Fault Detection for Non-Gaussian Stochastic Systems with Time-Varying Delay

Abstract

Fault detection (FD) for non-Gaussian stochastic systems with time-varying delay is studied. The available information for the addressed problem is the input and the measured output probability density functions (PDFs) of the system. In this framework, firstly, by constructing an augmented Lyapunov functional, which involves some slack variables and a tuning parameter, a delay-dependent condition for the existence of FD observer is derived in terms of linear matrix inequality (LMI) and the fault can be detected through a threshold. Secondly, in order to improve the detection sensitivity performance, the optimal algorithm is applied to minimize the threshold value. Finally, paper-making process example is given to demonstrate the applicability of the proposed approach.

1. Introduction

Fault detection for control systems has been of interest for many researchers during the past three decades (see [1–7] for surveys). Effective methodologies mainly include the filter-(or observer-) based approaches, the identification-based schemes, and the statistic approaches. However, most of the FD methodologies for stochastic systems only considered Gaussian systems. For example, the random process has been considered as Markov and Wiener processes in [8], respectively.

It has been shown that either the system variables are not Gaussian in [9, 10], existing methods may not be sufficient to characterize the non-Gaussian system behavior. Typical examples include fibre length distribution control in paper making, molecular weight distribution control, particle size distribution control in polymerization, and powder industries [8]. For such practical control problems, a new group of strategies that control the shape of PDFs for stochastic systems have been developed in the past few years (see [11, 12]), where the purpose is to design a controller so that the PDF of the system output can track a prespecified desired PDF, as close as possible.

To simplify system modeling, B-spline neural networks had been initially used to approximate the output PDF [9, 13]. The motivation of FD via the output PDFs from the retention system in paper making was first studied in [10], where the weight dynamical system was supposed to be a precise linear model. However, linear mappings cannot change the shape of output PDFs, which implies that the fault cannot be detected through the shape change of the PDFs. To meet the requirement in complex processes, nonlinearity should be considered in the weighting dynamic behavior. Recently, a kind of observer-based FD algorithm has been established in [14], where the nonlinear weighting system was considered. However, only the uniform boundedness of the estimation error could be guaranteed in [14], which leads to some conservative criteria.

On the other hand, time delay exists commonly in dynamic systems and is frequently a source of instability and poor performance [15–17]; many works had been done about time-delay systems along the development of stochastic theories [18]. Recently, FD problem has been studied for time-delay stochastic systems using PDF in [19–21]. But the criteria [19–21] are only available to systems with constant delay. Meanwhile, it is noted that in practice, time-varying delay is often encountered in dynamic systems. However, up to our knowledge, there have been few results in the literature of an investigation for the FD algorithm of dynamic systems with time-varying delay by using PDF.

In this paper, we provide a further contribution to FD for non-Gaussian stochastic systems with time-varying delay based on the method in [20, 21]. Firstly, by constructing an augmented Lyapunov functional, which involves some slack variables and a tuning parameter, a delay-dependent condition for the existence of FD observer is derived in terms of linear matrix inequality (LMI) and the fault can be detected through a threshold. Secondly, in order to improve the detection sensitivity performance, the optimal algorithm is applied to minimize the threshold value. Finally, paper-making process example is given to demonstrate the applicability of the proposed approach.

Notation. Throughout this paper,𝑅^𝑛denotes the𝑛-dimensional Euclidean space. The superscripts “𝜏” and “−1” stand for matrix transposition and matrix inverse, respectively;𝑃>(≥0)means that 𝑃 is real symmetric and positive definite (semidefinite). In symmetric block matrices or complex matrix expressions, diag{⋯}stands for a block-diagonal matrix, and ∗ represents a term that is induced by symmetry. For a vector 𝜈(𝑡), its norm is given by ‖𝜈(𝑡)‖²₂=∫^∞₀𝜈^⊤(𝑡)𝜈(𝑡)𝑑𝑡. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for related algebraic operations.

2. Problem Formulation and Preliminaries

In this section, firstly, we briefly review square-root B-spline expansion technique presented in [19, 21], which is used to formulate the output PDFs with the dynamic weight and is essential in solving our FD problem.

