Guaranteed cost finite-time control for positive switched delay systems with ADT

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1 Ieraioal Joural o dvaced Research i Comuer Egieerig & echology (IJRCE) Volume 6 Issue 9 Seember 7 ISSN: Guaraeed cos iie-ime corol or osiive swiched delay sysems wih D Xiagyag Cao Migliag Ma Hao Xig bsrac his aer ivesigaes he guaraeed cos iie-ime boudedess or osiive swiched delay sysems wih average dwell ime(d). he oio o guaraeed cos iie-ime boudedess is irs iroduced. Firs by usig he Lyauov-Krasovsii ucioal D aroach suicie codiios are obaied o guaraee ha he corresodig sysem is iie-ime boudedess. he a sae eedbac coroller is desiged o esure ha he closed-loo sysem is guaraeed cos iie-ime boudedess(gcfb). he obaied codiios ca be easily solved by liear rogrammig. Fially a racical examle is rovided o illusrae he eeciveess o he roosed mehod. Idex erms Posiive swiched delay sysems Guaraeed cos corol Fiie-ime boudedess verage dwell ime Liear rogrammig. I. INRODUCION Posiive swiched sysems which cosis o a umber o osiive subsysems a swichig sigal goverig he swichig amog his subsysems have bee aid much aeio i rece years see [-5] reereces herei. he roblems o sabiliy aalysis coroller syhesis o he sysems have bee ivesigaed by may researchers due o heir broad alicaios i commuicaio ewors [6] ewor emloyig CP[7] sysems heory[8-] so o. So ar may exisig resuls relaed o sabiliy aalysis ocus o he asymoic sabiliy or exoeial sabiliy i he area o osiive swiched sysems which relecs he behavior o he sysem i a iiie ime ierval. Bu i may racical codiios oe is more ieresed i wha haes over a iie-ime ierval. he coce o iie-ime sabiliy(fs) was irsly deied i []. he F. mao e.al [3] exeded his deiiio o iie ime boudedess (FB) whe hey deal wih he behavior o he sae i he resece o exeral disurbaces. Some relaed resuls ca be oud i [4-9]. Recely [] irsly exe he coce o FS o osiive swiched sysems gives some FS codiios o osiive swiched sysems. I [] he roblem o iie-ime sabiliy sabilizaio o racioal-order osiive swiched sysems is cosidered via mode-deede average dwell ime aroach. he roblem o iie-ime L corol or a class o osiive swiched Mauscri received ug 7. Xiagyag Cao College o Iormaio Egieerig Hea Uiversiy o Sciece echology Luoyag Chia. Migliag Ma College o Iormaio Egieerig Hea Uiversiy o Sciece echology Luoyag Chia. Hao Xig College o Iormaio Egieerig Hea Uiversiy o Sciece echology Luoyag Chia. liear sysems wih ime-varyig delays has bee ivesigaed i []. O he oher h guaraeed cos corol has he advaage o rovidig a uer boud o a give sysem erormace idex hus he sysem erormace degradaio icurred by he uceraiies or ime delays is guaraeed o be less ha his boud[3]. So i is ecessary o sudy he desig roblem o guaraeed cos iie-ime coroller. here are some resuls abou his roblem see [4-7] reereces herei. Bu here are ew resuls available o guaraeed cos iie-ime corol or osiive swiched sysems wih ime-varyig delays which moivaes our rese sudy. I his aer we are ieresed i ivesigaig he roblem o GCFB or osiive swiched delay sysems wih D. he mai coribuios o his aer ca be summarized as ollows: () he deiiio o guaraeed cos iie-ime boudedess is give i is diere rom he geeral oe i aes ull advaage o he characerisics o oegaive saes o osiive swiched sysems. () By usig he D aroach coosiive Lyauov- Krasovsii ucioal mehod a sae eedbac law is desiged suicie codiios are obaied o guaraee ha he closed-loo sysem is GCFB such codiios ca be easily solved by liear rogrammig. he aer is orgaized as ollows. I Secio roblem saemes ecessary lemmas are give. GCFB