estimationstrategiesfortheconditionmonitoringofabattery ...and the availability of reliable...

International Scholarly Research NetworkISRN Signal ProcessingVolume 2011, Article ID 120351, 17 pagesdoi:10.5402/2011/120351

Research Article

Estimation Strategies for the Condition Monitoring of a BatterySystem in a Hybrid Electric Vehicle

S. A. Gadsden, M. Al-Shabi, and S. R. Habibi

Department of Mechanical Engineering, McMaster University, Hamilton, ON, Canada L8S 4L7

Correspondence should be addressed to S. A. Gadsden, [email protected]

Received 4 January 2011; Accepted 6 February 2011

Academic Editors: L.-M. Cheng and F. Piazza

Copyright © 2011 S. A. Gadsden et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper discusses the application of condition monitoring to a battery system used in a hybrid electric vehicle (HEV). Batterycondition management systems (BCMSs) are employed to ensure the safe, efficient, and reliable operation of a battery, ultimatelyto guarantee the availability of electric power. This is critical for the case of the HEV to ensure greater overall energy efficiencyand the availability of reliable electrical supply. This paper considers the use of state and parameter estimation techniques forthe condition monitoring of batteries. A comparative study is presented in which the Kalman and the extended Kalman filters(KF/EKF), the particle filter (PF), the quadrature Kalman filter (QKF), and the smooth variable structure filter (SVSF) are usedfor battery condition monitoring. These comparisons are made based on estimation error, robustness, sensitivity to noise, andcomputational time.

1. Introduction

Condition monitoring is an essential process for faultdetection and diagnosis. It involves monitoring system statesor parameters over an operational period, where abnormalvalues or significant changes would indicate a fault. Quiteoften direct measurements of the states are not availabledue to limitations in design or cost. In these cases, stateand parameter estimation techniques can be used for infor-mation extraction. Condition monitoring of systems allowsproper maintenance to be scheduled, which helps reduceunscheduled downtime of manufacturing equipment, as wellas the cost to repair damaged systems [1, 2]. An importantarea for condition monitoring is energy management forhybrid electric (HEVs) and battery electric vehicles (BEVs).

In general, HEVs have two power sources: a gasolineengine and an electric motor. In full hybrid vehicles, theengine and the motor can operate separately or simultane-ously. The motor is used mainly during acceleration, startup,reverse mode, and in regenerative braking. A traction batterypack is used to provide power to the motor. It is recharged bya generator or during regenerative braking. The performanceof an HEV is largely dependent on a balance betweenthe gasoline engine and the electric motor, optimized with

respect to fuel consumption based on vehicle conditions[3]. Many different types of control methods have beenapplied to balance the power and energy requirements ofHEVs, including fuzzy logic [4–6], genetic algorithms [7],dynamic programming [8, 9], Pareto optimization [10], andintelligent mechanism designs [11]. These control strategiesrely heavily on the availability of battery power to balance theoperation of the gasoline engine versus the electric motor.The available battery power may be obtained from the state-of-charge (SOC) information [3, 12]. Further to the SOC,the battery state-of-health (SOH) is required in order to helpdetermine whether a battery would fail subject to a certainload [13]. The SOC cannot be measured directly with electricsignals, and as such it often, needs to be estimated throughcurrent and voltage relationships [3]. Poor estimation orcontrol of the SOC may lead to improper charging conditionsand can degrade the efficiency and reliability of the batteries[13]. Hence, proper condition monitoring of batteries plays apivotal role in the optimization of HEV performance, as wellas extending the lifetime and increasing the reliability of thebatteries [14].

State and parameter estimation techniques are an integralpart of condition monitoring and are used when directmeasurements of the states are not available. One of the most

2 ISRN Signal Processing

commonly studied methods for estimation is the Kalmanfilter (KF) for linear systems and its extended form (EKF)for nonlinear systems [15–19]. Operating conditions suchas battery SOC, power fade, capacity fade, resistance, andinstantaneous available power have been estimated wellusing the EKF [17]. Other methods such as the sigma-pointKalman filtering (SPKF) and support vector machine (SVM)have also been used for condition monitoring and faultdiagnosis [20–22].

In critical applications, such as automotive that requireadded safety and reliability, the choice of the estimationmethod is very important and should be selected basedon the linearity of the system, performance, robustness orsensitivity to noise and computational difficulty and time. Inrelation to battery condition monitoring, this paper presentsa quantitative and qualitative comparison of the followingmethods: the Kalman and extended Kalman filters (KF/EKF),the particle filter (PF), the quadrature Kalman filter (QKF),and the smooth variable structure filter (SVSF).

2. State and Parameter Estimation Techniques

State and parameter estimation is essential for sensing andinformation processing in model-based condition monitor-ing. Estimation theory involves information extraction bytracking changes in physical parameters or operational statesof the system. This paper studies four popular strategies:the Kalman and the extended Kalman filters (KF/EKF), theparticle filter (PF), the quadrature Kalman filter (QKF), andthe smooth variable structure filter (SVSF).

