Model's quality assurance using MSA and Petri Net

Transcription

1 Model's qulity ssurne using MSA nd Petri Net Dr. As M. Al-Bkry College of Computer Tehnology, University of Bylon, Irq Astrt: This work presents methodology for designing models in engineering nd edution fields for verifying the qulity ssurne onditions. The first model uses mtrix struturl nlysis (MSA) to lulte the expeted vlues for the model. The mtrix struturl nlysis (MSA) uses to design model in engineering field, this helps to expet the vlues whih is the nerest to the rel ones. The MSA verify the qulity ssurne onditions nd estimte priniples or d-ho priniples depending on the lulting the tul vlues. Petri nets uses in nother exmple (Edution field). Petri Nets provide very importnt feture whih is the dynmi property, so the model designing with Petri nets eomes look like s in rel. الخالصة: يعرض البحث طريقة لتصميم نماذج في حقلي الھندسة والتعليم الختيار شروط تأكيد ألجوده. تم استخدام تحليل ھيكل المصفوفة في النموذج األول لغرض احتساب القيم المتوقعة للنموذج إن استخدام تحليل ھيكل المصفوفة لنموذج ھندسي لغرض تأكيد جوده الموديل سھل في تخمين القيم الحقيقية والمفترض استخدامھا واقعيا. من الناحية األخرى تم استخدام مخططات ألبتري في الحقل التعليمي لنفس المھمة وھي تأكيد جودة الموديل وان من ابرز صفات مخططات ألبتري ھي الصفة الديناميكية والتي تشعرك بأن النموذج يعمل وكأنه في الواقع. Key words: Dt nlysis, Qulity mnge m ent, Petri Net, M trix Struturl nlysis. INTRODUCTION Dt nlysis is the proess of looking t nd summrizing dt with the intent to extrt useful informtion nd develop onlusions[wiki-en]. Dt nlysis n e divided into Explortory Dt Anlysis (EDA) nd Confirmtory Dt Anlysis (CDA), where the EDA fouses on disovering new fetures in the dt, nd CDA on onfirming or flsifying existing hypotheses in this pper fouses on CDA s effetive tools of uilding this models. Setion ١ provides rief review on Petri net. setion ٢ explins the speil struture nlysis. Setion ٣ highlight design of two different models with pplition to engineering nd edution. In setion ٤ disussion nd onludes the pper. 1. BASIC PETRI NETS A Petri net onsists of ples, trnsitions, nd direted rs. Ars run etween ples nd trnsitions not etween ples nd ples or trnsitions nd trnsitions. The ples from whih n r runs to trnsition re lled the input ples of the trnsition; the ples to whih rs run from trnsition re lled the output ples of the trnsition[koto84]. Ples my ontin ny numer of tokens. A distriution of tokens over the ples of net is lled mrking[ PATA4]. Trnsitions t on input tokens y proess known s firing. A trnsition is enled if it n fire, i.e., there re tokens in every input ple. When trnsition fires, it onsumes the tokens from its input ples, performs some proessing tsk, nd ples speified numer of tokens into eh of its output ples. It does this tomilly, i.e., in one noninterruptile step. Exeution of Petri nets is nondeterministi. This mens two things [PETE77]: - 1 -

