Optimal Design of Fault-Tolerant Petri Net Controllers

Transcription

1 Optial Design of Fault-Tolerant Petri Net ontrollers Yizhi Qu, Lingxi Li, Yaobin hen, an Yaping Dai Abstrat This paper proposes an approah for the optial esign of fault-tolerant Petri net ontrollers Given a syste ontroller that is oele as a Petri net, a fault-tolerant Petri net ontroller an be obtaine by ebeing the given Petri net ontroller into a larger Petri net ontroller that retains the funtionality of the original ontroller, an uses aitional plaes, tokens, an onnetions to allow the etetion an ientifiation of faults that ay our in the ontroller plaes An algorith is evelope to systeatially esign this faulttolerant Petri net ontroller in an optial sense The optiality is in ters of iniizing the su of ar weights of the input an output inient atries of the reunant ontroller Suh iniu su of ar weights is useful for haraterizing the iniu harware ost require to ipleent the reunant ontroller An exaple of the optial esign of a fault-tolerant Petri net ontroller for an autoate guie vehile AGV syste is also provie to illustrate our approah Inex Ters Petri nets, fault tolerane, optial esign, reunant ontrollers I INTRODUTION Faults in large-sale ynai systes an oproise their funtionality signifiantly an ight lea to evastating onsequenes 1, For this reason, fault tolerane has been stuie extensively In the fraework of isrete event systes, fault tolerane was stuie in 3 5 for finite state ahine oels base on enoing the state of the syste A siilar enoing approah was use in 6, 7 to ahieve fault etetion an ientifiation in systes that an be oele by Petri net oels The authors of 8 evelope approahes for the esign of reunant Petri net ontrollers that are able to provie tolerane against faults that ay our in a given syste ontroller that is oele by a Petri net This work was expane in 9 to eet the requireent that the reunant an original Petri net ontrollers nee to be bisiulation equivalent ie, any transition sequene enable in the original ontroller is also enable in the reunant one, an vie-versa A oplete haraterization of neessary an suffiient onitions for bisiulation equivalene was also obtaine in 9 but no optial esign riteria were isusse In this paper we onsier a setting siilar to 9 an ai at proviing fault etetion an ientifiation apabilities to This aterial is base upon work supporte by IUPUI RSFG Grant Yizhi Qu is with the Departent of Eletrial an oputer Engineering, Iniana University-Purue University Inianapolis, Inianapolis, USA, an the Departent of Autoation, Beijing Institute of Tehnology, Beijing, P R hina Lingxi Li an Yaobin hen are with the Departent of Eletrial an oputer Engineering, Iniana University-Purue University Inianapolis, Inianapolis, USA Yaping Dai is with Departent of Autoation, Beijing Institute of Tehnology, Beijing, P R hina orresponing author: Lingxi Li, 73 West Mihigan Street, Inianapolis, IN 460, USA Eail: LL7@iupuieu a given syste ontroller that is oele by a Petri net However, our ain goal is ifferent an is to exten the tehniques in 9 to aress the proble of optial esign of the fault-tolerant Petri net ontrollers In partiular, the neessary an suffiient onitions erive in 9 ight orrespon to a set of ifferent realizations of fault-tolerant Petri net ontrollers, epening on ifferent paraeters selete For pratial ipleentation onsierations, it is esirable that aitional riteria are propose in orer to isriinate between these esign aniates In this paper we onsier the riterion of iniizing the overall harware ost in orer to ipleent the fault-tolerant Petri net ontrollers In pratie, eah onnetion between the fault-tolerant ontroller an the plant is ipleente by a prograable evie an/or a iruit 10 Assuing that eah onnetion has the sae harware ost to be ipleente, the proble of iniizing the overall harware ost reues to the proble of iniizing the total nuber of onnetions between the ontroller an the plant Reall that in Petri net oels, eah of these onnetions orrespons to an ar in the input or output inient atrix of the net Thus, our goal in this paper is to iniize the su of ar weights of the input an output inient