# Formation Control for Multiple Quadrotor Aircraft via Fixed-time Consensus Algorithm

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

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## Abstract

This paper mainly studies the formation control problem of multiple quadrotor aircraft via fixed-time control theory. First, based on the bilimit homogeneous theory and the framework of multiagent theory, for multiaircraft, a fixed-time formation control strategy is proposed. Considering the external disturbance existing on the attitude loop of the aircraft, the corresponding fixed-time disturbance observer is designed with the observer technology. Then, a fixed-time attitude controller is designed based on the accurate observation and fast compensation from the disturbance observer. Finally, some simulations are performed to verify the effectiveness of the proposed theoretical method.


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## Body

## 1. Introduction

In the last few years, the quadrotor aircraft has received extensive attention and application. Different from the fixed-wing aircraft, there are some clear advantages for the quadrotor aircraft, such as simple structures, flexible operation, and vertical takeoff and landing [1]. Because of these characteristics, the quadrotor aircraft has good application scenarios in both military and civil fields, for example, environmental assessment and disaster relief [1–3].With the continuous in-depth development of research on quadrotor aircraft, the problem of multiaircraft formation coordination control has attracted more and more attention because multiple aircraft can accomplish more complex tasks than the separate aircraft. Actually, the cooperative work of multiple aircraft means that not only more equipment can be loaded to adapt to more complex and precise working environments but also the efficiency and robustness of the entire system can be significantly improved. Nevertheless, since the quadrotor system is an underactuated system with strong coupling, nonlinearity, etc., it is difficult to design a controller for a single aircraft [4, 5], not to mention further coordinating the control of multiple aircraft on the basis of controlling one aircraft, which is even more challenging.So far, the formation control has achieved a series of progress. Earlier studies [6, 7] realized the tracking control tasks for the time-varying formations. Based on the idea of distributed control, Lin et al. [8] proposed the formation control strategy about multiagent systems by using a complex Laplacian. Similarly for multiagent systems, considering the Lipschitz-type node dynamics, the consensus problem was studied by using the distributed method in [9]. Liu et al. [10] designed a distributed formation controller for mobile robots in consideration of sampling data and communication delay. For higher-order multileader multiagent systems, based on the observer and distributed method, Wen et al. [11] proposed a new class of containment protocols to weaken some irrational assumptions in the existing literature. For the coordinated control problem of mobile robots, Wang et al. [12] provided the corresponding solution by vision-based sensors. The global pinning synchronization problem was studied for a class of complex networks with switching directed topologies in [13]. Based on adaptive consensus theory and distributed method, a new kind of distributed dispatch algorithms were developed for smart grids subject to communication uncertainties in [14]. In addition, for the aircraft during the flight, the external disturbances are often unavoidable, such as wind disturbances. For the aircraft’s attitude system, Wang and Su [15] investigated the dynamic surface control of the nonlinear transport aircraft model during the process of continuous heavy cargo airdrop in case of disturbance and actuator saturation. Chwa [16] proposed the fuzzy adaptive output feedback tracking control method for VTOL (vertical takeoff and landing) aircraft.However, most of the existing results for formation controller algorithms only guarantee that the formation achievement is done with asymptotically stable. That is to say, the completion time of formation achievement will tend to be infinite. In practice, it is always hoped that the formation can be completed faster. To make up for the above shortcoming, the finite-time control method has been developed in [17–19] to complete the original intention of improving the convergence rate. As the name implies, the finite-time control means that, in a finite time, the states of the system will be stabilized to equilibrium. Combining with the nonlinear sliding mode control, Li et al. [20] proposed a finite-time formation control approach. For the spacecraft formation flying problem, Liu et al. [21] addressed the finite-time distributed orbit synchronization control strategy. Zhang et al. [22] realized the finite-time formation control for quadrotor aircraft with external disturbance. Although the finite-time control method has many advantages, its finite convergence time usually depends on the initial conditions. Considering this point, Polyakov [23] for the first time introduced the concept of fixed-time stability; in other words, the convergent time is independent of the initial conditions and can be predetermined. Wu et al. [24] considered the synchronization problem for a class of second-order master-slave nonlinear systems based on the fixed-time control theory and output feedback control method. Chu et al. [25] proposed the multirobot systems’ formation tracking problem with nonholonomic constraints under the fixed-time theoretical framework. Gao and Guo [26] investigated a fixed-time formation control method for AUVs with leader-following control. For common second-order multiagent systems, Chu et al. [27] designed a robust fixed-time consensus controller.Compared to the finite-time control method in [28], the fixed-time control has the advantage on the convergent time which is independent of the initial system states. Motivated by this consideration, this paper will study the formation control problem by using the fixed-time consensus algorithm. The main contribution of this paper is to realize the formation control of multiple aircraft with external disturbances in a fixed time. Firstly, the multiagent theory and the bilimit homogeneous theory are used to propose a fixed-time formation control algorithm. Then, the fixed-time attitude controller is designed for a single aircraft. At the same time, considering the negative influence of external disturbances on the aircraft, a fixed-time disturbance observer is designed for accurate/rapid observation of external disturbances. In this way, the external disturbances can be compensated by the observed values. Finally, the simulation results are provided to verify the dynamic- and steady-state performance of the system. Compared with the existing formation control algorithms, the main contribution of this paper is to propose a new fixed-time formation control algorithm in the presence of external disturbances, which can achieve attitude formation in a prescribed convergent time without depending on initial conditions. Hence, in an unknown environment, the proposed result in this paper can provide designer convergence information. To the best of the authors’ knowledge, the presented results in this paper are novel in the literature.


## 2. Prerequisite Knowledge

### 2.1. Problem Description

The formation withn four-rotor aircraft is considered, and let Γ=1,2,⋯,n. From the viewpoint of modelling, the aircraft can usually use six variables to uniquely determine itself the position and attitude in three-dimensional space. Without loss of generality, in the inertial coordinate system, let ϒi=ηi,ιi,κiT∈R3 be the aircraft’s position and Θi=pi,qi,riT∈R3 be the aircraft’s attitude based on Euler angles. As a result, the coordinates of the aircraft are(1)ηi,ιi,κi,pi,qi,riT∈R6,i∈Γ.


#### 2.1.1. Position Dynamical Model

As that in [29, 30], the mathematical model of the position of each quadrotor can be described as(2)η¨i=Timicospisinqicosri+sinpisinri,ι¨i=Timicospisinqisinri−sinpicosri,κ¨i=Timicospicosqi−g,i∈Γ,where mi denotes the i-th aircraft’s mass, Ti is the i-th aircraft’s thrust, and g is the gravitational acceleration which always exists.


#### 2.1.2. Attitude Dynamical Model

By Euler angles, the attitude dynamical equation is given in [30]:(3)Ji,1p¨i=liτi,1+di,1t,Ji,2q¨i=liτi,2+di,2t,Ji,3r¨i=ciτi,3+di,3t,i∈Γ,where τi=τi,1,τi,2,τi,3T denotes the i-th aircraft’s control torque, ci is the i-th aircraft’s force-to-moment factor, Ji=diagJi,1,Ji,2,Ji,3 is the inertia matrix, and li is the arm length of the i-th aircraft. In addition, the external disturbances are denoted by dit=di,1t,di,2t,di,3tT.