2 . multiple trnsitions n e enled t the sme time, ny one of whih n fire. none re required to fire they fire t will, etween time nd infinity, or not t ll (i.e. it is totlly possile tht nothing fires t ll). Sine firing is nondeterministi, Petri nets re well suited for modeling the onurrent ehvior of distriuted systems A forml definition [PETE77[REIS92] A Petri net is 5-tuple S is set of ples. T is set of trnsitions. S nd T re disjoint, i.e. no ojet n e oth ple nd trnsition. F is set of rs known s flow reltion. The set F is sujet to the onstrint tht no r my onnet two ples or two trnsitions, or more formlly: is n initil mrking, where for eh ple, there re tokens. is set of r weights, whih ssigns to eh r some denoting how mny tokens re onsumed from ple y trnsition, or lterntively, how mny tokens re produed y trnsition nd put into eh ple. A vriety of other forml definitions exist. Some definitions do not hve r weights, ut they llow multiple rs etween the sme ple nd trnsition, whih is oneptully equl to one r with weight of more thn one Bsi Mthemtil Properties The stte of Petri net n e represented s n M vetor, where the 1st vlue of the vetor is the numer of tokens in the 1st ple of the net, the 2nd is the numer of tokens in the 2nd ple, nd so on. Suh representtion fully desries the stte of Petri net. A stte-trnsitionlist,, whih n e shortened to simply is lled firing sequene if eh nd every trnsition stisfies the firing riteri (i.e. there re enough tokens in the input for every trnsition). In this se, the stte-trnsition list of is lled trjetory, nd is lled rehle from through the firing sequene of. Mthemtilly written:. The set of ll firing sequenes tht n e rehed from net N nd n initil mrking M re noted s L(N,M ). The stte-trnsition mtrix W is S y T lrge, nd represents the numer of tokens tken y eh trnsition from eh ple. Similrly, W + represents the numer of tokens given y eh trnsition to eh ple. The sum of the two, W = W + W n e used for lulting the ove mentioned eqution of whih n now e simply written s, where σ is vetor of how mny times eh trnsition fired in the sequene. Note tht just euse the eqution n e stisfied, does not men tht it n tully e rried out - for tht there should e enough tokens for eh trnsition to fire, i.e. the stisfiility of the eqution is required ut not suffiient to sy tht stte M n n e rehed from stte M [RIEM99]. Exmple:- Fig(1): Petri Net Exmple - 2 -

3 1.3. Petri Net Types There re mny extensions to Petri nets. Some of them re ompletely kwrds-omptile (e.g. oloured Petri nets) with the originl Petri net, some dd properties tht nnot e modelled in the originl Petri net (e.g. timed Petri nets). If they n e modelled in the originl Petri net, they re not rel extensions, insted, they re onvenient wys of showing the sme thing, nd n e trnsformed with mthemtil formuls k to the originl Petri net, without losing ny mening. Extensions tht nnot e trnsformed re sometimes very powerful, ut usully lk the full rnge of mthemtil tools ville to nlyse norml Petri nets[stor]. The term high-level Petri net is used for mny Petri net formlisms tht extend the si P/T net formlism. This inludes oloured Petri nets, hierrhil Petri nets, nd ll other extensions skethed in this setion. A short list of possile extensions: In stndrd Petri net, tokens re indistinguishle. In Coloured Petri net, every token hs vlue. In populr tools for oloured Petri nets suh s CPN Tools, the vlues of tokens re typed, nd n e tested nd mnipulted with funtionl progrmming lnguge. A susidiry of oloured Petri nets re the well-formed Petri nets, where the r nd gurd expressions re restrited to mke it esier to nlyse the net[zhou9]. Another populr extension of Petri nets is hierrhy: Hierrhy in the form of different views supporting levels of refinement nd strtion were studied y Fehling. Another form of hierrhy is found in so-lled ojet Petri nets or ojet systems where Petri net n ontin Petri nets s its tokens induing hierrhy of nested Petri nets tht ommunite y synhronistion of trnsitions on different levels[zhou98]. A Vetor Addition System with Sttes (VASS) n e seen s generlistion of Petri net. Consider finite stte utomton where eh trnsition is lelled y trnsition from the Petri net. The Petri net is then synhronised with the finite stte utomton, i.e., trnsition in the utomton is tken t the sme time s the orresponding trnsition in the Petri net. It is only possile to tke trnsition in the utomton if the orresponding trnsition in the Petri net is enled, nd it is only possile to fire trnsition in the Petri net if there is trnsition from the urrent stte in the utomton lelled y it. (The definition of VASS is usully formulted slightly differently.) Prioritised Petri nets dd priorities to trnsitions, wherey trnsition nnot fire, if higher-priority trnsition is enled (i.e. n fire). Thus, trnsitions re in priority groups, nd e.g. priority group 3 n only fire if ll trnsitions re disled in groups 1 nd 2. Within priority group, firing is still non-deterministi. The non-deterministi property hs een very vlule one, s it lets the user strt lrge numer of properties (depending on wht the net is used for). In ertin ses, however, the need rises to lso model the timing, not only the struture of model. For these ses, timed Petri nets hve evolved, where there re trnsitions tht re timed, nd possily trnsitions whih re not timed (if there re, trnsitions tht re not timed hve higher priority thn timed ones). A susidiry of timed Petri nets re the stohsti Petri nets tht dd nondeterministi time through djustle rndomness of the trnsitions. The exponentil rndom distriution is usully used to 'time' these nets. In this se, the nets' rehility grph n e used s Mrkov hin. There re mny more extensions to Petri nets, however, it is importnt to keep in mind, tht s the omplexity of the net inreses in terms of extended properties, the hrder it is to use stndrd tools to evlute ertin properties of the net. For this reson, it is good ide to use the most simple net type possile for given modeling tsk[zhou98]. 2. MATRIX STRUCTURE ANALYSIS(MSA). One of the responsiilities of the struturl design engineer is to devise rrngements nd proportions of memers tht n withstnd, eonomilly nd effiiently, the onditions ntiipted during the lifetime of struture. A entrl spet of this funtion is the lultion of the distriution of fores within the struture nd the displed stte of the system. Our ojetive is to desrie modern methods for performing these lultions in the prtiulr se of frmed strutures. The numer of strutures tht re tully simple frmeworks represents only prt of these whose - 3 -

