Applying set membership strategy in state of charge estimation for
Lithium-ion battery
WANG JIANHONG1 , RICARDO A. RAMIREZ-MENDOZA2
1School of Electronic Engineering and Automation, Jiangxi University of Science and Technology, Ganzhou,
343100, CHINA
2School of Engineering and Sciences, Tecnologico de Monterrey, Monterrey, Ave. Eugenio Garza Sada 2501,
Monterrey, N.L., 64849, MEXICO
Abstract: As state of charge is one important variable to monitor the later battery management system, and as
traditional Kalman filter can be used to estimate the state of charge for Lithium-ion battery on basis of probability
distribution on external noise. To relax this strict assumption on external noise, set membership strategy is
proposed to achieve our goal in case of unknown but bounded noise. External noise with unknown but bounded is
more realistic than white noise. After equivalent circuit model is used to describe the Lithium-ion battery charging
and discharging properties, one state space equation is constructed to regard state of charge as its state variable.
Based on state space model about state of charge, two kinds of set membership strategies are put forth to achieve
the state estimation, which corresponds to state of charge estimation. Due to external noise is bounded, i.e.
external noise is in a set, we construct interval and ellipsoid estimation for state estimation respectively in case of
external noise is assumed in an interval or ellipsoid. Then midpoint of interval or center of the ellipsoid are chosen
as the final value for state of charge estimation. Finally, one simulation example confirms our theoretical results.
Keywords: Lithium-ion battery; State of charge estimation; Set membership; Interval estimation; Ellipsoid
estimation
Received: May 19, 2021. Revised: December 4, 2021. Accepted: December 25, 2021. Published: January 6, 2022.
1. Introduction
Lithium-ion battery is the leading energy storage
technology for many fields, such as electric vehicle,
modern electric grids, transformation, etc. The main
features of Lithium-ion battery include energy
density, a long time and a lower self-discharge rate,
so many research on these main features of
Lithium-ion battery are carried out in recent years
from their own different points of view. One
interesting area of research is battery state
estimation, especially named as state of charge
(SOC) estimation, as State of charge can not only
reflect the remaining capacity of Lithium0ion battery,
but also embody the performance and endurance
mileage of electric vehicle. Furthermore State of
charge is the most important factor to be used in the
battery management system, which is critical for the
safety, efficiency and life expectancy of Lithium-ion
battery. Generally State of charge indicates the
remaining battery capacity to show how long the
battery will last. It helps the battery management
system to protect the battery from overcharging and
over-discharging, and makes energy management
system to determine an effective dispatching strategy.
But State of charge can not be directly measured
using physical sensors, then it must be estimated
using some newly developed methods with the aid
of measurable signals such as the voltage and
current of the battery. Then here in this paper, State
of charge estimation is our concerned problem for
Lithium-ion battery.
State of charge estimation has been widely studied
and lots of estimation algorithms have been
proposed to acquire precise state of charge
estimation. An improved extended Kalman filter
method is presented to estimate state of charge for
vanadium redox battery (Mohamed MR, 2015),
using a gain factor. Some unknown parameters from
state space model are identified by classical least
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squares method. The square root cubature Kalman
filter algorithm has been developed to estimate the
state of charge of battery (Guarnieri Massimo, 2016),
where 2n points are calculated to give the same
weight, according to cubature transform to
approximate the mean of state variables. To improve
the accuracy and reliability of state of charge
estimation for battery, an improved adaptive
cubature Kalman filter is proposed in (Hong WC,
2015), where the battery model parameters are
online identified by the forgetting factor recursive
least squares algorithm. An adaptive forgetting
recursive least squares method is exploited to
optimize the estimation alertness and numerical
stability (Petchsingh C, 2016), so as to achieve
online adaptation of model parameters. To reduce
the iterative computational complexity, a two stage
recursive least squares approach is developed to
identify the model parameters (Li X, Xiong J, 2018),
then the measurement values of the open circuit
voltage at varying relaxation periods and three
temperatures are sampled to establish the
relationships between state of charge and open
circuit voltage. In (Ngamsai Kittima, 2015), a
multi-scale parameter adaptive method based on
dual Kalman filters is applied to estimate multiple
parameters. Based on battery circuit model and
battery model state equation, the real time recursive
least squares method with forgetting factor is used to
identify unknown battery parameters (Ressel S, Bill
F, 2018). After introducing the concept of state of
health, the average error of the obtained state of
charge estimation is less than one given value. A
novel state and parameter co-estimator is developed
to concurrently estimate the state and model
parameters of a Thevenin model for Liquid mental
battery (Chou YS, Hsu NY, 2016), where the
adaptive unscented Kalman filter (UKF) is
employed for state estimation, including a battery
state of charge. After performing Lithium-ion battery
modelling and off-line parameter identification, a
sensitivity analysis experiment is designed to verify
which model parameter has the greatest influence on
state of charge estimation (Zhong Q, Zhong F, 2016).
