Robust 𝐻_∞ Filtering for Uncertain Discrete-Time Fuzzy Stochastic Systems with Sensor Nonlinearities and Time-Varying Delay

Abstract

The robust filtering problem for a class of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay is investigated. The parameter uncertainties are assumed to be time varying norm bounded in both the state and measurement equations. By using the Lyapunov stability theory and some new relaxed techniques, sufficient conditions are proposed to guarantee the robustly stochastic stability with a prescribed 𝐻_∞ performance level of the filtering error system for all admissible uncertainties, sensor nonlinearities, and time-varying delays. These conditions are dependent on the lower and upper bounds of the time-varying delays and are obtained in terms of a linear matrix inequality (LMI). Finally, two simulation examples are provided to illustrate the effectiveness of the proposed methods.

1. Introduction

Fuzzy systems in the Takagi-Sugeno (T-S) model can represent a lot of complex nonlinear systems in [1–3]. By using a T-S fuzzy plant model, one can describe a nonlinear system as a weighted sum of some simple linear subsystems. This fuzzy model is described by a family of fuzzy if-then rules that represent local linear input/output relations of the system. The overall fuzzy model of the system is achieved by smoothly blending these local linear models together through membership functions. Consequently, stability, control, and filtering problems for T-S fuzzy systems have attracted considerable attention, and many important results have been reported in [4–9].

It has been known that time delays are one of the main causes of instability and poor performance of control systems in [10]. Analysis and synthesis of such systems are of both theoretical and practical importance. Therefore, the study of time-delay systems has attracted great attention over the past few years in [11, 12]. For continuous-time systems, the obtained results can be generally classified into two types: delay-independent and delay-dependent ones. It has been understood that the latter is generally less conservative since the size of delays is considered, especially when time delays are small. Compared with continuous-time systems with time-varying delays, the discrete-time counterpart receives relatively less attention. See, for example, [13–16] and references therein.

In the past few years, much research effort has been paid to the control and filtering problems for nonlinear systems that have been widely applied in many fields such as communication network, image processing, and mobile robot localization [17, 18]. And the filtering for nonlinear stochastic systems has been of great interest since stochastic modeling has come to play an important role in many branches of engineering applications [19–22]. particularly for discrete-time stochastic systems, so far, a number of important results have been reported for linear discrete-time stochastic systems. The exponential filtering problem is studied for discrete time-delay stochastic systems with Markovian jump parameters and missing measurements in [23]. The robust fault detection filter problem for fuzzy 𝐼𝑡𝑜̂ stochastic systems is studied in [24]. The problem of robust 𝐻_∞ filtering for uncertain discrete-time stochastic systems with time-varying delays is considered in [25]. For nonlinear discrete-time stochastic systems, the filtering problem for a class of nonlinear discrete-time stochastic systems with state delays is considered in [26]. The robust 𝐻_∞ filtering problem for a class of nonlinear discrete time-delay stochastic systems is considered in [27]. The 𝐻_∞ filtering problem for a general class of nonlinear discrete-time stochastic systems with randomly varying sensor delays is considered in [28]. Recently, the filtering problem for discrete-time fuzzy stochastic systems with sensor nonlinearities is considered in [29]. And the problem of 𝐻_∞ filtering for discrete-time Takagi-Sugeno (T-S) fuzzy 𝐼𝑡𝑜̂ stochastic systems with time-varying delay is studied in [30]. In [27, 29, 31, 32], the nonlinearity for filtering problem of systems was assumed to satisfy sensor nonlinearities, which may included actuator saturation and sensor saturation. It is worth mentioning that, although the system in [29] is with sensor nonlinearities, the proposed filter design approach do not consider the discrete-time fuzzy stochastic systems with time delay, which is not applicable to stochastic delay systems. And in [30], the stochastic systems do not contain the sensor nonlinearities. To the best of our knowledge, no results on 𝐻_∞ filtering for the uncertain discrete-time fuzzy stochastic systems with both sensor nonlinearities and time-varying delay are available in the literature, which motivates the present study.

Motivated by the works in [25, 27, 29, 30], in this paper, a delay-dependent 𝐻_∞ performance analysis result is established for filtering error systems. A new uncertain discrete-time fuzzy stochastic systems model is proposed, and a new different Lyapunov functional is then employed to deal with systems with sensor nonlinearities and time-varying delay. As a result, the 𝐻_∞ filters are designed in terms of linear matrix inequalities (LMIs). The resulting filters can ensure that the filtering error system is robustly stochastic stable and the estimation error is bounded by a prescribed level for all possible bounded energy disturbances. Finally, two examples are given to show the effectiveness of the proposed method.

Throughout this paper, ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, and ℝ^𝑛×𝑚 is the set of 𝑛×𝑚 real matrices. 𝐼 is the identity matrix. |⋅| denotes Euclidean norm for vectors and ||⋅|| denotes the spectral norm of matrices. 𝑁 denotes the set of all natural numbers, that is, 𝑁={0,1,2,…}. (Ω,ℱ,{ℱ_𝑘}_𝑘∈𝑁,𝒫) is a complete probability space with a filtration {ℱ_𝑘}_𝑘∈𝑁 satisfying the usual conditions. 𝑀^𝑇 stands for the transpose of the matrix 𝑀. For symmetric matrices 𝑋 and 𝑌, the notation 𝑋>𝑌 (resp., 𝑋≥𝑌) means that the 𝑋−𝑌 is positive definite (resp., positive semidefinite). ∗ denotes a block that is readily inferred by symmetry. 𝐄{⋅} stands for the mathematical expectation operator with respect to the given probability measure 𝒫.

2. Problem Description

Consider a class of uncertain discrete-time stochastic systems that can be approximated by the following time-delay T-S fuzzy stochastic model with 𝑟 plant rules.

Plant rule 𝑖:

if 𝜃₁(𝑘) is 𝜂^𝑖₁ and … and 𝜃_𝑝(𝑘) is 𝜂^𝑖_𝑝,

then 𝑥(𝑘+1)=𝐴_𝑖+Δ𝐴_𝑖(𝑘)𝑥(𝑘)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑘)𝑥(𝑘−𝜏(𝑘))+𝐵_𝑖+Δ𝐵_𝑖(𝑘)𝑣(𝑘)+𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑥(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)𝑥(𝑘−𝜏(𝑘))+𝐺_𝑖+Δ𝐺_𝑖(𝑘)𝑣(𝑘𝑤(𝑘),𝑦(𝑘)𝐶)=𝜙_𝑖(𝑘)𝑥(𝑘)+𝐶_𝑑𝑖+Δ𝐶_𝑑𝑖(𝑘)𝑥(𝑘−𝜏(𝑘))+𝐷_𝑖+Δ𝐷_𝑖(𝑘)𝑣(𝑘),𝑧(𝑘)=𝐿_𝑖𝑥(𝑘),𝑖=1,2,…,𝑟, (2.1) where 𝜂^𝑖_𝑗 is the fuzzy set. 𝜃(𝑡)=[𝜃₁(𝑡),𝜃₂(𝑡),…,𝜃_𝑝(𝑡)]^𝑇 is the premise variable vector, 𝑥(𝑘)∈ℝ^𝑛 is the state vector, 𝑦(𝑘)∈ℝ^𝑞 is the measurable output vector, 𝑧(𝑘)∈ℝ^𝑟 is the state combination to be estimated, 𝑤(𝑘) is a real scalar process on a probability space (Ω,ℱ,𝒫) relative to an increasing family (ℱ_𝑘)_𝑘∈𝑁 of 𝜎-algebra ℱ_𝑘⊂ℱ generated by (𝑤(𝑘))_𝑘∈𝑁, and 𝑁 is the set of natural numbers. The stochastic process {𝑤(𝑘)} is independent, which is assumed to satisfy 𝐄{𝑤(𝑘)}=0,𝐄𝑤²(𝑘)=1,𝐄𝑤(𝑖)𝑤𝑗=0𝑖≠𝑗, (2.2) where the stochastic variables 𝑤(0),𝑤(1),𝑤(2),… are assumed to be mutually independent. The exogenous disturbance signal 𝑣(𝑘)∈ℝ^𝑝 is assumed to belong to 𝐿_𝑒2([0∞),ℝ^𝑝), and is ℱ_𝑘−1 measurable for all 𝑘∈𝑁, where 𝐿_𝑒2([0∞),ℝ^𝑝) denotes the space of 𝑘-dimensional nonanticipatory square summable stochastic process 𝑓(⋅)=(𝑓(𝑘))_𝑘∈𝑁 on 𝑁 with respect to ℱ_𝑘∈𝑁 satisfying 𝑓²_𝑒2=𝐄^∞_𝑘=0𝑓(𝑘)²=^∞_𝑘=0𝐄𝑓(𝑘)²<∞. (2.3) The time-varying delay 𝜏(𝑘) satisfies 𝜏₁≤𝜏(𝑘)≤𝜏₂, (2.4) where 𝜏₁ and 𝜏₂ are known positive integers representing the minimum and maximum delays, respectively.