For a dynamic stochastic system, its output PDFs is defined by 𝛾(𝑧,𝑢(𝑡),𝐹(𝑡)), where𝑢(𝑡)∈𝑅^𝑚is control input and𝐹(𝑡)is the fault vector to be detected. In [13, 19–21], the following square-root B-spline expansion model has been used to approximate 𝛾(𝑧,𝑢(𝑡),𝐹(𝑡)): 𝛾(𝑧,𝑢(𝑡),𝐹(𝑡))=^𝑛_𝑖=1𝑣_𝑖(𝑢,𝐹)𝑏_𝑖(𝑧), (1) where𝑏_𝑖(𝑧)(𝑖=1,2,…,𝑛)are prespecified basis functions defined on[𝑎,𝑏] and 𝑣_𝑖(𝑢(𝑡),𝐹(𝑡))(𝑖=1,2,…,𝑛) are the corresponding weights of such an expansion. Denote𝐵₀(𝑧)=𝑏₁(𝑧)𝑏₂(𝑧)⋯𝑏_𝑛−1(𝑧)^𝜏and 𝑣(𝑡)∶=𝑣(𝑢(𝑡),𝐹(𝑡))=𝑣₁𝑣₂⋯𝑣_𝑛−1^𝜏, and let Λ₁=∫^𝑏_𝑎𝐵₀(𝑧)𝐵^𝜏₀(𝑧)𝑑𝑧, Λ₂=∫^𝑏_𝑎𝐵^𝜏₀(𝑧)𝑏_𝑛(𝑧)𝑑𝑧, Λ₃=∫^𝑏_𝑎𝑏²_𝑛(𝑧)𝑑𝑧≠0, and Λ₀=Λ₁Λ₃−Λ₂Λ^𝜏₂. Furthermore, it can be verified that (1) can be rewritten as (see [19] for details) 𝛾(𝑧,𝑢(𝑡),𝐹)=𝐵^𝜏(𝑧)𝑣(𝑡)+ℎ(𝑣(𝑡))𝑏_𝑛(𝑧), (2) where 𝐵^𝜏(𝑧)=𝐵^𝜏₀(𝑧)−Λ₂Λ₃𝑏_𝑛(𝑧),ℎ(𝑣(𝑡))=Λ₃−𝑣^𝜏(𝑡)Λ₀𝑣^𝜏(𝑡)Λ₃. (3)

The relationship between the input and the weights related to the PDFs can be described by the following model:𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡(𝑥(𝑡(𝑡)𝑡−𝑑))+𝐺𝑔))+𝐻𝑢+𝐽_𝑓(𝑡),𝑣(𝑡)=𝐸𝑥(𝑡), (4) where𝑥(𝑡)∈𝑅^𝑚is the unmeasured state and𝐹(𝑡)∈𝑅^𝑚is the fault to be detected.𝐴∈𝑅^𝑚×𝑚, 𝐴_𝑑∈𝑅^𝑚×𝑚, 𝐺∈𝑅^𝑚×𝑚, 𝐻∈𝑅^𝑚×1, 𝐽_𝑓∈𝑅^𝑚×𝑚, and𝐸∈𝑅^𝑚×𝑚represent the known parametric matrices of the dynamic part of the weight system. In fact, these matrices can be estimated by using the scaling parameter identification algorithms in [22]. Similar model can be found in [21], However, only constant delay is considered in [21]. In this paper, the time delay 𝑑(𝑡) is a differential function, satisfying 0≤𝑑(𝑡)≤ℎ, 𝑑(̇𝑡)≤𝜇, and the initial condition is defined by 𝑥(𝑡)=𝜙(𝑡)(−ℎ≤𝑡≤0). In addition, similar to [19, 21], the following assumptions are needed.

Assumption 1. For any𝑥₁(𝑡)and𝑥₂(𝑡), 𝑔(𝑥(𝑡))satisfies 𝑔(0)=0 and 𝑔𝑥₁(𝑡)−𝑔𝑥₂(𝑡)≤𝑈₂𝑥₁(𝑡)−𝑥₂(𝑡), (5) where𝑈₂is a known matrix.

Assumption 2. There is a known matrix 𝑈₁; for any𝑣₁(𝑡)and𝑣₂(𝑡), ℎ(𝑣(𝑡))denoted by (3) satisfies the following condition: ℎ𝑣₁(𝑡)−ℎ𝑣₂(𝑡)≤𝑈₁𝑣₁(𝑡)−𝑣₂(𝑡), (6) where‖⋅‖is denoted as the Euclidean norm.

Inequalities (5) and (6) are typically required in the literature on FD for nonlinear systems, for example, [23, 24], which will help to simplify the design of algorithms later on.

Generally speaking a fault detection system consists of a residual generator, and a residual evaluator including an evaluation function and a threshold. We will consider two parts of fault detection systems by using the information of PDF in the following section.

2.1. Residual Generator

For the purpose of residual generation, we construct the following nonlinear observer: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡(𝑥(𝑡(𝑡)(𝑡),𝑡−𝑑))+𝐺𝑔))+𝐻𝑢+𝐿𝜉 (7) where𝑥(𝑡)is the estimated state and𝐿∈𝑅^𝑚×𝑝is the gain to be determined.𝜉(𝑡)is the residual signal which is defined in terms of output PDFs as 𝜉(𝑡)=^𝑏_𝑎𝜎(𝑧)𝛾(𝑧,𝑢(𝑡),𝐹)−𝛾(𝑧,𝑢(𝑡))𝑑𝑧,𝛾(𝑧,𝑢(𝑡))=𝐵^𝜏(𝑧)𝐸𝑥(𝑡)+ℎ(𝐸𝑥(𝑡))𝑏_𝑛(𝑧). (8)

Remark 3. The classical residual generator design methods (such as [25–27]) are formulated in Figure 1. Different from the classical residual generator, residual𝜉(𝑡)in (7) is formulated as an integral with respect to the difference of the measured PDFs and the estimated PDFs.

