Advanced Markov Chain Monte-Carlo Simulation For Gaussian
Filtering Based Inverse Problems
A report for the state of art seminar presentation
Submitted by
Arpita Ghosh
(196104102)
Under the supervision of
Prof. Anjan Dutta
Dr. Arunasis Chakraborty
DEPARTMENT OF CIVIL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI
June 25, 2024
Acknowledgements
I want to grab this opportunity to acknowledge the greatness of all those people who
help and support me to complete this report. First of all I want to thank both of my supervisor, Prof. Anjan Dutta and Prof. Arunasis Chakraborty , for giving endless important
suggestions and critical reviews of my work through their guidance. I also want to thank the
other members of my Doctoral Committee, Prof. Sajal K. Deb, and Dr. Sandip Das , Prof.
Debabrata Chakraborty for keeping their patience and believe over my work. And finally
I will not let go this chance without mentioning the infinite support and help of my senior
research fellows, my friends and family to keep me motivated for completing this report in
time.
Date:25-01-2021
Arpita Ghosh
(196104102)
IIT Guwahati,India
i
Abstract
Combining mathematical modeling techniques with experimental tools offers most powerful means to study existing structures. Such studies lie at the heart of problems of structural health monitoring. A major component in these problems consists of identification
of parameters of mathematical models of structural systems based on a limited set of measurements. The problem of structural system identification constitutes an important and
difficult class of inverse problems in structural engineering.Methods based on dynamic state
estimation offer powerful means to tackle these problems. These methods recognize that the
mathematical model as well as measurements are invariably imperfect in nature and employ
random processes to model these imperfections. Consequently, the problem of structural
system identification is tackled within a probabilistic framework and one seeks to determine
the joint probability density function of the system parameters conditioned on the measurements made. The solution to this problem becomes increasingly difficult as systems behave
nonlinearly, size of the problem increases, and noises become multiplicative and (or) nonGaussian. The advent of powerful computing tools and advances in sensing techniques has
spurred research activities to tackle these challenges. This report belongs to the literature
review of this area of research.
In the past two decades the development of Structural Health Assessment (SHA) techniques has become a booming research field in civil engineering community. Vibration based
SHA techniques can be broadly classified into two categories, namely frequency domain and
time domain. In frequency domain method, change in modal parameters (natural frequency,
mode shape, modal damping etc.) are taken as a signal of damage occurrence . Although
frequency domain techniques could successfully identify the global damaged condition of the
structure, but it is insensitive to local damages. Time domain approach provides better flexibility to locate and quantify damage at local level. Commonly used time domain approach
utilizes two techniques, least square and Bayesian filtering algorithm. Least square technique
estimate the unknown structural parameter by minimizing the sum of square error between
predicted and measured vibration response. Bayesian filter method for parameter estimation is more suitable in case vibration response contains ambient noise signatures. When
noise characteristics follow Gaussian distribution, Kalman filter based (Extended Kalman
filter, Ensemble Kalman Filter, Unscented Kalman filter etc.) methods are used. This limitation can be overcome using Particle filter method as it does not seek any approximate
density function in parametric form. However, Particle filter method is more computationally expense than Kalman filter based method. In comparison to Kalman filters, they are
more general and applicable to systems where model and measurement equations are highly
nonlinear.
ii
Contents
Acknowledgements
i
Abstract
ii
Contents
vi
List of Figures
vi
List of Abbreviations
viii
1 Introduction
1
1.1
Structural system identification . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Application of system identification . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Books and review papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Dynamic State Estimation
10
2.1
Bayesian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2
Bayesian Model Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2.1
Kalman filter (Kalman 1960)
. . . . . . . . . . . . . . . . . . . . . .
14
2.2.2
Extended Kalman filter (EKF) . . . . . . . . . . . . . . . . . . . . .
15
2.2.3
Unscented Kalman filter (UKF) . . . . . . . . . . . . . . . . . . . . .
16
2.2.4
Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2.4.1
Perfect Sampling and Sequential Importance Sampling (SIS)
17
2.2.4.2
Sequential Importance Resampling (SIR) and Bootstrap Filter 21
2.3
Hamiltonian Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3.1
General Description . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.3.2
Books and review papers . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.3.3
Overview of HMC Method . . . . . . . . . . . . . . . . . . . . . . . .
27
iv
Guides and tutorials
2.4
Overleaf
2.3.4
Time Evolution of the Hamiltonian System . . . . . . . . . . . . . . .
28
2.3.5
HMC for Model Updating . . . . . . . . . . . . . . . . . . . . . . . .
29
Brief model and analysis from the implementation of particle filter . . . . . .
30
2.4.1
System Identificatio of single-degree-of-freedom (SDOF) oscillator . .
30
2.4.2
System Identification of Linear Time Invariant (LTI) Synthetic model
31
3 Research proposal
40
3.1
Gap Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.2
Scope of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.3
Objectives of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
References
47
Page v
List of Figures
1.1
Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Schematic flowchart of system identification (Source:Soderstrom (2001)) . . .
4
2.-1 System identification of SDOF oscillator excited by Elcentro earthquake . . .
34
2.0
Parameter values for solving the forward problem . . . . . . . . . . . . . . .
36
2.1
System identification of LTI System . . . . . . . . . . . . . . . . . . . . . . .
39
vi
List of Abbreviations
SSI
SHM
KF
EKF
PCE
GA
RMS
MLS
MPP
HDMR
RSM
FEA
FEM
MC
MCMC
MCMC
MH
Structural System Identification
Structural Health monitoring
Kalman Filter
Extended Kalman Filter
Polynomial Chaos Expansion
Genetic Algorithm
Root Mean Square
Moving least square
Most probable failure point
High dimensional model representation
Response surface method
Finite Element Analysis
Finite Element method
Monte Carlo
Markov Chain Monte Carlo
Transitional Markov Chain Monte Carlo
Metropolis-Hastings
viii
Introduction
1.1
Structural system identification
Structural system identification (SSI) is the field of mathematical modeling of the inverse
problem from the experimental data. It consists of determination of properties of a structure
based on a set of measurements on system response and applied actions. It has acquired
widespread applications in several areas like controls and systems engineering where system
identification methods are used to get appropriate models for synthesis of a regulator, design
of prediction algorithm and in signal processing applications (such as in communications,
geophysical engineering and mechanical engineering). Models obtained by system identification are used for spectral analysis, fault detection, pattern recognition, adaptive filtering,
linear prediction and other purposes. These techniques are also successfully used in other
fields such as biology, environmental sciences and econometrics to develop models for increasing scientific knowledge on the identified object, or for prediction and control. These
system identification problems constitute an important class of inverse problems in structural mechanics and form a crucial step in structural health monitoring (SHM) A dynamic
system can be conceptually described in Fig 1.1. The system is driven by user controlled
input variables u(t) while disturbances v(t) cannot be controlled. The output y(t) provide
useful information about the system. There are several kinds of mathematical models used
Figure 1.1: Dynamic System
for solving the inverse problem which are mostly governed by the underlying differential
1
equations. The mathematical models can be segregated into two paradigms
• Modeling, which refers to derivation of models from the basic laws of physics. Often,
one uses fundamental balance equations for range of variables like energy, force, mass
etc.
• Identification, which refers to the determination of the model parameters from the experimental data. It includes the set up of identification experiment i.e data acquisition
and determination of a suitable form of the model which is fitted to the recorded data
by assigning suitable numerical values to its parameters.
Though SSI methods are useful for large and complex structures where it is difficult to obtain
the mathematical models directly, it has some limitations. They have a limited validity i.e
they are valid for a certain working point, a certain type of input, a certain process,etc. SSI
is not a foolproof methodology that can be used without interaction from the user. The
reasons for this are
• 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
Therefore these complicating features, such as, ill-posedness of governing equations, presence
of measurement noise, modeling imperfections, structural nonlinearities, spatially incomplete
measurements, lack of measurements on applied actions, and possible need for online identification of system parameters, make problems of SSI very challenging. It may be emphasized
that these problems are studied essentially with respect to structures that already exist
and the powers of both mathematical and experimental modeling are available for solving
such problems. The recent advances in computational and sensing hardwares have spurred
research interest in this class of problems.
The two modeling approaches have flourished in structural engineering practice with
extensive developments taking place in the areas of computational mechanics [essentially the
finite element method (FEM)] and progress in experimental testing and analysis methods.
These developments also have important bearing on studies on existing structures which provide the best opportunities to combine the two modeling approaches. Problems of structural
system identification sprawl at the forefront of this class of problems.
2
1.2
Application of system identification
In general terms an identification experiment is performed by exciting the system and observing its output over an interval of time. These signals are normally recorded in a computer
mass storage for subsequent information processing. We then try to fit a parametric model
of the process to the recorded input and output sequences. 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.
Fig 1.2 shows the schematic of the steps used in system identification.
Following the above discussion, there are two main purposes of model updating or
system identification of the structural system. The common goal is to identify the physical
parameters e.g. (stiffness) of a structural element. These identified parameters can further
be used as indicator for the status of the system. For example, stiffness parameter of a
structural member can be monitored from time to time and an abnormal reduction indicates
possible damage of the member.But this reduction may also be simply due to statistical uncertainty. Hence the quantification of uncertainty becomes important. Another purpose of
model updating can be to obtain a mathematical model to represent the underlying system
for future prediction. This is broadly known as Structural Health Monitoring. Another important area of application of system identification is structural vibration control which has
received great attention in the last several decades (Housner et al.,[1]). The various ingredients in problems of SSI are: excitation signals, response quantities measured, mathematical
model for the system being identified, and model validation. Depending on the manner in
which these ingredients are included in the identification procedure, different schemes for
SSI are possible.
1.3
Books and review papers
Structural System identification has remained an active area of research for several decades
with civil engineering researchers showing interest from 1970-s. Many researchers have come
up with various methods and have solved several problems ranging from experimental models
to real life large scale structures. There exists several monographs and review papers that
address various topics within this discipline of research and here we make brief mention
3
Figure 1.2: Schematic flowchart of system identification (Source:Soderstrom (2001))
4
about these works. The general approach can be divided in following categories.
• Conventional model-based approaches
• Time domain identification methods
• Biologically inspired approaches such as neural network and genetic algorithm
• Time-frequency based approaches using Wavelet, Hilbert Transform
• Chaos theory
Conventional model-based approaches for system identification typically use a computer model of the structure, such as a Finite-Element Method (FEM) model, to identify
structural parameters primarily from field or laboratory test data. Damage identification in
beams is a common theme in system identification. (Kim and Stubbs [2]) studied damage
identification of a two-span continuous beam using modal information. (Lee and Shin, [3])
detected the changes in the stiffness of beams based on a frequency-based response function. Model based system identification methods cannot be used effectively for large and
complicated real-world structures with nonlinear behavior.
For such cases, biologically-inspired or soft computing techniques such as Neural Networks[4], Genetic Algorithms (GA), or particle swarm optimization have been proposed as a
more effective approach. Nelles [5] has covered approaches based on neural networks, fuzzy
models, optimization and classical polynomial based methods (e.g., Volterra-series models).
(Franco et al., [6]) used an evolutionary algorithm to identify the structural parameters of a
10-DOF shear frame. (Raich and Liszkai [7]) used Genetic algorithm to identify the stiffness
changes in a steel beam and a 3-story, 3 bay frame. for nonlinear SSI problems.
In the past two decades because of their ability to retain both time and frequency
information, wavelets have been used increasingly to solve complicated time series pattern
recognition problems in different areas. (Liew and Wang [8]) used wavelets to identify cracks
in simply supported beams. Worden and Tomlinson [9] cover wide ranging topics in nonlinear structural dynamical systems; these topics cover higher order spectral analysis, Hilbert
transform technique, force state mapping, and time series models. The book also covers experimental studies in the area of nonlinear SSI.(Bao et al. [10]) employed the Hilbert-Huang
transform for system identification of concrete-steel composite beams. Mahato et al.[11]
developed a continuous wavelet transform based filtering strategy combined with Hilbert
transform for modal identification of reinforced concrete road bridge with vehicular excitation. The book by Bendat [12] focuses on frequency domain methods for nonlinear SSI and
5
discusses the use of one-dimensional spectral functions arising in multiple input and single
output nonlinear systems.
