# Event-Triggered Fault-Tolerant Control for Nonlinear Networked Control Systems

Authors: Xiaoyu Zhou; Xia Liu; Lu Wang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper addresses the communication congestion and actuator fault in a nonlinear networked control system. A weighted average event-triggered mechanism adopted the weighted average of data packets is proposed to alleviate the communication congestion and save the network communication resources. Meanwhile, based on the system state estimation and fault estimation obtained by a state-fault observer, a fault-tolerant controller is designed to compensate and reduce the influence of fault and nonlinear factors in the networked control systems. The stability of the closed-loop system is proved by the Lyapunov–Kroasovskii theory, and the gains of the observer and controller are obtained by linear matrix inequalities. The feasibility of the proposed scheme is verified by the networked motor control system. The proposed weighted average event-triggered fault-tolerant control scheme can reduce the data transmission without affecting system performance. Meanwhile, it not only has fault-tolerant control performance but also reduces the influence of nonlinear factors on the system output.


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## Body

## 1. Introduction

In recent decades, networked control systems (NCSs) have gradually attracted considerable attentions. Compared with conventional control systems with point-to-point wired communication, NCSs have more advantages. For example, the introduction of the network can reduce the complexity and cost of the control systems and improve the reliability and scalability of the control systems. Therefore, NCSs have been widely applied in the industrial process, automatic control, robotics, aircraft, and unmanned technology [1–3]. Nevertheless, some unfavorable factors also unavoidably appear in NCSs, such as communication congestion, actuator fault, nonlinear factors, network-induced delay, and so on, which inevitably affects the control performance [4–6].Due to limited communication bandwidth and imperfect communication link, the communication congestion may appear in NCSs [7–9]. In order to alleviate the communication congestion and save the network communication resources, the event-triggered mechanism is usually adopted in NCSs [10–13]. Different from the conventional time-triggered sampling, the event-triggered mechanism is based on data sampling. Therefore, an event-triggered mechanism can improve the efficiency of data transmission in the communication and calculation of NCSs. In [14], an adaptive event-triggered scheme was designed to alleviate the communication congestion. In [15], a self-triggered and event-triggered mixed sampling scheme was proposed to reduce the numbers of transmitted data packets in wireless NCSs. In order to get better quality of service for network communication, a hybrid method of random switching based on time-triggered and event-triggered was introduced in [16]. Nevertheless, the possible actuator fault in NCS is ignored in [14–16].Besides communication congestion, actuator fault is inevitable in practical NCSs which decreases the system performance. Therefore, the fault-tolerant control (FTC) of NCSs as well as event-triggered mechanism is widely studied. In [17], an adaptive event-triggered mechanism was proposed and a fault-tolerant controller based on the triggered output data was designed. In [18], a sliding mode fault-tolerant controller for NCSs was proposed under a dynamic event-triggered mechanism to ensure that the trajectories of the system can reach the sliding surface. In [19], an impulsive FTC was designed based on the estimated fault and an integral-based event-triggered mechanism was designed to alleviate communication congestion. In [20], a fault-tolerant controller based on an adaptive memory-based event-triggered mechanism was proposed to compensate for the influence of fault according to the fault estimation. In [21], based on the system state and fault estimation, a fault-tolerant controller was designed to compensate for the fault and the event-triggered mechanism was adopted to save communication resources. However, the NCSs considered in [17–21] are linear and the influence of nonlinear factors is ignored.Along with communication congestion and possible actuator fault, the modeling of NCSs is often accompanied with some complex nonlinear factors in practice [22–24], such as the nonlinear coupling relationship among different nodes in complex dynamic networks [25] and the communication topology structure of nonlinear multiagent systems [26]. Therefore, the research of nonlinear NCSs has further significance. In [27], a resilient event-triggered mechanism was developed, and the sufficient conditions were constructed to ensure that the switched control strategy subjected to actuator fault and nonlinear factors was exponentially stable. In [28], a period event-triggered sampling scheme was designed, and an FTC was proposed for nonlinear NCSs to guarantee its stability under the actuator fault. In [29], an adaptive event-triggered communication scheme was adopted to save the network communication resources by adjusting event-triggered threshold, and a state feedback controller was designed for nonlinear NCSs under fault. In [30], an event-based adaptive FTC using a fuzzy approximation mechanism was proposed for the nonlinear system, which could also be applied to NCSs. In [31], an adaptive event-triggered mechanism with an adjustable triggering threshold was proposed, and an FTC with stochastic event-driven actuator scheduling was investigated for nonlinear NCSs. However, some unexpected triggered events can be further avoided in [27–31].Inspired by the discussions above, in this paper, a weighted average event-triggered fault-tolerant control scheme is proposed for nonlinear NCSs subjected to communication congestion and actuator fault. The main contributions of this paper are as follows:(1)
Different from the conventional event-triggered mechanism, the weighted average event-triggered mechanism (WAETM) considers the weighted average of data packets, which can further alleviate the communication congestion and save the network communication resources.(2)
Based on the WAETM a state-fault observer is designed to obtain the system state estimation and fault estimation, which can guarantee the asymptotically stability of the observation error dynamic system.(3)
With the system state estimation and fault estimation, a fault-tolerant controller is designed, which can compensate for the influence of fault and reduce the influence of nonlinear factors in NCSs.The remainder of this paper is structured as follows. The model of nonlinear NCSs is presented in Section2. The design of the WAETM, state-fault observer, and fault-tolerant controller are presented, and the stability of the closed-loop system is proved in Section 3. The feasibility of the weighted average event-triggered FTC scheme for nonlinear NCSs is demonstrated by the networked motor control system in Section 4.Notations: the following notations are adopted in this paper.I denotes the identity matrix with an appropriate dimension. For matrix B, its inverse and transpose matrices are denoted by B† and BT, respectively. 0 denotes zero matrix. Rn denotes the n-dimension Euclidean space and Rn×n denotes the n×n matrix. ∗ denotes the symmetric term in a symmetric block matrix.


