Generalized Robust Diagnosability of Discrete Event Systems

Transcription

1 Prprints of th 18th IFAC World Congrss Milno (Itly) August 28 - Sptmr 2, 2011 Gnrlizd Roust Dignosility of Disrt Evnt Systms Lilin K. Crvlho Mros V. Morir João C. Bsilio Univrsidd Fdrl do Rio d Jniro, COPPE Progrm d Engnhri Elétri, Cidd Univrsitári, Ilh do Fundão, Rio d Jniro, , RJ, Brzil (-mils: lilin@op.ufrj.r, morir@d.ufrj.r, silio@d.ufrj.r). Astrt: W ddrss th prolm of roust dignosility of disrt vnt systms dsrid y lss of utomt, whr h utomton in th lss gnrts distint lngug. W introdu nw dfinition whih gnrlizs ll prvious dfinitions of roust dignosility; for this rson it is rfrrd hr to s gnrlizd roust dignosility. W lso prsnt nssry nd suffiint ondition for th gnrlizd roust dignosility nd propos polynomil tim lgorithm for its vrifition. Kywords: Disrt vnt systms, fult dignosis, finit utomt, roust dignosis. 1. INTRODUCTION Th usul pproh to fult dignosis of disrt vnt systms (DES) modld y utomt ssums tht th lngug gnrtd is uniquly dtrmind y th modl; s.g. Smpth t l. (1995), Douk t l. (2000), Zd t l. (2003), Qiu nd Kumr (2006) nd th rfrns thrin. Th noti of lngug unrtinty in fult dignosis of DES hs n introdud y Bsilio nd Lfortun (2009), initilly, in th ontxt of dntrlizd dignosility, ssuming unrlil ommunition twn lol sit nd th oordintor. Mor rntly, this notion hs n xtndd to lso nompss prmnnt nd intrmittnt snsor filurs (Lim t l., 2010; Crvlho t l., 2010), nd modl unrtinty du to inxt knowldg of th rl systm (Tki, 2010). Th dfinition of roust dignosility of disrt vnt systms (DES) sujt to prmnnt snsor filurs proposd in Lim t l. (2010) dploys th rdundny tht my xist in st of dignosis ss (st of vnts tht gurnt fult dignosility) with th viw to vrifying fult dignosility vn in th ourrn of prmnnt snsor filurs. A roust dignosr is otind using svrl prtil dignosrs, h on uilt for prtiulr dignosis sis nd onstrutd from th sm plnt G with spifi st of osrvl vnts tht is rltd to th dignoss sis of th prtil dignosr. Th roust dignosility prolm posd in Lim t l. (2010) n rgrdd s th prolm of idntifying th ourrn of n unosrvl fult vnt with unrtintis in th st of osrvl vnts. Th dfinition of roust dignosility introdud y Tki (2010) rquirs tht th systm dsrid y st of possil modls {G i : i I m } ovr ommon vnt Th rsrh works of L. K. Crvlho nd M. V. Morir hv n prtilly supportd y Crlos Chgs Foundtion (FAPERJ) nd th rsrh work of J. C. Bsilio hs n prtilly supportd y th Brzilin Rsrh Counil (CNPq), grnt / st Σ, whr I m := {1, 2,..., m} nd m N dnots th numr of systm modls. In this dfinition, h possil modl hs its own non-fulty spifition nd th lngug gnrtd y h modl is liv. Diffrntly from Lim t l. (2010), ll modls G i in Tki (2010) r ssumd to hv th sm osrvl vnt st. An lgorithm for th vrifition of th roust dignosility ondition sd on prviously proposd lgorithm y Qiu nd Kumr (2006) is lso prsntd in Tki (2010). W introdu hr nw dfinition of roust dignosility, lld gnrlizd roust dignosility, tht gnrlizs th roust dignosility dfinitions of Lim t l. (2010) nd Tki (2010), in th sns tht unrtintis r onsidrd oth