Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 402480, 25 pages
doi:10.1155/2012/402480
Research Article
Robust H∞ Filtering for Uncertain Discrete-Time
Fuzzy Stochastic Systems with Sensor
Nonlinearities and Time-Varying Delay
Mingang Hua,1, 2, 3 Pei Cheng,4 Juntao Fei,1
Jianyong Zhang,5 and Junfeng Chen1
1
College of Computer and Information, Hohai University, Changzhou 213022, China
Changzhou Key Laboratory of Sensor Networks and Environmental Sensing, Changzhou 213022, China
3
Jiangsu Key Laboratory of Power Transmission and Distribution Equipment Technology,
Changzhou 213022, China
4
School of Mathematical Sciences, Anhui University, Hefei 230601, China
5
Department of Mathematics and Physics, Hohai University, Changzhou 213022, China
2
Correspondence should be addressed to Mingang Hua, mghua@yahoo.cn
Received 10 October 2012; Revised 7 December 2012; Accepted 11 December 2012
Academic Editor: Hak-Keung Lam
Copyright q 2012 Mingang Hua et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The robust filtering problem for a class of uncertain discrete-time fuzzy stochastic systems with
sensor nonlinearities and time-varying delay is investigated. The parameter uncertainties are
assumed to be time varying norm bounded in both the state and measurement equations. By
using the Lyapunov stability theory and some new relaxed techniques, sufficient conditions are
proposed to guarantee the robustly stochastic stability with a prescribed H∞ performance level of
the filtering error system for all admissible uncertainties, sensor nonlinearities, and time-varying
delays. These conditions are dependent on the lower and upper bounds of the time-varying delays
and are obtained in terms of a linear matrix inequality LMI. Finally, two simulation examples are
provided to illustrate the effectiveness of the proposed methods.
1. Introduction
Fuzzy systems in the Takagi-Sugeno T-S model can represent a lot of complex nonlinear
systems in 1–3. By using a T-S fuzzy plant model, one can describe a nonlinear system
as a weighted sum of some simple linear subsystems. This fuzzy model is described by a
family of fuzzy if-then rules that represent local linear input/output relations of the system.
The overall fuzzy model of the system is achieved by smoothly blending these local linear
models together through membership functions. Consequently, stability, control, and filtering
problems for T-S fuzzy systems have attracted considerable attention, and many important
results have been reported in 4–9.
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Journal of Applied Mathematics
It has been known that time delays are one of the main causes of instability and poor
performance of control systems in 10. Analysis and synthesis of such systems are of both
theoretical and practical importance. Therefore, the study of time-delay systems has attracted
great attention over the past few years in 11, 12. For continuous-time systems, the obtained
results can be generally classified into two types: delay-independent and delay-dependent
ones. It has been understood that the latter is generally less conservative since the size of
delays is considered, especially when time delays are small. Compared with continuoustime systems with time-varying delays, the discrete-time counterpart receives relatively less
attention. See, for example, 13–16 and references therein.
In the past few years, much research effort has been paid to the control and filtering
problems for nonlinear systems that have been widely applied in many fields such as communication network, image processing, and mobile robot localization 17, 18. And the filtering
for nonlinear stochastic systems has been of great interest since stochastic modeling has come
to play an important role in many branches of engineering applications 19–22. particularly
for discrete-time stochastic systems, so far, a number of important results have been reported
for linear discrete-time stochastic systems. The exponential filtering problem is studied for
discrete time-delay stochastic systems with Markovian jump parameters and missing measurements in 23. The robust fault detection filter problem for fuzzy It
o stochastic systems
is studied in 24. The problem of robust H∞ filtering for uncertain discrete-time stochastic
systems with time-varying delays is considered in 25. For nonlinear discrete-time stochastic
systems, the filtering problem for a class of nonlinear discrete-time stochastic systems with
state delays is considered in 26. The robust H∞ filtering problem for a class of nonlinear discrete time-delay stochastic systems is considered in 27. The H∞ filtering problem for a general class of nonlinear discrete-time stochastic systems with randomly varying sensor delays
is considered in 28. Recently, the filtering problem for discrete-time fuzzy stochastic systems
with sensor nonlinearities is considered in 29. And the problem of H∞ filtering for discretetime Takagi-Sugeno T-S fuzzy It
o stochastic systems with time-varying delay is studied
in 30. In 27, 29, 31, 32, the nonlinearity for filtering problem of systems was assumed to
satisfy sensor nonlinearities, which may included actuator saturation and sensor saturation.
It is worth mentioning that, although the system in 29 is with sensor nonlinearities, the proposed filter design approach do not consider the discrete-time fuzzy stochastic systems with
time delay, which is not applicable to stochastic delay systems. And in 30, the stochastic systems do not contain the sensor nonlinearities. To the best of our knowledge, no results on H∞
filtering for the uncertain discrete-time fuzzy stochastic systems with both sensor nonlinearities and time-varying delay are available in the literature, which motivates the present study.
Motivated by the works in 25, 27, 29, 30, in this paper, a delay-dependent H∞ performance analysis result is established for filtering error systems. A new uncertain discrete-time
fuzzy stochastic systems model is proposed, and a new different Lyapunov functional is then
employed to deal with systems with sensor nonlinearities and time-varying delay. As a result,
the H∞ filters are designed in terms of linear matrix inequalities LMIs. The resulting filters
can ensure that the filtering error system is robustly stochastic stable and the estimation error
is bounded by a prescribed level for all possible bounded energy disturbances. Finally, two
examples are given to show the effectiveness of the proposed method.
Throughout this paper, Rn denotes the n-dimensional Euclidean space, and Rn×m is
the set of n × m real matrices. I is the identity matrix. | · | denotes Euclidean norm for vectors
and || · || denotes the spectral norm of matrices. N denotes the set of all natural numbers,
that is, N {0, 1, 2, . . .}. Ω, F, {Fk }k∈N , P is a complete probability space with a filtration
{Fk }k∈N satisfying the usual conditions. MT stands for the transpose of the matrix M. For
Journal of Applied Mathematics
3
symmetric matrices X and Y , the notation X > Y resp., X ≥ Y means that the X − Y is
positive definite resp., positive semidefinite. ∗ denotes a block that is readily inferred by
symmetry. E{·} stands for the mathematical expectation operator with respect to the given
probability measure P.
2. Problem Description
Consider a class of uncertain discrete-time stochastic systems that can be approximated by
the following time-delay T-S fuzzy stochastic model with r plant rules.
