#### Abstract

The delay-dependent exponential performance analysis and filter design are investigated for stochastic systems with mixed delays and nonlinear perturbations. Based on the delay partitioning and integral partitioning technique, an improved delay-dependent sufficient condition for the existence of the filter is established, by choosing an appropriate Lyapunov-Krasovskii functional and constructing a new integral inequality. The full-order filter design approaches are obtained in terms of linear matrix inequalities (LMIs). By solving the LMIs and using matrix decomposition, the desired filter gains can be obtained, which ensure that the filter error system is exponentially stable with a prescribed performance . Numerical examples are provided to illustrate the effectiveness and significant improvement of the proposed method.

#### 1. Introduction

Time delays are quite often encountered in various practical engineering systems, and they are regarded as one of the main sources causing instability and degrading performance of control systems [1–3]. Over the past decades, numerous results and various approaches on delay systems have been reported in the literatures. Many researchers have focused on the stability analysis, stabilization, and filtering for time-delay systems; see [4–9] and the references therein. Time delays are usually classified into discrete delays and distributed delays. In the existing literatures, discrete time-delay system [10–12], distributed time-delay system [13, 14], and mixed (including both discrete and distributed time delays) system [15–17] are considered.

Since certain unavoidable stochastic perturbations are widely existing in many engineering systems, stochastic systems have gained considerable research attention over the past few years [18–20]. Stochastic dynamic modeling has come to play an important role in many fields of science and engineering. In the past years, many researchers have focused on the problems of stability and stabilization of stochastic time-delay systems. For instance, robust stabilization for a class of large-scale stochastic systems was investigated in [21], delay-dependent stability results for stochastic systems were presented in [22–26], and state feedback control and dynamic output feedback control for uncertain stochastic time-delay systems were investigated in [27, 28], respectively.

In the field of stochastic dynamic system with time delays, the filtering problem, which is to estimate the unavailable state of variables of a given control system, is also an important issue. Kalman filtering scheme is a well-known effective way to deal with the filtering problem. However, it has some limitations in practical applications due to the fact that it assumes that the system and its disturbances are exactly known, that is, stationary Gaussian noised with known statistics. Under this view, recently, filtering, mixed filtering and filtering for stochastic time-delay systems have been widely studied [8, 9, 29–38]. In filtering, and filtering problems, the external disturbances are assumed to be bounded. In filtering problem, it requires that the filtering error systems satisfy not only a prescribed disturbance attenuation level but also the performance (minimum of the norm of transfer function of the filter error systems), while in filtering problem, it requires that the filtering error systems satisfy a prescribed disturbance attenuation level. filtering and mixed filtering problems of nonlinear stochastic systems are investigated in [30, 31]. In [32], a delay-independent robust filtering design approach for uncertain stochastic time-delay system is investigated. It is well known that the delay-independent results are generally more conservative than the delay-dependent ones. Authors in [33–35] developed delay-dependent filtering for stochastic time-delay systems. Authors in [36] proposed a delay-dependent filter design approach for stochastic time-delay systems, based on a delay partitioning technique presented in [37]. As the results showed, delay-partitioning can reduce conservatism to some extent. Authors in [38] investigated the problem of robust filtering for stochastic systems with both discrete and distributed delays. Although the filtering problems for stochastic systems with time delays have been well investigated in the aforementioned literatures, most of them are dealing with linear stochastic time-delay systems. To the authors’ knowledge, the filtering problems of stochastic time-delay systems with nonlinear perturbation are still insufficient. This motivates the authors to deal with the filtering problem of a class of nonlinear stochastic time-delay systems.

This paper focuses on the problems of delay-dependent filtering for stochastic systems with mixed delays and nonlinear perturbations. By Lyapunov-Krasovskii approach based on the delay partitioning and integral partitioning technique, we first develop a delay-dependent sufficient condition for performance analysis. And then, an improved delay-dependent sufficient condition is obtained for the existence of desired filter in the form of linear matrix inequalities (LMIs). The performance analysis and filter design of linear stochastic system with mixed delays are also investigated. Finally, numerical examples are provided to show that the proposed method is effective and less conservative than some existing literatures.

