Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. VII (2012), No. 1 (March), pp. 61-72

A New Approach to Nonlinear Tracking Control Based on Fuzzy
Approximation

Z. Du, T.-C. Lin, V.E. Balas

Zhenbin Du
Yantai University,
Yantai, Shandong 264005, China
E-mail: zhenbindu@yahoo.com.cn

Tsung-Chih Lin
Feng-Chia University
Taichung, 40724, Taiwan
E-mail: tclin@fcu.edu.tw

Valentina E. Balas
Aurel Vlaicu University
B-dul Revolutiei 77
310130 Arad, Romania
E-mail: balas@drbalas.ro

Abstract: The problem of tracking control is addressed for a class of nonlinear
systems with uncertainties. The original nonlinear systems are approximated
by a fuzzy T-S model based on which a state-feedback controller is constructed
by using the linear matrix inequalities. The approximating error is eliminated
by an adaptive compensator based on fuzzy logic systems. The effectiveness
of the proposed control scheme is demonstrated by a simulation example. The
main advantage is that the designer makes milder constraint assumption for
the approximation error and the uncertainties in nonlinear systems.
Keywords: fuzzy T-S model; fuzzy logic systems; nonlinear systems; uncer-
tainties; tracking control.

1 Introduction

The problem of controller design for the nonlinear systems with uncertainties is a challenging
work. One of effective tools used to solve the problem is fuzzy method. There are two frequently
used fuzzy models: fuzzy T-S model [1-13] and fuzzy logic systems [16-17]. Fuzzy T-S model is
usually used to approximate nonlinear systems, and it has been widely applied to analyze the
stability of nonlinear systems [1-4]. The contributions of these works are very important, but
these works could be further improved by a simpler and more practical control scheme. The
approximating error is neglected in [1-4], which impacts the stability of the system. Therefore,
the designed controller can’t always guarantee the stability of the original system. To overcome
the effect of the approximating error, some relaxed stability methods are developed in [5-8].
These methods improve the approximation accuracy for nonlinear system. However, the original
nonlinear system is still neglected in [5-8]. In order to further relax the effect of the approximating
error, it is assumed to satisfy the matching condition in [9-11] and have a upper bound in [12-
13]. However, the matching condition and the upper bound are not easy to be found in practice,
which adds some difficulties to the controller design. The uncertainties in nonlinear systems are
assumed to satisfy the constraint of the upper bound in [14-15]. However, the upper bound may
be too large or doesn’t exist. On the other hand, fuzzy logic systems have been proved to have

Copyright c⃝ 2006-2012 by CCC Publications



62 Z. Du, T.-C. Lin, V.E. Balas

the universal approximation property. By constructing a set of fuzzy "IF-THEN" rules, fuzzy
logic system is used to model uncertain nonlinear systems [16-17].

There exist some conservatism of pure fuzzy T-S model in dealing with the approximating
error. The pure adaptive fuzzy control also has some shortages of excessively depending on the
chosen membership functions. Therefore, it is more interesting to combine both fuzzy models to
overcome their shortages each other.

Based on the above discussions, fuzzy T-S model and fuzzy logic systems are combined to
design a new tracking-control scheme for a class of nonlinear systems with uncertainties in this
paper. A fuzzy T-S model is used to approximate the nonlinear system based on which a state-
feedback controller is constructed by use of the linear matrix inequalities. The approximating
error and the uncertain nonlinear parts are eliminated by a compensator based on fuzzy logic
systems. The main advantages are summarized as follows:

Firstly, fuzzy T-S model and fuzzy logic systems are combined to develop a controller. Com-
pared with the existing works based on fuzzy T-S model [1-13], the proposed method in this
paper makes milder constraint assumptions for the approximating error. Secondly, the dimen-
sion of the matrix inequalities is reduced, thus, the difficulty of solving the matrix equalities
is relaxed. Thirdly, the existing works based on fuzzy logic systems [16-17] excessively depend
on the chosen membership functions which are improved in the proposed method of this paper.
Finally, the developed controller makes full use of the advantages of two fuzzy models. As a
result, it is more convenient to implement the controller in practice.

