# Distributed Fixed-Time Coordinated Attitude Tracking Control with a Dynamic Leader for Spacecraft Formation Flying System

Authors: Zhanjie Zhou
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper studies the problem of distributed coordinated attitude tracking control with a dynamic leader for spacecraft formation systems in the presence of parameter uncertainties and external disturbances. Under the constraint that the time-varying attitude trajectory of the leader is only available for a subset of the followers, a fixed-time distributed estimator for each follower is proposed. Moreover, a fixed-time distributed attitude tracking control law combined with a fixed-time distributed estimator and distributed adaptive control algorithm is proposed. This control law is dedicated to reducing the effects of system uncertainty and external bounded disturbances, and the settling time of this algorithm is independent of the initial state of the system. The fixed-time stability of the spacecraft formation system is proven with the Lyapunov-based approach, and numerical simulations clearly verify the effectiveness of the proposed control algorithm.


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## Body

## 1. Introduction

With the development of science and technology, the scientific tasks undertaken by space technology have spread to many fields, including meteorology, geography, oceanography, communications, navigation, and space exploration. Moreover, scientific missions in the field of aerospace are becoming more complex and varied, and the traditional model of a single spacecraft working alone is no longer sufficient to satisfy the requirements of practical engineering. For this situation, the concept of spacecraft formation flying (SFF) was proposed. SFF carries several advantages in accomplishing space missions and creates new opportunities and applications for such missions [1]. As the basis for ensuring the realization of formation missions, coordinated attitude control, which can generally be categorized into centralized and distributed control, guarantees that the attitudes of spacecraft synchronize to a common value. Compared with traditional centralized coordination methods, distributed cooperative attitude control, which has the advantages of high efficiency, strong robustness, and high reliability, has received considerable attention.Distributed cooperative attitude control can be divided into two cases: with a leader spacecraft and without a leader spacecraft. First, the coordinated attitude control of leaderless spacecraft formation systems is studied in [2, 3], where the attitude of each spacecraft synchronizes to a common value, which is an unknown quantity that is not set in advance. Because the desired trajectory of each spacecraft is identical, predefined, and time varying, Wu et al. [4] proposed a decentralized adaptive sliding-mode control algorithm that maintains synchronization of spacecraft attitudes in formation. Considering the existence of communication delay in spacecraft formation systems, Abdessameud et al. [5] presented an attitude synchronization scheme that solves this problem. Second, the case when there is a leader spacecraft in a spacecraft formation system is discussed. Coordinated attitude tracking control was investigated in [6], and the control algorithm guaranteed that the attitude of a follower spacecraft was asymptotically synchronized to the specified attitude of the leader spacecraft. In [7], a robust adaptive sliding-mode control strategy supported by a neural network with radial basis functions was developed, and attitude tracking was achieved with this adaptive framework. However, all these methods in the mentioned literature only guarantee the asymptotic synchronization of a spacecraft’s attitude. In [8, 9], a control law that can make a system achieve asymptotic stability was designed for the multiagent dynamic cooperative control problem with nonvanishing disturbances and time-varying networks, respectively.Compared to asymptotic synchronization, finite-time synchronization implies faster application efficiency for SFF. The finite-time attitude synchronization problem for SFF is investigated in [10], and the control algorithm can ensure that the attitudes of spacecraft in the formation are synchronized in a finite time with zero final angular velocity. To improve the robustness of the system, a distributed attitude coordination control scheme using the terminal sliding mode (TSM) was proposed for a group of spacecrafts in the presence of external disturbances in [11], and the proposed control algorithm ensures that the system achieves finite-time stability. In [12–14], the finite-time algorithm was also applied, and the system obtained a faster convergence rate. The literature on finite-time cooperative attitude control algorithms for SFF is fruitful, but the settling time of finite-time control algorithms depends on the initial conditions of the system. The concept of fixed-time convergence, which ensures that the settling time is independent of the initial conditions, was proposed in [15]; thus, the settling times of fixed-time algorithms only depend on the design parameters. Based on fixed-time convergence theory, attitude control algorithms that can provide fixed-time convergence have been successively presented. Ning et al. [16] investigated the fixed-time coordinated tracking problem of nonlinear multiagent systems, and the settling time was directly estimated. A disturbance observer was employed in [17], and the observer combined with the fixed-time control algorithms effectively reduces the influence of external interference for a single spacecraft.In addition to the convergence rate of a system, parameter uncertainties and external disturbances that affect the control performance and even undermine the stability of the system are another problem of coordinated attitude tracking control of spacecraft formations. In the presence of parametric uncertainties, a distributed adaptive control algorithm combined with distributed sliding-mode estimators was proposed in [18], and this approach can effectively reduce the effect of parametric uncertainties. In [19], a distributed adaptive control algorithm for reducing the influence of external interference combined with a distributed fixed-time sliding-mode control algorithm, which proved that the attitude error converges to the regions containing the origin in a fixed time, was proposed. Considering both parameter uncertainty and bounded external disturbances, a distributed adaptive control algorithm was proposed in [20]; however, this method can only guarantee the asymptotic stability of SFF. It is noted that the literature is unable to guarantee fixed-time stability for an SFF system with a dynamic leader in the presence of system uncertainties and external disturbances.This study investigates the problem of distributed fixed-time attitude coordination tracking control with a dynamic leader for SFF in the presence of parameter uncertainties and bounded external disturbances. The objective is that the tracking attitude error of a group of follower spacecraft modeled by Euler–Lagrange equations converges to the regions containing the origin in a fixed time. The novel fixed-time controller designed in this study combined with fixed-time observers and adaptive algorithms that handle external disturbances and parameter uncertainties can effectively deal with the uncertainty of external disturbance and system inertia parameters. Since the convergence time of a fixed-time algorithm is independent of the initial system state, a fixed-time algorithm has wider application scenarios in practical engineering. Compared with existing works, the main contributions of this study are given as follows.(1)
In this study, only some follower spacecrafts can obtain the leader’s attitude information. A fixed-time distributed estimator is employed to estimate the time-varying attitude trajectory of the leader for each spacecraft by using its neighbors’ attitude information. The assumption that the leader’s information is available for all follower spacecraft in [21, 22] is unnecessary. Compared with the traditional estimator in [20], the fixed-time estimator, which is independent of the initial conditions, provides faster application efficiency. The fixed-time estimator also provides a basis for the design of subsequent fixed-time attitude tracking control algorithms.(2)
Distributed adaptive control algorithms for both parameter uncertainties and bounded external disturbances, which are more general in the actual working environment of spacecraft, are designed. Compared with the algorithms that model a system only for the case of uncertainties or external disturbances in [18, 19, 23], the proposed algorithms in this study are more adapted to the actual situation. In particular, compared with [19, 23], the controller designed in this study can effectively deal with the inertia uncertainty of spacecraft.(3)
Combined with a fixed-time distributed estimator and distributed adaptive control algorithms, a fixed-time distributed attitude tracking control law is proposed. Compared with those in [11, 24], the distributed coordinated attitude tracking control algorithms, which are independent of the initial conditions designed in this study, guarantee that the attitude tracking error converges to the regions containing the origin in a fixed time. Combined with the fixed-time stability theory, the settling time of the system only depends on the controller parameters, and by choosing the appropriate parameters, the settling time can be designed to satisfy the requirements of practical applications.Notations:⊗ represents the Kronecker product. Given a vector x=x1,...,xnT, we use x to denote the Euclidean norm and x1 represents the sum norm of the vector. For a vector x, xα=x1α,...,xnαT, xα=x1α,...,xnαT, and sigαx=sgnx1x1α,...,sgnxnxnαT. For x=x1,x2,x3T, define x×=0−x3x2x30−x1−x2x10. λminA and λmaxA represent the minimum and maximum eigenvalues of matrix A, respectively.


