# Asynchronous Adaptive Fault-Tolerant Control for Networked Stochastic Unmanned Surface Vehicles with Multiple Types of Actuator Faults

Authors: Ai-Min Wang; Jian-Ning Li; Xiao-Bin Xu
Journal: Complexity
 (2020)

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## Abstract

This paper aims to design an asynchronous adaptive fault-tolerant controller for the networked stochastic unmanned surface vehicles (NSUSVs) subject to multiple types of actuator faults and external disturbance. The partial fault and bias fault of the actuator are taken into consideration simultaneously. By estimating online the unknown bias fault of the actuator and the external disturbances, the proposed adaptive fault-tolerant controller can automatically compensate for these impacts produced by actuator faults and external perturbation while preserving the uniformly ultimate boundedness of the solutions. Both the faulty actuator and the designed controller are asynchronous with the NSUSVs. Moreover, a mode-dependent adaptive event-triggered mechanism (AETM) is introduced in order to facilitate network resources utilization. Finally, the effectiveness and correctness of the proposed design scheme are verified by a numerical example.


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## Body

## 1. Introduction

Happening along with the rapid development and continuous renewal of the AI, Big Data, and the Internet of Things, the marine vehicles trend toward advanced automation. Unmanned marine vehicles (such as unmanned surface vehicles (USVs)) are easier to deploy and operate than manned marine vehicles in some risky and extreme environments, so they have remarkable advantages in field operations, which makes them widely used in civil and military fields (see [1–4] and the references therein). Besides, due to the agile controllability and strong autonomy of USVs, a growing number of scholars have begun to conduct more in-depth research on various control strategies for USVs. Especially, the heading control strategy for USVs has been intensively addressed in [5, 6]. The maneuvering control method of USVs has been vastly investigated in [7, 8]. The trajectory tracking control approach has been extensively studied in [9–12].Note that USVs are usually dynamically located and remotely controlled by the land control station through advanced communication technologies such as Wi-Fi or radio modems. Therefore, USVs, communication network, and onshore control station constitute the networked unmanned surface vehicles (NUSVs). Since the communication network is introduced into USVs, the connectivity of the network might be neither constant nor simply switching between networks in a specific way, but randomly switching. The random switching of NUSVs may be caused by channel fading, packet dropouts, link failures, etc. Therefore, it is proper to model the randomly switching network topologies as a Markov process [13], which has not been taken into consideration in many of the existing works about USVs. The issue of networked stochastic unmanned surface vehicles (NSUSVs) has not yet received research attention and remains to be solved.Meanwhile, on account of USVs controlled in the network environment, they inevitably lead to communication problems in the proposed NSUSVs, such as time delay, packet loss, and quantization, which will result in the loss of information. In the NSUSVs, incomplete information transmission is responsible for the asynchronization phenomenon that the modes of the controller/filter, even the actuator, do not run synchronously with the modes of the original system. This problem has been way settled by many scholars as soon as it was put forward. For instance, the authors of [14–18] have portrayed the asynchronization phenomenon appears between system modes and controller modes as a so-called hidden Markov model (HMM). Analogically, the asynchronization phenomenon exists between system modes, and filter modes have been exhaustively discussed in [19–22]. Detailedly, in [22], an asynchronous filter is proposed for the original system with quantized measurements where the filter and quantizer are both mode-dependent, and their modes are asynchronous with that of the system, which is described by two corresponding HMMs. However, as far as the author knows, there is still a gap in the research of asynchronous problems among controllers, actuators, and systems. In addition to this problem, it is not difficult to find that the above references all adopt the proposed time-triggered mechanism (TTM), which inevitably leads to the waste of limited network resources. In order to limit the number of packets sent over the network, the event-triggered mechanism (ETM) is applied to HMM in [23, 24], but relevant approaches discussed in [23, 24] still have the drawback that the event-triggered threshold is assumed to be a preset fixed value, which makes it difficult to adapt to the changes of the system. Motivated by [25], a novel mode-dependent AETM with the adaptive threshold is designed. This problem is still a challenging and interesting research problem that promotes our current research work.All the references mentioned hereinabove assumes that the components of the system are running normally. However, in actual control systems, the actuator may suffer from uncertainties [26] or some failures, such as loss of effectiveness, bias, and outage fault, which will degrade the performance of the system or even crash the system. The fault-tolerant control (FTC) approach can automatically compensate for component failures while maintaining system stability. FTC can be classified into active fault-tolerant control (AFTC) and passive fault-tolerant control (PFTC). Note that their similarities and differences are elaborated at length in [27] and will not be repeated here because of limited space. The concentration on actuator failure has been increasing in recent decades. For instance, the study on partial failures of actuators has been vastly proposed in [28–34]. The problem regarding actuator bias faults has been discussed in [35–38]. Regrettably, only a single actuator failure model is considered in the aforementioned works about actuator faults. Although the passive fault-tolerant controller is designed in [39] for mixed actuator faults, which include the actuator bias fault and the actuator partial fault, the result is conservatism caused by the application of the PFTC method. However, it is gratifying that the adaptive fault-tolerant control approach can solve this kind of hybrid actuator faults [40–42]. As a matter of fact, the adaptive fault-tolerant control approach, as a branch of AFTC, can compensate for the failures well by reconstructing a new controller online by forming a suitable fault-tolerant control law (for details, see [29, 31, 33–37, 40–43] and references therein), which is the second motivation of this paper.The purpose of this paper is to design an adaptive fault-tolerant controller for NSUSVs that suffer from external disturbance and multiple types of actuator faults. First, two HMMs are used to describe the asynchronization among the controller, actuator, and the proposed system. By introducing a mode-dependent AETM, the resources transferred to the network can be limited effectively. An adaptive fault-tolerant control method using the adaptive algorithm is designed to estimate and compensate for the unknown actuator faults. Compared with existing work, the major contributions of our design method can be summarized as follows:(1)
In this work, a unified actuator faults model including the partial faults and the bias faults is taken into consideration. It is worth mentioning that we skillfully couple the unknown external disturbances and the unknown bias faults into the proposed augmented disturbances. Then, corresponding online adaptive protocols are constructed to track and estimate its upper and lower bounds simultaneously.(2)
The asynchronous phenomenon considered in this paper exists not only between the controller and the proposed NSUSVs but also between the actuator and the proposed NSUSVs. In other words, both the controller and the actuator are asynchronous with the controlled NSUSVs, which are described as two HMMs.(3)
To enhance the efficiency of network resources usage, a mode-dependent AETM with the network-induced delay is constructed, where the proposed triggered threshold is designed to be a time-varying variable.The remainder of this paper is structured as follows. The establishment of the NSUSVs model with actuator faults under a mode-dependent AETM is described inSection 2. Based on the constructed model, the main results are depicted in Section 3. A numerical example is given in Section 4. Eventually, there is a conclusion in Section 5.Notations in this work are as follows. Given symmetric square matricesX and Y, X≥Y (respectively, X≤Y) means that X−Y is positive semidefinite (respectively, negative semidefinite). Similarly, X>Y (respectively, X<Y) means that X−Y is positive definite (respectively, negative definite). AT is the transpose of the matrix A. diag⋯ represents a block-diagonal matrix. “∗” denotes the symmetric term in a symmetric matrix. The symbol λminX means the minimum eigenvalue of a invertible matrix X. For a square matrix X, one has HeX=X+XT.


