Global finite-time set stabilization of spacecraft attitude with disturbances using second-order sliding mode control

The performance of attitude stabilization control algorithms for rigid spacecraft can be limited by disturbances. In this paper, the global finite-time attitude stabilization problem with disturbances is investigated and handled by constructing a second-order sliding mode controller. Firstly, a virtual controller based on set stabilization idea is constructed to globally finite-time stabilize the system. Then, a relay polynomial second-order sliding mode controller is constructed to guarantee that the tracking error toward the virtual controller will converge to zero in finite-time. Finite-time Lyapunov theory is applied to support the proof and stability analysis. The global finite-time stability holds even with bounded disturbances. The effectiveness and feasibility of the controller are illustrated by the numerical simulations.


Introduction
In recent years, the attitude stabilization control of the spacecraft has attracted extensive attentions due to its prominent role in space missions [1][2][3], such as spacecraft pointing, maneuvering and alignment. It is well known that the linear controllers, e.g., the PID controllers [4][5][6], are widely adopted in spacecraft due to their simple structure and easy implementation. However, PID controllers are hard to satisfy the requirements of high performance owing to the existence of couplings, nonlinearities and disturbances in spacecraft dynamics. Thus, the ensuing research efforts are devoted to the nonlinear approaches. These available control schemes include, but not limited to, backstepping control [7,8], H ∞ control [9,10], inverse optimal control [11,12], sliding mode control (SMC) [13,14] and adaptive control [15,16]. These advanced control algorithms focus on improving the performance of spacecraft attitude control from different aspects. Worth mentioning, the afore-mentioned methods only derive the asymptotic stability of the system, which means that the convergence time of the system states to the origin is infinite. In order to improve the dynamic performance of the attitude control system, finite-time control schemes have attracted the investigate attentions from researchers.
Results have shown that faster dynamic responses, higher steady-state accuracies, and better robust performance can be achieved by applying finite-time con-trol approaches as compared with the corresponding asymptotically stable ones [17,18]. After years of research, significant progress has been made in finitetime control, among which numerous satisfactory approaches have been presented. These approaches can be divided into following categories: homogeneous control, adding a power integrator control (APIC), terminal sliding mode control (TSMC) and higherorder sliding mode control (HOSM). In [19], the actuator saturation is taken into account and the finitetime controller is constructed by using homogeneous method. The output feedback homogeneous controller is designed in [20], which considers the actuator saturation and rate as well. In [21,22], two different finitetime APIC controllers are constructed to guarantee the finite-time convergence of the system states. In [23], the disturbance observer-based APIC controller is constructed to handle the problem of finite-time attitude stabilization under mismatched disturbances. However, there exist some limitations when homogeneous control and APIC are applied in the attitude control problems. For the homogeneous controllers, disturbances in the spacecraft dynamics cannot be handled [24]. For the APIC approach, only the finite-time boundedness can be ensured when dealing with disturbances.
Among the algorithms investigated for finite-time attitude stabilization, finite-time SMC approaches have been proved to be effective owning to their robustness to disturbances. In [25], a TSMC method is proposed and the system states can be stabilized in finite time, while the unexpected singularity problem may occur in the implementation. In order to handle this problem, two different nonsingular TSMC (NTSMC) methods are proposed in [26,27]. To accelerate the convergence rates when the system states are far from the equilibrium, the fast NTSMC is proposed in [28]. Moreover, in [29], a fixed-time SMC is presented and the convergence time is independent of the initial states. Apart from the methods mentioned above, there exist various extensions of TSMC to improve the control performance from different aspects, e.g., observer-based output feedback TSMC [30][31][32], actuator saturated TSMC [33,34], adaptive TSMC [35,36], etc. In addition, the discontinuous control inputs lead to the chattering phenomenon. To this end, HOSM methods are applied in spacecraft to address chattering and guarantee the finite-time convergence as well. In [37], an integral second-order SMC (SOSM) is constructed, by which the finite-time stability of the system is guaranteed.
On this basis, a third-order SMC method is proposed in [38]. By hiding the discontinuous switching in the derivative of the control inputs, the continuous control inputs are derived. However, the above two papers involve the inverse of a time-varying matrix, which may be irreversible in some specific states; thus, the singularity problem may occur.
To sum up, the aforementioned papers provide various finite-time control methods for the attitude stabilization of spacecraft. However, there still exists one main problem that the results are not global. It is well known that the quaternion is usually used to describe the arbitrary attitude motion in the three-dimensional space globally. The attitude systems under quaternionbased descriptions all have two equilibria. For these systems, the stability involved is called set stability and its definition can be found in [39,40]. For the literature using quaternion-based attitude descriptions, the global result means that the system is set stable, specifically, both of the two equilibria should be designed to be stable. If only one of the equilibria is designed to be stable while the other equilibrium is designed to be unstable, the attractive domain of the stable equilibrium will cover the global state space excluding one particular point. In this case, the entire system states will converge to the stable equilibrium although some states are closer to the unstable one, which is called 'unwinding' phenomenon [41]. Hence, the controllers that only consider one equilibrium fail to derive set stability and their results can only be considered as almost global results like that in [42].
The global attitude stabilization methods can be seen in [43][44][45] and the references therein. In [43], the idea of set stability is combined with the SOSM to obtain global stability. However, since the sliding mode surface selected in this paper is linear, the system is asymptotically stable. In [44], an adaptive tripodal hybrid control scheme is raised to globally stabilize the system, but the system cannot ensure the finite-time convergence as well. In [45], by applying the set stabilization idea, the system states will converge to different equilibria depending on the initial states, however, only in the absence of disturbances. There are fewer controllers which can obtain global finite-time results with disturbances. Therefore, the motivation of this investigation is to design a controller which can derive global finitetime stability of the system when dealing with disturbances.
In this paper, inspired by the set stabilization idea, a relay polynomial SOSM (RPSOSM) controller is constructed by using a backstepping-like way to handle the global finite-time attitude stabilization problem. Firstly, a virtual controller is constructed to globally stabilize the system. Then the SOSM controller is designed. The finite-time tracking toward the virtual controller and the global finite-time stability of the system are proved based on Lyapunov theory. The major remarkable features are listed as follows: (1) Since the unit quaternion is adopted for the description of spacecraft attitude system, the set stability is introduced to design a global controller. In this control framework, the two equilibria of the spacecraft attitude closed-loop system can be stable, which avoiding the so-called unwinding phenomenon. (2) A discontinuous second-order sliding mode controller is designed to suppress the disturbances in the spacecraft dynamics. The controller guarantees the finite-time stability of the closed-loop system in the presence of disturbances. Meanwhile, to alleviative the chattering phenomenon, a modified continuous controller is also presented by using the saturation function. (3) Compared with the existing controllers, the presented controller in this paper has the advantages of fewer parameters and simple structure, which can reduce the difficulties in parameter adjustments and controller implementations.
The rest of the paper is organized as follows. In Sect. 2, we present some lemmas, after which the dynamic and kinematic model of spacecraft attitude are provided. Also, the problem formulation is established at the end of the section. In Sect. 3, the design progress of the controller as well as the stability analysis of the closed-loop system are presented. In Sect. 4, simulation results are displayed and analyzed. The conclusion part is included in Sect. 5.

