Finite-Time Second-Order Sliding Mode Controllers for Spacecraft Attitude Tracking

Abstract

The attitude tracking control problem of a spacecraft nonlinear system with external disturbances and inertia uncertainties is studied. Two robust attitude tracking controllers based on finite-time second-order sliding mode control schemes are proposed to solve this problem. For the first controller, smooth super twisting control is applied to quaternion-based spacecraft-attitude-tracking maneuvers. The second controller is developed by adding linear correction terms to the first super twisting control algorithm in order to improve the dynamic performance of the closed-loop system. Both controllers are continuous and, therefore, chattering free. The concepts of a strong Lyapunov function are employed to ensure a finite-time convergence property of the proposed controllers. Theoretical analysis shows that the resulting control laws have strong robustness and disturbance attenuation ability. Numerical simulations are also given to demonstrate the performance of the proposed control laws.

1. Introduction

In recent years, attitude tracking control of a rigid spacecraft has attached a great deal of attention. To improve the performance of closed-loop systems, various nonlinear control approaches have been proposed (see, e.g., [1–5]). The PD control approach was used in [6] to achieve asymptotic stability in the presence of model uncertainties, while passivity-based control was presented in [7] to guarantee the asymptotical convergence of attitude tracking without angular velocity measurements. In [8], a nonlinear controller was proposed for large-angle attitude control of a spacecraft. The nonlinear control parameters were determined using the linear matrix inequality (LMI) approach. Later, Luo et al. [9] developed an 𝐻_∞ inverse optimal adaptive controller for attitude tracking of spacecraft. Adaptive control and nonlinear 𝐻_∞ control were merged to design robust optimal controllers. These controllers can achieve asymptotic attitude tracking with an uncertain inertia matrix and can ensure boundedness. In [10], the attitude tracking and disturbance rejection problems of spacecraft were investigated. An adaptive controller was developed for the stabilization problem providing asymptotic rejection for a class of external disturbances by designing a compensator. The attitude tracking problem without angular velocity measurements was studied (see, e.g., [11, 12]). In [11], the authors developed an adaptive control law which incorporated a velocity filter from attitude measurements to ensure the asymptotic convergence of the attitude and angular velocity tracking errors. In [12], velocity filters were designed to compensate for the unmeasurable angular velocity and the proposed control law provided a semiglobal stability result. In [13], integrator backstepping-based control design was developed for application to the attitude maneuver problem.

In practical situations, the inertia matrix of the spacecraft cannot be exactly known, and the spacecraft is always subject to external disturbances. Sliding mode control (SMC) has been shown to be an effective method for the control of uncertain dynamical systems. Its ability to reject disturbances and parameter variations is useful for practical applications. Based on SMC schemes, the controller designs of attitude tracking control were proposed in [14, 15]. These control laws are asymptotically stable which means that the rotational motion of a rigid body can be driven to track a given desired trajectory as time trends to infinity. Obviously, it is more desirable to use finite-time control schemes, since these techniques ensure that tracking is achieved in finite time.

A recently developed technique for finite-time stabilization is terminal sliding mode (TSM) control [16, 17] which can guarantee that the system states converge to the equilibrium point in finite-time. In [18–20], controllers were designed using the TSM approach and a quaternion representation to achieve finite time convergence of both attitude and angular velocity tracking errors. However, SMC induces control chattering due to the discontinuous control law. In mechanical systems, such control signals are undesirable because they can cause damage and accelerate wear. The SMC theory has been extended in recent years to incorporate a new technique which is known as higher-order sliding mode control (HOSMC) [21, 22]. This technique preserves the main advantages of SMC and also yields improved accuracy and performance. Various real-life applications have been controlled in a practical implementation of HOSMC (see, e.g., [23–25]). However, higher-order sliding mode controllers (HOSMC) of nonlinear spacecraft systems have been rarely studied. Quasicontinuous second- and third-order sliding controllers have been presented in [26] to solve the attitude tracking control problem.

In this paper, we study spacecraft attitude tracking controller designs using HOSMC. Two finite-time second-order sliding mode control (SOSMC) algorithms are developed to design attitude tracking controllers of a rigid spacecraft. For the first controller, the smooth super twisting (ST) algorithm is applied to deal with quaternion-based spacecraft-attitude-tracking maneuvers. This control approach was proposed in [27] and the weak Lyapunov function was used to prove the stability. It is well known that a weak Lyapunov function ensures stability but cannot guarantee asymptotic stability, and finite-time convergence. The LaSalle theorem is needed to guarantee asymptotic stability and homogeneity properties are required to prove the finite-time convergence [28]. In this research, a strong Lyapunov design [29] is proposed which ensures the finite-time stability of this control law without the need for the LaSalle theorem and homogeneity properties, and even in the case when the bounded external perturbations are taken into account. In addition, a smooth modified super twisting (MST) sliding mode scheme is developed by adding linear correction terms to the first super twisting control algorithm to improve the dynamic performance of the closed-loop system. Hence, using strong Lyapunov designs, both proposed controllers can guarantee finite-time convergence to a given desired attitude motion of a rigid spacecraft.

This paper is organized as follows. Section 2 contains a description of the dynamic equations and the kinematics of attitude for the spacecraft attitude tracking problem ([30, 31]). Section 3 states the control problem. A finite-time attitude tracking controller is designed using the smooth ST algorithm. The finite convergence of the proposed controller for the resulting closed-loop system is proved theoretically. In Section 4, a controller design based on the smooth MST algorithm is provided. The stability of the closed-loop system is analyzed. Numerical simulations are given in Section 5 to demonstrate the performance of the proposed controllers. In Section 6, we present conclusions.