A few researchers have employed the chaos theory and fractal concept [13] to model
the complicated structural dynamics for system identification[14, 15]. However, the emphasis of discussion mainly focus on the literature review of the time domain dynamic state
estimation methods which directly utilizes the response time histories to detect structural
damage. These methods are particularly appropriate for situations with time-varying structural properties and where sequential data set is observed. The dynamic state estimation
methods derive their origin from the Bayesian Methods. Bayesian theory was originally discovered by the British researcher Thomas Bayes in a publication (Bayes[16]).The methods
have been widely used in many areas due to the pioneering work done by Thomas Bayes. The
modern form of the theory was rediscovered by French mathematician Simon de Laplace in in
Theorie Analy-tiquedes Probabilites. One of the earliest researches in iterative Bayesian estimation can be found in (Ho and Lee,[17]). (Spragins,[18]) discussed the iterative application
of Bayes rule to sequential parameter estimation and called it as ”Bayesian learning”. The
subject of dynamic state estimation has close links with problems of system identification.
The methods for dynamic state estimation can be categorized into two groups.
The first includes the well-known Kalman filter (Kalman[19]) and its variants[20,21],etc.
and the other is the Monte Carlo simulation based algorithms named as particle filters (Gordon et al.[22]). The works of Jazwinski [23], Maybeck [24], Kailath [25], Brown and Hwang
[26], Chui and Chen [27], and Grewal and Andrews [28] provide comprehensive accounts of
the Kalman filters (KF). The KF represents recursive algorithms to exactly compute the evolution of mean and covariance of system states conditioned on measurements for linear state
space models with additive Gaussian noise models. It provides an exact solution to the problem of state estimation for linear Gaussian state space models. The most popular variant of
Kalman filter is the Extended Kalman filter (EKF), where linearization of the process equation is done to provide a Gaussian approximation of what really is a non-Gaussian quantity
(Hoshiya and Saito[20]). Ghanem and Shinozuka[29] provided a review of methods of system identification by application to experimental data obtained on three and five-story steel
building structures subjected to seismic loading, including the EKF, maximum-likelihood
technique, recursive least-squares, and recursive instrumental-variable method. (Moaveni et
al., [30]) examined six variations of model-based approach including, data- driven stochastic
subspace identification, frequency domain decomposition, observer/Kalman filter identification, and general realization algorithm for system identification of a full-scale 7-story RC
building structure subjected to shake table loading, and concluded that probabilistic system
identification methods in connection with FE model updating provide the most desirable
6
results. Sen and Bhattacharya [31]coupled generalised Polynomial Coas with EKF algorithm in which the uncertainty propagation from parameter to measurement was described
through gPC expansion of parameters and outputs. Subsequently, the gPC coefficients of
the parameter expansion are estimated from available measurements employing EKF. Thus,
instead of selecting the system parameters as states, we consider the associated parameter
gPC coefficients as state variables which reduces the problem of estimating the complete
distribution of parameters down to identification of a few gPC coefficients. The proposed
method is tested on systems with either Gaussian or non-Gaussian parameters. The error
in estimating non-Gaussian parameters using KF based techniques is demonstrated.
The second group of methods known as the Monte Carlo methods begin by considering
the exact form of the recursive integral equations that govern the evolution of filtering
pdf and employ Monte Carlo simulation procedures to solve these equations in a recursive
manner by approximating the complex integrals. Monte Carlo (MC) methods are stochastic
computational algorithms and these are efficient for simulating the highly complex systems.
The MCB approach was conceived by Ulam [32], developed by Ulam and von Neumann [33],
and coined by Metropolis([34])(Candy[35]). The technique evolved during the Manhattan
project in 1940-s, when the scientists were investigating the calculations related to atomic
weapon designs. MC methods have wide variety of applications in engineering and finance.
It offers alternative approach to solve numerical integration and optimization problems.The
application of these methods to problems in structural mechanics is not yet widely explored.
This more general class of problems solving involving nonlinear process and measurement
equations, multiplicative/additive non-Gaussian noises implemented using MC methods has
been studied by Tanizaki [36, 37], Doucet et al.,[38], Liu and West (2001)[39] and Ristic
et al., [40]. Monte Carlo simulation methods are known variously as Monte Carlo filters,
Bayesian filters, population Monte Carlo algorithms, or particle filters (Gordon et al.,[22],
Tanizaki [36], Doucet [41], Liu and Chen [42], Pitt and Shephard [43], Doucet et al., [44], Iba
[45], Doucet et al., [38], Crisan and Doucet [46], Ristic et al., [40], Cappe et al., [47], Doucet
and Johansen [48]) and [49]. There are several variants of the particle filters available in
the literature as well (Chen[50]). These methods have been widely used in robotics and for
solving the tracking problems (Thrun[51]). The following passage describes the work done
by the researchers on implementation of these methods to structural mechanics.
Ching et al., [52]compared performance of EKF and particle filter by applying them
on planar four-story shear building with time-varying system parameters and non-linear
hysteretic damping system with unknown system parameters. The mass of the shear building
is assumed to be time invariant with time varying stiffness and damping parameters at each
floor level and applied synthetic data contaminated with noise. The non-linear SDOF Bouc7
Wen hysteretic damping system model was considered. He concluded that Particle filter is
advantageous over EKF and EKF may sometimes create misleading results. Further EKF is
not suitable for highly non-linear models. Manohar and Roy,[53] identified the parameters
of nonlinear structures using dynamic state estimation techniques. They considered two
single-degree of freedom nonlinear oscillators, namely, the Duffing oscillator and the one
with Coulomb friction damping. They identified of parameter alpha and mu based on noisy
observations using the density based MC filter, bootstrap filter and sequential importance
sampling filter. The basic objective of the study was to construct the posterior pdf of the
augmented state vector based on all available information.
Nasrellah and Manohar, [54] did the combined computational and experimental study
using multiple test and sensor data for structural system identification. They considered the
problem of identification of parameters of a beam with spatially varying density and flexural
rigidity as well as the identification of parameters of a rigidly jointed truss. It was concluded
that various factors affect the accuracy of identification like number of particles used in
filtering, closeness of the initial guess on system parameters to the true values, number
of global iterations, noise levels in measurements and model,imperfections, the number of
parameters to be identified and sensitivity of measurements with respect to the parameters
being identified.
Radhika and Manohar [55] The authors presented a approach for state estimation
based on both Kalman filter and Sequential Monte Carlo filter algorithm. Authors utilizes
the concept of substructure scheme to problems of hidden state estimation in structures with
local non-linearities, response sensitivity model updating in nonlinear systems,and characterization of residual displacements in instrumented inelastic structures. Author divided the
structure in linear and nonlinear substructure and solved the linear estimation problem with
Kalman filter and nonlinear problem with particle filter method. It was illustrated with the
help of numerical examples and the performance of the solution from the proposed scheme
is compared with exact solutions and with alternative approximate solutions which do not
employ sub-structuring, and the results are shown to be in satisfactory agreement.
Sen et al. [56] have developed a conceptual framework taking the input excitation
force to be an additional state that is estimated in parallel to the structural parameters
and employed two concurrent filters for parameters and force respectively. They used two
filters one for the parameters, an interacting Particle-Kalman filter to target systems with
correlated noise and another to estimate the seismic force acting on the structure. For
stability in the estimation the parameters and the inputs estimated as being conditional on
each other. They validated numerically the proposed algorithm on a 6-DOF mass-spring8
damper system and a five-story building structure. Thier estimation results confirmed the
applicability of the proposed algorithm.
Similar studies were carried out by (Namdeo and Manohar, [57]), (Ghosh et al.,[58])
and (Sajeeb and Roy,[59]) ,[60], Radhika and Manohar [61] and Sen et al. [62].The particle
filter algorithm has also been used for identification of fatigue cracks in vibrating beams
(Rangaraj[63]).[64] the following sections briefly outlines the development of some of above
strategies in the following sections. [65]
9
Dynamic State Estimation
The problem of dynamic state estimation (DSE) can be stated by considering three entities:
1. the postulated mathematical model for the system behavior encapsulated through a
process equation of the form
xk+1 = hk (xk , θ, vk );
k = 0, 1, 2, ....
(2.1)
where k is a independent variable (typically discretized time variable) along which the
n×1 system state vector system xk evolves; hk is a nx ×1 vector function: vk is a nv ×1
vector of random variables such that the sequence {vk },k = 1, 2, ... from an independent, identically distributed (iid) sequence of random variables with {vk } ∼ p(vk ); here
p(k )is the n dimensional joint probability density function (pdf ) of k ; thesymbol ∼ is
taken to mean ‘is distributed according to’; and θ is a p×1 vector of system parameters,
2. prior pdf of x0 , p(x0 ) at initial time step k = 0, and
3. a measurement equation of the form
yk = qk (xk , θ, wk );
2.1
k = 1, 2, ...
(2.2)
Bayesian Methods
In many of the engineering problems, modeling of uncertain parameters is necessary for various purposes. In this context, Bayes Theorem offers the framework of modeling and inferring
the uncertain models from the measurements. These methods have been applied to many
different disciplines of natural sciences, social sciences and engineering, especially in statistical physics, engineering hydrology, econometrics, archaeology, information sciences, medical
10
sciences, forensic sciences, marketing, mechanical engineering, computer science, engineering geology, aerospace engineering, finance, population migration, and many other areas.
In Structural Engineering , these methods have been used in system reliability, prediction
of concrete strength , structural dynamics and system identification. Bayesian inference is
very important for Structural engineering applications because of the wide variety of uncertainty associated with the structures. The examples of such uncertainties can be earthquake
ground motion or complete time varying description of the wind pressure, material properties which are difficult to determine for heterogenous materials like concrete and the number
and size of cracks present in concrete. Not only this, the modeling errors and uncertainties are also associated with the joints.Therefore, Bayesian statistics has wide application
in Structural engineering as well. In many scenarios, the solutions gained through Bayesian
inference are viewed as “optimal”.
2.2
Bayesian Model Updating
Many real world data analysis tasks involve estimating the unknown quantities from some
given observations. In such type of problems, generally the prior knowledge about the
phenomenon to be modeled is available. This knowledge can be used to formulate the
Bayesian models where prior knowledge about the state is updated using the likelihood
function to generate a posterior distribution. Often the measurements arrive sequentially
and it is possible to both carry out the offline as well as online inferences. Thus, one
of the most important steps of this process is to update the states recursively once the
measurements are available.The focus of dynamic state estimation techniques is to estimate
the state of the system using the measurement data. The governing equation can be written
as:
X(t) = q(P (t), t)
(2.3)
where X(t) is the response of the structure when an input force P (t) is given to the system
and q(.) relates the input to the output. Since the measurements are available at discrete
time steps, it becomes obvious to discretize the above model equation as
Xk+1 = qk (Xk , wk )
(2.4)
where Xk represents the state of the system at time t = k ; Xk+1 represents predicted state
at time t = k + 1 and wk represents the white noise.
The discretized measurement equation can be written as
Yk = hk (Xk , vk )
11
(2.5)
where Yk is the measurement at time t = k corresponding to the state Xk and vk is the
measurement noise similar to the model noise. However the model as well the measurement
noise has been assumed as uncorrelated.
The measurements from the sensors are sampled at a particular rate and can be denoted
as a vector
Mk = [Y1 , Y2 , ., Yk ]
(2.6)
The objective of this formulation is to estimate the current state Xk based on the measurement Yk . As the model and the measurements are corrupted with noise it is required the
problem of state estimate reduces to estimating the probability density function p(Xk |Mk ).
Since estimating p(Xk|M k) is itself not easy, so the more simplified problem is to determine
the moments of Xk . Mathematically, this can be written as:
Z
µ = Xk p(Xk |Mk )dXk
Z
σ=
(Xk − µ)T (Xk − µ)p(Xk |Mk )dXk
(2.7)
(2.8)
where µ and σ are the first moment or mean and the second moment or variance of the pdf
p(Xk |Mk ) respectively.
