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

Volume 2010, Article ID 392915,21pages doi:10.1155/2010/392915

Research Article

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

Jiyao An,

Guilin Wen,

and Wei Xu

1State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Institute of Space Technology, Hunan University, Changsha 410082, China

2School of Computer and Communication, Hunan University, Changsha 410082, China

3Faculty of Engineering and Information Technology, University of Technology Sydney, NSW 2007 Sydney, Australia

Correspondence should be addressed to Guilin Wen,wenguilin@yahoo.com.cn Received 12 July 2010; Accepted 30 July 2010

Academic Editor: Leonid Berezansky

Copyrightq2010 Jiyao An 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 fuzzy H∞ 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 varyingwith 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 fuzzyH∞filter is designed to ensure that the filter error system is asymptotically stable and has a prescribedH∞performance level. An improved delay- derivative-dependent condition for the existence of such a filter is derived in the form of linear matrix inequalitiesLMIs. Finally, numerical examples are given to show the effectiveness of the proposed method.

1. Introduction

During the last decades, the filtering problem has attracted many researchers to study through various methodologies, see, for example,1–20and the references therein, in which these methods mostly consist of two main approaches, namely, the Kalman filtering approach 1–3and theH∞filtering approach4–17. In contrast with the Kalman filtering, theH∞

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 signa statistics 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 technique18–20and theH∞filtering technique4–17.

On the other hand, Takagi-SugenoT-Sfuzzy 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 toH∞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 in21,29,30,33–35, while fuzzy controllers were designed in23–28. One set of fuzzyH∞filters for a class of T-S fuzzy systems was designed in32. 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.4and Su et al.5have concerned withH∞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.

36investigated the problem of delay-dependent robust stability andH∞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 inequalitiesLMIs. 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 fuzzy H∞ 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, andSection 6concludes 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 with rplant rules can be described by:

Plant rulei: IFθ1tisNi1and· · ·andθptisNip,THEN

xt ˙ Aixt Aτixt−τt Biwt, yt Cixt Cτixt−τt Diwt, zt Lixt Lτixt−τt Giwt,

xt φt, ∀t∈−hb,0,

2.1

where 0 ≤ha ≤τt≤ hb, andxt ∈Rⁿis the state vector;yt∈ R^mis the measurements vector;wt∈R^qis the disturbance signal vector which belongs toL20,∞;zt∈R^pis the signal vector to be estimated;φtis 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; θjt,j 1,2, . . . , p are the premise variables;Ni1, Ni2, . . . , Nipare the fuzzy sets. For the sake of convenience, we denote δhhb−ha.

The time-varying delayτtis assumed to be either differentiable with

d1≤τ˙t≤d2, 2.2

where d1 and d2 are given bounds, or fast-varying with no restrictions on the delay derivative.

The fuzzy system2.1is supposed to have singleton fuzzifier, product inference and centroid defuzzifier. The final output of the fuzzy system is inferred as follows:

xt ˙ ^r

i1

hiθtAixt Aτixt−τt Biwt,

yt ^r

i1

hiθtCixt Cτixt−τt Diwt,

zt ^r

i1

hiθtLixt Lτixt−τt Giwt, xt φt, ∀t∈−hb,0,

2.3

where fori1,2, . . . , r,

hiθt _rμiθt

i1μiθt, μiθt ^p

j1

Nij

θjt

2.4

andNijθjtis the membership function ofθjtinNij. Hereμiθt≥0. Here, we assume thatμiθt>0, and_r

i1hiθt 1.

Our aim is to design the following fuzzy filter.

Rulei: IFθ1tisNi1and· · ·andθptisNip,THEN

˙

xt Afixt Bfiyt, x0 0,

zt Cfixt Dfiyt, i1,2, . . . , r, 2.5

wherext ∈Rⁿis the filter state,zt ∈R^pis the estimation ofztin fuzzy system2.1, the constant matricesAfi∈R^n×n,Bfi∈R^n×m,Cfi ∈R^p×n,Dfi∈R^p×mare the filter matrices to be determined. The final fuzzy filter of fuzzy system2.1is thus inferred as follows

˙

xt ^r

i1

hiθt

Afixt Bfiyt , x0 0,

zt ^r

i1

hiθt

Cfixt Dfiyt .

