A Mutated Particle Filter Technique for System State Estimation and Battery Life Prediction

2034 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 63, NO. 8, AUGUST 2014

A Mutated Particle Filter Technique for SystemState Estimation and Battery Life Prediction

De Z. Li, Member, IEEE, Wilson Wang, Senior Member, IEEE, and Fathy Ismail

Abstract— When classical particle filter (PF) techniques areused for dynamic system state estimation, they have somelimitations: for example, when the weights of simulated samplesare not sufficiently large, these classical PFs may suffer fromsample impoverishment. In addition, the degraded diversity insampling particles will limit the estimation accuracy, since theparticles cannot capture the entire probability density function(pdf) effectively. To tackle these problems, a mutated PF (MPF)technique is proposed in this paper to approximate the posteriorpdf of system states. In the MPF, a novel mutation approachis proposed to search extended areas of the prior pdf usingmutated particles to make more comprehensive exploration ofthe posterior pdf. In addition, a particle selection scheme issuggested in the MPF to detect and process low-weight particlesso as to explore the high-likelihood area of the posterior pdf morethoroughly. The effectiveness of the developed MPF technique isverified by simulation tests using a benchmark test model. It isimplemented for predicting remaining useful life of batteries.Test results show that the developed MPF can capture a system’sdynamics effectively and track system characteristics accurately.

Index Terms— Lithium-ion batteries, particle filters (PFs), par-ticle mutation, remaining useful life (RUL) prediction, systemstate estimation.

I. INTRODUCTION

IN SYSTEM state estimation/prognosis, the system’s inter-nal states are usually inaccessible to sensors, and the

state estimation has to be inferred from noisy measurements.To make inferences about the characteristics of the measuredsystem, a dynamic state-space model is required, which usu-ally consists of a transition model and a measurement (sensor)model. The transition model performs system state prediction,and the measurement model links the predicted states tonoisy measurements. To carry out system state estimation,a posterior probability density function (pdf) has to be for-mulated based on system models and noisy measurements.The Kalman filter can be applied for state estimation of linearGaussian systems; however, it lacks the capacity to addresssystems with nonlinear/non-Gaussian properties [1]. Although

Manuscript received July 30, 2013; revised December 8, 2013; acceptedDecember 11, 2013. Date of publication February 14, 2014; date of currentversion July 8, 2014. This work was supported in part by the Natural Sciencesand Engineering Research Council of Canada and in part by eMech SystemsInc. The Associate Editor coordinating the review process was Dr. Kurt Barbe.

D. Z. Li and F. Ismail are with the Department of Mechanical andMechatronics Engineering, University of Waterloo, Waterloo, ON N2L 3G1,Canada (e-mail: [email protected]; [email protected]).

W. Wang is with the Department of Mechanical Engineering,Lakehead University, Thunder Bay, ON P7B 5E1, Canada (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2014.2303534

some advanced Kalman filter techniques can be used to modelnonlinear systems, such as first-order truncated Taylor seriesexpansion [2], [3], they apply a suboptimal implementationof the sequential Bayesian estimation framework for Gaussianrandom variables [4]. On the other hand, although the grid-based filter can approximate optimal Bayesian recursion byusing a uniform grid to explore the entire state space, itscomputational complexity limits its practical applications [5].

Particle filters (PFs) can be used to model systems with non-linear and non-Gaussian characteristics, which have been usedin system state tracking [6], [7] and prediction applications[8], [9]. A PF is a recursive Monte Carlo-based method thatconstructs pdf using a set of random particles with associ-ated weights. Among these available PF techniques, samplingimportance resampling PF (SIR-PF) is the fundamental PFtechnique, in which resampling is used to avoid the situationthat most of the particles’ weights are close to zero. However,SIR-PF conducts system state estimation without consideringcurrent measurement, which may degrade its performance.Although auxiliary PF (APF) can generate a new state estimateconditioned on current measurement with the use of somestatistical indicators to characterize transition density, it suffersfrom the loss of diversity among particles after resamplingbecause resampled particles are generated based on a discretedistribution, rather than a continuous one [1]. Regularized PFcan draw particles from continuous distribution to improveparticle diversity [10]; however, its estimated continuous dis-tribution may not be accurate if there are insufficient particleslocating at high-likelihood area of the distribution. RegularizedAPF (RAPF) can diversify the particles drawn by samplingparticles from a continuous distribution to improve systemstate estimation [11]; however, the high-likelihood area of itsposterior pdf may not be fully represented by particles aftercertain filtering iterations since some gaps could exist in thehigh-likelihood area of the estimated posterior pdf. These gapslack particles to represent them, and the information of theparticles over these gaps cannot be delivered to the followingiterations, which, in turn, would degrade inaccurate systemstate estimation. The unscented PF applies an unscentedKalman filter to generate an importance proposal distributionto improve estimation accuracy [12]; however, it may stillsuffer from inaccurate distribution representation caused byinsufficient high-weight particles. The Rao–Blackwellished PFcan marginalize some system states and conducts particlefiltering only to remaining system states, so as to improvecomputation efficiency [13]; although it could speed up thefiltering process, its distribution representation is not accurate.