In the following, a detailed derivation of the recursive Bayesian Estimation is presented,
which underlines the principles of sequential Bayesian filter. Two assumptions are used to
derive the recursive Bayesian Filter.
• The states follow a first order Markov process
p(Xk |X0:k−1 ) = p(Xk |Xk−1 );
(2.9)
• The observations are independent of the given states.
At any time t, the posterior is given by the Bayes theorem as
p(X0:t |Y1:t ) = R
p(Y1:t |X0:t )p(X0:t )
p(Y1:t |X0:t )p(X0:t )dX)
(2.10)
The recursive equation can be obtained as
p(X0:t+1 |Y1:t+1 ) = p(X0:t |Y1:t )
12
p(Yt+1 |Xt+1 )p(Xt+1 |Xt )
p( Yt+1 |Y1:t )
(2.11)
The following recursive relations are used for prediction and updating The prediction equation is given by
Z
p(Xt |Y1:t−1 ) =
p(Xt |Xt−1 )p(Xt−1 |Y1:t−1 )dXt−1
(2.12)
Based on this prediction the model updating equation is
p(Xt |Y1:t ) = R
p(Yt |Xt )p(Xt |Y1:t−1 )
p(Yt |Xt )p(Xt |Y1:t−1 )dXt
(2.13)
It is however difficult to compute the normalizing constant p(Y1:t ) and the marginal
of the posterior p(X0:t |Y1:t ) as it requires evaluation of complex high dimensional integrals.
The above expressions are modified in the following way when the system and the model
noise are also present.
Adopting the notations as p(Xk |Yk−1 )is the estimate of the state at time k based on
the measurements Yk−1 and p(Xk |Yk ) denotes the pdf of the state at time k based on the
measurements Yk . Therefore, the first one is the priori pdf or the prediction, while the latter
is the posteriori pdf or the correction to the state once the measurements are available at
time k. It is also assumed that
p(X1 |M0 ) = p(X1 )
is known. The prediction equation can be expressed as:
Z
p(Xk |Mk−1 ) = p(Xk |Xk−1 )p(Xk−1 |Mk−1 )dXk−1
(2.14)
(2.15)
Here p(Xk |Xk−1 ) can be derived from 2.6. The conditional density can be used to write the
following expressions.
Z
p(Xk |Xk−1 ) =
p(Xk |Xk−1 , wk−1 )p(wk−1 |Xk−1 )dwk−1
(2.16)
Since wk is independent of the state, it can be written that
p(wk−1 |Xk−1 ) ≡ p(wk−1 )
(2.17)
It can be clearly seen from the process equation that if Xk−1 and wk−1 are known, then
Xk can be obtained deterministically from the process equation 2.6. Therefore the pdf of
p(Xk |Xk−1 , wk−1 ) can be mathematically written as
p(Xk |Xk−1 , wk−1 ) ≡ δ(Xk − fk−1 (Xk−1 , wk−1 ))
(2.18)
where δ(.) is the Dirac-Delta function. Substituting in this in the above Eq. 2.16 we get
Z
p(Xk |Xk−1 , wk−1 ) = δ(Xk fk−1 (Xk−1 , wk−1 ))p(wk−1 |Xk−1 )dwk−1
(2.19)
13
The above expression can be substituted in Eq. 2.12
As soon as the measurement Yk is available at the time step k the prediction can be
updated using the Bayesian relation
p(Xk |Mk ) =
p(Yk |Xk )p(Xk |Mk−1 )
p(Yk |Mk−1
(2.20)
where the normalizing denominator is given by
Z
p(Yk |Mk−1 ) = p(Yk |Xk )p(Xk |Mk−1 )dXk
(2.21)
The only unknown in the Eq. 2.20 is p(Yk |Xk ) which can be obtained as:
Z
p(Yk |Xk ) = p(Yk |Xk , vk )p(vk )dvk
(2.22)
which again takes the form of the Dirac-Delta function if Xk and vk are known. The measurement Yk is obtained from the measurement Eq. 2.5. Thus the above equations form the
basis of the recursive Bayesian Model updating.
If the functions f (.) and h(.) are linear and the noise wk and vk are Gaussian; then
the closed form expressions of the above integrals are available and this leads to the wellknown Kalman Filter,[19]. However if the f (.) and h(.) are non-linear, then several other
methods have been prescribed in literature like EKF, [20]. However the most recent interest
is to exploit the cheap and faster computational facilities to develop methods based on the
Monte Carlo Simulations for approximating the integrals in the above equations.
2.2.1
Kalman filter (Kalman 1960)
Here we consider the following version of equations 2.12 and 2.13
xk+1 = Hk xk +k ;
k = 0, 1, ...
(2.23)
yk = Qk xk + wk ; k = 1, 2, ...
vk
0
Σv 0
∼N
,
wk
0
0 Σw
(2.24)
(2.25)
Here N (µ, Σ) denotes a multi-normal pdf with mean vector µ and covariance matrix Σ. The
sizes of the quantities xk , Hk ,k , yk , Qk ,and wk appearing in the above equation are respectively ,n × 1, nx × nx , nx × 1, ny × 1, ny × nx andny × 1. Accordingly, the covariance matrices
Σv and Σw have dimensions nx nx and ny ny , respectively. It is assumed that x0 ∼ N (µ0 , P0 )
14
is known and is independent of k and wk . It is clear that p(xk+1 |xk ) ∼ N (Hk xk , σ) and
p(yk |xk ) N (Qk , xk , Σ). Based on these facts, the functional recursive relations in equations
2.12 and 2.13 can be shown to lead to the following set of recursive relations for the evolution of conditional mean and covariance of the system states (see, for example, Tanizaki
[36], Brown and Hwang [26]).
Kk = Pk− Qtk [Qk Pk− QTk + Σw ]−1
x̂k = x̂k + Kk (yk − Qk x̂k )
Pk = [I − Kk Qk ]Pk−
(2.26)
x̂k+1 = Hk x̂k
−
Pk+1
= Hk Pk Hk + Σv
Here a hat (.̂) denotes the estimate and a superscript ‘−′ denotes that the estimate is evaluated prior to assimilation of data at time tk ; the superscript t denotes matrix transposition;
Pk denotes the covariance of estimation error ek = xk − x̂k ; Kk is the blending matrix (gain
matrix) that relates the updated estimate x̂k to the measurement yk through the relation
x̂k = x̂k + Kk [yk − Qk x̂k ] . In this model p(ck |y1:k ) ∼ N (x̂k , Pk )
2.2.2
Extended Kalman filter (EKF)
The extended Kalman filter (EKF) offers an approximate solution to the filtering problem
governed by the following equations
xk+1 = hk (xk ) + wk ; k = 0, 1, 2, ...
yk = qk (xk ) + vk ; k = 1, 2, ...
vk
0
Σv 0
∼N
,
wk
0
0 Σw
(2.27)
Here vk and wk are as in the preceding section. The basic idea here is to linearize the
functions hk (xk ) and qk (xk )around a reference state xR , that is, we approximate
h(xk ) = hk (xR + xk − xR ) ≈ hk (xR ) + ∇hk |xk =xR (xk − xR )
q ( xk ) = qk (xR + xk − xR ) ≈ qk (xR ) + ∇qk |xk =xR (xk − xr )(2.28) Accordingly, we replace
the model in Eq. 2.12 by the approximation
xk+1 = hk (xR ) + ∇hk |xk =xR + wk ;
yk = qk (xR ) + ∇qk |xk =xR (xk − xR )k = 1, 2, ...(2.29)
15
k = 0, 1, 2, ...
Here ∇hk and ∇qk are, respectively, nx × nx and ny × nx Jacobian matrices given by
∇hk (i, j) =
δqk (i)
δhk (i)
and ∇qk (i, j) =
δxk (j)
δxk (j)
(2.30)
The problem now is amenable for solution via the Kalman filtering. Typically, at the kt h step,
one selects xR = x̂k − 1 , that is, as the assimilation proceeds, we apply local linearization
on the fly around the evolving trajectory of x̂ˆk . The development of this method in DSE
problems has been discussed by several authors (see, for instance, the books by Jazwinski
[23], Maybeck [24], Brown and Hwang [26] and Grewal and Andrews [28]). The estimate
provided by EKF is biased and the accuracy of its performance, in a given problem, is difficult
to assess. The estimate does not permit interpretation as being minimum mean square error
estimator. The method assumes that the functions hk (xk ) and qk (xk ) are differentiable
with respect to xk - a condition that may not always be satisfied in problems of nonlinear
structural mechanics (e.g., in dealing with friction, gaps and hereditary nonlinearities). The
recent work of Ghosh et al., [58] presents an EKF method based on concept of transversal
linearization which avoids the computation of the Jacobian matrices. It may also be noted
that the idea of linearization can be extended to include higher order terms in Taylor’s
expansion in equation (1.19). Here one needs to make a Gaussian closure approximation to
handle the resulting recursions for x̂P and Pk Details of these formulations can be found in
the work of Tanizaki [36].
2.2.3
Unscented Kalman filter (UKF)
This algorithm is based on the notion of an unscented transform which is a deterministic
sampling technique to select realizations of random variables with a view to determine moments of functions of these random variables. In the context of nonlinear filtering problems,
this method avoids the linearization of the nonlinear functions that appear in process and
(or) measurement equations: instead, the method postulates Gaussian model for the posterior pdf and aims to estimate the mean and covariance of the transformed variables with
higher accuracy than is possible in extended Kalman filtering. The method thus avoids the
computation of gradients and, hence, could be used in dealing with nonlinearities that are
not differentiable. The deterministic points here are termed as sigma-points and the filtering
algorithm propagates these sigma points as measurements are assimilated. The posterior
mean and covariance matrix of the system states are computed based on these propagated
control points and associated weights. These response moments are shown to be accurate
upto the second order.
16
2.2.4
Monte Carlo Methods
The underlying principle of the MC methods is that they utilize Markov chain theory. The
resulting empirical distribution converges to the desired posterior distribution through random sampling. The method is widely used in signal processing where one is interested
in determining the moment of the stochastic signal f (X) with respect to some underlying
probabilistic distribution p(X). However the similar concept is used in system identification
problem where one is interested to estimate the expected values of the system parameters.
The methods have the great advantage since these are not subject to constraints of linearity and Gaussianity. The methods as well have appealing convergence properties. Several
variants of MC methods are available in the literature. This includes Perfect Monte Carlo
sampling,Sequential importance sampling, Sequential importance resampling and the Bootstrap particle filter. The following section presents the mathematical formulation of each of
the method. The concept has been illustrated by solving single degree of freedom oscillator
at the end of the chapter.
2.2.4.1
Perfect Sampling and Sequential Importance Sampling (SIS)
Monte Carlo methods use statistical sampling and estimation techniques to evaluate the
solutions to mathematical problems. The underlying mathematical concept of Monte Carlo
approximation is simple. Consider the statistical problem of estimating the expected value
of E[f (x)] with respect to some probabilistic distributionp(X):
Z
E[f (X)] = f (X)p(X)dX
(2.31)
Here the motivation is to integrate the above expression using stochastic sampling techniques
rather than using the numerical integration techniques. Such a practice is useful to estimate
complex integral where it is difficult to obtain the closed form solution. In MC approach, the
required distribution is represented by random samples rather than analytic function. The
approximation becomes better and more exact when the number of number of such random
samples increases. Thus, MC integration evaluates Eq. 2.31 by drawing samples X(i) from
p(X). Assuming perfect sampling, the empirical distribution is given by
N
1 X
p(x) =
δ(X − X(i))
N i=1
The above equation can be substituted to give
Z
N
1 X
E[f (x)] = f (X)p(X)dX ≃
f (X(i))
N i=1
17
(2.32)
(2.33)
Generalization of this approach is known as Importance sampling where the integral is written as
Z
I=
Z
p(x)dx =
where
p(x)
q(x)dx
q(x)
(2.34)
Z
q(x)dx = 1
(2.35)
Here q(X) is known as the importance sampling distribution since it samples p(X) nonuniformly giving more importance to some values of p(x). The Eq. 2.34 can be written
as
N
p(X)
1 X p(X(i))
I = Eq
=
q(X)
N i=1 qX(i))
(2.36)
where X(i) are drawn from the importance distribution q(.).
