Hybrid System Modeling Using Impulsive Differential Equations

Transcription

1 Hybrid Syse Modelig Usig Ipulsive Differeial Equaios ENDRA JOELIANO Depare of Egieerig Physics Isiu eologi Badug Jl Gaesha 1, Badug 413 INDONESIA Absrac: Hybrid syses have eerged as a rapidly expadig field for odelig ad aalyzig syses which have a ixure of coiuous ad discree valued sae variables he odelig fraewor of hybrid syses has received cosiderable aeio over he pas few years he ai is o build a aheaical odel which is suiable for coplex dyaical aalysis ad corol syhesis usig hybrid syses his paper shows ha he ipulsive differeial equaios ay provide a uified fraewor for hybrid syse odelig Key-words: Hybrid Syses, Ipulsive Differeial Equaios, Syse heory 1 Iroducio Hybrid syses are used for odelig ad aalyzig syses which have ieracig coiuous-valued ad discree-valued sae variables he coiuous sae variable ay be he value of he sae i coiuous ie, discree ie or a ixure of he wo he aheaical odel of he coiuous sae is described by a differeial or differece equaio he discree sae variable is geerally represeed by a fiie sae digial auoao or a ipu/oupu rasiio syse he behaviour of he hybrid syse is iflueced by sae variables which ierac a a eve or rigger ie which occurs wheever he evoluio of he syse saisfies a paricular codiio which he iiiaes chages i he sae variables Hybrid corol syses are corol syses where boh pla ad coroller cosis of coiuous ad discree sae variables Recely, a coo fraewor used for a hybrid corol has bee developed by separaig he syse copoes io hree sub copoes; aely, a pla wih a coveioal coroller, a discree sae variable coroller ad a ierface [1] he hree layered cofiguraio is show i Figure 1 he pla ad he coveioal coroller are usually odeled by differeial or differece equaios he discree sae variable coroller is desiged via a rule based decisio process which supervises he coveioal coroller his ierface faciliaes couicaio bewee he pla ad he discree coroller ad siulaeously covers he coiuous sae o discree sae variables C/D, ad vice versa D/C Exaples of hybrid corol syses are foud i auooive egie corol, auoaed highway syses, flexible aufacurig, cheical process corol, elecric power disribuio ad copuer couicaio ewors Soe exaples of hybrid corol syses are discussed i, for exaple, [1] Discree valued sae D/C Coiuous corol Discree Coroller Ierface Pla ad Cove Coroller Discree valued sae C/D Coiuous oupu Fig 1 hree Layers Fraewor he odelig fraewor for hybrid syses is aied o build a aheaical odel which is suiable for coplex dyaical aalysis ad corol syhesis usig hybrid syses he odels proposed i he lieraure reflec a wide rage of boh applicaios ad usificaios A overview of hybrid syse odelig ca be foud

2 i [] I ha paper, a uified approach is also proposed which is a geeralizaio of five odels of hybrid corol syses developed fro syse ad corol perspecives he uified aheaical odel ries o capure all possible ipora aspecs of coiuous ad discree valued variables as well as heir ieracio Currely, here are wo paradigs i he heoreical fraewor of hybrid syses: aggregaio ad coiuaio he aggregaio approach ca be raced bac o he earlier develope of hybrid syses fro copuer sciece i he coex of auoaa heory Laer, syse heory showed ha coplex syses ca be siplified as hybrid syses i which he radiioal fraewor deals wih coiuous sae variables of he uderlyig syse I he aggregaio approach, he syse sae variables are reaed as a discree eve dyaic syse or a fiie auoao by aggregaig he coiuous valued sae variables via cell o cell pariio A heory of cell o cell appig as a global ehod for aalyzig oliear syses is available [3] O he oher had, he coiuaio approach supposes he whole syse o be described by eiher differeial or differece equaios I his approach, he discree valued sae variables are cosidered as uceraiy or disurbaces, or ebeddig he as up acios i ordiary differeial or differece equaios I he firs case, a appropriae hybrid coroller ca be desiged usig robus corol I he secod case, if he up exhibis chagig dyaics, he coroller ca be desiged usig a uli coroller desig approach or he up liear quadraic JLQ ehod [4] I applicaios, he progress of boh paradigs have bee ipeded by he coservais which is ebedded i he desig ehodology [] he aggregaio approach, o he oe had, is ofe faced wih he proble of choosig a appropriae pariio ehod ad a o deeriis of auoaa ha lead o udecidabiliy ad copuaio coplexiy O he oher had, he coiuaio approach is ofe liied by a coproise of desig requirees foud i coveioal corol desig hus, verificaio is eeded whe he resulig coroller fro oe approach is ipleeed i a real syse Effor is beig ade o brig abou a uified desig for boh coiuous ad discree valued corollers Syses wih Ipulsive Effecs Physical syses are ofe subeced o disurbaces, chagig operaio codiios ad copoe failures, ad i ay cases, he chages ae place i a shor space of ie Exaples are foud for exaple i biological syses ad echaical syses subeced o shoc Such syses ca be odeled by differeial ad/or differece equaios which up isaaeously fro oe sae o aoher If here is o up over soe ie ierval, he he aheaical odel is described by he soluio of a differeial ad/or differece equaio he aalysis of a isaaeous chage i he sae of a syse is uch ore coplicaed Maheaical odels of syses ha udergo isaaeous chages i he sae are called ipulsive syses here are wo ai approaches for sudyig he behaviour of ipulsive differeial syses he firs approach uses a geeralized fucio o represe a up discoiuiy i he sae wih he help of he Dirac fucio his approach was developed i [5] I he secod approach, he up discoiuiy is represeed by a ipulsive vecor which was iiiaed by [6] where he firs sabiliy resuls were obaied ad furher developed i [7] [8] ad he refereces cied herei he ipulsive vecor represeaio provides a geeral characerizaio of exeral disurbaces, perurbaios or eve ipulsive corols he ipulsive ordiary differeial equaios of ieres are hose characerized by liear syses he equaios sudied here are cosequely referred o as liear ipulsive differeial equaios, or siply as liear ipulsive syses LIS Cosider a iial order coiuous ie liear ie ivaria pla wih sae oupu y l R, ad ipu u R x R, subeced o ipulsive vecors { d ; Z } o he pla sae as described by = A Bu y = C = d ; 1 where boh he ies ad values d are uow he ipulsive vecor ay represe he exeral disurbaces, failure of he syse s copoes or a ipulsive corol

