# Improved Results on FuzzyH∞ Filter Design for T-S Fuzzy Systems

Authors: Jiyao An; Guilin Wen; Wei Xu
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

The fuzzyH∞ filter design problem for T-S fuzzy systems with interval time-varying delay is investigated. The delay is considered as the time-varying delay being either differentiable uniformly bounded with delay derivative in bounded interval or fast varying (with no restrictions on the delay derivative). A novel Lyapunov-Krasovskii functional is employed and a tighter upper bound of its derivative is obtained. The resulting criterion thus has advantages over the existing ones since we estimate the upper bound of the derivative of Lyapunov-Krasovskii functional without ignoring some useful terms. A fuzzy H∞ filter is designed to ensure that the filter error system is asymptotically stable and has a prescribed H∞ performance level. An improved delay-derivative-dependent condition for the existence of such a filter is derived in the form of linear matrix inequalities (LMIs). Finally, numerical examples are given to show the effectiveness of the proposed method.


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## Body

## 1. Introduction

During the last decades, the filtering problem has attracted many researchers to study through various methodologies, see, for example, [1–20] and the references therein, in which these methods mostlyconsist of two main approaches, namely, the Kalman filtering approach [1–3] and the H∞ filtering approach [4–17]. In contrast with the Kalman filtering, the H∞ filtering approach does not require the exact knowledge of the statistics of the external noise signals and it is insensitive to the uncertainties both in the exogenous signastatistics and in dynamic models. This advantage renders the H∞ filtering approach very appropriate to some practical applications. Recently, the filter design contains two cases of filtering technique, that is, L2-L∞ filtering technique [18–20] and the H∞ filtering technique [4–17].On the other hand, Takagi-Sugeno (T-S) fuzzy model can provide an effective way to represent a complex nonlinear system into a weighted sum of some simple linear subsystems [8, 21, 22], which has been an increasing interest in the study of T-S fuzzy systems. In recent years, T-S fuzzy model approach has been extended to H∞ filter or controller design [4–6, 9, 10, 12, 15–21, 23–35]. For instance, the stability analysis and stabilization synthesis problems of T-S fuzzy systems were studied in [21, 29, 30, 33–35], while fuzzy controllers were designed in [23–28]. One set of fuzzy H∞ filters for a class of T-S fuzzy systems was designed in [32]. However, the above-mentioned works use common Lyapunov-Krasovskii functional, and the results under a common Lyapunov method are quite conservative. To reduce the conservatism, a fuzzy weighting-dependent Lyapunov method has been proposed in [6], which is effective in reducing conservatism of previous results on fuzzy systems. More recently, Lin et al. [4] and Su et al. [5] have concerned with H∞ filtering of nonlinear continuous-time state-space models with time-varying delays via T-S fuzzy model approach. However, some negative semidefinite terms are ignored and the lower bound of time delay is restricted to be zero, see, for example, [4–6] and the references therein. Qiu et al. [36] investigated the problem of delay-dependent robust stability and H∞ filtering design for a class of uncertain continuous-time nonlinear systems with time-varying state delay represented by T-S fuzzy models. However, there is room for further investigation to reduce the conservativeness of the filter design. This motivates the current research.In this paper, we discuss the fuzzyH∞ filter design problem for T-S fuzzy systems with interval time-varying delay. Our aim is to design a suitable fuzzy filter, which ensures both the fuzzy stability and a prescribed performance level of the filter error system. By constructing a Lyapunov-Krasovskii functional, estimating the time derivative of the Lyapunov-Krasovskii functional less conservatively, and adopting convex optimization approach, an improved delay-derivative-dependent condition for the solvability of fuzzy H∞ filter design problem is proposed in terms of linear matrix inequalities (LMIs). Two examples are used to compare with the previous literatures and demonstrate the effectiveness of the proposed method.The rest of this paper is organized as follows: The fuzzyH∞ filtering problem is formulated in Section 2; the fuzzy H∞ performance analysis is derived in Section 3; and fuzzy H∞ filter design is addressed in Section 4. Numerical examples are provided in Section 5, and Section 6 concludes this paper.


