Network-Based Robust 𝐻_∞ Filtering for the Uncertain Systems with Sensor Failures and Noise Disturbance

Abstract

The network-based robust 𝐻_∞ filtering for the uncertain system with sensor failures and the noise is considered in this paper. The uncertain system under consideration is also subject to parameter uncertainties and delay varying in an interval. Sufficient conditions are derived for a linear filter such that the filtering error systems are robust globally asymptotically stable while the disturbance rejection attenuation is constrained to a given level by means of the 𝐻_∞ performance index. These conditions are characterized in terms of the feasibility of a set of linear matrix inequalities (LMIs), and then the explicit expression is then given for the desired filter parameters. Two numerical examples are exploited to show the usefulness and effectiveness of the proposed filter design method.

1. Introduction

Networked control system (NCS) is a new control system structure where sensor-controller and controller-actuator signal link is through a shared communication network. Therefore, networked control systems have become an active research area in recent years in [1–3]. Recently, the filter design for networked systems become an active research area due to the advantages of using networked media in many aspects such as low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability in [4–10]. On the other hand, with the increasing of the working time in the domain of industry, some parts of the control system (e.g., actuator and sensor) can always be invalid. By time-scale decomposition, the reliable 𝐻_∞ control for linear time-invariant multiparities singularly perturbed systems against sensor failures is studied in [11]. For systems with both state and input time delays, a novel state and sensor fault observer is proposed to estimate system states and sensor faults simultaneously in [12]. A new robust 𝐻_∞ filtering problem is investigated for a class of time-varying nonlinear system with norm-bounded parameter uncertainties, bounded state delay, sector-bounded nonlinearity, and probabilistic sensor gain faults in [13]. The robust output feedback controller design for uncertain delayed systems with sensor failure and time delay is considered in [14]. The problems of robust fault estimation and fault-tolerant control for Takagi-Sugeno fuzzy systems with time delays and unknown sensor faults are addressed in [15]. The robust filtering problem for a class of discrete time-varying Markovian jump systems with randomly occurring nonlinearities and sensor saturation is studied in [16]. The robust 𝐻_∞ infinite-horizon filtering problem for a class of uncertain nonlinear discrete time-varying stochastic systems with multiple missing measurements and error variance constraints is considered in [17]. The problem of distributed 𝐻_∞ filtering in sensor networks using a stochastic sampled-data approach is investigated in [18]. The problems of stability analysis, 𝐻_∞ performance analysis, and robust 𝐻_∞ filter design for uncertain Markovian jump linear systems with time-varying delays are considered in [19].

In distributed industrial and military NCSs, sensors can be in a hostile environment and subject to failure and malfunction. Recently, the 𝐻_∞ filtering problem for NCSs has received considerable attention. The problem of designing 𝐻_∞ filters for a class of nonlinear networked control systems with transmission delays and packet losses is investigated in [20]. The control problems of networked control system with fault/failure of sensors and actuators are also received attention. The reliable control of a class of nonlinear NCSs via T-S fuzzy model with probabilistic sensor and actuator faults/failures, measurement distortion, time-varying delay, packet loss, and norm-bounded parameter uncertainties is investigated in [20]. Recently, based on T-S fuzzy model, the robust and reliable 𝐻_∞ filter design for a class of nonlinear networked control systems is investigated with probabilistic sensor failure in [21]. The reliable filtering problem for network-based linear continuous-time system with sensor failures has been studied in [22]. However, the proposed filer design approach [21, 22] do not consider the systems with uncertainty. The time delay has restriction when the rate of delay is differential, which is only applicable to unknown rate of time delay. No delay-dependent 𝐻_∞ filtering results on the uncertain networked control systems with sensor failures and disturbance noise are available in the literature, which motivates the present study.

In this paper, based on the delay-dependent stability criteria proposed in [23], a delay-dependent 𝐻_∞ performance analysis result is established for the filtering error systems. A new sensor failure model with uncertainties is proposed, and a new different Lyapunov functional is then employed to deal with systems with sensor failures and uncertainties. As a result, the 𝐻_∞ filter is designed in terms of linear matrix inequalities (LMIs), which involves fewer matrix variables but has less conservatism. The resulting filters can ensure that the filtering error system is asymptotically stable and the estimation error is bounded by a prescribed level for all possible bounded energy disturbances, which has advantages over the results of [22] in that it involves fewer matrix variables but has less conservatism. Meanwhile, the parameter uncertainties for system with sensor failures and the noise are considered in this paper, which are more general cases. Finally, two examples are given to show the effectiveness of the proposed method. This paper is organized as follows. Section 2 describes the system model and presents the definition and some lemmas. The robust 𝐻_∞ filter design method is derived in Section 3. Section 4 includes two simulation examples.

2. Problem Description

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, ||⋅|| stands for the induced matrix 2 norm, and 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.

