# Hybrid RobustH2/H∞ Fault-Tolerant Control with Random Delay NCS

Authors: Jun Wang; Miaomiao Ling
Journal: Journal of Control Science and Engineering
 (2020)

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## Abstract

In this paper, a hybrid fault-tolerant control method with off-line design and online scheduling is proposed for NCS with actuator faults, random delay, and external finite energy disturbance. The problem of less conservatism of robust generalizedH2/H∞ hybrid fault-tolerant control is studied. Firstly, a closed-loop fault model of the system with random delay parameters was established according to the Bernoulli 0-1 distribution; all possible prior faults are divided into a few intervals according to certain rules, and then an interval fault-tolerant controller is designed off-line according to the prior faults of each interval. Secondly, when the fault is estimated online, the corresponding interval fault-tolerant controller is called through the scheduling mechanism to achieve rapid fault tolerance of prior faults within the interval and mitigate the impact of other faults within the interval, which provides a guarantee for subsequent safe reconstruction control. Finally, the effectiveness of the proposed method is verified by Matlab simulation.


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## Body

## 1. Introduction

With the development of science and technology, a networked control system (NCS) has been widely used in aerospace, chemical industry, and other fields due to its convenient operation and strong integrated control. As one of the important components of an NCS, the actuator is susceptible to the influence of the external environment and long-term operation, which will inevitably affect the normal operation of the system, resulting in poor product quality and even system collapse. Therefore, fault-tolerant control for NCS actuator faults is a high concern in industry.Passive fault-tolerant control (PFTC) [1–3] has been studied by a large number of scholars due to its advantages of simple design. In addition, the active fault-tolerant control (AFTC) [4–6] method, which can design the controller online for each fault, is also discussed. From a large number of studies, it is not difficult to find that the above two classical control methods have their own shortcomings. Because PFTC contains a large number of fault modes, the solutions are often conservative, which cannot meet some performance requirements. AFTC requires a lot of computation time to estimate faults online and design controllers, so its fault-tolerant reliability is questionable. In order to further improve the reliability of fault-tolerant control, in recent years a few scholars have proposed a hybrid fault-tolerant control method combining their respective advantages [7–11]. Jiang and Yu[7] compare the advantages and limitations of active and passive fault-tolerant control methods and verifies them in the UAV fault calculation example. Su et al. [8] first combined AFTC and passive decentralized unconditional stability control technology to deduce the stability conditions of the FTC system in the case of sensor failure. Yu and Jiang [9] make use of the advantages of active and passive control to establish a hybrid control strategy for a UAV with control failure. The robustness of PFTC not only guarantees the stability of the system, but also provides favorable conditions for fault detection and diagnosis (FDD) to obtain accurate fault information and reconstruct faults. The above research on hybrid control methods is still in the preliminary exploration stage and has not been involved in the field of the NCS. However, network congestion, limited bandwidth, and other conditions widely exist in an NCS. Therefore, a discrete event-triggered communication scheme (DETCS) was established in reference [10] to alleviate the blocking of data transmission by screening effective data. Meanwhile, an active-passive hybrid fault-tolerant controller was integrated and designed for arbitrary actuator faults. In reference [11], under the action of DETCS, the hybrid robust fault-tolerant design problem with external interference and a parameter uncertainty system was discussed based on T-S fuzzy models. It is worth mentioning that, due to the existence of passive robustness, the above control methods are still some conservatism. How to use hybrid control to effectively reduce the conservatism of results and ensure the satisfactory performance of the system needs to be further studied.At the same time, the NCS is affected by the difference of transmission codes and topological structures, which causes the network-induced delay to be random. In practical systems, randomness usually manifests as random uncertainty, which will cause the delay to occur with a certain probability during the propagation process, resulting in partial loss of time-delay information. Therefore, based on the traditional delay rate change method, the system cannot be guaranteed to be stable. To overcome this difficulty, scholars have recently studied the stability problem under probabilistic delay. In reference [12], the Bernoulli 0-1 distribution is introduced to construct stochastic time-delay functions, and the robust stability conditions of stochastic systems are discussed. On the basis of [12], Li and Wu [13] use the state observer as the state feedback to establish the closed-loop augmented system and design the controller with exponential stability of mean square. In reference [14], the stability analysis of nonlinear disturbance and random time delay in the sampling process was studied. Obviously, none of the above random delay stability design problems take system failure into consideration. However, complex situations such as system failure under random delay cannot be ignored in the field of stability, and only a few scholars discuss this problem. For example, Wang et al. [15] study the synchronous design problem of network node coupling faults. Zhang et al. [16] discuss the design of the controller under actuator failure by considering relatively comprehensive random delay information. Therefore, the stability problem of NCS actuator faults with probabilistic delay has certain research value.In this paper, based on the Bernoulli 0-1 distribution, the random delay is applied to two different probability intervals, which effectively compensates for the shortcomings of the delay variation rate in the probabilistic delay problem. At the same time, the event-triggering conditions and scheduling mechanism are established. Aiming at network congestion and actuator failure, a hybrid robust fault-tolerant control method of “off-line design, online scheduling” with less conservative intervals is proposed based on DETCS. Under the action of FDD, when the interval prior fault occurs, the off-line designed interval fault-tolerant controller is quickly invoked for control. When other faults occur within the interval, FDD compensates the interval fault-tolerant controller for the purpose of fast fault tolerance and makes the system haveH2/H∞ performance.


## 2. Problem Description

### 2.1. Event-Triggered Communication Scheme

DETCSs are introduced to conserve network resources when network bandwidth is limited. Its system structure diagram is shown in Figure1.Figure 1
NCS structure diagram under event-triggering scheme.For the convenience of presentation, the following assumptions are made.Hypothesis 1.
The sampling period of the sampler ish, the set of sampling times ikh is ikhik∈N, and the set of transmission times tkh is tkhtk∈N.Hypothesis 2 (see [17]).
The system state is completely measurable.
It can be seen from Figure1 that the actuator, the controlled plant, the sensor, the controller, and the network jointly constitute the NCS, where the time delay exists in the sensor-to-controller τsc, controller-to-actuator τca, and calculation process τc. The data sampled by the sampler will be sent to the network through an event-triggering scheme to save network resources. According to system requirements, the following trigger conditions are created:(1)tk+1h=tkh+minlhexTikhΓexikh≥σxTikhΓxikh,where Γ is the positive definite symmetric matrix, σ∈0,1 is the trigger parameter, and the state error is(2)exikh=xikh−xtkh.


