# Compensation Control Strategy for Orbital Pursuit-Evasion Problem with Imperfect Information
## 4 Compensation Control Strategy for Delayed Information
### Time Delay Compensation Method Based on Uncertainty Set  
Few works have yet dealt with the orbital pursuit-evasion problem with delay information. However, several works on delay information have been done in other areas. To compensate for the delay information in the pursuit-evasion problem, [PERSON] [18] proposed the reachability set (uncertainty set) method and proved that the optimal strategy of the pursuer is the pursuit of the center of the reachability set of the evader. References [19; 20; 21; 22; 23; 24; 25] applied this method for missile guidance, which partially compensates for the information delay and effectively improves the guidance accuracy. However, the above works assumed two-dimensional situations, partial delay, and no gravity difference. These assumptions are invalid in the orbital pursuit-evasion problem. Thus, without the consideration of these assumptions, the calculation of the uncertainty set is more complicated. To get the geometric center of the spacecraft's uncertainty set, we analyze the uncertain set's characteristics based on the linear system theory rather than calculate the uncertainty set. Then, the delayed state is replaced by the uncertainty set's center. The detailed procedure is shown as follows. First, we give a definition of the uncertainty set. In the case delay time \(\Delta t\
eq 0\), the player's current state value is unknown to the opponent. The opponent can calculate the set of all possible state values based on the delayed state information he gets and the player's control constraint. This set is called the uncertainty set (denoted by \(S\)). Ignoring the subscripts, the spacecraft dynamics (3) can be expressed as follows:  
\[\dot{\mathbf{X}}(t)=\mathbf{A}\mathbf{X}(t)+\mathbf{B}\mathbf{U}(t), \tag{27}\]  
where \(\left\|\mathbf{U}(t)\right\|_{2}\leq\rho\). Let \(\Big{[}t_{0}\), \(t_{f}\Big{]}\) be the delay period. According to the linear system theory [29], the terminal state \(\mathbf{X}\Big{(}t_{f}\Big{)}\) satisfies  
\[\mathbf{X}\Big{(}t_{f}\Big{)}=\mathbf{\Phi}\Big{(}t_{f}-t_{0}\Big{)}\mathbf{X}(t_{0})+\int_{ t_{0}}^{t_{f}}\mathbf{\Phi}\Big{(}t_{f}-\tau\Big{)}\mathbf{B}\mathbf{U}(\tau)d\tau. \tag{28}\]  
According to the definition of the uncertainty set, for any given \(\mathbf{X}_{1}\Big{(}t_{f}\Big{)}\in\mathbf{S}\), there exists a corresponding control strategy \(\mathbf{U}_{1}(t)\), which satisfies \(\|\mathbf{U}_{1}(t)\|_{2}\leq\rho\) such that  
\[\mathbf{X}_{1}\Big{(}t_{f}\Big{)}=\mathbf{\Phi}\Big{(}t_{f}-t_{0}\Big{)}\mathbf{X}(t_{0})+ \int_{t_{0}}^{t_{f}}\mathbf{\Phi}\Big{(}t_{f}-\tau\Big{)}\mathbf{B}\mathbf{U}_{1}(\tau)d\tau. \tag{29}\]  
Let \(\mathbf{U}_{2}(t)=-\mathbf{U}_{1}(t)\), \(t\in\Big{[}t_{0}\), \(t_{f}\Big{]}\), where \(\|\mathbf{U}_{2}(t)\|_{2}\leq\rho\). The terminal state \(\mathbf{X}_{2}\Big{(}t_{f}\Big{)}\) corresponding to \(\mathbf{U}_{2}(t)\) can be expressed as  
\[\mathbf{X}_{2}\Big{(}t_{f}\Big{)}=\mathbf{\Phi}\Big{(}t_{f}-t_{0}\Big{)}\mathbf{X}(t_{0}) +\int_{t_{0}}^{t_{f}}\mathbf{\Phi}\Big{(}t_{f}-\tau\Big{)}\mathbf{B}\mathbf{U}_{2}(\tau)d\tau. \tag{30}\]  
According to the definition of the uncertainty set, we have \(\mathbf{X}_{2}\Big{(}t_{f}\Big{)}\in\mathbf{S}\). Combining Equations (29) and (30) yields  
\[\mathbf{X}_{1}\Big{(}t_{f}\Big{)}+\mathbf{X}_{2}\Big{(}t_{f}\Big{)}=2\mathbf{\Phi}\Big{(} t_{f}-t_{0}\Big{)}\mathbf{X}(t_{0}). \tag{31}\]  
Equation (30) can be further expressed as  
\[\mathbf{X}_{1}\Big{(}t_{f}\Big{)}-\mathbf{\Phi}\Big{(}t_{f}-t_{0}\Big{)}\mathbf{X}(t_{0}) =-\Big{(}\mathbf{X}_{2}\Big{(}t_{f}\Big{)}-\mathbf{\Phi}\Big{(}t_{f}-t_{0}\Big{)}\mathbf{ X}(t_{0})\Big{)}. \tag{32}\]  
According to Equation (32), states \(\mathbf{X}_{1}\Big{(}t_{f}\Big{)}\) and \(\mathbf{X}_{2}\Big{(}t_{f}\Big{)}\) are centrally symmetric about point \(\mathbf{\Phi}\Big{(}t_{f}-t_{0}\Big{)}\mathbf{X}(t_{0})\) in the state space.