Fault Detection Filter Design for Stochastic Systems with Mixed Time-Delays and Parameter Uncertainties

Abstract

This paper purposes the design of a fault detection filter for stochastic systems with mixed time-delays and parameter uncertainties. The main idea is to construct some new Lyapunov functional for the fault detection dynamics. A new robustly asymptotically stable criterion for the systems is derived through linear matrix inequality (LMI) by introducing a comprehensive different Lyapunov-Krasovskii functional. Then, the fault detection filter is designed in terms of linear matrix inequalities (LMIs) which can be easily checked in practice. At the same time, the error between the residual signal and the fault signal is made as small as possible. Finally, an example is given to illustrate the effectiveness and advantages of the proposed results.

1. Introduction

Stochastic systems have strong practical relevance in mechanical systems, economics, systems with human operators, and other engineering areas. Meanwhile, the filtering problem has many applications in the areas of signal processing, signal estimation, pattern recognition, and many practical control systems. Therefore, the filter design problems and the stochastic systems have become important areas of research and received great attention over the past few years [1–6].

With the rising demand for higher safety and increasing demand for higher performance in the modern industries, the research on fault detection for dynamic systems has received more and more attention during the past two decades, like model-based schemes [7], knowledge-based approaches [8], and signal-based methods [9], and so forth. Particularly, the problem of model-based fault detection has been an active research area among them. The basic objective of model-based fault detection is to construct the residual generator, and to determine the residual evaluation function and the threshold. An alarm of fault will be generated, when the value of the evaluation function is greater than the stated threshold. However, randomly occurring nonlinearities and the existence of unknown inputs may seriously affect the performance of model-based fault detection systems. Therefore, robustness issue plays an important role in the application of model-based fault detection schemes. Increasing the robustness of residual to unknown inputs and modelling errors and enhancing the sensitivity to faults are of prime importance in designing a model-based fault detection system. Among different methods for fault detection, the model-based approach has been widely used in recent years. So far, the problem of fault detection has been thoroughly investigated for a variety of systems including uncertain systems [2, 10–14], time-delay systems [15–17], and Markovian jump linear systems [18].

Besides, time-delay is one of the major sources of instability and poor performance of a practical control system. And many results on stochastic systems with time delays have been reported in the literature [10–12] and the references therein. There is a need to discuss the distributed delays that occur very often in practical systems. The engineering significance of distributed delays has been widely recognized, and a number of corresponding results have been published; see, for example, [15]. The distributed delays in the discrete-time setting, on the other hand, have received little attention. It is well known that nonlinearities exist universally in practical systems, and therefore nonlinear control has been an ever hot topic in the past few decades.

As far as we know, the delay-dependent criteria on fault detection filter design for delayed stochastic systems with parameter uncertainties have not been fully studied, which is still open. Motivated by the above discussion and in order to obtain less conservative results, we choose an appropriate new Lyapunov functional and establish a new integral inequality in the stochastic setting. What is more, because we have carefully considered the ranges for the time-varying delays, our criteria can be applicable to both fast and slow time-varying delays. The stability criteria obtained are in terms of linear matrix inequalities (LMIs) which can be checked efficiently via the LMI toolbox. Finally, a numerical example is also given to demonstrate the effectiveness and advantages of our theoretical results.

Notations. The notations used throughout the paper are fairly standard. The superscript “𝑇” stands for matrix transposition; 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space; 𝑅^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices; the notation 𝑃>0 means that 𝑃 is a positive definite matrix; 𝐼 and 0 represent identity matrix and zero matrix, respectively; and diag(⋅) denotes the diagonal matrix. The vector norm is taken to be Euclidean, and the matrix norm is the corresponding induced one. 𝐸{⋅} denotes the expectation operator. ℒ₂[0,+∞) is the space of square-integrable vector function over [0,+∞). In symmetric block matrices, we use an asterisk (∗) to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation and Some Preliminaries

The simplest and most fundamental case considered is the problem of quadratic stabilization for the following system: 𝑥()(𝑘)(𝑘)𝑦(𝑘)(𝑘),𝑘+1=𝐴𝑥+𝐵𝑢=𝐶𝑥 (1) where 𝑥(𝑘)∈𝑅^𝑛 is the state and 𝑢(𝑘) is the control input. Then, we add discrete time-delays 𝜏_𝑖(𝑘), (𝑖=1,2,…,𝑞), while 𝑑(𝑑=1,2,…,∞) describe the distributed time-delays. In addition, we are motivated to study the fault detection problem for a class of discrete-time systems involving stochastic time-delay, such that we consider the following stochastic time-delay system: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴₁^𝑞_𝑖=1𝛼_𝑖(𝑘)𝑥𝑘−𝜏_𝑖(𝑘(𝑘)𝐴)+𝛽₂^∞_𝑑=1𝜇_𝑑𝑥(𝑘−𝑑)+𝛾(𝑘)𝑔(𝑘,𝑥(𝑘))+𝐷₁𝑤(𝑘)(𝑘)𝑦(𝑘)(𝑘)+𝐺𝑓=𝐶𝑥+𝐷₂𝑤(𝑘)(𝑘)𝑥(𝑘)(𝑘)+𝐻𝑓=𝜙,∀𝑘∈ℤ⁻, (2) where 𝑥(𝑘)∈𝑅^𝑛 is the state; 𝑦(𝑘)∈𝑅^𝑚 is the process output; 𝑤(𝑘)∈𝑅^𝑝 is the unknown input which belongs to ℒ₂[0,∞); 𝑓(𝑘)∈𝑅^𝑙 is the fault to be detected; 𝐴, 𝐴₁, 𝐴₂, 𝐶, 𝐷₁, 𝐷₂, and 𝐺 are real known constant matrices with compatible dimensions. The discrete time-delay 𝜏_𝑖(𝑘)(𝑖=1,2,…,𝑛) is a time-varying differentiable function satisfying 𝜏_𝑚≤𝜏_𝑖(𝑘)≤𝜏_𝑀, while 𝑑(𝑑=1,2,…,∞) denotes the discrete time-delays, and the constant 𝜇_𝑑≥0 satisfies the following convergence conditions 𝜇∶=∑^∞_𝑑=1𝜇_𝑑≤∑^∞_𝑑=1𝑑𝜇_𝑑<∞, 𝜇∶=∑^∞_𝑑=1𝜇²_𝑑<∞.

