# Distributed Formation Control for Multiple Quadrotor Based on Multi-Agent Theory and Disturbance Observer

Authors: Xiaohua Zhang; Junli Gao; Wenfeng Zhang; Tao Zeng; Liping Ye
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper presents the disturbance observers-based distributed formation control for multiple quadrotor aircrafts with external disturbances and uncertain parameters using multi-agent theory and finite-time control method. Firstly, the finite-time disturbance observers are proposed to handle the external disturbances on the position-loop. Similarly, when there are both the uncertain parameters and external disturbances on the attitude-loop, the finite-time disturbance observers are designed to estimate the total lump disturbances. By skillfully using homogeneous system theory, Lyapunov theory, and multi-agent theory, the distributed formation control algorithms are developed. Finally, through simulations, the efficiency of the proposed method (including the convergence rate and disturbance rejection) is verified.


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## Body

## 1. Introduction

With the development and application of the quadrotor aircraft, nowadays, the formation control problem for multiple quadrotor aircrafts has become a hot topic and attracted a great deal of attention. Compared to the fixed-wing aircraft, the quadrotor aircraft has many different features, such as simple structures, easy maintenance, and maneuverability, etc [1]. Due to these advantages, the quadrotor aircraft has been widely used in the civilian field and the military field, such as aerial photography, geological survey, disaster relief, pipeline inspection, environmental assessment, etc. [2, 3]. Meanwhile, in some complex and specific environments, multiple quadrotor aircrafts have more advantages than a single aircraft. For example, multiple aircrafts can significantly improve the efficiency and the robustness of the whole system. In addition, more aircraft collaboration means more different equipment that can be carried on to accomplish more difficult tasks. From the control point of view, it is quite difficult and challenging to realize the formation control of multiple aircrafts because it is necessary to further coordinated control of multi-aircraft on the basis of controlling a single aircraft. Actually, even the design of control system and the analysis of stability for single quadrotor aircraft are also challenging since the model has highly nonlinear and strong coupling [4].Up to now, a series of achievements have been made in the formation control problem. For the heterogeneous multi-agent systems with nonlinear dynamics, the leader-following consensus problem has been considered in [5]. The work [6] studied a novel reliable consensus for uncertain nonlinear multi-agent systems in the presence of probabilistic time-varying delay. This article considers the leader-following consensus problem of heterogeneous multi-agent systems The time-varying formation tracking control has been realized for quadrotor aircrafts [7, 8]. Based on complex Laplacian, the work [9] studied the distributed formation control problem for multi-agent systems. Considering sampled-data and communication delays, the distributed formation controllers have been designed for mobile robots [10]. The work [11] discussed the problem of vision-based leader-follower formation control for mobile robots.However, note that most of the existing control algorithms are required to satisfy the Lipschitz continuity condition. In other words, the desired formation will be achieved at best exponential with an infinite time. As we all know, for the quadrotor aircraft, whether it is multiple formation control or single control, it is hoped that the convergence speed will be as fast as possible. To enhance the convergence rate and the ability of disturbance rejection, recently a nonlinear control method (i.e., finite-time control method) [12–14] was introduced. Theoretically, the finite-time control algorithm can guarantee that the system will stabilize in a finite time. Firstly, there have been already some results with regard to finite-time control for a single quadrotor aircraft [15–22]. Moreover, for the single quadrotor aircraft during the flight, the external disturbances are often unavoidable, such as wind disturbances. Some theoretical and experimental analysis were given in [13, 18, 23], to explain why finite-time control can offer better ability of disturbance rejection. Compared to the study of a single quadrotor aircraft, the results of finite-time research on multiple quadrotor aircrafts, especially the results of finite-time formation control, are still limited. Most of the existing finite-time formation control research results are mainly for multi-agents [24–26], mobile robots [27], etc. The high dimension and nonlinear coupling of multiple quadrotor aircraft systems make the controller design and stability analysis of the closed-loop system more difficult.Based on the advantages of finite-time control, this paper will employ it to design a distributed formation control for multiple quadrotor aircrafts. It is worth noting that the results in most of the previous papers on formation control problems did not consider the effect of external disturbances, such as payload changes (or mass changes), wind disturbance, inaccurate model parameters, and so on. The robustness is an important aspect in the controlling of quadrotors. To improve the system dynamical performances, the finite-time position and attitude controllers are proposed for the nominal systems based on homogeneous systems theory. A quadrotor controller must be robust enough in order to reject the effect of disturbances and cover the change in model parameter uncertainties and external disturbances. To handle the external disturbances for the position-loop subsystem, the finite-time disturbance observers are deigned to estimate the disturbances in a finite time. Similarly, when there are both the parameter uncertainties and external disturbances for the attitude-loop subsystem, the finite-time disturbance observers are deigned to estimate the lump disturbances in a finite time. Compared with the existing results, the main contributions of this paper are as follows: (1) Construct the finite-time observers to achieve accurate estimation and compensation of disturbances. (2) Based on the finite time control technology, the system dynamical performances can be improved (i.e., faster convergence speed and better anti-interference ability). Finally, based on the finite-time disturbance observers, the finite-time distributed formation control strategy is developed for multiple quadrotor aircrafts, whose efficiency (including the convergence rate and disturbance rejection) is verified through simulations.


## 2. Preliminaries

### 2.1. Problem Description

This paper mainly studies the formation control problem of a group of four-rotor aircraft. Then defineΓ=1,2,…,n. Generally speaking, the motion information to describe the an aircraft is mainly composed of six-degree-of-freedom variables, i.e., the position information and the attitude information. Specifically, the coordinates of the aircraft can be described as(1)pi,qi,ri,ϕi,θi,ψiT∈R6,i∈Γ,where χi=(pi,qi,ri)T∈R3 denotes the position information by inertial coordinate system, and Φi=(ϕi,θi,ψi)T∈R3 are Euler angles to describe the attitude information based on the inertial coordinate system.


#### 2.1.1. Position Dynamical Model

From [28, 29], the position dynamical model for i-th quadrotor aircraft can be described as(2)p¨i=Timicos⁡ϕisin⁡θicos⁡ψi+sin⁡ϕisin⁡ψi+δi,p,q¨i=Timicos⁡ϕisin⁡θisin⁡ψi-sin⁡ϕicos⁡ψi+δi,q,r¨i=Timicos⁡ϕicos⁡θi-g+δi,r,i∈Γ,where mi denotes the mass of the aircraft, δi=(δi,p,δi,q,δi,r)T represents the unknown external disturbances, Ti is the total thrust produced by the four rotors, and g is a positive constant, i.e., the gravitational acceleration.


#### 2.1.2. Attitude Dynamical Model

The description of i-th attitude dynamical equation is based on Euler angles [29], which is given as(3)Ji,1ϕ¨i=liτi,1+di,ϕt,Ji,2θ¨i=liτi,2+di,θt,Ji,3ψ¨i=ciτi,3+di,ψt,i∈Γ,where Ji=diagJi,1,Ji,2,Ji,3 is the inertia matrix in the body-fixed frame, τi=(τi,1,τi,2,τi,3)T represents three rotational forces produced by the four rotors, ci and li are the constants, respectively, for the force-to-moment factor and the arm length of the aircraft, and di(t)=(di,ϕ(t),di,θ(t),di,ψ(t))T denotes the unknown time-varying external disturbances.