The central theme of importance sampling is to choose importance distribution q(.)
which can approximate the target distribution p(.) as close as possible. Using the concept
of importance sampling, it is possible to approximate the posterior distribution. Since it is
generally not easy to sample from the posterior, we use importance sampling coupled with
an easy to sample proposal distribution q(Xt |Yt ).This is one of the most important steps
of the Bayesian importance sampling methodology. Using the importance sampling concept
the mean of f (Xt ) can be estimated as follows:
Z
E[f (Xt )] = f (Xt )p(Xt |Yt )dXt
(2.37)
where (Xt |Yt ) is the posterior distribution. Here, we insert the importance proposal density
function q(Xt |Yt ) such that the estimate becomes
Z
p(Xt |Yt )
F (t) = E[f (Xt )] = f (Xt )
q(Xt |Yt )dXt
q(Xt |Yt
(2.38)
Now using Eq. 2.20 (Bayes Rule) to the posterior distribution and defining the weighting
function as
p(Xt |Yt )
q(Xt |Yt )
p(Yt |Xt )p(Xt )
=
p(Yt )q(Xt |Yt )
W̃ (t) =
(2.39)
Calculation of W̃ (t) requires the knowledge of the normalizing constant p(Yt ) which is given
by
Z
p(Yt ) =
p(Yt |Xt )p(Xt )dXt
18
(2.40)
This normalizing constant is generally not available and hence the new weight W (t) can be
defined by substituting Eq. 2.39 into Eq. 2.38.
Z
1
p(Yt |Xt )p(Xt )
F (t) =
f (Xt )
q(Xt |Yt )dXt
p(Yt )
q(Xt |Yt )
Z
1
=
W (t)f (Xt )q(Xt |Yt )dXt
p(Yt )
1
Eq [W (t)f (Xt )]
=
p(yt )
(2.41)
The above equation can be also be written as:
W (t)q(Xt |Yt ) = p(Yt |Xt )p(Xt )
(2.42)
Thus, the normalizing constant in Eq.2.40 can be replaced by Eq.2.42
Eq [W (t)f (Xt )]
p(Yt )
Eq [W (t)f (Xt )]
= R
W (t)q(X|Y )
Eq [W (t)f (Xt )]
=
Eq [W (t)]
F (t) =
(2.43)
Now, if the samples are drawn from the distribution q(Xt|Y t), from perfect sampling distribution we have
N
1 X
q̃ =
δ(X − X(i))
N i=1
(2.44)
and therefore, the normalized weights w̃i of the it h sample can be written as
W i(t)
w̃i = PN
i=1 Wi (t)
where
W i(t) =
p(Yt |Xti )p(Xti )
p(Yt )q(Xti |Yt )
(2.45)
(2.46)
Therefore the final estimate of the Eq. 2.38 becomes
F (t) ≈
N
X
w̃i f (Xt (i))
(2.47)
i=1
As the number of samples (N → ∞), the approximation of posterior becomes
p(Xt |Yt ) ≈
N
X
w̃i δ(Xt − Xt (i))
i=1
19
(2.48)
With the above mathematical framework in place, we can derive the expressions for sequential interfacing of measurement data available at time instant t = k. One can write the
approximation of posterior as:
p(Xk |Y1:k ) ≈
N
X
w̃ki δ(Xk − Xk (i))
(2.49)
i=1
where δ(.) is the dirac delta function and w̃i is the normalized weight of the it h particle at
time k.
p(X0:k |Y1:k ) ∝ p(Yk |X0:k , Y1:k−1 )p(X0:k |Y1:k−1 )
= p(Yk |Xk )p(Xk |X0:k−1 , Y1:k−1 )p(X0:k−1 |Y1:k−1 )
(2.50)
= p(Yk |Xk )p(Xk |Xk−1 )p(X0:k−1 |Y1:k−1 )
i
We could now construct an importance distribution X0:k
∼ q(X0:k |Y1:k ) and compute
the corresponding (normalized) importance weights as
i
i
p(Yk |Xki )p(Xki |Xk−1
)p(X0:k−1
|Y1:k−1 )
i
w̃k ∝
1
q(X0:k
|Y1:k )
(2.51)
The recursive form of the importance distribution can be written as:
q(X0:k |Y1:k ) = q(Xk |X0:k−1 , Y1:k )q(X0:k−1 |Y1:k−1 )
(2.52)
Substituting Eq. 2.52 in Eq. 2.51 we obtain the following expression
w̃ki =
i
i
p(Yk |Xki )p(Xki |Xk−1
)p(X0:k−1
|Y1:k−1 )
i
i
i
q(Xk |X0:k−1 , Y1:k )q(X0:k−1 |Y1:k−1 )
(2.53)
Thus the recursive weight can be given as:
w̃ki ∝
i
p(Yk |Xki )p(Xki |Xk−1
) i
w̃
i
i
q(Xk |X0:k−1 , Y1:k ) k−1
(2.54)
So, the algorithm works the following way
• Initilization: Draw N samples X0i from the prior
X0i ∼ p(x0 )
(2.55)
• Prediction: Draw N new samples Xki from importance distribution
X ∼ q(Xk |X0:k−1i , Y1:k )
20
(2.56)
• Update: Calculate new weights according to Eq. 2.54. Once the weights are updated
the posterior can be calculated using Eq. 2.39
One of the major problems associated with SIS Filter is the degeneracy 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.
A suitable measure of the degeneracy of the algorithm is the effective sample size (Gordon
et al.,[22]) Nef f which can be defined as
Ne f f =
Ns
1 + V ar(wk∗i )
(2.57)
where wk∗i can be obtained from Eq.2.39 The estimate of Nef f is given by the following
relation
Ñef f = PN
1
i 2
i=1 (w̃k )
(2.58)
where w is the normalized weight obtained using the Eq. 2.54 When Nef f becomes less than
N ; it implies degeneracy and a small Nef f indicates severe degeneracy. Therefore to counter
this (Arulampalam et al.,[66]) suggested two ways
• Good choice of Importance density: This involves the choosing the importance density
such that the V ar(w∗i ) can be reduced and hence the value of Nef f increases.
• Resampling: This is another important step which differentiates SIR filter from SIS
filter and has been discussed in detail in the following section.
Both of the above issues form the basis of “Sequential Importance Resampling” also
known as “Adaptive Particle Filters” and have been discussed the following section.
2.2.4.2
Sequential Importance Resampling (SIR) and Bootstrap Filter
The SIR filter is an MC method which can be applied to recursive Bayesian filtering problems.
To use SIR algorithm, both the state dynamics Eq. 2.3 as well as the measurement equations
2.43 must be known. Further it is required to be able to sample from the noise distribution of
the process as well as from the prior. A likelihood functions p(Yk |Xk ) needs to be known for
computing the particle weights. SIR algorithm is very similar to SIS filter except the choice
of optimal importance density as well as Resampling step included in the SIR algorithm.
21
The SIR algorithm can be easily derived from SIS algorithm by appropriate choice of
the importance density. The optimal importance density used in SIR is
i
i
q(Xki |X0:k−1
, Y1:k ) = p(Xki |Xk−1
, Yk )
(2.59)
By substituting Eq. 2.59 in Eq. 2.54 the updated weight becomes
i
i
wki ∝ wk−1
p(Yk |Xk−1
),
(2.60)
This optimal importance distribution can be used when the state space is finite. The present
report also uses the similar assumption of importance density. However, the report deals
with the problem of system identification where we are more interested in identifying the
system parameters rather then tracking the sate vector. The algorithm can be implemented
in the following manner
• Draw particles Xi from the importance distribution
i
Xki ∼ q(Xk |X0:k−1
, Y1:k ), i = 1, ..., N
(2.61)
• The new weights can be calculated from Eq. 2.54 for all the particles an normalize
them to unity.
• If Nef f calculated in Eq. 2.58 becomes too low, perform the resampling step.
• Interpret each weight wki as the probability of obtaining the sample index i in the set
Xki for [i = 1, . . . ,N].
• Draw N samples from that discrete distribution and replace the old sample set with
this new one.
• Set all weights to the constant value wki = N1 .
The Bootstrap filter is a special case of SIR filter where the dynamic model is used
as importance distribution as in Eq. 2.60 and the resampling is done at each step. A brief
algorithm is presented here for a more clear illustration. However, the problem formulation section gives the detailed implementation of Bootstrap filter to System identification
problem.
• Draw point Xki from the dynamic model
i
) , i = 1, ..., N
Xki ∼ p(Xk |Xk−1
22
(2.62)
• Calculate new weights and normalize them to unity.
wki ∝ p(Yk |Xki ) , i = 1, ..., N
(2.63)
• Perform resampling after each iteration.
One of the important steps in the above algorithm is resampling from the discrete probability
mass function containing the normalized weights. Resampling ensures that particles with
larger weights are more likely to be preserved than particles with smaller weights. Although
the resampling solves the degeneracy, but it introduces sample impoverishment. There are
wide variety of resampling algorithms available in the literature (Li, 2013[67]).
2.3
Hamiltonian Monte Carlo Method
2.3.1
General Description
HMC is a non-random-walk based MCMC method which employs a deterministic mechanism derived from the Hamiltonian dynamics to simulate the samples according to a target
distribution. It uses the concept of canonical ensemble from statistical mechanics in conjunction with the Hamiltonian principles to give a physical interpretation of the problem
with useful intuitions. A canonical ensemble is a probability distribution which assigns a
finite probability to a particle attaining a particular state of the system based on the energy
of the state. For a system, where x represents the state and E(x) is the energy function, the
canonical ensemble is defined as
1
E(x)
f (x) = exp −
Z
T
(2.64)
where, T is the temperature of the system and Z is a normalizing constant. If we consider
a Hamiltonian system such that it is entirely determined by its position (β) and momentum
(γ), then the total energy of the system can be defined in terms of the Hamiltonian as
H(β, γ) = U (β) + K(γ)
(2.65)
where, U (β) represents the potential energy of the system, which depends only on the position and K(γ) is kinetic energy, which depends on the momentum. For this system, the
canonical ensemble becomes
1
f (β, γ) = exp
Z
23
H(β, γ)
−
T
(2.66)
If the potential energy and kinetic energy of a system are defined such that the canonical ensemble represents a target probability distribution then the solution of the governing
Hamiltonian equations, describes the time evolution of the system. A more detailed description of the properties of this evolution can be found in [68–70].
2.3.2
Books and review papers
The groundbreaking work in this domain is done by Beck and Katafygiotis [71] and Katafygiotis et al. [72] where the authors extended the central idea of Bayesian inference for updating
the continuous-valued uncertain parameters θ using the measurement of continuous-valued
structural response D [73]. Through their continual research, the authors have developed a
general Bayesian statistical formulation for updating the model parameters and quantitative
assessment of their relative accuracy using Bayesian inference [68, 72–85]. In this approach,
the values of uncertain parameters θ, attaining a specific value is considered as a hypothesis
while the initial degree of belief towards this hypothesis is updated using measured structural
response D in terms of posterior probability which is conditioned by the structural behavior
and a class of mathematical model C such that
p(D|θ, C)p(θ|C)
p(θ|D, C) =
= K0 p(D|θ, C)p(θ|C)
p(D|C)
(2.67)
The term p(θ|D, C) is the updated or posterior pdf of the uncertain model parameters θ
based on the findings D (i.e. measurement) which provides a quantitative judgment of the
plausibility of a set of uncertain parameter values while the term p(D|θ, C) is called the
Likelihood function which quantifies the likelihood of obtaining the structural response D
for a specified set of model parameter values θ. The term p(θ|C) is called the prior pdf, as
it gives the initial plausibility of model parameters θ for a mathematical model in its actual
1
state when the actual structural response is not available. K0 = p(D|C)
is a normalizing
constant, such that integrating the right-hand side of the above equation over the parameter
space Θ gives unity i.e.
