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

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 delaydependent 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.


Introduction
Fault detection for control systems has been of interest for many researchers during the past three decades (see [1][2][3][4][5][6][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][16][17]; many works had been done about time-delay systems along the development of stochastic theories [18].Recently, FD problem has been studied for timedelay stochastic systems using PDF in [19][20][21].But the criteria [19][20][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,   denotes the dimensional Euclidean space.The superscripts "" and "−1" stand for matrix transposition and matrix inverse, respectively;  > (≥ 0) means that  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 ](), its norm is given by ‖ Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for related algebraic operations.

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.
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.

Residual Generator.
For the purpose of residual generation, we construct the following nonlinear observer: where x() is the estimated state and  ∈  × is the gain to be determined.() is the residual signal which is defined in terms of output PDFs as Remark 3. The classical residual generator design methods (such as [25][26][27]) are formulated in Figure 1.Different from the classical residual generator, residual () in ( 7) is formulated as an integral with respect to the difference of the measured PDFs and the estimated PDFs.
In order to describe the dynamics of ( 7), we first consider () = () − x() and (); it can be shown that where Thus, the problem of designing an observer-based fault detection can be described as follows: (i) design matrix  such that error system (9) is asymptotically stable; (ii) the fault can be detected by residual generator.

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: where () is the residual evaluation function and  th > 0 is 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: where  is a selected weighting matrix.At this stage, we can consider the suboptimal guaranteed cost problem for  = ‖()‖ 2 2 subjected to error system (9) with x(0) = 0 and () = () (−ℎ ≤  ≤ 0).In this paper, we choose ‖()‖ and the bound  * =  1/2 as the residual evaluation function and threshold, respectively.
Furthermore, we denote an auxiliary cost function as Note that From ( 14),  < 0 holds.On the other hand, it can be shown that where (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 ė () and ( − ()) in the first term of (17).In addition, the importance of augmented Lyapunov functional is that it separates Lyapunov function matrix  1 from ,   ; that is, there are no terms containing the product of  1 and 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  3 = ,  5 = ,  2 = 0, and  4 =  6 = 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  * smaller.Theorem 7. Consider error system (9) with cost function (27).For given parameters   ( = 1, 2) and , if the following optimization problem: where Proof.By Theorem 4, (i) is clear.Also, it follows from the Schur complement that (ii), (iii), (iv), and (v) are equivalent to respectively.On the other hand Hence, it follows from (17) that  <  + tr (Ω 1 ) + tr (Ω 2 ) + tr (Ω 3 ) .
Thus, the minimization of  implies the minimization of the threshold values  * =  1/2 .

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: (()) = sin(()), and () is random number.By setting () = 1, then the following parametric matrices related to the B-spline approximation can be obtained: It can be verified that  1 = diag{0.05,0.05, . . ., 0.05} ∈  9 × 9 and it can be supposed that  2 = diag{1, 1, . . ., 1} ∈  9 × 9 .Corresponding to (11), it can be calculated that Γ 1 = 10 −2 ×[6.3 7.5 7.5 7.5 7.5 7.5 7.5 7.4 5.0], Γ 2 = 0.0063.In this case, () = 0.5 sin(),  = 0.5, ℎ = 0.5, and the initial value of observer (7) is selected as x() = 0 ∈  9 for all 0 ≤  ≤ +∞, while the initial value of (4) is selected as 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  * = 0.1137 by using Theorem 7 and the residual satisfies () ≥ 0.1143 >  * for  > 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.

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 delaydependent 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.

Figure 2 :
Figure 2: 3D mesh plot of the measured output PDFs.

Figure 3 :
Figure 3: The response of residual signal.

Figure 4 :
Figure 4: Threshold and the response of evaluation function.