain y in he inpu is 10 % in he low e-
quency ange, 100 % a w= 106Hz and 1000 % in he
high equency ange.
Figu e 9 compa es he uppe bounds o he s uc-
u ed singula alues, o he obus s abili y analysis
o he closed-loop sys ems wi h he h ee con olle s (1-
DOF PID, 2-DOF PID and MS-LS con ol). To achie e
obus s abili y, i is necessa y ha he µ- alues a e less
han 1 o e he equency ange [28] and [29]. I is clea
ha he con olle s achie e a obus s abili y. The bes
obus ness is ob ained by he MS-LS con olle .
100102104106108
0
0.02
0.04
0.06
0.08
0.1
0.12
F equency ( ad/sec)
µ uppe bounds
1DOF PID
2DOF PID
MS−LS
Fig. 9: QBC closed-loop obus s abili y.
100102104106108
F equency ( ad/sec)
0
0.2
0.4
0.6
0.8
1
1.2
7 uppe bound
1DOF PID
2DOF PID
MS-LS
Fig. 10: QBC closed-loop obus pe o mance.
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 556
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
Ou 1
e
PID(s)
PID Con olle Sa u a ion Add Relay Quad_Con e e 2
Quad_Con e e 1
Quad_Con e e 3
Relay1
Relay2
Add1
Add2
Sa u a ion1
Sa u a ion2
Repea ing
Sequence1
Scope6
Band-Limi ed
Whi e Noise
u
u
u 2
2
2
B
B
C
C
D
D1
A
A
In eg a o 1
In eg a o
1
s
1
s
1
T s+1·
D
b
b
b
b1
Ki
Ki
Kd
Kd
2 - DOF
PID
pwm
Mixed - Sensi i i y L.S. Con olle
Fig. 11: Simula ion block diag am.
The obus pe o mance is achie ed i and only i o
each equency compu ed o he closed-loop equency
esponse is less han 1. The obus pe o mance es s
o he h ee con olle s a e shown in Fig. 10. Again,
he MS-LS con olle shows la ge µ alues o e he low-
equency ange.
6. Simula ion Resul s
As shown in Fig. 11, he h ee con olle s de-
signed in he abo e sec ion a e implemen ed in Ma -
lab/Simulink. No e ha MS-LS con olle K (s)
is implemen ed using he s a e space ealiza ion
(A, B, C, D). To compa e he h ee con olle s’ pe -
o mances and obus ness, he ollowing es s a e pe -
o med.
6.1. Se Poin T acking
The QBC esponse o a 10 V cons an e e ence ol -
age is shown in Fig. 12. I can be obse ed ha QBC
se ling ime is 1 ms o he MS-LS con ol and 2-DOF
PID con ols, which is as e as compa ed o he 1.6 ms
se ling ime o he 1-DOF PID con ol. Ano he as-
pec is ha MS-LS con ol exhibi s an o e shoo o
16.5 %; while he o e shoo o he 1-DOF PID and
2-DOF PID is o he o de o 15.5 %.
In addi ion, when he e e ence inpu ol age
changes om 10 o 12 V, as shown in Fig. 13, an oscil-
la o y beha io is obse ed o MS-LS con ol. Com-
pa ison o ha PID con olle s p o ide a mo e dumped
beha io , in esponse o he e e ence ol age inc ease
01234
ime (sec) #10-3
0
2
4
6
8
10
12
14
16
18
2
Re e ence
1DOF PID
2DOF PID
MS-LS
0.8 1 1.2
#10-4
14
15
16
17
Fig. 12: QBC esponse o 10 V e e ence ol age.
o dec ease, as can be no iced om Fig. 14 and Fig. 15.
6.2. Load Va ia ion
A load a ia ion o 100 % ( om 10 o 20 Ω) is in o-
duced be ween 10 and 30 ms. QBC esponse shown
in Fig. 16 indica es ha all con ol me hods p o ide
almos he same pe o mance. The ou pu ol age ex-
hibi s an unde shoo o 6 % and an o e shoo o 8 %.
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 557
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
4 5 6 7 8 9 10
ime (sec) #10-3
9.5
10
10.5
11
11.5
12
12.5
2
Re e ence
1DOF PID
2DOF PID
MS-LS
Fig. 13: QBC esponse o e e ence ol age change om 10 o
12 V.
0.016 0.017 0.018 0.019 0.02
ime (sec)
9.5
10
10.5
11
11.5
12
12.5
2
Re e ence
1DOF PID
2DOF PID
MS-LS
Fig. 14: QBC esponse o e e ence ol age change om 12 o
10 V.
6.3. Supply Vol age Va ia ion
A ol age d op o 3 V is in oduced in he supply ol -
age be ween 5 and 10 ms. Figu e 17 shows ha he
h ee con ol me hods ob ained almos he same s abi-
lizing ime wi h same unde shoo (abou 7 % a 5 ms).
