Reliability analysis of existing structures
using dynamic state estimation methods
Technical Report
By
B Radhika and C S Manohar
April 2010
Department of Civil Engineering
Indian Institute of science
Bangalore 560 012 India
1
Abstract
The problem of time variant reliability analysis of existing structures subjected to stationary
random dynamic excitations is considered. The study assumes that samples of dynamic response
of the structure, under the action of external excitations, have been measured at a set of sparse
points on the structure. The utilization of these measurements in updating reliability models,
postulated prior to making any measurements, is considered. This is achieved by using dynamic
state estimation methods which combine results from Markov process theory and Bayes’
theorem. The uncertainties present in measurements as well as in the postulated model for the
structural behaviour are accounted for. The samples of external excitations are taken to emanate
from known stochastic models and allowance is made for ability (or lack of it) to measure the
applied excitations. The future reliability of the structure is modeled using expected structural
response conditioned on all the measurements made. This expected response is shown to have a
time varying mean and a random component that can be treated as being weakly stationary. For
linear systems, an approximate analytical solution for the problem of reliability model updating
is obtained by combining theories of discrete Kalman filter and level crossing statistics. For the
case of nonlinear systems, the problem is tackled by combining particle filtering strategies with
data based extreme value analysis. In all these studies, the governing stochastic differential
equations are discretized using the strong forms of Ito-Taylor’s discretization schemes. The
possibility of using conditional simulation strategies, when applied external actions are
measured, is also considered. The proposed procedures are exemplified by considering the
reliability analysis of a few low dimensional dynamical systems based on synthetically generated
measurement data. The performance of the procedures developed is also assessed based on
limited amount of pertinent Monte Carlo simulations.
Keywords
Dynamic state estimation; reliability of existing structures; extreme value analysis; Ito-Taylor
expansion.
2
1. Introduction
The present study considers the problem of reliability assessment of existing structural systems
which typically carry dynamic loads during their normal operation. We take that a realistic
representation of these loads demands the application of probabilistic models. One could
consider an existing railway bridge as an example of this class of structures. Here the formation
loads are dynamic in nature and, given the uncertainties that are invariably involved in train
speeds, formation lengths, payloads, and track unevenness, the loads could realistically be
modeled as being random in nature. The present study further considers that the reliability
assessment must take into account a set of measurements made on the existing structure in its
operating conditions and the availability of an acceptable mathematical model for the structure
and the applied loads. The measurements are typically taken to include time histories of
components of structural strains, displacements and (or) accelerations and to be, in general,
spatially incomplete and noisy in nature. In certain applications, such as earthquake response of
structures, it could be possible to measure external actions, while, in problems of vehicle
structure interactions or wind loaded structures, the applied forces could remain unmeasured.
The present study allows for the two contingencies of being able or not being able to measure the
external forces. Furthermore, the mathematical model for the structural system itself could be
nonlinear in nature and allowance is also made for possible imperfections in formulating the
mathematical model. The parameters of the mathematical model are assumed to have been
already identified through diagnostic measurements in a previous system identification step. The
reliability metric could be defined in terms of performance functions which are solely dependent
on measured response quantities, or, alternatively, and, more realistically, could involve a mix of
measured and unmeasured response states. In either case, the problem on hand consists of
updating the structural reliability model by assimilating the measured response. The present
study develops a framework, based on the Kalman and particle filtering strategies, to tackle this
problem.
The basic motivation for undertaking this study has arisen due to the present authors’
involvement in a project aimed at condition assessment of existing railway bridges in India with
a view to evaluate their ability to carry increased axle loads and faster and longer moving
3
formations. The field work conducted as a part of this investigation has resulted in large amount
of static and dynamic response data on structural strains, translations, rotations and accelerations
under various loading conditions. One of the data set that has been collected in these studies has
been the response of the bridge structure to ambient traffic load over a period of about 72-96
hours. Questions on the best utilization of such data towards establishing updated model for
reliability of the structure that takes into account the measurements made, do not have clear
answers and it is hoped that the issues addressed in the present study offer useful directions in
this regard. As a prelude to the development of the ideas, we briefly review the relevant
published literature.
2. A review of literature
The study of reliability of existing structures has been the subject of an early research
monograph (Yao 1985) and the topic is also covered in individual chapters in the books by
Ditlevsen and Madsen (1996) and Melchers (1999). Typically, the reliability analysis here
involves Bayesian updating of postulated models for the mechanical behaviour of the structure,
probability distribution functions (PDF-s) of the loads and structural resistance characteristics
based on a set of measurements on the structure under controlled (and hence measured) loads or
under (mostly unmeasured) ambient loads. Additional information based on non-destructive
testing could also be available. The main sources of uncertainties here include: (a) noise in
measurement of structural responses and applied actions, (b) imperfections in the mathematical
model governing the system states, (c) imperfections in mathematical model that relates
measured quantities to the system states, (d) assessment of present condition of the system that
includes models for structural resistance allowing for the degradation that may have taken place
during the completed life of the structure, and (e) mathematical model for external actions.
Moreover, a step involving structural system identification typically precedes the reliability
analysis which would have lead to optimal characterization of properties such as elastic
constants, boundary conditions, mass, and damping properties.
We briefly mention some of the studies in the existing literature in this area of research. The
work of Shah and Dong (1984) addresses questions on strengthening existing structures to meet
higher seismic demands and discusses a probabilistic framework to achieve this goal.
4
Diamantidis (1987) explores the application of first order reliability methods to assess reliability
of existing structures. The problem of combining system identification tools and reliability
methods with knowledge-based diagnostics has been investigated by Yao and Natke (1994).
Issues related to the development of reliability based condition assessment criteria have been
examined by Ellingwood (1996), Mori and Nonaka (2000) and Val and Stewart (2001). The
papers by Melchers (2001) and Catbas et al., (2008) provide overviews on current state of art in
this area of research. Recently Ching et al., (2007) have used Kalman filters for estimating
hidden states in randomly excited systems and have developed a simulation based strategy to
update the reliability of existing structures based on sparse measurements. Their study employs
importance sampling schemes and also includes a step involving identification of external forces.
The reliability here is essentially defined with respect to an unobserved response of the structure
exceeding a prescribed threshold during a loading episode that has already occurred. The major
differences between the study by Ching et al., (2007) and the present study are that (a) we are
focusing on updating the model for structural reliability against future episodes of loading, (b)
the study takes into account nonlinear models for structural behavior and measurement models,
and (c) the updating here is based on use of nonlinear filtering tools.
We consider linear/nonlinear multi degree of freedom (mdof) dynamical systems which are
randomly excited. It is assumed the measurements on response of this structure have been made
for a limited number of loading episodes and at a (possibly) sparse set of points. The study
allows for the likely inability to measure time histories of applied actions. A random process
model for the excitation that permits digital simulation of samples of excitation is assumed to be
available whether or not samples of applied excitations are measured. In case external excitations
are not measured, the study proceeds based on the basis of simulated samples from the assumed
stochastic model for excitations. On the other hand, if samples of excitation are measured, the
ensemble of future excitations is simulated using conditional simulation strategies. A validated
mathematical model for the structure is also taken to be available. The study proposes that
methods of dynamic state estimation, which employ Monte Carlo simulations, be used to
assimilate the measured responses into the mathematical model. This in turn facilitates the
simulation of sample realizations of the system states conditioned on the episodes of
5
measurements. These estimated trajectories are subsequently used to obtain updated reliability
models.
For the case of linear state space models with additive Gaussian noises, it is shown that an
approximate analytical solution to the problem of reliability model updating is obtainable. This is
based on the application Poisson counting process model to the threshold crossings of the
expected response conditioned on the measurements. For more general class of models,
involving nonlinear process and measurement equations and (or) non-Gaussian noises, we
propose a procedure that combines particle filtering strategy (to obtain updated system states)
with a data based extreme value analysis (to develop models for extreme responses). Here it is
postulated that the PDF of the highest response over a given duration agrees with one of the
classical asymptotic extreme value PDF-s, namely, Gumbel, Weibull or Frechet distributions.
This analysis involves two steps: (a) identification of basin of attraction to which the PDF of
extremes of the estimated response belongs to; this, in turn, is based on hypothesis tests such as
those discussed by Castillo (1988, attributed to Pickands and Galambos) or Hasofer and Wang
(1992), and (b) estimation of parameters of the extreme value distribution ascertained in the
previous step using limited number of samples of the estimated response. The proposed approach
is exemplified by considering a few low dimensional dynamical systems subjected to stochastic
excitations. The governing equations are taken to have cubic, tangent and (or) hysteretic
nonlinear terms. The excitations considered include white and band limited random excitations.
The efficacy of the results obtained is assessed with the help of limited large-scale Monte Carlo
simulation studies.
Thus, the present study draws on knowledge base available in the existing literature in the areas
of Kalman and particle filtering (Kalman 1960, Gordon et al., 1993, Tanizaki 1996, Doucet et
al., 2000 and Ristic et al., 2004), extreme value analysis (Castillo 1988, Kotz and Nadarajah
2000), numerical solution of stochastic differential equations (SDE-s) (Kloeden and Platen 1992)
and conditional simulation of random processes using techniques of dynamic state estimation
(Vanmarcke and Fenton 1991, Kameda and Morikawa 1994 and Hoshiya 1995). It may be noted
that particle filtering methods provide statistical solutions to problems of dynamic state
estimation and are capable of treating nonlinear and (or) non-Gaussian state space models. Their
6
applications in problems of structural engineering are recently being explored (Ching and Beck
2006, Manohar and Roy 2006, Sajeeb et al., 2007 Namdeo and Manohar 2007, and Ghosh et al.,
2008). The application of statistical methods to estimate the PDF-s of extremes of random
processes over specified duration is widely studied (Castillo 1988, Alves and Neves 2006).
Hasofer and Wang (1992) have proposed a test statistic to test the hypothesis that a sample
comes from a distribution in the domain of attraction of one of the classical extreme value
distributions. This test has been used by Dunne and Ghanbari (2001) to investigate the extremes
of response of nonlinear beams based on measured random response. Recently, Radhika et al.,
(2008) have investigated the problem of determining the model for extremes of nonlinear
structural response using data based extreme value analysis and Monte Carlo simulation method.
3. Problem statement
In this study we consider structural dynamical systems governed by the equation of the form
MU  F U (t ),U (t ), t   p(t )   (t ); U (0)  U 0 ; U (0)  U 0
…(1)
Here a dot represents differentiation with respect to the time variable t;
U (t ),U 0 , F , p(t ), and  (t ) are L  1 vectors and M is the L  L mass matrix; p(t ) is a vector random
process representing the external actions, with one realization of p(t ) corresponding to one
episode of loading (such as passage of a train formation or occurrence of one earthquake event).
The quantity  (t ) is a vector of random processes that accounts for modeling error in arriving at
equation 1. It is assumed that  (t ) is independent of p(t ) . Furthermore, we assume that a set of
measurements, denoted by y  tk  , have been made at discrete time instants tk   0, T k 1 and the
N
measured quantities are taken to be related to the system states U (t ) and U (t ) through the relation
y  tk   f k U  tk  ,U  tk   qk U  tk  ,U  tk  k ; k  1,2,
,N
…(2)
Here y  tk  and f k U  tk  ,U  tk  are s  1 vectors, qk U  tk  ,U  tk  is a s  r matrix and  k is a
r  1 vector of sequence of independent normal random variables with zero mean and
covariance  . The quantity  k accounts for the measurement noise and imperfections involved in
7
relating the measurement y  tk  , k  1,2,
N with the states U (t ) and U (t ) . We begin by assuming
that measurements on p(t ) have not been made. A random process model for p(t ) which enables
digital simulation of samples of p(t ) , is, however, taken to be known. The problem on hand
consists of determining the reliability of the system governed by equation 1 taking into account
the measurements made as in equation 2. To facilitate this, we define a measure of system
response denoted by h U  t  ,U (t ), t  . A few examples for this function could be the reaction
transferred to supports or von Mises’ stress metric or displacement at a critical point. We
consider the problem of determining the posterior probability Ps given by