## 2. Problem Statement

The model of nonlinear NCSs with actuator fault is as follows:(1)x˙t=Axt+But+Bfft+Fρxt,t,yt=Cxt,where xt∈Rn is the system state vector, ut∈Rm is the system control input, and yt∈Rr is the system measured output, ft∈Rm is the actuator fault, ρxt,t=ρ1x1t,t,…,ρnxnt,tT is the nonlinear function, A∈Rn×n,B∈Rn×m,C∈Rr×n,Bf∈Rn×m,F∈Rn×n are the system parameter matrices with appropriate dimensions.Assumption 1.
(See [32, 33]). It is assumed that the nonlinear function ρxt,t is continuous and bounded, and satisfies Lipschitz conditions. Therefore, the following condition is satisfied(2)ρx1t,t−ρx2t,t≤δx1t−x2t,where δ is the Lipschitz constant.Assumption 2.
Assume that the derivative of the fault is norm-bounded, i.e.f˙t≤fΔ, where fΔ is a constant.


## 3. Design of Weighted Average Event-Triggered Fault-Tolerant Control

The proposed weighted average event-triggered FTC scheme is shown in Figure1. In order to save the network communication resources in nonlinear NCSs, a WAETM is designed to reduce the measured output data from yt to ytkh. The zero-order holder ZOH guarantees the continuity of the successfully triggered data packet y˜t within the holding time-interval. The state-fault observer is designed to obtain the real-time system state estimation x^t and fault estimation f^t. Finally, the fault-tolerant controller ut is designed based on the system state estimation and fault estimation to compensate and reduce the influence caused by the fault and nonlinear factors.Figure 1
Weighted average event-triggered FTC scheme.


### 3.1. WAETM

Event-triggered mechanism is an effective way to alleviate the communication congestion and save the network communication resources. However, the conventional event-triggered mechanism depends on the difference between the values of the latest released data packet and the current sampling data packet. The conventional event-triggered mechanism is [21].(3)tk+1h=tkh+minl∈ℕ+lh|ecTtΩecTt>σyTtkh+lhΩytkh+lh,where tkh represents the release instant, tk is a positive integer, and h is the sampling period. l denotes the interval time between the latest triggered instant and the next triggered instant, σ is a constant, Ω is the constant weighted matrix, and ect=ytkh−ytkh+lh.In order to further decrease the triggered data error and save the network communication resources, the WAETM is designed as follows:(4)Ψ=l|eTtΩet−σy¯TtkhΩy¯tkh+ϕσy¯TtkhΩet+eTtΩy¯Ttkh≤0,where Ψ denotes a set of events satisfying the event trigger condition, y¯tkh=tkhytkh+tkh+lhytkh+lh/2tkh+lh denotes the weighted average value between the latest released data packet and the current sampling data packet, et=ytkh−y¯tkh denotes the difference between the latest released data packet and the weighted average data packet, and ϕ is a constant.Remark 1.
Compared with the conventional event-triggered mechanism (2), the proposed WAETM (3) can decrease the triggered data error et, avoiding unexpected triggered events. Meanwhile, the item ϕσy¯TtkhΩet+eTtΩy¯Ttkh in the WAETM (3) contains a triggered data packet ytkh. Therefore, the controller can obtain data packets from the network even when the system is subjected to others noises or disturbances. In fact, the WAETM in (3) can be converted into the conventional event-triggered mechanism in [21] when ϕ=0.
The release instant of WAETM is denoted astk+1h=tkh+lM+1h, where lM=maxl|l∈Ψ. When the data packets are successfully transmitted to the observer, the system measured output can be written as follows:(5)y˜t=ytkh,t∈tkh+τk,tk+1h+τk+1,where y˜t denotes the data successfully transmitted at the triggered instant and τk is the time delay.
Considering that the data packets transmitted by the WAETM are subjected to time delayτk=jk−tkh, where jk represents the instant when the data packets arrive at the ZOH. The ZOH deals with the data packets out-of-sequence and guarantees the continuity of control input within the holding time-interval. The subintervals are defined as follows:(6)I=I1=jk,jk+h,I2=jk+h,jk+2h,⋮In¯k=jk+n¯k−1h,jk+1,where n¯k=minl|jk+l−1h≥jk+1, and the time delay function can be written as follows:(7)τt=t−tkh,t∈I1,t−t+k1h,t∈I2,⋮t−tk+jk−1h,t∈In¯k,where 0≤τt≤τM, τ˙t≤τΔ, τM is the maximum value of the τt, and τΔ is a constant. Then the error function of ZOH can be expressed as follows:(8)eyt=ytkh−ytkh,t∈I1,ytkh−ytk+1h,t∈I2,⋮ytkh−ytkh+n¯k−1h,t∈In¯k,which implies that(9)y˜t=eyt+yt−τt,t∈jk,jk+1.


