MODEL UNCERTAINTY AND ROBUST CONTROL OF PARALLELED DC/DC CONVERTERS

Transcription

1 MODEL UNCERAINY AND ROBUS CONROL OF PARALLELED DC/DC CONVERERS I. Gaoura,. Suntio, an K. Zenger Helinki Univerity of echnology, Control Engineering Laboratory, Epoo, FINLAND Univerity of Oulu, Electronic Laboratory, Oulu, FINLAND ABSRAC Switch L r L v r In thi paper a ytem of parallele DC/DC converter operating in CICM with CMC i moelle an analye. he H control eign i ue in orer to guarantee the robut tability an robut performance of the ytem in pite of ifferent uncertaintie. he µ- analyi i ue to evaluate the robutne of the ytem. Simulation reult are preente to emontrate the control eign proceure. INRODUCION Baically a elecom power upply can be compoe of three part, AC/DC rectifier, back-up battery, an a loa. An important part that affect on the output behaviour of the whole ytem i parallel-connecte DC/DC converter, which i a ubytem of AC/DC rectifier tage. hee converter require categorical current haring mechanim to enure proper operation. Briefly, the current haring cheme can be achieve by inecting a ignal proportional to the eire converter output current into the voltage loop. he increae voltage loop error ignal force the uty cycle an then the output current of the converter to increae. In practice, the control i neee to enure proper current haring an many effective control cheme have been propoe. One common approach i to employ an active control cheme to force the current in parallel converter to follow the reference current. If the reference current i an average current of converter, it i o-calle a emocratic cheme. Moreover if the reference current i the output current of one converter, it i o-calle a mater-lave cheme. he eence of an active control i to monitor the ifference between the reference current an the output current of each converter an incorporate thi information into the control of voltage-loop. he neceity of reliable control ytem that offer robut tability for the overall ytem an robut performance for it ynamic in preence of uncertaintie i recommene, a preente in Buo (), Garabanic an Petrovic (5), an ymerki (). However a ingle moule ha only been coniere. he proceure of eigning a robut controller require a moel that take the uncertaintie of the ytem into conieration. here are many form of untructure uncertainty that can be ue to repreent the ytem uncertainty, uch a aitive uncertainty, multiplicative input uncertainty, an multiplicative output uncertainty a preente in Skogeta an Potlethwaite (9). In Gaoura et al (4), the H control eign i applie to v _ in v _ in v _ inn CPC Switch CPC Switch CPC L n i L i Ln i C _ v k v k v ref L r L kv n i L _ v n v ref i C _ v k v k v ref r Ln i Cn kn C r C C r C C n r Cn v v n i _ out i ref r i _ out i ref r n i _ outn iref r p r p r pn r pn r L r p ( n ) v out R Fig. - Parallele DC/DC converter with current-moe control cacaing voltage an current controller. he CPC i a current programming controller. the ytem of parallele DC/DC converter operating in continuou conuction moe (CICM) with voltagemoe control (VMC), however, in thi paper, the current-moe control (CMC) configuration i ue intea, a hown in Fig.. he uncertaintie, component of power converter, are coniere in orer to eign a robut controller that guarantee robut tability an performance of the ytem. H optimal control i ue to eign a tabilizing controller that can achieve robut tability an robut performance. hi paper i organize a follow. he moelling proceure i preente after introuction. hen, uncertainty moel for the plant an iturbance moel are obtaine a a multiplicative input uncertainty. H optimal controller i introuce an analye in orer to achieve robut tability an robut performance. _ i g

