Small-Signal z-domain Analysis of Digitally Controlled Converters

Transcription

1 004 35h Annual IEEE Power Elecrics Specialiss Cference Aachen, Germany, 004 Small-Signal z-domain Analysis of Digially Crolled Cverers David M. Van de Sype, Koen DePSfrag Gussemé, replacemens Alex P. Van den Bossche, and Jan A. Melkebeek Elecrical Energy Laboraory Deparmen of Elecrical Energy, Sysems and Auomai, Ghen niversiy S-Pieersnieuwsraa 4, B-9000 Gen, Belgium Absrac As he performance of digial signal processors has increased rapidly during he las decade, here is a growing ineres o replace he analog crollers in low power swiching cverers by more complicaed and flexible digial crol algorihms. Compared o high power cverers, he crol loop bandwidhs for cverers in he lower power range are generally much higher. Because of his, he dynamic properies of he uniformly-sampled pulse-widh modulaors used in low power applicais become an imporan resrici o he maximum achievable bandwidh of crol loops. Though frequency- and Laplace-domain models for uniformly-sampled pulse-widh modulaors are very valuable as hey improve he general percepi of he dynamic behavior of hese modulaors, he direc discree design of he digial compensaor requires a z- domain model for he combinai modulaor and cverer. For his purpose a new exac small-signal z-domain model is derived. In accordance wih he zero-order-hold equivalen commly used for regular digial crol sysems, his z-domain model gives rise o he developmen of a uniformly-sampled pulse-widhmodulaor equivalen of he cverer. This z-domain model is characerized by is capabiliy o quanify he differen dynamics of he cverer for differen modulaors, is ease of use and is abiliy o predic he values of he crol variables a he rue sampling insans of he real sysem. I. INTRODCTION For reass of price, crol circuis for low power swiching power supplies (< 3 kw) are almos always implemened using analog circuis. As he price/performance raio of digial signal processors has decreased rapidly during he las decade, he ineres for digial crol of swiching power supplies in he low power range has grown ], ]. When applying digial crol o a swiching power supply, he differen swiches in he supply are ofen crolled by a digial or uniformlysampled pulse-widh modulaor. Csequenly, he dynamics of a digially crolled swiching power supply are influenced by wo nlinear effecs: quanizai effecs and modulai effecs. As he effecs of quanizai in digial crol of swiching power supplies have been addressed before 3], 4], his sudy focuses modulai effecs. Frequencyand Laplace-domain models of he modulaors 5] provide insigh in he fundamenal dynamic behavior of uniformlysampled pulse-widh modulaors. However, he design of he compensaor mus be performed in he Laplace-domain. This Laplace ransfer funci is aferwards ranslaed ino a discree equivalen by approximai mehods such as he rapezoidal-rule, pole-zero mapping, ec. To avoid his indirec design mehod for he discree compensaor, a z-domain model for he combinai cverer-modulaor is required. Discreeime models have already been repored 6] bu heir use in crol is limied because of he presence of marix expenials and oher highly nlinear vecor funcis. Though Fig.. u u s ZOH v c A general uniformly-sampled pulse-widh modulaor approximais of his bilinear discree-ime model such as 7] provide a good solui o his problem, he usabiliy of hese models is furher resrained because hey are ly derived for an end-of--ime modulaor and because hey describe he behavior of he crol variable values a he beginning of he swiching cycle insead of a he sampling insans of he real digial crol sysem. Afer all, he sampling insans in a real digial crol sysem can be posiied anywhere in he swiching cycle. To overcome hese problems a new exac small-signal z- domain model is derived based he Laplace-domain analysis presened in 5]. In accordance wih he zero-order-hold equivalen commly used for regular digial crol sysems, his z-domain model gives rise o he developmen of a uniformlysampled pulse-widh-modulaor equivalen of he cverer. This z-domain model is characerized by