Filtering-Based Fault Detection for Stochastic Markovian Jump System with Distributed Time-Varying Delays and Mixed Modes

Abstract

The problem of fault detection for stochastic Markovian jump system is considered. The system under consideration involves discrete and distributed time-varying delays, Itô-type stochastic disturbance, and different system and delay modes. The aim of this paper is to design a fault detection filter such that the fault detection system is stochastically stable and satisfies a prescribed 𝐻_∞ disturbance attenuation level. By using a novel Lyapunov functional, a mix-mode-dependent sufficient condition is formulated in terms of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed main results.

1. Introduction

Fault detection received considerable attention over the past decades because of the increasing demand for higher performance, safety, and reliability standards. In recently, many effective methods have been developed for fault detection. To the best of the authors’ knowledge, the published results can be categorized into three approaches. The first category is the filter- or observer-based approaches, where filters are used to generate residual signals to detect and estimate the fault, for example, [1–9]. In the fault detection scheme based on filter or observer, a fault cannot only be detected but also be approximated, and the fault estimate can be further used in fault-tolerant control. The second category is the statistic approach, where the Bayesian theory and likelihood method are used to evaluate the fault signals [10]. The third category is the geometric approach. By utilizing the geometric framework, a set of residuals is generated such that each residual is affected by one fault and is partially decoupled from others [11]. In the framework of fault detection, faults are detected by setting a predefined threshold on residual signals. Once the value of residual evaluation function excesses the predefined threshold, an alarm of faults is generated. For example, by using Luenberger type observers, the authors of [6, 12] present an explicit expression of the filters for the fault such that both asymptotic stability and a prescribed level of disturbance attenuation are satisfied for all admissible nonlinear perturbations; by using the measured output probability density functions (PDFs), the authors of [13, 14] construct a stable filter-based residual generator.

Markovian jump systems (MJSs) are a special class of switched systems. The state vector of such system has two components 𝑥(𝑡) and 𝑟(𝑡). The first one is in general referred to as the state, and the second one is regarded as the mode. In its operation, the jump system will switch from one mode to another in a random way, based on a Markovian chain with finite state space. These systems are very common in economic systems, communication systems, robot manipulator systems and circuit systems, and so forth. Time delay is an inherent characteristic of many physical systems, which occurs due to signal transmission, inevitable defects of control equipment, and so on. The systems with or without time delays are convergent when time delays are close to zero. otherwise, they may be divergent. In other words, time delays, either constant or time varying, can degrade the performance of systems designed without considering the delays and can even destabilize the systems. Due to their extensive practical applications, considerable attention has been devoted to MJSs, see, for example, [15–18] for stability, [18–25] for control, and [25–32] for state estimation. More recently, the methods of fault estimation and fault detection have been extended successfully to MJSs [33–40]. From the published results, the delay mode is assumed to be the same as the system matrices mode. However, the assumption cannot always be satisfied in real applications. In some practical systems, variations of delay usually depend on phenomena which may not cause abrupt changes in other system parameters. For instance, in networked control systems, the randomness of delay is a result of communication network issues, but the process itself may contain separate sources of randomness which means that the system matrices mode may be different with the delay mode [41]. Therefore, it is important and necessary to pay attention to the study of Markovian jump systems with different system and delay modes. Furthermore, it appears that general results pertaining to fault detection for stochastic MJSs with discrete and distributed time delays, Itô-type stochastic disturbance and different system and delay modes are few and restricted, despite its practical importance, mainly due to the mathematical difficulties in dealing with such mixed modes. Research in this area should be interesting yet challenging as it involves the combination of two different jumping modes, which has motivated this paper.

This paper deals with the problem of fault detection for stochastic MJSs with discrete and distributed time-varying delays, Itô-type stochastic disturbance, and different system and delay modes. By using a novel mix-mode-dependent Lyapunov functional, a new sufficient condition on stochastic stability with an 𝐻_∞ performance is derived in terms of linear matrix inequalities (LMIs). Based on this, the existence condition of the fault detection filter which guarantees stochastic stability and the 𝐻_∞ performance of the corresponding augmented system is presented. A numerical example is provided to show the effectiveness of the proposed results.

Notation. Throughout this paper, ℝ^𝑛 denotes the 𝑛 dimensional Euclidean space. 𝜆_max(𝑄) and 𝜆_min(𝑄) denote, respectively, the maximal and minimal eigenvalue of matrix 𝑄. 𝐸{⋅} refers to the expectation operator with respect to some probability measure 𝒫. We use  diag{⋅,⋅,⋅} as a block-diagonal matrix. 𝐴>0(<0) means that 𝐴 is a symmetric positive (negative) definite matrix, 𝐴⁻¹ denotes the inverse of matrix 𝐴. 𝐴^𝑇 denotes the transpose of matrix 𝐴, and 𝐼 is the identity matrix with compatible dimension.

2. System Description and Definitions

Consider the following stochastic MJS with mode-dependent time-varying delays: d𝑥(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴₁𝑟_𝑡𝑥𝑡−𝜏𝑡,𝑠_𝑡+𝐴₂𝑟_𝑡^𝑡_{𝑡−𝜏𝑡,𝑠𝑡}𝑥(𝑠)d𝑠+𝐵₀𝑟_𝑡𝑢(𝑡)+𝐵₁𝑟_𝑡𝜈(𝑡)+𝐵₂𝑟_𝑡𝑓(𝑡)d𝑡+𝐺₁𝑟_𝑡𝑥(𝑡)(𝑡),(𝑡)=𝐶𝑟d𝜔d𝑦_𝑡𝑥(𝑡)+𝐶₁𝑟_𝑡𝑥𝑡−𝜏𝑡,𝑠_𝑡+𝐶₂𝑟_𝑡^𝑡_{𝑡−𝜏𝑡,𝑠𝑡}𝑥(𝑠)d𝑠+𝐷₁𝑟_𝑡𝜈(𝑡)+𝐷₂𝑟_𝑡𝑓(𝑡)d𝑡+𝐺₂𝑟_𝑡𝑥(𝑡)(𝑡),𝑥(𝑡)(𝑡)[],d𝜔=𝜙,𝑡∈−𝜏,0 (2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝑢(𝑡) is the exogenous disturbance input which belongs to 𝐿₂[0∞); 𝜈(𝑡) the unknown input; 𝑓(𝑡) is the fault to be detected; 𝑦(𝑡)∈ℝ^𝑝 is the measured output; 𝜔(𝑡) is a zero-mean one-dimensional Wiener process satisfying 𝐸{𝜔(𝑡)}=0 and 𝐸{𝜔²(𝑡)}=𝑡; 𝜙(𝑡) is a compatible vector-valued initial function defined on [−𝜏,0]; 𝐴(𝑟_𝑡), 𝐴₁(𝑟_𝑡), 𝐴₂(𝑟_𝑡), 𝐵₀(𝑟_𝑡), 𝐵₁(𝑟_𝑡), 𝐵₂(𝑟_𝑡), 𝐺₁(𝑟_𝑡), 𝐶(𝑟_𝑡), 𝐶₁(𝑟_𝑡), 𝐶₂(𝑟_𝑡), 𝐷₁(𝑟_𝑡), 𝐷₂(𝑟_𝑡), and 𝐺₂(𝑟_𝑡) are real constant matrices with appropriate dimensions. 𝜏(𝑡,𝑠_𝑡) is the mode-dependent time-varying delay. {𝑟_𝑡,𝑡⩾0} and {𝑠_𝑡,𝑡⩾0} are continuous-time Markovian processes with right continuous trajectories and taking values in finite sets 𝑆₁={1,2,…,𝑁}, 𝑆₂={1,2,…,𝑀} with the transition probability matrices Π=[𝜋_𝑖𝑙],  (𝑖,𝑙∈𝑆₁) and Λ=[𝜆_𝑗𝑘],  (𝑗,𝑘∈𝑆₂), respectively, given by Pr𝑟_𝑡+Δ=𝑙∣𝑟_𝑡=𝜋=𝑖^𝑖𝑙Δ+𝑜(Δ),𝑙≠𝑖,1+𝜋_𝑖𝑖(Δ)𝑠Δ+𝑜,𝑙=𝑖,Pr_𝑡+Δ=𝑘∣𝑠_𝑡=𝜆=𝑗^𝑗𝑘Δ+𝑜(Δ),𝑘≠𝑗,1+𝜆_𝑗𝑗Δ+𝑜(Δ),𝑘=𝑗, (2.2) where Δ>0 and lim_Δ→0(𝑜(Δ)/Δ)=0; 𝜋_𝑖𝑙⩾0 for 𝑖≠𝑙 is the transition rate from mode 𝑖 at time 𝑡 to mode 𝑙 at time 𝑡+Δ and 𝜋_𝑖𝑖=−∑^𝑁_{𝑙=1,𝑙≠𝑖}𝜋_𝑖𝑙; 𝜆_𝑗𝑘⩾0 for 𝑗≠𝑘 is the transition rate from mode 𝑗 at time 𝑡 to mode 𝑘 at time 𝑡+Δ and 𝜆_𝑗𝑗=−∑^𝑀_{𝑘=1,𝑘≠𝑗}𝜆_𝑗𝑘. The processes 𝑟_𝑡 and 𝑠_𝑡 are assumed to be independent throughout this paper. For simplicity, a matrix 𝑅(𝑟_𝑡) will be denoted by 𝑅_𝑖. For example, 𝐴(𝑟_𝑡) is denoted by 𝐴_𝑖, 𝐴₁(𝑟_𝑡) is denoted by 𝐴_1𝑖, (𝑖∈𝑆₁), 𝜏(𝑠_𝑡,𝑡) is denoted by 𝜏_𝑗(𝑡), (𝑗∈𝑆₂), and so on. When the mode is in 𝑠_𝑡=𝑗, the mode-dependent time-varying delay satisfies 0<𝜏_𝑗(𝑡)⩽𝜏_𝑗⩽𝜏,𝜏̇_𝑗(𝑡)⩽𝜇_𝑗, (2.3) where 𝜏=max{𝜏_𝑗}.

