Decentralized Sampled-Data Fuzzy Observer Design for Nonlinear Interconnected Systems

Abstract

This paper presents the decentralized sampled-data fuzzy observer design techniques for nonlinear interconnected systems, which are assumed to be composed of fuzzy subsystems and unknown interconnections. To design the decentralized fuzzy observer, the estimation error dynamics is obtained, and the performance function is defined. Based on the estimation error dynamics and the performance function, the continuous-time decentralized fuzzy observer is first proposed to minimize the interconnection bound to attenuation degree ratio. In addition, the decentralized sampled-data fuzzy observers are designed by using the approximate discretization and the exact discrete-time design approaches, respectively. Each proposed observer design technique is formulated into the optimal problems with the linear matrix inequalities. Finally, several simulation examples show the validity and superiority of the proposed observer design techniques by comparing with the conventional techniques.

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Decentralized sampled-data fuzzy observer design
for nonlinear interconnected systems
Geun Bum Koo, Jin Bae Park and Young Hoon Joo
Abstract—This paper presents the decentralized sampled-data
fuzzy observer design techniques for nonlinear interconnected
systems, which are assumed to be composed of fuzzy subsystems
and unknown interconnections. To design the decentralized fuzzy
observer, the estimation error dynamics is obtained and the
performance function is defined. Based on the estimation error
dynamics and the performance function, the continuous-time
decentralized fuzzy observer is firstly proposed to minimize the
interconnection bound to attenuation degree ratio (IAR). Also,
the decentralized sampled-data fuzzy observers are designed by
using the approximate discretization and the exact discrete-time
design approaches, respectively. Each proposed observer design
technique is formulated into the optimal problems with the linear
matrix inequalities (LMIs). Finally, several simulation examples
show the validity and superiority of proposed observer design
techniques by comparing with the conventional techniques.
Index Terms—Decentralized sampled-data fuzzy observer, non-
linear interconnected systems, interconnection bound to attenua-
tion degree ratio, approximate discretization, exact discrete-time
design.
I. INT ROD UC TI ON
TTHE observer design is one of the most important re-
search issues and has been expanded by many studies [1],
[2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], such
as observer-based control [14], [15], filter design [16], [17]
and fault detection [18], [19]. The previous observers were
designed by guaranteeing the convergence of the estimation
error. Thus, most of the previous techniques are restricted
to the systems which are exactly known [2], [4], [6], [8],
[10], [12] or satisfy the asymptotic stability [1], [16], [7]. In
the case of the oscillating or unstable uncertain systems, the
observers were designed by merging with the controller [14].
Especially, in the case of the interconnected systems, most
of the previous decentralized observer design techniques were
also proposed by merging with the control technique [9], [11],
[20] or for the asymptotically stable interconnected system [3],
[5], [13], because the system model is not able to be exactly
known due to the uncertain or unknown interconnection and all
measurements are not reflected in the decentralized observer.
Accordingly, the detailed study is needed for the decentralized
Manuscript received ? ??, ????; revised ? ??, ????. This work was supported
by the National Research Foundation of Korea (NRF) grant funded by the
Korea government (MEST) (NRF-2015R1A2A2A05001610) and Institute of
BioMed-IT, Energy-IT and Smart-IT Technology (BEST), a Brain Korea 21
plus program, Yonsei University.
G. B. Koo and J. B. Park are with the Department of Electrical and
Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail:
{milbam,jbpark}@yonsei.ac.kr).
Y. H. Joo is with the Department of Control and Robotics Engineering,
Kunsan National University, Kunsan, Chonbuk, 573-701, Korea, (e-mail:
yhjoo@kunsan.ac.kr).
observer design technique of the interconnected system with
unknown interconnection.
Apart from the decentralized observer issue, the sampled-
data observer design, which is for continuous-time systems
with sampled-data output, has attracted much attention due to
the rapid expansion of computer- or network-based estimators.
In linear systems, the sampled-data observer design is not
difficult due to the simple and exact discretization. However,
in nonlinear systems, the exact discretization is too hard
and complex. Thus, to solve this problem, many observer
design techniques have been proposed and can be categorized
into three approaches: approximate discretization, time-delay
conversion and exact discrete-time design approaches.
First, in the case of the approximate discretization approach,
the discrete-time observer is designed for the approximately
discretized nonlinear system. This approach is possible to
the direct design of the sampled-data observer through the
discrete-time observer design, but the discretization error can-
not be completely eliminated. The significant results for the
approximate discretization approach have been widely dis-
cussed in [21], [22], [23], [24], [25], [26]. Next, the time-delay
conversion approach is to design the continuous-time observer
by converting the sampled-data output into the time-delay
output and is able to perfectly eliminate the discretization
error. See references [27], [28], [29] for more information
about the time-delay conversion approach. The exact discrete-
time design approach is to use the exact discretized model
of the estimation error dynamics. In [30], [31], [32], various
control techniques are proposed using the exact discrete-time
design approach, but the observer design method has not been
studied in depth.
In summary of the aforementioned analysis, the previous
observer design techniques have some limitations in following
terms:
•There is no effective observer design method for the
oscillating or unstable uncertain systems.
•In the case of the interconnected systems with the uncer-
tain or unknown interconnection, it is hard to design the
decentralized observer.
•The more study is needed for the sampled-data observer
design by using the exact discrete-time design approach.
Motivated by the above problems of the previous tech-
niques, the decentralized observer design techniques are pro-
posed for the nonlinear interconnected system. To solve the
unknown interconnection problem, the interconnection bound
to attenuation degree ratio (IAR) is defined, and the estimation
error is minimized through the minimization of the IAR.
Based on the continuous-time decentralized fuzzy observer