Consider the following uncertain systems: 𝑥(𝑡)=((𝑡𝑥(𝑡)+((𝑡𝑤(𝑡),𝑦(𝑡)=((𝑡𝑥(𝑡),𝑧(𝑡)(𝑡),𝐴+Δ𝐴))𝐵+Δ𝐵))𝐶+Δ𝐶))=𝐿𝑥 (2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝑦(𝑡)∈ℝ^𝑞 is the measurable output vector, the noise disturbance 𝑤(𝑡)∈ℝ^𝑑 is external plant belongs to 𝐿₂[0∞], and 𝑧(𝑡) is a signal to be estimated. 𝐴, 𝐵, 𝐶, and 𝐿 are known real constant matrices, Δ𝐴(𝑡), Δ𝐵(𝑡), and Δ𝐶(𝑡) are unknown matrices representing time-varying parameter uncertainties, and the admissible uncertainties are assumed to be modeled in the form: Δ𝐴(𝑡)Δ𝐵(𝑡)=𝑀₁𝐹(𝑡)𝑁₁𝑁₂,Δ𝐶(𝑡)=𝑀₂𝐹(𝑡)𝑁₃, (2.2) where 𝑀₁,𝑀₂,𝑁₁,𝑁₂, and 𝑁₃ are known constant matrices, 𝐹(𝑡) is unknown time-varying matrices with Lebesgue measurable elements bounded by 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼. (2.3)

Assumption 1
The considered NCS consists of a time-driven sensor.

Assumption 2
There exist some sensor failures in the feedback channel.

Considering the effect of the common network on the data transmission, the filter can be expressed as 𝑥̇_𝑓(𝑡)=𝐴_𝑓𝑥_𝑓(𝑡)+𝐵_𝑓𝑦_𝑠𝑖_𝑘ℎ𝑡∈𝑖_𝑘ℎ+𝜏_𝑖𝑘,𝑖_𝑘+1ℎ+𝜏_𝑖𝑘+1,𝑧_𝑓(𝑡)=𝐶_𝑓𝑥_𝑓(𝑡)𝑘=1,2,…, (2.4) where 𝑥_𝑓(𝑡) represents the state estimate, 𝑦_𝑠(𝑖_𝑘ℎ) is the output with sensor failures, 𝑧_𝑓(𝑡) is the estimated output, 𝐴_𝑓,𝐵_𝑓, and 𝐶_𝑓 are the filter parameters to be designed. ℎ denotes the sampling period, and 𝑖_𝑘(𝑘=1,2,3,…) are some integers such that {𝑖₁,𝑖₂,𝑖₃,…}⊂{1,2,3,…}. 𝜏_𝑖𝑘 is the time from the instant 𝑖_𝑘ℎ when sensors sample from the plant to the instant when actuators send control actions to the plant. Obviously, ⋃^∞_𝑘=1[𝑖_𝑘ℎ+𝜏_𝑖𝑘,𝑖_𝑘+1ℎ+𝜏_𝑖𝑘+1)=[𝑡₀,∞), 𝑡≥0.

Remark 2.1. In (2.4), {𝑖₁,𝑖₂,𝑖₃,…} is a subset of {1,2,3,…}. Moreover, it is not required that 𝑖_𝑘+1>𝑖_𝑘. When {𝑖₁,𝑖₂,𝑖₃,…}={1,2,3,…}, it means that no packet dropout occurs in the transmission. If 𝑖_𝑘+1>𝑖_𝑘+1, there are dropped packets but the received packets are in ordered sequence. If 𝑖_𝑘+1<𝑖_𝑘+1, it means out-of-order packet arrival sequences occur. If 𝑖_𝑘+1=𝑖_𝑘+1, it implies that ℎ+𝜏_𝑘+1>𝜏_𝑘, which includes 𝜏_𝑘=𝜏 and 𝜏_𝑘<ℎ as special cases, where 𝜏 is a constant.

Remark 2.2. Since 𝑖_𝑘ℎ=𝑡−(𝑡−𝑖_𝑘ℎ), define 𝜏(𝑡)=𝑡−𝑖_𝑘ℎ,𝑡∈[𝑖_𝑘ℎ+𝜏_𝑖𝑘,𝑖_𝑘+1ℎ+𝜏_𝑖𝑘+1), which denotes the time-varying delay in the control signal. Obviously, 𝜏_𝑘≤𝜏(𝑡)≤𝑖_𝑘+1−𝑖_𝑘ℎ+𝜏_𝑘+1𝑡∈𝑖_𝑘ℎ+𝜏_𝑖𝑘,𝑖_𝑘+1ℎ+𝜏_𝑖𝑘+1 (2.5) which implies that 0≤𝜏₁≤𝜏(𝑡)≤𝜏₂, (2.6) where 𝜏₁ and 𝜏₂ denote infimum of 𝜏_𝑘 and supremum of [(𝑖_𝑘+1−𝑖_𝑘)ℎ+𝜏_𝑘+1], respectively.

