# Active Fault-Tolerant/Active Passive Intrusion-TolerantH∞ Cooperative Control of Discrete NCS under the Background of Big Data

Authors: Wang Jun; Meng Xiao-li
Journal: Wireless Communications and Mobile Computing
 (2021)

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## 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.


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## Body

## 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:(1)xk+1=Axk+Buk+Efk+Naaak+D1wk,yk=Cxk+Nsask+D2vk.Among them,xk∈Rn,yk∈Rm, and uk∈Rl are the state, output, and control input of the system, respectively; fk∈Rp is the time-varying fault vector of the actuator; aak∈Rq is the constant network attack vector suffered by the actuator side, referred to as the actuator attack vector; ask∈Rq is the constant network attack vector suffered by the sensor side, referred to as the sensor attack vector; A∈Rn×n,B∈Rn×l, C∈Rm×n,E∈Rn×p, and Ns∈Rn×q are the coefficient matrices of proper dimension; and wk∈Rp and vk∈Rp 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 Q∈Rp×p and R∈Rp×p; D1∈Rn×p and D2∈Rm×p are the perturbations of proper dimension.Referring to the idea of state augmentation, the system statexk, actuator attack aak, and sensor attack ask are augmented into a new state ηk, which is called the augmented state. The augmented system model is(2)ηk+1=A~ηk+B~uk+E~fk+D~1w~k,yk=C~ηk+D2vk,where(3)ηk=xTkaaTkasTk,w~k=wTkbaTkbsTk,A~=ANa0000000,B~=B00,E~=E00,D~1=D000I000I,C~=C0NsT.


### 2.2. DETCS Trigger Conditions

This paper adopts the most representative discrete event trigger condition [16]:(4)x∧k−x∧tkTΦx^k−x^tk≤σx∧TtkΦx^tk.Among them,x^k is the estimated value of the state at the current sampling time, x^tk is the latest estimated value of the state at the network transmission time that satisfies equation (4) at the previous time, Φ is the event trigger matrix with positive definite symmetry, and σ∈0,1 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 x^k 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 functionτk=k−tk,k∈tk,tk+1, and 0≤τk≤τM. Among them, τM≜Tkmax+τ~ are the maximum nonuniform transmission periods of data filtering, and τ~ is the upper bound of delay function τk at tk+1.


#### 2.2.1. Related Lemmas

Lemma 1 (Schur complement lemma [17]).
For a given symmetric matrix(5)Z=Z11Z12Z12TZ22,the following three conditions are equivalent:(6)Z<0,Z11<0,Z22−Z12TZ11−1Z12<0,Z22<0,Z11−Z12Z22−1Z12T<0.Lemma 2 (Wirtinger inequality in discrete form [18, 19]).
For a given positive definite matrixN∈Rn×n, scalar 0≤ρ1≤ρ2 and vector function η:−ρ2,ρ1⟶Rn×n satisfy the following inequalities:(7)−ρ2−ρ1∑s=k−ρ2k−ρ1−1ηTsNηs≤−Ω1Ω2TN003NΩ1Ω2.
Among them,(8)ηs=xs+1−xs,Ω1=xk−ρ1−xk−ρ2,Ω2=xk−ρ1+xk−ρ2−2ρ2−ρ1+1∑s=k−ρ2k−ρ1xs.