In addition, 𝐴_𝑖,𝐴_𝑑𝑖,𝐵_𝑖,𝐸_𝑖,𝐸_𝑑𝑖,𝐺_𝑖,𝐶_𝑖,𝐶_𝑑𝑖,𝐷_𝑖, and 𝐿_𝑖 are known real constant matrices, Δ𝐴_𝑖(𝑘), Δ𝐴_𝑑𝑖(𝑘),Δ𝐵_𝑖(𝑘), Δ𝐸_𝑖(𝑘),Δ𝐸_𝑑𝑖(𝑘),Δ𝐺_𝑖(𝑘),Δ𝐶_𝑑𝑖(𝑘), and Δ𝐷_𝑖(𝑘) are unknown matrices representing time-varying parameter uncertainties, and the admissible uncertainties are assumed to be modeled in the form Δ𝐴_𝑖(𝑘)Δ𝐸_𝑖(𝑘)=𝑀_1𝑖𝑀_2𝑖𝐹_1𝑖(𝑘)𝑁_1𝑖,Δ𝐴_𝑑𝑖(𝑘)Δ𝐸_𝑑𝑖(𝑘)Δ𝐶_𝑑𝑖(𝑘)=𝑀_3𝑖𝑀_4𝑖𝑀_5𝑖𝐹_2𝑖(𝑘)𝑁_2𝑖,Δ𝐵_𝑖(𝑘)Δ𝐺_𝑖(𝑘)Δ𝐷_𝑖(𝑘)=𝑀_6𝑖𝑀_7𝑖𝑀_8𝑖𝐹_3𝑖(𝑘)𝑁_3𝑖, (2.5)(2.6)(2.7) where 𝑀_1𝑖,𝑀_2𝑖,𝑀_3𝑖,𝑀_4𝑖,𝑀_5𝑖,𝑀_6𝑖,𝑀_7𝑖, and 𝑀_8𝑖 are known constant matrices and 𝐹_1𝑖(𝑘), 𝐹_2𝑖(𝑘), and 𝐹_3𝑖(𝑘) are unknown time-varying matrices with Lebesgue measurable elements bounded by 𝐹^𝑇_1𝑖(𝑘)𝐹_1𝑖(𝑘)≤𝐼,𝐹^𝑇_2𝑖(𝑘)𝐹_2𝑖(𝑘)≤𝐼,𝐹^𝑇_3𝑖(𝑘)𝐹_3𝑖(𝑘)≤𝐼,∀𝑘. (2.8) The defuzzified output of the T-S fuzzy system (2.1) is inferred as follows: 𝑥(𝑘+1)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑘))𝐴_𝑖+Δ𝐴_𝑖(𝑘)𝑥(𝑘)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑘)𝑥(𝑘−𝜏(𝑘))+𝐵_𝑖+Δ𝐵_𝑖(𝑘𝑣(𝑘)+𝐸_𝑖)+Δ𝐸_𝑖(𝑘)𝑥(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)𝑥((𝑘+𝐺𝑘−𝜏))_𝑖+Δ𝐺_𝑖(𝑘)𝑣(𝑘𝑤(𝑘,𝑦(𝑘)=))_𝑟𝑖=1ℎ_𝑖(𝜃(𝑘))𝜙𝐶_𝑖(𝑘)𝑥(𝑘)+𝐶_𝑑𝑖+Δ𝐶_𝑑𝑖(𝑘)𝑥(𝑘−𝜏(𝑘))+𝐷_𝑖+Δ𝐷_𝑖(𝑘𝑣(𝑘,𝑧(𝑘)=))_𝑟𝑖=1ℎ_𝑖(𝜃(𝑘))𝐿_𝑖𝑥(𝑘), (2.9) where ℎ_𝑖(𝜃(𝑘))=𝜇_𝑖(𝜃(𝑘))/∑^𝑟_𝑗=1ℎ_𝑗(𝜃(𝑘)), and 𝜇_𝑖(𝜃(𝑘))=∏^𝑝_𝑗=1𝜂^𝑖_𝑗(𝜃_𝑗(𝑘)), in which 𝜂^𝑖_𝑗(𝜃_𝑗(𝑘)) is the grade of membership of 𝜂_𝑗(𝑘) in 𝜂^𝑖_𝑗. According to the theory of fuzzy sets, we have 𝜇_𝑖(𝜃(𝑘))≥0,(𝑖=1,2,…,𝑟) and ∑^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑘))>0. Therefore, it is implied that ℎ_𝑖(𝜃(𝑘))≥0,(𝑖=1,2,…,𝑟) and ∑^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑘))=1.

For some given diagonal matrices 𝐾₁≥0 and 𝐾₂≥0 with 𝐾₂>𝐾₁, the nonlinear vector functions 𝜙∈[𝐾₁,𝐾₂], is assumed to represent sensor nonlinearities and satisfies the following sensor condition [29]: 𝜙(𝜉)−𝐾₁𝜉^𝑇𝜙(𝜉)−𝐾₂𝜉≤0,∀𝜉∈ℝ^𝑞. (2.10) Further, the nonlinear function 𝜙(𝜉) can be decomposed into a linear and a nonlinear part as follows: 𝜙(𝜉)=𝜙_𝑠(𝜉)+𝐾₁𝜉, (2.11) where the nonlinearity 𝜙_𝑠(𝜉) belongs to the set Φ_𝑠 given by Φ_𝑠=𝜙_𝑠∶𝜙^𝑇_𝑠(𝜉)𝜙_𝑠(𝜉)−𝐾𝜉≤0 (2.12) with 𝐾=𝐾₂−𝐾₁>0.

We consider the following fuzzy filter for the estimation of 𝑧_𝑘: 𝑥(𝑘+1)=^𝑟_𝑖=1ℎ_𝑖(𝜃(𝑘))𝐴_𝑓𝑖𝑥(𝑘)+𝐵_𝑓𝑖𝑦(𝑘,𝑧(𝑘)=_𝑟𝑖=1)ℎ_𝑖(𝜃(𝑘))𝐿_𝑖𝑥(𝑘), (2.13) where 𝑥(𝑘)∈ℝ^𝑛, 𝑧(𝑘)∈ℝ^𝑟, and the matrices 𝐴_𝑓𝑖 and 𝐵_𝑓𝑖(𝑖=1,2,…,𝑟) are to be determined.

