# Event-Based Time Varying Formation Control for Multiple Quadrotor UAVs with Markovian Switching Topologies

Authors: Zhen Zhou; Hongbin Wang; Zhongquan Hu
Journal: Complexity
 (2018)

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## Abstract

Time varying formation control problem for a group of quadrotor unmanned aerial vehicles (UAVs) under Markovian switching topologies is investigated through a modified dynamic event-triggered control protocol. The formation shape is specified by a time varying vector, which prescribes the relative positions and bearings among the whole agents. Instead of the general stochastic topology, the graph is governed by a set of Markov chains to the edges, which can recover the traditional Markovian switching topologies in line with the practical communication network. The stability proof for the state space origin of the overall closed-loop system is derived from the singular perturbation method and Lyapunov stability theory. An event-triggered formation control protocol in terms of a dynamically varying threshold parameter is delicately carried out, while acquiring satisfactory resource efficiency, and Zeno behavior of triggering time sequences is excluded. Finally, simulations on six quadrotor UAVs are given to verify the effectiveness of the theoretical results.


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## Body

## 1. Introduction

Along with the increasing applications in various areas, such as aerial photography, express delivery, and disaster relief, formation control of multiagent systems has attracted considerable attention from many researchers [1, 2]. In particular, as a typical class of physical systems with practical interest, the quadrotor UAVs is widely used in the military and civilian fields [3, 4]. Actually, due to the strong nonlinear coupling and limited communication resources [5], the control problem of multiple quadrotor UAVs will be very challenging and difficult [6]. Therefore, how to design the formation control protocol for multiple quadrotor UAVs subject to limited communication resources becomes a significant research focus.A defining feature of formation control problem is that multiple agents work together to accomplish a collaborative formation task [7]. Several classic formation control strategies, including leader-follower, virtual structure, and behavior based methods, were applied in the scientific community [8, 9]. For example, formation control of multiple quadrotor UAVs, based on position estimation [10], backstepping design technique [11], and finite time algorithms [12], respectively, was investigated so as to make a construct and keep the formation shape during flying. It should be pointed out that time varying formation tracking problems arise in some scenarios, such as source seeking and target enclosing. For example, time varying formation analysis and design problems for multiagent systems with switching topologies were solved in [13, 14]. Based on the fact that multiagent systems subject to random abrupt variations could be modeled as the switching systems, then some results have been obtained on it [14]. Compared with the previous works, time varying formation control results for multiagent systems with switching topologies were provided in [15, 16]. Besides, due to random link failures, variation meeting the need and sudden environmental disturbances [17], some dynamical systems could be modeled as Markovian switching systems, which were governed by a set of Markov chains [18, 19]. By considering the complex network as Markovian switching topologies, it plays a crucial role in the field of networked control system [20].In practice, under a limited bandwidth, it is necessary and important to consider the issues of energy waste and competition [21]. Therefore, event-triggered communication mechanism was born at the right moment [22, 23]. As a popular research topic, some latest event-triggered control results were provided in [24–28]. In particular, compared with the general event-triggered controller with a fixed threshold parameter, the authors in [24, 29] developed the dynamic/adaptive event-triggered control protocol of multiagent systems for acquiring satisfactory resource efficiency, respectively. Meanwhile, take the strong nonlinear coupling and underactuated of the quadrotor UAVs into account, time scaling based control method has also been recognized as a powerful tool in the analysis and design of controllers, which is with crucial importance in applications to the mobile inverted pendulum [30], the ball-beam system [31], and the quadrotor UAVs [32]. Therefore, it is of great importance to extend the event-triggered formation results to multiple quadrotor UAVs under Markovian switching topologies. In addition, it is difficult to obtain all the transition rates under the realistic communication environment [33]. So that randomly occurring control strategy is more realistic and meaningful to accomplish attitude stabilization and formation missions under the limited communication resources.Motivated by these observations, the contributions of this paper are proposing a novel time varying formation control strategy and an event-triggered communication scheme to solve the formation problem of multiple quadrotor UAVs with Markovian switching topologies. The main highlights of this paper are summarized as follows. First, a modified graph of entire system is governed by a set of Markov chains to the edges, and the traditional Markovian switching topologies can be recovered through adjusting the modes of edges and the transition rates. Second, the dynamic event-triggered controller is derived from a time scaling based control strategy, which consists of two parts: the closed-loop system stability analysis based on the framework of singularly perturbed theory and the event-triggered control scheme in terms of a new dynamically varying threshold parameter to guarantee time varying formation shape. Third, Markovian switching topologies involve partly unknown transition rates, which are of great importance to be considered and thus closer to the realistic communication environment. In addition, Zeno behavior can be excluded during the whole running process. Finally, several simulations can illustrate the theoretical results.The rest of this paper is organized as follows. The system dynamics and some preliminaries on graph theory are introduced in Section2; Section 3 provides main results on event-triggered formation control for multiple quadrotor UAVs. In Section 4, simulation results are given and this paper is concluded in Section 5.Notations 1.
Throughout this paper,∙ denotes L2-vector norm and ρ∙ stands for the spectral radius for matrices. The notation A⊗B means the Kronecker product of matrices A and B, and λmax∙ and λmin∙ represent its maximum and minimum eigenvalues.


## 2. Preliminaries and System Formulation

### 2.1. Graph Theory

Define a time varying random undirected graphGωt=ν,ξωijt,Aωt with a nonempty finite vertex set ν=ν1,…,νN and an edge set ξωijt⊆ν×ν. Different with the general ones, it consists of a time sequence of random graphs in which the edge set ξ varies with t. Namely, each edge νi,νj evolves according to a homogeneous Markov process ωijt, which takes values in S=1,2,…,s with the transition rate as(1)Prωijt+Δ=q∣ωijt=p=πpqΔ+oΔq≠p1+πppΔ+oΔq=p.Assume that ωijt do not change infinitely fast; thus, ωijt=ωijt+Δt if 0<Δt<Δ. It means that the total number of system modes is sN+1N/2 and the total transition rate is given by(2)Prωt+Δ∣ωt=∏i,j∈νPrωijt+Δ=q∣ωijt=p.The weighted adjacency matrixAω≜Aωt=aijωijt∈RN×N is associated with Gωt. Here aijω≜aijωijt>0 if νi,νj∈ξωijt and aijω=0 otherwise. Assumed that there is no self-loop in the graph, which implies that aiiω=0. In this paper, the set of neighbors with respect to the agent νi is Ωiω=νj∈ν∣νj,νi∈ξωijt. A graph Gωt is connected, if there is a path between any two vertices; otherwise, it is disconnected. A diagonal matrix Dω=diagdi∈RN×N with di=∑j∈Ωiωaijω being the ith row sum of Aω. Then, the Laplacian of the graph is defined as Lω=Dω-Aω. Consider the formation with a leader-follower structure by introducing a diagonal matrix Ll=diagaiil∈RN×N, which evolves according to the Markov process ω0t with a finite mode set S0=1,2,…,s0 and a time interval Δ0, where aiil>0 if vi is a leader and aiil=0 otherwise. Hence, the interaction matrix is given by LG=Lω+Ll.Assumption 2.
The undirected graphGωt is connected.Remark 3.
For simplicity, we just consider the undirected graph in this paper, that is,aijω=ajiω. Note that, if the graph is a general directed one, there will be some small differences. A possible approach to consider a directed graph is to introduce two Markov chains in each two agents; an alternative is to extend the state space set S, which could be defined according to the weight and direction of the graph. Both approaches will be addressed in our future work.Remark 4.
With a limited bandwidth, the switching ofaijω is caused by the sensing/detecting failure and communication failure, which is passive. In fact, it is difficult to obtain all the elements of the transition rate matrix, or some of the elements are not necessary to guarantee the system stability. Since that, the transition rate matrices are assumed to be partly accessed; even some of them are unknown completely, which could be descried as follows:(3)p11p12⋯???⋯p2s⋮⋮⋱⋮ps1?⋯pss,??⋯???⋯?⋮⋮⋱⋮??⋯?,where “?” represents the unknown transition rate.