### 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:(9)uk=unk+ufak.Among them,unk is the nominal control law when there is no fault and no attack, and ufak 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:(10)Sk=Gxk−τk+∑i=k−τkk−1GBui+Γk.Among them,G is a constant matrix with proper dimension, and GB is a nonsingular matrix; Γk is a sliding mode discrete compensator, which satisfies the following requirements: Γk+1=Γk−GA−BKxk−τk+Gxk−τk, where K is the gain matrix of the controller; the sliding mode of the system can be obtained by introducing ∑i=k−τkk−1GBui.Omittingwk and vk, combining formula (1) and formula (10), we can get(11)ΔSk=Sk+1−SkGxk+1−τk+1+∑i=k+1−τkkGBui+Γk+1‐Gxk−τk+∑i=k−τkk−1GBui+Γk=GAxk−τk+GBuk−τk+GEfk−τk+GNaaak−τk−Gxk−τk+GBuk−GBuk−τk+Gxk−τk−GAxk−τk+GBKxk−τk=GBKxk−τk+GBuk+GEfk−τk+GNaaak−τk.If the order isΔSk=0, there will be(12)uk=−GB−1GBKxk−τk+GEfk−τk+GNaaak−τk.Furthermore, the sliding mode control lawuk based on the reaching law method is(13)uk=−Kxk−τk−GB−1qSk+εsgnSk−GB−1GEfk−τk+GNaaak−τk+Nsask−τk,where q,ε is a constant scalar and q>0,ε>0.When there is no fault and no attack, the nominal control lawunk is(14)unk=−Kxk−τk−GB−1qSk+εsgnSk.When faults and attacks occur simultaneously, the compensation control lawufak is(15)ufa=−GB−1GEfk−τk+GNaaak−τk.Theorem 3.
When constantq,ε satisfies q>0,ε>0, respectively, the system (1) satisfies the existence and reachability conditions of the sliding mode.Proof.
Select Lyapunov function:Vk=STkSk.
Combining formulas (1) and (10), we can get(16)ΔVk=STkΔSk=STkSk+1−Sk=STkGBKxk−τk+GBuk+GEfk−τk+GNaaak−τk.
According to the control law (13),(17)ΔVk=STk−qSk−εsgnSk.
When the constantsq,ε satisfy q>0,ε>0, there are(18)ΔVk=−S2k−εSk≤0.
Therefore, 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.,Sk=0, the equivalent control term of the sliding mode control law can be obtained by substituting it into equation (13):(19)ueqk=−Kxk−τk−GB−1GEfk−τk+GNaaak−τk+Nsask−τk.
By substituting equation (19) into equation (1), the closed-loop model of NCS on a sliding surface can be obtained:(20)xk+1=Axk−Kxk−τk−BGB−1GEfk−τk+GNaaak−τk+BNsask−τk+Efk+Naaak+D1wk,yk=Cxk+Nsask+D2vk,xk=ϕk,−τM≤k≤0,where ϕk is a real valued initial function on −τM,0.☐


### 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:(21)η^k+1∣k=A~η^k+B~uk+E~f^k,Pk+1∣k=A~Pk−1∣k−1A~T+D~1QD~1T,y~k+1=yk+1−C~η^k+1∣k.Among them,P∈Rn×n is the state error covariance matrix, and y~k is the error between the system output and the estimated output.The gain matrix of adaptive Kalman filter fault/attack estimator is as follows:(22)Σk=C~Pk∣k−1C~T+D2RD2T,K1k=Pk∣k−1C~TΣ−1k.Among them,K1k is the gain matrix of Kalman filter.The error gain matrix of adaptive Kalman filter fault/attack estimator is as follows:(23)Ωk=C~γk+C~E~,γk+1=A~−K1kC~γk+E~,Fk=JTk−1ΩTkΛTk,Jk=1λI−Jk−1ΩTkΛkΩkJk−1,Λk=λΣk+ΩkJk−1ΩTk−1.Among them,γk,Fk is the error gain matrix, and λ∈0,1 is the forgetting factor.The update part of adaptive Kalman filter fault/attack estimator is as follows:(24)η^k+1=η^k+1∣k+K1ky~k+γk+1f^k+1−f^k,f^k+1=f^k+Fky~k,Pk+1∣k+1=I−K1kC~Pk+1∣k.Suppose 4. MatrixA~,C~ is completely observable, A~,Q1/2 is completely controllable, if matrix ξk satisfies(25)ξk+1=A~−K1kC~ξk.Then, for Kalman gain matrixK1k, ξk are exponentially convergent.Suppose 5. Under initial conditionJ0=ωIω>0, matrix Jk is strictly positive definite.System error definition:(26)η~k=ηk−η^k,f~k=fk−f^k,zk=η~k−γkf~k.From equations (2), (21), and (26),(27)zk+1=A~−K1kC~zk−K1kD2vk+D~1w~k.Further, we can get(28)Ezk+1=A~−K1kC~Ezk,where Ezk is the mean of zk.From hypothesis 1 and equation (28), we can see that 14 is exponentially convergent.From equations (21) and (24),(29)f~k+1=I−FkΩkf~k−FkC~zk+D2vk.Because ofEzk=0,Evk=0,(30)Ef~k+1=I−FkΩkEf~k.From hypothesis 2, we can see thatJk is strictly positive definite.Theorem 6.
When matrixJk is strictly positive definite and Eη~k exponentially approaches 0, the adaptive Kalman filter is convergent. Note Mk=J−1k.Proof.
Define Lyapunov function:(31)Vk+1=Ef~k+1TMkEf~k+1.
From equation (23), we can get(32)Jk=1λI−FkΩkJk−1.
Then,(33)Mk=λMk−1I−FkΩk−1.
From equations (32) and (33),(34)Vk+1=Ef~kTI−FkΩkTλMk−1Ef~k=λEf~kTMk−1Ef~k−λEf~kTΞEf~k.
In equation (34),(35)Ξk=ΩTkΓTkMk−1=ΩTkΛkΩkJk−1J−1k−1=ΩTkΛkΩk.
SinceΛk is a positive definite matrix, then Ξk is also a positive definite matrix. So,(36)Vk+1≤λEf~kTMkEf~k≤λVk.
It can be seen from equation (36) that Vk is exponentially close to 0. Since matrix Mk is strictly positive definite, Ef~k is exponentially close to 0.
From equation (26),(37)Eη~k=Ezk+γkEf~k.☐Then,Eη~k is also an index close to 0, so the adaptive Kalman filter is convergent.


