H _/ H ∞ fault detection filter design for discrete-time stochastic systems with limited communication

Abstract

This paper is concerned with the fault detection (FD) problem for discrete-time stochastic systems with limited communication. A filter structure is proposed to construct the residual model for fault detection. For the limited network resources, a novel event-triggered strategy is employed to decrease the amount of data that is transmitted from the sensor to the filter. With the consideration of stochastic model and limited network resources, a novel event-based method is designed to ensure the residual system is stochastically stable and satisfies the desired fault sensitivity level and disturbance attenuation level. Compared with the traditional FD method, the proposed design strategy can not only achieve the desired fault detection performance, but also save the limited network resources. The effectiveness of design strategy is verified by two simulation examples.

Keywords

Fault detection discrete-time systems event-triggered strategy limited network resources stochastic systems

Introduction

The fault detection (FD) techniques have been widely developed over the past few decades owing to increasing demands on the safety and reliability in practical industry processes (Hwang et al. 2010; Li al. 2015; Yin and Liu 2017; Zhong et al. 2003). Among the existing FD schemes, the model-based technology has received great attention for its effectiveness. A fault is that one or more parameters of control systems are quite different from the normal values, which means that systems cannot work normally. Common faults can be divided into component fault, actuator fault and sensor fault. The general idea of fault detection is to construct a residual signal with an observer or filter structure. In addition, an evaluation function and a predefined threshold are also essential to be designed. When a fault occurs, the output of evaluation function will exceed the threshold and thus an alarm is generated. Many design methods with different performance indexes have been studied in the literature. For example, Casavola et al. (2005) designed an optimal $H_{\infty}$ deconvolution filter to detect and isolate faults for linear uncertain systems. Su et al. (2016) proposed a novel observer-based fuzzy filter to deal with the FD problem for nonlinear switched stochastic systems via T-S fuzzy modelling technique. In Khosrowjerdi et al. (2004), a mixed $H_{2} / H_{\infty}$ approach was developed to solve the fault detection and control problem. As one of the most interesting performance indexes, the $H_{-} / H_{\infty}$ approach has been widely applied in different systems. The advantages include that it can exhibit the robustness to disturbances and the sensitivity for faults, simultaneously; the conservatism of obtained results can be reduced. In Wang and Yang (2008), the authors studied the FD problem with an observer structure in finite frequency domain for linear discrete-time systems. The FD design problem with a mixed $H_{-} / H_{\infty}$ index was investigated for switched systems (Li and Yang, 2014), nonlinear systems with parameter uncertainty (Aouaouda et al., 2015) and Takagi-Sugeno fuzzy systems (Chadli et al., 2013). However, very few results have been studied for discrete-time stochastic systems discussed in this paper.

Since the Brownian motion phenomenon is typically exhibited in many complex practical systems, the studies for stochastic modeling have received considerable attention in recent years and many results for deterministic systems have been developed in the stochastic systems successfully (Farina et al., 2015; Liu et al., 2015; Xu and Chen, 2003). For example, the stabilization problem was studied for stochastic systems with time-varying parameter uncertainties (Xie and Xie, 2000) and Markovian jumping parameter uncertainties (Wang et al., 2002). In Wang et al. (2009), the controller design problem was investigated for time-delay stochastic systems with missing measurements. In Wei et al. (2016), the authors investigated the disturbance observer-based disturbance attenuation control problems for stochastic systems with multiple disturbances. An elegant anti-disturbance control approach was studied for discrete-time stochastic systems subject to nonlinearity and multiple disturbances (Wei and Sun, 2018). The authors designed an adaptive fuzzy controller to deal with control problem for stochastic nonlinear switched systems (Li et al., 2017). In Wu et al. (2014), the asynchronous $l_{2} - l_{\infty}$ filtering problem was investigated for discrete-time stochastic Markov jump systems with sensor nonlinearity. The sensor FD problem was studied for It $\hat{o}$ stochastic systems (Wu and Ho, 2009).

On another research front, networked control systems (NCSs) have been widely applied in modern industry for many advantages such as flexibility, higher reliability, simplified installation and maintainability. For NCSs, all signals are transmitted via a shared communication network. In many practical situations, however, the limited communication resources are the main constraint for signal transmission and several design strategies have been proposed to deal with this challenging problem (Fu and Xie, 2005; Liu et al., 2017; Yin et al., 2018; Zhao et al., 2010). Among these strategies, the event-triggered one as one of the most powerful strategies has been frequently applied in many practical fields. To discuss a few, an event-triggered control design scheme was proposed to study the control problem for networked systems subject to state and input quantizations (Hu and Yue, 2012a). In Wu et al. (2017), the authors investigated the event-triggered sliding mode control problem for uncertain stochastic systems subject to limited communication capacity. The authors considered the event-based filter problem for networked systems (Hu and Yue, 2012b). The FD problem was studied for nonlinear discrete-time systems while an event-triggered strategy was applied for reducing communication data (Li et al., 2016). The main advantage of this scheme is to construct an event detector to decide and transmit the useful signals that are significantly different from the previous ones. Thus, the limited communication resources can be utilized reasonably and the network transmission pressure can be reduced.

Although the FD problem without the event-triggered strategy has been investigated for continuous-time stochastic systems (Li and Yang, 2012), the event-triggered fault detection filter design problem for discrete-time stochastic systems with limited communication still remains challenging, which motivates this study. This paper studies the $H_{-} / H_{\infty}$ performance approach to guarantee the stochastic stability of augmented residual system with desired disturbance attenuation performance and fault sensitivity performance, while an event-triggered scheme is applied to save network bandwidth. Sufficient conditions are developed and formulated as linear matrix inequalities (LMIs) to obtain fault detection filter and event-triggered parameters. The main contributions are shown as follows:

The event-triggered fault detection design strategy is firstly applied for discrete-time stochastic systems, while an event detector is proposed to save network communication resources.

For the disturbance attenuation level and fault sensitivity level, two different Lyapunov functions are applied to reduce the conservatism of obtained results.

A co-design approach is derived to obtain the parameters of fault detection filter and event-triggered strategy.