Assumption 1. The nonlinear function 𝑔(𝑘,𝑥(𝑘))∈𝑅^𝑛 satisfies the following condition: for any 𝑥, 𝑦∈𝑅, (𝑥≠𝑦), 𝑔⁻_𝑖≤𝑔_𝑖(𝑥)−𝑔_𝑖𝑦≤𝑔𝑥−𝑦^+𝑖, (3) where 𝑔⁻_𝑖, 𝑔⁺_𝑖(𝑖=1,2,…,𝑛) are constants.

Assumption 2. The stochastic variables 𝛼_𝑖(𝑘)(𝑖=1,2,…𝑛₁), 𝛽(𝑘), and 𝛾(𝑘) and mutually uncorrelated Bernoulli distributed white sequences that account for, respectively, the phenomena of randomly occurring discrete time-delays, distribute time-delays, and nonlinearities. A natural assumption on the sequences 𝛼_𝑖(𝑘)(𝑖=1,2,…𝑛₁), 𝛽(𝑘), and 𝛾(𝑘) is made as follows: Prob𝛼_𝑖(𝑘)=1=𝔼𝛼_𝑖(𝑘)=𝛼_𝑖,Prob𝛼_𝑖(𝑘)=0=1−𝛼_𝑖,Prob𝛽(𝑘)=1=𝔼𝛽(𝑘)=𝛽(𝑘)𝛽,Prob=0=1−𝛾(𝑘)𝛾(𝑘=𝛽,Prob=1=𝔼)𝛾(𝑘)𝛾,Prob=0=1−𝛾. (4)

In a networked environment, it is quite common that the measurements 𝑦(𝑘) of the system are quantized during the signal transmission. Let us denote the quantizer as ℎ(⋅)=[ℎ₁(⋅),ℎ₂(⋅),…,ℎ_𝑚(⋅)]^𝑇 which is symmetric, that is, ℎ_𝑗(−𝜈)=−ℎ_𝑗(𝜈), 𝑗=1,…,𝑚.

The map of the quantization process is 𝑦(𝑘)=ℎ(𝑦(𝑘))=[ℎ₁(𝑦⁽¹⁾(𝑘)),ℎ₂(𝑦⁽²⁾(𝑘)),…,ℎ_𝑚(𝑦^(𝑚)(𝑘))]^𝑇. In this paper, we are interested in the logarithmic static and time-invariant quantizer. For each, the set of quantization levels is described by 𝒰_𝑗={±𝜇_𝑖^(𝑗),𝜇_𝑖^(𝑗)=𝜒^𝑗_𝑗𝜇₀^(𝑗),𝑖=0,±1,±2,…}∪{0}, 0<𝜒_𝑗<1, 𝜇₀^(𝑗)>0, and each of the quantization levels corresponds to a segment, such that the quantizer maps the whole segment to this quantization level. According to [19], the quantizer is given by ℎ_𝑗𝑦^(𝑗)(𝑘)=𝜇_𝑖^(𝑗),11+𝛿_𝑗𝜇_𝑖^(𝑗)⩽𝑦^(𝑗)(𝑘)⩽11−𝛿_𝑗𝜇_𝑖^(𝑗)0,𝑦^(𝑗)(𝑘)=0−ℎ_𝑗−𝑦^(𝑗)(𝑘),𝑦^(𝑗)(𝑘)<0, (5) where 𝛿_𝑗=(1−𝜒_𝑗)(1+𝜒_𝑗), such that ℎ_𝑗(𝑦^(𝑗)(𝑘))=(𝐼+Δ_𝑘)𝑦^(𝑗)(𝑘) with |Δ_𝑘^(𝑗)|⩽𝛿_𝑗; the measurements with quantization effect can be expressed as 𝑦(𝑘)=𝐼+Δ_𝑘𝑦(𝑘)=𝐼+Δ_𝑘𝐶𝑥(𝑘)+𝐼+Δ_𝑘𝐷₂𝜔(𝑘)+𝐼+Δ_𝑘𝐻𝑓(𝑘), (6) with Δ_𝑘=diag(Δ_𝑘⁽¹⁾,…,Δ_𝑘^(𝑚)).

Consider the following full-order fault detection filter for system (2): 𝑥(𝑘+1)=𝐴_𝑓𝑥(𝑘)+𝐵_𝑓𝑦(𝑘),𝑟(𝑘)=𝐶_𝑓𝑥(𝑘)+𝐷_𝑓𝑦(𝑘), (7) where 𝑥(𝑘)∈𝑅^𝑛 is the filter state vector, 𝑟(𝑘)∈ℝ^𝑙 is the so called residual that is compatible with the fault vector 𝑓(𝑘), and 𝐴_𝑓, 𝐵_𝑓, and 𝐶_𝑓 are appropriately dimensioned filter matrices to be designed.

