## 2 Problem Analysis and Formulation  
### Description of the Problem  
Modern air combat is highly dependent on the pilot to manipulate the process of detection, tracking, occupying a superior situation, aiming, launching, and guidance. The aiming process as an important function of the aviation fire control system plays a leading role in giving full play to the combat effectiveness of airborne weapons to the greatest extent. BVR attack guidance is an important guidance mode for fighters in modern air combat. Based on the comprehensive consideration of the BVR AAM performance and motion state information of the fighter and the target, it makes the fighter's velocity vector always point in front of the target at a certain lead angle. Under the ideal condition that the target is moving in a straight line with uniform velocity, the missile can hit the target in a straight line trajectory, and can still destroy the target with a great probability when the target is maneuvering. The damage ability of the BVR AAM to the target is not only affected by the missile's performance but also largely determined by the aiming error angle at the launch time. If the aiming error angle is too large, the BVR AAM cannot meet the launching conditions, cannot capture the target, or cannot effectively destroy the target. On the contrary, the smaller the aiming error angle, the more favorable to improve the combat effectiveness of the AAM. Therefore, before launching the missile, it is particularly important to perform the process of attack guidance for the autonomous UCAV to achieve high-precision aiming. The following is an illustration of the problem through the attack guidance process of the manned fighter in the BVR air combat. The simulated situation of the manned fighter's head-up display (HUD) in the attack guidance state in the BVR air combat is shown in Figure 1. To pertinently analyze the problems involved in this research, the schematic diagram of the HUD in this situation is given in Figure 2.  
In Figure 2, the plus sign, the W-shaped symbol, and the square symbol represent the longitudinal axis direction of the BVR AAM, the longitudinal axis direction of the fighter, and the target, respectively. Furthermore, the large green circle is the allowable steering error circle (ASEC), which is always going to be a circle with a plus sign at the center and a variable radius. The small green circle is the attack steering cue (ASC), which is the best direction to launch the BVR AAM. \(\eta_{y}\) and \(\eta_{z}\) are the pitch angle and the azimuth angle of the target relative to the longitudinal axis of the fighter, respectively. \(\delta_{y}\) and \(\delta_{z}\) are the pitch angle and the azimuth angle of the ideal aiming point based on the longitudinal axis of the fighter, respectively. \(\xi\) refers to the off-axis angle of the AAM. \(\lambda\) is the lead angle for ideal aiming, which consists of the horizontal component \(\lambda_{z}\) and the vertical component \(\lambda_{y}\). Besides, the aiming error angle is \(\delta=\sqrt{\delta_{z}^{2}+(\delta_{y}-\delta_{w})^{2}}\), where \(\delta_{w}\) is the setting angle of the AAM. In the BVR attack guidance state shown in Figure 2, the mission of the pilot is to continuously control the fighter so that the ASC overlaps with the plus sign. In other words, let the ASC be located at the center of the ASEC, and then it is the ideal aiming state with zero aiming error angle. However, as the maneuvering magnitude of the target increases, the magnitude of ASC change also increases. It is inefficient for pilots to perform attack guidance for aiming by manually controlling the fighter, which is also easy to lose the best launch opportunity and obviously not suitable for autonomous UCAVs. To realize high-precision and autonomous attack guidance for UCAVs, it is necessary to make the designed attack guidance system has high dynamic response performance, and the aiming error angle needs to be stable within the specified milliradian range within a certain duration.