Plant rule i:
if θ1 k is η1i and . . . and θp k is ηpi ,
then
xk 1 Ai ΔAi kxk Adi ΔAdi kxk − τk Bi ΔBi kvk
Ei ΔEi kxk Edi ΔEdi kxk − τk Gi ΔGi kvkwk,
yk φCi kxk Cdi ΔCdi kxk − τk Di ΔDi kvk,
zk Li xk,
i 1, 2, . . . , r,
2.1
where ηji is the fuzzy set. θt θ1 t, θ2 t, . . . , θp tT is the premise variable vector,
xk ∈ Rn is the state vector, yk ∈ Rq is the measurable output vector, zk ∈ Rr is the state
combination to be estimated, wk is a real scalar process on a probability space Ω, F, P
relative to an increasing family Fk k∈N of σ-algebra Fk ⊂ F generated by wkk∈N , and N
is the set of natural numbers. The stochastic process {wk} is independent, which is assumed
to satisfy
E{wk} 0,
E w2 k 1,
E wiw j 0
i
/j ,
2.2
where the stochastic variables w0, w1, w2, . . . are assumed to be mutually independent.
The exogenous disturbance signal vk ∈ Rp is assumed to belong to Le2 0 ∞, Rp , and
is Fk−1 measurable for all k ∈ N, where Le2 0 ∞, Rp denotes the space of k-dimensional
nonanticipatory square summable stochastic process f· fkk∈N on N with respect to
Fk∈N satisfying
∞ ∞ 2
2 2
f E
fk E fk < ∞.
e2
k0
2.3
k0
The time-varying delay τk satisfies
τ1 ≤ τk ≤ τ2 ,
2.4
4
Journal of Applied Mathematics
where τ1 and τ2 are known positive integers representing the minimum and maximum delays, respectively.
In addition, Ai , Adi , Bi , Ei , Edi , Gi , Ci , Cdi , Di , and Li are known real constant matrices,
ΔAi k, ΔAdi k, ΔBi k, ΔEi k, ΔEdi k, ΔGi k, ΔCdi k, and ΔDi k are unknown matrices representing time-varying parameter uncertainties, and the admissible uncertainties are
assumed to be modeled in the form
ΔAi k ΔEi k M1i M2i F1i kN1i ,
ΔAdi k ΔEdi k ΔCdi k M3i M4i M5i F2i kN2i ,
ΔBi k ΔGi k ΔDi k M6i M7i M8i F3i kN3i ,
2.5
2.6
2.7
where M1i , M2i , M3i , M4i , M5i , M6i , M7i , and M8i are known constant matrices and F1i k,
F2i k, and F3i k are unknown time-varying matrices with Lebesgue measurable elements
bounded by
T
F1i
kF1i k ≤ I,
T
F2i
kF2i k ≤ I,
T
F3i
kF3i k ≤ I,
∀k.
2.8
The defuzzified output of the T-S fuzzy system 2.1 is inferred as follows:
xk 1 r
hi θk{Ai ΔAi kxk Adi ΔAdi kxk − τk Bi ΔBi kvk
i1
Ei ΔEi kxk Edi ΔEdi kxk − τk
Gi ΔGi kvkwk},
yk r
hi θk φCi kxk Cdi ΔCdi kxk − τk Di ΔDi kvk ,
i1
r
zk hi θkLi xk,
i1
where hi θk μi θk/
r
j1
hj θk, and μi θk 2.9
p
i
j1 ηj θj k,
in which ηji θj k is
the grade of membership of ηj k in ηji . According to the theory of fuzzy sets, we have
μi θk ≥ 0, i 1, 2, . . . , r and ri1 hi θk > 0. Therefore, it is implied that hi θk ≥ 0,
r
i 1, 2, . . . , r and i1 hi θk 1.
For some given diagonal matrices K1 ≥ 0 and K2 ≥ 0 with K2 > K1 , the nonlinear
vector functions φ ∈ K1 , K2 , is assumed to represent sensor nonlinearities and satisfies the following sensor condition 29:
φξ − K1 ξ
T φξ − K2 ξ ≤ 0,
∀ξ ∈ Rq .
2.10
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5
Further, the nonlinear function φξ can be decomposed into a linear and a nonlinear part as
follows:
φξ φs ξ K1 ξ,
2.11
where the nonlinearity φs ξ belongs to the set Φs given by
Φs φs : φsT ξ φs ξ − Kξ ≤ 0
2.12
with K K2 − K1 > 0.
We consider the following fuzzy filter for the estimation of zk :
xk
1 r
hi θk Afi xk
Bfi yk ,
i1
zk r
2.13
hi θkLi xk,
i1
where xk
∈ Rn , zk ∈ Rr , and the matrices Afi and Bfi i 1, 2, . . . , r are to be determined.
Remark 2.1. Similar to 27, 29, the sensor nonlinearities satisfying 2.10 are also considered
in this paper. Here, there exists the nonlinear function φξ in the system 2.1, which is
called sensor nonlinearities. It is noted that in the previous filter, the matrix Li is assumed
to be constant in order to avoid more verbosely mathematical derivation.
Defining xk
xk−xk,
zk zk−
zk, ek xT k xT kT , and augmenting
the model 2.9 to include the states of the filter 2.13, we obtain the following filtering error
system:
ek 1 r
r hi θkhj θk Aij ΔAi k ek Bfj φs Ci ek
i1 j1
Adij ΔAdij k Hek − τk Bij ΔBij k vk
Ei ΔEi k ek Edi ΔEdi k Hek − τk
Gi ΔGi k vk wk ,
2.14
zk r
r hi θkhj θkLij ek,
i1 j1
2.15
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Journal of Applied Mathematics
where
0
Ai
0
Bfj Ci Ci 0 ,
,
,
Ai − Afj − Bfj K1 Ci Afj
−Bfj
Adi
Bi
E 0
Adij Bij Ei i
,
,
,
Adi − Bfj Cdi
Bi − Bfj Di
Ei 0
E
Gi
Edi di ,
Gi Lij 0 Li − Lj ,
,
H I 0 ,
Edi
Gi
Aij ΔAi k MAi F1i kN 1i ,
ΔAdij k MAdij F2i kN2i ,
ΔBij k MBij F3i kN3i ,
ΔEi k MEi F1i kN 1i ,
ΔEdi k MEdi F2i kN2i ,
ΔGi k MGi F3i kN3i ,
M1i
M3i
M6i
MAi MAdij MBij ,
,
,
M1i
M3i − Bfj M5i
M6i − Bfj M8i
M2i
M4i
M7i
MEi MEdi MGi N 1i N1i 0 .
,
,
,
M2i
M4i
M7i
2.16
The H∞ filtering problem to be addressed in this paper can be formulated as follows.
Given discrete-time fuzzy stochastic systems 2.9, a prescribed level of noise attenuation
γ > 0, and any φ ∈ K1 K2 , find a suitable filter in the form of 2.13 such that the following
requirements are satisfied.
1 The filtering error system 2.14-2.15 with vk 0 is said to be robustly stochastic
stable if there exists a scalar c > 0 such that for all admissible uncertainties satisfying
2.5–2.8
E
∞
|xk|
2
≤ cE |x0|2 ,
2.17
k0
where xk denotes the solution of stochastic systems with initial state x0.
