exponential image filter design for uncertain takagi–sugeno fuzzy systems with time delay

8/10/2019 Exponential Image Filter Design for Uncertain TakagiSugeno Fuzzy Systems With Time Delay

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Engineering Applications of Artificial Intelligence 17 (2004) 645659

Exponential H1 filter design for uncertain TakagiSugeno fuzzysystems with time delay$

Shengyuan Xua,, James Lamb

aDepartment of Automation, Nanjing University of Science and Technology, Nanjing 210094, P.R. ChinabDepartment of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong

Received 23 September 2003; received in revised form 10 August 2004; accepted 10 August 2004

Available online 21 September 2004

Abstract

This paper considers the problem of robustH1filtering for state-space TakagiSugeno fuzzy systems with time delays and norm-

bounded parameter uncertainties. The problem we address is the design of an exponentially stable filter ensuring an exponential

stability and a prescribed H1 performance of the filtering error system for all admissible uncertainties. In terms of certain linear

matrix inequalities (LMIs), sufficient conditions for the solvability of this problem are obtained. When these LMIs are feasible, an

explicit expression of a desired H1filter is also given. Finally, a simulation example is provided to demonstrate the effectiveness and

applicability of the proposed design approach.

r 2004 Elsevier Ltd. All rights reserved.

Keywords: Fuzzy systems;H1 Filtering; Linear matrix inequality; Robust filtering; TakagiSugeno models; Time-delay systems; Uncertain systems

1. Introduction

In the past decades, there has been an increasing interest in the study of fuzzy systems in the TakagiSugeno (TS)

model since TS fuzzy models provide an effective way in representing complex nonlinear systems (Takagi and Sugeno

(1985)). In the fuzzy control setting, the TS fuzzy-model-based control technique has become popular recently. TS

fuzzy systems are defined by a set of IFTHEN fuzzy rules whose consequents are deterministic nonlinear dynamical

systems. Many control and filtering issues related to TS fuzzy systems have been studied in the literature. For

instance, the stability analysis and stabilization synthesis problems for these fuzzy systems were investigated in Cao et

al. (1997),Johansson et al. (1999),Tanaka and Sugeno (1992), and the references therein. In the case when parameter

uncertainty appears in a TS fuzzy system, the robust stability problem was addressed in Lee et al. (2001), where

several sufficient conditions for the robust stability were obtained in terms of linear matrix inequalities (LMIs); based

on these, the robust stabilization problem was solved and robust controllers were designed by the so-called parallel

distributed compensation (PDC) scheme (Tanaka et al., 1996). The observer design for TS fuzzy systems can be foundinKim (2002)andTanaka et al. (1998)and the H1 control problem was investigated inLee et al. (2001).

Since time delay is often a source of instability and encountered in various engineering systems such as chemical

processes, long transmission lines in pneumatic systems (Hale, 1977), the study of time-delay systems has received

much attention during the past years and various topics on time-delay systems have been addressed ( Hale, 1977;Jeung

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0952-1976/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engappai.2004.08.012

$This work is supported by RGC HKU 7103/01P, the Foundation for the Author of National Excellent Doctoral Dissertation of P.R. China

under Grant 200240, the National Natural Science Foundation of P.R. China under Grant 60304001 and the Fok Ying Tung Education Foundation

under Grant 91061.Corresponding author. Tel.: +852-28598982; fax: +852-28585415.

E-mail address: [email protected] (S. Xu).
http://www.elsevier.com/locate/engappaihttp://www.elsevier.com/locate/engappai


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et al., 1998;Xu et al., 2001a, b). Very recently, the TS fuzzy systems with time delays were introduced in Cao and

Frank (2000), in which some stability results were developed in both the continuous and discrete cases, and stabilizing

controllers and observers for such time-delay fuzzy systems were also designed by using the LMI approach. The robust

H1 control problem for time-delay fuzzy systems were considered in Lee et al. (2000), in which an LMI approach to

designing output feedback controllers were developed.