is capabiliy o quanify he differen dynamics of he cverer for differen modulaors, is ease of use as his approach uses ly he modified z-ransform raher han complex funcis, and is abiliy o predic he values of he crol variables a he rue sampling insans of he real sysem. The mehod is validaed and demsraed a digially crolled buck cverer by comparing he samples prediced by he model wih he simulaed waveforms rerieved from a Simulink model. II. NIFORMLY-SAMPLED PLSE-WIDTH MODLATORS The pulse-widh modulaors embedded in modern digial signal processors (e.g. TMS30CXX of Texas Insrumens, ADSP99X of Analog Devices, DSP568XX of Moorola, ec.) operae all in a similar fashi. If we disregard quanizai effecs, a model for he uniformly-sampled pulsewidh modulaor is shown in Fig.. The inpu u(), a cinuous funci of ime, is sampled wih a frequency ω s synchrously o he pulse-widh modulai (Fig. ). For he derivai of he z-domain model ly modulaors wih a swiching frequency equal o he sampling frequency ω s (single-updae-mode) are csidered. The sampled inpu u s () is sen o a zero-order-hold circui (ZOH). Finally, he PWM waveform is generaed by comparing he oupu of he ZOH () o he value of he carrier waveform v c (), a riangular waveform. Depending he shape of he carrier waveform y /04/$ IEEE. 499

2 004 35h Annual IEEE Power Elecrics Specialiss Cference Aachen, Germany, 004 PSfrag replacemens u PSfrag replacemens u PSfrag replacemens u PSfrag replacemens u Fig.. The single-updae-mode sawooh-carrier modulaors. : end-of-ime, : begin-of--ime Fig. 3. The single-updae-mode riangular-carrier modulaors. : symmeric-ime, : symmeric--ime v c () differen ypes of uniformly-sampled pulse-widh modulaors can be obained. In commercial digial crollers wo possibiliies are commly ered: he carrier waveform is eiher a sawooh-carrier Fig. or an isosceles-riangular-carrier Fig. 3. When he carrier v c is a sawooh carrier he commly available ypes are he end-of--ime modulaor Fig. and he begin-of--ime modulaor Fig.. For modulaors wih an isosceles-riangular-carrier waveform, wo modulaors can be idenified: he symmeric--ime modulaor Fig. 3 and he symmeric--ime modulaor Fig. 3. III. Z-DOMAIN ANALYSIS If we assume ha he respse of he oupu of he cverer q() o a change in he oupu of he uniformly-sampled pulsewidh modulaor can be described by he Laplace ransfer funci P (s)e sp, he schemaic of he crol loop can be represened as in Fig. 4. The crolled oupu of he cverer q() is cvered by a measuremen wih ransfer funci H(s)e sm ino he measured oupu q m (). The laer is sampled by he analog-o-digial cverer of he microcroller; he oupu of he sampler is qm(). Comparing his measured value o is desired value qw() yields he sampled error e (). The digial compensaor wih pulse-ransfer funci G(z) derives he crol oupu c () from his error. To accoun for he durai of he crol calculais (calculais for he oupu of G(z)) in he processor, a calculai delay d is inroduced. The delayed crol signal u () is he sampled inpu of he modulaor. The laer ransforms he sampled inpu u () ino he swiching funci y(), a cinuous funci of ime. If he pulse-widh modulaor can be modelled by a pulseo-cinuous ransfer funci G PWM (s) describing in he Laplace domain he change of he modulaor oupu y() as a funci of he sampled inpu u (), he model of Fig. 4 can be simplified o ha of Fig. 5. Noe ha his pulse-o-cinuous ransfer funci for he modulaor G PWM (s) is in accordance wih he model commly employed for a regular crol sysem wih zero-order-hold (G ZOH (s)=( )/s). e s If a hypoheical sampler wih sampling period ω s is inroduced he oupu of he cverer q(), he sampled oupu q () is obained. By using he z-ransform, he sampled oupu q () can be expressed as a funci of he commanded oupu qw(): Q(z) Q w (z) = G(z)Z e s( d + p) G PWM (s)p (s) + G(z)Z e s( d+ p+ m) G PWM (s)p (s)h(s), () wih Q(z) and Q w (z) he z-ransforms of q () and qw(), respecively. Equai () clearly shows ha he problem of calculaing he sampled oupu of a cverer wih a digial crol loop can be reduced o he problem of finding he z-ransform of G o (z) = Z e st G PWM (s)r(s) () or he calculai of he