Remark 2.1. Similar to [27, 29], the sensor nonlinearities satisfying (2.10) are also considered in this paper. Here, there exists the nonlinear function 𝜙(𝜉) in the system (2.1), which is called sensor nonlinearities. It is noted that in the previous filter, the matrix 𝐿_𝑖 is assumed to be constant in order to avoid more verbosely mathematical derivation.
Defining 𝑥(𝑘)=𝑥(𝑘)−𝑥(𝑘), 𝑧(𝑘)=𝑧(𝑘)−𝑧(𝑘), 𝑒(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇(𝑘)]^𝑇, and augmenting the model (2.9) to include the states of the filter (2.13), we obtain the following filtering error system: 𝑒(𝑘+1)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)𝑒(𝑘)+𝐵_𝑓𝑗𝜙_𝑠𝐶_𝑖𝑒(𝑘+)𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)𝐻𝑒(𝑘−𝜏(𝑘))+𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑘)𝑣(𝑘)+𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑒(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)((𝑘+𝐻𝑒𝑘−𝜏))𝐺_𝑖+Δ𝐺_𝑖(𝑘)𝑣(𝑘)𝑤(𝑘),𝑧(𝑘)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐿_𝑖𝑗𝑒(𝑘), (2.14)(2.15) where 𝐴_𝑖𝑗=𝐴_𝑖0𝐴_𝑖−𝐴_𝑓𝑗−𝐵_𝑓𝑗𝐾₁𝐶_𝑖𝐴_𝑓𝑗,𝐵_𝑓𝑗=0−𝐵_𝑓𝑗,𝐶_𝑖=𝐶_𝑖0,𝐴_𝑑𝑖𝑗=𝐴_𝑑𝑖𝐴_𝑑𝑖−𝐵_𝑓𝑗𝐶_𝑑𝑖,𝐵_𝑖𝑗=𝐵_𝑖𝐵_𝑖−𝐵_𝑓𝑗𝐷_𝑖,𝐸_𝑖=𝐸_𝑖0𝐸_𝑖0,𝐸_𝑑𝑖=𝐸_𝑑𝑖𝐸_𝑑𝑖,𝐺_𝑖=𝐺_𝑖𝐺_𝑖,𝐿_𝑖𝑗=0𝐿_𝑖−𝐿_𝑗,Δ,𝐻=𝐼0𝐴_𝑖(𝑘)=𝑀_𝐴𝑖𝐹_1𝑖(𝑘)𝑁_1𝑖,Δ𝐴_𝑑𝑖𝑗(𝑘)=𝑀_{𝐴𝑑𝑖𝑗}𝐹_2𝑖(𝑘)𝑁_2𝑖,Δ𝐵_𝑖𝑗(𝑘)=𝑀_𝐵𝑖𝑗𝐹_3𝑖(𝑘)𝑁_3𝑖,Δ𝐸_𝑖(𝑘)=𝑀_𝐸𝑖𝐹_1𝑖(𝑘)𝑁_1𝑖,Δ𝐸_𝑑𝑖(𝑘)=𝑀_𝐸𝑑𝑖𝐹_2𝑖(𝑘)𝑁_2𝑖,Δ𝐺_𝑖(𝑘)=𝑀_𝐺𝑖𝐹_3𝑖(𝑘)𝑁_3𝑖,𝑀_𝐴𝑖=𝑀_1𝑖𝑀_1𝑖,𝑀_{𝐴𝑑𝑖𝑗}=𝑀_3𝑖𝑀_3𝑖−𝐵_𝑓𝑗𝑀_5𝑖,𝑀_𝐵𝑖𝑗=𝑀_6𝑖𝑀_6𝑖−𝐵_𝑓𝑗𝑀_8𝑖,𝑀_𝐸𝑖=𝑀_2𝑖𝑀_2𝑖,𝑀_𝐸𝑑𝑖=𝑀_4𝑖𝑀_4𝑖,𝑀_𝐺𝑖=𝑀_7𝑖𝑀_7𝑖,𝑁_1𝑖=𝑁_1𝑖0. (2.16)
The 𝐻_∞ filtering problem to be addressed in this paper can be formulated as follows. Given discrete-time fuzzy stochastic systems (2.9), a prescribed level of noise attenuation 𝛾>0, and any 𝜙∈[𝐾₁𝐾₂], find a suitable filter in the form of (2.13) such that the following requirements are satisfied.(1) The filtering error system (2.14)-(2.15) with 𝑣(𝑘)=0 is said to be robustly stochastic stable if there exists a scalar 𝑐>0 such that for all admissible uncertainties satisfying (2.5)–(2.8) 𝐄^∞_𝑘=0|𝑥(𝑘)|²≤𝑐𝐄|𝑥(0)|², (2.17) where 𝑥(𝑘) denotes the solution of stochastic systems with initial state 𝑥(0).(2) For the given disturbance attenuation level 𝛾>0 and under zero initial conditions for all 𝑣(𝑘)∈𝐿_𝑒2([0∞),ℝ^𝑝), the performance index 𝛾 satisfies the following inequality: ‖𝑧(𝑘)−𝑧(𝑘)‖_𝑒2<𝛾‖𝑣(𝑘)‖_𝑒2. (2.18)
Before concluding this section, we introduce the following lemmas, which will be used in the derivation of our main results in the next section.

Lemma 2.2 (Gu et al. [33]). Given constant matrices Γ₁, Γ₂, and Γ₃ with appropriate dimensions, where Γ^𝑇₁=Γ₁ and Γ^𝑇₂=Γ₂>0, then Γ₁+Γ^𝑇₃Γ₂⁻¹Γ₃<0 (2.19) if and only if Γ₁Γ^𝑇₃Γ₃−Γ₂<0,or−Γ₂Γ₃Γ^𝑇₃Γ₁<0. (2.20)

Lemma 2.3 (see [34]). For given matrices 𝐻, 𝐸, and 𝐹(𝑡) with 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 and scalar 𝜀>0, the following inequality holds: 𝐻𝐹(𝑡)𝐸+𝐸^𝑇𝐹^𝑇(𝑡)𝐻^𝑇≤𝜀𝐻𝐻^𝑇+𝜀⁻¹𝐸^𝑇𝐸. (2.21)

3. Main Results

Theorem 3.1. Consider the uncertain discrete-time fuzzy stochastic systems in (2.1). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system (2.14)-(2.15) is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, 𝑄>0, any matrices 𝐴_𝐹𝑗,𝐵_𝐹𝑗, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟 such that the following LMI is satisfied: Σ₁₁000𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑋Σ₁₇𝐸^𝑇_𝑖𝑋𝐸^𝑇_𝑖𝑌000∗Σ₂₂0000𝐴^𝑇_𝐹𝑗00000∗∗Σ₃₃00𝐴^𝑇_𝑑𝑖𝑋Σ₃₇𝐸^𝑇_𝑑𝑖𝑋𝐸^𝑇_𝑑𝑖𝑌000∗∗∗Σ₄₄0𝐵^𝑇_𝑖𝑋Σ₄₇𝐺^𝑇_𝑖𝑋𝐺^𝑇_𝑖𝑌000∗∗∗∗−2𝐼0−𝐵^𝑇_𝐹𝑗00000∗∗∗∗∗−𝑋000𝑋𝑀_1𝑖𝑋𝑀_3𝑖𝑋𝑀_6𝑖∗∗∗∗∗∗−𝑌00𝑌𝑀_1𝑖Σ₇₁₁Σ₇₁₂∗∗∗∗∗∗∗−𝑋0𝑋𝑀_2𝑖𝑋𝑀_4𝑖𝑋𝑀_7𝑖∗∗∗∗∗∗∗∗−𝑌𝑌𝑀_2𝑖𝑌𝑀_4𝑖𝑌𝑀_7𝑖∗∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼00∗∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼0∗∗∗∗∗∗∗∗∗∗∗−𝜀_3𝑖𝐼<0, (3.1) where Σ₁₁=−𝑋+𝜏₂−𝜏₁+1𝑄+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_1𝑖,Σ₂₂=−𝑌+𝐿_𝑖−𝐿_𝑗^𝑇𝐿_𝑖−𝐿_𝑗,Σ₃₃=−𝑄+𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_2𝑖,Σ₄₄=−𝛾²𝐼+𝜀_3𝑖𝑁^𝑇_3𝑖𝑁_3𝑖,Σ₁₇=𝐴^𝑇_𝑖𝑌−𝐴^𝑇_𝐹𝑗−𝐶^𝑇_𝑖𝐾^𝑇₁𝐵^𝑇_𝐹𝑗,Σ₃₇=𝐴^𝑇_𝑑𝑖𝑌−𝐶^𝑇_𝑑𝑖𝐵^𝑇_𝐹𝑗,Σ₄₇=𝐵^𝑇_𝑖𝑌−𝐷^𝑇_𝑖𝐵^𝑇_𝐹𝑗,Σ₇₁₁=𝑌𝑀_3𝑖−𝐵_𝐹𝑗𝑀_5𝑖,Σ₇₁₂=𝑌𝑀_6𝑖−𝐵_𝐹𝑗𝑀_8𝑖. (3.2) Moreover, if the previous condition is satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by 𝐴_𝑓𝑗=𝑌⁻¹𝐴_𝐹𝑗,𝐵_𝑓𝑗=𝑌⁻¹𝐵_𝐹𝑗. (3.3)

