Abstract

For the linear discrete networked control system (NCS) which may suffer DoS attack on both sides of the controller, when the actuator has time-varying failure, the intelligent sensor unit uses wireless sensors to collect data. According to the large amount of data collected, the active fault tolerance/active passive capacity of linear discrete NCS under the discrete event-triggered communication mechanism (DETCS) is studied. The problem of cooperative controller design is discussed. Firstly, a linear discrete NCS model integrating DETCS, actuator fault, and network attack is established. Then, based on the idea of integral sliding mode control, an active fault-tolerant/attack active passive intrusion-tolerant cooperative controller is designed, and the actuator attack side network attack and sensor side network attack are extended to the state to obtain a new state vector. Then, an adaptive Kalman filter estimator (AKF) estimates the fault and attack information and then adjusts the initial fault-tolerant/intrusion-tolerant cooperative controller in real time according to the estimated information obtained by the adaptive Kalman filter estimator; finally, the MATLAB simulation example is used to verify the improvement of system performance by the designed control law and the saving of network resources by the introduction of DETCS.

1. Introduction

Compared with the traditional control system, NCS has the unique properties of remote operation, remote monitoring, security detection and resource sharing due to the existence of networks. However, due to the existence of networks, the structure of the control system is more complex, the data transmission will be affected to a certain extent, the system is more prone to component failure, and network transmission is more vulnerable to attack, so the physical fault tolerance is very important The research of intrusion control and network attack tolerance control has great practical significance and practical application value [1–3].

With the rapid development of various wireless sensor technologies, people can gradually perceive the real world through wireless sensors in every corner of the world, even in places where people rarely visit. These wireless sensors transmit the data to the computer on people's worktable through the network, which is convenient for scholars to carry out the corresponding research on all kinds of data. Generally speaking, fault-tolerant control (FTC) technology is divided into passive fault-tolerant control (PFTC) [4–6] and active fault-tolerant control (AFTC) [7–10]. PFTC uses a controller to make the closed-loop system insensitive to specific faults and maintain the stability and performance of the system. On the basis of acquiring fault information, AFTC designs a fault-tolerant controller to ensure the performance and stability of the system. For the actuator fault of the system, Zhang et al. proposed an adaptive observer [8, 9] by which the system state and fault can be obtained and the fault can be adjusted. Reference [10] studies the problem of adaptive fault-tolerant control for uncertain actuator fault compensation of linear time invariant NCS with unknown plant parameters and actuator fault parameters. Intrusion tolerance is developed on the basis of fault tolerance, but it is very different from fault tolerance. It is mainly reflected in that network attacks are malicious in motivation and difficult to judge in form. Denial of service (DoS) attack is a kind of attack means by weakening and preventing legitimate users from using legitimate network resources, which can affect the normal use of the network. Reference [11] studies the stability of control and measurement packets transmitted over a communication network under DoS attack. Reference [12] studies the event-triggered attack-tolerant tracking control problem of nonlinear NCS under sudden DoS attack. In Reference [13], the progress of security control and attack detection of industrial information physical fusion system is reviewed from the perspective of control theory. Although many achievements have been made in fault tolerance and attack tolerance of NCS, most of them are limited to independent design. For NCS with actuator failure and network attack coexisting, the research on comprehensive security control of fault tolerance and intrusion tolerance is rarely involved, and only literature [14] and literature [15] can be consulted at present. However, in actual NCS, it is inevitable that faults and attacks occur simultaneously.

In view of this, aiming at the actuator fault and DoS network attack on both sides of the controller, this paper establishes a closed-loop linear NCS fault/attack coexistence model by constructing the active fault-tolerant/active passive intrusion-tolerant system structure of NCS under DETCS with the help of the joint modeling method of network attack and controlled object. In addition, adaptive Kalman filter estimators are designed, respectively, for online monitoring of state, attack, and fault. The active fault-tolerant control is used to compensate the actuator network attack with the active intrusion-tolerant strategy, and the passive intrusion-tolerant mechanism is used to be robust to the sensor network attack, so as to realize the state feedback and active fault-tolerant and active passive intrusion-tolerant cooperative control.

2. Material and Method

2.1. System Model

Controlled object model:

Among them, , and are the state, output, and control input of the system, respectively; is the time-varying fault vector of the actuator; is the constant network attack vector suffered by the actuator side, referred to as the actuator attack vector; is the constant network attack vector suffered by the sensor side, referred to as the sensor attack vector; , , and are the coefficient matrices of proper dimension; and and are the perturbations of bounded energy. They are Gaussian white noise sequences with 0 mean value and independent of each other. Their covariance matrices are and ; and are the perturbations of proper dimension.