### 2.2. The System Model

Establish linear NCS fault model under the DETCS:(3)x˙=Axt+But+Eft+Dwt,yt=C11xt,zt=C12xt,where xt∈Rn and ut∈Rm are system states and control inputs; yt∈Rp is the measurement output, and zt∈Rp is the adjusted output; wt∈L2 is the external finite energy disturbance; the system matrices A, B, C11, C12, E, and D are constant matrices with appropriate dimensions; ft∈Rr is the additive fault of the actuator, and ft≤f0. Suppose the system has m actuators, then(4)ft=f1f2⋯fm,q=1,2,…,m.Whenfq=0, it means that the qth actuator works normally. When fq≠0, the fault of the qth actuator is indicated.For the convenience of subsequent proof, let the fault distribution matrixE=−B, and define the actuator fault ft as(5)ft=L^ut,L^=I−L,where L=diagl1,l2,…,lm is the fault degree matrix and lq∈0,1,q=1,2,…,m. When lq=1, the qth actuator is normal. When lq=0, the qth actuator completely fails. When lq∈1,0, the qth actuator partially fails.According to equations (3)–(5), Lut=ut−ft can be obtained; that is, the fault model (3) can be transformed into(6)x˙=Axt+BLut+Dwt,yt=C11xt,zt=C12xt.It can be known from the above that (3) and (6) are equivalent, so the state feedback control law is ut=Kxt−τt−exikh, t∈tkh+τtk,tk+1h+τtk+1, where τi is the delay corresponding to the transmission time of i. In the actual system, τt has randomness, so there is a constant d10≤d1≤d2 which makes the delay τt have a certain probability when it is distributed on 0,d1 and d1,d2. Therefore, Bernoulli sequence δt is defined to satisfy the following formula:(7)δt∼1,τ1∈0,d1,0,τ2∈d1,d2,and then the probability distribution of random delay τt is assumed to be(8)Pδt=1=P0≤τ1t≤d1=δ0,Pδt=0=Pd1<τ2t≤d2=1−δ0,δ0∈0,1,τ˙1t<μ1<∞,τ˙2t<μ2<∞.The closed-loop fault system model with random delay is as follows:(9)x˙=Ax+δ0BLKxτ1+1−δ0BLKxτ2+δ−δ0BLKxτ1−xτ2−BLKexikh+Dw,yt=C11xt,zt=C12xt.To facilitate further derivation and proof, the following lemma is introduced.Lemma 1 (see [18]).
Any matrixW>0, for a scalar b>a and a function w:a,b⟶Rn, has the following integral inequality:(10)∫abw¯sdsTW∫abw¯sds≤b−a∫abw¯TsWw¯sds.Lemma 2 (see [19] (Schur)).
Given the symmetric matrixZ=Z11Z12Z21Z22, following conditions are equivalent:(1)
Z<0(2)
Z11<0,Z22−Z12TZ11−1Z12<0(3)
Z22<0,Z11−Z12TZ22−1Z12<0Lemma 3 (see [20]).
For any matrixM=RU∗R≥0, if its scalar satisfies 0≤dt≤d for the vector x˙:−d,0⟶Rn, then the inequality is set up as −d∫t−dtx˙TaRx˙ada≤sTtZst, and(11)STt=xTtxTt−dtxTt−d,Z=−RR−UU∗−2R+U+UTR−U∗∗−R,where ∗ is the corresponding matrix block obtained by the symmetric matrix block.


## 2.1. Event-Triggered Communication Scheme

DETCSs are introduced to conserve network resources when network bandwidth is limited. Its system structure diagram is shown in Figure1.Figure 1
NCS structure diagram under event-triggering scheme.For the convenience of presentation, the following assumptions are made.Hypothesis 1.
The sampling period of the sampler ish, the set of sampling times ikh is ikhik∈N, and the set of transmission times tkh is tkhtk∈N.Hypothesis 2 (see [17]).
The system state is completely measurable.
It can be seen from Figure1 that the actuator, the controlled plant, the sensor, the controller, and the network jointly constitute the NCS, where the time delay exists in the sensor-to-controller τsc, controller-to-actuator τca, and calculation process τc. The data sampled by the sampler will be sent to the network through an event-triggering scheme to save network resources. According to system requirements, the following trigger conditions are created:(1)tk+1h=tkh+minlhexTikhΓexikh≥σxTikhΓxikh,where Γ is the positive definite symmetric matrix, σ∈0,1 is the trigger parameter, and the state error is(2)exikh=xikh−xtkh.


## 2.2. The System Model

Establish linear NCS fault model under the DETCS:(3)x˙=Axt+But+Eft+Dwt,yt=C11xt,zt=C12xt,where xt∈Rn and ut∈Rm are system states and control inputs; yt∈Rp is the measurement output, and zt∈Rp is the adjusted output; wt∈L2 is the external finite energy disturbance; the system matrices A, B, C11, C12, E, and D are constant matrices with appropriate dimensions; ft∈Rr is the additive fault of the actuator, and ft≤f0. Suppose the system has m actuators, then(4)ft=f1f2⋯fm,q=1,2,…,m.Whenfq=0, it means that the qth actuator works normally. When fq≠0, the fault of the qth actuator is indicated.For the convenience of subsequent proof, let the fault distribution matrixE=−B, and define the actuator fault ft as(5)ft=L^ut,L^=I−L,where L=diagl1,l2,…,lm is the fault degree matrix and lq∈0,1,q=1,2,…,m. When lq=1, the qth actuator is normal. When lq=0, the qth actuator completely fails. When lq∈1,0, the qth actuator partially fails.According to equations (3)–(5), Lut=ut−ft can be obtained; that is, the fault model (3) can be transformed into(6)x˙=Axt+BLut+Dwt,yt=C11xt,zt=C12xt.It can be known from the above that (3) and (6) are equivalent, so the state feedback control law is ut=Kxt−τt−exikh, t∈tkh+τtk,tk+1h+τtk+1, where τi is the delay corresponding to the transmission time of i. In the actual system, τt has randomness, so there is a constant d10≤d1≤d2 which makes the delay τt have a certain probability when it is distributed on 0,d1 and d1,d2. Therefore, Bernoulli sequence δt is defined to satisfy the following formula:(7)δt∼1,τ1∈0,d1,0,τ2∈d1,d2,and then the probability distribution of random delay τt is assumed to be(8)Pδt=1=P0≤τ1t≤d1=δ0,Pδt=0=Pd1<τ2t≤d2=1−δ0,δ0∈0,1,τ˙1t<μ1<∞,τ˙2t<μ2<∞.The closed-loop fault system model with random delay is as follows:(9)x˙=Ax+δ0BLKxτ1+1−δ0BLKxτ2+δ−δ0BLKxτ1−xτ2−BLKexikh+Dw,yt=C11xt,zt=C12xt.To facilitate further derivation and proof, the following lemma is introduced.Lemma 1 (see [18]).
Any matrixW>0, for a scalar b>a and a function w:a,b⟶Rn, has the following integral inequality:(10)∫abw¯sdsTW∫abw¯sds≤b−a∫abw¯TsWw¯sds.Lemma 2 (see [19] (Schur)).
Given the symmetric matrixZ=Z11Z12Z21Z22, following conditions are equivalent:(1)
Z<0(2)
Z11<0,Z22−Z12TZ11−1Z12<0(3)
Z22<0,Z11−Z12TZ22−1Z12<0Lemma 3 (see [20]).
For any matrixM=RU∗R≥0, if its scalar satisfies 0≤dt≤d for the vector x˙:−d,0⟶Rn, then the inequality is set up as −d∫t−dtx˙TaRx˙ada≤sTtZst, and(11)STt=xTtxTt−dtxTt−d,Z=−RR−UU∗−2R+U+UTR−U∗∗−R,where ∗ is the corresponding matrix block obtained by the symmetric matrix block.


