𝐻_∞ Filtering over Networks for a Class of Discrete-Time Stochastic System with Randomly Occurred Sensor Nonlinearity

Abstract

The 𝐻_∞ filtering problem for a class of discrete-time stochastic system with randomly occurred nonlinearity (RON) suffering from network packet loss is considered. Based on the Lyapunov-Krasovskii functional method, the asymptotical mean-square stability condition of the filtering system with a prescribed 𝐻_∞ level is derived. Besides, the filter parameters can be obtained simultaneously by solving the matrix inequalities we achieve. It is worth noting that no slack variable is introduced in the proposed conditions. The effectiveness of the theory developed is verified through a numerical example.

1. Introduction

In the past few decades, the 𝐻_∞ control problem has been extensively developed in many applications, for example, [1] in the state-space version [2], on output feedback 𝐻_∞ control for nonlinear systems, with [3, 4] for uncertain linear systems, since the 𝐻_∞ filtering was introduced in [5]. The objective of the 𝐻_∞ filtering is to design an estimator for a given system, such that the 𝑙₂ gain from the exogenous disturbance to the estimation error is less than a given level 𝛾>0, where the noise signals are assumed to be arbitrary but with bounded energy or bounded average power rather than just Gaussian. Without the assumption of the statistical properties in Kalman filtering, the 𝐻_∞ filtering has been widely applied; see [6–12] and the references therein.

The 𝐻_∞ filtering issue has been extended to a variety of complex dynamical systems, such as Markovian jumping systems [13], time-varying systems [14], fuzzy systems [15], and nonlinear systems [16]. Recently, stochastic 𝐻_∞ control and filtering problems for the systems expressed by Itô-type stochastic differential or difference equations have become a popular research area and have gained a lot of attention. In [17], an asymptotically stable (in some sense) observer was constructed which leads to a stable estimation error process whose 𝑙₂ gain with respect to uncertain disturbance signal is less than a prescribed level. The stochastic analysis presented in [18] was conducted to enforce the performance for the newly formulated NCS system under the usual stability requirement. The work of [19] formulated the filters for both continuous- and discrete-time Itô-type stochastic systems, with the nonlinear sensor and all admissible uncertainties under a prescribed 𝐻_∞ disturbance performance, respectively.

With the rapid development in the networked control technology, in practice, more and more systems are taking wired or wireless networked control system (NCS) as a solution, which has many advantages, such as low cost and easy maintenance and installation. However, due to the limited transmission capacity of the wired cable or wireless channel, issues like quantization [20–22], transmission delay [23], and packet dropouts [24] inevitably emerged. On the other hand, nonlinearity is a main source that complexes the control algorithms; hence, the 𝐻_∞ filtering problem has attracted a lot of research attention in the previous two areas. In [25], the nonlinearity was assumed to satisfy the sector bounded conditions, and for general stochastic systems, the nonlinear 𝐻_∞ filtering problem was investigated in [26]. Nevertheless, it is worth mentioning that one interesting problem that has been persistently overlooked is the so-called RON in the sensor parts or the transmission channel from sensor to controller. As is well known, the sensor parts of a wide class of practical systems are influenced by RON disturbances that are caused by environmental circumstances such as random failure stochastic fault on the linearization part of the sensor. Unfortunately, to the best of the authors’ knowledge, the 𝐻_∞ filtering problem for discrete-time stochastic systems with RON on sensor parts has not been fully studied, which motivates the work of this paper.

In this paper, the 𝐻_∞ filtering problem against randomly occurred sensor failures for a class of discrete time stochastic systems with norm-bounded noises suffering from network packet loss is considered. First of all, the RON model was used to describe the binary switches between the linear and nonlinear sensor by a Bernoli distribution with a known probability. Such a novel idea was first illustrated in [27, 28] to investigate the synchronization problem of stochastic delayed complex networks. The RON, also named as stochastic nonlinearity, has recently attracted many researchers’ interests. Readers interested in this area are suggested to refer to [27–29] and the references therein. Besides, the lossy network is also taken into consideration which is also modeled as a Bernoli process [29]. Next, the asymptotically mean-square stability condition of the filtering error dynamics with a prescribed 𝐻_∞ performance level is derived by using Lyapunov-Krasovskii functional technique. Finally, a simulation example is utilized to illustrate the effectiveness of the approach developed.