The rest of the paper is organized as follows. Section 2 provides preliminaries and the
formulation of the problem. Section3 develops a procedure of the controller design. Section 4
presents a simulation example of 2-link manipulator to illustrate the effectiveness of the proposed
method. These are followed by conclusions in Section 5.

2 Problem formulation

Consider the following nonlinear systems with uncertainties

ẋ1 = x2,

...

ẋ(β1−1) = xβ1,

ẋβ1 = f1(x,u) + f̃1(x,u) + d1,

ẋ(β1+1) = x(β1+2),

...

ẋn = fm(x,u) + f̃m(x,u) + dm,

(1)

where x,u are the system state vector, control input vector, respectively;
x = [x1, ...,x

(β1−1)
1 , ...,x(n−βm+1), ...,x

(βm−1)
(n−βm+1)

]T ∈ Rn , β1 + β2 + ... + βm = n, u =
[u1, ...,um]

T ∈ Rm, fi(i = 1, ...,m) are known smooth nonlinear functions, f̃i(i = 1, ...,m)
are unknown uncertain nonlinearities of the system, and di(i=1,2,...,m) denote the external dis-
turbances.

Remark 1: There are many practical physical systems which can be described by the model
(1), for example, the mass-spring-damper [18], the rotated inverted pendulum [19] and the n-link
manipulator [20].

A reference model is as follows:

ẋr(t) = Arxr(t) + r(t), (2)



A New Approach to Nonlinear Tracking Control Based on Fuzzy Approximation 63

where xr(t) is a reference state, r(t) is a bounded reference input, and Ar is an asymptotically
stable matrix.

Control objective: Design a controller to guarantee that the nonlinear system (1) is stable
and the state can track the reference state xr(t).

The known part of the system (1) can be approximated by a fuzzy T-S model composed of
L rules. For convenience of research, fuzzy T-S model includes the external disturbance d. The
ith rule of the fuzzy model is as follows:

IF z1(t) is F i1 and,...,and zs(t) is F
i
s, THEN

ẋ(t) = Aix(t) + Biu(t) + d, i = 1,2, ...,L, (3)

where z1(t), ...,zs(t) are the premise variables, F ij (j=1,2,...,s) are the fuzzy sets, L is the num-
ber of IF-THEN rules, Ai and Bi are some constant matrices with compatible dimensions, Bi =
[0, ...,bTi1, ...,0, ...,b

T
im]

T ∈ Rn×m with bi1 ∈ Rm, ...,bim ∈ Rm, and d = [0, ...,d1, ...,0, ...,dm]T .
The final output of the fuzzy system is inferred as follows:

ẋ(t) =
L∑
i=1

µiAix(t) +
L∑
i=1

µiBiu(t) + d, (4)

where

µi = νi(z(t))

/
L∑
i=1

νi(z(t)), νi(z(t)) =

S∏
j=1

F ij (zj(t)), (5)

and F ij (zj(t)) is the grade of membership of zj(t) in F
i
j . Therefore, the approximating error

for the nonlinear system (1) and the uncertainties of the nonlinear system (1) can be expressed
as B∆(x), where B = diag[B1, ...,Bm],Bi = [0, ...,0,1]T ∈ Rβi and ∆(x) = [∆1, ...,∆m]T .

Therefore, the nonlinear system (1) could be rearranged as

ẋ(t) =

L∑
i=1

µiAix(t) +

L∑
i=1

µiBiu(t) + B∆(x) + d. (6)

3 Design of controller and stability analysis

3.1 Design of controller

The controller is chosen as

u(t) = ul(t) − uf(t), (7)

where ul(t) denotes the state-feedback controller based on fuzzy T-S model, uf(t) is the
adaptive compensator based on fuzzy logic systems.

