Impulsive Tracking Control for Non-measurable State with Time-delay

Transcription

1 JOURNAL OF NEWORKS VOL 8 NO 4 APRIL 3 85 Impulsive racig Corol for No-measurable Sae wih ime-delay Yuaiag Che College Guizhou izu Uiversiy Guiyag555 Chia yuaiagc@6com Rebi ia Deparme of Educaio Sciece ogre Uiversiy ogre5543 Chai @com Absrac I his paper we discuss he problem by uilizig impulsive corol ad Lyapuov fucio meho which is abou he sae of disurbed sysems wih ime-delay racig he sae of referece sysems Sufficie codiios for he solvabiliy of he racig corol problem are give for he measurable sae ad he o-measurable sae of sysem respecively his impulsive corol law based o measured oupu isead of he sae iformaio is cosidered Fially a umerical example is preseed o illusrae he validiy of our resuls Idex erms Impulsive corol Sae racig Nomeasurable sae I INRODUCION he research of he racig corol wihou ime-delay []-[3] is uie maure I may evoluioary sysems here are wo commo pheomea: delay effecs ad impulsive effecs I implemeaio of elecroic sysem for example delays freuely appear because of he fiie swichig speed of amplifiers O he oher had he sae of elecroic sysem is ofe subjec o isaaeous perurbaios ad experiece abrup chage a cerai isas which may be caused by swichig pheomeo freuecy chage or oher sudde oise ha is do exhibi impulsive effecs Eve i biological sysem impulsive effecs are liely o exis For isace whe a simulus from he body or he exeral evirome is received by recepors he elecrical impulses will be coveyed o he eural e ad impulsive effecs arise aurally i he e herefore he racig corol sysem wih delays ad impulsive effecs should be more accurae o describe he evoluioary process of he sysems Sice delays ad impulses ca affec he dyamical behaviors of he sysem by creaig oscillaory ad usable characerisics i is ecessary o ivesigae boh delay ad impulsive effecs o he sabiliy of racig corol sysem he delayig ime which is he racig sigal s rasmissio ad ifluece o realizig of racig performace is very big herefore o cosider he racig problem wih ime-delay is more sigificace i has already bee researched recely for example [4]-[5] I fac he racig sysem is disurbed by exeral evirome So he racig corol problem which coais he delayig ime ad disurbace of he exeral evirome is more comprehesive Impulsive corol mehod [6]-[6] has araced cosiderable aeio because impulsive corol laws have fas respose ime low eergy cosumpio good robusess ad resisace o disurbaces Bu sudies of he racig corol specially he racig corol wih ime-delay ad disurbace by he impulsive corol is uie rare ow I his paper we give a referece liear sysem x r ( = Ar ( ( ( = Ad a racig liear sysem wih ime-delay ad disurbace x = Ax Bx ( hσ ( wx ( y ( = Cx( ( x = φ ( [ h ] Ad he sae racig performace idex lim x x = (3 where ( r x R is he sae variable of he referece sysem ad racig sysem respecively y R is oupu A ABad C are cosa marices m r of appropriae dimesios ( wx is bouded exeral disurbace which is he coiuous vecor valued fucio hi ( is he delay ime h h i = ad i x( hσ ( = x( h( x( h( x( h( We will adop he impulsive corol mehod ad he Lyapuov fucioal mehod ad he marix ieualiy echology o solve he sae racig sysem wih imedelay ad disurbace for he cases ha he sae of sysem is o-measurable ad give he sufficie codiios for he realizig of he racig corol performace he res of his paper is orgaized as 3 ACADEY PUBLISHER doi:434/jw