## 2.1. System Model

Controlled object model:(1)xk+1=Axk+Buk+Efk+Naaak+D1wk,yk=Cxk+Nsask+D2vk.Among them,xk∈Rn,yk∈Rm, and uk∈Rl are the state, output, and control input of the system, respectively; fk∈Rp is the time-varying fault vector of the actuator; aak∈Rq is the constant network attack vector suffered by the actuator side, referred to as the actuator attack vector; ask∈Rq is the constant network attack vector suffered by the sensor side, referred to as the sensor attack vector; A∈Rn×n,B∈Rn×l, C∈Rm×n,E∈Rn×p, and Ns∈Rn×q are the coefficient matrices of proper dimension; and wk∈Rp and vk∈Rp 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 Q∈Rp×p and R∈Rp×p; D1∈Rn×p and D2∈Rm×p are the perturbations of proper dimension.Referring to the idea of state augmentation, the system statexk, actuator attack aak, and sensor attack ask are augmented into a new state ηk, which is called the augmented state. The augmented system model is(2)ηk+1=A~ηk+B~uk+E~fk+D~1w~k,yk=C~ηk+D2vk,where(3)ηk=xTkaaTkasTk,w~k=wTkbaTkbsTk,A~=ANa0000000,B~=B00,E~=E00,D~1=D000I000I,C~=C0NsT.


## 2.2. DETCS Trigger Conditions

This paper adopts the most representative discrete event trigger condition [16]:(4)x∧k−x∧tkTΦx^k−x^tk≤σx∧TtkΦx^tk.Among them,x^k is the estimated value of the state at the current sampling time, x^tk is the latest estimated value of the state at the network transmission time that satisfies equation (4) at the previous time, Φ is the event trigger matrix with positive definite symmetry, and σ∈0,1 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 x^k 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 functionτk=k−tk,k∈tk,tk+1, and 0≤τk≤τM. Among them, τM≜Tkmax+τ~ are the maximum nonuniform transmission periods of data filtering, and τ~ is the upper bound of delay function τk at tk+1.


### 2.2.1. Related Lemmas

Lemma 1 (Schur complement lemma [17]).
For a given symmetric matrix(5)Z=Z11Z12Z12TZ22,the following three conditions are equivalent:(6)Z<0,Z11<0,Z22−Z12TZ11−1Z12<0,Z22<0,Z11−Z12Z22−1Z12T<0.Lemma 2 (Wirtinger inequality in discrete form [18, 19]).
For a given positive definite matrixN∈Rn×n, scalar 0≤ρ1≤ρ2 and vector function η:−ρ2,ρ1⟶Rn×n satisfy the following inequalities:(7)−ρ2−ρ1∑s=k−ρ2k−ρ1−1ηTsNηs≤−Ω1Ω2TN003NΩ1Ω2.
Among them,(8)ηs=xs+1−xs,Ω1=xk−ρ1−xk−ρ2,Ω2=xk−ρ1+xk−ρ2−2ρ2−ρ1+1∑s=k−ρ2k−ρ1xs.


## 2.2.1. Related Lemmas

Lemma 1 (Schur complement lemma [17]).
For a given symmetric matrix(5)Z=Z11Z12Z12TZ22,the following three conditions are equivalent:(6)Z<0,Z11<0,Z22−Z12TZ11−1Z12<0,Z22<0,Z11−Z12Z22−1Z12T<0.Lemma 2 (Wirtinger inequality in discrete form [18, 19]).
For a given positive definite matrixN∈Rn×n, scalar 0≤ρ1≤ρ2 and vector function η:−ρ2,ρ1⟶Rn×n satisfy the following inequalities:(7)−ρ2−ρ1∑s=k−ρ2k−ρ1−1ηTsNηs≤−Ω1Ω2TN003NΩ1Ω2.
Among them,(8)ηs=xs+1−xs,Ω1=xk−ρ1−xk−ρ2,Ω2=xk−ρ1+xk−ρ2−2ρ2−ρ1+1∑s=k−ρ2k−ρ1xs.


