PREDICTABILITY IN DISCRETE-EVENT SYSTEMS UNDER PARTIAL OBSERVATION 1. Sahika Genc, Stéphane Lafortune

Transcription

1 PREDICTABILITY IN DISCRETE-EVENT SYSTEMS UNDER PARTIAL OBSERVATION 1 Shik Gen, Stéhne Lfortune Dertment of Eletril Engineering n Comuter Siene, University of Mihign, 1301 Bel Avenue, Ann Aror, MI , USA sgen,stehne@ees.umih.eu Coyright 2006 IFAC Astrt: This er stuies the rolem of reiting ourrenes of signifint event in isrete-event system. The reitility of ourrenes of n event in system is efine in the ontext of forml lnguges. The reitility of lnguge is stronger onition thn the ignosility of the lnguge. An imlementle neessry n suffiient onition for reitility of ourrenes of n event in systems moele y regulr lnguges is resente. Keywors: Disrete-event systems, reition, ignosis. 1. INTRODUCTION This er resses the rolem of reiting ourrenes of signifint (e.g., fult) event in isrete-event system (DES). The system uner onsiertion is moele y lnguge over n event set. The event set is rtitione into oservle events (e.g., sensor reings, hnges in sensor reings) n unoservle events, i.e., the events tht re not iretly reore y the sensors tthe to the system. The ojetive is to reit ourrenes of ossily unoservle event in the system ehvior, se on the strings of oservle events. If it is ossile to reit ourrenes of n event in the system, then eening on the nture of the event the system oertor n e wrne n the oertor my eie to hlt the system or otherwise tke reventive mesures. 1 This reserh is suorte in rt y NSF grnt CCR n y ONR grnt N The first uthor wishes to knowlege suort from Brour Fellowshi from the Hore H. Rkhm Shool of Grute Stuies t the University of Mihign. To the est of our knowlege, the notion of reitility tht is introue n stuie in this er is ifferent from rior works on other notions of reitility in (Co, 1989; Buss et l., 1991; Shenging n Kumr, 2004; Fel n Hollowy, 1999). For instne, the reition rolem onsiere in (Co, 1989) is relte to the roerties of seil tye of rojetion etween two lnguges (sets of trjetories); this is is muh more generl thn our ojetive, whih is to reit ourrenes of seifi events, ut our work is not seil se. The stte reition of oule utomt stuie in (Buss et l., 1991) is formulte s omuting the stte vetor of n ientil utomt fter T stes in the oertion of the system; the system struture in this work is ifferent from ours. In our se the interest is on single utomton n event reition, not stte, uner rtil oservtion. The notion of reition onsiere in (Shenging n Kumr, 2004) iffers from the one in our work in the sense tht in (Shenging n Kumr, 2004) reitility of system is neessry onition for ignosility of the system while in our work ignosility is neessry onition for

2 reitility. The reition rolem stuie in (Fel n Hollowy, 1999) onsiers issuing wrning when it is likely for fult to hen in the future evolution of the system; in our work, if the ourrene of n event is reitle in lnguge, then it is ertin tht the event will our. Also, in (Fel n Hollowy, 1999), it is ossile tht flse fult reition wrnings re issue; in our work, no flse ositives re issue. The rolem of reition stuie in this er is insire y the rolem of fult ignosis for DES. The rolem of fult ignosis for DES hs reeive onsierle ttention in the lst ee (see the referenes in (Smth et l., 1996)) n ignosis methoologies se on the use of isrete-event moels hve een suessfully use in vriety of tehnologil systems rnging from oument roessing systems to intelligent