# General Periodic Cruise Guidance Optimization for Hypersonic Vehicles
## 4 Numerical Examples and Analysis
### Optimization by Using Augmented States  
In this part, we seek the optimal periodic cruising solutions by using the proposed method at the specified altitudes, as shown in Table 2. We select the altitude error tolerance, \(h_{\varepsilon}\), as 80 m by trial-and-error to ensure the fitting accuracy and convergence simultaneously. To facilitate the optimization process, we specify the initial altitude, \(h_{i}\) (also the terminal altitude, \(h_{t}\)), as the maximum altitude of each cycle, \(h_{\max}\). Thus, we have \(h_{i}=h_{t}=h_{max}\) and \(\gamma_{i}=\gamma_{t}=0\). According to the definition of a periodic cruise, the initial Mach number, \(M_{i}\), equals the terminal Mach number, \(M_{t}\). In the following study, all optimization procedures consistently use these constraints, as shown in Table 3, for the sake of fairness. \begin{table}
\begin{tabular}{c c c} \hline
**Variable** & **Lower Limit** & **Upper Limit** \\ \hline \(\alpha\) (deg) & 5 & 20 \\ \(s\) & 0 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Limits of control variables. \begin{table}
\begin{tabular}{c c c} \hline
**Altitude (km)** & **Mach Number** & **Fuel Consumption (kg/km)** \\ \hline
45 & 14.536 & 1.566 \\
44 & 13.980 & 1.564 \\
43 & 13.413 & 1.569 \\
42 & 12.830 & 1.580 \\ \hline \end{tabular}
\end{table} 
Table 2: Steady-state cruise optimization results. The proposed method was carried out utilizing the initial altitudes and Mach numbers as shown in Table 2, and the corresponding results are shown in Table 4. It can be seen that periodic cruise can improve fuel efficiency compared with the corresponding optimal steady-state cruise, which is more pronounced at higher altitudes. Compared with the results of [15; 36] of around 3% improvement, 2% fuel consumption improvement was achieved. Note that the unique advantage of the proposed method is its unified dealing with parameterized trajectory optimization with little programming work. In [15], the solving software NCONF (Visual Numerics, Houston, TX, USA) is rarely used nowadays and is hard to find; and in [36], the segmented optimization results have poor convergence and oversensitivity to parameter specifications. Therefore, we seek a general solving method for practitioners rather than a particular solution here. It should be also noted that the improvement degree is about 2%, like the results of 3% in [15; 36], which is not quite as robust to the uncertainties in the model. In other words, this result may be worse than that of the optimal steady state when there are uncertainties. In fact, the trajectory optimization is static, and the robustness is beyond the scope of this paper. For the 45 km case, the optimal periodic altitude is  
\[h=4595\cdot\cos(0.0072\cdot r_{\mathrm{km}})+40699, \tag{17}\]  
As shown in Figure 1, it can be seen that the discrete altitude nodes almost match the cosine curve. The Mach number is shown in Figure 2. The elevation angle is shown in Figure 3, wherein the line curve is obtained by using Equation (13) with the corresponding characteristic parameters. The fitting curve matches the optimal elevation angle and is relatively satisfactory, which can provide a proper guidance command in practice. The angle of attack is shown in Figure 4. The thrust throttle is shown in Figure 5, and we can see that the engine is turned on at around 500 km. It is clear that the thrust is switched off during the descending phase, such that the gravitational force can be fully utilized to reduce the drag effect; while the thrust is switched on again in the ascending phase to compensate for the velocity loss. In this way, the fuel efficiency can be improved further. According to this result, the engine can operate at only two modes, on or off, with \(s=1\) or \(s=0\), respectively. This fact is preferred due to its simplicity.