From (2), (5), and (7), we can obtain the following filtering error system: 𝑥()=𝑥(𝑘)+𝑘+1𝐴+Δ𝐴_𝑞𝑖=1𝐴_1𝑖+𝐴_1𝑖(𝑘)𝑥𝑘−𝜏_𝑖(𝑘)+𝐴₂+𝐴₂(𝑘)^∞_𝑑=1𝜇_𝑑𝑥()+𝑘−𝑑𝛾(𝑘)̃((𝑘+𝑣(𝑘)𝑟̃(𝑘)=𝑥(𝑘)+𝐷𝛾+𝐼𝑔𝑘,𝑥))𝐷+Δ𝐷𝐶+Δ𝐶_𝑓+Δ𝐷_𝑓𝑣(𝑘)𝑥(𝑘)=𝜙(𝑘),∀𝑘∈−𝜏_𝑀,0, (8) with 𝑥(𝑘)=𝑥^𝑇(𝑘),𝑥^𝑇(𝑘)^𝑇,𝜙^𝑇(𝑘)=𝜙^𝑇(𝑘),0^𝑇,𝑟̃(𝑘)=𝑟(𝑘)−𝑓(𝑘),𝑣(𝑘)=𝜔^𝑇(𝑘),𝑓^𝑇(𝑘)^𝑇,𝐵𝐴=𝐴0_𝑓𝐶𝐴_𝑓𝐵,Δ𝐴=00_𝑓Δ_𝑘,𝐴𝐶0_1𝑖=𝛼_𝑖𝐴₁0,𝐴00^1𝑖(𝑘)=𝛼_𝑖(𝑘)𝐴₁0,𝐴00₂=𝛽𝐴₂0,𝐴00²(𝑘)=𝛽(𝑘)𝐴₂0,̃𝐼0,𝐷00𝐼=𝐷=¹𝐺𝐵_𝑓𝐷₂𝐵_𝑓𝐻,𝐵Δ𝐷=00_𝑓Δ_𝑘𝐷₂𝐵_𝑓Δ_𝑘𝐻,𝐶=𝐷_𝑓𝐶𝐶_𝑓,Δ𝐶=𝐷_𝑓Δ_𝑘𝐶0,𝐷_𝑓=𝐷_𝑓𝐷₂𝐷_𝑓𝐻−𝐼,Δ𝐷_𝑓=𝐷_𝑓Δ_𝑘𝐷₂𝐷_𝑓Δ_𝑘𝐻,𝛼_𝑖(𝑘)=𝛼_𝑖(𝑘)−𝛼_𝑖,𝛽(𝑘)=𝛽(𝑘)−𝛽,𝛾(𝑘)=𝛾(𝑘)−𝛾. (9)

In order to conduct the stability analysis for the above systems, it is necessary to make the following definitions and lemmas.

Definition 3. Consider the system (8) with 𝑣(𝑘)=0; it is said to be mean-square exponentially stable if for any initial conditions, there exist a 𝛿>0 and 0<𝛼<1, such that 𝐸{|𝑥(𝑘)|²}≤𝛿𝛼^𝑘sup_𝑖∈𝑍⁻𝐸{|𝜙(𝑖)|²}, 𝑘≥0. Moreover, if lim_𝑡→∞𝐸{|𝑥(𝑘)|²}=0, then the system is said to be robust asymptotically mean-square stable.

Definition 4. Given a scalar 𝛾>0, the error system (8) is said to be mean-square robustly exponentially stable with the attenuation level 𝛾 if the error system (8) with 𝑣(𝑘)=0 is mean-square robustly exponentially stable, and under zero initial condition, the following is true for all nonzero 𝑣(𝑘): ^+∞_𝑘=0𝐸𝑟^𝑇̃(𝑘)𝑟̃(𝑘)≤𝛾^2∞_𝑘=0𝐸𝑣^𝑇(𝑘)𝑣(𝑘‖𝑒‖)i.e.,_𝐸∞≤𝛾‖𝑣‖₂,∀𝑤(𝑘)∈ℒ₂[0,+∞). (10) Correspondingly, system (7) is said to be stochastic ℒ₂−ℒ_∞ filter of system (2).

We further adopt a residual evaluation stage including an evaluation function 𝐽(𝑘) and a threshold 𝐽_th of the following form: 𝐽(𝑘)=∑^𝑘_ℎ=0𝑟(ℎ)𝑟(ℎ)^1/2 where 𝐿 denotes the maximum time step of the evaluation function. Based on (16), the occurrence of faults can be detected by comparing 𝐽(𝑘) with 𝐽_th according to the following rule: 𝐽(𝑘)>𝐽_th (with faults) alarm; 𝐽(𝑘)<=𝐽_th no faults.

Remark 5. The parametric uncertainties Δ𝐴, Δ𝐷, Δ𝐶, and Δ𝐷_𝑓 satisfy Δ𝐴Δ𝐷Δ𝐶Δ𝐷_𝑓=0𝐵_𝑓𝐷_𝑓𝐹(𝑘)Λ𝐶,0,Λ𝐷₂,Λ𝐻, (11) where Λ=diag{𝛿₁,𝛿₂,…,𝛿_𝑚} and 𝐹(𝑘)=Δ_𝑘Λ⁻, and 𝐹(𝑘) satisfies 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼.

Lemma 6. Let 𝑀, 𝑁, and 𝐹 be real matrices of appropriate dimensions with 𝐹^𝑇𝐹≤𝐼. Then, for any scalar 𝜖>0, one has 𝑀𝐹𝑁+(𝑀𝐹𝑁)^𝑇≤𝜖⁻¹𝑀𝑀^𝑇+𝜖𝑁𝑁^𝑇. (12)

Lemma 7 (Schur complement). Given constant matrices 𝑆₁, 𝑆₂, and 𝑆₃, where 𝑆₁=𝑆^𝑇₁ and 𝑆₂=𝑆^𝑇₂, then 𝑆₁+𝑆^𝑇₃𝑆₂⁻¹𝑆₃<0 if and only if 𝑆₁𝑆^𝑇₃𝑆₃−𝑆₂<0,𝑜𝑟−𝑆₂𝑆₃𝑆^𝑇₃𝑆₁<0. (13)

Lemma 8 (Liu et al. [5]). For any constant matrix 𝑀∈𝑅^𝑛×𝑛, 𝑀=𝑀^𝑇>0, 𝑥_𝑖∈𝑅^𝑛, and constants 𝑎_𝑖>0(𝑖=1,2,…). If the series concerned is convergent, then one has −^∞_𝑖=1𝑎_𝑖^∞_𝑖=1𝑎_𝑖𝑥^𝑇_𝑖𝑀𝑎_𝑖𝑥^𝑇_𝑖≤−^∞_𝑖=1𝑎_𝑖𝑥_𝑖^𝑇𝑀^∞_𝑖=1𝑎_𝑖𝑥_𝑖. (14)

3. Main Results

In this section, we will investigate a sufficient condition on the performance analysis for the filter error system (8). And then, the solution to the fault detection filter design for the system (2) is listed as follows.

