An algorithm for detection and isolation of faulty vibration modes based on a spectral decomposition of the plant time response

Transcription

1 VI Congreso Iberoaericano e Acústica - FIA 8 Buenos Aires, 5, 6 y 7 e noviebre e 8 FIA8-A9 An algorith for etection an isolation of faulty vibration oes base on a spectral ecoposition of the plant tie response José Maria Galvez Departent of Mechanical Engineering, Feeral University of Minas Gerais, Brazil Av Antonio Carlos 667, Papulha, 7-9 Belo Horizonte, MG, Brazil E-ail: jgalvez@ufgbr Abstract his wor presents an algorith for etection an isolation of faulty vibration oes base on the spectral ecoposition of the plant tie response A Hanel atri is built fro plant output easureents an its singular values are use to etect an ientify plant paraeters relate to faulty vibration oes It is shown that the etection (alar generation) an isolation (alar interpretation) tass are easily perfore base on the propose algorith Eaples are finally presente to illustrate the perforance an application of the propose algorith Resuen Este trabajo presenta un algorito para la etección y aislaiento e oos e vibración efectuosos, el algorito es basao en la escoposición espectral e la respuesta e tiepo e la planta Una atriz e Hanel se construye a partir e eiciones e la salia e la planta Los valores singulares e esta atriz se usan para etectar e ientificar los paráetros e la planta relacionaos con los oos e vibración efectuosos Se uestra coo la etección (generación e la alara) y el aislaiento (interpretación e la alara) e la falla son tareas fácilente realizaas con el algorito propuesto Finalente, se presentan ejeplos ue ilustran el esepeño y aplicación el algorito propuesto

2 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 Introuction he evelopent of safer an ore reliable control systes has been an increasingly nee in the last ecaes o full-fill the oern stanars, the control systes esign ust inclue fault etection an isolation issues at their very early esign stage, Delaire, Cassar an Staroswieci (995), Iseran (984) he ultiate goal of these systes is to reach a fault-tolerant control (FC) environent Fault etection an isolation (FDI) schees are ipleente as real-tie algoriths whose inputs are plant output observations hey are use for a) fault etection: to ecie whether the plant is in a noral operating conition or in a faulty one an b) fault isolation: to point out an ientify the in of the fault (if present) aong a given fault set Following the FDI iagnosis, on-line proceures are usually neee for FC purpose, while off-line proceures coul be use for aintenance purpose During the last ecaes, the international scientific counity has presente several fine wors, Iseran (984), Yu an Shiels (995) wo ain streas can be ientifie, control relate techniues an artificial intelligence base ethos Syste theory, signal processing or artificial intelligence approaches have been etensively use accoring to the available ata forat Most of the oel-base an non-oel-base techniues have been evelope base on the coparison of the ata prouce by the real-tie plant operation with soe previously obtaine nowlege of the syste his paper presents a novel FDI algorith base on the singular value ecoposition of a Hanel atri built fro plant output easureents he ain feature of the propose algorith is that it oes not rely on plant oels All it is reuire is a plant signature that can be eperientally obtaine he paper is organize as follows: Section inclues soe coents on the FDI proble; Section presents the basic forulation of the Eigensyste Realization Algorith (ERA); Section 4 introuces the singular values base fault etection an isolation (SVFDI) algorith; Section 5 eplores the SVFDI algorith features through eperiental results; an finally, Section 6 presents final coents an conclusions Soe coents on the FDI proble FDI algoriths use the plant input-output easureents to ipleent a two-steps proceure: the fault etection an the isolation tass he first step is the fault etection step or alar generation he proble of the alar generation is to ecie whether the syste is in a noral operating conition or not he set of output easureents along with a previously obtaine nowlege of the syste constitute the algorith inputs while a set of generate alars are the algorith outputs he secon step consists on the alars interpretation he ain issue in this case is to correctly ecie which faults are present (fault isolation) chosen fro a pre-efine fault set It is also of one s interest to establish their characteristics such as occurrence tie, fault size, class, conseuences, etc he input is the set of alars an the output is the faults isolation, characterization an iagnosis In the case of FC, further analysis is usually reuire to eterine whether the syste is still capable to perfor properly after the failure he algorith perforance is an iportant issue that ust always be consiere he ecisions taen at every step of the FDI proble solution ight inclue an accuulate evaluation errors he easure variables ay inclue noise an loa perturbations that ight obscure syste failures Also the nowlege one has about the syste noral operation ight inclue uncertainties Detection errors an false alars can be confire by their probability of occurrence Incoplete isolation an false isolation errors can be evaluate by coparison base on the faulty events probability of occurrence