2.1. Kalman and Extended Kalman Filters. Even after 50years, the Kalman filter (KF) remains the most studied andone of the most popular tools used in state estimation[23–26]. It may be applied to linear dynamic systems inthe presence of Gaussian white noise, and provides anelegant and statistically optimal solution by minimizing themean-squared error. The impact that the KF has had onestimation and control problems is considered by some sci-entists and engineers to be one of the greatest achievementin engineering and signal processing [26]. It is a methodthat utilizes measurements linearly related to the states, anderror covariance matrices, to generate a gain referred toas the Kalman gain. This gain is applied to the a prioristate estimate, thus creating an a posteriori estimate. Theestimation process continues in a predictor-corrector fashionwhile maintaining a statistically minimal state error covari-ance matrix for linear systems.

The following two equations describe the system dynam-ic model and the measurement model used in general forstate estimation. Refer to Appendix A for a description of thenomenclature

xk+1 = Akxk + Bkuk + wk,

zk+1 = Ck+1xk+1 + vk+1.(1)

The next five equations form the KF algorithm and areused in an iterative fashion. Equation (2) extrapolates the apriori state estimate, and (3) is the corresponding a priori

error covariance. The Kalman gain may be calculated by (4),and is used to update the state estimate and error covariance,described by (5) and (6), respectively.

xk+1|k = Akxk|k + Bkuk, (2)

Pk+1|k = AkPk|kATk +Qk, (3)

Kk = Pk+1|kCTk

[

CkPk+1|kCTk + Rk

]−1, (4)

xk+1|k+1 = xk+1|k + Kk[

zk − Ckxk+1|k]

, (5)

Pk+1|k+1 = [I − KkCk]Pk+1|k. (6)

The effects due to model uncertainties can have alarge impact on the stability and performance of the KF[26, 27]. For nonlinear systems, the EKF may be used. Itis conceptually similar to the KF process. The nonlinearsystem and measurement matrices are linearized accordingto their corresponding Jacobian, which is a first-order partialderivative. This linearization introduces uncertainties in theestimation process; such that overlooked nonlinearities inthe system may cause the EKF to become unstable [26].

2.2. Particle Filter. The particle filter (PF) has many forms:Monte Carlo filters, interacting particle approximations [28],bootstrap filters [29], condensation algorithm [30], andsurvival of the fittest [31], to name a few. Compared to theKF, it is newer, being introduced in 1993. Since then, thePF has become a very popular method for solving nonlin-ear estimation problems, ranging from predicting chemicalprocesses to target tracking. The PF takes the Bayesianapproach to dynamic state estimation, in which one attemptsto accurately represent the probability distribution function(PDF) of the values of interest [32]. The PDF contains all ofthe pertinent statistical information and may be consideredas holding the solution to the estimation problem [32]. Es-sentially, the distribution holds a probability of values for thestate being observed. The stronger or tighter the predictionPDF, the more accurate the state estimate.

The PF obtains its name from the use of weightedparticles or “point masses” that are distributed throughoutthe PDF to form an approximation. These particles are usedin an iterative process to obtain new particles and associatedimportance weights, with the goal of creating a more accurateapproximation of the PDF. In general, as the number ofimplemented particles becomes large, the PDF becomesmore accurate [32]. An important step in the PF is that ofresampling, which eliminates particles with low weights andmultiplies those with high weights [32]. This helps to avoidthe degeneracy problem with the PF, which refers to onlyone particle having a significant importance weight after alarge number of recursions. Furthermore, it also increases theaccuracy of the PDF approximation by replicating particleswith high weights. The sequential importance resampling(SIR) algorithm is a very popular form of the PF and may besummarized by (7) to (10). The first equation draws samplesor particles from the proposal distribution.

x(n)k ∼ π

(

xk | x(n)k−1, yk

)

. (7)

ISRN Signal Processing 3

The next equation updates the importance weights up to anormalizing constant

ω(n)k = ω(n)

k−1

p(

yk | x(n)k

)

p(

x(n)k | x(n)

k−1

)

π(

xk | x(n)k−1, yk

) . (8)

Next, the normalized weights are calculated for each particle

ω(n)k = ω(n)

k∑n

i=1 ω(i)k

. (9)

Finally, a constant known as the effective number ofparticles is calculated as shown in (10). Resampling isperformed if the effective number of particles is lower thansome design threshold

Neff = 1∑n

i=1

(

ω(i)k

)2 . (10)

