Exponential 𝐿₂-𝐿_∞ Filtering for a Class of Stochastic System with Mixed Delays and Nonlinear Perturbations

Abstract

The delay-dependent exponential 𝐿₂-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 delay-dependent sufficient condition for the existence of the 𝐿₂-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 𝐿₂-𝐿_∞ performance 𝛾. 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 𝐻_∞ state feedback control and 𝐻_∞ 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, 𝐻_∞ filtering, mixed 𝐻₂/𝐻_∞ filtering and 𝐿₂-𝐿_∞ filtering for stochastic time-delay systems have been widely studied [8, 9, 29–38]. In 𝐻₂/𝐻_∞ filtering, and 𝐿₂-𝐿_∞ filtering problems, the external disturbances are assumed to be bounded. In 𝐻₂/𝐻_∞ filtering problem, it requires that the filtering error systems satisfy not only a prescribed 𝐻_∞ disturbance attenuation level but also the 𝐻₂ performance (minimum of the 𝐻₂ norm of transfer function of the filter error systems), while in 𝐿₂-𝐿_∞ filtering problem, it requires that the filtering error systems satisfy a prescribed 𝐿₂-𝐿_∞ disturbance attenuation level. 𝐻_∞ filtering and mixed 𝐻₂/𝐻_∞ filtering problems of nonlinear stochastic systems are investigated in [30, 31]. In [32], a delay-independent robust 𝐿₂-𝐿_∞ 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 𝐿₂-𝐿_∞ 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 𝐿₂-𝐿_∞ 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 𝐿₂-𝐿_∞ filtering problems of stochastic time-delay systems with nonlinear perturbation are still insufficient. This motivates the authors to deal with the 𝐿₂-𝐿_∞ filtering problem of a class of nonlinear stochastic time-delay systems.

This paper focuses on the problems of delay-dependent 𝐿₂-𝐿_∞ 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 𝐿₂-𝐿_∞ 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 𝐿₂-𝐿_∞ 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, 𝑋>0(𝑋<0) means that the matrix 𝑋 is positive definite (negative definite). 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space; 𝐑^𝑚×𝑛 is the set of all 𝑚×𝑛 real matrices; 𝐋₂[0,∞) is the space of square-integrable vector functions over [0,∞). The superscript “𝑇” represents the transpose; “∗” denotes the symmetric terms in a matrix; diag() denotes a block-diagonal matrix; 𝜆_max() and 𝜆_min() denote the maximum eigenvalue and minimum eigenvalue, respectively. sym(𝑋)=𝑋+𝑋^𝑇; |⋅| denotes the Euclidean vector norm; ‖⋅‖₂ stands for the usual 𝐋₂[0,∞) norm. (𝛀,𝐅,𝐏) is a probability space with Ω the sample space, 𝐅 the 𝜎-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.

2. Problem Formulation

Consider the following stochastic systems with mixed delays and nonlinear perturbations: (𝑡)=(𝑡)𝑑𝑥𝐴𝑥+𝐴₁𝑥(𝑡−ℎ)+𝐴₂^𝑡_𝑡−𝑑𝑥(𝑠)𝑑𝑠+𝐴₃𝑓(𝑥(𝑡),𝑥(𝑡−ℎ),𝑡)+𝐴_𝑣𝑣(𝑡(𝑥(𝑡)())(𝑡),(𝑡)=(𝑡))𝑑𝑡+𝑔,𝑥𝑡−ℎ,𝑡𝑑𝜔𝑑𝑦𝐶𝑥+𝐶₁𝑥(𝑡−ℎ)+𝐶₂^𝑡_𝑡−𝑑𝑥(𝑠)𝑑𝑠+𝐶_𝑣𝑣(𝑡𝑧(𝑡)(𝑡),𝑥(𝑡)(𝑡)[],)𝑑𝑡,=𝐿𝑥=𝜑,∀𝑡∈−𝜏,0 (1) where 𝑥(𝑡)∈𝐑^𝑛 is the state; 𝑦(𝑡)∈𝐑^𝑚 is the measured output; 𝑧(𝑡)∈𝐑^𝑝 is the signal to be estimated; 𝑣(𝑡)∈𝐑^𝑞 is the disturbance input which belongs to 𝐋₂[0,∞), which is the space of square-integrable vector functions; 𝜔(𝑡) is a one-dimensional Brownian motion defined on a complete probability space (𝛀,𝐅,𝐏) satisfying 𝐄{𝑑𝜔(𝑡)}=0 and 𝐄{𝑑𝜔²(𝑡)}=𝑑𝑡; 𝜑(𝑡) is an initial function that is continuous on [−𝜏,0] with 𝜏=max{ℎ,𝑑}. ℎ and 𝑑 are discrete and distributed constant delays, respectively. 𝑓(⋅,⋅,⋅)∶𝐑^𝑛×𝐑^𝑛×𝐑→𝐑^𝑛 is a nonlinear function, which satisfies 𝑓𝑥,𝑦,𝑡²≤𝐹₁𝑥²+𝐹₂𝑦²,𝑓(0,0,0)=0, (2) where 𝐹₁∈𝐑^𝑛×𝑛 and 𝐹₂∈𝐑^𝑛×𝑛 are known constant matrices; 𝑔(⋅,⋅,⋅)∶𝐑^𝑛×𝐑^𝑛×𝐑→𝐑^𝑛 is a nonlinear perturbance input function, satisfying 𝑔𝑥,𝑦,𝑡²≤𝐺₁𝑥²+𝐺₂𝑦²,𝑔(0,0,0)=0, (3) where 𝐺₁∈𝐑^𝑛×𝑛 and 𝐺₂∈𝐑^𝑛×𝑛 are known constant matrices.

