Probabilistic Engineering Mechanics 38 (2014) 7787Contents lists available at ScienceDirectProbabilistic Engineering Mechanicshttp://d0266-89

n CorrE-mjournal homepage: www.elsevier.com/locate/probengmechIterated stochastic filters with additive updates for dynamic systemidentification: Annealing-type iterations and the filter bank

Tara Raveendran a, Debasish Roy b,n, Ram Mohan Vasu a

a Department of Instrumentation and Applied Physics, Indian Institute of Science, Bangalore, Indiab Computational Mechanics Lab, Department of Civil Engineering, Indian Institute of Science, Bangalore, Indiaa r t i c l e i n f o

Article history:Received 13 November 2013Received in revised form23 August 2014Accepted 17 September 2014Available online 20 September 2014

Keywords:Stochastic filteringIterated additive updateEnsemble square root filterGaussian sum approximationKushnerStratonovich equationDynamic system identificationx.doi.org/10.1016/j.probengmech.2014.09.00220/& Elsevier Ltd. All rights reserved.

esponding author.ail address: royd@civil.iisc.ernet.in (D. Roy).a b s t r a c t

A nonlinear stochastic filtering scheme based on a Gaussian sum representation of the filtering densityand an annealing-type iterative update, which is additive and uses an artificial diffusion parameter, isproposed. The additive nature of the update relieves the problem of weight collapse often encounteredwith filters employing weighted particle based empirical approximation to the filtering density. Theproposed Monte Carlo filter bank conforms in structure to the parent nonlinear filtering (KushnerStratonovich) equation and possesses excellent mixing properties enabling adequate exploration of thephase space of the state vector. The performance of the filter bank, presently assessed against a fewcarefully chosen numerical examples, provide ample evidence of its remarkable performance in terms offilter convergence and estimation accuracy vis--vis most other competing filters especially in higherdimensional dynamic system identification problems including cases that may demand estimatingrelatively minor variations in the parameter values from their reference states.

& Elsevier Ltd. All rights reserved.1. Introduction

Dynamic system identification aims at estimating the hiddenstate processes that solve the system or process model, often inthe form of stochastic ordinary differential equations (SDEs), givena set of noisy partial observations, which are typically character-ized by the observation SDEs whose drift fields are knownfunctions of the system (process) states. The 'estimate of a state'often stands for its mean (first moment) with respect to thefiltering probability density function (PDF) of the instantaneousstate conditioned on the observation history till the current time.Variants of Bayesian filtering, which provide a computationallyfeasible route in obtaining the filtering PDF, typically involve atwo-stage recursive procedure consisting of the prediction andupdate stages. While the prediction stage recursively propagatesthe process or system model in time, the predicted solution ismodified in the update stage in order to assimilate the currentlyavailable observation consistent with a recursive form of thegeneralized Bayes' formula [1] and thus characterize (marginalsof) the filtering PDF (also called the posterior PDF). The Kalmanfilter (KF) has been a major breakthrough [2], providing for ananalytical scheme to arrive at the exact posterior PDF for alinear Gaussian dynamic state space model. Nonlinear dynamicalsystems with non-Gaussian additive/multiplicative noises mayalso be dealt with, albeit sub-optimally, with the extended Kalmanfilter (EKF) that employs linearized approximations to the signal-observation dynamics. But the EKF and its variants [3] mayperform quite poorly where the dynamics are significantly non-linear due to the imprecise Gaussian approximation of the transi-tion law of the signal-observation process. Moreover, unless anextensive tuning operation for the process noise covariance isperformed, the evolution of the analytical error covariance in theKF/EKF may become divergent.

With the rapid emergence of cheaply available computingresources, sequential Monte Carlo (SMC) methods such as particlefilters (PFs), which provide asymptotically optimal estimates fornonlinear and non-Gaussian filtering problems, are being increas-ingly used. PFs rely on a first order Markov model for the time-discretized signal-observation processes and implement a recur-sive Bayesian update by Monte Carlo (MC) simulations [4]. Overa given time-step, they use particles, which are independentlysampled and weighted realizations of the random variables(representing the instantaneous filtered states) to approximatethe continuous filtering PDF by random (empirical) measures.Here the weights define the likelihood of the current observationgiven the predicted particles available through time-integration ofthe process dynamics. Being free from the approximations invol-ving linearizations, PFs are endowed with the universality thathave seen their applications in the context of a wide-rangingarray of noisy nonlinear dynamical systems encountered in target

www.sciencedirect.com/science/journal/02668920www.elsevier.com/locate/probengmechhttp://dx.doi.org/10.1016/j.probengmech.2014.09.002http://dx.doi.org/10.1016/j.probengmech.2014.09.002http://dx.doi.org/10.1016/j.probengmech.2014.09.002http://crossmark.crossref.org/dialog/?doi=10.1016/j.probengmech.2014.09.002&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.probengmech.2014.09.002&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1016/j.probengmech.2014.09.002&domain=pdfmailto:royd@civil.iisc.ernet.inhttp://dx.doi.org/10.1016/j.probengmech.2014.09.002

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 778778tracking, digital communications, chemical engineering etc. [57].Efforts to use a form of analyticity characteristic of the KF withinthe framework of PFs have also led to the development of semi-analytical PFs [8]. Such PFs transform the nonlinear system/observations to an ensemble of piecewise linearized equations sothat the KF can be used for each linearized system to yield a familyof conditionally Gaussian posterior PDFs whose weighted sumyield the filtering PDF. The accruing advantage of reduced sam-pling variance however comes at the cost of a substantivelyincreased computational overhead as the current observationmust be repetitively assimilated for each linearized system.

