CALIFORNIA INSTITUTE OF TECHNOLOGY

CALIFORNIA INSTITUTE OF TECHNOLOGY EARTHQUAKE ENGINEERING RESEARCH LABORATORY REAL-TIME BAYESIAN STATE ESTIMATION OF UNCERTAIN DYNAMICAL SYSTEMS BY JIANYE CHING, JAMES L. BECK, KEITH A. PORTER AND RUSTEM SHAIKHUTDINOV REPORT NO. EERL 4- A Repor on Research Suppored by CUREE-Kajma Jon Research Program, Phase V, and he Calforna Insue of Technology PASADENA, CALIFORNIA FEBRUARY 4

ACKNOWLEDGEMENTS The auhors wsh o acnowledge he fnancal suppor from he George W. Housner Posdocoral Fellowshp and he CUREE-Kajma Jon Research Program, Phase V.

Absrac The focus of hs repor s real-me Bayesan sae esmaon usng nonlnear models. A recenly developed mehod, he parcle fler, s suded ha s based on Mone Carlo smulaon. Unle he well-nown exended Kalman fler, s applcable o hghly nonlnear sysems wh non-gaussan unceranes. Recenly developed echnques ha mprove he convergence of he parcle fler smulaons are also nroduced and dscussed. Comparsons beween he parcle fler and he exended Kalman fler are made usng several numercal examples of nonlnear sysems. The resuls ndcae ha he parcle fler provdes conssen sae and parameer esmaes for hghly nonlnear sysems, whle he exended Kalman fler does no. The parcle fler s appled o a real-daa case sudy: a 7-sory hoel whose srucural sysem consss of non-ducle renforced-concree momen frames, one of whch was severely damaged durng he 994 Norhrdge earhquae. Two denfcaon models are proposed: a mevaryng lnear model and a smplfed me-varyng nonlnear degradaon model. The laer s derved from a nonlnear fne-elemen model of he buldng prevously developed a Calech. For he former model, he resulng performance s poor snce he parameers need o vary sgnfcanly wh me n order o capure he srucural degradaon of he buldng durng he earhquae. The laer model performs beer because s able o characerze hs degradaon o a ceran exen even wh s parameers fxed. Once agan, he parcle fler provdes conssen sae and parameer esmaes, n conras o he exended Kalman fler. I s concluded ha for a sae esmaon procedure o be successful, a leas wo facors are essenal: an approprae esmaon algorhm and a suable denfcaon model. Fnally, recorded moons from he 994 Norhrdge earhquae are used o llusrae how o do real-me performance evaluaon by compung esmaes of he repar coss and probably of componen damage for he hoel.

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Table of Conens Absrac.... Inroducon.... Applcaons of sae esmaon n cvl engneerng.... Developmen of Bayesan sae-esmaon algorhms....3 Scope of hs repor... 3.4 Organzaon of he repor... 4. Sae Esmaon... 5 3. Kalman Fler... 7 3. Basc Kalman fler... 7 3. Exended Kalman fler... 9 4. Parcle Flers... 3 4. Basc equaons... 3 4. Mone Carlo smulaon for sae esmaon... 4 4.. Imporance samplng echnque... 5 4.3 Reducng degradaon recursve resamplng and parallel parcle flers... 4.4 Reducng degradaon local random wal... 4.5 Advanages and dsadvanages of he PF echnque... 3 5. Numercal Examples... 5 5. A planar four-sory shear buldng wh me-varyng sysem parameers... 5 5.. Daa generaon... 5 5.. Idenfcaon model... 7 5..3 Resuls... 3 5. A nonlnear hyserec dampng sysem wh unnown sysem parameers... 36 5.. Daa generaon... 36 5.. Idenfcaon model... 37 5..3 Resuls... 4 5.3 Lorenz chaoc sysem... 46 5.3. Daa generaon... 46 5.3. Idenfcaon model... 47 5.3.3 Resuls... 47 5.4 Dscusson... 49 6. Real-daa Case Sudy... 5 6. Buldng descrpon... 5 6. Tme-varyng lnear denfcaon model... 55 6.. Resuls and dscusson... 57 6.3 Calech fne-elemen model of Van Nuys hoel... 63 6.4 Developmen of he smplfed nonlnear nonlnear degradaon model... 68 6.5 Tme-varyng nonlnear degradaon denfcaon model... 76 6.5. Prelmnary comparson of he wo denfcaon models... 78 6.5. Resuls and dscusson... 79 6.5.3 Comparson wh EKF resuls... 83 6.6 Dscusson... 87 6.7 Real-me performance evaluaon... 89 7. Concluson... 93 v

References... 95 Appendces... Appendx I. Esmang uncerany parameers n EKF... Appendx II. Choce of a good mporance samplng PDF... 7 Appendx III. Parcle flers for unceran me-varyng equvalen lnear sysems... 9 Appendx IV. Deermnaon of ner-sory sffness and dampng... 3 Appendx V. Evaluang momens of repar coss condoned on EDP... 5 V. Assembly-based-vulnerably framewor... 5 V. Momens of C R condoned on EDP... 8 v

. Inroducon. Applcaons of sae esmaon n cvl engneerng Ths sudy addresses how real-me sae esmaon can be used n he process of creang probablsc earhquae loss esmaes shorly afer he cessaon of srong moon, for he case of buldngs ha are equpped wh srong-moon nsrumens. Loss esmaon as used here refers o a hree-sep process: srucural analyss (calculaon of member forces and deformaons condoned on a srucural model and earhquae excaon); damage analyss (calculaon of requred repar effors condoned on srucural reponse); and loss analyss (calculaon of economc, operaonal, or lfe-safey losses condoned on physcal damage). Ths sudy deals wh he srucural analyss poron of hs process usng sae esmaon. In parcular, deals wh how o esmae he dsrbuon of srucural-response parameers ha have no been drecly observed, for example, floor acclereraons a levels ha lac srong-moon nsrumens. Sae esmaon s he process of usng dynamc daa from a sysem (such as a buldng subjeced o earhquae excaon) o esmae quanes ha gve a complee descrpon of he sae of he sysem accordng o some represenave model of. Sae esmaon can be used n varous cvl engneerng applcaons. Some applcaons of real-me sae esmaon nclude srucural healh monorng o deec changes of dynamcal properes of srucural sysems durng earhquaes; more generally, sudyng he nonlnear behavor of srucures subjec o sesmc loadng; and n developng and mplemenng effcen srucural-conrol echnques. Because of her wde applcably, real-me sae esmaon and denfcaon mehods have been suded n cvl engneerng for varous purposes. Bec (978) used an nvaranembeddng fler for modal denfcaon. Yun and Shnozua (98) used an exended Kalman

fler o sudy nonlnear flud-srucure neracon. Hoshya and Sao (984) used he exended Kalman fler for srucural sysem denfcaon. Ln e al. (99) developed a real-me denfcaon mehodology for beer undersandng of he degradng behavor of srucures subjec o dynamc loads. Ghanem and Shnozua (995) presened several dfferen adapve esmaon echnques (e.g. exended Kalman fler, recursve leas squares, recursve predcon error mehods) and verfed hem usng expermenal daa (Shnozua and Ghanem 995). Glaser (996) used he Kalman fler o denfy he me-varyng naural frequency and dampng of a lquefed sol o ge nsgh no he lquefacon phenomenon. Sao and Q (998) derved an adapve H fler and appled o me-varyng lnear and nonlnear srucural sysems n whch dsplacemens and veloces of he floors are measured. Smyh e al. (999) formulaed an on-lne adapve leas squares algorhm for denfyng mul-degree of freedom nonlnear hyserec sysems for he purpose of on-lne conrol and monorng.. Developmen of Bayesan sae-esmaon algorhms Among sae esmaon mehodologes, hose founded on he Bayesan framewor are powerful because of he followng facs: () hey are rgorously based on he probably, and hey herefore preserve nformaon; and () hey gve he probably densy funcon (PDF) of he sysem sae condoned on he avalable nformaon, whch may hen be used for any probably-based srucural healh monorng, sysem denfcaon, relably assessmen and conrol echnques. Wh he PDF avalable, one can esmae he sae and one can also gve a descrpon of he assocaed unceranes. For he Bayesan sae-esmaon algorhms, Kalman formulaed he well-nown Kalman fler (KF) (Kalman 96; Kalman and Bucy 96) for lnear sysems wh Gaussan unceranes. Laer, KF was modfed o gve he exended Kalman fler (EKF) (Jazwns 97) o accommodae lghly nonlnear sysems. EKF has been he domnan