## 3. Main Results

### 3.1. Failure Detection Observer Design

Considering the system model (3) for the NCS, a fault detection observer model with random delay is established:(12)x^˙=Ax^t+But+δ0Gyt−τ1−y^t−τ1+1−δ0Gyt−τ2−y^t−τ2+δ−δ0Gyt−τ1−y^t−τ1−yt−τ2−y^t−τ2,y^t=C11x^t,where G is the observer gain matrix.The system output residualrt, state error et, and fault estimation error ret are defined as follows:(13)rt=Wyt−y^t,(14)et=xt−x^t,(15)ret=rt−ft,where W is the system output residual weight matrix.By substituting equations (13)–(15) into equation (12), the observer error system of fault detection can be written as(16)e˙t=Aet−δ0GC11et−τ1−1−δ0GC11et−τ2−δ−δ0GC11et−τ1−et−τ2−Bft+Dwt,ret=WC11et−ft.Therefore, the observer design problem is equivalent to solvingG and W, making the error system (16) asymptotically stable and satisfying the following performance:(17)ret2≤γ12ft2+γ22wt2.Note: it is easy to write and remember(18)et=e,et−τt=eτ,et−τ1t=eτ1,et−τ2t=eτ2,xt=x,xt−τt=xτ,xt−τ1t=xτ1,xt−τ2t=xτ2,wt=w,ft=f,ret=re.Theorem 1.
For the error system16 and performance index (17), given positive numbers d1, d2, μ1, μ2, δ0, γ1, γ2, a, and b, if symmetric positive definite matrices P, V, W, and Qii=1,2,3,4 exist, the following matrix inequality (19) is feasible:(19)Φ=Φ11Φ120Φ140Φ16Φ17Φ18Φ19Φ110∗Φ22Φ2300000Φ29Φ210∗∗Φ33Φ34000000∗∗∗Φ44Φ45000Φ49Φ410∗∗∗∗Φ5500000∗∗∗∗∗Φ660Φ68Φ69Φ610∗∗∗∗∗∗Φ770Φ79Φ710∗∗∗∗∗∗∗Φ8800∗∗∗∗∗∗∗∗Φ990∗∗∗∗∗∗∗∗∗Φ1010<0,where(20)Φ11=ATP+PA+Q1+Q2−Q3,Φ12=−δ0VC11+Q3,Φ14=−1−δ0VC11,Φ16=−PB,Φ17=PD,Φ18=C11TWT,Φ19=d1ATP,Φ110=d2−d1ATP,Φ22=−1−μ1Q1−2Q3,Φ23=Q3,Φ29=−d1δ0C11TVT,Φ210=−d2−d1δ0C11TVT,Φ33=−Q3−Q4,Φ34=Q4,Φ44=−1−μ2Q2−2Q4,Φ45=Q4,Φ49=−d11−δ0C11TVT,Φ410=−d2−d11−δ0C11TVT,Φ55=−Q4,Φ66=−γ12I,Φ68=−I,Φ69=−d1BTP,Φ610=−d2−d1BTP,Φ77=−γ22I,Φ79=d1DTP,Φ710=d2−d1DTP,Φ88=−I,Φ99=−2aP+a2Q3,Φ1010=−2bP+b2Q4.
The designed fault detection observer can effectively detect faults and estimate the size of faults,G=P−1V.Proof.
construct the Lyapunov–Krasovskii functional as follows:(21)Vt=V1t+V2t+V3t+V4t+V5t,V1t=eTPe,V2t=∫t−τ1teTsQ1esds,V3t=∫t−τ2teTsQ2esds,V4t=∫−d10∫t+θtd1e˙TsQ3e˙sdθds,V5t=∫−d2−d1∫t+θtd2−d1e˙TsQ4e˙sdθds.
If we define(22)Eδ−δ0=0,Eδ−δ02=δ01−δ0,then taking the derivative of Vt, we can get(23)EV˙1t=E2eTtPe˙t,EV˙2t≤EeTtQ1et−1−μ1eτ1TQ1eτ1,EV˙3t≤EeTtQ2et−1−μ2eτ2TQ2eτ2,EV˙4t≤Ed12e˙TtQ3e˙t−d1∫t−d1te˙TsQ3e˙sds,EV˙5t≤Ed2−d12e˙TtQ4e˙t−d2−d1∫t−d2t−d1e˙TsQ4e˙sds.
According to Lemma1,(24)E−d1∫t−d1te˙TsQ3e˙sds≤−et−et−τ1tTQ3et−et−τ1t−et−τ1t−et−d1tTQ3et−τ1t−et−d1t.
In the same way,(25)E−d2−d1∫t−d2t−d1e˙TsQ4e˙sds≤−et−d1−et−τ2tTQ4et−d1−et−τ2t−et−τ2t−et−d2TQ4et−τ2t−et−d2.
Letε=A−δ0GC110−1−δ0GC110−BD and ξTt=eTeτ1Ted1Teτ2Ted2TfTwT.
Then,(26)EV˙t−γ12fTtft−γ22wTtwt≤ξTtΦ′+d12εTQ3ε+d2−d12εTQ4εξt,Φ′=Φ11′Φ12′0Φ14′0Φ16′Φ17′∗Φ22′Φ23′0000∗∗Φ33′Φ34′000∗∗∗Φ44′Φ45′00∗∗∗∗Φ55′00∗∗∗∗∗Φ66′0∗∗∗∗∗∗Φ77′<0,where(27)Φ11′=ATP+PA+Q1+Q2−Q3,Φ12′=−δ0PGC11+Q3,Φ14′=−1−δ0PGC11,Φ16′=−PB,Φ17′=PD,Φ22′=−1−μ1Q1−2Q3,Φ23′=Q3,Φ33′=−Q3−Q4,Φ34′=Q4,Φ44′=−1−μ2Q2−2Q4,Φ45′=Q4,Φ55′=−Q4,Φ66′=−γ12I,Φ77′=−γ22I.
According to equation (16), it is easy to obtain(28)reTtret=ξTtϒTϒξt,ϒ=WC110000−I0.
Then,(29)EV˙t+reTtret−γ12fTtft−γ22wTtwt≤ξTtΦ′+d12εTQ3ε+d2−d12εTQ4ε+ϒTϒξt,Φ″=Φ′+d12εTQ3ε+d2−d12εTQ4ε+ϒTϒ.
According to the performance index ofH∞,(30)J1=∫0treTtret−γ12fTtft−γ22wTtwtdt≤E∫0tV˙t+reTtret−γ12fTtft−γ22wTtwtdt≤0.
Whenft=0 and wt=0, if Φ″<0 exists, the system (16) is stable in mean square. When ft≠0 and wt≠0, the following integral is obtained for J1:(31)EVt−Vt0≤−∫t0treTtret−γ12fTtft−γ22wTtwtdt.
Under zero initial conditions,t0=0; when t⟶∞, ∫0∞reTtretdt≤∫0∞γ12fTtft+γ22wTtwtdt is reasonable; that is, the H∞ performance index is satisfied.
Furthermore, according to Lemma2, we can get(32)Φ″=Φ′Π∗Θ<0,Π=Π11Π12Π130Π22Π230000Π42Π43000Π61Π62Π630Π72Π73,Θ=Θ1100∗Θ220∗∗Θ33,where(33)Π11=C11TWT,Π12=d1ATQ3,Π13=d2−d1ATQ4,Π22=−d1δ0C11TGTQ3,Π23=−d2−d1δ0C11TGTQ4,Π42=−d11−δ0C11TGTQ3,Π43=−d2−d11−δ0C11TGTQ4,Π61=−I,Π62=−d1BTQ3,Π63=−d2−d1BTQ4,Π72=d1DTQ3,Π73=d2−d1DTQ4,Θ11=−I,Θ22=−Q3,Θ33=−Q4.
In addition, when the left and right sides ofΦ″ are multiplied by the diagonal matrix I,I,I,I,I,I,I,I,PQ3−1,PQ4−1 at the same time and then by the following inequality:(34)−α2Z1+2αP−PZ1−1P=−αZ1−PZ1−1αZ1−P≤0.
We can get(35)−PQ3−1P=−2aP+a2Q3,−PQ4−1P=−2bP+b2Q4.
So, the fault detection observer gain isG=P−1V. The theorem is proved.


