Finite-Time Orbit Control for Spacecraft Formation with External Disturbances and Limited Data Communication

This work addresses the ﬁ nite-time orbit control problem for spacecraft formation ﬂ ying with external disturbances and limited data communication. A hysteretic quantizer is employed for data quantization in the controller-actuator channel to decrease the communication rate and prevent the chattering phenomenon caused by the logarithmic quantizer. Combined with the adding one power integrator method and backstepping technique, a new ﬁ nite-time tracking control strategy with adaptation law is designed to ensure that the closed-loop system is practical ﬁ nite-time stable, and that the tracking errors of relative position and velocity are bounded within ﬁ nite-time despite with limited data communication and external disturbances. Finally, an example is shown to validate the e ﬀ ectiveness of the proposed ﬁ nite-time tracking controller.


Introduction
Spacecraft formation flying (SFF) is a concept that the functionality of traditional large spacecraft is replaced by a group of less-expensive, smaller, and cooperative spacecraft [1,2]. In recent years, the study of spacecraft formation control has gradually became an active area of research owing to the reason that SFF is a primary technology for modern space missions, such as deep space exploration, spacecraft on-orbit servicing, and deep space imaging [3][4][5][6]. So far, a rich body of orbit control strategies for SFF has been presented, and the relative dynamic model of these strategies can be roughly categorized into two types, namely, linear dynamic model and nonlinear dynamic model [7,8]. Considering the linearizing of relative dynamic model can induce model errors, various control schemes based on the nonlinear relative dynamic model for spacecraft formation flying have been proposed [9][10][11][12].
It is to be noticed that all aforementioned control schemes can only guarantee the asymptotical stability of the controlled systems. A fast convergence rate is an essen-tial demand for SFF. In the past few years, the finite-time control method has been widely adopted to design controllers for various nonlinear systems since it can guarantee the controlled systems have better disturbance rejection property, faster convergence rate, and higher precision control performance compared with the asymptotic control [13][14][15][16]. To date, the finite-time method has been found successfully applied to handle the spacecraft control problem [17,18]. The design approaches for finite-time control for spacecraft main include three type approaches, namely, the adding one power integrator technique, the terminal sliding mode method, and the homogeneity theorem. However, the few studies have focused on finite-time orbit control based on the adding one power integrator technique for spacecraft formation.
Despite the fact that the mentioned research results above can ensure the controlled system finite-time stability, most of the existing finite-time control strategies focused on traditional large spacecraft, in which the bandwidth of the communication channel is assumed to be not limited. However, in modern spacecraft formation control system, data communication between different modules is usually executed by wireless networks, which means that the bandwidth of the communication channel is limited [19,20]. Although the use of wireless networks brings many advantages, such as lighter weight, implementation, and installation with less cost, some new challenges have inevitably been induced, for example, data missing, communication time delay, and quantization effect [21][22][23][24]. It is well known that when the data of the control module is transmitted to the actuator module by limited communication networks, the quantization errors caused by signal quantization have unavoidably emerged. If the effect of quantization errors is not compensated effectively, the control performance may be degraded or even let the system unstable. Therefore, it is desirable to design a new controller considering limited data communication for network-based spacecraft formation control systems. Recently, the attitude control problem with limited data communication for spacecraft has been studied in [25]. Unfortunately, the study concentrated on finite-time control for SFF with limited data communication is scarce.
Motivated by the above discussion, the finite-time orbit control for SFF with limited data communication will be investigated. The main contribution of this work can be summarized as follows: (1) a novel finite-time tracking controller for SFF is proposed such that the tracking errors are bounded within finite time, in which the external disturbances are considered. (2) Compared with some existing finite-time controllers designed based on the terminal sliding mode method [26][27][28], the advantage of our method is that it can avoid noncontinuous and singular problem by using the backstepping approach and the adding one power integrator technique. (3) The needed communication rate is decreased by employing the hysteresis quantizer to quantify the control data, in which the chattering phenomenon induced by the logarithmic quantizer can be successfully prevented.
The rest of the paper is organized as follows. The modeling and preliminaries are given in Section 2. In Section 3, the main results are presented. In Section 4 and Section 5, the illustrative example and conclusions are presented, respectively.