trnsorttion systems. A isrete-event roess lle ignoser introue in (Smth et l., 1996) is of rtiulr relevne to the resent work. Lter in the er, the ignoser is use to erive neessry n suffiient onition for reitility in systems moele y regulr lnguges. The rest of the er is orgnize s follows. In Setion 2, the nottion n frequently use terms re introue. In Setion 3, the reitility of ourrenes of n event in system is efine in the ontext of forml lnguges. The reitility roerty of lnguge is stronger onition thn the ignosility of the lnguge s efine in (Smth et l., 1996). In Setion 4, it is shown tht in the se of regulr lnguges, there exists neessry n suffiient onition for reiting ourrenes of n event in the lnguge in the form of test on ignosers. In Setion 5, summry of the results in the er is resente, n onluing remrks re given. Omitte roofs re ville in (Gen, 2006). 2. PRELIMINARIES Let Σ e finite set of events. A string is finite-length sequene of events in Σ. The set of ll strings forme y events in Σ is enote y Σ. The set Σ is lso lle the Kleene-losure of Σ. Any suset of Σ is lle lnguge over Σ. The refix-losure of lnguge L is enote y L n efine s L = {s Σ : t L suh tht st L}. Given string s L, L/s is lle the ost-lnguge of L fter s n efine s L/s = {t Σ : st L}. L is live if every string in L n e extene to nother string in L. Let L e lnguge over Σ = Σ o Σ uo, where Σ o n Σ uo enote the oservle n unoservle events, resetively. The rojetion of strings from L to Σ o is enote y P. Given string s L, P (s) is otine y removing unoservle events (elements of Σ uo ) in s. The inverse rojetion of string s o Σ o with reset to L is the set of strings in L whose rojetion is equl to s o. Given n event σ Σ n string s Σ, we use the set nottion σ s to sy tht σ ers t lest one in s. Let L e refix-lose n live lnguge over Σ. Given n event σ Σ n L, S(σ, L) is the set of strings in L tht ens with σ. Formlly, S(σ, L) = {sσ L : s Σ, σ Σ}. 3. PROBLEM STATEMENT In this setion, we efine the rolem of reiting ourrenes of n event in system tht is uner rtil oservtion. We moel the system s lnguge L over n event set Σ. The event to e reite my e n unoservle event or n oservle one. First, we resent n illustrtive exmle to introue the notion of reitility. Then, we give the forml efinition for reitility of the ourrene of n event. We onlue the setion y omring the ignosility of lnguge L s efine in (Smth et l., 1996) to the reitility of L. Roughly seking, the ourrene of n event in lnguge is reitle if it is ossile to infer out future ourrenes of the event se on the oservle reor of strings tht o not ontin the event to e reite. Consier ny string s in S(σ, L) where σ is the event to e reite. We wish to fin refix t of s suh tht t oes not ontin σ n ll the long-enough ontinutions in L of the strings with the sme rojetion s t ontin σ. If there is t lest one suh t, then the ourrene of σ is reitle in L. Consier the refix-lose, live lnguge generte y the utomton shown in Fig. 1. The lnguge generte is L = + + +, (1) where Σ uo = {, } n Σ o = {, }. Let e the event to e reite. The set of strings tht en with is S(, L) = {,, }. (2) In orer to show tht is reitle in L, we must fin n n N n t s for ll s S(, L) suh tht / t n for ll u n its ontinutions v L/u if u reors the sme string of oservle events s t, i.e., P (t) = P (u), n u oes not ontin, i.e. / u, n v is of length greter thn n N, i.e. v n,