in th systm modl nd in th osrvl vnt st. In our pproh, th systm modl longs to st of possil modls ovr ommon vnt st Σ s in Tki (2010) with th diffrn tht, hr, h modl my hv distint st of osrvl vnts. This sitution my hppn, for instn, du to snsor filur tht hngs th hvior of systm undr suprvision, lding to ompltly nw utomton modl with distint sts of stts nd osrvl vnts, nd with diffrnt trnsition funtion; th rdr is rfrrd to th works y Lin (1993), Young nd Grg (1991), nd Soori nd Hshtrudi-Zd (1993), whih onsidr modl unrtintis in th ontxt of suprvisory ontrol. W lso propos polynomil tim lgorithm for th vrifition of th gnrlizd roust dignosility of DES whih hs lowr omputtionl omplxity thn tht proposd y Tki (2010). This ppr is orgnizd s follows: in Stion 2 w prsnt th prliminry onpts, nd in Stion 3 w introdu th dfinition of gnrlizd roust dignosility; in Stion 4, w propos polynomil tim lgorithm for th vrifition of roust dignosility nd, in th squl, w dtrmin its omputtionl omplxity; w prsnt n xmpl in Stion 5 to illustrt th lgorithm proposd in th ppr, nd drw onlusions in Stion 6. Copyright y th Intrntionl Fdrtion of Automti Control (IFAC) 8737

2 Prprints of th 18th IFAC World Congrss Milno (Itly) August 28 - Sptmr 2, PRELIMINARIES Lt G = (X, Σ, Γ, f, x 0 ) dnot th utomton modl of DES, whr X is th finit stt sp, Σ is th st of vnts, Γ is th fsil vnt funtion, f is th trnsition funtion, nd x 0 is th initil stt. Lt us dnot th lngug gnrtd y G s L(G) = L, nd lt pth dfind s squn of stts nd vnts (x k, σ 1, x k+1, σ 2,..., σ l, x k+l ), for l > 0, suh tht x k+i = f(x k+i 1, σ i ), i {1, 2,..., l}. A pth forms yl if x k+l = x k. Lt us prtition Σ s Σ = Σ o Σ uo, whr Σ o nd Σ uo r, rsptivly, th st of osrvl nd unosrvl vnts, nd lt Σ f Σ uo dnot th st of fult vnts. In ddition, ssum, without loss of gnrlity, tht Σ f = { }. Assum lso tht G modls th norml nd th fulty hvior of th systm nd lt H th suutomton of G tht rprsnts th non-fulty hvior of th systm. Thus, th lngug gnrtd y H, dnotd s K, is prfix-losd lngug formd with ll trs of L tht do not hv ny vnt from th fult vnt st Σ f. Th fult vnt is sid to dignosl if th ourrn of n dttd within finit numr of trnsitions ftr its ourrn using only trs formd with vnts in Σ o. Formlly, lngug dignosility is dfind s follows (Smpth t l., 1995). Dfinition 1. Lt L th prfix-losd lngug gnrtd y th systm G nd lt K L dnot th prfixlosd lngug gnrtd y H. Assum lso tht L is liv. Thn, L is dignosl with rspt to P o : Σ Σ o nd Σ f if nd only if ( n N)( s L \ K)( st L \ K, t n) ( w K, P o (st) P o (w)). 3. GENERALIZED ROBUST DIAGNOSABILITY A dfinition of roust dignosility of DES ginst prmnnt snsor filurs is prsntd in Lim t l. (2010), undr th following ssumptions: (i) snsor filurs n only our for th first ourrn of th vnt rordd y th snsors whos filurs r ing invstigtd; (ii) th non-osrvtion of th vnts dos not hng th lngug gnrtd y th systm; (iii) th roust dignosility is sttd in trms of distint sts of osrvl vnts Σ oi, i = 1,..., m, m N, whr Σ oi is sis for dignosility, i.., L is dignosl with rspt to projtions P oi : Σ Σ o i, i = 1,..., m, nd