# Decentralized Adaptive Fault-Tolerant Cooperative Control for Multiple UAVs with Input Saturation and State Constraints

Authors: Minrui Fu; Yiwei Xu; Ziquan Yu; Youmin Zhang
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

This paper proposes a fault-tolerant cooperative control (FTCC) scheme for multiple UAVs in a distributed communication network against input saturation, full-state constraints, actuator faults, and unknown dynamics. Firstly, by considering physical limitations, an auxiliary control signal is designed to simplify the analysis process. Secondly, to avoid the difficulty in the back-stepping design caused by full-state constraints, virtual control signals are constructed to transform constrained variables, while the dynamic surface control is adopted to avoid the phenomenon of “differential explosion.” Thirdly, a disturbance observer (DO) is designed to estimate the unknown uncertainty caused by parameter uncertainty and actuator fault. Moreover, a recurrent wavelet fuzzy neural network (RWFNN) is used to compensate for the estimation errors of DO. Finally, it is proved that all states are uniformly ultimately bounded (UUB) by Lyapunov and invariant set theory. The effectiveness of the proposed scheme is further demonstrated by the simulation results.


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## Body

## 1. Introduction

In recent years, the development of unmanned aerial vehicle (UAV) technology has led to wide applications. However, single UAV provides limited capabilities, which may not be applicable to some highly complex tasks. Inspired by multiagent technology, researchers begin to investigate the application technology of multiple UAVs (multi-UAVs). Compared with a single UAV, multi-UAVs have more benefits in terms of forest fire monitoring, area detection, and disaster assistance. Unlike a single UAV, the cooperative control of multi-UAVs need the information from neighboring UAVs, which significantly increasing the control design challenge.As the basis of multi-UAVs control, the cooperative control design is an important task. In the past few years, numerous research results of cooperative control have been reported. In [1], a cooperative control strategy for motion control of multiple unmanned vehicles was proposed, which can keep the formation during the motion. In [2], a novel distributed intermittent control framework for containment control of multiagent system was proposed, which can reduce the communication burden via a directed graph. In [3], the obstacle avoidance problem of multi-UAVs in multiple obstacle environment was studied. In [4], a robust adaptive control strategy for cooperative control of UAVs under the decentralized communication network was proposed against uncertainty. In [5], the authors investigated the cooperative transport control problem using multirotor UAVs. In [6], a system analysis method was proposed for the tracking control problem of multi-UAVs. The distributed framework was used to describe the dynamic model of UAV, and the information of nodes and networks were considered in the distributed control design. [7] studied the output feedback formation control of multi-UAVs without velocity and angular velocity sensors, which were obtained via the state observer. However, the above researches only focused on the distributed control of the first-order or second-order systems, and there exist few research on the cooperative control of fixed-wing UAVs with high-order nonlinear characteristics.In addition, the number of components in the multi-UAV system is more than that of a single UAV. Therefore, the probability of multi-UAVs suffering from the actuator, sensor, or component faults is higher than that of a single UAV. At present, many research results show that the incidence of actuator fault in flight is the highest. Therefore, many scholars mainly focus on actuator faults [8, 9]. The probability of actuator fault occurred in the multi-UAVs will also increase due to the fact that the number of actuators is significantly increased. When an actuator fault of a UAV in the communication network occurs and is not handled in time, it will reduce the stability and threaten the safety of all UAVs [10], making the investigation on fault-tolerant cooperative control (FTCC) a necessary task. In [11], based on the design of inner-outer-loop control and back-stepping control, an FTCC strategy was designed for multi-UAVs against permanent faults. In [12], Yu et al. further studied the FTCC design method of multi-UAVs by using a similar control framework of reference generator technology. It should be emphasized that the results [11, 12] are about the FTCC scheme of multi-UAVs under the master-slave framework, which cannot be directly applied to the distributed cooperative control design. Considering the diversity of research on cooperative control of multi-UAVs in the communication network, the FTCC scheme for multi-UAVs under distributed communication network needs to be further investigated.Moreover, the actual multi-UAV system often encounters some problems such as input saturation, inaccurate aerodynamic parameters, and external interference, which lead to system instability or performance degradation [13–16]. Recently, many results have been reported to solve the problem of input saturation. In [17], a piecewise auxiliary system was introduced to deal with the asymmetric input constraints for a class of uncertain multi-input and multi-output nonlinear systems. Then, the auxiliary system was further used to deal with the force and moment constraints on ship [18]. In [19], another auxiliary signal was constructed using the error between the desired control input and the saturation control input. In [20], to solve the disadvantages of conventional methods based on the hyperbolic tangent function, an nth-order auxiliary dynamic system was skillfully constructed to avoid the effect of input saturation.It should be emphasized that although numerous studies have been reported on the above literature, few results have studied the input saturation, inaccurate aerodynamic parameters, and external interference encountered by multi-UAVs in distributed communication networks at the same time. However, such factors are inevitable in the formation flight of multi-UAVs. If these factors are not solved in time, it may lead to the instability of the networked UAV system.Furthermore, due to physical limitations, UAV states should be constrained. However, control strategies developed recently for multi-UAVs in the distributed communication network rarely consider these constraints on the states. Based on the above discussion, this paper proposes a distributed FTCC scheme for multi-UAVs under the distributed communication network with input saturation, state constraints, actuator faults, and unknown dynamics. In this work, to avoid the difficulty in designing the control policy due to the input saturation, an auxiliary control signal is designed to transform the restricted input. To handle the full-state constraint problem, virtual control signals are defined to replace the constraints, which can simplify the back-stepping design. For the unknown nonlinear dynamics caused by actuator faults and other unknown uncertainties, disturbance observer (DO) is designed for providing the estimation, while a recurrent wavelet fuzzy neural network (RWFNN) is adopted to further compensate the estimation error. In the RWFNN, the online adaptive learning strategy of parameters is designed based on the Lyapunov theory. Compared with other existing works, the main contributions of this paper are as follows:(1)
In [21–23], actuator faults, input saturation, output constraints, and external disturbances were considered, while the state constraints were not taken into account. To obtain satisfactory control performance against such factors, the FTCC scheme is designed in this paper by simultaneously considering the state constraints, actuator faults, and external disturbances.(2)
Compared with [24–27], which assessed uncertainty dynamics by designing a DO without compensation of the DO estimation error, this work further adopts an RWFNN to offset the estimation error, in which the parameters are updated by the proposed online learning strategy.The organization of this paper is arranged as follows. Section2 describes the preliminaries and problem statement. The design process of the FTCC scheme and the stability analysis are given in Section 3. Section 4 shows the simulation results and analysis. Finally, the conclusion is drawn in Section 5.


## 2. Preliminaries and Problem Statement

### 2.1. System Dynamics

In this paper, the cooperative control ofN UAVs is investigated. The set of UAVs is denoted as Ω=1,2,⋯,N, and the position dynamics of the ith UAV is described as
(1)ẋi=Vicosγicosχi,ẏi=Vicosγisinχi,żi=Visinγi,where i∈Ω, xi, yi, and zi are the positions. Vi, γi, and χi are velocity, flight path angle, and heading angle, respectively.The aerodynamic force equations are given by(2)V̇i=1mi−Di+Ticosαicosβi−gsinγi,χ̇i=1miVicosγiLisinμi+Yicosμi+Tisinαisinμi−cosαisinβicosμi,γ̇i=1miViLicosμi−Yisinμi−gcosγiVi+TimiVicosαisinβisinμi+sinαicosμi,where i∈Ω, mi and g are the mass of the ith UAV and gravitational coefficient, respectively; Ti, Di, Li, Yi are the thrust, drag, lift, and lateral forces, respectively, and μi, αi, and βi are the bank angle, angle of attack, and sideslip angle, respectively.The attitude kinematic model is expressed as(3)α̇i=qi−tanβipicosαi+risinαi,β̇i=pisinαi−ricosαi+χ̇icosγi·cosμi−γ̇isinμi,μ̇i=picosαi+risinαicosβi+γ̇i·cosμitanhβi+χ̇isinγi+cosγisinμitanβi,where i∈Ω. pi, qi, and ri are the angular rates.The angular rate model is given as(4)ṗi=ci1ri+ci2piqi+ci3Li+ci4Ni,q̇i=ci5piri−ci6pi2−ri2+ci7Mi,ṙi=ci8pi−ci2riqi+ci4Li+ci9Ni,where i∈Ω. Li, Mi, and Ni are roll, pitch, and yaw moments, respectively.The forcesTi, Di, Li, and Yi and the aerodynamic moments Li, Mi, and Ni are expressed as
(5)Ti=TimaxδTi,Di=QisiCiD,Li=QisiCiL,Yi=QisiCiY,Li=QisibiCiL,Mi=QisiciCiM,Ni=QisibiCiN,where Qi=ρVi2/2 is the dynamic pressure and si, bi, and ci represent the wing area, wing span, and mean aerodynamic chord, respectively. Timax and δTi are the maximum thrust and instantaneous thrust throttle setting, respectively. CiL, CiD, CiY, CiL, CiM, and CiN are given by
(6)CiL=CiL0+CiLααi,CiD=CiD0+CiDααi+CiDα2αi2,CiY=CiY0+CiYββi,(7)CiL=CiL0+CiLββi+CiLδaδia+CiLδrδir+CiLpbipi+CiLrbiri2Vi,CiM=CiM0+CiMααi+CiMδeδie+CiMqciqi2Vi,CiN=CiN0+CiNββi+CiNδiaδia+CiNδrδir+CiNpbipi2Vi+CiNrbiri2Vi,where δia, δie, and δir are aileron, elevator, and rudder deflections, respectively. CiL0, CiLα, CiD0, CiDα, CiDα2, CiY0, CiYβ, CiL0, CiLβ, CiLδa, CiLδr, CiLp, CiLr, CiM0, CiMα, CiMδe, CiMq, CiN0, CiNβ, CiNδa, CiNδr, CiNp, and CiNr are aerodynamic derivatives. The definition of the inertial terms cijj=1,2,⋯,9 in (4) can be found in [28].