Ps  P  h U (t ),U (t ), t  t   0, T0  | y  tk  ; k  1, 2,

N

…(3)
Here  is the safe limit on h U  t  ,U (t ), t  and P  is the probability measure. For the sake of
simplicity we assume that T0  T . We further note that


Ps  P  h U (t ),U (t ), t  t   0, T  | y  tk  ; k  1, 2,

 P  max h U (t ),U (t ), t   | y  tk  ; k  1, 2, N 
 0t T


N


 P  hm   | y  tk  ; k  1, 2,
…(4)
N 
where hm  max hU (t ),U (t ), t . Thus, the problem of determining Ps reduces to the problem of
0t T
determining the extreme value distribution of h U  t  ,U (t ), t  conditioned on the
measurements y  tk  ; k  1, 2,
N . In case measurements on p(t ) have also been made, the
problem on hand consists of determining the conditional
PDF Ps  P hm   | y tk  , p tk  ; k  1, 2,
N  . In the following sections we propose a scheme for
solving this problem based on dynamic state estimation methods, data based extreme value
analysis, and conditional simulation strategy.
4. Models with time as a continuous variable
8
The basic issues in determining Ps  P hm   | y tk  , p tk  ; k  1, 2,
N  can be clarified by
considering the ideal situation in which time t is treated as a continuous variable.
4.1 Linear systems
It is instructive to begin with the case of linear state space models with additive Gaussian noises.
Equations 1 and 2 for this simplified situation can be recast into the following form
x  t   a  t  x  t   e  t  p  t   b  t  W  t  ; x  0   x0
z t 
…(5,6)
y (t )  c  t  x  t    c  t  ; y  0   y0
Here x  t   U  t  U  t  is the d 1 state vector (with d  2 L ); W  t  and  c  t  are random noise
t
vectors (of sizes m1 and s 1 respectively) representing, respectively, the process and
measurement noises; p  t  is the sample realization of L  1 excitation vector; a, b, c and e are time
dependent matrices of sizes d  d , d  m,, s  d and d  L, respectively. By interpreting the noise
terms W  t  and  c  t  as formal derivatives of Brownian motion process, the above equations can
be written as a pair of SDE-s as
dx  t   a  t  x  t  dt  e  t  p  t  dt  b  t  dBI  t  ; x  0   x0
dz  t   c  t  x  t  dt  dBII  t  ; z  0   z0
…(7,8)
Here dBI and dBII are m  1 and s  1 vectors of increments of Brownian motion processes. It is
assumed that dBI , dBII , x0 and y0 are stochastically independent. Furthermore, we denote
E  dBI  t  dBIt  t    Wc  t  dt and E  dBII  t  dBIIt  t     c  t  dt ; here the superscript t denotes
matrix transposition, E   denotes the expectation operator, and the subscript c emphasizes the
fact that we are treating time as being continuous.
In situations in which no measurements are available, it is well known that the transition
probability density function (pdf) satisfies the Fokker-Planck-Kolmagorov (FPK) equation and
there exists vast literature on strategies for its solution and characterization of response moments
(see, Lin and Cai 1995, for example). When additional information on measurements becomes
available, the FPK equation can be generalized and an equation for the evolution of
9
p  x; t | x0 , y   , 0    t  can be formulated. Such an equation is due to Kushner and Stratonovich
(Maybeck 1982) and, for the system in equation 7, one gets
n
p X  x; t | x0 , y   ,0    t 


p X  x; t | x0 , y   ,0    t  ai
t
i 1 xi
 

1
2
d

2
p X  x; t | x0 , y   ,0    t  bWc bt 
ij

x

x
i
j
j 1
d

i 1


 cx  E  cx(t )  c1  t  y (t )  E cx(t )  p X  x; t | x0 , y   ,0    t 
t
…(9)
where E  cx(t ) 

 cxp
X

 x; t | x0 , y   ,0    t  dx. The above equation is a stochastic partial
differential equation which provides a sample solution of p  x; t | x0 , y   , 0    t  for a given
sample of measurement y   , 0    t. It is important to note that, if the last term on the right
hand side is ignored (that is, when there are no measurements), the equation reduces to the FPK
equation for p  x; t | x0  .
It can be shown that the moments
xˆ  t  

 x p  x; t | x , y   ,0    t dx
X
0

P t  

  x  xˆ t   x  xˆ t 
t
…(10,11)
p X  x; t | x0 , y   ,0    t  dx

are governed by
xˆ  t   a  t  xˆ  t   c  t  p  t   P  t  c t  t  c1  y  t   c t  xˆ t  
P  t   a  t  P  t   P  t  a  t   b  t   wc b  t   P  t  c t 
t
t
t
c1
t  c t  P t 
The above pair of equations forms the continuous time Kalman filter.
10
…(12,13)
In the problem of determination of P hm   | y   ;0    T  , it must be noted that the above
equations have been obtained for a given realization of the process p  t  . If we restrict our
attention to performance functions involving one of the system states xi  t  , the problem on hand
consists of evaluating PS =P  xi t    | y   ,0    T ; t   0,T  . Since our interest lies in
knowing the future reliability of the system conditioned on the measurements y  t  , we propose
that PS be evaluated as
PS  P  xˆi  t    | y   ,0    T ; t   0, T 
…(14)
This can be considered as acceptable since xˆi  t  can be interpreted as the optimal estimate of
xi  t  given the measurements on y  t  . Based on this, we could now interpret equation 12 as
representing the equation governing the updated dynamics of the system which can be used to
investigate the structural behavior under future realizations of p  t  . Thus, if we model p  t  as a
random process, x̂  t  itself would be a random process and standard methods of random
vibration analysis of dynamical systems would lead to models for PS . An important feature of
x̂  t  is that, due to presence of the measured response y  t  on the right hand side, x̂  t  would
possess a time varying mean. Thus, if we denote mxˆ  t   E p  x t  , where E p  is the
expectation across ensemble of response due to ensemble of realizations of p  t  , and if we take
mean of p(t) to be zero, we get
mxˆ  t   a  t  mxˆ t   P t  ct t  c1  y t   c t  mxˆ t 
…(15)
Similarly, the equation for the random component Z xˆ  t   xˆ  t   mxˆ  t  can be obtained as
Z xˆ  t   a  t  Z xˆ  t   c  t  p  t   P  t  c t  t   cc  t  Z xˆ  t  ; Z xˆ  0   Z xˆ 0
…(16)
If we further assume p(t) to be stationary, given that the dynamical system in equation 1 is
damped, it may be expected that as t  , Z xˆ  t  reaches a stochastic steady state. In the
evaluation of PS we now use
11
PS  P  xˆi  t    | y   , t0    T ; t   t0 ,T  
 P  xˆi max   | y   , t0    T 
…(17)
where t0 is the time required for the dissipation of transients, and xˆi max  max xˆi  t  . To proceed
t t0 ,T 
further we make the approximation
xˆi max  max mxiˆ  t   max Z xiˆ  t 
t t0 ,T 
…(18)
t t0 ,T 
This approximation is expected to lead to a conservative result. An approximation to the PDF of
max Z xiˆ  t  , in the steady state, could be obtained by assuming Poisson’s models for the level
t t0 ,T 
crossing statistics of Z x̂i  t  , based on which, a Gumbel model for the extremes could be deduced
(Nigam 1983). This calculation requires the determination of the joint pdf


pZ ˆ , Z ˆ z , z; t | y   ,0    t of Z x̂i  t  and Z x̂i  t  at time t. From equation (16) it is clear that, when
xi
xi
p(t) is Gaussian, Z x̂i  t  and Z x̂i  t  would be jointly Gaussian. Based on this, further calculations
on outcrossing rates could be carried out.
4.2 Nonlinear systems
As one might expect, the analysis presented in the previous subsection no longer becomes
tenable if the system dynamics is governed by nonlinear models. To clarify this, we consider the
generalized version of equations 7 and 8 given by
dx  t   a  x  t  , p  t  , t  dt  b  x t  , t  dBI t  ; x  0   x0
dz  t   c  x  t  , t  dt  dBII  t  ; z  0   z0
…(19,20)
The functions a  x  t  , p t  , t  and b  x t  , t  are now the drift and diffusion matrices of sizes
 d  1 and  d  m respectively. The Kushner-Stratonovich equation can now be deduced as
(Maybeck 1982)
12
d
p X  x; t | x0 , y   ,0    t 


p X  x; t | x0 , y   ,0    t  ai
t
i 1 xi
 

1
2
d

2
p X  x; t | x0 , y   ,0    t  bWc bt 
ij

x

x
i
j
j 1
d

i 1






 c  x  t  , t   cˆ  x  t  , t  c1  t  y  t   cˆ  x  t  , t  p X  x; t | x0 , y   ,0    t 
t
…(21)
The equations for x̂  t  and P  t  (equations 10,11) can now be shown to be given by


ˆ  E  x  t  c t   xˆ  t  cˆt c1 dz  t   adt
ˆ 
dxˆ  t   adt
t
dP(t )  E  x  t   xˆ  t   a t  dt  E  a  x  t   xˆ  t    dt  E b wcb t  dt 


t
E  x  t   xˆ  t   a t  c1 E  a  x  t   xˆ  t    dt 


E  x  t   xˆ  t    x  t   xˆ  t    c  cˆ 

t
t
  1  dz  t   cdt
ˆ 
  c 
…(22,23)
with



  a  x, t  p
aˆ  E a  x  t  , t  


  c  x, t  p
cˆ  E c  x  t  , t  
X

X
 x; t | x0 , y   , 0    t  dx, and
 x; t | x0 , y   , 0    t  dx
…(24)
Here it is important to note that the above equations are essentially formal in nature since the
right hand sides in equations (22,23) contain higher order moments, such as,