## 2. Problem Formulation

Consider a nonlinear system with interval time-varying delay which could be approximated by a class of T-S fuzzy systems with interval time-varying delays. The T-S fuzzy model withr plant rules can be described by:Plant rulei: IF θ1(t) is Ni1 and⋯and θp(t) is Nip, THEN(2.1)ẋ(t)=Aix(t)+Aτix(t-τ(t))+Biw(t),y(t)=Cix(t)+Cτix(t-τ(t))+Diw(t),z(t)=Lix(t)+Lτix(t-τ(t))+Giw(t),x(t)=ϕ(t),∀t∈[-hb,0],
where 0≤ha≤τ(t)≤hb, and x(t)∈ℝn is the state vector; y(t)∈ℝm is the measurements vector; w(t)∈ℝq is the disturbance signal vector which belongs to L2[0,∞); z(t)∈ℝp is the signal vector to be estimated; ϕ(t) is the continuous initial vector function defined on [-hb,0]; The system coefficient matrices are constant real matrices with appropriate dimensions, where i=1,2,…,r and r is the number of IF-THEN rules; θj(t),(j=1,2,…,p) are the premise variables; Ni1,Ni2,…,Nip are the fuzzy sets. For the sake of convenience, we denote δh=hb-ha.The time-varying delayτ(t) is assumed to be either differentiable with(2.2)d1≤τ̇(t)≤d2,
where d1 and d2 are given bounds, or fast-varying (with no restrictions on the delay derivative).The fuzzy system (2.1) is supposed to have singleton fuzzifier, product inference and centroid defuzzifier. The final output of the fuzzy system is inferred as follows:
(2.3)ẋ(t)=∑i=1rhi(θ(t))[Aix(t)+Aτix(t-τ(t))+Biw(t)],y(t)=∑i=1rhi(θ(t))[Cix(t)+Cτix(t-τ(t))+Diw(t)],z(t)=∑i=1rhi(θ(t))[Lix(t)+Lτix(t-τ(t))+Giw(t)],x(t)=ϕ(t),∀t∈[-hb,0],
where for i=1,2,…,r,(2.4)hi(θ(t))=μi(θ(t))∑i=1rμi(θ(t)),μi(θ(t))=∏j=1pNij(θj(t))
and Nij(θj(t)) is the membership function of θj(t) in Nij. Here μi(θ(t))≥0. Here, we assume that  μi(θ(t))>0, and ∑i=1rhi(θ(t))=1.Our aim is to design the following fuzzy filter.Rulei: IF θ1(t) is Ni1 and⋯and θp(t) is Nip, THEN(2.5)x̂̇(t)=Afix̂(t)+Bfiy(t),x̂(0)=0,ẑ(t)=Cfix̂(t)+Dfiy(t),(i=1,2,…,r),
where x̂(t)∈ℝn is the filter state, ẑ(t)∈ℝp is the estimation of z(t) in fuzzy system (2.1), the constant matrices Afi∈ℝn×n, Bfi∈ℝn×m, Cfi∈ℝp×n, Dfi∈ℝp×m are the filter matrices to be determined. The final fuzzy filter of fuzzy system (2.1) is thus inferred as follows(2.6)x̂̇(t)=∑i=1rhi(θ(t))[Afix̂(t)+Bfiy(t)],x̂(0)=0,ẑ(t)=∑i=1rhi(θ(t))[Cfix̂(t)+Dfiy(t)].Defining the augmented state vectorx̃(t):=col{x(t)x̂(t)}, e(t):=z(t)-ẑ(t), from (2.3) and (2.6), we can then obtain the following filtering error system:(2.7)x̃̇(t)=Ã(t)x̃(t)+Ãτ(t)Ex̃(t-τ(t))+B̃(t)w(t),e(t)=C̃(t)x̃(t)+C̃τ(t)Ex̃(t-τ(t))+D̃(t)w(t),x̃(t)=[ϕT(t)0]T,∀t∈[-hb,0],
where  (2.8)Ã(t)=∑i=1rhi(θ(t))∑j=1rhj(θ(t))[Aj0BfiCjAfi]:=[A(t)0Bf(t)C(t)Af(t)],Ãτ(t)=∑i=1rhi(θ(t))∑j=1rhj(θ(t))[AτjBfiCτj]:=[Aτ(t)Bf(t)Cτ(t)],B̃(t)=∑i=1rhi(θ(t))∑j=1rhj(θ(t))[BjBfiDj]:=[B(t)Bf(t)D(t)],E=[I0],C̃(t)=∑i=1rhi(θ(t))∑j=1rhj(θ(t))[Lj-DfiCj-Cfi]:=[L(t)-Df(t)C(t)-Cf(t)],C̃τ(t)=∑i=1rhi(θ(t))∑j=1rhj(θ(t))[Lτj-DfiCτj]:=Lτ(t)-Df(t)Cτ(t),D̃(t)=∑i=1rhi(θ(t))∑j=1rhj(θ(t))[Gj-DfiDj]:=G(t)-Df(t)D(t).So far,the fuzzyH∞filter design problem for fuzzy system (2.3) can be stated as follows. Given a scalar γ>0, design a suitable fuzzy filter in the form of (2.5) such that the filtering error system (2.7) has a prescribed H∞ performance γ, and the following two purposes are satisfied: (i)
the system (2.7) with w(t)=0 is asymptotically stable;(ii)
theH∞ performance ∥e∥2<γ∥w∥2 is guaranteed for all nonzero w(t)∈L2[0,∞) and a prescribed γ>0 under the condition x̃(t)=0, for all t∈[-hb,0]. If this is the case, we say that the fuzzy H∞ filter design problem is solved.