K0−1 =
Z
Z
p(θ|D, C)dθ =
Θ
p(D|θ, C)(p(θ|C)dθ
(2.68)
Θ
In the above formulation, exact evaluation of the posterior pdf involves a multidimensional integral which cannot be performed using analytical procedure. In early research work,
since the exact nature of this Bayesian predictive pdf is not known a priori, it is approximated by a Gaussian distribution centered at the optimal points which globally minimize
the objective function [73] and an algorithm for efficient searching of parameter space to find
these optimum points has also been proposed [71].
24
However, the asymptotic Gaussian approximation is feasible only for the identifiable
cases where a non convex global optimization exists while for locally identifiable and unidentifiable cases, finding these optimal points involve a series of local optimization problems.
Therefore, the computational efforts required to solve this problem grows with the dimension
of the parameter space which makes it infeasible for high-dimensional cases [86]. These factors of model identifiability and asymptotic approximation of the posterior pdf are studied
by Katafygiotis et al. [71], where the authors have also addressed practical issues like noise
and incomplete measurement due to limited number of sensors.
Early work in model updating is carried out using ambient time histories for updating
model parameters. Subsequently, the formulation is extended to incorporate load-dependent
Ritz vectors [87] and eigenvalues [88] as the parameter to detect damage. Here, it may
be noted that the Metropolis-Hastings (MH) algorithm [80] is often used to simulate the
samples of the random variable distributed according to an arbitrary continuous pdf . Hence,
this method finds its way into Bayesian model updating where samples for the successive
iterations are simulated by this method. It has been successfully implemented for model
updating, bypassing the computationally exhaustive optimization algorithms [86, 89].
However, since the high probability region of target pdf p(θ|D, C) is concentrated
in narrow bands of probability space, direct application of the MH technique to simulate
samples distributed as per target pdf may lead to higher rejection rate of the generated
samples and form a chain with large number of repeated samples which may leave the
chain into nonergodic state. These problems are dealt with various adaptive MH algorithms
developed in the recent past. The adaption is done either on the proposal pdf used to
generate the candidate state [75, 83, 90] or the target posterior distribution [81, 86, 91] to
obtain the information about the manifold in an iterative manner.
The later case involving the adaption over the target pdf uses a sequence of intermediate
pdf s with their spread such that they gradually converge to posterior pdf while maintaining
the proper acceptance rate. In this context, use of intermediate pdf s solves the problem
of high rejection rate in Markov chain caused due to narrow high probability region which
cannot be anticipated before-hand. Beck and Au [86] have proposed an adaptive MetropolisHastings (AMH) method which uses predefined sequence of intermediate pdf s and requires
kernel density estimation to approximate them which makes it inefficient for high dimensional problem. Ching and Chen [81] have proposed a new adaptive approach called Transitional Markov Chain Monte Carlo (TMCMC) where it uses the resampling to bypass the
kernel density estimation. Unlike AMH method [86], TMCMC can automatically generate
intermediate pdfs. The recent developments for efficient simulation techniques include the
algorithms developed using Darwinian theory in the genetic algorithm [92], Gibbs-sampling
25
algorithm [78], enhanced MCMC method [93] and Online model updating [94].
All the above mentioned Markov Chain based simulation techniques exhibit random
walk behavior while generating the candidate state, which needs higher time to discover the
area of significant probability content. Duane et al. [68] have presented a new technique for
the numerical simulation of lattice field theory called Hybrid Monte Carlo (aka Hamiltonian
Monte Carlo) technique. It uses a deterministic mechanism inspired by the principles of
Hamiltonian dynamics for simulating samples following a target distribution. It alleviates
random-walk theory with a consistent exploration of the probability space. Its application in
Bayesian model updating is first studied by Cheung and Beck [79], where the authors have
used this method for finite element model updating of a ten-storey building using simulated
data. In general, HMC offers faster convergence as compared to conventional random walk
based MCMC methods [79]. Boulkaibet et al. [77] have proposed an improved version of the
HMC method called Shadow Hybrid Monte Carlo technique where the sampling is done from
a modified Hamiltonian function instead of the conventional Hamiltonian function. In both
these versions of HMC algorithm, the authors have used Leapfrog Algorithm to generate the
candidate state where the gradient of the logarithm of posterior pdf acts as a guide to explore
the high probability region of the target pdf . However, numerical evaluations of the gradient
over all the uncertain parameters are computationally exhaustive and even infeasible for a
multi-dimensional problem involving large scale complex finite element model. Mbalawata et
al. [95] developed a parameter estimation method for Stochastic differential equations(SDEs)
based on the HMC and utilized EKF for evaluating the (approximate) marginal likelihood of
the parameters. Septier and Peters[96] provided a unifying framework for Sequential Markov
Chain Monte Carlo (SMCMC) approache and proposed inclusion of the principle of Langevin
diffusion and Hamiltonian dynamics in order to cope with the increasing number of highdimensional applications. Thier simulation results show that the proposed algorithms are
advantageous over existing algorithms. In the recent work Wang et al. [70] have proposed a
modified version of HMC algorithms where the authors have derived a closed form solution
for the simulation of Standard Gaussian random variables. The authors have also studied its
application in reliability analysis using subset simulation where this method has shown great
improvements over the standard MCMC algorithm. Baisthakur and Chakraborty [97] developed a Hamiltonian Monte Carlo-based algorithm for finite element model updating under
Bayesian framework by proposing adaptive prior-based approach to generate the intermediate pdf s. The authors endorse efficiency of this method using synthetic experiments and
actual test data for updating the finite element model of a steel truss bridge, suggesting performance of this algorithm as advantageous over the standard MCMC algorithm. Nikbakht
et al.[98] They quantified rare events probabilities by introducing a gradient-based Hamil26
tonian Markov Chain Monte Carlo (HMCMC) framework, termed Approximate Sampling
Target with Post-processing Adjustment (ASTPA). The basic idea was to construct a relevant target distribution by weighting the high-dimensional random variable space through a
one-dimensional likelihood model, using the limit-state function. To sample from this target
distribution they utilize HMCMC algorithms that produce Markov chain samples based on
Hamiltonian dynamics rather than random walks. The performance of this Quasi-Newton
based mass preconditioned HMCMC algorithm was found faster and superior over the conventional scheme. They further examined the performance of the proposed methodology
against Subset Simulation in a series of static and dynamic low- and high-dimensional benchmark problems. Zhang et al.[99] analysed the applied effect of MCMC simulation method.
They proposed a reliability assessment process based on MCMC simulation method with
MH Algorithm. The advantage of this method is to improve the application efficiency and
accuracy of reliability assessment based on bridge health monitoring data.
2.3.3
Overview of HMC Method
Developing a deterministic Monte Carlo method using the concept of Hamiltonian dynamics
requires constructing a Hamiltonian system which is mathematically equivalent to the probability space of interest. To establish such a relation, the variable (x) is viewed as position β
of the Hamiltonian system, and an auxiliary momentum variable γ is introduced to complete
the position-momentum phase space. The potential energy, V (γ), of the system is expressed
in terms of the target pdf as a canonical ensemble in the following form
V (β) ≡ − log(π(β))
(2.69)
Also, the Kinetic energy K(γ) of the system is expressed in terms of the momentum as
follows
γ T M −1 γ
K(γ) ≡
(2.70)
2
where, M is a positive-definite symmetric matrix. For simplicity, this matrix is always
chosen as a scalar multiple of the identity matrix. Thus, the Hamiltonian of the equivalent
mathematical system becomes
H(β, γ) = V (β) + K(γ)
(2.71)
γ T M −1
γ
Setting the temperature T = 1, the canonical ensemble for this system in terms of position
= −log(π(β)) +
vector β and the momentum vector γ is given by
−γ T M −1 γ
1
1
1
2
f (x) = π(β, γ) = e−H(β,γ) = e−V (β) − e−K(γ) = π(β)e
Z
Z
Z
27
(2.72)
where, Z is a normalizing constant such that it makes the area under the pdf equal to unity.
Above equation shows that the position vector β and the momentum vector γ are statistically
independent of each other. Also, it is evident that the position vector β follows the target
probability distribution π(β) and the momentum vector γ follows a Gaussian distribution
with zero mean and covariance M . Therefore, if the samples are simulated as per the pdf
π(β, γ), then the momentum component γ can be easily projected out to get the samples
distributed as per the target probability distribution π(β).
2.3.4
Time Evolution of the Hamiltonian System
To simulate a random variable distributed as per the target distribution π(β), the equivalent Hamiltonian system can be assumed as a conservative system for which the governing
Hamiltonian equations are
dβ
∂H
=
(2.73)
dt
∂γ
dγ
∂H
=
(2.74)
dt
∂β
These equations are solved numerically and the solution of these equations generate the
trajectories of(β, γ) following the target pdf, π(β), and the Gaussian distribution for γ,
respectively. Generally leapfrog algorithm is used to solve the above mentioned differential
equations [[68, 79]]. However, when the target pdf follows Standard Gaussian distribution
and M becomes an identity matrix [i.e. K(γ) = γ T γ/2]. Hamiltonian for such a system
reduces to
H(z, γ) = V (z) + K(γ)
= −log(ϕ(z)) +
=
γT γ
2
(2.75)
zT z γT γ
+
2
2
The system in this form has an analytical solution which is given by [49]
z(t) = γin sin t + zin cos t
(2.76)
γ(t) = γin cos t − zin sin t
(2.77)
where, zin and γin denote the initial position and momentum, respectively. This closed
form solution describes the time evolution of the system and bypasses the computationally
exhaustive solution of leapfrog algorithm. This candidate proposal has been successfully
implemented for reliability analysis using subset simulation [[70]] . For generating the candidate state, the time parameter t is assigned with an arbitrary value at the beginning of the
28
chain which is then modified based on the acceptance ratio of the candidates and the limits
of acceptance [[70]] such that if the rate of acceptance (a) is less than minimum acceptance
rate (amin ) then t is decreased to
t = sin−1 sin t exp[(a − amin )/2]
(2.78)
whereas, if the acceptance ratio exceeds the maximum rate of acceptance (amax ), then t is
increased using the formula
t = sin−1 {sin t exp[(a − amax )/2]}
(2.79)
However, considering the circular behavior of Eq. 2.78 and 2.79 , it is reasonable to assign an
initial value to t such that t ∈ [−π/2, π/2]. Using the above formulation and the acceptance
-rejection criteria, HMC method can be applied for Bayesian model updating with Standard
Gaussian prior.
2.3.5
HMC for Model Updating
HMC algorithm is applied for model updating by Cheung and Beck [79] where the leapfrog
strategy is used to generate the candidate state. In this approach, generation of a new candidate state requires the evaluation of gradient of the Potential Energy function with respect
to each uncertain parameter. However, gradient evaluation using numerical techniques or
approximate allied methods are a cumbersome task and the computational cost grows with
the dimension and the complexity of the Potential Energy function. Cheung and Beck [79]
have used Simultaneous Perturbation Stochastic Approximation (SPSA) optimization algorithm to estimate the starting point θ0 closer to the high probability region of the posterior
pdf which itself requires large number of evaluations of the posterior pdf, further consuming
the computational resources. Also, the problem of low acceptance rate and multi-modal
posterior density are often encountered in model updating due to its inverse nature and
the narrow regions with high prob- ability concentration. These problems are addressed in
MCMC based simulation methods with the help of intermediate pdf s and adaptive proposal
distribution, respectively. However, in HMC based techniques, time parameter (δt) can be
tuned to maintain a consistent exploration of the probability space with a required rate
of acceptance. But, as the candidate state is proposed through a close form solution or a
numerical time-stepping method, the liberty of using the information from the simulated
posterior is not available. Therefore, this method fails to identify well-separated high probability regions leading to a multimodal posterior density function. These shortcomings limit
the application of this method to less complicated mathematical models having unimodal
29
posterior distribution. A new HMC based model updating algorithm is presented in the
following section that provides a way to overcome these drawbacks using the closed-form
solution in Eq. 2.76 and Eq. 2.77 for simulating the random variables following an adaptive
prior based formulation. It can be used to update more complicated finite element model
having multimodal posterior pdf.