Preliminaries
The expression and lemmas involved in subsequent derivations and proofs are listed here. For simplicity, we use the notation x α = sign(x)|x| α , x ∈ R.
Definition 1 [39,46] Suppose there exists a compact set M and a continuous function Lemma 1 [17] For the following systeṁ suppose there exists a positive definite function V (x) : U → R such that the following condition holds (i) There exists real numbers c > 0 , α ∈ (0, 1) and an open neighborhood U 0 ⊂ U of the origin such thaṫ Then the origin is a finite-time stable equilibrium of system (1). If U = U 0 = R n , the origin is a global finite-time stable equilibrium of system (1).

Lemma 2 [47]
If p 1 > 0 and 0 < p 2 ≤ 1, then for ∀x, y ∈ R, the following inequality holds Lemma 3 [48] Let c and d be positive constants, then ∀x, y ∈ R satisfy the inequality Lemma 4 [49] Let p be a real number with 0 < p < 1, then for ∀x i ∈ R, i = 1, . . . , n, we have Lemma 5 [50] Let V (x) be a continuous positivedefinite function with respect to a compact set M for system (1).

Attitude model and problem formulation
For the attitude stabilization problems, the dynamic model of spacecraft attitude can be described as [51] where J = J T is a positive-definite square inertia matrix whose dimension is 3, ω = [ω 1 , ω 2 , ω 3 ] T is the measured angular velocity, and The external disturbances are composed of solar radiation, magnetic effects or other uncertainties, which have the characteristics of small amplitudes and periodicity. Taking into account the above characteristics, sinusoidal disturbances can represent the disturbances in the attitude dynamics of the spacecraft well. Due to the fuel consumption, motivations of the devices as well as other factors, the inertia matrix may be perturbed. Denote the inertia matrix J as J = J 0 + J δ , where J 0 is the nominal value of the inertia matrix and J δ is the perturbed value. This fact leads to where d l (t) = J δω + a(ω) J δ ω + M(t) is the lumped disturbances. Using the symbol || • || to represent the Euclidean norm of a vector and the induced norm of a matrix, we have ||d l (t)|| ≤ || J δ || * ||ω|| + || J δ || * ||ω|| 2 + ||M(t)||.

