1720 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014

On Switched Regulator Design of Uncertain Nonlinear
Systems Using Takagi–Sugeno Fuzzy Models

Wallysonn A. de Souza, Member, IEEE,
Marcelo C. M. Teixeira, Member, IEEE, Rodrigo Cardim,

and Edvaldo Assunção

Abstract—This paper is concerned with the design of state-feedback
switched controllers for a class of uncertain nonlinear plants described by
Takagi–Sugeno (T–S) fuzzy models. The proposed methodology eliminates
the need to find the membership function expressions to implement the
control law, which is relevant in cases where the membership function
depends on uncertain parameters. The design of the switched controllers
is based on a minimum-type Lyapunov function and the minimization of
the time derivative of this Lyapunov function. The conditions of the new
stability criterion are represented by a kind of bilinear matrix inequalities
(BMIs) that has been solved by the path-following method. A numerical
example and the nonlinear control design of a magnetic levitator with
uncertainties illustrate the procedure.

Index Terms—Bilinear matrix inequalities (BMIs), control of uncer-
tain nonlinear systems, linear matrix inequalities (LMIs), switched control,
Takagi–Sugeno (T–S) fuzzy model.

I. INTRODUCTION

Currently, there are efficient control system design techniques for
nonlinear systems described by Takagi–Sugeno (T–S) fuzzy models,
based on parallel distributed compensation, as can be seen in the con-
siderable number of published papers in this well-established research
field (see, for instance, [1]–[15]). In the last decade, many researchers
have studied new relaxation methods for stability analysis and con-
troller synthesis, based on linear matrix inequalities (LMIs) or bilinear
matrix inequalities (BMIs), of T–S fuzzy models [3], [6]–[8], [10], [12],
[16], [17].

Nowadays, according to the authors’ knowledge, the conditions
stated in [8], [12], and [17] offer three of the most important con-
tributions of relaxation stability conditions for continuous T–S fuzzy
models. In [17], a switching fuzzy controller, based on a minimum-
type piecewise Lyapunov function, is proposed, and conditions more
relaxed than the known conditions are presented. Some of these con-
ditions are represented by a kind of BMIs, and the authors suggest the
path-following method as an adequate solution for this problem [18].

The study of switched systems grown a lot in the past decade,
initiating mainly with the control of linear systems [19]–[30], and
then, it has also been broadly used in the control of nonlinear systems
described by T–S fuzzy models, as can be seen in [4], [5], [13], [17],
and [31]–[38].

This paper proposes a new method to design switched control for
a class of uncertain nonlinear systems described by T–S fuzzy mod-
els. The proposed controller is defined by a switching law whose de-

Manuscript received July 11, 2013; revised October 23, 2013; accepted Jan-
uary 2, 2014. Date of publication January 24, 2014; date of current version
November 25, 2014. This work was supported in part by FAPESP - Fundação
de Amparo à Pesquisa do Estado de São Paulo (grant 2011/17610-0) and in part
by CNPq - Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico
(research fellowships) from Brazil.

W. A. de Souza is with the Department of Academic Areas of Jataı́, IFG -
Federal Institute of Education, Science, and Technology of Goiás, Jataı́, Goiás
75804-020, Brazil (e-mail: wallysonn@yahoo.com.br).

M. C. M. Teixeira, R. Cardim, and E. Assunção are with the De-
partment of Electrical Engineering, UNESP - Univ Estadual Paulista, Ilha
Solteira, São Paulo 15385-000, Brazil (e-mail: marcelo@dee.feis.unesp.br;
rcardim@dee.feis.unesp.br; edvaldo@dee.feis.unesp.br).

Digital Object Identifier 10.1109/TFUZZ.2014.2302494

sign consists of two stages. The first stage is based on [17] and [26]
and selects a positive definite matrix using a minimum-type piecewise
Lyapunov function, and the second stage chooses the controller gains
that minimize the time derivative of the Lyapunov function. For the
development of these control design methods, new stability conditions
were established, and some of these conditions are a kind of BMI. These
BMIs, which contain some bilinear terms as the product of a full ma-
trix and a scalar, have been solved by the path-following method [18].
Furthermore, the proposed switched controller can also operate even
with an uncertain reference control signal.

Due to the fact that the control law is switched, the main advantage
of this new procedure is its practical application because it eliminates
the need to find the explicit expressions of the membership functions,
which can often have long and/or complex expressions, or may not be
known, for instance, due to the plant uncertainties. Additionally, with
the proposed methodology, the closed-loop systems usually present a
settling time smaller than those obtained with fuzzy controllers without
using switching, which are widely studied in the literature. Moreover,
other constraints, for instance on the plant’s input and output, can also
be added to the control design specifications.

The proposed method was applied in the control design of a bench-
mark nonlinear T–S fuzzy system [17] that is used to compare re-
laxation stabilization criterions. Furthermore, simulation results of the
application of the procedure in the control of a magnetic levitator with
uncertainties is presented. The computational implementations were
carried out using the modeling language YALMIP [39] with the solver
SeDuMi [40].

The paper is organized as follows. Section II presents the preliminary
results on T–S fuzzy model and switching fuzzy controller. Section III
offers a new switching control method and new stability conditions for
a class of uncertain nonlinear systems described by T–S fuzzy models.
Examples illustrate the performance of the new proposed methods in
Section IV. Finally, Section V draws our conclusions.

For convenience, in some places, the following notation is used:

Kr = {1, 2, . . . , r}, r ∈ N, x(t) = x

αi (x(t)) = αi , V (x(t)) = V, ‖x‖2 =
√

xT x

(A, B, C, K)(α) =
r∑

i=1

αi (Ai , Bi , Ci , Ki )

with αi ≥ 0,
r∑

i=1

αi = 1 and αT = [α1 , α2 , . . . , αr ]. (1)

II. TAKAGI–SUGENO FUZZY SYSTEMS AND SWITCHING

FUZZY CONTROLLER

Consider the T–S fuzzy model as described in [41]:

Rule i : IF z1 (t) is Mi
1 and . . . and zp (t) is Mi

p

THEN

{
ẋ(t) = Aix(t) + Biu(t)

y(t) = Cix(t)
(2)

where i ∈ Kr , M i
j , j ∈ Kp is the fuzzy set j of rule i, x(t) ∈ Rn is

the state vector, u(t) ∈ Rm is the input vector, y(t) ∈ Rq is the output
vector, Ai ∈ Rn×n , Bi ∈ Rn×m , Ci ∈ Rq×n , and z1 (t), . . . , zp (t)
are premise variables, which, in this paper, are the state variables.

