Control Synthesis of Singularly Perturbed Fuzzy Systems

Abstract

This paper considers the problem of designing stabilizing and H_infin controllers for nonlinear singularly perturbed systems described by Takagi-Sugeno fuzzy models with the consideration of the bound of singular perturbation parameter e. For the synthesis problem of simultaneously designing the bound of e and stabilizing or H_infin controllers, linear matrix inequalities (LMI)- based methods are presented. For evaluating the upper bound of e subject to stability or a prescribed H_infin performance bound constraint for the resulting closed-loop system, sufficient conditions are developed, respectively. For the stabilizing and H_infin control synthesis without the consideration of improving the bound of e, new design methods are also given in terms of solutions to a set of LMIs. Examples are given to illustrate the efficiency of the proposed methods.

Full text

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008 615
Control Synthesis of Singularly Perturbed
Fuzzy Systems
Guang-Hong Yang, Senior Member, IEEE, and Jiuxiang Dong
Abstract—This paper considers the problem of designing stabi-
lizing and controllers for nonlinear singularly perturbed sys-
tems described by Takagi–Sugeno fuzzy models with the consider-
ation of the bound of singular perturbation parameter . For the
synthesis problem of simultaneously designing the bound of and
stabilizing or controllers, linear matrix inequalities (LMI)-
based methods are presented. For evaluating the upper bound of
subject to stability or a prescribed performance bound con-
straint for the resulting closed-loop system, sufficient conditions
are developed, respectively. For the stabilizing and control
synthesis without the consideration of improving the bound of ,
new design methods are also given in terms of solutions to a set
of LMIs. Examples are given to illustrate the efficiency of the pro-
posed methods.
Index Terms— performance, linear matrix inequalities
(LMIs), nonlinear control systems, singularly perturbed systems,
stabilizing control, state feedback control, Takagi–Sugeno (T-S)
fuzzy models.
I. INTRODUCTION
IN CONTROL engineering applications, it is well known that
the multiple time-scale systems or known as singularly per-
turbed systems often raise serious numerical problems. For the
purpose of avoiding the difficulties linked with the stiffness of
the equations involved in the design, the singular perturbation
design method has been developed [1], where singular pertur-
bation with a small parameter, say , is exploited to determine
the degree of separation between “slow” and “fast” modes of
the system, and the so-called reduction technique is proposed to
handle these systems. For the stabilization and control of
linear singularly perturbed systems, many important advances
have been achieved, see [1]–[6] and the references therein. In
particular, the fundamental results are given in [1], [4], and [5].
Manuscript received April 19, 2006; revised January 11, 2007. This work was
supported in part by the Program for New Century Excellent Talents in Univer-
sity (NCET-04-0283), by the Funds for Creative Research Groups of China (No.
60521003), by the Program for Changjiang Scholars and Innovative Research
Team in University (No. IRT0421), by the State Key Program of National Nat-
ural Science of China under Grant 60534010, by the Funds of National Science
of China under Grant 60674021, and by the Funds of the Ph.D. program of
MOE, China, under Grant 20060145019.
G.-H. Yang is with the College of Information Science and Engineering,
Northeastern University, Shenyang 110004, China, and also with the
Key Laboratory of Integrated Automation of Process Industry, North-
eastern University, Ministry of Education, Shenyang 110004, China (e-mail:
yangguanghong@ise.neu.edu.cn; yang_guanghong@163.com).
J. Dong is with the College of Information Science and Engineering, North-
eastern University, Shenyang 110004, China (e-mail: dongjiuxiang@ise.neu.
edu.cn; dong_jiuxiang@163.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TFUZZ.2007.905911
In recent years, there have been some attempts to address the
control problem for nonlinear singularly perturbed sys-
tems. In [7] and [8], the control for a class of singularly
perturbed systems with nonlinearity in the slow variables is ex-
amined. A local state feedback control problem for affine
nonlinear singularly perturbed systems is studied in [9]. How-
ever, the control design for general nonlinear singularly
perturbed systems still remains as an open research subject.
An important approach to the synthesis problems for non-
linear systems is to model the considered system as Takagi and
Sugeno (T-S) fuzzy systems [10], which are locally linear time-
invariant systems connected by IF-THEN rules. In [11] and [12],
it has shown that the T-S fuzzy systems can approximate any
continuous functions at any preciseness, which shows that the
T-S fuzzy models can approximate a wide class of nonlinear
systems. As a result, the conventional linear system theory can
be applied to analysis and synthesis of the class of nonlinear
control systems. In recent years, the T-S fuzzy control systems
have been studied extensively, and many significant advances
have been achieved (see [13]–[15] and the references therein).
For nonlinear singularly perturbed systems, some control syn-
thesis problems have been studied [16]–[19]. In [16] and [18],
design methods for the stabilization and control of non-
linear singularly perturbed systems via state feedback are given
in terms of solutions of linear matrix inequalities (LMIs) [20],
respectively. A robust state feedback control design is presented
in [17]. An LMI-based method of designing output feed-
back controllers for uncertain fuzzy singularly perturbed sys-
tems is presented in [19].
For the effective applications of the design methods for sin-
gularly perturbed control systems, the accurate knowledge of
the stability bound of a singularly perturbed system (i.e., the
system is stable for ) is very important. The charac-
terization and computation of the stability bound have attracted
considerable efforts for the past over two decades [21]–[28]. In
general, there are two classes of methods to characterize and
compute the stability bounds, one is based on frequency domain
transfer functions and another is based on state space models.
Both of the two methods can provide the exact bounds as shown
in [22], [25], and [26]. However, the issue of how to improve
the bound in controller designs has not been addressed in the
literature, which undoubtedly is very important for the applica-
tions of singularly perturbed system theory.
This paper is concerned with the problem of designing sta-
bilizing and controllers for nonlinear singularly perturbed
systems described by T-S fuzzy models with the considera-
tion of the bound of singular perturbation parameter .Two
LMI-based methods are presented for simultaneously designing
1063-6706/$25.00 © 2007 IEEE

