Stability criteria for T–S fuzzy systems with interval time-varying
delays and nonlinear perturbations based on geometric progression
delay partitioning method
$
Hao Chen
a,b,
n
, Shouming Zhong
a
, Min Li
b
, Xingwen Liu
b
, Fehrs Adu-Gyamfi
c,d
a
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
b
College of Electrical and Information Engineering, Southwest University for Nationalities, Chengdu 610041, China
c
School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
d
Faculty of Engineering, Koforidua Polytechnic, P.O.BOX KF981, Koforidua, Ghana
article info
Article history:
Received 21 December 2015
Received in revised form
20 March 2016
Accepted 8 April 2016
Available online 29 April 2016
This paper was recommended for publica-
tion by Dr. Jeff Pieper
Keywords:
Geometric sequence division
Interval time-varying delays
Nonlinear perturbations
Stability analysis
T–S fuzzy systems
abstract
In this paper, a novel delay partitioning method is proposed by introducing the theory of geometric
progression for the stability analysis of T–S fuzzy systems with interval time-varying delays and non-
linear perturbations. Based on the common ratio
α
, the delay interval is unequally separated into mul-
tiple subintervals. A newly modified Lyapunov–Krasovskii functional (LKF) is established which includes
triple-integral terms and augmented factors with respect to the length of every related proportional
subintervals. In addition, a recently developed free-matrix-based integral inequality is employed to avoid
the overabundance of the enlargement when dealing with the derivative of the LKF. This innovative
development can dramatically enhance the efficiency of obtaining the maximum upper bound of the
time delay. Finally, much less conservative stability criteria are presented. Numerical examples are
conducted to demonstrate the significant improvements of this proposed approach.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
In recent decades, the Takagi–Sugeno (T–S) fuzzy model, which
was introduced in [1], has attracted a number of research works
presenting stability analysis and control design of such a model
because of its practical application. Complex mathematical mod-
elling with higher order for physical system is frequently
encountered in many engineering applications, which could cause
nonlinearity in dynamic systems [2,3]. The T–S fuzzy theory can be
applied to flexibly approximate the complex nonlinear systems
into a unified framework [4–9]. Time delays are inevitable as a
source of instability in many dynamics systems such as chemical
processes, communication networks, and biological systems, due
to the material transfer requirement, processing time or
accumulation of time lags through system connections [10,11].In
these circumstances, stability analysis of time delayed T–S fuzzy
systems has received special attention and some remarkable
results have been achieved in recently developed research works
[12– 17].
On the other hand, the appearance of interval time varying
delays encountered in nonlinear dynamic systems can result in the
difficulty for stability analysis. Thus, some newly constructed
Lyapunov functionals have been established to present appro-
priate stability criteria to overcome this issue [4,18–24]. These
kinds of stability criteria can be commonly classified into two
different types: delay-independent and delay-dependent. As much
of information on the delay is commonly concerned, the delay-
dependent criteria are more useful for obtaining less conservative
results [6,24–26]. For further investigation of the stability issues of
time delayed systems, the T–S fuzzy systems with interval time
varying delay are considered as a common model for stability
analysis.
Delay partitioning technique, alternatively known as delay
fractionizing method, was first developed in [27]. It was proved
that delay partitioning approach can dramatically improve the
stability conditions and less conservative results can be obtained
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.04.005
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
☆
This work was partially supported by National Nature Science Foundation of
China (61273007, 61533006), SWUN Construction Projects for Graduate Degree
Programs (2015XWD-S0805), SWUN Fundamental Research Funds for Central
Universities (12NZYTH01), and Innovative Research Team of the Education
Department of Sichuan Province (15TD0050).
n
Corresponding author at: School of Mathematical Sciences, University of Elec-
tronic Science and Technology of China, Chengdu 611731, China.
E-mail address: haochen84@yahoo.co.uk (H. Chen).
ISA Transactions 63 (2016) 69–77