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Correntropy based IPKF filter for parameter estimationin presence of non-stationary noise process

Subhamoy Sen, Antoine Criniere, Laurent Mevel, Frédéric Cerou, JeanDumoulin

To cite this version:Subhamoy Sen, Antoine Criniere, Laurent Mevel, Frédéric Cerou, Jean Dumoulin. Correntropy basedIPKF filter for parameter estimation in presence of non-stationary noise process. SAFEPROCESS2018, 10th IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes, Aug2018, Varsovie, Poland. hal-01887557

Correntropy based IPKF filter for parameterestimation in presence of non-stationary

noise process

Subhamoy Sen ∗ Antoine Criniere ∗∗,∗∗∗∗ Laurent Mevel ∗∗,∗∗∗∗Frederic Cerou ∗∗∗ Jean Dumoulin ∗∗,∗∗∗∗

∗ Indian Institute of Technology Mandi, HP, India, (e-mail:subhamoy@iitmandi.ac.in)

∗∗ Inria, I4S, Rennes, France (e-mail: laurent.mevel@inria.fr)∗∗∗ Inria, Aspi, Rennes, France

∗∗∗∗ IFSTTAR, COSYS, SII, Bouguenais, France

Abstract: Existing filtering based structural health monitoring (SHM) algorithms assumeconstant noise environment which does not always conform to the reality as noise is hardlystationary. Thus to ensure optimal solution even with non-stationary noise processes, theassumed statistical noise models have to be updated periodically. This work incorporates amodification in the existing Interacting Particle-Kalman Filter (IPKF) to enhance its detectioncapability in presence of non-stationary noise processes. To achieve noise adaptability, theproposed algorithm recursively estimates and updates the current noise statistics using thepost-IPKF residual uncertainty in prediction as a measurement which in turn enhances theoptimality in the solution as well.Further, this algorithm also attempts to mitigate the ill effects of abrupt change in noise statisticswhich most often deteriorates/ diverges the estimation. For this, the Kalman filters (KF) withinthe IPKF have been replaced with a maximum Correntropy criterion (MCC) based KF that,unlike regular KF, takes moments beyond second order into consideration. A Gaussian kernel forMCC criterion is employed to define a correntropy index that controls the update in state andnoise estimates in each recursive steps. Numerical experiments on an eight degrees-of-freedomsystem establish the potential of this algorithm in real field applications.

Keywords: Particle filter, Noise estimation, Correntropy filter, Parameter change detection,Interacting Particle Kalman filter.

1. INTRODUCTION

Filtering based structural health monitoring (SHM) algo-rithms typically models noise processes as non-stationaryGaussian white noises with their statistics known a priori.However, in reality, statistical properties of noise processescan seldom be known in advance Zhang et al. (2012). Forsystems with unknown noise statistics, employing filteringbased SHM techniques thus more often than not resultsin a solution that might not be optimal. To ensure anoptimal solution, these noise statistics are thus required tobe estimated before even the SHM algorithm is put intooperation. Nevertheless, this also does not ensure that theoperational noise processes will follow the pre-estimatednoise models throughout the service life of the structuralsystem. The problem may further intensify in presenceof non-stationary noise processes undergoing an abruptchange in magnitude.A solution can, however, be attempted by estimating thenoise processes in parallel to the core SHM algorithmAlsuwaidan et al. (2008). This helps to provide the SHMalgorithm the required information about the current noisestatistics to enhance the optimal property of the solution.This article incorporates a noise estimation module within

