Kalman filter based Model Predictive Control of Boost DC-DC power converter

Transcription

1 Kalman flter base Moel Prete Control of Boost DCDC ower onerter Feth Ouns LGEA Laboratory, Eletral Engneerng Deartment, Larb Ben M h Unersty, 4 Oum ElBouagh, Algera. Emal: ouns_feth@yahoo.om Nourene Goléa LGEA Laboratory, Eletral Engneerng Deartment, Larb Ben M h Unersty, 4 Oum ElBouagh, Algera. Emal: nour_golea@yahoo.fr Abstrat Boost DCDC ower onerters are ery mortant ruts for moern ees, eseally battery oerate systems, ntegrate ruts, hybr eletr ehles an lghtng systems. In ths aer, the alaton of moel rete ontrol (MPC) for Boost onerter s emonstrate. hs MPC methoology reles on the use of the onerter sretetme aerage moel to ret the future alue of the loa urrent an oltage. he nremental ontrol whh mnmzes the ost funton s selete for the next samlng tme. Further, Kalman flter s use to estmate the onerter state etor an flterng sensor nose. he smulaton results show that MPC metho roues a fast ynam resonse an aurate oltage regulaton. Keywors Boost DCDC onerter, rete ontrol, MPC, ost funton, otmal ontrol, Kalman flter. I. INODUCION DCDC ower onerters are stanar omonents n a we arety of nustral alatons. In ths work we onser the Boost DCDC ower onerter ontrol roblem. Whh s use n Hybr Eletr Vehle (HEV), ortable eletron ees, battery oerate ntegrate ruts [][]. he objete of ths onerter s to onert a low oltage sule by a battery or hotoolta soures to a hgh outut oltage for rng hgh ower res. here are three ossble onuton moes for onerters: ontnuous onuton, sontnuous onuton an rtal onuton. In the resent work, ontnuous onuton moe s onsere. Sefally, n DCDC ower onerters, there s a nee to regulate the outut oltage subjet to sturbanes n the loa an/or nut oltage. Seeral ontrol methos were ale to DCDC onerters, suh as PID ontrollers [3] [4], nonlnear ontrol [5] [6], assty base ontrol [7] [8], slng moe [9] [], an rete ontrol [, ]. Uner the MPC label seeral ontrol tehnques hae been resente, eenng on the oeratng rnle an other haratersts. In general, rete ontrol use a reton moel on a fnte horzon to ret the future behaor of the ontrolle arables. hs nformaton s use by the ontroller n orer to obtan the otmal atuaton, aorng to a reefne otmzaton rteron. A man feature of MPC s that t an hanle general onstrane nonlnear systems wth multle nuts an oututs n a unfe an lear manner [, ]. hs work fouses on the use of the reenghorzon MPC to regulate the outut oltage of DCDC boost onerter. he Boost aerage moel s frst lnearze aroun the reefne oeratng state. hen, the obtane lnearze moel s sretze onserng an aequate samlng ero. Further, for the reton urose, an augmente nremental moel of the onerter s ere. Base on the reton nformaton, an otmal nremental ontrol nut s omute by mnmzng the ontrol effort an the oltage regulaton error. o reue the number of sensors use, a Kalman flter s ntroue to estmate the boost urrent from the measure outut oltage. heoretal an mlementaton asets are resente an analyze. In aton, smulaton results show fast an hgh erformane regulaton ynams. II. BOOS CONVEE DISCEEIME MODELLING E L C Fg.. Boost onerter rut. Assumng that the onerter s oeratng n CCM moe, then usng the nutor urrent an aator oltage as state arables, an the aerage funton Î [,] as nut, yels the onerter aerage moel: L = E ( ) t C = ( ) t he ontrol objete, n ower onerters, s to mantan the outut oltage at a esre leel. hus, the aboe task an be ewe as forng the onerter to onerge to ()