Bayesan sae-esmaon algorhm for nonlnear sysems and non-gaussan unceranes for he las 3 years. Alhough EKF has been wdely used, s only relable for sysems ha are almos lnear on he me scale of he updang nervals (Juler e al. ; Wan and van der Merwe ). However, cvl engneerng sysems are ofen hghly nonlnear when subjec o severe loadng evens. In such cases, he applcably of KF and EKF s quesonable. These older echnques have been used by cvl engneerng researchers for decades (Bec 978; Yun and Shnozua 98; Hoshya and Sao 984; Koh and See 994) alhough her applcably for nonlnear sysems and non-gaussan unceranes s seldom verfed emprcally or heorecally. Several mporan breahroughs (Alspach and Sorenson 97; Gordon e al. 993; Kagawa 996; Douce e al. ; Juler e al. ) have produced Bayesan sae-esmaon algorhms ha are applcable o hghly nonlnear sysems. Sae esmaon for general nonlnear dynamcal sysems s sll an acve research area, and novel echnques (e.g., van der Merwe e al. ; van der Merwe and Wan 3) can be found n he mos recen sgnal-processng leraure. Alhough hese breahroughs have had sgnfcan mpac n he area of sgnal processng, hey are rarely seen n he cvl engneerng leraure and have no been mplemened for cvl engneerng sysems..3 Scope of hs repor In hs repor, we nroduce some recen developmens n real-me Bayesan sae esmaon ha use Mone Carlo smulaon (MCS). An MCS echnque called he parcle fler (PF) s presened and dscussed. These MCS echnques have he followng advanages: () hey are applcable o hghly nonlnear sysems wh non-gaussan unceranes; () hey are no lmed o he frs wo momens as n KF and EKF; and (3) as he sample sze approaches nfny, he re- 3

sulng sae esmaes converge o her expeced values. However, he smulaon s usually compuaonally expensve, and somemes he sae esmaes can be naccurae due o nsuffcen samples. We nroduce several recen developmens ha address he afore-menoned dffcules and presen new echnques ha are useful o mprove he convergence. The performance of dfferen mehods (.e., EKF vs. PF) wll be compared hrough several numercal examples and a real-daa case sudy..4 Organzaon of he repor Ths chaper has nroduced he problem o be addressed. In Chaper, we defne he general problem of Bayesan sae esmaon for nonlnear dynamcal sysems. In Chaper 3, we revew he Kalman fler and exended Kalman fler algorhms. In Chaper 4, we nroduce mporance-samplng fler echnques. In Chaper 5, several numercal examples are conduced o compare he conssences of he exsng and new echnques. In Chaper 6, we presen a case sudy of a real buldng o demonsrae he applcaon of he dfferen mehods. We also apply he resuls o llusrae real-me esmaon of repar coss and probably of componen damage of he buldng durng an earhquae. 4

. Sae Esmaon Consder he followng dscree-me sae-space model of a dynamcal sysem: x = f ( x, u, w ) y = h ( x, u, v ) =,... T () The wo equaons n () are called, from lef o rgh, sae ranson (or evoluon) and observaon (or oupu) equaons, respecvely. In hs equaon, x n R, u p R and y q R are he sysem sae, npu (nown excaon) and observed oupu, respecvely, a me ; w l R and v m R are nroduced o accoun for unnown dsurbances, model errors and measuremen nose; f s he prescrbed sae ranson funcon a me ; and h s he prescrbed observaon funcon a me. The values of he varables x, y, modeled as random varables, whle u s consdered o be a nown excaon. w and v are unceran and so are For each me, he dynamcal sysem npu u and oupu yˆ are measured. (To avod confuson, we denoe he acual observed oupu value by y ˆ ). We denoe he me seres of observed oupu and npu { yˆ ˆ ˆ, y,..., y } and { u, u,..., u } by Y ˆ and U, respecvely. The goal of real-me Bayesan sae esmaon s o evaluae he condonal probably densy funcon (PDF) p( x ˆ Y ) for he sae x a every me n a real-me manner,.e., o sequenally updae hs condonal PDF usng he observed sysem npu and oupu up o he curren me, and on he probablsc models for w and v. From hs condonal PDF, px ( Y ˆ ), one can esmae mporan feaures of he sae, such as he condonal expecaon Ex ( Y ˆ ) and condonal covarance marx Cov( x Y ˆ ). Noe ha he condonng of every PDF on he npu excaon U s mplc. 5

The basc equaons for updang px ( ˆ Y ) o px ( ˆ Y ) are he predcor and correcor equaons ha follow from he Theorem of Toal Probably and Bayes Theorem, respecvely: px ( Yˆ ) = px ( x ) px ( Yˆ ) dx ˆ ˆ ˆ p( yˆ ) ( ˆ x p x Y ) p( y x) p( x Y ) px ( Y) = = py ( ˆ x) px ( Yˆ ) dx py ( ˆ Yˆ ) () where Yˆ s dropped n p( x x ) and p( yˆ x ) because he models for he sae ranson and observaon PDFs mae rrelevan. The man challenge n Bayesan sae esmaon for nonlnear sysems s ha hese basc equaons canno be readly evaluaed because hey nvolve hgh-dmensonal negraons. 6

3. Kalman Fler 3. Basc Kalman fler When f and h n () are boh lnear n u, x, w and v,.e., f ( x, u, w ) = A x + B u + G w h ( x, u, v ) = C x + D u + H v (3) and w and v are zero-mean ndependen Gaussan random varables, he condonal PDF s px ( Y ˆ ) also Gaussan and can be updaed analycally. Furhermore, s suffcen o updae he frs wo momens of x because hey compleely specfy hs condonal PDF. The updang algorhm s he well-nown Kalman fler (KF). I comprses wo seps, he predcor and correcor seps. In he predcor sep, he goal s o compue px (, ˆ y Y ) from px ( ˆ Y ). Frs, px ( Yˆ ) s compued based on px ( ˆ Y ) usng he followng momen equaons: x E( x Yˆ ) = A x + B u P Cov( x Yˆ ) = A P A + G G T T (4) Noe ha he values x and P mus be gven pror o he nalzaon of he algorhm. Second, p( y ˆ Y ) s compued based on p( x ˆ Y ) and u usng he followng momen equaons: y E( y Yˆ ) = C x + D u P Cov( y Yˆ ) = C P C + H H y T T (5) and fnally, he condonal covarance beween x and y s he n q marx compued as follows: 7

