This document presents a new method for designing piecewise state feedback controllers for affine fuzzy systems. The method uses dilated linear matrix inequalities (LMIs) to characterize the system in a way that separates the system matrix from the Lyapunov matrix. This allows the controller parameters to be independent of the Lyapunov matrix. The results provide less conservative LMI characterizations than existing methods and can be applied to more general systems. The method is also extended to H-infinity state feedback synthesis. Numerical examples demonstrate the effectiveness of the new approach.

1. Piecewise controller design for affine fuzzy systems via dilated linear matrix
inequality characterizations
Huimin Wang a
, Guang-Hong Yang a,b,n
a
College of Information Science and Engineering, Northeastern University, Shenyang 110004, PR China
b
State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, PR China
a r t i c l e i n f o
Article history:
Received 4 November 2011
Received in revised form
5 April 2012
Accepted 28 June 2012
Available online 21 July 2012
Keywords:
Affine fuzzy systems
Stabilizability
Slack variable
Linear matrix inequalities (LMIs)
State feedback
a b s t r a c t
This paper studies the problem of state feedback controller design for a class of nonlinear systems,
which are described by continuous-time affine fuzzy models. A convex piecewise affine controller
design method is proposed based on a new dilated linear matrix inequality (LMI) characterization,
where the system matrix is separated from Lyapunov matrix such that the controller parametrization is
independent of the Lyapunov matrix. In contrast to the existing work, the derived stabilizability
condition leads to less conservative LMI characterizations and much wider scope of the applicability.
Furthermore, the results are extended to H1 state feedback synthesis. Finally, two numerical examples
illustrate the superiority and effectiveness of the new results.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In the nonlinear control area, an important approach to nonlinear
control system design is to model the considered nonlinear systems
as Takagi and Sugeno (T–S) fuzzy systems [1]. By incorporating
linguistic information from human experts and ‘‘blending’’ some
locally linear systems, the T–S fuzzy model has been proved to be a
well universal approximator [2]. As a result, the conventional linear
system theory can be applied to the analysis and synthesis of the class
of nonlinear control systems (see [3–11] and the references therein).
Usually, T–S fuzzy systems can be classified into linear fuzzy
models and affine fuzzy models [12]. The main difference between
them is that the latter considers a constant bias term in each fuzzy
rule, which makes the function approximation capabilities of T–S
fuzzy systems be improved substantially [6,13]. Moreover, the
stability theory of affine fuzzy models can be extended to the linear
ones [14]. As such, the research on affine fuzzy systems is expected
to be interesting and significant.
Recently, an increasing amount of work has been devoted to
analysis of affine fuzzy systems [12,14,15]. These results considered
the structural information in the rule base and introduced S-proce-
dure to decrease the conservatism of the stability analysis. However,
due to the constant bias term and the introduced S-procedure, most
existing results on control synthesis for affine fuzzy systems are
obtained in the form of bilinear matrix inequalities (BMIs)
[12,16–18], and it is impossible to convert these BMIs to LMIs by
simply using the inverse of the Lyapunov matrix P. As we know, to
deal with such non-convex problems, we need to design an iterative
LMI algorithm and obtain an initial feasible solution, which is
usually conservative and even impracticable. Although a lot of
efforts have been spent on improving these weaknesses, the efficient
and effective method has not yet to be developed. In [19], Kim et al.
obtained a convex state feedback controller design condition
through limiting the Lyapunov matrix P to a diagonal structure.
Although the result is more solvable, this constraint may lead to
difficulty since such a Lyapunov matrix might not exist in many
cases, especially for highly nonlinear complex systems. In addition,
it is also noted that the results are obtained based on the
assumption that the input matric B is a common one, i.e.,
_xðtÞ ¼
Pr
i ¼ 1 hiðxðtÞÞðAixðtÞþ BuðtÞ þmiÞ. For the case Bi aB, the
convexifying techniques are not efficient. This is a much simpler
system, and leads to much narrower scope of the applicability. In
[20], the authors proposed two approaches to robust H1 output
feedback controller design for affine fuzzy systems. It is shown that
the synthesis conditions can be formulated in the terms of LMIs that
can be efficiently solved by interior-point methods. To the authors’
best knowledge, few efficient attempts have been made on convex
state feedback controller design for general continuous-time affine
fuzzy systems, which motivates us for this study.
In past years, the technique of decoupling the Lyapunov matrix
and the system matrix by adding slack variables has been developed,
and it can effectively decrease the conservatism in multi-objective
control problems [21–23]. In [24], by introducing slack variables, the
authors derived a convex stabilizability condition, and there is no
need to impose any structural constraint on the Lyapunov matrix.