2.3. Quadrature Kalman Filter. Similar to the PF, the quadra-ture Kalman filter (QKF) is a type of Bayesian filter that isable to model dynamic processes which are nonlinear andsubject to non-Gaussian noise. In 2007, it was proposed thata set of Gauss-Hermite quadrature points could be used toparameterize the PDF [33, 34]. When compared with theEKF, it was found that the QKF method provides a moreaccurate least-squares solution [33]. Figure 1 shows the PDFof a nonlinearly transformed Gaussian random variable. Themean of the EKF appears to be biased, and the covarianceis obviously far from the true covariance. However, the QKFmean and covariance match the true values quite well, withthe 5 point QKF working the best. The main drawback tothis method is the fact that the number of terms used inthe Gaussian sum grows exponentially, which means thatmore memory will be used over a longer period of time [33].The process of this filter is similar to the PF, in the sensethat it may be solved recursively using two stages (time andmeasurement update). Please refer to Appendix B for the fullQKF algorithm.

2.4. Smooth Variable Structure Filter. In 2002, the variablestructure filter (VSF) was introduced as a new predictor-corrector method used for state and parameter estimation[27, 35]. It is a type of sliding mode estimator, where gainswitching is used to ensure that the estimates converge totrue state values. An internal model of the system, eitherlinear or nonlinear, is used to predict an a priori stateestimate. A corrective term is then applied to calculate thea posteriori state estimate, and the estimation process isrepeated iteratively. The SVSF was later derived from theVSF and uses a simpler and less complex gain calculation[36]. In its present form, the SVSF is stable and robust tomodeling uncertainties and noise, given an upperbound onuncertainties [36]. The basic concept of the SVSF is shown inFigure 2. Assume that the solid line in Figure 2 is a trajectoryof some state (amplitude versus time). An initial value isselected for the state estimate. The estimated state is pushedtowards the true value until it reaches a subspace around the

150

100

50

0

− 50

− 100150100500− 50− 100

x 2

x1

5-point QKF covariance

EKF covariance

3-point QKF covariance

True covariance

3-point QKF mean: +

5-point QKF mean: o

EKF mean: ×

True mean: ∗

Figure 1: PDF of a Nonlinearly transformed gaussian randomvariable [33].

Am

plit

ude

Initial value of theestimated states

Existencesubspace

Estimated statetrajectory

Systemstate trajectory

Time

Figure 2: SVSF estimation concept [36].

actual state trajectory, referred to as the existence subspace.Once the value enters the existence subspace, the estimatedstate is forced to remain within it and into switching alongthe system state trajectory [36].

The SVSF method is model based and applies to smoothnonlinear dynamic equations. The estimation process maybe summarized by (11) to (16) and is repeated iteratively.An a priori state estimate is calculated using an estimatedmodel of the system. This value is then used to calculatean a priori estimate of the measurement defined by (12). Acorrective term, referred to as the SVSF gain, is calculated asa function of the error in the predicted output, as well as again matrix and the smoothing boundary layer width. The


corrective term calculated in (13) is then used in (14) to findthe a posteriori state estimate

xk+1|k = F(

xk|k ,uk)

, (11)

zk+1|k = Cxk+1|k , (12)

Kk+1 = C−1∣

∣

∣

(∣

∣

∣ezk+1|k

∣

∣

∣

ABS+ γ∣

∣

∣ezk|k∣

∣

∣

ABS

)∣

∣

∣

ABS

◦ sat(

ezk+1|k ,Ψ)

,(13)

xk+1|k+1 = xk+1|k + Kk+1, (14)

ezk|k = zk − zk|k, (15)

ezk+1|k = zk+1 − zk+1|k. (16)

Two critical variables in this process are the a priori anda posteriori output error estimates, defined by (15) and (16),respectively [36]. Note that (15) is the output error estimatefrom the previous time step and is used only in the gaincalculation.

3. Condition Monitoring of a Battery System ina Hybrid Electric Vehicle

A variety of batteries have been studied in literature, mostnotably lead-acid and lithium-ion batteries [13, 15, 19, 20,37]. Lead-acid batteries are the oldest type of rechargeablebatteries, and are most commonly found in motor vehicles.Lithium-ion batteries are also a form of rechargeable battery,which contain lithium in its positive electrode (cathode).These batteries are usually found in portable consumerelectronics (i.e., laptops or notebooks) due to particularlyhigh energy-to-weight ratios, slow self-discharge, and a lackof memory effect (i.e., where a battery loses its maximumenergy capacity over time) [16]. In recent years, lithium-ion batteries have slowly entered the hybrid electric vehiclemarket, due to the fact that they offer better energy densitycompared to standard batteries [38].