From [1], ẋ(t) given in (2) can be written as follows:

ẋ(t) =
r∑

i=1

αi (x(t))(Aix(t) + Biu(t)) = A(α)x(t) + B(α)u(t)

(3)

1063-6706 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications standards/publications/rights/index.html for more information.



IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014 1721

where αi (x(t)) is the normalized weight of each local model system
Aix(t) + Biu(t), i ∈ Kr that satisfies (1).

Assuming that the state vector x is available, from the T–S fuzzy
model (2), the control input of fuzzy regulators via parallel distributed
compensation has the following structure [1]:

Rule j : IF z1 (t) is Mj
1 and . . . and zp (t) is Mj

p

THEN u(t) = −Kj x(t). (4)

Similar to (3), from (4), one can consider the control law [1]

u(t) = uα = −
r∑

j=1

αj (x(t))Kj x(t) = −K(α)x(t). (5)

From (1), (3), and (5), one obtains

ẋ(t) = (A(α) − B(α)K(α))x(t)

=
r∑

i=1

r∑

j=1

αi (x(t))αj (x(t))
[
Ai − BiKj

]
x(t). (6)

Now, a switching fuzzy controller based on a minimum-type piece-
wise Lyapunov function proposed in [17] is presented. Let the piecewise
Lyapunov candidate function be as follows:

V (x) = min
k∈KN

(xT (t)Pk x(t)) (7)

where Pk , k ∈ KN are symmetric positive definite matrices.
Definition 1: Consider the index set ΩH (t) defined in the following:

ΩH (t) = arg min
i∈KN

xT (t)Hix(t)

=
{

j ∈ KN : xT (t)Hj x(t) = min
i∈KN

xT (t)Hix(t)
}

where Hi ∈ Rn×n , i ∈ KN , and x(t) ∈ Rn . The smallest index j ∈
ΩH (t) will be denoted by

arg min
i∈KN

∗{x(t)T Hix(t)} = min
j∈ΩH (t)

j.

In [17], the switching index minj∈ΩP ( t ) j was used to select, at
each instant of time, a positive definite matrix Pj , j ∈ KN , such that
the piecewise Lyapunov function (7) is equal to x(t)T Pj x(t). Thus,
considering (7) and from Definition 1, the switching fuzzy controller
proposed in [17] can be written as follows:

u(t) = uσ (t) = −
r∑

i=1

αi (x(t))Kiσ x(t)

where σ = arg min
k∈KN

∗{x(t)T Pk x(t)}. (8)

Therefore, from (1), the controlled system (3) and (8) is given by

ẋ(t) = A(α)x(t) + B(α)uσ (t)

=
r∑

i=1

r∑

j=1

αiαj

(
Ai − BiKjσ

)
x(t). (9)

In this context, considering the piecewise Lyapunov function (7)
and the switching fuzzy controller (8), in [17], a theorem was proposed
which established a relaxed stabilization criterion. However, some of
the proposed conditions are BMIs which contain some bilinear terms
as the product of a full matrix and a scalar. Fortunately, this kind of
problem has been solved by the path-following method [18].

III. MAIN RESULT

A. Switched Regulator Design of Uncertain Nonlinear Systems Using
Takagi–Sugeno Fuzzy Models

In this section, a new design method of switched controllers for the
T–S fuzzy systems described in (3) is presented. To find the feedback
gains the proposed controller uses two stages. The first stage is based
on the procedures presented in [17] and [26], and chooses an index
σ = arg mink∈KN

∗{xT Pk x}, where Pk , k ∈ KN are symmetric pos-
itive definite matrix. Note that the Lyapunov function given in (7) is
equal to V (x) = xT Pσ x. The basic idea of the second stage is the min-
imization of the time derivative of the Lyapunov function (7), through
the selection of the controller gain, which belongs to the set of gains
{Kjσ , j ∈ Kr }, where σ was obtained in the first stage. This stage
uses auxiliary symmetric matrices Qjk , j ∈ Kr , k ∈ KN , the index σ,
and selects an index ν = arg minj∈Kr

∗{xT Qjσ x}. Therefore, consid-
ering the indexes σ and ν cited above and Definition 1, the switched
controller is defined as follows:

u(t) = uν σ (t) = −Kν σ x(t)

σ = arg min
k∈KN

∗{xT Pk x}, ν = arg min
j∈Kr

∗{xT Qjσ x}. (10)

Therefore, from (1), the controlled systems (3) and (10) are given
by

ẋ(t) = A(α)x(t) + B(α)uν σ (t)

=
r∑

i=1

αi

(
Ai − BiKν σ

)
x(t). (11)

In this context, considering the piecewise Lyapunov function (7) and
the switching controller (10), the following theorem is proposed.

Theorem 1: Assume that there exist symmetric positive definite ma-
trices Xk ∈ Rn×n , symmetric matrices Zik , Rik , Yik ∈ Rn×n , ma-
trices Mik ∈ Rm ×n , and scalars λisk > 0, β < 0 such that, for all
i, j ∈ Kr and k, s ∈ KN :

−BiMjk − MT
jk BT

i − Zik − Rjk ≺ 0 (12)

Yik ≺ 0 (13)
⎡

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

Oik ∗ ∗ · · · ∗
λi1k Xk −λi1k X1 0 . . . 0

λi2k Xk 0 −λi2k X2 . . .
...

...
...