A 10 ms, simila o e shoo (35 %) is obse ed o he
h ee con ol me hods. Howe e , a 10 ms, PID con ol
0.02 0.0205 0.021 0.0215 0.022 0.0225 0.023 0.0235 0.024
ime (sec)
7.5
8
8.5
9
9.5
10
10.5
2
Re e ence
1DOF PID
2DOF PID
MS-LS
Fig. 15: QBC esponse o e e ence ol age change om 10 o
8 V.
0.01 0.015 0.02 0.025 0.03
ime (sec)
9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
10.8
11
2
Re e ence
1DOF PID
2DOF PID
MS-LS
Fig. 16: QBC esponse o load esis ance change.
me hods exhibi a null unde shoo compa ed o 3 % o
MS-LS con ol.
6.4. Dis u bance Rejec ion
The supply ol age is pe u bed by a sinusoidal com-
ponen o 100 Hz equency and 2 V peak- o-peak am-
pli ude. Fu he , senso whi e noise, o 108 ad·s−1
equency and 10−5powe , is assumed o be supe -
posed on he ou pu ol age. Figu e 18 shows ha
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 558
POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 5 |2016 |DECEMBER
4 5 6 7 8 9 10 11 12
ime (sec) #10-3
9
9.5
10
10.5
11
11.5
12
12.5
13
13.5
14
2
Re e ence
1DOF PID
2DOF PID
MS-LS
Fig. 17: QBC esponse o supply ol age change.
0 0.002 0.004 0.006 0.008 0.01
ime (sec)
0
2
4
6
8
10
12
14
16
18
2
Re e ence
1DOF PID
2DOF PID
MS-LS
6.5 7 7.5 8
#10-3
9.5
10
10.5
Fig. 18: QBC esponse o sinusoidal componen in powe sou ce
and whi e noise.
2-DOF PID con olle has be e dis u bance ejec ion
han 1-DOF PID and MS-LS con olle s in his band
o equencies.
7. Conclusion
A 2-DOF PID con olle is p oposed, designed and
simula ed o he quad a ic buck con e e . Fo com-
pa ison pu pose, 1-DOF PID and MS-LS con ol con-
olle s a e also es ed. E en i MS-LS con ol shows
a as e esponse, 2-DOF PID p o ides mo e duped
and accu a e esponse. Fu he , unde pe u ba ions
and unce ain ies, 2-DOF PID con ol exhibi s be e
obus ness in pe o mance and s abili y compa ed o
MS-LS con ol. Ano he p ac ically impo an ad an-
age o he 2-DOF PID is a lowe s uc u e complexi y
compa ed o MS-LS con ol.
Re e ences
[1] BACHA, S., I. MUNTEANU and A. I. BRATCU.
Powe elec onic con e e s modeling and con-
ol: wi h case s udies. London: Sp inge , 2014.
ISBN 978-1-4471-5477-8.
[2] TOLLE, T., T. DUERBAUM and R. ELFERICH.
Swi ching loss con ibu ions o synch onous ec i-
ie s in VRM applica ions. In: IEEE 34 h Annual
Powe Elec onics Specialis Con e ence 2003. Pis-
ca away: IEEE, 2003, pp. 144–149. ISBN 0-7803-
7754-0. DOI: 10.1109/PESC.2003.1218287.
[3] MATSUO, H. and K. HARADA. The cas-
cade connec ion o swi ching egula o s. IEEE
T ansac ions on Indus ial Applica ions. 1976,
ol. IA-12, iss. 2, pp. 192–198. ISSN 0093-9994.
DOI: 10.1109/TIA.1976.349401.
[4] MAKSIMOVIC, D. and S. CUK. Swi ching
con e e s wi h wide DC con e sion ange.
IEEE T ansac ions on Powe Elec onics. 1991,
ol. 6, iss. 6, pp. 151–157. ISSN 0885-8993.
DOI: 10.1109/63.65013.
[5] BARBOSA, L. R., J. B. VIEIRA, L. C. FRE-
ITAS, M. S. VILELA and V. J. FARIAS. A
buck quad a ic PWM so -swi ching con e e us-
ing a single ac i e swi ch. IEEE T ansac ions on
Powe Elec onics. 1999, ol. 14, iss. 3, pp. 445–
453. ISSN 0885-8993. DOI: 10.1109/63.761688.
[6] PACHECO, V. M., A. J. DO NASCIMENTO, V.
J. FARIAS, J. B. VIEIRA and L. C. DE FRE-
ITAS. A quad a ic buck con e e wi h lossless
commu a ion. IEEE T ansac ions on Indus ial
Elec onics. 2000, ol. 47, no. 2, pp. 264–272.
ISSN 0278-0046. DOI: 10.1109/41.836341.
[7] REYES-MALANCHE, J. A., N. VAZQUEZ
and J. LEYVA-RAMOS. Swi ched-capaci o
quad a ic buck con e e o wide con e sion
a ios. IET Powe Elec onics. 2015, ol. 8, iss. 12,
pp. 2370–2376. ISSN 1755-4535. DOI: 10.1049/ie -
pel.2014.0755.
c
2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 559