AUTOMATIC CONTROL SYSTEMS

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1 9 HE UO ONROL SYSES OSVE SLE RELZONS OF ONNUOUS-E LNER SYSES deuz Kzore trt: he rolem for exitee d determitio of the et of oitive ymtotilly tle reliztio of roer trfer futio of lier otiuou-time ytem i formulted d olved Neery d uffiiet oditio for exitee of the et of the reliztio re etlihed roedure for omuttio of the et of reliztio re rooed d illutrted y umeril exmle Keyword: oitive tle reliztio exitee roedure lier otiuou-time ytem trodutio etermitio of the tte e equtio for give trfer mtrix i lil rolem lled reliztio rolem whih h ee ddreed i my er d oo Fri d Rildi eveuti d Fri Kzore 99 9 Sher d ixo 977 overview o the oitive reliztio rolem i give i Fri d Rildi Kzore eveuti d Fri he reliztio rolem for oitive otiuou-time d direte-time lier ytem h ee oidered i Kzore d the oitive reliztio rolem for direte-time ytem with dely i Kzore 5 6 he frtiol oitive lier ytem h ee ddreed i Kzore 8 9 he reliztio rolem for frtiol lier ytem h ee lyzed i Kzore 8 d for oitive hyrid ytem i Kzore 8 method ed o imilrity trformtio of the tdrd reliztio to the direte oitive oe h ee rooed i Kzore oditio for the exitee of oitive tle reliztio with ytem etzler mtrix for trfer futio h ee etlihed i Kzore he rolem of the exitee d determitio of the

2 rtifiil telligee ethod d ehique for uie d Egieerig litio et of etzler mtrie for give tle olyomil h ee formulted d olved i Kzore t i well-ow tht Fri d Rildi Kzore 99 tht to fid reliztio for give trfer futio firt we hve to fid tte mtrix for give deomitor of the trfer futio thi er eery d uffiiet oditio for exitee of the et of oitive tle reliztio of roer trfer futio of lier otiuou-time ytem re etlihed d roedure for omuttio of the et of reliztio i rooed he er i orgized follow etio ome relimirie oerig oitive lier ytem re relled d the rolem formultio i give rolem olutio for ytem with rel egtive ole of the trfer futio i reeted i etio he rolem of the exitee d omuttio of the et of oitive ymtotilly tle reliztio for ytem with omlex ojugte ole i ddreed i etio oludig remr re give i etio 5 he followig ottio will e ued: - the et of rel umer m rel mtrie off-digol etrie - the - the et of idetity mtrix m rtiol mtrie i m - the et of m - the et of m mtrie with oegtive etrie d etzler mtrie rel mtrie with oegtive - the et of ymtotilly tle etzler mtrie - troe of the mtrix m - the et of relimirie d the rolem formultio oider the otiuou-time lier ytem where x t x t x t u t y t x t u t m u t m y t re the tte iut d outut vetor d m efiitio Fri d Rildi Kzore he ytem i lled iterlly oitive if x t x x d ll iut u t t y t t for y iitil oditio m heorem Fri d Rildi Kzore he ytem i oitive if d oly if

3 HE m m efiitio Fri d Rildi Kzore he oitive ytem i lled ymtotilly tle if t lim x t for y x heorem Fri d Rildi Kzore he oitive ytem i ymtotilly tle if d oly if ll oeffiiet of the olyomil det re oitive ie for i i efiitio Kzore mtrix i lled the moomil mtrix or geerlized ermuttio mtrix if it every row d it every olum oti oly oe oitive etry d it remiig etrie re zero Lemm Kzore he ivere mtrix - of the moomil mtrix i equl to the troe mtrix i whih every ozero etry i reled y it ivere Lemm f d the det for every moomil mtrie det 5 roof y Lemm if if t i ey to he tht the d det det det det det det{ det } 6 ie det det he trfer mtrix of the ytem i give y he trfer mtrix i lled roer if 7 d it i lled tritly roer if K = m lim K 8 efiitio trie re lled oitive reliztio of trfer mtrix if they tify the equlity 7

