Hybridization Based Reachability of Uncertain Planar Affine Systems

Transcription

1 Hybdzaton Based Reachablty of Uncetan Plana Affne Systems Othmane NASR Mae-Anne LEFEBVRE Heé GUEGUEN Spélec- ER BP 87 Cesson-Ségné Ced Fance Abstact he behao efcaton of hybd systems fo eample fo safety popetes s based on the comptaton of the eachable state space of a hybd atomaton modellng the system nde stdy n ths pape we pesent a method fo the comptaton of eachable set of ncetan hybd affne systems hs method etends peos woks whch compte the eachable set of cetan affne systems by abstactng the contnos dynamcs by a lnea hybd atomaton We show that the calcls of the eachable space of the ncetan system may be dedced fom the comptaton of the eachable sets of a fnte nmbe of cetan systems ntodcton n ode to ense the systems adeacy to the specfcaton aldatons ae necessay n ode to detect the eos ntodced dng the deelopment Valdatons ae geneally obtaned by smlaton bt ths does not allow to cetfy the absence of eos t s ths nteestng to se fomal efcaton n ode to nse the coect behao Fo hybd systems that m dscete eents contnos dynamcs t appeas that pedomnant popetes ae safety [] that they can be epessed by fobdden domans n the state space he efcaton s then based on the calcls of the eachable space fom an ntal egon n ode to gaanty that the system nee eaches ths fobdden doman Ecept fo ey specfc classes of hybd systems [9] eact comptaton of the eachable sets s mpossble he man dffclty les n the comptaton of the eachable sets of the contnos dynamcs he mpotance of the poblem has motated mch eseach on appomate eachablty analyss wo man appoaches hae been deeloped he fst one conssts n comptng by smlaton the eachable space at specfc nstants to dedce fom these eslts an appomaton of the eachable space [78] Uncetanty s then taken nto accont by consdeng an nknown petbaton comptng at each step the wose case ales of the petbaton n each decton wth optmsaton pogammng [ 6 ] he second appoach ses abstact models fo whch the eachable space may be compted [] n peos woks a method based on sch abstacton [ ] has been deeloped fo affne dynamcs Howee t only consdes systems that ae completely known ths appeas to be a geat estcton to apply the method to eal systems n ths pap we popose an etenson of ths method to some ncetan affne systems he pape s oganzed as follows Fst we ntodce the pncples of the abstacton fo cetan systems the hypothess fo modellng of ncetanty hen we eplan how the basc concepts ae sefl to take nto accont the ncetantes how the method can be etended to ncetan systems he basc dea of ths etenson s that the calcls fo contnos aaton of some paamete can be dedced fom the calcls fo a fnte nmbe of specfc ales of ths paamete Aftewads we llstate o appoach by an eample Pncples of the abstacton Pncple of the abstacton wthot ncetanty n ths secton we consde a non-sngla plana affne system n ode to ntodce the basc deas that may be etended to moe comple dynamcs see []) he dynamcs s then modelled by eaton ) whee b R A s an netble mat he elbm pont s then gen by ) X//$ EEE

