Fault detection of networked control systems in stochastic environments

Abstract

This paper deals with the problem of fault detection for networked control systems in the stochastic environments. The packet dropout, network-induced delay and out-of-order packets due to long delay are assumed to exist stochastically in the sensor-to-controller link and the controller-to-actuator link. By assuming that the stochastic environment could be described as a homogeneous Markovian chain, the networked control system is modelled as a discrete-time, non-uniform sampling stochastic parameter-varying Markov jump system. Based on the model, the residual generator is developed and the problem of fault detection is formatted as a filtering problem. By using the theory of a Markovian jump linear system, a fault detection filter which makes the residual generation system stochastically stable is considered, and a prescribed disturbance attenuation level is satisfied. A sufficient condition to solve this problem is given in terms of the linear matrix inequalities. A numerical example shows that the proposed fault detection filter is sensitive to the fault but robust to exogenous disturbance.

Keywords

Networked control systems fault detection linear matrix inequalities packet dropout stochastic processes

Introduction

With the rapid development of communication networks, conventional control system architectures have been evolving to modern networked control, and a great amount of effort has been devoted to the problems of networked control systems (NCSs). NCSs are control systems in which controller and plant are connected via a communication channel. A NCS integrates information, communications, and control into a single system in which control loops are closed over computer networks. Compared with conventional control systems, the networked control has a number of advantages, e.g. reduced cost, high resource utilization, simple installation and maintenance, increased system agility and reduced system wiring. Thus, it has received increasing interest in recent years. An introduction to networked control systems can be found (Tian and Levy, 2008; Yue et al., 2009).

However, owing to the limitation of the network resources, the introduction of a communication network also brings some new problems and challenges, such as network-induced delay, data packet dropout, network scheduling and quantization problems, which all will inevitably degrade the performance of the NCSs and even cause system instability, and make analysis and synthesis of NCSs complex. The researches on NCSs have become important as international investigation focused on the automatic control domain, such as the control problem (Peng et al., 2011; Wang and Yang, 2011), the stability problem and stabilization problem (Zhang and Fang, 2011) of NCSs, and so on. Among these researches focusing on NCSs, the fault diagnosis and fault-tolerant control of NCSs have received extensive attention. A review of work in the fault diagnosis domain of NCSs can be seen (Fang et al., 2007; Aubrun et al., 2008). A fault detection (FD) approach to NCSs with network-induced and unknown input based on eigendecomposition, adaptive evaluation and adaptive thresholds is proposed (Wang et al., 2007). A new FD scheme for networked control systems subject to uncertain time-varying delay is discussed (Wang et al., 2008). The FD problems of NCSs with random packet dropout are developed (Wang et al., 2009a, 2009b). Peng et al. (2010) addressed the problem of the FD for linear time-invariant systems over data networks with limited network quality of service (QoS) and the FD error dynamic systems are transferred to Markov jumping systems. A set of Kronecker delta functions is employed to characterize the random measurement delays and the stochastic data missing phenomenon, and the FD filter is designed in solving a certain set of linear matrix inequalities (LMIs) (He et al., 2008, 2010). Li and Tang (2010) presented a new design approach to the fault diagnostor for NCSs by constructing a novel reduced-order state observer of the augmented system, which can diagnose faults without using residual to embody faults.

To the best of the authors’ knowledge, much literature has been reported about FD problem for NCSs, but most of published results on fault detection of NCSs considers either delay or packet dropouts by employing robust FD methods (Ding, 2008). One category of existing work assumes that the statistics of delay or packet dropouts are known (Zheng et al, 2006; He et al., 2008; Wang et al., 2009b). Another category transforms the influence of delay or packet dropouts into model uncertainties and unknown inputs (Wang et al., 2008, 2009b). Only a few results consider random network environments with network delay, packet dropouts and out-of-order packets simultaneously. This motivates us to study this interesting and challenging problem, which has great potential in practical applications.

In this paper, the problems of modeling and fault detection for NCSs in a stochastic environment are considered, and the effect of the disturbance on the NCSs is also considered. Then the FD scheme is proposed and makes the residual generation system stochastically stable, and satisfies a prescribed disturbance attenuation level.

The rest of this paper is organized as follows. In the next section we give the system description and propose a mathematical model to describe the stochastic NCS. A FD filter is then considered. Next we give some main results and simulation results, which are followed by the conclusions.

Notation

The notation used throughout the paper is standard. Here $R$ and $R^{n \times m}$ denote n-dimensional Euclidean space and set of all $n \times m$ real matrices, respectively. For matrices or vectors, superscript T indicates transposition. For symmetric matrices, $X > 0 (< 0)$ indicates that $X$ is positive definite (negative definite). In some partitioned symmetric matrices, the symbol $*$ generically denotes each of its symmetric blocks. We use $l_{2}$ to denote the space of square summable sequences and $E (\cdot)$ to denote the expectation of $(\cdot)$ .

NCS model and problem statement

Consider a linear continuous time-invariant system as the controlled plant

{\begin{matrix} \overset{\cdot}{x} (t) = Ax (t) + Bu (t) + Dd (t) + Ff (t) \\ y (t) = Cx (t) \end{matrix}

(1)

where $x (t) \in R^{n}$ , $u (t) \in R^{s}$ , $y (t) \in R^{h}$ are the state vector, the control input vector and the measurement output vector, respectively. The unknown input $d (t) \in R^{p}$ denotes modelling errors, or exogenous disturbance input or random noise, $f (t) \in R^{q}$ is the directly immeasurable fault signal vector. Without loss of generality, the $l_{2}$ norms of $u (t)$ , $d (t)$ and $f (t)$ are assumed to be existing and bounded and $A$ , $B$ , $C$ , $D$ , $F$ are real constant matrices of appropriate dimensions.