Proof. We first establish the condition of robustly stochastic stability for the filtering error system (2.14)-(2.15). It can be shown that LMI (3.1) implies Σ₁₁00𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑋Σ₁₇𝐸^𝑇_𝑖𝑋𝐸^𝑇_𝑖𝑌00∗−𝑌000𝐴^𝑇_𝐹𝑗0000∗∗Σ₃₃0𝐴^𝑇_𝑑𝑖𝑋Σ₃₇𝐸^𝑇_𝑑𝑖𝑋𝐸^𝑇_𝑑𝑖𝑌00∗∗∗−2𝐼0𝐵^𝑇_𝐹𝑗0000∗∗∗∗−𝑋000𝑋𝑀_1𝑖𝑋𝑀_3𝑖∗∗∗∗∗−𝑌00𝑌𝑀_1𝑖Σ₇₁₁∗∗∗∗∗∗−𝑋0𝑋𝑀_2𝑖𝑋𝑀_4𝑖∗∗∗∗∗∗∗−𝑌𝑌𝑀_2𝑖𝑌𝑀_4𝑖∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼0∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼<0. (3.4) Define a matrix 𝑃>0 by 𝑃=,𝑋00𝑌 (3.5) where 𝑋>0 and 𝑌>0 satisfy the solvability of (3.1).
Then, it can be shown from (3.3) and (3.5) that LMI (3.4) can be rewritten as Σ₁₁0𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑗𝑃𝐸^𝑇_𝑖∗𝑃00Σ₂₂0𝐻^𝑇𝐴^𝑇_𝑑𝑖𝑗𝑃𝐻^𝑇𝐸^𝑇_𝑑𝑖𝑃00∗∗−2𝐼𝐵^𝑇_𝑓𝑗𝑃000∗∗∗−𝑃0𝑃𝑀_𝐴𝑖𝑃𝑀_{𝐴𝑑𝑖𝑗}∗∗∗∗−𝑃𝑃𝑀_𝐸𝑖𝑃𝑀_𝐸𝑑𝑖∗∗∗∗∗−𝜀_1𝑖𝐼0∗∗∗∗∗∗−𝜀_2𝑖𝐼<0, (3.6) where Σ₁₁=−𝑃+𝜏₂−𝜏₁+1𝐻^𝑇𝑄𝐻+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_1𝑖,Σ₂₂=−𝐻^𝑇𝑄𝐻+𝜀_2𝑖𝐻^𝑇𝑁^𝑇_2𝑖𝑁_2𝑖𝐻, (3.7) which is equivalent to the following inequality: Σ₁₁0𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑗𝑃𝐸^𝑇_𝑖𝑃∗−𝐻^𝑇𝑄𝐻0𝐻^𝑇𝐴^𝑇_𝑑𝑖𝑗𝑃𝐻^𝑇𝐸^𝑇_𝑑𝑖𝑃∗∗−2𝐼𝐵^𝑇_𝑓𝑗𝑃0∗∗∗−𝑃0∗∗∗∗−𝑃+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_1𝑖+𝜀⁻¹_1𝑖𝑀_1𝑖𝑀^𝑇_1𝑖+𝜀_2𝑖𝑁^𝑇_2𝑖𝑁_2𝑖+𝜀⁻¹_2𝑖𝑀_2𝑖𝑀^𝑇_2𝑖<0 (3.8) with Σ₁₁=−𝑃+𝜏₂−𝜏₁+1𝐻^𝑇𝑄𝐻,𝑀_1𝑖=000𝑀^𝑇_{𝐴𝑑𝑖𝑗}𝑃𝑀^𝑇_𝐸𝑑𝑖𝑃^𝑇,𝑁_1𝑖=𝑁_1𝑖0000,𝑀_2𝑖000𝑀^𝑇_𝐴𝑖𝑃𝑀^𝑇_𝐸𝑖𝑃^𝑇,𝑁_2𝑖=0𝑁_2𝑖𝐻000. (3.9) On the other hand, it is noted that the following equality holds: Σ₁₁0𝐶^𝑇_𝑖𝐾^𝑇𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐸_𝑖+Δ𝐸_𝑖(𝑘)^𝑇𝑃∗−𝐻^𝑇𝑄𝐻0𝐻^𝑇𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)^𝑇𝑃𝐻^𝑇𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)^𝑇𝑃∗∗−2𝐼𝐵^𝑇_𝑓𝑗𝑃0∗∗∗−𝑃0∗∗∗∗−𝑃=Σ₁₁0𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑗𝑃𝐸^𝑇_𝑖𝑃∗−𝐻^𝑇𝑄𝐻0𝐻^𝑇𝐴^𝑇_𝑑𝑖𝑗𝑃𝐻^𝑇𝐸^𝑇_𝑑𝑖𝑃∗∗−2𝐼𝐵^𝑇_𝑓𝑗𝑃0∗∗∗−𝑃0∗∗∗∗−𝑃+𝑀_1𝑖𝐹_1𝑖𝑁_1𝑖+𝑁^𝑇_1𝑖𝐹_1𝑖𝑀^𝑇_1𝑖+𝑀_2𝑖𝐹_2𝑖𝑁_2𝑖+𝑁^𝑇_2𝑖𝐹_2𝑖𝑀^𝑇_2𝑖. (3.10) By Lemmas 2.2 and 2.3, from (3.8)–(3.10), we have ΓΓ=¹¹Γ₁₂Γ₁₃∗Γ₂₂Γ₂₃∗∗Γ₃₃<0, (3.11) where Γ₁₁=Σ₁₁+𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)+𝐸_𝑖+Δ𝐸_𝑖(𝑘)^𝑇𝑃𝐸_𝑖+Δ𝐸_𝑖(𝑘),Γ₁₂=𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)𝐻+𝐸_𝑖+Δ𝐸_𝑖(𝑘)^𝑇𝑃𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)Γ𝐻,₁₃=𝐶^𝑇_𝑖𝐾^𝑇+𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐵_𝑓𝑗,Γ₂₂=𝐻^𝑇−𝑄+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)^𝑇𝑃𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)^𝑇𝑃𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)Γ𝐻,₂₃=𝐻^𝑇𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)^𝑇𝑃𝐵_𝑓𝑗,Γ₃₃=−2𝐼+𝐵^𝑇_𝑓𝑗𝑃𝐵_𝑓𝑗. (3.12) Then there exists a small scalar 𝛼>0 such that Γ<.−𝛼𝐼00000000 (3.13)
Consider the Lyapunov-Krasovskii functional candidate as follows: 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘), (3.14) where 𝑉₁(𝑘)=𝑒^𝑇(𝑘)(𝑘),𝑉𝑃𝑒₂(𝑘)=^𝑘−1_{(𝑘)𝑖=𝑘−𝜏}𝑒^𝑇(𝑖)𝐻^𝑇(𝑖),𝑉𝑄𝐻𝑒₃(𝑘)=^−𝜏1+1_{𝑗=−𝜏2+2}^𝑘−1_{𝑙=𝑘+𝑗−1}𝑒^𝑇(𝑙)𝐻^𝑇𝑄𝐻𝑒(𝑙). (3.15) Calculating the difference of 𝑉(𝑘) along the filtering error system (2.14) with 𝑣(𝑘)=0 and taking the mathematical expectation, we have 𝐄{Δ𝑉(𝑘)}=𝐄Δ𝑉₁(𝑘)+𝐄Δ𝑉₂(𝑘)+𝐄Δ𝑉₃(𝑘,𝐄)Δ𝑉₁(𝑘=_{𝑟𝑟𝑖=1𝑗})₌₁ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐄𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)𝑒(𝑘)+𝐵_𝑓𝑗𝜙_𝑠𝐶_𝑖𝑒(𝑘)+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)((𝑘+𝐻𝑒𝑘−𝜏))𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑒(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)((𝑘(𝑘𝐻𝑒𝑘−𝜏))×𝑤)^𝑇𝑃×𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)𝑒(𝑘)+𝐵_𝑓𝑗𝜙_𝑠𝐶_𝑖𝑒(𝑘)+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘((𝑘+)𝐻𝑒𝑘−𝜏))𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑒(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)×𝐻𝑒(𝑘−𝜏(𝑘))×𝑤(𝑘)−𝑒^𝑇(𝑘)𝑃𝑒(𝑘). (3.16) Noting (2.2) and (2.12), we have 𝐄Δ𝑉₁(𝑘≤_{𝑟𝑟𝑖=1𝑗})₌₁ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)𝑒(𝑘)+𝐵_𝑓𝑗𝜙_𝑠𝐶_𝑖𝑒(𝑘)+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)×𝐻𝑒(𝑘−𝜏(𝑘))^𝑇𝑃𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)𝑒(𝑘)+𝐵_𝑓𝑗𝜙_𝑠𝐶_𝑖𝑒(𝑘)+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘((𝑘+)𝐻𝑒𝑘−𝜏))𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑒(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)𝐻𝑒(𝑘−𝜏(𝑘))^𝑇𝑃×𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑒(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)((𝑘(𝑘)𝐻𝑒𝑘−𝜏))−𝑒^𝑇𝑃𝑒(𝑘)−2𝜙_𝑠𝐶_𝑖𝑒(𝑘)𝜙_𝑠𝐶_i𝑒(𝑘)−𝐾𝐶_𝑖𝑒(𝑘). (3.17) By