Second-Order Discrete-Time Sliding Mode Observer for State of Charge Determination Based on a Dynamic Resistance Li-Ion Battery Model

Abstract

A second-order discrete-time sliding mode observer (DSMO)-based method is proposed to estimate the state of charge (SOC) of a Li-ion battery. Unlike the first-order sliding mode approach, the proposed method eliminates the chattering phenomenon in SOC estimation. Further, a battery model with a dynamic resistance is also proposed to improve the accuracy of the battery model. Similar to actual battery behavior, the resistance parameters in this model are changed by both the magnitude of the discharge current and the SOC level. Validation of the dynamic resistance model is performed through pulse current discharge tests at two different SOC levels. Our experimental results show that the proposed estimation method not only enhances the estimation accuracy but also eliminates the chattering phenomenon. The SOC estimation performance of the second-order DSMO is compared with that of the first-order DSMO.

Full text

Energies 2013,6, 5538-5551; doi:10.3390/en6105538
OPEN ACCESS
energies
ISSN 1996-1073
www.mdpi.com/journal/energies
Article
Second-Order Discrete-Time Sliding Mode Observer for State of
Charge Determination Based on a Dynamic Resistance Li-Ion
Battery Model
Daehyun Kim 1, Keunhwi Koo 1, Jae Jin Jeong 1, Taedong Goh 1and Sang Woo Kim 1,2,*
1Department of Electrical Engineering, Pohang University of Science and Technology,
77 Cheongam-Ro, Nam-Gu, Pohang 790-784, Korea; E-Mails: daehyunkim@postech.edu (D.K.);
khkoo@postech.edu (K.K.); jin03jin@postech.edu (J.J.J.); ehd1116@postech.edu (T.G.)
2Department of Creative IT Excellence Engineering and Future IT Innovation Laboratory,
Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang 790-784, Korea
*Author to whom correspondence should be addressed; E-Mail: swkim@postech.edu;
Tel.: +82-54-279-2237; Fax: +82-54-279-2903.
Received: 1 August 2013; in revised form: 4 October 2013 / Accepted: 11 October 2013 /
Published: 22 October 2013
Abstract: A second-order discrete-time sliding mode observer (DSMO)-based method is
proposed to estimate the state of charge (SOC) of a Li-ion battery. Unlike the first-order
sliding mode approach, the proposed method eliminates the chattering phenomenon in
SOC estimation. Further, a battery model with a dynamic resistance is also proposed to
improve the accuracy of the battery model. Similar to actual battery behavior, the resistance
parameters in this model are changed by both the magnitude of the discharge current and
the SOC level. Validation of the dynamic resistance model is performed through pulse
current discharge tests at two different SOC levels. Our experimental results show that the
proposed estimation method not only enhances the estimation accuracy but also eliminates
the chattering phenomenon. The SOC estimation performance of the second-order DSMO
is compared with that of the first-order DSMO.
Keywords: Li-ion battery; second-order discrete sliding mode observer; dynamic resistance;
state of charge