The operation of batteries may be studied by using theadvanced vehicle simulator (ADVISOR), which was writtenin MATLAB and Simulink by the US Department of Energyand the National Renewable Energy Laboratory [39–41].ADVISOR is used for the analysis of performance and fueleconomy of three vehicle types: conventional, electric, andhybrid vehicles [39]. In 2001, the resistance-capacitance (RC)battery model was first implemented in ADVISOR [42]. Theelectrical model consists of three resistors (Re, Rc, and Rt)and two capacitors (Cb and Cc). The first capacitor (Cb)represents the capability of the battery to chemically store acharge, and the second capacitor (Cc) represents the surfaceeffects of a cell [41]. The resistances and capacitances varywith changing SOC and temperature (T) [41]. ADVISORoffers two different datasets for the RC battery model:lithium-ion and nickel-metal hydride chemistries. For thepurposes of this study, the lithium-ion chemistry was used inconjunction with the RC battery model. Figure 3 illustratesthe equivalent circuit diagram of the RC model.

Rt

Rc

Ic

Ib

Cb ,VCb Vo

IsRe

Cc ,VCc

Figure 3: RC battery circuit diagram.

A standard model of a parallel hybrid electric vehiclereferred to within ADVISOR as the Annex VII PHEVwas used for this study. This model has been developedby the International Energy Agency (IEA), which is aninternational research community for the development andcommercialization of hybrid and electric vehicles [43]. Themodel is based on data obtained from published sources andnational (U.S.) research test data [39]. The battery systemof the HEV represents the battery pack which stores energyon board the HEV. The system accepts a power request andreturns the available power from the battery, as well as theSOC, voltage and current [39].

The nonlinear equations that describe the system may bederived from the RC battery model of Figure 3, as shownin Appendix C. For the purposes of condition monitoringof the battery, two voltages VCb and VCc as well as twocapacitance parameters Cb and Cc need to be estimated.Should a fault exist in either of the battery capacitors, onewould be able to determine this given the correspondingchange in the parametric value. Further to the equationsfound in Appendix C, a discrete-time state space model ofthe Capacitor voltages may be defined as follows:

⎡

⎣

VCbk+1

VCck+1

⎤

⎦

=

⎡

⎢

⎢

⎢

⎣

− TsCb(Re + Rc)

+ 1Ts

Cb(Re + Rc)Ts

Cc(Re + Rc)− TsCc(Re + Rc)

+ 1

⎤

⎥

⎥

⎥

⎦

k

⎡

⎣

VCbk

VCck

⎤

⎦

+

⎡

⎢

⎢

⎢

⎣

TsRcCb(Re + Rc)

TsReCc(Re + Rc)

⎤

⎥

⎥

⎥

⎦

k

ISk .

(17)

For parameter estimation, (17) is used to formulate astate vector that would include parameters such that

x = [VCb VCc Cb Cc]T. (18)


0 5 10 15 20 25 30 35 40 450

0.5

1

1.5

2

2.5

3

3.5

Time (s)

Cu

rren

tin

put

(A)

Simulation input values

Figure 4: Supplied current input.

By rearranging (17), and further to Appendix C, thegeneral model used for state and parameter estimation isobtained as

xk+1 = f (xk,uk) + wk,

zk+1 = h(xk+1,uk+1) + vk+1.(19)

To implement the EKF method, the nonlinear system hadto be linearized using a first-order Taylor series approxima-tion. The linearized form of the system equation is providedin Appendix D.

4. Estimation Results

This paper presents a comparative performance for the appli-cation of the extended Kalman filter, the particle filter, thequadrature Kalman filter, and the smooth variable structurefilter for condition monitoring of RC batteries. The study isconducted by simulation using the ADVISOR battery model.The model parameters are documented in [39].

Figures 4 and 5 illustrate the input current (Is) and theoutput voltages VCb and VCc used in the simulation. The pa-rameter values Cb and Cc are made to vary in order to simu-late fault conditions as shown in Figures 6 and 7. ParametersCb and Cc as well as the states VCb and VCc are estimatedusing the four methods described in Section 2.

A comparison and discussion of the results is providedusing two simulated cases: one with noise and one with noiseas well as modeling errors. Note that the sampling time usedin the simulation was 0.01 seconds.

4.1. Extended Kalman Filter Results. For linear dynamicsystems in the presence of Gaussian white noise, the KFprovides an elegant and statistically optimal solution byminimizing the mean-squared estimation error. The EKFis used for nonlinear problems. The following covariance

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Vol

tage

s(V

)

Capacitance voltages

VCb

VCc

Figure 5: Capacitance voltages.

0 5 10 15 20 25 30 35 40 452.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

4.6

4.8× 104

Time (s)

Cb

valu

es

Cb simulation values

Figure 6: Cb simulation values.

matrices (error, process, and measurement, resp.) were usedfor the EKF and were obtained by trial-and-error:

P =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

1 30 30 30

30 1 30 30

30 30 1 30

30 30 30 1

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

Q = R =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

.

(20)


0 5 10 15 20 25 30 35 40 452000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Cc

valu

es

Cc simulation values

Figure 7: Cc simulation values.