Figure 1 

The classical residual generator.

In order to describe the dynamics of (7), we first consider𝑒(𝑡)=𝑥(𝑡)−𝑥(𝑡)and𝜉(𝑡); it can be shown that 𝑒̇(𝑡)=𝐴−𝐿Γ₁𝑒(𝑡)+𝐴_𝑑𝑒((𝑡𝑔(𝑥(𝑡(𝑥(𝑡𝑡−𝑑))+𝐺))−𝑔))−𝐿Γ₂[ℎ(𝐸𝑥(𝑡))−ℎ(𝐸𝑥(𝑡))]+𝐽_𝑓(𝑡),𝜉(𝑡)=Γ₁𝑒(𝑡)+Γ₂(ℎ(𝐸𝑥(𝑡))−ℎ(𝐸𝑥(𝑡))), (9)(10) where Γ₁=^𝑏_𝑎𝜎(𝑧)𝐵^𝜏(𝑧)𝐸𝑑𝑧,Γ₂=^𝑏_𝑎𝜎(𝑧)𝑏_𝑛(𝑧)𝑑𝑧. (11)

Thus, the problem of designing an observer-based fault detection can be described as follows:(i)design matrix𝐿such that error system (9) is asymptotically stable; (ii)the fault can be detected by residual generator.

2.2. Residual Evaluator

After designing of FD observer, the next task for FD is the evaluation of the generated residual. One of the widely adopted approaches is to use the following logical relationship for FD: ‖𝑟(𝑡)‖>𝐽_th‖𝑟(𝑡⟹faults⟹alarm,)‖≤𝐽_th⟹nofaults, (12) where𝑟(𝑡)is the residual evaluation function and 𝐽_th>0is the threshold. From the above logical relationship, it is clear that the threshold is important for FD sensitivity performance. In order to detect the fault more sensitively, the optimization techniques for the threshold are preferable. For this purpose, we construct a reference vector as follows: 𝑧(𝑡)=𝐶𝜉(𝑡)=𝐶Γ₁𝑒(𝑡)+𝐶Γ₂(ℎ(𝐸𝑥(𝑡))−ℎ(𝐸𝑥(𝑡))), (13) where 𝐶 is a selected weighting matrix. At this stage, we can consider the suboptimal guaranteed cost problem for 𝐽=‖𝑧(𝑡)‖²₂ subjected to error system (9) with 𝑥(0)=0 and 𝑒(𝑡)=𝜙(𝑡)(−ℎ≤𝑡≤0). In this paper, we choose ‖𝑧(𝑡)‖ and the bound 𝐽^∗=𝐽^1/2 as the residual evaluation function and threshold, respectively.

3. Main Result

In the section, an augmented Lyapunov-Krasovskii functional involving some slack variables and a tuning parameter is constructed and the following criterion is obtained.