Energies 2013,65539
1. Introduction
Li-ion batteries are used in a wide range of applications, such as in portable electronic devices and
electric vehicles, because of their advantages, e.g., high energy density, high electromotive force, and
low cost. To improve the performance of Li-ion batteries, battery management systems (BMSs) have
been studied [1,2]. In the various functions of BMSs, it is necessary to estimate the state of charge
(SOC) and model the battery dynamics [3–5]. In the present study, we focus on enhancing the accuracy
of not only the SOC estimation but also the battery model.
To accurately estimate the SOC, a variety of methods based on ampere-hour counting, a linear
SOC equation, artificial neural networks, impedance spectroscopy measurements, and battery modeling
have been proposed in recent papers. The ampere-hour counting method [6] is the most commonly
used SOC estimation method because of its simplicity. This method measures the charge and discharge
current and estimates the SOC using the integral of the current over time. However, there are several
problems associated with this method. First, integration errors caused by current measurement noise
or energy losses during charging/discharging, could accumulate. Moreover, determining the initial
SOC is difficult unless the battery is fully charged or conditionally discharged. The linear equation
method [7] calculates the variation of the SOC using the voltage output, current input, and previous
SOC data. In this equation, the coefficients are determined from the reference data of a specific
battery. Thus, if the battery is changed, then the coefficients must also be re-calculated. An artificial
neural network used for SOC estimation is presented in [8]. Fundamentally, because the battery
dynamics exhibit nonlinear behavior, this method usually produces a more accurate SOC estimate than
the other methods. However, the estimation process strongly depends on the training data (similar
to the case of the linear model method), and the learning process is difficult to implement on-line
owing to the burdensome computations required. The impedance spectroscopy method [9] examines
the electrochemical impedance of the battery to determine the SOC. However, sensors for impedance
measurement are expensive, and unexpected situations can arise when a signal is injected into the battery
to measure the impedance while the battery is powering electronic devices. The battery model-based
SOC estimation method [10,11] consists of a state equation derived from an equivalent circuit battery
model and a state observer. In this method, the SOC is considered a state variable; it is estimated
using a state observer. This method has been widely used because this approach enables to apply
existing observers. In [10], a Kalman filter (KF) was employed to estimate the SOC of a lead-acid
battery. However, the KF suffers from a well-known drawback in that no analytical selection methods
are available for either the process or measurement noise covariance matrices. Indeed, incorrectly
selected covariance matrices can reduce the estimation performance. A first-order sliding mode observer
(SMO) for SOC estimation was presented first in [11]. An SMO is widely used owing to its simplicity
and robustness to both parameter variations and external disturbances. In spite of these attractive
properties, the first-order SMO introduces a chattering phenomenon, which involves oscillations with
high frequency and finite amplitude [12]. This shortcoming is a major drawback of the sliding mode
approach in practical implementations. In order to reduce the chattering phenomenon, a second-order
sliding mode based-observer [13–15] has received a lot of attention recently, because the second-order
sliding mode approach can drive to zero not only the sliding variable, but also its derivative.

Energies 2013,65540
In the present paper, a novel SOC estimation method based on a second-order discrete-time SMO
(DSMO) is proposed to eliminate the chattering phenomenon. In addition, a dynamic resistance model
is proposed to more accurately describe the dynamics of the Li-ion battery. The resistance parameters
are updated on the basis of both the discharge current and the SOC level. The proposed SOC estimation
method and dynamic resistance model are validated by experimental results.
The remainder of this paper is organized as follows. Section 2 presents information on the equivalent
circuit-based battery model and the limitations of the existing schemes. Section 3 proposes an SOC
estimation method based on the second-order DSMO. Section 4 presents both our experimental results
for the SOC estimation performance using the second-order DSMO described in Section 3 and the
validation of the dynamic resistance model. Finally, Section 5 presents our conclusions.
2. Battery Modeling
The equivalent circuit-based battery model [10,11] is often used because it can be easily transformed
into a state-space equation form. In this paper, the circuit in Figure 1is employed in consideration of
both the model accuracy and complexity. This model consists of a nonlinear voltage source, a capacitor,
and three resistors.
Figure 1. Equivalent circuit-based battery model.
ScJ

SqDqSuWu

Wpd){*

Wq

JqJc
The voltage source Voc(Z)represents the relationship between the open circuit voltage (OCV) and
the SOC, as shown in Figure 2. The OCV-SOC curve is obtained from the conditional discharge
test data [16]. The nominal capacitance Cncharacterizes the capacity of the battery to store charge.
The polarization capacitance Cpdescribes the polarization effect of the battery. In addition, the terminal
resistor Rt, the diffusion resistor Rp, and the propagation resistor Rbrepresent the resistive components
in the battery.
Figure 2. Open circuit voltage (OCV)-state of charge (SOC) curve at room temperature.
0 20 40 60 80 100
3.2
3.6
4.0
4.4
SOC (%)
OCV (V)

