Special Issue

Automobile Electronics

Battery State Sensor~Curve Fitting of Lead-acid Battery’s Voltage Relaxation with Expanded Kalman-filter~
Noriyasu Iwane*

Stable open circuit voltage (OCV) prediction in a short time is required to detect
the state of the lead-acid battery for automobile. We have found in our pastdevelopment that the voltage relaxation behavior of the lead-acid battery could be expressed inhigher-order exponential function of 5 or higher order, and the trial result to detect OCV in ashort time was reported in Furukawa Denkou Jihou No.120 (Sept. 2007, in Japanese). In actual
usage of the function, the optimum coefficients of the function shall be obtained according to theactual voltage behavior. This time, the calculation method is changed from the general leastsquares method which requires high computational effort, to the expanded Kalman-filter calculation which is known as the less computational effort method, and the trial calculation was performed to assess the benefit.

ABSTRACT1.

INTRODUCTION

Because of the rapid popularization of start-stop systemetc, the necessity for accurate automotive battery statesensing is increasing these days. In addition, the conventional technique of managing only the amount of powerout of the battery by using a current sensor is no longer
enough and a high functionality sensor which can detect
the battery charge level and the degradation state in addition to the direct measurement values of such as current,
voltage and temperature, are becoming a requirement.
The development of the above described sensor was earlier promoted by German manufacturers such as Bosch,Hella and has started to be implemented in European
vehicles since 2007 approximately. In case of Japanese
car manufacturers, Honda started to equip with the Boschmade sensor in 2010. Furukawa Electric (FEC) has been
in a position to be able to share knowledge with The
Furukawa Battery Co., Ltd., and has been developing
technology related to battery state since around 2000.
From 2005, the development has focused on the similar
in-vehicle sensor. As a result, the developed in-vehicle
sensor was adapted, for the first time to Japanese car
manufacturers in the HONDA ACCORD launched in North
America on Sept. 19th 2012. Figure1 shows an example.

* Automotive System & Device Laboratories,
		 Research & Development Division

Figure 1 Furukawa Electric “Battery State Sensor”.

The typical technology developed by FEC is the technology to detect OCV, which is an index to estimate theState of Charge (SOC), in a short time using a higher
order exponential function. The bases of this technology
was reported in Furukawa Denkou Jihou No.120 1), (inJapanese). This technology can estimate OCV in a shorter time and with a higher accuracy in comparison to theconventional method, on the other hand a very high computational effort was required. Computational effort reduction was the issue for the in-vehicle sensor which has limitation in cost and dimension.
Until this point, prioritizing the certainty of the calculation, the development using mainly the least–squares
method based on Levenberg-Marquardt method2) which
requires a high computational effort. The sensor for
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Battery State Sensor ~Curve Fitting of Lead-acid Battery’s Voltage Relaxation With Expanded Kalman-filter~

2.

EQUILIBRIUM POTENTIAL OCV OF
LEAD-ACID BATTERIES AND ITSRELAXATION BEHAVIOR

The formula (1) shows chemical changes associated withcharging and discharging of the battery.
Pb+ PbO2+2H2SO4 ⇔ 2PbSO4+2H2O(1)

The left-hand side of the formula (1) indicates the discharging and the right –hand side indicates the charging.
Here, the important issue is as follow. As a peculiar feature of lead-acid batteries, in its progress of discharging
the sulfuric acid in the electrolyte is consumed and is
replaced by water and the sulfuric density is changed.
This shows that the electrolyte concentration can be adirect index to express the battery charging rate.
Meanwhile, the equilibrium potential in oxidation-reduction reaction is expressed by Nernst equation4) based on
the thermodynamic equilibrium by using the activity ofchemical elements for the reaction. Nernst equation forthe lead-acid battery is shown in the formula (2).
E=E0+(RT/F) · ln(aH+· aHSO4-/aH2O)