Σ f, s follows. Dfinition 2. (Roust dignosility ginst prmnnt snsor filurs) Lt L th liv prfix-losd lngug gnrtd y utomton G nd lt K L dnot th prfix-losd lngug gnrtd y H. In ddition, ssum tht Σ oi, i = 1,..., m, is dignosis sis for L. Thn, L is roustly dignosl ginst prmnnt snsor filurs with rspt to projtions P oi : Σ Σ o i, for i = 1,..., m, nd Σ f = { }, if nd only if ( n N)( s L \ K)( st L \ K, t n) ( i, j {1, 2,..., m}, i j)( w K, P oi (st) P oj (w)). Instd of using singl modl, Tki (2010) dploys st of possil utomt to modl th systm, nd introdus nothr dfinition of roust dignosility, similr to tht prsntd in Lim t l. (2010), ssuming tht: (i) th rl systm modl longs to st of possil modls G i = (X i, Σ, f i, x 0i ), i = 1, 2,..., m; (ii) h utomton modl G i gnrts diffrnt lngug L i nd hs distint non-fulty hvior, dsrid y nonmpty losd sulngug K i L i ; (iii) thy ll shr th sm osrvl vnt st Σ o. In th work y Lim t l. (2010), th systm is modld y uniqu utomton G nd th unrtintis r du to loss of osrvl vnts. On th othr hnd, th roust dignosility prolm formultd in Tki (2010) is only suitl whn th modl is sujt to hng nd nnot usd in strightforwrd wy whn th modl unrtintis r du to ithr prmnnt (Lim t l., 2010) or intrmittnt (Crvlho t l., 2010) loss of osrvl vnts. In ordr to nompss oth, distint osrvl vnt sts nd diffrnt utomton modls thrfor gnrlizing th roust dignosility dfinitions prsntd in Lim t l. (2010) nd Tki (2010) w prsnt th following dfinition. Dfinition 3. (Gnrlizd roust dignosility) Lt L i Σ th lngug gnrtd y G i, i = 1,..., m, nd ssum tht L i is liv. In ddition, ssum tht h modl i I m hs projtion P oi : Σ Σ o i, nd tht L i is dignosl with rspt to P oi nd Σ f. Lt us dnot H i s th suutomton of G i tht modls th non-fulty hvior of th orrsponding modl, nd K i L i th lngug gnrtd y H i. Thn, L = {L i : i I m }, th lss of ll possil lngugs gnrtd y th lss of utomt G = {G i : i I m }, is roustly dignosl with rspt to projtions P oi, i = 1,..., m, nd Σ f = { }, if nd only if ( i I m )( n i N)( s i L i \K i )( s i t i L i \K i, t i n i ) ( j I m, j i)( w j K j, P oi (s i t i ) P oj (w j )). 4. VERIFICATION OF GENERALIZED ROBUST DIAGNOSABILITY Aording to Dfinition 3, th prolm of vrifying th gnrlizd roust dignosility of DES n formultd s th prolm of srhing for trs s i t i L i \ K i nd w j K j, i j, whr s i L i \ K i nd t i is n ritrrily long tr, suh tht P oi (s i t i ) = P oj (w j ). If thr xist trs s i, t i, nd w j tht stisfy ths onditions, thn th lngug lss L will non-roustly dignosl with rspt to projtions P oi, i = 1, 2,..., m, nd Σ f. Lt G i G possil modl for th systm ovr th vnt st Σ nd ssum tht G i gnrts lngug L i L. Lt us prtition Σ s Σ = Σ oi Σ uoi whr Σ oi nd Σ uoi r th osrvl nd unosrvl vnt sts for modl G i, rsptivly, nd dfin Σ o = m i=1 Σ o i. W mk th following ssumption. A1. L i is dignosl with rspt to projtion P oi : Σ Σ o i nd Σ f, for i = 1, 2,..., m. 8738