z-ransform of he characerisic sysem as depiced in Fig. 6. Before he z-ransform for he characerisic sysem can be calculaed, he pulse-ocinuous ransfer funci of a uniformly-sampled pulsewidh modulaor mus be derived. The derivai of he pulse-o-cinuous ransfer funci G PWM (s) of he digial pulse-widh modulaor is based up he small-signal Laplace-domain analysis presened in 5]. This Laplace-domain analysis uses he waveforms of he general single-updae-mode modulaor (Fig. 7). This modulaor has a riangular waveshape as carrier waveform v c () deermined by he period T c and he raio α. The raio α is he durai of he falling edge of he riangle relaive o he period T c = (Fig. 7). Choosing α equal o 0, / and allows o obain he waveforms for he end-of--ime 4300

3 004 35h Annual IEEE Power Elecrics Specialiss Cference Aachen, Germany, 004 qw() Σ e () G(z) c () u () e sd ZOH () v c () PSfrag replacemens pulse-widh modulaor y() P (s)e sp q() q m() q m () H(s)e sm Fig. 4. A model for he crol loop of a digially crolled cverer wih he pulse-widh modulaor modelled by a pulse-o-cinuous ransfer funci NOH u q () H () v c () qw () e () c () u () y() q() Σ G(z) e sd G PWM (s) P (s)e sp frag replacemens q m () q m () H(s)e sm Fig. 5. replacemens A model for he crol loop of a digially crolled cverer wih he pulse-widh modulaor modelled by a pulse-o-cinuous ransfer funci w () x() x () e st G PWM (s) R(s) Fig. 6. The characerisic sysem csising of a linear sysem, a ransporai delay and he uniformly sampled pulse-widh modulaor Fig α u Y = T c n T 0 DT c T ŷ (n+) The key waveforms for a general single-updae-mode modulaor modulaor, he symmeric--ime modulaor and he beginof--ime modulaor respecively. Hence, hree ou of four differen single-updae-mode modulaors can be analyzed in a unified way. The inpu of he modulaor u() is separaed ino a seadysae par (a csan) and a small excursi o his seadysae û(), or u() = + û(). (3) The resuling oupu of he modulaor y() can also be separaed ino a seady-sae pori Y () (he respse o ) and a small excursi o he seady-sae ŷ(), or y() = Y () + ŷ(). (4) By using he Laplace-domain analysis of 5], he small-signal oupu ŷ() of he modulaor can be expressed as a funci of he sampled versi û () of he small-signal inpu û() of he modulaor (see 5]) Ŷ (s) = (αe st0 + ( α)e s( T)) Û (s), (5) wih T 0, T and α defined in Fig. 7. In accordance wih he analysis used for a zero-order-hold in a regular digial crol sysem 8], a pulse-o-cinuous Laplace ransfer funci can be derived from he equai above G PWM (s) = Ŷ (s) Û (s) = ( αe st0 + ( α)e s( T)). (6) If (6) is subsiued in (), he z-domain model for he characerisic sysem can be calculaed. A. The PWM-equivalen for he end-of--ime modulaor As an example he z-domain model of he characerisic sysem is derived for an end-of--ime modulaor. For his modulaor he following applies: α = 0 and T = ( D), wih D (= ) he average duy-raio. Wih hese parameer values, subsiui of (6) in () yields G o (z) = Z e st R(s)e sd = Z e s(ζ+d) R(s), (7) 430

4 004 35h Annual IEEE Power Elecrics Specialiss Cference Aachen, Germany, 004 TABLE I THE PWM-EQIVALENT OF THE CHARACTERISTIC SYSTEM: THE END-OF-ON-TIME MODLATOR, THE BEGIN-OF-ON-TIME MODLATOR case cdii G o (z) 0<ζ <( D) R(z, D ζ) ( D)<ζ < z R(z, D ζ) case cdii G o (z) 0<ζ <D R(z, D ζ) D <ζ < z R(z, +D ζ) TABLE II THE PWM-EQIVALENT OF THE CHARACTERISTIC SYSTEM: THE SYMMETRIC-ON-TIME MODLATOR, THE SYMMETRIC-OFF-TIME MODLATOR case cdii G o (z) 0<ζ < ( D) R ( z, ( + D) ζ) + R ( z, ( D) ζ)] ( D)<ζ < (+D) R ( z, ( + D) ζ) + z R ( z, + ( D) ζ)] 3 (+D)<ζ < z R ( z, + ( + D) ζ) + R ( z, + ( D) ζ)] case cdii G o (z) 0<ζ < D R ( z, ( D) ζ) + R ( z, D ζ)] D <ζ < ( D) R ( z, ( D) ζ) + z R ( z, + D ζ)] 3 ( D)<ζ < z R ( z, + ( D) ζ) + R ( z, + D ζ)] wih ζ =T. As in mos cases he oal delay T is smaller han he sampling period, ζ is comprised beween 0 and and wo cases can be disinguished G o (z) = Z e s(ζ+d) R(s) for ζ +D <. G o (z) = z Z e s(ζ+d ) R(s) for ζ +D > (8) If he modified z-ransform of R(s) is defined as R(z, m) = Z m R(s) = Z e s( m) R(s), (9) equai (8) can be ransformed ino G o (z) = R(z, D ζ) for