charge level of the lead-acid battery has been widelyused as a very common method. But, there was a difficult
problem to adapt to an application such as an automotive
battery. Because, the charge and the discharge current
always flows in driving the automobile, and even the battery voltage converges to stable OCV after the vehicle
stops, the unhomogeneity of the electrolyte generated ina dynamic environment takes very long time such as ten
and multiple tens of hours to eliminate this influence. Incase of an automobile, the resting period of the automobile is different from drivers and from time to time condition. And we can not avoid saying that ensuring the stable
and sufficient resting period is extremely difficult.
Figure 2 shows 86,400 s=24 hrs of the battery voltage
behavior observation result. The battery voltage, which
has received charging over-voltage, leads to a stable
OCV, under the severe condition of -20℃. AS this observation result shows, depending on the temperature, the battery voltage does not reach the stable OCV even after 24
hrs.

15.0

14.0

13.5

13.0

(2)

0

E0 : Standard electrode potential R : Gas constant
T : Temperature
F : Faraday constant
H+: Hydrogen ion activity
aH2SO4- : Sulfuric acid ion activity
aH2O : Water activity
Activity means the proportion of each molecule occupied in the total number of molecules, and the mole fraction is used normally. E0 is referred to as the standard
electrode potential1) and correspond to the case in which
the activity of all the chemical elements is defined as 1 in
the previously described Nernst equation. The value iscalculated uniquely from Gibbs standard free energy offormation. In case of the lead-acid battery, the value isapproximately 1.93 V4) which can be considered a constant.
From the above facts, the battery charge level isexpressed by the electrolyte concentration and the electromotive force of the batteries (in other words, the completely stable OCV without unhomogeneity in the electrolyte) can be expressed by mole fraction (≒electrolyte
concentration). In other words, the battery electromotive
force can be an index to estimate the battery charge level.
This idea of associating the OCV with the battery

Temperature : -20 ℃
Over-voltage : 15 V CV Charging
Base SOC : 90%

14.5
Voltage（V）

HONDA ACCORD, described before, based on this method. Meanwhile, (expanded) Kalman-filter method3) is
known to be used for the same purpose with less computational effort and, simple trial usage result has beenreported in Furukawa Denkou Jihou No.120, (in Japanese).
Here, full-fledged verification result of the capability for the
Kalman-filter method is reported.

20000

40000
60000
Time（s）

80000

100000Figure 2 Relaxation of charging over-voltage.

3.

FUNCTION TABLE OF RELAXATIONVEHAVIOR OF OVER-VOLTAGE

We considered a method to predict the stable OCV fromthe short resting period that can be secured from a stablecondition of the vehicle. The method expresses the relaxation behavior of over-voltage as a function of time andpredicts the stable OCV as the convergence of voltage at
an infinite time of this function. This study was conducted;
and as a result, we have managed to obtain the expected
results in 5-dimensional exponential function. This has
been reported in Furukawa Denkou Jihou No.120, (inJapanese). The formula (3) represents the general formula of the 5th-order exponential decay function.
y＝f (x)＝y0＋a1·exp(b1·x)＋a2·exp(b2·x)		 ＋a3·exp(b3·x)＋a4·exp(b4·x)＋a5·exp(b5·x)

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Battery State Sensor ~Curve Fitting of Lead-acid Battery’s Voltage Relaxation With Expanded Kalman-filter~

The relaxation behavior of the over-voltage shown in
Figure 2 was fit to a function by using the data analysis
software on the market OriginPro7.5J. From Figure 3, it isclear that the higher the exponential dimension, the closer the function becomes to the measurement. The correlation coefficient R2 of the 5-dimentional exponential function reaches 0.99998, and we can say that a good result
obtained explains the behavior almost completely.