4 rtifiil telligee ethod d ehique for uie d Egieerig litio he reliztio i lled ymtotilly tle if the mtrix i ymtotilly tle etzler mtrix Hurwitz etzler mtrix heorem Kzore he oitive reliztio i ymtotilly tle if d oly if ll oeffiiet of the olyomil det re oitive ie for i Lemm he mtrie i 9 m m = N re oitive ymtotilly tle reliztio of the roer trfer mtrix m for y moomil mtrix if d oly if the mtrie m m = N re oitive ymtotilly tle reliztio of roof y Lemm if i moomil mtrix the Hee if d oly if if d oly if Uig we oti { } m m if d oly if i lo moomil mtrix m d herefore the mtrie re oitive ymtotilly tle reliztio of if d oly if the mtrie re lo it oitive ymtotilly tle reliztio he rolem uder oidertio e tted follow: Give rtiol roer mtrix m fid et of it oitive ymtotilly tle reliztio thi er eery d uffiiet oditio for exitee of the et of the oitive ymtotilly tle reliztio for give will e etlihed d roedure for omuttio of the et of reliztio will e rooed

5 HE Sytem with rel egtive ole thi etio the igle-iut igle-outut lier otiuou-time lier ytem with the roer trfer futio hvig oly rel egtive ole ot eerily ditit α α α ie will e oidered Firt we hll ddre the rolem for = with the trfer futio > heorem here exit the et of oitive ymtotilly tle reliztio = for y oitive rmeter > d hvig oe of the form or 5 6 of the trfer futio if d oly if 7 roof t i ey to he tht the mtrie 5 re reliztio of he mtrix d if d oly if the oditio 7 re tified y Lemm the mtrie re oitive ymtotilly tle reliztio of for y > if d oly if the mtrie 5 re it oitive ymtotilly tle reliztio roof for mtrie 6 i imilr heorem here exit the et of oitive ymtotilly tle reliztio = 8

6 rtifiil telligee ethod d ehique for uie d Egieerig litio for y moomil mtrix d hvig oe of the form 9 9 of the trfer futio if d oly if d for roof he mtrix if d oly if it hrteriti olyomil det h egtive rel zero d thi i the e if d oly if the oditio i met d he mtrix lim if d oly if he tritly roer trfer futio h the form where umig = we oti

7 5 HE From omrio of d we hve 5 From 5 it follow tht if d oly if the oditio re tified he roof for 9 i imilr y Lemm the mtrie 8 re oitive ymtotilly tle reliztio for y moomil mtrix the mtrie 9 re it oitive ymtotilly tle reliztio if d oly if Exmle omute the et of oitive ymtotilly tle reliztio 8 of the trfer futio he trfer futio 6 tifie the oditio d ie d for 7 Uig 9 we oti d for the rmeter tifyig he deired et of oitive ymtotilly tle reliztio of 6 i give y d

8 rtifiil telligee ethod d ehique for uie d Egieerig litio where i y moomil mtrix heorem Let the trfer futio 9 hve oly rel egtive ole α α α ie d where here exit the et of oitive ymtotilly tle reliztio = for y moomil mtrix d hvig oe of the form or of the trfer futio 9 if d oly if the oditio re met roof he mtrix if d oly if for = he mtrix lim

9 7 HE if d oly if he tritly roer trfer futio h the form where umig we oti 5 From omrio of d 5 we hve 6 From 6 it follow tht if d oly if the oditio re met he roof for follow immeditely from the equlity tht 7 y Lemm the mtrie re oitive ymtotilly tle reliztio of 9 for y moomil mtrix if d oly if the mtrie re it oitive ymtotilly tle reliztio heorem here exit the et of oitive ymtotilly tle reliztio = 8 for y moomil mtrix d hvig oe of the form

10 rtifiil telligee ethod d ehique for uie d Egieerig litio 8 9 or 9 of the trfer futio with oly rel egtive ole α α α if d oly if the oditio where re met roof he mtrix if d oly for = he mtrix lim if d oly if he tritly roer trfer futio h the form where for = umig we oti