Despite its universality and algorithmic simplicity, a PF is besetwith the generic problem of particle impoverishment in applica-tions involving higher dimensional process models as the weightstend to collapse to a point mass [9]. Indeed, the necessary samplesize needed to counter such weight degeneracy could be practi-cally unattainable even with the best of computing resources. Away out of this degeneracy, which is also the primary focus of thisarticle, could be provided through additive updates that may becontrasted with the multiplicative, weight-based updates usedwith the PFs. One such prominent example, the ensemble Kalmanfilter (EnKF) that may be loosely viewed as an MC version of the KFimplementing additive gain-type updates, has indeed found ap-plications in higher dimensional filtering problems in oceano-graphic and atmospheric modeling [10]. The EnKF uses an en-semble of system states predicted through the process dynamics,thus avoiding the EKF-type Gaussian closure through linearizationin the prediction stage. However the additive update term, derivedbased on an MC-version of the Kalman gain formula, brings back aGaussian closure approximation to the empirical filtering density.

As a sequel to our recent work on an iterated gain-basedstochastic filter (IGSF) [14] incorporating an iterative form ofadditive updates on the predicted particles, our present aim is topropose a substantively modified version of the algorithm in orderto introduce an explicit non-Gaussian representation of the filter-ing density and an improved exploration of the process state spaceduring the iterated updating stage. As with the IGSF, the iterationsover a given time-step here are also aimed at driving the innova-tion term to a zero-mean random variable. This is consistent withthe original aim of a stochastic filter as described by the KushnerStratonovich (KS) equation [11], which is generally achieved bydesigning the temporal recursion such that the innovation processis reduced to a zero-mean martingale. The first part of the currentproposal is to develop the iterative and additive update through anannealing-type parameterization using an artificial diffusion para-meter (ADP). In addition, non-Gaussian representations of theprediction and filtering densities are now provided throughGaussian sums. Specifically, the iterations in the update stagerequire ADP-parameterized repetitive computations of gain-likecoefficient matrices Ki

l (i being the temporal recursion step and lthe iteration index for a fixed i), consistent with the nonlinear KSequation, with the initial guess Ki

0 evaluated on similar lines as inan ensemble square root filter (EnSRF) [15]. In addition to captur-ing non-Gaussianity in the posterior density, the Gaussian sumfilter bank [16] also helps exploring the phase space of the statevariables better and the added diversity in the particles enableseasier adaptation of the process dynamics with the measuredvariables. The ADP, which may be lowered to zero over successiveiterations at a much faster rate (allowing even for a discontinuousscheduling) than is feasible with the conventional simulatedannealing, also helps enhance the so called 'mixing property'[17] of the iterative update kernels. An attempt is made to provideadequate numerical evidence of the enhanced filter performancewith the introduction of some of these novel elements.2. Statement of the problem

Let F( , , ) be a complete probability space with F t, 0,t beingthe -algebra generated by all the noise processes involved in thepresentation to follow at a given time t. The collection of sets = F s t: { : 0 }t s defines the so called increasing 'filtration' as tincreases. Also the time interval of interest [0, ] is discretized as

= < < =t t t t0 ..... ....i L0 1 with = +t t t( , ]i i i 1 . The process modeldescribing the evolution of the so-called 'hidden' states of acontinuous-time dynamical system containing an additive Brow-nian noise term (which may, among others, account for modellingerrors) may be represented by the Ito stochastic differentialequation (SDE) [18]

= +dX t X t t t dt G t dB t( ) ( ( ), ( ), ) ( ) ( ) (2.1)

where the state vector X t( ) nx is a time-continuous signal, + : n n nx x is the system transition function,

= B t B t r q( ) { ( ): [1, ]}r( ) is a q-dimensional vector of independentlyevolving zero-mean Ft-Brownian motion processes with =B (0) 0r( )

and E = B t B s t s{( ( ) ( )) }r r( ) ( ) 2 , where E denotes the expectationwith respect to the probability measure , and + G: n rx is thediffusion or volatility co-efficient matrix. System identification typi-cally involves estimating the uncertain or inadequately knownparameters t( ) n in the system model and a solution, withinthe stochastic filtering framework, requires declaring t( ) as addi-tional states. The original state space model (SSM) is thus augmentedby allowing t( ) to artificially evolve as a vector Brownian motion, asdepicted through the following system of zero-drift SDEs:

= d t G dB t( ) ( ) (2.2)

where G n n is the diffusion coefficient matrix and B t( )

n , azero-mean Brownian noise vector process. In fact, restricting Eq. (2.2)over different time sub-intervals =+t t i{( , ], 0, 1, ... }i i 1 , t( ) may beinterpreted as a collection of local Brownian motions (i.e. differentmean vectors over different sub-intervals), or, more generally, as localmartingales (see [1] for a definition of local martingales). Theaugmented state vector (with parameters as additional states) is

now denoted as ~ = = ~ | = + X X X j J J n n: [ , ] { [1, ]} ;T T T j J

x( )

.The response of the dynamic system is partially observed throughthe noisy and continuous measurement process given by the SDEs(written below in the integral form):

= ~ + *Z t A X s ds G B t( ) ( , ) ( ) (2.3a)t

z z0

or more appropriately, since the measurements arrive in a time-discrete manner, by a discrete algebraic counterpart of the aboveequation:

= = ~ ++ + + +Z t Z X t G( ): ( , ) (2.3b)i i i i z1 1 1 1

Here~ = ~+ +X X t( )i i1 1 , = Z Z m d{ : [1, ]}

m d( ) denotes the vectorof measurements, is a d-dimensional vector of N(0, 1) indepen-dent normal random variables with coefficient matrix Gz

d d.Thus the covariance matrix of the discrete measurement noisevector Gz is given by

G Gz zT d d. The measurement vector

function:

= ~ + + H X t k d: { ( , ): ; [1, ]}k n n( ) x

maps the signal process~X t( ) tod. Let = Z Z Z: { , , }i i

T1: 1 denote the

set of measurement vectors till tti. The process Eqs. (2.1) and(2.2) may now be combined to yield the nonlinear SSM:

~~ = ~ + ~ ~dX t X t dt G t dB t( ) ( , ) ( ) ( ) (2.4)

where ~ ~= j J: { , [1, ]}j J( ) and ~ G t( ) J J are respectivelythe nonlinear drift vector and the diffusion coefficient matrix. The

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 7787 79required growth conditions for the existence of weak solutions[18] to the above SDEs are assumed to hold. The nonlinear systemof SDEs (2.4), whose solution yields the predicted response for thefilter, may be semi-analytically or numerically integrated by anyavailable method. We have presently used a derivative-free andexplicit local (piecewise in time) linearization [19] of ~ leading tothe following locally linearized form:

~~ = ~ ~ + ~ ~dX t X t X dt G t dB t( ) ( , ) ( ) ( ) (2.5)L

i iL

for +t t t( , ]i i 1 with~ ~X t X t( ) ( )L

(only in law, implying that weakstochastic solutions are admissible) as +t 0i . In view of the factthat the error due to local linearization may be weakly correctedvia a Girsanov change of measure [20,21] and in order to maintainnotational simplicity, we will not distinguish between the locallylinearized and true solutions to the process SDEs thereby denotingthe predicted solution generally as

~X t( ).

