Dynamic Event-Triggered Fault Detection for Discrete Networked Control System With Time-Delay

The dynamic event-triggered fault detection problem for discrete networked control systems with time-delay is investigated in this paper. Firstly, for reducing the triggering times, a new dynamic event-trigged is proposed. Secondly, an observer is constructed on the controller node to generate fault residual, and the mathematical model of the closed-loop systems is established by analyzing the sequence of the transmission signal. Thirdly, by constructing a suitable Lyapunov-Krasovskii functional, the sufficient stability conditions for the closed-loop networked control systems are derived. The computing method of the observer gain matrix, the controller gain matrix and the minimal disturbance repression index is provided. Finally, a casy study on the dynamic cart is employed to illustrate the merits of the proposed approach.


I. INTRODUCTION
Coupled by the progression of communication technology and the increasing scale of control systems, the introduction of computer communication networks into control systems has become a trend in the field of industrial control. The closed-loop feedback systems formed by the shared networks are networked control systems (NCSs) [1], [2], [3], [4]. NCSs own the strength of easily expandable structure, low installation and maintenance costs, etc., and have been widely used in the field of industrial automation [5], [6], [7], [8]. However, some inevitable issues like network-induced time-delay, data packet dropout have appeared due to the introduction of the networks, making system analysis more complicated, and making it more demanding on safety and reliability in engineering applications [9], [10], [11].
With the increasing structure and scale of NCSs, due to the aging of components, external disturbances and changes in the working environment, the NCSs will inevitably have various failures [12], [13]. Therefore, the problem of NCSs The associate editor coordinating the review of this manuscript and approving it for publication was Claudio Zunino. fault detection (FD) is a subject of theoretical and practical significance, which has received extensive attention from the academic community and has appeared many results.
The existing literature on FD of NCSs can be divided into three kinds. The first kind literature only considers time-delay. For example, the sum of the data transmission time-delay in the sensor-to-controller (S-to-C) link and the data transmission time-delay in the controller-to-actuator (C-to-A) link was taken as a discrete Markovain chain, thereby NCSs subject to time-delay were modeled by Markovain jump systems (MJSs) [14]. The stability conditions for the closed-loop NCSs were derived. Moreover, the FD observer design method was provided. The S-to-C timedelay and C-to-A time-delay were described with two irreverent discrete Markovain chains respectively, and the NCSs FD method was proposed [15]. The second kind literature only considers data packet dropout. For instance, assuming that data packet dropout satisfied the Bernoulli distribution character, the FD issue for NCSs subject to S-to-C and C-to-A data packet dropout was researched [16]. The stable conditions for the closed-loop system with a certain robust H ∞ disturbance repression performance were obtained and the FD filter design approach was given. The NCSs with S-to-C and C-to-A data packet dropout were described as MJSs owning four running modes, converting the NCSs FD problem into certain kind of H ∞ filter design issue [17]. The third kind literature takes both time-delay and packet dropout into consideration. For example, the conditions for NCSs subject to time-delay and packet dropout owning a certain level of H ∞ disturbance repression performance were obtained [18], [19].
Industrial control systems usually sample periodically to control and monitor the controlled plants. However, this traditional periodic sampling technique not only leads to a waste of energy but also increases the communication burden on the network. In this context, the event-triggered mechanisms (ETMs) are proposed, the idea of which is to sample or transmit data only when certain pre-specified conditions are satisfied. Therefore, ETMs can greatly save network bandwidth, thereby help to reduce network-induced time-delay and data packet dropout. Many ETMs have appeared for control or state estimation of various complex systems [20], [21], [22]. According to the different triggering conditions, the ETMs can be mainly divided into static and dynamic ETMs. The static ETMs are currently the most common ones, the triggering threshold of which is a constant [23], [24].
The common triggering condition is as follows: with y p i being the last transmitted system output, y l being the current system output and 0 < µ < 1 being the triggering threshold. The above condition is based on relative error, and is called relative ETM. The following triggering condition is called absolute event-triggering, because the triggering condition is based on absolute error.
With the purpose of further reducing the triggering times, some dynamic ETMs have been proposed, the characteristic of which is that the triggering threshold can be adjusted according to the evolution of the system, reducing the triggering times and thus saving system resources. Dynamic ETMs can be mainly divided into two types. One is to introduce auxiliary variables described by dynamic equations into the triggering conditions to adjust the triggering threshold [25], [26]. The scalar dynamic equation contains information about the system state, so the obtained threshold changes following the state of the system. The other is that the designed threshold does not evolve according to the dynamic equation, but decays at a certain rate until it reaches a given threshold [27], [28]. From the perspective of FD, the ETMs lose part of the NCSs information due to its non-uniform transmission mode, which increases the difficulty of FD for NCSs. Currently, there are few investigation on the FD of NCSs based on ETMs, and most literatures use the FD filter method. For example, considering the time-delay from the FD object to the FD filter, the problem of FD for NCSs under static ETM based on the FD filter was investigated [29], [30], [31], [32]. By expanding the states of the FD object and the FD filter, the FD issue was converted into a filtering problem, and the design method of the FD filter under ETM was proposed. However, the networks are not only in the S-C channel but also between the C-A channel for typical NCSs. The method proposed by [29], [30], [31], and [32] only took the time-delay or data packet dropout from the FD object to the FD filter into account, and only designed the FD filter, without realizing the co-design of the FD filter and the controller.
To the best of our knowledge, considering both S-to-C and C-to-A time-delay, the co-design of the FD filter and the controller for discrete NCSs under dynamic ETM has not been solved, which is the motivation of this paper. The contributions of this investigation can be exhibited as following points: 1) A new dynamic ETM is proposed, which can effectively reduce the triggering times. Compared with existing methods, the proposed method does not have any restriction on the sampling period. 2) An observer is constructed on the controller node to generate residuals and realize the output feedback control. Through the time sequence analysis of the signal, the closed-loop model of NCSs under the proposed dynamic ETM is established by using the time-delay system approach.