### 2.2. Control Objective

Letϒd=ηd,ιd,κdT be the referenced formation trajectory. As we all know, in fact, the three-dimensional relative position between two points can be expressed in the form of a three-dimensional vector. The formation geometry in 3D space is given by vector Δij=Δηij,Διij,ΔκijT, i,j∈Γ. Hence, the aircraft i and aircraft j are hoped to have the relative positional deviation:(4)Δij=Δi−Δj=Δηi,Διi,ΔκiT−Δηj,Διj,ΔκjT,where Δi=Δηi,Διi,ΔκiT and Δj=Δηj,Διj,ΔκjT represent the coordinates of two points in the same three-dimensional rectangular coordinate system. Based on these assumption, the control objective is to design a controller such that there is a fixed time T∗ which is independent of any initial condition such that for any i,j∈Γ,(5)limt⟶T∗ηit−ηjtιit−ιjtκit−κjt=Δij,t<T∗,ηit−ηjtιit−ιjtκit−κjt≡Δij,t≥T∗,limt⟶T∗1n∑i=1nηit,ιit,κitT=ηd,ιd,κdT,t<T∗,1n∑i=1nηit,ιit,κitT≡ηd,ιd,κdT,t≥T∗.To solve the proposed problem, we made some assumptions.Assumption 1.
We assume the desired formation trajectoryϒd=ηd,ιd,κdT, ϒ˙d, and ϒ¨d are bounded.Assumption 2.
For the external disturbance, we assume thatdit and d˙it are bounded. In other words, it can be found that there exist the known positive constants L1 and L2 such that dit≤L1 and d˙it≤L2.
In order to realize the formation task, this paper intends to adopt multiagent theory. So next, we give some relevant general knowledge about graph theory.


### 2.3. Graph Theory

In this paper, the multiaircraft system with master-slave structure will be considered. Each aircraft can be seen as a node. The information interaction amongn nodes, i.e., n follower agents, can be represented by the undirected graph GA=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 GA with nonnegative adjacency elements aij. If there is an edge from node j to node i, i.e., vj,vi∈E, then aij=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 degoutvi=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 the leader, then bi>0, otherwise, bi=0. Let B=diagb1,⋯,bn.Assumption 3.
The graph for all follower agents is connected in the above communication topology. Meanwhile, make sure that at least one follower is directly connected to the leader to get the leader’s information, i.e.,B≠0.


### 2.4. Related Definitions and Lemmas

Next, some corresponding definitions and lemmas about fixed-time control theory are presented. First of all, we provide the relevant concepts of fixed-time stability.Definition 1 [18, 23].
Consider the nonlinear system.(6)x˙=fx,f0=0,x∈Rn,where f⋅:Rn⟶Rn is a continuous vector function. The origin is finite-time stable equilibrium if it is Lyapunov stable and finite-time convergence. The finite-time convergence means that there is a function Tx0 such that limt⟶Tx0xt,x0=0 and xt,x0≡0,∀t≥Tx0. The system is said to be fixed-time stable if it is finite-time stable and fixed-time convergence if the convergent time satisfies sup∀x0Tx0<+∞.
Then, an important definition under the fixed-time theoretical framework is presented, i.e., the homogeneous definition.Definition 2 [18].
For system (6), if for any ε>0, there exists r1,⋯,rn with ri>0,i=1,⋯,n such that(7)fiεr1x1,⋯,εrnxn=εk+rifix,i=1,⋯,n,where k≥minri,i=1,⋯,n, and then fx is said to be homogeneous of degree k with respect to the dilation r1,⋯,rn.Definition 3.
Denotesigαx=signxxα, where α≥0, x∈ℛ, and sign⋅ is the standard sign function.
Finally, an important lemma for achieving fixed-time stability is presented.Lemma 1 [31].
For system (6), suppose that fx is a homogeneous vector field in the bilimit with associated triples r0,k0,f0 and r∞,k∞,f∞. If the origins of systems x˙=fx, x˙0=f0x, and x˙∞=f∞x are globally asymptotically stable, then the following statements hold:(1)
The origin of (6) is fixed-time stable when condition k∞>0>k0 holds.(2)
LetdV0 and dV∞ be the real numbers such that dV0>max1≤i≤nr0,i and dV∞>max1≤i≤nr∞,i. There exists a continuous, positive definite, and proper function Vx such that the function ∂V/∂xi is homogeneous in the bilimit with triples r0,dV0−r0,i,∂V0/∂xi and r∞,dV∞−r∞,i,∂V∞/∂xi, and the function ∂V/∂xifx is negative definite.


## 2.1. Problem Description

The formation withn four-rotor aircraft is considered, and let Γ=1,2,⋯,n. From the viewpoint of modelling, the aircraft can usually use six variables to uniquely determine itself the position and attitude in three-dimensional space. Without loss of generality, in the inertial coordinate system, let ϒi=ηi,ιi,κiT∈R3 be the aircraft’s position and Θi=pi,qi,riT∈R3 be the aircraft’s attitude based on Euler angles. As a result, the coordinates of the aircraft are(1)ηi,ιi,κi,pi,qi,riT∈R6,i∈Γ.


### 2.1.1. Position Dynamical Model

As that in [29, 30], the mathematical model of the position of each quadrotor can be described as(2)η¨i=Timicospisinqicosri+sinpisinri,ι¨i=Timicospisinqisinri−sinpicosri,κ¨i=Timicospicosqi−g,i∈Γ,where mi denotes the i-th aircraft’s mass, Ti is the i-th aircraft’s thrust, and g is the gravitational acceleration which always exists.


### 2.1.2. Attitude Dynamical Model

By Euler angles, the attitude dynamical equation is given in [30]:(3)Ji,1p¨i=liτi,1+di,1t,Ji,2q¨i=liτi,2+di,2t,Ji,3r¨i=ciτi,3+di,3t,i∈Γ,where τi=τi,1,τi,2,τi,3T denotes the i-th aircraft’s control torque, ci is the i-th aircraft’s force-to-moment factor, Ji=diagJi,1,Ji,2,Ji,3 is the inertia matrix, and li is the arm length of the i-th aircraft. In addition, the external disturbances are denoted by dit=di,1t,di,2t,di,3tT.


## 2.1.1. Position Dynamical Model

As that in [29, 30], the mathematical model of the position of each quadrotor can be described as(2)η¨i=Timicospisinqicosri+sinpisinri,ι¨i=Timicospisinqisinri−sinpicosri,κ¨i=Timicospicosqi−g,i∈Γ,where mi denotes the i-th aircraft’s mass, Ti is the i-th aircraft’s thrust, and g is the gravitational acceleration which always exists.