atries of the fault-tolerant ontroller Base on soe strutural analysis, an algorith is evelope to systeatially esign the paraeters of the fault-tolerant reunant ontroller in the optial sense An exaple of the optial esign of a fault-tolerant reunant ontroller for an autoate guie vehile AGV syste is also provie to illustrate our approah II NOTATION AND PRELIMINARIES This setion provies soe basi efinitions an terinology that will be use throughout the paper More etails about Petri nets an be foun in 11, 1 A Petri nets A Petri net struture is a irete weighte bipartite graph N P,T,A,W where P {p 1, p,, p n } is a finite set of n plaes rawn as irles, T {t 1,t,,t } is a finite set of transitions rawn as bars, A P T T P is a set of ars fro plaes to transitions an fro transitions to plaes, an W : A {1,,3,} is the weight funtion on the ars A arking is a vetor M : P N n that assigns to eah plae in the Petri net a nonnegative integer nuber of tokens rawn as blak ots We use M 0 to enote the initial arking of the Petri net an use Mp to enote the arking of plae p ie, the nuber of tokens in plae p A Petri net syste < N,M 0 > is a net N with an initial arking M 0

2 Let b i j enote the integer weight of the ar fro plae p i to transition t j, an b i j enote the integer weight of the ar fro transition t j to plae p i 1 i n, 1 j Note that b i j or b i j is taken to be zero if there is no ar fro plae p i to transition t j or vie versa We efine the input inient atrix B b i j respetively the output inient atrix B b i j to be the n atrix with b i j respetively b i j at its i th row, j th olun position The inient atrix of the Petri net is efine to be B B B The state or arking evolution of Petri net is given by Mk 1 Mk B B xk Mk Bxk, 1 where Mk is the arking of the Petri net at tie epoh k, an xk is the firing vetor that is restrite to have exatly one nonzero entry with value 1 when the j th entry is 1, transition t j fires at tie epoh k Note that transition t j is enable at tie epoh k if an only if Mk B :, j, where the inequality is taken eleent-wise an B :, j enotes the j th olun of B If transition t j is enable at arking Mk, it ay fire an yiel the new arking Mk 1 MkBx j k, where x j k iniates that the j th entry of xk is nonzero with value t j 1 This is enote by Mk Mk 1 an the arking Mk 1 is sai to be reahable fro the arking Mk B Matrix an vetor inequalities Let Z be the set of integer nubers Given atries A a i j an B b i j in Z n, A respetively B is sai to be nonnegative if A 0 respetively B 0, ie, if a i j 0 respetively b i j 0 for every i {1,,,n} an j {1,,,} Define A B if a i j b i j for every i {1,,,n} an j {1,,,}; A B is efine in a siilar way III DESIGN OF FAULT-TOLERANT REDUNDANT PETRI NET ONTROLLER In this setion we review the esign approah for the faulttolerant reunant Petri net ontroller an the proeure to perfor fault etetion an ientifiation that were propose in 9 The optial esign of suh reunant ontroller will be isusse in the next setion A fault-tolerant reunant ontroller Let be a given Petri net ontroller with n plaes, transitions an state evolution M k 1 M k B B xk M k B xk, where xk is the firing vetor, B is the output inient atrix of, B is the input inient atrix of, an B B B is the inient atrix of onsier a larger Petri net H with η n > 0 plaes, transitions an state evolution given by M h k 1 M h k B B xk M h k B xk, 3 where xk is the firing vetor, B is the output inient atrix of H, B is the input inient atrix of H, an B B B is the inient atrix of H Definition 1 9 H is a fault-tolerant reunant Petri net ontroller of a given ontroller if the state M h k satisfies In M h k M k for all tie epohs k where I n is }{{} G the n n ientity atrix an is a n atrix to be esigne learly, by left-ultiplying atrix G to equation an oparing it with equation 3, we obtain B M h k 1 M h k B B }{{ D B } D xk }{{} B B 4 where D is a atrix to be esigne Now the esign of fault-tolerant reunant ontroller H reues to the proble of hoosing atries an D appropriately In aition, in orer for H to be bisiulation equivalent to, the authors of 9 prove the following Lea Lea 1 9 A fault-tolerant reunant ontroller H is