Spacecraft Attitude Stabilization Control with Fault-Tolerant Capability via a Mixed Learning Algorithm

Abstract

The issue of active attitude fault-tolerant stabilization control for spacecrafts subject to actuator faults, inertia uncertainty, and external disturbances is investigated in this paper. To robustly and accurately reconstruct actuator faults, a novel mixed learning observer (MLO) is explored by combining the iterative learning algorithm and the repetitive learning algorithm. Moreover, to guarantee robust spacecraft attitude fault-tolerant stabilization, by synthesizing the mixed learning algorithm with the sliding mode controller, a novel mixed learning sliding-mode controller (MLSMC) is designed based on the separation principle, in which the mixed learning algorithm is used to update composite disturbances online, including fault errors, inertia uncertainty, and external disturbances. Finally, a numerical example is provided to demonstrate the effectiveness and superiority of our proposed spacecraft attitude fault-tolerant stabilization control approach.

1. Introduction

In order to achieve various advanced space missions, such as Earth observation, remote communication, navigation, and positioning, considerable research work has been performed on spacecraft attitude control, including attitude stabilization, attitude tracking, etc. [1,2]. For the above two attitude maneuvers, certain prescribed attitudes or attitude velocities within certain ranges are required for a spacecraft during its mission life. The diversity of space missions and the growing requirements for high performance unavoidably result in increasingly sophisticated attitude control systems (ACSs). On the other hand, the extremely harsh outer space environment and the aging of actuators raise the probability of malfunctions in spacecraft actuators to a certain extent, resulting in significant performance degradation or task paralysis [3]. Therefore, it is extremely necessary for the spacecraft ACS to possess the fault-tolerant capability that maintains the desired attitude performance properties. Plenty of research progress on spacecraft attitude control methods with fault tolerance has been reported, with recent progress published in [1,4,5] and the references therein.

It is widely acknowledged that fault-tolerant control (FTC) can be divided into two categories, namely, passive FTC and active FTC [6,7]. In the former FTC design, online fault information is not necessary, and it has the ability to deal with presumed faults, not all faults. However, this design has limited fault-tolerant capabilities, and the desired control performance cannot be guaranteed for unanticipated fault scenarios. Alternatively, an active FTC reacts to system faults actively by resorting to a reconfigurable controller [7]. The active FTC scheme can be further divided according to whether a fault detection and isolation (FDI) unit exists. The active FTC scheme involving an FDI unit requires fault diagnostic information on occurrence time and location. However, time-delay issues inevitably affect the overall system control performance under this FTC scheme. In contrast, the active FTC without FDI can compensate for system fault effects, and such FTC approaches mainly contain adaptive mechanism-based FTC [8] and fault reconstruction-based FTC. At present, various fault-reconstruction methods have been developed for spacecraft ACSs, e.g., adaptive observer [9,10], sliding mode observer [11,12], and neural network observer [13,14], just to name a few. Furthermore, based on different fault-reconstruction observers, recent progress on active attitude FTC strategies have been developed to actively compensate for actuator faults by utilizing reconfigurable fault-tolerant controllers; see, for example, [9,10,11,12,13,14,15,16].

In [9], an adaptive indirect estimator is proposed to reconstruct actuator faults, while a fault-tolerant steering logic design is investigated for spacecraft with a single-gimbal control moment gyro. In [10], an adaptive finite-time observer is designed to reconstruct the lumped faults, while an adaptive integral sliding mode control (SMC) method is proposed to achieve spacecraft active attitude FTC. In [11], an adaptive sliding mode observer is proposed to reconstruct actuator loss of effectiveness, and an integrated fault-reconstruction/FTC scheme is developed based on a fractional-order nonsingular terminal sliding mode method and backstepping technology. A robust nonlinear disturbance observer (RNDO)-based adaptive non-singular fast terminal SMC method is proposed in [12], where an RNDO with the sliding mode is proposed to reconstruct the lumped disturbance, including actuator faults, inertia uncertainties, and external disturbances. However, the chatting phenomenon resulting from the sliding modes [11,12] is inevitable. In [13], a neural network observer is used to reconstruct unknown faults while the sliding mode controller is designed to guarantee the attitude convergence in finite time. In [14], a neural network observer is developed to reconstruct the lumped disturbance, while an active event-based dual-channel control scheme is created to solve accurate attitude stabilization under limited communication. Based on the nonsingular terminal sliding-mode theory, an active reconfiguration FTC method is proposed in [15], where a Chebyshev neural network is used to update and compensate for the lumped disturbance. For robust attitude fault-tolerant stabilization, a neural network-based third-order SMC method is proposed in [16], where a neural network is suggested to reconstruct the lumped uncertainty. It is worth pointing out that integral-based adaptive learning algorithms [9,10] and various neural networks with multi-level nodes [13,14,15,16] require massive computational power.

Recently, learning algorithm-based fault reconstruction and active attitude FTC for spacecraft systems have attracted considerable attention, and fruitful results have been published. At present, various types of learning algorithms have been developed to provide system fault (or disturbance) information, such that they could be applied to fault reconstruction and accurate tolerant control for spacecraft systems. An iterative learning observer (ILO) is firstly designed to reconstruct thruster faults of spacecraft systems in [17], providing a contrasting analysis of computation burden between the ILO and the integrator-based adaptive observer. In [18], a repetitive learning observer (RLO) is developed for actuator fault detection, isolation, and reconstruction for satellite ACS. In [19], an ILO-based attitude sliding-mode fault-tolerant stabilization control approach is proposed for the satellite subjects to external disturbance and actuator constraints, and an ILO is designed to reconstruct loss of actuator effectiveness faults. In [20], a linear LO is designed for actuator fault reconstruction for a microsatellite ACS. However, a linear satellite ACS model inevitably results in modeling errors. Based on the iterative learning algorithm, an online-learning spacecraft attitude stabilization control approach is investigated in [21]. ILO-based adaptive learning control for spacecraft attitude stabilization is investigated in [22], where a barrier function-based ILO is proposed to estimate the lumped disturbance, including actuator faults, parameter uncertainty, and external disturbance. At present, two types of learning algorithms, including an iterative learning algorithm and a repetitive learning algorithm, have been separately developed for fault reconstruction and tolerant control for spacecraft systems. Considering the advantages of the two learning algorithms, a mixed learning observer (MLO)-based actuator fault reconstruction method for spacecraft system has been devised. In [23], an MLO is proposed by combining the iterative learning algorithm and the repetitive learning algorithm. Based on [23], an MLO-based active prescribed performance consensus control for multiagent systems is developed. An MLO-based closed-loop attitude proportional-derivative stabilization control approach is proposed in [24], where an MLO is adopted to compensate for the lumped disturbance. However, parameter selection and computation are not handled well. Therefore, motivated by practical considerations, mixed learning algorithm-based active attitude FTC for the spacecraft is a considerably interesting research topic. In [25], an adaptive LO is designed to reconstruct actuator faults, while an adaptive backstepping fault-tolerant controller is designed to guarantee accurate attitude tracking. In [26], an iterative learning observer with the sliding mode term is explored to robustly reconstruct actuator faults, while an iterative learning sliding mode controller is used to actively stabilize the spacecraft attitude. However, the robustness inevitably decreases fault convergence speed.