The nonlinear filtering problem is solved by computingthe conditional expectation (estimate) = | =^x E x t N( ): ( ( ) )t t

ZP

| < ^E x t Z s s t( ( ) { ( ); 0 })P , xJ , of the system states given the

measurement history till the current instant, or, more generally,

the associated conditional distribution ^ |P x t N( ( ) )tZ (or its density p,

called the filtering or posterior PDF, if it exists). Recall that thecontinuous filtration at time t generated by the measurement-algebra is given by = | N F s t: { }t

ZsZ . For a given t, the filtered

random variable drawn from (the numerical approximation to) the

posterior distribution P will be denoted by~X t( ), which may be

distinguished from the predicted process denoted by~X t( ). It is

assumed that the distributions P and P are absolutely continuouswith respect to each other.3. Filtering scheme

In order to maintain consistency of the filtering formulationwith the KS filtering equation, it is instructive to write the latter inan integral form for +t t t( , ]i i 1 as

= +

+ =

L ds

H x s s dI

( ) ( ) ( ( ))

{ (M ( )) ( ( ( ), )) ( )}(3.1)

t tt

t

s s

m

d

t

t

s sm

sm

s sm

1

( ) ( ) ( )

ii

i

where x( ( ))t , xJ , is the estimate of x( ) at time t and,

without a loss of generality, is taken to be a scalar-valuedfunction for a simpler exposition. Choosing, for instance, a family

of such functions ~ = ~ X X j J( ) , 1 ,jj( ) ( ) one can determine the

estimate of (components of) the augmented state vector~X .

= I x Z H x t{ ( )}: { ( ( , ))}tm

tm

tm( ) ( ) ( ) is referred to as the innovation

vector. The operators Lt and Mtm( ), m d[1, ], are defined through

~ = ~

+= = =

L x tx

x xx t

x

x( ( )):

12

( )( )

( , )( )

,(3.2a)

tk

J

j

Jkj

k jj

Jj

j1 1

( )2

( ) ( )1

( )

( )

and

=M x H x t x( ( )) ( , ) ( ) (3.2b)tm m( ) ( )

with

~ = ~ ~t G t G t( ): [ ( ) ( ) ] (3.2c)kj T kj( ) ( )

The mathematical problem of stochastic filtering may be statedas recursively updating the predicted stochastic process ~X t( ( )) to

the filtered stochastic process ~X t( ( )) such that Itm( )is reduced to a

zero-mean Brownian motion (or, more generally, a zero-meanmartingale) as t . This is also called the filtered martingaleproblem. Consistent with our recent work on iterative stochasticfiltering [14], the second term on the RHS of the KS Eq. (3.1) isapproximated as

= |^L ds L ds E L N ds( ( )) ( ( )) ( ( ) ) ,t

t

s s t

t

i s t

t

s iZ

Pi i i

where =(. ): (. )i ti and =N N:iZ

tZi

contains the measurementhistory only up to =t ti. By so uncoupling the prediction andupdating stages over the current time step +t t t( , ]i i 1 in theproposed filter via this approximation, the first two terms on theRHS of Eq. (3.1) yields the expectation ~E X t( ( ( ))P involving thepredicted process X t( ) that solves Eq. (2.5) with the initial

condition~ = ~X Xi i. Obtaining

~E X t( ( ( ))P in this way is therefore

the same as that determined via Dynkin's formula [18] applied to

the process SDE over +t t t( , ]i i 1 with = X Xi i. Subject to the aboveapproximation, the predicted solution X t( ( )) in the proposedfilter may thus be interpreted as being correspondent to the firsttwo terms on the RHS of the KS Eq. (3.1). Now, the third term onthe RHS involves a sum of integrals over weighted innovations,wherein each scalar innovation It

m( )is representative of the predic-tion error if the predicted process X t( ) is used to compute themeasurement function H X t( , )m( ) . Accordingly, the coefficientweights

= M H x t x( ) ( ( )) ( ( , )) ( ), ,tm

t tm

tm

tJ( ) ( ) ( )

may be looked upon as the coefficient gain terms (which, uponintegration, build up the gain matrix) that drive I{ }t

m( ) to a set of dzero-mean scalar Brownian motions. However, unlike a standardnon-iterative filter, the iterative filtering scheme additionallyintroduces an inner iteration parameter . Using iterations over , the aim is to drive the -parameterized innovation process

= I Z H x t{ : ( ( , ))}tm

tm

tm

,( ) ( )

,( ) to a zero-mean Brownian motion

with the statistical characteristics of the measurement noise fora given t (typically = +t ti 1). Ideally, one thus hopes at recoveringthe true estimate (one that satisfies the KS equation at time t) as =

( ) lim ( )t t, . This may be contrasted with the usual goal of

stochastic filtering wherein one merely aims at recovering ( )tonly for sufficiently large t. However, with the current unavail-ability of a mathematically consistent yet computationally feasiblestopping criterion, we typically stop the -iteration for maxwhere max is generally chosen in a problem-specific manner.

