A Second-Order Sliding Mode Controller Design for Spacecraft Tracking Control

Abstract

For spacecraft attitude tracking system, there exists the chattering phenomenon. In this paper, the spacecraft motion is decomposed into three-channel subsystems, and a second-order sliding mode control is proposed. This method has been proved to have good convergence and robustness. Combined with the proposed sliding surface, the three-channel controllers are designed. The control performance is confirmed by the simulation results, the approaching process is improved effectively, and a smooth transition is achieved without overshoot and buffeting.

1. Introduction

Sliding mode variable structure control (SMC) is very suitable for spacecraft control system design because of its unique advantages, such as the simple algorithm, no need of the precise mathematical model for the controlled object, and the invariability of the control system when the parameter perturbation exists and the external disturbance conforms exactly to the matching condition [1].

But classical variable structure control system has a fatal flaw, namely “chattering” phenomenon, which is the main obstacle for SMC to promote in the engineering field [2]. In order to eliminate or suppress the chattering, scholars have conducted a wide-ranging and in-depth study [3–5]. In the past, the solution was to use the continuous function instead of the discontinuous parts, which had played a certain role in weakening buffeting, but it also greatly reduced the control performance [6]. A variable structure terminal guidance law with adaptive quick-reaching law and a filter approach based on sliding mode observer was developed [7], and the simulation results demonstrated higher accuracy, better robustness, and easy realization of the designed guidance law. For the guidance of attacking target on the ground, a new robust nonlinear sliding mode guidance law with terminal angle constraint was proposed [8]. Wu et al. introduced some slack matrices and a delay-dependent sufficient condition to guarantee the sliding mode dynamics is generalized, quadratically stable, and robustly passive [9, 10]. And Wu et al. designed an integral sliding surface function and an observer to estimate the system states, which was proved to be attained in a finite time [11, 12]. An integral-type sliding surface function is designed for establishing a sliding mode dynamics, which can be formulated by a switched stochastic system with an external disturbance/uncertainty [13]. But the control methods used are based on a general variable structure control theory, which is very unfavorable for the normal operation of spacecraft control system due to the existence of the chattering phenomenon; it could easily lead to system instability. And in the application of general variable structure control theory, the controlled object must be a relative degree 1. When the controlled object relative degree is greater than 1, this theory cannot be used.

During the flight, the aerodynamic parameters change intensely, and the coupling between three channels is very serious, which is caused by the flight speed, dramatically changed air density, and other disturbances. It brings about great difficulties because the entire spacecraft control system presents a strong multivariable coupling, time-varying, nonlinearity and uncertainty. The traditional method of spacecraft control system design is based on the assumption of small perturbation and the coefficient of frozen [14, 15]. However, the large angle of aircraft maneuvering is inevitable. Therefore, it is difficult to reconcile three-channel controller design based on the classical control theory. BTT control technology is a future trend in this field, but it would make the coupling aggravate [16–18]. In the design of spacecraft control system, even if the spacecraft is regarded as a particle, the relative motion between spacecraft and target is strongly nonlinear [19, 20].

Due to the reasons above, it will encounter enormous difficulties to use classical control theory to design the spacecraft control system in future [21, 22]. Therefore, it is necessary to search a high-level sliding variable structure control method for the spacecraft control system, which cannot only keep strong robustness but also can achieve continuous control [4, 23–26]. Shtessel et al. proposed the second-order sliding mode variable structure control algorithm, which was proved to be good robustness and to eliminate effectively the chattering [27–29]. A second-order sliding mode controller is designed [30] for the aerodynamic missile, which was simple and effective. The results of the simulation indicated that this second-order sliding mode controller is robust for the systems with uncertainties and can effectively reduce the chatting.

However, there are certain limitations to the high-order sliding mode control algorithm, such as the demanding controlled object and the complexity of algorithm. The paper conducts a new research on the second-order sliding mode variable structure control system, and simulation results show that the method we proposed can effectively eliminate the chattering phenomenon and have good robustness in the spacecraft control systems. The research results have injected new idea for the sliding mode variable structure control algorithm and also have provided a new solution for the spacecraft control system design. Therefore, the study of this subject has not only theoretical significance but also has a very important application reference value.