Notation 1. The notations used in the paper is fairly standard. The superscript “𝑇” stands for matrix transposition; ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space; ℝ^𝑚×𝑛 is the set of all real matrices of dimension 𝑚×𝑛;𝐼 and 0 represent the identity matrix and zero matrix, respectively. The notation 𝑃>0 means that 𝑃 is a real symmetric and positive definite matrix; the notation ‖𝐴‖ refers to the norm of matrix 𝐴, defined by ‖𝐴‖=√tr(𝐴^𝑇𝐴), and ‖⋅‖₂ stands for the usual 𝑙₂ norm. In symmetric block matrices or complex matrix expressions, an asterisk (∗) is used to represent a term that is induced by symmetry. Besides, 𝔼{𝑥} and 𝔼{𝑥∣𝑦}, respectively, represent the expectation of 𝑥 and the expectation of 𝑥 conditional on 𝑦. The set of all nonnegative integers is denoted by 𝕀⁺ and the set of all nonnegative real numbers is represented by ℝ⁺. If 𝐴 is a matrix, then 𝜆_max(𝐴) (resp., 𝜆_min(𝐴)) means the largest (resp., smallest) eigenvalue of 𝐴. Matrices in this paper, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation

In this section, a class of discrete-time stochastic systems (Σ) with sensor nonlinearity occurred randomly is considered, which is represented by the following equations:(Σ)𝑦∶(𝑥𝑘)(𝑘=+𝛼1(𝑘))=𝜙𝐴(𝐶𝑥𝑥(𝑘𝑧()𝑘()+𝑘))+𝐵=𝑣(1𝐿(𝑘−𝑥)(+𝛼𝑘()[𝑘𝐸,)𝑥)𝐶(𝑘𝑥)(+𝑘)𝐺+𝑣𝐷(𝑘𝑣)](𝑘𝑤),(𝑘),(1) where 𝑥(𝑘)∈ℝ^𝑛 is the system state; 𝑦(𝑘)∈ℝ^𝑞 is the system output; 𝑧(𝑘)∈ℝ^𝑟 is the combination of the state to be estimated; 𝛼(𝑘) is a Bernoli process, taking different values to indicate that the output of the plant 𝑦(𝑘) is linear or not [29], with Pr{𝛼(𝑘)=1}=𝛼_𝑒, and Pr{𝛼(𝑘)=0}=1−𝛼_𝑒. 𝑤(𝑘) is a real scalar random process on a probability space (Ω,ℱ,𝒫) related to an increasing family (ℱ_𝑘)_𝑘∈ℕ of 𝜎-algebra ℱ_𝑘∈ℱ generated by (𝑤(𝑘))_𝑘∈ℕ with ℕ being the set of natural numbers and is assumed to satisfy 𝔼{𝑤(𝑘)}=0, 𝔼{𝑤(𝑘)²}=1 and 𝔼{𝑤(𝑖)𝑤(𝑗)}=0,forall𝑖≠𝑗. 𝑣(𝑘) is the exogenous disturbance signal, which belongs to 𝑙_𝑒2([0,∞);ℝ^𝑝), and is adapted and ℱ_𝑘−1 is measurable for all 𝑘∈ℕ, where 𝑙_𝑒2([0,∞)ℝ^𝑘) denotes the space of 𝑘-dimensional nonanticipatory square-summable stochastic process 𝑓(⋅)=(𝑓(𝑘))_𝑘∈ℕ with respect to (ℱ_𝑘)_𝑘∈ℕ satisfying‖𝑓‖^2𝑒2=𝔼^∞_𝑘=0||𝑓(𝑘)²||<∞.(2)