Theorem 4. If for the parameters 𝜆_𝑖>0(𝑖=1,2)and𝜀, there exist matrices 𝑃₁>0, 𝑄₁>0, 𝑄₂>0, 𝑇>0,  𝑃_𝑖(𝑖=2,…,6), and𝑅such that the following inequality holds: <0,^{Π11Π12Π13−𝑃𝜏4𝑃𝜏2G𝑅Γ2−ℎ𝑃𝜏3−ℎ𝑃𝜏4Γ𝜏1𝐶𝜏∗Π22𝜀𝑃𝜏2𝐴𝑑0𝜀𝑃𝜏2𝐺𝜀𝑅Γ2000∗∗Π33−𝑃𝜏600−ℎ𝑃𝜏5−ℎ𝑃𝜏60∗∗∗−𝑄2100000∗∗∗∗−𝜆221𝐼0000∗∗∗∗∗−𝜆21𝐼00Γ𝜏2𝐶𝜏∗∗∗∗∗∗−ℎ𝑇00∗∗∗∗∗∗∗−ℎ𝑇0∗∗∗∗∗∗∗∗−𝐼} (14) where∗denotes the symmetric terms in a symmetric matrix and Π₁₁=𝑃^𝜏₂𝐴−𝑅Γ₁+𝑃^𝜏₂𝐴−𝑅Γ₁^𝜏+1𝜆²₁𝐸^𝜏𝑈^𝜏₁𝑈₁𝐸+1𝜆²₂𝑈^𝜏₂𝑈₂+𝑄₁+𝑄₂+𝑃^𝜏₃+𝑃₃,Π₁₂=𝜀𝐴^𝜏𝑃₂−𝜀Γ^𝜏₁𝑅^𝜏+𝑃₁−𝑃^𝜏₂,Π₁₃=𝑃^𝜏₂𝐴_𝑑−𝑃^𝜏₃+𝑃^𝜏₄+𝑃₅,Π₂₂=−𝜀𝑃₂−𝜀𝑃^𝜏₂Π+ℎ𝑇,₃₃=−𝑃₅−𝑃^𝜏₅−𝑃₆−𝑃^𝜏₆−1−𝜇𝑄₁, (15) then in the absence of𝐹(𝑡), error system (9) with gain 𝐿=𝑃₂^−𝜏𝑅 is stable and the reference vector satisfies 𝐽=‖𝑧(𝑡)‖²₂≤𝑉(0)∶=𝜙^𝜏(0)𝑃₁𝜙(0)+⁰_−ℎ⁰_𝜃𝜙̇^𝜏(𝑠)𝑇𝜙̇(𝑠)+𝑑𝑠𝑑𝜃_0(0)−𝑑𝜙^𝜏(𝑠)𝑄₁𝜙(𝑠)𝑑𝑠+⁰_−ℎ𝜙^𝜏(𝑠)𝑄₂𝜙(𝑠)𝑑𝑠. (16) Moreover, the fault can be detected when ‖𝑧(𝑡)‖>𝐽^∗=𝑉^1/2(0) for 𝑡∈[−ℎ,∞).

