Mathematical Problems in Engineering

Volume 2008 (2008), Article ID 761342, 9 pages

http://dx.doi.org/10.1155/2008/761342

## Sliding Mode Control of Uncertain Neutral Stochastic Systems with Multiple Delays

^{1}Department of Automation, Shanghai Jiaotong University, Shanghai 200030, China^{2}Department of Mathematics, Shanghai Maritime University, Shanghai 200135, China

Received 31 August 2007; Accepted 4 March 2008

Academic Editor: Paulo Gonçalves

Copyright © 2008 Dilan Chen and Weidong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–6].

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 [7]. On the other hand, it has been shown that a lot of practical systems can be modeled by using functional differential equations of the neutral type [8, 9]. However, the mathematical model always contains some uncertain elements. Therefore, uncertain systems have been extensively studied in the past years [10–12].

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 [13–15].

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.

#### 2. Problem Formulation

Consider the following neutral stochastic system with uncertainties and multiple delays:where is the state, is the control input, is the constant delay, is the time-varying differentiable bounded delay satisfying , , is an -dimensional Brownian motion, . It is assumed that is the initial condition which is continuous, . In system (2.1), , , , , are known real constant matrices. , , , and represent the structured uncertainties in (2.1), which are assumed to be of the forms, , , , and are some given constant matrices, are unknown real time-varying matrices which have the following structure:

, , , , and , .

We define the sets aswhere , are invertible for , and , for .

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

Lemma 2.1 (See [1]). *The LMI , with , ,
is equivalent to*

Lemma 2.2 (See [11]). *Let , , be real matrices of appropriate dimensions and .
Then for any block-structured matrix ,*

Lemma 2.3 (See [11]). *Let , , , ,
and be real matrices of appropriate dimensions with and . Then for any block-structured matrix satisfying , one
has*

Lemma 2.4 (See [11]). *For any and for any symmetric positive-definite matrix , *

*Definition 2.5 (See [14]). *The nominal stochastic
time-delay system of form (2.1) with is said to be mean-square asymptotically
stable if

*Definition 2.6 (See [14]). *The uncertain time delay
system of the form (2.1) is robustly mean square stabilized if the nominal system
is mean-square asymptotically stable for all admissible uncertainties.

In order to simplify the treatment of the problem, the operator is defined to beThe stability of the operator is defined as follows.

*Definition 2.7 (See [9]). *The operator is said to be stable if the zero solution of
the homogeneous difference equationis uniformly asymptotically
stable.

If , then it is easy to find that there exist nonsingular constant matrices and , such thatwhere is an nonsingular matrix, and are and constant matrices, respectively.

Lemma 2.8 (See [9]). *The operator is stable if and , where , ,
and are defined as in (2.11) and is any matrix norm.*

#### 3. Switching Surface and Controller Design

In this work, we choose the switching function as follows:where the auxiliary variable satisfies the following:where and are constant matrices. The matrix is chosen such that the matrix is Hurwitz, and the matrix is to be designed later so that is nonsingular. As long as the system operates in the sliding mode, it satisfies the equations and [13].

Therefore, the equivalent control in the sliding manifold is given bySubstituting (3.3) into system (2.1), the following equivalent sliding mode dynamics can be obtained:Now, we proceed to the first task which is to analyze the robustly stochastic stability of the sliding motion described by (3.4), and derive a sufficient condition by means of the linear matrix inequality method.

#### 4. Robust Stabilization in the Mean Square Sense

Theorem 4.1. *Consider the equivalent sliding mode dynamics (3.4). If the operator is stable and there exist symmetric
positive-definite matrices , ,
, , , , , , and satisfying the following LMIs:
**where , , , , , **then the uncertain time delay
system of the form (2.1) with the switching surface (3.1) is robustly
stochastically stable and sliding mode matrix .
In the above LMIs, takes the form of for *

*Proof. *Choose a Lyapunov functional candidate asThen, the averaged derivative is
given by the following expression:Using Lemma 2.1, inequality (4.2)
is equivalent toHence, it follows from Lemma 2.3
thatNote that ,
and it follows form Lemma 2.4 thatSubstituting (4.7) and (4.8)
into (4.5), we obtainwhere , ,
Using Lemmas 2.1 and 2.2, we haveWith Lemma 2.1, we can see that is equivalent to LMIs (4.1)–(4.3).