aalysis coroller desig are develoed i Secio 3. racical examle is rovided i Secio 4. Fially Secio 5 cocludes his aer. Noaios. I his aer ( ) meas ha all elemes o marix are osiive(o-egaive). he suerscri deoes he rasose. R R R de- oe he -dimesioal o-egaive (osiive) vecor sace he -dimesioal Euclidea sace he se o real marices. -orm x is deied by x x wher x is he h eleme o x R i o exlicily saed e marices are assumed o have comaible dimesios. II. PROBLEM SEMENS ND PRELIMINRIES Cosider he ollowig osiive swiched delay sysems: x ( ) ( ) x( ) d ( ) x( d( )) G ( ) u( ) B ( ) w( ) () x( ) ( ) [ ]. m where x() R u() R rerese he sysem l sae corol iu. w() R is he disurbace iu which saisies ll Righs Reserved 7 IJRCE 33

2 34 Ieraioal Joural o dvaced Research i Comuer Egieerig & echology (IJRCE) Volume 6 Issue 9 Seember 7 ISSN: d : w( ) d d. () ( ):[ ) S { S} is he swichig sigal where S is he umber o subsysems. S d B G are cosa marices wih aroriae dimesios deoes h subsysem q deoes he q h swichig isa. ( ) is he iiial codiio o [ ] d () deoes he ime-varyig delay saisyig d ( ) d ( ) h where h are ow osiive cosas. he we will rese some deiiios lemmas or he ollowig osiive swiched sysems. x ( ) ( ) x( ) d ( ) x( d( )) B ( ) w( ) (3) x( ) ( ) [ ]. Deiiio []. Sysem (3) is said o be osiive i or ay iiial codiios ( ) [ ] disurbace iu w ( ) ay swichig sigals () he corresodig rajecory x ( ) hol or all. Deiiio []. is called a Mezler marix i he o-diagoal eries o marix are o-egaive. Lemma []. marix is a Mezler marix i oly i here exiss a osiive cosa such ha I. Lemma [3]. Sysem (3) is osiive i oly i S are Mezler marices S B G. Deiiio 3 []. For ay le N ( ) deoes he swichig umber o () over he ierval [ ). For give N i he iequaliy N ( ) N (4) hol he he osiive cosa is called a average dwell ime N is called a chaerig boud. Geerally seaig we choose N. Deiiio 4 (FS). For a give ime cosa vecors osiive swiched sysem (3) wih w ( ) is said o be FS wih resec o ( ( )) i su { ( ) } ( ) x x [ ]. i he above codiio is saisied or ay swichig sigals () sysem (3) is said o be uiormly FS wih resec o ( ). Deiiio 5 (FB). For a give ime cosa vecors osiive swiched sysem (3) is said o be FB wih resec o ( d ( )) where w () saisies () i d su { ( ) } ( ) x x [ ]. Now we give some deiiios abou GCFB or he osiive swiched delay sysems (). Deiiio 6. Deie he cos ucio o osiive swiched sysems () as ollows: [ ( ) J x R ( ) ] u R d (5) where R R are wo give vecors. Remar. I should be oed ha he roosed cos ucio is diere rom he geeral oe[3-6] his deiiio rovides a more useul descriio because i aes ull advaage o he characerisics o oegaive saes o osiive swiched sysems. Deiiio 7. (GCFB) For a give ime cosa vecors cosider he osiive swiched sysem () cos ucio (5) i here exis a corol law u () a osiive scalar J such ha he closed-loo sysem is FB wih resec o ( d ( )) he cos ucio J J saisies he he corresodig closed- loo sysem is called GCFB where J is a guaraeed cos value u () is a guaraeed cos iie-ime coroller. he aim o his aer is o desig a sae eedbac coroller u () id a class o swichig sigals () or osiive swiched sysem () such ha he corresodig closed-loo sysem is GCFB. III. MIN RESULS. Guaraeed cos iie-ime sabiliy aalysis I his subsecio we will ocus o he roblem o GCFB or osiive swiched sysem (3). he ollowig heorem gives suicie codiios o GCFB or sysem (3) wih D. heorem. Cosider he osiive swiched sysems (3) or give cosas vecors R i here exis a se o osiive vecors S osiive cosas 3 4 such ha he ollowig iequaliies hold: ' ' ' diag{ } (6) where 3 b r 3 (7) (8) e e d e (9) 3 4 a R r r r r r a ( h) max{ } ' r dr r S a ( a ) rereses he rh colum vecor o he r dr marix ( d ) [ ] [