### 3.2. State-Fault Observer

According to the data packetsy˜t successfully transmitted by the WAETM, the state-fault observer is designed as follows:(10)x^˙t=Ax^t+But+Bff^t+Fρx^t,t+Ly˜t−y^t−τt,f^˙t=Gf^t+Hy˜t−y^t−τt,y^t=Cx^t,where x^t∈Rn, y^t∈Rr, f^t∈Rm, ρx^t,t=ρ1x^1t,t,…,ρnx^nt,tT are the system state estimation vector, system measured output estimation, fault estimation signal, and nonlinear function estimation, respectively. L and H are observer gain matrices to be designed later.The estimation error of system state is defined asext=xt−x^t, the estimation error of fault is defined as ef=ft−f^t, and the estimation error of nonlinear function is defined as gext=ρxt−ρx^t. Then, the estimation error of system state and fault can be respectively rewritten as follows:(11)e˙xt=Aext+Fgexx+Bfeft−Leyt−LCext−τt,e˙ft=1−Gft+Geft−Heyt−HCext−τt.Defineξt=exteftT, then the observation error dynamic system can be given as follows:(12)ξ˙t=A¯ξt+B¯ξt−τt+L¯eyt+D¯ft+F¯gext,where A¯=ABf0G, B¯=−LC0−HC0, L¯=−L−H, D¯=01−G, F¯=F0.Up to now, we need some lemmas and definitions before proceeding the stability of observation error dynamic.Lemma 1.
(See [34]). For any positive definite constant matrix Y∈Rn×n, scalar α>0, and vector function x˙:−α,0⟶Rn, the following integration is defined:(13)−α∫t−αtx˙TθYx˙θdθ≤xtxt−αT−YY∗−Yxtxt−α.Definition 1.
(See [35]). System (12) satisfies the H∞ performance under initial condition, and the following performance index holds for all nonzero ft∈L20,∞.(14)∫0∞ξTsξsds≤λ1∫0∞fTsfsds,λ1>0.Theorem 1.
For given positive scalarsh, τM, ϕ, σ, λ1 and υ, there exists symmetric positive definite matrices P>0, R1>0, R2>0, R3>0, G>0, Q>0, and appropriate matrices M and Z such that the following condition holds.(15)Ξ=Ξ11Ξ12Ξ13QMTF¯ZMTD¯∗Ξ22MTB¯0MTF¯ZMTD¯∗∗Ξ3300Ξ360∗∗∗Ξ44000∗∗∗∗−υI00∗∗∗∗∗−Ω0∗∗∗∗∗∗−λ1I<0,where Ξ11=R1+R2−π2/4G−Q+MTA¯, Ξ12=2P−MT+A¯TMT,(16)Ξ13=π24G+π24GT+MTB¯,Ξ22=τM2G+τM2Q,Ξ33=σCTΩC+1−τR2−1−τR1−π24G,Ξ36=ϕσCTΩ+ΩC,Ξ44=−R2−R3−Q,then the observation error dynamic system (12) with the WAETM (4) is asymptotically stable. In addition, the state-fault observer gain matrices can be obtained by L¯=M−TZ.Proof.
Choose a Lyapunov–Krasovskii function as follows:(17)V1t=V11t+V12t+V13t,where(18)V11t=ξTtPξt+∫t−τttξTθR1ξθdθ+∫t−τMt−τtξTtR2ξt+∫t−τMtξTθR3ξθdθ,V12t=τM2∫ikhtξ˙TθGξ˙θdθ−π24∫ikhtξθ−ξikhTGξθ−ξikhdθ,V13t=τM∫t−τMt∫rtξ˙TθQξ˙θdθdr.
The derivative of (17) is obtained as follows:(19)V˙11t=2ξTtPξ˙t+ξTtR1ξt−1−τξTt−τtR1ξt−τt+1−τξTt−τtR2ξt−τt−ξTt−τMR2ξt−τM+ξTtR3ξt−ξTt−τMR3ξt−τM,V˙12t=τM2ξ˙TtGξ˙t−π24ξt−ξt−τtTGξt−ξt−τt,V˙13t=τM2ξ˙TtQξ˙t−τM∫t−τMtξ˙TθQξ˙θdθ.
According to Lemma1, we can obtain the following equation:(20)−τM∫t−τMtξ˙TθQξ˙θdθ≤ξtξt−τMT−QQ∗−Qξtξt−τM.
Therefore, from (19) we can get the following equation:(21)V˙1t≤2ξTtPξ˙t+ξTtR1+R3−π24G−Qξt−ξTt−τt1−τR2−1−τR1−π24Gξt−τt+ξTt−τM−R2−R3−Qξt−τM+ξ˙TtτM2G+τM2Qξ˙t+ξTtπ24G+π24GTξt−τt+ξTtQξt−τM.
In addition, the WAETM (3) is equivalent to the following equation:(22)eyTtΩeyt≤σyTt−τtΩyt−τt+ϕσyTt−τtΩeyt+eyTtΩyt−τt=σξTt−τtCTΩCξt−τt+ϕσξTt−τtCTΩeyt+eyTtΩCξt−τt.
It is worth noting that for any matrixM with appropriate dimensions, we have the following equation:(23)ξTtMT+ξ˙TtMT−ξ˙t+A¯ξt+B¯ξt−τt+L¯eyt+D¯ft+F¯gext=0.
According to Assumption1, the nonlinear function ρxt,t satisfies the Lipschitz condition, and for any positive number υ we have the following equation:(24)υξTtδ2ξt−υgTextgext≥0.
Define a new function as follows:(25)J1t=V˙1t+ξTtξt−λ1fTtft.
Integrating both sides of (25) under the zero initial condition gives us the following equation:(26)∫0tJ1sds=V1t+∫0tξTsξsds−λ1∫0tfTsfsds,where V1t>0. Then according to Definition 1, we can obtain the following equation:(27)∫0tξTsξsds≤λ1∫0tfTsfsds.
Define a variable asϖ1t=ξTtξ˙TtξTt−τtξTt−τMgTexeyTtfTt. Then according to (21) and (25) and (27), we can get the following equation:(28)J1t=V˙1t+ξTtξt−λ1fTtft≤ϖ1TtΞϖ1t.
According to (15) and (28), J1t<0 when Ξ<0. When ft=0, V˙1t<0. Therefore, the observation error dynamic system (12) is asymptotically stable with WAETM (3). The proof of Theorem 1 is completed.


