Fault Diagnosis in DC Motor Using Bond Graph Approach

Transcription

1 Fult Dignosis in DC Motor Using Bond Grph Approch González Contrers B. M. Theilliol D. VelVldés L. G. Universidd Autónom Tlxcl Université Henri Poincré CENIDET Tlxcl VndoeuvreLes Nncy Cuernvc, MOR. México Frnci México Abstrct. The Bond Grp (BG) shows direct correspondence between their component prmeters nd physicl phenomen to be modeled, not only system model is obtined, but lso component prmeter filure representtions, since the BG model obtined is useful like dignosis model too. Using like cse of study direct current (dc) motor to implement the ppliction, this pper shows how to obtin fult tree to define the first set of fult hypothesis, nd then, for using temporl grph to define the fult, i.e. in order to reduce the set of fult hypothesis to only one element. Key words. Fult dignosis, cuslity, dc motor, Bond Grph, quntittive nd qulittive modelbsed pproches. 1. Introduction Fult dignosis hs been presented in mny industril processes s n indispensble prt of the control systems to be ble to gurntee the relibility nd vilbility of the process (Isermn et l. 1996; vn Schrick Dirk 2000; Vergé 1994). It is stge of the supervisory system, which hs the objective of indicting undesired or forbidden process sttes nd to tke pproprite ctions in order to mintin the opertion nd to void dmges or ccidents. The supervisory system blocks re shown in figure 1. ws developed tht integrtes studies where the BG hs been used for fult dignosis, but here is presented new mnner to interpret the energy flux immeditely fter fult is presented. The pper is orgnized s follows. In section 2 the BG bsics re given nd then in section 3, the dc motor circuit nd BG model of it re presented. Section 4 shows the proposed method using BG elements nd its order to be pplied to the systems nd the proposed method is pplied to dc motor; fult in prmeter R is used s exmple. Then, in section 5, the proposed method is pplied to dc convertermotor set system. The conclusions re given in section Bond Grph Bsics Bond Grph (BG) pproch hs been used in mny senses: simultion, modeling nd control, show itself like very useful tool for ppliction in severl systems. This is good tool to representtion models bsed on physicl concepts using finite symbols for ppliction to ny systems; provides structured pproch to system dynmics modeling without loss of the physicl sense, tht tkes dvntge to dignosis. In tble 1 re shown the equivlences between elements from severl domins (Blundell 1982; Broenink 2001; Gwthrop et l. 1996; Krnopp 1990). Tble 1. Equivlences between domins Figure 1. Supervisory system The dignosis prt in fult dignosis systems shows hole between qulittive nd quntittive nlysis (vn Schrick Dirk 2000; Vergé 1994), it is necessry to pply, in mny cses, sttisticl, opertor experience, heuristic nd functionl informtion to complete the dignosis stge. In this pper method In BG common vribles effort nd flow re used in ech domin nd their product is power, s energy derivtive, is common concept relting these systems. From the viewpoint of energy flow, using BG, system components nd physicl phenomen re clssified into five bsic types: 1) energy conserving or storing elements (I nd C), 2) energy dissipting elements (R), 3) energy source elements (Se nd Sf), 4) energy 419

2 conversion elements (TF y GY), nd 5) junction elements (Blundell 1982; Broenink 2001; Gwthrop et l. 1996; Krnopp 1990). Ech element with chrcteristic reltionship re shown in tble 2. We cn think in junction s common effort (0) or common flow (1). Tble 2. Bond Grph elements flow integrl is displcement generlized (Blundell 1982; Broenink 2001; Gwthrop et l. 1996; Krnopp 1990). Tble 3. Generlized sttes 3. DC motor model A dc motor in independent excitement or permnents mgnets configurtion is presented in figure 2. Here, the rmture voltge, V, is constnt s well s the field current, i f. Figure 2. DC motor circuit It is necessry give to ech element cuslity, tht mens wht direction the effort tkes, nd therefore, the flow direction, due to in ech bond the conjugte floweffort re in opposite directions. To ssign the cuslity, cusl stroke indictes the effort direction, wht tell us which vrible is outside of the element (Blundell 1982; Broenink 2001; Gwthrop et l. 1996; Krnopp et l. 1990). Exist rules to ssign cuslity nd generte the equtions describing the system. It is preferred n integrl cuslity, tht it is determined for storing elements I nd C, by the reltionships: 1 t 1 t q = f ( τ ) dτ (1) p = C 0 e( τ ) dτ (2) I 0 clled integrl cuslity, nd mkes the equtions obtining n esy wy. From (1) nd (2) we cn obtin the generlized sttes combining the equtions obtined from the other elements, this vribles re the stte vribles from the BG model. Using the tble 3 we cn to obtin the more common vribles, for tht, it should be hs in mind tht n effort integrl is momentum generlized nd The mthemticl model of the system cn be described in dynmicl form strting from the bond grph model presented in figure 3. To obtin the equtions the generlized vribles re used: p 3 relted to electricl prt nd p 8 relted to mechnicl prt. Figure 3. BG of dc motor From figure 3 the equtions tht describes the cusl interctions re: p3 = e e e..(3) f3 =..(8) e f 1 = f 2 = f3 = f 4..(4) p8 f8 =..(9) J e8 = e5 e6 e7..(5) e6 = βf 6.(10) f 5 = f6 = f7 = f8..(6) e 4 = kf5.(11) e2 = R f 2..(7) e 5 = kf4...(12) L 420