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.06.014
n
Corresponding author at: College of Information Science and Engineering,
Northeastern University, Shenyang 110004, PR China. Tel.: þ86 24 83681939.
E-mail addresses: wanghuimin702@yahoo.cn (H. Wang),
yangguanghong@ise.neu.edu.cn, yang_guanghong@163.com (G.-H. Yang).
ISA Transactions 51 (2012) 771–777

2. However, the obtained method can only suit for the system with a
common input matrix. In this paper, a convex piecewise affine
controller design method is proposed based on a new dilated linear
matrix inequality (LMI) characterization, where the system matrix is
separated from Lyapunov matrix such that the controller parame-
trization is independent of the Lyapunov matrix. It is noted that the
controller design method is obtained in the formulation of LMIs in
conjunction with a search of scaling parameters, which can provide
more relaxed conditions and deal with more general systems than
the existing results. Furthermore, an extended H1 performance
analysis of a class of affine fuzzy systems is presented, and the H1
controller synthesis condition is derived.
The structure of this paper is as follows: following the introduc-
tion, the system description and the problem under consideration are
given in Section 2. In Section 3, lemmas which are used throughout
are proposed. A quadratic stabilizability condition in the form of LMIs
is presented in Section 4. Section 5 proposes an extended H1
controller design of affine fuzzy systems. Numerical examples are
given in Section 6 to show the superiority and effectiveness of
proposed method. Finally, conclusions are drawn in Section 7.
Notation: For a symmetric matrix M, M40 (Mo0) means that it
is positive definite (negative definite). MT
denotes the transpose of
matrix M. The symbol n is used in some matrix expressions to denote
the transposed elements in the symmetric positions of a matrix.
2. System description and problem statement
2.1. System description
The following continuous-time affine T–S system can be used
to represent a complex nonlinear system with both fuzzy infer-
ence rules and local analytic models as follows
Ri : If x1ðtÞ is Mi1 and . . . and xnðtÞ is Min;
Then _xðtÞ ¼ AixðtÞþBiuðtÞþmi ð1Þ
where xT
ðtÞ ¼ ðx1ðtÞ x2ðtÞ Á Á Á xnðtÞÞ; Mi1,Mi2, . . . ,Min are fuzzy vari-
ables, Ri ði ¼ 1; 2, . . . ,rÞ denotes the ith fuzzy rule and r is the
number of rules. Ai, Bi, mi are constant matrices with appropriate
dimensions. In (1), it can be seen that the local subsystem includes a
constant affine term mi, which can approximate the original non-
linear system more accurately [12]. By using the fuzzy inference
method with a singleton fuzzifier, product inference, and center
average defuzzifiers, the overall affine T–S model is represented as
_xðtÞ ¼
Pr
i ¼ 1 wiðxðtÞÞðAixðtÞþBiuðtÞþmiÞ
Pr
i ¼ 1 wiðxðtÞÞ
¼
Xr
i ¼ 1
hiðxðtÞÞðAixðtÞþBiuðtÞþmiÞ, ð2Þ
where
wiðxðtÞÞ ¼
Yn
j ¼ 1
MijðxjðtÞÞ:
Denote
hiðxðtÞÞ ¼
wiðxðtÞÞ
Pr
i ¼ 1 wiðxðtÞÞ
then
0rhiðxðtÞÞr1,
Xr
i ¼ 1
hiðxðtÞÞ ¼ 1:
MijðxjðtÞÞ is the grade of membership of xj(t) in Mij, and hiðxðtÞÞ is said
to be the normalized membership function.
In this paper, the membership functions of the fuzzy proposi-
tions are trapezoidal, and we will partition the state-space into
operating regions and interpolation regions [25]. In each operat-
ing region, there exists some l such that hlðxðtÞÞ ¼ 1, and all other
membership functions evaluate to zero and the dynamic of the
system is given by _xðtÞ ¼ AlxðtÞþBluðtÞþml. In between operating
regions, there are interpolation regions where 0ohlðxðtÞÞo1, and
the system dynamics are given by a convex combination of
several affine systems. This decomposition of the state-space into
operating and interpolation regions will be central in our analysis.
Here, fSigi AF DRs
denotes a polyhedral partition of the state, and
F denotes the set of cell indexes. For each cell Si, the set K(i) contains
the indexes for the system matrices used in the interpolation within
that cell. For operating regions, K(i) contains a single element.