## 2. Background and Problem Statement

### 2.1. Spacecraft Attitude Dynamics

Consider a multiple rigid spacecraft system consisting ofn followers, labeled spacecraft 1 to n, and one leader, labeled spacecraft 0. The attitude of the ith follower is described by modified Rodriguez parameters (MRPs) as(1)σ˙i=Gσiωi,where σi=σi1σi2σi3T∈R3 is the MRPs of the ith spacecraft denoting the attitude orientation of the body-fixed frame with respect to the inertial frame. ωi∈R3 is the angular velocity of the ith rigid body with respect to the inertial frame, and Gσi is expressed as(2)Gσi=121−σi22I3+σi×+σiσiT.The matrixGσi is provided with the following properties:(3a)detGσi=1+σiTσi34≠0,(3b)GTσiGσi=1+σiTσi42I3.The dynamics of theith spacecraft in the formation is described by(4)Jiω˙i=−ωi×Jiωi+ui+ti,where Ji∈R3×3, ui∈R3, and ti∈R3 are the inertia matrix, control torque, and external disturbance torque of the ith spacecraft, respectively.Then followers are represented with Euler–Lagrange equations by combining (1) and (4) as(5)Miσiσ¨i+Ciσi,σ˙iσ˙i=τi+di.where Ciσi,σ˙i=−G−TσiJiG−1σiG˙σiG−1σi−G−TσiJiωi×G−1(σi), τi=G−Tσiui, di=G−Tσiti, and Miσi=G−TσiJiG−1σi. According to (3a) and (3b), Miσi is the symmetric positive definite inertia matrix. Throughout the subsequent analysis, the following fundamental properties of system (5) are given in [25]Property 1 (parameter boundedness).
for anyith spacecraft, there exist positive constants km, kM, and kC, where 0<kmI3≤Miσi≤kMI3 and Ciσi,σ˙i≤kCσ˙i.Property 2.
(linearity in the dynamic parameters).Miσix+Ciσi,σ˙iy=Yiσi,σ˙i,x,yΘi for any x and y∈R3, where Θi is the constant parameters vector associated with the ith spacecraft and Yiσi,σ˙i,x,yΘi is a known regression matrix.Assumption 1.
The time-varying state quantities of the leader spacecraftσ0, σ˙0, and σ¨0 are bounded.We assume σ˙0m≤δ1 and σ¨0m≤δ2, where m=1,2,3.Assumption 2.
The external disturbance torquedi has an unknown upper bound; we assume dim≤ki, where i=1,2,…,n and m=1,2,3.


### 2.2. Graph Theory

To represent the topology of the information flow among spacecraft in the formation, graph theory is briefly introduced. An undirected weighted graph can be denoted asG=N,E,A, where N=n0,n1,…,nn is a finite nonempty set of nodes and E⊆N×N is a set of unordered pairs of nodes. An edge ni,nj∈E denotes that the jth spacecraft can obtain the information from the ith spacecraft and vice versa. The adjacency matrix A=aij∈Rn×n of graph G is defined such that adjacency elements aij satisfy aij=1 if ni,nj∈E, and aij=0 otherwise. The Laplacian matrix L=lij∈Rn×n associated with A is defined as lii=∑i≠jaij and lij=−aij, where i≠j. The leader adjacency matrix is defined as B=diaga10,a20,…,an0, where ai0=1 if the ith spacecraft can obtain the information from the leader, and ai0=0 otherwise. Here, H=L+B. Obviously, L and H are symmetric matrices.Lemma 1 (see [26]).
If the graphG¯ is an undirected connected graph, then all eigenvalues of the matrix L have an ascending order, where 0=λ1<λ2≤λ3≤⋯≤λn.Lemma 2 (see [27]).
If the graphG¯ is an undirected connected graph, then the matrix H is a symmetric matrix, and all eigenvalues of the matrix H are positive.


### 2.3. Mathematic Background

Consider the following system:(6)x˙=gt,x,x0=x0,where x∈Rn and g:R+×Rn⟶Rn is a nonlinear function that can be discontinuous. For system (6), the following definitions and lemmas are given.Definition 1 (see [15]).
The equilibrium of system (6) is fixed-time stable if it is finite-time stable, and the settling time Tx0 is uniformly bounded for any initial state, that is, ∃Tmax>0 such that Tx0≤Tmax, where ∀x0∈Rn.Lemma 3 (see [15]).
If there exists a continuous radially unbounded functionV:Rn⟶R+∪0 such that(1)
Vx=0⇔x=0.(2)
For any solutionxt of (6) satisfies the inequality D∗Vxt≤−αVpxt+βVqxtk for some α,β,p,q,k>0:pk<1,qk>1, then the origin is globally fixed-time stable for system (6) and the following estimate holds:(7)Tx0≤1αk1−pk+1βkqk−1∀x0∈Rn,where D∗ft=limh⟶0+supft+h−ft/h.Lemma 4 (see [28]).
Letξ1,ξ2,…,ξn≥0. Then,(8)∑i=1nξip≥∑i=1nξip,if0<p≤1,∑i=1nξip≥n1−p∑i=1nξip,ifp>1.Lemma 5 (see [29]).
For anya,b∈Rn and any symmetric positive definite matrix Φ∈Rn×n,(9)2aTb≤aTΦ−1a+bTΦb.


### 2.4. Problem Statement

Consider an SFF system consisting ofn followers, which are described by (5), and a leader spacecraft. The leader’s attitude information is only available for a some of the follower spacecraft under an undirected communication graph G¯. The attitude tracking error and its derivative for each follower spacecraft is defined as(10)e1i=σi−σ0,(11)e2i=σ˙i−σ˙0.This study aims to design a distributed attitude control algorithmτi for each follower spacecraft in the presence of parameter uncertainties and external disturbances such that all spacecraft attitudes achieve synchronization, which means that e1i and e2i converge to the regions containing the origin in a fixed time.