On the other hand, the problem ofH1filtering, which is concerned with the design of an estimator ensuring that the

L2-induced gain from the noise signal to the estimation error is less than a prescribed level, has received muchattention in the past years. Compared with the conventional Kalman filtering, the H1 filter technique has several

advantages. First, the noise sources in the H1 filtering setting are arbitrary signals with bounded energy or average

power, and no exact statistics are required to be known ( Nagpal and Khargonekar, 1991). Second, the H1 filter has

been shown to be much more robust to parameter uncertainty in a control system. These advantages render the H1filtering approach very appropriate to some practical applications. When parameter uncertainty arises in a system

model, the robust H1 filtering problem has been studied, and a great number of results on this topic have been

reported (de Souza et al., 1993;Jin and Park, 2001;Xie et al., 1996;Xu and Chen, 2003). In the case when parameter

uncertainty and time delays appear simultaneously in a system model, the robust H1filtering problem was dealt with

inde Souza et al. (2001)andXu and Van Dooren (2002)via LMI approach, respectively. The corresponding results for

uncertain discrete delay systems can be found in Xu (2002). However, it is noted that for TS fuzzy systems with

parameter uncertainty and time delays, the robust H1filtering problem has not been addressed, which is still open and

remains unsolved; this motivates the present study.

In this paper, we deal with the problem of exponential H1 filter design for uncertain TS fuzzy systems with time

delays. The parameter uncertainties are assumed to be time-varying but norm-bounded. The problem we address is to

design an exponentially stable filter which guarantees that the error system is exponentially stable and the L2-induced

gain from the noise signal to the estimation error is below a prescribed level. It is worth pointing out that the advantage

of a filter with exponential stability in comparison with that with asymptotic stability lies in that the former provides

fast convergence and desirable accuracy in terms of reasonable error covariance of the filtering process (Reif and

Unbehauen, 1999). Sufficient conditions for the solvability of this problem are obtained in terms of certain LMIs. A

desired filter can be constructed through a convex optimization problem, which can be solved by using standard

numerical algorithms, and no tuning of parameters is required (Boyd et al., 1994). Finally, a simulation example is

provided to demonstrate the effectiveness of the proposed approach.

Notation: Throughout this paper, for real symmetric matrices Xand Y; the notation XXY (respectively, X4Y)

means that the matrix XY is positive semi-definite (respectively, positive definite). I is an identity matrix with

appropriate dimension. The superscript T represents the transpose. L20;1is the space of square-integrable vectorfunctions over0;1:The notationj j refers to the Euclidean vector norm, while k k2 stands for the usual L20;1

norm. We use lminandlmax; respectively, to denote the minimum and maximum eigenvalue of a seal symmetric

matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions.

2. Definition and problem formulation

The continuous TakagiSugeno fuzzy dynamic model with parameter uncertainties and time delays is described by

the following fuzzy IFTHEN rules (Cao and Frank, 2000;Lee et al., 2000):

Plant Rule i: IF s1t is mi1 and s2t is m i2 and . . . and sgt is mig THEN

_xt Ai DAitxt Adi DAditxt t Diot; (1)

yt Ci DCitxt Cdi DCditxtt Eiot; (2)

zt Lixt; (3)

xt ft; t2 t; 0; i 1; 2; :. . .; r; (4)

where mij is the fuzzy set and r is the number of IFTHEN rules; xt 2 Rn is the state; yt 2 Rm is the

measured output; zt 2 Rp is the signal to be estimated; ot 2 Rq is the noise signal which is assumed to be an

arbitrary signal in L20;1;t40 is a scalar representing the time delay of the fuzzy system; s1t; s2t; ; sgtare the

premise variables. Ai;Adi;Ci;Cdi;Di;EiandLiare known real constant matrices; DAit; DAdit; DCitand DCdit

are real-valued unknown matrices representing time-varying parameter uncertainties, and are assumed to be

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of the form

DAit DAdit

DCit DCdit

M1i

M2i

FitN1i N2i; i 1; 2; :. . .; r; (5)

where M1i; M2i; N1i and N2i are known real constant matrices and Fi: N ! Rl1l2 are unknown time-varying

matrix function satisfying

FitTFitpI: (6)

The parameter uncertainties DAit; DAdit; DCit and DCdit are said to be admissible if both (5) and (6) hold.

By using a center-average defuzzier, product inference, and singleton fuzzifier, the dynamic fuzzy model in (1)(4)

can be represented by

_xt Xri1

histfAi DAitxt Adi DAditxt t Diotg; (7)

yt Xri1

histfCi DCitxt Cdi DCditxtt Eiotg; (8)

zt Xri1

histLixt; (9)

xt ft; t2 t; 0; (10)

where

hist $istPr

j1 $jst; $ist

Ygj1

mijsjt;

st s1t s2t sgt;

in which mijsjt is the grade of membership ofsjt inmij: Then, it can be seen that

$istX0; i 1; 2; :. . .; r;Xrj1

$jst40

for all t. Therefore, for all t,

histX0; i 1; 2; :. . .; r; (11)

Xrj1

hjst 1: (12)

Throughout the paper, we adopt the following exponential stability concept.

