Pin SS: 3-6; Online SS: 367-5357 DO: 55/ic-6-7 Peuion nlsis of he -sed Coninuous-ime ine udic egulo Polem fo Desco Ssems onchev Ke Wods: Coninuous-ime desco ssems; peuion nlsis; line qudic egulo sed conol polem sc his ppe consides n ppoch o pefom peuion nlsis of line qudic egulo conol polem fo coninuous-ime desco ssems he invesiged conol polem is sed on solving s ine i nequliies nd ppling punov funcions he ppe is concened wih oining line peuion ounds fo he coninuous-ime conol polem fo desco ssems he compued peuion ounds cn e used o sud he effec of peuions in ssem nd conolle on fesiili nd pefomnce of he consideed conol polem numeicl emple is lso pesened in he ppe noducion ine mi inequliies echniques e inspiing gowing inees in he conol communi nd e emeging s poweful numeicl ools fo he nlsis nd design of conol ssems [345] umeous polems in conol nd ssems heo cn e fomuled in ems of line mi inequliies his fc is hdl supising given h s e diec poducs of punov sed ciei nd h punov echniques pl n essenil ole in he nlsis nd conol of line ssems [] he conol polem fo desco/singul ssems [] is good illusion of wh ws discussed ove Desco ssems hve een of inees in he lieue since he hve mn impon pplicions in cicui ssems [] ooics [] nd ec n clssicl esuls in he usul se-spce heo s sili conollili nd osevili hve een eended o hese ssems [34] Peuion nlsis of some conol polems fo singul ssems is consideed in [89] he im of his ppe is o popose n ppoch o pefom line peuion nlsis of he sed conol polem fo desco ssems fe inoducing suile igh hnd p in he used mi inequliies houghou he ppe following noion is pplied: m n he spce of el m n mices; n n ; n he ideni n n mi; e n he uni n o; he nspose of ; he pseudo invese of ; σm he specl nom of whee σ m is he mimum singul vlue of ; m n he column-wise o epesenion of m n ; mn m n m n he -pemuion mi such h mn ; P he Koneke poduc of he mices nd P he noion : snds fo equl definiion he emining p of he ppe is ognized s follows n Secion we popose he polem se up nd ojecive n Secion 3 we descie he pefomed line peuion nlsis of he -sed coninuous-ime conol polems fo singul ssems n Secion 4 we pesen numeicl emple he oined esuls nd discussions he ppe concludes in Secion 5 wih some finl emks Polem Se Up nd Ojecive ine coninuous-ime desco ssems e genell descied he following se of diffeenil-lgeic equions E u whee n u m nd n e he ssem desco se inpu nd iniil condiions nd nd E e consn mices of compile size Definiion Ssem equivlence wo ssems E nd E e sid o e ssem equivlen denoed E E if hee eis nonsingul n n nsfomion mices such h he equions E E hold ue Definiion eguli he ssem is egul if he polnomil dese sisfies dese Definiion 3 Weiesss noml fom Fo n egul ssem hee eis wo non-singul mices n n such h he following decomposed epesenion cn e oined u u n Definiion 4 nde of nilpoence he inde of infomion echnologies nd conol 6 3
{ } nilpoence ν ie q ν : min q is sid o e inde of line desco ssem Ssems wih ν e clled high inde DE diffeenil lgeic ssems ssems he desco ssem hs soluion fo n iniil condiion nd sufficienl smooh inpu u is possile h he soluion migh show impulsive ehvio h is wh conside he ssem in Weiesss noml fom unde sufficienl smooh inpu sing fom n iniil condiion hen he se evoluion cn e descied ccoding o [3]: 3 e ν i δ i e τ i u dτ ν i Epession 3 fo se evoluion implies h inde one desco ssems ν nd will hve no impulsive soluions n his cse he ssem