Furthermore, we let F0 DF be the set of indexes for cells that
contain origin and F1 DF be the set of indexes of the cells that do
not contain the origin. Hence, in each cell, the global system in (2)
can be expressed by a blending of mAKðiÞ subsystems:
_xðtÞ ¼ ~AixðtÞþ ~BiuðtÞþ ~mi, xðtÞASi, iAF, ð3Þ
where
~Ai ¼
X
mAKðiÞ
hmðxðtÞÞAm, ~mi ¼
X
mA KðiÞ
hmðxðtÞÞmm,
~Bi ¼
X
mA KðiÞ
hmðxðtÞÞBm, hmðxðtÞÞ40,
X
mA KðiÞ
hmðxðtÞÞ ¼ 1:
2.2. Problem statement
In this paper, we assume that the states x(t) of system (3) can
be measured and used to design controllers. The piecewise state
feedback controller is designed as follows:
uðtÞ ¼ KixðtÞþsi, xðtÞASi, iAF: ð4Þ
The value of si in (4) is assumed to be 0 while the cells contain
origin. Applying the state feedback controller (4) to system (3),
the resulting closed-loop system is given by
_xðtÞ ¼ ð ~Ai þ ~BiKiÞxðtÞþð ~Bisi þ ~miÞ xðtÞASi, iAF: ð5Þ
Then, the purpose of this paper is to design a piecewise state
feedback controller described by (4) such that the resulting
closed-loop system, which is given in (5), is quadratically stable.
3. Preliminaries
To facilitate control system design, the following lemmas are
presented and will be used in the later developments.
Lemma 1. Let F be a symmetric matrix, P be a positive-definite
matrix. The following statements are equivalent:
(a) FþPAþAT
Po0.
(b) For a large enough constant a40, there exist matrices F, P such that
FÀ2aP PþðAþaIÞT
F
PþFT
ðAþaIÞ ÀFÀFT
" #
o0: ð6Þ
Proof. ðaÞ ) ðbÞ: If the statement (a) is satisfied, there exists a
sufficiently large positive scalar a1 such that 8a4a1
fþPAþAT
Pþ
1
2a
AT
PAo0:
H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777772

3. Rewrite the above inequality as
FÀ2aPþ
a
2
Pþ
1
a
ðAþaIÞT
P
PÀ1
Pþ
1
a
PðAþaIÞ
o0
by applying Schur complement, we get
FÀ2aP Pþ 1
a ðAþaIÞT
P
Pþ 1
a PðAþaIÞ À 2
a P
#
o0:
By taking F ¼ ð1=aÞP, then the statement (b) is obtained.
ðbÞ ) ðaÞ: Multiplying (6) by ½I ðAþaIÞT
Š from the left and by its
transpose from the right leads to the statement (a). Then the proof
is completed.
Remark 1. (1) Note that the statement (b) remains sufficient for
the statement (a) by using the property of congruence with the
matrix ½I ðAþaIÞT
Š while some constraints are imposed on F.
(2) This lemma provides a new dilated LMI characterization,
and while Fo0, through setting a ¼ 0, it can be reduced to the
result in [23].
Lemma 2. The continuous affine fuzzy system (3) is quadratically
stable with u 0 in the large if there exist a common positive definite
matrix P ¼ PT
and tiq Z0 ðiAF1; q ¼ 1; 2, . . . ,nÞ such that
AT
mPþPAm o0 ð7Þ
for iAF0, mAKðiÞ and
AT
mPþPAmÀ
Pn
q ¼ 1 tiqTiq PmmÀ
Pn
q ¼ 1 tiquiq
mT
mPÀ
Pn
q ¼ 1 tiquT
iq À
Pn
q ¼ 1 tiqviq
2
4
3
5o0 ð8Þ
for iAF1, mAKðiÞ, where Tiq, uiq, viq are defined in Appendix such
that
FiqðxðtÞÞ xT
ðtÞTiqxðtÞþ2uT
iqxðtÞþviq o0
while xðtÞASi, iAF.
Proof. The proof procedure is similar to Theorem 1 in [12], and it
is omitted here.
By Lemma 1, the following novel quadratic stability analysis
result is derived, which is equivalent to Lemma 2.
