Abstract - This paper introduces a new stability test and
controller design methodology for Takagi-Sugeno-Kang
(TSK) fuzzy systems. Unlike methods based on a common
Lyapunov function, our stability results can be applied to
systems with unstable consequents and our controllers can
be designed for systems with unstabilizable consequents.
Sufficient stability conditions are derived using the
comparison principle and the Dini derivative. The stability
results are used to design continuous-time fuzzy
proportional controllers and fuzzy PI controllers. We
provide examples to demonstrate our controller design. We
show that our results compare favorably with results
available in the literature and provide controllers where
earlier approaches fail.
I. INTRODUCTION
TSK controllers provide simple solutions with excellent
performance for a variety of control applications. These
controllers are often obtained using stability tests.
Sugeno reviewed the literature on the stability of TSK
system and provided some new stability conditions [1].
The main stability and controller design results for TSK
systems are based on a common quadratic Lyapunov
function and stability testing using linear matrix
inequalities (LMIs) [2], [3]. Unfortunately, there are
many stable TSK systems for which no common
Lyapunov function exists, including systems with one or
more unstable consequent. Moreover, a necessary
condition for the existence of common Lyapunov
functions is that all pair-wise products of the subsystem
state matrices result in a stable matrix [2], [4].
This paper proposes new tests for the exponential
stability analysis of continuous time TSK fuzzy systems
that do not require the use of a common Lyapunov
function and are therefore applicable to a larger class of
systems. We consider TSK fuzzy systems with linear
consequents and with membership functions of bounded
support. In addition, we present a new design
methodology for fuzzy proportional controllers. The
results are applicable even if some consequent dynamic
systems are not stabilizable.
Throughout the paper, we use the product t-norm for all
set operations and a notation similar to that of the text by
Wang [3]. The subscript ‡OIW· and ‡UJW· in the
inequalities stand for right and left respectively, and show
the side of the inequality in which the quantity is used.
II. FUNDAMENTALS
This section provides some preliminary definitions and
lemmas that are used throughout the paper. We begin with
the fuzzy systems and the membership functions
considered in this paper and their properties.
Figure 1- A membership function with bounded support.
Definition 1: Continuous Dynamic TSK Fuzzy Systems
A continuous TSK fuzzy system is a TSK fuzzy system
with rule base of the form
4
ª +( EO m
6*’0 6
L
:;
L >T
5
T
Æ
?
˝
Æ m
Lc#
5
#
Æ
g
˝
Æ
:; L >B
5
:; B
Æ
:;?
˝
Æ E L sÆÆ / ( 1 )
Lyapunov stability results are typically stated for the
equilibrium state at the origin. The following lemma
provides conditions for the equilibrium states of a
dynamic TSK system of Definition 1 to be at the origin.
Lemma 1: Equilibrium of continuous TSK Fuzzy Systems
If the system of Definition 1 satisfies the condition
:; L ( 2 )
for any rule 4
, where 4
is fired when L , then the
origin is an equilibrium point.
v
We assume the following
1. The origin is an equilibrium point for all our TSK
fuzzy systems.
2. The functions in the consequents of the rules are
linear, i.e. :; L ( Æ ( —9
ÆHÆ
3. The antecedent membership functions are complete
and normal.
4. The antecedent membership functions have bounded
support, i.e.
”
kT
oL P
JKJVANKÆ T
— B=
Æ =
C
rÆ T
B=
Æ =
C
( 3 )
5. Without loss of generality, we assume that
OCJk=
oL OCJ @=
A ( 4 )
where OCJ denotes the sign function.
Saeed Jafarzadeh, Member, IEEE, M. Sami Fadali, Senior Member
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2012 American Control Conference
Fairmont Queen Elizabeth, Montréal, Canada
June 27-June 29, 2012
978-1-4577-1094-0/12/$26.00 ©2012 AACC 5634