## 3. FuzzyH∞ Performance Analysis

In this section, we propose the sufficient criterion for the filter error system (2.7) satisfying a prescribed H∞ performance level for fuzzy system (2.1) or (2.3).Theorem 3.1.
Given scalars0≤ha≤hb, d1≤d2 and γ>0, the H∞ filter error system (2.7), for all differentiable delay τ(t)∈[ha,hb] with d1≤τ̇(t)≤d2, is asymptotically stable and has a prescribed H∞ performance level γ if there exist real symmetry matrices R0>0, Rδ>0, Q0>0, Qδ>0, P=[P1P2*P3]>0, Rτ≥0, Pτ≥0, and real matrices Xij(t), (i=1,2;j=1,2,…,6) with appropriate dimensions such that the two LMIs (3.1) where τ̇(t)=d1,d2, are feasible.
(3.1)Ξi(t):=[Ω(t)+[-IiTQδIi+δhXi(t)Ii+δhIiTXiT(t)]haΓ1T(t)Q0δhΓ1T(t)QδδhXi(t)Γ3T(t)*-Q0000**-Qδ00***-Qδ0****-I]<0,(i=1,2),
where
(3.2)Ω(t)=[φ11φ12Q00φ15φ16*φ2200φ25φ26**Rτ-R0+Rδ-Q0-Qδ0Qδ0***-Rδ-Qδ-PτQδ0****-(1-τ̇(t))(Rτ-Pτ)-2Qδ0*****-γ2I],φ11=P1A(t)+AT(t)P1+P2Bf(t)C(t)+CT(t)BfT(t)P2T+R0-Q0,φ12=P2Af(t)+AT(t)P2T+CT(t)BfT(t)P3,φ22=P3Af(t)+AfT(t)P3,φ15=P1Aτ(t)+P2Bf(t)Cτ(t),φ25=P2TAτ(t)+P3Bf(t)Cτ(t),φ16=P1B(t)+P2Bf(t)D(t),φ26=P2TB(t)+P3Bf(t)D(t),I1=[000-II0],I2=[00I0-I0],Xi(t):=col{Xi1(t)Xi2(t)Xi3(t)Xi4(t)Xi5(t)Xi6(t)},(i=1,2),(3.3)Γ1(t):=[A(t)000Aτ(t)B(t)],Γ2(t):=[Bf(t)C(t)Af(t)00Bf(t)Cτ(t)Bf(t)D(t)],Γ3(t):=[L(t)-Df(t)C(t)-Cf(t)00Lτ(t)-Df(t)Cτ(t)G(t)-Df(t)D(t)].Proof.
First, we show that the error system (2.7) with w(t)≡0 is asymptotically stable, and then prove that the second condition of the fuzzy H∞ filter design problem in the previous section can be achieved.
We introduce the following Lyapunov-Krasovskii Functional:(3.4)V(t,x̃t)=Vp(t,x̃t)+Vh(t,x̃t),
where x̃t denotes the function x̃(s) defined on [t-hb,t], Vp(t,x̃t)=x̃T(t)Px̃(t) and
(3.5)Vh(t,x̃t)=∫t-hatx̃T(s)ETR0Ex̃(s)ds+∫t-hbt-hax̃T(s)ETRδEx̃(s)ds+∫t-τ(t)t-hax̃T(s)ETRτEx̃(s)ds+∫t-hbt-τ(t)x̃T(s)ETPτEx̃(s)ds+ha∫-ha0∫t+θtx̃̇T(s)ETQ0Ex̃̇(s)dsdθ+δh∫-hb-ha∫t+θtx̃̇T(s)ETQδEx̃̇(s)dsdθ
with Rδ>0,Qδ>0, R0>0,Rτ≥0, Pτ≥0, Q0>0, P=[P1P2*P3]>0 being real symmetry matrices with appropriate dimensions.
We employ (3.4) and Jensen's inequality [40] to study the performance analysis for the filter error system (2.7). In doing so, for simplicity, we introduce the following vector:
(3.6)Υ:=col{x(t)x̂(t)x(t-ha)x(t-hb)x(t-τ(t))w(t)}
Then, rewrite error system (2.7) as
(3.7)x̃̇(t)=[Γ1(t)Γ2(t)]Υ,e(t)=Γ3(t)Υ,x̃(t)=[ϕT(t)0]T,∀t∈[-hb,0],
where Γi(t), (i=1,2,3) are defined in (3.3).
Now, taking the derivative of (3.4) with respect to t along the trajectory of (2.7) yields
(3.8)V̇p(t,x̃t)=2x̃T(t)P[Γ1(t)Γ2(t)]Υ,(3.9)V̇h(t,x̃t)=xT(t)R0x(t)-xT(t-ha)R0x(t-ha)+xT(t-ha)Rδx(t-ha)-xT(t-hb)Rδx(t-hb)+xT(t-ha)Rτx(t-ha)-(1-τ̇(t))xT(t-τ(t))Rτx(t-τ(t))+(1-τ̇(t))xT(t-τ(t))Pτx(t-τ(t))-xT(t-hb)Pτx(t-hb)+ha2ẋT(t)Q0ẋ(t)-ha∫t-hatẋT(s)Q0ẋ(s)ds+δh2ẋT(t)Qδẋ(t)-δh∫t-hbt-haẋT(s)Qδẋ(s)ds