To improve the state of charge estimation accuracy
under uncertain measurement noise statistics, a
variational Bayesian approximation based adaptive
dual extended Kalman filter is proposed in (Xiong B,
Zhao J, 2017), and the measurement noise variances
are simultaneously estimated in the state of charge
estimation process. Actually to the best of our
knowledge, these state of charge estimation
methodologies can be roughly divided into
data-driven methods and model-based methods (Wei
Z, Tseng KJ, 2017). In the model-based methods, the
Kalman filter based state of charge estimation
methods have the merits of self-correction, online
computation, and the availability of dynamic state of
charge estimation (Wei Z, Tseng KJ, 2016). Kalman
filter was firstly proposed to estimate the state of
linear system, then in order to apply it into nonlinear
system, the extended Kalman filter (EKF) and
unscented Kalman filter were developed. Meanwhile
the date-driven methods typically include the look
up table method, matching learning based method,
artificial neural networks and support vector
machine, etc (Wei Z, Bhattaraia A, 2018). the data
driven method means that in estimating the state
whatever in linear system or nonlinear system, no
mathematical model is needed, i.e. the state is
constructed only directly by observed data (Lin, C,
Mu, H, 2017), so a large number of training data
covering all of the operating conditions is collected
to improve the estimation accuracy of the considered
state.
From above mentioned papers or other literatures,
we see that it is only Kalman filter that is used to
achieve the state estimation. Here we regard all
kinds of Kalman filter’s extended forms as the same
category. To the best of our knowledge that no other
new strategy is proposed to estimate the unknown
state, except Kalman filter or its extended forms.
Furthermore through understanding Kalman filter
for state estimation carefully, roughly speaking,
Kalman filter holds for state estimation in case that
the considered external noise must be a zero mean
random signal, i.e. white and normal noise. This
condition corresponds to the classical probabilistic
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description on external noise. But this white noise is
an idea case, and it does not exist in reality. To relax
this strict probabilistic description on external noise,
we propose to apply set membership estimation
strategy in estimating our considered state in the
presence of unknown but bounded noise. Here our
considered unknown but bounded external noise is
more realistic than white noise in engineering or
other research field. It means that our goal is to
estimate the state in case of the unknown but
bounded external noise in this paper, where the
estimated state corresponds to the state of charge in
the constructed state space equation. Due to classical
Kalman filter or its extended forms are useless
within the framework of unknown but bounded
external noise, so it is necessary to propose another
estimation strategy to identify unknown state on the
basis of unknown but bounded external noise. The
idea of set membership estimation is from system
identification theory or adaptive control, and in
order to apply set membership estimation to deal
with the problem of estimate the state of charge for
Lithium-ion battery, firstly we need to reformulate
one state space equation for the state of charge
estimation, through using one equivalent circuit
model to replace the considered state of charge
estimation for Lithium-ion battery. Based on this
constructed state space equation corresponding to
the state of charge for Lithium-ion battery, then the
idea of set membership estimation strategy can be
easily applied here. More specially, due to the
external noise is unknown but bounded, i.e. it is
assumed to be in one set priori within the whole
framework of set membership estimation, then two
kinds of set membership estimation strategies are
proposed here based on the used set, which includes
the external noise. Without loss of generality,
according to the commonly used interval and
ellipsoid for the external noise, the interval
estimation and ellipsoid estimation are derived for
the considered state estimation respectively, which
corresponds to the state of charge estimation for
Lithium-ion battery. This correspondence is from
the equivalent state space equation. Based on our
obtained interval estimation or ellipsoid estimation,
the midpoint of the interval estimation can be chosen
as the final state estimation and similarly the center
of the ellipsoid estimation can be also chosen.
Equivalent circuit model
for Lithium-ion battery
Nonlinear state space
equation about SOC
External noise in
interval
External noise in
ellipsoid
Interval estimation for
state
Interval estimation for
output
Ellipsoid estimation for
state
Outer ellipsoidal
approximation
Inner ellipsoidal
approximation
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Figure 1 A flowchart of our paper
The paper is organized as follows. In section 2, the
battery modelling is addressed, further the definition
of state of charge and the state space models for
state of charge estimation are also described. In
section 3, consider the external noise is in one
interval, which is a special case of the unknown but
bounded, then the interval estimation for the state is
derived at every time instant by our own
mathematical derivation. Furthermore, for
completeness, the interval estimation for the output
is also obtained, though it is not our concern. In
section 4, consider the external noise is in one
ellipsoid, then we investigate to build ellipsoidal
approximation of the state estimation. The main
contribution in this section 4 is that given two
ellipsoids, we need to find the best inner and outer
ellipsoidal approximations of their arithmetic sum.
In section 5, one numerical example illustrates the
effectiveness of our proposed set membership
estimation in estimating the state of charge for
Lithium-ion battery. Section 8 ends the paper with
final conclusion and points out the next topic.
A flowchart of our two proposed set membership
strategies is given in Figure 1, where the yellow
parts are our two considered cases of external noise
with interval or ellipsoid. The main contributions in
these paper is to derive the interval estimation and
ellipsoid estimation for state, which correspond to
the above two considered unknown but bounded
noises respectively.
