Passive Fault-Tolerant Control for NCSs Using Event-Triggered Approach

In this paper, the passive event-triggered control problem for networked control systems with actuator faults is of concern. A sufficient condition is proposed by a Lyapunov-Krasovskii functional, under which such type continuous-time Networked control systems with actuator faults are asymptotically stable and passive. The designing method of controller is developed by using Wirtinger inequality and linear matrix inequalities (LMIs). The given strategy can reduce the number of transmissions, thus saving the communication resources. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method.


I. INTRODUCTION
With the development of the network, the plant, the sensor, the controller, and actuator usually located in different places. Consequently, the signals should be transmitted from one place to another. Therefore, networked control systems(NCSs), where the components are connected over networks, have been intensively studied in recent decades [1], [2], [3].
The time-triggering scheme and event-triggering scheme are the mainstream control strategies for NCSs [4], [5]. Theories and experiments have proved the event-triggered scheme can reduce the number of transmitted data and improve the utilization efficiency of limited network bandwidth to a large extent without degrading the desired system performance [6], [7]. When the network resource is limited, the time-trigger technique is inefficient. An important focus of NCSs' signal transmission and control is cost. Compared with the traditional time-triggered scheme, the eventtriggered scheme allows a considerable reduction of the The associate editor coordinating the review of this manuscript and approving it for publication was Haibin Sun . network resource occupancy while maintaining the control performance [8], [9], [11], [28]. In NCSs, there are schemes, such as event-triggered control based on time-delay method in [12]. It should be pointed out that the event-triggered control problems for NCSs are much more complicated. Therefore, investigating the analysis and design of NCSs is of fundamental importance in order to realize the functions of the real world.
On the other hand, actuator faults are also the distinguishing feature that can not be neglected in analysis and designing of NCSs. Because actuator faults can also lead to poor performance of the closed-loop systems and sometimes destabilizes the systems [13], [14]. Recently, a host of characters have paid attention to the study of various control problem of NCSs with actuator faults [14], [15], [16].
The event-triggered control design problem of NCSs with dynamic quantisation and fault is studied in [15], and stochastic actuator fault, which is modelled by the Bernoulli distributed white sequence is considered in this situation, and a robust feedback controller is designed. Reference [16] focuses on two novel event-triggered fault-tolerant control strategies for a class of stochastic systems with state delays. VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Passivity is closely related to bounded realness, which has been used in control problems to deal with robust stability problems for complex uncertain dynamic systems with disturbances. A passive system utilizes the product of input and output as the energy provision and embodies the energy attenuation character, which often links the stability problems.
The key point of passivity theory is that a passive system can keep itself internally stable. In fact, many applications based on passivity theory can be found in lots of areas, such as signal processing, chaos control [17], [18]. In recent years, researches on passive control of dynamic systems have achieved remarkable results [19], [20], [28]. The passive control problem for semi-Markov jump Takagi-Sugeno fuzzy systems based on an event-triggered mechanism is considered in [22]. Reference [23] proposed the asynchronous passive controller design for singular Markov jump systems.
Up to now, to the best of authors's knowledge, there are few results on the passive event-triggered control problem for NCSs with actuator faults and remains to be important and challenging. However, it is often required to reduce the transmission cost in practical engineering. Therefore, it becomes important to investigate the event-triggered controller design problems of continuous-time NCSs with actuator faults.
Motivated by the aforementioned considerations, in this paper, our attention is devoted to design the passive eventtriggered controller for NCSs with actuator faults. The contributions of this paper are as follows: • A class of more general control strategy is investigated, where we take actuator faults, event-triggered controller into consideration at the same time. The event-triggering scheme gives a unified framework that can include existing results for time-triggering scheme as special cases for NCSs. In addition, the criteria obtained in this paper are suitable for NCSs without actuator faults.
• Based on the Lyapunov-Krasovskii functional approach, a sufficient condition given in this paper makes NCSs with actuator faults and event-triggering scheme asymptotically stable and passive. Moreover, the designing method of controller is developed by using linear matrix inequalities (LMIs).
• The Wirtinger inequality is more tighter than Jensen inequality, and be used in the derivation of the sufficient criterion. In addition, the free weight matrix approach is also adopted in the criterion derivation process.
The outline of this paper is given as follows. In section II, the problem formulation and some preliminaries are presented. The controllers are given in section III. Section IV presents the numerical examples to demonstrate the proposed methodology. Conclusions are drawn in section V.
Notations: V > 0 means that V is positive definite; V T and V −1 denote the transpose and the inverse of any square matrix V ; ∥V ∥ means the spectral norm of the matrix V ; diag{V 1 , V 2 , . . . , V N } represents a block-diagonal matrix with diagonal elements V 1 , V 2 , . . . , V N . R n denote the n-dimensional Euclidean space, R n×m is the set of n × m real matrices; I is the identity matrix of appropriate dimensions. For asymmetric matrix * denotes the matrix entries implied by symmetry.