P Cov( x, y Yˆ ) = P C (6) xy T Ths complees he compuaon of all he momens needed o specfy he Gaussan PDF p( x, y Yˆ ). In he correcor sep, p( x Y ˆ ) s updaed based on (): px ( Yˆ ) = py ( ˆ ) ( ˆ x px Y ) py ( ˆ Yˆ ) = cons exp y C x D u H H y C x D u T exp ( x x ) ( P ) ( x x ) T T ( ˆ ) ( ) ( ˆ ) (7) where y ˆ agan denoes he observaon of y a me and cons s a normalzng quany (he nverse of he denomnaor) no dependng on x. Because p( x Y ˆ ) s Gaussan, has one mode ha s locaed a he mean value of x, so by dfferenang p( x ˆ Y ) wh respec o x and solvng for zero, we oban he expeced value x. The covarance P s equal o he negave of he nverse of he Hessan of log p( x ˆ Y) : T ( ) ( ) ( ) ( ˆ ) x = x + I + P C H H C P C H H y y T T T T T P = P C HH C + (8) Mang use of he followng lemmas ( ) ( ) ( ) ( ) T T T I + PQ P = P I + QP A + VC V = A AV C + V AV V A (9) where P and Q are conformable marces, A and C are posve defne, we conclude wh he followng equaons for he correcor sep: 8

x = x + P ( P ) ( yˆ y ) xy y P = P P ( P ) P xy y xy T () 3. Exended Kalman fler Many dynamcal sysems exhb nonlnear behavor. If one canno model he sysem ranson usng Equaon (3), one canno use he KF. However, f f and h are only slghly nonlnear, an approxmaon for KF can be derved by lnearzng he predcor and correcor seps. The resulng fler s he well-nown exended Kalman fler (EKF). To explan he lnearzaon echnque for he uncerany propagaon sep, we consder he followng general uncerany propagaon problem: Y = f (X ) () where X n R and Y m R are unceran vecors. Usng Taylor seres expanson around he mean value X E( X) =, we have ( ) ( ) ( ) x ( ( )) f X = f E X + f X E X + HOT () where s he Jacoban marx evaluaed a X E ( X ) X f = and where HOT denoes he hgher order erms. As a resul, he frs wo momens of Y are ( ) ( ) ( ) X ( ( )) E Y = f E X + f E X E X + HOT (3) and ( ) ( ) Cov Y = ( f ) Cov X ( f ) T + HOT (4) X Under he assumpon ha ( ) and E( Y ) and Cov( Y ) are approxmaed by ( ) ( ) X f X s nearly lnear near X E( X) =, he hgher order erms vansh ( ) ( ( )) ( ) ( ) E Y = f E X + f E X E X Cov Y = ( f ) Cov X ( f ) T (5) ln X ln X X 9

where E( Y ) ln and Cov( Y ) ln denoe he lnearzed approxmaons for E( Y ) and Cov( Y ), respecvely. The approxmaons E ( Y ) ln and Cov( Y ) ln are accurae esmaes of E ( Y ) and Cov( Y ) f f ( X ) s almos lnear on he suppor regon of he PDF of X and become exac when f ( X ) s lnear n X. The approxmaons are poor f f ( X ) s hghly nonlnear on he suppor regon of he PDF of X. For he predcor sep n EKF, he goal s o fnd lnear approxmaons of x, P, y y, xy P and P based on x and P. To smplfy he noaon, we defne T T T T T T T T n l m z = [ x w v ] and z = [ x ] R + +, so f( x, u, w) = f( z, u). When propagang from [ x, P ] o [ x, P ], we expand f ( z, u ) z he neghborhood of. Wh he lnear approxmaon, we ge ( ) ln n x = E f ( z, u ) f ( z, u ) x (6) and T ( ) P ( f ) E ( z z )( z z ) D ( f ) T z z ( ) ( f ) Cov z D ( f ) P T ln z z (7) where f R n ( n+ l+ m) z s he Jacoban marx evaluaed a z = z. I can be seen ha T ( ) ( ) ( ) ( ) ln ln ln ln ln T P = A P A + G G (8) where A f R ln n n x and z = z G f R ln n l w are he Jacoban marces. z = z y xy Smlarly, h s also lnearzed o ge he approxmaons for y, P and P :

y h ( x, u,) y ln ln T ( ) ( ) ( )( ) T, ( ) y ln ln ln ln ln T y, ln + P C P C H H P P P C P xy ln ln xy ln (9) where C h R ln q n x and x= x, v= H h R ln q m v. x= x, v= For he esmaon sep, () can sll be used as an approxmaon. If f and h are ndeed lnear, EKF s dencal o KF. The degree of accuracy of EKF reles on he valdy of he lnear approxmaon. EKF s no suable o rac mul-modal or hghly non-gaussan condonal PDFs because only updaes he frs wo momens of he sysem sae. When sysem parameers are unnown, s mporan o esmae hem. Unceran parameers can be augmened no sysem saes and esmaed usng EKF. However, he EKF algorhm s no suable for esmang unnown parameers used o parameerze he ampludes of he uncerany erms w and v. We dscuss hs ssue and provde soluons n Appendx I.

4. Parcle Flers We have seen ha EKF can only propagae and esmae he frs wo momens of he condonal PDF. For sysems wh non-gaussan unceranes, s ofen desrable o propagae and esmae he condonal PDF self, bu dong so effecvely requres an nfne number of parameers o represen he funconal form of he condonal PDF. An alernave s o conduc Mone Carlo smulaon (MCS) by drawng samples from he condonal PDF so ha he condonal expecaon of any funcon of x can be conssenly esmaed. We focus on he MCS echnque n hs chaper and use he erm parcle flers (PF) o denoe he resulng algorhms (followng van der Merwe e al. and Douce and Andreu ). Smlar PF algorhms have been called Mone Carlo flers by Kagawa (996) and sequenal Mone Carlo Bayesan flers by Douce and Godsll (998) and Douce e al. (). 4. Basc equaons We frs presen some basc equaons ha are useful hroughou hs chaper. Le x agan denoe he sysem sae a me, and le X { x, x,..., x } =,.e., he hsory of he sysem sae up o an ncludng me. Le y ˆ agan denoe he observaon a me (e.g., he floor acceleraons), and le Y ˆ { y ˆ, y ˆ,... y ˆ } =,.e., he hsory of observaons up o an ncludng me. Then accordng o Bayes rule, he PDF of X s gven by 3

px ( Yˆ ) = px (, ˆ Y) py ( ˆ ) px ( ˆ ˆ, x, Y, y) = py ( ˆ ) px ( ˆ ˆ ˆ, Y ) px (, y X, Y ) = py ( ˆ ) px ( ˆ ˆ ˆ Y ) px (, y X, Y ) = py ( ˆ Yˆ ) ˆ ˆ ˆ py ( ˆ x, X, Y ) px ( X, Y ) = px ( Y ) py ( ˆ Yˆ ) ˆ py ( ˆ x) px ( x ) = px ( Y ) py ( ˆ Yˆ ) () where we have used he fac ha p( yˆ, ˆ ˆ x X, Y ) = p( y x) and ha px ( ˆ X, Y ) p( x x ) based on () and he fac ha he PDFs for v and = w are prescrbed. Evaluang he recursve equaon n (), we ge px ( Yˆ ) = px ( ) px ( ) py ( ˆm xm) px ( m xm ) py ( ˆ Yˆ ) m= m m = ˆm m m m ( ˆ py x px x py) m= ( ) ( ) () 4. Mone Carlo smulaon for sae esmaon Our neres s o develop an MCS algorhm for he condonal PDF p( X Y ˆ ) ha s real-me. In oher words, f Xˆ s a sample from p( X ˆ Y ), he sample from p( X ˆ Y ) mus have he form X ˆ { ˆ ˆ X, x} =, where x ˆ s he new sample and X s he prevous sample from p( X ˆ Y ). However, such a real-me MCS algorhm canno be drecly mplemened. Ths s because p( X ˆ Y ) s dfferen from p( X ˆ ˆ ˆ Y) = p( X Y, y). ˆ 4