## 2.1.2. Attitude Dynamical Model

By Euler angles, the attitude dynamical equation is given in [30]:(3)Ji,1p¨i=liτi,1+di,1t,Ji,2q¨i=liτi,2+di,2t,Ji,3r¨i=ciτi,3+di,3t,i∈Γ,where τi=τi,1,τi,2,τi,3T denotes the i-th aircraft’s control torque, ci is the i-th aircraft’s force-to-moment factor, Ji=diagJi,1,Ji,2,Ji,3 is the inertia matrix, and li is the arm length of the i-th aircraft. In addition, the external disturbances are denoted by dit=di,1t,di,2t,di,3tT.


## 2.2. Control Objective

Letϒd=ηd,ιd,κdT be the referenced formation trajectory. As we all know, in fact, the three-dimensional relative position between two points can be expressed in the form of a three-dimensional vector. The formation geometry in 3D space is given by vector Δij=Δηij,Διij,ΔκijT, i,j∈Γ. Hence, the aircraft i and aircraft j are hoped to have the relative positional deviation:(4)Δij=Δi−Δj=Δηi,Διi,ΔκiT−Δηj,Διj,ΔκjT,where Δi=Δηi,Διi,ΔκiT and Δj=Δηj,Διj,ΔκjT represent the coordinates of two points in the same three-dimensional rectangular coordinate system. Based on these assumption, the control objective is to design a controller such that there is a fixed time T∗ which is independent of any initial condition such that for any i,j∈Γ,(5)limt⟶T∗ηit−ηjtιit−ιjtκit−κjt=Δij,t<T∗,ηit−ηjtιit−ιjtκit−κjt≡Δij,t≥T∗,limt⟶T∗1n∑i=1nηit,ιit,κitT=ηd,ιd,κdT,t<T∗,1n∑i=1nηit,ιit,κitT≡ηd,ιd,κdT,t≥T∗.To solve the proposed problem, we made some assumptions.Assumption 1.
We assume the desired formation trajectoryϒd=ηd,ιd,κdT, ϒ˙d, and ϒ¨d are bounded.Assumption 2.
For the external disturbance, we assume thatdit and d˙it are bounded. In other words, it can be found that there exist the known positive constants L1 and L2 such that dit≤L1 and d˙it≤L2.
In order to realize the formation task, this paper intends to adopt multiagent theory. So next, we give some relevant general knowledge about graph theory.


## 2.3. Graph Theory

In this paper, the multiaircraft system with master-slave structure will be considered. Each aircraft can be seen as a node. The information interaction amongn nodes, i.e., n follower agents, can be represented by the undirected graph GA=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 GA with nonnegative adjacency elements aij. If there is an edge from node j to node i, i.e., vj,vi∈E, then aij=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 degoutvi=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 the leader, then bi>0, otherwise, bi=0. Let B=diagb1,⋯,bn.Assumption 3.
The graph for all follower agents is connected in the above communication topology. Meanwhile, make sure that at least one follower is directly connected to the leader to get the leader’s information, i.e.,B≠0.


## 2.4. Related Definitions and Lemmas

Next, some corresponding definitions and lemmas about fixed-time control theory are presented. First of all, we provide the relevant concepts of fixed-time stability.Definition 1 [18, 23].
Consider the nonlinear system.(6)x˙=fx,f0=0,x∈Rn,where f⋅:Rn⟶Rn is a continuous vector function. The origin is finite-time stable equilibrium if it is Lyapunov stable and finite-time convergence. The finite-time convergence means that there is a function Tx0 such that limt⟶Tx0xt,x0=0 and xt,x0≡0,∀t≥Tx0. The system is said to be fixed-time stable if it is finite-time stable and fixed-time convergence if the convergent time satisfies sup∀x0Tx0<+∞.
Then, an important definition under the fixed-time theoretical framework is presented, i.e., the homogeneous definition.Definition 2 [18].
For system (6), if for any ε>0, there exists r1,⋯,rn with ri>0,i=1,⋯,n such that(7)fiεr1x1,⋯,εrnxn=εk+rifix,i=1,⋯,n,where k≥minri,i=1,⋯,n, and then fx is said to be homogeneous of degree k with respect to the dilation r1,⋯,rn.Definition 3.
Denotesigαx=signxxα, where α≥0, x∈ℛ, and sign⋅ is the standard sign function.
Finally, an important lemma for achieving fixed-time stability is presented.Lemma 1 [31].
For system (6), suppose that fx is a homogeneous vector field in the bilimit with associated triples r0,k0,f0 and r∞,k∞,f∞. If the origins of systems x˙=fx, x˙0=f0x, and x˙∞=f∞x are globally asymptotically stable, then the following statements hold:(1)
The origin of (6) is fixed-time stable when condition k∞>0>k0 holds.(2)
LetdV0 and dV∞ be the real numbers such that dV0>max1≤i≤nr0,i and dV∞>max1≤i≤nr∞,i. There exists a continuous, positive definite, and proper function Vx such that the function ∂V/∂xi is homogeneous in the bilimit with triples r0,dV0−r0,i,∂V0/∂xi and r∞,dV∞−r∞,i,∂V∞/∂xi, and the function ∂V/∂xifx is negative definite.


## 3. Main Results

In order to implement a fixed-time formation control algorithm, a two-step design procedure is applied, i.e., position and attitude control strategy design.


