Active Fractional-Order Sliding Mode Control of Flexible Spacecraft Under Actuators Saturation

In this article, a novel multi-purpose modiﬁed fractional-order nonsingular terminal sliding mode (MFONTSM) controller is designed for the ﬂexible spacecraft attitude control, assuming the control torque saturation in the system dynamics. The proposed controller is modiﬁed to be able to perform appendages passive vibration suppression. Furthermore, an active FONTSM controller is proposed separately to perform active vibration suppression of the ﬂexible appendages using piezoelectric actuators. The closed-loop system’s ﬁxed-time stability for both the passive and active controllers is analyzed and proved using the Lyapunov theorem. Finally, the proposed controllers’ performance has been tested in the presence of uncertainties, external distur-bances, and the absence of the damping matrix in order to study the eﬀectiveness of the proposed method.


Introduction
The emergence of new missions that usually require high-pointing accuracy, along with uncertainties and external disturbances, make the precise attitude control of spacecraft vital for mission accomplishment. Furthermore, cost reduction often requires the modern spacecraft to have large and lightweight, flexible appendages, which results in increased structural flexibility. Therefore, the problem of flexible spacecraft attitude control with appendages vibration suppression becomes more crucial and challenging than before.
In the past decades, for the flexible spacecraft vibration suppression, some active control methods have been suggested, such as component synthesis vibration suppression (CSVS) [1], modal velocity feedback compensator (MVFC) [2,3], strain rate feedback compensator (SRFC) [3], and positive position feedback (PPF) [4,5]. However, they usually concentrate on the attitude controller's innovation rather than the active controller [6]. Besides, most of them usually ignore the coupling term in the modal equation from the beginning of the attitude maneuver, despite its effect on the elastic mode vibrations. Hence, a desired active controller must either consider this term or be robust against its effects; otherwise, the system's stability is not fully guaranteed. Some passive vibration suppression methods usually consider flexible influences as external disturbance [7]. Although the above approaches can effectively reduce the appendages' vibrations, they rely solely on the attitude controller to reduce elastic vibrations' influence on the spacecraft attitude. In other words, the vibration itself is not suppressed but rather decayed through the damping of the structures. However, when the structures have low damping and high flexibility, these methods may not maintain the desired results.
In reality, since practical actuators' control output is always limited regardless of their type, the actuators' saturation, often as a dominant input nonlinearity, may severely limit the system performance and even lead to instability or undesirable inaccuracy. Thus far, various control schemes considering actuators saturation have been proposed. For instance, in [8], a tangent hyperbolic function and in [9,10] a saturation function was used in systems dynamics to describe actuators saturation. Then, [10] has been further studied by [11], and it was observed that the method was ineffective. Another approach is to augment a new equation, representing the actuators saturation, to the system dynamics [12]. However, this method results in prolonged settling time, reducing the practicality of the control approach. The majority of research on spacecraft attitude control generally ignores the actuators' saturation problem despite its inevitable effect on attitude maneuvers. Moreover, there is minimal research on attitude control design and stability analysis with explicit consideration of the flexible spacecraft's actuators' saturation.
In recent years, the 300-year-old fractional calculus concept has been combined with terminal sliding mode (TSM) and nonsingular terminal sliding mode (NTSM) methods. The resulting controllers have all the advantages of the NTSM controller with better transient performance and faster convergence due to the freedom on the integrator and differentiator order. These novel controllers have been used for various systems, including robot manipulator [13], cable-driven manipulator [14], and linear motor [15]. Nevertheless, in some of these works, the error signals' stability and boundedness after reaching the sliding surface are not mathematically studied. In the others with a complete stability analysis, the control law usually consists of a fractionalorder differentiator. Unlike the fractional-order integrator, the fractional-order differentiator's boundedness is not studied; thus, the control law's boundedness is not fully guaranteed.
Moreover, as mentioned so far, most of the research provides finite-time stability for the system where the exact upper bound of the settling time depends on the initial conditions. Therefore, to address this issue, the concept of fixed-time stability was introduced [16]. According to this concept, the settling time function is upper bounded by a predefined constant that depends on the design parameters rather than the system's initial state [17]. Consequently, the convergence time can be designed in a prescribed manner.
In this work, we propose two novel controllers for the flexible spacecraft's attitude control and vibration suppression to address the problems mentioned above. The main contributions of this paper are as follows: (i) A novel multi-purpose MFONTSM controller is designed to perform fast attitude stabilization and provide passive vibration suppression, considering the actuators' saturation in the system's dynamics. Furthermore, a new active FONTSM controller is introduced to perform active vibration suppression using the piezoelectric actuators. (ii) The closed-loop system's practical fixed-time stability under external disturbance, uncertainty, and actuators' saturation is analyzed and proved using the Lyapunov theorem. The proposed control method's effectiveness has been studied through comparative numerical simulations under the damping matrix's absence.
The rest of the paper is organized in the following manner: In Section 2, some preliminaries of the concepts of fractional calculus and fixed-time stability are given. Section 3 denotes the problem statement. In Section 4, the controller design and stability analysis are presented. The comparative numerical simulations are represented in Section 5, and Section 6 concludes the paper.
where α is the fractional order.
where h is the sampling interval, and x , x ∈ R is a flooring operator which gives the integer part of its input, also, Lemma 2 [10,21] Consider a nonlinear systeṁ where f : U 0 → R n is continuous in an open neighborhood U 0 of the origin, the unique solution of (1) is supposed to exist for any initial conditions. The nonlinear system (1) is practical fixed-time stable in case that there exists a Lyapunov function Therefore, the trajectory of the closed-loop system is bounded in fixed-time as where θ is scalar and satisfies 0 < θ ≤ 1. The time to reach the residual set i.e. the settling time T is bounded by The GL definition is commonly used to describe practical cases because of its clear expression and simplicity of implementation. As seen from Property 1 and 2, for a wide class of functions, important for the application, both RL and GL definitions are equivalent. Therefore, the RL definition can be used during problem formulation and then turn to GL definition for obtaining the numerical solutions [18]. Consequently, in this paper, a finite approximation of GL definition based on Power Series Expansion (PSE) of a generating function with h = 10 −3 has been used for implementation [22]. Furthermore, for the purpose of simplicity we use D α instead of t0 D α t , for the rest of this paper.

