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  1. 1. Sequential Markov Chain Monte Carlo Methods (MCMC) for Parameter Estimation of Linear Time Invariant (LTI) Systems subjected to Non-Stationary Seismic Excitations Anshul Goyal, Dept. of Civil, Architectural and Environmental Engineering, The University of Texas, Austin, TX 78712, USA Background & Objectives Methods for Analysis Numerical Studies The author gratefully acknowledges (i) Department of Civil Engineering, Indian Institute of Technology Guwahati for data. (ii) Dr. Arunasis Chakraborty of IIT Guwahati for supervising the project. Useful discussions with Prof. Lance Manuel of UT Austin helped in this presentation.  Arulampalam, M.S., Maskell, S., Gordon, N., and Clapp, T., “A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking,” Signal Processing, IEEE Transactions, 50(2):174-188, Feb 2002.  Namdeo, V., and Manohar, C.S., “Nonlinear Structural Dynamical System Identification Using Adaptive Particle Filters,” Journal Of Sound and Vibration, 306(35):524-563, 2007.  Rangaraj, R., “Identification of Fatigue Cracks in Vibrating Beam using Particle Filtering Algorithm,” Master’s Thesis, Indian Institute of Technology, Madras.  Goyal, A., and Chakraborty, A., “On Parameter Estimation of Linear Time Invariant Systems using Particle Filter,” Proc. SEC 2014, IIT Delhi. References Acknowledgements Conclusions Numerical Studies A. Single-Degree-of-Freedom Oscillator (Synthetic Data) B. Three-Story Shear Building (Synthetic Data) C. Multi-Story RC Framed Building (Field Data) (continued)  Tabulated results are for two simulation studies based upon El Centro and Lomaprieta ground motions as input excitations.  Stratified and Systematic Resamplings give the best identified values with smallest percent error.  Identification is robust even for a low SNR value.  The identified frequencies and mode shapes are very close to the original values.  MCMC methods give fairly good estimates of natural frequencies and mode shapes with both synthetic and field data.  All the three variants of the particle filter algorithm (SIS, SIR and BF) perform significantly well.  The SIS filter suffers from degeneracy which is overcome by setting a threshold and resampling.  Systematic and Stratified Resampling algorithms give the best values of the identified parameters.  The computational effort increases significantly as the number of parameters to be identified increases. This requires a larger pool of samples to get best results.  The sample impoverishment introduced as a result of resampling can be tackled by carefully training the algorithm through several test runs. This is also computationally expensive for a full-scale structure.  Effective sampling from the posterior in the direction of convergence can be further studied to improve the efficiency. Introduction High financial and environmental costs involved in demolishing and replacing aging civil engineering infrastructure, expensive structural systems such as aircrafts and reusable space vehicles, etc. have led to an increased focus on the possibility of extending the lifetime of such systems. An important step is monitoring which in turn requires system identification. This involves developing a mathematical model of the structure whose parameters are estimated from response measurements. This constitutes the so-called inverse problem, which is in contrast to the forward problem that involves estimating the structural response once the structural model is given. Topics of investigation in this study  To carry out a simulation study on the implementation of Sequential MCMC methods for synthetic as well as field data.  