2 For the given disturbance attenuation level γ > 0 and under zero initial conditions
for all vk ∈ Le2 0 ∞, Rp , the performance index γ satisfies the following
inequality:
zk − zke2 < γvke2 .
2.18
Before concluding this section, we introduce the following lemmas, which will be used
in the derivation of our main results in the next section.
Lemma 2.2 Gu et al. 33. Given constant matrices Γ1 , Γ2 , and Γ3 with appropriate dimensions,
where ΓT1 Γ1 and ΓT2 Γ2 > 0, then
Γ1 ΓT3 Γ−1
2 Γ3 < 0
2.19
Journal of Applied Mathematics
7
if and only if
Γ1 ΓT3
< 0,
Γ3 −Γ2
or
−Γ2 Γ3
< 0.
ΓT3 Γ1
2.20
Lemma 2.3 see 34. For given matrices H, E, and Ft with F T tFt ≤ I and scalar ε > 0, the
following inequality holds:
HFtE ET F T tH T ≤ εHH T ε−1 ET E.
2.21
3. Main Results
Theorem 3.1. Consider the uncertain discrete-time fuzzy stochastic systems in 2.1. A filter of form
2.13 and constants τ1 and τ2 , the filtering error system 2.14-2.15 is robustly stochastic stable
with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any matrices AFj , BFj , scalars
ε1i > 0, ε2i > 0 and ε3i > 0 for i, j 1, 2, . . . , r such that the following LMI is satisfied:
⎡
Σ11 0
0
0 CiT K T ATi X Σ17 EiT X EiT Y
⎢ ∗ Σ22 0
0
0
0
ATFj
0
0
⎢
⎢
T
T
T
0
Adi X Σ37 Edi X Edi Y
∗ Σ33 0
⎢∗
⎢
⎢
∗
∗ Σ44
0
BiT X Σ47 GTi X GTi Y
⎢∗
⎢
⎢
T
⎢∗
∗
∗
∗
−2I
0 −BFj
0
0
⎢
⎢∗
∗
∗
∗
∗
−X
0
0
0
⎢
⎢∗
∗
∗
∗
∗
∗
−Y
0
0
⎢
⎢
∗
∗
∗
∗
∗
∗
−X
0
⎢∗
⎢
⎢∗
∗
∗
∗
∗
∗
∗
∗
−Y
⎢
⎢∗
∗
∗
∗
∗
∗
∗
∗
∗
⎢
⎣∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0
0
0
0
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
0
0
0 ⎥
⎥
⎥
0
0
0 ⎥
⎥
XM1i XM3i XM6i ⎥
⎥ < 0,
Y M1i Σ711 Σ712 ⎥
⎥
⎥
XM2i XM4i XM7i ⎥
⎥
Y M2i Y M4i Y M7i ⎥
⎥
−ε1i I
0
0 ⎥
⎥
0 ⎦
∗
−ε2i I
∗
∗
−ε3i I
3.1
where
T
Σ11 −X τ2 − τ1 1Q ε1i N1i
N1i ,
T
Σ33 −Q ε2i N2i
N2i ,
T
Σ44 −γ 2 I ε3i N3i
N3i ,
T
Σ17 ATi Y − ATFj − CiT K1T BFj
,
Σ711 Y M3i − BFj M5i ,
T Σ22 −Y Li − Lj
Li − Lj ,
T T
Σ37 ATdi Y − Cdi
BFj ,
T
Σ47 BiT Y − DiT BFj
,
3.2
Σ712 Y M6i − BFj M8i .
Moreover, if the previous condition is satisfied, an acceptable state-space realization of the H∞ filter is
given by
Afj Y −1 AFj ,
Bfj Y −1 BFj .
3.3
8
Journal of Applied Mathematics
Proof. We first establish the condition of robustly stochastic stability for the filtering error
system 2.14-2.15. It can be shown that LMI 3.1 implies
⎡
Σ11
⎢∗
⎢
⎢
⎢∗
⎢
⎢∗
⎢
⎢∗
⎢
⎢∗
⎢
⎢
⎢∗
⎢
⎢∗
⎢
⎣∗
∗
0 0 CiT K T ATi X Σ17 EiT X EiT Y
0
0
0
0
−Y 0
0
0 ATFj
T
T
0
ATdi X Σ37 Edi
X Edi
Y
0
∗ Σ33
T
0
0
0
∗
∗
−2I
0
BFj
∗
∗
∗
−X
0
0
0 XM1i
∗
∗
∗
∗
−Y
0
0 Y M1i
∗
∗
∗
∗
∗
−X
0 XM2i
∗
∗
∗
∗
∗
∗
−Y Y M2i
∗
∗
∗
∗
∗
∗
∗
−ε1i I
∗
∗
∗
∗
∗
∗
∗
∗
⎤
0
0 ⎥
⎥
⎥
0 ⎥
⎥
0 ⎥
⎥
XM3i ⎥
⎥ < 0.
Σ711 ⎥
⎥
⎥
XM4i ⎥
⎥
Y M4i ⎥
⎥
0 ⎦
−ε2i I
3.4
Define a matrix P > 0 by
P
X 0
,
0 Y
3.5
where X > 0 and Y > 0 satisfy the solvability of 3.1.
Then, it can be shown from 3.3 and 3.5 that LMI 3.4 can be rewritten as
⎡
⎤
T
T
T
Σ11 0 Ci K T
Aij P
Ei P
0
0
⎢
⎥
T
T
⎢ ∗ Σ22
0
H T Adij P H T Edi P
0
0 ⎥
⎢
⎥
⎢
⎥
T
⎢∗
⎥
∗
−2I
B
P
0
0
0
fj
⎢
⎥
⎢
⎥ < 0,
∗
∗
−P
0
P MAi P MAdij ⎥
⎢∗
⎢
⎥
⎢∗
∗
∗
∗
−P
P MEi P MEdi ⎥
⎢
⎥
⎣∗
∗
∗
∗
∗
−ε1i I
0 ⎦
∗
∗
∗
∗
∗
∗
−ε2i I
3.6
where
T
Σ11 −P τ2 − τ1 1H T QH ε1i N 1i N 1i ,
T
Σ22 −H T QH ε2i H T N2i
N2i H,
3.7
which is equivalent to the following inequality:
⎡
⎤
T
T
T
11
Σ
0
Ci K T
Aij P
Ei P
⎢
T
T ⎥
⎢ ∗ −H T QH
0
H T Adij P H T Edi P ⎥
⎢
⎥
⎢
⎥
−1 T
T
T N
⎢∗
⎥ ε1i N
1i 1i ε1i M1i M1i
∗
−2I
B
P
0
⎢
⎥
fj
⎢
⎥
⎣∗
⎦
∗
∗
−P
0
∗
∗
∗
∗
−P
−1 T
T N
ε2i N
2i 2i ε2i M2i M2i < 0
3.8
Journal of Applied Mathematics
9
with
11 −P τ2 − τ1 1H T QH,
Σ
1i N 1i 0 0 0 0 ,
N
T
1i 0 0 0 MT P MT P ,
M
Adij
Edi
T
2i 0 0 0 MT P MT P ,
M
Ai
Ei
3.9
2i 0 N2i H 0 0 0 .