In view of this, the issue of active attitude fault-tolerant stabilization control for rigid spacecraft subject to actuator faults, inertia uncertainty, and external disturbances is investigated in this paper. A novel mixed learning observer (MLO) is proposed to robustly and accurately detect actuator faults, and its stability is proven in detail. Further, based on the mixed learning algorithm, a mixed learning sliding mode controller with a fault-tolerant ability is designed to accurately stabilize spacecraft attitude by compensating for the effect of the jointed disturbances. Finally, a satellite ACS example is simulated to show the effectiveness and superiority of the proposed spacecraft attitude fault-tolerant stabilization control approach. The main contributions of our research work can be highlighted in the following three aspects:

(1)

To accurately reconstruct bias actuator faults, we develop a novel mixed learning algorithm by synthesizing the iterative learning algorithm [27] and repetitive learning algorithm [28]. The proposed mixed learning algorithm inherits the advantages of both learning algorithms and can smooth the reconstruction signal, especially when an abrupt disturbance or noise exists.

(2)

Based on the mixed learning algorithm, a novel MLO is proposed such that it can both estimate attitude angular velocity and reconstruct bias torque resulting from actuator faults, robustly and accurately. Compared with existing work [29,30], it is more convenient for observer gain parameters to be computed.

(3)

To guarantee the robust and accurate attitude stabilization, a mixed learning sliding mode control (MLSMC)-based spacecraft active attitude fault-tolerant controller is designed by combining the proposed mixed learning algorithm and the SMC methodology. In the proposed MLSMC scheme, the mixed learning algorithm is first employed to update online and compensate for the jointed disturbance. Compared with the existing SMC [31], the proposed MLSMC approach has the ability to achieve higher attitude stabilization accuracy due to the mixed learning algorithm.

The remainder of this paper is organized as follows. In Section 2, spacecraft ACS is modeled, and the problem formulation is presented. Section 3 provides the MLO design for actuator fault reconstruction and its stability analysis. Next, the MLSMC-based spacecraft attitude fault-tolerant stabilization controller design is provided in Section 4. Simulation studies are provided in Section 5. Finally, some concluding comments are drawn in Section 6.

2. Spacecraft Modeling and Problem Formulation

Spacecraft Attitude Control Modeling

This paper considers the rigid spacecraft, whose attitude is represented in terms of unit quaternion [32]. The quaternion $Q=(q_0,q)\in {\mathbb{R}}^4$ has the scalar $q_0\in \mathbb{R}$ and the vector $q={[q_1,q_2,q_3]}^T\in {\mathbb{R}}^3$. The quaternion satisfies the unity operational law, e.g., $q_0^2+q^{T}q=1$.

Then, the ACS of a rigid spacecraft, including the unit quaternion-based attitude kinematics and attitude dynamics, can be characterized as [33]:

$$\left\{\begin{array}{cc}\hfill {\dot{q}}_0=& -\frac{1}{2}{q}^T\omega \hfill \\ \hfill \dot{q}=& \frac{1}{2}(q^{\times }+q_0{I}_3)\omega \hfill \end{array}\right.$$

(1)

and

$$(J_s+\Delta J)\dot{\omega }+{\omega }^{\times }(J_s+\Delta J)\omega =T_c+T_d,$$

(2)

where $\omega ={[{\omega }_x,{\omega }_y,{\omega }_z]}^T\in {\mathbb{R}}^3$ represents the attitude angular velocity; $T_c={[T_{cx},T_{cy},T_{cz}]}^T\in {\mathbb{R}}^3$ represents the control torque provided by all the actuators; and $T_d={[T_{dx},T_{dy},T_{dz}]}^T\in {\mathbb{R}}^3$ represents the external space disturbance torque. D denotes torque coefficient matrix, while $J_s$ and $\Delta J$ denote the nominal inertia moment matrix and its uncertainty, respectively.

Considering the inertia uncertainty as external disturbance, (2) can thus be transformed into:

$$J_s\dot{\omega }+{\omega }^{\times }{J}_s\omega =T_c+u_f+u_d,$$

(3)

where $u_{fb}$ denotes the bias actuator fault, and $u_d=-\Delta J\dot{\omega }+{\omega }^{\times }\Delta J\omega +T_d$ represents the jointed disturbance.

Assumption 1. 

There exist nonnegative constants $u_{Jdmax}$ and $u_{dmax}$ such that the jointed disturbances $u_{Jd}$ and $u_d$ are respectively bounded by $\parallel u_{Jd}\parallel \le u_{Jdmax}$ and $\parallel u_d\parallel \le u_{dmax}$.

Assumption 2 

([27]). Nonlinear term ${\omega }^{\times }J\omega$ in (2) satisfies:

$$\parallel {\omega }^{\times }J\omega -{\hat{\omega }}^{\times }J\hat{\omega }\parallel \le \eta \parallel \omega -\hat{\omega }\parallel$$

(4)

with a positive scalar η.

3. Mixed Learning Observer-Based Actuator Fault Reconstruction

3.1. Novel Mixed Learning Observer Construction

Based on the ILO design [30], a novel MLO is constructed for (2) in the following form:

$$\begin{array}{cc}\hfill J_s\dot{\widehat{\omega }}\left(t\right)+\widehat{\omega }{\left(t\right)}^{\times }{J}_s\widehat{\omega }\left(t\right)=& u_c\left(t\right)+{\widehat{u}}_f\left(t\right)+K_1\tilde{\omega }\left(t\right)\hfill \\ & +{\theta }_1\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right),\hfill \end{array}$$

(5)

and

$${\widehat{u}}_f\left(t\right)={\theta }_2{\widehat{u}}_f(t-\tau )+K_2({\sigma }_1\tilde{\omega }(t-\tau )+{\sigma }_2\tilde{\omega }\left(t\right)),$$

(6)

where $\widehat{\omega }\left(t\right)\in {\mathbb{R}}^3$ and ${\widehat{u}}_f\left(t\right)\in {\mathbb{R}}^3$ denote the attitude angular velocity estimation and the torque deviation reconstruction, respectively, and $\tilde{\omega }\left(t\right)=\omega \left(t\right)-\widehat{\omega }\left(t\right)$ denotes the estimation error of the attitude angular velocity. The signum function $\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right)$ is added in (5) for robustness performance against the jointed disturbance. Herein, $K_1\in {\mathrm{R}}^{3\times 3}$ and $K_2\in {\mathrm{R}}^{3\times 3}$ are observer gain matrices, and ${\theta }_i,i=1,2$ and ${\sigma }_i,i=1,2$ are positive observer gain scalars. These gain matrices and scalars are to be determined later.

Remark 1. 

By considering the advantages of the iterative learning algorithm [28] and the repetitive learning algorithm [27], we propose a novel mixed learning algorithm (6). It can be noted that the proposed mixed learning algorithm (6) will be degraded into the iterative learning algorithm [28] when ${\sigma }_1=1$ and ${\sigma }_2=0$ and into the repetitive learning algorithm [27] when ${\sigma }_1=0$ and ${\sigma }_2=1$. Therefore, positive scalars ${\sigma }_1$ and ${\sigma }_2$ can be viewed as weight coefficients for the two learning algorithms when ${\sigma }_1+{\sigma }_2=1$.

Remark 2. 