## 2.1. Spacecraft Attitude Dynamics

Consider a multiple rigid spacecraft system consisting ofn followers, labeled spacecraft 1 to n, and one leader, labeled spacecraft 0. The attitude of the ith follower is described by modified Rodriguez parameters (MRPs) as(1)σ˙i=Gσiωi,where σi=σi1σi2σi3T∈R3 is the MRPs of the ith spacecraft denoting the attitude orientation of the body-fixed frame with respect to the inertial frame. ωi∈R3 is the angular velocity of the ith rigid body with respect to the inertial frame, and Gσi is expressed as(2)Gσi=121−σi22I3+σi×+σiσiT.The matrixGσi is provided with the following properties:(3a)detGσi=1+σiTσi34≠0,(3b)GTσiGσi=1+σiTσi42I3.The dynamics of theith spacecraft in the formation is described by(4)Jiω˙i=−ωi×Jiωi+ui+ti,where Ji∈R3×3, ui∈R3, and ti∈R3 are the inertia matrix, control torque, and external disturbance torque of the ith spacecraft, respectively.Then followers are represented with Euler–Lagrange equations by combining (1) and (4) as(5)Miσiσ¨i+Ciσi,σ˙iσ˙i=τi+di.where Ciσi,σ˙i=−G−TσiJiG−1σiG˙σiG−1σi−G−TσiJiωi×G−1(σi), τi=G−Tσiui, di=G−Tσiti, and Miσi=G−TσiJiG−1σi. According to (3a) and (3b), Miσi is the symmetric positive definite inertia matrix. Throughout the subsequent analysis, the following fundamental properties of system (5) are given in [25]Property 1 (parameter boundedness).
for anyith spacecraft, there exist positive constants km, kM, and kC, where 0<kmI3≤Miσi≤kMI3 and Ciσi,σ˙i≤kCσ˙i.Property 2.
(linearity in the dynamic parameters).Miσix+Ciσi,σ˙iy=Yiσi,σ˙i,x,yΘi for any x and y∈R3, where Θi is the constant parameters vector associated with the ith spacecraft and Yiσi,σ˙i,x,yΘi is a known regression matrix.Assumption 1.
The time-varying state quantities of the leader spacecraftσ0, σ˙0, and σ¨0 are bounded.We assume σ˙0m≤δ1 and σ¨0m≤δ2, where m=1,2,3.Assumption 2.
The external disturbance torquedi has an unknown upper bound; we assume dim≤ki, where i=1,2,…,n and m=1,2,3.


## 2.2. Graph Theory

To represent the topology of the information flow among spacecraft in the formation, graph theory is briefly introduced. An undirected weighted graph can be denoted asG=N,E,A, where N=n0,n1,…,nn is a finite nonempty set of nodes and E⊆N×N is a set of unordered pairs of nodes. An edge ni,nj∈E denotes that the jth spacecraft can obtain the information from the ith spacecraft and vice versa. The adjacency matrix A=aij∈Rn×n of graph G is defined such that adjacency elements aij satisfy aij=1 if ni,nj∈E, and aij=0 otherwise. The Laplacian matrix L=lij∈Rn×n associated with A is defined as lii=∑i≠jaij and lij=−aij, where i≠j. The leader adjacency matrix is defined as B=diaga10,a20,…,an0, where ai0=1 if the ith spacecraft can obtain the information from the leader, and ai0=0 otherwise. Here, H=L+B. Obviously, L and H are symmetric matrices.Lemma 1 (see [26]).
If the graphG¯ is an undirected connected graph, then all eigenvalues of the matrix L have an ascending order, where 0=λ1<λ2≤λ3≤⋯≤λn.Lemma 2 (see [27]).
If the graphG¯ is an undirected connected graph, then the matrix H is a symmetric matrix, and all eigenvalues of the matrix H are positive.


## 2.3. Mathematic Background

Consider the following system:(6)x˙=gt,x,x0=x0,where x∈Rn and g:R+×Rn⟶Rn is a nonlinear function that can be discontinuous. For system (6), the following definitions and lemmas are given.Definition 1 (see [15]).
The equilibrium of system (6) is fixed-time stable if it is finite-time stable, and the settling time Tx0 is uniformly bounded for any initial state, that is, ∃Tmax>0 such that Tx0≤Tmax, where ∀x0∈Rn.Lemma 3 (see [15]).
If there exists a continuous radially unbounded functionV:Rn⟶R+∪0 such that(1)
Vx=0⇔x=0.(2)
For any solutionxt of (6) satisfies the inequality D∗Vxt≤−αVpxt+βVqxtk for some α,β,p,q,k>0:pk<1,qk>1, then the origin is globally fixed-time stable for system (6) and the following estimate holds:(7)Tx0≤1αk1−pk+1βkqk−1∀x0∈Rn,where D∗ft=limh⟶0+supft+h−ft/h.Lemma 4 (see [28]).
Letξ1,ξ2,…,ξn≥0. Then,(8)∑i=1nξip≥∑i=1nξip,if0<p≤1,∑i=1nξip≥n1−p∑i=1nξip,ifp>1.Lemma 5 (see [29]).
For anya,b∈Rn and any symmetric positive definite matrix Φ∈Rn×n,(9)2aTb≤aTΦ−1a+bTΦb.


## 2.4. Problem Statement

Consider an SFF system consisting ofn followers, which are described by (5), and a leader spacecraft. The leader’s attitude information is only available for a some of the follower spacecraft under an undirected communication graph G¯. The attitude tracking error and its derivative for each follower spacecraft is defined as(10)e1i=σi−σ0,(11)e2i=σ˙i−σ˙0.This study aims to design a distributed attitude control algorithmτi for each follower spacecraft in the presence of parameter uncertainties and external disturbances such that all spacecraft attitudes achieve synchronization, which means that e1i and e2i converge to the regions containing the origin in a fixed time.


## 3. Distributed Fixed-Time Coordinated Attitude Tracking Control

### 3.1. Fixed-Time Distributed Estimator Design

In this section, since only some of the following spacecraft can obtain the attitude information of the leader, to achieve attitude synchronization, state estimators dedicated to obtaining accurate estimates ofσ0 and σ˙0 for each follower based on fixed-time convergence theory are designed. The estimators are designed as(12)σ^˙i=−α1signr1∑j=1naijσ^i−σ^j+ai0σ^i−σ0−α2signr2∑j=1naijσ^i−σ^j+ai0σ^i−σ0=−β1sign∑j=1naijσ^i−σ^j+ai0σ^i−σ0,(13)v^˙i=−α1signr1∑j=1naijv^i−v^j+ai0v^i−σ˙0−α2signr2∑j=1naijv^i−v^j+ai0v^i−σ˙0=−β2sign∑j=1naijv^i−v^j+ai0v^i−σ˙0,where σ^i is the estimate of σ0 and v^i is the estimate of σ˙0, α1>0, α2>0, 0<γ1<1, γ2>1, β1≥δ1, and β2≥δ2.The errors of the estimated states are defined as(14)σ˜i=σ^i−σ0,(15)v˜i=v^i−σ˙0.Theorem 1.
Considering the spacecraft formation attitude system (5) under an undirected connected communication graph G¯. σ˜i and v˜i converge to zero in a fixed time with a settling time T1≤21−γ1/2/α1λminH1−γ1/21−γ1+2/3n1−γ2/2/α2λminHγ2+1/2γ2−1 by estimators (12) and (13) under Assumption 1.Proof.
Defineσ˜=σ˜1T,σ˜2T,…,σ˜nTT. Based on Lemma 2, the Lyapunov function is described as(16)V1=12σ˜TH⊗I3σ˜.
Taking the time derivative of the Lyapunov functionV1, we have(17)V˙1=σ˜TH⊗I3σ^˙−Ιn⊗σ˙0=−α1σ˜TH⊗I3sigr1H⊗I3σ˜−α2σ˜TH⊗I3sigr2H⊗I3σ˜=−β1σ˜TH⊗I3signH⊗I3σ˜−σ˜TH⊗I3Ιn⊗σ˙0.
DefineP=H⊗I3σ˜=p11,p12,p13,…,pn1,pn2,pn3T. Based on Lemma 4, we obtain(18)V˙1=−α1∑i=1n∑m=13pim2γ1+1/2−α2∑i=1n∑m=13pim2γ2+1/2−β1∑i=1n∑m=13pim−∑i=1n∑m=13pimσ˙0m≤−α1∑i=1n∑m=13pim2γ1+1/2−α23n1−γ2/2∑i=1n∑m=13pim2γ2+1/2−β1∑i=1n∑m=13pim−∑i=1n∑m=13pimσ˙0m≤−α1P2γ1+1/2−α23n1−γ2/2P2γ2+1/2−β1∑i=1n∑m=13pim−∑i=1n∑m=13pimσ˙0m≤−α1P2γ1+1/2−α23n1−γ2/2P2γ2+1/2−β1−δ1∑i=1n∑m=13pim≤−α12λminHγ1+1/2V1γ1+1/2−α23n1−γ2/22λminHγ2+1/2V1γ2+1/2.
According to Lemma3, σ˜i converges to zero in a fixed time with a settling time T1, which is described as(19)T1≤21−γ1/2α1λminHγ1+1/21−γ1+2/3n1−γ2/2α2λminHγ2+1/2γ2−1.
The proof process for the two estimators is similar. This completes the proof of Theorem1.Remark 1.
According to Lemma2, the matrix H is a positive definite symmetric matrix; then, there is certainly an orthogonal matrix U such that(20)P2=PTP=σ˜TH⊗I3H⊗I3σ˜=σ˜TUTdiagλ12,λ22,…,λ3n2Uσ˜=∑i=13nλi2bi2,(21)V1=12σ˜TH⊗I3σ˜=12σ˜TUTdiagλ1,λ2,…,λ3nUσ˜=12∑i=13nλibi2.
Then, we can obtain(22)2λminHV1≤P2≤2λmaxHV1,where b=Uσ˜=b1,b1,…,b3nT and λi is the eigenvalue of matrix H⊗I3, and based on the property of Kronecker product, λminH⊗I3=λminH and λmaxH⊗I3=λmaxH.