3 Prprints of th 18th IFAC World Congrss Milno (Itly) August 28 - Sptmr 2, 2011 Lt us now dfin th following on-to-on funtions R i : Σ Σ Ri, for i = 1, 2,..., m, s (Morir t l., 2010, 2011) { σ, if σ Σoi Σ R i (σ) = f. (1) σ Ri, if σ Σ uoi \ Σ f It is worth rmrking tht funtion R i just rnms th vnts in Σ uoi \ Σ f. Furthrmor, noti tht funtion R i n xtndd to domin Σ in th usul wy, s follows: R i (ε) = ε, nd R i (sσ) = R i (s)r i (σ), s Σ nd σ Σ. As onsqun, R i n lso xtndd to lngug L Σ y simply pplying it to ll strings in L. Bsd on R i, w n lso dfin th invrs rnming funtion, s follows: R 1 i : Σ Ri Σ σ Ri σ, whr σ Ri = R i (σ), with th following xtnsion to domin Σ R i : R 1 i (s Ri σ Ri ) = R 1 i (s Ri )R 1 i (σ Ri ) for ll s Ri Σ R i nd σ Ri Σ Ri, nd R 1 i (ε) = ε. In ordr to vrify th xistn of trs tht ld to violtion of th gnrlizd roust dignosility ondition givn in Dfinition 3, w will prsnt polynomil tim lgorithm nd, in th squl, thorm tht provs its orrtnss. W lso rry out th omputtionl omplxity nlysis of th lgorithm. 4.1 Vrifition Algorithm W now prsnt n lgorithm for th vrifition of th gnrlizd roust dignosility. Algorithm 1. Stp 1 For h modl G i = (X i, Σ, f i, Γ i, x i,0 ), uild utomton G Ri = (X i, Σ Ri, f Ri, Γ Ri, x i,0 ), whr Σ Ri = R i (Σ), Γ Ri (x) = R i [Γ i (x)], nd f Ri (x, R i (σ)) = f i (x, σ) for ll x X i nd σ Γ i (x). Stp 2 Comput th fulty hvior utomton s follows: Stp 2.1: Build th fulty ll utomton A l = (X Al, Σ f, f Al, x 0,Al ), whr X Al = {N, Y }, x 0,Al = N, nd f Al (N, ) = Y nd f Al (Y, ) = Y. Stp 2.2: Comput G Ri = G Ri A l nd mrk ll stts of G Ri tht hv th sond omponnt qul to Y. Stp 2.3: Comput th ossil prt of utomton G Ri nd dfin th fulty hvior utomton = (X Fi, Σ Ri, f Fi, x 0,Fi ), y unmrking ll mrkd stts of CoA( G Ri ). Stp 2.4: Rdfin th vnt st of s Σ Fi = Σ Ri Σ o. Stp 3 Build th non-fulty utomton H Ri s: Stp 3.1: Dfin Σ Zi = Σ Ri \ Σ f, nd uild utomton Z i = ({N}, Σ Zi, f Zi, N), omposd of singl stt with slf-loop lld with ll vnts in Σ Zi. Stp 3.2: Construt H Ri = G Ri Z i = (X HRi, Σ Ri, f HRi, Γ HRi, x 0,HRi ). Stp 3.3: Rdfin th vnt st of h H Ri s Σ HRi = Σ Ri \ Σ f. Stp 4 Construt th ugmntd utomton H i = (X Hi, Σ Hi, f Hi, x 0,Hi ) from utomton H Ri s follows: Stp 4.1: Dfin Σ Hi = Σ HRi Σ o. Stp 4.2: Dfin x 0,Hi = x 0,HRi. Stp 4.3: Add nw stt D i to th stt sp of H Ri. Thus, X Hi = X HRi {D i }. Stp 4.4: For h x Hi X HRi dfin: f HRi (x Hi, σ), if σ Γ HRi (x HRi ) f Hi (x Hi, σ) = D i, if σ Σ o \ Γ HRi (x HRi ), undfind, othrwis (2) nd for x Hi = D i dfin: { Di, for ll σ Σ f Hi (x Hi, σ) = o undfind, othrwis. (3) Stp 5 For i = 1, 2,..., m, omput vrifir V i whos jth stt x Vij X Fi ( m q=1,q i X H q ) nd th jth stt of X Fi is x Fij X i X Al, y mking omposition of, H 1,..., H i 1, H i+1,..., H m, following th sm produr s for th prlll omposition ( m j=1,j i H j), xpt tht if stt (x Fi, D 1,..., D i 1, D i+1,..., D m ) is rhd, whr x Fi X Fi, thn its fsil vnt st is ford to th mpty st 1, i., Γ Vi (x Fi, D 1,..., D i 1, D i+1,..., D m ) =. Th following rsult provids nssry nd suffiint ondition for gnrlizd roust dignosility. Thorm 1. Th lss L is not roustly dignosl with rspt to P oi, i = 1, 2,..., m, nd Σ f if nd only if thr xists yl l i = (x Vik, σ k, x Vik+1, σ k+1,..., σ l, x Vik ), whr