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This paper deals with the problem of stabilization for a class of networked control systems (NCSs) with random time delay via the state feedback control. Both sensor-to-controller and controller-to-actuator delays are modeled as Markov processes, and the resulting closed-loop system is modeled as a Markovian jump linear system (MJLS). Based on Lyapunov stability theorem combined with Razumikhin-based technique, a new delay-dependent stochastic stability criterion in terms of bilinear matrix inequalities (BMIs) for the system is derived. A state feedback controller that makes the closed-loop system stochastically stable is designed, which can be solved by the proposed algorithm. Simulations are included to demonstrate the theoretical result.

Feedback control systems in which the control loops are closed through a real-time network are called networked control systems (NCSs) [

One of the main issues in NCSs is network-induced delays, which are usually the major causes for the deterioration of system performance and potential system instability [

On the other hand, the study of stochastic systems has attracted a great deal of attention [

The aim of this paper is to consider a class of networked control systems with sensors and actuators connected to a controller via two communication networks in the continuous-time domain. Two Markov processes are introduced to describe sensor-to-controller transmission delay and the controller-to-actuator transmission delay. Based on Lyapunov stability theorem, a method for designing a mode-dependent state feedback controller that stabilizes this class of networked control systems is proposed. The existence of such a controller is given in terms of BMIs, which can be solved by the proposed algorithm.

This paper is organized as follows. In Section

Consider linear systems described by the differential equation

The plant is interconnected by a controller over a communication network, see Figure

Illustration of NCSs over communication network.

Throughout the paper, the following assumption is needed for the considered networked control systems.

The switching difference of consecutive delays is less than one sampling interval, that is,

Although Assumption

According to Figure

Define the time delay

Illustration of the time delay.

Then, we have

Applying controller (

We have the following stochastic stability concept for system (

The system (

The following lemmas will be essential for the proofs in Section

Given any real matrices

For the delay functional differential equation,

Suppose that

The following theorem provides sufficient conditions for existence of a mode-dependent state feedback controller for the system (

Consider the closed-loop system (

Consider the following Lyapunov candidate:

In case of constant transmission delay, that is,

It should be noted that the terms

Set

For

For

Return to step 2 until the convergence of

For a given

In this section, simulations of the position control for robotic manipulator ViSHaRD3 [

For simplicity, we only discuss the third joint of ViSHaRD3. Suppose that the sampling interval is

Associated with modes 1 and 2, let the system have time delay

The simulations of the state response and the control input for the closed-loop system are depicted in Figures

State response of closed-loop system.

Control input of closed-loop system.

In this paper, a technique of designing a mode-dependent state feedback controller for networked control systems with random time delays has been proposed. The main contribution of this paper is that both the sensor-to-controller and controller-to-actuator delays have been taken into account. Two Markov processes have been used to model these two time delays. Based on Lyapunov stability theorem combined with Razumikhin-based technique, some new delay-dependent stability criteria in terms of BMIs for the system are derived. A state feedback controller that makes the closed-loop system stochastically stable is designed, which can be solved by the proposed algorithm. Simulations results are presented to illustrate the validity of the design methodology.

This work is partially supported by the Youth Fund of Anhui Province no. 2010SQRL162 and the program of science and technology of Huainan no. 2011A08005.