3 then v ontins. Let us strt with s = S(, L). Then t. Suose tht t =. Then, P 1 () (Σ \ {}) L = {ɛ,, }. If u =, then L/u = + +. Sine /, there is ontinution of u tht oes not ontin. Then, there exists string whih reors the sme string of oservle events s t n not ll of its ontinutions ontin. Thus, t = is wrong hoie to rove the reitility of. Suose tht t =. For ll u P 1 () (Σ \ {}) L = {,, } n for ll v L/u suh tht v 2, then v ontins. Thus, t = is right hoie for s S(, L). Similrly, it n e verifie tht t = n t = work for s = n s = in S(, L), resetively Fig. 1. G N,8N,3N 9N,6F1,5F1 10F1,6F1,5F1 0N Fig. 2. D G. Bse on the ove isussion, we formlly efine the notion of reitility. Definition 1. Given L refix-lose, live lnguge over Σ, ourrenes of event σ Σ re reitle in L with reset to P if ( n N)( s S(σ, L))( t s)[(σ / t) P] where P : ( u L)( v L/u)[(P (u) = P (t)) (σ / u) ( v n) (σ v)]. 3.1 Dignosility vs. Preitility The reitility of ourrenes of n event σ in refix lose n live lnguge L is stronger thn the ignosility of L with reset to σ. We onsier the ignosility s efine in (Smth et l., 1996) in the ontext of forml lnguges. Roughly seking, L is ignosle with reset to σ if it is ossile to etet ourrenes of σ with finite ely. For the ske of omleteness, we rell in Definition 2 the forml efinition of ignosility. Definition 2. A refix-lose n live lnguge is ignosle with reset to P n σ if ( n N)( s S(σ, L))( t L/s)[ t n D] 11N where D : ω P 1 P (st) L σ ω. We now resent n illustrtive exmle where lnguge is ignosle with reset to n event ut the ourrene of the event is not reitle. We onsier the lnguge generte y the utomton shown in Fig. 3. The lnguge is L = e + e + + e (3) where Σ o = {,,, } n Σ uo = {e, }. In this se, the ourrene of is not reitle. Let s = e S(, L). Then, t e. For ny t e, we lwys hve hve the string n where n 0, whih oes not ontin. Thus, there oes not exist t so tht Definition 1 is stisfie for. However, the ourrene of (n unoservle event) n e etete with finite ely. After the oservtion of, we re ertin tht hs ourre t lest one. Thus, L is ignosle with reset to σ ut the ourrene of σ is not reitle in L e 8 Fig. 3. G. 0 e e N 6N,3N Fig. 4. D G. 0N 1N,10N 10N 8F1,5F1 The following roosition follows iretly from the ove efinitions. Proosition 3. Given refix-lose n live lnguge L Σ, if ourrenes of σ Σ re reitle in L with reset to P, then L is ignosle with reset to P n σ. 4. VERIFICATION OF PREDICTABILITY FOR REGULAR LANGUAGES In this setion, we onsier systems moele y regulr lnguges. Regulr lnguges re the lnguges tht re ete (or generte) y Finite Stte Automt (FSA). An FSA is four-tule G = (Q, Σ, δ, q 0 ) (4) where Q is the set of sttes, Σ is the finite set of events, δ : Q Σ Q is the stte trnsition funtion n q 0 is the initil stte.

4 The neessry n suffiient onition (resente lter in this setion) for reitility is se on isrete-event roess lle ignoser. The ignoser is n FSA uilt for the system with reset to rojetion P onto the set of oservle events n to given event. Let G = (Q, Σ, δ, q 0 ) e n FSA tht genertes lnguge L. We enote y D G the ignoser uilt for G n σ Σ. The ignoser D G is of the form D G = (Q D, Σ o, δ D, q D,0, σ ), (5) where Q D is the set of ignoser sttes, δ D : Q D Σ o Q D is the ignoser stte trnsition funtion, q D,0 Q D is the initil ignoser stte. The ignoser stte se Q D is suset of 2 Q {N,F 1}. Stte q D Q D is of the form q D = {(q 1, l 1 ),..., (q n, l n )}, (6) where q i Q n l i {N, F 1} for i = 1,..., n. Let q D n q D e two ignoser sttes in Q D suh tht q D is rehe from q D y σ o Σ o, i.e., q D = δ D(q D, σ o ) is