ζ +D <. (0) G o (z) = z R(z, D ζ) for ζ +D > This model G o (z) is he uniformly-sampled pulse-widhmodulaor equivalen or PWM-equivalen for an end-of-ime modulaor of he ransfer funci R(s)e st. A summary of he z-domain models of he characerisic sysem for an end-of--ime modulaor wih heir correspding cdiis is shown in Table I. The PWM-equivalen for he begin-of--ime modulaor, he oher sawooh-carrier modulaor, can be derived in a similar way. Though he calculai is omied, he resuls are abulaed in Table I. Noe ha he z-domain models for he begin-of--ime modulaor can be deduced from he z-domain models for he end-of--ime modulaor by subsiuing he average duy raio D in he laer wih he complemen of he average duy-raio D, and vice-versa (compare Tables I and I). B. The PWM-equivalen for he symmeric--ime modulaor As anoher example he PWM-equivalen of he characerisic sysem for a riangular-carrier modulaor, he symmeric--ime modulaor, is derived. The waveforms of he symmeric--ime modulaor can be obained by choosing α=/ in Fig. 7. For his modulaor he parameers T 0 and T are defined as T 0 =T = ( D). By using T =ζ he PWM-equivalen of he characerisic sysem can be expressed as G o (z) = T ( s Z D s(ζ+ e ) ) +D s(ζ+ + e ) R(s). () Depending wheher he differen delays in () are larger han he sampling period or no, and aking ino accoun ha ζ is almos always smaller han, hree differen cases can be disinghuised. In he firs case, defined by 0<ζ < ( D), equai () can be rewrien by using he definii for he 430

5 eplacemens h Annual IEEE Power Elecrics Specialiss Cference Aachen, Germany, 004 PSfrag replacemens S V in L S v s R v o Fig. 9. Block diagram for he digially crolled buck cverer Fig. 8. modified z-ransform of (9) G o (z) = A buck cverer wih a firs order filer ( R z, ( + D) ζ) + R ( z, ( D) ζ)]. () If he following applies ( D)<ζ < (+D), he secd ransporai delay in () becomes larger han he sampling period. Csequenly, he PWM-equivalen of he characerisic sysem becomes G o (z) = ( R z, ( + D) ζ) + z R ( z, + ( D) ζ)]. (3) In he las case were ζ is comprised beween (+D) and, boh delays in () are larger han he sampling period. Hence, he PWM-equivalen of he characerisic sysem can be expressed as G o (z) = z R ( z, + ( + D) ζ) + R ( z, + ( D) ζ)]. (4) The z-domain models of he characerisic sysem for he symmeric--ime modulaor () (4) wih heir correspding cdiis are recapiulaed in Table II. Similar calculais are performed o derive he z-domain models of he characerisic sysem for a symmeric--ime modulaor. Though he derivai of hese models is no explicily repeaed, he resuls are summarized in Table II. Similar o sawooh-carrier modulaors, he PWM-equivalens for he symmeric--ime modulaor can be obained by replacing D wih is complemen D in he PWM-equivalens for he symmeric--ime modulaor, and vice-versa (compare Tables II and II). IV. MODEL VALIDATION To show he validiy of he approach, he discree PWMequivalen is deduced for he buck cverer of Fig. 8 wih a firs order oupu filer. If he inpu volage V in of he cverer is a csan, he Laplace ransfer funci of is oupu can be wrien as (see also Figs. 4 and 5) P (s) = V in + sl/r = V in + s. (5) nder he assumpi ha he ransfer funci of he measuremen is given by H(s)e sm =, (6) he pulse-ransfer funci of he closed loop sysem can be wrien as Q(z) Q w (z) = V o(z) V o,w (z) = G(z)Z e s( d+ p) G PWM (s)p (s) + G(z)Z e s( d+ p) G (7) PWM (s)p (s). To calculae his ransfer funci, he delay d + p and he ype of modulaor used mus be known. Assume ha during normal operai he average duy raio D of he cverer is beween 0.5 and, and ha he following cdii applies for he delay d + p = ζ 0.35, (8) han he ype of modulaor and is correspding PWMequivalen can be chosen. If he differen cases for he z-domain models in Tables I and II are compared, i is clear ha case guaranees he fases dynamics of he sysem. Afer all, in he oher cases an exra pole appears in he origin, indicaing an exra delay of a sampling period. Hence, he fases respse of he closed loop sysem is achievable wih a modulaor for which he case cdii can be me. Keeping in mind he required duy-raio range and