Event-Triggered Adaptive Sliding Mode Attitude Containment Control for Microsatellite Cluster under Directed Graph

Abstract

In order to investigate the attitude containment control problem for a microsatellite cluster, an event-triggered adaptive sliding mode attitude containment control algorithm is proposed for the satellite cluster flight system under directed topology, so that attitude of followers asymptotically converges to the convex hull formed by the leaders’ orientations. At first, the event-triggered control strategy is introduced into the attitude containment control problem for the microsatellite cluster. The triggering condition consisting of state-dependent and time-dependent function is designed to adjust control period and avoid the Zeno behaviour. When the function value meets the triggering condition, the event is triggered, state information is sampled, control law is computed, and actuators are updated, while the control action performed in nontriggering time is the same as the previous triggering instant. Then, in the presence of model uncertainties and external disturbances, an event-triggered adaptive sliding mode attitude containment control algorithm is presented under directed topology, and sufficient and necessary conditions for the followers to enter into the target area formed by the leaders are given. Furthermore, cell partitions from graph theory are employed to investigate the influence of information topology on steady states of followers, which provides theoretical basis for orientation design of cluster satellites. Finally, simulation results show that the proposed control strategy could reduce control execution frequency, as well as ensure the similar control performance with the time-triggered one, and followers belonging to the same cell have the same steady states.

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1. Introduction

Satellite cluster has received growing attention in the recent years [1, 2] for its advantages of greater flexibility, faster response, higher reliability, lower cost, and better reconfigurability [3]. Contrary to satellite formation flight, cluster flight does not impose strict limits on the geometry of the cluster [4] and is hence more suitable for implementation by multiple microsatellite systems. Satellite cluster has been deployed by many institutes, such as ANTS [5], Breakthrough Starshot project [6], and so on.

Several technical barriers need to be broken down to pave the way for cluster satellites to come into being. Coordinated attitude control of cluster satellites has been identified as one of the enabling technologies. Although there has been lots of results on coordinated control problem for multiple satellites systems, we note that most of the existing research studies consider leaderless [7, 8] or one leader case [9, 10], where there exists no group objective or only a single group objective.

However, the presence of multiple leaders is more attractive for the satellite cluster system, owing to the fact that such strategies provide attractive solutions to cluster problems, both in terms of complexity and computational load. As a kind of extended consensus problem, the case with multiple leaders is what we call containment control [11]. The objective of containment control is to guarantee that all the followers asymptotically converge into the convex hull formed by the leaders through information interaction and coordinated control. The containment control problem received significant research interest due to its various applications, such as mobile robots [12], unmanned aerial vehicles (UAVs) [13], underwater vehicles [14], and satellite formation systems [15, 16].

By now, there has been lots of research studies on distributed containment control, and research studies could be divided into several types from different perspectives. According to the dynamics, the research studies include first-order systems [12, 17, 18], second-order systems [19–21], linear systems [12, 17, 18], nonlinear systems [15, 22–24], homogeneous and heterogeneous multiagent systems [25], etc. In view of information topology, fixed topology [15, 19, 22, 23], and switching topology [18], undirected graph [20] and directed graph [17, 23] are, respectively, considered. In many research studies, containment control is combined with other novel control strategies, such as finite-time control [14, 15, 22], adaptive control [14, 16], neural network [16], event-triggered control [26], and so on. For the design of leaders, there exists the case of stationary leaders [20, 24], dynamic leaders (constant velocity or time-varying velocity) [17], leaders formation [25, 27], and so on. In addition, other problems such as time delay [11, 21], model uncertainties [15, 16, 23], external disturbances [15, 16], collision avoidance [27], and unmeasured relative velocity [28] are also discussed.

Recently, distributed attitude containment control strategies have gained increased attention in satellite coordinated control community. In [22], the distributed finite-time attitude containment control problem for multiple rigid bodies was addressed. For multiple stationary leaders, a model-independent control law was proposed to guarantee the attitudes of followers converge to the stationary convex hull formed by leaders in finite time by using both the one-hop and two-hop neighbours’ information. Then, for dynamic leaders, a distributed sliding mode estimator and a nonsingular sliding surface were given to guarantee the attitudes and angular velocities of followers converge, respectively, to the dynamic convex hull formed by those of the leaders in finite time. Under undirected fixed connected graph, Weng et al. [29] investigated distributed robust finite-time attitude containment control for multiple rigid bodies with uncertainties including parametric uncertainties, external disturbances, and actuator failures. In [30], a distributed control strategy combined with finite-time command filtered backstepping (FTCFB) and an adaptive technique was established to solve the attitude containment control problem of spacecraft formation flying (SFF) with unknown external disturbances. The proposed novel distributed adaptive FTCFB approach could guarantee the containment errors of attitudes between leader spacecraft and follower spacecraft reach the desired neighbourhood in finite time under undirected topology. However, the information topology of cluster satellites may be directional in actual space missions. Because only a fraction of satellites was equipped with necessary sensors or communication equipment to measure relative state in cluster system, obviously, the directional information topology is more general. Ma et al. [31] studied the distributed finite-time attitude containment control problem for multiple rigid body systems with multiple stationary and dynamic leaders under directed graph. Based on sliding mode observer, adaptive attitude control algorithms were given, and the necessary and sufficient conditions were achieved which rendered all the followers converge to the convex hull spanned by the static and dynamic leaders in finite time. In [24], an attitude containment control algorithm was proposed in the case of undirected angle information topology and directed angular velocity information topology, and the case of unavailable relative angle velocity was also investigated.

Continuous or periodic sampled control scheme is usually applied to the aforementioned attitude containment control problem of satellite formation, whose results belong to time-triggered control. The state information of cluster satellites is usually sampled with a fixed and high sampling frequency, and the actuators are updated at each sampling instant, which increase the pressure of the whole network communication and lead to the wear of actuators and unnecessary energy consumption, thus seriously shortening the in-orbit operation life of cluster satellites. Moreover, in time-triggered control schemes, control action updates periodically even when the system has reached the desired state with satisfactory accuracy. Computation and communication pressure will be greatly increased, while resource and network bandwidth of the microsatellite cluster are extremely limited.

Efforts to overcome these problems have led to the proposal of event-triggered control strategy. Information interaction and controller updates are not determined by time, but by the triggering condition (event). Event-triggered mechanism consists of two types: state-dependent events [32] and time-dependent events [33]. In the event-triggered control strategy, the control tasks, consisting of sampling state information of satellites, computing control law, and updating actuators, are executed when the well-designed triggering condition is satisfied. Thus, communication and computation resources are utilized only when “needed” to preserve desired control performance [34]. It makes event-triggered control favourable, especially for satellite cluster missions with limited bandwidth and resources.

So far, event-triggered control has been investigated in the multi-rigid body system model with nonlinear characteristics. In [35], an event-triggered distributed adaptive controller was proposed to study the leader-follower consensus problem for a directed network of Euler–Lagrange agents. For the attitude control problem of spacecraft, Wu et al. [36] investigated the problem of spacecraft attitude stabilization control system with limited communication and external disturbances based on an event-triggered control scheme. Sun et al. [37] introduced an event-triggered control (ETC) strategy for the spacecraft attitude stabilization problem from the view of cyber-physical systems (CPSs); a new quaternion-based nonlinear control algorithm was proposed to ensure attitude dynamics systems’ exponential stability, and parameter selection of event function and controllers was discussed in this paper. There are also research studies combining event-triggered schemes with containment control. In [38], distributed event-triggered cooperative attitude control of multiple rigid bodies with leader-follower architecture was investigated; under the designed controllers with the event-triggered strategies, the orientations of followers converge to the convex hull formed by the desired leaders’ orientations with zero angular velocities. Xu et al. [26] studied the distributed event-triggered adaptive containment control problem for multiple Euler–Lagrange systems with stationary/dynamic leaders over directed communication networks.

Although various novel control strategies have been investigated for the attitude containment control problem of a satellite cluster, which enables cluster members to converge to the convex hull formed by leaders with a faster convergence rate, little attention has been paid to the relationship between system performance and information topology design of the satellite cluster system. Note that the interaction between satellites need not be bidirectional in practice due to communication bandwidth or sensor capability. Constrained by intersatellite distance and performance of sensors, only parts of followers can receive information from leaders directly. To the best of the authors’ knowledge, the event-triggered attitude containment control for the microsatellite cluster system under directed topology is worth studying and awaits a breakthrough. However, the existence of system uncertainties and unavoidable external disturbances of cluster system results in an unsatisfactory performance [39, 40]. Thus, the sliding mode control (SMC) strategy, which is robust to external disturbances and model uncertainties, is employed. The adaptive control method is also combined to realize the online estimation of uncertain parameters in real time, and it would not destroy the robustness properties of SMC [41].

In this paper, the attitude containment control problem and information topology structure design for the microsatellite cluster are investigated. First, the triggering condition consisting of the relative state error is given to adjust controller update period. If and only if the triggering condition is satisfied, state information is sampled, control law is computed, and actuators are updated. Then, an event-triggered adaptive sliding mode attitude containment control algorithm is proposed in the pressure of inertia uncertainties and external disturbances, which makes attitude of cluster members to enter asymptotically into the convex hull formed by leaders’ orientations. Furthermore, cell partitions in the view of graph theory are employed to investigate the influence of information topology on orientation of followers, which provides theoretical basis for information topology design of satellite cluster missions. Finally, simulation results show that the proposed event-triggered adaptive sliding mode attitude containment controller could drive the followers to enter into the convex hull formed by the leaders’ orientations in the presence of inertia uncertainties and external disturbances, and followers belonging to the same cell have the same orientation. Compared with the existing results, the proposed results in this paper have the following advantages.

(1) In the framework of the Euler–Lagrange system, this paper presents an event-triggered adaptive sliding mode attitude containment control algorithm for the microsatellite cluster system under directed topology, so that the followers asymptotically converge to the target area formed by the leaders’ orientation in the presence of inertia uncertainties and external disturbances. The controller is updated only when the triggering condition is satisfied, and the state information of the triggering instant is utilized at the nontriggering time. Therefore, the control input is a piecewise function and the controller update for each satellite is asynchronous. Compared with the time-triggered method, the event-triggered adaptive sliding mode attitude containment control algorithm not only ensures the similar control performance of cluster satellites but also effectively reduces information transmission and actuator update frequency of the satellite cluster system. Event-triggered attitude containment control is superior in microsatellite cluster missions with limited resource and lower precision.

(2) The triggering condition of time-varying threshold is given in this paper. Most researches only study state-dependent or time-dependent triggering condition. However, the triggering condition in this paper is the combination of state-dependent and time-dependent. The time-dependent function is introduced to avoid the Zeno behaviour, i.e., the controller does not update infinitely in finite time, nor does it update in a periodical manner. When the triggering condition is satisfied, state information is sampled, control law is computed, and actuator is updated, which can effectively reduce the computation and actuator update frequency while ensuring the control performance of cluster system.

(3) From the perspective of graph theory, the control algorithm is fully distributed in the sense that each satellite can select their control gains according to only local information. Then, the influence of information topology design on orientation of the microsatellite cluster is analysed. It is shown that in an ideal environment, the stable state of each follower is a convex combination of all leaders’ states it can access. Two kinds of cell partition of cluster information topology are given, and it is proved that satellites belonging to the same cell partition have the same stable state. It provides a theoretical basis for information topology design of microsatellite cluster missions with performance requirements such as full coverage of the target area.

The remainder of this paper is organized as follows. Mission scenarios, relative attitude dynamics of satellite cluster, basic knowledge of graph theory, and attitude containment control problem are briefly given in Section 2. In Section 3, the event-triggered adaptive sliding mode attitude containment control algorithm is proposed for the satellite cluster in the presence of inertia uncertainties and external disturbances. Then, sufficient and necessary conditions for asymptotical convergence of the cluster system and the absence of the Zeno behaviour are derived. Orientation of cluster satellites based on information topology design is given in Section 4, providing theoretical basis for information topology design of the microsatellite cluster. Numerical simulations to verify the effectiveness of the proposed control algorithm and information topology structure design are completed in Section 5, and concluding comments are given Section 6.

2. Preliminaries and Problem Formulation

In this section, some problem descriptions about mission scenarios are introduced, and then preliminary knowledge about containment control strategies is given.

2.1. Mission Scenarios

Traditional single leader consensus is a group of satellites aiming to achieve an agreement through information interaction and coordinated control, where leader’s trajectory will be tracked by followers of the cluster system. In contrast, containment control can drive the followers to enter into the target area formed by the multiple leaders. Containment control is more robust in the case of leader failure and more practical in microsatellite cluster missions with limited resource and lower precision. Two typical microsatellite cluster flying scenarios which are closely related to attitude containment control are firstly given and analysed as follows.

2.1.1. Earth Observation or Deep Space Exploration of Microsatellite Cluster

Microsatellite cluster has been employed in many observation missions, such as the Earth observation or deep space exploration. It is necessary for satellites to obtain and maintain a certain relative attitude [32]. In the observation missions, such as expanding observation view or searching observation target, members are not required to converge to the same orientation, but to enter into a target area formed by the leaders’ orientations, which is called attitude containment control. One of the basic objectives is that a subset of the satellite set (leaders) stabilizes to a specific relative attitude, and the orientation of the rest members (followers) enters into and remains within the specific attitude, determined by a convex hull formed by the leaders’ orientation. Meanwhile, each satellite is only allowed to communicate (attitude or angular velocity) with a specific member of the set, and these constraints limit the information interaction between satellites.