3) A proper Lyapunov-Krasovskii functional which can
deal with the S-to-C and C-to-A time-delay independently is constructed. The co-design algorithm of observer gain matrix, the controller gain matrix is provided. In this paper, through the signal timing analysis under the proposed dynamic ETM, the closed-loop systems subject to S-to-C and C-to-A time-delay are described with equivalent time-delay systems owning two time-delay variables. Furthermore, the co-design approach of the FD observer and the controller is provided by means of dealing with equation inequalities.
The reminder of this paper is arranged as follows. A new dynamic ETM is proposed and the mathematical model of NCSs is established in Section II. Section III provides the co-design approach of the FD observer and the controller. Section IV shows the merits and the effectiveness of the proposed approach by a numerical example. We conclude this paper in Section V.

II. PROBLEM FORMULATION
The considered NCSs'configuration is illustrated in Figure 1, in which an event generator is installed at the sensor node. The event generator's function is to decide whether to send system output to the controller at each sampling instant.
The following is the controlled plant's state space equation: where x l ∈ R n represents the plant's state, y l ∈ R g stands for the plant's output, u l ∈ R m means the plant's control input, The current sampled output of the system (3) is set as y l and the last transmitted sampled output is marked as y p i (i = 0, 1, 2, · · · , ∞). The event generator would send the current sampled output y l , only when y l and y p i (i = 0, 1, 2, · · · , ∞) satisfy the following proposed dynamic ETM.
where p i ∈ Z + is the triggering instant, µ ≥ 0 is the triggering threshold, 0 < σ < 0.5 is a known scalar and φ l represents an auxiliary variable with the following dynamic equation: with φ 0 ≥ 0 being the initial condition of φ l . Remark 1: Apparently we can see from (4) that, when the σ approaches zero, the dynamic ETM (4) becomes (1), and φ l ≥ 0 for l ∈ [0, ∞), which means that the triggering times can be decreased effectively compared to that of the static ETM (1).
where the internal dynamical variable φ l satisfies with δ ∈ (0, 1) being a given constant.
It can be obtained that φ l ≥ 0 from (6) and (8) on condition that δ > 1/σ and φ 0 ≥ 0. It should be pointed out that φ 0 ≥ 0 only holds for l ∈ [p i + s p i , p i+1 + s p i+1 − 1], rather that l ∈ [0, ∞) since (8) does not hold at the triggering instants. Let s l and c l represent S-to-C and C-to-A time-delay respectively, assuming that s l and c l are both bounded: where s M and c M are both nonnegative integers.
Considering the influence of the S-to-C time-delay s l , the instant when the data released by the event generator y p i (i = 0, 1, 2, · · · , ∞) reaches the observer is p i + s p i . Due to the function of the zero-order holder, the system output received by the observerỹ l can be expressed as: Similar to [35] and [36], consider two cases bellow: It can be seen that It is easy to see that where d ∈ 1, 2, · · · , N . Define η l as follows where, From (12), we can obtain From the above definition of h il , the following holds: Furthermore, At the controller node, construct an observer with the following structure: wherex l ∈ R n stands for the observer' state, r l ∈ R q means the residual signal andŷ l ∈ R g represents the observer' output. L ∈ R n×g and W ∈ R q×g are gain matrices of the observer and the residual, respectively. Adopt the following observer-based feedback control law: Remark 3: Result from the time-delay in C-to-A link, the controlled plant's control input in (3) differs from the observer control input in (17), and the following holds: Define the following state estimation error, residual error signal and augmentation vector: Based on the above analysis, when l ∈ [p i +s p i , p i+1 +s p i+1 − 1], the closed-loop NCSs can be expressed by the following time-delay system: where, The objective of this paper is to design FD observer (17) and the control law (18) under the proposed dynamic ETM (4) with the consideration of S-to-C and C-to-A time-delay, such that 1) When χ l = 0, the closed-loop NCS (20) is asymptotically stable; 2) The following H ∞ disturbance suppression performance holds under zero initial conditions.
∞ l=0 e T rl e rl < γ 2 with γ > 0 being the disturbance repression performance index. Choose the following evaluation function of residual: The corresponding FD threshold is as follows with ψ 0 being the initial instant of residual evaluation, and L 0 being the residual evaluation window length. The FD logic is designed as