3 Here, the stte vribles re electric momentum (mgnetic flux linkge) nd mechnicl momentum (ngulr momentum). The stte equtions from bove describing the system re: R k p3 p L J 1 0 V 3 = (13) k β p 0 1 p τ L 8 8 L J but we re interested in current nd velocity s the output vribles, which re the mesured vribles from dc motor, then: i 1 0 = L p 3 (14) ω 0 1 r p8 J Now, we use prllel model (Leitch 1993) to generte the residul tht shows the system chnges in the fult cse. Observing these residuls is possible to pply the dignosis method noting chnges in ech vrible mesured. We cn estblish threshold to decide when the residul is not null. This threshold depends on the vlue due to the 2% of stedy stte of the signl mesured, i.e. current nd velocity in this cse. If the vlue is 2% upper (or lower), then the residul is nonzero. 4. Fult dignosis method The fult dignosis method proposed here, is bsed using reserches tht hs been used BG to derive some results focused in dignosis. The stges of this method re shown in figure 4. Figure 4. Fult dignosis method The cusl grph is consequence of the cusl reltionships between elements, obtined in direct form from the BG of the system (Isermnn 1996; Krnopp et l. 1990; Mostermn et l. 1997; Mostermn et l. 1995). This grph is seemed to the sign flow grph used in clssic control. To construct the cusl grph reference nodes re necessry, these re selected from the junction 1 (or 0) nd using the eqution tht describes it. For the figure 3 model we use the equtions (3) nd (5), nd we relte ech component until fits the other equtions. Ech node constitutes blocks tht re the bses of the cusl grph. Arrows connect vribles nd prmeters, nd the reltion between vribles is indicted over the link (Mostermn et l. 1997; Mostermn et l. 1995). Figure 5. Nodes (blocks) to form cusl grph The cusl grph is done joining the reference nodes. Cusl grph for dc motor is shown in figure 6. Figure 6. Cusl grph of dc motor F rom this cusl grph we obtin the next elements of the fult dignosis method proposed. It follows the fult tree, which shows the qulittive vritions when n observed vrible hs chnged. A qulittive vrition refers to chnge from nominl vlue. In this work, the three symbolic terms low [], norml [0] nd high [] re used. Qulittive trnsitions occur t the boundry endpoints. Mthemticlly, the qulittive vlue of x, denoted [x], defined s the devition from reference point x 0 is given s [x]= sgn (x x 0 ). When the reference is rnge, the qulittive vlue of x is defined s [x] = when x < x0 [x] = 0 when x 0 x x 0 [x] = when x 0 < x where x 0 nd x 0 re the lower nd upper rnge endpoints, respectively. The fult tree derived from BG is obtined for ech vrible of interest nd cn be built using bck propgtion in the cusl grph (or ntecedents nd consequents tble), in the next form (Krnopp et l. 1990; Vergé et l. 1994): Strt from the vrible of interest with its qulittive vlue nd trck bck the sign through the pth ssigning the right qulittive vlue (from the vrible of interest, propgte the qulittive 421