2.6

Defining the augmented state vectorxt : col{xt xt}, et : zt−zt, from 2.3and2.6, we can then obtain the following filtering error system:

˙

xt At xt AτtExt −τt Btwt, et Ct xt CτtExt −τt Dtwt,

xt

φ^Tt 0 ^T, ∀t∈−hb,0,

2.7

where

At ^r

i1

hiθt^r

j1

hjθt

Aj 0 BfiCj Afi

:

At 0 BftCt Aft

,

Aτt ^r

i1

hiθt^r

j1

hjθt Aτj

BfiCτj

:

Aτt BftCτt

,

Bt ^r

i1

hiθt^r

j1

hjθt Bj

BfiDj

:

Bt BftDt

, E

I 0 ,

Ct ^r

i1

hiθt^r

j1

hjθt

Lj−DfiCj−Cfi :

Lt−DftCt −Cft ,

Cτt ^r

i1

hiθt^r

j1

hjθt

Lτj−DfiCτj :Lτt−DftCτt,

Dt ^r

i1

hiθt^r

j1

hjθt

Gj−DfiDj :Gt−DftDt.

2.8

So far, the fuzzyH∞filter design problem for fuzzy system2.3can be stated as follows.

Given a scalarγ >0, design a suitable fuzzy filter in the form of2.5such that the filtering error system2.7has a prescribedH∞performanceγ, and the following two purposes are satisfied:

ithe system2.7withwt 0 is asymptotically stable;

iitheH∞performancee₂ < γw₂is guaranteed for all nonzerowt ∈ L20,∞ and a prescribedγ >0 under the conditionxt 0, for allt∈−hb,0. If this is the case, we say that the fuzzyH∞filter design problem is solved.

3. Fuzzy H ∞ Performance Analysis

In this section, we propose the sufficient criterion for the filter error system2.7satisfying a prescribedH∞performance level for fuzzy system2.1or2.3.

Theorem 3.1. Given scalars 0≤ha≤hb,d1 ≤d2andγ >0,theH∞filter error system2.7, for all differentiable delayτt∈ha, hbwithd1 ≤τ˙t≤d2, is asymptotically stable and has a prescribed H∞performance levelγ if there exist real symmetry matricesR0 > 0,Rδ > 0,Q0 > 0,Qδ > 0,

P

P1P2

∗ P3

>0,Rτ ≥0,Pτ ≥0,and real matricesXijt,i1,2;j 1,2, . . . ,6with appropriate dimensions such that the two LMIs3.1where ˙τt d1, d2, are feasible.

Ξit

:

⎡

⎢⎢

⎢⎢

⎢⎣

Ωt

−I^T_iQδIiδhXitIiδhI^T_iX_i^Tt

haΓ^T₁tQ0 δhΓ^T₁tQδ δhXit Γ^T₃t

∗ −Q0 0 0 0

∗ ∗ −Qδ 0 0

∗ ∗ ∗ −Qδ 0

∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎥⎥

⎥⎦<0,

i1,2, 3.1

where

Ωt

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

ϕ11 ϕ12 Q0 0 ϕ15 ϕ16

∗ ϕ22 0 0 ϕ25 ϕ26

∗ ∗ Rτ−R0Rδ−Q0−Qδ 0 Qδ 0

∗ ∗ ∗ −Rδ−Qδ−Pτ Qδ 0

∗ ∗ ∗ ∗ −1−τtR˙ τ−Pτ−2Qδ 0

∗ ∗ ∗ ∗ ∗ −γ²I

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦ ,

ϕ11 P1At A^TtP1P2BftCt C^TtB^T_ftP₂^TR0−Q0, ϕ12P2Aft A^TtP₂^TC^TtB_f^TtP3, ϕ22 P3Aft A^T_ftP3,

ϕ15P1Aτt P2BftCτt, ϕ25P₂^TAτt P3BftCτt, ϕ16 P1Bt P2BftDt, ϕ26P₂^TBt P3BftDt,

I1

0 0 0 −I I 0 , I2

0 0 I 0 −I 0 , Xit:col

Xi1t Xi2t Xi3t Xi4t Xi5t Xi6t

, i1,2,

3.2 Γ1t:

At 0 0 0 Aτt Bt , Γ2t:

BftCt Aft 0 0 BftCτt BftDt , Γ3t:

Lt−DftCt −Cft 0 0 Lτt−DftCτt Gt−DftDt .