Additionally, in (1), 𝐴,𝐵,𝐸,𝐺,𝐶,𝐷,and𝐿 are real constant matrices with appropriate dimensions; the function 𝜙∈[𝐾₁,𝐾₂] for some given diagonal matrices 𝐾₁≥0 and 𝐾₂≥0 with 𝐾₂≥𝐾₁ satisfies the following sector condition:𝜙(𝑢)−𝐾₁𝑢^𝑇𝜙(𝑢)−𝐾₂𝑢≤0∀𝑢∈ℝ^𝑞.(3)

Remark 1. It is assumed here that when 𝛼(𝑘)=1, the output signal is the nonlinear function 𝜙(𝐶𝑥(𝑘)) other than the linear combination of the system states 𝐶𝑥(𝑘). Additionally, for the convenience of further discussion, we introduce the following two symbols: 𝜙_𝑠(𝑢)=𝜙(𝑢)−𝐾₁𝑢 and 𝐾=𝐾₂−𝐾₁.

Definition 2 (see [30]). The discrete-time stochastic system in the form of (1) with 𝑣(𝑘)=0 is said to be stochastic stable if there is a scalar 𝜇>0 such that for all admissible uncertainties, 𝔼^∞_𝑘=0||𝑥(𝑘)||²≤𝜇𝔼‖𝑥(0)‖²,(4) where 𝑥(𝑘) denotes the solution of stochastic systems with initial state 𝑥(0).

For discrete-time stochastic system (Σ), the estimation of 𝑧(𝑘) from measured output 𝑦(𝑘) is considered here. To accomplish this, we construct the following filter:(ℱ)∶𝑥̂(𝑘+𝑧̂(1𝑘))==𝐴𝐿_𝑓𝑥𝑥̂(̂(𝑘𝑘)),+𝐵_𝑓𝑦̂(𝑘),(5) where 𝑦(̂𝑘)=𝛽(𝑘)𝑦(𝑘), 𝑥̂(𝑘)∈ℝ^𝑛, 𝑧(̂𝑘)∈ℝ^𝑟. 𝐴_𝑓 and 𝐵_𝑓 are the matrices with appropriate dimension to be determined. The process {𝛽(𝑘),𝑘∈𝕀⁺} is a Bernoli process, indicating that the network packet is successfully received or not [29], with the probabilities Pr{𝛽(𝑘)=1}=𝛽_𝑒, Pr{𝛽(𝑘)=0}=1−𝛽_𝑒 relatively. It is also assumed here that the random processes 𝛼(𝑘), 𝛽(𝑘), 𝑤(𝑘) and 𝑣(𝑘), forall𝑘∈𝕀⁺ are mutually independent. Hence, the filtering problem can be stated as follows.

Discrete-Time Filtering
Given a disturbance attenuation level 𝛾>0, the parameters 𝐴_𝑓 and 𝐵_𝑓 of filter (5) are designed such that the resultant filtering error system is stochastically stable for 𝑣(𝑘)=0 and any 𝜙∈[𝐾₁,𝐾₂], satisfying ‖𝑧−𝑧‖̂_𝑒2<𝛾‖𝑣‖_𝑒2 under zero initial conditions for all 𝑣(𝑘)∈𝑙_𝑒2([0,∞);ℝ^𝑝).