3 he soluio of syses wih ipulsive effecs 1 i he exeded sae space begis fro he iiial codiio, ad oves alog he x raecory, If a ie isas here is a ipulsive up d, he he sae is isaaeously chaged o he ew sae x = d he sae he follows he raecory wih he ew iiial codiio uil he occurrece of he ex rasiio ie isa a ie 1 ha is, he soluios of ipulsive syses are characerized by hree copoes: he dyaics of ordiary differeial equaios, he rasiio ie isas ad he ipulsive vecors {d } 1 Ipulsive Vecors Ope loop ipulsive vecor I his case, he ipulsive vecor d is a arbirary vecor i R which occurs a ie, or he oupu of he syse his so called ope loop ipulsive vecor arises i aheaical odels of physical syses as a resul of: exogeous disurbaces, failure of syse copoes or ope loop ipulsive corols Applicaios of ope loop ipulsive corol ca be foud i drug aagee i he hua body [9] ad opial space raecories probles [1] Closed loop ipulsive vecor Z which is idepede of eiher he sae I his case, he ipulsive vecor d is depede o he sae or he oupu of he syse, ad ca be wrie i he for d = ψ where ψ is eiher a cosa or ie varyig arix his ype of ipulsive vecor is usually foud i corollig he behaviour of a liear syse i which iforaio abou he syse is beig used o effec he raecory Applicaios of closed loop ipulsive vecor ca be foud i he pulse frequecy odulaio syses sudied i [11] rasiio ie Isas he rasiio ie isas are defied as he ies whe ipulsive vecors occur he occurrece of ipulsive vecors ay be he resul of exeral disurbaces, syse copoe failures, cloc iig, or a logical decisio he ie isa also defies a eve or a rigger ie which represes a discoiuiy i he sae of he syse I addiio, a eve ay be used o cause aoher eve a soe ie i he fuure Followig [1], he ie isas will be classified i wo ways Scheduled ie isas he sequece of he scheduled ie isas is give by = 1 τ ; τ > where he value of τ is ow a priori for all Z I he siples case, τ = τ is cosa for all Z ha leads o uifor ie isas siilar o uifor saplig i digial syses Codiioed ie isas Codiioed ie isas occur if eiher he ie or he sae saisfies a paricular codiio he codiio ca be defied, for exaple, as = { : ς = ε, R } or = { : ς = ε,, R R } Aleraively, he codiio ca also be give i er of a se S R by = { : S, S R } 3 Ipulsive Dyaical Syses 31 Hybrid Syses Represeaio via LIS LIS odelig ca be exeded o cover he proble of syses wih swichig dyaics which is cooly foud i hybrid corol syse desig he aor develope of ipulsive dyaical syses is o capure he behaviour of a isaaeous up of sae of a dyaical syse However, ipulsive vecors ay also be used o describe a dyaical syse subeced o he oupu of a higher order odel Fro a syse ad corol perspecive, i is coo for he desig o be carried ou usig he coiuaio approach, sice corol syse desig requires a racable evoluio for boh syhesis ad assesse of he coroller I ers of discree variables, he coiuaio odel ca be characerized as follows: 1 Auooous or corolled swichig he vecor field of he coiuous dyaics chages discoiuously whe he sae saisfies soe cosrais Auooous or corolled ipulses he sae of he syse ups discoiuously o he saisfacio of soe give cosrais