2.4
Brief model and analysis from the implementation
of particle filter
2.4.1
System Identificatio of single-degree-of-freedom (SDOF) oscillator
This section presents the results from the implementation of particle filter for identifying the
stiffness of a single-degree-of-freedom oscillator excited by Elcentro ground motion. The SIS
filter have been used to solve this example. The measurement data has been synthetically
generated by solving the forward problem by assuming known values of the system parameters. Once the synthetic measurements are known, the inverse problem is solved using time
domain methods. A schematic diagram of the oscillator is shown in Fig 2.1a. The governing
equation of motion of SDOF oscillator is given by the second order differential equation as:
M ü(t) + C u̇(t) + Ku(t) = −M üg(t)
(2.80)
where , M is the Mass, C is the damping, K is the stiffness, üg(t) is the acceleration due to
the ground motion.The stiffness of the SDOF oscillator was identified in this problem. The
forward problem was solved using the β Newmark algorithm which is an implicit unconditionally stable time marching algorithm (Newmark, [100]). The ground excitation due to
the Elcentro earthquake has been plotted in Fig 2.1b. The overall duration of the excitation
is 40s. The time step considered in the analysis for solving the forward problem is 0.01 sec.
Hence the total number of data points are 4000. The SIS filter was then applied to identify
the stiffness value of the oscillator. The total number of particles considered are 50. The
initial values of the stiffness are generated in the domain of [10000, 90000] from the uniform
distribution. The algorithm is dependent on the parameter values generated at time t = 0.
The identified value over the entire time history is shown in the Fig 2.0c. Hence, the algorithm acts as a filter and returns the best value among all the values generated at t = 0. The
effect of domain dependency can be bypassed and the algorithm can be made more general
by mutating the particles so obtained by adding a small Gaussian noise with a controlled
30
value of σ which can be obtained by several test run of the algorithm. For simplicity and
clarity, only time history of 4 particles is given in Fig 2.0d. The evolution of the posterior
density with time is given by Fig 2.-1e. The estimated states and the states of the original
system are plotted in Fig 2.-1f.
2.4.2
System Identification of Linear Time Invariant (LTI) Synthetic model
The problem of identifying the system parameters using the Bootstrap Particle filter algorithm has been considered here. The natural frequency is calculated by solving the Eigen
value problem involving Mass and Stiffness Matrix. The general equation of motion for a
linear system can be written as
M ü(t) + C u̇(t) + Ku(t) = −M üg(t)
(2.81)
where, M is the mass matrix, C is the damping matrix, K is the stiffness matrix, üg(t) is
the ground excitation and u(t), u̇(t) and ü(t) are respectively the displacement, velocity and
acceleration of the nodes where the sensors are placed for recording. The system parameters
to be identified are represented by ϕ. Here ϕ could represent parameters such as stiffness,
damping, mass density etc. The above equation can be represented in continuous state space
form as
Ż(t) = τ (Z(t), ϕ(t), t)
(2.82)
where Ż(t) is the state vector of the vibrating system and τ (.) relates the state of the system
to its first order derivative with respect to time. Now the problem in hand is to identify
the system parameters ϕ. The particle filter algorithm simulates particles based on the
updated posteriori distribution of the state. More samples are generated from the region
where the likelihood is greater. To solve the problem for identification of parameters, ϕ, the
state vector can be augmented as Xk = [Zk ϕk ] and assuming model noise as the sequence of
i.i.d random variable wk , the above Eq 2.82 can be discretized in the form of Eq 2.82 The
dimension of the problem is equal to sum of vector Z and ϕ. Hence one is able to identify the
state vector as well as the parameters. In the system identification problem we are generally
interested in identifying the system parameters rather than tracking the state of the system.
(Nasrellah and Manohar, [54]) suggested that a larger computational effort can be reduced
by formulating the problem in terms of system parameters . Hence,the systems which remain
invariant with time, the system equation can be expressed as:
dϕ
=0
dt
31
(a) Schematic diagram of SDOF system
(b) Ground excitation due to Elcentro earthquake
32
(c) Estimation of ratio of identified stiffness to original stiffness as function of time
(d) Evolution of weights of particles over time
33
(e) Evolution of posterior density with time
(f) States estimation from the original and the identified system
Figure 2.-1: System identification of SDOF oscillator excited by Elcentro earthquake
34
ϕj(0) = ϕ0
j = 1, 2.........n(2.83)where ϕ0 is the value of the value of the system
parameters at time t = 0. The discrete version of the equation can be presented as
ϕk+1 = ϕk + wk
(2.84)
where ϕk is the system parameters at time k, wk is the model noise. The corresponding
measurement equation can be written as
Yk = hk (ϕk ) + vk
(2.85)
The Fig 2.1a shows the plan and elevation of three story shear building model kept
in the Structural Engineering laboratory of Department of Civil Engineering, IIT Guwahati.Synthetic measurement from the known value of system parameters have been considered and then validated using Bootsrap filter algorithm. The lumped mass for the slab (600
× 300 × 10mm) has been calculated for solving the forward problem. The density of the
steel taken is 8400 kg/m3. The classical damping matrix is obtained by considering the
Rayleigh Damping, (Chopra,[?]).
C = αM + βK
(2.86)
Here M ,K and C are the Mass, Stiffness and Damping matrix respectively, α and β are the
coefficients. For the given model, the mass and the stiffness matrix is given as:


m1 0
0
M =  0 m2 0 
0
0 m3


k1 + k2
−k2
0
k2 + k3 −k3 
K =  −k2
0
−k3
k3
(2.87)
where m1 , m2 and m3 are the lumped mass at the floor levels and k1 , k2 and k3 are the stiffness
of the each story. The coefficients α and β can be obtained from specified damping ratios i
and j for the it h and jt h mode respectively. If both the modes are assumed to have same
damping ratios then arpita
α=ζ
(2.88)
(2wi wj )
wi + wj
2
β = ζ wi +w
j
35
Figure 2.0: Parameter values for solving the forward problem
where wi and wj are the natural frequency of the system in it h and jt h modes respectively. The response of the model has been calculated using the values of the parameters
given in Table 2.0 below. This is known as solving the forward problem. The response of
the structure due to the ground motion excitation by the 1940 El-Centro Earthquake has
been considered.
The response of the model for the given excitation is shown in Fig 2.1b. The inverse
problem starts with simulating random values from the uniform distribution, for the parameters which are to be identified, at time t = 0. The simulation of random values for stiffness
and damping at all the floor levels is done firstly. The domain over which the stiffness
values are simulated is between 10000 to 90000 N/m and the damping values are between
0 to 50 N − s/m. The number of values generated at t = 0are 100. This number remains
constant for each and every iteration of the algorithm. Normal distribution has been used
to calculate the likelihood or the weights of the particles with error covariance been chosen
as 0.001. . Fig 2.1a shows the identified values of the stiffness using the Bootstrap filter.
The standard deviation of the parameters has been plotted in Fig 2.1b. The statistical
fluctuations die out once the parameters are identified and the standard deviation becomes
zero. The robustness of the algorithm is clearly depicted in Fig 2.1b and Fig 2.1a which
shows the mode shapes of the identified as well original system and the estimated states of
the identified and original system for El-Centro earthquake. The study suggests that the
bootstrap filter gives a good estimate of the stiffness values.
A detailed discussion of the various techniques used to m Based on the previous discussions the problem statement for the present work is formulated, which are explained
below.
36
(a) Plan and Elevation of Synthetic Model
(b) Ground excitation due to Elcentro earthquake
37
(a) Ratio of identified stiffness to original stiffness: Elcentro earthquake
(b) Standard deviation of stiffness: Elcentro earthquake
38
(a) Original and estimated states of model: El-Centro earthquake
(b) Mode shape of the original and identified structure
Figure 2.1: System identification of LTI System
39
Research proposal
A review of selected topics in the area of structural system identification has been presented
in the preceding sections. The focus of the review has been mainly on SSI methods based
on dynamic state estimation methods and bayesian updating. . These methods have roots
in Markov vector methods and Bayes’ theorem and provide a systematic framework to sequentially assimilate the measurements into mathematical models and they address several
features of SSI problems in a unified manner. The basis of these methods lies in the exact
functional recursive relations that the posterior pdf-s of the system states satisfy. These
equations can be solved in an exact manner for a limited class of problems but are amenable
for approximate solutions, via numerical methods, such as Monte Carlo simulation techniques, for, a more general class of problems. Some of the outstanding features of these
methods are : (a) they operate in time domain and hence they are most suited for study on
nonlinear SSI problems; (b) the method could also be suitably formulated to identify system
parameters even when the measurements originates from static tests, (c) both process and
measurement equations could be nonlinear; (d) issues related to specification of initial conditions can be taken care of; (e) spatial incompleteness of measurements does not pose any
special difficulties; the method does not need any model reduction methods to be invoked to
match measured and mathematically modeled dof-s; (f) imperfections in measurements and
in mathematical modeling are explicitly taken care of by modeling these imperfections as
white noise random processes;the governing equations can be interpreted as Ito- stochastic
differential equations and the power of Ito’s calculus based approaches can be brought to
bear on discretizing the mathematical models; (g) the method leads to probabilistic description of parameters to be identified and hence one could assess the confidence in the estimates
obtained; (h) the dual problem of state and parameter estimation can be tackled; (i) the
method could be applied in online or offline manners; and (j) the method can take advantage
of modern sensing and computing facilities.
The Kalman filter provides exact solution to the problem of DSE for linear state space
40
models with additive Gaussian noises. The method leads to exact recursive relations for evolution of posterior mean vector and covariance matrix. Ensemble Kalman filters are suited
when number of system states becomes very large. Extended Kalman filter provides a conceptually simple approach to deal with nonlinearities in process and measurement equations.
Here the system nonlinearities are linearized around the evolving expected trajectory of the
system states. The method is ad hoc in nature and leads to biased estimates for system
response moments. Unscented Kalman filter avoids linearization of the system nonlinearities
but instead assumes Gaussian models for posterior pdf-s and deduces mean and covariance
of system states with improved accuracy. Both the extended Kalman and unscented Kalman
filters provide Gaussian models for posterior pdf-s and they assume additive Gaussian noise
models. For more general class of problems, involving nonlinear state space models and
multiplicative/additive non-Gaussian noises, the particle filtering methods provide systematic framework to tackle the state estimation problems. These methods are based on Monte
Carlo simulation strategies and, hence, are computationally demanding; the existing computational power can however be exploited to obtain acceptable solutions. Various versions of
particle filtering methods exist and are widely used in modern engineering including applications to tracking and signal processing applications. The use of these methods in structural
engineering problems is of recent origin and their power has not yet been fully explored
in the existing literature. The challenge here lies in combining particle filtering methods
with commonly used structural modeling techniques, such as the finite element method,
to develop suitable identification tools. When fully developed, these methods can become
powerful tools for structural system identification based on response measurements under
ambient loads such as wind, earthquake and traffic loads.
3.1
Gap Areas
Based on the review of literature, we recognize the following as worthy of further research
and to contribute to their solution:
• The recently developed HMC algorithm has shown great potential in the Subset Simulation. Therefore, application of this algorithm in stochastic model updating problem is
expected to improve the efficiency of searching the probability space which is yet to be
studied. Further most of the studies involving Bayesian algorithms are focused on its
performance using synthetic experiments and/or small-scale laboratory applications.
Only a handful of studies are available for large-scale field structures. In particular,
the application of HMC algorithm for stochastic simulation and load carrying capacity
41
of bridges has never been studied before.
• Within the framework of dynamic state estimation methods, the problem of SSI and
estimation of hidden states are often solved as dual problems. Questions on the possibility of avoiding state estimation and focusing only on system parameter identification
require careful considerations.