...
. . . 0

λiN k Xk 0 . . . 0 −λiN k XN

⎤

⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

≺ 0 (14)

where Oik =Xk AT
i +AiXk +Zik +Rik − βXk −

∑N
s=1 λisk Xk −

Yik .
Then, the switched control law (10), where Qjk =

X−1
k Rjk X−1

k , Pk = X−1
k and controller gains are given by

Kjk = Mjk X−1
k , j ∈ Kr , k ∈ KN , makes the equilibrium point

x = 0 of the system (3) asymptotically stable in the large.
Proof: Consider a piecewise Lyapunov candidate function

given in (7). Suppose that V (x(t)) = mini∈KN
{x(t)T Pix(t)} =

x(t)T Pσ x(t), where σ is selected as described in (10) and Definition 1.
From the analysis presented in [17], considering that V (x(t+ )) ≤
Vσ (x(t+ )), then V̇ (x(t)) ≤ V̇σ (x(t)). This fact follows from

V̇ (x(t)) = lim
t+ →t

V (x(t+ )) − V (x(t))
t+ − t

= lim
t+ →t

V (x(t+ )) − Vσ (x(t))
t+ − t



1722 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014

and on the other hand

V̇σ (x(t)) = lim
t+ →t

Vσ (x(t+ )) − Vσ (x(t))
t+ − t

.

Consequently, V (x(t+ )) ≤ Vσ (x(t+ )) implies that

V̇ (x(t)) = lim
t+ →t

V (x(t+ )) − Vσ (x(t))
t+ − t

≤ lim
t+ →t

Vσ (x(t+ )) − Vσ (x(t))
t+ − t

= V̇σ (x(t)).

Thus, recalling that β < 0, then from the analysis above and from
(11), observe that

V̇ (x) − βV (x)

≤ V̇σ (x) − βVσ (x) = ẋT Pσ x + xT Pσ ẋ − βxT Pσ x

≤
r∑

i=1

αix
T (Pσ Ai + AT

i Pσ − Pσ BiKν σ − KT
ν σ BT

i Pσ − βPσ )x.

(15)

Considering the relaxing parameters λisσ > 0, i ∈ Kr , s, σ ∈ KN ,
note that from (8)

N∑

s=1

λisσ xT (Ps − Pσ )x ≥ 0.

Now, suppose that there exist symmetric matrices Z̄ik , Qjk ∈ Rn×n

such that

−
(
Pk BiKjk + KT

jk BT
i Pk

)
≺ Z̄ik + Qjk , ∀ i, j ∈ Kr , k ∈ KN .

(16)
Therefore, from (15) and (16) it follows that for x 
= 0

V̇ (x) − βV (x)

≤
r∑

i=1

αix
T

[

Pσ Ai + AT
i Pσ − Pσ BiKν σ − KT

ν σ BT
i Pσ

− βPσ +
N∑

s=1

λisσ (Ps − Pσ )
]

x

<

r∑

i=1

αix
T

[

Pσ Ai + AT
i Pσ + Z̄iσ + Qν σ

− βPσ +
N∑

s=1

λisσ (Ps − Pσ )
]

x

=
r∑

i=1

αix
T

[

Pσ Ai + AT
i Pσ + Z̄iσ

− βPσ +
N∑

s=1

λisσ (Ps − Pσ )
]

x + xT Qν σ x. (17)

From (1) and (10), note that Qν σ = mini∈Kr (xT Qiσ x) ≤∑r
i=1 αix

T Qiσ x. Then, from (17)

V̇ (x) − βV (x) ≤
r∑

i=1

αix
T

[

Pσ Ai + AT
i Pσ

+ Z̄iσ + Qiσ − βPσ +
N∑

s=1

λisσ (Ps − Pσ )
]

x. (18)

Now, suppose that there exist symmetric matrices Wik , i ∈ Kr , k ∈
KN , such that

Pk Ai +AT
i Pk +Z̄ik + Qik − βPk +

N∑

s=1

λisk (Ps − Pk ) − Wik � 0.

(19)
Therefore, from (18) and (19), assume that

V̇ (x) − βV (x) <
r∑

i=1

αix
T (Wiσ )x < 0, x 
= 0. (20)

Remembering that αi ≥ 0, i ∈ Kr and
∑r

i=1 αi = 1, from (20),
V̇ (x) − βV (x) < 0 (for x 
= 0) if for all i ∈ Kr and k ∈ Kr , the
following condition holds:

Wik ≺ 0. (21)

Now, define Xk = P −1
k , Zik = Xk Z̄ik Xk , Rik = Xk Qik Xk ,

Mjk = Kjk Xk , and Yik = Xk Wik Xk . Pre- and postmultiplying (16),
(21), and (19) by Xk , it follows (12), (13), and

AiXk + Xk AT
i + Zik + Rik − βXk

+
N∑

s=1

λisk (Xk PsXk − Xk ) − Yik � 0. (22)

Applying the Schur complement in (22), this condition is equivalent to
(14), and the proof is concluded. �

Remark 1: Nowadays, to the best knowledge of the authors, there
are no solvers that can find solutions for all kinds of BMIs. Note that
the conditions of Theorem 1 are given by two LMIs [see (12) and (13)]
and one BMI (14). However, this kind of problem has been solved by
the path-following method [18], whose steps are presented with details
in [17, App.]. In the examples of this paper, the path-following method
was implemented using the modeling language YALMIP [18] with the
solver SeDuMi [17].

Remark 2: The conditions of Theorem 1 can be rewritten when
N = 1, which is described as follows:

− BiMj 1 − MT
j 1B

T
i − Zi1 − Rj 1 ≺ 0, Yi1 ≺ 0

[
Oi1 ∗

λi11X1 −λi11X1

]

≺ 0, i, j ∈ Kr (23)

where Oi1 = X1A
T
i + AiX1 + Zi1 + Ri1 − βX1 − λi11X1 − Yi1 .

