Improved Extended Kalman Filter for Parameters Identifiation Peipei Zhao. Zhenyu Wang. ao Lu. Yulin Lu Institute of Disaster Prevention, Langfang,Hebei Provine, China SUMMARY: he extended Kalman filter (EKF) is a useful method for physial parameters identifiation in system with unnown parameters. A multipoint iteration method was proposed in EKF method, whih made full use of the present and the previous time information. he numerial results showed that the preision in lumped mass system ould be improved more highly by multipoint iteration. A more stable and onvergent solution ould be obtained by this method. In addition, the results ould be influened less by the hypothetial initial value. Keywords: Extended Kalman Filter Parameters Identifiation Multipoint Iteration. Introdution Under the effets of environmental loads inluding earthquae, wind, fire, Civil engineering struture may have damage in various degrees. As a result, great hanges may tae plae in physial and mehanial parameters. Furthermore, buildings may ollapse. herefore, how to obtain strutural parameters beame a ey problem, whih may judge the health of strutures. Among all the identifiation methods, extended alman filter was an effetive way, whih was similar with alman filter. he differene was that rigidity and damping was put into the state vetor. hrough multiple iterative, the physial parameters may onverge to the true value. his way ould have higher effiieny, but it depended on the seletion of its initial value. If it was hosen inappropriately, the identifiation results may be divergent. his paper proposed an improved way that ould mae parameters onverge and improve identifiation effiieny..introdution of extended alman filter he extended alman filter was proposed by Andrew H.Jazwinsi in 97. his paper demonstrated the filter proess systematially, and pointed that the inrement of state vetor x met the requirements of alman filter. Definition of extended state vetor x [4] x u x u x = = x x 4 (.) where = [ ] is damping array and [ ] dynami equation, derivative of x was got, whih is written as = is rigidity array. From
u u x u M f M Cu Ku M f M Cx Kx xt = = = (.) How to obtain the extended alman filter formula was demonstrated as follows. Firstly, partial differential of x t was written as Eq. (.) ould be written as δx t = A δx( t) where i x t A = x t d δx t i ( ) x t = x t δ x t (.) (.4). Under disrete state, Eq. (.) was written as A Δt dδ δ x = A x = + e δ x (.5) where A e I A t A t A t A t A Δt d = = + Δ + Δ + + Δ + = Δ!! =! δ y was inrement of observation vetor, whih was written as he model of inremental equation was written as δ y = y yˆ (.6) δx+ = Adδx δy = C δx + v d (.7) From the above model, it ould be seen that the inrement of extended state vetor met requirements of alman filter. Beause of this point, it ould be obtained by alman filter that was written as where Inrement of + ould be written as ( ˆ ) δxˆ = A δxˆ + K δy C A δx (.8) + d + + d d = + (.9) P A dp Ad Q ( ) ( ) P I K C P I K C K RK (.) = d d + ( ) K = P C C P C + R (.) d d d
δ xˆ A ˆ dδ x + = (.) whih was one-step estimation. his was optimum estimation with observed values, whih was the last alman estimation. Meanwhile δ xˆ+ ould also be gotten by the way as follows: ( + ) Δt δ xˆ = xˆ xˆ = xˆ dt x ˆ Δt (.) + + Δt hrough inremental equation, x ˆ + ould be written as ( δ ) xˆ = xˆ + A δxˆ + K y C A δxˆ + d + + d d (.4) where y = y yˆ = y C x (.5) δ + + + dˆ then ( δ ) xˆ = xˆ + A δxˆ + K y C xˆ C A xˆ + d + + d d d ( δ ) = xˆ + A δxˆ + K y C xˆ + A xˆ d + + d d ( δ ) = xˆ + δxˆ + K y C xˆ + xˆ + + + d + ( ) = xˆ ˆ ˆ ˆ + x Δ t+ K + y+ Cd x + x Δt (.6) Eq.(.6) was the last extended alman filter reursion formula, through whih x ˆ + ould be gotten by x ˆ. Before reursion, x and P needed to be supplied, where was variane of xˆ x P x and supposed that =.In order to aelerate rate of onvergene, the value of x was more aurate, the identifiation results were better. Extended alman filter was ideal in theory, but in fat how to hoose x was a ey problem. If x was hosen inappropriately, then it may inrease the amount of alulation, and even worse it may result in CP C R divergene. Furthermore, the matrix ( paper proposed a multipoint iteration method. Its priniple is distint. At first, x was given at random. hrough observation data and extended alman filter x + ould be obtained. Beause rigidity and damping were unhangeable in linear time-invariant systems, and of x + were the optimum estimation in this ase. Furthermore, and of x + that hosen as the initial value of x were applied into extended alman filter again. hrough multiple iteration, error may be ontrolled in a tolerane zone.if the identifiation results were divergent, the initial value of x needed to be modified and then repeated the multipoint iteration method. + ) may be singular. In order to solve this problem, this
. Numerial experiment Supposed a three DOF lumped mass system shown in Fig. exited by earth pulsation, whih was El-entro wave shown in Fig.. he mass, rigidity and damping of eah were g, Nmand 4 Nsm,Sampling frequeny was 5Hz and sampling point was 5. m m m Fig. three DOF lumped mass system Fig. El-entro wave Supposed that system was in stati state, and the displaement,veloity and aeleration were obtained by entral differene method. hen the alulated responses were onsidered as observation data, whih were used to identify physial parameters of struture. Before applying extended alman filter, x needed to be given at first. he initial value of rigidity and damping were shown in table.
