Fixed-switching frequency sliding mode control applied to power converters

Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s Thesis submi ed in pa ial ul illmen o he equi emen o he PhD Deg ee issued by he Uni e si a Poli ´ecnica de Ca alunya, in i s Elec onic Enginee ing P og am. V´ıc o Repecho Del Co al Ad iso : Domingo Biel Sol´e Ba celona, Decembe 2017 Aknowlegdemen s This hesis is he esul o almos ou yea s o in ensi e wo k, dedica ion and e o s. All he hou s spen in he labo a o y, he discussions a ound a blackboa d o he ealiza ion o scien i ic a icles ha e led o he p esen hesis, o which I am p oud o . I would like o show my g a i ude o all people ha ha e helped and suppo ed me du ing his exci ing ime, wi hou whom he de elopmen o his hesis would no ha e been possible. Fi s o all, i would like o exp ess my g a e ul hanks o my ad iso D . Domingo Biel, who no only ga e me he oppo uni y o become a pa o he elec onics and con ol labo a o y in he Uni e si y, bu also ha e managed his hesis wi h dedica ion, e o and pa ience. I eel uly g a e ul o Ra el Ca done , o his aluable ad ices abou powe elec onics implemen a ions and mic o con olle s p og amming, which ha e been indispensable o he expe imen a ion e alua ions o his wo k. Many hanks also o En ic Mi ´o o his help in he daily wo k a he labo a o y. I would like o exp ess my g a i ude owa ds Ra ael Ramos, who p o ided me his knowledge abou FGPA p og amming equi ed in one o he expe imen al es s. I am also g a e ul o he Ad anced Con ol o Ene gy Sys ems (ACES) g oup, which ha e con ibu ed o his wo k wi h he suppo o i s esea ch membe s. Speci ically, i would like o show my g a i ude o Josep Ma ´ıa Olm, An oni A ias, En ic Fossas, Robe G i˜n´o and A nau D`o ia o hei collabo a ion in se e al aspec s o he hesis de elopmen . Finally, i wan o hank my gi l Pili o he unde s anding and encou agemen and, mos o all, i would like o hank my amily and specially my pa en s, Isid o and Daniela, o whom his hesis is dedica ed. V´ıc o Repecho del Co al, Ba celona, Janua y 2018 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s iii Abs ac The applica ion o he sliding mode con ol in powe con e e s has a well-known incon- enien om he p ac ical poin o iew, which is o ob ain ixed swi ching equency implemen a ions. This hesis deals wi h he de elopmen o a hys e esis band con olle in cha ge o ixing he swi ching equency o he sliding mo ions in powe elec onics ap- plica ions. The p oposed con ol measu es he swi ching pe iod o he con ol signal and modi ies he hys e esis band o he compa a o in o de o egula e he swi ching equency o he sliding mo ion. The p oposed s uc u e becomes in an addi ional con ol loop aside om he main con ol loop implemen ing he sliding mode con olle . In he i s pa o he hesis, he swi ching equency con ol sys em is modelled and a design c i e ia o he con ol pa ame e s a e de i ed o gua an eeing closed-loop s abili y, unde di e en app oaches and aking in o accoun he mos expec able wo king scena ios. In he second pa o he hesis, he p oposed s a egies a e alida ed in se e al powe con e e s. Spe- ci ically, DC- o-DC and DC- o-AC powe con e e s a e assembled and he expe imen al esul s a e shown. In his pa , he p ocedu es used o implemen ing he con olle s a e also deeply discussed. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s Con en s Con en s ii I In oduc ion and P oblem S a emen 1 1 In oduc ion 3 1.1 The swi ched powe con e e s .......................... 3 1.2 Con ol echniques in Swi ched Powe Con e e s ............... 5 1.3 Ideal Sliding Mo ion ............................... 7 1.4 Real Sliding Mo ion ................................ 9 1.5 P oposed Solu ions o he Va iable Swi ching F equency P oblem ...... 11 1.5.1 Va iable Hys e esis Band .......................... 12 1.5.2 Ex e nal Synch oniza ion Signal ...................... 12 1.5.3 Ze o A e age Dynamics .......................... 13 1.5.4 PWM-Based SMC ............................. 15 1.5.5 Addi ional Con ol Loop .......................... 15 1.6 Thesis objec i es ................................. 16 1.7 Thesis s uc u e .................................. 18 II Theo e ical Analysis o he P oposed Solu ion and S udy o he es- ul ing Sliding Dynamics 21 2 Modelling and s abili y analysis o he swi ching equency con ol loop 23 2.1 Open loop case: The ixed hys e esis band compa a o ............ 23 2.2 Closed-loop case: A disc e e- ime modelling. .................. 25 2.2.1 The egula ion case ............................. 27 2.2.2 The acking case .............................. 29 2.3 Closed-loop case: A con inuous- ime modelling. ................ 34 3 A case o s udy: SFC design and simula ion esul s 39 3.1 Design o he sliding mode con olle ...................... 39 3.2 The egula ion case in he disc e e- ime app oach ............... 41 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s ii CONTENTS 3.3 The egula ion case in he con inuous- ime app oach .............. 43 3.4 The acking case ................................. 46 4 Real sliding dynamics in a swi ching equency con ol loop 53 4.1 The egula o m app oach ............................ 54 4.2 Fixed hys e esis band ............................... 56 4.3 Case s udy: he Buck con e e ......................... 59 4.3.1 Ma hema ical model ............................ 59 4.3.2 Simula ion esul s .............................. 60 4.4 Time- a ying hys e esis band .......................... 64 4.4.1 The egula ion case ............................. 64 4.4.2 The acking case .............................. 68 4.4.3 Simula ion esul s .............................. 70 III Applica ion o he P oposed Solu ion o Powe Con e e s. 77 5 Vol age Regula ion in a Buck Con e e . 79 5.1 The Buck Con e e ............................... 79 5.2 Sliding mode con ol o he Ou pu ol age ................... 80 5.2.1 Swi ching su ace design .......................... 80 5.2.2 Sliding dynamics .............................. 81 5.2.3 Con ol law ................................. 82 5.3 Swi ching equency egula ion .......................... 82 5.3.1 E alua ion o ρ± k.............................. 82 5.3.2 SFC design ................................. 83 5.4 Implemen a ion De ails .............................. 83 5.5 Expe imen al esul s ............................... 86 5.6 Conclusions .................................... 92 6 Vol age Regula ion in a Mul iphase Buck Con e e . 93 6.1 The mul iphase con e e ............................. 93 6.2 In e lea ed sliding mode con ol o he ou pu ol age ............. 95 6.2.1 Mas e swi ching su ace design ...................... 95 6.2.2 Sliding dynamics o he Mas e phase ................... 96 6.2.3 Mas e phase con ol law .......................... 97 6.2.4 Sla es swi ching su aces design. In e lea ed Sliding Mode ....... 97 6.3 Swi ching equency egula ion .......................... 98 6.3.1 E alua ion o ρ± k.............................. 98 6.3.2 SFC design ................................. 99 6.4 Implemen a ion De ails .............................. 100 6.5 Expe imen al esul s ............................... 102 6.6 Conclusions .................................... 108 iii Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CONTENTS 7 Vol age Regula ion in a Boos Con e e . 109 7.1 The Boos Con e e ............................... 109 7.2 Sliding mode con ol o he ou pu ol age ................... 110 7.2.1 Swi ching su ace design .......................... 110 7.2.2 Sliding dynamics .............................. 111 7.2.3 Con ol law ................................. 113 7.3 Swi ching equency egula ion .......................... 113 7.3.1 E alua ion o ρ± k.............................. 113 7.3.2 SFC design ................................. 113 7.4 Implemen a ion De ails .............................. 114 7.5 Expe imen al esul s ............................... 116 7.6 Conclusions .................................... 120 8 Vol age T acking in a Vol age Sou ce In e e . 121 8.1 The ol age sou ce in e e ............................ 121 8.2 Sliding mode acking o he ou pu ol age .................. 123 8.2.1 Swi ching su ace design .......................... 123 8.2.2 Ideal sliding dynamics ........................... 126 8.2.3 Con ol law ................................. 127 8.2.4 Sliding dynamics o pu e esis i e load .................. 127 8.2.5 Sliding dynamics o eac i e linea load ................. 131 8.2.6 Sliding dynamics o nonlinea load .................... 134 8.3 Swi ching equency egula ion .......................... 139 8.4 Implemen a ion de ails .............................. 141 8.4.1 E ec s o he hys e esis compa a o disc e iza ion ............ 141 8.4.2 Digi al emula ion o he ideal hys e esis compa a o ........... 143 8.4.3 Swi ching unc ion digi aliza ion issues .................. 145 8.4.4 Digi al emula ion o he SFC o a acking con ol ask ......... 146 8.4.5 Es ima ion o swi ching unc ion slopes .................. 147 8.4.6 Con olle implemen a ion ......................... 149 8.4.7 Assembled con e e and de ices employed ................ 150 8.5 Expe imen a ion esul s ............................. 152 8.5.1 SFC pe o mance .............................. 152 8.5.2 SMC pe o mance .............................. 154 8.5.3 Nonlinea load es ............................. 159 8.6 Conclusions .................................... 161 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s ix CHAPTER 1. INTRODUCTION Pin Swi ched Con ol Inpu Pou u d Powe Con e e c PWM Figu e 1.2: Swi ched powe con e e scheme, whe e he eal con ol inpu , u, and he a e aged con ol inpu , d, a e depic ed. Fo mos o he applica ions, he PWM-based linea con ol heo y is enough o p o ide a desi able pe o mance in he sys ems, bu , o speci ic applica ions, a di e en con ol echnique could be necessa y. Mainly, he d awbacks o he PWM-based linea con ol echniques come om he used linea ized model. Since he conside ed models o he con olle s design a e local, when he powe con e e mo es away om he nominal con- di ions he pe o mance can be comp omised. Addi ionally, he a e aged models assume known he elemen s a ec ing he sys ems dynamics as induc o s and capaci o s. Wha e e d i in hese alues om he nominal ones can also nega i ely a ec he con ol pe o m- ance. The Sliding Mode Con ol (SMC) [7] cons i u es an al e na i e o he classical PWM-based linea con olle s. This me hodology, which can be classi ied as a nonlinea con ol echnique [8,9], p o ides bene i s o he powe con e e s con ol. Besides he well- known obus ness, he o de educ ion and a as ansien esponse, he i s o de SMC uses as con ol ou pu s non-con inuous signals wi h wo allowable s a es, which pe ec ly ma ches wi h he physical s uc u e o he powe con e e s, ansis o s ac ing as swi ches. Mo eo e , he me hodology wo ks wi h he sys em s a e space equa ions wi hou linea - iza ion p ocess, and as a consequence, he p o ided s abili y condi ions a e, in gene al, mo e global. O cou se, i exis s an impo an incon enien ha his o ically has limi ed i s p ac ical applica ion: how o achie e a bounded swi ching equency o ope a ion. This chap e b ie ly in oduces he main idea o he sliding mode con ol heo y. This analysis no only help us o demons a e ha in a eal applica ion he swi ching equency unde sliding mo ion becomes a iable bu also o jus i y he basis o he new p oposal in o de o ge a ixed swi ching equency. Finally, a sho lis o he p oposed app oaches o sol ing his incon enien ound in he li e a u e un il now is e iewed. 6 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 1. INTRODUCTION 1.3 Ideal Sliding Mo ion Fi s o all, an a ine nonlinea sys em is conside ed ˙x= (x) + g(x)u(1.1) whe e he con ol signal u akes alues om a disc e e se wi h wo alues {u−, u+}. The s a e space equa ion ma ches wi h wha e e swi ched powe con e e wi h a single con ol inpu . Indeed, such equa ion could ep esen a bi a y sys ems, no only powe con e e s. The basis o he SMC is o design a swi ching unc ion, σ(x), which depends on he s a e ec o , and en o ces his unc ion o be in a s a e space egion whe e he desi ed sys em dynamics is achie ed, in gene al σ(x) = 0. The mos simple swi ching unc ion de ini ion is σ(x) = ex, whe e ex=x∗−xis he acking e o , being x∗ he desi ed alue o he s a e space ec o . I is immedia e o demons a e ha i he con ol gua an ees ha σ(x) = 0, he desi ed and eal s a e space ec o a e equal. The e o e, he con ol objec i e is o ensu e ha he unc ion σ(x) = x∗−xalways con e ges o he space egion de ined by σ(x) = 0. This condi ion is sa is ied when ˙σ(x)σ(x)<0 (1.2) is ul illed. The p e ious conside a ion can be p o ed jus aking he Lyapuno unc ion candida e V(x)=0.5σ(x)2, whe e he i s ime de i a i e yields ˙ V(x) = ˙σ(x)σ(x). As he unc ion V(x)=0.5σ(x)2 ul ils all he condi ions o be a Lyapuno unc ion [8], he condi ion de ined in (1.2) is enough o ensu e ha σ(x) = 0 is an a ac i e egion o σ(x). When he swi ching unc ion σ(x) is on he desi ed swi ching su ace σ(x) = 0, i is called ha he sys em is unde sliding mo ion. Ne e heless, acco ding o he sys em ela i e deg ee [7], he cons uc ion o he su ace may include highe o de s elemen s o he acking e o . I is simple o igu e ou ha he way o impose he condi ion ˙σ(x)σ(x)<0 will be h ough ˙σ(x), since σ(x) does no depend on con ol. In o he wo ds, in o de o en o ce sliding mo ion on σ(x) = 0, he con ol ac ion, u, has o appea in ˙σ(x). The equa ion (1.3) shows he case whe e he swi ching unc ion includes he acking e o and i s i s ime de i a i e σ(x) = φ1ex+φ2˙ex(1.3) whe e φ1, φ2a e s ic ly posi i e cons an s. In he ini ial case, whe e σ(x) = ex, when he sys em is unde sliding mo ion au oma ically implies ha ex= 0 and x acks x∗wi hou dynamics. This ac does no happen o he swi ching unc ion de ined in (1.3), since, when he swi ching unc ion is on he su ace σ(x) = 0, he ollowing i s o de linea di e en ial equa ion go e ns he e o dynamics: φ1ex+φ2˙ex= 0 (1.4) Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 7 CHAPTER 1. INTRODUCTION which has ex= 0 as unique asymp o ically s able equilib ium poin . The e o e, i is clea ha he e o dynamics unde sliding mo ion will be de e mined by he selec ed swi ching su ace. Rega dless o he swi ching unc ion o de , he con ol law o uhas o ensu e ha condi ion (1.2) always holds. The p e ious condi ion will be gua an eed by a discon inuous con ol law, aking he a ailable disc e e alues, p e iously de ined (u+and u−). I is e y common o use a sign unc ion as he con ol, since his unc ion gene a es he alues 1 o -1, which ma ch wi h alues o u+and u− o se e al con e e s. In gene al, he con ol law will be o he o m u=sign(σ(x)) (1.5) whe e he unc ion sign() p o ides 1 when σ≥0 and -1 when σ < 0. Ne e heless, some con e e s ha e con ol inpu s a es ha do no co espond o 1 o -1, being o ins ance 1 o 0. In hese cases, he con ol law will be modi ied acco dingly. In any case, since uis discon inuous, om a heo e ical poin o iew, his signal will swi ch a in ini e swi ching equency when σ(x) is ze o. This assump ion allows o analyse he ideal sliding dynamics, since a in ini e swi ching equency he swi ching unc ion will be placed exac ly on σ(x) = 0. S udying he ideal sliding dynamics is he p ocess o de i e he dynamics o he sys em de ined in (1.1) wi h he dynamics imposed by σ(x) = 0. This ask can be ca ied ou using he equi alen con ol me hod p oposed by P o esso Vadim U kin [7]. The equi alen con ol, ueq, is de ined as he solu ion o he equa ions σ= 0,˙σ(x, ueq) = 0.(1.6) Taking as swi ching unc ion an exp ession depending on he e o , ex=x−x∗, as: σ(x) = µ(ex),(1.7) he swi ching unc ion i s ime de i a i e is ound as: ˙σ(x) = ∂σ(x) ∂x (x) + ∂σ(x) ∂x g(x)u. (1.8) Acco ding o (1.6), he equi alen con ol can be ob ained sol ing he equa ion 0 = ∂σ(x) ∂x (x) + ∂σ(x) ∂x g(x)ueq,(1.9) which yields ueq =−∂σ(x) ∂x g(x)−1∂σ(x) ∂x (x).(1.10) No ice ha (1.10) implies a con inuous- ime solu ion, which ucanno a ain. Finally, u is eplaced by ueq in (1.1) in o de o ind he ideal sliding dynamics. Placing (1.10) in o (1.1), one ge s: ˙x= (x) + g(x)ueq.(1.11) 8 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 1. INTRODUCTION Fu he mo e, acco ding o [7], he space egion whe e he sliding mo ion occu s, which is called sliding domain, is cha ac e ized by he inequali y u−< ueq(x, )< u+.(1.12) 1.4 Real Sliding Mo ion The eal sliding mo ion can be unde s ood as he sys em dynamics ha a ises when he con ol ac ion, u, is o ced o ope a e a ini e swi ching equency. The employmen o a sign unc ion in eal sys ems leads o a ini e swi ching equency, due o unmodelled dynamics and delays. Howe e , his swi ching equency can be oo high o some sys ems, like powe con e e s. Subs i u e he sign unc ion on (1.5) by a compa a o wi h hys e esis, allows o bound he swi ching equency. The con ol law, he e o e, will be o he o m: u=u+i σ < −∆, u−i σ > ∆,(1.13) being ∆ he hys e esis wid h. This app oach can be ound in sliding mode con ol li e a u e in o de o s udy he eal sliding dynamics [10–12]. F om a igo ous poin o iew, he p e ious con ol law is no de ined wi hin he hys e esis bands, so an al e na i e exp ession can be used ins ead: u=u+i σ < −∆ o (|σ|<∆ & ˙σ > 0) u−i σ > ∆ o (|σ|<∆ & ˙σ < 0) .(1.14) The di e en beha iou o he sys em ajec o ies on he phase plane can be obse ed in Figu e 1.3. ex ˙ex ex ˙ex (a) (b) σ(x) = 0 σ(x) uu σ(x) = 0 σ(x) ∆ −∆ σ(x)σ(x) ∆ −∆ Figu e 1.3: Phase plane sys em ajec o ies. (a) Ideal sliding mo ion. (b) Real sliding mo ion wi hin a bounda y laye ∆. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 9 CHAPTER 1. INTRODUCTION As Figu e 1.3 shows in he igh side, he eal swi ching unc ion, σ(x), is no on he sliding su ace, σ= 0, bu cha e s in i s icini y. In his scena io, he condi ion |σ(x)|<∆ holds. I is a designe ask o keep he hys e esis band alue small enough, in he way ha his dynamics can be neglec ed wi h espec o he low equency dynamics, de e mined by he equi alen con ol. Analysing he high equency dynamics o σ(x) p o ides he exp ession ha allows o s udy and con ol he swi ching equency o he con ol ac ion. A good app oach consis s in de ining he con ol ac ion as a con inuous low equency componen , which is ueq, plus a high equency componen , uh , which akes alues in he se {u−, u+}. u=ueq +uh (1.15) Le us jus combine he exp essions (1.15) and (1.8). ˙σ(x) = ∂σ(x) ∂x (x) + ∂σ(x) ∂x g(x)ueq +∂σ(x) ∂x g(x)uh (1.16) By de ini ion, ˙σ(x, ueq) = 0, he e o e ˙σ(x) = ∂σ(x) ∂x g(x)uh (1.17) The mo ion o σ(x) in he icini y o σ(x) = 0 is go e ned by (1.17). F om his exp ession i is clea ha his dynamics depends on he sys em. O cou se, he dynamics o he s a e space ec o can be also s a ed as: ˙x= (x) + g(x)ueq +g(x)∂σ(x) ∂x g(x)−1 ˙σ(x).