Attitude Kinematics and Dynamics
As shown in Fig. 1, the simplified flexible spacecraft consists of a rigid hub and two flexible appendages.
Assuming that the thin, homogeneous, and isotropic piezoelectric films are attached on the surface of the flexible appendages as actuators; the attitude kinematic and dynamic model of the flexible spacecraft is given as [1]: Equation (2) is the kinematic equation, where [q 0 , q] T = [q 0 , q 1 , q 2 , q 3 ] T ∈ R 4 is the quaternion representation of the body frame attitude orientation with respect to an inertial frame, q 0 ∈ R and q = [q 1 , q 2 , q 3 ] T ∈ R 3 are scalar and vector components of the unit quaternion, respectively, and q T q+q 2 0 = 1. Also, I ∈ R 3×3 is the identity matrix, and ω ∈ R 3 denotes the angular velocity of the spacecraft expressed in the body frame relative to the inertial frame. Equation (3) is the attitude dynamic equation, where J ∈ R 3×3 is the positive definite symmetric inertia matrix of the spacecraft, u ∈ R 3 and d ∈ R 3 represent the control torque and external disturbance torque acting on the spacecraft, respectively. δ ∈ R 3×n denotes the coupling matrix between the rigid hub and flexible appendages, and η ∈ R n represents the n−dimensional modal displacement. Equation (4) is the modal equation, where δ p ∈ R n×m is a coupling matrix associated with the structural characteristics of the appendage and the mounting position of the n piezoelectric sensors and m piezoelectric actuators. u p ∈ R m denotes the m−dimensional piezoelectric input voltage. C = 2ζΩ ∈ R n×n and K = Ω 2 ∈ R n×n are the diagonal damping and stiffness matrices, respectively, where ζ ∈ R n×n and Ω ∈ R n×n represent the flexible appendages diagonal matrices of damping ratio and natural frequency, respectively. The notation x × ∈ R 3×3 represents the cross-product matrix operator of an arbitrary vector