Our aim in this paper is to design a robust 𝐻_∞ filter in the form of (2.4) such that(i)system (2.1) is said to be robust globally asymptotically stable, subject to 𝑤(𝑡)=0, for all admissible uncertainties satisfying (2.2)-(2.3);(ii)for the given disturbance attenuation level 𝛾>0 and under zero initial condition, the performance index 𝛾 satisfies the following inequality: ‖𝑧(𝑡)‖₂<𝛾𝑤(𝑡)₂₂. (2.7)

For an easy exposition of our results, we first consider the following systems with no uncertain parameters: 𝑥(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡),𝑧(𝑡)(𝑡).=𝐴𝑥+𝐵𝑤=𝐶𝑥=𝐿𝑥 (2.8)

The switch matrix G for filter (2.4) is introduced against sensor failures as follows: G=diag𝑔₁,𝑔₂,…,𝑔_𝑛, (2.9) where 𝑔_𝑖=1the𝑖thsensoriscompletenormal𝛼(0<𝛼<1)the𝑖thsensorpartiallyfails0,𝑖=1,2,…,𝑛the𝑖thsensorcompletelyfails. (2.10) From above analysis and (2.4), then we can get the output of the sensor failures 𝑦_𝑠(𝑡)=𝑦_𝑠𝑖_𝑘ℎ=G𝐶𝑥(𝑡−𝜏(𝑡))𝑡∈𝑖_𝑘ℎ+𝜏_𝑖𝑘,𝑖_𝑘+1ℎ+𝜏_𝑖𝑘+1 (2.11) so the filter (2.4) under consideration is of the following structure: 𝑥̇_𝑓(𝑡)=𝐴_𝑓𝑥_𝑓(𝑡)+𝐵_𝑓G((𝑡,𝑧𝐶𝑥𝑡−𝜏))_𝑓(𝑡)=𝐶_𝑓𝑥_𝑓(𝑡). (2.12)

Define 𝜍(𝑡)=_𝑥^𝑥(𝑡)_𝑓(𝑡) and the filter errors 𝑒(𝑡)=𝑧(𝑡)−𝑧_𝑓(𝑡), then the filtering error system can be represented as follows: 𝜍̇(𝑡)=𝐴𝜍(𝑡)+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+(𝑡),𝑒(𝑡)=𝐵𝑤𝐿𝜍(𝑡), (2.13) where 𝐴=𝐴00𝐴_𝑓,𝐴_𝑑=0𝐵_𝑓G𝐶,𝐵0𝐵=,𝐿=𝐿−𝐶_𝑓G𝐶. (2.14)

Throughout this paper, we use the following lemmas.

Lemma 2.3 (see [24]). 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 (2.15) if and only if Γ₁Γ^𝑇₃Γ₃−Γ₂<0,𝑜𝑟−Γ₂Γ₃Γ^𝑇₃Γ₁<0. (2.16)

Lemma 2.4 (see [25]). For any constant matrix 𝑀∈𝑅^𝑛×𝑛,𝑀=𝑀^𝑇>0, scalar 𝛾>0, vector function 𝜔∶[0,𝛾]→𝑅^𝑛 such that the integrations are well defined, the following inequality holds: ^𝛾₀𝑤(𝑠)𝑑𝑠^𝑇𝑀^𝛾₀𝑤(𝑠)𝑑𝑠≤𝛾^𝛾₀𝑤^𝑇(𝑠)𝑀𝑤(𝑠)𝑑𝑠. (2.17)

Lemma 2.5 (see [26]). 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. (2.18)

3. Main Results

Section 3.1 provides an 𝐻_∞ performance condition for the filtering error system (2.13). Design of 𝐻_∞ filter for the system (2.8) with no uncertainty will be developed in Section 3.2, and robust 𝐻_∞ filter design for the uncertain system (2.1) will be developed in Section 3.3.

3.1. Performance Analysis of 𝐻_∞ Filter

Theorem 3.1. Consider the system in (2.8). For a specified filter (2.12) and constants 𝜏₁ and 𝜏₂, the filtering error system (2.13) is globally asymptotically stable with performance 𝛾 if there exist real matrices 𝑃>0, 𝑄₁>0, 𝑄₂>0, 𝑄₃>0, 𝑅₁>0, and 𝑅₂>0, such that the following LMIs are satisfied: Σ₁₁𝑃𝐴_𝑑𝐻^𝑇𝑅₁0𝑃𝐵𝐴^𝑇𝐻^𝑇ℛ𝐿^𝑇∗Σ₂₂2𝑅₂𝑅₂000∗∗−𝑄₁−𝑅₁−2𝑅₂0000∗∗∗−𝑄₂−𝑅₂000∗∗∗∗−𝛾²𝐼𝐵^𝑇ℛ0∗∗∗∗∗−ℛ0∗∗∗∗∗∗−𝐼<0,Σ₁₁𝑃𝐴_𝑑𝐻^𝑇𝑅₁0𝑃𝐵𝐴^𝑇𝐻^𝑇ℛ𝐿^𝑇∗Σ₂₂𝑅₂2𝑅₂000∗∗−𝑄₁−𝑅₁−𝑅₂0000∗∗∗−𝑄₂−2𝑅₂000∗∗∗∗−𝛾²𝐼𝐵^𝑇ℛ0∗∗∗∗∗−ℛ0∗∗∗∗∗∗−𝐼<0, (3.1)(3.2) where Σ₁₁=𝑃𝐴+𝐴^𝑇𝑃+𝐻^𝑇𝑄₁+𝑄₂+𝑄₃−𝑅₁Σ𝐻,₂₂=−1−𝜇𝑄₃−3𝑅₂,ℛ=𝜏²₁𝑅₁+𝜏²₂₁𝑅₂,𝜏₂₁=𝜏₂−𝜏₁. (3.3)

