Prescribed-Performance-Bound-Based Adaptive Fault-Tolerant Control for Rigid Spacecraft Attitude Systems

Abstract

This paper investigates the attitude control problems of spacecraft subject to external disturbances and compound actuator faults, including both additive and multiplicative components. To address these problems, an improved learning observer (ILO) is proposed. Compared to traditional learning observers (TLOs), the improved learning observer incorporates the previous-step state estimation error as an iterative term. Based on the observer’s outputs, a robust adaptive fault-tolerant attitude control scheme is developed using the backstepping method, under a prescribed performance bound (PPB). This control framework guarantees that the attitude tracking error adheres to prescribed transient performance specifications, such as bounded overshoot and accelerated convergence. Unlike conventional control schemes, the proposed approach ensures that system trajectories remain strictly within the desired bound throughout the transient process. A comprehensive Lyapunov-based analysis rigorously demonstrates the global uniform ultimate boundedness of all closed-loop signals. Numerical simulations substantiate the efficacy of the proposed approach, highlighting the enhanced disturbance estimation capability of the ILO in comparison to the TLO, as well as the superior transient tracking performance of the PPB-based control strategy relative to existing methods.

1. Introduction

In the field of spacecraft control, with the continuous advancement of technology and the increasing complexity of space missions, autonomous control and precise navigation of spacecraft have become critical elements for achieving deep-space exploration [1,2], satellite deployment [3,4], and crewed space missions. Modern spacecraft executing critical operations such as orbital maneuvers and attitude adjustments in extreme space environments are continuously subjected to multiple threats, including cosmic radiation, thermal cycling, and space debris impacts. These threats pose a significant risk of directly leading to mission failure. Against this backdrop, developing control schemes with strong robustness and high fault tolerance have emerged as a core technical requirement to ensure stable spacecraft operation under extreme disturbances and partial failure conditions.

With the aim of improving the robustness and fault tolerance of spacecraft attitude control systems, researchers have explored various control strategies. These include classical methods such as PID control [5,6], as well as advanced techniques like backstepping control [7,8], sliding mode control [9,10], adaptive fuzzy control [11,12], hybrid neural network-based control [13,14], and model predictive control [15,16].

Backstepping is a systematic, recursive nonlinear methodology particularly well-suited for systems in strict-feedback form or those that can be transformed into such a form [17,18,19]. In spacecraft control, backstepping control has found widespread application in attitude control and trajectory tracking problems due to its simplicity and strong adaptability. Its core idea involves the step-by-step construction of virtual control laws and Lyapunov functions, decomposing complex nonlinear systems into a series of simpler subsystems for design. This approach effectively handles the inherent nonlinear behaviors and external disturbances in spacecraft dynamics. Consequently, it has garnered increasing attention for tasks like spacecraft attitude adjustment and trajectory tracking. As an illustration, ref. [20] presented an incremental backstepping control approach for attitude tracking that integrates a predefined-time disturbance observer to enhance the system’s robustness. Ref. [21] investigated robust finite-time controllers based on the rotation matrix capable of compensating for external disturbances with unknown bounds, achieving chattering-free continuous control. Furthermore, by combining adaptive control with backstepping, the study addressed uncertainties within spacecraft control systems, enhancing both robustness and adaptability. However, despite the significant successes achieved with backstepping control in spacecraft applications, it remains reliant on precise system modeling and the effective compensation of external disturbances and actuator faults. This dependence presents substantial challenges, particularly when spacecraft encounter sudden failures or actuator malfunctions, highlighting the need for more robust, adaptive, and fault-tolerant control strategies. Therefore, increasing research attention is being directed towards fault diagnosis techniques [22,23,24] to address failures and uncertainties within complex spacecraft systems.

Learning estimators, commonly referred to as learning observers (LOs), were initially introduced by Wen Chen and Mehrdad Saif [25] as an advanced approach for fault diagnosis. Unlike traditional adaptive disturbance observers, LOs are capable of estimating a wide range of fault types—including constant, time-varying, and periodic disturbances—without imposing restrictive assumptions on their structure. The core idea behind LOs is to exploit fault-related information from previous iterations along with the current state and output errors to iteratively refine the estimation of fault signals. In [26], an iterative learning disturbance observer was developed based on adaptive notch filtering to estimate and compensate for unknown multi-frequency disturbances in non-Gaussian singular stochastic distribution systems. This approach demonstrated effectiveness in handling both low- and high-frequency periodic disturbances. In [27], the authors investigated the closed-loop attitude stabilization of spacecraft affected by actuator failures, reaction wheel friction, and external perturbations. An LO-based fault estimation scheme was integrated with a fault-tolerant controller to ensure that the posterior probability density function could accurately track the desired distribution. Further developments were reported in [28], where an iterative learning disturbance observer was designed to estimate and mitigate lumped disturbances acting on spacecraft. A conventional proportional-derivative controller was then combined with the observer to achieve attitude stabilization under actuator degradations and system uncertainties. Additionally, Zhang et al. [29] proposed a novel event-triggered attitude control strategy incorporating an LO for actuator fault reconstruction. The observer output was used to reduce control update frequency while maintaining stabilization performance. While the aforementioned studies have demonstrated the capability of LO-based fault-tolerant control strategies to ensure stability under actuator faults and external disturbances, most of them focus primarily on asymptotic or steady-state performance. The transient behavior of spacecraft—such as overshoot and negative overshoot—has often been overlooked in the control design.

Prescribed performance bounds, initially proposed by Bechlioulis and Rovithakis [30], have been successfully applied in numerous fields. Inspired by [31], this paper designs an adaptive fault-tolerant control method based on PPB that adopts the error transformation function to formulate a control law with a prescribed performance bound to ensure that both transient and steady-state control errors meet the prescribed performance bounds, thereby ensuring the specified transient performance. To make this more convincing to journal readers, we have added the following to the Introduction section: unlike [31], which imposes conservative bounds $T_{Mi}$ on time-varying disturbances, our learning observer dynamically estimates perturbations through real-time error correction, circumventing the need for such unverifiable assumptions in complex space environments, where using learning observers enables real-time compensation for simultaneous actuator faults and cosmic radiation disturbances. First, an improved learning observer is designed to estimate lumped disturbances. Subsequently, an adaptive fault-tolerant control scheme incorporating prescribed performance bounds is proposed, guaranteeing the transient performance of the spacecraft attitude system under external disturbances and actuator failures. The stability of the control system is confirmed using Lyapunov stability criteria, and its effectiveness is validated through spacecraft simulations. The main contributions of this paper are outlined as follows:

(1) An improved learning observer design scheme is proposed. Compared to traditional learning observers, the learning estimation algorithm presented here updates the lumped disturbance estimate using the previous-step lumped disturbance estimate, the previous-step state estimation error, and the current state estimation error. This enables faster and more accurate estimation of system fault values.