2 y () () () G () () W () Wi () G () r () y () u p () u () G () pp W () r () Wip _ u () r () K() G () W () z( ) p r W () p _ n () Wn( ) y () W () u z () Fig. he control ytem of parallele DC/DC converter incluing all uncertaintie, which are preente a a multiplicative-input uncertainty. he moelling, analyi an control eign proceure preente i verifie by imulation reult on two 5W-buck converter connecte in parallel. MULI-LOOP OPERAION ic il () t () t i L m m c m i L In hottuvelil an Verghee (), Panov et al (6), Raagopalan et al (7), an Siri et al (8), the moelling technique of multiloop operation of parallele DC/DC converter are bae on loop-by-loop approach that cannot inclue the interaction among converter. Alo the converter cable ha ignificant impact on the performance of parallel-connecte converter an, therefore, houl be inclue in converter eign. Conier the ac mall-ignal repreentation of parallele buck converter with voltage-moe control (VMC) that i preente in Gaoura et al (4). Due to no pace to inclue all ynamic equation in thi paper, the reaer i referre to Gaoura et al (4). he tate-pace repreentation can be written a follow. xˆ = Axˆ Bu ˆ ˆVM Ew () yˆ = C xˆ DuˆVM F wˆ () where x i a vector of tate variable, i.e., the inuctor current an capacitor voltage, ŵ i a vector of iturbance, i.e., the line an loa iturbance, ŷ i an output vector, i.e., the output voltage an output current, an û VM i a vector of control comman, i.e., the uty-cycle. A mall-ignal repreentation of parallele DC/DC converter with current-moe control can be forme by coniering the inuctor current waveform, a hown in Fig. 3, where the uty-cycle contraint of converter can be written a k ( k ) Fig. 3. he waveform of the inuctor current i () t m () t = i () t i () t (3) c c L L From Fig. 3, the erivation of i L can be obtaine by olving the following governing equation: i i = m L L ( ) ( ) il il = m where m ( v ) = () () in t v t L an m = ( v () t ) L ( ) hen, il = v () in t (4) L By ubtituting equation (4) into (3), we can rewrite the uty-cycle contraint a: ( ) ic () t m () () () c t = il t v in t (5) L o erive the ac-mall ignal moel, each variable in equation (5) houl break to two component of ac an

3 ABLE I he power tage parameter Nominal Value Variation Output Voltage, V out 54V Input Voltage, V in 4V ±% Max. Output Power, P o 5W %~9% Reitive Loa, R Ω Switching Frequency, f khz Output Inuctor, L µh ±% Output Capacitor, C µf ±% ESR of Output Inuctor, r L 5mΩ 9% ESR of Output Capacitor, r C 5mΩ 9% Cable Reitance, r mω 9% Interconnection Reitance, r p mω 9% σ [e r ] Frequency (ra/ec) Fig. 4 he uncertainty boun of plant moel c, then the mall ignal of uty-cycle of converter can be written a ˆ DD () ˆ () ˆ () ˆ t = ic t i () L t v in t Vin L mc ( ) D L = F ˆ () ˆ () ˆ m i () c t i L t q vin t (6) σ [e r ] After ubtituting the matrix form of equation (6) into equation () an (), the new tate-pace repreentation of parallele DC/DC converter with current-moe control can be written a follow. xˆ = Axˆ Bu ˆ ˆCM Ew (7) yˆ = C xˆ DuˆCM F wˆ (8) where û CM i a vector of control comman, i.e., î c (t). All matrice preente in equation (7) & (8) are epene on the power tage component an obtaine from the ynamic equation of multiloop operation of parallele DC/DC converter. From the tate-pace repreentation, all tranfer function of the ytem incluing interaction can be erive a follow. y = G u G (9) p where, for implicity, the ymbol of ŷ, û CM, an ŵ have been replace by y, u, an, repectively. Again y i an output vector, u i a control vector, an i a iturbance vector. Alo G p i a plant moel an G i a iturbance moel. he cloe-loop tructure of parallele DC/DC converter incluing uncertainty moel an performance obective i hown in Fig. where the true plant an iturbance moel are repreente by ahe box. UNCERAINY MODEL he term uncertainty refer to the ifference or error between moel an real ytem, an whatever methoology i ue to preent thee error will be calle an uncertainty moel. For the tuy of robut tability an robut performance, we aume that the actual plant can be repreente by a tranfer matrix that Frequency (ra/ec) Fig. 5 he uncertainty boun of iturbance moel belong to an uncertainty moel et. A et of uncertainty moel of parallele converter ytem can be generate if we have taken into conieration the parameter variation in able I. he multiplicative input uncertainty ha been ue to repreent the moel uncertainty. he performance obective i that the -norm of tranfer function from w to z be mall for all poible uncertainty tranfer function. hee uncertaintie have been lumpe together into one block uncertainty at the input of a nominal moel, a hown in Fig.. he nominal plant G p an the uncertainty weight W ip parameterie an entire et of plant, G, which mut be uitably controlle by the robut controller K, Skogeta an Potlethwaite (9). { : i table an p p p } G Gp( Im Wip) = () he unknown tranfer function p () i ue to parameterie the ifference between the nominal moel G p an the actual behaviour of real ytem, G pp. o o thi, the weight W ip houl be choen o that the normalize perturbation atifie { ( )} max σ G G G w ω () Gp G p pp p ip Alo the weight W i houl be choen o that the normalize perturbation atifie { ( )} max σ G G G w ω () G G i