4.. Imporance samplng echnque Neverheless, we can sample from an mporance-samplng PDF qx ( Y ˆ ) ha adms a real-me samplng procedure,.e., qx ( ˆ Y ) s dencal o qx ( ˆ Y). In oher words, he srucure of qx ( Y ) s such ha X s ndependen of y condoned on Y. Drawng N samples { X ˆ :,..., N} = randomly from qx ( Y ˆ ) (a hs me, we assume ha qx ( Y ˆ ) can be easly sampled), he expecaon of any funcon of X, denoed by r( X ), condoned on Y ˆ can be esmaed usng he mporance samplng echnque as follows: ( ˆ N ) ˆ ( ) ( ˆ β ) ˆ, E r X Y r X r () N N = where ˆ ˆ ( ˆ ) ( ˆ β = p X Y q X Yˆ ) s he non-normalzed mporance wegh of he -h sample. Any quany of neres can be esmaed wh dfferen r() funcons n (); for nsance, f r( X ) = X, Er [( X ) Y ˆ ] s smply he condonal expecaon [ ˆ T EX Y ]; f r( X) = X X, E[( r X ) Y ˆ ] s he condonal second momen E[ X X T Y ˆ ]. Le { X :,..., N} = denoe he sae varables correspondng o N random samples from qx ( Y ˆ ) (before drawng he acual samples). I s readly shown ha he esmaor ( ) s an unbased esmaor of Er [( X ) ˆ Y ] f he suppor regon for N N, = β r X N = r p( X Y ˆ ) s a subse of ha for qx ( Y ˆ ): E r E r( X ) E r( X ) N N, = q β = q β N = = ( ˆ ) ( ˆ ) ( ) ( ˆ p X Y q X Y r X q X Y ) dx = r( X ) ( ˆ ) ( ) ˆ p X Y dx = E r X Y (3) 5

Accordng o he Cenral Lm Theorem, r N, converges (as N approaches nfny) o a Gaussan random varable wh mean equal o E[( r X ) Y ˆ ] and wh varance ha decays as / N. Therefore, r s a conssen esmaor of Er [( X ) Y ˆ ]. N, Alhough N, r s unbased and conssen, s no a feasble esmaor because he non- normalzed mporance weghs ( ˆ ) ( ˆ β = p X Y q X Y ) depend on p( X Y ˆ ), whch canno be compued easly snce n order o evaluae p( X Y ˆ ), we have o evaluae p( Y ˆ ), as shown by (), whch s a dffcul as. Neverheless, we show ha he followng esmaor s compuable whle s asympocally unbased and conssen: r r X r N N j N N, β ( ) N, N β = β = N j= (4) where β N N (5) N j = β j= Noe ha r ˆ, N, unle r ˆ, N, can be compued convenenly from samples { X ˆ :,..., N} = : N N N j ˆ ˆ ˆ rn, = β β r( X) = β r( X) = j= = where (6) N ˆ ˆ ˆ ˆ ˆ j ˆ ˆ j ( ) ( ) ( ) ( ˆ β = px Y qx Y px Y qx Y) j= N j px ( ˆ ˆ ) px ( ) j j j = ( ˆ ˆ ) ( ˆ ˆ ˆ ˆ ˆ ˆ m m m m ) ( m m) ( m m ) ( ˆ ˆ py x px x py x px x ) ˆ j qx ˆ Y m= j= qx ( Y) m= (7) 6

Therefore, he facor p( Y ˆ ) n () has been cancelled due o he use of he normalzed mporance weghs { β : =,..., N},.e. N β =. Also, he lelhood funcons p( yˆ ˆ m x m) and = px ( ˆ xˆ ) can be readly evaluaed usng he prescrbed PDFs for v m and m m w m f he mappngs n () unquely specfy v m and w m, gven m, m and m y x x. A hs me, we assume ha ˆ qx ( Y ˆ ) can be easly evaluaed oo. ha To sech he proof for he asympoc unbasedness and conssency of r N,, frs noe ( β ) ˆ ˆ ˆ Eq = p( X Y) q( X Y) q( X Y) dx = (8) N Therefore, β n (5) converges (as N approaches nfny) o a Gaussan random varable wh mean equal o and wh a varance ha decays as / N. As a consequence, N ( β ) lm = wp.. (9) N where w.p. sands for wh probably, and E r( X ) Yˆ w. p. N N N lm rn, = lm β r( X) β lm β r( X) lm rn, N N N = N = N N = = = (3) whch shows ha r N, s asympocally unbased and conssen as N approaches nfny because r s. N, The selecon of an mporance samplng PDF ha adms a real-me procedure s dscussed n Appendx II. The concluson s ha he followng mporance samplng PDF performs beer: 7

qx ( Yˆ ) = px ( ) px ( x, yˆ ) (3) m m m m= The correspondng modfed non-normalzed mporance wegh s seen o be from and (3): py ( ˆ x) px ( x ) py ( ˆ x) px ( x ) β = = (3) m m m m β m= px ( ˆ ˆ m xm, ym) px ( x, y) Douce and Godsll (998) and Lu and Chen (998) dscuss he opmaly of hs mporance samplng PDF. Due o he srucure of he algorhm, a any me, we are only requred o sore he sampled saes and weghs n he mos recen wo me seps,.e. and, f he quany of neres s rx ( ) and so depends on he curren sae (clearly, addonal dependence on he prevous sae x can also be reaed). As a resul, he followng recursve algorhm can be used: Algorhm 4.: Basc PF algorhm () Inalze he N samples: Draw xˆ from px ( ) and se β = N, =,..., N. () A me, sore he prevous samples and weghs x = xˆ β = β (33) For =,..., N, draw xˆ from px ( x ˆ = x, y) and updae he mporance wegh p( yˆ x = xˆ ) p( x = xˆ x = x ) β β = px ( ˆ ˆ = x x = x, y) (34) (3) For =,..., N, Erx [( ) Y ˆ ] can be approxmaed based on (6) and (3): N N [( ) ˆ j E rx ] ( ˆ Y β β rx) = j= (35) where r() s a funcon ha maps from x o any quany of neres. (4) Do Seps () and (3) for =,..., T. 8

Usually, p( x x ˆ = x, y ) n Sep s dffcul o sample. Noe ha esmang he frs wo momens of p( x x ˆ = x, y ) s a problem ha can be solved usng a sngle-me-sep EKF algorhm. The leas-nformave PDF (.e. he maxmum enropy PDF; see Jaynes (957)) gven he esmaed wo momens, whch s a Gaussan PDF (denoed by p ( x x = ˆ x, y ) ; LI LI subscrp means leas-nformave ), can be used for he mporance samplng PDF. The use of p ( x x ˆ = x, y ) s dscussed n Douce and Godsll (998) and van der Merwe e al. (). LI Algorhm 4.: Deermnng p ( x x = ˆ x, y ) LI () Uncerany propagaon: compue E ( x x = x ) = f ( x = x, u, w = ) x LN { } Cov ( x x = x ) = ( G ) Cov w ( G ) P LN LN T LN (36) where G f s he Jacoban marx, and LN w x = x, w = E ( y x = x ) = h ( x, u,) y LN T ( ) ( ) ( )( ) T, ( ) Cov y x x C P C H H P LN LN LN LN T y, LN ( = ) = + Cov ( x, y x = x ) = P C P LN xy LN (37) where C h and LN x x= x, v= H h. LN v x= x, v= () Esmaon: compue E ( x x = x, yˆ ) = x + P ( P ) ( yˆ y ) xy, y, LI Cov ( x x = x, yˆ ) = P P ( P ) P xy, y, xy, T LI (38) p ( x x = x, y ) s hen he Gaussan PDF wh he wo momens n (38). ˆ LI 9