The state-feedback controller ul(t) based on fuzzy T-S model is designed as

ul(t) =

L∑
i=1

µiKi(x(t) − xr(t)), (8)

where ul(t) is used to stabilize the linear part of the system (1), and Ki(i=1,2,...,L) are
matrices with proper dimensions and satisfy

A
T
ij + PAij +

1

ρ2
PP + Q < 0, i,j = 1,2, ...,L, (9)



64 Z. Du, T.-C. Lin, V.E. Balas

where Aij =

[
Ai + BiKj −BiKj

0 Ar

]
, Q = diag{2Q,2Q}, P and Q are some symmetric and

positive definite matrices, and ρ is a positive constant.
The adaptive compensator based on fuzzy logic systems is given by

uf(t) =

{
E−1û(x|Θ), if E is nonsingular
ET (I + EET )−1û(x|Θ) if E is singular,

(10)

which is used to compensate the approximating error and the uncertainties. In(10),

Ei = [b
T
i1, ...,b

T
im]

T ∈ Rm×m,E =
L∑
i=1

µiEi,

and û(x|Θ) is constructed by fuzzy logic systems. The updating law of Θ is as follows:

Θ̇ = η1Ψ
T (x)B

T
Px, (11)

where η1 is a positive constant, Ψ(x) is a fuzzy basis-function matrix, and the definition of
Ψ(x) is given in (16).

3.2 Stability analysis

Note that

L∑
i=1

µiBiuf(t) − B∆(x,x(t − τ)) = B(Euf(t) − ∆(x)) (12)

=

{
B(û(x|Θ) − ∆(x))
B(û(x|Θ) − (I + EET )−1û(x|Θ) − ∆(x))

∆B(û(x|Θ) − ∆(x)). (13)

Substituting (7) into (6) yields

ẋ(t) =

L∑
i=1

µiAix(t) +

L∑
i=1

L∑
j=1

µiµjBiKj(x(t) − xr(t)) − B(û(x|Θ) − ∆(x)) + d. (14)

Denote x̃(t) = [xT (t),xTr (t)]
T , and B = [BT 0]T . From (2) and (14), a new extended

closed-loop system is as follows:

˙̃x(t) =
L∑
i=1

L∑
j=1

µiµjAijx̃(t) + B(−(û(x|Θ) − ∆(x))) + d′, (15)

where d′ = [dT ,rT (t)]T . When fuzzy logic systems û(x|Θ) eliminate ∆(x), then the closed-
loop system (15) is stable. Thus, fuzzy logic systems are constructed to approximate the vector
function ∆(x) as follows:

∆̂(x|Θ) = Ψ(x)Θ, (16)

where



A New Approach to Nonlinear Tracking Control Based on Fuzzy Approximation 65

Ψ(x) = diag[ξT1 (x), ...,ξ
T
m(x)],Θ = [θ

T
1 ,θ

T
2 , ...,θ

T
m]

T

in which θi(i=1,2,...,m) are the column vectors, and the weight Θ is an adaptive parameter.
Define the optimal parameter estimation Θ∗ as follows:

Θ∗∆ arg min
Θ∈Ω

[sup
x∈U

||∆̂(x|Θ) − ∆(x)||], (17)

where U = {x ∈ Rn},Ω = {Θ ∈ Rpm×1}. U,Ω denote the sets of suitable bounds on x, Θ,
respectively. Then the estimation error for the vector function ∆(x) can be expressed as

∆̂(x|Θ) − ∆(x) = Ψ(x)Θ̃ + w, (18)

where w = [w1, ...,wm]T is a residual term, Θ̃ = Θ − Θ∗ = [(θ1 − θ∗1)
T , ...,(θm − θ∗m)T ]T .

Denote w′ = [wT ,rT (t)]t, and w = [0, ...,d1 − w1, ...,0, ...,dm − wm]T . Substituting (18) into
(15), (15) is rearranged as

˙̃x(t) =

L∑
i=1

L∑
j=1

µiµjAijx̃(t) + B(−Ψ(x)Θ̃) + w′. (19)

Theorem 1. For the nonlinear system (1), if the controller is chosen as (7) composed of the
fuzzy state-feedback controller (8) and the adaptive compensator(10), and the updating law for
the weight is chosen as (11), then the closed-loop system (15) is uniformly ultimately bounded
(UUB) and the following tracking performance is achieved as

∫ T
0

(x(t) − xr(t))T Q(x(t) − xr(t))dt ≤ x̃T (0)Px̃(0) +
1

η1
Θ̃T (0)Θ̃(0) + ρ2

∫ T
0

(w′T w′)dt, (20)

where ρ > 0, P, Q are some symmetric and positive definite matrices.
Proof. Consider the following functional

V =
1

2
x̃T Px̃ +

1

2η1
Θ̃T Θ̃ (21)

whose derivative can be computed as follows:

V̇ =
1

2
˙̃xT (t)Px̃(t) +

1

2
x̃T (t)P ˙̃x(t) +

1

η1
Θ̃T

˙̃
Θ = V̇1 + V̇2, (22)

where

V̇1 = (
L∑
i=1

L∑
j=1

µiµjAijx̃(t))
T Px̃(t)+x̃T (t)P(

L∑
i=1

L∑
j=1

µiµjAijx̃(t))+
1

2
w′T Px(t)+

1

2
xT (t)Pw′ (23)

V2 = [x̃
T PB(−(Ψ(x)Θ̃) +

1

η1
Θ̃T

˙̃
Θ. (24)

V̇1 ≤
1

2

L∑
i=1

L∑
j=1

µiµjx̃
T (t)(A

T
ijP + PAij +

1

ρ2
PP)x̃(t) +

1

2
ρ2w′T w′ (25)

Substituting (9) into (25) yields



66 Z. Du, T.-C. Lin, V.E. Balas

V̇1 ≤ −
1

2
x̃T (t)Qx̃(t) +

1

2
ρ2w′T w′. (26)

From (11),(24)

V2 = [x̃
tPB(−(Ψ(x)Θ̃) +

1

η1
Θ̃T

˙̃
Θ] = 0. (27)

Thus

V̇ = V̇1 + V̇2 ≤ −
1

2
x̃T (t)Qx̃(t) +

1

2
ρ2w′T w′. (28)

When ||e|| > ρ
λmin(Q)

||w||, V̇ < 0.Thus, the closed-loop system (15) is UUB.
Note that

∫ T
0

(x(t) − xr(t))T Q(x(t) − xr(t))dt =
∫ T
0

[xT (t) xTr (t)]

[
Q −Q
−Q Q

]
[xT (t) xTr (t)]

T dt

≤
∫ T
0

[xT (t) xTr (t)]diag{2Q,2Q}[x
T (t) xTr (t)]

T dt =

∫ T
0
x̃T (t)Qx̃(t)dt (29)

Integrating the above inequality (28) from t=0 to T yields (20).
By Schur complements, the inequalities (9) are transformed into the linear matrix inequalities.

Therefore, the common solution P and Kj(j=1,2,...,L) are required to be found. P is chosen
as the form P = diag{P1,P2}, where P1,P2 are some symmetric and positive definite matrices.
The inequalities (9) are equivalent to the following matrix inequalities



S11 −P1BiKj 0
−(BiKj)T P1 S22 P2
0 P2 −ρ2I


 < 0, i,j = 1,2, ...,L, (30)

where S11 = P1(Ai +BiKj)+(Ai +BiKj)T P1 + 1ρ2 P1P1 +2Q, and S22 = P2Ar +A
T
r P2 +2Q.

The matrix inequalities (30) imply S11 < 0. Denote W=P
−1
1 and Yj = KjW . S11 < 0 is

equivalent to the linear matrix inequalities

[
S W

W −(2Q)−1

]
< 0, i,j = 1,2, ...,L, (31)

where S = AiW + WATi + BiYj + (BiYj)
T + (ρ2)−1I.

P1 and Kj(j = 1,2, ...,L) are obtained by (31).And then, substituting P1 and Kj(j =
1,2, ...,L) into (30), P2 is obtained.

Remark 2: If a controlled system is fourth-order, the dimension of the matrix inequalities
(30) is 12. By use of the method in [12], the dimension of the matrix inequalities (58) is 20. By
use of the method in [9], the dimension of matrix inequalities in Theorem 1 is no less than 20.
Thus, the dimension of matrix inequalities is reduced.



A New Approach to Nonlinear Tracking Control Based on Fuzzy Approximation 67

4 Simulation example

Consider a 2-link manipulator system in [20]

M(q)q̈(t) + C(q, q̇)q̇(t) + G(q) = τ(t). (32)

Consider the existence of the uncertainties and external disturbances in system (32). Thus,
the plant is modified as follows:

q̈(t) + C(q, q̇)q̇(t) + g(q) = B(q)u(t) +

r∑
i=1

ξi(t)q(t) + d′, (33)

where C(q, q̇) = H−1(q)C′(q, q̇),g(q) = H−1(q)g′(q),B(q) = H−1(q),d′ = H−1(q)d,q =
[q1,q2]

T and u(t) = [u1,u2]T = τ(t). ξi(t)(i = 1,2, ...,r) are uncertain and bounded. d is random
noise with zero mean and variance 0.05, and d is bounded.