Now, applying the Schur complement in the third inequality of (23),
one obtains the simplified conditions given by

− BiMj 1 − MT
j 1B

T
i − Zi1 − Rj 1 ≺ 0, Yi1 ≺ 0

X1A
T
i + AiX1 + Zi1 + Ri1 − βX1 − Yi1 ≺ 0, i, j ∈ Kr . (24)

B. Switched Controller With Uncertainty in the Reference
Control Signal

In this case, it is assumed that the plant given by ˙̄x = f (x̄, ū) has
an equilibrium point x̄ = x0 and that the respective control input is
ū = u0 , such that f (x0 , u0 ) = 0. Suppose that x0 is known, u0 ∈ R is
uncertain because it depends on the plant uncertainties, but 0 < u0 ∈
[u0m in , u0m a x ], where u0m in and u0m a x are constants that are known,
and the plant can be described by the T–S fuzzy system (1)–(3)

ẋ(t) = A(α)x(t) + B(α)u (25)



IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014 1723

where
x(t) = x̄(t) − x0 , x̄(t) is the state vector of the plant;
u(t) = ū(t) − u0 , ū is the control input of the plant.
Now, consider that B(α) can be written as follows:

B(α) = Bg(x(t)) (26)

where B is a known constant matrix, and g(x(t)) > 0, for all x, is an
uncertain bounded nonlinear function. Thus, the system (25) can be
rewritten as follows:

ẋ(t) = A(α)x(t) + B(α)u = A(α)x(t) + Bg(x(t))u. (27)

Assume that the conditions (12)–(14) of Theorem 1 are feasible.
Then, one can obtain the gains Kjk = Mjk X−1

k , the matrices Pk =
X−1

k , and the matrices Qjk , j ∈ Kr , k ∈ KN . Now, given a constant
ξ > 0, define the control law

u(t) = u(ν ,σ ,ξ ) = ū(ν ,σ ,ξ ) (t) − u0

with ū(ν ,σ ,ξ ) (t) = −Kν σ x + γξ (28)

where

Kν σ ∈ {K11 , K21 , . . . , Kr 1 , . . . , KrN }

σ = arg min
k∈KN

∗{xT Pk x}

ν = arg min
j∈Kr

∗{xT Qjσ x}

γξ =
{

u0m a x , if xT Pσ B ≤ 0
u0m in , if xT Pσ B > 0.

(29)

Note that the control law (28) and (29) is only applicable to single input
systems. Based on the considerations above the following theorem is
proposed.

Theorem 2: Suppose that the conditions from Theorem 1 hold,
from the system (25) with the control law (10), and obtain Kjk =
Mjk X−1

k , Pk = X−1
k , and Qjk , j ∈ Kr , k ∈ KN . Then, the switched

control law (28) and (29) makes the equilibrium point x = 0 of the sys-
tem (25) asymptotically stable in the large.

Proof: Consider a piecewise Lyapunov candidate function V =
mini∈Kr {xT Pix} = xT Pσ x. Define V̇u ν σ and V̇u ( ν , σ , ξ ) as the time
derivatives of V for the system (25) and (26), with the control laws
(10) and (28) and (29), respectively. Then

V̇u ( ν , σ , ξ ) = 2xT Pσ ẋ = 2xT Pσ [A(α)x + B(α)u(ν ,σ ,ξ ) ]

= 2xT Pσ [A(α)x + B(α)(ū(ν ,σ ,ξ ) (t) − u0 )]

= 2xT Pσ [A(α)x + B(α)(−Kν σ x + γξ − u0 )]

= 2xT Pσ A(α)x − 2xT Pσ B(α)Kν σ x

+ 2xT Pσ B(α)(γξ − u0 )

= 2
r∑

i=1

αix
T (Pσ Ai − Pσ BiKν σ )x

+ 2g(x)xT Pσ B(γξ − u0 )

= V̇u ν σ + 2g(x)xT Pσ B(γξ − u0 ). (30)

Now, note that from (29), g(x)xT Pσ B(γξ − u0 ) ≤ 0. Thus, from
(30), V̇u ( ν , σ , ξ ) ≤ V̇u ν σ < 0 for x 
= 0, since from Theorem 1, the equi-
librium point x = 0 of the system (25) with the control law (10) is glob-
ally asymptotically stable, because V̇u ν σ < 0 for x 
= 0, and therefore,
the proof is concluded. �

Remark 3: Observe that the control input ū = −Kν σ + γξ defined
in (28) can be discontinuous due to the term γξ (29). Note also that
in some practical implementations, this phenomenon of discontinuity
may be inconvenient or cannot be allowed. Thus, as in [30], one can
modify the function γξ to make it continuous and ensure the uniform
ultimate boundedness of the system and smoothness of the control
input. Thus, for instance, consider a function γξ as follows [30]:

γξ =

⎧
⎪⎪⎨

⎪⎪⎩

u0m a x , if xT Pσ B < −ξ[
(u0m in − u0m a x )xT Pσ B

+ξ(u0m a x + u0m in )] /2ξ, if |xT Pσ B| ≤ ξ
u0m in , if xT Pσ B > ξ.

(31)

IV. NUMERICAL EXAMPLES

A. Example 1

The following example has been used in several papers and is consid-
ered as an index to compare the relaxation of the stabilization criterions
for T–S fuzzy systems [8], [17].

Let the continuous T–S fuzzy be defined by the following rules:

Rule i : IF x1 is Mi

Then ẋ(t) = Aix(t) + Biu(t)

x(t) = [x1 (t) x2 (t)]T , i ∈ K3 (32)

where [A1 |A2 |A3 |B1 |B2 |B3 ] =
[

1.59 −7.29
0.01 0

∣
∣
∣
∣
0.02 −4.64
0.35 0.21

∣
∣
∣
∣

∣
∣
∣
∣
−a −4.33
0 0.05

∣
∣
∣
∣
1
0

∣
∣
∣
∣
8
0

∣
∣
∣
∣
−b + 6
−1

]

.

Considering a = 2 and b ∈ [0, 7], the maximum value of b in
the discrete range 0 : 0.5 : 7 such that a given stabilization condi-
tion holds is calculated, in order to compare it with other stabilization
methods. From the conditions of Theorem 1, the maximum obtained
value was b = 6. Thus, by fixing a = 2 and b = 6, from the inequal-
ities (12)–(14) presented in the Theorem 1 for N = 4, by the path-
following method [18] considering in the initial step β0 = 0.4429 and
λisk (0), i ∈ Kr , s, k ∈ KN equal to random values between 0 and
1, a feasible solution was obtained for β = −0.0071. In this case, the
controller gains, the symmetric positive definite matrices of the piece-
wise Lyapunov function (7), and the symmetric matrices Qjk of this
feasible solution were the following:

K11 = [3.1964 1.5996], K21 = [3.1438 1.6354]

K31 = [3.1957 1.5990], K12 = [2.9822 1.4083]

K22 = [2.9814 1.4087], K32 = [2.9822 1.4072]

K13 = [88.7641 73.1443], K23 = [75.2306 63.7048]

K33 = [75.2309 63.6697], K14 = [1.9312 − 0.5863]

K24 = [0.8267 − 0.0527], K34 = [1.8368 − 0.5619] (33)

P1 =
[

169.1387 145.8907
145.8907 906.3213

]

P2 =
[

168.1260 141.6762
141.6762 900.0816

]

P3 =
[

171.0932 152.2472
152.2472 915.8180

]

P4 =
[

167.1593 131.3891
131.3891 892.2195

]

(34)



1724 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014

Fig. 1. State variables and control signal of the controlled system using the
switched controller (10) (solid line) and the switching fuzzy controller (8)
(dotted line), considering the initial condition [−2 1]T .