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LI et al.: MPF TECHNIQUE FOR SYSTEM STATE ESTIMATION AND BATTERY LIFE PREDICTION 2035

To tackle the aforementioned problems, a mutated PF (MPF)technique is proposed in this paper to explore both theextended area of the prior distribution and the entire distri-bution of the posterior pdf to improve state estimation ofdynamic systems. The proposed MPF has the following novelaspects: 1) a novel mutation approach is proposed to diversifythe particles and exploit the entire posterior pdf space; 2) anew particle selection mechanism is suggested to search thehigh-likelihood area of the posterior pdf; and 3) the proposedMPF technique is implemented for battery remaining usefullife (RUL) prediction.

The remainder of this paper is organized as follows. Theproposed MPF technique is discussed in Section II. Theeffectiveness of the MPF is demonstrated in Section III viasimulation tests; the MPF predictor is also implemented forbattery RUL prediction application. Some concluding remarksare summarized in Section IV.

II. MPF TECHNIQUE

To properly introduce the proposed MPF technique, a briefintroduction of the related RAPF technique is first given inSection II-A, and then the proposed mutation method andparticle selection scheme in the MPF technique are discussedin Section II-B.

A. Regularized APF

To formulate an RAPF, a state-space model with systemstates xt and observations yt is considered. Given an initialdensity p(x0), the states xt evolve over time as a partiallyobserved first-order Markov process based on probabilitytransition density f (xt |xt−1), where t = 1, 2, . . ., T , and T isthe number of observations. The measurements yt are con-ditionally independent, and are obtained through conditionalprobability density f (yt |xt ). Once the density f (xt−1|y1 : t−1)at time (t–1) is projected forward in time, the prior density ofthe state at time instant t can be estimated as

f (xt |y1 : t−1) =∫

f (xt |xt−1) f (xt−1|y1 : t−1)dxt−1. (1)

Applying the Bayesian rule, the updated posterior density canbe derived as

f (xt |y1 : t ) = f (yt |xt) f (xt |y1 : t−1)

f (yt |y1 : t−1)(2)

where the normalizing factor is given by

f ( yt | y1 : t−1) =∫

f ( yt | xt ) f ( xt | y1 : t−1) dxt . (3)

Equation (2) provides an optimal recursive solution to sys-tem state estimation. If the related system is linear Gaussian,Kalman filter techniques can provide formal recursion forthe density function of the system. However, nonlinear non-Gaussian systems do not have closed-form solutions, becausethe multidimensional integrals in (1) and (3) are usuallyintractable. Hence, the pdf, f ( xt | y1 : t ), can be evaluated usingsome suboptimal filters, such as the RAPF method.

In the RAPF [11], the first stage posterior densityf ( xt−1| y1 : t ) is characterized by a set of particles (i.e., ran-dom support points) xi

t−1 and their associated weights π it−1;

i = 1, 2, . . ., N , where N is the number of particles.The posterior density f ( xt−1| yt ) can be estimated with thefollowing expression [14]:

f (xt−1|yt) ≈N∑

i=1

π it−1 f

(yt |μi

t

)Rl

(xt−1 − xi

t−1

)

andN∑

i=1

π it−1 = 1 (4)

where μit can be the mean, the mode, or a random draw

associated with the density f (xt |xt−1). Rl (·) is the rescaledkernel function given by

Rl (x) = l−n R (x/ l) (5)

where l > 0 is the scalar kernel bandwidth and n is thedimension of the state vector x .

The kernel Rl(·) and bandwidth l are chosen to minimize themean integrated square error between the true posterior densityand the corresponding regularized empirical representationin (4). When all the samples have the same weight [10], theoptimal kernel would be the Epanechnikov kernel

R∗ ={ n+2

2cn(1 − ‖x‖2),

0,

if ‖x‖ < 1otherwise

(6)

where cn is the volume of the n-dimensional unit hyperspheregiven by

cn =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2,π,...2πcn−2/n,

n = 1n = 2

...n = n.