### 3.1. Position Control Scheme Design

For the brevity, let(8)μi,1=Timicospisinqicosri+sinpisinri,μi,2=Timicospisinqisinri−sinpicosri,μi,3=Timicospicosqi−g,i∈Γ.As a result, rewrite the position-loop dynamical equation (3) as(9)η¨i=μi,1,ι¨i=μi,2,κ¨i=μi,3.Theorem 1.
For thei-th aircraft’s position system (9), if we design the following controller:(10)μi=−∑j∈Niaijkpsigϱ1ϒi−ϒj−Δij+kdsigϱ2ϒ˙i−ϒ˙j+kpsigϱ1′ϒi−ϒj−Δij+kdsigϱ2′ϒ˙i−ϒ˙j−bikpsigϱ1ϒi−ϒd−Δi+1n∑i=1nΔi+kdsigϱ2ϒ˙i−ϒ˙d+kpsigϱ1′ϒi−ϒd−Δi+1n∑i=1nΔi+kdsigϱ2′ϒ˙i−ϒ˙d+ϒ¨d,where μi=μi,1,μi,2,μi,3T, 0<ϱ1<1,ϱ2=2ϱ1/1+ϱ1, ϱ1′>1,ϱ2′=2ϱ1′/1+ϱ1′, kp>0, and kd>0, and then the desired formation graphics can be completed in a fixed time.Proof.
Considering that the three-axis position model of aircraft is symmetrical, the form of their controllers is similar. Here, only the proof forη-axis is provided. First, we can perform a simple coordinate transformation to define the following error:(11)ei,η=ηi−ηd−Δηi+1n∑i=1nΔηi,i∈Γ.
Define the error vector asxi,η=ei,η,e˙i,ηT. Then, combining the position system (9) and the controller (1), the position error equation is(12)e˙i,η=η˙i−η˙d,e¨i,η=−∑j∈Niaijkpsigϱ1ei,η−ej,η+kdsigϱ2e˙i,η−e˙j,η+kpsigϱ1′ei,η−ej,η+kdsigϱ2′e˙i,η−e˙j,η−bikpsigϱ1ei,η+kdsigϱ2e˙i,η+kpsigϱ1′ei,η+kdsigϱ2′e˙i,η,i∈Γ.
The next proof process is mainly based on Lemma 1. First of all, we will show that the above error system (12) is globally asymptotically stable. Construct the following Lyapunov function as(13)V=V1+V2,where(14)V1=kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1ρdρ+2bi∫​0ei,ηsigϱ1ρdρ+kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1′ρdρ+2bi∫​0ei,ηsigϱ1′ρdρ,V2=∑i=1ne˙i,η2.
By calculation, the derivative ofV1 is(15)V˙1=kp∑i=1n∑j=1naijsigϱ1ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1ei,ηe˙i,η+kp∑i=1n∑j=1naijsigϱ1′ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1′ei,ηe˙i,η.
Furthermore, we know the fact thataij=aji,i,j∈Γ. Meanwhile, it is not difficult to find that the function sigα⋅ is the odd function. Therefore, the derivative of V2 can be written as(16)V˙2=−kp∑i=1n∑j=1naijsigϱ1ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1ei,ηe˙i,η−kp∑i=1n∑j=1naijsigϱ1′ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1′ei,ηe˙i,η−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2+2bie˙i,η1+ϱ2−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2′+2bie˙i,η1+ϱ2′.
Combined with the above derivatives ofV1 and V2, it can be seen that the derivative of V is(17)V˙=−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2+∑j=1naije˙i,η−e˙j,η1+ϱ2′+2bie˙i,η1+ϱ2+2bie˙i,η1+ϱ2′≤0.
Define setΞ=ei,η,e˙i,ηV˙≡0. It can be found that V˙≡0 implies e˙i,η≡0 and e¨i,η≡0, which also implies that ei,η≡0 from (12), ∀i∈Γ. As a result of LaSalle’s invariance principle [32], the system (12) is the globally asymptotically stable system.
Next, we explain that the closed-loop system (12) is homogeneous in the bilimit. Let fsxi,η=e˙i,η,μi,1−η¨dT. Then, the vectors(18)fs0xi,η=e˙i,η,−∑j∈Niaijkpsigϱ1ei,η−ej,η+kdsigϱ2e˙i,η−e˙j,η−bikpsigϱ1ei,η+kdsigϱ2e˙i,ηT,fs∞xi,η=e˙i,η,−∑j∈Niaijkpsigϱ1′ei,η−ej,η+kdsigϱ2′e˙i,η−e˙j,η−bikpsigϱ1′ei,η+kdsigϱ2′e˙i,ηTcan be considered as approximating functions for fs in 0 limit and ∞ limit, respectively. According to Definition 2, i.e., the homogeneous system definition, it is obvious that the vector field fs0 is homogeneous with the degree of ks0=ϱ1−1/2<0 with respect to the dilation rs0=s1,⋯,s1︸fore1,η,⋯,en,η,s2,⋯,s2︸fore˙1,η,⋯,e˙n,η, where s1=1 and s2=ϱ1+1/2. Similarly, the vector field fs∞ is homogeneous, and its homogeneous degree is ks∞=ϱ1′−1/2>0 about the dilation rs∞=s1′,⋯,s1′︸fore1,η,⋯,en,η,s2′,⋯,s2′︸fore˙1,η,⋯,e˙n,η, where s1′=1 and s2′=ϱ1′+1/2. Therefore, the closed-loop system (12) is the bilimit homogeneous system with rs0,ks0,fs0 and rs∞,ks∞,fs∞.
Finally, we will show that the systemsx˙i,η=fs0xi,η and x˙i,η=fs∞xi,η are globally asymptotically stable. For the system x˙i,η=fs0xi,η, the Lyapunov function can be chosen as(19)V3=kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1ρdρ+2bi∫​0ei,ηsigϱ1ρdρ+∑i=1ne˙i,η2.
Similar to the calculation for the above Lyapunov functionV, the derivative of V3 is(20)V˙3=−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2+2bie˙i,η1+ϱ2≤0.
Similarly, for the systemx˙i,η=fs∞xi,η, we choose(21)V4=kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1′ρdρ+2bi∫​0ei,ηsigϱ1′ρdρ+∑i=1ne˙i,η2,whose derivative is(22)V˙4=−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2′+2bie˙i,η1+ϱ2′≤0.
The LaSalle’s invariance principle ensures the global asymptotic stability of systemsx˙i,η=fs0xi,η and x˙i,η=fs∞xi,η.
In the end, based on Lemma 1, we can draw such a conclusion that the close-loop system (12) is globally fixed-time stable. Taking into account the previous definition of the error, it is obvious that multiple aircraft can accomplish position formation graphics in a fixed time. This completes the proof.