Proof. Consider the Lyapunov-Krasovskii functional candidate as follows: 𝑉𝑥_𝑡=𝜍^𝑇(𝑡)𝑃𝜍(𝑡)+^𝑡_𝑡−𝜏1𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏2𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−𝜏(𝑡)}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)+𝑑𝑠_{0−𝜏1𝑡𝑡+𝜃}𝜏₁𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^−𝜏1_−𝜏2^𝑡_𝑡+𝜃𝜏₂₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (3.4) Calculating the time derivative of along the trajectory of system (3.4), one has 𝑉̇𝑥_𝑡≤2𝑥^𝑇(𝑡)𝑃𝐴𝜍(𝑡)+𝐴_𝑑𝑥(𝑡−𝜏(𝑡))+𝑃𝐵𝑤(𝑡)+³_𝑖=1𝑥^𝑇(𝑡)𝑄_𝑖𝑥(𝑡)−²_𝑖=1𝑥^𝑇𝑡−𝜏_𝑖𝑄_𝑖𝑥𝑡−𝜏_𝑖+𝜂^𝑇(𝑡)[𝐻𝐴000𝐵]^𝑇ℛ[]𝜂(𝑡)−𝐻𝐴000𝐵_{𝑡𝑡−𝜏1}𝜏₁𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)𝑑𝑠−^𝑡−𝜏1_𝑡−𝜏2𝜏₂₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠, (3.5) where 𝜂^𝑇(𝑡)=𝜍^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏(𝑡))𝑥^𝑇𝑡−𝜏₁𝑥^𝑇𝑡−𝜏₂𝑤^𝑇(𝑡). (3.6) By using Lemma 2.3, we have that −^𝑡_𝑡−𝜏1𝜏₁𝑥̇^𝑇(𝑠)𝑅₁𝑥̇(𝑠)𝜍(𝑡)𝑥𝑑𝑠≤𝑡−𝜏₁^𝑇−𝐻^𝑇𝑅₁𝐻𝐻^𝑇𝑅₁𝑅₁𝐻−𝑅₁𝜍(𝑡)𝑥𝑡−𝜏₁. (3.7) On the other hand, −^𝑡−𝜏1_𝑡−𝜏2𝜏₂₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠=−^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏₂₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠−^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝜏₂₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠=−^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−𝜏2}𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠−^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝜏1(𝑡)𝑡−𝜏}𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠. (3.8) Set 𝜏(𝑡)𝛽=−𝜏₁𝜏₂₁. (3.9) Then −^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠=−𝛽^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏₂₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠≤−𝛽^{𝑡−𝜏(𝑡)}_𝑡−𝜏2𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠,−^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝜏₂−𝜏(𝑡)𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠=−1−𝛽^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝜏₂₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠.≤−1−𝛽^𝑡−𝜏1_{(𝑡)𝑡−𝜏}𝜏(𝑡)−𝜏₁𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑑𝑠. (3.10) Combining (3.8)–(3.10) and by Lemma 2.4, we have −^𝑡−𝜏1_𝑡−𝜏2𝜏₁₂𝑥̇^𝑇(𝑠)𝑅₂𝑥̇(𝑠)𝑥((𝑡𝑥𝑑𝑠≤−𝑡−𝜏))𝑡−𝜏₂^𝑇𝑅₂−𝑅₂−𝑅₂𝑅₂𝑥((𝑡𝑥𝑡−𝜏))𝑡−𝜏₂−𝑥𝑡−𝜏₁𝑥(𝑡−𝜏(𝑡))^𝑇𝑅₂−𝑅₂−𝑅₂𝑅₂𝑥𝑡−𝜏₁𝑥((𝑡𝑥((𝑡𝑥𝑡−𝜏))−𝛽𝑡−𝜏))𝑡−𝜏₂^𝑇𝑅₂−𝑅₂−𝑅₂𝑅₂𝑥((𝑡𝑥𝑡−𝜏))𝑡−𝜏₂−𝑥1−𝛽𝑡−𝜏₁𝑥(𝑡−𝜏(𝑡))^𝑇𝑅₂−𝑅₂−𝑅₂𝑅₂𝑥𝑡−𝜏₁𝑥((𝑡.𝑡−𝜏)) (3.11) Combining (3.5), (3.7), and (3.11) yields 𝑉̇𝑥_𝑡≤𝜂^𝑇(𝑡)1−𝛽Φ₁+𝛽Φ₂+[𝐻𝐴000𝐵]^𝑇ℛ[𝐻𝐴000𝐵]𝜂(𝑡), (3.12) where Φ₁=Σ₁₁𝑃𝐴_𝑑𝐻^𝑇𝑅₁0𝑃𝐵∗Σ₂₂2𝑅₂𝑅₂0∗∗−𝑄₁−𝑅₁−2𝑅₂00∗∗∗−𝑄₂−𝑅₂0,Φ∗∗∗∗0₂=Σ₁₁𝑃𝐴_𝑑𝐻^𝑇𝑅₁0𝑃𝐵∗Σ₂₂𝑅₂2𝑅₂0∗∗−𝑄₁−𝑅₁−𝑅₂00∗∗∗−𝑄₂−2𝑅₂0.∗∗∗∗0 (3.13) Under the zero-initial condition, one can obtain that 𝑉(𝑥_𝑡)|_𝑡=0=0 and 𝑉(𝑥_𝑡)≥0. Define 𝐽(𝑡)=^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡 (3.14) then, for any nonzero 𝑤(𝑡), 𝐽(𝑡)≤^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡+𝑉𝑥_𝑡|_𝑡→∞−𝑉(𝑥_𝑡)|_𝑡=0=^∞₀𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)+𝑉̇𝑥_𝑡≤𝑑𝑡_∞0𝜂^𝑇(𝑡)1−𝛽Φ₁+𝛽Φ₂+[𝐻𝐴000𝐵]^𝑇ℛ[𝐻𝐴000𝐵]𝜂(𝑡)𝑑𝑡, (3.15) where Φ₁=Σ₁₁+𝐿^𝑇𝐿𝑃𝐴_𝑑𝐻^𝑇𝑅₁0𝑃𝐵∗Σ₂₂2𝑅₂𝑅₂0∗∗−𝑄₁−𝑅₁−2𝑅₂00∗∗∗−𝑄₂−𝑅₂0∗∗∗∗−𝛾²𝐼,Φ₂=Σ₁₁+𝐿^𝑇𝐿𝑃𝐴_𝑑𝐻^𝑇𝑅₁0𝑃𝐵∗Σ₂₂𝑅₂2𝑅₂0∗∗−𝑄₁−𝑅₁−𝑅₂00∗∗∗−𝑄₂−2𝑅₂0∗∗∗∗−𝛾²𝐼. (3.16) Due to 0<𝛽<1, our elaborate estimation 𝐽(𝑡) induces a convex domain of matrices (1−𝛽)Φ₁+𝛽Φ₂, which are negative definite if and only if Φ₁<0 and Φ₂<0. By using Lemma 2.3, the LMIs (3.1) and (3.2) can guarantee (1−𝛽)Φ₁+𝛽Φ₂+[𝐻𝐴000𝐵]^𝑇𝑅[𝐻𝐴000𝐵]<0. Since 𝑤(𝑡)≠0, it implies that 𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)+𝑉̇(𝑥_𝑡)<0, and thus, 𝐽(𝑡)<0. That is, ||𝑧(𝑡)||₂<𝛾||𝑤(𝑡)||₂.
Second, we also can prove that under the condition of Theorem 3.1, the filtering error system (2.13) with 𝑤(𝑡)=0 is globally asymptotically stable. This completes the proof.