Relative Attitude Error
Assuming ϕ = [φ, θ, ψ] T describes the rotation angles of the spacecraft body frame axes with respect to the inertial frame.
the desired quaternion parameters, respectively. Using the standard coordinate transformation, one can reformulate the attitude tracking problem into a stabilization problem as [23]: where q e0 ∈ R and q e ∈ R 3 denote the error quaternion, ω d ∈ R 3 is the desired angular velocity with respect to the inertial frame expressed in the reference body frame, ω e denotes the relative angular velocity error and the matrix R = (1 − 2q T e q e )I + 2q e q T e − 2q e0 q × e is the rotation matrix with the conditions R = 1 anḋ R = −ω × e R in which x represents the L 2 norm of a vector or matrix. Substituting (4) in (3) and considerinġ ω =ω e + ω × e Rω d − Rω d , the relative attitude error equations can be obtained as where J 0 = J − δ T δ, and the term δ T δ is the effect of the flexible appendages on the total inertia matrix.

Controller Design and Stability Analysis
This section presents the main results regarding the controller design and stability analysis of the system.

Remark 2
The sliding surface given in most of the existing literature on FONTSM controllers, for example [13,14], can be written as whereq represents the error states of the system and the other parameters for (9) are similar to that of (8).
The single-purpose sliding surface (9) is a one-variable fractional-order equation that is only able to bring the error states on the sliding surface. Meanwhile, our multipurpose sliding surface (8) represents a multi-variable fractional-order equation that can bring the error quaternion, modal velocity error, and modal displacement error on the sliding surface at the same time. Furthermore, the term D α−1 in (9) is a fractional integrator which according to Lemma 1 is bounded. However, calculating the first derivative of (9) will result in the appearance of the term D α in the control input, which is a fractional differentiator, and therefore, the boundedness of the control input is not fully guaranteed. This problem has been solved with our proposed sliding surface.
Calculating the first derivative of (8), we obtaiṅ Calculating the derivative of (6) yields: Using (7) and (5) we can obtainq e as: (10) we can simply obtain: (13) To obtain a precise and fast control performance invariant to initial conditions, we propose a fixed-time fast TSM type reaching law as: where the matrices K 1 = diag(K 11 , K 12 , K 13 ) and K 2 = diag(K 21 , K 22 , K 23 ) are constant diagonal positive matrices to be designed. m and n are positive integers which 0 < m < 1 and n > 1.

Remark 3
The reaching law in the aforementioned papers can be written aṡ where all the parameters for (15) are similar to that of (14). Compared to (14), the reaching law (15) can only provide finite-time stability for the closed-loop system which heavily relies on the initial conditions. Now we can find the control input with substituting (14) in (13) and neglecting the disturbance term. Therefore, we obtain: where G and f are: Remark 4 If we set ∆ 1 = ∆ 2 = 0 in the sliding surface (8) and the controller (16), we get FONTSM sliding surface and controller. Therefore, for the purpose of comparison, we will study both FONTSM and MFONTSM controllers.

Actuators Saturation
In this section, we redesign the attitude controller to consider the actuators' saturation bound. The dynamic equation of the flexible spacecraft with actuators saturation can be expressed as [24]: Similarly the error dynamic equation is derived as: (18) where the saturation function sat(u) is defined as: where we have u max = u max [1 1 1] T , and u max is the upper bound of the control torque saturation. Now inspired by [25] we propose the saturated MFONTSM controller as: where µ is a positive scalar and z is the same as the control law u defined in (16). Equation (20) forms the proposed saturated MFONTSM attitude controller of this paper.

