# Event-Triggered Finite-Time Attitude Cooperative Control for Multiple Unmanned Aerial Vehicles

Authors: Qiang Han; Yongshuai Zhou; Xin Liu; Xianguo Tuo
Journal: Applied Bionics and Biomechanics
 (2022)

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## Abstract

The finite-time attitude cooperative control problem for a group of multiple unmanned aerial vehicle systems with external disturbances and uncertain parameters is discussed in this paper. The dynamics of the systems is described by quaternion avoiding the singularity. Based on the attitude error and angular velocity error, a novel nonsingular terminal sliding mode surface is proposed for the controller with event-triggered scheme. The lumped disturbances are estimated by neural networks with adaptive law. The communication frequency is decreased by the proposed distributed event-triggered based sliding mode controller. Lyapunov theory is utilized to analyze the stability of the systems, and the Zeno behavior is avoided by rigorous proof. Finally, simulation examples are presented to illustrate the efficiency of the proposed control algorithm.


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## Body

## 1. Introduction

Attitude cooperative control of multiple unmanned aerial vehicle systems (MUAVs) is significantly important in the formation flying missions. Compared to a single unmanned aerial vehicle (UAV), MUAVs can accomplish more complex and dangerous missions by collaboration, such as search and rescue, forest fire fighting, emergency rescue, low-attitude reconnaissance, and combat military missions [1–3]. Attitude cooperative control problem has been of growing interests in last several years due to its engineering and theoretical implications. Many scholars proposed different attitude control scheme to improve the accuracy and stability of the MUAVs. Variable structure control combined with decentralized communication scheme was proposed for spacecraft formation flying [4], based on the development of consensus theory, leader-follower was employed in the multiple aircrafts [5, 6]. As the amount of the MUAVs increases, the communication burden among each UAV will increase and may cause the network communication jam, and it would seriously affect the stability of the systems due to the band width is limited. It is significant to consider the network communication strategy of MUAVs when designing the attitude cooperative controller.Even-triggered scheme is employed in the multiagent systems for considering the limited band width and energy consumption, instead of continuous control input update, the controller updates the input depending on the event-triggered function, which is relevant to the measurement error, and when the estimation error comes up to the given threshold value the update of the controller will be updated [7]. An event-triggered-based controller was proposed in first-order multiagent systems (MAS) by introducing the event-triggered mechanism, and the triggered condition was designed associated with the states of agents [8]. Distributed rendezvous problem was investigated for second-order multiagent systems with combinational measurement by event-triggered mechanism [9]. Based on the measurement error, the event triggered function was built for linear MAS, and all the states of the agents reach to consensus [10, 11]. Event-triggered scheme was employed in many system dynamical model which can be described as second-order dynamics [12]. A distributed sliding mode controller based on event triggered finite time mechanism was designed for formation of multirobot systems [13]. The event triggered was widely used in attitude control of spacecrafts to save the communication resources [14–16]. However, the mentioned works with attitude control did not consider the uncertain parameters. Based on event-triggered strategy, time-varying formation problem was investigated for MUAVs under switching topology [17], attitude formation on SO (8) [18], and dynamical consensus formation problem which was limited by time-varying disturbances [19]. However, external disturbances and inertial matrix uncertainty cannot avoid in practical environment. The event-based formation control for MUAVs only guarantees asymptotically convergence in the aforementioned works [17–19]. Finite-time control is a useful tool which has high accuracy and robustness property, enabling the control systems to approach the stable region in finite time. Finite-time control has been extensively utilized in Euler-Lagrange systems [20–22]. Distributed attitude tracking problem of spacecrafts was proposed considering disturbances and uncertain parameters in finite time [20]. Adaptive control was introduced into the finite-time controller extended the mentioned work [20] for attitude tracking problem of spacecrafts [21]. Feedback control was employed in formation control for finite time convergence of nonholonomic wheeled mobile robots [22]. Distributed finite-time control (FTC) problem was studied for multiple quadrotor formation with the information of leader not available to all the followers [23]. However, no external disturbances were considered. The disturbance was estimated by the observer, and FTC was investigated for a single quadrotor [24, 25]. However, attitude cooperative problem was not considered. There is less work associated with the finite-time attitude cooperative or formation control with event-triggered mechanism for MUAVs. Most recently, FTC based on event-triggered was investigated for quadrotor flying control [26, 27]. However, attitude tracking problem was not considered, and the controller designed was limited to the specific UAV. So, attitude cooperative control with FTC theory and event-triggered mechanism is more interesting.Motivated by the aforementioned works and analysis, finite-time attitude cooperative control problem of MUAVs with event-triggered mechanism is investigated, and the network communication resources are reduced. The contribution of this paper is illustrated in the following aspects: (1) external disturbances and uncertain parameters are considered in the attitude dynamics, and the attitude cooperative problem is described by the quaternion avoiding the singularity. The attitude tracking consensus errors are measured by employing a positive error function, a novel integral sliding mode surface is proposed, the FTC is designed for the closed loop systems, and neural network is utilized to estimate the lumped uncertainties. (2) The communication frequency of the controller among the followers is reduced due to the event-triggered strategy which is employed in the controller, so the proposed novel event-triggered function saves the communication burden and energy of each UAV. The deduced lower bound between triggering intervals guarantee no Zeno behavior occurs. (3) Fast terminal sliding mode control is utilized in control law which guarantees that the attitude achieves the desired value in finite time.The rest of this paper is organized as follows. In Section2, preliminaries of graph theory, quaternion-based attitude dynamics of MUAVs, and some useful lemmas are given, and Section 3 gives main results. The performance of the controller is proved by numerical simulation examples in Section 4. Finally, conclusion is given in Section 5.