2 85 JOURNAL OF NEWORKS VOL 8 NO 4 APRIL 3 follows I Secio II some prelimiary lemmas are preseed I Secio III based o he impulsive corol mehod ad he Lyapuov fucioal mehod ad he marix ieualiy echology sufficie codiios for he solvabiliy of he racig corol problem are give for he measurable sae ad he o-measurable sae of sysem respecively oreover a umerical example is preseed i Secios IV Secio V cocludes he paper II PRELIINARIES < deoes a posiive defiie (semi-defiie egaive defiie semi-egaive defiie P λ P are respecively he I his paper P > ( marices P λ ad m larges ad he smalles eigevalue of P deoes he orm i R ad K deoes he se of coiuous fucios PC ( R R is he se of all piecewise coiuous fucios p: R R such ha p PC R R if P where p: R R is coiuous ( o R excep a he ime pois i he se { } ad is lef-coiuous ad has righ limi a for all We fis iroduce some prelimiary coceps which will be foud useful i he paper Cosider he followig impulsive corol sysem wih ime-delay: x = f ( x x ( h x = x x = u x = φ [ ] where ( f u C R R R x = h = (4 while < < < < wih as Defiiio For each ρ > defie Sρ { x R : x ρ} Ad for ( x ( ] R = < = le DV( x = limsup V( hx hfx ( V( x h h Defiiio Le V be he se coaiig all V : R S R which are coiuous fucios ρ o R S ρ of pois ad saisfy he followig wo codiios: ⅰ For each x S = excep possibly a a seuece { } lim V y = V x exiss; ( y ( x ⅱ V( x is locally Lipschiz i x he followig lemma gives sufficie codiios for asympoic sabiliy of sysem (4 Lemma[7] Assume ha here exis αβγ g K ( V( x V p PC R R ad σ > such ha he followig codiios are saisfied β x V x α x ( x [ h S ρ ; ⅰ ρ ( ϕ ϕ ( ( ϕ( R PC ( h S ρ ; ⅱ V u ( gv ( ϕ [ ] ⅲ ( ϕ( γ ( ( ϕ( d ϕ PC ([ h S ρ whe ( ϕ ( ϕ D V p V R a [ V g V s s s h ; ⅳ G = if > sup p ( s = G > γ s ; = ϕ { } g = sup < he he sysem (4 is asympoically sable Lemma[8] Le N be real marices of appropriae dimesios he for ay marix S > of appropriae dimesio ad ay scalar γ > he followig ieualiy hol N N γ S γn SN where ϕ( ( III AIN RESULS I his subsecio we fis cosider he case ha he sae of sysem ( is measurable Uder he impulsive u x u x = Q x Qx coroller { } where r r ad QQ r R he sysem ( ca be rewrie as x = Ax Bx h wx ( x Qx r r ( Qx = Cx( = φ [ ] = = = (5 y x h We have he followig heorem heorem Suppose ha P> ( x lx w µ x he he sae of sysem ( racs ad asympoically ad impulsively ha of he referece sysem ( if he followig eualiy hol where µ λ ( P < e < e < (6 ( G( l λ ( P λ = λ ( F m F = I Q P Q PQ G = Q PQ E = A P A P BP B P r r Proof Cosider he followig Lyapuov fucio cadidae ( V x = x Px (7 I clearly saisfies codiio (ⅰ of Lemma A = = we have ( = ( ( = ( x( x( P( x( x( V x x Px 3 ACADEY PUBLISHER

3 JOURNAL OF NEWORKS VOL 8 NO 4 APRIL ( ( ( = x P PQ Q PQ X ( ( x PQ Q PQ x x r r r ( Q PQ x ( r r r r ( ( x P PQ Q PQ x x ( Q PQ x ( r r r r ( G( l λ ( P λ λ ( F V ( x m = V ( x hus codiio (ii of Lemma is saisfied wih g( s = s By virue of he Lyapuov fucio (7 upper Dii derivaive alog he soluio of sysem (5 i follows ha a = we have ( = DV x x Px x Px = x AP PAx x PBx h x Pw x AP PAx x Pw x PBP B Px V µ x ( AP PAx P V x PBP B Px V µ λ V ( ( P hus codiio (iii is saisfied wih γ ( s = sad p( = λ µ ( P For ε < we have he followig wo iegrals ε µ G = sup p ( s = λ λ m ( P Ad G = if l > = g( γ ( s By he ieualiy (6 he followig ieualiy hol G > G hus codiio (iv of Lemma is also saisfied herefore he sysem (5 is asympoically sable lim x = By he ieualiy We have ( x lx we ow lim ( ( x = he he sae of sysem ( racs asympoically ad impulsively ha of he referece sysem ( For he followig racig sysem he each sae variable of which has o same delay ime x = Ax Bx ( hσ ( wx ( y ( = Cx( (8 x = φ ( [ h ] Where x h ( σ x( h( x( h( x( h( = We have he followig corollaries from Defiiio ad heorem Corollary Le hi ( h ( i = uder he impulsive corol law { Qr ( Qx( } he sae of sysem (8 racs asympoically ad impulsively ha of he referece sysem ( if he sysem corolled saisfies he codiios of heorem Proof Cosider he followig Lyapuov fucio cadidae V ( x = x Px I clearly saisfies codiio (ⅰ of Lemma A = = we have = = ( x( x( P( x( x( = x( ( P PQ Q PQ X ( x( ( PQr Q PQr ( ( Qr PQr ( x( ( P PQ Q PQ x( ( Qr PQr ( λ ( G( l λ ( P ( ( ( V x x Px F V x = V ( x hus codiio (ii of Lemma is saisfied wih g( s = s By virue of he Lyapuov fucio upper Dii derivaive alog he soluio of sysem (8 i follows ha a = we have DV x = x Px x Px ( σ = x AP PAx x PBx h x Pw x AP PAx x Pw x PBP B Px V µ x ( AP PAx V ( P 3 ACADEY PUBLISHER