Let 𝑥̃(𝑘)=𝑥(𝑘)−𝑥̂(𝑘), 𝑧̃=𝑧(𝑘)−𝑧(̂𝑘), and 𝑒(𝑘)=[𝑥(𝑘)^𝑇𝑥̃(𝑘)^𝑇]^𝑇. By means of the system (Σ) and filter (ℱ), we obtain the filtering error dynamics asΣ_𝑒∶𝑒(𝑘+1)=×𝐴𝑒(+𝑘)−𝛽_𝑒𝛽−(𝑘𝛽)(𝛼𝑘)(𝑘)𝐴𝐵+𝜙_𝑠𝛽_𝑒𝛼_𝑒−𝛽(𝑘)𝛼(𝑘)𝐴+(𝑢(𝑘))+𝐵+𝛽_𝑒−𝛽(𝑘)𝐵𝑣(𝑘)𝑧̃(𝑘)=𝐸𝑒(𝑘)+𝐺𝑣(𝑘)𝑤(𝑘),𝐿𝑒(𝑘),(6) where 𝑢(𝑘)=𝐶𝑥(𝑘), and𝐴=⎡⎢⎢⎣𝐴−𝐴_𝑓𝐴0−𝛽_𝑒𝐵_𝑓𝐶+𝛼_𝑒𝐵_𝑓𝐾₁𝐶−𝛼_𝑒𝐵_𝑓𝐶𝐴_𝑓𝐵=⎡⎢⎢⎣𝐵_𝑓0⎤⎥⎥⎦,𝐷⎤⎥⎥⎦,𝐺=⎡⎢⎢⎣𝐺𝐺⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣𝐵0_𝑓𝐴=⎡⎢⎢⎣𝐵00_𝑓⎤⎥⎥⎦,𝐾₁𝐶−𝐵_𝑓𝐶0⎤⎥⎥⎦,𝐴=⎡⎢⎢⎣𝐵0𝐶0_𝑓0⎤⎥⎥⎦,𝐿=0𝐿,𝐵=⎡⎢⎢⎣𝐵−𝛽𝐵_𝑒𝐵_𝑓𝐷⎤⎥⎥⎦,𝐸=⎡⎢⎢⎣𝐸𝐸00⎤⎥⎥⎦.(7)

3. Filter Design

The following theorem provides a sufficient condition for the solvability of discrete-time filtering problem for the system (Σ_𝑒).

Theorem 3. Consider the discrete-time stochastic systems (Σ_𝑒), for a given disturbance attenuation level 𝛾>0, if there exist matrices 𝑋=𝑋^𝑇>0,𝑌=𝑌^𝑇>0 satisfying the following matrix inequalities:⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎧⎪⎨⎪⎩Θ∗₁Θ0₆⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩Θ₂0⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩Θ0³⎫⎪⎬⎪⎭𝐴^𝑇𝛽_𝑒1−𝛽_𝑒𝐴^𝑇Θ₄𝐴^𝑇⎧⎪⎨⎪⎩Θ₅∗Θ₇0⎫⎪⎬⎪⎭0∗∗−20𝛼_𝑒𝐵^𝑇𝛽_𝑒1−𝛽_𝑒𝐵^𝑇000𝐼−𝛽_𝑒𝛼_𝑒𝐵^𝑇0Θ₄𝐵^𝑇0Θ₈∗∗∗∗∗∗−∗𝑃−0000∗∗∗∗∗𝑃−000∗∗∗∗∗∗𝑃−00∗∗∗∗∗∗∗𝑌−0𝑌⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0,(8)where Θ₁=Θ𝐸^𝑇𝑋𝐸+𝐸^𝑇𝑌𝐸−𝑋,Θ₂=𝐸^𝑇𝑋𝐺+𝐸^𝑇𝑌𝐺,₃Θ₅=𝛼_𝑒𝐶^𝑇𝐾^𝑇,Θ₄=𝛽_𝑒𝛼_𝑒1−𝛽_𝑒𝛼_𝑒,Θ₆=√𝛽_𝑒𝛼_𝑒1−𝛽_𝑒𝐵_𝑓𝐶+𝐵_𝑓𝐾₁𝐶^𝑇,Θ=𝐿^𝑇𝐿−𝑌,Θ₇=𝐺^𝑇𝑋𝐺+𝐺^𝑇𝑌𝐺−𝛾²𝐼,₈=𝛼_𝑒𝛽₄1−𝛽_𝑒𝐵^𝑇𝑓,𝑃=(diag{𝑋,𝑌})⁻¹,𝑌=𝑌⁻¹.(9) Then the systems (Σ_𝑒) is globally asymptotically stable with given disturbance attenuation level 𝛾>0.