## 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:(9)uk=unk+ufak.Among them,unk is the nominal control law when there is no fault and no attack, and ufak 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:(10)Sk=Gxk−τk+∑i=k−τkk−1GBui+Γk.Among them,G is a constant matrix with proper dimension, and GB is a nonsingular matrix; Γk is a sliding mode discrete compensator, which satisfies the following requirements: Γk+1=Γk−GA−BKxk−τk+Gxk−τk, where K is the gain matrix of the controller; the sliding mode of the system can be obtained by introducing ∑i=k−τkk−1GBui.Omittingwk and vk, combining formula (1) and formula (10), we can get(11)ΔSk=Sk+1−SkGxk+1−τk+1+∑i=k+1−τkkGBui+Γk+1‐Gxk−τk+∑i=k−τkk−1GBui+Γk=GAxk−τk+GBuk−τk+GEfk−τk+GNaaak−τk−Gxk−τk+GBuk−GBuk−τk+Gxk−τk−GAxk−τk+GBKxk−τk=GBKxk−τk+GBuk+GEfk−τk+GNaaak−τk.If the order isΔSk=0, there will be(12)uk=−GB−1GBKxk−τk+GEfk−τk+GNaaak−τk.Furthermore, the sliding mode control lawuk based on the reaching law method is(13)uk=−Kxk−τk−GB−1qSk+εsgnSk−GB−1GEfk−τk+GNaaak−τk+Nsask−τk,where q,ε is a constant scalar and q>0,ε>0.When there is no fault and no attack, the nominal control lawunk is(14)unk=−Kxk−τk−GB−1qSk+εsgnSk.When faults and attacks occur simultaneously, the compensation control lawufak is(15)ufa=−GB−1GEfk−τk+GNaaak−τk.Theorem 3.
When constantq,ε satisfies q>0,ε>0, respectively, the system (1) satisfies the existence and reachability conditions of the sliding mode.Proof.
Select Lyapunov function:Vk=STkSk.
Combining formulas (1) and (10), we can get(16)ΔVk=STkΔSk=STkSk+1−Sk=STkGBKxk−τk+GBuk+GEfk−τk+GNaaak−τk.
According to the control law (13),(17)ΔVk=STk−qSk−εsgnSk.
When the constantsq,ε satisfy q>0,ε>0, there are(18)ΔVk=−S2k−εSk≤0.
Therefore, 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.,Sk=0, the equivalent control term of the sliding mode control law can be obtained by substituting it into equation (13):(19)ueqk=−Kxk−τk−GB−1GEfk−τk+GNaaak−τk+Nsask−τk.
By substituting equation (19) into equation (1), the closed-loop model of NCS on a sliding surface can be obtained:(20)xk+1=Axk−Kxk−τk−BGB−1GEfk−τk+GNaaak−τk+BNsask−τk+Efk+Naaak+D1wk,yk=Cxk+Nsask+D2vk,xk=ϕk,−τM≤k≤0,where ϕk is a real valued initial function on −τM,0.☐