### 2.2. Control Objective

The control objective of this paper is to design a a distributed control law for a group of quadrotor aircraft (3)-(4) such that the formation control can be achievedin a finite time. Specifically, define the desired formation trajectory χd=(pd,qd,rd)T. At the same time, as we all know, the geometry in 3D space can be described by vector Δij=(Δpij,Δqij,Δrij)T,i,j∈Γ. In other words, the relative positional deviation between aircraft i and aircraft j is expressed as(4)Δij=Δi-Δj=Δpi,Δqi,ΔriT-Δpj,Δqj,ΔrjT.Based on mathematical expressions, the control objective of this paper can be described as that there is a finite time T∗ such that for any i,j∈Γ(5)limt→T∗pit-pjtqit-qjtrit-rjt=Δij,t<T∗,pit-pjtqit-qjtrit-rjt≡Δij,t≥T∗,and(6)limt→T∗1n∑i=1npit,qit,ritT=pd,qd,rdT,t<T∗,1n∑i=1npit,qit,ritT≡pd,qd,rdT,t≥T∗.To achieve the control objective, for each quadrotor aircraft, the following assumptions are imposed.Assumption 1.
Assume that the desired position trajectory isχd=(pd,qd,rd)T, where χ˙d and χ¨d are continuous and bounded.Assumption 2.
Assume that the external disturbances and the change rate of the disturbances are bounded; i.e.,di(t), d˙i(t), δi(t), δ˙i(t) are bounded.To achieve the finite-time formation control task, we will employ the multi-agent theory to design a distributed formation control algorithm. Next, we first review the knowledge of graph theory.


### 2.3. Graph Theory

Considering a group of quadrotor aircraft with leader-follower structure, assume that each aircraft is a node and the information exchange amongn follower agents is denoted by a directed graph G(A)=V,E,A. V=vi:i=1,…,n is the set of nodes, E⊆V×V is the set of edges, and A=[aij]∈Rn×n is the weighted adjacency matrix of the graph G(A) with non-negative adjacency elements aij. If there is an edge from node j to node i, i.e., (vj,vi)∈E, then aij>0, which means there exists an available information channel from node j to node i. The set of neighbors of node i is denoted by Ni=j:vj,vi∈E. The out-degree of node vi is defined as degout(vi)=di=∑j=1naij=∑j∈Niaij. Then the degree matrix of digraph G is D=diagd1,…,dn and the Laplacian matrix of digraph G is L=D-A.A path in graphG from vi1 to vik is a sequence of vi1,vi2,…,vik of finite nodes starting with vi1 and ending with vik such that (vil,vil+1)∈E for l=1,2,…,k-1. The graph G is connected if there is a path between any two distinct vertices.Assume that the reference state is represented by a leader. The connection weight between then agent and the leader is denoted by bi,i∈Γ. If the i-th agent has access to the information of leader, then bi>0; otherwise, bi=0. Let B=diagb1,…,bn.Assumption 3.
For the communication topology, it is assumed that the graph for all follower agents is connected. Meanwhile, at least one agent has access the signal directly from leader.


### 2.4. Related Definitions and Lemmas

Since the main objective of this paper is to design a finite-time formation control algorithm, we first introduce the concept of finite-time stability.Definition 4 (see [13]).
For a nonlinear system,(7)x˙=fx,f0=0,x∈Rn,where f(·) is a continuous function. If the system is Lyapunov stable and finite-time convergent, then it is called finite-time stable. The finite-time convergence is defined as that there is a finite time T(x0) such that limt→T(x0)x(t,x0)=0 and x(t,x0)=0 for all t≥T(x0).In addition, for brevity, we define the following notation.Definition 5.
Define function(8)sigαx=signxxα,α≥0,x∈R,where sign(·) is sign function.Finally, since the finite-time controller design in this paper is based on the homogeneous systems theory, we give the related definition and lemma about homogeneity.Definition 6 (see [13]).
For the system (7), if, for any ε>0, there exists (r1,…,rn) with ri>0,i=1,…,n, such that(9)fiεr1x1,…,εrnxn=εk+rifix,i=1,…,n,where k>-minri,i=1,…,n, then f(x) is said to be homogeneous of degree k with respect to the dilation (r1,…,rn).Lemma 7 (see [30]).
For system (7), if it has a continuous homogeneous vector space and is homogeneous of degree k<0 with respect to the dilation (r1,…,rn), then it is globally finite-time stable.


## 2.1. Problem Description

This paper mainly studies the formation control problem of a group of four-rotor aircraft. Then defineΓ=1,2,…,n. Generally speaking, the motion information to describe the an aircraft is mainly composed of six-degree-of-freedom variables, i.e., the position information and the attitude information. Specifically, the coordinates of the aircraft can be described as(1)pi,qi,ri,ϕi,θi,ψiT∈R6,i∈Γ,where χi=(pi,qi,ri)T∈R3 denotes the position information by inertial coordinate system, and Φi=(ϕi,θi,ψi)T∈R3 are Euler angles to describe the attitude information based on the inertial coordinate system.


### 2.1.1. Position Dynamical Model

From [28, 29], the position dynamical model for i-th quadrotor aircraft can be described as(2)p¨i=Timicos⁡ϕisin⁡θicos⁡ψi+sin⁡ϕisin⁡ψi+δi,p,q¨i=Timicos⁡ϕisin⁡θisin⁡ψi-sin⁡ϕicos⁡ψi+δi,q,r¨i=Timicos⁡ϕicos⁡θi-g+δi,r,i∈Γ,where mi denotes the mass of the aircraft, δi=(δi,p,δi,q,δi,r)T represents the unknown external disturbances, Ti is the total thrust produced by the four rotors, and g is a positive constant, i.e., the gravitational acceleration.


### 2.1.2. Attitude Dynamical Model

The description of i-th attitude dynamical equation is based on Euler angles [29], which is given as(3)Ji,1ϕ¨i=liτi,1+di,ϕt,Ji,2θ¨i=liτi,2+di,θt,Ji,3ψ¨i=ciτi,3+di,ψt,i∈Γ,where Ji=diagJi,1,Ji,2,Ji,3 is the inertia matrix in the body-fixed frame, τi=(τi,1,τi,2,τi,3)T represents three rotational forces produced by the four rotors, ci and li are the constants, respectively, for the force-to-moment factor and the arm length of the aircraft, and di(t)=(di,ϕ(t),di,θ(t),di,ψ(t))T denotes the unknown time-varying external disturbances.


## 2.1.1. Position Dynamical Model

From [28, 29], the position dynamical model for i-th quadrotor aircraft can be described as(2)p¨i=Timicos⁡ϕisin⁡θicos⁡ψi+sin⁡ϕisin⁡ψi+δi,p,q¨i=Timicos⁡ϕisin⁡θisin⁡ψi-sin⁡ϕicos⁡ψi+δi,q,r¨i=Timicos⁡ϕicos⁡θi-g+δi,r,i∈Γ,where mi denotes the mass of the aircraft, δi=(δi,p,δi,q,δi,r)T represents the unknown external disturbances, Ti is the total thrust produced by the four rotors, and g is a positive constant, i.e., the gravitational acceleration.


## 2.1.2. Attitude Dynamical Model

The description of i-th attitude dynamical equation is based on Euler angles [29], which is given as(3)Ji,1ϕ¨i=liτi,1+di,ϕt,Ji,2θ¨i=liτi,2+di,θt,Ji,3ψ¨i=ciτi,3+di,ψt,i∈Γ,where Ji=diagJi,1,Ji,2,Ji,3 is the inertia matrix in the body-fixed frame, τi=(τi,1,τi,2,τi,3)T represents three rotational forces produced by the four rotors, ci and li are the constants, respectively, for the force-to-moment factor and the arm length of the aircraft, and di(t)=(di,ϕ(t),di,θ(t),di,ψ(t))T denotes the unknown time-varying external disturbances.