Assumption 1. The sensor and actuator are clock driven and synchronized, the sampling period is T, and all measurement data packets are time-stamped. The actuator has a logic zero-order hold (ZOH).

Assumption 2. The controller is event driven and triggered by the arrival of a measurement data packet. Once the control data become available, it sends a control packet to the actuator of the control loop. The control packet carries the time-stamps of the measurement and control data.

Assumption 3. For any control loop, the measurement data of the plant is transmitted with a single packet and so is the control data.

Assumption 4. No new control command is generated if a data packet is dropped out since the controller is event driven. In this paper, the signal adopted by the actuator will keep the previous input if no new control commands are updated at the actuator.

Remark 1. Under the aforementioned assumptions, here is no distinction between packet dropouts that occur in the sensor-to-controller connection and the controller-to-actuator connection in the network. Indeed, for packet dropouts between the sensor and the controller (as shown the interval [(l+7)T, (l+8)T] in Figure 1) no new control update is computed by the controller, thus no new control input is sent to the actuator. In the case of packet dropouts between the controller and the actuator (as shown the interval [(l+3)T, (l+4)T] in Figure 1), the packet cannot be received by the actuator either.

Remark 2. Employing the time-stamping technique described in Assumptions 1 and 2, the logic ZOH at the actuator stores the latest control packet. This implies that the logic ZOH discards all control packets but the most recent valid one (as shown in Figure 1). The actuator keeps its control signal unchanged until the output of the logic ZOH gets updated to a new value (Peng et al., 2011).

An example of the timing diagram of data transmission of the considered NCS in a stochastic environment is shown in Figure 1, in which the two control signals shown in dashed lines are lost, and some signals are not used by the actuator due to the long delay and being out of order (as shown by (l+1) and (l+5)). It can be seen from the timing diagram that the control inputs acting on the actuator are different from one sampling interval to another, and thus the system models of the NCS vary from one sampling interval to another as the packet dropout situations change.

In Figure 1, the $t_{k} (k = 1, 2, 3, \dots)$ are some integers such that ${t_{1}, t_{2}, t_{3}, \dots} \subset {1, 2, 3, \dots}$ , and denote the time constant of the updating packet in the actuator side. Suppose that the time instants corresponding to two successively updating data packet are $t_{k} T$ and $t_{k + 1} T$ , respectively, with $t_{k + 1} T - t_{k} T = θ_{k} T (1 \leq θ_{k} \leq N)$ , then the results caused by delay, packet dropouts and being out of order can be regarded as non-uniform sampling with sampling intervals ${θ_{k} T}$ at the actuator (Suplin et al., 2007; Wang et al., 2009b), and $θ_{k}$ can be obtained from the time-stamp of the received control packet. Similar to the work of Wang et al. (2009b), the dynamics of the plant (1) at the time $t_{k}$ can be written as the following discrete-time Markov jump linear model

{\begin{matrix} x (k + 1) = A (θ_{k}) x (k) + B (θ_{k}) u (k) + D (θ_{k}) d (k) + F (θ_{k}) f (k) \\ y (k) = Cx (k) \end{matrix}

(2)

for k=0,1,2,…, where $t_{0} = 0$ , $t_{k} = \sum_{j = 0}^{k - 1} θ_{j} T (k \geq 1)$ , $A (θ_{k}) = e^{A (θ_{k} T)}$ , $B (θ_{k}) = \int_{0}^{θ_{k} T} e^{At} dtB$ $D (θ_{k}) = \int_{0}^{θ_{k} T} e^{At} dtD$ , $F (θ_{k}) = \int_{0}^{θ_{k} T} e^{At} dtF$ , $x (k) = x (t_{k})$ , $u (k) = u (t_{k})$ , $y (k) = y (t_{k})$ , ${θ_{k}}$ is a discrete homogeneous Markov chain taking values in a finite state space $Θ = {1, 2, \dots, N}$ with transition probability matrix $Π = [λ_{ij}]_{i, j \in Θ}$ , and $λ_{ij}$ is defined as

λ_{ij} = Pr {θ_{k + 1} = j | θ_{k} = i}

where $\sum_{j = 1}^{N} λ_{ij} = 1$ and $A (θ_{k})$ , $B (θ_{k})$ , $D (θ_{k})$ , $F (θ_{k})$ are known real constant matrices for all $θ_{k} = i \in Θ$ . The matrices $A_{i} : = A (θ_{k} = i)$ , $B_{i} : = B (θ_{k} = i)$ , $D_{i} : = D (θ_{k} = i)$ and $F_{i} : = F (θ_{k} = i)$ , are known real constant matrices of appropriate dimensions.

Remark 3. Although Equation (2) is obtained in a similar manner to the work of Wang et al. (2009b), the meaning of $θ_{k} T$ is different. In Equation (2), $θ_{k} T$ denotes the interval between two successfully received control packets at the actuator, and describes random character between the sensor–controller channel and controller–actuator channel, while $θ_{k} T$ is assumed as the interval between two successive successfully received measurements at the controller and just denotes the packet dropout of the sensor–controller channel of Wang et al. (2009b).

Fault detection filter design

A typical FD system consists of a residual generator and a residual evaluation stage including an evaluation function and a threshold. For the purpose of residual generation, the following fault detection filter is used

{\begin{matrix} \hat{x} (k + 1) = A_{fi} \hat{x} (k) + B_{fi} y_{f} (k) \\ r_{k} = C_{fi} \hat{x} (k) + D_{fi} y_{f} (k) \end{matrix}

(3)

where $\hat{x} (k) \in R^{m}$ is the state vector of the filter, $y_{f} (k)$ and $r_{k}$ are the input and output of the filter, respectively, the $r_{k}$ is the residual vector and $A_{fi}$ , $B_{fi}$ , $C_{fi}$ and $D_{fi}$ are appropriately dimensioned filter matrices to be determined.