(2) An adaptive fault-tolerant control strategy under PPB constraints is proposed to strictly ensure that the attitude tracking error always meets the preset transient and steady-state performance indicators, such as limiting the maximum overshoot within a specified bound and constraining the convergence rate to be no lower than a preset value, even in the presence of faults and disturbances.

(3) Through rigorous Lyapunov stability analysis, the global stability of the proposed observer and control scheme under compound faults is proved, and all closed-loop signals are uniformly bounded.

The structure of this paper is organized as follows. Section 2 introduces the spacecraft dynamics model and necessary preliminaries. Section 3 presents the designs of an improved learning observer and an adaptive fault-tolerant controller based on PPB. Section 4 provides the stability proofs for both the learning observer and the controller. Section 5 presents the simulation results of the proposed learning observer and control scheme. Section 6 concludes the paper.

• Notations: ℜ denotes the set of real numbers. ${\Re }^m$ and ${\Re }^{m\times m}$ represent the m-dimensional real space and the space of $m\times m$ real matrices, respectively. $E\in {\Re }^{m\times m}\mathrm{is}\mathrm{the}m\times m\mathrm{identity}\mathrm{matrix}$. $Q\in {\Re }^{3\times 3}$ represents a positive definite diagonal matrix. The notation $\parallel ·\parallel$ denotes the Euclidean norm for vectors or its induced spectral norm for matrices.

2. Problem Description and Preliminaries

This paper investigates spacecraft attitude dynamics subject to external disturbances

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\chi }}_1={\chi }_2\hfill \\ {\dot{\chi }}_2=f\left(\chi \right)+J\left(\chi \right)u_a+d\left(t\right)\hfill \\ y={\chi }_1\hfill \end{array}\right.\end{array}$$

(1)

The attitude dynamics (1) are described with respect to the inertial reference frame. The state ${\chi }_1={[\varphi ,\theta ,\psi ]}^{\mathrm{T}}\in {\Re }^3$ denotes the Euler angles (roll, pitch, and yaw, respectively), ${\chi }_2={[\dot{\varphi },\dot{\theta },\dot{\psi }]}^{\mathrm{T}}\in {\Re }^3$ represents their angular velocities, and $\chi ={[{\chi }_1,{\chi }_2]}^{\mathrm{T}}\in {\Re }^6$ represents the state vector of the attitude control system. The control input $u_a={[u_{a1},u_{a2},\dots ,u_{am}]}^{\mathrm{T}}\in {\Re }^m$ corresponds to actuator commands, where m represents the number of actuators incorporated in the system. Time-varying external disturbances are denoted by $d\left(t\right)\in {\Re }^3$. The configuration-dependent control torque distribution matrix $J\left(\chi \right)\in {\Re }^{3\times m}$ characterizes the spacecraft’s actuation structure. The smooth vector field $f\left(\chi \right)={[f_1\left(\chi \right),f_2\left(\chi \right),f_3\left(\chi \right)]}^{\mathrm{T}}\in {\Re }^3$ satisfies $f\left(0\right)=0$. The system output is given by $y\in {\Re }^3$.

This paper addresses actuator faults, as they represent both the most frequent and the most critical fault category encountered in control systems. Among the different actuator fault types, bias faults and efficiency degradation faults are particularly prevalent. In order to express efficiency reduction and bias faults in a generalized form, the control input in Equation (1) is expressed as follows:

$$\begin{array}{c}\hfill u_a=\Lambda u+\Delta u\end{array}$$

(2)

where $u={[u_1,u_2,\dots ,u_m]}^{\mathrm{T}}\in {\Re }^m$ represents the desired control input of the spacecraft. $\Lambda =\mathrm{diag}\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m\}$∈${\Re }^{m\times m}$, $0\le {\lambda }_i\le 1$ denotes the efficiency loss factor of the i-th actuator. Additionally, $\Delta u\in {\Re }^3$ represents the additive faults of the actuator.

Taking actuator faults into account, the state-space representation of the spacecraft attitude control system can be formulated as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\chi }}_1={\chi }_2\hfill \\ {\dot{\chi }}_2=f\left(\chi \right)+J\left(\chi \right)u+d_l\left(t\right)\hfill \\ y={\chi }_1\hfill \end{array}\right.\end{array}$$

(3)

with $d_l\left(t\right)=J\left(\chi \right)\Delta u+J\left(\chi \right)(\Lambda -E)u+d\left(t\right)$, and considering the $d_l\left(t\right)$ as the lumped disturbance of the system.

Control Objective. The objective of this paper is to design an adaptive fault-tolerant attitude control strategy for a rigid-body spacecraft that is subject to actuator faults and time-varying external disturbances. Specifically, both multiplicative and additive actuator faults are considered. Moreover, to ensure robust performance under these adverse conditions, the proposed controller is required to guarantee that the attitude tracking errors evolve within prescribed performance bounds, which impose strict specifications on the transient response (e.g., maximum overshoot, negative overshoot) and steady-state accuracy. The control law must ensure that all closed-loop signals remain bounded, the spacecraft attitude accurately tracks a time-varying reference trajectory, and the prescribed performance criteria are strictly satisfied despite the presence of uncertainties, actuator degradation, and external disturbances. For the purpose of this theoretical development, it is assumed that the desired control input u can be fully executed by the thrusters without exceeding their physical limits. The significant challenge of input saturation, which is particularly relevant in the context of actuator faults, is considered an important area for future investigation.

To support the subsequent analysis, the following assumptions and lemmas are adopted throughout this paper.

Assumption 1.

For all $t>0$, there exist ${\Omega }_i>0$ such that $\parallel y_d^{\left(i\right)}\left(t\right)\parallel \le {\Omega }_i,i=1,2$.

Remark 1.

This assumption bounds the range of the desired output and its derivatives, indicating that both the desired output and its rate of change are bounded. This is to ensure that, during the controller design, the system will not experience control failure due to unbounded variations in the desired output.

Lemma 1.

For any vectors $a,b\in {\Re }^n$ and $p,q>1$ such that ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, we have the following inequality:

$$a^{\mathrm{T}}b\le {\displaystyle \frac{1}{p}}{\parallel a\parallel }_p^p+{\displaystyle \frac{1}{q}}{\parallel b\parallel }_q^q$$

where ${\parallel a\parallel }_p$ and ${\parallel b\parallel }_q$ are the p-norm and q-norm of the vectors a and b, respectively.