4 vlue, tke this vrible s the consequent nd write the ntecedent s the next consequent). Join the sign by mens of rrow ( brnch) nd continue the trck bck for ll the signs cross the vribles (join the consequents to ntecedents by mens of rrows or brnches). Propgtion is terminted long pth when conflicting ssignment is reched. The fult tree cn be gotten from reltionships (3)( 12), more precisely from cusl reltionships given for the BG rules (Blundell 1982; Broenink 2001; Krnopp et l. 1990; Mostermn et l. 1997). In tble 4 this reltionships re presented, nmed ntecedents nd consequents tble. Tble 5. Antecedents nd consequents for dc motor cusl grph Tble 4. Antecedents nd consequents Using qulittive vlu es (, 0, ), strting from the vrible observed with its vlue obtined from residul, one propgtes this vlue through the brnches. With help of tble 4 is built the tble 5 for dc motor. The fult tree for the vribles of interest, e 3, relted to the current i, nd e 8, relted to the velocity ω r with qulittive vlue [], re shown in figure 7. In these trees, the qulittive vlue is propgted using logicl opertions with positive vlues. Figure 7. Fult trees for e 3 nd e 8 A suggested lgorithm to generte the first fult hypothesis set could be the following: 1. Construct n ntecedents nd consequents tble, or from cusl grph use bck propgtion. 2. Construct the fult tree for the vribles of interest. 3. Observe wht type of qulittive vrition presents the observed vribles through the residuls, nd which they re mtched with the qulittive chnges in the fult tree. 4. Form the hypothesis fult set with the prmeters tht mtch the sme qulittive comportment in fult trees. So fr, the blocks shown in figure 4 hs been developed by others reserchers, but the finl block bout using the temporl grph is suggested by Mostermn in ((Mostermn et l. 1997; Mostermn et l. 1995). One cn to predict how will be the comportment of the signl nd its derivtives before the system rech the new stedy stte fter the fult hs occur by mens 422

5 of the temporl grph ((Mostermn et l. 1997; Mostermn et l. 1995). This grph provides the signtures tht indicte the comportment before the new stedy stte is reched. The following steps to obtin the cusl grph re used: 1. Strt from one of the prmeters tht re included in fult hypothesis set nd propgte its qulittive vlue through the grph. 2. When the signl cross through link with differentil, then mrk this with or if qulittive vlue increse or decrese. This indictes the first derivtives nd the qulittive signture. 3. The propgtion continues until rech second derivtive nd ll the observed vribles hve been covered by the second derivtive. 4. Signtures ssocites to derivtives with smll time constnts re rejected, if there were two vribles with different signtures. discrded becuse is differentil relted, i.e. the fult comportment continue while time go on. Tble 6. Signtures for prmeters in hypothesis fult R,, 0,, e 3 e 8,,,, J e3 e 8 K 0,,,, e 3 e 8, discrded As it is necessry obtin derivtives, cn be used stte vrible filters, if the system is liner, like treted here, or use numericl methods to generte the derivtives. In figure 9 re shown the norml signl nd its derivtives for current nd velocity Derivtive Derivd (bove), (rrib), norml Corriente signl i (bjo) (below) e Derivtive Derivd (bove), (rrib), norml Velocidd signl ω r (bjo) (below) e5 Temporl grph for R is shown in figure A Time Tiempo(s) Time Tiempo (s) Figure 7. Temporl grph for R As exmple, it is presented the residul (figure 8) from R fu lt in the electric prt of dc motor. mgnitud A rd/s time (s) Figure 8. Residuls when R is fulty It is observed tht, current nd velocity decrese, then we only construct fult trees for e 3 nd e 8 with qulittive vlues (). These fult trees re shown in figure 7, nd cn be observed prmeters K, R, J hve the sme qulittive signture in both fult tree, then the first hypothesis fult set is formed by {R, J, K, }. The others prmeters re rejected becuse hve different qulittive vlue in ech tree. When use the temporl grph, it is obtined the signtures presented in tble 6. From figure 9, the norml signl nd first order derivtive tht mtch signtures ginst shown in tble 6 re R, J, but J is () Current i (b) Velocity ω r Figure 9. Derivtives nd norml signl of mesured vribles 5. DC convertermotor system set To model in BG dc converter it is introduced n element tht describes the discontinuous behvior elements like the switching devices included in the converter. First, is modeled reductor converter (buck) introducing the discontinuous element s TF bond grph element with trnsformer modulus m, which tkes 1 nd 0 vlues depending the conmuttions sttes of the switching device, nmely, duty cycle. The converter circuit is shown in the left side of figure 10. Figure 10. DC converter circuit nd BG model The BG model is shown in the right side of figure 10. When the full bridge dc converter work in one qudrnt, s shown in figure 11, the model cn be interpreted like the shown in figure 10, so the BG model is the sme shown in the right side of figure