4 I pracice, here igh oly be oe ype of discree variable prese If boh ypes are foud, we have he so called full power odelig of hybrid syses [] o illusrae, cosider he dyaics of a syse ha cosiss of swichig bewee wo dyaic syses accordig o = A ; i = 1, his syse ca be wrie i he for i z1 A1 z1 = z A z z z1 d1 1 = z z d = z z 1 where eiher d1 = z1 or d = z his choice of decisio vecors he iplies ha eiher x = z 1 or = z for < 1 Modelig swichig syses via syses wih ipulsive effec represeaio was firs observed i [] A auooous swichig ca be viewed as a special case of auooous ipulse by ebeddig he discree sae io a larger coiuous sae via he uiversal exesio propery of he phase space R [13] Rece papers o hybrid corol desig are relaed o probles of corollig syses wih swichig dyaics So far, lile aeio has bee paid o he proble of syses which are subeced o a isaaeous chage i he sae he capabiliy of a ipulsive differeial equaio o capure he odelig of swichig syses ay provide a uified fraewor for hybrid syses, ad prooe a ew direcio i aalyzig ad syhesizig hybrid corol syses 3 Fudaeal Properies Ipulsive differeial equaios are basically piecewise differeial equaios where he discoiuiies i he syse sae are caused by ups i he soluios Mos resuls of he heory of ipulsive ordiary differeial equaios have bee developed i [7] [8] ad he refereces cied herei bu he ivesigaio has bee liied o he case where he ipulsive vecors have a closed loop represeaio 31 Exisece ad Uiqueess of Soluios Le Ω R be a ope se, ad cosider a LIS of he for = A Bu ; = d ; = x = where he soluio x = ;, x Ω for he ipulsive vecor d ad ie isas, for each Z, are defied i he doai Ω which coais he se ς = {, x R Ω: < 1, Z ad x Ω} Auooous Liear Ipulsive Syses he syse is called a auooous syse if Bu = for all ad x R No auooous Liear Ipulsive Syses he syse is called a o auooous syses if Bu for all ad x R Noice ha he iiial codiio x = x is used raher ha x = x If he ie correspods o a rasiio ie isa he is udersood o be he iiial codiio of he ordiary differeial equaio he ie evoluio of liear ipulsive syses cosiss of coiuous ad up discoiuous fucios Codiio 1 Le Ω be a ope se where ad 1 < < < L < < L 1 Ω R d are bouded ipulsive vecors such ha x = d Ω he he oepy ς is defied by 1 ς = {, x R Ω : < 1, x Ω} he soluio x of is coiuous fro he lef o each ierval, ] ad has righ-had ad lefhad liis, x ad x 1 respecively his se defies he soluio of ipulsive syses i each regio, 1 Ω herefore ς = Uς he presece of ipulsive vecor d a ie eas ha he ipulsive syse is o auooous

5 For ay > >, he soluio = x ;, x ca be wrie i he iegral for ;, x = e A A σ e = e A σ Bu σ d δ σ dσ dσ 3 he rasiio equaio 3 gives he sae a ie i er of he sae x a ie he ipu over he ie ierval [, ad a sequece of ipulsive vecors {d } occurrig a ie isas, =,1,, where < 1 < L < < he uiqueess of he soluio is give by he followig heore which follows iediaely fro he exisece ad uiqueess of soluios of liear ordiary differeial equaios ODE, see for exaple [14] heore 1 Cosider he liear ie ivaria ipulsive syse Suppose Codiio 1 holds he for a give iiial value, x ς, here exis a uique soluio give by equaio 3 which is defied for all R 3 Coiuiy of Soluios Ulie he ODE, he coiuiy of he soluio of 3 wih respec o is iiial codiio cao solely be guaraeed by he iiial codiio x = x he soluio of 3 o he ierval, 1] wih he iiial codiio x is give by = e A A σ e Bu σ dσ he soluio follows he iiial value proble of ordiary differeial equaios for each ierval, 1] For he case of eves which occur a ie isas { } such ha 1 1 < < L < < <L 4 he exisece of a soluio as i 3 is guaraeed However, a proble ca arise whe codiio 4 cao be guaraeed ha is, suppose he iiial codiio x = x a ie is o o he hyperplae ς :{ : ς = ε,, R R Suppose also a he ie isa, he soluio, x ees he hyperplae ς such ha he soluio, lies o he hyperplae ς Hece, he par of he soluio o he ierval < 1 is also he soluio of he ierval 1 For exaple, suppose he pla oupu of he LIS is give by y = C 5 ad he rasiio ie isa is defied by he codiio: c = δ where c is he -h row of C i 5 he fro, c d = iplies c = δ i which case he soluio of he LIS does o leave he hyperplae a = Maheaically, his iplies he soluio x for > is o defied Moreover, if c d l = for all l >, he he soluio is also o coiuable o he righ of > If c d =, he exisece quesio ca be resolved [15] by assuig ha a appropriaely sall ie delay occurs bewee he occurrece of a eve a ie ad he chage i he sae of he referece odel a ie such ha he raecory oves off he hyperplae before swichig ha is, suppose = ad A x ce δ he x = d A = e d so ha c d = iplies A c = ce δ Aoher way o avoid he possibiliy of he aheaical o-exisece of a soluio is o ipose he cosrai o he decisio vecor such ha c d he oexisece proble ca also be resolved by usig he so called beaig codiio ha is cooly assued i he sudy of ipulsive dyaical syses, see [7] [8] he beaig codiio assues ha he soluio ees oe hyperplae o ore ha oce i order o }

