A COMBINED QUALITATIVE/QUANTITATIVE APPROACH FOR FAULT ISOLATION IN CONTINUOUS DYNAMIC SYSTEMS

Transcription

1 A COMBINED QUALITATIVE/QUANTITATIVE APPROACH FOR FAULT ISOLATION IN CONTINUOUS DYNAMIC SYSTEMS Eric-J. Manders Λ; Sriram Narasimhan Λ; Gauam Biswas Λ Pieer J. Moserman ΛΛ Λ Deparmen of Elecrical Engineering and Compuer Science Vanderbil Universiy, P.O. Box 679 Sa. B, Nashville, TN ΛΛ Insiue of Roboics and Mecharonics DLR Oberpfaffenhofen, P.O. Box 6, D-83 Wessling, Germany Absrac: The TRANSCEND sysem for faul deecion and isolaion in coninuous sysems uses qualiaive reasoning mehods o analyze ransiens caused by abrup fauls. Qualiaive ransien analysis avoids some of he compuaional difficulies associaed wih numerical schemes, bu hey lack discriminaing power. This paper presens he formal basis for qualiaive ransien analysis, and hen esablishes he limis of he discriminaory power of his mehodology. An inegraed scheme ha sars wih qualiaive faul isolaion o narrow down possible faul hypoheses, and hen uses a focused quaniaive parameer esimaion scheme o idenify he rue faul is developed. This approach provides a number of advanages over purely quaniaive FDI schemes. Copyrigh cfl IFAC Keywords: Faul deecion and isolaion, monioring, qualiaive analysis, ransien analysis, parameer esimaion.. INTRODUCTION Model-based approaches for faul deecion and isolaion (FDI) in coninuous dynamic sysems employ relaions imposed by he sysem configuraion and funcionaliy o compue residuals, ha capure he discrepancies beween nominal and observed behavior. Residual compuaion and analysis is non-rivial for complex sysems, primarily because of siffness, convergence, and inracabiliy problems in dealing wih he sysem s non-linear dynamics. To miigae his, an FDI framework has been developed ha derives residuals as qualiaive faul signaures, and analyzes hese residuals wih a faul isolaion observer mechanism based on a unique progressive monioring scheme (Moserman and Biswas 999). This paper Parially suppored by a gran from Agilen Laboraories Parially suppored by a gran from Xerox PARC demonsraes a combined qualiaive and quaniaive faul isolaion process where he qualiaive faul isolaion scheme is followed by a direced quaniaive parameer esimaion sep o resolve ambiguiies among he faul candidaes ha canno be disinguished by he qualiaive signaures alone. Fig. illusraes he archiecure of TRANSCEND, our qualiaive model-based approach o diagnosis (Moserman and Biswas 999). Variables u, x, and y, are he inpu, sae, and oupu vecors of he process under diagnosic scruiny. A sandard gain marix observer scheme (Brammer and Siffling 989) racks he residual, r = y ^y (^y is he prediced sysem behavior) o correc for small deviaions in he esimaed sae vecor ^x. A unique aspec of he qualiaive approach is he symbol generaion uni, ha uses robus mehods o compue symbolic values of he magniude deviaion and slope of signal ransiens, rs.

