## 2 Problem Statement and Preliminaries  
### Problem Statement  
Following [2], the control-oriented model of the rigid-body RLV is given as follows:  
\[\begin{cases}\dot{\mathbf{\Theta}}=\mathbf{R}\mathbf{\omega}\\ \mathbf{I}\dot{\mathbf{\omega}}=-\mathbf{\Omega}\mathbf{I}\mathbf{\omega}+\mathbf{M}+\Delta\mathbf{D}\end{cases} \tag{1}\]  
where \(\mathbf{\Theta}=[\alpha,\beta,\sigma]^{\top}\) represents the attitude angle vector, \(\mathbf{\omega}=[p,q,r]^{\top}\) denotes the angular rate vector, \(\mathbf{M}=[M_{x},M_{y},M_{z}]^{\top}\) is the control input vector, and \(M_{x},M_{y},M_{z}\) are limited to the interval \([-\bar{M},\bar{M}]\), \(\Delta\mathbf{D}=\Delta\mathbf{D}_{a}+\Delta\mathbf{D}_{e}\) is the unknown disturbance vector, \(\Delta\mathbf{D}_{a}\) is the aerodynamic torque vector, and \(\Delta\mathbf{D}_{e}\) is the external disturbance vector. \(\Delta\mathbf{D}_{a}\) can be formulated as:  
\[\Delta\mathbf{D}_{a}=\begin{bmatrix}\begin{pmatrix}\frac{m_{x}^{p}L_{P}}{V}+m_{x} ^{\sigma}\sigma\end{pmatrix}q_{V}S_{r}L_{r}\\ \begin{pmatrix}\frac{m_{y}^{p}L_{r}}{V}+m_{y}^{\beta}\beta\end{pmatrix}q_{V}S_{ r}L_{r}\\ \begin{pmatrix}\frac{m_{z}^{p}L_{P}}{V}+m_{z}^{\alpha}\alpha\end{pmatrix}q_{V}S_{ r}L_{r}\end{bmatrix} \tag{2}\]  
where \(m_{x}^{p}\), \(m_{y}^{\tau}\) and \(m_{z}^{q}\) represent the damping moment coefficients, \(m_{x}^{\sigma}\), \(m_{y}^{\beta}\) and \(m_{z}^{\alpha}\) are the static stability moment coefficients, \(V\) is the velocity, \(q_{V}\) is the dynamic pressure, and \(S_{r}\) and \(L_{r}\) are the cross-sectional area and the reference length of RLV, respectively.  
The skew-symmetric matrix \(\mathbf{\Omega}\), the inertia matrix \(\mathbf{I}\), and the coordinate transformation matrix \(\mathbf{R}\) are defined by  
\[\mathbf{\Omega} =\begin{bmatrix}0&-r&q\\ r&0&-p\\ -q&p&0\end{bmatrix},\mathbf{I}=\begin{bmatrix}I_{xx}&0&-I_{xz}\\ 0&I_{yy}&0\\ -I_{zx}&0&I_{zz}\end{bmatrix}, \tag{3}\] \[\mathbf{R} =\begin{bmatrix}-\cos\alpha\tan\beta&1&-\sin\alpha\tan\beta\\ \sin\alpha&0&-\cos\alpha\\ -\cos\alpha\cos\beta&-\sin\beta&-\sin\alpha\cos\beta\end{bmatrix}.\]  
Defining the guidance command vector \(\mathbf{\Theta}_{d}=[\alpha_{c},\beta_{c},\sigma_{c}]^{\top}\), the attitude tracking error vector \(\mathbf{e}_{1}=\mathbf{\Theta}-\mathbf{\Theta}_{d}=[e_{1\alpha},e_{1\beta},e_{1\sigma}]^{\top}\), and the angular rate tracking error vector \(\mathbf{e}_{2}=\mathbf{R}\mathbf{\omega}-\dot{\mathbf{\Theta}}_{d}\), the tracking error dynamics are given as:  
\[\begin{cases}\dot{\mathbf{e}}_{1}=\mathbf{e}_{2}\\ \dot{\mathbf{e}}_{2}=\mathbf{B}_{1}\mathbf{M}+\Delta\mathbf{D}_{1}\end{cases} \tag{4}\]  
where \(\mathbf{B}_{1}=\mathbf{R}\mathbf{I}^{-1}\) is the control matrix, \(\Delta\mathbf{D}_{1}=-\mathbf{R}\mathbf{I}^{-1}\mathbf{\Omega}\mathbf{I}\mathbf{\omega}-\dot{\mathbf{\Theta }}_{c}+\dot{\mathbf{R}}\mathbf{\omega}+\mathbf{R}\mathbf{I}^{-1}\Delta\mathbf{D}\) denotes the lumped disturbance vector.  
**Assumption 1** ([2]).: \(\Delta\mathbf{D}_{1}\) _is bounded by \(||\Delta\mathbf{D}_{1}||\leq D_{m}\)._  
**Assumption 2** ([2]).: _During the re-entry phase, \(\beta\
eq\pm 90\) deg, thus \(\mathbf{R}\) is always invertible._  
Control Objective: According to the tracking error dynamics (4), the control objective of this paper can be summarized as developing an RL-based optimal control scheme with guaranteed fixed-time prescribed performance such that the attitude tracking errors \(e_{1i}(i=\alpha,\beta,\sigma)\) can converge to a preset region within a preassigned fixed time.