Energies 2013,65541
In [11], the state-space representation of the battery model is given by Equation (1). These equations
consist of three state variables: the output terminal voltage Vt, the SOC Z, and the polarization
voltage Vp:
˙
Vt=−a11Vt+a12Voc(Z) + b1I
˙
Z=−a2Voc(Z) + a2Vp+b2I
˙
Vp=a3Voc(Z)−a3Vp+b3I
y=Vt= [1 0 0][VtZ Vp]T
(1)
where:
a11 =1
Rb(Rb+Rp)Rb
Cp−Rp
Cn
a12 =1
(Rb+Rp)2Rb
Cp−Rp
Cn−R2
p
CnRb+Rp
Cp
a2=1
Cn(Rb+Rp), a3=1
Cp(Rb+Rp)
b1=1
(Rb+Rp)2Rp+Rt+RpRt
RbRb
Cp−Rp
Cn+R2
p
Cn+R2
b
Cp
b2=Rp
Cn(Rb+Rp), b3=Rb
Cp(Rb+Rp)
(2)
In previous papers, the parameters of the battery were assumed to be constant during
discharge [10,11]. However, the actual internal resistance changes depending on the operating
conditions; this resistance variation causes modeling errors [17]. Therefore, we propose a dynamic
resistance battery model to represent the battery dynamics more accurately. The proposed model
employs a dynamic resistance that varies according to both the magnitude of the discharge current and
the SOC level. The modeling accuracy performance of the proposed model is verified in Section 4.
3. Second-Order DSMO for SOC Estimation
In this section, the second-order DSMO [13], which is used for the estimation of the SOC and the
terminal voltage, is introduced. We consider a system described by:
x(k+ 1) = Ax(k) + Bu(k) + ∆(k)
y(k) = Hx(k)
(3)
where x∈ <nis the state variable vector; u∈ < is the input; y∈ < is the system output; and
∆represents the uncertainties caused by parameter variations and non-linearities. A,B, and Hare the
system nominal matrices, and (A, H)is assumed to be observable. Then, the second-order DSMO of the
system [Equation (3)] has the form:
ˆx(k+ 1) = Aˆx(k) + Bu(k) + L(y(k)−ˆy(k)) + vd(k)
vd(k) = vd(k−1) + Msaty(k)−ˆy(k)
φ
ˆy(k) = Hˆx(k)
(4)

Energies 2013,65542
where ˆxis the estimated state vector; ˆyis the estimated output; φdetermines the boundary layer; and L
and Mare the observer gain vectors (which are related to ˜y=y−ˆyand the sign of ˜y, respectively).
In general, the sign function sign(·)in the SMO is replaced by the saturation function sat(·)in practical
implementations of sliding mode concepts.
On the other hand, linearization and discretization of the battery model in Equation (1) are required
because the second-order DSMO is developed on the basis of a discrete-time linear system. To obtain a
linear system, we assume that the value of the voltage source Voc is constant value αkover a sampling
interval [k, k + 1]. This assumption is valid because the rate of change of Zover the interval is negligible,
and errors caused by this assumption can be regarded as part of the uncertainty. Taking Voc (Z(t)) = αk,
t∈[k, k + 1], Equation (1) is rewritten as follows:
˙x=Aox+Bou(5)
where:
x=hVtZ VpiT
, u =I
Ao=


−a11 a12 ·αk0
0−a2·αka2
0a3·αk−a3



Bo=


b1
b2
b3



(6)
Then, using the Euler discretization method with a sampling period T, the discrete-time linear state
equation is given by:
x(k+ 1) '(I+AoT)x(k) + BoT u(k) + Λ(k)
=Adx(k) + Bdu(k) + Λ(k)
(7)
where Λ(k)represents modeling errors caused by parameter uncertainties, linearization, and discretization.
We assume that the variation of each Λi(k), which is an element of the uncertainty vector Λ(k),
is bounded by the known bound σi:
|Λi(k)−Λi(k−1)| ≤ σi, i = 1 . . . n (8)
the bound σiis determined experimentally by comparing the modeling output with the actual
battery output.
Remark 1: We note that the SOC at the sampling instant k+ 1,i.e.,Zk+1 , is updated according
to Equation (7), and the value of the voltage source is also reset as Voc(Z(t)) = Voc(Zk+1) = αk+1,
t∈[k+ 1, k + 2].
Finally, the proposed SOC estimation method based on the second-order DSMO is formed by:
ˆx(k+ 1) = Adˆx(k) + BdI+L(Vt(k)−ˆ
Vt(k)) + vd(k)
vd(k) = vd(k−1) + MsatVt(k)−ˆ
Vt(k)
φ
ˆy(k) = Hˆx(k) = ˆ
Vt(k)
(9)