6 guaraee he exisece of he soluio Isead, here i is assued ha he soluio will eveually leave he hyperplae oce he soluio reaches i, possibly avoidig he fas occurrece of ipulsive vecors ow as chaerig i corol heory More specifically, give a soluio of 3 which is defied o, where α >, a ~ [ α soluio x is a coiuaio o he righ of if, for ~ β > α, he soluio x is defied o ~ [, β ad = x for all [, α If is defied over he ierval, ad [ α o such coiuaio is possible for > α, he he ierval [, α is called he axial exisece of a soluio Le J, x be he axial ierval fro, ω i which he soluio x ;, x is defied he followig resul is he reforulaio of he codiio i [7] [8] which suarizes he coiuiy of he soluio of liear ipulsive syses heore Cosider he liear ie ivaria ipulsive syses 7 If he followig codiios 1 < < < L < < L 1 Q for J, x where Q is a copac subse of Ω hold he J, x =, Proof Codiio iplies ha here exiss a fiie η Q ad γ, η R Ω Suppose o he corary, i is assued ha J, x =, γ ad γ < he i follows ha li = γ provided codiio 1 is saisfied Bu, his coradics he assupio ha he lii li = η exiss ad is fiie γ Acowledges he auhor graefully acowledges he suppor fro hird World Acadey of Scieces WAS uder gra -1 RAG/MAHS/AS Refereces: [1] P J Asalis, W Koh, A Nerode, ad S Sasry, Hybrid Syses II, LNCS No 999, New Yor: Spriger, 1995 [] M S Braicy, V S Borar, ad S K Mier, A uified fraewor for hybrid corol: Model ad opial corol heory, IEEE ras Auoa Cor, Vol 43, Ja 1998, pp [3] C S Hsu, Cell-o-cell appig: a ehod of global aalysis for oliear syses, New Yor: Spriger-Verlag, 1987 [4] M Mario, Jup Liear Syses i Auoaic Corol, Marcel Deer, 199 [5] A Halaai ad D Wexler, Qualiaive heory of Ipulsive Syses, Bucuresi Mir, Moscow, 1971: Ediura Acadeiei Republici Soc Roaia, 1968 [6] V D Mil a ad A D Myshis, O he sabiliy of oio i he presece of ipulses i Russia, Sib Mah J, Vol 1, 196, pp [7] V Lashiaha, D D Baiov, ad P S Sieoov, heory of Ipulsive Differeial Equaios, Sigapore: World Scieific, 1989 [8] D D Baiov ad P S Sieoov, Syses wih Ipulse Effec: Sabiliy, heory, ad Applicaios, Chicheser-Wes Sussex: Ellis Horwood Liied, 1989 [9] J G Pierce ad A Schuizy, Opial ipulsive corol of copare odels, i : Qualiaive aspecs, J Opiz heory ad Applic, Vol 18, Apr 1976, pp [1] E Carer ad J Brie, Liearized ipulsive redezvous proble, J Opiz heory ad Applic, Vol 86, Sep 1995, pp [11] Pavlidis, Sabiliy of a class discoiuous dyaical syses, Iforaio ad Corol, Vol 9, 1966, pp 98-3 [1] M Adersso, OSi ad Oola uorial ad user s aual, versio 34, Dep of Auoaic Corol, Lud Isiue of echology, Mar 1995 [13] J R Mures, opology, Eglewood Cliffs, NJ: Preice-Hall, 1975 [14] E A Coddigo ad N Leviso, heory of Ordiary Differeial Equaios, New Yro: McGraw-Hill, 1955 [15] B D O Aderso ad J B Moore, Liear Opial Corol, Eglewood Cliffs, New Jersey: Preice-Hall Ic, 1971