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design, the nonlinear interconnected system is expanded into
the sampled-data system, and the decentralized sampled-data
fuzzy observers are designed using the approximate discretiza-
tion and the exact discrete-time design approaches, respec-
tively. The sufficient conditions of each proposed technique
are formulated into the linear matrix inequalities (LMIs).
Finally, by several examples, the performance of the proposed
techniques is compared with the previous observer design
techniques.
This paper is organized as follows: Section 2 describes the
decentralized fuzzy observer design for the nonlinear intercon-
nected system. The approximate discretization approach for
the decentralized sampled-data observer is presented in Section
3. In Section 4, the decentralized sampled-data observer design
using the exact discrete-time design approach is presented.
Some numerical examples are given for illustration in Section
5. Finally, the conclusions and the future works are given in
Section 6.
Notation: The subscripts iand jdenote fuzzy rule indices,
and the subscript kdenotes the subsystem index. (·)Tdenotes
the transpose of the argument. The notations ∗,He{A}
and λk
Akare used for the transposed element in symmetric
positions, A+ATand the maximum eigenvalue of kth matrix
AT
kAk, respectively. The observer gain LT hm.1
ij is described
the observer gain Lij of Theorem 1.
II. DECENTRALIZED FUZZY OBSERVER FOR NONLINEAR
INTERCONNECTED SYSTEMS
Consider an nonlinear interconnected system composed of
T–S fuzzy subsystems, which can be described by
˙xk(t) =Ak(t)xk(t) + hk(x(t))
yk(t) =Ckxk(t)(1)
where k∈ Iq:= {1,2, . . . , q};xk(t)∈Rnkand yk(t)∈
Rmkare the state and output variables of the kth subsys-
tem, respectively; x(t) = col{x1(t), x2(t), . . . , xq(t)}is the
whole state variable of the interconnected system; Ak(t) =
r
i=1 µki(zk(t))Aki and Ckare the system and output matri-
ces, respectively; µki(zk(t)) := ωki (zk(t))/r
i=1 ωki(zk(t))
and ωki(zk(t)) := s
p=1 Γkip(zk p(t));zkp (t)is the pth
premise variable and zk(t)is the premise variable vector with
zkp(t);Γk ip is a membership function of zkp(t); and hk(x(t))
is a piecewise continuous vector function with the following
assumption:
Assumption 1.The vector function hk(x(t)) is unknown but
satisfies the following quadratic inequality:
hk(x(t))Thk(x(t)) ≤α2
kx(t)THT
kHkx(t)
where αk>0is a bound constant of the interconnection, and
Hkis a constant matrix with appropriate dimension.
We also assume the observability and measurability of the
T–S fuzzy subsystems.
Assumption 2.All pairs (Aki, Ck)are observable for
(k, i)∈ Iq× Ir.
Assumption 3.The state variable xk(t)is not measurable,
but the premise variable zk(t)and the output variable yk(t)
are measurable.
Then, the decentralized fuzzy observer can be represented
in the following form:
˙
ˆxk(t) =Ak(t)ˆxk(t) + Lk(t)(yk(t)−ˆyk(t))
ˆyk(t) =Ckˆxk(t)(2)
where Lk(t) = r
i=1 µki(zk(t))Lk i, and Lki is the observer
gain matrix.
Remark 1.The fuzzy observer is one of the most efficient
nonlinear observer design techniques, because the various
linear techniques can be easily applied for nonlinear systems.
Thus, the fuzzy observer design has gathered much attention
and is classified into some categories: pure fuzzy observers
[33], [34], fuzzy observers with the fuzzy controller [35], [36],
[31], [37], [20], fuzzy filters [16], [3], [5], [13], [18], and
fuzzy observers with adaptive techniques [38], [11]. However,
most of the previous fuzzy observers can be only designed for
exactly known or asymptotically stable systems, and there are
few studies of the fuzzy observer design of the oscillating or
unstable uncertain systems. Thus, to design the fuzzy observer
for the uncertain systems, such as the nonlinear interconnected
system with the unknown interconnection, the novel approach
is needed.
By defining the estimation error ek(t) := xk(t)−ˆxk(t), we
can obtain the estimation error dynamics as follows:
˙ek(t) =Φk(t)ek(t) + hk(x(t)) (3)
where Φk(t) = Ak(t)−Lk(t)Ck.
The purpose of the decentralized fuzzy observer design
is to find the observer gain Lki such that i)the estimation
error dynamics (3) with hk(x(t)) = 0 is asymptotically
stable, ii)there exists the attenuation degree γ≥0satisfying
∥e(t)∥ ≤ γ∥x(t)∥, where e(t) = col{e1(t), e2(t), . . . , eq(t)}
is the whole estimation error, and iii)the IAR γ/αkis
minimized.
To achieve the design purpose, we define the following
performance function:
Definition 1.The performance function
J=∞
0
e(t)Te(t)−γ2x(t)Tx(t)dt
is said to be an IAR guaranteed function. If there exists
the fuzzy observer gain Lki such that the IAR guaranteed
function satisfies J≤0, we guarantee the existence of the
attenuation degree γand the decentralized fuzzy observer
satisfies ∥e(t)∥ ≤ γ∥x(t)∥.
Based on the above performance function, the sufficient
condition for the design purpose is summarized as follows:
Theorem 1.Consider the nonlinear interconnected system
(1) and the decentralized fuzzy observer (2). If there exist
scalar δkand matrices Pk=PT
k≻0and Lki such that
following optimal problem with the LMI is satisfied
min δksubject to
He{PkAki +NkiCk}+I∗
Pk−1
qλk
Hk
δkI≺0(4)

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for (k, i)∈ Iq×Ir, then the estimation error dynamics (3) is
asymptotically stable for hk(x) = 0 and satisfies J≤0, and
√δkis the minimum IAR.
Proof. Consider the Lyapunov function candidate as V(t) =
q
k=1 ek(t)TPkek(t), then
˙
V(t) +
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)
=
q

k=1 Φk(t)ek(t) + hk(x(t))TPkek(t)
+
q

k=1
ek(t)TPkΦk(t)ek(t) + hk(x(t))
+
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)
Because hk(x(t))TPkek(t) + ek(t)TPkhk(x(t)) ≤
σkek(t)TP2
kek(t) + σ−1
khk(x(t))Thk(x(t)) with σk≥0, we
have
˙
V(t) +
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)
≤
q

k=1
ek(t)TΦk(t)TPk+PkΦk(t) + I+σkP2
kek(t)
+
q

k=1 σ−1
kα2
kx(t)THT
kHkx(t)−γ2
qx(t)Tx(t)
based on Assumption 1.
Thus, the sufficient condition for ˙
V(t) +
q
k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)<0can be rewritten as
Φk(t)TPk+PkΦk(t) + I+σkP2
k≺0,(5)
σ−1
kα2
kλk
Hk−γ2
q= 0.(6)
Applying the schur complement to (5) yields
Φk(t)TPk+PkΦk(t) + I∗
Pk−σ−1
kI≺0.(7)
By substituting (6) into (7), denoting PkLki =Nki and
γ2/α2
k=δkand using the fuzzy property, we can obtain the
LMI (4). Thus, if the LMI (4) is satisfied, it follows
˙
V(t) +
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)<0.(8)
Then, we get J < 0by integrating inequality (8) from 0to
∞. Thus, the LMI (4) guarantees the existence of γ. Also, we
can easily guarantee that the LMI (4) satisfies He{PkAki +
NkiCk} ≺ 0, which is sufficient condition for the asymptotic
stability of the estimation error dynamics (3) with hk(x(t)) =
0.
Remark 2.Theorem 1 presents the decentralized fuzzy
observer design technique with the minimization of the IAR.
Because the interconnected system (1) contains an unknown
interconnection, the exact system model cannot be applied in
the observer design. Also, if the interconnected system (1) is
oscillating or unstable, the observer design techniques based
on the asymptotically stable system cannot provide the feasible
solution. To solve these problems, the novel observer design
technique is proposed through the minimization of the IAR
in Theorem 1. From Theorem 1, the minimized error can be
obtained in the fixed interconnection bound, or the maximum
interconnection bound can be found in the fixed attenuation
degree of the estimation error.
Example 1.Consider the nonlinear interconnected system
composed of two Van der Pol oscillators in [39] as follows:
¨zk(t)−φk1−zk(t)2˙zk(t) + zk(t)−hk(z(t)) = 0
yk(t) = zk(t)
where k∈ I2,φkis a scalar parameter indicating the strength
of the damping and hk(z(t)) is an unknown interconnection
function that is assumed as hk(z(t)) = αkHkz(t)where Hk=
[0 0.1 0 0.1], and αk≥0is an unknown constant value.
By choosing xk(t) = [xk1(t)xk2(t)]T= [zk(t) ˙zk(t)]T, the
kth fuzzy subsystem can be constructed as follows:
˙xk(t) =
2