Theorem 9. For nominal system of (8) with given filter parameters and the index 𝛾>0, the fault detection dynamics is robustly exponentially stable in mean square, if there exist constant 𝑙>0, positive matrices 𝑃, 𝑄, 𝑄_𝑖(𝑖=1,2,…,𝑞), and any matrices 𝑈 with appropriate dimensions, such that the following LMIs hold:

ΓΓ=¹¹0⋯00Γ₁₄0𝐶^𝑇𝐴^𝑇𝑃∗𝛼₁1−𝛼₁𝐴^𝑇₁𝑃𝐴₁−𝑄₁⋯00000𝐴^𝑇₁₁𝑃∗∗⋱⋮⋮⋮⋮⋮⋮∗∗∗𝛼_𝑞1−𝛼_𝑞𝐴^𝑇₁𝑃𝐴₁−𝑄_𝑞0000𝐴^𝑇_1𝑞𝑃𝑄∗∗∗∗−𝜇000𝐴^𝑇₂𝑃∗∗∗∗∗Γ₄₄00𝛾𝐼^𝑇̃𝑃∗∗∗∗∗∗Γ₅₅𝐷^𝑇_𝑓𝐷^𝑇𝑃Γ∗∗∗∗∗∗∗−𝐼0∗∗∗∗∗∗∗∗−𝑃<0,₁₁=−𝑃+^𝑞_𝑖=1𝜏_𝑀−𝜏_𝑚+1𝑄_𝑖+𝜇𝑄+𝜇𝛽1−𝛽𝐴^𝑇₂𝑃𝐴₂−𝑈𝐺₁0,Γ00₁₄=𝑈𝐺₂0,Γ₄₄=𝛾1−𝛾𝐼^𝑇̃𝑃𝐼̃−𝑈,Γ₅₅=−𝛾²𝐼,𝐴₁=𝐴₁0,𝐴00₂=𝐴₂000,𝐺₁=diag𝑔⁻₁𝑔⁺₁,…,𝑔⁻_𝑛𝑔⁺_𝑛,𝐺₂𝑔=diag⁻¹+𝑔⁺₁2𝑔,…,^−𝑛+𝑔⁺_𝑛2. (15)