In this work, if it is assumed that ${\sigma }_1=-1$ and ${\sigma }_2=1$ in (6), we have:

$$\begin{array}{cc}\hfill {\widehat{u}}_f\left(t\right)& ={\theta }_2{\widehat{u}}_f(t-\tau )+\tau K_2(\frac{\tilde{\omega }\left(t\right)-\tilde{\omega }(t-\tau )}{\tau })\hfill \\ & \approx {\theta }_2{\widehat{u}}_f(t-\tau )+\tau K_2\dot{\tilde{\omega }}\left(t\right)\hfill \end{array},$$

(7)

with a small learning interval τ. In view of this, (7) can be viewed as a derivative-type learning algorithm. As we all know, the single derivative term cannot be applied in the PID control strategy because it only adds the convergence speed and cannot ensure the reconstruction accuracy. Therefore, to guarantee the positive effect of the previous and current output estimation error on actuator fault reconstruction, ${\sigma }_1$ and ${\sigma }_2$ are chosen to be positive scalars.

The error dynamics of attitude angular velocity can be obtained by subtracting (2) from (5) such that:

$$\begin{array}{cc}& J_s\dot{\tilde{\omega }}\left(t\right)+\omega {\left(t\right)}^{\times }{J}_s\omega \left(t\right)-\widehat{\omega }{\left(t\right)}^{\times }{J}_s\widehat{\omega }\left(t\right)\hfill \\ \hfill =& {\tilde{u}}_f\left(t\right)-K_1\tilde{\omega }\left(t\right)-{\theta }_1\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right)+u_d\left(t\right)\hfill \end{array},$$

(8)

where ${\tilde{u}}_f\left(t\right)=u_f\left(t\right)-{\widehat{u}}_f\left(t\right)$ represents the reconstruction error of the actuator fault.

According to (6), the error dynamics for actuator fault reconstruction can be obtained as:

$${\tilde{u}}_f\left(t\right)={\theta }_2{\tilde{u}}_f(t-\tau )-K_2({\sigma }_1\tilde{\omega }(t-\tau )+{\sigma }_2\tilde{\omega }\left(t\right))+\Delta {\tilde{u}}_f\left(t\right),$$

(9)

where $\Delta {\tilde{u}}_f\left(t\right)=u_f\left(t\right)-{\theta }_2{u}_f(t-\tau )$.

According to (9), we have:

$$\begin{array}{cc}\hfill {\tilde{u}}_f^T\left(t\right){\tilde{u}}_f\left(t\right)=& {\theta }_2^2{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f(t-\tau )-2{\theta }_2{\sigma }_1{\tilde{u}}_f^T(t-\tau )K_2\tilde{\omega }(t-\tau )\hfill \\ & -2{\theta }_2{\sigma }_2{\tilde{u}}_f^T(t-\tau )K_2\tilde{\omega }\left(t\right)+2{\theta }_2{\tilde{u}}_f^T(t-\tau )\Delta {\tilde{u}}_f\left(t\right)\hfill \\ & +{\sigma }_1^2{\tilde{\omega }}^T(t-\tau )K_2^T{K}_2{\tilde{\omega }}^T(t-\tau )+{\sigma }_2^2{\tilde{\omega }}^T\left(t\right)K_2^T{K}_2{\tilde{\omega }}^T\left(t\right)\hfill \\ & +\Delta {\tilde{u}}_f^T\left(t\right)\Delta {\tilde{u}}_f\left(t\right)+2{\sigma }_1{\sigma }_2{\tilde{\omega }}^T(t-\tau )K_2^T{K}_2{\tilde{\omega }}^T\left(t\right)\hfill \\ & -2{\sigma }_1{\tilde{\omega }}^T(t-\tau )K_2^T\Delta {\tilde{u}}_f\left(t\right)-2{\sigma }_2{\tilde{\omega }}^T\left(t\right)K_2^T\Delta {\tilde{u}}_f\left(t\right).\hfill \end{array}$$

(10)

Assumption 3 

([26]). Assume that $\left|\right|\Delta {\tilde{u}}_f\left(t\right)=u_f\left(t\right)-{\theta }_2{u}_f(t-\tau )u_f\left|\right|\le {\overline{u}}_f$, where ${\overline{u}}_f$ is a small positive constant.

3.2. Stability Analysis of the Mixed Learning Observer

Theorem 1. 

Suppose that Assumptions 1–3 hold. If there exist positive-definite symmetrical matrix $P_1\in {\mathbb{R}}^{3\times 3}$, gain matrices $K_1\in {\mathbb{R}}^{3\times 3}$ and $K_2\in {\mathbb{R}}^{3\times 3}$, and positive gain parameters ${\theta }_i,i=1,2$ and ${\sigma }_i,i=1,2$ such that the following relations hold,

$$-2{\alpha }_1{\sigma }_2{K}_2^T{K}_1+({\alpha }_1{\eta }^2+1)I_3<0$$

(11)

$$P_1{J}_s^{-1}={\alpha }_1{\sigma }_2{K}_2^T$$

(12)

$${\alpha }_1{\theta }_2^2(1+{ϵ}_1+{ϵ}_2)-1<0$$

(13)

$${\alpha }_1{\sigma }_1^2(1+\frac{1}{{ϵ}_1}+{ϵ}_3)K_2^T{K}_2-I_3<0$$

(14)

$${\theta }_1>u_{dmax}$$

(15)

where ${ϵ}_i>0,i=1,2,3$ and ${\alpha }_1\ge 1$, then the proposed MLO involved in (5) and (6) can guarantee the uniform ultimate boundedness (UUB) of $\tilde{\omega }\left(t\right)$ and ${\tilde{u}}_f\left(t\right)$.

Proof. 

Consider a candidate Lyapunov function as:

$$\begin{array}{cc}\hfill V_1\left(t\right)=& {\tilde{\omega }}^T\left(t\right)P_1\tilde{\omega }\left(t\right)+{\int }_{t-\tau }^t{\tilde{\omega }}^T\left(\chi \right)\tilde{\omega }\left(\chi \right)\mathrm{d}\chi \hfill \\ & +{\int }_{t-\tau }^t{\tilde{u}}_f^T\left(\chi \right){\tilde{u}}_f\left(\chi \right)\mathrm{d}\chi \hfill \end{array}.$$

(16)

The time derivative of $V_1$ (16) along (8) and (9) can be derived as:

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& {\tilde{\omega }}^T\left(t\right)P_1\dot{\tilde{\omega }}\left(t\right)+{\dot{\tilde{\omega }}}^T\left(t\right)P\tilde{\omega }\left(t\right)+{\tilde{\omega }}^T\left(t\right)\tilde{\omega }\left(t\right)\hfill \\ & -{\tilde{\omega }}^T(t-\tau )\tilde{\omega }(t-\tau )+{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)\hfill \\ & -{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f^T(t-\tau )\hfill \\ \hfill =& 2{\tilde{\omega }}^T\left(t\right)P_1{J}_s^{-1}[-(\omega {\left(t\right)}^{\times }{J}_s\omega \left(t\right)-\widehat{\omega }{\left(t\right)}^{\times }{J}_s\widehat{\omega }\left(t\right))\hfill \\ & -K_1\tilde{\omega }+u_d\left(t\right)-{\theta }_1\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right)]+2{\tilde{\omega }}^T\left(t\right)P_1{J}_s^{-1}{\tilde{u}}_f\left(t\right)\hfill \\ & +{\tilde{\omega }}^T\left(t\right)\tilde{\omega }\left(t\right)-{\tilde{\omega }}^T(t-\tau )\tilde{\omega }(t-\tau )-{\alpha }_0{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)\hfill \\ & +{\alpha }_1{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)-{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f^T(t-\tau ),\hfill \end{array}$$

(17)

where ${\alpha }_1=1+{\alpha }_0$ with ${\alpha }_0\ge 0$.