3. Main Results

3.1. Design of Improved Learning Observer

In order to estimate the value of the lumped disturbance, an improved learning observer is designed based on the traditional learning observer:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\dot{\widehat{\chi }}}_2\left(t\right)& =f\left(\chi \right)+J\left(\chi \right)u+Q\widetilde{{\chi }_2}\left(t\right)+{\widehat{d}}_l\left(t\right)\hfill \\ \hfill {\widehat{d}}_l\left(t\right)& =l_1{\widehat{d}}_l\left(\vartheta \right)+l_2\widetilde{{\chi }_2}\left(t\right)+l_3\widetilde{{\chi }_2}\left(\vartheta \right)\hfill \end{array}\right.\end{array}$$

(4)

where $l_1,l_2,l_3$ are positive constant gains and ${\widehat{\chi }}_2\left(t\right)$ represents the state of the learning observer, which is the estimate value of the system state ${\chi }_2\left(t\right)$. ${\widehat{d}}_l\left(t\right)$ represents the estimate value of the lumped disturbance and $\vartheta =t-T_i$, in which $T_i$ is the learning interval of the observer. $Q\in {\Re }^{3\times 3}$ represents the positive definite diagonal matrix. ${\tilde{\chi }}_2\left(t\right)={\chi }_2\left(t\right)-{\widehat{\chi }}_2\left(t\right)$ and $\widetilde{d_l}\left(t\right)=d_l\left(t\right)-{\widehat{d}}_l\left(t\right)$ represent the observation error of the system state and the estimation error of the lumped disturbance, respectively.

Theorem 1.

Consider a general spacecraft attitude control system (1) subject to external disturbances and actuator faults. By adjusting the gain parameters $l_1$, $l_2$, $l_3$ of the learning observer (4), the state estimation error ${\tilde{\chi }}_2\left(t\right)$ and the lumped disturbance estimation error ${\tilde{d}}_l\left(t\right)$ of the observer can converge to any arbitrarily small domain radius. In other words, the state estimation error ${\tilde{\chi }}_2\left(t\right)$ and the lumped disturbance estimation error ${\tilde{d}}_l\left(t\right)$ of the observer are uniformly ultimately bounded.

3.2. Design of PPB-Based Adaptive Fault-Tolerant Controller

This subsection aims to design an adaptive fault-tolerant control strategy for spacecraft attitude tracking that enforces prescribed transient performance. Specifically, under simultaneous actuator faults and external disturbances, the attitude tracking error $e_1\left(t\right)=y\left(t\right)-y_d\left(t\right)$ is rigorously constrained within prescribed performance bounds for all $t\ge 0$. Following the framework in [31], the prescribed performance bound must be formally characterized. Then, a decreasing smooth function ${\phi }_i\left(t\right):{\Re }^{+}\to {\Re }^{+}\setminus \left\{0\right\}\mathrm{with}{lim}_{t\to \infty }{\phi }_i\left(t\right)={\phi }_{i\infty }>0$ is chosen as a performance function for each element of the tracking error vector $e_1$, where $i=1,2,3$. For example, ${\phi }_i\left(t\right)=({\phi }_{i0}-{\phi }_{i\infty })e^{-{\delta }_{i}t}+{\phi }_{i\infty }$, where ${\phi }_{i0}>{\phi }_{i\infty }$ and ${\delta }_i>0$.

Given the specified scalars $0<{\underline{\varphi }}_i\le 1$ and $0<{\overline{\varphi }}_i\le 1$, the guarantee for transient performance can be ensured provided that the following condition is satisfied:

$$-{\underline{\varphi }}_i{\phi }_i\left(t\right)<e_{1i}\left(t\right)<{\overline{\varphi }}_i{\phi }_i\left(t\right),\forall t\ge 0,i=1,2,3$$

(5)

where ${\underline{\varphi }}_i{\phi }_i\left(0\right)$ and ${\overline{\varphi }}_i{\phi }_i\left(0\right)$ define the undershoot lower bound and overshoot upper bound for $e_{1i}\left(t\right)$, respectively. Furthermore, the convergence rate of $e_{1i}\left(t\right)$ is lower-bounded by the decay rate of ${\phi }_i\left(t\right)$.

Considering the performance constraint (5) on the tracking error, this requirement can be reformulated as a bounded signal stabilization problem. The core design objective reduces to synthesizing an asymptotic stabilization controller for the transformed system to achieve asymptotic tracking of the spacecraft attitude system.

To facilitate this transformation, we introduce a smooth strictly monotonic function $M_i\left({\gamma }_i\right)$ with the following fundamental properties:

(i)

$$-{\underline{\varphi }}_i<M_i\left({\gamma }_i\right)<{\overline{\varphi }}_i$$

(6)

(ii)

$$\underset{{\gamma }_i\to +\infty }{lim}{M}_i\left({\gamma }_i\right)={\overline{\varphi }}_i,\underset{{\gamma }_i\to -\infty }{lim}{M}_i\left({\gamma }_i\right)=-{\underline{\varphi }}_i$$

(7)

(iii)

$$M_i\left(0\right)=0$$

(8)

By utilizing the characteristic properties (i)–(ii) of the transformation mapping $M_i\left({\gamma }_i\right)$, the performance constraint (5) admits the equivalent representation

$$e_{1i}\left(t\right)={\phi }_i\left(t\right)M_i\left({\gamma }_i\right)$$

(9)

By virtue of the strict monotonicity of the transformation mapping $M_i\left({\gamma }_i\right)$ and the property ${\phi }_i\left(t\right)\ne 0$, the inverse function can be expressed as

$${\gamma }_i=M_i^{-1}({\displaystyle \frac{e_{1i}\left(t\right)}{{\phi }_i\left(t\right)}})$$

(10)

Given the transformed error ${\gamma }_i$, if the inequality $-{\underline{\varphi }}_i{\phi }_i\left(t\right)<e_{1i}\left(t\right)<{\overline{\varphi }}_i{\phi }_i\left(t\right)$ holds and the controller guarantees boundedness of ${\gamma }_i\left(t\right)$ for all $t\ge 0$, then there exists $-{\underline{\varphi }}_i<e_{1i}\left(t\right)/{\phi }_i\left(t\right)<{\overline{\varphi }}_i$. Moreover, if ${lim}_{t\to +\infty }{\gamma }_i\left(t\right)=0$, then, according to property (iii) of $M_i\left({\gamma }_i\right)$, the tracking error $e_{1i}\left(t\right)$ asymptotically converges to zero, i.e., ${lim}_{t\to +\infty }{e}_{1i}\left(t\right)=0$.

In this paper, the transformed function $M_i\left({\gamma }_i\right)$ is designed as

$$M_i\left({\gamma }_i\right)={\displaystyle \frac{{\overline{\varphi }}_i{e}^{({\gamma }_i+{ϵ}_i)}-{\underline{\varphi }}_i{e}^{-({\gamma }_i+{ϵ}_i)}}{e^{({\gamma }_i+{ϵ}_i)}+e^{-({\gamma }_i+{ϵ}_i)}}}$$

(11)

where ${ϵ}_i={\displaystyle \frac{1}{2}}ln({\underline{\varphi }}_i/{\overline{\varphi }}_i)$. It can be shown that $M_i\left({\gamma }_i\right)$ satisfies properties (i)–(iii).

