Research Article  Open Access
YuLong Wang, TianBao Wang, WeiWei Che, "ActiveVarying SamplingBased Fault Detection Filter Design for Networked Control Systems", Mathematical Problems in Engineering, vol. 2014, Article ID 406916, 9 pages, 2014. https://doi.org/10.1155/2014/406916
ActiveVarying SamplingBased Fault Detection Filter Design for Networked Control Systems
Abstract
This paper is concerned with fault detection filter design for continuoustime networked control systems considering packet dropouts and networkinduced delays. The activevarying sampling period method is introduced to establish a new discretized model for the considered networked control systems. The mutually exclusive distribution characteristic of packet dropouts and networkinduced delays is made full use of to derive less conservative fault detection filter design criteria. Compared with the fault detection filter design adopting a constant sampling period, the proposed activevarying samplingbased fault detection filter design can improve the sensitivity of the residual signal to faults and shorten the needed time for fault detection. The simulation results illustrate the merits and effectiveness of the proposed fault detection filter design.
1. Introduction
Networked control systems (NCSs) are spatially distributed systems whose sensors, actuators, and controllers are connected via a communication network. NCSs have been the subject of intensive research, and many interesting research topics have been reported, such as packet dropouts and networkinduced delays [1–7], eventtriggered control design [8–10], networked predictive control systems [11, 12], finitehorizon filtering [13, 14], quantization effects [14–16], output feedback control [17, 18], and nonuniformly distributed delays [19], to mention a few.
In NCSs, the sensor is assumed to sample at a fixed nominal period. However, computer loads, networks, and sporadic faults may cause sampling period jitter. Then, it is of paramount importance to study timevarying sampling periods [20, 21]. It should be pointed out that the timevarying sampling periods considered in [20, 21] are induced by some external factors. Different from the timevarying sampling periods studied in [20, 21], the activevarying sampling periods were introduced in [22] to make full use of the network bandwidth.
For NCSs, the occurrence of faults is usually unavoidable. Then, it is interesting and important to study how to detect the occurrence of faults in time. Fault detection has been recognized as an important technique guaranteeing the safety and reliability of NCSs, and some nice results dealing with fault detection of NCSs have been reported [23, 24]. For example, Huang and Nguang [25] investigated robust fault estimation for NCSs with networkinduced delays. Based on likelihood ratios for networked predictive control systems with random networkinduced delays and clock asynchronism, Liu and Xia [26] proposed a fault detection and compensation scheme. Dong et al. [27] was concerned with the networkbased robust fault detection problem for TakagiSugeno fuzzy systems with stochastic mixed time delays and successive packet dropouts.
Notice that the constant sampling period is considered in [23–27]. When dealing with fault detection for NCSs, if a constant sampling period is adopted and the network is occupied by the most users, should be large enough to avoid network congestion. Then, the network bandwidth cannot be sufficiently used when the network is idle. However, if the sensor can adjust the length of the sampling period actively to make full use of the network bandwidth, the needed time for fault detection may be shortened. It is interesting and important to study fault detection for NCSs adopting the activevarying sampling periods. For the problem of fault detection of NCSs, considering the activevarying sampling periods will introduce some difficulty for system modelling, and the activevarying samplingbased fault detection for NCSs has been paid no attention in the existing literature, which motivates the current study.
For discretetime or discretized NCSs, suppose that is the sum of the number of consecutive packet dropouts and the length of networkinduced delays, and and are known constants with , , where is the largest integer smaller than or equal to . Then one can conclude that at any instant , or . On the other hand, for the specific instant , the events and cannot occur simultaneously; such phenomenon is named as mutually exclusive distribution in this paper. When dealing with fault detection for discretized NCSs, how to make full use of the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays to derive less conservative results is of paramount importance, which is the second motivation of this paper.
This paper is concerned with fault detection filter (FDF) design for continuoustime NCSs considering packet dropouts and networkinduced delays. By introducing the activevarying sampling period method, a new discretized model for fault detection NCSs is established. Based on the newly established model, the problem of fault detection filter design is studied. Even if the considered NCSs reduce to systems considering a constant sampling period, the proposed fault detection filter design is still applicable.
The contributions of this paper are summarized as follows.(i)The activevarying sampling period method is introduced to establish a new discretized model for fault detection NCSs.(ii)Fault detection filter design criteria for NCSs with packet dropouts and networkinduced delays are derived. The derived design criteria can guarantee the sensitivity of the residual signal to faults.(iii)The mutually exclusive distribution characteristic of packet dropouts and networkinduced delays is made full use of to deal with fault detection of the considered NCSs, and the newly derived fault detection filter design criterion is proved to be less conservative.
The remainder of this paper is organized as follows. By introducing the activevarying sampling period method, Section 2 establishes a new discretized model for continuoustime NCSs with faults. Section 3 is concerned with the full order fault detection filter design. The simulation results are presented in Section 4. Conclusions are drawn in Section 5.
Notation. and represent an identity matrix and a zero matrix with appropriate dimensions, respectively. denotes the entries of a matrix implied by symmetry. Matrices, if not explicitly stated, are assumed to have appropriate dimensions.