### 3.2. Design of Interval Fault-Tolerant Controller

An interval fault-tolerant controller is designed for an NCS closed-loop fault system (9) with random time delay. The state feedback control law with Bernoulli 0-1 distribution parameters is (36)ui=Kiδxt−τ1t+1−δxt−τ2t−exikh,i=1,2,3,4,which guarantees the stability of the system’s mean-square index and H2/H∞ performance.Theorem 2.
For the closed-loop fault system (9) with the DETCS, given positive numbers d1, d2, μ1, μ2, γ1′, γ2′, δ0, and σ∈0,1, if there are symmetric positive definite matrices X, Rjj=1,…,6 and matrices V, Yi, then the following matrix inequality (37), (38) are valid(37)Ω=Ω11Ω12Ω13Ω140Ω16Ω17Ω18Ω19Ω110∗Ω22Ω230000Ω28Ω290∗∗Ω33Ω34Ω3500000∗∗∗Ω44Ω4500Ω48Ω490∗∗∗∗Ω5500000∗∗∗∗∗Ω660Ω68Ω690∗∗∗∗∗∗Ω77Ω78Ω790∗∗∗∗∗∗∗Ω8800∗∗∗∗∗∗∗∗Ω990∗∗∗∗∗∗∗∗∗Ω1010<0,(38)XXC12T∗γ2′2I>0,where(39)Ω11=AX+XAT+2X−R1−R2+R3,Ω12=δ0BLYi−R3+R5,Ω13=2X−R5,Ω14=1−δ0BLYi,Ω16=−BLYi,Ω17=D,Ω18=d1XAT,Ω19=d2−d1XAT,Ω110=XC11T,Ω22=δ0σ2X−V−1−μ12X−R1+2R3−R5−R5T,Ω23=−R3+R5,Ω28=δ0d1YiTLTBT,Ω29=δ0d2−d1YiTLTBT,Ω33=−4X+R3+R4,Ω34=−R4+R6,Ω35=2X−R6,Ω44=σ1−δ02X−V−1−μ22X−R2+2R4−R6−R6T,Ω45=−R4+R6,Ω48=1−δ0d1YiTLTBT,Ω49=1−δ0d2−d1YiTLTBT,Ω55=−2X+R4,Ω66=−2X+V,Ω68=−d1YiTLTBT,Ω69=−d2−d1YiTLTBT,Ω77=−γ1′2I,Ω78=d1DT,Ω79=d2−d1DT,Ω88=−R3,Ω99=−R4,Ω1010=−I.
Then, there is the state feedback control law (36), which makes the closed-loop NCS fault system (9) mean-square exponential stable and have performance of γ1′ and γ2′, where the controller gain matrix Ki=YX−1, and event-triggered weight matrix Γ=V−1.Proof.
Construct a Lyapunov–Krasovskii functional:(40)Vt=V1t+V2t+V3t+V4t+V5t,where(41)V1t=xTPx,V2t=∫t−τ1txTsQ1xsds,V3t=∫t−τ2txTsQ2xsds,V4t=∫−d10∫t+θtd1x˙TsQ3x˙sdθds,V5t=∫−d2−d1∫t+θtd2−d1x˙TsQ4x˙sdθds.
Take the derivative ofVt and get the following inequalities from equation (22):(42)EV˙1t=E2xTtPx˙t,EV˙2t≤ExTtQ1xt−1−μ1xτ1TQ1xτ1,EV˙3t≤ExTtQ2xt−1−μ2xτ2TQ2xτ2,EV˙4t≤Ed12x˙TtQ3x˙t−d1∫t−d1tx˙TsQ3x˙sds,EV˙5t≤Ed2−d12x˙TtQ4x˙t−d2−d1∫t−d2t−d1x˙TsQ4x˙sds,EV˙t≤EV˙1t+EV˙2t+EV˙3t+EV˙4t+EV˙5t+exTikhΓexikh−exTikhΓexikh+yTtyt−γ1′2wTtwt+γ1′2wTtwt−yTtyt.
According to Lemma3, we have(43)E−d1∫t−d1tx˙TsQ3x˙sds≤xxτ1xd1T−Q3Q3−U1U1∗−2Q3+U1+U1TQ3−U1∗∗−Q3xxτ1xd1,E−d2−d1∫t−d2t−d1x˙TsQ4x˙sds≤xd1xτ2xd2T−Q4Q4−U2U2∗−2Q4+U2+U2TQ4−U2∗∗−Q4xd1xτ2xd2.
When(44)ξ′T=xTxτ1Txd1Txτ2Txd2TexTikhwT,ε′=Aδ0BLiKi01−δ0BLiKi0−BLiKiD,we can rewrite EV˙t as follows:(45)EV˙t≤ξ′TΩ1′+d12ε′TQ3ε′+d2−d12ε′TQ4ε′ξ′,Ω′=Ω11′Ω12′Ω13′Ω14′0Ω16′Ω17′∗Ω22′Ω23′0000∗∗Ω33′Ω34′Ω35′00∗∗∗Ω44′Ω45′00∗∗∗∗Ω55′00∗∗∗∗∗Ω66′0∗∗∗∗∗∗Ω77′<0,where (46)Ω11′=ATP+PA+Q1+Q2−Q3,Ω12′=δ0PBLiKi+Q3−U1,Ω13′=U1,Ω14′=1−δ0PBLiKi,Ω16′=−PBLiKi,Ω17′=PD,Ω22′=σδ0Γ−1−μ1Q1−2Q3+U1+U1T,Ω23′=Q3−U1,Ω33′=−Q3−Q4,Ω34′=Q4−U2,Ω35′=U2,Ω44′=σ1−δ0Γ−1−μ2Q2−2Q4+U2+U2T,Ω45′=Q4−U2,Ω55′=−Q4,Ω66′=−Γ,Ω77′=−γ1′2I.
The arrangement of equation (45) can be obtained as(47)EV˙t+yTtyt−γ1′2wTtwt≤ξ′TtΩ′+d12ε′TQ3ε′+d2−d12ε′TQ4ε′+ϕTϕξ′t,where ϕ=C11T000000.
According to Lemma2, Ω″ can be rewritten as(48)Ω″=Ω′+d12ε′TQ3ε′+d2−d12ε′TQ4ε′+ϕTϕ.
Before and after multiplyingΩ″ with diagP−1,P−1,P−1,P−1,P−1,P−1,I,Q3−1,Q4−1,I, we obtain Ω.
Consider theH2/H∞ performance indicators:(49)J2=∫0tyTtyt−γ1′2wTtwtdt≤E∫0tV˙t+yTtyt−γ1′2wTtwtdt≤0.
Whenwt=0, if Ω≤0, then E=V˙t≤0, and the system (9) is asymptotically stable in mean square. If wt∈0,∞, the integral of t0⟶t on both sides of J2 is expressed as(50)EVt−EVt0≤−∫t0tyTtyt−γ1′2wTtwtdt.
Under the zero initial condition,∫0∞yTtytdt≤∫0∞γ1′2wTtwtdt can be obtained; that is, the performance index yt2≤γ1′2wt2 is valid.
In addition,(51)ExTtPxt≤EVt<E∫0twTswsds,C12TC12<γ2′2P,can be obtained from equation (40) and Lemma 2, so(52)EzTtzt≤γ2′2ExTtPxt<γ2′2∫0∞wTswsds.
Namely, the generalizedH2/H∞ index is supportable. The theorem is proved.


