PID, 2-DOF PID and mixed sensitivity loop-shaping based robust voltage control of quadratic buck DC-DC converter

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. 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