• There resides almost no studies in the literature under review to develop a Hamiltonian
close from solution integrated with the Gaussian filters or particle filters to solve bridge
health monitoring problems applied to existing bridge structures. Developing such
scheme can be used to characterize fundamental modes and other associated parameters
of an existing bridge system.
• Within the framework of dynamic state estimation methods, the problem of SSI and
estimation of hidden states are often solved as dual problems. Questions on the possibility of avoiding state estimation and focusing only on system parameter identification
require careful considerations.
• The recently developed HMC algorithm in the literature has shown great potential
in the Subset Simulation. Therefore, application of this algorithm in model updating
problem is expected to improve the efficiency of searching the probability space which
is yet to be studied. The fusion of HMC methods and the state estimation filtering
techniques can be utilized for reliability assessment of existing bridge systems.
• The particle filtering methods are eminently suited for computation on parallel computers although very few researchers have explored this method for implementation
on such computers. The identification procedures can be efficiently implemented if
the particle filtering steps are embedded within professional FE softwares instead of
interfacing through batch files.
• Finite element methods are perhaps the most popular modeling tools that are currently
being used in structural engineering practice. A host of professional softwares are
readily available for simulating a wide range of structural behavior under static and
dynamic loading situations. In studies on condition assessment of large structures, such
as bridges, it would be most advantageous if the structural models are made to reside
on readily available finite element softwares and the identification tools are designed
to communicate with these models. To achieve this, we have to suitably formulate the
identification algorithm and develop the necessary suite of softwares.
42
• Most of the studies in literature has been limited to static behavior of linear systems.
Hence study for nonlinear static and dynamical systems w.r.t. synthetic models and
existing structures is needed. One of the interesting questions in this context would
be on identifying properties of degrading systems under severe dynamic loads. Such
problems are of significance in earthquake engineering problems.
• A very few literatures have been found where optimal sensor locations on the structure
have been identified in time domain approaches for acquiring responses of the structure.
The optimal number of sensors and their locations required for accurate identification
of structural parameters and input force need further development.
• The modern sensors, such as scanning laser vibrometres, distributed fibre optic sensors,
wireless sensors in collaborative mode, and image based distributed strain measurement
devices, provide measurements that are spatially extensive and their assimilation into
mathematical model for the purpose of system identification pose interesting challenges.
• Most of the studies in literature has assumed that the measurement and process noises
are additive and have zero mean, stationary and Gaussian distributed with a specified covariance matrix. This assumption has been found to be not restrictive, especially when the identification procedures have been applied on experimental/field data.
Notwithstanding this, a rational justification of the assumptions made on noise characteristics needs to be explored. One possibility would be to treat the parameters of the
noise process themselves to be unknowns to be estimated as a part of the overall identification procedure. Also, statistical processing of electronic noise from measurement
circuits would also enable to quantify a part of the model parameters. Characterizing
modeling errors, however, would require substantial efforts.
• In problems of bridge condition assessment, the moving vehicles serve as the simplest
means to excite the structures. The identification of bridge parameters here requires the
use of measurement data that originate from dynamic vehicle-structure interactions.
Methods to achieve this, based on particle filtering strategies, are presently not been
studied much in the existing literature.
3.2
Scope of Research
A part of the present study will be conducted in the context of an ongoing research project
at the Department of Civil Engineering, Indian Institute of Technology, Guwahati on in-situ
43
condition assessment of an existing bridge in West Bengal, India. The main objective of
this part of the project will be to analyse the test data/measurements for in-situ condition assessment of this bridge, so that appropriate measures can be taken, if the need be.
The emphasis of the present work will be mainly on time domain parametric methods for
structural system identification. Moreover, the focus will be on methods that can be generalized to include structural nonlinearities, finite element models for structures, measurements
on static and dynamic responses and imperfections in structural models and measurement
noises. Methods based on dynamic state estimation offer promise to achieve these ends.
Consequently, the review of literature on methods of SSI is basically limited to cover these
methods. Within this framework, methods that are applicable to linear state space models
and nonlinear systems are to be studied.
3.3
Objectives of Research
The following mentioned problems will be addressed during the course of the study to develop
generalised solutions:
• Formulation of a closed form HMC solution to carry out stochastic simulation and
model updating.
• Develop a framework for Bayesian parametric state estimation using fusion of the closed
form formulation and Gaussian filter for structural system identification.
• Extending the closed form formulation for stochastic simulation using particle filter to
solve load carrying capacity and rating factor for bridges.
• Using the above developed schemes for Reliability assessment and service life estimation
of existing bridge
The present work aims to tackle the above-mentioned problems. A satisfactory demonstration of identification tools thus developed requires measurement data from laboratory based
experiments and field studies. The work, accordingly, also aims to employ field data from
studies on an existing bridge.
44
Time Frame
45
References
[1] G Wea Housner, Lawrence A Bergman, T Kf Caughey, Anastassios G Chassiakos,
Richard O Claus, Sami F Masri, Robert E Skelton, TT Soong, BF Spencer, and
James TP Yao. Structural control: past, present, and future. Journal of engineering
mechanics, 123(9):897–971, 1997.
[2] J.-T. KIM and N. STUBBS. Improved damage identification method based on modal
information. Journal of Sound and Vibration, 252(2):223 – 238, 2002.
[3] U. Lee and Jinho Shin. A structural damage identification method for plate structures.
Engineering Structures, 24:1177–1188, 09 2002.
[4] Abhinav Subramanian and Sankaran Mahadevan. Bayesian estimation of discrepancy
in dynamics model prediction. Mechanical Systems and Signal Processing, 123:351–368,
2019.
[5] Oliver Nelles. Nonlinear dynamic system identification. In Nonlinear System Identification, pages 547–577. Springer, 2001.
[6] G. Franco, R. Betti, and H. LuÅŸ. Identification of structural systems using an
evolutionary strategy. Journal of Engineering Mechanics, 130(10):1125–1139, 2004.
[7] Anne Raich and Tamas Liszkai. Improving the performance of structural damage detection methods using advanced genetic algorithms. Journal of Structural Engineeringasce - J STRUCT ENG-ASCE, 133, 03 2007.
[8] K. M. Liew and Q. Wang. Application of wavelet theory for crack identification in
structures. Journal of Engineering Mechanics, 124(2):152–157, 1998.
[9] Keith Worden and Geoffrey R Tomlinson. Nonlinearity in experimental modal analysis.
Philosophical Transactions of the Royal Society of London. Series A: Mathematical,
Physical and Engineering Sciences, 359(1778):113–130, 2001.
47
[10] Chunxiao Bao, Hong Hao, Zhong-Xian Li, and Xinqun Zhu. Time-varying system identification using a newly improved hht algorithm. Computers Structures, 87(23):1611
– 1623, 2009.
[11] Swarup Mahato, Meda Vinay Teja, and Arunasis Chakraborty. Combined wavelet–
hilbert transform-based modal identification of road bridge using vehicular excitation.
Journal of Civil Structural Health Monitoring, 7(1):29–44, 2017.
[12] Julius S Bendat. Nonlinear systems techniques and applications. Wiley, 1998.
[13] Zhenyu He, Xinge You, Long Zhou, Yiuming Cheung, and Jianwei Du. Writer identification using fractal dimension of wavelet subbands in gabor domain. Integrated
Computer-Aided Engineering, 17(2):157–165, 2010.
[14] Franck Schoefs, Humberto Yáñez-Godoy, and Francesca Lanata. Polynomial chaos representation for identification of mechanical characteristics of instrumented structures.
Computer-Aided Civil and Infrastructure Engineering, 26(3):173–189, 2011.
[15] H Li, Y Huang, WL Chen, ML Ma, DW Tao, and JP Ou. Estimation and warning
of fatigue damage of frp stay cables based on acoustic emission techniques and fractal
theory. Computer-Aided Civil and Infrastructure Engineering, 26(7):500–512, 2011.
[16] Thomas Bayes and null Price. Lii. an essay towards solving a problem in the doctrine
of chances. by the late rev. mr. bayes, f. r. s. communicated by mr. price, in a letter to
john canton, a. m. f. r. s. Philosophical Transactions of the Royal Society of London,
53:370–418, 1763.
[17] Y. Ho and R. Lee. A bayesian approach to problems in stochastic estimation and
control. IEEE Transactions on Automatic Control, 9(4):333–339, 1964.
[18] J. Spragins. A note on the iterative application of bayes’ rule. IEEE Transactions on
Information Theory, 11(4):544–549, 1965.
[19] Rudolph Emil Kalman. A new approach to linear filtering and prediction problems.
Transactions of the ASME–Journal of Basic Engineering, 82(Series D):35–45, 1960.
[20] Masaru Hoshiya and Etsuro Saito. Structural identification by extended kalman filter.
Journal of Engineering Mechanics, 110(12):1757–1770, 1984.
[21] Simon J Julier and Jeffrey K Uhlmann. New extension of the kalman filter to nonlinear
systems. In Signal processing, sensor fusion, and target recognition VI, volume 3068,
pages 182–193. International Society for Optics and Photonics, 1997.
48
[22] N.J. Gordon, D.J. Salmond, and A.F.M. Smith. Novel approach to nonlinear/nongaussian bayesian state estimation. Radar and Signal Processing, IEE Proceedings F,
140:107 – 113, 05 1993.
[23] Andrew H Jazwinski. Mathematics in science and engineering. Stochastic processes
and filtering theory, 64, 1970.
[24] Peter S Maybeck, Any Form, Or Byany Means, Or Mechanical, et al. Stochastics
models, estimation, and control: Introduction. 1979.
[25] Thomas Kailath. Lectures on wiener and kalman filtering. In Lectures on Wiener and
Kalman Filtering, pages 1–143. Springer, 1981.
[26] R Brown and P Hwang. Introduction to random signals and applied kalman filtering
(john wiley & sons, new york). 1992.
[27] Charles K Chui, Guanrong Chen, et al. Kalman filtering. Springer, 2017.
[28] Mohinder S Grewal and Angus P Andrews. ”kalman filtering: Theory and practice
using matlab, ”john wiley &sons. New York, 2001.
[29] Masanobu Shinozuka and Roger Ghanem. Structural system identification. ii: Experimental verification. Journal of Engineering Mechanics, 121(2):265–273, 1995.
[30] Babak Moaveni, Xianfei He, Joel Conte, Jose Restrepo, and Marios Panagiotou. System identification study of a seven-story full-scale building slice tested on the ucsd-nees
shake table. Journal of Structural Engineering, 137, 01 2011.
[31] Subhamoy Sen and Baidurya Bhattacharya. Non-gaussian parameter estimation using
generalized polynomial chaos expansion with extended kalman filtering. Structural
Safety, 70:104–114, 2018.
[32] SM Ulam and John Von Neumann. Random ergodic theorems. Bull. Amer. Math.
Soc, 51:660, 1945.
[33] Stanislaw Ulam, Robert D Richtmyer, and John von Neumann. Statistical methods in
neutron diffusion. LAMS-551, Los Alamos National Laboratory, pages 1–22, 1947.
[34] Nicholas Metropolis and S. Ulam. The monte carlo method. Journal of the American
Statistical Association, 44(247):335–341, 1949.
49
[35] J. V. Candy, K. Sale, B. L. Guidry, E. Breitfeller, D. Manatt, D. Chambers, and
A. Meyer. Bayesian processing for the detection of radioactive contraband from uncertain measurements. In 2007 2nd IEEE International Workshop on Computational
Advances in Multi-Sensor Adaptive Processing, pages 49–52, 2007.
[36] Hisashi Tanizaki and Roberto S Mariano. Nonlinear filters based on taylor series
expansions. Communications in Statistics-Theory and Methods, 25(6):1261–1282, 1996.
[37] Hisashi Tanizaki. Nonlinear filters: estimation and applications. Springer Science &
Business Media, 2013.
[38] Arnaud Doucet, Nando De Freitas, and Neil Gordon. An introduction to sequential
monte carlo methods. In Sequential Monte Carlo methods in practice, pages 3–14.
Springer, 2001.