Substituting (9) and (10) into (17) leads to:

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& 2{\tilde{\omega }}^T\left(t\right)P_1{J}_s^{-1}[-(\omega {\left(t\right)}^{\times }{J}_s\omega \left(t\right)-\widehat{\omega }{\left(t\right)}^{\times }{J}_s\widehat{\omega }\left(t\right))\hfill \\ & -K_1\tilde{\omega }\left(t\right)+u_d\left(t\right)-{\theta }_1\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right)]\hfill \\ & +{\tilde{\omega }}^T\left(t\right)\tilde{\omega }\left(t\right)-{\tilde{\omega }}^T(t-\tau )\tilde{\omega }(t-\tau )\hfill \\ & +2{\tilde{\omega }}^T\left(t\right)P_1{J}_s^{-1}{\tilde{u}}_f\left(t\right)-{\alpha }_0{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)\hfill \\ & +{\alpha }_1{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)-{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f^T(t-\tau )\hfill \\ \hfill =& 2{\tilde{\omega }}^T\left(t\right)P_1{J}_s^{-1}[-(\omega {\left(t\right)}^{\times }{J}_s\omega \left(t\right)-\widehat{\omega }{\left(t\right)}^{\times }{J}_s\widehat{\omega }\left(t\right))\hfill \\ & -K_1\tilde{\omega }\left(t\right)+u_d\left(t\right)-{\theta }_1\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right)]\hfill \\ & -{\alpha }_0{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)+{\tilde{\omega }}^T\left(t\right)\tilde{\omega }\left(t\right)-{\tilde{\omega }}^T(t-\tau )\tilde{\omega }(t-\tau )\hfill \\ & +2{\theta }_2{\tilde{\omega }}^T\left(t\right)[P_1{J}_s^{-1}-{\alpha }_1{\sigma }_2{K}_2^T]{\tilde{u}}_f(t-\tau )\hfill \\ & -2{\sigma }_1{\tilde{\omega }}^T\left(t\right)[P_1{J}_s^{-1}-{\alpha }_1{\sigma }_2{K}_2^T]K_2\tilde{\omega }(t-\tau )\hfill \\ & -2{\sigma }_2{\tilde{\omega }}^T\left(t\right)[P_1{J}_s^{-1}{K}_2-{\alpha }_1{\sigma }_2{K}_2^T]\tilde{\omega }\left(t\right)\hfill \\ & +2{\tilde{\omega }}^T\left(t\right)[P_1{J}_s^{-1}-{\alpha }_1{\sigma }_2{K}_2^T]\Delta {\tilde{u}}_f\left(t\right)\hfill \\ & +{\alpha }_1({\theta }_2^2{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f(t-\tau )-2{\theta }_2{\sigma }_1{\tilde{u}}_f^T(t-\tau )K_2\tilde{\omega }(t-\tau )\hfill \\ & +2{\theta }_2{\tilde{u}}_f^T(t-\tau )\Delta {\tilde{u}}_f\left(t\right)+{\sigma }_1^2{\tilde{\omega }}^T(t-\tau )K_2^T{K}_2{\tilde{\omega }}^T(t-\tau )\hfill \\ & -2{\sigma }_1{\tilde{\omega }}^T(t-\tau )K_2^T\Delta {\tilde{u}}_f\left(t\right)+\Delta {\tilde{u}}_f^T\left(t\right)\Delta {\tilde{u}}_f\left(t\right)\hfill \\ & -{\sigma }_2^2{\tilde{\omega }}^T\left(t\right)K_2^T{K}_2{\tilde{\omega }}^T\left(t\right))-{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f^T(t-\tau ).\hfill \end{array}$$

(18)

With the aid of (12), (19) can be easily simplified into:

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& 2{\alpha }_1{\sigma }_2{\tilde{\omega }}^T\left(t\right)K_2^T[-(\omega {\left(t\right)}^{\times }{J}_s\omega \left(t\right)-\widehat{\omega }{\left(t\right)}^{\times }{J}_s\widehat{\omega }\left(t\right))\hfill \\ & -K_1\tilde{\omega }\left(t\right)+u_d\left(t\right)-{\theta }_1\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right)]-{\alpha }_0{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)\hfill \\ & +{\tilde{\omega }}^T\left(t\right)\tilde{\omega }\left(t\right)-{\tilde{\omega }}^T(t-\tau )\tilde{\omega }(t-\tau )\hfill \\ & +{\alpha }_1({\theta }_2^2{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f(t-\tau )-2{\theta }_2{\sigma }_1{\tilde{u}}_f^T(t-\tau )K_2\tilde{\omega }(t-\tau )\hfill \\ & +2{\theta }_2{\tilde{u}}_f^T(t-\tau )\Delta {\tilde{u}}_f\left(t\right)+{\sigma }_1^2{\tilde{\omega }}^T(t-\tau )K_2^T{K}_2{\tilde{\omega }}^T(t-\tau )\hfill \\ & -2{\sigma }_1{\tilde{\omega }}^T(t-\tau )K_2^T\Delta {\tilde{u}}_f\left(t\right)+\Delta {\tilde{u}}_f^T\left(t\right)\Delta {\tilde{u}}_f\left(t\right)\hfill \\ & -{\sigma }_2^2{\tilde{\omega }}^T\left(t\right)K_2^T{K}_2{\tilde{\omega }}^T\left(t\right))-{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f^T(t-\tau ).\hfill \end{array}$$

(19)

According to Young’s inequality [34], we have:

$$\begin{array}{cc}& -2{\sigma }_2{\tilde{\omega }}^T\left(t\right)K_2^T(\widehat{\omega }{\left(t\right)}^{\times }J\widehat{\omega }\left(t\right)-\omega {\left(t\right)}^{\times }J\omega \left(t\right))\hfill \\ \hfill \le & {\sigma }_2^2{\tilde{\omega }}^T\left(t\right)K_2^T{K}_2\tilde{\omega }\left(t\right)+{\eta }^2{\tilde{\omega }}^T\left(t\right)\tilde{\omega }\left(t\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}& -2{\theta }_2{\sigma }_1{\tilde{u}}_f^T(t-\tau )K_2\tilde{\omega }(t-\tau )\hfill \\ \hfill \le & {\theta }_2^2{ϵ}_1{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f(t-\tau )+\frac{{\sigma }_1^2}{{ϵ}_1}{\tilde{\omega }}^T(t-\tau )K_2^T{K}_2\tilde{\omega }(t-\tau ),\hfill \end{array}$$

(21)

$$\begin{array}{cc}& 2{\theta }_2{\tilde{u}}_f^T(t-\tau )\Delta {\tilde{u}}_f\left(t\right)\hfill \\ \hfill \le & {\theta }_2^2{ϵ}_2{\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f(t-\tau )+\frac{1}{{ϵ}_2}\Delta {\tilde{u}}_f^T\left(t\right)\Delta {\tilde{u}}_f\left(t\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}& -2{\sigma }_1{\tilde{\omega }}^T(t-\tau )K_2^T\Delta {\tilde{u}}_f\left(t\right)\hfill \\ \hfill \le & {\sigma }_1^2{ϵ}_3{\tilde{\omega }}^T(t-\tau )K_2^T{K}_2\tilde{\omega }(t-\tau )+\frac{1}{{ϵ}_3}\Delta {\tilde{u}}_f^T\left(t\right)\Delta {\tilde{u}}_f\left(t\right),\hfill \end{array}$$

(23)

where ${ϵ}_i>0,i=1,2,3$.