### 2.2. Problem Formulation

Consider a group ofN quadrotor UAVs as shown in Figure 1; the dynamics of agent i are given as the following form [12]:(4)x¨i=sin⁡ψisin⁡ϕi+cos⁡ψisin⁡θicos⁡ϕiu1imi-kxix˙imi,y¨i=-cos⁡ψisin⁡ϕi+sin⁡ψisin⁡θicos⁡ϕiu1imi-kyiy˙imi,z¨i=-g+cos⁡θicos⁡ϕiu1imi-kziz˙imi,ϕ¨i=liu2iIxi-likϕiϕ˙iIxi,θ¨i=liu3iIyi-likθiθ˙iIyi,ψ¨i=Kψiu4iIzi-kψiψ˙iIzi,where xi,yi,zi∈R are positions and ϕi,θi,ψi∈-π/2,π/2 denote the three Euler angles of rotation, representing pitch, roll, and yaw, respectively. kϕi,θi,ψi are the aerodynamic friction coefficients, and kxi,yi,zi are the coefficients of the translation drag forces. Ixi,yi,zi are the quadrotor moments of inertias, mi denotes the mass of the quadrotor, li is half length of the helicopter, Kψi is thrust to moment gain, and g is gravitational acceleration.Figure 1
Group of theN quadrotor UAVs.Each system dynamics consist of six degrees of freedom model, which can be separated into position dynamic and attitude dynamic. With the choice of state variablesx1i=xi, x2i=x˙i; y1i=yi, y2i=y˙i; z1i=zi, z2i=z˙i; ϕ1i=ϕi, ϕ2i=ϕ˙i; θ1i=θi, θ2i=θ˙i; ψ1i=ψi, ψ2i=ψ˙i. Then, it can be represented by(5)x˙1i=x2i,y˙1i=y2i,z˙1i=z2i,x˙2i=sin⁡ψisin⁡ϕi+cos⁡ψisin⁡θicos⁡ϕiu1imi-kxix2imi,y˙2i=-cos⁡ψisin⁡ϕi+sin⁡ψisin⁡θicos⁡ϕiu1imi-kyiy2imi,z˙2i=-g+cos⁡θicos⁡ϕiu1imi-kziz2imi,ϕ˙1i=ϕ2i,θ˙1i=θ2i,ψ˙1i=ψ2i,ϕ˙2i=liu2iIxi-likϕiϕ2iIxi,θ˙2i=liu3iIyi-likθiθ2iIyi,ψ˙2i=Kψiu4iIzi-kψiψ2iIzi.As is well known to us, the quadrotor is underactuated and differentially flat. Accordingly, choose four variables and specify the desired trajectory asRd=xdt,ydt,zdt,ψdtT.


### 2.3. Formation Definition

The formation shape can be described by a vectorδt=δ1Tt,…,δNTtT∈R4N, where δit=δixt,δiyt,δizt,δiψtT is the continuously differentiable formation vector, such that(6)limt→∞x1it-δixt-xdt=0,limt→∞y1it-δiyt-ydt=0,limt→∞z1it-δizt-zdt=0,limt→∞ψ1it-δiψt-ψdt=0.Define the desired position of agent i as xid,yid,zid,ψidT=δix+xd,δiy+yd,δiz+zd,δiψ+ψdT. For j∈Ωiω, there exists δij=δijx,δijy,δijz,δijψT=δjx-δix,δjy-δiy,δjz-δiz,δjψ-δiψT. Hence, there are NN-1/2 such shape vectors satisfying the following properties:(7)δik=δij+δjk,δii=0,0,0,0T,δji=-δij,∀i,j,k∈1,…,N.Remark 5.
The formation reference vectorRd is not available to all agents, but the desired interdistances of agent i and its neighbors are known. Note that the time varying formation vector δt is not unique, and the relative position between the reference vector Rd and the formation can be adjusted.Lemma 6 (see [34]).
For anyX,Y∈Rn and ρ1>0, it holds that XTY≤ρ1XTX/2+YTY/2ρ1.


## 2.1. Graph Theory

Define a time varying random undirected graphGωt=ν,ξωijt,Aωt with a nonempty finite vertex set ν=ν1,…,νN and an edge set ξωijt⊆ν×ν. Different with the general ones, it consists of a time sequence of random graphs in which the edge set ξ varies with t. Namely, each edge νi,νj evolves according to a homogeneous Markov process ωijt, which takes values in S=1,2,…,s with the transition rate as(1)Prωijt+Δ=q∣ωijt=p=πpqΔ+oΔq≠p1+πppΔ+oΔq=p.Assume that ωijt do not change infinitely fast; thus, ωijt=ωijt+Δt if 0<Δt<Δ. It means that the total number of system modes is sN+1N/2 and the total transition rate is given by(2)Prωt+Δ∣ωt=∏i,j∈νPrωijt+Δ=q∣ωijt=p.The weighted adjacency matrixAω≜Aωt=aijωijt∈RN×N is associated with Gωt. Here aijω≜aijωijt>0 if νi,νj∈ξωijt and aijω=0 otherwise. Assumed that there is no self-loop in the graph, which implies that aiiω=0. In this paper, the set of neighbors with respect to the agent νi is Ωiω=νj∈ν∣νj,νi∈ξωijt. A graph Gωt is connected, if there is a path between any two vertices; otherwise, it is disconnected. A diagonal matrix Dω=diagdi∈RN×N with di=∑j∈Ωiωaijω being the ith row sum of Aω. Then, the Laplacian of the graph is defined as Lω=Dω-Aω. Consider the formation with a leader-follower structure by introducing a diagonal matrix Ll=diagaiil∈RN×N, which evolves according to the Markov process ω0t with a finite mode set S0=1,2,…,s0 and a time interval Δ0, where aiil>0 if vi is a leader and aiil=0 otherwise. Hence, the interaction matrix is given by LG=Lω+Ll.Assumption 2.
The undirected graphGωt is connected.Remark 3.
For simplicity, we just consider the undirected graph in this paper, that is,aijω=ajiω. Note that, if the graph is a general directed one, there will be some small differences. A possible approach to consider a directed graph is to introduce two Markov chains in each two agents; an alternative is to extend the state space set S, which could be defined according to the weight and direction of the graph. Both approaches will be addressed in our future work.Remark 4.
With a limited bandwidth, the switching ofaijω is caused by the sensing/detecting failure and communication failure, which is passive. In fact, it is difficult to obtain all the elements of the transition rate matrix, or some of the elements are not necessary to guarantee the system stability. Since that, the transition rate matrices are assumed to be partly accessed; even some of them are unknown completely, which could be descried as follows:(3)p11p12⋯???⋯p2s⋮⋮⋱⋮ps1?⋯pss,??⋯???⋯?⋮⋮⋱⋮??⋯?,where “?” represents the unknown transition rate.