Interacting Particle-Kalman filter (IPKF) (Zghal et al.,2014) to enable it to estimate the noise statistics inparallel to damage detection and quantification. This noiseestimation module considers the residual uncertainty inIPKF prediction as an observation which is further used toupdate the current noise statistics. This recursive updatestrategy for noise statistics makes the algorithm noiseadaptive and also ensures optimal damage detection.This article also attempts to mitigate the ill effects ofunusually large noise in the measurement which shouldbe considered as an outlier. However, in common filteringbased algorithms, these outliers are consumed in the stateestimates which might cause a divergence in the estima-tion procedure. To avoid this undesired situation, the KFnested within IPKF filter is replaced with maximum cor-rentropy criterion based Kalman filter (MCC-KF). WithMCC-KF, a correntropy index measures the similarity be-tween the actual measurement and the predicted measure-ment based on current estimates of states and parameters(Cinar and Prıncipe, 2012). Eventually, while a small noisecontamination causes small deviation of the measurementfrom the actual response, an outlier causes a huge innova-tion. The correntropy index thus becomes important sinceit can effectively regulate the impact of each innovation in

the updated estimates. As a result, a large innovation dueto an outlier in the measurement yields significantly lowcorrentropy index which in turn leads to a small impacton the state update.Section 3 discusses the problem under consideration andSection 4 describes the proposed methodology. Under Sec-tion 4, IPKF algorithm has been demonstrated in Subsec-tion 4.1 followed by a brief description of the MCC-KFfilter in Subsection 4.2. The augmented noise estimationmodule is described in detail in Subsection 4.3. Finally, theproposed algorithm is tested on a typical SHM problemin which the system undergoes damage in presence ofvarying noise processes (Section 5). An appendix is alsoincluded at the end of this article (Appendix A) detailingthe derivation of the MCC-KF algorithm. However, priorto everything else, Section 2 briefly discusses the conceptof correntropy criterion.

2. CORRENTROPY CRITERION

In information theory, correntropy is used as a measureof similarity in statistics between two different randomvariables. Over the years, this measure has drawn sig-nificant attention in the fields of signal processing, ma-chine learning, and pattern recognition. The major qualityof this criterion involves its excellent performance withthe data containing large outliers. Correntropy, or cross-correntropy (for scalar random variables) between tworandom variables include information about higher ordermoments beyond the traditional first and second ordermoment information. The selection of the moment order,however, depends on the kernel selected for the correntropycalculation. For two scalar random variable x and y, thecross-correntropy information can be obtained as:

Cσ(x,y) =∫

Ωx

∫Ωy

kσ(x,y)fxy(x,y)dxdy (1)

where kσ is the selected kernel for this correntropy calcu-lation and fxy is the joint density function for the randomvariables x and y. Analytical integration over the entiredomain of x and y is, however, not possible and thecorrentropy information can, therefore, be approximatedas a summation over finite numbers of data points as:

Cσ(x,y) = 1N

N∑i=1

kσ(xi,yi) (2)

With a Gaussian kernel, the explicit description of corren-tropy can be presented as:

Cσ(x,y) = 1N

N∑i=1

e−(||xi−yi||

2

2σ2

)= 1N

N∑i=1

Gσ(||εi||) (3)

where, ||εi||= ||xi−yi|| defines the norm of error betweenthe two dataset in consideration. Gσ(||εi||) = e(−εiε

Ti /2σ2)

is the Gaussian kernel. Here, σ is the bandwidth of thekernel that defines how new data is adapted. In orderto be counted as a compatible kernel for the correntropymeasure, it has to be positive and bounded achieving itslimiting maximum value when its argument is zero. For aGaussian kernel, a Taylor series expansion demonstratesthat information about all the even order moments isstored in the correntropy measure.

Cσ(x,y) = 1√2πσ

∞∑n=0

(−1)n

2nσ2nn!E[(εiεTi )n] (4)

3. PROBLEM DEFINITION

The governing differential equation for the dynamics ofmechanical systems with mass, damping and stiffnessbeing M, C and K respectively can be described as:

Mq(t) +Cq(t) +Kq(t) = v(t) (5)where q(t), q(t) and q(t) are displacement, velocity andacceleration responses respectively. v(t) is the ambientforcing acting on the structure. The state space representa-tion of this dynamics can be constructed with the ambientforcing v(t) modeled as non-stationary process noise andw(t) as non-stationary measurement noise as:

x(t) = Fcx(t) +Bcv(t) (6a)y(t) = Hcx(t) +Dcv(t) +w(t) (6b)

