Modelling and Control of Bounded Hybrid Systems in Power Electronics

Transcription

1 Acta Unverstats Sapentae lectrcal and Mechancal ngneerng, 9 (27) DOI:.55/aseme-27-8 Modellng and ontrol o Bonded Hybrd Systems n Power lectroncs Áron FHÉR, Dénes Nmród KUASI Department o lectrcal ngneerng, Faclty o echncal and Hman Scences, Sapenta Unversty, g. Mreş, e-mal: ehera@ms.sapenta.ro, tas@ms.sapenta.ro Manscrpt receved October 22, 27; revsed December 8, 27. Abstract: In ths wor, an eplct Model Predctve ontrol algorthm s devsed and compared to classcal control algorthms appled to a seres resonant D/D converter crct. In the rst part, a model o the converter as a hybrd system s created and stded. In the second part, the predctve algorthm s appled and tested on the model. Fnally, the desgned control algorthm s compared to classcal PI and sldng mode controllers. Keywords: hybrd modellng, pecewse ane model, resonant converter, model predctve control. Introdcton hese days nearly every tme/response crtcal process s controlled wth an embedded system. he hgh demands to acheve relable perormance n comple systems reqred the development o new control methods. o control a process, the propertes o the sad process mst be nown. he term Hybrd Systems s relatvely yong n System theory. Systems whch belong to ths category cannot be categorsed as lly contnos, nor lly dscrete systems. From ths denton we can conclde that hybrd systems are a m o both, combnng contnos and dscrete events. Oten these systems contan an analoge contnos-tme process, some dscrete stml, and a dscrete controller. A Hybrd system can be descrbed as Pecewse Ane (PWA) system, whch orms a partclar class o nonlnear systems, where each state and otpt map are pecewse ane, or lnear on each polyhedral partton o the state-npt polytope. Formla () shows the descrpton o a PWA system, where

2 34 Á. Fehér, N. D. Ktas R n, yr p, R m are the state, otpt and npt vectors respectvely, A, B,, D, are the system matrces wth the approprate dmensons,, g are constant vectors and P s the npt-state polyhedron or dscrete states. P g D y B A,,, () Fgre : he gre shows the PWA appromaton o a non-lnear system contanng dscontntes sch as state changes or bondares Another descrpton method s the Med ogc Dynamc (MD) System, whch s compter orented and s wdely sed or controller synthess. An MD system can be wrtten as shown n qaton (2), where R n {,} n, y R p {,} p, R m {,} mb are the new concatenated state, npt and otpt vectors, whle δ {,} rb, and z R rr are alary varables z z D D D y z B B B A (2) 2. Seres Resonant onverter model In ths secton, the modellng o the Seres Resonant D/D onverter (SR) shown n Fg. 2 wll be presented.

3 Modelng and ontrol o Bonded Hybrd Systems n Power lectroncs 35 Fgre 2: he gre shows an SR crct. he resonant tan and the otpt load acts as a voltage dvder. he SR can operate wthot load, bt n that case, the otpt voltage can t be reglated An eqvalent crct mst be devsed or the transormer, to create the converter model. he transormer eqvalent crct s dedced neglectng the magnetzng and core loss crrents. Many descrpton methods were sccesslly sed to model resonant converters, or eample, dscrete tme model 2, contnos tme model based on averagng methods 3, or wth the progressve analyss o crct waveorms 4. he models are nonlnear. Hence t s common to se the small sgnal lnearzed appromaton o the model arond the operatng pont. he problem wth lnearzed model s the nvarance to pertrbaton, npt lctaton and load changes cannot be revealed. We have chosen the hybrd modellng as a descrpton technqe o the resonant converter. he SR can be represented as shown n Fg. 3 based on the eqvalent crct o the transormer. he H brdge swtches are operated symmetrcally (S and S4 are on, whle S2 and S3 are o; S and S4 are o, whle S2 and S3 are on). Fgre 3: D-D SR crct wth prmary sde appromate representaton o the transormer

4 36 Á. Fehér, N. D. Ktas et the state space vector be t ) (, the PWA representaton s shown n qaton (3), wth varables shown n able. as dscssed n 6. )) ( ), ( ( ) ( ) ( t t R t A t (3) he descrbed system mst be dscretzed to desgn an MP algorthm. he samplng perod mst be chosen accordng to Shannon s samplng theorem. he resonant reqency o the resonant tan can be calclated as n 2. In ths stdy, the samplng perod s chosen as s s. et s consder the pper mentoned D-D SR system wth =4.7µH, =56nF, R =Ω, =mf, =48V parameters. he resonant reqency o the system s n=55.47hz, so that the samplng perod wll be s=.8µs. Fg. 4 shows the open loop operaton o the D-D converter created wth Smscape Power Systems and Hysdel/MP toolbo smlated wth a Hz reqency and 5% dty cycle sqare wave npt sgnal. able -PWA partton o the SR based on the semcondctor states A R R R I (t)=&(t)> 2 R R I (t)=&(t)<

5 Modelng and ontrol o Bonded Hybrd Systems n Power lectroncs 37 3 R I (t)=&(t)= 4 R R I (t)=&(t)< 5 R R I (t)=&(t)> 6 R I (t)=&(t)= Fgre 4: omparson o the contnos SR model created n Smscape Power Systems wth the dscretzed verson n Hysdel/MP toolbo. he dscrete model closely ollows the state trajectory o the contnos model