Active Vibration Controller Design
To proceed further, the following basic assumption is required in this section: Assumption 1 The number of the piezoelectric actuators and sensors is equal, and they are mounted in such a way that the coupling matrix δ p is invertible [6].
Suppose that the attitude controller works in a way that the spacecraft reaches the desired attitude, in other words, as t → ∞, then q e0 → 1, q e → 0, ω e → 0 anḋ ω e → 0, that results in the elastic motion to asymptotically decouple from the rigid one. In this case spacecraft attitude has been successfully controlled, however, the bounded elastic oscillation might still persist in the flexible panels. With defining d p = δω as an external disturbance acting on the flexible panels, (4) can be written as following Remark 5 In most of the research regarding the active controller design, for example, [2,4], the controller is designed by settingω = 0 in the modal equation, which leads to a decoupled lower dynamics. Therefore, the effect of the term δω on the appendages' vibrations is ignored from the maneuver's beginning even though the angular velocity rate becomes zero only at the end of the attitude stabilization maneuvers. Now similar to the attitude controller design, we need to design a proper sliding surface for the active vibration controller as well. Therefore, we propose a second sliding surface s 2 ∈ R 3 as (22) where Γ 1 = diag(Γ 11 , Γ 12 , Γ 13 ) and Γ 2 = diag(Γ 21 , Γ 22 , Γ 23 ) are positive constant diagonal matrices. ρ 1 and ρ 2 are fractional-orders which 0 < ρ 1 , ρ 2 < 1. Furthermore, γ 1 and γ 2 are terminal fractions satisfying 0 < γ 1 , γ 2 < 1.
Calculating the first derivative of (22) and substituting (21) yieldṡ Now similar to (14), a fixed-time fast TSM type reaching law is designed as: where the matrices L 1 = diag(L 11 , L 12 , L 13 ) and L 2 = diag(L 21 , L 22 , L 23 ) are constant diagonal positive matrices to be designed. p and r are positive integers which 0 < p < 1 and r > 1. Now we can find the active control input with substituting (24) in (23). Thus, similar to (16), we obtain: where G p and f p are: Equation (25) forms the proposed active vibration controller of this paper.

Stability Analysis
In order to start the stability analysis of the proposed controllers, we need the following assumptions.

Assumption 2
The total disturbance d is bounded, but the upper bound is unknown: The inertia matrix J 0 with uncertainty is bounded, but the upper bound is unknown: Assumption 4 The matrix G in (16) is bounded and invertible: Considering Assumption 3 since the matrix J 0 is bounded we can deduce that, J −1 0 is bounded as well, therefore: and for the matrix (q e0 I + q × e ) we have [23]: we can calculate the bound of G as: thus, the matrix G is bounded as well. To ensure G is invertible, the matrix (q e0 I + q × e ) should be invertible as well. In other words [26]: to ensure (26) remains valid, attitude maneuver should be restricted that q e0 (0) = 0, where q e0 (0) represents the initial conditions of q e0 , and the subsequent control strategies should be designed to guarantee that q e0 = 0 for all time. Therefore, in this study, for the matrix G to be invertible and to avoid quaternion singularity, we limit the range of the attitude maneuver to (+π, −π). (2), (3) and (4), with the proposed sliding surface in (8) and the designed attitude controller with actuators saturation in (20), if the Assumptions 1, 2, 3 and 4 hold, the sliding surface of the attitude controller will reach a small neighborhood of origin within a fixed time, and all the corresponding signals of the closed-loop system will converge to a small neighborhood of origin in finite time, in the presence of unknown external disturbance and uncertainty.

Theorem 1 For the spacecraft attitude control and modal vibration systems in
Proof There are two main steps in this proof.
Step 1 shows the sliding surface s 1 , will reach in a small neighborhood of origin within a fixed time and proves the practical fixed-time stability of the system.
Step 2 ensures that, once the trajectory of the closed-loop system is driven onto the sliding surface, then q e ,q e ,η e and η e of the closed-loop system will remain bounded thereafter.
Step 1 The saturation function (20) can be rewritten as where Now consider the following Lyapunov function: taking the derivative of (29), we obtain: by replacing u with sat(u) in (13), we can rewrite (13) as: Substituting (31) in (30) yields: substituting the control law (16) in (32) yields: with defining ν = Gd, we obtain: now considering (28) we can write: considering (28) and the fact that |z| ≤ u max , one can obtain that z + u max ≥ 0 and z − u max ≤ 0. Moreover, the terms µs 1 − z > u max and µs 1 − z < −u max , result in s 1 > 0 and s 1 < 0, respectively. Since all the terms produce scalar values we have: and therefore: where λ min (K 1 ) > 0, λ min (K 2 ) > 0 and λ min (G) > 0 are the minimum eigenvalues of K 1 , K 2 and G, respectively. Now since the last term of (37) is always negative, the inequality (37) can be reduced to: Now we can rewrite (38) in the following two forms [13]: under the following conditions and considering ν ≤ G d ≤ 1 2J 0max d max , we obtain that the sliding surface is bounded and it will reach the following small neighborhood of s 1 = 0 as: Therefore, there exist a positive constant ϑ 1 such that the inequality ϑ 1 ≥ 1 2 ΥJ 0max d max holds. Thus, combining (29) with (38), we can obtain the following inequality: where 0 < a = m+1 2 < 1 and b = n+1 2 > 1. Therefore, according to Lemma 2, the equilibrium of system (8) is practical fixed-time stable and the upper bound of the settling time is obtained as Therefore, system states will persistently converge to the pre-described sliding surface given in (8) in fixed time from any initial condition, even in the presence of unknown external disturbance and uncertainty.
Step 2 Combining (43) with (8) yields where i = 1, 2, 3 stands for the i-th degree of freedom (DoF) and elastic mode. Equation (46) can be transformed into the following three forms: Equation (47) will remain the proposed sliding surface given in (8) if the following inequality holds Considering |s 1i | ≤ Υ , the system states will continue converging to the proposed sliding surface until we have [13]: Taking Lemma 1 into consideration and choosing p = ∞, we can obtain that where ess supf (x) represents the essential maximum values of function f (x). On the other hand, since we have Then, (53) can be transformed into following equality with a bounded time-varying factor σ ≥ 1 as with combining (51) and (54), we have Therefore, the theoretical control errors will be bounded and can be given as Applying the same analysis procedure to (48) and (49) yields and similarly we can calculate Then, combining (51), (57) and (58) with (46), we have Therefore, stability of the control system is ensured and the system errors will be bounded by (56), (59), (60) and (61).