The EKF worked very well in the absence of modelingerrors, as shown in Figure 8. Its performance degraded con-siderably when modeling errors were added, as illustrated inFigure 9.

As shown in Figure 10, the EKF was very sensitive tothe selection of the initial conditions; however, it oftenrecovered and performed very well. This is important tonote because without selecting good starting points, the filterperformed the worst. However, when good initial conditionswere selected, the EKF yields the best results (based on RMSEand the assumption that there were no uncertainties in themodel). When uncertainties (besides white measurementand process noise) were present in the filter model, in thisexample, the EKF became unstable (as shown in Figure 9)and failed to yield reasonable results. The simulation timefor the EKF was one of the fastest, and compared to the othermethods, it was computationally easy.

4.2. Particle Filter Results. For the PF method, a large numberof particles (500) were used. Its application results are shownin Figures 11 and 12. The PF was able to estimate fairly well inthe presence of noise. However, when modeling uncertaintieswere introduced, the PF had difficulty tracking the truevoltages. As shown in Figure 12(b), after about 20 seconds,the PF was able to recover and provide a good estimate.

The PF provided satisfactory results. However, the tech-nique had the highest RMSE (for the first case). That beingsaid, when compared to the EKF, no Jacobian matrix hadto be calculated to linearize the system matrix, as weightedparticles were used instead. This may be an attractive featureif the system is too difficult to linearize. A large numberof particles (which were required to accurately estimate thePDF), and the resampling feature of the PF, contributedto a slower computational time. When uncertainties wereadded to the filter model (as shown in Figure 12), the PF wasable to recover slowly from modeling errors and after sometime, accurately represent the first two states. There weresignificant errors in the third and fourth states; however this,

as previously mentioned, may be attributed to measurementsbeing only available for the first two states.

4.3. Quadrature Kalman Filter Results. For the QKF, the ini-tial values of the covariance matrices were set to the identitymatrix. The following results shown in Figures 13 and 14were generated using the QKF based on the simulation setup.The QKF performed very well for both cases. However, likethe previous methods, when uncertainties were introduced,it was unable to accurately track the two parameters associ-ated with the battery (Cb and Cc) as shown in Figure 14.

The QKF provided good results for the system. Likethe PF, there was no need to calculate the Jacobian, sinceweighted quadrature points were used instead. One of thedrawbacks of the QKF is the computational demand whena large number of states and quadrature points are used.This leads to a slower calculation time. For example, sincefour states were required, an array of 81 quadrature pointswas required (34). If a more accurate model was required,five quadrature points per state could be used. However,this would further increase the computational time. For thissystem, it was found that increasing the number of quadrat-ure points had a negligible effect on the accuracy. Note thatwhen uncertainties were added to the filter model, the QKFworked extremely well. However, the slow computation timeis its main hindrance.

4.4. Smooth Variable Structure Filter Results. The constantdiagonal matrix (γ) was set to 0.4, and the smoothing bound-ary layer thicknesses (ψ) were, respectively, set to 1 × 10−3

and 1 × 10−2 for the two states and the two parameters,respectively. The results are shown in Figures 15 and 16,and were generated using the SVSF based on the simulationsetup. The SVSF method is very robust to noise and modelinguncertainties, as demonstrated by the accurate estimation inboth cases.

The SVSF yielded very good results for condition mon-itoring of the battery system. Similar to the previous twomethods, it did not require calculation of the Jacobian ma-trix, and also required a time delay (signal extraction for thelast two states). The SVSF simulation time was just as fast asthe EKF, and the results were similar for the case with noise.When errors were added to the filter model, the SVSF workedthe best (in terms of RMSE). Although there were still largeestimation errors for the third and fourth states, they werefinite and stable. It is important to note that the SVSF wasrobust and not sensitive to changes in the filter parametersand the initial conditions.

4.5. Summary of Results. The results of the simulations werecompared based on estimation error, robustness, sensitivityto noise, and computational difficulty and time. As shownin Table 1, the EKF performed best in terms of estimationerror, in the case of only noise (referred to as Case (1)).In the presence of modeling uncertainties (Case (2)), theEKF’s performance severely degraded. The SVSF yielded themost accurate estimation for the case involving modelinguncertainties. Both the PF and QKF performed moderately


0 5 10 15 20 25 30 35 40 45−8000

−6000

−4000

−2000

0

2000

4000

Time (s)

Err

or

Error over time

Cb

Cc

VCb

VCc

(a) Estimation error over time

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Vol

tage

s(V

)

Capacitor voltages

VCb estimateVCc estimate

VCb actualVCc actual

(b) Capacitor voltages

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

× 104

Time (s)

Cb

valu

e


Cb estimateCb actual

(c) Cb simulation values

Cc

valu

e


Cc estimateCc actual

0 5 10 15 20 25 30 35 40 450

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

(d) Cc simulation values

Figure 8: Extended Kalman filter simulation results (only noise).

well, but at a cost of computational time, as shown in the lastrow of Table 1.

