Controller Design Based On Fuzzy Observers For T-S Fuzzy Bilinear Models

International Journal of Control Theory and Computer Modeling (IJCTCM) Vol.6, No.1/2, April 2016
DOI : 10.5121/ijctcm.2016.6201 1
CONTROLLER DESIGN BASED ON FUZZY
OBSERVERS FOR T-S FUZZY BILINEAR MODELS
Dhikra Saoudi
LSA - Laboratoire des Systèmes Avancés,
Polytechnic High School of Tunisia, University of Carthage,
BP 743, 2078 La Marsa, Tunisia.
ABSTRACT
This article is devoted to the design of a fuzzy-observer-based control for a class of nonlinear systems with
bilinear terms. The class of systems considered is the Takagi-Sugeno (T-S) fuzzy bilinear model. A new
procedure to design the observer-based fuzzy controller for this class of systems is proposed. The aim is to
design the fuzzy controller and the fuzzy observer of the augmented system separately in order to guarantee
that the error between the state and its estimation converges faster to zero. By using the Lyapunov function,
sufficient conditions are derived such that the closed-loop system is globally asymptotically stable.
Moreover, sufficient conditions for controller design based on state estimation for robust stabilization of T-
S FBS with parametric uncertainties is proposed. The observer and controller gains are obtained by
solving some linear matrix inequalities (LMIs). An example is given to illustrate the effectiveness of the
proposed approach.
Keywords
T-S fuzzy bilinear model, Fuzzy observer-based control, Lyapunov function, Parametric uncertainty, LMI.
1. INTRODUCTION
In recent years, the stability analysis and the synthesis of controllers/observers for nonlinear
systems have been the subject of many research works. In recent years, the stability analysis and
the synthesis of controllers/observers for nonlinear systems are still open problems owing to their
nature complexity and have been the subject of many research works. It is well known that
Takagi-Sugeno (T-S) fuzzy model is a modelling method [1] that can represent or approximate a
large class of nonlinear systems by a set of local linear dynamics [2]. Various results have been
devoted to the stability analysis and stabilization of T-S fuzzy systems [3]-[7]. These results use
different techniques such as the system parameters to be known but in reality the parameters of
the system can be either uncertain or time-dependent. Moreover, the problem of stabilization
remains an important issue in the controller designs of uncertain nonlinear systems [8], [9]. It is
also necessary to consider the robust stability of uncertain T-S fuzzy models in order to guarantee
both stability and the robustness with respect to the latter. The study of robustness in fuzzy
model-based control has been studied widely in the literature for nonlinear systems. For example,
fuzzy model-based control for T-S fuzzy models has been studied in [10]-[12] and for T-S fuzzy
models with parametric uncertainties has been given in [13]-[15]. Moreover, it is noted that all of
the aforementioned assume that the system states are measured, which is not true in many control
systems and real applications. For this reason, observer-based fuzzy controllers were considered

