## 3 Mixed Learning Observer-Based Actuator Fault Reconstruction  
### Novel Mixed Learning Observer Construction  
Based on the ILO design [30], a novel MLO is constructed for (2) in the following form:  
\[ \begin{split} J_{s}\dot{\omega}(t)+\hat{\omega}(t)^{\times}J_{s} \hat{\omega}(t)=& u_{c}(t)+\hat{u}_{f}(t)+K_{1}\hat{\omega}(t)\\ &+\theta_{1}\text{sgn}(\hat{\omega}(t)),\end{split}  \tag{5}\]  
and  
\[\hat{u}_{f}(t)=\theta_{2}\hat{u}_{f}(t-\tau)+K_{2}(\sigma_{1}\hat{\omega}(t- \tau)+\sigma_{2}\hat{\omega}(t)), \tag{6}\]  
where \(\hat{\omega}(t)\in\mathbb{R}^{3}\) and \(\hat{u}_{f}(t)\in\mathbb{R}^{3}\) denote the attitude angular velocity estimation and the torque deviation reconstruction, respectively, and \(\hat{\omega}(t)=\omega(t)-\hat{\omega}(t)\) denotes the estimation error of the attitude angular velocity. The signum function \(\text{sgn}(\hat{\omega}(t))\) is added in (5) for robustness performance against the jointed disturbance. Herein, \(K_{1}\in\mathbb{R}^{3\times 3}\) and \(K_{2}\in\mathbb{R}^{3\times 3}\) 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:_  
\[ \begin{split}\hat{u}_{f}(t)&=\theta_{2}\hat{u}_{f}(t- \tau)+\tau K_{2}(\frac{\tilde{\omega}(t)-\tilde{\omega}(t-\tau)}{\tau}),\\ &\approx\theta_{2}\hat{u}_{f}(t-\tau)+\tau K_{2}\tilde{\omega}( t)\end{split}  \tag{7}\]  
_with a small learning interval \(\tau\). 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:  
\[ \begin{split}& J_{s}\tilde{\omega}(t)+\omega(t)^{\times}J_{s} \omega(t)-\tilde{\omega}(t)^{\times}J_{s}\tilde{\omega}(t)\\ =&\bar{u}_{f}(t)-K_{1}\tilde{\omega}(t)-\theta_{1} \text{sgn}(\tilde{\omega}(t))+u_{d}(t)^{\prime}\end{split}  \tag{8}\]  
where \(\bar{u}_{f}(t)=u_{f}(t)-\hat{u}_{f}(t)\) represents the reconstruction error of the actuator fault. According to (6), the error dynamics for actuator fault reconstruction can be obtained as:  
\[\bar{u}_{f}(t)=\theta_{2}\bar{u}_{f}(t-\tau)-K_{2}(\sigma_{1}\tilde{\omega}(t -\tau)+\sigma_{2}\tilde{\omega}(t))+\Delta\bar{u}_{f}(t), \tag{9}\]  
where \(\Delta\bar{u}_{f}(t)=u_{f}(t)-\theta_{2}u_{f}(t-\tau)\). According to (9), we have:  
\[