## 2. Problem Formulation and Preliminaries

### 2.1. Stochastic Unmanned Surface Vehicles Model

As depicted in Figure1, υ,r,p,δ,φ, and ϕ represent the sway velocity, yaw velocity, roll velocity, rudder angle, heading angel, and roll angel, respectively, besides these variables, X0,Y0, and Z0 denote the longitudinal axis, transverse axis, and normal axis, respectively.Figure 1
Motion coordinate system for SUSVs.(1)υs=Kθtdυ1+Tυsδs,φs=11+TrsKθtdrδs+Kθtυrυs+ωφs,ϕs=ωn2s2+2ζωns+ωn2Kθtdpδs+Kθtυpυs+ωϕs,Using the similar technique as in [5], the transfer functions for the stochastic unmanned surface vehicles (SUSVs) model is derived as follows:where ωφ and ωϕ indicate the effect of the wave on φ and ϕ, respectively; ζ is the damping ratio, and ωn is the natural frequency under no damping; both Tυ and Tr denote the time constants; and Kθtυr,Kθtυp,Kθtdυ,Kθtdr, and Kθtdp stand for the uncertain parameters with the Markov random process θt.The parameterθt is the stochastic Markov process with right continuous trajectories and takes values in a finite set N=1,2,…,N, which is subject to the transition probability matrix S=Oij,∀i,j∈N, and the mode transition probability are defined as follows:(2)Prθt+Δt=j|θt=i=OijΔt+oΔt,i≠j,1+OijΔt+oΔt,i=j,where limΔt⟶0oΔt/Δt=0,Oij≥0i≠j,Oij=−∑j=1,j≠iNOij, and Oij denotes the probabilities from mode i at time t to mode j at time t+Δt.Remark 1.
Note that whenS=I, the proposed SUSVs model will be reduced to the traditional USVs model as introduced in [5, 44]. In other words, the considered model of USV in [5, 44] is a special case of our research.
Denotingxt=υtrtφtptϕtT,ωt=ωφtωϕtT, and Aθt,Bθt,W are given as follows:(3)Aθt=−1Tυ0000KθtυrTr−1Tr00001000ωn2Kθtυp00−2ζωn−ωn200010,Bθt=KθtdυTυKθtdrTr0ωn2Kθtdp0,W=001Tr0000ωn200,one can obtain the following state-space model:(4)x˙t=Aθtxt+Bθtδt+Wωt.
The corresponding structure of the proposed networked stochastic unmanned surface vehicles (NSUSVs) is shown in Figure2. Note that the proposed NSUSVs suffer from actuator failures including bias fault and partial fault, external disturbance, asynchronization problem among original system, controller, and actuator as well as AETM with adaptive threshold.Figure 2
The structure of NSUSVs.


### 2.2. The Mode-Dependent Adaptive Event-Triggered Mechanism

In this paper, the event detector is used to determine whether the present sampling data should be stored and sent out to the controller, and the event is triggered if the following inequality is satisfied:(5)xtkΔ+lΔ−xtkΔTΩθtxtkΔ+lT−xtkΔ>σtxTtkΔ+lΔΩθtxtkΔ+lΔ,where Δ is the sampling period of the sampler and the positive definite mode-dependent matrix Ωθt is the weighting matrix to be designed later. xkΔ is the current sampling state at sampling instant kΔkΔ∈R1≜0,1Δ,2Δ,…, and xtkΔ is the latest transmitted state at the releasing instant tkΔtkΔ∈R2≜t0Δ,t1Δ,t2Δ,…, as well as we assume the first releasing instant t0=0. On the other hand, σt is the event-triggered threshold variable, which meets the following adaptive law concerning event-triggered strategy:(6)σ˙t≜0,σt⟶0+,1σt1σt−ϑeTtΩθtet,otherwise,where 0<σt<1,ϑ>0.et=xtkΔ+lΔ−xtkΔ,∀l=1,2,…,ℓ. The term tkΔ+lΔ denotes the sampling instants from the current transmitted sampling instant tkΔ to the future transmitted sampling instant tk+1Δ. Note that the next releasing instant tk+1Δ=tkΔ+ℓΔ if condition (4) is satisfied.We denote(7)Ξtk=tkΔ+dtk,tk+1Δ+dtk+1=∪l=1ℓΞl,tk,where Ξl,tk=tkΔ+l−1Δ+dtk+l−1,tkΔ+lΔ+dtk+l.Define the network induced delay as(8)dt=t−tkΔ−lΔ,t∈∪l=1ℓΞl,tk.Noting that the network-induced time-delay is represented as the maximum allowable upper communication delay boundd¯ which is similar to [45], that is(9)dm=mindtk≤dt≤1+d¯=dM,∀t∈∪l=1ℓΞl,tk.Finally, the AETM is deduced as follows:(10)eTtΩθtet≤σtxTt−dtΩθtxt−dt.Figure3 gives an example to demonstrate the relationship between the sampling instant and the releasing instant, where the data are sampled discretely at instant kΔkΔ∈R1≜0,1Δ,2Δ,… and the data are chosen at instant tkΔtkΔ∈R2≜t0Δ,t1Δ,t2Δ,… to release into the network. Based on the above analysis, we know that R2⊆R1. Besides, the control input of the proposed NSUSVs is reserved by ZOH during the time interval Ξ1,0=0Δ+d0,3Δ+d3,Ξ2,3=3Δ+d3,5Δ+d5,Ξ3,5=5Δ+d5,t3Δ+4Δ+dt3+4,….Figure 3
An example of timing diagram for AETM.Remark 2.
Note that the mode-dependent AETM proposed in this paper relies on modes information of the system, which can efficiently reduce the loss of data in the network transmission, and it is not taken into account in most of the existing literature; see, for instance, [23, 24, 31, 45, 46].Remark 3.
Note that whenσ˙t=0, the supposed AETM can be degraded into the traditional ones, in which the event-triggered threshold is a preset constant, such as [23, 24, 31, 45] with the following form:(11)eTtΩθtet≤σ0xTt−dtΩθtxt−dt,where σ0∈0,1. And the above scheme is also applicable if σt⟶0+.Remark 4.
It is worth noting that the smallest interevent time can be regarded as the sampling periodΔ, so Zeno behavior will not exist in the related AETM proposed in this article.