4 ideliztion in the form of frmework is eptle for the purposes of nlysis. Building of vrious types, portions of erospe nd ship strutures, nd rdio telesopes nd the like n often e idelized s frmework [WILL]. In design, oth servieility limit sttes nd strength limit sttes should e onsidered. Strutures onsisting of two-or three-dimensionl omponents-pltes, memrnes, shells, solids re more omplited in tht rrely do ext solutions exist for the pplile prtil differentil equtions. One pproh to otining prtil, numeril solutions is the finite element method. The si onept of the method is tht the totl struture n e modeled nlytilly y its. Sudivision into regions(the finite elements) in eh of whih the ehvior y set of ssumed funtions representing the stresses or displements in tht region. This permits the prolem formultions to e ltered to one of the estlishment of system of lgeri equtions. Viewed in this wy, struturl nlysis my e roken down into five prts [MART2]. Bsi mehnis: the fundmentl reltionships of stress nd strin, omptiility nd equilirium. Finite element mehnis: the ext or pproximte solution of the differentil equtions of the element. Eqution formultion: the estlishment of the governing lgeri equtions of the system. Eqution solution: omputtionl methods nd lgorithms. Solution interprettion: the presenttion of results in form useful in design. This pper dels hiefly with prt 3,4 nd 5, it is on mtrix struturl nlysis. This is pproh to these prts tht urrently seems to e most suitle for utomtion of the eqution formultion proess nd for tking dvntge of the powerful pilities of the omputer in solving lrge order systems of equtions. The equtions of the mtrix (or finite) element pproh re of form so generlly pplile tht it is possile in the theory to write single omputer progrm tht will solve n lmost limitless vriety of prolems in struturl mehnis. Mny ommerilly ville generl purpose progrms ttempt to otin this ojetive, lthough usully on restrited sle. The dvntge of generl purpose progrms is not merely this pility ut the unity fforded in the instrution of prospetive users, input nd output dt interprettion proedures, nd doumenttion [JOHN6]. The four omponents in the flowhrt of figure 2 re ommon to virtully ll generl purpose, finite element nlysis progrms. 1. Input Definition of physil model geometry, mteril, loding nd oundry onditions 2. Lirry of elements Genertion of mthemtil models for struturl elements nd pplied lodings 3. Solution onstrution nd solutions of mthemtil model for struturl system 4. Output Disply of predited displements nd fores Fig. (2): Struturl nlysis omputtion flow - 4 -