The error in Table 1 refers to the root mean-squared error(RMSE), calculated as follows:

RMSE = 1N

√

√

√

√

√

N∑

i=1

(xi − xi)2. (21)

Table 2 gives a quantitative comparison of the four filters.The robustness was based on observations made on varyingthe noise levels and considering its impact on model errors.The sensitivities were determined by tuning the filters various

parameters. Computation (simulation) time was determinedfor each method and may be compared on a relative basis.

5. Conclusions

This paper discussed the application of condition monitoringto an RC battery system, typically found within a hybridelectric vehicle. State and parameter estimation techniquesare important as they are responsible for providing accurateestimates of the states when reliable measurements areunavailable, and hence ensure successful condition mon-itoring. A comparative study was presented in which theKalman and the extended Kalman filters (KF/EKF), the


Cb

Cc

0 5 10 15 20 25 30 35 40 45−1

0

1

2

3

4

5

Time (s)

Err

or

Error over time×104

VCb

VCc


0 5 10 15 20 25 30 35 40 45−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (s)

Vol

tage

s(V

)

Capacitor voltages




0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

Time (s)

Cb

valu

e



×104


0 5 10 15 20 25 30 35 40 450

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Cc

valu

e




Figure 9: Extended Kalman filter simulation results (including errors).

Table 1: Quantitative summary of results.

EKF PF QKF SVSF

States and parameters Case (1) Case (2) Case (1) Case (2) Case (1) Case (2) Case (1) Case (2)

RMSEVCb 2.25E− 09 5.61E− 01 2.34E− 04 9.50E− 03 2.07E− 05 4.19E− 05 2.82E− 07 1.24E− 08

VCc 1.10E− 08 9.23E− 02 2.95E− 04 1.30E− 03 2.07E− 05 4.19E− 05 6.58E− 07 9.62E− 09

Cb 12.02 597 41.19 1196 28.07 1196 18.48 1195

Cc 1.40 89 2.19 177 1.20 177 0.73 177

Simulation time (sec) 0.38 0.44 3.27 3.34 8.35 8.42 0.39 0.39


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time (s)

Cb

valu

e



×104

(a) Cb simulation values

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

−1000

0

1000

2000

3000

4000

5000

Time (s)

Cc

valu

e



(b) Cc simulation values

Figure 10: EKF Simulation results with poor initial conditions (only noise).

Table 2: Qualitative comparison of the filters.

Characteristic EKF PF QKF SVSF

Robustness Little Medium Good Good

Sensitivity toUncertainties

High High Low Low

Sensitivity to FilterParameters

High Low Low Low

Computation Time Fast Slow Very Slow Fast

Table 3: Ranking of the state estimation techniques.

RankingCase (1): Case (2):

Only noise Noise and model errors

1 EKF SVSF

2 SVSF QKF

3 QKF PF

4 PF EKF

particle filter (PF), the quadrature Kalman filter (QKF), andthe smooth variable structure filter (SVSF) were appliedfor condition monitoring. These estimation methods werecompared based on estimation error, robustness, sensitivityto noise, and computation time. Table 3 summarizes theresults of the comparison. For the case without modelingerrors, the EKF worked the best in terms of RMSE andcomputational speed. When modeling errors were present,which is common in physical applications, the SVSF wasshown to work significantly better than the other methodsin terms of stability and RMSE.

Appendices

A. List of Nomenclature

A: System matrix

B: Input matrix

C: Output matrix

e: State estimation error

f : Nonlinear system or process equation

h: Nonlinear measurement or output equation

k: Time step index

K : Gain value (KF, QKF, or SVSF)

m: Number of measurements

n: Number of states

Neff: Effective number of particles

P: Error covariance matrix

Q: System noise covariance matrix

R: Measurement noise covariance matrix

Sat: Saturation function

t: Simulation time

Ts: Sampling time

u: Input

v: Measurement noise

w: System noise

wl: Quadrature point weight


×104

0 5 10 15 20 25 30 35 40 45−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

Time (s)

Err

or

Error over time

Cb

Cc

VCb

VCc


0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Vol

tage

s(V

)

Capacitor voltages




0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

7×104

Time (s)

Cb

valu

e




0 5 10 15 20 25 30 35 40 450

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Cc

valu

e




Figure 11: Particle filter simulation results (only noise).

x: System states

y: Observed measurement or output

z: Measurement output

γ: Constant diagonal gain matrix with elements havingvalues between 0 and 1

ω: Particle weight

Ψ: Smoothing boundary layer

π: Probability distribution

τ: Sampling time

: Denotes an estimated value

∼ : Denotes an error value

· : On top of a parameter denotes a time derivative.

Furthermore, note that subscript k + 1 | k refers to an apriori time step and that the subscript k + 1 | k + 1 refers ana posteriori time step. A superscript of T denotes a matrixtranspose.