Fc, Bc, Hc and Dc are time dependent state, input, mea-surement and direct transmission matrices respectively de-fined in continuous time domain. Details of all the matricesare given in the following.

x(t) = q(t) q(t)T and Fc =[

0 I−M−1K −M−1C

];

Bc =[

0M−1

],Hc =

[−M−1K −M−1C

],and Dc = M−1

Since discrete measurements will be used for the esti-mation, the discrete time formulation of equation 6 canbe presented with xk, yk, F, B, H, D, vk and wk asthe discrete time counterparts against their correspondingcontinuous time entities.The system under consideration is, however, time-varyingand thus its model is defined with time-varying statematrix Fk and measurement matrix Hk. These system ma-trices are functions of a time-invariant mass matrix M andtime-varying physical system matrices Kk and Ck whichin turn depend on the component level health parametersθk. Accordingly, the mass matrix and its dependents, i.e.,B and D are assumed to be constant and known from nowon.

xk = F(θk)xk−1 +Bvk (7a)yk = H(θk)xk+Dvk+wk (7b)

where Fk = F(θk) and Hk = H(θk). The problem assumesthat the stochastic properties of process and measurementnoises i.e., vk and wk are not known but can be modeled asWGN with unknown covariance Qk and Rk respectively.

4. PRESENT APPROACH

It should be noted that with the typical filtering basedSHM problems, the state (system response) evolves intime through a linear process equation and thus KF canbe a good approach for such state estimation. However,the relationship between unobserved parameter states andits corresponding observation i.e., the measurements, isalways nonlinear. Bayesian belief propagation requiresan explicit analytical integration for the entire domainof states which is to be however straightforward if theproblem is linear and the states are assumed Gaussian. Onthe contrary, the current problem is nonlinear for which

explicit analytical integration over the entire parameterspace is not possible. Particle filter (PF) can approximatethis analytical integration by a numerical one and thuscan be considered to be a potential approach for nonlinearsystem estimation.Nevertheless, PF is computationally expensive and KFcannot be used for estimation of both states and pa-rameters. Employing two different filters for states andparameters, precision along with computational economycan although be achieved. With this in view, IPKF basedalgorithm given by Zghal et al. (2014) has been employedin his article that employs PF for parameter estimationwhile KF is used for linear state estimation.Within IPKF, a set of KF (for state estimation) runswithin an envelope of a PF (for parameter estimation).In this article, a modification has been proposed on theexisting IPKF targeting damage estimation in presenceof varying noise statistics and/or large outliers in themeasured signal. In the proposed algorithm, the KF isreplaced with MCC based KF or MCC-KF. An additionalmodule for noise estimation is augmented to recursivelyupdate the noise statistics. This methodology in detail isdiscussed in the following.

4.1 IPKF algorithm

IPKF algorithm, developed by Zghal et al. (2014) andfurther improved for computational efficiency by Sen et al.(2018) is an efficient approach to handle parameter estima-tion for a time-varying system. The strategy of decouplingthe estimation of states and parameters and employmentof two different filter types for each of them enhancesthe stability property of the algorithm while ensuringcomputational efficiency. In turn, this facilitates employingrelatively less expensive linear KF for linear state estima-tion while the costly PF is employed only for parameterestimation. Clearly, IPKF has two parts: i) a PF envelopto estimate the parameters as particles within which ii) aset of nested KFs estimates the states and both the partsare detailed in the following.

Envelop Particle filter: PF attempts a particleapproximation of the Chapman-Kolmogorov integrationby propagating the system uncertainty through a cloud ofN independent parameter particles Ξk = [ξ1

k, ξ2k, · · · , ξNk ].

