#### 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 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 and . 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 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 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, denotes the dimensional Euclidean space. and denote, respectively, the maximal and minimal eigenvalue of matrix . refers to the expectation operator with respect to some probability measure . We use as a block-diagonal matrix. means that is a symmetric positive (negative) definite matrix, denotes the inverse of matrix . denotes the transpose of matrix , and is the identity matrix with compatible dimension.

#### 2. System Description and Definitions

Consider the following stochastic MJS with mode-dependent time-varying delays: where is the state vector; is the exogenous disturbance input which belongs to ; the unknown input; is the fault to be detected; is the measured output; is a zero-mean one-dimensional Wiener process satisfying and ; is a compatible vector-valued initial function defined on ; , , , , , , , , , , , , and are real constant matrices with appropriate dimensions. is the mode-dependent time-varying delay. and are continuous-time Markovian processes with right continuous trajectories and taking values in finite sets , with the transition probability matrices , and , , respectively, given by where and ; for is the transition rate from mode at time to mode at time and ; for is the transition rate from mode at time to mode at time and . The processes and are assumed to be independent throughout this paper. For simplicity, a matrix will be denoted by . For example, is denoted by , is denoted by , , is denoted by , , and so on. When the mode is in , the mode-dependent time-varying delay satisfies where .

*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: where is the filter state vector. is its output which is sensitive to faults. 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 , that is, , where and denote, respectively, the Laplace transforms of and . One state-space realization of can be

Denoting , then the overall dynamic system can be governed by the following augmented system: where , , , and

For simplicity, let

Now, the problem of fault detection is transformed into an filtering problem for system (2.1), which is described as follows: given a prescribed level of disturbance attenuation , determine a series of filter matrices , , and 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 and a threshold are selected as where is the finite-time window, denotes the limited length, and denotes the initial evaluation time. The occurrence of fault can be detected by comparing and , according to the following logic:

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 , scalar , vector function such that the integrations concerned are well defined, the following inequality holds:
*

*Definition 2.5. *The filtering error system (2.6) with is said to be stochastically stable, if, for every system mode , every time-delay mode and all finite initial state , the following relation holds: .

*Definition 2.6. *Given a scalar , the filtering error system (2.6) is said to be stochastically stable with an performance , if, for every system mode and every time-delay mode , the filtering error system (2.6) with is stochastically stable, and, under zero initial condition, it satisfies for any nonzero .

It should be pointed out that the joint process is not Markovian. In order to cast our model into the frame work for a Markovion system, let us define a new Markovion process: , , and then is Markovian process with the initial state .

Let denote the family of all nonnegative functions on , which are continuously twice differentiable in and differentiable in . If , then, along the trajectory of system (2.6), we define an operator from to by where

#### 3. Main Results

In this section, we first propose a delay-dependent sufficient condition for stochastic stability with the performance of filtering error system (2.6). Now, define a stochastic Lyapunov functional candidate for systems (2.6) as where By Itô’s formula, we obtain the stochastic differential as where is the weak infinitesimal generator of the random process along the system (2.6).

Using the operator (2.12), we have The derivative of the first term in is given as follows: Following a similar method of (3.5), it is easy to obtain Using Lemma 2.4 and considering (2.3), we have Moreover, We define The following equations are true for any matrices , , , and with appropriate dimensions: where Considering (3.4)–(3.11), we obtain that where Therefore, we have the following result for the performance analysis.

Theorem 3.1. *Given scalars , , and , the fault detection system (2.6) is stochastically stable with an performance for any time delay satisfying (2.3), if there exist matrices , , , , , , , and matrices , , , denoted in (3.10)–(3.11) such that for each , **
where
*

*Proof. *Using Schur complement formula to (3.15), it can be seen that (3.15) is equivalent to
Now, we show that the filtering error system (2.6) with is stochastically stable. If , from (3.13) and (3.16)–(3.17), we can obtain
Inequality (3.20) implies that . Thus, we have
where . Therefore, for any , by Dynkin’s formula, we have
which means that . Thus, the filtering error system (2.6) with is stochastically stable by Definition 2.5.

In the sequel, we will deal with the performance of the filtering error system (2.6). Using (3.19) and performance, we have
Noting that the zero initial condition, then it follows from (3.23) that
Hence, if (3.15)–(3.17) hold, can be guaranteed. That is, for all nonzero . Therefore, the filtering error system (2.6) is stochastically stable with the performance 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 , , and . Also, the Lyapunov matrices , , , and depend on both the system mode and the delay mode . 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, can be extended to a wider range by dealing with the integral term in (3.5). Noting that , utilizing Lemma 2.4, one has
Further, deleting in (3.16) and adding to 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 , , and , the fault detection system (2.6) is stochastically stable with an performance for any time delay satisfying (2.3) if there exist matrices , , , , , , , , , , , and matrices , , , denoted in (3.10)–(3.11) such that for each , **
where
**
In this case, the parameters of the desired fault detection filter can be chosen by
*

*Proof. *For each , , we define a matrix by . Then, with the parameters in (3.28), it can be verified that, for each , , 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 in (3.15) includes , , and . Owing to the restrictions on the authors’ knowledge and the technique difficulties, is assumed to be diagonal matrices to obtain the parameters of the fault detection filter. Although this assumption may cause some conservativeness, considering complete 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:

The transition probability matrix is considered as
In this example, the weighting matrix in is supposed to be . Its state-space realization is given as (2.5) with , and . Also, we assume that , , , . For , by the Theorem 3.4 in this paper, the filter matrices are obtained as
For simulation purposes, we assume the initial condition . The time delays are , . The control input is chosen to be . The unknown input is assumed to be the band-limited white noise. The fault signal 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 and . Figure 3 shows the unknown input . Figure 4 shows the residual signal. Figure 5 is the simulation results of the evaluation function . Under the above conditions, with a selected threshold , the simulation of evaluation function with fault shows