## 2. Preliminaries and Problem Formulation

### 2.1. Notations

The following convenience notations are adopted throughout the paper:Rn denotes n×1 real column vector, In=1,...1T denotes n×1 column vector with each element being, and In denotes a n×n dimensional identity matrix. ⊗ stands for Kronecker product. . stands for the induced matrix 2-norm or the Euclidean vector norm. In addition, for a given vector x∈Rn, xi denotes the ith element of the vector x, sigrx=sgnxxr.Graph theory is utilized to describe the communication flow among the MUAVs. LetG=ν,ξ denotes the graph, in which, ν=ν1,ν2,⋯,νn is a nonempty set containing a group number of nodes which denotes the UAV, and ξ⊆ν×ν is called edge which is a set of nodes. If there any two nodes could communicate with each other, the graph G is called connected graph. A=aijN×N∈Rn×n is weighted adjacency matrix representing the communication between each node, in which, aii=0, otherwise, aij=aji≥0. D=diagd1,d2,⋯,n denotes the degree matrix of associated with weighted graph, the elements of the degree matrix are di=∑j=1naij. The Laplacian matrix of the weighted graph is denoted by L=D−A, and L is symmetric matrix.Throughout this paper, leader-follower MUAVs are considered which contains one leader andn followers. The followers are marked as ii=1,⋯,n, and the leader is marked as 0. Let G¯ denotes the topology graph associated with MUAVs containing one leader and n followers. A diagonal matrix B=diagb1,⋯,bn is utilized to denote the communication between the follower UAV and the leader UAV. If bi>0 means that the ith follower can obtain the communication flow of the leader, otherwise, bi=0.Assumption 1.
Consider the MUAVs consisting ofN followers and one leader, the topology of the MUAVs is described by G¯, and G¯ is directed connected graph.Based on Assumption1 and graph theory, we define a matrix C=L+B.Lemma 2 (see [28]).
If Assumption1 holds, then, matrix C is invertible.Lemma 3 (see [29]).
Consider a system modeled asż=fz,f0=0,x∈ℝn, a continuous function Vz∈ℂ1 which is defined on a neighbourhood of the origin. If the function Vz satisfies that it is positive definite and V̇z≤−∇1V∂z, where ∂∈0,1, ∇1 is positive parameters. The systems converge to the origin in finite-time, the converge time T which depends on the initial state of z0:
(1)Tz0≤V1−∂z0∇11−∂.Lemma 4 (see [30]).
If there is a real numberxi∈ℝ,i=1,⋯,n, α∈0,1, then
(2)∑i=1n∣xi∣α≤∑i=1nxiα≤n1−α∑i=1n∣xi∣α.
Forx∈ℝn, α∈0,1, then
(3)xα≤n1−αxα.


### 2.2. Attitude Dynamics Model of MUAVs

Throughout this paper, the attitude dynamics of UAV is described by quaternion which could avoid singular problem and analyze conveniently [31].
(4)Jiω̇i=ui−SωiJiωi+ϑi,Q̇i=12ϕQiωi,where Qi=qiqi0T represents the attitude of the ith UAV, qi∈ℝ3, qi0∈ℝ, Qi∈ℝ4, Qi∈ℝ4,Qi=1, and ωi∈ℝ3 is the angular velocity. Ji denotes the inertia matrix of the ith UAV and is positive definite; ui denotes the control torque of the ith UAV; ϑi denotes the external disturbances. ϕQi is given by
(5)ϕQi=qi0I3+Sqi−qiΤ.For a vector given asn=n1,n2,n3T∈R3×1, Sn is defined as
(6)Sn=0‐n3n2n30‐n1‐n2n10.LetRQi denote the rotation matrix which is given as RQi=2qi02−1I3+2qiqiT−2qi0Πqi, Qi=qiqi0T for the attitude control of UAV, the rotation matrix denotes the inertial frame of the ith UAV into the body frame. The multiplication between two unit quarternions is given by
(7)Q1⊙Q2=q10q2+q20q1+Sq1q2q10q20−q1Tq2,where Q1=q1q10T and Q2=q2q20T.In the leader-following MUAVs, the followers adjust itself attitude to be consistent with the attitude of the leader. The attitude tracking errors and angular velocity errors of theith UAV are given as follows
(8)Q~i=Qi−1⊙Qd,(9)ω~i=ωi−RQ~iωd,where Qd≜qd,ηdT denotes the desired attitude which is given Qd≜qd,ηdT, and the desired angular velocity is denoted by ωd. Based on the definition of the tracking errors, ω~i and Q~i represent the attitude velocity tracking error and angular tracking errors, respectively.The attitude tracking error systems can be obtained,(10)Jiω~̇i=−ωi×⋅Jiωi+ui+Ti⋅Πω~i⋅RQid⋅ωd−RQid⋅ω̇d,(11)Q~̇i=12ϕQ~i⋅ω~i.Assumption 5.
Three positive constantsdm, σ1,σ2 exist and are satisfying di≤dm, ωi≤σ1, and ω̇i≤σ2, respectively.Assumption 6.
The inertia matrixJ¯i is known and nonsingular. ΔJ denotes the uncertainties and is bounded.Lemma 7 (see [32]).
Considering the system formulated as eqs. (10) and (11), for sliding mode surface κi=ω~i+r1qi+r2qic, where 0<c<1,r1>0,r2>0, for i=1,⋯,n. If the sliding mode surface reaches zero, then, ω~i=0,q0,i=1 and qi=0 can be reached in finite time, respectively.