### 3.2. Attitude Control Scheme Design

In the above design process of the position formation control scheme, using the idea of backstepping, the attitude information (i.e., Euler angleΘi=pi,qi,riT) of the aircraft is treated as a virtual control input. Next, the fixed-time attitude controller for each aircraft will be designed. First, in regard to each quadrotor aircraft, the desired attitude can be denoted as Θi,d=pi,d,qi,d,ri,dT. According to (8), there is the following correspondence:(23)Ti,d=miμi,η2+μi,ι2+μi,κ+g2,pi,d=arcsinmiTi,dμi,ηsinri,d−μi,ιcosri,d,qi,d=arctan1μi,κ+gμi,ηcosri,d+μi,ιsinri,d,i∈Γ.At the same time, it is worth noting that the desired yaw angleri,d is an independent free variable. It can be set separately.To remove the effect of external disturbances, there are many disturbance observer methods [33, 34]. Considering the influence of external disturbances on the aircraft, the corresponding fixed-time disturbance observers will be designed in the next part to achieve accurate observation of external disturbances within a fixed time. Define di′=di,1′,di,2′,di,3′T=di,1/Ji,1,di,2/Ji,2,di,3/Ji,3T as the new disturbance states. Meanwhile, define d^i′=d^i,1′,d^i,2′,d^i,3′T=d^i,1/Ji,1,d^i,2/Ji,2,d^i,3/Ji,3T as the observers’ states.Lemma 2.
Considering the attitude system (4) and Assumption 2, the states d^i′ of the observers can be designed as(24)∂p˙^i∂t=liJi,1τi,1+d^i,1′+ρ1sig1/2p˙i−p˙^i+ρ2sigαp˙i−p˙^i,d^˙i,1′=ρ3signp˙i−p˙^i,∂q˙^i∂t=liJi,2τi,2+d^i,2′+ρ1sig1/2q˙i−q˙^i+ρ2sigαq˙i−q˙^i,d^˙i,2′=ρ3signq˙i−q˙^i,∂r˙^i∂t=ciJi,3τi,3+d^i,3′+ρ1sig1/2r˙i−r˙^i+ρ2sigαr˙i−r˙^i,d^˙i,3′=ρ3signr˙i−r˙^i,where ρ1, ρ2, and ρ3 are the gain parameters of the observers. Then, the estimated states will converge to the real states di=di,1,di,2,di,3 in a fixed time:(25)Tf≤1ρ2α−1εα−1+2ε1/2ρ11+1m1/M−hρ1/ρ1,where ε>0, m=ρ3−L2, M=ρ3+L2, hρ1=1/ρ1+2o/mρ11/3, and o is the base of natural logarithms, provided that the following conditions hold for control gains: ρ3>L2 h−1ρ1ρ1>M. The minimum value of Tfε is reached for ε=ρ1/ρ21/α+1/2.Proof.
First of all, let the corresponding disturbance observer error beσ1=p˙i−p˙^i and σ2=di,1′−d^i,1′. Then, the error dynamical equation can be obtained:(26)σ˙1=σ2−ρ1sig1/2σ1−ρ2sigασ1,σ˙2=d˙i,1′−ρ3signσ1.
On the basis of the result in [35], the system (26) is fixed-time stable, which means that the estimated state d^i,1′ from the above observers will converge to the real state di,1′ in a fixed time. The proof is completed.Since the disturbance observer mentioned above can accurately observe the disturbance in the aircraft’s attitude system in a fixed time, a composite attitude controller based on disturbance compensation is presented to track the desired attitude.Theorem 2.
About each aircraft’s attitude system (4), disturbance compensation (24) is added, and the attitude control algorithm is designed as(27)τi,1=Ji,1lip¨i,d+apsigβ1pi,d−pi+adsigβ2p˙i,d−p˙i+apsigβ1′pi,d−pi+adsigβ2′p˙i,d−p˙i−d^i,1′,τi,2=Ji,2liq¨i,d+apsigβ1qi,d−qi+adsigβ2q˙i,d−q˙i+apsigβ1′qi,d−qi+adsigβ2′q˙i,d−q˙i−d^i,2′,τi,3=Ji,3cir¨i,d+apsigβ1ri,d−ri+adsigβ2r˙i,d−r˙i+apsigβ1′ri,d−ri+adsigβ2′r˙i,d−r˙i−d^i,3′,where 0<β1<1,β2=2β1/1+β1, β1′>1,β2′=2β1′/1+β1′, ap>0,ad>0. So, in a fixed time, we can get the states pi,qi,riT⟶pi,d,qi,d,ri,dT.Proof.
Firstly, considering the results of the previous observer (2), we can see that when t≥Tf, there exists(28)d^i,1′=di,1′,d^i,2′=di,1′,d^i,3′=di,3′.
Letei,p=pi,d−pi be the system error about roll angle pi. When t≥Tf, substituting the controller (27) into the attitude system (4), the attitude error system can be written as(29)e˙i,p=p˙i,d−p˙i,e¨i,p=−apsigβ1ei,p−adsigβ2e˙i,p−apsigβ1′ei,p−adsigβ2′e˙i,p.
Obviously, the closed-loop system (31) and the system (12) have the similar structures. Hence, based on Lemma 1, the proof process is extremely similar, which is omitted here.Remark 1.
Note that the fixed-time control method is employed to design the position-loop controller, which implies that the desired Euler angles and rates may be nondifferentiable and nonsmooth. To this end, in order to obtain the differential signal for the attitude-loop controller design, here we can use the classical first-order differentiator (low-pass filter), i.e.:(30)Ys=sTcs+1Vs=1Tc1−1Tcs+1Vs,where Vt is the input signal, Yt is the output signal, and Tc is a proper small differential constant (in simulation, the time constant Tc is chosen as 0.01 sec).Remark 2.
Note that in Theorem 1 and Theorem 2, the fixed-time stability analysis is provided for only the two subsystems (i.e., position-loop subsystem and attitude-loop subsystem) rather than the total cascaded system. Since the two subsystems are both fixed-time convergent, the stability analysis for the total cascaded system can be simplified. We just prove the state boundedness before the fixed time.(1)
For the attitude subsystem, before the timeTf, the attitude error system (29) is(31)e˙i,p=p˙i,d−p˙i,e¨i,p=−apsigβ1ei,p−adsigβ2e˙i,p−apsigβ1′ei,p−adsigβ2′e˙i,p−d^i,1′−di,1′.
To prove the system state is bounded in the time interval0,Tf, choose the Lyapunov function(32)W=ap1+β1ei,p1+β1+ap1+β1′ei,p1+β1′+12e˙i,p2,which leads to(33)W˙=apsigβ1ei,pe˙i,p+apsigβ1′ei,pe˙i,p+e˙i,p−apsigβ1ei,p−adsigβ2e˙i,p−apsigβ1′ei,p−adsigβ2′e˙i,p−d^i,1′−di,1′≤−e˙i,pd^i,1′−di,1′≤12e˙i,p2+12d^i,1′−di,1′2.
Since the termd^i,1′−di,1′ is always bounded, there is a positive constant C such that(34)W˙≤C+W,which implies(35)Wt≤W0et+Ct≤W0eTf+CTf,∀t∈0,Tf.
Therefore, the attitude system’s states are bounded for any timet∈0,Tf.(2)
For the position system (2), it can be rewritten in the form of the cascaded system:(36)η¨i=Timicospi,d+pi−pi,dsinqi,d+qi−qi,dcosri,d+ri−ri,d+sinpi,d+pi−pi,dri,d+ri−ri,d=Timicospi,d−ei,psinqi,d−ei,qcosri,d−ei,r+sinpi,d−ei,psinri,d−ei,r.
Under the proposed fixed-time attitude controller, the attitude tracking errorei,p,ei,q,ei,r will converge to zero in a fixed time T∗. After the time instant T∗, the position tracking error system (36) will be fixed-time convergence. By using a similar proof as that in the previous attitude control system, the position states are bounded during the time interval 0,T∗.