4 v () ˆ () () w () P () z () u () y () K() y () Fig. 6 he general control ytem that i ue to eign the controller an analye the ytem in preence of uncertainty. he uncertaintie weight, a hown in Fig. 4 an Fig. 5, are erive by uing equation () an (). o enure goo performance with a mall input uage, i.e. z i mall, we woul like to have N mall. he H -norm i efine a the maximum ingular value over all frequencie. Uing an upper LF can erive the cloe-loop tranfer matrix from w to z. u (, ) ( ) F = F N = N N I N N (7) For robut tability, the ytem mut remain table for all plant in the uncertainty et. hi houl atify that F() i table for all where, which i equivalent to N < for all ω (8) w w ip i =.9 =. 3 ( 4.54 )( 7.6) 4 (.55 )( 58) 3 (.6 )( 5.3) ( 36.85)( 3.7) W ip = w ip. I an W i = w i. I 3 (3) Robut performance i achieve if robut tability i atifie an the tranfer matrix from w to z i mall for all plant in the uncertainty et. (4) F < for all where an ˆ (9) DESIGN EXAMPLE ROBUS CONROL he real problem in robut control eign for multivariable ytem, uch a parallele DC/DC converter, i to yntheize a control law that maintain ytem repone an error ignal within pre-pecifie tolerance in pite of the effect of uncertainty on the ytem. he general robut control problem can be ecribe mathematically, a preente in Chiang an Safonov (3) an Skogeta an Potlethwaite (9) a follow: Given a multivariable plant P, fin a tabilizing controller K uch that the cloe-loop tranfer matrix F atifie F <, where an ˆ an F i internally table. ω he cloe-loop tranfer matrix from [ ] [ ] (5) v w to y z can be erive by uing a lower LF (linear fractional tranformation) of Fig. 6 a follow. l (, ) ( ) N = F P K = P P K I P K P N N = N N [ ] (6) where v u,,, = w= [ r n] y y y an z = [ z z] = Wp( y Wr r) Wu u. = [ ] u he configuration of two-ientical parallel-connecte buck converter i coniere to verify the moelling proceure an the applicability of the H optimal control eign. he pecification are given in able I. he performance weight w p (ω) ha been choen a a low-pa filter..67 wp ( ω ) = () where W p = w p. I 4. he noie weight w n (ω) for twoientical parallele buck converter ha been choen a a high-pa filter.. wn ( ω ) = () where W n = w n. I 4. Alo the control weight w u (ω) ha been choen a a high-pa filter.. wu ( ω ) = () where W u = w u. I. However, W r an W are the reference an iturbance caling matrice, repectively. he Matlab function hinfyn i ue to yntheie the H controller by electing a value of γ that etermine if there exit a controller K uch that l (, ) F = F G K < γ, where G = F ( P, ) (3) hi value of γ i upate on a moifie biection algorithm, calle γ-iteration. It continuou until the u