2．Battery modelling
Our considered Lithium-ion battery has some merits
in energy density and life, further it is the leading
development direction of power batteries for electric
vehicles in the future. In order to give a brief
introduction on Lithium-ion battery, the internal
states of Lithium-ion battery are always divided as
four parts, i.e. SOC, temperature, rate of current,
state of health. These four states reflect the internal
relations of Lithium-ion battery with time variable.
Here our emphasis is on the internal structure of
Lithium-ion battery, which is shown in Figure 2,
whose cell generally comprises four parts: a polymer
positive electrode, a diaphragm, a negative electrode
and an electrolyte. The positive electrode of
Lithium-ion battery is generally composed of
Lithium-ion polymer. Common cathode Lithium-ion
polymer materials include lithium phthalate,
Lithium-ion phosphate, barium acid strontium,
Lithium-ion management, nickel diamond and
nickel-nickel aluminum ternary lithium. The
diaphragm is in the process of the first charge and
discharge of the liquid Lithium-ion battery. The
electrode material reacts with the electrolyte at the
solid-liquid phase interface to form a passivation
layer covering the surface of the electrode material
to isolate the electrode and the electrolyte, and the
Lithium-ion can finish chemical reaction with the
diaphragm.
Positive electrode
Negative electrode
Electrolyte
Lithium
Diaphragm
Figure 2. Battery internal structure
Actually in all of literatures on state of charge for
Lithium-ion battery, two commonly used battery
models exist, i.e. equivalent circuit model and
electrochemical model. As electrochemical model is
very complex, and it is very difficult to design the
latter Kalman filter in case of this electrochemical
model, so here in modelling Lithium-ion battery, the
equivalent circuit model is recently used. Equivalent
circuit model regards the battery internal reactions
as a circuit, containing some electronic components,
so equivalent circuit model consists of basic circuit
components such as resistors, capacitors and voltage
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sources. These four basic circuit components are widely explored due to their relatively simple
+
_
+_
0
R
p
R
p
C
p
U
OC
U
+
_
Ioad
U
I
Figure 3. Equivalent circuit model
mathematical structure and reduced computational
complexity.
Equivalent circuit model is shown in Figure 3,
which is simple and clear in physical meaning, and
will be applied to describe the battery charging and
discharging properties. Through balancing the
trade-off between model accuracy and
computational complexity, one Thevenin equivalent
circuit model is chosen for a Li-ion battery, which is
regarded as our battery model
Applying Kirchhoff law, variable
load
U
is defined
as that.
0load OC p
U U IR U
(1)
pp
p
p
U dU
IC
R dt
(2)
where in equation (1) and (2),
load
U
is the terminal
voltage,
I
is the load current,
0
R
is the internal
ohmic resistance,
p
R
and
p
C
are polarization
resistence and polarization capacitance of the battery,
p
U
is the polarization voltage.
OC
U
is the open
circuit voltage, which is monotonic with state of
charge. Further
OC
U
is rewritten as the following
polynomial form.
2 3 4
5 4 3 2 1OC
U x d d x d x d x d x
(3)
where
5
1
ii
d
are the coefficients of the polynomial
form, and
x
is the state of charge of the battery. The
state of charge is defined as a ratio of the remaining
capacity over the rated capacity. According to the
ampere hour counting method, state of charge can be
expressed as follows.
0
0
( ) ( ) t
tN
Idt
SOC t SOC t Q
(4)
where
t
is the sample time,
()SOC t
is the state of
charge of the battery at time instant
t
,
0
()SOC t
is
the initial
SOC
,
I
is the load current,
is the
coulombic efficiency, and
N
Q
is the nominal
capacity of battery.
1
, , 1
1
10
0 exp
(1 exp )
kk
s
p k p k
pp
k
s
p
pp
SOC SOC
T
UU
RC
I
T
RRC
(5)
, , 0load k OC k p k k
U U SOC U I R
(6)
where
k
is the sample time,
k
SOC
is the statue
value at the
k
th sample time,
s
T
is the specified
small sampling period.
OC k
U SOC
denotes a
nonlinear function of
k
SOC
.
The parameters in above state space equation (5) and
(6) can be identified by classical least squares
method, then our goal in this paper is to estimate
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state of charge
k
SOC
at time instant
k
.
3．Interval estimation for SOC
In this section we start to apply set membership
filter algorithm to estimate the SOC. by combining
equation (5) and (6), we see that
k
SOC
at time
instant
k
is one state variable in that state space
equation. Furthermore we also want to testify which
parameter will influence SOC estimation, then this
parameter will be added as the new state variables in
the extended state space equation.
3.1 Preliminary
As the main model parameter
0
R
is classified as a
new state variable with
p
U
and SOC, then an
extended state space equation for set membership
filter can be given as that.
