Distributed Attitude Synchronization Control of Switched Networked Satellite Formation Flying

: This paper addresses the leader-follower finite-time attitude synchronization and stabilization for satellite formation flying (SFF) under directed switching communication topologies. Specifically, a distributed attitude synchronization control scheme is proposed to guarantee finite-time convergence to leader’s attitude and desired angular velocity for SFF under time-varying but jointly connected switching communication topologies. First, finite-time consensus protocols are designed for leader’s attitude tracking. The protocols are developed based on non-smooth control techniques such as homogeneity with dilatation and LaSalle invariance principle. Then, a distributed finite-time angular velocity estimator is designed using super-twisting sliding-mode control. The estimator helps solving communication loss issues, reducing communication burden, and canceling chattering effect due to high-rate convergence. Simulations are shown to illustrate the effectiveness of the obtained theoretical results.


Introduction
The decentralized coordinated attitude control for Satellite Formation Flying (SFF) consists one of the main technical requirements for the success of SFF missions. The main challenge to maintain formation attitude alignment is the synchronization convergence rate.
The convergence speed of SFF synchronization consists one of the most important aspects of SFF control and stabilization. A high convergence rate is strongly recommended for highaccuracy, robust, and low-energy synchronization. In other words, the effectiveness of a synchronization algorithm is measured by its convergence time. Over the last decades, a variety of SFF attitude synchronization algorithms has been developed to satisfy SFF attitude synchronization requirements [1][2][3][4][5][6][7][8][9][10]. However, most existing results only show asymptotic convergence, which is slow and high-energy consuming.
To accelerate the SFF synchronization convergence rate and improve SFF robustness, finite-time attitude alignment has recently drawn the attention of SFF researchers as one of the promising solutions. Zhou and Hu [11] designed a SFF decentralized finite-time attitude synchronization controller and an angular speed estimator using first-order sliding mode. Liu et al. [12] addressed the problem of SFF attitude coordination in the presence of disturbances and noises based on the reference attitude trajectory. The authors combined reference model control, H method, sliding mode control, and linear matrix inequality method to ensure global robust attitude tracking and formation keeping. Integral sliding mode control was used in [13] to design adaptive attitude coordinated tracking protocols for a group of rigid spacecrafts. Authors designed the control system to handle formation control problems such as parameters uncertainties, external disturbances, and input saturations. A robust SFF control system was proposed by Liu et al. [14] for trajectory and attitude control to form the desired formation and align the satellite attitudes. Output feedback control was used in [15] to design a law for finite-time attitude synchronization of spacecraft formation flying. The control law was developed combining homogenous system theory with a nonlinear filter to avoid angular velocity measurement. The control scheme provided the attitude synchronization with zero final angular velocity. In [16], authors combined a distributed continuous protocol with an adaptive sliding mode observer to ensure finite-time synchronization of spacecraft formation without velocity measurement. Nonsingular fast terminal sliding mode was used in [17] to design a decentralized adaptive control law for attitude synchronization of a group of spacecrafts. On the other hand, inter-satellite communications (ISC) in results show fast convergence, the required control torques exhibit a chattering-like effect with high amplitude. Nonlinear fast terminal sliding mode control was combined with adaptive fuzzy control in [20] to design a distributed finite-time control algorithm for a team of spacecraft vehicles. Although the algorithm guarantees finite-time convergence of spacecraft formation under switching topologies, it presents high overshooting of angular speed tracking and sliding mode chattering effect. In [20], the attitude synchronization problem of switched networked spacecraft systems has been tackled within leader-following based formations. A distributed control law was developed based on asymptotic convergence analysis and using lemmas about exponential convergence, stability of switched systems, certainly equivalence principal control, and asymptotic distributed observation. An eventtriggered based coordinated control of SFF under limited communication has been proposed in [22]. By cancelling the continuous neighbor-to-neighbor velocity exchange, the coordinated controller conjointly with the event-triggered mechanism reduced the communication burden and the computational cost. Nevertheless, the simulation results reveal that the convergence time of the event-triggered controller is relatively long, and the control signals suffer from chattering effect, which can damage the actuation system of the individual spacecraft. A high-order sliding mode-based spacecraft formation control scheme was proposed in [23]. In this contribution, the super-twisting sliding mode control has been used to design a distributed observer and sufficient conditions for time-varying practical formation have been developed in term of average dwell time. Although, the proposed algorithm has shown good tracking, no indication was given about the control efforts magnitude and their vulnerability to chattering.
Motivated by the above results, this paper addresses the finite-time distributed control design for SFF systems subject to loss of communication. Specifically, our main contribution is the development of a robust distributed control scheme that would enable satellite system networks to achieve consensus and formation objectives in leader-follower switching topology architecture with fast rate convergence and chattering free control. The control scheme consists of a distributed formation synchronization controller and an angular velocity estimator. The distributed control law is developed using leader-follower consensus approach for undirected communication topologies and the estimator is designed using high-order sliding mode control.
The first part of the paper studies the finite-time distributed attitude synchronization problem of SFF under fixed and switched communication topologies. The theoretical aspect of the SFF consensus is investigated based on the graph theory, Lyapunov's direct method, homogeneity with dilation, and LaSalle's invariance principle. In the second part, and to maintain the SFF in case of communication loss between agents or with the leader, a distributed second order sliding mode observer based on the super-twisting algorithm is designed. Conjointly with the attitude synchronization controller, the observer allows the individual satellites to estimate the desired angular velocity in case of ISC links break. The super-twisting sliding mode control ensures the finite-time convergence of the sliding surface, cancels or alleviates the chattering effect of conventional sliding mode, and provides robustness against parameter uncertainties.
The rest of the paper is organized as follows. In section 2, satellite attitude kinematics and dynamics are formulated using quaternions, the basics of graph theory are given, and the necessary assumption and lemmas are made. Section 3 presents the main results of this work about the distributed SFF synchronization control protocols under undirected fixed and switched communication topologies. Section 4 presents the design of the distributed finite-time super-twisting estimator for individual satellites to obtain an accurate estimation of the leader angular velocity in case of break of ISC links or loss of communication with the leader. The effectiveness of the proposed control scheme is validated via numerical simulations in section 5. Finally, conclusions are made in section 6.

Preliminaries
Consider the case of a group of satellites in a SFF and use subscribe ∈ = {1,2, … , } to denote the ith agent satellite. The following preliminaries are given to solve the finite-time attitude synchronization problem of SFF.

Satellite attitude kinematics and dynamics
Let ℱ 0 , ℱ i , ℱ F being the inertial, satellite body, and formation frames, respectively. Consider the quaternion ̅ = [ 0 ] and the vector i to denote the orientation and the angular velocity of the ith satellite from the frame ℱ i to the frame ℱ 0 , respectively. and q 0i are the vector part and scaler part of the quaternion and ̅ * = [± 0 − ] denotes the inverse of the quaternion. The rigid-body kinematics and dynamics of the ith satellite from the frame ℱ i to the frame ℱ 0 can be described as follows [24] ̇= − 1 where i ∈ R 3×3 , i =∈ R 3×1 , and i =∈ R 3×1 denote the inertia tensor, the control vector of torques, and the external disturbance torque of the ith satellite, respectively.

Graph theory
Define = { , ℰ, } as the weighted graph that describes the communication topology among the SFF agents. = { 1 , 2 , … , } denotes the nonempty set of the nodes (satellites) and ℰ ⊆ × denotes the edge set (ISC links). The weighted adjacency matrix ∈ , is defined with nonnegative adjacency elements ≥ 0, ∀ , ∈ . The communication topology described by the graph is supposed to fulfil the following properties: • An edge = ( 1 , 2 ) ∈ ℰ indicates the communication link from the ith satellite (parent) to the jth satellite (child).

Assumptions and lemmas
Assumption 1: In the leader-follower SFF considered in this paper, the vertex represents the leader while the vertices 1 , 2 , … , represent the followers. Assumption 2: Only one or few followers communicate with the leader. If a follower satellite 'i' is connected to the leader, then the connection weight is denoted by > 0, otherwise = 0, ∀ ∈ . Assumption 3: The states of all followers can be affected directly or indirectly by the leader states under a set of protocols satisfying ≥ 0. Assumption 4: the switched graph is undirected and connected in each interval [ , +1 ). Lemma 1 [25,26]: If the graph is indirect and connected then the Laplacian matrix is semi-positive definite, its eigenvalues are real and satisfy • If > 0, = 0 is an asymptotically stable equilibrium of the system. • If = 0, the system equilibrium is exponentially stable. • If < 0, the system equilibrium is locally finite-time stable It results from Lemma 3 that there exists a C m -smooth homogeneous Lyapunov function V( ): → ℝ of degree of homogeneity > 0 where for any > 0 For the case < 0, the following condition holds when ≤ Then the equilibrium of system (7) is locally finite-time stable. where ∈ ℝ , ( 0 ) = .
Let Σ denotes the finite switching index set and assume that ∈ ≜ { }, ∈ Σ ≜ {1,2, … , } is continuous with respect to and the switching signal ∈ ℝ + × ℝ → Σ is a piecewise constant function of time. If system (9) is asymptotically stable and ̇= ( ) for fixed ∈ Σ is finitetime stable, then the switched system (9) where ( ), ( ) are the smallest and the largest eigenvalues of .