Proof. For simplicity, let ℎ₁(𝑘)=𝐴𝑥(𝑘)+^𝑞_𝑖=1𝐴_1𝑖𝑥𝑘−𝜏_𝑖(𝑘+𝐴_2∞𝑑=1)𝜇_𝑑𝑥(𝑘−𝑑)+𝛾̃((𝑘+(𝑘)ℎ𝐼𝑔𝑘,𝑥))𝐷𝑣₂(𝑘)=^𝑞_𝑖=1𝐴_1𝑖(𝑘)𝑥𝑘−𝜏_𝑖(𝑘)+𝐴₂(𝑘)^∞_𝑑=1𝜇_𝑑𝑥(𝑘−𝑑)+𝛾(𝑘)𝐼𝑔̃(𝑘,𝑥(𝑘)). (16) Then, we have the following equation: 𝑥(𝑘+1)=ℎ₁(𝑘)+ℎ₂(𝑘). (17) For this system, we construct the following Lyapunov functional: 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘), (18) with 𝑉₁(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘),𝑉₂(𝑘)=^𝑞_𝑖=1^𝑘−1_{𝑗=𝑘−𝜏𝑖(𝑘)}𝑥^𝑇𝑗𝑄_𝑖𝑥𝑗𝑉₃(𝑘)=^𝑞_𝑖=1^−𝜏𝑚_{𝑗=−𝜏𝑀+1}^𝑘−1_{𝑙=𝑘+𝑗}𝑥^𝑇(𝑙)𝑄_𝑖𝑥(𝑙)𝑉₄(𝑘)=^∞_𝑑=1^𝑘−1_{𝑖=𝑘−𝑑}𝑥^𝑇(𝑖)𝜇_𝑑𝑄+𝜇²_𝑑𝛽1−𝛽𝐴^𝑇₂𝑃𝐴₂𝑥(𝑖). (19)
Taking the difference of the functional along the solution of the system, we obtain 𝔼{Δ𝑉(𝑘)}=⁴_𝑖=1𝔼Δ𝑉_𝑖(𝑘𝔼)Δ𝑉₁(𝑘)=𝔼𝔼𝑉₁(𝑘+1)∣𝑥_𝑘−𝑉₁(𝑘𝑥)=𝔼^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=ℎ^𝑇₁(𝑘)𝑃ℎ₁(𝑘)+𝔼2ℎ^𝑇₁(𝑘)𝑃ℎ₂(𝑘)+𝔼^𝑞𝑞_𝑖=1𝑗=1𝑥^𝑇𝑘−𝜏_𝑖(𝑘)×𝐴^𝑇_1𝑖(𝑘)𝑃𝐴_1𝑗(𝑘)𝑥𝑘−𝜏_𝑗(𝑘)+^∞_𝑑1=1^∞_𝑑2=1𝜇_𝑑1𝜇_𝑑2𝑥^𝑇𝑘−𝑑₁×𝐴^𝑇₂(𝑘)𝑃𝐴₂(𝑘)𝑥𝑘−𝑑₂+𝛾(𝑘)²𝑔^𝑇(𝑘,𝑥(𝑘))𝐼^𝑇̃𝑃̃((𝑘−𝑥𝐼𝑔𝑘,𝑥))_𝑇(𝑘)𝑃𝑥(𝑘). (20)
It is obvious that 𝔼2ℎ^𝑇₁(𝑘)𝑃ℎ₂(𝑘𝔼𝐴)=0,_𝑇1𝑖(𝑘)𝑃𝐴_1𝑗(𝑘)=0𝑖≠𝑗𝛼_𝑖1−𝛼_𝑖𝐴^𝑇₁𝑃𝐴₁𝔼𝑖=𝑗,_{𝑞𝑞𝑖=1𝑗=1}𝑥^𝑇𝑘−𝜏_𝑖(𝑘)𝐴_1𝑖(𝑘)𝑃𝐴_1𝑗(𝑘)𝑥𝑘−𝜏_𝑗(𝑘)+^∞_𝑑1^∞_=1𝑑2=1𝜇_𝑑1𝜇_𝑑2𝑥^𝑇𝑘−𝑑₁𝐴^𝑇₂(𝑘)𝑃𝐴₂(𝑘)𝑥𝑘−𝑑₂+𝛾(𝑘)²𝑔^𝑇(𝑘,𝑥(𝑘))𝐼^𝑇̃𝑃̃((𝑘=𝐼𝑔𝑘,𝑥))_𝑞𝑖=1𝑥^𝑇𝑘−𝜏_𝑖(𝑘)𝛼_𝑖1−𝛼_𝑖𝐴^𝑇₁𝑃𝐴₁𝑥𝑘−𝜏_𝑖(𝑘)+^∞_𝑑=1𝑥^𝑇(𝑘−𝑑)𝛽1−𝛽𝜇²_𝑑𝐴^𝑇₂𝑃𝐴₂𝑥(𝑘−𝑑)+𝑔^𝑇(𝑘,𝑥(𝑘))𝛾1−𝛾𝐼^𝑇̃𝑃𝐼𝑔̃(𝑘,𝑥(𝑘)). (21) Then, we have the following equations: 𝔼Δ𝑉₁(𝑘)=ℎ^𝑇₁(𝑘)𝑃ℎ₁(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)+^𝑞_𝑖=1𝑥^𝑇𝑘−𝜏_𝑖(𝑘)𝛼_𝑖1−𝛼_𝑖×𝐴^𝑇₁𝑃𝐴₁𝑥𝑘−𝜏_𝑖(𝑘)+^∞_𝑑=1𝑥^𝑇(𝑘−𝑑)𝛽1−𝛽𝜇²_𝑑𝐴^𝑇₂𝑃𝐴₂𝑥(𝑘−𝑑)+𝑔^𝑇(𝑘,𝑥(𝑘))𝛾1−𝛾𝐼^𝑇̃𝑃̃((𝑘𝔼𝐼𝑔𝑘,𝑥))Δ𝑉₂(𝑘)=𝔼𝔼𝑉₂(𝑘+1)−𝑉₂(𝑘)=𝔼^𝑞𝑘_{𝑖=1𝑗=𝑘+1−𝜏𝑖(𝑘+1)}𝑥^𝑇𝑗𝑄_𝑖𝑥𝑗−^𝑞_𝑖=1^𝑘−1_{𝑗=𝑘−𝜏𝑖(𝑘)}𝑥^𝑇𝑗𝑄_𝑖𝑥𝑗=^𝑞_𝑖=1𝔼𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)−𝑥^𝑇𝑘−𝜏_𝑖(𝑘)𝑄_𝑖𝑔𝑘−𝜏_𝑖(𝑘)+^𝑘−1_{𝑗=𝑘−𝜏𝑖()𝑘+1+1}𝑥^𝑇𝑗𝑄_𝑖𝑥𝑗−^𝑘−1_{𝑗=𝑘−𝜏𝑖(𝑘)+1}𝑥^𝑇𝑗𝑄_𝑖𝑥𝑗≤^𝑞_𝑖=1𝔼𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)−𝑥^𝑇𝑘−𝜏_𝑖(𝑘)𝑄_𝑖𝑥𝑘−𝜏_𝑖(𝑘)+^{𝑘−𝜏𝑚}_{𝑗=𝑘−𝜏𝑀+1}𝑥^𝑇𝑗𝑄_𝑖𝑥𝑗,𝔼Δ𝑉₃(𝑘)=𝔼𝔼𝑉₃(𝑘+1)−𝑉₃(𝑘=_{𝑞𝑖=1−𝜏𝑚𝑗=−𝜏𝑀𝑘+1𝑙=𝑘+𝑗+1})𝑥^𝑇(𝑙)𝑄_𝑖𝑥(𝑙)−^𝑞_𝑖=1^−𝜏𝑚_{𝑗=−𝜏𝑀+1}^𝑘−1_{𝑙=𝑘+𝑗}𝑥^𝑇(𝑙)𝑄_𝑖𝑥(𝑙)=^𝑞_𝑖=1𝜏_𝑀−𝜏_𝑚𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)−^{𝑘−𝜏𝑚}_{𝑗=𝑘−𝜏𝑀−1}𝑥^𝑇𝑗𝑄_𝑖𝑥𝑗,𝔼Δ𝑉₄(𝑘)=𝔼𝔼𝑉₄(𝑘+1)−𝑉₄(𝑘=_∞𝑑=1)𝔼^𝑘_{𝑖=𝑘−𝑑+1}𝑥^𝑇(𝑖)×𝜇_𝑑𝑄+𝜇²_𝑑𝛽1−𝛽𝐴^𝑇₂𝑃𝐴₂𝑥(𝑖)−^𝑘−1_{𝑖=𝑘−𝑑}𝑥^𝑇(𝑖)×𝜇_𝑑𝑄+𝜇²_𝑑𝛽1−𝛽𝐴^𝑇₂𝑃𝐴₂𝑥(𝑖=𝑥_𝑇)(𝑘)𝜇𝑄+𝜇𝛽1−𝛽𝐴^𝑇₂𝑃𝐴₂𝑥(𝑘)−^∞_𝑑=1𝑥^𝑇(𝑘−𝑑)𝜇²_𝑑𝛽1−𝛽𝐴^𝑇₂𝑃𝐴₂𝑥×(𝑘−𝑑)−^∞_𝑑=1𝑥^𝑇(𝑘−𝑑)𝜇_𝑑𝑄𝑥(𝑘−𝑑). (22) By using Lemma 8, we have −^∞_𝑑=1𝑥^𝑇(𝑘−𝑑)𝜇_𝑑𝑄𝑥()1𝑘−𝑑≤−𝜇^∞_𝑑=1𝑥(𝑘−𝑑)^𝑇𝑄^∞_𝑑=1𝑥(𝑘−𝑑). (23) From Assumption 1, 𝑔⁻_𝑖≤(𝑔_𝑖(𝑥)−𝑔_𝑖(𝑦))/(𝑥−𝑦)≤𝑔⁺_𝑖, we have (𝑔(𝑘,𝑥(𝑘))−𝑔⁻_𝑖𝑥(𝑘))(𝑔(𝑘,𝑥(𝑘))−𝑔⁺_𝑖𝑥(𝑘))≤0.