1
, , 1
0, 0, 1
1, 1
1 2, 1
3, 1
1 0 0
0 exp 0
0 0 1
(1 exp )
0
kk
s
p k p k
pp
kk
k
s
p k k
pp k
SOC SOC
T
UU
RC
RR
w
T
R I w
RC w
(7)
, , 0load k OC k p k k k
U U SOC U I R v
(8)
Observing above equation (7) and (8), the problem
of state of charge for Lithium-ion battery is to
estimate the first state variable
k
SOC
at every time
instant
k
. Due to state of charge for Lithium-ion
battery is the first element of the state vector in
equation (7), then this problem is similar to the state
filtering in the modern control theory. So if the state
noise or external noise is a white noise, then the
classical Kalman filter can be well applied to deal
with the filter problem. But if the probability
distribution of the state noise or external noise is
unknown , then Kalman filter strategy is useless here,
due to the white noise is an idea case in reality. To
consider other more general case about the state
noise or external noise, the property of the state
noise or external noise is unknown but bounded. In
this whole paper, our contribution concern on deal
with the problem of state estimation in case of
unknown but bounded external noise.
Then in order to apply set membership algorithm
into above state space equation to estimate the first
state variable, we rewrite the above two equations (7)
and (8) as follows.
1 ( ) ( )
()
x k Ax k Bu k Dw k
y k Cx k v k
(9)
where
,
0,
1, 1
2, 1 1
3, 1
1 0 0
1 , 0 exp 0 ,
0 0 1
(1 exp ) ,
0
, ( ) ,
k
s
pk
pp
k
s
p
pp
k
kk
k
SOC T
x k U A RC
R
T
BR RC
w
w k w u k I D I
w
and
,0
,,,
OC k p k k
load k
k
d U SOC U I R
y k U C d SOC
where in equation (9)
k
is time instant,
xk
is the
state of this system at time instant
k
with its initial
state
0x
,
yk
is the observed output at time
instant
k
.
uk
is the control input,
wk
and
vk
are two unknown but bounded state noise and
observed noise respectively. All matrices
, , ,A B C D
are some matrices with compatible
dimensions, i.e.
, , , y
wnn
nn
n n n m
A R B R D R C R
(10)
Our considered linear discrete time invariant system
is one state space equation, whose structure can be
seen in Figure 4.
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Time delay
+
+
+++
1xk
()xk
wk
vk
()uk
A
B
C
D
Figure 4. The structure of state space equation
Let
x
and
x
be two vectors such that
xx
with
the inequality holding componentwise. An interval
,xx
is defined by.
n
, = R :x x x x x x
(11)
Then first of all, we give the assumptions about
initial state
0x
, state noise
wk
and observed
noise
vk
.
Assumption 1: There exist three kinds of unknown
but bounded signals
0 , 0 , , , ,x x w k w k v k v k
respectively, such that three uncertainties in state
space equation (1) be.
0 0 0x x x
w k w k w k for all k R
v k v k v k
(12)
where the inequalities are regarded as
componentwise.
As interval
,xx
can not be used in the latter
computational process, so its other equivalent form
is defined.
Definition 1: The interval
,xx
can be equivalently
represented by the following equivalent form.
, : , 1
n
x x x x x x x
C c p c P R
(13)
where
, ( ),
22
x x x x
x x x x
c P diag p p
(14)
Similarly
,
x x x x
x c p x c p
(15)
Here the notation
is the infinite norm of one
vector, and
()diag
is the notation section. Also in
Definition 1,
x
c
is the center of the interval
,xx
,
and
x
p
its radius, i.e.
,,
xx
C c p x x
.
Using the above Definition 1, the equivalent forms
in Assumption 1 are given as the following
Assumption 2.
Assumption 2: There exist three equivalent forms
for three intervals
0 , 0 , , , ,x x w k w k v k v k
respectively.
0 , 0 0 , 0
,,
,,
xx
ww
vv
x x C c p
w k w k C c k p k
v k v k C c k p k
(16)
Here the first contribution of our current paper is to
construct one interval
,x k x k
for state
estimation
xk
in equation (9), then after
substituting
,x k x k
into the observed
equation, the interval
,y k y k
corresponding
to the prediction output can be obtained, while
considering three uncertainties about initial state
0x
, state noise
wk
and observed noise
vk
.
3.2 Interval estimation
Firstly to obtain one interval
,x k x k
for state
estimation
xk
at time instant
k
, we take
z
transformation on both sides of the state equation,
i.e.
0zX z zx AX z BU z DW z
(17)
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where
z
is the variable in frequency domain, and
,,X z U z W z
are the transformation results in
frequent domain, corresponding to their forms in
time domain
,,x k u k w k
.
Formulating equation (17) to give that.
1 1 1
0X z zI A zx zI A BU z zI A DW z
(18)
Taking inverse
z
transformation on both sides of
equation (18), it holds that.
11
11
00
1
1
0
0
0
kk
k k i k i
ii
k
k k i
i
x k A x A Bu i A Dw i
A x A Bu i Dw i
(19)
Based on equation (19), we proceed to construct
intervals for state estimation and prediction output
respectively.
Observing equation (19), two uncertainties exist, i.e.
initial state
0x
and state noise
wi
. From
Assumption 2, two intervals about initial state
0x
and state noise
wi
are given as.