### 2.2. Control-Oriented Model

By definingXi1=μi,αi,βiT, Xi2=pi,qi,riT, and Ui=δ1a,δ1e,δ1rT and substituting (5), (6), and (7) into (1), (2), (3), and (4), then it follows that
(8)Ẋi1=Fi1+Gi1Xi2(9)Ẋi2=Fi2+Gi2Uiwhere Fi1, Gi1, Fi2, and Gi2 are given by
(10)Fi1=0sinγi+cosγisinμitanβicosμitanβi0−cosγisinμicosβi−cosμicosβi0cosγicosμi−sinμi−Di+Ticosαicosβimi−gsinγi1miVicosγiLisinμi+Yicosμi+Tisinαisinμi−cosαisinβicosμi−gcosγiVi+1miViLicosμi−Yisinμi+Ticosαisinβisinμi+sinαicosμi,Gi1=cosαicosβi0sinαicosβi−cosαitanβi1−sinαitanβisinαi0−cosαi,Fi2=Fi21,Fi22,Fi23TFi21=ci1qiri+ci2piqi+ci3q¯isibiCiL0+CiLβi+CiLpbipi2Vi+CiLrbiri2ViFi22=ci5piri−ci6pi2−ri2+ci7q¯isiciCiM0+CiMααi+CiMqciqi2ViFi23=ci8piqi−ci2qiri+ci4q¯isibiCiL0+CiLβ·βi+CiLpbipi2Vi+CiLrbiri2Vi+ci9q¯isibi·CiN0+CiNβi+CiNpbipi2Vi+CiNrbiri2ViGi2=a110a130a220a310a33,a11=ci3q¯isibiCiLδa+ci4q¯isibiCiNδa,a13=ci3q¯isibiCiLδr+ci4q¯isibiCiNδr,a22=ci7q¯isiciCiMδe,a31=ci4q¯isibiCiLδa+ci9q¯isibiCiNδa,a33=ci4q¯isibiCiLδr+ci9q¯isibiCiNδr.It can be seen thatFi2 and Gi2 have many aerodynamic parameters. However, it is difficult to obtain accurate aerodynamic parameters of UAVs in practical engineering applications. To facilitate the design of the controller, Fi2 and Gi2 can be decomposed into known items Fi20, Gi20 and unknown items ΔFi2, ΔGi2, respectively.Then, the attitude model can be described as(11)Ẋi1=Fi1+Gi1Xi2,(12)Ẋi2=Fi20+ΔFi2+Gi20+ΔGi2Ui,where ΔFi2, and ΔGi2 are unknown nonlinear functions caused by the uncertain parameters, while Fi1, Gi1, Fi20, and Gi20 are known functions.Remark 1.
Due to the physical constraints, the sideslip angleβi≠±π/2, and detGi1=−secβi, so Gi1 is invertible. In addition, Gi2 is related to aerodynamic parameters so that it is invertible in the flight envelope.


### 2.3. Actuator Fault and Input Saturation

In this paper, the actuator fault is considered, which includes gain and bias failures. Therefore, the fault model can be expressed as [29]
(13)Ui=ρiUi0+Uif,where i∈Ω, Ui0=ui01,ui02,ui03T represents the designed control signal, and Ui=δia,δie,δirT is the actual control signal. ρi=diagρi1,ρi2,ρi3 with 0<ρi1,ρi2,ρi3≤1 represents the gain fault matrix, and Uif∈ℝ3 represents bias fault vector.In the practical application, the output of the actuator is limited. In order to avoid the incredible phenomenon caused by actuator saturation, the designed control signalUi0 needs to satisfy the following constraint:
(14)ui0τmin≤ui0τ≤ui0τmax,τ=1,2,3,where τ=1,2,3, ui0τmax is a positive constant and ui0τmin is a negative constant, which is the maximum and minimum allowable values for the actuator, respectively.To solve input saturation problem, an auxiliary signalvi=vi1,vi2,vi3T is used to get control signal Ui0vi, and Ui0vi=ui01vi1,ui02vi2,ui03vi3T, which is expressed as
(15)ui0τviτ=ui0τmaxtanhviτui0τmax,viτ≥0,ui0τmintanhviτui0τmin,viτ<0.By substituting (13) and (15) into (12), then the attitude model can be expressed as
(16)Ẋi1=Fi1+Gi1Xi2,(17)Ẋi2=Fi20+Gi20Ui0vi+Di,where Di=ΔFi2+Gi20ρi−I3Ui0+Uif+ΔGi2Ui is an unknown nonlinear function. Due to Di being related to auxiliary control signal vi, designing the observer of unknown function Di for generating the control signal vi will cause the problem of “algebraic ring,” which is solved by introducing the following first-order filter:
(18)v̇i=−Λvi+ξ,where Λ is a diagnonal matrix with positive eigenvalue and ξ is an auxiliary control signal.Remark 2.
As shown in (13), the fault that occurs in the actuator will diminish its ability to provide control input. For example, the range of motion of the rudder surface can reach −25~25 deg in the normal state, while it may deteriorate into −20~20 deg after the fault occurs. It seems in the fault conditions the actuator is more likely to occur saturation, i.e., cannot reach to the expected control value. In this paper, the upper and lower boundaries of the control input are fixed to the values in the normal state of the actuator, and a hyperbolic function is used to prevent actuator saturation as shown in (15). Meanwhile, using virtual control signal vi and ξ to generate the expected control signal Ui0 and using RWFNN to evaluate uncertainty item Di which contains the actuator bias section, even though the fault could occur, the system can still keep stable.


### 2.4. State Constraints

The statesXi1=μi,αi,βiT and Xi2=p,q,rT generally have constraints in the practical application. In this paper, such a problem has been concerned. Due to the fact that states Xi1 and Xi2 have limits, inspired by works [30, 31], a transformation is used to convert the restricted states Xi1 and Xi2 to unrestricted states Zi1 and Zi2:
(19)Zi1τ=lnXi1τ−X¯i1τX¯i1τ−Xi1τ,(20)Zi2τ=lnXi2τ−X¯i2τX¯i2τ−Xi2τ,where τ=1,2,3. Xi1τ, Xi2τ, Zi1τ and Zi2τ represent the τth element of Xi1, Xi2, Zi1, and Zi2, respectively. X¯i1τ and X¯i2τ represent maximum allowable range of the τth element of Xi1 and Xi2, respectively, while the X¯i1τ and X¯i2τ represent minimum allowable range of the τth element of Xi1 and Xi2, respectively.Remark 3.
Since (19) and (20) are bijective, Xi1 and Xi2 will always stay in their own limits if Zi1 and Zi2 are bounded on ∀t≥0.


### 2.5. Basic Graph Theory

In this paper, an undirected graphG=V,C,A is used to describe the formation flight of N UAVs. The set of UAVs is described by V=v1,v2,⋯,vN, C⊆V×V represents the communication links between UAVs, and A∈RN×N is the adjacency matrix. If the link between the ith UAV and jth UAV exists, aij=aji=1, which are the elements of A. A path from the ith UAV to the kth UAV can be a sequence vi⟶vj⟶vk, where vi,vj,vj,vk∈C. If there exists a path between any two UAVs, then G is a connected graph. A set is defined as Ni=j:vi,vj∈C,i≠j, and the degree matrix is defined as D=diagdi∈ℝN×N, where di=∑j∈Niaij. The Laplacian matrix L∈ℝN×N is denoted as
(21)L=D−A.Assumption 4.
The undirected graphG containing N UAVs is connected.Lemma 5.
Under Assumption4, L and L+diagΩ are symmetric and positive definite [32].


### 2.6. Control Objective

In this paper, the control objective is to design an FTCC scheme forN UAVs, such that the attitude tracking error of each UAVs can be finally uniformly bounded, while the attitude Xi1 and Xi2 of all UAVs are always in limits, even when a portion of UAVs is subjected to actuator saturation and actuator faults.


## 2.1. System Dynamics

In this paper, the cooperative control ofN UAVs is investigated. The set of UAVs is denoted as Ω=1,2,⋯,N, and the position dynamics of the ith UAV is described as
(1)ẋi=Vicosγicosχi,ẏi=Vicosγisinχi,żi=Visinγi,where i∈Ω, xi, yi, and zi are the positions. Vi, γi, and χi are velocity, flight path angle, and heading angle, respectively.The aerodynamic force equations are given by(2)V̇i=1mi−Di+Ticosαicosβi−gsinγi,χ̇i=1miVicosγiLisinμi+Yicosμi+Tisinαisinμi−cosαisinβicosμi,γ̇i=1miViLicosμi−Yisinμi−gcosγiVi+TimiVicosαisinβisinμi+sinαicosμi,where i∈Ω, mi and g are the mass of the ith UAV and gravitational coefficient, respectively; Ti, Di, Li, Yi are the thrust, drag, lift, and lateral forces, respectively, and μi, αi, and βi are the bank angle, angle of attack, and sideslip angle, respectively.The attitude kinematic model is expressed as(3)α̇i=qi−tanβipicosαi+risinαi,β̇i=pisinαi−ricosαi+χ̇icosγi·cosμi−γ̇isinμi,μ̇i=picosαi+risinαicosβi+γ̇i·cosμitanhβi+χ̇isinγi+cosγisinμitanβi,where i∈Ω. pi, qi, and ri are the angular rates.The angular rate model is given as(4)ṗi=ci1ri+ci2piqi+ci3Li+ci4Ni,q̇i=ci5piri−ci6pi2−ri2+ci7Mi,ṙi=ci8pi−ci2riqi+ci4Li+ci9Ni,where i∈Ω. Li, Mi, and Ni are roll, pitch, and yaw moments, respectively.The forcesTi, Di, Li, and Yi and the aerodynamic moments Li, Mi, and Ni are expressed as
(5)Ti=TimaxδTi,Di=QisiCiD,Li=QisiCiL,Yi=QisiCiY,Li=QisibiCiL,Mi=QisiciCiM,Ni=QisibiCiN,where Qi=ρVi2/2 is the dynamic pressure and si, bi, and ci represent the wing area, wing span, and mean aerodynamic chord, respectively. Timax and δTi are the maximum thrust and instantaneous thrust throttle setting, respectively. CiL, CiD, CiY, CiL, CiM, and CiN are given by
(6)CiL=CiL0+CiLααi,CiD=CiD0+CiDααi+CiDα2αi2,CiY=CiY0+CiYββi,(7)CiL=CiL0+CiLββi+CiLδaδia+CiLδrδir+CiLpbipi+CiLrbiri2Vi,CiM=CiM0+CiMααi+CiMδeδie+CiMqciqi2Vi,CiN=CiN0+CiNββi+CiNδiaδia+CiNδrδir+CiNpbipi2Vi+CiNrbiri2Vi,where δia, δie, and δir are aileron, elevator, and rudder deflections, respectively. CiL0, CiLα, CiD0, CiDα, CiDα2, CiY0, CiYβ, CiL0, CiLβ, CiLδa, CiLδr, CiLp, CiLr, CiM0, CiMα, CiMδe, CiMq, CiN0, CiNβ, CiNδa, CiNδr, CiNp, and CiNr are aerodynamic derivatives. The definition of the inertial terms cijj=1,2,⋯,9 in (4) can be found in [28].