[39] Jane Liu and Mike West. Combined parameter and state estimation in simulationbased filtering. In Sequential Monte Carlo methods in practice, pages 197–223. Springer,
2001.
[40] Branko Ristic, Sanjeev Arulampalam, and Neil Gordon. Beyond the Kalman filter:
Particle filters for tracking applications, volume 685. Artech house Boston, 2004.
[41] Arnaud Doucet. On sequential simulation-based methods for bayesian filtering. 1998.
[42] Jun S Liu and Rong Chen. Sequential monte carlo methods for dynamic systems.
Journal of the American statistical association, 93(443):1032–1044, 1998.
[43] Michael K Pitt and Neil Shephard. Filtering via simulation: Auxiliary particle filters.
Journal of the American statistical association, 94(446):590–599, 1999.
[44] Arnaud Doucet, Simon Godsill, and Christophe Andrieu. On sequential monte carlo
sampling methods for bayesian filtering. Statistics and computing, 10(3):197–208, 2000.
[45] Yukito Iba. Extended ensemble monte carlo. International Journal of Modern Physics
C, 12(05):623–656, 2001.
[46] Dan Crisan and Arnaud Doucet. A survey of convergence results on particle filtering
methods for practitioners. IEEE Transactions on signal processing, 50(3):736–746,
2002.
50
[47] Olivier Cappé, Simon J Godsill, and Eric Moulines. An overview of existing methods
and recent advances in sequential monte carlo. Proceedings of the IEEE, 95(5):899–924,
2007.
[48] Adam M Johansen and Arnaud Doucet. A note on auxiliary particle filters. Statistics
& Probability Letters, 78(12):1498–1504, 2008.
[49] Simon Godsill. Particle filtering: the first 25 years and beyond. In ICASSP 2019-2019
IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),
pages 7760–7764. IEEE, 2019.
[50] Zhe Chen. Bayesian filtering: From kalman filters to particle filters, and beyond.
Statistics, 182, 01 2003.
[51] Sebastian Thrun. Probabilistic robotics. Commun. ACM, 45(3):52–57, March 2002.
[52] Jianye Ching, James L. Beck, and Keith A. Porter. Bayesian state and parameter estimation of uncertain dynamical systems. Probabilistic Engineering Mechanics, 21(1):81
– 96, 2006.
[53] Cs Manohar and Debasish Roy. Monte carlo filters for identification of nonlinear
systems. Sadhana, 31:399–427, 08 2006.
[54] H.A. Nasrellah and C.S. Manohar. Finite element method based monte carlo filters
for structural system identification. Probabilistic Engineering Mechanics, 26(2):294 –
307, 2011.
[55] B Radhika and CS Manohar. Nonlinear dynamic state estimation in instrumented
structures with conditionally linear gaussian substructures. Probabilistic engineering
mechanics, 30:89–103, 2012.
[56] Subhamoy Sen, Antoine Crinière, Laurent Mevel, Frédéric Cérou, and Jean Dumoulin.
Seismic-induced damage detection through parallel force and parameter estimation
using an improved interacting particle-kalman filter. Mechanical Systems and Signal
Processing, 110:231–247, 2018.
[57] Vikas Namdeo and C.S. Manohar. Nonlinear structural dynamical system identification
using adaptive particle filters. Journal of Sound and Vibration, 306(3):524 – 563, 2007.
[58] Debraj Ghosh and Charbel Farhat. Strain and stress computations in stochastic finite element methods. International Journal for Numerical Methods in Engineering,
74(8):1219–1239, 2008.
51
[59] R. Sajeeb, C.S. Manohar, and D. Roy. Use of particle filters in an active control
algorithm for noisy nonlinear structural dynamical systems. Journal of Sound and
Vibration, 306(1):111 – 135, 2007.
[60] HA Nasrellah and CS Manohar. A particle filtering approach for structural system
identification in vehicle–structure interaction problems. Journal of Sound and Vibration, 329(9):1289–1309, 2010.
[61] B Radhika and CS Manohar. Dynamic state estimation for identifying earthquake
support motions in instrumented structures. Earthquakes and Structures, 5(3):359–
378, 2013.
[62] Subhamoy Sen, Neha Aswal, Qinghua Zhang, and Laurent Mevel. Structural health
monitoring with non-linear sensor measurements robust to unknown non-stationary
input forcing. Mechanical Systems and Signal Processing, 152:107472, 2021.
[63] P. Rangaraj, Abhijit Chaudhuri, and Sayan Gupta. Polynomial Chaos in Bootstrap
Filtering for System Identification, pages 653–667. 01 2013.
[64] Simon Rouchier, Maria José Jiménez, and Sergio Castaño. Sequential monte carlo for
on-line parameter estimation of a lumped building energy model. Energy and Buildings,
187:86–94, 2019.
[65] Tijana Radivojević and Elena Akhmatskaya. Modified hamiltonian monte carlo for
bayesian inference. Statistics and Computing, 30(2):377–404, 2020.
[66] M. Arulampalam, Simon Maskell, Neil Gordon, and Tim Clapp. A tutorial on particle
filters for online nonlinear/non-gaussian bayesian tracking. Signal Processing, IEEE
Transactions on, 50:174 – 188, 03 2002.
[67] Tiancheng Li, Miodrag Bolic, and Petar Djuric. Resampling methods for particle
filtering: Classification, implementation, and strategies. Signal Processing Magazine,
IEEE, 32:70–86, 05 2015.
[68] Simon Duane, Anthony D Kennedy, Brian J Pendleton, and Duncan Roweth. Hybrid
monte carlo. Physics letters B, 195(2):216–222, 1987.
[69] Radford M Neal et al. Mcmc using hamiltonian dynamics. Handbook of markov chain
monte carlo, 2(11):2, 2011.
[70] Ziqi Wang, Marco Broccardo, and Junho Song. Hamiltonian monte carlo methods for
subset simulation in reliability analysis. Structural Safety, 76:51 – 67, 2019.
52
[71] Lambros S Katafygiotis, Costas Papadimitriou, and Heung-Fai Lam. A probabilistic
approach to structural model updating. Soil Dynamics and Earthquake Engineering,
17(7-8):495–507, 1998.
[72] LS Katafygiotis, NC Mickelborough, and KV Yuen. Statistical identification of structural modal parameters using ambient response data. In Proceedings 3rd International
Conference on Computational Stochastic Mechanics, pages 93–102, 1998.
[73] James L Beck and Lambros S Katafygiotis. Updating models and their uncertainties.
i: Bayesian statistical framework. Journal of Engineering Mechanics, 124(4):455–461,
1998.
[74] James L Beck and Ka-Veng Yuen. Model selection using response measurements:
Bayesian probabilistic approach. Journal of Engineering Mechanics, 130(2):192–203,
2004.
[75] Iman Behmanesh and Babak Moaveni. Probabilistic identification of simulated damage on the dowling hall footbridge through bayesian finite element model updating.
Structural Control and Health Monitoring, 22(3):463–483, 2015.
[76] Iman Behmanesh, Babak Moaveni, Geert Lombaert, and Costas Papadimitriou. Hierarchical bayesian model updating for structural identification. Mechanical Systems
and Signal Processing, 64:360–376, 2015.
[77] I Boulkaibet, L Mthembu, T Marwala, MI Friswell, and S Adhikari. Finite element
model updating using the shadow hybrid monte carlo technique. Mechanical Systems
and Signal Processing, 52:115–132, 2015.
[78] Sai Hung Cheung and Sahil Bansal. A new gibbs sampling based algorithm for bayesian
model updating with incomplete complex modal data. Mechanical Systems and Signal
Processing, 92:156–172, 2017.
[79] Sai Hung Cheung and James L Beck. Bayesian model updating using hybrid monte
carlo simulation with application to structural dynamic models with many uncertain
parameters. Journal of engineering mechanics, 135(4):243–255, 2009.
[80] Siddhartha Chib and Edward Greenberg. Understanding the metropolis-hastings algorithm. The american statistician, 49(4):327–335, 1995.
[81] Jianye Ching and Yi-Chu Chen. Transitional markov chain monte carlo method for
bayesian model updating, model class selection, and model averaging. Journal of
engineering mechanics, 133(7):816–832, 2007.
53
[82] Michael Friswell and John E Mottershead. Finite element model updating in structural
dynamics, volume 38. Springer Science & Business Media, 2013.
[83] Heikki Haario, Eero Saksman, Johanna Tamminen, et al. An adaptive metropolis
algorithm. Bernoulli, 7(2):223–242, 2001.
[84] François M Hemez and Scott W Doebling. Review and assessment of model updating for non-linear, transient dynamics. Mechanical Systems and Signal Processing,
15(1):45–74, 2001.
[85] Maryam Kalantarnia, Faisal Khan, and Kelly Hawboldt. Dynamic risk assessment
using failure assessment and bayesian theory. Journal of Loss Prevention in the Process
Industries, 22(5):600–606, 2009.
[86] James L Beck and Siu-Kui Au. Bayesian updating of structural models and reliability using markov chain monte carlo simulation. Journal of engineering mechanics,
128(4):380–391, 2002.
[87] Hoon Sohn and Kincho H Law. Application of load-dependent ritz vectors to bayesian
probabilistic damage detection. Probabilistic engineering mechanics, 15(2):139–153,
2000.
[88] Michael W Vanik, James L Beck, and SK2000 Au. Bayesian probabilistic approach
to structural health monitoring. Journal of Engineering Mechanics, 126(7):738–745,
2000.
[89] George Kuczera and Eric Parent. Monte carlo assessment of parameter uncertainty in
conceptual catchment models: the metropolis algorithm. Journal of hydrology, 211(14):69–85, 1998.
[90] Christophe Andrieu and Johannes Thoms. A tutorial on adaptive mcmc. Statistics
and computing, 18(4):343–373, 2008.
[91] Ka-Veng Yuen, James L Beck, and Siu Kui Au. Structural damage detection and
assessment by adaptive markov chain monte carlo simulation. Structural Control and
Health Monitoring, 11(4):327–347, 2004.
[92] Feng Qian and Wei Zheng. An evolutionary nested sampling algorithm for bayesian
model updating and model selection using modal measurement. Engineering Structures, 140:298–307, 2017.
54
[93] Heung-Fai Lam, Jiahua Yang, and Siu-Kui Au. Bayesian model updating of a coupledslab system using field test data utilizing an enhanced markov chain monte carlo simulation algorithm. Engineering Structures, 102:144–155, 2015.
[94] Roberto Rocchetta, Matteo Broggi, Quentin Huchet, and Edoardo Patelli. On-line
bayesian model updating for structural health monitoring. Mechanical Systems and
Signal Processing, 103:174–195, 2018.
[95] Isambi S Mbalawata, Simo Särkkä, and Heikki Haario.
Parameter estimation in
stochastic differential equations with markov chain monte carlo and non-linear kalman
filtering. Computational Statistics, 28(3):1195–1223, 2013.
[96] Francois Septier and Gareth W Peters. Langevin and hamiltonian based sequential
mcmc for efficient bayesian filtering in high-dimensional spaces. IEEE Journal of
selected topics in signal processing, 10(2):312–327, 2015.
[97] Shubham Baisthakur and Arunasis Chakraborty. Modified hamiltonian monte carlobased bayesian finite element model updating of steel truss bridge. Structural Control
and Health Monitoring, 27(8):e2556, 2020.
[98] Hamed Nikbakht and Konstantinos G Papakonstantinou. A direct hamiltonian mcmc
approach for reliability estimation. arXiv preprint arXiv:1909.03575, 2019.
[99] Liye Zhang, Lijuan Dong, Shoushan Cheng, Wanheng Li, Bingjian Wang, Hanyong Liu,
and Ke Chen. Efficient reliability assessment method for bridges based on markov chain
monte carlo (mcmc) with metropolis-hasting algorithm (mha). In IOP Conference
Series: Earth and Environmental Science, volume 580, page 012030. IOP Publishing,
2020.
[100] Nathan M Newmark. A method of computation for structural dynamics. Journal of
the engineering mechanics division, 85(3):67–94, 1959.
55