4.3 Reducng degradaon recursve resamplng and parallel parcle flers Noe ha s desrable o have he mporance weghs { β : =,,... N} be approxmaely unform so ha all samples conrbue sgnfcanly n (35), bu hey become far from unform as grows, whch s due o he recurson n (3) and he fac ha qx ( Y ˆ ) p( X Y ˆ ). Ulmaely, a few weghs become much larger han he res, so he effecve number of samples s small. Neverheless, he degradaon can be reduced, as descrbed n hs secon and he nex. Insead of leng he N samples evolve hrough me ndependenly (Algorhm 4.), we can resample he samples when he mporance weghs become hghly non-unform (Kagawa 996; Douce and Godsll 998; Lu and Chen 998; Douce and Andreu ). Afer he resamplng, he mporance weghs become unform, herefore he degradaon problem s allevaed. The resamplng sep ends o ermnae small-wegh samples and duplcae large-wegh samples and, herefore, forces he N samples o concenrae n he hgh probably regon of p( x Y ˆ ). Alhough he resamplng sep ses he weghs bac o unform, he prce o pay s ha he samples become dependen and herefore collecvely carry less nformaon abou he sae. As a resul, he resamplng procedure should only be execued when he mporance weghs become hghly non-unform. Ths can be done by monorng he coeffcen of varaon (c.o.v.) of he mporance weghs. The resamplng procedure s execued only when hs c.o.v. exceeds a ceran hreshold, ndcang ha he varably n he mporance weghs s large. Anoher way o allevae he dependency nduced by he resamplng sep s o conduc several ndependen PF algorhms and combne all of he obaned samples. Alhough he samples obaned n a sngle algorhm can be hghly dependen, he samples from dfferen algorhms are compleely ndependen. The resulng algorhm s as follows: Algorhm 4.3: Parallel PF algorhm wh resamplng

() Perform he followng ()-(6) for j =,..., L ndependenly. Snce he processes are compleely ndependen, hey can be conduced n parallel. () Inalze N samples: Draw, j x from px ( ) and se N = N., j β = for,..., (3) A me, sore he prevous samples and weghs ˆ, j, j, j, j x = x β = β (39) For =,..., N, draw x from p ( x x = x, y ) and updae he mporance wegh, j, j ˆ LI p( yˆ x = x ) p( x = x x = x ) β β, j, j, j, j, j =, j, j p ( ˆ LI x = x x = x, y ) (4), j (4) Compue he c.o.v. of { β : =,..., N}. If he c.o.v. s larger han he prescrbed hreshold, hen execue he resamplng sep for =,..., N : N, j, j, j, j xˆ = x w. p. β β (4) = and se, j β = N for,..., = N. Oherwse, for =,..., N : N, j, j, j, j, j xˆ = x β = β β = (5) Erx [( ) Y ˆ ] can be hen approxmaed by (4) N L ˆ, j, j Erx [( ) ] ( ˆ Y rx ) β L = j= (6) Do Seps o 5 for =,..., T.

4.4 Reducng degradaon local random wal Afer he resamplng sep (Algorhm 4.3), large-wegh samples are duplcaed; herefore, some samples are he same samples of p( x Y ˆ ), whch s no desrable from he pon of vew of prevenng degradaon. Andreu e al. (999) use he Marov chan Mone Carlo (MCMC) echnque o force he duplcaed samples o ae a random wal a each me sep, where p( x Y ˆ ) s he saonary PDF of he Marlov chan. Wha follows summarzes he procedure of he MCMC sep gven a proposal PDF p ( x x ˆ ) (hs algorhm s o be nsered n MC Sep 4 n Algorhm 4.3, rgh afer he resamplng sep): Algorhm 4.4: MCMC algorhm () Gven he curren sample, x ˆ j a me afer he resamplng sep n Algorhm 4.3, denoe he moher sae a me of he curren sample by where w ˆ s a sample from p( w ). x, j,.e., j, j xˆ = f (,, ˆ x u w), () Draw a random number r ha s dsrbued as unform on [,]. (3) Draw x from p ( x x ˆ )., j, j MC MC (4) Accep he move from, x ˆ j o x f, j MC, j, j, j, j, j py ( ˆ ) ( ˆ x = xmc px = xmc x = x ) pmc ( x xmc ), j, j > r, j, j, j py ( ˆ ˆ ) ( ˆ ˆ x = x px = x x = x ) pmc( xmc x ) (5) Go bac o sep (). (43) In hs repor, we consder an ndependence MCMC where he proposal PDF s seleced as: p ( x xˆ ) = p ( x x = x, yˆ ) (44), j, j MC LI

.e. p x x ˆ does no depend on, j MC ( ), x ˆ j. Due o hs ndependence, he cycle n Algorhm 4.4 only needs o be performed once o reach he saonary PDF; bu hs ndependence MCMC s only capable of explorng ( ˆ, j p x Y ) n he vcny of { x : =,..., N, j =,..., L}. Therefore, hs MCMC sep s called a local random-wal sep. 4.5 Advanages and dsadvanages of he PF echnque The advanages of he PF echnque nclude () as N (he number of samples per algorhm) approaches nfny, he value of any funcon of he sae x esmaed by PF converges o s expeced value; herefore, he PF echnque can be used o valdae oher mehodologes; and () parallel compuaons are possble for PF algorhms. A dsadvanage of he PF echnque s ha s compuaonally expensve, especally when he degradaon s severe so ha we need large N and L o have he algorhm converge. In general, he requred N and L grow wh he sze of he effecve suppor regon of p( x Y ˆ ). A smple es for convergence s o add parallel parcle flers unl he esmaed quanes of neres, rx ( ), do no sgnfcanly change. For lnear sysems wh me-varyng unnown parameers, we derve an effcen PF algorhm n Appendx III. Wh hs algorhm, he convergence rae can be sgnfcanly faser han he rae when sandard PF s used. 3

4

5. Numercal Examples We presen hree examples n hs Chaper. We generae daa ha s conamnaed by nose for hree smulaed dynamcal sysems. Wh he smulaed daa, we use denfcaon models ha are derved from hese dynamcal sysems o conduc EKF and PF and we compare her performance. The goal of hese examples s o see f hese echnques produce conssen resuls. 5. A planar four-sory shear buldng wh me-varyng sysem parameers 5.. Daa generaon We frs descrbe he sysem ha generaes he smulaed daa. Consder an dealzed planar four degrees of freedom (DOF) shear buldng sysem wh nown me-nvaran masses equal o m = m = m3 = m4 = 5, g (subscrp denoes he sory/floor number). The nersory sffnesses,, 3, and 4 change hrough me as shown n Fgure (, 3, and 4 drf around ceran values, whle sgnfcanly decreases and hen parally recovers). The ner-sory vscous dampng coeffcens are c, c, c 3, and c 4 and are also me-varyng (Fgure ). The me evoluons of, 3, 4, c, c, c 3, and c 4 are Brownan moons wh sandard devaon of he drf equal o % of her mean values durng each samplng nerval descrbed laer. The governng equaon of hs sysem s Mx + C x + K x = Fu (45) where 5

x x4, x 3, = x, x, M m4 = m 3 m m F m m = m m 4 3 C c4, c4, c4, c4, + c3, c3, = c3, c3, + c, c, c, c, + c, K 4, 4, 4, 4, + 3, 3, = 3, 3, +,,,, +, (46) x, s he dsplacemen of he ( + )-h floor (ffh floor s he roof) relave o he ground a me ; u s he acceleraon a he base (frs floor of he buldng); and c, and, are he nersory dampng coeffcen and sffness of he -h sory, respecvely, a me. We generae he daa usng whe-nose for he excaon u. The observed daa y ˆ s absolue acceleraon me hsores a he four sores: x, + u x u y + v M C x K x v, = + = [ + ] + x3, + u x 4, + u (47) where v 4 R ~ N(, Λ ) are he (saonary) measuremen unceranes for y ; 4 4 Λ R s dagonal such ha he overall sgnal/nose rms amplude raos for each channel s roughly equal o. Boh he excaon and observaon are sampled a a samplng nerval of. second and are shown n Fgure. Wh he excaon and observaon of he sysem,.e. { u :,..., T} { yˆ :,..., T} = and =, he goal s o esmae he sysem saes (dsplacemens and veloces) as well as he sysem parameers (dampngs, sffnesses, and he uncerany parameers) n real me. 6