2.1.2. Coordinated Attitude Control for Fractionated Spacecraft

Fractionated spacecraft distributes the functional capabilities of a monolithic spacecraft into multiple free-flying, wirelessly linking modules (service modules and different payloads) [42]. One of the main challenges of this architecture is cluster flight, keeping the various modules within bounded configurations. The fractionated spacecraft generally does not require precise relative orbit and attitude control, as long as the relative distance is within the range of communication and the relative attitude control enables power transmission links [43]. In this case, some (virtual) module spacecraft form an orientation area for the cluster members’ attitude, which makes modules to maintain communication and power transmission.

There is no precise requirement for the final attitude of cluster members in the aforementioned attitude control missions of the microsatellite cluster. The attitude of followers only needs to reach an area instead of the consensus state. It is necessary to form the target area using (virtual) leaders’ orientation and then control the followers to enter into the target area through intersatellite information interaction and properly designed coordinated attitude control protocol.

Constrained by unidirectional measurement characteristics of sensors or GPS-like radio frequency communication, as well as relative distance limitation between satellites, information topology of microsatellite cluster is unidirectional, asymmetric, and sparse. And the information topology structure generally remains fixed in a short time if there does not exist large disturbance. In the following, we will study the attitude containment control problem of the microsatellite cluster under fixed directed topology.

2.2. Relative Dynamics Model of Satellite Cluster

In this paper, the modified Rodriguez parameters (MRPs) are used to describe attitude motion of the satellite cluster. MRP is a kind of attitude description method without redundancy and singularity. Attitude vector $\sigma \in {ℝ}^3$ of MRPs is expressed by Euler axis $\mathbf{e}$ and Euler angle $\mathrm{\Phi }$ as follows:$$\begin{array}{}\text{(1)}& \mathbf{\sigma }=\mathbf{e}\tan \frac{\Phi }{4},\hspace{1em}-2\pi <\Phi <2\pi .\end{array}$$

The kinematic equation of follower $i$ is$$\begin{array}{}\text{(2)}& {\dot{\sigma }}_i=\mathbf{G}{\mathbf{\sigma }}_i{\mathbf{\omega }}_i,\end{array}$$where ${\sigma }_i\in {ℝ}^3$ and ${\omega }_i\in {ℝ}^3$, respectively, are MRPs and attitude angular velocity of the fixed-body coordinate system of satellite i relative to the inertia coordinate system, $\mathbf{G}{\sigma }_i=1/41-{\sigma }_i^T{\sigma }_i{\mathbf{I}}_3+2{\sigma }_i^{\times }+2{\sigma }_i{\sigma }_i^T$, ${\mathbf{I}}_3$ is the 3×3 identity matrix, and ${\sigma }^{\times }$ is the skew-symmetric matrix of ${\sigma }_i$ and is expressed as$$\begin{array}{}\text{(3)}& {\mathbf{\sigma }}^{\times }=\begin{array}{ccc}0& -{\mathbf{\sigma }}_3& {\mathbf{\sigma }}_2\\ {\mathbf{\sigma }}_3& 0& -{\mathbf{\sigma }}_1\\ -{\mathbf{\sigma }}_2& {\mathbf{\sigma }}_1& 0\end{array},\end{array}$$where ${\mathbf{v}}_k$ represents the $k$th component of vector $\mathbf{v}$.

Attitude dynamics equation of satellite i is expressed as$$\begin{array}{}\text{(4)}& {\mathbf{J}}_i{\dot{\omega }}_i=-{\mathbf{\omega }}_i^{\times }{\mathbf{J}}_i{\mathbf{\omega }}_i+{\mathbf{\tau }}_i+{\mathbf{\tau }}_{\text{d}i},\end{array}$$where ${\mathbf{J}}_i={\mathbf{J}}_i^T\in {ℝ}^3$ is the positive inertia matrix, ${\tau }_i\in {ℝ}^3$ represents the control torques, and ${\tau }_{\text{d}i}\in {ℝ}^3$ represents the external disturbance torques.

Attitude kinematic equation using MRPs as the attitude parameter can be converted into an unified Euler–Lagrange form by appropriate transformation [44]. The Euler–Lagrange equation can effectively introduce the linear relative attitude expression between different satellites, which plays a beneficial role in coordinated attitude control design of satellite cluster. Taking the derivative of (2) and substituting (3) into (2), the Euler–Lagrange form of attitude dynamics equation for satellite i could be obtained as follows:$$\begin{array}{}\text{(5)}& {\mathbf{M}}_i{\mathbf{\sigma }}_i{\ddot{\sigma }}_i+{\mathbf{C}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i{\dot{\sigma }}_i+{\mathbf{g}}_i{\mathbf{\sigma }}_i={\mathbf{u}}_i+{\mathbf{d}}_i,\end{array}$$where ${\sigma }_i={{\sigma }_i^1,{\sigma }_i^2,{\sigma }_i^3}^T$, ${\mathbf{M}}_i{\sigma }_i={\mathbf{G}}^{-T}{\sigma }_i{\mathbf{J}}_{0i}{\mathbf{G}}^{-1}{\sigma }_i$ is symmetric positive-definite inertia matrix of satellite i, ${\mathbf{C}}_i{\sigma }_i,{\dot{\sigma }}_i=-{\mathbf{G}}^{-T}{\sigma }_i{\mathbf{J}}_{0i}{\mathbf{G}}^{-1}{\sigma }_i\dot{G}{\sigma }_i{\mathbf{G}}^{-1}{\sigma }_i-{\mathbf{G}}^{-T}{\sigma }_i{{\mathbf{J}}_{0i}{\mathbf{G}}^{-1}{\sigma }_i{\dot{\sigma }}_i}^{\times }{\mathbf{G}}^{-1}{\sigma }_i$ is Coriolis and centrifugal torques matrix, ${\mathbf{J}}_i={\mathbf{J}}_{0i}+\mathrm{\Delta }{\mathbf{J}}_i$, ${\mathbf{J}}_{0i}$ is nominal inertia of follower i, $\mathrm{\Delta }{\mathbf{J}}_i$ is uncertainties part, ${\mathbf{g}}_i{\sigma }_i$ is uncertainties causing by model uncertainties, ${\mathbf{u}}_i={\mathbf{G}}^{-T}{\sigma }_i{\tau }_i$ is control torques, and ${\mathbf{d}}_i={\mathbf{G}}^{-T}{\sigma }_i{\tau }_{\text{d}i}$ is disturbance torques. According to Reference [45], the Euler–Lagrange equation has the following properties: (a) ${\mathbf{M}}_i{\sigma }_i\in {ℝ}^{3\times 3}$ is a symmetric positive-definite matrix; (b) here we assume that there exist positive $k_{\underset{¯}{m}},k_{\overline{m}},k_C,k_g$ satisfying $0<k_{\underset{¯}{m}}{\mathbf{I}}_n\le {\mathbf{M}}_i{\sigma }_i\le k_{\overline{m}}{\mathbf{I}}_n$, ${\mathbf{C}}_i{\sigma }_i,{\dot{\sigma }}_i<k_C{\dot{\sigma }}_i$, ${\mathbf{g}}_i{\sigma }_i\le k_g$; (c) for any ${\sigma }_i,{\dot{\sigma }}_i\in {ℝ}^3$, $1/2{\mathbf{M}}_i{\sigma }_i,{\dot{\sigma }}_i-{\mathbf{C}}_i{\sigma }_i,{\dot{\sigma }}_i$ is a skew-symmetric matrix, and there exist $\mathbf{x}\in {ℝ}^3$, ${\mathbf{x}}^{T}1/2{\dot{M}}_i{\sigma }_i-{\mathbf{C}}_i{\sigma }_i,{\dot{\sigma }}_i\mathbf{x}=0$; (d) we assume that the left-hand side of (5) can be parameterized in linear expression as ${\mathbf{M}}_i{\sigma }_{i}x+{\mathbf{C}}_i{\sigma }_i,{\dot{\sigma }}_{i}y+g_i={\mathbf{Y}}_i{\sigma }_i,{\dot{\sigma }}_i,x,y{\Theta }_i{.g}_i$, for any $x,y\in {ℝ}^3$, where ${\mathbf{Y}}_i{\sigma }_i,{\dot{\sigma }}_i,x,y$ is the regression function matrix and ${\Theta }_i$ is the unknown constant parameter vector.

Assumption 1.

All leaders’ states and state derivatives are bounded, that is, there exist constants ${\sigma }_{\text{LM}}$ and ${\dot{\sigma }}_{\text{LM}}$ such that ${\sigma }_{L}t\le {\sigma }_{\text{LM}}$,${\dot{\sigma }}_{L}t\le {\dot{\sigma }}_{\text{LM}}$. In this paper, we add an additional assumption that ${\lim }_{t⟶\infty }{\dot{\sigma }}_L⟶0$.

Assumption 2.

The external disturbance torque ${\mathbf{d}}_i$ is bounded, that is, there exists a positive constant $d_M$ such that ${\mathbf{d}}_i\le d_M$, where $d_M$ is bounded and unknown.

2.3. Graph Theory

Graph theory is introduced as an useful mathematical tool to describe intersatellite information interaction, providing theoretical basis for control performance analysis of the cluster system.

Suppose there are N + M satellites in the cluster, consisting of N followers (denoted by 1, 2, …, N) and M leaders (denoted by N + 1, …, N + M). We assume that the (virtual) satellites belong to either one of the two subsets, namely, the subset of followers $F=\mathrm{1,2},\dots ,N$ or the subset of leaders $L=N+1,\dots ,N+M$.

Information interaction topology of the cluster system can be modelled by a digraph $G=V,E$ with node set $VG=1,\dots ,N+M$ denoting satellite with dynamics or kinematic characteristics and edge set $EG=VG\times VG$ denoting intersatellite information interaction. Each edge $i,j\in EG$ means satellite j can access state information from satellite i, and satellite i is called a neighbour of satellite j. The neighbour set $N_i=j\in VG|j,i\in EG$ of satellite i is a collection of all the satellites from which satellite i can access state information. If $i,j\in EG\iff j,i\in EG$, bidirectional edge $i,j$ indicates that satellites i and j can access state information from each other. A directed path from node i to node j is a sequence of edges of the form (i, i₁), (i₁, i₂), …, (i_n, j), in a directed graph. A directed tree is a directed graph, where every node has exactly one parent except for one node, called the root, and the root has directed paths to every other node. A directed spanning tree of a directed graph is a direct tree that contains all nodes of the directed graph. A directed graph has or contains a directed spanning tree, if there exists a directed spanning tree as a subset of the directed graph.

Two matrices are frequently used to represent interaction topology: one is adjacency matrix $\mathbf{A}=a_{ij}$ with $a_{ij}\ge 0$, $a_{ij}>0$ if $i,j\in EG$. In this paper, we assume that self-edges are not allowed, i.e., $a_{ii}=0$. And the other is Laplacian matrix $\mathbf{L}=l_{ij}$ with $l_{ii}={\displaystyle {\sum }_{j=1,j\ne i}^N{a}_{ij}}$, $l_{ij}=-a_{ij},i\ne j$.

The system Laplacian matrix L can also be written as block matrix:$$\begin{array}{}\text{(6)}& \mathbf{L}=\begin{array}{cc}{\mathbf{L}}_F& {\mathbf{L}}_{\text{FL}}\\ 0& 0\end{array},\end{array}$$where ${\mathbf{L}}_F$ is the N × N submatrix composed by rows and columns in which the followers of L are located, and it represents information interaction relationship among followers. ${\mathbf{L}}_{FL}$ is the $N\times M$ submatrix composed by rows and columns in which the followers and leaders of L are, respectively, located, and it represents the information flow from leaders to followers. $G_F$ denotes followers and information interaction between followers.

Several graph theory tools are given to provide theoretical basis for information topology design of the cluster system.

Definition 1 (see [46]).

For directed graph $G$, a k partition $\pi$ of $VG$ is composed of k cells $C_1,\dots ,C_k$ with $C_i\cap C_j=\mathrm{\Phi }$, $i,j=1,\dots ,k,i\ne j$ and ${\displaystyle {\cup }_{i=1}^k{C}_i}=VG$. C_j is a neighbour of C_i$i,j=1,\dots ,k,i\ne j$ if there exist $i{}^{\prime }=c_i$ and $j{}^{\prime }=c_j$ such that $j^{\prime }$ is a neighbour of $i{}^{\prime }$.

Definition 2 (see [47]).

Suppose that $\pi =C_1,\cdots ,C_k$ is a k partition of $VG$; if any two distinct vertices in $C_i$ have the same number of neighbours in C_j for all $j\ne i$, then $\pi$ is called a ${\pi }^1$ partition. In particular, ${\pi }^1=H_1,\dots ,H_m$ denotes the ${\pi }^1$ cell partition containing $H_{i}i=1,\dots ,m$ as m cells of ${\pi }^1$.

Definition 3 (see [46]).

Suppose $\pi =C_1,\dots ,C_k$ is a k partition of $VG$; if for each vertex in $C_i$, it has the same number of neighbours in all neighbour cells of $C_{i}i=1,\dots ,m$, then $\pi$ is called a ${\pi }^2$ partition.

2.4. Containment Control Model

The following definitions, assumptions, and lemmas related to containment control strategy are needed to derive the main results of this paper.

2.4.1. Convex Hull

There may exist multiple leaders in microsatellite cluster missions, and all followers are required to enter into the target area formed by the leaders’ state instead of reaching a consensus state. Several vertices on the boundary of the target area are selected so that the target area can be approximately replaced by convex polygons which are formed by these vertices [48]. Generally, the more the vertices are selected, the higher the approximate accuracy is.

Suppose that the target area (moving or stationary) can be approximated by a convex hull formed by M vertices.

The definition of convex hull is given as follows.

Definition 4 (see [23]).