### 2.3. Actuator Faults Analysis

The control input used by the NSUSVs with single rudder is described as follows:(12)δt=Θηtut−γt,where Θηt represents the mode-dependent efficiency coefficient of actuator faults. Θηtmin and Θηtmax represent the unknown lower and upper bounds of Θηt, that is, 0≤Θηtmin≤Θηt≤Θηtmax≤1, and ut and γt are the applied control signal to be designed and the unknown time-varying bounded bias fault in the actuator, respectively. In addition, the jumps of actuator are under the control of Markov parameter ηt, which are affected by the mode of system through conditional probability matrix A=Pip,∀i∈N, and p∈L=1,2,…,L. The conditional probability Pip implies the possibility that the actuator runs in mode p given the mode i information of system, i.e.,(13)Prηt=p|θt=i=Pip,and for ∀i∈N,p∈L, there exist constraints Pip∈0,1 and ∑p=1LPip=1.For convenience, the subscriptp will be hereinafter employed to replace this Markov parameter ηt in the actuator, and the following multiple types of actuator faults model can be rewritten as(14)δt=Θput−γt.From Table1, one can see that when Θp=1 and γt=0, the actuator works normally. Both Θp=1 and γt≠0 mean the bias fault has appeared in the actuator. 0<Θp<1 and γt=0 denote the partial fault has existed in the actuator. 0<Θp<1 and γt≠0 represent the actuator that has occurred mixed failures including the partial fault and the bias fault. Regardless of whether γt is equal to 0 or not, the actuator outage fault has occurred in the actuator under Θp=0.Table 1
Fault models.
γt=0γt≠0Θp=1NormalBiasΘp∈0,1PartialPartial and biasΘp=0OutageOutageUsing a similar method as [32], that is, by defining Θp=I+ZpΘ¯p and Zp≤Hp≤I, the conservatism that Θp is not known in advance is moderated, where(15)Θ¯p=Θpmax+Θpmin2,Hp=Θpmax−ΘpminΘpmax+Θpmin,Zp=Θp−Θ¯pΘ¯p,∀p∈L.The following assumptions are necessary to facilitate the subsequent research.Assumption 1.
For state-space model (4), in order to make the disturbance and input signal satisfy the match condition, there are constant matrix Ci with appropriate dimensions such that(16)W=BiCi.Assumption 2.
For the actuator partial faultΘp,∀i∈N,p∈L, one has(17)rankBiΘp=rankBi.Assumption 3.
Define the augmented disturbance as follows:(18)ψt=−Θpγt+Ciωt,where ψ¯≤ψt≤ψ¯ and ψ¯ as well as ψ¯ are unknown constants.
One is defined as follows:(19)Ψ=ψ¯+ψ¯.


### 2.4. Asynchronous Adaptive Fault-Tolerant Controller

LetΘ^pt,Ψ^t,ψ¯^t,ψ¯^t denote the online estimation of Θp,Ψ,ψ¯,ψ¯, respectively.One can define(20)∂t=0,xTtPiBiΘ^ptμ−1t>0,1,xTtPiBiΘ^ptμ−1t≤0,where μt=Θ^pt+ρt and ρt is an exponential function and is defined as follows:(21)ρt=ae−bt,where both a and b are bounded positive bound constants to be designed. It can be figured out directly ∫0∞ρτdτ≤ρ¯<∞.Remark 5.
It is worth noting that the introduction ofρt aims to use variable μ−1t=Θ^pt+ρt−1 substitute for Θ^p−1t in order to compensate the actuator partial fault Θp. The reason for adopting this method is that we should consider the case of Θp=0 and also analyze the outage fault of the actuator (for detail, see Table 1), that is, Θ^pt=0. Hence, Θ^p−1twill be not defined well. So the introduction of parameter ρt is indispensable, which is not considered in [29, 34].
Next, we define the vector-valued function as follows:(22)ϖt=ψ¯^t+∂tψ¯^t−ψ¯^t.
The asynchronous adaptive fault-tolerant controller under AETM is designed as follows:(23)ut=K1ξt+K2ξtxtkΔ+K3ξtϖt,where K1ξt,K2ξt,andK3ξt are the gains of the proposed mode-dependent fault-tolerant controller, which will be designed in the sequel. Markov parameter ξt exerts influence on the variations of the controller among different modes; on the other hand, they are affected by the mode of system through conditional probability matrix ℂ=Qiq for ∀i∈N,q∈M=1,2,…,M. The conditional probability Qiq implies the possibility that the controller runs in mode q given the mode information i of system, i.e.,(24)Prξt=q|θt=i=Qiq,and it is extremely obvious that Qiq∈0,1 and ∑q=1MQiq=1 for ∀i∈N,q∈M.Remark 6.
The implication of mode-dependent in this paper is that the actuator and controller depend indirectly on the NSUSVs modeθt that is subject to transition probability (2) through conditional probabilities (13) and (24).Remark 7.
There exist problems such as packet loss, time delay, and quantization in NSUSVs, which will result in the incomplete information transfer. Therefore, the controller’s modes information which is represented byξt (subject to the conditional probability matrix ℂ) run asynchronously with the NSUSVs’ modes information which is represented by θt (subject to the transition probability matrix S), but the latter directly imposes influence on the former. The asynchronization between the NSUSVs and controller is described by the hidden Markov model θt,ξt,S,ℂ,. Similarly, the hidden Markov model θt,ηt,S,A is made up, which accounts for the asynchronization between the NSUSVs and actuator.Remark 8.
Note that similar to the description of HMM in [14], taking different values in the results will reduce to the following three cases:(1)
Whenℂ=I, the controller modes are completely synchronous with the NSUSVs modes.(2)
Similar with the description in [22], the Markov parameter θt of the system is grouped into several clusters, and in each cluster, the conditional probability Qiq for ξt solely relies on which cluster θt belongs to. One extreme case is that there is only one cluster and then the conditional probability matrix ℂ has identical rows.(3)
Whenξt∈1 and ℂ=1…1T, the mode information θt of system fail to work, in other word, the case is referred to as mode-independent.
For notational brevity again, the subscriptsi,j,q will be hereinafter employed to replace these Markov parameters θt,θt+Δt, and ξt in the system and controller. So, the dynamics of the closed-loop NSUSVs are obtained as follows:(25)x˙t=Aixt+BiΘpK1qxt−dt−BiΘpK1qet+BiΘpK2qxtkΔ+BiΘpK3qϖt+Biψt.
Next, the adaptive laws forΘ^pt is designed as follows:(26)Θ^⋅pt≜ProjminΘp,maxΘpLt=0,ifΘ^p=minΘp,andL≤0orΘ^p=minΘp,andL>0,Lt,otherwise,where Lt=−τΘxTtPiBiμ−1tϖt.
Moreover, the adaptive laws forΨ^⋅t,ψ¯^⋅t,ψ¯^⋅t are designed as follows:(27)Ψ^⋅t=ƛΨxtPiBiρtμ−1t,ψ¯^⋅t=gψ1−∂txTtPiBiΘ^ptμ−1t,ψ¯^⋅t=gψ∂txTtPiBiΘ^ptμ−1t,and ƛΨ,τΘ,gψ are the adaptive law gains to be designed in the sequel.
Besides those parameters,K2q and K3q are designed as follows:(28)K2q=−ρtμ−1t2PiBi2Ψ^2tβxtkΔρtμ−1tPiBiΨ^t+ρtBiTPiT,K3q=−μ−1t.
Our main purpose is to design an asynchronous adaptive fault-tolerant controller under AETM to ensure that uniformly ultimate boundedness of the solutions of the closed-loop NSUSVs (25) can be achieved with adaptive law (6), (26), (27).
Before presenting our results, it is necessary to introduce the following lemmas.Lemma 1 (Schur complement) (see [47]).
Given a symmetric matrix as follows:(29)S=S11S12S12TS22,
and the following statements are equivalent:(1)
S<0(2)
S11<0,S22−S12TS11−1S12<0(3)
S22<0,S11−S12S22−1S12T<0Lemma 2 (see [48]).
Given matricesΦ,Γ, and Λ of appropriate dimensions with Φ being symmetrical, then Φ+ΓFtΛ+ΛTFTtΓT<0 for all Ft satisfying FTtFt≤I, if and only if there exists a scalar α>0 such that Φ+αΓΓT+α−1ΛTΛ<0.Lemma 3 (see [43]).
For any partial fault parameterΘp, the matrix PiBiΘpBiTPiT is invertible, and there exists a positive constant β such that(30)PiBiΘpBiTPiT≥βI.Proof.
Settingr2=β=λminPiBiΘpBiTPiT in Lemma 2 in [43], the proof is obtained.