(7)

According to the density estimation theory, as stated in [19]and [20], when the underlying density is Gaussian with a unitcovariance matrix, the optimal bandwidth l∗ can be determinedas [10]

l∗ =[8c−1

n (n + 4)(2√

π)n

]1/(n+4)N1/(n+4) (8)

where N is the number of particles.In dealing with an arbitrary underlying density, it is nec-

essary to assume that the density is Gaussian. Then, itscovariance matrix can be replaced by the empirical covariancematrix D. If M is the square root matrix of the empiricalcovariance matrix D such that MMT = D, the kernel functionin (5) can be rewritten as

Rl (x) = (det M)−1 l−n R(M−1x/ l

)(9)

where det(M) is the determinant of matrix M .Once x j

t−1 are derived using (4), the second stage particles

x jt can be evaluated through f ( xt | xt−1); the second stage

weights are formulated as

πj

t ∝ f(

yt | x jt)

f(yt

∣∣μ jt

) (10)

where μjt = E(xt |x j

t−1). In this scenario, the superscriptj ∈ {1, 2, . . . , N } represents the index of particles after


Fig. 1. Illustration of the posterior distribution of a system state. (a) Withoutusing the mutation method. Blue crosses represent those posterior particleswith low weight factors. (b) Using the mutation method. High-weight pos-terior particles (red solid circles) are generated from those prior particlescorresponding to low-weight posterior particles.

resampling, while the superscript i ∈ {1, 2, . . . , N } in (4)denotes the index of particles before resampling.

B. Proposed Mutation PF Technique

The RAPF in [11] employs a single statistic value μit

(e.g., the mean value) to characterize f (xt |xt−1) in the APF.However, the approximation of f (xt |xt−1) may not be accuratewhen the process noise is large, which would lead to inaccurateestimation [1]. The large process noise problem could betackled using more particles to characterize f (xt |xt−1) ratherthan a single statistic value. The classical PF techniques, aswell as the RAPF, only evaluate f (xt |xt−1) based on priorparticles from the last iteration. If these particles cannot rep-resent the prior distribution accurately or the prior distributiondeviates from the posterior distribution after passing throughsystem state models, most of the generated posterior particlesmay have very low weights, which will result in sampleimpoverishment. To tackle this problem, the proposed MPFtechnique employs the mutated particles derived from priorparticles to explore a wider range of the prior distribution,rather than the range constrained by some prior particles. Itspurpose is to approximate the mapping between prior andposterior pdfs more accurately, and exploit the posterior pdfspace more comprehensively. A particle selection mechanismis suggested next to make posterior particles carry highweights.

Fig. 1(a) shows a typical posterior distribution with sampleimpoverishment. It can be seen that almost half of the gener-ated posterior particles (blue crosses) have very low weights,and part of the high-likelihood area lacks sufficient particles.Fig. 1(b) will be described in Section II-B2.

1) Mutation Mechanism: The proposed mutation method isdescribed as follows.

Given the particles at time instance t , {x1t , x2

t , . . . , x Nt }, the

respective upper and lower boundaries of these particles can

Fig. 2. Distribution of auxiliary particles ϕit and mutated particles x̂ i

t , withthe original particle (a) xi

t = 0.2 and (b) xit = 0.8. Blue solid line: the

distribution of ϕit . Red dotted line: the distribution of x̂ i

t .

be computed by

U(xit ) = max

(x1

t , x2t , . . . , x N

t

) + λt (11)

L(xit ) = min

(x1

t , x2t , . . . , x N

t

) − λt (12)

where λt is a positive constant, which defines the extendedsearching space of the prior distribution to accommodatemutated particles at time instant t .

For each particle in the particle set{x1

t , x2t , . . . , x N

t

}, a

mutated particle will be derived as

q = (U − xit )/(xi

t − L) (13)

γ = 7

⎧⎪⎨⎪⎩

q − q(

1 − rq

)b,

q + (1 − q)(

1 + r−q1−q

)b,

q ≥ r

otherwise(14)

φit = (1 − γ )L + γ U (15)

where φit is the auxiliary position around xi

t and r ∈ [0, 1] isa random number.

