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Sensors 2011, 11(1), 557572; doi:10.3390/s110100557
Abstract
: In this paper, the sensor data is transmitted only when the absolute value of difference between the current sensor value and the previously transmitted one is greater than the given threshold value. Based on this sendondelta scheme which is one of the eventtriggered sampling strategies, a modified fault isolation filter for a discretetime networked control system with multiple faults is then implemented by a particular form of the Kalman filter. The proposed fault isolation filter improves the resource utilization with graceful fault estimation performance degradation. An illustrative example is given to show the efficiency of the proposed method.1. Introduction
Over recent years, fault diagnosis for networked control system (NCS) using the modebased analytical redundancy method have received significant attention. In [1,2], the overviews of main ideas and results on fault diagnosis of NCS are given, including the fundamentals of fault diagnosis for NCS with information scheduling, fault diagnosis approaches based on the simplified timedelayed system models and the quasi TS fuzzy model, and fault diagnosis for linear and nonlinear NCS with long timedelay. However, most of the available results make use of timetriggered state estimation techniques by sampling the output of plant at an essentially equidistant time instant. Because the sampling period is determined according to the worst case operation conditions that rarely occur, the timetriggered sampling leads to a conservative usage of the communication bandwidth.
On the other hand, recent advances in computing and communication technologies enable the wireless networks (e.g., Bluetooth, wirelessHART and ZigBee) to rapidly replace wired networks in many applications, including industrial control and monitoring, home automation and consumer electronics, security and military sensing, and health monitoring [3,4]. Though the wireless channels are easier and cheaper to deploy and avoid cumbersome cabling, they also pose serious resource constraints. Therefore, applying the timetriggered sampling method to the wireless NCS may have some negative effects on the estimation and control performance of system, such as wasting the scarce communication resource and further shortening the lifetime of overall system. Since the eventtriggered sampling strategies present a number of potential advantages for NCS, such as clockfree operation, less traffic requirement, and better resource utilization, they have been regarded as the possible and important alternatives to the timetriggered sampling.
Until now, numerous eventtriggered sampling concepts have been proposed in the literature, such as sendondelta sampling [5,6], levelcrossing sampling [7], deadband sampling [8], Lebesgue sampling [9], sendonarea sampling [10], error energy sampling [11], selftriggered sampling [12], etc. Although these schemes have different terminologies, the same attribute is that the signal is sampled only when an a priori defined events occurs in the data monitored by sensors. For instance, the studies in [5–9] are concerned with the same sampling criterion where the event is defined as that the difference Δ between the current sensor value and the last transmitted one is greater than a given threshold. While in [10] and [11], the event is that the integral and energy of Δ is greater than a given threshold, respectively. Because of inherent distinctive benefits, the socalled eventtriggered state estimation and eventbased control for NCS with eventtriggered sampling schemes have gained increasing attention. In this paper, however, we focus our attention on the eventtriggered state estimation for the purpose of fault diagnosis. In relation to a parallel line of research on the eventbased control, we refer the readers to the literature, e.g., [12,13].
In the context of state estimation, although the timetriggered state estimation over networks with networkinduced effects taken into account have made great progress (see, e.g., [14–17]), research on the eventtriggered state estimation is relatively lacking apart from several works [6,10,18–26]. It is well known that utilizing more sensors can potentially improve the performance of the estimation algorithms. However, using too many sensors can in turn create bottlenecks in the communication resource when these sensors compete for bandwidth. As a result, the studies in [18–21] explore the tradeoff between communication and estimation performance. Rather than sending every raw measurement to the remote estimator via network, a socalled controlled communication policy was adapted, which firstly obtain the local estimate x̃_{kk} from the raw sensor measurements and then compare x̃_{kk} with the remote estimate to decide whether or not it is worth sending data x̃_{kk}. Also, Reference [21] proposes an optimal communication policy by dynamic programming and value iteration to minimize a longterm average cost function, which is related to the difference between the local and remote estimate. Based on the sendondelta method, Reference [6] proposes a modified Kalman filter where computed output with increased measurement noise covariance is used when there is no sensor data transmission. The authors also discuss how to choose the threshold which is a tradeoff parameter between the sensor data transmission rate and the estimation performance. Reference [22] extends the previous work [6] to address how to determine the measurement value at a sensor node if it does not send data. To avoid the inability of sendondelta method in detecting the signal oscillations or steadystate error, Reference [10] proposes a novel scheme called sendonarea and then formulates a networked estimator based on Kalman filter to estimate the states of the system. More recently, Reference [23] proposes a networked estimator for eventtriggered sampling systems with packet dropouts. Reference [24] develops an eventtriggered estimator which is updated both when an event occurs with a received measurement sample, as well as at sampling instants synchronous in time without receiving a measurement sample. However, to the authors’ knowledge, fault diagnosis of networked control systems making use of the eventtriggered state estimation method has not been addressed, which motivates the current study of this paper.