4 854 JOURNAL OF NEWORKS VOL 8 NO 4 APRIL 3 x PBP B Px V µ λ ( P V ( hus codiio (iii is saisfied wih p( = λ µ λ ( P ad ( s m γ = s For ε < we have he followig wo iegrals ε µ G = sup p ( s = λ λ m ( P Ad G = if = l γ > g( ( s By he ieualiy (6 he followig ieualiy hol G > G hus codiio (iv of Lemma is also saisfied herefore he sysem corolled is asympoically sable lim x = By he ieualiy We have ( x lx we ow lim ( ( x = he he sae of sysem (8 racs asympoically ad impulsively ha of he referece sysem ( Remar I view of heorem ad is corollary we see ha uder he corol law { Qr ( Qx( } if he sysem ( ad (8 saisfy he codiios of heorem he sae racig performace idex ca realize Now we ivesigae he possibiliy of desigig impulsive corol law based o measured oupu isead of he sae iformaio Cosider he sae esimaor of sysem ( give by x = Ax Bx h LC x x w x ( σ ( ( = Cx ( = φ [ ] y x h Where x( R is he sae esimaor of oupu feedbac gai marices C( RR (9 x L is he φ Defie he differece bewee he real sae ad he esimaor sae as e = x x From ( ad (9 we have e = A LCe Beh y = Ce e = φ φ [ ] e h ( For he asympoic sabiliy of sysem ( we have he followig heorem heorem he sysem ( ad sysem (9 is asympoically sychroous if here exiss oupu feedbac marix L such ha he followig eualiy hol J = A A I LC C L BB < Proof Cosider he followig Lyapuov fucio cadidae ( V e = e e es es h Obviously V( e > By virue of he Lyapuov fucio V( e upper Dii derivaive alog he soluio of sysem ( we have V e = e e e e e e ( e ( h e ( h = e A A ILCCL e e Beheh eh e A A ILC CL BB e From he codiio of heorem we have V e < ( hus he sysem ( is asympoically sable Now we will solve problem is he sae of esimaor sysem (9 racs asympoically he oe of referece sysem ( If he differece bewee he real sae ad he esimaor sae is see as he exeral disurbace of sysem (9 he (9 ca be rewrie as x = Ax Bx h w Where ( = Cx ( = φ [ ] y x h ( w = LCe w For sysem ( uder he impulsive corol law Q x ( Qx ( we have he followig resul { r r } heorem3 x x l x ha w µ x r Suppose x( x( m x( ad P > if here exiss oupu feedbac marices L such ha he followig wo eualiy hold ⅰ J = A A I LC C L BB < ( λ ⅱ < e < e E < ( he he sae of sysem (9 racs asympoically ad impulsively ha of he referece sysem ( Where λ ( G( l = λ ( F F = I Q P Q PQ ( P G = Q PQ r r λ µλ ( E = m C L PLC P P A P A P BP B P Proof Cosider he followig Lyapuov fucio cadidae 3 ACADEY PUBLISHER

5 JOURNAL OF NEWORKS VOL 8 NO 4 APRIL ( V x = x Px (3 I clearly saisfies codiio (ⅰ of Lemma A = = we have V ( x x ( ( P PQ Q PQ x ( x ( Q PQ x ( r r r r ( x ( G( l λ ( P λ λ ( F V x m = V ( hus codiio (ⅱ of Lemma is saisfied g s = s wih By virue of he Lyapuov fucio (3 upper Dii derivaive alog he soluio of sysem ( i follows ha a = we have DV( x ( λ µλ x A P PA PBP B P x x m C L PLC I P I x V λ ( E V ( γ s = sad hus codiio (ⅲ is saisfied wih p( = λ E For ε < we have he followig wo iegrals Ad ε G = sup p ( s = λ G = if = l γ > g( ( s By he ieualiy ( he followig ieualiy hol G > G hus codiio (ⅳ of Lemma is also saisfied herefore he sysem corolled ( is asympoically sable lim x = By he ieualiy We have ( x( l x( we ow lim ( ( x( = he he sae of sysem (9 racs asympoically ad impulsively ha of he