## 2.2. Control-Oriented Model

By definingXi1=μi,αi,βiT, Xi2=pi,qi,riT, and Ui=δ1a,δ1e,δ1rT and substituting (5), (6), and (7) into (1), (2), (3), and (4), then it follows that
(8)Ẋi1=Fi1+Gi1Xi2(9)Ẋi2=Fi2+Gi2Uiwhere Fi1, Gi1, Fi2, and Gi2 are given by
(10)Fi1=0sinγi+cosγisinμitanβicosμitanβi0−cosγisinμicosβi−cosμicosβi0cosγicosμi−sinμi−Di+Ticosαicosβimi−gsinγi1miVicosγiLisinμi+Yicosμi+Tisinαisinμi−cosαisinβicosμi−gcosγiVi+1miViLicosμi−Yisinμi+Ticosαisinβisinμi+sinαicosμi,Gi1=cosαicosβi0sinαicosβi−cosαitanβi1−sinαitanβisinαi0−cosαi,Fi2=Fi21,Fi22,Fi23TFi21=ci1qiri+ci2piqi+ci3q¯isibiCiL0+CiLβi+CiLpbipi2Vi+CiLrbiri2ViFi22=ci5piri−ci6pi2−ri2+ci7q¯isiciCiM0+CiMααi+CiMqciqi2ViFi23=ci8piqi−ci2qiri+ci4q¯isibiCiL0+CiLβ·βi+CiLpbipi2Vi+CiLrbiri2Vi+ci9q¯isibi·CiN0+CiNβi+CiNpbipi2Vi+CiNrbiri2ViGi2=a110a130a220a310a33,a11=ci3q¯isibiCiLδa+ci4q¯isibiCiNδa,a13=ci3q¯isibiCiLδr+ci4q¯isibiCiNδr,a22=ci7q¯isiciCiMδe,a31=ci4q¯isibiCiLδa+ci9q¯isibiCiNδa,a33=ci4q¯isibiCiLδr+ci9q¯isibiCiNδr.It can be seen thatFi2 and Gi2 have many aerodynamic parameters. However, it is difficult to obtain accurate aerodynamic parameters of UAVs in practical engineering applications. To facilitate the design of the controller, Fi2 and Gi2 can be decomposed into known items Fi20, Gi20 and unknown items ΔFi2, ΔGi2, respectively.Then, the attitude model can be described as(11)Ẋi1=Fi1+Gi1Xi2,(12)Ẋi2=Fi20+ΔFi2+Gi20+ΔGi2Ui,where ΔFi2, and ΔGi2 are unknown nonlinear functions caused by the uncertain parameters, while Fi1, Gi1, Fi20, and Gi20 are known functions.Remark 1.
Due to the physical constraints, the sideslip angleβi≠±π/2, and detGi1=−secβi, so Gi1 is invertible. In addition, Gi2 is related to aerodynamic parameters so that it is invertible in the flight envelope.


## 2.3. Actuator Fault and Input Saturation

In this paper, the actuator fault is considered, which includes gain and bias failures. Therefore, the fault model can be expressed as [29]
(13)Ui=ρiUi0+Uif,where i∈Ω, Ui0=ui01,ui02,ui03T represents the designed control signal, and Ui=δia,δie,δirT is the actual control signal. ρi=diagρi1,ρi2,ρi3 with 0<ρi1,ρi2,ρi3≤1 represents the gain fault matrix, and Uif∈ℝ3 represents bias fault vector.In the practical application, the output of the actuator is limited. In order to avoid the incredible phenomenon caused by actuator saturation, the designed control signalUi0 needs to satisfy the following constraint:
(14)ui0τmin≤ui0τ≤ui0τmax,τ=1,2,3,where τ=1,2,3, ui0τmax is a positive constant and ui0τmin is a negative constant, which is the maximum and minimum allowable values for the actuator, respectively.To solve input saturation problem, an auxiliary signalvi=vi1,vi2,vi3T is used to get control signal Ui0vi, and Ui0vi=ui01vi1,ui02vi2,ui03vi3T, which is expressed as
(15)ui0τviτ=ui0τmaxtanhviτui0τmax,viτ≥0,ui0τmintanhviτui0τmin,viτ<0.By substituting (13) and (15) into (12), then the attitude model can be expressed as
(16)Ẋi1=Fi1+Gi1Xi2,(17)Ẋi2=Fi20+Gi20Ui0vi+Di,where Di=ΔFi2+Gi20ρi−I3Ui0+Uif+ΔGi2Ui is an unknown nonlinear function. Due to Di being related to auxiliary control signal vi, designing the observer of unknown function Di for generating the control signal vi will cause the problem of “algebraic ring,” which is solved by introducing the following first-order filter:
(18)v̇i=−Λvi+ξ,where Λ is a diagnonal matrix with positive eigenvalue and ξ is an auxiliary control signal.Remark 2.
As shown in (13), the fault that occurs in the actuator will diminish its ability to provide control input. For example, the range of motion of the rudder surface can reach −25~25 deg in the normal state, while it may deteriorate into −20~20 deg after the fault occurs. It seems in the fault conditions the actuator is more likely to occur saturation, i.e., cannot reach to the expected control value. In this paper, the upper and lower boundaries of the control input are fixed to the values in the normal state of the actuator, and a hyperbolic function is used to prevent actuator saturation as shown in (15). Meanwhile, using virtual control signal vi and ξ to generate the expected control signal Ui0 and using RWFNN to evaluate uncertainty item Di which contains the actuator bias section, even though the fault could occur, the system can still keep stable.


## 2.4. State Constraints

The statesXi1=μi,αi,βiT and Xi2=p,q,rT generally have constraints in the practical application. In this paper, such a problem has been concerned. Due to the fact that states Xi1 and Xi2 have limits, inspired by works [30, 31], a transformation is used to convert the restricted states Xi1 and Xi2 to unrestricted states Zi1 and Zi2:
(19)Zi1τ=lnXi1τ−X¯i1τX¯i1τ−Xi1τ,(20)Zi2τ=lnXi2τ−X¯i2τX¯i2τ−Xi2τ,where τ=1,2,3. Xi1τ, Xi2τ, Zi1τ and Zi2τ represent the τth element of Xi1, Xi2, Zi1, and Zi2, respectively. X¯i1τ and X¯i2τ represent maximum allowable range of the τth element of Xi1 and Xi2, respectively, while the X¯i1τ and X¯i2τ represent minimum allowable range of the τth element of Xi1 and Xi2, respectively.Remark 3.
Since (19) and (20) are bijective, Xi1 and Xi2 will always stay in their own limits if Zi1 and Zi2 are bounded on ∀t≥0.


## 2.5. Basic Graph Theory

In this paper, an undirected graphG=V,C,A is used to describe the formation flight of N UAVs. The set of UAVs is described by V=v1,v2,⋯,vN, C⊆V×V represents the communication links between UAVs, and A∈RN×N is the adjacency matrix. If the link between the ith UAV and jth UAV exists, aij=aji=1, which are the elements of A. A path from the ith UAV to the kth UAV can be a sequence vi⟶vj⟶vk, where vi,vj,vj,vk∈C. If there exists a path between any two UAVs, then G is a connected graph. A set is defined as Ni=j:vi,vj∈C,i≠j, and the degree matrix is defined as D=diagdi∈ℝN×N, where di=∑j∈Niaij. The Laplacian matrix L∈ℝN×N is denoted as
(21)L=D−A.Assumption 4.
The undirected graphG containing N UAVs is connected.Lemma 5.
Under Assumption4, L and L+diagΩ are symmetric and positive definite [32].


## 2.6. Control Objective

In this paper, the control objective is to design an FTCC scheme forN UAVs, such that the attitude tracking error of each UAVs can be finally uniformly bounded, while the attitude Xi1 and Xi2 of all UAVs are always in limits, even when a portion of UAVs is subjected to actuator saturation and actuator faults.


## 3. Fault-Tolerant Cooperative Controller Design and Stability Analysis

In this section, the process of designing the FTCC scheme forN UAVs will be described. A main method adopted in the design is transforming the individual tracking error of each UAV to the synchronization tracking error.