### 3.3. Fault-Tolerant Controller Scheduling

Based on prior knowledge, all possible actuator faultsL are divided into four intervals 10.8, 0.80.5, 0.50.2, and 0.20 according to certain rules, and the interval controller set is designed off-line.Firstly, model setΛi=ΛjΛjT of actuator fault f is constructed, where Λj∈0,1,i∈1,2,3,4 is the scheduling table of the interval controller, and j∈1,2,…,m is the number of actuators. When Λj∩Λj<0.20.2T, the default strategy is adopted, and when Λj∪Λj≥0.20.2T, the switching strategy is started. Secondly, an online scheduling algorithm is established. When FDD detects the fault fi=fpfqT, min1,2Lj−fp, and min1,2′Lj−fq are calculated, respectively, and then maxmin1,2,min1,2′ is solved to classify the current fault interval. Finally, the corresponding interval controller is called to satisfy the fault tolerance rapidity. The scheduling principle is shown in Figure 2.Figure 2
Schematic diagram of scheduling mechanism.


### 3.4. Controller Compensation Design

It can be seen from Figure2 that, when the random delay with Bernoulli distribution exists in the system, the interval fault-tolerant controller designed by Theorem 2 is fault-tolerant for the interval prior faults, so as to ensure the stability of the mean-square exponential of the system and have certain dynamic performance. In case of other early faults in the interval, it can buffer the rate of system performance decline. Based on Theorem 1, the interval controller is reconstructed to compensate for the impact of other faults on the system.According to the delay closed-loop fault system (9), the compensation method for the interval controller upt is(53)ut=upt−F+Ef^t,where F+ is the right pseudo-inverse of B and satisfies the conditions of I−FF+E=0 and E=−B. Substitute equation (53) into equation (3) to get(54)x˙t=Axt+δ0BKxτ1+1−δ0BKxτ2+δ−δ0BKxτ1−xτ2−BKexikh+D⌢w⌢t,yt=C11xt,zt=C12xt.Letef=f−f^. Then, Theorem 1 can get ef asymptotic convergence, so there are D⌢=D−B and w⌢=wTefTT.Theorem 3.
For the closed-loop system (54), given positive numbers d1, d2, μ1, μ2, γ1′, γ2′, δ0, and σ∈0,1, if symmetric positive definite matrices X, V, Yii=1,2,3,4, and Rjj=1,…,6 exist to satisfy inequalities (55) and (56), then the system has both γ1′ and γ2′ performance and mean square stability.(55)Η=Η11Η12Η13Η140Η16Η17Η18Η19Η110Η111∗Η22Η2300000Η29Η2100∗∗Η33Η34Η35000000∗∗∗Η44Η45000Η49Η4100∗∗∗∗Η55000000∗∗∗∗∗Η6600Η69Η6100∗∗∗∗∗∗Η770Η79Η7100∗∗∗∗∗∗∗Η88Η89Η8100∗∗∗∗∗∗∗∗Η9900∗∗∗∗∗∗∗∗∗Η10100∗∗∗∗∗∗∗∗∗∗Η1111<0,(56)XXC12T∗γ2′2I>0,where(57)Η11=AX+XAT+2X−R1−R2+R3,Η12=δ0BYi−R3+R5,Η13=2X−R5,Η14=1−δ0BYi,Η16=−BYi,Η17=D,Η18=−B,Η19=d1XAT,Η110=d2−d1XAT,Η111=XC11T,Η22=δ0σ2X−V−1−μ12X−R1+2R3−R5−R5T,Η23=−R3+R5,Η29=δ0d1YiTBT,Η210=δ0d2−d1YiTBT,Η33=−4X+R3+R4,Η34=−R4+R6,Η35=2X−R6,Η44=σ1−δ02X−V−1−μ22X−R2+2R4−R6−R6T,Η45=−R4+R6,Η49=1−δ0d1YiTBT,Η410=1−δ0d2−d1YiTBT,Η55=−2X+R4,Η66=−2X+V,Η69=−d1YiTBT,Η610=−d2−d1YiTBT,Η77=−γ12I,Η79=d1DT,Η710=d2−d1DT,Η88=−γ1′2I,Η89=−d1DT,Η810=−d2−d1DT,Η99=−R3,Η1010=−R4,Η1111=−I.
The proof process is similar to Theorem2, and this section will not go into details.