referece sysem ( Remar I view of heorem 3 we see ha whe he sysem ( is asympoically sable we ca adop he esimaed sysem (9 o realize he sae racig performace idex by impulsive corol { r r } law Q x ( Qx ( Remar3 heorem ad 3 show ha our desiged impulsive coroller for he realizaio of sae racig performace relaes wih he referece sysem ad he racig sysem IV NUERICAL EXAPLE I his secio we will cosider wo examples o illusrae he resuls obaied i Secio 3 Cosider he referece sysem ( ad he racig sysem wih he followig daa: 5 5 A r = A = 3 3 B = 4 C = diag ( w( = x( x( x( x( 3 3 x = x r = ( x ( = ( 54 ( ( φ ( = ( 5 6 φ = ( ( Fig he ime seuece char for sysem ( x( x( Fig he ime seuece char for sysem ( Figure ad Figure is he ime seuece char of he referece liear sysem ( ad he racig liear sysem wih ime-delay ad disurbace ( respecively Figure3 is he ime seuece char of he error sysem bewee sysem ( ad sysem ( ad Figure4 is he ime seuece char of he error sysem bewee sysem ( ad sae esimaor sysem of sysem ( hey show ha sysem ( or sae esimaor sysem of sysem ( ca rac asympoically sysem ( wihou impulsive corol Figure5 is he ime seuece char of he error sysem bewee sysem ( ad is sae esimaor sysem 3 ACADEY PUBLISHER

6 856 JOURNAL OF NEWORKS VOL 8 NO 4 APRIL 3 Because his error sysem is asympoically sable our esimaig for sysem ( is very efficacious e( e( Fig3 he ime seuece char for error sysem of ( ad ( e( e( Fig4 he ime seuece char for error sysem of ( ad ( e( e( Fig5 he ime seuece char for error sysem of ( ad ( e( e( Fig6 he ime seuece char for corolled error sysem of ( ad (9 Whe he sae of racig sysem is measurable obviously µ = We ca ge he approximae value of l hrough simulaios we obai l = We le Qr = diag ( l ( l Q = diag 3 3 P = I = he we have = 7 λ ( E µ = Obviously = 7 < e = 935 Coseuely i follows from heorem ha he sae of sysem ( racs asympoically he oe of sysem ( uder he impulsive corol law { Qr ( Qx( } Whe he sae of racig sysem is o measurable obviously µ = m = we ca ge he approximae value of l hrough simulaios we obai l = We le Qr = diag 4 4 Q = diag 3 3 P = I = = ad he oupu feedbac gai marices cadidae L diag ( 63 5 have = he we ( J 5499 λ = < ad λ ( E Obviously 7 < e 786 Coseuely i follows from heorem ha he sae of sysem ( racs asympoically he oe of sysem ( uder he impulsive corol law { Qr ( Qx( } he sae racig char is show i Figure6 I is clear ha he sae of he sae esimaor sysem of sysem ( racs beer he sae of he sysem ( afer 5 secod V CONCLUSION We proposed a desig mehod for impulsive coroller for he realizaio of sae racig performace i his paper From he umerical example solved usig his desig mehod we see ha his impulsive coroller is effecive for he racig sysem wih ime-delay ad disurbace which sae is measurable or o-measurable ACKNOWLEDGEN his wor was parially suppored by he Naioal Naural Sciece Foudaio of Chia uder Gra 779 ad he Naural Sciece Foudaio of Guizhou Provice uder Gra LK []3 ad he Cosrucio Projecs of Key Laboraory abou Paer Recogiio & Iellige Sysems of Guizhou Provic uder Gra [9]4 ad he Graduae Educaio 3 ACADEY PUBLISHER

7 JOURNAL OF NEWORKS VOL 8 NO 4 APRIL Iovaio Bases abou Iformaio Processig & Paer Recogiio of Guizhou Provic REFERENCES [] G A Rovihais racig corol of muli-ipu affie oliear dyamical sysems wih uow olieariies usig dyamical eural ewors IEEE ras Sysems a ad Cybereics-Par B vol 9 pp [] W E Schmiedorf eho for obaiig robus racig corol laws Auomaicvol 3 pp [3] C J Zhao K Ogaa S Fuji Adapive swichig corol mehod ad is applicaio o racig corol of a robo i Proceedigs of he 996 IEEE IECON d Ieraioal Coferece o Idusrial Elecroics Corol ad Isrumeaio pp [4] J H Par J Robus sabilizaio for dyamic sysems wih muliple