Definition 1. The fuzzy system (7) is said to be exponentially stable if there exist constants c40 and d40 such that

jxtjpc suptpyp0

jfyjedt;

whenot 0:

Similar to the fuzzy observer design (Cao and Frank, 2000), we now consider the following fuzzy filter for the

estimate ofzt:

Filter Rule i: IF s1t is m i1 ands2t is m i2

and . . . and sgt is mig THEN

_xt Afixt Bfiyt; (13)

zt Lfixt; i 1; 2; :. . .; r; (14)

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where xt 2 Rn and zt 2 Rp; Afi; Bfi and Cfi are matrices to be determined. Then, the overall fuzzy filter can be

inferred by

_xt Xri1

histAfixt Bfiyt; (15)

zt Xri1

histLfixt: (16)

Let

et xtT xtTT; ~zt zt zt: (17)

Then, the filtering error dynamics from systems (7)(9) and (15) and (16) can be described by

~S: _et Xri1

Xrj1

histhjst ~Aijtet ~AdijtHett ~Dijot; (18)

~zt

Xr

i1 Xr

j1

hist hjst~Lijet; (19)

where

~Aijt ~Aij D ~Aijt; ~Adijt ~Adij D~Adijt; (20)

~AijAj 0

BfiCj Afi

; ~Adij

Adj

BfiCdj

; ~Dij

Dj

BfiEj

; (21)

H I 0; ~Lij Lj Lfi; (22)

D ~Aijt DAjt 0

BfiDCjt 0

; D ~Adij

DAdjt

BfiDCdjt

: (23)

The robustH1filtering problem to be addressed in this paper is formulated as follows: given the uncertain time-delayTS fuzzy system (7)(9) and a prescribed level of noise attenuationg40;determine an exponentially stable filter in the

form of (13) and (14) such that the following requirements are satisfied:

(I) The filtering error system ~S is exponentially stable;

(II) Under zero initial conditions, ~S satisfiesk~zk2ogkok2 (24)

for any nonzero o 2 L20; 1 and all admissible uncertainties.

3. Main results

In this section, we will develop an LMI approach to solve the robustH1 filtering problem for uncertain time-delayfuzzy systems formulated in the previous section. We first give the following results which will be used in the proof of

our main results.

Lemma 1 (Xie and de Souza, 1992). Given any matrices X; Y andZ with appropriate dimensions such that Y40:Then

we have

XTZZTXpXTYXZTY1Z:

Lemma 2 (Wang et al., 1992). Let D; H; and Ft be real matrices of appropriate dimensions with Ft satisfying

FtTFtpI: Then, for any scalar e40 and vectors x and y with appropriate dimensions, we have

xTDFtHyyTHTFtTDTxp1xTDDTx yTHTHy:

The following theorem will play a key role in the design of desired fuzzy H1 filters.

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Theorem 1. The uncertain system( ~S) is exponentially stable and(24) is satisfied if there exist matrices P40;and Q40;

and scalars eij40; 1pipjpr; such that the following LMIs hold for 1pipjpr:

Oij12~Lij ~Lji

T~Lij ~Lji 2HTQH Odij P ~Dij ~Dji P ~Mij P ~Mji

OTdij Xij 2Q 0 0 0

~Dij ~DjiTP 0 2g2I 0 0

~MT

ijP 0 0 ijI 0

~MT

jiP 0 0 0 ijI

2666666664

3777777775o0; (25)

where

OijP ~Aij ~Aji ~Aij ~AjiTP ij ~N

T

1i~N1i ~N

T

1j~N1j; (26)

OdijP~Adij ~Adji ij ~NT

1iN2i ~N

T

1jN2j; (27)

Xij ijNT2iN2i N

T2jN2j; (28)

~N1i N1i 0; ~Mij

M1j

BfiM2j

: (29)

Proof. Under the conditions of the theorem, we first show the exponential stability of system ~S: To this end, we

consider (18) with ot 0; that is,

_et Xri1

Xrj1

histhjst ~Aijtet ~AdijtHett: (30)