It can be deduced that there exists 𝐔=diag[𝑢₁,𝑢₂,…,𝑢_𝑛]>0 such that ^𝑛_𝑖=1𝑢_𝑖𝑥(𝑘)𝑔(𝑘)^𝑇𝑔⁻_𝑖𝑔⁺_𝑖𝑒_𝑖𝑒^𝑇_𝑖−𝑔⁻_𝑖+𝑔⁺_𝑖2𝑒_𝑖𝑒^𝑇_𝑖−𝑔⁻_𝑖+𝑔⁺_𝑖2𝑒_𝑖𝑒^𝑇_𝑖𝑒_𝑖𝑒^𝑇_𝑖×𝑥(𝑘)𝑔(𝑘)=𝑥(𝑘)𝑔(𝑘)^𝑇𝑈𝐺₁−𝑈𝐺₂−𝑈𝐺₂𝑈𝑥(𝑘)𝑔(𝑘)≤0, (24) where 𝑒_𝑖 denotes the unit column vector having one element on its 𝑟th row and zeros elsewhere.
Now, we are ready to prove the exponential stability of the system (8) with 𝑣(𝑘)=0. Combining (16)–(24), we can easily see that 𝔼{Δ𝑉(𝑘)}≤𝔼𝜁^𝑇₁(𝑘)Γ₁+Δ^𝑇𝑃Δ𝜁₁(𝑘), (25) where𝜁₁(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇𝑘−𝜏₁(𝑘)𝑥^𝑇𝑘−𝜏₂(𝑘)⋯𝑥^𝑇𝑡−𝜏_𝑞(𝑘)∑^∞_𝑑=1𝜇_𝑑𝑥(𝑘−𝑑)𝑔^𝑇(𝑘)^𝑇,Γ₁=𝜙₁₁∗0⋯000𝛼₁1−𝛼₁𝐴^𝑇₁𝑃𝐴₁−𝑄₁⋯000∗∗⋱⋮⋮⋮∗∗∗𝛼_𝑞1−𝛼_𝑞𝐴^𝑇₁𝑃𝐴₁−𝑄_𝑞𝑄00∗∗∗∗−𝜇0∗∗∗∗∗𝛾1−𝛾𝐼^𝑇̃𝑃̃𝐴𝐴𝐼−𝑙𝐼Δ=₁₁⋯𝐴_1𝑞𝐴₂𝛾𝐼̃. (26)
If Γ₁+Δ^𝑇𝑃Δ<0, we can obtain that Γ₂<0 by Schur complements (Lemma 7), whereΓ₂=𝜙₁₁0⋯000𝐴^𝑇𝑃∗𝛼₁1−𝛼₁𝐴^𝑇₁𝑃𝐴₁−𝑄₁⋯000𝐴^𝑇₁₁𝑃∗∗⋱⋮⋮⋮⋮∗∗∗𝛼_𝑞1−𝛼_𝑞𝐴^𝑇₁𝑃𝐴₁−𝑄_𝑞00𝐴^𝑇_1𝑞𝑃𝑄∗∗∗∗−𝜇0𝐴^𝑇₂𝑃∗∗∗∗∗𝛾1−𝛾𝐼^𝑇̃𝑃𝐼̃−𝑙𝐼𝛾𝐼^𝑇̃𝑃.∗∗∗∗∗∗−𝑃 (27)Then, the system (8) with 𝑣(𝑘)=0 is mean-square robustly asymptotically stable. Furthermore, along the same line of the proof for Theorem 1 in [15], the exponential stability of system (8) can be confirmed on the mean-square sense.
Next, we will establish the performance of the filtering error system (8) under the zero initial condition. To this end, we introduce 𝐽(𝑘)=𝐸^∞_𝑘=0𝑟^𝑇̃(𝑘)𝑟̃(𝑘)−𝛾^2∞_𝑘=0𝐸𝑣^𝑇(𝑘)𝑣(𝑘). (28) And, it is obvious to see that 𝐽(𝑛)=𝐸^𝑛_𝑘=0𝑟^𝑇̃(𝑘)𝑟̃(𝑘)−𝛾^2𝑛_𝑘=0𝐸𝑣^𝑇(𝑘)𝑣(𝑘)+𝐸^𝑛_𝑘=0{𝑉(𝑘+1)−𝑉(𝑘)}−𝑉(𝑛+1)+𝑉(0)≤𝐸^𝑛_𝑘=0𝑟^𝑇̃(𝑘)𝑟̃(𝑘)−𝛾^2𝑛_𝑘=0𝐸𝑣^𝑇(𝑘)𝑣(𝑘)+𝐸^𝑛_𝑘=0{Δ𝑉(𝑘)}. (29) Along the same line as the proof of the stability of system (7), 𝐽(𝑛)≤𝐸∑^𝑛_𝑘=0{𝜁^𝑇₂(𝑘)Γ𝜁₂(𝑘)}, where𝜁₂(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇𝑘−𝜏₁(𝑘)𝑥^𝑇𝑘−𝜏₂(𝑘)⋯𝑥^𝑇(𝑡−𝜏_𝑞(𝑘))∑^∞_𝑑=1𝜇_𝑑𝑥^𝑇(𝑘−𝑑)𝑔^𝑇(𝑘)𝑣^𝑇(𝑘)^𝑇, (30)we can show that 𝐽(𝑛)≤0.
Letting 𝑛→∞, we obtain ^+∞_𝑘=0𝐸𝑟^𝑇̃(𝑘)𝑟̃(𝑘)≤𝛾^2∞_𝑘=0𝐸𝑣^𝑇(𝑘)𝑣(𝑘). (31)
This completes the proof.

