Computation of an Over-Approximation of the Backward Reachable Set using Subsystem Level Set Functions

Transcription

1 Compuao o a Ove-Appomao o he Backwad Reachable Se usg Subsysem Level Se Fucos Duša M Spaov, Iseok Hwag, ad Clae J oml Depame o Aeoaucs ad Asoaucs Saod Uvesy Saod, CA , USA E-mal: {dusko, shwag, oml}@saodedu Fa: (650) Keywods: Reachable se; ove-appomaos; deeal games; ecoeced sysems Absac I hs pape, we pese a mehod o decompose he poblem o compug he backwad eachable se o a dyamc sysem a space o a gve dmeso, o a se o compuaoal poblems volvg level se ucos, each deed a lowe dmesoal (subsysem) space hs allows he poeal o gea educo compuao me he oveall sysem s cosdeed as a ecoeco o ehe dsjo o ovelappg subsysems he pojeco o he backwad eachable se o he subsysem spaces s oveappomaed by a level se o he coespodg subsysem level se uco I s show how hs mehod ca be appled o wo-playe deeal games Fally, esuls o he compuao o polyopc ove-appomaos o he usae se o he wo aca colc esoluo poblem ae peseed Ioduco Compuao o eachable ses o dyamc sysems has a mpoa applcao he auomac vecao o saey popees ad syhess o sae coolles o a ac sysems, 7] he eac eachable se bouday s kow o be he zeo level se o he vscosy soluo ] o a Hamlo-Jacob ype o paal deeal equao (PDE) 3] o he bes o ou kowledge, Lema was he s o ecogze he elaoshp bewee Bellma ucos ad he boudaes o he eachable ses 0] I ] was show how o appomae boudaes o eachable ses wh a abay accuacy usg smooh ucos I he sequece o papes by Khusalev 3, 4], locally Lpschz ucos ae used o descbe abaly accuae, ude- ad oveappomaos o eachable ses Fally, polyopc 8] ad ellpsodal appomaos 6-9] wee developed o appomae eachable ses o lea sysems (wh ad whou peubaos) he umecal soluos whch povde covege appomaos o eachable ses o dyamc sysems have compuaoal compley whch s epoeal he couous vaable dmeso, 3] o ovecome hs poblem, ] he auhos use a polyopc appomao mehod 8], based o opmal cool mehods (eg, 5, 0]), ad polyopc level se ucos, o ecely compue appomaos o owad ad backwad eachable ses o lea dyamc sysems hs appoach s eeded o eedback leazable olea sysems, lea dyamc games, ad om-bouded olea sysems ] I 9], Vce ad Wu cosde a appomao o he pojeco o he backwad eachable (coollable) se o a class o heachcal dyamc sysems Mchell ad oml 4] appomae he backwad eachable se o deeal games by solvg a se o lowe dmesoal Hamlo- Jacob-Isaacs PDEs pojeced spaces I hs pape, we popose a mehod whch decomposes he oveall poblem o compug a ove-appomao o he backwad eachable se o a geeal olea dyamc sysem, o a umbe o lowe dmesoal compuaoal poblems By dog so, he compuaoal compley o he poblem s deceased: geeal, we solve may eachably poblems o lowe dmeso ha he ogal eachably poblem Fo mplemeao, compuaoal load may be dsbued ove a ewok o paallel compues he oveall dyamc sysem s cosdeed as a se o ecoeced subsysems he, wh each subsysem we assocae a level se uco o Lpschz ype o whch a pacula level se s a ove-appomao o he bouday o he backwad eachable se he subsysem sae space (pojeco space) he ecoeco o hese ses s a ove-appomao o he backwad eachable se he pape s ogazed as ollows I Seco we pese he mahemacal backgoud o backwad eachable ses he aalyss o he compuao o ove-appomaos he subsysem (pojeced) spaces s povded Seco 3 he mehod s eeded o deeal games Seco 4, wh a applcao o he compuao o he usae se he wo aca collso avodace poblem Coclusos ae peseed Seco 5 Backwad Reachable (Coollable) Ses Le us cosde he ollowg dyamc sysem = (,, u), = 0, ], ( ) X ()