*Notations*. Throughout this paper, means that the matrix is positive definite (negative definite). denotes the -dimensional Euclidean space; is the set of all real matrices; is the space of square-integrable vector functions over . The superscript “” represents the transpose; “” denotes the symmetric terms in a matrix; denotes a block-diagonal matrix; and denote the maximum eigenvalue and minimum eigenvalue, respectively. ; denotes the Euclidean vector norm; stands for the usual norm. is a probability space with the sample space, the -algebra of subsets of , and the probability measure on . denotes the expectation operator with respect to some probability measure . and represent zero matrix and identity matrix with appropriate dimensions, respectively, unless we say otherwise.

#### 2. Problem Formulation

Consider the following stochastic systems with mixed delays and nonlinear perturbations: where is the state; is the measured output; is the signal to be estimated; is the disturbance input which belongs to , which is the space of square-integrable vector functions; is a one-dimensional Brownian motion defined on a complete probability space satisfying and ; is an initial function that is continuous on with . and are discrete and distributed constant delays, respectively. is a nonlinear function, which satisfies where and are known constant matrices; is a nonlinear perturbance input function, satisfying where and are known constant matrices.

For system (1), we are interested in constructing the following full-order linear filter: where is the filter state; , , and are filter matrices to be determined.

Define Then, the filtering error system can be written as where

The objective of this paper is to design full-order filter (4) for the stochastic time-delay system (1) such that the filtering error system (6) satisfies the following two requirements:(i)the filtering error system (6) with is exponentially stable [39];(ii)under the zero initial condition, the filtering error system (6) is stochastically asymptotically stable and achieves a prescribed attenuation level . The filtering error satisfies with , for any nonzero .

Before presenting the main results of this paper, we introduce the following lemmas, which will be essential to our derivation.

Lemma 1 (see [40]). *For a given symmetrical matrix , where , and , the linear matrix inequality is equivalent to
*

Lemma 2 (see [1]). *For any positive symmetric matrix , scalars and satisfying , a vector function , one has
*

Lemma 3 (see [14]). *For any positive symmetric matrix , scalars and satisfying , a vector function , one has
*

#### 3. Filtering Performance Analysis

In this section, a new delay-dependent condition of the filtering performance analysis for system (1) will be presented. A Lyapunov-Krasovskii functional is constructed; based on the idea of delay partitioning and integral partitioning, the conservatism will be reduced. For the convenience of expression, assume that the filter matrices (, , and ) are known.

Theorem 4. *Consider the stochastic time-delay system (1). For given scalars , , , , and and integers and , there exists a linear filter (4) such that the filtering error system (6) is stochastically asymptotically stable with a guaranteed performance γ, if there exist symmetrical positive definite matrices , , , , and and matrix satisfying
*

*where*

*Proof. *First, show the asymptotic stability of system (6) with . For simplicity of notations, rewrite the filtering error system (6) as
where

Next, denote , and choose the following Lyapunov-Krasovskii functional:
Then, by Itô differential formula, the stochastic differential along the trajectories of system (6) is
where
By Lemma 2, we have
By Lemma 3, we have
From (16), for any appropriately dimensioned matrix , we have
On the other hand, (2) implies that there exists such that
where we take for , for simplicity of notation.

Notice the fact of (3), and from (13), we have
Combine (20)–(26); then
where

Thus,

By Schur complement lemma, it is easy to show that implies . Combined with (29), these imply that, for any , we have .

By Dynkin’s formula, there exists , such that
Recalling the Lyapunov-Krasovskii functional in (18), notice the fact that there always exists satisfying
for any such that
where , , , , and .

On the other hand, from (18)
From (32) and (33), it can be easily obtained that
where and is the initial condition of filtering error system (6). Then by exponential stability definition of stochastic systems [39], the filtering error system (6) with is exponentially stable in the sense of mean square.

Now, we will establish the performance for the filtering error system (6). To this end, we assume the zero initial condition for . Under the initial condition, it is easy to see that, for any ,

Define
Then, for any nonzero and , combined with (29), (35)-(36), we have
where . ensuring that . Thus,

Moreover, by Schur complement, (14) holds if and only if
It follows from (38) and (39) that

Therefore, if (12)–(14) hold, the filtering error system (6) is mean-square exponentially stable with a prescribed performance under zero initial condition. This completes the proof.

In system (1), if and , then the linear stochastic system with mixed delays can be written as which is the same as the system in [38] with constant delays. Thus, following the similar lines in Theorem 4, a sufficient condition can be obtained guaranteeing that there exists a linear filter (4) such that the filtering error system is exponentially stable and achieves a prescribed performance .