Lemma 3. The conditions of Lemma2 are feasible if and only if there
exist a common positive definite matrix P ¼ PT
, matrices Wi, Fi (iAF)
with appropriate dimensions, and scalars tij Z0 ðiAF1; j ¼ 1; 2, . . . ,nÞ
such that, for large enough positive constants a,k
À2kP PþðAm þkIÞT
Wi
PþWT
i ðAm þkIÞ ÀWiÀWT
i
#
o0 ð9Þ
for iAF0, mAKðiÞ and
FiÀ2aP P þðbAm þaIÞT
Fi
P þFT
i ðbAm þaIÞ ÀFiÀFT
i
2
4
3
5o0 ð10Þ
for iAF1, mAKðiÞ, where
Fi ¼
À
Pn
j ¼ 1 tijTij À
Pn
j ¼ 1 tijuij
À
Pn
j ¼ 1 tijuT
ij À
Pn
j ¼ 1 tijvij
2
4
3
5, P ¼
P 0
0 1
,
bAm ¼
Am mm
0 0
: ð11Þ
Proof. Rewriting (8) as Fi þP bAm þ bA
T
mP o0, where Fi,bAm,P are
defined as (11), and applying Lemma 1, we can obtain this
conclusion immediately.
4. Quadratic stabilizing controller design
From Lemma 3, the novel sufficient condition under which the
closed-loop fuzzy system (5) is quadratically stable is given in the
following forms.
Lemma 4. The continuous affine fuzzy system (5) is quadratically
stable in the large if there exist P40, matrices Fi, Wi ðiAFÞ with
appropriate dimensions, and scalars tij Z0 ðiAF1; j ¼ 1; 2, . . . ,nÞ
such that, for large enough positive constants a,k
À2kP PþðAm þBmKi þkIÞT
Wi
PþW
T
i ðAm þBmKi þkIÞ ÀWiÀW
T
i
2
4
3
5o0 ð12Þ
for iAF0, mAKðiÞ and
FiÀ2aP P þðAim þaIÞT
Fi
P þFT
i ðAim þaIÞ ÀFiÀFT
i
#
o0 ð13Þ
for iAF1, mAKðiÞ, where
Aim ¼
Am þBmKi Bmsi þmm
0 0
ð14Þ
and Fi,P are defined as (11).
It can be seen that the conditions of Lemma 4 are with non-
convex forms because of controller gains on one side and free
matrices Fi (or Wi) on the other side. Due to the S-procedure, i.e.,
Fi, it is difficult to convert the non-convex conditions such as the
inequality (13) to convex ones by the equivalent transformation.
To overcome this difficulty, the following assumption is consid-
ered here:
Assumption 1. Assume that Bl,l ¼ 1; 2, . . . r, are of full column
rank, and let invertible matrices Tl,l ¼ 1; 2, . . . r, such that
TlBl ¼
ImÂm
0
for l ¼ 1; 2, . . . r: ð15Þ
Remark 2. For each Bl, the corresponding Tl generally is not
unique. A special Tl can be obtained by
Tl ¼
ðBT
l BlÞÀ1
BT
l
P
#
and
P
is a matrix composed of the rows which are mutually
independent and perpendiculars to the columns of Bl.
Based on Lemma 4 and Assumption 1, a new quadratic
stabilizability condition for the affine fuzzy system (3) is obtained
as follows.
Theorem 1. The continuous affine fuzzy system (3) is quadratically
stabilizable in the large by the piecewise controller (4) if there exist
P ¼ PT
40, matrices Wi, Fi1, Fi2, Fi3 with appropriate dimensions,
where
Wi ¼
Wi1 0
Wi2 Wi3
#
, Fi1 ¼
Fi11 0
Fi21 Fi22
#
H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777 773

4. and Yi ARmÂn
,Vi ARmÂ1
, scalars tij Z0 ðiAF; j ¼ 1; 2, . . . ,nÞ such
that, for large enough positive constants a,k
À2kP PþAT
mTT
mWi þkT
T
mWi þ
Yi
0
T
n ÀTT
mWiÀWT
i Tm
2
6
6
4
3
7
7
5o0 ð16Þ
for iAF0, mAKðiÞ, and
À
Xn
j ¼ 1
tijTijÀ2aP À
Xn
j ¼ 1
tijuij PþAT
mTT
mFi1 þ
Yi
0
T
þaTT
mFi1 0
n À
Xn
j ¼ 1
tijvijÀ2a
Vi
0
T
þmT
mTT
mFi1 þaFi2 1þaFi3
n n ÀTT
mFi1ÀFT
i1Tm ÀFT
i2
n n n ÀFi3ÀFT
i3
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
o0
ð17Þ
for iAF1, mAKðiÞ. The local gains of the fuzzy controller are given by
Ki ¼ WÀT
i1 Yi, iAF0; Ki ¼ FÀT
i11Yi, si ¼ FÀT
i11Vi, iAF1:
Proof. Assuming
Wi ¼
X
lA KðiÞ
flTT
l Wi, Fi ¼
P
lA KðiÞflTT
l Fi1 0
Fi2 Fi3
#
with appropriate dimensions, where
Wi ¼
Wi1 0
Wi2 Wi3
#
, Fi1 ¼
Fi11 0
Fi21 Fi22
#
,
and the function fl is defined as
fl ¼
1, l ¼ m,
0, else:
(
Then we have
TmBm ¼
ImÂm
0ðnÀmÞÂm
#
:
Substituting them to (12) and (13), the conditions (16) and (17)
are obtained.