3 Ieraioal Joural o dvaced Research i Comuer Egieerig & echology (IJRCE) Volume 6 Issue 9 Seember 7 ISSN: ] [ ] r r r rerese he rh elemes o he vecors resecively he uder he ollowig D scheme l l a a max l( e ) l( e 3 e 4 d) () he sysem (3) is GCFB wih resec o ( d ( )) where saisies q S q q q () he guaraeed cos value o sysem (3) is give by J x () s R J () e ( e e d) 3 4 Proo. Cosruc he ollowig co-osiive Lyauov- Krasovsii ucioal cidae or sysem (3): ( s) () ( ( )) ( ) ( ) d () V x x e x s where ( s) R e x () s d S. (3) For he sae o simliciy V ( ( )) () x is wrie as V () () i his aer. log he rajecory o sysem (3) we have V ( ) x ( ) x ( d( )) () d ( s) ( ) ( ) d () w B e x s d() x ( ) ( d ( )) e x ( d( )) ( s) e x ( s) d x ( ) e x ( ) d x ( ) x ( d( )) w () B d ( s) e x ( s) ( ) d () x ( h) x ( d( )) ( s) e x ( s) d x ( ) d () x () s From (7) (8) (3) (4) lea o V ( ) V ( ) x ( ) R ( ) ( ) x () R x ( d( )) ( h) w( ) d (4) (5) Subsiuig (6) io (5) yiel V ( ) V ( ) x ( ) R w( ) (6) I imlies ( ) ( ) (7) V ( ) V ( ) w( ) ( ) ( ) Iegraig boh sides o (7) durig he eriod [ ) or [ ) lea o ( ) ( ) ( ) ( ) ( ) s ( ) V ( ) e V ( ) e w ( s). For ay (8) le N be he swichig umber o () over [ ) deoe N as he swichig isas over he ierval [ ). he or [ ) V ( ) V ( ) is easily obaied rom ( ) ( ) (). From (7) we have V () () ( ) ( ) ( )( ) s ( ) e V ( ) e w ( s) ( )( ) e V ( ) ( )( s) ( ) ( s) ( ) e w () s ( ) ( ) ( ) ( ) e e V ( ) ( ) e w () s ( ) ( s) e N N ( s) () () ( ) N ( s) ( s) e w () s () N ( )( s ) ( s) e w () s () N N () () e w () s e V () e w ( s) w () s e V e w s N e V N e V () d Noig he deiiio o V () () (7) we have V x ( ) ( ) ( ). (9) () V () x () e su { x ( ) } () 3 e 4 x su { ( ) } ( e e ) su { x ( ) } 3 4 e e. 3 4 From (9)-() we obai l ( ) x ( ) e e 3 e 4 d. () ll Righs Reserved 7 IJRCE 35

4 36 Ieraioal Joural o dvaced Research i Comuer Egieerig & echology (IJRCE) Volume 6 Issue 9 Seember 7 ISSN: Subsiuig () io () oe has () x ( ). (3) ccordig o Deiiio 5 we coclude ha he sysem (3) is FB wih resec o ( d ( )). Nex we will give he guaraeed cos value o sysem (3). ( ) w( ) x ( ) R ccordig o (5) deoig iegraig boh sides o (6) rom o or [ ] i gives rise o ( ) ( s) ( ) ( ) V ( ) e V ( ) e ( s) (4) Similar o he roo rocess o (9) or ay [ ] we ca obai N ( )( ) ( ) () V ( ) e V () From (5) we ca ge N ( )( s ) ( s) ( s) N ( )( s ) e e x () s R () s N ( ) ( ) N ( ) ( s ) ( s) () e V () e w( s) Mulilyig boh sides o (6) by N ( )( s) ( s) e x () s R N ( ) ( ) N ( ) ( s) ( s) () lea o e V () e w( s) (5) (6) (7) Noig ha ( ) s l N () s we obai s s ha N () ( s) ha is e s l N ( ) ( s ). he (7) ca be ured io s ( s) N ( ) ( s ) ( s) ( ) ( ) Le e e x s R e x s R ( s) () () ( ) e V e w s lea o V he mulilyig boh sides o (8) by s s () () (8) e e x ( s) R V () e w( s) (9) () w( s) Subsiuig () io (9) yiel e x ( s) R V () () d (3) which ca be rewrie as x ( s) R e V () () d (3) Subsiuig () io (3) he guaraeed cos value o sysem (3) is give by () (3) e ( e 3 e 4 d) J x s R J hereore accordig o Deiiio 7 we ca coclude ha he claim o he heorem is rue. hus he roo is comleed. Remar. Geerally whe we cosider he asymoic sabiliy V () () is required o be egaive. s he dierece bewee asymoic sabiliy FB he resricio is relaxed i he roo rocess o heorem. Moreover i i () he oe ca obai which meas ha he swichig sigal ca be arbirary. B. Coroller desig I his secio we cocer wih he GCFB coroller desig o osiive swiched delay sysem (). Uder he coroller u( ) K () x( ) he corresodig