11 9 HE d d where d From omrio of d we hve From it follow tht if d oly if the oditio re met he roof for 9 follow immeditely from 7 y Lemm the mtrie 8 re oitive ymtotilly tle reliztio of for y moomil mtrix if d oly if the mtrie 9 re it oitive ymtotilly tle reliztio From ove oidertio we hve the followig roedure for omuttio of the et of oitive ymtotilly tle reliztio 8 of the trfer futio with rel egtive ole

12 rtifiil telligee ethod d ehique for uie d Egieerig litio roedure Ste he the oditio f the oditio re met go to Ste if ot the doe ot exit the et of reliztio Ste Uig 9 omute the mtrie for exmle for = or = Ste Uig 8 omute the deired et of reliztio Exmle omute the et of oitive ymtotilly tle reliztio of the trfer futio he trfer futio 5 h two rel doule ole Uig roedure we oti the followig Ste he oditio re tified ie 6 Ste thi e the mtrie 9 hve the form 7 Ste he deired et of reliztio of 5 i give y 8 for y moomil mtrix

13 HE Sytem with omlex ojugte ole thi etio the igle-iut igle-outut lier otiuou-time ytem with the trfer futio hvig t let oe ir of omlex ojugte ole will e oidered heorem here exit the et of oitive ymtotilly tle reliztio = for y moomil mtrix d the mtrie hvig oe of the form or of the trfer futio if d oly if the oeffiiet of the olyomil d tifie the oditio d 6 re where re oitive rmeter tifyig < + <

14 rtifiil telligee ethod d ehique for uie d Egieerig litio roof f the mtrix if d oly if it hrteriti olyomil det d 7 h the oeffiiet tifyig the oditio 5 9 d < + < he mtrix lim 8 if d oly if he tritly roer trfer futio h the form 9 where 9 umig we oti From omrio of 9 d we hve From it follow tht if d oly if the oditio 6 re met he roof for i imilr y Lemm the mtrie re oitive ymtotilly tle

15 HE reliztio for y moomil mtrix of if d oly if the mtrie re it oitive ymtotilly tle reliztio Remr he mtrix i heorem e reled y the mtrie 9 d the mtrix y For the mtrie d hve the form d for the mtrie d hve the form From ove oidertio we hve the followig roedure for omuttio of the et of oitive ymtotilly tle reliztio roedure Ste he the oditio 5 d 6 f the oditio re met go to Ste if ot the doe ot exit the et of reliztio Ste Uig omute the mtrie for exmle for = or = Ste Uig omute the deired et of reliztio Exmle omute the et of oitive ymtotilly tle reliztio of the trfer futio Uig roedure we oti the followig Ste he trfer futio tifie the oditio 5 d 6 ie d

16 rtifiil telligee ethod d ehique for uie d Egieerig litio 5 6 for < + < 9 Ste Uig d we oti where d re ritrry rmeter tifyig < + < 9 Ste he deired et of oitive tle reliztio i give y 6 for y moomil mtrix heorem here exit the et of oitive ymtotilly tle reliztio 7 for y moomil mtrix d the mtrie hvig oe of the form ~ ~ ~ ~ ~ ~ ~ ~ 8

17 5 HE d 8 or 8 of the trfer futio 9 if d oly if the oeffiiet of the olyomil d tifie the oditio d re where re oitive rmeter tifyig < + + < he roof i imilr to the roof of heorem Remr he mtrix i heorem e reled y the mtrie 5

18 rtifiil telligee ethod d ehique for uie d Egieerig litio 6 d the mtrix y the mtrie 5 geerl e let u oider the trfer futio with t let oe ir of omlex ojugte ole heorem here exit the et of oitive ymtotilly tle reliztio 5 for y moomil mtrix d hvig oe of the form 6 where - re oitive rmeter tifyig < < or 6 d 6 of the trfer futio if d oly if the oeffiiet of the olyomil d 7

19 7 HE tifie the oditio!!! 8 d 9 where 9 roof t i well-ow Kzore tht there exit if d oly if the oeffiiet of the olyomil 7 re oitive d tify the oditio 8 he mtrix lim if d oly if he tritly roer trfer futio h the form

20 rtifiil telligee ethod d ehique for uie d Egieerig litio 8 where for = umig we oti d d where d From omrio of d we hve

