1. Introduction
Fuzzy system models have been widely adopted to represent certain classes of nonlinear dynamic systems following the T-S fuzzy model [1]. Since then there have been several approaches for the study of stability analysis and robust controller synthesis using the so-called parallel distributed compensation (PDC) method for uncertain nonlinear systems [2,3]. Sufficient conditions have been derived based on the feasibility testing of a linear matrix inequality (LMI) in [4-7] and extended for classes of nonlinear discrete-time systems with time delays in [8-10] via different approaches. Recently, much attention has been paid to the stability issue of network based control systems [11]. Several results pertaining to the analysis and design of networked control systems (NCSs) enhanced their wide benefits such as reducing system wiring, ease of system diagnosis and maintenance, and increasing system agility, to name a few. However, communication network in the control loops gave rise to some new issues, especially the intermittent losses or delays of the communicated information due to use of a network, which imposes a challenge to system analysis and design. To address this challenge, many results have been developed in consideration of network-induced delay and packet dropout [12-18], with focus on stability analysis and controller design with random delays.
Further consideration of the communication of the NCSs over the channel emphasized the importance of signal quantization, which has significant impact on the performance of NCSs. In this regard, the problem of guaranteed cost control and quantized controller design were discussed in [17] by using two quantizers in the network both from sensor to controller and from controller to actuator, and the network-induced delay and data dropped were considered as well.
Recent advances converted the quantized feedback design problem into a robust control problem with sector bound uncertainties, [11] and [16-18]. Parallel investigations to the class of switched discrete-time systems with interval time-delays were developed in [19-23].
Despite the potential of these developments, the problem of how to analyze the stability of nonlinear NCSs with data drops still open. On the other hand, most industrial plants have severe nonlinearities, which lead to additional difficulties for the analysis and design of control systems. Though some issues on nonlinear NCSs have been investigated [23,24], limited work has been found on robust 
state feedback controller design of networks for fuzzy systems with consideration of both network conditions and signal quantization.
The guaranteed cost networked control and robust 
problem based on the T-S fuzzy model was treated in [25]. The results were derived by using a single Lyapunov function (SLF) method, which in general leas to a conservative result. Designing fuzzy controllers for a class of nonlinear networked control systems was considered in [26-28] by solving approximate uncertain linear networked Takagi-Sugeno (T-S) models with both network induced-delay and packet dropout. However, they do not quantize the signals. The foregoing facts motivate the present study.
In this research work, we address the robust 
state feedback control problem for discrete-time networked systems with state quantization and disturbances. The T-S fuzzy systems with norm-bounded uncertainties are utilized to characterize the nonlinear NCSs. Since the computation available is often limited, the quantized feedback controller is designed under consideration of effect of network-induced delay and data dropout, the employed quantizer is time-varying. By using a new fuzzy Lyapunov-Krasovskii functional (LKF), we provide a sufficient LMI-based condition for the existence of a fuzzy controller. A numerical example shows the feasibility of the developed technique.
Notations and facts: In the sequel, the Euclidean norm is used for vectors. We use 
and 
to denote the transpose and the inverse of any square matrix
, respectively. We use 
to denote a symmetric positive definite (positive semi-definite, negative, negative semi-definite matrix 
and 
to denote the 
identity matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. In symmetric block matrices or complex matrix expressions, we use the symbol (
) to represent a term that is induced by symmetry.
Fact 1: For any real matrices 
and 
with appropriate dimensions and
, it follows that
Sometimes, the arguments of a function will be omitted when no confusion can arise.
2. Problem Description
A typical networked control system typically has a clockdriven sampler and a quantizer, controller, a zero-order hold (ZOH) which is event-driven. The sampling period is assumed to be 
with the sampling instants as
. The plant belongs to class of uncertain discrete-time systems where the parametric uncertainties are norm-bounded.
In what follows, we consider that this class is represented by Takagi-Sugeno fuzzy model composed of a set of fuzzy implications, and each implication is expressed by a linear system model. The 
rule of this TakagiSugeno model has the following form:
(1)
where 
are the premise variables, each 
are the fuzzy sets, 
is the number of if-then rules and 
is the state vector, 
is the control input, 
is the output, 
is the disturbance input which belongs to 
and 
indicates the maximum allowable signal transmission delay. The uncertain matrices 
are represented by:
(2)
where the matrices 
describe the nominal dynamics and 
are known constant real matrices with appropriate dimensions. The matrices 
are unknown time-varying and satisfying
.