### 3.3. Fault-Tolerant Controller

With the system state estimation, fault estimation, and nonlinear function estimation obtained by the sate-fault observer, the fault-tolerant controller can be designed as follows:(29)ut=Kx^t−B†Bff^t−B∗ρx^t,where K is the controller gain. The fault-tolerant controller consists of three terms. The first term Kx^t is a state feedback control, the second term −B†Bff^t is used to compensate the influence of faults, and the third term −B∗ρx^t is used to reduce the influence of nonlinear factors.Combining the fault-tolerant controller (28) and the state-fault observer (10), we can obtain the following equation:(30)x^˙t=Ax^t+BKx^t−BB+Bff^t−BB∗ρx^t+Bff^t+Fρx^t+Ly˜t−y^t−τt=A+BKx^t+F−BB∗ρx^t+Leyt+LCext−τt.According to (10) and (30), the overall closed-loop system can be written as follows:(31)ε˙t=A˜εt+B˜εt−τt+L˜eyt+Bfeft+F˜φεt,where εt=x^TtexTtT, φεt=ρTx^tgexTtT, and(32)A˜=A+BK00A,B˜=0LC0−LC,L˜=L−L,F˜=F−BB∗0F0.There needs a lemma before the subsequent discussion.Lemma 2.
(See [36]). For a fixed constant matrix Q with full column rank, the unknown matrix Y can be calculated by the following equality for any real matrix X, Y and N with appropriate dimensions:(33)Y=QTXQ−1QTQN,when the following condition satisfies the following equation:(34)XQY=QN.Theorem 2.
For given positive scalarsh, τM, ϕ, σ, λ2 and μ, there exist symmetric positive definite matrices P1>0, S1>0, S2>0, S3>0, H>0, W>0. Then, suppose there exists matrix Λ, event-triggered weighted matrix Ω>0, and matrices M1 and Θ with appropriate dimensions such that the following condition holds:(35)Π=∏11∏12∏13WM1TF˜M1TL˜M1TBf∗∏22M1TB˜0M1TF˜M1TL˜M1TBf∗∗∏3300∏360∗∗∗∏44000∗∗∗∗−μI00∗∗∗∗∗−Ω0∗∗∗∗∗∗−λ2I<0,where ∏11=S1+S2−π2/4H−W+M1TA˜, ∏12=2P1−M1T+A˜TM1T,(36)∏13=π24H+π24HT+M1TB˜,∏22=τM2H+τM2W,∏33=σCTΩC+1−τS2−1−τS1−π24,∏36=φσCTΩ+ΩC,∏44=−S2−S3−W,then, the overall closed-loop system (31) is asymptotically stable under the WAETM with the controller gain designed as K=BTΘB−1BTBΛ.Proof.
According to Lemma2, if condition ΘBK=BΛ is satisfied, one can obtain that(37)K=BTΘB−1BTBΛ,whereafter, choose a Lyapunov-Krasovskii function as follows:(38)V2t=V21t+V22t+V23t,where(39)V21t=εTtP1εt+∫t−τttεTθS1εθdθ+∫t−τMt−τtεTtS2εt+∫t−τMtεTθS3εθdθ,V22t=τM2∫ikhtε˙TθHε˙θdθ−π24∫ikhtεθ−εikhTHεθ−εikhdθ,V23t=τM∫t−τMt∫rtε˙TθWε˙θdθdr,and the derivative of (39) is obtained as follows:(40)V˙21t=2εTtP1ε˙t+εTtS1εt−1−τεTt−τtS2εt−τt+1−τεTt−τtS2εt−τt−εTt−τMS2εt−τM+εTtS3εt−εTt−τMS3εt−τM,V˙22t=τM2ε˙TtHε˙t−π24εt−εt−τtTHεt−εt−τt,V˙23t=τM2ε˙TtWε˙t−τM∫t−τMtε˙TθWε˙θdθ.
According to Lemma1, we can obtain the following equation:(41)−τM∫t−τMtε˙TθWε˙θdθ≤εtεt−τMT−WW∗−Wεtεt−τM.
Then, from (40) we can get the following equation:(42)V˙2t≤2εTtP1ε˙t+εTtS1+S3−π24H−Wεt−εTt−τt1−τS2−1−τS1−π24Hεt−τt+εTt−τM−S2−S3−Wεt−τM+ε˙TtτM2H+τM2Wε˙t+εTtπ24H+π24HTεt−τt+εTtWεt−τM.
Again, the WAETM (4) is equivalent to the following equation:(43)eyTtΩeyt≤σyTt−τtΩyt−τt+ϕσyTt−τtΩeyt+eyTtΩyt−τt=ξTt−τtCTΩCξt−τt+ϕσξTt−τtCTΩeyt+eyTtΩCξt−τt.
It is worth noting that for any matrixM1 with appropriate dimension, we have the following equation:(44)εTtM1T+ε˙TtM1T−ε˙t+A˜εt+B˜εt−τt+L˜eyt+Bfeft+F˜φεt=0.
According to Assumption1, the nonlinear function ρxt,t satisfies the Lipschitz condition, and for any positive number μ we have the following equation:(45)μεTtδ2εt−μφTεtφεt≥0.
Define a new variable asϖ2t=εTtε˙TtεTt−τtεTt−τMφTεxeyTtefTt. Then according to (42) and (45), we can get the following equation:(46)J2t=V˙2t+εTtεt−λ2efTteft.
Similar to the proof of Theorem1, we can obtain the following equation:(47)J2t≤ϖ2Tt∏ϖ2t.
According to (47), for any small scalar β>0 under the initial condition, the following inequality holds:(48)J2t≤−βεt2.
According to (35) and (47), it is obvious that J2t<0when ∏<0. Also, (48) indicates limt⟶∞εt2=0 and hence the closed-loop system (31) is asymptotically stable with WAETM (4). The proof of Theorem 2 is completed.