cdii (8), he beginof--ime modulaor (Table I, case ) is he bes choice. If ly riangular-carrier modulaors are csidered (Table II), he case cdii can never be me. Hence, for hese ype of modulaors he bes solui is o look for a modulaor for which he case cdii is fulfilled. Csequenly, he fases usable riangular-carrier modulaor is he symmeric-ime modulaor (Table II, case ). For boh he beginof--ime modulaor and he symmeric--ime modulaor he discree PWM-equivalen of he buck cverer wih a firs order filer is derived and simulaed in his seci. A. The begin-of--ime modulaor To calculae he PWM-equivalen of he buck cverer of Fig. 8, an expressi for he modified pulse ransfer funci of he firs order sysem (5) is required: P (z, m) = Z P (s)e s( m) m = V in e z e. (9) Hence, he discree PWM-equivalen of he buck cverer wih a firs order filer and wih a begin-of--ime modulaor becomes (Table I, case ) (D ζ) G o (z) = V in e z e. (0) 4303

6 004 35h Annual IEEE Power Elecrics Specialiss Cference Aachen, Germany, 004 Im(z) Im(z) eplacemens Re(z) PSfrag replacemens z =a Re(z) Fig. 0. The roo locus for a buck cverer wih a firs order filer, a begin-of--ime modulaor and a discree croller Fig.. The roo locus for a buck cverer wih a firs order filer, a symmeric--ime modulaor and a discree croller V ] g replacemens V ] PSfrag replacemens v o (), sim. vo,w () vo (), z-mod. v o (), sim. vo,w () vo (), z-mod. ms ] ms ] Fig.. The closed-loop respse of he buck cverer wih a firs order filer and a begin-of--ime modulaor o a sep in he commanded oupu volage (vo,w ()): he simulaed model (vo(), sim.) and he z-domain model (vo(), z-mod.) Fig. 3. The closed-loop respse of he buck cverer wih a firs order filer and a symmeric--ime modulaor o a sep in he commanded oupu volage (vo,w ()): he simulaed model (vo(), sim.) and he z-domain model (vo(), z-mod.) Wih his pulse ransfer funci and wih (6) he crol circui of Fig. 5 can be simplified o he block diagram represened in Fig. 9. If he seady-sae error of he crol loop of Fig. 9 mus be zero, he croller G(z) requires a pole a z =. Furhermore, a zero of he croller can be used o compensae for he pole of he process (0) a z =e. The resuling croller is z e G(z) = K z. () The roo locus of he closed loop sysem for a variable gain K is depiced in Fig. 0. Apar from he hidden mode a z =e here remains ly e closed loop pole ha can be placed in he origin by choosing he gain of he croller as follows K = V in e(do ζ), () wih D o he design value for he average duy raio. Csequenly he closed-loop sysem will behave as a delay of e sampling period or, he respse is a dead-bea respse. If he average duy-raio D differs from is design value D o, he loop gain will increase or decrease (0). As a resul he posii of he closed-loop pole in he origin will aler depending he average value of he duy raio. Neverheless, wih a duyraio range of 0.5 up o and wih a design value for he average duy-raio of D o =0.75 his effec ly resuls in a sligh change in closed-loop respse. To verify he heoreical resuls he closed loop sep respse of he buck cverer is simulaed wih Simulink. The simulaed waveforms are compared o he resuls obained wih he z-domain model. For he simulai he following parameers were used V in = 400 V, = 0 µs, L = mh, R = 3 Ω, (3) ζ = 0.375, D o = The resul is depiced in Fig. (D =D o ). A comparis beween he waveforms obained wih he Simulink model (he solid lines) and he sep respse of he z-domain model (crosses) shows ha he z-domain model accuraely predics he closed-loop behavior of he digially crolled buck 4304

