Nonsingular Fast Terminal Adaptive Neuro-sliding Mode Control for Spacecraft Formation Flying Systems

Lin, Xiaohan; Shi, Xiaoping; Li, Shilun; Nguang, Sing Kiong; Zhang, Liruo

doi:https://doi.org/10.1155/2020/5875191

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Abstract Introduction Preliminaries Numerical Simulation Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 5875191 | https://doi.org/10.1155/2020/5875191

Nonsingular Fast Terminal Adaptive Neuro-sliding Mode Control for Spacecraft Formation Flying Systems

Xiaohan Lin,¹Xiaoping Shi,¹Shilun Li,²Sing Kiong Nguang,³and Liruo Zhang³

Academic Editor: Xianming Zhang

Received09 Mar 2020

Revised23 Apr 2020

Accepted11 May 2020

Published30 May 2020

Abstract

In this paper, a nonsingular fast terminal adaptive neurosliding mode control for spacecraft formation flying systems is investigated. First, a supertwisting disturbance observer is employed to estimate external disturbances in the system. Second, a fast nonsingular terminal sliding mode control law is proposed to guarantee the tracking errors of the spacecraft formation converge to zero in finite time. Third, for the unknown parts in the spacecraft formation flying dynamics, we proposed an adaptive neurosliding mode control law to compensate them. The proposed sliding mode control laws not only achieve the formation but also alleviate the effect of the chattering. Finally, simulations are used to demonstrate the effectiveness of the proposed control laws.

1. Introduction

With the increasing applications of the spacecraft formation flying system, the importance of improving system performance of complicated spacecraft with practical control design has gained much attention over recent years [1–5]. In the spacecraft formation flying system, stability is the most essential problem to be solved [6]. Many control methods have been applied to solve these problems in the literature, such as the adaptive control [7], sliding mode control [8], backstepping technique control [9], fuzzy control [10], optimal control [11], and control [12]. However, along with the goal of stabilization for the spacecraft, the disturbances and uncertainties should also be considered, which is often neglected in some research [13].

Sliding mode control is widely used due to its advantages of robustness against external disturbances [14–16] and parameter uncertainties [17], especially in spacecraft formation flying system. In sliding mode control, chattering is a common phenomenon that often degrades system performance and even causes instability. To deal with this problem, the terminal sliding mode control is developed, in which some nonlinear terms are introduced in the sliding surface. In [18], the case of spacecraft final approach, combining with the continuously differential collision avoidance state constraint function is studied and an improved terminal sliding surface is presented. In [19], based on the conditions of external disturbances and inertial uncertain parameters, Ran et al. have proposed the adaptive fuzzy terminal sliding mode control for the second-order spacecraft attitude manoeuvre system and the convergence time of the sliding surface has been reduced. In [20], a new nonsingular terminal sliding mode (NTSM) surface has been introduced to eliminate singularity within the finite time. An adaptive nonsingular terminal sliding mode control also has been presented for an attitude tracking of spacecraft with actuator faults to avoid singularity [21]. Although several drawbacks of the sliding mode control have been compensated by the modified control laws, due to the high-frequency reaching control term in the control input [22], the chattering problem still needs to be solved. In [23], a fractional order control law based on the nonsingular terminal supertwisting sliding mode control is proposed to achieve system stability and accurately estimate the unknown model. Consequently, the nonsingular fast terminal sliding mode surface has the ability to reduce the chattering and solve the singularity problem.

In practical spacecraft formation flying systems, internal and external disturbances are inevitable. Many control methods have been used to solve this problem [24, 25]. The disturbance observers have been widely considered to estimate the disturbances [26]. In [27], an extended state observer is introduced to estimate disturbances for the multiagent systems with unknown nonlinear dynamics and disturbances. Furthermore, by simultaneously taking the output measurement and delayed control input into account, the extended observer has been used to estimate the unmeasurable system states and the additive disturbances [28]. To have a smaller estimation error of the disturbance, high-order disturbance observers have been investigated in [29]. Among all types of high-order observer, the supertwisting disturbance observer stands out due to its ability of avoiding the excessively high order observer gains that can cause sensor noise within fixed bandwidth [30].