closed- loo sysem is give by x ( ) ( ( ) G ( ) K ( ) ) x( ) d ( ) x( d( )) B ( ) w( ) x( ) ( ) [ ]. (33) By Lemma o guaraee he osiiviy o sysem (33) G K should be Mezler marices S. he ollowig heorem gives some suicie codiios o guaraee ha he closed-loo sysem () is GCFB. heorem. Cosider he osiive swiched sysem (). For give cosas vecors R R 4 i here exis a se o osiive vecors S osiive cosas 3 such ha (7)-(9) he ollowig codiios hold: G K are Mezler marices. (34) diag{ } (35) where ' ' ' a R g r r r r r r a ( h) max{ } g K G + R S he rh elemes o vecor ' r dr r g r rereses g a ( a ) rereses he rh colum vecor o he marix ( d ) [ ] [ ] [ ] r r r rereses he rh elemes o he vecors resecively saisies () he uder he D scheme () he resulig closed-loo sysem (33) is GCFB wih resec o ( d ( )) he guaraeed cos value o sysem (33) is give by r dr

5 Ieraioal Joural o dvaced Research i Comuer Egieerig & echology (IJRCE) Volume 6 Issue 9 Seember 7 ISSN: J x ( s) R x ( s) K R ( ) 3 4 J e ( e e d) (36) Proo. By Lemma we ow ha S GK is a Mezler marix or each S. ccordig o Lemma he sysem (33) is osiive i oegaive. Relacig leig g K G + R d B G are all i (6) wih GK similar o heorem we easily obai ha he resulig closed-loo sysem (33) is GCFB wih resec o ( d ( )) he guaraeed cos value is give by (36). he roo is comleed. Nex a rocedure is reseed o obai he eedbac gai marices K S. Se. By adjusig he arameers solvig (7)-(9) () (35) via liear rogrammig osiive vecors g ca be obaied. Se. Subsiuig g K G +R K ca be obaied. Se 3. he gai K is subsiued io GK. I g io G K are Mezler marices he K are admissible. Oherwise reur o Se. IV. EXMPLE I his secio a racical rice dyamics examle is rovided o show he eeciveess o he roosed aroach. I a ecoomic sysem comosed o goo xˆ () deoes he rices o he -h good a ime suose ha he dem i he suly g i o good deed o he curre rices x ˆ( ) he revious rices xˆ( d( )) o all goo where xˆ ( ) [ xˆ ( ) xˆ ( )]. We have xˆ ( ) ( ˆ( ) ˆ( ( ))) ( ˆ( ) ˆ i x x d gi x x( d( ))) i is sigiica o lear wheher he rices ca ed o cosa values x. Here i is assumed ha dem suly ucios are liear he iluece coeicies o x () o ( g ) are egaive (oegaive) while he iluece i i coeicies o xˆ () o iq( g iq ) or q are oegaive (egaive); he iluece coeicies o xˆ ( d( )) o ( g ) are osiive (egaive) or all q. i i he by meas o a aroriae rasormaio rice dyamics ca be described by he sae equaios xˆ ( ) ( ˆ( ) ( )) ( ˆ i x x di x( d( ) x). Leig x( ) xˆ ( ) x he above sysem is described as x ( ) x( ) x( d( ). i which ca be see as osiive swiched sysem () wih ime-varyig delay. Cosider rice dyamic model described by swiched osiive sysem () wih he arameers: d.8.6 B..3 G d B G R R..3.. ssumig ha d( )..si( ) we ca ge.4 h.. Choosig he arameers = =..4 solvig he iequaliies i heorem by liear rogrammig we ge g g g K G + R we obai By K K di I is easy o veriy ha (34) is saisied. he accordig o () we ge Le si( ) () 3 w( ) ( ) ( ) () ohers. Choosig 3.5 he simulaio resuls are show i Figs. -3 where he iiial codiios o sysem () are x() [.5.] x( ) [ ) which mee he codiio su { x ( ) }. he sae rajecory o he closed-loo sysem is show i Fig.. he swichig sigal () is deiced i Fig.. Fig. 3 los he evoluio o x () which imlies ha he corresodig closed-loo sysem is GCFB wih resec o ( d ( )) he cos value J which ca be obaied by (36). ll Righs Reserved 7 IJRCE 37