3.3

Proof. First, we show that the error system2.7withwt ≡0 is asymptotically stable, and then prove that the second condition of the fuzzyH∞filter design problem in the previous section can be achieved.

We introduce the following Lyapunov-Krasovskii Functional:

Vt,xt Vpt,xt Vht,xt, 3.4

wherextdenotes the functionxs defined ont−hb, t,Vpt,xt x^TtPxt and

Vht,xt

_t

t−ha

x^TsE^TR0Exsds _t−ha

t−hb

x^TsE^TRδExsds

_t−ha

t−τtx^TsE^TRτExsds _t−τt

t−hb

x^TsE^TPτExsds

ha

₀

−ha

_t

tθ

˙

x^TsE^TQ0Exsds dθ˙ δh

_−ha

−hb

_t

tθ

˙

x^TsE^TQδExsds dθ˙ 3.5

withRδ > 0, Qδ > 0,R0 > 0, Rτ ≥ 0,Pτ ≥ 0, Q0 > 0, P

P1P2

∗ P3

> 0 being real symmetry matrices with appropriate dimensions.

We employ3.4and Jensen’s inequality40to study the performance analysis for the filter error system2.7. In doing so, for simplicity, we introduce the following vector:

Υ:col

xt xt xt−ha xt−hb xt−τt wt

3.6

Then, rewrite error system2.7as

˙ xt

Γ1t Γ2t

Υ,

et Γ3tΥ,

xt

φ^Tt 0 ^T, ∀t∈−hb,0,

3.7

whereΓit,i1,2,3are defined in3.3.

Now, taking the derivative of3.4with respect totalong the trajectory of2.7yields

V˙pt,xt 2x^TtP Γ1t

Γ2t

Υ, 3.8

V˙ht,xt x^TtR0xt−x^Tt−haR0xt−ha x^Tt−haRδxt−ha−x^Tt−hbRδxt−hb

x^Tt−haRτxt−ha−1−τtx˙ ^Tt−τtRτxt−τt

1−τ˙tx^Tt−τtPτxt−τt−x^Tt−hbPτxt−hb

h²_ax˙^TtQ0xt˙ −ha

_t

t−ha

x˙^TsQ0xsds˙ δ_h²x˙^TtQδxt˙

−δh

_t−ha

t−hb

x˙^TsQδxsds˙

3.9

Sinceτt∈ha, hb, and definingρt hb−τt/δh, we apply Jensen’s inequality to yield the following inequalities:

−ha

_t

t−ha

x˙^TsQ0xsds˙ ≤

xt xt−ha

T−Q0 Q0

∗ −Q0

xt xt−ha

3.10

−δh

_t−ha

t−hb

x˙^TsQδxsds˙

−δh

_t−τt

t−hb

x˙^TsQδxsds˙ −δh

_t−ha

t−τtx˙^TsQδxsds˙

−hb−τt _t−τt

t−hb

x˙^TsQδxsds˙ −τt−ha _t−ha

t−τtx˙^TsQδxsds˙

−

1−ρt δh

_t−τt

t−hb

x˙^TsQδxsds˙ −ρtδh

_t−ha

t−τtx˙^TsQδxsds˙

≤

⎡

⎣xt−ha xt−hb xt−τt

⎤

⎦

T⎛

⎝

⎡

⎣−Qδ 0 Qδ

∗ −Qδ Qδ

∗ ∗ −2Qδ

⎤

⎦

⎞

⎠

⎡

⎣xt−ha xt−hb xt−τt

⎤

⎦

−

1−ρt δh

_t−τt

t−hb

x˙^TsQδxsds˙ −ρtδh

_t−ha

t−τtx˙^TsQδxsds.˙

3.11

In addition, by the Leibniz-Newton formula, we obtain the following equation for any real matricesXijt,i1,2;j 1,2, . . . ,6 with appropriate dimensions:

02δh

1−ρt

Υ^TX1t

xt−τt−xt−hb− _t−τt

t−hb

xsds˙

,

02δhρtΥ^TX2t

xt−ha−xt−τt− _t−ha

t−τtxsds˙

,

Xit:col

Xi1t Xi2t Xi3t Xi4t Xi5t Xi6t

, i1,2.