Notations: The matrix transpose and inverse are defined by “T“ and “–1”, respectively. I and 0 represent the appropriate dimensional identity and zero matrix, respectively. $M > 0$ shows that M is a positive matrix. $(Ω, F, P)$ is a probability space, while $Ω$ , $F$ and $P$ , respectively, represent the sample space, $σ$ -algebra of subsets and probability measure. $E {x}$ is the expectation of x. $diag {\dots}$ represents the block-diagonal matrix. The sum of the matrix A and $A^{T}$ is shown as He(A). The $l_{2} ([0, \infty))$ norm is represented by $∥ \cdot ∥_{2}$ and $∥ f (k) ∥_{e 2}$ is the $l_{2}$ norm in $(Ω, F, P)$ and satisfies:

∥ f (k) ∥_{e 2} = E {\sum_{k = 0}^{N} ∥ f (k) ∥_{2}} = E {\sum_{k = 0}^{N} \sqrt{f^{T} (k) f (k)}} .

Problem statement and preliminaries

In this paper, the stochastic system with disturbance and fault is shown as follows

{\begin{matrix} x (k + 1) = Ax (k) + E_{1} v (k) + E_{2} f (k) \\ + [Bx (k) + E_{3} v (k) + E_{4} f (k)] ϖ (k) \\ y (k) = Cx (k) + F_{1} v (k) + F_{2} f (k) \\ + [Dx (k) + F_{3} v (k) + F_{4} f (k)] ϖ (k) \end{matrix}

(1)

where $x (k) \in R^{n}$ is state vector, $y (k) \in R^{q}$ is the measurement output vector of the plant, $v (k) \in R^{m}$ represents the unknown input, disturbance or modelling errors of the plant, $f (k) \in R^{l}$ is the fault signal. Both $v (k)$ and $f (k)$ are assumed to belong to $l_{2} [0, \infty)$ norm. The stochastic process $ϖ (k)$ is a zero-mean real constant and belongs to the probability space $(Ω, F, P)$ which is relative to an increasing family $(F_{k})_{k \in N}$ of $σ$ -algebras $F_{k} \subset F$ generated by $(ϖ (k))_{k \in N}$ , while the symbol $N$ represents a set of natural numbers. In this paper, the stochastic process $ϖ (k)$ is supposed to satisfy

E {ϖ (k)} = 0, E {ϖ (k)^{2}} = η, k = 0, 1, \dots

(2)

where $η$ is a known positive scalar. The matrices A, B, C, D, $E_{1}$ , $E_{2}$ , $E_{3}$ , $E_{4}$ , $F_{1}$ , $F_{2}$ , $F_{3}$ and $F_{4}$ are known parameters with appropriate dimensions.

The FD strategy proposed in this paper includes two contents: the residual generator and the evaluation function. The residual generator is constructed by a filter or observer structure, which has been widely applied in industrial applications. In this paper, the following filter structure is used to generate the residual signal

{\begin{matrix} \hat{x} (k + 1) = A_{f} \hat{x} (k) + B_{f} \hat{y} (k) \\ r (k) = C_{f} \hat{x} (k) + D_{f} \hat{y} (k) \end{matrix}

(3)

where the filter state vector is described by $\hat{x} (k) \in R^{n}$ , the generated residual signal is $r (k) \in R^{p}$ . $\hat{y} (k) \in R^{q}$ is the real input of filter. $A_{f}$ , $B_{f}$ , $C_{f}$ and $D_{f}$ are the fault detection filter (FDF) parameter matrices to be designed.

In the actual physical systems, the signal transmission capability of network device should be considered. In some conditions, the limited communication resources may not satisfy a large quantity of signal transmission. In order to overcome this issue and accomplish the fault detection objective at the same time, an event-triggered strategy is proposed in this paper. The main idea of this strategy is to decide and transmit the useful sensor data through the network with an event detector.

For the event-triggered FD strategy shown in Figure 1, we can obtain the sequence of data transmission instants ${k_{s}}_{s = 0, 1, 2 \dots}$ with the following event-triggered condition

k_{s + 1} = min_{k > k_{s}} {k | δ^{T} (k) Q δ (k) > ϵ y^{T} (k) Qy (k)},

(4)

where $ϵ > 0$ is the event-triggered threshold, $Q \in R^{q \times q}$ is a positive matrix to be designed properly. $δ (k) = y (i_{k}) - y (k)$ , for $k \in [k_{s}, k_{s + 1})$ . $y (k)$ and $y (k_{s})$ represent the current measured-data and the latest triggered one transmitted to the filter through the network, respectively. Once the condition (4) is satisfied, the current measured-data $y (k)$ will be defined as $y (k_{s + 1})$ and released to the filter through the network channel. The measured-data packets at the instants ${k_{s} + k, k_{s} + 2 k, \dots, k_{s + 1} - k}$ will be discarded purposely. Therefore, the resource utilization of network device will be significantly improved and the communication pressure will be reduced. The threshold $ϵ$ decides the transmission rates of the measured-data. The larger the threshold $ϵ$ is, the less measured-data will be transmitted to the filter.

As discussed above, the real input signal of the FDF can be described by

\hat{y} (k) = y (i_{k}) = y (k) + δ (k), k \in [i_{k}, i_{k + 1})

(5)

Substituting (5) into (3), one can obtain that

{\begin{matrix} \hat{x} (k + 1) = A_{f} \hat{x} (k) + B_{f} y (k) + B_{f} δ (k), \\ r (k) = C_{f} \hat{x} (k) + D_{f} y (k) + D_{f} δ (k), \end{matrix} k \in [i_{k}, i_{k + 1})

(6)

Define $ξ (t) = [\begin{matrix} x (k) \\ \hat{x} (k) \end{matrix}]$ , we can obtain the following augmented residual system by combining (1) and (6)

{\begin{matrix} ξ (k + 1) = A_{1} ξ (k) + A_{2} v (k) + A_{3} f (k) + A_{4} δ (k) \\ + [B_{1} ξ (k) + B_{2} v (k) + B_{3} f (k)] ϖ (k), \\ r (k) = C_{1} ξ (k) + C_{2} v (k) + C_{3} f (k) + C_{4} δ (k) \\ + [D_{1} ξ (k) + D_{2} v (k) + D_{3} f (k)] ϖ (k), \end{matrix} k \in [i_{k}, i_{k + 1})

(7)

where

\begin{matrix} A_{1} = [\begin{matrix} A & 0 \\ B_{f} C & A_{f} \end{matrix}], A_{2} = [\begin{matrix} E_{1} \\ B_{f} F_{1} \end{matrix}], A_{3} = [\begin{matrix} E_{2} \\ B_{f} F_{2} \end{matrix}], A_{4} = [\begin{matrix} 0 \\ B_{f} \end{matrix}], \\ B_{1} = [\begin{matrix} B & 0 \\ B_{f} D & 0 \end{matrix}], B_{2} = [\begin{matrix} E_{3} \\ B_{f} F_{3} \end{matrix}], B_{3} = [\begin{matrix} E_{4} \\ B_{f} F_{4} \end{matrix}], \\ C_{1} = [\begin{matrix} D_{f} C & C_{f} \end{matrix}], C_{2} = D_{f} F_{1}, C_{3} = D_{f} F_{2}, C_{4} = D_{f}, \\ D_{1} = [\begin{matrix} D_{f} D & 0 \end{matrix}], D_{2} = D_{f} F_{3}, D_{3} = D_{f} F_{4} \end{matrix}

The following definitions for augmented residual system (7) are essential for deriving the main results.