2 y(). Transien deecion and analysis Fig.. TRANSCEND archiecure. Faul deecion riggers a faul isolaion scheme ha consiss of hypohesis generaion and hypohesis refinemen. Hypohesis generaion uses he diagnosis model, m (implemened as a emporal causal graph (TCG) (Moserman and Biswas 999)), and he symbolic residuals, rs, o generae a se of hypohesized faul candidaes, f h, and o predic behavior, p, for each faul candidae. During hypohesis refinemen spurious candidaes are eliminaed from he se using progressive monioring o mach new observaions agains he predicions and derive he refined faul se, f r. This paper esablishes he basis for qualiaive signaures and progressive monioring in erms of he Taylor series expansion of he faul ransien signal. Based on his, he limiaions of he qualiaive faul isolaion mehod are derived, and a focused parameer esimaion scheme is inroduced o enable furher refinemen of he faul hypoheses. The reduced faul se obained by applying he qualiaive observers focuses and significanly reduces he compuaional complexiy in he parameer esimaion ask. The mehod is illusraed wih simulaion experimens on a hree-ank fluid sysem.. QUALITATIVE DIAGNOSIS FROM TRANSIENTS Model parameers in he TCG correspond direcly o sysem componens, and a faul is a parameer value ha deviaes from is nominal value. This paper focuses on he class of abrup fauls. Definiion. (Abrup faul). An insananeous and persisen change in a parameer value. The noion of insananeous change is a modeling absracion (e.g., see (Moserman and Biswas 998)), where he faul is defined as an insananeous parameer change in he model. In a physical sysem his change is never ruly insananeous bu he absracion eliminaes he seep nonlineariies and siffness ha occurs when simulaing he behavior of such sysems. An abrup faul resuls in ransien behaviors in sysem variables and he faul isolaion ask relies heavily on he characerizaion of he ransiens in he measuremen daa (Moserman and Biswas 999). Transien deecion implies a decision on wheher he residual r is deviaing significanly from. The need for sophisicaed deecion echniques is srongly dependen of he signal-o-noise raio of he residual signal. In simulaion sudies, a deecion scheme based on an insananeous signal value has been used by applying simple hreshold crossing deecors, for experimens where a small amoun of noise is added o he simulaion daa. For signals where he noise presens a larger problem, more sophisicaed mehods have been sudied ha employ saisical signal processing echniques and can be designed o obain desired sensiiviy and specificiy (Manders e al. 999). The FDI analysis in TRANSCEND assumes ha disconinuous changes in variable values can only occur a he poin of failure, hus sysem behavior is coninuously differeniable before and afer he occurrence of a faul. Therefore, he ransien response in a measuremen afer he ime poin of failure,, can be approximaed by he Taylor series expansion. If y( ) is he value of he residual signal jus afer he occurrence of he faul, he k h order Taylor series expansion for y(), is defined as: y() = y( )+y ( ) ( ) + y ( ) ( ) +!! + y (k) ( ) ( ) k + R k (); k! where R k () =y (k+) ( ) ( )k+ is a remainder erm and <. For mos well-behaved funcions he (k+)! series converges, i.e., R k ()! as k!(kreyszig 97). In paricular, if jy (k+) j is bounded, he Taylor series is a good approximaion of he rue signal y() when is close o. Consider he ransien signal and is firs hrough fourh order Taylor series approximaion shown in Fig.. Conforming o he definiion, as increases from, approximaions differ increasingly from he acual signal, bu higher order approximaions follow he signal for a longer ime inerval. The analysis of ransien dynamics by inerpreing he signal as a Taylor series approximaion is he basis for describing he faul ransien signal as a faul signaure. Definiion. (Faul signaure). The faul signaure f is he se of k +feaure values consising of he 3 Fig.. Transien signal for a nd order sysem (solid line) and s hrough h order Taylor series expansion (dashed lines) a = +.