II. PROBLEM FORMULATION AND PRELIMINARIES
Consider the NCSs subject to actuator faults with the following dynamics: A, B, B ω , C and D ω are known constant matrices with appropriate dimensions.
, then we can obtain that Then the control input can be described as There is an event generator in the network systems, which is constructed between sensor and controller, which is visualized in Fig 1. The event generator can determine which sampled signal should be sent out or not by utilizing the following judgement algorithm (4).
In this paper, we consider an event-triggered controller as where K is the matrix to be sought. Then, the closed-loop system can be described aṡ Remark 1: According to the above description of actuator failure, the following three special cases can be drawn, and the conclusions obtained in this paper are applicable to the following three special cases, which illustrates the universality of the model in this paper. .
• When f l − = f l + = 1, the actuator has not failed. Fault tolerant control in this paper means that the NCSs with actuator faults can still remain asymptotically stable, and can meet the performance indicator of passive. We are ready to state the passive fault-tolerant event-triggered control problem for system (1).
Definition 1: A linear control law of the form (5) is passive fault-tolerant event-triggered control for the NCSs (1) with the actuator faults represented by (2) and the event-triggering scheme (4), if the following two requirements are satisfied.
holds for all trajectories with zero initial condition. Lemma 1 [24]: For given constants r 1 , r 2 (0 ≤ r 1 < r 2 ) and an n × n real matrix Q = Q T > 0, a scalar continuous function r(t) satisfying r 1 r(t)r 2 and a vector-valued functionẋ → R n such that the integrations concerned below are well defined, then the following inequality is true for any matrices Z j ∈ R m j ×m j and S j ∈ R m j ×n satisfying Z j S j * Q ≥ 0 and any vectors c j ∈ R m j , j = 1, 2, 3, 4. Lemma 3 [26]: Let M , F, N and P be real matrices of appropriate dimensions with P > 0, F T F ≤ I and a scalar η > 0. Then Lemma 4 [27]: For a matrix R > 0 and a differentiable signal x in [α, β] → R n ,the following inequality holds: VOLUME 11, 2023

