## 3 Fault-Tolerant Control Design for the Attitude System of HSV
### Fault-Tolerant Control Design for Slow-Loop of HSV
#### 3.1.1 Baseline Controller under Nominal Condition  
In this work, the nonlinear generalized predictive control, one kind of optimal control is utilized to achieve the control target of the low-loop of HSV due to its excellent dynamic performance [22]. Initially, the receding horizon performance metric was introduced as  
\[J=\int_{0}^{T}\hat{e}_{s}^{T}(t+\hbar)\hat{e}_{s}(t+\hbar)d\hbar\bigg{/}2 \tag{15}\]  
where \(T\) is the predictive period and \(\hat{e}_{s}\) represents the predictive tracking error that is defined as  
\[\hat{e}_{s}(t+\hbar)=\hat{y}(t+\hbar)-\hat{y}_{r}(t+\hbar) \tag{16}\]  
where \(\hat{y}(t+\hbar)\) is the predictive output, and \(\hat{y}_{r}(t+\hbar)\) stands for the reference output. * **Obtaining future behaviors**. To this aim, by resorting to the Taylor series expansion, the future behavior of \(\hat{y}(t+\hbar)\) can be predicted by \[\hat{y}(t+\hbar) = \Gamma(\hbar)\bar{Y}(t)\] (17) where \(\Gamma(\hbar){=}\Big{[}1,\hbar,\cdots,\frac{\rho^{\rho+r}}{(\rho+r)! }\Big{]} \in R^{m\times m(\rho+r+1)}\), \(\hbar=diag\{\hbar,\cdots,\hbar\}\in R^{m\times m}\), \(\bar{Y}=[\hat{y},\)\(\hat{y},\cdots,\hat{y}^{[\rho+r]}]^{T}\), as shown in (A6), where \(\rho\) is the relative degree corresponding to the system output, and \(r\) is the control order [22]. Then, the index signal \(y_{r}\) is also predicted by \[\hat{y}_{r}(t+\hbar) = \Gamma(\hbar)\bar{Y}_{r}(t)\] (18) where \(\bar{Y}_{r}=[\hat{y}_{r},\hat{y}_{r},\cdots,\hat{y}_{r}^{[\rho+r]}]^{T}\), seen in (A6). From this, the tracking error associated with the predicted behavior is derived as \[\hat{e}(t+\hbar)=\Gamma(\bar{Y}-\bar{Y}_{r})\] (19) Moreover, by some algebraic manipulations and rearrangements of the resulting formulas, the prediction error in the performance index (15) is rewritten as \[J=\frac{1}{2}(\bar{Y}(t)-\bar{Y}_{r}(t))^{T}\int_{0}^{T}\Gamma^{T}(\hbar)\Gamma (\hbar)d\hbar(\bar{Y}(t)-\bar{Y}_{r}(t))\] (20) For analysis purpose, \(\bar{\Gamma}\) is defined as \(\bar{\Gamma}=\int_{0}^{T}\Gamma^{T}(\hbar)\Gamma(\hbar)d\hbar\) and its matrix blocks are split by \[\bar{\Gamma}=\left[ \begin{array}{cc}\bar{\Gamma}_{\rho\rho}&\bar{\Gamma}_{ \rho r}\\ \bar{\Gamma}_{\rho r}^{T}&\bar{\Gamma}_{rr}^{T}\end{array} \right]\] (21)* **Minimizing performance index**. In order to minimize the performance index (20) through the proper \(\hat{u}(z)\), the optimal \(\hat{u}(z)\) is solved by the following partial differential equation. \[\left.\frac{\partial J}{\partial\hat{u}_{z}}\right|_{u_{z}=u_{z}^{*}}=0\] (22) According to (20)-(22), it is further calculated as \[(\frac{\partial\hat{H}(\hat{u}_{z})}{\partial\hat{u}_{z}})^{T}[\Gamma_{pr}^{T }\ \hat{\Gamma}_{rr}]M+(\frac{\partial\hat{H}(\hat{u}_{z})}{\partial\hat{u}_{z}})^ {T}\Gamma_{rr}\hat{H}(\hat{u}_{z})=0\] (23) where \(M=[h(x)-y_{r}(t),L_{f_{s}}h(x)-\hat{y}_{r}(t),\cdots,L_{f_{s}}^{p+r}h(x)-y_{r} ^{p+r}(t)]^{T}\in R^{m(\rho+r+1)}\), and the details of \(\frac{\partial\hat{H}(\hat{u}_{z})}{\partial\hat{u}_{z}}\) are seen in (A7). Based on the assumption that \(L_{g_{s}}L_{f_{s}}^{\rho-1}h(x)\) and \(\frac{\partial\hat{H}(\hat{u}_{z})}{\partial\hat{u}_{z}}\) are reversible, it is obtained that \[\hat{H}(\hat{u}_{z})=-[\hat{\Gamma}_{rr}^{-1}\Gamma_{\rho r}^{T}\ \ \ I_{m(r+1)\times m(r+1)}]M\] (24) By invoking (A6), the front \(m\) rows of (24) can be described as follows \[L_{g_{s}}L_{f_{s}}^{\rho-1}h(x)\hat{u}_{z}(t)+KM_{\rho}+L_{f_{s}}^{\rho}h(x)-y_ {r}^{[\rho]}+L_{f_{s}}^{\rho-1}h(x)D_{s}=0\] (25) where \(K\) represents the front \(m\) rows of \(\Gamma_{rr}^{-1}\Gamma_{\rho r}^{T}\). \(M_{\rho}=[(h(x)-y_{r}(t))^{T},(L_{f_{s}}h(x)-\hat{y}_{r}(t))^{T},\cdots,(L_{ f_{s}}^{\rho-1}h(x)-y_{r}^{\rho-1}(t))^{T}]^{T}\in R^{m\rho}\). Eventually, the optimal control law \(\hat{u}_{z}(t)\) can be obtained as [33] \[\hat{u}_{z}(t)=-(L_{g_{s}}L_{f_{s}}^{\rho-1}h(x))^{-1}(L_{f_{s}}^{\rho}h(x)+ KM_{\rho}-y_{r}^{[\rho]}+L_{f_{s}}^{\rho-1}D_{s})\] (26)  
**Remark 2**.