The reference model is as follows:

ẋr(t) = Arxr(t) + r(t),

where

Ar = diag{Ar1,Ar2},Ar1 = Ar2 =

[
0 1

−6 −5

]
,r(t) = [0,r1(t),0,r2(t)]

T .

Denote x1 = q1,x2 = q̇1,x3 = q2,x4 = q̇2. Fuzzy T-S model is used to approximate the
nonlinear system at x1 = −π2 ,0,

π
2

and x3 = −π2 ,0,
π
2
. The membership functions are adopted

as triangle type. Fuzzy T-S model with Nine rules in the form (3) is given, where

A1 =




0 1 0 0

5.927 −0.001 −0.315 −0.0000084
0 0 0 1

−6.859 0.002 3.155 0.0000062


 ,A2 =




0 1 0 0

3.0428 −0.0011 −0.1791 −0.0002
0 0 0 1

−3.5436 0.0313 2.5611 0.0000114


 ,

A3 =




0 1 0 0

6.2728 0.003 0.4339 −0.0001
0 0 0 1

−9.1041 0.0158 −1.0574 −0.000032


 ,A4 =




0 1 0 0

6.5434 0.0017 1.2427 −0.0002
0 0 0 1

−3.1873 0.0306 −5.1911 −0.000018


 ,

A5 =




0 1 0 0

11.1336 0 −1.8145 0
0 0 0 1

−9.0918 0 9.1638 0


 ,A6 =




0 1 0 0

6.1702 −0.001 1.687 −0.0002
0 0 0 1

−2.3559 0.0314 4.5298 −0.000011


 ,

A7 =




0 1 0 0

6.1206 0.0041 0.6205 0.0001

0 0 0 1

8.8794 0.0193 −1.0119 0.000044


 ,A8 =




0 1 0 0

3.6421 −0.0018 0.0721 0.0002
0 0 0 1

2.429 −0.0305 2.9832 −0.000019


 ,



68 Z. Du, T.-C. Lin, V.E. Balas

A9 =




0 1 0 0

6.2933 −0.0009 0.2188 −0.000012
0 0 0 1

−7.4649 0.0024 3.2693 −0.0000092


 ,B1 =

[
0 1 0 −1
0 −1 0 2

]T
,B2 =

[
0 0.5 0 0

0 0 0 1

]T

B3 =

[
0 1 0 1

0 1 0 2

]T
,B4 =

[
0 0.5 0 0

0 0 0 1

]T
B5 =

[
0 1 0 −1
0 −1 0 2

]T
,B6 =

[
0 0.5 0 0

0 0 0 1

]T

B7 =

[
0 1 0 1

0 1 0 2

]T
,B8 =

[
0 0.5 0 0

0 0 0 1

]T
B9 =

[
0 1 0 −1
0 −1 0 2

]T
.

Using the LMI box in Matlab, Kj(j = 1,2, ...,L) are obtained

K1 =

[
−75.5707 −41.2895 −19.7728 −8.9886
4.5163 −0.5944 −49.7511 −24.5727

]
,K2 =

[
−76.4364 −41.3132 −13.7595 −5.9976
7.1834 1.0290 −49.0587 −24.2578

]
,

K3 =

[
−75.4503 −41.2461 −19.8614 −9.0371
4.3907 −0.6474 −49.6965 −24.54837

]
,K4 =

[
−76.4364 −41.3132 −13.7595 −5.9976
7.1834 1.0290 −49.0587 −24.2578

]
,

K5 =

[
−76.4364 −41.3132 −13.7595 −5.9976
7.1834 1.0290 −49.0587 −24.2578

]
,K6 =

[
−76.4364 −41.3132 −13.7595 −5.9976
7.1834 1.0290 −49.0587 −24.2578

]
,

K7 =

[
−76.4364 −41.9511 −6.2666 −2.2808
11.1565 3.4005 −48.7116 −24.1209

]
,K8 =

[
−76.4364 −41.3132 −13.7595 −5.9976
7.1834 1.0290 −49.0587 −24.2578

]
,

K9 =

[
lcl − 76.7763 −42.0400 −6.0715 −2.1821
−11.3875 3.4986 −48.8204 −24.1694

]
,

Then, the controller is given by

u(t) = ul(t) − uf(t),

where

ul(t) =
9∑

i=1

µiKi(x(t) − xr(t)) and uf = (
9∑

i=1

µiEi)
−1û(x|Θ)