### 3.1. Fault-Tolerant Cooperative Controller Design

Define the independent tracking error ofith UAV as Z~i1=Zi1−Zi1d, then the cooperative tracking error of ith UAV is defined as
(22)Ei1=λ1Z~i1+λ2∑j∈NiaijZ~i1−Z~j1,where Ei1=Ei11,Ei12,Ei13T, λ1 and λ2 are positive parameters, which are used to regulate the cooperative tracking performance.Using the Kronecker product “⊗”, and define E1=E11T,E21T,⋯,EN1TT, Z~1=Z~11T,Z~21T,⋯,Z~N1TT, then the cooperative tracking error of all UAVs can be expressed as
(23)E1=λ1IN+λ2L⊗I3Z~1.By recalling Lemma5, it yields Z~1=λ1+λ2L−1⊗I3E1≤1/σminλ1+λ2LE1, where σmin· represents the minimum singular value of matrix “·.” Therefore, Z~1⟶0 if E1⟶0.Using (22), the synchronization error of each UAV Ei1 can be expressed as
(24)Ei1=λ1+λ2∑j∈Nij≠iaijZ~i1−λ2∑j∈Nij≠iaijZ~j1.Differentiating (24) yields
(25)Ėi1=Aigi1Xi1Ẋi1−gi1Xi1dẊi1d−λ2∑j∈Nij≠iaijZ~̇j1=Aigi1Xi1Ẋi1−gi1Xi1dẊi1d−λ2∑j∈Nij≠iaijgi1Xj1Ẋj1−gi1Xj1dẊj1d,where Ai=λ1+λ2∑j∈Nij≠iaij, gi1x is ℝ3⟶ℝ3×3, x=x1,x2,x3T, which is expressed as
(26)gi1x=diaggi1τxτ,τ=1,2,3,gi1τxτ=X¯i1τ−X¯i1τxτ−X¯i1τX¯i1τ−xτ.Substituting (11) into (25) yields
(27)Ėi1=Aigi1Xi1Fi1+Gi1Xi2−Aigi1Xi1dẊi1d−λ2∑j∈Nij≠iaijZ~̇j1.Based on the back-stepping control architecture, (27) can be expressed as
(28)Ėi1=Aigi1Xi1Fi1−Aigi1Xi1dẊi1d−λ2∑j∈Nij≠iaijZ~̇j1+Aigi1Xi1Gi1·Xi2−Zi2+Ei2+Zi2d,where Ei2=Zi2−Zi2d and Zi2d is a virtual control signal.By using a low-pass filter, one has(29)Żi2d=−k∈1Zi2d−Z¯i2d,wherekϵ1 is a positive constant and Z¯i2d is an auxiliary signal, designed as
(30)Z¯i2d=Aigi1Xi1Gi1−1λ2∑j∈Nij≠iaijZ~̇j1−Aigi1Xi1Fi1+Gi1Xi2−Zi2+Aigi1Xi1dẊi1d−Ki1Ei1,where Ki1 is a positive diagonal matrix.Defining the filtering error asϵi1=Zi2d−Z¯i2d, one can obtain
(31)ϵ̇i1=−k∈i1ϵi1−Z¯̇i2d.By substituting (30) into (28), one can obtain
(32)Ei1TĖi1≤−Ei1TKi1Ei1+Ei1TAigi1Xi1Gi1·Ei2+12hi1Ei1TAigi1Xi1Gi12+hi12ϵi12,where hi1 is a positive constant and · represents 2-norm of vector.To estimate the unknown functionDi of each UAV, the following DO is designed for the ith UAV:
(33)D^i=k2Xi2−X^i2+k1k2∫Xi2−X^i2dt,(34)X^̇i2=Fi20+Gi20Ui0vi+D^i+k1Xi2−X^i2,where k1 and k2 are positive parameters, D^i is the estimate of Di.DefineeXi2=Xi2−X^i2 and D~i=Di−D^i, one can obtain
(35)ėXi2+k1eXi2=D~i.Taking the derivative of (35) and using (33) give
(36)D~̇i+k2D~i=D.̇From (36), it can be known that the estimation error D~i will not converge to zero since Ḋ≠0. In order to estimate the unknown function Di more accurate, a five-layer RWFNN is used to estimate the error D~i of the DO with defining Δi=D~i.The RWFNN structure is illustrated in Figure1, including five layers (input layer, membership layer, rule layer, composite layer, and output layer) [33]. The components of the RWFNN are introduced as follows:Figure 1
The structure of the RWFNN for each UAV.Layer 1–Input Layer: Input layer is the first layer, where ri=ri1,ri2,⋯,riv1T is the input features of RWFNN. The output of layer 1 is expressed as
(37)yij1=rij,where j∈1,2,⋯,v1, v1 is the dimension of input features, and yij1 represents the output of jth neuron of Layer 1.Layer 2–Membership Layer: Layer 2 has v1 rows and v2 columns, and its output can be described as
(38)yijk2=e−yij1−cijk2/σijk2,where j∈1,2,⋯,v1, k∈1,2,⋯,v2. v2 is a positive constant, depending on the number of neurons. yijk2 denotes the neuron of layer 2 in row j, column k.Layer 3–Rule Layer: Layer 3 has v2 neurons, and the output of layer 3 is described as
(39)yik3=∏j=1v1yijk2,where k∈1,2,⋯,v2, and yik3 denotes the kth neuron of Layer 3.Layer 4–Composite Layer: Layer 4 also has v2 neurons, and the input of Layer 4 consists the output of the wavelet layer, recurrent layer, and Layer 2, where the output of wavelet layer is described as
(40)ψik=∑j=1v1wijkFϕijkrij,ϕijkrij=1bijk1−rij−aijk2bijk2e−rij−aijk2/2bijk2,where j∈1,2,⋯,v1, k∈1,2,⋯,v2, ϕijk is the output of the jth neuron of Layer 1 to kth neuron of wavelet layer. ψik represents the output of kth neuron of wavelet. wijkF, aijk, and bijk represent the connecting weight, translation, and dilation variables, respectively.The output of Layer 4 is expressed as(41)yik4=yik3ψikwikryik4t−1,where yik4 is the output of kth neuron of Layer 4. wikr and yik4t−1 represent the recurrent weight and the output at the previous time step of the kth neuron, respectively.Layer 5–Output Layer: Layer 5 has v3 neurons, which determines the dimension of the final output. Each neuron’s output of this layer is given by
(42)yil5=∑k=1v2wikl5yik4,where l∈1,2,⋯,v3, wikl5 is the connecting weight, and yil5 denotes the lth neuron of Layer 5.Using (42), one can express yi5 in the following vector form:
(43)yi5=ωi5yi4ωir,ωiF,ci,σi, where yi5=dyi15,⋯,dyik5,⋯,div35T∈ℝv3×1, and yi4=dyi14,⋯,dyik4,⋯,div24T∈ℝv2×1. ωi5∈ℝv3×v2, ωir∈ℝv2×1, ωiF∈ℝv1v2×1, ci∈ℝv1v2×1, and σi∈ℝv1v2×1, which are expressed as
(44)ωi5=ωi15,ωi25,⋯,ωil5,⋯,ωiv35T,ωil5=ωi1l5,ωi2l5,⋯,ωikl5,⋯,ωiv2l5T,ωir=ωi1r,ωi2r,⋯,ωikr,⋯,ωiv2rT,ωiF=ωi1F,ωi2F,⋯,ωikF,⋯,ωiv2FT,ωikF=ωi1kF,ωi2kF,⋯,ωijkF,⋯,ωiv1kF,ci=ci1,ci2,⋯,cik,⋯,civ2T,cik=ci1k,ci2k,⋯,cijk,⋯,civ1k,σi=σi1,σi2,⋯,σik,⋯,σiv2T,σik=σi1k,σi2k,⋯,σijk,⋯,σiv1k.In this paper,v3 is set as 3 due to the fact that the estimated variable Δi is three-dimensional. Therefore, there exist optimal values ωi5∗, ωir∗, ωiF∗, ci∗, and σi∗, such that
(45)Δi=yi5∗+εi1=ωi5∗yi4∗ωir∗,ωiF∗,ci∗,σi∗+εi1,where εi1 is the approximation error.To design the adaptive law of weights for estimating the unknown item, it is needed to obtain the gradient ofyi5 of its variables firstly.Differentiating both sides ofyi5, one can obtain
(46)dyi5=dωi5·yi4+ωi5dyi4,To yielddyi4, by the same way, differentiating both sides of yik4, then
(47)dyik4=dyik3cik,σik+dψikωikF+dωikryik4t−1.For termdyik3, one can obtain it by combining (37), (38), and (39), that is
(48)dyik3=∑j=1v1dcijk·2yij1−cijkσijk2∏j=1v1yijk2+∑j=1v1dσijk·2yij1−cijk2σijk3∏j=1v1yijk2=2yi1T−cik⋄σik2·∏j=1v1yijk2.dcikT+2yi1T−cik2⋄σik3.∏j=1v1yijk2·dσikT,where “⋄” represents dot division between matrices, and “∗n” represent the aligned “∗” itself does n times dot product. And the cik and σik have been defined in (44).For the termdψik and dωikryik4t−1, there exists
(49a)dψik=d∑j=1v1ωijkFϕijkrij=∑j=1v1ϕijkrij·dωijkF=ϕik·dωikFT,(49b)dωikryik4t−1=yik4t−1·dωikr.Then, by combining (50), (48), and (49), dyik4 can be expressed as
(50)dyik4=ΓikcdcikT+ΓikσdσikT+ΓikFdωikFT+Γikrdωikr,where
(51)Γikc=2yi1T−cik⋄σik2·∏j=1v1yijk2,Γikσ=2yi1T−cik2⋄σik3·∏j=1v1yijk2,ΓikF=ϕik,Γikr=yik4t−1.Using (50) and (51), one can further write dyi4 as
(52)dyi4=Γicdci+Γiσdσi+ΓiFdωiF+Γirdωir,where
(53)Γic=Γi1c0⋯⋯⋯00Γi2c⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯Γikc⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2c,Γiσ=Γi1σ0⋯⋯⋯00Γi2σ⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯Γikσ⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2σ,ΓiF=Γi1F0⋯⋯⋯00Γi2F⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯ΓikF⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2F,Γir=Γi1r0⋯⋯⋯00Γi2r⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯Γikr⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2r.Moreover, using the property of Kronecker product, the termdωi5·yi4 in (58) can be transformed to the following column vector form:
(54)dωi5·yi4=In⊗yi4·dvecωi5T,where “⊗” represents Kronecker product, and “vec∗” represents the operation that converts the aligned “∗” to a column vector, that means
(55)vecωi5T=ωi15T,⋯,ωil5T,⋯,ωiv35TT.To simplify the representation, usingω⟶i5 to represent vecωi5T. Employing (58), (52), and (54) yeilds the following total differential equation:
(56)dyi5=Γi5dω⟶i5+ωi5Γicdci+ωi5Γiσdσi+ωi5ΓiFdωiF+ωi5Γirdωir,where
(57)Γi5=In⊗yi4.On the other hand, the total derivative ofdyi5 can be expressed as the following total differential form:
(58)dyi5=∂yi5∂ω⟶i5Tdω⟶i5+∂yi5∂ciTdci+∂yi5∂σiTdσi+∂yi5∂ωiFTdωiF+∂yi5∂ωirTdωir.Hence, one can derive that(59)∂yi5∂ω⟶i5T=Γi5,∂yi5∂ciT=ωi5Γic∂yi5∂σiT=ωi5Γiσ,∂yi5∂ωiFT=ωi5ΓiF,∂yi5∂ωirT=ωi5Γir.Taking into account (58) and (59) and using the Taylor expansion, yi5 can be expressed as
(60)yi5=yi5∗+Γi5ω⟶i5−ω⟶i5∗+ωi5Γicci−ci∗+ωi5Γiσσi−σi∗+ωi5ΓiFωiF−ωiF∗+ωi5Γirωir−ωir∗+εi2.UsingΔ^i to estimate the unknown item Δi, and it is expressed as
(61)Δ^i=yi5+diagsignEi2Tgi2Xi2H^i≜yi5+ΓiHH^i,where ΓiH=diagsignEi2Tgi2Xi2 and H^i is an estimated value and will be introduced later.DefiningΔ~i=Δ^i−Δi, ω⟶~i5=ω⟶i5−ω⟶i5∗, c~i=ci−ci∗, σ~i=σi−σi∗, ω~iF=ωiF−ωiF∗, and ω~ir=ωir−ωir∗ and combining (45), (60), and (61), it yeilds
(62)Δ~i=Γi5ω⟶~i5+ωi5Γicc~i+ωi5Γiσσ~i+ωi5ΓiFω~iF+ωi5Γirω~ir+εi2−εi1+ΓiHH^i.Taking the derivative ofEi2 and using (17) and (20), one can obtain
(63)Ėi2=Żi2−Żi2d=gi2Xi2Fi20+Gi20Ui0vi+Di−Żi2d,where gi2x and x=x1,x2,x3T are
(64)gi2x=diaggi2τxτ,τ=1,2,3,gi2τxτ=X¯i2τ−X¯i2τxτ−X¯i2τX¯i2τ−xτ.By using the back-stepping method and definingvid as a virtual control signal, then Ėi2 can be expressed as
(65)Ėi2=gi2Xi2Fi20+Gi20vid+Gi20Ui0−vi+Gi20Ei3+Di−Żi2d,where Ei3=vi−vid. In order to reduce computational burden of taking time derivative for virtual control signal vid, a filter is used to obtain vid, which is given by
(66)v̇id=−kϵ2vid−v¯id,where kϵ2 is a positive constant, and v¯id is an auxiliary signal, designed as
(67)v¯id=gi2Xi2Gi20−1−gi2Xi2Fi20+Gi20Ui0vi−vi+D^i+Δ^i+H^i+Żi2d−Ki2Ei2−Aigi1Xi1·Gi1Ei1,where Ki2 is a positive diagonal matrix.Define the filter error asϵi2=vid−v¯id, then one can obtain from (66) that
(68)ϵ̇i2=−kϵi2ϵi2−v¯̇id.Substituting (65) with (67) and (62), with considering εi1−εi2≤Hi in which ∗ represents the absolute value of the matrix “∗”, while defining H^i as the estimation of Hi and H~i=H−H^i as the estimation error, one can obtain
(69)Ei2TĖi2≤−Ei2TKi2Ei2−Ei2TAigi1Xi1Gi1Ei1+Ei2Tgi2Xi2Gi20Ei3+Gi20ϵi2−Γi5·ω⟶~i5−ωi5Γicc~i−ωi5Γiσσ~i−ωi5ΓiFω~iF−ωi5Γirω~ir+Ei2Tgi2Xi2H~i.Taking the time derivative ofEi3 and using (18), one can obtain
(70)Ėi3=−Λvi+ξ−v̇id.Design the auxiliary control signalξ as
(71)ξ=Λvi+v̇id−Ki3Ei3−gi2Xi2Gi20Ei2−1Ei32hi12ϵi12+hi22ϵi22Ei3,Ei32>μib,Λvi+v̇id−Ki3Ei3−gi2Xi2Gi20Ei2−1μibhi12ϵi12+hi22ϵi22Ei3,Ei32≤μib, where Ki3 is a positive diagonal matrix, and hi1 and hi2 are positive constants.By combining (70) and (71), one has
(72)Ei3TĖi3≤−Ei3TKi3Ei3−Ei3Tgi2Xi2Gi20Ei2.Finally, the adaptive laws of RWFNN for theith UAV are developed as
(73)ω⟶̇i5=ηi1−1−γi1ω⟶i5+Ei2Tgi2Xi2Γi5T,ċi=ηi2−1−γi2ci+Ei2Tgi2Xi2ωi5ΓicT,σ̇i=ηi3−1−γi3σi+Ei2Tgi2Xi2ωi5ΓiσT,ω̇iF=ηi4−1−γi4ωiF+Ei2Tgi2Xi2ωi5ΓiFT,ω̇ir=ηi5−1−γi5ωir+Ei2Tgi2Xi2ωi5ΓirT,H^̇i=ηi6−1−γi6H^i+Ei2Tgi2Xi2T,where γi1~γi5 are discontinuous switching constants to prevent the weights to infinity and γi6 is a positive constant, where the switching constants are designed as
(74a)γi1=0ifω⟶i52≤ω¯5,γ¯i1ifω⟶i52>ω¯5,(74b)γi2=0ifci2≤c¯,γ¯i2ifci2>c¯,(74c)γi3=0ifσi2≤σ¯,γ¯i3ifσi2>σ¯,(74d)γi4=0ifωiF2≤ω¯iF,γ¯i4ifωiF2>ω¯iF,(74e)γi5=0ifωir2≤ω¯ir,γ¯i5ifωir2>ω¯ir,where ω¯5, c¯, σ¯, ω¯iF, and ω¯ir are positive constants, which represent the boundness of weight ω⟶i5,ci,σi,ωiF,ωir, respectively.To this end, the proposed FTCC scheme is shown in Figure2 to better illustrate the design principle and functional components in the control system.Remark 6.
Many papers choose multiplication on input and recurrence data as an operation on the neuron of the composite layer in RWFNN. However, it is sometimes problematic. For example, when inputs from layer 3 are minuscule, the outputs of the composite neuron will also become exceedingly small under multiplication. Under the limitation of computational precision, the outputs are equal to zero. Since the outputs will loop to the next multiplication, the outputs will always be zero, which causes neuron inactivation. Therefore, this paper uses the addition operation as an alternative, and the back-propagation gradient is deduced in detail using vectorized expressions, i.e., (46)–(59).Figure 2
The proposed control scheme for theith UAV.


