1. Sequential MCMC Methods for
Parameter Estimation
of LTI Systems Subjected to Non-
stationary
Earthquake Excitations
Anshul Goyal
Under the guidance of
Dr. Arunasis Chakraborty
1
05-05-2014
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Contents
Contd.
• Bayesian and Monte Carlo Methods
Bayesian State Estimation
Sequential Importance Sampling (SIS Filter)
Sequential Importance Re-sampling (SIR Filter)
Bootstrap Filter (BF)
Re-sampling Algorithms
• Simulations for SDOF Oscillator
SIS (Degeneracy)
Bootstrap (Sample Impoverishment)
Handling Sample Impoverishment
• Introduction
System Identification- A brief review
Major research and contributions
Motivation & Scope of present study
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• Synthetic Study
Model (Mass, Stiffness and Damping Matrices)
Numerical Results
-Identified Parameters and Modal Frequencies
-SIS v/s SIR v/s BF
- Multinomial v/s Wheel v/s Stratified v/s Systematic
• BRNS Building
Model (Mass, Stiffness and Damping Matrices)
Numerical Results
-Identified Parameters and Modal Frequencies
-SIS v/s SIR v/s BF
- Multinomial v/s Wheel v/s Stratified v/s Systematic
• Conclusions and Future Studies
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What is System Identification?
System identification is the field of mathematical modeling of the inverse
problem from the experimental data.
Introduction
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What is System Identification?
System identification is the field of mathematical modeling of the inverse
problem from the experimental data.
How to apply System Identification?
• The first step is to determine an appropriate form of the model (typically
a differential equation of certain order).
• In the second step, several statistical approaches are used to estimate the
unknown parameters of the model. This estimation is often done
iteratively.
• The model obtained is then tested to see whether it is an appropriate
representation of the system. If this is not the case, some more complex
model structure is considered, its parameters estimated and validated
again.
Introduction
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Applications
There are two main purposes of model updating or system identification
of the structural system.
• Structural Health Monitoring
• Structural Control
Though system identification methods are useful for large and complex
structures where it is difficult to obtain the mathematical models directly,
it has some limitations.
• An appropriate model structure must be found. This can be a difficult
problem, particularly if the dynamics of structure is non-linear.
• The real life recorded data is not perfect always as these are always
disturbed by noises.
• The process may vary with time, which can cause problems if an
attempt is made to describe it with a time invariant model
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Dynamic State Estimation
State estimation is the process of using dynamic data from a system to
estimate quantities that give a complete description of the state according
to some representative model of it. Popular methods are:
KALMAN FILTERS
- 1960 by R Kalman
- Linear Gaussian
state space models
- Exact optimal
solutions
MONTE CARLO METHODS
- 1945 & 1949 by Ulam & Metropolis
- Linear as well as Non-linear systems
- Gaussian and Non- Gaussian state space
models
- Recursive estimation by approximating
integrals
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Major Research and Contributions
• Particle filter methods have been widely used in robotics and for
solving the tracking problems (Thrun, 2002). The application of
these methods to problems in structural mechanics is not yet
widely explored.
• (Ching et al., 2006) compared the performance of the Extended
kalman filter and particle filter by applying theses methods on
planar four-story shear building with time-varying system
parameters and non-linear hysteretic damping system with
unknown system parameters.
• (Manohar and Roy, 2006) identified the parameters of two single-
degree of freedom nonlinear oscillators, namely Duffing oscillator
& Coloumb oscillator.
• (Nasrellah and Manohar, 2011) did the combined computational
and experimental study using multiple test and sensor data for
structural system identification.
• (Rangaraj, 2012) used particle filter algorithm or identification of
fatigue cracks in vibrating beams.
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Motivation & Scope of present study
State estimation methods have wide scope in Structural Health
Monitoring as well as efficient control of structures. Models of
physical system always have uncertainties associated with them.
Hence, obtaining the parameters of the system optimally out of the
limited noise corrupted data is a challenge.
Scope of present study
• Implementation of the most common variants of Particle filter i.e
SIS, SIR and BF to both synthetic as well as field data.
• Parameter identification of a real life structure subjected to multi-
component non-stationary ground motion excitation using all the
three algorithms.
• The algorithm is very well able to identify the natural frequency
even in higher modes when the signal processing techniques are
generally not capable to do so.
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Bayesian and Monte Carlo Methods
The focus of dynamic state estimation techniques is to estimate the
state of the system using the measurement data.
Governing equation of system (Continuous form) :
X (t) = q (P(t),t)
X(t): response of the structure
P(t): Input force
q(.): relates the input to the output.