## 2.2. Control Objective

The control objective of this paper is to design a a distributed control law for a group of quadrotor aircraft (3)-(4) such that the formation control can be achievedin a finite time. Specifically, define the desired formation trajectory χd=(pd,qd,rd)T. At the same time, as we all know, the geometry in 3D space can be described by vector Δij=(Δpij,Δqij,Δrij)T,i,j∈Γ. In other words, the relative positional deviation between aircraft i and aircraft j is expressed as(4)Δij=Δi-Δj=Δpi,Δqi,ΔriT-Δpj,Δqj,ΔrjT.Based on mathematical expressions, the control objective of this paper can be described as that there is a finite time T∗ such that for any i,j∈Γ(5)limt→T∗pit-pjtqit-qjtrit-rjt=Δij,t<T∗,pit-pjtqit-qjtrit-rjt≡Δij,t≥T∗,and(6)limt→T∗1n∑i=1npit,qit,ritT=pd,qd,rdT,t<T∗,1n∑i=1npit,qit,ritT≡pd,qd,rdT,t≥T∗.To achieve the control objective, for each quadrotor aircraft, the following assumptions are imposed.Assumption 1.
Assume that the desired position trajectory isχd=(pd,qd,rd)T, where χ˙d and χ¨d are continuous and bounded.Assumption 2.
Assume that the external disturbances and the change rate of the disturbances are bounded; i.e.,di(t), d˙i(t), δi(t), δ˙i(t) are bounded.To achieve the finite-time formation control task, we will employ the multi-agent theory to design a distributed formation control algorithm. Next, we first review the knowledge of graph theory.


## 2.3. Graph Theory

Considering a group of quadrotor aircraft with leader-follower structure, assume that each aircraft is a node and the information exchange amongn follower agents is denoted by a directed graph G(A)=V,E,A. V=vi:i=1,…,n is the set of nodes, E⊆V×V is the set of edges, and A=[aij]∈Rn×n is the weighted adjacency matrix of the graph G(A) with non-negative adjacency elements aij. If there is an edge from node j to node i, i.e., (vj,vi)∈E, then aij>0, which means there exists an available information channel from node j to node i. The set of neighbors of node i is denoted by Ni=j:vj,vi∈E. The out-degree of node vi is defined as degout(vi)=di=∑j=1naij=∑j∈Niaij. Then the degree matrix of digraph G is D=diagd1,…,dn and the Laplacian matrix of digraph G is L=D-A.A path in graphG from vi1 to vik is a sequence of vi1,vi2,…,vik of finite nodes starting with vi1 and ending with vik such that (vil,vil+1)∈E for l=1,2,…,k-1. The graph G is connected if there is a path between any two distinct vertices.Assume that the reference state is represented by a leader. The connection weight between then agent and the leader is denoted by bi,i∈Γ. If the i-th agent has access to the information of leader, then bi>0; otherwise, bi=0. Let B=diagb1,…,bn.Assumption 3.
For the communication topology, it is assumed that the graph for all follower agents is connected. Meanwhile, at least one agent has access the signal directly from leader.


## 2.4. Related Definitions and Lemmas

Since the main objective of this paper is to design a finite-time formation control algorithm, we first introduce the concept of finite-time stability.Definition 4 (see [13]).
For a nonlinear system,(7)x˙=fx,f0=0,x∈Rn,where f(·) is a continuous function. If the system is Lyapunov stable and finite-time convergent, then it is called finite-time stable. The finite-time convergence is defined as that there is a finite time T(x0) such that limt→T(x0)x(t,x0)=0 and x(t,x0)=0 for all t≥T(x0).In addition, for brevity, we define the following notation.Definition 5.
Define function(8)sigαx=signxxα,α≥0,x∈R,where sign(·) is sign function.Finally, since the finite-time controller design in this paper is based on the homogeneous systems theory, we give the related definition and lemma about homogeneity.Definition 6 (see [13]).
For the system (7), if, for any ε>0, there exists (r1,…,rn) with ri>0,i=1,…,n, such that(9)fiεr1x1,…,εrnxn=εk+rifix,i=1,…,n,where k>-minri,i=1,…,n, then f(x) is said to be homogeneous of degree k with respect to the dilation (r1,…,rn).Lemma 7 (see [30]).
For system (7), if it has a continuous homogeneous vector space and is homogeneous of degree k<0 with respect to the dilation (r1,…,rn), then it is globally finite-time stable.


## 3. Main Results

The controller design method is mainly based on the backstepping design and disturbance estimation-compensation method. Specifically speaking, the design procedure is divided into two steps.(i)
Step 1: for the position subsystem, the attitudeΦi=(ϕi,θi,ψi) is taken as the virtual control input and is designed as the desired attitude such that all aircraft can converge to the desired 3D-pattern and move along the desired leader’s trajectoryin a finite time. For the external disturbance, a finite-time disturbance observer is used to estimate and compensate the disturbance, i.e., disturbance estimation-compensation method.(ii)
Step 2: for the attitude subsystem, by combining disturbance observer and finite-time controller, a control lawτi is designed such that the each quadrotor desired attitude can be tracked by the real attitudein a finite time.


### 3.1. Finite-Time Position Formation Controller Design

For the sake of statement, denote(10)ui,p=Timicos⁡ϕisin⁡θicos⁡ψi+sin⁡ϕisin⁡ψi,ui,q=Timicos⁡ϕisin⁡θisin⁡ψi-sin⁡ϕicos⁡ψi,ui,r=Timicos⁡ϕicos⁡θi-g,i∈Γ,under which the position system’s equation (3) can be rewritten as follows:(11)p¨i=ui,p+δi,p,q¨i=ui,q+δi,q,r¨i=ui,r+δi,r.In order to design a finite-time controller for system (11), we first consider the case without external disturbances.