2
in many researches. In [16]-[18], the authors have studied the controller designs based on fuzzy
observers for T-S fuzzy systems. Nonetheless, in [19]-[21], the authors have proposed sufficient
design conditions for robust stabilization of T-S fuzzy models with parametric uncertainties based
on state estimation.
In recent years, the fuzzy bilinear systems (FBS) based on the T-S fuzzy model with bilinear rule
consequence were considered in [22]-[24]. It is proved in these papers that often nonlinear
behaviours can be approximated by T-S bilinear multimodel description. This modeling method
is based on the bilinearization of the nonlinear system around some operating points and using
adequate weighting functions. This kind of T-S fuzzy model is especially suitable for a nonlinear
system with a bilinear term. Some topics of control are extended to T-S fuzzy bilinear model
(FBM). For example; in [22] proposed a fuzzy controller to stabilize the FBS and also for the
fuzzy bilinear systems with parametric uncertainties. In [23], [26], the authors studied the robust
∞H control problem for T-S FBS. In [25], the authors developed a static output feedback
controller for T-S discrete fuzzy bilinear systems (DFBS). Furthermore, the stabilization
problems of T-S FBS and T-S DFBS with time-delay have been proposed in [27],[28]. Moreover,
some studies have investigated the observer problem for T–S fuzzy bilinear systems [29]-[34] and
application to fault diagnosis and fault tolerant control, have been examined for this class of
systems in [35]-[37]. Unfortunately, to the best of our knowledge, no previous study has
investigated the problem of an observer-based control for T-S fuzzy bilinear models and a robust
stabilization for uncertain T-S fuzzy models.
In this paper, an observer-based control design for uncertain nonlinear systems in order to
guarantee the closed-loop stability is proposed. The T-S FBM is employed to represent a
nonlinear system with parametric uncertainties. This kind of T-S fuzzy model is especially
suitable for a nonlinear system with a bilinear term. The goal is to design the fuzzy controller and
the fuzzy observer of the augmented system separately in order to guarantee that the error
between the state and its estimation converges faster to zero. By employing the Lyapunov
function, design conditions are derived. The gains of the observer and the controller are obtained
by solving some linear matrix inequalities. So, this paper brings some results for the observer-
based control design dedicated to fuzzy bilinear models and uncertain fuzzy bilinear models.
This paper is organized as follows. In section 2, the considered structure of the T-S fuzzy bilinear
system is presented. In Section 3, the problem of observer-based control for the fuzzy bilinear
systems is developed. In section 4, the method of controller design based on state estimation for
robust stabilization of T-S FBS with parametric uncertainties is proposed. an illustrative example
is provided to show the effectiveness of the proposed approach in Section 5. Finally, Section 6
concludes the paper.
2. T-S FUZZY BILINEAR MODEL REPRESENTATION
In a fuzzy modelling framework, the fuzzy bilinear models based on the T-S fuzzy model with
bilinear rule consequence were an extension of the T-S fuzzy ordinary model. It is proved that
often nonlinear behaviours can be approximated by T-S fuzzy bilinear model description. This
technique is based on the bilinearization of the nonlinear system around some operating points
and using adequate weighting functions. This kind of T-S fuzzy model is especially suitable for a
nonlinear system with a bilinear term. Then, the fuzzy bilinear model is described by the
following fuzzy if-then rules:
1 1: if ( ) is and and ( ) isLi
i g igR t F t Fξ ξ

3
then
( ) ( ) ( ) ( )u(t)
( ) ( )
= + +

=
& i i ix t A x t Bu t N x t
y t Cx t (1)
where iR denotes the th
i fuzzy rule $ { }1, ,∀ = Li r , r is the number of if-then rules, ( )i tξ are
the premise variables which can be measurable or not measurable, and ( (t))ij jF ξ are fuzzy set.
( )∈ℜn
x t is the state vector, ( )∈ℜu t is the control input, and ( )∈ℜp
y t is the system output.
The matrices , , ,i i iA B N C are known matrices.
Then, the overall FBM can be described as follows:
( )
1
( ) ( ( )) ( ) ( ) ( )u(t)
( ) ( )
=

= + +

 =
∑&
r
i i i i
i
x t h t A x t Bu t N x t
y t Cx t
ξ
(2)
where (.)ih verify the following properties:
{ }1
( ( )) 1
1,2, ,
0 ( ( )) 1
=

=
∀ ∈
 ≤ ≤
∑ L
r
i
i
i
h t
i r
h t
ξ
ξ
(3)
Remark 1: Matrices , ,i i iA B N and C can be obtained by using the polytopic
transformation [2]. The advantage of this method is in one hand to lead to a bilinear
transformation of the nonlinear model without any approximation error, and in another hand to
reduce the number of local models compared to other methods [22].
3. CONTROL OF THE FUZZY BILINEAR MODEL
In this section, we will develop the observer-based control problem for the fuzzy bilinear systems.
Thereafter, we will propose sufficient conditions that guarantees the closed-loop stability by
using a Lyapunov function-based design approach. The goal is to design the fuzzy controller and
the fuzzy observer of the augmented system separately in order to guarantee that the error
between the state and its estimation converges faster to zero.
Many studies consider that all the state variables of the system are available. However in practice
this assumption often does not hold. Hence, we need to define a fuzzy observer to estimate the
states and make an analysis of the error system, from which we provides a design method of a
fuzzy observer for the fuzzy bilinear system (2).
Then, the proposed structure of the fuzzy bilinear observer has the following form:. Then, the
proposed structure of the fuzzy bilinear observer has the following form:
( )( )
1
ˆ ˆ ˆ ˆ( ) ( ( )) ( ) ( ) ( )u(t) ( ) ( )
ˆ ˆ( ) ( )
=