Proof. First, we would like to establish the stochastic stability of the filtering error system (Σ_𝑒), by choosing a stochastic Lyapunov functional candidate as 𝑉(𝑒(𝑘))=𝑒(𝑘)^𝑇𝑃𝑒(𝑘)(10) for the filtering error system (6). For 𝑣(𝑘)=0 we have 𝔼{𝑉≤(𝑒−𝔼(𝑘2{𝜙𝑉+_𝑇𝑠(1𝑒)(𝑒𝑘(+𝑘))1})−∣𝑒𝑉(𝑘()e)}(𝑘−))𝑉(𝑒(𝑘))=𝔼𝔼{𝛼(𝑘)}𝜙_𝑠−𝔼{𝛼(𝑘)}𝐾𝐶𝑥(𝑘)𝐴+𝛽−_𝑒𝛽−(𝛽𝑘)(𝑘𝛼)(𝑘𝐴)𝐵+_𝑓𝛽_𝑒𝛼_𝑒−𝛽(𝑘)𝛼(𝑘)𝐴𝑒(𝑘)𝜙_𝑠(𝑢(𝑘))^𝑇×𝑃𝐴++𝛽𝛽_𝑒−𝛽(𝑘)_𝑒𝐴−𝛽(𝑘𝛼)_𝑒𝛼−(𝑘𝛽)(𝐵𝑘)_𝑓𝛼(𝑘)𝐴𝑒(𝑘)+𝑒_𝑇𝜙_𝑠(𝑢(𝑘))−2𝜙_𝑇𝑠(𝑘)𝐸^𝑇𝑃𝐸𝑒(𝑘)=𝑒(𝑘)_𝑇𝔼{𝛼(𝑘)}𝜙_𝑠−𝔼{𝛼(𝑘)}𝐾𝐶𝑥(𝑘)−𝑉(𝑥(𝑘))𝜙^𝑇𝑠Ω_𝑠𝑒(𝑘)^𝑇𝜙^𝑇𝑠^𝑇,(11) where Ω_𝑠=⎡⎢⎢⎣Ω₁Ω₂∗𝛽_𝑒𝛼_𝑒𝐵^𝑇𝑃𝐵−2𝛼_𝑒𝐼⎤⎥⎥⎦,(12) with, Ω₁=+𝐴𝛽^𝑇_𝑒𝑃𝐴+𝛽_𝑒1−𝛽_𝑒𝐴^𝑇𝑃𝐴−𝑃+2𝛽𝛼_𝑒𝑒1−𝛽_𝑒𝛼_𝑒𝐴^𝑇𝑃𝐴𝛼_𝑒1−𝛽_𝑒𝐴^𝑇𝑃𝐴+𝐸^𝑇𝑃𝐸,Ω₂=+𝛽_𝑒𝛽𝛼_𝑒𝑒1−𝛽_𝑒𝐴^𝑇𝑃𝐵−𝛽_𝑒𝛼_𝑒𝐴^𝑇𝑃𝐵𝛼_𝑒1−𝛽_𝑒𝛼_𝑒𝐴^𝑇𝑃𝐵+𝛼_𝑒𝐶^𝑇𝐾^𝑇,𝐶=𝐶0.(13)
As (8) holds, one may have Ω_𝑠<0, which means that there is a scaler 𝜀>0 small enough, such thatΩ_𝑠<⎡⎢⎢⎣𝜀0𝐼00⎤⎥⎥⎦.(14) It follows from (14) that 𝔼{𝑉(𝑥(𝑘+1)𝑥(𝑘))}−𝑉(𝑥(𝑘))<−𝜀‖𝑒(𝑘)‖².(15) Taking expectations with respect to both sides of (15), we have 𝔼{𝑉(𝑥(𝑘+1))}−𝔼{𝑉(𝑥(𝑘))}<−𝜀𝔼‖𝑒(𝑘)‖².