Robust control of Takagi–Sugeno fuzzy systems with state and input time delays

Fuzzy Sets and Systems 160 (2009) 403–422www.elsevier.com/locate/fss

Robust H∞ control of Takagi–Sugeno fuzzy systems with stateand input time delays

Bing Chena,∗, Xiaoping Liub, Chong Lina, Kefu Liub

aInstitute of Complexity Science, Qingdao University, Qingdao 266000, PR ChinabFaculty of Electrical Engineering, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1

Received 13 April 2007; received in revised form 25 March 2008; accepted 28 March 2008Available online 8 April 2008

Abstract

This paper addresses the robust H∞ fuzzy control problem for nonlinear uncertain systems with state and input time delaysthrough Takagi–Sugeno (T–S) fuzzy model approach. The delays are assumed to be interval time-varying delays, and no restrictionis imposed on the derivative of time delay. Based on Lyapunov–Krasoviskii functional method, delay-dependent sufficient conditionsfor the existence of an H∞ controller are proposed in linear matrix inequality (LMI) format. Illustrative examples are given to showthe effectiveness and merits of the proposed fuzzy controller design methodology.Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.

Keywords: H∞ control; T–S fuzzy system; Time delay; Delay-dependent stabilization; Uncertainty

1. Introduction

During the last two decades, the problem of stability analysis and stabilization of nonlinear systems in Takagi andSugeno [21] fuzzy model has been extensively studied, and a lot of significant results on stabilization and H∞ controlvia linearmatrix inequality (LMI) approach have been reported, see [1–3,22,23,7,8,5,18,25,35,34,27], and the referencetherein. So far, Takagi–Sugeno (T–S) fuzzy model has become a popular and effective approach to control complexand ill-defined systems for which the application of conventional techniques is infeasible.

In view of time delays frequently occurring in practical dynamic systems, T–S fuzzy model was first used to dealwith the stability analysis and control synthesis of nonlinear time delay systems in [4]. Afterwards, many peoplehave devoted a great deal of effort to both theoretical research and implementation techniques for T–S fuzzy systemswith time delays. In these works, stability analysis and fuzzy controller design methods are provided in the sense ofdelay-independent stability [14,33], and in the sense of delay-dependent stability [9,15,16,27]. Generally speaking, thedelay-independent approach provides the stability conditions irrespective of the size of the delay. As a result, it maybe lead to comparatively conservative stability analysis results, particularly when the delay is small.

More recently, further delay-dependent stabilization results are presented for T–S fuzzy systems with interval timedelay in [13,24,12]. The robust stabilization problem has been addressed in [24], and LMI-based stability criteria andstabilization conditions have been proposed based on Lyapunov–Krasoviskii functional method, while the H∞ control

∗Corresponding author. Tel.: +8653285953607.E-mail address: [email protected] (B. Chen).

0165-0114/$ - see front matter Crown Copyright © 2008 Published by Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2008.03.024

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404 B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403–422

problem has been studied in [13,12], and some sufficient conditions for the existence of an H∞ controller have beenpresented by means of LMI format.

However, all the aforementioned results have not considered the effect of control input delay on the resulting systems.Hence, these approaches mentioned above may be invalid when they are used to control the systems with input delays.As well-known, input delays are often encountered in many industrial processes. The presence of input delay, if nottaken into account in the controller deign, may cause instability or serious deterioration in the performance of theresulting control systems. In addition, in modern industrial systems, sensors, controllers and plants are often connectedover a network medium. The controller signals are transmitted through a network, therefore, the network-induced timedelay is usually inevitable. In view of this, more recently, many controller design schemes have been presented forlinear systems with input delay [19,32,31]. Then, there are a few results on stabilization of T–S fuzzy systems with stateand input delays. To the best of author’s knowledge, the literature by Lin et al. [17] is the only one, which deals withthe stability and stabilization of T–S fuzzy systems with state and input delays. Based on Razumikhin Theorem, thedelay-dependent stability and stabilization conditions have been developed in terms of LMIs. However, uncertaintiesand external perturbations have not been considered in [17].

Motivated by the above concerns, we will consider the problem of robust H∞ control for T–S fuzzy systems withboth state and input delays. It is assumed that state and input delays are time-varying delays with both upper andlower bounds. there is no restriction imposed on the derivative of time delay. Delay-dependent sufficient conditions forstability and stabilization will be derived in terms of LMI format, and systematic design procedure for an H∞ fuzzycontroller will be proposed for uncertain T–S fuzzy systems with state and input delays. At last, three examples aregiven to illustrate the effectiveness and feasibility of our results.

Throughout this paper, identity matrix, of appropriate dimensions, will be denoted by I . The notation X > 0(respectively, X�0), for X ∈ Rn×n means that the matrix X is real symmetric positive definite (respectively, positivesemi-definite). If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations.The symbol “ *” in a matrix A ∈ Rn×n stands for the transposed elements in the symmetric positions. The superscripts“ T” and “ −1” denote the matrix transpose and inverse, respectively.

2. Problem statement

Consider a nonlinear time delay system represented by T–S fuzzy model as follows:Plant Rule i : IF �1(t) is Ni1 · · · �p(t) is Nip THEN

x(t)= (Ai + �A1i (t))x(t) + (A1i + �A1i (t))x(t − �(t)) + Bwiw(t)

+(Bi + �Bi (t))u(t) + (B1i + �B1i (t))u(t − �(t)),

z(t)=Ci x + C1i x(t − �(t)) + Diu(t) + D1i u(t − �(t)),

x(t)= �(t), t ∈ [−d, 0],

where Ni j is the fuzzy set, x(t) ∈ Rn stands for the state vector, u(t) ∈ Rm denotes the control input vector, z(t) ∈ Rq

is the controlled output, and w(t) ∈ Rp, which is assumed to belong to L2[0, ∞), denotes the external perturbation.Ai , A1i ∈ Rn×n , Bi , B1i ∈ Rn×m , Bwi ∈ Rn×d ,Ci andC1i ∈ Rq×s , Di and D1i ∈ Rq×m are known constant matrices.Constant d is the upper bound of �(t) and �(t). Scalar k is the number of IF–THEN rules. �1(t), �2(t), . . . , �p(t) are thepremise variables. It is assumed that the premise variables do not depend on the input u(t). �Ai (t), �A1i (t), �Bi (t)and �B1i (t) denote the time-varying uncertain matrices satisfying

[�Ai , �A1i , �Bi , �B1i ] = Gi Fi (t)[Ei , E1i , Ebi , Eb1i ],

where Gi , Ei , E1i , Ebi and Eb1i are known constant matrices of appropriate dimensions. F(t) is an unknown matrixfunction satisfying the inequality FT(t)F(t)� I . �(t) and �(t) denote state and input delays, respectively. In this paper,delays �(t) and �(t) are assumed to be interval type delay, i.e., there exist known positive constants �m , �M , �m and �M

such that

�m ��(t)��M , �m ��(t)��M ,

B. Chen et al. / Fuzzy Sets and Systems 160 (2009) 403–422 405

and let d = max{�M , �M }. Given a pair of (x(t), u(t)), the final output of the fuzzy system is inferred as follows:

x(t)=k∑

i=1

hi (�(t))[(Ai + �Ai )x(t) + (A1i + �A1i )x(t − �(t))

