Subspace Algorithms in Modal Parameter Estimation for Operational Modal Analysis: Perspectives and Practices

Transcription

1 Subsace Algorths n odal araeter Estaton for eratonal odal Analss: ersectves and ractces S. Chauhan Bruel & Kjær Sound and Vbraton easureent A/S Sodsborgvej 37, DK 85, Naeru, Denar Eal: schauhan@bsv.co Noenclature A C w and v Y H C Σ State vector Resonse vector State ranston atr utut atr rocess and easureent nose vectors Hanel data atr Hanel atr of covarance atrces Covarance atr Etended bservablt atr Etended Controllablt atr State Covarance atr atr olnoal coeffcents f rojecton of future outut resonses on ast outut resonses Xˆ Kalan flter state estate Abbrevatons A SS eratonal odal Analss Stochastc Subsace dentfcaton

2 SS-Cov SS-Data D ERA SVD Covarance-drven SS Data-drven SS brah e Doan Egensste Realzaton Algorth Sngular Value Decooston ABSRAC Subsace based algorths for estatng odal araeter have now becoe coon wthn odal analss doan. hs s esecall true for eratonal odal Analss, where Stochastc Subsace dentfcaton (SS) algorth s a well-nown and coonl used algorth. Deste ther ncreasng use and oulart, one often encounters basc questons such as (and not lted to). How are these algorths related to (or dfferent fro) tradtonal atr olnoal coeffcent based algorths le olreference e Doan (D) etc.?. What s the ln between covarance and data drven aroaches to SS? 3. What s the need for havng dfferent varants of SS (Covarance-drven and Data-drven)? n fact, even before addressng the questons lsted above, there s a fundaental need to loo at these algorths fro the ersectve of odal araeter estaton, whose requreents and deands dffer fro those of sste dentfcaton wthn Control Sstes Engneerng, where these algorths orgnated. hs aer as at addressng these ssues and eane subsace algorths fro a urel odal araeter estaton ersectve. he author eects that ths aer wll rovde readers wth a sle and clear understandng of these algorths towards ther utlzaton for odal araeter estaton. Kewords: Stochastc Subsace dentfcaton, Data-drven, Covarance-drven, state sace odellng, odal araeter estaton. ntroducton here are several alcatons, ncludng odal araeter estaton where araetrc odels are sought. hese nclude sgnal rocessng, defect detecton, desgn of control sstes and an others. Subsace algorths belong to the class of sste dentfcaton algorths that utlze state-sace odels of te seres for estatng araetrc odels. he use of these algorths for araeter estaton n odal analss s not new. Several oular algorths, such as brah e Doan (D) [, ] and Egensste Realzaton Algorth (ERA) [3, 4], use state sace forulatons for estatng odal araeters. he a of ths aer s to understand state sace odellng n contet of odal araeter estaton, wth a focus on one of the ost coonl used eratonal odal Analss (A) algorth, Stochastc Subsace dentfcaton (SS) [5-7], along wth ts two varants: Covarance-drven SS (SS-Cov) and Data-drven SS (SS-Data). n order to understand state-sace odels n the contet of odal analss, t s vtal to state the objectve of odal analss, whch s to estate odal araeters,.e. natural frequenc, dang and ode shae, of a structure. hus, unle sste dentfcaton roble n control sste desgn or forecastng roble n econoetrcs, the goal of a araeter estaton algorth s not estaton of atrces A, C (defned n secton ) or assocated quanttes such as state vectors, observablt or controllablt atr, etc. but odal araeters. he aer starts wth a general dscusson on SS algorth and descrbes sequentall ts two varants, Covarance-drven SS (SS-Cov) and Data-drven SS (SS-Data). n secton., two searate aroaches are rovded for the develoent of SS- Cov. Frst, tradtonal forulaton of SS-Cov s rovded along wth a odfed forulaton. hen an alternate forulaton s rovded that taes nsraton fro, and elores, the relatonsh between the olnoal odel and state-sace odel reresentaton of a dnac sste. Addtonall, ths secton elans how estaton of etended observablt atr n the conventonal forulaton s not a requreent fro odal analss ont of vew and desred results can be acheved n coaratvel sle stes. he secton concludes b notng how varous forulatons of SS-Cov sl dffer n how the covarance atrces are staced.

