Exponential L 2L ∞ Filtering for a Class of Stochastic System with Mixed Delays and Nonlinear Perturbations

The delay-dependent exponential L 2 -L ∞ 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 delaydependent sufficient condition for the existence of the L 2 -L ∞ 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 L 2 -L ∞ performance γ. Numerical examples are provided to illustrate the effectiveness and significant improvement of the proposed method.


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][2][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][5][6][7][8][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][11][12], distributed time-delay system [13,14], and mixed (including both discrete and distributed time delays) system [15][16][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][19][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][23][24][25][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  2 / ∞ filtering and  2 - ∞ filtering for stochastic time-delay systems have been widely studied [8,9,[29][30][31][32][33][34][35][36][37][38].In  2 / ∞ filtering, and  2 - ∞ filtering problems, the external disturbances are Notations.Throughout this paper,  > 0 ( < 0) means that the matrix  is positive definite (negative definite).R  denotes the -dimensional Euclidean space; R × is the set of all  ×  real matrices; L 2 [0, ∞) is the space of square-integrable vector functions over [0, ∞).The superscript "" represents the transpose; " * " denotes the symmetric terms in a matrix; diag( ) denotes a block-diagonal matrix;  max ( ) and  min ( ) denote the maximum eigenvalue and minimum eigenvalue, respectively.sym() =  +   ; | ⋅ | denotes the Euclidean vector norm; ‖ ⋅ ‖ 2 stands for the usual L 2 [0, ∞) norm.(Ω, F, P) is a probability space with Ω the sample space, F the -algebra of subsets of Ω, and P the probability measure on F. E{⋅} denotes the expectation operator with respect to some probability measure P. 0 and I represent zero matrix and identity matrix with appropriate dimensions, respectively, unless we say otherwise.

Problem Formulation
Consider the following stochastic systems with mixed delays and nonlinear perturbations: where () ∈ R  is the state; () ∈ R  is the measured output; () ∈ R  is the signal to be estimated; V() ∈ R  is the disturbance input which belongs to L 2 [0, ∞), which is the space of square-integrable vector functions; () is a onedimensional Brownian motion defined on a complete probability space (Ω, F, P) satisfying E{()} = 0 and E{ where  1 ∈ R × and  2 ∈ R × are known constant matrices; (⋅, ⋅, ⋅): where  1 ∈ R × and  2 ∈ R × are known constant matrices.
For system (1), we are interested in constructing the following full-order linear filter: where   () ∈ R  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  2 - ∞ 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 V() = 0 is exponentially stable [39]; (ii) under the zero initial condition, the filtering error system ( 6) is stochastically asymptotically stable and achieves a prescribed  2 - ∞ attenuation level .The filtering error () satisfies with Before presenting the main results of this paper, we introduce the following lemmas, which will be essential to our derivation.

Filtering Performance Analysis
In this section, a new delay-dependent condition of the  2 - ∞ 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  > 0, ℎ > 0,  > 0,  > 0, and  > 0 and integers  1 ≥ 1 and  2 ≥ 1, there exists a linear filter (4) such that the filtering error system (6) is stochastically asymptotically stable with a guaranteed  2 - ∞ performance , if there exist symmetrical positive definite matrices where Proof.First, show the asymptotic stability of system (6) with V() = 0.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 L (  , ) = 2  ()  () + trace (  ()  ()) 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  > 0 such that where we take  for ((), ( − ℎ), ), for simplicity of notation.
By Dynkin's formula, there exists  > 0, such that Recalling the Lyapunov-Krasovskii functional in (18), notice the fact that there always exists  > 0 satisfying for any − ≤  ≤ 0 such that where From ( 32) and ( 33), it can be easily obtained that where  = ∑ 5 =1   / min () and φ is the initial condition of filtering error system (6).Then by exponential stability definition of stochastic systems [39], the filtering error system (6) with V() = 0 is exponentially stable in the sense of mean square.Now, we will establish the  2 - ∞ performance for the filtering error system (6).To this end, we assume the zero initial condition () = 0 for  ∈ [−, 0].Under the initial condition, it is easy to see that, for any  > 0, Define Then, for any nonzero V() ∈ L 2 [0, ∞) and  > 0, combined with ( 29), ( 35)-( 36), we have where 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  2 - ∞ performance  under zero initial condition.This completes the proof.
In system (1), if  3 = 0 and ((), ( − ℎ), ) = () +  1 ( − ℎ) +  2 ∫  − () +  V V(), 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  2 - ∞ performance .

Filter Design
In this section, we will focus on the design of  2 - ∞ 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  2 - ∞ performance .
On the other hand, (44) implies Pre-and postmultiply (52) by Π − and Π −1 , respectively.Notice that  = ΠΛ −1 Π  , 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  2 - ∞ performance .This completes the proof.
Following the similar method in Theorem 6, one can obtain a result of filter design for linear stochastic time-delay system (41).

Numerical Examples
Example 1.Consider the stochastic time-delay system (1) with parameters Moreover, for the nonlinear functions, we let  1 =  2 = 0.1 and  1 =  2 = 0.1.Given ℎ = 1 and  = 0.2, from Theorem 6, one can obtain the upper bound of time delay , which is listed in Table 1.
In the case of  1 = 2 and  2 = 2, the desired filter parameters can be obtained: (59) Given  = 1 and  = 0.2, from Corollary 8, one can obtain the upper bound of time delay ℎ.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  1 = 2 and  2 = 2, the desired filter parameters can be obtained: (60) 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.

Conclusions
In this paper, a new approach has been developed to investigate the problems of delay-dependent  2 - ∞ 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  2 - ∞ 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.

Table 1 :
The upper bound of  max for ℎ = 1 and  = 0.2.