Robust Fault-Tolerant Control of Continuous Lurie Networked Control Systems Based on the Observer with the Event-Triggered Mechanism

For a class of uncertain continuously networked Lurie control systems with both sensor-to-controller time-delay and controllerto-actuator time-delay, the problem of codesigning its observer and fault-tolerant controller based on an event-triggered mechanism under actuator failure is investigated. Firstly, considering that the state of the system cannot be measured directly, an observer is constructed on the controller node. Secondly, to reduce the waste of network bandwidth resources and improve the performance of the network control system, a network control system approach based on event-triggered mechanism is proposed. By introducing the event-triggered mechanism and the actuator fault indication matrix, the Lurie networked control system is modeled as a Lurie system with time-delay using the state augmentation technique to obtain a model of the closed-loop system. Finally, based on Lyapunov stability theory, sucient condition for the stability of the closed-loop system is obtained, and the design method of the fault-tolerant controller and the observer is given.e obtained results are given in the form of linear matrix inequalities, which are easy to be solved by using the linearmatrix inequality toolbox. Finally, the feasibility and eectiveness of the method are illustrated by a simulation example.


Introduction
With the rapid development and wide application of computer, communication and control technologies, networked control systems (NCSs), and control systems, in which nodes (sensors, controllers, actuators, and controlled objects) transmit data through a shared communication network, have attracted much attention [1]. Compared with the traditional point-to-point control system, the NCS has the advantages of low installation cost, easy maintenance, high exibility, information resource sharing, and easy remote operation. It has been widely used in intelligent medical, intelligent transportation, and intelligent manufacturing elds [2][3][4][5]. However, due to the problems of time-delay, packet loss, and wasted communication resources derived from the application of NCS, the actual production process will be a ected by many uncertain factors, such as equipment wear and tear and production line aging, and the system will inevitably fail [6]. Once the system fails, it may cause catastrophic losses or even damage to personal safety, so it is of great economic and social signi cance to improve the reliability and stability of NCS operation with the help of fault-tolerant control.
Fault-tolerant control is the most common and e ective control method [7,8], which has received a lot of attention from academics and has achieved some results. e existing research literature on fault-tolerant control of NCS with time-delay can be divided into two categories. e rst category of literature focuses on fault-tolerant control of NCS with time-delay between the sensor and the controller. In Ref. [9], a fault-tolerant controller design method based on observer estimation is proposed for the time-delay between the sensor and the controller, and a closed-loop NCS is made asymptotically stable and satisfies the given control performance by combining Lyapunov stability theory and the linear matrix inequality method. e literature [10] models the time-delay from the sensor to the controller as a Markov chain with incomplete transfer probability and uses the state augmentation technique to transform the closed-loop system into a Markov jump system, gives sufficient conditions to guarantee the stability of the NCS, and designs fault-tolerant controllers. e second type of literature focuses on fault-tolerant control of NCS with time-delay between both sensors to controllers and controllers to actuators. e literature [11] combined the time-delay in the sensor-to-controller network and the time-delay in the controller-to-actuator network into a total time-delay, established a robust fault-tolerant control method for discrete NCS with time-delay, and combined Lyapunov stability theory and switching system theory, for the design of fault-tolerant controllers. e literature [12] describes the sum of the time-delay from the sensor to the controller and from the controller to the actuator in terms of random variables conforming to the Bernoulli distribution. A stability criterion for systems with stochastic time-varying time-delay is proposed to ensure that the designed faulttolerant controller can satisfy the asymptotic stability conditions and operability requirements when the state trajectory enters the slip surface.
With the development and integration of control science and computer technology, the demand for data and information transmission is also increasing. Due to the limitation of network communication bandwidth, communication resources are limited, and if the amount of data transmitted by the system exceeds the threshold value that can be carried, it will lead to network congestion or even system collapse. To address this problem, some scholars have conducted research. e traditional time-triggered mechanism is simple and easy to implement, but the selected sampling period is small, and frequent data transmission and update will lead to the waste of communication resources and easily cause network congestion [13,14]. In order to overcome the shortcomings of the time-triggered mechanism, many scholars have successively adopted the event-triggered mechanism to save network resources. e implementation method is to let the sampled signal reach an event generator, then the event generator decides whether the signal is transmitted to the network according to the triggering conditions. is mechanism of transmitting only necessary signals and reducing unnecessary signal transmission determined by the triggering conditions effectively reduces the occupation of limited communication resources and improves the efficiency of information transmission [15][16][17]. With the continuous development of NCS, many innovative results have been achieved for event-triggered mechanisms. For example, the adaptive event-triggered mechanism proposed in the literature [18,19] has attracted much attention. Compared with the traditional event-triggered mechanism, its triggering threshold is a time-variable related to the system state, so it can flexibly adjust the triggering conditions according to the system state to further reduce the data transmission and maintain a better system performance, which is significant for NCS. Scholars have introduced different kinds of event-triggered mechanisms into NCS to replace the traditional time-based periodic transmission method, which has become an important research element in NCS.