Since τ(t)∈[ha,hb], and defining ρ(t)=(hb-τ(t))/δh, we apply Jensen's inequality to yield the following inequalities:
(3.10)-ha∫t-hatẋT(s)Q0ẋ(s)ds≤[x(t)x(t-ha)]T[-Q0Q0*-Q0][x(t)x(t-ha)](3.11)-δh∫t-hbt-haẋT(s)Qδẋ(s)ds=-δh∫t-hbt-τ(t)ẋT(s)Qδẋ(s)ds-δh∫t-τ(t)t-haẋT(s)Qδẋ(s)ds=-(hb-τ(t))∫t-hbt-τ(t)ẋT(s)Qδẋ(s)ds-(τ(t)-ha)∫t-τ(t)t-haẋT(s)Qδẋ(s)ds-(1-ρ(t))δh∫t-hbt-τ(t)ẋT(s)Qδẋ(s)ds-ρ(t)δh∫t-τ(t)t-haẋT(s)Qδẋ(s)ds≤[x(t-ha)x(t-hb)x(t-τ(t))]T([-Qδ0Qδ*-QδQδ**-2Qδ])[x(t-ha)x(t-hb)x(t-τ(t))]-(1-ρ(t))δh∫t-hbt-τ(t)ẋT(s)Qδẋ(s)ds-ρ(t)δh∫t-τ(t)t-haẋT(s)Qδẋ(s)ds.
In addition, by the Leibniz-Newton formula, we obtain the following equation for any real matrices Xij(t), i=1,2; j=1,2,…,6 with appropriate dimensions:
(3.12)0=2δh(1-ρ(t))ΥTX1(t)[x(t-τ(t))-x(t-hb)-∫t-hbt-τ(t)ẋ(s)ds],0=2δhρ(t)ΥTX2(t)[x(t-ha)-x(t-τ(t))-∫t-τ(t)t-haẋ(s)ds],Xi(t):=col{Xi1(t)Xi2(t)Xi3(t)Xi4(t)Xi5(t)Xi6(t)},i=1,2.
By adding the right-hand side of (3.12) to (3.11), and combining  with (3.8)–(3.11) We yield the following inequality:
(3.13)V̇(t,x̃t)-γ2wT(t)w(t)≤ΥTΩτ̇(t)Υ-δh(1-ρ(t))∫t-hbt-τ(t)(ΥTX1+ẋT(s)Qδ)Qδ-1(X1TΥ+Qδẋ(s))ds-δhρ(t)∫t-τ(t)t-ha(ΥTX2+ẋT(s)Qδ)Qδ-1(X2TΥ+Qδẋ(s))ds,
where
(3.14)Ωτ̇(t)=(1-ρ(t))Ω1(t)+ρ(t)Ω2(t),Ωi(t):=Ω(t)+[-IiTQδIi+δhXi(t)Ii+δhIiTXiT(t)]+ha2Γ1T(t)Q0Γ1(t)+δh2Γ1T(t)QδΓ1(t)+δh2XiT(t)Qδ-1Xi(t),i=1,2.
with Xi(t),(i=1,2) and Ω(t), I1, I2 are defined in (3.12) and (3.2), respectively.
Notice that, sinceQδ>0,ρ(t)∈[0,1], (3.13) implies the following:
(3.15)V̇(t,x̃t)-γ2wT(t)w(t)≤ΥTΩτ̇(t)Υ.
Due to ρ(t)∈[0,1], Ωτ̇(t) is negative definite only if Ωi(t)<0,i=1,2. According to Schur’s complement, Ωi(t)<0,i=1,2 is equivalent to the following LMIs:
(3.16)Ξ̂i(t):=[Ω(t)+[-IiTQδIi+δhXi(t)Ii+δhIiTXiT(t)]haΓ1T(t)Q0δhΓ1T(t)QδδhXi(t)*-Q000**-Qδ0***-Qδ]<0,i=1,2.
And Ξ̂1(t)<0 leads for τ̇(t)=di, i=1,2 to the following:
(3.17)Ξ̂1i(t)=Ξ̂1(t)|τ̇(t)=di<0,i=1,2.
Notice that
(3.18)Ξ̂1(t)=d2-τ̇(t)d2-d1Ξ̂11(t)+τ̇(t)-d1d2-d1Ξ̂12(t).
Therefore, the two LMIs (3.17) imply (3.16), and Ξ̂1(t) is thus convex in τ̇(t)∈[d1,d2]. Similarly, Ξ̂2(t) is also convex in τ̇(t)∈[d1,d2]. Then if the two LMIs in (3.16) are feasible, then Ωτ̇(t)<0. It follows from (3.15) that
(3.19)V̇(t,x̃t)-γ2wT(t)w(t)<-λ∥x̃(t)∥2,∀x̃(t)≠0,
where λ=λmin(-Ωτ̇(t)).
From the above process, we can obtain the asymptotic stability of error system (2.7) with w(t)=0.
Next, assuming thatx̃(t)=0, for all t∈[-hb,0], we prove that the H∞ performance ∥e∥2<γ∥w∥2 is also guaranteed for all nonzero w(t)∈L2[0,∞) and a prescribed performance level γ>0.
Notice thateT(t)e(t)=ΥTΓ3T(t)Γ3(t)Υ, one rewrites (3.15) to the following:
(3.20)V̇(t,x̃t)≤ΥTΩ̂τ̇(t)Υ-eT(t)e(t)+γ2wT(t)w(t),
where
(3.21)Ω̂τ̇(t)=(1-ρ(t))Ω̂1(t)+ρ(t)Ω̂2(t),Ω̂i(t):=Ω(t)+[-IiTQδIi+δhXi(t)Ii+δhIiTXiT(t)]+ha2Γ1T(t)Q0Γ1(t)+δh2Γ1T(t)QδΓ1(t)+δh2XiT(t)Qδ-1Xi(t)+Γ3T(t)Γ3(t),(i=1,2).