3.12

By adding the right-hand side of3.12to3.11, and combining with3.8−3.11We yield the following inequality:

V˙t,xt−γ²w^Ttwt

≤Υ^TΩτt˙ Υ−δh

1−ρt ^t−τt

t−hb

Υ^TX1x˙^TsQδ

Q⁻¹_δ

X₁^TΥ Qδxs˙ ds

−δhρt _t−ha

t−τt

Υ^TX2x˙^TsQδ

Q_δ⁻¹

X₂^TΥ Qδxs˙ ds,

3.13

where

Ωτt˙

1−ρt

Ω1t ρtΩ2t, Ωit: Ωt

−I^T_iQδIiδhXitIiδhI^T_iX^T_it

h²_aΓ^T₁tQ0Γ1t δ_h²Γ^T₁tQδΓ1t δ²_hX_i^TtQ_δ⁻¹Xit, i1,2.

3.14

withXit,i1,2andΩt, I1, I₂are defined in3.12and3.2, respectively.

Notice that, sinceQδ>0, ρt∈0,1,3.13implies the following:

V˙t,xt−γ²w^Ttwt≤Υ^TΩτt˙ Υ. 3.15

Due toρt∈0,1,Ωτt˙ is negative definite only ifΩit< 0, i 1,2. According to Schur’s complement,Ωit<0, i1,2 is equivalent to the following LMIs:

Ξit:

⎡

⎢⎢

⎢⎢

⎢⎣ Ωt

−I^T_iQδI_iδhXitIiδhI^T_iX_i^Tt

haΓ^T₁tQ0 δhΓ^T₁tQδ δhXit

∗ −Q0 0 0

∗ ∗ −Qδ 0

∗ ∗ ∗ −Qδ

⎤

⎥⎥

⎥⎥

⎥⎦<0,

i1,2.

3.16

AndΞ1t<0 leads for ˙τt di,i1,2 to the following:

Ξ1it Ξ1t

τtd˙ i

<0, i1,2. 3.17

Notice that

Ξ1t d2−τt˙ d2−d1

Ξ11t τt˙ −d1

d2−d1

Ξ12t. 3.18

Therefore, the two LMIs 3.17 imply 3.16, and Ξ1t is thus convex in ˙τt ∈ d1, d2. Similarly,Ξ2tis also convex in ˙τt ∈d1, d2. Then if the two LMIs in3.16are feasible, thenΩτt˙ <0. It follows from3.15that

V˙t,xt−γ²w^Ttwt<−λxt ², ∀xt /0, 3.19

whereλλmin−Ωτt˙ .

From the above process, we can obtain the asymptotic stability of error system2.7 withwt 0.

Next, assuming thatxt 0, for allt ∈−hb,0, we prove that theH∞performance e₂< γw₂is also guaranteed for all nonzerowt∈L20,∞and a prescribed performance levelγ >0.

Notice thate^Ttet Υ^TΓ^T₃tΓ3tΥ, one rewrites3.15to the following:

V˙t,xt≤Υ^TΩτt˙ Υ−e^Ttet γ²w^Ttwt, 3.20

where

Ωτt˙

1−ρt Ω1t ρtΩ2t, Ωit: Ωt

−I^T_iQδI_iδhXitIiδhI^T_iX_i^Tt

h²_aΓ^T₁tQ0Γ1t δ_h²Γ^T₁tQδΓ1t δ_h²X^T_itQ⁻¹_δ Xit Γ^T₃tΓ3t, i1,2.

3.21

If the LMIs 3.1 are feasible, applying Schur’s complement yields Ωτt˙ < 0. Otherwise, similar to 3.16and 3.17, thenΞit,i 1,2are also convex in ˙τt ∈ d1, d2. So far,

one has the following:

V˙t,xt≤ −e^Ttet γ²w^Ttwt. 3.22

Integrating both sides of3.22from 0 to∞ont, and considering the zero initial condition, one obtains

_∞

0

e^Ttetdt < γ² _∞

0

w^Ttwtdt, 3.23

that is,e₂< γw₂. This completes the proof.

For unknown d1, only by substituting ˙τt d2 into 3.1-3.2, we can obtain the following Corollary.