2. Modelling for ActiveVarying SamplingBased NCSs
The continuoustime NCSs whose faults are to be detected are described by where , , , , and are the state vector, control input vector, measurement output, disturbance input, and fault signal, respectively; is assumed to belong to ; , , , , and are known constant matrices with appropriate dimensions.
In this paper, we assume that the system (1) is controlled through a onechannel network; packet dropouts and networkinduced delays occur in the planttoFDF channel; the sensor is both clockdriven and eventdriven, while the FDF and the actuator are eventdriven. It should be pointed out that the results in this paper can be extended to deal with NCSs considering both the planttoFDF channel and the FDFtoactuator channel.
When the network is idle and occupied by the most users, suppose that the planttoFDF networkinduced delays are and , respectively. If a constant sampling period is adopted and the network is occupied by the most users, one can choose as the sampling period to avoid network congestion. Suppose that is the latest sampling instant, is the networkinduced delays of the measurement output ( and is the plant state at the instant ), the instant that reaches the actuator is , and is the length of the th sampling period.
In the following, we will propose the activevarying sampling period method (see also [22] for a similar method) to improve the sensitivity of the residual signal to faults and shorten the needed time for fault detection.
Partitioning into equidistant small intervals ( is a positive integer), then the next sampling instant (which is denoted as ) after can be chosen as where , , . Then, that is, the sampling period switches in the finite set . If , the latest available measurement output will be used by the FDF, and will not be used even if it reaches the FDF eventually.
For a large enough and during the interval , one can find that the measurement output adopted by the FDF is approximately , where , and , while is the maximum number of consecutive packet dropouts, , , and . Then the discretized representation of (1) can be described as where , , and .
At the th sampling instant, define the measurement output received by the FDF as . Then one has . Based on the above statement, one can see that the fault detection filter can be described as where and are the state of the fault detection filter and the residual signal, respectively; , , and are to be determined.
A reference residual model is usually needed to describe the desired behavior of the residual signal . Introduce the following reference residual model, see also [28] for details: where and are the state and the output of the reference residual model, respectively; , , , and are known constant matrices of appropriate dimensions.
Define , , and . Then, one has where
Remark 1. Generally speaking, the shorter the sampling period in NCSs, the better the system performance. However, a short sampling period will increase the possibility of network congestion. In the closedloop system (7), the activevarying sampling period method is introduced to make full use of the network bandwidth, which will guarantee the system performance and avoid the occurrence of network congestion simultaneously.
If a constant sampling period is adopted, one can choose to avoid network congestion. In the case of adopting the constant sampling period , the system in (7) reduces to where , with , , and , , , , while , , and are the same as the corresponding items in (7).
To detect the occurrence of faults in time, one should construct a residual evaluation function. If the value of the residual evaluation function is larger than a given threshold, an alarm of faults will be generated. Define the residual evaluation function as Choose a threshold as follows: The fault detection logic is For the purpose of making full use of the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays, one can introduce a scalar , and where , , and are defined in Section 1.
Based on the established model and the fault detection logic, this paper is concerned with the problem of fault detection filter design for NCSs considering the activevarying sampling periods. The mutually exclusive distribution characteristic of packet dropouts and networkinduced delays is made full use of to derive a less conservative fault detection filter design criterion.
3. Fault Detection Filter Design for ActiveVarying SamplingBased NCSs
This section is concerned with fault detection filter design for NCSs considering the activevarying sampling periods. For this purpose, define , and choose the following Lyapunov functional: where , , , , and are symmetric positive definite matrices, and .
Then we state and establish the following result.
Theorem 2. For given scalars , , and , if there exist symmetric positive definite matrices , , , , , , , , , and and matrices , , , , , , , , , , and , such that the following inequalities hold for or : where then under the fault detection filter (5) with the residual system (7) is asymptotically stable with an norm bound .
Proof. Taking the time difference of the Lyapunov functional in (14) along the trajectory of the system (7), one has
where
Notice that . Considering the mutually exclusive distribution characteristic of , one has
Then,
Adopting the mutually exclusive distribution characteristic of and the Jensen integral inequality in [29], one has
and , .
Then, by combining (7) and (20)–(23) together, one has
where and
with
From (24), one can see that if , then . By using Schur complement, is equivalent to
where and .
Introduce a matrix , where the selection of and is discussed in Remark 3. Pre and postmultiplying both sides of (27) with , , , and its transpose, and considering that and , one can see that if the inequalities in (28) are satisfied for or , the inequalities in (27) are also satisfied as follows:
where
Suppose that . Pre and postmultiplying both sides of (28) with , , , and its transpose and defining , , , , , , , , , , , , one can see that the inequalities in (28) are equivalent to the ones in (16).
Then, by using the definition of performance, one can see that if the inequalities in (16) are satisfied for or , the system (7) is asymptotically stable with an norm bound . This completes the proof.
Remark 3. Notice that the feasibility of the inequalities in (16) implies the nonsingularity of and . Since , the nonsingularity of implies that and are also nonsingular. By using the singular value decomposition of , one can obtain the matrices and .