## 2.1. Stochastic Unmanned Surface Vehicles Model

As depicted in Figure1, υ,r,p,δ,φ, and ϕ represent the sway velocity, yaw velocity, roll velocity, rudder angle, heading angel, and roll angel, respectively, besides these variables, X0,Y0, and Z0 denote the longitudinal axis, transverse axis, and normal axis, respectively.Figure 1
Motion coordinate system for SUSVs.(1)υs=Kθtdυ1+Tυsδs,φs=11+TrsKθtdrδs+Kθtυrυs+ωφs,ϕs=ωn2s2+2ζωns+ωn2Kθtdpδs+Kθtυpυs+ωϕs,Using the similar technique as in [5], the transfer functions for the stochastic unmanned surface vehicles (SUSVs) model is derived as follows:where ωφ and ωϕ indicate the effect of the wave on φ and ϕ, respectively; ζ is the damping ratio, and ωn is the natural frequency under no damping; both Tυ and Tr denote the time constants; and Kθtυr,Kθtυp,Kθtdυ,Kθtdr, and Kθtdp stand for the uncertain parameters with the Markov random process θt.The parameterθt is the stochastic Markov process with right continuous trajectories and takes values in a finite set N=1,2,…,N, which is subject to the transition probability matrix S=Oij,∀i,j∈N, and the mode transition probability are defined as follows:(2)Prθt+Δt=j|θt=i=OijΔt+oΔt,i≠j,1+OijΔt+oΔt,i=j,where limΔt⟶0oΔt/Δt=0,Oij≥0i≠j,Oij=−∑j=1,j≠iNOij, and Oij denotes the probabilities from mode i at time t to mode j at time t+Δt.Remark 1.
Note that whenS=I, the proposed SUSVs model will be reduced to the traditional USVs model as introduced in [5, 44]. In other words, the considered model of USV in [5, 44] is a special case of our research.
Denotingxt=υtrtφtptϕtT,ωt=ωφtωϕtT, and Aθt,Bθt,W are given as follows:(3)Aθt=−1Tυ0000KθtυrTr−1Tr00001000ωn2Kθtυp00−2ζωn−ωn200010,Bθt=KθtdυTυKθtdrTr0ωn2Kθtdp0,W=001Tr0000ωn200,one can obtain the following state-space model:(4)x˙t=Aθtxt+Bθtδt+Wωt.
The corresponding structure of the proposed networked stochastic unmanned surface vehicles (NSUSVs) is shown in Figure2. Note that the proposed NSUSVs suffer from actuator failures including bias fault and partial fault, external disturbance, asynchronization problem among original system, controller, and actuator as well as AETM with adaptive threshold.Figure 2
The structure of NSUSVs.


## 2.2. The Mode-Dependent Adaptive Event-Triggered Mechanism

In this paper, the event detector is used to determine whether the present sampling data should be stored and sent out to the controller, and the event is triggered if the following inequality is satisfied:(5)xtkΔ+lΔ−xtkΔTΩθtxtkΔ+lT−xtkΔ>σtxTtkΔ+lΔΩθtxtkΔ+lΔ,where Δ is the sampling period of the sampler and the positive definite mode-dependent matrix Ωθt is the weighting matrix to be designed later. xkΔ is the current sampling state at sampling instant kΔkΔ∈R1≜0,1Δ,2Δ,…, and xtkΔ is the latest transmitted state at the releasing instant tkΔtkΔ∈R2≜t0Δ,t1Δ,t2Δ,…, as well as we assume the first releasing instant t0=0. On the other hand, σt is the event-triggered threshold variable, which meets the following adaptive law concerning event-triggered strategy:(6)σ˙t≜0,σt⟶0+,1σt1σt−ϑeTtΩθtet,otherwise,where 0<σt<1,ϑ>0.et=xtkΔ+lΔ−xtkΔ,∀l=1,2,…,ℓ. The term tkΔ+lΔ denotes the sampling instants from the current transmitted sampling instant tkΔ to the future transmitted sampling instant tk+1Δ. Note that the next releasing instant tk+1Δ=tkΔ+ℓΔ if condition (4) is satisfied.We denote(7)Ξtk=tkΔ+dtk,tk+1Δ+dtk+1=∪l=1ℓΞl,tk,where Ξl,tk=tkΔ+l−1Δ+dtk+l−1,tkΔ+lΔ+dtk+l.Define the network induced delay as(8)dt=t−tkΔ−lΔ,t∈∪l=1ℓΞl,tk.Noting that the network-induced time-delay is represented as the maximum allowable upper communication delay boundd¯ which is similar to [45], that is(9)dm=mindtk≤dt≤1+d¯=dM,∀t∈∪l=1ℓΞl,tk.Finally, the AETM is deduced as follows:(10)eTtΩθtet≤σtxTt−dtΩθtxt−dt.Figure3 gives an example to demonstrate the relationship between the sampling instant and the releasing instant, where the data are sampled discretely at instant kΔkΔ∈R1≜0,1Δ,2Δ,… and the data are chosen at instant tkΔtkΔ∈R2≜t0Δ,t1Δ,t2Δ,… to release into the network. Based on the above analysis, we know that R2⊆R1. Besides, the control input of the proposed NSUSVs is reserved by ZOH during the time interval Ξ1,0=0Δ+d0,3Δ+d3,Ξ2,3=3Δ+d3,5Δ+d5,Ξ3,5=5Δ+d5,t3Δ+4Δ+dt3+4,….Figure 3
An example of timing diagram for AETM.Remark 2.
Note that the mode-dependent AETM proposed in this paper relies on modes information of the system, which can efficiently reduce the loss of data in the network transmission, and it is not taken into account in most of the existing literature; see, for instance, [23, 24, 31, 45, 46].Remark 3.
Note that whenσ˙t=0, the supposed AETM can be degraded into the traditional ones, in which the event-triggered threshold is a preset constant, such as [23, 24, 31, 45] with the following form:(11)eTtΩθtet≤σ0xTt−dtΩθtxt−dt,where σ0∈0,1. And the above scheme is also applicable if σt⟶0+.Remark 4.
It is worth noting that the smallest interevent time can be regarded as the sampling periodΔ, so Zeno behavior will not exist in the related AETM proposed in this article.