In this paper, we are concerned with the chattering phenomenon in the spacecraft attitude tracking system. The spacecraft motion is decomposed into three-channel subsystems, and a second-order sliding mode control is proposed, which has good convergence and robustness. Combined with the proposed sliding surface, the three-channel controllers are designed. Furthermore, the control performance is confirmed by the simulation results, the approaching process is improved effectively, and a smooth transition is achieved without overshoot and buffeting. The rest of this paper consists of five sections about reaching condition and controller design. The spacecraft model is described in Section 2, and a three-channel decomposition model is proposed. Section 3 proposes a second-order sliding mode. Section 4 designs the three-channel controller. Section 5 validates control performance. Finally, we conclude this paper in Section 6.

2. Spacecraft Model

The spacecraft attitude motion equations introduced in [31] is considered here. The model are described as follows: 𝐽_𝑥𝑑𝜔_𝑥𝑑𝑡=𝑀_𝑥−𝐽_𝑧−𝐽_𝑦𝜔_𝑧𝜔_𝑦,𝐽_𝑦𝑑𝜔_𝑦𝑑𝑡=𝑀_𝑦−𝐽_𝑥−𝐽_𝑧𝜔_𝑧𝜔_𝑥,𝐽_𝑧𝑑𝜔_𝑧𝑑𝑡=𝑀_𝑧−𝐽_𝑦−𝐽_𝑥𝜔_𝑥𝜔_𝑦,𝜗̇=𝜔_𝑦sin𝛾+𝜔_𝑧̇1cos𝛾,𝜓=𝜔cos𝜗^𝑦cos𝛾−𝜔_𝑧,̇sin𝛾𝛾=𝜔_𝑥−tan𝜗𝜔_𝑦cos𝛾−𝜔_𝑧sin𝛾, (1) where the pitch angle 𝜗, the yaw angle 𝜓, and the roll angle 𝛾 are referred to the posture angle. 𝜔_𝑥, 𝜔_𝑦, and 𝜔_𝑧 are the angular velocity of the spacecraft around the shaft 𝑂𝑥, 𝑂𝑦, and 𝑂𝑧. 𝐽_𝑥, 𝐽_𝑦, and 𝐽_𝑧 are the moments of inertia, 𝑀_𝑥, 𝑀_𝑦, and 𝑀_𝑧 are the external torques on the body frame.

From (1), we can see that the spacecraft attitude motion equation is nonlinear. And because of the existence of the large attitude angle, the coupling between three channels cannot be ignored.

2.1. Decomposition of Mathematical Model

In order to make the process of decomposition clear, (1) can be broken down into three subsystems.

(i) Channel Subsystem of Pitch Angle. The state function for the channel subsystem of pitch angle can be obtained by Figure 1.

Figure 1 

The state function for the channel subsystem of pitch angle.

From Figure 1, we can obtain the following state function: 𝑥̇_1𝑧𝑥̇_2𝑧=01𝑥00𝑥^1𝑧_2𝑧+0cos𝛾𝐽_𝑧𝑢_𝑧+0𝑓_𝑧, (2) where 𝑢_𝑧=𝑀_𝑧, which is the controller of pitch channel subsystem.

And 𝑓_𝑧=𝜔̇_𝑦sin𝛾+𝜔_𝑦𝛾̇cos𝛾−(cos𝛾/𝐽_𝑧)(𝐽_𝑦−𝐽_𝑥)𝜔_𝑥𝜔_𝑦−𝜔_𝑧𝛾̇sin𝛾 is the coupling between the channels, which can be regarded as the external interference.

(ii) Channel Subsystem of Yaw Angle. The state function for the channel subsystem of yaw angle can be obtained by Figure 2.

Figure 2 

The state function for the channel subsystem of yaw angle.