Remark 2.1. In this work, we have assumed that the delay mode is different from the system mode. This is more powerful and desirable in modeling of real systems, because the reason for jump in delay value may not be the same as that for jump in other system parameters.

Remark 2.2. The generalized stochastic system (2.1) is quite general since it considers noise perturbations, discrete, and distributed time-varying delays and Markovian jump processes with different modes. To the best of our knowledge, the generalized stochastic system (2.1) has never been considered in the previous literature.

Remark 2.3. A fault detection system consists of a residual generator and an evaluation stage, including an evaluation function and a threshold. Therefore, the fault detection problem to be addressed in this paper can be stated as the following two steps. The first step is to design a suitable filter to reduce the effect of disturbances on residual signals and to enhance the influence of faults. The second step is to determine the residual evaluation function and an appropriate threshold.

In this study, the following full-order fault detection filter is considered: d𝑥_𝑓(𝑡)=𝐴_𝑓𝑖𝑗𝑥_𝑓(𝑡)d𝑡+𝐵_𝑓𝑖𝑗(𝑡),𝑟(𝑡)d𝑦=𝐶_𝑓𝑖𝑗𝑥_𝑓(𝑡),𝑥_𝑓(0)=0,𝑖∈𝑆₁,𝑗∈𝑆₂, (2.4) where 𝑥_𝑓(𝑡) is the filter state vector. 𝑟(𝑡) is its output which is sensitive to faults. (𝐴_𝑓𝑖𝑗𝐵_𝑓𝑖𝑗𝐶_𝑓𝑖𝑗) are appropriately dimensioned filter matrices to be determined.

To improve the sensitiveness of residual to fault, we add a weighting matrix function into the fault 𝑓(𝑡), that is, 𝐹_𝜔(𝑠)=𝑊(𝑠)𝐹(𝑠), where 𝐹(𝑠) and 𝐹_𝜔(𝑠) denote, respectively, the Laplace transforms of 𝑓(𝑡) and 𝑓_𝜔(𝑡). One state-space realization of 𝐹_𝜔(𝑠)=𝑊(𝑠)𝐹(𝑠) can be 𝑥̇_𝜔(𝑡)=𝐴_𝜔𝑥_𝜔(𝑡)+𝐵_𝜔𝑓(𝑡),𝑓_𝜔(𝑡)=𝐶_𝜔𝑥_𝜔(𝑡),𝑥_𝜔(0)=0. (2.5)

Denoting 𝑟_𝑒(𝑡)=𝑟(𝑡)−𝑓_𝜔(𝑡), then the overall dynamic system can be governed by the following augmented system: d𝜉(𝑡)=𝐴_𝑖𝑗𝜉(𝑡)+𝐴_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗(𝑡)+𝐴_2𝑖𝑗𝐾^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉(𝑠)d𝑠+𝐵_𝑖𝑗𝑤(𝑡+)d𝑡𝐺_𝑖𝑗𝐾𝜉(𝑡)d𝜔(𝑡),𝑟_𝑒(𝑡)=𝐶_𝑖𝑗𝜉(𝑡),𝜉(𝑡)=𝜙(𝑡),∀𝑡∈[−𝜏,0], (2.6) where 𝜉(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇_𝑓(𝑡)𝑥_𝜔(𝑡)]^𝑇, 𝑤(𝑡)=[𝑢^𝑇(𝑡)𝜈^𝑇(𝑡)𝑓^𝑇(𝑡)]^𝑇, 𝜙(𝑡)=[𝜙^𝑇(𝑡)0^𝑇0^𝑇]^𝑇, and 𝐴_𝑖𝑗=𝐴_𝑖𝐵00_𝑓𝑖𝑗𝐶_𝑖𝐴_𝑓𝑖𝑗000𝐴_𝜔,𝐴_1𝑖𝑗=𝐴_1𝑖𝐵_𝑓𝑖𝑗𝐶_1𝑖0,𝐴_2𝑖𝑗=𝐴_2𝑖𝐵_𝑓𝑖𝑗𝐶_2𝑖0,𝐵_𝑖𝑗=𝐵_0𝑖𝐵_1𝑖𝐵_2𝑖0𝐵_𝑓𝑖𝑗𝐷_1𝑖𝐵_𝑓𝑖𝑗𝐷_2𝑖00𝐵_𝜔,𝐺_𝑖𝑗=𝐺_1𝑖𝐵_𝑓𝑖𝑗𝐺_2𝑖0,𝐶_𝑖𝑗=0𝐶_𝑓𝑖𝑗−𝐶_𝜔,𝐾=𝐼00. (2.7)

For simplicity, let 𝜑(𝑡)=𝐴_𝑖𝑗𝜉(𝑡)+𝐴_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗(𝑡)+𝐴_2𝑖𝑗𝐾^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉(𝑠)d𝑠+𝐵_𝑖𝑗𝑤(𝑡),𝑔(𝑡)=𝐺_𝑖𝑗𝐾𝜉(𝑡). (2.8)

Now, the problem of fault detection is transformed into an 𝐻_∞ filtering problem for system (2.1), which is described as follows: given a prescribed level of disturbance attenuation 𝛾, determine a series of filter matrices 𝐴_𝑓𝑖𝑗, 𝐵_𝑓𝑖𝑗, and 𝐶_𝑓𝑖𝑗(𝑖∈𝑆₁,𝑗∈𝑆₂) such that the augmented system (2.6) is stochastically stable.