Definition 1: (Xu and Chen, 2003) With the condition of disturbance $v (k) = 0$ and fault $f (k) = 0$ , the augmented residual system (7) is stochastically stable if there exists a scalar $κ > 0$ such that

E {\sum_{k = 0}^{\infty} ∥ ξ (k) ∥^{2}} \leq κ ∥ ξ (0) ∥^{2} .

(8)

Definition 2: (Fault sensitivity performance): For a given scalar $γ > 0$ and the disturbance $v (k) = 0$ , the augmented residual system (7) is stochastically stable with a fault sensitivity level ( $H_{-}$ index) $γ$ if it is stochastically stable for $f (k) = 0$ and satisfies

∥ r (k) ∥_{e_{2}} \geq γ ∥ f (k) ∥_{e_{2}}

(9)

for all $f (k) \neq 0$ under the zero initial conditions.

Remark 1: $H_{-}$ index represents the lowest singular value of a transfer function matrix, which is widely applied for measuring the fault sensitivity for residual system (Li and Yang, 2012). In this paper, the definition of $H_{-}$ index for fault $f (k)$ to residual output $r (k)$ is shown as follows

inf_{v (k) = 0, f (k) = 0} \frac{∥ r (k) ∥_{e_{2}}}{∥ f (k) ∥_{e_{2}}} \geq γ .

Definition 3: (Disturbance attenuation performance): For a given scalar $β > 0$ and the fault $f (k) = 0$ , the augmented residual system (7) is stochastically stable with a disturbance attenuation level ( $H_{\infty}$ norm) if it is stochastically stable for $v (k) = 0$ and satisfies

∥ r (k) ∥_{e_{2}} \leq β ∥ v (k) ∥_{e_{2}}

(10)

for all $v (k) \neq 0$ under the zero initial conditions.

Problem: For the stochastic model (1) in this paper, the FD problem is formulated to design an event-triggered fault detection filter (ETFDF) such that the augmented residual system (7) is stochastically stable and satisfies (9)-(10) under the zero initial conditions.

Remark 2: The $H_{\infty}$ performance index $β$ shows the robustness of the disturbance signal on the residual output and (10) is a standard $H_{\infty}$ characterization for the filter design problem (Liu et al., 2005). The smaller the index $β$ is, the more robust performance becomes. The $H_{-}$ performance index $γ$ shows the sensitivity of the fault signal on the residual output. The larger the index $γ$ is, the more sensitive performance becomes. Therefore, the proposed event-triggered FD problem is developed by solving the multi-objective $H_{-} / H_{\infty}$ optimization problem.

For further discussion, the following lemma is helpful to derive the main results studied in this paper.

Lemma 1: (Projection Lemma (Pipeleers et al., 2009): Suppose $Θ \in R^{n \times n}$ , $M$ and $H$ are arbitrary matrices, we can find a matrix $F$ satisfying the following inequality

M^{T} F H + H^{T} F^{T} M + Θ < 0

(11)

if and only if the following two inequalities are satisfied

{\begin{matrix} N_{M}^{T} Θ N_{M} < 0 \\ N_{H}^{T} Θ N_{H} < 0 \end{matrix}

(12)

where $N_{M}$ and $N_{H}$ are arbitrary matrices that are constructed by all the column vectors of the nullspaces of matrices $M$ and $N$ , respectively.

Main results and proofs

In this section, a novel event-based approach is developed to solve the FD problem for discrete-time stochastic systems with limited communication. Two subsections are provided to obtain the conditions of fault sensitivity and disturbance attenuation, respectively. An optimization algorithm is formulated to obtain the desired parameters of filter and event detector.

Fault sensitivity condition

In this subsection, the following theorem is provided to obtain the stochastic stability of residual system (7) with the desired fault sensitivity condition (9).

Theorem 1: Consider the residual system (7) with the disturbance signal $v (k) = 0$ and suppose that the measured data satisfying the event-triggered condition (4) is transmitted to the FDF, the system is stochastically stable and satisfied the desired fault sensitivity condition described in (9) if there exist matrices $P_{11} > 0$ , $P_{13} > 0$ , $P_{12}$ , Q, $Y_{i}$ and $H_{i}$ for $i = 1 \dots 4$ , $M_{i}$ for $i = 1 \dots 2$ , N, R, $A_{F}$ , $B_{F}$ , $C_{F}$ and $D_{F}$ such that

[\begin{matrix} P Q_{11} - He (X_{1}) & - X_{3}^{T} & Ψ_{13} & - X_{2} + Ψ_{14} \\ * & - I_{2} & 0 & - X_{4} \\ * & * & - P Q_{12} & Ψ_{34} \\ * & * & * & Ψ_{44} \end{matrix}] < 0

(13)

where

\begin{matrix} P_{1} = [\begin{matrix} P_{11} & P_{12} \\ P_{12}^{T} & P_{13} \end{matrix}], X_{11} = [\begin{matrix} Y_{1} & M_{1} \\ N & N \end{matrix}], X_{12} = [\begin{matrix} Y_{2} & M_{2} \\ N & N \end{matrix}], X_{4} = [\begin{matrix} R \\ R \end{matrix}], \end{matrix}

\begin{matrix} I_{2} = [\begin{matrix} I & 0 \\ 0 & η I \end{matrix}], P Q_{12} = [\begin{matrix} P_{1} & 0 \\ 0 & Q \end{matrix}], X_{2} = {[\begin{matrix} X_{21}^{T} & X_{22}^{T} & H_{3}^{T} & H_{4}^{T} \end{matrix}]}^{T}, \\ C_{0} = [\begin{matrix} C & 0 \end{matrix}], D_{0} = [\begin{matrix} D & 0 \end{matrix}], X_{31} = X_{32} = [\begin{matrix} R L_{2} & R L_{3} \end{matrix}], \\ X_{21} = [\begin{matrix} H_{1} \\ N L_{1} \end{matrix}], X_{22} = [\begin{matrix} H_{2} \\ N L_{1} \end{matrix}], X_{3} = [\begin{matrix} X_{31} & 0 & R L_{2} & 0 \\ 0 & X_{32} & 0 & R L_{3} \end{matrix}], \end{matrix}