(1.18) The swi ching unc ion ajec o ies wi hin he hys e esis band a e go e ned by he las e m o he igh hand side o equa ion (1.18). Figu e 1.4 depic s he expec ed beha iou . σ(x)=0 ∆ −∆ 1 2 T σ(x) ˙σu=u+ ˙σu=u− Figu e 1.4: Swi ching unc ion beha iou wi hin he hys e esis band in he icini y o σ(x) = 0. 10 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 1. INTRODUCTION F om Figu e 1.4 he swi ching pe iod o he con ol ac ion is de i ed: T= 1+ 2=2 ∆ ˙σu=u+−2 ∆ ˙σu=u− (1.19) Exp ession (1.19) can be w i en in di e en ways. One in e es ing app oach is o o mula e i as a unc ion o he equi alen con ol. Recalling (1.15) and (1.17), one ge s: ˙σ(x) = ∂σ(x) ∂x g(x)(u−ueq) (1.20) and, he e o e, assuming ha u+= 1 and u−=−1, (1.19) boils down o T= 4 ∆ 1 (1 −u2 eq)∂σ(x) ∂x g(x)−1 .(1.21) No ice ha om (1.21) he hys e esis band o he compa a o can be calcula ed as a unc ion o he equi alen con ol and he desi ed swi ching pe iod, T∗, as: ∆ = ∂σ(x) ∂x g(x)T∗ 4(1 −u2 eq).(1.22) Howe e , his app oach has an impo an limi a ion, which will be discussed la e . Mo eo e , exp ession (1.21) con i ms he well-known p oblem o he SMC applied o swi ched powe con e e s, he swi ching equency is a iable and sys em dependen , as (1.21) s a es. A e y impo an aspec should be ema ked a his poin . In he p e ious me hod- ology, i has been assumed ha he slopes o σ(x) emain cons an along he swi ching in e al, ha is ¨σ(x) = 0. This is a e y easonable assump ion ha is o en aken in he sliding mode con ol li e a u e [11,13,14]. The basis o his hypo hesis elies in conside he swi ching pe iod o he con ol ac ion small enough wi h espec o he sys em ime con- s an s. I his conside a ion holds, i is easonable o assume ha he swi ching unc ion slopes a e locally cons an du ing a swi ching in e al. Indeed, he swi ching equency in he powe con e e s a e designed as high as possible, due o he eason in oduced in Sec ion 1.1, which pe ec ly i s wi h his assump ion. 1.5 P oposed Solu ions o he Va iable Swi ching F e- quency P oblem A his s age, some solu ions p oposed in he li e a u e in o de o se he swi ching e- quency in he powe con e e s unde SMC a e e iewed. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 11 CHAPTER 1. INTRODUCTION 1.5.1 Va iable Hys e esis Band An in ui i e solu ion is o adjus he compa a o hys e esis band using he exp ession (1.22), modi ying i s le el in acco dance wi h he sys em s a e. Such app oach has been p oposed by di e en au ho s [15–21]. An equi alen scheme is depic ed in Figu e 1.5. uEqua ion (1.1) x∗ x σ(x) x ∆ ∆ −∆ Equa ion (1.22)T∗ ueq Figu e 1.5: Sys em o ope a ing a ixed swi ching pe iod unde sliding mo ion based on a a iable hys e esis band. F om Figu e 1.5, and as equa ion (1.22) shows, he hys e esis compu a ion equi es a high le el o he plan knowledge since he e m ∂σ(x) ∂x g(x) should be known. Fu he mo e, he measu emen o he equi alen con ol o , al e na i ely, i s es ima ion based on exp ession (1.10) should be also pe o med, which en ails an e iden complexi y. The p ocedu e p o ides good esul s in gene al, al hough i is di icul o achie e he desi ed swi ching equency accu a ely, due o he complexi y o gene a ing he hys e esis. This complexi y is, in ac , i s main d awback. Addi ionally, i is e iden ha i in he hys e esis calcula ion some sys em pa ame e s a e assumed known and cons an , he sys em will no be obus in he ace o pa ame ic a ia ions. In o de o imp o e he obus ness, addi ional senso s and/o obse e s ha e o be included o ge a p ope adap a ion o he hys e esis band ampli ude, leading o a u he sys em complexi y, dec easing he eliabili y and inc easing he cos up o unmanageable le els. 1.5.2 Ex e nal Synch oniza ion Signal Fixed swi ching equency can also be achie ed by using an ex e nal signal o o ce he swi ching ins an s [22,23]. The idea is ske ched a Figu e 1.6, speci ically in he le side. The synch oniza ion signal, D( ), is added o he swi ching unc ion in he way ha such signal con ols he swi ching e en s. As i is depic ed in he Figu e, he peak alues o signal D( ) should be highe han he hys e esis alues, ∆, as D( ) can con ol he 12 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 1. INTRODUCTION swi ching e en s. Th ough a complica ed uning p ocess, i is possible o adjus he signal D( ) ixing he swi ching equency o he con ol ac ion. Howe e his p ocess is ex emely sensi i e, and i is usual ha s eady-s a e e o s appea in he sys em when he wo king condi ions change. The igh pa o Figu e 1.6 ies o show his phenomenon. No ice how he swi ching unc ion (g ey lines) does no each he uppe bound o he hys e esis band, en ailing an a e age alue di e en om ze o o σ(x), leading o he a o emen ioned s eady-s a e e o . ∆ σ( ) D( ) u Tk 2∆ -∆ Tk σ( ) D( ) Figu e 1.6: Ex e nal Signal Synch oniza ion scheme o a ixed equency sliding mode con ol. Mo eo e , his app oach needs some addi ional ha dwa e on he con olle in o de o gene a e he synch oniza ion signal. As a consequence, he usage o his me hod is, in gene al, no ecommended. 1.5.3 Ze o A e age Dynamics As i was in oduced in he p e ious Sec ion, and no ed in di e en wo ks a ailable in he li e a u e, a swi ching unc ion desc ibing piecewise linea beha iou wi hin a symme ic hys e esis band compa a o implies ha i s a e age alue, along he swi ching in e al, is ze o. This is he concep exploi ed by he Ze o A e age Dynamics (ZAD), which was p esen ed in [24]. The me hod compu es a du y cycle ha gua an ees ze o T-pe iodic mean alue o he swi ching unc ion, wi h Tdeno ing he swi ching pe iod. The Figu e 1.7 shows his idea. F om he Figu e, i is possible o de elop he o mula ion deli e ing he du y cycle o be used in he nex swi ching in e al ha leads o a ze o a e aged alue o σ(x). The esul s a e shown in Table 1.1. The e o e, ixed swi ching equency is eached in he s eady-s a e, and he a e aged beha iou is close o he ideal sliding mode one. The ZAD s a egy has been success ully implemen ed in [25]. The esul s p esen ed he ein show a good pe o mance o he ZAD, bu also poin ou he equi emen o a as digi al p ocesso o sol e he complex calcu- la ions in ol ed in he du y cycle compu a ion, as exp essions in Table 1.1 co obo a e. In ac , such compu a ion complexi y cons i u es he main d awback o ZAD-based SMC ixed swi ching equency implemen a ions. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 13 CHAPTER 1. INTRODUCTION T/2T/2 ˙σ(x, )u=u+ ˙σ(x, )u=u− dKT σ(x, ) Tk−1Tk Figu e 1.7: Ze o A e age Dynamics ajec o ies de ail. Table 1.1: Ze o A e age Dynamics con ol ac ion o mulas σ(x(Tk), Tk) and σ(x(Tk), Tk) + T 2˙σ|k,u+≥0u(Tk) = u+;dk= 1 σ(x(Tk), T k) and σ(x(Tk), Tk) + T 2˙σ|k,u+<0u(Tk0) = u+;dk= 1 − u u u ˙σ|(K,u+)−2|σ[x(Tk, Tk)]| T ˙σ|(K,u+)+˙σ|(K,u−) σ(x(Tk), Tk) and σ(x(Tk), Tk) + T 2˙σ|k,u+≤0u(Tk) = u−;dk= 1 σ(x(Tk), Tk) and σ(x(Tk), Tk) + T 2˙σ|k,u+>0u(Tk0) = u−;dk= 1 − u u u ˙σ|(K,u+)−2|σ[x(Tk, Tk)]| T ˙σ|(K,u+)+˙σ|(K,u−) 14 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 1. INTRODUCTION 1.5.4 PWM-Based SMC This p oposal is based on he use o PWM a ixed swi ching equency o implemen he so-called PWM-SMC. Ini ially p oposed in [26,27], he me hod di ec ly implemen s he equi alen con ol and ob ains he swi ching ins an s by compa ing he equi alen con ol wi h he ixed equency saw oo h wa e o m a he PWM. The me hod is equi alen o he adi ional PWM implemen a ion acco ding o a linea con olle design, bu using he equi alen con ol ueq ins ead o he du y cycle p o ided by he linea con olle . The co esponding scheme is depic ed in Figu e 1.8. + - ueq T T u Figu e 1.8: PWM-based sliding mode con olle . I should be no ed ha he exp ession o he equi alen con ol depends on he sys em s a e and on i s pa ame e s o a gi en wo king condi ions, as (1.10) s a es. I is impo an o no ice ha his equa ion holds om he ideal poin o iew. The esul s p esen ed in [28] show o e all good pe o mance, bu i should be highligh ed ha he same solu ion can also be de i ed by calcula ing he equi ed du y cycle o ob ain he desi ed sys em dynamics. Indeed, om ou poin o iew, he equi alen con ol is mo e a heo e ical concep han a p ac ical me hod. F om i s de ini ion, he equi alen con ol is he con inuous con ol ac ion ha places he sys em ajec o ies exac ly on he sliding su ace. In a p ac ical implemen a ion, his includes wha e e unmodelled dynamics, delays and unce ain ies in he powe con e e . Besides, powe con e e s commonly su e ex e nal dis u bances, which gene ally can only be bounded. In o he wo ds, he e is no way o de e mine he equi alen con ol a p io i. The equi alen con ol could be measu ed low pass il e ing he hys e e ic con ol ac ion en o cing sliding mo ion in a ce ain swi ching su ace (|σ(x)|<∆), bu no ice e sa. Hence, equa ion (1.10) should be used only o heo e ical issues. Mo eo e , in his me hod, some sliding mode p ope ies, such as o de educ ion o obus ness in he ace o dis u bances, could be los . 1.5.5 Addi ional Con ol Loop Hys e e ic con olle s a e o en applied o powe con e e s wi hou using SMC heo y. Howe e , like occu in SMC app oaches, he swi ching equency becomes a iable. Indeed, in mos o he hys e e ic con olle s, sliding mo ion na u ally occu s. Unde his esea ch ield, some in e es ing solu ions in o de o ix he swi ching equency we e p o ided [29–32]. These p oposals a e based on adding an addi ional loop in o de o p ope ly adjus he hys e esis band, as Figu e 1.9 shows. The sys em uses a pe iod senso o a Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 15 Chap e 2 Modelling and s abili y analysis o he swi ching equency con ol loop This chap e deals wi h he modelling and he analysis o he s uc u e o he SFC p esen- ed in he Chap e 1. As i was p e iously in oduced, he con ol me hodology is based on he adjus men o he hys e esis band alue o he compa a o h ough an addi ional con ol loop, as Figu e 1.10 shows. Since he con ol loop will a y such hys e esis band, he analysis o he swi ching pe iod p esen ed in Sec ion 1.4 needs o be e isi ed. Fu - he mo e, his s uc u e has an addi ional peculia i y: he SFC s uc u e used o adjus he hys e esis band alue will a ec he ela ion be ween he applied hys e esis and he co esponding swi ching pe iod. In his hesis wo di e en app oaches ha e been applied, namely: 1. A disc e e- ime app oach, whe e he hys e esis band can be upda ed jus once pe swi ching cycle. 2. A con inuous- ime app oach, assuming ha he hys e esis band will be a ime- a ying signal. The chap e is s uc u ed as ollows: In Sec ion 2.1, he analysis o he ime in a ian hys- e esis case done in Sec ion 1.4 is e isi ed, being used o de ine some essen ial pa ame e s o he analysis pe o med he ea e o he SFC s uc u e. In Sec ion 2.2 he disc e e- ime app oach is p esen ed and he wo mos expec able wo king scena ios in powe con e - e s, he egula ion and he acking case p oblems, a e analysed. Finally, in Sec ion 2.3 a con inuous- ime app oach o he SFC, pa icula izing i only o he egula ion con ol p oblem, is p esen ed. 2.1 Open loop case: The ixed hys e esis band com- pa a o The ime in a ian hys e esis band ampli ude case was al eady analysed in he Sec ion 1.4. Howe e , he esul ing exp ession o he swi ching pe iod (see equa ions (1.19), (1.21)) is Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 23 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP ede ined a his poin , including some impo an de ini ions. Assume ha a swi ching unc ion, σ(x), has been de ined in o de o en o ce sliding mo ion in he subspace σ(x) = 0, wi h he objec i e o con ol he dynamics o a powe con e e ( ol age, cu en ,..). Suppose also ha he con ol law p o ided by he SMC heo y, which gua an ees con e gence o σ(x) o σ(x) = 0, is implemen ed using a ixed band hys e esis compa a o , as (1.13) s a es. Once he sliding mo ion is eached, he expec ed beha iou o σ(x) wi hin he hys e esis band is shown in Figu e 2.1. No ice ha , as i was jus i ied in Chap e 1, he swi ching unc ion σ(x) has been ep esen ed by s aigh lines. Unde hese condi ions, he k- h swi ching pe iod, Tk, can be deduced om Figu e 2.1. The subindex kholds o any swi ching in e al ha will occu . Tk=T+ k+T− k= 2∆ ρ+ k−ρ− k,(2.1) whe e ρ+ kand ρ− ka e de ined as he in e ses o ˙σ(x) o each con ol inpu s a e: ρ+ k:= 1 ˙σ(x)+ k , ρ− k:= 1 ˙σ(x)− k .(2.2) σ(x)=0 ∆ −∆T+ kT− k Tk σ(x) ˙σku=u+ ˙σku=u− Figu e 2.1: Swi ching unc ion beha iou wi hin a ime in a ian hys e esis band. The ime de i a i e o σ(x) was al eady shown in he Chap e 1, speci ically by equa ion (1.20). F om he de ini ions o ρ± k, i is equi ed o e alua e ˙σ(x)±a any swi ching in e al k o he wo possible con ol inpu s a es, u+and u−. The e o e he exp essions o ˙σ(x)± k espond o ˙σ(x)+ k=∂σ(x) ∂x k g(x)ku+−ueq(x)k(2.3) ˙σ(x)− k=∂σ(x) ∂x k g(x)ku−−ueq(x)k(2.4) whe e he subindex kholds o he co esponding sampled coun e pa s o he o iginal 24 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP signals in he speci ic ime ins an . In Chap e 1, i was in oduced ha o he sliding mo ion o exis , he equi alen con ol, ueq, has o be wi hin u−and u+(see inequali y (1.12)). As a consequence, he igh hand sides o exp essions (2.3) and (2.4) a e always di e en om ze o. Fu he mo e, aking in o accoun he ul ilmen o he ans e sali y condi ion (see o de ails [33]), which de ines ∂σ(x) ∂x g(x)6= 0,(2.5) i is ob ious ha condi ion ˙σ(x)± k6= 0 ∀kholds unde sliding mo ion. The e o e, om (2.3), (2.4) and (2.5) i esul s ha he alues o ρ+ kand ρ− kalways exis unde sliding mo ion, acco ding o hei de ini ions in (2.2). Assuming ha (2.5) is no only non null bu posi i e, and u+>0 and u−<0, i ollows immedia ely ha ρ+ k>0, ρ− k<0,∀k≥0.(2.6) The analyses de eloped in he nex subsec ions assume ha ρ+ kis always posi i e de in- i e, meanwhile ρ− kis nega i e de ini e, as he inequali ies in (2.6) s a e. Howe e , in some applica ions he e m ∂σ(x) ∂x g(x) could esul nega i e de ini e. In hese cases, in o de o keep he me hodologies de eloped in his wo k, a sligh ly di e en de ini ion o ˙σ(x)± kmus be used as: ˙σ(x)+ k=∂σ ∂xk g(x)ku−−ueq(x)k(2.7) ˙σ(x)− k=∂σ ∂xk g(x)ku+−ueq(x)k.(2.8) Which such al e na i e de ini ions, he inequali ies de ined in (2.6) always hold. 2.2 Closed-loop case: A disc e e- ime modelling. Acco ding o (2.1), he swi ching pe iod becomes in a se ies o disc e e- ime measu emen s. Thus, i is easonable o modi y he hys e esis band ampli ude only a he beginning o each swi ching pe iod. Assuming ha he hys e esis band ampli ude is upda ed a he beginning o each swi ching in e al and emains cons an up o he nex swi ching e en , he expec ed swi ching unc ion beha iou wi hin he hys e esis band can be depic ed as he Figu e 2.2 shows. In o de o s ablish a s anda d me hodology, he beginning o he swi ching in e al will be conside ed when he slope o σ(x) changes om he nega i e alue o he posi i e one, as Figu es 2.1,2.2 display, ega dless o he con ol ac ion s a e. As i was commen ed in he p e ious Sec ion, he alues o ρ+ kwill be always s ic ly posi i e and ρ− ks ic ly Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 25 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP σ(x)=0 ∆k −∆k T+ kT− k Tk σ(x) ˙σku=u+ ˙σku=u− ∆k−1 −∆k−1 Figu e 2.2: Swi ching unc ion beha iou wi hin a ime- a ying hys e esis band. nega i e. The e o e he beginning o he swi ching in e al is ully de e mined. Since, in his app oach, he hys e esis alue will change be ween wo consecu i e swi ch- ing in e als, he exp ession ound in (2.1) should be e isi ed. F om Figu e 2.2 he new ela ion be ween ∆kand Tkcan be ound Tk=T+ k+T− k=ρ+ k(∆k+ ∆k−1)−2ρ− k∆k= ˆρk∆k+ (˜ρk−ˆρk) ∆k−1,(2.9) wi h ˆρk:= ρ+ k−2ρ− k, ˜ρk:= 2 ρ+ k−ρ− k. Exp ession (2.9) cons i u es he disc e e- ime model o he swi ching equency con ol loop, and ela es he k- h swi ching pe iod and he hys e esis band alue. Due o he used me hodology o upda e he hys e esis band, such ela ion includes he hys e esis band alues in he in e al kand k−1. In Figu e 2.3 he esul ing model o he SFC, which includes he model ound in (2.9), is shown. F om Figu es 2.2,2.1 an impo an aspec o be aken in o accoun can be deduced. I is easy o igu e ou ha he measu ed alue o Tk will no be a ailable un il he k- h swi ching in e al ends. This phenomenon is included o he sys em h ough an asynch onous disc e e delay, z−1(see bo om pa o Figu e 2.3). Tk ∆k Tk−1 Equa ion (2.9) z−1 T∗ SFC Figu e 2.3: De ail o he swi ching equency con ol loop. Disc e e- ime model. A his poin , using he loop shown in Figu e 2.3, he closed-loop beha iou o he 26 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP sys em can be s udied. Le us de ine he swi ching pe iod e o as e:= T∗−T, whe e T∗ is he e e ence swi ching pe iod. F om (2.9), he e o equa ion o he SFC can be easily ound as ek−ek−1= ˆρk(∆k−1−∆k) + ρ+ k−1(∆k−2−∆k−1) + (˜ρk−1−˜ρk) ∆k−1.(2.10) Once he disc e e- ime model is achie ed, he con ol objec i e will be he p ope design o he SFC in such a way ha ekcon e ges o ze o, hus implying Tk ends o T∗. The con olle design p ocedu e and he co esponding s abili y condi ion de i a ion will be ca ied ou in he nex subsec ions pa icula izing exp ession (2.10) in wo di e en wo king condi ions, namely: he egula ion case and he acking case. The analysis will be based on o subs i u e he exp ession p o ided by he SFC, which will be o he o m ∆k= (ek), in equa ion (2.10) and analyse he condi ions o ek o con e ge o ze o as k→ ∞. I is clea ha a speci ic con olle exp ession, ∆k= (ek), could deli e a bi a y alues in p esence o powe con e e ansien s, comp omising he SMC i sel (∆kcould become oo high o nega i e). Mo eo e , in Chap e 1i was explained ha in o de o app oxima e he eal sliding mo ion o he ideal one, i is equi ed ha he ∆k alue should be small enough (see Sec ion 1.4). As a consequence, i is impo an o de ine an allowable ange o he hys e esis band alues used by he SFC in o de o p ese e he good pe o mance o he SMC. This idea is exp essed in he Rema k 1. Rema k 1. A bi a ily hys e esis band alues may ake he sys em a away om he eal sliding egime. Hence, a speci ic ange I∆:= [∆min,∆max]such ha ∆k∈ I∆,∀k≥0, has o be de ined o p ese ing he exis ence o he sliding mo ion. In o de o es ablish he sui able hys e esis ange o a ce ain sys em, he ollowing design c i e ion is p oposed: ∆min may be ob ained om he maximum allowable swi ching equency, while he max- imum accep able ipple o he s a e a iables would be used o se ∆max and, consequen ly, he minimum swi ching equency. In u n, he swi ching equency e e ence should be acco dingly selec ed wi hin hese limi ing alues. 