3.2. Design of 𝐻_∞ Filter

Now based on the previous result, we are in a position to present the main result in this paper, which offers a new networked-based 𝐻_∞ filter design approach for the system (2.8).

Theorem 3.2. Consider the system in (2.8). A filter of form (2.12) and constants 𝜏₁ and 𝜏₂, the filtering error system is globally asymptotically stable with performance 𝛾, if there exist real matrices 𝑃₁>0, 𝑃₃>0, 𝑄₁>0, 𝑄₂>0, 𝑄₃>0, 𝑅₁>0,𝑅₂>0 and any matrices 𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓 such that the following LMIs are satisfied: Υ₁=Σ₁₁𝐴_𝑓+𝐴^𝑇𝑃₃𝐵_𝑓𝐺𝐶𝑅₁0𝑃₁𝐵𝐴^𝑇ℛ𝐿^𝑇∗𝐴_𝑓+𝐴^𝑇_𝑓𝐵_𝑓𝐺𝐶00𝑃₃𝐵0−𝐶^𝑇_𝑓∗∗Σ₃₃2𝑅₂𝑅₂000∗∗∗Σ₄₄₁0000∗∗∗∗Σ₅₅₁000∗∗∗∗∗−𝛾²𝐼𝐵^𝑇ℛ0∗∗∗∗∗∗−ℛ0∗∗∗∗∗∗∗−𝐼<0,Υ₂=Σ₁₁𝐴_𝑓+𝐴^𝑇𝑃₃𝐵_𝑓𝐺𝐶𝑅₁0𝑃₁𝐵𝐴^𝑇ℛ𝐿^𝑇∗𝐴_𝑓+𝐴^𝑇_𝑓𝐵_𝑓𝐺𝐶00𝑃₃𝐵0−𝐶^𝑇_𝑓∗∗Σ₃₃𝑅₂2𝑅₂000∗∗∗Σ₄₄₂0000∗∗∗∗Σ₅₅₂000∗∗∗∗∗−𝛾²𝐼𝐵^𝑇ℛ0∗∗∗∗∗∗−ℛ0∗∗∗∗∗∗∗−𝐼<0,𝑃₃−𝑃₁<0, (3.17)(3.18)(3.19) where Σ₁₁=𝑃𝐴₁+𝐴^𝑇₁𝑃+𝑄₁+𝑄₂+𝑄₃−𝑅₁,Σ₃₃=−1−𝜇𝑄₃−3𝑅₂,Σ₄₄₁=−𝑄₁−𝑅₁−2𝑅₂,Σ₄₄₂=−𝑄₁−𝑅₁−𝑅₂Σ₅₅₁=−𝑄₂−𝑅₂,Σ₅₅₂=−𝑄₂−2𝑅₂ (3.20) Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter is given by 𝐴_𝑓=𝐴_𝑓𝑃₃⁻¹,𝐵_𝑓=𝐵_𝑓,𝐶_𝑓=𝐶_𝑓𝑃₃⁻¹. (3.21)

