# A Novel Approach toℓ2-ℓ∞ Filter Design for T-S Fuzzy Systems with Multiple Time-Varying Delays

Authors: Xiaoyu Zhang; Cheng Gong; Zhen Zeng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper investigates theℓ2-ℓ∞ filtering problem of T-S fuzzy systems with multiple time-varying delays. First, by the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a delay-dependent sufficient condition on ℓ2-ℓ∞-disturbance attenuation is presented, in which both stability and prescribed ℓ2-ℓ∞ performance are required to be achieved for the filtering-error systems. Then, based on the condition, the full-order and reduced-order delay-dependent ℓ2-ℓ∞ filter design schemes for T-S fuzzy multiple time-varying delays systems are developed in terms of linear matrix inequality (LMI). Finally, an example is given to illustrate the effectiveness of the result.


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## Body

## 1. Introduction

Time delay arises frequently in many engineering areas of the real world, which is usually a source of instability. Therefore, the stability analysis and synthesis for time-delay system have been one of a most hot research area in the control community over the past years [1–9]. To research the nonlinear time-delay system, the scholars considered the Takagi-Sugeno (T-S) fuzzy time-delay model which is a kind of effective representation, and many analysis and synthesis methods for T-S fuzzy time-delay systems have been developed over the past years [10–13].Since the state variables in control systems are not always available, filtering or state estimation of a dynamic system through an available measurement state is one of the fundamental problems in signal processing, communications, and control application [14–22]. There are many works that appeared to cope with the nonlinear filtering problem for T-S fuzzy systems with time delays [23–31]. For example, in [23, 24], a delay-dependent ℋ∞ filter design via continuous-time T-S fuzzy model approach is proposed in terms of linear matrix inequalities. In [25], by using a basis-dependent Lyapunov function, a delay-dependent result on the ℋ∞ performance of the discrete filtering error system is presented. Based on the similar Lyapunov function combined with Finslers Lemma, [26] researched the delay-dependent robust ℋ∞ filtering problem for a class of uncertain discrete-time T-S fuzzy systems with interval-like time-varying state delay. Reference [27] proposed the delay-dependent approach to robust ℋ∞ and ℓ2-ℓ∞ filtering for a class of uncertain nonlinear time-delayed systems. References [28, 29] investigated delay-dependent ℋ∞ filter design problems for discrete-time T-S fuzzy time-delayed systems and continuous-time T-S fuzzy time-delayed systems, respectively, which were both based on a delay-dependent piecewise Lyapunov-Krasovskii functional.Severalℋ∞ filtering approaches for T-S fuzzy systems with multiple delays have been developed over the past few years [32–36]. For instance [32] studied the ℋ∞ filter design problem for discrete-time T-S fuzzy systems with multiple time delays. In [33], a robust mixed ℋ2/ℋ∞ filtering problem for continuous-time T-S fuzzy systems with multiple time-varying delays in state variables was addressed.While the time-varying delay functions above mentioned were all assumed slow-varying (the derivative of delay function is less than one) or fast-varying (the derivative of delay function is unknown). Reference [34] dealt with the fuzzy ℋ∞ filter design problem for discrete-time T-S fuzzy systems with multiple time delays in the state variables. Reference [35] introduced a decentralized ℋ∞ fuzzy filter design for nonlinear interconnected systems with multiple constant delays via T-S fuzzy models. Reference [36] addressed the problem of ℓ2-ℓ∞ filter design for T-S fuzzy systems with multiple time-varying delays, but the derivative of delay functions must be less than one.To the best of our knowledge, the problem ofℓ2-ℓ∞ filter design for T-S fuzzy systems with multiple time-varying delays has not been fully investigated in the literature. As is well known, time delays usually exist in many physical systems and result in unsatisfactory performance, and the derivative of delay function may vary from -∞ to +∞. So, the research on T-S fuzzy systems with multiple time-varying delays is of great practical and theoretical significance. This motivates the research in this paper.In summary, the purpose of this paper is to develop anℓ2-ℓ∞ filter for T-S fuzzy systems with multiple time-varying delays. Based on the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a delay-dependent sufficient condition on ℓ2-ℓ∞-disturbance attenuation is presented. Then, the full-order and reduced-order delay-dependent ℓ2-ℓ∞ filter design schemes for T-S fuzzy multiple time-varying-delays systems are developed in terms of LMI. The example illustrates the effectiveness of the result.This paper is organized as follows. In Section2, the T-S fuzzy model and corresponding filter are formulated. In Section 3 we give the sufficient condition to assure asymptotic stability and the ℓ2-ℓ∞ noise-attenuation level bound for the T-S fuzzy filtering-error systems. Based on the condition in Section 3, we present a stable fuzzy filter in terms of LMIs. Section 4 provides illustrative examples to demonstrate the effectiveness of the proposed method. Conclusions are given in Section 5.Notations. The notations used throughout this paper are fairly standard. The superscript “T” stands for matrix transpose, and the notation P>0  (P≥0) means that matrix P is real symmetric and positive (or being positive semidefinite). I and 0 and are used to denote identity matrix and zero matrix with appropriate dimension, respectively. The notation * in a symmetric matrix always denotes the symmetric block in the matrix. The parameter diag{⋯} denotes a block-diagonal matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions for algebraic operations.