## 3.1. WAETM

Event-triggered mechanism is an effective way to alleviate the communication congestion and save the network communication resources. However, the conventional event-triggered mechanism depends on the difference between the values of the latest released data packet and the current sampling data packet. The conventional event-triggered mechanism is [21].(3)tk+1h=tkh+minl∈ℕ+lh|ecTtΩecTt>σyTtkh+lhΩytkh+lh,where tkh represents the release instant, tk is a positive integer, and h is the sampling period. l denotes the interval time between the latest triggered instant and the next triggered instant, σ is a constant, Ω is the constant weighted matrix, and ect=ytkh−ytkh+lh.In order to further decrease the triggered data error and save the network communication resources, the WAETM is designed as follows:(4)Ψ=l|eTtΩet−σy¯TtkhΩy¯tkh+ϕσy¯TtkhΩet+eTtΩy¯Ttkh≤0,where Ψ denotes a set of events satisfying the event trigger condition, y¯tkh=tkhytkh+tkh+lhytkh+lh/2tkh+lh denotes the weighted average value between the latest released data packet and the current sampling data packet, et=ytkh−y¯tkh denotes the difference between the latest released data packet and the weighted average data packet, and ϕ is a constant.Remark 1.
Compared with the conventional event-triggered mechanism (2), the proposed WAETM (3) can decrease the triggered data error et, avoiding unexpected triggered events. Meanwhile, the item ϕσy¯TtkhΩet+eTtΩy¯Ttkh in the WAETM (3) contains a triggered data packet ytkh. Therefore, the controller can obtain data packets from the network even when the system is subjected to others noises or disturbances. In fact, the WAETM in (3) can be converted into the conventional event-triggered mechanism in [21] when ϕ=0.
The release instant of WAETM is denoted astk+1h=tkh+lM+1h, where lM=maxl|l∈Ψ. When the data packets are successfully transmitted to the observer, the system measured output can be written as follows:(5)y˜t=ytkh,t∈tkh+τk,tk+1h+τk+1,where y˜t denotes the data successfully transmitted at the triggered instant and τk is the time delay.
Considering that the data packets transmitted by the WAETM are subjected to time delayτk=jk−tkh, where jk represents the instant when the data packets arrive at the ZOH. The ZOH deals with the data packets out-of-sequence and guarantees the continuity of control input within the holding time-interval. The subintervals are defined as follows:(6)I=I1=jk,jk+h,I2=jk+h,jk+2h,⋮In¯k=jk+n¯k−1h,jk+1,where n¯k=minl|jk+l−1h≥jk+1, and the time delay function can be written as follows:(7)τt=t−tkh,t∈I1,t−t+k1h,t∈I2,⋮t−tk+jk−1h,t∈In¯k,where 0≤τt≤τM, τ˙t≤τΔ, τM is the maximum value of the τt, and τΔ is a constant. Then the error function of ZOH can be expressed as follows:(8)eyt=ytkh−ytkh,t∈I1,ytkh−ytk+1h,t∈I2,⋮ytkh−ytkh+n¯k−1h,t∈In¯k,which implies that(9)y˜t=eyt+yt−τt,t∈jk,jk+1.