Thus no consideration on Gaussianity of the parameterstates is enforced in the algorithm. The temporal evolutionof the system dynamics is defined by the evolution ofthe particle set in time. At any arbitrary time step k,the evolution of an arbitrary ith particle ξik is basicallya random perturbation around its current position ξik−1:

ξik = ξik−1 +N(δξk;σξk) (8)where a Gaussian blurring is performed on ξik−1 with ashift δξk and a spread of σξk.In IPKF, the state estimation is performed using a bank ofKFs within the PF environment where each of the KFs isassociated to one instance of the corresponding parameterparticles for which the state estimation is performed. Thusthe KF is employed to estimate the states xk while the PFestimates the parameters θk. For each particle, the nested

KF thus provides an optimal state estimates conditionedon the assumed parameter particle. Finally, based on theinnovation and its covariance produced by each of the KFsassociated to each of the parameter particles, the particlesare weighted. The weights are subsequently used to updatethe prior statistics of the particles. In the following, detailsof the nested KFs are described.

Nested Kalman filters: Each of the evolved param-eter particles is used to follow the system matrices asFik = F(θk = ξik) and Hi

k = H(θk = ξik). The estimationprocedure involves propagating the current state estimatexik−1|k−1

1 , through the system conditioned on currentparameter particles in order to predict xik|k−1. Predictedestimate is subsequently improved using current measure-ment yk to obtain the posterior estimate xik|k specificto the particle estimate. This process is repeated for allparticles yielding a set of posterior estimate for statesxik|k; for all i= 1,2, · · · ,N.

The prediction and correction steps of the KF for anarbitrary ith particle at kth time step are described in thefollowing.

Predictionxik|k−1 = Fikxik−1|k−1 (9a)

Pik|k−1 = FikPi

k−1|k−1FikT +BQk−1BT (9b)

Innovationεik = yk−Hi

kxik|k−1 (9c)Correction

xik|k = xik|k−1 +Kikεik (9d)

Pik|k = (I−Ki

kHik)Pi

k|k−1 (9e)

where Kik is the gain matrix. Pi

k1|k2is the estimate for

state error covariance at kth1 time step given measurementupto kth2 instance. The process noise Qk−1 is denoted herewith a • operator that defines that the process noise isnot known apriori. Instead, it is estimated in parallel toIPKF (but outside the particle filter) and Qk−1 signifiesthe available best current estimate for process noise Qk.

4.2 MCC-KF

Izanloo et al. (2016) developed a Kalman filter based onMCC criterion where they instead of considering the errorcovariance for the gain calculation, employed maximumcorrentropy criterion for the gain formulation. In order tomake a robust noise adaptive MCC based KF that canhandle measurement outliers, MCC-KF algorithm definesa cost function that incorporates correntropy informationinstead of error covariances. In the proposed algorithm,the KF within the IPKF filter is replaced by the MCC-KF was given by Cinar and Prıncipe (2012). A detaileddescription is presented in the Appendix A.For MCC-KF algorithm, the cost function, for the wellknown weighted least square approach for KF formulation1 xi|j represents estimate of the random variable x at the ith timeinstant provided the measurement including and upto time instantj.

targeting minimum variance in the estimates, is modifiedas:

J i = 1σ2Gσ

(||ε(x)i

k||Pik|k−1

−1

)+ 1σ2Gσ

(||ε(y)i

k||R−1k−1

)(10)

where ε(x)ik = xik|k−Fikxik−1|k−1, ε(y)i

k = yk−Hikxik|k and,

||•||w signifies the argument within the norm operator isweighted by index w. To minimize this objective functionpresented in equation (10), its gradient with respect toxik|k is equated to zero and a subsequent restructuringyields a Kalman like equation for state estimate updateas:

xik|k = xik|k−1 +Kik

yk−Hi

kxik|k−1

(11)

Pik|k = (I−Ki

kHik)Pi

k|k−1 (12)A detailed and generalized derivation for the Kalman gainand other relevant description is presented in Appendix A.