For system (1), we are interested in constructing the following full-order linear filter: 𝑑𝑥_𝑓(𝑡)=𝐴_𝑓𝑥_𝑓(𝑡)𝑑𝑡+𝐵_𝑓(𝑡),𝑧𝑑𝑦_𝑓(𝑡)=𝐶_𝑓𝑥_𝑓(𝑡), (4) where 𝑥_𝑓(𝑡)∈𝐑^𝑛 is the filter state; 𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓 are filter matrices to be determined.

Define 𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇_𝑓(𝑡)^𝑇,𝑒(𝑡)=𝑧(𝑡)−𝑧_𝑓(𝑡). (5) Then, the filtering error system can be written as 𝑑𝜉(𝑡)=𝐴𝜉(𝑡)+𝐴₁𝐻𝜉(𝑡−ℎ)+𝐴₂𝐻^𝑡_𝑡−𝑑𝜉(𝑠)+𝑑𝑠𝐴₃𝑓(𝜉(𝑡),𝜉(𝑡−ℎ),𝑡)+𝐴_𝑣𝑣(𝑡+)𝑑𝑡𝑔(𝜉(𝑡)())(𝑡)𝑒(𝑡)=,𝜉𝑡−ℎ,𝑡𝑑𝜔𝐿𝜉(𝑡), (6) where 𝐵𝐴=𝐴0_𝑓𝐶𝐴_𝑓,𝐴₁=𝐴₁𝐵_𝑓𝐶₁,𝐴₂=𝐴₂𝐵_𝑓𝐶₂,𝐴₃=𝐴₃0,00𝐴_𝑣=𝐴_𝑣𝐵_𝑓𝐶_𝑣,𝐻=𝐼_𝑛0_𝑛,𝑓(𝜉(𝑡)())=𝑓(𝑥(𝑡)())0,𝜉𝑡−ℎ,𝑡,,𝑥𝑡−ℎ,𝑡𝑔(𝜉(𝑡)())=𝑔(𝑥(𝑡)())0,𝜉𝑡−ℎ,𝑡,,𝑥𝑡−ℎ,𝑡𝐿=𝐿−𝐶_𝑓. (7)

The objective of this paper is to design full-order 𝐿₂-𝐿_∞ 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(𝑡)=0 is exponentially stable [39];(ii)under the zero initial condition, the filtering error system (6) is stochastically asymptotically stable and achieves a prescribed 𝐿₂-𝐿_∞ attenuation level 𝛾. The filtering error 𝑒(𝑡) satisfies ‖𝑒(𝑡)‖_𝐸∞<𝛾‖𝑣(𝑡)‖₂, (8)with ‖𝑒(𝑡)‖_𝐸∞=sup_𝑡𝐄{|𝑒(𝑡)|²}, ‖𝑣(𝑡)‖₂=∫^∞₀𝑣^𝑇(𝑡)𝑣(𝑡)𝑑𝑡 for any nonzero 𝑣(𝑡)∈𝐋₂[0,∞].

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 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}, where 𝑆₁₁=𝑆^𝑇₁₁, and 𝑆₂₂=𝑆^𝑇₂₂, the linear matrix inequality 𝑆<0 is equivalent to 𝑆₁₁<0,𝑆₂₂−𝑆₁₂𝑆⁻¹₁₁𝑆^𝑇₁₂𝑆<0,𝑜𝑟₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0. (9)

Lemma 2 (see [1]). For any positive symmetric matrix 𝑊∈𝐑^𝑛×𝑛, scalars 𝛿₁ and 𝛿₂ satisfying 𝛿₁<𝛿₂, a vector function 𝑥∶[𝛿₁,𝛿₂]→𝐑^𝑛, one has ^𝛿2_𝛿1𝑥^𝑇(𝑠)(𝑠)≥1𝑊𝑥𝑑𝑠𝛿₂−𝛿₁^𝛿2_𝛿1𝑥(𝑠)𝑑𝑠^𝑇𝑊^𝛿2_𝛿1𝑥(𝑠)𝑑𝑠. (10)

Lemma 3 (see [14]). For any positive symmetric matrix 𝑊∈𝐑^𝑛×𝑛, scalars 𝑎 and 𝑏 satisfying 𝑎<𝑏≤0, a vector function 𝑥∶[𝑎,𝑏]→𝐑^𝑛, one has ^𝑏_𝑎^𝑡_𝑡+𝜆𝑥^𝑇(𝑠)(𝑠)≥2𝑊𝑥𝑑𝑠𝑑𝜆𝑎²−𝑏^{2𝑏𝑎𝑡𝑡+𝜆}𝑥(𝑠)𝑑𝑠𝑑𝜆^𝑇𝑊^𝑏_𝑎^𝑡_𝑡+𝜆𝑥(𝑠)𝑑𝑠𝑑𝜆. (11)

3. Filtering Performance Analysis

In this section, a new delay-dependent condition of the 𝐿₂-𝐿_∞ 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 (𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓) are known.