7 004 35h Annual IEEE Power Elecrics Specialiss Cference Aachen, Germany, 004 cverer. Moreover, he sep respse is clearly a dead-bea respse. B. The symmeric--ime modulaor If a riangular-carrier modulaor is o be used, he bes choice is he symmeric--ime modulaor. The discree PWM-equivalen for he buck cverer wih a firs order filer and a symmeric--ime modulaor can be calculaed wih (9) and Table II, case (keeping in mind cdii (8)): G o = V in e ( (+D) ζ) z + e ( z ( D) z e ). (4) This pulse ransfer funci does no ly cain he same pole as he sysem of (0), bu i also cains an exra pole in he origin and an exra zero. A dead-bea croller for his sysem is of he following form ( ) z z e G(z) = K (z )(z a), (5) wih ( Do) a = e. (6) However, due o imporan changes in he posii of he ( D) sysem zero a z = e caused by he various values he average value of he duy raio D may adop (4), he closed-loop behavior may change drasically. To avoid his, i is beer o choose a somewha slower croller wih a seling ime of sampling periods. To achieve his he closed-loop sysem should have a double pole in he origin, or he roo locus should have a breakpoin in he origin. A roo locus has a breakpoin in he origin if (for D =D o ) d dz (G(z)G o(z)) = 0. (7) z=0 This equai yields a value for a a = e ( Do) + e ( Do). (8) The roo locus of he closed loop poles wih his value for a is depiced in Fig.. Besides he hidden mode a z =e, he wo closed-loop poles of he sysem coincide in he origin for ( + a) K = e ( (+Do) ζ). (9) V in The simulai is performed for ζ =0.5, while he oher parameers are chosen according o (3). The closed-loop sep respse for he buck cverer is shown in Fig. 3 (for D =D o ). The comparis beween he simulai resuls (solid line) and he resuls prediced by he z-domain model reveals he good agreemen. As indicaed above he seling ime for a sep respse is equal o sampling periods. V. CONCLSION As he performance of digial signal processors has increased rapidly during he las decade, here is a growing ineres o replace he analog crollers in low power swiching cverers by more complicaed and flexible digial crol algorihms. Compared o high power cverers, he crol loop bandwidhs for cverers in he lower power range are generally much higher. Because of his, he dynamic properies of he uniformly-sampled pulse-widh modulaors used in low power applicais become an imporan resrici o he maximum achievable bandwidh of crol loops. Though frequency- and Laplace-domain models for uniformly-sampled pulse-widh modulaors are very valuable as hey improve he general percepi of he dynamic behavior of hese modulaors, he direc discree design of he digial compensaor requires a z-domain model for he combinai modulaor and cverer. For his purpose a new exac small-signal z- domain model is derived. In accordance wih he zero-orderhold equivalen commly used for regular digial crol sysems, his z-domain model gives rise o he developmen of a uniformly-sampled pulse-widh-modulaor equivalen of he cverer. This z-domain model is characerized by is capabiliy o quanify he differen dynamics of he cverer for differen modulaors, is ease of use and is abiliy o predic he values of he crol variables a he rue sampling insans of he real sysem. The obained z-domain models are compared wih he resuls rerieved from simulai models of a buck cverer wih a firs order filer. REFERENCES ], Power supply Digial Crol -Real or Virual, RAP sessi, 8h Ann. Appl. Power Elecr. Cf. and Exp. (APEC003), Miami Beach, SA, Feb. 9-3, 003. ], Special issue digial crol in power elecrics, IEEE Trans. Power Elecr., Vol. 8, No., Par II of Two Pars, Jan ] D.A. Gran, M. Sevens, and J.A. Houldsworh, The effec of word lengh he harmic cen of microprocessor-based PWM waveform generaors, IEEE Trans. Ind. Applic., Vol., No., Jan./Feb. 985, pp ] A.V. Peerchev, S.R. Sanders, and J.A. Houldsworh, Quanizai resolui and limi cycling in digially crolled PWM cverers, Proceedings IEEE 3nd Annual Power Elecr. Spec. Cf., Vancouver, Canada, 00, cd-rom. 5] D.M. Van de Sype, K. De Gussemé, A.P. Van den Bossche, J.A. Melkebeek, Small-signal Laplace-domain analysis of uniformly-sampled pulsewidh modulaors, Proc. of he 35h Ann. Power Elecr. Spec. Cf., Aachen, Germany, 004, cd-rom. 6] G.C. Verghese, M.E. Elbuluk, and J.G. Kassakian, A general approach o sampled-daa modeling for power elecric circuis, IEEE Trans. Power Elecr., Vol., Apr. 986, pp ] V. Rajasekaran, J. Sun, and B.S. Heck, Bilinear discree-ime modeling for enhanced sabiliy predici and digial crol design, Trans. Power Elecr., Vol. 8, No., Jan. 003, pp ] G.F. Franklin, J.D. Powell, and M. Workman, Digial crol of dynamic sysems, Addis Wesley Lgman, California, SA, 3rd edii,