E x  t  c  x  t  , t  , which are yet not known. Any attempt to set up equations governing these
higher order moments leads to an unending hierarchy of equations, which at no stage yield
sufficient number of equations to obtain an acceptable solution. This difficulty can be overcome
in an ad hoc manner by invoking closure approximations and details of a few such studies in
existing literature are summarized by Maybeck (1982). It is interesting to note that such
13
approaches do not seem to have been yet investigated in structural mechanics applications,
although, in the context of traditional nonlinear random vibration problems, the closure
approximations are widely studied (Ibrahim 1985, Bolotin 1984). In a similar vein, it may also be
remarked that approximate solutions to equation 21 could also be developed based on method of
weighted residuals/finite element procedures; such a line of research has not yet been pursued in
structural engineering applications. Also, in dealing with the problem of determination of
P hm   | y   ;0    T  , it is not apparent how the approximation developed for linear systems
could be generalized. These issues are addressed in the present study by combining tools of
Monte Carlo simulation based filters and results from extreme value analysis. A first step in
developing the relevant details is to discretize, in time, the governing equations of motion.
5. Models with time as a discrete variable
In order to develop numerical solutions to the reliability problem on hand, it is required, as a first
step,to obtain discretized version of the governing process equation 19. Following Kloeden and
Platen (1992), we employ the order 1.5 strong Taylor’s scheme to achieve this. Thus, the
discretized version of equation 19 is obtained as
m
xkn1  xkn  akn    bkn , j W j 
j 1
1 0 n 2 m  j n j  n, j

L ak     L ak Z  bk W j   Z j 
2

t
j 1 

…(25)
with
W j   j , Z j 
1
1
p   2
12 2

2


 j  a j ,0 , a j ,0  
2

p
1
i
i 1
j ,i
 2  p  j , p ,
d
d
1 0


1 d m i, j l , j  2

i
j
,
L


a

b
b
,
L

bki , j i




k
k
k
2
i
i
l
tk i 1 xk 2 i ,l 1 j 1
xk xk
xk
i 1 i
i 1
p
…(26)
Here n  1, 2,..., d and k  0,1, 2,..., N denote the discretization in time with equal time steps of
  T N ;  j ,  j ,1 ,...,  j , p ,  j , p are independent standard Gaussian random variables; the notation
xkn and akn denote the nth element in the vectors x(t ) and a  x  t  , p  t  , t  , respectively,
evaluated at time tk; bkn , j denotes the  n, j  element of the matrix b  x  t  , t  evaluated at time
th
14
tk. In the numerical work p is taken to be 4. The random variables appearing in the above
equation have their origins in integrals of the type
 n1
I  j1 , j1 ,
, jl 

 n1
  dW

n
…(27)
dWsljl
j1
s1
n
which are known as multiple stochastic integrals (Kloeden and Platen 1992).
In the context of current sensing methods, the time variable in the measurement equation
invariably occurs in discrete form. If we assume that the time instants at which the process
equation is discretized and the time instants at which measurements are made coincide, the
governing equations for dynamic state estimation can be cast in the standard form as
xk 1  zk  xk   Fk  g k  xk Wk ; k  0,1, 2,..., N
…(28,29)
yk  f k  xk   qk  xk  k ; k  1, 2,..., N
Here xk , zk  xk  and Fk are d  1 vectors where Fk is the term containing the external actions;
gk  xk  is a d  m matrix; and Wk is a m  1 vector of normally distributed random variables with
zero mean and covariance W . In the further work we introduce the notations
y1:k   y1
y2
yk  and x0:k  x0
t
x1
xk 
t
…(30)
In the discrete time setting, the required reliability is given by
Ps  P  hk  xk    | yk0 :N ; k   k0 , N 
 P  max hk  xk    | yk0 :N 
 k0  k  N

…(31)
 P  hm   | yk0 :N 
where hm  max hk  xk  and k0 is the time required for dissipation of transients. It needs to be
k0  k  N
emphasized here that the original problem of determining
Ps  P  max h  x  t  , t    | y   ; t0    T  is now replaced by Ps  P  max hk  xk    | yk0 :N  .
 t0 t T

 k0  k  N

15
In this formulation the measurements y  t  ,0  t  T are replaced by its value y1:N at observation
time instants t  tk , k  1, 2,
, N . This clearly is consistent with the digital data acquisition that
underlies measurements. On the other hand the replacement of max h  x  t  , t  by max hk  xk  is
t0 t T
k0  k  N
an approximation arising from time discretization of the governing equations of motion. In the
context of current state of art in filtering methods, it appears possible to avoid this replacement;
see the works of Beskos et al., (2006) and Fearnhead et al., (2008) who have developed state
estimation schemes to treat continuous time process equations and discrete time measurements.
Here the time instants between two successive measurement episodes are not discretized in the
treatment of the process equation. While these developments indeed enhance the accuracy of
state estimation problems, in the present study, however, we are not considering these
refinements. We would like to point out that, in problems of structural mechanics, the equations
of motion for structural behaviour (equation 1) are typically obtained only after discretizing the
spatial variables using the finite element method. Here the choice of discretization intervals in
space and time are well known to be governed by interrelated issues. Thus, discretization of time
variable t in subsequent analysis can be deemed to be reasonable.
6. Discrete time dynamic state estimation techniques
The objective of discrete time dynamic state estimation is to determine the multi-dimensional
conditional probability density function p  x0:k | y1:k  , the marginal pdf p  xk | y1:k  (also known as
the filtering density function) and the corresponding first and second order moments given by
ak |k  E  xk | y1:k    xk p ( xk | y1:k )dxk
t
t
 k |k  E  xk  ak |k   xk  ak |k      xk  ak |k   xk  ak |k  p ( xk | y1:k )dxk


…(32,33)
Based on the application of Bayes’ theorem and utilizing the Markovian property of response
vector xk , it can be shown that the evolution of p  xk | y1:k  obeys the following recursive
relations
16

Prediction: p( xk | y1:k 1 )  p( xk | xk 1 ) p( xk 1 | y1:k 1 ) dxk 1
Updation: p( xk | y1:k ) 
…(34,35)
p ( yk | xk ) p ( xk | y1:k 1 )
 p( y
k
| xk ) p ( xk | y1:k 1 )dxk
Similarly, the evolution of the multi-dimensional posteriori pdf p  x0:k | y1:k  can be shown to be
governed by the recursive relation
p( x0:k 1 | y1:k 1 )  p ( x0:k | y1:k )
p( yk 1 | xk 1 ) p( xk 1 | xk )
; k  0,1,..., N
p( yk 1 | y1:k )
…(36)
For linear state space models and additive Gaussian noises, the above equations lead to the well
known discrete time Kalman filter which provides an exact set of recursive relations for the
evolution of the mean vector ak |k and covariance matrix  k |k . For this class of problems, the
procedure for the determination of Ps  P  max hk  xk    | yk0 :N  broadly follows the steps as
 k0  k  N

outlined in equations 14-18 for the case of continuous time models. These details are illustrated
through numerical examples later in the paper.
For more general class of problems, involving nonlinear mechanics and non-Gaussian noises,
equations 34 and 35 can be solved numerically using Monte Carlo simulations leading to a class
of methods known as particle filters. The efficacy of these simulation tools are often enhanced by
incorporating variance reduction schemes in the simulation steps. In the present study we
consider the application of two versions of particle filtering namely, the sequential importance
sampling (SIS) filter (Doucet 1998), which employs importance sampling to achieve variance
reduction, and the bootstrap filter as developed by Gordon et al., 1993. It may be noted that the
SIS filter is applicable for nonlinear process equations but linear measurement models, while, for
applying the bootstrap filter, the process equation as well as the measurement equation could be
nonlinear. The steps in implementing these filters are briefly summarized in Appendix A. These
steps lead to the determination of p  xk | y1:k  and hence the moments ak |k and  k |k . These, in turn,
serve as the starting point for estimating posterior PDF-s of extremes of system states based on a
statistical analysis of samples of time histories of system response.
17
7. Extreme value analysis of nonlinear system response
The analysis here is based on the assumption that Ps  P  max hk  xk    | yk0 :N  can be
 k0  k  N

approximated by Ps  P  max hk ak |k    | yk0 :N  . It must be noted that the probability measure
 k0  k  N

here is being assigned on the ensemble of response realizations corresponding to realizations of
excitation p(t). The realization of conditional expectation ak |k , corresponding to a realization of
p(t), is obtained by using one of the state estimation methods outlined in Appendix A. Thus, for a
set of realizations of p  t  , denoted by  p ( e )  t  , an associated ensemble of estimates of system
Ns
e 1
states, denoted by ak( e|k) 
Ns
e 1
, can be obtained. This, in turn, can be utilized to obtain the ensemble
  
of system performance function hk ak |k  . We denote this ensemble as h ak e|k
Ns
e 1
. The problem
of estimating the probability defined in equation 4 now could be addressed statistically based on
  
the available ensemble h ak e|k
Ns
e 1
. A brute force Monte Carlo simulation approach involving
simulations with sufficiently large value of sample size N s would, at least in principle, provide a
means to solve this problem. However, if the probability measure of interest is small, this
approach would require significant computational time. This difficulty could be mitigated, again,
in principle, by adopting Monte Carlo simulations with suitable variance reduction schemes. In
the present context, however, the method to choose the associated importance sampling density
functions is not evident. We proceed further in this study by adopting a data based extreme value
analysis procedure. This is based on the assumption that the peaks of system response lie in the
basin of attraction of one of the classical extreme value distributions, viz., Gumbel, Frechet, or
Weibull PDF. Based on the analysis of peaks in a few samples of system response we first
identify the actual basin of attraction to which the peaks belong to. This is followed by a further
analysis that estimates the parameters of the identified extreme value PDF.
A requirement for the application of this procedure is that the process hk ak |k  is stationary. This,
as has been seen for linear systems, is not true presently, since ak |k possesses a time varying
18
mean. We proceed further by restricting hk ak |k  to be a linear function of ak |k and assume that the
component
     
h s ak |k
e
Ns
e 1
 h ak |k
e
Ns
e 1
  

e
 E  h ak |k



e 1 
Ns
…(37)
 
e
can be treated as being stationary. The PDF of the peaks of the function h s ak |k is taken to lie in
the basin of attraction of Weibull, Gumbel or Frechet PDF-s. Based on a few samples of
 
h s ak |k , and using the hypothesis test as proposed by Hasofer and Wang (1992), we first
e
identify the extreme value PDF to whose basin of attraction the PDF of extremes of
 
h s ak |k belong to. Once the type of the extreme value PDF is identified, we use samples of
e
  
e
extremes over the specified duration of h s ak |k
Ns
e 1
and estimate the parameters of the relevant
extreme value PDF. The steps in implementing the Hasofer-Wang hypothesis test are as follows:
(a) Determine the number of extremes ( N ) in the given sample h s  ak |k  corresponding to
time duration T. Arrange the extremes in descending order hs1:N  hs 2:N 
 hs N:N . (see
figure 1 which shows a typical realization of a random process with the extremes marked
on it).
(b) Select the top k values hs1:N  hs 2:N 
 hs k:N with k  1.5 N , and determine the test
statistic W using
 k