Modeling and Preliminaries
2.1. Spacecraft Formation Flying Dynamic Model. The nonlinear relative motion dynamics of SFF can be described as follows [29]: where q 1 ðtÞ = ½x, y, z T is the relative position from the follower spacecraft to the leader spacecraft in the local coordinate frame, uðtÞ = ½u 1 , u 2 , u 3 T and u l ðtÞ = ½u l1 , u l2 , u l3 T are the control inputs of the follower spacecraft and leader spacecraft, respectively, m f and m l denote the mass of the follower spacecraft and leader spacecraft, respectively, wðtÞ = ½w 1 , w 2 , w 3 T denotes the bounded external disturbance, μ denotes the Earth gravitational constant, n denotes the orbit angular velocity of the leader spacecraft, and R = ð0, r, 0Þ T is the position vector from the inertial coordinate attached to the center of Earth to the leader spacecraft described in the local coordinate frame. The position tracking error e 1 ðtÞ and velocity tracking error e 2 ðtÞ are defined as where y d , _ z d T m/s are the relative velocity vector, desired position state, and desired velocity state, respectively.
Then, the error relative motion dynamics of SFF can be described as follows: where : Assumption 1. The desired states q 1d and q 2d are assumed to bounded, and their first two-order time derivatives are assumed to be bounded.
The solution of (3) can be regarded as finitetime bounded or practical finite-time stable if for all eð0Þ = e 0 , and there exists ν > 0 and T r ð ν, e 0 Þ < ∞, such that keðtÞ k < ν for t ≥ T r .

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Lemma 3 (see [30]). For Lemma 4 (see [31]). If α and β be positive constants, then ς ðy, zÞ > 0 is a real-valued function. We have 2.2. Fuzzy-Logic Systems. The IF-THEN rules of the FLSs are constructed: where x = ½x 1 , ⋯, x n T ∈ ℝ n is the input of FLSs, y ∈ ℝ is the output of FLSs, and F l i and G l are the fuzzy sets associate with the membership functions μ F l 2 ðx i Þ and μ G l ðyÞ, respectively. m = 1, 2, ⋯, k and k are the number of rules. Then, the FLSs can be expressed as where Λ m = max y∈ℝ μ G m ðyÞ, and the fuzzy basis functions can be expressed by Let Λ T = ½Λ 1 , Λ 2 , ⋯, Λ k and ξ T ðxÞ = ½ξ 1 , ξ 2 , ⋯, ξ k . Based on (9), we have ξ T ðxÞξðxÞ < 1. Then, considering (8) and (9), we can obtain Moreover, we can obtain the following property when the membership function is the Gaussian function.
Lemma 5 (see [32]). If f ðxÞ is a continuous function defined on a compact set Ω, then there exists an FLSs for any given positive constant ε * > 0 satisfying where Λ * is the optimal parameter vector.
It is well known that FLSs have been used to deal with the uncertainties of nonlinear control systems owing to their universal approximation ability. In this paper, the FLSs will be used to approximate ΩðR 2i Þ.

Hysteretic Quantizer.
In this paper, the hysteresis quantizer Qð·Þ is introduced to quantify the control signal, which can be expressed as follows [33]: where and u min > 0, 0 < ρ < 1, and η = 1 − ρ/1 + ρ. Qðu i Þ is in the set of U = f0,±u ik ,±u ik ð1 + δÞg. The size of the dead-zone for Qðu i Þ is determined by u min . The map of the hysteresis quantizer Qðu i Þ for u i > 0 is given in Figure 1.
Remark 6. The parameter ρ can be considered as a measure of quantization density. From (13), it is well known that the smaller ρ is, the coarser the quantizer becomes. Furthermore, η approaches 1 while ρ approaches zero, and then Q ðu i Þ will have fewer quantization levels since u i ranges over that interval [34]. Thus, the needed communication rate between the actuator module and control module is lower.
Remark 7. Unlike the logarithmic quantizer, additional quantization levels have been added in (12) to prevent oscillations. Besides, the parameter ρ chosen should be based on a principle that the controlled system is stable can be ensured.
To facilitate the next controller design, the hysteretic quantizer Qðu i Þ is decomposed into the following form: where ϕ i ðtÞ and d i ðtÞ are nonlinear functions.
Lemma 8. The nonlinear blue functions ϕ i ðtÞ and d i ðtÞ, respectively, satisfy 3 International Journal of Aerospace Engineering Proof. From Figure 2 and according to the sector-bound properties, for ju i j ≥ u min , we have For ju i j ≤ u min , when Qðu i Þ = 0, one has Define Then,

Main Results
In this section, a novel finite-time orbit tracking control strategy for formation spacecraft will be proposed. To facilitate the control strategy propose, the intermediate variables R 1i and R 2i is defined as follows: where α 1 = 1, 0 < α 2 = 1 + τ < 1, and τ = −m/n < 0 be the ration of an even integer m and an odd integer n. σ 1i and σ 2i are the virtual controller. Under the backstepping method framework, the control strategy develop procedure is described as follows: Step 1. Propose of virtual control law σ 1i and σ 2i . Intermediate variables (19) Virtual controllers (20), (23) Actual controller (31) Adaptive law (32) Desired states (2)  International Journal of Aerospace Engineering