= + + + ϒ −

 =
∑&
r
i i i i i
i
x t h t A x t Bu t N x t y t y t
y t Cx t
ξ
(4)

4
where ˆ( )x t is the estimated state vector of ( )x t , and , 1, ,ϒ = Li i r are observer parameters to
be determined.
In this paper, it is assumed that all of the premise variables do not depend on the state variables
estimated by a fuzzy observer.
Following the design concept in [38], the fuzzy control law for FBM is formulated as follows:
( )
( )
( )
1
1
1
ˆ( )
( ) ( )
ˆ ˆ1 ( ) ( )
ˆ( ) sin ( )
ˆˆ( ) ( )cos ( )
=
=
=
=
+
=
=
∑
∑
∑
r
i
i T T
i
i i
r
i i
i
r
i i i
i
D x t
u t h t
x t D D x t
h t t
h t D x t t
ρ
ξ
ξ ρ θ
ξ ρ θ
(5)
where:
ˆ( ) 1ˆ ˆsin ( ) , cos ( )
ˆ ˆ ˆ ˆ1 ( ) ( ) 1 ( ) ( )
ˆ ,
2 2
= =
+ +
 
∈ − 
 
i
i iT T T T
i i i i
i
D x t
t t
x t D D x t x t D D x t
θ θ
π π
θ
ρ is a given scalar, and 1
, 1, ,r×
∈ℜ = Ln
iD i are vectors to be determined.
Let us define the state estimation error:
( ) ( ) ( )ˆ= −e t x t x t (6)
Then, one can obtain the fuzzy observer error dynamics from (2), (4), and (5):
( ) ( ) ( )( )1 1
ˆ( ) ( ) ( ) + sin ( ) ( )
= =
= + ϒ∑∑&
r r
i j i i i j
i j
e t h t h t A C N t e tξ ξ ρ θ (7)
Let us define the augmented system containing both the fuzzy controller and observer
  
TT T
x e which dynamics is given by the augmented system:
( )
( ) ( )
( )
( ) ( )
   
=   
   
&
&
x t x t
A t
e t e t
ξ (8)

5
where the matrices ( )( )A tξ are given by:
( ) ( ) ( )
1 1
( ) ( ) ( )
= =
= ∑∑
r r
i j ij
i j
A t h t h t Aξ ξ ξ
with:
ˆ ˆ ˆsin cos cos
ˆ0 sin
 + +
=  
+ ϒ +  
i i j i j j i j j
ij
i i i j
A N B D B D
A
A C N
ρ θ ρ θ ρ θ
ρ θ
The goal is to design the observer-based fuzzy controller (5) for the fuzzy bilinear system (2)
such that the closed-loop augmented system (8) becomes globally asymptotically stable.
Therefore, the main result is then propounded in the following theorem.
Theorem 1: If there exist symmetric positive definite matrices 1P and 2P , matrices ,i iW V and
positive scalars 0 , 0> >ε ρ such that LMIs (9) and (10) are satisfied, then the augmented
system described by (8) is asymptotically stable 1 , 1∀ = ∀ =L Li r j r
1 1
1
* * * *
* * *
00 * *
0 0 *
0 0 0
 +
 
− 
  <−
 
− 
 − 
T
i i
i
i j
T
i
A P PA
N P I
BV I
I I
B I
ρ
ε
(9)
2 2
2
1
* * *
* *
0
0 *
0 0 −
 + + +
 
−  <
 −
 
−  
T T T
i i i i
i
j
P A A P WC C W
P N I
I I
D I
ρ
ε
(10)
The gains of the observer and the state feedback control law are given by:
1
1
−
=i iD V P (11)
1
2
−
ϒ =i iP W (12)
Proof: In order to ensure the global, asymptotic stability, a new procedure for designing the
observer-based fuzzy controller (5) for the fuzzy bilinear system (2) is proposed such that the
closed-loop system (8) becomes asymptotically robust stable by using a Lyapunov function-based
design approach. Then from (8), the sufficient conditions must be verified:
( )0 , , 0∃ = > = + <T D T
ij ij ij ijQ Q M A Q A Q QA (13)