  H i:N 
2
k ( H  H k :N )
i 1

W
, where H  
k

k
2
(k  1)   H i:N  H  
 i 1

…(38)
(c) The value of the test statistic W  104 W is compared with the upper and lower
percentage points (WU and WL) given in Table 1 for the chosen significance level.
(d) The domain of attraction is ascertained to be Gumbel or Weibull or Frechet at the chosen
significance level, as follows:
19
(i) W > WU: accept the hypothesis that the data belongs to the domain of attraction of
Weibull distribution.
(ii) WU >W >WL: accept the null hypothesis that the data belongs to the domain of
attraction of Gumbel distribution.
(iii) W < WL: accept the hypothesis that the data belongs to the domain of attraction of
Frechet distribution.
Once the basin of attraction is identified, we statistically estimate the parameters of the
 
s e 
e
associated extreme value PDF from the samples hmax
 max h s ak |k for e  1, 2, , N s . An
k0 k  N
approximation to the extreme of hk ak |k  is next obtained based on the premise that
max hk ak |K   max E  hk ak |K   max hks ak |K  .
k0  k  N
k0  k  N
k0  k  N
8. Numerical illustrations
For the purpose of illustration, we consider a few linear/nonlinear oscillators and apply the
procedure described in the preceding sections by using synthetically simulated measurement
data. The range of nonlinear behaviour considered includes memoryless and hereditary
nonlinearities. Measurements which could be nonlinear functions of the system states are
considered. The models for external excitations include white as well as mean-square bounded
random excitations. To validate the steps involved in extreme value analysis, a brute force Monte
Carlo simulation strategy is adopted in which a large ensemble of performance functions is
generated by assimilating the measurements into the relevant mathematical model using one of
the filtering tools. Simulation results on unconditional PDF of extremes are also included for
sake of reference; in a similar vein, where possible, the unconditional PDF-s for the extremes of
the response based on level crossing theory are also provided.
8.1 Linear single degree of freedom system subject to filtered white noise excitation
We first consider a linear single degree of freedom (sdof) system subject to filtered white noise
excitation F  t  . The governing equation here is given by
20
x  t   2 x  t    2 x  t   F  t     t 
…(39)
F t    F t   W t 
Here W  t  is a Gaussian white noise excitation with zero mean and E W  t W  t       12   ,
 is the damping coefficient,  is the natural frequency of the system, and   t  is the process
noise modeled as a Gaussian white noise with zero mean and E   t    t       22   . The
analysis in this section is based on the application of the Kalman filter for state estimation. By
recasting the above equation as a SDE, the discretized equations of motion, with
states  x1k
xk2
xk3
x1k 1  x1k  a1k  
xk21  xk2  ak2  
xk31  xk3  ak3  
  F t  x t  x t  , are obtained as
t
t
k
k
k
1 0 1 2
L ak   L1 a1k Z 1  L2 a1k Z 2
2
1
m1
W 1 
1 0 2 2
L ak   L1 ak2 Z 1  L2 ak2 Z 2
2
…(40)
1 0 3 2
L ak   L1 ak3 Z 1  L2 ak3 Z 2
2
with
L0 a1k    a1k , L1a1k    1 , L2 a1k  0
…(41)
L0 ak2  ak3 ; L1ak2  0; L2 ak2 ;
L0 ak3  a1k  ak2   2   ak3  2  , L1ak3   1 , L2 ak3   2  2 
In the numerical work we take
  0.2,   2 rad / s,   0.2,1  10 N , 2  0.1N , T  50.0s, t0  5s and   0.0250s .
8.1.1 Case 1
Here we take measurements to be made on x  t  and x  t  leading to the measurement equations
y1  tk   xk2   1k
…(42)
y2  t k   x   2 k
3
k
21
The applied excitation F(t) is taken to be not measured and the performance function is defined
in terms of permissible displacement of the system. It is assumed that standard deviations of
 1k and  2 k are 0.0419m and 0.0762m/s, respectively. Figure 2 shows the plots of E  ak(2)|k  and

 (3) 
E  ak(3)|k  ; similar plots for Var  ak(2)
|k  and Var  ak |k  are shown in figure 3. It is indicative from
(3)
these figures that ak(2)
|k and ak|k have time varying means and random components which become
stationary for large values of time. The time variation of the mean (figure 2) is of steady state
nature; that is, the function neither goes to zero nor grows to large values as time becomes large.
Also shown in figure 2 are the measured displacement and velocity responses and ak(2)
ak(3)|k
|k and
obtained during the measurement step. It is apparent from this figure that the time varying mean

 (3) 
components E  ak(2)
|k  and E  ak |k  arise out of measurements made on the system response.
Results on PDF of extremes are obtained using the following four alternatives: (a) method 1:
result based on level crossing approximation that employs analytical model for


pZ ˆ , Z ˆ z , z; t | y   ,0    t (see section 4.1); (b) method 2: result based on data based extreme
xi
xi
value analysis (section 7); (c) method 3: Monte Carlo simulations on 5000 samples of ak(2)
|k ; (d)
method 4: the unconditional extreme value PDF of x(t) when no measurements are available; this
is obtained using a 5000 samples Monte Carlo simulations. The results are shown in figure 4. In


implementing method 2, the W statistic was computed to be 641.106 N  105, k  15 and, when
compared with the reference values in table 1, the domain of attraction of extremes was
ascertained to be Gumbel at 5% significance level. The parameters of Gumbel distribution was
estimated using 500 samples of ak(2)
|k . Figure 5 shows the influence of sample size in estimation of
parameters of Gumbel distribution and it may be inferred that a sample size of 500 provides
satisfactory estimate of model parameters. The broad agreement between results from methods 1,
2 and 3 can be discerned from figure 4. More refined results from level crossing theory can be
obtained by considering effects due to one step memory of level crossings and correction due to
clumping of level crossings in realization of narrow band processes. Similarly, results from
Monte Carlo simulations could be improved by using larger number of samples.
22
8.1.2 Case 2
This case is similar to Case 1 except that here we assume that the sample time history of applied
force F(t) is also measured. Further samples of F(t) are now obtained using a conditional
simulation strategy. The process equation here remains similar to equation 40 but the
measurement equation reads
y1  tk   x1k   1k
…(43)
y2 (tk )  xk2   2 k
y3 (tk )  xk3   3k
The noise processes  1k , 2 k and  3k are taken to be zero mean, independent Gaussian sequences
with standard deviation of 1.7225N, 0.0427m and 0.0856m/s, respectively. The results on
extreme value PDF is obtained using data based extreme value analysis (section 7). The analysis
in this case yielded Gumbel model for the PDF
W  767.395, N  101, k  15
at 5%
significance level. Figure 6 compares resulting Gumbel PDF with corresponding results for the
case when F(t) was not measured (previous section).
8.1.3 Case 3
As has been noted already, one of the assumptions that has been made is that the sampling
interval used in measurements and step size used in discretizing the governing SDE-s coincide.
This assumption has simplified the presentation of the details of formulary. However, in
implementing the method, this is not a restrictive assumption. To illustrate this, we consider here
the problem in section 8.1.1 (case 1) with the time step in measurement equations being twice the
step size sued for discretizing the governing SDE. Figure 7 compares results on extreme value

PDF of displacement Gumbel model at 5% significance level: W  537.734, N  240, k  23

with corresponding result from Case 1. Both these results are based on data based extreme value
analysis.
23
8.2 Duffing’s system with 2-dofs
Here we consider the system shown in figure 8. When the springs are taken to have cubic forcedisplacement characteristics, the governing equation for the system can be obtained as
m1u1  k1u1  k2  u1  u2   c1u1  c2  u1  u2   1u13   2 u1  u2   p1 t   w1 t 
3
m2u2  k2  u2  u1   k3u2  c2  u2  u1   c3u2   2  u2  u1   3u23  p2 t   w2 t 
3
…(44)
Here p1 (t ) and p2 (t ) are externally applied forces that are modeled as a pair of independent, zero
mean, stationary, Gaussian random processes with auto-power spectral density functions (psdf-s)
and auto-covariance functions (acf-s) given respectively by
Sii   
 2
 2
Pi exp  
i
 4i

2
2
 ; Rii    Pi exp  i ; i  1, 2



…(45)
Clearly, the random processes p1 (t ) and p2 (t ) are mean-square bounded and, also, it can be
verified that the processes are differentiable in the mean-square sense to any desired order. The
functions w1 (t ) and w2 (t ) are zero mean, mutually independent Gaussian white noise processes
with E  wi t  wi t      i2   ; i  1, 2 . We assume that the system starts from rest. Also, we
take that, when the nonlinearities are absent, the damping matrix is classical so that the two
modes will have damping ratios of 1 and 2 . By interpreting equation 44 as a set of Ito SDE-s
with state vector given by x  u1 u1 u2 u2  , and by using the 1.5 order strong Taylor’s
t
scheme (equation 25), the following map is obtained
1
x1k 1  x1k  a1k   L0 a1k  2  L1a1k Z 1  L2 a1k Z 2
2
xk21  xk2  ak2  
1
1
W 1  L0 ak2  2  L1ak2 Z 1  L2 ak2 Z 2
m1
2
1
xk31  xk3  ak3  L0 ak3 2  L1ak3Z 1  L2 ak3Z 2
2
xk41  xk4  ak4  
2
1
W 2  L0 ak4  2  L1ak4 Z 1  L2 ak4 Z 2
m2
2
24
…(46)
with
L0 a1k  ak2 , L1a1k 
1
m1
, L2 a1k  0
1  dp1  t   a1k 
1 2
1
3 2

   k1  k2  31  xk   3 2  xk  xk  
m1  dt  m1
L0 ak2 

ak2
ak3 
ak4
1
3 2
c

c

k

3

x

x

 1 2
c 
2
2 k
k 
 m1 2
m1
m1 
 
  

L1ak2    12   c1  c2  , L2 ak2   2   c2  ; L0 ak3  ak4 , L1ak3  0, L2 ak3  2
m2
 m1 
 m2 m1 
ak2
1  dp2  t   a1k 
3
1 2

k

3

x

x

c 


2
2 k
k 
 m1 2
m2  dt  m1 
L0 ak4 

ak3 
ak4
3
1 2
3 2
k

k

3

x

x

3

x





2
3
2
k
k
3
k
 m  c2  c3 
m1 
1
  
 
L1ak4   1   c2  , L2 ak4    22   c2  c3 
 m2 m1 
 m2 
…(47)
The random variables Z i and W i ; i  1,2 are as defined in equation 26. In further work we take
that the samples of p1 (t ) and p2 (t ) are simulated using a truncated Fourier representation (Soong
and Grigoriu 1993) as
pi  t  
n
 a
ij
j 1
cos  j t  bij sin  j t ; i  1, 2
where aij , bij 
n
j 1
…(48)
; i  1,2 are a set of zero mean, mutually independent Gaussian random variables
 j 1
with
 aij2 
 bij2 
 
