A SYSTEMATIC APPROACH TO STATE FEEDBACK CONTROLLER DESIGN FOR DC/DC LINE-SIDE TRACTION CONVERTERS

Transcription

1 A SYSTEMATI APPROAH TO STATE FEEDBAK ONTROER DESIGN FOR D/D INE-SIDE TRATION ONVERTERS J. Böcker *,. Endrkat **, S. * Damler-Benz AG, Research and Technology, Berln, Germany * AEG Schenenfahrzege GmbH, Berln, Germany ** Abstract. The paper presents a systematc approach to state feedback controller desgn for D/D lne-sde tracton converters. The general approach s demonstrated on one partclar converter, whch can be regarded as a mlt-npt mlt-otpt system. It reslts n good closed-loop system dynamcs even when no resstances are present. Keywords. Mlt-npt mlt-otpt systems, D/D converter, lne-sde converter, tracton converter, state feedback INTRODUTION For modern tracton drves wth D lnks at D power spples D/D lne-sde converters, also known as choppers, are needed to set-p or set-down the voltage. A smple chopper s sally controlled by a cascaded strctre of PI controllers. However, applcaton of those smple choppers for locomotves operatng at hgh D voltage power spples s not yet possble de to the blockng voltage and maxmm crrent ratng of the GTOs or IGBTs avalable to date. For that reason seres or parallel connectons of chopper elements are reqred for hgh power converters at hgher spply voltages, c. f. for example Geßler and Unger-Weber []. One partclar practcal applcaton of those choppers n a tracton drve s mentoned n Weschta [2]. Some of those D/D converters are depcted n fgres to 4. Fgre : Bck/boost converter Fgre 3: Bck converter Fgre 2: Boost converter Fgre 4: Bck converter.34

2 From the control pont of vew those choppers are mltnpt mlt-otpt (MIMO) systems: For example, n fgres throgh 3 two otpt voltages are to be kept at specfed vales ndependent of load condtons, sng for gatng sgnals. At the same tme crrents n chokes arranged n parallel shold be eqal - at least nder symmetrcal load condtons - n order to ensre dentcal loads of chokes and phase modles. It s seen that deally no resstors are present so that n realty the crcts are weakly damped systems. Ths paper addresses control of those above-mentoned choppers. ookng at one partclar chopper, one mght thnk of sng some form of decoplng pls a nmber of PI controllers to acheve the control objectve. General strateges for desgnng decoplng controllers for MIMO-systems are descrbed n Tolle [3]. A specal decoplng strategy for one partclar of the choppers pctred below s gven n Tedtke [4] However, t s not obvos how sch decoplng controllers for weakly damped systems shold be generally strctred and parametrzed to gve the controlled system the reqred closed-loop dynamcs. Frthermore, t s not even obvos that sch an approach wll work for all D/D converters n fgres. throgh 4 and the many more that may exst. Hence n ths paper we propose to se state feedback controllers (see e. g. Kalath [5]) snce these controllers can be systematcally desgned for a varety of D/D converters. Frthermore, state feedback s well sted for controllng weakly damped systems. Algorthms for state feedback and related observer desgn are proven and are avalable n many commercal software packages. State feedback for D/D converters has been descrbed n the lteratre before, c. f. eng and Tam [6]. However, applcatons were lmted to sngle otpt systems sch as a k converter crct. ontrollers for tracton converters are sally mplemented on dgtal hardware. Specfc featres of dscrete-tme control of plse-wdth modlated systems sch as comptatonal delay between controller npts and otpts can be natrally ncorporated n the desgn: Usng the state-space model obtaned drng desgn, a predctor can be obtaned to compensate for that comptatonal delay. Also t s possble to nclde a state observer for a selected observable sb-system to dspense wth some measred varables (e. g. for reason of cost to measre those varables). From the state-space modellng - as a sort of by-prodct - mportant nformaton on certan system propertes, sch as controllablty of certan states as a fncton of operatng condtons and desgn parameters, can be gathered. ONTRO SYSTEM STRUTURE modlated (PWM) gatng sgnals as npts, dependng on the nmber of phase modls n the power crct. System otpts are sally two D lnk voltages and 2 or one D lnk voltage for the specal converter depcted n fgre 4. From the control pont of vew changes n npt voltage (lne voltage N ) and load crrents and 2 (or the sngle load crrent for the converter n fgre 4) are nterpreted as dstrbances of the control system. Note that notaton n fgre 5 s adapted to the specal case of the converter accordng to fgre. Fgre 5: State feedback control strctre The man task of the control system s to keep otpt voltages at fxed vales and 2 regardless of lne voltage and load crrents. To ensre zero steady-state error the system state s agmented by ntegrators for each D lnk voltage to be controlled. The state feedback controller conssts bascally of a (m n) matrx K of nmbers (m s the nmber of system npts, n s the system order) to be mltpled wth the system state vector. The steady state solton for the system npt vector s added for fast controller response. Addtonal featres of the proposed state feedback controller are, for example, easy dscrete-tme mplementaton, ncldng predcton, and possble se of observed state varables nstead of measred ones. Also, t s advantageos to synchronze controller operaton and plse-wdth modlaton. DESIGN PROEDURE As can be seen from fgres throgh 4, there s a varety of power crcts for hgh power D/D tracton converters. It s obvos that those choppers may behave rather dfferently from the control pont of vew. However, the advantage of the proposed state feedback approach to controller desgn s that t can be appled n more or less the same fashon to all those crcts. The reslt s a generc desgn procedre that can be descrbed n more detal as follows. The general control system proposed for D/D lnesde converters s as depcted n fgre 5. The D/D converter to be controlled has a nmber of plse-wdth.35