i=1
µki(zk(t))Ak ixk(t) + hk(x(t))
yk(t) =Ckxk(t)
where
Ak1=0 1
−1φk(1 −M2
k), Ak2=0 1
−1φk,
Ck=0 1,
µk1(zk(t)) =zk(t)2
M2
k
, µk2(zk(t)) = 1 −µk1(zk(t))
with φ1= 1,φ2= 0.5and M1=M2= 2.5.
By solving the corresponding LMIs, we obtain the observer
gains as follows:
LT hm.1
11 =−474.1620
50.8822 , LT hm.1
12 =−474.1620
57.1322 ,
LT hm.1
21 =−474.1620
53.5072 , LT hm.1
22 =−474.1620
56.6322 .
To show the effectiveness of the proposed technique, we
compare the performance of Theorem 1 and other techniques
by using the performance measure function
P=10
0P(t)dt =
q

k=1 10
0
r

i=1 |xki(t)−ˆxki(t)|dt (9)
the performances of all observers with the initial conditions
xk(0) = [1 1]Tand ˆxk(0) = [0 0]Tare compared in Table 1.
TABLE I
PERFORMANCE COMPARISON OF THE DECENTRALIZED FUZZY OBSERVER.
Interconnection bound αk0.1 0.5 1
Fuzzy observer 9.7380 12.8409 17.1506
H∞fuzzy filters Marginal Infeasibility
LMI (4) 2.7555 4.7092 7.4019
In Table 1, the first technique is obtained by the sufficient

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condition He{PkAki +NkiCk} ≺ 0and has the insufficient
performance due to the LMI condition without considering the
interconnection. Also, the H∞fuzzy filters proposed in [16],
[5], [13] are used. However, because Van der Pol oscillator
system is not satisfied the asymptotic stability condition, the
H∞fuzzy filter cannot provide the feasible solution. Thus,
we know that the proposed technique has better observing
performance than other techniques.
Remark 3.In Example 1, to demonstrate the effectiveness
of Theorem 1, the previous H∞fuzzy filters are used with
considering the unknown interconnection term as the distur-
bance. However, the sufficient condition of the H∞fuzzy filter
has the feasible solution when the fuzzy system satisfies the
asymptotic stability. Thus, in the case of the oscillating system
such as Example 1, the H∞fuzzy filter cannot present the
feasible solution.
Theorem 1 is based on the measurable premise variable
case. Now, we introduce the decentralized fuzzy observer
design technique for the non-measurable premise variable case
in the following corollary.
Corollary 1.Consider the nonlinear interconnected sys-
tem (1) and the decentralized fuzzy observer with the non-
measurable premise variable as follows:
˙
ˆxk(t) = ˆ
Ak(t)ˆxk(t) + ˆ
Lk(t)(yk(t)−ˆyk(t))
ˆyk(t) =Ckˆxk(t)(10)
where ˆ
Ak(t) = r
i=1 µki( ˆzk(t))Aki ,ˆ
Lk(t) =
r
i=1 µki( ˆzk(t))Lki , and ˆzk(t)is the estimated premise
variable. If there exist scalar δkand matrices Pk=PT
k≻0
and Lki such that the following optimal problem with the
LMI is satisfied
min δksubject to
He{PkAki +NkiCk}+I∗
Pk−1
q(λk
Hk+λk
Λkij )δkI≺0
(11)
where Λkij =Aki −Akj for (k, i, j)∈ Iq×Ir×Ir, then the
decentralized fuzzy observer (10) well estimates the nonlinear
interconnected system (1) and δk=γ2
α2
k+1 .
Proof. See Appendix A.
Remark 4.Unlike Theorem 1, δkis defined as γ2/(α2
k+ 1)
in Corollary 1. It means that, due to the difference of the firing
strength between the fuzzy system and the fuzzy observer, the
estimation error is not converged to 0, even if there does not
exist the interconnection by αk= 0.
III. DECENTRALIZED SAMPLED-DATA FUZ ZY O BS ERVER
VI A TH E AP PROXIMATE DISCRETIZATIO N AP PROACH
In the previous section, the sufficient condition is obtained
for the decentralized fuzzy observer design of the intercon-
nected nonlinear system with the unknown interconnection.
Now, we extend the proposed technique to the nonlinear inter-
connected system with the sample-data output yk(t) = yk(nT )
for t∈[nT, nT +T),n∈Z≥0where Tis a sampling period.
We firstly obtain the sufficient condition for the decentralized
sampled-data fuzzy observer design using the approximate
discretization approach with following assumption:
Assumption 4.The firing strength µki(zk(t)) and the un-
known interconnection function hk(x(t)) are approximated
by µki(zk(nT )) and hk(x(nT )) for t∈[k T, kT +T),
respectively.
Based on the above assumption, we obtain the approxi-
mately discretized model of the interconnected sampled-data
nonlinear system (1) as follows:
xk(nT +T) =Gk(nT )xk(nT ) + Fk(nT )hk(x(nT ))
yk(nT ) =Ckxk(nT )(12)
where Gk(nT ) = r
i=1 µki(zk(nT ))exp(AkiT)and
Fk(nT ) = r
i=1 µki(zk(nT ))A−1
ki (Gki −I).
From the discrete-time model (12), we design the discrete-
time decentralized fuzzy observer and obtain the discrete-time
estimation error dynamics as follows:
◦discrete-time decentralized fuzzy observer:
ˆxk(nT +T) = Gk(nT )ˆxk(nT ) + Lk(nT )(yk(nT )
−ˆyk(nT ))
ˆyk(nT ) = Ckˆxk(nT )(13)
◦discrete-time estimation error dynamics:
ek(nT +T) = Φk(nT )ek(nT ) + Fk(nT )hk(x(nT )) (14)
Also, we newly define the approximate IAR guaranteed
function
J≈
∞

n=0 nT +T
nT
e(nT )Te(nT )−γ2x(nT )Tx(nT )dt
=Jappox.(15)
Remark 5.The approximate IAR guaranteed function
Jappox is concerned only at sampling instants, but not during
the sampling interval. Thus, the optimization of γ≥0
based on Jappox ≤0does not provide the optimal observing
performance.
Then, the sufficient condition for the decentralized sampled-
data fuzzy observer design via approximate discretization
approach is addressed as the following theorem.
Theorem 2.Consider the approximately discretized nonlin-
ear interconnected system (12) and the discrete-time decentral-
ized fuzzy observer (13). If there exist some scalars δkand
βkand matrices Pk=PT
k≻0and Lki such that following
optimal problem with the LMIs is satisfied
min δksubject to



−Pk+T I ∗ ∗
PkAki +Nki Ck−Pk∗
PkAki +Nki Ck0−T
qλk
Fki λk
Hk
δk−βkI

≺0,
(16)
Pk−βkI≺0(17)
for (k, i)∈ Iq× Ir, then the discrete-time estimation error
dynamics (14) is asymptotically stable for hk(x(nT )) = 0
and satisfies Japrox ≤0, and √δkis the minimum IAR.