In recent years, some results have been achieved in the research about fault-tolerant control of NCS with timedelay under the event-triggered mechanism. For example, an event-triggered mechanism with the fault-tolerant control strategy is proposed for continuous uncertain NCS with time-delay from the sensor to thecontroller, which enables the system to remain stable in case of actuator failure [20]. e literature [21] investigated the asynchronous H ∞ fault-tolerant control of discrete Markov jump systems with actuator failure and time-delay, proposed a pattern-dependent event-triggered mechanism method based on the pattern-dependent Lyapunov-Krasovskii function, and obtained the performance of the closed-loop system with stochastic mean-square stability under actuator failure and the joint design method of the asynchronous fault-tolerant controller and event-triggered mechanism. However, all the above-mentioned literatures are studies done based on state feedback in the case of measurable system states. For the case where the system state is not measurable, literature [22] proposes a delaybased event-triggered fault estimation observer for a class of continuous NCSs with time-delay and faults, gives a fault-tolerant control method based on fault estimation to compensate the effects generated by system faults, and designs fault-tolerant controller gains and event-triggered parameters to ensure the required performance of the faulty system and to reduce the communication resources wastage. However, only the time-delay in the sensor-tocontroller network is considered in the paper. In the literature [23], a fault-tolerant control model for linear NCS with time-delay under the dynamic event-triggered mechanism is developed, an observer design method for simultaneously estimating the state vector and unknown actuator faults is proposed, and an observer-based faulttolerant controller is designed, but the time-delay from sensor-to-controller and controller-to-actuator is combined in the paper to deal with them.
Lurie systems are a class of nonlinear systems with typical structural features that can be more accurately modeled to reflect objective reality, and they represent many essential features of nonlinear systems. e absolute stability of Lurie systems has received much attention. However, few literature have been devoted to the problem of fault-tolerant control under actuator failure as an object of study.
It is well known that time-delay is the main cause of system instability in NCS. Most of the existing techniques have only studied the controller-to-actuator time-delay or have combined the sensor-to-controller and controller-to-actuator time-delay into one treatment. Based on the above discussion, the following questions naturally arise: Is it possible to build a unified model of NCS with bilateral delays based on eventtriggered mechanisms? How can the joint design of eventtriggered mechanism, observer, and fault-tolerant controller under this unified model make the closed-loop system with time-delay, uncertainty, and actuator failure stable?
In this paper, we address the codesign of its eventtriggered mechanism and fault-tolerant controller under actuator failure for a continuously networked Lurie control system with time-delay and uncertainties, with the following main contributions: (1) e influence of the time-delay from the sensor to the controller and from the controller to the actuator on the control system is taken into account, which is more interpretable and mechanistic (2) Also considering the time-delay, uncertainty of system parameters, and actuator failure, an observer is constructed on the controller node to establish the Lurie NCS model under the event-triggered mechanism, which enables the impact of time-delay, uncertainty, event-triggered mechanism, and actuator failure on system performance to be analyzed in the same framework (3) According to Lyapunov stability theory, the stability conditions of the closed-loop system are obtained, and the design methods for solving the event-triggered weight matrix, robust fault-tolerant controller, and observer are given e remainder of this paper is organized as follows. e modeling of a continuous Lurie NCS with actuator failures under an event-triggered mechanism is shown in Section 2. In Section 3, the system stability analysis and the codesign of the observer, fault-tolerant controller, and event-triggered mechanism are provided. Section 4 exhibits the construction of the simulation model and the verification of the algorithm. Section 5 summarizes the contents of the full paper.
Notations are as follows: R n represents the n-dimensional Euclidean space; * represents the transpose of the corresponding matrix block; if the matrix A is invertible, A − 1 represents the inverse of A; A T and A + represent the transpose and Moore-Penrose inverse of the matrix, respectively, and the real positive definite matrix X is represented as X > 0; I represents the unit matrix of appropriate dimensions; and diag a, b . . . { } represents the diagonal matrix with a, b as the main diagonal.