III. MAIN RESULTS
We need the following lemma to proceed further. Lemma 1 ( [37]): For scalars α, α 0 with α ≥ α 0 ≥ 1 and positive definite matrix G > 0, the following always holds The following theorem illustrates sufficient conditions for system (20) to be asymptotically stable.
Proof: Construct the following Lyapunov-Krasovskii functional: where , along the trajectory of the NCSs (20), we can obtain Making use of Lemma 1, we have From (26) - (32), under the condition that 0 < σ < 0.5, we can get where, Therefore, if (24) holds, then the NCS (20) is asymptotically stable, and this completes the proof. To obtain the K and L, the inequality constraints in Theorem 1 should be further processed and we get the following theorem.
Theorem 2: When χ l = 0, for given scalars 0 < µ < 1, 0 < σ < 0.5, and residual gain matrix W , if there exist matrices K , L and positive definite matrices P > 0, S 1 > 0, where, Using Schur's complement lemma,˜ < 0 is equivalent to (34) and (35) can be obtained. Besides, we have From (37) and (16), it is can be seen that For (38), summing up l from 0 to ∞, we have It can be obtained from the zero initial conditions of the system that ∞ l=0 e T rl e rl < γ 2 ∞ l=0 χ T l χ l , which ends the proof. Remark 4: Result from the inverse constraints in (35), we cannot obatain L, K , and γ min conveniently with the Matlab linear matrix inequality (LMI) toolbox. However, we can convert the inequality constrains in Theorem 2 into the following minimization problem with the aid of the cone complementarity linearization method [38].

Min Tr
The algorithm for computing L, K , and γ min is given in Algorithm 1.

IV. A CASE STUDY ON DYNAMIC CART SYSTEM
In this section, to show the advantage of the proposed dynamic event-triggered FD method, the obtained results will be used to the dynamic cart system [39] shown in Figure 2, where m means the cart mass, u stands for the force exerted on the cart, s represents the cart displacement, h refers to   the buffer viscous friction coefficient, and k s is the spring coefficient.
When a fault occurs, the triggering instant and triggering interval of the event generator are shown in Figure 4. The fault residual r l and function J l are illustrated respectively in Figure 7 and Figure 8, from which it can be seen that both r l and J l change dramatically when a fault occurs. Besides, we have J 19 = 0.8741 < J th = 0.8805 < J 20 = 1.0439, which indicates that the fault signal f l is detected at the instant l = 20.

V. CONCLUSION
The FD issue for NCSs is investigated with consideration of both the impact of the ETM and time-delay based on observer. With the purpose of reducing the triggering times, a new dynamic ETM is proposed. The NCSs subject to both S-to-C and C-to-A time-delay are converted into equivalent time-delay systems. By the method of Lyapunov-Krasovskii functional, the stability conditions of the closed-loop NCSs are established, and the FD observer and the controller gain matrices computation approach is provided, and the co-design of the FD observer and the controller is realized. The problem of FD for NCSs under possible network attacks subject to time-delay and data packet dropout in both S-to-C and C-to-A links with dynamic ETM will be researched in the future.