## 2.1. Notations

The following convenience notations are adopted throughout the paper:Rn denotes n×1 real column vector, In=1,...1T denotes n×1 column vector with each element being, and In denotes a n×n dimensional identity matrix. ⊗ stands for Kronecker product. . stands for the induced matrix 2-norm or the Euclidean vector norm. In addition, for a given vector x∈Rn, xi denotes the ith element of the vector x, sigrx=sgnxxr.Graph theory is utilized to describe the communication flow among the MUAVs. LetG=ν,ξ denotes the graph, in which, ν=ν1,ν2,⋯,νn is a nonempty set containing a group number of nodes which denotes the UAV, and ξ⊆ν×ν is called edge which is a set of nodes. If there any two nodes could communicate with each other, the graph G is called connected graph. A=aijN×N∈Rn×n is weighted adjacency matrix representing the communication between each node, in which, aii=0, otherwise, aij=aji≥0. D=diagd1,d2,⋯,n denotes the degree matrix of associated with weighted graph, the elements of the degree matrix are di=∑j=1naij. The Laplacian matrix of the weighted graph is denoted by L=D−A, and L is symmetric matrix.Throughout this paper, leader-follower MUAVs are considered which contains one leader andn followers. The followers are marked as ii=1,⋯,n, and the leader is marked as 0. Let G¯ denotes the topology graph associated with MUAVs containing one leader and n followers. A diagonal matrix B=diagb1,⋯,bn is utilized to denote the communication between the follower UAV and the leader UAV. If bi>0 means that the ith follower can obtain the communication flow of the leader, otherwise, bi=0.Assumption 1.
Consider the MUAVs consisting ofN followers and one leader, the topology of the MUAVs is described by G¯, and G¯ is directed connected graph.Based on Assumption1 and graph theory, we define a matrix C=L+B.Lemma 2 (see [28]).
If Assumption1 holds, then, matrix C is invertible.Lemma 3 (see [29]).
Consider a system modeled asż=fz,f0=0,x∈ℝn, a continuous function Vz∈ℂ1 which is defined on a neighbourhood of the origin. If the function Vz satisfies that it is positive definite and V̇z≤−∇1V∂z, where ∂∈0,1, ∇1 is positive parameters. The systems converge to the origin in finite-time, the converge time T which depends on the initial state of z0:
(1)Tz0≤V1−∂z0∇11−∂.Lemma 4 (see [30]).
If there is a real numberxi∈ℝ,i=1,⋯,n, α∈0,1, then
(2)∑i=1n∣xi∣α≤∑i=1nxiα≤n1−α∑i=1n∣xi∣α.
Forx∈ℝn, α∈0,1, then
(3)xα≤n1−αxα.


