On-Time Diagnosis of Discrete Event Systems

Transcription

1 Proeedings of the 9th Interntionl Workshop on Disrete Event Systems Göteorg, Sweden, My 28-30, 2008 FrM1.3 On-Time Dignosis of Disrete Event Systems Adity Mhjn nd Demosthenis Teneketzis Deprtment of EECS, University of Mihign, Ann Aror, MI 48109, USA. Astrt A formultion nd solution methodology for ontime fult dignosis in disrete event systems is presented. This formultion nd solution methodology ptures the timeliness spet of fult dignosis nd is therefore different from ll other pprohes to fult dignosis in disrete event systems whih re symptoti in nture. A monitor oserves projetion of the events tht our in the system. After eh oservtion it n either rise n lrm nd shut down the system or llow the system to ontinue. If the system is stopped when no fult hd ourred, flse lrm penlty is inurred; on the other hnd if fult hd ourred, delyed detetion penlty is inurred. Both these penlties re tre dependent. The on-time dignosis prolem is formulted s minimx optimiztion prolem where the ojetive is to hoose monitoring rule whih minimizes the worst se ost long ll tres of the lnguge desriing the disrete event system. An optiml dignosis rule is determined using dynmi progrmming lgorithm. An exmple is presented whih illustrtes our methodology nd highlights the differene etween our formultion of on-time dignosis with existing results on symptoti dignosis of disrete event systems. I. Introdution Fult dignosis is importnt in mny pplitions suh s trnsporttion systems [1], power systems [2], HVAC (heting ventiltion nd ir-onditioning) systems [3, 4] other industril systems [5] [8], nd ommunition networks [9] [14]. Fult dignosis n e formulted in two different wys: (i) symptoti fult dignosis; nd (ii) on-time fult dignosis. In symptoti fult dignosis the ojetive is to determine, s urtely s possile, the ourrene of fults; the dely in fult dignosis is not ritil. On the other hnd, on-time fult dignosis is onerned with situtions where the timeliness of fult detetion is importnt. Within the ontext of disrete event systems, symptoti fult dignosis hs een investigted in [15] [24], [25] nd referenes therein. To the est of our knowledge, on-time fult dignosis hs not een ddressed so fr within the ontext of DES. In this pper we formulte the on-time dignosis prolem in DES nd provide solution methodology. The rest of the pper is orgnized s follows. In Setion II we introdue some preliminry nottion used in the rest of the pper. In Setion III we formulte nd solve the on-time dignosis prolem. In Setion IV we present n exmple illustrting our solution methodology. We onlude in Setion V. A. Lnguges II. Preliminries Let Σ denote set of events. A lnguge defined over Σ is set of finite-length strings from events in Σ. Σ denotes the Kleene-losure of Σ, tht is, the set of ll finite strings of elements of Σ, inluding the empty stringε. The symol denotes the ontention opertion. Ift u =swith t,u Σ, thentis lled the prefix ofsnd is denoted y t s. Oserve tht othsndεre prefixes ofs. The prefix losure L oflonsists of ll the prefixes of ll the strings inl. A lnguge is lled prefix losed if ny prefix of ny string inlis lso n element ofl. For ny stringselonging to lngugel, the post lngugelfters, denoted yl/s, is given y L/s := t Σ:s t L For ny su-lngugel ofl, the post-lngugelfterl is given y L/L = s L L/s A stringselonging tolis lled terminl string if the post lngugelftersis empty (i.e.,l/s = ). Any lngugeln e prtitioned into the setl T of terminl strings nd the setl NT of non-terminl strings, i.e., L =L T L NT, wherel T := s L:L/s = ndl NT :=L\L T. IfL is prefix-losed, then L NT = L T \L T. For exmple, onsider lngugeldefined over Σ =,, given y then L := ε,,,,,,,,, (1) L T =,,,,, nd B. Prefix-preserving projetions L NT = ε,, Consider prefix-losed lnguge L, nd set vlued funtiono:l NT 2 Σ. For ny strings L,O(s) denotes the oservle set of oservle events immeditely followings. This funtionodefines projetionp s follows: for ny σ Σ, nd nys σ L, /08/$ IEEE 382