For this system, we choose a LyapunovKrasovskii functional as follows:

Vet etTPet

Z t

tt

esTHTQHes ds; (31)

where

et etb; b2 t; 0:

Using the expressions in (20) and taking the time-derivative ofV et along the solution of (30) give

_Vet 2Xri1

Xrj1

histhjstetTP~Aijtet ~AdijtHet t

etTHTQHet et tTHTQHett

1

2

Xri1

Xrj1

histhjstfetTP~Aijt ~Ajitet ~Adijt ~Adjitxtt

~Aijt

~Ajitet

~Adijt

~Adjitxt t

T

PetgetTHTQHet xt tTQxtt

1

2

Xri1

Xrj1

histhjstfetTP~Aij ~Ajiet ~Adij ~Adjixtt

~Aij ~Ajiet ~Adij ~AdjixttTPet 2etTHTQHet 2xt tTQxtt

etTPD~Aijt D ~Ajitet D~Adijt D~Adjitxtt

D~Aijt D ~Ajitet D ~Adijt D ~AdjitxttTPetg; 32

where the relationship

xtt Hett; (33)

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is used. By (5), it is easy to see that

D ~Aijt D ~Adijt ~MijFjt ~N1j N2j: (34)

Set

Mij ~Mij ~Mji;

N1ij~N1j

~N1i

" #; N2ij

N2j

N2i

" #;

Fijt Fjt 0

0 Fit

" #:

Then

FijtTFijtpI (35)

By considering (34) and (35) and applying Lemma 2, we have that for 1pipjpr;

etTP D ~Aijt D~Ajitet D ~Adijt D ~Adjitxt t

D ~

Aijt D~

Ajitet D ~

Adijt D ~

Adjitxtt

T

PetetTPMijFijt N1ijet N2ijxtt N1ijet N2ijxtt

TFijtTM

T

ijPet

p1ij et

TPMijMT

ijPet ijN1ijet N2ijxtt

T N1ijet N2ijxtt: 36

This together with (32) implies

_Vetp1

2

Xri1

hist2xtTGiixt

Xri;j1;ioj

histhjstxtTGijxt; (37)

where

xt et

xkt ; (38)

GijOij

1ij P

MijMT

ijP2HTQH Odij

OTdij Xij 2Q

24

35o0; 1pipjpr: 39

On the other hand, from (25), it is easy to see that for 1pipjpr;

Oij2HTQH Odij P ~Mij P ~Mji

OTdij Xij 2Q 0 0

~MT

ijP 0 ijI 0

~MT

jiP 0 0 ijI

2666664

3777775

o0;

which, by the Schur complement formula, implies that for 1pipjpr;

Gijo0:

This together with (37) gives

_Vetpajetj2

; (40)

where

a 12

minflminGij; 1pipjprg40:

Then, it can be shown that there always exists a scalar b40 such that

blmaxP abtlmaxHTQHebt 0: (41)

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Now, by (31) and (40), we have

d

dtebtVet e

btbVt _Vet

pebt blmaxHTQH

Z ttt

jesj2 ds blmaxP ajetj2

:

Integrating both sides from 0 to T40 gives

ebTVeT Ve0 pblmaxHTQH

Z T0

ebt dt

Z ttt

jesj2 ds blmaxP a

Z T0

ebtjetj2 dt: (42)

Since Z T0

ebt dt

Z ttt

es 2 ds Z 0

t

jesj2 ds

Z st0

ebt dt

Z Tt0

jesj2 ds

Z sts

ebt dt

Z TTt

jesj2 ds

Z Ts

ebt dt

p

Z 0t

tebstjxsj2 ds

Z Tt0

tetstjxsj2 ds

Z TTt

tebstjxsj2 ds

tebt Z T

t

ebsjesj2 ds;

we have

ebTVeT pVe0 blmaxP abtlmaxHTQHebt

Z T0

ebtjetj2 dtpk suptpyp0

jfyj2; (43)

where

k max lmaxP; lmaxHTQH

:

Taking into account that

VeTXlminPjeTj2

:

This together with (43) implies that for any T40;

jeTjp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik

lminP

s sup

tpyp0

jfyjebT:

Therefore, by Definition 1, we have that system ( ~S) is exponentially stable.