## 3.1. Failure Detection Observer Design

Considering the system model (3) for the NCS, a fault detection observer model with random delay is established:(12)x^˙=Ax^t+But+δ0Gyt−τ1−y^t−τ1+1−δ0Gyt−τ2−y^t−τ2+δ−δ0Gyt−τ1−y^t−τ1−yt−τ2−y^t−τ2,y^t=C11x^t,where G is the observer gain matrix.The system output residualrt, state error et, and fault estimation error ret are defined as follows:(13)rt=Wyt−y^t,(14)et=xt−x^t,(15)ret=rt−ft,where W is the system output residual weight matrix.By substituting equations (13)–(15) into equation (12), the observer error system of fault detection can be written as(16)e˙t=Aet−δ0GC11et−τ1−1−δ0GC11et−τ2−δ−δ0GC11et−τ1−et−τ2−Bft+Dwt,ret=WC11et−ft.Therefore, the observer design problem is equivalent to solvingG and W, making the error system (16) asymptotically stable and satisfying the following performance:(17)ret2≤γ12ft2+γ22wt2.Note: it is easy to write and remember(18)et=e,et−τt=eτ,et−τ1t=eτ1,et−τ2t=eτ2,xt=x,xt−τt=xτ,xt−τ1t=xτ1,xt−τ2t=xτ2,wt=w,ft=f,ret=re.Theorem 1.
For the error system16 and performance index (17), given positive numbers d1, d2, μ1, μ2, δ0, γ1, γ2, a, and b, if symmetric positive definite matrices P, V, W, and Qii=1,2,3,4 exist, the following matrix inequality (19) is feasible:(19)Φ=Φ11Φ120Φ140Φ16Φ17Φ18Φ19Φ110∗Φ22Φ2300000Φ29Φ210∗∗Φ33Φ34000000∗∗∗Φ44Φ45000Φ49Φ410∗∗∗∗Φ5500000∗∗∗∗∗Φ660Φ68Φ69Φ610∗∗∗∗∗∗Φ770Φ79Φ710∗∗∗∗∗∗∗Φ8800∗∗∗∗∗∗∗∗Φ990∗∗∗∗∗∗∗∗∗Φ1010<0,where(20)Φ11=ATP+PA+Q1+Q2−Q3,Φ12=−δ0VC11+Q3,Φ14=−1−δ0VC11,Φ16=−PB,Φ17=PD,Φ18=C11TWT,Φ19=d1ATP,Φ110=d2−d1ATP,Φ22=−1−μ1Q1−2Q3,Φ23=Q3,Φ29=−d1δ0C11TVT,Φ210=−d2−d1δ0C11TVT,Φ33=−Q3−Q4,Φ34=Q4,Φ44=−1−μ2Q2−2Q4,Φ45=Q4,Φ49=−d11−δ0C11TVT,Φ410=−d2−d11−δ0C11TVT,Φ55=−Q4,Φ66=−γ12I,Φ68=−I,Φ69=−d1BTP,Φ610=−d2−d1BTP,Φ77=−γ22I,Φ79=d1DTP,Φ710=d2−d1DTP,Φ88=−I,Φ99=−2aP+a2Q3,Φ1010=−2bP+b2Q4.
The designed fault detection observer can effectively detect faults and estimate the size of faults,G=P−1V.Proof.
construct the Lyapunov–Krasovskii functional as follows:(21)Vt=V1t+V2t+V3t+V4t+V5t,V1t=eTPe,V2t=∫t−τ1teTsQ1esds,V3t=∫t−τ2teTsQ2esds,V4t=∫−d10∫t+θtd1e˙TsQ3e˙sdθds,V5t=∫−d2−d1∫t+θtd2−d1e˙TsQ4e˙sdθds.
If we define(22)Eδ−δ0=0,Eδ−δ02=δ01−δ0,then taking the derivative of Vt, we can get(23)EV˙1t=E2eTtPe˙t,EV˙2t≤EeTtQ1et−1−μ1eτ1TQ1eτ1,EV˙3t≤EeTtQ2et−1−μ2eτ2TQ2eτ2,EV˙4t≤Ed12e˙TtQ3e˙t−d1∫t−d1te˙TsQ3e˙sds,EV˙5t≤Ed2−d12e˙TtQ4e˙t−d2−d1∫t−d2t−d1e˙TsQ4e˙sds.
According to Lemma1,(24)E−d1∫t−d1te˙TsQ3e˙sds≤−et−et−τ1tTQ3et−et−τ1t−et−τ1t−et−d1tTQ3et−τ1t−et−d1t.
In the same way,(25)E−d2−d1∫t−d2t−d1e˙TsQ4e˙sds≤−et−d1−et−τ2tTQ4et−d1−et−τ2t−et−τ2t−et−d2TQ4et−τ2t−et−d2.
Letε=A−δ0GC110−1−δ0GC110−BD and ξTt=eTeτ1Ted1Teτ2Ted2TfTwT.
Then,(26)EV˙t−γ12fTtft−γ22wTtwt≤ξTtΦ′+d12εTQ3ε+d2−d12εTQ4εξt,Φ′=Φ11′Φ12′0Φ14′0Φ16′Φ17′∗Φ22′Φ23′0000∗∗Φ33′Φ34′000∗∗∗Φ44′Φ45′00∗∗∗∗Φ55′00∗∗∗∗∗Φ66′0∗∗∗∗∗∗Φ77′<0,where(27)Φ11′=ATP+PA+Q1+Q2−Q3,Φ12′=−δ0PGC11+Q3,Φ14′=−1−δ0PGC11,Φ16′=−PB,Φ17′=PD,Φ22′=−1−μ1Q1−2Q3,Φ23′=Q3,Φ33′=−Q3−Q4,Φ34′=Q4,Φ44′=−1−μ2Q2−2Q4,Φ45′=Q4,Φ55′=−Q4,Φ66′=−γ12I,Φ77′=−γ22I.
According to equation (16), it is easy to obtain(28)reTtret=ξTtϒTϒξt,ϒ=WC110000−I0.
Then,(29)EV˙t+reTtret−γ12fTtft−γ22wTtwt≤ξTtΦ′+d12εTQ3ε+d2−d12εTQ4ε+ϒTϒξt,Φ″=Φ′+d12εTQ3ε+d2−d12εTQ4ε+ϒTϒ.
According to the performance index ofH∞,(30)J1=∫0treTtret−γ12fTtft−γ22wTtwtdt≤E∫0tV˙t+reTtret−γ12fTtft−γ22wTtwtdt≤0.
Whenft=0 and wt=0, if Φ″<0 exists, the system (16) is stable in mean square. When ft≠0 and wt≠0, the following integral is obtained for J1:(31)EVt−Vt0≤−∫t0treTtret−γ12fTtft−γ22wTtwtdt.
Under zero initial conditions,t0=0; when t⟶∞, ∫0∞reTtretdt≤∫0∞γ12fTtft+γ22wTtwtdt is reasonable; that is, the H∞ performance index is satisfied.
Furthermore, according to Lemma2, we can get(32)Φ″=Φ′Π∗Θ<0,Π=Π11Π12Π130Π22Π230000Π42Π43000Π61Π62Π630Π72Π73,Θ=Θ1100∗Θ220∗∗Θ33,where(33)Π11=C11TWT,Π12=d1ATQ3,Π13=d2−d1ATQ4,Π22=−d1δ0C11TGTQ3,Π23=−d2−d1δ0C11TGTQ4,Π42=−d11−δ0C11TGTQ3,Π43=−d2−d11−δ0C11TGTQ4,Π61=−I,Π62=−d1BTQ3,Π63=−d2−d1BTQ4,Π72=d1DTQ3,Π73=d2−d1DTQ4,Θ11=−I,Θ22=−Q3,Θ33=−Q4.
In addition, when the left and right sides ofΦ″ are multiplied by the diagonal matrix I,I,I,I,I,I,I,I,PQ3−1,PQ4−1 at the same time and then by the following inequality:(34)−α2Z1+2αP−PZ1−1P=−αZ1−PZ1−1αZ1−P≤0.
We can get(35)−PQ3−1P=−2aP+a2Q3,−PQ4−1P=−2bP+b2Q4.
So, the fault detection observer gain isG=P−1V. The theorem is proved.