Despite the external disturbances, uncertainties in practical formation flying systems are also inevitable; however, their impacts on the system’s performance have not been studied in most of the aforementioned results. In practice, parameter uncertainties may have direct impact on system performance [31]; hence, it is critical to deal with the problem of uncertainties. Due to the advantage of the neural networks in approximating nonlinear functions, it can be applied in dealing with the nonlinearity in the spacecraft formation systems as well [32–34]. Based on neural network and backstepping techniques, a distributed adaptive coordinated control algorithm has been proposed in [35], which effectively solves the chattering problem. To achieve robustness, in [36], a robust adaptive fuzzy PID-type sliding mode control has been designed with the neural network, with which the system stability is ensured and the transient performance is improved. In [37], the global adaptive neural control for a general class of nonlinear robot manipulators has been proposed, in which the transient performance and robustness are both improved. Compared with other neural networks [38–40], the Radial Basis Function (RBF) neural network stands out given its strong robustness and excellent performance of nonlinear approximation [41]. Therefore, it is desirable to use the RBF neural network to estimate and compensate for the nonlinear functions in the spacecraft formation flying system.

Motivated by the above observations, in this paper, a novel nonsingular fast terminal adaptive neurosliding mode control for the spacecraft formation flying system is proposed. The proposed control law guarantees that the tracking errors converge to zero in finite time. By using an adaptive RBF neural network, the uncertainties in the spacecraft formation flying system is compensated. The contributions of the paper can be summarized as follows:(1)External disturbances are estimated by a supertwisting disturbance observer in finite time. The supertwisting disturbance observer can avoid high observer gains that may amplify the noises of the sensors.(2)Compared with [42], our novel fast nonsingular terminal sliding mode control law achieves the stability of spacecraft formation flying system within finite time. Moreover, the singularity is avoided and the chattering problem is also alleviated.(3)If some of the nonlinear dynamics in the spacecraft formation system are unknown, an adaptive RBF neural network control law is proposed to approximate those nonlinear dynamics.

The remaining parts of the paper are arranged as follows. The background of the spacecraft formation flying system is introduced in Section 2. In Section 3, the proposed control laws and stability analysis are presented. Simulation results are given to verify the effectiveness of the proposed control laws in Section 4. Finally, some conclusions are given in Section 5.

1.1. Notations

The following notations are used throughout this paper. For a vector and , the standard sign function is defined as . The Euclidean norm of vector is denoted as . For a matrix , denotes the maximum absolute column sum norm. is the 2-norm, where means the maximum eigenvalue. denotes the trace of . denotes identity matrix. For a three-dimensional vector , the skew-symmetric matrix is defined as follows:

2. Spacecraft Formation Systems and Preliminaries

In this section, the spacecraft formation flying system and some useful lemmas are presented.

2.1. Spacecraft Formation Flying System

The relative orbit motion under the chaser spacecraft coordinate frame is shown in Figure 1. denotes the equatorial inertial coordinate frame (), is located at the centroid of the target spacecraft, and the coordinate axis of the target spacecraft coincides with the main inertia axis , the -axis points to the solar panel, the -axis is perpendicular to the longitudinal plane of -axis, and -axis satisfies the rule of the right hand. The chaser spacecraft coordinate frame is similar to the coordinate frame .

Assume that the active control force of the target spacecraft is zero, and the dynamic models of the target spacecraft and the chaser spacecraft are given as follows:where and denote the masses of the target spacecraft and the chaser spacecraft and and denote the external perturbation forces of the target spacecraft and the chaser spacecraft, respectively. is the control force of the chaser spacecraft. and , where and denote the position vectors of the target spacecraft and chaser spacecraft to the geocentric, respectively.

Define ; then, from (2) and (3) the relative acceleration vector is obtained as

Furthermore, let denote the projection vector in the chaser spacecraft body coordinate frame , which denotes

Then,where is the angular velocity of relative to the inertial coordinate frame under frame.

In this paper, the desired relative position and velocity under frame are assumed to be and 0, respectively. Then, the error vector can be expressed as ; then, (6) can be written as the Euler–Lagrange equation form to express the orbit relative motion:where

The Modified Rodrigues Parameters (MRPs) are used to describe the attitude dynamics of the chaser spacecraft which can be defined aswhere denotes attitude quaternion and and denote Euler axis and Euler angular of the attitude rotation, respectively.