2 P(ε) :=ε, σ, ifσ O(ε); P(σ) := ε, otherwise, P(s) σ, ifσ O(s); P(s σ) := P(s), otherwise. The projeted lngugep(l) is the set of projetions of ll strings of L, i.e., P(L) := P(s) :s L. By onstrution, the projeted lnguge is prefix-losed; hene, we ll suh projetions prefix-preserving projetions. The inverse projetionp 1 is mp fromp(l) to 2 L defined s follows P 1 (t) := s L:P(s) =t. These projetions re generliztion of the nturl projetions. In this pper we do not restrit ourselves to nturl projetions, euse, in the future, we wnt to investigte on-time dignosis of deentrlized systems with ommunition. In suh systems, ommunition rules n e tre dependent; so, it is neessry to understnd dignosis in entrlized systems with suh projetions. C. Some su-lnguges nd inverse mppings We define some su-lnguges oflnd some inverse mppings fromp(l) to 2 L tht will e useful in future nlysis. Definition 1: Define su-lngugesl C NT,LC ofls follows: L C NT := s σ L NT :σ O(s), L C :=L C NT L T Definition 2: Define inverse mpsq T,QfromP(L) to 2 L s follows: Q T (t) := s L T :t=p(s), Q(t) := s σ L:σ O(s) ndt =P(s σ). Notie thtq T (t) denotes the set of terminl strings of the lnguge tht give projetion t; Q(t) denotes the set of strings of the lnguge tht give projetiontnd whose lst event is oservle. We illustrte the onepts introdued in Setions B nd C y mens of the following exmple. D. An exmple Consider the lnguge L defined in (1). Let the oservle eventsoe given y: O(ε) =, O() =, nd O() =,. (2) This oservtion funtion is illustrted in Figure 1. Notie tht the set of oservle events is tre-dependent. After ε the eventis not oservle, ut it is oservle fter. Further, the projetions re given y Fig 1. A grphil representtion of the lnguge L of (1) nd the oservle events given y (2). An oservle event is denoted y solid line nd n unoservle event y dshed line. P(ε) =ε, P() =, P() =ε, P() =P() =P() =, P() =, P() =, ndp() =ε. (3) The projeted lngugep(l) = ε,,, nd the inverse projetionp 1 is given y P 1 (ε) = ε,,, P 1 () =,,,,,, nd P 1 () =. The su-lngugesl C NT ndlc re given y: L C NT = nd The mppingsq T ndqre given y: L C =,,,,,,. Q T (ε) =, Q T () =,,,, Q T () =, nd Q(ε) = ε, Q() =,, Q() =. A. The model III. The on-time dignosis prolem We onsider dynmi system represented y prefixlosed lngugelover n event set Σ. In this pper we restrit ttention to finite nd ounded lnguges (i.e., the lnguge hs finite numer of strings, nd the length of eh string in the lnguge is ounded.) Suppose Σ f Σ is the set of fult event. An oserver, whih we ll the monitor, oserves prefix-preserving projetion P with n oservtion funtion O. The monitor hs to sertin whether fult hs ourred in the system or not. After tking eh oservtion it n rise n lrm, in whih se the system is shut down, or it n deide to do nothing, in whih se the system ontinues to operte. The rule used to deide when to rise n lrm is lled monitoring rule nd is funtion fromp(l) to 0, 1, where 0 mens monitor does not rise n lrm nd 1 mens tht the monitor rises n lrm. When n lrm is rised, the system is shut down immeditely nd nnot exeute ny other event. This sttement n est e explined y n exmple. Consider the lnguge shown in Figure 2. Suppose on oserving the monitor deides to rise n lrm. Note thtp(l) = ε,,,, P 1 () =,,,,,,, ndq() =,. All strings inp 1 () led to the oservtion. 383