bisiulation equivalent to the given ontroller net if an only if an D satisfy onitions 1 an as follows 1 0 is a n atrix with nonnegative integer entries an; D 0 is a atrix with nonnegative integer entries suh that D inb,b onitions 1 an haraterize neessary an suffiient onitions for the esign of atries an D suh that the fault-tolerant reunant ontroller H is bisiulation equivalent to the original ontroller In the next setion we will isuss aitional onitions ipose on these atries to ahieve fault etetion an ientifiation B Fault etetion an ientifiation The faults we onsier in this paper are plae faults that an be ause by sensor failures or internal harware faults in the ontroller, whih ay orrupt the nuber of tokens in a partiular plae of the ontroller by inreasing or ereasing the nuber of tokens in that plae by an integer aount In partiular, a plae fault at tie epoh k results in an erroneous arking M f k that an be expresse as M f k M h k e pi, 5 where M h k GM k is the arking of the reunant ontroller that woul have been reahe uner fault-free onitions, an e pi is the plae error vetor with a single nonzero eleent at its i th entry If the i th entry of e pi is negative or positive, then the nuber of tokens in the i th plae has erease or inrease ue to the fault A possibly erroneous arking M f k an be heke by using the parity hek atrix P I, 6

3 where I is the ientity atrix, to verify whether the synroe, efine as sk PM f k, 7 is equal to 0 Note that uner fault-free onitions, PG 0 n where 0 n is a n atrix with all zero entries learly, for plae fault oel, the synroe at tie epoh k is given by sk PM f k PM h k e pi 8 PGM k e pi 0 Pe pi Pe pi, an fault etetion an ientifiation is exlusively eterine by atrix P For instane, to be able to etet an ientify a single plae fault, we an hoose atrix suh that any two oluns of atrix P are not rational ultiples 1 of eah other For sipliity, in this paper we only onsier the ase where we ai at eteting an ientifying a single plae fault In 7, the authors have shown that two plaes nee to be ae in orer to etet an ientify a single plae fault Therefore, we have, an atrix shoul satisfy onition 3 as follows for single fault etetion an ientifiation 3 the hoie of shoul guarantee that any two oluns of atrix P are not rational ultiples of eah other Note that there are any hoies of atries an D that satisfy onitions 1,, an 3 given above In the next setion, we will isuss our approah to esign these atries in ter of an optial riterion, ie, to iniize the su of ar weights of the input an output inient atries of the reunant ontroller IV OPTIMAL DESIGN OF FAULT-TOLERANT REDUNDANT A Proble forulation PETRI NET ONTROLLER The proble we eal with in this paper is the following onsier a Petri net ontroller with n plaes, transitions, an state evolution given in equation, esign atries an D suh that the resulting fault-tolerant reunant Petri net ontroller H with η n plaes, transitions, an state evolution given in equation 3 satisfy: i H is bisiultion equivalent to ie, to satisfy onitions 1 an ; ii H is able to etet an ientify a single plae fault ie, to satisfy onition 3; an iii the su of ar weights of the input an output inient atries of H is iniize Sine the input inient atrix of H is an η atrix enote by B an the output inient atrix of H is an η atrix enote by B, this proble an be forulate as an optiization proble as follows 1 One nees to ake sure that for all pairs of oluns of P there o not exist nonzero integers a, b suh that a P:,i b P:, j, where i, j {1,,,η} an i j η B i j B i j, 9 where B i j respetively B i j enotes the entry at the i th row, j th olun in atrix B respetively B, suh that atries an D satisfy onitions 1,, an 3 B Optial esign of reunant ontroller paraeters Note that aoring to equation 4, we have B B B D, B B B D Therefore, given the n input inient atrix B an the n output inient atrix B of the Petri net ontroller, the su of ar weights of the first n rows of atrix B respetively B is fixe Naely, the hoies of an D o not affet the solution of the proble in 9 for the first n rows but the rest rows reall that η n learly, the proble in 9 an be reue equivalent to 10 as follows