After designing a fault detection filter, the remaining important task is to evaluate the generated residual. One of widely adopted methods is to choose a residual evaluation function and a threshold. In this paper, residual evaluation function 𝑓(𝑟) and a threshold  𝐽th  are selected as 𝑓(𝑟)=^𝑡0_𝑡^+𝑇₀𝑟^𝑇(𝑡)𝑟(𝑡)d𝑡,𝐽th=sup_{𝜈(𝑡)∈𝐿2(𝑡),𝑓=0}𝐸^𝑡0_𝑡^+𝑇₀𝑟^𝑇(𝑡)𝑟(𝑡)d𝑡, (2.9) where [𝑡₀,𝑡₀+𝑇] is the finite-time window, 𝑇 denotes the limited length, and 𝑡₀ denotes the initial evaluation time. The occurrence of fault can be detected by comparing 𝑓(𝑟) and 𝐽th, according to the following logic: 𝑓(𝑟)𝑓(𝑟)>𝐽th⇒Faults⇒Alarm,<𝐽th⇒NoFaults. (2.10)

The following lemma and definitions are introduced, which will be used in the proof of the main results.

Lemma 2.4 (see [42]). For any matrix 𝑀>0, scalar 𝛾>0, vector function 𝜔∶[0,𝛾]→𝑅^𝑛 such that the integrations concerned are well defined, the following inequality holds: ^𝛾₀𝜔^𝑇(𝑠)d𝑠𝑀^𝛾₀𝜔(𝑠)d𝑠⩽𝛾^𝛾₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)d𝑠. (2.11)

Definition 2.5. The filtering error system (2.6) with 𝑤(𝑡)=0 is said to be stochastically stable, if, for every system mode 𝑟_𝑡, every time-delay mode 𝑠_𝑡 and all finite initial state 𝜙(𝑡), the following relation holds: lim_𝑡→∞𝐸{|𝜉(𝑡)|²}=0.

Definition 2.6. Given a scalar 𝛾>0, the filtering error system (2.6) is said to be stochastically stable with an 𝐻_∞ performance 𝛾, if, for every system mode 𝑟_𝑡 and every time-delay mode 𝑠_𝑡, the filtering error system (2.6) with 𝑤(𝑡)=0 is stochastically stable, and, under zero initial condition, it satisfies ‖𝑟_𝑒‖₂⩽𝛾‖𝑤‖₂ for any nonzero 𝑤(𝑡)∈𝐿₂[0,∞].

It should be pointed out that the joint process (𝜉(𝑡),𝑟_𝑡,𝑠_𝑡) is not Markovian. In order to cast our model into the frame work for a Markovion system, let us define a new Markovion process: 𝜉_𝑡(𝑠)=𝜉(𝑡+𝑠), −𝜏⩽𝑠⩽0, and then {(𝜉_𝑡,𝑟_𝑡,𝑠_𝑡),𝑡⩾0} is Markovian process with the initial state (𝜙(⋅),𝑟₀,𝑠₀).

Let 𝐂(ℝ^𝑛×ℝ^𝑛×ℝ⁺×𝑆₁×𝑆₂) denote the family of all nonnegative functions 𝑉(𝜉,𝜉_𝑡,𝑡,𝑖,𝑗) on ℝ^𝑛×ℝ^𝑛×ℝ⁺×𝑆₁×𝑆₂, which are continuously twice differentiable in 𝜉 and differentiable in 𝑡. If 𝑉∈𝐂(ℝ^𝑛×ℝ^𝑛×ℝ⁺×𝑆₁×𝑆₂), then, along the trajectory of system (2.6), we define an operator ℒ(⋅) from ℝ^𝑛×ℝ^𝑛×ℝ⁺×𝑆₁×𝑆₂ to ℝ by ℒ𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=𝑉_𝑡𝜉,𝜉_𝑡,𝑡,𝑖,𝑗+𝑉_𝜉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗𝜑(𝑡)+_𝑙∈𝑆1𝜋_𝑖𝑙𝑉𝜉,𝜉_𝑡,𝑡,𝑙,𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑉𝜉,𝜉_𝑡+1,𝑡,𝑖,𝑘2trace𝑔^𝑇(𝑡)𝑉_𝜉𝜉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗𝑔(𝑡), (2.12) where 𝑉_𝑡𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=𝜕𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗,𝑉𝜕𝑡_𝜉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=𝜕𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗𝜕𝜉₁,…,𝜕𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗𝜕𝜉_𝑛𝑉_𝜉𝜉,𝜉,𝜉_𝑡=𝜕,𝑡,𝑖,𝑗²𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗𝜕𝜉_𝑖𝜉_𝑗. (2.13)(2.14)(2.15)

3. Main Results

In this section, we first propose a delay-dependent sufficient condition for stochastic stability with the 𝐻_∞ performance of filtering error system (2.6). Now, define a stochastic Lyapunov functional candidate for systems (2.6) as 𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=⁶_𝑛=1𝑉_𝑛𝜉,𝜉_𝑡,𝑡,𝑖,𝑗, (3.1) where 𝑉₁𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=𝜉^𝑇(𝑡)𝑃𝑟_𝑡,𝑠_𝑡𝜉(𝑡),𝑉₂𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=^𝑡_{𝑡−𝜏𝑡,𝑠𝑡}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₁𝑟_𝑡,𝑠_𝑡(𝑠)+𝐾𝜉d𝑠_{𝑡𝑠𝑡−𝜏𝑡}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₂𝑟_𝑡,𝑠_𝑡(𝑠)𝑉𝐾𝜉d𝑠,₃𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=^𝑡_{𝑡−𝜏𝑡,𝑠𝑡}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅₁𝑟_𝑡,𝑠_𝑡^𝑡_𝜃(𝑠)𝑉𝐾𝜉d𝑠d𝜃,₄𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=⁰_−𝜏^𝑡_𝑡+𝜃𝜑^𝑇(𝑠)𝐾^𝑇(𝑠)𝑉𝑍𝐾𝜑d𝑠d𝜃,₅𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=⁰_−𝜏^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅₂(𝑠)𝑉𝐾𝜉d𝑠d𝜃,₆𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=^𝜏₀⁰_−𝜃^𝑡_𝑡+𝑠𝜉^𝑇(𝛼)𝐾^𝑇𝑅₃𝐾𝜉(𝛼)d𝛼d𝑠d𝜃. (3.2) By Itô’s formula, we obtain the stochastic differential as d𝑉=ℒ⁶_𝑛=1𝑉_𝑛𝜉,𝜉_𝑡,𝑡,𝑖,𝑗d𝑡+2𝜉^𝑇(𝑡)𝑃_𝑖𝑗𝑔(𝑡)d𝜔(𝑡), (3.3) where ℒ is the weak infinitesimal generator of the random process {𝜉_𝑡,𝑟_𝑡,𝑠_𝑡} along the system (2.6).