In this paper, we show our attention on the implementation problem of a modified fault isolation filter (FIF) for NCS with eventtriggered sampling and multiple faults. By the sendondelta scheme which is one of the eventtriggered sampling strategies, it means that the sensor data is transmitted only when the absolute value of difference between the current sensor value and the previously transmitted one is greater than the given threshold value. Based on this scheme, a modified FIF for a discretetime NCS with multiple faults is then implemented by a particular form of the Kalman filter. The rest of this paper is organized as follows. The modelling of NCS with eventtriggered sampling scheme is presented in Section 2. A modified FIF is proposed in Section 3. An illustrative example is presented in Section 4 to show the effectiveness of the result. The paper is concluded in Section 5.
Notations: In what follows, if not explicitly stated, matrices are assumed to have compatible dimensions. Z_{+} denotes the set of nonnegative integer numbers. $\mathcal{R}$^{n} and $\mathcal{R}$^{n×m} are, respectively, the ndimensional Euclidean space and the set of n × m real matrices. A^{T} denotes the transpose matrix or vector A. A^{−1} and A^{+} represent the inverse and pseudoinverse of A, respectively. diag(a_{1},…,a_{n}) refers to an n × n diagonal matrix with a_{i} as its ith diagonal entry. rank(A) stands for the rank operator of matrix A. $\mathcal{E}$(x) represents the mathematical expectation of random variable x. x ∼ $\mathcal{N}$(μ, Σ) means that the random vector satisfies the normal distribution with mean value μ and covariance matrix Σ. $\mathcal{P}$(xy) means the conditional probability distribution of x given y. Sign function is defined as $\mathit{sign}(x)=\{\begin{array}{ll}1,\hfill & x\ge 0;\hfill \\ 1,\hfill & x<0.\hfill \end{array}$
2. NCS with EventTriggered Sampling Scheme
The architecture of NCS with eventsampling discussed in this paper is shown in Figure 1, where the closedloop system consists of a plant with smart sensors and actuators, a remote FIF and a wireless network channel. The controller, FIF and actuator are assumed to be logically integrated. Thus, control commands do not need to experience any wireless transmission. This configuration represents a system, e.g., wireless sensor/actuator system, where actuation is inexpensive but sensor measurements are transmitted to the controller or FIF by sensors with a limited energy.
Since the event generator, the controller and the FIF have to be implemented on smart sensors and actuators by means of digital hardware, a discretetime plant model is considered as the alterative to the continuous plant together with a zeroorder hold and a sampler. The state evolution and sensor measurement equation are given as follows, respectively:
As shown in Figure 1, the smart sensor numbered i has a sampler which regularly samples the sensor measurement with period h and an event generator to decide whether or not to send new sensor measurement through the network. The event generator therein, also known as sendondelta scheme, is illustrated in Figure 2 where the sensor data ${y}_{k}^{i}$ is transmitted only when the absolute value of difference between the current sensor value ${y}_{k}^{i}$ and the previously transmitted one ${y}_{\mathit{sent}}^{i}$ is greater than the given threshold value δ_{i}, namely
Obviously, all the transmitted measurements ${y}_{\mathit{sent}}^{i}$ are the eventtriggered samplers which are subsequences of the raw measurement ${y}_{k}^{i}$. For instance, if the time instant when the previous event occurs is denoted as k_{j} ∈ Z_{+}, the eventtriggered sampling condition (2) is further formulated as
However, the time instant k_{j} (j = 1, 2....) can not be precisely determined because of the sampler, which is significantly different from the continuous one introduced in e.g., [5,6]. As shown in Figure 3, the dash line and real line represent the sensor measurement in the continuous time and discrete time, respectively. In the continuous time case, the time t_{c} when an event occurs can be exactly known since the measurement of plant is continuously updated by the sensor. While in the discrete time, the events have to be generated at the subsequent discrete time steps, e.g., k_{1} equivalent to kh, k_{2} equivalent to (k + 4)h, since the measurement is only updated at some discrete time instants and remains constant in the intersampling interval. Although producing the unsent measurements, e.g., the sample at the time instant (k + 1)h, wastes a bit of computing resource, the unsent measurements in turn save a lot of communication resources. In a sense of improving the resource utilization, applying the eventtriggered sampling schemes to discretetime plant is also meaningful. The reason is that wireless communication consumes more energy than information processing. As noted in [5], a sensor node can execute 3,000 instructions for the same energy cost of sending a single bit at the distance of 100 meters by radio.