l k > 0, in t lst on vrifir V i, i I m, stisfying th following ondition: j {k, k+1,..., l} s.t. (σ j Σ Ri ) (x Fij = {x ij,y }). (4) Proof. ( =) Suppos tht thr xists yl l i = (x Vik, σ k, x Vik+1, σ k+1,..., σ l, x Vik ), whr l k > 0, in vrifir V i tht stisfis ondition (4). Sin x Fij = {x ij, Y } for on j {k, k +1,..., l}, thn, y onstrution of V i, w hv tht x Fij = {x ij, Y } for ll j {k, k + 1,..., l}. Thus, thr xists tr s t L(V i ), suh tht s ontins th fult vnt, nd t = (σ k σ k+1... σ l ) n, n N. Aording to Algorithm 1, th stts of V i tht r qul to (x Fi, D 1,..., D i 1, D i+1,..., D m ) r ddlok stts. Thrfor, if vrifir V i posssss yl l i tht stisfis ondition (4), thn thr will xist q I mi := {1,..., i 1, i + 1,..., m} suh tht x Hq j D q for ll j {k, k + 1,..., l}. Lt Σ R = m ν=1σ Rν, nd dfin th following projtions: P Fi : Σ R Σ, (5) P Hp : Σ R Σ H p, (6) for p I mi. Aording to Algorithm 1, V i is otind through n utomton oprtion tht is prformd in th sm mnnr s th prlll omposition xpt for th ddlok stt. Thrfor [ ] L(V i ) P 1 [L( )] m p=1,p ip 1 H p [L(H p )], 1 This is quivlnt to sying tht stt (x Fi, D 1,..., D i 1, D i+1,..., D m ) is ford to ddlok stt. 8739

4 Prprints of th 18th IFAC World Congrss Milno (Itly) August 28 - Sptmr 2, 2011 whih implis tht s t P 1 [L( )] sin, y ssumption, s t L(V i ). Lt s t = [ P Fi (s t ), whr s = P Fi (s ) nd t = P Fi (t ). Sin P Fi P 1 (L( )) ] = L( ), it is not diffiult to s tht s t L( ). In ddition, sin t = (σ k σ k+1... σ l ) n, n N, nd, y ssumption, thr xists n vnt σ j Σ Ri for som j {k, k + 1,..., l}, w my onlud tht tr t = P Fi (t ) n md ritrrily long. At this point, it is worth rminding tht L( G Ri ) = L(G Ri ) nd L( ) L( G Ri ). As onsqun, s t L(G Ri ). In ddition, sin G Ri is otind from G i y rnming its vnts in Σ through th rnming funtion R i, dfind in Eqution (1), it is lr tht, ssoitd with s t L(G Ri ), thr xists squn st L i tht is otind s follows: st = R 1 i ( s t). Lt s q = P Hq (s t ). Thn s t P 1 H q [L(H q )] sin s t L(V i ). In ddition, using th ft tht P Hq [P 1 H q (L(H q ))] = L(H q ), thn s q L(H q ). By ssumption, stt x Hq j, is diffrnt from D q for ll j {k, k + 1,..., l}, nd thus s q L(H Rq ). Noti tht H Rq is otind from H q ftr rnming th vnts of th st Σ y using funtion R q. Thus, ssoitd with s q thr xists non-fulty tr s q L(G q ) whih is givn s: s q = R 1 q ( s q ). To onlud th if prt of th proof, noti tht, du to th onstrution of, H q nd V i Stps 2, 4 nd 5, rsptivly, of Algorithm 1 nd sd on th ft tht x Hq j D q for ll j {k, k + 1,..., l} in yl l i, thn ll vnts in Σ o tht r usd to form tr s t nd onsquntly, st nd s q r in Σ oi Σ oq. Thrfor, P oi (st) = P oq (s q ), whih lds to violtion of th roust dignosility ondition in Dfinition 3. (= ) Assum tht th lss L is not roustly dignosl with rspt to projtions P oi, i = 1,..., m, nd Σ f = { }. Thrfor, thr xists pir (i, q) I m I mi suh tht for som s i t i L i \ K i, with s i L i \ K i nd t i n i, n i N, nd for som w q K q (w q not nssrily ritrrily long), P oi (s i t i ) = P oq (w q ). Aording to Algorithm 1, G Ri nd G