21 9 HE From it follow tht if d oly if the oditio 9 re met he roof for 6 follow immeditely from 7 y Lemm the mtrie 5 re oitive ymtotilly tle reliztio of for y moomil mtrix if d oly if the mtrie 6 re it oitive ymtotilly tle reliztio Remr he mtrix i heorem e reled y the mtrie where - re oitive rmeter tifyig < < d the mtrix y the mtrie o omute the deired et of oitive ymtotilly tle reliztio 5 of roedure with light modifitio e ued oluio he rolem of exitee d omuttio of the et of oitive ymtotilly tle reliztio of roer trfer futio of lier otiuou-time ytem h ee formulted d olved Neery d uffiiet oditio for exitee of the et of reliztio hve ee etlihed heorem d roedure for omuttio of the et of reliztio for trfer futio with oly rel egtive ole d with t let oe ir of omlex ojugte ole hve ee rooed

22 rtifiil telligee ethod d ehique for uie d Egieerig litio roedure d he effetivee of the roedure hve ee demotrted o umeril exmle he reeted method e exteded to oitive ymtotilly tle direte-time lier ytem d lo to multi-iut multi-outut otiuou-time d direte-time lier ytem oe rolem i exitee of thee oidertio to frtiol lier ytem Kzore owledgmet hi wor w uorted uder wor S/WE// iliogrhy Fri d Rildi L Fri S Rildi oitive Lier Sytem heory d litio J Wiley New Yor eveuti d Fri L eveuti L Fri tutoril o the oitive reliztio rolem EEE r utom otrol vol 9 o Kzore 99 Kzore Lier otrol Sytem vol Reerh Studie re J Wiley New Yor 99 Kzore Kzore oitive d Sytem Sriger-Verlg Lodo Kzore Kzore Reliztio rolem for oitive direte-time ytem with dely Sytem Siee vol o 7- Kzore 5 Kzore oitive miiml reliztio for igulr direte-time ytem with dely i tte d dely i otrol ull ol d Si eh vol 5 o Kzore 6 Kzore reliztio rolem for oitive otiuou-time lier ytem with redued umer of dely t J l th om Si 6 Vol 6 No 5- Kzore 6 Kzore omuttio of reliztio of direte-time oe ytem ull ol d Si eh vol 5 o Kzore 6 Kzore Reliztio rolem for oitive multivrile direte-time lier ytem with dely i the tte vetor d iut t J l th om Si vol 6 o 6-6 Kzore 8 Kzore Frtiol oitive otiuou-time lier ytem d their rehility t J l th omut Si vol 8 o 8-8 Kzore 8 Kzore Reliztio rolem for frtiol otiuou-time ytem rhive of otrol Siee vol 8 o 8-58 Kzore 8 Kzore Reliztio rolem for oitive hyrid ytem OEL vol 7 o Kzore 9 Kzore Frtiol oitive lier ytem Kyerete: he tertiol Jourl of Sytem & yereti 9 vol 8 o 7/ Kzore 9 Kzore olyomil d Rtiol trie Sriger-Verlg Lodo 9 Kzore Kzore omuttio of oitive tle reliztio for lier otiuoutime ytem ull ol d Si eh vol 59 o 7-8 d ro th Euroe of iruit heory d eig ugut 9 to Liöig Swede Kzore Kzore oitive tle reliztio of frtiol otiuou-time lier ytem t J l th om Si Vol No Kzore Kzore oitive tle reliztio with ytem etzler mtrie rhive of otrol Siee vol o d ro of R -RO

23 HE Kzore Kzore Seleted rolem i Frtiol Sytem heory Sriger-Verlg Kzore Kzore Exitee d determitio of the et of etzler mtrie for give tle olyomil t J l omut Si i re Sher d ixo 977 U Sher ixo Geerlized miiml reliztio of trfer-futio mtrie t J otr vol 5 o uthor formtio deuz Kzore ilyto Uiverity of ehology; Fulty of Eletril Egieerig; Wiej ilyto old; e-mil: zore@iewedul jor Field of Sietifi Reerh: he theory of ytem d the utomti otrol ytem theory eilly igulr multidimeiol ytem oitive multidimeiol ytem d igulr oitive d ytem