able initial value of rigidity and damping parameters initial value was Beause system was in stati state, the displaement and veloity was zero. he variane of observed noise R= I, where I was unit matrix. he identifiation results were shown in table able identifiation results with no observed noise parameters initial value 4 4 4 identifiation results.75..84 4.8 4.6 7.94 relative error.8%.%.8% 9.5%.4% 7.65% he error matrix of initial state vetor P was written as P = 4 4 4 9 9 9 he responses after extended alman filter were shown in Fig., Fig.4 and Fig.5. It ould be learned that the filtered responses were the same as the truth-value, whih indiated that r extended alman filter ould identify responses of struture. hrough the results of numerial experiment shown in table, it ould be learned that extended alman filter may identify rigidity aurately and the identifiation of damping was relatively bad. Generally speaing, rigidity was a ommon parameter that was used to judge the health of struture, while damping was almost not be used. hen the identifiation auray of damping had no effets on strutural damage detetion.
8 6 4 位移 (m) - -4-6 -8 5 5 5 5 Fig. displaement of 8 6 4 位移 (m) - -4-6 -8 5 5 5 5 Fig.4 displaement of 5 5 位移 (m) -5 - -5 5 5 5 5 Fig.5 displaement of hen if the responses had observed noises, the method ould also get the physial parameters. he mean value of observed noises was zero and variane was 5,.he observed displaement were shown in Fig.6, Fig.7 and Fig.8.
hrough multiple iteration proposed by this paper, the filtered responses were shown in Fig.9, Fig. and Fig.. By ontrast, it ould be nown that the filtered responses were almost the same as the truth-value, whih indiated that extended alman filter ould also identify physial parameters even the data had ambient noises. 8 6 4 位移 (m) - -4-6 -8 5 5 5 5 Fig.6 displaement of with noise 8 6 位移 (m) 4 - -4-6 -8 5 5 5 5 Fig.7 displaement of with noise 5 5 位移 (m) -5 - -5 5 5 5 5
Fig.8 displaement of with noise 8 6 4 位移 (m) - -4-6 -8 5 5 5 5 Fig.9 omparison of 8 6 4 位移 (m) - -4-6 -8 5 5 5 5 Fig. omparison of 5 5 位移 (m) -5 - -5 5 5 5 5 he identifiation results were shown in table Fig. omparison of
able identifiation results with observed noise parameters initial value 4 4 4 identifiation results 998.57.5 4.8 44.7 9.9 7.57 relative error.4%.5%.4%.75%.% 6.% 4.Conlusion. Extended alman filter ould identify strutural rigidity and damping effetively even when the input of struture was unnown. he priniple of it was demonstrated in this paper.. Numerial experiment showed that through multipoint iteration in extended alman filter, it ould redue amount of alulation and aelerate the rapid of onvergene. he numerial results also showed that the identifiation had high auray. With the identifiation shown in table and, the auray of rigidity was far higher than damping. When measured data had observed noises, through multipoint iteration, the extended filter may also identify physial parameters aurately. Referenes [] Andrew H.Jazwinsi(97), Stohasti Proesses and Filtering heory, Aademi Press. [] Shuqian Cao, Wende Zhang, Longxiang Xiao(). Modal Analysis of Vibration struture.ianjin University Press. [] Van Overshee Peter, De Moor Bart(996). Subspae Identifiation for Linear Systems: heory, Implementation, Appliations. [4] Jianhua Xu, Guorui Bian, Chonguang Ni, Guoxing ang(98). State estimation and system identifiation. Siene Press.