### 3.2. Stability Analysis

Theorem 7.
Consider theN UAVs (1)–(4) under the distributed communication network against the actuator faults (13), states constraints, and input saturation (15), if the control laws are chosen as (30), (67), and (71), the disturbance observers are developed as (33), and (34), and the adaptive laws are constructed as (73), (74a); then, all the states in the system are ultimately uniformly bounded and strictly confined within the limits.Proof.
Choose a Lyapunov function as(75)V=12∑i=1NEi1TEi1T+Ei2TEi2T+Ei3TEi3T+ηi1·ω⟶~i5Tω⟶~i5+ηi2c~iTc~i+ηi3σ~iTσi+ηi4·ω~iFTω~iF+ηi5ω~irTω~ir+ηi6H~iTH~i+ϵi1T·ϵi1+ϵi2Tϵi2.
By taking the time derivative ofV, combining (32), (69), (72), (31), and (68) and using Young inequality, one has
(76)V̇≤∑i=1N−Ei1TKi1Ei1−Ei2TKi2Ei2−Ei3TKi3·Ei3+∑i=1N12hi1Ei1TAigi1Xi1Gi12+hi12ϵi12+12hi2Ei1Tgi2Xi2Gi202+hi22ϵi22−γi1ω⟶~i5Tω⟶i−γi2c~iTci−γi3·σ~iTσi−γi4ω~iFTωiF−γi5ω~irTωir−γi6H~iTH^i−kϵi1−12ϵi12−kϵi2−12ϵi22+12Z¯̇i2d2+12v¯̇i22.
Under the condition of (74a), the term γi1ω⟶~i5Tω⟶i has the following property:
(77)−γi1ω⟶~i5Tω⟶i5≤γ¯i1ω¯5+2ω⟶5∗2−12ω⟶~i52.
The reason is as follows:
Whenω¯5≥ω⟶i52, γi1=0, then
(78)−γi1ω⟶~i5Tω⟶i5=0≤γ¯i1ω¯5−ω⟶i52=γ¯i1ω¯5−ω⟶i5∗+ω⟶~i5∗2≤γ¯i1ω¯5−ω⟶i5∗−ω⟶~i5∗2≤γ¯i1ω¯5+2ω⟶i5∗Tω⟶~i5∗−ω⟶~i5∗2≤γ¯i1ω¯5+2ω⟶i5∗2−12ω⟶~i5∗2.
On the other hand, whenω¯5<ω⟶i52, γi1=γ¯i1, then,
(79)−γi1ω⟶~i5Tω⟶i5=−γ¯i1ω⟶~i5Tω⟶i5∗+ω⟶~i5≤−12γ¯i1ω⟶~i5T2−ω⟶i5∗2.
Hence, by combining (78) and (79), one can obtain (77). By the same way, one can conclude that
(80)−γi2c~iTci≤γ¯i2c¯+2c∗2−12c~i2,−γi3σ~iTσi≤γ¯i3σ¯+2σ∗2−12σ~i2,−γi4ω~iFTωiF≤γ¯i4ω¯F+2ωF∗2−12ω~iF2,−γi5ω~irTωir≤γ¯i5ω¯r+2ωr∗2−12ω~ir2.
Furthermore, the termγi6H~iTH^i in (76) satisfies that
(81)−γi6H~iTH^i≤−12γi6H~i2−Hi2.
Substituting (76) with (77), (80), and (81) then yeilds
(82)V̇≤∑i=1N−Ei1TKi1−12hi1Aigi1Xi1Gi12·Ei1−Ei2TKi2−12hi2gi2Xi2Gi202Ei2−Ei3TKi3Ei3+∑i=1N−12γ¯i1ω⟶~i52−12γ¯i2·c~i2−12γ¯i3σ~i2−12γ¯i4ω~iF2−12γ¯i5·ω~ir2−12γi6H~i2−kϵi1−1+hi12·ϵi12−kϵi2−1+hi22ϵi22+δ,where δ is
(83)δ=∑i=1N12Z¯̇i2d2+12v¯̇i22+γ¯i1ω¯5+2ω⟶5∗2+γ¯i2c¯+2c∗2+γ¯i3σ¯+2σ∗2+γ¯i4ω¯F+2ωF∗2+γ¯i5ω¯r+2ωr∗2+12γi6Hi2.
By choosing the parametersKi1 and Ki2 as
(84)Ki1=Ki10+12hi1Aigi1Xi1Gi12Ki2=Ki20+12hi2gi2Xi2Gi202, respectively, where Ki10 and Ki20 are positive diagonal matrices, (82) can be expressed as
(85)V̇≤−cV+δ,where c is
(86)c=minλmin2Ki10,λmin2Ki20,λmin2Ki30,ηi1−1γ¯i1,ηi2−1γ¯i2,ηi3−1γ¯i3,ηi4−1γ¯i4,ηi5−1γ¯i5,ηi6−1γ¯i6,2kϵi1−1−hi1,2kϵi2−1−hi2.
The expressionsZ¯̇i2d and v¯̇id can be respectively obtained from (30) and (67), given by
(87)Z¯̇i2d=∂Z¯i2d∂Xi1Ẋi1+∂Z¯i2d∂Ei1Ėi1+∂Z¯i2d∂Xi1dẊi1d+∂Z¯i2d∂Ẋi1dX¨i1d+∑j∈Nij≠iaij∂Z¯i2d∂Zj1Żj1+∂Z¯i2d∂Xi2Ẋi2+∂Z¯i2d∂Zi2Żi2,v¯̇id=∂v¯id∂Xi1Ẋi1+∂v¯id∂Xi2Ẋi2+∂v¯id∂viv̇i+∂v¯id∂D^iD^̇i+∂v¯id∂Δ^iΔ^̇i+∂v¯id∂H^iH^̇i+∂v¯id∂Żi2dZ¨i2d+∂v¯id∂Ei1Ėi1+∂v¯id∂Ei2Ėi2.
Define the following two vectorsXd and Ed(88)Xd=colX1d,⋯,Xid,⋯,XNd,Xid=colXi1d,Ẋi1d,X¨i1d,Ed=colE1d,⋯,Eid,⋯,ENd,Ei=colEi1,Ei2,Ei3.
For any constantsB¯Xd>0, B¯E>0, the sets X≔Xd:Xd2≤B¯Xd and E≔Ed:Ed2≤B¯E are compact ones. Then, P≔X×E is also compact. Due to the fact that Z¯̇i2d and v¯̇id both are continuous function, considering the continuous property, there exists a constant δM>0 when ∀Xd,Ed∈P such that
(89)Z¯̇i2d2+v¯̇id2≤δM.
Therefore, there exists a constantδ¯>0 satisfying
(90)δ≤δ¯,∀Xd,Ed∈P.
For any constantB¯E, there exists several parameters and initial condition satisfying
(91)V0≤δ¯/c<2δ¯/c≤B¯E,such that
(92)Vt≤δ¯c,∀t≥0,
andE is an invariant set.
The above result (92) can be approved by contradiction. Since the reference input Xd must be bounded, that leads to Xd∈X on ∀t≥0. Assuming ∃t1,Vt1>δ¯/c, due to V is a continuous function, considering the continuous property, there exists a moment t0 such that Vt0=δ¯/c,Vt0+>δ¯/c. Noticing Ed2≤2V in (75), there exists Edt02≤2δ¯/c. Considering (91) yields Xdt0,Edt0∈P, so δt0≤δ¯. Moreover, by taking account of (85), one can derive V̇t0≤−cVt0+δt0=−δ¯+δt0≤0. Hence, that causes Vt0+≤Vt0, which leads contradiction. This completes the proof of (91). Furthermore, considering Ed2≤2V and (92), (91), one can derive E is an invariant set.
Therefore, all the signals in the system are uniformly bounded, and the statesXi1, Xi2 are always within their constraints.