Given the additive nature of the gain-type updates yielding anun-weighted sequence of particle systems converging in measureto the solution of the KS equation [12,13], one may draw a directanalogy of the form of the KS equation with variants of theensemble Kalman filters, the EnKFs, even as most of the lattermethods are limited to Gaussian approximations to the filteringPDF. Nevertheless, since the EnKF provides a ready framework thatcombines Monte Carlo simulations with KF-like additive updates,it may be prudent to borrow part of the algorithmic setup/terminology whilst developing the iterative gain-based stochasticfilter bank (IGSF Bank) scheme. This is accomplished by iterativelyrefining a gain matrix (Ki

l, l being the iteration index correspond-ing to l, where = 0 ...0 1 max is an ordering of the iterationparameter ) consistent with the additively split prediction andupdate terms in the previously noted approximation to the KSequation. Over a given time step +t t( , ]i i 1 , the IGSF-Bank isimplemented in two stages. An EnKF-like prediction-cum-updatestep forms the first stage which is followed by an iterative updatestage employing an annealing-type ADP sequence ( l, l beingthe iteration index) and an iterated gain-sequence Ki

l within aGaussian sum approximation framework.

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 7787803.1. Gaussian sum approximation and filter bank

Gaussian sum approximation makes use of a weighted sum ofGaussian densities (the Gaussian mixture model) to approximatethe posterior PDF, wherein a tractable solution is arrived at fromthe sufficient statistics. [16] Let x MN( ; , ) denote a normal densityfor x J with the mean vector M and covariance matrix ,assumed to be non-singular.

The Gaussian mixture approximation theorem [22] suggests that, ifthe covariance matrices associated with temporally localized particlediffusion have relatively small norms, then the filtering and predictiondensities may be adequately approximated as Gaussian mixtures.Based on this approximation, one can construct a bank of NG Gaussianmixands wherein, at the start of recursion, i.e. at =t 0, equal numberof particles are drawn from each Gaussian PDF in the mixand (i.e. theset of particles is split into NG subsets, each containing = N N: / Gparticles drawn from each mixand in the mixture) so as to populatethe ensemble. This enables a tagging of subsets of particles with theappropriate terms in the Gaussian sum all through the recursion/iteration stages. Also, the mixands are assigned equal weights

= = w w t N: ( ) 1/ G0( ) ( )

0 to begin with.

3.2. Prediction and zeroth update

This stage of the algorithm consists of the conventional

propagation and the initial (zeroth) update steps. Let

=X{ }u i u( ),

( )

1be the sub-ensemble, consisting of realizations of the Gaussian

random variable Xi

( )

associated with the th mixand N( [1, ])Gin the Gaussian sum approximation of the last filtering density at=t ti. Let the sample mean and sample covariance of this sub-

ensemble be respectively denoted by < ~ >

Xi( )

and

i

( ). For arriving

at the predicted particles starting with Xi

( )

as the initial condition,numerical integration of the process SDEs may be accomplishedthrough any available numerical/semi-analytical scheme, e.g. thelocally transversal linearization (LTL) [23] or the phase spacelinearization (PSL) [24,25] etc. Specifically using the PSL, i.e. basedon the linearized SDE (2.5), the predicted solution corresponding

to the th mixand, X t( )

( ), +t t t( , ]i i 1 , is

~ ~

~

~ = ~ +

~ ~

X t t t X t t

s t G s dB s

( ) exp( ( )) ( ) exp( ( ))

[exp( ( )) ( ) ] ( )(3.4)

i i i

t

t

i

( ) ( )

i

The above solution at = +t ti 1 thus generates a sub-ensemble of

predicted particles + =X{ }u i u( ), 1( )

1, whose sample mean vector isgiven by

< > =

+ = +X X

1(3.5)i u u i1

( )

1 ( ), 1( )

In order to describe the zeroth update scheme, the so calledprediction error (anomaly) and predicted measurement anomalymatrices (corresponding to the thmixand) are respectively defined as

=

< > < >

+ + + + +S X X X X

11[( ), ... , ( )]

(3.6)i i i i i1( )

(1), 1( )

1( )

( ), 1( )

1( )

=

< >

< >

+ + + + +

+ + + +

S X t X t

X t X t

( )1

1[( ( , ) ( , ) ),...,

( ( , ) ( , ) )] (3.7)

z i i i i i

i i i i

( )1 (1), 1

( )1 1

( )1

( ), 1( )

1 1( )

1

The above are used in generating the zeroth iterate, which isinput to the iterative updating procedure outlined in the nextsection. The zeroth update (filtering) step, fashioned after thecurrently adopted particle-based form of the EnKF, takes in thecurrent observation and generates the updated particles at = +t ti 1via

~ = ~ + ~

=

=

+ + + + + +KX X Z X t

u

N

( ( , ))

[1, ],

[1, ], (3.8)

u i u i i i u i i

G

( ), 1

0( )

( ), 1( )

1( ),0

1 ( ), 1( )

1

Here +Ki 1( ),0 is the initial gain matrix (with the superscript '0'

without brackets indicating the zeroth update) defined through

= + + + + + +K S S S S( ) (( ) ( ) ) (3.9)i i z i

Tz i z i

TZ1

( ),01

( ) ( )1

( )1

( )1

1

Thus, if =

+ +X X( ( )): ( ( ))i i1,( )

10

( )

0denotes the mean of the

th filtered PDF component in the Gaussian sum following thezeroth iterate as above, then its sample approximation, uponaveraging over the associated sub-ensemble with elements, is

given by < >

+ +X X( ( )) ( )i u i10

( )

( ), 1

0( )

. Here < >. is the averagingoperator as in Eq. 3.5.