B. Quadrature Kalman Filter Algorithm

The following is a summary of the QKF algorithm, directlyas presented in [33, 34]. This process was used to obtain the


0 5 10 15 20 25 30 35 40 45−2

0

2

4

6

8

10

12

×104

Time (s)

Err

or

Error over time

Cb

Cc

VCb

VCc


0 5 10 15 20 25 30 35 40 45−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Vol

tage

s(V

)

Capacitor voltages




0 5 10 15 20 25 30 35 40 45−6

−4

−2

0

2

4

6×104

Time (s)

Cb

valu

e




0 5 10 15 20 25 30 35 40 45−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

×104

Time (s)

Cc

valu

e




Figure 12: Particle filter simulation results (including errors).

presented results. There are two updates at each step: the timeupdate and the measurement update.

Time Update Step.

(1) Assuming at time k the posterior density functionp(xk−1 | zk−1) = N(xk−1|k−1,Pk−1|k−1) is known, thenwe may factorize as follows:

Pk−1|k−1 =√

Pk−1|k−1

(√

Pk−1|k−1

)T. (B.1)

(2) Evaluate the quadrature points {Xl,k−1|k−1}ml=1 as

Xl,k−1|k−1 =√

Pk−1|k−1ξl + xk−1|k−1 . (B.2)

(3) Evaluate the propagated quadrature points{X∗l,k|k−1}ml=1as:

X∗l,k|k−1 = f(

Xl,k−1|k−1,uk−1, k − 1)

. (B.3)


0 5 10 15 20 25 30 35 40 45−1

−0.5

0

0.5

1

1.5×104

Time (s)

Err

or

Error over time

Cb

Cc

VCb

VCc


0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Vol

tage

s(V

)

Capacitor voltages




0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

7

× 104

Time (s)

Cb

valu

e




0 5 10 15 20 25 30 35 40 450

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Cc

valu

e




Figure 13: Quadrature Kalman filter simulation results (only noise).

(4) Estimate the predicted state:

xk|k−1 =m∑

l=1

wlX∗l,k|k−1 . (B.4)

(5) Estimate the predicted error covariance:

Pk|k−1 =m∑

l=1

wlX∗l,k|k−1X

∗Tl,k|k−1 − xk|k−1x

Tk|k−1 +Qk. (B.5)

Measurement Update Step.

(1) Factorize:

Pk−1|k−1 =√

Pk−1|k−1

(√

Pk−1|k−1

)T. (B.6)

(2) Evaluate the quadrature points {Xl,k−1|k−1}ml=1 as:

Xl,k−1|k−1 =√

Pk−1|k−1ξl + xk−1|k−1 . (B.7)


0 5 10 15 20 25 30 35 40 45− 2

0

2

4

6

8

10

12

× 104

Time (s)

Err

or

Error over time

Cb

Cc

VCb

VCc


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vol

tage

s(V

)

Capacitor voltages

0 5 10 15 20 25 30 35 40 45

Time (s)




− 6

− 4

− 2

0

2

4

6× 104

Cb

valu

e



0 5 10 15 20 25 30 35 40 45

Time (s)

(c) Cb Simulation values

0 5 10 15 20 25 30 35 40 45

Time (s)

− 1

− 0.8

− 0.6

− 0.4

− 0.2

0

0.2

0.4

0.6

0.8

1

× 104

Cc

valu

e



(d) Cc Simulation values

Figure 14: Quadrature Kalman filter simulation results (including errors).

(3) Evaluate the propagated quadrature points {Zl,k|k−1}ml=1as:

Zl,k|k−1 = h(

Xl,k|k−1,uk , k)

. (B.8)

(4) Estimate the predicted measurement:

zk|k−1 =m∑

l=1

wlZl,k|k−1 . (B.9)

(5) Estimate the innovation covariance matrix

Pzz,k|k−1 = Rk +m∑

l=1

wlZl,k|k−1ZTl,k|k−1 − zk|k−1z

Tk|k−1.

(B.10)

(6) Estimate the cross covariance matrix

Pxz,k|k−1 =m∑

l=1

wlXl,k|k−1ZTl,k|k−1 − xk|k−1 z

Tk|k−1. (B.11)


0 5 10 15 20 25 30 35 40 45− 8000

− 6000

− 4000

− 2000

0

2000

4000

6000

8000

10000

12000

Time (s)

Err

or

Error over time

Cb

Cc

VCb

VCc


0 5 10 15 20 25 30 35 40 45

Time (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vol

tage

s(V

)

Capacitor voltages




0 5 10 15 20 25 30 35 40 45

Time (s)

0

1

2

3

4

5

6

× 104

Cb

valu

e




0 5 10 15 20 25 30 35 40 45

Time (s)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Cc

valu

e




Figure 15: Smooth variable structure filter simulation results (only noise).