3 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 he basics of the Era Algorith his section presents the basic forulation of the Eigensyste Realization Algorith (ERA), as originally propose by Juang an Pappa in 985 Since then, the scientific counity has propose several oifications an iproveents he ERA algorith is a very reliable coputational proceure originally propose for oeling of ynaic systes For the sae of siplicity an without lost in the arguent, this wor focuses on the original algorith In the following, all posterior algorith iproveents, less iportant erivation steps an several results have been oitte Consier a state space realization for a linear tie-invariant iscrete-tie ynaic syste given by ( ) y( ) A( θ ) ( ) C( θ ) ( ) B( θ ) u( ) v( ) () where, [A, B, C] efines a iscrete-tie state space realization, is a n-iensional state vector, u an -iensional control input, y a p-iensional easureent vector an v represents easureent noise he syste ipulse response seuence is given by h( ) y( ) C A B () A Hanel atri can be constructe fro the ipulse response seuence as also with h( ) h( ) h( ) CA B CA B CA B h( ) h( ) h( 4) CA B CA B CA B H( ) h( ) h( 4) h( 5) 4 CA B CA B CA B CB CA B CA B CA B CA B CA B H ( ) 4 CA B CA B CA B [ C B () ] (4) C CA CA n B [ B AB A B A B] (5) n CA C where C an B are the observability an controlability atrices, respectively Also, it i H ( ) np Χ an H () shoul be note that, usually, H() is not suare an that ( ) n ran ( ) n Fro the singular value ecoposition (SVD)

4 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 then D In H( ) M Σ N M N M D[ In ] N (6) ( ) P DQ C B (7) H an H N Σ M Q D P (8) It is nown that, there eist atrices E p an E such that ( ) Ep H( ) E ; H( ) C A B (9) h an that then h( ) H( ) C B P DQ () Ep H( ) E Ep C A B E [ ][ ][ ][ ] [ ][ p E PD D P H( ) QD DQ ][ E] [ E ][ ] [ p / / PD D / P H() QD / D Q E] () finally, a inial orer realization can be foun as A [ / / ] [ / D P H( ) QD ; B D Q E] C [ Ep PD / ] () Besies that, Juang an Pappa have also propose two uantitative criteria to eliinate oal freuencies create by easureent noise, Juang an Pappa (986) 4 he singular values base fault etection an isolation algorith he propose SVFDI algorith can be seen as a generalization of the ERA algorith (originally applie for oel ientification) It will be shown later that in the case of the SVFDI proble there is no nee for a plant oel, all one nees is the singular values of the Hanel atri built fro the plant tie response, as shown in the previous section In the following an for the purposes of the SVFDI algorith we shall call an observable plant paraeter if any rift fro its noinal value can be etecte fro output easureents Also, it is assue that in a close neighborhoo of its noinal value a paraeter rift will frienly affect the singular values of the Hanel atri Finally, it is also clear that observability an controllability properties of the plant (as their stanar efinitions) also play iportant roles in the perforance of the SVFDI algorith In this contet, the set of singular values can be consiere a natural choice for etecting paraetric rifts an failures he singular values set can be interprete as an iage of the plant paraeters Assuing this fact, it can be establishe a relationship between the singular values an the plant paraeters using stanar correlation analysis an use these singular values as flags to inicate any paraeter rift fro its noinal value he choice of singular 4

5 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 values as a easure to etect paraetric rift is ue to the fact that its nature (real positive nubers) oes not change as natural freuencies an eigenvalues o when plant paraeters change Uner noral operational conitions any change of plant paraeters values woul affect the syste ynaics an in a final analysis the singular values of the Hanel atri In a statistic fraewor, correlation oes not iply causality eaning that correlation cannot be valily use to infer a causal relationship between variables Conseuently, correlation between variables is a necessary but not a sufficient conition to establish a causal relationship However, having establishe causality, an in a close-enough neighborhoo of the noinal plant, correlation can be taen as the natural choice for analysis he correlation analysis will eliver a apping of the plant paraeters rifts into the singular values set of the Hanel atri built fro the plant tie response he propose proceure for fault etection an isolation is epicte in the following section through eaples o illustrate the features of the propose techniue, two lupe paraeter oels have been chosen as shown net 5 Eperiental results Eaple I - Let us consier the spring-ass-ashpot syste shown in Figure u u u V R { } { } { } is the inetic energy, V is the potential energy an R is the Raleigh issipation function Paraeter Value Figure he Spring-Mass-Dashpot Syste for Eaple I hen, the ifferential euations of otions are given by in atri for, with u u u u, one has u u u 5

6 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 ; ; with F K D M F u K D M then, a state space representations can be written as [ ] [ ] [ ] [ ] ; with ; ; I C F M B D M K M I A C y u B A then with y u Eaple II Let us consier the syste presente in Figure u u u { } { } { } ) ( ) ( ) ( ) ( R V is the inetic energy, V is the potential energy an R is the Raleigh issipation function Paraeter Value Figure he Spring-Mass-Dashpot Syste for Eaple II 6