## 2.2. Problem Formulation

Consider a group ofN quadrotor UAVs as shown in Figure 1; the dynamics of agent i are given as the following form [12]:(4)x¨i=sin⁡ψisin⁡ϕi+cos⁡ψisin⁡θicos⁡ϕiu1imi-kxix˙imi,y¨i=-cos⁡ψisin⁡ϕi+sin⁡ψisin⁡θicos⁡ϕiu1imi-kyiy˙imi,z¨i=-g+cos⁡θicos⁡ϕiu1imi-kziz˙imi,ϕ¨i=liu2iIxi-likϕiϕ˙iIxi,θ¨i=liu3iIyi-likθiθ˙iIyi,ψ¨i=Kψiu4iIzi-kψiψ˙iIzi,where xi,yi,zi∈R are positions and ϕi,θi,ψi∈-π/2,π/2 denote the three Euler angles of rotation, representing pitch, roll, and yaw, respectively. kϕi,θi,ψi are the aerodynamic friction coefficients, and kxi,yi,zi are the coefficients of the translation drag forces. Ixi,yi,zi are the quadrotor moments of inertias, mi denotes the mass of the quadrotor, li is half length of the helicopter, Kψi is thrust to moment gain, and g is gravitational acceleration.Figure 1
Group of theN quadrotor UAVs.Each system dynamics consist of six degrees of freedom model, which can be separated into position dynamic and attitude dynamic. With the choice of state variablesx1i=xi, x2i=x˙i; y1i=yi, y2i=y˙i; z1i=zi, z2i=z˙i; ϕ1i=ϕi, ϕ2i=ϕ˙i; θ1i=θi, θ2i=θ˙i; ψ1i=ψi, ψ2i=ψ˙i. Then, it can be represented by(5)x˙1i=x2i,y˙1i=y2i,z˙1i=z2i,x˙2i=sin⁡ψisin⁡ϕi+cos⁡ψisin⁡θicos⁡ϕiu1imi-kxix2imi,y˙2i=-cos⁡ψisin⁡ϕi+sin⁡ψisin⁡θicos⁡ϕiu1imi-kyiy2imi,z˙2i=-g+cos⁡θicos⁡ϕiu1imi-kziz2imi,ϕ˙1i=ϕ2i,θ˙1i=θ2i,ψ˙1i=ψ2i,ϕ˙2i=liu2iIxi-likϕiϕ2iIxi,θ˙2i=liu3iIyi-likθiθ2iIyi,ψ˙2i=Kψiu4iIzi-kψiψ2iIzi.As is well known to us, the quadrotor is underactuated and differentially flat. Accordingly, choose four variables and specify the desired trajectory asRd=xdt,ydt,zdt,ψdtT.


## 2.3. Formation Definition

The formation shape can be described by a vectorδt=δ1Tt,…,δNTtT∈R4N, where δit=δixt,δiyt,δizt,δiψtT is the continuously differentiable formation vector, such that(6)limt→∞x1it-δixt-xdt=0,limt→∞y1it-δiyt-ydt=0,limt→∞z1it-δizt-zdt=0,limt→∞ψ1it-δiψt-ψdt=0.Define the desired position of agent i as xid,yid,zid,ψidT=δix+xd,δiy+yd,δiz+zd,δiψ+ψdT. For j∈Ωiω, there exists δij=δijx,δijy,δijz,δijψT=δjx-δix,δjy-δiy,δjz-δiz,δjψ-δiψT. Hence, there are NN-1/2 such shape vectors satisfying the following properties:(7)δik=δij+δjk,δii=0,0,0,0T,δji=-δij,∀i,j,k∈1,…,N.Remark 5.
The formation reference vectorRd is not available to all agents, but the desired interdistances of agent i and its neighbors are known. Note that the time varying formation vector δt is not unique, and the relative position between the reference vector Rd and the formation can be adjusted.Lemma 6 (see [34]).
For anyX,Y∈Rn and ρ1>0, it holds that XTY≤ρ1XTX/2+YTY/2ρ1.


## 3. Main Results

In this section, time varying formation control problem for multiple quadrotor UAVs is solved through an event-triggered control scheme. The system stability analysis and exclusion of Zeno behavior are also provided.


### 3.1. Singularly Perturbed System

Define the error vectors as(8)σ12e1ϕi=ϕ1i-ϕid,σ1e2ϕi=ϕ2i-ϕ˙id,σ12e1θi=θ1i-θid,σ1e2θi=θ2i-θ˙id,σ12ψ~1i=ψ1i-ψid,σ1ψ~2i=ψ2i-ψ˙id,x~1i=x1i-xid,x~2i=x2i-x˙id,y~1i=y1i-yid,y~2i=y2i-y˙id,z~1i=z1i-zid,z~2i=z2i-z˙id,where σ1 is positive constant denoted as perturbing parameter, which satisfies σ1≪1. The overall system can be written as(9)∑ϕi​≔σ1e˙1ϕi=e2ϕiσ1e˙2ϕi=liu2iIxi-likϕiϕ2iIxi-ϕ¨id,∑θi​≔σ1e˙1θi=e2θiσ1e˙2θi=liu3iIyi-likθiθ2iIyi-θ¨id,(10)∑ψi​≔σ1ψ~˙1i=ψ~2iσ1ψ~˙2i=Kψiu4iIzi-kψiψ2iIzi-δ¨iψ,(11)∑xi​≔x~˙1i=x~2ix~˙2i=sin⁡ψisin⁡ϕi+cos⁡ψisin⁡θicos⁡ϕiu1imi-kxix2imi-δ¨ix,∑yi​≔y~˙1i=y~2iy~˙2i=-cos⁡ψisin⁡ϕi+sin⁡ψisin⁡θicos⁡ϕiu1imi-kyiy2imi-δ¨iy,∑zi​≔z~˙1i=z~2iz~˙2i=-g+cos⁡θicos⁡ϕiu1imi-kziz2imi-δ¨iz.System (9)–(11) has the standard form of a singularly perturbed system with a two-time scale; that is, ai=e1ϕi,e2ϕi,e1θi,e2θi,ψ~1i,ψ~2iT represents the attitude states with fast time scale and pi=x~1i,x~2i,y~1i,y~2i,z~1i,z~2iT represents the position states with slow time scale. Roughly speaking, in order to maintain a predefined formation shape for multiple quadrotor UAVs, the attitude stabilization should be guaranteed firstly.


### 3.2. Controller Design

Dynamic event-triggered control scheme is introduced to the multiple quadrotor UAVs. Define the following measurement errors:(12)e1ψit=ψ~1itki-ψ~1it,e2ψit=ψ~2itki-ψ~2it,e1xit=x~1itki-x~1it,e2xit=x~2itki-x~2it,e1yit=y~1itki-y~1it,e2yit=y~2itki-y~2it,e1zit=z~1itki-z~1it,e2zit=z~2itki-z~2it.Let(13)ψpi=∑j∈Ωiωaijωψ~1itki-ψ~1jtk′j+aiilψ~1itki,ψvi=∑j∈Ωiωaijωψ~2itki-ψ~2jtk′j+aiilψ~2itki,xpi=∑j∈Ωiωaijωx~1itki-x~1jtk′j+aiilx~1itki,xvi=∑j∈Ωiωaijωx~2itki-x~2jtk′j+aiilx~2itki,ypi=∑j∈Ωiωaijωy~1itki-y~1jtk′j+aiily~1itki,yvi=∑j∈Ωiωaijωy~2itki-y~2jtk′j+aiily~2itki,zpi=∑j∈Ωiωaijωz~1itki-z~1jtk′j+aiilz~1itki,zvi=∑j∈Ωiωaijωz~2itki-z~2jtk′j+aiilz~2itki.Then, the auxiliary control variables are given by(14)vϕi=-αϕe1ϕi-βϕe2ϕi,vθi=-αθe1θi-βθe2θi,vψi=-αψψpi-βψψvi,vxi=-αxxpi-βxxvi,vyi=-αyypi-βyyvi,vzi=-αzzpi-βzzvi.Therefore, the choice of control inputs(15)u2i=Ixivϕili+kϕiϕ2i+Ixiϕ¨idli,u3i=Iyivθili+kθiθ2i+Iyiθ¨idli,u4i=IzivψiKψi+kψiψ2iKψi+Iziδ¨iψKψi,(16)u1i=mivzi+mδ¨iz+mig+kziz2icos⁡θidcos⁡ϕid,where the reference angles ϕid and θid are addressed as(17)θid=tan-1cos⁡ψidvxi+δ¨ix+kxix2i/mi+sin⁡ψidvyi+δ¨iy+kyiy2i/mivzi+g+δ¨iz+kziz2i/mi,(18)ϕid=tan-1cos⁡θidsin⁡ψidvxi+δ¨ix+kxix2i/mi-cos⁡ψidvyi+δ¨iy+kyiy2i/mivzi+g+δ¨iz+kziz2i/mi.Remark 7.
It should be stressed that (17) and (18) are needed to be nonsingular; in other words, the denominator of (17) or (18) cannot be zero. In practice, the quadrotor has to take a certain thrust to overcome gravity in order to maintain hovering; otherwise, it would sink vertically. It means that the denominator of (17) or (18) is approximate to g, and during the flight, it will also be greater than zero based on the defined domain of z dynamics z1i,z2i,z˙2iT∈Dz=R×z2i<dz2×z˙2i<dz˙2⊂R3, dz˙2≪g.