## 3.1. Position Control Scheme Design

For the brevity, let(8)μi,1=Timicospisinqicosri+sinpisinri,μi,2=Timicospisinqisinri−sinpicosri,μi,3=Timicospicosqi−g,i∈Γ.As a result, rewrite the position-loop dynamical equation (3) as(9)η¨i=μi,1,ι¨i=μi,2,κ¨i=μi,3.Theorem 1.
For thei-th aircraft’s position system (9), if we design the following controller:(10)μi=−∑j∈Niaijkpsigϱ1ϒi−ϒj−Δij+kdsigϱ2ϒ˙i−ϒ˙j+kpsigϱ1′ϒi−ϒj−Δij+kdsigϱ2′ϒ˙i−ϒ˙j−bikpsigϱ1ϒi−ϒd−Δi+1n∑i=1nΔi+kdsigϱ2ϒ˙i−ϒ˙d+kpsigϱ1′ϒi−ϒd−Δi+1n∑i=1nΔi+kdsigϱ2′ϒ˙i−ϒ˙d+ϒ¨d,where μi=μi,1,μi,2,μi,3T, 0<ϱ1<1,ϱ2=2ϱ1/1+ϱ1, ϱ1′>1,ϱ2′=2ϱ1′/1+ϱ1′, kp>0, and kd>0, and then the desired formation graphics can be completed in a fixed time.Proof.
Considering that the three-axis position model of aircraft is symmetrical, the form of their controllers is similar. Here, only the proof forη-axis is provided. First, we can perform a simple coordinate transformation to define the following error:(11)ei,η=ηi−ηd−Δηi+1n∑i=1nΔηi,i∈Γ.
Define the error vector asxi,η=ei,η,e˙i,ηT. Then, combining the position system (9) and the controller (1), the position error equation is(12)e˙i,η=η˙i−η˙d,e¨i,η=−∑j∈Niaijkpsigϱ1ei,η−ej,η+kdsigϱ2e˙i,η−e˙j,η+kpsigϱ1′ei,η−ej,η+kdsigϱ2′e˙i,η−e˙j,η−bikpsigϱ1ei,η+kdsigϱ2e˙i,η+kpsigϱ1′ei,η+kdsigϱ2′e˙i,η,i∈Γ.
The next proof process is mainly based on Lemma 1. First of all, we will show that the above error system (12) is globally asymptotically stable. Construct the following Lyapunov function as(13)V=V1+V2,where(14)V1=kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1ρdρ+2bi∫​0ei,ηsigϱ1ρdρ+kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1′ρdρ+2bi∫​0ei,ηsigϱ1′ρdρ,V2=∑i=1ne˙i,η2.
By calculation, the derivative ofV1 is(15)V˙1=kp∑i=1n∑j=1naijsigϱ1ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1ei,ηe˙i,η+kp∑i=1n∑j=1naijsigϱ1′ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1′ei,ηe˙i,η.
Furthermore, we know the fact thataij=aji,i,j∈Γ. Meanwhile, it is not difficult to find that the function sigα⋅ is the odd function. Therefore, the derivative of V2 can be written as(16)V˙2=−kp∑i=1n∑j=1naijsigϱ1ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1ei,ηe˙i,η−kp∑i=1n∑j=1naijsigϱ1′ei,η−ej,ηe˙i,η−e˙j,η+2bisigϱ1′ei,ηe˙i,η−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2+2bie˙i,η1+ϱ2−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2′+2bie˙i,η1+ϱ2′.
Combined with the above derivatives ofV1 and V2, it can be seen that the derivative of V is(17)V˙=−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2+∑j=1naije˙i,η−e˙j,η1+ϱ2′+2bie˙i,η1+ϱ2+2bie˙i,η1+ϱ2′≤0.
Define setΞ=ei,η,e˙i,ηV˙≡0. It can be found that V˙≡0 implies e˙i,η≡0 and e¨i,η≡0, which also implies that ei,η≡0 from (12), ∀i∈Γ. As a result of LaSalle’s invariance principle [32], the system (12) is the globally asymptotically stable system.
Next, we explain that the closed-loop system (12) is homogeneous in the bilimit. Let fsxi,η=e˙i,η,μi,1−η¨dT. Then, the vectors(18)fs0xi,η=e˙i,η,−∑j∈Niaijkpsigϱ1ei,η−ej,η+kdsigϱ2e˙i,η−e˙j,η−bikpsigϱ1ei,η+kdsigϱ2e˙i,ηT,fs∞xi,η=e˙i,η,−∑j∈Niaijkpsigϱ1′ei,η−ej,η+kdsigϱ2′e˙i,η−e˙j,η−bikpsigϱ1′ei,η+kdsigϱ2′e˙i,ηTcan be considered as approximating functions for fs in 0 limit and ∞ limit, respectively. According to Definition 2, i.e., the homogeneous system definition, it is obvious that the vector field fs0 is homogeneous with the degree of ks0=ϱ1−1/2<0 with respect to the dilation rs0=s1,⋯,s1︸fore1,η,⋯,en,η,s2,⋯,s2︸fore˙1,η,⋯,e˙n,η, where s1=1 and s2=ϱ1+1/2. Similarly, the vector field fs∞ is homogeneous, and its homogeneous degree is ks∞=ϱ1′−1/2>0 about the dilation rs∞=s1′,⋯,s1′︸fore1,η,⋯,en,η,s2′,⋯,s2′︸fore˙1,η,⋯,e˙n,η, where s1′=1 and s2′=ϱ1′+1/2. Therefore, the closed-loop system (12) is the bilimit homogeneous system with rs0,ks0,fs0 and rs∞,ks∞,fs∞.
Finally, we will show that the systemsx˙i,η=fs0xi,η and x˙i,η=fs∞xi,η are globally asymptotically stable. For the system x˙i,η=fs0xi,η, the Lyapunov function can be chosen as(19)V3=kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1ρdρ+2bi∫​0ei,ηsigϱ1ρdρ+∑i=1ne˙i,η2.
Similar to the calculation for the above Lyapunov functionV, the derivative of V3 is(20)V˙3=−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2+2bie˙i,η1+ϱ2≤0.
Similarly, for the systemx˙i,η=fs∞xi,η, we choose(21)V4=kp∑i=1n∑j=1naij∫​0ei,η−ej,ηsigϱ1′ρdρ+2bi∫​0ei,ηsigϱ1′ρdρ+∑i=1ne˙i,η2,whose derivative is(22)V˙4=−kd∑i=1n∑j=1naije˙i,η−e˙j,η1+ϱ2′+2bie˙i,η1+ϱ2′≤0.
The LaSalle’s invariance principle ensures the global asymptotic stability of systemsx˙i,η=fs0xi,η and x˙i,η=fs∞xi,η.
In the end, based on Lemma 1, we can draw such a conclusion that the close-loop system (12) is globally fixed-time stable. Taking into account the previous definition of the error, it is obvious that multiple aircraft can accomplish position formation graphics in a fixed time. This completes the proof.