6
Where
11
22
0
0
 
=  
 
Q
Q
Q
(14)
Let us define
( ) 1 2
, = +D D D
ij ij ij ijM A Q M M (15)
with
11 21
1 2
12 22
0 0
,
0 0
   
= =   
      
ij ijD D
ij ij
ij ij
D D
M M
D D
(16)
( )
( )
( )
11
11 11 11 11
11 11
12
22 22 22 22
22 22
21
22
22
22
ˆ ˆsin sin
ˆ ˆcos cos
ˆ ˆsin sin
ˆcos
ˆcos
= + + +
+ +
= + + +
+ ϒ + ϒ
=
=
T T
ij i i i j i j
T
i j j i j j
T T
ij i i i j i j
T
i i
ij i j j
T
ij i j j
D AQ Q A N Q Q N
Q B D B D Q
D AQ Q A N Q Q N
CQ Q C
D B D Q
D Q B D
ρ θ ρ θ
ρ θ ρ θ
ρ θ ρ θ
ρ θ
ρ θ
(17)
In order to investigate the stability criteria, the following lemma will be also used [39].
Lemma 1: For any matrices A and B with appropriate dimensions, the following property
holds for any positive scalar λ :
1−
+ ≤ +T T T T
A B B A A A B Bλ λ (18)
Let 11 1=Q P and 1
22 2
−
=Q P . From (16) and by using the separation Lemma (1) [40], one obtains:
( )
1
2
0
,
0
 
≤  
  
ijD
ij ij
ij
R
M A Q
R
(19)
In order to verify (13), we must have:
1
2
0
0
0
 
< 
  
ij
ij
R
R
(20)

7
Which implies :
1
2
0
0
<
<
ij
ij
R
R
(21)
where:
( )
( )
1
1 1 1 1 1
1
1
2 1 1 1 1
2 2 2 2
1 1 1 1
2 2 2 2
ˆ ˆ ˆsin sin cos
ˆcos
ˆ ˆsin sin
−
− − − −
− − − −
= + + + +
+ +
= + + +
+ ϒ + ϒ +
T T
ij i i i j i j i j j
T T
i j j i i
T T
ij i i i j i j
T T
i i j j
R A P PA N P PN B D P
P B D B B
R A P P A N P P N
CP P C P D D P
ρ θ ρ θ ρ θ
ρ θ ε
ρ θ ρ θ
ε
From the following variable changes 1=i iV D P , 2= ϒi iW P , by using Lemma (1) and using the
Schur’s complement [39] to (21) results in (9) and (10). This completes the proof of the theorem.
4. ROBUST CONTROL OF THE FUZZY BILINEAR MODEL WITH
UNCERTAINTIES
In this section, we want to design the robust control of the FBM. The objective is to consider
parametric uncertainties in the system for modelling the behaviour of complex nonlinear dynamic
systems in order to design of a stabilizing control law, based on state estimation. The nonlinear
system is modeled as a T-S fuzzy bilinear model. Then, the dynamic equation of the fuzzy
bilinear system can be formulated in the following form:
( ) ( )
( )1
( ) ( )
( ) ( ( ))
( ) ( )
( ) ( )
=
  + ∆ + + ∆
=  
 + + ∆  

=
∑&
r
i i i i
i
i i i
A A x t B B u t
x t h t
N N x t u t
y t Cx t
ξ
(22)
The parameter uncertainties considered here are norm-bounded and presented by the form
[ ] [ ]1 2 3∆ ∆ ∆ =i i i i i i i iA B N M F E E E (23)
where 1 2 3, , ,i i i iM E E E are known real constant matrices of appropriate dimensions, and
iF is an unknown matrix function satisfying ( ) ( ) ≤T
i iF t F t I , in which I is the identity matrix of
appropriate dimension.
The feedback controller and the state of fuzzy bilinear observer for T-S fuzzy bilinear model with
parametric uncertainties (22) is the same as that for (2). In order to examine the robustness of the
fuzzy controller (5), the augmented system is represented as follows:
( )ˆ( ) ( ) ( )=&
a aX t A t X tξ (24)
where the matrices ( )ˆ ( )A tξ and the augmented vector ( )aX t are respectively given by:

8
( ) ( ) ( )
1 1
ˆ ˆ( ) ( ) ( )
( )
( )
( )
= =
=
 
=  
 
∑∑
r r
i j ij
i j
a
A t h t h t A
x t
X t
e t
ξ ξ ξ
with
11 12
21 22
ˆ Ξ Ξ 
=  Ξ Ξ 
ijA
( ) ( ) ( )
( )
( )
11
12
21
22
ˆ ˆsin cos
ˆcos
ˆ ˆsin cos
ˆ ˆsin cos
Ξ = + ∆ + + ∆ + + ∆
Ξ = + ∆
Ξ = − ∆ + ∆ + ∆
Ξ = + + ϒ − ∆
i i i i j i i j j
i i j j
i i j i j j
i i j i i j j
A A N N B B D
B B D
A N B D
A N C B D
ρ θ ρ θ
ρ θ
ρ θ ρ θ
ρ θ ρ θ
The objective of this section is to seek a fuzzy observer-based controller of the form (5) such that
the closed-loop fuzzy augmented system (24) is globally asymptotically stable. Therefore, we
have a result that is summarized in the following theorem.
Theorem 2: If there exist symmetric positive definite matrices 1P and 2P , matrices ,i iW V and
positive scalars 0 , 0> >ε ρ such that LMIs (25) and (26) are satisfied, then the augmented
system described by (24) is asymptotically stable 1 , 1∀ = ∀ =L Li r j r
( )
1
1 1
2
3 1
1
1
2
2 1
1
* * * * * * * * * *
* * * * * * * * *
0 * * * * * * * *
0 0 * * * * * * *
0 0 0 * * * * * *
0 0 0 0 * * * * *
0 0 0 0 0 * * * *
1 2 0 0 0 0 0 0 * * *
0 0 0 0 0 0 0 * *
0 0 0 0 0 0 0 0 *
0 0 0 0 0 0 0 0 0
−
−
−
Π 
 −
 −

−
 −

−
 −

 + −

 −

−
 − 
i
i j
i
T
i
j
i j
T
i
i
i j
E P I
E V I
E P I
M I
V I
E V I
M I
N P I
BV I
I I
ε
ε
ε
ρ ε
ε
ε
ρ ε
ρ
0






 <








(25)

9
( )
2
1
2
3
2
2
1
2
2 1
2
2
* * * * * * * * *
* * * * * * * *
0 * * * * * * *
0 0 * * * * * *
0 0 0 * * * * *
00 0 0 0 * * * *
0 0 0 0 0 * * *
1 2 0 0 0 0 0 0 * *
0 0 0 0 0 0 0 *
0 0 0 0 0 0 0 0
−
−
Π 
 − 
 −
 
− 
 −
  <− 
 −
 
 + −
 
 −
 
−  
i
i j
i
T
i
T
i
i j
T
i
i
E I
E D I
E I
B P I
M P I
E D I
M P I
P N I
I I
ε
ε
ε
ρ ε
ρ ε
ε
ρ ε
ρ
(26)
where : 1 1 1 2 2 2,Π = + Π = + + +T T T T
i i i i i iA P PA P A A P WC C W
The gains of the observer and the state feedback control law are given by:
1
1
−
=i iD V P (27)
1
2
−
ϒ =i iP W (28)
Proof:
To prove the stability of the closed-loop for the augmented system (24), the sufficient conditions
must be verified:
( )ˆ ˆ ˆˆ0 , , 0∃ = > = + <T D T
ij ij ij ijQ Q M A Q A Q QA (29)
Where
11
22
0
0
 
=  
 
Q
Q
Q
(30)
Let us define
( ) 1 2ˆˆ ˆ ˆ, = +D D D
ij ij ij ijM A Q M M (31)
where
1 11 12
1 2
21 222
ˆ 0
ˆ ˆ,
ˆ0
   ∆ ∆
= =   
∆ ∆     
ij ij ijD D
ij ij
ij ijij
D
M M
D
(32)