From Equation (11), the transformed error ${\gamma }_i\left(t\right)$ can be written as

$${\gamma }_i={M_i}^{-1}\left({\beta }_i\left(t\right)\right)=0.5ln({\overline{\varphi }}_i{\beta }_i\left(t\right)+{\overline{\varphi }}_i{\underline{\varphi }}_i)-0.5ln({\overline{\varphi }}_i{\underline{\varphi }}_i-{\underline{\varphi }}_i{\beta }_i\left(t\right)),$$

(12)

where ${\beta }_i\left(t\right)=e_{1i}\left(t\right)/{\phi }_i\left(t\right)$. Considering Equation (12), the time derivative of ${\gamma }_i$ is given as

$$\begin{array}{cc}\hfill \dot{{\gamma }_i}={\displaystyle \frac{\partial M_i^{-1}}{\partial {\beta }_i}}\dot{{\beta }_i}& =0.5[{\displaystyle \frac{1}{{\beta }_i+{\underline{\varphi }}_i}}-{\displaystyle \frac{1}{{\beta }_i-{\overline{\varphi }}_i}}]({\displaystyle \frac{{\dot{e}}_{1i}}{{\phi }_i}}-{\displaystyle \frac{e_{1i}{\dot{\phi }}_i}{{{\phi }^2}_i}})\hfill \\ \hfill & ={\eta }_i(\dot{e_{1i}}-{\displaystyle \frac{e_{1i}\dot{{\phi }_i}}{{\phi }_i}})\hfill \\ \hfill & ={\eta }_i(\dot{y_i}-{\dot{y}}_{id}-{\displaystyle \frac{e_{1i}{\dot{\phi }}_i}{{\phi }_i}})\hfill \end{array}$$

(13)

where ${\eta }_i$ is defined as

$${\eta }_i={\displaystyle \frac{1}{2{\phi }_i}}[{\displaystyle \frac{1}{{\beta }_i+{\underline{\varphi }}_i}}-{\displaystyle \frac{1}{{\beta }_i-{\overline{\varphi }}_i}}]$$

(14)

By virtue of property (i) of the transformation function $M_i\left({\gamma }_i\right)$ and Equation (7), it follows that ${\eta }_i$ is well-defined and satisfies ${\eta }_i\ne 0$. Consequently, the prescribed performance bound can be formally embedded into the original attitude dynamics (3) through transformation (13).

To incorporate the prescribed performance bound into the controller design, the dynamics of the transformed error ${\gamma }_i$ must be expressed in terms of the system states. Recall that the system output is defined as $y={\chi }_1$, which implies that $\dot{y}={\dot{\chi }}_1$. This fundamental relationship allows us to rigorously reconfigure the attitude motion equations in terms of the transformed variable $\gamma$ for the subsequent controller synthesis.

Substituting the state derivative ${\dot{\chi }}_1$ with $\dot{\gamma }$, the attitude motion equations are reconfigured as

$$\begin{array}{cc}\hfill \dot{\gamma }& =\eta (\dot{y}-{\dot{y}}_d-{\phi }^{-1}\dot{\phi }{e}_1)=\eta ({\chi }_2-{\dot{y}}_d-{\phi }^{-1}\dot{\phi }{e}_1)\hfill \\ \hfill {\dot{\chi }}_2& =f\left(\chi \right)+J\left(\chi \right)u+d_l\left(t\right)\hfill \\ \hfill y& ={\chi }_1\hfill \end{array}$$

(15)

where $\eta =\mathrm{diag}\{{\eta }_1,{\eta }_2,{\eta }_3\},\gamma ={[{\gamma }_1,{\gamma }_2,{\gamma }_3]}^T,e_1={[e_{11},e_{12},e_{13}]}^T,\phi =\mathrm{diag}\{{\phi }_1,{\phi }_2,{\phi }_3\}$ and $\dot{\phi }=\mathrm{diag}\{\dot{{\phi }_1},\dot{{\phi }_2},\dot{{\phi }_3}\}$.

To develop the adaptive fault-tolerant controller based on PPBs that can handle actuator failures, the following variables are defined:

$$z_1=\gamma ,z_2={\chi }_2-{\alpha }_1$$

(16)

where the virtual control law ${\alpha }_1$ will be designed in Theorem 2.

Theorem 2.

Building upon the given faulty system (3) and learning observer (4), an adaptive fault-tolerant control law is subsequently developed as outlined below:

$$u=J^{†}[-\eta z_1-f\left(\chi \right)-{\widehat{d}}_l\left(t\right)+{\dot{\alpha }}_1-{\Lambda }_2{z}_2-\nu \mathrm{sgn}\left(z_2\right)]$$

(17)

where ${\alpha }_1$ is designed as

$${\alpha }_1=-{\Lambda }_1{\eta }^{-1}{z}_1+{\dot{y}}_d+{\phi }^{-1}\dot{\phi }{e}_1$$

(18)

in which $J^{†}=J^{\mathrm{T}}{\left(J{J}^{\mathrm{T}}\right)}^{-1}$ is the pseudo-inverse of J, $\nu ={\nu }_0+{\nu }_1>0$, where ${\nu }_0>\sqrt{\kappa /\zeta }$ and ${\nu }_1$ is the positive constant. By appropriate selection of the design matrix parameters ${\Lambda }_1\in {\Re }^{3\times 3}$ and ${\Lambda }_2\in {\Re }^{3\times 3}$, the proposed adaptive fault-tolerant control scheme guarantees uniformly asymptotic stability of the spacecraft’s tracking performance.

4. Stability Analysis

4.1. Stability Proof of the Improved Learning Observer

To prove the stability of the proposed improved learning observer (4), we first derive its error dynamics by combining the system state (3) with the observer state (4), as shown below:

$$\begin{array}{cc}\hfill {\dot{\tilde{\chi }}}_2& ={\dot{\chi }}_2-{\dot{\widehat{\chi }}}_2\hfill \\ \hfill & =f\left(\chi \right)+J\left(\chi \right)u+d_l\left(t\right)-f\left(\chi \right)-J\left(\chi \right)u-Q{\tilde{\chi }}_2\left(t\right)-{\widehat{d}}_l\left(t\right)\hfill \\ \hfill & ={\tilde{d}}_l\left(t\right)-Q{\tilde{\chi }}_2\left(t\right)\hfill \end{array}$$

(19)

The estimation error of the observer is given by

$$\begin{array}{cc}\hfill {\tilde{d}}_l\left(t\right)& =d_l\left(t\right)-{\widehat{d}}_l\left(t\right)\hfill \\ \hfill & =d_l\left(t\right)-l_1{\widehat{d}}_l\left(\vartheta \right)-l_2{\tilde{\chi }}_2\left(t\right)-l_3{\tilde{\chi }}_2\left(\vartheta \right)\hfill \\ \hfill & =l_1{\tilde{d}}_l\left(\vartheta \right)-l_2{\tilde{\chi }}_2\left(t\right)-l_3{\tilde{\chi }}_2\left(\vartheta \right)+k\left(t\right)\hfill \end{array}$$