Using a center average defuzzifier [1], product inference, and incorporating fuzzy “blending”, the fuzzy system under consideration can be cast into the form
(3)
where
(4)
where 
is the grade of membership of 
in
. In the sequel, we assume that
and therefore
Our objective in this paper is to design a fuzzy 
state feedback controller with state quantization.
3. Controlled Fuzzy System
In what follows, we proceed to consider establish the main result for the uncertain discrete-time fuzzy networked control systems described by (3) and design the quantized fuzzy 
state feedback controller. We consider a limited capacity communication channel and for reducing the amount of data rate of transmitting in the network, which led to the increase quality of service of the network, we assume that the state vector 
is measurable. The state signal from sensor to the controller is quantized via a quantizer, and then transmitted with a single packet. To reflect realty, network-induced time delay is modeled as an input delay and the packet dropout will be considered.
3.1. State-Feedback Control
In effect, we seek to design the state-feedback controller:
(5)
where 
is the feedback law to be defined in the sequel and 
are some integers such that
. Introduce 
which contains the information of packet dropouts and improper packet sequence in the control signal. Note that
.
It has been pointed out in [19] that when 
there would be no packets dropout and the case 
represents continuous packets lost. In addition, when 
the new packet reaches the destination before the old one. This case might lead to a less conservative result. In the sequel, we assume that 
and it is readily seen that
It should be observed that 
accounts for the time from the instant 
when sensor nodes sample the sensor data from the plant to the instant when actuator transfer data to the plant. Extending on this, we remark that
Consequently, we define
where 
are known finite integers.
3.2. Quantizer
Let the quantizer be described as
where 
is a symmetric, static and and time-invariant quantizer and the associated set of quantization levels is expressed as
(6)
Note that the quantization regions are quite arbitrary. In case of logarithmic quantizer, the set of quantization levels 
becomes
where 
is the initial state of the quantizer and 
is a parameter associated with the quantizer
. In this regard, a particular characterization of the quantizer is given by
where
. It follows from [19] that, for any
, a sector bound expression can be expressed as:
For simplicity in exposition, we use 
to denote
. Thus, 
can be written as
We assume henceforth that the updating signal at the instant 
has experienced signal transmission delay
, however the delay between the sensor and quantizer is neglected. In view of the limited capacity in communication channel, the state signal from sensor to the controller is quantized via a logarithmic quantizer 
for reducing the amount of data rate of transmitting in the network. When the static and time-invariant quantizer
, the state feedback controller would be in the form of
, which is the same as a traditional one.
Incorporating the notion of parallel distributed compensation, the following fuzzy state-feedback stabilizing control law is used:
(7)
where 
is the control gain for rule
. Accordingly, the overall fuzzy control law is expressed by
(8)
Applying controller (8) to system (3) with some mathematical manipulations, the resulting closed-loop system can be cast into the form:
(9)
which belongs to the class of switched time-delay system [15], where
(10)
4. Quantized Fuzzy Control Design
In this section, we seek to establish a sufficient condition for the solvability of the robust 
control problem. This condition will be expressed in an LMI framework to facilitate the design of the desired fuzzy state feedback controllers. Based on the so-called parallel distributed compensation scheme, the following theorem establishes a delay-dependent stabilization condition for the closedloop fuzzy networked control system (9):
Theorem 4.1 Consider system (9). Given the bounds 
and a scalar constant
, there exists a fuzzy controller in the form of (8), such that the uncertain closedloop fuzzy system (9) with an 
disturbance attention level 
is asymptotically stable, if there exist matrices 
matrices 
and scalars 
satisfying
(11)
(12)
(13)
Proof: In what follows, we adopt a parameterdependent approach [15]. Consider system (9) with 
and define
(16)
where 
are matrices of appropriate dimensions and 
are fuzzy weighting matrices that are directly include the membership functions instead of a single matrix, a fact that aims at relaxing the conservatism. For simplicity in notation, we let
(17)
In terms of the state increment 
and the time-span
we consider the Lyapunov-Krasovskii functional (LKF):
(18)
We focus initially on the case
. A straightforward computation gives the first-difference of
along the solutions of (17) with the help of (9) and (10) as:
(19)
To facilitate the delay-dependence analysis, we invoke the following identities
(20)
for some matrices
, and proceed to get
(21)
In terms of
we cast (31) with 
into the form:
(22)
where 
are given by (15). If 
for all admissible uncertainties satisfying (2), then by Schur complements it follows from (32) that 
for any 
guaranteeing the internal stability. Proceeding further and to assure the closed-loop stability with 
-disturbance attenuation, we follow [15] to get:
(23)
when 
where
(24)
Next, by applying Fact 1, we obtain
(25)
for some scalars
. Note that
The quantities 
correspond to 
given by (13) after deleting the last element, and
(26)
where 
are given by (15). Further convexification of 
in (35) yields
(27)
By Schur complements using the algebraic inequality 
for any matrix
, the desired stability condition can then be cast into the LMI (11), which concludes the proof.