0 0 , 0
,
xx
ww
x C c p
w i C c i p i
(20)
Then we can describe the uncertain initial state
0x
and state noise
wi
by.
0 0 0
x x x
w w w
x c P
w i c i P i
(21)
where
n
xR
and
w
n
wR
such that.
1, 1
xw
and
(22)
Substituting equation (20 ) into
xk
, then it holds
that.
1
1
0
1
1
0
1 1 1
1 1 1
0 0 0
00
0
0
k
k k i
x x x i
kki w w w
i
kx
k k k
k i k i k k i
w x x w w
i i i
x k A c P A Bu i
A D c i P i
Ac
A Bu i A Dc i A P A DP i
(23)
Define the following pair
,
xx
C c k p k
as.
11
11
00
1
1
0
0
0
kk
k k i k i
x x w
ii
k
k k i
x x w
i
c k A c A Bu i A Dc i
p k A p A D p i
(24)
where notation is the absolute value.
Then it holds that.
, , 1,2
xx
x k C c k p k k N
(25)
Generally the above derivations can be formulated
as the following Theorem 1.
Theorem 1: Set
0 , 0
xx
C c p
and
,
ww
C c i p i
be center-radius representations of
two uncertainties
0x
and
wi
, the interval
,x k x k
for state estimation
xk
in state
space equation (1) is constructed as.
11
11
00
1
1
0
11
11
00
1
1
0
00
00
xx
kk
k k i k i k
x w x
ii
kki w
i
xx
kk
k k i k i k
x w x
ii
kki w
i
x k c k p k
A c A Bu i A Dc i A p
A D p i
x k c k p k
A c A Bu i A Dc i A p
A D p i
(26)
where
x
ck
and
x
pk
are defined in equation
(24), then interval for state estimation is given that.
, , , 1, 2
xx
x k C c k p k x k x k k N
(27)
To analyze the recursive relation between the
k
th
interval
,x k x k
and its latter
1k
th interval
1 , 1x k x k
, we list their centers as follows.
11
11
00
1 1 1
00
0
10
kk
k k i k i
x x w
ii
kk
k k i k i
x x w
ii
c k A c A Bu i A Dc i
c k A c A Bu i A Dc i
(28)
Taking subtract operation, we find that.
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1
1
0
1
1
0
11
11
00
10
0
k
k k i
x x x i
kki ww
i
kk
k k i k i
xw
ii
w
xw
c k c k A c A I A Bu i A I
A Dc i A I Bu k Dc k
A c A Bu i A Dc i A I
Bu k Dc k
c k A I Bu k Dc k
(29)
Then it holds that
1
x x w
c k Ac k Bu k Dc k
(30)
Equation (30) is the recursive expression of the
centers. Similarly the recursive expression of the
radius is that.
1
x x w
p k A p k B p k
(31)
From these two recursive relations between the
adjacent interval for state estimation, we see that the
1k
th interval can be obtained from the
k
th
interval and the knowledge of control input and state
noise. The recursive computation for the interval for
state estimation is seen in Figure 5.
0 , 0
xx
C c p
0 , 0
ww
C c p
1 , 1
xx
C c p
+
2 , 2
xx
C c p
+
,
xx
C c k p k
…
1 , 1
ww
C c p
1 , 1
ww
C c k p k
1u
1uk
Figure 5. Recursive computation for interval
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But the most important element in model predictive
control is the prediction output, so the interval for
the prediction output can be obtained by substituting
interval
,x k x k
into the observed equation.
Due to
,
xx
x k C c k p k
and
,
vv
v k C c k p k
, i.e.
x x x
v v v
x k c k P k
v k c k P k
(32)
where
n
xR
and
v
n
vR
such that.
1, 1
xv
and
(33)
Substituting equation (32) into the observed
equation (9), we have that.
y
x x x v v v
x x v v
ck
y k C c k P k c k P k
Cc k c k CP k P k
(34)
Similarly define the center and radius as .
y x v
y x v
c k Cc k c k
p k Cp k p k
(35)
Then we have that
,
yy
y k C c k p k
(36)
Furthermore
y y x v x v
x x v v
y y x v x v
x x v v
y k c k p k Cc k c k Cp k p k
C c k p k c k p k
y k c k p k Cc k c k Cp k p k
C c k p k c k p k
(37)
Then it also means that
,,
yy
y k C c k p k y k y k
(38)
The above equation (38) is our interval for the
prediction output, which will be used for the next
robust model predictive control.
In order to simplify the latter exposition in robust
model predictive control, we need the explicit form
of interval
,,
yy
y k C c k p k y k y k
.
To achieve this goal, some notations are introduced
here.
11
11
00
1
12
0
1
1
0
1
11
12
0
0
1
0
0 , 1
kk
k k i k i
x x w
ii
k
i
k
k k i
x x w
i
k
k k i k i
xw
i
c k A c A Bu i A Dc i
c k c k i u i
p k A p A D p i
c k A c A Dc i c k i A B
(39)
Substituting notation (39) into the expressions
yk
and
yk
respectively, we obtain.