(20)

where $k\left(t\right)=d_l\left(t\right)-l_1{d}_l\left(\vartheta \right)$. From this, we have

$$\begin{array}{cc}\hfill {\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)& ={l_1}^2{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)+{l_2}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+{l_3}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)+k^{\mathrm{T}}\left(t\right)k\left(t\right)\hfill \\ \hfill & -2l_1{l}_2{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(t\right)-2l_1{l}_3{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)+2l_1{k}^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(\vartheta \right)\hfill \\ \hfill & +2l_2{l}_3{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(\vartheta \right)-2l_2{k}^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)-2l_1{l}_3{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \end{array}$$

(21)

Based on Lemma 1, it can be concluded that, for any vectors $x,y\in {\Re }^n$ and any constant $p>0$, the following bound holds: $-2p{x}^{\mathrm{T}}y\le p{x}^{\mathrm{T}}x+{\displaystyle \frac{1}{p}}{y}^{\mathrm{T}}y$. It follows that

$$\left\{\begin{array}{c}-2l_1{l}_2{\tilde{d}}_l^T\left(\vartheta \right){\tilde{\chi }}_2\left(t\right)\le c_1{l}_1^2{\tilde{d}}_l^T\left(\vartheta \right)d_l\left(\vartheta \right)+{\displaystyle \frac{l_2^2}{c_1}}{\tilde{\chi }}_2^T\left(t\right){\tilde{\chi }}_2\left(t\right)\hfill \\ \\ -2l_1{l}_3{\tilde{d}}_l^T\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\le c_2{l}_1^2{\tilde{d}}_l^T\left(\vartheta \right)d_l\left(\vartheta \right)+{\displaystyle \frac{l_3^2}{c_2}}{\tilde{\chi }}_2^T\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \\ \\ -2l_1{k}^T\left(t\right){\tilde{d}}_l\left(\vartheta \right)\le c_3{l}_1^2{\tilde{d}}_l^T\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)+{\displaystyle \frac{1}{c_3}}{k}^T\left(t\right)k\left(t\right)\hfill \\ \\ -2l_2{l}_3{\tilde{\chi }}_2^T\left(t\right){\tilde{\chi }}_2\left(\vartheta \right)\le c_4{l}_2^2{\tilde{\chi }}_2^T\left(t\right){\tilde{\chi }}_2\left(t\right)+{\displaystyle \frac{l_3^2}{c_4}}{\tilde{\chi }}_2^T\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \\ \\ -2l_2{\tilde{\chi }}_2^T\left(t\right)k\left(t\right)\le c_5{l}_2^2{\tilde{\chi }}_2^T\left(t\right){\tilde{\chi }}_2\left(t\right)+{\displaystyle \frac{1}{c_5}}{k}^T\left(t\right)k\left(t\right)\hfill \\ \\ -2l_2{\tilde{\chi }}_2^T\left(\vartheta \right)k\left(t\right)\le c_6{l}_2^2{\tilde{\chi }}_2^T\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)+{\displaystyle \frac{1}{c_6}}{k}^T\left(t\right)k\left(t\right)\hfill \end{array}\right.$$

Given the positive constants $c_i(i=1,2,\dots ,6)$, substitution of the preceding inequalities into Equation (21) leads to

$$\begin{array}{cc}\hfill {\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)& \le {l_1}^2{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)+{l_2}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+{l_3}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)+k^{\mathrm{T}}\left(t\right)k\left(t\right)\hfill \\ \hfill & +c_1{l_1}^2{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)+{\displaystyle \frac{{l_2}^2}{c_1}}{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+c_2{l_1}^2{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right)d_l\left(\vartheta \right)+{\displaystyle \frac{{l_3}^2}{c_2}}{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \\ \hfill & +c_3{l_1}^2{d}_l^{\mathrm{T}}\left(\vartheta \right)d_l\left(\vartheta \right)+{\displaystyle \frac{1}{c_3}}{k}^{\mathrm{T}}\left(t\right)k\left(t\right)+c_4{l_2}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+{\displaystyle \frac{{l_3}^2}{c_4}}{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \\ \hfill & +c_5{l_2}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+{\displaystyle \frac{1}{c_5}}{k}^{\mathrm{T}}\left(t\right)k\left(t\right)+c_6{l_3}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)+{\displaystyle \frac{1}{c_6}}{k}^{\mathrm{T}}\left(t\right)k\left(t\right)\hfill \end{array}$$

(22)

To facilitate the Lyapunov stability analysis, inequality (22) can be simplified as

$$\begin{array}{c}\hfill {\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)\le m_1{l_1}^2{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)+m_2{l_2}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+m_3{l_3}^2{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)+m_4{k}^{\mathrm{T}}\left(t\right)k\left(t\right)\end{array}$$

(23)

where $m_1=1+c_1+c_2+c_3$, $m_2=1+1/c_1+c_4+c_5$, $m_3=1+1/c_2+1/c_4+c_6$, $m_4=1+1/c_3+1/c_5+1/c_6$.

We choose the Lyapunov function

$$\begin{array}{c}\hfill V_L={\displaystyle \frac{1}{2}}{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+{\int }_{\vartheta }^t{\tilde{d}}_l^{\mathrm{T}}\left(\xi \right){\tilde{d}}_l\left(\xi \right)\mathrm{d}\xi +{\int }_{\vartheta }^t{\tilde{\chi }}_2^{\mathrm{T}}\left(\xi \right){\tilde{\chi }}_2\left(\xi \right)\mathrm{d}\xi \end{array}$$

(24)

By differentiating $V_L$ and substituting Equation (19), we obtain

$$\begin{array}{cc}\hfill {\dot{V}}_L& ={\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\dot{\tilde{\chi }}}_2\left(t\right)+{\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)-{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)+{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)-{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \\ & ={\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)-{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right)Q{\tilde{\chi }}_2\left(t\right)+{\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)-{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)+{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)-{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \end{array}$$

(25)

Based on Lemma 1, it follows that

$${\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)\le {\displaystyle \frac{c_7}{2}}{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)+{\displaystyle \frac{1}{2c_7}}{\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)$$

(26)

where $c_7$ is a positive constant. By substituting Equation (26) into Equation (25), such that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_L\le & -\left[{\lambda }_{min}\left(Q\right)-{\displaystyle \frac{c_7}{2}}-1\right]{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)-{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)-{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)\hfill \\ \hfill & -\zeta {\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)+\left[{\displaystyle \frac{1}{2c_7}}+\zeta +1\right]{\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)\hfill \end{array}\end{array}$$

(27)

where $\zeta$ is any positive constant, by substituting into inequality (23), ${\dot{V}}_L$ is represented by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_L\le & -\left[{\lambda }_{min}\left(Q\right)-{\displaystyle \frac{c_7}{2}}-(1+\zeta +{\displaystyle \frac{1}{2c_7}})m_2{l_2}^2\right]{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)-\left[1-(1+\zeta +{\displaystyle \frac{1}{2c_7}})m_1{l_1}^2\right]{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)\hfill \\ \hfill & -\left[1-(1+\zeta +{\displaystyle \frac{1}{2c_7}})m_3{l_3}^2\right]{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)-\zeta {\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)+m_4\left[{\displaystyle \frac{1}{2c_7}}+\zeta +1\right]k^{\mathrm{T}}\left(t\right)k\left(t\right)\hfill \end{array}\end{array}$$

