On the Implications of the Solvability of the Supervisory Control. Problem for Certain Innite-State Discrete Event Dynamic systems

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1 On he Imlicions of he olviliy of he uervisory Conrol Prolem for Cerin Innie-e Discree Even Dynmic sysems Rmvru. reenivs Dermen of Generl Engineering & CL Alie, Elecricl nd Comuer Engineering Universiy of Illinois Urn-Chmign Urn, IL E-mil: June 6, 1994 Asrc We show h cerin foridden-se, nd foriddensring [6, 5, 11] suervisory conrol rolems involving innie-se sysems hs soluion if nd only if corresonding rolem for every nie-se susysem of he originl innie-se sysem hs soluion. Using his oservion we rgue he exisence of soluion rocedure for cerin innie-se sysems (cf. [10, 8, 9]) imlies i is comuionlly enecil o model cerin lrge, nie-se sysems s innie-se sysems nd use he rocedures develoed o hndle innie-se sysems in lce of hose develoed o hndle nie-se sysems (cf.[]). We illusre his roch using wo exmles. 1 Inroducion uervisory conrol of Discree Even Dynmic ysems (DED) ws inroduced y Rmdge nd Wonhm [6, 5, 11, 4]. I involves discree se ln nd discree se suervisor usully modeled s uom. The ln nd suervisor hve n idenicl lhe se h is riioned ino conrollle nd unconrollle suses. The ln uomon generes lnguge nd he suervisor uomon cces he lnguge genered y he ln. The se of he suervisor is used o decide he conrollle symols h will e ermied o occur in he ln lnguge. The suervisor is ssumed o hve n inhiiing cion only on he conrollle symols. Given ln uomon, i is of ineres o synhesize suervisor h revens he occurrence of symols of he ln This reserch ws suored in r y grn RE-BRD- IC-REENIVA from he UIUC Cmus Reserch Bord in order o enforce secicions in he closed-loo sysem. The clsses of secicions h hve een considered in he lierure fll ino wo cegories: foridden-se, [5] nd foridden-sring rolems [6]. In his er we show h hese suervisory conrol rolems re solvle for innie-se sysems if nd only if corresonding rolem for every nie-se susysem of he originl innie-se sysem hs soluion. This oservion couled wih recen resuls in references [8, 9] imly h i is comuionlly enecil o reresen cerin lrge, nie-se sysems s innie-se sysems. This conce is loosely illusred in he nex rgrh in he conex of foridden-se rolems. In foridden se rolems we idenify suse of ln ses s desirle se. A suervisory conrol olicy is sic le h rovides lis of conrollle rnsiions o e disled for ech se rechle in he closed-loo. A suervisory conrol olicy solves he foridden-se rolem if he se of ses rechle in he closed-loo is suse of he desired se. Consider lrge nie se sysem where he ojecive is o void lrge se of foridden su-ses vi suervisory conrol. The roch we dvoce is o roxime his lrge nie se sysem s n innie se sysem, nd roxime he lrge se of foridden su-ses n innie suse of he in- nie se roximion. For he resen, ssume he solviliy of he suervisory conrol rolem for he innie se roximion (cf. our rior work on he solviliy of suervisory conrol rolems for non-regulr ehviorl secicions [8, 9]). As resul we imlicily roduce sic le of innie size h liss he disled evens for every rechle se of he innie se sysem h voids he foridden ses. I hels o icure his le s consising of wo enries for ech row, he rs enry idenies