## 2.2. Attitude Dynamics Model of MUAVs

Throughout this paper, the attitude dynamics of UAV is described by quaternion which could avoid singular problem and analyze conveniently [31].
(4)Jiω̇i=ui−SωiJiωi+ϑi,Q̇i=12ϕQiωi,where Qi=qiqi0T represents the attitude of the ith UAV, qi∈ℝ3, qi0∈ℝ, Qi∈ℝ4, Qi∈ℝ4,Qi=1, and ωi∈ℝ3 is the angular velocity. Ji denotes the inertia matrix of the ith UAV and is positive definite; ui denotes the control torque of the ith UAV; ϑi denotes the external disturbances. ϕQi is given by
(5)ϕQi=qi0I3+Sqi−qiΤ.For a vector given asn=n1,n2,n3T∈R3×1, Sn is defined as
(6)Sn=0‐n3n2n30‐n1‐n2n10.LetRQi denote the rotation matrix which is given as RQi=2qi02−1I3+2qiqiT−2qi0Πqi, Qi=qiqi0T for the attitude control of UAV, the rotation matrix denotes the inertial frame of the ith UAV into the body frame. The multiplication between two unit quarternions is given by
(7)Q1⊙Q2=q10q2+q20q1+Sq1q2q10q20−q1Tq2,where Q1=q1q10T and Q2=q2q20T.In the leader-following MUAVs, the followers adjust itself attitude to be consistent with the attitude of the leader. The attitude tracking errors and angular velocity errors of theith UAV are given as follows
(8)Q~i=Qi−1⊙Qd,(9)ω~i=ωi−RQ~iωd,where Qd≜qd,ηdT denotes the desired attitude which is given Qd≜qd,ηdT, and the desired angular velocity is denoted by ωd. Based on the definition of the tracking errors, ω~i and Q~i represent the attitude velocity tracking error and angular tracking errors, respectively.The attitude tracking error systems can be obtained,(10)Jiω~̇i=−ωi×⋅Jiωi+ui+Ti⋅Πω~i⋅RQid⋅ωd−RQid⋅ω̇d,(11)Q~̇i=12ϕQ~i⋅ω~i.Assumption 5.
Three positive constantsdm, σ1,σ2 exist and are satisfying di≤dm, ωi≤σ1, and ω̇i≤σ2, respectively.Assumption 6.
The inertia matrixJ¯i is known and nonsingular. ΔJ denotes the uncertainties and is bounded.Lemma 7 (see [32]).
Considering the system formulated as eqs. (10) and (11), for sliding mode surface κi=ω~i+r1qi+r2qic, where 0<c<1,r1>0,r2>0, for i=1,⋯,n. If the sliding mode surface reaches zero, then, ω~i=0,q0,i=1 and qi=0 can be reached in finite time, respectively.


## 3. Main Results

### 3.1. Event-Triggered Finite-Time Control Design

In this section, the control objective is to design a finite-time control law such that the angular velocity errorsω~i and the error quaternions Q~i of the closed-loop system (10) and (11) can converge to small regions in finite time, respectively.First, the sliding mode surfaces¯i is defined as
(12)s¯i=J¯iω~i+k1q~i+k2Τiq~i,with Τiq~iq~i=Τi1q~i1,Τi2q~i2,Τi3q~i3T∈R3×1,
(13)Τijq~ij=sigℓq~ij,ifsij∗=0orsij∗≠0,q~ij>ϒ,ϖ1q~ij+ϖ2sig2q~ij,ifsij∗≠0,q~ij≤ϒwhere i=1,⋯,n,j=1,2,3,si∗=si1∗,si2∗,si3∗T, and si∗=w~i+k1q~i+k2sigrq~i, where k1 and k2 are positive constants. Define sigιq~i=sigιq~i1,sigιq~i2,sigιq~i3T, ι∈0,1, ϖ1=2−rϒr−1, ϖ2=r−1ϒr−2, ϒ is a small positive constant.To develop the control law, the following equations are derived from (10) and (11):
(14)J¯iw~̇i+k1q~̇i+k2Τ̇iq~i=zi+δi+ui,where
(15)zi=−SwiJ¯iwi+J¯iSw~iRQ~iwd−RQ~iẇd+k1J¯iq~̇i+k2J¯iα̇iq~i,(16)α̇iq~i=rdiagq~ijr−1q~i,ifsij∗=0orsij∗≠0,q~ij>ϕ,l1q~̇i+2l2q~isgnq~iq~̇i,ifsij∗≠0,q~ij≤ϕ,(17)δi=ϑi−J~iw~̇i−SwiJ~iwi+J~iSω~iRQ~iωd−RQ~iω̇d.δi is the lumped disturbances containing model uncertainty and external disturbances.By (12) and (14), we can obtain
(18)s¯̇i=zi+δi+ui.Based on the sliding mode surfaces¯i=J¯iω~i+k1q~i+k2Τiq~i, a novel integral sliding mode surface is proposed which is given as follows
(19)si=s¯i−∫0txiηdt,where xi=−∑j∈Niaijs¯i−s¯j+bis¯i, and η∈0.5,1 is strictly the ratio of positive odd numbers. The derivative of (19) is
(20)ṡi=s¯̇i−xiη.An event-triggered finite-time sliding mode consensus controller is designed as follows(21)uit=xiηtki−k3signsitki−k4sitki−zitki−δ^itki,where k3 and k4 are positive constants, respectively. For t∈tki,tk+1i, tki is the latest event-triggered time for the ith UAV, and the UAV only updates the control protocol at its own event-triggered time.An adaptive radial basis function neural networks (RBFNNs) scheme is proposed for the unknown disturbanceδi, as RBFNNs can estimate the unknown continuous functions δi and ensure tracking error ultimately converges to an adequately small compact. Illustrated in Figure 1, the adaptive RBFNNs can be written as
(22)δ^i=W^iTHiXin,where δ^i∈ℝ3 is the RBFNNs output vector, Xin=q~iT,w~iTT∈ℝ6 is the input vector of the RBFNNs, W^i∈ℝJ×3 is the weight vector, J>1 is the nodes number of middle hidden layer, Hi=hi1,⋯,hiJT∈ℝJ is the basis function vector, and hiJ is being the commonly used Gaussian functions, which is simplified as
(23)hiJXin=exp−Xin−ciJ2σiJ,where ciJ is the center of the receptive field, and σiJ is the width of the Gaussian function. There exists an optimal vector Wi∗ such that
(24)δi=Wi∗THiXin+εi,where εmi denotes the maximum value of the RBFNNs estimation error εi.Figure 1
Adaptive RBFNN control.LetW~i=Wi∗−W^i denotes the vector of weight errors, and the adaptive weight update law are designed as
(25)W^̇i=AHiXinsiT,where A is a positive-definite symmetric matrix of gains.The measurement error of the event-triggered mechanism is defined as(26)eit=xiηtki−xiηt−k3signsitki+k3signsit−k4sitki+k4sit−zitki+zit−δ^itki+δ^it.


