Model Reduction of Uncertain Discrete Systems Having an Interval Structure Using Genetic Algorithms

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1 Poeegs of the 5th WSES/ISME It. Cof. o SYSTEMS THEORY SCIENTIFIC COMPUTTION Mlt Septeme pp3-37 Moel Reuto of Uet Dsete Systems Hvg Itevl Stutue Usg Geet lgothms Che-Che Hsu * Tsug-Ch Lu * We-Ye Wg # * Deptmet of Eleto Egeeg St. Joh's Uvesty 499 Tm Kg R. Se. 4 Tm-Su Tpe Tw 535. # Deptmet of Eleto Egeeg Fu-Je Cthol Uvesty 5 Chug Cheg R. Hs-Chug 45 Tpe Tw. stt: - I ths ppe evolutoy ppoh s popose to ot eue-oe sete tevl moel fo uet sete-tme systems hvg tevl uettes se o esemle of sete sequee eegy etwee the ogl eue systems. System pefome of the sete tevl moel ote y usg the popose evolutoy ppoh s the vefe se o tme esposes of the esultg moel ompso to exstg methos to emostte the effetveess of the popose ppoh. Key-Wos: - Moel euto sete-tme systems geet lgothms uet systems tevl plt sete sequee eegy.. Itouto Most ptl systems suh s flght vehles elet motos oots e fomulte otuous-tme uet settgs. The uettes these systems se fom umoelle yms pmete vto seso oses tuto ostts et. These vtos o ot follow y of the ow polty stutos geel e most ofte qutfe tems of mpltue /o fequey ous. Hee ptl systems o plts e most sutly epesete y otuous-tme pmet tevl moels ste of etemst mthemtl moels. I my stutos t s lso esle to eple the hgh-oe system y lowe oe moel. Typl methos fo moel euto lue ggegto metho momet mthg tehque Pe ppoxmto Routh ppoxmto et. Reet evelopmets of moel euto hve ee me tow the eto to hle uet tevl systems 3-6 se o vts of the Routh ppoxmto methos whee tevl thmet s pefome to eve Routh α β o γ δ ol otue-fto expso veso 7. It hs ee show howeve tht some tevl Routh ppoxmts my ot e oustly stle eve though the ogl tevl system s oustly stle 67. Futhemoe the eue moels ote v these methos e geelly ot sutle fo oust otolle esg euse of the poo fequey esposes ompso to the ogl outept 7. s f s moel euto of sete tevl systems s oee vey lmte sussos hve ee fou ltetue 7-4. mog them thee ws metho of moel oe euto usg Pe ppoxmto to et omt poles 8 whee the eomto of the eue moel s fome y etg the omt poles of the gve sete tevl system whle the umeto s ote y mthg the fst momets of the moel wth tht of the system. se o multpot Pe ppoxmto 9 eue moel ws ote s Pe ppoxmt of the ogl system out pots. Howeve system pefomes of the eue-oe moels v the ove-metoe methos e geelly ot stsftoy. Moe effetve ppohes ee to e evelope so tht htests of the eue moel sutly ppoxmtes those of ts ogl system. Reet evelopmets of evolutoy lgothms hve pove pomsg ltetve to ess the ove-metoe polems ffultes euse of the pltes of ete om seh fo glol optmto 34. Ths motvtes the use of geet lgothms to eve eue sete tevl moel wth ese oe fo the ogl system se o the egee of esemle of the sete sequee eegy. System pefome of the sete tevl

2 moel ote y usg the popose evolutoy ppoh s the vefe se o tme esposes of the esultg moel ompso to exstg methos to emostte the effetveess of the popose ppoh.. Polem of moel euto of sete tevl systems Cose the stle sete-tme system wth tevl uettes: G The polem ow e fomulte to eteme ese th-oe stle moel C D G so tht the pefome of the eue moel G sutly ppoxmtes tht of the ogl system G tem of esemle of sete sequee eegy. 3. Dsete sequee eegy DSE of sete tevl systems s ompso ss we ee to evlute the sete sequee eegy DSE fo oth the eue ogl tevl moels espetvely. 3. Dsete sequee eegy DSE of etemst systems Dsete sequee eegy of sequee T h whh plys vey mpott ole my otol ppltos s efe s: T h 3 I evlutg Eq.3 Psevl s theoem 5 sttes tht sequee eegy s elte to otou tegto log the ut le postve eto h T HH π 4 whee H s the -tsfom of the sequee ht. Wth eusve lge lgothm 6 t s esy to ompute the sete sequee eegy Eq.4. Tht s fo sete tsfe futo H 5 whee e polyomls wth el oeffets - > 6-7 hs ll ts eos se the ut le we ot the sete sequee eegy s C T h π 8 whee oeffets e eusvely efe 6 s: m m 9 m m 3. Dsete sequee eegy DSE of tevl systems Fo sete tevl tsfe futo H I whee e polyomls wth tevl oeffets I I. 3 The DSE of the sete tevl system H I of Eq. e ote v tevl thmet mpultos. Tht s f hs ll ts eos se the ut le h I T epesets the mpulse espose of the sete tevl moel H I we ot the sete sequee eegy s: α α π C I T h 4 Poeegs of the 5th WSES/ISME It. Cof. o SYSTEMS THEORY SCIENTIFIC COMPUTTION Mlt Septeme pp3-37