Proof. Defining 𝑃𝑃=¹𝑃^𝑇₂𝑃₂𝑃₃,𝐽=𝐼00𝑃^𝑇₂𝑃₃⁻¹ (3.22) then 𝑃₁>0 and 𝑃₁−𝑃₃>0. Choose the LKF candidate as (3.4), and set Δ=diag{𝐽,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼},𝐴_𝑓=𝑃^𝑇₂𝐴_𝑓𝑃₃⁻¹𝑃₂,𝐵_𝑓=𝑃^𝑇₂𝑃₃⁻¹𝐵_𝑓,𝐶_𝑓=𝑃^𝑇₂𝑃₃^−𝑇𝑃₂,𝑃₃=𝑃^𝑇₂𝑃₃^−𝑇𝑃₂. (3.23) Pre- and postmultiplying (3.1) and (3.2) by Δ and Δ^𝑇, respectively, we can obtain that (3.1) is equivalent to (3.17), and (3.2) is equivalent to (3.18). Thus, we can conclude from Theorem 3.2 that the error systems are globally asymptotically stable with the 𝐻_∞ attenuation level 𝛾. In addition, the filter matrices 𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓 can be constructed from (3.21). This completes the proof.

Remark 3.3. When 𝜏₁,𝜏₂, and 𝜇 are given, matrix inequalities (3.17) and (3.18) are linear matrix inequalities in matrix variables 𝑃₁, 𝑃₃, 𝑄₁, 𝑄₂, 𝑄₃, 𝑅₁, 𝑅₂, 𝑃₂, 𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓, which can be efficiently solved by the developed interior point algorithm [24]. Meanwhile, it is esay to find the minimal attenuation level 𝛾.

Remark 3.4. From the proof process of Theorem 3.1, one can clearly see that neither model transformation nor bounding technique for cross terms is used. Therefore, the obtained filter design method is expected less conservative. It is well known that the number of variables has a great influence on the computation burden. The number of variables involved in the LMIs (3.17)–(3.19) is (9/2)𝑛²+(7/2)𝑛+2𝑛. However, the numbers of variables in [22] is 15𝑛²+5𝑛. With much fewer matrix variables Theorem 3.2 also saves much computation than Theorem  2 in [22].

3.3. Robust 𝐻_∞ Filter

On the basis of the result of Theorem 3.2, it is easy to obtain the network-based robust 𝐻_∞ filter design for the uncertain systems (2.1) with uncertainties Δ𝐴(𝑡), Δ𝐵(𝑡), and Δ𝐶(𝑡) satisfying (2.2)-(2.3).

Theorem 3.5. Consider the uncertain system in (2.1). A filter of form (2.12) and constants 𝜏₁ and 𝜏₂, the filtering error system is robust globally asymptotically stable with performance 𝛾, if there exist real matrices 𝑃₁>0, 𝑃₃>0, 𝑄₁>0,  𝑄₂>0, 𝑄₃>0, 𝑅₁>0, and 𝑅₂>0 and any matrices 𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓, and scalars 𝜀₁>0 and 𝜀₂>0 such that the following LMIs and (3.19) are satisfied: Σ₁₁𝐴_𝑓+𝐴^𝑇𝑃₃𝐵_𝑓𝐺𝐶𝑅₁0Σ₁₆𝐴^𝑇ℛ𝐿^𝑇𝑃₁𝑀₁𝐵_𝑓𝐺𝑀₂∗𝐴_𝑓+𝐴^𝑇_𝑓𝐵_𝑓𝐺𝐶00𝑃₃𝐵0−𝐶^𝑇_𝑓𝑃₃𝑀₁𝐵_𝑓𝐺𝑀₂∗∗Σ₃₃2𝑅₂𝑅₂00000∗∗∗Σ₄₄₁000000∗∗∗∗Σ₅₅₁Σ00000∗∗∗∗∗₆₆𝐵^𝑇ℛ000∗∗∗∗∗∗−ℛ0ℛ𝑀₁0∗∗∗∗∗∗∗−𝐼00∗∗∗∗∗∗∗∗−𝜀₁𝐼0∗∗∗∗∗∗∗∗∗−𝜀₂𝐼Σ<0,₁₁𝐴_𝑓+𝐴^𝑇𝑃₃𝐵_𝑓𝐺𝐶𝑅₁0Σ₁₆𝐴^𝑇ℛ𝐿^𝑇𝑃₁𝑀₁𝐵_𝑓𝐺𝑀₂∗𝐴_𝑓+𝐴^𝑇_𝑓𝐵_𝑓𝐺𝐶00𝑃₃𝐵0−𝐶^𝑇_𝑓𝑃₃𝑀₁𝐵_𝑓𝐺𝑀₂∗∗Σ₃₃2𝑅₂𝑅₂00000∗∗∗Σ₄₄₂000000∗∗∗∗Σ₅₅₂Σ00000∗∗∗∗∗₆₆𝐵^𝑇ℛ000∗∗∗∗∗∗−ℛ0ℛ𝑀₁0∗∗∗∗∗∗∗−𝐼00∗∗∗∗∗∗∗∗−𝜀₁𝐼0∗∗∗∗∗∗∗∗∗−𝜀₂𝐼<0, (3.24) where Σ₁₁=𝑃𝐴₁+𝐴^𝑇₁𝑃+𝑄₁+𝑄₂+𝑄₃−𝑅₁+𝜀₁𝑁^𝑇₁𝑁₁,Σ₁₆=𝑃₁𝐵+𝜀₁𝑁^𝑇₂𝑁₁,Σ₃₃=−1−𝜇𝑄₃−3𝑅₂+𝜀₁𝑁^𝑇₃𝑁₃,Σ₆₆=−𝛾²𝐼+𝜀₁𝑁^𝑇₂𝑁₂. (3.25) Moreover, if the previous conditions are satisfied, an acceptable state-space realization of the 𝐻_∞ filter 𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓 are given by (3.21).