3 magniude and he s hrough k h order derivaive values compued a from he residual y: f = fy( );y ( );y ( ); ;y (k) ( )g The noion of disconinuous change warrans addiional explanaion. Similar o he concep of an abrup change in a model parameer, a disconinuous change is an absracion in he observaion of a physical sysem. Wha is considered as a disconinuous change is direcly relaed o he sampling rae of he discree ime sampled signal. Definiion. (Disconinuiy). A change in a measured variable ha exhibis a ransien response faser han he sampling rae of he signals. Faul isolaion is hen based on he comparison of he faul signaures wih subsequen measuremens made on he sysem. Performing his analysis quaniaively is an inracable problem. When a faul occurs, he exac magniude of parameer value changes is unknown, so derivaive values in he faul signaure have o be compued from subsequen measuremens. For complex nonlinear sysems, his is a very difficul problem o solve, eiher by closed form analyic echniques or by numeric ieraion. To address his a qualiaive consrain analysis scheme, discussed in he nex secion, was developed for he faul isolaion ask. In he qualiaive framework, individual measuremens are labeled as normal (), above normal (+) and below normal ( ). Similarly, derivaives ake on values, increasing (+), seady (), and decreasing( ). The faul signaure in he qualiaive framework hen is he sequence of +,, or magniude and k derivaive values compued a he poin of failure,. This faul signaure is he basis for qualiaive ransien analysis using he progressive monioring scheme. Lemma. (Qualiaive ransien analysis). Transien dynamics are capured by evaluaion of he direcion of abrup change a he poin of failure (if i occurs), and he signs of he derivaives of he signal afer he onse of a faul. In his paper, all ransien signals are considered idealized signals. This means ha daa is sampled a appropriae sampling raes and ha noise does no play a role in deermining he componen values of a signaure. Oher work (Manders e al. ) describes saisical signal analysis algorihms for exracing ransien feaures from noisy signals.. Faul Isolaion Faul isolaion using residuals is radiionally achieved by designing muliple faul observers wih a oneo-one correspondence beween he individual faul hypoheses and he observers (Paon and Chen 997). In our work, we define each observer in erms of he faul signaures. Faul deecion riggers he faul isolaion mechanism. The hypohesis generaion algorihms, implemened as a wo sep process, faul hypohesis generaion followed by faul signaure generaion for each hypohesis, is described in deail in (Moserman and Biswas 999). An observer, defined in erms of a se of faul signaures, one for each measuremen, is designed for each faul hypohesis. The faul signaure is of order k when i includes derivaive values of up o order k. The minimal pracical faul signaure consiss of magniude and firs order derivaive, he slope, of he signal. The choice of k is direcly relaed o he concep of diagnosabiliy (Moserman and Biswas 999). Ideally, he faul signaure order and he se of measured variables are seleced such ha ha all possible fauls ha can be hypohesized by he model can also be uniquely deermined. Comparing he faul signaure wih he feaure vecor obained from he evolving ransien daa is he basis of a progressive monioring scheme for racking signal ransiens (Moserman and Biswas 999). Lemma. (Progressive monioring). Qualiaive magniude and slope of a faul ransien are mached a- gains a qualiaive faul signaure by saring in a sequence from a disconinuous magniude and firs order change o a succession of higher order derivaives. Comparing he i h and (i+) h erms in he Taylor series, one can esablish jy (i) ( )j > jy (i+) ( )j ( ) (i+) for some period of ime. As increases, a some poin in ime he inequaliy reverses. From ha poin in ime, he higher order derivaive dominaes he lower one. Lemma provides he basis for progressive monioring of signal dynamics using higher order derivaives. Saring from he poin of failure,, he signal magniude in response o he faul y( ) deermines he signal value. Immediaely afer ha he firs derivaive of he signal dominaes he dynamic behavior because small values of ( ) dominae higher powers ( ) i in he Taylor series. As increases, higher order derivaives in succession increasingly conribue o he dynamics of he signal. In dynamic ransien analysis, a curren normal measuremen or slope value canno be used o eliminae a faul candidae, because, here is no guaranee ha his measuremen will no deviae a some laer ime. The excepion o his is he case when a disconinuous change can be inferred from he signaure, because any disconinuous change in he measuremens should manifes iself a he poin of failure. Therefore he abiliy o reliably deec disconinuous changes in he measuremen daa enhances faul isolaion.