\begin{matrix} Ψ_{44} = r^{2} I + He (E_{2}^{T} H_{1} + F_{2}^{T} B_{F}^{T} L_{1} + E_{4}^{T} H_{2} + F_{4}^{T} B_{F}^{T} L_{1} \\ + F_{2}^{T} H_{3} + F_{4}^{T} H_{4} + F_{2}^{T} D_{F}^{T} + F_{4}^{T} D_{F}^{T}), \\ Ψ_{341} = A^{T} H_{1} + C^{T} B_{F}^{T} L_{1} + B^{T} H_{2} + D^{T} B_{F}^{T} L_{1} + C^{T} H_{3} \\ + D^{T} H_{4} + C^{T} D_{F}^{T} + D^{T} D_{F}^{T}, \\ Ψ_{34} = [\begin{matrix} Ψ_{341} \\ A_{F}^{T} L_{1} + C_{F}^{T} \\ B_{F}^{T} L_{1} + D_{F}^{T} \end{matrix}], P Q_{11} = [\begin{matrix} P_{1} & 0 & 0 & 0 \\ * & η P_{1} & 0 & 0 \\ * & * & ϵ Q & 0 \\ * & * & * & ϵ η Q \end{matrix}], \\ Ψ_{14} = [\begin{matrix} Y_{1}^{T} E_{2} + B_{F} F_{2} + L_{2}^{T} D_{F} F_{2} \\ M_{1}^{T} E_{2} + B_{F} F_{2} + L_{3}^{T} D_{F} F_{2} \\ Y_{2}^{T} E_{4} + B_{F} F_{4} + L_{2}^{T} D_{F} F_{4} \\ M_{2}^{T} E_{4} + B_{F} F_{4} + L_{3}^{T} D_{F} F_{4} \\ Y_{3}^{T} F_{2} + L_{2}^{T} D_{F} F_{2} \\ Y_{4}^{T} F_{4} + L_{3}^{T} D_{F} F_{4} \end{matrix}], X_{1} = [\begin{matrix} X_{11} & 0 & 0 & 0 \\ * & X_{12} & 0 & 0 \\ * & * & Y_{3} & 0 \\ * & * & * & Y_{4} \end{matrix}], \\ Ψ_{13} = [\begin{matrix} Y_{1}^{T} A + B_{F} C + L_{2}^{T} D_{F} C & A_{F} + L_{2}^{T} C_{F} & B_{F} + L_{2}^{T} C_{F} \\ M_{1}^{T} A + B_{F} C + L_{3}^{T} D_{F} C & A_{F} + L_{3}^{T} C_{F} & B_{F} + L_{3}^{T} C_{F} \\ Y_{2}^{T} B + B_{F} D + L_{2}^{T} D_{F} D & 0 & 0 \\ M_{2}^{T} B + B_{F} D + L_{3}^{T} D_{F} D & 0 & 0 \\ Y_{3}^{T} C + L_{2}^{T} D_{F} C & L_{2}^{T} C_{F} & L_{2}^{T} D_{F} \\ Y_{4}^{T} D + L_{3}^{T} D_{F} D & 0 & 0 \end{matrix}] \end{matrix}

and the auxiliary matrices $L_{1}$ , $L_{2}$ , $L_{3}$ are determined with appropriate dimensions beforehand.

Proof: The obtained residual system (7) with $v (k) = 0$ can be described by

ξ (k + 1) = A_{1} ξ (k) + A_{3} f (k) + A_{4} δ (k) + [B_{1} ξ (k) + B_{3} f (k)] ϖ (k)

(14)

Let

\begin{matrix} A B_{1} = {[\begin{matrix} A_{1}^{T} & B_{1}^{T} & C_{0}^{T} & D_{0}^{T} \\ A_{4}^{T} & 0 & 0 & 0 \end{matrix}]}^{T}, C D_{1} = [\begin{matrix} C_{1} & C_{4} \\ D_{1} & 0 \end{matrix}], \\ A B_{3} = {[\begin{matrix} A_{3}^{T} & B_{3}^{T} & F_{2}^{T} & F_{4}^{T} \end{matrix}]}^{T}, C D_{3} = [\begin{matrix} C_{3} \\ D_{3} \end{matrix}] \end{matrix}

According to (13), we have

\begin{matrix} [\begin{matrix} P Q_{11} & 0 & 0 & 0 \\ * & - I_{2} & 0 & 0 \\ * & * & - P Q_{12} & 0 \\ * & * & * & r^{2} I \end{matrix}] \\ + He ([\begin{matrix} - I & 0 \\ 0 & - I \\ {AB}_{1}^{T} & {CD}_{1}^{T} \\ {AB}_{3}^{T} & {CD}_{3}^{T} \end{matrix}] X [\begin{matrix} I & 0 & 0 & 0 \\ 0 & 0 & 0 & I \end{matrix}]) < 0 \end{matrix}

(15)

where $X = [\begin{matrix} X_{1} & X_{2} \\ X_{3} & X_{4} \end{matrix}]$ .

Based on Lemma 1, one can obtain the inequality (15) only when the following two LMIs are satisfied

{[\begin{matrix} A B_{1} & A B_{3} \\ C D_{1} & C D_{3} \\ I & 0 \\ 0 & I \end{matrix}]}^{T} Ξ_{1} [\begin{matrix} A B_{1} & A B_{3} \\ C D_{1} & C D_{3} \\ I & 0 \\ 0 & I \end{matrix}] < 0

(16)

{[\begin{matrix} 0 & 0 \\ I & 0 \\ 0 & I \\ 0 & 0 \end{matrix}]}^{T} Ξ_{1} [\begin{matrix} 0 & 0 \\ I & 0 \\ 0 & I \\ 0 & 0 \end{matrix}] < 0

(17)

where $Ξ_{1} = [\begin{matrix} P Q_{11} & 0 & 0 & 0 \\ * & - I_{2} & 0 & 0 \\ * & * & - P Q_{12} & 0 \\ * & * & * & r^{2} I \end{matrix}]$ and r is the designed fault sensitivity index.