To identify parameters (stiffness and damping), mode shapes, and natural frequencies of a fully functional fixed-base RC frame building subjected to seismic excitation.  To compare traditional resampling algorithms.  To compare variants of particle filter algorithms such as SIS, SIR and Bootstrap on the basis of the required number of convergence steps.  To study the effect of measurement noise on the performance of the algorithms. Mathematical Modelling 𝑥 𝑘+1 = (𝑥 𝑘, 𝑤 𝑘) 𝑦 𝑘 = (𝑥 𝑘, 𝑣 𝑘) 𝑀 𝑘 = [𝑦1, 𝑦2, , 𝑦 𝑘] 𝜇 = ∫ 𝑥 𝑘 𝑝 𝑥 𝑘 𝑀 𝑘 𝑑𝑥 𝑘 𝜎 = ∫ (𝑥 𝑘−𝜇)(𝑥 𝑘 − 𝜇) 𝑝(𝑥 𝑘 𝑀 𝑘 𝑑𝑥 𝑘 Target 𝑝 𝑥 𝑘 𝑥 𝑘−1 = ∫ 𝑝 𝑥 𝑘 𝑥 𝑘−1, 𝑤 𝑘−1 𝑝 𝑤 𝑘−1 𝑥 𝑘−1 𝑑𝑤 𝑘−1 Recursive Bayesian Estimate 𝑝 𝑥 𝑘 𝑥0:𝑘−1 = 𝑝 𝑥 𝑘 𝑥 𝑘−1) States follow the first-order Markov Process and current observation depends upon current state 𝑝 𝑥0:𝑘 𝑦0:𝑘 = 𝑝 𝑥 𝑘 𝑥 𝑘−1 𝑝 𝑦 𝑘 𝑥 𝑘 𝑝 𝑦 𝑘 𝑦0:𝑘−1 (𝑝(𝑥0:𝑘−1|𝑦0:𝑘−1)) Recursive Bayesian Estimation 𝑝 𝑦 𝑘 𝑥0:𝑘 = 𝑝 𝑦 𝑘 𝑥 𝑘) Prior 𝑝 𝑦 𝑘 𝑥 𝑘 = ∫ 𝑝 𝑦 𝑘 𝑥 𝑘, 𝑣 𝑘 𝑑𝑣 𝑘 Normalizing Factor 𝑝 𝑦 𝑘 𝑦0:𝑘−1 = ∫ 𝑝 𝑦 𝑘 𝑥 𝑘 𝑝 𝑥 𝑘 𝑦 𝑘−1 𝑑𝑥 𝑘 Particle Filter/Sequential Markov Chain Monte Carlo Particle filter generates a set of samples that approximates the filtering distribution 𝑝 𝑥 𝑘 𝑦0, 𝑦1, … 𝑦 𝑘 also known as the posterior distribution. The expectation with respect to the posterior distribution can be approximated by N, the number of particles. However it may not be beneficial to sample each time from the posterior distribution, instead this may be doe using an importance distribution. ∫ 𝑓 𝑥 𝑘 𝑝 𝑥 𝑘 𝑦0, 𝑦1, … 𝑦 𝑘 𝑑𝑥 𝑘 ≈ 1 𝑁 𝑖=1 𝑁 𝑓(𝑥 𝑘 𝑖 ) ∫ 𝑓 𝑥 𝑘 𝑝 𝑥 𝑘 𝑦0, 𝑦1, … 𝑦 𝑘 𝑑𝑥 𝑘 ≈ 𝑖=1 𝑁 𝑓 𝑥 𝑘 𝑖 𝑤 𝑘 𝑖 𝑤 𝑘 𝑖 = 𝑝(𝑥0:𝑘|𝑦0:𝑘) 𝜋(𝑥0:𝑘|𝑦0:𝑘) 𝑤 𝑘 𝑖 ∝ 𝑝 𝑦 𝑘 𝑥 𝑘 𝑖 𝑝 𝑥 𝑘 𝑖 𝑥 𝑘−1 𝑖 𝜋 𝑥 𝑘 𝑖 𝑥0:𝑘−1 𝑖 , 𝑦1:𝑘 𝑤 𝑘−1 𝑖 Recursive Importance Sampling Recursive Weights 𝑤 𝑘 𝑖 ∝ 𝑝 𝑦 𝑘 𝑥 𝑘 𝑖 𝑝 𝑥 𝑘 𝑖 𝑥 𝑘−1 𝑖 𝑝(𝑥0:𝑘−1 𝑖 |𝑦1:𝑘−1) 𝜋 𝑥0:𝑘|𝑦1:𝑘 This is Sequential Importance Sampling. (Cumulating) Degeneracy of Algorithm Control 𝑁𝑒𝑓𝑓 = 1 𝑖=1 𝑁 (𝑤 𝑘 𝑖 2 ) Smaller value of 𝑁𝑒𝑓𝑓 implies more severe degeneracy !! Resampling Stratified Systematic Multinomial Wheel Resampling % Sequential Importance Resampling (SIR)Bootstrap Filter 𝜋 𝑥 𝑘 𝑖 𝑥0:𝑘−1 𝑖 , 𝑦1:𝑘 = 𝑝(𝑥 𝑘 𝑖 |𝑥 𝑘−1 𝑖 )100% 0% < N < 100% Recursive Bayesian Estimation is the underlying principle of the dynamic state estimation. The target is to estimate the posterior distribution based on sequential measurements at discrete time steps. The state and the system are updated based on the recent measurements and prior knowledge. SDOF Oscillator Input Excitation Posterior evolution (Sample Impoverishment) Weights evolution Mean of Identified Stiffness Parameter Mutating Samples by noise addition: Mean & Std. Deviation 𝑀1 𝐾1 𝐶1 𝑀2 𝐾2 𝐶2 𝑀3 𝐾3 𝐶3 Shear Building Model Acceleration Response Time History Identified Damping and Stiffness Comparison Resampling Algorithms Original and Identified Mode Shapes  Input excitation is the El Centro earthquake ground acceleration time series. The original stiffness parameter chosen is 60,000 KN/m.  Degeneracy of algorithm and the solution is resampling by setting equal weights.  Resampling leads to Sample Impoverishment which is avoided by mutation. Multi-Story Building- Mass and Stiffness Matrix Multi-Component Seismic Excitation Response Time History Identified Parameters AVOIDING SAMPLE IMPOVERISHMENT: MUTATION Results  Stratified and Systematic Resampling algorithms perform best for the field data as well.  All the three variants of the algorithm: SIS, SIR and BF perform equally well.  Identified mode shapes are close to the original.

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