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Proof. Consider the discrete-time Lyapunov function candi-
date as V(nT ) = q
k=1 ek(nT )T×Pkek(nT ), then
∆V(nT ) +
q

k=1 T ek(nT )Tek(nT )−Tγ2
qx(nT )Tx(nT )
=
q

k=1 Φk(nT )ek(nT ) + Fk(nT )hk(x(nT ))TPk
×Φk(nT )ek(nT ) + Fk(nT )hk(x(nT ))
−
q

k=1
ek(nT )TPkek(nT )
+
q

k=1 T ek(nT )Tek(nT )−Tγ2
qx(nT )Tx(nT )
≤
q

k=1
ek(nT )TΦk(nT )T˜
PkΦk(nT )−Pk+T I ek(nT )
+
q

k=1 (Fk(nT )hk(x(nT )))T(σ−1
kI+Pk)
×Fk(nT )hk(x(nT )) −Tγ2
qx(nT )Tx(nT )
where ˜
Pk=Pk+σkP2
k.
If there exists some scalar βksuch that Pk≺βkI,
then the sufficient condition for ∆V(nT ) +
q
k=1 T ek(nT )Tek(nT )−Tγ2
qx(nT )Tx(nT )<0is
rewritten as
Φk(nT )T˜
PkΦk(nT )−Pk+T I ≺0,(18)
(σ−1
k+βk)α2
kλk
Fki λk
Hk−Tγ2
q= 0.(19)
Applying the schur complement to (18), we have


−Pk+T I ∗ ∗
PkΦk(nT )−Pk∗
PkΦk(nT ) 0 −σ−1
kI
≺0.(20)
By substituting (19) into (20), denoting PkLki =Nki and
γ2/α2
k=δk, and using the fuzzy property, we can determine
that, if the LMI (16) is satisfied, it follows that
∆V(nT ) +
q

k=1 T ek(nT )Tek(nT )−Tγ2
qx(nT )Tx(nT )
<0.(21)
Then, Japprox <0is guaranteed by summing the inequality
(21) from 0to ∞. Also, the LMI (16) satisfies the asymptotic
stability of the discrete-time estimation error dynamics (14)
with hk(x(nT )) = 0.
Remark 6.Theorem 2 presents the decentralized sampled-
data observer design technique using the approximate dis-
cretization approach. Also, Theorem 2 can be directly applied
in the decentralized observer design for the discrete-time inter-
connected nonlinear system, because the observer is designed
based on the discretized system.
Similarly to Corollary 1, we address the sufficient condition
for the discrete-time decentralized fuzzy observer design of
the approximately discretized model with the non-measurable
premise variable as follows:
Corollary 2.Consider the approximately discretized non-
linear interconnected system (12) and the discrete-time de-
centralized fuzzy observer with the non-measurable premise
variable. If there exist some scalars δkand βkand matrices
Pk=PT
k≻0and Lki such that following optimal problem
with the LMIs is satisfied
min δksubject to



−Pk+T I ∗ ∗
PkAki +Nki Ck−Pk∗
PkAki +Nki Ck0−1
λk
Fki λk
Hk+λk
Λkij
ξkiI

≺0,
(22)
Pk−βkI≺0,(23)
Pk−λk
Fki λk
Hk
λk
Λkij
βkI≺0(24)
where ξki =T
qδk−2λk
Fki λk
Hkβkfor (k, i, j)∈ Iq× Ir×
Ir, then the discrete-time decentralized fuzzy observer well
estimates the nonlinear interconnected system (1) and δk=
γ2
α2
k+1 .
Proof. The proof of Corollary 2 is similar to the process of
the proofs for Corollary 1 and Theorem 2 and so is omitted
here.
Remark 7.In Section 3, we cannot prevent the occurrence
of the discretized error due to the use of the approximate
discretization approach. Also, because the approximate IAR
guaranteed function is used in Theorem 2, the IAR is mini-
mized only at sampling instants. Thus, to solve these problems,
another approach is needed for the sampled-data observer
design.
IV. DEC EN TR AL IZ ED S AM PL ED -DATA FUZ ZY O BS ERV ER
VIA THE EXACT DISCRETE-TIM E DE SI GN A PP ROAC H
In this section, the new decentralized sampled-data observer
design technique is proposed to conquer the problem of
Section 3 by using the exact discrete-time design approach. We
consider the exact decentralized sampled-data fuzzy observer,
which is assumed that the premise variable is measurable in
the continuous-time sense, and the estimation error dynamics
as follows:
◦decentralized sampled-data fuzzy observer:
ˆxk(t) = Ak(t)ˆxk(t) + Lk(t)(yk(nT )−ˆyk(nT ))
ˆyk(nT ) = Ckˆxk(nT )(25)
◦estimation error dynamics:
˙ek(t) = Φk(t)ek(nT ) + Ak(t)˜ek(t) + hk(x(t)) (26)
where ˜ek(t) = ek(t)−ek(nT ).
Before addressing the sufficient condition, the following
lemmas and proposition have to be considered for the proof
of the theorem.