is clled impulse fee nd inde one Conside he line coninuous-ime desco ssem whee hee is no diec elion eween he inpu nd he oupu signl houghou he ppe we ssume he desco ssem is n inde one ssem hee eiss n equivlen ssem E n in Weiesss cnonicl fom whee is sle mi he nsfomed ssem is given s 4 u i u i he nsfomed ssem 4 is oined using he epession 3 fo se evoluion ine qudic egulo polem fo desco ssems [] mens fo given iniil se o find conol lw which minimizes he cos funcion p u pu d should e found qudic punov funcion V P P> such h d V d p p [ K K ] > 5 [ K P P K ] < K K P Using Schu complemen gumen [5] he epession ove is equivlen o K P P K K 6 K P < P > p < p p o solve he polem fo desco ssems nd o ensue closed-loop sili nd specified pefomnce i is necess o design sefeedck conol uk n he ppe we ppl n ppoch o solve he line qudic egulo conol polem fo singul ssems s sed in [] We pe- nd pos-mull epession 6 dig[p ] Fuhe we will use new viles P > nd K P o oin he following ssem: 7 P p < > he min ojecive of he ppe is o conduc line sensiivi nlsis of he ssem 7 needed o solve he coninuous-ime line qudic egulo conol polem fo desco ssems Fuhe in he ppe we will use he following noion: p p p p houghou he e we ssume h he mices e sujec o peuions nd ssume h he do no chnge he sign of he ssem 7 he sensiivi nlsis of he coninuous-ime sed line qudic egulo polem fo singul ssems is imed deemining peuion ounds of he ssem 7 s funcions of he peuions in he d 4 6 infomion echnologies nd conol
6 5 infomion echnologies nd conol 3 ine Peuion nlsis n his secion peuion nlsis of he 7 whee fo he coninuous-ime desco inde one ssem 4 is pefomed 8 < is essenil o invesige he effec of he peuions on he peued soluions nd whee nd e he nominl soluion of he inequli 7 nd he peuions especivel he imponce of ou ppoch is o pefom sensiivi nlsis of he inequli 7 simill o peued mi equion fe inoducing slighl peued suile igh hnd p n his w 9 is oined 9 < whee nd nd is compued using he nominl elow < he mi includes infomion egding d nd closed-loop pefomnce peuions he ounding eos nd he sensiivi of he ineio poin mehod h is used o solve he s Using he elion he peued equion 9 cn e wien in he following w whee Due o he fc h we conduc line peuion nlsis hee he ems of second nd highe ode will e elimined fewds he oized fom of epession is epessed elow
6 6 infomion echnologies nd conol whee [ ] : q [ ] : 5 4 3 Π Π Π Π m m m m n n he mhemicl epesenions give us he possiili o oin he elion 3 5 4 3 q Finll he elive peuion ound fo he soluion of he 7 is oined 4 5 4 3 q hee
3 4 : : : 3 4 5 : : : 5 e clled he individul elive condiion numes of he 7 wih espec o he peuions nd ppling simil deivion seg he elive peuion ounds fo he soluion of he 7 cn e oined We use he following epession 5 whee Since we pefom line sensiivi nlsis hee he ems of second nd highe ode e negleced hen he oized fom of elion 5 cn e oined whee 6 [ Π Π ] : n m m m Π n Π m infomion echnologies nd conol 6 7
[ ] : 3 4 5 q q he deivions mde llow us o oin he epession 7 3 4 5 he end he elive peuion ound fo he soluion of he 7 is oined whee 8 q q 4 q q 5 q 3 q q3 q4 : : : 3 4 q q5 : : : 5 e he individul elive condiion numes of he 7 wih espec o he peuions nd 4 umeicl Emple [3] Conside he coninuous-ime inde one desco ssem given in Weiesss noml fom ie E Due o he fc h we would like o oin line ounds peuions in he ssem mices e chosen in ode o elimine second nd highe ode ems in he deivion pocedue ie 8 6 infomion echnologies nd conol