some calculations, we have 𝐄Δ𝑉₂(𝑘)=^𝑘−𝜏1_{()𝑖=𝑘+1−𝜏𝑘+1}𝑒^𝑇(𝑖)𝐻^𝑇𝑄𝐻𝑒(𝑖)−𝑒^𝑇(𝑘−𝜏(𝑘))𝐻^𝑇𝑄𝐻𝑒(𝑘−𝜏(𝑘))+𝑒^𝑇(𝑘)𝐻^𝑇𝑄𝐻𝑒(𝑘)+^𝑘−1_{𝑖=𝑘+1−𝜏1}𝑒^𝑇(𝑖)𝐻^𝑇𝑄𝐻𝑒(𝑖)−^𝑘−1_{(𝑘)𝑖=𝑘+1−𝜏}𝑒^𝑇(𝑖)𝐻^𝑇𝑄𝐻𝑒(𝑖),𝐄Δ𝑉₃(𝑘)=^−𝜏1+1_{𝑗=−𝜏2+2}𝑒^𝑇(𝑘)𝐻^𝑇𝑄𝐻𝑒(𝑘)+^𝑘−1_{𝑙=𝑘+𝑗}𝑒^𝑇(𝑙)𝐻^𝑇𝑄𝐻𝑒(𝑙)−^𝑘−1_{𝑙=𝑘+𝑗−1}𝑒^𝑇(𝑙)𝐻^𝑇(𝑙)=𝜏𝑄𝐻𝑒₂−𝜏₁𝑒^𝑇(𝑘)𝐻^𝑇𝑄𝐻𝑒(𝑘)−^𝑘−𝜏1_{𝑖=𝑘+1−𝜏2}𝑒^𝑇(𝑖)𝐻^𝑇𝑄𝐻𝑒(𝑖). (3.18)(3.19) Since 𝜏₁≤𝜏(𝑘)≤𝜏₂, we have ^𝑘−1_{𝑖=𝑘+1−𝜏1}𝑒^𝑇(𝑖)𝐻^𝑇𝑄𝐻𝑒(𝑖)−^𝑘−1_{(𝑘)𝑖=𝑘+1−𝜏}𝑒^𝑇(𝑖)𝐻^𝑇𝑄𝐻𝑒(𝑖)≤0,_{𝑘−𝜏1()𝑖=𝑘+1−𝜏𝑘+1}𝑒^𝑇(𝑖)𝐻^𝑇𝑄𝐻𝑒(𝑖)−^𝑘−𝜏1_{𝑖=𝑘+1−𝜏2}𝑒^𝑇(𝑖)𝐻^𝑇Q𝐻𝑒(𝑖)≤0. (3.20)(3.21) Combining (3.18)–(3.21), we have 𝐄Δ𝑉₂(𝑘)+Δ𝑉₃(𝑘)≤𝜏₂−𝜏₁+1𝑒^𝑇(𝑘)𝐻^𝑇𝑄𝐻𝑒(𝑘)−𝑒^𝑇(𝑘−𝜏(𝑘))𝐻^𝑇𝑄𝐻𝑒(𝑘−𝜏(𝑘)). (3.22) Combining (3.17) and (3.22) yields 𝐄{(𝑘≤Δ𝑉)}_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)𝑒(𝑘)+𝐵_𝑓𝑗𝜙_𝑠𝐶_𝑖𝑒(𝑘)+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)×𝐻𝑒(𝑘−𝜏(𝑘))^𝑇𝑃𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)𝑒(𝑘)+𝐵_𝑓𝑗𝜙_𝑠𝐶_𝑖𝑒(𝑘)+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)((𝑘+𝐻𝑒𝑘−𝜏))𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑒(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)𝐻𝑒(𝑘−𝜏(𝑘))^𝑇𝑃×𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑒(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)𝐻𝑒(𝑘−𝜏(𝑘))+𝑒^𝑇(𝑘)𝜏₂−𝜏₁+1𝐻^𝑇𝑄𝐻−𝑃𝑒(𝑘)−2𝜙_𝑠𝐶_𝑖𝑒(𝑘)×𝜙_𝑠𝐶_𝑖𝑒(𝑘)−𝐾𝐶_𝑖𝑒(𝑘)−𝑒^𝑇(𝑘−𝜏(𝑘))𝐻^𝑇((𝑘=𝑄𝐻𝑒𝑘−𝜏))_{𝑟𝑟𝑖=1𝑗=1}ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝜁^𝑇(𝑘)Γ𝜁(𝑘), (3.23) where 𝜁^𝑇(𝑘)=𝑒^𝑇(𝑘)𝑒^𝑇(𝑘−𝜏(𝑘))𝜙^𝑇_𝑠𝐶_𝑖𝑒(𝑘), (3.24) and Γ is given in (3.11). Thus, it follows from (3.13) and (3.23) that 𝐄{𝑉(𝑘+1)}−𝐄{𝑉(𝑘)}<−𝛼𝐄|𝑒(𝑘)|². (3.25) Hence, by summing up both sides of (3.25) from 0 to 𝑁 for any integer 𝑁>1, we have 𝐄{𝑉(𝑁+1)}−𝐄{𝑉(0)}<−𝛼𝐄^𝑁_𝑘=0|𝑒(𝑘)|², (3.26) which yields 𝐄^𝑁_𝑘=0|𝑒(𝑘)|²<1𝛼[𝐄{𝑉(0{𝑉(≤1)}−𝐄𝑁+1)}]𝛼𝐄{𝑉(0)}≤𝑐𝐄|𝑥(0)|², (3.27) where 𝑐=(1/𝛼)𝜆_max(𝑃). Taking 𝑁→∞, it is shown from (2.17) and (3.27) that the filtering error system (2.14) is robustly stochastic stable for 𝑣(𝑘)=0.
Next, we will show that the filtering error system (2.14)-(2.15) satisfies ‖𝑧(𝑘)‖_𝑒2<𝛾‖𝑣(𝑘)‖_𝑒2 (3.28) for all nonzero 𝑣(𝑘)∈𝐿_𝑒2([0∞),ℝ^𝑝). To this end, define 𝐽(𝑁)=𝐄^𝑁_𝑘=1|𝑧(𝑘)|²<𝛾²|𝑣(𝑘)|² (3.29) with any integer 𝑁>0. Then for any nonzero 𝑣(𝑘), we have 𝐽(𝑁)=𝐄^𝑁_𝑘=1|𝑧(𝑘)|²−𝛾²|𝑣(𝑘)|²≤𝐄^𝑁_𝑘=1|𝑧(𝑘)|²−𝛾²|𝑣(𝑘)|²+Δ𝑉(𝑘)−𝐄{𝑉(𝑁+1)}=𝐄^𝑁_𝑘=1|𝑧(𝑘)|²−𝛾²|𝑣(𝑘)|²+Δ𝑉(𝑘)=𝐄^𝑁𝑟_𝑘=1^𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝜁̂^𝑇(𝑘)Γ_𝑖𝑗𝜁̂(𝑘), (3.30) where 𝜁̂^𝑇(𝑘)=𝑒^𝑇(𝑘)𝑒^𝑇(𝑘−𝜏(𝑘))𝑣^𝑇(𝑘)𝜙_𝑠𝐶_𝑖𝑒(𝑘)^𝑇,Γ_𝑖𝑗=Γ₁₁Γ₁₂Γ₁₃Γ₁₄∗Γ₂₂Γ₂₃Γ₂₄∗∗Γ₃₃Γ₃₄∗∗∗Γ₄₄,Γ₁₁=Σ₁₁+𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)+𝐸_𝑖+Δ𝐸_𝑖(𝑘)^𝑇𝑃𝐸_𝑖+Δ𝐸_𝑖(𝑘)+𝐿_𝑖−𝐿_𝑗^𝑇𝐿_𝑖−𝐿_𝑗,Γ₁₂=𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)𝐻+𝐸_𝑖+Δ𝐸_𝑖(𝑘)^𝑇𝑃𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)Γ𝐻,₁₃=𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑘)+𝐸_𝑖+Δ𝐸_𝑖(𝑘)^𝑇𝑃𝐺_𝑖+Δ𝐺_𝑖(𝑘),Γ₁₄=𝐶^𝑇_𝑖𝐾^𝑇+𝐴_𝑖𝑗+Δ𝐴_𝑖(𝑘)^𝑇𝑃𝐵_𝑓𝑗,Γ₂₂=𝐻^𝑇−𝑄+𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)^𝑇𝑃𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)+𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)^𝑇𝑃𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)Γ𝐻,₂₃=𝐻^𝑇𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖𝑗(𝑘)𝑃𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑘)+𝐻^𝑇𝐸_𝑑𝑖+Δ𝐸_𝑑𝑖(𝑘)𝑃𝐺_𝑖+Δ𝐺_𝑖(𝑘),Γ₂₄=𝐻^𝑇𝐴_𝑑𝑖𝑗+Δ𝐴_𝑑𝑖(𝑘)^𝑇𝑃𝐵_𝑓𝑗,Γ₃₃=−𝛾²𝐼+𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑘)^𝑇𝑃𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑘)+𝐺_𝑖+Δ𝐺_𝑖(𝑘)^𝑇𝑃𝐺_𝑖+Δ𝐺_𝑖(𝑘),Γ₃₄=𝐵_𝑖𝑗+Δ𝐵_𝑖𝑗(𝑘)^𝑇𝑃𝐵_𝑓𝑗,Γ₄₄=−2𝐼+𝐵^𝑇_𝑓𝑗𝑃𝐵_𝑓𝑗. (3.31) It can be shown that if there exist real matrices 𝑋>0, 𝑌>0, 𝑄>0, any matrices 𝐴_𝐹𝑗,𝐵_𝐹𝑗, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟 satisfying LMI (3.1). Since 𝑣(𝑘)∈𝐿_𝑒2([0∞),ℝ^𝑝)≠0 and by Lemma 2.2, it is implied that Γ_𝑖𝑗<0, and, thus, 𝐽(𝑁)<0. That is, ||𝑧(𝑘)||_𝑒2<𝛾||𝑣(𝑘)||_𝑒2. This completes the proof.