Equations (13)–(15) represent a mutation mechanism [21]that generates a random number φi

t over the feasible rangeof the particle set

{x1

t , x2t , . . . , x N

t

}with an approximately

uniform distribution. To enable the mutated particle to deviatefrom the neighborhood of the original particle xi

t , a mutatedparticle will be generated as

x̂ it = U + L − xi

t − η(φi

t − xit

)(16)

where η ∈ [0], [1] is a random number, x̂ it is the mutated

particle, and b ∈ [0.5, 1] is a strength factor that estimates thevariance of the location of the mutated particles. The largerthe factor b, the wider the area in which the mutated particlesmay appear.

Randomly generate 104 possible data of the auxiliaryparticles φi

t and their corresponding 104 data of mutatedparticles x̂ i

t , from a fixed value of original particle xit ∈ [0], [1]

(xit = 0.2 and 0.8 in Fig. 2). The distributions of φi

t and x̂ it

can be estimated using kernel density estimation, as shown inFig. 2. It can be observed from Fig. 2 that the auxiliary particleφi

t has an approximately uniform distribution, irrespective ofthe value of the original particle xi

t ; on the other hand, themutated particle x̂ i

t has higher probability locating in theremote area from the original particle xi

t .In using the proposed mutation approach, if one particle

xit lies in a low-value area of particle space, its mutated

counterpart will be located in the high value region in a


probabilistic form, and vice versa. The mutated particles{x̂1

t , x̂2t , . . . , x̂ N

t

}with associated weights π̂ i

t can be used toenrich the representation of the high-likelihood area of thedensity function f (xt | y1:t ), so as to enhance the posteriorpdf and improve the system state estimation.

2) Selection Scheme: When posterior particles are gener-ated, those particles with very low weights usually haveless contribution to the system state estimation, while theparticles with high weight may not be sufficient to representthe posterior pdf accurately. To enhance the posterior pdfrepresentation, a threshold ξ is introduced to characterize thecontribution of different particles. After the weights of allposterior particles are normalized, the particles with weightslarger than (or equal to) ξ will be accepted, while those withweights less than ξ will be replaced by their mutated particles.The threshold ξ is usually a constant, which can be selectedbased on particular applications (e.g., ξ = 0.01 in this case).If a mutated prior particle generates posterior particles withweights less than ξ , the above process will be repeated untilthe weight factor of the resulting new particle becomes greaterthan (or equal to) ξ .

Using the proposed mutation method and particle selec-tion mechanism, the prior particles corresponding to thoseposterior particles with low weights can be replaced bytheir mutated counterparts. Therefore, the representation ofthe high-likelihood area in the posterior distribution can beimproved, as shown in Fig. 1(b).

3) Implementation of the MPF Technique: In the implemen-tation, N prior particles and their mutated N particles are usedto explore the posterior pdf. By testing the derived posteriorparticles using the proposed selection scheme, each posteriorparticle with a weight less than the threshold ξ will be replacedby a posterior particle with higher weight using the mutationmechanism. The following summarizes the implementationprocedure of the proposed MPF technique.

a) Draw samples xit from f ( xt | xi

t−1); i = 1, 2, …, N .b) Compute the mutated particles of xi

t , x̂ it ; i = 1, 2, …, N ,

using (11)–(16).c) Calculate weights π i

t ∝ f (yt |xit ) of these 2N par-

ticles (i.e., xit and x̂ i

t ) and conduct normalization:∑2Ni=1 π i

t = 1.d) Apply the selection scheme to the weights π i

t . If π it < ξ ,

a new mutated particle x̂ it will be generated from its cor-

responding prior particle xit ; this process is repeated until

the weight of the generated posterior particle π it ≥ ξ .

e) Calculate the empirical covariance matrix Dt of{xi

t , πit

}2Nj=1 and the root-square matrix Mt using

Cholesky decomposition Mt MTt = Dt .

f) Carry out resampling using appropriate resamplingalgorithms (e.g., multinomial resampling [22]).

g) Draw ε from Epanechnikov kernel and calculate l∗using (8). Then, compute new particles [11]:Xi

t = xit + l∗Mtε.

The mutation method at steps b) and d) aims to explore anextended range of prior distribution so as to improve posteriordistribution approximation. The selection scheme at step b)will replace those posterior particles with very low weights

TABLE I

AVERAGED MEAN AND STANDARD DEVIATION OF

RMSE WITH DIFFERENT PARTICLE NUMBERS

by appropriate posterior particles. Therefore, these particleswill explore the high-likelihood area of the posterior pdf morethoroughly.

III. PERFORMANCE EVALUATION AND

BATTERY RUL PREDICTION

The effectiveness of the proposed MPF technique will beevaluated in this section by simulation tests based on abenchmark model. Then, it will be implemented for batteryRUL prediction. The related SIR-PF and RAPF techniqueswill be used for comparison.