Q11 = 103
[

0.9411 3.1029
3.1029 3.0112

]

, Q21 = 103
[

2.9040 4.5544
4.5544 4.0824

]

Q31 = 103
[

0.9444 3.0926
3.0926 3.0520

]

, Q12 = 103
[

0.8452 2.8836
2.8836 2.5406

]

Q22 = 103
[

2.6570 4.1871
4.1871 3.4788

]

, Q32 = 103
[

0.8452 2.8835
2.8835 2.5453

]

Q13 = 105
[

0.4798 1.0200
1.0200 1.3921

]

, Q23 = 105
[

0.7306 1.2216
1.2216 1.5526

]

Q33 = 105
[

0.2294 0.7877
0.7877 1.1771

]

, Q14 = 103
[

0.6052 1.1802
1.1802 1.1775

]

Q24 = 103
[

2.6153 1.6752
1.6752 0.2856

]

, Q34 =
[

662.1756 967.9830
967.9830 988.6938

]

.

(35)

For a simulation, the same conditions from the numerical example
presented in [17] were considered, namely, the initial condition x(0) =
[−2 1]T and the membership functions

α1 (t) =
cos(10x1 (t)) + 1

4
, α2 (t) =

sin(10x1 (t)) + 1
4

α3 (t) =
sin(10x1 (t)) + cos(10x1 (t)) + 2

4
.

The simulation results of the controlled systems (10), (32), (33)–(35),
and (8), (32)–(34) are shown in Fig. 1.

The proposed methodology (see Theorem 1) presented a good re-
sult because it was feasible for the maximum value b = 6, which is
the same value of the relaxation methods described in [6] and [7].
Note that, according to the comparative analysis described in [17], this
value was only smaller than those presented in [8] (b = 6.5) and [17]
(b = 7.0). However, it is worth remembering that the main goal of this
paper is not to establish a new relaxed stabilization criterion but to
propose a new switched control design method for uncertain nonlinear
plants described by T–S fuzzy models that does not use the membership
functions to implement the control law. Additionally, this new control
design method can be mainly useful to control plants with uncertain pa-
rameters when the membership functions depend on these parameters
and are unknown. For instance, the methods given in [6]–[8] and [17]
cannot be directly used when the membership functions are unknown,
because for their implementations, the membership functions are
necessary.

B. Example 2

In this example, the proposed nonlinear control method is applied
in a magnetic levitator, whose mathematical model [42, p. 24] is given

by

mÿ = −kẏ + mg − λμi2

2(1 + μy)2 (36)

where
g = 9.8 m/s2 is the gravity acceleration;
λ = 0.460 H, μ = 2 m−1 and k = 0.001 Ns/m are positive

constants;
m is the mass of the ball that is an uncertain parameter;
i is the electric current, and y is the position of the ball.
Define the state variables x̄1 = y and x̄2 = ẏ. Then, (36) can be

written as follows [9]:

˙̄x1 = x̄2 , ˙̄x2 = g − k

m
x̄2 − λμi2

2m(1 + μx̄1 )2 . (37)

Consider that during the required operation, [x̄1 x̄2 ]T ∈ D1 , where

D1 = {[x̄1 x̄2 ]T ∈ R2 : 0 ≤ x̄1 ≤ 0.15}. (38)

The objective in this example is to design a controller that keeps
the ball at a desired position y = x̄1 = y0 , after a transient response.
Thus, the equilibrium point of the system (37) is x̄e = [x̄1e x̄2e ]T =
[y0 0]T .

From the second equation in (37), observe that in the equilibrium
point, ˙̄x2 = 0 and i = i0 , where

i20 =
2mg

λμ
(1 + μy0 )2 . (39)

Note that the equilibrium point is not at the origin [x̄1 x̄2 ]T =
[0 0]T . Thus, for the stability analysis, the following change of coor-
dinates is necessary:

x1 = x̄1 − y0 , x2 = x̄2 , u = i2 − i20 , i.e.,

x̄1 = x1 + y0 , x̄2 = x2 , i2 = u + i20 .

Therefore, ẋ1 = ˙̄x1 , ẋ2 = ˙̄x2 and, from (39), i2 = u + i20 = u +
2m g
λμ

(1 + μy0 )2 .
Hence, the system (37) can be rewritten as

[
ẋ1

ẋ2

]

=
[

0 1
f21 (z) f22 (z)

] [
x1

x2

]

+
[

0
g21 (z)

]

u (40)

where

f21 (z) =
gμ(μx1 + 2μy0 + 2)
(1 + μ(x1 + y0 ))2 , f22 (z) = − k

m

g21 (z) =
−λμ

2m(1 + μ(x1 + y0 ))2 (41)

where z = [x1 x2 y0 m]T ∈ R4 . Thus, to find the local models, the
maximum and minimum values of the functions f21 , f22 , and g21 must
be obtained. In this case, the methodology proposed in [9] will be used.
Then, suppose that the desired position belongs to set y0 ∈ [0.05, 0.1],
the mass m ∈ [0.06, 0.1], and the domain D1 stated in (38), and
consider y0 and m as new variables for the specification of the domain
D2 of the nonlinear functions f21 , f22 , and g21 :

D2 = {z = [x1 x2 y0 m]T ∈ R4 : −0.1 ≤ x1 ≤ 0.1

0.05 ≤ y0 ≤ 0.1, 0.06 ≤ m ≤ 0.1}. (42)

Observe that the system (40) can be rewritten as in (27), i.e.,

ẋ = A(α)x + Bg(x)u (43)



IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014 1725

where B = [0 − 1]T and g(z) = −g21 (z) = λμ
2m (1+μ (x 1 + y 0 ))2 .