## 3.2. Design of Interval Fault-Tolerant Controller

An interval fault-tolerant controller is designed for an NCS closed-loop fault system (9) with random time delay. The state feedback control law with Bernoulli 0-1 distribution parameters is (36)ui=Kiδxt−τ1t+1−δxt−τ2t−exikh,i=1,2,3,4,which guarantees the stability of the system’s mean-square index and H2/H∞ performance.Theorem 2.
For the closed-loop fault system (9) with the DETCS, given positive numbers d1, d2, μ1, μ2, γ1′, γ2′, δ0, and σ∈0,1, if there are symmetric positive definite matrices X, Rjj=1,…,6 and matrices V, Yi, then the following matrix inequality (37), (38) are valid(37)Ω=Ω11Ω12Ω13Ω140Ω16Ω17Ω18Ω19Ω110∗Ω22Ω230000Ω28Ω290∗∗Ω33Ω34Ω3500000∗∗∗Ω44Ω4500Ω48Ω490∗∗∗∗Ω5500000∗∗∗∗∗Ω660Ω68Ω690∗∗∗∗∗∗Ω77Ω78Ω790∗∗∗∗∗∗∗Ω8800∗∗∗∗∗∗∗∗Ω990∗∗∗∗∗∗∗∗∗Ω1010<0,(38)XXC12T∗γ2′2I>0,where(39)Ω11=AX+XAT+2X−R1−R2+R3,Ω12=δ0BLYi−R3+R5,Ω13=2X−R5,Ω14=1−δ0BLYi,Ω16=−BLYi,Ω17=D,Ω18=d1XAT,Ω19=d2−d1XAT,Ω110=XC11T,Ω22=δ0σ2X−V−1−μ12X−R1+2R3−R5−R5T,Ω23=−R3+R5,Ω28=δ0d1YiTLTBT,Ω29=δ0d2−d1YiTLTBT,Ω33=−4X+R3+R4,Ω34=−R4+R6,Ω35=2X−R6,Ω44=σ1−δ02X−V−1−μ22X−R2+2R4−R6−R6T,Ω45=−R4+R6,Ω48=1−δ0d1YiTLTBT,Ω49=1−δ0d2−d1YiTLTBT,Ω55=−2X+R4,Ω66=−2X+V,Ω68=−d1YiTLTBT,Ω69=−d2−d1YiTLTBT,Ω77=−γ1′2I,Ω78=d1DT,Ω79=d2−d1DT,Ω88=−R3,Ω99=−R4,Ω1010=−I.
Then, there is the state feedback control law (36), which makes the closed-loop NCS fault system (9) mean-square exponential stable and have performance of γ1′ and γ2′, where the controller gain matrix Ki=YX−1, and event-triggered weight matrix Γ=V−1.Proof.
Construct a Lyapunov–Krasovskii functional:(40)Vt=V1t+V2t+V3t+V4t+V5t,where(41)V1t=xTPx,V2t=∫t−τ1txTsQ1xsds,V3t=∫t−τ2txTsQ2xsds,V4t=∫−d10∫t+θtd1x˙TsQ3x˙sdθds,V5t=∫−d2−d1∫t+θtd2−d1x˙TsQ4x˙sdθds.
Take the derivative ofVt and get the following inequalities from equation (22):(42)EV˙1t=E2xTtPx˙t,EV˙2t≤ExTtQ1xt−1−μ1xτ1TQ1xτ1,EV˙3t≤ExTtQ2xt−1−μ2xτ2TQ2xτ2,EV˙4t≤Ed12x˙TtQ3x˙t−d1∫t−d1tx˙TsQ3x˙sds,EV˙5t≤Ed2−d12x˙TtQ4x˙t−d2−d1∫t−d2t−d1x˙TsQ4x˙sds,EV˙t≤EV˙1t+EV˙2t+EV˙3t+EV˙4t+EV˙5t+exTikhΓexikh−exTikhΓexikh+yTtyt−γ1′2wTtwt+γ1′2wTtwt−yTtyt.
According to Lemma3, we have(43)E−d1∫t−d1tx˙TsQ3x˙sds≤xxτ1xd1T−Q3Q3−U1U1∗−2Q3+U1+U1TQ3−U1∗∗−Q3xxτ1xd1,E−d2−d1∫t−d2t−d1x˙TsQ4x˙sds≤xd1xτ2xd2T−Q4Q4−U2U2∗−2Q4+U2+U2TQ4−U2∗∗−Q4xd1xτ2xd2.
When(44)ξ′T=xTxτ1Txd1Txτ2Txd2TexTikhwT,ε′=Aδ0BLiKi01−δ0BLiKi0−BLiKiD,we can rewrite EV˙t as follows:(45)EV˙t≤ξ′TΩ1′+d12ε′TQ3ε′+d2−d12ε′TQ4ε′ξ′,Ω′=Ω11′Ω12′Ω13′Ω14′0Ω16′Ω17′∗Ω22′Ω23′0000∗∗Ω33′Ω34′Ω35′00∗∗∗Ω44′Ω45′00∗∗∗∗Ω55′00∗∗∗∗∗Ω66′0∗∗∗∗∗∗Ω77′<0,where (46)Ω11′=ATP+PA+Q1+Q2−Q3,Ω12′=δ0PBLiKi+Q3−U1,Ω13′=U1,Ω14′=1−δ0PBLiKi,Ω16′=−PBLiKi,Ω17′=PD,Ω22′=σδ0Γ−1−μ1Q1−2Q3+U1+U1T,Ω23′=Q3−U1,Ω33′=−Q3−Q4,Ω34′=Q4−U2,Ω35′=U2,Ω44′=σ1−δ0Γ−1−μ2Q2−2Q4+U2+U2T,Ω45′=Q4−U2,Ω55′=−Q4,Ω66′=−Γ,Ω77′=−γ1′2I.
The arrangement of equation (45) can be obtained as(47)EV˙t+yTtyt−γ1′2wTtwt≤ξ′TtΩ′+d12ε′TQ3ε′+d2−d12ε′TQ4ε′+ϕTϕξ′t,where ϕ=C11T000000.
According to Lemma2, Ω″ can be rewritten as(48)Ω″=Ω′+d12ε′TQ3ε′+d2−d12ε′TQ4ε′+ϕTϕ.
Before and after multiplyingΩ″ with diagP−1,P−1,P−1,P−1,P−1,P−1,I,Q3−1,Q4−1,I, we obtain Ω.
Consider theH2/H∞ performance indicators:(49)J2=∫0tyTtyt−γ1′2wTtwtdt≤E∫0tV˙t+yTtyt−γ1′2wTtwtdt≤0.
Whenwt=0, if Ω≤0, then E=V˙t≤0, and the system (9) is asymptotically stable in mean square. If wt∈0,∞, the integral of t0⟶t on both sides of J2 is expressed as(50)EVt−EVt0≤−∫t0tyTtyt−γ1′2wTtwtdt.
Under the zero initial condition,∫0∞yTtytdt≤∫0∞γ1′2wTtwtdt can be obtained; that is, the performance index yt2≤γ1′2wt2 is valid.
In addition,(51)ExTtPxt≤EVt<E∫0twTswsds,C12TC12<γ2′2P,can be obtained from equation (40) and Lemma 2, so(52)EzTtzt≤γ2′2ExTtPxt<γ2′2∫0∞wTswsds.
Namely, the generalizedH2/H∞ index is supportable. The theorem is proved.


## 3.3. Fault-Tolerant Controller Scheduling

Based on prior knowledge, all possible actuator faultsL are divided into four intervals 10.8, 0.80.5, 0.50.2, and 0.20 according to certain rules, and the interval controller set is designed off-line.Firstly, model setΛi=ΛjΛjT of actuator fault f is constructed, where Λj∈0,1,i∈1,2,3,4 is the scheduling table of the interval controller, and j∈1,2,…,m is the number of actuators. When Λj∩Λj<0.20.2T, the default strategy is adopted, and when Λj∪Λj≥0.20.2T, the switching strategy is started. Secondly, an online scheduling algorithm is established. When FDD detects the fault fi=fpfqT, min1,2Lj−fp, and min1,2′Lj−fq are calculated, respectively, and then maxmin1,2,min1,2′ is solved to classify the current fault interval. Finally, the corresponding interval controller is called to satisfy the fault tolerance rapidity. The scheduling principle is shown in Figure 2.Figure 2
Schematic diagram of scheduling mechanism.