## 2. System Descriptions and Preliminaries

Consider the nonlinear system with multiple state delays that is described by the following T-S model.


### 2.1. Plant Form

Rule
i. IF s1(t) is Fi1, s2(t) is Fi2, and sn(t) is Fin, then
(1)x˙(t)=Aix(t)+∑k=1q‍Adikx(t-dk(t))+Biω(t),y(t)=Cix(t)+∑k=1q‍Cdikx(t-dk(t))+Diω(t),z(t)=Eix(t),x(t)=ϕ(t),t∈[-h,0],h=max{hk},k=1,2,…,q,
where s1(t),s2(t),…,  and  sn(t) are the premise variables that are measurable, and each Fij  (j=1,2,…,n) is a fuzzy set. x(t)∈Rn is the state variables. y(t)∈Rm is the measured output of the system. z(t)∈Rp is the signal to be estimated. ω(t)∈Rb is the disturbance input. r is the number of IF-THEN rules. Also 0≤dk(t)≤hk is the time-varying delay in the state, and it is assumed that d˙k(t)≤dk. That is, the derivative of time-varying delay function is continuous and bounded. q is the number of time delays. ϕ(t) is a vector-valued initial continuous function.By using center-average defuzzifier, product inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model:
(2)x˙(t)=∑i=1r‍ui(s(t))[Aix(t)+∑k=1q‍Adikx(t-dk(t))+Biω(t)]=A(t)x(t)+∑k=1q‍Adkx(t-dk(t))+B(t)ω(t),y(t)=∑i=1r‍ui(s(t))[Cix(t)+∑k=1q‍Cdikx(t-dk(t))+Diω(t)]=C(t)x(t)+∑k=1q‍Cdkx(t-dk(t))+D(t)ω(t),z(t)=∑i=1r‍ui(s(t))[Eix(t)]=E(t)x(t)
with
(3)ui(s(t))=αi(s(t))∑i=1rαi(s(t)),αi(s(t))=∏j=1n‍Fij(sj(t)),
in which Fij(sj(t)) is the grade of membership of sj(t) in Fij. It is assumed that αi(s(t))≥0,  i=1,2,…,r,   ∑i=1r‍αi(s(t))>0 for all t. Therefore, ui(s(t))≥0 and ∑i=1r‍ui(s(t))=1 for all t. In this paper, we study the following filter form of order l (l=n for full-order filter, and 1≤l<n for reduced-order filter):
(4)x˙f(t)=∑i=1r‍ui(s(t))[Afix(t)+Bfiy(t)],zf(t)=∑i=1r‍Cfixf(t),
where the Afi,Bfi,andCfi are the filter parameters to be designed. Combining (2) and (4) and defining ξ(t)=[xT(t),xfT(t)]T, e(t)=z(t)-zf(t), and K=[In×n0n×l], we have the filtering-error system:
(5)ξ˙(t)=A-(t)ξ(t)+∑k=1q‍A-dk(t)Kξ(t-dk(t))+B-(t)ω(t),e(t)=E-(t)ξ(t),
where ξ(t)≐[ϕT(t),0]T for t∈[-h,0] and
(6)A-(t)=[A(t)0Bf(t)C(t)Af(t)]=∑i=1r‍∑j=1r‍ui(s(t))uj(s(t))[Aj0BfiCjAfi],A-dk(t)=[Adk(t)Bf(t)Cd(t)]=∑i=1r‍∑j=1r‍ui(s(t))uj(s(t))[AdjkBfiCdj],B-(t)=[B(t)Bf(t)D(t)]=∑i=1r‍∑j=1r‍ui(s(t))uj(s(t))[BjBfiDj],E-(t)=[E(t)-Cf(t)]=∑i=1r‍∑i=1r‍ui(s(t))uj(s(t))[Ej-Cfi].Before ending this section, we introduce the following definition, which will be used in the derivation of our main results.Definition 1 (ℓ2-ℓ∞ performance).
Given a scalarγ>0, the system (1) is said to be with ℓ2-ℓ∞ performance if the system (1) is asymptotically stable and the output z(t) satisfies
(7)∥z(t)∥∞≤γ∥ω(t)∥2
for all nonzero ω∈ℓ2[0,+∞] under zero-initial condition, where,
(8)∥ω(t)∥2=∫0∞‍ωT(t)ω(t),∥z(t)∥∞=supt{zT(t)z(t)}.Here, we want to design a suitable filter (4) for the system (1) with a ℓ2-ℓ∞ performance.