Then, the attitude kinematics and the dynamics of the chaser spacecraft are given aswhere denotes the inertia matrix of the chaser spacecraft and and denote the control torque and the external disturbance torque of the chaser spacecraft, respectively.

Defining the error attitude and the error angular velocity aswhere denotes the desired attitude and denotes the desired angular velocity. Furthermore, the attitude model of the chaser spacecraft can be expressed as the Euler–Lagrange form:where

In order to describe the attitude-orbit coupling dynamics motion of the spacecraft formation uniformly, define

Then, based on (5) and (22), the attitude-orbit coupling model is given aswhere

Remark 1. It can be seen in (12), the attitude and orbit in the spacecraft formation flying system are coupled. Therefore, it is necessary to build the spacecraft coupling motion model (15) and (16) to enhance the control accuracy and efficiency. From (15) and (16), it can be seen that the coupling of attitude and orbit is reflected by the matrix and the vector simultaneously. represents the attitude effect on orbit. In addition, the gravity gradient torque contains orbit information, and the disturbance torque can reflect the influence of the orbit on the attitude, thus can reflect the influence of the orbit on the attitude.

Assumption 1. The external disturbance vector in the spacecraft formation flying system (15) and (16) is assumed to satisfy and , where and .

Assumption 2. All the state variables of the spacecraft formation flying system are assumed to be measurable.

2.2. Preliminary Knowledge

In this section, some related lemmas and definitions in this paper are given below.

Lemma 1 (see [43]). For , and , where is continuous with respect to on an open neighborhood of the origin . Suppose that there are a positive-definite function (defined on , where is a neighborhood of the origin), real numbers and , such that is negative semidefinite on . Then, is locally finite-time convergent, or equivalently, and becomes 0 locally in finite time, with its setting time for a given initial condition in a neighborhood of the origin in .

Lemma 2 (see [44]). Suppose Lyapunov function is defined on a neighborhood of the origin, and in which , . Then, there exists an area such that any that start from can reach within finite time in which is the initial time of .

Definition 1. (see [45]). For the autonomous system,where and , if the Lyapunov function satisfies the following conditions:(a) is a positive continuous differentiable function(b)There exist , , , and an open neighborhood containing the origin, such that the inequality holds; then, it can be concluded that the origin is the finite time stable equilibrium of system (20)

3. Control Law Design

In this section, first, we employ a supertwisting observer to estimate the external disturbances in the system. Second, we propose a fast nonsingular terminal sliding mode control law along with the supertwisting disturbance observer for the spacecraft formation flying system. In the last section, the nonlinear matrices and in (18) and (19), respectively, are assumed to be unknown, and an adaptive neurosliding model control law for the spacecraft formation flying system is proposed.

3.1. The Fast Nonsingular Terminal Sliding Mode Control Law with a Supertwisting Disturbance Observer

The block diagram of the fast nonsingular terminal sliding mode control law with the supertwisting disturbance observer is shown in Figure 2. In Figure 2, it can be seen that the overall control law consists of two parts: a fast nonsingular terminal sliding mode control law along with the supertwisting disturbance observer. The supertwisting disturbance observer is applied to estimate the disturbances for the spacecraft formation flying system if the system dynamics are known.

3.1.1. Supertwisting Disturbance Observer

In practical applications, the disturbances in the spacecraft formation flying system are fast time-varying disturbances. Hence, it is difficult for conventional observers to estimate those disturbances. In this section, a supertwisting disturbance observer is employed to estimate those disturbances. The supertwisting disturbance observer can estimate the disturbance in finite time and also it avoids high-order observer gains with fixed bandwidth.

According to the attitude-orbit coupling system (15) and (16), the disturbance vector can be rewritten as follows:

Define the state variables as follows:

Then, the disturbance dynamic system for the spacecraft formation flying is given as

For each disturbance in the spacecraft formation flying system, the disturbance’s model is given as follows:

The supertwisting disturbance observer is given as follows:where and denote the positive observer gains, and are the estimations of the state vectors and , and and are the estimation errors, respectively.

According to the attitude-orbit coupling system (15) and (16) and the supertwisting state observer (26), the estimation errors can be rewritten as

Define the state vectors , where , .