Remark 3.2. When 𝜏₁ and 𝜏₂ are given, matrix inequality (3.1) is linear matrix inequalities in matrix variables 𝑋>0, 𝑌>0, 𝑄>0, 𝐴_𝐹𝑗,𝐵_𝐹𝑗, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟, which can be efficiently solved by the developed interior point algorithm [33]. Meanwhile, it is esay to find the minimal attenuation level 𝛾.

Remark 3.3. Theorem 3.1 is suggested to the 𝐻_∞ filter design for uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay. This approach is called fuzzy-rule-dependent approach, which is applicable to the case that the information of the premise variable 𝜃_𝑖(𝑘)(𝑖=1,2,…,𝑟) is available; for example, see [24]. For the special case that the premise variables are unavailable, the fuzzy-rule-independent approach as in [24, 29] is adopted, which may result in lager conservativeness due to lack of the information of the premise variables.
In the following, we will present the fuzzy-rule-dependent 𝐻_∞ filter design for uncertain discrete-time fuzzy stochastic system with time-varying delay, which are less conservative than Theorem 3.1.

Theorem 3.4. Consider the uncertain discrete-time fuzzy stochastic systems in (2.1). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system (2.14)-(2.15) is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, 𝑄>0, any matrices 𝐴_𝐹𝑖,𝐵_𝐹𝑖, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗, such that the following LMIs are satisfied: Σ_𝑖𝑖<0,𝑖=1,2,…,𝑟,Σ_𝑖𝑗+Σ_𝑗𝑖<0,𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗, (3.32)(3.33) where Σ_𝑖𝑗=Σ₁₁000𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑋Σ₁₇𝐸^𝑇_𝑖𝑋𝐸^𝑇_𝑖𝑌000∗Σ₂₂0000𝐴^𝑇_𝐹𝑗00000∗∗Σ₃₃00𝐴^𝑇_𝑑𝑖𝑋Σ₃₇𝐸^𝑇_𝑑𝑖𝑋𝐸^𝑇_𝑑𝑖𝑌000∗∗∗Σ₄₄0𝐵^𝑇_𝑖𝑋Σ₄₇𝐺^𝑇_𝑖𝑋𝐺^𝑇_𝑖𝑌000∗∗∗∗−2𝐼0−𝐵^𝑇_𝐹𝑗00000∗∗∗∗∗−𝑋000𝑋𝑀_1i𝑋𝑀_3𝑖𝑋𝑀_6𝑖∗∗∗∗∗∗−𝑌00𝑌𝑀_1𝑖Σ₇₁₁Σ₇₁₂∗∗∗∗∗∗∗−𝑋0𝑋𝑀_2𝑖𝑋𝑀_4𝑖𝑋𝑀_7𝑖∗∗∗∗∗∗∗∗−𝑌𝑌𝑀_2𝑖𝑌𝑀_4𝑖𝑌𝑀_7𝑖∗∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼00∗∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼0∗∗∗∗∗∗∗∗∗∗∗−𝜀_3𝑖𝐼, (3.34) where Σ₁₁,Σ₂₂,Σ₃₃,Σ₄₄,Σ₁₇,Σ₃₇,Σ₄₇,Σ₇₁₁, and Σ₇₁₂ are defined in Theorem 3.1. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

Proof. Considering (2.2) and (3.1), it follows from (3.30) that 𝐽(𝑁)≤𝐄^𝑁𝑟_𝑘=1^𝑟_𝑖=1𝑗=1ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))𝜁̂^𝑇(𝑘)Γ_𝑖𝑗𝜁̂(𝑘)=𝐄^𝑁_𝑘=1𝜁̂^𝑇(𝑘)^𝑟_𝑖=1ℎ²_𝑖(𝜃(𝑘))Γ_𝑖𝑖+^𝑟𝑟_{𝑖=1𝑖<𝑗}ℎ_𝑖(𝜃(𝑘))ℎ_𝑗(𝜃(𝑘))Γ_𝑖𝑗+Γ_𝑗𝑖𝜁̂(𝑘). (3.35) Our elaborate estimation 𝐽(𝑁) is negative definite if and only if Γ_𝑖𝑖<0 and Γ_𝑖𝑗+Γ_𝑗𝑖<0. By Lemmas 2.2 and 2.3, we can obtain that (3.32) is equivalent to Γ_𝑖𝑖<0, and (3.33) is equivalent to Γ_𝑖𝑗+Γ_𝑗𝑖<0. Thus, we can conclude that the LMIs (3.32) and (3.33) can guarantee 𝐽(𝑁)<0. That is, ||𝑧(𝑘)||_𝑒2<𝛾||𝑣(𝑘)||_𝑒2. This completes the proof.

Theorem 3.5. Consider the uncertain discrete-time fuzzy stochastic systems in (2.1). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system (2.14)-(2.15) is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, 𝑄>0, any matrices 𝐴_𝐹𝑖,𝐵_𝐹𝑖, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗, such that the following LMIs are satisfied: Σ_𝑖𝑖<Ω_𝑖𝑖,𝑖=1,2,…,𝑟,Σ_𝑖𝑗+Σ_𝑗𝑖<Ω_𝑖𝑗+Ω^𝑇_𝑖𝑗,𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗,ΩΩ=¹¹Ω₁₂⋯Ω_1𝑟∗Ω₂₂⋯Ω_2𝑟⋮⋮⋱⋮∗∗⋯Ω_𝑟𝑟<0, (3.36)(3.37)(3.38) where Σ_𝑖𝑗,Σ₁₁,Σ₂₂,Σ₃₃,Σ₄₄,Σ₁₇,Σ₃₇,Σ₄₇,Σ₇₁₁, and Σ₇₁₂ are defined in Theorems 3.4 and 3.1, respectively. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

Proof. This theorem can be proved by employing the same techniques as in the proof of Theorem 3.4; hence, the detailed procedure is omitted here.

Remark 3.6. It is noted that the convexifying procedure proposed in this paper is based on the relaxation inequality (3.36), and extensions of the current derivations based on the more powerful relaxation techniques presented in [8, 24, 30] are straightforward. In this way, the design conservatism can be further reduced but the computation complexity will also increase. By considering the information on the premises [8, 24, 30], Theorems 3.4 and 3.5 relaxed the conservatism of the previous works [29] by representing the interactions among the fuzzy subsystems in a matrix and solving it in a numerical manner.
For comparison, if there is no time-varying delay exit in (2.1), we consider the following discrete-time stochastic systems [29]: 𝑥(𝑘+1)=𝐴_𝑖+Δ𝐴_𝑖(𝑘)𝑥(𝑘)+𝐵_𝑖+Δ𝐵_𝑖(𝑘)𝑣(𝑘)+𝐸_𝑖+Δ𝐸_𝑖(𝑘)𝑥(𝑘)+𝐺_𝑖+Δ𝐺_𝑖(𝑘)𝑣(𝑘𝑤(𝑘),𝑦(𝑘)𝐶)=𝜙_𝑖(𝑘)𝑥(𝑘)+𝐷_𝑖+Δ𝐷_𝑖(𝑘)𝑣(𝑘),𝑧(𝑘)=𝐿_𝑖𝑥(𝑘),𝑖=1,2,…,𝑟. (3.39)

Remark 3.7. This class of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities has been considered in [29]; a new different Lyapunov functional is then employed to deal with systems with sensor nonlinearities. Then from Theorems 3.1, 3.4, and 3.5, we can get the following corollaries, which are less conservative than Theorem  3.1 in [29].