3 Strctre Obtan the basc power-electronc crct for the partclar D/D converter ncldng parameters for chokes and capactors. Operatng condtons sch as possble npt voltage, desred D lnk voltage and load condtons are also necessary. Modellng Set p the contnos-tme system model: The gatng sgnals for the GTOs or IGBTs wll be plse-wdth modlated (PWM). Assmng that the swtchng freqency s hgh as compared to the system egenfreqences, the method of state-space averagng can be appled. Hence the contnos-tme state space model can be set p from the varos system dfferental eqatons. It can be agmented by ntegrators to acheve zero steady-state error of the system otpts f necessary. Also, t trns ot to be sefl to ncorporate the lne flter nto the system model. De to ts comparatvely low egenfreqences the lne flter s the converter element most lmtng to the overall system dynamcs.the state-space model wll generally be a nonlnear one wth dty cycles of the gatng sgnals enterng mltplcatvely,. e. as elements of the system matrx. Determne the systems steady-state and lnearze the state-space model arond a general steady-state solton. Here system propertes sch as local controllablty or observablty can be nvestgated. ompte the dscrete-tme state space model for varos selected operatng ponts by dscretzaton sng the samplng freqency. Usally the samplng tme wll be an nteger mltple of the plse tme. State feedback desgn ompte a state feedback controller va pole placement or QR desgn. The latter approach s advantageos, snce the system dynamcs can be adjsted by means of only a few desgn parameters, c. f. e. g. Kwakernaak and Svan [7]. Here algorthms of commercal control engneerng software packages are readly applcable. If necessary, determne an observer (fll-order or redced-order enberger observer) for an observable sbsystem n order to redce the nmber of varables to be measred. Here also Q desgn or a stable pole placement strategy can be appled. ontrol system performance can be mproved by ncorparatng a predctor for system varables to cope wth comptatonal delays wthn the control system and wth delays cased by the PWM converter operaton. Also, snce for tracton applcaton system operatng condtons may vary n a wde range, there shold be dfferent controller desgns for those dfferent operatng ponts. Smlaton and testng Smlaton of the D/D converter, ncldng control, permts frst nsght nto closed-loop system behavor. ontroller desgn can possbly be mproved as a reslt of smlatons by modfyng, e. g., desgn parameters for QR state feedback controller desgn. After mplementaton on dgtal hardware the system shold be tested sng an analog or qas-analog smlaton of the power electronc crct. ontrol commssonng wth the real power electronc crct s the fnal step. EXAMPE In what follows the above desgn algorthm wll be appled to desgn a state feedback controller for the bck/boost converter depcted n fgre. ontrol npts of the converter are PWM gatng sgnals for each phase modl. The converter wll be operated n sch a way that all for phase modls are plsng all the tme. Ths makes sre the converter wll operate contnosly when the lne voltage s close to the desred D lnk voltage. Here semcondctors ndexed 2, 4, 5 and 7 are sed for control, gatng sgnals for ndces, 3, 6 and 8 are the respectve nverses. Interpretng, for the tme beng, 2, 4, 5 and 7 as contnos-tme sgnals n the range between and, the crct can be modelled by the followng set of dfferental eqatons: d dt N d dt N d dt d dt + () d dt + N N N N d dt + N N 2 N Steady-state solton of ths system s N N N N N + 2 (2) learly, when both D lnk voltages are to be kept at dentcal vales (say 2 ), and when for reason.36