(16) Hence, by summing up both sides of (16), from 0 to 𝑁 for any integer 𝑁>1, we can get 𝔼{𝑉(𝑥(𝑁+1))}−𝔼{𝑉(0)}<−𝜀𝔼^𝑁_𝑘=0‖𝑒(𝑘)‖²,(17) which leads to 𝔼^𝑁_𝑘=0‖𝑒(𝑘)‖²≤<11𝜀[𝔼{𝑉(0)}−𝔼{𝑉(𝑥(𝑁+1))}]≤𝜇𝜀𝔼𝔼{𝑉‖𝑥((00))}‖₂,(18) where 𝜇=1/𝜀𝜆_max{𝑃}. Taking 𝑁→∞, it is shown from (18) and Definition 2 that the filtering error system (7) is stochastically stable for 𝑣(𝑘)=0.
Next, we will present that the filtering error system (6) satisfies‖𝑧(̃𝑘)‖_𝑒2<𝛾‖𝑣(𝑘)‖_𝑒2(19) for all 𝑣(𝑘)∈𝑙_𝑒2([0,∞);ℝ^𝑝). To accomplish this, first of all, we would introduce the filtering error index: 𝐽(𝑁)=𝔼^𝑁_𝑘=1‖𝑧̃(𝑘)‖²−𝛾²‖𝑣(𝑘)‖²(20) for any 𝑁≥1. Thus, under zero initial conditions, forall𝑣(𝑘)∈𝑙_𝑒2([0,∞);ℝ^𝑝), there is 𝐽(𝑁)=𝔼^𝑁_𝑘=1||𝑧̃(𝑘)||²≤𝔼_𝑁+−𝔼𝔼{{𝑉𝑉−((𝑥𝑥𝛾((²𝑘𝑁||𝑣+(+𝑘1)1)||²)∣)}𝑥(𝑘))}−𝑉(𝑥(𝑘))_𝑘=1||𝑧̃(𝑘)||²≤𝔼𝑒(𝑘)_𝑇+𝔼{𝑉−(𝑥𝛾(²𝑘||𝑣+(𝑘1))||²∣𝑥(𝑘))}−𝑉(𝑥(𝑘))𝑣(𝑘)^𝑇𝜙_𝑠(𝐶𝑥(𝑘))^𝑇Ω_𝑓×𝑒(𝑘)^𝑇𝑣(𝑘)^𝑇𝜙_𝑠(𝐶𝑥(𝑘))^𝑇^𝑇,(21) whereΩ_𝑓=⎡⎢⎢⎢⎢⎢⎢⎣Ω₁+𝐿^𝑇𝐿Ω₃Ω₂∗⎧⎪⎨⎪⎩𝛽^𝑒+1−𝛽_𝑒𝐵^𝑇𝑃𝐵−𝛾²𝐼𝐵^𝑇𝑃𝐵+𝐺^𝑇𝑃𝐺⎫⎪⎬⎪⎭𝛽^𝑒∗∗𝛼_𝑒1−𝛽_𝑒𝛽_𝑒𝐵^𝑇𝑃𝐵−𝛽_𝑒𝛼_𝑒𝐵^𝑇𝑃𝐵𝛼_𝑒𝐵^𝑇𝑃𝐵−2𝛼_𝑒𝐼⎤⎥⎥⎥⎥⎥⎥⎦(22)with Ω₃=𝐴^𝑇𝑃𝐵+𝛽_𝑒(1−𝛽_𝑒)𝐴^𝑇𝑃𝐵+𝛽_𝑒𝛼_𝑒(1−𝛽_𝑒)𝐴^𝑇𝑃𝐵+𝐸^𝑇𝑃𝐺. As Ω_𝑓<0 means Ω_𝑠<0 which can be deduced from (8), one may conclude that if there are matrices 𝑋>0and𝑌>0, with appropriate dimensions, ‖𝑧(̃𝑘)‖_𝑒2<𝛾‖𝑣(𝑘)‖_𝑒2 holds for forall𝑣(𝑘)∈𝑙_𝑒2([0,∞);ℝ^𝑝).