Note that g(z) > 0, for all z ∈ D2 .
After the calculations, the maximum and minimum values of the

functions f21 , f22 , g21 , and u0 = i20 , in the domain D2 , one obtains

a211 = max
z∈D 2

{f21 (z)} = 48.3951

a212 = min
z∈D 2

{f21 (z)} = 26.0000

a221 = max
z∈D 2

{f22 (z)} = −0.0100

a222 = min
z∈D 2

{f22 (z)} = −0.0167

b211 = max
z∈D 2

{g21 (z)} = −2.3469

b212 = min
z∈D 2

{g21 (z)} = −9.4650

max u0 = max
z∈D 2

{i20 (z)} = 3.0678

min u0 = max
z∈D 2

{i20 (z)} = 1.5467. (44)

From (44), the local models of the plant (40) and (41) are the following:

A1 =
[

0 1
a211 a221

]

, A3 =
[

0 1
a211 a222

]

A5 =
[

0 1
a212 a221

]

, A7 =
[

0 1
a212 a222

]

B1 =
[

0
b211

]

, B2 =
[

0
b212

]

(45)

where A1 = A2 , A3 = A4 , A5 = A6 , A7 = A8 , B1 = B3 = B5 =
B7 , and B2 = B4 = B6 = B8 .

It is worth observing that this system presents no problems with
feasibility. Thus, to solve the inequalities (12)–(14) presented in the
Theorem 1 for N = 2, β0 = −2, and λisk (0), i ∈ K8 , s, k ∈ K2 ran-
domly chosen between 0 and 1, a feasible solution was obtained. In
this case, the controller gains, the positive definite matrices of the
piecewise Lyapunov function (7), and the matrices symmetric matrices
Qjk , j ∈ K8 , k ∈ K2 were the following:

K11 = [−28.2676 − 3.3652], K12 = [−27.5830 − 3.5562]

K21 = [−12.5336 − 1.2352], K22 = [−12.5957 − 1.3727]

K31 = [−28.3030 − 3.3638], K32 = [−27.5806 − 3.5367]

K41 = [−12.5093 − 1.2289], K42 = [−12.6320 − 1.3664]

K51 = [−23.5195 − 2.9365], K52 = [−22.4997 − 3.0800]

K61 = [−10.0612 − 1.2000], K62 = [−10.5085 − 1.3822]

K71 = [−23.5313 − 2.9361], K72 = [−22.5533 − 3.0798]

K81 = [−10.4454 − 1.2359], K82 = [−10.1927 − 1.3458]

(46)

P1 = 103
[

2.0288 0.2755
0.2755 0.0672

]

, P2 = 103
[

1.6511 0.2549
0.2549 0.0607

]

(47)

Q11 =
[

0.0053 −0.1359
−0.1359 0.3891

]

, Q12 =
[

0.0077 −0.1876
−0.1876 0.6888

]

Q21 =
[

0.0033 −0.0776
−0.0776 0.3173

]

, Q22 =
[

0.0045 −0.1098
−0.1098 0.5256

]

Fig. 2. Position (y(t) = x̄1 (t)), velocity (x̄2 (t)), and electric current
(i(ν ,σ ,ξ ) (t)) of the controlled system, considering y0 = 0.05 m and m =
0.06 Kg, y0 = 0.1 m, and m = 0.1 Kg, and y0 = 0.07 m and m = 0.1 Kg,
for t ∈ [0 1), t ∈ [1 2) and t ≥ 2, respectively.

Q31 =
[

0.0054 −0.1364
−0.1364 0.3923

]

, Q32 =
[

0.0080 −0.1894
−0.1894 0.7016

]

Q41 =
[

0.0033 −0.0776
−0.0776 0.3193

]

, Q42 =
[

0.0046 −0.1108
−0.1108 0.5372

]

Q51 =
[

0.0040 −0.1114
−0.1114 0.2148

]

, Q52 =
[

0.0053 −0.1467
−0.1467 0.3896

]

Q61 =
[

0.0044 −0.0586
−0.0586 0.1472

]

, Q62 =
[

0.0060 −0.0802
−0.0802 0.2481

]

Q71 =
[

0.0040 −0.1116
−0.1116 0.2160

]

, Q72 =
[

0.0054 −0.1477
−0.1477 0.3962

]

Q81 =
[

0.0043 −0.0595
−0.0595 0.1517

]

, Q82 =
[

0.0061 −0.0793
−0.0793 0.2435

]

.

(48)

Therefore, the control law (28) and (29) for the levitator is given by

u(t) = u(ν ,σ ,ξ ) (t) = i2(ν ,σ ,ξ ) (t) − i20

with i2(ν ,σ ,ξ ) (t) = −Kν σ x(t) + γξ (49)

Kν σ ∈ {K11 , K21 , . . . , K81 , . . . , K82}

σ = arg min
k∈KN

∗{xT Pk x}, ν = arg min
j∈Kr

∗{xT Qjσ x},

γξ =

{
3.0678, if xT Pσ B ≤ 0

1.5467, if xT Pσ B > 0

where Kik , i ∈ K8 , k ∈ K2 are given in (46).
For the simulation illustrated in Fig. 2, at t = 0 s, the initial condition

x̄(0) = [0.11 0]T , y0 = 0.05 m, and m = 0.06 Kg were adopted. In
t = 1 s, from Fig. 2, the system is practically at the point x̄(1) =
[0.05 0]T . After changing y0 from 0.05 to 0.1 m and m from 0.06 to
0.1 Kg at t = 2 s, one can see that the system is practically at the point
x̄(2) = [0.1 0]T , which will be the new initial condition. Finally, y0

was changed from 0.1 to 0.07 m at t = 2 s and m = 0.01 Kg for t ≥ 2 s.
Thus, observe that in Fig. 2, x(∞) = [0.07 0]T .

Note that in this case, it is not possible to directly obtain the mem-
bership functions, since the mass is uncertain, but the proposed method
overcomes this problem, because it does not depend on such functions.
Observe also that even with uncertainty in the reference control signal
(because u = i2 − i20 and i20 given in (39) is uncertain considering that



1726 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014

m is uncertain), the proposed methodology was efficient and provided
an appropriate transient response, as shown in Fig. 2.