In the sequel, we further assume that all sensor measurements are timestamped and the network is communication link without packet losses and time delays. For the purpose of reducing communication, only parts of the raw measurements will be communicated to the remote FIF by the sendondelta scheme. Through the communication link, all the eventtriggered samplers ${\{{y}_{{k}_{j}}^{i}\}}_{j=1}^{\infty}$ are then stored in an infinite buffer. If sensor measurement ${y}_{k}^{i}$ does not yet arrive at the buffer at time instant k, it means that the current value ${y}_{k}^{i}$ has not significantly changed in contrast to the previous eventtriggered sampler ${y}_{{k}_{j}}^{i}$. In this case, the previous buffer value ${z}_{k1}^{i}$ whose value is equivalent to ${y}_{{k}_{j}}^{i}$ will be stored in the kslot of the buffer, as illustrated in Figure 4.
Furthermore, the arrival of the measurement ${y}_{k}^{i}$ at time k is defined as a binary variable ${\theta}_{k}^{i}$, namely ${\theta}_{k}^{i}=1$ when ${y}_{k}^{i}$ arrives at the buffer at time instant k, otherwise ${\theta}_{k}^{i}=0$. Rather than considering the design problem where only the probabilities of ${\theta}_{k}^{i}$ at each time instant is known and the research interest lies in studying the effect of loss and delay probabilities, we address the implementation problem where the value of ${\theta}_{k}^{i}$ at each time instant is known in advance. The last received value of ith sensor output at time instant k_{j} is denoted as ${y}_{{k}_{j}}^{i}$. If there is no sensor data received for k > k_{j}, the estimator node considers that the measurement value of the ith sensor output ${y}_{k}^{i}$ is still equal to ${y}_{{k}_{j}}^{i}$, but the measurement noise is increased from ${\upsilon}_{k}^{i}$ to ${\overline{\upsilon}}_{k}^{i}={\upsilon}_{k}^{i}+{\Delta}_{k,kj}^{i}$ where ${\Delta}_{k,{k}_{j}}^{i}={y}_{{k}_{j}}^{i}{y}_{k}^{i}$ satisfies ${\Delta}_{k,{k}_{j}}^{i}\le {\delta}_{i}$.
From the existing literature [6,22], the assumption that ${\Delta}_{k,{k}_{j}}^{i}$ has a uniform distribution with zero mean and a variance ${\delta}_{i}^{2}/3$ is valid only if the measurement covariance R > δ^{2}. Otherwise, the mean and variance of Δ_{i}(k, k_{j}) is $\mathit{sign}({y}_{{k}_{j}}^{i}{y}_{{k}_{j}1}^{i})*{\delta}_{i}/2$ and δ_{i}/12 respectively. Thus, the measurements ${z}_{k}^{i}$ which the FIF will use for fault diagnosis are formulated as
Moreover, the following selector is designed to flexibly determine whether (5) or (6) is applied to the FIF:
By the selector (7) and compensating some values for the unsent measurements, we can also regularly implement some existing fault isolation filter algorithms, e.g., [27] to NCS with eventtriggered sampling even though the measurements are transmitted irregularly.
3. Modified Fault Isolation Filter
Fault isolation filter, a special dynamic observer which generates the directional residuals in response to a particular fault, is an attractive way for enhancing the fault isolability [28,29]. It was first developed by Beard [28] and [29] and later revisited by Massoumnia [30] in the geometric framework and by White and Speyer [31] and Park and Rizzoni [32] in the context of eigenstructure assignment. Further improvements were suggested by Liu and Si [33], and Keller [27]. For linear continuous timeinvariant system, Liu and Si [33] have proposed a fault isolation filter such that faults can be asymptotically detected and isolated. To guarantee that the ith component of the output residual is decoupled from all but the ith fault, the columns of the fault detectability matrix are assigned as the eigenvectors of the filter’s transition matrix with a set of fixed eigenvalues. Keller extended this approach to discretetime stochastic linear systems. A new fault isolation filter has been developed to isolate q faults with at least q output measurements. More recently, Reference [34] addressed the fault detection problem for a class of linear networked control systems by extending the FIF proposed in [27].