4 () (MN/m) 3 () (MN/m) () (MN/m) () (MN/m) 6 4 6 4 6 4 6 4 5 5 5 5 5 5 5 5 Tme (sec) c () (N/m*sec) c () (N/m*sec) c () (N/m*sec) c () (N/m*sec) 3 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 Tme (sec) Fgure. Tme evoluons of he acual ner-sory sffnesses and dampngs 5.. Idenfcaon model Now we descrbe he denfcaon model. We use he followng equvalen-lnear (see Appendx III) sae-space model as our denfcaon model: d x x x M K x M C x M F u w d θ G y = M K( θ ) x M C( θ ) x + H( θ ) v = ( θ) ( θ) + + (48) 7

where θ R s he vecor conanng sysem parameers (ncludng four sffnesses, four dampngs, and four uncerany parameers); w ~ N(, I ) ; v ~ N(, I ) ; G R s a dagonal marx whose dagonals we have o specfy (he reason ha we have o specfy wll be explaned laer); H( θ ) 4 4 R marx s also aen o be dagonal, and he dagonals are unnown parameers (.e. he four uncerany parameers). The dmenson of he sae of he denfcaon model s weny (four dsplacemens, four veloces, four sffnesses, four dampngs and four uncerany parameers) alhough he dmenson of he sae n (45) s only egh. roof - 4 6 8 4 6 8 4h floor Acceleraon daa (m/sec ) - 4 6 8 4 6 8 3rd floor - 4 6 8 4 6 8 nd floor - 4 6 8 4 6 8 base (npu) - 4 6 8 4 6 8 Tme (sec) Fgure. The smulaed excaon and observaon daa 8

To complee he probablsc denfcaon model, we mus also specfy he pror PDF T T T T for he enre (augmened) sae rajecory { x x θ : =,..., T}. More specfcally for he model n (48), we mus specfy he followng: he pror PDF of x and x, he pror PDF of θ, and he dagonals of he G marx. Noe ha he denfcaon model n (48) uses a Brownan-moon pror PDF for he parameer evoluon { :,..., } dynamcal equaon n (48): θ = T due o he followng θ = G w (49) A dagonal G marx means ha all sysem parameers are nown a pror o drf ndependenly. The effec of he G marx s smlar o he forgeng facor ofen used n adapve flerng (Ljung and Gunnarsson 99). When he enres of G are large, he sysem parameers are allowed o drf more freely, relaxng he dependency beween parameer values of adjacen me seps; herefore, he denfed parameers only reflec mos recen daa. The converse s rue when he enres of G are small; n hs case, he denfed parameers can reflec remoe daa. In hs example, he pror PDF for x and x s aen o be zero-mean Gaussan wh large varances; he pror PDF of θ s aen o be Gaussan wh mean equal o he acual value of θ and large varances; he dagonal enres of G are chosen such ha n each me sep, each parameer drfs wh a coeffcen of varaon (c.o.v., defned by he sandard devaon dvded by he mean value) equal o %. Recall ha for, 3, 4, c, c, c 3, and c 4, her acual evoluons (see Fgure ) are Brownan moons wh he same % drf c.o.v.,.e. here s no modelng error for he evoluons of, 3, 4, c, c, c 3, and c 4. Bu for and he four uncerany parameers, he acual evoluons are no Brownan moons (he acual evoluon of s shown n Fg- 9

ure ; he four uncerany parameers are acually consan), whle he denfcaon model uses a Brownan-moon pror on her evoluons. Before we can proceed, we frs conver (48) o he followng dscree-me sysem usng numercal negraon (negrae over me sep): x x x x = f x, u, w y h x =, u, v (5) θ θ θ where f s evaluaed usng numercal negraon. Wh he above dscree-me denfcaon model, EKF can be conduced. Snce EKF s no suable for drecly racng he uncerany parameers (.e. marx H n Appendx I), we mplemen Algorhm A. wh (83) and γ =.95 for he esmaon of uncerany parameers. For PF, he uncerany parameers are no separaely reaed. 5..3 Resuls The sffness, dampng, and uncerany parameer esmaes and he assocaed 95% confdence nervals from EKF and PF (wh N = and L = and he mporance wegh c.o.v. hreshold = % usng Algorhms 4.3 and 4.4) are shown n Fgures 3-8 (here s no confdence nerval shown for he EKF uncerany parameer esmaes). For hs example, usng more samples n PF han N L= gves lle mprovemen n he convergence of PF, ndcang ha he resuls are close o convergence. We rea he resuls from PF as a comparson sandard snce asympocally gves conssen esmaes for he condonal means and varances. In Fgures 3-8, he hc lnes ndcae he acual parameer evoluons whle he hn dashed lnes are he condonal means of he denfed sysem parameers and he hn doed lnes ndcae he 95% confdence nervals. The resuls from EKF are smlar o hose of PF. 3

Boh algorhms successfully rac he sysem parameers; for mos parameers, he acual parameer evoluons le whn he 95% confdence bounds. Noce ha alhough he Brownan moon pror for and he uncerany parameers does no exacly mach her acual evoluons, boh Bayesan algorhms can sll appropraely rac and he uncerany parameers. Compared o he accuracy of he sffnesses, he esmaes of he dampng and uncerany parameers are worse and he assocaed unceranes are larger. Alhough EKF and PF perform roughly equally n hs example, here s a noceable dfference n he varances of he denfed dampng from PF, whch are slghly larger han hose from EKF. Sffness (MN/m) 8 6 4 Acual, EKF mean esmae Sffness (MN/m) 8 6 4 Acual, EKF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) Sffness (MN/m) 8 6 4 Acual 3, EKF mean esmae Sffness (MN/m) 8 6 4 Acual 4, EKF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) Fgure 3. The EKF esmaes of ner-sory sffness 3

Sffness (MN/m) 8 6 4 Acual, PF mean esmae Sffness (MN/m) 8 6 4 Acual, PF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) Sffness (MN/m) 8 6 4 Acual 3, PF mean esmae Sffness (MN/m) 8 6 4 Acual 4, PF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) Fgure 4. The PF esmaes of ner-sory sffness 3

5 5 Dampng (N/m*sec) 5 5 Acual c, EKF mean esmae Dampng (N/m*sec) 5 5 Acual c, EKF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) 5 5 Dampng (N/m*sec) 5 5 Acual c 3, EKF mean esmae Dampng (N/m*sec) 5 5 Acual c 4, EKF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) Fgure 5. The EKF esmaes of ner-sory dampng 33

5 5 Dampng (N/m*sec) 5 5 Acual c, PF mean esmae Dampng (N/m*sec) 5 5 Acual c, PF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) 5 5 Dampng (N/m*sec) 5 5 Acual c 3, PF mean esmae Dampng (N/m*sec) 5 5 Acual c 4, PF mean esmae 5 5 Tme (sec) 5 5 Tme (sec) Fgure 6. The PF esmaes of ner-sory dampng 34