2 rechle se nd he second enry idenies se of disled evens. This le lso solves he foridden se rolem for ny nie susysem of he in- nie se roximion. Loosely, he fc h we cn solve he innie se roximion is reson enough o elieve here is lrge nie se susysem eyond which roximing he sysem s n innie se sysem is comuionlly suerior. To mke his cler, we imgine comlexiy lo where he x-xis reresens he size of he rolem nd he y- xis he comuionl eor. Any nie se mehod h is no O(1) will symoiclly hve osiive sloe, while he comlexiy lo of our mehod is rllel o he x-xis s he soluion o he innie se roximion is lso soluion o every nie susysem. Oviously, he lo of he nie se sed mehod wih he osiive sloe should cross he rllel line some oin, hus idenifying rolem size eyond which n innie-se roximion is comuionlly suerior o nie se model. In secions nd we resen deiled licion of he ove menioned conce o he foridden sring, nd foridden se rolems. Finlly, in secion 4 we conclude wih some direcions for fuure reserch. Foridden ring Prolem Consider wo monooniclly incresing sequences fl i g 1, fk i=1 ig 1 i=1 of rex-closed, regulr lnguges (i.e. L i L i+1 nd K i K i+1 ) wih he ddiionl requiremen h 8 i, K i+1 \ L i = K i \ L i. This requires h once sring is illegl i remins illegl for ll rolem insnces. We consider he foridden sring rolem where i is required h we enforce K i given ln lnguge L i. This rolem is solvle if nd only if K i is conrollle wih resec o L i (cf. [6] for deils). If he index (cf. secion.4, [1]) of he regulr lnguge L i (K i ) is m i (n i ), he comlexiy of he es for he conrolliliy of K i wih resec o L i is O(m i n i ) []. We show h he rex-closed lnguge K = 1 K i=1 i is conrollle wih resec o L = 1 L i=1 i, if nd only if K i is conrollle wih resec o L i, for ll i. The following lemm is useful in eslishing heorem.1. Lemm.1 Given wo monooniclly incresing sequences fl i g 1 i=1, fk ig 1 i=1 of rex-closed, regulr lnguges such h 8 i, K i+1 \ L i = K i \ L i, nd riion of he symol se = u [ c, hen K i is unconrollle wih resec o L i, hen K j is unconrollle wih resec o L j, 8 j i. Proof: (By conrdicion) If K i is unconrollle wih resec o L i hen 9! L i \ K i, 9 u u, such h! u L i,! u = K i. ince! u L i nd L i L i+1, we infer! u L j, 8 j i. Now, le! u K m for some m > i, while! u = K m?1. This, ogeher wih he fc h! u L j, 8 j i, suggess h! u = K m?1 \ L m?1 nd! u K m \ L m?1. conrdicing he requiremen h 8 i, K i+1 \ L i = K i \ L i. Hence he resul. Theorem.1 Given wo monooniclly incresing sequences fl i g 1 i=1, fk ig 1 i=1 of rex-closed, regulr lnguges such h 8 i, K i+1 \ L i = K i \ L i, nd riion of he symol se = u [ c, hen K = 1 K i=1 i is conrollle wih resec o L = 1 L i=1 i if nd only if K i is conrollle wih resec o L i, 8 i 1. Proof: (If r, y conrdicion) Le K i e conrollle wih resec o L i, 8 i 1, nd le K e unconrollle wih resec o L. The ler requiremen sugges h 9! K \ L, 9 u u, such h! u L,! u = K. ince! u = K, we infer! u = K i, 8 i 1. Also, since! K, we cn infer h 9 j such h! K i, 8 i j. Likewise, since! u L, we cn infer h 9 k such h! u L i 8 i k. Also,! L i, 8 i k s ech L i is rex-closed. Leing m = mx(k; j), we infer h! K i \ L i,! u L i nd! u = K i, 8 i m. This is conrdicion s i suggess h K i is no conrollle wih resec o L i, 8 i m. (Only if r) We show h if K i is unconrollle wih resec o L i hen K is lso unconrollle wih resec o L. By lemm.1 we know h if K i is unconrollle wih resec o L i hen K j is unconrollle wih resec o L j, 8 j i. Th is, 9! K j \ L j, 9 u u, such h! u L j nd! u = K j, 8 j i )! K \ L,! u L,!u = K. o, K is no conrollle wih resec o L. Hence he resul. The ove heorem leds o he oservion h he comlexiy of synhesizing suervisor i h enforces K i for ln lnguge of L i, is ounded ove y he comlexiy of synhesis rocedure for he suervisor h enforces K for ln lnguge of L. The es for conrolliliy nd herefore he suervisor synhesis rolem is known o e solvle when (i) K nd L re regulr lnguges [6] nd (ii) K nd L elong o resriced clss of non-regulr lnguges [8, 9].