## 3.2. State-Fault Observer

According to the data packetsy˜t successfully transmitted by the WAETM, the state-fault observer is designed as follows:(10)x^˙t=Ax^t+But+Bff^t+Fρx^t,t+Ly˜t−y^t−τt,f^˙t=Gf^t+Hy˜t−y^t−τt,y^t=Cx^t,where x^t∈Rn, y^t∈Rr, f^t∈Rm, ρx^t,t=ρ1x^1t,t,…,ρnx^nt,tT are the system state estimation vector, system measured output estimation, fault estimation signal, and nonlinear function estimation, respectively. L and H are observer gain matrices to be designed later.The estimation error of system state is defined asext=xt−x^t, the estimation error of fault is defined as ef=ft−f^t, and the estimation error of nonlinear function is defined as gext=ρxt−ρx^t. Then, the estimation error of system state and fault can be respectively rewritten as follows:(11)e˙xt=Aext+Fgexx+Bfeft−Leyt−LCext−τt,e˙ft=1−Gft+Geft−Heyt−HCext−τt.Defineξt=exteftT, then the observation error dynamic system can be given as follows:(12)ξ˙t=A¯ξt+B¯ξt−τt+L¯eyt+D¯ft+F¯gext,where A¯=ABf0G, B¯=−LC0−HC0, L¯=−L−H, D¯=01−G, F¯=F0.Up to now, we need some lemmas and definitions before proceeding the stability of observation error dynamic.Lemma 1.
(See [34]). For any positive definite constant matrix Y∈Rn×n, scalar α>0, and vector function x˙:−α,0⟶Rn, the following integration is defined:(13)−α∫t−αtx˙TθYx˙θdθ≤xtxt−αT−YY∗−Yxtxt−α.Definition 1.
(See [35]). System (12) satisfies the H∞ performance under initial condition, and the following performance index holds for all nonzero ft∈L20,∞.(14)∫0∞ξTsξsds≤λ1∫0∞fTsfsds,λ1>0.Theorem 1.
For given positive scalarsh, τM, ϕ, σ, λ1 and υ, there exists symmetric positive definite matrices P>0, R1>0, R2>0, R3>0, G>0, Q>0, and appropriate matrices M and Z such that the following condition holds.(15)Ξ=Ξ11Ξ12Ξ13QMTF¯ZMTD¯∗Ξ22MTB¯0MTF¯ZMTD¯∗∗Ξ3300Ξ360∗∗∗Ξ44000∗∗∗∗−υI00∗∗∗∗∗−Ω0∗∗∗∗∗∗−λ1I<0,where Ξ11=R1+R2−π2/4G−Q+MTA¯, Ξ12=2P−MT+A¯TMT,(16)Ξ13=π24G+π24GT+MTB¯,Ξ22=τM2G+τM2Q,Ξ33=σCTΩC+1−τR2−1−τR1−π24G,Ξ36=ϕσCTΩ+ΩC,Ξ44=−R2−R3−Q,then the observation error dynamic system (12) with the WAETM (4) is asymptotically stable. In addition, the state-fault observer gain matrices can be obtained by L¯=M−TZ.Proof.
Choose a Lyapunov–Krasovskii function as follows:(17)V1t=V11t+V12t+V13t,where(18)V11t=ξTtPξt+∫t−τttξTθR1ξθdθ+∫t−τMt−τtξTtR2ξt+∫t−τMtξTθR3ξθdθ,V12t=τM2∫ikhtξ˙TθGξ˙θdθ−π24∫ikhtξθ−ξikhTGξθ−ξikhdθ,V13t=τM∫t−τMt∫rtξ˙TθQξ˙θdθdr.
The derivative of (17) is obtained as follows:(19)V˙11t=2ξTtPξ˙t+ξTtR1ξt−1−τξTt−τtR1ξt−τt+1−τξTt−τtR2ξt−τt−ξTt−τMR2ξt−τM+ξTtR3ξt−ξTt−τMR3ξt−τM,V˙12t=τM2ξ˙TtGξ˙t−π24ξt−ξt−τtTGξt−ξt−τt,V˙13t=τM2ξ˙TtQξ˙t−τM∫t−τMtξ˙TθQξ˙θdθ.
According to Lemma1, we can obtain the following equation:(20)−τM∫t−τMtξ˙TθQξ˙θdθ≤ξtξt−τMT−QQ∗−Qξtξt−τM.
Therefore, from (19) we can get the following equation:(21)V˙1t≤2ξTtPξ˙t+ξTtR1+R3−π24G−Qξt−ξTt−τt1−τR2−1−τR1−π24Gξt−τt+ξTt−τM−R2−R3−Qξt−τM+ξ˙TtτM2G+τM2Qξ˙t+ξTtπ24G+π24GTξt−τt+ξTtQξt−τM.
In addition, the WAETM (3) is equivalent to the following equation:(22)eyTtΩeyt≤σyTt−τtΩyt−τt+ϕσyTt−τtΩeyt+eyTtΩyt−τt=σξTt−τtCTΩCξt−τt+ϕσξTt−τtCTΩeyt+eyTtΩCξt−τt.
It is worth noting that for any matrixM with appropriate dimensions, we have the following equation:(23)ξTtMT+ξ˙TtMT−ξ˙t+A¯ξt+B¯ξt−τt+L¯eyt+D¯ft+F¯gext=0.
According to Assumption1, the nonlinear function ρxt,t satisfies the Lipschitz condition, and for any positive number υ we have the following equation:(24)υξTtδ2ξt−υgTextgext≥0.
Define a new function as follows:(25)J1t=V˙1t+ξTtξt−λ1fTtft.
Integrating both sides of (25) under the zero initial condition gives us the following equation:(26)∫0tJ1sds=V1t+∫0tξTsξsds−λ1∫0tfTsfsds,where V1t>0. Then according to Definition 1, we can obtain the following equation:(27)∫0tξTsξsds≤λ1∫0tfTsfsds.
Define a variable asϖ1t=ξTtξ˙TtξTt−τtξTt−τMgTexeyTtfTt. Then according to (21) and (25) and (27), we can get the following equation:(28)J1t=V˙1t+ξTtξt−λ1fTtft≤ϖ1TtΞϖ1t.
According to (15) and (28), J1t<0 when Ξ<0. When ft=0, V˙1t<0. Therefore, the observation error dynamic system (12) is asymptotically stable with WAETM (3). The proof of Theorem 1 is completed.