+(Bi + �Bi )u(t) + (B1i + �B1i )u(t − �(t)) + Bwiw(t)],

z(t)=k∑

i=1

hi (�(t))[Ci x + C1i x(t − �(t)) + Diu(t) + D1i u(t − �(t))],

x(t)= �(t), t ∈ [−d, 0], (1)

where hi (�(t)) = �i (�(t))/∑k

i=1 �i (�(t)), �i (�(t)) = ∏pj=1 Ni j (� j (t)) and Ni j (� j (t)) is the degree of the membership

of � j (t) in fuzzy set Ni j . In this paper, it is assumed that �i (�(t))�0 for i = 1, 2, . . . , k and∑k

i=1 �i (�(t)) > 0 for allt . Therefore, hi (�(t))�0 (for i = 1, 2, . . . , k) and

∑ki=1 hi (�(t)) = 1. For simplicity, the following notations will be

used:

Ai j = Ai j + �Ai j , Ai j = Ai + Bi K j , �Ai j = �Ai + �Bi K j ,

A1i = A1i + �A1i , B1i = B1i + �B1i , hi = hi (�(t)),

hi (�) = hi (�(t − �)), � = �(t), � = �(t).

To design a state feedback fuzzy controller, the ith fuzzy control rule is presented as follows:Control Rule i : IF �1(t) is Ni1 · · · �p(t) is Nip THEN

u(t) = Ki x(t), i = 1, 2, . . . , k.

Hence, the overall fuzzy control law is represented by

u(t) =k∑

i=1

hi (�(t))Ki x(t), (2)

where Ki (i = 1, 2, . . . , k) are the local control gains.

Remark 1. The existence of input delay leads to the term∑k

i=1 hi (�(t−�))Ki x(t−�). Therefore, a natural and essentialassumption is that all hi (�(t))’s are well defined for t ∈ [−�M , 0], and also satisfy the equality

∑ki=1 hi (�(t)) = 1 with

hi (�(t))�0 for t ∈ [−�M , 0] for i = 1, 2, . . . , k.

Associated with the control law (2), the resulting closed-loop system can be expressed as follows:

x(t)=k∑

i=1

hi (�(t))

⎧⎨⎩

⎛⎝Ai + �Ai +

k∑j=1

h j (�(t))(Bi + �Bi )K j

⎞⎠ x(t)

+(A1i + �A1i )x(t − �) +k∑

s=1

hs(�(t − �))(B1i + �B1i )Ksx(t − �) + Bwiw(t)

⎫⎬⎭

=k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)( Ai j x(t) + A1i x(�) + B1i Ks x(t − �) + Bwiw(t)), (3)

z(t)=k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)((Ci + Di K j )x + C1i x(t − �) + D1i Ks x(t − �)). (4)


Definition. Systems (3)–(4) is said to be robustly asymptotically stable with an H∞ norm bound � > 0 if there existsa feedback control law u(t) such that the following holds:

(1) System (3) with w(t) ≡ 0 is asymptotically stable.(2) Under the assumption of zero initial condition, the controlled output z(t) satisfies ‖z(t)‖2��‖w(t)‖2 for any

nonzero w(t) ∈ L2[0, ∞).

The objective of this paper is to determine the feedback control law u(t) = ∑ki=1 Ki x(t) such that systems (3)–(4)

to be robust asymptotically stable with an H∞ norm bound � > 0. To state our main results, the following lemmas areuseful in the proofs of our results.

Lemma 1 (Park and Kwon [20]). For any constant matrix M > 0, any scalars a and b with a < b, and any vectorfunction x(t) : [a, b] → Rn such that the integrals concerned are well defined, then

[∫ b

ax(s) ds

]TM

[∫ b

ax(s) ds

]�(b − a)

∫ b

axT(s)Mx(s) ds.

Lemma 2 (Wang et al. [26]). Let D, E , and F(t) be real matrices of appropriate dimensions, and F(t) satisfying

FT(t)F(t)� I.

Then, the following inequality holds for any constant � > 0:

DF(t)E + ETFT(t)DT��DDT + �−1ETE .

3. H∞ performance analysis

In this section, we will derive the delay-dependent LMI conditions for H∞ performance analysis for the systems(3)–(4). To this end, define

�0 = 12 (�M + �m), = 1

2 (�M − �m),

�0 = 12 (�M + �m), = 1

2 (�M − �m). (5)

Then, by the above notations, we have that �(t) ∈ [�0−, �0+], �0�, �(t) ∈ [�0−, �0+] and �0 �. Apparently,when = 0, i.e., �m = �M , �(t) becomes a constant delay. If = �0, then the routine case for the time delay, i.e.,0��(t)��M , is covered. Obviously, the similar results also hold for �(t) in the cases of = 0 and = �0.

In the following, it is assumed that the feedback gain matrices Ki (i = 1, 2, . . . , k) are known. Then, the followingconclusion can be obtained.

Theorem 1. For given scalars �0 > 0, > 0, �0 > 0, > 0 and � > 0, as well as the given matrices Ki

(i = 1, 2, . . . , k), systems (3)–(4) are robustly asymptotically stable with an H∞ norm bound � if there exist matricesTl > 0, Hl > 0, P11, P12, P13, P22, P23, P33, Qq , Nq , Lq , Mq and Wq (l = 1, 2, 3, 4, q = 1, 2, . . . , 6) so that thefollowing LMIs hold, for 1� i , j , s�k:⎡

⎢⎣�(�rl ) + �� + �T� QBwi

∗ −� 0

∗ ∗ −�2 I

⎤⎥⎦ < 0, (6)

⎡⎢⎣

P11 P12 P13

∗ P22 P23

∗ ∗ P33

⎤⎥⎦ > 0, (7)

where

� = diag

(1

�0T2,

1

�0T3,

1

T4,

1

�0H2,

1

�0H3,

1

H4

),


= [Z1 − N − L Z2 − M − W ], � = [Ci + Di K j C1i 0 0 D1i Ks 0],

ZT1 = [PT

22 + P23 0 − PT22 PT

12 0 − P23], QT = [QT1 QT

2 QT3 QT

4 QT5 QT

6 ],

ZT2 = [PT

23 + PT33 0 − PT

23 PT13 0 − PT

33], NT = [NT1 NT

2 NT3 NT

4 NT5 NT

6 ],

LT = [LT1 LT

2 LT3 LT

4 LT5 LT

6 ], MT = [MT1 MT

2 MT3 MT

4 MT5 MT

6 ],

WT = [WT1 WT

2 WT3 WT

4 WT5 WT

6 ], �� = QGi Fi (t)Ei j + (QGi Fi (t)Ei j )T,

Ei j = [Ei + Ebi K j E1i 0 0 Eb1i Ks 0],

and �(�rl ) is a symmetric block-matrix with �rl being its element at the position (r, l), and defined as follows for 1�r ,l�6:

