70 S. Hu et al. / Nonlinear Analysis: Hybrid Systems 8 (2013) 69–82
years. Up until now, under consideration of network-induced delays and/or data losses in transmission, a number of valuable
results on networked T–S fuzzy systems have been reported in the literature. To mention a few, in [29], a novel guaranteed
cost networked control method was presented for the T–S fuzzy systems with constant bounded time-delay. In [30], the
robust H
∞
control problem for T–S fuzzy networked systems with uncertainties was addressed. A new iterative algorithm
for the controller design was proposed in [31]. The modeling and stabilization problem for a class of nonlinear NCSs was
investigated in [32]. More recently, a piecewise analysis method has been proposed to investigate the robust H
∞
control
problem for networked T–S fuzzy systems in [33]. Reliable guaranteed cost sampling control was studied in [34]. Under
an assumption that the probability distribution of communication delays (network-induced delays) is known or specified
priori, robust networked control for a class of T–S fuzzy systems was discussed in [35]. However, unfortunately, most of
the above mentioned results on nonlinear NCSs are dedicated to the continuous-time cases, few results on the discrete-
time cases are available. On the other hand, the approximation errors between the fuzzy model and the nonlinear system
have not been considered except [31], where only the continuous-time case was discussed. Actually, it has been shown that
the approximation errors can degrade control performance of systems and even destabilize the system [36]. Therefore, it is
necessary to consider the approximation errors. Nevertheless, up until now, little progress has been reported for the stability
analysis and control synthesis of nonlinear discrete-time NCSs in the presence of network-induced delays and data packet
dropouts, which motivates the present study.
In this paper, we will consider the stabilization problem for a class of nonlinear discrete-time NCSs. More specifically,
the nonlinear plant is represented by a T–S fuzzy model, and the approximation errors between the nonlinear NCSs and T–S
fuzzy models are also included. Considering the network-induced delays and data packet dropouts and using input delay
approach, a new NCSs model based on the updating instants of the actuator is formulated. Sufficient conditions are derived
for the existence of a fuzzy controller based on the Lyapunov–Krasovskii functional. Since the obtained conditions are not
strict LMIs, two approaches are then developed to solve the nonlinear convex feasibility problem. Finally, two examples are
given to demonstrate the effectiveness of the proposed design method.
The remainder of this paper is organized as follows. Section 2 formulates the problem under consideration. Section 3
analyzes the robust stability of the networked T–S fuzzy systems that are formulated in Section 2. Section 4 develops a
controller design method. Two numerical examples are given in Section 5 to illustrate the effectiveness of the developed
method.
Notation: R
n
denotes the n-dimensional Euclidean space, R
n×m
is the set of real n × m matrices. Z
+
is the set of positive
integers. The notation X > 0 (X < 0) for any X ∈ R
n×n
means that the matrix X is a real symmetric positive definite (negative
definite). For a real matrix B and two real symmetric matrices A and C of appropriate dimensions,
A B
∗ C
denotes a real
symmetric matrix, where ∗ denotes the entries implied by symmetry. The superscript ‘‘T ’’ stands for matrix transposition.
Throughout this paper, if not explicitly stated, matrices are assumed to have compatible dimensions.
1. Problem formulation
Consider the following nonlinear system
x(k + 1) = f (x(k)) + g(x(k))u(k) (1)
where x(k) ∈ R
n
is the state vector and u(k) ∈ R
m
is the input vector. We assume that f (x(k)) and g(x(k)) are unknown
nonlinear function vectors depending on x(k). The initial condition of the system (1) is given by
x(0) = φ. (2)
Throughout this paper, system (1) and (2) are assumed to be controlled through a network, whose general structure is shown
in Fig. 1. From [27], we know the nonlinear system (1) can be represented by a T–S fuzzy plant model, which expresses the
nonlinear system as some simple local linear dynamic systems. The ith rule is of the following format.
Plant Rule i: IF θ
1
(k) is M
i1,
θ
2
(k) is M
i2,
. . . , θ
g
(k) is M
ig
THEN
x(k + 1) = A
i
x(k) + B
i
u(k) (3)
where M
ij
is a fuzzy set (j = 1, 2, . . . , g); r is the number of IF–THEN rules; A
i
∈ R
n×n
and B
i
∈ R
n×m
are the known
parameter matrices of appropriate dimensions. θ
1
(k), θ
2
(k), . . . , θ
g
(k) are the premise variables. By using a center average
defuzzifier, product inference, and a singleton fuzzifier, the global dynamics of the T–S fuzzy systems (3) are described by
x(k + 1) =
r
i=1
µ
i
(θ(k))
[
A
i
x(k) + B
i
u(k)
]
(4)
where
µ
i
(θ(k)) =
ϖ
i
(θ(k))
r
i=1
ϖ
i
(θ(k))
, ϖ
i
(θ(k)) = Π
g
j=1
M
ij
(θ(k)),
θ(k) =
θ
1
(k), θ
2
(k), · · · , θ
g
(k)
,