Rq r otind y rnming th unosrvl vnts of G i nd G q, rsptivly, through th rnming funtions R i nd R q. Thrfor, th following onlusions n drwn: C1. Thr xists tr s i t i = R i (s i t i ), with s i = R i (s i ) nd t i = R i (t i ), s i t i L(G Ri ). In ddition, sin s i t i L i \ K i, i.. s i t i is fulty tr, s i t i L( ). It is importnt to rmrk tht sin t i is ritrrily long, so is t i. C2. Thr xists tr w q L(G Rq ) suh tht w q = R q (w q ). In ddition, sin w q is non-fulty tr of G q, w q L(H Rq ) nd, s onsqun, w q L(H q ). C3. f Fi (x 0,Fi, s i t) = {x i, Y }, whr x i X i, for ll t pr( t i ), whr pr(.) dnots prfix-losur. C4. f Hq (x 0,Hq, w) D q for ll w pr( w q ). Du to Assumption A1, L i is dignosl with rspt to projtion P oi nd Σ f, for i = 1, 2,..., m. Thrfor, it is not diffiult to s tht P oi (s i t i ) ε, nd sin P oi (s i t i ) = Tl 1. Computtionl omplxity of Algorithm 1 No. of stts G i X i X i Σ A l 2 2 G Ri 2 X i 2 X Σ 2 X i 2 X i Σ No. of trnsitions Z i 1 Σ Σ f H Ri X i X i ( Σ Σ f ) H i X i + 1 X i ( Σ uoi Σ f + Σ o )+ Σ o [ ] V i N Vi =2 X i j i ( X j +1) N Vi m( Σ Σf ) + Σ f ( ) Complxity O m X i j i X j ( Σ Σ f ) P oq (w q ), w my onlud tht Σ oi Σ oq. Thrfor, sin L(V i ) P 1 [L( )] [ m p=1,p i P 1 H p [L(H p )]], [ P Fi P 1 (L( )) ] = L( ) nd P Hq [P 1 H q (L(H q ))] = L(H q ), it is possil to find tr s t L(V i ) suh tht s i t i = P Fi (s t ) nd w q = P Hq (s t ), whr s i = P Fi (s ) nd t i = P Fi (t ), with t ritrrily long. To onlud th only if prt of th proof, noti tht vrifir V i is finit stt utomton nd thrfor, sin t is ritrrily long, it n writtn s t = t f (σ kσ k+1... σ l ) n, n N, whr t f is finit lngth tr nd σ k σ k+1... σ l r th vnts of yl l i = (x Vik, σ k, x Vik+1, σ k+1,..., σ l, x Vik ), l k > 0, in whih, σ j Σ Ri for som j {k, k + 1,..., l}. In ddition, sd on onlusions C3) nd C4) ov, it is not hrd to s tht for ll trs s t f t, t pr((σ kσ k+1... σ l ) n ) thn f Vi (x 0,Vi, s t f t) = x V ij, whr th first omponnt of x Vij is {x ij, Y }, x ij X i, nd th q-th omponnt of x Vij is diffrnt from D q. 4.2 Computtionl omplxity of lgorithm 1 Tl 1 shows th mximum numr of stts nd trnsitions of ll utomt tht must omputd in ordr to otin V i ording to Algorithm 1, ssuming tht thr r m possil modls G i G. It n hkd tht th numr of stts nd trnsitions of V i r, in th worst s, qul to 2 X i m j=1,j i ( X j + 1) nd [2 X i m j=1,j i ( X j + 1)][m( Σ Σ f ) + Σ f ], rsptivly. Thrfor, th omplxity of Algorithm 1 is O(m X i m j=1,j i X j ( Σ Σ f )). Th intrmdit stps tht ld to th worst s ound ov r prsntd in Crvlho (2011). Rmrk 1. Noti tht th omputtionl omplxity of Algorithm 1 is O(m X i m j=1,j i X j ( Σ Σ f )), whih is smllr thn th omplxity of th lgorithm proposd y Tki (2010), onsidring th numr of stts of H Ri nd G i r qul, whih is O( X i 3 m j=1,j i X j Σ m+1 ). It is lso importnt to rmrk tht th siz of th vrifir utomton V i is, in gnrl, smllr thn th worst s prsntd in Tl 1 sin th lgorithm srhs only for thos trs in L i \ K i nd K j, i j, tht my ld to violtion of th roust dignosility ondition. 8740