Proof. Defineℎ=ℎ(𝑥(𝑡)), ℎ=ℎ(𝑥(𝑡)), 𝑔=𝑔(𝑥(𝑡)), and 𝑔=𝑔(𝑥(𝑡))and denote the Lyapunov function candidate as follows: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡), (17) where 𝑉₁(𝑡)=𝜉^𝜏₁(𝑡)𝑀𝑃𝜉₁(𝑡),𝑉₂(𝑡)=⁰_−ℎ^𝑡_𝑡+𝜃𝑒^𝜏̇(𝑠)𝑇𝑒̇(𝑠)𝑉𝑑𝑠𝑑𝜃,₃(𝑡)=^𝑡_{(𝑡)𝑡−𝑑}𝑒^𝜏(𝑠)𝑄₁𝑒(𝑠)𝑑𝑠+^𝑡_𝑡−ℎ𝑒^𝜏(𝑠)𝑄₂𝑒(𝑠)𝑉𝑑𝑠,₄(𝑡)=1𝜆²₁^𝑡0𝑈₁𝐸𝑒(𝑠)²−ℎ−ℎ²+1𝑑𝑠𝜆²₂^𝑡0𝑈₂𝑒(𝑠)²−𝑔−𝑔²𝑑𝑠 (18) with 𝑃𝑀=𝐼00000000000,𝑃=¹𝑃00₂𝜀𝑃₂0𝑃₃0𝑃₅𝑃₄0𝑃₆,𝜉₁(𝑡)=𝑒(𝑡)𝑒̇(𝑡)𝑒((𝑡,𝑡−𝑑)) (19) and with𝑃₁>0,  𝑇>0, 𝑄₁>0,  𝑄₂>0. It is noted that 𝑉₁(𝑡) is actually𝑒^𝜏(𝑡)𝑃₁𝑒(𝑡). Then following (5) and (6) gives 𝑉(𝑡)≥0. On the other hand, from the Leibniz-Newton formula, the following equations are true: 𝛼₁∶=𝑒(𝑡)−𝑒(𝑡−𝑑(𝑡))−^𝑡_{(𝑡)𝑡−𝑑}𝑒̇(𝑠)𝛼𝑑𝑠=0,₂∶=𝑒(𝑡−𝑑(𝑡))−𝑒(𝑡−ℎ)−^{𝑡−𝑑(𝑡)}_𝑡−ℎ𝑒̇(𝑠)𝑑𝑠=0. (20) Hence, calculating the time derivative of 𝑉₁(𝑡)along the solution of (9) in the absence of 𝐹(𝑡) and using (20) yields 𝑉̇₁(𝑡)=2𝜉₁(𝑡)𝑃^𝜏×𝑒̇(𝑡)−𝑒̇(𝑡)+𝐴−𝐿Γ₁𝑒(𝑡)+𝐴_𝑑𝑒((𝑡𝑔𝑡−𝑑))+𝐺𝑔−−𝐿Γ₂ℎ𝛼ℎ−₁𝛼₂=2𝑒^𝜏(𝑡)𝑃₁𝑒̇(𝑡)−2𝑒^𝜏(𝑡)𝑃^𝜏₂𝑒̇(𝑡)+𝑒^𝜏(𝑡)𝑃^𝜏₂𝐴−𝑅Γ₁+𝑃^𝜏₂𝐴−𝑅Γ₁^𝜏𝑒(𝑡)+2𝑒^𝜏(𝑡)𝑃^𝜏₂𝐴_𝑑𝑒(𝑡−𝑑(𝑡))+2𝑒^𝜏(𝑡)𝑃^𝜏₂𝐺𝑔−𝑔−2𝑒^𝜏(𝑡)𝑅Γ₂ℎ−ℎ−2𝜀𝑒^𝜏̇(𝑡)𝑃^𝜏₂𝑒̇(𝑡)+2𝜀𝑒^𝜏̇(𝑡)𝑃^𝜏₂𝐴−𝑅Γ₁𝑒(𝑡)+2𝜀𝑒^𝜏̇(𝑡)𝑃^𝜏₂𝐴_𝑑(𝑡−𝑑(𝑡))+2𝜀𝑒^𝜏̇(𝑡)𝑃^𝜏₂𝐺𝑔𝑒̇𝑔−−2𝜀^𝜏(𝑡)𝑅Γ₂ℎ−ℎ+2𝑒^𝜏(𝑡)𝑃^𝜏₃𝑒(𝑡)−2𝑒^𝜏(𝑡)𝑃^𝜏₃𝑒(𝑡−𝑑(𝑡))−2𝑒^𝜏(𝑡)𝑃^𝜏₃^𝑡_{(𝑡)𝑡−𝑑}𝑒̇(𝑠)𝑑𝑠+2𝑒^𝜏(𝑡−𝑑(𝑡))𝑃^𝜏₅𝑒(𝑡)−2𝑒^𝜏(𝑡−𝑑(𝑡))𝑃^𝜏₅𝑒(𝑡−𝑑(𝑡))−2𝑒^𝜏(𝑡−𝑑(𝑡))𝑃^𝜏₅^𝑡_{(𝑡)𝑡−𝑑}𝑒̇(𝑠)𝑑𝑠+2𝑒^𝜏(𝑡)𝑃^𝜏₄𝑒(𝑡−𝑑(𝑡))−2𝑒^𝜏(𝑡)𝑃^𝜏₄𝑒(𝑡−ℎ)−2𝑒^𝜏(𝑡)𝑃^𝜏₄^{𝑡−𝑑(𝑡)}_𝑡−ℎ𝑒̇(𝑠)𝑑𝑠+2𝑒^𝜏(𝑡−𝑑(𝑡))𝑃^𝜏₆𝑒(𝑡−𝑑(𝑡))−2𝑒^𝜏(𝑡−𝑑(𝑡))𝑃^𝜏₆𝑒(𝑡−ℎ)−2𝑒^𝜏(𝑡−𝑑(𝑡))𝑃^𝜏₆^{𝑡−𝑑(𝑡)}_𝑡−ℎ𝑒̇(𝑠)𝑑𝑠=𝜂^𝜏(𝑡)Θ₁𝜂(𝑡), (21) where