4 f3 Qualiaive ransien analysis based on he progressive monioring scheme esablished in Lemma becomes he basis for racking sysem behavior and eliminaing inconsisen faul hypoheses ill he rue faul is isolaed. As an example, he mehodology is applied o a hreeank fluid sysem in a simulaion. The ank sysem and corresponding TCG model are shown in Fig. 3. The ank capaciies, C, C, and C 3, and pipe resisances, R R 3, and R b, consiue he se of model parameers of he physical componens and hus he possible faul candidaes. The e i and f i verices shown in he TCG correspond o ank pressure and pipe flow rae variables, respecively. Circled verices in he TCG indicae he measured variables. In his example he pressure in he hird ank (e ) and he flow rae in he pipe connecing ank and ank (f 3 ) are measured. Consider he faul siuaion where he capaciy of ank decreases abruply (indicaed wih C ), as migh happen when an objec falls ino he ank. The resuling ransiens in he measuremens are shown in Fig. (a). The iniial deviaion ha riggers he isolaion scheme is an abrup increase in f 3, indicaed by f + 3. Faul deecion and hypohesis generaion based on his deviaion produces he se of faul candidaes wih prediced faul signaures for boh measured variables, as shown in Fig. (b). This figure liss he complee qualiaive faul isolaion sequence. Sep is defined as he iniial faul isolaion sep. Abrup change deecion allows he eliminaion of hree of he six iniial faul hypoheses in he firs sep. A sep, e and f 3 are repored as (; ) and (+; ), respecively. All hree faul hypoheses are sill consisen wih he observaions. A sep, e crosses he hreshold and is repored o be (+; +). Applicaion of progressive monioring based on Lemma resuls in faul candidaes C and R sill remaining consisen wih he observaions bu C + does no. Furher refinemen of he faul hypoheses is no possible wih he given measuremens. firs sep, he occurrence of an abrup faul can have one of hree disinc effecs on a measured signal: (i) an observed posiive disconinuiy in he signal, (ii) an observed negaive disconinuiy in he signal, and (iii) no disconinuiy observed in he measured signal. The example for he hree-ank sysem discussed above shows ha reliable disconinuiy deecion plays a major role in pruning he faul hypohesis. Following he abrup change, he following observaion paerns can occur: (a) (+; +), (b) (+; ), (c) ( ; ), and (d) ( ; +), implying ha here are a leas four disinc faul signaures ha can be recognized afer an abrup change. Case (iii) above implies he presence of inegraing energy sorage elemens in he direc pah from he componen parameer o he measuremen signal in he TCG. Depending on he number of such inegraing elemens in he pah, a corresponding number of derivaive erms ( s, nd, ) may be in he faul signaure. Applying progressive monioring, he iniial direcion of change for his signal will be he firs nonzero erm in he faul signaure. Given he assumpion ha he ransien is appropriaely sampled, he observed signal will necessarily exhibi a (+; +) or a ( ; ) paern. Therefore, iniially, here are only wo disinc faul signaures associaed wih a measured signal ha does no undergo a disconinuous change in response o an abrup faul. The limiaions of he sricly qualiaive analysis can now be summarized in he following lemma. e.3 Analysis of he Qualiaive Faul Isolaion Scheme The Taylor series expansion example above illusraes he wo primary assumpions of his scheme: () he signal sampling rae is fas enough o pick up all significan qualiaive changes in signal magniudes and slopes, and () abrup changes in signal magniudes can be reliably deeced. A faul signaure of order k resuls on k disinc signaures. Combining his wih he above assumpions and Lemma, one may come o he conclusion ha for a sysem wih K possible fauls, complee diagnosabiliy can be achieved wih dlog kke measuremens. However, careful analysis of he progressive monioring framework reveals ha his is no he case. As a (a) Transien daa for measuremens e (op) and f 3, wih TRANSCEND diagnosis seps indicaed. Sep acual e: f3: + R e: + C + e: + R 3 e: + + f3: + + C + 3 e: + + f3: + + R e: + b f3: + C e: + Sep acual e: f3: + R e: + C + e: + C e: + (b) Diagnosis resuls. Sep acual e: + f3: + R e: + C e: + Fig.. Faul deecion and isolaion for a faul C.