Using the operator (2.12), we have ℒ𝑉₁𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=2𝜉^𝑇(𝑡)𝑃_𝑖𝑗𝐴_𝑖𝑗𝜉(𝑡)+𝐴_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗(𝑡)+𝐴_2𝑖𝑗𝐾^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉(𝑠)d𝑠+𝐵_𝑖𝑗𝑤(𝑡)+𝜉^𝑇(𝑡)_𝑙∈𝑆1𝜋_𝑖𝑙𝑃_𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑃_𝑖𝑘𝜉(𝑡)+𝑔^𝑇(𝑡)𝑃_𝑖𝑗𝑔(𝑡). (3.4) The derivative of the first term in 𝑉₂(𝜉,𝜉_𝑡,𝑡,𝑖,𝑗) is given as follows: ℒ^𝑡_{𝑡−𝜏𝑡,𝑠𝑡}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₁𝑟_𝑡,𝑠_𝑡𝐾𝜉(𝑠)d𝑠=lim_Δ→0+1Δ𝐸_𝑠^𝑡+Δ_{𝑡+Δ−𝜏𝑡+Δ,𝑡+Δ}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₁𝑟_𝑡+Δ,𝑠_𝑡+Δ(𝑠)−𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑗(𝑠)=𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇_𝑙∈𝑆1𝜋_𝑖𝑙𝑄_1𝑙𝑗𝐾𝜉(𝑠)d𝑠+_𝑘∈𝑆2𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑘(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑘𝐾𝜉(𝑠)d𝑠+𝜉^𝑇(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉(𝑡)−1−𝜏̇_𝑗(𝑡)𝜉^𝑇𝑡−𝜏_𝑗(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗(𝑡)⩽𝜉^𝑇(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉(𝑡)−1−𝜇_𝑗𝜉^𝑇𝑡−𝜏_𝑗(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗(𝑡)+^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇_𝑙∈𝑆1𝜋_𝑖𝑙𝑄_1𝑙𝑗𝐾𝜉(𝑠)d𝑠+𝜆_𝑗𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏}𝜉^𝑇(𝑠)𝐾^𝑇_𝑘≠𝑗𝜆_𝑗𝑘𝑄_1𝑖𝑘𝐾𝜉(𝑠)d𝑠. (3.5) Following a similar method of (3.5), it is easy to obtain ℒ^𝑡_{𝑠𝑡−𝜏𝑡}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₂𝑟_𝑡,𝑠_𝑡𝐾𝜉(𝑠)d𝑠=𝜉^𝑇(𝑡)𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉(𝑡)−𝜉^𝑇𝑡−𝜏_𝑗𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗+^𝑡_{𝑡−𝜏𝑗}𝜉^𝑇(𝑠)𝐾^𝑇_𝑙∈𝑆1𝜋_𝑖𝑙𝑄_2𝑙𝑗𝐾𝜉(𝑠)d𝑠+_𝑘∈𝑆2𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑘}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_2𝑖𝑘𝐾𝜉(𝑠)d𝑠⩽𝜉^𝑇(𝑡)𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉(𝑡)−𝜉^𝑇𝑡−𝜏_𝑗𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗+^𝑡_{𝑡−𝜏𝑗}𝜉^𝑇(𝑠)𝐾^𝑇_𝑙∈𝑆1𝜋_𝑖𝑙𝑄_2𝑙𝑗𝐾𝜉(𝑠)d𝑠+𝜆_𝑗𝑗^𝑡_{𝑡−𝜏𝑗}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_2𝑖𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏}𝜉^𝑇(𝑠)𝐾^𝑇_𝑘≠𝑗𝜆_𝑗𝑘𝑄_2𝑖𝑘𝐾𝜉(𝑠)d𝑠,ℒ𝑉₃𝜉,𝜉_𝑡𝜏̇,𝑡,𝑖,𝑗=−1−_𝑗(𝑡)^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝜉(𝑠)d𝑠+2^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑡)𝐾^𝑇𝑅_1𝑖𝑗×^𝑡_𝜃𝐾𝜉(𝑠)d𝑠d𝜃+_𝑙∈𝑆1𝜋_𝑖𝑙^𝑡_{𝑡−𝜏𝑗(𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑙𝑗^𝑡_𝜃(𝑠)+𝐾𝜉d𝑠d𝜃_𝑘∈𝑆2𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑘(𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)d𝑠d𝜃. (3.6) Using Lemma 2.4 and considering (2.3), we have ℒ𝑉₃𝜉,𝜉_𝑡,𝑡,𝑖,𝑗⩽−1−𝜇_𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝜉(𝑠)d𝑠+𝜉^𝑇(𝑡)𝐾^𝑇12𝜏²_𝑗𝑅_1𝑖𝑗(𝑡)+𝐾𝜉_{𝑡𝑡−𝜏𝑗(𝑡)𝑡𝜃}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗𝐾𝜉(𝑠)d𝑠d𝜃+_𝑙≠𝑖𝜋_𝑖𝑙^𝑡_{𝑡−𝜏𝑗(𝑡)}(𝑡−𝜃)^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑙𝑗𝐾𝜉(𝑠)d𝑠d𝜃+_𝑘≠𝑗𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑘(𝑡)}(𝑡−𝜃)^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)d𝑠d𝜃⩽−1−𝜇_𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝜉(𝑠)d𝑠+𝜉^𝑇(𝑡)𝐾^𝑇12𝜏²_𝑗𝑅_1𝑖𝑗𝐾𝜉(𝑡)+^𝑡_{𝑡−𝜏𝑗(𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗𝐾𝜉(𝑠)d𝑠d𝜃+_𝑙≠𝑖𝜋_𝑖𝑙^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑙𝑗(𝑠)×𝐾𝜉_{𝑡𝑡−𝜏𝑗(𝑡)}(𝑡−𝜃)d𝜃d𝑠+_𝑘≠𝑗𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑘(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)^𝑡_{𝑡−𝜏𝑘(𝑡)}(𝑡−𝜃)d𝜃d𝑠⩽−1−𝜇_𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝜉(𝑠)d𝑠+𝜉^𝑇(𝑡)𝐾^𝑇12𝜏²_𝑗𝑅_1𝑖𝑗(𝑡)+𝐾𝜉_{𝑡𝑡−𝜏𝑗(𝑡)𝑡𝜃}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗(𝑠)1𝐾𝜉d𝑠d𝜃+2𝜏²_{𝑗𝑙≠𝑖}𝜋_𝑖𝑙^𝑡_𝑡−𝜏𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑙𝑗(𝑠)+1𝐾𝜉d𝑠2_𝑘≠𝑗𝜏²_𝑘𝜆_𝑗𝑘^𝑡_𝑡−𝜏𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)d𝑠. (3.7) Moreover, ℒ𝑉₄𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=𝜏𝜑^𝑇(𝑡)𝐾^𝑇𝑍𝐾𝜑(𝑡)−^𝑡_𝑡−𝜏𝜑^𝑇(𝑠)𝐾^𝑇𝑍𝐾𝜑(𝑠)d𝑠,ℒ𝑉₅𝜉,𝜉_𝑡,𝑡,𝑖,𝑗=𝜏𝜉^𝑇(𝑡)𝐾^𝑇𝑅₂𝐾𝜉(𝑡)−^𝑡_𝑡−𝜏𝜉^𝑇(𝑠)𝐾^𝑇𝑅₂𝐾𝜉(𝑠)d𝑠,ℒ𝑉₆𝜉,𝜉_𝑡=1,𝑡,𝑖,𝑗2𝜏²𝜉^𝑇(𝑡)𝐾^𝑇𝑅₃𝐾𝜉(𝑡)−^𝑡_𝑡−𝜏^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅₃𝐾𝜉(𝑠)d𝑠d𝜃. (3.8) We define 𝜂_𝑗(𝑡)=𝜉^𝑇(𝑡)𝜉^𝑇𝑡−𝜏_𝑗(𝑡)𝐾^𝑇𝜉^𝑇𝑡−𝜏_𝑗𝐾^𝑇𝜑^𝑇(𝑡)𝐾^𝑇𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠^𝑇. (3.9) The following equations are true for any matrices 𝐋, 𝐌, 𝐍, and 𝐘 with appropriate dimensions: 0=2𝜂^𝑇_𝑗(𝑡)𝐋𝐾𝜉(𝑡)−𝐾𝜉𝑡−𝜏_𝑗(𝑡)−^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝜑(𝑠)d𝑠−^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝑔(𝑠)d𝜔(𝑠),0=2𝜂^𝑇_𝑗(𝑡)𝐌𝐾𝜉𝑡−𝜏_𝑗(𝑡)−𝐾𝜉𝑡−𝜏_𝑗−^{𝑡−𝜏𝑗(𝑡)}_{𝑡−𝜏𝑗}𝐾𝜑(𝑠)d𝑠−^{𝑡−𝜏𝑗(𝑡)}_{𝑡−𝜏𝑗}𝐾𝑔(𝑠)d𝜔(𝑠),0=2𝜂^𝑇_𝑗(𝑡)𝐍−𝐾𝜑(𝑡)+𝐾𝐴_𝑖𝑗𝜉(𝑡)+𝐾𝐴_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑗(𝑡)+𝐾𝐴_2𝑖𝑗𝐾^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉(𝑠)d𝑠+𝐾𝐵_1𝑖𝑗𝜈(𝑡),0=𝜏_𝑗𝜂^𝑇_𝑗(𝑡)𝐘𝜂_𝑗(𝑡)−^{𝑡−𝜏𝑗(𝑡)}_{𝑡−𝜏𝑗}𝜂^𝑇_𝑗(𝑡)𝐘𝜂_𝑗(𝑡)d𝑠−^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜂^𝑇_𝑗(𝑡)𝐘𝜂_𝑗(𝑡)d𝑠, (3.10)(3.11) where 𝐋=𝐿^𝑇₁𝐾𝐿^𝑇₂𝐿^𝑇₃𝐿^𝑇₄𝐿^𝑇₅^𝑇,𝐌=𝑀^𝑇₁𝐾𝑀^𝑇₂𝑀^𝑇₃𝑀^𝑇₄𝑀^𝑇₅^𝑇,𝐍=000𝑁^𝑇0^𝑇,𝐾𝐘=^𝑇𝑌₁₁𝐾𝐾^𝑇𝑌₁₂𝐾^𝑇𝑌₁₃𝐾^𝑇𝑌₁₄𝐾^𝑇𝑌₁₅∗𝑌₂₂𝑌₂₃𝑌₂₄𝑌₂₅∗∗𝑌₃₃𝑌₃₄𝑌₃₅∗∗∗𝑌₄₄𝑌₄₅∗∗∗∗𝑌₅₅. (3.12) Considering (3.4)–(3.11), we obtain that ℒ𝑉𝜉,𝜉_𝑡⩽𝜂,𝑖,𝑗_𝑗(𝑡)𝑤(𝑡)^𝑇Σ_𝑖𝑗Φ_1𝑖𝑗𝜂∗0^𝑗(𝑡)𝑤(𝑡)+^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇_{𝑙∈𝑆1,𝑙≠𝑖}𝜋_𝑖𝑙𝑄_1𝑙𝑗+𝜏²_𝑗2𝑅_1𝑙𝑗+𝜋_𝑖𝑖𝑄_1𝑖𝑗+𝜆_𝑗𝑗𝑄_1𝑖𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏}𝜉^𝑇(𝑠)𝐾^𝑇_{𝑘∈𝑆2,𝑘≠𝑗}𝜆_𝑗𝑘𝑄_1𝑖𝑘+𝑄_2𝑖𝑘+𝜏²_𝑘2𝑅_1𝑖𝑘−𝑅₂𝐾𝜉(𝑠)d𝑠+^𝑡_{𝑡−𝜏𝑗}𝜉^𝑇(𝑠)𝐾^𝑇_𝑙∈𝑆1𝜋_𝑖𝑙𝑄_2𝑙𝑗+𝜆_𝑗𝑗𝑄_2𝑖𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑡𝜃}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗−𝑅₃(𝑠)−𝐾𝜉d𝑠d𝜃_{𝑡𝑡−𝜏𝑗(𝑡)}𝜂_𝑗(𝑡)𝐾𝜑(𝑠)^𝑇𝐘𝐋𝜂∗𝑍^𝑗(𝑡)(𝑠)−𝐾𝜑d𝑠_{𝑡−𝜏𝑗(𝑡)𝑡−𝜏𝑗}𝜂_𝑗(𝑡)𝐾𝜑(𝑠)^𝑇𝐘𝐌𝜂∗𝑍^𝑗(𝑡)(𝑠)𝑓(𝑡),𝐾𝜑d𝑠+ (3.13) where 𝑓(𝑡)=2𝜉^𝑇(𝑡)𝑃_𝑖𝑗𝑔(𝑡)d𝜔(𝑡)−2𝜂^𝑇_𝑗(𝑡)𝐋^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝑔(𝑠)d𝜔(𝑠)−2𝜂^𝑇_𝑗(𝑡)𝐌^{𝑡−𝜏𝑗(𝑡)}_{𝑡−𝜏𝑗}(𝑠)(𝑠),Σ𝐾𝑔d𝜔_𝑖𝑗=Σ₁₁Σ₁₂Σ₁₃Σ₁₄Σ₁₅∗Σ₂₂Σ₂₃Σ₂₄Σ₂₅∗∗Σ₃₃Σ₃₄Σ₃₅∗∗∗Σ₄₄Σ₄₅∗∗∗∗Σ₅₅,Σ₁₁=𝑃_𝑖𝑗𝐴_𝑖𝑗+𝐴^𝑇_𝑖𝑗𝑃_𝑖𝑗+𝐾^𝑇𝐺^𝑇_𝑖𝑗𝑃_𝑖𝑗𝐺_𝑖𝑗𝐾+_𝑙∈𝑆1𝜋_𝑖𝑙𝑃_𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑃_𝑖𝑘+𝐾^𝑇𝑄_1𝑖𝑗+𝑄_2𝑖𝑗+𝜏𝑅₂+12𝜏²𝑅₃+12𝜏²_𝑗𝑅_1𝑖𝑗+𝐿₁+𝐿^𝑇₁+𝜏_𝑗𝑌₁₁Σ𝐾,₁₂=𝑃_𝑖𝑗𝐴_1𝑖𝑗+𝐾^𝑇−𝐿₁+𝐿^𝑇₂+𝑀₁+𝜏_𝑗𝑌₁₂,Σ₁₃=𝐾^𝑇𝐿^𝑇₃−𝑀₁+𝜏_𝑗𝑌₁₃,Σ₁₄=𝐾^𝑇𝐿^𝑇₄+𝜏_𝑗𝑌₁₄+𝐴^𝑇_𝑖𝑗𝐾^𝑇𝑁^𝑇,Σ₁₅=𝑃_𝑖𝑗𝐴_2𝑖𝑗+𝐾^𝑇𝐿^𝑇₅+𝜏_𝑗𝑌₁₅,Σ₂₂=−1−𝜇_𝑗𝑄_1𝑖𝑗−𝐿₂−𝐿^𝑇₂+𝑀₂+𝑀^𝑇₂+𝜏_𝑗𝑌₂₂,Σ₂₃=−𝐿^𝑇₃−𝑀₂+𝑀^𝑇₃+𝜏_𝑗𝑌₂₃,Σ₂₄=−𝐿^𝑇₄+𝑀^𝑇₄+𝐴^𝑇_1𝑖𝑗𝐾^𝑇𝑁^𝑇+𝜏_𝑗𝑌₂₄,Σ₂₅=−𝐿^𝑇₅+𝑀^𝑇₅+𝜏_𝑗𝑌₂₅,Σ₃₃=−𝑄_2𝑖𝑗−𝑀₃−𝑀^𝑇₃+𝜏_𝑗𝑌₃₃,Σ₃₄=−𝑀^𝑇₄+𝜏_𝑗𝑌₃₄,Σ₃₅=−𝑀^𝑇₅+𝜏_𝑗𝑌₃₅,Σ₄₄=𝜏𝑍−𝑁−𝑁^𝑇+𝜏_𝑗𝑌₄₄,Σ₄₅=𝑁𝐾𝐴_2𝑖𝑗+𝜏_𝑗𝑌₄₅,Σ₅₅=−1−𝜇_𝑗𝑅_1𝑖𝑗+𝜏_𝑗𝑌₅₅,Φ_1𝑖𝑗=𝐵^𝑇_𝑖𝑗𝑃_𝑖𝑗00𝐵^𝑇_𝑖𝑗𝐾^𝑇𝑁^𝑇0^𝑇. (3.14) Therefore, we have the following result for the 𝐻_∞ performance analysis.

