## 4 Simulation  
Considering that the HSV is carrying out a hypersonic re-entry flight with the speed of \(3\) km\(/\)s, there is an altitude of \(30\) km and a mass of \(m=136,820\) kg. The initial altitude vector is \(\Omega=[0.017,0.026,0.017]\) rad and the angular rate is \(\omega=[0,0,0]^{T}\mathrm{rad}/\mathrm{s}\). The reference signals are set as \(\alpha_{\mathrm{c}}=0.087\) rad during \(1\) s\(\sim 8\) s, \(\alpha_{\mathrm{c}}=0.02\) rad during \(8\) s\(\sim\)\(20\) s and \(\alpha_{\mathrm{c}}=0.05\) rad during \(t>20\) s, besides, \(\beta\) is always \(0\) rad and \(\mu\) is keeping \(0.035\) rad. In addition, the loss of efficiency fault of the actuator was set to \(\lambda=diag\{0.2,0.17,0.2\}\) as the 5 th second. The external perturbations of low-loop and quick-loop systems of HSV are described as follows \(\mathrm{d_{s1}}=-0.005\cdot(\sin(t)+1)\mathrm{rad}/\mathrm{s}\), \(\mathrm{d_{s2}}=0.003\cdot(\cos(t)+1)\mathrm{rad}/\mathrm{s}\), \(\mathrm{d_{s3}}=-0.002\cdot\sin(t+1)\mathrm{rad}/\mathrm{s}\), \(\mathrm{d_{f1}}=2\times 10^{5}\sin(4t+0.2)N\cdot m\), \(\mathrm{d_{f2}}=4\times 10^{5}\sin(11t-0.6)N\cdot m\), \(\mathrm{d_{f3}}=4\times 10^{5}\sin(5t+0.2)N\cdot m\). The design parameters of the low-loop controller of HSV are as follows \(K_{s}=\mathrm{diag}\{k_{1},k_{2},k_{3}\},k_{i}=3/(2*T),i=1,2,3\), \(T=2.4\). \(P_{s}=\mathrm{diag}\{5,5,5\}\), \(\theta=25\mathrm{diag}\{3,3,3\},\xi=16\mathrm{diag}\{2,2,2\},l_{1}=3.5,l_{2}=2, p=5,q=3\). As for the fast-loop controller, \(K_{f}=\mathrm{diag}\{5,6.5,6.5\}\), \(Q=50*\mathrm{diag}\{2.5,2.5,2.5,3,3,6\},\psi_{1}=\mathrm{diag}\{2.5,2.5,2.5\}, \psi_{2}=\mathrm{diag}\{3.2,3.2,3.2\},P=\mathrm{diag}\{4.5,4.5,4.5\}\). To study the effect of the centroid shift on the HSV motion and validate the effectiveness of the designed fault-tolerant controller, simulations were performed. For analyzing the effects of unexpected centroid shift on the longitudinal movement of HSV, comparedwith the tracking effects of the FTC strategy consisted of sliding mode control combined with nonlinear disturbance observer and the FTC controller based on RBFNN [10], the simulation results demonstrate the effectiveness of the designed FTC strategy. As shown in Figures 3-11, \((\cdot)_{mml}\) is the expected index of HSV's slow and fast loops. \((\cdot)_{ob}\) represents the fault-tolerant control effect of sliding mode controller (SMC) combined with the nonlinear disturbance observer (NDO) [35] by moving the uncertainties of \(J^{*}\) on the right hands of attitude dynamics, and \((\cdot)_{m}\) represents the simulation results of a control scheme that is composed of nonlinear generalized predictive control (NGPC) and radial basis function neural network (RBFNN), as shown in [10]. In addition, \((\cdot)_{ad}\) denotes the control effect of the controller designed in this paper. On the basis of Figures 3-8, in case of actuator failure only at the fifth second, attitude angles \((\alpha,\beta,\mu)\) with \((\cdot)_{ob}\) and \((\cdot)_{ad}\) tracking effect, \((\cdot)_{m}\) has faster convergence speed. The main reason may be the heavy line load calculation of RBFNN and the horizontal regression principle of NGPC. Before the centroid shift occurs, it is straightforward to verify that the above controller presents an acceptable FTC tracking effect based on the arguments above. When the unforeseeable centroid displacement occurs in the 10 th second, the eccentric movement and variation of inertial matrix combined with the LOE fault of the actuator give rise to the fluctuation of the control surface of \((\cdot)_{ob}\) compared with that of \((\cdot)_{m}\) and \((\cdot)_{ad}\), as shown in Figures 9-11. However, this does not affect the steady flight of the HSV considerably for the time being, as shown by the trajectories in Figures 3-8 during the 10 th and 20 th seconds.