Remark 4. The design idea of the FD filter is similar to the work of Yao et al. (2011). The filter of Yao et al. (2011) is designed to detect the faults of singular systems, while our paper deals with the problem of modelling and FD for networked control systems in stochastic environments. So the results of Yao et al. (2011) and our paper are different.

Data packet dropout happens unavoidably in the network environment, so the measurements $y (k)$ will drop randomly. In this paper, it is assumed that the data packet dropout is described by a Bernoulli stochastic variable $α_{k}$ with $α_{k} = 0$ denoting the measurements dropout condition, $α_{k} = 1$ the correct receiving condition. Here $α_{k}$ can be defined as

Pr {α_{k} = 1} = E {α_{k}} = α, Pr {α_{k} = 0} = 1 - E {α_{k}} = 1 - α

where $α \in R$ is a known positive scalar. The relationship between $y (k)$ and $y_{f} (k)$ can be established, i.e.

y_{f} (k) = α_{k} y (k)

(4)

By the second expression of Equations (2), (3) and (4), the FD filter is to be the following form

{\begin{matrix} \hat{x} (k + 1) = A_{fi} \hat{x} (k) + α_{k} B_{fi} Cx (k) \\ r_{k} = C_{fi} \hat{x} (k) + α_{k} D_{fi} Cx (k) \end{matrix}

(5)

The objective of FD is to identify the fault $f (k)$ when it appears. Similar to the cases presented by Gao et al. (2008), Zhong et al. (2005) and Yao et al. (2010), the introduction of a suitable weighting matrix $G (s)$ is used to limit the frequency interval, in which the fault should be identified, and the system performance could be improved. Here we have the equation $\tilde{f} (s) = G (s) f (s)$ , where $f (s)$ and $\tilde{f} (s)$ denote the Laplace transforms of $f (t)$ and $\tilde{f} (t)$ , respectively. One state space realization of $\tilde{f} (s) = G (s) f (s)$ can be

{\begin{matrix} \tilde{x} (k + 1) = A_{w} \tilde{x} (k) + B_{w} f (k) \\ \tilde{f} (k) = C_{w} \tilde{x} (k) + D_{w} f (k) \end{matrix}

(6)

where $\tilde{x} (k) \in R^{r}$ is the state vector, and $A_{w}$ , $B_{w}$ , $C_{w}$ and $D_{w}$ are chosen correspondingly.

We use the notation $η (k) = {[\begin{matrix} x^{T} (k) & {\tilde{x}}^{T} (k) & {\hat{x}}^{T} (k) \end{matrix}]}^{T}$ , $w (k) = {[\begin{matrix} u^{T} (k) & d^{T} (k) & f^{T} (k) \end{matrix}]}^{T}$ , $e (k) = r_{k} - \tilde{f} (k)$ and ${\tilde{α}}_{k} = α_{k} - α$ , here $α$ ( $0 \leq α \leq 1$ ) is a positive constant.

Based on (2), (5) and (6), the residual system is given by

{\begin{matrix} η (k + 1) = ({\tilde{A}}_{1 i} + {\tilde{α}}_{k} {\tilde{A}}_{2 i}) η (k) + {\tilde{B}}_{i} w (k) \\ e (k) = ({\tilde{C}}_{1 i} + {\tilde{α}}_{k} {\tilde{C}}_{2 i}) η (k) + {\tilde{D}}_{i} w (k) \end{matrix}

(7)

where

{\tilde{A}}_{1 i} = [\begin{matrix} A_{i} & 0 & 0 \\ 0 & A_{w} & 0 \\ α B_{f i} C & 0 & A_{f i} \end{matrix}], {\tilde{A}}_{2 i} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ B_{f i} C & 0 & 0 \end{matrix}], {\tilde{B}}_{i} = [\begin{matrix} B_{i} & D_{i} & F_{i} \\ 0 & 0 & B_{w} \\ 0 & 0 & 0 \end{matrix}]

{\tilde{C}}_{1 i} = [\begin{matrix} α D_{fi} C & - C_{w} & C_{fi} \end{matrix}], {\tilde{C}}_{2 i} = [\begin{matrix} D_{fi} C & 0 & 0 \end{matrix}], {\tilde{D}}_{i} = [\begin{matrix} 0 & 0 & - D_{w} \end{matrix}]

In the residual evaluation stage, an evaluation function and a threshold should be provided for the purpose of FD. Here, as in most contributions, we adopt the following residual evaluation function

J (k) = {\sum_{i = 1}^{k} r_{i}^{T} r_{i}}^{\frac{1}{2}}

(8)

The threshold is selected as

J_{th} = sup_{0 \neq d (k), u (k) \in l_{2}, f = 0} \sqrt{\sum_{i = 1}^{ρ} r_{i}^{T} r_{i}}

(9)

where $ρ$ denotes the maximum time step of the evaluation function. The decision logic can be defined as

\begin{matrix} J (k) > J_{th}, fault and alarm for fault \\ J (k) \leq J_{th}, no fault \end{matrix}

(10)

Main results

For formulating some practically computable criteria to obtain the filter parameters described by (3), the following definition is useful in deriving the criteria.

Definition 1 (Yao et al., 2011). When $w (k) = 0$ , system (7) is said to be stochastically Markovian jump stable, if there exists a piecewise quadratic Lyapunov function

V (η (k), θ_{k}) = η^{T} (k) P (θ_{k}) η (k)

(11)

with $P_{i} = P (θ_{k} = i) > 0, i \in Θ$ , and its difference $Δ V = E {V (η (k + 1), θ_{k + 1})} - V (η (k), θ_{k})$ is negative decreasing.

