MINIMUM ENERGY CONTROL OF FRACTIONAL POSITIVE ELECTRICAL CIRCUITS. Tadeusz Kaczorek

Transcription

1 MINIMUM ENEGY CONO OF FACIONA POSIIVE EECICA CICUIS ausz Kaczor alyso Uvrsy o chology Faculy o Elcrcal Egrg jsa 45D 5-5 alyso -al: aczor@sppwupl ASAC Mu rgy corol probl or h racoal posv lcrcal crcus s orula a solv Suc coos or h sc o soluo o h probl ar sablsh A procur or solvg o h probl s propos a llusra by a apl o racoal posv lcrcal crcu I rs racoal posv lcrcal crcus u rgy corol bou pus procur INODUCION A yacal sys s call posv s rajcory sarg ro ay ogav al sa ras orvr h posv orha or all ogav pus A ovrvw o sa o h ar posv hory s gv h oographs [ 7 Vary o ols havg posv bhavor ca b ou grg coocs socal sccs bology a c c Mahacal uaals o h racoal calculus ar gv h oographs [6-8 h posv racoal lar syss hav b vsga [6 9 Sably o racoal lar couous- syss has b vsga h paprs [ h oo o praccal sably o posv racoal lar syss has b rouc [ So rc rsg rsuls racoal syss hory a s applcaos ca b ou [ 9- h u rgy corol probl or saar lar syss has b orula a solv by J Klaa [- 4 a or D lar syss wh varabl cocs [ h corollably a u rgy corol probl o racoal scr- lar syss has b vsga by Klaa [5 h u rgy corol o racoal posv couous- lar syss has b arss [4 8 a or scrpor posv scr- lar syss [ I hs papr h u rgy corol probl or posv lcrcal crcus wll b orula a solv h papr s orgaz as ollows I sco h basc os a hors o h racoal posv lcrcal crcus ar rcall a h cssary a suc coos or h rachably o h lcrcal crcus ar gv h a rsul o h papr s gv sco whr u rgy corol probl s orula suc coos or s soluo ar sablsh a a procur s propos Illusrag apl o racoal posv lcrcal crcus s gv sco 4 Coclug rars ar gv sco 5 h ollowg oao wll b us: - h s o ral ubrs - h s o ral arcs - h s o arcs wh ogav rs a M - h s o arcs wh ogav o-agoal rs y ar PEIMINAIES Mzlr arcs ral I - h h ollowg Capuo o o h racoal rvav wll b us [ D < N {} whr s h orr o racoal rvav a s h gaa uco a Cosr a racoal lcrcal crcus copos o rssors cols capacors a volag curr sourcs hs wor was suppor ur wor S/E//

2 Usg h Krchho s laws w ay scrb h ras sas h lcrcal crcus by sa quaos [ D A u < whr u ar h sa a pu vcors a A hor [ h soluo o quao s gv by whr u A E A 4 A [ a E A s h Mag-lr ar uco [ Do [ h racoal sys s call h rally posv racoal sys a oly or or ay al coos a all pus u hor [ h couous- racoal sys s rally posv a oly h ar A s a Mzlr ar a A M 6 Do h sa 5 o h racoal sys s call rachabl hr s a pu u [ whch srs h sa o sys ro zro al sa o h sa A ral squar ar s call ooal ach row a ach colu coas oly o posv ry a h rag rs ar zro hor h posv racoal sys s rachabl [ a oly h ar A M s agoal a h ar s ooal Proo Succy I s wll-ow [7 ha s agoal h s ooal h I hs cas h ar s also ooal a A M s also agoal a s also ooal 7 h pu u 8 srs h sa o h sys ro o sc usg or a 5 w oba u h proo o cssy s gv [4 POEM FOMUAION AND IS SOUION Cosr h racoal posv lcrcal crcu wh A M a ooal I h sys s rachabl h usually hr s ay r pus [ u 9 ha sr h sa o h sys ro o Aog hs pus w ar loog or u pu ha zs h prorac [ I u u Qu whr Q s a syrc posv ar a Q h u rgy corol probl or h racoal posv lcrcal crcu ca b sa as ollows Gv h arcs A M h prorac ar pu u a Q o a > a or ha srs h sa vcor o [ h sys ro o a zs h prorac o solv h probl w h ar Q whr s by 5 Fro a hor ollows ha h ar s ooal a oly h racoal posv lcrcal crcu s rachabl [

3 I hs cas w ay h pu ˆ Q u or No ha h pu sass h coo [ hor 4 Q u a [ u or or b a pu ha [ srs h sa o h racoal posv lcrcal crcu ro o sa o h sys ro o h h pu also srs h a zs h prorac I uˆ I u h al valu o h prorac s qual o I uˆ 4 Proo I h coos ar h h pu s wll a ˆ or shall show u [ ha h pu srs h sa o h sys ro o yls Subsuo o o or a uˆ Q 5a sc hols y assupo h pus u a u ˆ srs h sa o h sys ro o [ a u uˆ 5b [ u uˆ 5c y rasposo o 5c a posulplcao by w oba [ u uˆ 6 Subsuo o o 6 yls [ u uˆ [ u uˆ Quˆ Usg 7 s asy o vry ha u Qu uˆ Quˆ [ u uˆ Q[ u uˆ 7 8 Fro 8 ollows ha I uˆ < I u sc h sco r h rgh-ha s o h qualy s ogav o h al valu o h prorac w subsu o a w oba I uˆ uˆ Qu uˆ Q 9 sc hols Fro h abov cosraos w hav h ollowg procur or copuao o h opal pus ha sr h sa o h sys ro o a z h prorac Procur Sp Kowg A M a usg 5 copu Sp Usg copu h ar or gv A Q a so Sp Usg copu h sr u ˆ or gv Sp 4 Usg 4 copu h aal valu o h prorac 4 EXAMPE Cosr h racoal lcrcal crcu show Fgur wh gv rssacs ucacs a sourc volags