Acual uncerany magnude for y, Acual uncerany magnude for y,.8 EKF esmae.8 EKF esmae.6.6.4.4.. 5 5 Tme (sec) 5 5 Tme (sec) Acual uncerany magnude for y 3, Acual uncerany magnude for y 4,.8 EKF esmae.8 EKF esmae.6.6.4.4.. 5 5 Tme (sec) 5 5 Tme (sec) Fgure 7. The EKF esmaes of uncerany parameers 35

.8 Acual uncerany magnude for y, PF mean esmae.8 Acual uncerany magnude for y, PF mean esmae.6.6.4.4.. 5 5 Tme (sec) 5 5 Tme (sec).8 Acual uncerany magnude for y 3, PF mean esmae.8 Acual uncerany magnude for y 4, PF mean esmae.6.6.4.4.. 5 5 Tme (sec) 5 5 Tme (sec) Fgure 8. The PF esmaes of uncerany parameers 5. A nonlnear hyserec dampng sysem wh unnown sysem parameers 5.. Daa generaon The prevous example s a me-varyng lnear sysem. In he curren example, we consder a me-varyng nonlnear sysem conssng of a sngle DOF (SDOF) Bouc-Wen hyserec dampng sysem (Wen 98). The purpose of hs example s o compare he performances of dfferen mehods for racng he sae and unnown parameers of a nonlnear sysem. The sysem ha generaes he daa can be descrbed by he followng governng equaon: 36

x x d x / m r / m u d = + r θ x θ x r r + θ x r y = / m r + / m u + v θ4, θ4,,, 3, (5) where r s he resorng force of he SDOF sysem; m s he mass, whch s se o uny durng he daa generaon; u s a whe-nose excaon force on he mass; y s he acceleraon measured on he mass; v s saonary such ha he overall sgnal/nose amplude rao s ;,, 3, 4, θ, θ, θ, θ are me-varyng sysem parameers (her acual flucuaons are shown n Fgures -, and hey are Brownan moons wh drf c.o.v. equal o % durng each samplng nerval): θ, s he sffness, θ,, θ 3, and θ 4, are parameers ha fne une he shape of he hyserec loop. Noe ha Bouc-Wen hyserec dampng sysem s Marovan n he sense ha we can defne a sysem sae such ha he curren sysem saus s compleely characerzed by he sae. Boh he excaon u and observaon y ˆ (shown n Fgure 9) are sampled a a samplng nerval of.5 second (roughly fve sample pons per oscllaon cycle of he sysem). The acual hyserec loops are shown n Fgure. 5.. Idenfcaon model Gven he daa u and y ˆ, he goal s o esmae he means and varances of he denfed sysem sae and sysem parameers usng he followng denfcaon model: 37

x x / mr / mu x + θ4, θ4, r θ, x θ, x r r + θ3, x r d θ, = + w d θ, θ 3, G θ 4, h y = / m r + / m u + h v (5) where w R 5 ~ N(, I) ; v R~ N(,) ; we assume ha m s nown and so s no consdered as one of he unceran parameers. The pror PDF for x and x s aen o be zero-mean Gaussan wh large varances; he pror PDF of θ,, θ,, θ3,, θ 4,,h s aen o be Gaussan wh mean equal o her acual value a me zero and large varances; he G marx n (5) s aen o be dagonal. The dagonals of G are chosen such ha n each me sep, each parameer s allowed o drf wh a c.o.v. equal o %,.e. no modelng error for he evoluons of θ,,, θ, θ 3,, and θ 4 ; bu modelng error exss for he evoluon of h n (5) (he acual h s consan nsead of a Brownan moon). For EKF, we mplemen Algorhm A. wh (83) and γ =.95 for he esmaon of he uncerany parameers h. For PF, he uncerany parameers are no separaely reaed. The connuous-me model s numercally negraed o ge he dscree-me verson of hs model smlar o (5) wh samplng rae equal o.5 second. 38

Inpu force 3 - - -3-4 3 4 5 6 7 8 9 Measured Acceleraon 6 4 - -4-6 3 4 5 6 7 8 9 Tme (sec) Fgure 9. The excaon force u and he observed acceleraon y ˆ 4 3 r : resorng force - - -3-4 -.8 -.6 -.4 -...4.6.8 x : dsplacemen Fgure. The acual hyserec loops of he smulaed sysem 39

5..3 Resuls Fgures -6 show he resuls of denfcaon of EKF and PF ( N = and L = and he mporance wegh c.o.v. hreshold = % usng Algorhms 4.3 and 4.4). For hs example, usng more samples n PF han N L= gves lle mprovemen n he convergence of PF, ndcang ha he resuls are close o convergence. We rea he resuls from PF as a comparson sandard. As before, he 95% confdence nervals on he parameers and saes are ndcaed by hn doed lnes n Fgures -6, excep for Fgure 5. We fnd ha EKF performs less effecvely han PF: a some me nsans, he EKF esmaes of he sffness parameer θ, oscllae around he acual evoluon (Fgure ), whle hs s no seen for PF (Fgure ). Also, he EKF esmaes for θ, (Fgure ) sgnfcanly devae from hose of PF (Fgure ). For he esmaon of dsplacemen, velocy and resorng force, he performances from he hree mehods are smlar (Fgures 3 and 4). However. PF esmaes he uncerany parameers h much beer han EKF (Fgures 5 and 6). 4

7.5.3 7 Acual hea EKF mean esmae. Acual hea EKF mean esmae 6.5. 6 5.5.9.8 5.7 4.5.6 4 4 6 8 Tme (sec).5 4 6 8 Tme (sec) -..4 -.4 -.6 Acual hea 3 EKF mean esmae.3.. Acual hea 4 EKF mean esmae -.8 -. -..9.8.7 -.4 4 6 8 Tme (sec).6 4 6 8 Tme (sec) Fgure. The EKF esmaes of he sysem parameers 4

7.5.3 7 Acual hea PF mean esmae. Acual hea PF mean esmae 6.5. 6 5.5.9.8 5.7 4.5.6 4 4 6 8 Tme (sec).5 4 6 8 Tme (sec) -..4 -.4 -.6 Acual hea 3 PF mean esmae.3.. Acual hea 4 PF mean esmae -.8 -. -..9.8.7 -.4 4 6 8 Tme (sec).6 4 6 8 Tme (sec) Fgure. The PF esmaes of he sysem parameers 4

- Acual dsplacemen EKF mean esmae - 3 4 5 6 7 8 9 Tme (sec) Acual velocy EKF mean esmae - - 3 4 5 6 7 8 9 5 Tme (sec) Acual resorng force EKF mean esmae -5 3 4 5 6 7 8 9 Tme (sec) Fgure 3. The EKF esmaes of he sysem saes 43

- - 3 4 5 6 7 8 9 Tme (sec) Acual dsplacemen PF mean esmae Acual velocy PF mean esmae - - 3 4 5 6 7 8 9 5 Tme (sec) Acual resorng force PF mean esmae -5 3 4 5 6 7 8 9 Tme (sec) Fgure 4. The PF esmaes of he sysem saes 44

.6 Acual uncerany magnude EKF esmae.5.4.3.. 3 4 5 6 7 8 9 Tme (sec) Fgure 5. The EKF esmaes of he uncerany parameer.6 Acual uncerany magnude PF mean esmae.5.4.3.. 3 4 5 6 7 8 9 Tme (sec) Fgure 6. The PF esmaes of he uncerany parameer 45