with the updating law (11). In (11), the symmetric and positive definite matrix

P =




0.0070 0.0035 −0.0004 −0.0002
0.0035 0.0021 0.0004 0.0001

−0.0004 0.0004 0.0073 0.0032
−0.0002 0.0001 0.0032 0.0017


 .



A New Approach to Nonlinear Tracking Control Based on Fuzzy Approximation 69

Seven fuzzy rules are defined in adaptive fuzzy logic systems.
R(j): if x1 is F

j
1 ,..., x4 is F

j
4 , then y is G

j (j=1,2,...,7),

µF 1i
(xi) =

1

1 + exp[5(xi + 0.8)]
(i = 1,2, ...,4), µF 1i

(xi) = exp[−(xi + 0.6)2](i = 1,2, ...,4),

µF 3i
(xi) = exp[−(xi + 0.4)2](i = 1,2, ...,4), µF 4i (xi) = exp[−(xi)

2](i = 1,2, ...,4),

µF 5i
(xi) = exp[−(xi − 0.4)2](i = 1,2, ...,4), µF 6i (xi) = exp[−(xi − 0.6)

2](i = 1,2, ...,4),

µF 7i
(xi) =

1

1 + exp[5(xi − 0.8)]
(i = 1,2, ...,4).

Denote S1 =
7∑

j=1

4∑
i=1

µ
F

j
i
(xi), then

ξ(x) =

[
4∏

i=1

µF 1i
(xi)/S1, ...,

4∏
i=1

µF 7i
(xi)/S1

]
= [ξ1, ...,ξ7],Ψ(x) = diag[ξ

T (x),ξT (x)].

The initial condition is set to be

(x1(0),x2(0),x3(0),x4(0),xr1(0),xr2(0),xr3(0),xr4(0)) = (0.4,0,−0.4,0,0,0,0,0).

Choose r1(t) = r2(t) = 4sin(t),r = 2,ξ1(t) = 1 + 20 sin(t),ξ2(t)=2(1-exp(-t))/(1+exp(-t))
and the parameter η1 = 20. Simulation results are shown in Fig.1- Fig.3.

Figure 1: State responses of x1,x2,x3 and x4 (dotted line), xr1,xr2,xr3 and xr4 (solid line)

By only using the fuzzy controller based on T-S model [20], simulation result is shown in
Fig.4 and the tracking performance comparison between proposed method and approach in [20]
is given in table 1.



70 Z. Du, T.-C. Lin, V.E. Balas

Figure 2: The control input u1

Figure 3: The control input u2

Figure 4: State responses of x1,x2,x3 and x4 (dotted line), xr1,xr2,xr3 and xr4 (solid line)



A New Approach to Nonlinear Tracking Control Based on Fuzzy Approximation 71

Table 1: The tracking performance comparison between proposed method and
approach in [20].

Ei =
∫ t
0
(xi − xri)2dt E1 E2 E3 E4

The proposed method 0.0677 0.1338 0.0587 0.2177
The method in [20] 0.1380 0.1760 0.1244 0.3343

5 Conclusion

The developed controller makes full use of the advantages of two fuzzy models. Theory anal-
ysis verifies the feasibility of the proposed control scheme and simulation results demonstrate the
effectiveness of the proposed control scheme.

Acknowledgements
This work is supported by the National Natural Science Foundation of China (60974028).

Bibliography

[1] K.Tanaka, M.Sugeno. Stability analysis and design of fuzzy control systems. Fuzzy Sets and
Systems,1992,45(2):135-156.

[2] Bong-Jae Rhee, Sangchul Won.A new fuzzy Lyapunov function approach for a Takagi -Sugeno
fuzzy control system design. Fuzzy Sets and Systems, 2006, 157(9): 1211 - 1228.

