Stability and stabilization of positive Takagi-Sugeno fuzzy continuous
systems with delay
Abdellah Benzaouia, Rkia Oubah, Ahmed El Hajjaji and Fernando Tadeo
Abstract— This paper deals with the problem of stability and
stabilization of Takagi-Sugeno (T-S) fuzzy systems with a fixed
delay by linear programming (LP) while imposing positivity in
closed-loop. The stabilization conditions are derived using the
single Lyapunov-Krasovskii Functional (LKF). An example of
a real plant is studied to show the advantages of the design
procedure.
Key-words: T-S fuzzy systems, positive systems,
Lyapunov-Krasovskii functional, stabilization, Linear
programming.
I. I NTRODUCTION
The problem concerns a special class of nonlinear systems
called Takagi-Sugeno models (T-S) [7]. From the history of
the approach, this class can be interpreted as a collection of
linear models interconnected by nonlinear functions, called
membership functions, which are dependent variables. The
most delicate problem is the choice of premise variables that
partition the space [6], [8].
Positive systems have been of great interest to researchers in
recent years [9], [1], [4], [5] and [10]. The class of positive
T-S fuzzy systems was considered for the first time in [2].
The obtained results were presented using LMIs.
In this paper, the conditions of stability and stabilization of
such systems are studied by using linear programming (LP).
An application on the model of a real process is considered.
A comparison of the obtained results with those of [3] is
proposed. The rest of this paper is organized as follows: In
section 2, the description of T-S fuzzy models with fixed
state delay and fuzzy control law based on PDC structure
is given. New delay independent stabilization conditions are
established for positive systems in section 3. In section 4, an
example of a real plant is given to show the need for such
controllers. Some conclusions are given in section 5.
Notation:
• M
T
denotes the transpose of a real matrix M.
• F is called a positive matrix denoted by F 0 if all
its elements are positive and there is a strictly positive
element ( f
ij
≥ 0, ∀ (i, j) , ∃ (i, j) : f
ij
0).
• A matrix A ∈ ℜ
n×n
is called a Metzler matrix if its
off-diagonal elements are nonnegative. That is, if A =
a
ij
n
i, j=1
, A is Metzler if a
ij
≥ 0 whenever i 6= j.
Benzaouia and Oubah are with LAEPT URAC 28, University Cadi
Ayyad, Faculty of Science Semlalia, BP 2390, Marrakech, Morocco.
benzaouia@ucam.ac.ma,rkia.oubah@gmail.com
El Hajjaji is with University of Picardie Jules Vernes (UPJV), 7, Rue de
Moulin Neuf 8000 Amiens, France. ahmed.hajjaji@u-picardie.fr
Tadeo is with Universidad de Valladolid, Depart. de
Ingenieria de Sistemas y Automatica, 47005 Valladolid, Spain.
fernando@autom.uva.es
II. PROBLEM FORMULATION AND PRELIMINARY RESULTS
Specifically, the Takagi-Sugeno fuzzy system is described
by fuzzy IF-THEN rules, which locally represent linear
input-output relations of a system. The fuzzy system is of
the following form:
Rule i: IF z
1
(t ) is F
1
i
and ··· and z
p
(t ) is F
p
i
Then:
˙ x(t ) = A
i
x(t )+ A
i1
x(t - τ )+ B
i
u(t ) (1)
x(t ) = Ψ(t ) 0, t ∈ [-τ , 0] (2)
where x(t ) ∈ IR
n
is the state, u(t ) ∈ IR
m
is the control input,
τ is a fixed delay, with i = 1, 2, ..., r, r is the number of
IF-THEN rules, z
1
(t ) ··· z
p
(t ) and F
j
i
are respectively the
premise variable and the fuzzy sets.
The control law is chosen to be a state feedback one given
by:
u(t )= K
i
x(t ), (3)
Systems (1) will be represented by T-S fuzzy models de-
scribed by:
˙ x(t )=
r
∑
i=1
h
i
(z(t ))(A
i
x(t )+ A
i1
x(t - τ )+ B
i
u(t )) (4)
The control used in this work is the so called PDC control:
u(t )=
r
∑
i=1
h
i
(z(t ))K
i
x(t ), (5)
where h
i
(z(t )) =
w
i
(z(t ))
2011 50th IEEE Conference on Decision and Control and
European Control Conference (CDC-ECC)
Orlando, FL, USA, December 12-15, 2011
978-1-61284-799-3/11/$26.00 ©2011 IEEE 8279