Theorem 3.1. Given scalars 𝜏, 𝜏_𝑗, and 𝜇_𝑗, the fault detection system (2.6) is stochastically stable with an 𝐻_∞ performance 𝛾 for any time delay 𝜏_𝑗(𝑡) satisfying (2.3), if there exist matrices 𝑃_𝑖𝑗>0, 𝑍>0, 𝑄_1𝑖𝑗>0, 𝑄_2𝑖𝑗>0, 𝑅_1𝑖𝑗>0, 𝑅₂>0, 𝑅₃>0, and matrices 𝐋, 𝐌, 𝐍, 𝐘 denoted in (3.10)–(3.11) such that for each 𝑖∈𝑆₁, 𝑗∈𝑆₂Σ_𝑖𝑗Φ_1𝑖𝑗Φ_2𝑖𝑗Φ_3𝑖𝑗∗−𝛾²𝐼00∗∗−𝐼0∗∗∗−𝑃_𝑖𝑗<0,_{𝑙∈𝑆1,𝑙≠𝑖}𝜋_𝑖𝑙𝑄_1𝑙𝑗+𝜏²_𝑗2𝑅_1𝑙𝑗+𝜋_𝑖𝑖𝑄_1𝑖𝑗+𝜆_𝑗𝑗𝑄_1𝑖𝑗<0,_{𝑘∈𝑆2,𝑘≠𝑗}𝜆_𝑗𝑘𝑄_1𝑖𝑘+𝑄_2𝑖𝑘+𝜏²_𝑘2𝑅_1𝑖𝑘−𝑅₂<0,_𝑙∈𝑆1𝜋_𝑖𝑙𝑄_2𝑙𝑗+𝜆_𝑗𝑗𝑄_2𝑖𝑗𝑅<0,_1𝑖𝑗−𝑅₃<0,𝐘𝐋∗𝑍⩾0,𝐘𝐌∗𝑍⩾0, (3.15)(3.16)(3.17) where Σ_𝑖𝑗=Σ₁₁−𝐾^𝑇𝐺^𝑇_𝑖𝑗𝑃_𝑖𝑗𝐺_𝑖𝑗𝐾Σ₁₂Σ₁₃Σ₁₄Σ₁₅∗Σ₂₂Σ₂₃Σ₂₄Σ₂₅∗∗Σ₃₃Σ₃₄Σ₃₅∗∗∗Σ₄₄Σ₄₅∗∗∗∗Σ₅₅,Φ_2𝑖𝑗=𝐶_𝑖𝑗0000^𝑇,Φ_3𝑖𝑗=𝑃_𝑖𝑗𝐺_𝑖𝑗𝐾0000^𝑇. (3.18)