Remark 3. It is noted that in [19], the Lyapunov matrix P is
limited to a diagonal structure. In this paper, by applying the
proposed decoupling technique, the derived result does not need
to impose any constraint on the Lyapunov matrix P. In addition,
the results in [19] are obtained based on the assumption that the
input matric B is a common one, this is a much simpler system,
and leads to much narrower scope of the applicability. By
introducing extra slack variables Fi, Wi, the proposed method in
this paper can deal with more general systems whose input
matrices are uncommon, and the controller design conditions
are obtained in the formulation of linear matrix inequalities.
5. H1 controller design
In this section, we will investigate the H1 control synthesis
problem for a class of affine fuzzy systems. The objective of this
section is to design a suitable controller based on the approach
proposed in the above section, such that the induced L2-norm of
the operator from disturbance w(t) to the controlled output z(t) is
less than under zero initial conditions
JzðtÞJ2 ogJwðtÞJ2
for all nonzero wðtÞAL2.
Consider the following affine fuzzy system:
_xðtÞ ¼
Xr
l ¼ 1
hlðxðtÞÞðAlxðtÞþml þB1luðtÞþB2lwðtÞÞ,
zðtÞ ¼ LxðtÞ, ð18Þ
where xðtÞARn
is the state vector, uðtÞARp
is the control input,
wðtÞARm
is the disturbance input which is assumed to belong
to L2½0,1Þ, and zðtÞARq
is the controlled output. ½AlŠnÂn, ½mlŠnÂ1,
½B1lŠnÂp, ½B2lŠnÂm, ½LŠqÂn ðl ¼ 1; 2, . . . ,rÞ are constant matrices, and
B1ls are of full column rank.
Similar to the partition in Section 2, the global system in (18)
can be expressed by a blending of mAKðiÞ subsystems
_xðtÞ ¼ ~AixðtÞþ ~mi þ ~B1iuðtÞþ ~B2iwðtÞ,
zðtÞ ¼ LxðtÞ, xðtÞASi, iAF, ð19Þ
where
~Ai ¼
X
mAKðiÞ
hmðxðtÞÞAm, ~mi ¼
X
mA KðiÞ
hmðxðtÞÞmm,
~B1i ¼
X
mAKðiÞ
hmðxðtÞÞB1m
~B2i ¼
X
mAKðiÞ
hmðxðtÞÞB2m,
hmðxðtÞÞ40,
X
mAKðiÞ
hmðxðtÞÞ ¼ 1:
Lemma 5. The affine fuzzy system (19) with u 0 is quadratically
stable with a guaranteed H1 disturbance attenuation level g if
there exist a symmetric matrix P40 and scalars tiq Z0 ðiAF; q ¼
1; 2, . . . ,nÞ such that
AT
mPþPAm þLT
L PB2m
BT
2mP Àg2
I
#
o0 ð20Þ
for iAF0, mAKðiÞ, and
AT
mPþPAm þLT
LÀ
Pn
q ¼ 1 tiqTiq PB2m PmmÀ
Pn
q ¼ 1 tiquiq
n Àg2
I 0
n n À
Pn
q ¼ 1 tiqviq
2
6
6
4
3
7
7
5o0
ð21Þ
for iAF1, mAKðiÞ; where Tiq, uiq, viq are defined as in Appendix.
Proof. Consider the following Lyapunov function:
VðxðtÞÞ ¼ xT
ðtÞPxðtÞ:
It is well known that it suffices to show the following inequality:
_V ðxðtÞÞþzT
ðtÞzðtÞÀg2
wT
ðtÞwðtÞo0
to prove that the affine fuzzy system (19) is asymptotically stable
with a given H1 performance g under zero initial conditions.