2. Mathematical Model of Spacecraft Attitude Tracking Control

2.1. Kinematics of the Attitude Error

We now briefly explain the use of quaternions for description of the attitude error. We define the quaternion 𝑄=[𝑞^𝑇𝑞₄]^𝑇∈ℛ³×ℛ with 𝑞=[𝑞₁𝑞₂𝑞₃]^𝑇∈ℛ³ and 𝑄_𝑟=𝑞^𝑇_𝑟𝑞_4𝑟^𝑇, (1) where 𝑞_𝑟=[𝑞_1𝑟𝑞_2𝑟𝑞_3𝑟]^𝑇∈ℛ³ is the desired reference attitude. The quaternion for the attitude error is 𝑄_𝑒=[𝑞^𝑇_𝑒𝑞_4𝑒]^𝑇∈ℛ³×ℛ with 𝑞_𝑒=[𝑞_1𝑒𝑞_2𝑒𝑞_3𝑒]^𝑇∈ℛ³. Using the multiplication law for quaternions, we then obtain [32] 𝑄_𝑒=𝑞_4𝑟𝑞−𝑞₄𝑞_𝑟−𝑞^×_𝑟𝑞𝑞₄𝑞_4𝑟+𝑞^𝑇𝑞_𝑟 (2) subject to the constraint 𝑄^𝑇_𝑒𝑄_𝑒=𝑞^𝑇𝑞+𝑞²₄𝑞^𝑇_𝑟𝑞_𝑟+𝑞²_4𝑟=1. (3) Note that a quaternion consists of the scalar 𝑞₄ and the three-dimensional vector 𝑞, so it has four components. The scalar term is used for avoidance of singular points in the attitude representation [33]. The quaternion kinematics equation is required to be solved for all four components. However, to indicate the maneuver of the spacecraft, it is sufficient to use only the vector 𝑞 because this vector properly represents both Euler axis and Euler angle. Furthermore, the scalar 𝑞₄ can be calculated easily using the vector 𝑞 and the condition ‖𝑄‖=1. For more details of quaternion and other attitude representation, see [31, 32].

The kinematic equation for the attitude error can then be expressed as (see, e.g., [10, 31]) 𝑄̇_𝑒=12𝑇𝑄_𝑒−𝑞^𝑇_𝑒𝜔_𝑒, (4) where 𝐼₃ is the 3×3 identity matrix and 𝑇(𝑄_𝑒)=𝑞^×_𝑒+𝑞_4𝑒𝐼₃.

To avoid the singularity of 𝑇(𝑄_𝑒) that will occur at 𝑞_4𝑒=0, we let the attitude of the spacecraft be restricted to the workspace 𝑊 defined by [23] 𝑊=𝑄_𝑒∣𝑄_𝑒=𝑞^𝑇_𝑒𝑞_4𝑒^𝑇,𝑞_𝑒𝑞≤𝛽<1,_4𝑒≥1−𝛽²>0, (5) where 𝛽 is a positive constant.

2.2. Dynamic Equations of the Error Rate

In [31], the dynamic equation for a rigid spacecraft rotating under the influence of body-fixed devices is given as 𝐽𝜔̇=−𝜔^×𝐽𝜔+𝑢+𝑑, (6) where 𝜔=[𝜔₁𝜔₂𝜔₃]^𝑇 is the angular rate of the spacecraft, 𝑢=[𝑢₁𝑢₂𝑢₃]^𝑇 represents the control vector, 𝑑=[𝑑₁𝑑₂𝑑₃]^𝑇 are bounded disturbances, 𝐽 is the inertia matrix, and the skew-symmetric matrix 𝜔^× is defined by 𝜔^×=0−𝜔₃𝜔₂𝜔₃0−𝜔₁−𝜔₂𝜔₁0. (7)

We denote 𝜔_𝑟=[𝜔_1𝑟𝜔_2𝑟𝜔_3𝑟]^𝑇 as the desired reference rate and 𝜔_𝑒=𝜔−𝐶𝜔_𝑟 with 𝐶=(𝑞²_4𝑒−2𝑞^𝑇_𝑒𝑞_𝑒)𝐼₃+2𝑞_𝑒𝑞^𝑇_𝑒−2𝑞_4𝑒𝑞^×_𝑒 as the error rate. Then, substitution into (6) gives the dynamic equation of the error rate in the form [32] 𝐽𝜔̇_𝑒=−𝜔_𝑒+𝐶𝜔_𝑟^×𝐽𝜔_𝑒+𝐶𝜔_𝑟𝜔+𝑢+𝑑+𝐽^×_𝑒𝐶𝜔_𝑟−𝐶𝜔̇_𝑟. (8)

Then, taking the following coordinate transformation suggested in [10] 𝑥=𝜔_𝑒+𝜆𝑞_𝑒 (9) gives 𝑞̇_𝑒=12𝑞_4𝑒𝐼₃+𝑞^×_𝑒𝜔_𝑒,𝑞̇_4𝑒1=−2𝑞^𝑇_𝑒𝜔_𝑒,𝐽𝑥̇=−𝜔_𝑒+𝐶𝜔_𝑟^×𝐽𝜔_𝑒+𝐶𝜔_𝑟+𝐽𝜔^×_𝑒𝐶𝜔_𝑟−𝐶𝜔̇_𝑟×12𝜆𝐽𝑞_4𝑒𝐼₃+𝑞^×_𝑒𝜔_𝑒+𝑢+𝑑, (10)(11) where 𝜆 is a positive constant.