3 ounting for not stopping fulty system. Thus the ost funtionc is: fors L C NT (n τ(s)), ifsontins fult C(s) = H NT, otherwise () () Fig 2. () A lngugel, () monitored su-lngugel g. forsinl C \L C NT (n τ(s)) +HT, ifsontins fult C(s) = 0, otherwise whereτ(s) denotes the first time fult ours in strings, is onstnt,h NT is the flse lrm penlty, ndh T is the terminl penlty. Now, the performne of monitoring rule n e quntified y the worst se ost when the system stops (either shut down due to n lrm, or finishes ll its tsks) nd is given y However, the moment the monitor oserves fter, the system hs to e in strings inq(). The monitor rises n lrm nd the system is shut down immeditely. So, when the system stops, it would e in one of the strings inq(). The strings elonging top 1 ()\Q() re never rehed. For ny monitoring rule, the system will lwys stop in string elonging tol C. A monitoring poliyg:p(l) 0, 1 restrits the lnguge L to su-lnguge. We ll this restrited lnguge the monitored su-lnguge nd denote it yl g. For the lnguge shown in Figure 2, the monitoring ruleg given y g() = 1,g(ε) =g() =g() = 0 gives the monitored su-lnguge shown in Figure 2. We now wnt to define metri to mesure the qulity of monitoring rule. We wnt this metri to pture the notion of timeliness of fult dignosis. In order to define suh metri we first introdue ost funtionc( ) on ll strings inl C. For strings elonging tol C NT, if the string ontins fult, the ost orresponds to the dmge done y llowing the system to run in fulty stte; if the string does not ontin fult, the ost orresponds to the flse lrm penlty. For strings elonging tol T, if the string ontins fult, the ost orresponds to the dmge done y llowing the system to omplete ll its tsks in fulty stte; if the (terminl) string does not ontin fult, the ost is zero. The hoie of the ost funtionc depends on the pplition. For exmple, onsider system where we wnt to detet fult s soon s possile fter its ourrene (i.e., fter s few s possile events hve ourred fter the fult) nd the flse lrm penlty is fixed. In this sitution, suppose the system is stopped fter non-terminl treshs ourred: ifsdoes not ontin fult, the ost is equl to the flse lrm penlty; ifsontins fult, the ost is proportionl to the numer of events tht hve ourred fter the fult. If the system is not-stopped nd exeutes terminl tres, then ifsdoes not ontin fult the ost is zero; if s ontins fult the ost is proportionl to the numer of events tht hve ourred fter the fult plus fixed penlty J (g) := B. Prolem formultion nd solution mx s (L g ) T C(s). (4) We re interested in the following optimiztion prolem. Prolem 1 (The on-time dignosis prolem): Given prefix-losed, finite nd ounded lnguge L, prefixpreserving projetionp with n oservtion funtiono, nd ost funtionc defined onl C, hoose monitoring rule g elonging to the fmily G of ll funtions fromp(l) to 0, 1, whih minimizes the worst se ost given y (4), i.e. J (g ) =J := min g G mx C(s). (5) s (L g ) T Sine there re only finite numer of monitoring rules, Prolem 1 is well defined nd n optiml rule lwys exists. Prolem 1 is entrlized minimx optimiztion prolem nd n e solved y dynmi progrmming. An optiml solution n e otined s follows. Theorem 1: An optiml monitoring rule for Prolem 1 n e otined y solving the following set of reursive equtions. t (P(L)), T V (t) = min mx t (P(L)) NT, V (t) = min mx C(s), C(s), mx C(s) (6) mx mx C(s), mx V (t e),(7) whereo C (t) := e Σ:t e P(L). Furthermore, n optiml monitoring rule is desried s follows: t (P(L)) T, g 1, if mx C(s)<mx (t) = s QT (t)c(s), 0, otherwise, (8) 384