### 3.3. Dynamic Event-Triggered Communication

Figure2 depicts the control law running in the ith quadrotor. Generally speaking, each agent updates its controller whenever the designed trigger condition is reached, called the triggered event. Based on local information, it decides when to broadcast its current state over the network. In other words, the key problem is to find a triggering rule that determines when agent i has to broadcast the new state information to its neighbors.Figure 2
Block diagram of the event-based control strategy.In contrast to most of the existing works, a new dynamic event-triggered communication mechanism is developed to schedule interagent communication. The threshold parameter in the proposed event triggering condition will not be fixed permanently but vary with time by following a dynamic rule. The detailed dynamic rule is provided in the following section. The event-triggered control strategy works as follows: define a trigger functionfit for each agent, which depends on local information only; an event is triggered as soon as the trigger condition fit>0 is fulfilled, while each agent recomputes its control law in accordance with the measurement error, such that all the agents could reach and keep the predesigned formation shape. It will be shown numerically that the dynamic event-triggered communication mechanism can achieve a better tradeoff between reducing data transmissions and preserving favorable formation performance.


### 3.4. System Stability via Two-Time Scale

The stability analysis for each subsystem will be done by starting from the faster one to the slower one. The main results of this paper are presented next with the help of the following theorems.Theorem 8.
Consider theith quadrotor dynamics, which is also a singularly perturbed system. Then, there always exists σ1∗>0 such that the state space origin of the closed-loop system (9)–(11) is exponentially stable with σ1∗>σ1>0.Proof.
The proof is set down in the following five items.
( 1 ) System (9)–(11) has a unique equilibrium point at aiT,piTT=0.
( 2 ) The quasi-steady-state solution of closed-loop system (9)-(10) is ai=a∗=h(t,pi). Hence, by substituting σ1=0 into (9)-(10), the isolated root is given by a∗=0. It is noteworthy that the isolated root evaluated in (16)–(18) gives the specific values ϕid∗, θid∗, and ψid∗.
( 3 ) System (9)–(11) and the isolated root have bounded partial derivatives in compact sets.
( 4 ) By considering σ1=0 and using the isolated root into (11), the following slow dynamics system is obtained:(19)∑xi​≔x~˙1i=x~2ix~˙2i=sin⁡ψid∗sin⁡ϕid∗+cos⁡ψid∗sin⁡θid∗cos⁡ϕid∗u1imi-kxix2imi-δ¨ix,∑yi​≔y~˙1i=y~2iy~˙2i=-cos⁡ψid∗sin⁡ϕid∗+sin⁡ψid∗sin⁡θid∗cos⁡ϕid∗u1imi-kyiy2imi-δ¨iy,∑zi​≔z~˙1i=z~2iz~˙2i=-g+cos⁡θid∗cos⁡ϕid∗u1imi-kziz2imi-δ¨iz.The local stability of the slow dynamics (11) can be derived by using event-triggered control method. For the sake of simplicity, it will be given in Theorem 12 later.
( 5 ) To obtain the boundary layer system, by deriving a-a∗ with respect to scaled time κ=t/σ1 and setting σ1=0, the boundary layer system is obtained:(20)ddκe1ϕi=e2ϕi,ddκe2ϕi=-αϕe1ϕi-βϕe2ϕi,ddκe1θi=e2θi,ddκe2θi=-αθe1θi-βθe2θi,(21)ddκψ~1i=ψ~2i,ddκψ~2i=-αψψpi-βψψvi.Thus, for the ϕi and θi subsystems(22)ddκeϕi=Aϕeϕi⟹ddκe1ϕie2ϕi=01-αϕ-βϕe1ϕie2ϕi,ddκeθi=Aθeθi⟹ddκe1θie2θi=01-αθ-βθe1θie2θi.It is obvious that Aϕ and Aθ are Hurwitz, where system (22) is exponentially stable. Meanwhile, the stability result of ψi subsystem will be established in Theorem 10 as follows.
According to [32], there are sufficient conditions to claim that there exists σ1∗ such that σ1∗>σ1>0 guarantees that system (9)–(11) achieves the following limit limt→∞⁡aiT,piTT=0.Remark 9.
Based on the system stability analysis by the singular perturbation method, instead of using the feedback anglesϕi, θi, ψi, the control input u1i is related to the reference angles ϕid, θid, ψid, and the coupling between the attitude subsystem and the position subsystem is diminished.Theorem 10.
Givenψ~pi=∑j∈Ωiωaijωe1ψi-e1ψj+aiile1ψi and ψ~vi=∑j∈Ωiωaijωe2ψi-e2ψj+aiile2ψi, then define Eψi=ψ~pi,ψ~viT and Ψ~i=ψpi-ψ~pi,ψvi-ψ~viT. Consider the yaw subsystem (21); the system stability could be obtained through the distributed control law (14) and (15) and the following event-triggered communication condition:(23)fit=B2TQEψi-σtB2TQΨ~i,where σt=e-μes is the dynamic threshold parameter with es=∫0tkieψiTeψids, eψi=e1ψi,e2ψiT, μ>0, and the initial value σ0=σ0∈0,1, which ensures that fit≤0 always holds.Proof.
Obviously,σt is a monotone nonincreasing function;thus, σt≤σ0 always holds. Particularly, if the threshold parameter is preset as a constant σ0 before dynamic event-triggered controller is implemented, this is called the static event-triggered controller. Define B1=1,0T, B2=0,1T, and ψ~i=ψ~1i,ψ~2iT. Based on (13), (21) becomes(24)ddκψ~i=B1B2Tψ~i+B2-αψψpi-βψψvi.Hence, it follows from [35] that there exist positive constants α0, αψ, and βψ such that(25)12QB1B2T+B2B1TQ-ρ0QB2TB2Q+2I2≤0Q>0,where Q=α0,αψ;αψ,βψ, B2TQ=αψ,βψ, and(26)0<ρ0≤1-ρ12-σ022ρ1λminLG.Let Ψ=ψ~1,…,ψ~NT and Eψ=eψ1,…,eψNT. Consider the following Lyapunov function:(27)Vψ=12ΨTIN⊗QΨ.Then, its derivative is(28)ddκVψ=ΨTIN⊗QB1B2T+B2B1TQ2Ψ-ΨTLG⊗QB2B2TQΨ-ΨTLG⊗QB2B2TQEψ.Based on Lemma 6, the following inequality holds:(29)ΨTLG⊗QB2B2TQEψ≤ρ12ΨTLG⊗QB2B2TQΨ+12ρ1EψTLG⊗QB2B2TQEψ.Based on (23) and (25), (28) can be written as(30)ddκVψ≤ΨTIN⊗QB1B2T+B1B2TQT2Ψ-λminLGΨTIN⊗QB2B2TQΨ+ρ12ΨTLG⊗QB2B2TQΨ+12ρ1EψTLG⊗QB2B2TQEψ≤ΨTIN⊗QB1B2T+B1B2TQT2-1-ρ12-σ2t2ρ1λminLGQB2B2TQΨ≤-2ρQVψ.Hence, Vψ≤σ1Vψ0e-2ρQt, and limt→∞⁡ψ1it-δiψt-ψdt=0 exponentially.
Next, Zeno behavior is excluded. For anyt>tki, it follows from (23) that(31)ddκLG⊗B2TQEψ≤αψ2+βψ2LGEψ+αψ2+βψ2LGΨ+αψ2+βψ2LGEψ≤σt+1+σtαψ2+βψ2LGαψ2+βψ2LGΨ,where Ψ≤2σ1ρQVψ0e-ρQt. Hence,(32)ddκLG⊗B2TQEψ≤σt+1+σtαψ2+βψ2LGαψ2+βψ2LG2σ1ρQVψ0e-ρQt.Thus,(33)LG⊗B2TQEψ≤σ1σt+1+σtαψ2+βψ2LGαψ2+βψ2LG2σ1ρQVψ0e-ρQtt-tk.The next event tk+1 will not be triggered before LG⊗B2TQEψ=σtLG⊗B2TQΨ. Thus, a lower bound on the interevent intervals is given by τ=t-tk that solves the following equation:(34)στLG⊗B2TQΨ=σ1τστ+1+σταψ2+βψ2LGαψ2+βψ2LG2σ1ρQVψ0e-ρQt.Therefore, for all tk≥0 the solutions τtk are greater than or equal to τ given by(35)στ=σ1τστ+1+σταψ2+βψ2LGwhich is strictly positive, so Zeno behavior is excluded. This completes the proof.Remark 11.
It should be pointed out that the vectorsϕid and θid are obtained inside of the controller; as shown in Figure 2, the pitch and roll attitude dynamics are not event-triggered. In other words, only the yaw angle in the attitude subsystem is controlled through an event-triggered strategy. In addition, compared with some existing works, in which assuming ψ1i=0 in the position control loop, the yaw angle is not always zero in practice. Since that the control method is more suitable in this paper.
After the attitude angles reach quasi-steady states, hierarchically,σ1=0, ϕ1i→ϕid, θ1i→θid, ψ1i→ψid. Substituting (14), (16)–(18) into (19), it is possible to show that the closed-loop position subsystem can be written as(36)x~˙1i=x~2i,x~˙2i=-αxxpi-βxxvi,y~˙1i=y~2i,y~˙2i=-αyypi-βyyvi,z~˙1i=z~2i,z~˙2i=-αzzpi-βzzvi.Without loss of generality, we only provide the proof along x-subsystem in the following.Theorem 12.
Givenx~pi=∑j∈Ωiωaijωe1xi-e1xj+aiile1xi and x~vi=∑j∈Ωiωaijωe2xi-e2xj+aiile2xi, then let Exi=x~pi,x~viT and X~i=xpi-x~pi,xvi-x~viT. Considering the position subsystem (19), the system stability could be obtained through the distributed control law (16)–(18) and the following event-triggered communication condition:(37)fit=B2TQ0Exi-σtB2TQ0X~i,where σt=e-μes is the dynamic threshold parameter with es=∫0tkiexiTexids, exi=e1xi,e2xiT, μ>0 and the initial value σ0=σ0∈0,1.Proof.
Letx~i=x~1i,x~2iT; the x-subsystem could be rewritten as(38)x~˙i=B1B2Tx~i+B2-αxxpi-βxxvi.Similarly, there exist positive constants α0, αx, and βx such that(39)12Q0B1B2T+B2B1TQ0-ρ0Q0B2TB2Q0+2I2≤0Q0>0,where Q0=α0,αx;αx,βx. Note that B2TQ0=αx,βx, and let X=x~1,…,x~NT and EX=ex1,…,exNT. Design the following Lyapunov function:(40)Vx=12XTIN⊗Q0X.Then, its derivative is(41)V˙x=XTIN⊗Q0B1B2T+B2B1TQ02X-XTLG⊗Q0B2B2TQ0X-XTLG⊗Q0B2B2TQ0EX.Based on Lemma 6, the following inequality holds:(42)XTLG⊗Q0B2B2TQ0EX≤ρ12XTLG⊗Q0B2B2TQ0X+12ρ1EXTLG⊗Q0B2B2TQ0EX.Based on (37) and (39), (41) can be written as(43)V˙x≤XTIN⊗Q0B1B2T+B1B2TQ0T2-1-ρ12-σ2t2ρ1λminLGQ0B2B2TQ0X≤-2ρQ0Vx.Hence, Vx≤Vx0e-2ρQ0t, and limt→∞⁡x1it-δixt-xdt=0 exponentially.
Next, Zeno behavior is excluded. For anyt>tki, one has(44)ddtLG⊗B2TQEX≤αx2+βx2LGEX+αx2+βx2LGX+αx2+βx2LGEX≤σt+1+σtαx2+βx2LGαx2+βx2LGX,where X≤2ρQ0Vx0e-ρQ0t. Hence,(45)ddtLG⊗B2TQEX≤σt+1+σtαx2+βx2LGαx2+βx2LG2ρQ0Vx0e-ρQ0t.Thus,(46)LG⊗B2TQEX≤σt+1+σtαx2+βx2LGαx2+βx2LG2ρQ0Vx0e-ρQ0tt-tk.The next event tk+1 will not be triggered before LG⊗B2TQEX=σtLG⊗B2TQX; it means(47)σtLG⊗B2TQX≤σt+1+σtαx2+βx2LGαx2+βx2LG2ρQ0Vx0e-ρQ0ttk+1-tk.By the same graphical argument as in the proof of Theorem 10, it can be concluded that a lower bounded on the interevent times is given by τ, so Zeno behavior is excluded.
This completes the proof.Remark 13.
Compared with the existing works, the threshold parameter of event triggering condition will not be fixed permanently but vary with time by following the measurement errors. Since that it can achieve a better tradeoff between reducing event times and preserving favorable formation performance.