A. Testing Using a Benchmark Model

The following is a benchmark model that is commonlyused in PF testing because of its specific properties, such asnonlinear and non-Gaussian [23]

xk = 1

2xk−1 + 25xk−1

1 + x2k−1

+ 8 cos [1.2 (k − 1)] + ωk (17)

yk = 1

20x2

k + υk (18)

where ωk and υk are Gaussian white noise signals with zeromeans. The following conditions are used in this testing: thenumber of time steps k = 50, the variance of the measurementnoise υk = 1, the variance of the process noise ωk = 10, andthe initial state x0 = 0.1. Fifty particles (i.e., N = 50) will beused in the related PFs (i.e., SIR-PF, RAPF, and MPF). In theproposed MPF, λt is selected as the value of standard deviationof the data; the threshold ξ is selected based on applicationrequirements (ξ = 0.01 in this case); the strength factor b isdetermined by a trail-and-error process (b = 0.8 in this case).

To examine the parameter sensitivity of these three PFs, threedifferent particle numbers of 50, 100, and 150, are used forsimulation tests with the variance of the process noise ωk = 1and strength factor b = 0.8. In general, the more particles used,the higher the estimation accuracy of the PFs (but the longertime they will consume in modeling). It is observed from thesimulation results listed in Table I that the averaged mean ofthe root-mean-squares error (RMSE) becomes smaller as thenumber of particles increases in all three PFs; however, theproposed MPF gives the least averaged variance when particlenumbers N = 50. On the other hand, Table II summarizes thesimulation results of the proposed MPF with different valuesof strength factor b, when the process noise ωk = 1 and theparticle number N = 50. It is observed that the proposed MPF


TABLE II

AVERAGED MEAN AND STANDARD DEVIATION OF RMSE WITH

DIFFERENT STRENGTH FACTOR USING THE PROPOSED MPF

Fig. 3. Performance comparison of the related PF techniques over 30 runsby (a) SIR-PF, (b) RAPF, and (c) MPF. Red solid line: the true states. Bluedotted lines: the estimated states at different runs.

gives the best performance when b = 0.8 with the comparisonof 0.7 and 0.9, respectively.

Fig. 3 shows the test results of 30 random runs using thesame observation data set, as the multinomial resampling isimplemented at each time step. It can be observed that theRAPF generates less variance of the estimation error thanthe SIR-PF, because of its more effective particle-diversifyingmechanism. However, the proposed MPF provides more accu-rate estimation (i.e., with less variance of the estimationerror) than the RAPF, because its mutation mechanism canenrich the particle species and capture the distribution morecomprehensively and accurately.

To further verify the effectiveness of the proposed MPF,three test scenarios corresponding to different variancesωk = 1, 4, and 10, are tested. In each scenario, 100 data

Fig. 4. Test results of three PFs when the variance of the process noise isone. (a) Mean of RMSE. (b) Standard deviation of RMSE. Pink dashed line:the results of SIR-PF. Red dashed dotted line: the results of RAPF. Blue solidline: the results of the proposed MPF. Black solid line: the mean of the data.

Fig. 5. Test results of three PFs when the variance of the process noise isfour. (a) Mean of RMSE. (b) Standard deviation of RMSE. Pink dashed line:the results of SIR-PF. Red dashed dotted line: the results of RAPF. Blue solidline: the results of the proposed MPF. Black solid line: the mean of the data.

sets in total are randomly generated using (17) and (18).For each data set, these three PF techniques (i.e., SIR-PF,RAPF, and MPF) are tested over 100 runs. The RMSE betweenthe true states and the estimated states are computed for eachrun. For comparison, the mean and standard deviation of theRMSE over 100 runs in each data set are calculated and shownin Figs. 4–6. The results are summarized in Table III; it canbe observed that the averaged mean and standard deviationof RMSE become larger as the process noise increases inall three PF estimation scenarios. The RAPF provides loweraveraged mean and standard deviation of the RMSE than thoseof the SIR-PF, because of its diversified particle representation.The proposed MPF, however, provides the best estimationaccuracy (i.e., with the lowest mean and the smallest variance)compared with both SIR-PF and RAPF. This is because theMPF can approximate the distribution more accurately, andexplore the distribution more thoroughly.


Fig. 6. Test results of three PFs when the variance of the process noise isten. (a) Mean of RMSE. (b) Standard deviation of RMSE. Pink dashed line:the results of SIR-PF. Red dashed dotted line: the results of RAPF. Blue solidline: the results of the proposed MPF. Black solid line: the mean of the data.