Energies 2013,65543
by Theorem 1 in [13], if the observer gain vector Msatisfies the inequalities:
σi≤Mi, i = 1 . . . n (10)
then the estimation error ˜x(k) = x(k)−ˆx(k)converges to a finite bound when k→ ∞. A stability
analysis of the second-order DSMO is presented in Appendix.
4. Experimental Results
Two types of experiments are performed to show the following:
•improvement of the battery modeling accuracy with the dynamic resistance varied with the
operating conditions;
•the SOC estimation method using the second-order DSMO for the elimination of chattering.
The battery test bench is composed of an electrical load, a power supply, and an National Instruments
Data Acquisition device. The electrical load and the power supply are used to discharge/charge the
battery, and the output voltage is measured using LabView, a PC-based program. In the experiment,
a fresh 18650-type Li-ion battery is used; it has a 3.0 A h nominal capacity and a 3.7 V nominal voltage.
All experiments are carried out at room temperature.
4.1. Parameter Extraction
The parameters of the battery model are determined from the battery characterization test data.
The initial value of the nominal capacitance is obtained from the energy stored in the capacitor and
the OCV at 100% and 0% SOC:
Energy Cn=1
2CnV2=1
2Cn(V2
100%SOC −V2
0%SOC )
=AmpSec ×V100%S OC
∴Cn init =AmpSec rated×V100%SOC
1
2(V2
100%SOC −V2
0%SOC )
(11)
where AmpSec rated is the rated battery capacity.
The polarization capacitance is derived from high-frequency excitation tests, which apply a 1.0C
discharge pulse in 0.5 s intervals. Using the results in Figure 3, we approximate the time constant using
the following equation:
Vno load =V1=V3+ (V4−V3)(1 −e−t/τ )
∴τ=−∆tln(1 −V4−V3
V1−V3)
(12)
The time constant associated with Cpis represented by its associated resistance:
τ=Cp(Rb+Rp)
∴Cp init =τ
Rb+Rp
(13)

Energies 2013,65544
The internal resistance Rint is obtained using the direct current-internal resistance (DC-IR) method:
Rint =V3−V2
Idis
(14)
where Idis represents the discharge current. Generally, it is assumed that Rband Rpare taken to be 75%
of Rint and Rtis equivalent to 25% of Rint [10].
Figure 3. Output voltage when a discharge current of 1.0C is applied in 0.5 s intervals.
0.0 0.4 0.8 1.2 1.6
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
time (s)
Output Voltage (V)
9

9

9

9

4.2. Dynamic Resistance
As mentioned in Section 2, previous papers assumed that the parameters used in the battery model
are constant. However, this assumption leads to large modeling errors because the internal resistance
of the battery depends on the operating conditions such as the magnitude of the discharge current and
the SOC level. To examine this change in the internal resistance, our first experiment is performed.
The current profile used in this experiment is composed of a 3 s discharge pulse and a 3 s rest, as shown
in Figure 4.
Figure 4. Pulse discharge current used to examine the variation in the internal resistance.
0 10 20 30 40 50 60
0.0
0.6
1.2
1.8
2.4
3.0
time (s)
Discharge current (A)

Energies 2013,65545
To verify the effect of the discharge current and the SOC on the internal resistance, the discharge
current is increased from 0.1C to 1.0C in intervals of 0.1C; this profile is repeated from 100% to 0% SOC.
The experimental results in Figure 5show the SOC versus Rint curves for each discharge current.
These results are obtained from 10 independent trials. As expected, the internal resistance has a higher
value at higher discharge currents and at both ends of the SOC level. Therefore, allowing the resistance
to vary with the discharge current and the SOC level is a reasonable way of more accurately representing
the battery dynamics.
Figure 5. Internal resistance variation for different discharge currents and different
SOC levels.
0 20 40 60 80 100
0.185
0.19
0.195
0.2
0.205
0.21
0.215
0.22
0.225
0.23
SOC (%)
Internal Reistance (Ω)
0.3 A
0.6 A
0.9 A
1.2 A
1.5 A
1.8 A
2.1A
2.4 A
2.7 A
3.0 A
To validate the dynamic resistance model, the current profile in Figure 4is applied repeatedly
over the 100%–0% SOC range. Figures 6shows the output voltage modeling results at 100% and 50%
SOC, respectively.
Figure 6. Comparison of the modeling performance: at (a) 100% SOC; and (b) 50% SOC.
0 10 20 30 40 50 60
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
time (s)
Output Voltage (V)
Measurement
Constant resistance model
Dynamic resistance model
(a)
0 10 20 30 40 50 60
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
time (s)
Output Voltage (V)
Measurement
Constant resistance model
Dynamic resistance model
(b)