�11 = P12 + PT12 + P13 + PT

13 + T1 + H1 + �0T2 + �0H2 + N1 + NT1 + M1 + MT

1 + Q1Ai j + ATi j Q

T1 ,

�12 = NT2 + MT

2 − L1 + Q1A1i + ATi j Q

T2 ,

�13 = −P12 − N1 + NT3 + MT

3 + L1 + ATi j Q

T3 ,

�14 = P11 + NT4 + MT

4 − Q1 + ATi j Q

T4 ,

�15 = NT5 + MT

5 − W1 + Q1B1i Ks + ATi j Q

T5 ,

�16 = −P13 + NT6 − M1 + MT

6 + W1 + ATi j Q

T6 ,

�22 = −L2 − LT2 + Q2A1i + AT

1i QT2 ,

�23 = −N2 + L2 − LT3 + AT

1i QT3 ,

�24 = −LT4 − Q2 + AT

1i QT4 ,

�25 = −LT5 − W2 + Q2B1i Ks + AT

1i QT5 ,

�26 = −M2 − LT6 + W2 + AT

1i QT6 ,

�33 = −N3 − NT3 + L3 + LT

3 − T1,

�34 = −NT4 + LT

4 − Q3,

�35 = −NT5 + LT

5 − W3 + Q3B1i Ks,

�36 = −NT6 − M3 + LT

6 + W3,

�44 = �0T3 + �0H3 + 2T4 + 2H4 − Q4 − QT4 ,

�45 = −W4 + Q4B1i Ks − QT5 ,

�46 = −M4 + W4 − QT6 ,

�55 = −W5 − WT5 + Q5B1i Ks + KT

s BT1i Q

T5 ,

�56 = −M5 + W5 − WT6 + KT

s BT1i Q

T6 ,

�66 = −H1 − M6 − MT6 + W6 + WT

6 .

Proof. Consider the following Lyapunov function candidate:

V = V1 + V2 + V3, (8)

where

V1 = xTP11x + 2xTP12

∫ t

t−�0x(s) ds + 2xTP13

∫ t

t−�0

x(s) ds +∫ t

t−�0xT(s) dsP22

∫ t

t−�0x(s) ds

+2∫ t

t−�0xT(s) dsP23

∫ t

t−�0

x(s) ds +∫ t

t−�0

xT(s) dsP33

∫ t

t−�0x(s) ds,

V2 =∫ t

t−�0xT(s)T1x(s) ds +

∫ 0

−�0

∫ t

t+sxT(�)T2x(�) d� ds

+∫ t

t−�0xT(s)H1x(s) ds +

∫ 0

−�0

∫ t

t+sxT(�)H2x(�) d� ds,


V3 =∫ 0

−�0

∫ t

t+sxT(�)T3 x(�) d� ds +

∫ −�0+

−�0−

∫ t

t+sxT(�)T4 x(�) d� ds

+∫ 0

−�0

∫ t

t+sxT(�)H3 x(�) d� ds +

∫ −�0+

−�0−

∫ t

t+sxT(�)H4 x(�) d� ds.

Differentiating V gives

V = xT(t)(P12 + PT12 + P13 + PT

13 + T1 + H1 + �0T2 + �0H2)x(t)

+2xT(t)P11 x(t) − 2xT(t)P12x(t − �0) − 2xT(t)P13x(t − �0)

−xT(t − �0)T1x(t − �0) − xT(t − �0)H1x(t − �0)

+xT(t)(�0T3 + 2T4 + �0H3 + 2H4)x(t)

+2xT(t)(P22 + PT23)

∫ t

t−�0x(s) ds + 2xT(t)(P23 + P33)

∫ t

t−�0x(s) ds

−2xT(t − �0)P22

∫ t

t−�0x(s) ds − 2xT(t − �0)P23

∫ t

t−�0x(s) ds

−2xT(t − �0)PT23

∫ t

t−�0x(s) ds − 2xT(t − �0)P33

∫ t

t−�0x(s) ds

+2xT(t)P12

∫ t

t−�0x(s) ds + 2xT(t)P13

∫ t

t−�0x(s) ds

−∫ t

t−�0xT(s)T2x(s) ds −

∫ t

t−�0xT(s)H2x(s) ds

−∫ t

t−�0xT(s)T3 x(t) ds −

∫ t−�0+

t−�0−xT(s)T4 x(t) ds

−∫ t

t−�0xT(s)H3 x(t) ds −

∫ t−�0+

t−�0−xT(s)H4 x(t) ds. (9)

Define

eT(t) = [xT(t) xT(t − �) xT(t − �0) xT(t) xT(t − �) xT(t − �0)],

ZT1 = [PT

22 + P23 0 − PT22 PT

12 0 − P23],

ZT2 = [PT

23 + PT33 0 − PT

23 PT13 0 − PT

33].

Then, (9) can be rewritten as follows:

V = eT(t)

[�0 �1∗ �2

]e(t)

+2e(t)TZ1

∫ t

t−�0x(s) ds + 2e(t)TZ2

∫ t

t−�0x(s) ds −

∫ t

t−�0xT(s)T2x(s) ds

−∫ t

t−�0xT(s)H2x(s) ds −

∫ t

t−�0xT(s)T3 x(t) ds −

∫ t−�0+

t−�0−xT(s)T4 x(t) ds

−∫ t

t−�0xT(s)H3 x(t) ds −

∫ t−�0+

t−�0−xT(s)H4 x(t) ds (10)

with

�0 = P12 + PT12 + P13 + PT

13 + T1 + H1 + �0T2 + �0H2,

�1 = [0 − P12 P11 0 − P13],

�2 = diag(0, −T1, �, 0, −H1),

� = �0T3 + 2T4 + �0H3 + 2H4.


To establish the delay-dependent stability conditions, the free-weighting matrix approach [10,11] will be used in thefollowing. By Newton–Leibniz formula, we have the equality below:

0 = x(t) − x(t − �0) −∫ t

t−�0x(s) ds.

Furthermore, define a matrix NT = [NT1 NT

2 NT3 NT

4 NT5 NT

6 ] with appropriate dimensions. Then, the followingequality can be verified easily:

0= 2eT(t)N

(x(t) − x(t − �0) −

∫ t

t−�0x(s) ds

)

= 2eT(t)N [I 0 − I 0 0 0]e(t) − 2eT(t)N∫ t

t−�0x(s) ds

= eT(t)(�N + �TN )e(t) − 2eT(t)N

∫ t

t−�0x(s) ds (11)

with �N = [N 0 − N 0 0 0].Similarly, the following equalities (12)–(15) can be obtained.For matrix LT = [LT

1 LT2 LT

3 LT4 LT

5 LT6 ] with Li having compatible dimensions,

0= 2eT(t)L

(x(t − �0) − x(t − �) −

∫ t−�0

t−�x(s) ds

)

= eT(t)(�L + �TL )e(t) − 2eT(t)L

∫ t−�0

t−�x(s) ds, (12)

where �L = [0 − L L 0 0 0].For matrix MT = [MT

1 MT2 MT

3 MT4 MT

5 MT6 ] with Mi being of compatible dimensions,

0= 2eT(t)M

(x(t) − x(t − �0) −

∫ t

t−�0

x(s) ds

)