## 2.3. Actuator Faults Analysis

The control input used by the NSUSVs with single rudder is described as follows:(12)δt=Θηtut−γt,where Θηt represents the mode-dependent efficiency coefficient of actuator faults. Θηtmin and Θηtmax represent the unknown lower and upper bounds of Θηt, that is, 0≤Θηtmin≤Θηt≤Θηtmax≤1, and ut and γt are the applied control signal to be designed and the unknown time-varying bounded bias fault in the actuator, respectively. In addition, the jumps of actuator are under the control of Markov parameter ηt, which are affected by the mode of system through conditional probability matrix A=Pip,∀i∈N, and p∈L=1,2,…,L. The conditional probability Pip implies the possibility that the actuator runs in mode p given the mode i information of system, i.e.,(13)Prηt=p|θt=i=Pip,and for ∀i∈N,p∈L, there exist constraints Pip∈0,1 and ∑p=1LPip=1.For convenience, the subscriptp will be hereinafter employed to replace this Markov parameter ηt in the actuator, and the following multiple types of actuator faults model can be rewritten as(14)δt=Θput−γt.From Table1, one can see that when Θp=1 and γt=0, the actuator works normally. Both Θp=1 and γt≠0 mean the bias fault has appeared in the actuator. 0<Θp<1 and γt=0 denote the partial fault has existed in the actuator. 0<Θp<1 and γt≠0 represent the actuator that has occurred mixed failures including the partial fault and the bias fault. Regardless of whether γt is equal to 0 or not, the actuator outage fault has occurred in the actuator under Θp=0.Table 1
Fault models.
γt=0γt≠0Θp=1NormalBiasΘp∈0,1PartialPartial and biasΘp=0OutageOutageUsing a similar method as [32], that is, by defining Θp=I+ZpΘ¯p and Zp≤Hp≤I, the conservatism that Θp is not known in advance is moderated, where(15)Θ¯p=Θpmax+Θpmin2,Hp=Θpmax−ΘpminΘpmax+Θpmin,Zp=Θp−Θ¯pΘ¯p,∀p∈L.The following assumptions are necessary to facilitate the subsequent research.Assumption 1.
For state-space model (4), in order to make the disturbance and input signal satisfy the match condition, there are constant matrix Ci with appropriate dimensions such that(16)W=BiCi.Assumption 2.
For the actuator partial faultΘp,∀i∈N,p∈L, one has(17)rankBiΘp=rankBi.Assumption 3.
Define the augmented disturbance as follows:(18)ψt=−Θpγt+Ciωt,where ψ¯≤ψt≤ψ¯ and ψ¯ as well as ψ¯ are unknown constants.
One is defined as follows:(19)Ψ=ψ¯+ψ¯.


