# Compensation Control Strategy for Orbital Pursuit-Evasion Problem with Imperfect Information
## 1 Introduction  
Recently, the orbital pursuit-evasion has attracted increasing attention in space rendezvous and proximity operations (RPOs) [1; 2; 3; 4]. It is usually considered a zero-sum differential game, whose capture time or distance is shortened by the pursuer and increased by the evader. Many researchers worked on this game with the assumption of perfect information [5]. However, this assumption violates real scenarios with noise (caused by measurement) and time delay (caused by observation and data processing). The games with these uncertain states are called imperfect information games [6]. For this situation, [PERSON] et al. [6] adopted the unscented Kalman filter (UKF) to estimate the players' relative states and proposed adaptive linear-quadratic feedback control strategies. [PERSON] et al. [7] built a linear regression model from a large data set to analyze the effect of the pursuer's uncertainty start position. The orbital pursuit-evasion problem with noise or uncertainty start conditions is well solved by these works. However, there exist two shortcomings in scenario and strategy. For the scenario, the time required for measurement and state estimation is ignored, which leads to the deviation of control strategy in the actual situation. For the strategy, the shortcomings depend on its characters. In the linear-quadratic feedback control strategy, the players' control without boundary constraints is unrealistic, while in the near-optimal feedback control, the solving process based on the numerous open-loop solutions is time-consuming and capture unguaranteed. To overcome the above shortcomings, we need to focus on the following two aspects when studying the orbital pursuit-evasion problem with imperfect information. One is thefeedback control strategy with bounded control. The other is the compensation method for the delayed information. It is difficult to obtain an analytic feedback solution by solving the Hamilton-Jacobin-Isaacs (HJI) partial differential equation for the orbital pursuit-evasion problem with bounded control [8]. Therefore, many works are devoted to generating open-loop equilibrium solutions. The process for obtaining the open-loop equilibrium solution is as follows. First, the orbital pursuit-evasion problem is converted to a two-point boundary value problem by using the necessary conditions [9] for the existence of solutions. Then various kinds of methods are designed to solve the problem, such as the semidirect collocation with nonlinear programming algorithm [5], the hybrid approach combining the semidirect nonlinear programming and the multiple shooting method [10], the sensitivity method [11], the semidirect control parameterization method [12], the indirect optimization method [13], and the hybrid numerical algorithm combining the differential evolution algorithm and [PERSON]'s iteration method [14]. However, the open-loop equilibrium solution only relies on the initial state and current time, which cannot deal with interference errors during the game. Therefore, some researchers have studied the near-optimal closed-loop feedback solutions. [PERSON] et al. [15] proposed a new extremal-field approach for synthesizing nearly optimal feedback controllers. In this approach, a large number of open-loop solutions were first generated offline. Then the online nearly optimal feedback control was obtained by interpolating these open-loop solutions. [PERSON] [16] obtained the near-optimal feedback controls for spacecraft pursuit-evasion problems by periodically resolving the differential game using a modified differential dynamic programming algorithm. However, the near-optimal closed-loop feedback solutions generated by these methods are based on solving a large number of open-loop solutions, which is time-consuming and capture unguaranteed. To overcome the drawbacks in the existing feedback controls for the orbital pursuit-evasion problem, we applied the optimal missile guidance method proposed by [PERSON] [17] to the orbital pursuit-evasion problem.