## 3.1. Fault-Tolerant Cooperative Controller Design

Define the independent tracking error ofith UAV as Z~i1=Zi1−Zi1d, then the cooperative tracking error of ith UAV is defined as
(22)Ei1=λ1Z~i1+λ2∑j∈NiaijZ~i1−Z~j1,where Ei1=Ei11,Ei12,Ei13T, λ1 and λ2 are positive parameters, which are used to regulate the cooperative tracking performance.Using the Kronecker product “⊗”, and define E1=E11T,E21T,⋯,EN1TT, Z~1=Z~11T,Z~21T,⋯,Z~N1TT, then the cooperative tracking error of all UAVs can be expressed as
(23)E1=λ1IN+λ2L⊗I3Z~1.By recalling Lemma5, it yields Z~1=λ1+λ2L−1⊗I3E1≤1/σminλ1+λ2LE1, where σmin· represents the minimum singular value of matrix “·.” Therefore, Z~1⟶0 if E1⟶0.Using (22), the synchronization error of each UAV Ei1 can be expressed as
(24)Ei1=λ1+λ2∑j∈Nij≠iaijZ~i1−λ2∑j∈Nij≠iaijZ~j1.Differentiating (24) yields
(25)Ėi1=Aigi1Xi1Ẋi1−gi1Xi1dẊi1d−λ2∑j∈Nij≠iaijZ~̇j1=Aigi1Xi1Ẋi1−gi1Xi1dẊi1d−λ2∑j∈Nij≠iaijgi1Xj1Ẋj1−gi1Xj1dẊj1d,where Ai=λ1+λ2∑j∈Nij≠iaij, gi1x is ℝ3⟶ℝ3×3, x=x1,x2,x3T, which is expressed as
(26)gi1x=diaggi1τxτ,τ=1,2,3,gi1τxτ=X¯i1τ−X¯i1τxτ−X¯i1τX¯i1τ−xτ.Substituting (11) into (25) yields
(27)Ėi1=Aigi1Xi1Fi1+Gi1Xi2−Aigi1Xi1dẊi1d−λ2∑j∈Nij≠iaijZ~̇j1.Based on the back-stepping control architecture, (27) can be expressed as
(28)Ėi1=Aigi1Xi1Fi1−Aigi1Xi1dẊi1d−λ2∑j∈Nij≠iaijZ~̇j1+Aigi1Xi1Gi1·Xi2−Zi2+Ei2+Zi2d,where Ei2=Zi2−Zi2d and Zi2d is a virtual control signal.By using a low-pass filter, one has(29)Żi2d=−k∈1Zi2d−Z¯i2d,wherekϵ1 is a positive constant and Z¯i2d is an auxiliary signal, designed as
(30)Z¯i2d=Aigi1Xi1Gi1−1λ2∑j∈Nij≠iaijZ~̇j1−Aigi1Xi1Fi1+Gi1Xi2−Zi2+Aigi1Xi1dẊi1d−Ki1Ei1,where Ki1 is a positive diagonal matrix.Defining the filtering error asϵi1=Zi2d−Z¯i2d, one can obtain
(31)ϵ̇i1=−k∈i1ϵi1−Z¯̇i2d.By substituting (30) into (28), one can obtain
(32)Ei1TĖi1≤−Ei1TKi1Ei1+Ei1TAigi1Xi1Gi1·Ei2+12hi1Ei1TAigi1Xi1Gi12+hi12ϵi12,where hi1 is a positive constant and · represents 2-norm of vector.To estimate the unknown functionDi of each UAV, the following DO is designed for the ith UAV:
(33)D^i=k2Xi2−X^i2+k1k2∫Xi2−X^i2dt,(34)X^̇i2=Fi20+Gi20Ui0vi+D^i+k1Xi2−X^i2,where k1 and k2 are positive parameters, D^i is the estimate of Di.DefineeXi2=Xi2−X^i2 and D~i=Di−D^i, one can obtain
(35)ėXi2+k1eXi2=D~i.Taking the derivative of (35) and using (33) give
(36)D~̇i+k2D~i=D.̇From (36), it can be known that the estimation error D~i will not converge to zero since Ḋ≠0. In order to estimate the unknown function Di more accurate, a five-layer RWFNN is used to estimate the error D~i of the DO with defining Δi=D~i.The RWFNN structure is illustrated in Figure1, including five layers (input layer, membership layer, rule layer, composite layer, and output layer) [33]. The components of the RWFNN are introduced as follows:Figure 1
The structure of the RWFNN for each UAV.Layer 1–Input Layer: Input layer is the first layer, where ri=ri1,ri2,⋯,riv1T is the input features of RWFNN. The output of layer 1 is expressed as
(37)yij1=rij,where j∈1,2,⋯,v1, v1 is the dimension of input features, and yij1 represents the output of jth neuron of Layer 1.Layer 2–Membership Layer: Layer 2 has v1 rows and v2 columns, and its output can be described as
(38)yijk2=e−yij1−cijk2/σijk2,where j∈1,2,⋯,v1, k∈1,2,⋯,v2. v2 is a positive constant, depending on the number of neurons. yijk2 denotes the neuron of layer 2 in row j, column k.Layer 3–Rule Layer: Layer 3 has v2 neurons, and the output of layer 3 is described as
(39)yik3=∏j=1v1yijk2,where k∈1,2,⋯,v2, and yik3 denotes the kth neuron of Layer 3.Layer 4–Composite Layer: Layer 4 also has v2 neurons, and the input of Layer 4 consists the output of the wavelet layer, recurrent layer, and Layer 2, where the output of wavelet layer is described as
(40)ψik=∑j=1v1wijkFϕijkrij,ϕijkrij=1bijk1−rij−aijk2bijk2e−rij−aijk2/2bijk2,where j∈1,2,⋯,v1, k∈1,2,⋯,v2, ϕijk is the output of the jth neuron of Layer 1 to kth neuron of wavelet layer. ψik represents the output of kth neuron of wavelet. wijkF, aijk, and bijk represent the connecting weight, translation, and dilation variables, respectively.The output of Layer 4 is expressed as(41)yik4=yik3ψikwikryik4t−1,where yik4 is the output of kth neuron of Layer 4. wikr and yik4t−1 represent the recurrent weight and the output at the previous time step of the kth neuron, respectively.Layer 5–Output Layer: Layer 5 has v3 neurons, which determines the dimension of the final output. Each neuron’s output of this layer is given by
(42)yil5=∑k=1v2wikl5yik4,where l∈1,2,⋯,v3, wikl5 is the connecting weight, and yil5 denotes the lth neuron of Layer 5.Using (42), one can express yi5 in the following vector form:
(43)yi5=ωi5yi4ωir,ωiF,ci,σi, where yi5=dyi15,⋯,dyik5,⋯,div35T∈ℝv3×1, and yi4=dyi14,⋯,dyik4,⋯,div24T∈ℝv2×1. ωi5∈ℝv3×v2, ωir∈ℝv2×1, ωiF∈ℝv1v2×1, ci∈ℝv1v2×1, and σi∈ℝv1v2×1, which are expressed as
(44)ωi5=ωi15,ωi25,⋯,ωil5,⋯,ωiv35T,ωil5=ωi1l5,ωi2l5,⋯,ωikl5,⋯,ωiv2l5T,ωir=ωi1r,ωi2r,⋯,ωikr,⋯,ωiv2rT,ωiF=ωi1F,ωi2F,⋯,ωikF,⋯,ωiv2FT,ωikF=ωi1kF,ωi2kF,⋯,ωijkF,⋯,ωiv1kF,ci=ci1,ci2,⋯,cik,⋯,civ2T,cik=ci1k,ci2k,⋯,cijk,⋯,civ1k,σi=σi1,σi2,⋯,σik,⋯,σiv2T,σik=σi1k,σi2k,⋯,σijk,⋯,σiv1k.In this paper,v3 is set as 3 due to the fact that the estimated variable Δi is three-dimensional. Therefore, there exist optimal values ωi5∗, ωir∗, ωiF∗, ci∗, and σi∗, such that
(45)Δi=yi5∗+εi1=ωi5∗yi4∗ωir∗,ωiF∗,ci∗,σi∗+εi1,where εi1 is the approximation error.To design the adaptive law of weights for estimating the unknown item, it is needed to obtain the gradient ofyi5 of its variables firstly.Differentiating both sides ofyi5, one can obtain
(46)dyi5=dωi5·yi4+ωi5dyi4,To yielddyi4, by the same way, differentiating both sides of yik4, then
(47)dyik4=dyik3cik,σik+dψikωikF+dωikryik4t−1.For termdyik3, one can obtain it by combining (37), (38), and (39), that is