### 3.2. Fixed-Time Distributed Coordinated Adaptive Attitude Tracking Control Algorithm Design

In this section, the fixed-time distributed coordinated adaptive attitude tracking control algorithm is designed based on the fixed-time distributed estimator designed in the previous section.The attitude state errors between theith and jth spacecraft are defined by(23)e1ij=σi−σj,(24)e2ij=σ˙i−σ˙j.The lumped attitude state errors are defined as(25)θ1i=∑j=1naije1ij+e1i=∑j=1naijσi−σj+σi−σ0=∑j=1naijσi−σ0−σj−σ0+σi−σ0,(26)θ2i=∑j=1naije2ij+e2i=∑j=1naijσ˙i−σ˙j+σ˙i−σ˙0=∑j=1naijσ˙i−σ˙0−σ˙j−σ˙0+σ˙i−σ˙0.Here,e1=e11T,e12T,...,e1nTT, e2=e21T,e22T,...,e2nTT, θ1=θ11T,θ12T,...,θ1nTT=L+In⊗I3e1, and θ2=θ21T,θ22T,...,θ2nTT=L+In⊗I3e2. Based on Lemma 2, L+In is a symmetric positive definite inertia matrix.An auxiliary variableσ˙ri is constructed as(27)σ˙ri=v^i−∑j=1naijσ˙i−σ˙j−μ1sigφ1∑j=1naijσi−σ^i−σj−σ^i+σi−σ^i,=−μ2sigφ2∑j=1naijσi−σ^i−σj−σ^i+σi−σ^i.It was proven that the statesv^i are bounded in (18); moreover, v^˙i, σ˙ri, and σ¨ri are bounded from (13) and (27).Construct another auxiliary variablesi as(28)si=σ˙i−σ˙ri.By Property2, we obtain(29)Miσiσ¨ri+Ciσi,σ˙iσ˙ri=Yiσi,σ˙i,σ¨ri,σ˙riΘi.AndYi=Yiσi,σ˙i,σ¨ri,σ˙ri is bounded for bounded states σ¨ri and σ˙ri.Combined with (5), the following is obtained:(30)Miσis˙i+Ciσi,σ˙isi=τi+di−YiΘi.The coordinated attitude tracking control law for theith follower spacecraft is designed as(31)τi=Ciσi,σ˙isi+YiΘ^i−κ1Miσisigpsi−κ2Miσisigqsi−k^isgnsiTMi−1σiT,(32)Θ^˙i=YiTMi−Tσisi,(33)k^˙i=siTMi−1σi1,where κ1>0, κ2>0, 0<p<1, and q>1. Θ^i and k^i are the estimates of Θi and ki, respectively.Remark 2.
It is proven in Theorem1 that σ˜i and v˜i converge to zero in a fixed time, which means σ^i=σ0 and v^i=σ˙0 in a fixed time; thus, after T1, σ˙ri is equivalent to(34)qi=σ˙0−∑j=1naijσ˙i−σ˙j−μ1sigφ1∑j=1naijσi−σ0−σj−σ0+σi−σ0−μ2sigφ2∑j=1naijσi−σ0−σj−σ0+σi−σ0=σ˙0−∑j=1naije2ij−k1sigφ1θ1−k2sigφ2θ1.
AfterT1, si is equivalent to(35)ci=θ2i+μ1sigφ1θ1i+μ2sigφ2θ1i.Theorem 2.
Considering the spacecraft formation attitude system (5) under an undirected connected communication graph G¯, control law (31) combined with estimators (12) and (13) guarantee the attitude errors e1 and e2 converge to the regions e1≤3∑i=1nΔθ1i2/λminL+InTL+In and e2≤3∑i=1nΔθ2i2/λminL+InTL+In in the fixed time T=T1+T2+T3, respectively.Proof.
First, we prove the boundedness of the system-related variables and defineΘ˜i=Θ^i−Θi and k˜i=ki−k^i. τ, s, d, k^, and Θ˜ are column stack vectors of τi, si, di, k^i, and Θ˜i, respectively. Cσ,σ˙, Mσ, and Y are block diagonal matrices of Ciσi,σ˙i, Miσi, and Yi, respectively. Substituting (31), (32), and (33) into (30), the system can be expressed in the following form:(36)Mσs˙=−κ1Mσsigps−κ2Mσsigqs−k^⊗I3signsTM−1σT+YΘ˜+d.
The Lyapunov function candidateV2 is denoted as(37)V2=12sTs+12Θ˜TΘ˜+12∑i=1nk˜i2.
Taking the time derivative of the Lyapunov functionV2, we have(38)V˙2=sTs˙+Θ˜TΘ˜˙+∑i=1nk˜ik˜˙i=sTM−1σ−κ1Mσsigps−κ2Mσsigqs−k^⊗I3sgnsTM−1σT+YΘ˜+d−Θ˜TΘ^˙−∑i=1nk˜ik^˙i=−κ1sTsigps−κ2sTsigqs+Θ˜TYTM−Tσs−Θ^˙−∑i=1nk^isiTMi−1σi1−∑i=1nk˜ik^˙i+sTM−1σd=−κ1sTsigps−κ2sTsigqs−∑i=1nk^isiTMi−1σi1−∑i=1nk˜isiTMi−1σi1+sTM−1σd≤−κ1sTsigps−κ2sTsigqs.
This completes the proof of the boundedness ofs, Θ˜, and k˜.
Second, we prove the fixed-time stability of (36). The Lyapunov function V3 is described as(39)V3=12sTs.
Taking the time derivative of the Lyapunov functionV3, based on Lemma 4 and Lemma 5, we have(40)V˙3=sTs˙=sTM−1σ−κ1Mσsigps−κ2Mσsigqs−k^⊗I3sgnsTM−1σ+YΘ˜+d=−κ1sTsigps−κ2sTsigqs+sTM−1σYΘ˜−∑i=1nk^isiTMi−1σi1+sTM−1σd=−κ1sTsigps−κ2sTsigqs−∑i=1nsiTMi−1σi12+sTM−1σYΘ˜+d≤−κ1sTsigps−κ2sTsigqs−∑i=1nsiTMi−1σi12+sTM−1σTsTM−1σ+YΘ˜+dTYΘ˜+d≤−κ1sTsigps−κ2sTsigqs−∑i=1nsiTMi−1σi12+∑i=1nsiTMi−1σi2+∑i=1nYiΘ˜i+di2≤−κ1sTsigps−κ2sTsigqs+∑i=1nYiΘ˜i+di2=−κ1∑i=1n∑m=13sim2p+1/2−κ2∑i=1n∑I=13sim2q+1/2+∑i=1nYiΘ˜i+di2≤−κ1∑i=1n∑m=13sim2p+1/2−κ23n1−q/2∑i=1n∑m=13sim2q+1/2+∑i=1nYiΘ˜i+di2.
SinceYi, Θ˜i, and di are bounded, there exist a constant η such that(41)∑i=1nYiΘ˜i+di2≤η.
Then, we obtain(42)V˙3≤−κ12p+1/2V3p+1/2−κ22q+1/23n1−q/2V3q+1/2+η.
Here, (42) is rewritten as the following two equations:(43)V˙3≤−1−ς1κ12p+1/2−ηV3p+1/2V3p+1/2−ς1κ12p+1/2V3p+1/2−κ22q+1/23n1−q/2V3q+1/2,(44)V˙3≤−κ12p+1/2V3p+1/2−1−ς2κ22q+1/23n1−q/2−ηV3q+1/2V3q+1/2−ς2κ22q+1/23n1−q/2V3q+1/2,where ς1=ς2=0.5.
From (43), if 1−ς1κ12p+1/2−η/V3p+1/2>0, then the fixed-time stability is guaranteed. By Lemma 3, auxiliary variable s converges to the region s≤η/1−ς1κ11/p+1 in fixed-time T21≤21−p/2/ς1κ11−p+2/3n1−q/2/κ2q−1. From (44), if 1−ς2κ22q+1/23n1−q/2−η/V3q+1/2>0, auxiliary variable s converges to the region s≤η/1−ς2κ23n1−q/21/q+1 in fixed time T22≤21−p/2/κ11−p+2/3n1−q/2/ς2κ2q−1. Then, the auxiliary variable s converges to the region:(45)s≤Δ=minη1−ς1κ11/1+p,η1−ς2κ23n1−q/21/q+1,which means sim≤Δ, where i=1,2,...,n and m=1,2,3, in a fixed time as(46)T2=maxT21,T22=max21−p/2ς1κ11−p+2/3n1−q/2κ2q−1,21−p/2ς1κ11−p+2/3n1−q/2κ2q−1.
Therefore, we obtain the following equations afterT1+T2:(47)sim=cim=θ2im+μ1sigφ1θ1im+μ2sigφ2θ1im=ϕim,ϕim≤Δ,(48)θ2im=−μ1sigφ1θ1im−μ2sigφ2θ1im+ϕim≤−μ1sigφ1θ1im−μ2sigφ2θ1im+Δ.
The Lyapunov function candidateV4 is denoted as(49)V4=12θ1im2.
Taking the time derivative of the Lyapunov functionV4, we have(50)V˙4=θ1imθ2im=−μ12φ1+1/212θ1im2φ1+1/2−μ22φ2+1/212θ1im2φ2+1/2+ϕimθ1im≤−μ12φ1+1/2V4φ1+1/2−μ22φ2+1/2V4φ2+1/2+Δθ1im.
Similar to the analysis ofV3, V˙4≤−μ12φ1+1/2V4φ1+1/2−μ22φ1+1/2V4φ2+1/2+Δθ1im can be rewritten as the following two equations:(51)V˙4≤−1−ς1μ12φ1+1/2−Δθ1imV4φ1+1/2V4φ1+1/2−ς1μ12φ1+1/2V4φ1+1/2−μ22φ2+1/2V4φ2+1/2,(52)V˙4≤−1−ς2μ22φ2+1/2−Δθ1imV4φ2+1/2V4φ2+1/2−μ12φ2+1/2V4φ2+1/2−ς2μ22φ2+1/2V4φ2+1/2.
From (51), if 1−ς1μ12φ1+1/2−Δθ1im/V4φ1+1/2>0, then the fixed-time stability is guaranteed. By Lemma 3, the lumped attitude state error θ1im converges to the region θ1im≤Δ/1−ς1μ11/φ1 in fixed time T31≤21−φ1/2/ς1μ11−φ1+21−φ1/2/u2φ2−1. From (52), if 1−ς2μ22φ1+1/2−Δθ1im/V4φ2+1/2>0, the lumped attitude state error θ1im converges to the region θ1im≤Δ/1−ς2μ21/φ2 in fixed time T32≤21−φ1/2/μ11−φ1+21−φ1/2/ς2u2φ2−1. Then, the lumped attitude state error θ1im converges to the region:(53)θ1im≤Δθ1i=minΔ1−ς1μ11/φ1,Δ1−ς2μ21/φ2.
Based on (48), the lumped attitude state error θ2im converges to the region:(54)θ2im≤μ1θ1imφ1+μ2θ1imφ2+Δ=μ1Δθ1iφ1+μ2Δθ1iφ2+Δ=Δθ2i.
Moreover, the settling time ofθ1im and θ2im is(55)T3=maxT31,T32=max21−φ1/2ς1μ11−φ1+21−φ2/2u2φ2−1,21−φ1/2μ11−φ1+21−φ2/2ς2u2φ2−1.
Therefore, afterT1+T2+T3, we obtain(56)λminL+InTL+Ine1Te1≤e1TL+In⊗I3TL+In⊗I3e1=θ1Tθ1≤3∑i=1nΔθ1i2,(57)λminL+InTL+Ine2Te2≤e2TL+In⊗I3TL+In⊗I3e2=θ2Tθ2≤3∑i=1nΔθ2i2.
Then, the attitude state errorse1 and e2 converge to the regions:(58)e1≤3∑i=1nΔθ1i2λminL+InTL+In,(59)e2≤3∑i=1nΔθ2i2λminL+InTL+In,in fixed time T=T1+T2+T3, respectively. This completes the proof of Theorem 2.Remark 3.
The controller proposed in this study contains several parameters, where0<γ1<1, 0<p<1, 0<μ1<1, and 0<φ1<1 affect the convergence accuracy of the system. The smaller the values of γ1, p, μ1, and φ1, the higher the accuracy of the estimator and the control performance; however, smaller values cause the system to produce larger chattering. γ2>1, q>1, μ2>1, and φ2>1 affect the control system response speed; the larger the values of γ2, q, μ2, and φ2 are, the higher the obtained system response speed is; however, large values will cause a large overshoot.
As shown in Figure1, the control system designed in this study includes fixed-time state observers, fixed-time attitude tracking controllers, and adaptive disturbance estimators for the ith spacecraft, where j=0,...,nandj≠i represents the spacecraft that is available to provide the ith spacecraft with the attitude information.Figure 1
The control system schematic diagram of theith spacecraft.