4.3 Adaptive estimation of noise

Targeting optimal estimates for health parameters, theproposed algorithm additionally estimates the noise pro-cesses recursively. After each iteration of the IPKF, theparticle approximations for states xk|k and parameters θkare obtained as:

xk|k =N∑i

w(ξik)xik|k and, θk =N∑i

w(ξik)θik; (13)

where N is the number of particles and w(ξik) is the weightof ith particle. This approximated estimates are furtherused to define the particle approximation for the systemmatrices Fik and Hi

k. Matrix B and D being particleindependent can be used directly.

Fk =N∑i

w(ξik)Fik; and, Hk =N∑i

w(ξik)Hik; (14)

The current process noise is further observed based onthese particle approximated entities. For this, the devia-tion between predicted output and actual output i.e., δykis computed first as:

δyk = yk− Hkxk|k (15)This deviation δyk is due to uncertainty in state estimatesand noise processes in process and measurement. Withthe prior estimates of state error covariance Pk|k (particleapproximated) and measurement noise covariance Rk−1,a measure for the process noise covariance can be observedfrom the innovation as:

cov(εk) = δykδyTk − HkPk|kHTk − Rk−1 (16)

This measure is further mapped in the domain of state xkand process noise Qk. This provides an observation of theinstantaneous estimate of the process noise as:

Q∗k = (HkB+D)†cov(εk)(HkB+D)†T (17)Here ∗ signifies that an observation has been made onprocess noise Qk at time step k. This observation is thencombined with the previous estimate in a moving averagesense as following:

Qk = Qk−1 + 1LQ

(Q∗k− Qk−1) (18)

Fig. 1. Numerical test setupLQ is the window length variable that controls how fastnew estimates are adapted with the prior estimates. Inturn, this defines how new data is adapted in to the currentestimates of the noise processes. A high value ensuressmooth and stable estimate of the noise covariance, while alow value puts significant weight on the new measurement.A compromise between rapid update and stable estimationshould thus be desired.Subsequently, an estimate vk for the process noise vk isobtained as:

vk = (HkB+D)†δyk (19)With this estimate, current state estimate is updated as:

x∗k|k = xk|k+Bvk (20)Correspondingly, the output can be predicted as:

y∗k|k = Hkx∗k|k+Dvk (21)Thus, the deviation between actual measurement and theupdated measurement prediction y∗k|k can be attributedto measurement noise. Evidently, an observation can bemade about the current measurement noise as:

R∗k = δy∗k|kδy∗k|k

T (22)where δy∗k|k = yk − y∗k|k. Finally, similar to the processnoise, the measurement noise estimate is updated as:

Rk = Rk−1 + 1LR

(R∗k− Rk−1) (23)

Again, LR is the window parameter defining the windowfor the moving averaging of measurement noise estimate.As it has been previously discussed, this work intendsto estimate a system for which abrupt change in noisestatistics is possible. During the transition from one noisestatistics to the other, the observations for the process andmeasurement noise should be assimilated with caution. Inthis attempt, the correntropy information is employed todefine a weight for the update based on current observa-tion. To obtain this weight, a particle approximation forall Lik corresponding to nested MCC-KFs is obtained as:

Lk =N∑i=1

w(ξik)Lik (24)

The weights LQ and LR for the moving averaging windowcan then be defined based on this weight Lk as:

LQ = LkLQ0 and LR = LkLR0 (25)where LQ0 and LR0 are the base values for the window.This ensures that if the data correntropy is severelypoor, the algorithm puts significantly less belief on thenew observations and consequently less update. On theother hand, if the data correntropy is high, the currentobservation is assimilated with more belief.