3 Sae aroach s taen whle dscussng SS-Data (Secton.). Gven the fraewor of odal araeter estaton, the a s to connect the two varants of SS. n ths contet, t s elaned how the need to estate state vectors necesstates the avalablt of raw outut data for SS-Data algorth and t s ehaszed that these requreents do not for a art of odal araeter estaton. he forulaton of SS-Data s then elaned n lght of ths nowledge and t s shown how SS-Data s not uch dfferent fro SS-Cov n the absence of the need to estate state vectors.. Stochastc Subsace dentfcaton algorth Stochastc Subsace dentfcaton (SS) s a well-nown oeratonal odal analss (A) algorth [5-7]. t s based on state-sace reresentaton of a dscrete lnear te nvarant () sste descrbed as A C where s the vector of easured resonses, s the vector of state varables atr, A s the state transton atr, C s the outut atr and w and v are rocess and easureent nose vectors. As s clear fro Eqn., SS algorth oerates on easured outut resonse data onl (nut ectaton s not easured n A). Fro the ont of vew of odal araeter estaton, t s the estaton of state transton atr A that s ost ortant as egenvalue decooston of ths atr reveals odal araeters. here are two varants of ths algorth that are oular n ractce: data-drven (SS-Data) and covarance-drven (SS-Cov) [6]. hese varants dffer n ters of the data on whch the oerate. SS-Data oerates drectl on easured outut resonse data wthout rocessng t. n the other hand, SS-Cov requres that covarance functons are frst estated fro raw outut te hstores and t s these covarance functons that SS-Cov utlzes for the urose of odal araeter estaton. he dscusson n ths secton centers on understandng the fner asects of these two varants. n ths contet, tradtonal develoent of these algorths s studed to understand the reasons that led to the develoent of these two varants. call, the outut resonse data s assebled n a Hanel data atr, whch s further dvded nto a artton of ast ( Y ) and future resonses ( Y f ) (see [6, 7] for ore detals). hs arrangeent ads n theoretcal forulaton of SS and s gven as he two varants of SS are now dscussed n secton. and.. w v () Y Y () Yf. Covarance drven Stochastc Subsace dentfcaton algorth SS-Cov algorth, as the nae suggests, oerates on covarance functons. call, the raw resonse data s rocessed nto covarance functons, whch are then utlzed n SS-Cov algorth. hs rocessng of raw resonse data to covarance (or correlaton) functons can be done n several was, ncludng obtanng covarance functons drectl fro resonse data wthout nvolvng an nteredar sgnal rocessng stes. hs s n contrast to, for e.g. Welch erdogra ethod [8], where data s rocessed to estate ower sectra, whch s then nverse Fourer transfored to obtan covarance functons, and nvolves sgnal-rocessng technques le averagng, wndowng etc. n lterature, SS-Cov s generall develoed usng the Hanel (or oeltz) atr of covarance functons. he algorth then utlzes the fact that the Hanel atr equals the covarance between future resonses and ast resonses,.e. H Y f Y. n case of SS-Cov, t s tcall assued that covarance functons are calculated a ror and access to raw outut resonses s not avalable. hs secton s arranged n the followng anner. Frst, the tradtonal forulaton of SS-Cov, based on estaton of etended observablt and controllablt atrces, s resented. t s then shown that estaton of etended observablt and controllablt atrces s redundant fro the ersectve of odal araeter estaton and how state transton atr A can be drectl obtaned fro Hanel atr of covarance functons. Fnall, a new forulaton of SS-Cov s suggested n secton