Proof. Replace 𝐴, 𝐵 and 𝐶 in the LMIs (3.17) and (3.18) with 𝐴+Δ𝐴(𝑡), 𝐵+Δ𝐵(𝑡) and 𝐶+Δ𝐶(𝑡), respectively, then the LMIs (3.17) and (3.18) can be rewritten as Υ₁+𝜒₁𝐹₁(𝑡)𝜒₂+𝜒^𝑇₂𝐹^𝑇₁(𝑡)𝜒^𝑇₁+𝜒₃𝐹₂(𝑡)𝜒₄+𝜒^𝑇₄𝐹^𝑇₂(𝑡)𝜒^𝑇₃Υ<0,₂+𝜒₁𝐹₁(𝑡)𝜒₂+𝜒^𝑇₂𝐹^𝑇₁(𝑡)𝜒^𝑇₁+𝜒₃𝐹₂(𝑡)𝜒₄+𝜒^𝑇₄𝐹^𝑇₂(𝑡)𝜒^𝑇₃<0, (3.26) where 𝜒₁=𝑀^𝑇₁𝑃₁𝑀^𝑇₁𝑃₃0000𝑀^𝑇₁ℛ^𝑇,𝜒₂=𝑁₁0000𝑁₂,𝜒00₃=𝑀^𝑇₂𝐺𝐵^𝑇_𝑓𝑀^𝑇₂𝐺𝐵^𝑇_𝑓000000^𝑇,𝜒₄=00𝑁₃00000. (3.27) By Lemma 2.5, there exist scalars 𝜀₁>0 and 𝜀₂>0 such that Υ₁+𝜀₁⁻¹𝜒₁𝜒^𝑇₁+𝜀₁𝜒^𝑇₂𝜒₂+𝜀₂⁻¹𝜒₃𝜒^𝑇₃+𝜀₂𝜒^𝑇₄𝜒₄Υ<0,₂+𝜀₁⁻¹𝜒₁𝜒^𝑇₁+𝜀₁𝜒^𝑇₂𝜒₂+𝜀₂⁻¹𝜒₃𝜒^𝑇₃+𝜀₂𝜒^𝑇₄𝜒₄<0, (3.28) then by Lemma 2.3, (3.24) follows directly.

Remark 3.6. When the 𝜇 is unknown, by setting 𝑄₃=0, Theorem 3.2 and Theorem 3.5 reduce to a delay-dependent and rate-independent network-based robust 𝐻_∞ filter design condition for the uncertain systems (2.1) with uncertainties Δ𝐴(𝑡), Δ𝐵(𝑡), and Δ𝐶(𝑡) satisfying (2.2)-(2.3).

4. Numerical Example

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

Example 4.1. Consider the system (2.8) with [22] 𝐴=.0.53−2−5,𝐵=−0.50.9,𝐶=01,𝐿=11 (4.1) This example has been considered in [22], and assume that 𝜏(𝑡) satisfies 0.01≤𝜏(𝑡)≤0.2, 𝑤(𝑡)=0.1sin𝑒^−0.1𝑡 and the sensor has a probabilistic distort, that is, the distort matrix G=0.6. Note that different 𝜏₁ and 𝜏₂ yield different 𝛾_min, to compare with the existing results [22]; we assume that 𝜇 is unknown, and by setting 𝑄₃=0 in Theorem 3.2, the computation results of 𝛾_min under different 𝜏₁ and 𝜏₂ are listed in Tables 1 and 2. Minimum index 𝛾 for different 𝜇 and 𝜏₁=0 is listed in Table 3. From Tables 1–3, it can be seen that the value of 𝛾_min grows for 𝜏₂→20 and for given 𝜏₁, which tends to be 0.2143.
To get minimum index 𝛾, the approach in [22] needs 70 decision variables; however, the number of decision variables involved in Theorem 3.2 is only 29, which sufficiently demonstrates the efficiency of the proposed method. With fewer matrix variables the minimum index 𝛾 obtained in this paper are less conservative than those in [22].

Table 1 

Minimum index 𝛾 for unknown 𝜇 and 𝜏₁ = 0.01.