## 2.1. Plant Form

Rule
i. IF s1(t) is Fi1, s2(t) is Fi2, and sn(t) is Fin, then
(1)x˙(t)=Aix(t)+∑k=1q‍Adikx(t-dk(t))+Biω(t),y(t)=Cix(t)+∑k=1q‍Cdikx(t-dk(t))+Diω(t),z(t)=Eix(t),x(t)=ϕ(t),t∈[-h,0],h=max{hk},k=1,2,…,q,
where s1(t),s2(t),…,  and  sn(t) are the premise variables that are measurable, and each Fij  (j=1,2,…,n) is a fuzzy set. x(t)∈Rn is the state variables. y(t)∈Rm is the measured output of the system. z(t)∈Rp is the signal to be estimated. ω(t)∈Rb is the disturbance input. r is the number of IF-THEN rules. Also 0≤dk(t)≤hk is the time-varying delay in the state, and it is assumed that d˙k(t)≤dk. That is, the derivative of time-varying delay function is continuous and bounded. q is the number of time delays. ϕ(t) is a vector-valued initial continuous function.By using center-average defuzzifier, product inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be expressed by the following global model:
(2)x˙(t)=∑i=1r‍ui(s(t))[Aix(t)+∑k=1q‍Adikx(t-dk(t))+Biω(t)]=A(t)x(t)+∑k=1q‍Adkx(t-dk(t))+B(t)ω(t),y(t)=∑i=1r‍ui(s(t))[Cix(t)+∑k=1q‍Cdikx(t-dk(t))+Diω(t)]=C(t)x(t)+∑k=1q‍Cdkx(t-dk(t))+D(t)ω(t),z(t)=∑i=1r‍ui(s(t))[Eix(t)]=E(t)x(t)
with
(3)ui(s(t))=αi(s(t))∑i=1rαi(s(t)),αi(s(t))=∏j=1n‍Fij(sj(t)),
in which Fij(sj(t)) is the grade of membership of sj(t) in Fij. It is assumed that αi(s(t))≥0,  i=1,2,…,r,   ∑i=1r‍αi(s(t))>0 for all t. Therefore, ui(s(t))≥0 and ∑i=1r‍ui(s(t))=1 for all t. In this paper, we study the following filter form of order l (l=n for full-order filter, and 1≤l<n for reduced-order filter):
(4)x˙f(t)=∑i=1r‍ui(s(t))[Afix(t)+Bfiy(t)],zf(t)=∑i=1r‍Cfixf(t),
where the Afi,Bfi,andCfi are the filter parameters to be designed. Combining (2) and (4) and defining ξ(t)=[xT(t),xfT(t)]T, e(t)=z(t)-zf(t), and K=[In×n0n×l], we have the filtering-error system:
(5)ξ˙(t)=A-(t)ξ(t)+∑k=1q‍A-dk(t)Kξ(t-dk(t))+B-(t)ω(t),e(t)=E-(t)ξ(t),
where ξ(t)≐[ϕT(t),0]T for t∈[-h,0] and
(6)A-(t)=[A(t)0Bf(t)C(t)Af(t)]=∑i=1r‍∑j=1r‍ui(s(t))uj(s(t))[Aj0BfiCjAfi],A-dk(t)=[Adk(t)Bf(t)Cd(t)]=∑i=1r‍∑j=1r‍ui(s(t))uj(s(t))[AdjkBfiCdj],B-(t)=[B(t)Bf(t)D(t)]=∑i=1r‍∑j=1r‍ui(s(t))uj(s(t))[BjBfiDj],E-(t)=[E(t)-Cf(t)]=∑i=1r‍∑i=1r‍ui(s(t))uj(s(t))[Ej-Cfi].Before ending this section, we introduce the following definition, which will be used in the derivation of our main results.Definition 1 (ℓ2-ℓ∞ performance).
Given a scalarγ>0, the system (1) is said to be with ℓ2-ℓ∞ performance if the system (1) is asymptotically stable and the output z(t) satisfies
(7)∥z(t)∥∞≤γ∥ω(t)∥2
for all nonzero ω∈ℓ2[0,+∞] under zero-initial condition, where,
(8)∥ω(t)∥2=∫0∞‍ωT(t)ω(t),∥z(t)∥∞=supt{zT(t)z(t)}.Here, we want to design a suitable filter (4) for the system (1) with a ℓ2-ℓ∞ performance.