4 .. underlnng the fact that SS-Cov can be forulated n ultle was; each dfferng n the anner the covarance functons are arranged... SS-Cov based on tradtonal forulaton radtonal forulaton of SS-Cov begns wth forulaton of a Hanel (or oeltz) atr of covarance atrces, whch equals H Y f Y ( Y f, Y are defned n Eqn. ). H Y Y f 3 N o N o (3) As elaned revousl, for odal araeter estaton uroses, one s nterested n estaton of state transton atr A. call, estaton of state transton atr A s done on the bass of etended bservablt atr, whch s obtaned usng sngular value decooston (SVD) of H. H U S V C (4) Note that C n the Eqn. 4 reresents etended Controllablt atr. he etended observablt atr s obtaned n followng anner, and state transton atr A s estated as ½ U S (5) - A r (6) where r s the bloc shfted verson of etended observablt atr (obtaned after reovng the frst bloc of ) and - reresent seudo-nverse of frst (-) rows of. he reason for tang ths aroach n conventonal sste dentfcaton fraewor s that dentfcaton of observablt and controllablt atrces s often art of the overall dentfcaton roble. hs however, s not the case wth odal araeter estaton, where the a s to obtan odal araeters. Shown below s another aroach of estatng state transton atr A, usng the Hanel atr eressed n Eqn. 3. hs forulaton s based on the wor resented n [9], where t s shown that state transton atr A, can be drectl obtaned fro Hanel atr H n followng anner. where H s the bloc shfted verson of Hanel atr and H H - A (7) H - reresent seudo-nverse of frst (-) rows of H. t should be noted that state transton atr estated usng Eqn. 7 s related to that obtaned usng the tradtonal aroach (descrbed b Eqn. 4-6) b eans of a slart transforaton (see [9] for detals) and hence ossesses the sae Egen structure. n other words, odal araeters estated on the bass of ether state transton atr wll stll be the sae... Alternate forulaton B defnng state vectors, n Eqn., n followng anner, t s eas to understand the relatonsh between state-sace reresentaton and the olnoal odel reresentaton of a dnac sste. he state transton atr n ths case becoes a coanon atr.

5 , A (8) Defnng covarance atr between outut resonses as ] [ E and state covarance atr as ] [ E Σ, t s eas to show, usng Eqn., that ΣC CA (9) Eandng state equaton n Eqn. and usng Eqn. 8, () Snce the state transton atr A s constant (the sste s te nvarant), a nuber of equatons slar to Eqn. can be fored as shown below. 3 ) ( () For the sae of slct, Eqn. can be wrtten n a coact for as A F () whch can be solved for A n a least squares anner as, [ ] F A (3) he two roducts n the above equaton, F, are two oeltz atrces corsng covarance functons ) ( ) ( ) (, F t s recognzable that the two atrces F, have slar structure as, H H, descrbed n Eqn. 7. hs elans the connecton between the tradtonal aroach (based on Y f Y H ) and the aroach suggested n ths aer.

6 Based on above observatons, t can be argued that there s no fundaental dfference between varous forulatons of SS- Cov. he bggest dfference s erhas how the covarance atrces are fored and staced, and as shown n ths aer, ths can be done n several was wthout havng an act on the fnal outcoe n ters of estaton of odal araeters.. Data drven Stochastc Subsace dentfcaton algorth Forulaton of SS-Cov algorth assues that covarance functons are avalable and raw outut data does not la an role rresectve of whether t s avalable or not. Varous forulatons shown n revous secton assue that raw data s avalable, but t s easl notceable that ths s not a requreent and, startng wth Eqn. 3 (or Eqn. 3) the forulatons can be arrved at wthout uch dffcult b sl usng the covarance functons n case the are avalable a ror. SS-Data, on the contrar, aes t andator to have the raw data avalable. hs s the oft-quoted dfference between the two oular varants of SS. However, t can be argued that ths requreent s drven ore b the need to estate state vectors wthn controls engneerng doan than b requreents assocated wth odal araeter estaton. Estaton of state vectors s tcall done b eans of Kalan flter [5-7], whch rovdes otal redcton of state vector. t s ths requreent that necesstates the avalablt of raw outut te data. SS-Data s based on the concet of rojecton [5, ], where future oututs are rojected on the ast oututs. hs rojecton s defned as he Kalan flter state estates are gven as X ˆ f [ Yf Y ][ YY ] Y (4) [ YY ] Y C (5) Usng Eqns. 3-4, t s eas to follow that rojecton defned n Eqn. 4 can be wrtten as f Xˆ hs aves the wa for decoosng rojecton C [ Y Y ] can be estated. hs s done b eans of sngular value decooston of observablt atr reans sae. U S Xˆ S Y f such that etended observablt atr and Kalan flter states ½ ½ V (6) f. t s notceable that estaton of etended he state transton atr can now be easl obtaned usng the eresson rovded n Eqn. 6. However, t s coon to rovde an alternate eresson based on the estated Kalan flter state and ts shfted verson X ˆ. Usng Eqn. t s eas to obtan A, n ters of estated state vectors, as Xˆ A X Xˆ (7) ˆ (8) t s clear fro above dscusson that the reason for usng SS-Data s estaton of state vectors. However, as has been entoned, ths s not a requreent fro odal araeter estaton ersectve. ore ortantl, even after estaton of Kalan flter states, one can stll utlze the etended observablt atr for estatng state transton atr [7]. hus, f estatng the state vectors s not the goal, t s eas to follow that SS-Data can be forulated sl on the bass of etended observablt atr. n that case, there s no dfference between ths aroach and the one eressed n revous secton that descrbed SS-Cov. Note that basc forulaton of SS-Cov s based on covarance functons and one cannot estate state vectors onl on the bass of covarance functons. he ajor dstncton between the two varants of SS s n ters of leentaton; unle SS-Cov, SS-Data can be leented such that calculaton of covarance atrces s not requred [6]. therwse, the onl dstncton between the two