6 38 Ieraioal Joural o dvaced Research i Comuer Egieerig & echology (IJRCE) Volume 6 Issue 9 Seember 7 ISSN: Fig.. Sae rajecories o closed-loo sysem () Fig.. Swichig sigal o sysem () Fig. 3. he evoluio o x () o sysem () V. CONCLUSION I his aer we have sudied he roblem o guaraeed cos iie-ime corol or osiive swiched delay sysems wih D. ovel guaraeed cos erormace idex is iroduced. Based o he D aroach a sae eedbac coroller is cosruced o guaraee ha he closed-loo sysem is GCFB he obaied suicie codiios ca be solved by liear rogrammig. Fially a examle is give o illusrae he eeciveess o he roosed mehod. REFERENCES [] M. Xiag Z. Xiag Robus aul deecio or swiched osiive liear sysems wih ime-varyig delays IS rasacios vol. 53 o [] Z. Zhao L. Zhag P. Shi Sabiliy o a class o swiched osiive liear ime-delay sysems I J Robus Noliear Corol vol. 3 o [3] J. Dog Sabiliy o swiched osiive oliear sysems I J Robus Noliear Corol vol. 6 o [4] M. Xiag Z. Xiag Sabiliy -gai corol syhesis or osiive swiched sysems wih ime-varyig delay Noliear alysis: Hybrid Sysems vol. 9 o [5] X. Liu C. Dag Sabiliy aalysis o osiive swiched liear sysems wih delays IEEE ras. uoma. Corol vol. 56 o [6] M. Xiag Z. Xiag Observer desig o swiched osiive sysems wih ime-varyig delays Circuis Sys Sigal Process vol [7] R. Shore F. Wirh D. Leih osiive sysems model o CP-lie cogesio corol: asymoic resuls IEEE/CM ras. New. vol. 4 o [8] O. Maso R. Shore O liear coosiive Lyauov ucios he sabiliy o swiched osiive liear sysems IEEE ras. uoma. Corol vol. 5 o [9] J. Liu J. Lia Y. Zhuag Ouu eedbac iie-ime corol o swiched osiive delayed sysems wih MDD Noliear alysis: Hybrid Sysems vol [] S. Liu Z. Xiag Exoeial ouu racig corol or osiive swiched liear sysems wih ime-varyig delays Noliear alysis: Hybrid Sysems vol [] L. Gurvis R. Shore O. Maso O he sabiliy o swiched osiive liear sysems IEEE ras. uoma. Corolvol. 5 o [] D. Peer Shor ime sabiliy i liear ime-varyig sysems I Proc. IRE I. Coveio Record Par [3] F. mao M. riola P. Dorao Fiie-ime corol o liear syems subjec o arameric uceraiies disurbaces uomaica vol. 37 o [4] Z. Xiag Y. N. Su M. S. Mahmoud Robus iie-ime corol or a class o ucerai swiched eural sysems Commu Noliear Sci Numer Simula vol. 7 o [5] H. Liu Y. She Fiie-ime corol or swiched liear sysems wih ime-varyig delay Iel Corol uo vol. o [6] Y. Che Q. Liu R. Lu. Xue Fiie-ime corol o swiched sochasic delayed sysems Neurocomuig 6 9: [7] Y. Ma L. Fu Y. Jig Q. Zhag Fiie-ime corol or a class o discree-ime swiched sigular ime-delay sysems lied Mahemaics Comuaio vol [8] L. Zhu Y. She C. Li Fiie-ime corol o discree ime sysems wih ime-varyig exogeous disurbace Comm Noliear Sci Numer Simula vol [9] Q. Meg Y. She Fiie-ime corol or liear coiuous sysem wih orm-bouded disurbace Comm Noliear Sci Numer Simula vol [] G. Che Y. Yag Fiie-ime sabilizaio o swiched osiive liear sysems I J Robus Noliear Corol vol. 4 o [] J. Zhag X. Zhao Y. Che Fiie-ime sabiliy sabilizaio o racioal order osiive swiched sysems Circuis Sysems Sigal Processig vol. 35 o [] M. Xiag Z. Xiag Fiie-ime corol or osiive swiched liear sysems wih ime-varyig delay Commu. Noliear Sci. Numer. Simula vol. 8 o [3] L. Wu Z. Wag Guaraeed cos corol o swiched sysems wih eural delay via dyamic ouu eedbac Ieraioal Joural o Sysems Sciece vol. 4 o [4] P. Niamsu K. Rachagi V. N. Pha Novel crieria or iie-ime sabilizaio guaraeed cos corol o delayed eural ewors Neurocomuig vol [5] L. Liu M. Cheg M. Xu Guaraeed cos iie-ime corol or sochasic diereial iclusio sysems IEEE ICI [6] L. Zhag X. Wag K. Zhag No-ragile iie-ime guaraeed cos uzzy corol or coiuous-ime oliear sysems Ieraioal Joural o Comuaioal Ielligece Sysems vol. 7 o [7] X. Cao L. Liu Z. Fu e al. Guaraeed cos iie-ime corol or osiive swiched liear sysems wih ime-varyig delays Joural o Corol Sciece Egieerig ID