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Lemma 1 ([40]).Given any function vector x, matrix P=
PT≻0, and t0, tf∈Rwith t0< tf, we have
tf
t0
x(τ)dτT
Ptf
t0
x(τ)dτ
≤(tf−t0)tf
t0
x(τ)TP x(τ)dτ.
Lemma 2 ([32]).Suppose the nonlinear system ˙x=f(t, x),
where f: [nT, nT +T)×Rnis piecewise continuous in tand
locally Lipschitz in x, and the matrix P=PT≻0, then the
following inequality is always satisfied
nT +T
nT x(t)−x(nT )TPx(t)−x(nT )dt
≤T2nT +T
nT
˙x(t)TP˙x(t)dt.
Proposition 1.In the estimation error dynamics (26), there
exists some constant υk>0such that
∥ek(t)∥ ≤υk∥ek(nT )∥
with hk(x(t)) = 0 for t∈[nT , nT +T).
Proof. The estimation error dynamics (26) with hk(x(t)) = 0
can be rewritten as follows:
˙xk(t) = Ak(t)ek(t)−Lk(t)Ckek(nT ).(27)
Integrating from nT to tand taking the norm on both sides
of the dynamics (27) yields
∥ek(t)∥=

ek(nT ) + t
nT
Ak(τ)ek(τ)−Lk(τ)Ckek(nT )dτ


≤(1 + T bk)∥ek(nT )∥+akt
kT ∥ek(τ)∥dτ
where ak= supi∈Ir∥Aki∥and bk= supi∈Ir∥LkiCk∥.
Then, an application of the Gronwall-Bellman inequality to
∥ek(t)∥results in
∥ek(t)∥ ≤(1 + T bk) exp (akT)∥ek(nT )∥=υk∥ek(nT )∥.
Based on the above lemmas and proposition, the sufficient
condition is summarized for the decentralized sampled-data
fuzzy observer design via the exact discrete-time design ap-
proach in the following theorem.
Theorem 3.Consider the nonlinear interconnected system
(1) and the decentralized sampled-data fuzzy observer (25). If
there exist some scalars δkand βkand matrices Pk=PT
k≻0
and Lki such that the following optimal problem with the
LMIs is satisfed
min δksubject to






He{Φki}+I∗ ∗ ∗ ∗
−AT
kiPk+I−ρk
TPk+I∗ ∗ ∗
Φki PkAki −ϱkPk∗ ∗
1
TPk+ Φki PkAki 0−ηkI∗
ρkΦki ρkPkAki 0 0 −ηkI






≺0,(28)
Pk−βkI≺0(29)
where Φki =PkAki +Nki Ck,ϱk=1
(1+ρk)T,ηk=
1
2T1
qT λk
Hk
δk−(1 + ρk)βkand ρk>0is a given constant
for (k, i)∈ Iq× Ir, then the estimation error dynamics (26)
is asymptotically stable for hk(x(t)) = 0 and satisfies J≤0,
and √δkis the minimum IAR.
Proof. By integrating the estimation error dynamics (26) from
nT to nT +T, we have the exact discretized estimation error
dynamics as follows:
ek(nT +T) =ek(nT ) + nT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)
+hk(x(t))dt. (30)
Based on the exact discretized model (30), we consider
the discrete-time Lyapunov function candidate as V(nT) =
q
k=1 ek(nT )TPkek(nT ), then
∆V(nT ) + nT +T
nT
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)dt
=
q

k=1 ek(nT ) + nT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)
+hk(x(t))dtT
Pkek(nT ) + nT +T
nT
Φk(t)ek(nT )
+Ak(t)˜ek(t) + hk(x(t))dt−
q

k=1
ek(t)TPkek(t)
+nT +T
nT
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)dt. (31)
By applying Lemma 2, the equation (31) becomes
∆V(nT ) + nT +T
nT
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)dt
=
q

k=1 ek(nT ) + nT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)
+hk(x(t))dtT
Pkek(nT ) + nT +T
nT
Φk(t)ek(nT )
+Ak(t)˜ek(t) + hk(x(t))dt−
q

k=1
ek(t)TPkek(t)
+nT +T
nT
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)dt
+
q

k=1
1
TnT +T
nT nT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)
+hk(x(t))dtT
ρkPknT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)
+hk(x(t))dtdt −
q

k=1
1
TnT +T
nT
˜ek(t)TρkPk˜ek(t)dt
≤
q

k=1 nT +T
nT
ek(nT )TPkΦk(t)ek(nT ) + Ak(t)˜ek(t)dt

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. ?, NO. ?, ?? 2015 7
+
q

k=1 nT +T
nT Φk(t)ek(nT ) + Ak(t)˜ek(t)TPkek(nT )dt
−
q

k=1 nT +T
nT
˜ek(t)Tρk
TPk˜ek(t)
+
q

k=1 nT +T
nT ek(nT ) + ˜ek(t)Tek(nT ) + ˜ek(t)dt
+
q

k=1 nT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)dtT
×(1 + ρk)PknT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)dt
+σk
q

k=1 nT +T
nT
Ψk(t)ek(nT ) + Ak(t)˜ek(t)dtT
P2
k
×nT +T
nT
Ψk(t)ek(nT ) + Ak(t)˜ek(t)dt
+σk
q

k=1 nT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)dtT
ρ2
kP2
k
×nT +T
nT
Φk(t)ek(nT ) + Ak(t)˜ek(t)dt
+
q

k=1 nT +T
nT
hk(x(t))dtT2σ−1
kI+ (1 + ρk)Pk
×nT +T
nT
hk(x(t))dt−
q

k=1 nT +T
nT
γ2
qx(t)Tx(t)dt
(32)
where Ψk(t) = 1
TI+ Φk(t)and ρk>0is a given scalar.
Based on Lemma 1 and Assumption 1, we can further
majorize inequality (32) as follows:
∆V(nT ) + nT +T
nT
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)dt
≤
q

k=1 nT +T
nT
ϵk(t)TΥk(t) + ˜
Ψk(t)TσkT P 2
k˜
Ψk(t)
+˜
Φk(t)T(1 + ρk)T Pk+σkρ2
kT P 2
k˜
Φk(t)ϵk(t)dt
+
q

k=1 nT +T
nT
T2σ−1
k+ (1 + ρk)βkα2
kx(t)THT
kHkx(t)
−γ2
qx(t)Tx(t)dt
where
Υk(t) = PkΦk(t)+Φk(t)TPk+I∗
Ak(t)TPk+I−ρk
TPk+I,
˜
Φk(t) = Φk(t)Ak(t),˜
Ψk(t) = Ψk(t)Ak(t)
and ϵk(t) = col{ek(nT ),˜ek(t)}.
If there exists some scalar βksuch that
Pk≺βkI, then the sufficient condition for
∆V(nT ) + nT +T
nT q
k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)<0
is represented as
PkΦk(t)+Φk(t)TPk+I∗
Ak(t)TPk+Iρk
TPk+I
+Φk(t)Ak(t)T(1 + ρk)T PkΦk(t)Ak(t)
+Ψk(t)Ak(t)TσkT P 2
kΨk(t)Ak(t)
+Φk(t)Ak(t)Tσkρ2
kT P 2
kΦk(t)Ak(t)≺0,(33)
T2σ−1
k+ (1 + ρk)βkα2
kλk
Hk−γ2
q= 0.(34)
Applying the schur complement to (32) yields






Υ11
k(t)∗ ∗ ∗ ∗
Υ21
k(t) Υ22
k∗ ∗ ∗
PkΦk(t)PkAk(t)−ϱkPk∗ ∗
PkΨk(t)PkAk(t) 0 −1
Tσ−1
kI∗
ρkPkΦk(t)ρkPkAk(t) 0 0 −1
Tσ−1
kI