Theorem 1. Consider the system (7) with zero initial conditions, let $γ > 0$ be a given scalar, then the residual error $e (k)$ satisfies the following performance index

‖ e ‖_{2, E} < γ ‖ w ‖_{2}

(12)

and when $w (k) = 0$ the system is stochastically stable, if there exist matrices $P_{i} > 0, i \in Θ$ such that the following LMIs

[\begin{matrix} - P_{i} & * & * & * & * & * \\ 0 & - γ^{2} I & * & * & * & * \\ {\tilde{C}}_{1 i} & {\tilde{D}}_{i} & - I & * & * & * \\ u {\tilde{C}}_{2 i} & 0 & 0 & - I & * & * \\ u {\bar{P}}_{i} {\tilde{A}}_{2 i} & 0 & 0 & 0 & - {\bar{P}}_{i} & * \\ {\bar{P}}_{i} {\tilde{A}}_{1 i} & {\bar{P}}_{i} {\tilde{B}}_{i} & 0 & 0 & 0 & - {\bar{P}}_{i} \end{matrix}] < 0

(13)

hold for $i \in Θ$ , where

{\bar{P}}_{i} = \sum_{j = 1}^{N} λ_{ij} P_{j}, u = \sqrt{α (1 - α)}, ‖ e ‖_{2, E} = E {\sqrt{\sum_{k = 0}^{\infty} e^{T} (k) e (k)}}, ‖ w ‖_{2} = \sqrt{\sum_{k = 0}^{\infty} w {(k)}^{T} w (k)}

Proof. An index is introduced as

J_{Δ} = E {η^{T} (k + 1) P (θ_{k + 1}) η (k + 1) | η (k), θ_{k}} - η^{T} (k) P (θ_{k}) η (k) + E {e^{T} (k) e (k)} - γ^{2} w (k)^{T} w (k)

Then, along the residual system (7),

\begin{matrix} J_{Δ} = E {η^{T} (k + 1) P (θ_{k + 1}) η (k + 1) | η (k), θ_{k}} - η^{T} (k) P (θ_{k}) η (k) \\ + E {e^{T} (k) e (k)} - γ^{2} w (k)^{T} w (k) \\ = E {(({\tilde{A}}_{1 i} + {\tilde{α}}_{k} {\tilde{A}}_{2 i}) η (k) + {\tilde{B}}_{i} w (k))^{T} {\bar{P}}_{i} (({\tilde{A}}_{1 i} + {\tilde{α}}_{k} {\tilde{A}}_{2 i}) η (k) \\ + {\tilde{B}}_{i} w (k))} - η (k)^{T} P_{i} η (k) \\ + E {(({\tilde{C}}_{1 i} + {\tilde{α}}_{k} {\tilde{C}}_{2 i}) η (k) + {\tilde{D}}_{i} w (k))^{T} (({\tilde{C}}_{1 i} + {\tilde{α}}_{k} {\tilde{C}}_{2 i}) η (k) \\ + {\tilde{D}}_{i} w (k))} - γ^{2} w (k)^{T} w (k) \\ = E {η^{T} (k) {\tilde{A}}_{1 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{1 i} η (k) + {\tilde{α}}_{k} η^{T} (k) {\tilde{A}}_{1 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{2 i} η (k) + η^{T} (k) {\tilde{A}}_{1 i}^{T} {\bar{P}}_{i} {\tilde{B}}_{i} w (k) \\ + {\tilde{α}}_{k} η^{T} (k) {\tilde{A}}_{2 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{1 i} η (k) + {\tilde{α}}_{k}^{2} η^{T} (k) {\tilde{A}}_{2 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{2 i} η (k) \\ + {\tilde{α}}_{k} η^{T} (k) {\tilde{A}}_{2 i}^{T} {\bar{P}}_{i} {\tilde{B}}_{i} w (k) \\ + w^{T} (k) {\tilde{B}}_{i}^{T} {\bar{P}}_{i} {\tilde{A}}_{1 i} η (k) + {\tilde{α}}_{k} w^{T} (k) {\tilde{B}}_{i}^{T} {\bar{P}}_{i} {\tilde{A}}_{2 i} η (k) + w^{T} (k) {\tilde{B}}_{i}^{T} {\bar{P}}_{i} {\tilde{B}}_{i} w (k)} \\ - η (k)^{T} P_{i} η (k) + E {η^{T} (k) {\tilde{C}}_{1 i}^{T} {\tilde{C}}_{1 i} η (k) + {\tilde{α}}_{k} η^{T} (k) {\tilde{C}}_{1 i}^{T} {\tilde{C}}_{2 i} η (k) \\ + η^{T} (k) {\tilde{C}}_{1 i}^{T} {\tilde{D}}_{i} w (k) \\ + {\tilde{α}}_{k} η^{T} (k) {\tilde{C}}_{2 i}^{T} {\tilde{C}}_{1 i} η (k) + {\tilde{α}}_{k}^{2} η^{T} (k) {\tilde{C}}_{2 i}^{T} {\tilde{C}}_{2 i} η (k) + {\tilde{α}}_{k} η^{T} (k) {\tilde{C}}_{2 i}^{T} {\tilde{D}}_{i} w (k) \\ + w^{T} (k) {\tilde{D}}_{i}^{T} {\tilde{C}}_{1 i} η (k) + {\tilde{α}}_{k} w^{T} (k) {\tilde{D}}_{i}^{T} {\tilde{C}}_{2 i} η (k) + w^{T} (k) {\tilde{D}}_{i}^{T} {\tilde{D}}_{i} w (k)} \\ - γ^{2} w (k)^{T} w (k) \end{matrix}

(14)

Note the fact that $E {{\tilde{α}}_{k}} = E {α_{k} - α} = 0$ and $E {{\tilde{α}}_{k}^{2}} = α (1 - α)$ , we have