Corollary 3.2. Given scalars 0 ≤ ha ≤ hb, d2 and γ > 0,the H∞filter error system 2.7, for all differentiable delayτt ∈ha, hbwith ˙τt ≤ d2, is asymptotically stable and has a prescribed H∞performance levelγ if there exist matricesR0 > 0, Rδ > 0, Q0 > 0, Qδ > 0, P

P1P2

∗ P3

> 0, Rτ ≥0, Pτ ≥ 0, and real matricesXijt,i 1,2;j 1,2, . . . ,6with appropriate dimensions such that two LMIs3.1where ˙τt d2, with notations in3.2and3.3, are feasible.

Moreover, if the above LMIs are feasible withRτ 0,Pτ 0,then theH∞filter error system 2.7, for all fast-varying delayτt∈ha, hb, is also asymptotically stable and has a prescribedH∞

performance levelγ.

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:

xt ˙ Axt Adxt−τt, t >0,

xt φt, t∈−hb,0, 3.24

where

τt∈ha, hb, d1 ≤τt˙ ≤d2. 3.25

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 scalars 0 ≤ ha ≤ hb,d1 ≤ d2, the system3.24, for all differentiable delay τt ∈ ha, hbwithd1 ≤ τ˙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

τt ˙ di,i1,2, are feasible:

⎡

⎢⎣ Ω

I^t_iQδI_i

haΓ^T₁Q0 δhΓ^T₁Qδ

∗ −Q0 0

∗ ∗ −Qδ

⎤

⎥⎦<0, i1,2, 3.26

whereI₁: 0 0 −I I,I₂ : 0 I 0 −I,Γ1: A 0 0 Adand

Ω

:

⎡

⎢⎢

⎣

P AA^TPR0−Q0 Q0 0 P Ad

∗ −R0RδRτ−Q0−Qδ 0 Qδ

∗ ∗ −Rδ−Pτ−Qδ Qδ

∗ ∗ ∗ −1−τ˙tRτ−Pτ−2Qδ

⎤

⎥⎥

⎦ 3.27

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−ha

t−hb x^TsE^TRδExsds and!_t−ha

t−τtx^TsE^TRτExsds 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. Fuzzy H ∞ Filter Design

It is worth mentioning that the problem in this paper essentially aims at designing a filter to estimatedztbased onH∞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 matricesAfi,Bfi, Cfi, andDfi,i1,2, . . . , r.

Theorem 4.1. Given scalars 0≤ha≤hb,d1≤d2andγ >0,the fuzzyH∞filter design problem, for all differentiable delayτt∈ha, hbwithd1≤τ˙t≤d2, is solvable if there exist matricesP1 >0, U >0,Rτ ≥0, Pτ≥0,R0 >0,Rδ>0,Q0 >0, andQδ>0, and real matricesN1i, N2i, N3i, N4i,i 1,2, . . . , r,X^k_i :col{X_i1^k X_i2^k X^k_i3 X_i4^k X^k_i5 X_i6^k},andi 1,2; k 1,2, . . . , r with appropriate dimensions such that the following LMIs: where ˙τt d1, d2, are feasible:

U−P1 <0, 4.1

Πim, n Πin, m<0, m≤n, m, n1,2, . . . , r, i1,2, 4.2

where I1,I2is defined in3.2, and Πim, n

:

⎡

⎢⎢

⎢⎢

⎢⎢

⎣ Ωmn

−I^T_iQδIiδhX_i^mIiδhI^T_i X_i^m_T

ha

Γ^m_1T Q0 δh

Γ^m_1T

Qδ δhX_i^m Γ^m_3T

∗ −Q0 0 0 0

∗ ∗ −Qδ 0 0

∗ ∗ ∗ −Qδ 0

∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎥⎥

⎥⎥

⎦ ,

Ωmn

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

ϕ11 ϕ12 Q0 0 ϕ15 ϕ16

∗ ϕ22 0 0 ϕ25 ϕ26

∗ ∗ Rτ−R0Rδ−Q0−Qδ 0 Qδ 0

∗ ∗ ∗ −Rδ−Qδ−Pτ Qδ 0

∗ ∗ ∗ ∗ −1−τ˙tRτ−Pτ−2Qδ 0

∗ ∗ ∗ ∗ ∗ −γ²I

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦ ,

ϕ11 P1AmA^T_mP1N2nCmC_m^TN_2n^T R0−Q0,

ϕ12 N1nA^T_mUC^T_mN_2n^T , ϕ22N1nN_1n^T,

ϕ15 P1AτmN2nCτm, ϕ25UAτmN2nCτm

ϕ16 P1BmN2nDm, ϕ26 UBmN2nDm. Γ^m₁ :