Theorem 4. Consider the stochastic time-delay system (1). For given scalars 𝛾>0, ℎ>0, 𝑑>0, 𝜌>0, and 𝜀>0 and integers 𝑟₁≥1 and 𝑟₂≥1, there exists a linear filter (4) such that the filtering error system (6) is stochastically asymptotically stable with a guaranteed 𝐿₂-𝐿_∞ performance γ, if there exist symmetrical positive definite matrices 𝑃∈𝐑^2𝑛×2𝑛, 𝑄_𝑖∈𝐑^𝑛×𝑛, 𝑅_𝑖∈𝐑^𝑛×𝑛(𝑖=1,2,…,𝑟₁), 𝑊_𝑗∈𝐑^𝑛×𝑛, and 𝑍_𝑗∈𝐑^𝑛×𝑛(𝑗=1,2,…,𝑟₂) and matrix 𝑀∈𝐑^𝑛×𝑛 satisfying ΦΦ=¹¹Φ₁₂Φ₁₃𝑃𝐴₃𝑃𝐴_𝑣𝐴^𝑇𝐻^𝑇𝑀∗Φ₂₂000Φ₂₆∗∗Φ₃₃00Φ₃₆∗∗∗−𝜀𝐼0𝐴^𝑇₃𝐻^𝑇𝑀∗∗∗∗−𝐼𝐴^𝑇_𝑣𝐻^𝑇𝑀∗∗∗∗∗Φ₆₆<0,𝑃≤𝜌𝐼,𝑃Γ=𝐿^𝑇∗𝛾²𝐼>0, (12)(13)(14) where Φ₁₁=𝑃𝐴+𝐴^𝑇𝑃+𝐻^𝑇𝑄₁𝐻−𝐻^𝑇𝑅₁𝐻−^𝑟2_𝑗=1𝐻^𝑇2𝑍2𝑗−1^𝑗𝐻+𝑑𝑟₂^2𝑟2_𝑗=1𝐻^𝑇𝑊_𝑗𝐻+𝜌𝐻^𝑇𝐺^𝑇₁𝐺₁𝐻+𝜀𝐻^𝑇𝐹^𝑇₁𝐹₁Φ𝐻,₁₂=𝐻^𝑇𝑅₁𝐾+𝑃𝐴₁𝐾_𝑟1,Φ₁₃=𝑃𝐴₂𝐾_𝑟2+𝐻^𝑇Ζ,𝑍=2𝑚𝑑𝑍₁13𝑍₂⋯12𝑟₂𝑍−1^𝑟2,Φ₂₂=𝑄₂−𝑄₁−𝑅₂−𝑅₁𝑅₂∗𝑄⋯00₃−𝑄₂−𝑅₃−𝑅₂𝑄⋯00⋮⋮⋱⋮⋮∗∗⋯^𝑟1−𝑄_𝑟1−1−𝑅_𝑟1−𝑅_𝑟1−1𝑅_𝑟1∗∗⋯∗Φ₂₂,Φ₂₂=−𝑄_𝑟1−𝑅_𝑟1+𝜌𝐺^𝑇₂𝐺₂+𝜀𝐹^𝑇₂𝐹₂,Φ₂₆=𝐾^𝑇_𝑟1𝐴^𝑇₁𝐻^𝑇Φ𝑀,₃₃=diag−𝑊₁−2𝑚²𝑑²𝑍₁,−𝑊₂−2𝑚²3𝑑²𝑍₂,…,−𝑊_𝑟2−2𝑚²2𝑟₂−1𝑑²𝑍_𝑟2,Φ₃₆=𝐾^𝑇_𝑟2𝐴^𝑇₂𝐻^𝑇Φ𝑀,₆₆=ℎ𝑟₁^2𝑟1_𝑖=1𝑅_𝑖+𝑑𝑟₂^2𝑟2_𝑗=12𝑗−12𝑍_𝑗−𝑀−𝑀^𝑇,𝐾=𝐼_𝑛0_{𝑛×(𝑟1−1)𝑛},𝐾_𝑟1=0_{𝑛×(𝑟1−1)𝑛}𝐼_𝑛,𝐾_𝑟2=𝐼𝐼⋯𝐼_{𝑛×𝑟2𝑛}. (15)