2 whee 0 <, s he sae, u m U s he cool pu wh U a compac se depede o he sae A compac se o al saes s deoed as X, X he uco : U s assumed o be Lpschz couous he cool pu uco u( ) s a measuable uco, ad he ajecoy o he sysem ( ) s a absolue couous uco such ha equao () s sased almos eveywhee We wll ee o he se o al saes as he age se he backwad eachable (coollable) se (om a gve age se X ) o he sysem () s deed as ollows: Deo he backwad eachable se ( ) a me ( ), o he sysem () om he age se X, s he se o all saes o whch hee ess a admssble cool pu u( τ ) ( τ, ] ) ad a coespodg absolue couous ajecoy ( τ ) ( τ, ] ) o he sysem (), such ha ( ) X ad = ( ) We dee he Hamloa o he backwad popagao o he sysem () as H (,, p) = m{ p (,, u)} wh p beg he adjo sae veco he e poduc o wo vecos a, b, s deoed as a b I s well kow 3] ha o ay, he eac bouday o he backwad eachable se ca be compued as he zeo level se o he vscosy soluo v(, ) o he Hamlo-Jacob equao, v + H (,, v / ) = 0 wh he al codo v(, ) havg zeo level se whch s he bouday o he age se X Le us ewe sysem () decomposed om as a se o subsysems = (,, u),, = {,, } (4) whee () (3) s he -h subsysem sae such ha =,, ], ha s, he subsysems ae dsjo he, om equaos (5) ad (6) ollows ha H (,, p) H (,, p ) (7) = I we assume ha each subsysem has s ow depede pu, ha s, (,, u) = ˆ (,, u ) (8) wh u U m, m = m, (ha s, decomposo o he = pu veco s dsjo) he usg (5), (6), ad (8) we oba he ollowg: H (,, p ) = H (,, p ) (9) = hus, whe he subsysems ca be chose o have depede pus, equaly (7) becomes equaly (9) o coclude hs seco, le us sae ha he movao o devg equaly (7) ad equaly (9) s he dea o coveg he hgh dmesoal poblem o compug he backwad eachable se o he oveall sysem o a se o lowe dmesoal poblems volvg subsysem Hamloas 3 Ove-Appomaos o he Subsysem Pojecos o he Backwad Reachable Ses I ode o compue a ove-appomao o he backwad eachable se, deoed as R ( ), ( ( ) R( ) ), o each subsysem we assocae a level se uco (, ) v (, ) whch s assumed o be Lpschz couous wh posve Lpschz cosa k By compug s devave alog he soluo o sysem () we oba dv (, ) v v d () = + ˆ v v = + (,, u) (,, u) Fo p = v /, om equao (6) we oba (0) v v (,, u) H (,, ) () H (,, p) = m{ p (,, u)} p u = = = m{ (,, )} m{ p (,, u)} By deg he subsysem Hamloas as H (,, p ) = m{ p (,, u)} (5) (6) ow, om equaos (0) ad () ollows: dv v v + H (,, ) d () () I hee ess a measuable uco µ ( ) such ha (see 3, 4, 7] o applcao o a sgle sysem)

3 ! " 3 v v + H (,, ) µ ( ) (3) v v + H g (, ( ), ) ( ) µ (8) he om () ad (3) ollows: v (, ( )) µ ( τ ) dτ + ma { v (, ( ))} ( ) ( X ) (4) whee ( X ) s he pojeco o he age se o he subsysem space, whee I s mpoa o oe ha he choce o a appopae µ ( ) equaly (3) s vey much poblem depede, ad o suggesos o how o compue he se o measuable ucos { µ ( )} = o he geeal sysem ca be gve We wll addess hs ssue moe deal Seco 3 by cosdeg ecoeced sysems as a eample o sysems wh a specal sucue he ove-appomao o he eachable se o he subsysem wh sae s gve by he ollowg omula 3, 4, 7]: ( ) ( X ) R ( ) = { v (, ) µ ( τ ) dτ + ma { v (, ( ))}} (5) oce ha om equao (5) ollows ha we ca compue a ove-appomao o he backwad