5 Prprints of th 18th IFAC World Congrss Milno (Itly) August 28 - Sptmr 2, EXAMPLE Lt G = {G 1, G 2, G 3 } th lss of utomt shown in Figur 1, nd ssum tht Σ = {,,, d,, σ u, } is th st of ll vnts usd in th modling of th systm. In ddition, lt Σ o1 = {,, }, Σ o2 = {,,, } nd Σ o3 = {,, d, }, rsptivly, th osrvl vnt sts of G 1, G 2 nd G 3. Th ojtiv hr is to vrify if th lss L of lngugs gnrtd y th utomt in G is roustly dignosl with rspt to P oi, i = 1, 2,..., m, nd Σ f = { }. 1N 2N 3Y 4Y () 1N 2N 3Y () 1N 2N 3Y 4Y () R1 Fig. 2. Fulty utomt F 1 (), F 2 (), nd F 3 () d 1 σ u () () () Fig. 1. Clss of utomt G = {G 1, G 2, G 3 }: () G 1 with Σ o1 = {,, }; () G 2 with Σ o2 = {,,, }; () G 3 with Σ o3 = {,, d, }. Initilly, not tht Σ o = 3 i=1 Σ o i = {,,, d, }. Now, ording to Algorithm 1, th first stp is to otin utomton G Ri, i = 1, 2, 3, y rnming th vnts in Σ uoi \Σ f. Thrfor vnts d, nd σ u should rnmd, rsptivly, s, R1 nd σ ur1, in G 1, nd vnt s R3 in G 3. Noti tht no vnt nds to rnmd in G 2. Th stt trnsition digrm of utomt G Ri, i = 1, 2, 3, r not shown sin thy r idntil to thos of G i, i = 1, 2, 3, xpt for th ov rnming. Th nxt stp of Algorithm 1 is to omput th fulty utomt, i = 1, 2, 3. Following Stps , utomt F 1, F 2 nd F 3, shown in Figur 2, r otind. Noti tht, lthough only vnts,, R1 nd ppr in th stt trnsition digrm of F 1, its vnt st is Σ F1 = Σ R1 Σ o = {,,, d,,, R1, σ ur1, }. Sm nlysis n rrid out for F 2 nd F 3 lding to Σ F2 = {,,, d,, d R2, σ ur2, } nd Σ F3 = {,,, d,, R3, σ ur3, }. Th nxt stp of Algorithm 1 is to otin th non-fulty utomt H R1, H R2 nd H R3 tht ounts for th nonfulty hvior of G R1, G R2 nd G R3, rsptivly, nd, in th squl to otin th ugmntd utomt H 1, H 2 nd H 3, whos stt trnsition digrms r dpitd in Figur 3. It is worth rmrking tht Σ H1 = Σ F1 \ { }, Σ H2 = Σ F2 \ { } nd Σ H3 = Σ F3 \ { } Proding in ordn with Stp 5 of Algorithm 1, th vrifir utomt V 1, V 2 nd V 3 must omputd. Figurs 4() nd 4() show th stt trnsition digrm of vrifirs V 1 nd V 3, rsptivly; th stt trnsition digrm of vrifir V 2 hs n omittd sin it lds to onlusion similr to tht drwn from V 3. Th vrifition of roust dignosility of L with rspt to P oi, i = 1, 2,..., m, nd Σ f = { } is rrid out in ordn with Thorm 1. W first onsidr vrifir V 1 shown in Figur 4(). Noti tht sin yl (4Y D 2 6N, R1, 4Y D 2 6N) is formd with n vnt in Σ R1, w my onlud tht L is not roustly dignosl with rspt to P oi, i = 1, 2,..., m, nd Σ f = { }, in spit of th ft tht vrifirs V 2 (not shown in th ppr) nd V 3 (dpitd in Figur 4()) do not hv ny yls with vnts in Σ R2 nd Σ R3, rsptivly, whos first omponnts of thir stts hv fult lls. A los xmintion of vrifir V 1 rvls th trs rsponsil for th non-roust dignosility. Noti tht tr s V1 = R3 n R 1, whr n n ritrrily lrg, tks V 1 from its initil stt to stt 4Y D 2 6N nd yls ovr this stt. Sin s F1 = P F1 (s V1 ) = n R 1 nd s H3 = P H3 (s V1 ) = R3, thn ftr invrs rnm w otin s 1 = d n L 1 nd s 3 = L 3. Furthrmor, P o1 (s 1 ) = P o3 (s 3 ) =, whih implis tht whn tr s 1 ours, it is not possil to onlud tht th systm is ithr in stt 4 of G 1, whih is ftr th ourrn of fult vnt, or in stt 6 of G 3, whih