B i j D i j B i j D i j, 10 suh that atries an D satisfy onitions 1,, an 3 Note that onition requires that atrix D is a atrix with nonnegative integer entries suh that 0 D inb,b Next we will show that the solution of 10 orrespons to the hoie of atrix D suh that D satisfies D i j inb i j,b i j for every i {1,,,}, j {1,,,} Lea The solution of 10 subjet to onition orrespons to the hoie of atrix D suh that D satisfies D i j inb i j,b i j for every i {1,,,}, j {1,,,} Proof: We prove by ontraition Suppose that atrix D satisfies D inb,b exept a single entry D i j 0 D i j < inb i j,b i j at its i th row, j th olun position, where i {1,,,}, j {1,,,} learly, D i j < inb i j,b i j iplies that an D i j > inb i j,b i j B i j D i j > B i j inb i j,b i j, B i j D i j > B i j inb i j,b i j Suing up two inequalities above, we have: B i j D i j B i j D i j > B i j inb i j,b i j B i j inb i j,b i j,

4 whih iplies that 10 B i j D i j B i j D i j is not iniize ontraition! Therefore, D i j inb i j,b i j for every i {1,,,}, j {1,,,} an the result follows onition is satisfie with the hoie of atrix D given in Lea Note that we an rewrite proble given in 10 as argin argin B i j D i j B i j D i j B i j B i j D i j B B i j D i j, 11 where atrix satisfies onitions 1 an 3, an atrix D satisfies onitions given in Lea In suary, fro the analysis above so far, we have eterine the hoie of atrix D suh that the proble given in 10 is iniize We have also shown that the proble in 10 an be reue equivalently to the proble given in 11 with atrix satisfying onitions 1 an 3, an atrix D satisfying onitions given in Lea Next we will present an algorith for the optial esign of atrix suh that the proble in 11 is iniize First of all, reall that atrix is a n atrix For single fault etetion an ientifiation, we have, whih iplies that is a n atrix In aition, onition 3 requires that the hoie of shoul guarantee that any two oluns of atrix P are not rational ultiples of eah other It iplies that one shoul esign atrix suh that any two oluns of are not rational ultiples of eah other whih are, of ourse, not rational ultiples of vetors 1 0 T an 0 1 T in ientity atrix I by inspetion of equation 6 Also, it is not iffiult to show that the vetor 0 0 T annot be a olun vetor of atrix sine it violates onition 3 Base on the observations above, we propose an algorith that is able to esign atrix ie, by eterining oluns of atrix suh that the proble in 11 is iniize This algorith is base on the partial-orer tree of twoiensional olun vetors with nonnegative integer entries shown in Fig 1 as follows In Fig 1, every noe is a two-iensional olun vetor with nonnegative integer entries Ars between noes in ifferent levels of the tree apture a partial orer of the, ie, if there is an ar fro noe M to noe M, then we have M M eleent-wise Nuber n at eah level enotes the su of all eleents of iniviual noe note that the su is the sae for eah noe in the sae level Therefore, as 1 3 T 1 T 1 4 T 3 T 1 1 T 1 5 T 5 1 T 1 T 3 1 T 4 1 T 3 T Fig 1 Partial-orer tree of two-iensional vetors with nonnegative integer entries the level goes fro top to botto in the tree, the su n inreases Note that every noe in Fig 1 satisfies onition 1 beause they are all vetors with nonnegative integer entries; in aition, any two vetors in Fig 1 are not rational ultiple of eah other beause the vetor whih is a rational ultiple of any vetor in Fig 1 has alreay been reove fro the tree Therefore, the partial-orer tree in Fig 1 provies all vali vetors for the hoies of oluns of atrix an our algorith is propose base on it Reall that atrix is a n atrix In aition, we have erive optial esign onitions on atrix D in Lea Now we isuss how to solve equation 11 by only onsiering ters assoiate with atrix in the objetive funtion In other wors, we onsier objetive funtion fro now on instea of B B i j 1 B B i j D i j With the forulation above, we an rewrite equation 1 as argin B B i j 11 1n 1 n b 11 b 11 b 1 b 1 b 1 b 1 b b b n 1 b n 1 b n b n Note that we onstrut the partial-orer tree in a top-own fashion We start fro vetor 1 1 T following the analysis before an erive vetors in the next level by inreasing the su n by 1, an so forth Vetors in the for of k 0 T respetively 0 k T where k {,3,, } annot be in the tree beause they are rational ultiples of 1 0 T respetively 0 1 T, ie, rational ultiples of oluns of ientity atrix I

