In the Battery Management System (BMS) the State of Charge (SOC) is closely related to the reliable and safe operation of lithium-ion (Li-ion) batteries. Kalman Filter is an effective algorithm for estimating SOC with a battery modeling. This presentation will briefly describe about battery modeling and Kalman Filter for SOC estimation.

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Understanding Kalman Filter for
SOC estimation
Reported by：Ratul
Research And Teaching Assistant at
Southwest University of Science and Technology.

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• Origin of Kalman Filter
• Mathematical Model
• State Observer and Kalman Filter
• Kalman Filter equation and derivation

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Origin of Kalman Filter
In statistics and control theory, Kalman filtering, also known as
linear quadratic estimation (LQE), is an algorithm that uses a series of
measurements observed over time.
It can containing statistical noise and other inaccuracies, and produces
estimates of unknown variables that tend to be more accurate than those
based on a single measurement alone,
The filter is named after Rudolf E. Kálmán, one of the primary developers
of its theory.

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Basic concept of Kalman filtering

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When we use Kalman Filter
We can not measure the energy inside the battery, but we can get an related
measurement of the energy inside the battery by its Voltage. So we have a measurement
From a different structure and we need to estimate some calculation from an another
Environment. In this situation Kalman Filter is the more efficient way to estimation.

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Mathematical Model for KF
The Kalman filter uses a system's dynamic model which is known as Mathematical
Model, and consist of control inputs to that system, and multiple sequential
measurements (such as from circuit resister/capacitor) to form an estimate of the
system's varying quantities, that is better than the estimate obtained by using only one
measurement alone.
For SOC estimation we need to develop a model for the battery circuit which is as much
like as real system. In this part it is known as Battery Modeling and Parameter
identification. There are lots of methods for Battery Modeling.
Compared to all models, ECMs are much easier for the understanding of the electrical
characteristic of the battery. Moreover, due to the plentiful circuit components and their
combinations, ECM gives researchers sufficient freedom to design a suitable structure
for the application.

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Battery modeling for KF

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State Observer and KF
u y
x
y Cx
x Ax Bu 
The concept of state observer will help to understand what KF is and how its work.
For SOC estimation of a battery we only know the terminal voltage but we need to
calculate the internal charge amount. For a battery we know the basic equation how the
charge turns into energy (voltage), and from the test experiment on the battery model we
will get more additional information about the circuit parameter.
But there is a problem, the battery model we are using it is not the100% as the real
circuit. That’s why KF use and State Estimator for estimate the internal state. So we are
using a feedback control system with a State Observer K
1 0.t ocU U U I R  

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State Observer and KF
obse x x 
u y
x +_
y
∧
x
∧
y Cx
x Ax Bu 
x Ax Bu 
y Cx
K
x Ax Bu  y Cx
( )x Ax Bu k y y    y Cx
( )x x Ax Ax k Cx Cx    
( ) ( ( ))x x A x x k C x x    
( )obs obse A KC e 
State Observer

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Kalman Filter
( 1) ( 1)( ( )k kk k k kx Ax Bu k y C Ax Bu     
We can say Kalman Filter as a state observer of an Stochastic System
( 1)k kAx Bu 
Prediction Step
In prediction state it predict the recent state by using state estimator from previous (k-1)
time step and recent input kU
We can denote: ( 1)k kx Ax Bu
 
( )k kx x k y Cx 
  So the equation became:

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Kalman Filter
Update Step
In the Update Step of the equation we uses the measurement and incorporate with the
prediction to update the prior estimation.
( 1)k kx Ax Bu
 
( )k kk y Cx

Error covariance
( 1)
T
k kP AP A Q
 
T
k
k T
k
P C
K
CP C R




( )k kx x k y Cx 
  
( )k k kP I K C P
 
Error covariance

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If the Prior error covariance is close to zero then the Kalman gain will be,
0 0
lim lim
k k
T
k
k T
P P
k
P C
k
CP C R 


 


So plugin the value of Kalman gain on Kalman Filter equation
0
0
0 R
 

( )k k k kx x k y Cx 
  
0( )k k k kx y Cx x  
   
So the estimate is come from a prior estimate.
Once we get the posterior estimate then it will
use for next step as prior estimate. That’s why
KL filter algorithm is an Recursive Algorithm

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The basic Kalman filter is limited to a linear assumption. More complex systems,
however, can be nonlinear. The non-linearity can be associated either with the
process model or with the observation model or with both.
 Extended Kalman filter
 Unscented Kalman filter
When the state transition and observation models that is, the predict and update
functions and are highly non-linear, the extended Kalman filter can give particularly
poor performance.

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Any Question?
Quora : https://www.quora.com/profile/Shopno-Karigor
Email: ratul.sg2018@gmail.com

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References:
1. Wikipedia.org [CrossRef]
2. Jinhao Meng, Guangzhao Luo, Mattia Ricco, Maciej Swierczynski, Daniel-
Ioan Stroe and Remus Teodorescu; Overview of Lithium-Ion Battery Modeling
Methods for State-of-Charge Estimation in Electrical Vehicles.
3. MATLAM, Understanding Kalman Filters. [CrossRef]