616 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008
the bound of and stabilizing or controllers for a fuzzy
singularly perturbed system, respectively. For the issue of com-
puting the bound of singularly perturbed parameter , sufficient
conditions are derived for evaluating the upper bound of
subject to stability or a prescribed performance bound con-
straint for the resulting closed-loop system for , re-
spectively, where the upper bound can be obtained by solving
a generalized eigenvalue problem (GEVP) [20]. For the problem
of designing stabilizing and controllers without the consid-
eration of improving the bound of , design methods are also
given in terms of solutions to LMIs. The paper is organized as
follows. In Section II, the system description, the considered
problems, and preliminary lemmas are presented. Section III
considers the problem of designing stabilizing controllers and
the bound of , and the case for controller design is studied
in Section IV. Section V gives examples to illustrate the effec-
tiveness of the new proposed methods. Finally, Section VI con-
cludes this paper.
Notation: For a matrix , is defined as the largest
singular value of . For a square matrix , is defined
as
The symbol within a matrix represents the symmetric entries
.
.
..
.
.....
.
.
II. SYSTEM DESCRIPTION AND PROBLEM STATEMENT
A. System Description
The class of nonlinear singularly perturbed systems under
consideration is described by the following fuzzy system model:
IF is and is is
THEN
(1)
where are fuzzy sets, are the premise
variables, and are the state vectors,
is the control input, is the disturbance,
is the controlled output, the matrices , , ,
, , , , , , , and are of appropriate
dimensions, is the number of IF-THEN rules, and is a
small constant.
Denote
is the grade of membership of in , where
it is assumed that
Let
then
(2)
is said to be normalized membership functions. The
T-S fuzzy model of (1) is inferred as follows:
(3)
The system (3) can be rewritten as follows:
(4)
where
In this paper, the concept of parallel distributed compensation
(PDC) is used to design fuzzy controller, i.e., the designed fuzzy
controller shares the same fuzzy sets with the fuzzy model in

YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 617
the premise parts (more details can be found in [13]). For the
fuzzy model (1), the following state feedback controller [13] is
adopted:
IF is and is is
THEN (5)
Because the controller rules are same as plant rules, we obtain
the state feedback controller as follows:
(6)
Combining (6) and (4), then the resulted closed-loop system is
given as follows:
(7)
B. Problem Statement
In this paper, the following problems will be addressed.
Controller Design With Consideration of Bound of :
(i) Find , and an as big as possible
such that the closed-loop system (7) with is
asymptotically stable for any and all
satisfying (2).
(ii) Let be a given constant. Find
and an as big as possible such that the closed-loop
system (7) is asymptotically stable and with an -norm
less than or equal to for any and all
satisfying (2).
Evaluation of Bound of :
(iii) Let be given. Find an as
big as possible such that the closed-loop system (7) with
is asymptotically stable for any and
all satisfying (2).
(iv) Let and be given. Find an
as big as possible such that the closed-loop system (7) is
asymptotically stable and with an -norm less than or
equal to for any and all satisfying
(2).
Remark 1: Problems (i) and (ii) are concerned with simul-
taneously designing and finding the upper
bound of with guaranteeing the stability and performance
of the closed-loop system (7), respectively. Problems (iii) and
(iv) are related to the problem of finding the upper bound of
subject to that the closed-loop system (7) is asymptotically
stable or with the constraint of an performance bound when
are given. Moreover, the controller design
problems without the consideration of bound of will also be
studied in Sections III and IV, respectively.
C. Preliminaries
The following preliminaries will be used in the sequel.
For the fuzzy control system (7), let be a constant.
If (7) is asymptotically stable, and for any
(the space of square integrable functions) and , the
following inequality holds:
then the system (7) is said to be with an -norm less than or
equal to [29], [30].
Denote
then the closed-loop system (7) can be rewritten as follows:
(8)
The following lemma from [29] gives a sufficient condition
for the system (8) to be with an -norm less than or equal to
.
Lemma 1: [29] Consider the system (8). If there exists a pos-
itive definite matrix such that
holds. Then the system (8) is asymptotically stable and with an
-norm less than or equal to .
Lemma 2: If there exist symmetric matrices , ,
, , and matrices , ,
such that the following LMIs hold:
(9)
(10)
(11)
where
then
.
.
..
.
.(12)

618 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008
Proof:
.
.
..
.
.
.
.
..
.
.....
.
.
From (9) to (11), it follows (12).
Lemma 3: [16] If there exist matrices and , ,
with
where and are symmetric matrices, satisfying the fol-
lowing LMIs:
(13)
(14)
then there exists a scaler such that, for , the
state feedback controller (6) with
renders the singularly perturbed fuzzy system (7) asymptoti-
cally stable.
Lemma 4: [18] For given , if there exist matrices
and , , with
where and are symmetric matrices, satisfying the fol-
lowing LMIs, shown in (15) and (16) at the bottom of this page,
then there exists a scaler such that, for , the
state feedback controller (6) with
renders the singularly perturbed fuzzy system (7) with an
norm less than .
III. STABILITY BOUND AND STABILIZATION
In this section, a method of simultaneously designing the
upper bound of and stabilizing controller gains is derived
where the upper bound of singularly perturbed parameter can
be improved, which provides a solution to Problem (i). For
solving Problem (iii), a method of computing the upper bound
of singularly perturbed parameter subject to the stability of
the closed-loop system is presented. Moreover, a technique for
designing stabilizing controllers for singularly perturbed fuzzy
systems without consideration of improving the bound of sin-
gularly perturbed parameter is also presented in Section III-C.
A. Design of Stability Bound of and Stabilizing Controllers
The following theorem presents a method of simultaneously
designing the upper bound of and stabilizing controller gains.
Theorem 1: If there exist matrices , , , , ,
, , , , and positive scalars ,
, , , with
where , , , , , are symmetric
matrices, satisfying the following LMIs:
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
where
(15)
He
(16)

YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 619
and denote
(27)
where
then for , the state feedback controller (6) with
renders the singularly perturbed fuzzy system (7) asymptoti-
cally stable.
Proof: See Appendix A.
Remark 2: Theorem 1 presents sufficient conditions under
which an upper bound of singularly perturbed parameter
and stabilizing controller gains can be obtained. From (27) and
, , can be minimized by solving the following
optimization problem:
minimize
subject to (17)–(26) (28)
where , are positive weighting constants to be
chosen. However, the optimization problem cannot be solved
directly due to the term . Consider
From (25) and (26), we have
then it follows that
Combining it with , , (28) can be minimized
by solving the optimization problem
minimize
subject to (17)–(26) (29)
where , are positive weighting constants to
be chosen. Since the constraints (17)–(26) are of LMIs, the
optimization problem can be effectively solved via LMI Con-
trol Toolbox [31]. Regarding the issue of how to choose the
weighting scalars , generally, one can
choose bigger for rendering smaller .It
should be pointed out that the upper bound obtained by
solving the previous optimization problem may be conservative.
After obtaining stabilizing controller gains, a less conservative
bound of can be obtained by Theorem 2 in Section III-B.
B. Computation of Stability Bound of
In this subsection, assume that the controller has been de-
signed. The following theorem gives a technique to estimate the
upper bound of singularly perturbed parameter subject to the
stability of the closed-loop system.
Theorem 2: If there exist matrices , , , ,
, and a constant with
where , , , are symmetric matrices,
satisfying (9)–(11), and the following LMIs:
(30)
(31)
(32)
(33)
where , , , .
Then, for , the singularly perturbed closed-loop
fuzzy system (7) is asymptotically stable, where
(34)
(35)
Proof: From condition (30) and (31), we have
(36)
By , and (36), it follows that
for (37)

620 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008
Pre- and post multiplying (37) by
and its transpose, then the following inequality holds:
for
Consider the following Lyapunov function:
then
.
.
..
.
.
(38)
where , are given by (34) and (35).
On the other hand, from (32) and (33), we have
for (39)
Applying (9)–(11), (39) and Lemma 2, it follows
.
.
..
.
.
combined with (38), we have
for
Thus, for , the closed-loop singularly perturbed
fuzzy system (7) is asymptotically stable.
Remark 3: By Theorem 2, an upper bound of can be ob-
tained by solving inequalities (9)–(11), (30)–(33) for . The op-
timization problem
Minimize subject to (9), (10), (11), (30)–(33)
is a generalized eigenvalue problem (GEVP) [20], which can
be effectively solved using LMI Control Toolbox [31]. The
problem of computing the bound of was considered in [17],
where a method of finding an interval so that the system
is stable for was derived. However, to search for
small is related to -dependent computation, which cannot
avoid the difficulties linked with the stiffness of the equations
involved in the design.
C. Stabilizing Controller Design Without Considering Stability
Bound
In this section, a technique for designing stabilizing con-
trollers for singularly perturbed fuzzy systems without con-
sideration of improving the bound of singularly perturbed
parameter , is given as follows.
Theorem 3: If there exist matrices , , , ,
, , with
where , , , are symmetric matrices,
satisfying the following LMIs:
(40)
(41)
(42)
(43)
(44)

YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 621
where
then there exists a scaler such that, for , the
state feedback controller (6) with
renders the singularly perturbed fuzzy system (7) asymptoti-
cally stable.
Proof: By
and , , we choose , then by matrix
invertible formula, it follows that
Since and , there exists a scaler , such
that for , which implies
Choose Lyapunov function
then
(45)
On the other hand, substitute
(46)
for in (44) and pre- and post multiplying (44) by
and , then we can obtain
(47)
where
Now, pre- and post multiplying (41)–(43) by and , then it
follows that
(48)
(49)
(50)
From (47), we have that there exist a scalar such that
for
(51)
where ,
. By (48)–(50), (51) and Lemma 2, it follows
that
i.e.,
combined with (45), then we have
for
which implies that the closed-loop singularly perturbed fuzzy
system (7) is asymptotically stable, and from (46), we obtain
Thus, the proof is complete.
Remark 4: Theorem 3 presents a method for designing state
feedback stabilizing controllers for singularly perturbed fuzzy
systems. However, the upper bound of singularly perturbed pa-
rameter is not addressed in the design.
The following theorem shows that the new method given in
Theorem 3 is less conservative than that given in Lemma 3 [16].
Theorem 4: If the condition in Lemma 3 holds, then the con-
dition in Theorem 3 holds.
Proof: Assume that the condition in Lemma 3 holds, then
choose ,
, , . Then it follows that
conditions (40)–(43) hold from (13) and (14). Moreover, from
(14) and the chosen , the following inequalities hold:
.
.
..
.
.....
.
.

622 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008
which implies that (44) holds. Therefore, the condition of The-
orem 3 holds. Thus, the proof is complete.
IV. BOUND OF AND CONTROL
In this section, the results in Section III are extended to the
case that the closed-loop system is required to be with an
performance bound. Solutions to Problem (ii) and (iv) formu-
lated in Section II-B are, respectively, given in Section IV-A and
IV-B. Moreover, Section IV-C also presented a new method for
designing controllers without consideration of improving
the bound of singularly perturbed parameter .
A. Design for Bound of and Performance
In this subsection, a method is given for designing con-
trollers with consideration of improving the bound of singularly
perturbed parameter .
Theorem 5: For given , if there exist matrices , ,
, , , , , , , and
positive scalar variables , , , , , with
where , , , , , are symmetric
matrices, satisfying the following LMIs:
(52)
(53)
(54)
(55)
(56)
(57)
(58)
(59)
(60)
(61)
where , are the same as in Theorem 3, and
Denote
(62)
where
(63)
then for , the singularly perturbed closed-loop
fuzzy system (8) via state feedback controller
where
is asymptotically stable and with an norm less than .
Proof: See Appendix A.
Remark 5: If the conditions of Theorem 5 hold, then
is an upper bound of singularly perturbed parameter , and
The problem of minimizing can be reduced to solving the
following optimization problem with LMI constraints:
Minimize
subject to (52)–(61)
where , , , , and are positive weighting constants to
be chosen.
B. Computation of Upper Bound of
In this subsection, we assume that the state feedback gains are
given. Then the following theorem gives a method to estimate
the upper bound of singularly perturbed parameter subject to
the closed-loop system with an performance bound.
Theorem 6: For a given , if there exists , , ,
, , and a constant with
where , , , are symmetric matrices
satisfying (9)–(11) and the following LMIs:
(64)
(65)
(66)
(67)

YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 623
where
where
then, for , the singularly perturbed closed-loop
fuzzy system (8) is asymptotically stable and with an norm
less than .
Proof: From (64) to (67), it follows that, for any
(68)
and
By (9)–(11), and Lemma 2, which further implies that
.
.
..
.
.(69)
On the other hand
.
.
..
.
.
Thus, from Lemma 1 and (69), the conclusion follows.
Remark 6: When the state feedback gains are given, The-
orem 6 gives a method to estimate the upper bound of singularly
perturbed parameter subject to that the closed-loop system
is required to be with a prescribed performance bound.
An upper bound of can be obtained by solving inequalities
(9)–(11) and (64)–(67) for . The optimization problem
Minimize subject to (9), (10), (11), (64)–(67)
is also a generalized eigenvalue problem (GEVP) [20], which
can be effectively solved using LMI Control Toolbox [31].
C. Controller Design
In this subsection, a new method is given for designing
controllers, but without consideration of improving the bound
of singularly perturbed parameter .
Theorem 7: For given , if there exist matrices , ,
, , , , with
where , , , and are symmetric ma-
trices satisfying the following LMIs:
where
then there exists a scaler such that, for ,
the singularly perturbed fuzzy system (7) via the state feedback
controller (6) with
is asymptotically stable and with an norm less than .
Proof: It is similar to Theorem 3, and omitted here.
The following theorem shows that the new method given in
Theorem 7 is less conservative than that given in Lemma 4 [18].
Theorem 8: If the condition of Lemma 4 holds, then the con-
dition of Theorem 7 holds.
Proof: It is similar to Theorem 4 and omitted here.
V. E XAMPLE
We consider an inverted pendulum controlled by a motor via
a gear train. It can be described by the following state equa-
tions [32]:

624 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008
TABLE I
UPPER BOUNDS OF

VIA THEOREM 6
where , ,
, is the control input, is the disturbance input,
is the motor torque constant, is the back emf constant,
and is the gear ratio. The parameters for the plant are given as
9.8 m/s , 1m, 1 kg, , 0.1 Nm/A,
0.1 Vs/rad, 1 and mH. Note that the
inductance represents the small “parasitic”parameter in the
system. Then, we get
Choose the membership functions of the fuzzy sets as
This fuzzy model exactly represents the dynamics of the non-
linear mechanical system under . A T-S fuzzy
model can be obtained as follows
Plant Rule 1:
IF is
THEN
Plant Rule 2:
IF is
THEN
where
Lemma 4, Theorems 5 and 7 are applicable for designing fuzzy
controllers for the system.
By using Lemma 4 and Theorem 7, the obtained optimal
performance indexes are and , re-
spectively, and the corresponding controller gains are given as
follows:
(70)
(71)
Now we apply Theorem 5 to design a fuzzy controller for
the system. Choose weighting scalars as
and , then the controller gains are given as follows:
(72)
By using Theorem 6, the upper bounds of subject to guaran-
teeing performance bound can be estimated, and shown
in Table I.
From Table I, it is easy to see that the new proposed design
given by Theorem 5 gives a considerable improvement of upper
bounds of . The upper bounds of given by Lemma 4 and
Theorem 7 are very small.
Assume the initial states are zero, and the disturbance input
signal is shown in Fig. 1. The simulation results of the
output with the controller gains (70)–(72) are given in
Figs. 2–4, respectively. The simulations for the square root of
ratio of the regulated output energy to the disturbance input
noise energy are depicted in Figs. 5–7. From the simulation
results, it can be seen that the fuzzy controller (72) guarantees
good performance of the resulting closed-loop system,
while the controllers (70) and (71) give poor system responses.

YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 625
Fig. 1. Disturbance input
w
(
t
)
used during simulation.
Fig. 2. Trajectory of
z
(
t
)
via controller (70).
Fig. 3. Trajectory of
z
(
t
)
via controller (71).
Fig. 4. Trajectory of
z
(
t
)
via controller (72).
Fig. 5.
z
(
t
)
z
(
t
)
dt= w
(
t
)
w
(
t
)
dt
via controller (70).
Fig. 6.
z
(
t
)
z
(
t
)
dt= w
(
t
)
w
(
t
)
dt
via controller (71).

626 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008
Fig. 7.
z
(
t
)
z
(
t
)
dt= w
(
t
)
w
(
t
)
dt
via controller (72).
VI. CONCLUSION
In this paper, we have studied the problem of designing sta-
bilizing and controllers for nonlinear singularly perturbed
systems described by T-S fuzzy models with the consideration
of improving the bound of singular perturbation parameter .
The main contribution is as follows. For the synthesis problem
of simultaneously designing the bound of and stabilizing or
controllers, LMI-based methods are presented. For evalu-
ating the upper bound of subject to stability or a prescribed
performance bound constraint for the resulting closed-loop
system, LMI-based sufficient conditions are developed, respec-
tively. For the stabilizing and control synthesis without the
consideration of improving the bound of , new design methods
are also given in terms of solutions to a set of LMIs. The exam-
ples have shown the efficiency of the proposed methods.
APPENDIX A
PROOF OF THEOREM 1
Proof: It consists of three parts. In Parts 1 and 2, we prove
that (78) and (92) hold, respectively. In Part 3, the proof is com-
pleted by using (78) and (92).
Part 1: First, from conditions (25) and (26), we have
(73)
From (27) and (73), it follows
(74)
then
(75)
Pre- and postmultiplying (75) by and its transpose and
applying the Schur complement, we obtain
which further implies that
for (76)
Let , then can be expressed as follows:
So (76) is equivalent to
(77)
Pre- and postmultiplying (77) by
and its transpose, then the following inequality holds:
(78)
Part 2: From (25), we can obtain
(79)
From (27), we have
(80)
By (17)–(19), (23), and Lemma 2, it follows that
i.e.,
(81)
Applying the Schur complement to (81), we have
i.e.,
(82)

YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 627
Notice that
(83)
where are the same as in (34). Applying (82) to (83), it
follows that
(84)
By (80) and (84), we have
which implies that
(85)
Pre- and postmultiplying (24) by and its trans-
pose, it follows that
.
.
..
.
.(86)
Multiplying (86) by , and summing them, it
follows
.
.
..
.
.
Combining it with (79), we can obtain
.
.
..
.
.
Applying Lemma 2 to the previous inequality, then
which implies that
(87)
(88)
From (85) and (88), we have
(89)
Substituting into (89), then we have
(90)
Pre- and postmultiplying (90) by , then we can obtain
(91)
where are same as in (35). From (87), then
Combining it and (91), that yields
for (92)
Part 3: Choose Lyapunov function
Then, by (78) and (92), it follows that and
.
.
..
.
.
for
for . Thus, the proof is complete.

628 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 3, JUNE 2008
APPENDIX B
PROOF OF THEOREM 5
Proof: From (58), (62), and Part 1 of the proof of Theorem
3, we have
for
where
. By (62), it follows that
(93)
On the other hand, from (60), we have
(94)
Substituting , , then (94) becomes
(95)
where
For brief expression, we also denote
Applying the Schur complement to (95), we obtain
which implies that
(96)
From (63), it follows that
(97)
Combining (96), (97), and (93), we have
which further implies that
for (98)
where
On the other hand, from (59), we have
(99)
Thus, from (98) and (99), it follows:
(100)
By (61) and Lemma 2, we can obtain (101) shown at the bottom
of the page. Applying the Schur complement to (101), it follows
that
(102)
where
(101)

YANG AND DONG: CONTROL SYNTHESIS OF SINGULARLY PERTURBED FUZZY SYSTEMS 629
and
By (99), (100), and (102), we have
(103)
Pre- and postmultiplying (103) by and , then we obtain
i.e.,
Then, by Lemma 1, the proof is completed.
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Guang-Hong Yang (SM’03) received the B.S.
and M.S. degrees in mathematics from Northeast
University of Technology, Shenyang, China, in 1983
and 1986, respectively, and the Ph.D. degree in
control engineering from Northeastern University,
Shenyang, China (formerly, Northeast University of
Technology), in 1994.
He is currently a Professor with the College
of Information Science and Engineering, North-
eastern University. From 1986 to 1995, he was
a Lecturer/Associate Professor with Northeastern
University. In 1996, he was as a Postdoctoral Fellow with the Nanyang
Technological University, Singapore. From 2001 to 2005, he was a Research
Scientist/Senior Research Scientist with the National University of Singapore,
Singapore. His current research interests include fault tolerant control, fault
detection and isolation, nonfragile control systems design, and robust control.
Dr. Yang is an Associate Editor for the International Journal of Control, Au-
tomation, and Systems (IJCAS), and an Associate Editor on the Conference Ed-
itorial Board of the IEEE Control Systems Society.
Jiuxiang Dong received the B.S. degree in mathe-
matics and applied mathematics and the M.S. degree
in applied mathematics from Liaoning Normal
University, Dalian, China, in 2001 and 2004, respec-
tively. He is currently pursing the Ph.D. degree at
Northeastern University, Shenyang, China.
His research interests include fuzzy control, robust
control, and fault-tolerant control.