Then, the derivatives of the state vectors and can be written as

Similar to [46], the following Lyapunov function for (28) is constructed:where .

Then, the derivative of Lyapunov function with respect to the time is given as

Furthermore, (30) can be written as follows:

Since , (31) can be rewritten as

Define the following matrix as

Then, (32) can be concluded as

From (33), by selecting appropriate observer gains, the matrix can be a positive definite matrix. Then, (34) becomeswhere is the minimum eigenvalue of .

must to be a positive definite matrix to be Lyapunov matrix; hence, the observer gains and need to be selected appropriately so that . Let us define the minimal and maximal eigenvalues of as and , respectively. Then, we have

Furthermore, by observing from the state vectors and , the following can be concluded:

Finally, combining (36) with (37), (35) can be derived as

Based on (37) and Lemma 1, we prove that and with . Furthermore, the estimations errors of the state vector and can converge in finite time, and the convergence time is .

Remark 2. To guarantee that the trajectory errors converge to zero within finite time, the supertwisting disturbance observer gains and should be selected appropriately so that and are positive definite matrices.

3.1.2. Fast Nonsingular Terminal Sliding Mode Control Law with Supertwisting Disturbance Observer

To design a fast nonsingular terminal sliding mode control law, the following sliding mode surface is chosen:where , , and are all positive constants.

If the spacecraft formation flying system state errors reach the sliding mode , then

From (15), substituting the equation into (40), we have

In order to analyze the fast convergence of the nonsingular fast terminal sliding mode surface, assume that the integration time of the state variable denotes as and , respectively. Then, integrating the time along both sides of (41),

Furthermore, it can be seen from (42) that the tracking errors of the spacecraft formation flying system can converge to zero in a limited time .

Remark 3. The proposed nonsingular fast terminal sliding mode surface (39) indicates that when the system state variable is far from the equilibrium point, the term plays the major role in making the system trajectory converge at a fast rate. When approaches to the equilibrium point, the term plays the dominant role to achieve rapid convergence of the state trajectory.

A design procedure for a fast nonsingular terminal sliding mode control law with a supertwisting disturbance observer is given in the following theorem.

Theorem 1. Consider the spacecraft formation flying system described by (15) and (16) withwhereand is the reaching law with , , and . Then, the proposed control law along with the supertwisting disturbance observer (26) can guarantee the tracking errors of the spacecraft formation flying system converge to zero in finite time.

Proof. From (39), the derivative of the sliding surface is given as follows:Consider the following Lyapunov function:Taking the time derivative of (48), we haveSubstituting the proposed control law (44)–(46) into (49),Define , , and , thenDefine and select ; we haveFinally, define , then (52) can be rewritten as

Remark 4. When the system state variable is far from the fast nonsingular terminal sliding mode surface, the term in the reaching law plays the major role. On the contrary, the term in the reaching law dominates. By choosing different and and selecting appropriate parameters in the fast terminal sliding mode surface, the convergence speed can be controlled. Moreover, compared with the conventional reaching law in [47], the proposed reaching law has faster convergence and better sliding performance. When and , the speed of the state variable approaching the sliding surface is reduced to zero, which can alleviate the chattering phenomenon.

3.2. Fast Nonsingular Terminal Sliding Mode Control Law Based on the Adaptive Neural Network

In this section, an adaptive RBF neural network is employed to approximate the unknown nonlinear and in the spacecraft formation flying system (15) and (16), which is not easy to be precisely estimated in practical situations. The block diagram of a fast nonsingular terminal sliding mode control law based on the adaptive neural network is depicted in Figure 3.

The structure of RBF neural network with multiple inputs and multiple outputs is shown in Figure 4.

It can be seen that the RBF neural network has three layers including input layer, output layer, and hidden layer. Denote as the input, and is the Gaussian function of the th neuron in the hidden layer. Then, the Gaussian function can be expressed as below:where is the th neural radial basis function and is the width of the th neural radial basis function. In this section, we choose to approximate and compensate the unknown nonlinear parts in the spacecraft formation flying system, where is the unknown nonlinear term. We rewrite as follows:where denotes the ideal weighting matrix and denotes the approximation error, .

The estimation of can be expressed as follows:where denotes the estimation of .