3.3. ADP-based iterative update scheme

The second stage involves repetitive updates over the zerothfiltered particles via iterations. The iterative updates entail, forevery given time +i 1( i.e. +ti 1), an iterating index =l 1, 2, ... thatmay be construed as being correspondent to a discrete iterationparameter set < < ={ : ... }l 1 max , with = 00 (corresponding tothe zeroth update) and denoting the maximum number ofiterations. Let = (. ): (. )t t

l, l

denote the estimate at the end ofthe l th iterate. Then the iterative updates, upon averaging overprocess noise, may be written at = +t ti 1 as

=

+ +

+

+

+ +

+K

X X

Z X t

( ( )) E ( ( ( )))

(1 ) ( ( ( ), )) (3.10a)

il

i

li

li i

li

1

( )

P( )

1( ), 1

1 11

( )

1

wherei stands for the explicit time marching map (as obtainablefrom the locally linearized solution Eq. 3.4) upon integrating theprocess SDE such that

= = +X X X( ) ( ( )) ( ( )).i i i1( ) ( ) ( )

Moreover +Kil

1( ), 1 denotes the l 1th iterative update of the gain

matrix corresponding to the th mixand in the Gaussian sum and =: ( )l l is the iteration dependent ADP that can be likened to theannealing parameter typically used with simulated annealing (SA)applied to a Markov chain [26]. Unlike the SAwhere the temperatureis recursively reduced to zero whilst evolving a single Markov chain,the sequence | +{ }l l l1 is used in the present filter to evolve anensemble of =N NG pseudo-Markov chains in so that, for a given t ,the chains proceed in a controlled way to finally arrive at anensemble that drives the innovation process It, to a zero-meanBrownian martingale. Analogous to the SA, one must then supple-ment the iterative schemewith 1, the initial ADP, and an appropriateschedule to drive l to zero over successive iterations. However,given that a Monte Carlo scheme (based on an ensemble of

approximations to Xt, ) supplemented with a Gaussian sum repre-sentation is in itself a means to efficaciously explore the phasespace, the conservative or even geometric annealing schedules,typically used with standard SA algorithms or MCMC filters [27]and involving a very large number of iterations, need not beadhered to here. Indeed, as the numerical experiments confirm,considerable flexibility with the scheduling of the ADP (as well as inthe choice of 1) is possible with the proposed filter. Specifically,the exponentially decaying schedule adopted here is given by

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 7787 81 = =+ + l: ( ) /exp ( )l

ll1

1 , with 1 determined in a problem-specific

manner through a few trial runs (alternatively, it may be computedas the empirical average of an instantaneously defined cost func-tional evaluated at the particle locations corresponding to the zerothupdate). The scheduled sequence of the ADP { }l is intended toprovide an additional handle in controlling the mixing property ofthe iterative update kernels and ensure that the process variablesvisit every finite subset of the phase space of interest sufficientlyfrequently [17]. Finally, the uncoupled nature of the prediction anditerative update, as adopted here, is reflected in the fact that thelatter affects only the third term on the RHS of the KS Eq. (3.1) whilstleaving unaltered the prediction part of the estimate (i.e. the first twoterms on the RHS of Eq. 3.1).

Replacing the expectation operators appearing in +il1, +

il11 and

EP by appropriate sub-ensemble averages, Eq. (3.10a) may bemodified to go with MC simulations involving finite ensembles as

< > = < > + +

< >

+ +

+

+

+

KX X Z

X t

( ) ( ( )) (1 ) (

( , ) ) (3.10b)

u i

l

i ul

il

i

u i

l

i

( ), 1

( )

( )( )

1( ), 1

1

( ), 1

1( )

1

A particle based version of the above, as implemented in thiswork and applicable to the th sub-ensemble, becomes:

=

+ +

+

+

+ +

+K

X X

Z X t

u

( ) ( ( ))

(1 ) ( ( , ))

; [1, ] (3.11)

u i

l

i u

li

li u i

l

i

( ), 1

( )

( )( )

1( ), 1

1 ( ), 1

1( )

1

+X{ }u il

( ), 1

( )

denotes the ensemble of filtered particles = +t ti 1following the lth update. Now define the sub-ensemble wiseupdated prediction and measurement anomaly matrices respec-tively as

^ =

+

+

+

+

+

S X X

X X

11[( ),...,

( )] (3.12)

i

l

i

l

i

i

l

i

1

( ), 1

(1), 1

1( )

(1), 1( )

( ), 1

1( )

( ), 1( )

^ =

+ +

+ +

+

+ +

S X t Z

X t Z

( )1

1[( ( , ) ), ...,

( ( , ) )] (3.13)

z

l

i i

l

i i

i

l

i i

( ), 1

1 (1), 1

1( )

1 1

( ), 1

1( )

1 1

These are employed for evaluating the updated gain matrix+

Kil

1( ), 1 (used in Eq. 3.11) as

= ^ ^ ^ ^

+

+

+

+

+K S S S S( ) (( ) ( ) ) (3.14)i

li

l

z

l

iT

z

l

i z

l

iT

1( ), 1

1

( ), 1 ( ), 1

1

( ), 1

1

( ), 1

11

where the conventions ^ =

+ +S S:i i1( ),0

1( ) and ^ =

+ +S S( ) : ( )z i z i( ),0

1( )

1 areadopted. The mixand weights are then updated and normalized

using the particles

+X{ }u i( ), 1( )

, available after the last (i.e. th )iteration, as

+

+

~ =

< ~ > ^ ^

< ~ > ^ ^

=

+ +

+ + + + +

= + + + + +

w w

N Z X t S S

N Z X t S S

N

( ; ( , ) , ( ) ( ) )

( ; ( , ) , ( ) ( ) )

,

1,..., (3.15)

i i

i i i z i z iT

Z

Ni i i z i z i

TZ

G

1( )

1( )

1 1

( )

1

( ),

1

( ),

1

1 1 1

( )

1

( ),

1

( ),

1G

This is followed by weight normalization to re-define +wi 1( ) as:

=

+

+

= +

ww

w (3.16)i

iN

i1

( ) 1( )

1 1( )G

Also, by convention, the last updated particles corresponding to

the th mixand in the Gaussian sum are denoted as =

+X{ }:u i( ), 1( )

+X{ }u i( ), 1( )

before carrying them over to the next time step.Statistics estimation may now be performed based on the empiri-cal posterior PDF so obtained. Specifically, the sample estimate ofthe state vector at = +t ti 1 is given by

^ ^ = <

+ = + +

wX X (3.17)iN

i i1 1 1( ) ( )

1G

It may be noted that the current filter bank (henceforth calledthe IGSF Bank) reduces to the previously reported IGSF, [14] if

=N 1G and the ADP = 0l for l [1, ].