If the LMIs (3.1) are feasible, applying Schur’s complement yields Ω̂τ̇(t)<0. Otherwise, similar to (3.16) and (3.17), then Ξ̂i(t), (i=1,2) are also convex in τ̇(t)∈[d1,d2]. So far, one has the following:
(3.22)V̇(t,x̃t)≤-eT(t)e(t)+γ2wT(t)w(t).
Integrating both sides of (3.22) from 0 to ∞ on t, and considering the zero initial condition, one obtains
(3.23)∫0∞eT(t)e(t)dt<γ2∫0∞wT(t)w(t)dt,
that is, ∥e∥2<γ∥w∥2. This completes the proof.For unknownd1, only by substituting τ̇(t)=d2 into (3.1)-(3.2), we can obtain the following Corollary.Corollary 3.2.
Given scalars0≤ha≤hb,d2 and γ>0, the H∞ filter error system (2.7), for all differentiable delay τ(t)∈[ha,hb] with τ̇(t)≤d2, is asymptotically stable and has a prescribed H∞ performance level γ if there exist matrices R0>0,Rδ>0,Q0>0,Qδ>0,P=[P1P2*P3]>0, Rτ≥0,Pτ≥0, and real matrices Xij(t), (i=1,2;j=1,2,…,6) with appropriate dimensions such that two LMIs (3.1) where τ̇(t)=d2, with notations in (3.2) and (3.3), are feasible.
Moreover, if the above LMIs are feasible withRτ=0, Pτ=0, then the H∞ filter error system (2.7), for all fast-varying delay τ(t)∈[ha,hb], is also asymptotically stable and has a prescribed H∞ performance level γ.In addition, when the number of IF-THEN rules is one, and the system is reduced to a simple time delay systems, that is, the system can be described as follows:(3.24)ẋ(t)=Ax(t)+Adx(t-τ(t)),t>0,x(t)=ϕ(t),t∈[-hb,0],
where(3.25)τ(t)∈[ha,hb],d1≤τ̇(t)≤d2.
According to the similar line of Theorem 3.1, without using the free-weighting matrices technique, one derives the following Corollary.Corollary 3.3.
Given scalars0≤ha≤hb, d1≤d2, the system (3.24), for all differentiable delay τ(t)∈[ha,hb] with d1≤τ̇(t)≤d2, is asymptotically stable if there exist real symmetry matrices R0>0, Rδ>0, Q0>0, Qδ>0, P>0, Rτ≥0, Pτ≥0 such that the following LMIs, where τ̇(t)=di,(i=1,2), are feasible:
(3.26)[Ω̂+[ÎitQδÎi]haΓ̂1TQ0δhΓ̂1TQδ*-Q00**-Qδ]<0,(i=1,2),
where Î1:=[00-II], Î2:=[0I0-I], Γ̂1:=[A00Ad] and
(3.27)Ω̂:=[PA+ATP+R0-Q0Q00PAd*-R0+Rδ+Rτ-Q0-Qδ0Qδ**-Rδ-Pτ-QδQδ***-(1-τ̇(t))(Rτ-Pτ)-2Qδ]Remark 3.4.
It is worth mentioning that in the previous studies (see [37–39, 41]), some negative terms are ignored when estimating the time derivative of the Lyapunov-Krasovskii functional, which may lose a great amount of useful information and lead to conservative results. Instead, in this paper, those negative terms are effectively used in (3.11). In addition, when constructing the Lyapunov-Krasovskii functional candidate, the information on the lower bound of the delay is taken full advantage of by introducing the terms ∫t-hbt-hax̃T(s)ETRδEx̃(s)ds and ∫t-τ(t)t-hax̃T(s)ETRτEx̃(s)ds in the Lyapunov-Krasovskii functional. From Example 5.3 below, it is clear to see that our approach is less conservative than the existing ones.