Let $C$ be a set in a real vector space $V\subseteq {\mathbf{R}}^p$. The set $C$ is convex if, for any x and y in $C$, the point $1-tx+ty\in C$ for any $t\in \mathrm{0,1}$. The convex hull for a set of points $X=x_1,\dots ,x_M$ in $V$ is the minimal convex set containing all points in $X$. We use $\text{CO}X$ to denote the convex hull of X, that is, $\text{CO}X={\displaystyle {\sum }_{i=1}^M{\alpha }_i{x}_i}|x_i\in X,{\alpha }_i\in \mathbf{R},{\alpha }_i\ge 0,{\displaystyle {\sum }_{i=1}^M{\alpha }_i=1}$.

Vertex information of the target area could be provided by the Earth station or could be autonomously generated by cluster members which have strong sense, communication, and information processing capability.

Once the convex hull which approximates the target area is selected, vertex information of the convex hull can be seen as (virtual) leaders of the cluster system, while cluster members are regarded as followers.

2.4.2. Distributed Attitude Containment Control Formulation

Followers need to generate control decisions based on absolute state of itself and relative state information of its neighbours, which makes attitude of followers to converge to the convex hull formed by the leaders’ orientation.

Containment control protocol for followers could be written in the following form:$$\begin{array}{}\text{(7)}& {\mathbf{u}}_i=h_i{\mathbf{\sigma }}_F,{\mathbf{\omega }}_F,{\mathbf{\sigma }}_L,{\mathbf{\omega }}_L,\hspace{1em}i\in F,\end{array}$$where ${\sigma }_F={{\sigma }_1^T,\dots ,{\sigma }_N^T}^T$, ${\omega }_F={{\omega }_1^T,\dots ,{\omega }_N^T}^T$, ${\sigma }_L={{\sigma }_{N+1}^T,\dots ,{\sigma }_{N+M}^T}^T$, and ${\omega }_L={{\omega }_{N+1}^T,\dots ,{\omega }_{N+M}^T}^T$, respectively, are the attitudes and angular velocities of followers and leaders.

We assume that there is no information interaction between leaders. The trajectories of leaders are not affected by other members, while followers need to generate control instructions with neighbours’ information [49]. To drive all the states of followers to enter into the convex hull formed by the leaders’ orientation, it must be ensured that each follower can receive information from leaders directly or indirectly. Otherwise, there will exist followers whose motion is not affected by any leader, nor will converge to the convex hull formed by the leaders’ orientation. Thus, the information topology of the cluster system needs to meet the following assumption.

Assumption 3 (see [20]).

Suppose that for each follower i, there exists at least one leader $j$ that has a path to the follower i.

Lemma 1 (see [20]).

If Assumption 3 holds, then ${\mathbf{L}}_F$ is invertible; each entry of $-{\text{L}}_F^{\mathrm{-1}}{\text{L}}_{\text{FL}}$ is nonnegative and each row sum of $-{\mathbf{L}}_F^{\mathrm{-1}}{\mathbf{L}}_{\mathrm{FL}}$ is equal to one.

Lemma 2 (see [23]).

Let $\mathbf{L}$ be the Laplacian matrix associated with a directed graph $G$. Then, $\mathbf{L}$ has a single zero eigenvalue and all other eigenvalues have positive real parts if and only if $G$ has a directed spanning tree.

Lemma 3 (see [50]).

Consider the system $\dot{x}=ft,x,u$, where $ft,x,u$ is continuously differentiable and globally uniform Lipschitz in $x,u$ for all t. If the unforced system $\dot{x}=ft,x,0$ has a globally exponentially stable equilibrium point at the origin $x=0$, then the system $\dot{x}=ft,x,u$ is input-to-state stable.

According to Definition 4 and Lemma 1, the desired state is convex weighted average of the leaders' attitude and angular velocity, which can be written as$$\begin{array}{c}\text{(8)}& {\mathbf{\sigma }}_d=-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}{\mathbf{\sigma }}_L,{\mathbf{\omega }}_d=-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}{\mathbf{\omega }}_L,\end{array}$$where ${\sigma }_d$ and ${\omega }_d$, respectively, are the desired attitude and desired angular velocity.

Our aim in this paper is to propose appropriate distributed attitude control algorithm for the followers (i.e., those indexed from 1 to N), so that in an asymptotic manner, they can travel into the convex hull formed by the leaders (i.e., those satellites indexed from N + 1 to N + M). We will also analyse under what conditions the containment behaviours can be guaranteed and perform rigorous convergence with less sampled information and control action, that is, ${\lim }_{t⟶\infty }{\sigma }_{F}t⟶-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\sigma }_L$, ${\lim }_{t⟶\infty }{\omega }_{F}t⟶0$, $\forall i\in F$. On this basis, the influence of information topology design on orientation of followers is analysed, which provides a theoretical reference for orientation design of cluster members.

3. Distributed Attitude Containment Control for Microsatellite Cluster

For attitude containment control problem of leader-follower satellite cluster, an event-triggered adaptive sliding mode control algorithm is proposed in the presence of inertia uncertainties and external disturbances. The triggering condition based on relative attitude error is set for each follower, that is, the entire system is triggered asynchronously and triggering time of each follower is different. State information is sampled, control law is computed, and actuators are updated if and only if triggering conditions are met. At nontriggering time, controller of followers uses the state information of triggering instant.

The event-triggered adaptive sliding mode controller and triggering condition are designed to make attitude of followers asymptotically enter into the convex hull formed by the leaders and attitude angular velocity converge to 0; meanwhile, the update frequency of control tasks is reduced.

3.1. Event-Triggered Formulation for Attitude Containment Control

The traditional time-triggered control scheme is usually updated periodically, and periodic sampling may be a better control scheme in the view of control scheme design and problem analysis. However, when the system is working normally, periodic control will cause unnecessary energy consumption and actuator update.

To reduce the pressure of computation, communication, and actuators and meet the constraints of mass, volume, and power on the microsatellites, the event-triggered mechanism is introduced to solve coordinated attitude control of the microsatellite cluster, while to ensure the control performance of the microsatellite cluster, the triggering condition is used to adjust update period of controller and reduce the amount of computation and communication pressure. The design of the event-triggered coordinated attitude controller could be divided into two main steps. First is the setting of triggering condition. The triggering function with relative attitude error is designed to adjust update period of controller, state information is sampled, control law is computed, and actuator is updated if and only if the event is triggered. Second, the event-triggered distributed attitude controller is designed in the presence of model uncertainties and external disturbances, which makes followers to converge to the convex hull formed by the leaders’ orientation. And the Zeno behaviour is excluded. Triggering frequency is reduced and resource is saved as well as control performance of cluster system is ensured. The triggering condition is given in this section, and distributed attitude containment control protocol will be proposed in the next section.

First, to rewrite the dynamics equation of the satellite cluster into the form which is convenient for stability analysis, auxiliary variable ${\dot{\sigma }}_{ri}$ is introduced:$$\begin{array}{}\text{(9)}& {\dot{\sigma }}_{ri}=-\alpha {\displaystyle \sum _{j=1}^{N+M}{a}_{ij}{\mathbf{\sigma }}_{i}t-{\mathbf{\sigma }}_{j}t},\hspace{1em}i\in F,j\in F\cup L.\end{array}$$

Design the sliding variable ${\mathbf{s}}_i$:$$\begin{array}{}\text{(10)}& {\mathbf{s}}_i={\dot{\sigma }}_{i}t-{\dot{\sigma }}_{ri}t={\dot{\sigma }}_{i}t+\alpha {\displaystyle \sum _{j=1}^{N+M}{a}_{ij}{\mathbf{\sigma }}_{i}t-{\mathbf{\sigma }}_{j}t},\end{array}$$where α is a positive constant and $a_{ij}$ is the weight of adjacency matrix $\mathbf{A}$ of directed graph $G$.

According to the properties of equations (5) and (9), the dynamics system could be rewritten as$$\begin{array}{}\text{(11)}& {\mathbf{M}}_i{\mathbf{\sigma }}_i{\ddot{\sigma }}_{ri}+{\mathbf{C}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i{\dot{\sigma }}_{ri}+{\mathbf{g}}_i={\mathbf{Y}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i,{\ddot{\sigma }}_{ri},{\dot{\sigma }}_{ri}{\Theta }_i+{\mathbf{\epsilon }}_i,\hspace{1em}i=1,\dots ,N.\end{array}$$

In coordinated attitude control of the satellite cluster, ${\Theta }_i$ is the inertia matrix with unknown parameters, ${\Theta }_i={J_{11},J_{22},J_{33},J_{12},J_{13},J_{23}}^T$. For any $\mathbf{x}\in {ℝ}^3$, there exists ${\mathbf{J}}_i\mathbf{x}=\mathbf{L}\mathbf{x}{\Theta }_i$. ${\epsilon }_i$ is the approximation error for the ith follower, ${\epsilon }_i\le {\epsilon }_M$. Linear operator $\mathbf{L}x$ is shown as follows:$$\begin{array}{}\text{(12)}& \mathbf{L}\mathbf{x}=\begin{array}{cccccc}{x}_1& 0& 0& x_2& x_3& 0\\ 0& x_2& 0& x_1& 0& x_3\\ 0& 0& x_3& 0& x_1& x_2\end{array}.\end{array}$$

Regression function matrix:$$\begin{array}{c}\text{(13)}& {\mathbf{Y}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i,{\dot{\sigma }}_{ri},{\ddot{\sigma }}_{ri}={\mathbf{G}}^{-T}{\mathbf{\sigma }}_i\mathbf{L}{\mathbf{G}}^{-1}{\mathbf{\sigma }}_i{\ddot{\sigma }}_{ri}-{\mathbf{G}}^{-T}{\mathbf{\sigma }}_i\mathbf{L}{\mathbf{G}}^{-1}{\mathbf{\sigma }}_i\dot{G}{\mathbf{\sigma }}_i{\mathbf{G}}^{-1}{\mathbf{\sigma }}_i{\dot{\sigma }}_{ri}\\ +{\mathbf{G}}^{-T}{\mathbf{\sigma }}_i{{\mathbf{G}}^{-1}{\mathbf{\sigma }}_i{\dot{\sigma }}_{ri}}^{\times }\mathbf{L}{\mathbf{G}}^{-1}{\mathbf{\sigma }}_i{\dot{\sigma }}_i.\end{array}$$

Let the triggering time sequence of follower i be $t_0^i,t_1^i,\dots ,t_k^i,\dots$, where $k=\mathrm{0,1},\dots ,$ and $t_0^i=0$. $t_k^i$ denotes the kth event time of the ith satellite.

Two types of state errors are defined to facilitate the design of the triggering condition:$$\begin{array}{c}\text{(14)}& {\mathbf{e}}_{si}{t}^i={\mathbf{s}}_i{t}_k^i-{\mathbf{s}}_i{t}^i;t^i\in t_k^i,t_{k+1}^i,{\mathbf{e}}_{Ji}{t}^i={\mathbf{Y}}_i{t}_k^i{\widehat{\Theta }}_i{t}_k^i-{\mathbf{Y}}_i{t}^i{\widehat{\Theta }}_i{t}^i,\end{array}$$where ${\widehat{\Theta }}_i{t}^i$ is the estimation of ${\Theta }_i$ at time $t^i$.

According to Reference [35], the triggering condition can be defined as follows:$$\begin{array}{}\text{(15)}& {\mathbf{e}}_{Ji}{t}^i+{\lambda }_{\max }{\mathbf{K}}_i{\mathbf{e}}_{\text{si}}{t}^i-\frac{{\gamma }_i}{2}{\lambda }_{\min }{\mathbf{K}}_i{\mathbf{s}}_i{t}^i-{\mu }_i{t}^i\ge 0,\end{array}$$where ${\mathbf{K}}_i$ is the symmetric positive-definite matrix, $0<{\gamma }_i<1$, ${\mu }_i{t}^i={\rho }_i\sqrt{{\lambda }_{\min }{\mathbf{K}}_i}\exp -{\nu }_i{t}^i$, ${\rho }_i>0$, and $0<{\nu }_i<1$.

Remark 1.

${\mu }_i{t}^i$ is a time-dependent function. The introduction of ${\mu }_i{t}^i$ can completely avoid the Zeno behaviour. The initial attitude and angular velocity ${\sigma }_i{t}_0^i$, ${\omega }_i{t}_0^i$, $i\in F$ of followers and attitude and angular velocity ${\sigma }_i{t}_0^i$, ${\omega }_i{t}_0^i$, $i\in L$ of leaders are given at initial time $t_0^i$. The system continuously monitors the auxiliary state ${\mathbf{s}}_i{t}^i$ and uncertain parameter ${\widehat{\Theta }}_i{t}^i$ of each follower. Once the triggering condition of satellite i is satisfied, that is, $f_i{\mathbf{e}}_{\text{si}}{t}^i,{\mathbf{e}}_{\text{Ji}}{t}^i\ge 0$, the event of satellite i is triggered immediately, triggering time is $t^i=t_k^i$, state information is sampled, control law is computed, and actuators are updated. The state errors ${\mathbf{e}}_{\text{Ji}}{t}^i$ and ${\mathbf{e}}_{si}{t}^i$ are reset to 0. In nontriggering time, the controller uses state information of the last triggering instant, i.e., for $t^i\in t_k^i,t_{k+1}^i$, ${\mathbf{u}}_i{t}^i={\mathbf{u}}_i{t}_k^i$. Control law is not computed until the next event is triggered, and thus the control input is piecewise constant. Under the action of the event-triggered attitude containment controller, the attitude of followers asymptotically converges to the convex hull formed by the leaders’ orientation.