### 3.2. Stability Analysis

In the following, the stability of the attitude cooperative under event-triggered adaptive RBFNNs control law is analyzed in detail.Theorem 8.
On the basis of Assumptions1, and considering that the system (10) and (11) under the action of the controller (21) and the adaptive weight update law (25), the following event-triggered function is given as follows
(27)Yt=ei−k3−k4si+ρi,where εmi<ρi<k3, and when the event-triggered function Yt>0, that is, ei>k3+k4si−ρi, the event is triggered. The ith UAV performs information interaction and update the control protocol at its own event-triggered time. And the system can achieve finite-time consensus under this action.Proof.
First, selecting the Lyapunov function as follows(28)V1=12siTsi+12trW~iTA−1W~i.
Taking the derivative of (28), substituting (21) and (26) into the derivative, we can obtain
(29)V̇1=siTei−k3signsi−k4si+δi−δ^i+tr−W~iTA−1W^̇i≤siei−k3si−k4si2+siTW~iTHi+εi+tr−W~iTA−1W^̇i≤siei−k3−k4si+εmi+trW~iTHisiT+tr−W~iTA−1W^̇i.
By the adaptive weight update law (25), it is obtained that
(30)V̇1≤siei−k3−k4si+εmi.
When the event-triggered functionYt≤0, ei≤k3+k4si−ρi,
(31)V̇1≤−siρi−εmi<0.
According to the Lyapunov stability theory, it can be seen that under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), the sliding mode surface si can realize si=0 and ṡi=0. By lemma 2, the reaching time is given as follows
(32)tr=2V11/20ρi−εmi.
Then, select the Lyapunov function as(33)V2=12S¯ΤL+B⊗I3ΤL+B⊗I3S¯,where S¯=s¯1T,⋯,s¯nTT.
Define(34)pi=∑j∈niaijs¯i−s¯j+bis¯i,P=p1Τ,⋯,pnΤΤ=L+B⊗I3⋅S¯.
Then, we can get(35)V2=12PΤP.
LetS=s1T,⋯,snTT, when ṡi=0, we know s¯̇i=xiη, thenS¯̇=−L+B⊗I3⋅S¯η.
Under Assumption1 and Lemma 2, it can be obtained that λminL+B>0. And taking the derivative of (35), we can obtain
(36)V̇2=−PΤL+B⊗I3L+B⊗I3⋅S¯η=−PΤL+B⊗I3Pη≤−λminL+BPΤPη,
Since the positive odd ratio parameterη∈0.5,1, and combined with Lemma 3, we can find
(37)V̇2≤−λminL+B∑i=13npi1+η≤−λminL+BP21+η/2≤−21+η/2λminL+BV21+η/2.
According to Lemma 1, under the action of the controller (21), the state of the system can reach and remain on the sliding mode surface S¯=0 within finite time. The settling time is tf.
(38)tf=V21−η/221+η/2λminL+B1−η/2.
WhenS¯=0, then s¯i=J¯iw~i+k1q~i+k2αiq~i=0, so w~i+k1q~i+k2αiq~i=0. According to Lemma 4, we know w~i⟶0, q~i⟶0 in finite time will be satisfied.
Next, we need to analyze whether the system has a minimum event-triggered time interval strictly greater than zero, which means that there is no Zeno behavior. When the event-triggered function (27) satisfies Yt>0, the event is triggered. Combining (26), we can see that between any two adjacent event-triggered moments, ei increases from zero to k3+k4si−ρi, therefore, when the growth rate is the fastest, the event-triggered time interval is the smallest. In this case, when the minimum time interval is a value greater than zero, it can be guaranteed that there is no Zeno behavior.☐Theorem 9.
Based on Assumption1, the system (10) and (11) under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), the system does not have Zeno behavior under any initial conditions.Proof.
Letβit=k3signsit+k4sit+zit+δ^it, since the system (10) and (11) can achieve consensus under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), β̇it have upper bounds, which are taken as β̇itmax. And combining (26) to derive ei as follows
(39)deidt≤deidt≤d−xiηt+βitdt≤ddtxiηt+β̇it≤ηxiη−1ẋi+β̇itmax.Let X=x1T,⋯,xnTT, we can obtain Ẋ=−L+B⊗I3⋅S¯̇. When Ṡ=0, S¯̇=Xη will be satisfied, so Ẋ=−L+B⊗I3⋅Xη. And according to Lemma 3, we know
(40)xiη−1≤Xη−1≤3n2−ηXη−1,ẋi≤Ẋ≤L+B⊗I3⋅Xη≤L+BXη≤3n1−ηL+BXη.
By (38) and (39), we can obtain
(41)deidt≤η3n3−2ηL+BX2η−1+β̇itmax.
Due toX=−P and P=2V21/2t≤2V21/20, we have
(42)deidt≤22η−1/2η3n3−2ηL+BV22η−1/20+β̇itmax.
For anyt∈tki,tk+1i, tki is the latest event-triggered time for the ith UAV, the time interval Tmi=tk+1i−tki, and eitki=0 at the event-triggered moment, and let λi=22η−1/2η3n3−2ηL+BV22η−1/20+β̇itmax, we can obtain
(43)eit−eitki=eit≤t−tkiλi≤Tmiλi.
When the event-triggered function (27) satisfies Yt>0, the event is triggered, we have
(44)eit>k3+k4si−ρi>k3−ρi.
Combining (41) and (42), we can know
(45)Tmi>k3−ρiλi.
It can be concluded from (43) that the event-triggered time interval is strictly greater than zero, so there is no Zeno behavior.☐