## 3.1. Fixed-Time Distributed Estimator Design

In this section, since only some of the following spacecraft can obtain the attitude information of the leader, to achieve attitude synchronization, state estimators dedicated to obtaining accurate estimates ofσ0 and σ˙0 for each follower based on fixed-time convergence theory are designed. The estimators are designed as(12)σ^˙i=−α1signr1∑j=1naijσ^i−σ^j+ai0σ^i−σ0−α2signr2∑j=1naijσ^i−σ^j+ai0σ^i−σ0=−β1sign∑j=1naijσ^i−σ^j+ai0σ^i−σ0,(13)v^˙i=−α1signr1∑j=1naijv^i−v^j+ai0v^i−σ˙0−α2signr2∑j=1naijv^i−v^j+ai0v^i−σ˙0=−β2sign∑j=1naijv^i−v^j+ai0v^i−σ˙0,where σ^i is the estimate of σ0 and v^i is the estimate of σ˙0, α1>0, α2>0, 0<γ1<1, γ2>1, β1≥δ1, and β2≥δ2.The errors of the estimated states are defined as(14)σ˜i=σ^i−σ0,(15)v˜i=v^i−σ˙0.Theorem 1.
Considering the spacecraft formation attitude system (5) under an undirected connected communication graph G¯. σ˜i and v˜i converge to zero in a fixed time with a settling time T1≤21−γ1/2/α1λminH1−γ1/21−γ1+2/3n1−γ2/2/α2λminHγ2+1/2γ2−1 by estimators (12) and (13) under Assumption 1.Proof.
Defineσ˜=σ˜1T,σ˜2T,…,σ˜nTT. Based on Lemma 2, the Lyapunov function is described as(16)V1=12σ˜TH⊗I3σ˜.
Taking the time derivative of the Lyapunov functionV1, we have(17)V˙1=σ˜TH⊗I3σ^˙−Ιn⊗σ˙0=−α1σ˜TH⊗I3sigr1H⊗I3σ˜−α2σ˜TH⊗I3sigr2H⊗I3σ˜=−β1σ˜TH⊗I3signH⊗I3σ˜−σ˜TH⊗I3Ιn⊗σ˙0.
DefineP=H⊗I3σ˜=p11,p12,p13,…,pn1,pn2,pn3T. Based on Lemma 4, we obtain(18)V˙1=−α1∑i=1n∑m=13pim2γ1+1/2−α2∑i=1n∑m=13pim2γ2+1/2−β1∑i=1n∑m=13pim−∑i=1n∑m=13pimσ˙0m≤−α1∑i=1n∑m=13pim2γ1+1/2−α23n1−γ2/2∑i=1n∑m=13pim2γ2+1/2−β1∑i=1n∑m=13pim−∑i=1n∑m=13pimσ˙0m≤−α1P2γ1+1/2−α23n1−γ2/2P2γ2+1/2−β1∑i=1n∑m=13pim−∑i=1n∑m=13pimσ˙0m≤−α1P2γ1+1/2−α23n1−γ2/2P2γ2+1/2−β1−δ1∑i=1n∑m=13pim≤−α12λminHγ1+1/2V1γ1+1/2−α23n1−γ2/22λminHγ2+1/2V1γ2+1/2.
According to Lemma3, σ˜i converges to zero in a fixed time with a settling time T1, which is described as(19)T1≤21−γ1/2α1λminHγ1+1/21−γ1+2/3n1−γ2/2α2λminHγ2+1/2γ2−1.
The proof process for the two estimators is similar. This completes the proof of Theorem1.Remark 1.
According to Lemma2, the matrix H is a positive definite symmetric matrix; then, there is certainly an orthogonal matrix U such that(20)P2=PTP=σ˜TH⊗I3H⊗I3σ˜=σ˜TUTdiagλ12,λ22,…,λ3n2Uσ˜=∑i=13nλi2bi2,(21)V1=12σ˜TH⊗I3σ˜=12σ˜TUTdiagλ1,λ2,…,λ3nUσ˜=12∑i=13nλibi2.
Then, we can obtain(22)2λminHV1≤P2≤2λmaxHV1,where b=Uσ˜=b1,b1,…,b3nT and λi is the eigenvalue of matrix H⊗I3, and based on the property of Kronecker product, λminH⊗I3=λminH and λmaxH⊗I3=λmaxH.