4 nd t (P(L)) NT, where g (t) = A = mx C(s) 1, ifa<b 0, otherwise. B = mx mx C(s), mx V (t e). s e O C (t) The reursive equtions (6) nd (7) hve the following interprettion: Eq (6):For terminl stringst, lulte the worst ost of stopping (i.e., mx C(s)) nd the worst ost of ontinuing (i.e., mx s QT (t)c(s)) nd hoose the tion whih inurs the smller ost. Eq (7): For non-terminl strings t, lulte the worst ost of stopping (i.e., mx C(s)) nd the worst ost of ontinuing whih is omputed s follows: if the monitor llows the system to ontinue, the tre tht is eing exeuted my () terminte efore the monitor otins nother oservtion, or () its ontinution my led to nother oservtion. The worst ost for () is mx s QT (t)c(s), the worst ost for () is mx e OC (t)v(t e). The mximum of these two osts gives the worst ost of ontinuing. The monitor tkes the tion (stop or ontinue) tht inurs the smller ost. Proof of Theorem 1. We first estlish the following three lemms. Lemm 1: Let J(t; g) denote the worst se ontinution ost fter the monitor hs oservedtnd is using the monitoring ruleg. Then t ((P(L)) T, mx C(s), ifg(t) = 1, J(t;g) = (10) mx s QT (t)c(s), ifg(t) = 0, nd t ((P(L)) NT, ifg(t) = 1 J(t;g) = mx C(s), while t ((P(L)) NT, ifg(t) = 0 J(t;g) = mx mx C(s), mx J(t e;g) Lemm 2: For ny monitoring ruleg nd nyt P(L g ), (9) J(t;g) V (t) (11) Lemm 3: The monitoring ruleg defined y (8) stisfies J(t;g ) =V (t) (12) The ove three lemms re proved in Appendix A. Oserve tht for ny monitoring rule g, ε P(L g ) nd J (g) =J(ε;g). Therefore, Lemms 2 nd 3 imply tht for ny monitoring poliy g, J (g ) =J(ε;g ) J(ε;g) =J (g). (13) Thus,g is n optiml monitoring rule. IV. An exmple In this setion we one lnguge with three instnes of ost funtions. These instnes pture the senrios with high penlty for flse lrm, nd high penlty for delyed detetion, nd the senrio where tstrophi event n our. For eh of these three instnes we find n optiml monitoring rule. The detils of the omputtions re ville t [26]. C 3 C 7 C 11 d C 1 C 8 f d C 4 d C 5 C 2 C 6 C 9 C 10 Fig 3. An exmple of on-time dignosis prolem. The eventf denotes the fult event. The projetion in this se is nturl projetion where eventsndre oservle, while events,d, ndf re unoservle. Consider the lnguge shown in Figure 3. The projetion is nturl projetion where eventsndre oservle while events, d, nd f re unoservle. The event f is fult event. Thus, the oserved lngugep(l) is ε,,,,,. There re 11 strings tht elong tol C ; we denote the ost of stopping t these strings yc i s shown in Figure 3. We onsider the following instnes for the ost funtions. Instne 1 (High flse lrm penlty): Suppose the osts for the lnguge of Figure 3 rec 1 = 1,C 2 = 10,C 3 = C 4 = 3,C 5 = 4,C 6 = 10,C 7 =C 8 = 4,C 9 =C 10 = 10, ndc 11 = 5. The ost of stopping t non-fulty sttes (,,, nd) is muh higher thn the ost of stopping t other sttes, thus, these osts orresponds to sitution where the flse lrm penlty is high. The optiml monitoring rule in this se is g(ε) =g() =g() =g() =g() = 0 g() = 1 Due to the high flse lrm penlty, the monitor does not stop the system until it is sure tht fult hs ourred. In generl, if the flse lrm penlty is infinite, optiml monitoring rule would e the sme s the symptoti dignoser of [15]. Instne 2 (High penlty of delyed detetion): Suppose 385