## 3. Main Results

In this section, the conditions to assure the system (1) asymptotically stable with ℓ2-ℓ∞ performance γ for the T-S fuzzy filtering-error systems are presented (Lemmas 2 and 3). Then, based on the conditions, a filter is given in terms of LMIs.Lemma 2.
Givenγ>0, if there exist common matrices 0<P∈R(n+l)×(n+l), 0<Qk∈Rn×n, 0<Rk∈Rn×n, Yk(t)∈Rn×n, Tk(t)∈Rn×n, Uk∈Rb×n, k=1,2,…,q, and Af(t), Bf(t), and Cf(t) satisfying
(9)[Ξ11Ξ12Ξ13*-Ξ220**-Ξ33]<0,(10)[PE-T(t)*γ2I]>0,
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,then the system (5) is asymptotically stable with an ℓ2-ℓ∞ performance γ.Proof.
Choose a Lyapunov-Krasovskii functional candidate as(12)V(t)=V1(t)+V2(t)+V3(t),
where
(13)V1(t)=ξT(t)Pξ(t),V2(t)=∑j=1q‍∫t-dj(t)t‍ξT(s)KTQjKξ(s)ds,V3(t)=∑j=1q‍∫-hj0‍∫t+θt‍ξ˙T(s)KTRjKξ˙(s)ds.
The time derivative ofV(t) along the solution of (5) is computed as follows:
(14)V˙1(t)=ξ˙T(t)Pξ(t)+ξT(t)Pξ˙(t)=(A-(t)ξ(t)+∑k=1q‍A-dkKξ(t-dk(t))+B-(t)ω(t)∑k=1q‍)TPξ(t)+ξT(t)P(A-(t)ξ(t)+∑k=1q‍A-dkKξ(t-dk(t))+B-(t)ω(t)∑k=1q‍),(15)V˙2(t)≤∑k=1q‍{(t-dk(t))KTξT(t)KTQkKξ(t)≤∑k=1q‍==-(1-dk)ξT(t-dk(t))KTQkKξ(t-dk(t))},(16)V˙3(t)=∑k=1q‍hkξ˙T(t)KTRkKξ˙(t)-∑k=1q‍∫t-dk(t)t‍ξ˙T(s)KTRkKξ˙(s)ds.
Applying free-weighting matrix method [37], using the Newton-Leibniz formula that
(17)0=2[ξT(t)KTYk(t)+ξT(t-dk(t))KTTk(t)+ωT(t)Uk(t)]×[Kξ(t)-Kξ(t-dk(t))-∫t-dk(t)t‍Kξ˙(s)ds],×[Kξ(t)-Kξ(t-dk(t))---]k=1,2,…,q,
and defining(18)ηkT(t)=[ξT(t),ξT(t-dk(t))KT,ωT(t)],ηkT(t)=[ξT(t),ξT(t-==)]k=1,2,…,q,ηT(t)=[ξT(t),ξT(t-d1(t))KT,…,ξT(t-dq(t))KT,ωT(t)],MT(t)=[YT(t),TT(t),UT(t)],
we can know that(19)-∫t-dk(t)t‍ξ˙T(s)KTRkKξ˙(s)ds=-∫t-dk(t)t‍ξ˙T(s)KTRkKξ˙(s)ds+2[ξT(t)KTYk(t)+ξT(t-dk(t))KTTk(t)+ωT(t)Uk(t)]×[Kξ(t)-Kξ(t-dk(t))-∫t-dk(t)t‍Kξ˙(s)ds]≤hkηkTM(t)Rk-1MT(t)ηk-∫t-dk(t)t‍[ηkTM(t)+ξ˙T(s)KTRk]×Rk-1[MT(t)ηk+RkKξ˙(s)]ds+2[ξT(t)KTYk(t)+ξT(t-dk(t))KTTk(t)+ωT(t)KTUk(t)ξT(t-dk(t))]×[Kξ(t)-Kξ(t-dk(t))].
Consider the index(20)J≜V(t)-∫0t‍ωT(s)ω(s)ds.
Then for any nonzero ω∈ℓ2[0,+∞] under zero-initial condition,
(21)J≜∫0t‍(V˙(s)-ωT(s)ω(s))ds.
After substitution ofξ˙(t) into (16) with (5). and taking into consideration (19), one has from (14), (15), and (16) that
(22)J≤ηT(t){Ξ11+Ξ12Ξ22-1Ξ12T+Ξ13Ξ33-1Ξ13T}η(t).
Applying the Schur complement to (22), we know that (9) guarantees J<0, which implies that
(23)ξT(t)Pξ(t)≤V(t)<∫0t‍ωT(s)ω(s)ds.