Next, consider the inertia matrix containing the parameter uncertainty in the form 𝐽=𝐽₀+Δ𝐽, where 𝐽₀ is the known constant matrix which is selected nonsingular, and Δ𝐽 denotes the unmatched uncertainty. Thus, the dynamic equation (11) can be written as 𝐽₀+Δ𝐽𝑥̇=−𝜔_𝑒+𝐶𝜔_𝑟^×𝐽₀+Δ𝐽𝜔_𝑒+𝐶𝜔_𝑟+𝐽₀+Δ𝐽𝜔^×_𝑒𝐶𝜔_𝑟−𝐶𝜔̇_𝑟×𝜆2𝐽₀+Δ𝐽𝑞_4𝑒𝐼₃+𝑞^×_𝑒𝜔_𝑒+𝑢+𝑑. (12) Note that (𝐽₀+Δ𝐽)⁻¹ can be expressed as 𝐽₀+Δ𝐽⁻¹=𝐽₀⁻¹+Δ𝐽,̃ (13) where Δ𝐽̃ is also an uncertainty that can be found by expanding the first term of the expression below: Δ𝐽̃=𝐽₀+Δ𝐽⁻¹−𝐽₀⁻¹=𝐽₀⁻¹𝐼₃+𝐽₀⁻¹Δ𝐽⁻¹−𝐽₀⁻¹ (14) leading to Δ𝐽̃=𝐼₃+𝐽₀⁻¹Δ𝐽⁻¹−𝐼₃𝐽₀⁻¹ (15) provided that ‖Δ𝐽‖<1/‖𝐽₀⁻¹‖. This is a sufficient condition for the existence of Δ𝐽̃. From Banach perturbation lemma, one can obtain ‖(𝐼₃+𝐽₀⁻¹Δ𝐽)⁻¹‖≤1/(1−‖𝐽₀⁻¹‖‖Δ𝐽‖).

Thus, with some simple algebraic manipulations to (12), one obtains [19] 𝑥̇=𝐹+𝐵+𝐽₀⁻¹𝑢+𝑑, (16) where 𝐹=𝐽₀⁻¹−𝜔_𝑒+𝐶𝜔_𝑟^×𝐽₀𝜔_𝑒+𝐶𝜔_𝑟+𝐽₀𝜔^×_𝑒𝐶𝜔_𝑟−𝐶𝜔̇_𝑟+12𝜆𝐽₀𝑞_4𝑒𝐼₃+𝑞^×_𝑒𝜔_𝑒,𝐵=𝐽₀⁻¹−𝜔_𝑒+𝐶𝜔_𝑟^×Δ𝐽𝜔_𝑒+𝐶𝜔_𝑟+Δ𝐽𝜔^×_𝑒𝐶𝜔_𝑟−𝐶𝜔̇_𝑟+12𝜆Δ𝐽𝑞_4𝑒𝐼₃+𝑞^×_𝑒𝜔_𝑒+Δ𝐽̃−𝜔_𝑒+𝐶𝜔_𝑟^×𝐽𝜔_𝑒+𝐶𝜔_𝑟+𝐽𝜔^×_𝑒𝐶𝜔_𝑟−𝐶𝜔̇_𝑟+12𝜆𝐽𝑞_4𝑒𝐼₃+𝑞^×_𝑒𝜔_𝑒+Δ𝐽𝑢̃,𝑑=𝐽₀⁻¹𝑑+Δ𝐽𝑑̃. (17) The system (16) includes both parameter uncertainty 𝐵 and external disturbance 𝑑. In this section, the uncertainty and disturbance are lumped together as the total disturbances 𝑑, and thus the simplified system is obtained: 𝑥̇=𝐹+𝐵₀𝑢+𝑑, (18) where 𝑑=𝐵+𝑑 and 𝐵₀=𝐽₀⁻¹.

2.3. Finite-Time Stability

We now restate the concepts related to finite-time stability [30, 34].

Definition 1 (see [30]). Consider a time invariant system in the form of 𝑥̇=𝑓(𝑥),𝑓(0)=0,𝑥∈ℛ^𝑛, (19) where 𝑓∶𝐷→ℛ^𝑛 is continuous on an open neighborhood 𝐷 of the origin. The equilibrium 𝑥=0 of the system is (locally) finite-time stable if (i) it is asymptotically stable, in 𝑈, an open neighborhood of the origin, with 𝑈⊆𝐷; (ii) it is finite-time convergent in 𝑈; that is, for any initial condition 𝑥₀∈𝑈⧵{0}, there is a settling time 𝑇>0 such that every solution 𝑥(𝑡,𝑥₀) of system (19) is defined with 𝑥(𝑡,𝑥₀)∈𝑈⧵{0} for 𝑡∈[0,𝑇) and satisfies lim_{𝑡→𝑇(𝑥0)}𝑥𝑡,𝑥₀=0 (20) and 𝑥(𝑡,𝑥₀)=0, if 𝑡≥𝑇. Moreover, if 𝑈=ℛ^𝑛, the origin is globally finite-time stable.

Definition 2. Consider a controlled system 𝑥̇=𝑓(𝑥)+𝑔(𝑥)𝑢,𝑥∈ℛ^𝑛,𝑢∈ℛ^𝑚 (21) with 𝑔(𝑥)≠0. It is finite-time stabilizable if there is a feedback law 𝑢(𝑥) such that 𝑥=0 is a (locally) finite-time stable equilibrium of closed-loop system.

Lemma 3 (see [34]). Consider the nonlinear system described in (19). Suppose that there is a 𝐶¹ function 𝑉(𝑥)∶𝐷→ℛ defined on a neighborhood 𝑈⊆𝐷 of the origin such that the following conditions hold: (i) 𝑉(𝑥) is positive definite on 𝐷⊆ℛ^𝑛; (ii) there are real numbers 𝜚>0 and 0<𝜄<1, such that 𝑉̇+𝜚𝑉^𝜄≤0,𝑥∈𝑈⧵{0}. (22) Then, the origin of system (19) is finite-time stable. If 𝐷=ℛ^𝑛 and 𝑉(𝑥) is radially unbounded, then the origin of system (19) is globally finite-time stable.

3. Smooth ST Algorithm

In this section, the problem of attitude tracking control for a rigid spacecraft is studied. The control objective is to design the control input 𝑢_𝑓 so that the trajectory tracking errors of the state with respect to the desired state will converge to a desired region in finite time. A proof of finite convergence of the closed-loop system is provided based on the strong Lyapunov function.