efine. Let q D = {(q 1, l 1 ),..., (q m, l m )} n q D = {(q 1, l 1),..., (q n, l n)}. For ll i {1,..., n}, there exists j {1, 2,..., m} suh tht q i = δ(q j, s), (7) where s = tσ o n t Σ uo, n { l i F 1, if lj = F 1 or (σ = s), (8) N, if l j = N n (σ / s). We sy tht ignoser stte q D = {(q 1, l 1 ),..., (q m, l m )} Q D for m N is norml if l j = N for ll j = 1,..., m; ertin if l j = F 1 for ll j = 1,..., m; n unertin if there exist l j = N n l i = F 1 for some i, j {1,..., m}. We enote y Q N D Q D the set of ignoser sttes tht re norml, y Q U D Q D the set of ignoser sttes tht re unertin, n y Q C D Q D the set of ignoser sttes tht re ertin. Consier FSA G in Fig. 1. Let Σ uo = {, }. The ignoser 2 for G n is s shown in Fig. 2. The ignoser stte {1N, 8N, 3N} is norml, {9N, 6F 1, 5F 1} is unertin, n {10F 1, 6F 1, 5F 1} is ertin. We efine n essiility oertion on n FSA to fin the essile rt of n FSA from stte. Definition 4. Let G = (Q, Σ, δ, q 0 ) n q Q. The essile rt of G with reset to q is enote y A(G, q) n is A(G, q) = (Q, Σ, δ, q), (9) where Q = {q Q : ( s Σ )(δ(q, s) = q is efine)}, n δ = δ Q Σ Q. 2 Dignosers shown in this er re uilt using UMDES- LIB softwre (Lfortune, n..). Let G = (Q, Σ, δ, q 0 ). We sy tht set of sttes {q 1, q 2,..., q n } Q n string σ 1 σ 2... σ n Σ form yle if q i+1 = δ(q i, σ i ), i = 1, 2,..., n 1 n q 1 = δ(q n, σ n ). In the rest of this setion, we ssume the system stisfies the following: If {q 1, q 2,..., q n } Q n σ 1 σ 2... σ n Σ form yle, then there exists t lest one oservle event σ j in {σ 1,..., σ n } Σ. Tht is, G oes not ontin yle in whih sttes re onnete with unoservle events only. Lemm 5 elow sttes tht if there is yle in D G tht ontins ertin ignoser stte, then ll the ignoser sttes in the yle re ertin (sine the F 1 lel rogtes). Lemm 6 sttes tht if there is yle in D G tht is forme y unertin or norml sttes, then there exists orresoning yle in G suh tht ll the sttes in the yle hve norml lels in the yle in D G. Lemm 5. Let G = (Q, Σ, δ, q 0 ) e n FSA tht genertes L suh tht L is refix-lose n live n let D G = (Q D, Σ o, δ D, q D,0, σ ) e the ignoser for G n σ. Suose {q D,1,..., q D,n } Q D n σ o,1... σ o,n Σ o form yle in D G where n N. If there exists i {1, 2,..., n} suh tht q D,i Q C D, then q D,j Q C D for ll j = 1, 2,..., n. Lemm 6. Let G = (Q, Σ, δ, q 0 ) e n FSA tht genertes L suh tht L is refix-lose n live, n let D G = (Q D, Σ o, δ D, q D,0, σ ) e the ignoser for G n σ. Suose {q D,1,..., q D,n } Q D n σ o,1... σ o,n Σ o form yle in D G where n N n q D,i is in Q U D or QN D for ll i = 1, 2,..., n. Then, there exists (q i, l i ) q D,i for i = 1, 2,..., n, suh tht q i+1 = δ(q i, s i ) for i = 1, 2,..., n 1 n q 1 = δ(q n, s n ) where s i Σ, P (s i ) = σ o,i, n l i = N for i = 1, 2,..., n. Let F D e the set of norml ignoser sttes tht ossess n immeite suessor tht is not norml. Formlly, F D = {x D Q N D : y D = δ D (x D, σ o ) suh tht σ o Σ o n y D / Q N D }. Lemm 7 sttes tht ny unertin or ertin ignoser stte is rehe from ignoser stte in F D. Lemm 7. Let G = (Q, Σ, δ, q 0 ) e n FSA tht genertes L suh tht L is refix-lose n live, n let D G = (Q D, Σ o, δ D, q D,0, σ f ) e the ignoser for G n σ. Let x D,i = δ D (x D,i 1, σ o,i ) for i = 1, 2,..., m where m N, x D,i is ignoser stte, σ o,i is n oservle event for i = 1, 2,..., m, n x D,0 is the initil ignoser stte. If x D,m is in Q U D or QC D, then there exists M m suh tht x D,M F D. Proof 8. The roof is y inution on the sequene of oservle events.

