## 4 MLO-Based Spacecraft Attitude Fault-Tolerant Stabilization Control
### Stability Analysis of the Mixed Learning Sliding-Mode Controller
\begin{split}\tilde{V}_{2}(t)=& S^{T}(t)(-2a_{3} \sigma_{4}K_{3}^{T}+I_{3})S(t)\\ &+(\alpha_{3}(1+\epsilon_{4}+\epsilon_{5})\theta_{3}^{2}-1) \tilde{u}_{df}^{T}(t-\tau)\tilde{u}_{df}(t-\tau)\\ &+S^{T}(t-\tau)[\alpha_{3}\sigma_{3}^{2}(1+\frac{1}{\epsilon_{4}} +\epsilon_{6})K_{3}^{T}K_{3}-1]S(t-\tau)\\ &+\alpha_{3}(1+\frac{1}{\epsilon_{5}}+\frac{1}{\epsilon_{6}}) \Delta\tilde{u}_{df}^{T}(t)\Delta\tilde{u}_{df}(t)\\ &-\alpha_{2}\tilde{u}_{df}^{T}(t)\tilde{u}_{df}(t).\end{split}  \tag{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 \(\omega(t)\) and \(q(t)\). 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 \(\Delta\tilde{u}_{df}(t)\) is sufficiently small. In the other hand, a sufficiently small term \(\Delta\tilde{u}_{df}(t)\) can be obtained when appropriately small scalars \(\epsilon_{i},i=4,5\) 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, \(\delta\) 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 \(\epsilon_{i},i=4,5\) and compute parameter \(\theta_{3}\) using (35). **Step 5** Choose an appropriate scalar \(\epsilon_{6}\) and compute parameter \(\sigma_{3}\) using (36). **Step 6** Choose appropriate positive scalars \(\delta,\kappa_{1}\), and \(\kappa_{2}\). 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 \(\omega(t)\) and \(q(t)\) are uniformly ultimately bounded. 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].