≺0(35)
where Υ11
k(t) = PkΦk(t)+Φk(t)TPk+I,Υ21
k(t) =
Ak(t)TPk+Iand Υ22
k=−ρk
TPk+I.
By substituting (34) into (35), denoting PkLki =Nki and
γ2/α2
k=δkand using the fuzzy property, we can determine
that, if the LMI (28) is satisfied, it follows that
∆V(nT ) + nT +T
nT ek(t)Tek(t)−γ2
qx(t)Tx(t)<0.
(36)
Then, we get J < 0by summing the inequality (36) from
0to ∞. Also, by Proposition 1, the LMI (28) satisfies the
asymptotic stability of the estimation error dynamics (26) with
hk(x(t)) = 0.
We also address the sufficient condition for the non-
measurable premise variable case in the following corollary.
Corollary 3.Consider the nonlinear interconnected sys-
tem (1) with the sampled-data output and the decentralized
sampled-data fuzzy observer with the non-measurable premise
variable. If there exist some scalars δkand βkand matrices
Pk=PT
k≻0and Lki such that the following optimal
problem with the LMIs is satisfied:
min δksubject to






He{Φki}+I∗ ∗ ∗ ∗
−AT
kiPk+I−ρk
TPk+I∗ ∗ ∗
Φki PkAki −ϱkPk∗ ∗
1
TPk+ Φki PkAki 0−ωkI∗
ρkΦki ρkPkAki 0 0 −ωkI






≺0,(37)
Pk−βkI≺0,(38)
Pk−λk
Hk
λk
Λkij
βkI≺0(39)
where ωk=1
T1
4qT (λk
Hk+λk
Λkij )δk−(1+ρk)λk
Hk
λk
Hk+λk
Λkij
βkfor
(k, i, j)∈ Iq× Ir× Ir, then the decentralized sampled-data
fuzzy observer well estimates the nonlinear interconnected
system (1), and δk=γ2
α2
k+1 .
Remark 8.It is noted that
•By defining the IAR, the effective observer design tech-
nique is provided for oscillating or unstable uncertain
systems.

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. ?, NO. ?, ?? 2015 8
•The decentralized fuzzy observer is designed for the
nonlinear interconnected system with the unknown inter-
connection. Also, the decentralized fuzzy observer design
problem is converted to the optimal problem for the
minimization of the IAR.
•The decentralized fuzzy observers are designed for not
only the measurable premise variable case, but also the
non-measurable premise variable case.
•Through the approximate discretization and the ex-
act discrete-time design approaches, the decentralized
sampled-data observer design techniques are proposed,
respectively.
•The proposed decentralized fuzzy observer can be used
in the various interconnected systems, such as power
systems, transportation systems, industrial process. Also,
by using the proposed decentralized sampled-data fuzzy
observer design techniques, the effective observer can
be designed for the various computer- or network-based
systems.
V. SI MU LATI ON E XA MP LE
To validate the proposed decentralized sampled-data fuzzy
observer design techniques, we consider three simulation ex-
amples.
A. Example 2
First, we consider two Van der Pol oscillators used in Exam-
ple 1. To design the decentralized sampled-data fuzzy observer,
we assume a sampling period T= 0.1and the constant
ρk= 1 for Theorem 3. By solving the corresponding LMIs,
we obtain the decentralized sampled-data fuzzy observer gains,
respectively:
LT hm.2
11 =−8.2651
1.2392 , LT hm.2
12 =−8.2278
1.9794 ,
LT hm.2
21 =−3.0380
1.0407 , LT hm.2
22 =−3.0218
1.3676 ,
LT hm.3
11 =−10.7635
1.9895 , LT hm.3
12 =−10.7599
9.0007 ,
LT hm.3
21 =−6.4539
3.9023 , LT hm.3
22 =−6.4539
7.2604 .
Time responses of each subsystem are shown in Figs. 1, 2,
3 and 4 with αk= 0.1. From the figures, especially Figs 1
and 3, we can guarantee that both Theorems 2 and 3 guar-
antee the prominent observing performance for the nonlinear
interconnected system composed of fuzzy subsystems and the
unknown interconnection. Also, to highlight the superiority of
the proposed techniques, the results of performance measure
function (9) for each proposed technique and other techniques
are shown in Table 2. In Table 2, the first is the discrete-
time fuzzy observer design technique for the approximately
discretized model without considering the interconnection and
obtained by the sufficient condition
−Pk∗
PkGki +Nki Ck−Pk≺0,
and the second is sampled-data observer techniques with
approximate discretization. Also, the third is the sampled-
data fuzzy observer technique with exactly discretized model.
As shown in the table, the proposed techniques have better
performances that other techniques. Also, Theorem 3 has
better result than Theorem 2, because the discretization error
exists in Theorem 2 but does not exist in Theorem 3.
TABLE II
PER FOR MA NCE C OMPA RIS ON F OR TW O VAN DER POL OSCILLATORS.
Interconnection bound αk0.1 0.5 1
Discrete-time fuzzy observer 10.8311 10.7471 10.6608
Theorem 2 in [23] 10.5807 13.8111 17.7640
Theorem 1 in [41] Marginal Infeasibility
Theorem 2 8.8096 8.0794 8.3160
Theorem 3 4.3042 5.7898 8.0179
0246810
−8
−6
−4
−2
0
2
4
time
x11(t)
Fig. 1. The state variable x11(t)for two Van der Pol oscillators: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), and the
discrete-time fuzzy observer (circled).
0246810
−3
−2
−1
0
1
2
3
time
x12(t)
Fig. 2. The state variable x12(t)for two Van der Pol oscillators: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), and the
discrete-time fuzzy observer (circled).

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI
10.1109/TFUZZ.2015.2470564, IEEE Transactions on Fuzzy Systems
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. ?, NO. ?, ?? 2015 9
0246810
−3
−2
−1
0
1
2
3
time
x21(t)
Fig. 3. The state variable x21(t)for two Van der Pol oscillators: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), and the
discrete-time fuzzy observer (circled).
0246810
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
time
x22(t)
Fig. 4. The state variable x22(t)for two Van der Pol oscillators: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), and the
discrete-time fuzzy observer (circled).
B. Example 3
In order to further emphasize the advantage of the proposed
techniques, we simulate the double mass-spring systems [42],
[43] connected by a spring as follows:
¨
ζk(t) = −fk(ζk(t)) −hk(ζ(t)) (40)
yk(t) =ζk(nT )(41)
where
f1(ζ1(t)) =0.01ζ1(t)+0.67ζ1(t)3,
f2(ζ2(t)) =0.02ζ2(t)+0.63ζ2(t)3,
and hk(ζ(t)) = αk(0.1ζk(t)−0.1ζl(t)), which is the in-
terconnection function by the spring constant αk/10, with
ζ(t) = [ζ1(t)Tζ2(t)T]Tfor {(k, l)∈ I2|k̸=l}.
Then, by choosing xk(t)=[xk1(t)xk2(t)]T=
[ζk(t)˙
ζk(t)]Tand using the membership function,
µk1(ζk(t)) = 1 −ζk(t)2and µk2(ζk(t)) = ζk(t)2the
double mass-spring system can be represented by the
following fuzzy model:
˙xk(t) =
2