## 3.2. Fixed-Time Distributed Coordinated Adaptive Attitude Tracking Control Algorithm Design

In this section, the fixed-time distributed coordinated adaptive attitude tracking control algorithm is designed based on the fixed-time distributed estimator designed in the previous section.The attitude state errors between theith and jth spacecraft are defined by(23)e1ij=σi−σj,(24)e2ij=σ˙i−σ˙j.The lumped attitude state errors are defined as(25)θ1i=∑j=1naije1ij+e1i=∑j=1naijσi−σj+σi−σ0=∑j=1naijσi−σ0−σj−σ0+σi−σ0,(26)θ2i=∑j=1naije2ij+e2i=∑j=1naijσ˙i−σ˙j+σ˙i−σ˙0=∑j=1naijσ˙i−σ˙0−σ˙j−σ˙0+σ˙i−σ˙0.Here,e1=e11T,e12T,...,e1nTT, e2=e21T,e22T,...,e2nTT, θ1=θ11T,θ12T,...,θ1nTT=L+In⊗I3e1, and θ2=θ21T,θ22T,...,θ2nTT=L+In⊗I3e2. Based on Lemma 2, L+In is a symmetric positive definite inertia matrix.An auxiliary variableσ˙ri is constructed as(27)σ˙ri=v^i−∑j=1naijσ˙i−σ˙j−μ1sigφ1∑j=1naijσi−σ^i−σj−σ^i+σi−σ^i,=−μ2sigφ2∑j=1naijσi−σ^i−σj−σ^i+σi−σ^i.It was proven that the statesv^i are bounded in (18); moreover, v^˙i, σ˙ri, and σ¨ri are bounded from (13) and (27).Construct another auxiliary variablesi as(28)si=σ˙i−σ˙ri.By Property2, we obtain(29)Miσiσ¨ri+Ciσi,σ˙iσ˙ri=Yiσi,σ˙i,σ¨ri,σ˙riΘi.AndYi=Yiσi,σ˙i,σ¨ri,σ˙ri is bounded for bounded states σ¨ri and σ˙ri.Combined with (5), the following is obtained:(30)Miσis˙i+Ciσi,σ˙isi=τi+di−YiΘi.The coordinated attitude tracking control law for theith follower spacecraft is designed as(31)τi=Ciσi,σ˙isi+YiΘ^i−κ1Miσisigpsi−κ2Miσisigqsi−k^isgnsiTMi−1σiT,(32)Θ^˙i=YiTMi−Tσisi,(33)k^˙i=siTMi−1σi1,where κ1>0, κ2>0, 0<p<1, and q>1. Θ^i and k^i are the estimates of Θi and ki, respectively.Remark 2.
It is proven in Theorem1 that σ˜i and v˜i converge to zero in a fixed time, which means σ^i=σ0 and v^i=σ˙0 in a fixed time; thus, after T1, σ˙ri is equivalent to(34)qi=σ˙0−∑j=1naijσ˙i−σ˙j−μ1sigφ1∑j=1naijσi−σ0−σj−σ0+σi−σ0−μ2sigφ2∑j=1naijσi−σ0−σj−σ0+σi−σ0=σ˙0−∑j=1naije2ij−k1sigφ1θ1−k2sigφ2θ1.
AfterT1, si is equivalent to(35)ci=θ2i+μ1sigφ1θ1i+μ2sigφ2θ1i.Theorem 2.
Considering the spacecraft formation attitude system (5) under an undirected connected communication graph G¯, control law (31) combined with estimators (12) and (13) guarantee the attitude errors e1 and e2 converge to the regions e1≤3∑i=1nΔθ1i2/λminL+InTL+In and e2≤3∑i=1nΔθ2i2/λminL+InTL+In in the fixed time T=T1+T2+T3, respectively.Proof.
First, we prove the boundedness of the system-related variables and defineΘ˜i=Θ^i−Θi and k˜i=ki−k^i. τ, s, d, k^, and Θ˜ are column stack vectors of τi, si, di, k^i, and Θ˜i, respectively. Cσ,σ˙, Mσ, and Y are block diagonal matrices of Ciσi,σ˙i, Miσi, and Yi, respectively. Substituting (31), (32), and (33) into (30), the system can be expressed in the following form:(36)Mσs˙=−κ1Mσsigps−κ2Mσsigqs−k^⊗I3signsTM−1σT+YΘ˜+d.
The Lyapunov function candidateV2 is denoted as(37)V2=12sTs+12Θ˜TΘ˜+12∑i=1nk˜i2.
Taking the time derivative of the Lyapunov functionV2, we have(38)V˙2=sTs˙+Θ˜TΘ˜˙+∑i=1nk˜ik˜˙i=sTM−1σ−κ1Mσsigps−κ2Mσsigqs−k^⊗I3sgnsTM−1σT+YΘ˜+d−Θ˜TΘ^˙−∑i=1nk˜ik^˙i=−κ1sTsigps−κ2sTsigqs+Θ˜TYTM−Tσs−Θ^˙−∑i=1nk^isiTMi−1σi1−∑i=1nk˜ik^˙i+sTM−1σd=−κ1sTsigps−κ2sTsigqs−∑i=1nk^isiTMi−1σi1−∑i=1nk˜isiTMi−1σi1+sTM−1σd≤−κ1sTsigps−κ2sTsigqs.
This completes the proof of the boundedness ofs, Θ˜, and k˜.
Second, we prove the fixed-time stability of (36). The Lyapunov function V3 is described as(39)V3=12sTs.
Taking the time derivative of the Lyapunov functionV3, based on Lemma 4 and Lemma 5, we have(40)V˙3=sTs˙=sTM−1σ−κ1Mσsigps−κ2Mσsigqs−k^⊗I3sgnsTM−1σ+YΘ˜+d=−κ1sTsigps−κ2sTsigqs+sTM−1σYΘ˜−∑i=1nk^isiTMi−1σi1+sTM−1σd=−κ1sTsigps−κ2sTsigqs−∑i=1nsiTMi−1σi12+sTM−1σYΘ˜+d≤−κ1sTsigps−κ2sTsigqs−∑i=1nsiTMi−1σi12+sTM−1σTsTM−1σ+YΘ˜+dTYΘ˜+d≤−κ1sTsigps−κ2sTsigqs−∑i=1nsiTMi−1σi12+∑i=1nsiTMi−1σi2+∑i=1nYiΘ˜i+di2≤−κ1sTsigps−κ2sTsigqs+∑i=1nYiΘ˜i+di2=−κ1∑i=1n∑m=13sim2p+1/2−κ2∑i=1n∑I=13sim2q+1/2+∑i=1nYiΘ˜i+di2≤−κ1∑i=1n∑m=13sim2p+1/2−κ23n1−q/2∑i=1n∑m=13sim2q+1/2+∑i=1nYiΘ˜i+di2.
SinceYi, Θ˜i, and di are bounded, there exist a constant η such that(41)∑i=1nYiΘ˜i+di2≤η.
Then, we obtain(42)V˙3≤−κ12p+1/2V3p+1/2−κ22q+1/23n1−q/2V3q+1/2+η.
Here, (42) is rewritten as the following two equations:(43)V˙3≤−1−ς1κ12p+1/2−ηV3p+1/2V3p+1/2−ς1κ12p+1/2V3p+1/2−κ22q+1/23n1−q/2V3q+1/2,(44)V˙3≤−κ12p+1/2V3p+1/2−1−ς2κ22q+1/23n1−q/2−ηV3q+1/2V3q+1/2−ς2κ22q+1/23n1−q/2V3q+1/2,where ς1=ς2=0.5.
From (43), if 1−ς1κ12p+1/2−η/V3p+1/2>0, then the fixed-time stability is guaranteed. By Lemma 3, auxiliary variable s converges to the region s≤η/1−ς1κ11/p+1 in fixed-time T21≤21−p/2/ς1κ11−p+2/3n1−q/2/κ2q−1. From (44), if 1−ς2κ22q+1/23n1−q/2−η/V3q+1/2>0, auxiliary variable s converges to the region s≤η/1−ς2κ23n1−q/21/q+1 in fixed time T22≤21−p/2/κ11−p+2/3n1−q/2/ς2κ2q−1. Then, the auxiliary variable s converges to the region:(45)s≤Δ=minη1−ς1κ11/1+p,η1−ς2κ23n1−q/21/q+1,which means sim≤Δ, where i=1,2,...,n and m=1,2,3, in a fixed time as(46)T2=maxT21,T22=max21−p/2ς1κ11−p+2/3n1−q/2κ2q−1,21−p/2ς1κ11−p+2/3n1−q/2κ2q−1.
Therefore, we obtain the following equations afterT1+T2:(47)sim=cim=θ2im+μ1sigφ1θ1im+μ2sigφ2θ1im=ϕim,ϕim≤Δ,(48)θ2im=−μ1sigφ1θ1im−μ2sigφ2θ1im+ϕim≤−μ1sigφ1θ1im−μ2sigφ2θ1im+Δ.
The Lyapunov function candidateV4 is denoted as(49)V4=12θ1im2.
Taking the time derivative of the Lyapunov functionV4, we have(50)V˙4=θ1imθ2im=−μ12φ1+1/212θ1im2φ1+1/2−μ22φ2+1/212θ1im2φ2+1/2+ϕimθ1im≤−μ12φ1+1/2V4φ1+1/2−μ22φ2+1/2V4φ2+1/2+Δθ1im.
Similar to the analysis ofV3, V˙4≤−μ12φ1+1/2V4φ1+1/2−μ22φ1+1/2V4φ2+1/2+Δθ1im can be rewritten as the following two equations:(51)V˙4≤−1−ς1μ12φ1+1/2−Δθ1imV4φ1+1/2V4φ1+1/2−ς1μ12φ1+1/2V4φ1+1/2−μ22φ2+1/2V4φ2+1/2,(52)V˙4≤−1−ς2μ22φ2+1/2−Δθ1imV4φ2+1/2V4φ2+1/2−μ12φ2+1/2V4φ2+1/2−ς2μ22φ2+1/2V4φ2+1/2.
From (51), if 1−ς1μ12φ1+1/2−Δθ1im/V4φ1+1/2>0, then the fixed-time stability is guaranteed. By Lemma 3, the lumped attitude state error θ1im converges to the region θ1im≤Δ/1−ς1μ11/φ1 in fixed time T31≤21−φ1/2/ς1μ11−φ1+21−φ1/2/u2φ2−1. From (52), if 1−ς2μ22φ1+1/2−Δθ1im/V4φ2+1/2>0, the lumped attitude state error θ1im converges to the region θ1im≤Δ/1−ς2μ21/φ2 in fixed time T32≤21−φ1/2/μ11−φ1+21−φ1/2/ς2u2φ2−1. Then, the lumped attitude state error θ1im converges to the region:(53)θ1im≤Δθ1i=minΔ1−ς1μ11/φ1,Δ1−ς2μ21/φ2.
Based on (48), the lumped attitude state error θ2im converges to the region:(54)θ2im≤μ1θ1imφ1+μ2θ1imφ2+Δ=μ1Δθ1iφ1+μ2Δθ1iφ2+Δ=Δθ2i.
Moreover, the settling time ofθ1im and θ2im is(55)T3=maxT31,T32=max21−φ1/2ς1μ11−φ1+21−φ2/2u2φ2−1,21−φ1/2μ11−φ1+21−φ2/2ς2u2φ2−1.
Therefore, afterT1+T2+T3, we obtain(56)λminL+InTL+Ine1Te1≤e1TL+In⊗I3TL+In⊗I3e1=θ1Tθ1≤3∑i=1nΔθ1i2,(57)λminL+InTL+Ine2Te2≤e2TL+In⊗I3TL+In⊗I3e2=θ2Tθ2≤3∑i=1nΔθ2i2.
Then, the attitude state errorse1 and e2 converge to the regions:(58)e1≤3∑i=1nΔθ1i2λminL+InTL+In,(59)e2≤3∑i=1nΔθ2i2λminL+InTL+In,in fixed time T=T1+T2+T3, respectively. This completes the proof of Theorem 2.Remark 3.
The controller proposed in this study contains several parameters, where0<γ1<1, 0<p<1, 0<μ1<1, and 0<φ1<1 affect the convergence accuracy of the system. The smaller the values of γ1, p, μ1, and φ1, the higher the accuracy of the estimator and the control performance; however, smaller values cause the system to produce larger chattering. γ2>1, q>1, μ2>1, and φ2>1 affect the control system response speed; the larger the values of γ2, q, μ2, and φ2 are, the higher the obtained system response speed is; however, large values will cause a large overshoot.
As shown in Figure1, the control system designed in this study includes fixed-time state observers, fixed-time attitude tracking controllers, and adaptive disturbance estimators for the ith spacecraft, where j=0,...,nandj≠i represents the spacecraft that is available to provide the ith spacecraft with the attitude information.Figure 1
The control system schematic diagram of theith spacecraft.