5 3. THE PROPOSED MODELS 3.1 The First Exmple in engineering field This pper dels with some of the methods for trining prolems of lrge size. One wy to hndle lrger strutures ideliztion, tht is to disregrd, suppress, or pproximte the effet of degrees of freedom tht in the opinion of the nlyst, hve only minor ering on the result. The mny different wys in whih this my e done re so dependent upon the individul struture tht they nnot e disussed usefully in generl text. Here we presents sheme for reduing the order of the system of equtions tht hve e solved t ny one time one the struture hs een idelized. This mens tht generlly we will disussing method for reduing the order of the stiffness mtries to e inverted. The flowhrt of this model explins in figure 3. Fig.(3): Flowhrt of the Engineering Models If we eginning with the following vlues: u1 =, v1 =, Q1 =, u3 =, v3 =, Q3 =, fx2 = 5, fy2 = -16, m2 = , re = 6, e1 = 2, l=8. The Computtion: ii = 2 * (1 ^ 6) (1) = ( re * e1) ( l) (2) = (12 * e1 * ii) (3) (l ^ 3) = (6*e1*ii) (l^2) (4) - 5 -

6 - 6 - d= (l) (4*e1*ii) (5) e = (1 * 1 ^ 7) (6) The Mtrix n e uilds s follow: = d e e d Mtrix Vertil = d e e d Mtrix Horizontl Exmple:- Find Fx1,Fy1,M1,Fx3,Fy3,M3. Depend on the ove informtion. Generl Mtrix =

7 Fig.(4): The resulted vlues of the Engineering Model U 1 fx1 1 1 V fy Q1 m1 2 2 U fx V 2 fy 2 = [ overllmt rix ]* (7) 2 2 Q m 3 3 U fx V 3 fy Q m By Appling this model, we find: U2=.344, V2=.1187, Q2=.5, Fx1= , Fy1= , M1= 629, Fx3= -51.6, Fy3= -18, M3= From this result, we n see the urte nd fst of the proposed model nd lso we n use this model with different ngles. 4.2 The Seond Exmple in Edution field This model dels with qulity ssurne in one deprtment of higher edution, so we need to hek the interesting thinks suh tht( Stff vilility, lssroom suitility, student lssifitions, ls. suitility,...et.). When we strt to design the model we need to hek the qulity ssurne requirements for this deprtment, if this requirements ville whih n e represented y onditions (C1,C2,..Cn). If ll onditions hieved the trnsition requirement hek fire nd token in requirement ple trnsfer to mthed requirements ple. At the seond model stge ll hieved onditions n mthed with the tul limittions if it mth the mthed trnsition fire nd the token trnsfer to the environment requirements ple. If it is not mthed we need to refine feed k. The mthed hieved when the mthed trnsition fire nd the token trnsferred to Aepted Qulity ple nd the proess finished. When mthed not hieved we need refine feed k

8 Fig.(5): Petri net of the Edution Model 5. CONCLUSIONS The min enefits of use Mtrix Struturl Anlysis (MSA) tehnique nd Petri net for modeling prolems is to get the est model qulity sed on qulity ssurne onditions. As the min ontriution, we introdued MSA to improve the designed model to get fst nd rue results: Eqution formultion: the estlishment of the governing lgeri equtions of the system. Eqution solution: omputtionl methods nd lgorithms. Solution interprettion: the presenttion of results in form useful in design. In MSA pproh the prts urrently seems to e most suitle for utomting the eqution formulizing proess nd for tking dvntge of the powerful pilities of the omputer to solve lrge order of systems of equtions. Petri nets provide dynmi struture for the model this fility mke the resulted model like in rel world. Petri net used to test the qulity ssurne in edution field exmple. As result, using MSA nd Petri Net for designing models in (engineering nd edution exmples) gives ler, understndle, nd very good result whih is very nerest to the rel

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