Next, we are in a position to deal with the design of the filter output feedback controller for the system (8) and obtain the main result of this paper in the following theorem.

Theorem 10. For given constants 0≤𝜏_𝑚<𝜏_𝑀, 𝜀>0, and a scalar 𝜆>0, the fault detection filter (7) exists such that error system (8) is robustly asymptotically exponentially stable in mean square, if there exist constant 𝑙>0, positive matrices 𝑃, 𝑅, and 𝑅_𝑖(𝑖=1,2,…,𝑞), and any matrices 𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓, and 𝐷_𝑓 with appropriate dimensions, respectively,

ΣΣ=¹¹0⋯00Σ₁₄0Σ₁₆Σ₁₇∗𝛼₁1−𝛼₁𝐴₁−𝑅₁⋯00000Σ_2,17∗∗⋱⋮⋮⋮⋮⋮⋮∗∗∗𝛼_𝑞1−𝛼_𝑞𝐴₁−𝑅_𝑞0000Σ_2,𝑞7𝑅∗∗∗∗−𝜇000Σ₃₇∗∗∗∗∗Σ₄₄00Σ₄₇∗∗∗∗∗∗Σ₅₅Σ₅₆Σ₅₇∗∗∗∗∗∗∗−𝐼0∗∗∗∗∗∗∗∗Σ₇Σ<0,₁₁=−+𝑋𝑌𝑌𝑌^𝑞𝑖=1𝜏_𝑀−𝜏_𝑚+1𝑅_𝑖+𝜇𝑅+𝜇𝛽1−𝛽𝐴^𝑇₂−𝑈,𝑈=𝑈𝐺₁𝑈𝐺₁𝑈𝐺₁𝑈𝐺₁,𝐴₁=𝐴^𝑇₁𝑋𝐴₁𝐴^𝑇₁𝑋𝐴₁𝐴^𝑇₁𝑋𝐴₁𝐴^𝑇₁𝑋𝐴₁,𝐴₂=𝐴^𝑇₂𝑋𝐴₂𝐴^𝑇₂𝑋𝐴₂𝐴^𝑇₂𝑋𝐴₂𝐴^𝑇₂𝑋𝐴₂,Σ₁₄=𝑈𝐺₂𝑈𝐺₂,Σ₁₆=𝐶^𝑇𝐷^𝑇_𝑓𝐶^𝑇𝐷^𝑇_𝑓+𝐶^𝑇_𝑓,Σ₁₇=𝐴^𝑇𝑋+𝐶^𝑇𝐵^𝑇_𝑓𝐴^𝑇𝑌𝐴^𝑇𝑋+𝐶^𝑇𝐵^𝑇_𝑓+𝐴^𝑇_𝑓𝐴^𝑇𝑌,Σ_2,𝑖7=𝛼_𝑖𝐴^𝑇₁𝑋𝛼_𝑖𝐴^𝑇₁𝑌𝛼_𝑖𝐴^𝑇₁𝑋𝛼_𝑖𝐴^𝑇₁𝑌,Σ𝑖=1,2,…,𝑞₃₇=𝛽𝐴₂𝑋𝛽𝐴₂𝑌𝛽𝐴₂𝑋𝛽𝐴₂𝑌,Σ₄₄=𝛾1−𝛾𝑋−𝑈,Σ₅₅=−𝛾²𝐼,Σ₄₇=,Σ𝛾𝑋𝛾𝑌₅₆=𝐷^𝑇₂𝐷^𝑇_𝑓𝐻^𝑇𝐷^𝑇_𝑓−𝐼,Σ₅₇=𝐷^𝑇₁𝑋+𝐷^𝑇₂𝐵^𝑇_𝑓𝐷^𝑇₁𝑌𝐺^𝑇𝑋+𝐻^𝑇𝐵^𝑇_𝑓𝐺^𝑇𝑌,Σ₇₇=−.𝑋𝑌𝑌𝑌 (32)(33)

If (𝑋,𝑌,𝑄,𝑄_𝑖) is a feasible solution of (15), there exist nonsingular matrices 𝑈 and 𝑉 satisfying 𝑈𝑉=𝐼−𝑋𝑌⁻¹. The fault detection filter parameters in the form of (7) are given as follows: 𝐴_𝑓=𝑈⁻¹𝐴_𝑓𝑌⁻¹𝑉⁻¹,𝐵_𝑓=𝑈⁻¹𝐵_𝑓,𝐶_𝑓=𝐶_𝑓𝑌⁻¹𝑉⁻¹,𝐷_𝑓=𝐷_𝑓. (34)

Proof. The condition in (31) pledges that ^𝑋𝑌_∗𝑌>0.
This inequality implies that 𝐼−𝑋𝑌⁻¹ and 𝑋−𝑌 are nonsingular by Schur complement. If the condition in (32) is feasible, there exist nonsingular matrices 𝑈, 𝑉 satisfying 𝑈𝑉=𝐼−𝑋𝑌⁻¹.
Then, we introduce the following matrices: Ω₁=𝑈𝑋𝑌_𝑇0,Ω₂=𝐼𝐼0𝑉𝑌. (35)
Obviously Ω₁, Ω₂ are nonsingular.
Let 𝑃≜Ω₁Ω₂⁻¹=𝑈𝑋𝑈_𝑇−𝑈^𝑇𝑌⁻¹𝑉⁻¹; (36) then Ω^𝑇₂𝑃Ω₂=_𝑌^𝑋𝑌_𝑇𝑌.
Pre- and postmultiplying inequality (32) by diag{Ω₂⁻¹,Ω₂⁻¹,…,Ω₂⁻¹_𝑞,Ω₂⁻¹,𝐼,𝐼,𝐼,Ω₂⁻¹},  and  letting Ω^𝑇₂𝑄_𝑖Ω₂=𝑅_𝑖, Ω^𝑇₂𝑄Ω₂=𝑅, we can obtain (8). This completes the proof.

When there are parameters uncertainties, we have the following corollary.