Consider a Lyapunov function candidate as
Design the virtual controller σ 1i as follows: The derivative of V 1 is obtained as follows: and then applying Lemmas 3 to 5 yields where ς = α 2 ð2 − α 2 Þ ð2−α 2 Þ/α 2 > 0 is a positive constant. The virtual control scheme is proposed as Then substituting (22) and (23) into above (21) Step 2. Propose of controller u i . Consider the Lyapunov function as follows: where 0 < α 2 + τ < 1 The time derivative of ϖ is derived as Note that Based on (28), we have Substituting (29) into the time derivative of V, it yields The controller u i is designed as The adaptive law of b ψ i is designed as where ℓ i > 0 and μ i > 0 are designed parameters. Substituting (31) into (30) yields 5 International Journal of Aerospace Engineering 2i ðð1/mÞðd i ðtÞ + w i ÞÞ + χðe 2i , σ 2i ÞÞ. Then, (33) can be rewritten as By using the approximation ability of FLSs in Lemma 5, we have For convenience, let where φ * T i is the optimal fuzzy weight vector, jε 1 j ≤ ε * 1 , with ε * 1 being a positive constant.

According to Lemmas 3 to 4, and noting that
The main theorem of this paper is presented as follows.
Theorem 9. Consider the dynamic model of SFF described by (1) with the hysteretic quantizer given in (14). If the controller is designed as (31), and the adaptation law is given in (32), then the tracking errors e 1i and e 2i , i = 1, 2, 3 will converge into a region of the origin within finite-time.

Proof. Denoting
Choosing 0 < γ = 2/2 − τ < 1, and according to Lemma 3, one then has According to the definition of V and V, we can obtain According to (38) and (39), one has Moreover, applying Lemma 4 where Then, International Journal of Aerospace Engineering (44), we can obtain that _ V ≤ −12κV γ if and only if V γ > 2ϑ/1 − κ, which means V < 0 out the set fV γ ≤ 2ϑ/1 − κg. The required time of reach the bounded set is T r ≤ 2V 1−γ ðe 0 Þ/κð1 − γÞ. Thus, as the analysis in [16], we can conclude that V can reach the set within finite time based on Definition 2. Furthermore, we can obtain that the tracking errors e 1i and e 2i , i = 1, 2, 3 converge into a region of the origin with finite-time.
The design procedure of the controller could be visualized from the bloc diagram shown in Figure 2.
Remark 10. By employing the property 0 < ξ T i ξ i ≤ 1 of FLSs [16], only one adaptation law for _ b ψ i is designed to proposed the controller. Furthermore, the effects of quantization errors and external disturbances can be eliminated by the designed adaptation law.
Remark 11. Compared with the finite-time controllers using the terminal sliding mode method, the finite-time controller (31) is proposed by using the adding one power integrator technique and backstepping approach to guarantee the finite-time convergence and overcome the noncontinuous and singular problem.

Illustrative Example
In this section, an illustrative example is shown to illustrate the efficiency of the proposed finite-time control scheme.
The leader spacecraft and follower spacecraft mass are m l = 1 kg and m f = 1 kg, respectively. For simplicity, the leader spacecraft is assumed in a circular reference orbit of radius 6728 km. The input control force is limited as u i ≤ u max = 1 N, i = 1, 2, 3.
The tracking errors e 1i and e 2i , ði = 1, 2, 3Þ of the finitetime controller (31) are shown Figures 3 and 4, respectively. As observed from Figure 2, the position tracking errors can converge to almost zero with 150 s, and the position tracking errors is within je 1i j < 0:01 m at 500 s. The control force is given in Figure 5, which force magnitude is limited to 1 N. Figure 6 shows the quantized control force for the finite-time controller (31). The response curves of b ψ i are shown in Figure 7. Clearly, the simulation results verify the validity of the hysteresis quantizer and illustrate the efficiency of the proposed finite-time control scheme.

Conclusion
In this paper, the problem of finite-time orbit control for SFF with limited data communication and external disturbances was studied. We have applied the hysteretic quantizer to quantize the data of the controller-actuator channel to decrease the communication rate. By combining with the adding one power integrator method and the backstepping technique, a finite-time adaptive controller has been developed. Under the proposed control strategy, the effects of quantization errors and external disturbances have been eliminated. Moreover, the finite-time stability of the controlled system and bounded of tracking errors of position and velocity within finite time are guaranteed by the developed controller. Finally, an example has been shown to illustrate the effectiveness of the proposed control strategy. Future work will focus on dealing with the finite-time tracking control problem for SFF with actuator faults.

Data Availability
The data used to support the findings of this study are included within the article.