Further, substituting (20)–(23) into (19) leads to:

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& {\tilde{\omega }}^T\left(t\right)[-2{\alpha }_1{\sigma }_2{K}_2^T{K}_1+({\alpha }_1{\eta }^2+1)I_3]\tilde{\omega }\left(t\right)\hfill \\ & +2{\alpha }_1{\sigma }_2{\tilde{\omega }}^T\left(t\right)K_2^T[u_d\left(t\right)-{\theta }_1\mathrm{sgn}\left(\tilde{\omega }\left(t\right)\right)]-{\alpha }_0{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)\hfill \\ & +({\alpha }_1{\theta }_2^2(1+{ϵ}_1+{ϵ}_2)-1){\tilde{u}}_f^T(t-\tau ){\tilde{u}}_f(t-\tau )\hfill \\ & +{\tilde{\omega }}^T(t-\tau )({\alpha }_1{\sigma }_1^2(1+\frac{1}{{ϵ}_1}+{ϵ}_3)K_2^T{K}_2-I_3)\tilde{\omega }(t-\tau )\hfill \\ & +{\alpha }_1(1+\frac{1}{{ϵ}_2}+\frac{1}{{ϵ}_3})\Delta {\tilde{u}}_f^T\left(t\right)\Delta {\tilde{u}}_f\left(t\right).\hfill \end{array}$$

(24)

If conditions (13) and (14) hold, it can be noted from (24) that:

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)=& {\tilde{\omega }}^T\left(t\right)[-2{\alpha }_1{\sigma }_2{K}_2^T{K}_1+({\alpha }_1{\eta }^2+1)I_3]\tilde{\omega }\left(t\right)\hfill \\ & -{\alpha }_0{\tilde{u}}_f^T\left(t\right){\tilde{u}}_f^T\left(t\right)+{\alpha }_1(1+\frac{1}{{ϵ}_2}+\frac{1}{{ϵ}_3}){\overline{u}}_f^2\hfill \end{array}.$$

(25)

If condition (11) holds, it can be noted from (25) that the error dynamics (8) and (9) are stable and convergent. Therefore, the designed MLO can guarantee that $\tilde{\omega }\left(t\right)$ and ${\tilde{u}}_f\left(t\right)$ are uniformly ultimately bounded. This completes the proof. □

Remark 3. 

Compared with the existing ILOs and RLOs [18,24], the proposed MLO will have a certain smoothing effect of noises and abrupt disturbances on fault results, and it will have the potential to achieve higher fault-reconstructing accuracy. Compared with the existing LOs [29,30], the proposed MLO design is more convenient for observer parameter computation. Compared with the existing MLO [25], the proposed MLO design requires fewer computational resources. Therefore, the proposed MLO design is more suitable to be implemented in spacecraft systems.

Remark 4. 

Compared to integral-based adaptive learning algorithms, the algebraic iterative learning algorithm requires less online computing power and can handle discontinuous measurement signals [34,35]. Therefore, compared with the adaptive observers [9,10], the proposed MLO requires less computational resources. In addition, the norm-boundedness of the fault derivative is required in adaptive observers [9,10] and in sliding-mode observers (SMOs) [11,12], while the proposed MLO only requires Assumption 3 on the fault indirect increment. Therefore, the proposed MLO has a wider application in spacecraft systems.

Remark 5. 

To simplify the parameter computation of the proposed MLO, ${\alpha }_1$ can be selected to be 1, and $P_1$ can be selected to be a positive-definite diagonal matrix with the same element or the integral multiple of the term $J_s^{-1}$. According to (25), it can be seen that there is high fault-reconstructing accuracy if $\Delta {\tilde{u}}_f\left(t\right)$ is sufficiently small. In the other hand, a sufficiently small term $\Delta {\tilde{u}}_f\left(t\right)$ can be obtained when appropriately small scalars ${ϵ}_i,i=1,2$ are given such that ${\theta }_2$, which is close to 1, can be obtained according to (13). In addition, ${\sigma }_1$ and ${\sigma }_2$ can be chosen to satisfy ${\sigma }_1+{\sigma }_2=1$ with $0<{\sigma }_i<1,i=1,2$.

Following Theorem 1, the MLO design is given as follows:

• Step 1 Choose a sufficiently small positive scalar ${\alpha }_0$, then compute ${\alpha }_1$.
• Step 2 Choose an appropriate matrix $P_1$ and positive scalar ${\sigma }_2$, then compute gain matrix $K_2$ based on (12).
• Step 3 Calculate gain matrix $K_1$ according to (11).
• Step 4 Choose an appropriate scalar ${ϵ}_i,i=1,2$ and compute parameter ${\theta }_2$ using (13).
• Step 5 Calculate parameter ${\sigma }_1$ to satisfy (14).
• Step 6 Compute ${\theta }_1$ according to (15).

Corollary 1. 

Suppose that ${\sigma }_1=0$ and that Assumptions 1 and 2 hold. If there exist positive-definite diagonal matrix $P_1\in {\mathbb{R}}^{3\times 3}$, gain matrices $K_1\in {\mathbb{R}}^{3\times 3}$ and $K_2\in {\mathbb{R}}^{3\times 3}$, and scalars ${\theta }_i,i=1,2,$ such that the conditions (11)–(13) and (15) hold, then the proposed MLO involved in (5) and (6) can guarantee the UUB of $\tilde{\omega }\left(t\right)$ and ${\tilde{u}}_f\left(t\right)$.

Remark 6. 

If ${\sigma }_1=0$, it can be noted that condition (14) always holds. Therefore, the proposed MLO will degrade into the ILO proposed in [27]. Further, if ${\theta }_2=1$ and constant actuator faults are taken into consideration, the ILO designed using Corollary 1 can achieve asymptotic reconstruction of constant actuator faults because $\Delta {\tilde{u}}_f\left(t\right)\equiv 0$ [27].

4. MLO-Based Spacecraft Attitude Fault-Tolerant Stabilization Control

First, the attitude dynamics of the rigid spacecraft can be rewritten as:

$$J\dot{\omega }\left(t\right)+\omega {\left(t\right)}^{\times }J\omega \left(t\right)=D_0{u}_c\left(t\right)+{\widehat{u}}_f\left(t\right)+u_{df},$$

(26)

where $u_{df}={\tilde{u}}_f\left(t\right)+u_d\left(t\right)$. In this paper, it is assumed that fault reconstruction error ${\tilde{u}}_f\left(t\right)$ satisfies $\parallel {\tilde{u}}_f\left(t\right)\parallel \le {\overline{u}}_f$.

4.1. Design of a Mixed Learning Sliding-Mode Controller

To design a mixed learning sliding-mode controller, a linear sliding-mode manifold and the reaching law [31] are suggested as follows:

$$S=\omega +\delta q$$

(27)

and

$$\dot{S}=-{\kappa }_1\mathrm{sgn}\left(S\right)-{\kappa }_{2}S,$$

(28)

where $\delta >0$ and ${\kappa }_i>0,i=1,2$.