V. CONCLUSION

This paper proposed a new switched control design method and
establishes a new criterion of stability for uncertain nonlinear plants
described by Takagi–Sugeno fuzzy models. The methodology elimi-
nates the need to obtain the explicit expressions of the membership
functions to implement the control law. This is relevant in cases where
the membership functions depend on uncertain parameters or are dif-
ficult to implement. Some conditions of the stabilization criterion are
represented by a class of BMIs which has been solved by the path-
following method. The controller gain is chosen by a switching law
that minimizes a piecewise Lyapunov function and its time derivative.
Simulating this new procedure, the controlled system presented an ap-
propriate transient response, as displayed in Figs. 1 and 2. Thus, it
is considered that the proposed method may be an useful procedure
in practical applications for the control design of uncertain nonlinear
systems. Future research on the subject include the design of robust
discrete-time switched T–S controllers [43] for plants with uncertain-
ties or subjected to structural failures [44]–[46].

ACKNOWLEDGMENT

The authors acknowledge the Editor, the Associate Editor, and the
reviewers for their valuable comments.

REFERENCES

[1] K. Tanaka, T. Ikeda, and H. O. Wang, “Fuzzy regulators and fuzzy ob-
servers: Relaxed stability conditions and LMI-based designs,” IEEE Trans.
Fuzzy Syst., vol. 6, no. 2, pp. 250–265, May 1998.

[2] M. C. M. Teixeira and S. H. Żak, “Stabilizing controller design for un-
certain nonlinear systems using fuzzy models,” IEEE Trans. Fuzzy Syst.,
vol. 7, no. 2, pp. 133–142, Apr. 1999.

[3] T. Taniguchi, K. H. Ohatake, and H. O. Wang, “Model construction, rule
reduction, and robust compensation for generalized form of Takagi–
Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 525–
537, Aug. 2001.

[4] B. Yang, D. Yu, G. Feng, and C. Chen, “Stabilisation of a class of nonlinear
continuous time systems by a fuzzy control approach,” Proc. IEE Control
Theory Appl., vol. 153, no. 4, pp. 427–436, Jul. 2006.

[5] N. S. D. Arrifano, V. A. Oliveira, and L. V. Cossi, “Synthesis of an
LMI-based fuzzy control system with guaranteed cost performance: A
piecewise Lyapunov approach,” Sba: Controle & Automação Sociedade
Brasileira de Automatica, vol. 17, pp. 213–225, Jun. 2006

[6] C.-H. Fang, Y.-S. Liu, S.-W. Kau, L. Hong, and C.-H. Lee, “A new LMI-
based approach to relaxed quadratic stabilization of T-S fuzzy control
systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 3, pp. 386–397, Jun. 2006.

[7] F. Delmotte, T. M. Guerra, and M. Ksantini, “Continuous Takagi–
Sugeno’s models: Reduction of the number of LMI conditions in various
fuzzy control design technics,” IEEE Trans. Fuzzy Syst., vol. 15, no. 3,
pp. 426–438, Jun. 2007.

[8] V. F. Montagner, R. C. L. F. Oliveira, and P. L. D. Peres, “Convergent LMI
relaxations for quadratic stabilizability and H∞ control of Takagi–Sugeno
fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 17, no. 4, pp. 863–873, Aug.
2009.

[9] M. P. A. Santim, M. C. M. Teixeira, W. A. Souza, E. Assunção, and R.
Cardim, “Design of a Takagi–Sugeno fuzzy regulator for a set of operation
points,” Math. Problems Eng., vol. 2012, art. no. 731298, pp. 1–17, 2012.

[10] F. A. Faria, G. N. Silva, and V. A. Oliveira, “Reducing the conservatism
of LMI-based stabilisation conditions for TS fuzzy systems using fuzzy
Lyapunov functions,” Int. J. Syst. Sci., vol. 44, no. 10, pp. 1956–1969,
2013.

[11] M. Chadli and H. R. Karimi, “Robust observer design for unknown inputs
Takagi–Sugeno models,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 158–
164, Feb. 2013.

[12] V. C. S. Campos, F. O. Souza, L. A. B. Torres, and R. M. Palhares, “New
stability conditions based on piecewise fuzzy Lyapunov functions and

tensor product transformations,” IEEE Trans. Fuzzy Syst., vol. 21, no. 4,
pp. 748–760, Aug. 2013.

[13] Z. Ding, J. Ma, and A. Kandel, “Petri net representation of switched
fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 16–29, Feb.
2013.

[14] W. Zhao, K. Li, and G. Irwin, “A new gradient descent approach for local
learning of fuzzy neural models,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1,
pp. 30–44, Feb. 2013.

[15] C. Peng, Q.-L. Han, and D. Yue, “To transmit or not to transmit: A
discrete event-triggered communication scheme for networked Takagi–
Sugeno fuzzy systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 164–
170, Feb. 2013.

[16] M. C. M. Teixeira, E. Assunção, and R. G. Avellar, “On relaxed LMI-
based designs for fuzzy regulators and fuzzy observers,” IEEE Trans.
Fuzzy Syst., vol. 11, no. 5, pp. 613–623, Oct. 2003.

[17] Y.-J. Chen, H. Ohtake, K. Tanaka, W.-J. Wang, and H. O. Wang, “Relaxed
stabilization criterion for T-S fuzzy systems by minimum-type piecewise-
Lyapunov-function-based switching fuzzy controller,” IEEE Trans. Fuzzy
Syst., vol. 20, no. 6, pp. 1166–1173, Dec. 2012.

[18] A. Hassibi, J. How, and S. Boyd, “A path-following method for solving
BMI problems in control,” in Proc. Amer. Control Conf., 1999, vol. 2,
pp. 1385–1389.

[19] M. A. Wicks, P. Peleties, and R. A. DeCarlo, “Construction of piecewise
Lyapunov functions for stabilizing switched systems,” in Proc. IEEE 33rd
Conf. Decision Control, Dec. 1994, vol. 4, pp. 3492–3497.