On the other hand, the following Lyapunov function is applied for the system (14)

V (k, ξ (k)) = ξ (k)^{T} P_{1} ξ (k)

(18)

Then, we have

\begin{matrix} Δ V (k, ξ (k)) = E {V (k + 1, ξ (k + 1) | ξ (k))} - V (k, ξ (k)) \\ = {[\begin{matrix} A_{1} ξ (k) + A_{3} f (k) + A_{4} δ (k) \end{matrix}]}^{T} P_{1} [\begin{matrix} A_{1} ξ (k) + A_{3} f (k) + A_{4} δ (k) \end{matrix}] \\ + η {[\begin{matrix} B_{1} ξ (k) + B_{3} f (k) \end{matrix}]}^{T} P_{1} [\begin{matrix} B_{1} ξ (k) + B_{3} f (k) \end{matrix}] - ξ {(k)}^{T} P_{1} ξ (k) \end{matrix}

(19)

According to the event-triggered strategy (4), we have

\begin{matrix} Δ V (k, ξ (k)) + E {γ^{2} f {(k)}^{T} f (k) - r {(k)}^{T} r (k))} \\ < Δ V (k, ξ (k)) + E {γ^{2} f {(k)}^{T} f (k) - r {(k)}^{T} r (k))} \\ + ϵ y^{T} (k) Qy (k) - δ^{T} (k) Q δ (k) \\ = α {(k)}^{T} Θ α (k) \end{matrix}

(20)

where $α (k) = {[\begin{matrix} ξ (k) & f (k) & δ (k) \end{matrix}]}^{T}$ , $Θ = {[\begin{matrix} A B_{1} & A B_{3} \\ C D_{1} & C D_{3} \\ I & 0 \\ 0 & I \end{matrix}]}^{T} Ξ_{1} [\begin{matrix} A B_{1} & A B_{3} \\ C D_{1} & C D_{3} \\ I & 0 \\ 0 & I \end{matrix}]$ .

According to the inequality (16), one can obtain that $Θ < 0$ . Subsequently, there exists a constant $κ > 0$ such that $Δ V (k, ξ (k)) < κ ∥ ξ (0) ∥^{2}$ when $f (k) = 0$ . Thus, based on the Definition 1, one can obtain that the residual system is stochastically stable.

When $f (k) \neq 0$ , summing both sides of (20) over $[0, N]$ , one can get

\begin{matrix} E {\sum_{k = 0}^{N} (γ^{2} f {(k)}^{T} f (k) - r {(k)}^{T} r (k))} \\ + V (N + 1, ξ (N + 1)) - V (0, ξ (0)) < 0 \end{matrix}

(21)

where $V (N + 1, ξ (N + 1)) > 0$ and $V (0, ξ (0)) = 0$ under the zero initial condition, we have

E {\sum_{k = 0}^{N} (γ^{2} f {(k)}^{T} f (k) - r {(k)}^{T} r (k))} < 0

(22)

therefore, fault sensitivity performance (9) is satisfied and the proof is completed.

Disturbance attenuation condition

In this subsection, the following theorem is provided to obtain the stochastic stability of residual system (7) with the desired disturbance attenuation condition (10).

Theorem 2: Consider the system (7) with the fault signal $f (k) = 0$ and suppose that the measured data satisfying the event-triggered condition (4) is transmitted to the FDF, the system is stochastically stable and satisfied the desired disturbance attenuation condition described in (10) if there exist matrix variables $P_{21} > 0$ , $P_{23} > 0$ , $P_{22}$ , Q, and $Y_{i}$ for $i = 5 \dots 8$ , $M_{i}$ for $i = 3, 4$ , N, R, $A_{F}$ , $B_{F}$ , $C_{F}$ and $D_{F}$ such that

[\begin{matrix} P Q_{21} - He (X) & Ψ_{12} & Ψ_{13} & 0 \\ * & - P Q_{22} & 0 & Ψ_{24} \\ * & * & - β^{2} I & Ψ_{34} \\ * & * & * & I_{2} - He (R_{2}) \end{matrix}] < 0

(23)

where

\begin{matrix} P_{2} = [\begin{matrix} P_{21} & P_{22} \\ P_{22}^{T} & P_{23} \end{matrix}], X_{1} = [\begin{matrix} Y_{5} & M_{3} \\ N & N \end{matrix}], X_{2} = [\begin{matrix} Y_{6} & M_{4} \\ N & N \end{matrix}], \\ P Q_{22} = [\begin{matrix} P_{2} & 0 \\ 0 & Q \end{matrix}], R_{2} = [\begin{matrix} R & 0 \\ 0 & R \end{matrix}], \\ P Q_{21} = [\begin{matrix} P_{2} & 0 & 0 & 0 \\ * & η P_{2} & 0 & 0 \\ * & * & ϵ Q & 0 \\ * & * & * & ϵ η Q \end{matrix}], X = [\begin{matrix} X_{1} & 0 & 0 & 0 \\ * & X_{2} & 0 & 0 \\ * & * & Y_{7} & 0 \\ * & * & * & Y_{8} \end{matrix}], \end{matrix}

\begin{matrix} Ψ_{12} = [\begin{matrix} Y_{5}^{T} A + B_{F} C & A_{F} & B_{F} \\ M_{3}^{T} A + B_{F} C & A_{F} & B_{F} \\ Y_{6}^{T} B + B_{F} D & 0 & 0 \\ M_{4}^{T} A + B_{F} D & 0 & 0 \\ Y_{7}^{T} C & 0 & 0 \\ Y_{8}^{T} D & 0 & 0 \end{matrix}], Ψ_{13} = [\begin{matrix} Y_{5}^{T} E_{1} + B_{F} F_{1} \\ M_{5}^{T} E_{1} + B_{F} F_{1} \\ Y_{6}^{T} E_{3} + B_{F} F_{3} \\ M_{6}^{T} E_{3} + B_{F} F_{3} \\ Y_{7}^{T} F_{1} \\ Y_{8}^{T} F_{3} \end{matrix}], \\ I_{2} = [\begin{matrix} I & 0 \\ 0 & η I \end{matrix}], \\ Ψ_{24} = [\begin{matrix} C^{T} D_{F}^{T} & D^{T} D_{F}^{T} \\ C_{F}^{T} & 0 \\ D_{F}^{T} & 0 \end{matrix}], Ψ_{34} = [\begin{matrix} F_{1}^{T} D_{F}^{T} & F_{3}^{T} D_{F}^{T} \end{matrix}] \end{matrix}

Proof: The augmented residual system (7) with $f (k) = 0$ can be described by

ξ (k + 1) = A_{1} ξ (k) + A_{2} v (k) + A_{4} δ (k) + [B_{1} ξ (k) + B_{2} v (k)] ϖ (k)

(24)