Combination of
Step 1 and 2 completes the proof of Theorem 1.
Theorem 2 For the spacecraft attitude control and modal vibration systems in (2), (3) and (4), with the proposed sliding surface in (22) and the designed active vibration controller in (25), if the hypothesis of Theorem 1 is satisfied and Assumptions 1, 2, 3 and 4 hold, the sliding surface of the active controller will reach a small neighborhood of origin within a fixed time, and all the corresponding signals of the closed-loop system will converge to a small neighborhood of origin in finite time in the presence of unknown external disturbance.
Proof Applying the same analysis as Theorem 1 we obtain that the sliding surface (22) is bounded as and therefore, 2 > 1 and ϑ 2 ≥ δ max ω . Therefore, according to Lemma 2, the equilibrium of system (22) is practical fixed-time stable and the upper bound of the settling time is obtained as Furthermore, all the signals in s 2 i.e.η and η, will remain bounded thereafter. When the attitude stabilization maneuver is complete, as t → ∞, we haveω → 0. Therefore, the active controller can provide bounded vibration suppression as the maneuver completes, and it can provide full vibration suppression at the end of the stabilization maneuver onceω = 0. Thus, the proof of Theorem 2 is completed.
Remark 6 Note that according to Theorem 1 and 2, one treats the question of spacecraft attitude control and appendages elastic mode stabilization separately where the coupling effect is treated as an external disturbance. Furthermore, Theorem 1 proves that the designed attitude controller can bring the modal displacement and velocity to a small neighborhood of zero, while, according to Theorem 2, once the active controller is in the loop, the modal displacement and velocity can reach zero in finite time.
In the next section, numerical simulation and comparison are given to verify the proposed attitude controller's success in conjunction with the active vibration control technique.

Numerical Simulation
In this section, the performance of the proposed controllers is evaluated using computer simulations in MAT-LAB software. The FONTSM and MFONTSM controllers' performance in passive/active vibration suppression of the flexible appendages are compared. Furthermore, the effectiveness of the proposed active controller (25) has been compared to the active MVFC controller ((81) in [2]) to study its superiority over the other methods.

Control Parameters
In order to get the best performance and robustness, the corresponding controller gains in (16) and (25) should be tuned appropriately. The controller gains selected by the trial-and-error method to reach the optimal response and performance of the closed-loop system are listed in Table 1.