In this section, we develop an attitude tracking control based on the ST algorithm. The goal is to enforce the sliding mode on the manifold 𝑠=𝑥, (23) where 𝑠=[𝑠₁𝑠₂𝑠₃]^𝑇∈ℛ³.

We now ignore the disturbance term in (18). Letting 𝑢=𝑢_𝑓, the proposed controller is given by the following: 𝑢_𝑓=𝐵₀⁻¹−𝐹−𝐾₁sign(𝑠)^{(𝑝−1)/𝑝}−𝐾₂^𝑡₀sign(𝑠(𝜏))^{(𝑝−2)/𝑝}𝑑𝜏, (24) where 𝑝≥2. The function sign(𝑠)^𝑟 is defined as sign(𝑠)^𝑟=𝑠₁^𝑟sign𝑠₁𝑠₂^𝑟sign𝑠₂,…,𝑠_𝑚^𝑟sign𝑠_𝑚^𝑇 (25) with 0<𝑟<1. The gain matrices 𝐾₁=diag(𝑘₁₁,𝑘₁₂,𝑘₁₃) and 𝐾₂=diag(𝑘₂₁,𝑘₂₂,𝑘₂₃), with 𝑘_1𝑖 and 𝑘_2𝑖, are positive gains. Differentiating (23) and substituting the result and (24) into (18), the closed-loop dynamics are obtained as 𝑠̇=−𝐾₁sign(𝑠)^{(𝑝−1)/𝑝}−𝐾₂^𝑡₀sign^{(𝑝−2)/𝑝}(𝑠(𝜏))𝑑𝜏+𝑑. (26) Let us define 𝑧=[𝑧_1𝑖𝑧_2𝑖]^𝑇 with 𝑧_1𝑖=𝑠_𝑖,𝑧_2𝑖=−𝑘_2𝑖^𝑡₀sign^{(𝑝−2)/𝑝}𝑠_𝑖(𝜏)𝑑𝜏+𝑑_𝑖,̇𝑑_𝑖(𝑡)=𝛿_𝑖(𝑡); (27) then (26) can be written in scalar form (𝑖=1,2,3) as 𝑧̇_1𝑖=−𝑘_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖+𝑧_2𝑖,𝑧̇_2𝑖=−𝑘_2𝑖𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖+𝛿_𝑖. (28) Note that (28) is in the form of the smooth super twisting algorithm [27]. Next, the proof of finite-time convergence to the origin is given.

Theorem 4. Assume that the first-time derivative of disturbance 𝑑_𝑖 is bounded and |𝛿_𝑖(𝑡)|≤𝐿, where 𝐿 is a positive constant. With 𝑘_1𝑖>0,𝑘_2𝑖>0(𝑖=1,2,3), all trajectories of the system (28) converge in finite time to the region ‖𝜗‖≤𝐿𝑞𝜀=𝜎_min𝑄^{(𝑝−1)/(𝑝−2)}, (29) where 𝜗=[|𝑧_1𝑖|^{(𝑝−1)/𝑝}sign(𝑧_1𝑖)𝑧_2𝑖] and ‖𝜗‖ denotes the Euclidean norm of 𝜗. 𝜎_min(𝑄) denote the minimum singular value of the matrix 𝑄 given by 𝑘𝑄=^1𝑖22𝑘_2𝑖+𝑘²_1𝑖2𝑝−2𝑝−𝑘_1𝑖2𝑝−2𝑝−𝑘_1𝑖2𝑝−2𝑝2𝑝−2𝑝,2𝑞=−𝑘_1𝑖. (30)