J_{Δ} = {[\begin{matrix} η (k) \\ w (k) \end{matrix}]}^{T} Λ_{i} [\begin{matrix} η (k) \\ w (k) \end{matrix}]

where

Λ_{i} = [\begin{matrix} - P_{i} + {\tilde{C}}_{1 i}^{T} {\tilde{C}}_{1 i} + {\tilde{A}}_{1 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{1 i} + α (1 - α) {\tilde{A}}_{2 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{2 i} + α (1 - α) {\tilde{C}}_{2 i}^{T} {\tilde{C}}_{2 i} & {\tilde{A}}_{1 i}^{T} {\bar{P}}_{i} {\tilde{B}}_{i} + {\tilde{C}}_{1 i}^{T} {\tilde{D}}_{i} \\ {\tilde{B}}_{i}^{T} {\bar{P}}_{i} {\tilde{A}}_{1 i} + {\tilde{D}}_{i}^{T} {\tilde{C}}_{1 i} & - γ^{2} I + {\tilde{D}}_{i}^{T} {\tilde{D}}_{i} + {\tilde{B}}_{i}^{T} {\bar{P}}_{i} {\tilde{B}}_{i} \end{matrix}]

Applying Schur complement to (13), we have $Λ_{i} < 0$ , and which leads to

E {η^{T} (k + 1) P (θ_{k + 1}) η (k + 1) | η (k), θ_{k}} - η^{T} (k) P (θ_{k}) η (k) + E {e^{T} (k) e (k)} - γ^{2} w (k)^{T} w (k) < 0

(15)

Summing up both sides of the above inequality from 0 to $\infty$ , and considering zero initial condition and

E {η^{T} (k + 1) P (θ_{k + 1}) η (k + 1)} |_{k \to \infty} > 0

we have

E {\sum_{k = 0}^{\infty} e^{T} (k) e (k)} - γ^{2} \sum_{k = 0}^{\infty} w {(k)}^{T} w (k) < 0

so the condition (12) is satisfied.

Next, we proved that system (7) is stochastically Markovian jump stable. Given the Lyapunov function (11), when $w (k) = 0$ , we can obtain from the deducing process of Equation (14)

Δ V = E {V (η (k + 1), θ_{k + 1})} - V (η (k), θ_{k}) = η^{T} (k) (- P_{i} + {\tilde{A}}_{1 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{1 i} + α (1 - α) {\tilde{A}}_{2 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{2 i}) η (k)

from (13), and applying the Schur complement to (13), we have

- P_{i} + {\tilde{A}}_{1 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{1 i} + α (1 - α) {\tilde{A}}_{2 i}^{T} {\bar{P}}_{i} {\tilde{A}}_{2 i} < 0

so $Δ V < 0$ . We can obtain that system (7) is stochastically Markovian jump stable. The proof is completed.□

Lemma 1 (Zhong et al., 2005). Consider system (7) and let $γ > 0$ be a given scalar, then LMIs (13) with $P_{i} > 0, i \in Θ$ are feasible, if and only if there exist matrices $P_{i} > 0$ and $W_{i}$ such that the following LMIs

[\begin{matrix} - P_{i} & * & * & * & * & * \\ 0 & - γ^{2} I & * & * & * & * \\ {\tilde{C}}_{1 i} & {\tilde{D}}_{i} & - I & * & * & * \\ u {\tilde{C}}_{2 i} & 0 & 0 & - I & * & * \\ u W_{i} {\tilde{A}}_{2 i} & 0 & 0 & 0 & - W_{i} - W_{i}^{T} + {\bar{P}}_{i} & * \\ W_{i} {\tilde{A}}_{1 i} & W_{i} {\tilde{B}}_{i} & 0 & 0 & 0 & - W_{i} - W_{i}^{T} + {\bar{P}}_{i} \end{matrix}] < 0

(16)

hold for $i \in Θ$ .

Theorem 2 Consider the residual system (7). Let $m = n + r$ . For a given positive constant $γ$ , there exist $Q_{i} = [\begin{matrix} Q_{1 i} & Q_{2 i} \\ Q_{2 i}^{T} & Q_{3 i} \end{matrix}] > 0$ and ${\bar{W}}_{i} = [\begin{matrix} U_{i} & Γ_{i} \\ V_{i} & Γ_{i} \end{matrix}]$ , matrices ${\bar{A}}_{fi}$ , ${\bar{B}}_{fi}$ , ${\bar{C}}_{fi}$ and ${\bar{D}}_{fi}$ satisfying the following LMIs for $i \in Θ$

[\begin{matrix} - Q_{i} & * & * & * & * & * \\ 0 & - γ^{2} I & * & * & * & * \\ [\begin{matrix} α {\bar{D}}_{fi} \underline{C} - {\underline{C}}_{w} & {\bar{C}}_{fi} \end{matrix}] & {\tilde{D}}_{i} & - I & * & * & * \\ [\begin{matrix} u {\bar{D}}_{fi} \underline{C} & 0 \end{matrix}] & 0 & 0 & - I & * & * \\ [\begin{matrix} u {\bar{B}}_{fi} \underline{C} & 0 \\ u {\bar{B}}_{fi} \underline{C} & 0 \end{matrix}] & 0 & 0 & 0 & - {\bar{W}}_{i} - {\bar{W}}_{i}^{T} + {\bar{Q}}_{i} & * \\ [\begin{matrix} U_{i} {\underline{A}}_{i} + α {\bar{B}}_{fi} \underline{C} & {\bar{A}}_{fi} \\ V_{i} {\underline{A}}_{i} + α {\bar{B}}_{fi} \underline{C} & {\bar{A}}_{fi} \end{matrix}] & [\begin{matrix} U_{i} {\underline{B}}_{i} \\ V_{i} {\underline{B}}_{i} \end{matrix}] & 0 & 0 & 0 & - {\bar{W}}_{i} - {\bar{W}}_{i}^{T} + {\bar{Q}}_{i} \end{matrix}] < 0