The pseudo-code for the implementation of the proposedfiltering scheme is given below for further expositional clarity.

3.3.1. InitializationStep 1 Choose an ensemble size, N.

Step 2 Discretize the time interval of interest T[0, ] into L time-

steps: = < < < =t t t T0 ... L0 1 .

Step 3 At =t t ,0 construct a Gaussian sum filter bank with NG

mixands and draw = N N: / G particles from each Gaus-sian PDF in the mixand, thereby generating the sub-

ensembles, =

=X{ }u t u( ),

( )

1

=X{ }u u( ),0

( )

1 where N[1, ]G .

Assign equal weights =w :0( ) =w t( )

N( )

01

Gto the mix-

ands. For each i L[1, ] (i.e. over every time step), thefollowing steps are recursively followed over all thesamples in the ensemble.3.3.2. Prediction and zeroth update

At =t ti,

=X{ }u i u( ),

( )

1 is the sub-ensemble, consisting of

realizations of the Gaussian random variable Xi

( )

associated

with the th mixand ( N[1, ]G ) in the Gaussian sum approx-imation of the last filtering density.Step 4. Obtain the sub-ensemble of predicted particles

+ =X{ }u i u( ), 1

( )1 using the predicted solution at = +t ti 1 via

Eq. (3.4).Step 5. Construct the prediction error (anomaly) and predictedmeasurement anomaly matrices as per Eqs. (3.6) and (3.7).Step 6. Evaluate the initial gain matrix +Ki 1

( ),0using Eq. (3.9)Step 7. Using the zeroth update (filtering) step as in Eq. (3.8),generate the updated particles corresponding to the zerothiterate at = +t ti 1.

3.3.3. ADP based iterative update schemeFor every given time = +t ti 1 and for =l 1, 2, ... , choose the

ADP vector { }l .

Step 8. Generate the sub-ensemble-wise updated prediction andmeasurement anomaly matrices according to Eqs. (3.12) and (3.13).Step 9. Evaluate the updated gain matrix +

Kil

1( ), 1 by Eq. (3.14)

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 778782Step 10. Generate

+X u N{ : [1, ], [1, ]}u il

G( ), 1

( )

, the ensem-

ble of filtered particles at = +t ti 1 following the lth update basedon Eq. (3.11).Step 11. If l , go to step 8, else continue to the next step.Step 12. Update and normalize the mixand weights using the

particles

+X{ }u i( ), 1( )

, available after the th iteration making useof Eqs. (3.15) and (3.16).Step 13. Evaluate the sample estimate of the state vector

^+Xi 1

at = +t ti 1 using Eq. (3.17).4. Numerical illustrations

For state estimation, the performance of the proposed filter isfirst illustrated via a 1-dimensional nonlinear system and a targettracking problem. Towards assessing the performance of the filterfor combined state-parameter estimation and by way of high-lighting its efficacy in resolving higher dimensional nonlinearfiltering problems, a multi-degree-of-freedom (MDOF) shearframe model is made use of as a final example.

4.1. 1-Dimensional nonlinear system with additive gaussian noise

A univariate nonstationary growth model with additiveGaussian noise is chosen as the first numerical example, where-in both the process and measurement equations are nonlinear.Similar examples have been widely used [5,2830] for theperformance evaluation of various particle filters. A discreteversion of the governing equation of the nonlinear system (or,equivalently, the predicted solution) considered here may bewritten as

= + + + +X X X i h G B( 8 cos( )) (4.1)i i i i1 1 22

where 1,2 and are scalar system parameters. Here = +h t ti i1 ,the time-step size, is uniformly chosen as 1 s and the referenceparameter values are given by = 0.21 , = 0.012 and = 1.2.Process noise variance is G2 and = +B B t B t: ( ) ( )i i i1 is theBrownian increment over h where B t( ) denotes a standardBrownian noise starting at zero. The state is estimated from theFig. 1. Time histories of RMSE of the estimates via IGmeasurement equation given by

= ++ +Z X G B( ) (4.2)i i z z i1 12

where the measurement noise variance is Gz2and B t( )z is another

standard Brownian motion. The performance of the IGSF Bank(always ADP-based, unless explicitly noted otherwise) is comparedwith the Gaussian sum particle filter (GSPF) [31] for state estima-tion. The root-mean-squared-error (RMSE) via the IGSF Bank andthe GSPF over 100 independent Monte Carlo runs are computedusing an ensemble size =N 1000 and with =N 10G . Here, theassumptions as in [28] are followed in that the reference initialstate is assumed to be a uniformly distributed random variable in

[0, 1] and prior state at =t 00 is taken as~ = ~ ~X X N(0.5, 2)0 0 . The

initiating ADP 1 and the number of updating iterations arechosen as 1 and 5 respectively. Fig. 1(a) and (b) show the timehistories of estimate RMSEs by the two filters for =G 0.01z

2 and 10respectively. Both use a constant process noise variance =G 102 .

The observed performance of the IGSF Bank, as evidenced fromthe results in Fig. 1, is indicative of improved estimation accuracyirrespective of the measurement noise level when compared tothe GSPF.4.2. Target tracking

In a target tracking problem, one typically estimates thetrajectory of a maneuvering target (i.e. position and velocity) fromthe noise-corrupted sensor bearing and range data. The dynamicmodel of the maneuvering target (not the process model, ratherthe one used to synthetically generate the measurement) adoptedhere in a discretized form is

= + ++ a m[ ] (4.3)i i i i1

where

=

1 0 00 1 0 00 0 10 0 0 1

and

=

0

0

0

0

12

2

12

2

= X X Y Yi v v iTis the state vector with X and Xv respectively

being the position and velocity of the moving target along theCartesian x-axis and Y , Yv denoting similar variables along theSF Bank and GSPF: (a) =G 0.012 and (b) =G 102 .