Proof. We first construct the Lyapunov function by extending the ideas of Moreno and Osorio [35]. Let the Lyapunov function be chosen as 𝑉(𝑧)=𝑘_2𝑖𝑝𝑧𝑝−1^{1𝑖2(𝑝−1)/𝑝}+12𝑧²_2𝑖+12𝑘_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖−𝑧_2𝑖², (31) which can be written as 𝑉(𝑧)=𝜗^𝑇𝑃𝜗, (32) where 𝜗=[|𝑧_1𝑖|^{(𝑝−1)/𝑝}sign(𝑧_1𝑖)𝑧_2𝑖] and 1𝑃=22𝑘_2𝑖𝑝𝑝−1+𝑘²_1𝑖−𝑘_1𝑖−𝑘_1𝑖2. (33) Note that 𝑉(𝑧) is continuous but is not differentiable at 𝑧_1𝑖=0. It is positive definite and radially unbounded if 𝑘_2𝑖>0; that is, 𝜎_min(𝑃)‖𝜗‖²≤𝑉≤𝜎_max(𝑃)‖𝜗‖², (34) where ‖𝜗‖²=|𝑧_1𝑖|^{2(𝑝−1)/𝑝}+𝑧²_2𝑖. 𝜎_min(𝑃) and 𝜎_max(𝑃) denote the minimum and maximum singular values of the matrix 𝑃. The derivative of 𝑉 can be written as 𝑉̇=𝑘_2𝑖𝑝+1𝑝−12𝑘²_1𝑖2𝑝−2𝑝𝑧_1𝑖^{(𝑝−2)/𝑝}×sign𝑧_1𝑖𝑧̇_1𝑖+2𝑧_2𝑖𝑧̇_2𝑖−𝑘_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖𝑧̇_2𝑖−𝑘_1𝑖𝑝−1𝑝𝑧_2𝑖𝑧_1𝑖^−1/𝑝𝑧̇_1𝑖. (35) Substituting (28) into (32), we obtain 𝑉̇=𝑘_2𝑖𝑝+1𝑝−12𝑘²_1𝑖2𝑝−2𝑝𝑧_1𝑖^{(𝑝−2)/𝑝}×sign𝑧_1𝑖𝑧_2𝑖−𝑘_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖+2𝑧_2𝑖−𝑘_2𝑖𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖+𝛿_𝑖−𝑘_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}×sign𝑧_1𝑖−𝑘_2𝑖𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖+𝛿_𝑖−𝑘_1𝑖𝑧_2𝑖𝑝−1𝑝𝑧_1𝑖^−1/𝑝×𝑧_2𝑖−𝑘_2𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖,𝑉̇=−𝑧_1𝑖^−1/𝑝−𝑘²_1𝑖2𝑝−2𝑝𝑧_1𝑖^{(𝑝−1)/𝑝}×sign𝑧_1𝑖𝑧_2𝑖+𝑘_1𝑖𝑘_2𝑖𝑧_1𝑖^{2(𝑝−1)/𝑝}×𝑝−1𝑝𝑘³_1𝑖𝑧_1𝑖^{2(𝑝−1)/𝑝}+𝑘_1𝑖𝑝−1𝑝𝑧²_2𝑖+2𝑧_2𝑖𝛿_𝑖−𝑘_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖𝛿_𝑖. (36) Also, 𝑉̇ can be rearranged as 𝑉̇≤−𝑧_1𝑖^−1/𝑝𝜗^𝑇𝑄𝜗+𝐿𝑞^𝑇𝜗=−𝑧_1𝑖^−1/𝑝𝜎_min𝑄‖𝜗‖²+𝐿𝑞^𝑇𝜗. (37) From (34), it follows by |𝑧_1𝑖|^−1/𝑝≥‖𝜗‖^{−1/(𝑝−1)} and (37) that 𝑉̇≤−‖𝜗‖^{−1/(𝑝−1)}𝜎_min𝑄‖𝜗‖²+𝐿𝑞‖𝜗‖≤−𝜎_min𝑄‖𝜗‖^{(2𝑝−3)/(𝑝−1)}𝑞𝜗‖𝜎+𝐿‖≤−_min𝑄‖𝜗‖^{(𝑝−2)/(𝑝−1)}−𝐿𝑞‖𝜗‖. (38) Using (34), we know that 𝑉/𝜎_max(𝑃)≤‖𝜗‖²≤𝑉/𝜎_min(𝑃). We obtain 𝑉̇≤−𝜎_min𝑄‖𝜗‖^{(𝑝−2)/(𝑝−1)}−𝐿𝑞𝑉𝜎_max(𝑃). (39) If 𝜎_min(𝑄)‖𝜗‖^{(𝑝−2)/(𝑝−1)}>𝐿‖𝑞‖, the error system (28) will finite-time converge to the region ‖𝜗‖≤𝐿𝑞𝜀=𝜎_min𝑄^{(𝑝−1)/(𝑝−2)}. (40)
In fact, we can choose 𝑘_𝑖1 and 𝑘_2𝑖 such that 𝐿‖𝑞‖/𝜎_min(𝑄)<1. With 𝑝>2,(𝑝−1)/(𝑝−2) is sufficiently large; ‖𝜗‖ is sufficiently small in finite time.

4. Smooth MST Controller Design

Next, we study the modified version of the ST algorithm. To improve the dynamical performance and reduce chattering phenomenon, a controller is developed by adding linear correction terms to the nonlinear one.

The proposed controller is given by the following: 𝑢_𝑓=𝐵₀⁻¹×−𝐹−Λ₁sign(𝑠)^{(𝑝−1)/𝑝}−Λ₂𝑠−Λ₃^𝑡₀sign(𝑠(𝜏))^{(𝑝−2)/𝑝}𝑑𝜏−Λ₄^𝑡₀𝑠(𝜏)𝑑𝜏. (41) The gain matrices Λ₁=diag(𝜆₁₁,𝜆₁₂,𝜆₁₃), Λ₂=diag(𝜆₂₁,𝜆₂₂,𝜆₂₃), Λ₃=diag(𝜆₃₁,𝜆₃₂,𝜆₃₃), and Λ₄=diag(𝜆₄₁,𝜆₄₂,𝜆₄₃) with 𝜆_1𝑖,𝜆_2𝑖,𝜆_3𝑖, and 𝜆_4𝑖 are positive gains.

On substituting (41) into (18) the closed-loop dynamics are obtained as 𝑠̇=−Λ₁sign(𝑠)^{(𝑝−1)/𝑝}−Λ₂𝑠−Λ₃^𝑡₀sign(𝑠(𝜏))^{(𝑝−2)/𝑝}𝑑𝜏−Λ₄^𝑡₀𝑠(𝜏)𝑑𝜏+𝑑. (42) Using the same procedure as in Section 3, (42) can be written as 𝑧̇_1𝑖=−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖+𝑧_2𝑖−𝜆_2𝑖𝑧_𝑖1𝑧̇_2𝑖=−𝜆_3𝑖𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖−𝜆_4𝑖𝑧_𝑖1+𝛿_𝑖. (43) The finite-time stability of the closed-loop system (43) is ensured by using the following theorem.