3 Poeegs of the 5th WSES/ISME It. Cof. o SYSTEMS THEORY SCIENTIFIC COMPUTTION Mlt Septeme pp3-37 whee oeffets eusvely efe s: e m m 5 whee w e the weghtg oeffets epesetg the eltve mpote of the oetve futos. Comg the oetves to ot optme soluto hs the vtge of poug sgle soluto euse the esg polem t h eques o teto wth the eso mg mog the pmetes eve y the popose ppoh s f s the evto of the eue-oe sete tevl moel s oee. m m 6 Smlly the DSE of the eue-oe sete tevl moel G of Eq. e ote extly the sme wy. G G DSE of the mpulse espose DSE of the mpulse espose G α α β β - 4. Moel euto of sete tevl systems se o DSE Fgue shows the fmewo of the G-se moel euto sheme to eve eue moel G fo the ogl system G se o the loseess of sete sequee eegy DSE of the mpulse esposes etwee these two tevl systems whh α α β β st fo the lowe uppe ous of the DSE of the mpulse esposes fo the ogl eue sete tevl moels espetvely. The tole of the popose ppoh s to seh fo optml set of pmetes fo the uppe lowe ous of the uet oeffets of the eue moel G so tht the oetve futos J α β 7 J α β 8 e mme. y og so the htests of the eue-oe sete-tme tevl moel G ppoxmtes those of ts ogl system G f oth oetves J J e mme. Wth the multoetve polem pesete Eqs. 7-8 the smplest tul wy s to hve t efomulte s moo-oetve optmto polem y mes of ggegtg futo of the fom m J w α β w α β 9 C D Fg. Fmewo of the G-eve eue tevl moel. 5. Illustte exmple Cose the sete tevl system gve y G The st oe eue moel s G v multpot Pe ppoxmto s evele 9. y usg the popose G-se ppoh we ot the eue moel G g fte geetos of evoluto. The G pmetes opte lue populto se of 5 p.8 pm.. Toumet seleto s use wth toumet se of 4. G opetos lue two-pot ossove o-ufom mutto. Fo ompso pupose Tle shows the DSE of vous tevl moels of ths Exmple. It s le tht the eue-oe moel G g eve v the popose ppoh possesses extly the sme DSE s tht of the ogl system G. The tevl moel G eve v 9 howeve hs seous evto tems of DSE fom ts ogl outept. Theefoe we expet ette system

4 Poeegs of the 5th WSES/ISME It. Cof. o SYSTEMS THEORY SCIENTIFIC COMPUTTION Mlt Septeme pp3-37 pefome fom the tevl moel G g ompso to the tevl moel G. Fgues -7 show mpulse step esposes of the espetve tevl moels whh ofm the effetveess of the popose ppoh. To show the osstey of the evoluto poess Fg. 8 llusttes the smulto esults of the evoluto poess to eve the eue-oe tevl moel fo 5 us. s emostte Fg. 8 stsftoy osstey of the evoluto poess e ote v the popose G-se ppoh. Tle DSE of vous tevl moels. Dsete Itevl Moels Dsete Sequee Eegy Fg. 4 Impulse esposes of the eue-oe sete tevl moel G 9. G G..447 G g Fg. 5 Step esposes of the eue-oe sete tevl moel G 9. Fg. Impulse esposes of the ogl sete tevl system G. Fg. 6 Impulse esposes of the eue-oe sete tevl moel G g ote v the popose ppoh. Fg. 3 Step esposes of the ogl sete tevl system G.