3 The uer-ound menioned ove is riculrly imressive in he cse when L nd K elong o he resriced clss of non-regulr lnguges where he es for conrolliliy, nd herefore he suervisory synhesis rolem, is known o e solvle [8, 9]. In his cse i is ossile o exrc susequences of monoone, rex-closed, regulr lnguges fe Lj g 1 nd j=1 f Kj e g 1 j=1 wih incresing indices (cf. secion.4 [1]) from he sequences fl i g 1 nd fk i=1 ig 1 i=1 such h ek i+1 \ L e i = K e i \ L e i, 1 e j=1 L j = 1 e i=1 L i = L, nd 1 K e j=1 j = 1 K e i=1 i = K. Essenilly, he es for conrolliliy of Kj e wih resec o Lj e ecomes incresing dicul s he j increses if one were o use he lgorihm oulined in reference [] for regulr secicions. This in urn rnsles o he fc h he synhesis of he suervisor e j h enforces Kj e for ln lnguge of L e j ecomes incresingly dicul s j increses. Oviously, eyond criicl vlue of j i is comuionlly suerior o use he suervisor. To mke his cler, we imgine comlexiy lo where he x-xis reresens he size of he rolemem j en j, nd he y-xis he comuionl eor involved in synhesizing he suervisor e j. If we were o use he es for conrolliliy of ir of regulr lnguges oulined in reference [] s one of he ses in he synhesis rocedure, we would hve comlexiy lo h grows wih size. On he oher hnd if we used he suervisor o enforce K e j for ln lnguge of Lj e, we would hve comlexiy lo h is rllel o he x-xis, s lso enforces K e j for ln lnguge of Lj e for ll j. Oviously, he lo of he nie se sed mehod wih he osiive sloe should cross he rllel line some oin, hus eslishing he fc h here is rolem-size eyond which i is comuionlly suerior o use he suervisor. Th is, he suervisor is comuionlly suerior o he nie-se sed, or regulr-lnguge sed suervisors e j for lrge enough j. This is es illusred y exmle. Consider symol se u = fg c = fg, nd he monoone sequences fl i g 1, fk i=1 ig 1, where L i=1 = f n m j 0 n ; 0 m g nd K = f n m j 0 m n g. Figure 1 shows miniml recognizer for L nd K. I is no hrd o see h he miniml recognizers for L nd K will ech conin + 1 ses. As increses he synhesis of he suervisor h enforces K given ln lnguge of L will ecome incresingly dicul. L = 1 L =1 i = f g nd K = 1 =1 K i = f n m j 0 m ng. The wo lnguges L nd K re Deerminisic Peri ne Lnguges (DPNL), clss of forml lnguges where he suervisory conrol rolem is known o e solvle [7]. The DPNL soluion for his suervisory conrol rolem is illusred in gure. Noe h he DPNL soluion will enforce K for ln lnguge of L for ny. Foridden e Prolem In his secion we ddress he foridden se rolem. Consider sequence of non-deerminisic, nie uom (NDFA) fm i g 1 i=1. Where M i = (Q i,, i, q 0 ), nd Q i is he nie se of ses, (= u [ c ) is he lhe se, q 0 Q i is he iniil-se nd i Q i Q i is he single-se rnsiion relion. If (q i ; ; q j ) i, hen here exiss h of uni lengh in he grhicl reresenion of M i h origines verex (se) q i nd ermines verex (se) q j wih lel of on he edge connecing q i nd q j. We require Q i Q i+1 nd i i+1. I is lso imorn o oin ou h every NDFA in he sequence fm i g 1 i=1 hs he sme iniil se q 0. Le fp i g 1 i=1 e monoonic sequence of legl su-ses, h is, P i Q i nd P i P i+1. As