2 some reefne esre equlbrum state. o omute ths esre state; we set the esre alue for the outut (usually the referene alue for the outut oltage = ae wth a > ), an solng the onerter equatons (), for the steay state regme, yels the esre steaystate oeratng ont arameterzaton: = ae, a E/ = an = ( a / ) a () he lnearzaton of the aerage moel () aroun the esre equlbrum ont (), usng the frstorer aylor seres exanson, roues the lnear moel gen by: = + Bu éx éx A ê x ú êx ú A é é = = ê ú E a ê alú L, B ê E ê a ac Cú C an x =, x =, u =. Sne sretetme rete ontrol wll be onsere n ths work, t s neessary to omute the onerter sretetme moel. For a gen samlng ero, the sretzaton of (3) roues the followng lnearze sretetme moel: x( k + ) = Ax + B u (4) éx A A x =, A = e, B = A ( e I) B x. Conserng that the ontrolle outut s the nutor urrent, we get: yk () = x = Cx (5) C = é ê ë ú û. k+ ( ) PWM Control nut omutaton xk+ ( ) DCDC onerter Preton moel Loa Measurements xk ( ) Fg.. Gener DCDC ower onerter MPC agram. III. MPC algorthm esgn Fg. llustrates the MPC ontrol sheme for the DCDC boost ower onerter. It s lear that MPC s base on two key omonents: the frst s the mathematal moel use n the reton of the onerter future state; the seon (3) omonent omutes the ontrol nut that mnmzes a gen ost funton. A. Preton moel o embe ntegrators n the reton moel, we onser the nremental state sae moel gen by: D x( k + ) = AD x + BDu yk ( + ) yk ( ) = CA D x + CB Duk ( ) the nrements of the arables are, D x = x( k + ) Dx, D u = u u( k ) Further, the augmente state moel use for the reton s gen by xa ( k + ) = Axa + BDu y = Cxa (7) édxk () é A é B xa() k =, A =, B, C é yk () CA = = CB ëê ûú Note that, sne the sretetme moel (4) has no zero at z =, the augmente moel (7) s ontrollable an obserable. hs rorety s nsensable for the ontrol feebak an obserer esgn. Uon formulaton of the augmente moel (7), the next ste n the MPC esgn s to alulate the rete outut oltage wth the future ontrol sgnal as the ajustable arables. For the gen onerter atual state x( k ) an the ontrol horzon N, the future ontrol nut trajetory s efne as: D U = é u u( k ) u( k )... u( k N ) êd D + D + D + ë Base on the moel ( ABC,, ) an ontrol nut trajetory D U, the rete outut samles are gen by: Y = Fx() k +FD U (8) Y = é y( k ) y( k ) y( k 3)... y( k N ) ê ë éca é CB CA CAB CB 3 F = CA ; F = CA B CAB CB N N N N3 NN êca ú CA B CA B CA B... CA B an N s the gen rete horzon. (6)

3 Note that all rete outut samles are formulate n terms of urrent state arable nformaton x( k ) an the future ontrol trajetory D U. B. Cost funton mnmzaton he objete s to fn the best ontrol etor D U suh that an error funton between the setont an the rete outut s mnmze. he ost funton hosen to reflet ths ontrol objete s gen by J Y Y U Q U = +D D (9) Q = qi N wth q > s esgn arameter. he frst term n (9) s lnke to the objete of mnmzng errors between the rete outut an the setont sgnal; whle the seon term reflets the onseraton gen to the sze of D U when the objete funton J s mae to be as small as ossble. he mnmzaton of J leas to sole the followng equaton J DU = hen, the soluton of () yels the otmal ontrol etor: () D U =F ( F+ ) F Fx () Due to the rnle of reeng horzon ontrol, only the frst element of () s onsere for the alaton n the next ste. hs yels the aerage ontrol nut: = uk ( ) +Duk () k ( + ) = + uk ( ) () IV. MPC WIH KALMAN FILE o reue the esgn omlexty, we onser that only the nutor urrent s sense. hen, to estmate the outut oltage an reue the nosy enronment effets, a sretetme Kalman flter s ntroue. For ths en, we onser the nosy sretetme boost moel gen by x( k + ) = Ax + Bu + x yk () = Cxk () + wk () (3) where x() k = éx() k x() k ê ë ú s the state sturbane, an û wk () s the urrent sensor nose. x () k an wk () are assume to be entere whte noses, wth oaranes efne, resetely, by E éx() k x () l = xi( k l) Ewkwl [ () ()] = w( kl) For some onstants x, w an ( k l) = for k = l an ( k l) = for k ¹ l. Base on (3), the struture of the Kalman flter s gen by xk ˆ( + ) = Axk ˆ( ) + Buk ( ) + K( yk ( ) Cxk ˆ( )) (4) K s the otmal gan etor, soluton of the algebra at equaton: ( ) P = A PA A PC C PC + w C PA + x I (5) gen by ( ) K = PC C PC + w (6) V. SIMULAION ESULS he ere MPC s erfe through smulaton on the boost onerter wth the followng arameters: L = 5.9mH, C = 5mF, = 5W, E = V. he esre oeratng state s efne as: =.5E = 4V, = 9.654A an =.8. he Boost onerter oen loo resonse s gen n Fg. 3. (A) (V) t(se) Fg. 3. Boost oen loo resonse. As an be remarke, the oen loo resonse exhbts large k an osllatons before the steay state regme. For the hosen oeratng state, the lnearze Boost moel s gen by: é 4.9 é885.6 A =, B = he lnearze sretetme moel s: é é.89 A =, B =