2.2.1 The egula ion case In he powe elec onics ield, i is e y common o egula e he ou pu ol age o a gi en con e e o a DC alue. In some applica ions he ou pu cu en o a con e e is also con olled o a ixed alue, ac ing he con e e as a cons an cu en sou ce. These wo cases co espond o a egula ion case. In he egula ion case he s a e ec o e e ence, x∗, is cons an . Assume ha a SMC has been designed o egula e such ol age o cu en , and ha h ough a p ope design o he I∆(see Rema k 1), he oscilla ions o σ(x) in he icini y o σ(x) = 0 a e small, such ha x≃x∗holds. The e o e, conside ing he s a e ec o xcons an , he swi ching unc ion de i a i es and hei in e ses a e also cons an unde s eady-s a e sliding mo ion. Hence, om a ce ain disc e e- ime ins an k0 he ollowing ela ions a e ul illed: ρ± k=ρx∗, u±:= ρ± ∗,ˆρk:= ˆρ∗,˜ρk:= ˜ρ∗,∀k≥k0,(2.11) Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 27 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP o a gi en s eady-s a e wo king poin . Wi h hese de ini ions, (2.10) can be simpli ied up o he ollowing exp ession: ek−ek−1= ˆρ∗(∆k−1−∆k) + ρ+ ∗(∆k−2−∆k−1).(2.12) Now he goal is o p ope ly design he SFC gene a ing he alue o ∆kas a unc ion o he measu ed swi ching pe iod e o , ek, in o de o ge he con e gence o he e o o ze o. Le us y as SFC con olle a pu e disc e e- ime in eg a o , which co esponds o: ∆k= ∆k−1+γek−1,(2.13) being γ > 0 he in eg al cons an . Replacing he con ol ac ion, (2.13), in he closed-loop e o dynamics, (2.12), one ge s ek= (1 −γˆρ∗)ek−1−γρ+ ∗ek−2.(2.14) The a isen equa ion in (2.14) is an homogeneous linea di e ence equa ion, which implies ha he only solu ion o he equilib ium is ek= 0. The in eg al gain γ, is he design pa ame e which should be selec ed o ge he equilib ium poin o (2.14) s able. In o de o igo ously de ine he condi ions o γ o achie e a s able beha iou o he equilib ium poin ek= 0, he Assump ion Ais de ined. Assump ion A. A eal sliding mo ion has been en o ced o e a speci ic swi ching su ace, σ(x) = 0, in such a way |σ(x)|<∆wi h a gi en con ol law de ined as (1.5)depic s. Mo eo e , a e he sliding mode ansien , he sys em eaches he s eady-s a e whe e x≃ x∗, and he ela ions shown in equa ions (2.11)a e ul illed. Once he Assump ion Ais de ined, he Theo em 1s a es he condi ion o γ o achie e s abili y o he SFC. Theo em 1. Le Assump ion A be ul illed, and le he hys e esis band ampli ude, ∆k, be upda ed acco ding o (2.13). I γ ul ils ha 0< γ < min nρ+ ∗ −1,ρ− ∗ −1o,(2.15) wi h ρ± ∗de ined in (2.11), hen he swi ching pe iod, Tk, con e ges asymp o ically o he e e ence alue, T∗, in he s eady-s a e. P oo . I ollows applying Ju y s abili y c i e ion [34] o he cha ac e is ic polynomial associa ed o he di e ence equa ion (2.14), which esul s in p(z) = z2+z(γˆρ∗−1) + γρ+ ∗.(2.16) The condi ions o be ul illed, acco ding o he c i e ion, a e: 1. p(1) >0→γ > 0 28 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP 2. p(−1) >0→γ < |ρ− ∗|−1 3. |γˆρ∗−1|<|γρ+ ∗| → γ < |ρ+ ∗|−1 In his case, since he alues o ρ± ∗can be conside ed pa ame e s, he z ans o m has been used in o de o ind he s abili y condi ions o (2.14). Finally, he equi alen model in he zdomain is shown in Figu e 2.4. No ice how he inhe en delay p oduced when he swi ching pe iod is measu ed has been conside ed o he p ope implemen a ion o (2.13). Tk eq (2.9) T∗ γ∆k Tk−1z−1 z−1 Equi alen o (2.13) Figu e 2.4: Equi alen model in he zdomain o he con ol loop o he disc e e- ime app oach in he egula ion case. 2.2.2 The acking case The powe con e e s usually ha e o wo k connec ed o he AC g id, some imes injec ing powe om a enewable plan , some imes consuming powe om he g id. In hese cases, he powe con e e s a e wo king wi h e e ences (o ol age o cu en s) ha a y wi h ime. These applica ions a e classi ied as acking con ol p oblems. When he SMC is acking a ime- a ying e e ence, x∗= ( ), i exis s a a ia ion o he swi ching unc ion ime de i a i es wi h k, i.e. ρ+ k6=ρ+ k−1,ρ− k6=ρ− k−1. As a consequence, he simpli ica ion made in (2.11) does no apply. Be o e p og essing wi h he acking case s abili y analysis, an impo an conside a ion abou he a ia ions o ρ± kwi h kis discussed in Rema k 2. Rema k 2. Al hough in he acking case i is assumed ha he alues o ρ± k a y wi h k, such a ia ions mus be su icien ly slow so ha hese alues can be conside ed locally cons an du ing a swi ching in e al. This ac will be achie ed when he e ec i e swi ching pe iod o he con ol ac ion is small enough wi h espec o he sys em ime cons an s. This hypo heses is equi alen o assume ha he beha iou o σ(x)in Figu es 2.1,2.2 can be conside ed piecewise linea e en when he sys em is acking a ime- a ying e e ence x∗= ( ). The e o e, he swi ching pe iod has o be small enough as he condi ion: d ( ) d ≈ (Tk)− (Tk−1) Tk→0 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 29 CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP is me . No ice how in his case he las e m on he igh side in equa ion (2.10)does no anish. The chosen SFC s uc u e o he egula ion case was a pu e disc e e- ime in eg a o (equa ion (2.13)), which was cha ac e ized by: ∆k= ∆k−1+γek−1. Keeping such s uc u e as SFC, and wi hou he a o emen ioned simpli ica ions o ρ± k, he new exp ession o he e o dynamics in he acking case is ound. The dynamics is go e ned by (2.17): ek= (1 −γˆρk)ek−1−γρ+ k−1ek−2+ ∆k−1(˜ρk−1−˜ρk).(2.17) The di e ence be ween exp ession (2.12) and (2.17) is he las e m on he igh side o exp ession (2.17). I should be highligh ed ha his e m makes he exp ession (2.17) non-homogeneous and, as a consequence, i losses ek= 0 as equilib ium solu ion. Thus, in o de o eco e he homogeneous cha ac e is ic o exp ession (2.14), his wo k p oposes he design o a eed o wa d ac ion in o de o cancel his e m in he closed-loop dynamics. This idea is ske ched in Figu e 2.5. Tk eq (2.9) T∗ γ∆k Tk−1z−1 z−1 Equi alen o (2.19) Ψk Ωk Figu e 2.5: Swi ching equency egula ion con ol loop wi h eed o wa d ac ion. The inhe en ime delay due o he swi ching pe iod measu emen is ep esen ed by z−1. The analysis is simple, he new con ol ac ion includes he eed o wa d e m Ωk, as ∆k= Ψk+ Ωk,(2.18) whe e Ψkkeeps he s uc u e o a disc e e- ime in eg a o Ψk= Ψk−1+γek−1.(2.19) The alue o Ωkis de i ed placing (2.18) in (2.10) and equalling all he e ms ha do no depend on ek o ze o. The esul o he p oposed p ocedu e is gi en by equa ion (2.20): Ωk=ˆρk−1−ρ+ k ˆρk Ωk−1+ρ+ k−1 ˆρk Ωk−2+˜ρk−1−˜ρk ˆρk Ψk−1.(2.20) 30 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 2. MODELLING AND STABILITY ANALYSIS OF THE SWITCHING FREQUENCY CONTROL LOOP The p oposed modi ied s uc u e o he acking con ol p oblem is depic ed in Figu e 2.6, acco ding o (2.18), (2.19) and (2.20), whe e he de ail o Ωkcompu a ion has been highligh ed. Tk eq (2.9) T∗ γ∆k Tk−1z−1 z−1 Equi alen o (2.19) Ψk Ωk ρ± k eq (2.20) Figu e 2.6: Swi ching equency egula ion con ol loop wi h eed o wa d ac ion. De ail o new con olle s uc u e including he eed o wa d ac ion. F om he ob ained exp ession o Ωk, some conside a ions should be aken in o accoun , which a e mainly no ed in Rema k 3. Rema k 3. An impo an ema k should be made a his poin wi h ega ds o he exp es- sion ound o Ωkin (2.20). Looking ca e ully he exp ession, i is simple o no e how he k- h alue o Ωkdepends on he k- h alues o ˆρ,˜ρand ρ+. Such esul is no ealiz- able om a p ac ical poin o iew since i depends on samples ha a e no a ailable ye . Besides, he alues ˆρ,˜ρand ρ+should be p ope ly es ima ed. As a consequence, some app oxima ions will be equi ed when he p oposed con olle is implemen ed, bo h in he simula ion and in he expe imen a ion cases, which will be discussed la e . Assuming ha he exp ession o Ωkcan be p ope ly ob ained, he inal e o dynamics o he p oposed con olle wi h eed o wa d ac ion can be e alua ed. I ollows eplacing (2.18), (2.19) and (2.20) in (2.10), which p o ides he esul ing closed-loop e o dynamics in (2.21). ek= (1 −γˆρk)ek−1−γρ+ k−1ek−2.(2.21) Now he equa ion o he swi ching pe iod e o boils down o an homogeneous ime- a ying disc e e- ime linea sys em, eco e ing ek= 0 as he equilib ium solu ion. I should be no ed ha ˆρk,ρ+ k−1canno be ea ed as cons an pa ame e s in his case. Al hough he ρ± ka e no cons an wi h k, i can be de ined an expec able ange o alues ela ed o a ce ain wo king condi ions. Unde s eady-s a e sliding mo ion, he s a e ec o p o ile x∗= ( ) will p oduce he ollowing ime- a ying alues: ρ+ k=ρk(x∗( ), u+) = ρ+ ∗k, ρ− k=ρk(x∗( ), u−) = ρ− ∗k, ˆρk= ˆρk(x∗( )) := ˆρ∗ k, ˜ρk= ˜ρk(x∗( )) := ˜ρ∗ k, (2.22) Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 31 Chap e 3 A case o s udy: SFC design and simula ion esul s In his Sec ion, a e y simple sys em is in oduced in o de o show he comple e p ocess o he SFC loop design, including he a o emen ioned p ac ical app oaches. The design and simula ion o he p oposed s a egies will be es ed in a linea sys em. Fi s ly, an sliding mode con olle will be designed using he classical app oach, h ough he equi alen con ol me hod, as i was in oduced in Sec ions 1.3 and 1.4. Such SMC con olle will be applied o bo h egula ion and acking asks. The nex s ep shows he de i a ion o he exp essions o he ρ±, allowing o p ope ly design he SFC in he di e en cases, namely: he disc e e- ime app oach o egula ion and acking cases, and he con inuous- ime app oach in a egula ion scena io. Finally, he simula ions esul s o all he designed app oaches a e p esen ed. 3.1 Design o he sliding mode con olle Le us in oduce he single-inpu single-ou pu linea sys em ˙x1=−x1+x2,(3.1) ˙x2=−x1+Mu, (3.2) whe e Mis a sys em pa ame e , which could be in e p e ed as he sys em gain and uis he discon inuous inpu , aking alues in he disc e e se {−1,1}. The con ol objec i e is o con ol he dynamics o x2 o a desi ed beha iou , x∗ 2( ). Taking in o accoun he ela i e deg ee o he sys em (3.1), (3.2) (see Sec ion 1.3), a swi ching su ace based on he acking e o is enough. The e o e, he swi ching su ace is chosen as (3.3) s a es. σ(x, ) := x2−x∗ 2( ) = 0.(3.3) Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 39 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS The desi ed dynamics o x2is x∗ 2( ) := A+Bsin ω , (3.4) wi h A, B ≥0. I should be no iced ha he swi ching unc ion can c ea e a egula ion con ol case, (B= 0), o a acking con ol case, (B6= 0). The sliding mo ion will be en o ced in |σ(x, )|<∆ wi h a hys e e ic con ol law o he ype shown in equa ion (1.13), being u+= 1 and u−=−1. As i was p esen ed in Sec ion 1.3, he sliding mode con ol design is based on he i s ime de i a i e o he swi ching unc ion. Once he i s ime de i a i e is ound, he equi alen con ol can also be de i ed (see exp essions (1.9) and (1.10)). These ope a ions esul in: ˙σ=Mu −(x1+ ˙x∗ 2),(3.5) and, hence, ueq =1 M(x1+ ˙x∗ 2).(3.6) F om (3.5) he con ol law can be de i ed. The con ol law en o cing σ˙σ < 0 is: u=u+i σ < −∆, u−i σ > ∆,(3.7) Recalling he sliding mode domain condi ion de ined in (1.12), he ideal sliding mo ion on σ= 0 (o he eal sliding mo ion on |σ(x, )|<∆) can be en o ced i −1<1 M(x1+ ˙x∗ 2)<1 holds. The ideal sliding dynamics is de e mined placing he equi alen con ol in he s a e space equa ions. The e o e using (3.6) in (3.1), (3.2) one ge s x2=x∗ 2( ),(3.8) ˙x1=−x1+x∗ 2( ).(3.9) The dynamic o x2is di ec ly imposed by he sliding mo ion, which is equal o x∗ 2( ). The emaining dynamics cha ac e izing x1on sliding mo ion, x∗ 1, is asymp o ically s able. Hence, i is also possible o ind he s eady-s a e solu ion o he di e en ial equa ion (3.9) in he ime domain, x∗ 1( ), which is x∗ 1( ) = A+B 1 + ω2(sin ω −ωcos ω ).(3.10) Once he ideal s eady-s a e sliding egime x∗= (x∗ 1, x∗ 2)>has been eached, he swi ching unc ion ime de i a i es allow o ind he alues o ρ± ∗, which can be de i ed using (3.4), 40 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS (3.5) and (3.10), esul ing in ρ± ∗( ) = ±M−A−B 1 + ω2sin ω +ω3cos ω −1 (3.11) o he acking case. The egula ion case is de e mined by B= 0, and he e o e ρ± ∗=1 ±M−A.(3.12) No ice ha he p e ious alues, ρ± ∗, a e he key o he SFC design, as i was p esen ed in Chap e 2. The eupon, he simula ion esul s o he sys em de ined in (3.1),(3.2) wi h he di e en app oaches p esen ed in Chap e 2 o he SFC will be shown. Speci ically, he egula ion con ol p oblem is simula ed o he disc e e- ime app oach (Sec ion 2.2.1) and o he con inuous- ime one (Sec ion 2.3). Analogously, he acking case is simula ed o he disc e e- ime app oach using he eed o wa d ac ion (Sec ion 2.2.2). The simula ions ha e been pe o med wi h Ma lab-Simulink using he ollowing pa ame e s: M= 3, A= 1, and ω= 2π·0.02, while he desi ed swi ching pe iod is T∗= 0.1 s. The pa ame e Bwill be selec ed depending on he used app oach. 3.2 The egula ion case in he disc e e- ime app oach Fi s o all, he design o he SFC o he disc e e- ime egula ion con ol case is pe o med. In his case, he alue o Bis se o 0. Fo he design o he γ alue o he SFC, i is equi ed o use he alues o ρ±a he co esponding s eady-s a e sliding mo ion. As i was de i ed in (3.12), he alues o ρ±a he speci ic s eady-s a e sliding mo ion, ρ± ∗, can be e alua ed wi h he gi en da a o Mand A. The e o e, i s ems om (3.12) ha ρ+ ∗= 0.5 and ρ− ∗=−0.25. Recalling Theo em 1, he cha ac e is ic polynomial de ailed in (2.16), o he case o s udy is ound: p(z) = z2+z(γ−1) + 0.5γ. (3.13) Hence, ollowing Theo em 1and equa ion (2.15), unde he SFC wi h he s uc u e gi en by (2.13), he closed-loop sys em o he SFC is s able o alues in he ange 0 < γ < 2. Fu he mo e, using he equa ion (3.13), he oo locus o di e en alues o γcan be s udied. Figu e 3.1 depic s he poles placemen in he complex plane o di e en γ alues, oge he wi h he esponses, in he ime domain, o he sys em in on o a s ep e e ence ( om T∗= 0.05 s o T∗= 0.1 s). F om he Figu e 3.1, i can be seen how alues below γ= 0.3 deli e esponses wi h he poles in he eal axis, being such esponses unde damped o alues om 0.3 o 2. Fo γ > 2 he SFC loop eaches he uns able egion. No ice ha he esponse o γ= 2 is omi ed due o he oscilla ing beha iou , showing he one o γ= 1.8 ins ead. I is ob ious ha , o his case, he sys em is close o he s abili y limi . In Figu e 3.2 i can be seen he esponse o he SFC and he SMC o wo alues o γand T∗, speci ically o γ= 0.1 and γ= 1 wi h a e e ence alue ha changes, again, Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 41 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS -1 01 -1 1 Real Pa Imagina y Pa γ– + γ γ= 0.1 γ= 0.3 γ= 1 γ= 2 γ= 0.1 γ= 0.3 γ= 1 γ= 1.8 0.1 0.05 11.512 12.5 13 13.5 14 14.5 15 15.5 (s) T∗, Tk 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -0.5 0.5 16 Figu e 3.1: Le Plo : Roo locus o p(z) o he simula ion case o s udy. Righ Plo : esponse in he ime domain o di e en alues o γ. 0.5 0 2 4 6 8 10 12 14 16 6810 12 14 16 0.995 1 Tk sx1su σ(x), ∆k (s) γ= 0.1 Tk Tm k x1 γ= 0.1γ= 1 T∗= 0.1sT∗= 0.05 sT∗= 0.1s 1.005 0.4 0.2 0 1 0.5 0 0.2 0 -0.2 Figu e 3.2: Regula ion: pe o mance o he SFC, wi h he disc e e- ime app oach, o di e en alues o γand T∗. 42 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS be ween T∗= 0.1 s and T∗= 0.05 s. The alues ha ake T∗and γdu ing he simula ion a e depic ed in he op o Figu e 3.2. The op plo o he Figu e con ains he swi ching pe iod o he simula ed sys em (3.1),(3.2), i.e. Tk, as well as he swi ching pe iod a ising om he model de eloped in Sec ion 2.2.1 and de ailed in Figu e 2.4, which is labelled as Tm k. The mid plo shows he esul ing dynamics o x1, while he x2one is di ec ly de i ed om σ(see equa ion (3.3)), which is plo ed a he bo om plo o he Figu e. No ice in he zoomed iew o his mid plo how he ipple o x1changes as he swi ching pe iod does. This ipple is he consequence o he di e en alue o ∆ gene a ed by he SFC o di e en desi ed swi ching pe iods. Howe e , in all he cases such ipples a e small wi h espec o he DC alue o x1, as expec ed. The hi d plo illus a es how ∆kis upda ed un il Tka ains he e e ence alue T∗, con i ming ha he desi ed swi ching pe iod is achie ed in s eady-s a e. Mo eo e , he model esponse, Tm k, always i s wi h he eal swi ching pe iod, Tk, excep o he s a -up, whe e he s eady-s a e sliding mo ion is s ill no eached and, as a consequence, ρ±6=ρ± ∗. Once he sys em eaches he s eady-s a e sliding mo ion, i.e. ρ±=ρ± ∗, bo h esponses pe ec ly ma ch, e en unde swi ching pe iod e e ence a ia ions occu ing a =8s and = 12 s; hus alida ing he ma hema ical model o he swi ching pe iod beha iou p esen ed in Sec ion 2.2. Finally, i has o be no ed ha he hys e esis band ampli ude achie es a cons an alue in he s eady-s a e, as i can be in e ed om (2.13). The asymp o ic con e gence o he s a e a iable x1 o he e e ence one x∗ 1is illus a ed in he mid pa o he Figu e 3.2, hus co obo a ing an o e all good pe o mance o he ull sys em (SMC+SFC). 3.3 The egula ion case in he con inuous- ime ap- p oach In he con inuous- ime app oach, he main di e ence wi h espec o he analysis pe o med in he p e ious sec ion a e ocused in he SFC implemen a ion. The analysis o he SMC is sha ed by he wo app oaches, since he con ol goal is he same o bo h, ha is o egula e x2 o x∗ 2. E en he e alua ion o he pa ame e s ρ± ∗a e exac ly he same, because such alues a e ela ed o he same s eady-s a e sliding mo ion. The analysis di e ences ely on he SFC app oach. In his case, he s abili y condi ions and he design guideline o he SFC we e de ined in Sec ion 2.3. I should be emembe ed ha in he con inuous- ime app oach, an impo an assump ion was aken (Assump ion B) in o de o de elop he equi alen model o he SFC loop. Thus, he equi alen diag am in he sdomain shown in Figu e 2.9 only can be used unde ce ain condi ions, when Assump ion Bis me . A his s age, we assume ha he e o o he swi ching pe iod in he simula ion pe o med he eunde , will be limi ed by a known alue, being his bound |ek|<0.05 s. Le us s udy he g ade o ul ilmen o he Assump ion B, which is: γL|ek|<< min nρ+ ∗ −1,ρ− ∗ −1o,∀k≥k0. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 43 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS In his case, a ac o o 20 is p oposed. This implies he ollowing ela ion 20 ·|ek|γL20 = min nρ+ ∗ −1,ρ− ∗ −1o. Acco ding o Sec ion 3.2, he alues o ρ± ∗a e ρ+ ∗= 0.5 and ρ− ∗=-0.25, he e o e: 20 ·|ek|γL20 = 2. Hence, γL20 =2 20 ·0.05 = 2. Thus, ensu ing ha he designed alue o γLis wi hin he ange 0 < γL< γL20 , he model o Figu e 2.9 holds. Addi ionally, Theo em 3allows o check i he equi alen model is s able. Since in he simula ions p esen ed he ea e he swi ching pe iod senso is ideal, he s abili y condi ions mus be de e mined by (2.43). This s abili y condi ion is s a ed as: γL<2 λT∗, being λ= 2 (ρ+ ∗−ρ− ∗), yielding γL<2 2 (0.5+0.25) ·0.1= 13.3. In he ollowing simula ion he swi ching pe iod e e ence will be also a ied o T∗= 0.05 s, hus he co esponding s abili y condi ion is also checked γL<2 2 (0.5+0.25) ·0.05 = 26.667. F om his esul , he s abili y o alues o γL≤2 a e gua an eed. No ice ha in his case, i does no make sense o show he esul ing oo locus o di e en alues o γLsince he model no always applies. The simula ion shown in Figu e 3.3 co esponds o he same simula ion se -up o he p e ious Sec ion, bu in his case, he SFC is implemen ed by means o a con inuous- ime in eg a o , as i has al eady commen ed. In he es , wo di e en alues o γLa e used, speci ically γL= 1 and γL= 10, co esponding one o a alue wi hin he alues whe e Assump ion Bis ul illed, and ano he ou side his ange. In he simula ion, he ideal esponse o he swi ching pe iod acco ding o Figu e 2.9, labelled as Tm k, is shown. Du ing he simula ions, wo ansien s in he e e ence swi ching pe iod a e applied, speci ically om T∗=0.1 s o T∗= 0.05 s and ice e sa. In ac , om hese ansien s a ise he bound o he pe iod e o , p e iously used in he analysis pe o med o e he γL alues ul illing Assump ion B. Du ing he s a -up, he eal pe iod, Tk, and he ideal one, Tm k, do no ma ch since he s eady-s a e sliding mo ion has no been es ablished, and ρ± k6=ρ± ∗. Once he sys em 44 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS 0 0 1 0 2 4 6 8 10 12 14 16 0 Tk σ(x) (s) γL= 1 x1 T∗= 0.1sT∗= 0.1s T∗= 0.05 s Tk Tm k 6810 12 14 16 1 12 14 13 1.005 0.995 0.05 1.05 1 γL= 1 γL= 10 8910 0.1 0.06 -0.2 0.2 0.5 Figu e 3.3: Regula ion: pe o mance o he SFC, wi h he con inuous- ime app oach, o di e en alues o γand T∗. eaches he s eady-s a e, he esponses Tk,Tm kcon e ge. Despi e o he s a -up, he beha iou du ing he ansien s a ime ins an = 8 s and = 12 s dese e a special a en ion. In he i s ansien ( e e ence change a = 8 s), he alue o in eg al gain is γL= 1. Du ing his ansien , i is e iden ha Tkand Tm kma ch, hus alida ing he model and also con i ming ha , in his case, he Assump ion Bis me . I should be no ed ha since he esponse is unde damped, he ul ilmen o he hypo hesis in he ini ial condi ion (|e0|<0.05 s) implies ha he ansien s a e exac ly he same o Tkand Tm k. Howe e , his does no happen in he second ansien ( e e ence change a = 12 s), whe e a gain o γ= 10 is used, and, as expec ed, he esponses o Tkand Tm kdo no o e lap. Indeed, since he ini ial e o does no mee he Assump ion B, he esponses Tk and Tm kwill no i along he ansien , as i could be obse ed om he zoomed iew o he op plo . As a conclusion, wi h a easonable known bound o |ek|∀k, i is possible o success ully design he SFC in he con inuous- ime app oach, ob aining he expec ed dynamics o Tk. No ice ha he no ul ilmen o Assump ion Bdoes no imply an uns able esponse, i jus means ha he de eloped model is no alid. Despi e o he ange o applica ion o he model de eloped in Sec ion 2.3, he swi ching pe iod is p ope ly egula ed (e en when γL=10) in he en i e es , being negligible he impac o e he SMC which egula es x2, and as a consequence, x1. In he zoomed a ea o he mid plo , i can be app ecia ed he di e en ipple le els in x1acco ding o he a ia ion in he swi ching pe iod, being in bo h cases negligible when compa ing wi h he DC alue. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 45 CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS 3.4 The acking case A acking pe o mance is ob ained when B6= 0. I should be no iced again ha he SMC design is he same as he one p esen ed in Sec ion 3.1 so, i is omi ed he e o he sake o b e i y. Since he e e ence o x∗ 2is a ime- a ying signal, he alues o ρ∗ kwill also ha e dependency wi h k. I is simple o e alua e he expec ed alues o ρ± ∗( ) jus eplacing he gi en alues o M,A,Band ωin (3.11). The alues o he ρ± ∗kwill be he co esponding ones o ρ± ∗( ) a he speci ic ime ins an k. As a esul , he unc ions ˜ρ∗ kand ˆρ∗ k, can be de i ed, ollowing hei de ini ions in Sec ion 2.2 (equa ion (2.9)). The exp ession o ρ± ∗( ) a e: ρ± ∗( ) = ±M−A−B 1 + ω2sin ω +ω3cos ω −1 The alues o M,Aand ωwe e al eady de ined, being he selec ed ampli ude o he e e ence signal B= 0.5. The e o e: ρ± ∗( ) = ±3−1−0.5 1 + (2π·0.02)2sin (2π·0.02) + (2π·0.02)3cos (2π·0.02) −1 . The p e ious exp essions a e he ools equi ed o he SFC design in he acking case. F om hese exp essions ˜ρ∗ kand ˆρ∗ ka e compu ed, and he s able ange o γacco ding o Theo em 2is ound. Wi h he goal o cla i ying he esul p o ided by Theo em 2, and how o apply i , a g aphical app oach is shown in Figu e 3.4. In his plo can be seen, om op o bo om, he e e ence signal, x∗ 2( ), he esul ing signals ρ+ ∗k,ρ− ∗kand he se o solu ions ob ained applying Theo em 2. The mid plo co obo a es he expec ed a ia ions o ρ± ∗kwi h kand also shows ha he ime e olu ion o such alues a e slow wi h espec o he desi ed swi ching pe iod. Once such signals a e plo ed, he solu ion acco ding o Theo em 2can be g aphically ound, inding he minimum alue and he maximum one o he co esponding solu ions (see bo om plo o Figu e 3.4). No ice how in he bo om plo he minimum alue, γm, and he maximum one, γM, delimi ing he s able ange o γa e g aphically de e mined. These alues esul in 0.314 < γ < 1.0315. I is wo h ema king ha nume ical simula ions show ha s abili y is indeed gua an eed o 0 < γ < 1.6. The eade has o keep in mind ha he s abili y condi ion p o ided by Theo em 2is su icien bu no necessa y. Once he s able ange o γis ound, he emaining ask is o implemen he eed o wa d ac ion. Recalling he Subsec ion 2.2.2, he SFC o acking cases includes a eed o wa d ac ion Ω (see equa ion (2.18)). As i was commen ed in Rema k 3, he implemen a ion o such unc ion implies an impo an limi a ion, since due o he inhe en delay o he swi ching pe iod measu emen , i is no possible o exac ly implemen he heo e ical alue ound o Ωk. Besides, he eed o wa d ac ion needs o es ima e he alues o ρ± ∗k(see equa ion (2.20)) which also inhe i he p oblem o he measu ing delay o Tk. In o de o 46 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 3. A CASE OF STUDY: SFC DESIGN AND SIMULATION RESULTS o e come hese p oblems he ollowing app oxima ion is used: Ωk≃Ωk−1,(3.14) whe e he co esponding alues o ρ± k−1a e es ima ed using he exp essions: ρ+ k−1=T+ k−1 ∆k−1+ ∆k−2 ;ρ− k−1=T+ k−1 2 ∆k−1 . The basis o his app oxima ion is again he idea exp essed in Rema k 2, in he sense ha he sys em dynamics is slow enough wi h espec o he swi ching pe iod, and, hence, he a ia ion be ween consecu i e ksamples is small. Such phenomena will be highligh ed la ely in he simula ion esul s. 100 110 120 130 140 150 160 170 180 190 200 0 1 2 0 1 0 1 -1 1.5 0.5 (s) ρ+ k∗( ), ρ− k∗( ) Solu ions o Theo em 2x∗ 2 ρ+ k∗( ) ρ− k∗( ) o γM o γm γMγm Figu e 3.4: F om op o bo om: Re e ence signal x∗ 2; empo al e olu ion o he signals ρ+ k, ρ− k, measu ed om he sys em; solu ions acco ding o Theo em 2. Fo he e alua ion o he pe o mance o he designed SFC o he acking case, di e - en es s a e pe o med. The esul s a e de ailed in he simula ions shown in Figu es 3.5, 3.6 and 3.7. The selec ed alue o he in eg al gain o he SFC is γ= 0.4. Figu e 3.5 compa es he pe o mance o he sys em wi hou and wi h SFC ac ion, and highligh s he e ec o he SFC. The SMC ope a es wi h ixed hys e esis band o he i s 150 s. The SFC wi h γ=0.4 is enabled a =150 s. I should be no iced ha i is no un il he ac i a ion o he SFC ha Tkis able o a ain he e e ence alue T∗= 0.1 s. Indeed, he swi ching pe iod wi h a ixed band alue is ime- a ying, as expec ed om (1.19), aking in o accoun ha he alues o ˙σ± k a y wi h k. No ice how, in he same way, Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 47 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 1.4, equa ion (1.18)). Such desc ip ion, based on he egula o m, will be used in his Chap e in o de o analyse he impac o a hys e esis band on he eal sliding dynamics. The chap e is s uc u ed as ollows. In Sec ion 4.1, he app oach based on he egula o m is b ie ly in oduced. In Sec ion 4.2, he case ∆ 6→ 0 is conside ed, leading o an in- e es ing esul ega ding he mean alue o he swi ching unc ion and i s piecewise linea beha iou . Some simula ion esul s a e in oduced in Sec ion 4.3 wi h he aim o alida e he esul p o ided in Sec ion 4.2. Finally, in Sec ion 4.4 he case o ime- a ying ∆ is conside ed, leading o some conclusions abou he eal sliding mode in sys ems using SFC, which will be co obo a ed h ough nume ical simula ions as well. 4.1 The egula o m app oach Conside he ollowing linea sys em shown in (4.1). ˙x=Ax +Bu +g( ),(4.1) whe e x∈Rn,A∈Mn(R), B∈Mn×1(R), g∈Rnis a ec o unc ion ep esen ing a smoo h ex e nal dis u bances, and u∈Ris he con ol ac ion. An associa ed swi ching su ace o his sys em would be: σ(x) = Cx = 0,(4.2) whe e C∈M1×n(R). Le he s a e space equa ion o he sys em gi en by (4.1) be exp essed as i is shown in (4.3),(4.4) h ough a speci ic s a e ans o ma ion: ˙x1=A11x1+A12x2+g1( ),(4.3) ˙x2=A21x1+A22x2+B1u+g2( ).(4.4) This desc ip ion is call he egula o m ( [37]). Taking in o accoun he pe o med change o a iables, he swi ching unc ion, σ(x), can also be w i en in he ans o med s a e space as σ(x1, x2) = C1x1+C2x2,(4.5) whe e x1∈Rn−m,x2∈Rm, and ma ices Aij, Ci, B1ha e app op ia e dimensions. Then, he compu a ion o ˙σ(x1, x2) ( om now on ˙σ) yields: ˙σ=C1˙x1+C2˙x2=C1(A11x1+A12x2+g1( )) + +C2(A21x1+A22x2+B1u+g2( )) , 54 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP which allows u o be exp essed as ollows: u=−B−1 1C−1 2C1(A11x1+A12x2+g1( )) + −B−1 1(A21x1+A22x2+g2( )) + B−1 1C−1 2˙σ. (4.6) No ice ha he p e ious exp ession o he con ol law includes bo h he low equency componen ( ela ed o he equi alen con ol) and he high equency one ( ela ed o he eal sliding mo ion). The equi alen con ol is easily ound assuming ha ˙σ= 0, acco ding o i s de ini ion in Sec ion 1.3, equa ion (1.6), as: ueq =−B−1 1C−1 2C1(A11x1+A12x2+g1( )) + −B−1 1(A21x1+A22x2+g2( )) .(4.7) Then, (4.6) can be upda ed up o: u=ueq +B−1 1C−1 2˙σ. (4.8) As a consequence, he eal dynamics is go e ned by he educed o de sys em: ˙x1=A11x1+A12x2+g1( ), σ=C1x1+C2x2, I is ob ious o see ha i (4.6) is eplaced in (4.3)-(4.4), he second s a e space equa ion becomes he desi ed dynamics imposed by (4.5), since i consis s o a linea combina ion o x1and x2. Hence, he p e ious sys em can also be exp essed as: ˙x1=A11 −A12C−1 2C1x1+g1( ) + A12C−1 2σ, (4.9) x2=−C−1 2C1x1+C−1 2σ. (4.10) The ideal dynamics can be ob ained se ing σ=0in(4.9),(4.10), which esul s in he ideal sliding dynamics o x1, x2, deno ed as x? 1, x? 2: ˙x? 1=A11 −A12C−1 2C1x? 1+g1( ),(4.11) x? 2=−C−1 2C1x? 1.(4.12) No ice ha now he inpu signal in he eal dynamics is σ=σ( ), whe e σ( ) is he eal e olu ion o he swi ching unc ion cha e ing in he icini y o σ= 0. Meanwhile, in he app oach p esen ed in Sec ion 1.4 his ole was played by ˙σ(see equa ion (1.18)). Rema k 5. No ice ha ini ially, in Sec ion 1.4 a mo e gene al nonlinea sys em was conside ed in equa ion (1.1). In his chap e a linea sys em has been conside ed in he analysis. As a consequence, he esul s ound he ea e only hold o linea sys ems. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 55 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 4.2 Fixed hys e esis band Bo h on he egula app oach and on he classical one, i has been p o ed ha he eal and he ideal ajec o ies o he sliding mode a e ela ed in such a way ha kx( )−x?( )k ≤ N∆, being Na posi i e numbe (see [38], [37] o de ails). Mo eo e , as he swi ching unc ion is en o ced o cha e a ound he desi ed space egion σ(x) = 0, i is easonable o hink ha x( ) will cha e a ound he ideal esponse x?( ). Assuming ha in he eal sliding dynamics x( ) = x?( ) canno be a ained, le us see i a leas , unde ce ain condi ions, he a e age alues o bo h esponses ma ch. One could hink ha i he a e age alue o he swi ching unc ion, σ(x), is ze o, he a e age alue o x( )−x?( ) should con e ge o ze o as well. The analysis o such hypo heses is de eloped a his s age using he egula o m app oach shown in he p e ious sec ion. Fi s o all, he a e age alue o he swi ching unc ion in he case o a ixed hys e esis band is analysed h ough he a e aging ope a o de ined in (4.13). h(·)i=1 TZ −T (·)d . (4.13) Recalling Figu e 2.1, and assuming a piecewise linea beha iou o σ, i s ime e olu ion can be modelled as: σ( ) = σ(0) + ˙σ(x)+ k o 0 ≤ < T+ k σ( ) = σ(T+ k) + ˙σ(x)− k( −T+ k) o T+ k≤ < Tk(4.14) Applying he a e aging ope a o , he mean alue o σ( ) is ound in eg a ing o he wo ime in e als: 0 ≤ < T+ kand T+ k≤ <Tk: hσi=1 Tk" ZT+ k 0σ(0) + ˙σ(x)+ k d !+ ZTk T+ kσ(T+ k) + ˙σ(x)− k −T+ kd !#.(4.15) Sol ing he p e ious in eg als one ge s: hσi=1 Tk σ(0) +˙σ(x)+ k 2 2T+ k 0 +σ(T+ k) +˙σ(x)− k 2( −T+ k)2Tk T+ k!.(4.16) Acco ding o Figu e 2.1 he ollowing ela ions hold: σ(0) = −∆, σ(T+ k) = ∆ and σ(Tk) = −∆. Mo eo e , i is also ob ious ha T+ k= 2∆/˙σ(x)+ k,T− k=−2∆/˙σ(x)− kand Tk= T+ k+T− k. The e o e, he esul ing mean alue is: hσi=1 Tk−2∆2 ˙σ(x)+ k +2∆2 ˙σ(x)+ k +2∆2 ˙σ(x)+ k−2∆2 ˙σ(x)− k +2∆2 ˙σ(x)− k−2∆2 ˙σ(x)+ k= 0.(4.17) Exp ession (4.17) con i ms he sugges ed esul wi h ega d o he a e age alue o σ 56 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP when i is con ined wi hin a ixed and symme ic hys e esis band. No ice ha his esul holds assuming a piecewise linea beha iou o σ. Once he null a e age alue o he swi ching unc ion is con i med, he nex s ep is o see i hσi= 0 en ails ha hxi=hx?iin s eady-s a e. Fo such pu pose, he ollowing assump ion is made: Assump ion C. Once he sliding mode s eady-s a e has been eached, he swi ching unc- ion, σ, becomes T-pe iodic, wi h T∈R+, om a ce ain ime ins an . Theo em 4. Assume ha equa ions (4.9),(4.10)and (4.11),(4.12)cha ac e ize he eal sliding dynamics and he ideal sliding dynamics, espec i ely. Conside also ha Assump- ion Cis ul illed and ha Ma ix A1, de ined as A1=A11 −A12C−1 2C1(see equa ions (4.9),(4.11)), is a Hu wi z ma ix. Then, bo h sys ems admi asymp o ically s able, T- pe iodic solu ions ˜x,˜x?, espec i ely, such ha h˜x?i= Γ?hg1( )i,(4.18) h˜xi=h˜x?i+ Γ hσ(x)i,(4.19) whe e Γ?=−A−1 1 C−1 2C1A−1 1, Γ = −A−1 1A12C−1 2 C−1 2C1A−1 1A12C−1 2+I. hen, since hσi= 0,h˜xi=h˜x?iholds, wi h h˜x?igi en by (4.18). The p oo o Theo em 4is ou lined he eunde [38]. P oo . As A1is Hu wi z and Assump ion Cis ul illed, he exis ence o T-pe iodic, asymp- o ically s able solu ions ˜x>=˜x> 1,˜x> 2, ˜x?>=˜x?> 1,˜x?> 2 o (4.9), (4.10) and (4.11), (4.12), espec i ely, is ensu ed by basic linea sys ems heo y ( [39]). In u n, hese solu- ions sa is y: ˙ ˜x1=A1˜x1+g1( ) + A12C−1 2σ, (4.20) ˜x2=−C−1 2C1˜x1+C−1 2σ, (4.21) and ˙ ˜x? 1=A1˜x? 1+g1( ),(4.22) ˜x? 2=−C−1 2C1˜x? 1.(4.23) Applying he a e aging ope a o (4.13) o (4.20), (4.22) while aking in o accoun i s linea i y and he ac ha ˙ ˜x1= 0, ˙ ˜x? 1= 0, because nei he ˙ ˜x1no ˙ ˜x? 1ha e con inuous Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 57 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP componen bu jus ze o a e aged e ms, i esul s ha 0 = A1h˜x1i+hg1( )i+A12C−1 2hσi, 0 = A1h˜x? 1i+hg1( )i. Then, i ollows immedia ely om he Hu wi z cha ac e o A1 ha h˜x? 1i=−A−1 1hg1( )i, h˜x1i=h˜x? 1i−A−1 1A12C−1 2hσi and, subsequen ly om (4.21),(4.23), h˜x? 2i=C−1 2C1A−1 1hg1( )i, h˜x2i=h˜x? 2i+C−1 2C1A−1 1A12C−1 2+Ihσi. Now, ga he ing e ms app op ia ely, (4.18), (4.19) ollow immedia ely. The p e ious esul con i ms ha i hσi= 0, and σis T-pe iodic, o a linea sys em he ela ion hxi=hx?iis ul illed in s eady-s a e. This esul can be applied as long as he a e age alue o σbecomes null in a T-pe iodic window. I is e iden ha o egula ion cases, i.e. x∗=c , once he s eady-s a e sliding mo ion has been achie ed and he swi ching pe iod has been success ully egula ed o he desi ed swi ching pe iod T∗, appea s a T∗-pe iodic beha iou o σ(x). Since unde hese condi ions ∆ becomes cons an , he mean alue o σis null and he p e ious esul applies. Equi alen ly, in he acking case scena io, his esul could also be applied, e en when he swi ching unc ion does no ha e a T∗-pe iodic beha iou i exis s a T-pe iodic beha iou o σin a la ge ime window, Tw. In o de o i he condi ions o Theo em 4, a s eady-s a e sliding mo ion and wi h he SFC p ope ly egula ing he swi ching pe iod, Twmus be a mul iple o T∗and he mean alue o σdu ing Twshould be also ze o. Howe e , no ice ha he p e ious esul does no apply in ansien s, nei he unde egula ion ask no in he acking case. Sec ion 4.4 deals wi h his case, in o de o see i s impac in he eal sliding mode. Rema k 6. The null mean alue o σob ained in (4.17)and used in Theo em 4 akes an impo an ele ance in his hesis, due o he ac ha such null mean alue was achie ed unde he assump ion o piecewise linea beha iou o he swi ching unc ion wi hin a ixed hys e esis band. In all he heo e ical de elopmen s shown in Chap e 2, he piecewise linea cha ac e is ic o σwas he hypo hesis suppo ing he alidi y o he de eloped models. Hence, wi h he las esul , a swi ching unc ion wi h cons an slopes in s eady-s a e is no only use ul o he alidi y o he SFC de elopmen s bu also o gua an ee ha x?