## 4 Numerical Simulations  
In this section, numerical simulations for the re-entry phase of RLV are carried out to illustrate the effectiveness and improved performance of the proposed control scheme. The parameters of the RLV are based on [31]. The initial states of the re-entry phase are set as \(\alpha_{0}=47\) deg, \(\beta_{0}=-2\) deg, \(\sigma_{0}=2\) deg, \(p_{0}=0\) deg/s, \(q_{0}=0\) deg/s and \(r_{0}=0\) deg/s. The guidance commands are designed as \(\alpha_{c}=(45-0.5t)\) deg, \(\beta_{c}=0\) deg and \(\sigma_{c}=0\) deg. The uncertainties of the inertia parameters, the aerodynamic parameters and the air density are each set to +20% bias. The external disturbance \(\Delta D_{e}\) is given as [32]:  
\[\Delta D_{e}= \begin{bmatrix}(0.5+\cos(\pi t/2)+\sin(\pi t/3))\times p\\ (0.5+\cos(\pi t/2)+\sin(\pi t/4))\times r\\ (0.5+\cos(\pi t/3)+\sin(\pi t/2))\times q\end{bmatrix} \times 10^{4}(\mathbb{N} \cdot\mathbb{m}). \tag{52}\]  
The saturation bound of the control torques is set as \(\bar{M}=1\times 10^{5}\)\((\mathbb{N}\cdot\mathbb{m})\). The attitude tracking errors are required to be less than \(0.5\) deg for \(t\geq 2\) s. Therefore, the parameters of the FTPPF are selected as \(\rho_{0i}=\pi/36,\rho_{\text{o}i}=\pi/360\) and \(T=2\). Comprehensively considering the satisfactory transient and steady-state performance, the parameters of the proposed control scheme are rigorously chosen as \(\mathbf{c}=2\mathbf{I}_{3\times 3},\mathbf{\omega}=\mathbf{I}_{3\times 3},\mathbf{Q}=\mathbf{I}_{3 \times 3}\lambda_{1i}=2,\lambda_{2i}=0.1\). Inspired by the work of [12; 14], the suitable basis vectors can be selected as polynomial combinations of the concerned state variables in the performance index (16). The base vectors of the actor and critic networks are adjusted in repeated trials to balance the approximation error and the computational burden. They are identically selected as:  
\[\sigma[\mathbf{s}(t)]= \begin{bmatrix}\frac{s_{a}^{4}}{4},\frac{s_{a}^{3}}{3}, \frac{s_{a}^{2}}{2},\frac{s_{b}^{4}}{4},\frac{s_{b}^{3}}{3},\frac{s_{b}^{2}}{2},\frac{s_{a}^{4}}{4},\frac{s_{b}^{3}}{3},\frac{s_{b}^{2}}{2}\end{bmatrix} ^{ \top}. \tag{53}\]  
The initial values of \(\hat{\mathbf{W}}_{a}\) and \(\hat{\mathbf{W}}_{c}\) are randomly generated in \((0,1)\). The user-defined parameters of the weight adaptation laws are designed as \(\mathbf{A}_{1}=\mathbf{I}_{9\times 9},\mathbf{A}_{2}=0.1\mathbf{I}_{9\times 9},p_{c}=0.001\), \(p_{a}=1,\kappa=0.001\). Furthermore, in order to demonstrate the superiority of the proposed control scheme, the RL-based finite-time control (RLFTC) with input constraints in [17], and the robust adaptive backstepping control (RABC) method in [33], are implemented. To provide a fair  
Figure 4: Time histories of the attitude tracking error \(e_{1\beta}\). Figure 5: Time histories of the attitude tracking error \(e_{1\sigma}\). Figure 6: Time histories of the angular rates. Figure 8: Time histories of the control torques. Figure 7: Time histories of the sliding manifolds. Figure 9: Weights of the actor network. To make a clear comparison, the maximum overshoot and the adjustment time are introduced to evaluate the transient performance of these controllers. Moreover, the integral absolute control effort (IACE) index \(\int_{0}^{30}(|M_{i}|)\mathrm{d}\tau,(i=x,y,z)\) and the integral of the time and absolute error (ITAE) \(\int_{0}^{30}\tau(|e_{i1}|)\mathrm{d}\tau,(i=\alpha,\beta,\sigma)\) index of three control schemes are calculated to evaluate the tracking accuracy and control effort. The performance indices of the three channels are summarized in Tables 1-3. From the foregoing simulation results, it can be concluded that the proposed control scheme outperforms RLFTC and RABC in terms of transient performance, tracking accuracy and control effort. Furthermore, by synthesizing the AC structure and the fixed-time PPC paradigm, the proposed control scheme offers an online RL-based model-free solution for controlling RLV and other complex industrial systems.