5 f C C C3 f3 f8 f e R e6 R3 e Rb Fig. 3. Three-ank fluid sysem and is TCG model represenaion. Lemma 3. (Discriminaory power). In a purely qualiaive framework, only he following characerisics of a signal can be used o discriminae among fauls: () if here is an abrup change, he direcion of abrup change plus he direcion of change immediaely following he abrup change. This implies ha here are four disinc faul signaures. () if here is no abrup change, he firs direcion of change in he signal. Therefore, his case has wo unique faul signaures. For he case where he signal does no undergo an abrup change, higher order derivaives beyond he firs non zero derivaive have no discriminaory power. Consider wo fauls wih second order signaures < ; +; + > and < ; +; >, respecively for a paricular measuremen. In boh cases, he signal shows no disconinuous change a he poin of failure, and subsequenly maches a < +; +; > signaure, where can be a + or. Subsequenly, even if he signal slope is measured o be, he < +; +; + > canno be eliminaed, because a higher order derivaive effec ha is no capured in he second order signaure could be. This problem can only be overcome by modeling more quaniaive informaion abou he signal ime consans, and hus he sysem parameers. 3. PARAMETER ESTIMATION Qualiaive mehods for diagnosis are robus in ha hey apply in uncerain environmens, and avoid he compuaional difficulies associaed wih he siffness and convergence problems of numerical schemes. However, for he reasons discussed in Sec..3 he inabiliy o incorporae ime consan informaion adversely affecs heir abiliy o discriminae fauls ha show no qualiaive differences, or differ only in higher order ransien effecs. Fig. shows ha he qualiaive faul isolaion scheme is unable o disinguish beween faul hypoheses, C and R for he hreeank sysem. To achieve higher resoluion, a quaniaive analysis approach, illusraed in Fig. 5 is inroduced ino he faul isolaion scheme. The idea is o express he ransien behavior as a funcion of he hypohesized faul parameers derived from f r, and esimae he value of hese parameers from he available measuremens. The bond graph modeling approach (Rosenberg and Karnopp 983) is he saring poin for deriving boh he sae equaion based observer model and he TCG based diagnosis model. This paradigm provides a direc mapping from physical componen parameers o he sandard sae equaion form: _x = Ax + Bu () y = Cx + Du: () The quaniaive parameer esimaion scheme is implemened by expressing he coefficiens of he marices, A, B, C, and D, in erms of he single parameer corresponding o he hypohesized faul, and using he nominal (known) values for all oher componen parameer values. Like he qualiaive faul observers, a separae parameer esimaor is iniiaed for each faul hypohesis in f r. Given he observaion vecor y() ( ), a sandard leas squares esimaion mehod is applied o derive he faul parameer values. Faul parameers for which he error erm, i.e., he difference beween he prediced ( ^y) and observed measuremens (y), do no converge o zero are eliminaed. The decision es for convergence o zero is implemened as a saisical hypohesis esing scheme. Prediced measuremen values are compued from he esimaed faul parameer values. The parameer esimaion is execued only for he coefficiens of he sae marices in which he faul parameers appear. As a resul, he form of he nonlinear funcions used for he esimaion ask are simplified, which in urn reduces he complexiy of he leas squares esimaion, and numerical convergence is easier o achieve. The sae vecor x for he hree-ank sysem is defined by he pressure variables in he hree anks, i.e., x = [e e 6 e ] T. The inpu vecor, u =[f ] and he oupu vecor y in he example are he measured variables, [e f 3 ] T. The symbolic form of he marices A, B, Fig. 5. Exending faul isolaion wih quaniaive parameer esimaion mehods.