In this paper, the event-triggered distributed attitude control algorithm could be written in the following form:$$\begin{array}{}\text{(16)}& \begin{array}{c}{\mathbf{u}}_i{t}_k^i=h_i{\mathbf{s}}_i{t}_k^i,{\mathbf{Y}}_i{t}_k^i{\widehat{\Theta }}_i{t}_k^i,\\ {\mathbf{u}}_i{t}^i=\mathbf{u}{t}_k^i,\hspace{1em}{t}^i\in t_k^i,t_{k+1}^i.\end{array}\end{array}$$

It is noteworthy that the control law does not need be computed until the next event.

Attitude dynamics equation of followers could be rewritten in the following form:$$\begin{array}{}\text{(17)}& {\mathbf{M}}_i{\mathbf{\sigma }}_i{\dot{s}}_i+{\mathbf{C}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i{\mathbf{s}}_i={\mathbf{u}}_i+{\mathbf{d}}_i-{\mathbf{Y}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i,{\ddot{\sigma }}_r,{\dot{\sigma }}_r{\Theta }_i-{\mathbf{\epsilon }}_i,i=1,\dots ,N.\end{array}$$

Correspondingly, triggering time is defined as$$\begin{array}{}\text{(18)}& t_{k+1}^i=\inf t^i>t_k^i|f_i{\mathbf{e}}_{\text{si}}{t}^i,{\mathbf{e}}_{\text{Ji}}{t}^i\ge 0.\end{array}$$

Once $t^i=t_k^i$, the event is triggered, state information is sampled, control law is computed and actuators are updated for satellite i, and the corresponding triggering function is less than 0 again.

The event-triggered control scheme is shown in Figure 1. The scenarios we consider include sampling time and triggering time; the sampling time is determined by the fixed clock frequency of the sensor, and the latter is controlled by the triggering condition based on the state. It is noteworthy that the time interval between two consecutive events is usually variable and can be equal to the minimum interval (that is, the sampling period). When the interval between two consecutive events is a fixed sampling period, the system degenerates into a traditional periodic sampling control. For the triggering conditions in this paper, continuous detection is necessary, which may require additional equipment and is a waste of communication and computing resources.

[figure omitted; refer to PDF]

3.2. Event-Triggered Adaptive Sliding Mode Attitude Containment Control

For each follower i, the event-triggered adaptive sliding mode control protocol is designed as$$\begin{array}{}\text{(19)}& {\mathbf{u}}_i{t}^i=-{\mathbf{K}}_i{\mathbf{s}}_i{t}_k^i+{\mathbf{Y}}_i{t}_k^i{\widehat{\Theta }}_i{t}_k^i-{\widehat{k}}_i{t}_k^i\mathrm{sgn}{\mathbf{s}}_i{t}_k^i,\hspace{1em}i\in F,t^i\in t_k^i,t_{k+1}^i.\end{array}$$

Adaptation law of ${\widehat{\Theta }}_i{t}^i$ and ${\widehat{k}}_i$ is defined as follows:$$\begin{array}{}\text{(20)}& {\dot{\widehat{\Theta }}}_{i}t=-{\mathbf{\Lambda }}_i{\mathbf{Y}}_i^T{t}^i{\mathbf{s}}_i{t}^i,\text{(21)}& {\dot{\widehat{k}}}_i{t}^i={\Gamma }_i{{\mathbf{s}}_i{t}^i}_1,\end{array}$$where ${\mathrm{\Lambda }}_i$ is a symmetric positive-definite matrix and ${\mathrm{\Gamma }}_i$ is a positive constant.

According to (9) and (10), system (17) can be rewritten as$$\begin{array}{}\text{(22)}& {\mathbf{M}}_i{\mathbf{\sigma }}_i{\dot{s}}_i{t}^i+{\mathbf{C}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i{\mathbf{s}}_i{t}^i=-{\mathbf{K}}_i{\mathbf{s}}_i{t}_k^i+{\mathbf{Y}}_i{t}_k^i{\widehat{\Theta }}_i{t}_k^i-{\widehat{k}}_i\mathrm{sgn}{\mathbf{s}}_i{t}_k^i+{\mathbf{d}}_i-{\mathbf{Y}}_i{t}^i{\Theta }_i-{\mathbf{\epsilon }}_i,i\in F,t^i\in t_k^i,t_{k+1}^i.\end{array}$$

The following conclusion shows that event-triggered adaptive sliding mode attitude control protocol (19), adaption laws (20) and (21), and triggering condition (15) can realize attitude containment control of microsatellite cluster system (5) under directed information topology. When $t⟶\infty$, the attitude and angular velocity of followers asymptotically enter into the convex hull formed by leaders, that is, ${\lim }_{t⟶\infty }{\sigma }_{F}t⟶-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\sigma }_L$ and ${\lim }_{t⟶\infty }{\omega }_{F}t⟶0$. Moreover, it can be further proved that the Zeno behaviour will not occur in the microsatellite cluster system under the action of the proposed event-triggered attitude control strategy.

Theorem 1.

If Assumptions 1 and 2 hold, under the action of triggering condition (15), event-triggered adaptive sliding mode attitude control protocol (19), and adaption laws (20) and (21), the attitude of followers asymptotically enters into the convex hull formed by leaders’ orientation, that is, ${\lim }_{t⟶\infty }{\sigma }_{F}t⟶-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\sigma }_L$ and ${\lim }_{t⟶\infty }{\omega }_{F}t⟶0$, and then the sufficient and necessary condition is that Assumption 3 holds.

(1) Proof. Sufficiency: The proof includes the following two consecutive steps. (i) State trajectory asymptotically converges to the sliding surface, that is, ${\lim }_{t^i⟶\infty }{\mathbf{s}}_i=0$. (ii) In the case of ${\mathbf{s}}_i=0$, the state of followers will reach ${\lim }_{t⟶\infty }{\sigma }_{F}t⟶-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\sigma }_L$, ${\lim }_{t⟶\infty }{\omega }_{F}t⟶0$ asymptotically.

First, consider the following Lyapunov candidate:$$\begin{array}{}\text{(23)}& V_1=\frac{1}{2}{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{M}}_i{\mathbf{\sigma }}_i}{\mathbf{s}}_i+\frac{1}{2}{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^T{\mathbf{\Lambda }}_i^{-1}{\tilde{\Theta }}_i}+\frac{1}{2}{\displaystyle \sum _{i=1}^N{k_i-{\widehat{k}}_i}^2{\Gamma }_i^{-1}},\end{array}$$where $k_i\ge {\epsilon }_M+d_M$ and the estimation error is$$\begin{array}{}\text{(24)}& {\tilde{\Theta }}_i={\Theta }_i-{\widehat{\Theta }}_i.\end{array}$$

Taking the derivative of $V_1$, we can obtain$$\begin{array}{c}\text{(25)}& {\dot{V}}_1=\frac{1}{2}{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\dot{M}}_i{\mathbf{\sigma }}_i{\mathbf{s}}_i}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{M}}_i{\mathbf{\sigma }}_i{\dot{s}}_i}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^T{\mathbf{\Lambda }}_i^{-1}{\dot{\tilde{\Theta }}}_i}-{\displaystyle \sum _{i=1}^N{k}_i-{\widehat{k}}_i{\Gamma }_i^{-1}{\dot{\widehat{k}}}_i}={\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T\frac{1}{2}{\dot{M}}_i{\mathbf{\sigma }}_i-{\mathbf{C}}_i{\mathbf{\sigma }}_i,{\dot{\sigma }}_i}{\mathbf{s}}_i-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{s}}_i{t}_k^i}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{Y}}_i{t}_k^i{\widehat{\Theta }}_i{t}_k^i}\\ -{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\widehat{k}}_i\mathrm{sgn}{\mathbf{s}}_i{t}_k^i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{Y}}_i{\Theta }_i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{\epsilon }}_i}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{d}}_i}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^T{\mathbf{Y}}_i^T{\mathbf{s}}_i}-{\displaystyle \sum _{i=1}^N{k}_i-{\widehat{k}}_i{\mathbf{s}}_i}.\end{array}$$

According to property (c) of the Euler–Lagrange equation, $1/2{\dot{M}}_i{\sigma }_i-{\mathbf{C}}_i{\sigma }_i$ is a skew-symmetric matrix; substituting (24) into (25), we can obtain that$$\begin{array}{c}\text{(26)}& {\dot{V}}_1=-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{s}}_i{t}_k^i}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{Y}}_i{t}_k^i{\widehat{\Theta }}_i{t}_k^i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\widehat{k}}_i\mathrm{sgn}{\mathbf{s}}_i{t}_k^i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{Y}}_i{\tilde{\Theta }}_i+{\widehat{\Theta }}_i}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^T{\mathbf{Y}}_i^T{\mathbf{s}}_i}-{\displaystyle \sum _{i=1}^N{k}_i{\mathbf{s}}_i}+{\displaystyle \sum _{i=1}^N{\widehat{k}}_i{\mathbf{s}}_i}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{d}}_i-{\mathbf{\epsilon }}_i}\\ =-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{s}}_i{t}_k^i}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{Y}}_i{t}_k^i{\widehat{\Theta }}_i{t}_k^i-{\mathbf{Y}}_i{\widehat{\Theta }}_i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{Y}}_i{\tilde{\Theta }}_i}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^T{\mathbf{Y}}_i^T{\mathbf{s}}_i}\\ -{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\widehat{k}}_i\mathrm{sgn}{\mathbf{s}}_i{t}_k^i}+{\displaystyle \sum _{i=1}^N{\widehat{k}}_i{\mathbf{s}}_i}-{\displaystyle \sum _{i=1}^N{k}_i{\mathbf{s}}_i+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{d}}_i-{\mathbf{\epsilon }}_i}.}\end{array}$$

Substituting definitions of state errors (14) into (26), it can be obtained that$$\begin{array}{c}\text{(27)}& {\dot{V}}_1=-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{s}}_i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{e}}_{\text{si}}}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{e}}_{Ji}}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\widehat{k}}_i\mathrm{sgn}{\mathbf{s}}_i{t}_k^i}+{\displaystyle \sum _{i=1}^N{\widehat{k}}_i{\mathbf{s}}_i}-{\displaystyle \sum _{i=1}^N{k}_i{\mathbf{s}}_i+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{d}}_i-{\mathbf{\epsilon }}_i}.}\end{array}$$

According to Assumption 2, we can get ${\mathbf{s}}_i^T{\mathbf{d}}_i-{\epsilon }_i\le {\mathbf{s}}_i{\mathbf{d}}_i+{\mathbf{s}}_i{\epsilon }_i\le k_i{\mathbf{s}}_i$. Thus,$$\begin{array}{}\text{(28)}& {\dot{V}}_1\le -{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{s}}_i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{e}}_{\text{si}}}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{e}}_{Ji}}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\widehat{k}}_i\mathrm{sgn}{\mathbf{s}}_i{t}_k^i}+{\displaystyle \sum _{i=1}^N{\widehat{k}}_i{\mathbf{s}}_i}.\end{array}$$

In the time interval $t^i\in t_k^i,t_{k+1}^i$, we can know that the system trajectories start from a region where $\mathrm{sgn}{\mathbf{s}}_i{t}_k^i=\mathrm{sgn}{\mathbf{s}}_i{t}^i$. And ${\mathbf{K}}_i$ is a symmetric positive-definite matrix; then, the upper bound for ${\dot{V}}_1$ can be expressed as$$\begin{array}{c}\text{(29)}& {\dot{V}}_1\le -{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{s}}_i}-{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{K}}_i{\mathbf{e}}_{\text{si}}}+{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{e}}_{\text{Ji}}}\le -{\displaystyle \sum _{i=1}^N{\lambda }_{\min }{\mathbf{K}}_i{{\mathbf{s}}_i}^2}+{\displaystyle \sum _{i=1}^N{\lambda }_{\max }{\mathbf{K}}_i{\mathbf{s}}_i{\mathbf{e}}_{\text{si}}+}{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i{\mathbf{e}}_{\text{Ji}}}.\end{array}$$

It is worth noting that triggering condition (15) guarantees ${\mathbf{e}}_{\text{Ji}}+{\lambda }_{\max }{\mathbf{K}}_i{\mathbf{e}}_{\text{si}}\le {\gamma }_i/2{\lambda }_{\min }{\mathbf{K}}_i{\mathbf{s}}_i+{\mu }_{i}t$ holds throughout the evolution of cluster system (5). Thus, we can get$$\begin{array}{}\text{(30)}& {\dot{V}}_1\le -{\displaystyle \sum _{i=1}^N{\lambda }_{\min }{\mathbf{K}}_i{{\mathbf{s}}_i}^2}+{\displaystyle \sum _{i=1}^N\frac{{\gamma }_i}{2}{\lambda }_{\min }{\mathbf{K}}_i{{\mathbf{s}}_i}^2}+{\displaystyle \sum _{i=1}^N\sqrt{{\lambda }_{\min }{\mathbf{K}}_i}{\rho }_i{e}^{-{\nu }_{i}t}{\mathbf{s}}_i}.\end{array}$$

Because $\forall x,y\in ℝ$, $xy\le {\gamma }_i/2x^2+1/2{\gamma }_i{y}^2$ holds. For $0<{\gamma }_i<1$, analysing the right-hand side of (24), the upper bound of ${\dot{V}}_1$ can be expressed as$$\begin{array}{}\text{(31)}& {\dot{V}}_1\le -{\displaystyle \sum _{i=1}^{N}1-{\gamma }_i{\lambda }_{\min }{\mathbf{K}}_i{{\mathbf{s}}_i}^2}+{\displaystyle \sum _{i=1}^N\frac{{\rho }_i^2}{2{\gamma }_i}{e}^{-2{\nu }_{i}t}}.\end{array}$$