N
On the other hand, it is noted that the following equality holds:
⎡
⎤
T
T
T
11
Σ
0
Ci K T
Aij ΔAi k P
Ei ΔEi k P
⎢
T
T ⎥
⎢
⎥
T
T
⎢ ∗ −H T QH
0
H Adij ΔAdij k P H Edi ΔEdi k P ⎥
⎢
⎥
⎢
⎥
T
⎢∗
⎥
Bfj P
0
∗
−2I
⎢
⎥
⎢
⎥
⎣∗
⎦
∗
∗
−P
0
∗
∗
∗
∗
−P
⎡
⎤
T
T
T
11
Σ
0
Ci K T
Aij P
Ei P
⎢
T
T ⎥
⎢ ∗ −H T QH
0
H T Adij P H T Edi P ⎥
⎢
⎥
⎢
⎥ T
T F1i M
1i N
T
⎢ ∗
⎥ M1i F1i N
1i
1i
∗
−2I
B
P
0
⎢
⎥
fj
⎢
⎥
⎣∗
⎦
∗
∗
−P
0
∗
∗
∗
∗
−P
3.10
2i F2i N
T F2i M
2i N
T .
M
2i
2i
By Lemmas 2.2 and 2.3, from 3.8–3.10, we have
⎤
⎡
Γ11 Γ12 Γ13
Γ ⎣ ∗ Γ22 Γ23 ⎦ < 0,
∗ ∗ Γ33
3.11
where
T T 11 Aij ΔAi k P Aij ΔAi k Ei ΔEi k P Ei ΔEi k ,
Γ11 Σ
T T Γ12 Aij ΔAi k P Adij ΔAdij k H Ei ΔEi k P Edi ΔEdi k H,
T
T
Γ13 Ci K T Aij ΔAi k P Bfj ,
Γ22
T H −Q Adij ΔAdij k P Adij ΔAdij k
T
T Edi ΔEdi k P Edi ΔEdi k H,
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Journal of Applied Mathematics
T
Γ23 H T Adij ΔAdij k P Bfj ,
T
Γ33 −2I Bfj P Bfj .
3.12
Then there exists a small scalar α > 0 such that
⎡
⎤
−αI 0 0
Γ < ⎣ 0 0 0⎦.
0 0 0
3.13
Consider the Lyapunov-Krasovskii functional candidate as follows:
V k V1 k V2 k V3 k,
3.14
where
V1 k eT kP ek,
V2 k k−1
eT iH T QHei,
ik−τk
V3 k −τ
1 1
k−1
3.15
eT lH T QHel.
j−τ2 2 lk
j−1
Calculating the difference of V k along the filtering error system 2.14 with vk 0 and
taking the mathematical expectation, we have
E{ΔV k} E{ΔV1 k} E{ΔV2 k} E{ΔV3 k},
E{ΔV1 k}
r
r hi θkhj θkE
Aij ΔAi k ek Bfj φs Ci ek
i1 j1
Adij ΔAdij k Hek − τk
Ei ΔEi k ek Edi ΔEdi k Hek − τk
× wk
T
P×
Aij ΔAi k ek Bfj φs Ci ek
Adij ΔAdij k Hek − τk
Ei ΔEi k ek Edi ΔEdi k
× Hek − τk × wk − eT kP ek .
3.16
Journal of Applied Mathematics
11
Noting 2.2 and 2.12, we have
E{ΔV1 k}
≤
r
r hi θkhj θk
Aij ΔAi k ek Bfj φs Ci ek Adij ΔAdij k
i1 j1
T × Hek − τk P Aij ΔAi k ek Bfj φs Ci ek
Adij ΔAdij k Hek − τk
T
Ei ΔEi k ek Edi ΔEdi k Hek − τk P
× Ei ΔEi k ek Edi ΔEdi k Hek − τk
− ekT P ek − 2φs Ci ek φs Ci ek − KCi ek .
3.17
By some calculations, we have
k−τ
1
E{ΔV2 k} eT iH T QHei − eT k − τkH T QHek − τk
ik
1−τk
1
eT kH T QHek k−1
eT iH T QHei −
ik
1−τ1
E{ΔV3 k} −τ
1 1
eT iH T QHei,
ik
1−τk
⎡
k−1
⎣eT kH T QHek j−τ2 2
k−1
eT lH T QHel −
lk
j
k−τ
1
τ2 − τ1 eT kH T QHek −
k−1
3.18
⎤
eT lH T QHel⎦
lk
j−1
eT iH T QHei.
ik
1−τ2
3.19
Since τ1 ≤ τk ≤ τ2 , we have
k−1
ik
1−τ1
k−τ
1
k−1
eT iH T QHei −
eT iH T QHei ≤ 0,
3.20
ik
1−τk
eT iH T QHei −
ik
1−τk
1
k−τ
1
ik
1−τ2
eT iH T QHei ≤ 0.
3.21
12
Journal of Applied Mathematics
Combining 3.18–3.21, we have
E{ΔV2 k ΔV3 k} ≤ τ2 − τ1 1eT kH T QHek − eT k − τkH T QHek − τk.
3.22
Combining 3.17 and 3.22 yields
E{ΔV k}
≤
r
r hi θkhj θk
Aij ΔAi k ek Bfj φs Ci ek Adij ΔAdij k
i1 j1
T × Hek − τk P Aij ΔAi k ek Bfj φs Ci ek
Adij ΔAdij k Hek − τk
T
Ei ΔEi k ek Edi ΔEdi k Hek − τk P
×
Ei ΔEi k ek Edi ΔEdi k Hek − τk
eT k τ2 − τ1 1H T QH − P ek − 2φs Ci ek
× φs Ci ek − KCi ek
− eT k − τkH T QHek − τk
r r
hi θkhj θkζT kΓζk,
i1 j1
3.23
where
ζT k eT k eT k − τk φsT Ci ek ,
3.24
and Γ is given in 3.11. Thus, it follows from 3.13 and 3.23 that
E{V k 1} − E{V k} < −αE |ek|2 .
3.25
Hence, by summing up both sides of 3.25 from 0 to N for any integer N > 1, we have
N
2
E{V N 1} − E{V 0} < −αE
|ek| ,
k0
3.26
Journal of Applied Mathematics
13
which yields
N
E
|ek|2
<
k0
1
1
E{V 0} − E{V N 1} ≤ E{V 0} ≤ cE |x0|2 ,
α
α
3.27
where c 1/αλmax P . Taking N → ∞, it is shown from 2.17 and 3.27 that the filtering
error system 2.14 is robustly stochastic stable for vk 0.