eachable se o oly a poo o he sae space, ha s, he oveappomao o he eachable se o he oveall sae s gve by R ( ) = R ( ) R ( ), ( ) R( ) whee symbol deoes he Caesa poduc 3 Iecoeced Sysems (6) I hs seco we cosde a class o olea sysems wh a specal sucue, ha s, ecoeced sysems, o whch we ca aalyze he pocedue o deemg a boud, epeseed by equaly (3), moe deal Le us assume ha ucos { } = equao (4) ae o he ollowg om: (,, u) = g (,, u) + h (,, u) (7) # # whee g : U s a Lpschz couous uco ha epeses subsysem dyamcs, ad $ $ h : U s a Lpschz couous uco ha epeses ecoecos bewee subsysems, o all, Le us assume ha o each hee s a measuable uco µ ( ) ove he e me hozo, such ha whee he subsysem Hamloa H g (,, ) (as deed (6)) s compued wh espec o he uco g (,, ) deed (7) he, dv (, ) v v v v = + ) * (,, u) + H (,, ) d () % & ' ( v v v + H g (,, ) + m{ ) * h (,, )} u (9) % & ' ( v µ ( ) + m{ ) * h (,, )} u % & ' ( We assume ha he ecoecos sasy seco bouds h (,, u)} + β µ ( ) (0) j j j= o some β j s Sce v (, ) s a Lpschz uco wh posve Lpschz cosa k, we dee such ha β ( ) =, k β µ ( ) () j j j= - / 0 v m{ h (,, u)} β ( ) Fom (9) ad () ollows ha: dv (, ) d () () µ ( ) β ( ) (3) By equag µ = µ ( ) β ( ), ad egag (3), he ove-appomao o he backwad eachable se o gve by ( ) ( X ) s R ( ) = { v (, ) µ ( τ ) dτ + ma { v (, ( ))}} (4) whch has he same om as (5), ad he ove-appomao R ( ) o he backwad eachable se o he sae s obaed usg (6) 3 Ovelappg Ove-Appomaos We have cosdeed decomposg he oveall sae o dsjo subsysems I hs seco, we allow paoed subsysem saes o ovelap Cosde equaos (4) ad (5), ad o smplcy o he peseao (ad whou loss o

4 E 7 B 7 B H P IM K P M R Q L Q S W V I K P M J M U MO L Q J U M geealy), assume ha we have oly wo subsysems wh saes, 4, ad, 5, such ha hey ovelap, ha s, ad shae compoes, meag =, 6, oce ha + = We dee eeded ove-appomaos as :9<;=9?> R ( ) = R ( ) mes B B (5) ( ) 8:9@;A9?> ( ) mes R = R whee R ( ) ad R ( ) ae compued as (5) o, he case o ecoeced sysems, as (4) he, he ollowg s ue: D D D ( ( ) R ( )) ( ( ) R ( )) C ( ) R ( ) R ( ) (6) whee symbol deoes he ad logc opeao I ohe wods, we ca compue a ove-appomao as a eseco o ay se o eeded lowe dmesoal oveappomaos he subspaces ha cove he whole space he eeso o subsysems wh ovelappg saes s clea hs gves us moe eedom, sce we ae o ay moe esced oly o dsjo paos o he sae space 4 Deeal Games wh Applcao o Colc Resoluo I hs seco, we show how he aalyss peseed Seco 3 caes ove o accommodae he compuao o he backwad eachable se o he wo playe deeal game Le us cosde a dyamc sysem wh wo depede pus u ad d as = g ( ) + q ( ) u + q ( ) d + h ( ),, u U, d D (7) u whee U, U F m md, ad D, D G, ae compac ses Fucos u( ) ad d( ) ae assumed o be measuable ucos he gh-had sde o equao (7) s Lpschz couous, ad he soluo ajecoes ( ) ae assumed o be absolue couous he pao { } = o he sae s a se o subsysem saes ha ae ehe dsjo o ovelappg he age se s deoed as X he backwad eachable se o he sysem (7) s deed as ollows Deo he backwad eachable se ( ) a me ( ), o he sysem (7) om he age se X, s he se o all saes o whch hee ess a admssble cool pu u( τ ) ( τ, ] ) such ha o ay admssble pu d( τ ) ( τ, ] ), a absolue couous ajecoy ( τ ) ( τ, ] ) o he sysem () sases ( ) X ad = ( ) he epeao o Deo s ha u dves he sysem o X despe d Fo a gve Lpschz level se uco (, ) v(, ) we compue