Problem Description
Consider the following Lurie system with uncertainty and actuator failure: where x(t) ∈ R n , u(t) ∈ R m , y(t) ∈ R q are the system state vector, input vector, and output vector, respectively; A, B, C, D are real constant matrices of appropriate dimensions; ΔA, ΔB, Δ D are uncertainty matrices with bounded parametrization and satisfy ΔA ΔB Δ D � 3 are known matrices with appropriate dimensions, and Ξ(t) is an unknown matrix satisfying Ξ(t) T Ξ(t) < I; F is the fault indication matrix which we define under actuator failure and satisfies where f j � 0 means the jth actuator is completely failed, f j � 1 indicates normal operation of the j th actuator, and f j ∈ (0, 1) indicates partial failure of the j th actuator. ψ(·): R q ⟶ R q is a memoryless nonlinear function, satisfying the local Lipschitz condition, ψ(0) � 0, and for any y(t) ∈ R q satisfies, the sector condition is where Θ is the real diagonal matrix. Such a nonlinear function ψ(·) is usually said to belong to the sector , it can always be transformed into the case of ψ(·) ∈ ℓ[0, Θ] by loop transformation, so only the case of ψ(·) ∈ ℓ[0, Θ] is considered in this paper. e NCS structure diagram shown in Figure 1 consists of an actuator, a controlled object, a sensor, a sampler, a zeroorder retainer, an observer, an event generator, and a controller.
e sampler acquires the output signal of the system with a fixed period h. e signal collected by the sampler is directly transmitted to the event generator connected to it. en, whether the event generator can send the sampled data y(t k h + jh) to the controller needs to be determined according to the following conditions: where V is a positive definite matrix and j ∈ Z + ,σ ∈ [0, 1].