1
12
0
1
12
0
1
1
y y x v x v
x x v v
k
v v x
i
k
i
y k c k p k Cc k c k Cp k p k
C c k p k c k p k
C c k c k i u i c k p k Cp k
a k a k i u i
(40)
where
11
22
11
v v x
a k Cc k c k p k Cp k
a k i Cc k i
(41)
Similarly
1
32
0
1
x x v v
k
i
y k C c k p k c k p k
a k a k i u i
(42)
where
31v v x
a k Cc k c k p k Cp k
(43)
The advantage of reformulating
yk
and
yk
is
that the explicit form can be divided as one linear
affine function of the control input
ui
. Based on
equation (40) and (43), we rewrite equation (38) as.
11
3 2 1 2
00
,,
1 , 1
yy
kk
ii
y k C c k p k y k y k
a k a k i u i a k a k i u i
(44)
Then equation (44) will be used in the detailed
computation about the other research field. From
equation (32), the center or midpoint
x
ck
can be
chosen as the final state estimation, corresponding to
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our considered state of charge. Also combing
equation (32) and (44), we derive not only the
interval estimation for the state, but also the interval
estimation for the prediction output. As the emphasis
here is only the interval estimation for the state, so
that interval estimation for the prediction output is
not deep studied here, and it is applied in learning
model predictive control.
4．Ellipsoid estimation for SOC
In section 3, we assume the state noise and the initial
state are in one interval (12). As in the research field,
there are other sets to be used to denote the
uncertainty, such as ellipsoid. It means that interval
and ellipsoid are two commonly used sets in set
membership estimation. So for completeness and
comparison, here in this section we consider the
ellipsoid estimation for state, which corresponds to
the state of charge for Lithium-ion battery.
Observing only the first state equation in equation (9)
again, matrices
,,A B D
can be identified by using
least squares method. Based on this state equation,
our problem is to estimate the state
xk
at different
time instant
1, 2 1kN
. It is similar to section 3
that control input
uk
is determined by researcher,
and
uk
is a deterministic value, not an uncertainty.
So for convergence, we neglect this term
()Bu k
in
the latter derivation. It means that if one ellipsoid
estimation for the state is derived by our own
derivation, then we can apply translation
transformation to give the true ellipsoid estimation
for the state. Then we rewrite the considered state
equation as follows.
0
1 ( )
0, 0,1, 1
x k Ax k Dw k
x k N
(45)
This special state equation is driven by
wk
,
satisfying the following norm bound.
1, 0,1, 1w k k N
(46)
Our goal here is to build ellipsoid approximation of
the state recursively. Let
k
X
be the set of all states
where the system can be driven in time instant
kN
,
and assume that we have build inner and outer
ellipsoidal approximations
k
in
E
and
k
out
E
of the
set
k
X
.
kk
in k out
E X E
(47)
Let also
/1
T
E x Dw w w
(48)
Then the set
1
1 2 1 2
/,
k k k
in in in
F AE E x Aw w w E w E
(49)
clearly cover
1k
X
, and a natural recurrent way to
define an outer ellipsoidal approximation of
1k
X
is to take as
1k
out
E
the smallest volume ellipsoid
containing
1k
out
F
. Note that the sets
1k
in
F
and
1k
out
F
are of the same structure: each of them is the
arithmetic sum
1 2 1 1 2 2
/,x w w w W w W
of two
ellipsoids
1
W
and
2
W
. Thus we come to the problem
as follows: Given two ellipsoids
1
W
and
2
W
, find
the best inner and outer ellipsoidal approximations
of their arithmetic sum
12
WW
. In fact, it makes
sense to consider a problem.
Given two ellipsoids
1
W
and
2
W
, find the best inner
and outer ellipsoidal approximations of their
arithmetic sum
1 2 1 1 2 2
/,W x w w w W w W
(50)
of two ellipsoids
1
W
and
2
W
.
4.1 Outer ellipsoidal approximation
Let the ellipsoids
1
W
and
2
W
be represented as.
/1
T
ii
W x x D x
(51)
Our strategy to approximate is that, we want to build
a parametric family of ellipsoids in such a way that,
first, every ellipsoid from the family contains the
arithmetic sum
12
WW
of two given ellipsoids, and
second, the problem of finding the smallest volume
ellipsoid within the family is a simple problem.
Let us start with the observation that an ellipsoid.
/1
T
W Z x x Zx
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contains
12
WW
if and only if the following
implication holds.
2
1
1 2 1 2
/ 1, 1, 2
1
T
i i i
i
i
T
x x D x i
x x Z x x
(52)
Let
i
D
be one block diagonal matrix, such that all
diagonal blocks, except the
i
th one, are zero, let
MZ
be that.
1
12
2
00
0,,
0
00
D Z Z
D D M Z
DZZ
(53)
Due to the fact that for every symmetric positive
semidefinite matrix
X
such that
1, 1, 2
i
Tr D X i m
, one has
1Tr M Z X
.
Then we arrive at the following result.