## 3.2. Attitude Control Scheme Design

In the above design process of the position formation control scheme, using the idea of backstepping, the attitude information (i.e., Euler angleΘi=pi,qi,riT) of the aircraft is treated as a virtual control input. Next, the fixed-time attitude controller for each aircraft will be designed. First, in regard to each quadrotor aircraft, the desired attitude can be denoted as Θi,d=pi,d,qi,d,ri,dT. According to (8), there is the following correspondence:(23)Ti,d=miμi,η2+μi,ι2+μi,κ+g2,pi,d=arcsinmiTi,dμi,ηsinri,d−μi,ιcosri,d,qi,d=arctan1μi,κ+gμi,ηcosri,d+μi,ιsinri,d,i∈Γ.At the same time, it is worth noting that the desired yaw angleri,d is an independent free variable. It can be set separately.To remove the effect of external disturbances, there are many disturbance observer methods [33, 34]. Considering the influence of external disturbances on the aircraft, the corresponding fixed-time disturbance observers will be designed in the next part to achieve accurate observation of external disturbances within a fixed time. Define di′=di,1′,di,2′,di,3′T=di,1/Ji,1,di,2/Ji,2,di,3/Ji,3T as the new disturbance states. Meanwhile, define d^i′=d^i,1′,d^i,2′,d^i,3′T=d^i,1/Ji,1,d^i,2/Ji,2,d^i,3/Ji,3T as the observers’ states.Lemma 2.
Considering the attitude system (4) and Assumption 2, the states d^i′ of the observers can be designed as(24)∂p˙^i∂t=liJi,1τi,1+d^i,1′+ρ1sig1/2p˙i−p˙^i+ρ2sigαp˙i−p˙^i,d^˙i,1′=ρ3signp˙i−p˙^i,∂q˙^i∂t=liJi,2τi,2+d^i,2′+ρ1sig1/2q˙i−q˙^i+ρ2sigαq˙i−q˙^i,d^˙i,2′=ρ3signq˙i−q˙^i,∂r˙^i∂t=ciJi,3τi,3+d^i,3′+ρ1sig1/2r˙i−r˙^i+ρ2sigαr˙i−r˙^i,d^˙i,3′=ρ3signr˙i−r˙^i,where ρ1, ρ2, and ρ3 are the gain parameters of the observers. Then, the estimated states will converge to the real states di=di,1,di,2,di,3 in a fixed time:(25)Tf≤1ρ2α−1εα−1+2ε1/2ρ11+1m1/M−hρ1/ρ1,where ε>0, m=ρ3−L2, M=ρ3+L2, hρ1=1/ρ1+2o/mρ11/3, and o is the base of natural logarithms, provided that the following conditions hold for control gains: ρ3>L2 h−1ρ1ρ1>M. The minimum value of Tfε is reached for ε=ρ1/ρ21/α+1/2.Proof.
First of all, let the corresponding disturbance observer error beσ1=p˙i−p˙^i and σ2=di,1′−d^i,1′. Then, the error dynamical equation can be obtained:(26)σ˙1=σ2−ρ1sig1/2σ1−ρ2sigασ1,σ˙2=d˙i,1′−ρ3signσ1.
On the basis of the result in [35], the system (26) is fixed-time stable, which means that the estimated state d^i,1′ from the above observers will converge to the real state di,1′ in a fixed time. The proof is completed.Since the disturbance observer mentioned above can accurately observe the disturbance in the aircraft’s attitude system in a fixed time, a composite attitude controller based on disturbance compensation is presented to track the desired attitude.Theorem 2.
About each aircraft’s attitude system (4), disturbance compensation (24) is added, and the attitude control algorithm is designed as(27)τi,1=Ji,1lip¨i,d+apsigβ1pi,d−pi+adsigβ2p˙i,d−p˙i+apsigβ1′pi,d−pi+adsigβ2′p˙i,d−p˙i−d^i,1′,τi,2=Ji,2liq¨i,d+apsigβ1qi,d−qi+adsigβ2q˙i,d−q˙i+apsigβ1′qi,d−qi+adsigβ2′q˙i,d−q˙i−d^i,2′,τi,3=Ji,3cir¨i,d+apsigβ1ri,d−ri+adsigβ2r˙i,d−r˙i+apsigβ1′ri,d−ri+adsigβ2′r˙i,d−r˙i−d^i,3′,where 0<β1<1,β2=2β1/1+β1, β1′>1,β2′=2β1′/1+β1′, ap>0,ad>0. So, in a fixed time, we can get the states pi,qi,riT⟶pi,d,qi,d,ri,dT.Proof.
Firstly, considering the results of the previous observer (2), we can see that when t≥Tf, there exists(28)d^i,1′=di,1′,d^i,2′=di,1′,d^i,3′=di,3′.
Letei,p=pi,d−pi be the system error about roll angle pi. When t≥Tf, substituting the controller (27) into the attitude system (4), the attitude error system can be written as(29)e˙i,p=p˙i,d−p˙i,e¨i,p=−apsigβ1ei,p−adsigβ2e˙i,p−apsigβ1′ei,p−adsigβ2′e˙i,p.
Obviously, the closed-loop system (31) and the system (12) have the similar structures. Hence, based on Lemma 1, the proof process is extremely similar, which is omitted here.Remark 1.
Note that the fixed-time control method is employed to design the position-loop controller, which implies that the desired Euler angles and rates may be nondifferentiable and nonsmooth. To this end, in order to obtain the differential signal for the attitude-loop controller design, here we can use the classical first-order differentiator (low-pass filter), i.e.:(30)Ys=sTcs+1Vs=1Tc1−1Tcs+1Vs,where Vt is the input signal, Yt is the output signal, and Tc is a proper small differential constant (in simulation, the time constant Tc is chosen as 0.01 sec).Remark 2.
Note that in Theorem 1 and Theorem 2, the fixed-time stability analysis is provided for only the two subsystems (i.e., position-loop subsystem and attitude-loop subsystem) rather than the total cascaded system. Since the two subsystems are both fixed-time convergent, the stability analysis for the total cascaded system can be simplified. We just prove the state boundedness before the fixed time.(1)
For the attitude subsystem, before the timeTf, the attitude error system (29) is(31)e˙i,p=p˙i,d−p˙i,e¨i,p=−apsigβ1ei,p−adsigβ2e˙i,p−apsigβ1′ei,p−adsigβ2′e˙i,p−d^i,1′−di,1′.
To prove the system state is bounded in the time interval0,Tf, choose the Lyapunov function(32)W=ap1+β1ei,p1+β1+ap1+β1′ei,p1+β1′+12e˙i,p2,which leads to(33)W˙=apsigβ1ei,pe˙i,p+apsigβ1′ei,pe˙i,p+e˙i,p−apsigβ1ei,p−adsigβ2e˙i,p−apsigβ1′ei,p−adsigβ2′e˙i,p−d^i,1′−di,1′≤−e˙i,pd^i,1′−di,1′≤12e˙i,p2+12d^i,1′−di,1′2.
Since the termd^i,1′−di,1′ is always bounded, there is a positive constant C such that(34)W˙≤C+W,which implies(35)Wt≤W0et+Ct≤W0eTf+CTf,∀t∈0,Tf.
Therefore, the attitude system’s states are bounded for any timet∈0,Tf.(2)
For the position system (2), it can be rewritten in the form of the cascaded system:(36)η¨i=Timicospi,d+pi−pi,dsinqi,d+qi−qi,dcosri,d+ri−ri,d+sinpi,d+pi−pi,dri,d+ri−ri,d=Timicospi,d−ei,psinqi,d−ei,qcosri,d−ei,r+sinpi,d−ei,psinri,d−ei,r.
Under the proposed fixed-time attitude controller, the attitude tracking errorei,p,ei,q,ei,r will converge to zero in a fixed time T∗. After the time instant T∗, the position tracking error system (36) will be fixed-time convergence. By using a similar proof as that in the previous attitude control system, the position states are bounded during the time interval 0,T∗.