Next, we will show that the filtering error system 2.14-2.15 satisfies
3.28
zke2 < γvke2
for all nonzero vk ∈ Le2 0 ∞, Rp . To this end, define
JN E
N 2
zk| < γ |vk|
|
2
2
3.29
k1
with any integer N > 0. Then for any nonzero vk, we have
N
2
2
2
JN E
zk| − γ |vk|
|
k1
N 2
2
2
≤E
zk| − γ |vk| ΔV k − E{V N 1}
|
k1
N 2
2
2
E
zk| − γ |vk| ΔV k
|
k1
⎧
⎫
N r r
⎨
⎬
ij ζk
E
,
hi θkhj θkζT kΓ
⎩ k1 i1 j1
⎭
where
T ζT k eT k eT k − τk vT k φs Ci ek
,
ij
Γ
⎡
11
Γ
⎢∗
⎢
⎢
⎣∗
∗
12 Γ
13
Γ
22 Γ
23
Γ
33
∗ Γ
∗
∗
⎤
14
Γ
24 ⎥
Γ
⎥
⎥,
Γ34 ⎦
44
Γ
3.30
14
Journal of Applied Mathematics
T 11 Σ
11 Aij ΔAi k P Aij ΔAi k
Γ
T T Ei ΔEi k P Ei ΔEi k Li − Lj
Li − Lj ,
T T 12 Aij ΔAi k P Adij ΔAdij k H Ei ΔEi k P Edi ΔEdi k H,
Γ
T T 13 Aij ΔAi k P Bij ΔBij k Ei ΔEi k P Gi ΔGi k ,
Γ
T
14 CTi K T Aij ΔAi k P Bfj ,
Γ
T 22 H T −Q Adij ΔAdij k P Adij ΔAdij k
Γ
T Edi ΔEdi k P Edi ΔEdi k H,
23 H T Adij ΔAdij k P Bij ΔBij k H T Edi ΔEdi k P Gi ΔGi k ,
Γ
T
24 H T Adij ΔAdi k P Bfj ,
Γ
T T 33 −γ 2 I Bij ΔB ij k P Bij ΔB ij k Gi ΔGi k P Gi ΔGi k ,
Γ
T
34 Bij ΔBij k P Bfj ,
Γ
44 −2I BTfj P Bfj .
Γ
3.31
It can be shown that if there exist real matrices X > 0, Y > 0, Q > 0, any matrices AFj , BFj ,
scalars ε1i > 0, ε2i > 0 and ε3i > 0 for i, j 1, 2, . . . , r satisfying LMI 3.1. Since vk ∈
ij < 0, and, thus, JN < 0. That is,
0 and by Lemma 2.2, it is implied that Γ
Le2 0 ∞, Rp /
||
zk||e2 < γ||vk||e2 . This completes the proof.
Remark 3.2. When τ1 and τ2 are given, matrix inequality 3.1 is linear matrix inequalities
in matrix variables X > 0, Y > 0, Q > 0, AFj , BFj , scalars ε1i > 0, ε2i > 0 and ε3i > 0 for
i, j 1, 2, . . . , r, which can be efficiently solved by the developed interior point algorithm
33. Meanwhile, it is esay to find the minimal attenuation level γ.
Remark 3.3. Theorem 3.1 is suggested to the H∞ filter design for uncertain discrete-time fuzzy
stochastic systems with sensor nonlinearities and time-varying delay. This approach is called
fuzzy-rule-dependent approach, which is applicable to the case that the information of the
premise variable θi k i 1, 2, . . . , r is available; for example, see 24. For the special case
that the premise variables are unavailable, the fuzzy-rule-independent approach as in 24, 29
is adopted, which may result in lager conservativeness due to lack of the information of the
premise variables.
Journal of Applied Mathematics
15
In the following, we will present the fuzzy-rule-dependent H∞ filter design for uncertain discrete-time fuzzy stochastic system with time-varying delay, which are less conservative than Theorem 3.1.
Theorem 3.4. Consider the uncertain discrete-time fuzzy stochastic systems in 2.1. A filter of form
2.13 and constants τ1 and τ2 , the filtering error system 2.14-2.15 is robustly stochastic stable
with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any matrices AFi , BFi , scalars
ε1i > 0, ε2i > 0 and ε3i > 0 for i, j 1, 2, . . . , r, i < j, such that the following LMIs are satisfied:
Σii < 0,
Σij Σji < 0,
i 1, 2, . . . , r,
3.32
i, j 1, 2, . . . , r, i < j,
3.33
where
⎡
Σ11 0
0
0 CiT K T ATi X Σ17 EiT X EiT Y
⎢ ∗ Σ22 0
0
0
0
ATFj
0
0
⎢
⎢
T
T
T
0
Adi X Σ37 Edi
X Edi
Y
∗ Σ33 0
⎢∗
⎢
⎢
∗
∗ Σ44
0
BiT X Σ47 GTi X GTi Y
⎢∗
⎢
⎢
T
⎢∗
∗
∗
∗
−2I
0 −BFj
0
0
⎢
Σij ⎢
∗
∗
∗
∗
∗
−X
0
0
0
⎢
⎢∗
∗
∗
∗
∗
∗
−Y
0
0
⎢
⎢
∗
∗
∗
∗
∗
∗
−X
0
⎢∗
⎢
⎢∗
∗
∗
∗
∗
∗
∗
∗
−Y
⎢
⎢∗
∗
∗
∗
∗
∗
∗
∗
∗
⎢
⎣∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0
0
0
0
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
0
0
0 ⎥
⎥
⎥
0
0
0 ⎥
⎥
XM1i XM3i XM6i ⎥
⎥,
Y M1i Σ711 Σ712 ⎥
⎥
⎥
XM2i XM4i XM7i ⎥
⎥
Y M2i Y M4i Y M7i ⎥
⎥
−ε1i I
0
0 ⎥
⎥
0 ⎦
∗
−ε2i I
∗
∗
−ε3i I
3.34
where Σ11 , Σ22 , Σ33 , Σ44 , Σ17 , Σ37 , Σ47 , Σ711 , and Σ712 are defined in Theorem 3.1. Moreover, if the
previous conditions are satisfied, an acceptable state-space realization of the H∞ filter is given by LMI
3.3.