Proposition 1: Let a positive definite matrix
Z
be
such that the optimal value in the semidefinite
program.
max / 1, 1,2, 0
i
XTr M Z X Tr D X i X
(54)
is
1
. Then the ellipsoid
/1
T
W Z x x Zx
contains
12
WW
of two ellipsoids
/1
T
ii
W x x D x
.
The above proposition is the first step to build a
parametric family of ellipsoids, which contains the
arithmetic sum
12
WW
. Then the second problem
of finding the smallest volume ellipsoid within the
parametric family can be reduced to one
semidefinite program as that.
Proposition 2: Given two centered at the origin full
dimensional ellipsoids.
/ 1 , 1,2
T
ii
W x x D x i
Let us associate with these two ellipsoids the
semidefinite program.
1
12
12
1 2 1 2
,,
/,
max 0, 0, 1
0
x
n
tZ
t t Det Z D D M Z
Z
(55)
Every feasible solution
,,tZ
to this semidefinite
program with positive value of the objective
produces ellipsoid
/1
T
W Z x x Zx
.
which contains
12
WW
, and the smallest volume
ellipsoid is given by optimal solution of the
semidefinite program (55).
4.2 Inner ellipsoidal approximation
Let us represent the given centered at the origin
ellipsoids
i
W
as.
/ / 1 , 1, 2
T
ii
W x x Aw w w i
Due to the fact that an ellipsoid
/1
T
E Z x Zw w w
is contained in the sum
12
WW
of the ellipsoids
i
W
if and only if one has.
2
22
1
:TT
i
i
x Z x A x
(56)
A natural way to generate ellipsoids satisfying
equation (56) is to note that whenever matrix
i
X
satisfying the following the property about its
special norms.
max 22
max / 1, 1, 2
T
i i i i
x
X X X X x x i
(57)
Then the matrix
1 2 1 1 2 2
,Z Z X X A X A X
(58)
satisfies equation (56).
2
1 1 2 2 2
22
1
22
22
11
T T T
ii
i
T T T
i i i
ii
Z x A X A X x X A x
X A x A x
(59)
Thus every collection of square matrices
i
X
with
spectral norms not exceeding 1 produces an ellipsoid
satisfying equation (56) and thus contained in
W
.
Similarly the largest volume ellipsoid within the
parametric family can also be reduced to the
following semidefinite program.
Proposition 3: Let
/ / 1 , 0, 1,2
T
i i i
W x x Aw w w A i
, consider
the following semidefinite program.
1
2
1
2
1
max
1;
2
0;
0, 1, 2
x
n
TT
ii
i
TT
ii
i
T
ni
in
t
subject to t Det X A x
X A x
IX i
XI
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(60)
with design variables
12
,XX
. Every feasible
solution
12
,,XXt
to this problem produces the
ellipsoid.
2
12
1
, / 1
T
ii
i
E X X x A X w w w
contained in the arithmetic sum
12
WW
of the
original ellipsoids, and the largest volume ellipsoid
which can be obtained in this way is associated with
optimal solution to the semidefinite program (60).
After solving these two semidefinite programs (55)
and (60), then we build inner and outer ellipsoidal
approximations
k
in
E
and
k
out
E
of the set
k
X
, i.e.
kk
in k out
E X E
Based on these two inner and outer ellipsoidal
approximations
k
in
E
and
k
out
E
of the set
k
X
, then the
final state estimation at time instant
xk
can be
chosen as the midpoint between the two centers of
inner and outer ellipsoidal approximations
k
in
E
and
k
out
E
.
So generally whatever the state noise or external
noise is included in an interval or an ellipsoid, firstly
we apply our mentioned interval estimation or
ellipsoid estimation to obtain the state estimation set,.
Secondly the center or midpoint can be chosen as
the final state estimation value, which corresponds
to the state of charge for Lithium-ion battery.
5．Simulation example
Here we do not have yet the experimental platform,
so this simulation example is based on references in
the open literatures. To acquire experimental data
such as current, voltage and temperature from the
battery, a battery test bench was established. The
configuration of the battery test bench is shown in
Figure 6.
Thermostat
Lithium-ion battery
Battery test system Host computer
AC Power
Temperature control
Current
voltage
control
Charging/
Discharging
Current voltage control
Current command file
Figure 6. Lithium battery test platform
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For convenience in the latter simulation example,
Lithium battery test needs to charge and discharge
the lithium ion battery at different temperatures and
different rates. Therefore, the equipment required for
the experimental bench includes a thermostat, a
battery charging and discharging device, a ternary
neon battery, and a host computer. Lithium battery
test platform is plotted in Figure 6, where the
detailed processes are described as follows.
Step1. The charging and discharging positive and
negative terminals of the battery are
respectively connected to the positive and
negative electrodes of the battery through
the wire harness, and the wire harness of the
appropriate diameter is selected according
to the allowable charging and discharging
ratio of the battery to avoid the burning of
the wire harness. One end of the voltage
sampling line to the other end of the battery
is connected to the voltage sampling and
wiring port of the battery charging and
discharging device. Finally, the temperature
measuring line of the thermistor is attached
to the surface of the battery, and the other
side of the temperature detecting line is
connected to the temperature detecting
terminal of the battery charging and
discharging device.