Theorem 2. Consider the spacecraft formation flying system described by (15) and (16), with the following control law and the adaptive updating law of the RBF neural network:wherewhere is ith element of the term , , and is a positive symmetrical matrix. The tracking errors of the spacecraft formation flying system will converge to zero.

Proof. Define the error of the weighting vector as . Consider the following Lyapunov function:Taking the derivative of (63) with respect to time and substituting the proposed control law (57)–(62) and the adaptive updating law (62), then (63) can be converted asSinceTherefore,Since(66) can be rewritten as follows:Define , then we havewhere .
Therefore, it can be concluded that the tracking errors of the spacecraft formation flying system converge to zero and the proof is completed.

4. Numerical Simulation

In this section, the simulation result that shows the effectiveness of the supertwisting disturbance observer, the nonsingular fast terminal sliding mode control law, and the adaptive neurosliding mode control law are presented, respectively, in each section.

The parameters in the formation flying system are given as follows: the masses of the target and the chaser spacecraft are , the initial orbit parameters of the target spacecraft are set as the semimajor axis , the eccentricity , the orbit inclination , and the ascending node is , and apogee and perigee are both .

The inertial matrix of the chaser spacecraft is , and the disturbance torque . The desired attitude and position are given as and .

4.1. Supertwisting Disturbance Observer

The gains of the supertwisting disturbance observer are chosen as and . In order to demonstrate the advantage of the proposed supertwisting disturbance observer, we compare its performance with the traditional linear observer as below:

The and in (70) are selected as

The disturbance estimation error given by the linear disturbance observer (70) and the proposed supertwisting disturbance observer (26) are depicted in Figures 5(a) and 5(b), respectively.

(a)

(b)

Remark 5. It can be seen from Figure 5 that the supertwisting disturbance observer estimation error converges to zero in finite time, whereas the estimation error from the linear disturbance observer takes longer time to converge to zero. Therefore, the proposed supertwisting observer can estimate the disturbance in finite time, which demonstrates its advantage over the linear one in the spacecraft formation flying system.

4.2. The Nonsingular Fast Terminal Sliding Mode Control Law with Supertwisting Disturbance Observer

The parameters for the nonsingular fast terminal sliding mode surface are set as , , , and . The parameters in the control law are set as , , , and .

The state responses of and under the proposed control sliding model control law (43) are given in Figures 6 and 7, respectively.

Furthermore, to illustrate the effectiveness of the proposed control law, in this part, we compare it with a widely used nonlinear PD control law given as follows:where the coefficient and are calculated as follows:

The state responses of vectors and under the nonlinear PD control law are given in Figures 8 and 9.

Remark 6. From Figures 6–9, it can be seen that the proposed control law (43) yields better transient responses as compared to the nonlinear PD control law. Also, it is obvious that the proposed control law (43) guarantees the tracking errors converge to zero in a finite time.

4.3. Adaptive Neurosliding Mode Control Law

The parameters of the RBF neural network, is evenly selected over the space and . 200 neurons are used to approximate each . .

The state responses of the system under the RBF neural network control law (57) are given in Figures 10 and 11.

Remark 7. Figures 10 and 11 indicate that the tracking errors converge to a small region of zero with a good adaptive performance. This concludes that the RBF neural network control law (57) is able to compensate for the unknown dynamics in the spacecraft formation flying system.

5. Conclusion

This paper has proposed the nonsingular fast terminal sliding mode control laws to solve the tracking control problem for spacecraft formation flying systems. First, a supertwisting disturbance observer has been introduced to estimate the disturbances in finite time. Then, based on a nonsingular fast terminal sliding mode surface, a novel control law has been presented to guarantee the tracking errors of the spacecraft formation converge to zero in finite time. Furthermore, for the unknown nonlinear parts in the spacecraft formation flying system, a RBF neural network has been proposed to approximate them and then compensate them. Finally, via simulations the effectiveness of the proposed control laws has been demonstrated. In our future work, we will investigate (1) the spacecraft formation system subject to the time-delay on terminal adaptive neurosliding mode control [48, 49] and (2) nonlinear observer design for the spacecraft formation system.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

All authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Scholarship from China Scholarship Council.

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Copyright © 2020 Xiaohan Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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