## 4. FuzzyH∞ Filter Design

It is worth mentioning that the problem in this paper essentially aims at designing a filter to estimatedz(t) based on H∞ norm constraint. The following theorem provides sufficient condition for the existence of fuzzy H∞ filter for fuzzy system (2.3) with interval time-varying delay. And a suitable filter design is obtained from the parameter matrices  Afi, Bfi, Cfi, and Dfi, (i=1,2,…,r).Theorem 4.1.
Given scalars0≤ha≤hb, d1≤d2 and γ>0, the fuzzy H∞ filter design problem, for all differentiable delay τ(t)∈[ha,hb] with d1≤τ̇(t)≤d2, is solvable if there exist matrices P1>0, U>0, Rτ≥0,Pτ≥0, R0>0, Rδ>0, Q0>0, and Qδ>0, and real matrices N1i,N2i,N3i,N4i,(i=1,2,…,r), X̂ik:= col {Xi1kX̂i2kXi3kXi4kXi5kXi6k},andi=1,2;k=1,2,…,r with appropriate dimensions such that the following LMIs: where τ̇(t)=d1,d2, are feasible:
(4.1)U-P1<0,(4.2)Πi(m,n)+Πi(n,m)<0,m≤n,(m,n=1,2,…,r),(i=1,2),
where I1,I2 is defined in (3.2), and
(4.3)Πi(m,n):=[Ω̂mn+[-IiTQδIi+δhX̂imIi+δhIiT(X̂im)T]ha(Γ1m)TQ0δh(Γ1m)TQδδhX̂im(Γ̂3m)T*-Q0000**-Qδ00***-Qδ0****-I],Ω̂mn=[φ̂11φ̂12Q00φ̂15φ̂16*φ̂2200φ̂25φ̂26**Rτ-R0+Rδ-Q0-Qδ0Qδ0***-Rδ-Qδ-PτQδ0****-(1-τ̇(t))(Rτ-Pτ)-2Qδ0*****-γ2I],φ̂11=P1Am+AmTP1+N2nCm+CmTN2nT+R0-Q0,φ̂12=N1n+AmTU+CmTN2nT,φ̂22=N1n+N1nT,φ̂15=P1Aτm+N2nCτm,φ̂25=UAτm+N2nCτmφ̂16=P1Bm+N2nDm,φ̂26=UBm+N2nDm.Γ1m:=[Am000AτmBm],Γ̂3m:=[Lm-N4nCm-N3n00Lτm-N4nCτmGm-N4nDm].
Moreover, a suitable filter in the form of (2.5) is given by
(4.4)Afi=N1iU-1,Bfi=N2i,Cfi=N3iU-1,Dfi=N4i(i=1,2,…,r).Proof.
Set(4.5)Nk(t):=∑i=1rhi(θ(t))[Nki],k=1,2,3,4,(4.6)X̂i(t):=col{Xi1(t)X̂i2(t)Xi3(t)Xi4(t)Xi5(t)Xi6(t)},i=1,2,
where
(4.7)Xik(t):=∑j=1rhj(θ(t))[Xikj(t)],i=1,2;k=1,3,…,6,X̂i2(t):=∑j=1rhj(θ(t))[X̂i2j(t)],Γ1(t):=∑m=1rhm(θ(t))[Γ1m],Γ̂3(t):=∑m=1rhm(θ(t))[Γ̂3m].
Thus, from (2.8) and the above definition, we have
(4.8)Πi(t)=∑m=1rhm2(θ(t))[Πi(m,m)]+∑m<nrhm(θ(t))hn(θ(t))[Πi(m,n)+Πi(n,m)],(i=1,2),
where
(4.9)Πi(t):=[Ψ(t)+[-IiTQδIi+δhX̂i(t)Ii+δhIiTX̂iT(t)]haΓ1T(t)Q0δhΓ1T(t)QδδhX̂i(t)Γ̂3T(t)*-Q0000**-Qδ00***-Qδ0****-I]<0(i=1,2)
with
(4.10)Ψ(t)=[Ψ11Ψ12Q00Ψ15Ψ16*Ψ2200Ψ25Ψ26**Rτ-R0+Rδ-Q0-Qδ0Qδ0***-Rδ-Qδ-PτQδ0****-(1-τ̇(t))(Rτ-Pτ)-2Qδ0*****-γ2I],Ψ11=P1A(t)+AT(t)P1+N2(t)C(t)+CT(t)N2T(t)+R0-Q0,Ψ12=N1(t)+AT(t)U+CT(t)N2T(t),Ψ22=N1(t)+N1T(t),Ψ15=P1Aτ(t)+N2(t)Cτ(t),Ψ25=UAτ(t)+N2(t)Cτ(t),Ψ16=P1B(t)+N2(t)D(t),Ψ26=UB(t)+N2(t)D(t).
Next, based on Theorem 3.1, we calculate the feasibility of the LMIs Πi(t)<0,(i=1,2).
Due toU>0, there exist a nonsingular real n×n matrix P2 and a real n×n matrix P3>0 such that U=P2P3-1P2T.Let us define
(4.11)J:=diag{IP2-TP3IIIIIIII}
left- and right-multiply Πi(t), i=1,2 defined in (4.8) by JT and J, respectively, and take Xi2(t):=P3P2-1X̂i2(t), i=1,2 and
(4.12)A̅(t)=P2-1N1(t)U-1P2,B̅(t)=P2-1N2(t),C̅(t)=N3(t)U-1P2,D̅(t)=N4(t),