5 argin i1 b 1 j b 1 j i1 in b n j b n j 13 Note that the iension of atrix is n an the iension of atrix B B is n To siplify our notation, we efine b 1 b 1 j b 1 j,, b n b n j b n j, 14 where b i is the su of all eleents in the i th row of B B i {1,,,n } Siilarly, we efine 1 i1 i1,, n in, 15 where j is the su of all eleents in the j th olun of j {1,,,n } With these efinitions, we siplify equation 13 as 1b 1 b n b n argin n k1 k b k, 16 where k is the su of all eleents in the k th olun of atrix an b k is the su of all eleents in the k th row of atrix B B Reark 1 Note that eah su k k {1,,,n } an be seen as orresponing to a partiular su n shown in Fig 1 Our task is how to pik up vetors ie, oluns of atrix fro the partial-orer tree in Fig 1 suh that 16 is iniize given the input an output inient atries B an B Reark Before we present our algorith, it is iportant to briefly isuss the iea as follows Given B an B note that B 0 an B 0, we opute b k ie, the su of all eleents of their k th rows where k {1,,,n } Base on these n sus, we an orer the in the esening orer Without loss of generality, we assue that b 1 b b n learly, in orer to iniize equation 16, we will pik up the vetor fro the topost level in Fig 1 ie, hoose the sallest su n to be ultiplie with b 1 ; then hoose one vetors fro the seon topost level to be ultiplie with b, an so forth Reark 3 Note that there are two vetors in the seon topost level with the sae su n, therefore the hoie of any of the for b will not affet the result of equation 16 but will result in ifferent realizations of atrix The siilar analysis an be applie to the rest of b k where k {3,4,,n } Now we present our algorith in etail as follows Algorith 1 Input: A Petri net ontroller N with n plaes, transitions, the input inient atrix B, the output inient atrix B 1 opute su b k k {1,,,n } fro equation 14; Re-arrange b k k {1,,,n } in the esening orer; 3 Pik up vetors fro the partial-orer tree as we isusse in Reark ; 4 Obtain atrix ay be ultiple aoring to vetors hosen in Step 3; 5 opute atrix D as D in{b,b } as we have shown in Lea Reark 4 In Step 3 of Algorith 1, there ay exist ultiple hoies of vetors that we an pik up fro the partialorer tree for a partiular su b k where k {1,,,n }; we treat all of these possible hoies as possible solutions for atrix We will illustrate this through our running exaple in the next setion V AN ILLUSTRATIVE EXAMPLE In this setion, we use an autoate guie vehile AGV syste propose in 13 as our illustrative exaple By enforing onstraints Mp 1 an Mp 4 1 on the original syste rawn in soli ars, a Petri net ontroller is erive base on the plae invariant approah propose in 14 an is shown in Fig Plaes p 7 an p 8 are ontroller plaes an the ashe ars are onnetions between these ontroller plaes an the transitions in the plant to satisfy onstraints entione above We o not raw the initial arking of the ontroller plant here beause it is not involve in our oputation p 1 p 7 t 1 p t p 4 t4 t 3 Fig p 3 t 5 p5 An AGV syste with its ontroller Now we ai at esigning the optial fault-tolerant reunant ontroller for the Petri net ontroller shown in Fig The input output inient atrix B B of the Petri net ontroller an be erive as follows B , B Note that there are two ontroller plaes, ie, n ; an there are five transitions, ie, 5 We opute b 1 an b fro equation 14 as b 1 5 b 5 1 j b 5 1 j 3, b b 5 j b j 3 p 8 p 6