## 3.4. Controller Compensation Design

It can be seen from Figure2 that, when the random delay with Bernoulli distribution exists in the system, the interval fault-tolerant controller designed by Theorem 2 is fault-tolerant for the interval prior faults, so as to ensure the stability of the mean-square exponential of the system and have certain dynamic performance. In case of other early faults in the interval, it can buffer the rate of system performance decline. Based on Theorem 1, the interval controller is reconstructed to compensate for the impact of other faults on the system.According to the delay closed-loop fault system (9), the compensation method for the interval controller upt is(53)ut=upt−F+Ef^t,where F+ is the right pseudo-inverse of B and satisfies the conditions of I−FF+E=0 and E=−B. Substitute equation (53) into equation (3) to get(54)x˙t=Axt+δ0BKxτ1+1−δ0BKxτ2+δ−δ0BKxτ1−xτ2−BKexikh+D⌢w⌢t,yt=C11xt,zt=C12xt.Letef=f−f^. Then, Theorem 1 can get ef asymptotic convergence, so there are D⌢=D−B and w⌢=wTefTT.Theorem 3.
For the closed-loop system (54), given positive numbers d1, d2, μ1, μ2, γ1′, γ2′, δ0, and σ∈0,1, if symmetric positive definite matrices X, V, Yii=1,2,3,4, and Rjj=1,…,6 exist to satisfy inequalities (55) and (56), then the system has both γ1′ and γ2′ performance and mean square stability.(55)Η=Η11Η12Η13Η140Η16Η17Η18Η19Η110Η111∗Η22Η2300000Η29Η2100∗∗Η33Η34Η35000000∗∗∗Η44Η45000Η49Η4100∗∗∗∗Η55000000∗∗∗∗∗Η6600Η69Η6100∗∗∗∗∗∗Η770Η79Η7100∗∗∗∗∗∗∗Η88Η89Η8100∗∗∗∗∗∗∗∗Η9900∗∗∗∗∗∗∗∗∗Η10100∗∗∗∗∗∗∗∗∗∗Η1111<0,(56)XXC12T∗γ2′2I>0,where(57)Η11=AX+XAT+2X−R1−R2+R3,Η12=δ0BYi−R3+R5,Η13=2X−R5,Η14=1−δ0BYi,Η16=−BYi,Η17=D,Η18=−B,Η19=d1XAT,Η110=d2−d1XAT,Η111=XC11T,Η22=δ0σ2X−V−1−μ12X−R1+2R3−R5−R5T,Η23=−R3+R5,Η29=δ0d1YiTBT,Η210=δ0d2−d1YiTBT,Η33=−4X+R3+R4,Η34=−R4+R6,Η35=2X−R6,Η44=σ1−δ02X−V−1−μ22X−R2+2R4−R6−R6T,Η45=−R4+R6,Η49=1−δ0d1YiTBT,Η410=1−δ0d2−d1YiTBT,Η55=−2X+R4,Η66=−2X+V,Η69=−d1YiTBT,Η610=−d2−d1YiTBT,Η77=−γ12I,Η79=d1DT,Η710=d2−d1DT,Η88=−γ1′2I,Η89=−d1DT,Η810=−d2−d1DT,Η99=−R3,Η1010=−R4,Η1111=−I.
The proof process is similar to Theorem2, and this section will not go into details.


## 4. Simulation Examples and Results

In order to verify the effectiveness of the control method described in this paper, the numerical model of reference [21] is cited, and the correlation matrix is as follows:(58)A=−1.3−0.50.7−1.8,B=10.501,C11=1001,C12=2002,D=0.40,wt=cos2∗pi∗t∗exp−0.2∗t,6≤t≤12,0,else.Assuming the initial state of the system isx0=2−2T, the scalar parameters selected in the theorem are d1=0.5, d2=0.8, μ1=0.1, μ2=0.6, δ0=0.75, a=1, b=0.5, γ1=1.1, γ2=0.6, γ1′=1.9, and γ2′=0.6.Considering the minor faultL0=0.9000.9 and unknown fault f0=00.5, the correlation matrix of the fault detection observer can be obtained as(59)G=0.1941−0.0307−0.01280.0199,W=−0.1141−0.1163−0.1163−0.0589,according to Theorem 1.Similarly, select interval prior faults(60)L1=1001,L2=0.8000.8,L3=0.5000.5,L4=0.2000.2,to obtain the event-triggered weight matrix Γ and interval fault-tolerant controller Kii=1,2,3,4 according to Theorem 2 as follows:(61)Γ=5.13680.15290.15294.0107,K1=−0.5648−0.1028−0.10460.0113,K2=−0.7060−0.1284−0.13080.0141,K3=−1.1297−0.2055−0.20930.0226,K4=−2.8242−0.5138−0.52320.0566.The following is the simulation diagram of the observer estimation fault and the effect of different control strategies for the prior faults and other faults within the interval.If the system runs tot=6s, a minor fault L0 occurs; when it runs to t=12s, an unknown fault f0 occurs. It can be seen from Figure 3 that the observer designed in this paper can effectively detect and estimate the fault size, laying a foundation for the subsequent start-up scheduling mechanism and controller reconstruction. Figures 4 and 5, respectively, show the fault-tolerant effects of state x1 and state x2 in three control modes (T1 is a traditional passive fault-tolerant controller, T2 is an interval fault-tolerant controller, and T3 is an interval hybrid fault-tolerant controller). When t=6s, a prior fault occurs within the interval, and all three control methods have good robustness to the fault. However, since T2 requires the online dispatching controller and T3 to compensate for the fault, these two control methods will consume part of the time, so the stable trend is slowed down. When t=12s, other faults occur in the interval. Since T1 cannot effectively tolerate faults for unknown faults, it is finally in a state of instability. By contrast, T2 reduces conservatism and slows down the rate of system deterioration, providing a reliable guarantee for T3’s compensation. Based on the above analysis, interval hybrid fault-tolerant control T3 enables the system to have certain dynamic performance and meet the performance index H2/H∞.Figure 3
Fault estimation diagram.Figure 4
Response curve of statex1.Figure 5
Response curve of statex2.Figure6 is the partial interception diagram of data transmission time and transmission interval under the sampling period h=0.05s and σ=0.36 of the NCS. The system sampled data for a total of 400 times under the periodic time-triggered communication scheme (PTTCS), and the data in the figure were transmitted for only 245 times. It follows that the DETCS acting on the NCS can effectively save network space.Figure 6
Transmission time and transmission interval diagram of NCS under DETCS.


## 5. Conclusion

In this paper, an interval fault-tolerant control method with less conservatism is proposed based on the hybrid control idea. Firstly, a closed-loop fault model with Bernoulli 0-1 distribution is constructed according to the delay information. On this basis, a fault-tolerant controller set with little conservatism is designed off-line by using prior interval faults. Secondly, the fault size is estimated by the fault detection observer so as to dispatch the corresponding interval fault-tolerant controller rapidly. Finally, the interval compensation algorithm is designed to satisfy the system’s mean square stability andH2/H∞ performance index in the case of failure. The simulation example shows that if the prior fault occurs within the interval, the pre-established interval fault-tolerant controller can be quickly invoked, and if other faults occur within the interval, the control method of compensating the interval fault-tolerant controller is effective. In addition, compared to the PTTCS, the DETCS can better meet the requirements of efficient data transmission when network resources are limited to improve the utilization of network space.



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*Source: 1328068-2020-01-21.xml*