Discretized version:
Xk+1 =qk (Xk,wk)
Xk : state of the system at time t = k
wk: Model white noise
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Discretized Measurement equation:
Yk = hk(Xk, vk)
Yk : measurement at time t = k
vk : measurement noise
Measurements from the sensors:
Mk = [Y1, Y2, …., Yk]
Target:
Estimation of p(Xk |Mk)
This is equivalent to determining moments of Xk
μ = ∫ Xk p(Xk |Mk) dXk
σ = ∫(Xk − μ)T (Xk − μ)p(Xk |Mk)dXk
μ: first moment or mean
σ: second moment or variance
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Assumptions in Bayesian modeling :
• The sates follow a first order Markov process
p(Xk |X0:k-1) = p(Xk |Xk-1)
• The observations are independent of the given states.
At any time t using Bayes Theorem
Posterior:
p(X0:t|Y1:t) =
p(Y1:t|X 0:t)p(X 0:t)
∫ p(Y 1:t |X0:t)p(X 0:t)dX 0:t
Recursive Posterior:
p(X0:t+1|Y1:t+1) = p(X0:t|Y1:t)
p(Yt+1|Xt+1)p(Xt+1|Xt)
p(Yt+1|Y1:t)
Recursive Prediction:
p(Xt |Y1:t-1) = ∫p(Xt |Xt-1)p(Xt-1|Y1:t-1)dXt-1
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Recursive Prediction:
p(Xt |Y1:t-1) = ∫p(Xt|Xt-1)p(Xt-1|Y1:t-1)dXt-1
The above equations are modified when the model and the process noise
wk and vk are present.
Closed form expression of above equations are available when f (.) and
h(.) are linear and the noise wk and vk are Gaussian and this leads to well
known KALMAN FILTERS.
Difficulty: Computation of marginal of the posterior p(X0:t |Y1:t) as it
requires evaluation of complex high dimensional integrals.
What to do?
Exploit the cheap and faster computational facilities to develop methods
based on the Monte Carlo Simulations for approximating the integrals
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Sequential Importance Sampling (SIS Filter)
Target: Statistical problem of estimating the expected value of E[f(x)]
w.r.t some probabilistic distribution p(X)
E [f (X)] =∫f (X)p (X)dX
• In MC methods distribution is represented by random samples
rather than analytic function.
• Approximation is better for increased number of particles.
Difficulty
• Sometimes it becomes difficult to sample from p (X)
What to do?
• Sample from the importance distribution
I = ∫p(x)dx = ∫
𝑝(𝑥)
𝑞(𝑥)
. 𝑞(𝑥)𝑑𝑥
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∫ 𝑞(𝑥)𝑑𝑥 = 1
𝐼 = 𝐸 𝑞[
𝑝 𝑋
𝑞 𝑋
] =
1
𝑁
𝑖=1
𝑁
𝑝(𝑋 𝑖 )
𝑞(𝑋(𝑖))
Thus the above expectation E [f (X) represented as F(t) becomes
where the posterior can be approximated as:
Here wk
i is the normalized weight of the particles which
can be computed using an importance distribution
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SIS Filter
Initiation:
• Draw N samples from prior distribution at t = 0
• Set the weights of all the particles as
1
𝑁
Prediction:
• Calculate the likelihood 𝑝 𝑌𝑘 𝑋 𝑘)
Weight updating:
• Update weights of all the particles using
Estimation:
• Perform the estimation using
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• One of the major problems associated with SIS filter is the degeneracy
What is Degeneracy?
Degeneracy is a process where all the particles have negligible weight
except one particle after few iterations. The variance of the importance
weights increases with time and it becomes impossible to control the
degeneracy phenomenon.
How to measure it?
Gordon,1993 proposed effective sample size Neff as the criteria of
degeneracy
• A very small value of Neff indicates severe degeneracy.
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How to control degeneracy of the samples?
(Arulampalam et al., 2002) suggested two ways
• Good choice of Importance density such that variance of weights
is reduced.
• Resampling: One of the most important step which differentiate
SIS to SIR and Bootstrap. Incorporation of resampling leads to
“Adaptive particle filters.”
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SIR , Bootstrap and Re-sampling
Algorithms
• Choice of optimal importance density
• Therefore the updated weights become:
• If Neff falls below the threshold resample the particles and set the
weights as
1
𝑁
• The Bootstrap filter is a special case of SIR filter where the
dynamic model is used as importance distribution and the
resampling is done at each step.
Key steps
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Simulations for SDOF Oscillator
• Example to illustrate the mathematics behind the algorithms.
• Governing equation of motion of SDOF oscillator
Mu¨ (t) + C u˙ (t) + K u (t) = −Mu¨g (t)
• Ground motion considered is 1940 El-centro earthquake.