Energies 2013,65546
In the constant resistance model, the initial parameters are extracted from the battery with 100% SOC,
and they are fixed as, which are independent of the SOC level, Cn= 1.491 ×104F, Cp= 17.6735 F,
Rb= 0.165 Ω,Rp= 0.165 Ω, and Rt= 0.055 Ω. On the other hand, the resistance is updated using the
SOC versus Rint curves according to the operating conditions in the dynamic resistance model. Here, the
capacitance is assumed to be constant, because its variation is negligibly small and the sampling period
Tis 0.01 s. The results show that the accuracy of both models is similar at 100% SOC. However, at 50%
SOC, the dynamic resistance model shows better accuracy than the constant resistance model. Similarly,
in other SOC levels, modeling errors are caused by differences between actual and constant resistance
value. It represents that the proposed battery model is more suitable for the modeling of Li-ion batteries.
4.3. Random Current Discharge Test
The second experiment is designed to verify the estimation performance of the proposed SOC
estimation method. In this experiment, a random discharge current (shown in Figure 7) is applied
repeatedly over the 100%–0% SOC range to simulate the actual battery usage.
Figure 7. (a) Random discharge current; and (b) measured output voltage.
0 50 100 150 200
0.0
0.5
1.0
1.5
2.0
2.5
3.0
time (s)
Discharge Cunrret (A)
(a)
0 50 100 150 200
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
time (s)
Output Voltage (V)
(b)
Then, the second-order DSMO is used in the dynamic resistance model to estimate both the SOC and
the output voltage of the battery. The gains of the second-order DSMO, are chosen as follows:

Energies 2013,65547
L=h0.03 0.03 0.01iT
M=h0.015 0.01 0.015iT(15)
Remark 2: To compare the second-order DSMO with the first-order DSMO under the same
conditions, the gain Mis applied equally to each one and the sign function sign(·)is used for
second-order DSMO [Equation (9)] instead of the saturation function sat(·).
The SOC estimation results are shown in Figure 8. For the desired SOC, the ampere-hour counting
method that takes into account the Coulombic efficiency is used. Here, the battery is initially fully
charged using the constant-current/constant-voltage method. The results show that the estimated
SOC using the first-order DSMO exhibits the undesirable chattering phenomenon, and the SOC
estimation error increases gradually to about 5%. In contrast, the proposed SOC estimation method
enhances the estimation performance in terms of chattering elimination and SOC estimation precision.
Additionally, for the clear view, we present the one-cycle estimation results of the SOC in Figure 8b.
Figure 8. SOC estimation results for the proposed method.
0 2000 4000 6000 8000
20
30
40
50
60
70
80
90
100
time (s)
SOC (%)
Ampere−Houre Counting
First−Order DSMO
Second−Order DSMO
(a)
1200 1240 1280 1320 1360 1400
87
88
89
90
91
time (s)
SOC (%)
Ampere−Houre Counting
First−Order DSMO
Second−Order DSMO
(b)
Figure 9shows an improvement in the chattering elimination when the second-order DSMO is used
for output voltage estimation. In contrast, the chattering phenomenon occurred in the results for output
voltage estimation when the first-order DSMO is used. It is possible to verify the improvement in the
accuracy by comparing the estimation errors over a single cycle, as shown in Figure 9b. It is clear that
the estimation error is reduced drastically by the second-order DSMO.
To investigate the robustness of the second-order DSMO to the situation where the initial SOC
is unknown, we repeat the same experiment above but the initial value of estimated SOC is set to be 90%.
As shown the results in Figure 10, we can find that the both estimated SOC values converges to the actual
one within 70 s. This implies that the proposed SOC estimation method based on the second-order
DSMO can reduce the chattering phenomenon while maintaining the robustness properties of the sliding
mode approach.