6 38 Á. Fehér, N. D. Ktas 3. Applcaton o the eplct MP he Model Predctve ontrol (MP) s a partclar case o constraned optmal control whch predcts the optmal control sgnal or a gven system or a gven horzon. An nnte horzon sb-optmal controller can be desgned by repeatedly solvng nte tme optmal control problems n recedng horzon ashon. he resltng controller s reerred to as Recedng Horzon ontroller (RH). An RH where the nte tme optmal control law s compted by solvng the optmzaton problem onlne s called MP 5. Solvng onlne the Mlt Parametrc Qadratc Program (MPQP) at each tme sample s a compte-ntensve operaton. he eplct verson o the MP developed n order to decrease the comptaton reqrements. he state bondares o the hybrd SR were dedced wth a step ncton as an npt sgnal. he HYSD PWA model was mported to MP (Mlt Parametrc oolbo) and was transormed to MD model, wth the bondares attached to the states, and npt as shown n Fg. 7. he onlne MP was generated by solvng the Mlt Integer MPQP. For reerence tracng, we sed lnear cost ncton, becase the qadratc one aled to translate to S-ncton. he eplct verson o the controller was generated wth the help o the MP toolbo, and the generated control algorthm was optmzed by concatenatng polyhedral parttons wth the same control sgnal. A seres o tests were made wth the desgned controller. In Fg. 5 the closed loop system s tested wth a constant load o 3Ω, whle the reerence voltage s changed rom V to 2V (he converter parameters are the same as n the last chapter). We can see, that steady state error s n every case, bt the overshoot s present. In Fg. 6 the reerence voltage s ept at a constant V whle the load s changed. We can see, that the transent overshoot s present, bt the controller tracs the reerence wth zero steady state error. Also, by ncreasng the predcton horzon, the overshoot decreases and the response tme o the controller decreases.

7 Modelng and ontrol o Bonded Hybrd Systems n Power lectroncs 39 Fgre 5: MP appled to the SR. In ths scenaro the reerence voltage s pertrbed, whle the load s constant. he otpt voltages correspond to a controller wth N = 3, 5, and predcton and control horzons respectvely Fgre 6: MP appled to the SR. In ths scenaro the reerence voltage s constant, whle the load s changed. he otpt voltages correspond to a controller wth N = 3, 5, and predcton and control horzons respectvely. Fgre 7: he gres show the reachable set o the SR n the state space. Wth an npt voltage o 48V, and a transormer rato o 7:

8 4 Á. Fehér, N. D. Ktas 4. omparson o derent control methods For comparson prposes, we desgned two controllers or the converter. he rst controller was a PI controller tned to the lnearzed model o the SR devsed wth the help o the Forer seres epanson o the nonlnear terms, eepng only the ndamental ones 7. Fg. 8 shows the behavor o the controlled system, rom where we can conclde, the PI controller can be sed or a gven operatng pont. Fgre 8: SR controlled wth PI controller. he operatng pont o the crct was 48V npt voltage, 2V otpt voltage at Ω otpt load he second appled controller was a Sldng Mode ontroller (SM) wth the rst two states as the sldng plane. In Fg. 9 the load pertrbaton, whle n Fg. the reerence pertrbaton s shown. he response tme o the closed loop system can be measred wth the tme constant o the controlled system. In case o the SM ths tme s SM = 3,2426ms.

9 Modelng and ontrol o Bonded Hybrd Systems n Power lectroncs 4 Fgre 9: he load stablty testng o the SR Fgre : he reerence tracng dynamcs o the SR 5. Reslts and conclsons In ths research we ve stded the novel modellng technqe named hybrd modellng, whch ncorporates contnos and dscrete event desgn, to create a more precse representaton o the descrbed system. Wth the hybrd modellng technqe, we sccesslly created a precse representaton o the SR crct, whch was tested and compared aganst the contnos model (Smscape/Power systems). Wth the help o the hybrd model, we created controller strctres o basc PI, SM, and MP, whch were sccesslly tested both on the hybrd systems and the contnos models. As nal note we can state that the model predctve controller s deal to trac the reerence voltage, and to create mmnty to parameter pertrbatons, bt not very adeqate or ast transent response, and control wthot overshoot. I we want to acheve the same response tme n case o the MP as n case o the SM the control horzon tme shold match the control tme o the SM, ths tme can be epressed n horzon samples ( s* SM=8) whch wold reqre an neasble generaton tme and memory.

10 42 Á. Fehér, N. D. Ktas Reerences azar, M., Heemels, W. P. M. H., Weland S., and Bemporad, A., Stablzng Model Predctve ontrol o Hybrd Systems, n I ransactons on Atomatc ontrol, vol. 5, no., Nov. 26, pp Wtls, A. F., Hernandez, A. F., and rcson, R. W., Small sgnal eqvalent crct modelng o resonant converters, n I ransactons on Power lectroncs, vol. 6, no., Jan 99, pp Sn, J., and Grotstollen, H., Averaged modelng and analyss o resonant converters, Power lectroncs Specalsts onerence, 993. PS '93 Record, 24th Annal I, Seattle, WA, 993, pp hng, H. S. H., Ionovc, A., and Zhang, J., Descrbng nctons o power electroncs crcts sng progressve analyss o crct waveorms, n I ransactons on rcts and Systems I: Fndamental heory and Applcatons, vol. 47, no. 7, Jl 2, pp Borrell, F., Bemporad, A., and Morar, M., Predctve control or lnear and hybrd systems, ambrdge Unversty Press, Ashang, H., aham, F., and Molla-Ahmadan, H., A novel hybrd modelng o D-D seres resonant converters, ION 23-39th Annal onerence o the I Indstral lectroncs Socety, Venna, 23, pp Salem, M., Jsoh, A., Idrs, N. R. N., and Alhamron, I., Modelng and smlaton o generalzed state space averagng or seres resonant converter, n 24 Astralasan Unverstes Power ngneerng onerence (AUP),, Sept 24, pp -5.