By replacing (Af(t),Bf(t),Cf(t),Df(t)) in Ξi(t),i=1,2 defined in (3.1) with (A̅(t),B̅(t),C̅(t),D̅(t)), one yields
(4.13)Ξi(t)=JTΠi(t)J,i=1,2
Note that if LMIs (4.1) and (4.2) hold, from (4.8), we arrive at Πi(t)<0, then Ξi(t)<0.
On the other hand, from (4.1), notice that P1-U=P1-P2P3-1P2T>0, applying Schur complement yields [P1P2*P3]>0.
So far, we conclude from Theorem3.1 that the filter, that is,
(4.14)x̅̇(t)=A̅(t)x̅(t)+B̅(t)y(t),x̅(0)=0,z̅(t)=C̅(t)x̅(t)+D̅(t)y(t),
with (A̅(t),B̅(t),C̅(t),D̅(t)) defined in (4.12), guarantees that the H∞ filter error system (2.7) is asymptotically stable and has a prescribed H∞ performance level γ.
And, performing an irreducible linear transformationx̂(t)=P2x̅(t) in (4.14) yields
(4.15)x̂̇(t)=N1(t)U-1x̂(t)+N2(t)y(t),x̂(0)=0,ẑ(t)=N3(t)U-1x̂(t)+N4(t)y(t).
Therefore, the desired filter (2.5) with the filter matrices in (4.4) is readily obtained from (4.15). This completes the proof.Similar to Corollary3.2, when d1 is unknown, by substituting τ̇(t)=d2 into (4.2), the following result is then obtained.Corollary 4.2.
Given scalars0≤ha≤hb,d2 and γ>0, the fuzzy H∞ filter design problem, for all differentiable delay τ(t)∈[ha,hb] with τ̇(t)≤d2, is solvable if there exist matrices R0>0, Rδ>0, Q0>0, Qδ>0, P1>0, U>0, Rτ≥0, Pτ≥0, and real matrices N1i,N2i,N3i,N4i, (i=1,2,…,r), X̂ik:=col {Xi1kX̂i2kXi3kXi4kXi5kXi6k}, i=1,2; k=1,2,…,r with appropriate dimensions such that the LMIs: (4.1) and (4.2) where τ̇(t)=d2, are feasible. Meanwhile, a desired filter in the form of (2.5) is given by the filter matrices in (4.4).
Moreover, if the above LMIs are feasible withRτ=0,Pτ=0, then the fuzzy H∞ filter design problem, for all fast-varying delay τ(t)∈[ha,hb], is solvable in which a desired filter in (2.5) is given by the filter matrices in (4.4).Remark 4.3.
Notice that for any scalarσ, if (σZ-P)Z-1(σZ-P)≥0, then -PZ-1P≤-2σP+σ2Z. The fact played a key role in the existing results in [4, 5], respectively. But there existed some coupled matrix variables in the LMIs in [4, 5].Therefore, to solve filter design problem, [4, 5] must use decoupling technique similar to [42] to convert the conditions in [4, 5] into another form, respectively. These decoupling approaches were shown as [4, 5], respectively. Furthermore, because of a scalar being predescribed, the constraint may lead to considerable conservativeness of these results. Examples below show that for different δ yields different γmin. From simulation results in Table 2, we can see that if δ=0.7 or δ=20, the conditions in [4, 5] are unsolvable when hb=1.0, while our result works. Meanwhile, the scalar is not needed in this paper. Examples 5.1 and 5.2 below show that our approach yields less conservative results.