4 Fg Elcrcal crcu Usg h Krchho s laws w ca wr h quaos a a whr < < whch ca b wr h or A a whr A b h racoal lcrcal crcu s posv sc h ar A s Mzlr ar a h ar has ogav rs No ha h saar par b s rachabl sc bu s o rachabl as a posv par shall show ha h racoal posv lcrcal crcu s rachabl I hs cas A a usg 5 w oba [ [ A h ar a h ar by b ar ooal a by hor h racoal posv lcrcal crcu s rachabl a ay valus o h u rgy corol probl o h racoal posv rachabl lcrcal crcu ca b sa as ollows: Copu h pu ˆ u ha srs h sa o h lcrc al crcu ro zro sa o [ os h raspos a zs h prorac wh Q 4 Usg h Procur w oba h ollowg: Sp h ar has h or Sp Usg b a 4 w g Q Q [ 5 Sp Usg 4 a 5 w oba [ [ ˆ Q u 6 Sp 4 Fro 4 a 5 w hav h al valu o h prorac u I ˆ 7 h cosraos ca b o syss cossg o subsyss wh r racoal orrs s gv [

5 6 CONCUDING EMAKS Ncssary a suc coos or h rachably o h racoal posv couous- lar syss hav b sablsh hor h u rgy corol probl or h racoal posv lcrcal crcus has b orula a solv Suc coos or h sc o a soluo o h probl has b gv hor 4 a a procur or copuao o opal pu a h al valu o prorac has b propos h cvss o h procur has b osra o h apl o racoal posv lcrcal crcu h prs ho ca b o posv scr- lar syss a o racoal posv scr- lar syss wh bou pus EFEENCES [ M usłowcz Sably o lar couous racoal orr syss wh lays o h rar yp ull Pol Aca Sc ch vol 56 o [ A Dzlńs D Srocu a G Sarwas Ulracapacor parars cao bas o racoal orr ol Proc ECC 9 uaps 9 [ Fara a S al Posv ar Syss; hory a Applcaos J ly Nw Yor [4 Kaczor Polyoal approach o racoal scrpor lcrcal crcus Sub o 4 [5 Kaczor Asypoc sably o posv racoal D lar syss ull Pol Aca Sc ch vol 57 o [6 Kaczor Fracoal posv couous- syss a hr achably I J Appl Mah Copu Sc vol 8 o 8-8 [7 Kaczor Posv D a D syss Sprgr Vrlag oo [8 Kaczor Corollably a obsrvably o lar lcrcal crcus Elcrcal vw vol 87 o 9a [9 Kaczor Posvy a rachably o racoal lcrcal crcus Aca Mchaca Auoaca vol 5 o 4-5 [ Kaczor Posv lar syss cossg o subsyss wh r racoal orrs IEEE ras Crcus a Syss vol 58 o 6 - [ Kaczor Praccal sably o posv racoal scr- lar syss ull Pol Aca Sc ch vol 56 o [ Kaczor achably a corollably o zro ss or saar a posv racoal scr- syss Joural Europé s Sysès Auoasés JESA vol 4 o [ Kaczor ar Corol Syss sarch Sus Prss a Jly Nw Yor 99 [4 Kaczor Mu rgy corol o racoal posv couous- lar syss Proc Co MMA [5 Kaczor Chcg o h posvy o scrpor lar syss by h us o h shulalgorh Archv o Corol Sccs vol o [6 Kaczor Mu rgy corol o scrpor posv scr- lar syss Copl vol o [7 Kaczor Mu rgy corol o racoal posv scr- lar syss wh bou pus I J Appl Mah Copu Sc 4 Prss [8 Kaczor Mu rgy corol o posv couous- lar syss wh bou pus I J Appl Mah Copu Sc 4 Prss [9 Kaczor A so o Klaa s ho o u rgy corol o racoal posv scr- lar syss wh bou pus ull Pol Aca Sc ch 4 Prss [ Kaczor Slc Probls o Fracoal Syss hory Sprgr-Vrlag rl [ Kaczor J Klaa Mu rgy corol o D lar syss wh varabl cocs I J o Corol vol 44 o [ J Klaa Corollably o Dyacal Syss Kluwr Acac Prss Dorrch 99 [ J Klaa Mu rgy corol o D syss Hlbr spacs Sys Sccs vol 9 o [4 J Klaa lav corollably a u rgy corol o lar syss wh srbu lays corol IEEE ras Auo Cor vol o [5 J Klaa Corollably a u rgy corol probl o racoal scr- syss Chapr Nw rs Naochology a Fracoal Calculus Es alau D Guvc Z rro Machao JA Sprgr- Vrlag Nw Yor 5-59 [6 K Olha a J Spar h Fracoal Calculus Acac Prss Nw Yor 974 [7 P Osalczy Epo o h racoal calculus: hory a s Applcaos Auoacs yawcwo Polch Łózj Łóź 8 Polsh [8 I Poluby Fracoal Dral Equaos Acac Prss Sa Dgo 999 [9 AG awa AM Sola AS Elwal a A S O h sably o lar syss wh racoal-orr ls Chaos Solos a Fracals vol 4 o [ EJ Solro Prs JA rro Machao P Moura Olvra Fracoal yacs gc algorhs orshop o Fracoal Drao a s Applcao vol [ M Vagr CA Moj AJ Calro Fracoal orr syss a racoal orr corol acos cur IEEE CDC #: Fracoal calculus Applcaos Auoac Corol a obocs