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This paper is concerned with the sliding mode control for uncertain stochastic neutral systems with multiple delays. A switching surface is adopted first. Then, by means of linear matrix inequalities (LMIs), a sufficient condition is derived to ensure the global stochastic stability of the stochastic system in the sliding mode for all admissible uncertainties. The synthesized sliding mode controller guarantees the existence of the sliding mode.

Time delay
occurs due to the finite capabilities of information processing and data
transmission among various parts of the system. The phenomena of time delay are
often encountered in various relevant systems, such as HIV infection with drug
therapy, aircraft stabilization, chemical engineering systems, inferred
grinding model, manual control, neural network, nuclear reactor, population
dynamic model, rolling mill, ship stabilization, and systems with lossless
transmission lines. It is well known that time delay factors always lead to
poor performance. Hence, problems of stability analysis and stabilization of
dynamical systems with time delays in the state variables and/or control inputs
have received considerable interest for more than three decades [

In practice, systems are almost always innately
“noisy”. Therefore, in order to model a system realistically, a degree of
randomness must be incorporated into the model. Thus, a class of stochastic
systems has received great attention in the past decade [

To cope with the problem of stability of uncertain stochastic neutral delay systems, most of the research focused on the retarded functional differential equations and it also seems that few results are available on the variable structure control.

Sliding mode control (SMC) is a particular type of
variable structure control. It provides an effective alternative to deal with
the nonlinear dynamic systems. The main feature of SMC is its easy realization,
control of independent motion, insensitivity to variation in plant parameters
or external perturbations, and wide variety of operational models [

The purpose of this paper lies in the design of SMC for a class of uncertain stochastic neutral delay systems. A switching surface, which makes it easy to guarantee the stability of the uncertain stochastic neutral delay systems in the sliding mode, is first proposed. By means of linear matrix inequalities (LMIs), a sufficient condition is given such that the stochastic dynamics in the specified switching surface is globally stochastically stable. And then, based on this switching surface, a synthesized SMC law is derived to guarantee the existence of the composite sliding motion. Finally, a numerical example is illustrated to demonstrate the validity of the proposed SMC.

Consider the
following neutral stochastic system with uncertainties and multiple
delays:

We define the sets

The following useful lemmas will be used to derive the desired LMI-based stability criteria.

The LMI

Let

Let

For any

The nominal stochastic
time-delay system of form (

The uncertain time delay
system of the form (

In order to simplify the treatment of the problem, the
operator

The operator

If

The operator

In this work,
we choose the switching function as follows:

Therefore, the equivalent control

Consider the equivalent sliding mode dynamics (

Choose a Lyapunov functional candidate

According to It

We now design an SMC law such that the reachability of the specified switching surface is ensured.

Consider the uncertain stochastic time delay system (

A Lyapunov functional candidate

Consider
neutral stochastic systems (

In this paper, we have investigated the sliding mode control problem for uncertain stochastic neutral systems with multiple delays. The stability criteria are expressed by means of LMIs, which can be readily tested by some standard numerical packages. Therefore, the developed result is practical.

This work is supported in part by National Science Foundation of China (60474031), NCET (04-0383), National 973 Key Fundamental Research Program (2002cb312200-3), and Australia-China Special Fund for Scientific and Technological Cooperation.