Table 2 

Minimum index 𝛾 for unknown 𝜇 and 𝜏₁ = 0.

Table 3 

Minimum index 𝛾 for different 𝜇 and 𝜏₁ = 0.

The initial conditions 𝑥(𝑡) and 𝑥_𝑓(𝑡) are [0.1−0.1]^𝑇 and [0.2−0.5]^𝑇, respectively, for an appropriate initial interval. For given 𝜏₁=0.01,𝜏₂=0.2 with 𝛾_min=0.1361, according to Theorem 3.2, we can obtain the desired 𝐻_∞ filter parameters as follows: 𝐴_𝑓=74.8448−136.6475146.4634−261.6214,𝐵_𝑓=,𝐶−0.9327−1.3543_𝑓=127.2527−228.1007. (4.2) Next, we apply the filter (2.12) with the filter matrices (3.21) to the system (2.8) and obtain the simulation results as in Figures 1–3. Figure 1 shows the state response 𝑥(𝑡) under the initial condition. Figure 2 shows error response 𝑒(𝑡)=𝑧(𝑡)−𝑧_𝑓(𝑡). Figure 3 shows the the output 𝑧(𝑡) and 𝑧_𝑓(𝑡). From these simulation results, we can see that the designed 𝐻_∞ filter can stabilize the system (2.8) with sensor failures and noise disturbance.

Figure 1 

The state response of system 𝑥(𝑡) in Example 4.1.

Figure 2 

The error response 𝑒(𝑡)=𝑧(𝑡)−𝑧𝑓(𝑡) in Example 4.1.

Figure 3 

The output 𝑧(𝑡) and 𝑧𝑓(𝑡) in Example 4.1.

The example conclusively shows that our results are less conservative than the previous ones in [22].

Example 4.2. Consider the uncertain system (2.1) with 1𝐴=,−5010−4110−3,𝐵=−0.50.1,𝐶=101010,𝐿=111 (4.3) and the uncertainties of the system are of the forms (2.2) and (2.3) with 𝑀₁=0.1−0.20.1,𝑁₁=0.10.10.1,𝑁₂𝑀=1,₂=0.1−0.2,𝑁₃=0.10.10.1. (4.4)
We assume 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤1, 𝑤(𝑡)=0.1sin𝑒^−0.1𝑡, and the sensor has a probabilistic distort, that is, the distort matrix 𝐺=.0.5000.3 (4.5) When 𝜇 is unknown, by setting 𝑄₃=0 in Theorem 3.5, the minimum achievable noise attenuation level is given by 𝛾=0.24 and the correspond filter parameters as follows: 𝐴_𝑓=53.9212−73.080738.4644132.0705−156.619487.3469−321.6563355.4126−233.8653,𝐵_𝑓=,𝐶−0.2813−0.7511−0.4865−1.5153−0.4195−1.0039_𝑓=−192.4663205.3754−136.1696. (4.6) When 𝜇=0.5, by Theorem 3.5, the minimum achievable noise attenuation level is given by 𝛾=0.24 and the correspond filter parameters as follows: 𝐴_𝑓=41.5346−62.093426.4966104.1741−131.564566.4086−218.6961248.5441−162.7258,𝐵_𝑓=,𝐶−0.2547−0.7358−0.5284−1.6716−0.5576−1.3325_𝑓=−132.7246143.3911−97.2733. (4.7)
The initial conditions 𝑥(𝑡) and 𝑥_𝑓(𝑡) are [0.3−0.1−0.2]^𝑇 and [−0.1−0.1−0.3]^𝑇, respectively, for an appropriate initial interval. Next, we apply the filter (2.12) with the filter matrices (3.21) to the uncertain system (2.1) and obtain the simulation results as in Figures 4–6. Figure 4 shows the state response 𝑥(𝑡) under the initial condition. Figure 5 shows error response 𝑒(𝑡)=𝑧(𝑡)−𝑧_𝑓(𝑡). Figure 6 shows the the output 𝑧(𝑡) and 𝑧_𝑓(𝑡). From these simulation results, we can see that the designed 𝐻_∞ filter can stabilize the system (2.1) with sensor failures and noise disturbance.

Figure 4 

The state response of system 𝑥(𝑡) in Example 4.2.

Figure 5 

The error response 𝑒(𝑡)=𝑧(𝑡)−𝑧𝑓(𝑡) in Example 4.2.

Figure 6 

The output 𝑧(𝑡) and 𝑧𝑓(𝑡) in Example 4.2.

5. Conclusions

In this paper the network-based robust 𝐻_∞ filtering for the uncertain system with sensor failures and noise disturbance has been developed. A new type of Lyapunov-Krasovskii functional has been constructed to derive a less conservative sufficient condition for a linear full-order filter in terms of LMIs, which guarantees a prescribed 𝐻_∞ performance index for the filtering error system. Two numerical examples have shown the usefulness and effectiveness of the proposed filter design method. Finally, our future study will focus mainly on the following two issues: (1) to further improve our results by using the delay decomposition LKF. (2) When the noise is stochastic, that is to say, the network-based robust 𝐻_∞ filtering for the uncertain system with sensor failures and stochastic noise could be considered.