Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 382859, 9 pages
http://dx.doi.org/10.1155/2015/382859

Research Article

Control for Nonlinear Stochastic Systems with Time-Delay and Multiplicative Noise

Ming Gao,1,2 Weihai Zhang,² and Zhengmao Zhu³

¹College of Information and Control Engineering, China University of Petroleum (East China), Qingdao 266580, China
²College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China
³Institute of Science and Technology, North China Electric Power University, Beijing 102206, China

Received 8 August 2014; Accepted 26 September 2014

Academic Editor: Quanxin Zhu

Copyright © 2015 Ming Gao et al. 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 studies the infinite horizon control problem for a general class of nonlinear stochastic systems with time-delay and multiplicative noise. The exponential/asymptotic mean square control design of delayed nonlinear stochastic systems is presented by solving Hamilton-Jacobi inequalities. Two numerical examples are provided to show the effectiveness of the proposed design method.

1. Introduction

It is well known that control is one of the most effective approaches to eliminate the effect of the external disturbance [1]. For deterministic linear systems, norm is defined by a norm of the transfer function, which cannot be extended to stochastic or nonlinear systems directly. In 1989, Doyle et al. found that, from the view point of time-domain, the norm of a transfer function was the -induced norm of the input-output operator [2], which made it possible to develop the nonlinear or stochastic theory [3, 4]. Following along the lines of [4], Zhang and Chen developed infinite and finite horizon nonlinear stochastic control designs by means of Hamilton-Jacobi equations [5]. Moreover, the mixed control has also received much attention due to its important significance in practical applications [6].

The phenomena of time-delay are frequently encountered in many engineering applications owing to the finite speed of information processing [7]. Time-delay, nonlinearity, and stochasticity are arguably three of the main sources in reality which result in the complexity of a system. Over the past years, the stability of delayed nonlinear stochastic systems (DNSSs) has gained significant research interests [8–15]. In [8], Mao established the LaSalle-type theorems for the solutions of stochastic differential delay equations, which was applied to establish sufficient criteria for the stochastically asymptotic stability of the delay equations. In [10], the problem of exponential stability for a class of impulsive nonlinear stochastic differential equations with mixed time-delays was investigated, and some interesting results were derived. In [13], the delay-dependent stability conditions for DNSSs were derived based on the convergence theorem for semimartingale inequalities.

Although many results for the stability analysis of DNSSs have been published, the control problem of DNSSs has received relatively little attention [16–18]. In [16], the analysis problem was studied for a general class of nonlinear stochastic systems with time-delay by using the Razumikhin-type method. In [17], the problem of robust output feedback control was studied for a class of uncertain discrete-time DNSSs with missing measurements. In [18], the quantized control problem was investigated for delayed nonlinear stochastic network-based systems with data missing. However, most of the above literatures only considered the stochastic systems with state-dependent noise. As pointed in [19], the control input and external disturbance may also be corrupted by noise. Therefore, it is necessary to study the stochastic systems with state, control, and disturbance-dependent noise [20, 21].

Motivated by the preceding discussion, this paper will investigate the infinite horizon control for a class of nonlinear stochastic state-delayed systems with multiplicative noise. Compared with [16–18, 22], the considered system in this paper is more general since state, control, and disturbance enter into the diffusion term simultaneously. By means of Hamilton-Jacobi inequalities (HJIs), a sufficient condition is derived for the exponential and asymptotic mean square control of DNSSs, respectively. In contrast to the conditions for delay-free control [20, 21], the current HJIs depend on more variables owing to the appearance of time-delay. Finally, two numerical examples are given to demonstrate the effectiveness of the obtained results.

Throughout this paper, the following notations will be used. is -dimensional Euclidean space. is the set of all real matrices. is the transpose of a matrix .   : is a positive definite (positive semidefinite) symmetric matrix. is the mathematical expectation. is the Euclidean norm of a vector . is the space of nonanticipative stochastic processes with respect to an increasing -algebras    satisfying . is the class of functions twice continuously differential with respect to and once continuously differential with respect to , except possibly at the point . is a vector space of all continuous -valued functions defined on . : .

2. Definitions and Preliminaries

Consider the following delayed nonlinear stochastic system with multiplicative noise:

where , , , and represent the system state, control input, exogenous disturbance, and regulated output, respectively. is the one-dimensional standard Wiener process defined on a complete filtered space , a filtration satisfying usual conditions. denotes all -measurable bounded -valued random variable . Assume that , , , , , , and satisfy the local Lipschitz condition and the linear growth condition, which guarantee system (1) has a unique strong solution [23]. Moreover, suppose that ; hence is an equilibrium point of (1). For simplicity, we denote and .

For each , an operator associated with (1) is defined as follows [8]:

where , , and .

To deal with the infinite horizon control of system (1), the following internal stability is needed.

Definition 1 (see [23]). The delayed nonlinear stochastic system,

is exponentially mean square stable, if there exist positive constants and such that every solution of (3) satisfies

where .

Lemma 2 (see [24]). System (3) is exponentially mean square stable, if there exist a positive definite Lyapunov function and with such that

(i), ,(ii), .