= eT(t)(�M + �TM )e(t) − 2eT(t)M

∫ t

t−�0

x(s) ds, (13)

where �M = [M 0 0 0 0 − M].For matrix WT = [WT

1 WT2 WT

3 WT4 WT

5 WT6 ] with Wi having compatible dimensions,

0= 2eT(t)W

(x(t − �0) − x(t − �) −

∫ t−�0

t−�x(s) ds

)

= eT(t)(�W + �TW )e(t) − 2eT(t)W

∫ t−�0

t−�x(s) ds, (14)

where �W = [0 0 0 0 − W W ].Finally, for matrix QT = [QT

1 QT2 QT

3 QT4 QT

5 QT6 ] with Qi having compatible dimensions,

0= 2eT(t)Q

⎛⎝ k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)( Ai j x(t) + A1i x(t − �) + B1i Ks x(t − �) + Bwiw(t)) − x(t)

⎞⎠

=k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)2eT(t)Q[Ai j A1i 0 − I B1i Ks 0]e(t)

+k∑

i=1

k∑j=1

k∑s=1

hih j hs(�)2eT(t)QMi Fi (t)[Ei + Ebi K j E1i 0 0 Eb1i Ks 0]e(t)

−k∑

i=1

k∑j=1

k∑s=1

hih j hs(�)2QeT(t)Bwiw(t)

=k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)eT(t)(� + ��)e(t) −

k∑i=1

k∑j=1

k∑s=1

hih j hs(�)2QeT(t)Bwiw(t), (15)


where �� = QGi Fi (t)Ei j + (QGi Fi (t)Ei j )T with Ei j = [Ei + Ebi K j E1i 0 0 Eb1i Ks 0], and � = �(�rl ) is asymmetric block-matrix and its block-elements �rl are defined as follows for 1�r , l�6:

�11 = Q1Ai j + ATi j Q

T1 , �12 = Q1A1i + AT

i j QT2 , �13 = AT

i j QT3 ,

�14 = −Q1 + ATi j Q

T4 , �15 = Q1B1i Ks + AT

i j QT5 , �16 = AT

i j QT6 ,

�22 = Q2A1i + AT1i Q

T2 , �23 = AT

1i QT3 , �24 = −Q2 + AT

1i QT4 ,

�25 = Q2B1i Ks + AT1i Q

T5 , �26 = AT

1i QT6 , �33 = 0, �34 = −Q3,

�35 = Q3B1i Ks, �36 = 0, �44 = −Q4 − QT4 , �45 = Q4B1i Ks − QT

5 ,

�46 = −QT6 , �55 = −Q5B1i Ks − KT

s BT1i Q

T5 , �56 = KT

s BT1i Q

T6 , �66 = 0.

Then, combining (10)–(15) yields the following equality:

V =k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)eT(t)(�(�rl ) + ��)e(t)

+2e(t)TZ1

∫ t

t−�0x(s) ds + 2e(t)TZ2

∫ t

t−�0x(s) ds − 2eT(t)N

∫ t

t−�0x(s) ds

−2eT(t)L∫ t−�0

t−�x(s) ds − 2eT(t)M

∫ t

t−�0

x(s) ds − 2eT(t)W∫ t−�0

t−�x(s) ds

−∫ t

t−�0xT(s)T2x(s) d −

∫ t

t−�0

xT(s)H2x(s) ds −∫ t

t−�0xT(s)T3 x(s) ds

−∫ t

t−�0xT(s)H3 x(s) ds −

∫ t−�0+

t−�0−xT(s)T4 x(s) ds −

∫ t−�0+

t−�0−xT(s)H4 x(s) ds

−k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)2eT(t)QBwiw(t), (16)

where matrix �(�rl ) and �� are the same ones defined in (6).Note that T4 > 0. Thus when � − �0�0, we have

−∫ t−�0+

t−�0−xT(s)T4 x(s) ds = −

∫ t−�0

t−�0−xT(s)T4 x(s) ds −

∫ t−�

t−�0xT(s)T4 x(s) ds

−∫ t−�0+

t−�xT(s)T4 x(s) ds

� −∫ t−�

t−�0xT(s)T4 x(s) ds. (17)

Since � ∈ [�0 − , �0 + ], applying Lemma 1 to − ∫ t−�t−�0

xT(s)T4 x(s) ds yields

−∫ t−�

t−�0xT(s)T4 x(s) ds � −1

(∫ t−�

t−�0xT(s) ds

)T4

(∫ t−�

t−�0x(s) ds

)(18)

= −1

(∫ t−�0

t−�xT(s) ds

)T4

(∫ t−�0

t−�x(s) ds

).

Consequently, (17) and (18) imply that for � − �0�0,

−∫ t−�0+

t−�0−xT(s)T4 x(s) ds� − 1

(∫ t−�0

t−�xT(s) ds

)T4

(∫ t−�0

t−�x(s) ds

). (19)

On the other hand, it can be clearly seen that (19) is also true for the case of � − �0 > 0.Similar to (19), the following inequality holds for any � ∈ [�0 − , �0 + ]:

−∫ t−�0+

t−�0−xT(s)H4 x(s) ds� − 1

(∫ t−�0

t−�xT(s) ds

)H4

(∫ t−�0

t−�x(s) ds

). (20)


In addition, by applying Lemma 2 to the corresponding integral terms in (16), the following inequalities are obtained:

−∫ t

t−�0xT(s)T2x(s) ds� − 1

�0

(∫ t

t−�0xT(s) ds

)T2

(∫ t

t−�0x(s) ds

), (21)

−∫ t

t−�0xT(s)H2x(s) ds� − 1

�0

(∫ t

t−�0xT(s) ds

)H2

(∫ t

t−�0

x(s) ds

), (22)

−∫ t

t−�0xT(s)T3 x(s) ds� − 1

�0

(∫ t

t−�0xT(s) ds

)T3

(∫ t

t−�0x(s) ds

), (23)

−∫ t

t−�0xT(s)H3 x(s) ds� − 1

�0

(∫ t

t−�0

xT(s) ds

)H3

(∫ t

t−�0x(s) ds

). (24)

Then, substituting (19)–(24) into (16) results in that

V �

k∑i=1

k∑j=1

k∑s=1

hi h j hs(�)

[e(t)�(t)

]T [�(�rl ) + ��

∗ −�

] [e(t)�(t)

]

+k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)eT(t)QBwi B

Twi Qe(t)�−2 + �2wT(t)w(t), (25)

where the inequality 2eT(t)QBwiw(t)�eT(t)QBwi BTwi Qe(t)�−2 + �2wT(t)w(t) is used, �(t) is defined by

�T(t) =[∫ t

t−�0xT(s) ds

∫ t

t−�0xT(s) ds

∫ t−�0

t−�xT(s) ds

∫ t

t−�0

xT(s) ds∫ t

t−�0xT(s) ds

∫ t−�0

t−�xT(s) ds

],

and the matrices and � are the same ones defined in (6). Let � = [Ci + Di K j C1i 0 0 D1i Ks 0]. Then, accordingto the definition of z(t), we have

zT(t)z(t) =⎛⎝ k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)eT(t)�T

⎞⎠ ×

⎛⎝ k∑

l=1

k∑p=1

k∑q=1

hlh phq (�)�e(t)

⎞⎠

�

k∑i=1

k∑j=1

k∑s=1

hi h j hs(�)eT(t)�T�e(t).