4 of symmetry two gatng sgnals are chosen eqal n steady-state operaton ( 2 7, 4 5 ), then ( ) N (3) Hence the sm s defned by D lnk and lne voltages, whereas ndvdal vales for 2 and 4 mst be determned from load crrents and crrents n converter chokes. learly, nder realstc condtons load crrents and 2 wll never be dentcal. It s seen from eqaton (3) that the matrx consstng of the gatng sgnals has to be nonsnglar n order to ensre fnte crrents and 2 n both chokes nder those realstc load condtons. Hence the straght-forward steady state solton of the system wth s not feasable. Instead steady state gatng sgnals have to be chosen n an asymmetrcal way, for example as 2 7 ( ) γ + γ 4 5 ( γ) ( ) ( γ) 2 4 N N (4) for some γ n the range between.5 and. hosng γ close to.5 means that varos phase modls wll have near eqal load, whch s clearly desrable n terms of semcondctor losses. However, n case of asymmetrcal load crrents ths desgn wll reslt n hgh crrents throgh converter chokes. hosng γ close to nty means good system behavor n terms of crrents n chokes when loads are asymmetrcal bt wll reslt n an neven dstrbton of losses among phase modles. As a compromse, n what follows γ was chosen as γ 2/3. Next the lnearzed state space model s compted as d dt x A x + B + B z z (5) where x N N N N z N N (6) 2 2 and the system matrces are defned as follows: 2 4 A B B z N N N N 2 2 N (7) Two ntegrators for the D lnk voltages are easly added to the lnearzed state-space model to ensre zero steady state error of the otpt voltages. The bck/boost converter accordng to fgre s to be operated at a plse freqency of 25 Hz. Hence the samplng tme for dscretzaton of the lnearzed system model s chosen as T 2 ms,. e. samplng and control s done twce n one plse perod. For the resltng dscrete-tme system model a state feedback controller s desgned sng the QR approach wth stable weghtng matrces [7]. Both above steps can be easly performed by commercally avalable control engneerng software. Note that the lne flter s nclded n the model sed for controller desgn. The resltng controller was mplemented on dgtal hardware ncldng a predctor to compensate for the comptatonal delay. Fgres 6 throgh 8 show some reslts for step-lke load changes ( MW to MW per D lnk) and for varatons of desred D lnk voltages (2 V to 24 V) and lne voltage (5 V to 8 V) showng that the system can be controlled n a dynamcally acceptable way..37

5 2 A 3 V -2 A V 5 A N, 2 N demonstrated on one partclar bck/boost converter. However, the proposed control method s applcable to a varety of other D/D converters. The approach yelds good closed-loop system dynamcs even f no dampng by resstances s present n the system. - 5 A 3 V 2 A, 2 REFERENES V -2 A, 2 s.2 s Fgre 6: System response to load changes of MW to MW per D lnk [] Geßler, W., Unger-Weber, F, 99, EPE 9, vol. 4, [2] Weschta, A., 99, ZEV+DET Glas. Ann. 4, [3] Tolle, H., 983, "Mehrgrößen-Regelkressysteme", Oldenborg Verlag, München Wen 2 A 3 V -2 A V 5 A - 5 A 3 V 2 A V -2 A, 2 N N, 2, 2, 2 s.2 s Fgre 7: System response to step-lke changes of desred D lnk voltages from 2 V to 24 V [4] Tedtke, J., 994. German Patent Specfcaton DE [5] Kalath, T., 98, "near Systems", Prentce Hall, Englewood lffs [6] eng, F., Tam, P., 99, IEEE Trans. Ind. Electr., 65-7 [7] Kwakernaak, H., Svan, R., 972, "near Optmal ontrol Systems", Wley-Interscence, New York ADDRESSES OF AUTHORS 2 A 3 V -2 A V 5 A N N, 2 Dr.-Ing. Joachm Böcker, Dr.-Ing. Steven Damler-Benz AG, Research and Technology Insttte of Drve Systems and Power Electroncs lemeyerstrasse D-2277 Berln, Germany Phone: Fax A 3 V 2 A V -2 A, 2, 2 s.2 s Fgre 8: System response to step-lke change of lne voltage from 5 V to 8 V Dr.-Ing. hrstan Endrkat AEG Schenenfahrzege GmbH ASF TE/63 Nonnendammallee 5-2 D-3599 Berln, Germany Phone Fax ONUSIONS The paper proposes to se state feedback for control of D/D converters. Here specal attenton s gven to those choppers that are - from the control pont of vew - mlt-npt mlt-otpt systems. The paper presents a systematc approach to desgn of state feedback controllers for a varety of sch D/D converters. The advantage of the proposed approach s that t can be appled more or less n the same fashon to a nmber of D/D converters. From a varety of choppers gven as possble applcatons, the approach has been.38