Energies 2013,65548
Figure 9. One-cycle estimation results for the proposed method: (a) output voltage
estimation; and (b) estimation errors.
1200 1240 1280 1320 1360 1400
3.4
3.6
3.8
4.0
4.2
time (s)
Output Voltage (V)
Measurement
First−Order DSMO
Second−Order DSMO
(a)
1200 1240 1280 1320 1360 1400
−0.05
−0.025
0
0.025
0.05
time (s)
Output Voltage Estimation Error (V)
First−Order DSMO
Second−Order DSMO
(b)
Figure 10. SOC estimation results with unknown the initial SOC.
0 50 100 150 200
90
92
94
96
98
100
time (s)
SOC (%)
Ampere−Houre Counting
First−Order DSMO
Second−Order DSMO
5. Conclusions
The chattering phenomenon in SOC estimated by a first-order SMO deteriorates estimation accuracy.
To overcome this drawback and improve accuracy, a second-order DSMO-based SOC estimation method
is proposed. Furthermore, a dynamic resistance battery model is also proposed to more accurately
represent the dynamics of a Li-ion battery. The improvement of model accuracy is achieved with
the resistance parameters which are varied with both the magnitude of the discharge current and the
SOC level. This approach is suitable for on-line implementation because the SOC vs. Rint curves for

Energies 2013,65549
each discharge current can be obtained in advance through a simple experiment. Our experimental results
show that the proposed battery model has a better modeling accuracy than the constant resistance model.
In addition, elimination of chattering in both the SOC and output voltage estimates is achieved using the
second-order DSMO.
Appendix: Stability Analysis
The stability analysis of the second-order DSMO has been investigated by Mihoub et al. [13].
From Equations (7) and (9), the estimation error dynamics given by:
˜x(k+ 1) = (Ad−LH)˜x(k)−vd(k) + Λ(k)
vd(k) = vd(k−1) + MsatVt(k)−ˆ
Vt(k)
φ
˜y(k) = H˜x(k) = ˜
Vt(k)
(16)
where Λ(k)and Msatisfy the conditions in Equations (8) and (10), respectively.
Then we have:
˜x(k+ 1) −˜x(k) = (Ad−LH)(˜x(k)−˜x(k−1)) −M satH˜x(k)
φ+ Λ(k)−Λ(k−1) (17)
which can be written as:
˜x(k+ 1) −(Ad−LH +I)˜x(k)+(Ad−LH)˜x(k−1) + M sat˜
Vt(k)
φ−(Λ(k)−Λ(k−1)) = 0 (18)
Consider now the following two cases:
•Case 1 : suppose that |˜
Vt(k)| ≥ φ.
In this case, we have:
Λ(k)−Λ(k−1) −Msat˜
Vt(k)
φ=−|M±(Λ(k)−Λ(k−1))|sign(˜
Vt(k)) (19)
By Lemma 1 in [13], for |˜y(k)| ≥ φ, there exists a bounded function F(k)≥0such that:
−F(k)|˜
Vt(k)|= Λ(k)−Λ(k−1) −Msat˜
Vt(k)
φ(20)
where:
0≤F(k),|M±(Λ(k)−Λ(k−1))|sign(˜y(k))
˜y(k)≤σ+M
φ(21)
Thus, Equation (18) can be written as follows:
˜x(k+ 1) −(A−LH +I+F H )˜x(k)+(A−LH )˜x(k−1) = 0 (22)

Energies 2013,65550
From Equation (22), we obtain the state equations:
Z(k+ 1) = AZ(k)(23)
where:
Z(k) = ˜x(k−1)
˜x(k),A=0I
−(A−LH)A−LH +I+F H (24)
It is easily to ensure the convergence of Z(k)when eigenvalues’ modules of matrix Aare smaller
than one. Therefore, the convergence of the estimation error is also guaranteed.
•Case 2:|˜
Vt(k)|< φ.
In this case, Equation (18) can be expressed as:
˜x(k+ 1) −A−LH +I−MH
φ˜x(k)+(A−LH )˜x(k−1) −Λ(k) + Λ(k−1) = 0 (25)
Then we have:
Z0(k+ 1) = A0Z0(k) + 0
Λ(k)−Λ(k−1) (26)
where:
Z0(k) = ˜x(k−1)
˜x(k),A0=0I
−(A−LH)A−LH +I+F H (27)
Consequently, if the uncertainty Λ(k)satisfies the condition in Equation (8) and eigenvalues’
modules of matrix A’ are smaller than one, Z(k)0is bound. Also, the estimation error ˜x(k)
is bounded.
Thus, the convergence of the estimation error can be guaranteed.
Acknowledgments
This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the
“IT Consilience Creative Program” support program supervised by the NIPA (National IT Industry
Promotion Agency) (C1515-1121-0003).
Conflicts of Interest
The authors declare no conflict of interest.
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