−x, has a null mean alue unde ce ain condi ions. 58 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 4.3 Case s udy: he Buck con e e In o de o highligh he impo ance o he esul p o ided by Theo em 4, some simula ions a e included below using a simila sys em han he p oposed in (3.1), (3.2). 4.3.1 Ma hema ical model The p oposed sys em is gi en by (4.24), (4.25), whe e he ac o βhas been included and he con ol ac ion, u, akes alues in he se {0,1}, being u+= 1, u−= 0. ˙x1=−βx1+x2,(4.24) ˙x2=−x1+Mu, (4.25) whe e Mand βa e eal posi i e sys em pa ame e s. Unlike he simula ion shown in Chap e 3, in his case he egula ed s a e a iable is x1ins ead o x2. Acco ding o he ela i e deg ee, he swi ching unc ion includes he egula ion e o and i s i s ime de i a i e: σ=σ(x) = α(x∗ 1−x1) + d d (x∗ 1−x1),(4.26) being α > 0 a su ace pa ame e and x∗ 1 he e e ence alue. Assume a his poin ha , ollowing he p ocedu e discussed in Sec ion 3.1, he con ol law o uensu ing sliding mo ion on |σ|<∆ has been p ope ly designed. Le us di ec ly ollow wi h he applica ion o he egula o m me hod in oduced in Sec ion 4.1. The new a iables a e based on he e o s e= (e1, e2)>, de ined as: e1=x∗ 1−x1, e2=βx∗ 1−x2.(4.27) The selec ion o he new a iables a ise om x1,x2 ela ed o he s eady-s a e equilib ium poin o (4.24), (4.25), which a e x1=x∗ 1and x2=βx∗ 1. The e o e, he e o dynamics can be ound as: ˙e1=e2−βe1(4.28) ˙e2=−e1−M u +x∗ 1,(4.29) and he swi ching unc ion becomes: σ(e1, e2)=(α−β)e1+e2.(4.30) Sys em (4.28), (4.29) is al eady in egula o m and ma ches (4.3), (4.4), and so does he swi ching unc ion (4.30) wi h (4.5). Hence, ollowing he p ocedu e de ailed in Sec ion 4.1, he eal dynamics (4.9), (4.10) eads in his case as: ˙e1=−αe1+σ, (4.31) e2= (β−α)e1+σ, (4.32) Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 59 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP while he ideal dynamics (4.11), (4.12) boils down o ˙e? 1=−αe? 1,(4.33) e? 2= (β−α)e? 1.(4.34) No ice ha , in his case, g( ) = 0 and α > 0; consequen ly, when σ=σ( ) is T- pe iodic wi hin he hys e esis bandwid h (|σ|<∆), he hypo heses o Theo em 4, including Assump ion C, a e ul illed. Hence, he solu ions o (4.31), (4.32) end asymp o ically o he pe iodic solu ion ˜e= (˜e1,˜e2)>, wi h ˜e1,˜e2 ela ed by (4.32). As o ˜e1, a T-pe iodic solu ion o (4.31) is gi en by (see, o example, [40]): ˜e1( ) = e−α eαT −1ZT 0 eατ σ(τ)dτ +e−α Z 0 eατ σ(τ)dτ. (4.35) In u n, he solu ions o (4.33), (4.34) end asymp o ically o he equilib ium poin ˜e?= 0. Then, acco ding o Theo em 4, since he dis u bance ec o g( ) is null in his case, (4.18) becomes h˜e?i= 0, while h˜eiis ul illing (4.19), i.e. h˜e1i=1 αhσi,(4.36) h˜e2i=β αhσi.(4.37) The e o e, in case ha hσi= 0, (4.36), (4.37) yield h˜ei= 0. This implies ha hx1i=x∗ 1 and hx2i=βx∗ 1, meaning ha he a e age alue o he eal signal is exac ly equal o he e e ence one. 4.3.2 Simula ion esul s A nume ical analysis o sys em (4.24), (4.25) has been ca ied ou wi h he pa ame e alues M=β= 3. Fo he swi ching unc ion in (4.26) we ha e chosen α= 1, while he e e ence alue o x1has been se o x∗ 1= 1. In acco dance wi h (4.27), his selec ion en ails ha he ideal s eady-s a e alue o x2is x∗ 2= 3. The simula ion consis s in implemen ing he con ol law o he o m deno ed in equa ion (1.13), wi h h ee di e en alues o he hys e esis bandwid h, ∆ =    0.1 when 0 ≤ < 16s, 0.04 when 16s≤ < 30s, 0.01 when 30s≤ < 40s, and checking ha (4.36), (4.37) a e ul illed, i.e. ha he a e age s eady-s a e e o s he1i, he2i e i y: he1i=hσi,he2i= 3 hσi. Le us assume ha du ing he las ins an s o he ime windows in which each o he h ee 60 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP alues o ∆ is ac i e he a iables ha e achie ed a s eady-s a e. The es s ha e ca ied ou wi h he so wa e package MATLAB/Simulink (R2016b) using an ODE 5 sol e wi h a ixed s ep o 10−5. As o he a e age alues, hey ha e been ex ac ed using a low-pass il e wi h ans e unc ion H(s) = 100 s2+ 18s+ 10010 . 0 5 10 15 20 25 30 35 40 -0.2 0 0.2 0 5 10 15 20 25 30 35 40 0 0.5 1 0 5 10 15 20 25 30 35 40 0.99 0.995 1 ∆ =0.1 ∆ =0.04 ∆ =0.01 ime (s) x? 1 x1 x? 1 x1 σ Figu e 4.1: Sys em esponse wi h di e en hys e esis band alues. Top plo : swi ching unc ion. Mid plo : s a e a iable x1and i s e e ence, x? 1. Bo om plo : zoom o he mid plo . The op plo in Figu e 4.1 depic s he swi ching unc ion, σ. As expec ed, he cha e ing ampli ude is highe when ∆ = 0.1, and dec eases while ∆ does. In u n, he mid plo depic s he beha iou o he s a e a iable x1wi h espec o i s e e ence alue and he ideal sliding dynamics one, x? 1. The zoom in he bo om plo e eals ha x1s abilizes close o he e e ence, i.e. wi h less a e age s eady-s a e e o , and also wi h dec easing cha e ing ampli ude, o lowe alues o ∆. F om he e i is con i med how x1→x? 1as ∆→0. Acco ding o Theo em 4, i his exis s a egula ion e o in x1wi h espec o x∗ 1 (as Figu e 4.1 depic s, mainly when ∆ = 0.1), he mean alue o σshould no be ze o. Le us see such esul wi h de ailed zoomed iews o he h ee cases appea ed in Figu e 4.1. The op plo s in Figu es 4.2,4.3 and 4.4 show he swi ching unc ion and i s mean alue o ∆ = 0.1, ∆ = 0.04 and ∆ = 0.01. No ice ha he signal en elope allows an easy Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 61 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 15 15.5 16 -0.2 0 0.2 15 15.5 16 3 4 510-3 15 15.5 16 0 1 2 ∆ = 0.1 ime(s) 10-2 hσi he1i 3hσi he2i σ Figu e 4.2: Pe o mance o ∆ = 0.1 in he s eady-s a e. Top plo : swi ching unc ion. Mid plo : he1ima ching hσi. Bo om plo : he2ima ching 3 hσi. iden i ica ion o he cu en hys e esis alue. I is also wo h emphasizing ha , as we a e in a egula ion con ol p oblem and he hys e esis band is symme ic wi h espec o ze o, he swi ching pe iod o he con ol ac ion achie es a cons an alue, T, in he s eady-s a e, and he swi ching unc ion becomes T-pe iodic, hus mee ing Assump ion C. I is clea om he op plo in Figu e 4.2 ha , o he highes alue o he hys e - esis, namely ∆ = 0.1, he swi ching unc ion does no show a piecewise linea beha iou . Consequen ly, i s mean alue is no ze o, as con i med by he mid plo , whe e i is shown o ma ch ha o e1, his esul ing in he s eady-s a e e o o x1obse ed in he i s pa o he bo om plo in Figu e 4.1. In u n, one can obse e in Figu es 4.3 and 4.4 ha lowe alues o ∆ en o ce he piecewise linea cha ac e o σwi hin he hys e esis band, his yielding lowe mean alues o he espec i e swi ching unc ions and also o he s eady-s a e e o s o x1a ising in Figu e 4.1. In all hese cases he1ima ches hσi. In u n, he2ialways coincides wi h 3 hσi, as expec ed. Hence, his con i ms he heo e ical p edic ions o Theo em 4. When he piecewise linea assump ion o σ(x) is close o be ul illed, he a e age alues o σ end o ze o, and so do he a e age s a e e o s, he1i,he2i. Con e sely, s eady-s a e e o s appea in he s a e a iables when he piecewise linea assump ion o σdoes no hold and hσi 6= 0. In any case he1i,he2i, ma ch he expec ed alues hσiand 3 hσi, espec i ely. 62 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 29.5 29.75 30 -0.1 0 0.1 29.5 29.75 30 5 5.5 610-4 29.5 29.75 30 0 1 2 310-3 ime (s) hσi he1i 3hσi he2i ∆ = 0.04 σ Figu e 4.3: Pe o mance o ∆ = 0.04 in he s eady-s a e. Top plo : swi ching unc ion. Mid plo : he1ima ching hσi. Bo om plo : he2ima ching 3 hσi. 39.9 39.95 40 -0.02 0 0.02 39.9 39.95 40 2.5 3 3.5 10 39.9 39.95 40 0 1 210 ime (s) hσi he1i 3hσi he2i ∆ = 0.01 σ -4 -5 Figu e 4.4: Pe o mance o ∆ = 0.01 in he s eady-s a e. Top plo : swi ching unc ion. Mid plo : he1ima ching hσi. Bo om plo : he2ima ching 3 hσi. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 63 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP dynamics o x. As a consequence, wha e e mean alue appea ing on xdue o he a ia ion o he hys e esis band, as (4.49)s a es, will be negligible wi h espec o x∗. 4.4.3 Simula ion esul s In o de o show he e ec o he SFC ansien s in he SMC pe o mance, a simple sim- ula ion is done. Using he sys em in oduced in Sec ion 4.3.2, a sudden a ia ion o he swi ching pe iod e e ence is applied en o cing a SFC ansien . The alues o γ(γL) will be designed in o de o show how highe alues o γ(γL) will imply a highe impac in he s a e a iables beha iou s. The simula ion depic s he ansien de ail o Tk ollowing he a ia ion o T∗. This ansien will be es ed wi h wo di e en alues o γ(γL), designed so ha one o hem p oduces a so ansien in Tk, and he o he an unde damped one. Le us calcula e he alues o ρ± ∗( ela ed o he s eady-s a e sliding mo ion) o he simula ed sys em in Sec ion 4.3.2. Using (4.26) he i s ime de i a i e o σ o he s eady-s a e sliding mo ion can be ound assuming ha x1=x∗ 1and x2=βx∗ 1: ˙σ∗=x∗ 1−Mu. (4.51) Using he alues p o ided in Sec ion 4.3.2 o x∗ 1and M, and acco ding o he ρ± ∗de ini ion in (2.2), one ge s: ρ+ ∗:= 1 ˙σ∗|u=0 , ρ− ∗:= 1 ˙σ∗|u=1 . Hence, ρ+ ∗= 1, ρ− ∗=−0.5. Regula ion case: he disc e e- ime app oach Fi s ly, he disc e e- ime app oach is simula ed (Sec ion 2.2.1, equa ion (2.13)).The cha - ac e is ic polynomial o he SFC o his case is (acco ding o (2.16)): p(z) = z2+z(2γ−1) + γ, and he selec ed alues o γ o he a o emen ioned pu pose a e: γ= 0.05 and γ= 0.5, which lead o he ollowing closed-loop poles: γ= 0.05; pz1= 0.8405, pz2= 0.0595 γ= 0.5; pz1= 0.7071i, pz2=−0.7071i. No ice how he chosen alues o γp o ide poles in he eal axis o a so ansien esponse o he case γ= 0.05, becoming complex o γ= 0.5. The esul ing poles a e wi hin he uni ci cle in bo h cases, hus con i ming s able SFC esponses. The simula ion o bo h cases a e shown in Figu es 4.7,4.8. The signals depic ed in he Figu es a e s uc u ed as ollows. In he op plo he swi ching unc ion, σ, and he hys e esis bandwid h gene a ed 70 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP by he SFC a e depic ed. In he mid plo he swi ching pe iod e e ence, T∗, and eal swi ching pe iod, Tk, a e shown. Finally, he bo om plo p esen s he beha iou o x1and he mean alue o σ. In his case, he low pass il e de ailed in Sec ion 4.3.2 measu ing he a e age alue is subs i u ed by he ope a o shown in (4.13) compu ed one ime pe swi ching pe iod. As a consequence, such mean alue is delayed one swi ching in e al. 14 15 16 17 18 19 20 0.05 0.1 14 15 16 17 18 19 20 -0.05 0 0.05 14 15 16 17 18 19 20 -2 0 210-3 T∗ Tk x∗ 1−x1 hσi σ ∆ Time (s) Figu e 4.7: Mean alue impac on σin he ace o a swi ching pe iod ansien wi h he disc e e- ime app oach. γ= 0.05. Fi s o all, i should be no ed ha sliding mo ion is p ese ed du ing he en i e es , since σis pe ec ly con ined wi hin he hys e esis bandwid h du ing he ansien s (see op plo s o Figu es 4.7 and 4.8). Simila ly, he p ope egula ion o Tk o T∗in he ansien s is also con i med om he mid plo s o Figu es 4.7,4.8. F om he esul s on he bo om plo s, i is ob ious ha he signal x1is highly pe u bed when he ansien in he SFC is as e (Figu e 4.8,γ=0.5), being such pe u ba ion smalle when γ=0.05 is used (Figu e 4.7). This esul coincides wi h he esul ob ained in (4.39), as he mean alue o σis p opo ional o he used γ. Fu he mo e, i is wo h ema king ha he simula ions in Figu es 4.7 and 4.8 co obo a e he esul o Theo em 4, since he mean alues o σand x∗ 1−x1only ma ch a s eady-s a e, when ∆ is cons an and σbecomes T∗-pe iodic. Regula ion case: he con inuous- ime app oach The same simula ion has been pe o med using he SFC in he con inuous- ime app oach (see Sec ion 2.3, equa ion (2.35)). The p e ious calculus o ρ± ∗hold o his app oach, which yields α= 2 ( ˙σ+ ∗= 1, ˙σ− ∗= -2). The ini ial condi ion o ek−1is 0.05 and he alues Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 71 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP Time (s) 14 15 16 17 18 19 20 0.05 0.1 14 15 16 17 18 19 20 -0.05 0 0.05 14 15 16 17 18 19 20 -2 0 210-3 T∗ Tk x∗ 1−x1 hσi σ ∆ Figu e 4.8: Mean alue impac on σin he ace o a swi ching pe iod ansien wi h he disc e e- ime app oach. γ= 0.5. o γLin o de o p oduce a smoo h and a as ansien in Tka e γL= 0.5 and γL= 2.5, espec i ely. F om he di ec compa ison o he esul s in Figu es 4.9,4.10, he analysis made in Sec ion 4.4.1 can be alida ed, since he use o a highe alue o γLinc eases he mean alue o σdu ing he ansien . The acking case The sys em simula ed in Sec ion 4.3.2 is now es ed wi h a ime- a ying e e ence o x1, in o de o check he mean alue o σ. In his case, he simula ion esul s a e ocused in he s eady-s a e ope a ion, as he ansien e ec s a e essen ially he same han he ones shown in Figu es 4.7,4.8, because he app oach o adjus he hys e esis alues is he same. The objec i e he e is o check i he mean alue o σdu ing a la ge ime in e al, Tw, p esen s a pe iodic beha iou , and i could become ze o when a e aging i in Tw. The ime- a ying e e ence o his case is: x∗ 1= 2 sin (2 ·0.02 π ), keeping T∗= 0.1 s as he desi ed swi ching pe iod. Fo he acking case, he sys em is sligh ly modi ied, including a con ol ac ion aking u+= 1 and u−=−1 (see (4.24), (4.25)). Wi h his con igu a ion, he sys em is able o ack e e ences wi hou o se . The SFC s uc u e is he co esponding o a acking app oach, (Sec ion 2.2.2, Figu e 2.6, Equa ions 72 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP 14 15 16 17 18 19 20 14 15 16 17 18 19 20 14 15 16 17 18 19 20 10 -0.5 1 0.05 -0.05 0 -3 2 -2 0 Time (s) T∗ Tk x∗ 1−x1 hσi Figu e 4.9: Mean alue impac on σin he ace o a swi ching pe iod ansien wi h he con inuous- ime app oach. γL= 0.5. 14 15 16 17 18 19 20 14 15 16 17 18 19 20 14 15 16 17 18 19 20 10 -0.5 1 0.05 -0.05 0 -3 2 -2 0 Time (s) T∗ Tk x∗ 1−x1 hσi Figu e 4.10: Mean alue impac on σin he ace o a swi ching pe iod ansien wi h he con inuous- ime app oach. γL= 2.5. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 73 CHAPTER 4. REAL SLIDING DYNAMICS IN A SWITCHING FREQUENCY CONTROL LOOP (2.18), (2.19) and (2.20)). The con ol gain is se o γ= 0.5 acco ding o Theo em 2. 100 120 140 160 180 200 0 100 120 140 160 180 200 0 100 120 140 160 180 200 -2 0 2 x10-3 x∗ 1−x1 hσi ∆ σ T∗ T Time (s) 0.1 0.2 0.1 -0.1 Figu e 4.11: S eady-s a e mean alue o σin he acking case wi h SFC, γ= 0.5. Figu e 4.11 shows: he swi ching unc ion and he hys e esis band in he op plo , he desi ed and eal swi ching pe iods in he mid plo and he acking e o oge he wi h he a e age alue o σin he bo om plo . I should be no iced ha he mean alue o σ has been calcula ed using he ope a o shown in (4.13) a any swi ching pe iod, T. F om Figu e 4.11, he p ope SFC unc ion is con i med, since T∗and Tpe ec ly i (mid plo ). The in e es ing esul is loca ed in he bo om plo , whe e a pe iodic beha iou o hσiis obse ed ( he showed ime in e al co esponds exac ly o a wo pe iods o Tw= 50 s). Due o he symme y o he signal, i is no un easonable o assume ha he mean alue o σalong a ime in e al o Tw= 50 s is null. No ice how hσiis close o he a e aged acking e o de ined as e =x∗ 1−x1, since he exp ession en o ced by he sliding mo ion (see (4.26)): σ=αe + ˙e , always holds. Pe o ming he Laplace ans o m, he p e ious exp ession e eals ha hσi is he esul o low-pass il e ing e . Finally, i has been conside ed in e es ing o show he same esul in Figu e 4.11 bu wi h a ixed hys e esis band. The esul s a e shown in Figu e 4.12. The di e ence, aside ha a ixed hys e esis alue p oduces ime- a ying swi ching pe iods, elies in he di e en mean alues p oduced in σ. By di ec compa ison wi h Figu e 4.11, i can be no ed ha he maximum alue o he mean alue is sligh ly highe when he hys e esis band is ixed. Again, he mean alue o σma ches he a e aged acking e o . 74 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 100 120 140 160 180 200 100 120 140 160 180 200 100 120 140 160 180 200 10-3 x∗ 1−x1 hσi 0.1 -0.1 0 0.1 0.2 0 x 2 -2 0 T∗ T ∆ σ Time (s) Figu e 4.12: S eady-s a e mean alue o σin he acking case wi h ixed hys e esis band, ∆ = 0.05. Pa III Applica ion o he P oposed Solu ion o Powe Con e e s. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 77 Chap e 5 Vol age Regula ion in a Buck Con e e . A his s age, he disc e e- ime app oach o he SFC is expe imen ally es ed in a DC-DC Buck con e e o e alua ion pu pose. The s a e space equa ions modelling he Buck con e e beha iou co espond o a linea sys em. In his case, he SMC will egula e he con e e ou pu ol age, becoming a egula ion ask o a ime in a ian linea sys em. The SFC will be implemen ed by a mid- ange mic o-con olle (µC), whe eas he SMC will be assembled by means o analog ci cui y. The Chap e is s uc u ed as ollows: in he i s Sec ion he powe con e e is p esen ed, whe e i s s a e space equa ions and pa ame ic alues a e in oduced. Then, he SMC con olle is designed o egula ing he ou pu ol age. The nex s ep de elops he s udies equi ed o p ope ly uning he SFC. Finally, he implemen a ion issues and he expe imen al esul s a e p esen ed. 5.1 The Buck Con e e The Buck con e e ci cui scheme is shown in Figu e 5.1. Table 5.1 lis s he pa ame e alues o he expe imen al p o o ype. E M1 u= 1 u= 0 L CR c il M2 Figu e 5.1: Buck con e e . Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 79 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. In ha case, he compu ing ime esul s in c= 1.4 µs (empi ically measu ed using an oscilloscope), while he expec ed alue o T+ kis 2.5 µs, hus con i ming he use ulness o (2.1). Finally, and addi ional aspec abou his implemen a ion me hodology is discussed. I exis s a ela ion be ween he esolu ions o he used TIM and DAC de ices [29]. I he esolu ion o he DAC is highe han he esolu ion o he TIM, he hys e esis alues could oscilla e be ween wo alues in s eady-s a e. Fo his eason, is ad isable o use a TIM wi h a highe esolu ion han he DAC one in o de o ensu e ha he hys e esis alue becomes cons an in s eady-s a e. This e ec is also ela ed wi h he in eg al gain γ, since he g ade o change o ∆ ( ∆k−∆k−1) is, om (2.13), ekγ, and ekinhe i s he ime esolu ion. The e o e, in o de o a oid oscilla ions in he sys em due o he esolu ions o he used pe iphe al, he inequa ion (5.14) has o be ul illed: DACR> γ TIMR,(5.14) being DACRand TIMR he co esponding pe iphe al esolu ions. 5.5 Expe imen al esul s The expe imen al esul s a e p esen ed a his s age. In he ollowing oscilloscope cap u es, he swi ching pe iod, Tk, appea s con e ed o ol age wi h a a e o 0.35 V/µs. Due o implemen a ion aspec s, σhas an o se o 2.5 V (as i was shown in Figu e 5.2). Unless o he wise no ed, he SFC gain is se o γ= 2 ·104. 