5 the osts for the lnguge of Figure 3 rec 1 = 10,C 2 = 1, C 3 =C 4 = 11,C 5 = 12,C 6 = 1,C 7 =C 8 = 12,C 9 = C 10 = 1,C 11 = 13. These ost orrespond to sitution where the delyed detetion penlties re muh higher thn flse lrm penlties. The optiml monitoring rule in this se is g(ε) =g() =g() = 0 g() =g() =g() = 1 Due to the high delyed detetion penlty, the monitor rises n lrm nd stops the system s soon s there is possiility tht fult ould hve ourred. Instne 3 (Ctstrophi event): Suppose the osts for the lnguge of Figure 3 rec 1 = 1,C 2 = 10,C 3 = 2,C 4 = 4, C 5 = 100,C 6 = 10,C 7 = 3,C 8 = 12,C 9 = 10,C 10 = 15, C 11 = 12. These osts orrespond to sitution when the ourrene of twodevents fter fult re tstrophi. The optiml monitoring rule in this se is g(ε) =g() =g() = 0 g() =g() =g() = 1 Due to the tstrophi nture offdd, the monitor rises n lrm when there is possiility tht ontinuing the system ould led to the tstrophi string. V. Conlusion We hve formulted nd solved fult dignosis prolem within the ontext of disrete event models. Our formultion tkes into ount the timeliness of fult dignosis. The key ide in the formultion is to penlize flse lrm s well s the string-dependent mount of dely in fult detetion. With these penlties the on-time dignosis prolem n e formulted s minimx optimiztion prolem. We provided n lgorithm for the solution of this prolem nd illustrted vi exmples the nture of the solution nd its differenes from the solution of the symptoti fult dignosis prolem tht hs ppered in the literture. Aknowledgements This reserh ws supported in prt y NSF Grnt CCR Appendix A. Proof of the three lemms 1. Proof of Lemm 1 Ifg(t) = 1, then n lrm is rised nd the system is shutdown. The system n hve exeuted ny of the strings in Q(t) when the lst event intis oserved. So, the worst se ontinution ost is the mximum ost inurred mongst ll strings in Q(t). Thus, J(t;g) = mxc(s), wheng(t) = 1. (14) If g(t) = 0 then the monitor llows the system to ontinue. Ift ((P(L)) T, then the monitor will not see ny other event 386 in the future. The system will ultimtely stop in one of the strings inq T (t) nd will inur orresponding ost. Thus, the worst se ontinution ost in this se is J(t;g) = mx C(s), wheng(t) = 0 ndt (P(T)) T. (15) Ift (P(L)) NT, the system my either end up in terminl string inq T (t), in whih se the monitor will not see nything in the future, or the system will not end inq T (t), in whih se the monitor will oserve some event ino C (t) in the future. So, the worst se ontinution ost in this se is the higher of these osts, hene J(t;g) = mx mx C(s), mx mx C(s), mx V (t e) wheng(t) = 0 ndt (P(L)) NT. (16) Equtions (14), (15), nd (16) omplete the proof of the lemm. 2. Proof of Lemm 2 PrtitionP(L g ) into disjoint setsm 0,M 1,...suh tht M 0 = t (P(L g )) T M 1 = t (P(L g )) NT : σ 1 Σ suh thtt σ 1 (P(L g )) T = M i = t (P(L g )) NT : σ 1,,σ i Σ suh thtt σ 1 σ i (P(L g )) T Note tht y onstrution, for llt M i,i 0,g(t) = 0. We wnt to show tht for llt P(L g ) J(t;g) V (t) (17) We will show this y indution on the setsm i. Fort M 0, J(t;g) = mxc(s) or mx C(s) min mxc(s), mx C(s) =:V(t) (18) This is the sis for indution. Now ssume tht for ll t M 0 M i, (17) is true. We will show tht (17) is lso true for llt M i+1. IfM i+1 =, the lim is vuous truth. Otherwise, onsidert M i+1. As noted oveg(t) = 0. Thus, J(t;g) = mx mx C(s), mx J(t e;g) Oserve tht for lle O C (t),t e elongs tom 0 M i. By the indution hypothesisj(t e;g) V (t e). Thus,

6 J(t;g) = mx mx C(s), mx J(t e;g) mx mx C(s), mx V (t e) =:V(t) (19) Thus, y the priniple of indution, the lemm is true. 3. Proof of Lemm 3 This lemm n e proved long the sme lines s the proof of Lemm 2. By the definition ofg (t), the reltion of (18) holds with equlity. As onsequene of this, the reltion in (19) lso holds with equlity. Therefore, y the priniple of indution, Lemm 3 is true. Referenes [1] R. Sengupt, Disrete event dignosis of utomted vehiles nd highwys, in Pro. of 2001 Amerin Control Conf., June [2] T.