From Figure 2, we can obtain the following state function: 𝑥̇_1𝑦𝑥̇_2𝑦=01𝑥00𝑥^1𝑦_2𝑦+0cos𝛾cos𝜗𝐽_𝑦𝑢_𝑦+0𝑓_𝑦, (3) where 𝑢_𝑦=𝑀_𝑦 is the controller of yaw channel subsystem, and 𝑓_𝑦=cos𝛾cos𝜗𝐽_𝑦𝐽_𝑧−𝐽_𝑥𝜔_𝑥𝜔_𝑧+cos𝛾sin𝜗𝜗̇−sin𝛾𝛾̇cos𝜗cos²𝜗𝜔_𝑦−𝜔̇_𝑧sin𝛾−̇̇cos𝜗cos𝛾cos𝜗𝛾+sin𝛾𝜗sin𝜗cos²𝜗𝜔_𝑧 (4) can be regarded as the external interference.

(iii) Channel Subsystem of Roll Angle. The state function for the channel subsystem of roll angle can be obtained by Figure 3.

Figure 3 

The state function for the channel subsystem of roll angle.

From Figure 3, we can obtain the following state function: 𝑥̇_1𝑥𝑥̇_2𝑥=01𝑥00𝑥^1𝑥_2𝑥+01𝐽_𝑥𝑢_𝑥+0𝑓_𝑥, (5) where 𝑢_𝑥=𝑀_𝑥 is the controller of roll channel subsystem, and 𝑓_𝑥=𝐽_𝑦−𝐽_𝑧𝐽_𝑥𝜔_𝑧𝜔_𝑦−sec²𝜗⋅𝜗̇𝜔_𝑦cos𝛾−𝜔_𝑧𝜔̇sin𝛾−tan𝜗_𝑦cos𝛾−𝜔_𝑦𝛾̇sin𝛾−𝜔̇_𝑧sin𝛾−𝜔_𝑧𝛾̇cos𝛾 (6) can be regarded as the external interference.

The process of deduction regards the coupling between three channels as the external interference. When the spacecraft carries a maneuver of large angle attitude, the coupling will be exacerbated, which will affect the work of three subsystems seriously. Therefore, in the design of controller, we should make the control system have better robustness.

3. Reaching Condition Design and Proof

The system (2) can be expressed as: 𝑥̇(𝑡)=𝐴(𝑥)+𝐵(𝑥)𝑢+𝑓(𝑥), (7) where 𝑥∈𝑅^𝑛, 𝑢∈𝑅^𝑚, 𝐴∈𝑅^𝑛, 𝐵∈𝑅^𝑛×𝑚, and 𝐵(𝑥)=(𝑏₁,𝑏₂,…,𝑏_𝑚); each component of the vector function 𝐴(𝑥) and 𝑏_𝑖(𝑖=1,…,𝑚) are sufficiently smooth scalar functions. 𝑓(𝑥) is the external interference. The phase trajectory of the system (2) will be significantly affected due to the existence of 𝑓(𝑥).

3.1. Model System Description and Simplification

Assume that 𝑓(𝑥)=0, and the system description can be simplified as 𝑥̇(𝑡)=𝐴(𝑥)+𝐵(𝑥)𝑢. (8) Designing the sliding mode surface 𝜎=𝑥₁, due to the theory [32] of second-order sliding mode, we can get the following function: 𝜎̇=𝑥₂,𝜎̈=𝑎(𝑥)+𝑏(𝑥)𝑢. (9)

3.2. Reaching Conditions

Define the following reaching condition: 𝑠̈=−𝑘₁𝑠̇−𝑘₂𝑠−𝜀|𝑠|^𝛼sat(𝑠), (10) where 𝑘₁>0, 𝑘₂>0, 𝜀>0, 𝛼>1, and sat(⋅) indicates the saturation function, which is used to eliminate chattering (𝑠)=|𝑠|1sat1𝑠>Δ𝑘𝑠≤Δ−1𝑠<−Δ𝑘=Δ. (11)

Theorem 1. By satisfying the reaching condition (10), the phase trajectory of the system (8) can reach the set points Ω={𝑥∣𝑠(̇𝑥)=𝑠(𝑥)=0} at a limited time and stay at this state.