1. SMC pe o mance The s a -up o he con e e is illus a ed o wo di e en ini ial alues o ∆ in Figu es 5.5 and 5.6. In hese cases he ou pu ol age is egula ed o 12 V. The ini ial alues o ∆ ha e been selec ed as hey become smalle and highe han he s eady-s a e one. No ice ha bo h cand Tka ain hei e e ences wi h a good ansien esponse, while he hys e esis band is adap ing ill Tk eaches T∗. F om bo h esponses, he expec ed o e damped cha ac e is ic can also be con i med, acco ding o he designs de eloped a Sec ions 5.2,5.3. The swi ching pe iod, Tk, can be easily measu ed obse ing he swi ching unc ion in he zoomed windows (g een signals a he bo om pa s o he Figu es). As i is no ed in (2.13), once Tk eaches he desi ed alue, he hys e esis band ∆kbecomes cons an . In Figu e 5.7 a ol age egula ion om 12 o 24 V and ice- e sa, wi h R= 4 Ω, is es ed. In his case ∗ cis s ep changed and, consequen ly, he swi ching unc ion suddenly d ops he hys e esis ampli ude alue and eco e s i again in less han one swi ching pe iod, wi h a b ie and smoo h ansien o Tk. Again, he expec ed o e damped beha iou o cis con i med. 86 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. cTk σ ∆ Figu e 5.5: S a -up o ∗ c= 12 V wi h R= 2 Ω and ∆0lowe han he s eady-s a e alue. c: blue; σ: g een; |∆k|: magen a; Tk: ed. cTk σ ∆ Figu e 5.6: S a -up o ∗ c= 12 V wi h R= 2 Ω and ∆0highe han he s eady-s a e alue. c: blue; σ: g een; |∆k|: magen a; Tk: ed. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 87 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. Tk c ∗ c σ Figu e 5.7: Vol age e e ence, ∗ c, a ia ion om 12 V o 24 V and ice- e sa, wi h R= 4 Ω. c: blue; σ: g een; ∗ c: magen a; Tk: ed. 2. SFC pe o mance In o de o e alua e he p ope pe o mance o he SFC, se e al e e ence s ep a i- a ions a e es ed in he ollowing. Fi s o all, Figu e 5.8 shows he esul o a s ep change o T∗be ween 12.5 µs and 8.3 µs wi h a desi ed ou pu ol age o 12 V and R=4 Ω. The Figu e con i ms a good pe o mance o he SFC wi h an o e damped beha iou , as expec ed wi h he used alue o γ(γ= 2 ·104). Tk c ∆ σ Figu e 5.8: O e damped esponses o γ= 2 ·104wi h R=4 Ω and a T∗ a ia ion om 12.5 µs o 8.3 µs. c: blue; σ: g een; |∆k|: magen a; Tk: ed. Figu e 5.9 shows he esul o he same es (s ep change o T∗), when he in eg al gain is selec ed close o he uns able alues (γ= 2 ·105, being he s abili y limi o 88 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. 2.07 ·105). In his case T∗is a ied om 12.5 µs o 14 µs and Ris kep a 4 Ω. The modi ica ion o he s ep alues is equi ed o a oid he e ec s o ∆ sa u a ion and he compu ing ime in luence ha would modi y he expec ed dynamics (see Sec ion 5.4). As γis now close o he uppe s abili y limi , bo h Tkand ∆kexhibi unde damped ansien esponses wi h e y low damping a io, hus con i ming he heo e ical p edic ion. I is e iden ha such esponse i s wi h he expec ed one, as Figu e 3.2 deno ed in Sec ion 3.1 Tk c ∆ σ Figu e 5.9: Unde damped esponses o γ= 2 ·105wi h R=4 Ω and a T∗ a ia ion om 12.5 µs o 14 µs. c: blue; σ: g een; |∆k|: magen a; Tk: ed. Finally, he p e ious es is epea ed wi h di e en alues o γ, in o de o con i m he alidi y o he model de eloped in Chap e 2. In Figu e 5.10 he oo locus o he SFC in he case o 12 V is shown, whe e he poles placemen o di e en alues o γa e depic ed. Rela ed wi h such poles, he co esponding esponses in he ime domain a e also shown. All he es a e made wi h a s ep o T∗ om 12.5 µs and 8.3 µs, excep o he γ= 2 ·105case, which is om 12.5 µs o 14 µs. The swi ching pe iod ( ed signal in Figu e 5.10) is measu ed wi h an analog senso , which adds some dynamics o he measu e. The pe o mance has o be analysed om he ∆ esponse (magen a signal), which is gene a ed h ough a DAC by he µC, wi hou any delay. F om he esul s shown in Figu e 5.10, he SFC model de eloped a Chap e 2is ully co obo a ed. 3. Sys em obus ness Finally, a load ansien es is pe o med a he con e e ou pu , in o de o con i m ha one o he main bene i s o he SMC (i s obus ness) is p ese ed unde he SFC ope a ion. The es consis s o suddenly a ia ions o he linea load applied, om 0 o 6 A in he 12 V and 24 V cases. In he Figu es 5.11,5.12, he a iable io esponds Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 89 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. γ= 2 ·105 γ= 1 ·105 γ= 6 ·104 γ= 3 ·104 γ= 2 ·104 γ= 1 ·104 γ= 0 Imagina y Pa Real Pa 1 0.4 0.8 0.6 0.2 0 1 -0.4 -0.8 -0.6 -0.2 1 -0.6 0 -1 -0.2 0.2 0.6 Roo locus zp1= 0.86, zp2= 0.05 zp1= 0.7, zp2= 0.14 zp1,2= 0.38 ±0.025i zp1,2= 0.26 ±0.47i zp1,2= 0.1±0.69i zp1,2= 0.3±0.93i Figu e 5.10: Poles placemen in he complex plane and he co esponding ime domain s ep esponses, wi h R=4 Ω. c: blue; σ: g een; |∆k|: magen a; Tk: ed. 90 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. o: io= c R. c io Tk σ Figu e 5.11: Load T ansien : esponses o γ= 2 ·104wi h R om no load o 2 Ω, T∗=10 µs and ∗ c=12 V. c: blue; σ: g een; io: magen a; Tk: ed. F om Figu es 5.11 and 5.12, he good ansien esponse o he SMC is con i med, since he egula ed ou pu ol age is ha dly dis u bed. In he same way, i is also con i med ha in his design he alues o ρ± ∗do no depend on he ou pu cu en , since be o e and a e he ansien , he s eady-s a e hys e esis alues a e essen ially he same. This ac con i ms he heo e ical alues ound o ρ± ∗in Sec ion 5.3.1, as exp ession (5.12) does no depend on he ou pu load R. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 91 CHAPTER 5. VOLTAGE REGULATION IN A BUCK CONVERTER. c io Tk σ Figu e 5.12: Load T ansien : esponses o γ= 2·104wi h R om no load o 2Ω, T∗= 10µs and ∗ c=24 V. C: blue; σ: g een; io: magen a; Tk: ed. 5.6 Conclusions In his Chap e he design and implemen a ion s eps o he SMC and SFC ha e been de- sc ibed, and se e al expe imen al esul s ha e been p esen ed co obo a ing he alidi y o he p oposed p ocedu es, and he models de eloped in Sec ion 2.2.1. Addi ionally, a digi al implemen a ion o he SFC has been de eloped using a µC om ST Mic oelec- onics (STM32F407) allowing o demons a e he expec ed beha iou o he SFC in he disc e e- ime app oach. The expe imen al esul s con i m he ou pu ol age egula ion, he ope a ion a ixed swi ching pe iod, and he sys em obus ness wi h espec o load and ou pu ol age a ia ions. 92 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s Chap e 6 Vol age Regula ion in a Mul iphase Buck Con e e . The con inuous- ime app oach o he SFC, ound in Sec ion 2.3, is applied o a mul iphase synch onous Buck con e e in he ollowing. The Chap e is s uc u ed as ollows, in he i s Sec ion he powe con e e da a and he sliding mode in e lea ing ope a ion a e b ie ly in oduced. Then, he SMC con olle is designed o egula ing he ou pu ol age wi h in e lea ing ope a ion, ollowed by he design o he SFC in he con inuous- ime app oach. Finally, he implemen a ion de ails and he expe imen al esul s a e p esen ed. 6.1 The mul iphase con e e The mul iphase synch onous con e e is made up by he pa allel connec ion o mBuck con e e s. Such opology is shown in Figu e 6.1. Since his opology is based on he connec ion o se e al synch onous Buck con e e s wi h hei ou pu s joined in pa allel, he co esponding s a e space equa ions a e equi alen o he ones p esen ed in Chap e 5, ex ended o a mul i-inpu case. E Mm M1 Mm M1 L1 Lm mC R il1 ilm io m m c Figu e 6.1: Ci cui scheme o he m-phase synch onous Buck con e e . Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 93 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. The e o e, o he mul iphase s uc u e he equa ions esul in: Ldik d =− c+E uk;k= 1, .., m (6.1) Cd c d = m X k=1 ik− c R(6.2) whe e he con ol ac ions uk ake alues om he se {0,1},Eis he inpu ol age, c he ou pu ol age, ikis he cu en lowing h ough he k- h phase and L,C,Ra e he induc ance, he capaci ance and he esis i e load, espec i ely. An impo an bene i o he mul iphase opology is he possibili y o implemen in e - lea ing ope a ion. This echnique is based on phase shi ing he con ol ac ions o each con e e in such a way ha he high equency cu en ipple o each induc o a e can- celled in he common ou pu connec ion, hus gene a ing an ideally ee ipple cu en o he ou pu ol age. This echnique allows o educe conside ably he ou pu capaci o alue, since, in he ideal case, he e is no high equency cu en ipple o be il e ed a he ou pu . Mo eo e , he dis ibu ion o he powe h ough di e en con e e s pe mi s he educ ion o he componen ea u es, as he allowable conduc ion cu en o he powe swi ches, o he equi ed hea sinks o losses dissipa ion, which in some cases can be di ec ly emo ed om he sys em. The educ ion o he cu en lowing by he swi ches also allows o inc ease he swi ching equency, which in u n, would lead o an addi ional educ ion o he alue o he eac i e componen s. As a consequence, he mul iphase s uc- u es ha e gained in e es wi hin he indus ial communi y o di e en applica ions due o hei high e iciency, good powe densi y, as ansien esponse, and abili y o in e lea ing ope a ion [47–51]. The con ol objec i e is again he ou pu ol age egula ion, wi h he addi ional ask o gua an eeing he in e lea ing among he phases, which is also con olled by a sliding mode echnique. The SMC and he SFC will be implemen ed by means o analog ci cui y, meanwhile he in e lea ing con ol is implemen ed wi h an FPGA. The pa ame ic da a o he assembled mul iphase con e e a e de ailed in Table 6.1. 94 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. Table 6.1: Mul iphase Buck Con e e Pa ame e s Pa ame e Symbol Value Inpu Vol age E48 V Desi ed Ou pu Vol age Range ∗ c12-24 V Ou pu Capaci o C100 µF Phase Induc ance L22 µH±10% Numbe o phases m8 Load Range Io0-65 A Desi ed swi ching pe iod T∗10 µs Cu en ans o me pa ame e s Lx,M800 µH, 6.4 µH Cu en ans o me bu den esis o Rb10 Ω 6.2 In e lea ed sliding mode con ol o he ou pu ol age The in e lea ed sliding mode con ol co esponds o a Mas e -Sla e s a egy. One o he Buck con e e s egula es he ou pu ol age, he Mas e con e e (o Mas e phase), while he es o he phases ack he con ol signal o he Mas e one wi h he p ope phase shi ing. 6.2.1 Mas e swi ching su ace design The Mas e swi ching su ace σM o ou pu ol age egula ion is designed as: σM:= ψ1( c− ∗ c) + ψ2xM= 0,(6.3) whe e ψ1,2a e he swi ching su ace cons an s and ∗ cis he desi ed ou pu ol age. Com- pa ing i wi h he su ace designed o he single Buck con e e (see Sec ion 5.2), he i s ime de i a i e o ecis eplaced by he signal xM. As i was discussed in he Sec ion 5.4, i is usual o employ he measu ed cu en lowing by he ou pu capaci o ins ead o ime di e en ia e he ou pu ol age in he implemen a ion se -up. In his case, i is no possible o use he cu en ipple o he ou pu capaci o since i is ideally cancelled by he in e lea ing ope a ion. As a consequence, a cu en ans o me is placed in se ies wi h he Mas e phase induc o . The signal xMis he ou pu o his cu en ans o me , which Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 95 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. ci cui s, one o hem is in cha ging o synch onizing he cap u e o he ol age alue o he capaci o in a ising edge o he con ol signal u, while he second one con ols he swi ch ha ese s his ol age jus a e he alue has been acqui ed. This beha iou is depic ed in he le block o Figu e 6.5, called Pe iod Senso . The sys em wo ks as a sample and hold ci cui synch onized wi h uM, holding he las measu emen un il he ci cui is igge ed again. Al hough he sys em deli e s disc e e- ime measu emen s, he ime used o cap u e he ol age alue (1 µs) oge he wi h he ini e alue o he sample and hold capaci o (CSH = 1 nF) add dynamics o he measu emen , which can be modelled by a i s o de esponse. This model is cha ac e ized by τ, which was empi ically ound o 65 ·10−6and used in he p e ious s abili y analysis. Such designed ci cui is in ended o gene a e 5 V when he swi ching pe iod is o 10 µs. The pa s called In eg a o and Pe iod E o make up he SFC. Indeed, as i can be in e ed o m he Figu e, hey use s anda d con igu a ion based on AO o compu ing he pe iod e o and he in eg al ac ion. No ice how due o he elec onics used a e unipola , all he ci cui s a e pola ized wi h a 5 V o se . Despi e o he in eg al ac ion i sel , such ci cui includes a ha dwa e sa u a o o he hys e esis alue, con o med by a Scho ky diode in an i se ies wi h a zene diode. This s uc u e implemen s he maximum inc emen o ∆, ∆max, being he block called Hys e esis gene a o who ixes he minimum alue o ∆, ∆min. Again using AO he ∆ and −∆ a e gene a ed, ensu ing hei symme ical cha ac e is ic. 6.5 Expe imen al esul s Finally, he expe imen al esul s ob ained in he labo a o y wi h he buil p o o ype a e shown a his Sec ion. Fi s ly, he measu ed sys em ea u es as e iciency, line egula ion and load egula ion a e summa ized in Table 6.2. Table 6.2: Expe imen al esul s o he Mul iphase con e e E iciency (%) Load Regula ion (%) Line Regula ion (%) Line Regula ion (%) @65A, c=12/24. E=48 V, c=12/24. @ c=24 V, Pou =1 kW. @ c=24 V, No load. E=48 V 0 - 65 A E=36 o 55 V E=36 o 55 V 94.8/97.1 ≤0.97/1.1 ≤1.7 ≤0.5 The da a shown a Table 6.2 ce i y a good pe o mance o he p o o ype, achie ing good le els o e iciency. I should be ema ked ha such e iciency includes all he powe consump ion o he sys em (including, FPGA, d i e s, e c). The sys em also p esen s a good obus ness, which can be in e ed om he line and load egula ion esul s. In o de o check he expec ed ea u es p o ided by he designed con olle s, se e al es s ha e been pe o med in he ollowing. They a e o ganized in ou g oups, namely: 102 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. SMC pe o mance, in e lea ing, SFC pe o mance and sys em obus ness. 1. SMC pe o mance The i s es consis s in s ep-change a ia ions o he e e ence ol age om 12 V o 24 V and e e sely, wi h a esis i e load o 1 Ω connec ed a he ou pu . Figu e 6.6 po ays he esponses o he ou pu ol age, he load cu en , he swi ching unc ion and he measu ed swi ching pe iod (scaled by 0.5 V/µs). The bo om windows show a zoom iew o he ansien beha iou when he ou pu ol age e e ence changes om 12 V o 24 V (le window) and when i dec eases om 24 V o 12 V ( igh window). No ice how he ou pu ol age beha es wi h a smoo h ansien esponse, which co esponds o he ideal sliding mo ion, and he hys e esis alues a e adap ed such ha he swi ching pe iod eaches he desi ed alue a s eady-s a e. c Tk σM io Figu e 6.6: Re e ence change 12-24-12 V wi h a load o 1 Ω a he ou pu . 2. In e lea ing Figu e 6.7 shows he beha iou o he cu en ans o me signals o he 8 phases in he s a -up, when he con e e supplies a load o 21 A and he ou pu ol age is egula ed o 24 V. As i can be seen in he oscilloscope cap u e, he in e lea ing ope a ion is s a ed om he second swi ching pe iod (see cu en wa e o ms on he le bo om window) and achie es in e lea ing a he desi ed swi ching equency o 100 kHz in he s eady-s a e (see igh bo om window). Figu e 6.8 shows he s eady- s a e beha iou s o he cu en ans o me signals o he 8 phases o a load o 65 A wi h an ou pu ol age o 24 V. F om he p e ious Figu es, he p ope in e lea ing ope a ion is ully co obo a ed. 3. SFC pe o mance The wo ollowing es s a e designed wi h he pu pose o alida e he SFC ope a ion. In Figu e 6.9 he s a -up o he mul iphase con e e o an ou pu ol age e e ence Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 103 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. Figu e 6.7: S a -up o he cu en ans o me signals o he 8 phases wi h a load o 21 A o an ou pu ol age o 24 V. Figu e 6.8: S eady-s a e o he cu en ans o me signals o he 8 phases wi h a load o 65 A o an ou pu ol age o 24 V. 104 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. o 24 V deli e ing 21 A o he load (a s eady-s a e) is p esen ed. No ice ha he ini ial alue o ∆ is a away om he s eady-s a e one. Again, he Figu e shows he beha iou s o he ou pu ol age, he swi ching unc ion, he Mas e con ol signal and he measu ed swi ching pe iod. The bo om windows de ail he wa e o ms in he ansien s a e (le window) and in he s eady-s a e ( igh window). As i can be seen in he Figu e, he ou pu ol age eaches he desi ed ol age wi h a smoo h ansien and wi h a small o e shoo . Fu he mo e, he hys e esis bands a e adap ed such ha he s eady-s a e swi ching equency achie es he desi ed alue o 100 kHz wi h he heo e ically p edic ed o e damped esponse in he swi ching pe iod ansien (see Sec ion 6.3). c Tk σM uM Figu e 6.9: S a -up wi h a load o 21 A o a desi ed ou pu ol age o 24 V. The second es is de o ed o highligh he swi ching pe iod acking o a s ep- ype e e ence. The swi ching pe iod e e ence a ies om 8 µs o 12 µs and ice- e sa. The ou pu ol age is egula ed o 24 V and he e is no load a he ou pu . Figu e 6.10 shows he beha iou s o he ou pu ol age ipple, ∆ c, he swi ching unc ion, he swi ching pe iod, and he swi ching pe iod e e ence. The SFC adjus s he hys e esis band alue in o de o achie e he desi ed s eady-s a e swi ching pe iod wi h he expec ed mo ion acco ding o he model de i ed in Sec ion 6.3. F om he Figu e, i can be seen how he expec ed se ling ime o a ound 2.5 ms o Tkis quali a i ely ul illed in he eal sys em, alida ing he de eloped models and assump ions aken. Besides, he ou pu ol age is no a ec ed by he swi ching pe iod e e ence a ia ion, implying ha he eal sliding mode is no being pe u bed by he ac ion o he SFC. This e ec can be also in e ed om he low ou pu ol age ipple, ∆ c, obse ed du ing he en i e es . The wa e o ms de ailed in he bo om windows co espond o he s eady-s a e dynamics a 8 µs (le window) and a 12 µs ( igh window). Such Figu es also con i m he assump ion o he piecewise linea beha iou o he swi ching unc ion. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 105 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. ∆ c Tk σM T∗ Figu e 6.10: Swi ching pe iod a ia ion om 8 µs o 12 µs wi h a desi ed ou pu ol age o 24 V and no load. 4. Sys em obus ness Finally, he obus ness o bo h con olle s a e e alua ed h ough sudden a ia ions o he load applied a he ou pu . The ollowing Figu es depic he esponses o he ou pu ol age, he swi ching unc ion and he swi ching pe iod (scaled by 0.5 V/µs) when he load changes om 21 A o 65 A (Figu e 6.11) and om 65 A o 21 A (Figu e 6.12). In bo h cases, he ou pu ol age e e ence is se o 24 V. F om hese Figu es i can be in e ed how he con e e eco e s he desi ed ou pu ol age a e a smoo h ansien and he swi ching pe iod is no a ec ed by he load changes. Mo eo e , i is con i med ha he alues o ρ± ∗and λa e almos insensi i e o load changes, since he hys e esis alue, ∆, is nea ly he same in bo h cases. 106 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. c Tk σM io Figu e 6.11: Load change om 21 A o 65 A o a egula ed ou pu ol age o 24 V. c Tk σM io Figu e 6.12: Load change om 65 A o 21 A o a egula ed ou pu ol age o 24 V. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 107 CHAPTER 6. VOLTAGE REGULATION IN A MULTIPHASE BUCK CONVERTER. 6.6 Conclusions Th ough he expe imen al esul s shown abo e, he p ope pe o mance o he SMC o- ge he wi h he SFC is con i med in a mul i inpu linea sys em. The expec ed p ope - ies o he ull sys em as ou pu ol age egula ion, in e lea ing ope a ion, s eady-s a e ixed swi ching pe iod and obus ness wi h espec o load a ia ions ha e been con i med. Mo eo e , he con inuous- ime app oach o he SFC has been success ully implemen ed by analog ci cui y, hus con i ming he expec ed dynamics acco ding o he de eloped model in Sec ion 2.3. 108 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s Chap e 7 Vol age Regula ion in a Boos Con e e . The con e e s implemen ed in he p e ious Chap e s (Chap e s 5and 6) espond o linea sys ems wi h espec o he con ol inpu . Howe e , he e a e powe con e e s ha do no i wi h his desc ip ion, as he Boos con e e . The equa ions desc ibing he Boos con e e espond o a nonlinea ones, and i is in e es ing o e alua e he pe o mance o he SFC in his ype o s uc u e. The e o e, he SFC is implemen ed o egula ing he swi ching pe iod o a Boos con e e . The Chap e is s uc u ed as ollows: i s ly he nonlinea equa ions o he con e e and he pa ame ic da a o he Boos con e e a e p esen ed. Then, he SMC will be designed o egula e i s ou pu ol age. Nex , he design o he SFC in he con inuous- ime app oach is de ailed. Finally, he implemen a ion de ails and he expe imen al esul s will be shown. 7.1 The Boos Con e e The ci cui scheme o a Boos con e e is shown in Figu e 7.1. The Boos con e e can be unde s ood like a Buck con e e whe e he inpu is used as he ou pu , and his one as he inpu . Ne e heless, he s a e space equa ions de i ed om his s uc u e become nonlinea . The equa ions desc ibing he dynamics o he Boos con e e a e shown in (7.1), (7.2). Ldil d =E− c(1 −u),(7.1) Cd c d =il(1 −u)− c R,(7.2) whe e Eis he inpu ol age, c he ou pu ol age and L,C,Ra e he induc ance, he capaci ance and he esis i e load, espec i ely. In he p e ious equa ions, he discon inu- ous con ol inpu , u, akes again he alues {0,1}.M1 and M2 wo k in a complemen a y Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 109 CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. E M1 u= 0 u= 1 L CR c + - il M2 io Figu e 7.1: Boos con e e . manne , emaining one closed while he o he one is open and ice e sa. The pa ame ic da a o he con e e buil in he labo a o y a e shown in Table 7.1. Table 7.1: Boos con e e pa ame e s Pa ame e Symbol Value Inpu ol age E12 V Ou pu ol age e e ence ∗ c48 V Ou pu capaci o C132 µF Induc ance L20 µH Nominal esis i e load R20 Ω Swi ching pe iod e e ence T∗10 µs The Boos con e e , due o i s s ep-up ol age con e sion p ope y, is ex ensi ely employed in di e en indus ial applica ions [54–57]. In gene al, he con ol o his opology (and o he simila ones) in ol es a complex ask due o i s nonlinea cha ac e is ics, which becomes a challenge om he con ol poin o iew. 7.2 Sliding mode con ol o he ou pu ol age 7.2.1 Swi ching su ace design The sliding mode con olle is designed o egula ing he ou pu ol age o he Boos con e e . Due o he non minimum phase p ope y o he Boos con e e , he di ec ou pu ol age egula ion using he na u al swi ching unc ion σ( c) := ∗ c− cis no possible, since i esul s in an uns able beha iou o he induc o cu en in sliding mo ion 110 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. [10]. Al e na i ely, an indi ec ou pu ol age egula ion is p oposed wi h he sliding su ace: σ( c, il) := κ1e +κ2Ze (τ)dτ −κ3il= 0,(7.3) whe e he ol age e o has been de ined as e = ∗ c− cand he swi ching su ace pa a- me e s κ1,2,3a e assumed posi i e. Some eade s could iden i y he p e ious su ace as he ypical con ol s uc u e wi h wo nes ed loops, he inne one egula ing he induc o cu en , and he ou e one gene a ing he p ope induc o cu en e e ence o he inne loop, in o de o keep he ou pu ol age a he desi ed le el. In ha sense, he e ms in (7.3) depending on e cons i u e he ou e con olle , deli e ing he a o emen ioned cu - en e e ence. In hese ypes o con ol s uc u es, in o de o s udy he sys em s abili y, he inne loop is assumed o be much mo e as e han he ou e one [7]. The di e ence be ween ha analysis and he one pe o med he e esides in he ac ha such hypo hesis is no employed. 7.2.2 Sliding dynamics The ime de i a i e o he swi ching unc ion is ˙σ( c, il) = −ψ1( c, il) + (1 −u)ψ2( c, il),(7.4) whe e ψ1( c, il) = Eκ3 L−κ1 R C c−κ2e , ψ2( c, il) = κ3 L c−κ1 Cil.(7.5) F om (7.4) i is clea ha he sliding mode exis s when |ψ2( c, il)|>|ψ1( c, il)|. Using he equi alen con ol me hod [7], i is possible o ind he ideal sliding dynamics. The ueq, which is ound wi h ˙σ( c, il) = 0 and σ( c, il) = 0, is de e mined by: ueq =ψ2( c, il)−ψ1( c, il) ψ2( c, il),(7.6) leading o an exis ence ange o he sliding mode as: 0<ψ1( c, il) ψ2( c, il)<1.(7.7) The sliding mode dynamics a e de i ed eplacing he equi alen con ol in (7.1)-(7.2): Ldil d =E− c ψ1( c, il) ψ2( c, il),(7.8) Cd c d =il ψ1( c, il) ψ2( c, il)− c R,(7.9) Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 111 CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. u σ Tk ∆ c Figu e 7.5: Swi ching pe iod egula ion o a s ep change om T∗= 8 µs o T∗= 12 µs o ∗ c= 48 V and R = 20 Ω. ∆ c: ed, T: blue, σ: magen a, u: g een. 3. Sys em obus ness The wo las esul s show he obus ness o he con olle s in on o load ansien s a he con e e ou pu . Again, he esponses o he ou pu ol age, c, he load cu en , io= c R, he swi ching unc ion, σ( c, il), and he swi ching pe iod (scaled by 0.5 V/µs), Tk, a e shown in Figu es 7.6 and 7.7, when he esis i e load changes om R= 20 Ω o R= 100 Ω, and om R= 100 Ω o R= 20 Ω, espec i ely. In hese Figu es, one can see how he con e e eco e s he desi ed ou pu ol age a e a e y smoo h ansien (wi h a maximum de ia ion o 2 V a ound he desi ed alue o 48 V), as expec ed om he design o Sec ion 7.2, while he swi ching equency is only sligh ly a ec ed by he load change. 118 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. io σ Tk c Figu e 7.6: Load a ia ion om R= 20 Ω o R= 100 Ω o ∗ c= 48 V. c: ed, io: blue, σ: magen a, T: g een. c io σ Tk Figu e 7.7: Load a ia ion om R= 100 Ω o R= 20 Ω o ∗ c= 48 V. c: ed, io: blue, σ: magen a, T: g een. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 119 CHAPTER 7. VOLTAGE REGULATION IN A BOOST CONVERTER. 7.6 Conclusions The expe imen al esul s con i m an o e all good pe o mance o he sys em unde he con ol o he SMC and SFC, and mos impo an ly, hey co obo a e ha he SFC is able o egula e he swi ching equency in s eady-s a e wi hou deg ading he well-known ea u es o he SMC, as he obus ness o he high ansien esponse. The dynamics obse ed in he labo a o y esul s, highligh he use ulness o he heo e ical de elopmen s desc ibed along he Chap e , o bo h SMC and SFC con olle s. Mo eo e , he de eloped model in Sec ion 2.3 is alida ed o a nonlinea sys em as he Boos con e e is. 120 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s Chap e 8 Vol age T acking in a Vol age Sou ce In e e . The applica ions o he SFC p esen ed in he p e ious chap e s belong o cases whe e he SMC is unde a egula ion con ol ask. In his Chap e , he applica ion o he swi ching equency egula ion s a egy in a acking con ol p oblem is se ou . Speci ically, a ol age sou ce in e e (VSI) is assembled o expe imen al e alua ion in he labo a o y. A sliding mode con ol will be designed in o de o gene a e a sinusoidal ol age a he con e e ou pu , hus leading o a acking con ol p oblem. In his expe imen a ion, bo h con olle s, he SMC and he SFC, a e digi ally implemen ed by a mic o-con olle . Mo eo e , wi h he pu pose o showing esul s as ealis ic as possible, he VSI de eloped in he labo a o y has been designed wi h a medium powe -handling capabili y (up o 2.2 kW). Ano he impo an cha ac e is ic o he wo k p esen ed he ea e is ha he SMC has been designed wi h he aim o ope a ing wi h linea and nonlinea loads connec ed a he VSI ou pu . Such p ope y allows us o es u he he pe o mance o he SFC unde a new wo king scena io. The Chap e is o ganized as ollows: in he i s Sec ion he VSI s uc u e, i s pa ame - ic da a and he co esponding s a e space equa ions a e in oduced. The second Sec ion ackles he sliding mode gene a ion o he AC signal (220 V RMS / 50 Hz) a he VSI ou pu , suppo ing linea and nonlinea loads. Then, he SFC is designed acco ding o he heo y de eloped a Sec ion 2.2.2. Subsequen ly, he implemen a ion o bo h con olle s in a digi al pla o m is add essed and discussed. Las ly, he main expe imen al esul s a e shown in he las Sec ion. 8.1 The ol age sou ce in e e The VSI ci cui scheme is depic ed in Figu e 8.1. This ci cui is commonly employed o gene a e a sinusoidal signal a i s ou pu and i is classi ied as DC/AC con e e . Wi h ega d o i s s uc u e, he VSI can be unde s ood as a adi ional Buck s uc u e wi h a ull b idge o swi ches. As a consequence, he VSI is able o gene a e ol ages a i s ou pu Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 121 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. be ween E o −E. The VSI dynamics a e desc ibed by he ollowing s a e space equa ions: Eu= 1 L CZ + - il M4 c u=−1 M3 M2 M1 u=−1u= 1 io Figu e 8.1: Vol age sou ce in e e s uc u e. Cd c d =il−io,(8.1) Ldil d =− c+E u, (8.2) whe e ilis he induc o cu en , cis he ou pu ol age, iois he load cu en , Lis he induc ance, Cis he capaci o and Eis he inpu ol age. The discon inuous con ol inpu u akes alues in he disc e e se {−1,1}. This s a egy co esponds o he well-known wo le el modula ion in he pulse wid h modula ion (PWM) echniques [60]. The powe swi ches a e ep esen ed by M1, M2, M3,and M4. As i is shown in Figu e 8.1,M1and M4 a e sho ci cui ed when u= 1, and emain open when u=−1, whe eas M2and M3wo k in a complemen a y way. Table 8.1 p esen s he speci ic alues o he con e e pa ame e s used in he expe imen al se up. The ol age sou ce in e e is he mos used DC/AC con e e in he indus y [61–65]. Because o i s simple s uc u e, he con e e has he capabili y o wo k wi h high ol age and manage high powe s. As an example o applica ion, VSIs a e used in he pho o ol aic plan s injec ing he powe gene a ed by he sola cells o he AC powe g id. Ano he applica ion o VSIs can be ound in he unin e up ible powe supply (UPS) sys ems, whe e is common he employmen o back- o-back s uc u es [66], being he VSI one o i s mos impo an pa s. Simila ly, he VSI con e e is he p e e able op ion o AC machine d i es. The AC mo o s a e used, among o he , in ai condi ione ’s comp esso s, e ige a o s, wa e pumps, elec ic saw, con eyo bel s, elec ic ac ion in ains and, inc easingly, in he g owing ma ke o he elec ic ehicle. 122 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. Table 8.1: Vol age Sou ce In e e pa ame e s. Pa ame e Symbol Value Inpu ol age E420 V Desi ed ou pu ol age ampli ude A220√2 V Ou pu ol age equency 50 Hz Induc o L440 µH Ou pu capaci o C100 µF Nominal ou pu powe (Linea Load) P2.2 kW Peak ou pu powe (Linea Load) Pp3.3 kW Swi ching pe iod e e ence T∗50 µs 8.2 Sliding mode acking o he ou pu ol age 8.2.1 Swi ching su ace design In his case, he con ol objec i e is o ack a ime- a ying e e ence ol age a he ou pu . The signal o be acked is: ∗ c=Asin ω . (8.3) Once he unc ionali y o he VSI has been de ined, he ollowing ask is o design a swi ching unc ion ha ul ils he desi ed pe o mance. In he adi ional SMC schemes applied o his con e e , since he ela i e deg ee o he ou pu ol age, c, wi h espec o he con ol, u, is wo, he ollowing i s o de linea swi ching su ace is ypically used [46]: σ( c,˙ c) = ψ1e +ψ2C˙e = 0,(8.4) whe e e = c− ∗ c. No ice how (8.4) con ains he i s ime de i a i e o he ou pu ol age. As i was explained in Sec ion 5.4, i is usual o ake he cu en lowing by he ou pu capaci o as he i s ime de i a i e o he ou pu ol age, a oiding he di ec di - e en ia ion o he measu ed ol age which always b ings noise p oblems (e en in digi al di e en ia ion). Such measu ed cu en is p ope ly eplaced in he swi ching unc ion w i - en in (8.4), leading o an equi alen exp ession. Howe e , his echnique usually p o ides good esul s only wi h linea loads, deg ading i s pe o mance wi h o he ypes o loads. Mo eo e , om he sliding mode con ol design, he subs i u ion o he i s ime de i a - i e by he ou pu capaci o cu en can comp omise he piecewise linea beha iou o he swi ching unc ion when he load is no pu e esis i e. As a consequence, in o de o design a swi ching unc ion less sensi i e o he ype o load applied a he ou pu , an al e na i e swi ching unc ion is p oposed. The new swi ching unc ion uses he ou pu signal gene - Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 123 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. a ed by a Cu en T ans o me (CT), measu ing he induc o cu en , il. Speci ically, he p oposed swi ching unc ion is: σ( c, xM) := −ψ1e +ψ2C˙ ∗ c−ψ2 Lx MRb xM(8.5) whe e again ∗ cis he desi ed ou pu ol age, xM he signal coming om he cu en ans o me ou pu and ψ1, ψ2>0 a e he swi ching unc ion pa ame e s. Lx,Mand Rb a e CT pa ame e s, and Cis he capaci ance o he ou pu VSI il e . As i was al eady in oduced in Sec ion 6.2, he s a e space equa ion gene a ed by he CT inclusion is: Lx dxM d =−RbxM+RbMdil d .(8.6) In o de o illus a e he lack o piecewise linea beha iou in he swi ching unc ion i ic is used as C˙ c, a simula ion o he sys em designed and expe imen ed in his Chap e is ea ly in oduced a his s ep. Figu e 8.2 depic s he simula ion esul o a nonlinea load, whe e he signal icand ˆxMa e shown, being ˆxM: ˆxM=xM Lx MRb . The e o e, exp ession (8.5) can be ew i en as: σ( c, xM) := −ψ1e +ψ2C˙ ∗ c−ψ2ˆxM. The beha iou o he CT can be unde s ood as a high pass il e , since, om (8.6), i can be de i ed ha : Il(s) = sLx+Rb sRbMXM(s). The gain o he p e ious il e in he band pass can be ound sol ing he ollowing limi : lim s→∞ sLx+Rb sRbM=Lx RbM, and hence ˆxMis p opo ional o he high equency ipple o il. The nonlinea load used in he simula ion co esponds o a diode ec i ie wi h a il e ing capaci o supplying a esis o (see Figu e 8.10). When he diodes a e closed (| c| ≥ Vdc), he load connec ed a he VSI is a capaci o in pa allel wi h a esis o (assuming ideal diodes). When he diodes s ay open (| c| ≤ Vdc), he VSI is in no load condi ion. F om he esul in Figu e 8.2, i is clea ha when he diodes a e swi ched on and he e is cu en lowing o he load, icloses he piecewise linea beha iou while ˆxMdoes no . Such esul s jus i y he selec ion o he swi ching unc ion shown in (8.5). 124 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. 0.0658 0.066 -14 -10 -6 -2 0 4 8 0.0298 0.03 -30 -20 -10 0 10 20 -400 0 400 0 0.02 0.04 0.06 0.08 -40 0 40 c 10 ·io (s) ˆxM ic ˆxM ic ˆxM ic Figu e 8.2: De ails o he signals icand ˆxMwhen he in e e is loaded by a nonlinea load. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 125 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. 8.2.2 Ideal sliding dynamics The SMC design is de i ed om he i s ime de i a i e o he swi ching unc ion: ˙σ( c, xM) = −ψ1˙e +ψ2C¨ ∗ c+ψ2 MxM−ψ2 L[E u − c].(8.7) Mo eo e , he dynamics en o ced by he sliding mo ion on σ( c, xM) = 0 can be ound om (8.5) as: xM=MRb ψ2Lx [ψ1( ∗ c− c) + ψ2C˙ ∗ c],(8.8) and, he e o e ˙σ( c, xM) = −ψ1˙e +ψ2C¨ ∗ c−β(ψ1e −ψ2C˙ ∗ c)−ψ2 L[E u − c] (8.9) whe e β=Rb Lx. Thanks o he inclusion o xMin (8.5), he ela i e deg ee be ween he swi ching unc ion and he con ol inpu is one, as i co obo a es he ac ha uappea s in ˙σ( c, xM). A anging e ms in (8.9) one ge s (no ice ha e = c− ∗ c): ˙σ( c, xM) = ∗+ e−ψ2 LEu (8.10) whe e ∗=ψ2C¨ ∗ c+βψ2C˙ ∗ c+ψ2 L ∗ c, e=ψ2 L−βψ1e −ψ1˙e . F om (8.10) i can be igu ed ou ha a sliding mo ion can be en o ced in σ( c, xM) = 0 when he e m depending on con ol domina es he es o e ms, o ψ2 LE > | ∗+ e|. F om (8.10) he equi alen con ol is easily ound: ueq = ( ∗+ e)L ψ2E.(8.11) Once he equi alen con ol and he sliding mode equa ion ha e been ound, i is ime o analyse he esul ing sliding dynamics in o de o check i he desi ed acking o ∗ cby c is achie ed. Replacing he equi alen con ol, (8.11), in he o iginal sys em (8.1), (8.2), he s a e space equa ions ha a ise a e: ψ2 dil d = ∗+ e−ψ2 L c,(8.12) Cd c d =il−io.(8.13) Le us analyse he sliding dynamics o he acking e o e . Combining equa ions 126 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. (8.12) and (8.13), he esul ing dynamics o he acking e o , e = c− ∗ c, can be ound: C¨e +ψ1 ψ2 ˙e +ψ1Rb ψ2Lx e =RbC Lx ˙ ∗ c−˙ io.(8.14) As expec ed, he sliding mode dynamics o he ou pu ol age includes he e olu ion o he load cu en . In his hesis, h ee ypes o loads a e s udied, namely: esis i e loads, eac i e loads and nonlinea loads. Each o hose cases a e analysed sepa a ely in he coming Sec ions 8.2.4,8.2.5 and 8.2.6 o u he unde s anding. 8.2.3 Con ol law The con ol law ha ensu es sliding mo ion in σ( c, xM) = 0 ega dless o he ype o load applied o he VSI, is ob ained om (8.10) as: u=−1 i σ < −∆ko (|σ|<∆k& ˙σ > 0) 1 i σ > ∆ko (|σ|<∆k& ˙σ < 0) .(8.15) 8.2.4 Sliding dynamics o pu e esis i e load The speci ic case o a pu e esis i e load is cha ac e ized by io= c R, which, using (8.14), boils down o he equa ion: C¨e +ψ1 ψ2 ˙e +ψ1Rb ψ2Lx e =RbC Lx ˙ ∗ c−˙ c R.