(28)

If the observer gains satisfies the following inequalities:

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{k}_1={\lambda }_{min}\left(Q\right)-{\displaystyle \frac{c_7}{2}}-(1+\zeta +{\displaystyle \frac{1}{2c_7}})m_2{l_2}^2>0$$

(29)

$$k_2=1-(1+\zeta +{\displaystyle \frac{1}{2c_7}})m_1{l_1}^2\ge 0$$

(30)

$$k_3=1-(1+\zeta +{\displaystyle \frac{1}{2c_7}})m_3{l_3}^2\ge 0$$

(31)

then $k_1$, $k_2$, and $k_3$ are not direct tuning gains but are positive scalars derived from the stability analysis. Conditions (29)–(31) guide the selection of observer gains $l_1$, $l_2$, $l_3$ and constants $c_i$, $m_i$ to ensure stability.

Consequently, we have

$${\dot{V}}_L\le -k_1{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right){\tilde{\chi }}_2\left(t\right)-k_2{\tilde{d}}_l^{\mathrm{T}}\left(\vartheta \right){\tilde{d}}_l\left(\vartheta \right)-k_3{\tilde{\chi }}_2^{\mathrm{T}}\left(\vartheta \right){\tilde{\chi }}_2\left(\vartheta \right)-\zeta {\tilde{d}}_l^{\mathrm{T}}\left(t\right){\tilde{d}}_l\left(t\right)+\kappa$$

(32)

where $\kappa =m_4({\displaystyle \frac{1}{2c_7}}+\zeta +1)k^{\mathrm{T}}\left(t\right)k\left(t\right)$.

According to reference [32], the estimation error of the learning observer is ultimately uniformly bounded, satisfying

$$\underset{t\to \infty }{lim}{\left[\begin{array}{cc}{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right)& {\tilde{d}}_l^{\mathrm{T}}\left(t\right)\end{array}\right]}^{\mathrm{T}}\in {\Re }_c,{\Re }_c\triangleq \left\{\left[\begin{array}{c}{\tilde{\chi }}_2\left(t\right)\\ {\tilde{d}}_l\left(t\right)\end{array}\right]|\parallel {\tilde{\chi }}_2\parallel \le \sqrt{{\displaystyle \frac{\kappa }{k_1}}},\parallel {\tilde{d}}_l\parallel \le \sqrt{{\displaystyle \frac{\kappa }{\zeta }}}\right\}.$$

(33)

Clearly, the set ${\Re }_c$ constitutes a neighborhood containing the origin ${\left[\begin{array}{cc}{\tilde{\chi }}_2^{\mathrm{T}}\left(t\right)& {\tilde{d}}_l^{\mathrm{T}}\left(t\right)\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{c}00\end{array}\right]}^{\mathrm{T}}$. Through gain parameterization with sufficiently large $k_1$, $\zeta$ or sufficiently small $\kappa$, the estimation error converges to an arbitrarily small neighborhood of the origin. This completes the proof of Theorem 1.

4.2. Stability Proof of a PPB-Based Adaptive Fault-Tolerant Controller

In this section, we present a step-by-step stability proof based on the Lyapunov criterion.

Step 1. By taking the derivative of Equation (16), it follows that

$$\begin{array}{c}\hfill \dot{z_1}=\dot{\gamma }=\eta ({\chi }_2-{\dot{y}}_d-{\phi }^{-1}\dot{\phi }{e}_1)\end{array}$$

(34)

We choose the Lyapunov function as

$$V_1={\displaystyle \frac{1}{2}}{z_1}^{\mathrm{T}}{z}_1$$

(35)

Invoking Equation (34), the time derivative of $V_1$ can be written as

$$\begin{array}{cc}\hfill {\dot{V}}_1& =z_1^{\mathrm{T}}{\dot{z}}_1=z_1^{\mathrm{T}}\eta (z_2+{\alpha }_1-{\dot{y}}_d-{\phi }^{-1}\dot{\phi }{e}_1)\hfill \end{array}$$

(36)

Substituting (18),

$$\begin{array}{c}\hfill {\dot{V}}_1={z_1}^{\mathrm{T}}\eta (z_2-{\Lambda }_1{\eta }^{-1}{z}_1)=-{z_1}^{\mathrm{T}}{\Lambda }_1{z}_1+{z_1}^{\mathrm{T}}\eta z_2\end{array}$$

(37)

It is evident that the first term in Equation (37) is stable. Hence, the subsequent analysis will concentrate on the second term.

Step 2. Differentiating $z_2$ to obtain its derivative and then substituting Equations (15) and (16) into this derivative, we obtain

$$\begin{array}{c}\hfill \dot{z_2}={\dot{\chi }}_2-\dot{{\alpha }_1}=f\left(\chi \right)+J\left(\chi \right)u+d_l\left(t\right)-{\dot{\alpha }}_1\end{array}$$

(38)

We choose the Lyapunov function as

$$\begin{array}{c}\hfill V_2=V_1+{\displaystyle \frac{1}{2}}{z}_2^{\mathrm{T}}{z}_2\end{array}$$

(39)

Taking the time derivative of $V_2$ yields

$${\dot{V}}_2={\dot{V}}_1+z_2^{\mathrm{T}}{\dot{z}}_2$$

(40)

Applying Equations (37) and (38), we derive

$${\dot{V}}_2=-{z_1}^{\mathrm{T}}{\Lambda }_1{z}_1+{z_1}^{\mathrm{T}}\eta z_2+z_2^{\mathrm{T}}\left[f\left(\chi \right)+J\left(\chi \right)u+d_l\left(t\right)-{\dot{\alpha }}_1\right]$$

(41)

Substituting the adaptive fault-tolerant control input (17), Equation (41) can be expressed as

$$\begin{array}{cc}\hfill {\dot{V}}_2& =-z_1^{\mathrm{T}}{\Lambda }_1{z}_1+z_1^{\mathrm{T}}\eta z_2+z_2^{\mathrm{T}}\left[{\tilde{d}}_l\left(t\right)-\eta z_1-\nu \mathrm{sgn}\left(z_2\right)-{\Lambda }_2{z}_2\right]\hfill \\ & =-z_1^{\mathrm{T}}{\Lambda }_1{z}_1-z_2^{\mathrm{T}}{\Lambda }_2{z}_2+z_2^{\mathrm{T}}{\tilde{d}}_l\left(t\right)-({\nu }_0+{\nu }_1)z_2^{\mathrm{T}}\mathrm{sgn}\left(z_2\right)\hfill \end{array}$$

(42)