Satellite Attitude Synchronization
Consider the case of a group of satellites labelled ′1, . . ′ and a virtual leader labelled ′0′. Since the SFF control aims to enforce satellites agents to track desired dynamics (virtual leader states), we define and as the desired states for each agent relative to the inertial frame ℱ 0 .
Definition 1: Assuming that the SFF attitude is described by Eq. (1), the finite-time closed-loop system stability and attitude synchronization can be achieved if, for any initial values of the system states, there exists a synchronization time ∈ [0, +∞) such that the solution of Eq. (1) satisfies The synchronization time depends on the initial values of the system states ( 0 ) = 0 , ( 0 ) = 0 , ∀ ∈ (12) where 1 , 2 > 0 are the control gain, with 0 < < 1, ≥ 1 and the skew-symmetric matrix × ( ) is defined as

FSS Attitude Synchronization with Fixed Network
For each i-th satellite, the first term in the control protocol The objective here is to prove that the consensus (11) can be achieved within a finite-time synchronization T. To do so, consider a positive definite Lyapunov function candidate as Tacking the time derivative of V along the trajectories of system (15) Equality (22) implies that ̂=̂= 0 ∀ , ∈ . From Lemma 2, − 0 → 0, − 0 → 0, ∀ ∈ , as → ∞. Define the homogeneity degree of system (15) as = − 1 < 0, according to Lemma 3 the equilibrium of system (15) is globally asymptotically stable with local finite-time convergence. It results that the origin is a globally finitetime stable equilibrium of system (15) and the consensus (11) is reached in finite-time. This completes the proof.

SFF Synchronization with Switching Network
Consider the case of networked systems with switching communication topologies. Let = { , ℰ, ( ) } denotes the finite set of all possible topologies, ℳ = {1,2, … , } denotes the index set, and denotes the number of switching topologies. We define a switching signal ( ): + → ℳ and a switching sequence of bounded non-overlapping time intervals [ , +1 ). The control objective is now to reach following switched state consensus in finite-time.

Finite-time SFF control design with HOSM Estimator
In this section, a high-order sliding mode angular velocity estimator is designed based on the super-twisting algorithm. The distributed estimator allows satellites to estimate accurately the desired angular velocity in the presence of disturbances and in case of loss of communication among SFF agents or with the leader. The distributed sliding mode observer is obtained, for each agent 'i', as follows with , being the observer states and the sliding mode surface defined as 1 , 1 ∈ ℝ + are the observer parameters. Considering ̂= as the angular velocity estimator, the distributed control law (13) is replaced by the following control law End of proof.

Numerical Simulations
In this section, numerical examples are given to illustrate the obtained theoretical results and demonstrate the effectiveness and performance of the proposed distributed SFF attitude synchronization protocols. The different torque control laws and the angular velocity estimator are simulated to achieve finite-time attitude synchronization of a given SFF. The satellites orbit under the undirected graph , = 1, . . ,4 as shown in Fig. 1.
For illustration, two scenarios are considered. In the first scenario, the control laws (13) and (24) are us to achieve unperturbed SFF attitude synchronization in finite-time. In the second scenario, the observer-controller scheme given by Eqs. (29) and (31) is used to enable the formation satellites to track a desired time-varying angular velocity in the presence of external disturbances and under switched communication topology.
For fair comparison, the data of the three-satellite SFF given in [11] is used in the simulations. The case where the three satellites orbit with a switching communication topology composed of four digraphs 1 -4 as shown in Fig  1 is considered. For simplicity, the adjacency matrix elements are 0 or 1. Table 1 gives the SFF parameters and initial conditions.    and control torques, respectively. The SFF attitude synchronization accuracy is determined using the relative attitude errors ̅ = ̅ * ̅ for the three satellites. As shown in Fig. 7, a threshold of 10 −9 occurs at 22 s. Figure 8 depicts the attitude tracking path for the three satellites using the observer-controller scheme (29)- (31).
The main contribution of the proposed control scheme is twofold. First, both controllers (24) and (31) enable finitetime SFF consensus, despite satellite-to-satellite or satellite-to-leader communication losses. It is worth noting that finite-time formation control under loss of communication is highly recommended for performance improvement rather than asymptotic formation control. Second, the controller (24) conjointly with the observer (29) provide more stable and chattering free SFF attitude tracking as compared to relevant recent works (e.g., [11]).

Conclusion
The finite-time attitude synchronization problem for SFF under switching communication topology is investigated in this paper. First, SFF finite-time consensus problem under fixed and switching communication topologies was solved based upon graph and matrix theories, local finite-time convergence in homogeneous systems, and nonsmooth LaSalle's invariance principle. Then, to reduce the number of communication links, supper-twisting sliding mode control theory was used to design a finite-time observer for estimating the desired angular velocity (i.e., leader velocity). Numerical simulations showed the outer performance and effectiveness of the proposed design.