Based on (23), it is easily to observe that $I_{2} - He (R_{2}) < 0$ and $R_{2}$ is a nonsingular matrix. As $I_{2} > 0$ , we can obtain that $(I_{2} - R_{2})^{T} I_{2}^{- 1} (I_{2} - R_{2}) \geq 0$ which implies that $- R_{2}^{T} I_{2}^{- 1} R_{2} \leq I_{2} - He (R_{2})$ . Then, the following matrix inequality can be obtained from (23)

[\begin{matrix} P Q_{21} - He (X) & Ψ_{12} & Ψ_{13} & 0 \\ * & - P Q_{22} & 0 & Ψ_{24} \\ * & * & - β^{2} I & Ψ_{34} \\ * & * & * & - R_{2}^{T} I_{2}^{- 1} R_{2} \end{matrix}] < 0

(25)

Define

\begin{matrix} A B_{1} = {[\begin{matrix} A_{1}^{T} & B_{1}^{T} & C_{0}^{T} & D_{0}^{T} \\ A_{4}^{T} & 0 & 0 & 0 \end{matrix}]}^{T}, C D_{1} = [\begin{matrix} C_{1} & C_{4} \\ D_{1} & 0 \end{matrix}], \\ A B_{2} = {[\begin{matrix} A_{2}^{T} & B_{2}^{T} & F_{1}^{T} & F_{3}^{T} \end{matrix}]}^{T}, C D_{2} = [\begin{matrix} C_{2} \\ D_{2} \end{matrix}] \end{matrix}

where $C_{0}$ and $D_{0}$ have been defined in Theorem 1.

Based on Lemma 1, one can obtain the inequality (25) only when the following two LMIs are satisfied

{[\begin{matrix} A B_{1} & A B_{2} \\ I & 0 \\ 0 & I \end{matrix}]}^{T} Ξ_{2} [\begin{matrix} A B_{1} & A B_{2} \\ I & 0 \\ 0 & I \end{matrix}] < 0

(26)

{[\begin{matrix} 0 & 0 \\ I & 0 \\ 0 & I \end{matrix}]}^{T} Ξ_{2} [\begin{matrix} 0 & 0 \\ I & 0 \\ 0 & I \end{matrix}] < 0

(27)

where $Ξ_{2} = [\begin{matrix} P Q_{21} & 0 & 0 \\ * & - P Q_{22} + {CD}_{1}^{T} \cdot I_{2} \cdot C D_{1} & {CD}_{1}^{T} \cdot I_{2} \cdot C D_{2} \\ * & * & - β^{2} I + {CD}_{2}^{T} \cdot I_{2} \cdot C D_{2} \end{matrix}]$ .

On the other hand, the following Lyapunov function is applied for the system (24)

V (k, ξ (k)) = ξ (k)^{T} P_{2} ξ (k)

(28)

Similar to the proof 1, we have

\begin{matrix} Δ V (k, ξ (k)) + E {r {(k)}^{T} r (k)) - β^{2} v {(k)}^{T} v (k)} \\ < Δ V (k, ξ (k)) + E {r {(k)}^{T} r (k)) - β^{2} v {(k)}^{T} v (k)} \\ + ϵ y^{T} (k) Qy (k) - δ^{T} (k) Q δ (k) \\ = α {(k)}^{T} Θ α (k) \end{matrix}

(29)

where $α (k) = {[\begin{matrix} ξ (k) & v (k) & δ (k) \end{matrix}]}^{T}$ , $Θ = {[\begin{matrix} A B_{1} & A B_{2} \\ I & 0 \\ 0 & I \end{matrix}]}^{T} Ξ_{2} [\begin{matrix} A B_{1} & A B_{2} \\ I & 0 \\ 0 & I \end{matrix}]$ .

According to inequality (26), one can obtain that $Θ < 0$ . Subsequently, there exists a constant $κ > 0$ such that $Δ V (k, ξ (k)) < κ ∥ ξ (0) ∥^{2}$ when $v (k) = 0$ . Thus, based on the Definition 1, one can obtain that the residual system is stochastically stable.

When $v (k) \neq 0$ , summing both sides of (29) over $[0, N]$ , one can get

\begin{matrix} E {\sum_{k = 0}^{N} (r {(k)}^{T} r (k) - β^{2} v {(k)}^{T} v (k))} \\ + V (N + 1, ξ (N + 1)) - V (0, ξ (0)) < 0 \end{matrix}

(30)

where $V (N + 1, ξ (N + 1)) > 0$ and $V (0, ξ (0)) = 0$ under the zero initial condition, we have

E {\sum_{k = 0}^{N} (r {(k)}^{T} r (k) - β^{2} v {(k)}^{T} v (k))} < 0

(31)

therefore, disturbance attenuation performance (10) is satisfied and the proof is completed.

Solutions of the event-triggered fault detection filter

Based on the Theorem 1 and 2, we can obtain the event-triggered fault detection filter parameter matrices by solving the following minimization problem

\begin{matrix} min β \\ s . t . (13), (23) \end{matrix}

(32)

where $γ$ is a predetermined positive scalar (Li and Yang, 2012).

Then, the following procedure is applied to obtain the parameters of desired FD filter:

Step 1: For given parameters $η$ , $ϵ$ , $γ$ and auxiliary matrices $L_{1}$ , $L_{2}$ , $L_{3}$ , the matrix parameters $A_{F}$ , $B_{F}$ , $C_{F}$ , $D_{F}$ , N and R can be obtained by computing the optimization problem (32).

Step 2: the desired FD filter parameters can be computed by

A_{f} = N^{- T} A_{F}, B_{f} = N^{- T} B_{F}, C_{f} = R^{- T} C_{F}, D_{f} = R^{- T} D_{F}

Remark 3: In the normal condition, the minimization problem that minimizes $(β - γ)$ subject to the minimization problem (32) can obtain better results. However, in order to obtain the relationship between the disturbance attenuation level $β$ and the measured data transmission rates, we predefine the fault sensitivity level $γ$ . We can also predefine the disturbance attenuation level $β$ to obtain the relationship between the fault sensitivity level $γ$ and the measured data transmission rates. The desired fault sensitivity level $γ$ can be obtained by soloving the following optimization problem

\begin{matrix} max γ \\ s . t . (13), (23) \end{matrix}

(33)

where $β$ is a predetermined positive scalar.