From the condition in Equation (33), it follows that $\parallel {\tilde{d}}_l\parallel \le \sqrt{\kappa /\zeta }$. Furthermore, the controller design specifies ${\nu }_1>0$ and ${\nu }_0>\sqrt{\kappa /\zeta }$, yielding

$$\begin{array}{cc}\hfill {\dot{V}}_2& =-z_1^{\mathrm{T}}{\Lambda }_1{z}_1-z_2^{\mathrm{T}}{\Lambda }_2{z}_2-{\nu }_1\parallel z_2\parallel +z_2^{\mathrm{T}}{\tilde{d}}_l\left(t\right)-{\nu }_0{z}_2^{\mathrm{T}}\mathrm{sgn}\left(z_2\right)\hfill \\ & \le -z_1^{\mathrm{T}}{\Lambda }_1{z}_1-z_2^{\mathrm{T}}{\Lambda }_2{z}_2-{\nu }_1\parallel z_2\parallel \le 0\hfill \end{array}$$

(43)

Since ${\Lambda }_1$ and ${\Lambda }_2$ are designed as positive definite matrices, it follows that $-z_1^{\mathrm{T}}{\Lambda }_1{z}_1\le 0$ and $-z_2^{\mathrm{T}}{\Lambda }_2{z}_2\le 0$. Furthermore, the condition ${\nu }_1>0$ holds by design. Consequently, Theorem 2 is proven.

5. Simulation

To validate the effectiveness of the proposed novel learning observer and PPB-based adaptive fault-tolerant control scheme, simulations are conducted for both components. Comparative studies incorporating a conventional iterative learning observer are performed, with subsequent analysis highlighting the quantitative performance metrics of the PPB-enhanced controller.

This study adopts a mathematical model of the attitude control system of a spacecraft operating in a circular orbit, as shown in [33,34]. The control torque of the system is provided by four thrusters, and the smooth vector field $f\left(\chi \right)={[f_1\left(\chi \right),f_2\left(\chi \right),f_3\left(\chi \right)]}^{\mathrm{T}}$, where the specific expressions for $f\left(\chi \right)$ are as follows:

$$\begin{array}{cc}\hfill f_1\left(\chi \right)=& {\omega }_0\dot{\psi }cos\psi cos\theta -{\omega }_0\dot{\theta }sin\psi sin\theta +\frac{I_y-I_z}{I_x}\times [\dot{\theta }\dot{\psi }+{\omega }_0\dot{\theta }cos\varphi sin\psi sin\theta +{\omega }_0\dot{\theta }cos\psi sin\varphi \hfill \\ \hfill & +{\omega }_0\dot{\psi }cos\psi cos\varphi +\frac{1}{2}{\omega }_0^{2}sin\left(2\psi \right){cos}^2\varphi sin\theta +\frac{1}{2}{\omega }_0^2{cos}^2\psi sin\left(2\varphi \right)-{\omega }_0\dot{\psi }sin\psi sin\theta sin\varphi \hfill \\ & -\frac{1}{2}{\omega }_0^2{sin}^2\theta {sin}^2\psi sin\left(2\varphi \right)-\frac{3}{2}{\omega }_0^2{cos}^2\theta sin\left(2\varphi \right)-\frac{1}{2}{\omega }_0^{2}sin\left(2\psi \right)sin\theta {sin}^2\varphi ]\hfill \end{array}$$

$$\begin{array}{cc}\hfill f_2\left(\chi \right)=& {\omega }_0\dot{\psi }sin\psi cos\varphi +{\omega }_0\dot{\varphi }cos\psi sin\varphi +{\omega }_0\dot{\psi }cos\psi sin\theta sin\varphi +{\omega }_0\dot{\theta }sin\psi cos\theta sin\varphi \hfill \\ \hfill & +{\omega }_0\dot{\varphi }sin\psi sin\theta cos\varphi +\frac{I_z-I_x}{I_y}\times [\dot{\varphi }\dot{\psi }+{\omega }_0\dot{\varphi }cos\varphi sin\psi sin\theta +{\omega }_0\dot{\varphi }cos\left(\psi \right)sin\varphi \hfill \\ \hfill & -{\omega }_0\dot{\psi }sin\psi cos\theta -\frac{1}{2}{\omega }_0^{2}sin\left(2\theta \right){sin}^2\psi cos\varphi -\frac{1}{2}{\omega }_0^{2}cos\theta sin\varphi sin\left(2\psi \right)+\frac{3}{2}{\omega }_0^{2}sin\left(2\theta \right)cos\varphi ]\hfill \end{array}$$

$$\begin{array}{cc}\hfill f_3\left(\chi \right)=& {\omega }_0\dot{\varphi }sin\varphi sin\psi sin\theta -{\omega }_0\dot{\psi }cos\varphi cos\psi sin\theta -{\omega }_0\dot{\theta }cos\varphi sin\psi cos\theta +{\omega }_0\dot{\psi }sin\psi sin\varphi \hfill \\ \hfill & -{\omega }_0\dot{\varphi }cos\psi cos\varphi +\frac{I_x-I_y}{I_z}\times [\dot{\varphi }\dot{\theta }+{\omega }_0\dot{\varphi }cos\psi cos\varphi -{\omega }_0\dot{\varphi }sin\left(\psi \right)sin\theta sin\varphi -{\omega }_0\dot{\theta }sin\psi cos\theta \hfill \\ \hfill & -\frac{1}{2}{\omega }_0^{2}sin\left(2\psi \right)cos\theta cos\varphi +\frac{1}{2}{\omega }_0^2{sin}^2\psi sin\varphi sin\left(2\theta \right)-\frac{3}{2}{\omega }_0^{2}sin\left(2\theta \right)sin\varphi ]\hfill \end{array}$$

where the orbital angular velocity is ${\omega }_0=1.0312\times {10}^{-1}\mathrm{rad}/\mathrm{s}$, and the moments of inertia about the three principal axes are $I_x=I_z=200\mathrm{N}·\mathrm{m}·{\mathrm{s}}^2$, $I_y=400\mathrm{N}·\mathrm{m}·{\mathrm{s}}^2$. The control allocation matrix $J\in {\Re }^{3\times 4}$ is explicitly given by

$$J=\left(\begin{array}{cccc}0.67& 0.67& 0.67& 0.67\\ 0.69& -0.69& -0.69& 0.69\\ 0.28& 0.28& -0.28& -0.28\end{array}\right)$$

Remark 2.

Although the control torque allocation matrix $J\left(\chi \right)$ is usually state-dependent, in this simulation, the thrusters are assumed to be fixed relative to the spacecraft frame and are therefore treated as a constant matrix [33,34].