i i i i i i i i fo i 874 he peued soluions nd e compued ppling he mehod pesened in [7] nd using he sofwe [4] Pefoming he poposed ppoch he line elive peuion ounds fo he soluions nd Peuion ounds of he ssem 7 e clculed using epessions 4 nd 8 especivel he esuls oined fo diffeen size of peuions e pesened in he les q ound4 ound8 8 7387-8 45-7 6479-8 876-7 7 7387-7 45-6 6479-7 876-6 6 7387-6 45-5 6479-6 876-5 5 7387-5 45-4 6479-5 876-4 4 7387-4 45-3 6479-4 876-3 sed on he suggesed soluion mehod o pefom peuion nlsis he coninuous-ime sed line qudic egulo conol polem fo desco ssems we oin he peuion ounds 4 nd 8 hese ounds e close o he el elive peuion ounds q nd h he e good in sense h he e igh 5 Conclusion which mens We poposed n ppoch o compue he line peuion ounds of he coninuous-ime sed line qudic egulo conol polem fo desco ssems We lso suggesed how o clcule he esimes of he individul condiion numes fo he consideed s igh line peuion ounds wee oined fo he mi inequliies deemining he polem soluion he compued peuion ounds cn e used o nlze he fesiili nd pefomnce of he consideed conol polem in pesence of peuions in he ssem nd he conolle Hving in mind he oined heoeicl esuls we hve pesened numeicl emple o vividl epess he pplicili nd pefomnce of he poposed soluion ppoch o invesige he sensiivi of he sed line qudic egulo conol polem fo singul ssems efeences od S El Ghoui E Feon nd V lkishnn ine i nequliies in Ssem nd Conol heo Phildelphi S 994 onchev Konsninov nd P Pekov ine i nequliies in Conol heo Sofi Deme 5 S 954:956-3- in ulgin 3 esoov nd P Ghine neio-poin Polnomil lgoihms in Conve Pogmming Phildelphi S P 994 4 Ghine P emiovski u nd Chilli Conol oolo fo Use wih he hwoks nc 5 Pucelle D D Henion i nd K iz Use s Guide fo SeDui nefce 4 S-CS 6 enon nd D Smih on eive sed lgoihm fo ous Oupu Feedck Silizion n J Conol 7 999 4 3-33 7 Ghine P nd P pkin ine i nequli ppoch o H Conol n J ous nd online Conol 4 994 4-448 8 onchev ine Peuion ounds of he Coninuous-ime -sed H-infini udic Sili Polem fo Desco Ssems Jounl of Ceneics nd nfomion echnologies infomion echnologies nd conol 6 9
o 4 Sofi 3-4 SS 3-97 9 onchev Fis Ode Peuion ounds of he Disceeime -sed H-infini udic Sili Polem fo Desco Ssems Jounl of Ceneics nd nfomion echnologies o Sofi 3- SS 3-97 ende D J nd J u he ine udic Opiml egulo fo Desco Ssems EEE ns uom Con C-3 987 o 8 67-688 ewcom W nd Dziul Some Cicuis nd Ssems pplicions of Semi-se heo Cicus Ss Sig Poc 8 989 35-6 ills J K nd Goldeneg Foce nd Posiion Conol of nulos duing Consined oion sks EEE ns oo uom 38 989 3-46 3 Di Singul Conol Ssems elin Spinge Velg 989 4 Co D Conollili Osvili nd Duli in Singul Ssems EEE ns uom Con C-9 984 o 76-8 5 Dulleud G E F Pgnini Couse in ous Conol heo ew ok Spinge-Velg nusc eceived on 986 ssoc Pof D nde onchev eceived his Sc nd PhD degees in Fcul of uomics Depmen of Ssems nd Conol fom echnicl Univesi Sofi ulgi Since 7 D onchev woks in U Sofi nd ecme n ssoc Pof in His pofessionl ineess nd esech civiies e in he es of ssems nd conol heo ous nd opiml conol model pedicive conol line mi inequliies in conol peuion nlsis opimizion poducion nd opeions mngemen decision mking unde isk nd uncein opeions esech nd ec Concs: echnicl Univesi in Sofi Fcul of uomics Depmen of Ssems nd Conol Klimen Ohidski lvd l Sofi ulgi e-mil: onchev@u-sofig 6 infomion echnologies nd conol