Based on the above Lyapunov function, we get
_V ðxðtÞÞþzT
ðtÞzðtÞÀg2
wT
ðtÞwðtÞ
¼
xðtÞ
wðtÞ
1
2
6
4
3
7
5
T
~A
T
i PþP ~Ai þLT
L P ~B2i P ~mi
n Àg2
I 0
n n 0
2
6
6
4
3
7
7
5
xðtÞ
wðtÞ
1
2
6
4
3
7
5:
Then, by taking into consideration the partition information with
iAF1, we have
_V ðxðtÞÞþzT
ðtÞzðtÞÀg2
wT
ðtÞwðtÞr
xðtÞ
wðtÞ
1
2
6
4
3
7
5
T
H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777774

5. ~A
T
i PþP ~Ai þLT
LÀ
Pn
q ¼ 1 tiqTiq P ~B2i P ~miÀ
Pn
q ¼ 1 tiquiq
n Àg2
I 0
n n À
Pn
q ¼ 1 tiqviq
2
6
6
6
4
3
7
7
7
5
xðtÞ
wðtÞ
1
2
6
4
3
7
5:
By expanding the fuzzy-basis functions, then the condition (21) is
obtained. Similarly, for the case iAF0, the condition (20) can be easily
obtained where ~mi ¼ 0 and the S-procedure is not involved.
Theorem 2. The affine fuzzy system (19) with the state feedback
control law (4) is quadratically stabilizable with a guaranteed H1
disturbance attenuation level g if there exist P40, matrices Wi, Fi11,
Fi21, Fi22, Fi31, Fi32, Fi33 with appropriate dimensions, where
Fi11 ¼
Fi1 0
Fi2 Fi3
#
, Wi ¼
Wi1 0
Wi2 Wi3
#
,
Yi ARmÂn
, Vi ARmÂ1
and scalars tiq Z0 ðiAF; q ¼ 1; 2, . . . ,nÞ such
that, for large enough positive constants a,k
xi
11 n n n n n
0 Àg2
IÀ2aI n n n n
xi
31 0 xi
33 n n n
xim
41 xim
42 xim
43 ÀTT
mFi11ÀFT
i11Tm n n
0 IþaFT
i22 aFT
i32 ÀFi21 ÀFi22ÀFT
i22 n
0 0 1þaFT
i33 ÀFi31 ÀFi32 ÀFi33ÀFT
i33
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
o0
ð22Þ
for iAF1, mAKðiÞ, and
LT
LÀ2kP n n n
0 Àg2
IÀ2kI n n
Lim
31 WT
i1TmB2m þkW
T
i2 ÀTT
mWi1ÀWT
i1Tm n
0 IþkW
T
i3 ÀWi2 ÀWi3ÀWT
i3
2
6
6
6
6
6
4
3
7
7
7
7
7
5
o0
ð23Þ
for iAF0, mAKðiÞ, where
xi
11 ¼ LT
LÀ
Xn
q ¼ 1
tiqTiqÀ2aP,
xi
31 ¼ À
Xn
q ¼ 1
tiquT
iq,
xi
33 ¼ À
Xn
q ¼ 1
tiqviqÀ2a,
xim
41 ¼ PþaFT
i11Tm þFT
i11TmAm þ
Yi
0
,
xim
43 ¼ FT
i11Tmmm þ
Vi
0
þaFT
i31,
xim
42 ¼ FT
i11TmB2m þaFT
i21,
Lim
31 ¼ PþkW
T
i1Tm þWT
i1TmAm þ
Yi
0
:
The controller gains are given by Ki ¼ WÀT
i1 Yi, iAF0; Ki ¼ FÀT
i1 Yi,
si ¼ FÀT
i1 Vi, iAF1.
Proof. The proving procedure is similar to Theorem 1, thus its
details are omitted here.
Remark 4. When k and a in (16), (17) and (22), (23) are set to be
fixed parameters, the problems become convex and can be solved
by employing the LMI Toolbox. To find the optimal values of
corresponding parameters, in this paper, we will first solve the
feasibility problem of LMIs (16), (17) and (22), (23) by using LMI
Toolbox under a set of initial scaling parameters. Then, for the
optimization problem, applying a numerical optimization algo-
rithm, such as the program fminsearch, and then a locally
convergent solution to the problem is obtained.
Remark 5. The results given in this paper are obtained based on
the assumption that the input matric Bls are of full column rank,
which leads to much wider scope of the applicability than the
common input matrix. Nevertheless, if there exists invertible
matrices Tl,l ¼ 1; 2, . . . r, such that
TlBl ¼
IncÂnc
0
0 0
for l ¼ 1; 2, . . . ,r,
where nc om, which means that the input matrices are not full
column rank, and then a novel piecewise affine state feedback
controller is desirable with the form
uðtÞ ¼ ~KixðtÞþ ~si, xðtÞASi, iAF, ð24Þ
where
~Ki ¼
Knc
i
0
#
, ~si ¼
snc
i
0
#
:
The corresponding state feedback control synthesis method is
proposed as follows.