Proof. Using Schur complement formula to (3.15), it can be seen that (3.15) is equivalent to Σ_𝑖𝑗+Φ_2𝑖𝑗Φ^𝑇_2𝑖𝑗Φ_1𝑖𝑗∗−𝛾²𝐼<0. (3.19) Now, we show that the filtering error system (2.6) with 𝑤(𝑡)=0 is stochastically stable. If 𝑤(𝑡)=0, from (3.13) and (3.16)–(3.17), we can obtain 𝐸ℒ𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗⩽𝐸𝜉^𝑇(𝑡)Σ_𝑖𝑗𝜉(𝑡). (3.20) Inequality (3.20) implies that Σ_𝑖𝑗<0. Thus, we have ℒ𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗⩽−𝛼₁𝜉^𝑇(𝑡)𝜉(𝑡), (3.21) where 𝛼₁=min_{𝑖∈𝑆1,𝑗∈𝑆2}{𝜆_min(−Σ_𝑖𝑗)}>0. Therefore, for any 𝑇>0, by Dynkin’s formula, we have ^𝑇₀𝐸𝜉^𝑇(𝑠)𝜉(𝑠)d𝑠⩽𝛼₁⁻¹𝑉𝜙(0),𝑟₀,𝑠₀, (3.22) which means that lim_𝑡→∞𝐸{|𝜉(𝑡)|²}=0. Thus, the filtering error system (2.6) with 𝑤(𝑡)=0 is stochastically stable by Definition 2.5.
In the sequel, we will deal with the 𝐻_∞ performance of the filtering error system (2.6). Using (3.19) and 𝐻_∞ performance, we have 𝐸ℒ𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗+𝑟^𝑇_𝑒(𝑡)𝑟_𝑒(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡⩽𝜂)^𝑗(𝑡)𝑤(𝑡)^𝑇Σ_𝑖𝑗+Φ_2𝑖𝑗Φ^𝑇_2𝑖𝑗Φ_1𝑖𝑗∗−𝛾²𝐼𝜂_𝑗(𝑡)𝑤(𝑡)<0. (3.23) Noting that the zero initial condition, then it follows from (3.23) that 𝐽_𝐻=𝐸^∞₀𝑟^𝑇_𝑒(𝑡)𝑟_𝑒(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)d𝑡⩽𝐸^∞₀𝑟^𝑇_𝑒(𝑡)𝑟_𝑒(𝑡)−𝛾²𝑤^𝑇(𝑡)𝑤(𝑡)+ℒ𝑉𝜉,𝜉_𝑡,𝑡,𝑖,𝑗d𝑡<0. (3.24) Hence, if (3.15)–(3.17) hold, 𝐽_𝐻<0 can be guaranteed. That is, ‖𝑟_𝑒‖₂⩽𝛾‖𝑤‖₂ for all nonzero 𝑤(𝑡). Therefore, the filtering error system (2.6) is stochastically stable with the 𝐻_∞ performance 𝛾 by Definition 2.6. This completes the proof.

Remark 3.2. Theorem 3.1 presents a new stochastic stability criterion by employing a novel mixed mode-dependent Lyapunov functional. The Lyapunov functional in this paper uses all information about 𝑟_𝑡, 𝑠_𝑡, and 𝜏(𝑡,𝑠_𝑡). Also, the Lyapunov matrices 𝑃(𝑟_𝑡,𝑠_𝑡), 𝑄₁(𝑟_𝑡,𝑠_𝑡), 𝑄₂(𝑟_𝑡,𝑠_𝑡), and 𝑅₁(𝑟_𝑡,𝑠_𝑡) depend on both the system mode 𝑟_𝑡 and the delay mode 𝑠_𝑡. Hence, the Lyapunov functional in this paper is more general, and the condition on stability is more applicable. In the most published papers about Markovian jump systems with mixed time delays, the authors choose the mode-independent Lyapunov matrices which may lead to some conservativeness, such as [15, 17, 23, 31, 33, 37–39], to name a few among many important results in the literature. But, the selected mode-dependent Lyapunov matrices in this paper can reduce some conservativeness because they allow more freedom in choosing feasible solutions of LMIs.

