This paper investigates the ℓ2-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.
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.
Several ℋ∞ 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 of ℓ2-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 an ℓ2-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 Section 2, 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 scalar γ>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.
Given γ>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,
where
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 of V(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 of ξ˙(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 over t≥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.
Lemma 2 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.
Given γ>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,
where
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 postmultiply D 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.
Given γ>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,
then ℓ2-ℓ∞ filter parameters in (4) are given by
(36)Afi=F-1A^fi,Bfi=F-1B^fi,Cfi=C^fi,
where
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 Lemmas 2 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 set h1=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.
ℓ2-ℓ∞ gain bound from Theorem 4 with h1=0.5 and h2=0.3.
Then, we set d1=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.
ℓ2-ℓ∞ gain bound from Theorem 4 with d1=0.4 and d2=0.4.
Now, We choose both time-varying delays to be 0.4sin(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 disturbance ω(t) as follows:
(43)ω(t)={1,0.5≤t≤1-1,1.5≤t≤20,otherwise.
The simulation result of signal e(t) is given in Figure 3.
The signal e(t).
The resulting output ℓ∞-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 on ℓ2-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.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (61203005), in part by Harbin Engineering University Central University Foundation Research Special Fund (HEUCFR1024), and in part by Postdoctoral Science-Research Developmental Foundation of Heilongjiang Province (LBH-Q12130) and Heilongjiang Province Natural Science Foundation (F201221).
CaoY.-Y.SunY.-X.ChengC.Delay-dependent robust stabilization of uncertain systems with multiple state delays19984311160816122-s2.0-003220410410.1109/9.728880MR1652855GaoH.LamJ.WangC.WangY.Delay-dependent output-feedback stabilization of discrete-time systems with time-varying state delay20041516691698WuL.ZhengW. X.Weighted ℋ∞ model reduction for linear switched systems with time-varying delay200945118619310.1016/j.automatica.2008.06.024MR2531510LiuP.Delay-dependent robust exponential stabilization criteria for uncertain time-varying delay singular systems201391165178WuL.SuX.ShiP.Sliding mode control with bounded ℓ2 gain performance of Markovian jump singular time-delay systems20124881929193310.1016/j.automatica.2012.05.064MR2950452ShiP.BoukasE.-K.ShiY.AgarwalR. K.Optimal guaranteed cost control of uncertain discrete time-delay systems200315724354512-s2.0-004252103610.1016/S0377-0427(03)00433-3MR1998344WuL.SuX.ShiP.QiuJ.Model approximation for discrete-time state-delay systems in the T-S fuzzy framework20111923663782-s2.0-7995364472410.1109/TFUZZ.2011.2104363WuL.SuX.ShiP.QiuJ.A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems20114112732862-s2.0-7955168884610.1109/TSMCB.2010.2051541YangR.ShiP.LiuG.-P.GaoH.Network-based feedback control for systems with mixed delays based on quantization and dropout compensation20114712280528092-s2.0-8115514819810.1016/j.automatica.2011.09.007MR2886956CaoY.-Y.FrankP. M.Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach2000822002112-s2.0-003372118910.1109/91.842153SongX.LuJ.XuS.ShenH.LuJ.Robust stabilization of state delayed T-S fuzzy systems with input saturation via dynamic anti-windup fuzzy design2011712666566762-s2.0-80455137246WuL.SuX.ShiP.QiuJ.Model approximation for discrete-time state-delay systems in the T-S fuzzy framework20111923663782-s2.0-7995364472410.1109/TFUZZ.2011.2104363WuL.SuX.ShiP.QiuJ.A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems20114112732862-s2.0-7955168884610.1109/TSMCB.2010.2051541LiF.ZhangX.Delay-range-dependent robust ℋ∞ filtering for singular LPV systems with time variant delay201391339353WilsonD. A.Convolution and Hankel operator norms for linear systems198934194972-s2.0-002447949010.1109/9.8655MR970936GaoH.WangC.Robust ℓ2-ℓ∞ filtering for uncertain systems with multiple time-varying state delays200350459459910.1109/TCSI.2003.809816MR1983029WangZ.YangF.Robust filtering for uncertain linear systems with delayed states and outputs20024911251302-s2.0-003612441610.1109/81.974887QiuJ.FengG.YangJ.Improved delay-dependent ℋ∞ filtering design for discrete-time polytopic linear delay systems20085521781822-s2.0-5024917026610.1109/TCSII.2007.910962QiuJ.FengG.YangJ.New results on robust ℋ∞ filtering design for discrete-time piecewise linear delay systems20098211831942-s2.0-5494915491310.1080/00207170802036196MR2523742ChenX.LamJ.GaoH.ZhouS.Stability analysis and control design for 2-D fuzzy systems via basis-dependent Lyapunov functions20111212-s2.0-8155523100210.1007/s11045-011-0166-zYaoX.WuL.ZhengW. X.WangC.Robust ℋ∞ filtering of Markovian jump stochastic systems with uncertain transition probabilities20114271219123010.1080/00207720903513350MR2817856YaoX.WuL.ZhengW. X.Fault detection filter design for Markovian jump singular systems with intermittent measurements2011597309931092-s2.0-7995918705310.1109/TSP.2011.2141666MR2839337LinC.WangQ.-G.LeeT. H.ChenB.ℋ∞ filter design for nonlinear systems with time-delay through T-S fuzzy model approach20081637397462-s2.0-4554909963110.1109/TFUZZ.2007.905915SuY.ChenB.LinC.Delay-dependent ℋ∞ filtering for nonlinear systems via T-S fuzzy model approachProceedings of the Chinese Control and Decision Conference (CCDC '09)June 20092212262-s2.0-7044941388210.1109/CCDC.2009.5195117LiJ.ZhouS.ChaiL.ZhangY.ℋ∞ filter design for discrete-time-delay T-S fuzzy systemsProceedings of the 16th IEEE International Conference on Control Applications, (CCA '07)October 2007155715602-s2.0-4304916572810.1109/CCA.2007.4389458QiuJ.FengG.YangJ.A new design of delay-dependent robust ℋ∞ filtering for discrete-time T-S fuzzy systems with time-varying delay2009175104410582-s2.0-7035007476310.1109/TFUZZ.2009.2017378GaoH.WangC.Delay-dependent robust ℋ∞ and ℓ2-ℓ∞ filtering for a class of uncertain nonlinear time-delay systems20034891661166610.1109/TAC.2003.817012MR2000128ChenM.FengG.MaH.A delay-dependent approach to ℋ∞ filtering for fuzzy time-varying delayed systemsProceedings of the IEEE International Conference on Fuzzy Systems (FUZZY '07)July 20072-s2.0-5024910537710.1109/FUZZY.2007.4295440LiuH.ChenC.GuanX.WuX.ℋ∞ piecewise filtering for continuous T-S fuzzy systems with time delaysProceedings of the IEEE International Conference on Fuzzy Systems (FUZZ '08)June 2008101210172-s2.0-5524912383910.1109/FUZZY.2008.4630493FengG.Robust ℋ∞ filtering of fuzzy dynamic systems20054126586702-s2.0-2494453882110.1109/TAES.2005.1468755WuL.WangZ.Fuzzy filtering of nonlinear fuzzy stochastic systems with time-varying delay2009899173917532-s2.0-6734918795910.1016/j.sigpro.2009.03.011LunS.ZhangH.LiuD.Fuzzy ℋ∞ filtering of discrete-time nonlinear systems with multiple time delaysProceedings of the 20th IEEE International Symposium on Intelligent Control, ISIC '05 and the13th Mediterranean Conference on Control and Automation (MED '05)June 2005102310282-s2.0-3374518447210.1109/.2005.1467154LinY.-C.LoJ.-C.Robust mixed ℋ2/ℋ∞ filtering for time-delay fuzzy systems2006548289729092-s2.0-3374650620410.1109/TSP.2006.875380ZhangH.LunS.LiuD.Fuzzy ℋ∞ filter design for a class of nonlinear discrete-time systems with multiple time delays20071534534692-s2.0-3425082039710.1109/TFUZZ.2006.889841ZhangH.DangC.ZhangJ.Decentralized fuzzy ℋ∞ filtering for nonlinear interconnected systems with multiple time delays2010404119712032-s2.0-7795475737910.1109/TSMCB.2010.2042956GongC.SuB.Robust ℓ2-ℓ∞ filtering of convex polyhedral uncertain time-delay fuzzy systems2008447938022-s2.0-48249156941HeY.WuM.SheJ.-H.LiuG.-P.Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties20044958288322-s2.0-294259446110.1109/TAC.2004.828317MR2057826