4 an, for the reton urose, the onerter augmente moel s: é.4 é.89 A= , B = he MPC esgn arameters are: N =, N = 4 an 4 q =. he urrent an oltage sensors base MPC struture s shown n Fg. 4. From fg. 6, t s lear that the LPC realzes ratally aetable setont trakng; howeer, the sensors nose (assume of. arane) affets sensbly the ontrol erformane. o aress the nosy ase, we assume whte nose state an outut sturbanes; wth the oaranes w =. an.i. he esgne Kalman flter s gen by (4) wth: é.947 K =.47 he Kalman flter base MPC struture s shown n Fg. 6. Kalman flter Control nut omutng Preton moel Control nut omutng ˆ î Preton moel Fg. 4. Full nformaton MPC boost onerter ontrol. (A) Fg. 6. Kalman flter base boost onerter ontrol. (A) (V) t(se) Fg. 5. Full nformaton MPC boost onerter resonse (V) t(se) Fg. 7. Kalman flter base boost onerter resonse. 4

5 .5 (A) (V) t(se) Fg. 8. Kalman flter base boost urrent an oltage estmaton. estmaton, real. he smulaton result n fg. 7 reflets the beneft of ang the Kalman flter on both setont trakng an the nose effet reuton. he fg. 8 shows the real an estmate urrent an oltage when the Kalman flter s use. It s remarkable that n aton to the estmaton erformane, the nose effet s reue. [6] F. Garofalo, P. Marno, S. Sala, an F. Vasa, Control of DCDC onerters wth lnear otmal feebak an nonlnear fee forwar, IEEE rans. Power Eletron., ol. 9, no. 6, , No [7] W.C. So, C. K. se, an Y.S. Lee, Deeloment of a fuzzy log ontroller for DC/DC onerters: Desgn, omuter smulaton, an exermental ealuaton, IEEE rans. Power Eletron., ol., no.,. 4 3, Jan [8] P. Mattaell, L. ossetto, G. Sazz, an P. ent, Generalurose fuzzy ontroller for DCDC onerters, IEEE rans. Power Eletron., ol., no., , Jan [9] P. Mattaell, L. ossetto, an G. Sazz, Smallsgnal analyss of DC DC onerters wth slng moe ontrol, IEEE rans. Power Eletron., ol., no.,. 96, Jan [] J. Maha, A. Ema, an H. A. olyat, Alaton of state sae aeragng metho to slng moe ontrol of PWM DCDC onerters, n Conf. e. 3n IEEE IAS Annu. Meetng, Ot. 5 9, 997, ol., []. Kennel an A. Lner, Prete ontrol of nerter sule eletral res, n Pro. IEEE PESC, Galway, Irelan, Jun., CDOM. [] A. Lner an. Kennel, "Moel rete ontrol for eletral res," n IEEE 36th Power Eletrons Sealsts Conferene, PESC'5. 5, [3] A. Shru, H. Kharaz, A. S. Ashur, A. Fars an M. Benamar, A stuy of Moelng an Smulaton for Interleae Buk Conerter, st Power Eletron, Dre Systems & ehnologes Conferene, : 8 35,. IV. CONCLUSION hs aer resents the alaton of the MPC ontrol tehnque to the ontrol of boost onerter. Base on the reton moel an ery flexble ost funton, otmal ontrol sgnals an be generate at eah samlng ste. hs ontrol metho has roen fast an smle to the mlementaton. Smulaton results onte out that ang a Kalman flter reues the sensors ost an effet on the MPC trakng erformane. Future work wll aress the ratal mlementaton of the eeloe MPC struture. EFEENCES [] S. Kouro, P. Cortés,. Vargas, U. Ammann, an J. oríguez, "Moel Prete Control A Smle an Powerful Metho to Control Power Conerters".IEEE trans. In. eletron., ol. 56, no. 6, jun. 9. [] F. M. Oettmeer, J. Neely, S. Pekarek,. DeCarlo an K. Uthahana, "MPC of Swthng n a Boost Conerter Usng a Hybr State Moel Wth a Slng Moe Obserer" IEEE trans. In. eletron, ol. 56, no. 9, se 9. [3] A. J. Forsyth an S. V. Mollo, Moellng an ontrol of DCDC onerters, Power Eng. J., ol., no. 5,. 9 36, Ot [4] P.. Kren, Elements of Power Eletrons. Lonon, U.K. Oxfor Un. Press, 998. [5] M. K. Kazmerzuk an A. Massarn, Fee forwar ontrol of PWM boost onerter, IEEE rans. Cruts Syst. I, Funam. heory Al., ol. 44, no., , Feb