Remark 4.1 It is significant to observe that Theorem 4.1 provides a delay-dependent condition for the design of robust 
for fuzzy NCS in terms of feasibility testing of a family of strict LMIs with a total number of LMIvariables as
. The key feature is that the matrix gain 
is treated as a direct LMI variable. This will eventually lessen the conservatism in robust fuzzy control design.
Remark 4.2 It is worthy to note that the number of LMIs increases linearly with the number of rules 
which limits the applicability of the method for very large values of
. Had we used
then Theorem 4.1 reduces to the following corollary:
Corollary 4.1 Given the bounds 
and a scalar constants
, there exists a fuzzy controller in the form of (8), such that the uncertain closed-loop fuzzy system (9) with an 
disturbance attention level 
is asymptotically stable, if there exist matrices 
matrices 
and scalars 
satisfying
(28)
(29)
(30)
(31)
and the number of LMI variables would be
. The price paid is that the LKF becomes non-fuzzy.
5. Special Cases
In this section, we seek to derive a sufficient condition for the solvability of the robust 
control problem for NCS without quantizer.
NCS without Quantizer
In this case, the resulting closed-loop fuzzy system can be expressed as:
(32)
The corresponding control design is given by the following corollary:
Corollary 5.1 Given the bounds 
and a scalar constants
, there exists a fuzzy controller in the form of (8), such that the uncertain closed-loop fuzzy system (23)
with an 
disturbance attention level 
is asymptotically stable, if there exist matrices 
matrices 
and scalars 
satisfying
(33)
(34)
where the various terms are as in (13)-(15).
6. Example
In what follows, a typical simulation example is considered to illustrate the fuzzy controller design procedure developed in Theorem 4.1. A class of discrete-time fuzzy networked control systems model with state quantization is described by:
The membership functions for the rules 1, 2, 3 are
For the purpose of implementation, we consider the fuzzy system to be controlled through a network. A quantizer 
is selected to be of of logarithmic type with 
leading to 
. The bounds on data packet dropout are selected as 
respectively. Using the Matlab LMI solver, the feasible solution of
Theorem 4.1 yields the fuzzy 
state-feedback controller gains of the form:
The simulation results of the state and controlledoutput trajectories are plotted in Figures 1-3. It is quite evident that all the state and output variables of the fuzzy system settle at the equilibrium level within 20 sec.
7. Conclusion
We have addressed the problem of robust 
statefeedback controller design for discrete-time TakagiSugeno (T-S) fuzzy networked control systems including state quantization. A quantized feedback fuzzy controller has been designed under consideration of effect of network-induced delay and data dropout, and the timevarying quantizer has been selected to be logarithmic. By employing a fuzzy Lyapunov-Krasovskii functional, we have derived some LMI-based sufficient conditions for the existence of fuzzy controller. A numerical example has been given to illustrate the efficiency of the theoretic results.
8. Acknowledgements
The author would like to thank the deanship of scientific research (DSR) at KFUPM for financial support through research group project RG1105-1.