Integrating both sides of (25), for any $t^i>0$, we can get$$\begin{array}{}\text{(32)}& V_{1}t+{\displaystyle \sum _{i=1}^{N}1-{\gamma }_i{\lambda }_{\min }{\mathbf{K}}_i{\displaystyle {\int }_0^{t^i}{{\mathbf{s}}_i\tau }^2\text{d}\tau }}\le V_{1}0+{\displaystyle \sum _{i=1}^N\frac{{\rho }_i^2}{4{\gamma }_i{\nu }_i}.}\end{array}$$

Because $0<{\gamma }_i<1$, ${\displaystyle {\sum }_{i=1}^{N}1-{\gamma }_i{\lambda }_{\min }{\mathbf{K}}_i{\displaystyle {\int }_0^{t^i}{{\mathbf{s}}_i\tau }^2\text{d}\tau }}>0$, we obtain that $V_{1}t\le V_{1}0+{\displaystyle {\sum }_{i=1}^N{\rho }_i^2/4{\gamma }_i{\nu }_i}$, and thus $V_1$ is bounded. According to (23), ${\mathbf{s}}_i$ and ${\tilde{\Theta }}_{i}t$ are bounded for every $i=1,\dots ,N$. When Assumption 3holds, we can know ${\mathbf{L}}_F^{-1}$ exists according to Lemma 1. Then, (10) can be rewritten as follows:$$\begin{array}{}\text{(33)}& {\dot{\sigma }}_F=-\alpha {\mathbf{L}}_F\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_F-\alpha {\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_L+{\mathbf{s}}_F.\end{array}$$

Let ${\overline{\sigma }}_F={\sigma }_F+{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\sigma }_L$, and equation (33) can be written as$$\begin{array}{}\text{(34)}& {\dot{\overline{\sigma }}}_F=-\alpha {\mathbf{L}}_F\otimes {\mathbf{I}}_3{\overline{\sigma }}_F+{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F.\end{array}$$

According to Lemma 2, all eigenvalues of ${\mathbf{L}}_F$ have positive real parts. According to Assumption 1, ${\dot{\sigma }}_L$ is bounded and ${\lim }_{t⟶\infty }{\dot{\sigma }}_L⟶0$. It thus follows that when ${\mathbf{s}}_F=0$, (34) is globally exponentially stable at the origin ${\overline{\sigma }}_F=0$. We can conclude from Lemma 3 that cluster system (34) is input-to-state stable with respect to the input ${\mathbf{s}}_F$ and the state ${\overline{\sigma }}_F$. According to Lemma 3, because input ${\mathbf{s}}_F$ is bounded, state ${\overline{\sigma }}_F$ is bounded. Thus, we can obtain that ${\dot{\overline{\sigma }}}_F$ is bounded from (34). Equation (9) can be written as ${\dot{\sigma }}_r=-\alpha {\mathbf{L}}_F\otimes {\mathbf{I}}_3{\overline{\sigma }}_F$, and thus ${\dot{\sigma }}_r$ is bounded. It also follows from (10) that ${\dot{\sigma }}_F$ is bounded. Differentiating (9), we can get that ${\ddot{\sigma }}_r$ is bounded.

According to property (b) of the Euler–Lagrange equation, ${\mathbf{M}}_i{\sigma }_i$, ${\mathbf{C}}_i{\sigma }_i,{\dot{\sigma }}_i{\dot{\sigma }}_r$, ${\mathbf{g}}_i{\sigma }_i{\dot{\sigma }}_r$, ${\ddot{\sigma }}_r$, and ${\epsilon }_i$, $i\in F$ are all bounded. Then, according to equation (11), we can obtain that ${\mathbf{Y}}_i{\sigma }_i,{\dot{\sigma }}_i,{\dot{\sigma }}_{ri},{\ddot{\sigma }}_{ri}$ is bounded. Because ${\mathbf{C}}_i{\sigma }_i,{\dot{\sigma }}_i{\mathbf{s}}_i<k_C{\dot{\sigma }}_i{\mathbf{s}}_i$, we can obtain from (17) that ${\dot{s}}_F$ is bounded.

Because $V_1$(t) ≥ 0, and ${\mathbf{s}}_i,{\dot{s}}_i\in L_{\infty }$, then it can be derived from (32) that$$\begin{array}{}\text{(35)}& {\displaystyle \sum _{i=1}^{N}1-{\gamma }_i{\lambda }_{\min }{\mathbf{K}}_i{\displaystyle {\int }_0^t{{\mathbf{s}}_i\tau }^2\text{d}\tau }\le V_{1}0+{\displaystyle \sum _{i=1}^N\frac{{\rho }_i^2}{4{\gamma }_i{\nu }_i}},}\end{array}$$which implies that ${\displaystyle {\int }_0^t{{\mathbf{s}}_i\tau }^2\text{d}\tau }$ is bounded, that is, ${\mathbf{s}}_i\in L_2$.

${\mathbf{s}}_i,{\dot{s}}_i\in L_{\infty }$, ${\mathbf{s}}_i\in L_2$, and thus ${\mathbf{s}}_{i}t$ is uniformly continuous. According to Barbalat’s lemma [51], ${\lim }_{t⟶\infty }{\mathbf{s}}_F⟶0$. Because (34) is input-to-state stable with respect to the input ${\mathbf{s}}_F$ and state ${\overline{\sigma }}_{F}t$, it can be concluded that ${\lim }_{t⟶\infty }{\overline{\sigma }}_{F}t⟶0$. As a result, it follows that ${\lim }_{t⟶\infty }{\sigma }_{F}t⟶-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\sigma }_L$, so similarly ${\lim }_{t⟶\infty }{\dot{\sigma }}_F⟶-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L$; according to Assumption 1, ${\lim }_{t⟶\infty }{\dot{\sigma }}_L⟶0$, and thus ${\lim }_{t⟶\infty }{\dot{\sigma }}_F⟶0$. According to ${\dot{\sigma }}_i=\mathbf{G}{\sigma }_i{\omega }_i$, we can obtain that ${\lim }_{t⟶\infty }{\omega }_F⟶0$. The attitude of followers asymptotically converges to the convex hull formed by leaders’ orientation.

According to Reference [52], we see that the system trajectories are attracted towards the sliding manifold as long as $\mathrm{sgn}{\mathbf{s}}_i{t}_k^i=\mathrm{sgn}{\mathbf{s}}_i{t}^i$. This holds for all triggering instants whenever $\mathrm{sgn}{\mathbf{s}}_i{t}_k^i=\mathrm{sgn}{\mathbf{s}}_i{t}^i$. As a result, the trajectories reach the neighbourhood of the sliding manifold due to ${\lim }_{t⟶\infty }{\mathbf{s}}_F⟶0$ and enter the region where the sign of ${\mathbf{s}}_i$ changes before next triggering occurs. So, in the time interval, the decrement of $V_1$ cannot be guaranteed because zero crossing of ${\mathbf{s}}_i$ occurs, and hence the trajectory increases. However, the sliding trajectories are ultimately bounded within this region. Therefore, the ultimate band is the region where $\mathrm{sgn}{\mathbf{s}}_i{t}_k^i\ne \mathrm{sgn}{\mathbf{s}}_i{t}^i$. The size of this band can be calculated by finding the maximum deviation of sliding trajectory with zero crossing.

This ultimate region can now be derived as follows. We know that for any $t^i\in t_k^i,t_{k+1}^i$,$$\begin{array}{}\text{(36)}& {\mathbf{s}}_i{t}_k^i-{\mathbf{s}}_i{t}^i={\mathbf{e}}_{\text{si}}{t}^i\le \frac{1}{{\lambda }_{\max }{\mathbf{K}}_i}\frac{{\gamma }_i}{2}{\lambda }_{\min }{\mathbf{K}}_i{\mathbf{s}}_{i}t+{\mu }_{i}t-{\mathbf{e}}_{Ji}t.\end{array}$$

This gives the maximum deviation of sliding variable from its immediate sampled value. Then, the maximum value of band can be obtained for the case ${\mathbf{s}}_i{t}_k^i=0$ and is given in (15).

According to (32), $V_{1}t\le V_{1}0+{\displaystyle {\sum }_{i=1}^N{\rho }_i^2/4{\gamma }_i{\nu }_i}$; then, substituting (23) into it, we can obtain$$\begin{array}{}\text{(37)}& {\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{M}}_i{\mathbf{s}}_i}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^T{\mathbf{\Lambda }}_i^{-1}{\tilde{\Theta }}_i}+{\displaystyle \sum _{i=1}^N{\tilde{k}}_i^2{\Gamma }_i^{-1}}\le {\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^{T}0{\mathbf{M}}_{i}0{\mathbf{s}}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^{T}0{\mathbf{\Lambda }}_i^{-1}{\tilde{\Theta }}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{k}}_i^{2}0{\Gamma }_i^{-1}}+{\displaystyle \sum _{i=1}^N\frac{{\rho }_i^2}{2{\gamma }_i{\nu }_i}}.\end{array}$$

Because ${\displaystyle {\sum }_{i=1}^N{\tilde{\Theta }}_i^T{\mathrm{\Lambda }}_i^{-1}{\tilde{\Theta }}_i}+{\displaystyle {\sum }_{i=1}^N{\tilde{k}}_i^2{\mathrm{\Gamma }}_i^{-1}}\ge 0$, we can obtain that$$\begin{array}{}\text{(38)}& {\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{M}}_i{\mathbf{s}}_i}\le {\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^{T}0{\mathbf{M}}_{i}0{\mathbf{s}}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^{T}0{\mathbf{\Lambda }}_i^{-1}{\tilde{\Theta }}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{k}}_i^{2}0{\Gamma }_i^{-1}}+{\displaystyle \sum _{i=1}^N\frac{{\rho }_i^2}{2{\gamma }_i{\nu }_i}}.\end{array}$$

Further,$$\begin{array}{}\text{(39)}& {\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^T{\mathbf{s}}_i}\le {{\lambda }_{\min }{\mathbf{M}}_i}^{-1}{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^{T}0{\mathbf{M}}_{i}0{\mathbf{s}}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^{T}0{\mathbf{\Lambda }}_i^{-1}{\tilde{\Theta }}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{k}}_i^{2}0{\Gamma }_i^{-1}}+{\displaystyle \sum _{i=1}^N\frac{{\rho }_i^2}{2{\gamma }_i{\nu }_i}}.\end{array}$$

That is,$$\begin{array}{c}\text{(40)}& {\mathbf{s}}_F\le \sqrt{{{\lambda }_{\min }{\mathbf{M}}_i}^{-1}{\displaystyle \sum _{i=1}^N{\mathbf{s}}_i^{T}0{\mathbf{M}}_{i}0{\mathbf{s}}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{\Theta }}_i^{T}0{\mathbf{\Lambda }}_i^{-1}{\tilde{\Theta }}_{i}0}+{\displaystyle \sum _{i=1}^N{\tilde{k}}_i^{2}0{\Gamma }_i^{-1}}+{\displaystyle \sum _{i=1}^N\frac{{\rho }_i^2}{2{\gamma }_i{\nu }_i}}}=s_M.\end{array}$$

Second, consider the following Lyapunov function candidate:$$\begin{array}{}\text{(41)}& V_2={\overline{\sigma }}_F^T\mathbf{D}\otimes {\mathbf{I}}_3{\overline{\sigma }}_F.\end{array}$$

Taking the derivative of V₂ along (34) gives$$\begin{array}{c}\text{(42)}& {\dot{V}}_2={\dot{\overline{\sigma }}}_F^T\mathbf{D}\otimes {\mathbf{I}}_3{\overline{\sigma }}_F+{\overline{\sigma }}_F^T\mathbf{D}\otimes {\mathbf{I}}_3{\dot{\overline{\sigma }}}_F=2{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F\mathbf{D}\otimes {\mathbf{I}}_3{\overline{\sigma }}_F-\alpha {\overline{\sigma }}_F^T\mathbf{D}{\mathbf{L}}_F+{\mathbf{L}}_F^T\mathbf{D}{\overline{\sigma }}_F\\ \le \frac{2\max d_i{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F}{\sqrt{\min d_i}}\sqrt{V_2}-\frac{\alpha {\lambda }_{\min }Q}{\max d_i}{V}_2.\end{array}$$

There exists a diagonal matrix $D=\text{diag}{d}_1,\dots ,d_N$ with $d_i>0$, $\forall i=1,\cdots ,N$, such that $\mathbf{Q}=\mathbf{D}{\mathbf{L}}_F+{\mathbf{L}}_F^T\mathbf{D}$ is symmetric positive definite. Because $\forall x,y\in ℝ$, $2xy\le {\gamma }_i{x}^2+1/{\gamma }_i{y}^2$ holds. Then, (42) can be written as$$\begin{array}{c}\text{(43)}& {\dot{V}}_2\le 2\sqrt{\frac{\alpha {\lambda }_{\min }Q}{2\max d_i}{V}_2}\cdot \sqrt{\frac{2\max d_i}{\alpha {\lambda }_{\min }Q}}\cdot \frac{\max d_i{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F}{\sqrt{\min d_i}}-\frac{\alpha {\lambda }_{\min }Q}{\max d_i}{V}_2\le \frac{2\max {d_i}^3{{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F}^2}{\alpha {\lambda }_{\min }Q\min d_i}-\frac{\alpha {\lambda }_{\min }Q}{2\max d_i}{V}_2.\end{array}$$

Furthermore,$$\begin{array}{}\text{(44)}& {\dot{V}}_2+\frac{\alpha {\lambda }_{\min }Q}{2\max d_i}{V}_2\le \frac{\max {d_i}^3{{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F}^2}{\alpha {\lambda }_{\min }Q\min d_i}.\end{array}$$

It can be obtained that$$\begin{array}{}\text{(45)}& V_{2}t\le V_{2}0e^{-\alpha {\lambda }_{\min }Q/2\max d_{i}t}+\frac{4{\max d_i}^4{{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F}^{2}1-e^{-\alpha {\lambda }_{\min }Q/2\max d_{i}t}}{{\alpha }^2{{\lambda }_{\min }Q}^2\min d_i}.\end{array}$$

Let $c_M=2{\max d_i}^2/\alpha {\lambda }_{\min }Q\min d_i$. Because ${\lim }_{t⟶\infty }{e}^{-\alpha {\lambda }_{\min }Q/2\max d_{i}t}⟶0$, then according to (41), we can get$$\begin{array}{c}\text{(46)}& \underset{t⟶\infty }{\lim \sup }{\overline{\sigma }}_F\le c_M\underset{t⟶\infty }{\lim \sup }{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F\le c_M\underset{t⟶\infty }{\lim \sup }{\mathbf{s}}_F+c_M\underset{t⟶\infty }{\lim \sup }-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L.\end{array}$$

According to the aforementioned analysis, ${\lim }_{t⟶\infty }{\mathbf{s}}_F⟶0$. According to Assumption 1,${\dot{\sigma }}_L$ is bounded, and thus $\lim {\sup }_{t⟶\infty }-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L$ is bounded. Moreover, we assume that ${\lim }_{t⟶\infty }{\dot{\sigma }}_L⟶0$, and thus we can obtain that ${\lim }_{t⟶\infty }{\overline{\sigma }}_F⟶0$.

Substituting (41) into equation (45), we can get$$\begin{array}{c}\text{(47)}& {\overline{\sigma }}_{F}t\le \sqrt{\frac{V_{2}0}{\min d_i}+c_M^2{{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}\otimes {\mathbf{I}}_3{\dot{\sigma }}_L+{\mathbf{s}}_F}^2}\le \sqrt{\frac{\max d_i{\overline{\sigma }}_F^{2}0}{\min d_i}+c_M^2{{\lambda }_{\max }{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}{\dot{\sigma }}_{\text{LM}}+s_M}^2}\\ ={\overline{\sigma }}_{\text{FM}}.\end{array}$$

From definitions of ${\sigma }_F$ and ${\dot{\sigma }}_F$, we can also obtain$$\begin{array}{c}\text{(48)}& {\mathbf{\sigma }}_F\le {\overline{\sigma }}_F+{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_L\le {\overline{\sigma }}_{\text{FM}}+{\lambda }_{\max }{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}{\sigma }_{\text{LM}}\\ ={\sigma }_{\text{FM}},\\ {\dot{\sigma }}_F\le \alpha {\mathbf{L}}_F\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_F+\alpha {\mathbf{L}}_{\text{FL}}\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_L+{\mathbf{s}}_F\\ \le \alpha {\lambda }_{\max }{\mathbf{L}}_F{\sigma }_{\text{FM}}+\alpha {\lambda }_{\max }{\mathbf{L}}_{\text{FL}}{\sigma }_{\text{LM}}+s_M\\ ={\dot{\sigma }}_{\text{FM}}.\end{array}$$

Therefore, for any bounded initial states and $t\ge 0$, the states of (5) using equations (15) and (19)–(21) will always lie in the compact set ${\sigma }_{i}t,{\dot{\sigma }}_{i}t|{\sigma }_{F}t\le {\sigma }_{\text{FM}},{\dot{\sigma }}_{F}t\le {\dot{\sigma }}_{\text{FM}}$.

(2) Proof. Necessity: Necessity is proven by contradiction. If Assumption 3 does not hold, then there must exist parts of followers which cannot receive information from any leader directly or indirectly (through other followers). According to Reference [53], the followers can be divided into two subsets, namely, one set with the followers that can receive the information from the leaders directly or indirectly, denoted by $F_1$, and the follower set, which cannot receive information from leaders denoted by $F_2$. The system Laplacian matrix L can also be in the following form:$$\begin{array}{}\text{(49)}& \mathbf{L}=\begin{array}{ccc}{\mathbf{L}}_{F_1{F}_1}& {\mathbf{L}}_{F_1{F}_2}& {\mathbf{L}}_{F_{1}L}\\ 0& {\mathbf{L}}_{F_2{F}_2}& 0\\ 0& 0& 0\end{array}.\end{array}$$

Let the attitude of $F_1$ and $F_2$ be ${\sigma }_{F_1}$ and ${\sigma }_{F_2}$. Auxiliary variables ${\mathbf{s}}_{F_1}$,${\mathbf{s}}_{F_2}$ are given with respect to ${\sigma }_{F_1}$ and ${\sigma }_{F_2}$. Then, we can obtain$$\begin{array}{c}\text{(50)}& {\dot{\sigma }}_{F_1}=-\alpha {\mathbf{L}}_{F_1{F}_1}\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_{F_1}-\alpha {\mathbf{L}}_{F_1{F}_2}\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_{F_2}-\alpha {\mathbf{L}}_{F_{1}L}\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_L+{\mathbf{s}}_{F_1},{\dot{\sigma }}_{F_2}=-\alpha {\mathbf{L}}_{F_2{F}_2}\otimes {\mathbf{I}}_3{\mathbf{\sigma }}_{F_2}+{\mathbf{s}}_{F_2}.\end{array}$$

Because ${\mathbf{s}}_{F_1}$ and ${\mathbf{s}}_{F_2}$ are proved to converge to zero, the final states of ${\sigma }_{F_1}$ and ${\sigma }_{F_2}$ will depend on the equilibrium points. We can get from equation (50) that the trajectories of the followers in $F_2$ are independent of the leaders, and thus these followers cannot always converge to the convex hull formed by the leaders from any initial state. For the followers in $F_1$, Assumption 3 holds. From the analysis of the sufficiency in this proof, the motion of followers in $F_1$ may also be affected by its neighbours including leaders and followers in $F_2$, and thus there might exist some followers in $F_1$ that cannot converge to the convex hull formed by the leaders.

The proof is completed.

Remark 2.

In order to avoid the chattering problem caused by the sign function, controller (19) can be substituted by saturation function$$\begin{array}{}\text{(51)}& \text{sat}x,\delta =\begin{array}{cc}1,& x>\delta ,\\ \frac{x}{\delta },& x\le \delta ,\\ -1,& x<-\delta ,\end{array}\end{array}$$where $\delta$ is a small positive constant.

Remark 3.

The attitude containment control problem with multiple leaders has been studied in the context. Obviously, if there exists one leader in control protocol (19), the multiple leader-follower coordinated attitude containment control problem becomes the coordinated attitude tracking problem with single leader.

3.3. The Absence of Zeno Behaviour

Zeno behaviour means the minimum time interval between two consecutive events is 0 and the event triggers infinite times in a finite time, which is forbidden in control tasks. Let $T$ denote interevent time interval of event-triggered attitude control protocol (19)–(21) and triggering condition (15). $T$ needs to be strictly positive to exclude the Zeno behaviour.

${\lim }_{t^i⟶\infty }{\mathbf{s}}_i⟶0$ is satisfied in the context. But at $t_{k+1}^i$, there may exist the case of ${\mathbf{s}}_i{t}^i=0,{\dot{s}}_i{t}^i\ne 0$. The attitude containment control is not reached at this moment. Therefore, the event is triggered in the following two cases:

(1) If ${\mathbf{s}}_i{t}_{k+1}^i\ne 0$, the interevent time interval of this case is denoted as $T_1$ and triggering condition is ${\mathbf{e}}_{\text{Ji}}{t}_{k+1}^i+{\lambda }_{\max }{\mathbf{K}}_i{\mathbf{e}}_{\text{si}}{t}_{k+1}^i\ge {\gamma }_i/2{\lambda }_{\min }{\mathbf{K}}_i{\mathbf{s}}_i{t}_{k+1}^i+{\mu }_i{t}_{k+1}^i$.

(2) If ${\mathbf{s}}_i{t}_{k+1}^i=0$, the interevent time interval of this case is denoted as $T_2$ and triggering condition is ${\mathbf{e}}_{Ji}{t}_{k+1}^i+{\lambda }_{\max }{\mathbf{K}}_i{\mathbf{e}}_{si}{t}_{k+1}^i\ge {\mu }_i{t}_{k+1}^i$.

Comparing the aforementioned two cases, we can know that for any ${\mathbf{s}}_i{t}_{k+1}^i\ne 0$, there exists ${\mathbf{s}}_i{t}_{k+1}^i>0$, so it can be concluded that case (1) takes longer time, which implies $T_2<T_1$. We just need to prove that there exists strictly positive $T_2$ to exclude the Zeno behaviour.

Theorem 2.

If conditions of Theorem 1 hold, then the interevent time interval $T$ of triggering condition (15) has positive lower bound.

Proof.

According to Reference [35], for any $k\ge 0$ and $t^i\in t_k^i,t_{k+1}^i$, $i=1,\dots ,N$, the derivative of ${\mathbf{e}}_{\text{si}}{t}^i$ with respect to time satisfies$$\begin{array}{}\text{(52)}& \frac{\text{d}}{\text{d}t}{\mathbf{e}}_{\text{si}}{t}^i\le {\dot{s}}_i{t}^i,\end{array}$$where ${\dot{s}}_i{t}^i$ has been proved to be bounded. Let positive constant $B_s$ denote the maximum of ${\dot{s}}_i{t}^i$, and we obtain $\text{d}/\text{d}t{\mathbf{e}}_{\text{si}}{t}^i\le B_s$.

Similar to equation (42), the derivative of ${\mathbf{e}}_{Ji}{t}^i$ satisfies$$\begin{array}{}\text{(53)}& \frac{\text{d}}{\text{d}t}{\mathbf{e}}_{\text{Ji}}{t}^i\le \frac{\text{d}}{\text{d}t}{\mathbf{Y}}_i{t}^i{\widehat{\Theta }}_i{t}^i={\dot{Y}}_i{t}^i{\widehat{\Theta }}_i{t}^i+{\mathbf{Y}}_i{t}^i{\dot{\widehat{\Theta }}}_i{t}^i.\end{array}$$

It can be seen from the Section 3.1 that ${\mathbf{Y}}_i{t}^i$ and ${\widehat{\Theta }}_i{t}^i$ are both bounded; according to Assumption 1 in Reference [35], we know that ${\dot{Y}}_i$ is bounded. Then, because ${\mathbf{Y}}_i{t}^i$ and ${\mathbf{s}}_i{t}^i$ are bounded, ${\dot{\widehat{\Theta }}}_i{t}^i$ is bounded according to adaption update law. Let positive constant $B_J$ denote the upper bound of $\text{d}/\text{d}t{\mathbf{Y}}_i{t}^i{\widehat{\Theta }}_i{t}^i$, and $\text{d}/\text{d}t{\mathbf{e}}_{\text{Ji}}{t}^i\le B_J$ can be obtained.

Let $B=\max B_J,{\lambda }_{\max }{\mathbf{K}}_i{B}_s$; it follows that$$\begin{array}{}\text{(54)}& {\mathbf{e}}_{\text{Ji}}{t}^i+{\lambda }_{\max }{\mathbf{K}}_i{\mathbf{e}}_{\text{si}}{t}^i\le 2{\displaystyle {\int }_{t_k^i}^{t^i}B\text{d}t}=2B{t}^i-t_k^i,\hspace{1em}{t}^i\in t_k^i,t_{k+1}^i.\end{array}$$

According to case (2), the lower bound of interevent time interval $T_2$ can be derived:$$\begin{array}{}\text{(55)}& 2B{T}_2\ge {\rho }_i\sqrt{{\lambda }_{\min }{\mathbf{K}}_i}\exp -{\nu }_i{T}_2,\end{array}$$where B > 0. $T_2$ can get positive solution from equation (55). Thus, there exists time interval between two consecutive events, and the system avoids continuous control. It means that the interevent time interval of triggering condition (15) has positive lower bound.

4. The Influence of Information Topology Design on Followers’ Orientation

Compared with single leader/leaderless case, containment control has lower accuracy requirement for the final state of cluster members. However, in practical space missions, the orientation of satellites in the target area needs to meet certain constraints, such as multiple satellites observing the same orientation simultaneously, the observation field covering the entire target area, and so on.

It indicates in [46] that the steady state of each follower is a convex combination of all leaders’ states it can access, and the combination coefficient is a quantity related to the system Laplacian matrix. It can be concluded that the orientation of followers in the target area is determined by system information topology (including the weights that are assigned to edges).

In this section, approaches from graph theory to investigate influence of information topology on the distributions of followers are presented to provide a reference for orientation design of the microsatellite cluster.

4.1. The Constraints of Leader Reachable Set on Followers’ Orientation

Denote $\mathbf{C}=-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{\text{FL}}$ as the coefficient matrix of steady state of followers with respect to the state of leaders, where $0\le c_{\mathrm{ij}}\le 1$, ${\displaystyle {\sum }_{j=1}^M{c}_{ij}=1}$. It reflects the relationship between system information topology and steady state of followers. According to the definition of reachable set [46], (1) if follower i is reachable from multiple leaders including $N+j$, $0<c_{\mathrm{ij}}<1$, the motion of i is affected by multiple leaders, and its stable state is correspondingly determined by the states of these leaders. (2) If follower i is only reachable from leader $N+j$, $c_{ij}=1$, there exists a directed path from $N+j$ to i, that is, the motion of follower i will be only affected directly or indirectly by leader $N+j$. (3) If follower i is not reachable from leader $N+j$, $c_{ij}=0$, the motion of i will not be affected by leader $N+j$, and thus the steady state of follower i is not related to $N+j$.

$R_{j}j=\mathrm{1,2},\dots ,m$ is denoted as the reachable set of leader $N+j$. $H_j=R_j/{\cup }_{j\ne i}{R}_i$ is a set of all followers which is only reachable from leader $N+j$. $Y_j=F/R_j$ represents the unreachable set from leader $N+j$. $P_j=R_j/H_j$ represents other follower set in $R_j$ without $H_j$, and ${\tilde{H}}_j=H_j/N+j$ represents the subset without $N+j$ in $H_j$.

Then, ${\mathbf{L}}_F$ and ${\mathbf{L}}_{\text{FL}}$ can be written in the following block matrix forms:$$\begin{array}{c}\text{(56)}& {\mathbf{L}}_F=\begin{array}{ccc}{\mathbf{L}}_{{\tilde{H}}_j}& 0& 0\\ {\mathbf{L}}_{P_j{\tilde{H}}_j}& {\mathbf{L}}_{P_j}& {\mathbf{L}}_{P_j{Y}_j}\\ 0& 0& {\mathbf{L}}_{Y_j}\end{array},{\mathbf{L}}_{FL}={\delta }_1,\dots ,{\delta }_N,\end{array}$$where ${\delta }_j={{\mathbf{L}}_{{\tilde{H}}_j,N+j}^T\begin{array}{cc}{\mathbf{L}}_{P_j,N+j}^T& 0\end{array}}^T$.

Theorem 3.

Assume all leaders have converged to the steady states ${\sigma }_{N+1}^e,{\sigma }_{N+2}^e,\dots ,{\sigma }_{N+M}^e$. Then, under the action of triggering condition (15), adaptive sliding mode attitude containment control algorithm (19), and adaption laws (20) and (21), all followers of cluster system (5) will asymptotically converge to the steady state ${\sigma }_i^e={\displaystyle {\sum }_{j=1}^M{c}_{ij}{\sigma }_{N+j}^e},i\in F,j\in L$.

Proof.

Since$$\begin{array}{}\text{(57)}& {\mathbf{L}}_{{\tilde{H}}_j}{1}_{{\tilde{H}}_j}+{\mathbf{L}}_{\tilde{H},N+j}=0,\end{array}$$we have$$\begin{array}{}\text{(58)}& {\mathbf{L}}_{{\tilde{H}}_j}^{-1}{\mathbf{L}}_{\tilde{H},N+j}=-1_{{\tilde{H}}_j},\end{array}$$and we can obtain that$$\begin{array}{}\text{(59)}& -{\mathbf{L}}_F^{-1}{\mathbf{\delta }}_j=-\begin{array}{c}{\mathbf{L}}_{{\tilde{H}}_j}^{-1}{\mathbf{L}}_{{\tilde{H}}_{j,N+j}}\\ -{\mathbf{L}}_{P_j}^{-1}{\mathbf{L}}_{P_j{\tilde{H}}_{j,N+j}}+{\mathbf{L}}_{P_j}^{-1}{\mathbf{L}}_{P_{j,N+j}}\\ 0\end{array}=\begin{array}{c}{1}_{{\tilde{H}}_j}\\ {\mathbf{L}}_{P_j}^{-1}{\mathbf{L}}_{P_j{\tilde{H}}_j}{1}_{{\tilde{H}}_j}+{\mathbf{L}}_{P_{j,N+j}}\\ 0\end{array}.\end{array}$$

Further, each entry of ${\mathbf{L}}_{P_j}^{-1}{\mathbf{L}}_{P_j{\tilde{H}}_j}{1}_{{\tilde{H}}_j}+{\mathbf{L}}_{P_{j,N+j}}$ is positive. Otherwise, ${\mathbf{L}}_{P_j{\tilde{H}}_j}{1}_{{\tilde{H}}_j}+{\mathbf{L}}_{P_{j,N+j}}=0$, which means ${\mathbf{P}}_j$ does not have neighbours in ${\mathbf{H}}_j$, which contradicts the definition of ${\mathbf{P}}_j$.

Now Theorem 3 has been proved.

4.2. The Constraints of Graph Differentiation on Followers’ Orientation

In microsatellite cluster flying missions, each member obtains the information from its neighbours through communication or relative state measurement. Due to the performance difference of sensors and communication equipment, as well as relative distance between members, neighbour satellite sets of each satellite are different. However, there may exist some commonalities among cluster members in information interaction, according to which cluster members can be divided into several subsets, and dynamic behaviour of cluster members belonging to the same subset may also have commonalities.

Although steady orientation of followers can be roughly estimated based on leader reachable sets, in some observation missions, there exist more constraints on followers’ orientation. The steady state of followers can be described using ${\pi }^1$ partition and ${\pi }^2$ partition of cluster information topology. These two graph theory tools can divide the cluster member into several subsets, and the number of neighbours in other subsets of cluster members belonging to the same subset has a certain commonality, which provides theoretical basis for information topology design of the microsatellite cluster.

At first, we prove that the satellites in the same cell partition belonging to ${\pi }_{H_1,\dots ,H_m}^1$ partition have same steady state.

Theorem 4.

For event-triggered adaptive sliding mode attitude containment control protocol (19), triggering condition (15), and adaption laws (20) and (21) of the microsatellite cluster system which satisfies ${\sigma }_F^e=-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}{\sigma }_L$, if system information topology $G$ has a cell partition ${\pi }_{H_1,\cdots ,H_m}^1=C_1,\cdots ,C_k$, then all the followers that belong to the same cell $C_{i}i=1,\cdots k$ have the same steady state.

Proof.

According to Reference [46], suppose ${\pi }_{H_1,\cdots ,H_m}^1=C_1,\dots ,C_k$ is a cell partition of $VG$ with $C_1=H_1,\dots ,C_m=H_m$; then, according to the cell partition of $VG$, by ordering the vertices appropriately, the graph Laplacian can be written in the following form:$$\begin{array}{}\text{(60)}& L=\begin{array}{cccccc}{L}_{C_1{C}_1}& \cdots & 0& 0& \cdots & 0\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & L_{C_m{C}_m}& 0& \cdots & 0\\ L_{C_{m+1}{C}_1}& \cdots & L_{C_{m+1}{C}_m}& L_{C_{m+1}{C}_{m+1}}& \cdots & L_{C_{m+1}{C}_k}\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ L_{C_k{C}_1}& \cdots & L_{C_k{C}_m}& L_{C_k{C}_{m+1}}& \cdots & L_{C_k{C}_k}\end{array}.\end{array}$$

The block matrix at the lower left corner and the lower right corner is denoted by A, B, respectively. Since $\mathbf{L}{\sigma }_F^e=0$, $\mathbf{A}{\sigma }_H^e+\mathbf{B}{\sigma }_{\overline{F}}^e=0$ is obtained. Then, by Lemma 2 in [46], ${\sigma }_{\overline{F}}^e=-{\mathbf{B}}^{-1}\mathbf{A}{\sigma }_H^e$. Assuming that each vertex in C_i has s_ij number of neighbours in $C_{j}i,j=1,\dots ,k,i\ne k$, then, according to the definition of ${\pi }^1$ partition, each row sum of the submatrix $L_{C_i{C}_j}i,j=1,\dots ,k,i\ne j$ equals s_ij. Combining with Corollary 1 in [46], it is given that$$\begin{array}{}\text{(61)}& \mathbf{A}{\mathbf{\sigma }}_H^e=\begin{array}{c}-s_{m+\mathrm{1,1}}{1}_{rm+1}\\ ⋮\\ -s_{k,1}{1}_{rk}\end{array}{\mathbf{\sigma }}_{N+1}^e+\cdots +\begin{array}{c}-s_{m+1,m}{1}_{rm+1}\\ ⋮\\ -s_{k,m}{1}_{rk}\end{array}{\mathbf{\sigma }}_{N+M}^e,\end{array}$$where $r_i$ is the cardinality of $C_{i}i=m+1,\dots ,k$.

It follows that$$\begin{array}{}\text{(62)}& {\mathbf{\sigma }}_{\overline{F}}^e={\mathbf{B}}^{-1}\begin{array}{c}{s}_{m+\mathrm{1,1}}{1}_{rm+1}\\ ⋮\\ s_{k,1}{1}_{rk}\end{array}{\mathbf{\sigma }}_{N+1}^e+\cdots +{\mathbf{B}}^{-1}\begin{array}{c}{s}_{m+1,m}{1}_{rm+1}\\ ⋮\\ s_{k,m}{1}_{rk}\end{array}{\mathbf{\sigma }}_{N+M}^e.\end{array}$$

By using Lemma 3 in [46], it can be concluded that all the followers that belong to the same cell C_i(i = 1, …, k) have the same steady state.

Now Theorem 4 has been proved.

Then, another cell partition, ${\pi }_{H_1,\dots ,H_m}^2$ partition, is provided to prove that satellites in the same cell partition belonging to ${\pi }_{H_1,\dots ,H_m}^2$ partition have the same steady state.

Theorem 5.

For event-triggered adaptive sliding mode containment control protocol (19)–(21) of microsatellite cluster which satisfies ${\sigma }_F^e=-{\mathbf{L}}_F^{-1}{\mathbf{L}}_{FL}{\sigma }_L$, if the system information topology $G$ has a cell partition ${\pi }^2=C_1,\dots ,C_k$ with $C_i\subseteq H_j$ or $C_i\subseteq F$, $i=1,\dots ,k,j=1,\dots ,m$, then all the followers that belong to the same cell $C_{i}i=1,\dots ,k$ have the same steady state.

Proof.

According to Reference [46], construct the deduced unweighted graph ${\overline{G}}^2$ of $G$ as follows: use a follower $\overline{i}$ to represent cell $C_i$ and draw an edge from $\overline{j}$ to $\overline{i}$ is a neighbour cell of $C_i$. Then, ${\overline{G}}^2$ has m reaches according to Assumption 3. Let ${\overline{L}}^2$ be the Laplacian matrix of the deduced graph ${\overline{G}}^2$; it can be derived that ${\overline{L}}^2{\overline{\sigma }}_F=0$ with ${\overline{\sigma }}_i={\sigma }_{N+j}^e$ if $C_i\subseteq H_j$ has unique solution. Finally, it can be verified that ${\sigma }_{F{C}_i}^e={\overline{\sigma }}_i{1}_{C_i}$ is the solution of $\overline{L}{\overline{\sigma }}_F^e=0$.

Now Theorem 5 has been proved.

5. Simulation Results

5.1. Simulation Results and Analysis of Attitude Containment Control

In this section, simulations for multiple leader-follower satellite cluster are presented to illustrate the effectiveness of the proposed control protocol and information topology design. Suppose that in microsatellite cluster observation mission, six satellites, denoted by $F=\mathrm{1,2},\dots ,6$, are needed to obtain observation information from three different directions. Suppose the target area is defined by three attitude orientations, and the leaders $L=\mathrm{7,8,9}$ are stationary relative to the reference frame. The attitude of leaders, respectively, is$$\begin{array}{c}\text{(63)}& {\mathbf{\sigma }}_7={\begin{array}{ccc}0.09& 0.06& -0.08\end{array}}^T\text{rad},{\mathbf{\sigma }}_8={\begin{array}{ccc}0.06& -0.1& 0.05\end{array}}^T\text{rad},\\ {\mathbf{\sigma }}_9={\begin{array}{ccc}-0.08& 0.1& 0.09\end{array}}^T\text{rad.}\end{array}$$

The nominal inertia of six followers, respectively, is$$\begin{array}{c}\text{(64)}& {\mathbf{J}}_1=\begin{array}{ccc}20& 0& 2\\ 0& 25& 0\\ 2& 0& 29\end{array}{\mathrm{kgm}}^2,{\mathbf{J}}_2=\begin{array}{ccc}22& 1& 0.5\\ 1& 24& 3\\ 0.5& 3& 22\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_3=\begin{array}{ccc}25& 0.8& 2\\ 0.8& 29& 1\\ 2& 1& 21\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_4=\begin{array}{ccc}23& 0.4& 0\\ 0.4& 26& 0.8\\ 0& 0.8& 28\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_5=\begin{array}{ccc}22& 0.2& 0\\ 0.2& 26& 0.6\\ 0& 0.6& 24\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_6=\begin{array}{ccc}23& 0.4& 0\\ 0.4& 24& 0.8\\ 0& 0.8& 28\end{array}{\mathrm{kgm}}^2.\end{array}$$

The initial estimation parameter is ${\widehat{\Theta }}_{i}0={\mathrm{22,26,25,0,0,0}}^T$.

The initial state of followers is, respectively, shown in Table 1.

Table 1

The initial attitude information of followers.

Followers	Attitude ${\sigma }_i$(rad)	Angular velocity ${\omega }_i$(rad/s)
1	${\begin{array}{ccc}0.1& 0.4& -0.3\end{array}}^T$	${\begin{array}{ccc}0.01& 0& 0\end{array}}^T$
2	${\begin{array}{ccc}0.3& 0.2& 0.3\end{array}}^T$	${\begin{array}{ccc}0& 0.01& 0\end{array}}^T$
3	${\begin{array}{ccc}-0.2& 0.1& -0.1\end{array}}^T$	${\begin{array}{ccc}0& 0& 0.01\end{array}}^T$
4	${\begin{array}{ccc}-0.4& 0.2& -0.1\end{array}}^T$	${\begin{array}{ccc}0& 0& -0.01\end{array}}^T$
5	${\begin{array}{ccc}0.3& -0.5& 0.1\end{array}}^T$	${\begin{array}{ccc}0& -0.015& 0\end{array}}^T$
6	${\begin{array}{ccc}0.2& 0.2& -0.3\end{array}}^T$	${\begin{array}{ccc}0& 0& 0.015\end{array}}^T$

System information topology is shown in Figure 2.