N∑ exp( b1 )･ {Yn - f ( Xn ) } = 0
n=1N∑ exp( b2 )･ {Yn - f ( Xn ) } = 0
n=1N∑ exp( b3 )･ {Yn - f ( Xn ) } = 0
n=1N∑ exp( b4 )･ {Yn - f ( Xn ) } = 0
n=1N∑ exp( b5 )･ {Yn - f ( Xn ) } = 0
n=1N

Voltage（V）

14.8

2 terms : R 2=0.99531
3 terms : R 2=0.99954
4 terms : R 2=0.9998
5 terms : R 2=0.99998
measured

14.4
14

			
∑ Xn ･exp( b1)･{ Yn - f ( Xn ) } = 0

(5)

n=1
N

∑ Xn ･exp( b2 )･{Yn - f ( Xn ) } = 0
n=1N∑ Xn ･exp( b3)･{Yn - f ( Xn ) } = 0
n=1
N

13.6

∑ Xn ･exp( b4 )･{Yn - f ( Xn ) } = 0
n=1
N

13.2

∑ Xn ･exp( b5 )･{Yn - f ( Xn ) } = 0

12.8
0

40000

80000

120000 160000 200000

240000 280000

Time（s）

n=1
N∑{Yn - f ( Xn ) } = 0
n=1

Figure 3 Results of high order exponential functions (72h).

4.

THE LEAST-SQARES METHOD AND
THE KALMAN-FILTER CALCULATION

To adapt the above-mentioned knowledge for the prediction of the stable OCV, which is our objective, the relaxation behavior of the over-voltage shall be measured byin-vehicle a battery state detection sensor and the sensor
is required to perform the similar curve fitting from theabove mentioned OriginPro7.5. The least-squares calculation by the Levenberg-Marquardt method is used in the
curve fitting engine of OriginPro7.5. The basic idea of theleast-squares method is intended to obtain a minimum
solution of a sum of squared deviation between measured data Yn and function value f (Xn) as shown in theformula (4).
N

2

			
∑ {Yn - f ( Xn )} = min

(4)

n =1

In case of the 5-dimentional exponential function, which
is targeted in this study, solving the formula (4) means tosolve the following 11 simultaneous equations (5) whichcorrespond to 11 coefficients included in the formula (3).

This type of nonlinear simultaneous equation does nothave analysis solving method, except for a special exception. Iteration calculation, which gradually approaches the
optimal solution from an initial value of an assumed solution, is a practical solving method. The LevenvergMarqardt method is used most widely in these days forthis iteration calculation. The general formula for updating
the solution by the Levenberg-Marquardt method isshown in the formula (6).
( k -1)( k -1)			u( k ) = u( k -1) − ( H u+ cD [ H u ]) −1 × ∇u J (k -1)(6)Here, u(k) is the vector of the coefficient to be obtained
and the updating is repeated until the solution converges
to the optimum value. Hu(k) is the Hessian matrix, and acalculation formula is shown in the next formula (7). In
this example, this is a matrix with 11x11 elements,
11x11matrix elements are calculated by partial differentiation in accordance with the 1 to N of individually obtained
data and all of those are added.
N

H u = ∑ { ( ∂f ( Xn ) / ∂u( k ) ) × ( ∂f ( Xn ) / ∂u( k ) ) T }
			
(k)

(7)

n =1

D[Hu(k)] can be obtained easily by extracting only the
diagonal elements, by completing the calculation of Hu(k).
But, an inverse matrix of Hu(k)+cD[Hu(k)] which is the summation of both matrixes is obtained. The calculation ofthis inverse matrix is not easy. It is necessary to solve asimultaneous equation by using a high computational
effort method such as the Gauss-Jordan method etc.
∇uJ(k) is a gradient and calculating formula is explained by
the formula (8). This is also a vector with 11 elements
including partial differentiation.