## 5. Numerical Examples

In this section, three examples are given to show the effectiveness of the proposed method in this paper.Example 5.1.
Consider the following fuzzy system borrowed from [4, 5]:
(5.1)ẋ(t)=∑i=12hi(θ(t))[Aix(t)+Aτix(t-τ(t))+Biw(t)],y(t)=∑i=12hi(θ(t))[Cix(t)+Cτix(t-τ(t))+Diw(t)],z(t)=∑i=12hi(θ(t))[Lix(t)+Lτix(t-τ(t))+Giw(t)],
where
(5.2)A1=[-2.10.11-2],A2=[-1.90-0.2-1.1],Aτ1=[-1.10.1-0.8-0.9],Aτ2=[-0.90-1.1-1.2],B1=[1-0.2],B2=[0.30.1],C1=[10],C2=[0.5-0.6],Cτ1=[-0.80.6],Cτ2=[-0.21]D1=0.3,D2=-0.6,L1=[1-0.5],L2=[-0.20.3],Lτ1=[0.10],Lτ2=[00.2],G1=0,G2=0.
For d2=0.3 and γ=0.5, choosing d1 and ha in Table 1 and applying Theorems 4.1, the results are d1-dependent (see Table 1). Moreover, for unknown d1 and d2, that is, fast-varying delay case, according to Corollary 4.2, by setting Rτ=0,Pτ=0,ha=0,andhb=0.5, we get the optimal attenuation level γopt=0.230 after 38 iterations.
Forha=0, d1unknown and d2=0.2, to compare with the recently developed fuzzy H∞ filter, it is worthwhile to point out that a given scalar δ is needed in [4, 5] while the scalar δ is any value in our results. Thus, we consider different hb and δ to find the minimum index γ. The results obtained by various methods in the literature and in this paper are listed in Table 2. Moreover, for the case of no additional prescribed scalar, in order to demonstrate the advantages of the proposed approach over the existing results, a detailed comparison between the minimum H∞ performance levels obtained by the methods in [4, 36, 37] and in this paper for different cases is summarized in Table 3. From Tables 2 and 3, it can be seen that stability conditions obtained in this paper are less conservative than the existing ones.
As an example, for givenha=0,hb=0.5,d1=0,ansd2=0.3, according to Theorem 4.1, solve LMIs in (4.1) and (4.2), and get the minimum performance level γopt=0.206 after 32 iterations, and then compute the fuzzy H∞ filter matrices from (4.4) as follows
(5.3)Af1=[-7.1207-5.3463-0.7273-4.5289],Bf1=[-0.19320.2146],Cf1=[-6.3345-2.6742],Df1=0.2486,Af2=[-3.5662-1.3711-7.4183-10.6811],Bf2=[-0.17650.1956],Cf2=[-2.3625-5.2980],Df2=0.2498.Table 1
Maximum values ofhb for d2=0.3.
ha∖d10−0.1−0.3−0.5−0.7−1ha=12.3582.3572.3552.3532.3492.351ha=02.0112.0122.0122.0112.0112.012Table 2
Minimum indexγ for d2=0.2 (d1 unknown and ha=0).
Methodδ=0.7δ=1δ=2δ=10δ=20Anyδ[4][5][4][5][4][5][4][5][4][5]Our resultshb=0.50.590.420.380.270.350.250.340.240.370.260.2054hb=0.61.030.740.430.310.360.250.350.250.450.320.2100hb=0.811.988.540.830.590.380.270.370.261.010.700.2204hb=1——2.221.570.410.290.450.32——0.2324Table 3
Minimum indexγ for different cases (d1 unknown and hb=1.25).
hamethodd2=0.4d2=0.6d2=0.8d2≥10[4]0.442.77∞∞[37]0.421.41∞∞[36]0.320.490.841.14Our results0.290.410.791.030.8[37]0.400.891.061.06[36]0.320.400.400.40Our results0.240.240.240.241.0[37]0.370.380.380.38[36]0.280.280.280.28Our results0.200.200.200.20In order to further show the merit of our method, let us consider the following numerical example.Example 5.2.
Consider the following fuzzy system with interval time-varying delay:(5.4)ẋ(t)=∑i=12hi(θ(t))[Aix(t)+Aτix(t-τ(t))+Bw(t)],y(t)=∑i=12hi(θ(t))[Cix(t)+Cτix(t-τ(t))+Dw(t)],z(t)=∑i=12hi(θ(t))[Lix(t)+Lτix(t-τ(t))+Giw(t)],
where
(5.5)A1=[-1000-0.900-0.5-1],A2=[-0.90.20-0.2-0.500-0.1-0.8],Aτ1=[-0.80.2-0.10.1-0.80-0.40.25-1],Aτ2=[-10.50.10.5-10-0.80.9-0.25],B=[000.5],C1=[0.50.40],C2=[0.5-10],Cτ1=[1-0.50.5],Cτ2=[10.1-0.5],D=0.25,L1=[0.500],L2=[1-0.50],Lτ1=[0.10.50.5],Lτ2=[0.100.5],G1=0,G2=0.
To compare with the ones existing in [4, 5], we assumed that d1is unknown and ha=0. According to Corollary 4.2, choose d2=0.2 and the simulations are run for two cases. In the first case, we compute the minimum index γ for the given different hb and δ in [4, 5] or any δ in this paper. In the second case, we compute the maximum values of hb for the given different γ and δ in [4, 5] or any δ in this paper. The simulation results are shown by Tables 4 and 5, respectively. It can also be clearly seen that our approach has less conservative results than the results in the literatures.
As an example, we assume thatha=0,hb=1.0,d1=0,d2=0.2,γ=0.5, the solutions can be obtained after 20 iterations in which the fuzzy H∞ filter in the form of (2.5) is given by the following filter matrices as
(5.6)Af1=[-3.35793.2252-0.32865.2755-12.386312.6979-0.4419-0.1659-7.8342],Bf1=[4.17412.2921-0.2958],Cf1=[0.25970.05472.2850],Df1=0.4963,Af2=[-1.6047-1.0030-4.28611.0643-4.94690.2980-0.24510.1274-3.3907],Bf2=[5.11352.0674-0.4810],Cf2=[-0.71620.2062-6.4570],Df2=-0.2080.Table 4
Minimum performance levelγ.
Methodδ=0.7δ=1δ=2δ=4Anyδ[4][5][4][5][4][5][4][5]Our resultshb=0.50.370.260.350.240.360.240.380.260.218hb=0.60.440.310.380.270.380.270.410.290.241hb=0.80.630.450.490.340.440.310.550.390.300Table 5
Maximum values ofhb.
Methodδ=0.7δ=1δ=2δ=5Anyδ[4][5][4][5][4][5][4][5]Our resultsγ=0.30.330.590.330.690.250.780.120.590.80γ=0.40.550.760.640.910.701.080.520.701.16γ=0.50.690.840.821.020.971.130.670.701.19Next, we will give another example to illustrate that our methods are reduced more conservative than the existing results.Example 5.3.
Consider the linear system (3.24) with the following parameters
(5.7)A=[01-1-2],Ad=[00-11].
To compare with those results in the previous literatures, assume that d1 is unknown. For ha=1.0, d2 unknown or d2=0.3, the result of Corollary 3.3 coincides with the one in [41] (the latter are less conservative than those of [39]). Comparison with various existing methods in the literature for the admissible upper-bound hb, which guarantee the stability of the system (3.24) is listed in Table 6. It is clear that our results are much less conservative than those in [37–39].Table 6
Maximum values ofhb for ha=1.0 (d1 unknown).
d2∖Method[38][37][39]Our resultsunknown1.501.51871.61691.70010.3—2.21252.24742.3076


## 6. Conclusion

This paper deals with the problem of fuzzyH∞ filter design for T-S fuzzy systems with interval time-varying delay through T-S fuzzy models. By constructing a novel Lyapunov-Krasovskii functional and estimating the time derivative of the Lyapunov-Krasovskii functional less conservatively, an improved H∞ filter design scheme is proposed. Three numerical examples are used to illustrate the design procedure and the merit of the proposed method.



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