## 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:(21)η^k+1∣k=A~η^k+B~uk+E~f^k,Pk+1∣k=A~Pk−1∣k−1A~T+D~1QD~1T,y~k+1=yk+1−C~η^k+1∣k.Among them,P∈Rn×n is the state error covariance matrix, and y~k is the error between the system output and the estimated output.The gain matrix of adaptive Kalman filter fault/attack estimator is as follows:(22)Σk=C~Pk∣k−1C~T+D2RD2T,K1k=Pk∣k−1C~TΣ−1k.Among them,K1k is the gain matrix of Kalman filter.The error gain matrix of adaptive Kalman filter fault/attack estimator is as follows:(23)Ωk=C~γk+C~E~,γk+1=A~−K1kC~γk+E~,Fk=JTk−1ΩTkΛTk,Jk=1λI−Jk−1ΩTkΛkΩkJk−1,Λk=λΣk+ΩkJk−1ΩTk−1.Among them,γk,Fk is the error gain matrix, and λ∈0,1 is the forgetting factor.The update part of adaptive Kalman filter fault/attack estimator is as follows:(24)η^k+1=η^k+1∣k+K1ky~k+γk+1f^k+1−f^k,f^k+1=f^k+Fky~k,Pk+1∣k+1=I−K1kC~Pk+1∣k.Suppose 4. MatrixA~,C~ is completely observable, A~,Q1/2 is completely controllable, if matrix ξk satisfies(25)ξk+1=A~−K1kC~ξk.Then, for Kalman gain matrixK1k, ξk are exponentially convergent.Suppose 5. Under initial conditionJ0=ωIω>0, matrix Jk is strictly positive definite.System error definition:(26)η~k=ηk−η^k,f~k=fk−f^k,zk=η~k−γkf~k.From equations (2), (21), and (26),(27)zk+1=A~−K1kC~zk−K1kD2vk+D~1w~k.Further, we can get(28)Ezk+1=A~−K1kC~Ezk,where Ezk is the mean of zk.From hypothesis 1 and equation (28), we can see that 14 is exponentially convergent.From equations (21) and (24),(29)f~k+1=I−FkΩkf~k−FkC~zk+D2vk.Because ofEzk=0,Evk=0,(30)Ef~k+1=I−FkΩkEf~k.From hypothesis 2, we can see thatJk is strictly positive definite.Theorem 6.
When matrixJk is strictly positive definite and Eη~k exponentially approaches 0, the adaptive Kalman filter is convergent. Note Mk=J−1k.Proof.
Define Lyapunov function:(31)Vk+1=Ef~k+1TMkEf~k+1.
From equation (23), we can get(32)Jk=1λI−FkΩkJk−1.
Then,(33)Mk=λMk−1I−FkΩk−1.
From equations (32) and (33),(34)Vk+1=Ef~kTI−FkΩkTλMk−1Ef~k=λEf~kTMk−1Ef~k−λEf~kTΞEf~k.
In equation (34),(35)Ξk=ΩTkΓTkMk−1=ΩTkΛkΩkJk−1J−1k−1=ΩTkΛkΩk.
SinceΛk is a positive definite matrix, then Ξk is also a positive definite matrix. So,(36)Vk+1≤λEf~kTMkEf~k≤λVk.
It can be seen from equation (36) that Vk is exponentially close to 0. Since matrix Mk is strictly positive definite, Ef~k is exponentially close to 0.
From equation (26),(37)Eη~k=Ezk+γkEf~k.☐Then,Eη~k 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 x^k, actuator fault f^k, and network attack a^ak on the actuator side, considering the event triggering conditions, substituting the estimated value satisfying the conditions into control law uk (13), the adjusted fault-tolerant/intrusion-tolerant control law uk is(38)uk=−Kx^tk−GB−1qSk+εsgnSk−GB−1GEf^tk+GNaa^atk+Nsastk.Suppose 7. Actuator faultfk is bounded, i.e., fk≤ρ1; aak is bounded actuator attack, i.e., aak≤ρ2, where ρ1,ρ2 are known functions greater than 0.The existence condition and global reachability condition of sliding mode are(39)STkΔSk<0.By substituting