Am 0 0 0 Aτm Bm , Γ^m₃ :

Lm−N4nCm −N3n 0 0 Lτm−N4nCτm Gm−N4nDm .

4.3

Moreover, a suitable filter in the form of 2.5is given by

AfiN1iU⁻¹, BfiN2i, CfiN3iU⁻¹, DfiN4i i1,2, . . . , r. 4.4 Proof. Set

Nkt:^r

i1

hiθtNki, k1,2,3,4, 4.5

Xit:col"

Xi1t Xi2t Xi3t Xi4t Xi5t Xi6t#

, i1,2, 4.6

where

Xikt:^r

j1

hjθt X_ik^jt

, i1,2; k1,3, . . . ,6, Xi2t:^r

j1

hjθt X_i2^jt

, Γ1t:^r

m1

hmθt

Γ^m₁ , Γ3t: ^r

m1

hmθt Γ^m₃

.

4.7

Thus, from2.8and the above definition, we have

Πit ^r

m1

h²_mθtΠim, m ^r

m<n

hmθthnθtΠim, n Πin, m, i1,2, 4.8

where Πit

:

⎡

⎢⎢

⎢⎢

⎢⎣ Ψt

−I^T_iQδIiδhXitIiδhI^T_iX_i^Tt

haΓ^T₁tQ0 δhΓ^T₁tQδ δhXit Γ^T₃t

∗ −Q0 0 0 0

∗ ∗ −Qδ 0 0

∗ ∗ ∗ −Qδ 0

∗ ∗ ∗ ∗ −I

⎤

⎥⎥

⎥⎥

⎥⎦<0

i1,2 4.9

with

Ψt

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎢

⎢⎣

Ψ11 Ψ12 Q0 0 Ψ15 Ψ16

∗ Ψ22 0 0 Ψ25 Ψ26

∗ ∗ Rτ−R0Rδ−Q0−Qδ 0 Qδ 0

∗ ∗ ∗ −Rδ−Qδ−Pτ Qδ 0

∗ ∗ ∗ ∗ −1−τ˙tRτ−Pτ−2Qδ 0

∗ ∗ ∗ ∗ ∗ −γ²I

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎥

⎥⎦ ,

Ψ11 P1At A^TtP1N2tCt C^TtN₂^Tt R0−Q0, Ψ12N1t A^TtUC^TtN₂^Tt, Ψ22 N1t N₁^Tt,

Ψ15 P1Aτt N2tCτt, Ψ25 UAτt N2tCτt, Ψ16P1Bt N2tDt, Ψ26 UBt N2tDt.

4.10

Next, based onTheorem 3.1, we calculate the feasibility of the LMIsΠit<0, i1,2.

Due toU > 0, there exist a nonsingular realn×nmatrix P2 and a realn×nmatrix P3>0 such thatUP2P₃⁻¹P₂^T.Let us define

J:diag

I P₂^−TP3 I I I I I I I I

4.11

left- and right-multiply Πit, i 1,2 defined in 4.8byJ^T andJ, respectively, and take Xi2t:P3P₂⁻¹Xi2t,i1,2 and

At P₂⁻¹N1tU⁻¹P2, Bt P₂⁻¹N2t,

Ct N3tU⁻¹P2, Dt N4t, 4.12

By replacing Aft, Bft, Cft, Dft in Ξit, i 1,2 defined in 3.1 with At, Bt, Ct, Dt, one yields

Ξit J^TΠitJ, i1,2 4.13

Note that if LMIs4.1and4.2hold, from4.8, we arrive atΠit<0, thenΞit<0.

On the other hand, from4.1, notice thatP1−UP1−P2P₃⁻¹P₂^T >0, applying Schur complement yields

P1P2

∗ P3

>0.