According to It's formula, system (2.1) is robustly
stochastically stable. This completes the proof.

#### 5. Sliding Mode Control

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

Theorem 5.1. *Consider the uncertain stochastic time delay system (2.1). Suppose that the
switching function is given as (3.1) with ,
where is the solution of LMIs (4.1)–(4.3). Then the
reachability of the sliding surface can be guaranteed by the following SMC
law:**where the switching gain is given as**with .*

*Proof. *A Lyapunov functional candidate is defined asHence we haveThis completes the proof.

#### 6. An Illustrative Example

Consider neutral stochastic systems (2.1) withWe select matrix . Using Matlab LMI control toolbox to solve the LMIs (4.1)–(4.3), we obtain the following:

#### 7. Conclusions

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.

#### Acknowledgments

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.

#### References

- M. S. Mahmoud,
*Robust Control and Filtering for Time-Delay Systems*, vol. 5 of*Control Engineering*, Marcel Dekker, New York, NY, USA, 2000. View at Zentralblatt MATH · View at MathSciNet - J. Sun, “Delay-dependent stability criteria for time-delay chaotic systems via time-delay feedback control,”
*Chaos, Solitons & Fractals*, vol. 21, no. 1, pp. 143–150, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao, “Global stability conditions for delayed CNNs,”
*IEEE Transactions on Circuits and Systems I*, vol. 48, no. 11, pp. 1330–1333, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Zhang, “Stability analysis of linear neutral systems with multiple time delays,”
*Mathematical Problems in Engineering*, vol. 2005, no. 2, pp. 175–183, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao and J. Wang, “Delay-dependent robust stability of uncertain nonlinear systems with time delay,”
*Applied Mathematics and Computation*, vol. 154, no. 1, pp. 289–297, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ren and J. Cao, “Novel $\alpha $-stability criterion of linear systems with multiple time delays,”
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 282–290, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - E.-K. Boukas,
*Stochastic Hybrid Systems: Analysis and Design*, Birkhäuser, Boston, Mass, USA, 2005. - D. Ya. Khusainov, “Investigation of interval stability of linear systems of neutral type of Lyapunov function method,”
*Journal of Applied Mathematics and Stochastic Analysis*, vol. 15, no. 1, pp. 71–81, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Li, H.-B. Li, and S.-M. Zhong, “Stability of neutral type descriptor system with mixed delays,”
*Chaos, Solitons & Fractals*, vol. 33, no. 5, pp. 1796–1800, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - M. S. Mahmoud and L. Xie, “Passivity analysis and synthesis for uncertain time-delay systems,”
*Mathematical Problems in Engineering*, vol. 7, no. 5, pp. 455–484, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Lee and J. G. Lee, “Robust stability and stabilization of linear delayed systems with structured uncertainty,”
*Automatica*, vol. 35, no. 6, pp. 1149–1154, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Yu and J. Cao, “Robust control of uncertain stochastic recurrent neural networks with time-varying delay,”
*Neural Processing Letters*, vol. 26, no. 2, pp. 101–119, 2007. View at Publisher · View at Google Scholar - C. Edwards and S. K. Spurgeon,
*Sliding Mode Control: Theory and Applictions*, Taylor & Francis, New York, NY, USA, 1998. - Y. Niu, D. W. C. Ho, and J. Lam, “Robust integral sliding mode control for uncertain stochastic systems with time-varying delay,”
*Automatica*, vol. 41, no. 5, pp. 873–880, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Boutat-Baddas, J. P. Barbot, D. Boutat, and R. Tauleigne, “Sliding mode observers and observability singularity in chaotic synchronization,”
*Mathematical Problems in Engineering*, vol. 2004, no. 1, pp. 11–31, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet