# General Periodic Cruise Guidance Optimization for Hypersonic Vehicles
## 1 Introduction
However, it is quite difficult to solve these equations because such a two-point boundary value problem (TPBVP) is well known for its difficulty. So far, the shooting method [30] is one of the few methods that can handle TPBVP by iteratively correcting the initial values through the terminal discrepancy. Nevertheless, the shooting method is too sensitive to the choice of the initial guess. Therefore, successful application of the shooting method heavily relies on a refined understanding of the plant and an accurate initial guess. Besides, the existing experiments showed that the convergence behavior of the first-order optimization schemes is poor, relative to a class of aircraft cruise problems [30]. In short, there is currently no effective indirect numerical method to obtain the optimal periodic solution. Due to the complicated calculation process and special expertise requirement, the above-mentioned methods lack easy utility, in general, for fast design and evaluation. To date, there is no professional software that can handle periodic optimization problems in a unified framework. Therefore, we must apply existing and mature optimization software to deal with this problem. In recent years, direct methods, especially pseudo-spectral methods (PSM), have become increasingly popular in the preliminary design phase of the aerospace industry with the development of computers and the corresponding optimization software, GPOPS (Gauss Pseudo-spectral OPtimization Software), which is based on some effective nonlinear programming methods, such as SQP (Sequential Quadratic Programming) [31; 32; 33; 34; 35]. GPOPS has been successfully applied to many trajectory optimization problems. Although PSM is widely used in trajectory optimization problems because of its maturity and module standardization, there are no reports on its application in the periodic cruise optimization to our best knowledge, except [36] which partitioned a cycle into four segments to impose different kinds of constraints in a tiresome manner. The main difficulty is that the parameterized periodic trajectory is beyond the framework of PSM, which is based on the polynomial fitting in terms of Gauss nodes to handle infinite-DOF curve optimization problems with independent variables to be optimized. In other words, PSM cannot be directly applied to optimize trajectory with a specified form that implies strong interdependence of the variables. At present, it is hard to find other means, other than PSM, to solve this problem. Based on the current popularity and acceptance of GPOPS, the use of PSM to optimize periodic cruise trajectories can provide a general, convenient, and effective methodology for practitioners. Since the periodic cruise cannot be embedded directly into the GPOPS software, the problem is how to achieve this flexibly. As a matter of fact, many trajectory optimization problems are parameterized to facilitate implementation. We hope for a general solving method for this wide range of problems, by using the GPOPS rather than a particular solution. In this paper, the Gauss pseudospectral method (GPM) is used to obtain the optimal periodic trajectory of the HL-20 hypersonic aircraft in a general manner, different to the method in [36]. The periodic form is firstly specified, and then the characteristic parameters are considered as augmented states. The originally physical states plus these augmented states establish an augmented system. The periodic form can be added as a path constraint embedded in the GPOPS, with acceptable accuracy. Mathematical simulations are performed to verify the proposed approach. The remaining parts of the paper are organized as follows. The preliminary knowledge is presented in Section 2. The solving strategy is offered in detail in Section 3. The simulation results are presented in Section 4. The conclusions are given in Section 5.