Proof. First, show the asymptotic stability of system (6) with 𝑣(𝑡)=0. For simplicity of notations, rewrite the filtering error system (6) as 𝑑𝜉(𝑡)=𝑢(𝑡)𝑑𝑡+𝜋(𝑡)𝑑𝜔(𝑡), (16) where 𝑢(𝑡)∶=𝐴𝜉(𝑡)+𝐴₁𝐻𝜉(𝑡−ℎ)+𝐴₂𝐻^𝑡_𝑡−𝑑𝜉(𝑠)+𝑑𝑠𝐴₃𝑓(𝜉(𝑡),𝜉(𝑡−ℎ),𝑡)+𝐴_𝑣𝑣(𝑡),𝜋(𝑡)∶=𝑔(𝜉(𝑡),𝜉(𝑡−ℎ),𝑡). (17)
Next, denote 𝜂(𝑡)𝑑𝑡=𝑑𝜉(𝑡), and choose the following Lyapunov-Krasovskii functional: 𝑉𝜉_𝑡,𝑡=𝜉^𝑇(𝑡)𝑃𝜉(𝑡)+^𝑟1_𝑖=1^{𝑡−((𝑖−1)/𝑟1)ℎ}_{𝑡−(𝑖/𝑟1)ℎ}𝜉^𝑇(𝑠)𝐻^𝑇𝑄_𝑖(𝑠)+𝐻𝜉𝑑𝑠_𝑟1𝑖=1ℎ𝑟₁^{−((𝑖−1)/𝑟1)ℎ}_{−(𝑖/𝑟1)ℎ}^𝑡_𝑡+𝜃𝜂^𝑇(𝑠)𝐻^𝑇𝑅_𝑖(𝑠)+𝐻𝜂𝑑𝑠𝑑𝜃_𝑟2𝑗=1𝑑𝑟₂^{−((𝑗−1)/𝑟2)𝑑}_{−(𝑗/𝑟2)𝑑}^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝐻^𝑇𝑊_𝑗(𝑠)+𝐻𝜉𝑑𝑠𝑑𝜃_{𝑟2𝑗=1−((𝑗−1)/𝑟2)𝑑−(𝑗/𝑟2)𝑑0𝜃𝑡𝑡+𝛽}𝜂^𝑇(𝑠)𝐻^𝑇𝑍_𝑗𝐻𝜂(𝑠)𝑑𝑠𝑑𝛽𝑑𝜃. (18) Then, by Itô differential formula, the stochastic differential along the trajectories of system (6) is 𝑑𝑉𝜉_𝑡,𝑡=𝐋𝑉𝜉_𝑡𝑑𝑡+2𝜉(𝑡)𝑃𝜋(𝑡)𝑑𝜔(𝑡), (19) where 𝐋𝑉𝜉_𝑡,𝑡=2𝜉^𝑇(𝑡)𝑃𝑢(𝑡)+trace𝜋^𝑇(𝑡)(𝑡+𝑃𝜋)_𝑟1𝑖=1𝜉^𝑇𝑡−𝑖−1𝑟₁ℎ𝐻^𝑇𝑄_𝑖𝐻𝜉𝑡−𝑖−1𝑟₁ℎ−^𝑟1_𝑖=1𝜉^𝑇𝑖𝑡−𝑟₁ℎ𝐻^𝑇𝑄_𝑖𝑖𝐻𝜉𝑡−𝑟₁ℎ+ℎ𝑟₁^2𝑟1_𝑖=1𝜂^𝑇(𝑡)𝐻^𝑇𝑅_𝑖(𝑡)−𝐻𝜂_𝑟1𝑖=1ℎ𝑟₁^{𝑡−((𝑖−1)/𝑟1)ℎ}_{𝑡−(𝑖/𝑟1)ℎ}𝜂^𝑇(𝑠)𝐻^𝑇𝑅_𝑖(𝑠)+𝑑𝐻𝜂𝑑𝑠𝑟₂^2𝑟2_𝑗=1𝜉^𝑇(𝑡)𝐻^𝑇𝑊_𝑖(𝑡)−𝐻𝜉_𝑟2𝑗=1𝑑𝑟₂^{𝑡−((𝑗−1)/𝑟2)𝑑}_{𝑡−(𝑗/𝑟2)𝑑}𝜉^𝑇(𝑠)𝐻^𝑇𝑊_𝑗(𝑠)+𝑑𝐻𝜉𝑑𝑠𝑟₂^2𝑟2_𝑗=12𝑗−12𝜂^𝑇(𝑡)𝐻^𝑇𝑍_𝑗(𝑡)−𝐻𝜂_{𝑟2𝑗=1−((𝑗−1)/𝑟2)𝑑−(𝑗/𝑟2)𝑑𝑡𝑡+𝜃}𝜂^𝑇(𝑡)𝐻^𝑇𝑍_𝑗𝐻𝜂(𝑡)𝑑𝑠𝑑𝜃. (20) By Lemma 2, we have −^𝑟1_𝑖=1ℎ𝑟₁^{𝑡−((𝑖−1)/𝑟1)ℎ}_{𝑡−(𝑖/𝑟1)ℎ}𝜂^𝑇(𝑠)𝐻^𝑇𝑅_𝑖𝐻𝜂(𝑠)𝑑𝑠≤−^𝑟1_𝑖=1^{𝑡−((𝑖−1)/𝑟1)ℎ}_{𝑡−(𝑖/𝑟1)ℎ}𝜂(𝑠)𝑑𝑠^𝑇×𝐻^𝑇𝑅_𝑖𝐻^{𝑡−((𝑖−1)/𝑟1)ℎ}_{𝑡−(𝑖/𝑟1)ℎ}𝜂(𝑠)𝑑𝑠=−^𝑟1_𝑖=1𝜉𝑡−𝑖−1𝑟₁ℎ𝑖−𝜉𝑡−𝑟₁ℎ^𝑇×𝐻^𝑇𝑅_𝑖𝐻𝜉𝑡−𝑖−1𝑟₁ℎ𝑖−𝜉𝑡−𝑟₁ℎ,−^𝑟2_𝑗=1𝑑𝑟₂^{𝑡−((𝑗−1)/𝑟2)𝑑}_{𝑡−(𝑗/𝑟2)𝑑}𝜉^𝑇(𝑠)𝐻^𝑇𝑊_𝑗𝐻𝜉(𝑠)𝑑𝑠≤−^𝑟2_𝑗=1^{𝑡−((𝑗−1)/𝑟2)𝑑}_{𝑡−(𝑗/𝑟2)𝑑}𝜉(𝑠)𝑑𝑠^𝑇×𝐻^𝑇𝑊_𝑗𝐻^{𝑡−((𝑗−1)/𝑟2)𝑑}_{𝑡−(𝑗/𝑟2)𝑑}𝜉(𝑠)𝑑𝑠. (21)(22) By Lemma 3, we have −^𝑟2_𝑗=1^{−((𝑗−1)/𝑟2)𝑑}_{−(𝑗/𝑟2)𝑑}^𝑡_𝑡+𝜃𝜂^𝑇(𝑡)𝐻^𝑇𝑍_𝑗𝐻𝜂(𝑡)𝑑𝑠𝑑𝜃≤−^𝑟2_𝑗=12𝑟2𝑗−1𝑑^{22−((𝑗−1)/𝑟2)𝑑}_{−(𝑗/𝑟2)𝑑}^𝑡_𝑡+𝜃𝜂(𝑠)𝑑𝑠𝑑𝜃^𝑇×𝐻^𝑇𝑍_𝑗𝐻^{−((𝑗−1)/𝑟2)𝑑}_{−(𝑗/𝑟2)𝑑}^𝑡_𝑡+𝜃𝜂(𝑠)𝑑𝑠𝑑𝜃=−^𝑟2_𝑗=12𝑟2𝑗−1𝑑²²𝑑𝑟₂𝜉(𝑡)−^{𝑡−((𝑗−1)/𝑟2)𝑑}_{𝑡−(𝑗/𝑟2)𝑑}𝜉(𝑠)𝑑𝑠^𝑇×𝐻^𝑇𝑍_𝑗𝐻𝑑𝑟₂𝜉(𝑡)−^{𝑡−((𝑗−1)/𝑟2)𝑑}_{𝑡−(𝑗/𝑟2)𝑑}𝜉(𝑠)𝑑𝑠. (23) From (16), for any appropriately dimensioned matrix 𝑀, we have 2𝜂^𝑇(𝑡)𝐻^𝑇𝑀^𝑇𝐻𝑢(𝑡)𝑑𝑡+𝜋(𝑡)𝑑𝜔(𝑡)−𝜂(𝑡)𝑑𝑡=0. (24) On the other hand, (2) implies that there exists 𝜀>0 such that 𝜉^𝑇(𝑡)𝜀𝐻^𝑇𝐹^𝑇₁𝐹₁𝐻𝜉(𝑡)+𝜉^𝑇(𝑡−ℎ)𝜀𝐻^𝑇𝐹^𝑇₂𝐹₂𝐻𝜉(𝑡−ℎ)−𝜀𝑓^𝑇𝑓≥0, (25) where we take 𝑓 for 𝑓(𝑥(𝑡),𝑥(𝑡−ℎ),𝑡), for simplicity of notation.
Notice the fact of (3), and from (13), we have trace𝜋^𝑇(𝑡)𝑃𝜋(𝑡)≤𝜉^𝑇(𝑡)𝜌𝐻^𝑇𝐺^𝑇₁𝐺₁𝐻𝜉(𝑡)+𝜉^𝑇(𝑡−ℎ)𝜌𝐻^𝑇𝐺^𝑇₂𝐺₂𝐻𝜉(𝑡−ℎ). (26) Combine (20)–(26); then 𝐋𝑉𝑥_𝑡,𝑡≤𝜁^𝑇(𝑡)Φ𝜁(𝑡)+2𝜂^𝑇(𝑡)𝐻^𝑇𝑀^𝑇𝐻𝜋(𝑡)𝑑𝜔(𝑡), (27) where𝜁^𝑇(𝑡)=𝜉^𝑇_𝑝1(𝑡)𝜉^𝑇_𝑝2(𝑡)𝑓(𝑡)𝜂^𝑇(𝑡)𝐻^𝑇,𝜉^𝑇_𝑝1=𝜉^𝑇(𝑡)𝜉^𝑇1𝑡−𝑟₁ℎ𝐻^𝑇𝜉^𝑇2𝑡−𝑟₁ℎ𝐻^𝑇⋯𝜉^𝑇(𝑡−ℎ)𝐻^𝑇,𝜉^𝑇_𝑝2(𝑡)=∫^𝑡_{𝑡−(1/𝑟2)𝑑}𝜉^𝑇(𝑠)𝐻^𝑇𝑑𝑠∫^{𝑡−(1/𝑟2)𝑑}_{𝑡−(2/𝑟2)𝑑}𝜉^𝑇(𝑠)𝐻^𝑇𝑑𝑠⋯∫^{𝑡−((𝑟2−1)/𝑟2)𝑑}_𝑡−𝑑𝜉^𝑇(𝑠)𝐻^𝑇𝑑𝑠,ΦΦ=¹¹Φ₁₂Φ₁₃𝑃𝐴₃𝐴^𝑇𝐻^𝑇𝑀∗Φ₂₂00Φ₂₆∗∗Φ₃₃0Φ₃₆∗∗∗−𝜀𝐼𝐴^𝑇₃𝐻^𝑇𝑀∗∗∗∗Φ₆₆. (28)
Thus, 𝐄𝐋𝑉𝜉_𝑡,𝑡≤𝐄𝜁^𝑇(𝑡)Φ𝜁(𝑡). (29)
By Schur complement lemma, it is easy to show that Φ<0 implies Φ<0. Combined with (29), these imply that, for any 𝜁(𝑡)≠0, we have 𝐄{𝐋𝑉(𝜉_𝑡,𝑡)}<0.
By Dynkin’s formula, there exists 𝛽>0, such that 𝑒^𝛽𝑡𝐄𝑉𝜉_𝑡,𝑡≤𝐄𝑉𝜉₀,0. (30) Recalling the Lyapunov-Krasovskii functional in (18), notice the fact that there always exists 𝜅>0 satisfying 𝜂(𝑡)²≤𝜅𝜉(𝑡)² (31) for any −𝜏≤𝑡≤0 such that 𝐄𝑉𝜉₀,0≤⁵_𝑖=1𝛼_𝑖sup_{−𝜏≤𝑠≤0}𝐄𝜉(𝑠)², (32) where 𝛼₁=𝜆_max(𝑃), 𝛼₂=ℎmax_𝑖{‖𝑄_𝑖‖}, 𝛼₃=(𝜅ℎ³/2𝑟₁)max_𝑖{‖𝑅_𝑖‖}, 𝛼₄=(𝑑³/2𝑟₂)max_𝑗{‖𝑊_𝑗‖}, and 𝛼₅=(𝜅𝑑³/6)max_𝑗{‖𝑍_𝑗‖}.
On the other hand, from (18) 𝐄𝑉𝜉_𝑡,𝑡≥𝜆_min(𝑃)𝐄𝜉(𝑡)². (33) From (32) and (33), it can be easily obtained that 𝐄𝜉𝑡;𝜑²≤𝛼𝑒^−𝛽𝑡sup_{−𝜏≤𝑠≤0}𝐄𝜉(𝑠)², (34) where 𝛼=∑⁵_𝑖=1𝛼_𝑖/𝜆_min(𝑃) 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(𝑡)=0 is exponentially stable in the sense of mean square.
Now, we will establish the 𝐿₂-𝐿_∞ performance for the filtering error system (6). To this end, we assume the zero initial condition 𝜁(𝑡)=0 for 𝑡∈[−𝜏,0]. Under the initial condition, it is easy to see that, for any 𝑡>0, 𝐄𝑉𝜉_𝑡,𝑡=𝐄^𝑡₀𝐋𝑉𝜉_𝑠,𝑠𝑑𝑠. (35)
Define 𝐽(𝑡)=𝐄𝑉𝜉_𝑡,𝑡−^𝑡₀𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (36) Then, for any nonzero 𝑣(𝑡)∈𝐋₂[0,∞) and 𝑡>0, combined with (29), (35)-(36), we have 𝐽(𝑡)=𝐄^𝑡₀𝐋𝑉𝜉_𝑠,𝑠−𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠≤𝐄^𝑡₀𝜗^𝑇(𝑠)Φ𝜗(𝑠)𝑑𝑠, (37) where 𝜗^𝑇(𝑡)=𝜉^𝑇_𝑝1(𝑡)𝜉^𝑇_𝑝2(𝑡)𝑓^𝑇𝑣^𝑇(𝑡)𝜂^𝑇(𝑡)𝐻^𝑇. Φ<0 ensuring that 𝐽(𝑡)≤0. Thus, 𝐄𝜉^𝑇(𝑡)𝑃𝜉(𝑡)≤𝐄𝑉𝜉_𝑡,𝑡≤^𝑡₀𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (38)
Moreover, by Schur complement, (14) holds if and only if 𝐿^𝑇𝐿<𝛾²𝑃. (39) It follows from (38) and (39) that 𝐄|𝑒(𝑡)|²=𝐄𝜉^𝑇(𝑡)𝐿^𝑇𝐿𝜉(𝑡)<𝛾²𝐄𝜉^𝑇(𝑡)𝑃𝜉(𝑡)≤𝛾²𝐄𝑉𝜉_𝑡,𝑡≤𝛾^2𝑡₀𝑣^𝑇(𝑠)𝑣(𝑠)𝑑𝑠. (40)
Therefore, if (12)–(14) hold, the filtering error system (6) is mean-square exponentially stable with a prescribed 𝐿₂-𝐿_∞ performance 𝛾 under zero initial condition. This completes the proof.