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Battery State Sensor ~Curve Fitting of Lead-acid Battery’s Voltage Relaxation With Expanded Kalman-filter~

N

(k)
			
∇u J = ∑ { ( Yn − f(Xn )) × ( ∂f ( Xn ) / ∂u( k ) ) }(8)

Partial differential value of 11x11 elements for eachobserved data of the N shall be calculated and added
together, and perform an inverse matrix calculation for the
finally summed matrix of 11x11. Further more, the above
calculation is repeated until a solution converges to the
optimum value. As described, the Levenberg-Marquardt
method is a very severe calculation to implement to thein-vehicle sensor which has a limitation in the manufacturing cost and the product dimension.
Meanwhile, the Kalman-filter calculation is well known
as requiring much less computational effort method
which can be adapted to the optimum coefficient estimation similar to the Levenberg-Marquardt method. The
Kalman-filter calculation method was proposed by
Hungarian-American Rudolf Kalman and used for the
rocket orbital calculation of the Apollo program. This is
the famous technology that contributed to the successful
landing on the moon in Apollo 11. Currently, this method
is widely used as the basic technology for the airplane
automatic navigation and the car navigation systems.
Initially, the filter theory proposed by Rudolf Kalman could
only be adapted to the linear system (linear Kalman filter),and could not be adapted to non-linear system such as
exponential function. In Furukawa Denkou Jihou No.120,(in Japanese), the linear Kalman-filter was simplified to beable to be used with assumption that power coefficientsof exponential function can be calculated individually, andpossible to be treated as constants in a function fitting,and the trial proceeded. The filter was improved to the
expanded Kalman-filter which could be adapted to a nonlinear system by Stanley Schmidt in NASA, in the stagesof the Apollo program application. Nowadays, the filter isused more widely. Here, we tried to optimize all the coefficient of the high-order exponential function by using the
expanded Kalman-filter. The general formula of the
expanded Kalman-filter is shown in the following.
One step forward estimation:xˆ n− = f ( xˆ n+−1, uˆ n+−1 )

			−

(9)

∑ xˆ ,n = An∑ +xˆ ,n−1 AnT + ∑ n

Jacoby matrix calculation:∂f| +∂xˆ xˆ n−1
			
∂h
Cn =
| −
∂xˆ xˆ n
An =xˆ ：State vector
y ：Observed value

n =1

(10)

A ：Jacoby matrix in time evolution model
Cn ：Jacoby matrix of observed model∑ xˆ ,n：Covariance matrix

Ln ：Kalman gain
∑ w ：System noise∑ v ：Observation noise

un ：Input vector

When observed value yn can be represented by the formula (12) with the state vector, and the state vector xn is inspace state representation as shown in the formula (13),yn＝h(xn)
xn＝f (xn-1)

(12)
(13)

The purpose of this calculation is to proceed filtering (=optimization of state vectors) by using the formula (10)(11) for more optimum prediction with further observationby proceeding the one-step forward estimation using theformula (9).
Similar to the least-squares method, an initial value
shall be set prior to the calculation starting; the largest difference is as follow. In case of the least-squares method,
the N predetermined observation data shall be arranged
and the optimization starts at the time of the buffering
completion for all the data, but the Kalman-filter method
proceeds the optimization at each observation procedure.
In other words, the Kalman-filter method, in principle,
does not need k repetition, and the buffers for observation data from 1 to N are not necessary. This method is
the infinite response filter that is not subjected to restriction of N.
In comparison to the computational effort, the strict
comparison is not possible since the computational effort
is greatly different depending on the number of repetitionk until convergence. With assumption that from 30 to 50
repetition is required, if optimization is completed by a setof learning from 1 to N that is the same observation number of the least-squares method, about two digits of computational effort reduction can be expected.
As a matter of fact, the least-squares method has excellent components. The least-squares method is superior inreliability and robustness as it can reach the optimum
value more certainly. But, if the same accuracy of calculation is obtained by the Kalman-filter method, a significant
reduction of computational effort can be expected as
described above.

