## 4 Test Example
### Glider Dynamic Saring  
This case is a modified optimization problem of the glider dynamic soaring. The optimization objective is the maximum \(\overline{\rho}\), which is the fundamental parameter describing the performance of a wind gradient profile for a given glider [32]. The mathematical expression of the MDSDO problem for this case is shown in Formula (15), containing state variables \([\mathrm{x},y,h,V,\gamma,\psi]\) and two control variables \([C_{L},\mu]\). \[ \begin{array}{ll}\underset{\omega,x,y,z,V,\gamma,\psi,C_{L},\mu}{max}&\phi=- \overline{\rho}=-\frac{\rho\overline{\rho}\overline{\rho}\overline{\rho}}{2m \omega^{2}}\\ \text{subject to}&\dot{x}=Vcos(\gamma)sin(\psi)+\omega Vsin(\gamma)\\ \dot{y}=cos(\gamma)\\ \dot{z}=Vsin(\gamma)\\ \mathrm{V}=-\frac{\rho\overline{\rho}}{2m}(C_{D_{0}}+KC_{L}^{2})\mathrm{V}^{ 2}-gsin(\gamma)-\omega Vsin(\gamma)cos(\gamma)sin(\psi)\\ \dot{\gamma}=\frac{\rho\overline{\rho}}{2m}C_{L}Vcos(\mu)-\frac{1}{V}\big{(} gcos(\gamma)+\omega Vsin(\gamma)^{2}sin(\psi)\big{)}\\ \dot{\psi}=\frac{1}{cos(\gamma)}\Big{(}\frac{\rho\overline{\rho}}{2m}C_{L}Vsin (\mu)-\omega sin(\gamma)cos(\psi)\Big{)}\\ \mathrm{g}_{\mathrm{i}}(\cdot)\geq 0\ \mathrm{i}=1,2,3,4,5,6\\ \frac{L_{L}}{mg}-5=\frac{\rho\overline{\rho}}{mg}C_{L}V^{2}-5\leq 0\\ \mathbf{X}=[\mathrm{x},y,z,V,\gamma,\psi],\mathbf{U}=[C_{L},\mu]\\ \omega_{min}\leq\omega\leq\omega_{max}\\ \mathrm{X}_{min}\leq\mathbf{X}\leq X_{max}\\ \mathrm{U}_{min}\leq\mathbf{U}\leq U_{max}\end{array}  \tag{15}\]  
where \(x\), \(y\) are east and north positions, respectively; \(z\) is the altitude, \(V\) is the airspeed, \(\gamma\) is the air-relative flight path angle, \(\psi\) is the heading angle measured clockwise from the north, \(C_{L}\) is the lift coefficient, and \(\mu\) is the glider bank angle. In addition, \(\mathrm{g}_{\mathrm{i}}\) represents the boundary conditions of state variables, \(\omega\) is the mean of wind gradient slope, \(C_{D_{0}}\) is the parasitic drag coefficient, \(L\) is the lift force, \(S\) is the windward area, and \(m\) is the mass. In the OCP, the value of the parameter \(\omega\) is always constant and is calculated using Formula (16). \[\omega=\frac{W_{\mathrm{x},max}}{h_{tr}} \tag{16}\]  
where \(W_{\mathrm{x},max}\) is the maximum horizontal wind, and \(h_{tr}\) is the transition altitude at which the horizontal wind becomes constant. In the practical situation, the parameter \(\omega\) obeys the normal distribution [32]. Therefore, in reliability-based MDSDO research, this parameter is treated as the design variable and obeys the normal distribution \(\omega\sim N(h_{\omega},0.005)\). Moreover, the constraints of state variables are also transformed into reliability constraints, and the objective reliability index is \(\beta_{obj}=2\). The boundary conditions of all the variables are listed in Table 3. Furthermore, the initial and the final condition of the state and control variables are listed in the same table. Angle-related parameters were calculated using the radian system (rad). In addition, the glider is required to return to the starting point within the specified time interval, and the time endpoint condition is \(t_{f}\in[0,\ 30]\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Design Variable** & **Objective** & **Failure Probability** \\ \hline MDSDO & \([0.0657,1.0266]\) & \(1.9693\) & \(43.7\%\) \\ RB-MDSDO [30] & \([-0.008,0.014]\) & \(1.9780\) & \(3.10\%\) \\ C-MDSDO & \([-0.0101,0.0039]\) & \(1.9781\) & \(0.03\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The optimization result comparison between MDSDO and C-MDSDO. Employing the same MCS-based presentation method as in the previous case, the state trajectories of deterministic MDSDO and the state trajectories of the design variables subject to uncertainty are shown in Figure 9. In the 1e5 MCS samples, 4967 samples did not satisfy all the constraints. Thus, the optimization result obtained using MDSDO did not meet the reliability requirements.