In system (1), if 𝐴₃=𝟎 and 𝑔(𝑥(𝑡),𝑥(𝑡−ℎ),𝑡)=𝐵𝑥(𝑡)+𝐵₁𝑥(𝑡−ℎ)+𝐵₂∫^𝑡_𝑡−𝑑𝑥(𝑠)𝑑𝑠+𝐵_𝑣𝑣(𝑡), then the linear stochastic system with mixed delays can be written as 𝑑𝑥(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ)+𝐴₂^𝑡_𝑡−𝑑𝑥(𝑠)𝑑𝑠+𝐴_𝑣𝑣(𝑡+(𝑡))𝑑𝑡𝐵𝑥+𝐵₁𝑥(𝑡−ℎ)+𝐵₂^𝑡_𝑡−𝑑𝑥(𝑠)𝑑𝑠+𝐵_𝑣𝑣(𝑡(𝑡),(𝑡)=(𝑡))𝑑𝜔𝑑𝑦𝐶𝑥+𝐶₁𝑥(𝑡−ℎ)+𝐶₂^𝑡_𝑡−𝑑𝑥(𝑠)𝑑𝑠+𝐶_𝑣𝑣(𝑡𝑧(𝑡)(𝑡),𝑥(𝑡)(𝑡)[])𝑑𝑡,=𝐿𝑥=𝜑,∀𝑡∈−𝜏,0 (41) 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 𝐿₂-𝐿_∞ performance 𝛾.