On the other hand, using the Schur complement to (10), we can know that E-T(t)E-(t)<γ2P. Then it can be easily got that for all t≥0(24)eT(t)e(t)=ξT(t)E-T(t)E-(t)ξ(t)<γ2ξT(t)Pξ(t)<γ2∫0t‍ωT(s)ω(s)ds<γ2∫0∞‍ωT(s)ω(s)ds.
Taking the supremum overt≥0 yields ∥e(t)∥∞2<γ2∥ω(t)∥22 for all nonzero ω∈ℓ2[0,+∞].
Next, we prove the asymptotic stability of system (5) when ω=0. Choose a Lyapunov-Krasovskii functional V(ξt) as in (12), where ξt=ξ(t-α), α∈[-h,0]. It is easy to find that there exist two scalars c1>0 and c2>0 such that
(25)c1|ξ|2≤V≤c2supα∈[-h,0]|ξt|2.
Similar to the above deduction, we can know from (9) that the time derivative of V along the solution of (5) with ω=0 satisfies V˙<0. This proves the asymptotic stability of system (5) with ω=0 according to the same method of [19]. This completes the proof.Lemma2 is the sufficient condition for the ℓ2-ℓ∞ filter design which contains the coupled matrix variables in the matrix inequality. Using the decoupling technique as follows, we can transform Lemma 2 into another form.Lemma 3.
Givenγ>0, if there exist common matrices 0<P∈R(n+l)×(n+l), 0<Qk∈Rn×n, 0<Rk∈Rn×n, Yk(t)∈Rn×n, Tk(t)∈Rn×n, Uk∈Rb×n, k=1,2,…,q, and Af(t), Bf(t), and Cf(t), such that (9), (10) hold if and only if there exist matrices 0<Ω, 0<F, 0<Qk, Yk(t), Tk(t), Uk(t), k=1,2,…,q, and A^f(t), B^f(t), and C^f(t) such that the following inequalities hold:
(26)Σ(t)<0,(27)[ΩEFET(t)*F-C^fT(t)**γ2I]>0,
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Φ
0
=
[
Ω
A
(
t
)
+
E
B
^
f
(
t
)
C
(
t
)
E
A
^
f
(
t
)
F
E
T
A
(
t
)
+
B
^
f
(
t
)
C
(
t
)
A
^
f
(
t
)
]
+
[
Ω
A
(
t
)
+
E
B
^
f
(
t
)
C
(
t
)
E
A
^
f
(
t
)
F
E
T
A
(
t
)
+
B
^
f
(
t
)
C
(
t
)
A
^
f
(
t
)
]
T
+
∑
k
=
1
q
‍
(
K
T
Q
k
K
+
K
T
Y
k
(
t
)
K
+
K
T
Y
k
T
(
t
)
K
)
,
Φ
1
=
[
Ω
A
d
1
(
t
)
+
E
B
^
f
(
t
)
C
d
1
(
t
)
F
E
T
A
d
1
(
t
)
+
B
^
f
(
t
)
C
d
1
(
t
)
]
-
K
T
Y
1
(
t
)
+
K
T
T
1
T
(
t
)
,
Φ
q
=
[
Ω
A
d
q
(
t
)
+
E
B
^
f
(
t
)
C
d
q
(
t
)
F
E
T
A
d
q
(
t
)
+
B
^
f
(
t
)
C
d
q
(
t
)
]
-
K
T
Y
q
(
t
)
+
K
T
T
q
T
(
t
)
,
Φ
q
+
1
=
[
Ω
B
(
t
)
+
E
B
^
f
(
t
)
D
(
t
)
F
E
T
B
(
t
)
+
B
^
f
(
t
)
D
(
t
)
]
+
K
T
U
q
T
(
t
)
,
Σ
12
=
[
h
1
A
-
T
(
t
)
K
T
R
1
⋯
h
q
A
-
T
(
t
)
K
T
R
q
h
1
A
-
d
1
T
(
t
)
R
1
⋯
h
q
A
-
d
1
T
(
t
)
R
q
⋮
⋮
⋮
h
1
A
-
d
q
T
(
t
)
R
1
⋯
h
q
A
-
d
q
T
(
t
)
R
q
h
1
B
-
T
(
t
)
R
1
⋯
h
q
B
-
T
(
t
)
R
q
]
,
Σ
13
=
[
h
1
K
T
Y
1
(
t
)
⋯
h
q
K
T
Y
q
(
t
)
h
1
T
1
(
t
)
⋯
0
⋮
⋱
⋮
0
⋯
h
q
T
q
(
t
)
h
1
U
1
(
t
)
⋯
h
q
U
q
(
t
)
]
,
Σ
22
=
Σ
33
=
diag