i=1
µki(xk1(t))Ak ixk(t) + αkHk(x(t))
yk(t) =Ckxk(nT )
where
A11 =0 1
−0.01 0, A12 =0 1
−0.68 0,
A21 =0 1
−0.02 0, A22 =0 1
−0.65 0,
H1=0 0 0 0
−0.100.1 0,
H2=0 0 0 0
0.1 0 −0.1 0,
Ck=1 0
By assuming a sampling period T= 0.05 and the constant
ρk= 0.18, choosing the initial condition x1(0) = [0.8 0]T,
x2(0) = [0.7 0]Tand ˆxk(0) = [0 0]T, and solving the corre-
sponding LMIs in Theorem 2 and 3, we get the decentralized
sampled-data fuzzy observer gains:
LT hm.2
11 =1.6602
13.2083, LT hm.2
12 =1.6591
13.1630,
LT hm.2
21 =1.6585
13.1722, LT hm.2
22 =1.6574
13.1296,
LT hm.3
11 =33.1177
100.5884, LT hm.3
12 =33.1208
98.7771,
LT hm.3
21 =33.0713
100.4322, LT hm.3
22 =33.0722
98.7304.
Time responses of each subsystem are shown in Figs. 5, 6, 7
and 8 with αk= 10 for the previous and proposed techniques.
As shown in figures, especially Figs. 6 and 8, we can guarantee
not only that both Theorems 2 and 3 guarantee the prominent
observing performance than the previous observer design tech-
niques, but also that Theorem 3 has better performance than
Theorem 2. The results of the performance measure function
with the time period from 1sto 20sis shown in Table 3.
TABLE III
PER FOR MA NCE C OMPA RIS ON F OR TH E DO UBL E MA SS-S PR ING S YS TEM S.
Interconnection bound αk1 5 10
Discrete-time fuzzy observer 2.1478 2.3202 2.5784
Theorem 2 in [23] 4.6137 5.1896 5.7446
Theorem 1 in [41] 1.5247 1.9052 2.3393
Theorem 2 1.2367 1.2424 1.2470
Theorem 3 0.2270 0.4883 0.8680
C. Example 4
Now, we simulate the double Chua’s circuit systems [44]
connected by a resistor as follows:
˙vk1(t) = 1
Ck11
Rk
(vk2(t)−vk1(t)) −fk(vk1(t))

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0 5 10 15 20
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time
x11(t)
Fig. 5. The state variable x11(t)for the double mass-spring systems: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), the
discrete-time fuzzy observer (circled), and [41] (starred).
0 5 10 15 20
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
time
x12(t)
Fig. 6. The state variable x12(t)for the double mass-spring systems: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), the
discrete-time fuzzy observer (circled), and [41] (starred).
+1
Ra
(vl1(t)−vk1(t))
˙vk2(t) = 1
Ck21
Rk
(vk1(t)−vk2(t)) −ik(t)
˙
ik(t) = −1
Lk
vk2(t)
yk(t) =yk(nT ) = vk1(nT )
where {(k, l)∈ I2|k̸=l},vk1(t),vk2(t)and ik(t)are
voltages and current of the kth Chua’s circuit, Rkis a resistor
with R1= 100mΩand R2= 83.33mΩ,Ck1and Ck2are
capacitors with Ck1=Ck2= 1F,Lkis an inductor with
L1= 67.25mH and L2= 59.28mH,Rais an unknown
resistor connecting the first and second Chua’s circuits, and
fk(vk1(t)) = gb
Rkvk1(t) + (ga−gb)
2Rk|vk1(t) + 1|−|vk1(t)−1|
is a Chua’s diode with ga=−1.27 and gb=−0.68.
By choosing xk(t) = [xk1(t)xk2(t)xk3(t)]T=
[vk1(t)vk2(t)ik(t)]T, the T–S fuzzy system of the kth
0 5 10 15 20
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
time
x21(t)
Fig. 7. The state variable x21(t)for the double mass-spring systems: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), the
discrete-time fuzzy observer (circled), and [41] (starred).
0 5 10 15 20
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
time
x22(t)
Fig. 8. The state variable x22(t)for the double mass-spring systems: original
(solid), Theorem 2 (dashed), Theorem 3 (dash-dotted), [23] (dotted), the
discrete-time fuzzy observer (circled), and [41] (starred).
subsystem can be constructed as follows:
˙xk(t) =
2

i=1
µki(xk1(t))Akixk(t) + hk(x(t))
yk(t) =Ckxk(nT )
where
Ak1=

(d−1)σk1σk10
1−1 1
0−σk20
,
Ak2=
−(d+ 1)σk1σk10
1−1 1
0−σk20
,
Ck=1 0 0,
µk1(xk1(t)) = 


1
21−fk(xk1(t))
dxk1(t), xk1(t)̸= 0,
1
21−ga
d, xk1(t) = 0,
µk2(xk1(t)) =1 −µk1(xk1(t)),

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. ?, NO. ?, ?? 2015 11
and σ11 = 10,σ12 = 14.87,σ21 = 12,σ22 = 16.87 and
d= 1.8.
Also, the interconnection function hk(x(t)) is represented as
hk(x(t)) = αkHkx(t)where αkis the inverse of an unknown
resistor and
H1=

−100100
0 00000
0 00000
,
H2=

100−100
0 0 0 0 0 0
0 0 0 0 0 0
.
By assuming a sampling period T= 0.01 and choosing
the constant ρk= 1 and solving the corresponding LMIs in
Theorem 2 and 3, we get the decentralized sampled-data fuzzy
observer gains:
LT hm.2
11 =

1.2886
1.9827
9.4046
, LT hm.2
12 =

0.9286
1.9809
9.4049
,
LT hm.2
21 =

1.3408
1.8997
9.9389
, LT hm.2
22 =

0.9080
1.8937
9.9191
,
LT hm.3
11 =

89.0382
112.2865
500.5009
, LT hm.3
12 =

46.0523
126.7138
500.5009
,
LT hm.3
21 =

89.0382
112.2865
471.0810
, LT hm.3
22 =

41.4931
112.2713
471.0212
.
The results of the performance measure function (9) are shown
in Table 4 and Figure 9.
TABLE IV
PERFORMANCE COMPARISON FOR THE DOUBLE CHUA’S CIRCUIT
SY STE MS W IT H T= 0.01.
Interconnection bound αk1 5 10
Discrete-time fuzzy observer 40.0166 38.5414 39.3113
Theorem 2 in [23] 47.1717 64.1046 79.8577
Theorem 1 in [41] Marginal Infeasibility
Theorem 2 14.6220 21.5924 20.1650
Theorem 3 3.7043 7.5373 8.9492
Also, with T= 0.02, we get
LT hm.2
11 =