Corollary 3.8. Consider the uncertain discrete-time fuzzy stochastic systems in (3.39). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, any matrices 𝐴_𝐹𝑗,𝐵_𝐹𝑗, scalars 𝜀_1𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟 such that the following LMI is satisfied: ΠΠ=¹¹00𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑋Π₁₆𝐸^𝑇_𝑖𝑋𝐸^𝑇_𝑖𝑌00∗Π₂₂000𝐴^𝑇_𝐹𝑗0000∗∗Π₃₃0𝐵^𝑇_𝑖𝑋Π₃₆𝐺^𝑇_i𝑋𝐺^𝑇_𝑖𝑌00∗∗∗−2𝐼0−𝐵^𝑇_𝐹𝑗0000∗∗∗∗−𝑋000𝑋𝑀_1𝑖𝑋𝑀_6𝑖∗∗∗∗∗−𝑌00𝑌𝑀_1𝑖Π₆₁₀∗∗∗∗∗∗−𝑋0𝑋𝑀_2𝑖𝑋𝑀_7𝑖∗∗∗∗∗∗∗−𝑌𝑌𝑀_2𝑖𝑌𝑀_7𝑖∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼0∗∗∗∗∗∗∗∗∗−𝜀_3𝑖𝐼<0, (3.40) where Π₁₁=−𝑋+𝜀_1𝑖𝑁^𝑇_1𝑖𝑁_1𝑖,Π₂₂=−𝑌+𝐿_𝑖−𝐿_𝑗^𝑇𝐿_𝑖−𝐿_𝑗,Π₃₃=−𝛾²𝐼+𝜀_3𝑖𝑁^𝑇_3𝑖𝑁_3𝑖,Π₁₆=𝐴^𝑇_𝑖𝑌−𝐴^𝑇_𝐹𝑗−𝐶^𝑇_𝑖𝐾^𝑇₁𝐵^𝑇_𝐹𝑗,Π₃₆=𝐵^𝑇_𝑖𝑌−𝐷^𝑇_𝑖𝐵^𝑇_𝐹𝑗,Π₆₁₀=𝑌𝑀_6𝑖−𝐵_𝐹𝑗𝑀_8𝑖. (3.41) Moreover, if the previous condition is satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

Corollary 3.9. Consider the uncertain discrete-time fuzzy stochastic systems in (3.39). Give a filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, any matrices 𝐴_𝐹𝑗,𝐵_𝐹𝑗, scalars 𝜀_1𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟 such that the following LMIs are satisfied: Π_𝑖𝑖Π<0,𝑖=1,2,…,𝑟,_𝑖𝑗+Π_𝑗𝑖Π<0,𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗,_𝑖𝑗=Π₁₁00𝐶^𝑇_𝑖𝐾^𝑇𝐴^𝑇_𝑖𝑋Π₁₆𝐸^𝑇_𝑖𝑋𝐸^𝑇_𝑖𝑌00∗Π₂₂000𝐴^𝑇_𝐹𝑗0000∗∗Π₃₃0𝐵^𝑇_𝑖𝑋Π₃₆𝐺^𝑇_𝑖𝑋𝐺^𝑇_𝑖𝑌00∗∗∗−2𝐼0−𝐵^𝑇_𝐹𝑗0000∗∗∗∗−𝑋000𝑋𝑀_1𝑖𝑋𝑀_6𝑖∗∗∗∗∗−𝑌00𝑌𝑀_1𝑖Π₆₁₀∗∗∗∗∗∗−𝑋0𝑋𝑀_2𝑖𝑋𝑀_7𝑖∗∗∗∗∗∗∗−𝑌𝑌𝑀_2𝑖𝑌𝑀_7𝑖∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼0∗∗∗∗∗∗∗∗∗−𝜀_3𝑖𝐼, (3.42) where Π₁₁,Π₂₂,Π₃₃,Π₁₆,Π₃₆,Π₆₁₀ are defined in Corollary 3.8. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

Corollary 3.10. Consider the uncertain discrete-time fuzzy stochastic systems in (3.39). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, any matrices 𝐴_𝐹𝑗,𝐵_𝐹𝑗, scalars 𝜀_1𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟 such that the following LMIs are satisfied: Π_𝑖𝑖<Ω_𝑖𝑖Π,𝑖=1,2,…,𝑟,_𝑖𝑗+Π_𝑗𝑖<Ω_𝑖𝑗+Ω^𝑇_𝑗𝑖Ω,𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗,_𝑖𝑗=Ω₁₁Ω₁₂⋯Ω_1𝑟∗Ω₂₂⋯Ω_2𝑟⋮⋮⋱⋮∗∗⋯Ω_𝑟𝑟<0, (3.43) where Π₁₁,Π₂₂,Π₃₃,Π₁₆,Π₃₆,Π₆₁₀ are defined in Corollary 3.8. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

In the sequel, special results for the case of linear sensor, that is, 𝜙(𝐶𝑥(𝑘))=𝐶𝑥(𝑘), may be obtained from Theorems 3.1, 3.4, and 3.5.

Corollary 3.11. Consider the uncertain discrete-time fuzzy stochastic systems in (2.1) with 𝜙(𝐶𝑥(𝑘))=𝐶𝑥(𝑘). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, 𝑄>0, any matrices 𝐴_𝐹𝑗,𝐵_𝐹𝑗, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟 such that the following LMI is satisfied: Σ₁₁000𝐴^𝑇_𝑖𝑋Σ₁₇𝐸^𝑇_𝑖𝑋𝐸^𝑇_𝑖𝑌000∗Σ₂₂000𝐴^𝑇_𝐹𝑗00000∗∗Σ₃₃0𝐴^𝑇_𝑑𝑖𝑋Σ₃₇𝐸^𝑇_𝑑𝑖𝑋𝐸^𝑇_𝑑𝑖𝑌000∗∗∗Σ₄₄𝐵^𝑇_𝑖𝑋Σ₄₇𝐺^𝑇_𝑖𝑋𝐺^𝑇_𝑖𝑌000∗∗∗∗−𝑋000𝑋𝑀_1𝑖𝑋𝑀_3𝑖𝑋𝑀_6𝑖∗∗∗∗∗−𝑌00𝑌𝑀_1𝑖Σ₇₁₁Σ₇₁₂∗∗∗∗∗∗−𝑋0𝑋𝑀_2𝑖𝑋𝑀_4𝑖𝑋𝑀_7𝑖∗∗∗∗∗∗∗−𝑌𝑌𝑀_2𝑖𝑌𝑀_4𝑖𝑌𝑀_7𝑖∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼00∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼0∗∗∗∗∗∗∗∗∗∗−𝜀_3𝑖𝐼<0, (3.44) where Σ₁₇=𝐴^𝑇_𝑖𝑌−𝐴^𝑇_𝐹𝑗−𝐶^𝑇_𝑖𝐵^𝑇_𝐹𝑗. (3.45) Moreover, if the previous condition is satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

Corollary 3.12. Consider the uncertain discrete-time fuzzy stochastic systems in (2.1) with 𝜙(𝐶𝑥(𝑘))=𝐶𝑥(𝑘). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, 𝑄>0, any matrices 𝐴_𝐹𝑖,𝐵_𝐹𝑖, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗, such that the following LMIs are satisfied: Σ_𝑖𝑖<0,𝑖=1,2,…,𝑟,Σ_𝑖𝑗+Σ_𝑗𝑖<0,𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗, (3.46) where Σ_𝑖𝑗=Σ₁₁000𝐴^𝑇_𝑖𝑋Σ₁₇𝐸^𝑇_𝑖𝑋𝐸^𝑇_𝑖𝑌000∗Σ₂₂000𝐴^𝑇_𝐹𝑗00000∗∗Σ₃₃0𝐴^𝑇_𝑑𝑖𝑋Σ₃₇𝐸^𝑇_𝑑𝑖𝑋𝐸^𝑇_𝑑𝑖𝑌000∗∗∗Σ₄₄𝐵^𝑇_𝑖𝑋Σ₄₇𝐺^𝑇_𝑖𝑋𝐺^𝑇_𝑖𝑌000∗∗∗∗−𝑋000𝑋𝑀_1𝑖𝑋𝑀_3𝑖𝑋𝑀_6𝑖∗∗∗∗∗−𝑌00𝑌𝑀_1𝑖Σ₇₁₁Σ₇₁₂∗∗∗∗∗∗−𝑋0𝑋𝑀_2𝑖𝑋𝑀_4𝑖𝑋𝑀_7𝑖∗∗∗∗∗∗∗−𝑌𝑌𝑀_2𝑖𝑌𝑀_4𝑖𝑌𝑀_7𝑖∗∗∗∗∗∗∗∗−𝜀_1𝑖𝐼00∗∗∗∗∗∗∗∗∗−𝜀_2𝑖𝐼0∗∗∗∗∗∗∗∗∗∗−𝜀_3𝑖𝐼, (3.47) where Σ₁₁,Σ₂₂,Σ₃₃,Σ₄₄,Σ₃₇,Σ₄₇,Σ₇₁₁,Σ₇₁₂, and Σ₁₇ are defined in Theorem 3.1 and Corollary 3.11, respectively. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

Corollary 3.13. Consider the uncertain discrete-time fuzzy stochastic systems in (2.9) with 𝜙(𝐶𝑥(𝑘))=𝐶𝑥(𝑘). A filter of form (2.13) and constants 𝜏₁ and 𝜏₂, the filtering error system is robustly stochastic stable with performance 𝛾, if there exist real matrices 𝑋>0, 𝑌>0, 𝑄>0, any matrices 𝐴_𝐹𝑖,𝐵_𝐹𝑖, scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0 and 𝜀_3𝑖>0 for 𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗, such that the following LMIs are satisfied: Σ_𝑖𝑖<Ω_𝑖𝑖,𝑖=1,2,…,𝑟,Σ_𝑖𝑗+Σ_𝑗𝑖<Ω_𝑖𝑗+Ω^𝑇_𝑖𝑗,𝑖,𝑗=1,2,…,𝑟,𝑖<𝑗,Ω=Ω₁₁Ω₁₂⋯Ω_1𝑟∗Ω₂₂⋯Ω_2𝑟⋮⋮⋱⋮∗∗⋯Ω_𝑟𝑟<0, (3.48) where Σ_𝑖𝑗,Σ₁₁,Σ₂₂,Σ₃₃,Σ₄₄,Σ₃₇,Σ₄₇,Σ₇₁₁,Σ₇₁₂, and Σ₁₇ are defined in Corollary 3.12, Theorem 3.1, and Corollary 3.11, respectively. Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by LMI (3.3).

