## 3 Process of Design Optimization
### Deterministic Optimization Design Based on DK Model  
Based on the performance goals of pulse correction projectiles, deterministic optimization design is conducted using the DK surrogate model. At the design point where  
Figure 8: Drag coefficient versus _Ma_.  
Figure 7: Drag coefficient versus \(\delta\).  
_aero_factor_ = 0.93, flight dynamic stability during the optimization process must be ensured The mathematical optimization model can be expressed as follows:  
\[\begin{array}{c}\mathit{Objective}\;Function:\quad\max\{\hat{c}(X),\hat{r}(X )\}\quad\min\{\hat{a}(X)\}\\ \mathit{S.T.}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\delta_{max}<15^{ \circ}\\ \mathit{Variable}:\qquad\qquad\qquad\qquad X=[N,\varepsilon,R,S,\mathit{l}_{ imp},\mathit{L}_{imp}]\end{array}. \tag{4}\]  
In this equation, \(c\), \(r\), and \(a\) correspond to the representative lateral correction distance, maximum range, and maximum angle of attack of the target, respectively. The symbol (\") denotes that this value is predicted by the DK surrogate model. In the design process, it is not required for the three performance criteria to be strictly mutually exclusive. In the design process, 600 sample points were generated using Latin hypercube design (LHD) as training samples and an additional 30 test samples were generated. The number of samples to select is based on the expected prediction accuracy and computation. The distribution of performance indicators, namely lateral correction distance, maximum range, and maximum angle of attack, categorized according to the number of fins, is shown in Figures 9 and 10. The red dots in the figures indicate unstable points in the design (\(\delta_{\max}>15^{\circ}\)). It can be observed that reducing the number of fins can increase the range effectively. However, an improper design of the number and location of fins may cause the center of pressure and center of gravity to be too close, resulting in the static stability reaching the critical value. Moreover, when the projectile is subject to pulse forces during its controlled trajectory, it becomes more challenging to meet the dynamic stability requirements. Additionally, larger correction capabilities often require more significant pulse energy consumption, which puts higher demands on the projectile's flight stability.  
Figure 11 shows that the DK surrogate model constructed has good agreement with the 6 DOF simulation results for the predicted values of most sample points; the prediction of maximum range and lateral correction distance are more accurate than the maximum angle of attack. This indicates that the surrogate model can predict the performance variation trend of the test samples, and its accuracy meets the needs of the subsequent optimization processes. However, it should be noted that the selection of hyperparameters and regression functions in the DK model can affect the model's accuracy, and the selection of model parameters is somewhat empirical. Therefore, different target performances may require appropriate model parameters. Additionally, the flight stability constraint of the projectile is handled in the MOSSA optimization process by imposing a penalty term on individuals who violate the constraint on the amplitude of the attack angle. Finally, to eliminate accidental errors, each algorithm runs independently for 30 times. The optimization results will be compared and analyzed in Section 3.4.  
Figure 9: LHD data with respect to lateral correction distance and max range.