2 A b e ) A b ) Fo all ponts of a gen lne that stats at the elbm pont the deate ectos ae collnea So fo all ponts of a secto defned by two sch lnes the ecto feld s n the secto defned by the two deate ectos on the bonday as llstated Fge Fo a lne defned by the fst eaton of ) t s possble to chaacteze the ecto feld by the second eaton of ) fo moe detals see []) e ) wth A ) ) Moeo f the bode of sch a secto s defned by a eal left egen ecto of mat A t can t be cossed by contnos taectoy Othewse t s always cossed n the same decton S a) b) Fge Sectos dffeental nclsons fo plana systems t s then possble to etact fom a patton of the state space wth sch sectos an abstacton sch as each locaton s the soce the taget of at most one tanston as llstated Fge Fo a system wth comple egen ales) ts contnos dynamcs s gen by the nclson ) whee the two ectos ae gen by the second eaton of ) e S' ) ) ) t may be noted that ) mposes constants only on the decton not on the nom of the ecto feld bt as the abstacton s sed to calclate the eachable space ths s not a poblem moeoe t makes ths calclaton ey easy ndeed the eachable space fom a pont belongng to a secto s gen by the ntesecton of the secto wth the constants : a) b) ) ) ) ) ) S S S S S S S e S S Fge he patton of the state space a) the assocated atomaton b) Assmptons on ncetanty n ode to take nto accont the modellng ncetanty t s consdeed that the ecto b of eaton ) s not completely known bt can take a fed ale n a cone doman defned by ts etces Howee wthot loss of genealty t wll be consdeed fo smplcty eason that ths doman s a segment he eaton that defnes the stdy dynamcs s then: A b S S 8 S 8 S 7 S 6 S 7 S 6 6) Whee b ) b b ) b b ae known ectos he elbm pont fthe efes ) Reachablty fo ncetan system O am s then to compte an oe-appomaton of the eachable spac fom an ntal pont ald whatee s the ale of n eaton 6) n ode to eplan how ths can be acheed let s ntodce some notatons

3 { / n } s a set of ectos sch as ae lnealy ndependent { R / ) } s the soclne gong thogh the elbm pont othogonal to a ecto t cold be notced that fo all ponts of ths lne the ecto feld s othogonal to A ) that does not depend on S / ) ) { } the secto delmted by the half soclnes assocated wth an elbm pont t s possble to dedce fom ) the peos emak that the dffeental nclson assocated wth sch a secto does not depend on bt only on ) denotes the ntesecton of the eachable R space fom fo the abstacton the secto S And R ) denotes the epected oe-appomaton of the eachable space fom t can be notced that: R ) R ) [] n 7) he dffcltes of comptng R ) wth eaton 7) ae lnked to the contnos aaton of n ode to oecome these dffcltes let s consde a set of eal nmbes sch as fo all m m denote by : R ) R ) the non of the [ ] eachable spaces n S fo all t s then easy to see that eaton 8) holds: R ) R ) 8) he fndamental dea on whch the appoach s based s that a good choce of the allows to compte R ; ) then R ) by 8) Calcls of R ; ) We fst consde that ae two eal sch as the pont belongs to a same secto : a [ S s ] 9) hee ests a eal β sch as ) β β β whee y y y Fge dynamc system n the elately sectos As the dynamcs n the sectos S S S ae the sam fom ) the accessblty spaces fom hae the same fontes We denote by y y y the ntesecton ponts of one fonte wth the coespondng soclnes hese ponts ae algned Moeo we hae: y y β ) y β ) Let s fo defne by : y ) ) y ) ) ) hese ponts belong to the ntesecton of the accessble feld fom y wth see Fge )

4 y y y Fge Accessble space n the elately sectos he pont combnaton of the two ponts s the eslt of a cone : β ) β ) β β ) ) hogh ths easonng we hae shown that the etces of the eachable space n the secto S may be wtten as cone combnaton of the coespondng etces of the eachable space n S n S he same easonng may be ecently appled to the followng sectos then: R ) Con _ hll R ) R )) 6) f o ae non-ndependent then the ponts o may not be defned bt ths s not a lmtaton as ethe the ete of the eachable egon s the elbm pont ethe t s nfnte as shown Fge ) n both cases ths can be consdeed as specal cases of eaton ) hs can be poed by ewtng ): So β y β e efes : β ) y β ) )) )) S Fom 9) β y β y y β ) y β ) )) )) efy espectely: ) ) ) ) ) ) Accodng to ) ) we hae the followng eslt: β β β ) β ) )) )) Fge Accessblty space no lmted When the pont does not belong to the same sectos elate to the thee elbm ponts e the ponts y y y do not efy the eaton 9) they ae not algned) t s then nteestng to consde the patcla ale sch as : - [ ] S - [ ] S hs ale s defned as the ale sch as the pont belongs to see Fge 6) hs wll lead to the choce of the that wll be ntodced n the eachablty calcls f the ponts ests then ae ndependent ths poes ) n the same mann we hae:

5 yb k * yb y * y y Fge 6 Accessblty space n two dffeent sectos Smmng p of the eachablty pocess n the peos pat we hae shown that t s necessay sffcent that ae sch as 9) holds n ode to be able to apply 6) hs s the basc pont of o pocess that hae fo stages: Fst we compte the elbm ponts the ales k k belongs smltaneosly to the sectos S S fo k sch as so that belongs to t s clea that the pont k We then compte the sets R ) fo all ales { / k } petnent ales of by a classcal eachablty comptaton hen we can compte : R ) Con _ hll R ) R )) And fnally we compte R ) by 8) llstate eample n ode to eemplfy the appoach let s consde the system defned by A b b let s compte the eachable 9 space fom he lmt ales of b leads to lmt elbm ponts 7 / he patton of the state space n ode to defne the abstacton s based on ten sectos that ae sch as: S S he applcaton of the fst stage of the appoach 7 / leads to consde e : / / whch s assocated to b 97 / 7 he second stage ges the eachable space fo the thee specfc ales of that ae epesented fge7 - - Fge 7 Reachable space elate to thee elbm ponts Fom these comptatons t s possble to compte R ) see Fge8) R ) see Fge9) fnally R )

6 - - Fge 8 Global eachable space fo [ ] - - Fge9 Global eachable space fo [ ] hese eslts may seem ey ogh bt ths s becase we hae chosen a geat ncetanty on the ales of b n ths eample n ode that the ntemedate eslts may be ead easly f we consde 8 that b b whch s a moe ealstc ncetanty the appoach leads to the fnal eslt that s gen Fge - - Fge Global eachable space % new b ) Conclson pespectes n ths pap we hae pesented a method fo eachablty analyss of ncetan affne systems We etended a peos wok whch has the adantage to allow to compte easly the eachable space when the dynamc systems nde stdy s completely known hs appoach also poed to be well adapted to ncetan systems as t allows to compte the oeappomaton of the eachable space fo ncetan systems whee a paamete takes ales n a contnos space wth a fnte nmbe of patcla ales n ths wok t has been consdeed that the paamete was nknown bt f fte wok wll consde the case whee these paamete may eole wth tme Fnally an othe pont that hae to be consdeed s the adaptaton of ths wok fo systems wth constaned naant domans Refeences [] H Gégen J Zaytoon On the fomal efcaton of hybd systems Contol Engneeng Pactc Else [] M-ALefeb Abstactons po la éfcaton de sûeté des systèmes hybdes ER-Spelec PhD thess Un Rennes n Fench) [] M-ALefeb H Gegen Hybd Abstactons of Affne Systems FAC wold Conges [] -A Henzng P-H Ho H Wong-o Algothmc analyss of non-lnea hybd systems Specal sse on hybd system P-JAntsakls A Node Eds) EEE ansacton on Atomatc Contol ) pp [] A Gad Reachablty of Uncetan Lnea Systems Usng Zonotopes Hybd Systems: Comptaton Contol M Moa L hele Eds) no n LNCS Spng pp 9 [6] Dang Véfcaton et synthèse des systèmes hybdes NPG Genobl PhD thess [7] E Asan O Bonez Dang O Mal Apomate eachablty analyss of pecewse lnea dynamcal systems Hybd Systems: Comptaton Contol N Lynch B H Kogh Eds) no 79 n LNCS Spng pp [8] A Chtnan BH Kogh Vefcaton of polyedal naant hybd atomata sng flow ppe appomatons Hybd Systems: Comptaton Contol F Vaag J Van Schppen Eds) no 69 n LNCS Spng pp 76-9 [9] R Al Dang F ancc Reachablty analyss of hybd systems a pedcate abstacton Hybd Systems: Comptaton Contol CJ omln MR Geensteet Eds) no 89 n LNCS Spng pp - 8 [] A wa GKhanna Geometc pogammng elaaton fo lnea system eachablty Poc Amecan Contol Confeenc [] A Kzhansk P Vaaya Ellpsodal technes fo eachablty analyss Hybd Systems: Comptaton Contol N Lynch B H Kogh Eds) no 79 n LNCS Spng pp -