6 Note that for single fault etetion an ientifiation, we have Sine n, atrix in this exaple is a atrix reall that is a n atrix Therefore, we have to pik up a vetor fro the partial-orer tree shown in Fig 1 for b 1 an a vetor for b Sine b 1 b 3, we an hoose the vetor 1 1 T fro the topost level of the tree for b 1 or b an hoose one of the vetors 1 T an 1 T fro the seon topost level of the tree for b or b 1 Sine the positions of these vetors in o not affet our goal of iniization, we have four possible hoies for atrix as follows eah of the an be a solution in our ase , 1 1, , To siplify the iagra, we only hoose 1 an for further oputation an isussion In orer to opute atrix D, first we obtain B an B fro iniviual 1 an as 1 B , B , B B Sine D in{b,b } as shown in Lea, we obtain D 1 an D as D , D Now we have obtaine oplete haraterization of two possible optial ipleentations of fault-tolerant reunant ontrollers in ters of atries an D To oplete our isussion, next we will show that base on these two ifferent hoies, the su of ar weights of the input an output inient atries of resulting reunant ontrollers is the sae an of ourse, the sallest aong all possible esign hoies B B ,,B ,B It is not iffiult to see that the su of ar weights of two ifferent fault-tolerant reunant ontroller ipleentations is the sae an is equal to 11 3 Note that all these possible hoies of are feasible solutions of the optiization algorith However, with ifferent hoies of, one an have ifferent topologial onnetions between the ontroller an the plant, VI ONLUSIONS AND FUTURE WORK In this paper we propose an approah for the optial esign of fault-tolerant Petri net ontrollers Given a syste ontroller that is oele by a Petri net, we present an approah of obtaining a fault-tolerant reunant Petri net ontroller that is able to retain the funtionality an properties of the original ontroller an enable the fault etetion an ientifiation in a systeati anner We evelope an algorith that is able to esign this fault-tolerant reunant ontroller that has the iniu su of ar weights of the input an output inient atries An exaple of the optial esign of fault-tolerant reunant ontroller for an autoate guie vehile AGV syste was also provie to illustrate our approah Future extensions of this work inlue the evelopent of algoriths for the optial esign of fault-tolerant reunant ontroller base on other optiization riterion, eg, in ters of the total nuber of plaes, onnetions, an tokens neee for the esign of the reunant ontroller Another iportant future iretion is to evelop the optial esign approah for reunant ontroller by onsiering etetion an ientifiation of ultiple faults REFERENES 1 B Johnson, Design an Analysis of Fault-Tolerant Digital Systes, Aison-Wesley, Reaing, Massahusetts, 1989 D P Siewiorek an R S Swarz, Reliable oputer Systes: Design an Evaluation, 3r eition, A K Peters, Lt, Natik, Massahusetts, A Sengupta, D K hattopahyay, A Palit, A K Banyopahyay, an A K houhury, Realization of fault-tolerant ahines linear oe appliation, IEEE Trans oputers, vol -30, no 3, pp 37 40, Marh G R Reinbo, Finite fiel fault-tolerant igital filtering arhitetures, IEEE Trans oputers, vol -36, no 10, pp , Otober N Hajiostis, Finite-state ahine ebeings for nononurrent error etetion an ientifiation, IEEE Trans Autoati ontrol, vol 50, no, pp , February N Hajiostis an G Verghese, Monitoring isrete event systes using Petri net ebeings, in Proeeings of Appliation an Theory of Petri Nets 1999 Series Leture Notes in oputer Siene, vol 1639, pp , Springer-Verlag, Y Wu an N Hajiostis, Algebrai approahes for fault ientifiation in isrete-event systes, IEEE Trans Autoati ontrol, vol 50, no 1, pp , Deeber L Li, N Hajiostis, an R S Sreenivas, Fault etetion an ientifiation in Petri net ontrollers, in Pro of 43r IEEE onf Deision an ontrol, pp , Deeber L Li, N Hajiostis, an R S Sreenivas, Designs of bisiilar Petri net ontrollers with fault tolerane apabilities, IEEE Trans Systes, Man, an ybernetis, Part A, vol 38, no 1, pp 07 17, January S Bulah, A Brauhle, H-J Pfleierer, an Z Kuerovsky, Design an ipleentation of isrete event ontrol systes: A Petri net base harware approah, Disrete Event Dynai Systes: Theory an Appliations, vol 1, no 3, pp , July T Murata, Petri nets: properties, analysis an appliations, Pro of the IEEE, vol 77, no 4, pp , April G assanras an S Lafortune, Introution to Disrete Event Systes, Kluwer Aaei Publishers, Boston, MA, S Hsieh an Y-J Shih, Autoate guie vehile systes an their Petri-net properties, Journal of Intelligent Manufaturing, vol 3, no 6, pp , Deeber K Yaaliou, J Mooy, M Leon an P J Antsaklis, Feebak ontrol of Petri nets base on plae invariants, Autoatia, vol 3, no 1, pp 15 8, January 1996