Filtering calculation:∑ −xˆ ,n = An−1∑ +xˆ , n−1 AnT−1+ ∑ w[]
			xˆ = xˆ + L [ y − h ( xˆ ,0) ]∑ = [ 1− L C ] ∑Ln = ∑ −xˆ ,nCnT Cn∑ −xˆ ,nCnT + ∑ v
+
n

+
xˆ ,n

−
n

n

n

(11)

−
n

n

n

−1

−
xˆ ,n

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Battery State Sensor ~Curve Fitting of Lead-acid Battery’s Voltage Relaxation With Expanded Kalman-filter~

APPLICATION OF THE KALMAN-FILTER
TO THE EXPORNENTIAL CURVE FITTING

When applying the expanded Kalman-filter to the exponential curve fitting, the formula (3) of the continuity function is rewritten to the discrete-time representation byusing sampling interval (observation interval) dt as shownin the formula (14), this treatment is the same for the
least-squares method.
f (n)＝Y0＋A1· exp(−dt· n/T1)＋A2·exp(−dt· n/T2)＋
A3·exp(−dt·n/T3)＋A4·exp(−dt·n/T4)＋
A5·exp(−dt·n/T5)(14)

Voltage（V）

5.

15.0

Temperature condition: -20℃
Polarization condition : 15 V-CV Charged for 10 m
Sample battery : Furukawa Battery FPX1255(VRLA)

14.5

Sampling interval dt=20 s
N=901(Total observation time: 5 h)

14.0

13.5

13.0
－2000

0

2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Time
（s）
Figure 4 Relaxation data for trial.

The state vector xn is set as follow for convenience tocalculate.
xnT ＝(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)＝(A1, A2, A3, A4, A5, exp(−dt /T1,)
exp(−dt/T2), exp(−dt/T3), exp(−dt/T4),exp(−dt/T5), Y0)(15)

With these treatments, one-step forward estimation of xn
can be simplified a lot as there is no input for this calculation, and computational effort can be significantly
reduced.
−nxˆ = f ( xˆ
			

+
n−1

) = xˆ

+n−1

yˆ n− = h( xˆ n−1) nnn= ( xˆ 1− × xˆ 6n− + xˆ 2n− × xˆ 7-n + xˆ 3−n × xˆ 8−nnnn+ xˆ 4−n × xˆ 9−n + xˆ 5−n × xˆ 10−n + xˆ 11−n )(17)Based on the above established formulae, we tried toverify the adequacy of the expanded Kalmen-filter by performing the exponential curve fitting with the expanded
Kalman-filer method. The result was compared with thecalculation by the Levenberg-Marquardt method under
the same condition.

6.

A1

Table 1Result of least square fitting.

A2

A4

A3

A5

T1

T2

T3

T4

T5

Y0

Optimization
0.18237 0.4074 0.58806 0.34749 0.36972 55.83597 352.1912 1985.511 1985.694 11750.23 13.07268
coefficient(16)Here, observation value yn can be expressed by the formula (17).

n			The least-squares method that serves as a reference forcomparison used data analysis software on the market
OriginPro8.5 implementing the Levenberg-Marquardt
method.
The learning result by OriginPro8.5 is shown in Table 1.

COMPARISON OF THE CALCULATIONRESULT OF THE LEAST-SQUARESMETHOD AND THE KALMAN-FILTER

In making comparative verification of the calculation
results of the least-squares method and with those of the
Kalman-filter, next relaxation data of over-voltage dt=20 s
and N=901 (total 18,000 s=5 hrs) are as shown in figure
4.

In performing the expanded Kalmen-filter calculation,
our self-made tool uses Excel and VBA macro programs.
Both the least-squares method and the Kalman- filter
have initial value dependence in calculation results. In thisverification, following initial values are set, A1=0.3,
A2=0.3, A3=0.3, A4=0.3, A5=0.3, T1=10, T2=100,
T3=100, T4=1000, T5=10000, Y0=135. At first, as astraightforward way, the calculation result of one series ofobservation, started from n=1 to n=901 is shown in Table
2.
Table 2

Result of Kalman-filter without iteration.

A1

A3

A2

A4

A5

T1

T2

T3

T4

T5

Y0

Optimization
0.243151 0.537924 0.477655 0.551855 0.125053 63.86299 423.646 4190.568 5308.926 6181.159 13.12579
coefficientComparing Table1 and Table 2, it is difficult to say thatsimilar results were obtained. To see the optimization
progress in the process of observation, changes in Y0,
which is the coefficient to represent OCV, is shown in
Figure 5.