Furthermore,

zT(t)z(t) − �2wT(t)w(t)

�

k∑i=1

k∑j=1

k∑s=1

hi h j hs(�)eT(t)�T�e(t) − �2wT(t)w(t) + V − V

�

k∑i=1

k∑j=1

k∑s=1

hi h j hs(�)eT(t)�T�e(t) − �2wT(t)w(t)

+k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)

[e(t)�(t)

]T [�(�rl ) + ��

∗ �

] [e(t)�(t)

]

+k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)�−2eT(t)QBwi B

Twi Qe(t) + �2wT(t)w(t) − V

=k∑

i=1

k∑j=1

k∑s=1

hi h j hs(�)

[e(t)�(t)

]T [� ∗ �

] [e(t)�(t)

]− V (26)

with � = �(�rl ) + �� + �T� + �−2QBwi BTwi Q

T.


At present stage, by applying Schur complement to (6), we have that[� ∗ �

]< 0 (27)

which means that

zT(t)z(t) − �2wT(t)w(t)� − V . (28)

Consequently, integrating both sides of (28) from 0 to T gives∫ T

0zT(t)z(t) dt −

∫ T

0�2wT(t)w(t) dt� − V (T ) + V (0) (29)

which, together with zero initial condition, implies that∫ ∞

0zT(t)z(t) dt�

∫ ∞

0�2wT(t)w(t) dt.

As a result, ‖z(t)‖2��‖w(t)‖ holds.When w(t) ≡ 0, it follows from (25) that V < 0, which ensures the asymptotical stability of the closed-loop system.

The proof is thus completed. �

Because of the existence of the parameter uncertainty �� in (6), Theorem 1 cannot be directly used to determine theperformance of the closed-loop system. However, the following theorem provides a sufficient condition for the systemto be robustly asymptotically stable with an H∞ norm bound �.

Theorem 2. For given scalars �0 > 0, > 0, �0 > 0, > 0 and � > 0, as well as the given matrices Ki

(i = 1, 2, . . . , k), systems (3)–(4) are robustly asymptotically stable with an H∞ norm bound � > 0 if there existmatrices Tl > 0, Hl > 0, P11, P12, P13, P22, P23, P33, Qq , Nq , Lq , Mq and Wq (l = 1, 2, 3, 4, q = 1, 2, . . . , 6), andscalars �i js > 0 so that the following LMIs hold, for i = 1� i , j , s�k:⎡

⎢⎢⎣�(�rl ) + �T� + �i js ET

i j Ei j QBwi QGi

∗ −� 0 0∗ ∗ −�2 I 0∗ ∗ ∗ −�i js I

⎤⎥⎥⎦ < 0, (30)

⎡⎣ P11 P12 P13

∗ P22 P23∗ ∗ P33

⎤⎦ > 0, (31)

where �(�i j ), Ei j , and � are defined as in Theorem 1.

Proof. In light of the structure of ��, we have

�(�rl ) + �� = �(�rl ) + QGi F(t)Ei j + ETi j F

T(t)GTi Q. (32)

Applying Lemma 2 to (32) produces

�(�rl ) + ��(�rl ) + �−1i js QGiG

Ti Q

T + �i js ETi j Ei j .

Therefore, a sufficient condition for (6) holding is⎡⎣ �(�rl ) + �T� + �−1

i js QGiGi QT + �i js ETi j Ei j QBwi

∗ � 0∗ ∗ −�2 I

⎤⎦ < 0. (33)

Obviously, by Schur complement, (30) is equivalent to (33). This means that under conditions (30) and (31), (6) and(7) hold. Thus, the proof is completed by Theorem 1. �


4. Robust H∞ fuzzy controller design

Based on Theorem 2, we will present a systematic procedure for designing the robust H∞ fuzzy controller. Thesuggested fuzzy controller (2) can guarantee systems (3)–(4) to be robustly asymptotically stable with an H∞ normbound � > 0.

Theorem 3. For given scalars �0 > 0, > 0, �0 > 0, > 0, � > 0 and ap (p = 2, . . . , 6, a4 � 0), systems (3)–(4) isrobust asymptotically stable with an H∞ norm bound � > 0, and the feedback gain matrices are given by

Ki = Fi X−1, i = 1, 2, . . . , k,

if there exist matrices Tl > 0, Hl > 0, P11, P12, P13, P22, P23, P33, Nq , Lq , Mq , Wq (l = 1, 2, 3, 4, q = 1, 2, . . . , 6),and X, as well as matrices Fi and scalars �i js > 0 so that the following LMIs hold, for 1� i, j, s�k:

⎡⎢⎢⎣

� � �1

∗ −� 0 0∗ ∗ −�2 I 0∗ ∗ ∗ −�1

⎤⎥⎥⎦ < 0, (34)

⎡⎣ P11 P12 P13

∗ P22 P23∗ ∗ P33

⎤⎦ > 0, (35)

where

�T = [BT

wi a2BTwi a3B

Twi a4B

Twi a5B

Twi a6B

Twi ],

�T1 =

[Ei X + Ebi Fj E1i X 0 0 Eb1i Fs 0Ci X + Di Fj C1i X 0 0 D1i Fs 0

],

� = diag

(1

�0T2,

1

�0T3,

1

T4,

1

�0H2,

1

�0H3,

1

H4

),

� = diag(�i js I, I ),

= [Z1 − N − L Z2 − M − W ],

ZT1 = [PT

22 + P23 0 − PT22 PT

12 0 − P23],

ZT2 = [PT

23 + PT33 0 − PT

23 PT13 0 − PT

33],

NT = [NT1 NT

2 NT3 NT

4 NT5 NT

6 ],

LT = [LT1 LT

2 LT3 LT

4 LT5 LT

6 ],

MT = [MT1 MT

2 MT3 MT

4 MT5 MT

6 ],

WT = [WT1 WT

2 WT3 WT

4 WT5 WT

6 ]

and � = �(�rl ) is a symmetric block-matrix with �rl being its element at the position (r, l), and defined as follows for1�r, l�6:

�11 = P12 + PT12 + P13 + PT

13 + T1 + H1 + �0T2 + �0 H2 + N1 + NT1

+M1 + MT1 + Ai X + Bi Fj + X AT

i + FTj B

Ti + �i jsGiG

Ti ,

�12 = NT2 + MT

2 − L1 + A1i X + a2X ATi + a2F

Tj B

Ti + a2�i jsGiG

Ti ,

�13 = −P12 − N1 + NT3 + MT

3 + L1 + a3X ATi + a3F

Tj B

Ti + a3�i jsGiG

Ti ,

�14 = P11 + NT4 + MT

4 − X + a4X ATi + a4F

Tj B

Ti + a4�i jsGiG

Ti ,

�15 = NT5 + MT

5 − W1 + B1i Fs + a5X ATi + a5F

Tj B

Ti + a5�i jsGiG

Ti ,

�16 = −P13 + NT6 − M1 + MT

6 + W1 + a6X ATi + a6F

Tj B

Ti + a6�i jsGiG

Ti ,

�22 = −L2 − LT2 + a2A1i X + X AT

1i a2 + a2a2�i jsGiGTi ,


�23 = −N2 + L2 − LT3 + X AT


�24 = −LT4 − a2X + X AT


�25 = −LT5 − W2 + a2B1i Fs + X AT


�26 = −M2 − LT6 + W2 + X AT


�33 = −N3 − NT3 + L3 + LT

3 − T1 + a3a3�i jsGiGTi ,

�34 = −NT4 + LT

4 − a3X + a3a4�i jsGiGTi ,

�35 = −NT5 + LT

5 − W3 + a3B1i Fs + a3a5�i jsGiGTi ,

�36 = −NT6 − M3 + LT

6 + W3 + a3a6�i jsGiGTi ,

�44 = �0T3 + �0 H3 + 2T4 + 2H4 − a4X − a4X + a4a4�i jsGiGTi ,

�45 = −W4 + a4B1i Fs − a5X + a4a5�i jsGiGTi ,

�46 = −M4 + W4 − a6X + a4a6�i jsGiGTi ,

�55 = −W5 − WT5 + a5B1i Fs + FT

s BT1i a5 + a5a5�i jsGiG

Ti ,

�56 = −M5 + W5 − WT6 + FT

s BT1i a6 + a5a6�i jsGiG

Ti ,

�66 = −H1 − M6 − MT6 + W6 + WT

6 + a6a6�i jsGiGTi .

Proof. Notice that (34) implies that X is a nonsingular matrix. So, let Q1 = X−1. Thus, to prove Theorem 3, we canshow that (34) and (35) imply (30) and (31). To this end, let Qp = apQ1 (p = 2, . . . , 6), �i js = �−1

i js , and define thefollowing variables:

T1 = Q1T1QT1 , T2 = Q1T2Q

T1 , T3 = Q1T3Q

T1 , T4 = Q1T4Q

T1 ,

H1 = Q1 H1QT1 , H2 = Q1 H2 Q1, H3 = Q1 H3Q

T1 , H4 = Q1 H4Q

T1 ,

P11 = Q1 P11QT1 , P12 = Q1 P12Q

T1 , P13 = Q1 P13Q

T1 , P22 = Q1 P22Q

T1 ,

P23 = Q1 P23QT1 , P33 = Q1 P33Q

T1 , Ki = Fi Q1, i = 1, 2, . . . , k,

Nl = Q1 Nl QT1 , Ll = Q1 Ll Q

T1 , Ml = Q1Ml Q

T1 , Wl = Q1Wl Q

T1 , l = 1, 2, . . . , 6.

Then, pre- and post-multiplying both sides of (34) by � = diag

⎛⎝Q1, · · · Q1,︸︷︷︸

12

I, I, I

⎞⎠ and its transpose, respec-

tively, shows that �(34)�T < 0. Then, it can be verified by Schur complement that �(34)�T < 0 is equivalent to (33).Consequently, from the proof of Theorem 2, (33) ensures that (30) is true. In addition, (31) is immediately obtained bypre- and post-multiplying both sides of (35) with diag(Q1, Q1, Q1) and its transpose. Therefore, by Theorem 2, theproof is completed. �

Remark 2. Unlike in Theorem 2, in order to determine the feedback gain matrices, we have to set Qi = ai Q1(i = 2, 3, 4, 5, 6) to obtain the LMI conditions. The parameters ai (i = 2, 3, 4, 5, 6) should be given prior to solveLMI (34). How to choose these design parameters to optimize is still an open problem. If the common form for Qi

(i = 1, 2, 3, 4, 5, 6) are retained (i.e., a′i s are not introduced), one has to employ nonlinear programming to solve

nonrestrict LMI conditions which brings much computational burden and may lead to conservative results.

Remark 3. To reduce the conservatism of the stability criteria, some slack variables are introduced. Then, it should bepointed out that the works in [28–30] show that when the slack variables are introduced, some ones among themmaybenot useful for improving the results. And the redundant slack variables will lead to the burdensome computation.Therefore, it is important to known which slack variables are redundant to reduce the conservatism of the stabilitycriteria. However, to expressly known the effect of each slack variable to improvement of the stability criteria is verydifficult, and is still an opened problem which will be studied further.

In what follows, we consider the robust stabilization problem of system (3) with w(t) ≡ 0. And the followingcorollary can be derived from Theorem 3 directly.


Corollary 1. For given scalars �0 > 0, > 0, �0 > 0, > 0, and scalars a� (� = 2, 3, 4, 5, 6, a4 � 0), if there existmatrices Tl > 0, Hl > 0, P11, P12, P13, P22, P23, P33, Nq , Lq , Mq , Wq (l = 1, 2, 3, 4, q = 2, 3, 4, 5, 6) and X, aswell as matrices Fi and scalars �i js > 0 such that for 1� i, j, s�k,

⎡⎣ � �

∗ −� 0∗ ∗ −�i js I

⎤⎦ < 0, (36)

⎡⎣ P11 P12 P13

∗ P22 P23∗ ∗ P33

⎤⎦ > 0, (37)

where �T = [Ei X + Ebi Fj E1i X 0 0 Eb1i Fs 0], and �, and � are defined as in Theorem 3, then the feedback gain

matrices Ki are given by

Ki = Fi X−1, i = 1, 2, . . . , k

and the resulting closed-loop system (3) is asymptotically stable.

In addition, another especial case of (3) is u(t − �) ≡ 0. For this case, the following corollary can be obtained fromthe proof of Theorems 1–3 by setting P13 = 0, P23 = 0, P33 = 0, Hl = 0 (for l = 1, 2, 3, 4), M = 0, W = 0, as wellas Ni = 0, Li = 0 and Qi = 0 for i = 5, 6.

Corollary 2. For given scalars �0 > 0, > 0, � > 0 and ap (p = 2, 3, 4), systems (3)–(4) are robust asymptoticallystable with an H∞ norm bound � > 0 and the feedback gain matrices are given by

Ki = Fi X−1, i = 1, 2, . . . , k,

if there exist matrices Tl > 0, P11, P12, P22, Nq , Lq (l = 1, 2, 3, 4, q = 1, 2, 3, 4), and X, as well as matrices Fi andscalars �i j > 0 so that the following LMIs hold, for 1� i < j�k:⎡

⎢⎢⎣� I Bwi �(i, j)∗ −� 0 0∗ ∗ −�2 I 0∗ ∗ ∗ −�

⎤⎥⎥⎦ < 0, 1� i�k, (38)

⎡⎢⎢⎢⎢⎣

�(i, j) + �( j, i)

2 I

Bwi + Bw j

2

�(i, j) + �( j, i)

2∗ −� 0 0∗ ∗ −�2 I 0∗ ∗ ∗ −�

⎤⎥⎥⎥⎥⎦ < 0, 1� i < j�k, (39)