1.5477
1.7563
6.3948
, LT hm.2
12 =

0.8372
1.7464
6.3842
,
LT hm.2
21 =

1.6692
1.7526
7.1414
, LT hm.2
22 =

0.8197
1.7435
7.1407
,
LT hm.3
11 =

40.8508
20.5834
10.7443
, LT hm.3
12 =

4.7681
20.5834
10.7445
,
LT hm.3
21 =

42.1618
16.9057
19.1499
, LT hm.3
22 =
−1.3544
16.9052
19.1508 
.
and the results of the performance measure function in Table
5 and Figure 10.
0246810
0
1
2
3
4
5
6
7
8
9
10
time
P(t)
Fig. 9. Time response of the performance measure function P(t)with
T= 0.01: the discrete-time fuzzy observer (solid), Theorem 2 (dashed),
and Theorem 3 (dash-dotted).
TABLE V
PERFORMANCE COMPARISON FOR THE DOUBLE CHUA’S CIRCUIT
SYSTEMS WITH T= 0.02.
Interconnection bound αk1 5 10
Discrete-time fuzzy observer 86.8068 68.9134 77.8810
Theorem 2 in [23] 53.4539 58.3127 84.4254
Theorem 1 in [41] Marginal Infeasibility
Theorem 2 25.5928 32.4931 29.8237
Theorem 3 3.2401 7.5603 8.8276
From the tables and figures, we know that the proposed
techniques are suitable for the oscillating interconnected sys-
tem, and Theorem 3 has better performance than Theorem 2.
VI. CO NC LU SI ON S
In this paper, the decentralized sampled-data fuzzy observer
design techniques have been proposed for the nonlinear inter-
connected system. Prior to the design of the sampled-data ob-
server, the decentralized fuzzy observer design technique was
presented by using IAR concept, and it has been shown that
the proposed technique has better performance than previous
techniques through a simple example. To obtain the decentral-
ized sampled-data observer, the approximate discretization and
the exact discrete-time design approaches were used, and their
sufficient conditions were derived in the optimal problems with
LMI format, respectively. Finally, some numerical examples
have been provided to demonstrate the validity of the proposed
methods by comparing with previous techniques.
Future works: To approve the IAR performance, the de-
centralizes sampled-data observer will be designed with the
relaxed sufficient condition by using the Lyapunov-Krasovskii
functional approach [45], [46] and non-PDC methods [47],
[48]. Also, the proposed methodologies are applicable in other
decentralized technique fields, such as the observer-based
control, filter design, and fault detection.
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APP EN DI X
From the nonlinear interconnected system (1) and the de-
centralized fuzzy observer (10), the estimation error dynamics
with the non-measurable premise variable is represented as
follows:
˙ek(t) = ˆ
Φk(t)ek(t) + ˜
Λk(t)x(t) + hk(x(t)) (42)
where ˆ
Φk(t) = ˆ
Ak(t)−ˆ
Lk(t)Ckand ˜
Λk(t) = [0 ··· Ak(t)−
ˆ
Ak(t) 0].
Based on the estimation error dynamics (42), we
consider the Lyapunov function candidate as V(t) =
q
k=1 ek(t)TPkek(t), then
˙
V(t) +
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)
=
q

k=1 ˆ
Φk(t)ek(t) + ˜
Λk(t)x(t) + hk(x(t))TPkek(t)
+
q

k=1
ek(t)TPkˆ
Φk(t)ek(t) + ˜
Λk(t)x(t) + hk(x(t))
+
q

k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)
≤
q

k=1
ek(t)Tˆ
Φk(t)TPk+Pkˆ
Φk(t) + I
+σk1 + λk
Λkij
λk
HkP2
kek(t)
+
q

k=1 σ−1
kα2
kx(t)THT
kHkx(t)
+σ−1
k
λk
Hk
λk
Λkij
x(t)T˜
Λk(t)T˜
Λk(t)x(t)−γ2
qx(t)Tx(t).
(43)
From (43), the sufficient condition for ˙
V(t) +
q
k=1 ek(t)Tek(t)−γ2
qx(t)Tx(t)<0is arranged
as
ˆ
Φk(t)TPk+Pkˆ
Φk(t) + I+σk1 + λk
Λkij
λk
HkP2
k≺0,(44)
σ−1
kα2
kλk
Hk+σ−1
k
λk
Hk
λk
Λkij
λk
Λkij −γ2
q= 0.(45)
By applying the schur complement to (44), substituting (45)
into (44), denoting PkLki =Nki and γ2
α2
k+1 =δkand using
the fuzzy property, we can obtain the LMI (11). Also, we can
easily know that the LMI (11) satisfies the asymptotic stability
of the estimation error dynamics (42) with hk(x(t)) = 0 and
˜
Λk(t) = 0.
Geun Bum Koo received the B.S. and Ph.D. de-
grees in Electrical and Electronic Engineering from
Yonsei University, Seoul, Korea in 2007 and 2015,
respectively.
His current research interests include large-scale
systems, decentralized control, sampled-data control,
digital redesign, nonlinear control, and fuzzy sys-
tems.
Jin Bae Park received the B.S. degree in electrical
engineering from Yonsei University, Seoul, Korea,
and the M.S. and Ph.D. degrees in electrical engi-
neering from Kansas State University, Manhattan,
KS, USA, in 1977, 1985, and 1990, respectively.
Since 1992, he has been with the Department
of Electrical and Electronic Engineering, Yonsei
University, where he is currently a Professor. His
major research interests include robust control and
filtering, nonlinear control, intelligent mobile robot,
drone, fuzzy logic control, neural networks, adaptive
dynamic programming, chaos theory, and genetic algorithms.
Dr. Park served as the Editor-in-Chief for the International Journal of
Control, Automation, and Systems (IJCAS) (2006-2010) and the President
for the Institute of Control, Robot, and Systems Engineers (ICROS) (2013).
He is currently serving as the Senior Vice-President for Yonsei University.
Young Hoon Joo received his B.S., M.S., and
Ph.D. degrees in Electrical Engineering from Yonsei
University, Seoul, Korea, in 1982, 1984, and 1995,
respectively.
He worked with Samsung Electronics Company,
Seoul, Korea, from 1986 to 1995, as a project
manager. He was with the University of Houston,
Houston, TX, from 1998 to 1999, as a visiting
professor in the Department of Electrical and Com-
puter Engineering. He is currently a professor in the
Department of Control and Robotics Engineering,
Kunsan National University, Korea. His major interest is mainly in the field
of intelligent robot, robot vision, intelligent control, human-robot interaction,
wind-farm control, and intelligent surveillance systems. He served as President
for Korea Institute of Intelligent Systems (KIIS) (2008-2009) and the Vice-
President for the Korean Institute of Electrical Engineers (KIEE) (2013-2014),
and is serving as Editor-in-Chief for the International Journal of Control,
Automation, and Systems (IJCAS) (2014-present).