Remark 3.14. In the previous results, there still exists conservativeness; for these, we can also use the delay-dependent stability results of discrete-time systems in [13–16] to derive the less conservative theorems and corollaries for filtering problem of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay.

4. Numerical Example

In this section, two simulation examples are given to illustrate the effectiveness and benefits of the proposed approach.

Example 4.1. Consider the system (3.39) with parameters as follows: 𝐴₁=−0.20.10.1−0.3,𝐵₁=0.10.2,𝐸₁=−0.20.30.10.3,𝐺₁=,𝐶−0.10.3₁=−0.20.30.10.4,𝐷₁=0.20.1,𝐿₁=,𝑀0.30.20.20.4₁₁=0.10.1,𝑀₂₁=−0.4−0.5,𝑀₆₁=−0.20.1,𝑀₇₁=,𝑀−0.20.3₈₁=−0.20.3,𝑁₁₁=0.20.1,𝑁₃₁=0.2,𝐾₁{},𝐴=diag0.4,0.6₂=−0.1−0.20.20.3,𝐵₂=0.2−0.2,𝐸₂=−0.2−0.20.30.1,𝐺₂=,𝐶−0.20.2₂=−0.1−0.20.30.3,𝐷₂=0.40.3,𝐿₂=,𝑀−0.10.30.20.3₁₂=−0.20.2,𝑀₂₂=0.30.1,𝑀₆₂=0.30.1,𝑀₇₂=,𝑀0.10.3₈₂=0.1−0.2,𝑁₁₂=0.20.3,𝑁₃₂=0.2,𝐾₂=diag{0.1,0.8}. (4.1) This system has been considered in [29]. For comparison with the Theorem  3.1 in [29], the computation results of 𝛾_min under Corollaries 3.8, 3.9, and 3.10 are listed in Table 1.
This example conclusively shows that our results are less conservative than Theorem  3.1 in [29].

Table 1 

Minimum index 𝛾 for different methods.

Example 4.2. Consider the system (2.1) with parameters as follows: 𝐴₁=−0.100.1−0.4,𝐴_𝑑1=0.20−0.1−0.1,𝐵₁=0,𝐸0.3₁=−0.110.100.06,𝐸_𝑑1=0.110.100.06,𝐺₁=,𝐶−0.20.1₁=−0.10.050.20.05,𝐶_𝑑1=−0.110.050.60.3,𝐷₁=,𝑀0.1−0.1₁₁=−0.030.02,𝑀₂₁=−0.03−0.02,𝑀₃₁=−0.03−0.02,𝑀₄₁=,𝑀−0.03−0.02₅₁=−0.03−0.02,𝑀₆₁=0.060.02,𝑀₇₁=0.010.05,𝑀₈₁=,𝑁0.08−0.03₁₁=0.020.02,𝑁₂₁=0.020.02,𝑁₃₁𝐿=0.01,₁=0.10.10.120.1,𝐹₁₁=0.2sin(𝑡),𝐹₂₁=0.3sin(𝑡),𝐹₃₁(𝑡),𝐴=0.3sin₂=−0.2−0.30.50.4,𝐴_𝑑2=0.1−0.10.10.2,𝐵₂=0,𝐸−0.3₂=−0.12−0.110.110.04,𝐸_𝑑2=0.12−0.110.110.04,𝐺₂=,𝐶−0.10.1₂=−0.2−0.10.50.2,𝐶_𝑑2=−0.2−0.10.50.2,𝐷₂=,𝑀0.10.2₁₂=0.070.01,𝑀₂₂=0.020.12,𝑀₃₂=−0.03−0.02,𝑀₄₂=,𝑀−0.03−0.02₅₂=−0.03−0.02,𝑀₆₂=0.080.07,𝑀₇₂=0.040.07,𝑀₈₂=,𝑁0.03−0.07₁₂=0.010.02,𝑁₂₂=0.010.01,𝑁₃₂𝐿=0.01,₂=0.20.190.10.1,𝐹₁₂=0.2sin(𝑡),𝐹₂₂=0.3sin(𝑡),𝐹₃₂=0.3sin(𝑡) (4.2) and 𝑣(𝑡)=10𝑒^−𝑡, the membership functions are ℎ₁(𝜃)=(1−sin(𝑥₂))/2, ℎ₂(𝜃)=(1−sin(𝑥₁))/2, and the nonlinear vector function is 𝜙(𝑢)=(𝐾₂+𝐾₁)/2+(𝐾₂−𝐾₁)/2sin(𝑢), 𝐾₁=diag{0.1,0.3}, 𝐾₂=diag{0.2,0.4}.

Note that different 𝜏₁ and 𝜏₂ yield different 𝛾_min, and assume 𝜏(𝑘) satisfies 1≤𝜏(𝑘)≤5.

By Theorem 3.1, the minimum achievable noise attenuation level is given by 𝛾_min=0.3474 and we can obtain the corresponding filter parameters as follows: 𝐴_𝑓1=−0.3876−0.65040.49670.8339,𝐵_𝑓1=,𝐴1.45380.8653−4.9933−1.7834_𝑓2=−0.3600−0.60510.45930.7720,𝐵_𝑓2=.1.20170.7808−4.6656−1.6742 (4.3)

By Theorem 3.4, the minimum achievable noise attenuation level is given by 𝛾_min=0.3419 and we can obtain the corresponding filter parameters as follows: 𝐴_𝑓1=0.50590.4421−0.7283−0.6452,𝐵_𝑓1=,𝐴−0.37950.2226−2.1600−0.9045_𝑓2=−0.5060−0.44200.72880.6429,𝐵_𝑓2=.1.69331.0131−3.1230−1.1367 (4.4)

By Theorem 3.5, the minimum achievable noise attenuation level is given by 𝛾_min=0.2563 and we can obtain the corresponding filter parameters as follows: 𝐴_𝑓1=−0.0002−0.00110.0016−0.0066,𝐵_𝑓1=,𝐴−0.17080.3000−2.8286−1.0410_𝑓2=−0.4489−0.39420.65160.5707,𝐵_𝑓2=.3.23361.6093−5.2772−1.9714 (4.5)

The relaxation technique is adopted in this paper, so Theorems 3.4 and 3.5 relaxed the conservatism more than Theorem 3.1. There has been a decrease in the minimum achievable noise attenuation level 𝛾_min.

With the initial conditions 𝑥(𝑡) and 𝑥(𝑡) are [2−1.5]^𝑇 and [2−1.5]^𝑇, respectively, for an appropriate initial interval. For given 𝜏₁=1,𝜏₂=5 with 𝛾_min=0.2563, according to Theorem 3.5, we apply the filter parameters mentioned previously to the system (2.1) and obtain the simulation results as in Figures 1–3. Figure 1 shows the state response 𝑥(𝑘) under the initial condition. Figure 2 shows the estimation of filter 𝑥(𝑘). Figure 3 shows error response 𝑥(𝑘). From these simulation results, we can see that the designed 𝐻_∞ filter can stabilize the system (2.1) with sensor nonlinearities and time-varying delay.

Figure 1 

The state response 𝑥(𝑘).

Figure 2 

The estimation of filter 𝑥(𝑘).

Figure 3 

The error response 𝑥(𝑘).

5. Conclusions

In this paper the robust filtering problem for a class of uncertain discrete-time fuzzy stochastic systems with sensor nonlinearities and time-varying delay has been developed. A new type of Lyapunov-Krasovskii functional has been constructed to derive some sufficient conditions for the filters in terms of LMIs, which guarantees a prescribed 𝐻_∞ performance index for the filtering error system. A numerical example has shown the usefulness and effectiveness of the proposed filter design method.