Next, we show that for any nonzero o2 L20; 1; the uncertain system in ( ~S) satisfies (24) under the zero initial

condition. To this end, we introduce

JT

Z T0

~ztT ~zt g2otTot

dt; (44)

where the scalarT40:Noting the zero initial condition, it can be shown that for any nonzero o 2 L20;1and T40;

JT Z T

0

~ztT ~zt g2otTot _Vet dtVeTp

Z T0

~ztT ~zt g2otTot _Vet

dt; 45

where Vet is defined in (31). By some calculations, it can be verified that


Xri1

Xrj1

Xru1

Xrv1

histhjst husthvstetT ~L

T

ij~Luvet g

2otTot

2Xri1

Xrj1

histhjstetT P ~Aijtet ~AdijtHet t ~Dijot

etTHTQHet xttTQxt t: 46

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By (36), it can be seen that

2Xri1

Xrj1

histhjstetTP~Aijtet ~AdijtHet t ~Dijot

1

2Xr

i1X

r

j1

histhjstfetTP~Aijt ~Ajitet ~Adijt ~Adjitxtt ~Dij ~Djiot

~Aijt ~Ajitet ~Adijt ~Adjitxtt ~Dij ~DjiotTPetg

1

2

Xri1

Xrj1

histhjstfetTP~Aij ~Ajiet ~Adij ~Adjixt t ~Dij ~Djiot

~Aij ~Ajiet ~Adij ~Adjixt t ~Dij ~DjiotTPet

etTPD~Aijt D~Ajitet D ~Adijt D~Adjitxt t D ~Aijt D ~Ajitet

D~Adijt D ~AdjitxttTPetg

p1

2

Xri1

Xrj1

histhjstZtT ~OijZt; 47

where

Zt etT xt tT otT T

;

~Oij

Oij 1ij P

MijMT

ijP Odij P~Dij ~Dji

OTdij Xij 0

~Dij ~DjiTP 0 0

26664

37775:

Using Lemma 1, we have

Xr

i1 Xr

j1 Xr

u1 Xr

v1

histhjst husthvstetT ~L

T

ij~Luvet

1

4

Xri1

Xrj1

Xru1

Xrv1

histhjst husthvstetT~Lij ~Lji

T~Luv ~Lvuet

p1

4

Xri1

Xrj1

histhjstetT ~Lij ~Lji

T~Lij ~Ljiet: 48

Then, it follows from (46)(48) that


p1

2

Xri1

hist2ZtT ~GiiZt

Xri;j1;ioj

histhjstZtT ~GijZt; 49

where

~Gij ~Oij

12~Lij ~Lji

T~Lij ~Lji 2HTQH 0 0

0 2Q 0

0 0 2g2I

264

375; 1pipjpr:

Applying the Schur complement formula to the LMI in (25) results in

~Gijo0; 1pipjpr:

This together with (45) and (49) givesJTo0 for anyT40;which impliesk ~zk2ogkok2 for any nonzero o 2 L20;1:

This completes the proof. &

Now, we are in a position to give the main result on the solvability of the robust H1 filtering problem.

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Theorem 2. Consider the uncertain time-delay fuzzy systems (1)(4),and letg40be a given scalar.Then the robust H1filtering problem is solvable if there exist scalars ij40; 1pipjpr; and matrices X40; Y40; Q40; Yi; Fi and Ci;

1pipr; such that the following LMIs

J1ij H1ij H2ij H3ij H3ji H4ij

HT1ij J2ij 0 0 0 0

HT2ij 0 2g2I 0 0 0

HT3ij 0 0 ijI 0 0

HT3ji 0 0 0 ijI 0

HT4ij 0 0 0 0 2I

26666666666664

37777777777775o0; 1pipjpr 50

and

XY40; (51)

hold, where

J1ij

J11ij J11ji ijNT1iN1i N

T1jN1j

J12ij J12jiT ijN

T1iN1i N

T1jN1j

J12ij J12ji ijNT1iN1i NT1jN1j J13ij J13ji ijN

T1iN1i N

T1jN1j

" #;

J11ijYAj AT

jYQ;

J12ijXAj AT

jYFiCj Ci Q;

J13ijAT

jXXAj FiCj CT

jFTi Q;

J2ij ijNT2iN2i N

T2jN2j 2Q;

H1ijYAdi Adj ijN

T1iN2i N

T1jN2j

XAdi Adj FiCdj FjCdi ijNT1iN2i N

T1jN2j

" #;

H2ijYDi Dj

XDi Dj FiEj FjEi

;