equation (38) into equation (39), we find that(40)STkΔSk=STkGBKxtk+GBuk+GEftk+GNaaatk=STkGBKxtk+GEftk+GNaaatk+GB−Kx^tk−GB−1qSk+εsgnSk+GEf^tk+GNaa^atk=STkGBKxtk−x^tk+GEftk−f^tk+GNaaatk−a^atk−qSk−εsgnSk.From hypothesis 3,(41)ftk−f^tk≤ρ1,aatk−a^atk≤ρ2can be obtained.Then,(42)STkΔSk≤‐qS2k−εSTk+STkGBKe+GEρ1+GNaρ2.When the constantq,ε satisfies the following inequality:(43)ε≥GBKe+GEρ1+GNaρ2;q>0.Then,STkΔSk≤‐qS2k≤0.Therefore, the existence and reachability conditions of the sliding mode of the system (20) are satisfied, which means that the starting point from any initial state other than the sliding mode surface Sk=0 can return to the sliding mode surface in a finite time.When the system is on the sliding surface, the equivalent control term of the controller can be obtained by combining equation (54) with equation Sk=0:(44)ueqk=−Kx^tk+Nsastk−GB−1GEf^tk+GNaa^atk.By substituting equation (44) into equation (1), the state space model of the closed-loop system on the sliding surface can be obtained as follows:(45)xk+1=Axk+Efk+Naaak+D1wk+B,−Kx^tk−GB−1GEf^tk+GNaa^atk+Nsastk,yk=Cxk+Nsask+D2vk,xk=ϕk,k∈−τM,0,where ϕk is a real valued initial function on −τM,0.Since the system has satisfied the reachability condition of the sliding mode surface, in order to ensure the stability, the sliding mode described by system (45) can be stabilized by designing an appropriate gain matrix K:(46)X1∗∗∗∗∗∗∗∗∗∗∗∗Y1−Q1∗∗∗∗∗∗∗∗∗∗∗2S10−4S1∗∗∗∗∗∗∗∗∗∗6S106S1−12S1∗∗∗∗∗∗∗∗∗τM2BKT000−γ2I∗∗∗∗∗∗∗∗τM2BKT0000−γ2I∗∗∗∗∗∗∗τM2BKT00000−γ2I∗∗∗∗∗∗τM2BKT000000−γ2I∗∗∗∗∗00000000σΦ∗∗∗∗000000000−Φ∗∗∗τMAS1−τMBK100−τMBK1−τME−τMNaτMD100−S1∗∗AS1−BK100−BK1−E−NaD1000−2S1+P1∗S100000000000−I<0.Among them,(47)Y1=τM2K1TBT−2S1,X1=−P1+Q1+R1+τM2−4S1−τM2S1AT−τM2AS1,and K1=KS~.Theorem 8.
Given scalarσ>0,τM>0,γ>0, if there are matrix K, symmetric positive definite matrix P1,Q1,R1,S1, and event triggering matrix Φ satisfying the following matrix inequalities (46), then the sliding mode (45) of the closed-loop system is asymptotically stable at H∞ disturbance rejection level γ.It is proved that the stability of mode (45) of the closed-loop system is investigated in the presence of external disturbance wk, actuator failure fk and network attack aak. The asymptotic stability condition and robust performance index of the closed-loop system are as follows(48)∑k=0∞xTkxk<γ2∑k=0∞wTkwk+tk+1−tkexTkexk+efTkefk+eaTkeak.The Lyapunov-Krasovskii function is constructed as(49)Vk=∑i=14Vik.Among them,(50)V1k=xTkP~xk,V2k=∑i=k−τkk−1xTiQ~xi,V3k=∑i=k−τMk−1xTiR~xi,V4k=τM∑i=−τM−1∑j=k+ik−1ξTjS~ξj,ξk≜xk+1‐xk.Fork∈tk,tk+1, define ΔVk=∑i=14ΔVik, where(51)ΔV1k=xTk+1P~xk+1−xTkP~xk,(52)ΔV2k=∑i=k+1−τk+1kxTiQ~xi−∑i=k−τkk−1xTiQ~xi=xTkQ~xk−xTk−τkQ~xk−τk,(53)ΔV3k=∑i=k+1−τMkxTiR~xi−∑i=k−τMk−1xTiR~xi=xTkR~xk−xTk−τMR~xk−τM,(54)ΔV4k=τM∑i=−τM−1∑j=k+1+ikξTjS~ξj−τM∑i=−τM−1∑j=k+ik−1ξTjS~ξj=τM∑i=−τM−1∑j=k+1+ikξTjS~ξj−∑j=k+ik−1ξTjS~ξj=τM∑i=−τM−1ξTkS~ξk−ξTk+iS~ξk+i=τM2ξTkS~ξk−τM∑j=k−τMk−1ξTjS~ξj.According to Wirtinger’s