10
( )
( )
1
11 11 11 11 11 11
2
22 22 22 22 22 22
11
11 11 11 11
11
ˆ ˆ ˆ ˆˆ sin sin cos cos
ˆ ˆˆ sin sin
ˆ ˆsin sin
= + + + + +
= + + + + ϒ + ϒ
∆ = ∆ + ∆ + ∆ + ∆
+ ∆
TT T
ij i i i j i j i j j i j j
TT T
ij i i i j i j i i
T T
ij i i i j i j
D AQ Q A N Q Q N Q B D B D Q
D AQ Q A N Q Q N CQ Q C
AQ Q A N Q Q N
Q B
ρ θ ρ θ ρ θ ρ θ
ρ θ ρ θ
ρ θ ρ θ
ρ ( )
( )
( ) ( )
11
12
22 22 22
11 11
21
22 22 22
11 11
22
ˆ ˆcos cos
ˆ ˆsin cos
ˆ ˆcos cos
ˆ ˆsin cos
ˆ ˆcos cos
+ ∆
∆ = − ∆ − ∆ − ∆
+ ∆ +
∆ = −∆ − ∆ − ∆
+ + ∆
∆
T
i j j i j j
TT T
ij i i j i j j
i j j i j j
ij i i j i j j
T T
i j j i j j
ij
D B D Q
Q A Q N Q B D
B D Q B D Q
AQ N Q B D Q
Q B D Q B D
θ ρ θ
ρ θ ρ θ
ρ θ ρ θ
ρ θ ρ θ
ρ θ ρ θ
( )22 22
ˆ ˆcos cos= − ∆ − ∆
T
i j j i j jB D Q Q B Dρ θ ρ θ
(33)
The equation 2ˆ D
ijM can be reformulated as follows
2 1 2 3ˆ = Ψ + Ψ + ΨD
ij ij ij ijM (34)
with:
12 12 11
1 21 2 3
21 21 22
1 2
0 0 0
, ,
0 0 0
     Ψ Ψ ∆
Ψ = Ψ = Ψ =     
Ψ Ψ ∆          
ij ij ij
ij ij ij
ij ij ij
(35)
( )
( ) ( )
12
1 22 22 22
21
1 22 22 22
12
2 11 11
21
2 11 11
ˆ ˆsin cos
ˆ ˆsin cos
ˆ ˆcos cos
ˆ ˆ= cos cos
Ψ = − ∆ − ∆ − ∆
Ψ = −∆ − ∆ − ∆
Ψ = ∆ +
Ψ + ∆
TT T
ij i i j i j j
ij i i j i j j
ij i j j i j j
T T
ij i j j i j j
Q A Q N Q B D
AQ N Q B D Q
B D Q B D Q
Q B D Q B D
ρ θ ρ θ
ρ θ ρ θ
ρ θ ρ θ
ρ θ ρ θ
(36)
Let 11 1=Q P and 1
22 2
−
=Q P . Using the uncertainties structure and by using the separation
Lemma (1) [40], one obtains:
1
2
2
ˆ 0
ˆ
ˆ0
 
≤  
  
ijD
ij
ij
T
M
T
(37)

11
where:
( )1 2
1 1 1 2 2 1
1 2 1 1
1 1 1 1 1 2 2 1
1
1 3 3 1
2 1 1 1 1 1 1 1 1 1
2 1 1 2 2 3 3 2 2 2 2 2
ˆ 1 2
ˆ
− − −
−
− − − − − − − − −
= + + +
+ + +
+
= + +
T T T T T
ij i i i i j j j i i j
T T T T T
i i i i i i j i i j
T
i i
T T T T
ij i i i i j i i j
T M F F M PD D P PD E E D P
M F F M PE E P PD E E D P
PE E P
T P E E P P E E P P D E E D P
ρ ε ε ε
ε ρ ε ε
ε
ε ε ε
( )1 2 1 2 2
1 1
2 2 2 2
1 2− −
− −
+ + + +
+
T T T T T
i i i i i i i i i i
T T
j i i j
B B M F F M M FF M
P D E E D P
ε ρ ε ρ ρ ε
ε
(38)
From (31), (33) and (37), we have:
( )
1 1
2 2
ˆ ˆ 0
ˆˆ ,
ˆ ˆ0
 +
≤  
+  
ij ijD
ij ij
ij ij
D T
M A P
D T
(39)
In order to verify (29), we must have:
1
2
ˆ 0
0
ˆ0
 