So far, we conclude fromTheorem 3.1that the filter, that is,

xt ˙ Atxt Btyt, x0 0,

zt Ctxt Dtyt, 4.14

withAt, Bt, Ct, Dtdefined in4.12, guarantees that theH∞filter error system2.7 is asymptotically stable and has a prescribedH∞performance levelγ.

And, performing an irreducible linear transformationxt P2xtin4.14yields ˙

xt N1tU⁻¹xt N2tyt, x0 0,

zt N3tU⁻¹xt N4tyt. 4.15

Therefore, the desired filter2.5with the filter matrices in4.4is readily obtained from4.15. This completes the proof.

Similar toCorollary 3.2, whend1is unknown, by substituting ˙τt d2into4.2, the following result is then obtained.

Corollary 4.2. Given scalars 0≤ha≤hb,d₂andγ >0,the fuzzy H∞filter design problem, for all differentiable delayτt∈ha, hbwith ˙τt≤d2, is solvable if there exist matricesR0 >0,Rδ>0, Q0>0,Qδ>0,P1>0,U >0, R_τ ≥0, P_τ ≥0,and real matricesN1i, N2i, N3i, N4i,i1,2, . . . , r, X_i^k :col{X_i1^k X_i2^k X^k_i3 X_i4^k X^k_i5 X_i6^k},i 1,2;k 1,2, . . . , r with appropriate dimensions such that the LMIs:4.1and4.2where ˙τt d2, are feasible. Meanwhile, a desired filter in the form of 2.5is given by the filter matrices in4.4.

Moreover, if the above LMIs are feasible withRτ 0, Pτ 0,then the fuzzy H∞filter design problem, for all fast-varying delayτt∈ha, hb, is solvable in which a desired filter in2.5is given by the filter matrices in4.4.

Remark 4.3. Notice that for any scalarσ, ifσZ−PZ⁻¹σZ−P≥0, then−P Z⁻¹P ≤ −2σP σ²Z. The fact played a key role in the existing results in4,5, Lemma 1, respectively. But there existed some coupled matrix variables in the LMIs in4,5.Therefore, to solve filter design problem, 4,5 must use decoupling technique similar to42to convert the conditions in 4,5, Lemma 1into another form, respectively. These decoupling approaches were shown as4,5, Lemma 2, respectively. Furthermore, because of a scalar being predescribed, the constraint may lead to considerable conservativeness of these results. Examples below show that for differentδyields differentγmin. From simulation results inTable 2, we can see that ifδ 0.7 orδ 20, the conditions in4,5are unsolvable whenhb 1.0, while our result works. Meanwhile, the scalar is not needed in this paper. Examples5.1and5.2below 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 from4,5:

xt ˙ ²

i1

hiθtAixt Aτixt−τt Biwt,

yt ²

i1

hiθtCixt Cτixt−τt Diwt,

zt ²

i1

hiθtLixt Lτixt−τt Giwt,

5.1

where

A1

−2.1 0.1 1 −2

, A2

−1.9 0

−0.2 −1.1

, Aτ1

−1.1 0.1

−0.8 −0.9

, Aτ2

−0.9 0

−1.1 −1.2

,

B1 1

−0.2

, B2 0.3

0.1

,

C1

1 0 , C2

0.5 −0.6 , Cτ1

−0.8 0.6 , Cτ2

−0.2 1 D10.3, D2−0.6,

L1

1 −0.5 , L2

−0.2 0.3 , Lτ1

0.1 0 , Lτ2 0 0.2 , G10, G20.

5.2

Ford20.3 andγ0.5, choosingd1andhainTable 1and applying Theorems4.1, the results ared1-dependentseeTable 1. Moreover, for unknownd1andd2, that is, fast-varying delay

Table 1: Maximum values ofhbford20.3.

ha\d1 0 −0.1 −0.3 −0.5 −0.7 −1

ha1 2.358 2.357 2.355 2.353 2.349 2.351

ha0 2.011 2.012 2.012 2.011 2.011 2.012

Table 2: Minimum indexγford20.2d1unknown andha0.