Corollary 11. For given constants 0≤𝜏_𝑚<𝜏_𝑀, 𝜀>0 and a scalar 𝜆>0, the fault detection filter (7) exists such that error system (8) is robustly asymptotically exponentially stable in mean square, if there exist constant 𝑙>0, 𝜖>0, positive matrices 𝑃, 𝑅, and 𝑅_𝑖(𝑖=1,2,…,𝑞), and any matrices 𝐴_𝑓, 𝐵_𝑓, 𝐶_𝑓, and 𝐷_𝑓 with appropriate dimensions, respectively,ΣΣ=¹¹0⋯00Σ₁₄0Σ₁₆Σ₁₇0Σ₁₉∗𝛼₁1−𝛼₁𝐴₁−𝑅₁⋯00000Σ_2,1700⋮⋮⋱⋮⋮⋮⋮⋮⋮⋮⋮∗∗∗𝛼_𝑞1−𝛼_𝑞𝐴₁−𝑅_𝑞0000Σ_2,𝑞7𝑅00∗∗∗∗−𝜇000Σ₃₇00∗∗∗∗∗Σ₄₄00Σ₄₇00∗∗∗∗∗∗Σ₅₅Σ₅₆Σ₅₇0Σ₅₉∗∗∗∗∗∗∗−𝐼0Σ₆₈0∗∗∗∗∗∗∗∗Σ₇Σ₇₈0∗∗∗∗∗∗∗∗∗−𝜖𝐼0∗∗∗∗∗∗∗∗∗∗−𝜖𝐼<0, (37)where, Σ₁₁, Σ₁₆, Σ₁₇, Σ_2,17,…, Σ_2,𝑞7, 𝐴₁, Σ₃₇, Σ₄₄, Σ₄₇, Σ₅₅, Σ₅₆, Σ₅₇, and Σ₇ have been defined in Theorem 9: Σ₁₉=𝐶^𝑇Λ^𝑇𝐶^𝑇Λ^𝑇,Σ₅₉=𝐷^𝑇₂Λ^𝑇𝐻^𝑇Λ^𝑇,Σ₆₈=𝜖𝐷_𝑓,Σ₇₈𝜖𝐵=−^𝑓0. (38) If (𝑋,𝑌,𝑄,𝑄_𝑖) is a feasible solution of (15), there exist nonsingular matrices 𝑈 and 𝑉 satisfying 𝑈𝑉=𝐼−𝑋𝑌⁻¹. The fault detection filter parameters in the form of (7) are given as follows: 𝐴_𝑓=𝑈⁻¹𝐴_𝑓𝑌⁻¹𝑉⁻¹,𝐵_𝑓=𝑈⁻¹𝐵_𝑓,𝐶_𝑓=𝐶_𝑓𝑌⁻¹𝑉⁻¹,𝐷_𝑓=𝐷_𝑓. (39)

Proof. When considering the system with uncertain parameters, we need to replace 𝐴, 𝐷, 𝐶, 𝐷_𝑓 with 𝐴+Δ𝐴, 𝐷+Δ𝐷, 𝐶+Δ𝐶, 𝐷_𝑓+Δ𝐷_𝑓 in (32). Then, (32) changes into Σ+𝑀𝐹(𝑘)𝑁+𝑁^𝑇𝐹^𝑇(𝑘)𝑀^𝑇<0, where 𝑀=0,0,…,0,𝐷^𝑇_𝑓,𝐵^𝑇_𝑓,0^𝑇; 𝑁=Λ𝐶,Λ𝐶,0,…,0,Λ𝐷₂,Λ𝐻,0,0,0.
Using Schur complement and lemma, we can derive (37). This completes the proof.

4. Illustrative Example

In this section, a numerical example will be presented to illustrate the effectiveness of our results.

Example 1. Consider the system (2) with the parameters listed as follows [13]: 𝐴=0.9719−0.0013−0.03400.8628,𝐴₁=,𝐴0.140.200.2₂=0000,𝐷₁=,𝐷0.1000.3₂=00.1,𝐶=10.1,𝐺=−0.08390.0761^𝑇,𝐵=0,𝐻=0,𝛼₁=0.9,𝛼₂=0.7,𝛾=0.8,𝑔₁(𝑘,𝑥(𝑘))=0.5𝑥₁(𝑘)sin𝑥₂(𝑘),𝑔₂(𝑘,𝑥(𝑘))=0.5𝑥₂(𝑘)sin𝑥₁(𝑘)(𝑘)𝐺,𝜀=1,₁=0.25000.26,𝐺₂=.0.45000.45 (40)
Let the time-varying communication delays satisfy 1≤𝜏_𝑖(𝑘)≤3(𝑖=1,2). For the measurement quantization, the parameters of the logarithmic quantizer are set as 𝜇₀=2 and 𝜒=0.8. Then, the fault detection filter parameters can be obtained from Corollary 11 as follows.
For the parameters listed previously, by Corollary 11 in our paper, we can obtain the following feasible filtering parameters: 𝐴_𝑓=−0.2524−0.0325−0.0325−0.1975,𝐵_𝑓=,𝐶−3.4252−0.4137_𝑓=0.0065−0.0045,𝐷_𝑓=0.0122, (41) and the performance index between the robustness and sensitivity is 𝛾=3.8382.

Example 2. To show the usefulness and effectiveness of the designed fault detection filter, let the external disturbance be 𝑤(𝑘)=0. Figures 1 and 2 show the residual signal 𝑟(𝑘) and evolution of residual function 𝐽(𝑘), respectively, when the fault signal 𝑓(𝑘) is given as 𝑓(𝑘)=1,40≤𝑘≤800,else. (42) It indicates that the designed filter can detect the fault effectively. In [13], the fault can be detected in 6 time steps after its occurrence. However, from Figure 2, we can see that the fault can be detected in 4 time steps after its occurrence. Therefore, it can be seen that the residual can not only reflect the fault in time but also detect the fault without confusing it with the disturbance 𝑤(𝑘).

Figure 1 

Residual signal without 𝑤(𝑘).

Figure 2 

Evolution of residual function 𝐽(𝑘) without 𝑤(𝑘).

5. Conclusion

The fault detection filter design for stochastic systems with time-varying delays has been investigated in this research. Based on the Lyapunov functional method, sufficient conditions are obtained to ensure that the error systems are mean-square robustly asymptotically stable, and then the filters are designed in terms of LMIs. Numerical example has been given to illustrate the effectiveness of the proposed main results. The foregoing results have the potential to be useful for the study of stochastic systems.

Acknowledgments

This research was supported by the National Basic Research Program of China (2010 CB732501), the Fund of Sichuan Provincial Key Laboratory of Signal and Information Processing (SZJJ2009-002), and the Fund of Sichuan Provincial Key Laboratory of Signal and Information Processing (SGXZD0101-10-1).