Corollary 1. The continuous affine fuzzy system (3) is quadratically
stabilizable in the large by the piecewise controller (24) if there exist
P ¼ PT
40, matrices Wi, Fi1, Fi2, Fi3 with appropriate dimensions,
where
Wi ¼
Wi1 0
Wi2 Wi3
#
, Fi1 ¼
Fi11 0
Fi21 Fi22
#
and Yi ARncÂn
, Vi ARncÂ1
, scalars tij Z0 ðiAF; j ¼ 1; 2, . . . ,nÞ such
that, for large enough positive constants a,k, the linear matrix
inequalities in (16) and (17) are satisfied, and the local gains of the
fuzzy controller are given by
Knc
i ¼ WÀT
i1 Yi, iAF0; Knc
i ¼ FÀT
i11Yi, snc
i
¼ FÀT
i11Vi, iAF1:
6. Examples
6.1. Example 1
The following example illustrates the merits of the new stability
condition (Theorem 1) compared to the approach in [19]. Consider a
continuous affine fuzzy plant composed of the following three rules:
Ri: If x3ðtÞ is Mi, then _xðtÞ ¼ AixðtÞþBuðtÞþmi; i ¼ 1; 2,3 where
A1 ¼ A3 ¼
0 1þb 4À 2
p ðEþ1Þ
À1 À1 À1þ 2
p E
0 1 1À 2
p E
2
6
6
4
3
7
7
5,
m1 ¼ Àm3 ¼
À2ðEþ1Þ
2E
À2E
2
6
4
3
7
5, B ¼
0
0
1
2
6
4
3
7
5,
A2 ¼
0 1þa 4þ 2
p ðEþ1Þ
À1 À1 À1À 2
p E
0 1 1þ 2
p E
2
6
6
4
3
7
7
5, m2 ¼
0
0
0
2
6
4
3
7
5,
xðtÞ ¼ ðx1ðtÞ x2ðtÞ x3ðtÞÞT
, and E is set to 0.5. The membership
functions M1,M2 and M3 are depicted in Fig. 1. The parameters a
in A2, and b in A1,A3, will take values in a prescribed grid, in order
H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777 775

6. to check the feasibility of the associated fuzzy control synthesis
problem under two different approaches.
With the space partition defined in this paper, we can have the
cells
S1 ¼ xðtÞ9À
3p
2
rxðtÞrÀ
p
2
, S2 ¼ xðtÞ9À
p
2
rxðtÞrÀ
2p
5
,
S3 ¼ xðtÞ9À
2p
5
rxðtÞr
2p
5
, S4 ¼ xðtÞ9
2p
5
rxðtÞr
p
2
,
S5 ¼ xðtÞ9
p
2
rxðtÞr
3p
2
:
As stated earlier, the piecewise controller is designed as follows:
uðtÞ ¼ KixðtÞþsi ði ¼ 1; 2,3; 4,5Þ
and s3 ¼ 0 because of F0 ¼ f3g in this example.
Fig. 2 shows the parameter regions where the stability of the
fuzzy control system is ensured by respectively using the controller
synthesis method in [19] and the method proposed in this paper
with a ¼ k ¼ 1. In this figure, the  mark indicates the existence of
feasible stabilizing regulators proved by [19] (and, of course, also by
Theorem 1 in this paper); the J mark indicates parameter values for
which stabilizability is proved in this paper, but not in [19]. It is
apparently seen that our stabilization region is larger than the results
in [19].
6.2. Example 2
In this section, the following affine fuzzy system with uncom-
mon input matrices is considered. It is noted that, the approaches
given in [19,24] cannot be applied for this class of systems.
Ri: If x1ðtÞ is Mi, Then
_xðtÞ ¼ AixðtÞþmi þB1iuðtÞþB2iwðtÞ,
zðtÞ ¼ LxðtÞ, i ¼ 1; 2,3,
where
A1 ¼ A3 ¼
0:9817 1 0
À0:5317 À1 À1
3:0451 1:7320 0
2
6
4
3
7
5, m1 ¼ Àm3 ¼
À0:9831
1:0347
À2:5836
2
6
4
3
7
5,
A2 ¼
1:3183 1 0
À1:2883 À1 À1
4:9549 1 0
2
6
4
3
7
5, m2 ¼
0
0
0
2
6
4
3
7
5,
B11 ¼ B13 ¼
1
0
0
2
6
4
3
7
5, B12 ¼
1
0:5
0
2
6
4
3
7
5,
L ¼ ½0 1 0Š, B21 ¼
0:2
0
0:1
2
6
4
3
7
5, B22 ¼
0:1
0:2
0
2
6
4
3
7
5, B23 ¼
0:3
0:1
1
2
6
4
3
7
5:
xðtÞ ¼ ðx1ðtÞ x2ðtÞ x3ðtÞÞT
. The membership functions M1,M2 and M3
are depicted in Fig. 3.