## 3.1. Event-Triggered Finite-Time Control Design

In this section, the control objective is to design a finite-time control law such that the angular velocity errorsω~i and the error quaternions Q~i of the closed-loop system (10) and (11) can converge to small regions in finite time, respectively.First, the sliding mode surfaces¯i is defined as
(12)s¯i=J¯iω~i+k1q~i+k2Τiq~i,with Τiq~iq~i=Τi1q~i1,Τi2q~i2,Τi3q~i3T∈R3×1,
(13)Τijq~ij=sigℓq~ij,ifsij∗=0orsij∗≠0,q~ij>ϒ,ϖ1q~ij+ϖ2sig2q~ij,ifsij∗≠0,q~ij≤ϒwhere i=1,⋯,n,j=1,2,3,si∗=si1∗,si2∗,si3∗T, and si∗=w~i+k1q~i+k2sigrq~i, where k1 and k2 are positive constants. Define sigιq~i=sigιq~i1,sigιq~i2,sigιq~i3T, ι∈0,1, ϖ1=2−rϒr−1, ϖ2=r−1ϒr−2, ϒ is a small positive constant.To develop the control law, the following equations are derived from (10) and (11):
(14)J¯iw~̇i+k1q~̇i+k2Τ̇iq~i=zi+δi+ui,where
(15)zi=−SwiJ¯iwi+J¯iSw~iRQ~iwd−RQ~iẇd+k1J¯iq~̇i+k2J¯iα̇iq~i,(16)α̇iq~i=rdiagq~ijr−1q~i,ifsij∗=0orsij∗≠0,q~ij>ϕ,l1q~̇i+2l2q~isgnq~iq~̇i,ifsij∗≠0,q~ij≤ϕ,(17)δi=ϑi−J~iw~̇i−SwiJ~iwi+J~iSω~iRQ~iωd−RQ~iω̇d.δi is the lumped disturbances containing model uncertainty and external disturbances.By (12) and (14), we can obtain
(18)s¯̇i=zi+δi+ui.Based on the sliding mode surfaces¯i=J¯iω~i+k1q~i+k2Τiq~i, a novel integral sliding mode surface is proposed which is given as follows
(19)si=s¯i−∫0txiηdt,where xi=−∑j∈Niaijs¯i−s¯j+bis¯i, and η∈0.5,1 is strictly the ratio of positive odd numbers. The derivative of (19) is
(20)ṡi=s¯̇i−xiη.An event-triggered finite-time sliding mode consensus controller is designed as follows(21)uit=xiηtki−k3signsitki−k4sitki−zitki−δ^itki,where k3 and k4 are positive constants, respectively. For t∈tki,tk+1i, tki is the latest event-triggered time for the ith UAV, and the UAV only updates the control protocol at its own event-triggered time.An adaptive radial basis function neural networks (RBFNNs) scheme is proposed for the unknown disturbanceδi, as RBFNNs can estimate the unknown continuous functions δi and ensure tracking error ultimately converges to an adequately small compact. Illustrated in Figure 1, the adaptive RBFNNs can be written as
(22)δ^i=W^iTHiXin,where δ^i∈ℝ3 is the RBFNNs output vector, Xin=q~iT,w~iTT∈ℝ6 is the input vector of the RBFNNs, W^i∈ℝJ×3 is the weight vector, J>1 is the nodes number of middle hidden layer, Hi=hi1,⋯,hiJT∈ℝJ is the basis function vector, and hiJ is being the commonly used Gaussian functions, which is simplified as
(23)hiJXin=exp−Xin−ciJ2σiJ,where ciJ is the center of the receptive field, and σiJ is the width of the Gaussian function. There exists an optimal vector Wi∗ such that
(24)δi=Wi∗THiXin+εi,where εmi denotes the maximum value of the RBFNNs estimation error εi.Figure 1
Adaptive RBFNN control.LetW~i=Wi∗−W^i denotes the vector of weight errors, and the adaptive weight update law are designed as
(25)W^̇i=AHiXinsiT,where A is a positive-definite symmetric matrix of gains.The measurement error of the event-triggered mechanism is defined as(26)eit=xiηtki−xiηt−k3signsitki+k3signsit−k4sitki+k4sit−zitki+zit−δ^itki+δ^it.