(17)

where ${\bar{Q}}_{i} = \sum_{j = 1}^{N} λ_{ij} Q_{j}, j \in Θ$ , ${\underline{A}}_{i} = [\begin{matrix} A_{i} & 0 \\ 0 & A_{w} \end{matrix}]$ , ${\underline{B}}_{i} = [\begin{matrix} B_{i} & D_{i} & F_{i} \\ 0 & 0 & B_{w} \end{matrix}]$ , $\underline{C} = [\begin{matrix} C & 0 \end{matrix}]$ and ${\underline{C}}_{w} = [\begin{matrix} 0 & C_{w} \end{matrix}]$ . There exists a FD filter in the form of (3) such that residual system (7) is stochastically stable with the guaranteed performance index $γ$ . Moreover, if the above conditions are feasible, then the matrices for a desired filter in the form of (3) are given

[\begin{matrix} A_{fi} & B_{fi} \\ C_{fi} & D_{fi} \end{matrix}] = [\begin{matrix} Γ_{i}^{- 1} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} {\bar{A}}_{fi} & {\bar{B}}_{fi} \\ {\bar{C}}_{fi} & {\bar{D}}_{fi} \end{matrix}]

(18)

Proof. Inspired by Gao et al. (2008) and Yao et al. (2011), let the matrices $P_{i}$ and $W_{i}^{T}$ be partitioned as

P_{i} = [\begin{matrix} P_{1 i} & P_{2 i} \\ P_{2 i}^{T} & P_{3 i} \end{matrix}], W_{i}^{T} = [\begin{matrix} W_{1 i} & W_{2 i} \\ W_{4 i} & W_{3 i} \end{matrix}]

(19)

where $P_{1 i} \in R^{(n + r) \times (n + r)}$ , $P_{2 i} \in R^{(n + r) \times m}$ , $P_{3 i} \in R^{m \times m}$ , $W_{1 i} \in R^{(n + r) \times (n + r)}$ , $W_{2 i} \in R^{(n + r) \times m}$ , $W_{3 i} \in R^{m \times m}$ and $W_{4 i} \in R^{m \times (n + r)}$ . The $W_{3 i}$ and $W_{4 i}$ are non-singular. Define the following matrices

J_{i} = [\begin{matrix} I & 0 \\ 0 & W_{3 i}^{- 1} W_{4 i} \end{matrix}], Q_{i} = J_{i}^{T} P_{i} J_{i} = [\begin{matrix} Q_{1 i} & Q_{2 i} \\ Q_{2 i}^{T} & Q_{3 i} \end{matrix}]

Thus $J_{i}$ is non-singular. By pre- and post-multiplying the resulting expression with matrices $diag {J_{i}^{T}, I, I, I, J_{i}^{T}, J_{i}^{T}}$ and $diag {J_{i}, I, I, I, J_{i}, J_{i}}$ to (16) respectively, we have

[\begin{matrix} - J_{i}^{T} P_{i} J_{i} & * & * & * & * & * \\ 0 & - γ^{2} I & * & * & * & * \\ {\tilde{C}}_{1 i} J_{i} & {\tilde{D}}_{i} & - I & * & * & * \\ u {\tilde{C}}_{2 i} J_{i} & 0 & 0 & - I & * & * \\ {uJ}_{i}^{T} W_{i} {\tilde{A}}_{2 i} J_{i} & 0 & 0 & 0 & J_{i}^{T} (- W_{i} - W_{i}^{T} + {\bar{P}}_{i}) J_{i} & * \\ J_{i}^{T} W_{i} {\tilde{A}}_{1 i} J_{i} & J_{i}^{T} W_{i} {\tilde{B}}_{i} & 0 & 0 & 0 & J_{i}^{T} (- W_{i} - W_{i}^{T} + {\bar{P}}_{i}) J_{i} \end{matrix}] < 0

(20)

thus, the following equations can be obtained

{\tilde{C}}_{1 i} J_{i} = [\begin{matrix} α D_{fi} \underline{C} - {\underline{C}}_{w} & C_{fi} W_{3 i}^{- 1} W_{4 i} \end{matrix}], u {\tilde{C}}_{2 i} J_{i} = u [\begin{matrix} D_{fi} \underline{C} & 0 \end{matrix}], {uJ}_{i}^{T} W_{i} {\tilde{A}}_{2 i} J_{i} = u [\begin{matrix} W_{4 i}^{T} B_{fi} \underline{C} & 0 \\ W_{4 i}^{T} B_{fi} \underline{C} & 0 \end{matrix}], J_{i}^{T} W_{i} {\tilde{B}}_{i} = [\begin{matrix} W_{1 i}^{T} {\underline{B}}_{i} \\ W_{4 i}^{T} W_{3 i}^{- T} W_{2 i}^{T} {\underline{B}}_{i} \end{matrix}]

