# Superior Control of Spacecraft Re-Entry Trajectory
## 1 Introduction  
In the design of control systems, traditional control methods evaluate the quality based on a given single index. If multiple aspects are required to be good in this process, it is impossible to achieve this using this single index control method. The most common method is to clarify the main and secondary optimization requirements of the system, assign corresponding weights to each index, and obtain a comprehensive index after weighting. By optimizing only this single comprehensive index, the optimization result can be obtained. However, in a huge system engineering, the complexity of optimization problems increases exponentially. It is not applicable to design control methods based on a single index, and assigning weights based on experience often ignores secondary contradictions. When the index with a large weight reaches an optimal value, the secondary indexes cannot be satisfied at all. However, secondary needs also need to be met in large systems [1]. Some control algorithms are also trying to reduce the presence of weighting factors, such as PID control, which does not directly use weighting factors but achieves control by adjusting the three parameters of proportional integral and differential [2]. Although the Linear Quadratic Regulator (LQR) involves the selection of weight matrices, once the design is completed, the control law itself does not involve the weighting process [3]. Model predictive control (MPC) may use weights to balance different performance metrics during the design process, but its core algorithm does not directly rely on weighting factors but instead calculates control inputs by optimizing a performance metric within a prediction horizon [4]. Sliding mode control designs a sliding surface and corresponding control law to ensure that the system state can reach and remain on the sliding surfacewithout deviation, without involving weighting factors and relying on the invariance and robustness of the system [5; 6]. Adaptive control can automatically adjust its parameters to adapt to dynamic changes in the system, usually without using weighting factors, but relies on an adaptive law to adjust the control action [7]. However, adaptive laws also select one or some indexes for adaptive design, which has the same problem as weighted control, that is, it cannot simultaneously meet multiple index requirements. A robust controller is designed to deal with model uncertainty. By modeling the uncertainty of the system, it usually does not use weighting factors [8]. It designs a controller by ensuring that all allowed uncertainties are modeled, making the closed-loop system stable for all allowed uncertainties and meeting preset performance metrics. Heo control theory designs a controller to minimize the maximum gain from external disturbances to the system output [9]. Although weights may be involved in the design process, the final control law does not contain weighting factors. When it is desired to optimize the performance indexes of each subsystem and its subsystems in the system simultaneously, superior control without weight coefficients is required. Initially, [PERSON] proposed the idea and theory of superior control in Engineering Cybernetics [1] but did not provide specific simulation modeling methods. This article first theoretically refines the theoretical content of superior control and then proposes a simulation algorithm for superior control. During the re-entry flight of the lunar landing, due to the proportional relationship between the aerodynamic heating rate and the flight speed raised to the power of 3.15, when the re-entry point speed is 10.8 km/s, the aerodynamic heating, overload, and dynamic pressure are much more severe than when the speed is 7.8 km/s [10].