In this section, the initial values of the attitude angle and the attitude angular rate for the spacecraft system are set to ${\chi }_1\left(0\right)={[0.5,-0.15,1.2]}^{\mathrm{T}}$ and ${\chi }_2\left(0\right)={[0.5,1,-0.7]}^{\mathrm{T}}$, respectively. The desired tracking command signal ${\chi }_d\left(t\right)$ is defined as ${[1.8,0.1,0.6cos\left(1.5t\right)]}^{\mathrm{T}}$. External disturbances $d\left(t\right)$ are characterized by ${[1.5sin\left(t\right),2cos\left(0.5t\right),1]}^{\mathrm{T}}$. Consider a scenario in the spacecraft attitude system where actuators experience both multiplicative and additive faults due to overheating conditions. Specifically, the first and fourth actuators develop faults while the second and third actuators maintain nominal operation. The actuator efficiency loss factor matrix is designed as $\Lambda =\mathrm{diag}\{0.5,0,0,0.3\}$, with the additive fault vector defined as $\Delta u={[2,3,0,0]}^{\mathrm{T}}$. The observer parameter gains are set as $Q=\mathrm{diag}\{16,16,16\}$, $l_1=0.88,l_2=9,l_3=13$. The learning interval of the observer is designed as $T_i=0.02$. The initial and steady-state boundary values of the performance function parameters ${\phi }_i\left(t\right)$, where $i=1,2,3$, are set as ${\phi }_{i0}=2.9$ and ${\phi }_{i\infty }=0.1$, respectively. The decay rates are set to ${\delta }_i=0.2$ ($i=1,2,3$); this specific value governs the lower bound of the convergence rate for tracking errors $e_{1i}\left(t\right)$ by regulating the contraction speed of performance functions ${\phi }_i\left(t\right)$. The boundary scale factors are set as $\underline{\varphi }=1$, $\overline{\varphi }=1$, and are used to constrained transient performance (overshoot and undershoot).

5.1. Learning Observer Estimation Performance

Figure 1, Figure 2 and Figure 3 present a comparative analysis of the lumped disturbance estimation performance, contrasting the traditional learning observer with the improved learning observer scheme against the ground-truth lumped disturbance in the system. Specifically, the black solid line corresponds to the ground-truth lumped disturbance, the red dashed line represents the estimation result of the proposed improved learning observer, and the blue dash–dot line denotes that of the traditional learning observer.

Figure 4, Figure 5 and Figure 6 illustrate the estimation errors of both the improved learning observer and the traditional learning observer, providing a clearer comparison of their respective convergence rates. The red dash–dot line represents the estimation error of the external lumped disturbance as obtained by the improved learning observer proposed in this study, while the blue dashed line indicates the estimation error associated with the traditional learning observer.

As illustrated in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, the improved learning observer developed in this paper differs from the traditional learning observer [32] by incorporating an additional iterative term, which involves the estimated system state error from the previous time step. This modification enhances the observer’s ability to iteratively refine the state estimation accuracy over time. Relative to the estimation performance of the traditional observer, the proposed scheme enables rapid reconstruction of the lumped disturbance in the system and achieves significantly smaller estimation errors.

Comparative simulations reveal that the convergence time of the proposed improved learning observer is, on average, reduced by 94.1% when compared to the traditional learning observer. Specifically, the ILO achieves convergence in 0.3 s, 0.42 s, and 0.49 s for each axis, whereas the TLO requires 5.1 s, 0.98 s, and 5.3 s under identical convergence conditions.

5.2. Control Scheme Performance

To further validate the dynamic performance of the proposed adaptive fault-tolerant controller under prescribed performance bounds, numerical simulations are conducted on the spacecraft attitude control system under actuator faults and external disturbances. For comparison, we employ a conventional backstepping control method without prescribed performance bounds (denoted as ‘without PPB’), which is a standard approach in the spacecraft attitude control literature [35]. This method utilizes the same observer and control law framework as our proposed scheme but excludes the PPB constraints in Equations (5) and (18). The corresponding results are illustrated in Figure 7, Figure 8 and Figure 9, which present the tracking error curves for the roll, pitch, and yaw channels.

Based on the analysis of Figure 7, Figure 8 and Figure 9, it can be concluded that the proposed adaptive fault-tolerant controller, which is based on the PPB, not only ensures the system’s stable operation under actuator failures and external disturbances but also guarantees that the tracking error meets the prescribed performance specifications. This approach achieves predictability and controllability of the system response process. Compared to traditional control methods, the proposed scheme demonstrates significant advantages in dynamic performance and robustness.

The simulation results indicate that the control scheme incorporating the PPB exhibits a reduced overshoot, with the roll, pitch, and yaw angles converging to the desired tracking attitude in 5.3 s, 5.8 s, and 5.7 s, respectively. In contrast, the control scheme without the PPB displays a higher overshoot, with the roll, pitch, and yaw angles converging to the target attitude in 7.0 s, 7.1 s, and 7.2 s, respectively. These findings demonstrate that the control scheme with the PPB effectively mitigates overshoot, thereby enhancing the stability and precision of the system.

To provide a more rigorous quantitative comparison of the tracking performance, the root mean square error (RMSE) for each channel is calculated over the entire simulation duration. The results, presented in

Table 1. RMSE comparison: proposed method and that without PPB.

Posture Channel	Proposed Method	Without PPB
Phi ($\varphi$)	0.0094	0.0213
Theta ($\theta$)	0.0152	0.0387
Psi ($\psi$)	0.0081	0.0195

, clearly show that the proposed PPB-based controller achieves a significantly lower tracking error compared to the conventional backstepping controller without the PPB, quantitatively confirming its superior accuracy and robustness.

6. Conclusions

This paper proposed an adaptive fault-tolerant attitude control strategy for rigid-body spacecraft subject to multiplicative and additive actuator faults as well as time-varying external disturbances. To guarantee prescribed transient and steady-state performance, a control framework incorporating prescribed performance bounds was developed. An improved learning observer was designed by introducing the estimation error from the previous time step as an iterative term, thereby enhancing the observer’s convergence rate and estimation accuracy under faulty conditions. The proposed controller integrates fault estimation and PPB-constrained control design, and theoretical stability of the closed-loop system was rigorously proved via Lyapunov analysis. Comprehensive simulations were conducted on a rigid-body spacecraft model to validate the effectiveness of the proposed scheme. Comparative studies were performed between the proposed and conventional learning observers, as well as between the control schemes with and without PPB enforcement. The results clearly demonstrate that the proposed method achieves faster convergence, improved fault estimation accuracy, and superior tracking performance while strictly satisfying the prescribed performance bounds, even in the presence of external disturbances and actuator faults.

Despite the demonstrated effectiveness of the proposed scheme, this study has limitations that present clear avenues for future research. A primary limitation is the assumption that the control inputs do not saturate. In practice, severe actuator faults can demand control efforts that exceed physical limits—a condition that poses a significant challenge for PPB-based methods and can potentially lead to performance degradation. Future work will focus on integrating anti-windup compensation or developing modified PPB frameworks to explicitly handle input saturation. Additionally, while the observer gains were chosen to satisfy stability conditions, a more rigorous sensitivity analysis could lead to a systematic tuning methodology. Finally, further validation through hardware-in-the-loop simulations or on a physical testbed would be crucial to assess the controller’s performance in a real-world environment.