2
ij
 S  d . In the first step of the present analysis, we need to assimilate
ii
j
the measured data into the mathematical model for given realizations of p1 (t ) and p2 (t ) . This can
be achieved in implementation by carrying out the assimilation for fixed values
of aij , bij 
n
j 1
; i  1,2 . In the numerical work it is assumed that
25
m1  10kg, m2  15kg, k1  1kN/m, k2  2kN/m, k3  1.5kN/m,1  2  0.05, P1  100N, P2  200N,
1  2 N / m3 , 2  4 N / m3 ,3  5N / m3 , 1  10, 2  20, 1  0.01N, 2  0.02N,
T =32.0s, t0  10s and =0.0210s .
8.2.1 Case 1
Here we consider the system in equation 46 and take the measurement model to be given
by y  tk   x1k   k . The measurement noise is assumed to have a standard deviation of 0.0310 m.
Thus, in this example, we have nonlinear process equation and linear measurement model with
additive Gaussian noises. The filtering problem here is not amenable for solution via the Kalman
filter, while the SIS filter and the bootstrap filters are applicable. We consider the problem of
estimating the extreme value PDF of u2  t  conditioned on the measurements on u1  t  using the
method described in section 7. In implementing the filters, 200 particles were used for SIS and
bootstrap filters. Figure 9 shows the time variation of E  ak(3)|k  . The estimate of the correlation
coefficient of ak(3)|k  E  ak(3)|k  , denoted by   tr , tr    , is shown in figure 10 for different values
of tr and  . These estimates are obtained using 5000 sample Monte Carlo simulations. Based on
the results in figures 9 and 10, it may be inferred that the process ak(3)|k has a time varying mean
and a random component that becomes stationary as time becomes large. Using the Hasofer and
Wang hypothesis test on peaks in ak(3)|k  E  ak(3)|k  , the domain of attraction of the extremes was
determined to be Gumbel at 5% significance level
(data from SIS filter: N  35, k  9,W  1224.916;
data from bootstrap filter : N  73, k  13,W  1060.412) . The parameters of the PDF-s were
estimated using 1000 samples of ak(3)|k  E  ak(3)|k  and the resulting PDF-s for displacement
response are shown in figure 11. For sake of reference the unconditional PU2,max   using 5000
sample brute force Monte Carlo simulation is also shown on this figure. It may be noted that the
estimation of PU2,max  | y1:N  based on brute force Monte Carlo approach in this case requires
prohibitive computational effort and this has not been attempted here.
26
8.2.2 Case 2
Here we consider the system as in the case 1(section 8.2.1) except that we now consider the
reaction transferred to the left support as the quantity being measured. Thus the measurement
equation is obtained as y  tk   k1x1k  c1xk2  1  x1k   k ; k  1,2,
3
, N where, standard deviation
of  k is 31.1134N. This would mean that both the process and measurement equations here are
nonlinear in nature and, consequently, we employ the bootstrap filter to estimate the hidden
variables. Again, we focus attention on estimating the extreme value PDF of u2  t  now
conditioned on the measurements on the reaction R1  t  . It was verified that the response ak(3)|k had
a time varying mean and a random component that can be taken to be stationary for large times.
The W-statistic of the Hasofer-Wang test on data on ak(3)|k  E  ak(3)|k  was obtained as 1044.046
 N  73, k  13 leading to the acceptance of the Gumbel model for the extremes at 5%
significance level. Following the estimation of parameters of the Gumbel PDF using 1000
samples of estimates of ak(3)|k  E  ak(3)|k  , the conditional PDF of extremes of ak(3)|k is obtained and
this is shown in figure 12 along with the unconditional PU2,max   estimated using 5000 samples.
8.3 Hysteretic nonlinear mdof system under white noise excitations
Here we consider systems with hereditary nonlinearities and demonstrate how measurements
from more than one sensor could be assimilated within particle filtering strategy. Towards
realizing this goal we again consider the system shown in figure 8 with the modification that the
spring k3 is now modeled as possessing memory dependent nonlinear characteristics fashioned
after Bouc’s model (Wen 1976). The springs k1 and k2 are taken to possess cubic force
displacement characteristics. Thus, this system can be considered to be archetypal of a structure
with both geometric and material nonlinearities. The excitations p1 (t ) and p2 (t ) are modeled as a
pair of zero mean, stationary, Gaussian random processes with auto-psd-s and acf-s given
respectively by
27
Sii   
 2pi
 i2   2


; Rii     2pi 2i exp  i |  | ; i  1, 2
…(49)
We obtain samples of p1 (t ) and p2 (t ) as steady state outputs of linear systems driven by white
noise inputs. Accordingly, we get the governing equations as
m1u1  c1u1  c2  u1  u2   k1u1  1u13  k2  u1  u2    2  u1  u2   p1  t   w1  t 
3
m2u2  c2  u2  u1   c3u2  k2  u2  u1    2  u2  u1   k3 u2  k3 z 1     p2  t   w2  t 
3
z   | u2 | z | z |n 1   u2 | z |n  Au2  w3  t 
…(50)
p1  1 p1  w4  t 
p2   2 p2  w5  t 
Here z is an internal state variable which imparts memory dependence to the force displacement
characteristic of spring k3 and, by assigning different values to the model
parameters  ,  , n ,  and A , different shapes of the hysteresis loops could be obtained (Wen,
1976). The noise terms wi  t i 1 are taken to be independent and to have zero mean and
5
 wi (t ) wi (t   )   i2   ; i  1, 2,
,5 . We now interpret equation 50 as an Ito SDE with the state
vector given by x  t   u1  t  u1  t  u2  t  u2  t  z  t  p1  t  p2  t  and subsequently obtain
t
the discretized set of equations as:
28
1
x1k 1  x1k  a1k   L0 a1k  2  L1a1k Z 1  L2 a1k Z 2  L3a1k Z 3  L4 a1k Z 4  L5 a1k Z 5
2
xk21  xk2  ak2  
1
1
W 1  L0 ak2  2  L1ak2 Z 1  L2 ak2 Z 2  L3ak2 Z 3  L4 ak2 Z 4  L5 ak2 Z 5
m1
2
1
xk31  xk3  ak3   L0 ak3 2  L1ak3Z 1  L2 ak3Z 2  L3ak3Z 3  L4 ak3 Z 4  L5 ak3Z 5
2
xk41  xk4  ak4  
2
1
W 2  L0 ak4  2  L1ak4 Z 1  L2 ak4 Z 2  L3ak4 Z 3  L4 ak4 Z 4  L5 ak4 Z 5
m2
2
1
xk51  xk5  ak5    3W 3  L0 ak5  2  L1ak5 Z 1  L2 ak5 Z 2  L3ak5 Z 3  L4 ak5 Z 4  L5ak5Z 5
2
1
xk61  xk6  ak6    4 W 4  L0 ak6  2  L1ak6 Z 1  L2 ak6 Z 2  L3 ak6 Z 3  L4 ak6 Z 4  L5 ak6 Z 5
2
1
xk71  xk7  ak7    5 W 5  L0 ak7  2  L1ak7 Z 1  L2 ak7 Z 2  L3ak7 Z 3  L4 ak7 Z 4  L5ak7 Z 5
2
with
29
…(51)
L0 a1k  ak2 , L1 a1k 
L0 ak2  
1
m1
, L2 a1k  0, L3 a1k  0, L4 a1k  0, L5 a1k  0
a1k 
ak2
ak3 
ak4
ak5
1 2
1
3 2
1
3 2
k

k

3

x

3

x

x

c

c


k

3

x

x


c





1
2
1 k 
2  k
k  
2  k
k  
2
 m1 1 2 m1  2
 m1
m1 
m1
 
  

L1 ak2    12   c1  c2  , L2 ak2    2   c2  , L3 ak2  0, L4 ak2  4 , L5 ak2  0
m1
 m2 m1 
 m1 
L0 ak3  ak4 , L1 ak3  0, L2 ak3 
L0 ak4  
a1k
m2
2
m2
, L3 ak3  0, L4 ak3  0, L5 ak3  0
2
3
 k  3  x 3  x1 2   ak  c   ak
2
k
k
2
 2
 m
m2
2
4
5
 k  k   3  x 3  x1 2   ak c  c   ak  k 1    
3
2
k
k

 2
 m 2 3 m  3
1
2
 
  


L1 ak4    1   c2  , L2 ak4    22   c2  c3  , L3 ak4  3 k3 1    , L4 ak4  0, L5 ak4  5
m2
m2
 m2 m1 
 m2 
L0 ak5  ak4   xk5 | xk5 |n 1 sgn  xk4    | xk5 |n  A
 ak5   | xk4 || xk5 |n 1   n  1 | xk4 | xk5 | xk5 |n  2 sgn  xk5    nxk4 | xk5 |n 1 sgn  xk5  

 22
 2  xk4  xk5 | xk5 |n 1 

2m 
2
2

   n  1 | x 4 || x 5 |n  2 sgn x 5    n  1 n  2  | x 4 | x 5 | x 5 |n 3 sgn x 5
 k
 k
k
k
k
k
k

2 

 3    n  1 | xk4 || xk5 |n  2 sgn  xk5   2  n  1 | xk4 | xk5 | xk5 |n  2   xk5 
2 

2
4
5 n 2
sgn  xk5   2  nxk4 | xk5 |n 1   xk5 
   n  n  1 xk | xk |


L1 ak5  0, L2 ak5   22
 m2

 
2






5
5 n 1
4
5 n
   xk | xk | sgn  xk    | xk |  A ,

L3 ak5   3   | xk4 || xk5 |n 1   n  1 | xk4 | xk5 | x 5 |n  2 sgn  xk5    nxk4 | xk5 |n  2 sgn  xk5   , L4 ak5  0, L5 ak5  0
L0 ak6  ak61 , L1 ak6  0, L2 ak6  0, L3 ak6  0, L4 ak6   41 , L5 ak6  0
L0 ak7  ak7 2 , L1 ak7  0, L2 ak7  0, L3 ak7  0, L4 ak7  0, L5 ak7   4 2
…(52)
The random variables Z i and W i ; i  1, 2, ,5 are as defined in equation 26. The random
variables Z i , W i 
5
i4
originate from the discretization of the random processes p1 (t ) and
30
p2 (t ) . In the first step of the solution procedure, we need to assimilate the measurements for
given realizations of the processes p1 (t ) and p2 (t ) . This is achieved in the numerical work by
assimilating measurements for fixed values of numerically simulated Z i , W i 
5
i4
. It may
also be noted that the right hand side of the above equations contain Dirac’s delta functions.
Their presence is indicative of the fact that the drift term in equation 50 has modulus function of
the form | z | which is non-differentiable at z  0 . Even though the event z  0 , has a measure
zero of occurrence, its occurrence is ideally required to be detected when the solutions are sought
in the strong form. The theory behind detection of such events, in the context of SDE-s, appears
to be still an open area of research and we have not addressed this issue in our work. Therefore,
in the numerical work, we either altogether drop these terms or, alternatively, replace the Dirac
delta function by a Gaussian probability density function with zero mean and arbitrarily small
standard deviation. It was observed in the numerical studies that the estimates of the states and
the estimate of parameter were not notably influenced by this choice. In the numerical work we
take m1  1.0kg, m2  1.5kg, k1  0.1kN / m, k2  0.2kN/m, k3  0.15kN/m, 1  2 0.0 5,
1  2, 2  4,   0.05,   0.5,   0.5, A  1, n  3,1  10, 2  20, T  15.6s, t0  2s,   0.0042s
 1  0.01N, 2  0.02N, 3  0.01N, 4  10.0N and  5  20.0N . The measurements are assumed
to be made on u1  t  and R1  t  leading to
y1  tk   x1k  1k
y2  tk   k x  k1  x
1
1 k