(48)dyik3=∑j=1v1dcijk·2yij1−cijkσijk2∏j=1v1yijk2+∑j=1v1dσijk·2yij1−cijk2σijk3∏j=1v1yijk2=2yi1T−cik⋄σik2·∏j=1v1yijk2.dcikT+2yi1T−cik2⋄σik3.∏j=1v1yijk2·dσikT,where “⋄” represents dot division between matrices, and “∗n” represent the aligned “∗” itself does n times dot product. And the cik and σik have been defined in (44).For the termdψik and dωikryik4t−1, there exists
(49a)dψik=d∑j=1v1ωijkFϕijkrij=∑j=1v1ϕijkrij·dωijkF=ϕik·dωikFT,(49b)dωikryik4t−1=yik4t−1·dωikr.Then, by combining (50), (48), and (49), dyik4 can be expressed as
(50)dyik4=ΓikcdcikT+ΓikσdσikT+ΓikFdωikFT+Γikrdωikr,where
(51)Γikc=2yi1T−cik⋄σik2·∏j=1v1yijk2,Γikσ=2yi1T−cik2⋄σik3·∏j=1v1yijk2,ΓikF=ϕik,Γikr=yik4t−1.Using (50) and (51), one can further write dyi4 as
(52)dyi4=Γicdci+Γiσdσi+ΓiFdωiF+Γirdωir,where
(53)Γic=Γi1c0⋯⋯⋯00Γi2c⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯Γikc⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2c,Γiσ=Γi1σ0⋯⋯⋯00Γi2σ⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯Γikσ⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2σ,ΓiF=Γi1F0⋯⋯⋯00Γi2F⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯ΓikF⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2F,Γir=Γi1r0⋯⋯⋯00Γi2r⋯⋯⋯0⋮⋮⋱⋮⋮⋮0⋯⋯Γikr⋯0⋮⋮⋮⋮⋱⋮0⋯⋯⋯⋯Γiv2r.Moreover, using the property of Kronecker product, the termdωi5·yi4 in (58) can be transformed to the following column vector form:
(54)dωi5·yi4=In⊗yi4·dvecωi5T,where “⊗” represents Kronecker product, and “vec∗” represents the operation that converts the aligned “∗” to a column vector, that means
(55)vecωi5T=ωi15T,⋯,ωil5T,⋯,ωiv35TT.To simplify the representation, usingω⟶i5 to represent vecωi5T. Employing (58), (52), and (54) yeilds the following total differential equation:
(56)dyi5=Γi5dω⟶i5+ωi5Γicdci+ωi5Γiσdσi+ωi5ΓiFdωiF+ωi5Γirdωir,where
(57)Γi5=In⊗yi4.On the other hand, the total derivative ofdyi5 can be expressed as the following total differential form:
(58)dyi5=∂yi5∂ω⟶i5Tdω⟶i5+∂yi5∂ciTdci+∂yi5∂σiTdσi+∂yi5∂ωiFTdωiF+∂yi5∂ωirTdωir.Hence, one can derive that(59)∂yi5∂ω⟶i5T=Γi5,∂yi5∂ciT=ωi5Γic∂yi5∂σiT=ωi5Γiσ,∂yi5∂ωiFT=ωi5ΓiF,∂yi5∂ωirT=ωi5Γir.Taking into account (58) and (59) and using the Taylor expansion, yi5 can be expressed as
(60)yi5=yi5∗+Γi5ω⟶i5−ω⟶i5∗+ωi5Γicci−ci∗+ωi5Γiσσi−σi∗+ωi5ΓiFωiF−ωiF∗+ωi5Γirωir−ωir∗+εi2.UsingΔ^i to estimate the unknown item Δi, and it is expressed as
(61)Δ^i=yi5+diagsignEi2Tgi2Xi2H^i≜yi5+ΓiHH^i,where ΓiH=diagsignEi2Tgi2Xi2 and H^i is an estimated value and will be introduced later.DefiningΔ~i=Δ^i−Δi, ω⟶~i5=ω⟶i5−ω⟶i5∗, c~i=ci−ci∗, σ~i=σi−σi∗, ω~iF=ωiF−ωiF∗, and ω~ir=ωir−ωir∗ and combining (45), (60), and (61), it yeilds
(62)Δ~i=Γi5ω⟶~i5+ωi5Γicc~i+ωi5Γiσσ~i+ωi5ΓiFω~iF+ωi5Γirω~ir+εi2−εi1+ΓiHH^i.Taking the derivative ofEi2 and using (17) and (20), one can obtain
(63)Ėi2=Żi2−Żi2d=gi2Xi2Fi20+Gi20Ui0vi+Di−Żi2d,where gi2x and x=x1,x2,x3T are
(64)gi2x=diaggi2τxτ,τ=1,2,3,gi2τxτ=X¯i2τ−X¯i2τxτ−X¯i2τX¯i2τ−xτ.By using the back-stepping method and definingvid as a virtual control signal, then Ėi2 can be expressed as
(65)Ėi2=gi2Xi2Fi20+Gi20vid+Gi20Ui0−vi+Gi20Ei3+Di−Żi2d,where Ei3=vi−vid. In order to reduce computational burden of taking time derivative for virtual control signal vid, a filter is used to obtain vid, which is given by
(66)v̇id=−kϵ2vid−v¯id,where kϵ2 is a positive constant, and v¯id is an auxiliary signal, designed as
(67)v¯id=gi2Xi2Gi20−1−gi2Xi2Fi20+Gi20Ui0vi−vi+D^i+Δ^i+H^i+Żi2d−Ki2Ei2−Aigi1Xi1·Gi1Ei1,where Ki2 is a positive diagonal matrix.Define the filter error asϵi2=vid−v¯id, then one can obtain from (66) that
(68)ϵ̇i2=−kϵi2ϵi2−v¯̇id.Substituting (65) with (67) and (62), with considering εi1−εi2≤Hi in which ∗ represents the absolute value of the matrix “∗”, while defining H^i as the estimation of Hi and H~i=H−H^i as the estimation error, one can obtain
(69)Ei2TĖi2≤−Ei2TKi2Ei2−Ei2TAigi1Xi1Gi1Ei1+Ei2Tgi2Xi2Gi20Ei3+Gi20ϵi2−Γi5·ω⟶~i5−ωi5Γicc~i−ωi5Γiσσ~i−ωi5ΓiFω~iF−ωi5Γirω~ir+Ei2Tgi2Xi2H~i.Taking the time derivative ofEi3 and using (18), one can obtain
(70)Ėi3=−Λvi+ξ−v̇id.Design the auxiliary control signalξ as
(71)ξ=Λvi+v̇id−Ki3Ei3−gi2Xi2Gi20Ei2−1Ei32hi12ϵi12+hi22ϵi22Ei3,Ei32>μib,Λvi+v̇id−Ki3Ei3−gi2Xi2Gi20Ei2−1μibhi12ϵi12+hi22ϵi22Ei3,Ei32≤μib, where Ki3 is a positive diagonal matrix, and hi1 and hi2 are positive constants.By combining (70) and (71), one has
(72)Ei3TĖi3≤−Ei3TKi3Ei3−Ei3Tgi2Xi2Gi20Ei2.Finally, the adaptive laws of RWFNN for theith UAV are developed as
(73)ω⟶̇i5=ηi1−1−γi1ω⟶i5+Ei2Tgi2Xi2Γi5T,ċi=ηi2−1−γi2ci+Ei2Tgi2Xi2ωi5ΓicT,σ̇i=ηi3−1−γi3σi+Ei2Tgi2Xi2ωi5ΓiσT,ω̇iF=ηi4−1−γi4ωiF+Ei2Tgi2Xi2ωi5ΓiFT,ω̇ir=ηi5−1−γi5ωir+Ei2Tgi2Xi2ωi5ΓirT,H^̇i=ηi6−1−γi6H^i+Ei2Tgi2Xi2T,where γi1~γi5 are discontinuous switching constants to prevent the weights to infinity and γi6 is a positive constant, where the switching constants are designed as
(74a)γi1=0ifω⟶i52≤ω¯5,γ¯i1ifω⟶i52>ω¯5,(74b)γi2=0ifci2≤c¯,γ¯i2ifci2>c¯,(74c)γi3=0ifσi2≤σ¯,γ¯i3ifσi2>σ¯,(74d)γi4=0ifωiF2≤ω¯iF,γ¯i4ifωiF2>ω¯iF,(74e)γi5=0ifωir2≤ω¯ir,γ¯i5ifωir2>ω¯ir,where ω¯5, c¯, σ¯, ω¯iF, and ω¯ir are positive constants, which represent the boundness of weight ω⟶i5,ci,σi,ωiF,ωir, respectively.To this end, the proposed FTCC scheme is shown in Figure2 to better illustrate the design principle and functional components in the control system.Remark 6.
Many papers choose multiplication on input and recurrence data as an operation on the neuron of the composite layer in RWFNN. However, it is sometimes problematic. For example, when inputs from layer 3 are minuscule, the outputs of the composite neuron will also become exceedingly small under multiplication. Under the limitation of computational precision, the outputs are equal to zero. Since the outputs will loop to the next multiplication, the outputs will always be zero, which causes neuron inactivation. Therefore, this paper uses the addition operation as an alternative, and the back-propagation gradient is deduced in detail using vectorized expressions, i.e., (46)–(59).Figure 2
The proposed control scheme for theith UAV.