J_{i}^{T} W_{i} {\tilde{A}}_{1 i} J_{i} = [\begin{matrix} W_{1 i}^{T} {\underline{A}}_{i} + α W_{4 i}^{T} B_{fi} \underline{C} & W_{4 i}^{T} A_{fi} W_{3 i}^{- 1} W_{4 i} \\ W_{4 i}^{T} W_{3 i}^{- T} W_{2 i}^{T} {\underline{A}}_{i} + α W_{4 i}^{T} B_{fi} \underline{C} & W_{4 i}^{T} A_{fi} W_{3 i}^{- 1} W_{4 i} \end{matrix}],

let

{\bar{W}}_{i} = J_{i}^{T} W_{i} J_{i} = {[\begin{matrix} I & 0 \\ 0 & W_{3 i}^{- 1} W_{4 i} \end{matrix}]}^{T} [\begin{matrix} W_{1 i}^{T} & W_{4 i}^{T} \\ W_{2 i}^{T} & W_{3 i}^{T} \end{matrix}] [\begin{matrix} I & 0 \\ 0 & W_{3 i}^{- 1} W_{4 i} \end{matrix}] = [\begin{matrix} W_{1 i}^{T} & W_{4 i}^{T} W_{3 i}^{- 1} W_{4 i} \\ W_{4 i}^{T} W_{3 i}^{- T} W_{2 i}^{T} & W_{4 i}^{T} W_{3 i}^{- 1} W_{4 i} \end{matrix}]

{\bar{Q}}_{i} = J_{i}^{T} {\bar{P}}_{i} J_{i} = \sum_{j = 1}^{N} λ_{ij} Q_{j}, (i = 1, \dots, N)

then (20) can be rewritten as

[\begin{matrix} Π_{1} & * \\ Π_{3} & Π_{2} \end{matrix}] < 0

(21)

here

Π_{1} = [\begin{matrix} - Q_{i} & * & * \\ 0 & - γ^{2} I & * \\ [\begin{matrix} α D_{fi} \underline{C} - {\underline{C}}_{w} & C_{fi} W_{3 i}^{- 1} W_{4 i} \end{matrix}] & {\tilde{D}}_{i} & - I \end{matrix}] Π_{2} = [\begin{matrix} - I & * & * \\ 0 & - {\bar{W}}_{i} - {\bar{W}}_{i}^{T} + {\bar{Q}}_{i} & * \\ 0 & 0 & - {\bar{W}}_{i} - {\bar{W}}_{i}^{T} + {\bar{Q}}_{i} \end{matrix}]

Π_{3} = [\begin{matrix} [\begin{matrix} u D_{fi} \underline{C} & 0 \end{matrix}] & 0 & 0 \\ [\begin{matrix} {uW}_{4 i}^{T} B_{fi} \underline{C} & 0 \\ {uW}_{4 i}^{T} B_{fi} \underline{C} & 0 \end{matrix}] & 0 & 0 \\ [\begin{matrix} W_{1 i}^{T} {\underline{A}}_{i} + α W_{4 i}^{T} B_{fi} \underline{C} & W_{4 i}^{T} A_{fi} W_{3 i}^{- 1} W_{4 i} \\ W_{4 i}^{T} W_{3 i}^{- T} W_{2 i}^{T} {\underline{A}}_{i} + α W_{4 i}^{T} B_{fi} \underline{C} & W_{4 i}^{T} A_{fi} W_{3 i}^{- 1} W_{4 i} \end{matrix}] & [\begin{matrix} W_{1 i}^{T} {\underline{B}}_{i} \\ W_{4 i}^{T} W_{3 i}^{- T} W_{2 i}^{T} {\underline{B}}_{i} \end{matrix}] & 0 \end{matrix}]

define

{\bar{W}}_{i} = [\begin{matrix} W_{1 i}^{T} & W_{4 i}^{T} W_{3 i}^{- 1} W_{4 i} \\ W_{4 i}^{T} W_{3 i}^{- T} W_{2 i}^{T} & W_{4 i}^{T} W_{3 i}^{- 1} W_{4 i} \end{matrix}] = [\begin{matrix} U_{i} & Γ_{i} \\ V_{i} & Γ_{i} \end{matrix}]

[\begin{matrix} {\bar{A}}_{fi} & {\bar{B}}_{fi} \\ {\bar{C}}_{fi} & {\bar{D}}_{fi} \end{matrix}] = [\begin{matrix} W_{4 i}^{T} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} A_{fi} & B_{fi} \\ C_{fi} & D_{fi} \end{matrix}] [\begin{matrix} W_{3 i}^{- 1} W_{4 i} & 0 \\ 0 & I \end{matrix}] = [\begin{matrix} W_{4 i}^{T} A_{fi} W_{3 i}^{- 1} W_{4 i} & W_{4 i}^{T} B_{fi} \\ C_{fi} W_{3 i}^{- 1} W_{4 i} & D_{fi} \end{matrix}]

(22)

from (21), one readily obtains (17).

From Equation (22), the FD filter matrices $A_{fi}$ , $B_{fi}$ , $C_{fi}$ and $D_{fi}$ in Equation (3) can be written as

\begin{array}{l} [\begin{matrix} A_{f i} & B_{f i} \\ C_{f i} & D_{f i} \end{matrix}] = [\begin{matrix} W_{4 i}^{- T} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} {\bar{A}}_{f i} & {\bar{B}}_{f i} \\ {\bar{C}}_{f i} & {\bar{D}}_{f i} \end{matrix}] [\begin{matrix} {(W_{3 i}^{- 1} W_{4 i})}^{- 1} & 0 \\ 0 & I \end{matrix}] \\ = [\begin{matrix} {(W_{3 i}^{- 1} W_{4 i})}^{- 1} Γ_{i}^{- 1} & 0 \\ 0 & I \end{matrix}] [\begin{matrix} {\bar{A}}_{f i} & {\bar{B}}_{f i} \\ {\bar{C}}_{f i} & {\bar{D}}_{f i} \end{matrix}] [\begin{matrix} {(W_{3 i}^{- 1} W_{4 i})}^{- 1} & 0 \\ 0 & I \end{matrix}] \end{array}

which implies that $(W_{3 i}^{- 1} W_{4 i})^{- 1}$ can be viewed as a similarity transformation on the state space realization of the filter and has no effect on the filter mapping form $y_{f} (k)$ to $r_{k}$ . Without loss of generality, we may set $(W_{3 i}^{- 1} W_{4 i})^{- 1} = I$ , thus leading to (18). The proof is complete.□

From Theorem 2, the parameters of the FD filter (3) can be obtained. Based on the performance index (12), the design of the FD filter (3) is formulated as an $H_{\infty}$ -filtering problem, which can be solved by the following optimization problem

min λ, s . t . (17)

(23)

Remark 5. Based on the filter in the present paper, it is easy to resolve the fault-tolerant control problem by means of feedback control. When some faults are detected, we can reconfigure the control law to compensate for the faults based on the Markovian jump system model. Meanwhile, by using the guaranteed cost control approach according to the performance index (12) and online controller switching based on the system model, the fault-tolerant control strategies based on the present filter in our paper can be designed to ensure stability of the closed-loop networked control system at all times. The issue of fault-tolerant control may become the topic of our future work.