5 Bse (m = 1): In this se, x D,m = x D,1 / Q N D n x D,1 = δ D (x D,0, σ o,1 ). Sine x D,0 is the initil ignoser stte, y efinition it is norml. If the immeite suessor x D,1 of x D,0 is not norml ignoser stte, then x D,0 F D. This omletes the roof of inution se. Hyothesis (m = M ): If x D,M / Q N D, then there exists M M suh tht x D,M F D. Ste (m=m +1): We nee to show tht if x D,M +1 / Q N D, then there exists M M + 1 suh tht x D,M F D. We onsier two ses: (i) x D,M Q N D, n (ii) x D,M / QN D. In the first se, if x D,M Q N D, then x D,M is in F D y efinition. For the other se, if x D,M / Q N D, then y the inution hyothesis there exists M M < M + 1 suh tht x D,M is in F D. This omletes the roof of the inution ste. In the following theorem, we stte the neessry n suffiient onition for reitility of ourrenes of n event. The onition is se on nlyzing the yles in the ignoser. Theorem 9. Let G = (Q, Σ, δ, q 0 ) e n FSA tht genertes L where L is refix-lose n live. Let D G = (Q D, Σ o, δ D, q D,0, σ ) e the ignoser for G n σ. The ourrenes of σ re reitle in L with reset to P iff for ll q D F D, onition C hols, where C : ll yles in A(D G, q D ) re yles of ertin ignoser sttes. Proof 10. The roof is in two rts. ( ): We rove tht if σ is reitle in L, then for ll q D F D the only yles in A(D G, q D ) re yles of ertin ignoser sttes. The roof is y ontrition. Suose tht there exists q D F D suh tht A(D G, q D ) ontins yle forme y {x D,1,..., x D,m } n σ o,1... σ o,m Σ o where x D,i / Q C D for some i {1, 2,..., m}. By Lemm 5, if there exists ignoser stte x D,i in the yle suh tht x D,i is not ertin ignoser stte, then none of the other ignoser sttes in the yle re ertin. Thus, x D,i / Q C D for ll i = 1, 2,..., m. By Lemm 6, orresoning to the yle of ignoser sttes in the ignoser, there exists yle in G suh tht eh stte in tht yle is lele with N in the yle in the ignoser. Suose tht the yle in G is forme y {x 1,..., x m } n s 1... s m Σ where (x i, N) x D,i n s i Σ suh tht P (s i ) = σ o,i for i = 1, 2,..., m. Let q D F D e rehe from the initil ignoser stte q D,0 y s o Σ o. Sine q D is in F D, then there exists s S(σ, L) suh tht P (s) = s o. Moreover, sine σ is reitle in L, then y efinition of reitility, there exists t s suh tht (σ / t) P. We now rove tht there exists u suh tht P (u) = P (t) n σ / u, n for ll ontinutions v of u if v is of length greter thn ny n N, then v oes not ontin σ. Pik ignoser stte in the yle. Without loss of generlity ik x D,1. Then, we ik the stte in the ignoser stte whih hs lel N n is rt of the orresoning yle in G. Let (x 1, l 1 ) e tht stte in x D,1, with l 1 = N. Suose tht x D,1 is rehe from q D y exeuting s o Σ o. Then, x D,1 = δ D (q D,0, s o s o). Let u n u L/u e suh tht P (uu ) = s o s o n x 1 = δ(q 0, uu ). Sine l 1 = N, then neither u nor u ontin σ. Sine x 1 is in the orresoning yle in G, then x 1 = δ(q 0, uu (s 1 s 2... s m ) k ) for k N. Let v = u (s 1 s 2... s m ) k where k is n integer suh tht v n. Then, there exists u n v L/u suh tht P (u) = P (t) (sine P (t) = s o ), σ / u, v n n σ / v (sine l 1 = N). This violtes the onition P in the efinition of reitility. Thus, there is ontrition. This omletes one rt of the roof. ( ): We rove tht if for ll q D F D the only yles in A(D G, q D ) re yles of ertin ignoser sttes, then σ is reitle in L. Pik ny s S(σ, L). Let q = δ(q 0, s) Q. Then, ik ny s uo σ o L/s suh tht s uo Σ uo n σ o Σ o. Let y = δ(q, s uo σ o ) Q. Suose tht P (s) = s o Σ o. Then, let x D = δ D (q D,0, s o ) n y D = δ D (x D, σ o ) in Q D. Then, there exists (y, l y ) y D where l y = F 1. Thus, y D Q U D Q C D. We now onsier the following two ses: (i) x D Q N D, thus, x D F D, n (ii) x D Q U D QC D. Cse (i). Sine x D Q N D n y D / Q N D, then x D F D. We hoose t = s. For ll u suh tht P (u) = P (t), P (u) = s o. Sine the only yles in A(D G, x D ) re yles of ertin sttes, then for ll v L/u, v ontins σ. Cse (ii). If x D Q U D QC D, i.e., x D is not norml, then we wish to fin norml ignoser stte in F D from whih x D is rehe. By Lemm 7, there exists ignoser stte w D rehle from the initil ignoser stte, x D is essile from w D, n w D is in F D. Then, sine F D onsists of norml ignoser sttes, w D is in Q N D. Thus, the roof of Cse (ii) reues to the se of (i) in whih we sustitute w D Q N D for x D Q N D. This omletes the seon rt of the roof. Consier the FSA in Fig. 1 n the orresoning ignoser in Fig. 2 where Σ uo = {, } n Σ o = {, }, n F D = {{1N, 8N, 3N}}. The