H3ijYM1j

XM1j FiM2j

;

H4ijLi Lj

T Yi YjT

Li LjT

" #:

In this case, a desired exponentially stable fuzzy H1 filter is given in the form of (13) and (14) with parameters

as follows:AfiS

1CiY1WT; BfiS

1Fi; Lfi YiY1WT; (52)

where S and W are any nonsingular matrices satisfying

SWT I XY1: (53)

Proof. We first note that (51) implies thatI XY1 is nonsingular; therefore, there always exist nonsingular matrices

Sand Wsuch that (53) holds. Now, set

Y Y1; P1 Y I

WT 0

" #; P2

I X

0 ST

(54)

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and

P P2P11 : (55)

Then, by some calculations, we have

PX S

ST U ;

where

U W1 Y XY Y WT40: (56)

Noting

XSU1ST Y40: (57)

Then, we have that P40:Pre- and post-multiplying the first row and the first column of the LMI in (50) by diag Y; I;

we have that for 1pipjpr;

PT1Oij 2HTQHP1 P

T1Odij P

T1 P

~Dij ~Dji PT1 P

~Mij PT1 P

~Mji PT1

~Lij ~LjiT

OTdijP1 Xij2Q 0 0 0 0

~Dij ~DjiTPP1 0 2g

2I 0 0 0

~MT

ijPP1 0 0 ijI 0 0

~MT

jiPP1 0 0 0 ijI 0

~Lij ~Lji

P1 0 0 0 0 2I

2666666666664

3777777777775o0; (58)

whereO ij; Odijare given in (26)(29) and (22) with Pbeing replaced by P; and Afi; Bfi andLfiin (52). Now, pre- and

post-multiplying (58) by diagPT1 ; I; I; I; I; I and diagP11 ; I; I; I; I; I; respectively, result in

Oij2HTQH Odij P ~Dij ~Dji P ~Mij P ~Mji ~Lij ~Lji

T

OTdij Xij 2Q 0 0 0 0

~Dij ~DjiTP 0 2g2I 0 0 0

~MT

ijP 0 0 ijI 0 0

~MT

jiP 0 0 0 ijI 0

~Lij ~Lji 0 0 0 0 2I

266666666664

377777777775o0; 1pipjpr;

which, by the Schur complement formula, implies that for 1pipjpr;

Oij12~Lij ~Lji

T~Lij ~Lji 2HTQH Odij P ~Dij ~Dji P ~Mij P ~Mji

OTdij Xij 2Q 0 0 0

~Dij ~DjiTP 0 2g2I 0 0

~MTijP 0 0 ijI 0

~MT

jiP 0 0 0 ijI

2666666664

3777777775o0: (59)

By (59) and Theorem 1, the desired result follows immediately. &

Remark 1. Theorem 2 presents a sufficient condition for the solvability of the robust H1 filtering problem for

uncertain time-delay TS fuzzy systems. A desired H1 filter can be constructed by solving the LMIs in (50) and (51),

which can be implemented by resorting to some standard numerical algorithms, and no tuning of parameters will be

involved (Boyd et al., 1994).

Remark 2. It is noted that the LMI conditions provided in Theorem 1 are independent of the delay size. Therefore,

they can be applicable to the case when no prior knowledge about the size of the time delay is available.

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4. Simulation example

In this section, we provide a simulation example to illustrate theH1filter design approach developed in the previous

section. The uncertain time-delay TS fuzzy system considered in this example is with two rules:

Plant Rule 1: IF x1t is m1 e:g: small THEN

_xt A1

D

A1txt Ad1D

Ad1txt1:

5 D1o

t;

yt C1 DC1txt Cd1 DCd1txt1:5 E1ot;

zt L1xt

and

Plant Rule 2: IF x1t is m2 e:g: big THEN

_xt A2 DA2txt Ad2 DAd2txt1:5 D2ot;

yt C2 DC2txt Cd2 DCd2txt1:5 E2ot;

zt L2xt;

where

A1

1:2 0:8

1 2" #

; Ad1

0:5 0:2

1 0" #

; D1

1

0:2" #

;

C1 1 0; Cd1 0:8 0:6; E1 0:3; L1 1 0:5;

A2 1:6 0:2

0:7 1

; Ad2

0:6 0:2

0:5 0:8

; D2

0:3

0:1

;

C2 0:5 0:6; Cd2 0:2 1; E2 0:6; L2 0:2 0:3:

The membership functions ofm1 and m2 are, respectively, described by

h1x1t

1 for x1o1;12 1

2x1 for jx1jp1;

0 for x141;

8>:

and

h2x1t

0 for x1o1;12 1

2x1 for jx1jp1;

1 for x141:

8>:

The parameter uncertainties DAit; DAdit; DCit and DCdit; i 1; 2; are assumed to satisfy (5) and (6) with

M11 0

0:5

; M21 0:8; N11 0 0:3; N21 0:2 0;

M12

0

0:3

; M22 0:6; N12 0:5 0; N22 0 0:2:

Then, the final outputs of the fuzzy systems are inferred as follows:

_xt X2i1

hix1tfAi DAitxt Adi DAditxtt Diotg; (60)

yt X2i1

hix1tfCi DCitxt Cdi DCditxt t Eiotg; (61)

zt

X2

i1

hix1tLixt: (62)

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The purpose of this example is to design an exponentially stable fuzzy filter such that the filtering error system is

exponentially stable and satisfies a prescribed H1 performance level. In this example, the H1 performance level is

specified to beg 0:8:For this purpose, we resort to the Matlab LMI Control Toolbox to solve the LMIs in (50) and

(51), and obtain the solution as follows:

X 1:8536 0:6848

0:6848 1:3291 ; Y 0:3917 0:0271

0:0271 0:3908 ; Q0:5094 0:1667

0:1667 0:2935 ;Y1 1:0017 0:4944; Y2 0:1968 0:2985;

F1 1:0774

0:9802

; F2

0:8742

0:9654

;C1

3:1581 0:5266

2:2911 2:1262

;

C2 3:0772 1:5642

2:2103 1:6686

; 11 0:5562; 12 2:2393; 11 0:5172:

Therefore, by Theorem 2, it is to see that the robust H1 filtering problem is solvable. To construct such a desired

exponentially stable fuzzy filter, we choose nonsingular matrices Sand Was

S1 0:5

0:8 1 ; W 2:0523 3:1620

0:2021 2:4576 : (63)

It can be verified that the matrices Sand Win (63) satisfy the equality in (53); thus, by Theorem 2, a desired fuzzyH1filter can be constructed as

Filter Rule 1: IF x1t is m1 THEN

_xt Af1xt Bf1yt;

zt Lf1xt

and

Filter Rule 2: IF x1t is m2 THEN

_xt Af2xt Bf2yt;

zt Lf2xt;

where

Af1 2:9870 0:7508

1:8256 1:5045

" #; Af2

1:1822 0:3462

1:1459 1:8463

" #;

Bf1 0:4195

1:3158

" #; Bf2

0:2797

1:1891

" #;

Lf1 0:6002 0:3954; Lf2 0:2737 0:3206:

The simulation results of the state response of the plant and the filter are given in Fig. 1, where the initial conditions are

1 0:8

T

and 0 0

T

; respectively, and the exogenous disturbance input ot is set as

ot 1

2t; tX0;

which belongs to L20;1: Fig. 2 shows the simulation results of the signal zt and zt: Figs. 3 and 4 show the

simulations of the measured output ytand the error response zt zt;respectively. From these simulation results,

it can be seen that the designed fuzzy H1 filter meet the specified requirements.

5. Conclusions

The problem of robust H1 filtering for continuous TS fuzzy systems with time delays and time-varying norm-

bounded parameter uncertainties has been studied. A sufficient condition for the existence of a full-order exponentially

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0 2 4 6 8 10 12 14 16 18 20

1.5

1

0.5

0

0.5

1

Fig. 1. State response ofxt () and xt :

0 2 4 6 8 10 12 14 16 18 20

0

0.5

0.5

1.5

1

Fig. 2. Response of the signalzt () and zt :

0 2 4 6 8 10 12 14 16 18 20

0

0.5

0.5

1

1.5

Fig. 3. Measured outputyt:



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stable filter, which ensures that the error system is exponentially stable and the L2-induced gain from the noise signal

to the estimation error is below a prescribed level, has been obtained. An explicit expression of a desired H1filter has

been given. A simulation example has demonstrated the effectiveness of the proposed approach.

References

Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., 1994. Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied

Mathematics. SIAM, Philadelphia, PA.

Cao, Y.-Y., Frank, P.M., 2000. Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach. IEEE Transactions on Fuzzy

Systems 8, 200211.