Numerical example

In this section, we illustrate the effectiveness of our proposed methods with a numerical example. We consider a linear time-invariant system (1) with the following parameters described by Peng et al. (2010)

A = [\begin{matrix} 0 & 1 \\ - 3 & - 4 \end{matrix}], B = [\begin{matrix} 2 \\ 1 \end{matrix}], D = [\begin{matrix} 0.1 \\ 0.05 \end{matrix}], F = [\begin{matrix} - 2 \\ - 1 \end{matrix}], C = [\begin{matrix} 1 & 1 \end{matrix}]

Assume that the sampling period of the random NCS is $T = 0.01$ , the finite state space of the Markov chain is $Θ = {1, 2, 3}$ . By some calculation, the Markovian jump system model is obtained as (2) with the following parameters in different modes

A 1 = [\begin{matrix} 0.9999 & 0.0098 \\ - 0.0294 & - . 9606 \end{matrix}], B 1 = [\begin{matrix} 0.0200 \\ 0.0095 \end{matrix}], D 1 = [\begin{matrix} 0.0010 \\ 0.0005 \end{matrix}], F 1 = [\begin{matrix} - 0.0200 \\ - 0.0095 \end{matrix}]

A 2 = [\begin{matrix} 0.9994 & 0.0192 \\ - 0.0577 & 0.9225 \end{matrix}], B 2 = [\begin{matrix} 0.0400 \\ 0.0200 \end{matrix}], D 2 = [\begin{matrix} 0.0020 \\ 0.0010 \end{matrix}], F 2 = [\begin{matrix} - 0.0400 \\ - 0.0200 \end{matrix}]

A 3 = [\begin{matrix} 0.9987 & 0.0283 \\ - 0.0848 & 0.8857 \end{matrix}], B 3 = [\begin{matrix} 0.0598 \\ 0.0316 \end{matrix}], D 3 = [\begin{matrix} 0.0030 \\ 0.0016 \end{matrix}], F 3 = [\begin{matrix} - 0.0598 \\ - 0.0316 \end{matrix}]

The transition probability matrix is given as follows (Peng et al., 2010; He et al., 2008)

Π = [\begin{matrix} 0.5 & 0.4 & 0.1 \\ 0.2 & 0.6 & 0.2 \\ 0.2 & 0.3 & 0.5 \end{matrix}]

Figure 2 indicates the switching mode with random environment satisfying the Markov transition probability matrix $Π$ . The unknown input or disturbance signal is supposed to be randomly uniformly distributed over $[- 0.5 0.5]$ , and Figure 3 shows the signal pattern. The fault signal $f (k)$ is given as

f (k) = {\begin{matrix} 2 + 0.5 \cos (4 π k) for k = 100, 101, \dots, 200 \\ 0 others \end{matrix}

Figure 4 gives its corresponding wave shape.

With the given parameters and based on the optimal filter design problem (23), Table 1 shows the minimum guaranteed performances $γ$ in the terms of the feasibility of (23) for different values of $α$ , from which one can see that the smaller the value of the $α$ , the larger the value of $γ$ . This is reasonable, as smaller $α$ implies a higher chance of measurements missing, and thus worse disturbance attenuation performance $γ$ .

Table 1.

Minimum $γ$ for different values of $α$

$α$	0.6	0.65	0.7	0.75	0.8	0.85	0.9	0.95	0.99
$γ$	0.1337	0.1333	0.1327	0.1320	0.1312	0.1300	0.1283	0.1250	0.1152

From Table 1, we can see that $γ = 0.1283$ when $α = 0.9$ , under this situation, Figure 5 shows the generated residual signal $r_{k}$ , and the evolution in (8) is presented in Figure 6. We select the threshold as $J_{th} = sup_{f = 0} \sqrt{\sum_{i = 1}^{500} r_{i}^{T} r_{i}}$ , after the 500 times simulations, the value $J_{th} = 0.4251$ is obtained. The simulation result in Figure 6 shows that $J (101) = 0.3734$ and $J (102) = 0.4823$ , and $J (102) > J_{th}$ . Thus, the appeared fault can be detected after two time steps.

Conclusions

In this paper, the discussion has been based on NCSs under stochastic environments. A FD approach for NCSs has been presented. Considering the stochastic characters of NCSs with packet dropout, network-induced delay, and out-of-order packets, we deal with them by modelling the NCSs as a discrete-time, non-uniform sampling stochastic parameter-varying Markov jump system. Then by using a design method for the FD filter, which makes the residual generation system stochastically stable and a sufficient condition is obtained by solving this problem in terms of LMI. At last, a numerical simulation example is given to illustrate our results, and which can detect the occurrence of a fault in a very short time.

There are some further research topics. Similar to linear systems, new fault diagnosis theory for nonlinear NCSs should also be developed.

Footnotes

Funding

This work was supported by the National Natural Science Foundation of China (project number 61074009) and supported by Shanghai Leading Academic Discipline Project (project number B004).

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