rllel o he foridden sring rolem, le us require P i \ K i = P i+1 \ K i. We dene he non-deerminisic, innie-se uomon M c = ( Q,,, q 0 ), where Q = 1 K i=1 i nd = 1 i=1 i. imilrly, le P = 1 P i=1 i. For given NDFA M i nd legl suse of ses P i such h q 0 P i, we dene he rechle suse of P i, R(M i, P i ) s he smlles se oined y using he following rule - if q i R(M i, P i ), nd 9 q j Q i such h (q i ; ; q j ) i nd q j P i, hen q j R(M i, P i ). We lso dene he wekes lierl recondiion of P i [5], (P i ) s follows- 6 (P i ) = fq Q i j (8 u u ; 8eq Q i ; if (q; u ; eq) i ; heneq P i ) or (8 u u ; 9eq Q i ; s.. (q; u ; eq) i )g In he generl cse, R(M i, P i ) P i. Li nd Wonhm [] use he erm conrolliliy o joinly refer o he roery of rechiliy nd conrol-invrince. Th is, P i is sid o e conrollle if (i) P i = R(M i, P i ), nd (ii) P i \ (P i ) = P i. We relx he requiremen of rechiliy y noing h here is sic-feedck h enforces R(M i, P i ) if nd only if P i is conrolinvrin. The following lemm lys criicl role in he roof heorem.1. Lemm.1 For he condiions descried ove, if P i Q i is no conrol-invrin, hen P j Q j is no conrol-invrin, 8 j i. Proof: (By conrdicion) If P i is no conrol invrin, 9 q P i, 9 u u, 9 eq Q i such h (q; u ; eq) i nd eq = P i. ince Q i Q i+1, eq

4 Q i ) eq Q j 8 j i. Likewise, since P i P i+1, q P i ) q P j, 8 j i. Now, if eq P m for some m > i, such h eq = P m?1, hen eq = P m?1 \ Q m?1, u eq P m \ Q m. This violes he requiremen 8 i, P i \ Q i = P i+1 \ Q i. Hence he resul. Theorem.1 For he sequence of NDFA's fm i g 1 i=1 nd legl su-ses fp i g 1 i=1 s dened ove, P = 1 i=1 P i is conrol-invrin if nd only if P i is conrol-invrin, 8 i 1. Proof: (If r) We show h if P is no conrolinvrin hen P i will no e conrol invrin 8 i n for some n. If P is no conrol-invrin hen 9 q P, 9 u u, 9 eq Q, such h (q; u ; eq) nd eq = P. ince eq = P, eq = P i, 8 i. Also, q P ) 9 j such h q P i, 8 i j. ince Q = 1 Q i=1 i, = 1 i=1 i, (q; u ; eq), nd eq Q ) 9 m such h eq Q i, nd (q; u ; eq) i, 8 i m. Leing n = mx(m; j), we ge 8 i n, q P i, (q; u ; eq) i nd eq = P i. Th is, 8 i n, P i will no e conrol-invrin. (Only If r) If P i is no conrol-invrin hen lemm.1 suggess h P j is no conrolinvrin 8 j i. o, 8 j i, 9 q P j, 9 u u, 9 eq Q j such h (q; u ; eq) j nd eq = P j ) 9 q P, 9 u u, 9 eq Q such h (q; u ; eq) nd eq = P. Th is, P is no conrol-invrin. Hence he resul. The oservions mde in he conex of he foridden sring rolem fer heorem.1 re eqully relevn here. We refrin from resening hem gin. I is relevn o oin ou h he soluion of he foridden se rolem h enforces R(c M, P ) will lso enforce R(M i, P i ), 8 i. We close his secion wih n exmle h illusres his conce. For n ineger n N, dene DFA M n = (Q n,, n, q 0 ), s follows: Q n = f1; : : :; ng f1; : : :; ng, u = fg, c = fg, q 0 = (0; 0). The single-se rnsiion relion n is dened s n = f((n 1 ; n ); ; (n 1 + 1; n )) j 0 n 1 < n; 0 n ng [ f((n 1 ; n ); ; (n 1 + 1; n )) j 0 n 1 n; 0 n < ng: Le he legl suse of ses P n = f(n 1 ; n ) j n n 1 n 0g. Figure shows he grhicl reresenion of he DFA for M. The se of ses in P = f(0; 0); (1; 0); (; 0); (1; 1); (; 1); (; )g re highlighed y drk circles. P cn e shown o e conrol-invrin s ech P i cn e shown o e conrol-invrin. Furhermore, P cn e reresened s Free-leled Peri ne Lnguge (FLPNL), noher rdigm where suervisory conrol rolems re known o e solvle [8]. The sic feedck olicy sed on he se of FLPNL for his rolem is shown in gure 4. 