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 7787 83y-axis. Moreover,

= =

m

if t s

0

{ , }

otherwisei

iT

i1 2

is the acceleration vector that brings in maneuvering at the chosentime instants s t t[0, , ... , , ... , ]i1 , ai is the random accelerationvector of the target, currently characterized as Brownian, is aconstant sampling interval (state update period) and 0 2. Thesensor is situated at the origin x y( , )0 0 of the plane with the bearingangle and the distance from the sensor taken as the measurementsaccording to the measurement equation:

= +

++

+ ++

+

+

( )Z

Y Y X XY

tan

{( ) ( )}(4.4)

i

Y YX X

i i

i1

1

1 02

1 01

i

i

1 0

1 0

=+ +v v v{ , }i z z iT

1 1 2 1 is a zero mean white Gaussian sequence with

covariance G Gz zT diag G G([( ) ( ) ])z z112 222 . The actual initial state of

the target (based on which the noise-corrupted synthetic mea-surement is generated) is chosen to be at [0.5 m 3 ms1 1 mFig. 2. Estimated target tracks in the x-y plane by the IGSF Bank and the ASIR alongwith the true trajectory as reference.

Fig. 3. Evolutions of the RMSE of the (a) x-coordinate and (b) y-co1 ms1] in the Cartesian coordinates. From here it undergoes 3-legmanoeuvring sequences by taking sharp turns at 20 s, 30 s and60 s with respective accelerations [40ms2 40 ms2], [25 ms225 ms2] and [25 ms2 25 ms2] whilst moving alongstraight lines with constant velocities during the intervals till thetrajectory ends at 80 s. Assuming that the dynamic model (4.3)of the manoeuvring target is unknown, we consider a simplermotion model

= ++ wi i i1

for tracking, where wi, the random acceleration of the target, istaken as a zero-mean Gaussian process noise sequence withcovariance diag ([8m2s4 8 m2s4 ]). For initiating the filter, theprior state at =t 00 is taken as Gaussian with the mean vector [0 m40 ms1 0.2 m 0.075 ms1] (which is far away from the true state)and the sampling interval is set to 0.1 s. The measusrement noisecovariance is chosen as diag([0.2 rad2 35 m2 ]). Filter parametersfor the IGSF Bank are chosen as 110, = 10 and =N 5G . Theperformance of the proposed filter with an ensemble size of

=N 200 particles is compared with that of auxiliary samplingordinate of the tracked target via the IGSF Bank and the ASIR.

Fig. 4. An n-DOF shear frame model.

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 778784importance resampling filter (ASIR) [32] in Fig. 2, which reportsthe estimated tracks of the target states.

A comparison of the RMSEs of the estimated x and y positionsof the target is given in Fig. 3. The significantly improvedperformance of the IGSF bank over the ASIR in terms of fasterconvergence and estimation accuracy is evident, despite thesampling fluctuations inevitable with a small ensemble size. Asexpected, with a larger ensemble size the performance of ASIR alsoimproves.

.4.3. An MDOF shear frame model

A mechanical oscillator in the form of a multi-degree-of-free-dom (MDOF) shear frame model [34], schematically depicted inFig. 4, is chosen to evaluate the performance of the IGSF Bank forhigher dimensional state-cum-parameter identification problems.The governing differential equation of the model, after pre-multi-plying with the inverse of the mass matrix and with an additiveBrownian diffusion term, is formally represented as

' + + = +X CX SX F t G B t.

( ).( ) (4.5)

Denoting by * * *M C S, , the original mass, damping and stiffnessmatrices for the frame, = * *S M S[ ] 1 and = * *C M C[ ] 1 respectivelyFig. 5. Estimates of stiffness parameters for the 5-DOF shear frame mdenote the n n (mass-normalized) stiffness and viscous dampingmatrices and they are presently of the form

=

+ +

+

S

s s s

s s s s

s s s s

s s

0 0 0

0 0

0 ... .. ... 00 0

0 0 0 (4.6)n n n n

n n

1 2 2

2 2 3 3

1 1

=

+ +

+

C

c c c

c c c c

c c c c

c c

0 0 0

0 0

0 ... .. ... 00 0

0 0 0 (4.7)n n n n

n n

1 2 2

2 2 3 3

1 1

The mass-normalized deterministic force vector =F t( )* * = M F t f t[ ] ( ) { ( )}j n1 ( ) is derived [34] from an excitation

acting at the supports. The support excitation is here assumedto be harmonic, yielding nodal force components of the form

= f t f t j n( ) cos ( ) [1, ]j f( )

0 . Also, the mass-normalized noiseintensity matrix G is presently an n n diagonal matrix. Thedamping matrix C is constructed based on Rayleigh'sodel via (a) EnKF, (b) IGSF, (c) IGSF with ADP and (d) IGSF Bank.

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 7787 85(proportional) damping [34], i.e., = * + *C a M a S1 2 . a1 , a2 arethe proportionality constants so chosen to obtain an appropriatedamping mechanism for the example shear frame. Prior to theaugmentation with the parameter states, the state vector is given

by =

X X X X X X X,

., ,

., .. , ,

.n nT

(1) (1) (2) (2) ( ) ( ) and the initial condi-

tion is = X 0 R n02 . Note that the combined state-parameter

estimation problem is here a nonlinear filtering problem (e.g.nonlinear in the terms that contain the stiffness and dampingparameters as they are considered as additional states), eventhough the process dynamics (Eq. 4.5) would be strictly linear ifthe parameters were precisely known. Including the unknown(mass-normalized stiffness and damping) parameters as addi-tional states, the augmented process state vector is given by

~ = ~ X X X: [ , ] ,T T T n4 , with =n n2 denoting the parameter di-mension so that =J n4 . Given the ability of proposed filters (theIGSF Bank as well as the ADP-based IGSF) to work with lowmeasurement noise levels (a scenario commonly encounteredwith sophisticated measuring devices), the data (herein consistingonly of the noisy displacement DOFs) is synthetically generatedby adding a very low noise intensity (less than 1%) to allthe elements of displacement vector X . Owing to quicker weightcollapse, particle filters [33] typically perform very poorly withFig. 6. Estimates of stiffness parameters for a 20-DOF shear frame mlow-intensity measurement noises, employed here to improve theestimation quality (i.e. to avoid random oscillations owing tolarger variance in the measurement noise). For the record, itmay however be emphasized that the proposed filter works wellover a fairly broad range of noise intensities, from very small tolarge.