[figure omitted; refer to PDF]

Control gain coefficients and adaptive parameters are$$\begin{array}{c}\text{(65)}& {\alpha }_i=0.02,{\mathbf{K}}_i={\begin{array}{ccc}1& 1& 1\end{array}}^T,\\ {\mathbf{\Lambda }}_i=\text{diag}\begin{array}{ccc}2& 2& 2\end{array},\\ {\Gamma }_i=\mathrm{0.001.}\end{array}$$

Triggering parameters are ${\gamma }_i=0.6,{\rho }_i=2,{\nu }_i=0.5$.

A microsatellite cluster in LEO is mainly affected by the gravity gradient torque, while the disturbances such as the solar radiation pressure torque will be dominant for a cluster in high-Earth orbits such as the geostationary orbit. All these torques are slowly varying and can be treated as signals composed of constants and periodic trigonometric functions. Taking into account these factors, the disturbances are chosen as$$\begin{array}{}\text{(66)}& {\mathbf{d}}_i={\mathbf{d}}_{\text{li}}+{\mathbf{d}}_{\text{uni}},\end{array}$$where$$\begin{array}{c}\text{(67)}& {\mathbf{d}}_{l1}={\begin{array}{ccc}0.0012& -0.0018& 0.0012\end{array}}^T\text{N.m},{\mathbf{d}}_{l2}={\begin{array}{ccc}0.001& 0.0014& -0.0017\end{array}}^T\text{N.m},\\ {\mathbf{d}}_{l3}={\begin{array}{ccc}-0.0013& 0.0016& -0.001\end{array}}^T\text{N.m,}\\ {\mathbf{d}}_{l4}={\begin{array}{ccc}0.0015& -0.0014& -0.0013\end{array}}^T\text{N.m,}\\ {\mathbf{d}}_{l5}={\begin{array}{ccc}-0.001& -0.001& 0.0015\end{array}}^T\text{N.m},\\ {\mathbf{d}}_{l6}={\begin{array}{ccc}-0.0012& 0.0013& 0.0014\end{array}}^T\text{N.m}\end{array}$$denote the constant disturbing torques and$$\begin{array}{}\text{(68)}& {\mathbf{d}}_{\text{uni}}=\frac{1}{5}\begin{array}{c}{d}_{li1}\sin \frac{t}{12}\\ d_{li2}\cos \frac{t}{15}\\ d_{\text{li}3}\sin \frac{t}{10}\cos \frac{t}{15}\end{array}\end{array}$$is used to simulate the residual disturbances including aerodynamic torques, solar radiation torques, and similar effects.

5.1.1. Event-Triggered Attitude Containment Control

Simulation results of satellite cluster are shown in Figure 3. Figures 3(a) and 3(b) are the curves of relative attitude and relative angular velocity over time, respectively. It can be seen that the relative attitude converges to the convex hull formed by the leaders at about 600 s, and relative angular velocity converges to 0 within 700 s. The followers can converge to the convex hull even though there exists large disturbance torque. The interevent time of each follower is shown in Figures 3(c) and 3(d). At the initial stage, the state of cluster members is far from the desired state, then the event is triggered frequently, and the update of control input is relatively frequent, but when the system asymptotically converges to the desired state, fewer events are triggered, and interevent time increases and tends to be stable finally. We can find that if satellites are influenced by periodic disturbance, the interevent time also changes periodically. The control torques acting on each satellite are shown in Figure 3(e). It can be seen that control torques acting on the followers are piecewise function, and the control input is only updated at the next triggering instant. The control torques are limited within the range of $-\mathrm{1,1}\text{\hspace{0.17em}N}\cdot \text{m}$. Attitude trajectories of followers are shown in Figure 3(f), from which we can see that followers asymptotically converge to the convex hull formed by the leaders. Figures 3(g) and 3(h) show the event error and triggering threshold for each follower in 200 s. It can be seen that if event error exceeds threshold, the event is triggered and the state is sampled.

[figures omitted; refer to PDF]

In a word, under the action of the event-triggered adaptive sliding mode attitude controller, the attitude of each satellite asymptotically converges to the convex hull formed by the leaders, and angular velocity asymptotically converges to 0.

5.1.2. Traditional Time-Triggered Attitude Containment Control

According to Reference [53], the time-triggered adaptive sliding mode attitude containment control algorithm is$$\begin{array}{c}\text{(69)}& {\mathbf{u}}_{i}t=-{\mathbf{K}}_i{\mathbf{s}}_i+{\mathbf{Y}}_i{\widehat{\Theta }}_i-{\widehat{k}}_i\mathrm{sgn}{\mathbf{s}}_i,\hspace{1em}i\in F,{\dot{\widehat{\Theta }}}_{i}t=-{\mathbf{\Lambda }}_i{\mathbf{Y}}_i^{T}t{\mathbf{s}}_{i}t,\\ {\dot{\widehat{k}}}_{i}t={\Gamma }_i{{\mathbf{s}}_{i}t}_1.\end{array}$$

The simulation results of time-triggered distributed attitude adaptive sliding mode controller are shown in Figure 4. The attitude of the cluster system converges to the convex hull formed by the leaders at about 600 s, and attitude angular velocity converges to 0 within 700 s. The controller is continuously updated while the convergence rate and control accuracy are not better than the event-triggered one.

[figures omitted; refer to PDF]

It can be clearly seen from Figures 3 and 4 that both time-triggered and event-triggered control strategies can realize attitude containment. In addition, it is noteworthy that event-triggered containment control is updated in an aperiodic manner, while time-triggered control is updated at a fixed interval of 0.01 s. Within 1200 s, the sampling and control input update times of the time-triggered control method are 120,000, while the event of each follower, respectively, is 391, 445, 405, 354, 402, and 480, from which the update frequency of control action is greatly reduced by the event-triggered control strategy. Compared with time-triggered attitude containment control protocol, event-triggered one can effectively reduce the control input update frequency while ensuring the similar control performance. Through the reasonable selection of control parameters and triggering function, event-triggered control can guarantee the convergence rate and control accuracy and reduce the amount of computation and communication, as well as the update frequency of actuators.

5.2. The Influence of Information Topology on Follower’s Orientation

Suppose in the Earth observation mission of the microsatellite cluster, twelve satellites are used to obtain the observation information of three different orientations. In order to meet the accuracy requirement, two satellites can be used to observe the same azimuth at the same time. Suppose the target area is formed by three attitude orientations. The initial state of leaders, respectively, is$$\begin{array}{c}\text{(70)}& {\mathbf{\sigma }}_{10}={\begin{array}{ccc}0.09& 0.06& -0.08\end{array}}^T\text{rad},{\mathbf{\sigma }}_{11}={\begin{array}{ccc}0.06& -0.1& 0.05\end{array}}^T\text{rad},\\ {\mathbf{\sigma }}_{12}={\begin{array}{ccc}-0.08& 0.1& 0.09\end{array}}^T\text{rad}.\end{array}$$

Nominal inertia of followers, respectively, is$$\begin{array}{c}\text{(71)}& {\mathbf{J}}_1=\begin{array}{ccc}20& 0& 2\\ 0& 25& 0\\ 2& 0& 29\end{array}{\mathrm{kgm}}^2,{\mathbf{J}}_2=\begin{array}{ccc}22& 1& 0.5\\ 1& 24& 3\\ 0.5& 3& 22\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_3=\begin{array}{ccc}25& 0.8& 2\\ 0.8& 29& 1\\ 2& 1& 21\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_4=\begin{array}{ccc}23& 0.4& 0\\ 0.4& 26& 0.8\\ 0& 0.8& 28\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_5=\begin{array}{ccc}22& 0.2& 0\\ 0.2& 26& 0.6\\ 0& 0.6& 24\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_6=\begin{array}{ccc}23& 0.4& 0\\ 0.4& 24& 0.8\\ 0& 0.8& 28\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_7=\begin{array}{ccc}22& 0.2& 0.3\\ 0.2& 29& 0\\ 0& 0.3& 25\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_8=\begin{array}{ccc}21& 0.2& 1\\ 0.2& 26& 0.2\\ 1& 0.2& 24\end{array}{\mathrm{kgm}}^2,\\ {\mathbf{J}}_9=\begin{array}{ccc}24& 0.6& 0.3\\ 0.6& 25& 0\\ 0& 0.3& 27\end{array}{\mathrm{kgm}}^2.\end{array}$$

The initial estimation parameter of followers is $\mathrm{\Theta }0={\mathrm{22,26,25,0,0,0}}^T$.

The constant disturbance acting on the followers 1–6 and periodic disturbances are the same as Section 5.1, while constant disturbance acting on followers 7–9 is$$\begin{array}{c}\text{(72)}& {\mathbf{d}}_{l7}={\begin{array}{ccc}-0.0016& 0.001& 0.0012\end{array}}^T\text{N}\cdot \text{m},{\mathbf{d}}_{l8}={\begin{array}{ccc}-0.0018& 0.001& -0.0012\end{array}}^T\text{N}\cdot \text{m},\\ {\mathbf{d}}_{l9}={\begin{array}{ccc}0.0011& -0.0015& -0.0014\end{array}}^T\text{N}\cdot \text{m.}\end{array}$$

The initial state of followers is, respectively, shown in Table 2.

Table 2

The initial attitude information of followers.

Followers	Attitude ${\sigma }_i$(rad)	Angular velocity ${\omega }_i$(rad/s)
1	${\begin{array}{ccc}0.1& 0.4& -0.3\end{array}}^T$	${\begin{array}{ccc}0.01& 0& 0\end{array}}^T$
2	${\begin{array}{ccc}0.3& 0.2& 0.3\end{array}}^T$	${\begin{array}{ccc}0& 0.01& 0\end{array}}^T$
3	${\begin{array}{ccc}-0.2& 0.1& -0.1\end{array}}^T$	${\begin{array}{ccc}0& 0& 0.01\end{array}}^T$
4	${\begin{array}{ccc}-0.4& 0.2& -0.1\end{array}}^T$	${\begin{array}{ccc}0& 0& -0.01\end{array}}^T$
5	${\begin{array}{ccc}0.3& -0.5& 0.1\end{array}}^T$	${\begin{array}{ccc}0& -0.015& 0\end{array}}^T$
6	${\begin{array}{ccc}0.2& 0.2& -0.3\end{array}}^T$	${\begin{array}{ccc}0& 0& 0.015\end{array}}^T$
7	${\begin{array}{ccc}0.2& -0.1& -0.4\end{array}}^T$	${\begin{array}{ccc}0.01& -0.01& 0\end{array}}^T$
8	${\begin{array}{ccc}-0.4& -0.3& 0.1\end{array}}^T$	${\begin{array}{ccc}0& 0.015& -0.01\end{array}}^T$
9	${\begin{array}{ccc}-0.2& -0.3& 0.2\end{array}}^T$	${\begin{array}{ccc}0.01& 0& -0.015\end{array}}^T$

Information topology of the cluster system is shown in Figure 5.

[figure omitted; refer to PDF]

The information topology structure of the microsatellite cluster system is shown in Figure 5. It can be seen that Figure 5 belongs to ${\pi }^1$ partition. Satellites 1, 2, 7, and 8 belong to the same cell, 3 and 4 belong to the same cell, and 5 and 6 belong to the same cell. According to Theorem 4, satellites belonging to the same cell converge to the same azimuth under the action of control torques in the presence of model uncertainties and external disturbances. This property can be used to observe the same azimuth. Simulation results are shown in Figure 6. The attitude of follower converges to the convex hull formed by the leaders within 600 s, and attitude angular velocity converges to 0 within 700 s. The control torques are piecewise constant. Satellites 1 and 2 converge to the same azimuth, and 7 and 8 converge to same azimuth, respectively. Because satellites in C₅ only receive information from C₃, 3, 4, 5, and 6 converge to the same azimuth. Within 1200 s, the sampling and control input update times of each follower, respectively, are 232, 247, 522, 420, 162, 142, 196, 288, and 315.

[figures omitted; refer to PDF]

6. Conclusion

In this paper, an event-triggered adaptive sliding mode attitude containment control protocol is proposed in the framework of the Euler–Lagrange model for the attitude containment control problem of the microsatellite cluster system. Considering the constraints of resource and power on the microsatellite cluster system, the event-triggered control strategy is introduced into the attitude containment control problem of the microsatellite cluster. The triggering condition consisting of state-dependent function and the time-dependent function is given to adjust controller update period and exclude the Zeno behaviour. If and only if the triggering condition is satisfied, state information is sampled, control law is computed, and actuators are updated. Then, under directed topology, the event-triggered adaptive sliding mode attitude containment control algorithm is proposed, which makes attitude of followers asymptotically enter into the convex hull formed by leaders’ orientation in the presence of inertia uncertainties and external disturbances. Numerical simulation is carried out to verify the effectiveness of the proposed event-triggered distributed attitude containment control algorithm. Then, compared with the time-triggered one, it can be seen that while ensuring the control performance of the cluster system, the designed event-triggered attitude containment controller only updates control law in the triggering instant, which effectively reduces the amount of computation and communication and update frequency of actuators and saves resources on board.

Furthermore, the influence of cluster information topology structure design on the stable state of containment control algorithm is also studied. An appropriate information topology is designed to meet the attitude orientation requirements of containment control. It is shown that the steady state of each follower is a convex combination of all leaders’ states it can access. The cell partition of cluster information topology is given based on the number of satellite’s neighbours, and it is proved that the cluster members belonging to the same cell have the same stable state. However, there is no requirement for the information interaction between satellites inside the cell, and the information links inside the cell will not affect the stable state of cluster members. It provides a theoretical basis for the design of information topology of the microsatellite cluster system. Numerical simulation is conducted to verify the influence of information topology on steady state of the microsatellite cluster system.

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