# Rapid Parametric Modeling and Robust Analysis for the Hypersonic Ascent Based on Gap Metrics
## 2 Model Description
### Polynomial Fitting Modeling
After that, the sampling matrix \(\mathbf{Q}_{k}\) and response matrix \(\mathbf{Y}_{k}=y\) are determined for the \(k\)-th iteration wherein \(\mathbf{Q}_{k}\) is expressed by  
\[\mathbf{Q}_{k}=\mathbf{J^{\prime}}_{n_{i}}\cdot diag\big{[}\mathbf{O}_{n_{k}}(\cdot,k) \big{]} \tag{16}\]  
For IFFD screening analysis, one must simulate the experiments according to the sampling matrix. The main effect of the \(i\)-th parameter is identified by the difference in average response between the two levels (\(1\) and \(-1\)), which is assumed to be a linear effect:  
\[ME(i,\mathbf{Y})=\frac{\sum\limits_{k=1}^{n_{k}}\mathbf{Q}_{k}(\cdot,i)\mathbf{Y}_{k}}{ \sum\limits_{k=1}^{n_{k}}\left\lvert\mathbf{O}_{k}(i,k)\right\rvert} \tag{17}\]The analysis of the interaction effect of parameters is similar to the main effect. Nonlinear effects of the \(i\)-th parameter are identified by the difference in average response between the median (0) and the boundary (1, \(-\)1), which is assumed to be a quadratic effect:  
\[ME(i,\mathbf{Y})=\frac{\sum\limits_{k=1}^{n_{k}}\sum\limits_{j=1}^{2^{\pi i}}(1- |\mathbf{Q}_{k}(j,i)|)\mathbf{Y}_{k}(j)}{\sum\limits_{k=1}^{n_{k}}(1-|\mathbf{ Q}_{k}(i,k)|)}-\frac{\sum\limits_{k=1}^{n_{k}}\sum\limits_{j=1}^{2^{\pi i}}| \mathbf{Q}_{k}(j,i)|\mathbf{Y}_{k}(j)}{\sum\limits_{k=1}^{n_{k}}|\mathbf{O}_{k }(i,k)|} \tag{18}\]  
The factors in polynomial approximation are sequentially selected from the sensitivity analysis based on IFFD. The number of factors is increased until the fitting performance is satisfied. Following that, the structure of the polynomial expression of \(C_{L}\) is yielded. Similarly, we can obtain the polynomial interpolation expressions regarding \(C_{D}\), \(C_{T}\), and \(C_{M}\). Furthermore, the matching functions, including the goodness of fit \(VAF\), mean square error \(RMSE\), and maximum standard error \(MSR\), are applied to evaluate the accuracy of the SM, and they are provided by  
\[\left\{ \begin{array}{l}VAF=\left(1-\frac{\|y_{\text{obs}}-y_{\text{fit}}\|_{2 }^{2}}{\|y_{\text{obs}}\|_{2}^{2}}\right)\times 100\%\\ RMSE=\frac{\|y_{\text{obs}}-y_{\text{fit}}\|_{2}}{N_{s}}\\ MSR=\max|y_{\text{obs},j}-y_{\text{fit},i}|\end{array} \right. \tag{19}\]  
The structure of the SM regarding the aerodynamic and thrust coefficients is identified as the following fitting expressions:  
\[\left\{ \begin{array}{l}C_{L}=-0.0182+2.5138a+0.5255\delta_{e}+0.3347 Ma^{-1}+ 17.9879\alpha Ma^{-1}+\\ 2.4639\delta_{e}Ma^{-1}+0.2754\delta_{e}^{2}-3.4 Ma^{-2}+0.0324\phi\\ C_{D}=0.0066+0.0336\alpha-0.00045\delta_{e}+2.225\alpha\delta_{e}+6.3216\alpha ^{2}+0.8627\delta_{e}^{2}+\\ 0.7869 Ma^{-2}+7.5902\delta_{e}^{2}Ma^{-2}-0.0146\phi\\ C_{m}=0.1699+1.2296\alpha-0.8943\delta_{e}-0.7209\alpha\delta_{e}+8.1079\alpha ^{2}-0.363\delta_{e}^{2}+\\ 4.4097\alpha Ma^{-1}-4.3520\delta_{e}Ma^{-1}+4.9149 Ma^{-2}-0.0392\phi\\ C_{T}=1.0771-0.2034 Ma+1.7564\alpha-0.2389\alpha Ma-5.1418\alpha^{2}+0.0096 Ma^{2}+ \\ -1.4256\epsilon^{-6}h-\phi(1.5368-0.5322 Ma-1.1112\alpha-0.2319\alpha Ma+\\ 5.0981\alpha^{2}+0.0310 Ma^{2})\end{array} \right. \tag{20}\]  
Once the matching degree between the SM and original model is satisfactory, the model features can be further analyzed to understand the internal dynamics and to design the applicable control law. In particular, Equation (20) shows that the lift coefficient, drag coefficient, and pitch moment coefficient are relevant to the angle of attack and flight Mach with which the propulsive coefficient and thrust moment are correlated. As a result, when the changes in the angle of attack and flight Mach lead to the change in the engine's thrust, the variation in the propulsive force will affect the flight states such that the aerodynamics and propulsion are inseparable. The goodness of fit VAF and mean square error RMSE of the polynomial model in Equation (20) are given in Table 1. In addition, the comparative results between the original model and the polynomial model at the flight condition of Ma 4 and 25.9 km are provided in Figures 2 and 3.