## 3.2. Stability Analysis

In the following, the stability of the attitude cooperative under event-triggered adaptive RBFNNs control law is analyzed in detail.Theorem 8.
On the basis of Assumptions1, and considering that the system (10) and (11) under the action of the controller (21) and the adaptive weight update law (25), the following event-triggered function is given as follows
(27)Yt=ei−k3−k4si+ρi,where εmi<ρi<k3, and when the event-triggered function Yt>0, that is, ei>k3+k4si−ρi, the event is triggered. The ith UAV performs information interaction and update the control protocol at its own event-triggered time. And the system can achieve finite-time consensus under this action.Proof.
First, selecting the Lyapunov function as follows(28)V1=12siTsi+12trW~iTA−1W~i.
Taking the derivative of (28), substituting (21) and (26) into the derivative, we can obtain
(29)V̇1=siTei−k3signsi−k4si+δi−δ^i+tr−W~iTA−1W^̇i≤siei−k3si−k4si2+siTW~iTHi+εi+tr−W~iTA−1W^̇i≤siei−k3−k4si+εmi+trW~iTHisiT+tr−W~iTA−1W^̇i.
By the adaptive weight update law (25), it is obtained that
(30)V̇1≤siei−k3−k4si+εmi.
When the event-triggered functionYt≤0, ei≤k3+k4si−ρi,
(31)V̇1≤−siρi−εmi<0.
According to the Lyapunov stability theory, it can be seen that under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), the sliding mode surface si can realize si=0 and ṡi=0. By lemma 2, the reaching time is given as follows
(32)tr=2V11/20ρi−εmi.
Then, select the Lyapunov function as(33)V2=12S¯ΤL+B⊗I3ΤL+B⊗I3S¯,where S¯=s¯1T,⋯,s¯nTT.
Define(34)pi=∑j∈niaijs¯i−s¯j+bis¯i,P=p1Τ,⋯,pnΤΤ=L+B⊗I3⋅S¯.
Then, we can get(35)V2=12PΤP.
LetS=s1T,⋯,snTT, when ṡi=0, we know s¯̇i=xiη, thenS¯̇=−L+B⊗I3⋅S¯η.
Under Assumption1 and Lemma 2, it can be obtained that λminL+B>0. And taking the derivative of (35), we can obtain
(36)V̇2=−PΤL+B⊗I3L+B⊗I3⋅S¯η=−PΤL+B⊗I3Pη≤−λminL+BPΤPη,
Since the positive odd ratio parameterη∈0.5,1, and combined with Lemma 3, we can find
(37)V̇2≤−λminL+B∑i=13npi1+η≤−λminL+BP21+η/2≤−21+η/2λminL+BV21+η/2.
According to Lemma 1, under the action of the controller (21), the state of the system can reach and remain on the sliding mode surface S¯=0 within finite time. The settling time is tf.
(38)tf=V21−η/221+η/2λminL+B1−η/2.
WhenS¯=0, then s¯i=J¯iw~i+k1q~i+k2αiq~i=0, so w~i+k1q~i+k2αiq~i=0. According to Lemma 4, we know w~i⟶0, q~i⟶0 in finite time will be satisfied.
Next, we need to analyze whether the system has a minimum event-triggered time interval strictly greater than zero, which means that there is no Zeno behavior. When the event-triggered function (27) satisfies Yt>0, the event is triggered. Combining (26), we can see that between any two adjacent event-triggered moments, ei increases from zero to k3+k4si−ρi, therefore, when the growth rate is the fastest, the event-triggered time interval is the smallest. In this case, when the minimum time interval is a value greater than zero, it can be guaranteed that there is no Zeno behavior.☐Theorem 9.
Based on Assumption1, the system (10) and (11) under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), the system does not have Zeno behavior under any initial conditions.Proof.
Letβit=k3signsit+k4sit+zit+δ^it, since the system (10) and (11) can achieve consensus under the action of the controller (21), the adaptive weight update law (25), and the event-triggered function (27), β̇it have upper bounds, which are taken as β̇itmax. And combining (26) to derive ei as follows
(39)deidt≤deidt≤d−xiηt+βitdt≤ddtxiηt+β̇it≤ηxiη−1ẋi+β̇itmax.Let X=x1T,⋯,xnTT, we can obtain Ẋ=−L+B⊗I3⋅S¯̇. When Ṡ=0, S¯̇=Xη will be satisfied, so Ẋ=−L+B⊗I3⋅Xη. And according to Lemma 3, we know
(40)xiη−1≤Xη−1≤3n2−ηXη−1,ẋi≤Ẋ≤L+B⊗I3⋅Xη≤L+BXη≤3n1−ηL+BXη.
By (38) and (39), we can obtain
(41)deidt≤η3n3−2ηL+BX2η−1+β̇itmax.
Due toX=−P and P=2V21/2t≤2V21/20, we have
(42)deidt≤22η−1/2η3n3−2ηL+BV22η−1/20+β̇itmax.
For anyt∈tki,tk+1i, tki is the latest event-triggered time for the ith UAV, the time interval Tmi=tk+1i−tki, and eitki=0 at the event-triggered moment, and let λi=22η−1/2η3n3−2ηL+BV22η−1/20+β̇itmax, we can obtain
(43)eit−eitki=eit≤t−tkiλi≤Tmiλi.
When the event-triggered function (27) satisfies Yt>0, the event is triggered, we have
(44)eit>k3+k4si−ρi>k3−ρi.
Combining (41) and (42), we can know
(45)Tmi>k3−ρiλi.
It can be concluded from (43) that the event-triggered time interval is strictly greater than zero, so there is no Zeno behavior.☐