Corollary 5. Consider the stochastic time-delay system (41). For given scalars 𝛾>0, ℎ>0, and 𝑑>0 and integers 𝑟₁≥1 and 𝑟₂≥1, there exists a linear filter (4) such that the corresponding filtering error system is exponentially stable with a guaranteed 𝐿₂-𝐿_∞ performance 𝛾, if there exist symmetrical positive definite matrices 𝑃∈𝐑^2𝑛×2𝑛, 𝑄_𝑖∈𝐑^𝑛×𝑛, 𝑅^𝑇_𝑖∈𝐑^𝑛×𝑛(𝑖=1,2,…,𝑟₁), 𝑊_𝑗∈𝐑^𝑛×𝑛, and 𝑍_𝑗∈𝐑^𝑛×𝑛(𝑗=1,2,…,𝑟₂) and matrix 𝑀∈𝐑^𝑛×𝑛 satisfying (14) and ΦΦ=¹¹Φ₁₂Φ₁₃𝑃𝐴_𝑣𝐴^𝑇𝐻^𝑇𝑀∗Φ₂₂00Φ₂₆∗∗Φ₃₃0Φ₃₆∗∗∗−𝐼𝐴^𝑇_𝑣𝐻^𝑇𝑀∗∗∗∗Φ₆₆+𝐵^𝑇_𝜉𝑃𝐵_𝜉<0, (42) where Φ₁₁=Φ₁₁−𝜌𝐻^𝑇𝐺^𝑇₁𝐺₁𝐻−𝜀𝐻^𝑇𝐹^𝑇₁𝐹₁𝐻, Φ₂₂=Φ₂₂−𝐾^𝑇_𝑟1(𝜌𝐺^𝑇₂𝐺₂+𝜀𝐹^𝑇₂𝐹₂)𝐾_𝑟1, and 𝐵_𝜉=𝐵𝐵₁𝐾_𝑟1𝐵₂𝐾_𝑟2𝐵_𝑣.

4. Filter Design

In this section, we will focus on the design of 𝐿₂-𝐿_∞ 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 𝐿₂-𝐿_∞ performance 𝛾.