#### 3.1.1. Case 1: No External Disturbance

Lemma 8.
For the position motion model (11) without the external disturbances, i.e., δi(t)=0, if the controller is designed as(12)ui,p=-∑j∈Niaijkpsigα1pi-pj-Δpij+kdsigα2p˙i-p˙j-bikpsigα1pi-pd-Δpi+1n∑j=1nΔpi+kdsigα2p˙i-p˙d+p¨d,ui,q=-∑j∈Niaijkpsigα1qi-qj-Δqij+kdsigα2q˙i-q˙j-bikpsigα1qi-qd-Δqi+1n∑j=1nΔqi+kdsigα2q˙i-q˙d+q¨d,ui,r=-∑j∈Niaijkpsigα1ri-rj-Δrij+kdsigα2r˙i-r˙j-bikpsigα1ri-rd-Δri+1n∑j=1nΔri+kdsigα2r˙i-r˙d+r¨d,where 0<α1<1,α2=2α1/(1+α1), kp>0,kd>0, then the desired formation position control can be achieved in a finite time.Proof.
Since the three axis controllers are similar, without loss of generality, we will only give the proof process aboutp-axis. Let(13)ei,p=pi-pd-Δpi+1n∑j=1nΔpi.be the coordinate changes for i-th quadrotor aircraft. Under the proposed controller (12), it can be concluded from (11) that the position tracking error equation is(14)e˙i,p=p˙i-p˙d,e¨i,p=-∑j∈Niaijkpsigα1ei,p-ej,p+kdsigα2e˙i,p-e˙j,p-bikpsigα1ei,p+kdsigα2e˙i,p,i∈Γ.In the sequel, we will prove that the system (14) is globally finite-time stable and the proof is divided into two steps.
At the first step, we will show that system (14) is globally asymptotically stable. The Lyapunov function is chosen as(15)V=V1+V2,where(16)V1=kp∑i=1n∑j=1naij∫0ei,p-ej,psigα1ρdρ+2bi∫0ei,psigα1ρdρ,V2=∑i=1ne˙i,p2.Clearly, the Lyapunov function V is positive definite and radially unbounded. On one hand, the derivative of V1 along system (14) is(17)V˙1=kp∑i=1n∑j=1naijsigα1ei,p-ej,pe˙i,p-e˙j,p+2bisigα1ei,pe˙i,p.On the other hand, based on aij=aji,i,j∈Γ and the function sig(·) being an odd function, the derivative of V2 along system (14) is(18)V˙2=-2∑i=1ne˙i,p∑j∈Niaijkpsigα1ei,p-ej,p+kdsigα2e˙i,p-e˙j,p+bikpsigα1ei,p+kdsigα2e˙i,p=-kp∑i=1n∑j=1naije˙i,p-e˙j,psigα1ei,p-ej,p+2bie˙i,psigα1ei,p-kd∑i=1n∑j=1naije˙i,p-e˙j,psigα2e˙i,p-e˙j,p+2bie˙i,psigα2e˙i,p.In summary, the derivative of V along system (14) is(19)V˙=-kd∑i=1n∑j=1naije˙i,p-e˙j,p1+α2+2bie˙i,pα2+1.Since aij≥0, bi≥0, kd>0, it is obvious that V˙≤0. Define set Ψ=ei,p,e˙i,p∣V˙≡0. It can be concluded that V˙≡0 implies e˙i,p≡0 and e¨i,p≡0, which also implies that ei,p≡0 from (14), ∀i∈Γ. Using LaSalle’s invariant principle [31] leads to (ei,p,e˙i,p)→0 as t→0; i.e., system (14) is globally asymptotically stable.
Furthermore, according to the definition of homogeneous system (i.e., Definition6), it can be found that system (14) is homogeneous with degree of k=(α1-1)/2<0 with respect to the dilation Δ=(s1,…,s1︸fore1,p,…,en,p,s2,…,s2︸fore˙1,p,…,e˙n,p), where s1=1, s2=(α1+1)/2. In summary, based on Lemma 7, it can be concluded that the closed-loop system (14) is globally finite-time stable.However, in practice, the external disturbance is usually unavoidable. Next, we will consider this case.Assumption 9.
Assume that there are known positive constantsM1, M2, such that |δi|≤M1, |δ˙i|≤M2.


#### 3.1.2. Case 2: The Existence of External Disturbance

Lemma 10.
For system (11) under Assumption 9, the disturbance estimation (δ^i,p,δ^i,q,δ^i,r) from the observers(20)∂p˙^i∂t=ui,p+δ^i,p+λ1sig1/2p˙i-p˙^i,δ^˙i,p=λ2signp˙i-p˙^i,∂q˙^i∂t=ui,q+δ^i,q+λ1sig1/2q˙i-q˙^i,δ^˙i,q=λ2signq˙i-q˙^i,∂r˙^i∂t=ui,r+δ^i,r+λ1sig1/2r˙i-r˙^i,δ^˙i,r=λ2signr˙i-r˙^i,will converge to the real states (δi,p,δi,q,δi,r) in a finite time, where(21)λ2>M2,λ1>2λ2-M2·λ2+M21+h11-h1,with 0<h1<1.Proof.
Similarly, here, only the proof about x-axis is given. Define the disturbance observer error asη1=p˙i-p˙^i,η2=δi,p-δ^i,p, under which the error dynamical equation can be obtained from (11) and (20) that(22)η˙1=η2-λ1sig1/2η1,η˙2=-λ2signη1+δ˙i,p.By Assumption 9, we have -M2<δ˙i,p<M2. As a result, it follows from [32] that system (22) is finite-time stable under the gain condition (21), which means that δ^i,p can converge to δi,p in a finite time. The proof is completed.Based on the precise estimations for the external disturbances, a disturbance-observer-based composite controller is given to achieve finite-time position tracking control.Theorem 11.
For thei-th position motion model (11) with the disturbance, if the controller is designed as(23)ui,p=-∑j∈Niaijkpsigα1pi-pj-Δpij+kdsigα2p˙i-p˙j-bikpsigα1pi-pd-Δpi+1n∑j=1nΔpi+kdsigα2p˙i-p˙d+p¨d-δ^i,p,ui,q=-∑j∈Niaijkpsigα1qi-qj-Δqij+kdsigα2q˙i-q˙j-bikpsigα1qi-qd-Δqi+1n∑j=1nΔqi+kdsigα2q˙i-q˙d+q¨d-δ^i,q,ui,r=-∑j∈Niaijkpsigα1ri-rj-Δrij+kdsigα2r˙i-r˙j-bikpsigα1ri-rd-Δri+1n∑j=1nΔri+kdsigα2r˙i-r˙d+r¨d-δ^i,r,where 0<α1<1,α2=2α1/(1+α1), kp>0,kd>0, then the desired formation position control can be achieved in a finite time.Proof.
The proof is straightforward since the results in Lemmas8 and 10 are about finite-time convergence. In other word, there is a finite time T∗, for any i∈Γ,(24)limt→T∗ei,pt,e˙i,pt=0,t<T∗,ei,pt,e˙i,pt≡0,t≥T∗.Therefore, considering the coordination changes (13), for any i,j∈Γ, it can be concluded that(25)limt→T∗pit-pjt=Δpij,limt→T∗1n∑i=1n=pd,t<T∗,pit-pjt≡Δpij,limt→T∗1n∑i=1n≡pd,t≥T∗.Therefore, the formation control of a group of quadrotor aircraft can be achieved in a finite time. The proof is completed.


### 3.2. Attitude Controller Design

In the previous section, the attitude (i.e., Euler angleΦi=(ϕi,θi,ψi)T) is taken as a virtual control input. It is designed such that the desired position trajectory can be tracked in a finite time. Next, in this section, the attitude dynamical system will be considered. According to the relation (10), the desired attitude can be generated from the virtual control input ui,p,ui,q,ui,r. Denote Φi,d=(ϕi,d,θi,d,ψi,d)T as the desired attitude for i-th quadrotor aircraft; then it follows from (10) that(26)Ti,d=miui,p2+ui,q2+ui,r+g2,ϕi,d=arcsin⁡miTi,dui,psin⁡ψi,d-ui,qcos⁡ψi,d,θi,d=arctan⁡1ui,r+gui,pcos⁡ψi,d+ui,qsin⁡ψi,d,i∈Γ.Since the variable ψi,d is a free variable, the desired yaw angle can be set as ψi,d=0 for the convenience of analysis.Due to both uncertain parameters and external disturbances, the unknown parameters are broken into two parts, i.e., known nominal parts and unknown uncertainty parts. Specifically, let(27)Ji,j=Ji,jo+△Ji,j,j=1,2,3,where Ji,jo represents the known part and △Ji,j is the uncertainty part. With the help of the above notations, the attitude equation (4) can be rewritten as(28)ϕ¨i=liJi,1oτi,1+di,ϕ′,θ¨i=liJi,2oτi,2+di,θ′,ψ¨i=ciJi,3oτi,3+di,ψ′,where(29)di,ϕ′=1Ji,1-1Ji,1oliτi,1+di,ϕtJi,1,di,θ′=1Ji,2-1Ji,2oliτi,2+di,θtJi,2,di,ψ′=1Ji,3-1Ji,3ociτi,3+di,ψtJi,3.Remark 12.
Note that the uncertainty functions, i.e.,di′=(di,ϕ′,di,θ′,di,ψ′)T, defined above include not only the external disturbances but also the parameter variations, which are usually called the lumped disturbances in the literature [33].To deal with the lump disturbances in system (29), a disturbance estimation and then compensation method will be employed. As that in [33–35], the following assumption is given on the lump disturbances.Assumption 13.
Assume that there are known positive constantsL1, L2, such that |di′|≤L1, |d˙i′|≤L2.Under this assumption, motivated by the work [32], a finite-time disturbance observer is proposed to estimate the lump disturbances in a finite time.Lemma 14.
For the system (28) under Assumption 13, the states d^i′=(d^i,ϕ′,d^i,θ′,d^i,ψ′) of the observers(30)∂ϕ˙^i∂t=liJi,1oτi,1+d^i,ϕ′+ρ1sig1/2ϕ˙i-ϕ˙^i,d^˙i,ϕ′=ρ2signϕ˙i-ϕ˙^i,∂θ˙^i∂t=liJi,2oτi,2+d^i,θ′+ρ1sig1/2θ˙i-θ˙^i,d^˙i,θ′=ρ2signθ˙i-θ˙^i,∂ψ˙^i∂t=ciJi,3oτi,3+d^i,ψ′+ρ1sig1/2ψ˙i-ψ˙^i,d^˙i,ψ′=ρ2signψ˙i-ψ˙^i,will converge to the states di′=(di,ϕ′,di,θ′,di,ψ′) in a finite time, where(31)ρ2>L2,ρ1>2ρ2-L2·ρ2+L21+h21-h2,with 0<h2<1.Proof.
The proof process is the same as that of the position loop disturbance observer, i.e., Lemma10. The proof is straightforward.Based on the precise estimations for the lump disturbances, a disturbance-observer-based composite controller is given to achieve finite-time attitude tracking control.Theorem 15.
For the attitude subsystem (4) of i-th quadrotor aircraft in the presence of unknown parameters and external disturbances, if the controller is designed as(32)τi,1=Ji,1oliϕ¨i,d+apsigβ1ϕi,d-ϕi+adsigβ2ϕ˙i,d-ϕ˙i-d^i,ϕ′,τi,2=Ji,2oliθ¨i,d+apsigβ1θi,d-θi+adsigβ2θ˙i,d-θ˙i-d^i,θ′,τi,3=Ji,3ociψ¨i,d+apsigβ1ψi,d-ψi+adsigβ2ψ˙i,d-ψ˙i-d^i,ψ′,where 0<β1<1,β2=2β1/(1+β1), ap>0,ad>0, then the desired attitude can be tracked in a finite time, i.e., (ϕi,θi,ψi)T→(ϕi,d,θi,d,ψi,d)T in a finite time.Proof.
Since the disturbance observers (30) are about finite-time convergence, then there is a finite time To such that(33)d^i,ϕ′=di,ϕ′,d^i,θ′=di,θ′,d^i,ψ′=di,ψ′,∀t≥To.Similarly, only the proof about roll angle ϕi is provided. Define the attitude tracking error as ei,ϕ=ϕi,d-ϕi, when t≥To, whose dynamical equation can be obtained from (28), i.e.,(34)e˙i,ϕ=ϕ˙i,d-ϕ˙i,e¨i,ϕ=ϕ¨i,d-liJi,1τi,1.Under the proposed controller (32), the closed-loop system is(35)e˙i,ϕ=ϕ˙i,d-ϕ˙i,e¨i,ϕ=-apsigβ1ei,ϕ-adsigβ2e˙i,ϕ.It can be found that the closed-loop system (35) has the same structure as that of system (14). Hence, based on Lemma 7, it is easy to prove that system (35) is globally finite-time stable, which is omitted here.


## 3.1. Finite-Time Position Formation Controller Design

For the sake of statement, denote(10)ui,p=Timicos⁡ϕisin⁡θicos⁡ψi+sin⁡ϕisin⁡ψi,ui,q=Timicos⁡ϕisin⁡θisin⁡ψi-sin⁡ϕicos⁡ψi,ui,r=Timicos⁡ϕicos⁡θi-g,i∈Γ,under which the position system’s equation (3) can be rewritten as follows:(11)p¨i=ui,p+δi,p,q¨i=ui,q+δi,q,r¨i=ui,r+δi,r.In order to design a finite-time controller for system (11), we first consider the case without external disturbances.


### 3.1.1. Case 1: No External Disturbance

Lemma 8.
For the position motion model (11) without the external disturbances, i.e., δi(t)=0, if the controller is designed as(12)ui,p=-∑j∈Niaijkpsigα1pi-pj-Δpij+kdsigα2p˙i-p˙j-bikpsigα1pi-pd-Δpi+1n∑j=1nΔpi+kdsigα2p˙i-p˙d+p¨d,ui,q=-∑j∈Niaijkpsigα1qi-qj-Δqij+kdsigα2q˙i-q˙j-bikpsigα1qi-qd-Δqi+1n∑j=1nΔqi+kdsigα2q˙i-q˙d+q¨d,ui,r=-∑j∈Niaijkpsigα1ri-rj-Δrij+kdsigα2r˙i-r˙j-bikpsigα1ri-rd-Δri+1n∑j=1nΔri+kdsigα2r˙i-r˙d+r¨d,where 0<α1<1,α2=2α1/(1+α1), kp>0,kd>0, then the desired formation position control can be achieved in a finite time.Proof.
Since the three axis controllers are similar, without loss of generality, we will only give the proof process aboutp-axis. Let(13)ei,p=pi-pd-Δpi+1n∑j=1nΔpi.be the coordinate changes for i-th quadrotor aircraft. Under the proposed controller (12), it can be concluded from (11) that the position tracking error equation is(14)e˙i,p=p˙i-p˙d,e¨i,p=-∑j∈Niaijkpsigα1ei,p-ej,p+kdsigα2e˙i,p-e˙j,p-bikpsigα1ei,p+kdsigα2e˙i,p,i∈Γ.In the sequel, we will prove that the system (14) is globally finite-time stable and the proof is divided into two steps.
At the first step, we will show that system (14) is globally asymptotically stable. The Lyapunov function is chosen as(15)V=V1+V2,where(16)V1=kp∑i=1n∑j=1naij∫0ei,p-ej,psigα1ρdρ+2bi∫0ei,psigα1ρdρ,V2=∑i=1ne˙i,p2.Clearly, the Lyapunov function V is positive definite and radially unbounded. On one hand, the derivative of V1 along system (14) is(17)V˙1=kp∑i=1n∑j=1naijsigα1ei,p-ej,pe˙i,p-e˙j,p+2bisigα1ei,pe˙i,p.On the other hand, based on aij=aji,i,j∈Γ and the function sig(·) being an odd function, the derivative of V2 along system (14) is(18)V˙2=-2∑i=1ne˙i,p∑j∈Niaijkpsigα1ei,p-ej,p+kdsigα2e˙i,p-e˙j,p+bikpsigα1ei,p+kdsigα2e˙i,p=-kp∑i=1n∑j=1naije˙i,p-e˙j,psigα1ei,p-ej,p+2bie˙i,psigα1ei,p-kd∑i=1n∑j=1naije˙i,p-e˙j,psigα2e˙i,p-e˙j,p+2bie˙i,psigα2e˙i,p.In summary, the derivative of V along system (14) is(19)V˙=-kd∑i=1n∑j=1naije˙i,p-e˙j,p1+α2+2bie˙i,pα2+1.Since aij≥0, bi≥0, kd>0, it is obvious that V˙≤0. Define set Ψ=ei,p,e˙i,p∣V˙≡0. It can be concluded that V˙≡0 implies e˙i,p≡0 and e¨i,p≡0, which also implies that ei,p≡0 from (14), ∀i∈Γ. Using LaSalle’s invariant principle [31] leads to (ei,p,e˙i,p)→0 as t→0; i.e., system (14) is globally asymptotically stable.
Furthermore, according to the definition of homogeneous system (i.e., Definition6), it can be found that system (14) is homogeneous with degree of k=(α1-1)/2<0 with respect to the dilation Δ=(s1,…,s1︸fore1,p,…,en,p,s2,…,s2︸fore˙1,p,…,e˙n,p), where s1=1, s2=(α1+1)/2. In summary, based on Lemma 7, it can be concluded that the closed-loop system (14) is globally finite-time stable.However, in practice, the external disturbance is usually unavoidable. Next, we will consider this case.Assumption 9.
Assume that there are known positive constantsM1, M2, such that |δi|≤M1, |δ˙i|≤M2.


### 3.1.2. Case 2: The Existence of External Disturbance

Lemma 10.
For system (11) under Assumption 9, the disturbance estimation (δ^i,p,δ^i,q,δ^i,r) from the observers(20)∂p˙^i∂t=ui,p+δ^i,p+λ1sig1/2p˙i-p˙^i,δ^˙i,p=λ2signp˙i-p˙^i,∂q˙^i∂t=ui,q+δ^i,q+λ1sig1/2q˙i-q˙^i,δ^˙i,q=λ2signq˙i-q˙^i,∂r˙^i∂t=ui,r+δ^i,r+λ1sig1/2r˙i-r˙^i,δ^˙i,r=λ2signr˙i-r˙^i,will converge to the real states (δi,p,δi,q,δi,r) in a finite time, where(21)λ2>M2,λ1>2λ2-M2·λ2+M21+h11-h1,with 0<h1<1.Proof.
Similarly, here, only the proof about x-axis is given. Define the disturbance observer error asη1=p˙i-p˙^i,η2=δi,p-δ^i,p, under which the error dynamical equation can be obtained from (11) and (20) that(22)η˙1=η2-λ1sig1/2η1,η˙2=-λ2signη1+δ˙i,p.By Assumption 9, we have -M2<δ˙i,p<M2. As a result, it follows from [32] that system (22) is finite-time stable under the gain condition (21), which means that δ^i,p can converge to δi,p in a finite time. The proof is completed.Based on the precise estimations for the external disturbances, a disturbance-observer-based composite controller is given to achieve finite-time position tracking control.Theorem 11.
For thei-th position motion model (11) with the disturbance, if the controller is designed as(23)ui,p=-∑j∈Niaijkpsigα1pi-pj-Δpij+kdsigα2p˙i-p˙j-bikpsigα1pi-pd-Δpi+1n∑j=1nΔpi+kdsigα2p˙i-p˙d+p¨d-δ^i,p,ui,q=-∑j∈Niaijkpsigα1qi-qj-Δqij+kdsigα2q˙i-q˙j-bikpsigα1qi-qd-Δqi+1n∑j=1nΔqi+kdsigα2q˙i-q˙d+q¨d-δ^i,q,ui,r=-∑j∈Niaijkpsigα1ri-rj-Δrij+kdsigα2r˙i-r˙j-bikpsigα1ri-rd-Δri+1n∑j=1nΔri+kdsigα2r˙i-r˙d+r¨d-δ^i,r,where 0<α1<1,α2=2α1/(1+α1), kp>0,kd>0, then the desired formation position control can be achieved in a finite time.Proof.
The proof is straightforward since the results in Lemmas8 and 10 are about finite-time convergence. In other word, there is a finite time T∗, for any i∈Γ,(24)limt→T∗ei,pt,e˙i,pt=0,t<T∗,ei,pt,e˙i,pt≡0,t≥T∗.Therefore, considering the coordination changes (13), for any i,j∈Γ, it can be concluded that(25)limt→T∗pit-pjt=Δpij,limt→T∗1n∑i=1n=pd,t<T∗,pit-pjt≡Δpij,limt→T∗1n∑i=1n≡pd,t≥T∗.Therefore, the formation control of a group of quadrotor aircraft can be achieved in a finite time. The proof is completed.


## 3.1.1. Case 1: No External Disturbance

Lemma 8.
For the position motion model (11) without the external disturbances, i.e., δi(t)=0, if the controller is designed as(12)ui,p=-∑j∈Niaijkpsigα1pi-pj-Δpij+kdsigα2p˙i-p˙j-bikpsigα1pi-pd-Δpi+1n∑j=1nΔpi+kdsigα2p˙i-p˙d+p¨d,ui,q=-∑j∈Niaijkpsigα1qi-qj-Δqij+kdsigα2q˙i-q˙j-bikpsigα1qi-qd-Δqi+1n∑j=1nΔqi+kdsigα2q˙i-q˙d+q¨d,ui,r=-∑j∈Niaijkpsigα1ri-rj-Δrij+kdsigα2r˙i-r˙j-bikpsigα1ri-rd-Δri+1n∑j=1nΔri+kdsigα2r˙i-r˙d+r¨d,where 0<α1<1,α2=2α1/(1+α1), kp>0,kd>0, then the desired formation position control can be achieved in a finite time.Proof.
Since the three axis controllers are similar, without loss of generality, we will only give the proof process aboutp-axis. Let(13)ei,p=pi-pd-Δpi+1n∑j=1nΔpi.be the coordinate changes for i-th quadrotor aircraft. Under the proposed controller (12), it can be concluded from (11) that the position tracking error equation is(14)e˙i,p=p˙i-p˙d,e¨i,p=-∑j∈Niaijkpsigα1ei,p-ej,p+kdsigα2e˙i,p-e˙j,p-bikpsigα1ei,p+kdsigα2e˙i,p,i∈Γ.In the sequel, we will prove that the system (14) is globally finite-time stable and the proof is divided into two steps.
At the first step, we will show that system (14) is globally asymptotically stable. The Lyapunov function is chosen as(15)V=V1+V2,where(16)V1=kp∑i=1n∑j=1naij∫0ei,p-ej,psigα1ρdρ+2bi∫0ei,psigα1ρdρ,V2=∑i=1ne˙i,p2.Clearly, the Lyapunov function V is positive definite and radially unbounded. On one hand, the derivative of V1 along system (14) is(17)V˙1=kp∑i=1n∑j=1naijsigα1ei,p-ej,pe˙i,p-e˙j,p+2bisigα1ei,pe˙i,p.On the other hand, based on aij=aji,i,j∈Γ and the function sig(·) being an odd function, the derivative of V2 along system (14) is(18)V˙2=-2∑i=1ne˙i,p∑j∈Niaijkpsigα1ei,p-ej,p+kdsigα2e˙i,p-e˙j,p+bikpsigα1ei,p+kdsigα2e˙i,p=-kp∑i=1n∑j=1naije˙i,p-e˙j,psigα1ei,p-ej,p+2bie˙i,psigα1ei,p-kd∑i=1n∑j=1naije˙i,p-e˙j,psigα2e˙i,p-e˙j,p+2bie˙i,psigα2e˙i,p.In summary, the derivative of V along system (14) is(19)V˙=-kd∑i=1n∑j=1naije˙i,p-e˙j,p1+α2+2bie˙i,pα2+1.Since aij≥0, bi≥0, kd>0, it is obvious that V˙≤0. Define set Ψ=ei,p,e˙i,p∣V˙≡0. It can be concluded that V˙≡0 implies e˙i,p≡0 and e¨i,p≡0, which also implies that ei,p≡0 from (14), ∀i∈Γ. Using LaSalle’s invariant principle [31] leads to (ei,p,e˙i,p)→0 as t→0; i.e., system (14) is globally asymptotically stable.
Furthermore, according to the definition of homogeneous system (i.e., Definition6), it can be found that system (14) is homogeneous with degree of k=(α1-1)/2<0 with respect to the dilation Δ=(s1,…,s1︸fore1,p,…,en,p,s2,…,s2︸fore˙1,p,…,e˙n,p), where s1=1, s2=(α1+1)/2. In summary, based on Lemma 7, it can be concluded that the closed-loop system (14) is globally finite-time stable.However, in practice, the external disturbance is usually unavoidable. Next, we will consider this case.Assumption 9.
Assume that there are known positive constantsM1, M2, such that |δi|≤M1, |δ˙i|≤M2.


## 3.1.2. Case 2: The Existence of External Disturbance

Lemma 10.
For system (11) under Assumption 9, the disturbance estimation (δ^i,p,δ^i,q,δ^i,r) from the observers(20)∂p˙^i∂t=ui,p+δ^i,p+λ1sig1/2p˙i-p˙^i,δ^˙i,p=λ2signp˙i-p˙^i,∂q˙^i∂t=ui,q+δ^i,q+λ1sig1/2q˙i-q˙^i,δ^˙i,q=λ2signq˙i-q˙^i,∂r˙^i∂t=ui,r+δ^i,r+λ1sig1/2r˙i-r˙^i,δ^˙i,r=λ2signr˙i-r˙^i,will converge to the real states (δi,p,δi,q,δi,r) in a finite time, where(21)λ2>M2,λ1>2λ2-M2·λ2+M21+h11-h1,with 0<h1<1.Proof.
Similarly, here, only the proof about x-axis is given. Define the disturbance observer error asη1=p˙i-p˙^i,η2=δi,p-δ^i,p, under which the error dynamical equation can be obtained from (11) and (20) that(22)η˙1=η2-λ1sig1/2η1,η˙2=-λ2signη1+δ˙i,p.By Assumption 9, we have -M2<δ˙i,p<M2. As a result, it follows from [32] that system (22) is finite-time stable under the gain condition (21), which means that δ^i,p can converge to δi,p in a finite time. The proof is completed.Based on the precise estimations for the external disturbances, a disturbance-observer-based composite controller is given to achieve finite-time position tracking control.Theorem 11.
For thei-th position motion model (11) with the disturbance, if the controller is designed as(23)ui,p=-∑j∈Niaijkpsigα1pi-pj-Δpij+kdsigα2p˙i-p˙j-bikpsigα1pi-pd-Δpi+1n∑j=1nΔpi+kdsigα2p˙i-p˙d+p¨d-δ^i,p,ui,q=-∑j∈Niaijkpsigα1qi-qj-Δqij+kdsigα2q˙i-q˙j-bikpsigα1qi-qd-Δqi+1n∑j=1nΔqi+kdsigα2q˙i-q˙d+q¨d-δ^i,q,ui,r=-∑j∈Niaijkpsigα1ri-rj-Δrij+kdsigα2r˙i-r˙j-bikpsigα1ri-rd-Δri+1n∑j=1nΔri+kdsigα2r˙i-r˙d+r¨d-δ^i,r,where 0<α1<1,α2=2α1/(1+α1), kp>0,kd>0, then the desired formation position control can be achieved in a finite time.Proof.
The proof is straightforward since the results in Lemmas8 and 10 are about finite-time convergence. In other word, there is a finite time T∗, for any i∈Γ,(24)limt→T∗ei,pt,e˙i,pt=0,t<T∗,ei,pt,e˙i,pt≡0,t≥T∗.Therefore, considering the coordination changes (13), for any i,j∈Γ, it can be concluded that(25)limt→T∗pit-pjt=Δpij,limt→T∗1n∑i=1n=pd,t<T∗,pit-pjt≡Δpij,limt→T∗1n∑i=1n≡pd,t≥T∗.Therefore, the formation control of a group of quadrotor aircraft can be achieved in a finite time. The proof is completed.


## 3.2. Attitude Controller Design

In the previous section, the attitude (i.e., Euler angleΦi=(ϕi,θi,ψi)T) is taken as a virtual control input. It is designed such that the desired position trajectory can be tracked in a finite time. Next, in this section, the attitude dynamical system will be considered. According to the relation (10), the desired attitude can be generated from the virtual control input ui,p,ui,q,ui,r. Denote Φi,d=(ϕi,d,θi,d,ψi,d)T as the desired attitude for i-th quadrotor aircraft; then it follows from (10) that(26)Ti,d=miui,p2+ui,q2+ui,r+g2,ϕi,d=arcsin⁡miTi,dui,psin⁡ψi,d-ui,qcos⁡ψi,d,θi,d=arctan⁡1ui,r+gui,pcos⁡ψi,d+ui,qsin⁡ψi,d,i∈Γ.Since the variable ψi,d is a free variable, the desired yaw angle can be set as ψi,d=0 for the convenience of analysis.Due to both uncertain parameters and external disturbances, the unknown parameters are broken into two parts, i.e., known nominal parts and unknown uncertainty parts. Specifically, let(27)Ji,j=Ji,jo+△Ji,j,j=1,2,3,where Ji,jo represents the known part and △Ji,j is the uncertainty part. With the help of the above notations, the attitude equation (4) can be rewritten as(28)ϕ¨i=liJi,1oτi,1+di,ϕ′,θ¨i=liJi,2oτi,2+di,θ′,ψ¨i=ciJi,3oτi,3+di,ψ′,where(29)di,ϕ′=1Ji,1-1Ji,1oliτi,1+di,ϕtJi,1,di,θ′=1Ji,2-1Ji,2oliτi,2+di,θtJi,2,di,ψ′=1Ji,3-1Ji,3ociτi,3+di,ψtJi,3.Remark 12.
Note that the uncertainty functions, i.e.,di′=(di,ϕ′,di,θ′,di,ψ′)T, defined above include not only the external disturbances but also the parameter variations, which are usually called the lumped disturbances in the literature [33].To deal with the lump disturbances in system (29), a disturbance estimation and then compensation method will be employed. As that in [33–35], the following assumption is given on the lump disturbances.Assumption 13.
Assume that there are known positive constantsL1, L2, such that |di′|≤L1, |d˙i′|≤L2.Under this assumption, motivated by the work [32], a finite-time disturbance observer is proposed to estimate the lump disturbances in a finite time.Lemma 14.
For the system (28) under Assumption 13, the states d^i′=(d^i,ϕ′,d^i,θ′,d^i,ψ′) of the observers(30)∂ϕ˙^i∂t=liJi,1oτi,1+d^i,ϕ′+ρ1sig1/2ϕ˙i-ϕ˙^i,d^˙i,ϕ′=ρ2signϕ˙i-ϕ˙^i,∂θ˙^i∂t=liJi,2oτi,2+d^i,θ′+ρ1sig1/2θ˙i-θ˙^i,d^˙i,θ′=ρ2signθ˙i-θ˙^i,∂ψ˙^i∂t=ciJi,3oτi,3+d^i,ψ′+ρ1sig1/2ψ˙i-ψ˙^i,d^˙i,ψ′=ρ2signψ˙i-ψ˙^i,will converge to the states di′=(di,ϕ′,di,θ′,di,ψ′) in a finite time, where(31)ρ2>L2,ρ1>2ρ2-L2·ρ2+L21+h21-h2,with 0<h2<1.Proof.
The proof process is the same as that of the position loop disturbance observer, i.e., Lemma10. The proof is straightforward.Based on the precise estimations for the lump disturbances, a disturbance-observer-based composite controller is given to achieve finite-time attitude tracking control.Theorem 15.
For the attitude subsystem (4) of i-th quadrotor aircraft in the presence of unknown parameters and external disturbances, if the controller is designed as(32)τi,1=Ji,1oliϕ¨i,d+apsigβ1ϕi,d-ϕi+adsigβ2ϕ˙i,d-ϕ˙i-d^i,ϕ′,τi,2=Ji,2oliθ¨i,d+apsigβ1θi,d-θi+adsigβ2θ˙i,d-θ˙i-d^i,θ′,τi,3=Ji,3ociψ¨i,d+apsigβ1ψi,d-ψi+adsigβ2ψ˙i,d-ψ˙i-d^i,ψ′,where 0<β1<1,β2=2β1/(1+β1), ap>0,ad>0, then the desired attitude can be tracked in a finite time, i.e., (ϕi,θi,ψi)T→(ϕi,d,θi,d,ψi,d)T in a finite time.Proof.
Since the disturbance observers (30) are about finite-time convergence, then there is a finite time To such that(33)d^i,ϕ′=di,ϕ′,d^i,θ′=di,θ′,d^i,ψ′=di,ψ′,∀t≥To.Similarly, only the proof about roll angle ϕi is provided. Define the attitude tracking error as ei,ϕ=ϕi,d-ϕi, when t≥To, whose dynamical equation can be obtained from (28), i.e.,(34)e˙i,ϕ=ϕ˙i,d-ϕ˙i,e¨i,ϕ=ϕ¨i,d-liJi,1τi,1.Under the proposed controller (32), the closed-loop system is(35)e˙i,ϕ=ϕ˙i,d-ϕ˙i,e¨i,ϕ=-apsigβ1ei,ϕ-adsigβ2e˙i,ϕ.It can be found that the closed-loop system (35) has the same structure as that of system (14). Hence, based on Lemma 7, it is easy to prove that system (35) is globally finite-time stable, which is omitted here.