Step 2. Set the lithium battery in the incubator , and
set the experimental ambient temperature.
Step 3. Start battery charging and discharging
equipment and incubator.
Step 4. In the online machine, we edit the charge
and discharge test step or import the edited
current test file into the host computer to
automatically generate the test step, then set
the sampling time and output file save
address, start the test.
Based on the experimental platform, the open-circuit
voltage of the battery has a monotonic relationship
with the state of charge. The relation between
open-circuit voltage and state of charge is
established by running test on the considered
lithium-ion battery. Let all batteries are fully charged
and rested for 3 hours, such that the internal
chemical reactions attain a desired equilibrium state.
Moreover the discharge test includes a sequence of
pulse current of 1 C with 6-min discharge and
10-min rest, then the discharge test can make the
battery to return back to its expected equilibrium
state before running the next cycle , which is shown
in Figure 7.
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Figure 7. Voltage and current curves
Figure 8. Polynomial form for
OC
U
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As
OC
U
is rewritten as the following polynomial
form
2 3 4
5 4 3 2 1OC
U x d d x d x d x d x
. To
identify these unknown parameters in this
polynomial form, Least squares method is used to
achieve this goal. Then the identification result for
this polynomial form is given in Figure 8, which
shows the relation between the true data point and
its identified polynomial form.
In the whole simulation process, the true parameters
can be identified by using some system
identification strategy, for example least squares
method, instrumental variable method, maximum
likelihood method etc. Then identified parameter are
obtained as follows.
00.0994 , 0.030 , 20773 ;
1.10 ; 0.3
pp
s
R R C KF
I A T s
Substituting the above values into the equation (7),
(8) and (9), then each matrix is given as follows.
1 0 0 0.8
0 0.68 0 , 0.064 , 2.5 1 1.2 , 1
0 0 1 0.1
A B C D
Consider the unknown but bounded noise in
simulation, these two kinds of unknown but bounded
signals in equation (9) are formulated as.
0.5 0 0.5
11
11
x
w k for all k R
vk
In simulation we consider not only the state
estimation, but also the output estimation, and the
state estimation corresponds to the state of charge.
The interval estimation on observed output with
bounded noise
vk
can be used in another research
field, such as robust control, interval model
predictive control etc.
Firstly we apply equation (27) to obtain the interval
estimation for state. The state trajectory can be
easily obtained by using equation (19) in Matlab,
where some priori information about initial state and
bounded noise are used. The simulation results are
shown in Figure 9, where the black curve is the true
state trajectory and the two red curves denote the
estimated curves. One curve is consisted by upper
bound, and the other curve is the lower bound. From
Figure 9, we see that the true state trajectory lies in
between the two red curves., so at each time instant,
the midpoint of the upper bound and lower bound
can be chosen as the final state estimation at the
considered time instant. Similarly the interval
estimation for output is also given in Figure 10,
where the true output trajectory lies in between the
two estimated curves. The interval estimation for
output is obtained based on equation (44), and the
final output estimation can also selected as the
midpoint at each time instant.
For the sake of completeness, the ellipsoid
estimation for state or state of charge is given in
Figure 11, where the true state trajectory is same
with that curve in Figure 9. Twelve data points are
sampled in the true state trajectory, and we need to
construct twelve ellipsoids to include these twelve
data points as their own interior points. As ellipsoid
is used to denote the uncertainty, so at every time
instant we will obtain one ellipsoid to include the
true data point. From the simulation result in Figure
11, twelve ellipsoids are constructed through using
relations (47), and these twelve ellipsoids include
these twelve data points as their own interior points
exactly. When the center of the corresponding
ellipsoid is chosen as the final state estimation, we
find that the error exists yet, i.e. the center of the
considered ellipsoid is not equal to the true state
value at the considered time instant.
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Figure 9. Interval state estimation
Figure 10. Interval output estimation
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Figure 11. Ellipsoid state estimation
6．Conclusion
In this paper, set membership strategy is applied to
estimate the state of charge for Lithium-ion battery,
so that the state estimation can be dealt with in case
of unknown but bounded external noise. The goal of
introducing set membership strategy is to alleviate
the shortcoming of the traditional Kalman filter
algorithm. After formulating one state space
equation for the state of charge estimation, through
using one equivalent circuit model to replace the
considered state of charge estimation for
Lithium-ion batter. According to the commonly used
interval and ellipsoid for the external noise, the
interval estimation and ellipsoid estimation are
derived for the considered state estimation
respectively, which corresponds to the state of
charge estimation for Lithium-ion battery. But here
we only propose the detailed strategy, and the
accuracy or convergence of our considered strategy
can be regarded as our future work.
Acknowledgements
This work is partially supported by the Grants from
the Mexico National Science Foundation (No.
20202BAL202009).
Data Availability
The data used to support the findings of this study
are available from the corresponding author upon
request.
Conflict of interest
The authors declare that there is no conflict of
interests regarding the publication of this paper.
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