Theorem 6. Consider the stochastic time-delay system (1). For given scalars 𝛾>0, ℎ>0, 𝑑>0, 𝜌>0, and 𝜀>0 and integers 𝑟₁≥1 and 𝑟₂≥1, there exists a linear filter (4) such that the filtering error system (6) is stochastically asymptotically stable with a prescribed 𝐿₂-𝐿_∞ performance 𝛾, if there exist symmetrical positive definite matrices 𝑋∈𝐑^𝑛×𝑛, 𝑌∈𝐑^𝑛×𝑛, 𝑄_𝑖∈𝐑^𝑛×𝑛, 𝑅_𝑖∈𝐑^𝑛×𝑛(𝑖=1,2,…,𝑟₁), 𝑊_𝑗∈𝐑^𝑛×𝑛, and 𝑍_𝑗∈𝐑^𝑛×𝑛(𝑗=1,2,…,𝑟₂) and matrices 𝑀∈𝐑^𝑛×𝑛, 𝐴_𝑓∈𝐑^𝑛×𝑚, 𝐵_𝑓∈𝐑^𝑛×𝑛, and 𝐶_𝑓∈𝐑^𝑛×𝑝 satisfying ΥΥ=¹¹Υ₁₂Υ₁₃Υ₁₄Υ₁₅Υ₁₆∗Φ₂₂000Υ₂₆∗∗Φ₃₃00Υ₃₆∗∗∗−𝜀𝐼0𝐴^𝑇₃𝑀∗∗∗∗−𝐼𝐴^𝑇_𝑣𝑀∗∗∗∗∗Φ₆₆<0,𝑌𝑌𝑋−𝜌𝐼𝑌1−𝜌<0,Π₂≤𝐼,Λ=𝑋𝑌𝐿^𝑇∗𝑌−𝐶^𝑇_𝑓∗∗𝛾²𝐼>0, (43)(44)(45)(46) where Υ₁₁=Υ¹₁₁𝐴_𝑓+𝐴^𝑇𝑌+𝐶^𝑇𝐵^𝑇_𝑓∗𝐴_𝑓+𝐴^𝑇_𝑓,Υ₁₂=𝑅₁𝐾₁+𝑋𝐴₁+𝐵_𝑓𝐶₁𝐾_𝑟1𝑌𝐴₁+𝐵_𝑓𝐶₁𝐾_𝑟1,Υ₁₃=𝑋𝐴₂+𝐵_𝑓𝐶₂𝐾_𝑟2+𝑍𝑌𝐴₂+𝐵_𝑓𝐶₂𝐾_𝑟2,Υ¹₁₁=sym𝑋𝐴+𝐵_𝑓𝐶+𝑄₁−𝑅₁+𝑑𝑟₂^2𝑟2_𝑗=1𝑊_𝑗−^𝑟2_𝑗=12𝑍2𝑗−1^𝑗+𝜌𝐺^𝑇₁𝐺₁+𝜀𝐹^𝑇₁𝐹₁,Υ₁₄=𝑋𝐴₃𝑌𝐴₃,Υ₁₅=𝑋𝐴_𝑣+𝐵_𝑓𝐶_𝑣𝑌𝐴_𝑣+𝐵_𝑓𝐶_𝑣,Υ₁₆=𝐴^𝑇𝑀0_𝑛,𝑛,Υ₂₆=𝐾^𝑇_𝑟1𝐴^𝑇₁𝑀,Υ₃₆=𝐾^𝑇_𝑟2𝐴^𝑇₂𝑀. (47)
In this case, the parameters of a desired filter in the form of (4) can be taken as 𝐴_𝑓=Π₁⁻¹𝐴_𝑓Π₁^−𝑇Π₂,𝐵_𝑓=Π₁⁻¹𝐵_𝑓,𝐶_𝑓=𝐶_𝑓Π₁^−𝑇Π₂, (48) where Π₁ and Π₂ are nonsingular matrices satisfying 0<Π₂=Π^𝑇₂≤𝐼, 0<𝑌=Π₁Π₂⁻¹Π^𝑇₁, and 𝑋>𝑌>0.