In this section, we further construct a modified Keller’s FIF to detect and isolate the multiple faults in NCS with eventtriggered sampling.
Recalling the definitions of fault detectability indexes and matrice introduced in [27]:
Definition 1. The linear stochastic system (1) is said to have fault detectability indexes ρ = {ρ_{1}, ρ_{2}, …, ρ_{q}} if ρ_{i} = min{ν: CA^{ν−1} f_{i} ≠ 0, ν = 1,2,…}
Definition 2. If the linear stochastic system (1) has finite fault detectability indexes, the fault detectability matrix D is defined as:
Now, the following filter is presented as the residual generator of discretetime plant (1):
From Equations (1) and (9), the state estimation error e_{k} = x_{k} − x̂_{k} and the output of the filter α_{k} propagate as
Let G_{nα}(z) be the transfer function from n_{k} to the output residual α_{k}. Then the following theorem is presented to design K_{k} and L_{k} such that
Theorem 1. Under the condition rank(D) = q, the solutions of (11) can be parameterized as K_{k} = ωΠ+K̄_{k}Σ, L_{k} = Π, with Σ = β(I − DΠ), Π = D^{+}, ω = AΨ and D = CΨ, where K̄_{k} ∈ $\mathcal{R}$^{n×m−q} is the free parameters to be designed, D^{+} is the pseudoinverse of D and β is an arbitrary matrix chosen so that rank(Σ) = m − q
From Theorem 1, the fault isolation filter (9) is rewritten from the free parameter K̄_{k} as
From Equation (14), the following theorem is then proposed to design the free parameter K̄_{k} which minimizes the trace of the faultfree state estimation error covariance matrix ${\overline{P}}_{k+1}=\mathcal{E}\{{\overline{e}}_{k+1}{\overline{e}}_{k+1}^{T}\}$.
Theorem 2. The proposed fault isolation filter described by the following relations:
Based on Theorem 2 and the measurement noise shown in Equations (5) and (6), the modified FIF for NCS with eventtriggered sampling is proposed as follows:

4. Illustrative Example
In this section, we will present an example to illustrate the implementation approach proposed in this paper. The modified example is borrowed from [27] described by Equation (1), where the parameters are as follows:
By Theorem 1, we have D = CΨ and Ψ = F since rank(CF) = q (ρ_{1} = 1, ρ_{2} = 1). The parametrization of the FIF’s gain K_{k} and projector L_{k} is then given by
Furthermore, we choose $\frac{1}{30}$fold of the maximal amplitude of ${y}_{k}^{i}$ (i = 1,2,3). Then, the threshold values in Equation (3) are determined as δ_{1} = 4.8406, δ_{2} = 2.0382, δ_{3} = 1.0009. The sufficiently small threshold ε_{i} (i = 1,2,3) in Equation (7) is given as 10^{−4}. Then, Figure 5(a)–7(a) show all the transmitted measurements of sensors with eventtriggered scheme. Figures 5(b)–7(b) indicate all the transmitted measurements of sensors with timetriggered scheme. By comparison, the sensors with eventtriggered scheme transmit only 66%, 67%, and 58% of samples produced by timetriggered scheme, respectively. In other words, the resource utilization by the eventtriggered scheme can be obtained 34%, 33% and 42% improvement, respectively.
The measurements used in the simulation are indicated in Figure 8, where ${z}_{k}^{i}$ (i = 1,2,3) and ${({z}_{k}^{i})}^{\u2605}$ (i = 1,2,3) represent the measurements with compensating and without compensating some values for the unsent ones, respectively. By Algorithm 1 and the measurements ${z}_{k}^{i}$, a modified FIF is implemented to the NCS with eventtriggered sampling. Figure 9 shows the innovation sequences of residual ${\alpha}_{k}={\left[\begin{array}{cc}{({\alpha}_{1})}_{k}& {({\alpha}_{2})}_{k}\end{array}\right]}^{T}$, ${\alpha}_{k}^{\#}={\left[\begin{array}{cc}{({\alpha}_{1}^{\#})}_{k}& {({\alpha}_{2}^{\#})}_{k}\end{array}\right]}^{T}$ and ${\alpha}_{k}^{\u2605}={\left[\begin{array}{cc}{({\alpha}_{1}^{\u2605})}_{k}& {({\alpha}_{2}^{\u2605})}_{k}\end{array}\right]}^{T}$, which correspond to the residual obtained by the modified FIF described in Algorithm 1, the Keller’s FIF in [27] using the timetriggered samples and the Keller’s FIF using the measurements ${({z}_{k}^{i})}^{\u2605}$, (i = 1,2,3), respectively.