For the attitude dynamics (26), a novel MLSM-based spacecraft attitude fault-tolerant stabilization controller is designed as follows:

$$u_c=u_{smc}+u_{rec},$$

(29)

where $u_{smc}$ denotes sliding-mode control law and $u_{rec}$ denotes the mixed learning algorithm-based jointed disturbance compensation part. In the MLSM controller, $u_{smc}$ and $u_{rec}$ are respectively designed as follows:

$$u_{smc}={\omega }^{\times }{J}_s\omega -\frac{1}{2}\delta J_s(q^{\times }+q_0{I}_3)\omega -{\kappa }_{1}S-{\kappa }_2\mathrm{sgn}\left(S\right)$$

(30)

and

$$\begin{array}{cc}\hfill u_{rec}=& -{\widehat{u}}_{df}-{\widehat{u}}_f\left(t\right)\hfill \end{array}$$

(31a)

$$\begin{array}{cc}\hfill {\widehat{u}}_{df}\left(t\right)=& {\theta }_3{\widehat{u}}_{df}(t-\tau )+K_3({\sigma }_{3}S(t-\tau )+{\sigma }_{4}S\left(t\right)),\hfill \end{array}$$

(31b)

where $K_3\in {\mathbb{R}}^{3\times 3}$ represents controller gain matrix and ${\theta }_3$, $\varphi$, ${\sigma }_3$, and ${\sigma }_4$ are positive scalars that will be determined later.

According to (31a) and (31b), the estimation error dynamics of ${\widehat{u}}_{df}$ can be obtained as:

$${\tilde{u}}_{df}\left(t\right)={\theta }_3{\tilde{u}}_{df}(t-\tau )-K_3({\sigma }_{3}S(t-\tau )+{\sigma }_{4}S\left(t\right))+\Delta {\tilde{u}}_{df}\left(t\right),$$

(32)

where $\Delta {\tilde{u}}_{df}\left(t\right)=u_{df}\left(t\right)-{\theta }_3{u}_{df}(t-\tau )$.

4.2. Stability Analysis of the Mixed Learning Sliding-Mode Controller

Theorem 2. 

Suppose that Theorem 1 holds. If there exist positive-definite symmetric matrices $P_2\in {\mathbb{R}}^{3\times 3}$ as well as matrix $K_3\in {\mathbb{R}}^{3\times 3}$, positive scalars ${\sigma }_i,i=3,4$, and ${\theta }_3$ such that:

$$-2P_2{J}_s^{-1}+I_3<0$$

(33)

$$P_2{J}_s^{-1}={\alpha }_3{\sigma }_4{K}_3^T$$

(34)

$${\alpha }_3(1+{ϵ}_4+{ϵ}_5){\theta }_3^2-1\le 0$$

(35)

$${\alpha }_3{\sigma }_3^2(1+\frac{1}{{ϵ}_4}+{ϵ}_6)K_3^T{K}_3-I_3\le 0,$$

(36)

where ${ϵ}_i>0,i=4,5,6$ and ${\alpha }_3\ge 1$, then the proposed MLSM controller involved in (29), (30), (31a) and (31b) can guarantee the UUB of $\omega \left(t\right)$ and $q\left(t\right)$.

Proof. 

Consider a candidate Lyapunov function as:

$$V_2\left(t\right)=S^T{P}_{2}S+{\int }_{t-\tau }^t{S}^T\left(\chi \right)S\left(\chi \right)\mathrm{d}\chi +{\int }_{t-\tau }^t{\tilde{u}}_{df}^T\left(\chi \right){\tilde{u}}_{df}\left(\chi \right)\mathrm{d}\chi ,$$

(37)

where $P_2$ is a positive-definite symmetric matrix.

Then, the time derivative of ${\dot{V}}_2\left(t\right)$ can be deduced as follows:

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(t\right)=& S^T{P}_2\dot{S}+{\dot{S}}^T{P}_{2}S+S^T\left(t\right)S\left(t\right)\hfill \\ & -S^T(t-\tau )S(t-\tau )-{\alpha }_2{\tilde{u}}_{df}^T\left(t\right){\tilde{u}}_{df}\left(t\right)\hfill \\ & +{\alpha }_3{\tilde{u}}_{df}^T\left(t\right){\tilde{u}}_{df}\left(t\right)-{\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau ),\hfill \end{array}$$

(38)

where ${\alpha }_3=1+{\alpha }_2$ with ${\alpha }_2\ge 0$.

Substituting (29) and (32) into (38) leads to:

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(t\right)=& 2S^T{P}_2{J}_s^{-1}(-{\kappa }_{1}S-{\kappa }_2\mathrm{sig}\left(S\right))+S^T\left(t\right)S\left(t\right)\hfill \\ & -S^T(t-\tau )S(t-\tau )-{\alpha }_2{u}_{df}^T\left(t\right){\tilde{u}}_{df}\left(t\right)\hfill \\ & +2{\theta }_3{S}^T(P_2{J}_s^{-1}-{\alpha }_3{\sigma }_4{K}_3^T){\tilde{u}}_{df}\left(t\right)\hfill \\ & -2{\sigma }_3{S}^T(P_2{J}_s^{-1}-{\alpha }_3{\sigma }_4{K}_3^T)K_{3}S(t-\tau )\hfill \\ & -2{\theta }_4{S}^T(P_2{J}_s^{-1}-{\alpha }_3{\sigma }_4{K}_3^T)S\left(t\right)\hfill \\ & -{\alpha }_3{\sigma }_4^2{S}^T\left(t\right)K_3^T{K}_{3}S\left(t\right)\hfill \\ & +2S^T\left(t\right)(P_2{J}_s^{-1}-{\alpha }_3{\sigma }_4{K}_3^T)\Delta {\tilde{u}}_{df}^T\left(t\right)\hfill \\ & +{\alpha }_3({\theta }_3{\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau )\hfill \\ & -2{\theta }_3{\sigma }_3{\tilde{u}}_{df}^T{K}_{3}S(t-\tau )\hfill \\ & +2{\theta }_3{\tilde{u}}_{df}^T(t-\tau )\Delta {\tilde{u}}_{df}^T\left(t\right)\hfill \\ & +{\theta }_3^2{S}^T(t-\tau )K_3^T{K}_{3}S(t-\tau )\hfill \\ & -2{\sigma }_3{S}^T(t-\tau )K_3^{T}S\left(t\right)\Delta {\tilde{u}}_{df}^T\left(t\right)\hfill \\ & +\Delta {\tilde{u}}_{df}^T\left(t\right)\Delta {\tilde{u}}_{df}\left(t\right)-{\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau ).\hfill \end{array}$$

(39)

With the aid of (34), (44) can be easily simplified to:

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(t\right)=& 2S^T{P}_2{J}_s^{-1}(-{\kappa }_{1}S-{\kappa }_2\mathrm{sig}\left(S\right))+S^T\left(t\right)S\left(t\right)\hfill \\ & -S^T(t-\tau )S(t-\tau )-{\alpha }_3{\sigma }_4^2{S}^T\left(t\right)K_3^T{K}_{3}S\left(t\right)\hfill \\ & -{\alpha }_2{u}_{df}^T\left(t\right){\tilde{u}}_{df}\left(t\right)+{\alpha }_3({\theta }_3{\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau )\hfill \\ & -2{\theta }_3{\sigma }_3{\tilde{u}}_{df}^T{K}_{3}S(t-\tau )+2{\theta }_3{\tilde{u}}_{df}^T(t-\tau )\Delta {\tilde{u}}_{df}^T\left(t\right)\hfill \\ & +{\theta }_3^2{S}^T(t-\tau )K_3^T{K}_{3}S(t-\tau )\hfill \\ & -2{\sigma }_3{S}^T(t-\tau )K_3^{T}S\left(t\right)\Delta {\tilde{u}}_{df}^T\left(t\right)\hfill \\ & +\Delta {\tilde{u}}_{df}^T\left(t\right)\Delta {\tilde{u}}_{df}\left(t\right))-{\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau ).\hfill \end{array}$$