TABLE III

AVERAGED MEAN AND STANDARD DEVIATION OF

RMSE OVER 100 DATA SETS

It should also be realized that, after several iterations ofimplementing PF, gaps may develop in the high-likelihoodarea of the estimated posterior pdf, which is an indicationof information loss. These gaps may degrade the estimationaccuracy in the current iteration, which can, in turn, propagateto the subsequent iterations and continue to degrade estimationaccuracy. Using the proposed mutation method, the pdf canbe explored more thoroughly, especially in the pdf gaps, andtherefore the estimation accuracy can be improved.

B. Battery RUL Prediction

Lithium-ion batteries are widely used in industrial anddomestic applications. An effective prognostic tool is veryuseful to predict the future state of the battery, so as todiagnose the battery’s health condition, and estimate its RULinformation. Reliable RUL information is critically needed inmany applications, such as electric vehicles and aircraft, toschedule battery recharging operations and prevent malfunc-tion of the related equipment. PF-based battery RUL prognos-tics have been investigated in several works. For example, Sahaet al. [15] used a hybrid method of relevance vector machineand Rao-Blackwellized PF to predict the RUL of lithium-ionbatteries. Daigle and Geobel [16] proposed a fixed-lag PF topredict RUL information of lithium-ion batteries. He et al. [8]used Dempster–Shafer theory and Bayesian Monte Carlo

Fig. 7. Schematic illustration of the battery prognostics tests, adoptedfrom [24]. BHM: the BHM modules. DAQ: the DAQ system.

to conduct battery remaining useful performance analysis.Liu et al. [11] proposed an RAPF technique to forecast the bat-tery life. Xian et al. [9] employed the Verhulst model, particleswarm optimization, and PF to estimate the RUL of batteries.Chen and Pecht [17] integrated model-based and data-drivenmethods in the PF framework to conduct prognostics oflithium-ion batteries. Liu et al. [18] presented a fusion prog-nostic framework with autoregressive model and PF methodsto perform batteries RUL estimation.

To test the proposed MPF technique for battery RULprediction, the data sets of battery 6 and battery 18 from theNASA Prognostics Center of Excellence are employed [25].Fig. 7, adopted from [24], schematically shows the batteryprognostics tests. The load bank and environmental chamberwere used to adjust load and environmental conditions ofthe battery cells, respectively. The electrochemical impedancespectroscopy (EIS) measurement was conducted in the batteryhealth monitoring (BHM) module, which was used to measurebattery impedance. The three states of batteries (i.e., charge,discharge, and impedance) were controlled by a switchingcircuitry. The sensors’ signals were collected by a data acquisi-tion (DAQ) system. The aim of the experiments was to identifydifferent health states of battery cells with similar terminalvoltages, and then to predict the RUL of the cells.

To generate these data sets, a lithium-ion battery was runthrough three different operational profiles: charge, discharge,and impedance, at room temperature. The charge processwas conducted by feeding a 1.5-A constant current to thebattery until its voltage reached 4.2 V, then the charge processcontinued in a constant voltage mode, until the charge currentdropped to 20 mA. The discharge process was carried outat a constant-current mode (2 A) until the battery voltagedropped to 2.5 V. The aging process of batteries can beaccelerated by repeatedly charging and discharging the batter-ies. The experiments were stopped when the battery reachedend-of-life criterion, which was set at 30% fade in ratedcapacity.

Fig. 8 shows the lumped-parameter model, where RE andRCT represent the electrolyte resistance and charge transferresistance, respectively, RW is the Warburg impedance, andCDL is the dual-layer capacitance. Although the battery capac-ity is usually inaccessible for measurement, RE + RCT isinversely proportional to the capacity C/1, and can be used


Fig. 8. Lumped-parameter model of a lithium-ion battery. RE and RCTrepresent the respective electrolyte resistance and charge transfer resistance,RW is the Warburg impedance, and CDL is the dual-layer capacitance.