Θ₁=^{Θ11Π12Π13−𝑃𝜏4𝑃𝜏2𝐺𝑅Γ2−𝑃𝜏3−𝑃𝜏4∗Θ22𝜀𝑃𝜏2𝐴𝑑0𝜀𝑃𝜏2𝐺𝜀𝑅Γ200∗∗Θ33−𝑃𝜏600−𝑃𝜏5−𝑃𝜏6∗∗∗00000}_{∗∗∗∗0000∗∗∗∗∗000∗∗∗∗∗∗00∗∗∗∗∗∗∗0},𝜂(𝑡)=𝑒(𝑡)𝑒̇(𝑡)𝑒((𝑡𝑒()𝑔ℎ∫𝑡−𝑑))𝑡−ℎ𝑔−ℎ−_{𝑡(𝑡)𝑡−𝑑}̇∫𝑒𝑑𝑠_{𝑡−𝑑(𝑡)𝑡−ℎ}̇,𝑒𝑑𝑠 (22) where Θ₁₁=𝑃^𝜏₂𝐴−𝑅Γ₁+𝑃^𝜏₂𝐴−𝑅Γ₁^𝜏+𝑃₃+𝑃^𝜏₃,Θ₂₂=−𝜀𝑃₂−𝜀𝑃^𝜏₂,Θ₃₃=−𝑃₅−𝑃^𝜏₅−𝑃₆−𝑃^𝜏₆, (23) with 𝑅=𝑃^𝜏₂𝐿andΠ₁₂, Π₁₃ being defined in (14). It follows from Jensen integral inequality with 𝑑(𝑡)≤ℎ that one can obtain 𝑉̇₂(𝑡)=ℎ𝑒^𝜏̇𝑇𝑒̇−^𝑡_𝑡−ℎ𝑒^𝜏̇𝑇̇𝑒̇𝑒𝑑𝑠=ℎ^𝜏𝑇𝑒̇−^{𝑡−𝑑(𝑡)}_𝑡−ℎ𝑒^𝜏̇𝑇𝑒̇𝑑𝑠−^𝑡_{𝑡−𝑑(𝑡)}𝑒^𝜏̇𝑇̇𝑒̇𝑒𝑑𝑠≤ℎ^𝜏(𝑡)𝑇𝑒̇(𝑡)−1ℎ^{𝑡−𝑑(𝑡)}_𝑡−ℎ𝑒̇(𝑠)𝑑𝑠^𝜏𝑇^{𝑡−𝑑(𝑡)}_𝑡−ℎ𝑒̇(𝑠)−1𝑑𝑠ℎ^𝑡_{(𝑡)𝑡−𝑑}𝑒̇(𝑠)𝑑𝑠^𝜏𝑇^𝑡_{(𝑡)𝑡−𝑑}𝑒̇(𝑠)𝑑𝑠. (24) On the other hand, it can be verified that 𝑉̇₃(𝑡)=𝑒^𝜏(𝑡)𝑄₁+𝑄₂𝑒(𝑡)−1−𝜇𝑒^𝜏(𝑡−𝑑(𝑡))×𝑄₁𝑒(𝑡−𝑑(𝑡))−𝑒^𝜏(𝑡−ℎ)𝑄₂𝑒(),𝑉̇𝑡−ℎ₄(𝑡)=1𝜆²₁𝐸^𝜏𝑈^𝜏₁𝑈₁1𝐸+𝜆²₂𝑈^𝜏₂𝑈₂−1𝜆²₁ℎ−ℎ^𝜏ℎ−1ℎ−𝜆²₂𝑔−𝑔^𝜏𝑔−𝑔. (25) It then follows from (21)–(25) that 𝑉̇(𝑡)≤𝜂^𝜏(𝑡)Ξ₀𝜂(𝑡), (26) whereΞ₀=Θ₁+diag{𝑄₁+𝑄₂+(1/𝜆²₁)𝐸^𝜏𝑈^𝜏₁𝑈₁𝐸+(1/𝜆²₂)𝑈^𝜏₂𝑈₂,ℎ𝑇,−(1−𝜇)𝑄₁,−𝑄₂,−(1/𝜆²₂)𝐼,−(1/𝜆²₁)𝐼,−(1/ℎ)𝑇,−(1/ℎ)𝑇}. Applying the Schur complement to (14) gives Ξ₀<0, which implies 𝑉̇(𝑡)<0 and error system (9) is asymptotically stable.
Furthermore, we denote an auxiliary cost function as 𝐽=^∞₀𝑧^𝜏(𝜏)𝑧(𝜏)+𝑉̇(𝜏)𝑑𝜏. (27) Note that 𝑧^𝜏(𝑡)𝑧(𝑡)=𝑒^𝜏(𝑡)Γ^𝜏₁𝐶^𝜏𝐶Γ₁𝑒(𝑡)+2𝑒^𝜏(𝑡)Γ^𝜏₁𝐶^𝜏𝐶Γ₂ℎ+ℎℎ−ℎ−^𝜏Γ^𝜏₂𝐶^𝜏𝐶Γ₂ℎ−ℎ. (28) From (14), 𝐽<0 holds. On the other hand, it can be shown that 𝐽=^∞₀𝑧^𝜏(𝜏)𝑧(𝜏)𝑑𝜏+lim_𝑇→∞(𝑉(𝑇)(0≤‖𝑧(𝑡−𝑉)))‖²−𝑉(0), (29) where 𝑉(0) is defined in (16). Therefore (16) is satisfied. This completes the proof.