Theorem 5. Suppose that 𝜆_1𝑖,𝜆_2𝑖,𝜆_3𝑖,𝜆_4𝑖>0   (𝑖=1,2,3) are selected such that 4𝜆_3𝑖𝜆_4𝑖>2𝜆_3𝑖𝑝𝑝−1+𝜆²_1𝑖4𝑝−2𝑝²𝜆²_2𝑖 (44) and the first-time derivative of disturbance 𝑑_𝑖 is bounded and |𝛿_𝑖(𝑡)|≤𝐿, where 𝐿 is a positive constant. Then, all trajectories of the system (43) converge in finite time to the region 𝜉≤Δ=minΔ₁,Δ₂,Δ₁=𝐿‖Γ‖2𝜎_minΩ₂,Δ₂=𝐿‖Γ‖𝜎_min(Ω₁)^{(2𝑝−3)/(2𝑝−2)}, (45) where Δ₁,Δ₂>0 are the results from the chosen gains

Ω₁=𝜆_1𝑖22𝜆_3𝑖+𝜆²_1𝑖2𝑝−2𝑝0−𝜆_1𝑖2𝑝−2𝑝02𝜆_4𝑖+𝜆²_2𝑖6𝑝−2𝑝−𝜆_2𝑖4𝑝−2𝑝−𝜆_1𝑖2𝑝−2𝑝−𝜆_2𝑖4𝑝−2𝑝2𝑝−1𝑝,Ω₂=𝜆_2𝑖𝜆_3𝑖+𝜆²_1𝑖3𝑝−220𝜆00_4𝑖+𝜆²_2𝑖−𝜆_2𝑖0−𝜆_2𝑖1,Γ=−𝜆_1𝑖−𝜆_2𝑖2. (46)