is in norml pth. 6. CONCLUSION W hv proposd nw dfinition of roust dignosility tht nompsss prviously introdud dfinitions of roust dignosility. Instd of singl lngug, w hv onsidrd lss of lngugs gnrtd y lss of utomt tht modl th systm hvior. W hv givn nssry nd suffiint onditions for roust dignosility nd dvlopd polynomil-tim lgorithm for th vrifition of th roust dignosility of lss of lngugs. REFERENCES Bsilio, J.C. nd Lfortun, S. (2009). Roust odignosility of disrt vnt systms. In Prodings of th Amrin Control Confrn, St. Louis, Missouri. 8741

6 Prprints of th 18th IFAC World Congrss Milno (Itly) August 28 - Sptmr 2, N 2N σ ur1,,, d, 5N,,, d, 6N,, d, D 1,,, d, (), d,,, d, 8N 7N,, d, 2N1N1N 1N1N1N C R3 3Y 1N1N 4Y D 2D 3 C R3 3Y 1N5N C R3 1N1N5N 2N1N5N 1N1N1N 2ND 1 2N 1N5N1N 3ND 1 2N σ ur1 1N2N1N 1N 2N 4N 5N 4Y D 26N R1 () 4Y D 1 D 2 () 1N 2N,,, d,, d,,, d, R3,,, d,,, d, D 2,,, d, () () 5N D 3,, d,,,, d,,, d, 6N,, d, Fig. 3. Augmntd non-fulty utomt H 1 (), H 2 (), nd H 3 (). Crvlho, L.K. (2011). Roust dignosility of disrt vnt systms (in Portugus). Ph.D. thsis, Fdrl Univrsity of Rio d Jniro, COPPE-Eltril Enginring Progrm, Rio d Jniro, Brzil. Crvlho, L.K., Bsilio, J.C., nd Morir, M.V. (2010). Roust dignosility of disrt vnt systms sujt to intrmittnt snsor filurs. In Prprints of th 10th Intrntionl Workshop on Disrt Evnt Systms, Brlin, Grmny. Douk, R., Lfortun, S., nd Tnktzis, D. (2000). Coordintd dntrlizd protools for filur dignosis of disrt vnt systms. Disrt Evnt Dynmi Systms: Thory nd Applitions, 10, Lim, S.T.S., Bsilio, J.C., Lfortun, S., nd Morir, M.V. (2010). Roust dignosis of disrt-vnt systms Fig. 4. Vrifir utomt V 1 () nd V 3 (). sujt to prmnnt snsor filurs. In Prprints of th 10th Intrntionl Workshop on Disrt Evnt Systms, Brlin, Grmny. Lin, F. (1993). Roust nd dptiv suprvisory ontrol of disrt vnt systms. IEEE Trnstions on Automti Control, 38(12), Morir, M.V., Jsus, T.C., nd Bsilio, J.C. (2010). Polynomil tim vrifition of dntrlizd dignosility of disrt vnt systms. In Prodings of th 2010 Amrin Control Confrn, Bltimor, MD. Morir, M.V., Jsus, T.C., nd Bsilio, J.C. (2011). Polynomil tim vrifition of dntrlizd dignosility of disrt vnt systms. IEEE Trnstions on Automti Control (to ppr). Qiu, W. nd Kumr, R. (2006). Dntrlizd filur dignosis of disrt vnt systms. IEEE Trnstions on Systms, Mn nd Cyrntis, Prt A, 36(2), Soori, A. nd Hshtrudi-Zd, S. (1993). Roust nd dptiv suprvisory ontrol of disrt vnt systms. IEEE Trnstions on Automti Control, 38(12), Smpth, M., Sngupt, R., Lfortun, S., Sinnmohidn, K., nd Tnktzis, D. (1995). Dignosility of disrt-vnt systms. IEEE Trns. on Automti Control, 40, Tki, S. (2010). Roust filur dignosis of prtilly osrvd disrt vnt systms. In Prprints of th 10th Intrntionl Workshop on Disrt Evnt Systms, Brlin, Grmny. Young, S. nd Grg, V.K. (1991). Trnsition unrtinty in disrt vnt systms. In Prodings of th IEEE Intrntionl Symposium on Intllignt Control, Arlington, Virgini. Zd, S.H., Kwong, R.H., nd Wonhm, W.M. (2003). Fult dignosis in disrt-vnt systms: frmwork nd modl rdution. IEEE Trnstions on Automti Control, 48(7),