The performance of the IGSF Bank is compared with the EnKF(which is known to tackle relatively higher dimensional filteringproblems) and IGSF with ADP, i.e. the degenerate case of the IGSFbank with =N 1G . A realistic example of a 5-DOF shear frame model(equivalent to a 20-dimensional system for filtering) with mass andstiffness parameters respectively chosen as * = * = = *=m m m...1 2 5

2.1012 10 kg6 . and * = * = = *= s s s... 2.61251 2 5 10 N/m8 . The

Rayleigh damping mechanism as assumed for the example system(using the proportionality constants =a 0.051 and =a 0.022 ) yieldsthe damping coefficients * = * = = * c c c... 5.3 101 2 5

6 N s/m. Theshear frame (Fig. 4) being a stick model, the harmonic supportexcitation (in acceleration units) is transferred to the nodal DOF witha uniform amplitude. Thus each nodal force so obtained is

= f t t j( ) 0.1 cos (2 ) [1, 5]j( ) with the frequency of oscillationbeing 1 rad/s. While the ensemble size is consistently given by

=N 400, the other algorithmic parameters of relevance are fixed as=N 10G , = 10 and = 2

1 . In order to demonstrate the possibleodel via (a) EnKF, (b) IGSF, (c) IGSF with ADP and (d) IGSF Bank.

Fig. 7. Estimates of damping parameters for an 20-DOF shear frame model by (a) EnKF, (b) IGSF, (c) IGSF with ADP and (d) IGSF Bank.

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 778786flexibility in scheduling the sequence { }l , 1 for this example is notexponentially decreased as in the previous examples, but kept con-stant till the ( 1)thiterative update and discontinuously reduced tozero in the last iterate. The estimates of stiffness parameters via EnKF,IGSF (as reported in [14]), IGSF with ADP and the IGSF Bank are shownin Fig. 5. The superior performance features of the last two filters areagain clearly brought forth.

The contrast in the performance the proposed filters vis--vis afew existing ones may be further highlighted by considering thecombined state-parameter estimation of a 20-DOF shear frame( =n 20)-in fact an extension of the 5 DOF model consideredearlier-yielding an 80-dimensional =J( 80) nonlinear filteringproblem. In order to verify if the new filtering schemes cansuccessfully treat 'incipient' damage/degradation scenarios oftencharacterized by small local changes in (some of) the para-meter profiles, the stiffness parameters *s19 and *s20 in the refer-ence shear frame model (whose response, following corruptionby low intensity noise, provides the synthetic measurement)are taken as 2.4819108 N/m with *s1 through *s18 remaining

2.6125 108 N/m as in the 5-DOF example. While all other model/numerical parameters are kept the same as in the last example, = 31 (slightly higher with respect to the 5-DOF case owing to themeasurement noise intensity being lower) and = 8 are presentlyused. In addition to the higher dimensional nature of the problem,wherein particle filters often fail to work, low-intensity measure-ment noises contribute to an additional performance barrier,needed nevertheless if small variations in the estimates wereto be detected. Performance of the IGSF Bank is compared withthe EnKF, IGSF and IGSF with ADP. Specifically, the estimates ofstiffness and damping parameters via all these filters are shown inFigs. 6 and 7 respectively. As observed in Fig. 6, while the IGSFwith ADP continues to perform better than the EnKF and the IGSF,only the IGSF Bank appears to resolve the problem with a goodmeasure of success. Fig. 7 also reveals a substantively superiorresolution of the damping parameters, probably being reported forthe first time to the authors' knowledge, through the IGSF Bank.While the details are omitted for the sake of brevity, the estimatesreported in these figures have been tested for convergence byvarying the dimension d of the measurement vector. Thus, for the20-DOF model, varying d [17, 20] is seen to yield nearly identicalestimates via the IGSF Bank.

Finally, it may be noted that our efforts at applying the ASIRfilter for the identification of the MDOF shear frame model havefailed to yield any convergent estimates for both the casesinvolving =J 20 and =J 80.

T. Raveendran et al. / Probabilistic Engineering Mechanics 38 (2014) 7787 875. Conclusions

Motivated through a discretized and iterative approximation tothe KushnerStratonovich nonlinear filtering equations, a recur-sive-iterative Monte Carlo filtering approach employing a Gaussiansum approximation to the filtering density is proposed in thiswork. Using additive gain-based iterated updates whilst assimilat-ing the current measurement within the predicted solution, theproposed filters also make use of an additional annealing-typescalar parameter in order to boost the diffusion provided by theinnovation term. The proposed additive updates not only enableapplications to higher dimensional nonlinear filtering problems,but also provide a relief from the curse of weight collapseencountered with most weight-based particle filters. Unlike thecomputationally costly simulated annealing or MCMC filters, theschedule as well as the initial value of the annealing-type para-meter (the ADP) may be chosen far more flexibly in the presentfiltering schemes. For instance, schedules with very steep expo-nential or even discontinuous decay, thereby implying only a fewiterations, are admissible without requiring any burn-in periods.Whilst keeping the predicted particle locations unchanged at agiven time, the iterative updates aim at so refining the particlelocations as to drive the so-called measurement error to a zero-mean Brownian martingale, i.e. the measurement noise term itself.Using the proposed filters (especially the IGSF Bank), strikingimprovement in filter convergence as well as estimation accuracy(involving lower sampling variance) is observed consistentlyacross all the numerical examples considered here on nonlinearstate estimation and/or system identification.Acknowledgement

The authors are grateful to Dr. G V Rao for useful discussions inpreparing the revised manuscript.References

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Iterated stochastic filters with additive updates for dynamic system identification: Annealing-type iterations and the...IntroductionStatement of the problemFiltering schemeGaussian sum approximation and filter bankPrediction and zeroth updateADP-based iterative update schemeInitializationPrediction and zeroth updateADP based iterative update scheme

Numerical illustrations1-Dimensional nonlinear system with additive gaussian noiseTarget trackingAn MDOF shear frame model

ConclusionsAcknowledgementReferences