Proof. Considering 2.2 and 3.1, it follows from 3.30 that
⎫
⎧
r r
N ⎬
⎨
ij ζk
hi θkhj θkζT kΓ
JN ≤ E
⎭
⎩ k1 i1 j1
⎫
⎧
⎡
⎤
r
N
r
r ⎬
⎨
ji ⎦ζk
ii ij Γ
.
hi θkhj θk Γ
ζT k⎣ h2i θkΓ
E
⎭
⎩ k1
i1
i1 i<j
3.35
ii < 0 and Γ
ij Γ
ji < 0. By
Our elaborate estimation JN is negative definite if and only if Γ
Lemmas 2.2 and 2.3, we can obtain that 3.32 is equivalent to Γii < 0, and 3.33 is equivalent
ij Γ
ji < 0. Thus, we can conclude that the LMIs 3.32 and 3.33 can guarantee JN < 0.
to Γ
That is, ||
zk||e2 < γ||vk||e2 . This completes the proof.
16
Journal of Applied Mathematics
Theorem 3.5. Consider the uncertain discrete-time fuzzy stochastic systems in 2.1. A filter of form
2.13 and constants τ1 and τ2 , the filtering error system 2.14-2.15 is robustly stochastic stable
with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any matrices AFi , BFi , scalars
ε1i > 0, ε2i > 0 and ε3i > 0 for i, j 1, 2, . . . , r, i < j, such that the following LMIs are satisfied:
Σii < Ωii ,
i 1, 2, . . . , r,
Σij Σji < Ωij ΩTij ,
⎡
Ω11 Ω12
⎢ ∗ Ω22
⎢
Ω⎢ .
..
⎣ ..
.
∗
∗
3.36
i, j 1, 2, . . . , r, i < j,
3.37
⎤
· · · Ω1r
· · · Ω2r ⎥
⎥
.. ⎥ < 0,
..
. . ⎦
· · · Ωrr
3.38
where Σij , Σ11 , Σ22 , Σ33 , Σ44 , Σ17 , Σ37 , Σ47 , Σ711 , and Σ712 are defined in Theorems 3.4 and 3.1, respectively. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the H∞
filter is given by LMI 3.3.
Proof. This theorem can be proved by employing the same techniques as in the proof of
Theorem 3.4; hence, the detailed procedure is omitted here.
Remark 3.6. It is noted that the convexifying procedure proposed in this paper is based on
the relaxation inequality 3.36, and extensions of the current derivations based on the more
powerful relaxation techniques presented in 8, 24, 30 are straightforward. In this way, the
design conservatism can be further reduced but the computation complexity will also
increase. By considering the information on the premises 8, 24, 30, Theorems 3.4 and 3.5
relaxed the conservatism of the previous works 29 by representing the interactions among
the fuzzy subsystems in a matrix and solving it in a numerical manner.
For comparison, if there is no time-varying delay exit in 2.1, we consider the following discrete-time stochastic systems 29:
xk 1 Ai ΔAi kxk Bi ΔBi kvk
Ei ΔEi kxk Gi ΔGi kvkwk,
yk φCi kxk Di ΔDi kvk,
zk Li xk,
3.39
i 1, 2, . . . , r.
Remark 3.7. This class of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities has been considered in 29; a new different Lyapunov functional is then employed
to deal with systems with sensor nonlinearities. Then from Theorems 3.1, 3.4, and 3.5, we can
get the following corollaries, which are less conservative than Theorem 3.1 in 29.
Corollary 3.8. Consider the uncertain discrete-time fuzzy stochastic systems in 3.39. A filter of
form 2.13 and constants τ1 and τ2 , the filtering error system is robustly stochastic stable with
Journal of Applied Mathematics
17
performance γ, if there exist real matrices X > 0, Y > 0, any matrices AFj , BFj , scalars ε1i > 0 and
ε3i > 0 for i, j 1, 2, . . . , r such that the following LMI is satisfied:
⎡
Π11 0
0 CiT K T ATi X
⎢ ∗ Π22 0
0
0
⎢
⎢
T
0
Bi X
∗ Π33
⎢ ∗
⎢
⎢ ∗
∗
∗
−2I
0
⎢
⎢ ∗
∗
∗
∗
−X
Π⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎣ ∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Π16 EiT X EiT Y
0
ATFj
0
0
0
Π36 GTi X GTi Y
0
T
0
0
0
−BFj
0
0
0 XM1i
−Y
0
0 Y M1i
∗
−X
0 XM2i
∗
∗
−Y Y M2i
∗
∗
∗
−ε1i I
∗
∗
∗
∗
0
0
0
0
XM6i
Π610
XM7i
Y M7i
0
−ε3i I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥ < 0,
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
3.40
where
T
Π11 −X ε1i N1i
N1i ,
T Π22 −Y Li − Lj
Li − Lj ,
T
Π16 ATi Y − ATFj − CiT K1T BFj
,
T
Π36 BiT Y − DiT BFj
,
T
N3i ,
Π33 −γ 2 I ε3i N3i
Π610 Y M6i − BFj M8i .
3.41
Moreover, if the previous condition is satisfied, an acceptable state-space realization of the H∞ filter is
given by LMI 3.3.
Corollary 3.9. Consider the uncertain discrete-time fuzzy stochastic systems in 3.39. Give a filter of
form 2.13 and constants τ1 and τ2 , the filtering error system is robustly stochastic stable with
performance γ, if there exist real matrices X > 0, Y > 0, any matrices AFj , BFj , scalars ε1i > 0 and
ε3i > 0 for i, j 1, 2, . . . , r such that the following LMIs are satisfied:
Πii < 0,
Πij Πji < 0,
i 1, 2, . . . , r,
i, j 1, 2, . . . , r, i < j,
⎡
Π11 0
0 CiT K T ATi X
⎢ ∗ Π22 0
0
0
⎢
⎢
0
BiT X
∗ Π33
⎢ ∗
⎢
⎢ ∗
∗
∗
−2I
0
⎢
⎢ ∗
∗
∗
∗
−X
Πij ⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎢ ∗
∗
∗
∗
∗
⎢
⎣ ∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
Π16 EiT X EiT Y
0
T
AFj
0
0
0
Π36 GTi X GTi Y
0
T
0
0
0
−BFj
0
0
0 XM1i
−Y
0
0 Y M1i
∗
−X
0 XM2i
∗
∗
−Y Y M2i
∗
∗
∗
−ε1i I
∗
∗
∗
∗
0
0
0
0
XM6i
Π610
XM7i
Y M7i
0
−ε3i I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
3.42
where Π11 , Π22 , Π33 , Π16 , Π36 , Π610 are defined in Corollary 3.8. Moreover, if the previous conditions
are satisfied, an acceptable state-space realization of the H∞ filter is given by LMI 3.3.
18
Journal of Applied Mathematics
Corollary 3.10. Consider the uncertain discrete-time fuzzy stochastic systems in 3.39. A filter of
form 2.13 and constants τ1 and τ2 , the filtering error system is robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, any matrices AFj , BFj , scalars ε1i > 0 and ε3i > 0
for i, j 1, 2, . . . , r such that the following LMIs are satisfied:
Πii < Ωii ,
i 1, 2, . . . , r,
Πij Πji < Ωij ΩTji ,
⎡
Ω11 Ω12
⎢ ∗ Ω22
⎢
Ωij ⎢ .