7 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 In this case, the ifferential euations of otions are given by ( ) ( ( ) in atri for, with u u u u, one has ) u ( ) u u M D K F u with D K an with M, F, C an efine as in the previous eaple, then 6 y [ C ] with 8 4 u It shoul be notice that in both eaples the plant observability atri is ill conitione as shown in able Despite of that the propose techniue still elivers goo results able presents the systes eigenvalues an the noinal singular values of the plants able Conitioning Nubers for Eaples I an II Uncouple Syste Couple Syste Observability Matri Conitioning Nuber Observability Matri Conitioning Nuber γ 49 γ able Syste Eigenvalues an Singular Values for Eaples I an II Uncouple Syste Eigenvalues Couple Syste Eigenvalues Uncouple Syste Singular Values Couple Syste Singular Values - j j j 9999 j 4495 j 4495 j 65 j j 49 j 974 j 974 j 875 j

8 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 Figures through 5 present several ynaic results of the plants use in the Eaples I an II hey are place sie by sie for coparison purposes he Hanel atri in both cases was built fro the ipulse responses results shown in Figures a an b, respectively 5 Uncouple Syste - Ipulse Responses 5 Couple Syste - Ipulse Responses o: Out() o: Out() Aplitue o: Out() -5 Aplitue o: Out() o: Out() ie (sec) Figure a Ipulse Responses for Eaple I o: Out() ie (sec) Figure b Ipulse Responses for Eaple II Figures 4a an 4b present the freuency responses for Eaples I an II, respectively Uncouple Syste - Freuency Responses 5 Couple Syste - Freuency Responses o: Out() o: Out() Magnitue (B) o: Out() -5 Magnitue (B) o: Out() o: Out() -5 o: Out() Freuency (ra/sec) Figure 4a Freuency Responses for Eaple I - - Freuency (ra/sec) Figure 4b Freuency Responses for Eaple II 8

9 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 Figures 5a an 5b present the power spectru ensities for Eaples I an II, respectively Uncouple Syste - Power Spectru Densities (B) Couple Syste - Power Spectru Densities (B) Output Y() - Output Y() Output Y() - Output Y() Output Y() - Output Y() Freuency (ra/sec) Figure 5a Spectru Densities for Eaple I Freuency (ra/sec) Figure 5b Spectru Densities for Eaple II Using stanar regression analysis techniues, the correlation coefficients between plant paraeters an the singular values of the Hanel atri were calculate an noralize such that it has been assigne the value of to the greatest coefficient an to the sallest he results are epicte on able A one woul ean strong correlation an a zeros a wea or ineistent correlation able can be use to select the best singular values to be use as flags base on their correlation with plant paraeters Fro able one can built able 4 that presents the structural sensitivity of the singular values with respect to paraeter rifts able Correlation Coefficients for Eaples I an II Correlation Coefficients for the Uncouple Syste SV SV SV SV4 SV5 SV Correlation Coefficients for the Couple Syste SV SV SV SV4 SV5 SV

10 VI Congreso Iberoaericano e Acústica - FIA 8 FIA8-A9 able 4 Structural Sensitivity Coefficients for Eaples I an II Structural Sensitivity Coefficients for the Uncouple Syste SV SV SV SV4 SV5 SV6 Structural Sensitivity Coefficients for the Couple Syste SV SV SV SV4 SV5 SV6 6 Final coents an conclusions his paper presente a fault etection an isolation algorith base on plant output easureents In a close neighborhoo of the noinal plant values regression analysis has shown to be the proper choice to lin the Hanel atri singular values with plant paraeters An iportant feature of the SVFDI algorith is that its forulation oes not reuire a plant oel Having obtaine a noinal plant iage through the singular values of the Hanel atri; this iage can be use to eterine, by coparison, any value rift of the plant paraeters wo functional levels of SVFDI proceure can be istinguishe, naely alar generation an alar interpretation At the alar generation level (etection) the SVFDI algorith naturally isplays plant failure through the change of the singular values structure an values At the alar interpretation level (isolation) the SVFDI algorith elivers an iage of the plant paraeters through the singular values allowing the ientification of the faulty paraeter Finally, the propose SVFDI algorith was applie to ill conitioning plants showing outstaning perforance in solving both, etection an isolation probles 7 References Delaire, G; Cassar, JPh; Staroswieci, M (995) Coparison of Generalise Least Suare Ientification an Parity Space echniues for FDI Purpose in SISO Systes Proceeings of the r European Control Conference (ECC 95), Roe Iseran, R (984) Process Fault Detection Base on Moeling an Estiation Methos, A Survey Autoatica, vol, pp Juang, JN; Pappa, RS (985) Eigensyste Realization Algorith, Journal of Guiance an Control, vol 8, n, pp 6-67 Juang, JN; Pappa, RS (986) Effects of Noise on Moal Paraeters Ientifie by the Eigensyste Realization Algorith Journal of Guiance an Control, vol 8, pp 94- Patton, RJ; Fran, PM; Clarc, RN (989) Fault Diagnosis in Dynaical Systes, heory an Application Prentice-Hall, Englewoo Cliffs, NJ, USA Yu, DN; Shiels, DN (995) Fault Diagnosis in Bilinear Systes - A Survey Proceeings of the r European Control Conference (ECC 95), Roe