In order to compare the simulation results obtained by different schemes, we propose the root mean square (RMS) of the fault estimation errors as a performance index. This error for the scalar variable n_{i} with respect to its estimate α_{i} for N_{s} simulation steps is defined as
From Figure 9 and Table 1, the performance of the modified FIF using less sensor data transmission experiences graceful degradation in contrast to the Keller’s FIF using timetriggered samples. By graceful degradation, it means that a system degenerates in such a manner that it continues to operate, but provides a reduced level of service rather than causing total breakdown. On the other hand, the Keller’s FIF using the measurements ${({z}_{k}^{i})}^{\u2605}$ (i = 1,2,3) fails to work completely, while the number of the total sensor data transmission is same as the modified FIF used. From these results, we can draw a conclusion that the modified FIF has an advantage in the tradeoff between communication cost and fault estimation performance.
In order to establish the relationships between the number of the sensor data transmissions, the RMS of the fault estimation errors by different methods and the threshold values δ_{i} (i = 1,2,3), some further simulations are done by adjusting the threshold values δ_{i} (i = 1,2,3) from 1/20 to 1/80 of the maximal amplitude of ${y}_{k}^{i}$ (i = 1,2,3). Figure 10 shows the percentage of transmitted samples to total samples in relation to δ_{i} by eventtriggered scheme, wherein the real line, dash line and dashdotted line represent the percentage for sensor ${y}_{k}^{1}$, ${y}_{k}^{2}$ and ${y}_{k}^{3}$, respectively. From Figure 10, it can be seen that the sensor data transmission rate is inversely proportional to δ_{i}, namely the communication cost reduced by the eventtriggered scheme will increase when δ_{i} increases, and vice versa.
In Figure 11, the real line, dash line and dashdotted line represent the fault estimation performance of the modified FIF, the Keller’s FIF with timetriggered sampling and the FIF with the measurements ${({z}_{k}^{i})}^{\u2605}$, respectively. It can be seen that the fault estimation performance of the modified FIF is improved and eventually approaches the performance of Keller’s FIF with timetriggered sampling as δ_{i} decreases. However, the performance of the FIF with the measurement ${({z}_{k}^{i})}^{\u2605}$ (i = 1,2,3) is very poor even by flexibly adjusting the threshold δ_{i}.
5. Conclusions
This paper is concerned with the implementation problem of fault isolation filter for networked control system with the sendondelta scheme which is one of the eventtriggered sampling strategies. By sendondelta, the sensor data is transmitted only when the absolute value of difference between the current sensor value and the previously transmitted one is greater than the given threshold value. Based on this scheme, a modified fault isolation filter for a discretetime networked control system with multiple faults is then implemented by a particular form of the Kalman filter. In contrast to the Keller’s fault isolation filter using timetriggered samples, the proposed fault isolation filter improves the resource utilization with graceful fault estimation performance degradation. Also, we can improve the performance of the modified FIF by flexibly adjusting the threshold values with taking resource utilization into account.
Throughout the paper, no networkinduced packet losses are taken into account in the model of the networked control system. Further study of the fault isolation filter for networked control system with eventtriggered sampling and packet losses is encouraged.
Acknowledgments
The authors would like to acknowledge the NSFCGuangdong Joint Foundation Key Project under Grant No. U0735003, the Scientific Research Foundation for Returned Scholars, the Ministry of Education of China, Fundamental Research Funds for the Central Universities under Grant No. 2009ZM0076, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20100172120028 and the French research program of Agence Nationale de la Recherche (ANR) under Grant No. SSIA_NV_15 for funding the research.
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α  (α_{1})_{k}  $({\alpha}_{1}^{\#})k$  $({\alpha}_{1}^{\u2605})k$  (α_{2})_{k}  $({\alpha}_{2}^{\#})k$  $({\alpha}_{2}^{\u2605})k$ 

RMS  1.6258  0.9368  18.0693  1.7998  1.6235  14.2398 
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