Remark 4. When 𝛼_𝑒=1,𝛽_𝑒=1,forall𝑘∈ℕ, that is, the output of the sensor is always nonlinear, the problem formulated previously equals to the case that discussed in [19] without considering the uncertainty and network packets loss.

4. Numerical Example

In this section, a numerical example is presented to illustrate the approach proposed in this work. The discrete-time stochastic system parameters are described as follows: 𝐿=⎡⎢⎢⎣𝐴00𝐷..00=𝐸31=61⎡⎢⎢⎢⎢⎣=79000⎡⎢⎢⎣.00..1⎡⎢⎢⎢⎢⎣−13..50000130....0000.−00100245020.3..2.529.63000153...⎤⎥⎥⎦00000,132−0..032.03.033000..0⎤⎥⎥⎥⎥⎦00..004𝐶3,151512=0⎤⎥⎥⎥⎥⎦⎤⎥⎥⎦,⎡⎢⎢⎣,𝐵00..00=36𝐺𝐾⎡⎢⎢⎢⎢⎣−−=₁00000..00...⎡⎢⎢⎢⎢⎣2001800016...222−−−−0−−−0000.0....00010007...2431133⎤⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎦⎤⎥⎥⎦,,,𝐾=⎡⎢⎢⎣00.600.5₂⎤⎥⎥⎦,=⎡⎢⎢⎣1000.8⎤⎥⎥⎦,𝜙(𝑢)=𝐾¹+𝐾₂2𝑢+𝐾²−𝐾₁2sin(𝑢).(23) With the initial conditions being 𝑥(0)=[1.50.40.1]^𝑇 and 𝑥̂(0)=[201]^𝑇 for the system and the filter, respectively.

By Theorem 3 and the techniques presented in [31] one can obtain the following matrices through solving LMI (8) for 𝛾=0.8,𝛼_𝑒=0.8, and 𝛽_𝑒=0.95; the filter parameters in (5) can be obtained as 𝐴_𝑓=𝐵⎡⎢⎢⎢⎢⎣−−000.0..00606282036−000..01.0306498499000...110665535810_𝑓⎤⎥⎥⎥⎥⎦,=⎡⎢⎢⎢⎢⎣−−−000...110923415122−−−000...100121768229⎤⎥⎥⎥⎥⎦.(24)

Under the condition that the external disturbance is taken as 𝑣(𝑘)=[1/(1+𝑘²)1/(1+𝑘²)1/(1+𝑘²)]^𝑇, the trajectories of the system state 𝑥(𝑘), filtering system state 𝑥̂(𝑘), and the difference between them 𝑥(𝑘)−𝑥̂(𝑘) are shown in Figures 1, 2, and 3, respectively.

Figure 1 

The trajectories of state 𝑥(𝑘).

Figure 2 

The trajectories of filtering state 𝑥̂(𝑘).

Figure 3 

The trajectories of 𝑥(𝑘)−𝑥̂(𝑘).

5. Conclusion

The filtering problem for a class of discrete-time stochastic systems with RON on the sensor parts over communication channels is studied in this paper. Both the theoretical analysis and numerical simulations are introduced to verify the effectiveness of the proposed method in this work. These conditions are obtained by introducing a simple Lyapunov function, so the computation process is greatly simplified, and the filter parameters can be obtained simultaneously by solving the sufficient conclusion. Moreover, 𝐻_∞ performance analysis has been satisfied.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC: 61134007), National Natural Science Foundation of China (NSFC: 60736021), The National High Technology Research and Development Program of China (863 Program 2008AA042902), Open Project of the State Key Laboratory of Industrial Control Technology under Grant no. ICT1002, National Natural Science Foundation of China (61104102), and China Postdoctoral Science Foundation (20100480086).