[P11 P12∗ P22

]> 0, (40)

where

= [Z − N − L], �T =

[Ei X + Ebi Fj E1i X 0 0Ci X + Di Fj C1i X 0 0

],

ZT = [PT22 0 − PT

22 PT12], NT = [NT

1 NT2 NT

3 NT4 ],

LT = [LT1 LT

2 LT3 LT

4 ], � = diag

(1

�0T2

1

�0T3

1

T4

),

IT = [I a2 I a3 I a4 I ], � = diag(�i j I, I )


and

�(i, j) =

⎡⎢⎢⎣

�11 �12 �13 �13∗ �22 �23 �24∗ ∗ �33 �34∗ ∗ ∗ �44

⎤⎥⎥⎦

with

�11 = P12 + PT12 + T1 + �0T2 + N1 + NT

1 + Ai X + Bi Fj + X ATi + FT

j BTi + �i j GiG

Ti ,

�12 = NT2 − L1 + a2A1i X

T + a2X ATi + a2F

Tj B

Ti + a2�i j GiG

Ti ,

�13 = −P12 − N1 + NT3 + L1 + a3X AT

i + a3FTj B

Ti + a3�i j GiG

Ti ,

�14 = P11 + NT4 − X + a4X AT

i + a4FTj B

Ti + a4�i j GiG

Ti ,

�22 = −L2 − LT2 + a2A1i X

T + X AT1i a2 + a2a2�i j GiG

Ti ,

�23 = −N2 + L2 − LT3 + X AT

1i a3 + a2a3�i j GiGTi ,

�24 = −LT4 − a2X + X AT

1i a4 + a2a4�i j GiGTi ,

�33 = −N3 − NT3 + L3 + LT

3 − T1 + a3a3�i j GiGTi ,

�34 = −NT4 + LT

4 − Q3 + a3a4�i j GiGTi ,

�44 = �0T3 + 2T4 − a4X − a4X + a4a4�i j GiGTi .

5. Computer simulation

In this section, some examples are used to illustrate the effectiveness of the proposed results, and to compare withthe existing results.

Example 1. Consider the following uncertain T–S fuzzy system with state and input delays:

Rule 1: IF �(t) = x2(t) + a(vt/2L)x1(t) + (1 − a)(vt/2L)x1(t − �) is about 0, THEN

x(t) = (A1 + �A1)x(t) + (A11 + �A11)x(t − �(t)) + (B1 + �B1)u(t) + (B11 + �B11)u(t − �(t)).

Rule 2: IF �(t) = x2(t) + a(vt/2L)x1(t) + (1 − a)(vt/2L)x1(t − �) is about � or −�, THEN

x(t) = (A2 + �A2)x(t) + (A21 + �A21)x(t − �(t)) + (B2 + �B2)u(t) + (B21 + �B21)u(t − �(t)),

where

A1 =

⎡⎢⎢⎢⎢⎢⎣

−avt

Lt00 0

avt

Lt00 0

−av2 t2

2Lt0

vt

t00

⎤⎥⎥⎥⎥⎥⎦ , A11 =

⎡⎢⎢⎢⎢⎢⎣

−(1 − a)vt

Lt00 0

(1 − a)vt

Lt00 0

(1 − a)v2 t2

2Lt00 0

⎤⎥⎥⎥⎥⎥⎦ ,

A2 =

⎡⎢⎢⎢⎢⎢⎣

−avt

Lt00 0

avt

Lt00 0

−adv2 t2

2Lt0

dvt

t00

⎤⎥⎥⎥⎥⎥⎦ , A21 =

⎡⎢⎢⎢⎢⎢⎣

−(1 − a)vt

Lt00 0

(1 − a)vt

Lt00 0

(1 − a)dv2 t2

2Lt00 0

⎤⎥⎥⎥⎥⎥⎦ ,

B1 =

⎡⎢⎢⎣

vt

lt000

⎤⎥⎥⎦ , B2 =

⎡⎢⎢⎣

vt

lt000

⎤⎥⎥⎦ , B11 = 0.1B1, B21 = 0.1B2,


0 10 20 30 40 50 60−10

0

10

20

30

40

50

60

70

80

Time (Sec)

x1x2x3

Fig. 1. Response of state for Example 1.

Gi = [0.255 0.255 0.255]T, Ei = Ei1 = [0.1 0 0],

Ebi = 0.15, Ebi1 = 0.05, i = 1, 2

and l = 2.8, L = 5.5, v = −1.0, t = 2.0, t0 = 0.5, a = 0.7, d = 10t0/�. When all the uncertainties are removed fromthe above system, this system is just Example 5.2 in [17]. According to [17], take

h1 =(

1

1 + exp(−3(�(t) + 0.5�))

), h2 = 1 − h1.

For the case where �Ai = 0, �A1i = 0, �Bi = 0, and �B1i = 0, the following fuzzy control law is proposed in [17]for � = � = 0.1:

u(t) = h1[3.6857 − 9.1238 0.8632]x(t) + h2[3.7138 − 9.7537 0.9086]x(t), (41)

which guarantees the asymptotic stability of the resulting closed-loop system. Because the uncertainties have not beentaken into account, the control law (41) cannot, theoretically, be used to stabilize this uncertain system. Then, Corollary1 can be used to solve the robust stabilization problem for the above system. By setting the parameter as ap = 0.0125(p = 2, 3, 5, 6) and a4 = 1.5, and �0 = 1.2135 and �0 = 1, solving LMIs (36)–(37) gives max = 1.2135, max = 1and the feedback gains as follows:

K1 = [3.6776 − 0.8520 0.0154], K2 = [3.6793 − 0.8500 0.0155],

which implies that the regarding fuzzy controller guarantees the asymptotic stability of the closed-loop system for �(t) ∈[0, 2.4270] and �(t) ∈ [0, 2]. The simulation is run under the initial conditions as �(t) = [4, −1, 2] for t ∈ [−2.4270,0], and u(t) = 0 for t ∈ [−2, 0] with �(t) = 1 + sin(t) and �(t) = 1.2 + 1.2 sin(t). The simulation results are shownby Figs. 1, 2. Fig. 1 shows the state response of the system. Fig. 2 displays the control input signal.

In addition, for given �m and �m by using Corollary 1 in this paper, we can get �M , �M and the feedback gains asshown in Table 1.

Example 2. Consider the following T–S fuzzy system:

Plant Rule i: IF x2(t) is hi , Then

x(t)= (Ai + �Ai )x + (A1i + �A1i )x(t − �(t)) + (Bi + �Bi )u(t)

+(B1i + �B1i )u(t − �(t)) + Bwiw(t),

z(t)=Ci x(t) + C1i x(t − �(t)) + Diu(t) + D1i u(t − �(t)), i = 1, 2, (42)


0 10 20 30 40 50 60−6

−5

−4

−3

−2

−1

0

1

Time (Sec)

u

Fig. 2. Control input curve for Example 1.