## 2.4. Asynchronous Adaptive Fault-Tolerant Controller

LetΘ^pt,Ψ^t,ψ¯^t,ψ¯^t denote the online estimation of Θp,Ψ,ψ¯,ψ¯, respectively.One can define(20)∂t=0,xTtPiBiΘ^ptμ−1t>0,1,xTtPiBiΘ^ptμ−1t≤0,where μt=Θ^pt+ρt and ρt is an exponential function and is defined as follows:(21)ρt=ae−bt,where both a and b are bounded positive bound constants to be designed. It can be figured out directly ∫0∞ρτdτ≤ρ¯<∞.Remark 5.
It is worth noting that the introduction ofρt aims to use variable μ−1t=Θ^pt+ρt−1 substitute for Θ^p−1t in order to compensate the actuator partial fault Θp. The reason for adopting this method is that we should consider the case of Θp=0 and also analyze the outage fault of the actuator (for detail, see Table 1), that is, Θ^pt=0. Hence, Θ^p−1twill be not defined well. So the introduction of parameter ρt is indispensable, which is not considered in [29, 34].
Next, we define the vector-valued function as follows:(22)ϖt=ψ¯^t+∂tψ¯^t−ψ¯^t.
The asynchronous adaptive fault-tolerant controller under AETM is designed as follows:(23)ut=K1ξt+K2ξtxtkΔ+K3ξtϖt,where K1ξt,K2ξt,andK3ξt are the gains of the proposed mode-dependent fault-tolerant controller, which will be designed in the sequel. Markov parameter ξt exerts influence on the variations of the controller among different modes; on the other hand, they are affected by the mode of system through conditional probability matrix ℂ=Qiq for ∀i∈N,q∈M=1,2,…,M. The conditional probability Qiq implies the possibility that the controller runs in mode q given the mode information i of system, i.e.,(24)Prξt=q|θt=i=Qiq,and it is extremely obvious that Qiq∈0,1 and ∑q=1MQiq=1 for ∀i∈N,q∈M.Remark 6.
The implication of mode-dependent in this paper is that the actuator and controller depend indirectly on the NSUSVs modeθt that is subject to transition probability (2) through conditional probabilities (13) and (24).Remark 7.
There exist problems such as packet loss, time delay, and quantization in NSUSVs, which will result in the incomplete information transfer. Therefore, the controller’s modes information which is represented byξt (subject to the conditional probability matrix ℂ) run asynchronously with the NSUSVs’ modes information which is represented by θt (subject to the transition probability matrix S), but the latter directly imposes influence on the former. The asynchronization between the NSUSVs and controller is described by the hidden Markov model θt,ξt,S,ℂ,. Similarly, the hidden Markov model θt,ηt,S,A is made up, which accounts for the asynchronization between the NSUSVs and actuator.Remark 8.
Note that similar to the description of HMM in [14], taking different values in the results will reduce to the following three cases:(1)
Whenℂ=I, the controller modes are completely synchronous with the NSUSVs modes.(2)
Similar with the description in [22], the Markov parameter θt of the system is grouped into several clusters, and in each cluster, the conditional probability Qiq for ξt solely relies on which cluster θt belongs to. One extreme case is that there is only one cluster and then the conditional probability matrix ℂ has identical rows.(3)
Whenξt∈1 and ℂ=1…1T, the mode information θt of system fail to work, in other word, the case is referred to as mode-independent.
For notational brevity again, the subscriptsi,j,q will be hereinafter employed to replace these Markov parameters θt,θt+Δt, and ξt in the system and controller. So, the dynamics of the closed-loop NSUSVs are obtained as follows:(25)x˙t=Aixt+BiΘpK1qxt−dt−BiΘpK1qet+BiΘpK2qxtkΔ+BiΘpK3qϖt+Biψt.
Next, the adaptive laws forΘ^pt is designed as follows:(26)Θ^⋅pt≜ProjminΘp,maxΘpLt=0,ifΘ^p=minΘp,andL≤0orΘ^p=minΘp,andL>0,Lt,otherwise,where Lt=−τΘxTtPiBiμ−1tϖt.
Moreover, the adaptive laws forΨ^⋅t,ψ¯^⋅t,ψ¯^⋅t are designed as follows:(27)Ψ^⋅t=ƛΨxtPiBiρtμ−1t,ψ¯^⋅t=gψ1−∂txTtPiBiΘ^ptμ−1t,ψ¯^⋅t=gψ∂txTtPiBiΘ^ptμ−1t,and ƛΨ,τΘ,gψ are the adaptive law gains to be designed in the sequel.
Besides those parameters,K2q and K3q are designed as follows:(28)K2q=−ρtμ−1t2PiBi2Ψ^2tβxtkΔρtμ−1tPiBiΨ^t+ρtBiTPiT,K3q=−μ−1t.
Our main purpose is to design an asynchronous adaptive fault-tolerant controller under AETM to ensure that uniformly ultimate boundedness of the solutions of the closed-loop NSUSVs (25) can be achieved with adaptive law (6), (26), (27).
Before presenting our results, it is necessary to introduce the following lemmas.Lemma 1 (Schur complement) (see [47]).
Given a symmetric matrix as follows:(29)S=S11S12S12TS22,
and the following statements are equivalent:(1)
S<0(2)
S11<0,S22−S12TS11−1S12<0(3)
S22<0,S11−S12S22−1S12T<0Lemma 2 (see [48]).
Given matricesΦ,Γ, and Λ of appropriate dimensions with Φ being symmetrical, then Φ+ΓFtΛ+ΛTFTtΓT<0 for all Ft satisfying FTtFt≤I, if and only if there exists a scalar α>0 such that Φ+αΓΓT+α−1ΛTΛ<0.Lemma 3 (see [43]).
For any partial fault parameterΘp, the matrix PiBiΘpBiTPiT is invertible, and there exists a positive constant β such that(30)PiBiΘpBiTPiT≥βI.Proof.
Settingr2=β=λminPiBiΘpBiTPiT in Lemma 2 in [43], the proof is obtained.


