## 3 Mixed Learning Observer-Based Actuator Fault Reconstruction
### Stability Analysis of the Mixed Learning Observer
Further, substituting (20)-(23) into (19) leads to:  
\[-K_{1}\tilde{\omega}(t)+u_{d}(t)-\theta_{1}\text{sgn}(\tilde{ \omega}(t))]-\alpha_{0}\tilde{u}_{f}^{T}(t)\tilde{u}_{f}^{T}(t)\]5 \[-K_{1}\tilde{\omega}(t)+u_{d}(t)-\theta_{1}\text{sgn}(\tilde{ \omega}(t))]-\alpha_{0}\tilde{u}_{f}^{T}(t)\tilde{u}_{f}^{T}(t)\]6 \[-K_{1}\tilde{\omega}(t)+u_{d}(t)-\theta_{1}\text{sgn}(\tilde{ \omega}(t))]-\alpha_{0}\tilde{u}_{f}^{T}(t)\tilde{u}_{f}^{T}(t)\]7 \[-K_{1}\tilde{\omega}(t)+u_{d}(t)-\theta_{1}\text{sgn}(\tilde{ \omega}(t))]-\alpha_{0}\tilde{u}_{f}^{T}(t)\tilde{u}_{f}^{T}(t)\]8 \[-K_{1}\tilde{\omega}(t)+u_{d}(t)-\theta_{1}\text{sgn}(\tilde{ \omega}(t))]-\alpha_{0}\tilde{u}_{f}^{T}(t)\tilde{u}_{f}^{T}(t)\]9  
If conditions (13) and (14) hold, it can be noted from (24) that:  
\[+\tilde{\omega}^{T}(t)\tilde{\omega}(t)-\tilde{\omega}^{T}(t-\tau )\tilde{\omega}(t-\tau)\]0 \[+\tilde{\omega}^{T}(t)\tilde{\omega}(t)-\tilde{\omega}^{T}(t-\tau )\tilde{\omega}(t-\tau)\]1  
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 \(\tilde{\omega}(t)\) and \(\tilde{u}_{f}(t)\) 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 \(I_{s}^{-1}\). According to (25), it can be seen that there is high fault-reconstructing accuracy if \(\Delta\tilde{u}_{f}(t)\) is sufficiently small. In the other hand, a sufficiently small term \(\Delta\tilde{u}_{f}(t)\) can be obtained when appropriately small scalars \(e_{i},i=1,2\) 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 \(e_{i},i=1,2\) 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.