This paper is concerned with the state feedback stabilization problem for a class of Takagi-Sugeno (T-S) fuzzy networked control systems (NCSs) with random time delays. A delay-dependent fuzzy networked controller is constructed, where the control parameters are ndependent on both sensor-to-controller delay and controller-to-actuator delay simultaneously. The resulting NCS is transformed into a discrete-time fuzzy switched system, and under this framework, the stability conditions of the closed-loop NCS are derived by defining a multiple delay-dependent Lyapunov function. Based on the derived stability conditions, the stabilizing fuzzy networked controller design method is also provided. Finally, simulation results are given to illustrate the effectiveness of the obtained results.

During the past decades, Fuzzy control technique has been widely developed and used in many scientific applications and engineering systems. Especially, the so-called Takagi-Sugeno (T-S) fuzzy model has been well recognized as an effective method in approximating complex nonlinear system and has been widely used in many real-world physical systems. In T-S fuzzy model, local dynamics in different state space regions are represented by different linear models, and the overall model of the system is achieved by fuzzy “blending” of these fuzzy models. Under this framework, the controller design of nonlinear system can be carried out by utilizing the well-known parallel distributed compensation (PDC) scheme. As a result, the fruitful linear system theory can be readily extended to the analysis and controller synthesis of nonlinear systems. Therefore, the last decades have witnessed a rapidly growing interest in T-S fuzzy systems, with many important results reported in the literature. For more details on this topic, we refer the readers to [

However, it is worth noting that in traditional T-S fuzzy control systems, system components are located in the same place and connected by point-to-point wiring, where an implicit assumption is that the plant measurements and the control signals transmitted between the physical plant and the controller do not exhibit time delays. However, in many modern control systems, it is difficult to do so, and thus the plant measurements and control signals might be transmitted from one place to another. In this situation, communication networks such as Internet are used to connect the spatially distributed system components, which gives rise to the so-called networked control systems (NCSs) [

Among the aforementioned problems, time delay is one of the most important ones, since time delay is usually the major cause for NCSs performance deterioration and potential system instability. Therefore, the analysis and synthesis of NCSs with time delays have been the focus of some research studies in recent years, with many interesting results reported in the literature; see [

Therefore the intention of this paper is to investigate the two-mode-dependent for a class of nonlinear NCSs with time delays, where the remote controlled plant is described by T-S fuzzy model. A

^{T}” denotes matrix transposition, and

In this paper, we consider the state feedback stabilization problem for a class of discrete-time nonlinear NCSs, where the corresponding system framework is depicted in Figure

The structure of the considered NCSs.

In the NCSs under study, the dynamics of the controlled plant are described by the T-S fuzzy model and can be represented by the following form:

By using the fuzzy inference method with a center-average defuzzifier, product inference, and singleton fuzzifier, the controlled plant in (

It is assumed that

It is worth mentioning that the sensor in NCSs is time-driven, and it is assumed that the full state variables are available. At each sampling period, the sampled plant state and its timestamp (i.e., the time the plant state is sampled) are encapsulated into a packet and sent to the controller via the network.

Networks exist in both channels from the sensor to the controller and from the controller to the actuator. The sensor packet will suffer a sensor-to-controller (S-C) delay during its transmission from the sensor to the controller, while the control packet will suffer a controller-to-actuator (C-A) delay during its transmission from the controller to the actuator. For notation simplicity, let

Please note that the control signal in NCSs suffers the S-C delay

The networked controller is time-driven. At each sampling period, it calculates the control signals with the most recent sensor packet available. Immediately after the calculation, the new control signals and the timestamp of the used plant state are encapsulated into a packet and sent to the actuator via the network. The timestamp will ensure that the actuator selects the appropriate control signal to control the plant.

The actuator in NCS is time-driven. The actuator and the sensor have the same sampling period

It is worth noting that when the networked controller (

The objective of this paper is to design the fuzzy networked controller (

For the convenience of notation, we let

One can readily infer from

For notation convenience, we define the following matrix variable:

Apparently, the most appealing advantage of the proposed networked controller (

Before proceeding further, we introduce the following definition, and it will be used throughout this paper.

The delays in NCSs are called arbitrary bounded delays, if

In the following theorem, the stability conditions are derived for NCS (

The closed-loop NCS (

For NCS (

Let

On the other hand, by applying Schur complement to (

For (

Then, it follows from (

Therefore, if the conditions (

Now, we are in a position to present the stabilizing controller design method. To this end, we proposed equivalent stability conditions for NCSs in the following theorem.

The closed-loop NCS (

Condition (

Substituting (

Note that the conditions stated in Theorem

It has been demonstrate that delay-dependent strategy is an effective way to improve the control performance and reduce the conservatism of NCSs. Therefore, the stabilization of NCSs with time delays and/or packet losses, either under sensor-to-controller (SCC) delay-dependent strategy or under two sides delay-dependent strategy (i.e., the control parameter depends on sensor-to-controller (S-C) delay and controller-to-actuator (C-A) delay simultaneously), has received a lot of attentions [

It is not difficult to see that if we consider a fuzzy controller with delay-independent gains and define the following matrix variable:

Then by following similar lines in proof of Theorem

The closed-loop NCS (

One can readily infer that, by remaining the control parameter constant (i.e.,

To make our idea more lucid, in this paper, we only consider the stabilization case under a simple framework. However, it is worth mentioning that the previous derived results can be easily extended to the robust control case or

In this section, an illustrative example will be presented to demonstrate the effectiveness of the proposed approach. To this end, let us consider an NCS shown in Figure

In this scenario, the random delays are set to

With the initial state

Typical simulation results using the proposed networked controller.

The corresponding network conditions.

Sensor-controller random delays

Controller-actuator random delays

Then to further illustrate the advantage of the proposed method, let us consider the networked system with the delay-independent controller. To this end, we applied Corollary

Typical simulation results using the proposed networked controller.

This paper presents a delay-dependent state feedback stabilization method for a class of T-S fuzzy NCSs with random time delays. A two-mode-dependent fuzzy controller is constructed, and the resulting NCSs is transformed into discrete-time fuzzy switched system. Under this framework, the stability conditions are derived for the closed-loop NCS, and the corresponding stabilizing controller design method is also provided. The main advantage of the proposed method is that the control signal computation can effectively employ most recent delay information, and therefore better control performance of NCSs could be obtained. Simulation and experimental results are given to illustrate the effectiveness of the obtained results. In the future work, we will consider more performance requirements such as

The authors would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. The work of H. Li was supported by National Basic Research Program of China (973 Program) under Grant 2012CB821206, the National Natural Science Foundation of China under Grant 61004021, and Beijing Natural Science Foundation under Grant 4122037. The work of Z. Sun was supported in part by the National Natural Science Foundation of China under Grants 61174069, 61174103, and 61004023.