Definition 3. For given , is said to be an exponential mean square control of system (1), if

(i)for any nonzero and , , one always has

(ii)system (1) with and is internally stable; that is, the system

is exponentially mean square stable.

Equation (5) is equivalent to , where the perturbation operator is defined by as

Definition 4. In (ii) of Definition 3, if the equilibrium point of system (6) is asymptotically mean square stable, that is,

and (5) holds, then is called an asymptotic mean square control.

Lemma 5 (see [25]). For a positive definite symmetric matrix and any matrices (or vectors) and , one has

Lemma 6 (see [21]). For any vectors and symmetric matrix , exists, and one has

3. Infinite Horizon Stochastic Control

In this section, several sufficient conditions are presented for the infinite horizon control of system (1) by using inequality technique.

Theorem 7. Assume that there exist a positive function and with such that

(i), ,(ii), .

For given , if   solves the Hamilton-Jacobi inequalities (HJIs)

then

is an exponential mean square control of (1).

Proof. Applying Itô’s formula to , we have

Taking mathematical expectation on both sides of (14), we obtain

where

Considering and Lemma 5, we have

Therefore,

where

Set

According to Lemma 6, and can be rewritten as

Implementing (21) and into (18) yields

According to (11), we have

Considering (12) and taking , (23) leads to

Let , and then (5) of Definition 3 is proved.
Next, we will prove system (6) to be exponentially mean square stable. Let be the infinitesimal generator of the system (6), and then

Setting

and implementing

into and , it yields

Substituting (28) into (25) and considering conditions (i), (ii), and (11) in Theorem 7, it follows that

From Lemma 2, system (6) is exponentially mean square stable. This theorem is proved.

The following theorem is derived for the asymptotic mean square control, which is weaker than the exponential mean square control.

Theorem 8. Assume that    has an infinitesimal upper limit; that is, and for some . If   solves HJIs (11)-(12), then (13) is an asymptotic mean square control of (1).

Proof. It only needs to prove that system (6) is asymptotically mean square stable when . We know that from (29), which implies that system (6) is globally asymptotically stable in probability 1 [26]. According to Itô formula and the property of stochastic integration, we obtain

Let , and then (30) leads to

which means that is a nonnegative supermartingale with respect to . According to Doob’s convergence theorem [27] and   a.s., we have   a.s. Furthermore, . Since for some , it yields that . The proof is completed.

Remark 9. In [24], Zhang et al. studied the robust filtering problem of nonlinear stochastic systems with time delay. However, the control problem was not tackled in [24], mainly due to mathematical difficulties in dealing with the case that state, control, and disturbance enter into the diffusion term simultaneously. In this paper, Lemma 6 is applied to solve this problem, and two sufficient conditions for control of delayed nonlinear stochastic systems are obtained in Theorems 7 and 8.

Remark 10. A further development of the present issue is twofold. On the one hand, in order to avoid solving HJIs (11) and (12), the global linearization approach [25] or fuzzy approach based on Takagi-Sugeno model [28] can be used to design control for delayed nonlinear stochastic systems. On the other hand, Lévy noise is more versatile and interesting with a wider range of applications in comparison to the standard Gaussian noise [29, 30]. Therefore, the control of stochastic differential equations with Lévy noise is another valuable research topic.

4. Numerical Examples

In this section, two numerical examples are given to illustrate the proposed control design.

Example 1. Consider the following one-dimensional nonlinear stochastic state-delayed system:

Set , to be determined, and then HJIs (11)-(12) become

Given , the above inequalities have a solution . From Theorem 7, the control of system (32) is .

The initial condition is chosen as for any with and . Applying the Euler-Maruyama method [31], the state responses of the unforced system and the controlled system and the control input are shown in Figures 1, 2, and 3, respectively. It is found that the controlled system can achieve stability and attenuation performance by using the proposed control.

Figure 1: The state of the unforced system in Example 1.

Figure 2: The state of the controlled system in Example 1.

Figure 3: The control input in Example 1.

Example 2. Consider a two-dimensional system (1) with the following parameters:

Take with to be determined. For a given disturbance attenuation level , is a solution to (11)-(12). According to Theorem 7, is an control of system (1). The initial condition is chosen as for any , and take . By using a similar method in Example 1, the states of the controlled system and the control input are shown in Figures 4-5, which show the effectiveness of the designed controller.

Figure 4: The states and of the controlled system in Example 2.

Figure 5: The control input in Example 2.

5. Conclusions

For general delayed nonlinear stochastic systems with state, control, and disturbance-dependent noise, this paper has presented a sufficient condition for exponential/asymptotic mean square control problem in terms of HJIs. There still remain many interesting topics, for example, how to derive delay-dependent conditions or how to design control for delayed nonlinear stochastic systems. These issues deserve further research.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (nos. 61174078, 61203053, and 61403420), China Postdoctoral Science Foundation Funded Project (no. 2013M531635), Special Funds for Postdoctoral Innovative Projects of Shandong Province (no. 201203096), Research Fund for the Taishan Scholar Project of Shandong Province of China, and SDUST Research Fund (no. 2011KYTD105).

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