5. NUMERICAL EXPERIMENTS

Three sets of numerical experiments are undertaken: i)first one with a varying process noise, ii) second one with

a varying measurement noise and iii) the third one withprocess and measurement noises varying simultaneously.For each of the cases, the system has been induced witha damage. The proposed IP-MCC-KF is then employedto detect the damage while estimating the noise statisticssimultaneously.The numerical experiments detailed in the following dealwith an eight degrees-of-freedom mass-spring-damper sys-tem (see Fig. 1). The springs are considered to be ofstiffness 8000N/m while the dampers are assumed to beof 8N − s/m. The mass blocks are assumed to be of 10kgeach. Damage is introduced in the third and fourth springsand their stiffness values are reduced to 2000N/m. Thesystem is excited with an ambient vibration at all its freenodes. This ambient vibration has been considered in thisformulation as process noise. Finally, the system is simu-lated for time series length of 3072 sampled measurementscollected at a sampling frequency of 50 Hz. The simulatedacceleration response is collected as measurements whichare further contaminated with a Gaussian white noise asa replication of the sensor noise contamination. A value of1000 has been assumed as the bandwidth of the Gaussiankernel (i.e. σ). Also, based on our previous experience withIPKF, a set of 2000 particles are employed for the PF forall case studies in this article. Further details pertainingto each example case are presented in the following.

5.1 Case 1: Varying process noise

The first case discusses a system for which the processnoise is varying with time. For the first 10.24 seconds, theprocess noise is assumed to be a zero-mean Gaussian whitenoise with a standard deviation of 100 N. This standarddeviation is further changed to four and eight folds of itsinitial value. The assumed process noise is presented inFig. 2. The initial weight for the moving averaging windowi.e., LQ0, has been assumed to be 0.01. The measurementnoise statistics is considered to be stationary and known. Avalue of 5m/s2 has been selected as the standard deviationof the zero mean Gaussian white noise and employed as themeasurement noise.The results for the parameter estimation are presentedin Fig. 3 where it can be observed that the parameterestimation and damage detection as prompt and precise.The parallel estimation of the process noise statistics isalso presented in Fig. 4. It can be seen that the initialestimates are to some extent turbulent. This is due tothe error in the estimates of the parameter for which theinnovation uncertainty contained the uncertainty due toinexact parameter estimates. However, once the parameterestimates converged to their true value, the turbulence inthe process noise estimates settled down and become morestable.The correntropy index for this estimation is also presentedin Fig. 5. It should be noticed that whenever the systemunderwent a change in process noise statistics, the corren-tropy index decreased in a sense to reduce the effect ofthe measurement in the parameter update. This, in turn,damps the effect of the sudden change in the system andfacilitates a smooth and stable estimation.

Fig. 2. Assumed process noise variation

Fig. 3. Case study 1: Parameter estimation

Fig. 4. Case study 1: Estimation of process noise statistics

5.2 Case 2: Varying measurement noise

For the second case, the process noise statistics are keptconstant with a standard deviation of 100 N as before.However, the measurement noise is kept changing up to 5to 10 folds of its initial value. The initial value of standarddeviation for the measurement noise has been assumed forthis case study as 5m/s2. The measurement noise used forthis case study is presented in Fig. 6. In a similar way,the initial weight for the moving averaging window formeasurement noise estimation i.e., LR0, has been assumedto be 0.01.The results for the parameter estimation are presented inFig. 7 where again it can be observed that the parameter

Fig. 5. Case study 1: Correntropy evolution

Fig. 6. Assumed measurement noise variation

estimates are stable and prompt. The estimation of themeasurement noise statistics is presented in Fig. 8. It canbe seen that estimation of measurement noise statisticsis not as accurate as for the process noise. This can beattributed to the very small magnitude of the measure-ment noise. Both process and measurement noise statis-tics are estimated from the innovation uncertainty in aprocess that might be termed as a systematic reduction ofuncertainty. In this process, the process noise uncertaintyis first removed followed by reduction of uncertainty dueto the improper state estimates. Finally, the residual un-certainty is mapped to the measurement space in orderto estimate the measurement uncertainty. Obviously, allthese uncertainties are different in their scales and thereis a wide possibility that the smallest of them will suffer aloss of information. This will obviously cause the estimateof measurement noise statistics to be poorer than theother two, i.e., process and state estimates. However, thecurrent proposal can successfully identify the change in themeasurement noise and the estimate of the measurementnoise statistics is sufficiently accurate.

5.3 Case 3: Varying process and measurement noise

For the third case study, both the measurement andprocess noise are allowed to vary simultaneously. Thevariation assumed for this particular experiment is thesame as the ones that have been used in the earlier casestudies. LQ0 and LR0 are assumed to be 0.01 for this casestudy.

Fig. 7. Case study 2: Parameter estimation

Fig. 8. Case study 2: Estimation of measurement noisestatistics

For this case study, the results for the parameter es-timation is presented in Fig. 9. Also, in this case, theparameter estimates are sufficiently accurate and stable.The estimates for process and measurement noise statisticsare presented in Fig. 10 and Fig. 11. It has been foundthat while both measurement and process noise statisticsare estimated simultaneously, the estimation result forthe measurement noise statistics is interestingly better.However, due to the scale effect of the process noise in theinnovation uncertainty, the measurement noise estimatesare influenced by the process uncertainty. While in theprevious case studies, the measurement uncertainty is sig-nificantly underestimated, estimation for the current casestudy is precision wise much better. This is due to the factthat with no flexibility in the process uncertainty, the stateestimates are sometimes over fitted which often underesti-mates measurement uncertainty. Simultaneous estimates,on the other hand, balance the estimation of process andmeasurement uncertainty which cause the measurementestimates to be better.Finally, we experimented with a case where the noisestatistics are actually varying over time but not estimatedin the process of parameter estimation. The parameterevolution for this case study has been demonstrated inFig. 12 where it can be seen that with inaccurate noisemodel values, the parameter estimates are very turbulentand it ultimately diverges resulting failed estimation. Itcan also be noticed that at some point in time the esti-mates even went below zero stiffness. The system modelemployed in this article is simplistic for which simulation is

Fig. 9. Case study 3: Parameter estimation

Fig. 10. Case study 3: Estimation of process noise statistics

Fig. 11. Case study 3: Estimation of measurement noisestatistics

still proceeded without any hindrance. However, for costlyand complicated system models, such abnormal parameterestimates will stop the parameter estimation. On the con-trary, the same data-set when put through the proposedIP-MCC-KF algorithm yielded perfect estimation of theparameters for the third case study (cf. Fig. 9). Thisestablishes the relevance of noise estimation in parallel tothe parameter estimation.

6. CONCLUSION

In this article, a novel methodology has been presentedbased on IPKF filter in which the Kalman filter compo-nent of IPKF filter has been replaced with correntropy

Fig. 12. Parameter estimation without noise estimationbased Kalman filter (MCC-KF) that helps to stabilize theestimation in presence of an abrupt change in the systemdue to damage or noise statistics. Numerical experimentsestablished that incorporation of MCC-KF facilitates im-parting less updates in cases of high mismatch betweenpredicted and actual measurement caused due to eitherdamage or change in noise statistics. It has also been ob-served that measurement noise statistics when estimatedsolely underestimates the noise covariance. However, whenthe statistics of measurement noise are estimated along-side the statistics of the process noise, the estimates aremuch better than that with the solo estimation strategy.Eventually, this simultaneous estimation strategy for noisestatistics bettered the parameter estimation. In turn, thisstrategy enhanced the robustness of the proposed IP-MCC-KF algorithm.

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Appendix A. DERIVATION OF MCC-KF

Of the many formulations through which the Kalmanfilter can be derived, the one that uses optimization basedapproach is relevant in the derivation of MCC-KF. Thederivation of the MCC-KF presented in the following isfollowed for each MCC-KF filter nested within the PF. Theparticle indices (•i) for each MCC-KF and the variablesinvolved are however dropped here in this appendix forthe sake of simplicity. For MCC-KF, a quadratic costfunction needs to be set up prior to a recursive estimationto minimize it,

(A.1)J = 1σ2Gσ

(||yk −Hkxk||R−1

k

)+ 1σ2Gσ

(||xk − xk||P−1

k

)where xk = Fkxk−1 is the prediction on xk. Pk is the co-variance of error between actual state xk and its predictionxk. Theoretically, this cost function needs to be minimizedwith xk as the argument. Finally, xk|k will be obtained asan estimate of xk:

xk|k = argminxkJ(xk) (A.2)The required gain matrix Kk can be derived by analyti-cally solving ∂J(xk)

∂xk= 0 which yields the following equa-

tion:

− 1σ2Gσ

(||yk −Hkxk||R−1

k

)HTkR−1

k (yk −Hkxk)

+ 1σ2Gσ

(||xk − xk||P−1

k

)P−1k (xk − xk) = 0

(A.3)

Upon simplification, this equation demonstrates the in-terrelation between state error covariance and the corren-tropy information as:

(A.4)P−1k (xk − xk)

=Gσ

(||yk −Hkxk||R−1

k

)Gσ

(||xk − xk||P−1

k

) HTkR−1

k (yk −Hkxk)

Plugging in the best estimates available for xk, xk and Pk

as xk|k, xk|k−1 and Pk|k−1, the estimate of the states canbe restated as:

P−1k|k−1xk|k = P−1

k|k−1xk|k−1

+Gσ

(||yk −Hkxk|k||R−1

k

)Gσ

(||xk|k − xk|k−1||P−1

k|k−1

)HTkR−1

k

(yk −Hkxk|k

)(A.5)

The update equation is therefore further simplified as:P−1k|k−1xk|k = P−1

k|k−1xk|k−1 +LkHTkR−1

k

(yk−Hkxk|k

)(A.6)

where Lk =Gσ

(||yk−Hkxk|k||R−1

k

)Gσ

(||xk|k−xk|k−1||P−1

k|k−1

) .

As it can be seen from the expression of Lk, that the indexhas xk|k term in its denominator which makes the esti-mation of Lk an implicit problem and to be solved usingcostly optimization problem. Chen et al. (2017) employed afixed point algorithm to solve for the posterior which is al-though accurate but increases the computational expense.Since the proposed approach itself is recursive in nature,we incorporated a little modification in the algorithm byreplacing xk|k by its best available estimate before themeasurement correction, i.e., xk|k−1 (similar to Cinar andPrıncipe (2012)). Thus the correntropy can be calculatedas:

Lk =Gσ

(||yk−Hkxk|k−1||R−1

k

)(A.7)

This, in turn, make the problem explicit and therefore lesscomputationally expensive.Finally, to give this update equation a Kalman likeappearance we proceed with adding and subtractingLkHT

kR−1k Hkxk|k−1 in the right hand sides of equation

A.6: (P−1k|k−1 +LkHT

kR−1k Hk

)xk|k =

P−1k|k−1xk|k−1 +LkHT

kR−1k yk

+LkHTkR−1

k Hkxk|k−1−LkHTkR−1

k Hkxk|k−1

(A.8)

which when arranged gives:

(A.9)

(P−1k|k−1 + LkHT

kR−1k Hk

)xk|k =(

P−1k|k−1 + LkHT

kR−1k Hk

)xk|k−1

+ LkHTkR−1

k

(yk −Hkxk|k−1

)or in compact form:

(A.10)xk|k = xk|k−1 + Kk(yk −Hkxk|k−1)

where Kk =(P−1k|k−1 +LkHT

kR−1k Hk

)−1LkHT

kR−1k . The

corresponding update equation for the error covariancePk|k−1 can be derived as:

Pk|k = (I−KkHk)Pk|k−1(I−KkHk) +KkRkKTk

(A.11)This is the generalized expression for update equations(also called as Joseph’s form). However, if the Kalman gainKk is optimal, this form takes even a simpler description(see Kulikova (2017) for the alternative form) as:Kk = Pk|k−1LkHk

T (HkPk|k−1LkHkT +Rk)−1 (A.12)

and corresponding error covariance update equation canbe given as:

Pk|k = (I−KkHk)Pk|k−1 (A.13)