## 4.Numerical Simulations

The formation task for a leader and three followers in aircraft is studied. Figure1 shows the corresponding communication topology which is to be undirected connected graph. The weights are, respectively, given as a12=a21=1, a23=a32=1, and b1=1. The desired formation graphics is set to a regular triangle on the p-q plane. According to Figure 1, the relative position deviation can be expressed as(37)Δηi=0,1,0T,Διi=cos5π6,sin−π6,0T,Δκi=cos−π6,sin−π6,0T.Figure 1
Referenced desired formation and the topology graph.


### 4.1. System and Controller Parameters

The virtual leader’s position trajectory will be set as(38)ηd,ιd,κdT=5×sin0.2×t,5×cos0.2×t,5T.In addition, the aircraft’s system initial states are(39)η10,ι10,κ10,p10,q10,r10=−5,2,1,0,0,0,η20,ι20,κ20,p20,q20,r20=2,−4,0,0,0,0,η30,ι30,κ30,p30,q30,r30=1,3,0,0,0,0.As that in Table1, the system parameter values about some aircraft required in the simulation are provided. Meanwhile, by Theorem 1 and Theorem 2, the control parameters can be set as kp=ap=3.5, kd=ad=4.5, α1=β1=3/4, ρ1=6, and ρ2=3.Table 1
i-th aircraft’s system parameter values.
Descriptions
Parameter values
Inertia matrix
J
i
=
diag
0.0020
,
0.0022
,
0.0030
Force-to-moment factor
c
i
=
2.2
Mass
m
i
=
0.468
kg
Arm length of the aircraft
l
i
=
0.35
m
Gravitational acceleration
g
=
9.81
m
/
s
2
Attitude-loop disturbances
d
i
,
1
t
=
0.003
sin
0.8
t
+
5
d
i
,
2
t
=
0.002
cos
1.5
t
+
3
d
i
,
3
t
=
0.005
sin
0.5
t
+
2
,
i
=
1,2,3Through the fixed-time control scheme described in this paper, the flight trajectory curves of three aircraft formation in three-dimensional space are shown in Figure2. Clearly, the proposed fixed-time control scheme can enable multiple aircraft to form a desired formation pattern within a fixed time and cause the entire aircraft formation to track the virtual leader’s trajectory. In addition, the correlation curves of the aircraft’s position and attitude are shown in Figures 3 and 4. The response curves for the velocity and the angular velocity are, respectively, shown in Figures 5 and 6. As can be seen from Figure 2, under the fixed-time formation control scheme, the multiaircraft system with external disturbances can achieve better formation flying effect. Finally, in order to verify the statement that the convergence time is independent of initial state, Figure 7 shows the convergence time under different initial conditions, where X0=∑​i=13ηi0−ηd0+ιi0−ιd0+κi0−κd0. At the same time, the convergence time of the proposed method is compared with that of the finite-time control scheme in Figure 7. From Figure 7, it can be concluded that compared with the finite-time formation control scheme, the fixed-time formation control scheme has the better convergence performance, and the upper level of convergence time is independent of the initial state.Figure 2
Formation flight path of multiple aircraft in 3D space.Figure 3
Position curves of aircraft.Figure 4
Attitude curves of aircraft.Figure 5
Velocity curves of aircraft.Figure 6
Angular velocity curves of aircraft.Figure 7
Convergence time for the different initial conditions.


## 4.1. System and Controller Parameters

The virtual leader’s position trajectory will be set as(38)ηd,ιd,κdT=5×sin0.2×t,5×cos0.2×t,5T.In addition, the aircraft’s system initial states are(39)η10,ι10,κ10,p10,q10,r10=−5,2,1,0,0,0,η20,ι20,κ20,p20,q20,r20=2,−4,0,0,0,0,η30,ι30,κ30,p30,q30,r30=1,3,0,0,0,0.As that in Table1, the system parameter values about some aircraft required in the simulation are provided. Meanwhile, by Theorem 1 and Theorem 2, the control parameters can be set as kp=ap=3.5, kd=ad=4.5, α1=β1=3/4, ρ1=6, and ρ2=3.Table 1
i-th aircraft’s system parameter values.
Descriptions
Parameter values
Inertia matrix
J
i
=
diag
0.0020
,
0.0022
,
0.0030
Force-to-moment factor
c
i
=
2.2
Mass
m
i
=
0.468
kg
Arm length of the aircraft
l
i
=
0.35
m
Gravitational acceleration
g
=
9.81
m
/
s
2
Attitude-loop disturbances
d
i
,
1
t
=
0.003
sin
0.8
t
+
5
d
i
,
2
t
=
0.002
cos
1.5
t
+
3
d
i
,
3
t
=
0.005
sin
0.5
t
+
2
,
i
=
1,2,3Through the fixed-time control scheme described in this paper, the flight trajectory curves of three aircraft formation in three-dimensional space are shown in Figure2. Clearly, the proposed fixed-time control scheme can enable multiple aircraft to form a desired formation pattern within a fixed time and cause the entire aircraft formation to track the virtual leader’s trajectory. In addition, the correlation curves of the aircraft’s position and attitude are shown in Figures 3 and 4. The response curves for the velocity and the angular velocity are, respectively, shown in Figures 5 and 6. As can be seen from Figure 2, under the fixed-time formation control scheme, the multiaircraft system with external disturbances can achieve better formation flying effect. Finally, in order to verify the statement that the convergence time is independent of initial state, Figure 7 shows the convergence time under different initial conditions, where X0=∑​i=13ηi0−ηd0+ιi0−ιd0+κi0−κd0. At the same time, the convergence time of the proposed method is compared with that of the finite-time control scheme in Figure 7. From Figure 7, it can be concluded that compared with the finite-time formation control scheme, the fixed-time formation control scheme has the better convergence performance, and the upper level of convergence time is independent of the initial state.Figure 2
Formation flight path of multiple aircraft in 3D space.Figure 3
Position curves of aircraft.Figure 4
Attitude curves of aircraft.Figure 5
Velocity curves of aircraft.Figure 6
Angular velocity curves of aircraft.Figure 7
Convergence time for the different initial conditions.


## 5. Conclusion

To solve the formation control problem of a group of quadrotor aircraft, a distributed consensus algorithm has been proposed. To enhance the formation convergent rate, a fixed-time formation control scheme is proposed. Next, the external disturbance is introduced and a fixed-time disturbance observer is designed to compensate it. Some simulation results are presented to test the effectiveness of the proposed theoretical result. In future work, we will further study the directed connectivity of neighbor graphs and the antidisturbance control of aircraft.



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*Source: 6250870-2019-09-25.xml*