1 3
k
…(53,54)
 c x  2 k
2
1 k
The question of estimating the conditional extreme value PDF of ak(3)|k (i.e., displacement at 2nd
dof) over duration of 13.6s is considered. The standard deviations of the two measurement noises
are taken to be 0.0037m and 0.3741N, respectively. The problem on hand is tackled using
bootstrap filter with 200 particles. The plot of E  ak(3)|k  and covariance of ak(3)|k  E  ak(3)|k  are
shown in figures 13 and 14. The results in these figures again indicate that ak(3)|k could be
construed as a random process with time varying mean and a random component that becomes
stationary as time becomes large. Based on the Hasofer and Wang test, the hypothesis that the
31
domain of attraction of ak(3)|k  E  ak(3)|k  is Gumbel was acceptable at 5% significance level
 N  893, k  45,W  239.703 . Figure 15 shows the conditional PDF of extremes of
ak(3)|k obtained using 800 samples. For sake of comparison, the results on PDF of extremes of
ak(3)|k are also provided when measurements are taken to include only displacement (equation 53)
and only the reaction transferred to the left support (equation 54). Clearly, the estimate of the
conditional extreme value PDF, and hence the updated reliability of the system, is significantly
influenced by the details of measurements made.
8.4 System with tangent stiffness under white noise excitation
In the previous examples it was observed that the Gumbel model emerged as the acceptable PDF
for modeling the extremes. This may not always be the case. In order to illustrate this we
consider an sdof system with tangent stiffness with the governing process equation of the form
u  2u 
2d0 2

 u 
tan 
  p(t )  w(t )
 2d0 
…(55)
where  is the damping coefficient and  is a frequency-like parameter. The motion of such
system is bounded between (-d0,d0) and, consequently, Gumbel models are unlikely to be
acceptable for the extremes of displacement response especially for large levels of excitation. In
the present study we consider the excitation p  t  also to be a white noise and obtain the
governing discretized equations for the system states x  u u as
t
1
x1k 1  x1k  a1k   L0 a1k  2  L1a1k Z 1  L2 a1k Z 2
2
0 1
2
1 1
L ak  ak , L ak   , L2 a1k  
xk21  xk2  ak2   W 1  W 2 
1 0 2 2 1 2 1 2 2 2
L ak   L ak Z  L ak Z
2
L0 ak2  a1k  2 sec 2  x1k 2d    ak2  2  , L1ak2  2 , L2 ak2  2
In the rest of the formulation the following notations are used:
32
…(56)
4d02
2
 p(t ) p(t   )      ;  w(t ) w(t   )      ; 
and n0  2 2
4 2
 0
2
2
2
0
…(57)
The response of this system under white noise excitations has been studied earlier by Klein
(1964) by using Fokker-Planck equation approach and the results on the steady state pdf of
u  t  for different values of system parameters are shown in figure 16. The bounded nature of
system response is clearly evident from this figure. This may also be inferred from the phase
plane plot of sample of the system response shown in figure 17. It can be shown that as n0  ,
the pdf of the displacement approaches a uni-modal Gaussian-like distribution, and, as n0  0, it
approaches a near rectangular distribution (figure 16). It may thus be expected that the extremes
of the displacement response here could be Gumbel for large values of n0 and Weibull for
smaller values of n0 . In the present study we consider the problem of estimating the extreme
value PDF of ak(1)|k (i.e., displacement) when measurements have been made on the velocity
response u  t  . As in the previous example, we note that the random variables Z 2 and
W 2 originate from the excitation p(t) and, in the first step of the solution procedure, we
assimilate the measurements for fixed values of Z 2 and W 2 . Also, it was verified that E  ak(1)|k 
had a time varying mean and ak(1)|k  E  ak(1)|k  could be approximated as being weakly stationary. In
the numerical work it is assumed that   0.05,   2 rad/s,   10.0 N,
  0.01N, n0  1, T  30.0s and t0  5s ; the standard deviation of the measurement noise and is
taken to be 1.0908 m/s. The state estimation problem here is tackled using bootstrap filter with
200 particles. The results of Hasofer and Wang hypothesis test on ak(1)|k  E  ak(1)|k  , depending on
the samples used, favored the possibility that the domain of attraction of the extremes of ak(1)|k to
be Gumbel or Weibull. Consequently, based on a sample of 800 realizations of extremes of
ak(1)|k over 25 s, both Gumbel and Weibull models were fitted for the extreme value PDF and these
results are shown in figure 18. For sake of reference, the estimate of unconditional pdf of
extreme of u  t  over 25 s using 5000 samples Monte Carlo simulation is also shown in this
figure.
33
9. Discussion and closing remarks
Monte Carlo simulation based Bayesian filtering tools offer effective framework to assimilate
measured data into postulated mathematical models for existing structures. These tools have
wide ranging capabilities that include capability to treat structural nonlinearities, nonlinear
measurement models, imperfections in mathematical models and measurements, non-Gaussian
nature of uncertainties, and spatially incomplete measurements. They have great promise for
application in the area of structural health monitoring - a potential that is yet to be fully explored
in the field of structural engineering research.
The present paper utilizes these tools, in conjunction with statistical methods for extreme value
analysis, to develop a strategy to update the time variant reliability of existing nonlinear
dynamical systems based on assimilation of noisy measurements. The limit states considered in
the study depend on measured and (or) unmeasured response quantities and are taken to be linear
functions of system states. The study treats the governing differential equations as a set of
stochastic differential equations and employs Ito-Taylor expansion to discretize the process
equations. This results in acceptable modeling of strong forms of response random processes – a
feature that is crucial for application of data based extreme value analysis. The external
excitations are modeled as random processes with known model parameters. This enables
simulation of ensemble of these excitations. When the excitations are also measured, the samples
of excitations are obtained based on conditional simulations. The illustrative examples
considered include systems with geometric and (or) material nonlinearities. The models for
random excitations considered include filtered white noise models as well as excitations that are
differentiable to any desired order.
The following are the main findings of this study:
1. The conditional expectation of the response ( ak |k ) can be viewed as a random process
with each of its realization corresponding to a realization of the external forces. This
random process is shown to have a time varying mean and a random component which
can be approximated as being weakly stationary. The time varying nature of the mean
essentially originates from the assimilation of the measured responses.
34
2. A conservative bound on the updated model for the PDF of extreme response can be
obtained by making the assumption max hk ak |K   max E  hk ak |K   max hks ak |K  .
k0  k  N
k0  k  N
k0  k  N
3. For linear state space models with additive Gaussian noises an approximate analytical
solution to the problem of reliability model updating can be obtained by using theory of
level crossings of Gaussian random processes.
4. Gumbel models for extremes are found to be generally acceptable although the
illustration in section 8.4 presents an example in which response is bounded and hence
Weibull models for extremes become appropriate.
The present study has involved illustrations in which the measured data are synthetically
simulated. While this strategy helps to demonstrate the acceptable performance of the proposed
methods, it is clearly required to test the methods by applying them to laboratory and field
applications. This also calls for treatment of more elaborate models for structural behaviour
based, for example, on finite element method. These aspects of the work are being currently
researched upon by the present authors. The present study has assumed that the user would be
able to specify the numerical values for the covariance matrices of process and measurement
noises which appear in the state estimation problem. The selection of these covariances is indeed
a practical problem and the success of the dynamic state estimation technique depends upon the
correct tuning of these parameters. This calls for engineering judgment on confidence that can be
placed on the mathematical model, assessment of electronic noise in the sensor and accuracy of
sensor calibration. In principle, these parameters could also be identified in the identification step
which is assumed to precede the considered problem of reliability model updating. This, again
calls for further work.
35
References
1. I F Alves and C Neves, 2006, Testing extreme value conditions - an overview and recent
approaches, International Conference on Mathematical Statistical Modeling in honor of E
Castillo, June 28-30, 1-16.
2. A Beskos, O Papaspiliopoulos, G Roberts and P Fearnhead, 2006, Exact and
computationally efficient likelihood-based inference for discretely observed diffusion
process, Journal of Royal Society, B, 68(2), 1-29.
3. V V Bolotin, 1984, Random vibrations of elastic systems, Martinus Nijhoff Publishers,
Hague.
4. E Castillo, 1988, Extreme value theory in engineering, Academic Press, Boston.
5. F N Catbas, M Susoy, and D M Frangopol, 2008, Structural health monitoring and
reliability estimation: Long span truss bridge application with environmental monitoring
data, Engineering Structures, 30, 2347-2359.
6. J Ching, J L Beck and K A Porter, 2006, Bayesian state and parameter estimation of
uncertain dynamical systems, Probabilistic Engineering Mechanics, 21, 81–96.
7. J Ching and J L Beck, 2007, Real-time reliability estimation for serviceability limit states
in structures with uncertain dynamic excitation and incomplete output data, Probabilistic
Engineering Mechanics 22, 50–62.
8. D Diamantidis, 1987, On the Reliability Assessment of Existing Structures, Engineering
Structures, 9, 177-182.
9. Ditlevsen and H O Madsen, 1996, Structural reliability methods. John Wiley & Sons,
Chichester.
10. A Doucet, 1998, On sequential simulation-based methods for Bayesian filtering,
Technical report CUED/F-INFENG/TR.310 (1998), Department of Engineering,
University of Cambridge, UK.
36
11. A Doucet, S Godsill and C Andrie, 2000, On sequential Monte Carlo sampling methods
for Bayesian filtering. Stat. Comput. 0, 197–208.
12. J F Dunne and M Ghanbari, 2001, Efficient extreme value prediction for nonlinear beam
vibrations using measured random response histories, Nonlinear dynamics, 24, 71-101.
13. B Ellingwood, 1996, Reliability-based condition assessment and LRFD for existing
structures, Structural Safety 18(2), 67-80.
14. P Fearnhead, O Papaspiliopoulos and G Roberts, 2008, Particle filters for partially
observed diffusions, Journal of Royal Statistics, B, 70(2), 755-777.
15. S Ghosh, C S Manohar and D Roy, 2008, Sequential importance sampling filters with a
new proposal distribution for parameter identification of structural systems, Proceedings
of Royal Society of London, A, 464, 25-47.
16. N J Gordon, D J Salmond and A F M Smith, 1993, Novel approach to nonlinear/nonGaussian Bayesian state estimation, IEE Proceedings. F140: 107-113.
17. A M A Hasofer and Z Wang, 1992, A test for extreme value domain of attraction, Journal
of the American Statistical Association, 87(417), 171-177.
18. M Hoshiya, 1995, Kriging and conditional simulation of Gaussian field, Journal of
Engineering Mechanics, 121(2), 181-186.
19. R.A. Ibrahim, 1985, Parametric Random Vibration, John Wiley, New York.
20. R E Kalman, 1960, A new approach to linear filtering and prediction problems,
Transactions of ASME, Journal of Basic Engineering, 82 (Series D), 35-45.
21. H Kameda and H Morikawa, 1994, Conditioned stochastic processes for conditional
random fields, Journal of Engineering Mechanics, 4, 855-875.
22. G H Klein, 1964, Random excitation of a nonlinear system with tangent elasticity
characteristics, Journal of Acoustical Society of America, 36(11), 2095-2105.
37
23. P E Kloeden and E Platen, 1992, Numerical solution of stochastic differential equations,
Springer, Berlin.
24. S Kotz and S Nadarajah, 2000, Extreme value distributions, Imperial College Press,
London.
25. Y K Lin and C Q Cai, 1995, Probabilistic Structural Dynamics: Advanced Theory and
Applications, McGraw Hill, New York.
26. J A Lechner, E Simiu, and N A Heckert, 1993, Assessment of ‘peaks over threshold’
methods for estimating extreme value distribution tails, Structural Safety, 12, 305-314.
27. C S Manohar and D Roy, 2006, Nonlinear structure system identification using Monte
Carlo filters, Sadhana, Academy Proceedings in Engineering, Indian Academy of
Science, 31(4), 399-427.
28. R E Melchers, 1999, Structural reliability analysis and prediction, Second edition, John
Wiley, Chichester.
29. R E Melchers, 2001, Assessment of existing structures-approaches and research needs,
Journal of Structural Engineering ASCE, 127(4), 406-411.
30. Y Mori and M Nonaka, 2001, LRFD for assessment of deteriorating existing structures,
Structural Safety, 23, 297-313.
31. V Namdeo and C S Manohar, 2007, Nonlinear structural dynamical system identification
using adaptive particle filters, Journal of Sound and Vibrations, 306, 524-563.
32. N C Nigam, 1983, Introduction to random vibrations, MIT Press, Cambridge.
33. A Naess and P H Clausen, 2001, Combination of peaks-over-threshold and bootstrapping
methods for extreme value prediction, Structural Safety, 315-330.
34. B Radhika, S S Panda and C S Manohar, 2008, Time variant reliability analysis using
data based extreme value analysis, Computer Modeling in Engineering and Sciences,
27(1-2),79-110.
38
35. B Ristic, S Arulampallam and N Gordon, 2004, Beyond the Kalman filter: Particle filters
for tracking applications, Artech House, Boston.
36. R Sajeeb, C S Manohar and D Roy, 2007, Control of nonlinear structural dynamical
systems with noise using particle filters, Journal of Sound and Vibration, 306, 25, 111135.
37. H C Shah and W M Dong, 1984, Reliability assessment of existing buildings subjected to
probabilistic earthquake loadings, Soil Dynamics and Earthquake Engineering, 3(1), 3541.
38. A F M Smith and A E Gelfand, 1992, Bayesian statistics without tears: a samplingresampling perspective. American Statistician, 46, 84-88.
39. T T Soong and M Grigoriu, 1993, Random vibration of mechanical and structural
systems, PTR Prentice-Hall, New Jersey.
40. H Tanizaki, 1996, Nonlinear filters: estimation and applications 2nd edition, Springer
Verlag, Berlin.
41. D V Val and M G Stewart, 2002, Safety factors for assessment of existing structures,
ASCE Journal of Structural Engineering, 128(2), 258-265.
42. E H Vanmarcke and A G Fenton, 1991, Conditional simulation of local fields of
earthquake ground motion, Structural Safety, 10(1-3), 247-264.
43. Y K Wen, 1976, Method for random vibration of hysteretic systems, ASCE Journal of
Engineering Mechanic, 102(2), 249–263.
44. J T P Yao, 1985, Safety and reliability of existing structures. Pitman, Boston.
45. J T P Yao and H G Natke, 1994, Damage detection and reliability evaluation of existing
structures, Structural Safety, 15, 3-16.
39
Appendix A: State estimation algorithms used in the study.
A.1 Kalman filter
The Kalman filter provides a Gaussian estimate for the filtering pdf p  xk | y1:k  for linearGaussian state space models obtained as simplified form of equations 28 and 29 as
xk 1  zk xk  Fk  Wk ; k  0,1, 2,..., N
…(A1,2)
yk  f k xk  k ; k  1, 2,..., N
The sizes of various quantities appearing in the above equation are as follows:
xk : d 1; zk : d  d ;Wk : d 1; yk : s 1; f k : s  d ; and  k : s 1 . The filtering algorithm provides a set of
recursive relations for the evolution of the mean ak |k and covariance  k |k given by (Hwang and
Brown 1997)
K k   k|k f kt  f k  k|k f kt   
1
ak|k  ak|k 1  K k  yk  f k ak|k 1 
 k|k   I  K k f k   k|k
…(A3-7)
ak1|k  zk ak|k  Fk
 k 1|k  zk  k|k zkt  W
Here a superscript ‘-’ indicates the prior estimate; K k is the Kalman gain matrix and I is an s  s
identity matrix. The implementation of the algorithm requires the specification of  0 and a0 .
Typically, one takes  0  I and a0  0 . It may be noted that the above set of equations are exact in
nature.
The estimate of conditional mean ak |k k 1 in equation A4 corresponds to a sample of Fk k 1 . For
N
an ensemble of