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Battery State Sensor ~Curve Fitting of Lead-acid Battery’s Voltage Relaxation With Expanded Kalman-filter~

13.5
13.45

Y0
（V）

13.4
13.35
13.3
13.25
13.2
13.15
13.1

0

200

400

600

800

1000

AS shown in Figure 6, Y0 converged by repeating calculation for 70 to 80 times. But, the feature of the Kalmanfilter, infinite response filter, was broken. Furthermore, theeffect of computational effort reduction would almost vanish because of the 70 to 80 times of repetitive calculations. Further consideration was made looking for an
improvement method and the influence by a linear simplification in the sampling of the discrete-time interval of the
expanded Kalman-filter. The image of the linear simplification in the expanded Kalman-filter is shown in Figure 7.

ｎFigure 5 Optimization behavior of Y0 parameter.

Exponential Curve

Y0（ V）

From Figure 5, unfortunately in 5 hours of observationfrom n=1 to 901, T0 is still in the process of value changing and not converged yet. As a matter of fact, with longer
time of learning the calculation result goes towards theconvergence, 5 hours of learning is almost the limit of thestable secured time for vehicles. From the original objective point of view, the stable OCV prediction in a shortwhile, a longer time of observation than this time is notrealistic any more. As a result of survey to find out the
problem solving method, we have reached to the study
used in the construction field. The optimization study forthe full elastic-plastic spring approximation model of theground in rapid loading test of piles, after N-th of the lastfiltering completion, the return to the initial observation
value by using the obtained optimum values as the initialvalue, and repeat the calculation like the least-squares
method5). This method kills the original feature of the infinite response filter which does no require buffering of theobserved value. Also computational effort is increased.
Even computational effort is significantly increased by therepetition of k; we decided to carry out the similar calculation considering that the most important thing is to see if
the aimed curve fitting can be successfully performed byusing the Kalman-filter. In the former study, the method toincrease the covariance matrix by 20 times was taken at
the repetition to return to the initial observation point. We
repeated this treatment. The study result is shown in
Figure 6 as the change followed to the repetitive calculation of Y0.

Error

V（n）

dt

Ｔ Ｔ+ dt（n+1）（n）
Figure 7 Image of linear simplification for non-linear system.

Even in a non-linear system, to explain behaviors in the
short enough time dt, the linear simplification (Taylor
series simplification of the primary) is possible as shownin Figure 7; this is widely used in general engineering
field. The expanded Kalman- filter uses this method aswell. In other words, the expanded Kalman-filter is the calculation method which can be satisfied only in the shortenough sampling interval that satisfies the above simplification. Then, the shorter the sampling interval the less thelinear simplification error, but the computational effort isincreased as a matter of fact because of the increment in
the observation points. And this would defy the purpose.
The degree of error generated by the sampling interval forprimary exponential function X(n)=exp(-dt · n/T0) is
shown in Table 3, as an example.
Table 3

Error caused by linear simplification.
dt

13.18
13.16
13.14
13.12
13.1
13.08
13.06
13.04
13.02
13
12.98
12.96

Linear simplification
around the observation
point

Error

0

20

40

60

80

100

Number of repetition k

120

T0

T0/2

T0/5

T0/10

0.37 × X(n)0.107 × X(n)0.019 × X(n)0.005 × X(n)As shown in Table 3, the linear simplification error of the
exponential function can be calculated uniquely as a ratio
to the power. The sampling interval dt of this study is 20 s.
On the other hand, the smallest power coefficient T1 of
the optimization coefficient is 55.83597, this makes
dt=T1/2.791799 by the expression according to Table 3,

Figure 6 Optimization of Y0 parameter through iteration.