## 3.1. Singularly Perturbed System

Define the error vectors as(8)σ12e1ϕi=ϕ1i-ϕid,σ1e2ϕi=ϕ2i-ϕ˙id,σ12e1θi=θ1i-θid,σ1e2θi=θ2i-θ˙id,σ12ψ~1i=ψ1i-ψid,σ1ψ~2i=ψ2i-ψ˙id,x~1i=x1i-xid,x~2i=x2i-x˙id,y~1i=y1i-yid,y~2i=y2i-y˙id,z~1i=z1i-zid,z~2i=z2i-z˙id,where σ1 is positive constant denoted as perturbing parameter, which satisfies σ1≪1. The overall system can be written as(9)∑ϕi​≔σ1e˙1ϕi=e2ϕiσ1e˙2ϕi=liu2iIxi-likϕiϕ2iIxi-ϕ¨id,∑θi​≔σ1e˙1θi=e2θiσ1e˙2θi=liu3iIyi-likθiθ2iIyi-θ¨id,(10)∑ψi​≔σ1ψ~˙1i=ψ~2iσ1ψ~˙2i=Kψiu4iIzi-kψiψ2iIzi-δ¨iψ,(11)∑xi​≔x~˙1i=x~2ix~˙2i=sin⁡ψisin⁡ϕi+cos⁡ψisin⁡θicos⁡ϕiu1imi-kxix2imi-δ¨ix,∑yi​≔y~˙1i=y~2iy~˙2i=-cos⁡ψisin⁡ϕi+sin⁡ψisin⁡θicos⁡ϕiu1imi-kyiy2imi-δ¨iy,∑zi​≔z~˙1i=z~2iz~˙2i=-g+cos⁡θicos⁡ϕiu1imi-kziz2imi-δ¨iz.System (9)–(11) has the standard form of a singularly perturbed system with a two-time scale; that is, ai=e1ϕi,e2ϕi,e1θi,e2θi,ψ~1i,ψ~2iT represents the attitude states with fast time scale and pi=x~1i,x~2i,y~1i,y~2i,z~1i,z~2iT represents the position states with slow time scale. Roughly speaking, in order to maintain a predefined formation shape for multiple quadrotor UAVs, the attitude stabilization should be guaranteed firstly.