[3] H. K. Lam, Lakmal D. Seneviratne. Stability Analysis of Interval Type-2 Fuzzy-Model- Based
Control Systems. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernet-
ics,2008,38(3): 617-628.

[4] Sung Hyun Kim ,Poo Gyeon Park.Observer-Based Relaxed H∞ Control for Fuzzy Systems
Using a Multiple Lyapunov Function. IEEE Transactions on Fuzzy Systems, 2009,17(2):476-
484.

[5] Antonio Sala, Carlos Ariño.Relaxed Stability and Performance LMI Conditions for Takagi-
Sugeno Fuzzy Systems With Polynomial Constraints on Membership Function Shapes. IEEE
Transactions on Fuzzy Systems,2008,16(5):1328-1336.

[6] Miguel Bernal, Thierry Marie Guerra,Alexandre Kruszewski.A membership-function- depen-
dent approach for stability analysis and controller synthesis of Takagi-Sugeno models. Fuzzy
Sets and Systems,2009,160(19):2776-2795.

[7] Yan-Wu Wang,Zhi-Hong Guan,Hua O.Wang.Impulsive synchronization for Takagi-Sugeno
fuzzy model and its application to continuous chaotic system.Physics Letters A, 2005,339(3-
5):325-332.

[8] Shinn-Horng Chen, Wen-Hsien Ho, Jyh-Horng Chou.Robust Controllability of T-S Fuzzy-
Model-Based Control Systems With Parametric Uncertainties. IEEE Transactions on Fuzzy
Systems,2009,17(6):1324-1335.

[9] J.C.Lo, M.L.Lin. Robust nonlinear modeling and control via uncertain fuzzy systems. Fuzzy
Sets and Systems,2004,143(2):189-209.



72 Z. Du, T.-C. Lin, V.E. Balas

[10] JunYoneyama.Robust H∞ control analysis and synthesis for Takagi-Sugeno general uncer-
tain fuzzy systems. Fuzzy Sets and Systems 2006,157(16):2205- 2223.

[11] Fuwen Yang,Yongmin Li,Set-Membership Fuzzy Filtering for Nonlinear Discrete- Time Sys-
tems. IEEE Transactions on Fuzzy Systems,2010,40(1):116-123.

[12] B.S.Chen, C.S.Tseng, H.J.Uang. Mixed fuzzy output feedback control design for nonlinear
dynamic systems: an LMI approach. IEEE Transactions on Fuzzy Systems,2000,8(3): 249-
265.

[13] C.M.Park.LMI-based robust stability analysis for fuzzy feedback linearization regulators with
its application.Information Sciences,2003,152:287-301.

[14] Changchun Hua, Qing-Guo Wang , Xinping Guan.Robust Adaptive Controller Design
for Nonlinear Time-Delay Systems via T-S Fuzzy Approach. IEEE Transactions on Fuzzy
Systems,2009,17(4):901-910.

[15] Guoliang Wei, Gang Feng, Zidong Wang . Robust H∞ Control for Discrete-Time Fuzzy
Systems With Infinite-Distributed Delays.IEEE Transactions on Fuzzy Systems,2009,17(1)
224-232.

[16] Li-Xin Wang. Stable adaptive fuzzy control of nonlinear systems. IEEE Transactions on
Fuzzy Systems, 1993,1(3): 146-155.

[17] Weisheng Chen, Zhengqiang Zhang. Globally stable adaptive backstepping fuzzy control for
output-feedback systems with unknown high-frequency gain sign. Fuzzy Sets and Systems,
2010, 161(6): 821-836.

[18] Huai-Ning Wu, Kai-Yuan Cai.H2 guaranteed cost fuzzy control for uncertain nonlinear sys-
tems via linear matrix inequalities. Fuzzy Sets and Systems, 2004,148(3):411-429.

[19] Y. Liu, S.S. Hu. Fuzzy Robust Tracking Control for Uncertain Nonlinear Systems. Acta
Automatica Sinica. 2004,30(6):949-953.

[20] C. S. Tseng, B. S. Chen, H.J. Uang. Fuzzy tracking control design for nonlinear dynamic
systems via T-S fuzzy model. IEEE Transactions on Fuzzy Systems,2001,9(3): 381-392.