6 Figure 11. Full bridge dc converter circuit Looking the figure 6, e 1 corresponds to dc converter output, nd it is possible to dd the cusl grph of dc converter lone, shown in figure 12 ) to obtin the csul grph of ll the system, s is presented in figure 12 b). ) for buck converter b) for convertermotor system Figure 12. Cusl grph of dc converter The next tble 7, summrize the simulted fults pplying the proposed method. Tble 7. Summry of fults Fult Dignosis Residul (minimum hypothesis set) comportment J {J } Trnsitory J {J } T rnsitory K {R, K, m } Permnent K {R, K, m } Permnent R {R, m } Permnent R {R, m } Permnent L {L } Trnsitory L {L } Trnsitory β {β } Permnent β {β } Permnent m {R, K, m } Permnent m {m } Permnent 6. Conclusion The fult dignosis method presented in this pper use the BG pproch. The finl stge of fult dignosis needs to nlyze the comportment over the time, nd mtch this comportment ginst the provided by the temporl grph to decide where is the fult exctly by mens of reduction of the fult hypothesis set. Although the method cnnot reduce the fult hypothesis set to only one element in some cses, shows it s new ttempt to unify the qulittive nd quntittive resoning in fult detection nd cn pply the integrtion of fult detection methods becuse the equtions llow it. The method is esy to pply nd sequentil to perform, does not require nother tool more thn only BG elements. However, the limits of this method re reched when the system is non liner. Some interesting questions remin open in the frmework of this cse, for exmple the selection between: ) Anlyticl redundncy reltions (ARR) with prity equtions for nonliner systems (Stroswiecki et. l 1991). b) Multiple models pproch (Chen t l. 1999, Boskovic et l. 2003). References Blundell A. J., Bond grphs for modeling engineering systems, John Wiley nd Sons, Gret Britin, Boskovic J. D., Mehr R. M., Filure detection, identifiction nd reconfigurtion in flight control, in: Fult Dignosis nd Fult Tolernce for Mechtronic Systems: Recents Advences, pp , SpringerVerlg, Berlin, Broenink J.F., Introduction to physicl systems modeling with Bond Grphs, 27/07/2001. Chen J., Ptton R. J., Robust modelbsed fult dignosis for dynmic systems, chpter 9, pp , Kluwer Acdemic Publishers, London, 1999 Gwthrop P., Smith L., Metmodelling: For Bond Grphs nd dynmic systems, Prentice Hll, E. U., Isermnn Rolf, Bllé Peter, Trends in the ppliction of model bsed fult detection nd dignosis of technicl process, in: IFAC SAFEPROCESS '96, vol. N, Sn Frncisco, U.S., June 1996, pp Krnopp D.C., Mrgolis D.L., Rosenberg R.C., Systems dynmics: unified pproch, John Wiley nd Sons, New York, 2nd ed., Leitch R., Engineering dignosis: Mtching problems to solutions, en: Interntionl Conference on Fult Dignosis, vol. 3, Tolouse, Frnce, April, 1993, pp Mostermn P.J., Bisws G., Monitoring, prediction, nd fult isoltion in dynmic physicl systems, in: Proceedings of AAAI97, Rhode Islnd, July 1997, pp Mostermn P.J., Kpdi R., Bisws G., Model Bsed Dignosis of Dynmic Systems, in: Proceedings of DX95, the Sixth Interntionl Workshop on Principles of Dignosis, Goslr, Germny, October 24, 1995, pp Stroswiecki M, CometVrg G. Anlyticl redundncy reltions for fult detection nd isoltion in lgebric dynmic systems, in Automtic, vol 37, pp , Pergmon, vn Schrick Dirk, Conceptul system drft for sfe process: reltions nd definitions, in: IFAC SAFEPROCESS 2000, vol. 1, Budpest, Hungry, June 2000, pp Vergé M., Jume D., Fult detection nd Bond Grph modeling, in: IFAC SAFEPROCESS '94, vol. 2, Espoo, Finlnd, June 1994, pp