Referring to the idea of state augmentation, the system state , actuator attack , and sensor attack are augmented into a new state , which is called the augmented state. The augmented system model iswhere

2.2. DETCS Trigger Conditions

This paper adopts the most representative discrete event trigger condition [16]:

Among them, is the estimated value of the state at the current sampling time, is the latest estimated value of the state at the network transmission time that satisfies equation (4) at the previous time, is the event trigger matrix with positive definite symmetry, and is the event trigger parameter scalar, which can be set in advance and is related to the expected performance of the system. Therefore, only when the sampling data meets the trigger condition of equation (4) can it be transmitted from the sensor network to the controller. A lot of network resources is saved, so as to improve the utilization of the network. The system control structure is shown in Figure 1.

Figure 1 

System control structure diagram.

Define delay function , and . Among them, are the maximum nonuniform transmission periods of data filtering, and is the upper bound of delay function at .

2.2.1. Related Lemmas

Lemma 1 (Schur complement lemma [17]). For a given symmetric matrixthe following three conditions are equivalent:

Lemma 2 (Wirtinger inequality in discrete form [18, 19]). For a given positive definite matrix , scalar and vector function satisfy the following inequalities:Among them,

2.3. Preliminary Design of Active Fault-Tolerant/Active Passive Intrusion-Tolerant Cooperative Control Law

For system (1), the control law can be designed as follows:

Among them, is the nominal control law when there is no fault and no attack, and is the compensation control law when fault and attack occur simultaneously.

Considering the time delay problem, the sliding mode function can be designed as follows:

Among them, is a constant matrix with proper dimension, and is a nonsingular matrix; is a sliding mode discrete compensator, which satisfies the following requirements: , where is the gain matrix of the controller; the sliding mode of the system can be obtained by introducing .

Omitting and , combining formula (1) and formula (10), we can get

If the order is , there will be

Furthermore, the sliding mode control law based on the reaching law method iswhere is a constant scalar and .

When there is no fault and no attack, the nominal control law is

When faults and attacks occur simultaneously, the compensation control law is

Theorem 3. When constant satisfies , respectively, the system (1) satisfies the existence and reachability conditions of the sliding mode.

Proof. Select Lyapunov function: .
Combining formulas (1) and (10), we can getAccording to the control law (13),When the constants satisfy , there areTherefore, the reachability condition of the sliding surface is satisfied. So, the system state can reach the sliding surface in finite time.
When the system is on the sliding mode surface, i.e., , the equivalent control term of the sliding mode control law can be obtained by substituting it into equation (13):By substituting equation (19) into equation (1), the closed-loop model of NCS on a sliding surface can be obtained:where is a real valued initial function on .☐

2.4. Design of Adaptive Kalman Filter Fault/Attack Estimator

For augmented system (2), the following adaptive Kalman filter fault/attack estimator is designed.

The prediction part of the adaptive Kalman filter fault/attack estimator is as follows:

Among them, is the state error covariance matrix, and is the error between the system output and the estimated output.

The gain matrix of adaptive Kalman filter fault/attack estimator is as follows:

Among them, is the gain matrix of Kalman filter.

The error gain matrix of adaptive Kalman filter fault/attack estimator is as follows:

Among them, is the error gain matrix, and is the forgetting factor.

The update part of adaptive Kalman filter fault/attack estimator is as follows:

Suppose 4. Matrix is completely observable, is completely controllable, if matrix satisfies

Then, for Kalman gain matrix , are exponentially convergent.

Suppose 5. Under initial condition , matrix is strictly positive definite.

System error definition:

From equations (2), (21), and (26),

Further, we can getwhere is the mean of .

From hypothesis 1 and equation (28), we can see that 14 is exponentially convergent.

From equations (21) and (24),

Because of ,

From hypothesis 2, we can see that is strictly positive definite.

Theorem 6. When matrix is strictly positive definite and exponentially approaches 0, the adaptive Kalman filter is convergent. Note .

Proof. Define Lyapunov function:From equation (23), we can getThen,From equations (32) and (33),In equation (34),Since is a positive definite matrix, then is also a positive definite matrix. So,It can be seen from equation (36) that is exponentially close to 0. Since matrix is strictly positive definite, is exponentially close to 0.
From equation (26),☐

Then, is also an index close to 0, so the adaptive Kalman filter is convergent.

3. Analysis

The online fault-tolerant/intrusion-tolerant control for system (1) can be described as follows: using the estimated value of system state , actuator fault , and network attack on the actuator side, considering the event triggering conditions, substituting the estimated value satisfying the conditions into control law (13), the adjusted fault-tolerant/intrusion-tolerant control law is

Suppose 7. Actuator fault is bounded, i.e., ; is bounded actuator attack, i.e., , where are known functions greater than 0.

The existence condition and global reachability condition of sliding mode are

By substituting equation (38) into equation (39), we find that