Remark 3.3. In Theorem 3.1, 𝜇_𝑗<1 can be extended to a wider range 𝜇_𝑗<∞ by dealing with the integral term 𝜆_𝑗𝑗∫^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉(𝑠)d𝑠 in (3.5). Noting that 𝜆_𝑗𝑗<0, utilizing Lemma 2.4, one has 𝜆_𝑗𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑗(𝑠)𝜆𝐾𝜉d𝑠⩽^𝑗𝑗𝜏_𝑗^𝑡_{𝑡−𝜏𝑗}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑄_1𝑖𝑗^𝑡_{𝑡−𝜏𝑗(𝑡)}𝐾𝜉(𝑠)d𝑠. (3.25) Further, deleting 𝜆_𝑗𝑗𝑄_1𝑖𝑗 in (3.16) and adding (𝜆_𝑗𝑗/𝜏_𝑗)𝑄_1𝑖𝑗 to Σ₅₅ in (3.15), we can obtain a more general stability condition.

Based on Theorem 3.1, the fault detection filter synthesis problem can be developed in terms of LMIs for the system (2.1) with different system and delay modes.

Theorem 3.4. Consider the system (2.1). Given scalars 𝜏, 𝜏_𝑗, and 𝜇_𝑗, the fault detection system (2.6) is stochastically stable with an 𝐻_∞ performance 𝛾 for any time delay 𝜏_𝑗(𝑡) satisfying (2.3) if there exist matrices 𝑉_𝑖𝑗>0, 𝑊_𝑖𝑗>0, 𝑈_𝑖𝑗>0, 𝑍>0, 𝑄_1𝑖𝑗>0, 𝑄_2𝑖𝑗>0, 𝑅_1𝑖𝑗>0, 𝑅₂>0, 𝑅₃>0, 𝐴_𝑓𝑖𝑗, 𝐵_𝑓𝑖𝑗, 𝐶_𝑓𝑖𝑗 and matrices 𝐋, 𝐌, 𝐍, 𝐘 denoted in (3.10)–(3.11) such that for each 𝑖∈𝑆₁, 𝑗∈𝑆₂Σ_𝑖𝑗Φ_1𝑖𝑗Φ_2𝑖𝑗Φ_3𝑖𝑗∗−𝛾²𝐼00∗∗−𝐼0∗∗∗−Φ_4𝑖𝑗<0,_{𝑙∈𝑆1,𝑙≠𝑖}𝜋_𝑖𝑙𝑄_1𝑙𝑗+𝜏²_𝑗2𝑅_1𝑙𝑗+𝜋_𝑖𝑖𝑄_1𝑖𝑗+𝜆_𝑗𝑗𝑄_1𝑖𝑗<0,_{𝑘∈𝑆2,𝑘≠𝑗}𝜆_𝑗𝑘𝑄_1𝑖𝑘+𝑄_2𝑖𝑘+𝜏²_𝑘2𝑅_1𝑖𝑘−𝑅₂<0,_𝑙∈𝑆1𝜋_𝑖𝑙𝑄_2𝑙𝑗+𝜆_𝑗𝑗𝑄_2𝑖𝑗𝑅<0,_1𝑖𝑗−𝑅₃<0,𝐘𝐋∗𝑍⩾0,𝐘𝐌∗𝑍⩾0, (3.26) where Σ_𝑖𝑗=Σ₁₁Σ₁₂0Σ₁₄Σ₁₅Σ₁₆Σ₁₇∗Σ₂₂0Σ₂₄00Σ₂₇∗∗Σ₃₃0000∗∗∗Σ₄₄Σ₄₅Σ₄₆Σ₄₇∗∗∗∗Σ₅₅Σ₅₆Σ₅₇∗∗∗∗∗Σ₆₆Σ₆₇∗∗∗∗∗∗Σ₇₇,Φ_1𝑖𝑗=𝑉_𝑖𝑗𝐵_0𝑖𝑉_𝑖𝑗𝐵_1𝑖𝑉_𝑖𝑗𝐵_2𝑖0𝐵_𝑓𝑖𝑗𝐷_1𝑖𝐵_𝑓𝑖𝑗𝐷_2𝑖00𝑈_𝑖𝑗𝐵_𝜔000000𝑁𝐵_0𝑖𝑁𝐵_0𝑖𝑁𝐵_0𝑖,000Φ_2𝑖𝑗=0𝐶^𝑇_𝑓𝑖𝑗−𝐶^𝑇_𝜔0000,Φ_3𝑖𝑗=𝐺^𝑇_1𝑖𝑉^𝑇_𝑖𝑗𝐺^𝑇_2𝑖𝐵^𝑇_𝑓𝑖𝑗0,000000000000000000Φ_4𝑖𝑗=𝑉_𝑖𝑗00∗𝑊_𝑖𝑗0∗∗𝑈_𝑖𝑗,Σ₁₁=𝑉_𝑖𝑗𝐴_𝑖+𝐴^𝑇_𝑖𝑉_𝑖𝑗+_𝑙∈𝑆1𝜋_𝑖𝑙𝑉_𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑉_𝑖𝑘+𝑄_1𝑖𝑗+𝑄_2𝑖𝑗+𝜏𝑅₂+12𝜏²𝑅₃+12𝜏²_𝑗𝑅_1𝑖𝑗+𝐿₁+𝐿^𝑇₁+𝜏_𝑗𝑌₁₁,Σ₁₂=𝐶^𝑇_𝑖𝐵^𝑇_𝑓𝑖𝑗,Σ₁₄=𝑉_𝑖𝑗𝐴_1𝑖−𝐿₁+𝑀₁+𝐿^𝑇₂+𝜏_𝑗𝑌₁₂,Σ₁₅=𝐿^𝑇₃−𝑀₁+𝜏_𝑗𝑌₁₃,Σ₁₆=𝐿^𝑇₄+𝜏_𝑗𝑌₁₄+𝐴^𝑇_𝑖𝑁^𝑇,Σ₁₇=𝑉_𝑖𝑗𝐴_2𝑖+𝐿^𝑇₅+𝜏_𝑗𝑌₁₅,Σ₂₂=𝐴_𝑓𝑖𝑗+𝐴^𝑇_𝑓𝑖𝑗+_𝑙∈𝑆1𝜋_𝑖𝑙𝑊_𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑊_𝑖𝑘,Σ₂₄=𝐵_𝑓𝑖𝑗𝐶_1𝑖,Σ₂₇=𝐵_𝑓𝑖𝑗𝐶_2𝑖,Σ₃₃=𝑈_𝑖𝑗𝐴_𝜔+𝐴^𝑇_𝜔𝑈_𝑖𝑗+_𝑙∈𝑆1𝜋_𝑖𝑙𝑈_𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑈_𝑖𝑘,Σ₄₄=−1−𝜇_𝑗𝑄_1𝑖𝑗−𝐿₂−𝐿^𝑇₂+𝑀₂+𝑀^𝑇₂+𝜏_𝑗𝑌₂₂,Σ₄₅=−𝐿^𝑇₃−𝑀₂+𝑀^𝑇₃+𝜏_𝑗𝑌₂₃,Σ₄₆=−𝐿^𝑇₄+𝑀^𝑇₄+𝜏_𝑗𝑌₂₄+𝐴^𝑇_1𝑖𝑁^𝑇,Σ₄₇=−𝐿^𝑇₅+𝑀^𝑇₅+𝜏_𝑗𝑌₂₅,Σ₅₅=−𝑄_2𝑖𝑗−𝑀₃−𝑀^𝑇₃+𝜏_𝑗𝑌₃₃,Σ₅₆=−𝑀^𝑇₄+𝜏_𝑗𝑌₃₄,Σ₅₇=−𝑀^𝑇₅+𝜏_𝑗𝑌₃₅,Σ₆₆=𝜏𝑍−𝑁^𝑇−𝑁+𝜏_𝑗𝑌₄₄,Σ₆₇=𝑁𝐴_2𝑖+𝜏_𝑗𝑌₄₅,Σ₇₇=−1−𝜇_𝑗𝑅_1𝑖𝑗+𝜏_𝑗𝑌₅₅. (3.27) In this case, the parameters of the desired fault detection filter can be chosen by 𝐴_𝑓𝑖𝑗=𝑊⁻¹_𝑖𝑗𝐴_𝑓𝑖𝑗,𝐵_𝑓𝑖𝑗=𝑊⁻¹_𝑖𝑗𝐵_𝑓𝑖𝑗,𝐶_𝑓𝑖𝑗=𝐶_𝑓𝑖𝑗. (3.28)

