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This paper is devoted to investigating sliding mode control (SMC) for Markovian switching singular systems with time-varying delays and nonlinear perturbations. The sliding mode controller is designed to guarantee that the nonlinear singular system is stochastically admissible and its trajectory can reach the sliding surface in finite time. By using Lyapunov functional method, some criteria on stochastically admissible are established in the form of linear matrix inequalities (LMIs). A numerical example is presented to illustrate the effectiveness and efficiency of the obtained results.

The sliding mode control (SMC) theory has made rapid progress since it was proposed by Utkin [

Singular systems, also referred to as descriptor systems, generalized state-space systems, differential-algebraic systems, or semistate systems, are more appropriate to describe the behavior of some practical systems, such as economic systems, power systems, and circuits systems, because singular systems mix up dynamic equations and static equations. Basic control theory for singular systems has been widely studied, such as stability and stabilization [

In practice, many physical systems may happen to have abrupt variations in their structure, due to random failures or repair of components, sudden environmental disturbances, changing subsystem interconnections, and abrupt variations in the operating points of a nonlinear plant. Therefore, Markovian jump systems have received increasing attention, see [

Motivated by the above discussion, we consider exponential stabilization for Markovian switching nonlinear singular systems via sliding mode control. First, we develop two lemmas. Based on these lemmas, delay-dependent sufficient condition on exponential stabilization for singular time-varying delay systems is given in terms of nonstrict LMIs. Some specified matrices are introduced and the non-strict LMIs are translated into strict LMIs which are easy to check by MATLAB LMI toolbox. Second, a sliding surface is derived using an equivalent control approach. A sliding mode controller is developed to drive the systems to the sliding surface in finite time and maintain a sliding motion thereafter. Finally, a numerical example is provided to show the effectiveness of the proposed result.

Consider the following singular system with time-varying delays, nonlinear perturbations, and Markovian switching:

Let

The nominal Markovian jump singular and time-delay system of system (

The initial Markovian jump singular system in (

Recall that the Markovian process

(i) The system (

(ii) The system (

For a given scalar

The system (

The system (

We will assume the followings to be valid.

The perturbation term

with

Lemma

Let

There exists symmetric matrix X such that

Let

Let

We give the following result for the stochastic admissibility of the system (

The Markovian jump singular system (

For a prescribed scalars

First to prove the system

From (

By pre- and postmultiplying (

Based on Lemma

Second, to prove the system (

Then, let

With (

By Dynkin's formula, we get

Therefore by Definition

SMC design involves two basic steps: sliding surface design and controller design. For every

Therefore, by

Substituting

In this section, we pay attention to establishing

Given scalars

Note that the conditions in Theorem

Given scalars

Let

Using Lemma

By pre- and postmultiplying (

In light of Lemma

It is easy to know that

Using Shur's complement, the following inequality can ensure (

There exists a matrix

The above inequality is equivalent to the following inequality:

And the following inequality can guarantee that (

By Shur's complement, the right of (

From the proof of the Theorem

After switching surface design, the next important part of sliding mode control is to design a slide mode controller to guarantee the existence of a sliding mode. Now, we design an SMC law, by which the trajectories of singular system (

With the constant matrix

Choose

Differentiating

In this section, a numerical example is presented to illustrate the effectiveness of the main results in this paper.

Let us consider the system (

In addition,

For

Thus the sliding surface function is

Take

State trajectories of the open-loop system.

Switching surface

State trajectories of the closed-loop system.

Controller input

State trajectories of sliding mode.

Obviously, our results include Markovian switching and nonlinear perturbation effects, and this model can not be dealt with by the results of [

In this paper, the stochastically admissible using sliding mode control for singular system with time-varying delay and nonlinear perturbations is studied by LMI method. The sliding mode control is designed to ensure that the closed-loop system is stochastically admissible. A numerical example demonstrates the effectiveness of the method mentioned above.

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and constructive suggestions. This research is supported by the Natural Science Foundation of Guangxi Autonomous Region (no. 2012GXNSFBA053003), the National Natural Science Foundations of China (60973048, 61272077, 60974025, 60673101, 60939003), National 863 Plan Project (2008 AA04Z401, 2009AA043404), the Natural Science Foundation of Shandong Province (no. Y2007G30), the Scientific and Technological Project of Shandong Province (no. 2007GG3WZ04016), the Science Foundation of Harbin Institute of Technology (Weihai) (HIT(WH) 200807), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF. 2001120), the China Postdoctoral Science Foundation (2010048 1000), and the Shandong Provincial Key Laboratory of Industrial Control Technique (Qingdao University).