inequality, it can be concluded that(55)−∑j=k−τMk−1τMxj+1−xjTS~xj+1−xj≤xk−xk−τMTS~xk−xk−τM−3xk+xk−τM−2τM+1∑j=k−τMkxjTS~xk+xk−τM−2τM+1∑j=k−τMkxj.The event triggers the condition, whenk∈tk+τtk,tk+1+τtk+1, there is(56)elTkΦelk≤σx∧TtkΦx^tk.Among them,elk=x^k−x^tk is the state estimation error.NoteφTk=xTkxTtkxTk−τMψTkexTtkefTtkeaTtkwkx∧TtkelTk.Among them,ψk=1/τM+1∑j=k−τMkxj can be obtained by combining formulas (51)–(55) and (45):(57)ΔVk+xTkxk−γ2wTkwk+∑k=0∞tk+1−tkexTtkextk+efTtkeftk+eaTtkeatk+σx∧TtkΦx^tk−elTkΦelk≤φTkΘφk.Among them,(58)Θ=Θ11∗∗∗∗∗∗∗∗∗Θ21Θ22∗∗∗∗∗∗∗∗2S~0−4S~∗∗∗∗∗∗∗6S~06S~−12S~∗∗∗∗∗∗Θ51Θ52Θ53Θ54Θ55∗∗∗∗∗Θ61Θ62Θ63Θ64Θ65Θ66∗∗∗∗Θ71Θ72Θ73Θ74Θ75Θ76Θ77∗∗∗Θ81Θ82Θ83Θ84Θ85Θ86Θ87Θ88∗∗00000000σΦ∗000000000−Φ,Θ11=ATP~+τM2S~A−P~+Q~+R~+τM2S~−τM2ATS~−τM2S~A−4S~+I,Θ21=−BKTP~+τM2S~A+τM2BKTS~−2S~,Θ51=−BKTP~+τM2S~A+τM2BKTS~,Θ61=−ETP~+τM2S~A+τM2ETS~,Θ71=−NaTP~+τM2S~A+τM2NaTS~,Θ71=D1TP~+τM2S~A−τM2D1TS~,Θ22=BKTP~+τM2S~BK−Q~,Θ52=BKTP~+τM2S~BK,Θ62=ETP~+τM2S~BK,Θ72=NaTP~+τM2S~BK,Θ82=D1TP~+τM2S~BK,Θ53=Θ63=Θ73=Θ83=0,Θ54=Θ64=Θ74=Θ84=0,Θ55=BKTP~+τM2S~BK−γ2I,Θ65=ETP~+τM2S~BK,Θ75=NaTP~+τM2S~BK,Θ85=−D1TP~+τM2S~BK,Θ66=ETP~+τM2S~E−γ2I,Θ76=NaTP~+τM2S~E,Θ86=−D1TP~+τM2S~E,Θ77=NaTP~+τM2S~Na−γ2I,Θ87=−D1TP~+τM2S~Na,Θ88=−D1TP~+τM2S~D1−γ2I,supposing that ΔVk+xTkxk−γ2wTkwk+∑k=0∞tk+1−tkexTtkextk+efTtkeftk+eaTtkeatk<0,Θ<0. By using Lemma 1 twice, we can get the following results:(59)X∗∗∗∗∗∗∗∗∗∗∗Y−Q~∗∗∗∗∗∗∗∗∗∗2S~0−4S~∗∗∗∗∗∗∗∗∗6S~06S~−12S~∗∗∗∗∗∗∗∗τM2BKTS~000−γ2I∗∗∗∗∗∗∗τM2ETS~0000−γ2I∗∗∗∗∗∗τM2NaTS~00000−γ2I∗∗∗∗∗τM2D1TS~000000−γ2I∗∗∗∗00000000σΦ∗∗∗000000000−Φ∗∗τMS~A−τMS~BK00−τMS~BK−τMS~E−τMS~NaτMS~D100−S~∗P~A−P~BK00−P~BK−P~E−P~NaP~D1000−P~<0.Among them,(60)Y=τM2BKTS~−2S~,X=−P~+Q~+R~+τM2−4S~−τM2ATS~−τM2S~A+I.If (59) is multiplied by diagS~−1,S~−1,S~−1,S~−1,S~−1,S~−1,S~−1,I,I,I,P~−1,P~−1 on the left and the right, then Lemma 1 is applied again to the inequality obtained. If K1=KS~−1,S1=S~−1,P1=S~−1P~S~−1,Q1=S~−1Q~S~−1, R1=S~−1R~S~−1 is used at the same time, then(61)X1∗∗∗∗∗∗∗∗∗∗∗∗Y1−Q1∗∗∗∗∗∗∗∗∗∗∗2S10−4S1∗∗∗∗∗∗∗∗∗∗6S106S1−12S1∗∗∗∗∗∗∗∗∗τM2BKT000−γ2I∗∗∗∗∗∗∗∗τM2ET0000−γ2I∗∗∗∗∗∗∗τM2NaT00000−γ2I∗∗∗∗∗∗τM2D1T000000−γ2I∗∗∗∗∗00000000σΦ∗∗∗∗000000000−Φ∗∗∗τMAS1−τMBK100−τMBK1−τME−τMNaτMD100−S1∗∗AS1−BK100−BK1−E−NaD1000−P~−1∗S100000000000−I<0.Among them,(62)Y1=τM2K1TBT−2S1,X1=−P1+Q1+R1+τM2−4S1−τM2S1AT−τM2AS1.Similarly, for symmetric positive definite matrixP~,S~, we have(63)S~P~−1S~−2S~+P~=S~−P~P~−1S~−P~≥0.Further, we can get(64)−P~−1≤−2S1+P1.Combining formulas (61) and (64), formula (46) holds.If (46) is true, then(65)ΔVk+xTkxk−γ2wTkwk+∑k=0∞tk+1−tkexTtkextk+efTtkeftk+eaTtkeatk<0.We can go further:(66)∑k=0∞xTkxk−γ2wTkwk+tk+1−tkexTtkextk+efTtkeftk+eaTtkeatk<−∑k=0∞ΔVk=−Vk≤0.Obviously,(67)∑k=0∞xTkxk<γ2∑k=0∞wTkwk+tk+1−tkexTtkextk+efTtkeftk+eaTtkeatk.Namely,(68)xt2<γminwt2+tk+1‐tkextk2+eftk2+eatk2.In other words, when equation (46) holds, sliding mode (45) of the closed-loop system is asymptotically stable and has H∞ disturbance rejection performance.Theorem8 is proved.