Remark 1
The same assumptions on the lumped disturbances d l can be found in many papers such as [ Using above dynamic model, with the quaternion being used to describe the spacecraft attitude, the kinematic model is [53] The (4) is the unit quaternion. It satisfies where θ denotes the principal angle and e = [e 1 , e 2 , e 3 ] T denotes the Euler axis with e 2 1 + e 2 2 + e 2 3 = 1. Thus, the unit quaternion satisfies where I 3×3 represents a 3 × 3 identity matrix. Based on (6), one can easily obtain that Using the nominal value J 0 to counteract the structural items in (3) and letting where As presented in (5) and (6), to derive set stability, there should be two equilibria (1, 0, 0, 0) T and (−1, 0, 0, 0) T . The difference in principal angle between them is 360 • , and it seems to be exactly the same from a geometric point of view, based on which many papers claim themselves global stable though they only consider one equilibrium. Obviously, this is not entirely true for the reason that if the equilibrium (1, 0, 0, 0) T is designed to be stable while (−1, 0, 0, 0) T is unstable, the entire system states will converge to the equilibrium (1, 0, 0, 0) T although some states are closer to (−1, 0, 0, 0) T , engendering the so-called unwinding phenomenon. To avoid this phenomenon, a SOSM control scheme based on set stability is introduced in subsequent sections.

Second-order sliding mode controller design
In this section, a SOSM controller is designed by using the idea of APIC and set stability. The design process and stability analysis are divided into two steps. At first, we select the sliding variables as follows: where q 0 (0) denotes the initial condition of q 0 .
Remark 2 As the q 0 is always available throughout the control process, the value of q 0 (0) can be obtained directly. By introducing the sign (q 0 (0)) term into the sliding variable, it will be proved later that whether the system states converge to (1, 0, 0, 0) T or (−1, 0, 0, 0) T depends on the initial value of q 0 . In other words, if the above sliding variables can be driven to the origin in finite time, the global finite-time stability can be achieved.

Proof
Step 1 Choosing a C 1 Lyapunov function of the form: Then taking derivative of (11) along system (8), it provideṡ T is the virtual control law which is designed as Using Lemma 4, supposing that the tracking error toward the virtual controller has converged to zero in finite-time and considering the condition that q 0 (0) ≥ 0, then (12) can be expressed aṡ From system (8), we haveq 0 = − 1 2 (q 1 ω 1 + q 2 ω 2 + q 3 ω 3 ) . This, together with virtual control law (13), implieṡ As k 2 is a positive constant, the conditionq 0 > 0 and q 0 > 0 will be reached in a finite time t 1 . This fact leads tȯ Another The same proof turns out to be correct for the case q 0 (0) < 0. Using Lemma 1, it is not complicated to conclude that system (8) under the designed virtual controller (13) is globally finite-time stable. One may consider that, based on backstepping control, a controller that drives the tracking error to zero in finite time can be constructed directly. However, the derivative operation in the design process will bring the control input with the negative fractional power, which may cause singularity problem. For this reason, we use a generalized 'APIC' [54] to guarantee the finite-time tracking of the virtual controller and the establishment of SOSM.
Remark 3 As mentioned in the introduction, the controller designed in this paper owns a simple structure and only one parameter needs to be tuned. From (34), it can be concluded that only parameter k 3 needs to be selected because the parameter k 2 is totally decided by k 3 , while k 1 is decided by k 2 and k 3 . Actually, the parameter k 3 determines the convergence rates of the system states. When a faster convergence rate is required, k 3 needs to be adjusted to a larger value. If the over large control torques are to avoided, then k 3 needs to be adjusted smaller. In addition, owing to the conservative derivation process of APIC, the controller gain seems to be bigger than a overlarge constant to guarantee the stability of the system. For the possible problem of excessive parameters, in practical, the controller gain can be adjusted appropriately small to avoid saturation of the actuator and energy waste without affecting the stability of the system.
Remark 5 Since the attitude system is a secondorder system, the discontinuous switching term of the second-order sliding mode controller still appears in the actual control torques. However, the ideas and derivation methods of this paper make it possible to extend the second-order sliding mode controller to the third-order, so as to completely address the problem of discontinuous control torques. In addition, our controller is of the form relay polynomial, it is simple to transform it into a continuous controller. This is also one of the methods to completely address the chattering problem.
It can be derived from the aforementioned analysis that the spacecraft attitude stabilization system (8) under the SOSM controller (34) is globally finite-time stable with respect to set M. Remark 6 In practice, the discontinuous control inputs caused by the symbolic function in (34) may lead to chattering phenomenon. To avoid this, the continuous function |s| |s|+φ can be used as a substitute for the discontinuous symbolic function sign(s). Although the stability of the system under the continuous control inputs degrades from finite-time stable to finite-time boundedness, the boundary can be reduced to a tolerable range by choosing a appropriately small φ. The continuous controller is of the form where k 1 , k 2 , k 3 are the same as in (34), φ is chosen as a small positive constant. The effectiveness of the controller will be verified by simulations in Sect. 4.