Method δ0.7 δ1 δ2 δ10 δ20 Anyδ

4 5 4 5 4 5 4 5 4 5 Our results

hb0.5 0.59 0.42 0.38 0.27 0.35 0.25 0.34 0.24 0.37 0.26 0.2054 hb0.6 1.03 0.74 0.43 0.31 0.36 0.25 0.35 0.25 0.45 0.32 0.2100 hb0.8 11.98 8.54 0.83 0.59 0.38 0.27 0.37 0.26 1.01 0.70 0.2204

hb1 — — 2.22 1.57 0.41 0.29 0.45 0.32 — — 0.2324

Table 3: Minimum indexγfor different casesd1unknown andhb1.25.

ha method d20.4 d20.6 d20.8 d2≥1

0

4 0.44 2.77 ∞ ∞ 37 0.42 1.41 ∞ ∞ 36 0.32 0.49 0.84 1.14

Our results 0.29 0.41 0.79 1.03

0.8

37 0.40 0.89 1.06 1.06 36 0.32 0.40 0.40 0.40

Our results 0.24 0.24 0.24 0.24

1.0

37 0.37 0.38 0.38 0.38 36 0.28 0.28 0.28 0.28

Our results 0.20 0.20 0.20 0.20

Table 4: Minimum performance levelγ.

Method δ0.7 δ1 δ2 δ4 Anyδ

4 5 4 5 4 5 4 5 Our results

hb0.5 0.37 0.26 0.35 0.24 0.36 0.24 0.38 0.26 0.218

hb0.6 0.44 0.31 0.38 0.27 0.38 0.27 0.41 0.29 0.241

hb0.8 0.63 0.45 0.49 0.34 0.44 0.31 0.55 0.39 0.300

case, according toCorollary 4.2, by settingRτ 0, Pτ 0, ha 0, andhb 0.5, we get the optimal attenuation levelγopt 0.230 after 38 iterations.

Forha 0,d1unknown andd2 0.2, to compare with the recently developed fuzzy H∞filter, it is worthwhile to point out that a given scalar δ is needed in4,5 while the scalarδis any value in our results. Thus, we consider differenthbandδto find the minimum indexγ. The results obtained by various methods in the literature and in this paper are listed inTable 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 minimumH∞performance levels obtained by the methods in4,36,37and in this paper for different cases is summarized inTable 3. From Tables2and3, it can be seen that stability conditions obtained in this paper are less conservative than the existing ones.

As an example, for given ha 0, hb 0.5, d1 0,ans d2 0.3, according to Theorem 4.1, solve LMIs in4.1and 4.2, and get the minimum performance levelγopt 0.206 after 32 iterations, and then compute the fuzzyH∞filter matrices from4.4as follows

Af1

−7.1207 −5.3463

−0.7273 −4.5289

, Bf1

−0.1932 0.2146

,

Cf1

−6.3345 −2.6742 , Df10.2486,

Af2

−3.5662 −1.3711

−7.4183 −10.6811

, Bf2

−0.1765 0.1956

,

Cf2

−2.3625 −5.2980 , Df20.2498.

5.3

In 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:

xt ˙ ²

i1

hiθtAixt Aτixt−τt Bwt,

yt ²

i1

hiθtCixt Cτixt−τt Dwt,

zt ²

i1

hiθtLixt Lτixt−τt Giwt,

5.4

where

A1

⎡

⎢⎢

⎣

−1 0 0 0 −0.9 0 0 −0.5 −1

⎤

⎥⎥

⎦, A2

⎡

⎢⎢

⎣

−0.9 0.2 0

−0.2 −0.5 0 0 −0.1 −0.8

⎤

⎥⎥

⎦, Aτ1

⎡

⎢⎢

⎣

−0.8 0.2 −0.1 0.1 −0.8 0

−0.4 0.25 −1

⎤

⎥⎥

⎦,

Aτ2

⎡

⎢⎢

⎣

−1 0.5 0.1 0.5 −1 0

−0.8 0.9 −0.25

⎤

⎥⎥

⎦, B

⎡

⎢⎢

⎣ 0 0 0.5

⎤

⎥⎥

⎦, C1

0.5 0.4 0 ,

C2

0.5 −1 0 , Cτ1

1 −0.5 0.5 , Cτ2

1 0.1 −0.5 , D0.25, L1

0.5 0 0 , L2

1 −0.5 0 , Lτ1

0.1 0.5 0.5 , Lτ2

0.1 0 0.5 , G10, G20.

5.5