K L M M v v v µ ( ) = + g ( ) + m q ( ) u( ) O R S R S v + ma q ( ) d( ) + β ( ) d D (8) alog he soluos ( ) o (7), o all We assume ha ( ) v / h ( ( )) β ( ) holds o all he, he ollowg se s he ove-appomao o he backwad eachable pojeco o, he sae space o he subsysem wh sae : R ( ) = { v (, ) µ ( τ ) dτ + ma { v (, ( ))}} ( ) ( X ) (9) he oly deece bewee equaos (5) ad (9) s ha compug µ ( ) we ea d( ) as he peubao hus, he case o compug he backwad eachable se o he dyamc sysem (7) whch descbes wo playe deeal game ollows decly om he aalyss peseed Seco 3 4 Colc Resoluo bewee wo Aca o demosae he poposed pocedue we cosde he wo aca collso avodace poblem 3] whch s modeled as deeal game (7) I Fgue we show he elave coguao o he wo aca whee aca es o avod he collso egadless o he behavo o aca I hs poblem we wsh o compue he backwad eachable (usae) se om he age se (poeced zoe), ha s, he se whch cludes all he saes ( elave coodaes) om whch aca ca choose a cool ha wll lead o loss o sepaao o ay cool saegy o aca Fgue Relave coodae sysem o he wo aca colc esoluo poblem

5 g e e ^ g b d We use he plaa kemac model o each aca, ad ae dyamc eeso ad eedback leazao o he model 5], we oba he ollowg lea model ems o elave coodaes o he wo aca 6]: whee Y X Y X Y ] \ ] \ ] = + u u (30) 4 _ s he sae veco, u U `, ad U a, ae cool pus o he wo aca, u especvely Fom he deo o he usae se oce ha u coespods o d, ad u coespods o u equao (7) We decompose he sae veco as =, ], whee =, ] y, ad he wo emag saes om I ohe wods, he oveall sae veco s decomposed o wo dsjo subsysems, each o dmeso wo he, sysem (30) a decomposed om ca be we as wh, c, = b (3) = u u u U, ad u U As level se ucos we choose polyopc ucos descbed as esecos o he suppog hypeplaes o he om 8]: v (, ) = ( ), d (3) v (, ) = ( ) he poeco zoe s he age se Usg (8) ad he pocedue poposed, 8], we compue dv = + d ; { } µ ( ) = ma ( )( u u ) o ( ) ( ), dv = + ( u u); d { } { } µ ( ) = m ( ) u m ( ) u o ( ) ( ) (33) Usg equaos (9), (3), ad (33) we oba he oveappomao R o, ha s, ecagula elave coodaes ad y oce ha hs pocedue, he em = ( u u ) s eaed as a ecoeco (ha s, peubao h( ) (7)) By compug he backwad eachable appomao R he -space, he oveall ove-appomao s compued as R = R R I Fgue, he e se s he eac usae se (, y, ψ ) -space, ψ beg he elave headg agle as Fgue, ad he oue se s he ove-appomao obaed by pojecg R = R R o (, y, ψ ) -space he eac se s compued 3], usg he covege appomao desged hee hs eac compuao ook appomaely 5 mues o a Su UlaSpac wh 50 gd odes each dmeso he ove-appomao compuao (usg MALAB o a 700 MHz Peum III PC) ook 0 secods (whch cludes plog he gue) Fgue A ove-appomao o he eac usae se (wo -dmesoal subsysems) I we choose he s hee coodaes o he sae space veco as ou subsysem, whch us ou o be a e decomposo, ad compue he ove-appomao hs hee dmesoal space, we oba a bee oveappomao o he eac se, as show Fgue 3 Fgue 3 A ove-appomao o he eac usae se (oe 3-dmesoal subsysem) he ove-appomao compuao ploed Fgue 3 (aga, usg MALAB o a 700 MHz Peum III PC), ook 03 secods (cludg plog he gue) Fuhe eemes ca be obaed by esecos o vaous dsjo o ovelappg ove-appomaos, ad a ece mehodology o dog hs s he subjec o ou cue eseach 5 Coclusos A mehod o solvg he poblem o compuao o he backwad eachable se, as poposed hs pape, decomposes he poblem o a se o smalle dmesoal