..
⎣ ..
.
∗
∗
i, j 1, 2, . . . , r, i < j,
⎤
· · · Ω1r
· · · Ω2r ⎥
⎥
. ⎥ < 0,
..
. .. ⎦
3.43
· · · Ωrr
where Π11 , Π22 , Π33 , Π16 , Π36 , Π610 are defined in Corollary 3.8. Moreover, if the previous conditions
are satisfied, an acceptable state-space realization of the H∞ filter is given by LMI 3.3.
In the sequel, special results for the case of linear sensor, that is, φCxk Cxk,
may be obtained from Theorems 3.1, 3.4, and 3.5.
Corollary 3.11. Consider the uncertain discrete-time fuzzy stochastic systems in 2.1 with
φCxk Cxk. A filter of form 2.13 and constants τ1 and τ2 , the filtering error system is
robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any
matrices AFj , BFj , scalars ε1i > 0, ε2i > 0 and ε3i > 0 for i, j 1, 2, . . . , r such that the following LMI
is satisfied:
⎡
Σ11 0
0
0 ATi X Σ17 EiT X EiT Y
⎢∗ Σ
0
0
0 ATFj
0
0
⎢
22
⎢
T
T
T
⎢∗
0
A
X
Σ
E
X
E
Y
∗
Σ
33
37
di
di
di
⎢
⎢
⎢∗
∗
∗ Σ44 BiT X Σ47 GTi X GTi Y
⎢
⎢
⎢∗
∗
∗
∗
−X
0
0
0
⎢
⎢∗
∗
∗
∗
∗
−Y
0
0
⎢
⎢∗
∗
∗
∗
∗
∗
−X
0
⎢
⎢
∗
∗
∗
∗
∗
∗
−Y
⎢∗
⎢
⎢∗
∗
∗
∗
∗
∗
∗
∗
⎢
⎣∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0
0
0
0
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
0
0
0 ⎥
⎥
⎥
XM1i XM3i XM6i ⎥
⎥ < 0,
Y M1i Σ711 Σ712 ⎥
⎥
XM2i XM4i XM7i ⎥
⎥
⎥
Y M2i Y M4i Y M7i ⎥
⎥
−ε1i I
0
0 ⎥
⎥
0 ⎦
∗
−ε2i I
∗
∗
−ε3i I
3.44
where
T
Σ17 ATi Y − ATFj − CiT BFj
.
3.45
Moreover, if the previous condition is satisfied, an acceptable state-space realization of the H∞ filter is
given by LMI 3.3.
Journal of Applied Mathematics
19
Corollary 3.12. Consider the uncertain discrete-time fuzzy stochastic systems in 2.1 with
φCxk Cxk. A filter of form 2.13 and constants τ1 and τ2 , the filtering error system is
robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any
matrices AFi , BFi , scalars ε1i > 0, ε2i > 0 and ε3i > 0 for i, j 1, 2, . . . , r, i < j, such that the following
LMIs are satisfied:
Σii < 0,
Σij Σji < 0,
i 1, 2, . . . , r,
3.46
i, j 1, 2, . . . , r, i < j,
where
⎡
Σ11 0
0
0 ATi X Σ17 EiT X EiT Y
⎢∗ Σ
0
0
0 ATFj
0
0
⎢
22
⎢
T
T
T
⎢∗
∗ Σ33 0 Adi X Σ37 Edi X Edi Y
⎢
⎢
⎢∗
∗
∗ Σ44 BiT X Σ47 GTi X GTi Y
⎢
⎢
⎢∗
∗
∗
∗
−X
0
0
0
Σij ⎢
⎢∗
∗
∗
∗
∗
−Y
0
0
⎢
⎢∗
∗
∗
∗
∗
∗
−X
0
⎢
⎢
∗
∗
∗
∗
∗
∗
−Y
⎢∗
⎢
⎢∗
∗
∗
∗
∗
∗
∗
∗
⎢
⎣∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
∗
0
0
0
0
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
0
0
0 ⎥
⎥
⎥
XM1i XM3i XM6i ⎥
⎥,
Y M1i Σ711 Σ712 ⎥
⎥
XM2i XM4i XM7i ⎥
⎥
⎥
Y M2i Y M4i Y M7i ⎥
⎥
−ε1i I
0
0 ⎥
⎥
0 ⎦
∗
−ε2i I
∗
∗
−ε3i I
3.47
where Σ11 , Σ22 , Σ33 , Σ44 , Σ37 , Σ47 , Σ711 , Σ712 , and Σ17 are defined in Theorem 3.1 and Corollary 3.11,
respectively. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of
the H∞ filter is given by LMI 3.3.
Corollary 3.13. Consider the uncertain discrete-time fuzzy stochastic systems in 2.9 with
φCxk Cxk. A filter of form 2.13 and constants τ1 and τ2 , the filtering error system is
robustly stochastic stable with performance γ, if there exist real matrices X > 0, Y > 0, Q > 0, any
matrices AFi , BFi , scalars ε1i > 0, ε2i > 0 and ε3i > 0 for i, j 1, 2, . . . , r, i < j, such that the following
LMIs are satisfied:
Σii < Ωii ,
Σij Σji < Ωij i 1, 2, . . . , r,
T
Ωij ,
⎡
Ω11 Ω12
⎢
⎢ ∗ Ω22
Ω⎢
..
⎢ ..
⎣ .
.
∗
∗
i, j 1, 2, . . . , r, i < j,
⎤
· · · Ω1r
⎥
· · · Ω2r ⎥
⎥ < 0,
.
..
⎥
. .. ⎦
· · · Ωrr
3.48
20
Journal of Applied Mathematics
Table 1: Minimum index γ for different methods.
Methods
γmin
Theorem3.1 29
0.2931
Corollary 3.8
0.2030
Corollary 3.9
0.1869
Corollary 3.10
0.1376
where Σij , Σ11 , Σ22 , Σ33 , Σ44 , Σ37 , Σ47 , Σ711 , Σ712 , and Σ17 are defined in Corollary 3.12, Theorem 3.1,
and Corollary 3.11, respectively. Moreover, if the previous conditions are satisfied, an acceptable statespace realization of the H∞ filter is given by LMI 3.3.
Remark 3.14. In the previous results, there still exists conservativeness; for these, we can also
use the delay-dependent stability results of discrete-time systems in 13–16 to derive the less
conservative theorems and corollaries for filtering problem of uncertain discrete-time fuzzy
stochastic systems with sensor nonlinearities and time-varying delay.
4. Numerical Example
In this section, two simulation examples are given to illustrate the effectiveness and benefits
of the proposed approach.