Fig. 9. Relationship between RE + RCT and C /1 in battery 6. Red circles:measured RE + RCT versus C /1. Blue solid line: a linear fit.

to predict battery capacity drop [14]. RE and RCT can be esti-mated from EIS tests. The battery model can be formulated as

�k = �k−1 + vk (19)

k = k−1 exp (�kk) + wk (20)

Yk = k−1 + ρk (21)

where k is the state vector (i.e., RE or RCT) at time step k,�k is the exponential growth model parameter, and Yk is themeasurement vector containing battery parameters inferredfrom measured data. The state vector 1 at the first time steptakes the initial value of RE or RCT. The exponent �1 canbe calculated from training data using least square estimate.vk , wk , and ρk are Gaussian noise signals. The data sets ofRE , RCT, and C in battery 6 and battery 18 are smoothedto improve the RUL prediction using the battery modelrepresented by (19)–(21). The approximate linear relationshipbetween RE + RCT and C/1 in battery 6 is shown in Fig. 9.

The SIR-PF, RAPF, and the proposed MPF techniques wereimplemented for battery state estimation and RUL prediction.The first part of the trajectory (i.e., RE or RCT) was employedto estimate the battery model parameters. Then, the identifiedmodel for each technique was applied to predict the remainingpart of the trajectory. In each iteration, 1000 particles wereused to estimate the posterior pdf. The time to trigger theprediction depends on application requirements. Fig. 10 showsboth the state tracking and the future state prediction of dataRE and RCT, with the prognosis starting at cycle 30. In Fig. 10,

Fig. 10. State tracking and future state prediction at cycle 30 for the batteryparameters (a) RE and (b) RCT using three PFs: SIR-PF (◦—black line),RAPF (�—blue line), and MPF (∇—red line). Green solid line: the truestates of RE and RCT.

Fig. 11. Battery RUL prediction at cycle 30 using three different PFs:SIR-PF (◦—black line), RAPF (�—blue line), and MPF (∇—red line). Greensolid line: the real measurement of C /1.

the system model is identified in the estimation period;multiple-steps-ahead forecast is conducted in the predictionperiod, i.e., the nth estimated data in the prediction periodare the n-steps-ahead forecast using the identified model. It isobserved that RAPF outperforms SIR-PF in both RE and RCTpredictions, while the MPF provides even better predictionperformance than the RAPF in these two prediction scenarios(RE and RCT). By a linear transformation, the tracking ofthe capacity C/1 is shown in Fig. 11; it is observed thatRAPF generates less RUL prediction error (15.69 cycles) thanSIR-PF (17.39 cycles) because of its diversified resampledparticles. The MPF yields the minimum RUL prediction error(10.01 cycles) compared with SIR-PF and RAPF techniques,because it can explore the entire distribution with the help ofthe mutated particles.

Fig. 12 shows the prognostics results of the battery parame-ters RE and RCT when the prediction is triggered at cycle 54.



Fig. 13. Battery RUL prediction at cycle 54 using three different PFs:SIR-PF (◦—black line), RAPF (�—blue line), and MPF (∇—red line). Greensolid line: the real measurement of C /1.

To clarify the forecasting distinction, only the estimationperiod from cycle 40 to 54 and prediction period from cycle55 to 69 are shown in Fig. 12. The prediction performancesof these three PFs (i.e., SIR-PF, RAPF, and MPF) becomemore accurate in this case, because more data are used formodel parameter estimation. It can be observed from Fig. 12that the proposed MPF can predict the trend of RE and RCTmore accurately than both RAPF and SIR-PF, because of itsadvanced mutation mechanism. Fig. 13 shows the predictionof the corresponding capacity C/1; it can be observed that theMPF yields the minimum RUL prediction error (0.58 cycles).On the other hand, at the end of the prediction, RAPF andSIR-PF cannot reach end-of-life threshold, and generate largeerrors of 0.008 and 0.026 A h, respectively, at the last cycle.The MPF outperforms both RAPF and SIR-PF because of itsefficient mutation mechanism to diversify the particles andenhance model parameter identification.

The uncertainty of the classical PF algorithms can be repre-sented by the pdf of the estimated system states [18]. The pdfs

Fig. 14. Uncertainty representation of battery RUL estimation correspondingto (a) prediction at cycle 30 and (b) prediction at cycle 54, corresponding topdf of the SIR-PF (blue solid line), the RAPF (black dashed line), and theMPF (red dotted line).

Fig. 15. Relationship between RE + RCT and C /1 in battery 18. Red circles:measured RE + RCT versus C /1. Blue solid line: a linear fit.

of the estimated state (i.e., capacity) using SIR-PF, RAPF, andMPF are shown in Fig. 14, where their mean values are allset as zero to compare their spread (or uncertainty). Fig. 14(a)shows the pdfs of these three PFs at the cycle when theirmean values just reach failure threshold, as shown in Fig. 11.Fig. 14(b) shows the pdfs of three PFs at the cycle when theirmean values just reach failure threshold (i.e., MPF) or theend of prediction cycle (i.e., SIR-PF and RAPF), as shownin Fig. 13. It is observed that the MPF outperforms bothSIR-PF and RAPF with least uncertainty because of its capa-bility to fully explore high-likelihood area and prevent thepossible deviations by low-weight particles.