## 3. Main Results

This section will present a sufficient uniformly ultimate boundedness (for details, see definition 4.6 in [49]) condition for the closed-loop NSUSVs (25).Theorem 1.
For given positive scalarsdM,dm,ϑ, and matrix K1q, if there exist positive definite matrices Ωi,G, a positive definite symmetric matrix Pi,∀i∈N,p∈L,q∈M, such that the following conditions hold:(31)Π=Π11Πipq−Πipq∗Ωi−G0∗∗−ϑΩi<0,where(32)Π11=PiAi+AiTPi+dM−dmG+∑j=1NOijPj,Πipq=∑p=1LPip∑q=1MQiqPiBiΘpK1q.The solutions of (25) are uniformly ultimate bounded with adaptive law (6), (26), (27).Proof.
Define the following error variables:(33)Θ˜pt=Θ^pt−Θp,ψ¯˜t=ψ¯^t−ψ¯,ψ¯˜t=ψ¯^t−ψ¯,Ψ˜t=Ψ^t−Ψ,Ψ=ψ¯+ψ¯.
We introduce the following Lyapunov–Krasovskii functional:(34)Vt=∑ℏ6Vℏt,where(35)V1t=xTtPixt,V2t=∫t−dMt−dm∫t−1+ztxTyGxydydz,V3t=12σ2t,V4t=Θ˜p2tτΘ,V5t=Ψ˜2tƛΨ,V6t=1gψψ¯˜2t+ψ¯˜2t.
Based on the weak infinitesimal operatorA, one obtains the differential of Vt along with the solution of (25).
One has firstly,(36)AV1t=xTtAiTPi+PiAixt+xTt∑j=1NOijPjxt+2xTtPiBiψt+2∑p=1LPip∑q=1MQiqxTtPiBiΘpK1qxt−dt−2∑p=1LPip∑q=1MQiqxTtPiBiΘpK1qet+2∑p=1LPip∑q=1MQiqxTtPiBiΘpK2qxtkΔ+2∑p=1LPip∑q=1MQiqxTtPiBiΘpK3qϖt,where(37)2xTtPiBiψt=2xTtPiBiΘ^ptΘ^pt+ρtψt+2xTtPiBiρtΘ^pt+ρtψt.
Secondly,(38)AV2t=xTkdM−dmGxk−∫−dM+1−dm+1xTt−1+zGxt−1+zdz.We can easily obtain that(39)∫−dM+1−dm+1xTt−1+zGxt−1+zdz=∫t−dMt−dmxTzGxzdz≥xTt−dtGxt−dt,so(40)AV2t≤xTkdM−dmGxk−xTt−dtGxt−dt.
Thirdly, based on the adaptive law (6),(41)AV3t=σtσ˙t=1σteTtΩiet−ϑeTtΩiet≤xTt−dtΩixt−dt−ϑeTtΩiet.
Furthermore, according to (26) and (27), we can obtain(42)AV4t=2Θ˜ptΘ˜⋅ptτΘ=−2Θ˜ptxTtPiBiμ−1ϖt,(43)AV5t=2Ψ˜tΨ˜⋅tƛΨ=2Ψ˜txtPiBiρtμ−1t.
Eventually, one derives(44)AV6t=2ψ¯˜tψ¯˜⋅tgψ+ψ¯˜tψ¯˜⋅tgψ=2ψ¯˜tgψ1−∂txTtPiBiΘ^ptμ−1tgψ+ψ¯˜tgψ∂txTtPiBiΘ^ptμ−1tgψ=2Θ^ptμ−1tψ¯˜t+∂tψ¯˜t−ψ¯˜txTtPiBi=2Θ^ptμ−1tψ¯^t+∂tψ¯^t−ψ¯^txTtPiBi−2Θ^ptμ−1tψ¯t+∂tψ¯t−ψ¯txTtPiBi.
On the one hand, according to Lemma3 and reminding that Ψ=ψ¯+ψ¯, one obtains from (36), (37), and (43)(45)2xTtPiBiρtΘ^pt+ρtψt+2∑p=1LPip∑q=1MQiqxTtPiBiΘpK2qxtkΔ+AV5t=2xTtPiBiρtμ−1tψtβxtkΔρtμ−1tPiBiΨ^t+ρtβxtkΔρtμ−1tPiBiΨ^t+ρt−2∑p=1LPipxTtPiBiΘpρtμ−1t2PiBi2Ψ^2tβxtkΔρtμ−1tPiBiΨ^t+ρtBiTPiTxtkΔ+2Ψ˜txtPiBiρtμ−1tβxtkΔρtμ−1tPiBiΨ^t+ρtβxtkΔρtμ−1tPiBiΨ^t+ρt≤2Ψ^txtPiBiρtμ−1tρtβxtkΔρtμ−1tPiBiΨ^t+ρt=2Ψ^txtkΔPiBiρtμ−1tρtβxtkΔρtμ−1tPiBiΨ^t+ρt≤2β−1ρt.
On the other hand, from (36), (37), and (44), one gets(46)2xTtPiBiΘ^ptΘ^pt+ρtψt+2∑p=1LPip∑q=1MQiqxTtPiBiΘpK3pϖt+AV6t≤2xTtPiBiΘ^ptμ−1tψt−2∑p=1LPip∑q=1MQiqxTtPiBiΘ^ptμ−1tψ¯^t+∂tψ¯^t−ψ¯^t+2∑p=1LPip∑q=1MQiqxTtPiBiΘ˜ptμ−1tψ¯^t+∂tψ¯^t−ψ¯^t+2Θ^ptμ−1tψ¯^t+∂tψ¯^t−ψ¯^txTtPiBi−2Θ^ptμ−1tψ¯t+∂tψ¯t−ψ¯txTtPiBi≤2xTtPiBiΘ^ptμ−1tψ¯t+∂tψ¯t−ψ¯t+2∑p=1LPipxTtPiBiΘ˜ptμ−1tψ¯^t+∂tψ¯^t−ψ¯^t−2∑p=1LPipΘ^ptμ−1tψ¯t+∂tψ¯t−ψ¯txTtPiBi=2∑p=1LPipxTtPiBiΘ˜ptμ−1tψ¯^t+∂tψ¯^t−ψ¯^t.
We introduce notationχt=xTtxTt−dteTtT to facilitate the following analysis. By combining (26) and (33)–(46), we endeavor to derive the following inequality:(47)AVt=A∑ℏ6Vℏt≤χTtΠχt−2Θ˜ptxTtPiBiμ−1ϖt+2β−1ρt+2∑p=1MψipxTtPiBiΘ˜ptμ−1tψ¯^t+∂tψ^¯t−ψ¯^t≤χTtΠχt+2β−1ρt.
Therefore,AVt<0 as long as it meets χt>2β−1ρt/−λminΠ=℘. Hence, AVt is negative definite outside the compact set Σχ≜χ|χt≥℘, which represents that whatever χt leaves the set Σχ,Vt will decrease and χt will still be bounded. Thus, one can conclude on the uniformly ultimate boundedness of the solutions of (25).Remark 9.
In Theorem1, a sufficient condition is derived by introducing Lyapunov–Krasovskii functional (34) to guarantee the uniformly ultimate boundedness of the solutions of NSUSVs (25) with the multiple types of actuator faults and external disturbance. Moreover, the designed controller is based on the releasing state of adaptive event-triggered mechanism (5) and the online estimation of proposed augmented disturbance (18).
Next, we will design the proposed fault-tolerant controller gainK1q for (25). The theorem is provided as follows.Theorem 2.
For given positive scalarsdM,dm,ϑ,ε1,ε2,ε3, and a matrix Yq, if there exist positive definite matrices G¯,Ω¯i,P¯j,ℒ with appropriate dimensions, such that the following LMI condition holds:(48)Φipq=Φ11Φ12∗−Φ22<0,where(49)Φ11=Φ¯11+E1Φ11ipq−Φ11ipq∗Ω¯i−G¯+E20∗∗−ϑΩ¯i+E3,Φ¯11=Aiℒ+ℒAiT+dM−dmG¯+∑j=1NOijP¯j,Φ11ipq=∑p=1LPip∑q=1MQiqBiΘ¯pYq,E1=ε1BiHpHpTBiT,E2=ε2BiHpHpTBiT,E3=ε3BiHpHpTBiT,Φ12=ΛT=0Φ12ipq−Φ12ipq000000,Φ12ipq=∑q=1MQiq∑p=1LPipYqTΘ¯pT,Φ22=diagε1,ε2,ε3.Then,K1q can be parameterized as follows:(50)K1q=Yqℒ−1.Proof.
To begin with, we denote(51)L=Pi−1,Yq=K¯1qℒ,G¯=ℒGℒ,Ω¯i=ℒΩiℒ,P¯j=ℒPjℒ,we denote L=diagℒ,ℒ,ℒ, pre- and postmultiply (31) by L, respectively, and then we get(52)Φ¯ipq=Φ¯11Φ¯12ipq−Φ¯12ipq∗Ω¯i−G¯0∗∗−ϑΩ¯i<0,where Φ¯12ipq=∑p=1LPip∑q=1MQiqBiΘpYq.
BecauseΘp=I+ZpΘ¯p,Zp≤Hp≤I, so the following inequality is true:(53)Φ¯ipq=Φ¯˜ipq+ΓFtΛ+ΛTFtΓT,ΓΓT≤Γ¯Γ¯T.
Thus,(54)Φ¯ipq≤Φ¯˜ipq+HeΓ¯FtΛ,where(55)Φ¯˜ipq=Φ¯11Φ11ipq−Φ11ipq∗Ω¯i−G¯0∗∗−ϑΩ¯i,ΛT=0Φ12ipq−Φ12ipq000000,ΓT=diagZpTBiT,ZpTBiT,ZpTBiT,Γ¯T=diagHpTBiT,HpTBiT,HpTBiT,Ft=I.
In accordance with Lemma2, for FTtFt≤I if there exist positive scalar matrices Φ22=diagε1,ε2,ε3 such that(56)Φ¯˜ipq+Φ22Γ¯Γ¯T+Φ22−1ΛTΛ<0,meets.
Therefore, inequality (54) can be obtained further. Eventually, based on Lemma 1, inequality (48) can be derived from (51)–(56). This completes the proof.Remark 10.
Note that Theorem2 provides a sufficient condition for the mode-dependent adaptive event-triggered matrix Ωθt in the triggering condition (5) and the asynchronous adaptive fault-tolerant controller gain matrix K1ξt in equation (23). If the linear matrix inequalities (LMI) (48) is feasible, both Ωθt and K1ξt can be easily figured out.