III. MAIN RESULTS
In this section, we first give a condition to guarantee the system (6) with ω (t) = 0 is asymptotically stable.
Theorem 1: For the given scalars τ , a 1 , and a 2 , the closedloop system (6) with ω (t) = 0 is asymptotically stable, if there exist symmetric matrices X > 0,P 0 > 0,Q > 0, > 0, and matrices Y ,S j ,Z j , such that Proof: First we choose a Lyapunov−Krasovskii functional candidate as Taking the time derivative along with the trajectory of the system (6), it yieldṡ , we now consider the following two cases: Consider the relaxation factor as follows 2a 1ẋ Apply Lemma 1 to get the fact that where S = S 1 (e 2 − e 3 )+(e 2 − e 3 ) T S 1 T + S 2 (e 1 −e 2 )+(e 1 −e 2 ) T S 2 T + π 2 S 3 (e 2 +e 3 −2e 5 )+ π 2 (e 2 +e 3 −2e 5 ) T S 3 T + π 2 S 4 (e 1 +e 2 −2e 6 )+ π 2 (e 1 +e 2 −2e 6 ) T S 4 Then, we can obtain thaṫ where Then, pre-and post multiply inequality 0 by 1, 2, 3, 4). considering the fact By Lemma 2 and 3, the LMIs (7), (8) and (9) We also consider the relaxation factor as follows 2a 2ẋ Apply Lemma 4 to get the fact that where Considering the event-triggered condition, we can geṫ Then, pre-and post multiply inequality 1 by X 0 O = diag {X , X , X , X , X } and its transposition, considering the fact By Lemma 3, the condition (10) . Still now, this completes the proof. Remark: The event-triggering scheme in this paper can avoid the Zeno phenomenon. In the special case δ = 0, it comes into a time-triggering method.
Next, a sufficient condition is provided to guarantee asymptotical stability and passivity of system (1).
Taking the time derivative along with the trajectory of the system (6), it yieldṡ The relaxation factor is considered as We can get the following condition by Lemma 1, where By integrating (24) over the time period 0 to T 0 , Then, Pre-and post multiply 0 th2 by X O th2 = diag {X , X , X , X , X , X , I } and its transposition. Then, the same proof process can be obtained for inequalities (20), (21) and (22).
Based on the conditioṅ The same proof process can be obtained for inequality (23). This completes the proof.

IV. NUMERICAL EXAMPLE
In this section, we provide two numerical examples to show the effectiveness of the proposed conditions. Example 1: Consider the NCSs system with ω (t) = 0 and the following coefficient matrices: Let ε 1 = 0.1, ε 2 = 0.2, a 1 = 1.5, a 2 = 1.5, τ = 0.2, δ = 0.4, the fault function 0.08 sin t + 0.8. Solving the LMIs (7-10) in Theorem 1, it is obtained that   Now, we assume that the initial condition is In the following, we provide some simulation results.    Then, the feedback gain Initial state has been chosen as Fig 5 shows the state responses of the closed-loop system, which shows that the system stability is guaranteed. Fig 6 displays the trajectory of control input. Based on the judgement algorithm (4), the release instants and the time intervals between any two adjacent release instants are presented in Fig 7. The average release interval can be calculated as 1.25 s. It is easy to obtain that event-triggered control can efficiently reduce the number of control task execution. The communication resources can be saved significantly while retaining satisfactory closed-loop performance. Table 1 shows the maximum allowable value of τ for different β.

V. CONCLUSION
Stability analysis and passive event-triggered controller designing method for continuous-time NCSs with actuator faults are investigated in this paper. A sufficient condition is proposed by a Lyapunov-Krasovskii functional, which can guarantee the NCSs with actuator faults are asymptotically stable and passive. The event-triggering scheme has been used in the designing of the feedback controller. The effectiveness of the proposed theorem is illustrated by two numerical examples. The closed-loop systems are modeled as dynamics with time-varying delay, so event-triggered faulttolerant control in this paper can be extended to NCSs with time delay. In addition, the observer-based event-triggered fault-tolerant control of NCSs with time-delay is of great interest and will be studied in the future [28]. Besides, the event-triggered control for stochastic nonlinear systems can be considered in future work. JIACHANG SHI received the bachelor's degree in information engineering from Hebei University, in 2021, where he is currently pursuing the master's degree with the College of Electronic and Information Engineering.
His main research interests include dissipative control and fault tolerant control of T-S fuzzy systems.
FEI LI received the Ph.D. degree from the Beijing University of Posts and Telecommunications (BUPT), China, in 2019.
He is currently a Lecturer with the College of Electronic and Information Engineering, Hebei University. He has accumulated rich experience in the field of 5G. His research interests include wireless communication, such as new waveform and channel estimation, and communication problems in control systems. VOLUME 11, 2023