{
R
1
,
…
,
R
q
}
,
E
=
[
I
l
×
l
0
l
×
(
n
-
l
)
]
,then the system (5) is asymptotically stable with an ℓ2-ℓ∞ performance γ.Proof.
Necessity. Suppose (9), (10) hold. Partition as
(29)P=[ΩESSTETW],
where Ω>0, W>0, and S is invertible. Let
(30)H=[I00SW-1].
We pre- and postmultiplyD and its transpose to (9) and (10), respectively, where
(31)D=diag{H,I,I,I,I,I,I,I,I,I,I}.
Apply the changes of variables such that(32)F=SW-1ST,A^f(t)=SAf(t)W-1ST,B^f(t)=SBf(t),C^(t)=Cf(t)W-1ST.
Then we obtain (26) and (27).
Sufficiency. Suppose that (26) and (27) hold for Ω>0, F>0,   Qk>0,  Yk(t), Tk(t), Uk(t), k=1,2,…,q, and A^f(t), B^f(t), and C^f(t). Choose two matrices with W>0 and S being invertible such that F=SW-1ST. Let P and H be defined as in (29) and (30). Then P>0 is concluded from (27). Pre- and postmultiply D-1 and D-T to (26) and (27), respectively. We can get (9) and (10) with the changes of variables as
(33)Af(t)=S-1A^f(t)S-TW,Bf(t)=S-1B^f(t),Cf(t)=C^f(t)S-TW.
This completes the proof.Theorem 4.
Givenγ>0, if there exist common matrices Ω>0, F>0, Qk>0, Tki, Yki, Uki, k=1,2,…,q, and A^fi, B^fi, and C^fi such that the following inequalities hold:
(34)Θij+Θji<0,i≤j≤r,(35)[ΩEFEiT*F-C^fiT**γ2I]>0,
thenℓ2-ℓ∞ filter parameters in (4) are given by
(36)Afi=F-1A^fi,Bfi=F-1B^fi,Cfi=C^fi,
where(37)
Θ
i
j
=
[
Ψ
11
Ψ
12
Ψ
13
*
-
Ψ
22
0
*
*
-
Ψ
13
]
,
Ψ
11
=
[
Φ
0
Φ
1
⋯
Φ
q
Φ
q
+
1
*
-
(
1
-
d
k
)
Q
1
-
T
1
i
-
T
1
i
T
0
0
-
U
1
i
T
*
*
⋱
⋮
⋮
*
*
*
-
(
1
-
d
q
)
Q
q
-
T
q
i
-
T
q
i
T
-
U
q
i
T
*
*
*
*
-
I
]
,
Φ
0
=
[
Ω
A
i
+
E
B
^
f
j
C
i
E
A
^
f
j
F
E
T
A
i
+
B
^
f
j
C
i
A
^
f
j
]
+
[
Ω
A
i
+
E
B
^
f
j
C
i
E
A
^
f
j
F
E
T
A
i
+
B
^
f
j
C
i
A
^
f
j
]
T
+
∑
k
=
1
q
‍
(
K
T
Q
k
K
+
K
T
Y
k
i
K
+
K
T
Y
k
i
T
K
)
,
Φ
1
=
[
Ω
A
d
1
i
+
E
B
^
f
j
C
d
1
i
F
E
T
A
d
1
i
+
B
^
f
j
C
d
1
i
]
-
K
T
Y
1
i
+
K
T
T
1
i
T
,
Φ
q
=
[
Ω
A
d
q
i
+
E
B
^
f
j
C
d
q
i
F
E
T
A
d
q
i
+
B
^
f
j
C
d
q
i
]
-
K
T
Y
q
i
+
K
T
T
q
i
T
,
Φ
q
+
1
=
[
Ω
B
i
+
E
B
^
f
j
D
i
F
E
T
B
i
+
B
^
f
j
D
i
]
+
K
T
U
q
i
T
,
Ψ
12
=
[
h
1
(
A
i
K
)
T
R
1
⋯
h
q
(
A
i
K
)
T
R
q
h
1
A
d
1
i
T
R
1
⋯
h
q
A
d
1
i
T
R
q
⋮
⋮
⋮
h
1
A
d
q
i
T
R
1
⋯
h
q
A
d
q
i
T
R
q
h
1
B
i
T
R
1
⋯
h
q
B
i
T
R
q
]
,
Ψ
13
=
[
h
1
K
T
Y
1
i
⋯
h
q
K
T
Y
q
i
h
1
T
1
i
⋯
0
⋮
⋱
⋮
0
⋯
h
q
T
q
i
h
1
U
1
i
⋯
h
q
U
q
i
]
,
Ψ
22
=
Ψ
33
=
diag