## 3.3. Fault-Tolerant Controller

With the system state estimation, fault estimation, and nonlinear function estimation obtained by the sate-fault observer, the fault-tolerant controller can be designed as follows:(29)ut=Kx^t−B†Bff^t−B∗ρx^t,where K is the controller gain. The fault-tolerant controller consists of three terms. The first term Kx^t is a state feedback control, the second term −B†Bff^t is used to compensate the influence of faults, and the third term −B∗ρx^t is used to reduce the influence of nonlinear factors.Combining the fault-tolerant controller (28) and the state-fault observer (10), we can obtain the following equation:(30)x^˙t=Ax^t+BKx^t−BB+Bff^t−BB∗ρx^t+Bff^t+Fρx^t+Ly˜t−y^t−τt=A+BKx^t+F−BB∗ρx^t+Leyt+LCext−τt.According to (10) and (30), the overall closed-loop system can be written as follows:(31)ε˙t=A˜εt+B˜εt−τt+L˜eyt+Bfeft+F˜φεt,where εt=x^TtexTtT, φεt=ρTx^tgexTtT, and(32)A˜=A+BK00A,B˜=0LC0−LC,L˜=L−L,F˜=F−BB∗0F0.There needs a lemma before the subsequent discussion.Lemma 2.
(See [36]). For a fixed constant matrix Q with full column rank, the unknown matrix Y can be calculated by the following equality for any real matrix X, Y and N with appropriate dimensions:(33)Y=QTXQ−1QTQN,when the following condition satisfies the following equation:(34)XQY=QN.Theorem 2.
For given positive scalarsh, τM, ϕ, σ, λ2 and μ, there exist symmetric positive definite matrices P1>0, S1>0, S2>0, S3>0, H>0, W>0. Then, suppose there exists matrix Λ, event-triggered weighted matrix Ω>0, and matrices M1 and Θ with appropriate dimensions such that the following condition holds:(35)Π=∏11∏12∏13WM1TF˜M1TL˜M1TBf∗∏22M1TB˜0M1TF˜M1TL˜M1TBf∗∗∏3300∏360∗∗∗∏44000∗∗∗∗−μI00∗∗∗∗∗−Ω0∗∗∗∗∗∗−λ2I<0,where ∏11=S1+S2−π2/4H−W+M1TA˜, ∏12=2P1−M1T+A˜TM1T,(36)∏13=π24H+π24HT+M1TB˜,∏22=τM2H+τM2W,∏33=σCTΩC+1−τS2−1−τS1−π24,∏36=φσCTΩ+ΩC,∏44=−S2−S3−W,then, the overall closed-loop system (31) is asymptotically stable under the WAETM with the controller gain designed as K=BTΘB−1BTBΛ.Proof.
According to Lemma2, if condition ΘBK=BΛ is satisfied, one can obtain that(37)K=BTΘB−1BTBΛ,whereafter, choose a Lyapunov-Krasovskii function as follows:(38)V2t=V21t+V22t+V23t,where(39)V21t=εTtP1εt+∫t−τttεTθS1εθdθ+∫t−τMt−τtεTtS2εt+∫t−τMtεTθS3εθdθ,V22t=τM2∫ikhtε˙TθHε˙θdθ−π24∫ikhtεθ−εikhTHεθ−εikhdθ,V23t=τM∫t−τMt∫rtε˙TθWε˙θdθdr,and the derivative of (39) is obtained as follows:(40)V˙21t=2εTtP1ε˙t+εTtS1εt−1−τεTt−τtS2εt−τt+1−τεTt−τtS2εt−τt−εTt−τMS2εt−τM+εTtS3εt−εTt−τMS3εt−τM,V˙22t=τM2ε˙TtHε˙t−π24εt−εt−τtTHεt−εt−τt,V˙23t=τM2ε˙TtWε˙t−τM∫t−τMtε˙TθWε˙θdθ.
According to Lemma1, we can obtain the following equation:(41)−τM∫t−τMtε˙TθWε˙θdθ≤εtεt−τMT−WW∗−Wεtεt−τM.
Then, from (40) we can get the following equation:(42)V˙2t≤2εTtP1ε˙t+εTtS1+S3−π24H−Wεt−εTt−τt1−τS2−1−τS1−π24Hεt−τt+εTt−τM−S2−S3−Wεt−τM+ε˙TtτM2H+τM2Wε˙t+εTtπ24H+π24HTεt−τt+εTtWεt−τM.
Again, the WAETM (4) is equivalent to the following equation:(43)eyTtΩeyt≤σyTt−τtΩyt−τt+ϕσyTt−τtΩeyt+eyTtΩyt−τt=ξTt−τtCTΩCξt−τt+ϕσξTt−τtCTΩeyt+eyTtΩCξt−τt.
It is worth noting that for any matrixM1 with appropriate dimension, we have the following equation:(44)εTtM1T+ε˙TtM1T−ε˙t+A˜εt+B˜εt−τt+L˜eyt+Bfeft+F˜φεt=0.
According to Assumption1, the nonlinear function ρxt,t satisfies the Lipschitz condition, and for any positive number μ we have the following equation:(45)μεTtδ2εt−μφTεtφεt≥0.
Define a new variable asϖ2t=εTtε˙TtεTt−τtεTt−τMφTεxeyTtefTt. Then according to (42) and (45), we can get the following equation:(46)J2t=V˙2t+εTtεt−λ2efTteft.
Similar to the proof of Theorem1, we can obtain the following equation:(47)J2t≤ϖ2Tt∏ϖ2t.
According to (47), for any small scalar β>0 under the initial condition, the following inequality holds:(48)J2t≤−βεt2.
According to (35) and (47), it is obvious that J2t<0when ∏<0. Also, (48) indicates limt⟶∞εt2=0 and hence the closed-loop system (31) is asymptotically stable with WAETM (4). The proof of Theorem 2 is completed.