## 4. Example Simulation

Considering the system composed of four UAVs includes three follower UAVs and one leader UAV, and the leader node is marked as 0. The directed communication topology is shown in Figure2. Hence, we have
(46)L=1−1001−1−101B=100010001.Figure 2
Communication topology.The actual inertia matrices are assumed to be(47)J1=15112160.500.514J2=130.501150.501.514J3=141201302115.With the existence of model uncertainties and external disturbances, the nominal inertia matrices of the UAV are given byJ¯1=J¯2=J¯3=diag202020T. Take the disturbances as
(48)d1=0.1sint,0.2cos0.5t,0.15cos0.7tT,d2=0.1cost,0.2sin0.5t,0.15sin0.7tT,d3=0.1cost,0.2cos0.5t,0.15sin0.7tT.The initial quaternions of the follower UAVs are selected asQ10=1,0,0,0T, Q20=0,1,0,0T, and Q30=0,0,1,0T, the initial angular velocities of the follower UAVs are selected as w10=0,0,1T, w20=0,0,1T, and w30=0,0,1T. The initial quaternion of the leader UAV is selected as Q00=0,0,0,1T, and the angular velocity of the leader UAV is given asw0t=0.1cos0.2t,−0.1sin0.2t,−0.1cos0.2tT.The controller parameters are chosen withk1=1, k2=0.001, k3=0.01, k4=1, r=0.6, ϕ=0.1, η=0.1, and ρi=0.009. The adaptive RBFNN controller parameters are adjusted as J=7, ci=−1.5,−1,−0.5,0,0.5,1,1.5, σi=5, and A=diag0.50.50.5T.Figures3 and 4, respectively, show the attitude tracking Qi and the attitude tracking errors Q~i of the ith UAV,i=1,2,3.Figure 3
Attitude tracking.
(a)(b)(c)(d)Figure 4
Attitude tracking errors.
(a)(b)(c)Figures5 and 6, respectively, show the angular velocity tracking wi and the attitude tracking errors w~i of the ith UAV. From Figures 3–6, it can be seen that the attitude and the angular velocity of all follower UAVs can accurately track the leader UAV over time under the action of the controller (17) and the event-triggered function (23).Figure 5
Angular velocity tracking.
(a)(b)(c)Figure 6
Angular velocity errors.
(a)(b)(c)Figure7 shows the evolution process of the measurement error norm of the system, where threshold eimax=k3+k4si−ρi. When the value of ei increases from zero to eimax, the event is triggered.Figure 7
Variation trend of measurement error normei and thresholdeimax.
(a)(b)(c)(d)(e)(f)Figures8–10 show the event-triggered time of the UAV i, i=1,2,3, and the denser part marked by the rectangular box is enlarged. At the event-triggered time of the UAV i, the UAV i interacts with information and updates the controller. Figure 11 shows the control torque ui of the UAV i.Figure 8
Event-triggered time for UAV1.
(a)(b)(c)(d)Figure 9
Event-triggered time for UAV2.
(a)(b)(c)(d)Figure 10
Event-triggered time for UAV3.
(a)(b)(c)(d)Figure 11
Control torques.
(a)(b)(c)From Figures7–11, it can be seen that the superior performance of the proposed event-triggered control strategy in reducing the energy dissipation of the system and the update frequency of the controller.The approximation errorεi of the RBFNNs to unknown lumped disturbance δi is shown in Figure 12. It can be seen that the RBFNNs can approach δi at a faster speed under the action of the adaptive weight update law (23).Figure 12
The RBFNN estimation errorεi.


## 5. Conclusions

In this paper, a distributed finite time event-triggered control strategy with RBFNNs is proposed for attitude cooperative control of MUAVs. Under the leader-following framework, the tracking errors of attitude converge to zero in finite time, the communication resources is saved and the Zeno behavior is excluded by utilizing the event-triggered scheme. Finally, theory and numerical simulation proof is given for the proposed control law. In the future, we will consider actuator saturation problem by fault-tolerant technology and self-triggered scheme to be used in finite-time control.



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*Source: 5875004-2022-02-21.xml*