7 aroaches s that n case of SS-Cov, covarance functons are suosed to be re-calculated, whch s not the case wth SS- Data. SS-Cov starts drectl fro the foraton of Hanel atr H (Eqn. 3) where as n SS-Data the raw outut data s arranged n ters of ast and future resonses (Eqn. ) so that the covarance functons are calculated as a art of the algorth. 3. Conclusons hs aer revews the Stochastc Subsace dentfcaton (SS) algorth and ts two varants (SS-Cov and SS-Data) wthn the fraewor of odal araeter estaton. ne underlnng asect of SS onted out n ths aer s the fact that the goals of odal araeter estaton stage of oeratonal odal analss,.e. estaton of natural frequenc, dang and unscaled ode shae, can be acheved through several forulatons of SS. hs fact s hghlghted b eans of an alternate forulaton of SS-Cov resented n ths aer. hs s done b elorng the relatonsh between state-sace odel and hgh order olnoal odel reresentaton of a dnac sste. he aer further ehaszes that, when vewed wthn the fraewor of odal araeter estaton, there s not uch dfference between SS-Cov and SS-Data. hs s due to the fact that the a of these varants dffer fro when the are aled n Controls Engneerng doan (where these algorths were orgnall develoed) to ther alcaton n odal analss doan. t s understandable fro the forulaton of SS-Data that one of ts rar goal s state estaton, a goal that s not shared b odal araeter estaton. hat state transton atr can be obtaned usng the estated states s an aular ste, as estaton of state transton atr does not rel solel on state vector. State transton atr can be obtaned sl on the bass of etended observablt atr, n whch case, there s not uch dstncton between SS-Data and SS- Cov. References. brah, S.R., ulc, E.C., A ethod for drect dentfcaton of vbraton araeters fro the free resonse, Shoc and Vbraton Bulletn, Vol. 47, art 4, , aa, R.S., Soe statstcal erforance characterstcs of the D odal dentfcaton algorth, AAA aer Nuber 8-768,. 9, Juang, J.N., aa, R.S., An Egensste Realzaton Algorth for odal araeter dentfcaton and odel reducton, AAA Journal of Gudance, Control and Dnacs 8 (4) (985) ongan, R.W., Juang, J.N., Recursve for of the egensste realzaton algorth for sste dentfcaton, AAA Journal of Gudance, Control and Dnacs (5) (989) Van verschee,., De oor, B., Subsace dentfcaton for near Sstes: heor-leentatons-alcatons. Kluwer Acadec ublshers, Dordrecht, Netherlands, eeters B and De Roec G., Stochastc Sste dentfcaton for eratonal odal Analss: A Revew, ASE Journal of Dnac Sstes, easureent, and Control, 3, ,. 7. Brncer R., Andersen., Understandng Stochastc Subsace dentfcaton, n roceedngs of 4th nternatonal odal Analss Conference (AC), St. ous (), USA, Ka, S.., odern Sectral Estaton. Englewood Clffs, NJ: rentce Hall, Chauhan, S., Usng the Unfed atr olnoal Aroach (UA) for the develoent of the Stochastc Subsace dentfcaton (SS) algorth, Journal of Vbraton and Control, 9(3), 95-96, 3.. Strang, G., ntroducton to near Algebra, Wellesle Cabrdge ress, Wellesle A, USA, 9.