## 4. Simulation Results

In this section, to demonstrate the validity of the proposed control algorithms, numerical simulations for a spacecraft formation system are conducted, and the communication topology is described in Figure2. A system consisting of four follower spacecraft and one leader spacecraft satisfies Assumption 1. Table 1 lists the inertia matrix and initial conditions of the follower spacecraft. To obtain better control performance in the simulation process, after several debugging operations, the parameters of the distributed estimator in (12) and (13) and the auxiliary variable σ˙ri and the control law (31) are set as α1=1, α2=1, κ1=1, κ2=0.8, γ1=0.8, γ2=1.2, μ1=0.4, μ2=1.2, φ1=0.5, φ2=1.5, p=0.4, and q=1.3, respectively. The additional variation inertia matrix for each follower spacecraft is set as −0.03×sin0.1t0.04×cos0.1t0.05×sin0.1tT N ⋅m. The additional variation inertia matrix for each follower spacecraft is set as 0.1sin0.1t0.01sin0.1t0.01sin0.1t0.01sin0.1t0.15sin0.1t0.01sin0.1t0.01sin0.1t0.01sin0.1t0.1cos0.1t kg ⋅m2.Figure 2
Communication topology.Table 1
Initial parameters of follower spacecrafts.
Spacecraft no.iInertia matrixJikg⋅m2Initial attitudeσi0Initial attitude derivativeσ˙i01150.41;0.4141.5;11.512−0.0462,0.0215,0.0500,0,02140.31;0.3161.3;11.3150.0542,0.0114,−0.05480,0,03180.52;0.5131.6;21.6140.0495,−0.0641,0.00670,0,04160.61;0.6151.4;11.4170.0335,0.0414,−0.03220,0,0Figure3 shows that the attitude tracking error ei converges to eim<4×10−4 in four seconds in the presence of external disturbing torques and periodic changes in the inertia matrix. The estimation errors σ˜i and v˜i are illustrated in Figures 4 and 5, respectively. σ˜i and v˜i can converge to regions σ˜im<4×10−4 and v˜im<2×10−5 with a dynamic leader, respectively. As illustrated in Figure 6, the auxiliary variable si converges to region sim<2×10−4. Figure 7 shows the control torques of the follower spacecraft. As the desired attitude is dynamic, the control torques will never be zero, which is illustrated in Figure 7. To reflect the performance of the controller proposed in this study, comparisons are made with the controllers in [19] (the controller for comparison 1) and [23] (the controller for comparison 2). The station-keeping attitude error metric (SKAEM) is defined as SKEAM=∑i=1n∑m=13eim2. The control performance comparison is shown in Figure 8. The controller proposed in this study can provide a faster convergence rate and a more accurate attitude tracking performance.Figure 3
Attitude tracking errorse1i.Figure 4
Estimation errorsσ˜i.Figure 5
Estimation errorsv˜i.Figure 6
Auxiliary variablesi.Figure 7
Control torquesui.Figure 8
The control performance comparison of SKAEM.


## 5. Conclusion

The distributed coordinated attitude tracking control problem for a spacecraft formation system with a dynamic leader is investigated in the presence of external disturbing torques and parameter uncertainty in this study. Considering that the attitude information of the leader is only available to some of the follower spacecraft, fixed-time distributed estimators that only utilize the neighbors’ information are proposed to estimate the leader’s attitude information. Based on the estimated information, auxiliary variables are designed for the subsequent design. By employing the fixed-time control method, a fixed-time control algorithm combined with an adaptive control algorithm, which is used to cope with external disturbances and parameter uncertainties, is proposed. The algorithm achieves fixed-time convergence of the auxiliary variable, and the convergence time is independent of the initial condition. By employing the Lyapunov approach, the proposed control algorithms demonstrate that the attitude of each follower spacecraft can be synchronized with the attitude of the dynamic leader in a fixed time. The fixed-time control algorithm can be used to reduce the convergence time in the case of tracking a dynamic target, as the convergence time is influenced by the initial conditions, which change at all times due to the dynamic target in conventional control methods and finite-time control algorithms. The effectiveness of the proposed control algorithms is demonstrated by numerical simulations. In future work, based on this research, I will continue to deeply study the fixed-time attitude tracking control of multispacecraft formations and strive to focus on the state estimator that can be used in the case of directed graph communication.



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*Source: 3425073-2022-06-17.xml*