Proof. Define that ℎ(𝑠)=𝑘₂𝑠+𝜀|𝑠|^𝛼sat(𝑠), and we can get 𝑠ℎ(𝑠)>0. Here, we select the Lyapunov function below 𝑉=0.5𝑠²̇+^𝑠₀ℎ𝑦𝑑𝑦 (12) and 𝑉>0. The time derivative of formula (12) can be obtained as 𝑉̇=𝑠̇−𝑘₁𝑠̇−𝑘₂𝑠−𝜀|𝑠|^𝛼sat(𝑠)+ℎ(𝑠)𝑠.̇ (13) By substituting ℎ(𝑠)=𝑘₂𝑠+𝜀|𝑠|^𝛼sat(𝑠) into (13), 𝑉̇ can hold as 𝑉̇=−𝑘₁𝑠²̇. (14)
Next, we will make some analysis about 𝑉̇.(i)When 𝑠̇≠0, 𝑉̇<0; that is to say that 𝑉→0. And in order to make lim_{|𝑠|→+∞}∫^𝑠₀ℎ(𝑦)𝑑𝑦=0, there must be lim_𝑡→+∞𝑠=0 for an arbitrary initial position of the system.(ii)When 𝑠̇=0, 𝑠̈=0. At this time, ℎ(𝑠)=𝑘₂𝑠+𝜀|𝑠|^𝛼sat(𝑠)=0. Because of 𝑠ℎ(𝑠)>0, 𝑠=0 is the only zero for this function. Therefore, 𝑠≡0.
According to (i) (ii) above, 𝑠 will be zero in the end, and the phase trajectory of system (8) will reach the set points Ω={𝑥∣𝑠(̇𝑥)=𝑠(𝑥)=0} asymptotically and stay at this state.

Explanation. In the reaching condition 𝑠̈=−𝑘₁𝑠̇−𝑘₂𝑠−𝜀|𝑠|^𝛼sat(𝑠) above, the reaching speed will become faster when 𝑘₂𝑠+𝜀|𝑠|^𝛼sat(𝑠) gets bigger, but the phase trajectory will resuilt in larger chattering in neighborhood of 𝑠=0. However, if we make 𝑘₂𝑠+𝜀|𝑠|^𝛼sat(𝑠) smaller, the reaching time will become longer with the decrease of speed. Besides, the reaching speed will slow down if the coefficient 𝑘₁ is too large, and the chattering in neighborhood of 𝑠=0 will be weakened in this condition. On the contrary, if the coefficient 𝑘₁ is too small, the reaching speed will enlarge and the phase trajectory will result in larger chattering in neighborhood of 𝑠=0.

3.3. Robustness Prove

Combining (9) with (10), we can get 𝑎(𝑥)+𝑏(𝑥)𝑢=−𝑘₁𝑠̇−𝑘₂𝑠−𝜀|𝑠|^𝛼sat(𝑠), and the expression of controller can hold as 𝑢=𝑏⁻¹(𝑥)−𝑘₁𝑠̇−𝑘₂𝑠−𝜀|𝑠|^𝛼sat(𝑠)−𝑎(𝑥). (15) Here, the equivalent external interference is considered, namely, 𝑓(𝑥)≠0. Then, we can get 𝜎̇=𝑥₂,𝜎̈=𝑎(𝑥)+𝑏(𝑥)𝑢+𝑓(𝑥). (16) Thus, by substituting (15) into (16), the sliding mode surface can be formulated as 𝑠̈=−𝑘₁𝑠̇−𝑘₂𝑠−𝜀|𝑠|^𝛼sat(𝑠)+𝑓(𝑥). (17)

Assumption 2. 𝑓(𝑥) is bounded, there exists |𝑓|≤𝑀, 𝑖=1,…,𝑚.

Theorem 3. By satisfying the Assumption 2 and 𝜀>𝑀(𝑖=1,2,…,𝑚), the phase trajectory of the system (7) can reach the set points Ω={𝑥∣𝑠(̇𝑥)=𝑠(𝑥)=0} under the control law (15) and stay at this state.