Cao, S.-G., Rees, N.W., Feng, G., 1997. Analysis and design for a class of complex control systems Part II: fuzzy controller design. Automatica 33,

10291039.de Souza, C.E., Xie, L., Wang, Y., 1993. H1 filtering for a class of uncertain nonlinear systems. Systems & Control Letter 20, 419426.

de Souza, C.E., Palhares, R.M., Peres, P.L.D., 2001. Robust H1 filter design for uncertain linear systems with multiple time-varying state delays.

IEEE Transactions on Signal Processing 49, 569576.

Hale, J.K., 1977. Theory of Functional Differential Equations. Springer, New York.

Jeung, E.T., Kim, J.H., Park, H.B., 1998. H1-output feedback controller design for linear systems with time-varying delayed state. IEEE

Transactions on Automatic Control 43, 971974.

Jin, S.H., Park, J.B., 2001. Robust H1 filtering for polytopic uncertain systems via convex optimisation. IEE ProceedingsControl Theory

Applications 148, 5559.

Johansson, M., Rantzer, A., A rzen, K.-E., 1999. Piecewise quadratic stability of fuzzy systems. IEEE Transactions on Fuzzy Systems 7, 713721.

Kim, E., 2002. A fuzzy disturbance observer and its application to control. IEEE Transactions on Fuzzy Systems 10, 7784.

Lee, K.R., Kim, J.H., Jeung, E.T., Park, H.B., 2000. Output feedback robust H1 control of uncertain fuzzy dynamic systems with time-varying

delay. IEEE Transactions on Fuzzy Systems 6, 657664.

Lee, K.R., Jeung, E.T., Park, H.B., 2001. Robust fuzzy H1control for uncertain nonlinear systems via state feedback: an LMI approach. Fuzzy Sets

and Systems 120, 123134.

Nagpal, K.M., Khargonekar, P.P., 1991. Filtering and smoothing in an H1 setting. IEEE Transactions on Automatic Control 36, 152166.Reif, K., Unbehauen, R., 1999. The extended Kalman filter as an exponential observer for nonlinear systems. IEEE Transactions on Signal

Processing 47, 23242328.

Takagi, T., Sugeno, M., 1985. Fuzzy identification of systems and its applications to modelling and control. IEEE Transactions on Systems Man and

Cybernetics 15, 116132.

Tanaka, K., Sugeno, M., 1992. Stability analysis and design of fuzzy control systems. Fuzzy Sets and Systems 45, 135156.

Tanaka, K., Ikeda, T., Wang, H.O., 1996. Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability,

H1 control theory, and linear matrix inequalities. IEEE Transactions on Fuzzy Systems 4, 113.

Tanaka, K., Ikeda, T., Wang, H.O., 1998. Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE

Transactions on Fuzzy Systems 6, 250264.

Wang, Y., Xie, L., de Souza, C.E., 1992. Robust control of a class of uncertain nonlinear systems. Systems & Control Letters 19, 139149.

Xie, L., de Souza, C.E., 1992. RobustH1control for linear systems with norm-bounded time-varying uncertainty. IEEE Transactions on Automatic

Control 37, 11881191.

Xie, L., de Souza, C.E., Wang, Y., 1996. Robust filtering for a class of discrete-time uncertain nonlinear systems: an H1 approach. International

Journal of Robust and Nonlinear Control 6, 297312.

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0 2 4 6 8 10 12 14 16 18 201

0.8

0.6

0.4

0.2

0

0.2

Fig. 4. Error response ofzt zt:



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Xu, S., 2002. RobustH1 filtering for a class of discrete-time uncertain nonlinear systems with state delay. IEEE Transactions on Circuits Systems I

49, 18531859.

Xu, S., Chen, T., 2003. Robust H1 filtering for uncertain impulsive stochastic systems under sampled measurements. Automatica 39, 509516.

Xu, S., Van Dooren, P., 2002. RobustH1filtering for a class of non-linear systems with state delay and parameter uncertainty. International Journal

of Control 75, 766774.

Xu, S., Lam, J., Yang, C., 2001a. H1 and positive real control for linear neutral delay systems. IEEE Transactions on Automatic Control 46,

13211326.

Xu, S., Lam, J., Yang, C., 2001b. Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay. Systems & ControlLetters 43, 7784.

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exponential image filter design for uncertain takagi–sugeno fuzzy systems with time delay

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