5 Poeegs of the 5th WSES/ISME It. Cof. o SYSTEMS THEORY SCIENTIFIC COMPUTTION Mlt Septeme pp3-37 Fg. 7 Step esposes of the eue-oe sete tevl moel G g ote v the popose ppoh. Fg. 8 Evoluto poess to eve the eue moel G g fo 5 us. 6. Colusos Thee e vey few sussos o the evto of eue-oe tevl moel fo uet sete-tme systems hvg tevl uettes. Exstg ppohes ot pove stsftoy esults geel. To fltte the evto poess ths ppe hs pesete G-se ppoh s omputtolly smple yet ptl wy to ot optml eue moel fo hgh-oe tevl systems wth lest evto of DSE fom ts ogl outept. I ompso to the vlle tehques the G-se moel euto ppoh poves stsftoy pefome. Smulto esults hve emostte tht the eue moel ote v the popose ppoh out-pefoms the exstg methos tems of tme esposes sete sequee eegy. owlegemets Ths wo ws ptlly suppote y the Ntol See Coul Tw ue Gt NSC 9-3-E-9-8. Refeees. msh New Tools fo Roustess of Le Systems Mmll Pulhg Compy New Yo 994. L. Sheh W. Wg J. Suel Dgtl eesg of se logue otolles fo smple-t tevl systems Poeegs of the Isttuto of Eletl Egees Pt D Vol yophyy O. Isml R. Goe Routh-Pe ppoxmto fo Itevl Systems IEEE Tstos o utomt Cotol Vol. 39 pp yophyy vsh Uphye O. Isml γ-δ Routh ppoxmto fo tevl systems IEEE Tstos o utomt Cotol Vol. 4 No. 8 pp G.V.K.R. Ssty R. R Ro P. Mllu Ro Lge sle tevl system moellg usg Routh ppoxmts Eletos Lettes Vol. 36 No. 8 pp pl. 6 Y. Dolg E. Zehe O Routh-Pe moel euto of tevl systems IEEE Tstos o utomt Cotol Vol. 48 No. 9 pp. 6-6 Sept Chy Hwg Shh-Feg Yg Commets o the omputto of tevl Routh ppoxmts IEEE Tstos o utomt Cotol Vol. 44 No. 9 pp Sept O. Isml. yophyy R. Goe Dsete tevl system euto usg Pe ppoxmto to llow eteto of omt poles IEEE Tstos o Cuts Systems Vol. 44 No. pp Nov O. Isml O multpot Pe ppoxmto fo sete tevl systems Poeegs of the Twety-Eghth Southeste Symposum o System Theoy pp pl 996. D. Goleg Geet lgothms Seh Optmsto Mhe Leg so-wesley 989. Z. Mhlew Geet lgothms Dt Stutue Evoluto Pogm Spge-Velg 996. Che-Che Hsu Chh-Yug Yu Desg of optml otolle fo tevl plt fom sgl eegy pot of vew v evolutoy ppohes IEEE Tstos o Systems M Cyeets-Pt Vol. 34 No. 3 pp J. Rees S. Flsse Hy methos usg geet lgothms fo glol optmto IEEE Tstos o Systems M Cyeets-Pt Vol. 6 No. pp P.J. Flemg R.C. Pushouse Evolutoy lgothms otol systems egeeg: suvey Cotol Egeeg Pte Vol. pp R. Roets C. Mulls Dgtl sgl poessg so-wesley K. J. stom E. I. Juy R. gel umel

6 Poeegs of the 5th WSES/ISME It. Cof. o SYSTEMS THEORY SCIENTIFIC COMPUTTION Mlt Septeme pp3-37 metho fo the evluto of omplex tegls IEEE utomt Cotol pp ugust Dl Wg l Zlouh Moel euto of sete le systems v fequey om le stutue Poeegs of me Cotol Cofeee Phlelph US Vol. Jue 998 pp S. ou P. ehoet P. Vle L. C. Clve ew sete mpulse espose Gm ts pplto to moel euto IEEE Tstos o utomt Cotol Vol. 45 No. 3 pp Mh. 9 O. Sehy M.N ly Dsete-tme moel euto wth optml eo lotos y om mmto Poeegs of the 998 IEEE Itetol Cofeee o Cotol ppltos Teste Itly Vol. Sept. 998 pp G. Lm V. Seem E. Zehe tehque fo euto of uet FIR fltes Poeegs of the 3 Itetol Symposum o Cuts Systems Vol. 4 My 3 pp. IV IV-659. Y. Dolg E. Zehe Moel euto of uet FIR sete-tme systems IEEE Tstos o Cuts Systems Vol. 5 No. 8 pp ug. 4. Do Chg Oh Hog e P Dsete Moel Reuto Pesevg oue Reless utomt Vol. 33 No. 4 pp pl S. Muheee Stsh R. C. Mttl Dsete system oe euto usg multpot step espose mthg Joul of Computtol pple Mthemts Vol. 7 No. pp Septeme Somth P Jyt Pl Reue oe moellg of sete-tme systems pple Mthemtl Moellg Vol. 9 No. 3 pp Mh 995.

Noise propagation simulation in and around buildings using improved integral energy equations

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