(8.16) The dynamics in (8.16) depends on he VSI pa ame e s and he con olle gains. The selec ion p ocedu e o Land C alues is omi ed o he sake o b e i y, bu , summa izing, i ollows se ing he cu -o equency o he ou pu LC il e a leas a decade below he swi ching equency, hus educing he ou pu ol age ipple [2]. The e o e, he pa ame e s o design a e ψ1,ψ2,Lxand Rb. No ice ha he CT pa ame e s a e ea ed as swi ching su aces gains, including he senso dynamics in he design. Le us de ine he ollowing pa ame e s in o de o cla i y he u u e de elopmen s: α=ψ1 ψ2 , β =Rb Lx , γ =1 RC , Replacing hose de ini ions in he sliding mode dynamics, (8.16) esul s in: C¨e + [α+γ C] ˙e +αβe =C˙ ∗ c[β−γ].(8.17) F om (8.17) i can be easily de i ed ha i β=γa pe ec acking o he ou pu ol age is achie ed, since (8.17) has e = 0 as an asymp o ic s able equilib ium poin . The p oblem is ha he ou pu load is no ixed and can a y unde di e en si ua ions. As a consequence, he s a egy ollowed in his s udy is o adjus β o he wo s load case, which co esponds Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 127 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. wi h posi i e coe icien s, implying ha G(s) is Hu wi z [69] and he condi ions 1 and 2 o De ini ion 2a e mee . Le us check he hi d condi ion in De ini ion 2. The e alua ion o he eal pa o G(jω) is p esen ed as ollows: G(jω) = jω −ω2C+jωψ1 ψ2+ψ1Rb ψ2Lx = ω2ψ1 ψ2+jω ψ1Rb ψ2Lx−ω2C ψ1Rb ψ2Lx−ω2C2+ω2ψ2 1 ψ2 2 (8.27) being he eal pa o he complex impedance, Re G(jω) = ω2ψ1 ψ2 ψ1Rb ψ2Lx−ω2C2+ω2ψ2 1 ψ2 2 ,(8.28) always posi i e. The e o e, he condi ion 3 o he De ini ion 2is also me and G(s) is PR. As a consequence, he phase o G(s)H(s), ul ils |a gG(jω)H(jω)| ≤ π. The s abili y o he esul ing sys em can also be ound h ough he Nyquis s abili y c i e ion. Since |a gG(jω)H(jω)| ≤ π, he esul ing pa h o G(jω)H(jω) in he complex plane desc ibing he con ou −jω o jω, wi h 0 ≤ω≤ ∞, canno c oss om he second o he hi d quad an o ice e sa, and i is he e o e impossible ha he poin −1 + j0 be enci cled [6]. In Figu e 8.7 a able wi h di e en con igu a ions o loads, Z(s), connec ed o he VSI is shown. The objec i e is o demons a e ha he Nyquis diag ams o G(s)H(s) wi h hese loads do no enci cle he poin −1+j0. Such esul s a e depic ed in Figu e 8.8. Wi h he aim o assigning alues o he di e en load con igu a ions in Figu e 8.7, he loads ha e been designed o p o ide an app oxima e appa en powe be ween 0.5 and 1.5 kVA a 50 Hz in all he cases. Figu e 8.8 con i ms ha he Nyquis cu es in he complex plane do no enci cle he poin −1 + j0. Fo he de i a ion o he plo s shown in Figu e 8.8, he designed alues o ψ1,ψ2and βob ained in Sec ion 8.2.4 ha e been used. No ice, howe e , ha such alues could be edesigned in o de o achie e a op imized acking pe o mance o a speci ic impedance connec ed a he VSI. Finally, he bode diag ams o he equency esponses o he ans e unc ions, T(s), de ined in (8.25) a e shown in Figu e 8.9. F om such equency esponses, i can be co obo a ed ha in all he cases he acking e o s a e small in magni ude, being a ound o 2.5 % in he wo s case o he load Z4(s). 8.2.6 Sliding dynamics o nonlinea load The e a e di e en ypes o loads ha exhibi a nonlinea consump ion om he powe sou ce hey a e connec ed. In he powe elec onics ield, some ec i ie s (con e e s om AC o DC ol ages) p oduce nonlinea cu en s, as he uncon olled ec i ie (diode ec- i ie ), which is he nonlinea load s udied in his Sec ion. The diode ec i ie , shown in Figu e 8.10, is widely used in indus y, appliances, e c. The ec i ie p o ides a DC ol age 134 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. + - cio s RL + - cio s RL CL LL + - cio RL CL + - c ioLL RL Z1(s) = RL+ s+sCLRL s 1 + sCLRL Z2(s) = sRL+sLL( s+RL) RL+sLL Z3(s) = 1 + sCL s sCL Z4(s) = s+sLL + - c io LL RL Z5(s) = s2CLLL+sRLLL+ 1 sCL CL RL= 8 Ω s= 30 Ω CL= 0.1mF RL= 10 Ω s= 30 Ω LL= 2 mH RL= 60 Ω CL= 0.1mF RL= 50 Ω LL= 50 mH RL= 40 Ω LL= 2 mH CL= 6 mF Figu e 8.7: Se e al con igu a ions o ou pu impedance, Z(s), connec ed a he VSI ou pu wi h hei pa ame ic alues. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 135 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. Nyquis Diag am Real Axis Imagina y Axis -0.01 0 0.01 0.02 0.03 0.04 -0.02 -0.01 0 0.01 0.02 Zo1 Zo2 Zo3 Zo4 Zo5 Zo6 Figu e 8.8: Nyquis diag am o he di e en condi ions o he ou pu impedance shown in Figu e 8.7. The case o Zo6is he pu e esis i e case wi h R= 25 Ω. 0 0.02 0.04 0.06 0.08 Magni ude (abs) 100101102103104105 -90 -45 0 45 90 Phase (deg) Bode Diag am F equency (Hz) 50 T1(s) T2(s) T3(s) T4(s) T5(s) T6(s) Figu e 8.9: Bode esponses o he acking e o s o he SMC wi h he ou pu impedances, Z(s), shown on he able o Figu e 8.7. The case o T6(s) is he case o he pu e esis i e load o R= 25 Ω. 136 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. om an AC inpu ol age, which in his case will be he ou pu ol age o he VSI designed in his Chap e . CLRL + - Vdc io + - c D1D3 D2D4 s Figu e 8.10: Diode ec i ie opology. The nonlinea cha ac e is ic o his s uc u e comes om he beha iou o he diodes oge he wi h he ou pu capaci o , CL. The diodes ac as swi ches, aking wo disc e e s a es, closed o open. Acco ding o his beha iou , he ec i ie dynamics can be di ided in wo al e na ing opologies: one when he load is connec ed o he ou pu o he VSI h ough he diodes (diodes ON) and o he when he diodes emain in open ci cui and he VSI is in no load condi ion (diodes OFF). The esul ing opologies o he VSI wi h he ec i ie connec ed a i s ou pu o he wo s a es o he diodes a e shown in Figu e 8.11. When he diodes a e closed (ON), he esul ing load is linea and eac i e, and he analysis de eloped in Sec ion 8.2.5 can be applied. When he diodes emain open (OFF), he equi alen ou pu esis o is R=∞, which i s wi h he analysis o Sec ion 8.2.4. Since he gene a ed ou pu ol age, c, is a sinusoidal wa e o m, he a o emen ioned opologies occu wo imes pe cycle. Speci ically, he e exis 4 di e en ime in e als, wo o hem co esponding o a no load condi ion (io= 0 A) and wo co esponding o a eac i e load (see op plo o Figu e 8.2). F om he s abili y condi ions s a ed in Sec ions 8.2.4 and 8.2.5, i is clea ha he closed- loop sys ems o he esul ing opologies a e s able. The e o e, he nonlinea acking e o in s eady-s a e will be bound by known alues i he se ling imes o each opology, wi h espec o he du a ion o he expec ed ime in e al, a e as enough. In he case ha only one o he se ling imes is as enough o achie e he s eady-s a e e o , such acking e o a leas will each a known alue wo imes pe cycle, co esponding o a acking e o which can be also bounded. In he case when bo h opologies p esen slow se ling imes wi h espec o hei ime in e als, he s abili y o he sys em is no gua an eed. No ice ha he ound s abili y condi ions depend on se e al aspec s as he expec ed opologies, hei co esponding se ling imes and he ime in e al applica ion o each Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 137 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. CLRL + - Vdc + - c s Diodes ON Eu= 1 L C il M4 u=−1 M3 M2 M1 u=−1u= 1 io CLRL + - Vdc + - c s Diodes OFF Eu= 1 L C il M4 u=−1 M3 M2 M1 u=−1u= 1 io= 0 Figu e 8.11: Resul ing opologies wi h he VSI and a diode ec i ie connec ed as ou pu load. 138 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. opology. In ha sense, he s abili y condi ions should be checked o a ce ain sys em. Le us analyse he ec i ie used in he labo a o y. The alues o he nonlinea load a e (see Figu e 8.10): RL= 132 Ω, s= 1 Ω, CL= 6.6mF. Fo he calcula ions, he diodes a e assumed ideal. Acco ding o he equi alen sys ems in sliding mo ion ob ained in (8.18), (8.25) he expec ed acking e o s, e , and se ling imes a 95 % o he inal alue, s95% , o each s a e a e ound (i should be no iced ha he pa ame e s αand βha e been al eady designed o α=1, β=680): Diode OFF →e = 2.7% s95% = 4.1 ms, Diode ON →e = 30% s95% = 30 ms. The case o he diodes connec ed p oduces a e y slow ansien and a conside able acking e o , due o he load capaci o , CL, is eally big in alue. As a consequence, aking in o accoun ha he pe iod o he ou pu ol age will be 20 ms, i is clea ha his s a e will no each he s eady-s a e beha iou . The co esponding case o he diodes swi ched o p o ides a as e se ling ime, p oducing also a smalle ampli ude e o . This second se ling ime is o 4.1 ms, and assuming ha he signal pe iod is o 20 ms, i is easonable o igu e ou ha he esul ing ime in e al o his opology would be la ge han 4.1 ms, making possible o achie e he s eady-s a e ampli ude e o o 2.7%. This esul will be co obo a ed in he expe imen al pa . 8.3 Swi ching equency egula ion The nex s ep is o design he SFC con olle in o de o p o ide s eady-s a e ixed swi ching equency o he VSI. Again, he pa ame e s ρ±should be de i ed o he s eady-s a e sliding mo ion. Since in his case he con ol p oblem belongs o a acking scheme, he SFC s uc u e de ined in Sec ion 2.2.2 applies. Fi s ly, he exp ession yielding he e olu ion o ρ± k o he s eady-s a e sliding mo ion, ρ± k∗, a e ound om (8.10). In he p e ious Sec ions, i has been explained ha he pe ec acking pe o mance can be only achie ed when a esis i e load o a speci ic alue, Rp, is connec ed a he ou pu . Anyhow, i was also demons a ed ha al hough he load a ies, e en including elemen s as capaci o s and induc o s a he ou pu load, i is possible o adjus he con ol pa ame e s such as he acking e o becomes negligible. As a consequence, in he ollowing s udy, i is assumed ha , unde s eady-s a e sliding mo ion, c= ∗ cholds. Recalling (8.10), he i s ime de i a i e o he swi ching unc ion a s eady-s a e sliding mo ion becomes: ˙σ( ∗ c, xM) = ωβCA cos(ω )−ω2ψ2CA sin(ω ) + ψ2 L(Asin(ω )−Eu).(8.29) F om (8.29), i is immedia e o de i e he exp essions o ρ± ∗jus eplacing he wo Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 139 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. possible alues o he con ol signal u, which a e u+= 1 and u−=−1. No ice ha once he discon inuous con ol inpu is eplaced by one o i s possible s a es, he exp essions o ρ± ∗become con inuous unc ions o ime, ρ( )± ∗. The sampling o hese unc ions a any swi ching pe iod in e al, T∗, yields: ρ+ ∗k=˙σ( ∗ c, xM)u=−1−1=ωβCA cos(ωkT∗)−ω2ψ2CA sin(ωkT ∗) + ψ2 L(Asin(ωkT ∗) + E)−1 ρ− ∗k= [ ˙σ( ∗ c, xM)u=1]−1=ωβCA cos(ωkT∗)−ω2ψ2CA sin(ωkT∗) + ψ2 L(Asin(ωkT ∗)−E)−1 . Once he exp essions o ρ± k∗a e ound, he s abili y alues o γcan be ound applying ime (s) ∗ c( ) ρ( )± ρ( )+ ρ( )− γm Roo s o γM o γM o γm (2.24), (2.25) Figu e 8.12: F om op o bo om. 1- Desi ed ou pu ol age, ∗ c. 2- Dynamic e olu ion o ρ+ ∗kand ρ− ∗k. 3- Roo s o he condi ions se in Theo em 2. Theo em 2(Sec ion 2.2.2), which is ske ched in Figu e 8.12. Speci ically, in he op plo , Figu e 8.12 shows he desi ed ou pu ol age, ∗ c, and he dynamic e olu ion o ρ( )+, ρ( )− in he mid plo . Finally, he se o solu ions o he condi ion s a ed a Theo em 2 o he esul ing alues o ρ+ ∗k, ρ− ∗ka e p esen ed in he bo om plo . Wi h such signals, i is s aigh o wa d o ind he maximum and minimum alues gua an eeing s abili y o he SFC. The exac alues ha de ine he s abili y ma gin a e nume ically ound in γM= 1.76 ·107and γm= 9.98 ·106, i.e. 9.98 ·106< γ < 1.76 ·107. Howe e , h ough simula ions and expe imen al es ing, he s abili y o he sys em wi h alues below he minimum alue, γm, has been con i med. I is wo h ema king he e ha 140 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. Theo em 2gi es su icien bu no necessa y s abili y condi ions. Indeed, γ alues below he ange p o ide a much mo e eliable pe o mance in p ac ise. As a consequence, in he expe imen al e alua ion, some di e en alues o γwill be es ed, including alues wi hin and ou side o he ange. Rema k 8. In he ea ly de eloped analysis, i has been assumed ha unde s eady-s a e sliding mo ion he ou pu ol age, c, pe ec ly acks he e e ence signal ∗ c. As i was explained in he p e ious Sec ion, jus o a ce ain load condi ion such pe ec acking occu s. Howe e , he swi ching su ace pa ame e s a e designed o ensu e ha he ou pu ol age acking e o can be neglec ed in all he load condi ions, as Figu es 8.5,8.9 depic . As a consequence, he applied load does no in luence in he swi ching unc ions slopes, and he a o emen ioned s abili y condi ion applies o wha e e load applied o he VSI. 8.4 Implemen a ion de ails In his Sec ion, he me hodologies used o implemen he designed con olle s a e explained. I is impo an o ema k he e ha he implemen a ion is based on a mic o-con olle (µC). The digi aliza ion o he swi ching unc ion and o he hys e esis compa a o will dese e a special a en ion. Hence, his Sec ion will be o ganized in se e al pa s di ided as ollows: in he i s Subsec ion he e ec s o sampling he hys e esis compa a o a e p esen ed; in he second Subsec ion, a s a egy able o emula e he ope a ion o he ideal hys e esis compa a o using digi al de ices is in oduced, ollowed by he hi d Subsec ion, whe e he disc e iza ion o he swi ching unc ion is de ailed. The ou h Subsec ion deals wi h he digi al implemen a ion o he SFC in he acking case. Fi hly, a Subsec ion desc ibing he a ailable p ocedu es in o de o es ima e he swi ching unc ions slopes (and hei in e se alues, ρ± k) is p esen ed. Finally, he de ails o he assembled sys em in he labo a o y a e gi en. 8.4.1 E ec s o he hys e esis compa a o disc e iza ion As i has been explained in Sec ion 2.1, a sliding mo ion en o ced in a ixed hys e esis band compa a o p o ides a known swi ching equency. F om he implemen a ion poin o iew, he hys e esis compa a o can be accu a ely buil using analog ci cui y, as i was made in he expe imen al e alua ions o Chap e s 5,6and 7. Howe e , he implemen a ion in digi al p ocesso s deg ades i s pe o mance i he sampling e ec is no aken in o accoun . Figu e 8.13 shows he beha iou o a swi ching unc ion, σ, wo king wi h an ideal hys e esis compa a o , being Ti s ela ed swi ching pe iod. Analogously, he expec ed pe o mance o he same swi ching unc ion unde he e ec o disc e iza ion is ske ched h ough σz, and i s co esponding swi ching pe iod, Tz. No ice ha he sampling p ocess includes a leas a delay equal o he sampling ime, s, since he equi ed ac ions o be pe o med canno be execu ed up o he nex sampling pe iod. This con empla es ha he compu ing ime, c, which is he ime spen by he µC Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 141 CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. σ= 0 ∆ −∆ σ ˙σ+˙σ− TTz σz T+ ˙σ+ z˙σ− z −∆−δ T− ∆ + δ s ∆z Figu e 8.13: Hys e esis compa a o disc e iza ion e ec on he swi ching su ace and e- la ed swi ching pe iod. o e alua e a ce ain algo i hm acco ding o he sampled signals, is lowe han s. I his cis highe han s, a highe delay should be conside ed. This e ec p oduces ha he swi ching unc ion is no longe con ined wi hin he hys- e esis band in sliding mo ion, inducing s a iona y e o s in he s a e a iables. In his scena io, as i is illus a ed in Figu e 8.13, a new and la ge bounda y laye appea s, de ined as |σz|<∆ + δ. Acco ding o Figu e 8.13, in he bes case he sys em will swi ch wi h a delay o one sampling pe iod (jus one sample wi hin δ). In he wo s scena io he delay will be o 2 s. No ice ha he expec ed swi ching pe iod will no be cons an e en a s eady-s a e, due o such disc e iza ion e ec . Howe e , i is possible o quali a i ely analyse hese e o s. The maximum and minimum alues o δa e de ined in (8.30). δ+ min = s˙σ+;δ− min =− s˙σ− δ+ max = 2 s˙σ+;δ− max =−2 s˙σ−,(8.30) whe e ˙σ= ˙σzhas been assumed. Then, he maximum and he minimum ∆z alues will be: ∆zmax = 2∆ + δ+ max −δ− max ∆zmin = 2∆ + δ+ min −δ− min.(8.31) Replacing (8.30) in (8.31) he maximum and minimum alues a e: ∆zmax = 2∆ + 2 s[ ˙σ++ ˙σ−] ∆zmin = 2∆ + s[ ˙σ++ ˙σ−].(8.32) Using he p e ious exp essions, he maximum and minimum swi ching pe iods in a ixed hys e esis band could be ound applying equa ion (2.1), placing ∆zmax and ∆zmin ins ead o ∆, espec i ely. E en hough i is possible o bound he swi ching pe iods, such bounds 142 Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s CHAPTER 8. VOLTAGE TRACKING IN A VOLTAGE SOURCE INVERTER. depend on ˙σ+,˙σ+, which a e ime- a ying signals in a acking con ol case, leading o a nondesi able si ua ion. Rema k 9. F om now on, he subindex nwill iden i y he sampling ela ed wi h he µC ope a ion’s, s, keeping k o he ones ela ed o he swi ching pe iods e en s, T∗. 8.4.2 Digi al emula ion o he ideal hys e esis compa a o In o de o eco e he ideal swi ching pe iod de i ed in Sec ion 2(see exp essions (2.1), (2.9)) o an ideal hys e esis compa a o , an emula ion o his ideal beha iou is de eloped using a disc e e- ime algo i hm. The con ol law is p ope ly modi ied such ha he esponse shown in Figu e 2.1 is eco e ed in a disc e ized sys em. The p ocedu e is able o deli e he p ope con ol ac ion u( ), in such a way he swi ching unc ion, σ, changes i s slope sign jus when i hi s he hys e esis band ∆. The desi ed pe o mance is illus a ed in Figu e 8.14. Since he µC is able o pe o m ac ions only a he sampling ime ins an s, n+1, n+2, .., in o de o make he swi ching ac ion a ime ins an = 1+ n+1, a pulse wid h modula ed (PWM) con ol signal is applied. The de ini ion o he desi ed alues o u( ) in he ime in e al n o n+3 a e (see Figu e 8.14): u( ) =        −1 o n< < n+1 −1 o n+1 < < 1+ n+1 1 o 1+ n+1 < < n+2 1 o n+2 < < n+3 .(8.33) The ollowing du y cycles, compu ed acco ding o he alues p esen ed in ((8.33)), ha e o be upda ed in he pulse wid h modula o a ime ins an s ( n, n+1, n+2), as: d=   dn= 0 a = n dn+1 = 1/ sa = n+1 dn+2 = 1 a = n+2 (8.34) The du y a io dn+1 upda ed a = n+1 makes possible he commu a ion a he desi ed ime ins an = n+1 + 1. This alue, acco ding o Figu e 8.14, is de ined as: dn+1 =∆−σn+1 σn+2 −σn+1 .(8.35) I is wo h ema king ha dn+1 depends on he u u e sample σn+2. A his poin , a p edic ion o he alue o σn+2 is equi ed. Fu he mo e, an addi ional delay should be aken in o accoun , due o he compu ing ime c. This means ha he du y cycle which can be applied a = n+1, has o be calcula ed wi h he a ailable in o ma ion a = n. Re e ed o he equa ion (8.35), his implies a p edic ion o σn+1 and σn+2 a ime ins an = n. Fixed-Swi ching F equency Sliding Mode Con ol Applied To Powe Con e e s 143