6 poblems o he bass o he decomposo o he oveall dyamc sysem o a se o subsysems he he aalyss s caed ove usg a se o subsysem level se ucos o whch he level ses povde he ove-appomaos o he pojecos o he eachable se o subsysem spaces he mehod accommodaes compuao o he eachable ses o wo playe dyamc games ad he applcao o compuao o he usae se o he wo aca collso avodace poblem poduced ecouagg esuls Ou uue wok s o cosde compug eachable ses o hybd sysems usg subsysem level se ucos, ad o wok o hghe dmesoal applcaos Fally, s mpoa o oe ha he aalyss peseed hs pape caes ove o he compuao o he owad eachable se o dyamc sysems a obvous way Reeeces ] M G Cadall, L C Evas, ad P-L Los Some popees o vscosy soluos o Hamlo-Jacob equaos asacos o Ameca Mahemacal Socey, Vol 8, o, pp , 984 ] I Hwag, D M Spaovh, ad C J oml Applcaos o polyopc appomaos o eachable ses o lea dyamc games ad a class o olea sysems Poceedgs o he Ameca Cool Coeece, Deve, Coloado, Jue 4-6, 003, pp ] M M Khusalev Eac descpo o eachable ses ad global opmaly codos Avomaka elemekhaka, o 5, pp 6-70, 988 4] M M Khusalev Eac descpo o eachable ses ad global opmaly codos II: codos o global opmaly Avomaka elemekhaka, o 7, pp 70-80, 988 5] V F Koov Global Mehods Opmal Cool heoy Macel Dekke, Ic, ew Yok, Y, 996 6] A Kuzhask ad P Vaaya Ellpsodal echques o eachably aalyss I Hybd Sysems: Compuao ad Cool (B Kogh ad Lych, eds), o 790 Lecue oes Compue Scece, pp 0-4, Spge Velag, 000 7] A Kuzhask ad P Vaaya Dyamc opmzao o eachably poblems Joual o Opmzao heoy ad Applcaos, Vol 08, o, pp 7-5, 00 8] A Kuzhask ad P Vaaya O eachably ude uceay SIAM Joual o Cool ad Opmzao, Vol 4, o, pp 8-6, 00 9] A Kuzhask ad P Vaaya Reachably aalyss o ucea sysems-he ellpsodal echque Dyamcs o Couous, Dscee & Impulse Sysems, Sees B: Applcaos & Algohms, Vol 9, o 3, , 00 0] G Lema A Ioduco o Opmal Cool Mc- Gaw-Hll, ew Yok, Y, 967 ] J Lygeos, C J oml, ad S Sasy Coolles o eachably speccaos o hybd sysems Auomaca, Vol 35, o 3, pp , 999 ] I Mchell, A M Baye, ad C J oml Valdag a Hamlo-Jacob appomao o hybd sysem eachable ses I Hybd Sysems: Compuao ad Cool (M D D Beedeo ad A Sagova- Vceell, eds), o 034 Lecue oes Compue Scece, pp 48-43, Spge Velag, 00 3] I Mchell, A M Baye, ad C J oml Compug eachable ses o couous dyamc games usg level se mehods Submed o publcao he IEEE asacos o Auomac Cool, Decembe 00 4] I M Mchell ad C J oml Oveappomag eachable ses by Hamlo-Jacob pojecos Acceped o publcao he Joual o Scec Compug, specal ssue dedcaed o Saley Oshe o he occaso o hs 60 h bhday, 003 5] S Sasy olea Sysems: Aalyss, Sably, ad Cool Spge-Velag, ew Yok, Y, 999 6] D M Spaovh, G alha, R eo, ad C J oml Decealzed Ovelappg Cool o a Fomao o Umaed Aeal Vehcles Poceedgs o he 00 IEEE Coeece o Decso ad Cool, Las Vegas, evada, Decembe 0-3, 00, pp ] C J oml, J Lygeos, ad S Sasy A game heoec appoach o coolle desg o hybd sysems Poceedgs o he IEEE, Vol 88, o 7, pp , 000 8] P Vaaya Reach se compuao usg opmal cool I Poceedgs o he KI Wokshop o Vecao o Hybd Sysems, Vemag, Geoble, Face, 998 9] L Vce ad Y Wu Esmag pojecos o he coollable se Joual o Gudace, Cool, ad Dyamcs, Vol 3, o 3, pp , 990 0] L Vce ad W J Gaham olea ad Opmal Cool Sysems Joh Wley & Sos, Ic, ew Yok, Y, 997 ] R Ve A chaacezao o he eachable se o olea cool sysems SIAM Joual o Cool ad Opmzao, Vol 8, o 6, pp , 980