Proof. Define that ℎ(𝑠)=𝑘₂𝑠+𝜀|𝑠|^𝛼sat(𝑠)−𝑓, and we can get 𝑠ℎ(𝑠)=𝑘₂𝑠²+𝜀|𝑠|^𝛼+1−𝑓𝑠≥𝑘₂𝑠²+𝜀|𝑠|^𝛼+1−𝑀𝑠=𝑘₂𝑠²+𝜀|𝑠|^𝛼−𝑀𝑠. (18)(i)When |𝑠|>^𝛼𝑀/𝜀, 𝑠ℎ(𝑠)≥0.(ii)When |𝑠|<^𝛼𝑀/𝜀, |𝑠|^𝛼+1→0, 𝛼→1, 𝑘₂>𝜀. 𝑠ℎ(𝑠)≥𝑘₂𝑠²−𝑀𝑠=𝑘₂𝑀𝜀−𝑀>0. (19)Here, we select the Lyapunov function below 𝑉=0.5𝑠²̇+^𝑠₀ℎ𝑦𝑑𝑦. (20) The process of proof is the same as Theorem 1. Then, we can get the conclusions of 𝑉̇=−𝑘₁𝑠²̇ and lim_𝑡→+∞𝑠=0.

From Theorems 1 and 3, formula (17) can be regarded as the system controller regardless of the equivalent external interference if only it meets Assumption 2. So, the robustness of the controller can be reflected. Theoretically, we can make 𝑘₁ and 𝜀 large enough to achieve the better robustness and dynamic performance. However, in practical system, we will encounter limited energy of the controller; thus, it is impossible to make 𝑘₁ and 𝜀 infinite.

4. Controller Design

Combined with the proposed second order sliding mode control law (10), we can get the following controllers.

4.1. Controller Design of Pitch Channel

Assume that pitch angle 𝜗 tracks a given angle 𝜗_𝑑 and 𝜗_𝑑 is second-order derivative. Meanwhile, the equivalent external interference 𝑓_𝑧 is bounded.

Let 𝑥_𝑧=𝑥_1𝑧𝑥_2𝑧^𝑇, and in order to use the sliding mode variable structure control of second-order conveniently, we select 𝑠_𝑧=𝑒=𝑥_1𝑧−𝜗_𝑑. And then, we can get the following functions: 𝑠_𝑧̇=𝑥_2𝑧−𝜗̇_𝑑,𝑠_𝑧̈=𝑥̇_2𝑧−𝜗̈_𝑑=cos𝛾𝐽_𝑧𝑢_𝑧−𝜗̈_𝑑. (21)(22) And if 𝑠_𝑧 is regarded as the output of system (2), its relative order is 2, which satisfies the condition of Theorem 1. Then, we select the reaching condition as follows: 𝑠_𝑧̈=−𝑘_1𝑧𝑠_𝑧̇−𝑘_2𝑧𝑠_𝑧−𝜀_𝑧|𝑠|^𝛼sat𝑠_𝑧, (23) where 𝑘_1𝑧>0, 𝑘_2𝑧>0, and 𝜀_𝑧>0.

Combining formula (22) with formula (23), we can get −𝑘_1𝑧𝑠_𝑧̇−𝑘_2𝑧𝑠_𝑧−𝜀_𝑧|𝑠|^𝛼sat𝑠_𝑧=cos𝛾𝐽_𝑧𝑢_𝑧−𝜗̈_𝑑. (24) Therefore, 𝑢_𝑧=𝐽_𝑧̈𝜗cos𝛾^𝑑−𝑘_1𝑧𝑠_𝑧̇−𝑘_2𝑧𝑠_𝑧−𝜀_𝑧|𝑠|^𝛼sat𝑠_𝑧. (25)

From the deduction above, we know that 𝑠_𝑧 will converge to zero in a finite time. That is to say that the pitch angle 𝜗 will be on track for the given angle 𝜗_𝑑 in a limited time.

4.2. Controller Design of Yaw Channel

The process of deduction is the same as pitch channel. For the system (3), assume that the yaw angle 𝜓 tracks a given angle 𝜓_𝑑 and 𝜓_𝑑 is second-order derivative. Meanwhile, the equivalent external interference 𝑓_𝑦 is bounded.