## 3.2. Stability Analysis

Theorem 7.
Consider theN UAVs (1)–(4) under the distributed communication network against the actuator faults (13), states constraints, and input saturation (15), if the control laws are chosen as (30), (67), and (71), the disturbance observers are developed as (33), and (34), and the adaptive laws are constructed as (73), (74a); then, all the states in the system are ultimately uniformly bounded and strictly confined within the limits.Proof.
Choose a Lyapunov function as(75)V=12∑i=1NEi1TEi1T+Ei2TEi2T+Ei3TEi3T+ηi1·ω⟶~i5Tω⟶~i5+ηi2c~iTc~i+ηi3σ~iTσi+ηi4·ω~iFTω~iF+ηi5ω~irTω~ir+ηi6H~iTH~i+ϵi1T·ϵi1+ϵi2Tϵi2.
By taking the time derivative ofV, combining (32), (69), (72), (31), and (68) and using Young inequality, one has
(76)V̇≤∑i=1N−Ei1TKi1Ei1−Ei2TKi2Ei2−Ei3TKi3·Ei3+∑i=1N12hi1Ei1TAigi1Xi1Gi12+hi12ϵi12+12hi2Ei1Tgi2Xi2Gi202+hi22ϵi22−γi1ω⟶~i5Tω⟶i−γi2c~iTci−γi3·σ~iTσi−γi4ω~iFTωiF−γi5ω~irTωir−γi6H~iTH^i−kϵi1−12ϵi12−kϵi2−12ϵi22+12Z¯̇i2d2+12v¯̇i22.
Under the condition of (74a), the term γi1ω⟶~i5Tω⟶i has the following property:
(77)−γi1ω⟶~i5Tω⟶i5≤γ¯i1ω¯5+2ω⟶5∗2−12ω⟶~i52.
The reason is as follows:
Whenω¯5≥ω⟶i52, γi1=0, then
(78)−γi1ω⟶~i5Tω⟶i5=0≤γ¯i1ω¯5−ω⟶i52=γ¯i1ω¯5−ω⟶i5∗+ω⟶~i5∗2≤γ¯i1ω¯5−ω⟶i5∗−ω⟶~i5∗2≤γ¯i1ω¯5+2ω⟶i5∗Tω⟶~i5∗−ω⟶~i5∗2≤γ¯i1ω¯5+2ω⟶i5∗2−12ω⟶~i5∗2.
On the other hand, whenω¯5<ω⟶i52, γi1=γ¯i1, then,
(79)−γi1ω⟶~i5Tω⟶i5=−γ¯i1ω⟶~i5Tω⟶i5∗+ω⟶~i5≤−12γ¯i1ω⟶~i5T2−ω⟶i5∗2.
Hence, by combining (78) and (79), one can obtain (77). By the same way, one can conclude that
(80)−γi2c~iTci≤γ¯i2c¯+2c∗2−12c~i2,−γi3σ~iTσi≤γ¯i3σ¯+2σ∗2−12σ~i2,−γi4ω~iFTωiF≤γ¯i4ω¯F+2ωF∗2−12ω~iF2,−γi5ω~irTωir≤γ¯i5ω¯r+2ωr∗2−12ω~ir2.
Furthermore, the termγi6H~iTH^i in (76) satisfies that
(81)−γi6H~iTH^i≤−12γi6H~i2−Hi2.
Substituting (76) with (77), (80), and (81) then yeilds
(82)V̇≤∑i=1N−Ei1TKi1−12hi1Aigi1Xi1Gi12·Ei1−Ei2TKi2−12hi2gi2Xi2Gi202Ei2−Ei3TKi3Ei3+∑i=1N−12γ¯i1ω⟶~i52−12γ¯i2·c~i2−12γ¯i3σ~i2−12γ¯i4ω~iF2−12γ¯i5·ω~ir2−12γi6H~i2−kϵi1−1+hi12·ϵi12−kϵi2−1+hi22ϵi22+δ,where δ is
(83)δ=∑i=1N12Z¯̇i2d2+12v¯̇i22+γ¯i1ω¯5+2ω⟶5∗2+γ¯i2c¯+2c∗2+γ¯i3σ¯+2σ∗2+γ¯i4ω¯F+2ωF∗2+γ¯i5ω¯r+2ωr∗2+12γi6Hi2.
By choosing the parametersKi1 and Ki2 as
(84)Ki1=Ki10+12hi1Aigi1Xi1Gi12Ki2=Ki20+12hi2gi2Xi2Gi202, respectively, where Ki10 and Ki20 are positive diagonal matrices, (82) can be expressed as
(85)V̇≤−cV+δ,where c is
(86)c=minλmin2Ki10,λmin2Ki20,λmin2Ki30,ηi1−1γ¯i1,ηi2−1γ¯i2,ηi3−1γ¯i3,ηi4−1γ¯i4,ηi5−1γ¯i5,ηi6−1γ¯i6,2kϵi1−1−hi1,2kϵi2−1−hi2.
The expressionsZ¯̇i2d and v¯̇id can be respectively obtained from (30) and (67), given by
(87)Z¯̇i2d=∂Z¯i2d∂Xi1Ẋi1+∂Z¯i2d∂Ei1Ėi1+∂Z¯i2d∂Xi1dẊi1d+∂Z¯i2d∂Ẋi1dX¨i1d+∑j∈Nij≠iaij∂Z¯i2d∂Zj1Żj1+∂Z¯i2d∂Xi2Ẋi2+∂Z¯i2d∂Zi2Żi2,v¯̇id=∂v¯id∂Xi1Ẋi1+∂v¯id∂Xi2Ẋi2+∂v¯id∂viv̇i+∂v¯id∂D^iD^̇i+∂v¯id∂Δ^iΔ^̇i+∂v¯id∂H^iH^̇i+∂v¯id∂Żi2dZ¨i2d+∂v¯id∂Ei1Ėi1+∂v¯id∂Ei2Ėi2.
Define the following two vectorsXd and Ed(88)Xd=colX1d,⋯,Xid,⋯,XNd,Xid=colXi1d,Ẋi1d,X¨i1d,Ed=colE1d,⋯,Eid,⋯,ENd,Ei=colEi1,Ei2,Ei3.
For any constantsB¯Xd>0, B¯E>0, the sets X≔Xd:Xd2≤B¯Xd and E≔Ed:Ed2≤B¯E are compact ones. Then, P≔X×E is also compact. Due to the fact that Z¯̇i2d and v¯̇id both are continuous function, considering the continuous property, there exists a constant δM>0 when ∀Xd,Ed∈P such that
(89)Z¯̇i2d2+v¯̇id2≤δM.
Therefore, there exists a constantδ¯>0 satisfying
(90)δ≤δ¯,∀Xd,Ed∈P.
For any constantB¯E, there exists several parameters and initial condition satisfying
(91)V0≤δ¯/c<2δ¯/c≤B¯E,such that
(92)Vt≤δ¯c,∀t≥0,
andE is an invariant set.
The above result (92) can be approved by contradiction. Since the reference input Xd must be bounded, that leads to Xd∈X on ∀t≥0. Assuming ∃t1,Vt1>δ¯/c, due to V is a continuous function, considering the continuous property, there exists a moment t0 such that Vt0=δ¯/c,Vt0+>δ¯/c. Noticing Ed2≤2V in (75), there exists Edt02≤2δ¯/c. Considering (91) yields Xdt0,Edt0∈P, so δt0≤δ¯. Moreover, by taking account of (85), one can derive V̇t0≤−cVt0+δt0=−δ¯+δt0≤0. Hence, that causes Vt0+≤Vt0, which leads contradiction. This completes the proof of (91). Furthermore, considering Ed2≤2V and (92), (91), one can derive E is an invariant set.
Therefore, all the signals in the system are uniformly bounded, and the statesXi1, Xi2 are always within their constraints.


## 4. Simulation Results and Analysis

To illustrate the effectiveness and the superiority of the proposed FTCC scheme in this paper, using MATLAB/Simulink to simulate four UAVs whose communication topology is shown in Figure3.Figure 3
Communication topology.Figure3 shows the communication network, and the parameters of UAVs are referred to [28]. The element aij of the adjacency matrix A is defined as 1 if the link between ith UAV and jth UAV existed; otherwise, aij=0. The initial attitudes of all UAVs are presented in Table 1, and the initial angular rates are defined opposite as values of attitudes. The initial values of Vi, χi, and γi of all UAVs are set as 30 m/s, 0.01 rad and 0.01 rad, respectively.Table 1
Initial attitudes of all UAVs.
μi0 (rad)αi0 (rad)βi0 (rad)UAV 10.010.010.01UAV 2-0.015-0.015-0.015UAV 30.020.020.02UAV 4-0.025-0.025-0.025Assuming the safe range of the attitudesμi,αi,andβi of all UAVs are −8~8 deg, and the corresponding angular velocity pi,qi,ri are −0.25~0.25 rad/s. In addition, the maximum operation range of control surfaces δia,δie,andδir are −25~25 deg.Therefore, for anyith UAV, the upper bound X¯i1 and the lower bound X¯i1 of states Xi1 are defined as 8,8,8T deg and −8,8,8T deg, respectively. Similarly, the upper bound X¯i2 and the lower bound X¯i2 of states Xi2 are 0.25,0.25,0.25T rad/s and −0.25,0.25,0.25T rad/s, respectively. The upper bound ui0τmax and the lower bound ui0τmin are −25 deg and 25 deg, respectively.The main control parameters are chosen asKi10=diag5,5,5, Ki20=diag20,20,20, Ki3=diag30,30,30, ηi1=ηi2=ηi3=ηi4=ηi5=1.5, ηi6=1.8, kϵi1=kϵi2=100, hi1=hi2=20, and μib=0.001.To verify the effectiveness of the proposed FTCC scheme under actuator fault, the following fault signals are chosen:(93)ρ1=ρ2=ρ3=ρ4=1,U1f=U2f=U3f=U4f=0∘t<8ρ1=diag0.7,0.9,0.8,ρ2=ρ3=ρ4=0,U1f=3∘,2∘,4∘T,U2f=U3f=U4f=0∘,8≤t<10,ρ1=ρ2diag0.7,0.9,0.8,ρ3=ρ4=0,U1f=U2f=3∘,2∘,4∘T,U3f=U4f=U4f=0∘,10≤t<12,ρ1=ρ2=ρ3=diag0.7,0.9,0.8,ρ4=0,U1f=U2f=U3f=3∘,2∘,4∘T,U4f=0∘,12≤t<14,ρ1=ρ2=ρ3=ρ4=diag0.7,0.9,0.8,U1f=U2f=U3f=U4f=3∘,2∘,4∘T.t≥14.Remark 8.
The reason of choosing the main control parameter briefly describes as follows. The parameterKi10 decides the closed-loop dynamics of Ėi1, which can be approximated as Ėi1≈−Ki1Ei1+Ei1TAigi1Xi1Gi1Ei2+hi1/2·ϵi12 according to (32) and (84), and the term Ei1TAigi1Xi1Gi1Ei2+hi1/2·ϵi12will rapidly reduce due to the response of inner-loop dynamics Ei2and ϵi1 are converge rapidly, then, it can be further represented as Ėi1≈−Ki10Ei1, so that the settling time of the dynamic Ei1 approximately equal to 3/λminKi10. There exists a tradeoff – a larger value Ki10 helps to reduce the settling time while it needs larger change of the state Xi2, which may cause saturation, so Ki10 is taken as diag5,5,5. Similarly, parameters Ki20 and Ki3 decide the convergence rate of the inner-loop control errors Ei2 and Ei3, respectively, and their values normally take several times as the Ki10 to get faster response. Moreover, for parameters ηi1~ηi6, their inverses serve as learning rates of neural network, which usually take as a value among 1~1000 for fixed rate learning algorithms. In addition, parameters kϵi1 and kϵi2 are adjustment factors of dynamic surface filter, which are usually taken as a large value. Finally, for μib, it can adjust the astringency of the error Ei3, which is the smaller the better, but the too small value will tend to cause virtual control signal “explosion”.The response of bank angleμi, angle of attack αi, and sideslip angle βi of each UAV are shown in Figures 4–6, respectively. It can be seen that all UAVs can track their references μid, αid, and βid. Although the fault occurs to UAV 1, UAV 2, UAV 3, and UAV 4 at 8 s, 10 s, 12 s, and 14 s, respectively, all UAVs can quickly track individual references again.Figure 4
Bank anglesμi and references μid of all UAVs (i=1,2,3,4).Figure 5
Attack anglesαi and references αid of all UAVs (i=1,2,3,4).Figure 6
Sideslip anglesβi and references βid of all UAVs (i=1,2,3,4).The response of stateXi2 is shown in Figure 7. It can be seen that the states pi, qi, and ri of all UAVs never exceed its upper or lower bound. Meanwhile, if the state constraints are not considered in the FTCC scheme, the states q1 and q3 will exceed the lower bound, which is shown in Figure 8. In addition, the control input signals δia, δie, and δir are presented in Figure 9. Since the actuator constraint scheme (15) is adopted, the input signals never exceed their upper and lower bounds.Figure 7
The time responses ofpi, qi, and ri with the developed control scheme (i=1,2,3,4).Figure 8
The time responses ofpi, qi, and ri without the constraint mechanism (i=1,2,3,4).Figure 9
Control input signalsδia, δie, and δir of each UAVs (i=1,2,3,4).


## 5. Conclusion and Future Work

This paper has explored an FTCC scheme for multi-UAVs under the distributed communication network, in which the issues including input saturation, state constraints, actuator faults, and unknown disturbances have all been taken into account.It can be noted that the proposed FTCC scheme only considers fixed and undirected communication network. In addition, communication delay and communication interferences are not considered, and finite-time convergence technology has not been considered in the FTCC scheme, so the control performance cannot be achieved in finite time. Moreover, compared to the Euler attitude angles, the airflow attitude angles are necessary and easy to combine with the UAV’s outer loop for position control, hence in this paper, it is directly used in the attitude control. However, using the airflow attitude angles for feedback control is less reliable than the former. Furthermore, sensor fault may occur at the same time, which perhaps outweigh the risk of actuator fault, so it deserves more attention and investigation. Finally, state measurements have been directly used in the control law without considering noise filtering, so that the performance may be degraded when sensor measurements have severe noises. Taking into account the noise filtering algorithms and sensor faults simultaneously will significantly increases the difficulty of proving the closed-loop system stability, which makes the issue challenging.Therefore, in future work, the essence of communication delays, finite-time convergence technology, the reliability of using airflow attitude angle, sensor fault, and noise filtering will be taken into account on the basis of existing research. Besides, based on the simulation results from MATLAB/Simulink, the hardware-in-the-loop verification scheme will be adopted to further verify the proposed control scheme towards more practical applications.



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*Source: 8385913-2022-10-05.xml*