Simulation results
In this section, the simulation results are presented in the form of figures and tables. Choosing the nominal value of the inertia matrix as The controllers involved in the following comparison section are designed as follows: The integral SOSM (ISOSM) controller is designed as [37] where is the integral sliding mode surface.
According to the expression of the matrix B, we can get the determinant of B, which is q 0 . On this basis, we know that the matrix B is irreversible when the q 0 is equal to zero. By the definition of unit quaternion, the equation q 0 = 0 satisfies when the rotation angle around the axis is π . Hence, the singularity problem of the ISOSM has been verified from a theoretical point of view.
The global finite time control (GFTC) controller is designed as [45] where k 1 , k 2 are all positive constants.

Validation of set stability
In order to verify the global finite-time stability mentioned above, the RPSOSM algorithm proposed in this paper is simulated separately. The initial conditions of q, w and d(t) are chosen as q(0) T  As is shown in Figs. 1, 2 and 3, when q 0 (0) satisfies q 0 (0) ≥ 0, q T converges to (1, 0, 0, 0) T , which indicates that (1, 0, 0, 0) T is one of the equilibria of the system. Also, though there exist discontinuous items in actual control input, the accuracy of the controller remains at a satisfying level. Therefore, we can reasonably claim that the RPSOSM controller can provide a great steady-state attitude accuracy.

Comparisons of the control performance
In this section, the performance of the proposed RPSOSM algorithm (34), the ISOSM (43) algorithm and the GFTC (44) algorithm are compared in the absence/presence of disturbances.
The comparisons are divided into two cases: Case 1 considers the system without disturbances, while in Case 2 the external disturbances are cho- Noting that in order to ensure the fairness of comparisons, suitable parameters are adjusted to ensure the performance of ISOSM and GFTC, and the control inputs are uniformly limited to 65N .m. The parameters of the above-mentioned controllers are shown in Table 1 When q 0 (0) is selected as q 0 < 0, q T converges to (−1, 0, 0, 0) T . Therefore, (−1, 0, 0, 0) T is also the equilibrium of the system, thus verifying the global  Fig. 6 Response curves of control torque without disturbances. a RPSOSM (34). b ISOSM (43). c GFTC (44) stability of the designed controller. In the absence of disturbance, RPSOSM has a faster convergence speed than ISOSM and GFTC, and the discontinuity existing in the control does not affect the accuracy of the controller. Subsequently, when the system is subject to sinusoidal disturbances of different frequencies, it is obvious that RPSOSM still owns a faster convergence rate  Table 2.

Validation of the continuous controller
In this part, the proposed continuous controller (42) is simulated to verify its effectiveness. Since the main difference between the controller (42) and controller (34) is the disturbance rejection abilities, the simulation is carried out with disturbances. The parameters

Conclusion
The global finite-time attitude stabilization of the rigid spacecraft by using SOSM control has been investigated in this article. A RPSOSM controller has been a CT (convergence time) denotes the time after which |q i − q i * | < 5e−3, i = 0, 1, 2, 3 holds b QE (quaternion error) denotes the maximum error of the quaternion after the convergence time, which is obtained by Q E = max |q i − q * i |, i = 0, 1, 2, 3 c The QE is zero here because the actual error in the simulation is small enough, whose amplitude is less than 1e−7. This error may be caused by the discretization of the continuous-domain sliding mode controller designed to obtain a finite-time result. The set stabilization idea has been applied to derive set stability for the system. The global finite-time stability has been proved based on a Lyapunov function. Simulation results reveal that the controller retain a satisfactory performance when dealing with bounded disturbances. Our future work is to further eliminate the discontinuity in the control input of the SOSM controller by using HOSMC.