## 3.2. Controller Design

Dynamic event-triggered control scheme is introduced to the multiple quadrotor UAVs. Define the following measurement errors:(12)e1ψit=ψ~1itki-ψ~1it,e2ψit=ψ~2itki-ψ~2it,e1xit=x~1itki-x~1it,e2xit=x~2itki-x~2it,e1yit=y~1itki-y~1it,e2yit=y~2itki-y~2it,e1zit=z~1itki-z~1it,e2zit=z~2itki-z~2it.Let(13)ψpi=∑j∈Ωiωaijωψ~1itki-ψ~1jtk′j+aiilψ~1itki,ψvi=∑j∈Ωiωaijωψ~2itki-ψ~2jtk′j+aiilψ~2itki,xpi=∑j∈Ωiωaijωx~1itki-x~1jtk′j+aiilx~1itki,xvi=∑j∈Ωiωaijωx~2itki-x~2jtk′j+aiilx~2itki,ypi=∑j∈Ωiωaijωy~1itki-y~1jtk′j+aiily~1itki,yvi=∑j∈Ωiωaijωy~2itki-y~2jtk′j+aiily~2itki,zpi=∑j∈Ωiωaijωz~1itki-z~1jtk′j+aiilz~1itki,zvi=∑j∈Ωiωaijωz~2itki-z~2jtk′j+aiilz~2itki.Then, the auxiliary control variables are given by(14)vϕi=-αϕe1ϕi-βϕe2ϕi,vθi=-αθe1θi-βθe2θi,vψi=-αψψpi-βψψvi,vxi=-αxxpi-βxxvi,vyi=-αyypi-βyyvi,vzi=-αzzpi-βzzvi.Therefore, the choice of control inputs(15)u2i=Ixivϕili+kϕiϕ2i+Ixiϕ¨idli,u3i=Iyivθili+kθiθ2i+Iyiθ¨idli,u4i=IzivψiKψi+kψiψ2iKψi+Iziδ¨iψKψi,(16)u1i=mivzi+mδ¨iz+mig+kziz2icos⁡θidcos⁡ϕid,where the reference angles ϕid and θid are addressed as(17)θid=tan-1cos⁡ψidvxi+δ¨ix+kxix2i/mi+sin⁡ψidvyi+δ¨iy+kyiy2i/mivzi+g+δ¨iz+kziz2i/mi,(18)ϕid=tan-1cos⁡θidsin⁡ψidvxi+δ¨ix+kxix2i/mi-cos⁡ψidvyi+δ¨iy+kyiy2i/mivzi+g+δ¨iz+kziz2i/mi.Remark 7.
It should be stressed that (17) and (18) are needed to be nonsingular; in other words, the denominator of (17) or (18) cannot be zero. In practice, the quadrotor has to take a certain thrust to overcome gravity in order to maintain hovering; otherwise, it would sink vertically. It means that the denominator of (17) or (18) is approximate to g, and during the flight, it will also be greater than zero based on the defined domain of z dynamics z1i,z2i,z˙2iT∈Dz=R×z2i<dz2×z˙2i<dz˙2⊂R3, dz˙2≪g.


## 3.3. Dynamic Event-Triggered Communication

Figure2 depicts the control law running in the ith quadrotor. Generally speaking, each agent updates its controller whenever the designed trigger condition is reached, called the triggered event. Based on local information, it decides when to broadcast its current state over the network. In other words, the key problem is to find a triggering rule that determines when agent i has to broadcast the new state information to its neighbors.Figure 2
Block diagram of the event-based control strategy.In contrast to most of the existing works, a new dynamic event-triggered communication mechanism is developed to schedule interagent communication. The threshold parameter in the proposed event triggering condition will not be fixed permanently but vary with time by following a dynamic rule. The detailed dynamic rule is provided in the following section. The event-triggered control strategy works as follows: define a trigger functionfit for each agent, which depends on local information only; an event is triggered as soon as the trigger condition fit>0 is fulfilled, while each agent recomputes its control law in accordance with the measurement error, such that all the agents could reach and keep the predesigned formation shape. It will be shown numerically that the dynamic event-triggered communication mechanism can achieve a better tradeoff between reducing data transmissions and preserving favorable formation performance.