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Battery State Sensor ~Curve Fitting of Lead-acid Battery’s Voltage Relaxation With Expanded Kalman-filter~

and we can see that close to 10% of error is generated.
The feature of the Kalman-filter is established based on
the assumption that the distribution of the error follows aGaussian distribution in consideration of the influence ofthe error. But, in this case, the absolute value is large andin addition the above written error does not follow theGaussian distribution. Meanwhile, in this case, the error
caused from the power coefficient can be calculated asshown above Table 3. Here, we made the following trial
for convergence improvement. The predictive calculationof the error was calculated from the power coefficient in
the learning process. And the influenced part was subtracted by the observed noise.
In addition to the above matter, based on the 5 hours ofdata that was used this time, it was suggested that the
5th-order exponential decay function is in over-fitting
because of the excessive number of terms, as T3 and T4
have the same number in the learning result of the leastsquares method shown in Table 1. So, the change to the
4th-order exponential decay function was made together
with the reduction of the number of terms. The recalculation result based on the above written two improvements
is shown in Figure 8 and Table 4.

13.16
13.15

Y0（V）

13.14
13.13
13.12
13.11
13.1
13.09

method. We consider that meaningfulness of the Kalmanfilter calculation was confirmed as an optimized value as
well.

7.

CONCLUSION

In this report, we confirmed the effectiveness of the
expanded Kalman-filter method as one mathematicalmethod for purely curve fitting, and showed that the
expanded Kalman-filter method can function as an effective tool by adding some ingenuity and improvement in
the application. As a matter of fact, this study is not the
final and we believe still there is room for improvement
and optimization.
This is the same for the least-squares method, enoughstudies are necessary not only from a point of view ofcomputational effort, but also from a stability in calculation of results.
Both the least-squares method and the Kalman-filter
method have in common a dependence on the initial values of the solution, to be set at the beginning of the calculation, in the computational effort and stability. Of
course, it is more advantageous when the initial values
are closer to the optimum solution. In this study, the initialvalues were initially fixed, but initial value setting is a very
important subject in the actual algorithm.
In addition, tools for curve fitting and coefficient optimization have been studied by many researchers on a dailybases, and many methods have been devised and proposed. We believe, it is very important to absorb andadopt these techniques widely for the establishment of an
ideal algorithm.

13.08
13.07

0

20

40

60

80

100

120

Repetition number k
Figure 8 Optimization of Y0 in the modified calculation.

Table 4Comparison of fitted coefficients of exponentialfunction.
A1

A2

A3

Least- squares
0.18237
method

0.4076

0.93555 0.36972 55.83597 352.1912 1985.579 11750.23 13.07268

Kalman-filter

A4

T1

T2

T3

T4

Y0

0.304729 0.456292 0.950762 0.381907 31.15547 302.3189 1909.384 10775.69 13.08112

REFERENCES
1) Noriyasu Iwane: “Battery Monitoring Sensor”, Furukawa Denkou
Jihou No.120 (in Japanese)
2) Kenichi Kanaya: “Optimum Mathematics Can be Understood Now
on” KYORITSU SHUPPAN CO., LTD. (in Japanese)3) Toru Katayama: “New Edition Applied Kalman-filter” Asakura
Publishing Co., Ltd. (in Japanese)
4) Watanabe (Tadashi), Kanemura, Masuda, Watanabe (Masayoshi):“Electrochemistry, Basic Science Course” MARUZEN CO., LTD.
(in Japanese)5) Toshihiko Asou, Shinji Aramaki, Kiyoshi Uno, Tadayoshi Soda,
Hidetoshi Ochiai: “ A Study on the Application of the Kalman-filter
to the Rapid Load Test of Piles” Journal of Japan Society of Civil
Engineers No.673/Ⅲ-54, 133-141. (in Japanese)

As shown in Figure 8, the learned value of Y0 is mostly
converging by about 20 times of repetitive calculation; asignificant repetition number reduction effect was
obtained in comparison to Figure 6. With this degree ofrepetition number, a meaningful computational effort
reduction is expected in comparison to the least-squares
method. Furthermore, as shown in Table 4, the learned
value of each coefficient was obtained as almost thesame value of the calculated result by the least-squares
Furukawa Review, No. 44

20138