ime-varyig delays ad oliear uceraiies J Opim heory Appl vol 8 pp55-74 [5] H L Xu H P Che X JDig Impulsive sablilizaio of cellular eural ewors wih ime delays Advaces i Neural Newors vol 4 pp [6] Yag Impulsive corol IEEE rasacios o Auomaic Corol vol 44 pp [7] Yag L Yag C Yag Impulsive corol of Lorez sysem Physica D vol pp [8] R Z Luo Impulsive corol ad sychroizaio of a ew chaoic sysem Aca Physica Siica vol 56 pp [9] X Z Liu K L eo Impulsive corol of chaoic sysem Ieraioal Joural of Bifurcaio ad Chaos vol pp8-9 [] G Q Liu S X Yag W Fu New Robus Sabiliy of Ucerai Neural-ype Neural Newors wih Discree Ierval ad Disribued ime-varyig Delays Joural of Compuers vol7 pp64-7 [] D G Yag C Y Hu Global Asympoic Sabiliy Crieria of Bam Neural Newors wih ime Delays Ieraioal Joural of Advacemes i Compuig echology vol4 pp9-6 [] Y Q Zhag C X Liu H Corol of Fuzzy Impulsive Sysems wih Quaized Feedbac Joural of Sofware vol4 pp [3] X F Wu C Xu J Feg ea sychroizaio of piig complex ewors wih liearly ad oliearly ime-delay couplig Ieraioal Joural of Digial Coe echology ad is Applicaios vol 5 pp [4] Y Q Ji J w Lei Y Liag racig of Super Chaoic Sysem wih Saic Ucerai Fucios ad Uow Parameers Joural of Compuers Vol 7 pp [5] W Li J Cao D Wu uli-feaure Fusio racig Based o A New Paricle Filer Joural of Compuers Vol 7 pp [6] J Wu Y G Wag J Huag H Y Zhou Noliear Ieral odel Corol Usig Echo Sae Newor for Peumaic uscle Sysem Joural of Compuers Vol 7 pp [7] J J Du C Y Sog uli-pi Corol for Blocsrucured Noliear Sysems Joural of Compuers Vol 7 pp [8] B Whiehead C H Lug P Rabiovich racig Per- Flow Sae Bied Duraio Flow racig Joural of Newors Vol 7 pp [9] X Wu A Disribued rus Evaluaio odel for obile PP Sysems Joural of Newors Vol 7 pp [] Y Gao Wu W F Du Performace Research of odulaio for Opical Wireless Commuicaio Sysem Joural of Newors Vol 6 pp 99-8 [] X Wu P H Lou D B ag uli-objecive Geeic Algorihm for Sysem Ideificaio ad Coroller Opimizaio of Auomaed Guided Vehicle Joural of Newors Vol 6 pp [] H Li Y HShe K Xu Neural Newor wih omeum for Dyamic Source Separaio ad is Covergece Aalysis Joural of Newors Vol 6 pp [3] A W Zheg Y H Gao Y P a J P Zhou Sofware Desig ad Developme of Chag E- Faul Diagosis Sysem Joural of Sofware Vol 7 pp [4] S b Hu H Shu c Li Aalysis o Sabiliy of a Newor Based o RED Scheme Joural of Newors Vol 6 pp [5] J Shao Z Jia Z P Li F Q Liu J W Zhao P Y Peg A Closed-loop Bacgroud Subracio Approach for uliple odels based uliple Objecs racig Joural of ulimedia Vol 6 pp [6] Gao P Wag C S Wag Z J Yao Feaure Paricles racig for ovig Objecs Joural of ulimedia Vol 7 pp [7] X Liu G Balliger Uiform asympoic sabiliy of impulsive delay differeial euaios Compuer ad ahemaics wih Applicaios vol 4 pp [8] Y Cao Y Su C Cheg Delay depede robus sabilizaio of ucerai sysems wih muliple sae delays IEEE ras Auoma Cor vol 43 pp Yuaiag Che bor i 976 a Associae Professor wih he College of Guizhou izu Uiversiy Chia He is a seior member of he Guizhou Provic Isiue of Sysems Egieerig He received he BSdegree i ahemaics ad Applied ahemaics from Guizhou izu Uiversiy Chia i 999 he Sdegree i Operaios research ad Corol heory from Guizhou Uiversiy Chia i 7 He curre research fiel focus o complex dyamical sysems smar power sysems oliear impulsive corol ad eural ewors He has over papers published i ieraioal peer reviewed jourals ad presided five research projecs Rebi ia bor i 978 lecurer wih he Deparme of Educaio Sciece of ogre Uiversiy Chia She is a member of he Basic educaio research ceer of Guizhou Provic She received he BS degree i ahemaics ad Applied ahemaics from Guizhou izu Uiversiy Chia i 3 Her mai research ieres icludes sesor ewors ad oliear corol 3 ACADEY PUBLISHER