## 3.4. System Stability via Two-Time Scale

The stability analysis for each subsystem will be done by starting from the faster one to the slower one. The main results of this paper are presented next with the help of the following theorems.Theorem 8.
Consider theith quadrotor dynamics, which is also a singularly perturbed system. Then, there always exists σ1∗>0 such that the state space origin of the closed-loop system (9)–(11) is exponentially stable with σ1∗>σ1>0.Proof.
The proof is set down in the following five items.
( 1 ) System (9)–(11) has a unique equilibrium point at aiT,piTT=0.
( 2 ) The quasi-steady-state solution of closed-loop system (9)-(10) is ai=a∗=h(t,pi). Hence, by substituting σ1=0 into (9)-(10), the isolated root is given by a∗=0. It is noteworthy that the isolated root evaluated in (16)–(18) gives the specific values ϕid∗, θid∗, and ψid∗.
( 3 ) System (9)–(11) and the isolated root have bounded partial derivatives in compact sets.
( 4 ) By considering σ1=0 and using the isolated root into (11), the following slow dynamics system is obtained:(19)∑xi​≔x~˙1i=x~2ix~˙2i=sin⁡ψid∗sin⁡ϕid∗+cos⁡ψid∗sin⁡θid∗cos⁡ϕid∗u1imi-kxix2imi-δ¨ix,∑yi​≔y~˙1i=y~2iy~˙2i=-cos⁡ψid∗sin⁡ϕid∗+sin⁡ψid∗sin⁡θid∗cos⁡ϕid∗u1imi-kyiy2imi-δ¨iy,∑zi​≔z~˙1i=z~2iz~˙2i=-g+cos⁡θid∗cos⁡ϕid∗u1imi-kziz2imi-δ¨iz.The local stability of the slow dynamics (11) can be derived by using event-triggered control method. For the sake of simplicity, it will be given in Theorem 12 later.
( 5 ) To obtain the boundary layer system, by deriving a-a∗ with respect to scaled time κ=t/σ1 and setting σ1=0, the boundary layer system is obtained:(20)ddκe1ϕi=e2ϕi,ddκe2ϕi=-αϕe1ϕi-βϕe2ϕi,ddκe1θi=e2θi,ddκe2θi=-αθe1θi-βθe2θi,(21)ddκψ~1i=ψ~2i,ddκψ~2i=-αψψpi-βψψvi.Thus, for the ϕi and θi subsystems(22)ddκeϕi=Aϕeϕi⟹ddκe1ϕie2ϕi=01-αϕ-βϕe1ϕie2ϕi,ddκeθi=Aθeθi⟹ddκe1θie2θi=01-αθ-βθe1θie2θi.It is obvious that Aϕ and Aθ are Hurwitz, where system (22) is exponentially stable. Meanwhile, the stability result of ψi subsystem will be established in Theorem 10 as follows.
According to [32], there are sufficient conditions to claim that there exists σ1∗ such that σ1∗>σ1>0 guarantees that system (9)–(11) achieves the following limit limt→∞⁡aiT,piTT=0.Remark 9.
Based on the system stability analysis by the singular perturbation method, instead of using the feedback anglesϕi, θi, ψi, the control input u1i is related to the reference angles ϕid, θid, ψid, and the coupling between the attitude subsystem and the position subsystem is diminished.Theorem 10.
Givenψ~pi=∑j∈Ωiωaijωe1ψi-e1ψj+aiile1ψi and ψ~vi=∑j∈Ωiωaijωe2ψi-e2ψj+aiile2ψi, then define Eψi=ψ~pi,ψ~viT and Ψ~i=ψpi-ψ~pi,ψvi-ψ~viT. Consider the yaw subsystem (21); the system stability could be obtained through the distributed control law (14) and (15) and the following event-triggered communication condition:(23)fit=B2TQEψi-σtB2TQΨ~i,where σt=e-μes is the dynamic threshold parameter with es=∫0tkieψiTeψids, eψi=e1ψi,e2ψiT, μ>0, and the initial value σ0=σ0∈0,1, which ensures that fit≤0 always holds.Proof.
Obviously,σt is a monotone nonincreasing function;thus, σt≤σ0 always holds. Particularly, if the threshold parameter is preset as a constant σ0 before dynamic event-triggered controller is implemented, this is called the static event-triggered controller. Define B1=1,0T, B2=0,1T, and ψ~i=ψ~1i,ψ~2iT. Based on (13), (21) becomes(24)ddκψ~i=B1B2Tψ~i+B2-αψψpi-βψψvi.Hence, it follows from [35] that there exist positive constants α0, αψ, and βψ such that(25)12QB1B2T+B2B1TQ-ρ0QB2TB2Q+2I2≤0Q>0,where Q=α0,αψ;αψ,βψ, B2TQ=αψ,βψ, and(26)0<ρ0≤1-ρ12-σ022ρ1λminLG.Let Ψ=ψ~1,…,ψ~NT and Eψ=eψ1,…,eψNT. Consider the following Lyapunov function:(27)Vψ=12ΨTIN⊗QΨ.Then, its derivative is(28)ddκVψ=ΨTIN⊗QB1B2T+B2B1TQ2Ψ-ΨTLG⊗QB2B2TQΨ-ΨTLG⊗QB2B2TQEψ.Based on Lemma 6, the following inequality holds:(29)ΨTLG⊗QB2B2TQEψ≤ρ12ΨTLG⊗QB2B2TQΨ+12ρ1EψTLG⊗QB2B2TQEψ.Based on (23) and (25), (28) can be written as(30)ddκVψ≤ΨTIN⊗QB1B2T+B1B2TQT2Ψ-λminLGΨTIN⊗QB2B2TQΨ+ρ12ΨTLG⊗QB2B2TQΨ+12ρ1EψTLG⊗QB2B2TQEψ≤ΨTIN⊗QB1B2T+B1B2TQT2-1-ρ12-σ2t2ρ1λminLGQB2B2TQΨ≤-2ρQVψ.Hence, Vψ≤σ1Vψ0e-2ρQt, and limt→∞⁡ψ1it-δiψt-ψdt=0 exponentially.
Next, Zeno behavior is excluded. For anyt>tki, it follows from (23) that(31)ddκLG⊗B2TQEψ≤αψ2+βψ2LGEψ+αψ2+βψ2LGΨ+αψ2+βψ2LGEψ≤σt+1+σtαψ2+βψ2LGαψ2+βψ2LGΨ,where Ψ≤2σ1ρQVψ0e-ρQt. Hence,(32)ddκLG⊗B2TQEψ≤σt+1+σtαψ2+βψ2LGαψ2+βψ2LG2σ1ρQVψ0e-ρQt.Thus,(33)LG⊗B2TQEψ≤σ1σt+1+σtαψ2+βψ2LGαψ2+βψ2LG2σ1ρQVψ0e-ρQtt-tk.The next event tk+1 will not be triggered before LG⊗B2TQEψ=σtLG⊗B2TQΨ. Thus, a lower bound on the interevent intervals is given by τ=t-tk that solves the following equation:(34)στLG⊗B2TQΨ=σ1τστ+1+σταψ2+βψ2LGαψ2+βψ2LG2σ1ρQVψ0e-ρQt.Therefore, for all tk≥0 the solutions τtk are greater than or equal to τ given by(35)στ=σ1τστ+1+σταψ2+βψ2LGwhich is strictly positive, so Zeno behavior is excluded. This completes the proof.Remark 11.
It should be pointed out that the vectorsϕid and θid are obtained inside of the controller; as shown in Figure 2, the pitch and roll attitude dynamics are not event-triggered. In other words, only the yaw angle in the attitude subsystem is controlled through an event-triggered strategy. In addition, compared with some existing works, in which assuming ψ1i=0 in the position control loop, the yaw angle is not always zero in practice. Since that the control method is more suitable in this paper.
After the attitude angles reach quasi-steady states, hierarchically,σ1=0, ϕ1i→ϕid, θ1i→θid, ψ1i→ψid. Substituting (14), (16)–(18) into (19), it is possible to show that the closed-loop position subsystem can be written as(36)x~˙1i=x~2i,x~˙2i=-αxxpi-βxxvi,y~˙1i=y~2i,y~˙2i=-αyypi-βyyvi,z~˙1i=z~2i,z~˙2i=-αzzpi-βzzvi.Without loss of generality, we only provide the proof along x-subsystem in the following.Theorem 12.
Givenx~pi=∑j∈Ωiωaijωe1xi-e1xj+aiile1xi and x~vi=∑j∈Ωiωaijωe2xi-e2xj+aiile2xi, then let Exi=x~pi,x~viT and X~i=xpi-x~pi,xvi-x~viT. Considering the position subsystem (19), the system stability could be obtained through the distributed control law (16)–(18) and the following event-triggered communication condition:(37)fit=B2TQ0Exi-σtB2TQ0X~i,where σt=e-μes is the dynamic threshold parameter with es=∫0tkiexiTexids, exi=e1xi,e2xiT, μ>0 and the initial value σ0=σ0∈0,1.Proof.
Letx~i=x~1i,x~2iT; the x-subsystem could be rewritten as(38)x~˙i=B1B2Tx~i+B2-αxxpi-βxxvi.Similarly, there exist positive constants α0, αx, and βx such that(39)12Q0B1B2T+B2B1TQ0-ρ0Q0B2TB2Q0+2I2≤0Q0>0,where Q0=α0,αx;αx,βx. Note that B2TQ0=αx,βx, and let X=x~1,…,x~NT and EX=ex1,…,exNT. Design the following Lyapunov function:(40)Vx=12XTIN⊗Q0X.Then, its derivative is(41)V˙x=XTIN⊗Q0B1B2T+B2B1TQ02X-XTLG⊗Q0B2B2TQ0X-XTLG⊗Q0B2B2TQ0EX.Based on Lemma 6, the following inequality holds:(42)XTLG⊗Q0B2B2TQ0EX≤ρ12XTLG⊗Q0B2B2TQ0X+12ρ1EXTLG⊗Q0B2B2TQ0EX.Based on (37) and (39), (41) can be written as(43)V˙x≤XTIN⊗Q0B1B2T+B1B2TQ0T2-1-ρ12-σ2t2ρ1λminLGQ0B2B2TQ0X≤-2ρQ0Vx.Hence, Vx≤Vx0e-2ρQ0t, and limt→∞⁡x1it-δixt-xdt=0 exponentially.
Next, Zeno behavior is excluded. For anyt>tki, one has(44)ddtLG⊗B2TQEX≤αx2+βx2LGEX+αx2+βx2LGX+αx2+βx2LGEX≤σt+1+σtαx2+βx2LGαx2+βx2LGX,where X≤2ρQ0Vx0e-ρQ0t. Hence,(45)ddtLG⊗B2TQEX≤σt+1+σtαx2+βx2LGαx2+βx2LG2ρQ0Vx0e-ρQ0t.Thus,(46)LG⊗B2TQEX≤σt+1+σtαx2+βx2LGαx2+βx2LG2ρQ0Vx0e-ρQ0tt-tk.The next event tk+1 will not be triggered before LG⊗B2TQEX=σtLG⊗B2TQX; it means(47)σtLG⊗B2TQX≤σt+1+σtαx2+βx2LGαx2+βx2LG2ρQ0Vx0e-ρQ0ttk+1-tk.By the same graphical argument as in the proof of Theorem 10, it can be concluded that a lower bounded on the interevent times is given by τ, so Zeno behavior is excluded.
This completes the proof.Remark 13.
Compared with the existing works, the threshold parameter of event triggering condition will not be fixed permanently but vary with time by following the measurement errors. Since that it can achieve a better tradeoff between reducing event times and preserving favorable formation performance.