Simulation Setup
System description of the flexible spacecraft for simulation studies are presented in Table 2 [6]. Assuming that the spacecraft is orbiting the planet at the altitude of 500 km, the total external disturbance is assumed as where ω 0 ≈ 0.0011 rad/s represents the orbital angular velocity [6]. Furthermore, to study the robustness of the controllers, 20% uncertainty on the inertia matrix and a zero damping matrix C = 0 is considered in all the simulations. The actuators' saturation bound is set as u max = 10 N.m, and since we aim to perform attitude stabilization and vibration suppression,

Simulation Results
The simulation results of the FONTSM controller ( (20) where ∆ 1 = ∆ 2 = 0) and MFONTSM controller (20) with active OFF, active MVFC ((81) in [2]) and active FONTSM (25) are shown in Fig. 2 and 3, respectively. For the proposed controllers, all the control gains are presented in Table 1, and for the MVFC controller, everything is as in [2], except the gain F = diag(10, 10, 10), which is chosen for better performance. It can be observed from Fig. 2 that there is a negligible difference for the FONTSM controller once the active controller is OFF with active MVFC and active FONTSM; in Euler angles, attitude quaternion, angular velocity, and the sliding surface plots as all the graphs overlap. The upper bound of the settling time for the attitude FONTSM controller can be calculated using (45) and Table 1 as T 1max = 49.6040 s, independent of the initial conditions, therefore all the signals in the sliding surface should be settled before T 1max = 49.6040 s is reached. The tracking errors in Euler angles for all controllers are less than 0.02 degree when t ≥ T 1max . Predictably, the main difference can be seen in modal displacement and velocity plots. The attitude FONTSM controller cannot provide any vibration suppression, and therefore, since there is no damping matrix, the vibration amplitude increases over time. However, once the active controllers are in the loop, modal vibrations are properly controlled and neutralized. Furthermore, comparing the performance of the active MVFC with active FONTSM, a noticeable difference can be observed in the settling time, maximum overshoot, and the steady-state error of appendages modal displacement and velocity. The upper bound of the settling time for the active FONTSM controller can be calculated using (64) and Table 1 as T 2max = 51.9931 s, independent of the modal initial conditions. It can be observed from analyzing the plots that the modal displacement error for the active MVFC and active FONTSM are under 5 × 10 −6 and 5 × 10 −9 , respectively, when t ≥ T 2max .
Furthermore, it can be observed in Fig. 3 that there is a slight difference for the MFONTSM controller once the active controller is OFF with active MVFC and active FONTSM; in Euler angles, attitude quaternion, angular velocity, and the sliding surface plots. Comparing the modal displacement and velocity plots from Fig. 2 and 3, it can be seen that despite the FONTSM controller, the MFONTSM controller can provide passive vibration suppression once the active controller is OFF. The upper bound of the settling time for the attitude MFONTSM controller is T 1max = 49.6040 s, and the tracking errors in Euler angles for all controllers are less than 0.02 degree when t ≥ T 1max as before. However, despite the FONTSM controller, since the modal displacement and velocity exist in the modified sliding surface, both modal displacement and velocity are settled before T 1max = 49.6040 s is reached. The upper bound of the active FONTSM controller's settling time is again calculated as T 2max = 51.9931 s. It can be observed from analyzing the plots that the modal displacement error for the MFONTSM controller with active OFF, active MVFC, and active FONTSM are under 5 × 10 −5 , 5 × 10 −7 , and 5 × 10 −10 , respectively, when t ≥ T 2max . Comparing the control torque plots in Fig. 2 and 3 shows that the passive MFONTSM controller experiences more chattering than the FONTSM controller at the beginning of the maneuver, which is an expected consequence of the sliding surface modification. It can also be deduced from the piezoelectric voltage plots that the active FONTSM uses a bit more voltage comparing to the active MVFC controller.

Conclusion
This paper studies the problem of flexible spacecraft attitude control and appendages vibration suppression by introducing a new fractional-order nonsingular terminal sliding mode (FONTSM) controller. A modified FONTSM (MFONTSM) controller is introduced by designing a novel multi-purpose sliding surface that can perform passive vibration suppression. Furthermore, an active FONTSM controller is proposed separately for active vibration suppression using piezoelectric actuators. The total practical fixed-time stability of the closed-loop system and the boundedness of all the subsequent error signals have been analyzed and proved using the Lyapunov theorem. The proposed control scheme's performance has been studied through numerical simulations. It is observed that the proposed control scheme can provide fast convergence with short settling time and small overshoot/undershoot with passive vibration suppression ability. Furthermore, the proposed controllers' effectiveness is studied under uncertainty, unexpected external disturbance, and the damping matrix's absence. It is observed that the proposed active controller can provide faster convergence with far less steady-state error compared to previous methods.