Remark 2.
Only the output sampled data y(t k h + jh) satisfying equation (3) will be transmitted, and obviously this event-triggered mechanism will reduce the communication load in the network and reduce the computation of the controller. ere is a special case that if σ � 0 in Eq. (3) holds for all sampled output data y(t k h + jh), then the eventtriggered mechanism will degrade to a time-triggered mechanism. In addition, the triggering instant of the eventtriggered mechanism used in this paper is based on the system sampling period h, which can be triggered only at the sampling instant, ensuring that the minimum event interval time is positive and avoiding Zeno behavior. After the event generator has released the current sample signal, the sample signal is transmitted directly to the zeroorder keeper via the network and then to the actuator via the observer. In this process, the time-delay from the sensor to the controller is denoted by τ sc i k , and the time-delay from the controller to the actuator is denoted by τ ca i k . To facilitate the following analysis, the following hypotheses are proposed: . .. In addition, the period of the event generator sending data can be expressed as Based on the above analysis of the system output received at the observer side, we get We can obtain Accordingly, define an error vector as it is easy to prove the existence of a positive integer ε ≥ 1 such that To simplify the expression, let en the interval [i k h + τ sc i k , i k+1 h + τ sc i k+1 ) can be divided into the following ε + 1 subintervals Define the segmentation function as From (9) and (11), we can obtain At this point, define the error vector From the definition of ϑ k (t), the following equation y i k h � ϑ k (t) + y(t − ρ(t)), (14) ϑ T k (t)Vϑ k (t) ≤ σy T (t − ρ(t))Vy(t − ρ(t)). (15) Construct an observer of the following form on the controller side as where x(t) represents the state of the observer, y(t) represents the output of the observer, and L is the gain matrix of the observer to be designed. e following observer-based feedback control law is used as where K is the controller gain matrix.
Remark 3. Due to the time-delay τ ca i k in the controller-toactuator network, the input of the control object is different from the input of the observer, and the following equation holds true: Define M . Based on the above analysis, the closed-loop network control system can be written in the following form: At this point, condition (2) becomes where Θ � ΘC I 0 .
Remark 4. In a real industrial system, the actuator failure indication matrix F � diag 0, 0 { } means the actuator is completely failed and the controller will not be able to realize the control of the controlled object. erefore, this paper only considers the case of partial actuator failure, and the actuator can still work normally within a certain range even though it has failed.

System Stability Analysis.
e following lemmas will be used in the derivation of the expected result.

Proof. Construct a Lyapunov-Krasovskii function
By taking derivation of V(t, z(t)) for t along the system (19), it can be obtained that Applying Lemma 1 and Lemma 2 to the integral term in (29), the following equations hold true: where Mathematical Problems in Engineering Substituting (32), (33), (34), (35) into (29) and combining (15) and (21), we can obtain where . From Schur's complement lemma and Definition 1, if (36) is less than 0, it is known that the Lurie system (19) is globally asymptotically stable for any ℓ[0, Θ] and therefore is absolutely stable. where Due to the presence of nonlinear terms in Λ 11 − Λ 12 Λ −1 22 Λ T 12 , it is inconvenient to solve the controller, so take W � diag W, W , and define X � , M � XMX, N � XNX, P i � XP i X, S i � XS i X, i � 1, 2, 3, 4. Pre and postmultiply diag X, X, X, X, X, X, X, CXC T , I, P −1 1 , P −1 2 , P −1 3 , P −1 4 , I, I} on both sides of the matrix Λ 11 Λ 12 * Λ 22 , noting that since P 1 > 0, P 2 > 0, P 3 > 0, P 4 > 0, there must be at is, we can get the linear matrix inequality (38), where K � Y 1 X − 1 , L � Y 2 X − 1 C + and V � C +T X − 1 YX − 1 C + . □ Remark 5. Since the matrix C is a row full rank matrix of m × n, the matrix C + of n × m exists such that CC + � I, C + is called the right inverse of C, and the observer gain matrix L � Y 2 X − 1 C + is obtained from Y 2 � LCX. Remark 6. Since X > 0, V > 0, CX ≠ 0,C is a row full rank matrix, there must exist (CX) T V(CX) > 0; Y � XC T VCX is a nonsingular matrix, which yields the event-triggered weight matrix V � C +T X − 1 YX − 1 C + .

Remark 7.
e upper and lower bounds of time-delay are introduced in the proof, which makes the conclusion less conservative. In addition to the decision variables required for constructing Lyapunov function, no other free-weighting-matrix is introduced, which avoids the computational burden caused by too many decision variables and the possible conservatism caused by the optimization of too many decision variables. A modified Jensen's inequality is used in the paper. is inequality has a tighter integration bound, which reduces the computational complexity while reducing the conservatism of the conclusion.

Simulation Example
To verify the effectiveness and usability of the method, consider the following Lurie system, where