[20] D. Liberzon and A. S. Morse, “Basic problems in stability and design of
switched systems,” IEEE Control Syst., vol. 19, no. 5, pp. 59–70, Oct.
1999.

[21] R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennartson, “Per-
spectives and results on the stability and stabilizability of hybrid systems,”
Proc. IEEE, vol. 88, no. 7, pp. 1069–1082, Jul. 2000.

[22] J. P. Hespanha and A. S. Morse, “Switching between stabilizing con-
trollers,” Automatica, vol. 38, no. 11, pp. 1905–1917, Nov. 2002.

[23] Z. Ji, L. Wang, G. Xie, and F. Hao, “Linear matrix inequality approach
to quadratic stabilisation of switched systems,” Proc. IEE Control Theory
Appl., vol. 151, no. 3, pp. 289–294, May 2004.

[24] J. P. Hespanha, “Uniform stability of switched linear systems: Extensions
of LaSalle’s invariance principle,” IEEE Trans. Autom. Control, vol. 49,
no. 4, pp. 470–482, Apr. 2004.

[25] G. Xie and L. Wang, “Periodical stabilization of switched linear systems,”
J. Comput. Appl. Math, vol. 181, no. 1, pp. 176–187, Sep. 2005.

[26] J. C. Geromel and P. Colaneri, “Stability and stabilization of continuous-
time switched linear systems,” SIAM J. Control Optim., vol. 45, no. 5,
pp. 1915–1930, Dec. 2006.

[27] R. Cardim, M. C. M. Teixeira, and E. Assunção, M. R. Covacic,
“Variable-structure control design of switched systems with an application
to a DC-DC power converter,” IEEE Trans. Ind. Electron., vol. 56, no. 9,
pp. 3505–3513, Sep. 2009.

[28] G. S. Deaecto, J. C. Geromel, F. S. Garcia, and J. A. Pomilio, “Switched
affine systems control design with application to dc-dc convert-
ers,” IET Control Theory Appl., vol. 4, no. 7, pp. 1201–1210, Jul.
2010.

[29] G. S. Deaecto, J. C. Geromel, and J. Daafouz, “Switched state-feedback
control for continuous time-varying polytopic systems,” Int. J. Control,
vol. 84, no. 9, pp. 1500–1508, Sep. 2011.

[30] W. A. Souza, M. C. M. Teixeira, M. P. A. Santim, R. Cardim, and
E. Assunção, “On switched control design of linear time-invariant sys-
tems with polytopic uncertainties,” Math. Problems Eng., vol. 2013, art.
no. 595029, pp. 1–10, 2013.

[31] K. Tanaka, M. Iwasaki, and H. O. Wang, “Stable switching fuzzy control
and its application to a hovercraft type vehicle,” in Proc. IEEE Ninth Int.
Conf. Fuzzy Syst., 2000, vol. 2, pp. 804–809.

[32] K. Tanaka, M. Iwasaki, and H. O. Wang, “Stability and smoothness con-
ditions for switching fuzzy systems,” in Proc. Amer. Control Conf., 2000,
vol. 4, pp. 2474–2478.

[33] G. Feng, “H∞ controller design of fuzzy dynamic systems based on piece-
wise Lyapunov functions,” IEEE Trans. Syst., Man, Cybern. B, vol. 34,
no. 1, pp. 283–292, Feb. 2004.

[34] J. Dong and G.-H. Yang, “State feedback control of continuous-time T-S
fuzzy systems via switched fuzzy controllers,” Inf. Sci., vol. 178, no. 6,
pp. 1680–1695, Mar. 2008.

[35] J. Dong and G.-H. Yang, “Dynamic output feedback control synthesis for
continuous-time T-S fuzzy systems via a switched fuzzy control scheme,”
IEEE Trans. Syst., Man, Cybern. B, vol. 38, no. 4, pp. 1166–1175, Aug.
2008.



IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 6, DECEMBER 2014 1727

[36] S. Yan and Z. Sun, “Study on separation principles for T-S fuzzy sys-
tem with switching controller and switching observer,” Neurocomputing,
vol. 73, no. 13-15, pp. 2431–2438, Aug. 2010.

[37] G.-H. Yang and J. Dong, “Switching fuzzy dynamic output feedback H∞
control for nonlinear systems,” IEEE Trans. Syst., Man, Cybern. B, vol. 40,
no. 2, pp. 505–516, Apr. 2010.

[38] D. Jabri, K. Guelton, N. Manamanni, A. Jaadari, and C. C. Duong, “Robust
stabilization of nonlinear systems based on a switched fuzzy control law,”
J. Control Eng. Appl. Informat., vol. 14, no. 2, pp. 40–49, 2012.

[39] J. Lofberg, “YALMIP: A toolbox for modeling and optimization in
MATLAB,” in Proc. IEEE Int. Symp. Comput. Aided Control Syst. Design,
Sep. 2004, pp. 284–289.

[40] J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization
over symmetric cones,” Optimization Methods Softw., vol. 11–12, pp. 625–
653, 1999.

[41] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its ap-
plications to modeling and control,” IEEE Trans. Syst., Man, Cybern.,
vol. SMC-15, no. 1, pp. 116–132, Feb. 1985.

[42] H. J. Marquez, Nonlinear Control Systems—Analysis and Design. Wiley,
2003

[43] X. Xie, H. Ma, Y. Zhao, D. Ding, and Y. Wang, “Control synthesis
of discrete-time T–S fuzzy systems based on a novel non-PDC control
scheme,” IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp. 147–157, Feb. 2013.

[44] E. Assunção, M. C. M. Teixeira, F. A. Faria, N. A. P. da Silva, and
R. Cardim, “Robust state-derivative feedback LMI-based designs for mul-
tivariable linear systems,” Int. J. Control, vol. 80, no. 8, pp. 1260–1270,
Aug. 2007.

[45] F. A. Faria, E. Assunção, M. C. M. Teixeira, R. Cardim, and N. A. P. da
Silva, “Robust state-derivative pole placement LMI-based designs for
linear systems,” Int. J. Control, vol. 82, no. 1, pp. 1–12, Jan. 2009.

[46] M. Liu, X. Cao, and P. Shi, “Fault estimation and tolerant control for fuzzy
stochastic systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 2, pp. 221–229,
Apr. 2013.