Remark 5. Theorem 4 is based on a newly proposed augmented Lyapunov functional of form (17), which contains a structure more general than the traditional ones as those in [19, 20] for involving 𝑒(̇𝑡) and 𝑒(𝑡−𝑑(𝑡)) in the first term of (17). In addition, the importance of augmented Lyapunov functional is that it separates Lyapunov function matrix 𝑃₁ from 𝐴,  𝐴_𝑑; that is, there are no terms containing the product of𝑃₁and any of them, which makes the design of the fault detection observer more easier.

Remark 6. Recently, Li et al. [21] provided a fault detection observer design criterion by using PDFs for stochastic system with constant delay. Compared with the criterion in [21], the advantage in our paper is that time-varying delay is considered and the design criterion includes more delay information. Moreover, it is noted that setting 𝑃₃=𝑌,   𝑃₅=𝑊, 𝑄₂=0, and 𝑃₄=𝑃₆=0 in Theorem 4 yields precisely Theorem  1 in [21]. So, Theorem 4 in this paper is an extension of the result in [21].

In order to improve fault detection sensitivity, the following result provides an optimization algorithm in order to make the threshold 𝐽^∗ smaller.

Theorem 7. Consider error system (9) with cost function (27). For given parameters 𝜆_𝑖(𝑖=1,2) and 𝜀, if the following optimization problem:min𝛼+trΩ₁+trΩ₂+trΩ₃ (30) subject to (i)(),()𝐿𝑀𝐼14ii−𝛼𝜙^𝜏(0)𝑃₁𝑃₁𝜙(0)−𝑃₁()<0,iii−Ω₁𝑌^𝜏₁𝑄₁𝑄₁𝑌₁−𝑄₁()<0,iv−Ω₂𝑌^𝜏₂𝑄₂𝑄₂𝑌₂−𝑄₂(v)<0,−Ω₃𝑌^𝜏₃𝑇𝑇𝑌₃−𝑇<0, (31) where ⁰_(0)−𝑑𝜙(𝑠)𝜙^𝜏(𝑠)𝑑𝑠=𝑌₁𝑌^𝜏₁,_0−ℎ𝜙(𝑠)𝜙^𝜏(𝑠)𝑑𝑠=𝑌₂𝑌^𝜏₂,_0−ℎ0𝜃𝜙̇(𝑠)𝜙̇^𝜏(𝑠)𝑑𝑠𝑑𝜃=𝑌₃𝑌^𝜏₃, (32) has solutions with positive definite matrices𝑃₁, 𝑄₁, 𝑄₂, 𝑇, Ω₁, Ω₂, and Ω₃; matrices 𝑃_𝑖(𝑖=2,…,6) and  𝑅; and positive scalar𝛼, then in the absence of 𝐹(𝑡), error system (9) with gain 𝐿=𝑃₂^−𝜏𝑅 is stable and the smallest threshold satisfies 𝐽^∗=𝛼+trΩ₁+trΩ₂+trΩ₃^1/2, (33) for 𝑡∈[−ℎ,∞).

Proof. By Theorem 4, (i) is clear. Also, it follows from the Schur complement that (ii), (iii), (iv), and (v) are equivalent to 𝜙^𝜏(0)𝑃₁𝜙(0)𝑌<𝛼,_𝜏1𝑄₁𝑌₁<Ω₁,𝑌^𝜏₂𝑄₂𝑌₂<Ω₂,𝑌^𝜏₃𝑇𝑌₃<Ω₃, (34) respectively. On the other hand ⁰_(0)−𝑑𝜙^𝜏(𝑠)𝑄₁𝜙(𝑠)=𝑑𝑠_0(0)−𝑑tr𝜙^𝜏(𝑠)𝑄₁𝜙(𝑠)𝑄𝑑𝑠=tr₁^1/20_(0)−𝑑𝜙(𝑠)𝜙^𝜏(𝑠)𝑑𝑠𝑄₁^1/2=tr𝑄₁^1/2𝑌₁𝑌^𝜏₁𝑄₁^1/2=tr𝑌^𝜏₁𝑄₁𝑌₁<trΩ₁,_0−ℎ𝜙^𝜏(𝑠)𝑄₂𝜙(𝑠)=𝑑𝑠_0−ℎtr𝜙^𝜏(𝑠)𝑄₂𝜙(𝑠)𝑄𝑑𝑠=tr₂^1/20_−ℎ𝜙(𝑠)𝜙^𝜏(𝑠)𝑑𝑠𝑄₂^1/2=tr𝑄₂^1/2𝑌₂𝑌^𝜏₂𝑄₂^1/2=tr𝑌^𝜏₂𝑄₂𝑌₂<trΩ₂,_0ℎ0𝜃𝜙̇^𝜏(𝑠)𝑇𝜙̇(𝑠)=𝑑𝑠𝑑𝜃_0ℎ0𝜃tr𝜙̇^𝜏(𝑠)𝑇𝜙̇(𝑠)𝑇𝑑𝑠𝑑𝜃=tr^1/20_ℎ⁰_𝜃𝜙̇(𝑠)𝜙̇^𝜏(𝑠)𝑑𝑠𝑑𝜃𝑇^1/2=tr𝑇^1/2𝑌₃𝑌^𝜏₃𝑇^1/2=tr𝑌^𝜏₃𝑇𝑌₃<trΩ₃. (35) Hence, it follows from (17) that 𝐽<𝛼+trΩ₁+trΩ₂+trΩ₃. (36) Thus, the minimization of 𝐽 implies the minimization of the threshold values 𝐽^∗=𝐽^1/2.