< 
  
ij
ij
R
R
(40)
where:
1 1 1
2 2 2
ˆ ˆ ˆ
ˆ ˆ ˆ
= +
= +
ij ij ij
ij ij ij
R D T
R D T
(41)
Which implies :
1
2
ˆ 0
ˆ 0
<
<
ij
ij
R
R
(42)
Using the following variable changes 1=i iV D P , 2= ϒi iW P , and using the Schur’s complement
[39] to (42) results in (25) and (26). This completes the proof of the theorem.
To illustrate the theoretical development and the design algorithm, numerical example is
proposed in the following section.
5. SIMULATION EXAMPLE
In this section, an example is given to illustrate the effectiveness of the proposed design
conditions. Consider a nonlinear system represented by a T-S fuzzy bilinear model with
uncertainties of two local models:

12
( ) ( )
( )
2
1
( ) ( )
( ) ( ( ))
( ) ( )
( ) ( )
=
  + ∆ + + ∆
=  
 + + ∆  

=
∑&
i i i i
i
i i i
A A x t B B u t
x t h t
N N x t u t
y t Cx t
ξ
The numerical values of those parameters are defined by:
[ ]
1 1 1
2 2 2
0 0.8 1 1 0
, ,
1.5 3.2 1 0 1
0 0.5 2 1 0
, ,
1.5 3.4 1 0 1
1 0
−     
= = =     − − − −     
−     
= = =     − − − −     
=
A B N
A B N
C
Uncertainties are also defined by the following matrices:
1 11 21 31
2 12 22 32
0.1 0 0 0.5 1 0 0
, , ,
0 0.1 0 0.3 0.5 0 0
0.1 0 0 0.5 1 0 0
, , ,
0 0.1 0 0.6 0.5 0 0
       
= = = =       
       
       
= = = =       
       
M E E E
M E E E
The weighting functions are defined by:
( ) ( ) ( )
2
1
1 2 1
51( ) exp , ( ) 1 ( )
2 2
 + 
= = −     
x
h x t h x t h x t
In order to design robust observer-based controller for this uncertain T-S fuzzy bilinear system,
we solve LMIs of theorem 2. So, we obtain the following controller and observer gain matrices
respectively with 0.7=ρ :
[ ] [ ]
[ ] [ ]
2
2
1.94 0.18 , 1.35 0.08 ,
7.74 80.83 , 7.79 82.25
= − − = − −
ϒ = − − ϒ = − −
i
T T
i
D D
The simulation results are shown in figures (1)-(3). Figures (1) and (2) show respectively the
trajectories of states 1( )x t and 2 ( )x t of the considered system and their corresponding observer
variables. One can see that the estimated states can closely track the original states despite of the
presence of norm-bounded parametric uncertainties. Figure (3) shows the evolution of the control
signal.

13
Figure 1. The trajectories of the state 1( )x t and its estimate .
Figure 2. The trajectories of the state 2 ( )x t and its estimate .

14
Figure 3. Control signal ( )u t
6. CONCLUSION
In this paper, we have developed the fuzzy observer-based control problem and the robust
stabilization for continuous nonlinear systems. Such design is based on a T-S fuzzy bilinear
model representation, particularly suitable for a nonlinear system with a bilinear term. Design
conditions for an observer-based controller have been formulated and solved using a LMI
framework. The gains of the control and observer can be obtained by solving this set of LMIs.
Simulation results have checked the effectiveness of our approach in controlling nonlinear
systems described by a fuzzy bilinear model with parametric uncertainties. Based on the results in
this paper, interesting future studies may be extended the proposed techniques to uncertain T-S
fuzzy bilinear systems for discrete-time and to the robust ∞H fuzzy observer-based control
design for discrete-time uncertain T-S fuzzy bilinear systems.

15
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