Residual evaluation

In order to evaluate the effect of the obtained event-triggered fault detection filter parameter matrices, a residual evaluation function is essential to be designed. Moreover, the fault can be detected by comparing with the predefined threshold $J_{th}$ . The structure of residual evaluation function adopted in this paper can be described by

J_{r} (k) = {(E {\frac{1}{N} \sum_{k = k_{1}}^{k_{2}} r^{T} (k) r (k)})}^{\frac{1}{2}}, N = k_{2} - k_{1} + 1

(34)

where $k_{1}$ represents the initial evaluation time instant and N represents the length of evaluation time instant.

The predefined threshold is as the following structure

J_{th} = sup_{f = 0, v \neq 0} J_{r} (k)

(35)

Then, the fault signal can be detected successfully with the following relationship comparing the evaluation function $J_{r}$ with the threshold $J_{th}$

J_{r} > J_{th} \Rightarrow faults \Rightarrow alarm

J_{r} \leq J_{th} \Rightarrow no faults

Illustrative example

In this section, two simulation experiments are provided to illustrate the effectiveness and applicability of the proposed ETFDF design strategy.

Example 1

Consider the discrete-time stochastic system in $(1)$ with the model parameters given as follows

\begin{matrix} A = [\begin{matrix} - 0.2 & 0.3 \\ 0.5 & 0.1 \end{matrix}], B = [\begin{matrix} 0 & 0.1 \\ - 0.1 & 0.3 \end{matrix}], \\ E_{1} = [\begin{matrix} 0.1 \\ - 0.2 \end{matrix}], E_{2} = [\begin{matrix} 0.2 \\ - 0.3 \end{matrix}], \\ C = [\begin{matrix} 1 & 0 \\ 0.2 & - 0.2 \end{matrix}], D = [\begin{matrix} 0.2 & 0.5 \\ 0 & - 0.7 \end{matrix}], \\ F_{1} = [\begin{matrix} 0.2 \\ 0.1 \end{matrix}], F_{2} = [\begin{matrix} - 0.1 \\ - 0.2 \end{matrix}], \\ E_{3} = [\begin{matrix} 0 \\ - 0.1 \end{matrix}], E_{4} = [\begin{matrix} - 0.2 \\ 0.1 \end{matrix}], F_{3} = [\begin{matrix} 0.1 \\ 0 \end{matrix}], F_{4} = [\begin{matrix} 0.2 \\ - 0.3 \end{matrix}] \end{matrix}

In this example, we set the stochastic process $ω (k)$ satisfying (2) with $η = 0.1$ , the threshold of the event-triggered condition satisfying (4) with $ϵ = 0.05$ , the fault sensitivity level $γ = 1$ , the auxiliary matrices $L_{1} = {[\begin{matrix} 0.3 & 0.1 \end{matrix}]}^{T}$ , $L_{2} = [\begin{matrix} 3.4 & - 0.3 \end{matrix}]$ and $L_{3} = [\begin{matrix} 0 & - 0.1 \end{matrix}]$ . By solving the minimization problem (32) with the MATLAB LMI Tools, we can calculate the desired disturbance attenuation level $β = 1.5527$ , and the fault detection filter parameter matrices can be calculated by

\begin{matrix} A_{f} = [\begin{matrix} 0.4945 & 0.9090 \\ - 0.1115 & - 0.8584 \end{matrix}], B_{f} = [\begin{matrix} 0.3645 & 0.9753 \\ - 0.5341 & 0.0426 \end{matrix}], \\ C_{f} = [\begin{matrix} 0.9248 & 0.6625 \end{matrix}], D_{f} = [\begin{matrix} - 3.3344 & 8.8817 \end{matrix}] \end{matrix}

Meanwhile, the event-triggered parameter matrix can be obtained by

Q = [\begin{matrix} 42.0286 & - 114.4655 \\ - 114.6950 & 340.9541 \end{matrix}]

In addition, we choose the following external disturbance signal $v (k)$ and the fault signal $f (k)$ to exhibit the effectiveness of the proposed design strategy

v (k) = {\begin{matrix} 0.1 * \sin (k), & 30 \leq k \leq 170 \\ 0, & else \end{matrix}

(36)

f (t) = {\begin{matrix} 0.3, & 50 \leq k \leq 150 \\ 0, & else \end{matrix}

(37)

The simulation results are shown in Figures 2 –4. Among them, the residual signal under the event-triggered strategy is shown in Figure 2. The detection threshold and the residual evaluation curves under fault case and fault free case are shown in Figure 3. Based on (34)–(35), we can conclude the designed detection threshold $J_{th} = 6.7256 \times 10^{- 3}$ and $J_{r} (50) = 1.1238 \times 10^{- 2} > J_{th}$ . Thus, the fault will be detected when it occurs at k=50. Figure 4 demonstrates the release instants of measured data. From which, we can see clearly that only 120 data packets satisfying the event-triggered condition $(4)$ are successfully transmitted to the filter and the data transmission rate is 60%. Above all, the proposed design strategy can achieve the desired fault detection objective taking the event-triggered strategy into account (see Figure 2 and Figure 3). In addition, the amount of measured data transmitted through the network are decreased efficiently (see Figure 4). The energy and network bandwidth can be saved significantly.

For different event-triggered threshold $ϵ$ and the stochastic process parameter $η$ , the disturbance attenuation level $β$ and the transmission rate $α$ are different. Their relationships are shown in Table 1 and Table 2, respectively. From Table 1, it is easily to see that the smaller the $ϵ$ is, the better the disturbance attenuation performance becomes and the more data packages are transmitted to the filter. From Table 2, we can see that the smaller the stochastic process parameter $η$ is, the better the disturbance attenuation performance becomes and the less data packages are transmitted to the filter.

Table 1.

Minimum $β$ and transmission rate $α$ for different $ϵ$ when $η = 0.1$ .

$ϵ$	0.05	0.10	0.15	0.20	0.25
$β$	1.5527	1.8485	2.1702	2.5535	3.0521
α(%)	60	50	43.5	40	36.5

Table 2.

Minimum $β$ and transmission rate $α$ for different $η$ when $ϵ = 0.1$ .

$η$	0.10	0.15	0.20	0.25	0.30
$β$	1.8485	1.9267	2.0620	2.2394	2.4583
α(%)	50	54.5	56	61.5	66.5

By solving the optimization problem (33), we can also calculate the fault sensitivity level $γ$ and the measured-data transmission rate under different event-triggered threshold $ϵ$ and stochastic process parameter $η$ . Their relationships are shown in Table 3 and Table 4, respectively. From Table 3, we can observe that that the smaller event-triggered threshold $ϵ$ is, the better fault sensitivity performance becomes and the more measured-data should be transmitted to the desired filter. Table 4 shows that the smaller stochastic process parameter $η$ is, the better fault sensitivity performance becomes the less measured-data are transmitted to the filter. Table 5 and Table 6 give the comparison results between the proposed method and existing reference (Li and Yang, 2012), which has been cited in References section.