Proof. Let the Lyapunov function be chosen as 𝑉(𝑧)=𝜆_3𝑖𝑝𝑧𝑝−1^{1𝑖2(𝑝−1)/𝑝}+𝜆_4𝑖𝑧²_1𝑖+12𝑧²_2𝑖+12𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖+𝜆_2𝑖𝑧_1𝑖−𝑧_2𝑖². (47) The selected Lyapunov function can be expressed as 𝑉(𝑧)𝜆=2^3𝑖𝑝𝑧𝑝−1^{1𝑖2(𝑝−1)/𝑝}+12𝜆²_1𝑖+𝑧²_2𝑖+𝜆_4𝑖+12𝜆²_2𝑖𝑧²_1𝑖+𝜆_1𝑖𝜆_2𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}×sign𝑧_1𝑖𝑧_1𝑖−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}×sign𝑧_1𝑖𝑧_2𝑖−𝜆_2𝑖𝑧_1𝑖𝑧_2𝑖, (48) which can be written as 𝑉(𝑧)=𝜉^𝑇Π𝜉, (49) where 𝜉=[|𝑧_1𝑖|^{(𝑝−1)/𝑝}sign(𝑧_1𝑖)𝑧_1𝑖𝑧_2𝑖]^𝑇 and 1Π=22𝜆_3𝑖𝑝𝑝−1+𝜆²_1𝑖𝜆_1𝑖𝜆_2𝑖−𝜆_1𝑖𝜆_1𝑖𝜆_2𝑖2𝜆_4𝑖+𝜆²_2𝑖−𝜆_2𝑖−𝜆_1𝑖−𝜆_2𝑖2. (50) It satisfies 𝜎_min(Π)𝜉²≤𝑉≤𝜎_max(Π)𝜉², (51) where ‖𝜉‖²=|𝑧_1𝑖|^{2(𝑝−1)/𝑝}+𝑧²_1𝑖+𝑧²_2𝑖.
Consider 𝑉̇≤2𝜆_3𝑖+𝜆²_1𝑖𝑝−1𝑝𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖𝑧̇_1𝑖+2𝑧_2𝑖𝑧̇_2𝑖+2𝜆_4𝑖+𝜆²_2𝑖𝑧_1𝑖𝑧̇_1𝑖+2𝑝−1𝑝𝜆_1𝑖𝜆_2𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖𝑧̇_1𝑖−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖𝑧̇_2𝑖−𝜆_1𝑖𝑝−1𝑝𝑧_1𝑖^−1/𝑝𝑧_2𝑖𝑧̇_1𝑖−𝜆_2𝑖𝑧_1𝑖𝑧̇_2𝑖−𝜆_2𝑖𝑧_2𝑖𝑧̇_1𝑖. (52) Substituting 𝑧̇_1𝑖 and 𝑧̇_2𝑖 into (52), we obtain 𝑉̇≤2𝜆_3𝑖+𝜆²_1𝑖𝑝−1𝑝𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖×𝑧_2𝑖−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖−𝜆_2𝑖𝑧_2𝑖+2𝑧_1𝑖−𝜆_3𝑖𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖−𝜆_4𝑖𝑧_1𝑖+𝛿_𝑖+2𝜆_4𝑖+𝜆²_2𝑖𝑧_1𝑖×𝑧_2𝑖−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖−𝜆_2𝑖𝑧_1𝑖+𝜆_1𝑖𝜆_2𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖×𝑧_2𝑖−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖−𝜆_2𝑖𝑧_1𝑖+𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖×−𝜆_3𝑖𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖−𝜆_4𝑖𝑧_1𝑖+𝛿_𝑖−𝜆_1𝑖𝑧_2𝑖𝑝−1𝑝𝑧_1𝑖^−1/𝑝𝑧_1𝑖×𝑧_2𝑖−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖−𝜆_2𝑖𝑧_1𝑖−𝜆_1𝑖𝑧_1𝑖−𝜆_3𝑖𝑧_1𝑖^{(𝑝−2)/𝑝}sign𝑧_1𝑖𝜆_4𝑖𝑧_1𝑖+𝛿_𝑖−𝜆_2𝑖𝑧_2𝑖𝑧_2𝑖−𝜆_1𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖−𝜆_2𝑖𝑧_1𝑖. (53)
After lengthy algebraic manipulation, one obtains 𝑉̇≤−𝑧_1𝑖^−1/𝑝×𝜆_1𝑖𝜆_3𝑖+𝜆³_1𝑖𝑝−1𝑝𝑧_1𝑖^{2(𝑝−1)/𝑝}−2𝜆²_1𝑖𝑝−1𝑝𝑧_1𝑖^{(𝑝−1)/𝑝}×sign𝑧_1𝑖𝑧_2𝑖+𝜆_1𝑖𝜆_4𝑖+12𝜆_1𝑖𝜆²_1𝑖6𝑝−2𝑝𝑧²_1𝑖−2𝜆_1𝑖𝜆_2𝑖4𝑝−2𝑝𝑧_1𝑖𝑧_2𝑖+𝑝−1𝑝𝜆_1𝑖𝑧²_2𝑖−𝜆_2𝑖𝜆_3𝑖+𝜆³_2𝑖3𝑝−2𝑝𝑧_1𝑖^{2(𝑝−1)/𝑝}+𝜆_2𝑖𝜆_4𝑖+𝜆³_2𝑖𝑧²_1𝑖+2𝜆²_2𝑖𝑧_1𝑖𝑧_2𝑖−𝜆_2𝑖𝑧²_2𝑖−𝛿_𝑖𝑧_1𝑖^{(𝑝−1)/𝑝}sign𝑧_1𝑖−𝛿_𝑖𝜆_2𝑖𝑧_1𝑖+2𝛿_𝑖𝑧_2𝑖, (54) which can be written as 𝑉̇=−𝑧_1𝑖^−1/𝑝𝜉^𝑇Ω₁𝜉−𝜉^𝑇Ω₂𝜉+𝛿^𝑇Γ𝜉. (55) Note that with positive values of 𝜆_1𝑖, 𝜆_2𝑖, 𝜆_3𝑖, and 𝜆_4𝑖, it can be ensured that Ω₂ is positive definite. Also, the condition (44) is required to guarantee that Ω₁ is positive definite.
Therefore, we have 𝑉̇≤−𝑧_1𝑖^−1/𝑝𝜎_minΩ₁𝜉²−𝜎_minΩ₂𝜉²+𝐿‖Γ‖𝜉. (56) Using |𝑧_1𝑖|^−1/𝑝≥‖𝜉‖^{−1/(𝑝−1)}, one obtains 𝑉̇≤−𝑧_1𝑖^−1/𝑝𝜎_minΩ₁𝜉²−𝜎_minΩ₂𝜉²+𝐿‖Γ‖𝜉≤−𝜎_minΩ₁𝜉^{(2𝑝−3)/(𝑝−1)}−𝜎_minΩ₂𝜉²+𝐿‖Γ‖𝜉. (57) We can change (57) into the following forms: 𝑉̇≤−𝜉𝜎_minΩ₁𝜉^{(𝑝−2)/(𝑝−1)}−𝐿‖Γ‖−𝜎_minΩ₂𝜉²𝑉̇≤−𝜎_minΩ₁𝜉^{(2𝑝−3)/(𝑝−1)}−𝜉𝜎_minΩ₂𝜉−𝐿‖Γ‖. (58) Therefore, if we choose the gains such that 𝜎_minΩ₁−𝐿‖Γ‖𝜉^{(2𝑝−3)/(𝑝−1)}𝜎>0,_minΩ₂−𝐿‖Γ‖𝜉>0, (59) then the finite-time stability is guaranteed. Let positive scalars 𝜂₁ and 𝜂₂ be defined as 𝜂₁=𝜎_minΩ₁−𝐿‖Γ‖𝜉^{(2𝑝−3)/(𝑝−1)},𝜂₂=𝜎_minΩ₂−𝐿‖Γ‖𝜉. (60) Then, (58) can be respectively written as 𝑉̇≤−𝜂₁𝜉^{(2𝑝−3)/(𝑝−1)}−𝜎_minΩ₂𝜉²𝑉̇≤−𝜎_minΩ₁𝜉^{(2𝑝−3)/(𝑝−1)}−𝜂₁𝜉². (61)
From (51), we get 𝑉/𝜎_max(Π)≤‖𝜉‖²≤𝑉/𝜎_min(Π). Therefore, we obtain ̇𝜂𝑉≤−¹𝜎_max(Π)^{(2𝑝−3)/(2𝑝−2)}𝑉^{(2𝑝−3)/(2𝑝−2)}−𝜎_minΩ₂𝜎_max(Π)𝑉𝑉̇≤−𝜎_minΩ₁𝜎_max(Π)^{(2𝑝−3)/(2𝑝−2)}𝑉^{(2𝑝−3)/(2𝑝−2)}−𝜂₂𝜎_max(Π)𝑉. (62)
Therefore, if we choose 𝜆_1𝑖, 𝜆_2𝑖, 𝜆_3𝑖, and 𝜆_4𝑖 such that the conditions (59) are satisfied, the finite-time convergence property can be obtained. Meanwhile, the gains guarantee the system trajectory will finite-time converge to the region 𝜉≤Δ=minΔ₁,Δ₂,Δ₁=𝐿‖Γ‖2𝜎_minΩ₂,Δ₂=𝐿‖Γ‖𝜎_min(Ω₁)^{(2𝑝−3)/(2𝑝−2)}. (63) In fact, we can choose 𝜆_1𝑖, 𝜆_2𝑖, 𝜆_3𝑖, and 𝜆_4𝑖 such that (𝐿‖Γ‖)/(𝜎_min(Ω₁))<1. With 𝑝>2, (2𝑝−3)/(2𝑝−2) is sufficiently large, so that Δ₂ can be greatly reduced with Δ₂≪Δ₁. This means that the proposed control has strong robustness and disturbance rejection ability.

For (56), we can find that when the system state is far from the origin, −|𝑧_1𝑖|^−1/𝑝𝜎_min(Ω₁)‖𝜉‖² is small and −𝜎_min(Ω₂)‖𝜉‖² is large, so the behavior of −𝜎_min(Ω₂)‖𝜉‖²is strong. On the other hand around, the origin, −|𝑧_1𝑖|^−1/𝑝𝜎_min(Ω₁)‖𝜉‖² is large and −𝜎_min(Ω₂)‖𝜉‖² is small, so the behavior of −|𝑧_1𝑖|^−1/𝑝𝜎_min(Ω₁)‖𝜉‖² is strong. Hence, the behaviors are strong during the whole convergence.