Example 4.1. Consider the system 3.39 with parameters as follows:
−0.2 0.1
0.1
−0.2 0.3
−0.1
A1 ,
B1 ,
E1 ,
G1 ,
0.1 −0.3
0.2
0.1 0.3
0.3
−0.2 0.3
0.2
0.3 0.2
,
D1 ,
L1 ,
C1 0.1 0.4
0.1
0.2 0.4
0.1
−0.4
−0.2
−0.2
M11 ,
M21 ,
M61 ,
M71 ,
0.1
−0.5
0.1
0.3
−0.2
M81 N31 0.2,
K1 diag{0.4, 0.6},
,
N11 0.2 0.1 ,
0.3
−0.1 −0.2
0.2
−0.2 −0.2
−0.2
A2 ,
B2 ,
E2 ,
G2 ,
0.2 0.3
−0.2
0.3 0.1
0.2
−0.1 −0.2
0.4
−0.1 0.3
C2 ,
D2 ,
L2 ,
0.3 0.3
0.3
0.2 0.3
−0.2
0.3
0.3
0.1
,
M22 ,
M62 ,
M72 ,
M12 0.2
0.1
0.1
0.3
0.1
M82 N32 0.2,
K2 diag{0.1, 0.8}.
,
N12 0.2 0.3 ,
−0.2
4.1
This system has been considered in 29. For comparison with the Theorem 3.1 in 29, the
computation results of γmin under Corollaries 3.8, 3.9, and 3.10 are listed in Table 1.
This example conclusively shows that our results are less conservative than Theorem 3.1 in 29.
Journal of Applied Mathematics
21
Example 4.2. Consider the system 2.1 with parameters as follows:
−0.1 0
,
0.1 −0.4
Ad1 0.2
0
,
−0.1 −0.1
B1 −0.11 0.1
,
0
0.06
Ed1 0.11 0.1
,
0 0.06
G1 −0.11 0.05
,
0.6 0.3
D1 A1 E1 C1 −0.1 0.05
,
0.2 0.05
Cd1 M11 −0.03
,
0.02
M21 M51 −0.03
,
−0.02
M61 N11 0.02 0.02 ,
L1 0.1 0.1
,
0.12 0.1
0.06
,
0.02
M71 Ad2
−0.2
,
0.1
0.1
,
−0.1
−0.03
,
−0.02
0.01
,
0.05
M41 −0.03
,
−0.02
M81 0.08
,
−0.03
N31 0.01,
F31 0.3 sint,
4.2
0.1 −0.1
,
0.1 0.2
Ed2
F21 0.3 sint,
−0.12 −0.11
E2 ,
0.11 0.04
F11 0.2 sint,
0
,
0.3
M31 N21 0.02 0.02 ,
−0.2 −0.3
A2 ,
0.5 0.4
−0.03
,
−0.02
0
B2 ,
−0.3
0.12 −0.11
,
0.11 0.04
−0.1
G2 ,
0.1
−0.2 −0.1
−0.2 −0.1
0.1
,
Cd2 ,
D2 ,
C2 0.5 0.2
0.5 0.2
0.2
0.07
0.02
−0.03
−0.03
,
M22 ,
M32 ,
M42 ,
M12 0.01
0.12
−0.02
−0.02
M52 −0.03
,
−0.02
N12 0.01 0.02 ,
L2 0.2 0.19
,
0.1 0.1
M62 0.08
,
0.07
M72 N22 0.01 0.01 ,
F12 0.2 sint,
0.04
,
0.07
M82 0.03
,
−0.07
N32 0.01,
F22 0.3 sint,
F32 0.3 sint
and vt 10e−t , the membership functions are h1 θ 1−sinx2 /2, h2 θ 1−sinx1 /2,
and the nonlinear vector function is φu K2 K1 /2 K2 − K1 /2 sinu, K1 diag{0.1, 0.3}, K2 diag{0.2, 0.4}.
Note that different τ1 and τ2 yield different γmin , and assume τk satisfies 1 ≤ τk ≤ 5.
22
Journal of Applied Mathematics
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
5
10
15
t (s)
x1
x2
Figure 1: The state response xk.
By Theorem 3.1, the minimum achievable noise attenuation level is given by γmin 0.3474 and we can obtain the corresponding filter parameters as follows:
Af1
Af2
−0.3876
0.4967
−0.3600
0.4593
−0.6504
,
0.8339
−0.6051
,
0.7720
1.4538 0.8653
Bf1 ,
−4.9933 −1.7834
1.2017 0.7808
Bf2 .
−4.6656 −1.6742
4.3
By Theorem 3.4, the minimum achievable noise attenuation level is given by γmin 0.3419 and we can obtain the corresponding filter parameters as follows:
Af1
Af2
0.5059 0.4421
,
−0.7283 −0.6452
−0.5060 −0.4420
,
0.7288 0.6429
Bf1
Bf2
−0.3795 0.2226
,
−2.1600 −0.9045
1.6933 1.0131
.
−3.1230 −1.1367
4.4
By Theorem 3.5, the minimum achievable noise attenuation level is given by γmin 0.2563 and we can obtain the corresponding filter parameters as follows:
−0.0002 −0.0011
,
0.0016 −0.0066
−0.4489 −0.3942
,
0.6516 0.5707
Af1 Af2
−0.1708 0.3000
,
−2.8286 −1.0410
3.2336 1.6093
.
−5.2772 −1.9714
Bf1 Bf2
4.5
Journal of Applied Mathematics
23
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
5
10
15
t (s)
x̂1
x̂2
Figure 2: The estimation of filter xk.
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
5
10
15
t (s)
x̃1
x̃2
Figure 3: The error response xk.
The relaxation technique is adopted in this paper, so Theorems 3.4 and 3.5 relaxed the
conservatism more than Theorem 3.1. There has been a decrease in the minimum achievable
noise attenuation level γmin .
With the initial conditions xt and xt
are 2 − 1.5T and 2 − 1.5T , respectively,
for an appropriate initial interval. For given τ1 1, τ2 5 with γmin 0.2563, according
to Theorem 3.5, we apply the filter parameters mentioned previously to the system 2.1
and obtain the simulation results as in Figures 1–3. Figure 1 shows the state response xk
24
Journal of Applied Mathematics
under the initial condition. Figure 2 shows the estimation of filter xk.
Figure 3 shows error
response xk.
From these simulation results, we can see that the designed H∞ filter can
stabilize the system 2.1 with sensor nonlinearities and time-varying delay.
5. Conclusions
In this paper the robust filtering problem for a class of uncertain discrete-time fuzzy stochastic
systems with sensor nonlinearities and time-varying delay has been developed. A new type
of Lyapunov-Krasovskii functional has been constructed to derive some sufficient conditions
for the filters in terms of LMIs, which guarantees a prescribed H∞ performance index for the
filtering error system. A numerical example has shown the usefulness and effectiveness of
the proposed filter design method.
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