Table 3.

Minimum $γ$ and transmission rate $α$ for different $ϵ$ when $η = 0.1$ .

$ϵ$	0.02	0.04	0.06	0.08	0.10
$γ$	0.6880	0.5909	0.5062	0.4244	0.3382
α(%)	71.5	63	56.5	52.5	45.5

Table 4.

Minimum $γ$ and transmission rate $α$ for different $η$ when $ϵ = 0.06$ .

$η$	0.10	0.15	0.20	0.25	0.30
$γ$	0.5062	0.4967	0.4536	0.3927	0.3162
α(%)	56.5	63.5	65.5	67.5	69.5

Table 5.

The comparison between the proposed method and the existing one.

Design methods	The proposed method	The method in Li and Yang (2012)
Release number of sensor data	120	200
Disturbance attenuation level $β$	1.5527	1.2365

Table 6.

The comparison between the proposed method and the existing one.

Design methods	The proposed method	The method in Li and Yang (2012)
Release number of sensor data	133	200
Disturbance attenuation level $β$	4.2191	3.6365

For further discussion, we compare the proposed result with the existing time-triggered scheme designed in (Li and Yang, 2012). The comparison results are shown in Table 5, from which we can observe that: the designed fault detection filter can achieve the desired fault detection objective by transmitting 120 sensor data and the transmission rate is 60%, while all 200 sensor data should be transmitted to achieve the similar objective with the approach designed in (Li and Yang, 2012). Therefore, the proposed strategy in this paper is helpful to reduce computation and increase the lifetime of the overall system.

Example 2

Consider the uncertain stochastic system (1), in which model parameters are taken from Xu et al. (2013)

\begin{matrix} A = [\begin{matrix} - 0.25 & - 0.11 & - 0.12 \\ - 0.01 & - 0.30 & - 0.09 \\ 0.13 & - 0.12 & - 0.23 \end{matrix}], E_{1} = [\begin{matrix} - 2.5 \\ 2.0 \\ 1.4 \end{matrix}], \\ B = [\begin{matrix} - 0.10 & 0.13 & 0.31 \\ 0.00 & 0.16 & - 0.17 \\ 0.10 & - 0.10 & - 0.23 \end{matrix}], E_{3} = [\begin{matrix} - 0.21 \\ 0.61 \\ 0.15 \end{matrix}], \\ C = [\begin{matrix} - 0.31 & 0.3 & - 0.63 \\ 0.70 & 0.40 & 0.50 \end{matrix}], F_{1} = [\begin{matrix} 0.3 \\ - 0.3 \end{matrix}], \end{matrix}

The fault matrices and other model parameters are assumed to be

\begin{matrix} E_{2} = [\begin{matrix} - 2.00 \\ 3.00 \\ 2.40 \end{matrix}], E_{4} = [\begin{matrix} - 0.42 \\ 1.20 \\ 0.30 \end{matrix}], \\ F_{2} = [\begin{matrix} - 0.60 \\ - 0.80 \end{matrix}], F_{4} = [\begin{matrix} 0.02 \\ - 0.03 \end{matrix}], \\ D = [\begin{matrix} - 0.01 & 0.03 & 0.30 \\ 0.20 & 0.10 & - 0.20 \end{matrix}], F_{3} = [\begin{matrix} - 0.03 \\ 0.05 \end{matrix}], \end{matrix}

In this example, we set the stochastic process $ω (k)$ satisfying (2) with $η = 0.1$ , the threshold of the event-triggered condition satisfying (4) with $ϵ = 0.1$ , the fault sensitivity level $γ = 1$ , the auxiliary matrices $L_{1} = {[\begin{matrix} 0.3 & 0.1 & 4.2 \end{matrix}]}^{T}$ , $L_{2} = [\begin{matrix} 0.4 & - 0.3 & 0.1 \end{matrix}]$ and $L_{3} = [\begin{matrix} 0 & - 0.1 & 0.2 \end{matrix}]$ . According to the optimization problem (32), one can deduce the desired disturbance attenuation level $β = 4.2191$ , and the fault detection filter parameter matrices can be calculated by

\begin{matrix} A_{f} = [\begin{matrix} 0.2453 & - 0.3310 & 0.1928 \\ 0.3026 & 1.4651 & - 0.4422 \\ 1.0113 & 4.7644 & - 1.2304 \end{matrix}], \\ B_{f} = [\begin{matrix} - 1.7027 & - 1.4674 \\ 1.7189 & 3.0466 \\ 3.5684 & 6.9833 \end{matrix}], \\ C_{f} = [\begin{matrix} - 0.1294 & - 0.6215 & 0.1801 \end{matrix}], \\ D_{f} = [\begin{matrix} - 5.6165 & - 3.5850 \end{matrix}] \end{matrix}

Meanwhile, the event-triggered parameter matrix can be obtained by

Q = [\begin{matrix} 74.9674 & 47.4111 \\ 47.4111 & 37.8305 \end{matrix}] .

The residual evaluation curves and threshold $J_{r}$ are depicted in Figure 5 and Figure 6 shows the release instants and release interval taking the event-triggered scheme. From which, one can observe that the desired event-triggered fault detection filter can detect the fault in time. Moreover, 133 measured-data packets are successfully transmitted to the the filter and the data transmission rate is 66.5%.

Similarly, we compare the proposed event-triggered fault detection design scheme with the previous one (Li and Yang, 2012). Table 6 shows the comparison results, which indicates that the proposed design scheme can achieve the desired fault detection performance by transmitting 133 measured-data through the network, while all 200 measured-data should be transmitted to achieve the similar performance with the previous one designed in (Li and Yang, 2012).

Conclusion

In this paper, the fault detection problem has been investigated for discrete-time stochastic systems with limited communication. With the consideration of stochastic model and limited network resources, an event-triggered fault detection filter has been designed to ensure the stochastic stability of the residual system and the desired $H_{\infty} / H_{-}$ performance index. In addition, only the necessary data are transmitted to the filter, which can lead to a more rational utilization of limited network resources. Then, an optimization solution has been derived to co-design the parameters of the filter and the event detector. Finally, an available example is proposed to demonstrate the validity of the design strategy.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflict of interests with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partially supported by the National Natural Science Foundation of China (61673133).

ORCID iD

Zhaoke Ning

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