5. Simulation Results

An example of a rigid-body satellite [10] is presented with numerical simulations to compare the performance of the developed controllers (24) and (41) with the controller (1) in [20]. The spacecraft is assumed to have the nominal inertia matrix 𝐽=201.20.91.2171.40.91.415kg⋅m² (64) and the parameter uncertainties Δ𝐽=diagsin(0.1𝑡)2sin(0.2𝑡)3sin(0.2𝑡)kg⋅m². (65) The attitude control problem is considered in the presence of external disturbance 𝑑(𝑡). The external disturbances are described as 𝑑(𝑡)=()()()0.1sin0.1𝑡0.2sin0.2𝑡0.3sin0.3𝑡N-m. (66) In this numerical simulation, we assume that the desired angular velocity is given by 𝜔_𝑟(𝑡)=0.05sin𝜋𝑡100sin2𝜋𝑡100sin3𝜋𝑡100rad/s. (67) The initial condition of the unit quaternion is 𝑄(0)=[0.3−0.2−0.30.8832]^𝑇 and the initial target unit quaternion is 𝑄_𝑟(0)=[0001]^𝑇. The initial value of the angular velocity is 𝜔(0)=[000]^𝑇 rad/s.

For the smooth ST controller (24), the chosen gains are given as 𝐾₁=diag(2.0,2.0,2.0) and 𝐾₂=diag(2.5,2.5,2.5). On the other hand, for the smooth MST controller (41), the control gains are selected to be Λ₁=diag(2.0,2.0,2.0), Λ₂=diag(0.5,0.5,0.5), Λ₃=diag(2.0,2.0,2.0), and Λ₄=diag(1.5,1.5,1.5). For both control algorithms, we use the same sliding manifold (23). The corresponding parameters in controller (1) in [20] are selected.

Simulation results with the controller (1) in [20] are presented in Figures 1–4. As shown in Figures 1 and 2, attitude and angular velocity tracking errors converge to zero in about 15 seconds. Figure 3 shows very the fast convergence property of the controller (1) in [20]. From Figure 4, it can be seen that chattering cannot be avoided. The boundary layer ‖𝑠‖≤1.9×10⁻⁴ is reached in finite time. Regarding the accuracy, the bounds on ‖𝑞_𝑒‖ and ‖𝜔_𝑒‖ are ‖𝑞_𝑒‖≤7.1×10⁻⁵ and ‖𝜔_𝑒‖≤1.1×10⁻⁴ with the sampling time ℎ=0.005.

Figure 1 

Quaternion tracking errors—Controller (1) in [20].

Figure 2 

Angular velocity tracking errors—Controller (1) in [20].

Figure 3 

Switching function—Controller (1) in [20].

Figure 4 

Control torques—Controller (1) in [20].

For simulation results with the controllers (24) and (41), the attitude quaternion tracking errors are shown in Figures 5 and 9, and angular velocity tracking errors are illustrated in Figures 6 and 10. The smooth MST controller (41) gives smoother angular velocity tracking outputs than the controller in (24). From Figures 7 and 11, it can be seen that the sliding vectors are on the sliding surface 𝑠=0 after 5 seconds. As shown in Figures 8 and 12, the response of the control variables obtained by the controller (24) has a quick change in a short time for the first 5 seconds, while the controller (41) provides smoother responses of control torques. For the controller (24), the boundary layer ‖𝑠‖≤5×10⁻⁷ is achieved in finite time. Regarding the accuracy, the bounds on ‖𝑞_𝑒‖ and ‖𝜔_𝑒‖ are ‖𝑞_𝑒‖≤2×10⁻⁷ and ‖𝜔_𝑒‖≤6×10⁻⁷ with the sampling time ℎ=0.005. Also, for the controller (41), the boundary layer ‖𝑠‖≤3.2×10⁻⁷ is reached in finite time. Regarding the accuracy, the bounds on ‖𝑞_𝑒‖ and ‖𝜔_𝑒‖ are ‖𝑞_𝑒‖≤9.9×10⁻⁸ and ‖𝜔_𝑒‖≤2×10⁻⁷ with the sampling time ℎ=0.005.

Figure 5 

Quaternion tracking errors —smooth ST controller (24).

Figure 6 

Angular velocity tracking errors—smooth ST controller (24).

Figure 7 

Switching function—smooth ST controller (24).

Figure 8 

Control torques—smooth ST controller (24).

Figure 9 

Quaternion tracking errors—smooth MST controller (41).

Figure 10 

Angular velocity tracking errors—smooth MST controller (41).

Figure 11 

Switching function—smooth MST controller (41).

Figure 12 

Control torques—smooth MST controller (41).

A comparison of the simulation results obtained by the smooth ST controller, smooth MST controller, and the controller (1) in [20] shows the following. Although the controller (1) in [20] gives the fastest convergence rate, it can be seen that the undesired chattering is not avoided. Moreover, it cannot provide high-precision tracking performance. The smooth MST controller gives outstanding accuracy and provides smoother response of control torques when compared with the smooth ST approach. In view of these simulation results, the smooth MST controller (41) seems to give the best overall control for practical attitude tracking control of a rigid spacecraft.

6. Conclusion

The proposed finite robust attitude tracking controllers have been successfully applied to spacecraft tracking maneuvers. For the first controller, the smooth ST control is applied to deal with quaternion-based spacecraft-attitude-tracking maneuvers. The second controller is developed by adding the linear correction terms to the first super twisting control algorithm to improve the dynamic performance of the closed-loop system. The concepts of the strong Lyapunov function are employed to ensure a finite-time property of the proposed controllers. The presented methods provide very good accuracy of tracking results and reduce the undesired chattering. Moreover, strong robustness ability is obtained and is shown by using proposed Lyapunov functions. Numerical simulations are also given to demonstrate the performance of the proposed control laws.