4 Conclusions In his er we hve shown h cerin foriddense nd foridden sring rolems h involve innie-se sysems hs soluion if nd only if corresonding rolem for every nie-susysem of he originl innie-se sysem hs soluion. This oservion is hen used o eslish he exisence of symoiclly ecien soluions o he suervisory conrol rolem using recen resuls on he solviliy of he suervisory conrol rolem for cerin nonregulr secicions [8, 9]. We rgue h even in hose cses h involve nie-se sysems, here is rolem-size eyond which i is comuionlly ene- cil o reresen he nie-se sysem s n inniese sysem. This oservion couled wih recen osiive resuls on deciding conrolliliy holds lo of romise in coming he comlexiy h lgues suervisory conrol. References [1] J.E. Hocrof nd J.D. Ullmn. Inroducion o Auom Theory, Lnguges, nd Comuion. Addison Wesley, Reding, Msschuses, [] R. Kumr, V. Grg, nd.i. Mrcus. On he conrolliliy nd normliy of discree even dynmicl sysems. ysems & Conrol Leers, 17:. 157{168, [] Y. Li nd W.M. Wonhm. Conrol of vecor discree-even sysems I- The se model. IEEE Trns. on Auomic Conrol, 8(8):114{17, Augus 199. [4] P.J. Rmdge nd W.M. Wonhm. Modulr suervisory conrol of discree even sysems. In Proceedings of he evenh Inernionl Conference on he Anlysis nd Oimizion of ysems, ges 0{14, Anies, June Lecure Noes in Conrol nd Oimizion of ysems, Vol.8 (A. Bensoussn nd J.L. Lions, eds.), ringer-verlg, New York, 1986.

5 [5] P.J. Rmdge nd W.M. Wonhm. Modulr feedck logic for discree even sysems. IAM J. Conrol nd Oimizion, 5(5):10{118, eemer [6] P.J. Rmdge nd W.M. Wonhm. uervisory conrol of clss of discree even rocesses. IAM J. Conrol nd Oimizion, 5(1):06{0, Jnury [7] R.. reenivs. Deerminisic -free eri ne lnguges nd heir licion o he suervisory conrol of discree even dynmic sysems. In Proc. of he 6h Midwes ymosium on Circuis nd ysems. IEEE, Augus 199. Deroi, MI. [8] R.. reenivs. A noe on deciding he conrolliliy of lnguge K wih resec o lnguge L. IEEE Trns. on Auomic Conrol, 8(4):658{66, Aril 199. Conrol Acion α( )= 4 α( 1)= 4 α( )= α( )= α( )= α( )= 6 uervisor Pln G Θ L( Θ G) [9] R.. reenivs. On weker noion of conrolliliy of lnguge K wih resec o lnguge L. IEEE Trns. on Auomic Conrol, 8(9):1446{1447, eemer 199. [10] R.. reenivs nd B.H. Krogh. On eri ne models of innie se suervisors. IEEE Trns. on Auomic Conrol, 7():. 74{77, Ferury 199. [11] W.M. Wonhm nd P.J. Rmdge. On he sureml conrollle sulnguge of given lnguge. IAM J. Conrol nd Oimizion, 5():67{ 659, My Miniml Recognizer for L_ Miniml Recognizer for K_ Figure 1: Miniml Recognizers for L nd K of secion. Figure : A DPN-sed soluion o he suervisory conrol rolem of secion. (1,0) (,1) (0,0) (,0) (1,1) (0,) (,) (0,1) (1,) Figure : The uomon for M of secion. 1 Conrol-Lw: o "" whenever 1 is emy Figure 4: A FLPN-sed soluion o he suervisory conrol rolem of secion.