(7) Estimate the Kalman gain:

Wk = Pxz,k|k−1P−1zz,k|k−1 . (B.12)

(8) Estimate the updated step:

xk|k = xk|k−1 +Wk(

zk − zk|k−1)

. (B.13)

(9) Estimate the corresponding error covariance:

Pk|k = Pk|k−1 −WkPzz,k|k−1WTk . (B.14)

C. Derivation of the RC BatteryModel Equations

The output voltage may be calculated by summing thevoltages of each element in the circuit. Summation of theouter loop and inner loop voltages yield two equations forthe output voltage, respectively:

VO = ISRt + IbRe +VCb, (C.1)

VO = ISRt + IcRc +VCc. (C.2)


0 5 10 15 20 25 30 35 40 45− 2

0

2

4

6

8

10

12× 104

Time (s)

Err

orError over time

Cb

Cc

VCb

VCc


0 5 10 15 20 25 30 35 40 45

Time (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vol

tage

s(V

)

Capacitor voltages




0 5 10 15 20 25 30 35 40 45

Time (s)

− 6

− 4

− 2

0

2

4

6× 104

Cb

valu

e




0 5 10 15 20 25 30 35 40 45

Time (s)

− 1

− 0.8

− 0.6

− 0.4

− 0.2

0

0.2

0.4

0.6

0.8

1

Cc

valu

e



× 104


Figure 16: Smooth variable structure filter simulation results (including errors).

Recall Kirchhoff ’s current law, which states that the totalcurrent entering a node must equal the total current leaving,such that the supply current is defined by

IS = Ib + Ic. (C.3)

Rearranging for Ic in (C.3) and substituting into (C.2) yields

VO = ISRt + (IS − Ib)Rc +VCc. (C.4)

Equating (C.4) with the first output voltage (C.1) and solvingfor Ib yields

Ib = ISRcRe + Rc

+VCc

Re + Rc− VCb

Re + Rc. (C.5)

Note that VCb = Ib/Cb such that (C.5) becomes

VCb = ISRcCb(Re + Rc)

+VCc

Cb(Re + Rc)− VCb

Cb(Re + Rc). (C.6)

Similarly, the above approach may be used to solve for thesecond capacitor voltage rate of change:

VCc = ISReCc(Re + Rc)

+VCb

Cc(Re + Rc)− VCc

Cc(Re + Rc). (C.7)


The output voltage may be solved by manipulating (C.1)through (C.3). Setting Ic = IS − Ib in (C.2), and rearrangingfor Ib, results in the following:

Ib = Rt + RcRc

IS +VCc

Rc− VO

Rc. (C.8)

Rearranging (C.1) for Ib, and setting the result equal to (C.8)yields

VO

Re− VCb

Re− RtReIS = Rt + Rc

RcIS +

VCc

Rc− VO

Rc. (C.9)

Manipulating (C.9) and simplifying for the output voltageresults in

VO = RcRe + Rc

VCb +Re

Re + RcVCc +

[

Rt +ReRcRe + Rc

]

IS.

(C.10)

D. Linearization of the Battery System

The battery model described in Section 3 requires lineariza-tion for the EKF to be implemented. The linearization wasbased on the Jacobian matrix and the four states (where forneatness, WCb = 1/Cb and WCc = 1/Cc)

φk =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

∂ f1∂VCb

∂ f1∂VCc

∂ f1∂WCb

∂ f1∂WCc

∂ f2∂VCb

∂ f2∂VCc

∂ f2∂WCb

∂ f2∂WCc

∂ f3∂VCb

∂ f3∂VCc

∂ f3∂WCb

∂ f3∂WCc

∂ f4∂VCb

∂ f4∂VCc

∂ f4∂WCb

∂ f4∂WCc

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

k

, (D.1)

where the functions in (D.1) are described by

f1 = −TSVCbWCb

Re + Rc+VCb

+TSVCcWCb

Re + Rc+TSRcWCb

Re + RcIS,

f2 = −TSVCbWCc

Re + Rc− TSVCcWCb

Re + Rc

+VCc +TSReWCc

Re + RcIS,

f3 =WCb,

f4 =WCc.

(D.2)

The linearized form (φ) of the system matrix, using thestate vector described by (18), is as follows:

φk =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

φ11 φ12 φ13 0

φ21 φ22 0 φ24

0 0 1 0

0 0 0 1

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

k

, (D.3)

where the elements are described by:

φ11 = −TSWCbk

Rek + Rck+ 1,

φ12 = TSWCbk

Rek + Rck,

φ13 = TSRek + Rck

(−VCbk +VCck + RckISk),

φ21 = TSWCck

Rek + Rck,

φ22 = −TSWCck

Rek + Rck+ 1,

φ24 = TSRek + Rck

(VCbk −VCck + RekISk).

(D.4)

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