- Duration of excitation: 40sec
- Time step for forward problem, dt = .01sec
• Assumed parameter for forward problem
- Mass: 40 kg
- Stiffness:60000N/m
- Damping : 15 N-s/m
- Natural frequency of oscillator: 38.72 rad/s
• Algorithm used for forward problem: β Newmark
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Response of SDOF Oscillator
Forward Problem
Inverse problem
SIS Filter: Ratio of Identified
stiffness to original
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Evolution of weights of particles
over time using SIS Filter
Evolution of posterior density over
time using SIS Filter
Degeneracy !!
How to tackle ??
Resample it !!
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Posterior evolution of distribution at iteration number a) initial, b) intermediate
(100) and c) final using Bootstrap Filter
Sample
Impoverishment !!
How to tackle it???
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Add Noise to tackle Sample Impoverishment !!
Expected value of stiffness by addition of
noise (2% expected value)
Convergence of the stiffness due to
addition of noise (2% expected value)
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Synthetic Study
Model Details:
Dimensions of slab: 600×300×10 mm
Lumped mass of slab: 15.2 kg
Height of each story : 600mm
Column cross-section : 30 ×100 mm
Mass matrix
M =
Stiffness Matrix
K =
M1 0 0
0 M2 0
0 0 M3
k1 + k2 -k2 0
-k2 K2 + k3 -k3
0 -k3 k3
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Damping matrix is obtained by considering Rayleigh damping
C = α M + β K
α = ζ
2wiwj
wi + wj
β = ζ
2
wi + wj
m1,m2 and m3 : lumped mass at the floor levels
k1, k2 and k3 : stiffness of the each story.
α and β : Damping Coefficients
ζi : Damping ratio in ith mode
wi : natural frequency of the system in ith mode
Different earthquakes have been considered for synthetic study
.
• 1940 El-Centro
• 1989 Lomaprieta
• 1995 Kobe
• 1999 Chichi
• 2004 Parkfield
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Forward Problem Parameters
Mass (kg) Stiffness (N/m) Damping (N-s/m) Natural Frequency (Hz)
15.2 41987 19.032 4.1565
15.2 76842 34.173 12.8093
15.2 74812 33.630 19.8511
Inverse Problem Details
• Domain for stiffness values [10000,90000 N/m]
• Domain for damping values [0, 50 N-s/m]
• Number of particles taken 100
• Likelihood function: Normal distribution [measurement, 0.001]
• Algorithms used : SIS, SIR and Bootstrap
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Inverse Problem Solution: Bootstrap Filter with a comparison
between the traditional re-sampling algorithms
Ratio of identified parameters to original
parameters: Elcentro Earthquake
Standard Deviation of identified stiffness:
Elcentro Earthquake
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Ratio of identified parameters to original
parameters: Lomaprieta Earthquake
Standard Deviation of identified stiffness:
Lomaprieta Earthquake
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BRNS Building
• Implementation of SIS, SIR and Bootstrap filter for parameter estimation of
BRNS building subjected to non-stationary recorded earthquake excitations
• Measurement data available for top and first story in both x and y
directions.
• Two set of data recording on 3rd and 21st September, 2009.
• Identified parameters:
• Stiffness values in both the directions at each
of the floor level
• coefficients α and β of the modal damping
matrix
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• Domain for a & b
Damping coefficients Domain
a 0.0005, 0.0015
b 0.01,0.03
• Number of particles taken 150
• Likelihood function: Normal distribution [measurement, 0.01]
• Algorithms used : SIS, SIR and Bootstrap
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Original and estimated states of BRNS building a) first story x direction,
b)First story y direction, c) top story x direction and d) top story y
direction
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Conclusions & Future work
• Fairly good results with the natural frequency and mode shapes of the
identified systems is close to the original system in case of both synthetic
as well as field data.
• Algorithm is very robust and is able to pick up the best value among the
samples generated at t = 0.
• Similar results obtained from SIS, SIR and Bootstrap when a similar pool of
simulated particles is used. However, the results vary slightly for the field
data.
• Sensitivity analysis using Bootstrap also proves the robustness for the
algorithm converging even for very low SNR values.
• Stratified and Systematic resamplings give better estimates than
Multinomial and Wheel resamplings for both the studies i.e. synthetic
model as well as BRNS building.
• Convergence is achieved faster when Multinomial and wheel re-samplings
are used.
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• Resampling leads to sample impoverishment. It becomes an issue to tackle
this for complex, large dynamical systems. The present study is dependent on
the particles simulated at t = 0. Also the domain dependency is reflected.
• The problem becomes trivial if one is able to sample from particles every
time from the updated posterior distribution.
Future works
• Improving the sampling strategies such that one is able to sample from
posterior with less computational effort.
• Implementation on Base isolated building with nonlinear characteristics.
• Large scale real life structures like bridges. One such bridge is the railway
overhead bridge connecting to the main campus of IIT Guwahati.