The objective is to design a piecewise state feedback controller in
the form of (4) such that the resulting closed-loop system is
asymptotically stable with H1 performance g under wðtÞ ¼ 15eÀ0:1t
sinð20ptÞ and zero initial conditions. By applying Theorem 2 and the
fminsearch, we can get a piecewise state feedback controller with the
controller gains are given as follows. Moreover, the optimal scaling
Fig. 1. Membership functions.
-2 -1 0 1 2 3 4
0
1
2
3
4
5
6
a
b
Fig. 2. Stabilization region based on Theorem 3 in [19] ( Â ) and Theorem 1 in this
paper with a ¼ k ¼ 1 ( Â ,J).
Fig. 3. Membership functions.
0 5 10 15 20 25 30 35 40 45 50
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
x(1)
x(2)
x(3)
Fig. 4. State responses of the closed-loop system.
H. Wang, G.-H. Yang / ISA Transactions 51 (2012) 771–777776

7. parameters are obtained as a ¼ 1:3710 and k¼0.8275 with the initial
values a ¼ k ¼ 0:5, and the optimal H1 performance g ¼ 0:8449.
K1 ¼ ½À7:8209À1:3782À0:1890Š, s1 ¼ 0:9831,
K2 ¼ ½À21:7245À2:3324À0:8507Š, s2 ¼ 0:6985,
K3 ¼ ½À7:9316À1:3810À0:1591Š, s3 ¼ 0,
K4 ¼ ½À21:8990À2:3496À0:8616Š, s4 ¼ À0:7012,
K5 ¼ ½À7:9193À1:3906À0:2011Š, s5 ¼ À0:9831:
To show the effectiveness of the obtained results, simulations
have been carried out. Fig. 4 shows the state responses of the
corresponding closed-loop system under initial conditions xð0Þ ¼
½À1:5 2 3ŠT
.
7. Conclusion
In this paper, the problem of control synthesis for a class of
continuous-time affine fuzzy systems has been discussed. First, a
lemma is provided to decouple the system matrix and the Lyapunov
matrix, and then a new stability analysis condition based on the
lemma is presented and the corresponding controller design method
is obtained in the formulation of LMIs in conjunction with a search
of scaling parameters, which can provide more relaxed conditions
and deal with more general systems than the existing results.
Furthermore, an extended H1 performance analysis of a class of
affine fuzzy systems is presented, and the H1 controller synthesis
condition is derived. Finally, two examples are used to illustrate the
superiority and effectiveness of the proposed method.
Although this paper addressed the convex quadratic stability
and stabilizability of the affine fuzzy system, the common
Lyapunov function could lead to very conservative results. To
reduce the conservatism, the results on dynamic output feedback
controller design for discrete-time affine fuzzy systems based on
piecewise Lyapunov function have been obtained in [26], and the
works on analogous design and synthesis method based on
piecewise Lyapunov function for continuous-time affine fuzzy
system are under investigation.
Acknowledgement
This work was supported in part by the Funds for Creative
Research Groups of China (no. 60821063), National 973 Program
of China (Grant no. 2009CB320604), the Funds of National Science
of China (Grant no. 60974043), the Funds of Doctoral Program of
Ministry of Education, China (20100042110027).
Appendix
The range of x(t) is represented by the following system of at
most n linear inequalities while xðtÞASi:
for x1ðtÞ, x1ðtÞrai1 or x1ðtÞZbi1 or ai1 rx1ðtÞrbi1
for x2ðtÞ, x2ðtÞrai2 or x2ðtÞZbi2 or ai2 rx2ðtÞrbi2
^
for xn(t), xnðtÞrain or xnðtÞZbin or ain rxnðtÞrbin
Thus, the S-procedure is obtained with FiqðxðtÞÞ xT
ðtÞTiqxðtÞþ
2uT
iqxðtÞþviq r0, and the structure of matrices Tiq, uiq, viq is similar
to [12].
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