The approximate linear relationship between RE + RCTand C/1 in battery 18 is shown in Fig. 15. Fig. 16 showsthe prognosis of the battery parameters RE and RCT whenthe prediction is triggered at cycle 5. It is observed fromFig. 16 that the proposed the MPF outperforms both the RAPFand the SIR-PF in predicting RE and RCT because of itseffective posterior pdf estimation using the proposed mutation.Fig. 17 shows the prediction of the corresponding capacityC/1; it is observed that the MPF yields less RUL predictionerror (0.34 cycles) than that of RAPF (1.21 cycles) and SIR-PF(1.78 cycles), when they reach the end-of-life criterion. TheMPF outperforms both the RAPF and the SIR-PF, because itseffective mutation can improve posterior pdf estimation so asto enhance model identification.



Fig. 17. Battery RUL prediction at cycle 5 using three different PFs: SIR-PF(◦—black line), RAPF (�—blue line), and MPF (∇—red line). Green solidline: the real measurement of C /1.

IV. CONCLUSION

An MPF technique has been developed in this paper toimprove the accuracy of system state estimation. In this MPFtechnique, a new mutation method is proposed to explore theentire posterior space with focus on the remote area fromthe original particle, in which the original particle has lowweight. A novel particle selection scheme is suggested to directthe mutated particles to exploit high-likelihood area of thedistribution. As a result, the posterior pdf can be estimatedmore accurately, and the model identification accuracy can beimproved. The effectiveness of the developed MPF techniquehas been verified by simulation tests with a nonlinear non-Gaussian benchmark model. It has also been implemented fora battery RUL prediction application. Test results have shownthat the MPF can capture the dynamic behavior of a complexsystem quickly and accurately. It has a potential to be used forRUL prediction of batteries for real-world applications. In thefuture work, a more efficient particle selection mechanism willbe proposed to further improve the accuracy of the posterior

pdf estimation. Systematic research will also be carried out toinvestigate the effect of the process noise and measure noiseon PF result.

ACKNOWLEDGMENT

The authors would like to thank Dr. K. Goebel from thePrognostics Center of Excellence at NASA Ames ResearchCenter, for his assistance to this paper.

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De Z. Li (M’10) received the B.Sc. degree inelectrical engineering from Shandong University,Jinan, China, and the M.Sc. degree in control engi-neering from Lakehead University, Thunder Bay,ON, Canada, in 2008 and 2010, respectively. He iscurrently pursuing the Ph.D. degree with the Depart-ment of Mechanical and Mechatronics Engineering,University of Waterloo, Waterloo, ON.

He was a Research Associate with Lakehead Uni-versity from 2010 to 2011. His current researchinterests include signal processing, machinery con-

dition monitoring, mechatronic systems, linear/nonlinear system control, andartificial intelligence.

Wilson Wang (M’04–SM’07) received the M.Eng.degree in industrial engineering from the Universityof Toronto, Toronto, ON, Canada, and the Ph.D.degree in mechatronics engineering from the Univer-sity of Waterloo, Waterloo, ON, in 1998 and 2002,respectively.

He was a Senior Scientist with Mechworks Sys-tems, Inc., Waterloo, from 2002 to 2004. He joinedthe Lakehead University, Thunder Bay, ON, in2004, where he is currently a Professor with theDepartment of Mechanical Engineering. His current

research interests include signal processing, artificial intelligence, machinerycondition monitoring, intelligent control, mechatronics, and bioinformatics.

Fathy Ismail received the B.Sc. and M.Sc. degreesin mechanical and production engineering fromAlexandria University, Alexandria, Egypt, and thePh.D. degree from McMaster University, Hamilton,ON, Canada, in 1970, 1974, and 1983, respectively.

He joined the University of Waterloo, Waterloo,ON, in 1983, where he is currently a Professor withthe Department of Mechanical and MechatronicsEngineering. He has served as the Chair of theDepartment and the Associate Dean of the Facultyof Engineering. His current research interests include

machining dynamics, high-speed machining, modeling structures from modalanalysis testing, and machinery health condition monitoring and diagnosis.

A Mutated Particle Filter Technique for System State Estimation and Battery Life Prediction

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