It should be pointed out that the activevarying sampling period method is adopted in Theorem 2 to deal with fault detection filter design. Even if the constant sampling period is adopted, the fault detection filter design in Theorem 2 is still applicable. For the purpose of comparison, we establish the following fault detection filter design criterion for the system (9) with a constant sampling period.
Corollary 4. For given scalars , , and , if there exist symmetric positive definite matrices , , , , , , , , , and and matrices , , , , , , , , , , and , such that the following inequalities hold for or : where and are the same as the corresponding items in (16), is derived from in (16) by substituting and with and , respectively, and and are the same as the corresponding items in (9), then, under the fault detection filter (5) with the residual system (9) is asymptotically stable with an norm bound .
In the following, we will analyze the merits for considering the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays.
If the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays is neglected, the result in Theorem 2 is described as shown in the following corollary.
Corollary 5. For given scalars , , and , if there exist symmetric positive definite matrices , , , , , , , , , and and matrices , , , , , , , , , , and , such that where is derived from in (16) by deleting all the items multiplied by and , and are the same as the corresponding items in (16), then, under the fault detection filter (5) with the residual system (7) is asymptotically stable with an norm bound .
The following theorem establishes the relationship between Theorem 2 and Corollary 5.
Theorem 6. Consider the system (7). If the fault detection filter design criterion presented in Corollary 5 is satisfied, then the fault detection filter design criterion in Theorem 2 is also satisfied.
Proof. Define the matrices in (16) and in (32) as and , respectively. Then, where and . From (34), one can see that if is satisfied, is also satisfied. This completes the proof.
Remark 7. It has been proved theoretically that the fault detection filter design criterion in Theorem 2 is easier to be satisfied than the fault detection filter design criterion in Corollary 5, which illustrates the merits for considering the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays. Notice that the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays is adopted to deal with bounding inequalities for products of vectors. Even for the systems in [30, 31] without considering the occurrence of faults, adopting the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays will also introduce better results. The corresponding results are omitted here for briefness.
In the following, we will show the effectiveness of the proposed fault detection filter design by the simulation results.
4. Simulation Results and Discussion
To illustrate the effectiveness of the proposed fault detection filter design, we consider the following linear model for the motion of ships, see also [32–34] for similar models: where is the state vector, is the control input vector, is the unknown disturbance input, and is the fault signal. The physical meaning for , , , , , and can be found in [34] and the references therein.
The system matrices in (35) are similar to the ones in [34] with The parameters for the fault weighting system (6) are chosen as , , , and . The activevarying sampling periods are considered in this paper, and suppose that the sampling periods switch in the finite set , , . Then one has . Define and . To avoid that some elements of the obtained matrix are close to zero, we assume that , . Discretizing the system (35), constructing the closedloop system, and solving the fault detection filter design criterion presented in Theorem 2, one has
Suppose that the initial state of the augmented closedloop system (7) is . For , will switch between 1 and 2 in cycles. For and , , , and are assumed to be zero. For , suppose that , , and in the case that a fault occurs, suppose that , where , , and denote the values of , , and at the th sampling instant, respectively.
It should be pointed out that the fault detection filter design method in Theorem 2 is applicable for , switched sampling periods (i.e., switches between and in cycles), and . The residual evaluation function response for , switched sampling periods, and is presented in Figures 1, 2, and 3, respectively. Considering the fault detection logic presented in (12) and from Figures 1, 2, and 3, one can see that the newly proposed fault detection scheme can not only reflect the occurrence of faults in time but also recognize the faults without confusing faults with the disturbance .
Define as the number of needed sampling periods for fault detection. Table 1 shows the numbers of needed sampling periods for fault detection corresponding to different cases. As one can see in Table 1, the shorter the sampling period, the smaller the number of needed sampling periods for fault detection, which illustrates the effectiveness of the newly proposed activevarying samplingbased fault detection scheme.

5. Conclusions
The fault detection filter design for continuoustime NCSs considering packet dropouts and networkinduced delays has been studied in this paper. The activevarying sampling period method has been introduced to establish a new discretized model and improve the fault detection performance for the considered NCSs. The fault detection filter design criterion which considers the mutually exclusive distribution characteristic of packet dropouts and networkinduced delays has been proved theoretically to be less conservative than the criterion without considering such a mutually exclusive distribution characteristic. The merits and effectiveness of the proposed activevarying samplingbased fault detection filter design have been verified by the simulation results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by the National Science Foundation of China (Grant no. 61004025, Grant no. 61210306066, Grant no. 61374063, Grant no. 61104106, and Grant no. 61104029), the “333 Project” in Jiangsu Province, China, the “Qing Lan Project” in Jiangsu Province, China, the Key Project for International Science and Technology Cooperation and Inviting Experts in Universities of Jiangsu Province, China, the Natural Science Foundation of Liaoning Province, China (Grant no. 201202156), and the Program for Liaoning Excellent Talents in University (LNET), China (Grant no. LJQ2012100).
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Copyright © 2014 YuLong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.