{
R
1
,
…
,
R
q
}
,
E
=
[
I
l
×
l
0
l
×
(
n
-
l
)
]
.Proof.
By considering (6),we know that Σ(t) in (26) and the Θij in (34) satisfy
(38)Σ(t)=∑i=1r‍∑j=1r‍ui(s(t))uj(s(t))Θij=∑i=1r‍ui2(s(t))Θii+∑i<jr‍ui(s(t))uj(s(t))(Θij+Θji).
Based on Lemmas2 and 3, the ℓ2-ℓ∞ filter matrices are given by (33). Under the transformation S-TWxf(t), the filter matrices functions can be of the following forms:
(39)Af(t)=S-TW(S-1A^f(t)S-TW)W-1ST,Bf(t)=S-TW(S-1B^f(t)),Cf(t)=(C^f(t)S-TW)W-1ST.
Hence, the filter in (4) can be got by (36). This completes the proof.


## 4. Simulation

In this section, we give a numerical example to illustrate the use of the present method. Consider the system of the form (1) with two plants (r=2) and two delays (q=2), where
(40)A1=[-2.10.11-2],Ad11=[-1.10.1-0.8-0.9],Ad21=[-0.10.1-0.1-0.3],A2=[-1.90-0.2-1.1],Ad12=[-0.90-1.1-1.2],Ad22=[-0.10-0.2-0.2],B1=[1-0.2],C1=[10],Cd11=[-0.80.6],Cd21=[-0.10.2],D1=0.3,E1=[1-0.5],B2=[0.30.1],C2=[0.5-0.6],Cd12=[-0.10.1],Cd22=[-0.10.1],D2=-0.6,E2=[-0.20.3].Here, we only consider the full-order filter design. First, we seth1=0.5, h2=0.3. Figure 1 shows the ℓ2-ℓ∞ gain bound got from Theorem 4 in pointwise manner, where the derivatives of time-varying delay function bound d1∈[01.9], d2∈[01.9]. We can find that the maximum ℓ2-ℓ∞ gain is γ=0.3714 at the point d1=1.9, d2=1.9. The minimum ℓ2-ℓ∞ gain is γ=0.3358 at the point d1=0, d2=0.Figure 1
ℓ
2-ℓ∞ gain bound from Theorem 4 with h1=0.5 and  h2=0.3.Then, we setd1=0.4, d2=0.4. Figure 2 shows the ℓ2-ℓ∞ gain bound got from Theorem 4 in pointwise manner, where the time-varying delay function bound h1∈[01.1], h2∈[01.1]. We can find that the maximum ℓ2-ℓ∞ gain is γ=0.5257 at the point h1=1.1, h2=1.1. The minimum ℓ2-ℓ∞ gain is γ=0.0055 at the point h1=0, h2=0.Figure 2
ℓ
2-ℓ∞ gain bound from Theorem 4 with d1=0.4  and d2=0.4.Now, We choose both time-varying delays to be0.4sin(t)+0.7 which gives h1=1.1, h2=1.1, d1=0.4, and d2=0.4, and we get a set of feasible solutions to Theorem 4 with the ℓ2-ℓ∞ gain γ=0.5257 and
(41)F=[3.8096-1.2607-1.26070.6116],A^f1=[-12.90711.85165.1902-1.1454],A^f2=[-10.02764.26533.0519-1.5986],B^f1=[-5.02661.9199],B^f2=[-4.19591.8120],C^f1=[0.8524-0.3426],C^f2=[-0.26290.1428].Therefore, we can solve the corresponding filter from (36) as
(42)Af1=[-1.8241-0.42064.7259-2.7398],Af2=[-3.08580.8012-1.3706-0.9622],Bf1=[-0.88291.3191],Bf2=[-0.38062.1781],Cf1=[0.8524-0.3426],Cf2=[-0.26290.1428].To illustrate the performance of the designed filter, we assume zero initial condition and the disturbanceω(t) as follows:
(43)ω(t)={1,0.5≤t≤1-1,1.5≤t≤20,otherwise.The simulation result of signale(t) is given in Figure 3.Figure 3
The signale(t).The resulting outputℓ∞-norm of the filtering-error system is about 0.15, while ∥ω(t)∥2=1. Simulation result for the ratio of the output ℓ∞-norm to the disturbance ℓ2-norm is 0.15, and 0.15<γ=0.5257 with h1=1.1, h2=1.1, d1=0.4, and d2=0.4.


## 5. Conclusion

The problem onℓ2-ℓ∞ filter design has been addressed for a class of TCS fuzzy-model-based systems with multiple time-varying delays. Based on the Lyapunov-Krasovskii functional approach and free-weighting matrix method, a sufficient condition for the existence of ℓ2-ℓ∞ filter, which stabilizes the T-S fuzzy-model-based filtering-error systems and guarantees a prescribed level on disturbance attenuation, has been obtained in terms of LMI form. The numerical example has shown the effectiveness of the proposed method. In addition, the basis-dependent Lyapunov-Krasovskii functional approach for filtering problems of T-S fuzzy delayed systems is also challenging, and could be our further work.



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*Source: 298268-2013-09-30.xml*