5.3 Lorenz chaoc sysem The Lorenz sysem s a chaoc sysem dscovered by Lorenz (963) when he solved a smplfed problem regardng wo-dmensonal flud moon drven by emperaure dfference, gravy, buoyancy, ec. The resulng smplfed se of dfferenal equaons consderng he frs few modes of he sysem s x = σ( x x ),,, x = r x x x x,,,, 3, x = x x bx 3,,, 3, (53) where x, specfes he me evoluon of he sream funcon of he frs mode, whose conours are he sreamlnes; x, and x 3, specfy he me evoluons of he emperaure of he frs wo modes of he sysem; he parameer σ depends on he properes of he flud (for waer he value s ypcally beween and 4); he number b depends on he scales of he modes; r s he emperaure dfference: for small r, he sysem s asympocally sable,.e. x, = x, = x3, =. For large r, chaos occurs wh he so-called buerfly aracor where he ulmae fae of a rajecory of he sysem s o wander around wo unsable equlbrum pons and he rajecory s exremely sensve o s nal condon. 5.3. Daa generaon In hs example, he values of σ, b, and r are se o be 3,, and 6 ( r s large so ha he buerfly aracor occurs), and we observe x, (conamnaed by nose) wh samplng nerval of.5 second: y = x + h v (54), 46

where h s chosen such ha he overall sgnal/nose rao s. Fgure 7 shows he observed value y ˆ, whch clearly shows ha he rajecory of x, swches several mes beween oscllang around he wo unsable equlbrum pons ha occur a x = 5 and x = 5 (especally durng -5 second). 5.3. Idenfcaon model The goal s o esmae he rajecory of he hree sysem saes based on y ˆ usng EKF and PF. When applyng EKF and PF, we assume ha we are very unceran abou he poson of he nal sae (.e. large varances for he pror PDF of he hree saes); we also assume ha σ, b, r, and h are nown, and her acual values are used durng he denfcaon. Equaons (53) and (54) are drecly used n he denfcaon model. y 8 6 4 - -4-6 -8 5 5 5 3 35 4 45 5 Tme (sec) Fgure 7. The plo for y ˆ 5.3.3 Resuls Fgures 8-9 show he esmaes made by EKF and PF (wh N = and L = and he mporance wegh c.o.v. hreshold = % usng Algorhms 4.3 and 4.4). For hs example, 47

usng N L= samples n PF s found o be suffcen for convergence. We rea he resuls from PF as a comparson sandard. As before, he 95% confdence nervals on he saes are ndcaed by hn doed lnes n Fgures 8-9. I s clear ha PF can successfully rac all hree sysem saes, whle EKF can only relably rac x, (snce y ˆ drecly measures x,, s possble ha an napproprae flerng algorhm can sll rac x, perfecly) as well as par of x, and x 3,. Bu EKF canno rac he begnnng porons of x, and x 3,. Acual x, EKF mean esmae x 3, - - 5 5 5 3 35 4 45 5 3 Acual x, EKF mean esmae x, - - 5 5 5 3 35 4 45 5 6 4 Acual x 3, EKF mean esmae x, - 5 5 5 3 35 4 45 5 Tme (sec) Fgure 8. The EKF esmaes of he sysem saes 48

x 3, - Acual x, PF mean esmae - 5 5 5 3 35 4 45 5 3 Acual x, PF mean esmae x, - - 5 5 5 3 35 4 45 5 6 4 Acual x 3, PF mean esmae x, - 5 5 5 3 35 4 45 5 Tme (sec) Fgure 9. The PF esmaes of he sysem saes 5.4 Dscusson In hs chaper, we have presened hree numercal examples o verfy he conssency of EKF and PF. Bascally, he hree examples represen hree dfferen classes of dynamcal sysems: he lnear model wh me-varyng sysem parameers (Secon 5.) can be consdered as a lghly nonlnear model, he nonlnear hyserec model (Secon 5.) can be consdered as a moderaely nonlnear model, whle he Lorenz chaoc model (Secon 5.3) s a hghly nonlnear model. 49

We beleve ha PF has performed sasfacorly for all examples, judgng from he fac ha PF always provde esmaes for he sysem sae and unnown parameers wh assocaed confdence nervals ha are conssen wh her acual values. In heory, PF should provde esmaes ha asympocally converge o he expeced values. I urns ou ha EKF can only rac he sysem sae and unnown parameers for he frs example, s performance for he second example s only far, and fals badly for he Lorenz chaoc example. Ths s conssen wh he expecaon ha EKF s no suable for hghly nonlnear models. 5

6. Real-daa Case Sudy 6. Buldng descrpon The seleced buldng for he case sudy s a 7-sory, 66, square fee (6, m ) hoel locaed a 844 Oron Ave, Van Nuys, CA, a 34. norh laude, 8.47 wes longude, n he San Fernando Valley of Los Angeles Couny. We refer o hs buldng as he Van Nuys hoel. I was bul n 966 accordng o he 964 Los Angeles Cy Buldng Code. The laeral force-ressng sysem s a renforced concree momen frame n boh drecons. The buldng was lghly damaged by he M6.6 97 San Fernando even, approxmaely m o he norheas, and severely damaged by he M6.7 994 Norhrdge Earhquae, whose epcener was approxmaely 4.5 m o he souhwes. The buldng has been suded exensvely, e.g., by Jennngs (97), Scholl e al. (98), Islam (996a, 996b), Islam e al. (998), L and Jrsa (998) and Bec e al. (). The buldng s 63 f by 5 f n plan, conssng of 3 bays by 8 bays, wh he long drecon orened eas-wes. I s approxmaely 65 f all: he frs sory s 3 f, 6 nches; sores hrough 6 are 8 f, 6-½ nches; he 7h sory s 8 f, 6 nches. Floors hrough 7 have hoel sues each, for a oal of 3 sues. The ground floor conans he recepon area, he hoel offce, a banque room, avern, dnng room, chen, lnen room, and oher hoel servces. The srucural desgner s Rssman and Rssman Assocaes (965). The srucural sysem s a 7-sory cas-n-place renforced-concree momen-frame buldng wh nonducle column dealng. Laeral force ressance s provded prmarly by he permeer momen frames, alhough he neror columns and slabs also conrbue o laeral sffness. The gravy sysem comprses -way renforced-concree fla slabs suppored by square columns a he neror and by he recangular col- 5

umns of he permeer frame. Slabs are -nch deep a he nd floor, 8½ nches a he 3rd hrough 7h floors, and 8 nches a he roof. The roof also has lghwegh concree oppng varyng n hcness beween 3-/4 nches and 8 nches. The column plan (wh he desgner s column numbers) s shown Fgure. The frame s regular n elevaon, as shown n Fgure. The fgure shows he desgner s noaon for beam and column numberng. Columns n he souh frame are 4 nch wde by nch deep,.e., orened o bend n her wea drecon when ressng laeral forces n he plane of he frame. Spandrel beams n he souh frame are generally 6 nch wde by 3 nch deep a he nd floor, 6 nch wde by -½ nch deep a he 3rd o 7h floors, and 6 nch wde by nch deep a he roof. 3 4 5 6 7 8 9 8 @ 8'-9" = 5'-" N Ca C8 C9 C3 C3 C3 C33 C34 C35 C36 C9 C C C C3 C4 C5 C C C C3 C4 C5 C6 C7 C8 C6 C7 C6a C7a 3'-5 4'- 3'-5 D '-" C '-" B '-" Ca C 8'-8 C C3 C4 C5 C6 C7 C8 C9 8'-9 A Fgure. Column plan of he Van Nuys hoel Column concree has nomnal srengh of 5 s for he frs sory, 4 s for he second sory, and 3 s from he hrd sory o he sevenh. Beam and slab concree s nomnally 4 s a he second floor and 3 s from he hrd floor o he roof. Column renforcemen seel s sched- 5