-S. Yoo nd H. Gri, Event dignosis of disrete event systems with uniformly nd non-uniformly ounded dignosis delys, in Pro of 2004 Amerin Control Conf, 2004, pp [3] K. Sinnmohideen, Disrete-event dignosis of heting, ventiltion, nd ir-onditioning systems, in Pro. of 2001 Amerin Control Conf., June [4] M. Smpth (1995). Disrete event systems sed dignostis for vrile ir volume terminl ox pplition. Tehinl Report Advned Development Tem, Johnsons Control In.. [5] M. Smpth, A. Godme, E. Jkson, nd E. Mllow, Comining qulittive nd quntittive resoning hyrid pproh to filure dignosis of industril systems, in IFAC Sveproesses, 2000, pp [6] M. Smpth (1999). Emedded print engine dignostis: The DC265 projet nd eyond. Tehnil Report X , Xerox Corportion. [7] E. Gri, F. Mornt, R. Blso-Giménez, nd E. Quiles, Centrlized modulr dignosis nd the phenomenon of oupling, in Pro of the 6th Interntionl Workshop on Disrete Event Systems (WODES 02), 2002, pp [8] Y.-L. Chen nd G. Provn, Modelling nd dignosis of timed disrete event systems ftory utomtion exmple, in Pro. of 1997 Amerin Control Conf., 1997, pp [9] Y. Penolé, Dignosti déentrlisé de systèmes à évènements disrets: pplition ux résux de téléommunitions, Ph.D. Thesis, Université de Rennes, Frne, [10], Deentrlized dignoser pproh: Applition to teleommunition networks, in Pro. DX 00: Eleventh Interntionl Workshop on Priniples of Dignosis, 2000, pp [11] Y. Penolé, M.-P. Cordier, nd L. Rozé, A deentrlized model-sed dignosti tool for omplex systems, in Pro of 13th IEEE Int. Conf. on Tools with Arif. Intel. (IC-TAI 01), 2001, pp [12] L. Rozé, Supervision de réseux de téléommunitions: Une pprohe à se de modèles, Ph.D. Thesis, Université de Rennes I, Frne, [13] L. Rozé nd M.-O. Cordier, Dignosing disrete-event systems: An experiment in teleommunition netowrks, in Pro of the 1998 Interntionl Worksho[ on Disrete Event Systems (WODES 98), 1998, pp [14], Dignosing disrete-event systems: Extending the dignoser pproh to del with teleommunition netowrks, Disrete Event Dynmi Systems: Thoery nd Applitions, vol. 12, pp , [15] M. Smpth, R. Sengupt, S. Lfortune, K. Sinnmohideen, nd D. Teneketzis, Dignosility of disrete event systems,, vol. 40, no. 9, pp , Sep [16], Filure dignosis using disrete event models,, vol. 4, no. 2, pp , Mr [17] M. Smpth, S. Lfortune, nd D. Teneketzis, Ative dignosis of disrete event systems,, vol. 43, no. 7, pp , Jul [18] M. Lrson, Dignosis nd nlysis of dignosis properties using disrete event systems, in Pro of 37th IEEE Confrerene on Deision nd Control, 1998, pp [19] T. Chun (1996). Dignosti supervisory ontrol: A disrete event system pproh. Mster s thesis, University of Toronto, Dept. of Ele. Eng.. [20] S. Jing, R. Kumr, nd H. Gri, Dignosis of repeted/intermittent filures in disrete event systems, Trnsporttion Reserh, vol. 19, no. 2, pp , Apr [21] J. Ashley nd L. E. Hollowy, Qulittive dignosis of ondition systems, Disrete Event Dynmi Systems: Theory nd Applitions, vol. 14, no. 4, pp , Ot [22] P. Broni, G. Lmperti, P. Poglino, nd M. Znell, Dignosis of lss of distriuted disrete-event systems, Systems, Mn nd Cyernetis, Prt A, IEEE Trnstions on, vol. 30, no. 6, pp , Nov [23] S. Bvishi nd E. K. P. Chong, Automted fult dignosis using disrete event systems frmework, in Intelligent Control, 1994., Proeedings of the 1994 IEEE Interntionl Symposium on, 1994, Columus, OH, pp [24] O. Contnt, S. Lfortune, nd D. Teneketzis, Filure dignosis of disrete event systems: The se of intermittent filures, in Pro of 41st IEEE Conferene on Deision nd Control, [25] S. Lfortune, D. Teneketzis, M. Smpth, R. Sengupt, nd K. Sinnmohideen, Filure dignosis of dynmi systems: n pproh sed on disrete event systems, in Amerin Control Conferene, Proeedings of the 2001, vol. 3, 2001, Arlington, VA, pp [26] A. Mhjn nd D. Teneketzis, On-time dignosis of disrete event systems some exmples, Aville t ditym/pulitions/onferenes/wodes2008/ppendix.pdf. 387