(40)

According to Young’s inequality [34], we have:

$$\begin{array}{cc}& -2{\theta }_3{\sigma }_3{\tilde{u}}_{df}^T(t-\tau )K_{3}S(t-\tau )\hfill \\ \hfill \le & {ϵ}_4{\theta }_3^2{\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau )+\frac{{\sigma }_3^2}{{ϵ}_4}{S}^T(t-\tau )K_3^T{K}_{3}S(t-\tau ),\hfill \end{array}$$

(41)

$$\begin{array}{cc}& 2{\theta }_3{\tilde{u}}_{df}^T(t-\tau )\Delta {\tilde{u}}_{df}\left(t\right)\hfill \\ \hfill \le & {ϵ}_5{\theta }_3^2{\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau )+\frac{1}{{ϵ}_5}\Delta {\tilde{u}}_{df}^T\left(t\right)\Delta {\tilde{u}}_{df}\left(t\right),\hfill \end{array}$$

(42)

$$\begin{array}{cc}& -2{\sigma }_3{S}^T(t-\tau )K_3^T\Delta {\tilde{u}}_{df}\left(t\right)\hfill \\ \hfill \le & {ϵ}_6{\sigma }_3^2{S}^T(t-\tau )K_3^T{K}_{3}S(t-\tau )+\frac{1}{{ϵ}_6}\Delta {\tilde{u}}_{df}^T\left(t\right)\Delta {\tilde{u}}_{df}\left(t\right),\hfill \end{array}$$

(43)

where ${ϵ}_i>0,i=4,5,6$.

Further, substituting (41)–(43) into (40) leads to:

$$\begin{array}{cc}\hfill {\dot{V}}_2\left(t\right)=& S^T\left(t\right)(-2{\alpha }_3{\sigma }_4{K}_3^T+I_3)S\left(t\right)\hfill \\ & +({\alpha }_3(1+{ϵ}_4+{ϵ}_5){\theta }_3^2-1){\tilde{u}}_{df}^T(t-\tau ){\tilde{u}}_{df}(t-\tau )\hfill \\ & +S^T(t-\tau )[{\alpha }_3{\sigma }_3^2(1+\frac{1}{{ϵ}_4}+{ϵ}_6)K_3^T{K}_3-1]S(t-\tau )\hfill \\ & +{\alpha }_3(1+\frac{1}{{ϵ}_5}+\frac{1}{{ϵ}_6})\Delta {\tilde{u}}_{df}^T\left(t\right)\Delta {\tilde{u}}_{df}\left(t\right)\hfill \\ & -{\alpha }_2{\tilde{u}}_{df}^T\left(t\right){\tilde{u}}_{df}\left(t\right).\hfill \end{array}$$

(44)

If conditions (33), (35), and (36) hold, S and ${\tilde{u}}_{df}$ are uniformly ultimately bounded. Therefore, the proposed MLSM controller (29) can guarantee the UUB of $\omega \left(t\right)$ and $q\left(t\right)$. □

Remark 7. 

To simplify the parameter computation of the proposed MLSMC, ${\alpha }_3$ can be selected to be 1, and $P_2$ can be selected to be a positive-definite diagonal matrix with the same element or the integral multiple of the term $J_s^{-1}$. According to (44), it can be seen that there is a high fault reconstructing accuracy if $\Delta {\tilde{u}}_{df}\left(t\right)$ is sufficiently small. In the other hand, a sufficiently small term $\Delta {\tilde{u}}_{df}\left(t\right)$ can be obtained when appropriately small scalars ${ϵ}_i,i=4,5$ are given such that ${\theta }_3$, which is close to 1, can be obtained according to (35). ${\sigma }_3$ and ${\sigma }_4$ can be chosen to satisfy ${\sigma }_3+{\sigma }_4=1$ with $0<{\sigma }_i<1,i=3,4$. In addition, δ can be chosen to be a small positive number.

Following Theorem 2, the MLSM controller design is given as follows:

• Step 1 Compute $P_2$ to satisfy condition (33).
• Step 2 Choose a sufficiently small positive scalar ${\alpha }_2$, then compute ${\alpha }_3$.
• Step 3 Choose a positive scalar ${\sigma }_4$ and compute $K_3$ based on (34).
• Step 4 Choose an appropriate scalar ${ϵ}_i,i=4,5$ and compute parameter ${\theta }_3$ using (35).
• Step 5 Choose an appropriate scalar ${ϵ}_6$ and compute parameter ${\sigma }_3$ using (36).
• Step 6 Choose appropriate positive scalars $\delta$, ${\kappa }_1$, and ${\kappa }_2$.

Corollary 2. 

Assume ${\sigma }_3=0$. If there exist positive-definite symmetric matrix $P_2\in {\mathbb{R}}^{3\times 3}$, matrices $K_3\in {\mathbb{R}}^{3\times 3}$, and positive parameter ${\sigma }_4$ such that conditions (33)–(35) hold, then the proposed MLSM controller (29) can guarantee that $\omega \left(t\right)$ and $q\left(t\right)$ are uniformly ultimately bounded.

Remark 8. 

In the MLSM controller (29), the mixed learning algorithm is used to update and compensate for the jointed disturbance, including fault reconstruction error, inertia uncertainty, and space disturbance torque. Therefore, compared with the existing SMC approach [31], the proposed MLO-based MLSMC approach could achieve higher attitude stabilization accuracy under more computational sources resulting from the mixed learning algorithm. Compared with the adaptive SMC methods [15,16,36], the proposed MLSMC has a lower computational burden because the P-type learning algorithm has this advantage compared with adaptive learning algorithms [36] and adaptive neural network models [15,16].

5. Simulation Studies

5.1. Simulation Inputs

In the satellite example, $J_s$ and $\Delta J$ are respectively assumed to be as follows:

$J_s=\left[\begin{array}{ccc}18.73& 1.0& 0.090\\ 1.0& 20.77& 0.092\\ 0.090& 0.092& 23.63\end{array}\right]\mathrm{kg}·{\mathrm{m}}^2$, $\Delta J=\left[\begin{array}{ccc}1.873& 0.1& 0.09\\ 0.1& 2.077& 0.0092\\ 0.009& 0.0092& 2.363\end{array}\right]\mathrm{kg}·{\mathrm{m}}^2$.

Space disturbance torque [27] is considered in the following form:

$$d\left(t\right)=\left[\begin{array}{c}{A}_0{J}_x(3\cos \left(0.0012t\right)+1)\\ A_0{J}_y(1.5\sin \left(0.0012t\right)+3\cos \left({\omega }_{0}t\right))\\ A_0{J}_z(3\sin \left(0.0012t\right)+1)\end{array}\right],$$

where $A_0=1.5\times {10}^{-5}$$\mathrm{N}·\mathrm{m}$. The initial actual value and the initial estimated value of attitude angular velocity are respectively chosen as $\omega \left(0\right)={[0.16-0.12-0.1]}^T\mathrm{rad}/\mathrm{s}$ and $\widehat{\omega }\left(0\right)={\left[000\right]}^T\mathrm{rad}/\mathrm{s}$. The initial value of the attitude quaternion is chosen as $Q\left(0\right)=[0.51500.7221-0.37600.2681]$. The actuator fault scenarios are considered as follows:

• Constant fault

  $$u_{fz}\left(t\right)=\left\{\begin{array}{cc}0.1& t\ge 100\mathrm{s}\\ 0& \mathrm{other}\end{array}\right..$$
• Time-varying fault

  $$u_{fx}\left(t\right)=\left\{\begin{array}{cc}0.1+0.08\sin (\pi /5t)& t\ge 50\mathrm{s}\\ 0& \mathrm{other}\end{array}\right..$$

  .