Corollary 5. *Consider the stochastic time-delay system (41). For given scalars , , and and integers and , there exists a linear filter (4) such that the corresponding filtering error system is exponentially stable with a guaranteed performance , if there exist symmetrical positive definite matrices , , , , and and matrix satisfying (14) and
**
where , , and .*

#### 4. Filter Design

In this section, we will focus on the design of filter for stochastic system (1). Based on Theorem 4, a delay-dependent sufficient condition will be obtained in the forms of LMI, which ensures that the filtering error system (6) is stochastically asymptotically stable and achieves a prescribed performance .

Theorem 6. *Consider the stochastic time-delay system (1). For given scalars , , , , and and integers and , there exists a linear filter (4) such that the filtering error system (6) is stochastically asymptotically stable with a prescribed performance , if there exist symmetrical positive definite matrices , , , , , and and matrices , , , and satisfying
**
where
**In this case, the parameters of a desired filter in the form of (4) can be taken as
**
where and are nonsingular matrices satisfying , , and .*

*Proof. *From (46), it can be seen that , and . For any positive definite and symmetric matrix , one can always find nonsingular matrix and , such that .

Set
Then , and .

Define
Then

Substitute , , , and into (43), (44), and (46) and then pre- and postmultiply (43) by and its transpose, respectively. Premultiply and postmultiply (46) by and its transpose, respectively. Noticing that , using Schur complement Lemma, one can obtain (12) and (14).

On the other hand, (44) implies
Pre- and postmultiply (52) by and , respectively. Notice that , one can obtain
By (45), it is easy to see that
So, (13) is satisfied.

Therefore, by Theorem 4, the suitable filter parameters can be constructed by (48), which ensures the filtering error system (6) to be stochastically asymptotically stable with performance . This completes the proof.

*Remark 7. *When deriving the results in Theorem 6 based on Theorem 4, considering dealing with the LMI (13), we give a method to avoid nonlinear terms emerging. Using Matlab LMI toolbox, one can solve linear matrix inequalities (43)-(44) and (46). Then, by matrix diagonalization approach, one can easily find that diagonally positive matrix and nonsingular matrix satisfy . If the obtained matrix does not satisfy (45), one can take for and for . Thus, the desired filter parameters can be obtained by (48).

Following the similar method in Theorem 6, one can obtain a result of filter design for linear stochastic time-delay system (41).

Corollary 8. *Consider the stochastic time-delay system (41). For given scalars , , and and integers and , there exists a linear filter (4) such that the corresponding filtering error system is stochastically asymptotically stable with a prescribed performance γ, if there exist symmetrical positive definite , , , , , and and matrices , , , and satisfying (46) and
*

*where*

*Remark 9. *The results presented in Theorem 6 and Corollary 8 can be easily extended to the systems with only discrete or distributed delays and also to the robust performance analysis for uncertain stochastic systems with mixed delays.

#### 5. Numerical Examples

*Example 1. *Consider the stochastic time-delay system (1) with parameters

Moreover, for the nonlinear functions, we let and . Given and , from Theorem 6, one can obtain the upper bound of time delay , which is listed in Table 1.

In the case of and , the desired filter parameters can be obtained:

*Example 2. *Consider the stochastic time-delay system (36)–(39) with parameters

Given and , from Corollary 8, one can obtain the upper bound of time delay . Table 2 lists the results of Corollary 8 and [38] with constant delays. It is easy to see that the proposed filter design method in this paper is less conservative than [38].

From Corollary 8, in the case of and , the desired filter parameters can be obtained:

*Remark 3. *It can be seen from the results that the conservatism can be reduced with the increase of partition integers. However, it is necessary to point out that the less conservatism is at the cost of a higher computational complexity.

#### 6. Conclusions

In this paper, a new approach has been developed to investigate the problems of delay-dependent filter design for stochastic system with mixed delays and nonlinear perturbations. Based on the idea of delay partitioning and integral partitioning, using Lyapunov-Krasovskii functional approach, a delay-dependent sufficient condition has been established that ensures the filtering error system is exponentially stable with performance . By solving the LMIs, one can get the desired filter gain matrices. The results also depend on the partition integers with the increase of partition integers, the conservatism can be decreased. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed approach.

#### Acknowledgments

The project is supported by Nature Science Foundation of China (NSFC, Grant no. 51277022). The authors would also like to thank the editor and the anonymous reviewers for their valuable comments and suggestions which helped to improve the quality of this paper.