Proof. From (46), it can be seen that ^𝑋𝑌_𝑌𝑌>0, and 𝑋>𝑌>0. For any positive definite and symmetric matrix 𝑌, one can always find nonsingular matrix Π₁∈𝐑^𝑛×𝑛 and 0<Π₂=Π^𝑇₂∈𝐑^𝑛×𝑛, such that 𝑌=Π₁Π₂⁻¹Π^𝑇₁.
Set 𝑃=𝑋Π₁Π^𝑇₁Π₂. (49) Then 𝑋−Π₁Π₂⁻¹Π^𝑇₁=𝑋−𝑌>0, and 𝑃>0.
Define 𝐼Π=^𝑛0_𝑛,𝑛0_𝑛,𝑛Π₂⁻¹Π^𝑇₁. (50) Then Π⁻¹=𝐼_𝑛0_𝑛,𝑛0_𝑛,𝑛Π₁^−𝑇Π₂. (51)
Substitute 𝐴_𝑓=Π₁𝐴_𝑓Π₂⁻¹Π^𝑇₁, 𝐵_𝑓=Π₁𝐵_𝑓, 𝐶_𝑓=𝐶_𝑓Π₂⁻¹Π^𝑇₁, and 𝑌=Π₁Π₂⁻¹Π^𝑇₁ into (43), (44), and (46) and then pre- and postmultiply (43) by diag{Π^−𝑇,𝐼_𝑟1𝑛,𝐼_𝑟2𝑛,𝐼_𝑛,𝐼_𝑛,𝐼_𝑛} and its transpose, respectively. Premultiply and postmultiply (46) by diag{Π^−𝑇,𝐼_𝑝} and its transpose, respectively. Noticing that 𝑃=^𝑋Π1_Π𝑇1Π2, using Schur complement Lemma, one can obtain (12) and (14).
On the other hand, (44) implies .𝑋𝑌𝑌𝑌<𝜌𝐼00𝑌 (52) Pre- and postmultiply (52) by Π^−𝑇 and Π⁻¹, respectively. Notice that 𝑌=ΠΛ⁻¹Π^𝑇, one can obtain 𝑋Π₁Π^𝑇₁Π₂<𝜌𝐼00Π₂. (53) By (45), it is easy to see that 𝑋Π₁Π^𝑇₁Π₂<𝜌.𝐼00𝐼 (54) 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 𝐿₂-𝐿_∞ performance 𝛾. 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 Π₂ and nonsingular matrix Π₁ satisfy Π₂=Π^𝑇₁𝑌⁻¹Π₁. If the obtained matrix Π₂ does not satisfy (45), one can take Π₂ for Π₂/max{eig(Π₂)} and Π₁ for max{eig(Π₂)}Π₁. 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 𝛾>0, ℎ>0, and 𝑑>0 and integers 𝑟₁≥1 and 𝑟₂≥1, there exists a linear filter (4) such that the corresponding filtering error system is stochastically asymptotically stable with a prescribed 𝐿₂-𝐿_∞ performance γ, if there exist symmetrical positive definite 𝑋∈𝐑^𝑛×𝑛, 𝑌∈𝐑^𝑛×𝑛, 𝑄_𝑖∈𝐑^𝑛×𝑛, 𝑅_𝑖∈𝐑^𝑛×𝑛(𝑖=1,2,…,𝑟₁), 𝑊_𝑗∈𝐑^𝑛×𝑛, and 𝑍_𝑗∈𝐑^𝑛×𝑛(𝑗=1,2,…,𝑟₂) and matrices 𝑀∈𝐑^𝑛×𝑛, 𝐴_𝑓∈𝐑^𝑛×𝑚, 𝐵_𝑓∈𝐑^𝑛×𝑛, and 𝐶_𝑓∈𝐑^𝑛×𝑝 satisfying (46) and ΥΥ=¹¹Υ₁₂Υ₁₃Υ₁₅Υ₁₆Υ₁₇∗Φ₂₂00Υ₂₆Υ₂₇∗∗Φ₃₃0Υ₃₆Υ₃₇∗∗∗−𝐼𝐴^𝑇_𝑣𝑀Υ₄₇∗∗∗∗Φ₆₆0∗∗∗∗∗Υ₇₇<0, (55) where Υ₁₁=Υ¹₁₁𝐴_𝑓+𝐴^𝑇𝑌+𝐶^𝑇𝐵^𝑇_𝑓∗𝐴_𝑓+𝐴^𝑇_𝑓,Υ¹₁₁=Υ¹₁₁−𝜌𝐺^𝑇₁𝐺₁−𝜀𝐹^𝑇₁𝐹₁,Υ₁₇=𝐵^𝑇𝑋𝐵^𝑇𝑌0_𝑛,𝑛0_𝑛,𝑛,Υ₂₇=𝐾^𝑇_𝑟1𝐵^𝑇₁𝑋𝐵^𝑇₁𝑌,Υ₃₇=𝐾^𝑇_𝑟2𝐵^𝑇₂𝑋𝐵^𝑇₂𝑌,Υ₄₇=𝐵^𝑇_𝑣𝑋𝐵^𝑇_𝑣𝑌,Υ₇₇=.−𝑋−𝑌∗−𝑌 (56)

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 𝐴=−1.50.5−1−3,𝐴₁=,𝐴−0.80.20.2−0.5₂=0.2000.2,𝐴₃=,𝐴0.1000.1_𝑣=,𝐶0.200−0.2,𝐶=2−0.5−1.50.5₁=0.150.1−0.10.1,𝐶₂=,𝐶0.5−0.20.60_𝑣=,0.10.20.10.03,𝐿=0.1−0.200.1𝜀=1,𝜌=7. (57)
Moreover, for the nonlinear functions, we let 𝐺₁=𝐺₂=0.1𝐼 and 𝐹₁=𝐹₂=0.1𝐼. Given ℎ=1 and 𝛾=0.2, from Theorem 6, one can obtain the upper bound of time delay 𝑑, which is listed in Table 1.
In the case of 𝑟₁=2 and 𝑟₂=2, the desired filter parameters can be obtained: 𝐴_𝑓=−6.14602.26440.6168−4.5679,𝐵_𝑓=,𝐶−3.87491.88680.7066−0.1266_𝑓=.−0.05680.00520.02020.0057 (58)

Table 1 

The upper bound of 𝑑_max for ℎ=1 and 𝛾=0.2.

Example 2. Consider the stochastic time-delay system (36)–(39) with parameters 𝐴=−1.50.5−1−3,𝐴₁=,𝐴−0.80.20.2−0.5₂=0.2000.2,𝐴_𝑣=,0.200−0.2𝐵=−0.80.20.5−0.5,𝐵₁=,𝐵0.50.50.20.3₂=0.2000.2,𝐵_𝑣=,−0.2000.5𝐶=2−0.5−1.50.5,𝐶₁=,𝐶0.150.1−0.10.1₂=0.5−0.20.60,𝐶_𝑣=,.0.10.20.10.03𝐿=0.1−0.200.1 (59)
Given 𝑑=1 and 𝛾=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 𝑟₁=2 and 𝑟₂=2, the desired filter parameters can be obtained: 𝐴_𝑓=−0.0000−4.1570−0.0000−2.5527,𝐵_𝑓=,𝐶10.1507−11.12366.0149−6.1484_𝑓=.0.00000.0065−0.0000−0.0027 (60)

Table 2 

The upper bound of ℎ_max for 𝑑=1 and 𝛾=0.2.

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 𝐿₂-𝐿_∞ 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 𝐿₂-𝐿_∞ performance 𝛾. 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.