Table 1

�m �M �m �M Feedback gain matrices

0 2 0 2.4270 [3.6776 − 0.8520 0.0154], [3.6793 − 0.8500 0.0155]0.5 2.9205 0.5 3.8210 [2.6259 − 0.2866 0.0022], [2.6239 − 0.2859 0.0022]1 4.7610 1 4.8209 [2.2239 − 0.0830 0.0002], [2.2238 − 0.0830 0.0002]

where h1 = (1 − 1/(1 + exp(−6x2 + 1.5�)))(1/(1 + exp(−6x2 − 1.5�))) and h2 = 1 − h1, and the system matricesare

A1 =[

0 10.1 −2

], A11 =

[0.1 00.1 −0.2

], B1 =

[01

], B11 =

[0

0.25

],

A2 =[

0 10.1 −0.75

], A12 =

[0.1 0.10 −0.15

], B2 = B1, B12 = B11,

Bw1 = Bw2 =[01

], C1 = [0.5 0.15], C2 = [0.35 0.25], D1 = D2 = 0.1,

C11 = [0.05 0.015], C12 = [0.035 0.025], D11 = D12 = 0.01,

G1 = G2 =[ −0.3 0

0 0.3

], E1 = E2 =

[ −0.15 0.20 0.04

],

E11 = E12 =[ −0.05 −0.35

0.08 −0.45

], Eb1 = Eb2 = [0.05 0.15],

Eb11 = Eb12 = [0.025 0.075].

Note that both the state delay and the input delay appear in the system dynamics and the control output equations.Given � = 1, then for �m = 0 and �m = 0, we apply Theorem 3 with ap = 0.0125 (p = 2, 3, 5, 6) and a4 = 1.5to solve (34) and (35), and it was found that the maximal upper bounds of �(t) and �(t) are �M = 1.0996 and�M = 1.0324, and the feedback gain matrices are K1 = [−0.9735 −0.9788], K2 = [−0.9679 −0.9748], for �m = 1and �m = 1, we have that �M =1.6841 and �M =1.3782 and the feedback matrices are K1 = [−1.0421 − 1.2718],K2 = [−1.0384 − 1.2698]. Thus for the case of 1��(t)�1.6841 and 1��(t)�1.3782, take the disturbance input asw(t) = 4e−0.1t sin(t), �(t) = 1 + |0.37 cos(2t)| and �(t) = 1 + |0.68 cos(2t)|, the simulation is carried out under theinitial conditions �(t) = [4, −3] and u(t) = 0 for t�0. Fig. 3 shows the state response of the closed-loop systems (42)and Fig. 4 displays the control input curve. It is seen from Fig. 3 that the closed-loop system is asymptotically stable.


0 5 10 15 20 25 30 35 40 45 50−3

−2

−1

0

1

2

3

4

Time (Sec)

x1x2


0 5 10 15 20 25 30 35 40 45 50−4

−3

−2

−1

0

1

2

3

Time (Sec)

u


Example 3. For the special case of u(t−�(t)) ≡ 0, the robust H∞ fuzzy control problem is studied in [13], in which theH∞ fuzzy controller design procedure is proposed. The following example is taken from [13]. Consider the uncertainnonlinear time-delay system described as follows:

x1 = −x1(2 + sin2 x2) + x2 + 0.1x1(t − �(t)) + 0.2x2(t − �(t))

+c(t)x1 cos2 x2 + u1 + (1 + sin2(x2))w(t),

x2 = x12 − x2(1 − cos2 x2) + 0.2x1(t − �(t)) sin2 x2 − 0.5x2(t − �(t)) + 0.5u2 + 0.1c(t)x2,

where c(t) is an uncertain parameter satisfying c(t) ∈ [−0.2, 0.2]. According to [13], by selecting the membershipfunctions as

M1(x2) = sin2 x2, M2(x2) = cos2 x2,

then the above nonlinear system can be represented by the following T–S fuzzy model.


0 5 10 15 20 25 30 35 40 45 50

−2

−1

0

1

Time (Sec)

x1x2

1.5

0.5

−0.5

−1.5

−2.5


0 5 10 15 20 25 30 35 40 45 50−1

0

1

2

3

4

5

6

7

8

Time (Sec)

u1u2


Rule 1. IF x2 is M1, THEN

x = (A1 + �A1)x + (A11 + �A11)x(t − �(t)) + (B1 + �B1)u + Bw1w(t),

z = C1x + D1u(t),

Rule 2. IF x2 is M2, THEN

x = (A2 + �A2)x + (A21 + �A21)x(t − �(t)) + (B2 + �B2)u + Bw2w(t),

z = C2x + D2u(t),

where

A1 =[ −3 1

1 −1

], A11 =

[0.1 00.2 −0.5

], B1 =

[1 00 0.5

], Bw1 =

[20

],

A2 =[ −2 1

1 0

], A21 =

[0.1 0.20 −0.5

], B2 =

[1 00 0.5

], Bw2 =

[10

],


Table 2

�m 0 2 4 6 8 10

�M 1.7435 3.7005 5.6999 7.7005 9.6995 11.6983

C1 = C2 =[0.1 00 0.1

], D1 = D2 = I, G1 = G2 =

[1 00 0

],

E1 =[0 0.20 0

], E2 =

[0.2 00 0

], E1i = 0, Ebi = 0, i = 1, 2.

For this system, setting � = 1 and �0 = 1, the method in [13] suggests the maximum allowed bound = 0.2358. Thisimplies that the allowable interval for �(t) is [0.7642, 1.2358].

Then, applying Corollary 2 with ai = 0.015 (i = 2, 3) and a4 = 0.25, it was found that the maximal allowable valueof is 0.8536. When setting = 0.2358, the maximum allowed bound of �0 is 9.8799 and the feedback gain matricesare

K1 =[ −1.0176 −1.1074

−0.6989 −2.1621

], K2 =

[ −0.7408 −1.5285−0.6989 −2.0718

],

which means that the regarding fuzzy controller guarantees the asymptotic stability of the closed-loop system for�(t) ∈ [9.6441 10.1157]. Take the disturbance input as w(t) = 2e−0.1t sin(t), and time-varying delay as �(t) =9.65 + |0.36 sin(2t)| which satisfies 9.6441��(t)�10.1157. Then, the simulation is carried out under the initialconditions �(t) = [1.5, −2.5] for t < 0. Fig. 5 shows the state response of the resulting closed-loop systems and Fig.6 displays the control input curve. In addition, for given different �m , by using Corollary 2 we have the corresponding�M , as shown in Table 2.

6. Conclusion

In this paper, we have studied the H∞ control problem for T–S fuzzy systems with state and input delays. Both thestate time delay and input time delay are time-varying delays and no restriction is imposed on the derivative of timedelays. Based on Lyapunov–Krasoviskii functional method, a new design scheme of H∞ fuzzy controller is derived.The main contribution lies in that delay-dependent conditions are presented in terms of LMI format. The effectivenessof the proposed results has been illustrated by some examples.

Acknowledgment

This work is partially supported by the Natural Sciences and Engineering Research Council of Canada, the NationalNatural Science Foundation of China (Nos. 60674055, 60774047), the Taishan Scholar programme and the NaturalScience Foundation of Shandong Province (No. Y2006G04).

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Robust control of Takagi–Sugeno fuzzy systems with state and input time delays