Fke

N

N
Ns
k 1 e 1
where, N s is the number of samples, an ensemble of conditional mean
40
a
e
k |k

N

Ns
k 1 e 1
is obtained. From the expression for conditional mean in equation A4 the quantities
t
aˆk |k  EF  ak |k  and ˆ k |k  EF  ak |k  aˆk |k  ak |k  aˆk |k   can be shown to be given by



  z ˆ

aˆk |k   I  K k f k  zk aˆk |k 1  Fˆk  K k yk ; Fˆk  E  Fk 
ˆ k |k   I  K k f k
k
t
ˆF
k |k 1 zk  Pk
  I  Kk f k t ; PˆkF

 E  Fk  Fˆk


Fk  Fˆk

t
…(A8,9)


In the above equation EF   denotes expectation across the ensemble of excitation
A.2 Sequential importance sampling (SIS) filter
When the dynamic state estimation problem is characterized by nonlinear process equation,
linear measurement model and additive Gaussian process and measurement noises, the
estimation of p  xk | y1:k  could be based on the sequential importance sampling filter that
employs an ideal importance sampling density function as proposed by Doucet (1998). Thus,
equations 28 and 29 here have the form
xk 1  zk  xk   Wk ; k  0,1, 2,..., N
…(A10,11)
yk  f k xk  k ; k  1, 2,..., N
The sizes of various quantities appearing in the above equation are as follows:
xk : d 1; zk  xk  : d 1;Wk : d 1; yk : s 1; f k : s  d ; and  k : s 1 We focus attention on evaluating the
expectation of the form
I [ k ]  E p  ( x0:k ) | y1:k    ( x0:k ) p( x0:k | y1:k )dx0:k
…(A12)
Here E p  denotes the expectation operator defined with respect to the pdf p( x0:k | y1:k ) . The
basic idea here consists of choosing an importance sampling pdf, denoted by  ( x0:k | y1:k ) , such
that, if p( x0:k | y1:k )  0   ( x0:k | y1:k )  0. In this case the integral in equation A12 can be
written as
41
I [ k ]   ( x0:k )
p ( x0:k | y1:k )
 ( x0:k | y1:k )dx0:k
 ( x0:k | y1:k )

  ( x0:k )w ( x0:k ) ( x0:k | y1:k )dx0:k  E  ( x0:k ) w ( x0:k ) | y1:k
*
*
…(A13)


Here E  denotes expectation with respect to the pdf  ( x0:k | y1:k ) . If {xi ,0:n }iN1 are samples
drawn from  ( x0:k | y1:k ) , and

1 N
IˆN [ 0:k ]    ( xi ,0:k ) w* ( xi ,0:k ) with
N i 1
…(A14)
p( xi ,0:k | y0:k ) p( y0:k | xi ,0:k ) p( xi ,0:k )
p( y0:k | xi ,0:k ) p( xi ,0:k )
*
w ( xi , 0:k ) 


 ( xi ,0:k | y1:k )  ( xi ,0:k | y1:k ) p( y1:k )  ( xi ,0:k | y1:k )  p( y0:k | x0:k ) p( x0:k )dx0:k
IˆN [ k ]  I [ k ] almost surely (Doucet 1998). It may be emphasized
it can be shown that lim

N 
that the evaluation of the normalization constant p( y1:k )   p( y 0:k | x0:k ) p ( x0:k )dx0:k in equation
A14 presents a major difficulty in implementing the above procedure since it is seldom possible
to express this constant in a closed form. However, if one evaluates this integral too via
importance sampling, one gets,

p( y0:k | xi ,0:k ) p( xi , 0:k )
1 N
  ( xi , 0:k )
N i1
 ( xi ,0:k | y1:k )

IˆN [ 0:k ] 

N p( y
1
0:k | x j , 0:k ) p ( x j , 0:k )

N j 1
 ( x j ,0:k | y1:k )

N
 ( x
i 1
i , 0:k
) w( xi , 0:k )

N
 w( x
j 1
j , 0:k

N
~ ( x ) …(A15)
  ( xi , 0:k ) w
i , 0:k
)
i 1
~ ( x ) given by
with the new weights w
i , 0:k
~( x ) 
w
i , 0: k
w( xi , 0:k )
with w( xi , 0:k ) 

N
 w( x
j 1
j , 0:k
)
p ( y1:k | xi , 0:k ) p ( xi , 0:k )
 ( xi ,0:k | y1:k )
.
… (A16)
If one selects the importance sampling density function of the form
 ( x0:k | y1:k )   ( x0:k 1 | y1:k ) ( xk | x0:k 1 , y1:k )
…(A17)
42
it would then be possible to compute the importance weights in a recursive manner. To see this,
k
k
j 1
j 1
note that p( y1:k | x0:k )   p( y j | x j ) and p( x0:k )  p( x0 ) p( x j | x j 1 ) . Using these facts along
with equation A17 in second of equation A16 we get
w( x0:k ) 
k
k
j 1
j 1
 p( y j | x j ) p( x0 ) p( x j | x j 1 )
 ( x0:k 1 | y1:k ) ( x k | x0:k 1 , y1:k )
k 1

 p( y
k 1
j
j 1
| x j ) p( x0 ) p( x j | x j 1 )
j 1
 ( x0:k 1 | y1:k )
 w( x0:k 1 )
p( y k | x k ) p( x k | x k 1 )
 ( x k | x0:k 1 , y1:k )
…(A18)
p( y k | x k ) p( x k | x k 1 )
 ( x k | x0:k 1 , y1:k )
One of the difficulties in implementing this algorithm in practice is that after a few time steps,
most weights become small in comparison to a few which become nearly equal to unity. This
implies that most of the computational effort gets wasted on trajectories which do not eventually
contribute to the final estimate. This problem is widely discussed in the existing literature and
one way to remedy this problem is to introduce the notion of an effective sample size given by
N eff 