6 essile FSA from {1N, 8N, 3N} ontins only one yle forme y {10F 1, 6F 1, 5F 1} whih is ertin ignoser stte. Thus, the ourrene of is reitle. If we onsier the FSA in Fig. 3 n the orresoning ignoser in Fig. 4 where Σ o = {,,, } n Σ uo = {e, }, then, F D = {{6N, 3N}}. The essile FSA from {6N, 3N} ontins two yles one of whih ontins norml ignoser stte. Here, the ourrene of is not reitle. We now show tht it is suffiient to test onition C in Theorem 9 on ertin susets of F D to gurntee tht this onition hols for ll sttes in F D. Corollry 11. Let x D, y D F D suh tht y D = f D (x D, s o ) is efine for some s o Σ o. Then, onition C hols for ll q D F D iff C hols for ll q D F D \ {y D }. F D, thus resulting in omuttionl svings one Min(S ED ) hs een omute. 5. CONCLUSION We hve efine the new roerty of reitility of the ourrene of signifint event (e.g., fult) se on the urrent reor of oservle events. We hve shown neessry n suffiient onition for reitility in the se of systems moele y regulr lnguges. We hve resente test to verify the reitility roerty se on ignosers. An lternte test of olynomiltime omlexity (in the numer of system sttes) is resente in (Gen, 2006). The stuy of reitility is insire n motivte y the stuy of fult ignosis. Our long term gol is to form n integrte theory of ignosis n reition in the frmework of forml lnguges. In view of Corollry 11, let us ll suset of F D C-suffiient if testing onition C in Theorem 9 on this suset is suffiient to gurntee tht C hols for ll q D F D. Denote y S FD the set of ll C-suffiient susets of F D. Let Min(S FD ) enote ll susets of F D in S FD tht hve minimum rinlity. Proosition 12. Min(S FD ) is not singleton in generl. Define reltion etween x D n y D in F D s follows: x D y D s o, t o o suh tht y D = δ D (x D, s o ) n x D = δ D (y D, s o ). Tht is, two sttes in F D re relte if oth of them er in yle in the ignoser. Proosition 13. The reltion is n equivlene reltion. We now work on the equivlene lsses (inue y ) in F D inste of the sttes in F D. Let E D e the equivlene lsses of F D for the reltion. Denote y S ED the set of ll C-suffiient susets of E D. Let Min(S ED ) enote ll sets in S ED tht hve minimum rinlity. Theorem 14 sttes tht there is only one C-suffiient suset of E D with the minimum rinlity. Theorem 14. Min(S ED ) is singleton. We hve eveloe n lgorithm for fining this unique element in Min(S ED ). In view of Corollry 11 n Theorem 14, the neessry n suffiient onition for reitility in Theorem 9 eomes: Conition C hols for ll q D Min(S ED ). In generl, Min(S ED ) REFERENCES Buss, Smuel R., Christos Pimitriou n John Tsitsiklis (1991). On the reitility of oule utomt: n llegory out hos. Comlex Systems 5, Co, Xi-Ren (1989). The reitility of isrete event systems. IEEE Trns. Automti Control 34(11), Fel, H.K. n L.E. Hollowy (1999). Using SPC n temlte monitoring metho for fult etetion n reition in isrete event mnufturing systems. In: Proeeings of the 1999 IEEE Interntionl Symosium on Intelligent Control/Intelligent Systems n Semiotis Gen, Shik (2006). On ignosis n reitility of rtilly-oserve isreteevent systems. Ph.D. Thesis. University of Mihign, Eletril Engineering: Systems. Lfortune, S. (n..). Umes-li softwre lirry. htt:// tooloxes.html. Smth, M., R. Sengut, S. Lfortune, K. Sinnmohieen n D. Teneketzis (1996). Filure ignosis using isrete event moels. IEEE Trns. Control Systems Tehnology 4(2), Shenging, Jing n Rtnesh Kumr (2004). Filure ignosis of isrete-event systems with liner-time temorl logi seifitions. IEEE Trns. Automti Control 49(6),