## 4. Results

In this section, the networked motor control system shown in Figure2 is adopted to verify the effectiveness of the proposed weighted average event-triggered FTC scheme. The mathematical model of the brushless direct current motor is(49)A=01.000−5.867−3.038,B=0.3404.868,C=10,F=−0.818−1.65−0.2−0.02,Bf=−4.811−25.852.Figure 2
Brushless direct current networked motor control system.The control input of the motor is the voltage value. The system measured output of the motor is the speed (r/min). The reference input function is set asyrt=400sint+1200, and the nonlinear function is ρxt,t=40sinxt0T. The delay is τt=0.2sinπt, and hence τM=0.2, τΔ=0.2. In addition, parameters in the event-triggered mechanism are set as σ=0.0015, and ϕ=0.12. The sampling time is h=0.02s. The event-triggered weighted matrix here is set as Ω=I.According to Theorem1 and Theorem 2, the gains of the state-fault observer and the fault-tolerant controller can be obtained as follows:(50)L=49.522−26.896,H=−10.982,K=−2.5−0.264.Other parameters of the state-fault observer and the fault-tolerant controller are(51)G=0.2,B†=0.0140.204,B∗=−0.271−0.232.Besides, the actuator fault signal is considered as follows:(52)ft=25,5≤t<15,50−25e−t+15,15≤t<30,80+80cos0.1t−30,30≤t<50,0,otherwise.The scheme proposed in this paper (denoted by WAETM-FTC) is compared with the scheme in [21] (denoted by ETM-FTC). In [21], an FTC scheme was proposed, however, the conventional event-triggered mechanism was utilized and the influence of the nonlinear factors in NCSs were not considered.Figures3 and 4 show the fault estimation and fault estimation error, respectively. Compared with the ETM-FTC in [21], when WAETM-FTC is employed it is obvious that the fault can be estimated much better in the networked motor control system.Figure 3
Fault estimation (a) ETM-FTC in [21]; (b) WAETM-FTC.
(a)(b)Figure 4
Fault estimation error.As shown in Figure5, when WAETM-FTC is employed the nonlinear function can also be effectively estimated. In contrast, the ETM-FTC in [21] does not consider nonlinear factors and thus the nonlinear function estimation is not shown here.Figure 5
Nonlinear function estimation.Figure6(a) shows the measured output (speed) with the ETM-FTC in [21]. During 0-5second, although there is no fault in the system, the measured output can not track the reference input well because there are still nonlinear factors in the system. After 5 seconds, the fault occurs and the system measured output suffers from both fault and nonlinear factors. Therefore, the error between the measured output and the reference input becomes even larger as shown in Figure 7. Figure 6(b) shows the measured output (speed) with the WAETM-FTC scheme. During 0–5 seconds when the fault does not occur while there are nonlinear factors, the measured output can track the reference input. After 5 seconds, even though there are both fault and nonlinear factors in the system, the measured output can still track the reference input well. Therefore, the error between the measured output and the reference input can converge in Figure 7.Figure 6
Measured output and reference input (a) ETM-FTC in [21]; (b) WAETM-FTC.
(a)(b)Figure 7
Measured output error.The release intervals are shown in Figure8, when the ETM-FTC in [21] is adopted, most of the release intervals are the same with the sampling time of 0.02 seconds, which may lead to the communication congestion. In contrast, when WAETM-FTC is adopted, it is clearly that most of the release intervals are larger than 0.02 seconds, which indicates that the unnecessary transmitted data packets are reduced. Moreover, the triggered times are 1688 in Figure 8(a), and the triggered times are 837 in Figure 8(b), which indicates that the proposed WAETM can decrease the triggered data packets by 34.04%. Therefore, the communication congestion is effectively alleviated.Figure 8
Release intervals (a) ETM-FTC in [21]; (b) WAETM-FTC.
(a)(b)


## 5. Conclusions

In this paper, a weighted average event-triggered fault-tolerant control scheme is proposed for nonlinear NCSs with the communication congestion and actuator fault. The weighted average event-triggered mechanism, which adopts the weighted average of data packets, is to alleviate the communication congestion and save the network communication resources. The state-fault observer based on the weighted average event-triggered mechanism is to estimate the system state and fault. With the system state estimation and fault estimation, the fault-tolerant control law is obtained. The proposed weighted average event-triggered fault-tolerant control scheme can effectively save the network communication resources in nonlinear NCSs, while compensating the influence of the fault and reducing the influence of nonlinear factors. Adaptive event-triggered based fault-tolerant control for nonlinear networked control systems will be investigated in our future work.



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*Source: 5482290-2022-08-25.xml*