5 5.4 Log Magnitue -5. Phae (egree) Frequency (raian/ec) - σ Frequency (raian/ec) Fig. 7 he boe plot of H robut controller Frequency (ra/ec) Fig. 8 he ingular value of the cloe-loop ytem with H robut controller RS.8 RP µ(n ) an µ(n ) µ(f) NP Frequency (ra/ec) Fig. 9 he µ-evaluation for nominal performance (NP) an robut tability (RS) Frequency /ra/ec) Fig. he µ-evaluation for robut performance (RP) he output voltege (V) he line iturbance ha beenapplie to channel # at.4ec by V an to channel # at ame time by -V. - he loa iturbance ha been applie at time.7ec by 5A. he output current (A) he line iturbance ha beenapplie to channel # at.4ec by V an to channel # at ame time by -V. - he loa iturbance ha been applie at time.7ec by 5A ime (ec) Fig. he output voltage of firt moule when the line an loa iturbance are applie. magnitue of the ifference between the mallet γ value that ha pae an the larget γ value that ha faile i mall; ee Bala et al (). he achieve γ i.6 a hown in Fig. 8 of ingular value of cloeloop ytem. From Fig. 7, the H controller how to be table, however it ha an orer of 8 that mae it very ifficult to implement in reality. Moel reuction technique can be applie to reuce the 8 th -orer controller to n -orer controller without looing the robutne propertie. he balance realization of the ime (ec) Fig. he output current when the line an loa iturbance are applie. ytem that help to remove all unobervable an/or uncontrollable moe, an the Hankel norm approximation are ue to prouce a low-orer controller. he Matlab function ybal an hankmr are performe to achieve a econ-orer controller that i ue in imulation of Fig. an Fig.. he controlle ytem achieve nominal performance an robut tability, a hown in Fig. 9. Alo the robut performance i achieve a hown in Fig. In Fig., the imulation reult of output voltage of two-buck

6 converter in parallel how goo reection to line an loa iturbance. Alo the output current of two moule are hown in Fig., however both repone are bit lowly. CONCLUSION he tate-pace repreentation of parallel-connecte DC/DC converter operating in CICM with CMC ha been preente in thi paper. he uncertaintie for both plant an iturbance moel are attaine an the H controller i eigne in orer to achieve robut tability an robut performance. he imulation reult emontrate the effectivene of H controller that provie robut tability an robut performance an how goo tracking performance an iturbance reection capability in the preence of uncertaintie. REFERENCES. Bala, G. J., J. C. Doyle, K. Glover, A. Packar, an R. Smith, 993, µ-analyi an Synthei oolbox, MUSYN Inc., an he Math Work, Inc.. Buo, Simone, 999, Deign of a Robut Voltage Controller for a Buck-Boot Converter Uing µ Synthei, IEEE ran. on Control Sytem echnology, 7,, Chiang, R., an M. Safonov, 99, Robut Control oolbox For Ue with Matlab, he Mathwork, Inc., USA. 4. Gaoura, I.,. Suntio, an K. Zenger,, Dynamic Sytem Moeling an Analyi for Multiloop Operation of Parallele DC/DC Converter, Proc. of the International Conference on Power Electronic an Intelligent Motion, Nuremberg, Germany, Garabanic, D. S., an. B. Petrovic, 995, Moeling Parallel Operating PWM DC/DC Power Supplie, IEEE ran. on Inutrial Electronic, 4, 5, Panov, Y., J. Raagopalan, an F. C. Lee, 997, Analyi an Control Deign of N Parallele DC-DC Converter with Mater-Slave Current Sharing Control, Proc. of the IEEE Applie Power Electronic Conference, Raagopalan J., K. Xing, Y. Guo, F. C. Lee, an B. Manner, 996, Moeling an Dynamic Analyi of Parallele c/c Converter with Mater-Slave Current Sharing Control, Proc. of the IEEE Power Electronic Specialit Conference, Siri, K., C.Q. Lee, an. F. Wu, 99, Current Ditribution For Parallel Connecte Converter: Part I & Part II, IEEE ran. on Aeropace an Electronic Sytem, 8, 3, 89-84, Skogeta, S. an I. Potlethwaite, 998, Multivariable Feeback Control: Analyi an Deign, John Wiley & Son Lt., Wet Suex PO9 UD, Englan.. hottuvelil, V. J., an G. C. Verghee, 998, Analyi an Control Deign of Parallele DC/DC Converter with Current Sharing, IEEE ran. on Power Electronic, 3, 4, ymerki, R., 996, Wort-Cae Stability Analyi of Switching Regulator Uing the Structure Singular Value, IEEE ran. on Power Electronic,, 5,