Abstract
This work addresses the finite-time orbit control problem for spacecraft formation flying 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 finite-time tracking control strategy with adaptation law is designed to ensure that the closed-loop system is practical finite-time stable, and that the tracking errors of relative position and velocity are bounded within finite-time despite with limited data communication and external disturbances. Finally, an example is shown to validate the effectiveness of the proposed finite-time tracking controller.
1. 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 [3–6]. 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 [9–12].
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 essential 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 [13–16]. 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 [21–24]. 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 [26–28], 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.
2. Modeling and Preliminaries
2.1. Spacecraft Formation Flying Dynamic Model
The nonlinear relative motion dynamics of SFF can be described as follows [29]: where is the relative position from the follower spacecraft to the leader spacecraft in the local coordinate frame, and are the control inputs of the follower spacecraft and leader spacecraft, respectively, and denote the mass of the follower spacecraft and leader spacecraft, respectively, denotes the bounded external disturbance, denotes the Earth gravitational constant, denotes the orbit angular velocity of the leader spacecraft, and 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 and velocity tracking error are defined as where , m, and 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 and are assumed to bounded, and their first two-order time derivatives are assumed to be bounded.
Definition 2. The solution of (3) can be regarded as finite-time bounded or practical finite-time stable if for all , and there exists and , such that for .
Lemma 3 (see [30]). For , if , we have
Lemma 4 (see [31]). If and be positive constants, then is a real-valued function. We have
2.2. Fuzzy-Logic Systems
The IF-THEN rules of the FLSs are constructed: where is the input of FLSs, is the output of FLSs, and and are the fuzzy sets associate with the membership functions and , respectively. and are the number of rules. Then, the FLSs can be expressed as where , and the fuzzy basis functions can be expressed by
Let and . Based on (9), we have . 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 is a continuous function defined on a compact set , then there exists an FLSs for any given positive constant satisfying where 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 .
2.3. Hysteretic Quantizer
In this paper, the hysteresis quantizer is introduced to quantify the control signal, which can be expressed as follows [33]: where and , , and . is in the set of . The size of the dead-zone for is determined by . The map of the hysteresis quantizer for is given in Figure 1.
Remark 6. The parameter can be considered as a measure of quantization density. From (13), it is well known that the smaller is, the coarser the quantizer becomes. Furthermore, approaches 1 while approaches zero, and then will have fewer quantization levels since 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 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 is decomposed into the following form: where and are nonlinear functions.
Lemma 8. The nonlinear blue functions and , respectively, satisfy
Proof. From Figure 2 and according to the sector-bound properties, for , we have For , when , one has
Define
Then, holds, where and satisfy (15).
2.4. Control Objective
The objective of this paper is to develop a finite-time adaptive tracking controller such that the position state and velocity state can track the desired state and in finite-time, and that the tracking error states and are finite-time bounded with limited data communication and external disturbances.
3. 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 and is defined as follows: where , , and be the ration of an even integer and an odd integer . and 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 and .
Consider a Lyapunov function candidate as .
Design the virtual controller as follows:
The derivative of is obtained as follows:
and then applying Lemmas 3 to 5 yields
where is a positive constant. The virtual control scheme is proposed as
Then substituting (22) and (23) into above (21)
Step 2. Propose of controller .
Consider the Lyapunov function as follows:
where The time derivative of is derived as
Note that
Based on (28), we have
Substituting (29) into the time derivative of , it yields
The controller is designed as The adaptive law of is designed as where and are designed parameters.
Substituting (31) into (30) yields
Let . Then, (33) can be rewritten as
By using the approximation ability of FLSs in Lemma 5, we have
For convenience, let where is the optimal fuzzy weight vector, , with being a positive constant. and . According to Lemmas 3 to 4, and noting that , one has where , , and are positive design parameters, , and is the estimate of the parameter . Substituting (32) and (37) into (34) yields
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 and will converge into a region of the origin within finite-time.
Proof. Denoting , one has Choosing , and according to Lemma 3, one then has According to the definition of and , we can obtain According to (38) and (39), one has Moreover, applying Lemma 4where .
Then, where . From (44), we can obtain that if and only if , which means out the set . The required time of reach the bounded set is . Thus, as the analysis in [16], we can conclude that can reach the set within finite time based on Definition 2. Furthermore, we can obtain that the tracking errors and 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 of FLSs [16], only one adaptation law for 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.
4. 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 kg and kg, respectively. For simplicity, the leader spacecraft is assumed in a circular reference orbit of radius km. The input control force is limited as , .
The initial relative position and velocity are m and m/s, respectively. The desired relative position and relative are m and m/s, respectively. The external disturbance is . The parameters of the finite time control scheme (31) and the adaptation law (33) are chosen as , , , , , and , and the quantization parameters are designed as , .
The tracking errors and of the finite-time 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 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 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.
5. 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.