Let 𝑥_𝑦=𝑥_1𝑦𝑥_2𝑦^𝑇, and we select 𝑠_𝑦=𝑒=𝑥_1𝑦−𝜓_𝑑. And then, we can get the following functions: 𝑠_𝑦̇=𝑥_2𝑦−𝜓̇_𝑑,𝑠_𝑦̈=𝑥̇_2𝑦−𝜓̈_𝑑=cos𝛾cos𝜗𝐽_𝑦𝑢_𝑦−𝜓̈_𝑑. (26)(27) Regarding 𝑠_𝑦 as the output of system (3), we select the reaching condition as follows: 𝑠_𝑦̈=−𝑘_1𝑦𝑠_𝑦̇−𝑘_2𝑦𝑠_𝑦−𝜀_𝑦|𝑠|^𝛼sat𝑠_𝑦, (28) where 𝑘_1𝑦>0, 𝑘_2𝑦>0, and 𝜀_𝑦>0.

Combining formula (27) with formula (28), we can get 𝑢_𝑦=cos𝜗𝐽_𝑦𝜗̈cos𝛾^𝑑−𝑘_1𝑦𝑠_𝑦̇−𝑘_2𝑦𝑠_𝑦−𝜀_𝑦|𝑠|^𝛼sat𝑠_𝑦. (29)

Thus, 𝑠_𝑦 will converge to zero in a finite time. That is to say that the yaw angle 𝜓 will be on track for the given angle 𝜓_𝑑 in a limited time.

4.3. Controller Design of Roll Channel

For the system (5), assume that the roll angle 𝛾 tracks a given angle 𝛾_𝑑 and 𝛾_𝑑 is second-order derivative. Meanwhile, the external outside interference 𝑓_𝑥 is bounded.

Let 𝑥_𝑥=[𝑥_1𝑥𝑥_2𝑥]^𝑇, and we select 𝑠_𝑥=𝑒=𝑥_1𝑥−𝛾_𝑑. Then, we can get the following functions: 𝑠_𝑥̇=𝑥_2𝑥−𝛾̇_𝑑,𝑠_𝑥̈=𝑥̇_2𝑥−𝛾̈_𝑑=1𝐽_𝑥𝑢_𝑥−𝛾̈_𝑑. (30) Regarding 𝑠_𝑥 as the output of system (5), we select the reaching condition as follows: 𝑠_𝑥̈=−𝑘_1𝑥𝑠_𝑥̇−𝑘_2𝑥𝑠_𝑥−𝜀_𝑥|𝑠|^𝛼sat𝑠_𝑥, (31) where 𝑘_1𝑥>0, 𝑘_2𝑥>0, and 𝜀_𝑥>0.

And we can get 𝑢_𝑥=𝐽_𝑥𝛾̈_𝑑−𝑘_1𝑥𝑠_𝑥̇−𝑘_2𝑥𝑠_𝑥−𝜀_𝑥|𝑠|^𝛼sat𝑠_𝑥. (32) Thus, 𝑠_𝑥 will converge to zero in a finite time. That is to say that the yaw angle 𝛾 will be on track for the given angle 𝛾_𝑑 in a limited time.

In the spacecraft control system, we want roll angle to calm to zero, and in order to achieve target tracking, generally, we let the pitch angle and yaw angle track a given angle.