N
1
…(A19)
2
w~i,0:k 
i 1

It may be noted that if all weights are equal, Neff  N and if all weights excepting one are
zero, N eff  1 . Thus, when the effective sample at any time step falls below a threshold N thres , a
step involving resampling is implemented with a view to mitigate the effect of depletion of
samples. Furthermore, the choice of the importance sampling density function plays a crucial
role in the success of the algorithm. It can be shown that  ( x k | xi ,0:k 1, y1:k )  p( x k | xi ,k 1 , y k ) is
the importance function, which minimizes the variance of the non-normalized weights,
wi*,k conditioned upon xi , 0:k 1 and y1:k . In general, it is not feasible to deduce the ideal importance
43
sampling density function p( xk | yk , xi ,k 1 ) . However, when the process and measurement
equations assume the form
xk 1  zk ( xk )  Wk ; Wk  N 0 nW 1
W 
…(A20,21)
yk  f k xk   k ;  k  N 0 n 1  
it can then be shown that the ideal importance density function is a normal density function with
mean m k and covariance  given by

mk 1   W1 zk ( xk )  f kt 1 yk )

…(A22,23)
1
1  W
zk ( xk )  f kt 1 f k
It may be noted that x  N (a, B) denotes that x is a vector of normal random variables with
mean vector a and covariance matrix B. When dealing with more general forms of process and
measurement equations, as in equations 28 and 29, more elaborate strategies such as the one
proposed by Ghosh et al., (2008) could be used.
The following steps are adopted to implement the SIS filter:
(a) Set k  0 ; draw samples  xi ,0 i 1 from an assumed initial density function p  x0  .
N
(b) Set k  k 1.
(c) Sample  xˆi ,0 i 1 from   xk | xi ,0:k 1 , y1:k  and set xˆi ,0:k   xi ,0:k 1 , xˆi , k  for i [1, N ].
N
(d) For i [1, N ] compute the importance weights
wi*,k  wi*,k 1  p  yk | xˆi ,k  p  xˆi ,k | xˆi ,k 1    xˆi ,k | xˆi ,0:k 1 , y1:k 
(e) For i  [1, N ], normalize the weights wi , k  wi*, k
N
w
*
j ,k
j 1
(f) Evaluate the expectations of interest,
44
.
N
Mean: ak |k   xi ,k wi ,k
i 1
Covariance:  k |k    xi ,k  ak |k  xi ,k  ak |k  wi ,k .
N
t
i 1
 w  .
N
(g) Evaluate the effective sample size N eff  1
i 1
2
i ,0:k
(h) If N eff  N thres , where Nthres is the threshold of the sample size, set xi ,0:k  xˆi ,0:k , go to step
(b) if k  N , otherwise end.
(i) If N eff  N thres , implement the following:
(i) For i  [1, N ], sample an index j  i  distributed according to the discrete
distribution with N elements satisfying P  j  i   l   wl , k for l  1, 2,..., N .
(ii) For i  [1, N ], set xi ,0:k  xˆ j i ,0:k and wi ,k  1 N .
(iii) Go to step (b) if k  N , otherwise end.
In implementing this method it is assumed that p  x0  is known.
A.3 Bootstrap filter
Here we consider the governing equations in their general form (equations 28 and 29) and permit
nonlinearity in both process and measurement equations. The noises could be additive and (or)
multiplicative and they could be non-Gaussian as well. One alternative to deal with this class of
problems is to employ the bootstrap filtering method as developed by Gordon et al., (1993). This
method is based on an earlier result by Smith and Gelfand (1992). This result can be stated as

follows: suppose that samples xi* (i ) : i  1,2,, N are available from a continuous density


function G  x  and that samples are required from the pdf proportional to L  x  G  x  , where
L  x  is a known function. The theorem states that a sample drawn from the discrete distribution
45



N

over x (i ) : i  1,2,, N with probability mass function L( xk* (i )) /  L( xk* ( j )) on x k* (i ) , tends in
*
i
j 1

distribution to the required density as N  . The algorithm for implementing the filter is as
follows:
 
1. Set k  0 . Generate xi*, 0



N
*
i , k 1 i 1
2. Obtain x

N
from the initial pdf p( x0 ) and wi ,0 i 1 from p(w).
i 1

N
by using the process equation 28.
3. Consider the k measurement y k . Define qi 
th

N
p( yk | xi*, k )
 p( y
j 1

k
.
*
j,k
|x )



4. Define the probability mass function P xk ( j )  xi*,k  qi ; generate N samples {xi , k }iN1
from this discrete distribution.

5. Evaluate ak |k

1 N
1 N
   xi , k and k |k  
( xi , k  ak |k )t ( xi , k  ak |k ) .

N i 1
N  1 i 1
6. Set k  k 1 and go to step 2 if k  N ; otherwise, stop.
The central contention in the formulation of this algorithm is that the samples xi , k i 1 drawn in

N
step 4 above are approximately distributed as per the required pdf p( xk | y 0:k ). The justification
for this is provided by considering the updation equation 35 in conjunction with the result due to
Smith and Gelfand with G  x  identified with p( xk | y1:k 1 ) and L(x) with p( xk | y k ) . This
method has the advantage of being applicable to nonlinear process and measurement equations,
and, non-Gaussian noise models. It is also important to note that the samples in this algorithm
naturally get concentrated near regions of high probability densities.
46
Figure 1. Sample realization of a random process with the extremes marked.
47
(a)
(b)
Figure 2. Example in section 8.1.1; Expectation of conditional mean of system responses (a)
displacement (b) velocity; ak(i|k)  i  2,3 refers to the estimate of the system responses
corresponding to the measurements yi  tk  ; i  1,2. .
48
(a)
(b)
Figure 3. Example in section 8.1.1; Variance of conditional mean of system responses (a)
displacement; (b) velocity.
49
Figure 4. Example in section 8.1.1; Conditional PDF of extremes of the hidden displacement
variable using methods 1-4.
50
(a)
(b)
Figure 5. Example in section 8.1.1; Estimation of parameters of Gumbel distribution with
different sample sizes; (a) parameter  : 1 for displacement;  2 for velocity ; (b) parameter  : 1 for
displacement; 2 for velocity.
51
Figure 6. Example in section 8.1.2; Conditional PDF of extremes of the hidden displacement
variable; Case: 1 – when the force is not measured (section 8.1.1); Case: 2 - when the force is
measured (section 8.1.2).
52
Figure 7. Example in section 8.1.3; Conditional PDF of extremes of the hidden displacement
variable; Case: 1 – when the force is not measured (section 8.1.1); Case: 3 - when the
measurements are sampled at twice the discretization time interval of the process equation
(section 8.1.3).
f1(t)
1 k1
2 k2
k3 3
m2
m1
R1(t)
c1
Left
support
f2(t)
R2(t)
c2
u1(t)
c3
u2(t)
Figure 8. System considered in examples 8.2 and 8.3.
53
Right
support
Figure 9. Example in 8.2.1; Expectation of conditional mean of displacement response u2  t  .
Figure 10. Example in section 8.2.1; Correlation coefficient of random component of conditional
mean of displacement response u2 (t ) ;  0  0.0210s .
54
Figure 11. Example in section 8.2.1; Conditional PDF of extremes of the hidden variable
u2  t  based on SIS and bootstrap filter estimates; Uncond-MCS refers to PDF of extremes of
u2  t  when no measurements are available.
55
Figure 12. Example in section 8.2.2; Conditional PDF of extremes of the hidden variable
u2  t  based on bootstrap filter estimates; Uncond-MCS refers to PDF of extremes of u2  t  when
no measurements are available.
56
Figure 13. Example in section 8.3; Expectation of conditional mean of displacement response
u2  t  .
57
Figure 14. Example in section 8.3; Correlation coefficient of random component of conditional
mean of displacement response u2 (t ) ;  0  0.0042s .
58
Figure 15. Example in section 8.3; Conditional PDF of extremes of the hidden variable
u2  t  based on bootstrap filter estimates; model 1: measurement on R1  t  and u1  t  ; model 2:
measurement on u1  t  alone; model 3: measurement on R1  t  alone.
59
Figure 16. Example in section 8.4; The exact pdf of the displacement in the steady state for
system under white noise excitation (based on Klein 1964).
Figure 17. Example in section 8.4; Phase plane plot of sample response of the system under
white noise excitation.
60
Figure 18. Example in section 8.4; Conditional PDF of extremes of the hidden variable
u2  t  based on bootstrap filter estimates; Uncond-MCS refers to PDF of extremes of u2  t  when
no measurements are available.
61
Table.1 Upper and lower percentage points for W in the Hasofer and Wang hypothesis test (from
Hasofer and Wang 1992)
Sample
size
k
13
14
15
16
17
18
19
20
21
22
25
30
40
50
60
80
100
200
500
0.01
333.8
305.6
288.1
272.8
258.6
247.4
237.1
226.9
217.7
210.1
189.4
165.0
131.7
110.4
96.4
76.5
63.6
35.4
16.3
Lower tail (WL)
0.025 0.05
395.6 457.7
361.2 416.3
339.4 389.3
321.3 368.8
306.0 349.6
290.2 332.0
277.6 316.6
265.4 302.5
255.0 289.1
244.9 278.3
219.3 248.8
189.2 212.6
149.5 166.1
124.1 136.6
106.9 116.6
84.2 91.0
69.2 74.3
37.6 39.9
16.9 17.4
Percentage level
Upper tail (WU)
0.10
0.10
0.05
0.025
538.4 1599.5 1827.5 1998.5
489.6 1406.3 1589.4 1749.6
456.5 1275.6 1450.4 1596.2
431.5 1183.3 1334.9 1469.4
407.7 1095.0 1234.7 1357.9
386.5 1017.1 1143.9 1259.7
368.5 950.4 1064.5 1171.1
350.7 888.8 997.3 1096.5
333.6 836.0 938.1 1027.2
321.0 787.3 881.9 968.7
285.4 672.1 747.4 821.1
241.7 535.5 593.6 650.7
186.4 377.1 413.6 451.5
151.8 289.5 316.5 340.2
128.8 232.9 253.8 270.8
98.9 168.4 179.9 190.7
80.2 129.6 138.4 146.4
41.9
59.4
62.2
64.8
18.0
22.6
23.2
23.8
62
0.01
2204.6
1934.4
1768.1
1629.5
1508.3
1397.7
1301.4
1220.9
1145.8
1077.2
910.8
719.9
497.0
371.1
293.3
205.5
155.5
68.1
24.5