6 Predicion Error Predicion Error f3 e 3 5 (a) C 5 5 f3 e 3 5 (b) R Fig. 6. Parameer esimaion for remaining candidaes afer qualiaive faul isolaion C and D, derived from he bond graph model as a funcion of he componen parameers are: A = R C R C ( ) R C C R 3 R 6 B = C 3 " 5 ; C = R 3 C ( + ) R 3 C 3 C 3 R 3 R b R R # ; D =» Consider faul hypohesis, C. Inspecion of he A marix shows ha his parameer appears only in wo of he marix coefficiens, a and a. This simplifies he parameer esimaion ask for his faul hypohesis, and sysem idenificaion echniques have o be applied o derive he new values for a and a. In he simulaion experimens, he normal values for all parameers are se o. Two ses of parameerized marices, A, B, C, and D, are consruced for he wo faul hypoheses, C, and R. The ransien response for measuremens e and f 3 (illusraed in Fig. (a)), are used o compue he numeric values of C and R. The derived values are hen used o predic he values for he same measuremens, and he predicion error, e = ^y y, is ploed in Fig. 6. For predicions wih parameer, C, he error e converges oward. However, for predicion wih parameer R, he error e diverges for boh measuremens, indicaing ha C is he rue faul. Currenly, he decision procedure is implemened as a saisical hypohesis esing algorihm ha checks for he convergence of e o a a predeermined confidence level.. DISCUSSION AND CONCLUSIONS This paper presens a sysemaic analysis of an approach o FDI ha combines qualiaive and quaniaive analysis for robus faul isolaion. The Taylor series expansion of ransien signals provides he basis for he consrucion of qualiaive faul signaures and he progressive monioring scheme for racking faul ; : ransiens. The limis of he discriminaory abiliy of he qualiaive scheme could hen be esablished based on a formal analysis. To improve he isolaion ask, a focused parameer esimaion mehod is developed ha works in conjuncion wih he qualiaive scheme o enable isolaion of he rue faul candidae. This mehodology allows a he sae equaions of he sysem o be parameerized in erms of he hypohesized faul parameers, and in he process creaes a simpler formulaion for quaniaive analysis. This miigaes a number of compuaional problems ha arise wih radiional numeric schemes. Simulaion experimens conduced on a hree-ank fluid sysem demonsrae he effeciveness of he mehodology. In fuure work, he quaniaive analysis will be applied o more complex sysems, such as he auomobile engine es bed used in previous experimens (Manders e al. ). The parameer esimaion problem for nonlinear dynamic sysems mus also be addressed, and he challenge here will be o derive simplified parameerized inpu-oupu represenaions (cf. (Zhang e al. 998)) from he sae equaions for he parameer esimaion ask. 5. REFERENCES Brammer, K. and G. Siffling (989). Kalman-Bucy Filers. Arec House, Norwood MA. Kreyszig, E. (97). Advanced Engineering Mahemaics, Third Ed. John Wiley, New York. Manders, E.-J., G. Biswas, P.J. Moserman, L.A. Barford and R.J. Barne (). Signal inerpreaion for monioring and diagnosis, a cooling sysem esbed. IEEE Trans. on Insrumenaion and Measuremen. To Appear. Manders, E.-J., P.J. Moserman and G. Biswas (999). Signal o symbol ransformaion echniques for robus diagnosis in TRANSCEND. In: Tenh Inernaional Workshop on Principles of Diagnosis. Loch Awe, Scoland. pp Moserman, P. J. and G. Biswas (998). A heory of disconinuiies in physical sysem models. Journal of he Franklin Insiue: Engineering and Applied Mahemaics 335B(3), 39. Moserman, P.J. and G. Biswas (999). Diagnosis of coninuous valued sysems in ransien operaing regions. IEEE Trans. on Sysems, Man and Cyberneics 9(6), Paon, R.J. and J. Chen (997). Observer-based faul deecion and isolaion: Robusness and applicaions.. Conrol Engineering Pracice 5(5), Rosenberg, R. C. and D. Karnopp (983). Inroducion o Physical Sysem Dynamics. McGraw-Hill Publishing Company. New York, New York. Zhang, Q., M. Basseville and A. Benvenise (998). Faul deecion and isolaion in nonlinear dynamic sysems: A combined inpu-oupu and local approach. Auomaica 38(), 9.