5.2. Simulation on MLO-Based Actuator Fault Reconstruction

According to the design procedure of the MLO, related parameters are chosen and computed in

Table 1. Parameter selection and computation of the MLO.

Parameters	Value
Variables	${ϵ}_1=0.01,{ϵ}_2=0.01,{ϵ}_3=1,P_1=I_3,{\alpha }_1=1$
Gain matrices	$K_1=\left[\begin{array}{ccc}0.2159& 0.0062& 0.0006\\ 0.0062& 0.2285& 0.0006\\ 0.0006& 0.0006& 0.2462\end{array}\right]$ $K_2=\left[\begin{array}{ccc}4.8176& -0.2319& -0.0174\\ -0.2319& 4.3444& -0.0160\\ -0.0174& -0.0160& 3.8088\end{array}\right]$
Scalars	${\sigma }_1=0.1,{\sigma }_2=0.9,{\theta }_1=0.005,$ ${\theta }_2=0.99,\tau =0.001$

. Based on the MLO design, simulation studies on MLO-based actuator fault reconstruction are performed, and related simulation results are provided in Figure 1, Figure 2, Figure 3 and Figure 4. The reconstruction signals and the reconstruction error of actuator faults are respectively demonstrated in Figure 1 and Figure 2, while Figure 3 and Figure 4 illustrate the estimation signals and the estimation error signals of the satellite attitude angular velocity, respectively. From Figure 1 and Figure 2, it can be seen that the MLO can accurately reconstruct actuator faults. Through a series of computations, the reconstruction accuracy of actuator faults reaches 0.005 $\mathrm{N}·\mathrm{m}$. From Figure 4, it can be observed that the estimation error signals of the satellite attitude angular velocity converge to zero regions. In other words, the MLO designed using Theorem 1 can accurately estimate the satellite angular velocity. Through the computation, the estimation accuracy of the satellite attitude angular velocity is less than $5\times {10}^{-4}\mathrm{rad}/\mathrm{s}$.

To illustrate the superiority of the proposed MLO-based actuator fault reconstruction approach, a comparison between the proposed MLO and the iterative learning SMO (ILSMO) [26] is provided in detail. Comparative simulation results for reconstructing actuator faults in the x axis and in the z axis are respectively given in Figure 5 and Figure 6. Figure 5 and Figure 6 reveal that the proposed MLO has higher accuracy than the ILSMO [26], despite having the robustness against space disturbance. This is mainly because the space disturbance torque is small within the normal space environment and has little influence on actuator fault reconstructing results. In addition, the proposed MLO updates the fault reconstructing law using both current and previous estimation of attitude angular velocity, such that it can accurately and quickly track actuator faults regardless of whether they are constant. Therefore, it can be concluded that the designed MLO can accurately estimate attitude angular velocity and reconstruction of actuator faults.

5.3. Simulation of MLSMC-Based Satellite Attitude Fault-Tolerant Stabilization Control

Related parameters are chosen and computed in

Table 2. Parameter selection and computation of the MLSMC.

Parameters	Value
Middle variables	${ϵ}_4=0.01,{ϵ}_5=0.01,{ϵ}_6=1,P_2=J_s,{\alpha }_3=1$
Gain matrices	$K_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
Parameter scalars	$\begin{array}{c}{\sigma }_3=0.01,{\sigma }_4=0.99,{\theta }_3=0.99,\\ \tau =0.001,{\kappa }_1=0.01,{\kappa }_2=3,\delta =0.1\end{array}$

. Based on the MLSMC design, simulation studies on MLSMC-based satellite attitude fault-tolerant stabilization control are performed, and related simulation results are demonstrated in Figure 7, Figure 8, Figure 9 and Figure 10. Additionally, a comparison between the proposed MLSMC approach and the SMC-based approach [31] is also provided. Under both approaches, the curves of satellite attitude angular velocity are illustrated in Figure 7, while the curves of the satellite attitude quaternion are illustrated in Figure 10. The stabilization control errors of satellite attitude angular velocity and of attitude quaternion under both approaches are computed and given in

Table 3. The stabilization control error of attitude angular velocity.

Error Term	MLSMC	SMC [31]
${\omega }_x$	$5\times {10}^{-4}$	$1\times {10}^{-2}$
${\omega }_y$	$5\times {10}^{-6}$	$1\times {10}^{-4}$
${\omega }_z$	$3\times {10}^{-4}$	$8\times {10}^{-3}$

and

Table 4. The stabilization control error of attitude quaternion.

Error Term	MLSMC	SMC [31]
$q_1$	$5\times {10}^{-3}$	$0.35$
$q_2$	$2\times {10}^{-5}$	$6\times {10}^{-3}$
$q_3$	$7\times {10}^{-3}$	$0.3$

. From Figure 7, Figure 8, Figure 9 and Figure 10, it can be noted that the MLSMC and the SMC can guarantee the actual satellite attitude angular velocity and that the satellite attitude quaternion can converge to small neighborhoods of zero. However, from Table 3 and Table 4, it can be found that, compared with the SMC-based approach proposed in [31], it has higher stabilization control accuracy on the satellite attitude angular velocity and the satellite attitude quaternion. This is mainly because the actuator faults are accurately reconstructed using the MLO, and the mixed learning algorithm in the MLSMC strategy is adopted to compensate for the composite disturbance. In addition, the curves of the sliding mode surface under both approaches are demonstrated in Figure 9, while the curves of the compensation term for the composite disturbance are illustrated in Figure 10. Therefore, we can conclude that the proposed attitude stabilization control approach has actuator fault-tolerant capability and is effective and applicable for rigid spacecraft with actuator faults.

6. Conclusions

This paper investigated the issues of active attitude fault-tolerant stabilization control for rigid spacecraft subject to space disturbance and inertial uncertainty. A novel MLO was designed to accurately reconstruct actuator faults with low computational resources. Moreover, to guarantee robust spacecraft attitude fault-tolerant stabilization, a novel MLO-based MLSMC was designed by synthesizing the mixed learning algorithm into the sliding mode controller based on the separation principle, in which the mixed learning algorithm was used to update composite disturbances online. Compared with the existing SMC methods, the proposed MLSMC has higher attitude stabilization accuracy due to its mixed learning algorithm. Finally, a numerical example was provided to demonstrate the validity and superiority of our proposed approach.

It is worth noting that the linear sliding mode algorithm in the MLSMC design is only suggested so that the stability and convergence of the closed-loop spacecraft attitude can be guaranteed. At present, various nonlinear terminal SMC strategies have been developed for fast spacecraft attitude control, and fast convergence performances (e.g., finite-time, fixed-time, and prescribed time) have been identified. Therefore, combined with the mixed learning algorithm and fast terminal SMC strategies, MLSMC-based spacecraft attitude fault-tolerant stabilization or tracking control will be the focus of our future work.