## 4. Numerical Simulations

To verify the proposed theoretical results in Theorems11 and 15, some numerical simulations are given. The network system considered in this section consists of one leader and three followers. The undirected topology graph topology is shown in Figure 1. Specifically, the weights of the undirected edges are, respectively, given as b1=1, a12=a21=1, a23=a32=1. Meanwhile, the desired formation pattern is to form a regular triangle on in the p-q plane. And from Figure 1, the relative position deviations can been given as(36)Δpi=0,1,0T,Δqi=cos5π6,sin⁡-π6,0T,Δri=cos-π6,sin⁡-π6,0T.Figure 1
The communication topology graph and the desired formation pattern.


### 4.1. Model Parameters and Controllers Gains

The desired formation trajectory, i.e., leader’s trajectory, is(37)xd,yd,zdT=5sin⁡0.2t,5cos⁡0.2t,0.5tT.And the initial conditions for each quadrotor aircraft are as follows:(38)p10,q10,r10,ϕ10,θ10,ψ10=-3,5,0,0,0,0,p20,q20,r20,ϕ20,θ20,ψ20=1,-4,0,0,0,0,p30,q30,r30,ϕ30,θ30,ψ30=2,3,0,0,0,0.The model parameter values are given in Table1. Meanwhile, by Theorems 11 and 15, the controllers’ and observers’ parameters are selected as kp=ap=3.5, kd=ad=4.5, α1=β1=3/4, λ1=4, λ2=1, ρ1=5, ρ2=2.Table 1
System’s parameters.
Descriptions Parameter Values mass mi=0.468kg  gravitational acceleration g=9.81m/s2  inertia matrix J i = J i o + △ J i ,   i = 1,2 , 3, with J i o = d i a g ( 0.0020,0.0022,0.0030 ) △Ji=diag(0.0003,0.0002,-0.0004)  position-loop disturbances δ i , p ( t ) = 0.002 sin ⁡ t + 5 δ i , q ( t ) = 0.002 cos ⁡ 2 t + 3   i = 1,2 , 3 δ i , r ( t ) = 0.003 sin ⁡ 3 t + 2  attitude-loop disturbances d i , ϕ ( t ) = 0.8 sin ⁡ 0.4 t + 5 d i , θ ( t ) = cos ⁡ 0.3 t + 3   i = 1,2 , 3 d i , ψ ( t ) = 0.5 sin ⁡ 0.3 t + 2


### 4.2. Numerical Simulation Results

Under the proposed control algorithm, Figure2 shows the position trajectory of each quadrotor aircraft in 3D space. It can be found that the desired formation control task can be achieved under the proposed finite-time controller. The tracking curve for each aircraft’s position and attitude are given in Figures 3 and 4. And the adjacent distance of all quadrotor aircrafts is shown in Figure 5. The response curves for the estimated position disturbances and the estimated attitude lump disturbances are, respectively, shown in Figures 6 and 7. It can be found that the proposed methods of finite-time disturbances estimation are effective.Figure 2
The response curves of each aircraft’s trajectory in 3D space.Figure 3
The response curves of the position for each aircraft.Figure 4
The response curves of the attitude for each aircraft.Figure 5
The adjacent distance of all quadrotor aircrafts.Figure 6
The response curves of the estimated position disturbances for each aircraft.Figure 7
The response curves of the estimated attitude lump disturbances for each aircraft.


## 4.1. Model Parameters and Controllers Gains

The desired formation trajectory, i.e., leader’s trajectory, is(37)xd,yd,zdT=5sin⁡0.2t,5cos⁡0.2t,0.5tT.And the initial conditions for each quadrotor aircraft are as follows:(38)p10,q10,r10,ϕ10,θ10,ψ10=-3,5,0,0,0,0,p20,q20,r20,ϕ20,θ20,ψ20=1,-4,0,0,0,0,p30,q30,r30,ϕ30,θ30,ψ30=2,3,0,0,0,0.The model parameter values are given in Table1. Meanwhile, by Theorems 11 and 15, the controllers’ and observers’ parameters are selected as kp=ap=3.5, kd=ad=4.5, α1=β1=3/4, λ1=4, λ2=1, ρ1=5, ρ2=2.Table 1
System’s parameters.
Descriptions Parameter Values mass mi=0.468kg  gravitational acceleration g=9.81m/s2  inertia matrix J i = J i o + △ J i ,   i = 1,2 , 3, with J i o = d i a g ( 0.0020,0.0022,0.0030 ) △Ji=diag(0.0003,0.0002,-0.0004)  position-loop disturbances δ i , p ( t ) = 0.002 sin ⁡ t + 5 δ i , q ( t ) = 0.002 cos ⁡ 2 t + 3   i = 1,2 , 3 δ i , r ( t ) = 0.003 sin ⁡ 3 t + 2  attitude-loop disturbances d i , ϕ ( t ) = 0.8 sin ⁡ 0.4 t + 5 d i , θ ( t ) = cos ⁡ 0.3 t + 3   i = 1,2 , 3 d i , ψ ( t ) = 0.5 sin ⁡ 0.3 t + 2


## 4.2. Numerical Simulation Results

Under the proposed control algorithm, Figure2 shows the position trajectory of each quadrotor aircraft in 3D space. It can be found that the desired formation control task can be achieved under the proposed finite-time controller. The tracking curve for each aircraft’s position and attitude are given in Figures 3 and 4. And the adjacent distance of all quadrotor aircrafts is shown in Figure 5. The response curves for the estimated position disturbances and the estimated attitude lump disturbances are, respectively, shown in Figures 6 and 7. It can be found that the proposed methods of finite-time disturbances estimation are effective.Figure 2
The response curves of each aircraft’s trajectory in 3D space.Figure 3
The response curves of the position for each aircraft.Figure 4
The response curves of the attitude for each aircraft.Figure 5
The adjacent distance of all quadrotor aircrafts.Figure 6
The response curves of the estimated position disturbances for each aircraft.Figure 7
The response curves of the estimated attitude lump disturbances for each aircraft.


## 5. Conclusion

In this paper, observers-based distributed formation control law has been proposed to solve the formation control problem for multiple quadrotor aircrafts in the presence of parameter uncertainties and external disturbances. Rigorous stability analysis and some simulation results have been presented to show that the formation task can be achieved using the proposed control algorithm.



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