4. Simulations

An application of paper-making process is given to demonstrate the applicability of our proposed approach. The basis functions are selected in a similar way to [20] as follows: 𝐵(𝑧)=𝑏₁(𝑧),𝑏₂(𝑧),…,𝑏₉(𝑧)^𝜏,𝑏_𝑖(𝑧)=exp−𝑧−𝑧_𝑖²𝜎_𝑖⁻²,(),𝑧𝑖=1,2,…,10_𝑖=0.003+0.006(𝑖−1),𝜎_𝑖=0.003,(𝑖=1,2,…,10). (37) Consider the following weighting system: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡(𝑥(𝑡(𝑡)𝑡−𝑑))+𝐺𝑔))+𝐻𝑢+𝐽_𝑓(𝑡),𝑣(𝑡)=𝑥(𝑡), (38) where 𝐴=diag{−0.83,−0.83,…,−0.83}∈𝑅^9×9,𝐴_𝑑=diag{0.2,0.2,…,0.2}∈𝑅^9×9,𝐺=diag{1,1,…,1}∈𝑅^9×9,𝐻=00.010.010.010.010.010.010.010^𝜏,𝐽_𝑓=diag{1.18,0,1.56,0,1.56,0,1.56,1.56,1.56}, (39)𝑔(𝑥(𝑡))=sin(𝑥(𝑡)), and 𝑢(𝑡) is random number. By setting 𝜎(𝑧)=1, then the following parametric matrices related to the B-spline approximation can be obtained: Λ₀=10⁻⁵×^{2.4000.9560.0480000000.9562.6070.9590.047000000.0480.9592.6050.9560.0490.00100000.0470.9562.6010.9600.048000}_{000.0490.9602.6050.9580.0480.0010000.0010.0480.9582.6050.9600.049000000.0480.9602.6010.9560.04700000.0010.0490.9562.6040.9390000000.0470.9392.226},Λ₁=10⁻³×,Λ4.920.100000025.320.1000000.125.320.1000000.125.320.1000000.125.320.1000000.125.320.1000000.125.320.1000000.125.320000000.125.3₂=,Λ00000000.00010.0020₃=0.0049. (40)

It can be verified that 𝑈₁=diag{0.05,0.05,…,0.05}∈𝑅^9×9 and it can be supposed that 𝑈₂=diag{1,1,…,1}∈𝑅^9×9. Corresponding to (11), it can be calculated that Γ₁=10⁻²×6.37.57.57.57.57.57.57.45.0,  Γ₂=0.0063. In this case, 𝑑(𝑡)=0.5sin(𝑡), 𝜇=0.5, ℎ=0.5, and the initial value of observer (7) is selected as 𝑥(𝑡)=0∈𝑅⁹ for all 0≤𝑡≤+∞, while the initial value of (4) is selected as 𝑥₁(𝑡)=2.05+exp(𝑡−2),  𝑥₂(𝑡)=1.05+exp(𝑡−2), 𝑥₅(𝑡)=−2.05+exp(𝑡−2), 𝑥₉(𝑡)=3.0+exp(𝑡−2), and 𝑥₃(𝑡)=𝑥₄(𝑡)=𝑥₆(𝑡)=𝑥₇(𝑡)=𝑥₈(𝑡)=0, (−0.5≤𝑡≤0). The fault is supposed to be 𝐹(𝑡)=0,𝑡<20,20.220.20.220.220.2,20≤𝑡≤40,0,𝑡≥40. (41)

Figure 2 shows the changes of the output PDF when the fault occurs. With the detection observer, Figure 3 demonstrates the responses of the residual signal. The threshold can be computed to give 𝐽^∗=0.1137 by using Theorem 7 and the residual satisfies 𝑧(𝑡)≥0.1143>𝐽^∗ for 𝑡>26. However, it is infeasible by using Theorem  2 in [21]. Thus, a better FD algorithm may more rapidly detect the occurrence of faults. From Figure 4, we can see that the fault can be detected 6 s after its occurrence.

Figure 2 

3D mesh plot of the measured output PDFs.

Figure 3 

The response of residual signal.

Figure 4 

Threshold and the response of evaluation function.

5. Conclusion

The FD problem is studied for a class of non-Gaussian stochastic systems with time-varying delay using augmented Lyapunov functional approach, where only the output PDFs can be measured rather than an output signal. A new delay-dependent FD observer design criterion is obtained. And the fault can be detected with an optimal threshold, where the guaranteed cost optimization algorithm is applied to minimize it. Further work will focus on the fault diagnosis and fault tolerant for these systems.

Acknowledgments

This work is supported by the NSF of China (nos. 60904025 and 61104103) and in part by Qing Lan project of Jiang Su province and the scholarship from China Scholarship Council. The authors are very grateful to Professor Qing Zhao in the Department of Electrical and Computer Engineering, University of Alberta, for her helpful suggestions which have helped them in improving the quality of the paper.