## 4. Simulation Results

In this section, simulations are given to demonstrate the effectiveness of the theoretical results. Consider a group of agents, which consists of six quadrotor UAVs, and the dynamics can be written as (4). The system parameters are shown in Table 1. Furthermore, the initial mode of undirected topology Gωt is given in Figure 3. A special case is considered, where ω0 takes the switch value in S0=1,2,3,4,5,6 with an equal probability 1/6 and the time interval Δ0=5s. Meanwhile, assume that each Markov chain ωij takes values in a finite state space S=0,1, and the time interval Δ=1s. Otherwise, each undirected edge has a Markov chain, and the six agents have 15 chains totally. The transition rate matrices are considered as Table 2.Table 1
The system parameters of quadrotor UAVs.
Parameters Nominal value Parameters Nominal value l i 0.20  m k x i 0.010 N/m⋅s−1 m i 1.79 kg k y i 0.010 N/m⋅s−1 I x i 0.03 kg⋅m2 k z i 0.010 N/m⋅s−1 I y i 0.03 kg⋅m2 k ϕ i 0.012 N/rad⋅s−1 I z i 0.04 kg⋅m2 k θ i 0.012 N/rad⋅s−1 g 9.81 m/s2 k ψ i 0.012 N/rad⋅s−1 K ψ i 0.40 N⋅mTable 2
Transition rate matrices of the Markov chains in graph.
Edges v 1 , v 2; v2,v4; v4,v5  v3,v5; v3,v6 v 1 , v 3; v2,v3; v2,v6  v5,v6; v1,v5 v 1 , v 4; v1,v6; v2,v5  v3,v4; v4,v6 Transition rate matrices 0.2 0.8 0.4 0.6 0.5 0.5 ? ? ? ? ? ?Figure 3
Initial mode of interaction topology.As an example, the time evolutions of some Markov chains are depicted in Figure4.Figure 4
Evolutions of some Markov chains.Let the initial states of six quadrotor UAVs be(48)x10,y10,z10,ϕ10,θ10,ψ10T=1,1,0,0,0,0T,x20,y20,z20,ϕ20,θ20,ψ20T=3,-1,0,0,0,0T,x30,y30,z30,ϕ30,θ30,ψ30T=-3,2,0,0,0,0T,x40,y40,z40,ϕ40,θ40,ψ40T=-2,-2,0,0,0,0T,x50,y50,z50,ϕ50,θ50,ψ50T=3.5,2,0,0,0,0T,x60,y60,z60,ϕ60,θ60,ψ60T=-3,1,0,0,0,0T.The desired trajectory is given by Rd=0,0,0.05t,π/30T, and the formation shape specified by the desired interagent distances is described as(49)δit=δixtδiytδiztδiψt=3sin⁡0.2t+i-1π33cos⁡0.2t+i-1π300which means that if the quadrotor system achieves the desired time varying formation, then the six quadrotor UAVs will follow a circle while keeping a phase separation of π/3. In order to verify the effectiveness and advantage of the developed formation control protocol, the following two cases are evaluated:Case 1.
The distributed event-triggered function (23), (37) and the controller are implemented with the parameters as follows:(50)σ0=0.6,σ1=0.1,μ=0.3,αϕ=αθ=αψ=4.0,βϕ=βθ=βψ=3.2,αx=αy=αz=2.0,βx=βy=βz=1.8.
Simulation results are shown in Figures5 and 6. The response curves of formation trajectory and attitude are given in Figure 5, which reveal that positions of six agents successfully achieve and maintain a predesigned formation shape. As depicted in Figure 6, it demonstrates the snapshots of position trajectories of the six robots at t=0s, t=5s, t=20s, and t=40s, respectively.Figure 5
Trajectories of the agents within 50 s in Case1.
(a)
Evolution of the position states (b)
Evolution of the attitude states (c)
Evolution of the velocity states (d)
Evolution of the angular velocityFigure 6
Position snapshots of the six agents in Case1.Case 2.
The same formation task is carried out by implementing the event-triggered controller with a constant threshold parameterσt=σ0=0.6. The aforementioned results derived from theorems indicate that the formation control problem is feasible under the general event-triggered controller. Figure 7 shows the trajectories of positions and attitude angles within 50 s and Figure 8 shows the snapshots of position trajectories at t=0s, t=5s, t=20s, and t=40s, respectively.
The actual event release instants for all agents are shown in Figures9 and 10. A quantitative comparison of how many events on each agent are actually transmitted is provided in Table 3, and Table 4 shows the mean and standard deviation for each error signal in 20s≤t≤50s.Table 3
Event times of all agents.
Agent ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 Case1 131 123 109 126 143 115 Case2 62 48 67 60 61 54Table 4
Comparison results in Cases1 and 2.
Signals Case1 Case2 Mean e1ϕi - 0.0281 - 0.0856 e1θi - 0.0257 - 0.0849 ψ~1i - 0.0319 - 0.1147 x~1i 0.0156 0.0664 y~1i 0.0221 0.0702 z~1i - 0.0202 - 0.0445 Standard deviation e1ϕi 0.1075 0.2490 e1θi 0.1196 0.2772 ψ~1i 0.1402 0.3128 x~1i 0.0809 0.1824 y~1i 0.0873 0.1905 z~1i 0.0894 0.1943Figure 7
Trajectories of the agents within 50 s in Case2.
(a)
Evolution of the position states (b)
Evolution of the attitude statesFigure 8
Position snapshots of the six agents in Case2.Figure 9
Events times of all agents in Case1.Figure 10
Events times of all agents in Case2.Remark 14.
Some conclusions could be summarized as follows:
( 1 ) Simulation results in both Cases 1 and 2 show that the formation shape could be guaranteed under Markovian switching topologies with partially unknown transition rates.
( 2 ) In Figure 5, the convergence time of ψ and ψ˙ (i.e., 0.80 s) is shorter than that of x, x˙, y, y˙, z, and z˙ (i.e., 7.20 s); these plots have different time horizons; it confirms the two-time-scale structure controller derived from the perturbing parameter σ1. Particularly, the reason why the convergence time of ϕ and θ in Figure 5 is as long as that of the position states is that ϕ and θ are tracking the reference ϕid and θid rather than zero.
( 3 ) As shown in Figures 6 and 8, at t=0s, t=20s, and t=40s, the real shape of six agents is always uniform. When t=5s, there is a little difference because the graph is randomly governed by Markov chains.
( 4 ) Compared with Case 2, there are more events for all agents in Case 1, which means that it has more potential to reduce the occupancy of limited communication resources in Case 2. That is because the threshold parameter σt is fixed as σ0 all the time, while in Case 1, σt is monotonically nonincreasing (shown in Figure 11). Generally, the larger the value of the threshold parameter is, the less events are derived.
( 5 ) Based on most of the computed performance indexes in Table 4, the controller in Case 1 can achieve better formation performance than that in Case 2. Although less events are derived by using a fixed threshold parameter, the formation performance is compromised.Figure 11
The threshold parameters in Cases1 and 2.


## 5. Conclusion

In this paper, formation control problem of multiple quadrotor UAVs under Markovian switching topologies is addressed through an event-triggered control strategy. Markovian switching topologies are redesigned through utilizing the Markov chains to the edge set, which could recover the traditional ones by adjusting the modes of edges and the transition rates. Then, a predesigned formation shape can be kept along with a distributed formation controller constructed with a reasonably event-triggered updated rule, even subject to the unknown transition rates. Rigorous analyses of the convergence results are obtained based on singular perturbations theory and Lyapunov stability theory. In addition, Zeno behavior is excluded for the triggered time sequences. Simulation results have been given to illustrate the effectiveness of the proposed control strategy. In some of the cases, the velocity states are unmeasurable. Since that, event-triggered formation control problem for multiple quadrotor UAVs without collision subject to output feedback will be considered in our future work.



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