5. Simulation Results

In this section, experimental simulations will be carried out to evaluate the effectiveness of the proposed sliding mode controller of second order: 𝑥̇_1𝑧𝑥̇_2𝑧𝑥̇_1𝑦𝑥̇_2𝑦𝑥̇_1𝑥𝑥̇_2𝑥=010000𝑥000000000100000000000001000000^1𝑧𝑥_2𝑧𝑥_1𝑦𝑥_2𝑦𝑥_1𝑥𝑥_2𝑥+000cos𝑥_1𝑥𝐽_𝑧000000cos𝑥_1𝑥cos𝑥_1𝑧𝐽_𝑦0100000𝐽_𝑥𝑢_𝑧𝑢_𝑦𝑢_𝑥+0𝑓_𝑧0𝑓_𝑦0𝑓_𝑥. (33) Detailed parameters of this control system are presented as follows. The initial angles are 𝜗𝜓𝛾=𝜋/6𝜋/3𝜋/4; the angular accelerations are 𝜗̇𝜓̇𝛾̇=0.20.40.5. And the desired tracking angles are 𝜗_𝑑𝜓_𝑑𝛾_𝑑=𝜋/4𝜋/60. The related parameters of system are 𝐽_𝑥=224kg⋅m², 𝐽_𝑦=284kg⋅m², and 𝐽_𝑧=324kg⋅m². The equivalent interference 𝑓_𝑧=𝑓_𝑦=𝑓_𝑥=3sin𝑡. Then, formula (33) can be reduced as follows: 𝑥̇_1𝑧𝑥̇_2𝑧𝑥̇_1𝑦𝑥̇_2𝑦𝑥̇_1𝑥𝑥̇_2𝑥=010000𝑥000000000100000000000001000000^1𝑧𝑥_2𝑧𝑥_1𝑦𝑥_2𝑦𝑥_1𝑥𝑥_2𝑥+000cos𝑥_1𝑥032400000cos𝑥_1𝑥284cos𝑥_1𝑧0100000𝑢224^𝑧𝑢_𝑦𝑢_𝑥+000.3sin𝑡3sin𝑡3sin𝑡 (34) The three sliding surface are selected as 𝑠_𝑧=𝑥_1𝑧−𝜋/4, 𝑠_𝑦=𝑥_1𝑦−𝜋/6, and 𝑠_𝑥=𝑥_1𝑥. Firstly, we assume that 𝑓_𝑧=𝑓_𝑦=𝑓_𝑥=0, and we can get the following functions: 𝑠_𝑧̇=𝑥_2𝑧,𝑠_𝑧̈=𝑥̇_2𝑧=cos𝑥_1𝑥𝑢324^𝑧,𝑠_𝑦̇=𝑥_2𝑧,𝑠_𝑦̈=𝑥̇_2𝑧=cos𝑥_1𝑥284cos𝑥_1𝑧𝑢_𝑦,𝑠_𝑥̇=𝑥_2𝑥,𝑠_𝑥̈=𝑥̇_2𝑥=1𝑢224^𝑥. (35)(36)(37)(38)(39)(40) And we select the same reaching conditions: 𝑠_𝑧̈=−8𝑠_𝑧̇−5𝑠_𝑧−5𝑠_𝑧^𝛼sat𝑠_𝑧,𝑠_𝑦̈=−8𝑠_𝑦̇−5𝑠_𝑦−5𝑠_𝑦^𝛼sat𝑠_𝑦,𝑠_𝑥̈=−8𝑠_𝑥̇−5𝑠_𝑥−𝑠_𝑥^𝛼sat𝑠_𝑥. (41) By taking formulas (36), (38), (40), and (41), the control law of each channel can hold as 𝑢_𝑧=324cos𝑥_1𝑥−8𝑠_𝑧̇−5𝑠_𝑧−5𝑠_𝑧^𝛼sat𝑠_𝑧,𝑢_𝑦=284cos𝑥_1𝑧cos𝑥_1𝑥−8𝑠_𝑦̇−5𝑠_𝑦−5𝑠_𝑦^𝛼sat𝑠_𝑦,𝑢_𝑥=224−8𝑠_𝑥̇−5𝑠_𝑥−𝑠_𝑥^𝛼sat𝑠_𝑥. (42)

Under the control law (42), the phase trajectory of three-channel will reach the sliding mode in a finite time. Simulation diagrams are as shown in Figures 4, 5, and 6.

Figure 4 

The angle tracking of the three subsystems.

Figure 5 

The switching function curve of the three subsystems.

Figure 6 

The control output of the three subsystems.

From Figure 4, we can see that the proposed control law achieves a better angle tracking, and no overshoot occurs. From Figure 5, it can be seen that the proposed sliding surface achieves a smooth transition in the whole process of approaching, and there is no chattering. By Figure 6, we can see that chattering of the controllers reduces significantly during the approaching process under a certain external interference.

6. Conclusion

A three-channel model of spacecraft has been analysed in this paper. By simplifying the model, a second-order sliding surface was proposed, and its conditions of convergence and robustness were analyzed. Combined with the proposed sliding surface, three-channel controllers were designed, respectively. The control performance was confirmed by the simulation results, the approaching process was improved effectively, and a smooth transition was achieved without overshoot and buffeting. The coupling between the three channels was fully taken into account in the design of the controller, and it was equivalent to external interference, which provided a high practical value.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 60904087 and 51109045), and the Fundamental Research Funds for the Central Universities of China (nos. HEUCF110419 and HEUCFX41302).