## 4. Numerical Example

In this section, the validity and availability of the proposed methods will be elaborated by a numerical example, where system modesθt∈1,2, actuator modes ηt∈1,2, and controller modes ξt∈1,2 are considered, and the relevant parameters are available from [44]:(57)U=5.6ms,Tv=1.8U,Tr=2U,ωn=1.86rads,ζ=0.62+0.46U,K1vr=−0.52,K1vp=0.21U,K1dv=0.01U,K1dr=−0.003U,K1dp=−0.002U2,K2vr=−0.58,K2vp=0.16U,K2dv=0.016U,K2dr=−0.0036U,K2dp=−0.0022U2,where U is the forward speed of the USVs. Then, one educes(58)A1=−3.1110000−1.456−2.8000010004.06800−11.889−3.46000010,B1=0.174−0.0470−0.2170,A2=−3.1110000−1.624−2.8000010003.10000−11.889−3.46000010,B2=0.279−0.0560−0.2390,W=002.800003.46000.The transition probability matrixS, along with conditional probability matrices A and ℂ, is selected as follows:(59)S=−0.750.750.45−0.45,A=0.650.350.50.5,ℂ=0.550.450.60.4.Select the unknown partial failures as0.3≤Θ1≤0.9,0.2≤Θ2≤0.8, and the bias faults γt are as follows:(60)γt=0,0≤t<5,0.1sin0.35t−5,t≥5.In addition to those parameters, the network induced time-delay is chosen randomly from value 0.01 or 0.21, that is, dM−dm=0.2 and ϑ=0.2.By solving the LMI in Theorem2, we can obtain the adaptive fault-tolerant controller gain matrices(61)K11=0.2234−0.20220.4622−0.3122−0.0773,K12=0.2426−0.21950.5018−0.3389−0.0839,and the mode-dependent event-triggered matrices(62)Ω1=7.9307−0.00300−0.01280−0.00307.916000.00250007.915400−0.01280.002507.9259000007.9154,Ω2=7.9195−0.00130−0.00590−0.00137.915800.00170007.915400−0.00590.001707.9235000007.9154.To perform some simulations about performances of NSUSVs, we suppose the initial conditionx0=0.1−0.50.4−0.30.6T and the external interference input imposed on the system is ωt=ωφtωϕtT=0.2sin3t0.4cos4tT. Besides, the adaptive gains in (26) and (27) are set to be τΘ=0.8,gψ=0.5,andƛΨ=0.3, respectively. The parameters a and b are designated as a=1 and b=0.01, so, we achieve ρt=e−0.01t.Based on the aforementioned parameters, we can obtain the simulation results demonstrated in Figures4–11. The possible modes evolution of the NSUSVs, controller, and actuator is depicted in Figure 4, which obviously proves that controller and actuator run asynchronously with the NSUSVs since their variations among different modes are governed by three different parameters ξt,ηt, and θt. The release instants and release intervals of the AETM (5) are displayed in Figure 5, which illustrates the availability and validity of the proposed AETM. Furthermore, the threshold σt of the AETM is described in Figure 6, and as can be seen from Figure 6 that σt eventually converges to 0.043, which corresponds to the case σt⟶0+.Figure 4
Modes evolution of the NSUSVs, controller, and actuator.Figure 5
Release instants and release intervals.Figure 6
The thresholdσt of AETM.Figure 7
The state response of open-loop NSUSVs (25).Figure 8
The state response of closed-loop of NSUSVs (25) by using our method.Figure 9
The state response of closed-loop of USVs by using the proposed method in [44].Figure 10
Actuator partial fault efficiency factors and the estimations.
(a)(b)Figure 11
Actuator bias fault and estimations of its upper and lower bounds.Moreover, the state response of open-loop and closed-loop of NSUSVs (25) is depicted in Figures 7 and 8, respectively. It is not difficult to find that the primary open-loop of NSUSVs is unstable from Figure 7. Comparably, it can be seen from Figure 8 that the state feedback trajectory of the closed-loop of NSUSVs gradually converges to the stable level despite both the partial fault and the bias fault that exist in the actuator when the proposed fault-tolerant controller of Theorem 2 is employed.On the other hand, Figures10 and 11 demonstrate the estimations fault efficiency coefficients of actuator partial fault and the estimations of the upper and lower bounds of actuator bias fault, respectively. More exactly, it can be known from Figure 10 that when t<2 seconds, the actuator is functioning normally, that is, Θ1=Θ2=1. When t=2 seconds, the actuator partial fault begins to occurs, and when t=5 seconds, the faults efficiency factors start to reach equilibrium, that is, Θ1=0.6 and Θ2=0.5. The bias fault of actuator emerges from t=5 seconds, and the tracking performances of actuator bias faults have achieved an ideal performance from Figure 11.It is worth noting that the work [44] addresses the codesign problem of a fault detection filter and controller for a networked-based USVs system subject to external disturbance and actuator fault under a conventional ETM (its limitation has been discussed above). Since the asynchronization problem, multiple types of actuator faults, and the proposed AETM are not taken into consideration in this work, the controller designed by Ma et al. cannot guarantee that our proposed NSUSVs are stable (for detail, see Figure 9); in other words, the effectiveness of the proposed adaptive event-triggered asynchronous fault-tolerant controller for NSUSVs is self-evident.


## 5. Conclusion

In this paper, an adaptive fault-tolerant controller has been designed for the NSUSVs suffering from external disturbance and actuator faults including the partial fault and bias fault. To boost the utilization ratio of network resources, a mode-dependent AETM is carried out. Moreover, the asynchronizations among the controlled system, adaptive fault-tolerant controller, and faulty actuator are taken into consideration, which are described by two corresponding HMMs. Under this framework, a sufficient condition is provided in order to guarantee the uniformly ultimate boundedness of the solutions by Lyapunov–Krasovskii functional. Finally, an example and a comparison are presented to demonstrate the validity and feasibility of the proposed design techniques.In the future, we will pay close attention to the double event-triggered mechanisms, in the sensor-to-controller and controller-to-actuator channels, where both the communication resources consumption in the network and control updates can be reduced. It broadens the trade-off option between NSUSV dynamic performance and network resource utilization.



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