Proof. For each 𝑟_𝑡=𝑖∈𝑆₁, 𝑠_𝑡=𝑗∈𝑆₂, we define a matrix 𝑃_𝑖𝑗>0 by 𝑃_𝑖𝑗=diag[𝑉_𝑖𝑗𝑊_𝑖𝑗𝑈_𝑖𝑗]. Then, with the parameters in (3.28), it can be verified that, for each 𝑖∈𝑆₁, 𝑗∈𝑆₂, the LMI (3.26) can be rewritten as (3.15). Then, we can obtain the results in Theorem 3.1. This completes the proof.

Remark 3.5. Noting that the first diagonal element Σ₁₁ in (3.15) includes 𝑃_𝑖𝑗, 𝑃_𝑙𝑗, and 𝑃_𝑖𝑘. Owing to the restrictions on the authors’ knowledge and the technique difficulties, 𝑃_𝑖𝑗 is assumed to be diagonal matrices to obtain the parameters of the fault detection filter. Although this assumption may cause some conservativeness, considering complete 𝑃_𝑖𝑗 results in bilinear matrix inequalities and not LMIs which are more conservative.

4. A Numerical Example

In this section, a numerical example will be presented to show the validity of the main results derived above.

Example 4.1. Let us consider the stochastic system (2.1) with the following system of matrices:
𝐴(1)=−1000.6−12,𝐴₁(1)=−10.32−1,𝐴₂(1)=0.5𝐼,𝐵₀(1)=,𝐵0.10.2₁(1)=0.70.1,𝐵₂(1)=00.6,𝐺₁(1)(1)=,𝐶=𝐼,𝐶22.1₁(1)=1.50,𝐶₂(1)=0.10.1,𝐷₁(1)=0.1,𝐷₂(1)𝐺=0.2,₂(1)=,𝐴(2)=−0.5−0.5−121−2−14.3,𝐴₁(2)=−11.30.7−1.1,𝐴₂(2)=0.1𝐼,𝐵₀(2)=,𝐵0.10.2₁(2)=0.10.3,𝐵₂(2)=00.3,𝐺₁(2)=(2)=,𝐶1011,𝐶22₁(2)=1.20.7,𝐶₂(2)=0.10.1,𝐷₁(2)=0.2,𝐷₂(2)𝐺=0.2,₂(2)=−0.5−0.5. (4.1) The transition probability matrix is considered as Π=−553−3,Λ=−0.60.60.5−0.5. (4.2) In this example, the weighting matrix 𝑊(𝑠) in 𝐹_𝜔(𝑠)=𝑊(𝑠)𝐹(𝑠) is supposed to be 𝑊(𝑠)=5/(𝑠+5). Its state-space realization is given as (2.5) with 𝐴_𝜔=−5, 𝐵_𝜔=5 and 𝐶_𝜔=1. Also, we assume that 𝜏₁=0.6, 𝜏₂=0.4, 𝜇₁=0.4, 𝜇₂=0.3. For 𝛾=2.0, by the Theorem 3.4 in this paper, the filter matrices are obtained as 𝐴_𝑓11=−3.17241.1747−1.1416−0.5011,𝐵_𝑓11=,𝐶0.6403−0.0123_𝑓11=,𝐴4.3588−0.1614_𝑓12=−2.3595−0.9045−4.1454−3.4820,𝐵_𝑓12=,𝐶0.51990.8936_𝑓12=,𝐴5.53963.8431_𝑓21=−0.36480.0560−0.1076−2.8163,𝐵_𝑓21=,𝐶0.01611.0228_𝑓21=,𝐴−0.00774.4208_𝑓22=−3.41330.02100.0135−0.5002,𝐵_𝑓22=,𝐶0.6410−0.0114_𝑓22=4.8422−0.0192. (4.3) For simulation purposes, we assume the initial condition 𝑥(0)=[0.6−0.6]^𝑇. The time delays are 𝜏₁(𝑡)=0.2+0.4sin(𝑡), 𝜏₂(𝑡)=0.1+0.3cos(𝑡). The control input 𝑢(𝑡) is chosen to be sin(𝑡)𝑒^−2𝑡. The unknown input 𝜈(𝑡)(𝑡∈[030]) is assumed to be the band-limited white noise. The fault signal 𝑓(𝑡) is simulated as a square wave signal with unit amplitude that occurred from the 10 s to 20 s. Figures 1–5 illustrate the simulation results. The possible realizations of the Markovian jumping modes of system and delay are plotted in Figures 1 and 2, respectively, where the initial modes are assumed to be 𝑟₀=1 and 𝑠₀=1. Figure 3 shows the unknown input 𝜈(𝑡). Figure 4 shows the residual signal. Figure 5 is the simulation results of the evaluation function 𝑓(𝑟). Under the above conditions, with a selected threshold 𝐽th=0.1486, the simulation of evaluation function 𝑓(𝑟) with fault shows that ∫₀^10.9𝑟^𝑇(𝑡)𝑟(𝑡)d𝑡=0.1492>𝐽th. Thus, the appeared fault can be detected after 0.9 s. The simulation results demonstrate that the designed fault detection filter is feasible and effective.

Figure 1 

System jumping mode.

Figure 2 

Delay jumping mode.

Figure 3 

The unknown input 𝜈(𝑡).

Figure 4 

Generated residual signal 𝑟(𝑡).

Figure 5 

Evolution of residual evaluation function 𝑓(𝑟).

Remark 4.2. In this study, the fault signal 𝑓(𝑡) is assumed to be a square wave signal that occurred from the 10 s to 20 s. Figure 4 shows that the generated residual signal is sensitive to the fault and possesses robustness to exogenous disturbance. Furthermore, if no less than one fault appears in the systems, the designed filter is also effective to estimate fault.

5. Conclusion

The problem of fault detection for a class of stochastic MJS is investigated in this paper. Different system mode and delay mode are considered in the model. By using the Lyapunov functional, a mixed mode-dependent sufficient condition is developed to design the stable filter. A numerical example demonstrates the effectiveness of the given method.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (103.1.2E022050205) and the National Natural Science Foundation of China (Grant no. 61004046).