## 4. Result and Discussion

In this paper, the control system model given in Reference [20] is taken as the research object:(69)A=0.98790.0098−0.08370.9908,B=−0.0029−0.0005−0.1919−0.0378,E=−0.0029−0.1919,Na=−0.1341−0.1243,Ns=−0.0051−0.0078,Dw=0.11,Dv=0.11,C=1001.In this simulation, we make the following assumptions about actuator failure and network attack.Actuator failure:(70)fk=2sin0.01πk+2,230≤k≤500.Network attack:(71)aak=2,100≤k≤500,ask=−2,100≤k≤500.The expanded coefficient matrices are as follows:(72)A11=0.98790.0098−0.00291−0.08370.9908−0.1919100100001,B11=−0.0029−0.0005−0.1919−0.03780000,E11=−0.0029−0.191900,Dw11=diagDw,I,I=0.100100010001,C11=10000100,Dv11=Dv=0.11.Let the initial condition of the system beP0∣0=I4,λ=0.9, x0=x^0=−30202−2T,R=0.0025,rk=02×1,Q=0.0001.From equation (26),(73)K1=1.1555−0.11681.6243−0.15980.4433−0.0447−0.0156−0.0018.In Theorem6, let the maximum delay upper bound τM=1.2, trigger parameter σ=0.85, take q=5,ε=2,T=0.1. Based on the sliding mode trigger matrix, we can get the event gain matrix G and Φ, respectively,(74)G=−5.69951.0126−0.6560.8053,Φ=1.58120.2811−0.26572.6766.The state feedback gain matrices are(75)Kc1=63.72−4823.2823,Kc2=−273.6−998.25832.23389.58.The simulation results are shown in Figures2–11. Figures 2 and 3 show the attack and its estimation curve and attack estimation error diagram, respectively. Figures 4 and 5 show the actuator fault and its estimation curve and error diagram, respectively. Figures 6–9 show the state of the system and its estimation curve and state estimation curve, respectively. Figure 10 shows the output response curve of the system when actuator failure and network attack occur simultaneously. Figure 11 shows the data transmission time and the transmission interval of the system nonuniform transmission NCS.Figure 2
Actuator attack and its estimation for active fault-tolerant/active passive intrusion-tolerant systems.Figure 3
Actuator attack estimation error of active fault-tolerant/active passive intrusion-tolerant system.Figure 4
Fault estimation of active fault-tolerant/active passive intrusion-tolerant system.Figure 5
Fault estimation error of active fault-tolerant/active passive intrusion-tolerant system.Figure 6
State 1 and estimation of active fault-tolerant/active passive intrusion-tolerant system.Figure 7
State 1 estimation error of active fault-tolerant/active passive intrusion-tolerant system.Figure 8
State 2 and estimation of active fault-tolerant/active passive intrusion-tolerant system.Figure 9
State 2 estimation error of active fault-tolerant/active passive intrusion-tolerant system.Figure 10
Output response of active fault-tolerant/active passive intrusion-tolerant system.Figure 11
Transmission time and interval of nonuniform transmission.It can be seen from Figures2–9 that the adaptive Kalman filter fault/attack estimator can well estimate the state, fault, and attack of the system. It can be seen from Figure 10 that when the system encounters actuator failure and network attack, the vibration is gradually attenuated and tends to balance by using the cooperative controller designed in this paper and the compensation strategy of failure and attack. Simulation results show that the active fault-tolerant/active passive intrusion-tolerant cooperative controller designed in this paper is effectively fault-tolerant and active and passive intrusion-tolerant for network attacks and also suppresses the influence of disturbance and noise.According to the analysis in Figure11, when the event trigger parameter is σ=0.85, compared with the traditional PPTCS which needs to transmit 500 data in the simulation time of 50 s, 249 data are transmitted in the DETCS, the data transmission rate is 49.8%, the average transmission cycle is T¯=0.2006s, and the maximum transmission cycle is Tmax=0.6s. This shows that the control method proposed in this paper not only ensures the excellent performance of the system but also effectively saves the network communication resources and improves the efficiency of network utilization.


## 5. Conclusion

In this paper, an active fault-tolerant/active passive intrusion-tolerant cooperative controller design method is proposed for linear discrete NCS with time-varying delay based on discrete event triggering mechanism when actuator failure and network attack occur. In this method, the attack is extended to the state, and the adaptive Kalman filter is used to estimate the state, fault, and attack. Then, the integral sliding mode control method is used to design the active fault-tolerant/active passive intrusion-tolerant cooperative controller, so that the system can keep normal operation and haveH∞ disturbance rejection performance in the case of actuator failure and network attack. Finally, a simulation example is given to illustrate the effectiveness and applicability of the proposed method.



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*Source: 9258411-2021-07-19.xml*